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+{"text":"\\section*{Introduction}\nThe theory of Lie groupoids can be viewed as a blend of geometry and Lie theory, and plays an \nimportant role in several branches of mathematics. Lie groupoids can be viewed as the global \nintegrations of geometrically defined Lie brackets and as such posses a natural deformation theory. \nIn \\cite{cms}, the authors wrote down an explicit cochain complex controlling such deformations. \nAs expected, this ``deformation cohomology'' is isomorphic to the cohomology of the adjoint representation (up to homotopy as in \\cite{ac}), but the point of \\cite{cms} is that the latter involves the choice of a connection, whereas the deformation complex is intrinsically defined. Any deformation\nof Lie groupoids defines a class in this deformation cohomology.\n\nLie groupoids also play an important role in noncommutative geometry where they give prime examples in the form of their associated convolution algebras. As is well-known, the deformation theory of an algebra is controlled by its Hochschild cohomology. Since a deformation of the underlying Lie groupoid induces a deformation of its convolution algebra, this strongly suggests a \nrelationship between Hochschild cohomology and the adjoint representation.\n\nThe aim of the present article is to shed light on this issue by exhibiting an explicit morphism of \ncochain complexes between the deformation complex of a Lie groupoid and the Hochschild complex \nof its convolution algebra which relates the deformation classes associated to a deformation of the \nunderlying Lie groupoid. We expect this morphism to be part of a larger picture computing the \nHochschild cohomology in terms of higher powers of the adjoint representation.\n\nA classical theme in the theory of Lie groupoids is its relation with the infinitesimal theory of Lie \nalgebroids. For the deformation complex this is highlighted by the ``van Est'' morphism to the \ndeformation complex of the Lie algebroid of Lie groupoid, a complex first considered in \\cite{cm}.  \n\nFrom the point of view of noncommutative geometry, the relation with the infinitesimal theory follows \nfrom ``quantization and the classical limit'': we show that our cochain morphism can be extended to the adiabatic groupoid \nof \\cite{connes} interpolating between a Lie groupoid and its Lie algebroid. The van Est-map is \nthen obtained by constructing a quantization map on the dual of the Lie algebroid using an exponential map on the Lie groupoid. The picture is completed by the relationship between \nthe deformation cohomology a Lie algebroid  and the Poisson cohomology of its dual. \n\nThis article is organized as follows: in \\cref{prelim} we recall the basic set-up of Lie groupoids and their convolution algebras. In \\cref{chainmap} we construct the morphism between the deformation complex of a Lie groupoid and the Hochschild complex of its convolution algebra, and explore some of its properties. Finally, \\cref{quant} is devoted to obtaining the van Est map using the adiabatic groupoi together with a quantization.\n\n\\section{Preliminaries}\\label{prelim}\n\\subsection{Densities along the fibers of a submersion}\n\\label{das}\nLet $V$ be an n-dimensional vector space. A \\textit{density} of $V$ is a map $a:\\Lambda^n V\\to\\mathbb{R}$ such that for every invertible map $A\\in\\text{GL}(n,\\mathbb{R})$ it holds that $a(Av_1,...,Av_n)=|\\det(A)|a(v_1,...,v_n)$.\n\nMore generally, from a vector bundle $E\\to M$, one constructs a bundle of densities $\\mathcal{D}_E$. Then if one has a vector bundle isomorphism $\\Psi: E\\to E$ covered by a diffeomorphism $\\Phi: M\\to M$, one obtains an action on the section of $\\mathcal{D}_E$, defined by\n\\begin{equation*}\n(\\Psi^\\ast a)_x(v_1,...,v_n)=a_{\\Phi(x)}(\\Psi v_1,...,\\Psi v_n)\n\\end{equation*}\nThe case where $E=TM$ is of particular interest because the integral $\\int_Ma$ is \ncanonically defined for a compactly supported density of $TM$. In this case one obtains an action of a vector field $X\\in\\mathfrak{X}(M)$ on the densities on $TM$, namely:\n\\begin{equation}\n\\label{apvf}\nXa=\\left.\\frac{d}{dt}\\right|_{t=0} (\\Phi^t_X)^\\ast a,\n\\end{equation}\nwhere $\\Phi^t_X$ denotes the flow of $X$. \n\nWe will be mostly interested in densities along the fibers of a submersion. For this, let $f: M\\to N$ be a submersion, and denote by $\\mathcal{D}_f$ the bundle of densities of the vector bundle $\\ker df$. In this case the fiber integral\n\\[\n\\int_f:\\Gamma_c(M,\\mathcal{D}_f)\\to C^\\infty_c(N)\n\\]\nis canonically defined. A vector field $X$ acts on sections of $\\mathcal{D}_f$ provided that the flow preserves the fibers of $f$. This is equivalent to there being a vector field $Y\\in\\mathfrak{X}(N)$ such that $df\\circ X=Y\\circ f$. In this case $X$ is called \\textit{$f$-projectable}, and since $\\Phi^t_X\\circ f=f\\circ\\Phi^t_Y$ the flow of $X$ preserves the fibers of $f$ and in turn acts on $\\ker df$, and we obtain an action of $X$ on $\\Gamma(\\mathcal{D}_f)$ by formula \\eqref{apvf}.\nWe denote by $\\text{Diff}_f(M)$ the diffeomorphisms of $M$ that preserve the fibers of $f$, and by $\\mathfrak{X}_f(M)$ the $f$-projectable vector fields of $M$.\n\nIn the following we shall consider $f$-projectable vector fields defined on only a single fiber of $f$ and let it act on densities to get a density on that one fiber. This is similar to the fact that the directional derivative $X(f)(p)$ of a function $f$ along a vector field $X$ in a point $p$ only depends\non $X(p)$. \n\\begin{lem}\n\\label{nl}\nLet $a\\in\\Gamma(\\mathcal{D}_f)$, let $y\\in N$, and let $X\\in\\mathfrak{X}_f(M)$ be an $f$-projective vector field. If $X$ vanishes along $f^{-1}(y)$, then $(Xa)_x=0$ for all $x\\in f^{-1}(y)$.\n\\end{lem}\n\\begin{proof}\nIf $X$ vanishes along $f^{-1}(y)$ we have $\\Phi^t_X(x)=x$ for all $t$ and all $x\\in f^{-1}(y)$. In particular $d(\\Phi^t_X)_x(v)=v$ for all $v\\in\\text{ker}(df)\\subset T_xM$. This means that $((\\Phi^t_X)^\\ast a)_x=a_x$ and hence $(Xa)_x=0$.\n\\end{proof}\n\\begin{rmk}\nThe previous Lemma allows us to define $(Xa)_x$ for $x\\in f^{-1}(y)$, $a\\in\\Gamma(\\mathcal{D}_f)$ and $X\\in\\mathfrak{X}_f(M)|_{f^{-1}(y)}$. Indeed, we can choose $Y\\in\\mathfrak{X}_f(M)$ to be an extension of $X$ to a global vector field and define $(Xa)_x=(Ya)_x$. The previous Lemma is then used to show that this definition is independent of the choice of $Y$.\n\\end{rmk}\n\\subsection{The convolution algebra of a Lie groupoid}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid. For an introduction to the theory of Lie groupoids we refer to \\cite{mm}. Here we denote source and target maps by $s,t:\\mathcal{G}\\to M$ and will think of arrows $g\\in\\mathcal{G}$ as pointing from right to left, so that the product $g_1g_2$ is defined whenever $s(g_1)=t(g_2)$. The Lie algebroid  $A(\\mathcal{G})$ is defined as $A(\\mathcal{G})=\\ker(ds)|_M$. We will concern ourselves with the convolution algebra of $\\mathcal{G}$. To define the convolution product, we need entities which can be integrated, and this is where densities come into play. To this end we look at densities along the source-fibers, where we note that there is a canonical isomorphism between $\\ker ds$ and $t^\\ast A(\\mathcal{G})$ using right translations. In this way we can define the convolution product for two compactly supported densities $a_1, a_2\\in\\Gamma_c(\\mathcal{D}_s)$ by\n\\begin{equation*}\n(a_1\\ast a_2)_g(v_1,...,v_n)=\\int_{h\\in s^{-1}(s(g))}(a_1)_{gh^{-1}}(v_1,...,v_n)(a_2)_h\n\\end{equation*}\nIn this notation $v_1,...,v_n\\in A_{t(g)}=A_{t(gh^{-1})}$ so that the product in the integrand yields a well-defined compactly supported density along $s^{-1}(h)=s(g)$ that can be integrated. Colloquially this product will be written as:\n\\begin{equation*}\n(a_1\\ast a_2)(g)=\\int_{g_1g_2=g}a_1(g_1)a_2(g_2)=\\int_{h\\in s^{-1}(s(g))}a_1(gh^{-1})a_2(h)\n\\end{equation*}\nWe define the convolution algebra $\\mathcal{A}_\\mathcal{G}$ of $\\mathcal{G}$ to be $\\mathcal{A}_\\mathcal{G}=(\\Gamma_c(\\mathcal{D}_s),\\ast)$. This definition of the convolution algebra differs slightly (but is isomorphic as a complex algebra) from the more usual one in e.g. \\cite{connes} using $1\/2$-densities along source {\\em and} target fibers.\n\\subsection{The deformation complex of a Lie groupoid}\n Let $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid and write $\\overline{m}$ for the map $\\overline{m}(g,h)=gh^{-1}$. Note that $\\overline{m}$ has as domain $\\mathcal{G}\\hspace*{1mm}^s\\!\\times^s\\mathcal{G}:=\\{(g_1,g_2)\\in\\mathcal{G}\\times\\mathcal{G},~s(g_1)=s(g_2)\\}$. Furthermore we write $\\mathcal{G}^{(k)}$ for the $k$'th nerve of $\\mathcal{G}$:\n\\begin{equation*}\n\\mathcal{G}^{(k)}=\\{(g_1,...,g_k)\\in\\mathcal{G}^k|s(g_i)=t(g_{i+1})\\}\n\\end{equation*}\nIn \\cite{cms} the deformation complex is defined as follows:\n\\begin{defi}\nFor $k\\geq 1$ define \n$C^k_\\text{def}(\\mathcal{G})$ to be the set of smooth maps $c:\\mathcal{G}^{(k)}\\to T\\mathcal{G}$ such that $c(g_1,...,g_k)\\in T_{g_1}\\mathcal{G}$ and such that there is a section $s_c$ of tthe vector bundle $t^*TM$ over $\\mathcal{G}^{(k-1)}$ such that \n\\[\nds(c(g_1,...,g_k))=s_c(g_2,...,g_k).\n\\]\nThe differential $\\delta: C^k_\\text{def}(\\mathcal{G})\\to C^{k+1}_\\text{def}(\\mathcal{G})$ is defined by setting:\n\\begin{align*}\n(\\delta c)(g_1,...,g_{k+1})=&-d\\overline{m}(c(g_1g_2,g_3,...,g_{k+1}),c(g_2,...,g_{k+1}))\\\\\n&+\\sum_{i=2}^k(-1)^ic(g_1,...,g_ig_{i+1},...,g_{k+1})\n+(-1)^{k+1}c(g_1,...,g_k).\n\\end{align*}\nThe {\\em deformation complex} is defined by the graded vector space $C^\\bullet_\\text{def}(\\mathcal{G}):=\\bigoplus_{k\\geq 1} C^k_\\text{def}(\\mathcal{G})$ equipped with the differential $\\delta$, its cohomology is denoted $H^\\bullet_\\text{def}(\\mathcal{G})$. \n\\end{defi}\n\\begin{rmk}\n\\label{deg0}\nIt is possible, as in \\cite{cms}, to extend the deformation complex in degree zero by putting $C^0_\\text{def}(\\mathcal{G})=\\Gamma(M,A(\\mathcal{G}))$ \nwith differential defined for $\\alpha\\in\\Gamma(M,A(\\mathcal{G}))$ by\n\\begin{equation*}\n(\\delta \\alpha)(g)=(dr_g)(\\alpha(t(g))+(d(l_g\\circ\\iota))(\\alpha(s(g))\n\\end{equation*}\nWe exclude these elements in degree $0$ because, as we will see, these element cannot correspond to Hochschild $0$-cochains.\n\\end{rmk}\n\n\\begin{rmk}\n\\label{mv}\nIt follows from the definition above that the closed elements in degree $1$ are exactly the multiplicative vector fields, c.f.\\ \\cite[\\S 4.3]{cms}. These are vector fields $X\\in\\mathfrak{X}(\\mathcal{G})$ that are $s$ and $t$-projectable to the same image in $\\mathfrak{X}(M)$, satisfying the following equation:\n\\begin{equation*}\ndm_{(g,h)}(X(g),X(h))=X(gh)\n\\end{equation*}\n\\end{rmk}\n\nFor certain purposes, most importantly applying the Van Est map, it often necessary to impose more strict relations on elements $c\\in C^k_\\text{def}(\\mathcal{G})$ and their symbol $s_c$. To this end we also introduce the normalized deformation complex:\n\n\\begin{defi}\nThe \\textit{normalized deformation complex} is the subcomplex $\\hat{C}^\\bullet_\\text{def}(\\mathcal{G})$ of $C^\\bullet_\\text{def}(\\mathcal{G})$ consisting of those elements $c\\in C^k_\\text{def}(\\mathcal{G})$ which satisfy\n\\begin{equation*}\nc(1_x,g_2,...,g_k)=du(s_c(g_2,...,g_k))\n\\end{equation*}\nand\n\\begin{equation*}\ns_c(g_2,...,1_x,...,g_k)=0\n\\end{equation*}\nwhere the unit is put in any of the $k-1$ slots.\n\\end{defi}\nIt is shown in \\cite[Prop 11.8]{cms} that the inclusion of the normalized deformation complex into the whole deformation complex is a quasi-isomorphism.\n\\section{From deformation to Hochschild cohomology}\\label{chainmap}\n\\subsection{The cochain map}\n\\label{cmh}\nIn this section we define a cochain map from the deformation complex of $\\mathcal{G}$ to the Hochschild complex of the convolution algebra $\\mathcal{A}_\\mathcal{G}$. As a first hint for the existence of such a morphism, we make the following observation:\n\\begin{prop}\\label{multvfderiv}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid. \nMultiplicative vector fields on $\\mathcal{G}$  act as derivations on the convolution algebra.\n\\end{prop}\n\\begin{proof}\nRecall the definition of a multiplicative vector field from \\cref{mv}. Since a multiplicative vector field on $\\mathcal{G}$ is by definition $s$-projectable to $M$, its action on an $s$-density is well-defined by \nthe discussion in \\cref{das}, c.f.\\ equation \\eqref{apvf}.\n\nThe key ingredient in the proof is the observation that the flow of a multiplicative vector field is a groupoid map, that is if $X\\in\\mathfrak{X}(\\mathcal{G})$ is a multiplicative vector field then $\\Phi^t_X(gh^{-1})=\\Phi^t_X(g)\\Phi^t_X(h)^{-1}$. A simple calculation then shows\n\\begin{align*}\nX(a_1 \\ast a_2)(g)&=\\left.\\frac{d}{dt}\\right|_{t=0}(a_1\\ast a_2)(\\Phi^t_Xg)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(\\Phi^t_X g))}a_1((\\Phi^t_X g)h^{-1})a_2(h)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1((\\Phi^t_X g)(\\Phi^t_X h)^{-1})a_2(\\Phi^t_Xh)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1(\\Phi^t_X(gh^{-1}))a_2(\\Phi^t_Xh)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1(\\Phi^t_X(gh^{-1}))a_2(h)\\\\ &\\hspace{2.5cm}+\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1(gh^{-1})a_2(\\Phi^t_Xh)\\\\\n&=(Xa_1\\ast a_2)(g)+(a_1\\ast Xa_2)(g)\n\\end{align*}\nwhich proves the proposition.\n\\end{proof}\nIn the following we write $C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ for the Hochschild complex of the convolution algebra $\\mathcal{A}_\\mathcal{G}$ with values in the bimodule $\\mathcal{A}_\\mathcal{G}$ with differential $\\delta_{\\rm Hoch}$.\nWe now describe the cochain map $C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$. \n\\begin{defi}\n\\label{chainmap}\nThe map $\\Phi:C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ is defined by \n\\begin{equation*}\n(\\Phi c)(a_1,...,a_k)(g)=\\int_{g_1\\cdots g_k=g}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_k(g_k),\\qquad\\mbox{for}~c\\in C^k_\\text{def}(\\mathcal{G}).\n\\end{equation*}\nThis formula should be read as an inductive convolution (first over $g_1g_2=h_1$, then over $h_1g_3=h_2$, et cetera).\n\\end{defi}\n\\begin{rmk}\nThe formula for $\\Phi$ above is justified by \\cref{nl}: $c\\in C^k_\\text{def}(\\mathcal{G})$ is $s$-projectable and therefore the action of $c(-,g_2,...,g_k)$ on  $a\\in \\mathcal{A}_\\mathcal{G}$ along $s^{-1}(t(g_2))$ is well-defined. In particular $(c(-,g_2,...,g_k)a)(g_1)$ is a well-defined density at $g_1$ for $(g_1,...,g_k)\\in\\mathcal{G}^{(k)}$.\n\\end{rmk}\nShowing that $\\Phi$ is a chain map is done by a calculation similar to the one in \\cref{multvfderiv}. In particular we need to deal with the term $\\Phi^t_X(g)(\\Phi^t_X(h))^{-1}$ for divisible $g$ and $h$ when $t$ goes to $0$. In the mulitplicative case this is precisely $\\Phi^t_X(gh^{-1})$, but for general deformation elements we need a more general description.\n\nFor this we abbreviate the term in $\\delta c$ involving $d\\overline{m}$ by $\\overline{m}c$, that is:\n\\begin{equation*}\n(\\overline{m}c)(g_1,...,g_{k+1})=d\\overline{m}(c(g_1g_2,...,g_{k+1}),c(g_2,...,g_{k+1}))\n\\end{equation*}\nWe remark that this notation commutes with keeping the last all-but-two entries fixed, i.e.:\n\\begin{equation*}\n\\overline{m}(c(-,g_3,...,g_{k+1}))(g_1,g_2)=(\\overline{m}c)(g_1,...,g_{k+1})\n\\end{equation*}\nThe key Lemma is then as follows:\n\\begin{lem}\nLet $x\\in M$, $X\\in\\mathfrak{X}_s(\\mathcal{G})|_{s^{-1}(x)}$ and $a_1,a_2\\in \\mathcal{A}_\\mathcal{G}$. Then for all $h\\in s^{-1}(x)$ we have $\\overline{m}X(-,h)\\in\\mathfrak{X}_s(\\mathcal{G})|_{s^{-1}(t(h))}$ and for $g\\in s^{-1}(x)$:\n\\begin{equation*}\nX(a_1\\ast a_2)(g)=(a_1\\ast Xa_2)(g)+\\int_{h\\in s^{-1}(x)}((\\overline{m}X(-,h))a_1)(gh^{-1})a_2(h)\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nBy definition we have:\n\\begin{equation*}\n\\overline{m}X(gh^{-1},h)=d\\overline{m}(X(g),X(h))\\in T_{gh^{-1}}\\mathcal{G}\n\\end{equation*}\nwith $s$-projection\n\\begin{equation*}\nds(\\overline{m}X(gh^{-1},h))=dt(X(h))\n\\end{equation*}\nso indeed $\\overline{m}X(-,h)\\in\\mathfrak{X}_s(\\mathcal{G})|_{s^{-1}(t(h))}$.\n\nNext we assume that $X$ is a globally defined $s$-projectable vector field (otherwise, we choose an extension at this point). Then we know that $\\overline{m}X(-,h)$ is generated by the path $\\Phi_t$ through $\\text{Diff}_s(\\mathcal{G})$, which along $s^{-1}(t(h))$ looks like:\n\\begin{equation*}\n\\Phi_t(gh^{-1})=\\Phi^t_X(g)(\\Phi^t_X(h))^{-1}\n\\end{equation*}\nso that we see that:\n\\begin{equation*}\n\\int_{h\\in s^{-1}(x)}((\\overline{m}X(-,h))a_1)(gh^{-1})a_2(h)=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1(\\Phi^t_X(g)(\\Phi^t_X(h))^{-1})a_2(h)\n\\end{equation*}\nUsing this we calculate $X(a_1\\ast a_2)(g)$:\n\\begin{align*}\nX(a_1\\ast a_2)(g)=&\\left.\\frac{d}{dt}\\right|_{t=0} (a_1\\ast a_2)(\\Phi^t_Xg)\\\\\n=&\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(\\Phi^t_Xg))}a_1((\\Phi^t_Xg)h^{-1})a_2(h)\\\\\n=&\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1((\\Phi^t_Xg)(\\Phi^t_Xh)^{-1})a_2(\\Phi^t_X(h))\\\\\n=&\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1((\\Phi^t_Xg)(\\Phi^t_Xh)^{-1})a_2(h)+\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1(gh^{-1})a_2(\\Phi^t_Xh)\\\\\n=&\\int_{h\\in s^{-1}(x)}((\\overline{m}X(-,h))a_1)(gh^{-1})a_2(h)\\\\\n&+(a_1\\ast Xa_2)(g)\n\\end{align*}\nwhich finishes the proof.\n\\end{proof}\n\\begin{prop}\nThe map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ is a morphism of cochain complexes.\n\\end{prop}\n\\begin{proof}\nThis proof is essentially writing out all the parts of the Hochschild differential and applying some bookkeeping. We start with $c\\in C^k_\\text{def}(\\mathcal{G})$ for $k\\geq 1$, and write down the definition of the various parts of $\\delta_\\text{Hoch}(\\Phi c)$.\n\\begin{equation*}\n(a_1\\ast(\\Phi c)(a_2,...,a_{k+1}))(g)=\\int_{g_1\\cdots g_{k+1}=g}a_1(g_1)(c(-,g_3,...,g_{k+1})a_2)(g_2)a_3(g_3)\\cdots a_{k+1}(g_{k+1})\n\\tag{$\\star$}\n\\end{equation*}\n\\begin{equation*}\n-(\\Phi c)(a_1\\ast a_2,a_3,...,a_{k+1})(g)=-\\int_{h\\cdot g_3\\cdots g_{k+1}=g}(c(-,g_3,...,g_{k+1})(a_1\\ast a_2))(h)a_3(g_3)\\cdots a_{k+1}(g_{k+1})\\tag{$\\star\\star$}\n\\end{equation*}\n\\begin{multline*}\n\\sum_{i=2}^k(-1)^i(\\Phi c)(a_1,...,a_i\\ast a_{i+1},...,a_{k+1})(g)=\\\\\n\\sum_{i=2}^{k}\\int_{g_1\\cdots g_{k+1}=g}(-1)^i (c(-,g_2,...,g_ig_{i+1},...,g_{k+1})a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\n\\end{multline*}\n\\begin{equation*}\n(-1)^{k+1}((\\Phi c)(a_1,...,a_k)\\ast a_{k+1})(g)=(-1)^{k+1}\\int_{g_1\\cdots g_{k+1}=g}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\n\\end{equation*}\nThe latter two terms we recognize from the differential of the deformation complex, while the first two terms can be rewritten to:\n\\begin{multline*}\n(\\star)+(\\star\\star)=\\int_{hg_3\\cdots g_{k+1}=g}\\left((a_1\\ast (c(-,g_3,...,g_{k+1}) a_2)-c(-,g_3,...,g_{k+1})(a_1\\ast a_2)\\right)(h)a_3(g_3)\\cdots a_{k+1}(g_{k+1})\n\\end{multline*}\nThen by the key Lemma we can rewrite this to\n\\begin{align*}\n(\\star)+(\\star\\star)=&-\\int_{g_1\\cdots g_{k+1}=g}((\\overline{m}c)(-,g_2,...,g_{k+1})a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\n\\end{align*}\nPutting this all together we conclude that:\n\\begin{align*}\n(\\delta_\\text{Hoch}(\\Phi c))(a_1,...,a_{k+1})(g)\n&=(\\Phi(\\delta c))(a_1,...,a_{k+1})(g)\n\\end{align*}\nSo we see that $\\Phi$ is indeed a chain-map.\n\\end{proof}\n\\subsection{Comparing deformation classes}\nIn this section we compare the deformation classes in $H^2_\\text{def}(\\mathcal{G})$ and $H^2_{\\rm Hoch}(\\mathcal{A}_\\mathcal{G})$ coming from deformations of the Lie groupoid $\\mathcal{G}$.\nRecall from \\cite[\\S 5.2]{cms} that an $s$-constant deformation of $\\mathcal{G}$ is a smooth family $\\overline{m}_\\epsilon: \\mathcal{G}\\hspace*{1mm}^s\\!\\times^s\\mathcal{G}\\to\\mathcal{G}$ of division maps parameterized by $\\epsilon$ in an open interval in $\\mathbb{R}$ containing $0$, such that $\\overline{m}_0=\\overline{m}$. This induces a deformation cocycle $\\beta\\in C^2_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ by deforming the associative algebra that is the convolution algebra:\n\\begin{equation*}\n\\beta(a_1,a_2)(g)=\\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0}\\int_{h\\in s^{-1}(s(g))}a_1(\\overline{m}_\\epsilon(g,h))a_2(h)\n\\end{equation*}\nOn the other hand the deformation also induces a deformation element $\\xi\\in C^2_\\text{def}(\\mathcal{G})$ set for $(g,g')\\in\\mathcal{G}^{(2)}$ by:\n\\begin{equation*}\n\\xi(g,g'):=\\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0}\\overline{m}_\\epsilon(gg',g').\n\\end{equation*}\nBy \\cite[Lemma 5.3]{cms}, this cochain is is closed: $\\delta\\xi=0$.\n\\begin{prop}\nThe chain map $\\Phi$ sends $\\xi$ to $\\beta$.\n\\end{prop}\n\\begin{proof}\nThis follows from the observation that if $s(h)=s(g)$, then\n\\begin{equation*}\n\\xi(gh^{-1},h)=\\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0}\\overline{m}_e(g,h)\n\\end{equation*}\nWith this we see that\n\\begin{equation*}\n\\beta(a_1,a_2)(g)=\\int_{h\\in s^{-1}(s(g))}(\\xi(-,h)a_1)(gh^{-1})a_2(h)=\\Phi(\\xi)(a_1,a_2)(g),\n\\end{equation*}\nexactly as needed.\n\\end{proof}\n\\begin{rmk}\nIn \\cite[Prop 5.12]{cms} a deformation cocycle $\\xi\\in C^2_\\text{def}(\\mathcal{G})$ is assigned to any deformation (in particular those who are not $s$-constant), whose cohomology class is canonical. Then $\\Phi(\\xi)$ induces a Hochschild cohomology class of degree 2, which is not immediately linked to a deformation of the convolution product, since if the source map changes the underlying space of the convolution algebra also changes as it consists of densities along the $s$-fibers. Indeed, in \\cite{cms} the authors need an auxillary choice of a vector field on the larger deformation space to define the cocycle. This choice of an auxillary vector field is precisely what is needed to compare the various convolution algebras when the source map varies, and in this way $\\Phi$ maps $[\\xi]\\in H^2_\\text{def}(\\mathcal{G})$  to the Hochschild class of the deformation of the convolution product thus defined. \n\\end{rmk}\n\n\\subsection{Compatibility with the characteristic map to cyclic cochomology}\nDenote by $(C^\\bullet_{\\rm diff}(\\mathcal{G}),\\delta)$ the cochain complex of inhomogeneous groupoid cochains given by $C^k_{\\rm diff}(\\mathcal{G}):=C^\\infty(\\mathcal{G}^{(k)})$ with differential\n\\begin{align*}\n\\delta\\varphi(g_1,\\ldots,g_{k+1})&=\\varphi(g_2,\\ldots,g_{k+1})+\\sum_{i=1}^k(-1)^i\\varphi(g_1,\\ldots,g_ig_{i+1},\\ldots, g_k)+(-1)^{k+1}\\varphi(g_1,\\ldots, g_k).\n\\end{align*}\nWe can turn this cochain complex into a DGA by introducing the product $\\cup:C^k_{\\rm diff}(\\mathcal{G})\\times C^l_{\\rm diff}(\\mathcal{G})\\to C^{k+l}_{\\rm diff}(\\mathcal{G})$ given by\n\\[\n(\\varphi\\cup\\psi)(g_1,\\ldots,g_{k+l}):=\\varphi(g_1,\\ldots,g_k)\\psi(g_{l+1},\\ldots, g_{k+l}).\n\\]\nIn \\cite{cms} it is shown that by replacing $\\varphi$ by a deformation cochain $c\\in C^k_{\\rm def}(\\mathcal{G})$ in the above formula, $C^\\bullet_{\\rm def}(\\mathcal{G})$ becomes a right module over $C^\\bullet_{\\rm diff}(\\mathcal{G})$. On the other hand, in \\cite{ppt}, the smooth groupoid cohomology was used to construct cyclic cocycles. In this section we shall that these two structures are compatible with each others under the cochain map $\\Phi$ to Hochschild cohomology of \\cref{cmh}. We start by re-writing the map to cyclic cohomology of \\cite{ppt} in the following way.\n\nFirst recall that the Hochschild cochain complex $C^\\bullet(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})$ can be given a DGA structure by introducing the product $\\cup:C^k(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})\\times C^l(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})\\to C^{k+l}(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})$\n\\[\n(D\\cup E)(a_1,\\ldots,a_{k+l}):=D(a_1,\\ldots,a_k)*E(a_{k+1},\\ldots,a_{k+l}).\n\\]\nConstruct a map $\\Phi_0:C^\\bullet_{\\rm diff} (\\mathcal{G})\\to C^\\bullet(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ by\n\\begin{equation}\n\\label{dgm}\n\\Phi_0(\\varphi)(a_1,\\ldots,a_k)(g):=\\int_{g_1\\cdots g_k=g}\\varphi(g_1,\\ldots,g_k)a_1(g_1)\\cdots a_k(g_k).\n\\end{equation}\n\\begin{lem}\nThe map $\\Phi_0:(C^\\bullet_{\\rm diff} (\\mathcal{G}),\\delta,\\cup)\\to (C^\\bullet(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G}),\\delta_{\\rm Hoch},\\cup)$ is a morphism of DGA's.\n\\end{lem}\n\\begin{proof}\nThis is a straightforward computation.\n\\end{proof}\nWith this Lemma we can also equip the Hochschild complex $C^\\bullet(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ with a module structure over $C^\\bullet_{\\rm diff}(\\mathcal{G})$ by using the cup-product on Hochschild cochains:\n\\[\n(D\\cup E)(a_1,\\ldots, a_{k+l}):=D(a_1,\\ldots,a_k)E(a_{k+1},\\ldots,a_{k+l}).\n\\] \nExplicitly, this module structure is given by\n\\[\nD\\cdot\\varphi:=D\\cup\\Phi_0(\\varphi).\n\\]\nWe then have:\n\\begin{prop}\nThe cochain map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ is a morphism of $C^\\bullet_{\\rm diff}(\\mathcal{G})$-modules.\n\\end{prop}\n\\begin{proof}\nLet us start with the following case: For $c\\in C^k_\\text{def}(\\mathcal{G})$ and $f \\in C^\\infty(\\mathcal{G})=C^1_{\\rm diff}(\\mathcal{G})$ we have\n\\begin{equation*}\n\\Phi(c\\cup f)(a_1,...,a_{k+1})=\\Phi(c)(a_1,...,a_k)\\ast (f\\cdot a_{k+1})\n\\end{equation*}\nThe claim follows by carefully writing out the definition\n\\begin{align*}\n\\Phi(c\\cup f)(a_1,...,a_{k+1})(g)&=\\int_{g_1\\cdots g_{k+1}=g}((c\\cup f)(-,g_2,...,g_{k+1})a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\\\\\n&=\\int_{g_1\\cdots g_{k+1}=g}(f(g_{k+1})c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\\\\\n&=\\int_{g_1\\cdots g_{k+1}=g}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_k(g_k)\\left(f(g_{k+1})a_{k+1}(g_{k+1})\\right)\\\\\n&=\\int_{hg_{k+1}=g}\\int_{g_1\\cdots g_k=h}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_k(g_k)\\left(f(g_{k+1})a_{k+1}(g_{k+1})\\right)\\\\\n&=\\int_{hg_{k+1}=g}\\Phi(c)(a_1,...,a_k)(h)\\left(f(g_{k+1})a_{k+1}(g_{k+1})\\right)\\\\\n&=\\left(\\Phi(c)(a_1,...,a_k)\\ast (f\\cdot a_{k+1})\\right)(g)\n\\end{align*}\nHence by induction we obtain\n\\begin{equation*}\n\\Phi(c\\cup (f_1\\otimes\\cdots\\otimes f_l))(a_1,...,a_{k+l})=\\Phi(c)(a_1,...,a_k)\\ast (f_1\\cdot a_{k+1})\\ast\\cdots\\ast (f_l\\cdot a_{k+l})\n\\end{equation*}\nWriting $f\\ast a=\\Phi_0(f)(a)$, we can rewrite this as\n\\[\n\\Phi(c\\cup (f_1\\otimes\\cdots\\otimes f_l))=\\Phi (c)\\cup\\Phi_0(f_1)\\cup\\ldots\\cup\\Phi_0(f_l).\n\\]\nFrom this the general statement of the proposition follows. \n\\end{proof}\n\n Now, analogous to the action of vector fields on differential forms in geometry, the Hochschild cochains act on Hochschild chains by contraction:\n\\[\nC^k(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})\\times C_l(\\mathcal{A}_\\mathcal{G})\\longrightarrow C_{l-k}(\\mathcal{A}_\\mathcal{G}),\\qquad (D,a)\\mapsto\\iota_Da,\n\\]\ngiven explicitly by\n\\[\n\\iota_D(a_0\\otimes\\ldots\\otimes a_{k+l}):=a_0D(a_1,\\ldots,a_k)\\otimes a_{k+1}\\otimes\\ldots\\otimes a_{k+l}.\n\\]\nThis action satisfies the properties\n\\begin{align*}\n\\iota_D\\circ\\iota_E&=\\iota_{D\\cup E}\\\\\n[b,\\iota_D]&=\\iota_{\\delta D}.\n\\end{align*}\nThe analogue of the Cartan formula for the ``Lie derivative''  $L_D:=B\\circ \\iota_D+\\iota_D\\circ B$ in noncommutative geometry also holds true on the level of Hochschild homology.\n\nNext, recall from \\cite{ppt} that when $\\mathcal{G}$ is unimodular we can define a trace on the convolution algebra $\\mathcal{A}_\\mathcal{G}$ by \n\\[\n\\tau(a):=\\int_Ma\\Omega,\n\\]\nwith on the right hand side $\\Omega$ a $\\mathcal{G}$-invariant section of the bundle $\\mathcal{D}_{A^*}\\otimes\\mathcal{D}_{TM}$, and we use the duality $\\mathcal{D}_A\\times\\mathcal{D}_{A^*}\\to\\mathbb{R}$ together with the isomorphism $\\mathcal{D}_s|_M=\\mathcal{D}_A$, to obtain a density on $M$ that can be integrated. With this trace (a degree $0$ cyclic cocycle), the cochain map \n\\begin{equation}\n\\label{chain-diff}\n\\Psi_\\tau:(C^\\bullet_{\\rm diff}(\\mathcal{G}),\\delta)\\longrightarrow (C^\\bullet(\\mathcal{A}_\\mathcal{G}),b_{\\rm Hoch}),\n\\end{equation}\nconstructed in \\cite{ppt} is simply given by $\\Psi_\\tau(c):=\\iota_{\\Phi_0(c)}\\tau$.\n\\begin{cor}\nLet $c\\in C^k_{\\rm def}(\\mathcal{G})$ and $f\\in C^l_{\\rm diff}(\\mathcal{G})$. Then the following\nidentity holds true:\n\\[\n\\iota_{\\Phi(c\\cup f)}\\tau=\\iota_{\\Phi(c)}\\Psi_\\tau(f).\n\\]\n\\end{cor}\nWith this Corollary, we can construct new cyclic cocycles on the convolution algebra. First of all, if we start with a smooth \ngroupoid cocycle $\\varphi\\in C^k_{\\rm diff}(\\mathcal{G})$, we obtain a Hochschild cocycle by applying $\\Psi_\\tau$ as in \\eqref{chain-diff}. A small computation shows that this cocycle is closed under the $B$-differential, i.e., $B\\Psi_\\tau(\\varphi)=0$, when $\\varphi$ is cyclic:\n\\[\n\\varphi(g_1,\\ldots,g_k)=(-1)^k\\varphi((g_1\\cdots g_k)^{-1},g_1,\\ldots,g_{k-1})\n\\]\nWe can work out similar conditions for elements $c\\in C^k_\\text{def}(\\mathcal{G})$, but they are more involved. For example, for $k=2$ we find\n\\begin{equation*}\n(d\\iota)(c(g,g^{-1}))=-c(g^{-1},g).\n\\end{equation*}\n\n\n\\begin{rmk}\nIt is proved in \\cite[\\S 9]{cms} that $H^\\bullet_\\text{def}(\\mathcal{G})\\cong H^\\bullet(\\mathcal{G},{\\rm Ad})$, where ${\\rm Ad}$ denotes the adjoint representation up to homotopy constructed in \\cite{ac}. Taking into account the morphism \\eqref{dgm}, this strongly suggests to relabel the morphism  of \\cref{chainmap} as $\\Phi_1$ and conjecture the existence of a map $\\Phi_p:H^\\bullet(\\mathcal{G},{\\rm Sym}^p({\\rm Ad}))\\to H^\\bullet_{\\rm Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ extending the cases $p=0,1$ described in this paper. This would naturally fit with the infinitesimal theory (see also the next section) and the computation in \\cite{blom} of the Hochschild cohomology of the universal enveloping algebra $\\mathcal{U}(A)$ of the Lie algebroid $A$:\n\\[\nH^\\bullet_{\\rm Hoch}(\\mathcal{U}(A),\\mathcal{U}(A))\\cong \\bigoplus_{p\\geq 0} H^\\bullet_{CE}(A,{\\rm Sym}^pA).\n\\]\n\\end{rmk}\n\\subsection{The case $k=0$}\nFor the chain map between $C^\\bullet_\\text{def} (\\mathcal{G})$ and $C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ we have just defined, a natural question is whether it can be extended to degree $k=0$, c.f. \\cref{deg0}. For this, one must find a  a map  $\\Phi^0: \\Gamma(A)\\to \\mathcal{A}_\\mathcal{G}$ which extends the chain map $\\Phi$. This is only possible if $\\Phi(\\delta(\\alpha))\\in\\text{Der}(\\mathcal{A}_\\mathcal{G})$ is an inner derivation for every $\\alpha\\in \\Gamma(A)$.\n\nIntuitively it is clear that is should not be always possible, since the derivation $\\Phi(\\delta(\\alpha))$ includes taking derivatives, while an inner derivation $\\partial_H(a)$ only includes integrations. The following example presents a concrete counterexample:\n\n\\begin{ex} Consider the pair groupoid $\\mathbb{R}\\times\\mathbb{R}\\rightrightarrows \\mathbb{R}$. For this groupoid, a bundle of densities is trivialized by $|dx|$, so that every compactly supported density is of the form $f|dx|$ for a compactly supported smooth function $f$.  Furthermore, a section of the algebroid is simply a vector field $X\\in\\mathfrak{X}(\\mathbb{R})$ and for this example we take $X=\\frac{\\partial}{\\partial x}$. We have\n\\begin{equation*}\n\\delta(X)(x,y)=(X(x),X(y))\n\\end{equation*}\nso that in this case $\\delta(X)=\\frac{\\partial}{\\partial x}+\\frac{\\partial}{\\partial y}$. This vector field has flow\n\\begin{equation*}\n\\Phi^t_{\\delta(X)}(x,y)=(x+t,y+t)\n\\end{equation*}\nNext we consider $\\Phi\\left(\\delta\\left(\\frac{\\partial}{\\partial x}\\right)\\right)$, so we look at the action of $\\frac{\\partial}{\\partial x}+\\frac{\\partial}{\\partial y}$ on a density $f(x,y)|dx|\\in \\mathcal{A}_{\\mathbb{R}\\times\\mathbb{R}}$. We see\n\\begin{equation*}\n(\\Phi^t_{\\delta(X)})^\\ast (f(x,y)|dx|)=f(x+t,y+t)|d(x+t)|=f(x+t,y+t)|dx|\n\\end{equation*}\nSo that:\n\\begin{equation*}\n\\Phi(\\delta(X))(f|dx|)=\\left(\\frac{\\partial f}{\\partial x}+\\frac{\\partial f}{\\partial y}\\right)|dx|\n\\end{equation*}\nNow suppose that there is some $g|dx|\\in \\mathcal{A}_{\\mathbb{R}\\times\\mathbb{R}}$, such that $\\Phi(\\delta(X))=\\partial_H(g|dx|)$. Then since always $\\partial_H(g|dx|)(g|dx|)=0$, we see that:\n\\begin{equation*}\n\\frac{\\partial g}{\\partial x}+\\frac{\\partial g}{\\partial y}=0\n\\end{equation*}\nso that\n\\begin{equation*}\ng(x+t,y+t)=g(x,y)\n\\end{equation*}\nSince $g$ has to be compactly supported, the only possibility is that $g=0$, which is obviously not a solution to $\\Phi(\\delta(X))=\\partial_H(g|dx|)$. We conclude that $\\Phi(\\delta(X))$ is not an inner derivation.\n\\end{ex}\nIn fact, using supports as an argument, we can deduce that $\\Phi(X)$ can never be an inner derivation for any $X\\in\\mathfrak{X}_s(\\mathcal{G})$.\n\\begin{prop}\nLet $D\\in\\text{Hom}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ be a non-zero Hochschild-1-cochain. If $D$ satisfies $\\text{supp}(Da)\\subset\\text{supp}(a)$, then there is no $b\\in \\mathcal{A}_\\mathcal{G}$ such that $D=[-,b]$.\n\\end{prop}\n\\begin{proof}\nSuppose by contrary that there is a $b$ such that $D=[-,b]$. Let $g\\in\\mathcal{G}$ and let $a\\in \\mathcal{A}_\\mathcal{G}$ be supported arbitrarily close to $g$. For $h\\in t^{-1}(s(g))$ outside of the isotropy of $s(g)$ we obtain:\n\\begin{equation*}\n(a\\ast b)(gh)=\\int_k a(gk^{-1})b(kh)\\sim a(g)b(h)\n\\end{equation*}\nwhere we use that $a$ is only non-zero close enough to $g$. For the other part of the commutator we have\n\\begin{equation*}\n(b\\ast a)(gh)=\\int_k b(gk^{-1})a(kh)=0\n\\end{equation*}\nSince there is no way to let $kh$ come arbitrarily close to $g$ since $h$ is not in the isotropy of $s(g)$.\n\nSince $\\text{supp} (Da)\\subset \\text{supp}(a)$ we see that $(a\\ast b)(gh)$ also has to be supported arbitrarily close to $g$, so that $b$ is identically zero outside of the isotropy of $\\mathcal{G}$.\n\nIf we look at $h$ an isotropy element of $\\mathcal{G}$ we see that the second term acts like $b(ghg^{-1})a(g)$, so that we see that $b$ is invariant under conjugation. However, if $b$ is invariant under conjugation we conclude that $b\\in Z(\\mathcal{A}_\\mathcal{G})$, which is in contradiction to the fact that $D$ is non-zero. We conclude that there is no $b$ that solves $D=[-,b]$.\n\\end{proof}\n\\begin{rmk}\nIt is possible to define the map $\\Phi^0$ if one allows for distributions to be cochains of degree $0$, that is if one defines $C^0_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G}):=\\Gamma^{-\\infty}_c(\\mathcal{D}_s)$.\n\\end{rmk}\n\\begin{cor} If $X\\in C^1_\\text{def}(\\mathcal{G})$ is non-zero, then $\\Phi(X)$ can never be an inner derivation.\n\\end{cor}\n\\begin{proof}\nThis follows from the previous proposition by the observation that $\\Phi(X)$ is local since it involves taking derivatives and the fact that $\\Phi$ is easily observed to be injective.\n\\end{proof}\n\\subsection{Examples}\nIn this section we discuss how the chain map $\\Phi$ links the deformation cohomology of $\\mathcal{G}$ and the Hochschild cohomology of $\\mathcal{A}_\\mathcal{G}$ in certain examples.\n\n\\begin{ex}[Trivial groupoid]\nWe consider the trivial groupoid $\\mathcal{G}=M\\rightrightarrows M$. On the density side we simply have $(\\mathcal{A}_\\mathcal{G},\\ast)=(C^\\infty_c(M),\\cdot)$, with $H^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})=\\Lambda^\\bullet\\mathfrak{X}(M)$. At the side of the deformation complex we note that the $k$-nerve of the trivial groupoid is $M$ for every $k$ and $s$-projectability is a void property, so that for $k>0$ we have $C^k_\\text{def}(\\mathcal{G})=\\mathfrak{X}(M)$, with differential alternating between the identity and the zero map:\n\\begin{equation*}\nC^\\bullet_\\text{def}(\\mathcal{G})=\\left[ 0\\to \\mathfrak{X}(M)\\xrightarrow{0}\\mathfrak{X}(M)\\xrightarrow{\\rm id}\\mathfrak{X}(M)\\to\\cdots\\right]\n\\end{equation*}\nSo the deformation cohomology equals:\n\\begin{equation*}\nH^k_\\text{def}(\\mathcal{G})\\cong\\left\\{\\begin{matrix}\n\\mathfrak{X}(M) &\\text{ if } k=1\\\\\n0 & \\text{ else}\n\\end{matrix}\\right.\n\\end{equation*}\nThe chain map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ simply becomes:\n\\begin{equation*}\n\\Phi(X)(f_1,...,f_k)=(Xf_1)\\cdot f_2\\cdots f_k\n\\end{equation*}\nand we simply see that:\n\\begin{equation*}\nH^k(\\Phi)=\\left\\{\\begin{matrix}\n\\text{id} &\\text{ if }k=1\\\\\n0 & \\text{ else}\n\\end{matrix}\\right.\n\\end{equation*}\nWe should also remark for this example that using the classical Hochschild--Kostant--Rosenberg theorem, we see that taking exterior powers of deformation elements we retrieve the whole Hochschild cohomology of $C^\\infty_c(M)$.\n\\end{ex}\n\\begin{ex}[\\'Etale groupoids]\nIn the case of an \\'Etale groupoid $\\mathcal{G}\\rightrightarrows M$, we have $\\mathcal{A}_\\mathcal{G}=C^\\infty_c(\\mathcal{G})$, since the distribution $\\ker(ds)$ is the trivial distribution. The convolution product in this case is commonly written as\n\\begin{equation*}\n(f_1\\ast f_2)(g)=\\sum_{g_1g_2=g}f_1(g_1)f_2(g_2)\n\\end{equation*}\nIn this case the action of vector fields on densities is just the normal action of vector fields on functions, and the map $\\Phi$ reduces to\n\\begin{equation*}\n\\Phi(c)(f_1,...,f_k)(g)=\\sum_{g_1\\cdots g_k=g}(c(g_1,...,g_k)f_1)\\cdot f_2(g_2)\\cdots f_k(g_k)\n\\end{equation*}\nSince the source map of $\\mathcal{G}$ is a local diffeomorphism, we see that there is a 1-1 correspondence between deformation elements $c\\in C^k_\\text{def}(\\mathcal{G})$ and their symbols $s_c\\in\\Gamma(t^\\ast TM\\to\\mathcal{G}^{(k-1)})$ since we have\n\\begin{equation*}\nc(g_1,...,g_k)=(ds_{g_1})^{-1}(s_c(g_2,...,g_k))\n\\end{equation*}\nIn fact, the correspondence establishes an isomorphism between $C^\\bullet_\\text{def}(\\mathcal{G})$ and $C^\\bullet(\\mathcal{G},TM)[-1]$ where we see $TM$ as a representation of $\\mathcal{G}$ where $g$ acts $T_{s(g)}M\\to T_{t(g)}M$ as\n\\begin{equation*}\ng\\cdot v=dt_g((ds_g)^{-1})(v))\n\\end{equation*}\nThe shift by 1 we see here also serves as a justification of why the case $k=0$ is a tricky thing (although for \\'Etale groupoids of course we have $C^0_\\text{def}(\\mathcal{G})=0$).\n\nIn the case that we have a proper \\'Etale groupoid (over a connected base $M$) we can calculate the cohomologies in both sides of the equation. On the side of the deformation complex we use \\cite[Thm 6.1]{cms} to obtain:\n\\begin{align*}\nH^0_\\text{def}(\\mathcal{G})&\\cong\\{0\\}\\\\\nH^1_\\text{def}(\\mathcal{G})&\\cong\\mathfrak{X}(M)_\\text{inv}\\\\\nH^k_\\text{def}(\\mathcal{G})&\\cong\\{0\\}\\hspace*{1cm}(k\\geq 2)\n\\end{align*}\nFor the Hochschild cohomology of the convolution algebra we refer to \\cite[Thm 3.11]{nppt} to obtain\n\\begin{align*}\nH^k(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})\\cong \\bigoplus_{\\mathcal{O}\\in\\text{Sec}(\\mathcal{G})}\\Gamma_\\text{inv}(\\Lambda^{k-\\text{codim}(\\mathcal{O})}T\\mathcal{O})\n\\end{align*}\nwhere the sum is over the sectors $\\mathcal{O}$ of $\\mathcal{G}$. The action of the chain map $\\Phi$ on the cohomology of degree $1$ is the inclusion of $\\mathfrak{X}(M)_\\text{inv}$ into this sum as the term for the sector $\\mathcal{O}=M$.\n\\end{ex}\n\\section{Deformation quantization and the van Est map}\\label{quant}\n\\subsection{The adiabatic groupoid}\\label{adiabatic}\n\nIn the theory of deformation quantizations and applications thereof, there is an inherent place for replacing a groupoid with its adiabatic groupoid, as first described in \\cite{connes}. In view of our discussion of the deformation complex, we describe it using the division map:\n\\begin{defi}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid, with Lie algebroid $A\\xrightarrow{\\pi} M$. We define the \\textit{adiabatic groupoid} $\\mathcal{G}_{\\text{ad}}\\to M\\times\\mathbb{R}$ by:\n\\begin{equation*}\n\\mathcal{G}_{\\text{ad}}=A\\times\\{0\\}\\sqcup \\mathcal{G}\\times\\mathbb{R}^\\ast\n\\end{equation*}\nThe source and target are defined by:\n\\begin{equation*}\ns(v,0)=(\\pi(v),0)\n\\end{equation*}\n\\begin{equation*}\ns(g,\\tau)=(s(g),\\tau)\n\\end{equation*}\n\\begin{equation*}\nt(v,0)=(\\pi(v),0)\n\\end{equation*}\n\\begin{equation*}\nt(g,\\tau)=(t(g),\\tau)\n\\end{equation*}\nThen we define the inversion map by\n\\begin{equation*}\n\\iota(v,0)=(-v,0)\n\\end{equation*}\n\\begin{equation*}\n\\iota(g,\\tau)=(\\iota(g),\\tau)\n\\end{equation*}\nLastly, to define we division map, we note that pairs of divisible arrows come in 2 shapes, namely pairs $(v,0)$ and $(w,0)$ with $\\pi(v)=\\pi(w)$, and pairs $(g,\\tau)$ and $(h,\\tau)$ where $g$ and $h$ are divisible. We then define the division map by:\n\\begin{equation*}\n\\overline{m}((v,0),(w,0))=(v-w,0)\n\\end{equation*}\n\\begin{equation*}\n\\overline{m}((g,\\tau),(h,\\tau))=(\\overline{m}(g,h),\\tau)\n\\end{equation*}\n\\end{defi}\nThis is just the set-theoretical description, but the remarkable feature is that the adiabatic groupoid can be given a smooth\nstructure. Here we briefly recall this smooth structure and show how to extend normalized deformation elements to deformation elements of the adiabatic groupoid. Both will be done in the context of the procedure known as the {\\em deformation to the normal cone}.\n\\subsubsection{Deformation to the normal cone}\nThe part of the discussion below concerning the smooth structure and the smooth maps on the deformation to the normal cone is after \\cite[\\S 4]{higson} and \\cite[\\S 1.1]{ds}.\n\\begin{defi}\\label{nms}\nLet $S\\hookrightarrow M$ be a submanifold with normal bundle $N\\to S$. The \\textit{deformation to the normal cone} $N(M,S)$ is the manifold defined by:\n\\begin{equation*}\nN(M,S)=N\\times\\{0\\}\\sqcup M\\times\\mathbb{R}^\\ast\n\\end{equation*}\n\\end{defi}\nThe deformation to the normal cone can be given a topology and smooth structure in two ways. Either it is characterized by the fact that the following two types of maps\n\\begin{itemize}\n\\item The map $N(M,S)\\to M\\times\\mathbb{R}$ that sends $(x,\\tau)$ for $\\tau\\neq  0$ to $(x,\\tau)$ and sends $(v,0)$ with $v\\in N_x$ to $(x,0)$.\n\\item For every $f\\in C^\\infty(M)$ such that $f|_S=0$, the map $\\delta f: N(M,S)\\to\\mathbb{R}$ defined by\n\\begin{align*}\n(\\delta f)(x,\\tau)&=\\frac{f(x)}{\\tau}\\,\\,\\,(x\\in M,\\,\\tau\\neq 0),\\\\\n(\\delta f)(v,0)&=d_nf(v)\\,\\,\\,(v\\in N)\n\\end{align*}\n\\end{itemize}\nare smooth. Here by $d_nf$ we mean the smooth map on $N$ that for $v\\in TM|_S$ sends $[v]$ to $df(v)$ and which is well-defined since $f|_S=0$.\n\nEquivalently, one uses an exponential map, that is a map $\\theta: U\\to M$ from an open neighbourhood $U\\subset N$ of the zero-section, with the property that for all $p\\in S$ and $v\\in N_p$ it holds that\n\\begin{equation*}\n\\theta(0_p)=p,\\qquad \n\\left.\\frac{d}{d\\tau}\\right|_{\\tau=0}\\theta(\\tau v)=v ~\\text{ mod }T_pS\n\\end{equation*}\nThe smooth structure on $N(M,S)$ can then also be characterized by the fact that the maps\n\\begin{equation*}\ni_1: M\\times\\mathbb{R}^\\ast\\to N(M,S):\\,\\,(x,\\tau)\\mapsto (x,\\tau)\n\\end{equation*}\n\\begin{equation*}\ni_2: U'=\\{(v,\\tau)\\in N\\times\\mathbb{R}: \\tau v\\in U\\}\\to N(M,S):\\,\\,\\begin{matrix}(v,\\tau)&\\mapsto& (\\theta(\\tau v),\\tau)\\\\\n(v,0)&\\mapsto& (v,0)\\end{matrix}\n\\end{equation*}\nare open smooth embeddings.\n\nImportant in considering deformations to normal cones is the action of $\\mathbb{R}^\\ast$ on $N(M,S)$, which is given by:\n\\begin{align*}\n\\lambda\\cdot (x,\\tau)&=(x,\\lambda\\tau),\\\\\n\\lambda\\cdot (v,0)&=\\left(\\frac{v}{\\lambda},0\\right)\n\\end{align*}\nwhere $\\lambda,\\tau\\in\\mathbb{R}^\\ast$, $x\\in M$ and $v\\in N$.\n\nWe will describe how to extend a vector field on $M$, that is parallel to $S$, to a vector field on $N(M,S)$ that is invariant under the $\\mathbb{R}^\\ast$-action.\n\nThis will be done by writing down a vector field on the normal bundle and combining it with a vector field over $M\\times\\mathbb{R}^\\ast$ to a discrete vector field on $N(M,S)$, and using an explicit description of the smooth functions on $N(M,S)$ to show that this is in fact  a {\\em smooth} vector field.\n\\begin{defi} \\cite{higson}\nLet $X$ be a set and $\\mathcal{F}=\\{f_\\alpha: X\\to V_\\alpha\\}$ be a family of functions from $X$ into smooth manifolds. We say that a function $f: X\\to\\mathbb{R}$ is \\textit{smoothly composed from the family $\\mathcal{F}$} if there is a finite collection $(f_{\\alpha_1},...,f_{\\alpha_n})\\subset\\mathcal{F}$ and a smooth map $h: V_{\\alpha_1}\\times\\cdots V_{\\alpha_n}\\to\\mathbb{R}$ such that\n\\begin{equation*}\nf(x)=h(f_{\\alpha_1}(x),...,f_{\\alpha_n}(x))\n\\end{equation*}\n\\end{defi}\nThe smooth structure of $N(M,S)$ then means that all smooth functions on $N(M,S)$ are smoothly composed of type of functions as described after \\cref{nms}. If we then apply Taylors theorem we conclude the following.\n\\begin{lem}\\label{smoothvfnms}\nA discrete vector field $X$ on $N(M,S)$ is smooth if and only if for every $f\\in C^\\infty(M)$ with $f|_s=0$ and every $g\\in C^\\infty(M\\times\\mathbb{R})$ the maps $\\delta f$ and $\\tilde{g}\\in C^\\infty(N(M,S))$ defined by:\n\\begin{equation*}\n\\begin{matrix}\n(\\delta f)(x,\\tau)=\\frac{f(x)}{\\tau}&\\,\\,\\,(\\tau\\neq 0)\\\\\n(\\delta f)(v,0)=d_nf(v)&\\,\\,\\,(v\\in N)\\\\\n\\\\\n\\tilde{g}(x,\\tau)=g(x,\\tau)&\\,\\,\\,(\\tau\\neq 0)\\\\\n\\tilde{g}(v,0)=g(x,0)&\\,\\,\\,(v\\in N_x)\n\\end{matrix}\n\\end{equation*}\nsatisfy that $X(\\delta f),X(\\tilde{g})\\in C^\\infty(N(M,S))$.\n\\end{lem}\nWe start with writing down the vector field over $N$. This is the {\\em linearization}, as also in \\cite[\\S 4.1]{az}, that we describe in detail below:\n\\begin{prop}\\label{vectorfieldnormalbundle}\nLet $S\\hookrightarrow M$ be a submanifold with normal bundle $\\pi:N\\to S$ and $X\\in\\mathfrak{X}(M)$ a vector field that is parallel to $S$. Then:\n\\begin{itemize}\n\\item[\\textbf{a)}] The map that sends a smooth function $f\\in C^\\infty(M)$ satisfying $f|_S=0$ to the map $d_nf\\in C^\\infty_\\text{lin} (N)$ is a surjection onto $C^\\infty_\\text{lin}(N)$\n\\item[\\textbf{b)}] If $f\\in C^\\infty(M)$ satisfies that $f|_S=0$ and $d_nf=0$, then $Xf$ satisfies that $d_n(Xf)=0$.\n\\item[\\textbf{c)}] The maps $(X_N)_{\\text{lin}}: C^\\infty_\\text{lin}(N)\\to C^\\infty(N)$ and $(X_N)_{\\text{cst}}: C^\\infty(S)\\to C^\\infty(N)$ defined by\n\\begin{equation*}\n(X_N)_\\text{lin}(d_nf)=d_n(Xf)\n\\end{equation*}\n\\begin{equation*}\n(X_N)_\\text{cst}(g)=X|_S (g)\\circ \\pi\n\\end{equation*}\ndefine a smooth vector field $X_N\\in\\mathfrak{X}(N)$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof} Working down the list:\n\\begin{itemize}\n\\item[\\textbf{a)}] By using a partition of unity this reduces to the local case $M=\\mathbb{R}^m\\times\\mathbb{R}^n$ with $S=\\mathbb{R}^m\\times\\{0\\}$. In this local case there is a canonical diffeomorphism between $M$ and $N$ and pushing a linear map on $N$ through this canonical diffeomorphism yields a smooth map on $M$ which normal derivative equals the linear map on $N$ we started with.\n\\item[\\textbf{b)}] This is again a computation in the local case $M=\\mathbb{R}^m\\times\\mathbb{R}^n$ with $S=\\mathbb{R}^m\\times\\{0\\}$. Write\n\\begin{equation*}\nX=\\sum_{i=1}^m \\alpha_i(x,y)\\frac{\\partial}{\\partial x_i}+\\sum_{j=1}^n\\beta_j(x,y)\\frac{\\partial}{\\partial y_j}\n\\end{equation*}\nThe fact that $X$ is parallel to $S$ means that $\\beta_j(x,0)=0$ for all $j=1,...,n$. The fact that $d_nf=0$ is equivalent to the fact $\\frac{\\partial f}{\\partial y_j}(x,0)=0$ for all $j=1,...,n$. Then we have\n\\begin{equation*}\nXf=\\sum_{i=1}^m \\alpha_i\\frac{\\partial f}{\\partial x_i}+\\sum_{j=1}^n \\beta_j\\frac{\\partial f}{\\partial y_j}\n\\end{equation*}\nSo that for $k=1,...,n$ we have\n\\begin{equation*}\n\\frac{\\partial (Xf)}{\\partial y_k}=\\sum_{i=1}^m\\frac{\\partial \\alpha_i}{\\partial y_k}\\frac{\\partial f}{\\partial x_i}+\\sum_{i=1}^m\\alpha_i\\frac{\\partial^2 f}{\\partial y_k\\partial x_i}+\\sum_{j=1}^n\\frac{\\partial\\beta_j}{\\partial y_k}\\frac{\\partial f}{\\partial y_j}+\\sum_{j=1}^n\\beta_j\\frac{\\partial^2 f}{\\partial y_k\\partial y_j}\n\\end{equation*}\nThen since respectively $\\frac{\\partial f}{\\partial x_i}(x,0)=0$ (since $f(x,0)=0$), $\\frac{\\partial^2 f}{\\partial y_k\\partial x_i}(x,0)=\\left(\\frac{\\partial}{\\partial x_i}\\frac{\\partial f}{\\partial y_k}\\right)(x,0)=0$ (since $\\frac{\\partial f}{\\partial y_k}(x,0)=0$), $\\frac{\\partial f}{\\partial y_j}(x,0)=0$ (by assumption) and $\\beta_j(x,0)=0$ (by assumption), we see that\n\\begin{equation*}\n\\frac{\\partial(Xf)}{\\partial y_k}(x,0)=0\n\\end{equation*}\nwhich implies that $d_n(Xf)=0$.\n\\item[\\textbf{c)}] First note that (by restriction) a smooth vector field $Y\\in\\mathfrak{X}(E)$ on a vector bundle $\\pi: E\\to M$ is the same as a pair of maps $Y_\\text{lin}: C^\\infty_\\text{lin}(E)\\to C^\\infty(E)$ and $Y_\\text{cst}: C^\\infty(M)\\to C^\\infty(E)$ such that for all $f,g\\in C^\\infty(M)$ and $h\\in C^\\infty_\\text{lin}$ it holds that\n\\begin{equation*}\nY_\\text{cst}(fg)=(f\\circ\\pi)\\cdot Y_\\text{cst}(g)+(g\\circ\\pi)\\cdot Y_\\text{cst}(f)\n\\end{equation*}\n\\begin{equation*}\nY_\\text{lin}((f\\circ\\pi)\\cdot h)=(f\\circ\\pi)\\cdot Y_\\text{lin}(h)+h\\cdot Y_\\text{cst}(f)\n\\end{equation*}\nWe show that these properties hold for the maps $(X_N)_\\text{cst}$ and $(X_N)_\\text{lin}$.\n\nFirst we note that $(X_N)_\\text{lin}$ is well-defined by parts a) and b). To show that they define a smooth vector field we check for $f,g\\in C^\\infty(S)$\n\\begin{align*}\n(X_N)_\\text{cst}(fg)&=(X|_S(fg))\\circ\\pi=(f\\cdot X|_S(g)+g\\cdot X|_S(f))\\circ\\pi\\\\\n&=(f\\circ\\pi)\\cdot (X|_S(g)\\circ\\pi)+(g\\circ\\pi)\\cdot (X|_S(f)\\circ\\pi)\\\\\n&=(f\\circ\\pi)X_\\text{cst}(g)+(g\\circ\\pi)X_\\text{cst}(f)\n\\end{align*}\nSecondly let $f\\in C^\\infty(S)$ and $h\\in C^\\infty_\\text{lin}(N)$ given by $h=d_ng$ with $g\\in C^\\infty(M)$ such that $g|_S=0$. Then first we need to find $g'\\in C^\\infty(M)$ with $g'|_S=0$ such that $fh=d_n(g')$. This can be done by choosing an extension of $f$ which is `constant in the normal direction', which is only well-defined locally or if we choose an exponential map.\n\nWe resort to the local case $M=\\mathbb{R}^m\\times\\mathbb{R}^n$ with $S=\\mathbb{R}^m\\times\\{0\\}$. Then the map $g'(x,y)=f(x)g(x,y)$ clearly satisfies that $d_ng'=fh$. Then writing $X$ in coordinates as\n\\begin{equation*}\nX=\\sum_{i=1}^m\\alpha_i\\frac{\\partial}{\\partial x_i}+\\sum_{j=1}^n\\beta_j\\frac{\\partial}{\\partial y_j}\n\\end{equation*}\nwe have\n\\begin{equation*}\n(Xg')(x,y)=\\sum_{i=1}^m\\alpha_i(x,y)\\frac{\\partial f}{\\partial x_i}(x)g(x,y)+f(x)(Xg)(x,y)\n\\end{equation*}\nso that we see\n\\begin{equation*}\n\\frac{\\partial (Xg')}{\\partial y_k}(x,0)=\\sum_{i=1}^m\\alpha_i(x,0)\\frac{\\partial f}{\\partial x_i}(x)\\frac{\\partial g}{\\partial y_k}(x,0)+\\sum_{i=1}^m\\frac{\\partial \\alpha_i}{\\partial y_k}(x,0)\\frac{\\partial f}{\\partial x_i}(x)g(x,0)+f(x)\\frac{\\partial (Xg)}{\\partial y_k}(x,0)\n\\end{equation*}\nThen $g(x,0)=0$ so that the middle term vanishes. Then recognizing terms we obtain\n\\begin{equation*}\nd(Xg')_{(x,0)}\\left(\\frac{\\partial}{\\partial y_k}\\right)=X|_S(f)(x)\\cdot(dg)_x\\left(\\frac{\\partial}{\\partial y_k}\\right)+f(x)\\cdot d(Xg)_{(x,0)}\\left(\\frac{\\partial}{\\partial y_k}\\right)\n\\end{equation*}\nso that globalizing we have\n\\begin{align*}\n(X_N)_\\text{lin}((f\\circ \\pi)\\cdot d_ng)&=(X_N)_\\text{lin}(d_ng')\\\\\n&=d_n(Xg')\\\\\n&=(X|_S(f)\\circ\\pi)d_ng+(f\\circ\\pi)d(Xg)\\\\\n&=(X_N)_\\text{cst}(f)d_ng+(f\\circ\\pi)(X_N)_\\text{lin}(d_ng)\n\\end{align*}\nSo we see that we obtain a smooth vector field $X_N\\in\\mathfrak{X}(N)$.\n\\end{itemize}\nThis completes the proof.\n\\end{proof}\nWe are now ready to define the $\\mathbb{R}^\\ast$-invariant extension of the vector field $X$.\n\\begin{prop}\\label{vfonnms}\nLet $S\\hookrightarrow M$ be a submanifold with normal bundle $N\\to S$. Let $X\\in\\mathfrak{X}(M)$ be a vector field that is parallel to $S$. Then the discrete vector field $X_\\text{inv}$ on $N(M,S)$ defined by\n\\begin{align*}\nX_\\text{inv}(x,\\tau)&=X(x),\\quad(\\tau\\neq 0)\\\\\nX_\\text{inv}|_{N\\times\\{0\\}}&=X_N\n\\end{align*}\nis a smooth vector field $X_\\text{inv}\\in\\mathfrak{X}(M,S)$ which is the unique vector field on $N(M,S)$ which equals $X$ on $M\\times\\mathbb{R}^\\ast$ and the unique $\\mathbb{R}^\\ast$-invariant vector field on $N(M,S)$ which equals $X$ along $M\\times\\{1\\}$.\n\\end{prop}\n\\begin{proof}\nThe invariance and uniqueness is clear assuming that $X_\\text{inv}$ is smooth. To show that it is smooth, by \\cref{smoothvfnms} the only thing we have to check is that $X_\\text{inv}(\\delta f)$ and $X_\\text{inv}(\\tilde{g})$ are smooth for $f\\in C^\\infty(M)$ with $f|_S=0$ and $g\\in C^\\infty(M\\times\\mathbb{R})$. The definition of $X_N$ makes sure that the result is\n\\begin{equation*}\nX_\\text{inv}(\\delta f)=\\delta(Xf)\n\\end{equation*}\n\\begin{equation*}\nX_\\text{inv}(\\tilde{g})=\\tilde{Xg}\n\\end{equation*}\nwhere in the second equation $X$ acts on $C^\\infty(M\\times\\mathbb{R})$ as the vector field $X(x,\\tau)=X(x)$ on $M\\times\\mathbb{R}$. By definition $\\delta(Xf)$ and $\\tilde{Xg}$ are smooth and so the result follows.\n\\end{proof}\n\\subsubsection{The adiabatic groupoid as a deformation to the normal cone}\nWe can now apply this to the case $M\\hookrightarrow\\mathcal{G}$ with normal bundle $A=\\ker ds|_M$. The fact that the source, target and division maps are smooth, follows from the fact that away from $\\tau=0$ they are just the respective maps of the original groupoid, while along $\\tau=0$ they are the normal derivatives of the respective maps. A general principle of deformations to normal cones then means they are smooth. We note that an exponential map can be obtained by choosing a connection on $A$, see \\cite{nwx} and \\cite{landsmanboek}.\n\nNext we want to describe the nerve of the adiabatic groupoid. As a set it equals $(\\mathcal{G}_{\\text{ad}})^{(k)}=\\mathcal{G}^{(k)}\\times\\mathbb{R}^\\ast\\sqcup A^{\\oplus k}\\times\\{0\\}$. From the view point of trying to define vector fields on the nerve of the adiabatic groupoid, this set-theoretic description leads to searching for a connection between $A^{\\oplus k}$ and the normal bundle of $M$ inside $\\mathcal{G}^{(k)}$ as the diagonal of units.\n\n\\begin{lem}\\label{normalbundenerve}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid with $\\Delta: M\\to\\mathcal{G}^{(k)}$ the diagonal inclusion via the units. The vector bundle map $\\nu: A^{\\oplus k}\\to\\Delta^\\ast T\\mathcal{G}^{(k)}$ given by\n\\begin{equation*}\n\\nu(v_1,...,v_k)=(v_1+\\sum_{i=2}^k du(dt(v_i)),v_2+\\sum_{i=3}^k du(dt(v_i)),...,v_{k-1}+du(dt(v_k)),v_k)\n\\end{equation*}\ninduces an isomorphism between $A^{\\oplus k}$ and the normal bundle of $M$ inside $\\mathcal{G}^{(k)}$.\n\\end{lem}\n\\begin{proof}\nFirst one checks that $\\nu$ indeed maps into the tangent space of $\\mathcal{G}^{(k)}\\subset\\mathcal{G}^{\\times k}$, which is a simple calculation. Next to show that it induces an isomorphism to the normal bundle to $\\Delta$, we first use the decomposition $T_{1_x}M=A_x\\oplus T_xM$ to see that if $\\nu(v_1,...,v_k)\\in T_xM\\subset T_{\\Delta(x)}\\mathcal{G}^{(k)}$ then $(v_1,...,v_k)=0$, so that the map into the normal bundle is injective. A simple case of dimension counting then implies that it the induced map is an isomorphism.\n\\end{proof}\n\\begin{cor}\nThere is a natural isomorphism between $N(\\mathcal{G}^{(k)},M)$ and $\\mathcal{G}_{\\text{ad}}^{(k)}$ which away from $\\tau=0$ links $((g_1,...,g_k),\\tau)$ and $((g_1,\\tau),...,(g_k,\\tau))$.\n\\end{cor}\n\\subsubsection{Haar systems on the adiabatic groupoid}\nWe intend to link deformation quantizations of the Poisson manifold $A^\\ast$ with the Van Est map $\\mathcal{V}:\\tilde{C}^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{def}(A)$. To make the syntax line up, we need to explicitely write down isomorphisms between smooth functions on $\\mathcal{G}$ and elements of the convolution algebra. This is done via Haar systems, which we will describe here in terms of densities.\n\n\\begin{defi}A Haar system on a groupoid $\\mathcal{G}\\rightrightarrows M$ is a collection $\\lambda=\\{\\lambda_x\\}_{x\\in M}$ of positive sections $\\lambda_x\\in\\Gamma(\\mathcal{D}_s|_{s^{-1}(x)})$ that are invariant under right translations $R_g: s^{-1}(t(g))\\to s^{-1}(s(g))$ and such that for every compactly supported function $f\\in C^\\infty_c(\\mathcal{G})$ the map $\\lambda(f): M\\to\\mathbb{R}$ given by\n\\begin{equation*}\n\\lambda(f)(x)=\\int_{s^{-1}(x)}f(g)\\lambda_x(g)\n\\end{equation*}\nis smooth.\n\\end{defi}\nWe know that every Lie groupoid admits a Haar system (\\cite[Prop 3.4]{landsman}) and if we have a Haar system $\\lambda$ on a Lie groupoid $\\mathcal{G}\\rightrightarrows M$ with $s$-fibers of dimension $d$, we can (\\cite[p.19]{landsman}) induce a Haar system $\\hat{\\lambda}$ on $\\mathcal{G}_{\\text{ad}}$ given by\n\\begin{equation*}\n\\hat{\\lambda}(g,\\tau)=|\\tau|^d\\lambda(g)\n\\end{equation*}\n\\begin{equation*}\n\\hat{\\lambda}(v,0)=\\lambda(\\pi(v))\n\\end{equation*}\nHere $\\pi: A\\to M$ is the projection and we take the canonical isomorphism $\\ker(d\\pi)\\cong\\pi^\\ast(A)$ as a given.\n\nNote that in particular we obtain a Haar system on the vector bundle $A\\to M$, seen as a groupoid in the canonical way.\n\nThe choice of a Haar system induces an isomorphism between the sheaf of smooth functions on $\\mathcal{G}$ and the sheaf of densities along the source fibers, and hence we can transport the convolution product over to the compactly supported functions where it is given by:\n\\begin{equation*}\n(f_1\\ast f_2)(g)=\\int_{s^{-1}(s(g))}f_1(gh^{-1})f_2(h)\\lambda_{s(g)}(h)\n\\end{equation*}\nIn particular on the adiabatic groupoid $\\mathcal{G}_{\\text{ad}}$ if we have two compactly supported functions $f_1,f_2$ we obtain:\n\\begin{equation*}\n(f_1\\ast f_2)(g,\\tau)=|\\tau|^{-d}\\int_{s^{-1}(s(g))}f_1(gh^{-1},\\tau)f_2(h,\\tau)\\lambda_{s(g)}(h)\\,\\,\\,\\,(\\tau\\neq 0)\n\\end{equation*}\n\\begin{equation*}\n(f_1\\ast f_2)(v,0)=\\int_{A_{\\pi(v)}}f_1(v-w,0)f_2(w,0)\\lambda_{\\pi(v)}(w)\n\\end{equation*}\nAt this point we notice that the convolution at $\\tau=0$ does not require the functions to be compactly supported on $A_x$, being Schwartz is enough (c.f. the usual theory of Fourier transform in $\\mathbb{R}^n$). This allows us, in the case of $\\mathcal{G}_{\\text{ad}}$, to enlarge the type of functions\/densities on which we let the deformation complex act.\n\nTo this end we refer to the work of \\cite{cr}, where a Fr\\'ech\\`et algebra $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ is constructed with evaluations\n\\[\n\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})_t=\\begin{cases} \\mathscr{S}_c(A)& t=0\\\\ C_c^\\infty(\\mathcal{G})&t\\not = 0.\\end{cases}\n\\]\nHere $\\mathscr{S}_c(A)$ denotes the space of functions that are Schwartz along the fibers of the Lie algebroid and have compact support along $M$. This Schwartz type algebra should be thought of as a dense subalgebra the reduced $C^*$-algebra $C^*_r(\\mathcal{G}_{\\text{ad}})$.\n\nBy the discussion above, the convolution product is perfectly well-defined on $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ and we can extend our viewpoint of the map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G}_{\\text{ad}})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_{\\mathcal{G}_{\\text{ad}}})$ to let $\\Phi(c)$ (for $c\\in C^k_\\text{def}(\\mathcal{G}_{\\text{ad}})$) act on functions in $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$. At this point it should be remarked that the isomorphism between functions and densities induced by a Haar system does not preserve the action of vector fields (indeed on the level of densities one also needs to compare $\\mathcal{L}_X\\lambda$ with $\\lambda$!). So really we should introduce in parallel to $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ the notion of densities with are of Schwartz-type along $\\tau=0$, but for the sake of not being overly pedantic we will not do this and just be careful when writing down the action of $\\Phi(c)$.\n\nIn what follows for a smooth family $\\{f_t\\}_{t\\neq 0}$ of compactly supported functions on $\\mathcal{G}$ and $f'\\in\\mathscr{S}_c(A)$ we will use the notation\n\\begin{equation*}\n\\lim_{t\\to 0} f_t=f'\n\\end{equation*}\nif the function $F:\\mathcal{G}_{\\text{ad}}\\to\\mathbb{R}$ given by\n\\begin{equation*}\nF(g,t)=f_t(g)\n\\end{equation*}\n\\begin{equation*}\nF(v,0)=f'(v)\n\\end{equation*}\nis an element of $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$.\n\\subsection{Fourier transform on vector bundles}\nWe briefly discuss the notion of Fourier transform on a vector bundle $E\\to M$ under the choice of a Haar system on $E$. This discussion follows the results of Landsman and Ramazan \\cite[\\S 7]{landsman}. Recall that a vector bundle $\\pi: E\\to M$ can be seen as a groupoid over $M$ where both the source and the target map are the projection $\\pi$ and the multiplication is the fiberwise addition. Since $\\ker(d\\pi)\\cong\\pi^\\ast E$ a choice of a Haar system is at every $v\\in E$ a choice of a density on $E_{\\pi(v)}$ that is invariant, where invariance in this case means that the choice is constant along the fiber.\n\nIf we choose such a Haar system $\\{\\mu_x\\}_{x\\in M}$, in \\cite{landsman} the Fourier transform $\\mathcal{F}_\\mu:\\mathscr{S}(E)\\to\\mathscr{S}(E^\\ast)$ was defined by\n\\begin{equation*}\n(\\mathcal{F}_\\mu f)(\\xi_x)=\\int_{E_x} f(v)e^{-i\\langle\\xi_x,v\\rangle}d\\mu_x(v)\n\\end{equation*}\nFurthermore, it was shown that this map is a linear isomorphism which intertwines the $\\mu$-convolution product on $E$ and the pointwise product on $E^\\ast$, and when $(x,v)$ are coordinates on $E$ induced by a frame with dual coordinates $(x,\\xi)$ we have for $f\\in\\mathscr{S}(E)$, $g\\in\\mathscr{S}(E^\\ast)$ and $a\\in C^\\infty(M)$ that\n\\begin{align*}\n\\mathcal{F}_\\mu((a\\circ\\pi)f)&=(a\\circ \\pi)\\mathcal{F}_\\mu\\\\\n\\frac{\\partial\\mathcal{F}_\\mu(f)}{\\partial x_j}&=\\mathcal{F}_\\mu\\left(\\frac{\\partial f}{\\partial x_j}\\right)+\\left(\\frac{\\partial\\text{log}(\\mu_e)}{\\partial x_j}\\circ\\pi\\right)\\mathcal{F}_\\mu(f)\\\\\n\\frac{\\partial\\mathcal{F}_\\mu(f)}{\\partial \\xi_j}&=-i\\mathcal{F}_\\mu(v_j f)\\\\\n\\frac{\\partial\\mathcal{F}_\\mu^{-1}(g)}{\\partial v_j}&=i\\mathcal{F}_\\mu^{-1}(\\xi_jg)\n\\end{align*}\nNote that after the choice of a Haar system $\\mu$ we obtain an isomorphism between the algebra of functions $C^\\infty_c(E)$ with the $\\mu$-convolution product and the convolution algebra $\\mathcal{A}_E$ of densities with the (intrinsic) convolution product. In particular if $X\\in\\mathfrak{X}(E)$, we can see $\\Phi(X)$ as defined on functions (which is, again, not equal to the usual action of vector fields on functions), and we can extend the action to Schwartz functions.\n\nNow using the Fourier transform, we can transport the action on the convolution algebra of $E$ to an action on the usual algebra with the pointwise product on $E^\\ast$.\n\\begin{prop}\nLet $X$ be a linear vector field on $E$. Then the map $\\hat{X}:\\mathscr{S}(E^\\ast)\\to\\mathscr{S}(E^\\ast)$ given by\n\\begin{equation*}\n\\hat{X}(f)=\\mathcal{F}_\\mu(\\Phi(X)(\\mathcal{F}_\\mu^{-1}(f)))\n\\end{equation*}\ndefines a linear vector field on $E^\\ast$. Here $\\Phi$ is the natural chain map we defined before, applied to the vector bundle $E$ seen as a groupoid.\n\\end{prop}\n\\begin{proof}\nFirst we show that $\\hat{X}$ is indeed a vector field, i.e. a derivation with respect to the pointwise product. Since $\\hat{X}$ is the conjugation of $\\Phi(X)$ with an isomorphism which intertwines the convolution product on $\\mathscr{S}(E)$ and the pointwise product on $\\mathscr{S}(E^\\ast)$ this is equivalent to showing that $\\Phi(X)$ is a derivation for the convolution product. When we see $E\\to M$ as a groupoid, this is equivalent to showing that $X$ is a multiplicative vector field, and it is easy to see that on a vector bundle the multiplicative vector fields are precisely the linear vector fields.\n\nTo see that $\\hat{X}$ is a linear vector field we do a local computation on a trivial vector bundle $E=\\mathbb{R}^m_x\\times\\mathbb{R}^n_v\\to\\mathbb{R}^m_x$ with Haar system $f(x)dv_1\\wedge\\cdots\\wedge dv_n$. Using the properties of the Fourier transform stated before it follows that if\n\\begin{equation*}\nX(x,v)=\\sum_{i=1}^mX_i(x)\\frac{\\partial}{\\partial x_i}+\\sum_{j=1}^n\\sum_{k=1}^n Y_{jk}(x)v_j\\frac{\\partial}{\\partial v_k}\n\\end{equation*}\nthen\n\\begin{equation*}\n\\hat{X}(x,\\xi)=\\sum_{i=1}^mX_i(x)\\frac{\\partial}{\\partial x_i}-\\sum_{j=1}^n\\sum_{k=1}^nY_{jk}(x)\\xi_k\\frac{\\partial}{\\partial\\xi_j}\n\\end{equation*}\nwhich indeed shows that $\\hat{X}$ is a linear vector field.\n\\end{proof}\nRecall that a linear vector field $X\\in\\mathfrak{X}(E)$ is the same as a linear map $X:\\Gamma(E^\\ast)\\to\\Gamma(E^\\ast)$ with a symbol $s_X\\in\\mathfrak{X}(M)$ such that\n\\begin{equation*}\nX(f\\alpha)=fX(\\alpha)+s_X(f)\\alpha\\qquad(f\\in C^\\infty(M),~\\alpha\\in\\Gamma(E)).\n\\end{equation*}\nFurthermore, recall the canonical pairing $\\langle-,-\\rangle:\\Gamma(E^\\ast)\\times\\Gamma(E)\\to C^\\infty(M)$. Then for a linear vector field $X$, the local calculation from the proof above generalizes to the following.\n\\begin{prop}\\label{xhat}\nLet $X\\in\\mathfrak{X}(E)$ be a linear vector field, then the linear vector field $\\hat{X}\\in\\mathfrak{X}(E^\\ast)$ is uniquely determined by the fact that for $\\beta\\in\\Gamma(E^\\ast)$ and $\\alpha\\in\\Gamma(E)$\n\\begin{equation*}\n\\langle\\beta,\\hat{X}(\\alpha)\\rangle+\\langle X(\\beta),\\alpha\\rangle=s_X(\\langle\\beta,\\alpha\\rangle)\n\\end{equation*}\n\\end{prop}\nWe can play a similar game, albeit slightly more involved in notation, for higher order deformation elements of the vector bundle. So consider an element $X\\in\\tilde{C}^k_\\text{def}(E)$ given by\n\\begin{equation*}\nX(v_1,...,v_n)=X_1(v_1)\\langle \\beta_2,v_2\\rangle\\cdots\\langle \\beta_k,v_k\\rangle\n\\end{equation*}\nwhere $X_1$ is a linear vector field on $E$ and $\\beta_2,...,\\beta_k\\in\\Gamma(E^\\ast)$. One immediately checks that this is a closed element of $\\tilde{C}^k_\\text{def}(E)$, so that the Fourier transform\n\\begin{equation*}\n\\hat{X}(f_1,...,f_k)=\\mathcal{F}_\\mu(\\Phi(X)(\\mathcal{F}_\\mu^{-1}(f_1),...,\\mathcal{F}_\\mu^{-1}(f_k)))\n\\end{equation*}\nis a closed element of the Hochschild complex of $C^\\infty(E^\\ast)$. By the specific form of $X$ is it easy to see that\n\\begin{equation*}\n\\Phi(X)(a_1,...,a_k)=\\Phi(X_1)(a_1)\\ast (\\beta_2a_2)\\ast\\cdots\\ast(\\beta_k a_k)\n\\end{equation*}\nwhere we see the $s_i$ as fiberwise linear maps on $E$. In particular we see that\n\\begin{equation*}\n\\hat{X}=\\hat{X_1}\\otimes\\hat{\\beta_2}\\otimes\\cdots\\otimes\\hat{\\beta_k}\n\\end{equation*}\nwhere for $\\beta\\in\\Gamma(E^\\ast)$, $\\hat{\\beta}$ is the vector field on $E^\\ast$ given by\n\\begin{equation*}\n\\hat{\\beta}(f)=\\mathcal{F}_\\mu(\\beta\\mathcal{F}_\\mu^{-1}(f))\n\\end{equation*}\nA local computation shows that $\\hat{\\beta}$ is identically zero on fiberwise constant maps and for the map induced by a section $\\alpha\\in\\Gamma(E)$ we have\n\\begin{equation*}\n\\hat{\\beta}(\\alpha)=\\frac{1}{i}\\langle \\beta,\\alpha\\rangle\n\\end{equation*}\nIn particular, we see that if we anti-symmetrize, we obtain the linear multivectorfield $\\hat{X_1}\\wedge\\hat{\\beta_2}\\wedge\\cdots\\wedge\\hat{\\beta_k}$ on $E^\\ast$.\n\\subsection{Deformation quantization of $A^\\ast$ and the Van Est map}\nNow, fix a choice of a Haar system of $\\mathcal{G}$, which by the discussion above induces a Haar system on $\\mathcal{G}_{\\text{ad}}$ and a Haar system $\\mu$ on $A\\to M$. The last one makes sure that we can talk about a Fourier transform $\\mathcal{F}_\\mu:\\mathscr{S}(A)\\to\\mathscr{S}(A^\\ast)$.\n\nSlightly tweaking the results of \\cite{landsman} we obtain {\\em quantization maps} $q_t:\\mathscr{S}_c(A^*)\\to C^\\infty_c(\\mathcal{G}),~t\\not = 0$ given by\n\\[\nq_t(f)(g):=\\chi(g)\\mathcal{F}_\\mu^{-1}(f)(\\frac{1}{t}\\exp^{-1}(g)),\n\\]\nwhich satisfy\n\\begin{equation}\n\\label{quant}\n\\lim_{t\\to 0}(q_t(f_1 f_2)-q_t(f_1)\\ast q_t(f_2))=0,\\quad \\lim_{t\\to 0}(\\frac{1}{it}[q_t(f_1),q_t(f_2)]-q_t(\\{f_1,f_2\\}))=0.\n\\end{equation}\nHere $\\chi\\in C^\\infty_c(\\mathcal{G})$ is a cut-off function that equals $1$ in a neighborhood of $M\\subset\\mathcal{G}$ \nwith support inside an open neighbourhood of the units onto which the exponential map is a diffeomorphism. The Poisson \nbracket $\\{~,~\\}$ is the bracket associated to the so-called Lie--Poisson structure on $A^*$.\n\nExplicitely, one of the differences with the results of \\cite{landsman} is that we do not need the property $q_t(f^\\ast)=q_t(f)^\\ast$ for which the Weyl exponential map $\\text{exp}^{\\text{W}}$ is used, and instead we can use the normal exponential map. Secondly, we do not need to restrict to Paley-Wiener functions, as we allow for Schwarz-type functions at $t=0$ and use the cut-off function on the level of $\\mathcal{G}$ instead of $A$, the deviation vanishing as $t$ approaches $0$. Lastly, as the relevant calculations on the local forms in $A$ and $A^\\ast$ are valid for all Schwarz functions and not just Paley-Wiener functions, the relevant propositions in \\cite{landsman} still hold in this situation. The variety of quantizations by using different types of exponential maps is also reflected on the more algebraic level in \\cite{nw} by using different orderings in the Fedossov construction of {\\em formal} deformation quantizations of $A^*$.\n\nWe now briefly recall the van Est-map as given in \\cite[\\S 10]{cms}. First the deformation complex of the algebroid $C^k_\\text{def}(A)$ is given by antisymmetric multilinear maps $D: \\Gamma(A)^k\\to\\Gamma(A)$ that have a symbol $s_D:\\Gamma(A)^{k-1}\\to\\mathfrak{X}(M)$ such that\n\\begin{equation*}\nD(\\alpha_1,...,f\\alpha_k)=fD(\\alpha_1,...,\\alpha_k)+s_D(\\alpha_1,...,\\alpha_{k-1})(f)\\alpha_k\n\\end{equation*}\nNote that we can, and will, see elements of $C^k_\\text{def}(A)$ as linear multivectorfields on $A^\\ast$ (by noting that sections of $A$ are the same as fiberwise linear maps on $A^\\ast$) and in turn see the deformation complex of $A$ as the linear Poisson complex of the Poisson manifold $A^\\ast$.\n\nThen for $\\alpha\\in\\Gamma(A)$ there are maps $R_\\alpha:\\tilde{C}^k_\\text{def}(\\mathcal{G})\\to\\tilde{C}^{k-1}_\\text{def}(\\mathcal{G})$ which are given for $k=1$ by\n\\begin{equation*}\nR_\\alpha(c)=[c,\\overrightarrow{\\alpha}]|_M\n\\end{equation*}\nand for $k>0$ by\n\\begin{equation*}\nR_\\alpha(c)(g_1,...,g_{k-1})=(-1)^{k-1} \\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0} c(g_1,...,g_{k-1},\\Phi^\\epsilon_{\\overrightarrow{\\alpha}}(s(g_{k-1}))^{-1})\n\\end{equation*}\nThe van Est-map $\\mathcal{V}:\\tilde{C}^k_\\text{def}(\\mathcal{G})\\to C^k_\\text{def}(A)$ is then given by\n\\begin{equation*}\n\\mathcal{V}(c)(\\alpha_1,...,\\alpha_k)=\\sum_{\\sigma\\in S_k}(-1)^\\sigma (R_{\\alpha_{\\sigma(k)}}\\circ\\cdots\\circ R_{\\alpha_{\\sigma(1)}})(c)\n\\end{equation*}\nThe connection between the van Est-map and the quantization maps is then as follows.\n\\begin{thm}\nLet $k\\geq 1$ and $c\\in \\tilde{C}^k_{\\text{def}}(\\mathcal{G})$ and suppose the choice of a Haar system on $\\mathcal{G}$ inducing a Haar system $\\mu$ on the algebroid $A$. Given $f_1,\\ldots,f_k\\in  \\mathscr{S}_c(A^*)$, the following \nequality holds true:\n\\[\n\\mathcal{V}(c)(f_1,\\ldots,f_k)=\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_k}(-1)^\\sigma\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\]\n\\end{thm}\n\\begin{rmk}\nNote that the right hand side of the equation above is well-defined, since the sum of which we take the limit is a function on the groupoid. The limit is then a Schwartz function on the algebroid and so if we take the Fourier transform we obtain a Schwartz function on the dual of the algebroid.\n\\end{rmk}\n\\begin{proof}\nWe start with the case $k=1$. First note that for $f\\in\\mathscr{S}_c(A^\\ast)$ the map $q(f):\\mathcal{G}_{\\text{ad}}\\to\\mathbb{R}$ given by\n\\begin{align*}\nq(f)(g,t)&=q_t(f)(g)\\\\\nq(f)(v,0)&=\\mathcal{F}_\\mu^{-1}(f)(v)\n\\end{align*}\nis an element of $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$. Then note that the family $\\{c\\}_{t\\neq 0}$ is a family of vector fields on $\\mathcal{G}$ which can be extended to a vector field on $\\mathcal{G}_{\\text{ad}}$, namely to the vector field $c_\\text{inv}$ obtaines by \\cref{vfonnms}. Then notice that $\\Phi(c_\\text{inv})(q(f))$ is an element of $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ consisting of\n\\begin{align*}\n\\Phi(c_\\text{inv})(q(f))_t&=\\Phi(c)(q_t(f)),\\qquad(t\\neq 0)\\\\\n\\Phi(c_\\text{inv})(q(f))_0&=\\Phi(c_0)(\\mathcal{F}_\\mu^{-1}(f))\n\\end{align*}\nwhere $c_0$ is the linear vector field that is the restriction of $c_\\text{eqv}$ to $t=0$. Note that it is linear, since it is the application of \\cref{vectorfieldnormalbundle} to the vector field $c$ on $\\mathcal{G}$. In particular we see that\n\\begin{equation*}\n\\lim_{t\\to 0}\\Phi(c)(q_t(f))=\\Phi(c_0)(\\mathcal{F}_\\mu^{-1}(f))\n\\end{equation*}\nand so we need to show that $\\mathcal{V}(c)=\\hat{c_0}$.\n\nBy \\cref{xhat} this means that we need to show that for $\\beta\\in\\Gamma(A^\\ast)$ and $\\alpha\\in\\Gamma(A)$ we have\n\\begin{equation*}\n\\langle\\beta,\\mathcal{V}(c)(\\alpha)\\rangle+\\langle c_0(\\beta),\\alpha\\rangle=s_c(\\langle\\beta,\\alpha\\rangle)\n\\end{equation*}\nWe note two things. First that every $\\beta\\in\\Gamma(A^\\ast)=C^\\infty_\\text{lin}(A)$ can be written as $d_nh$ for $h\\in C^\\infty(\\mathcal{G})$ with $h|_M=0$. Second that since we have an explicit inclusion of $A$ into the tangent bundle of $\\mathcal{G}$ this means that:\n\\begin{equation*}\n\\langle d_nh,\\alpha\\rangle(x)=\\alpha(x)(h)\n\\end{equation*}\nWe are now ready to show the equality. First we have\n\\begin{align*}\n\\langle d_nh,\\mathcal{V}(c)(\\alpha)\\rangle(x)&=[c,\\overrightarrow{\\alpha}](1_x)(h)\\\\\n&=c(1_x)(\\overrightarrow{\\alpha}(h))-\\alpha(x)(c(h))\n\\end{align*}\n\\begin{equation*}\n\\langle c_0(d_n h),\\alpha\\rangle(x)=\\langle d_n(ch),\\alpha\\rangle(x)=\\alpha(x)(c(h))\n\\end{equation*}\nand since $c_{1_x}=du(s_c(x))$ combined with $\\overrightarrow{\\alpha}|_M=\\alpha$ we have\n\\begin{equation*}\ns_c(\\langle d_nh,\\alpha\\rangle)(x)=s_c(\\overrightarrow{\\alpha}(h)|_M)(x)=c(1_x)(\\overrightarrow{\\alpha}(h))\n\\end{equation*}\nFor $k>1$ we restrict to the case where $c=c_1\\otimes h_2\\otimes\\cdots\\otimes h_k$ with $c_1\\in\\mathfrak{X}(\\mathcal{G})$ and $h_2,...,h_k\\in C^\\infty(\\mathcal{G})$. For $c$ to be an element of $\\tilde{C}^k_\\text{def}(\\mathcal{G})$ it is necessairy and sufficient to have $c_1\\in\\tilde{C}^1_\\text{def}(\\mathcal{G})$ and $h_i|_M=0$. Similar to the case $k=1$ we note that\n\\begin{equation*}\n\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_1),...,q_t(f_k))=\\Phi(\\frac{1}{(it)^{k-1}}c)(q_t(f_1),...,q_t(f_k))\n\\end{equation*}\nwhich, as $t\\to 0$, converges\n\\begin{equation*}\n\\Phi(c_0)(\\mathcal{F}_\\mu^{-1}(f_1),...,\\mathcal{F}_\\mu^{-1}(f_k))\n\\end{equation*}\nif we find a vector field $c_0$ on $A$ that together with the family $\\{\\frac{1}{(it)^{k-1}}c\\}_{t\\neq 0}$ defines a smooth deformation element of $\\mathcal{G}_{\\text{ad}}$.\n\nTo calculate this localization we remark that we can do the calculation in $\\mathcal{G}^k$ using the cartesian product of the exponential map $A\\to\\mathcal{G}$, in stead of working in $\\mathcal{G}^{(k)}$ and using the machinery of the previous section. This is for two reasons: firstly our definition of $c$ extends to $\\mathcal{G}^k$. Secondly the difference of $(v_1,...,v_k)\\in A^{\\oplus k}$ seen as tangent vectors on $\\mathcal{G}^k$ and $(v_1,...,v_k)\\in A^{\\oplus k}$ seen as tangent vectors in $\\mathcal{G}^{(k)}$ which are normal to the units, using the isomorphism of \\cref{normalbundenerve}, are tangent vectors in $\\mathcal{G}^k$ which are along the units. Since $c$ vanishes along the units, we can neglect this.\n\nNow to do the actual calculation we consider the chart $\\theta: A^{\\oplus k}\\times\\mathbb{R}^\\ast\\to\\mathcal{G}^k\\times\\mathbb{R}^\\ast$ given by\n\\begin{equation*}\n\\theta(v_1,...,v_k,t)=(\\exp(tv_1),...,\\exp(tv_k),t)\n\\end{equation*}\nThen if we look at the family $\\{\\frac{1}{(it)^{k-1}}c\\}_{t\\neq 0}$, we see that if we take the pullback along $\\theta$ we obtain:\n\\begin{equation*}\n\\theta^\\ast(\\{\\frac{1}{(it)^{k-1}}c\\}_{t\\neq 0})(v_1,...,v_k,t)=\\frac{1}{(it)^k}c_1(\\exp(tv_1))h_2(\\exp(tv_2))\\cdots h_k(\\exp(tv_k))\n\\end{equation*}\nDistributing the $k$ powers of $\\frac{1}{t}$ over the $k$ different terms we see that\n\\begin{equation*}\nc_0(v_1,...,v_k)=\\frac{1}{i^{k-1}}(c_1)_0(v_1)d_nh_2(v_2)\\cdots d_nh_2(v_k)\n\\end{equation*}\nsince\n\\begin{equation*}\n\\frac{1}{t}c_1(\\exp(tv_1))\\to (c_1)_0(v_1)\n\\end{equation*}\n\\begin{equation*}\n\\frac{1}{t}h(\\exp(tv))\\to d_nh(v)\n\\end{equation*}\nas $t\\to 0$, so we see that $c_0=\\frac{1}{i^{k-1}}(c_1)_0\\otimes d_nh_2\\otimes\\cdots\\otimes d_nh_k$, which is a linear deformation element, and we want to show that $\\mathcal{V}(c)$ is the anti-symmetrization of the Fourier transform $\\hat{c_0}$. By the discussion at the end of the previous subsection we see that $\\hat{c_0}$ is determined for $\\alpha_1,...,\\alpha_k\\in\\Gamma(A)$ by\n\\begin{equation*}\n\\hat{c_0}(\\alpha_1,...,\\alpha_k)=\\frac{1}{i^{2(k-1)}}\\hat{(c_1)_0}(\\alpha_1)\\langle d_nh_2,\\alpha_2\\rangle\\cdots\\langle d_nh_k,\\alpha_k\\rangle\n\\end{equation*}\nNext we investigate $R_\\alpha(c)$, we obtain:\n\\begin{align*}\nR_\\alpha(c)(g_1,...,g_{k-1})&=(-1)^{k-1}\\frac{d}{d\\epsilon}|_{\\epsilon=0}c_1(g_1)h_2(g_2)\\cdots h_{k-1}(g_{k-1})h_k(\\Phi^\\epsilon_{\\overrightarrow{\\alpha}}(s(g_k))^{-1})\\\\\n&=(-1)^{k-1}c_1(g_1)h_2(g_2)\\cdots h_{k-1}(g_{k-1})dh_k(d\\iota(\\alpha(s(g_k)))\n\\end{align*}\nThen since $f_k|_M=0$ and for $v\\in A_x$ we have $d\\iota v=-v+d(u\\circ t)(v)$ we obtain\n\\begin{equation*}\nR_\\alpha(c)(g_1,...,g_{k-1})=(-1)^k c_1(g_1)h_2(g_2)\\cdots h_{k-1}(g_{k-1})d_nh_k(\\alpha(s(g_{k-1}))\n\\end{equation*}\nDoing this inductively, and using that the flow of $\\overrightarrow{\\alpha}$ preserves source fibers, we see\n\\begin{equation*}\n(R_{\\alpha_2}\\circ\\cdots\\circ R_{\\alpha_k})(c)(g)=(-1)^{\\frac{(k-1)(k-2)}{2}}c_1(g)d_nh_2(\\alpha_2(s(g))\\cdots d_nh_k(\\alpha_k(s(g))\n\\end{equation*}\nThen since this is simply $c_1$ multiplied with a function that is constant along the $s$-fibers, we obtain:\n\\begin{align*}\n(R_{\\alpha_1}\\circ\\cdots \\circ R_{\\alpha_k})(c)&=(-1)^{\\frac{(k-1)(k-2)}{2}}\\mathcal{V}(c_1)(\\alpha_1)\\langle d_nh_2,\\alpha_2\\rangle\\cdots\\langle d_nh_k,\\alpha_k\\rangle\\\\\n&=i^{(k-1)(k-2)}\\mathcal{V}(c_1)(\\alpha_1)\\langle d_nh_2,\\alpha_2\\rangle\\cdots\\langle d_nh_k,\\alpha_k\\rangle\n\\end{align*}\nSince already know by the calculation in the case $k=1$ that $\\mathcal{V}(c_1)(\\alpha_1)=\\hat{(c_1)_0}(\\alpha_1)$ we see that\n\\begin{equation*}\n(R_{\\alpha_1}\\circ\\cdots \\circ R_{\\alpha_k})(c)=i^{k(k-1)}\\hat{c_0}(\\alpha_1,...,\\alpha_k)\n\\end{equation*}\nThen note that there is a mismatch in the summation over $S_k$ in $\\mathcal{V}(c)$ and in the right hand side of the theorem. In particular the right hand side in the last equation corresponds to the identity permutation in the statement of the theorem, while the right hand side corresponds to the permutation in the definition of $\\mathcal{V}(c)$ that sends $j$ to $k-j$. This sign of this permutation is $(-1)^{\\frac{k(k-1)}{2}}$, for which we have to correct, so that we obtain\n\\begin{align*}\n\\mathcal{V}(c)(\\alpha_1,...,\\alpha_k)&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma (R_{\\alpha_{\\sigma(k)}}\\circ\\cdots R_{\\alpha_{\\sigma(1)}})(c)\\\\\n&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma i^{k(k-1)}(R_{\\alpha_{\\sigma(1)}}\\circ\\cdots\\circ R_{\\alpha_{\\sigma(k)}})(c)\\\\\n&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma i^{2k(k-1)}\\hat{c_0}(\\alpha_{\\sigma(1)},...,\\alpha_{\\sigma(k)})\\\\\n&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma \\hat{c_0}(\\alpha_{\\sigma(1)},...,\\alpha_{\\sigma(k)})\n\\end{align*}\nSo we see that $\\mathcal{V}(c)$ equals the linear multivector field that is the antisymmetrization of $\\hat{c_0}$. In particular this means that for $f_1,...,f_k\\in\\mathscr{S}_c(A^\\ast)$ we have\n\\begin{align*}\n\\mathcal{V}(c)(f_1,...,f_k)&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma\\hat{c_0}(f_{\\sigma(1)},...,f_{\\sigma(k)})\\\\\n&=\\frac{1}{i^{k-1}}\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_k}(-1)^\\sigma\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\end{align*}\nThis completes the proof.\n\\end{proof}\n\n\\begin{rmk}\nThis theorem, restricted to multiplicative vector fields, can be viewed as a statement about the ``classical limit'' of certain derivations of the convolution algebra, and looks very similar to certain aspects of the proof of the Atiyah--Singer index theorem given in \\cite{enn}. Indeed, it would be interesting to investigate its use in index theory for Lie groupoids, as it \nexactly fits into the framework of relating the van Est map to the classical limit, as shown in the index theorem of \\cite{ppt} \nfor smooth groupoid cohomology $H^\\bullet_{\\rm diff}(\\mathcal{G})$.\n\\end{rmk}\n\nIn the previous proof we have only used the fact that $q_t(f)$ converges to $\\mathcal{F}_\\mu^{-1}(f)$ in $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ as $t$ goes to $0$, we have not used the properties which makes the family $\\{q_t\\}_{t\\neq 0}$ a family of quantization maps, namely their compatibility with the Poisson bracket. However, we have not introduced these specific maps without reason, since we will use the fact that\n\\begin{equation*}\n\\lim_{t\\to 0}(\\frac{1}{it}[q_t(f_1),q_t(f_2)])=\\lim_{t\\to 0}q_t(\\{f_1,f_2\\}))\n\\end{equation*}\nto give an alternative proof of the fact that the Van Est map is a {\\em chain map}, i.e, compatible with the differentials:\n\\begin{cor}\nThe van Est map $\\mathcal{V}:\\tilde{C}_\\text{def}^\\bullet(\\mathcal{G})\\to C^\\bullet_{\\text{Pois,lin}}(A^\\ast)$ is a chain map.\n\\end{cor}\n\\begin{proof}\nLet $c\\in\\tilde{C}^k_\\text{def}(\\mathcal{G})$ for $k\\geq 1$ and we start by dissecting $\\mathcal{V}(\\partial c)$. Using the previous theorem we obtain\n\\small\n\\begin{align*}\n\\mathcal{V}(\\delta c)(f_1,...,f_{k+1})=&\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\frac{1}{(it)^{k}}\\Phi(\\delta c)(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k+1)}))\\right)\\right)\\\\\n=&\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\frac{1}{(it)^{k}}(\\delta_\\text{Hoch}\\Phi(c))(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k+1)}))\\right)\\right)\\\\\n=&\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\frac{1}{(it)^k}[q_t(f_{\\sigma(1)}),\\Phi(c)(q_t(f_{\\sigma(2)}),...,q_t(f_{\\sigma(k+1)}))]\\right)\\right)\\\\\n&+\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{j=1}^k\\sum_{\\substack{\\sigma\\in S_{k+1}\\\\\\sigma^{-1}(j)<\\sigma^{-1}(j+1)}}(-1)^\\sigma (-1)^j\\frac{1}{(it)^k}\\Phi(c)(q_t(f_{\\sigma(1)}),...,[q_t(f_{\\sigma(j)}),q_t(f_{\\sigma(j+1)})],...,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\end{align*}\n\\normalsize\nNow the relation between the commutator, the Poisson bracket and the quantization maps, we can use 1 power of $\\frac{1}{it} $ to turn the commutators into Poisson brackets. Also using the fact that $q_t(f)\\to \\mathcal{F}_\\mu^{-1}(f)$ as $t\\to 0$ this results in\n\\small\n\\begin{align*}\n\\mathcal{V}(\\delta c)(f_1,...,f_{k+1})=&\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\left\\{f_{\\sigma(1)},\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(2)}),...,q_t(f_{\\sigma(k+1)}))\\right)\\right)\\right\\}\\\\\n&+\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{j=1}^k\\sum_{\\substack{\\sigma\\in S_{k+1}\\\\\\sigma^{-1}(j)<\\sigma^{-1}(j+1)}}(-1)^\\sigma (-1)^j\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(1)}),...,q_t(\\{f_{\\sigma(j)},f_{\\sigma(j+1)}\\}),...,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\end{align*}\n\\normalsize\nThen using the previous Theorem in reverse order we see that this leads to\n\\begin{align*}\n\\mathcal{V}(\\delta c)(f_1,...,f_{k+1})&=\\sum_{j=1}^{k+1}(-1)^{j+1}\\left\\{f_j,\\mathcal{V}(c)(f_1,...,\\hat{f_j},...,f_{k+1})\\right\\}\\\\\n&+\\sum_{j_1<j_2}(-1)^{j_1+j_2}\\mathcal{V}(c)(\\{f_{j_1},f_{j_2}\\},f_1,...,\\hat{f_{j_1}},\\hat{f_{j_2}},...,f_{k+1})\n\\end{align*}\nwhich shows that the Van Est map is a chain map.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\nRecent advances in deep reinforcement learning\\cite{mnih2013playing, van2016deep, wang2015dueling} have triggered subgoal and option research within reinforcement learning using deep neural networks. Here we investigate Micro-Objective Learning, a new technique to discover important states for a complex task and exploit them to accelerate learning.\n\nMany techniques have been investigated with respect to subgoals. \\cite{kulkarni2016deep} uses a Successor map, which approximates the count of the successive states given (state, action) pairs. It successfully extracted subgoals in the simple grid world and 3D Doom task. However, it did not exploit the subgoals while discovering them online to accelerate learning. \\cite{kulkarni2016hierarchical} used heuristically defined subgoals and two neural networks: One to maximize the external reward, and the other to maximize the internal reward to achieve subgoals. This \\cite{kulkarni2016hierarchical} paper has had promising results in Montezuma's Revenge, an Atari game notorious for its difficult exploration. \\cite{bacon2016option} has incorporated option learning into policy gradient theorem to directly do end-to-end learning of the options for semi-MDP (Markov Decision Process) and has been tested on four Atari games\\cite{bellemare2013arcade} but left the number of options as a hyper-parameter.\n\nDespite the promising results by \\cite{kulkarni2016hierarchical, bacon2016option}, several drawbacks still exist. 1) Although subgoals and options are inseparable, discovering the subgoals online while learning options has been extremely difficult as they need to be learned in addition to original reinforcement learning algorithms. As far as we know, online learning of subgoals while exploiting them to accelerate learning in large state-space, such as the Atari domain, has not been accomplished. In \\cite{bacon2016option}, they did not use the idea of subgoals to learn options.\n\nAdditionally, 2) the definition of subgoal has been somewhat ambiguously defined. In \\cite{murata2007introduction}, a subgoal was defined as a set of states that are known to be visited when reaching the goal. However, every state can be considered a subgoal. For example, when a robot is learning to move from one room to another and a door is in between, the door can be viewed as a subgoal in a cognitive way. Also, we can consider a desk or a chair that one can pass (but does not need to pass) to get to the other room as a subgoal with lower importance than the door.\n\nIn this paper, we present a hierarchical reinforcement learning algorithm, Micro-Objective Learning(MOL). We first define micro-objectives, a set of states with a continuous measurement of importance assigned to each element, which is a micro-version of the subgoal. We induce the logic that frequently visited states can be considered as subgoals, from \\cite{mcgovern2001accelerating}, however only successful trajectories are used to estimate the importance of micro-objectives. Similar to the empirical results in \\cite{mcgovern2001accelerating}, simply counting all states results in distracted subgoals, while counting the states only when visited for the first time gives clearer results.\n\nThere are drawbacks of first-visit counting as mentioned in \\cite{mcgovern2001accelerating}: It is noisy, and impractical in large or continuous state-space domains. Though it appears noisy from the traditional subgoal point of view, it does help to approximate the importance of micro-objective states. Also, noisy subgoals are normally discouraged because learning option policy takes time. However, since we give an additional reward to the original MDP rather than explicitly creating macro-actions, estimating the importance does not need to be precise. Additionally, we use recent pixel differences to sample dissimilar states and count them from successful trajectories instead of using first-visit counts in non-tabular domains. For justification of first-visit sampling, we analyzed the difference between theoretical importance, approximated importance achieved with first-visit counting, and approximated importance achieved with every-state counting of successful trajectories. In MOL, we use pseudo-count from \\cite{bellemare2016unifying} for fast learning of importance.\n\nBy using first-visit pseudo-counts for approximating the importance of micro-objectives and sampling dissimilar states, we achieved significantly better results than the baselines in two Atari games: Montezuma's Revenge and Seaquest, where we used \\cite{bellemare2016unifying} as a baseline for Montezuma's Revenge, the state-of-the-art model as far as we know, and Deep Q-learning for Seaquest.\n\nOverall, our main contributions are:\n\\begin{itemize}\n\\item Defined micro-objectives with precise measurement of their importance using visited counts combining the notions of the subgoal and the option.\n\\item Automatic discovery and utilization of the micro-objectives at the same time resulting in accelerated learning.\n\\item Micro-Objective Learning scales up to large and continuous states, such as the Atari domain.\n\\end{itemize}\n\n\\section{Background}\n\\subsection{Double Deep Q-learning}\nIn Double Deep Q-learning (DDQN)\\cite{van2016deep}, one operator calculates the action-value and chooses the action simultaneously. This results in a positive bias. Double Deep Q-learning uses two parameters for the action-value function to solve this problem.\n\\begin{equation}\ny_i^{DDQN} = r + \\gamma \\, Q(s', \\arg\\max_{a'} Q(s', a';\\theta_i);\\theta^-)\n\\end{equation}\n\n\\subsection{Pseudo-Count Exploration}\nPseudo-Count Exploration (PSC)\\cite{bellemare2016unifying} extends traditional count-based approach for efficient exploration. PSC uses a sequential density model to approximate the count of the state in large and continuous state-space. Specifically, it uses a CTS density model for each pixel to get the probability $\\rho_{n}(x)$ of the state and define the recording probability $\\rho'_{n}(x)$ to get the pseudo-count $\\hat{N}_{n}(x)$ and pseudo-count total $\\hat{n}$.\n\\begin{equation}\n\\rho_{n}(x) = \\frac{\\hat{N}_{n}(x)}{\\hat{n}}, \\;\\;\n\\rho'_{n}(x) = \\frac{\\hat{N}_{n}(x)+1}{\\hat{n}+1}\n\\end{equation}\nPSC calculates the pseudo-count by solving the equations between $\\rho'_{n}(x)$ and $\\rho_n$.\n\\begin{equation}\n\\hat{N}_n(x) = \\frac{\\rho_n(x)(1-\\rho'_n(x))}{\\rho'_n(x)-\\rho_n(x)}\n\\end{equation}\nBy pseudo-counting the states the agent has visited, it gives an additional reward to the states the agent has not seen before. \n\\begin{equation}\nR_{new} = R + \\beta \\ (\\hat{N}_n(x)+0.01)^{-1\/2}\n\\end{equation}\nDirect usage of an exploration bonus results in a destabilized Q function. PSC mixed Double Q target with Monte Carlo return as below:\n\\begin{equation}\n\\begin{aligned}\n\\Delta Q(x_t, a_t) :=\n(1-\\eta)\\Delta Q_{DOUBLE}(x_t, a_t) + \\\\ \\eta[\\sum_{s=0}^\\infty \\gamma^s((R+R_n^+)(x_{t+s}, a_{t+s}))-Q(x_t, a_t)]\n\\end{aligned}\n\\end{equation}\n\n\\section{Micro-Objective Learning}\nIn Micro-Objective Learning (MOL), we approximate how important a state is given the current policy and give an additional reward to the original MDP that is proportional to the importance, as below.\n\\begin{equation}\n    R_{obj} = \\alpha \\cdot min(R_{max}, \\frac{(1-R_{exp})}{R_{c-max}})\n\\end{equation}\n\\begin{equation}\n    R_{new} = R + R_{obj}\n\\end{equation}\nwhere $R_{max}$ is a constant value to limit the maximum of $R_{obj}$, $\\alpha$ is a coefficient for $R_{obj}$, and $R_{c-max}$ is the current maximum value of (1-$R_{exp}$).\n\\begin{equation}\n    R_{exp}=\\frac{0.1}{\\sqrt{N(s)+0.01}}\n\\end{equation}\n\n$R_{exp}$ is an additional reward function used in  \\cite{bellemare2016unifying} for exploration.\n\nLet us define a successful trajectory as a trajectory that has its last state as the only goal state in the trajectory which is given in MDP. A goal state is dependent on the task, but typically we can define a goal state as a positive rewarding state. The visit-counts of the states in successful trajectories were used to estimate the importance of the states. However, counting all of the states in the successful trajectories results in poor estimation of the importance. We use dissimilar sampling, which is an extension of first-visit sampling, to solve this issue. First-visit sampling is a simple method that samples the states that have been visited for the first time when moving along the trajectory.\n\nGiving an additional reward to the original MDP has several benefits over the option framework. By giving an additional reward, we do not need to learn the option policies, a task which is as expensive as the original RL. Also, the options need to find an initiation set and learn the termination condition to decide where to start and terminate the options, which MOL does not require. Because micro-objectives are smaller units than the subgoal or initiation set in the option framework, MOL can be viewed as a process that learns a micro-version of the option policies, but in a flexible manner.\n\n\\subsection{Empirical explanation of Micro-Objective Learning}\nIntuitively, giving an additional reward to frequently visited states in successful trajectories accelerates the learning process. One possible approach to calculating importance is to count all of the states from the successful trajectories and use this count as an importance of the states. Giving an additional reward, proportional to the importance, to every state when the agent visits can encourage the agent to go to important states. However, this simple approach has two critical flaws: It creates overly-important states and it encourages the agent to revisit a certain state that gives a substantial additional reward.\n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=0.7\\columnwidth]{grid_with_trajectory}}\n\\caption{A simple Grid World MDP with $s_0$ as an initial state and $s_8$ as a goal, the terminal state. Figures (a) and (b) show two different successful trajectories. Figures below show the estimated importance of states using 1) every-visit counting ((c)), and 2) first-visit counting ((d)). Redder areas mean increased importance.}\n\\label{simple-mdp}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\\begin{itemize}\n \\item{\\bf Overly-important states}\\\\\nLook at the simple MDP in figure \\ref{simple-mdp}. Assume that the agent has succeeded in achieving the goal while exploring, as in figure \\ref{simple-mdp}(c). Because the successful trajectory has a loop, states $s_1, s_3, s_6$, the states that have no relationship with achieving the goal, receive the same importance as the states $s_0, s_2, s_7$ which actually contributed to achieving the goal. If the successful trajectory has many loops, which is the normal case in early stages of exploration, the importance becomes high in the states that are included in the loops. Since we are going to give an additional reward proportional to the importance, this will result in encouraging the agent to follow the loop many times, which is definitely not what we desire.\n \\item{\\bf Revisiting states}\\\\\nLet us say we have a good estimate of the importance. In the case of figure \\ref{simple-mdp}(c), huge importance would be given to states $s_0, s_2, s_4, s_7, s_8$ which are the states that have contributed to earning the goal. However, if we give an additional reward to every state the agent has visited, it results in the agent going back to the states that have large importance. It is because the estimated Q value of (state, action) pairs that have states with large importance as the next state, will explode as substantial reward is continuously given. This is also not what we desire.\n\\end{itemize}\n\nWe solve these two problems by sampling the first-visit states from the trajectories. By using first-visit sampling of the successful trajectories before counting, we can avoid giving too much importance to any specific state because it limits the maximum count for each state in a single successful trajectory to 1. An example is shown in figure \\ref{simple-mdp}. When using every-visit counting on two successful trajectories in figure \\ref{simple-mdp}(c), and \\ref{simple-mdp}(d), we get more importance on the states that are near the initial state because the agent has explored them frequently. However, with first-visit counting, we get a clear path to the goal state. Also, if we use first-visit sampling when we give an additional reward, the additional reward will only be given once to the state. This prevents the agent from revisiting a certain state because it will only get the reward when it first visits that state. Using first-visit sampling when giving an additional reward has one additional benefit; because we give the reward only once even if the agent visits that state several times, the expected additional reward given by visiting that state becomes $1\/n$ (n is the number of times the agent has visited that certain state) times the additional reward it gets when it visits that state for the first time. We can assume the current policy of the agent has a bias for a certain state if the agent visits that state often. However, because the additional reward decreases on subsequent visits, it naturally forces the agent to go to the next state rather than forming unnecessary loops.\n\nDefining what first-visit is in a large and continuous space is difficult because every state is different. To create the same effect as first-visit sampling, we use dissimilar sampling. Dissimilar sampling is a method that samples dissimilar states from the trajectories. This is a natural extension of first-visit sampling because the main idea of first-visit sampling is to avoid reusing the same states when calculating the importance and giving an additional rewards, and, the 'same states' can be extended to 'similar states' in large and continuous domains. To define similar states, we use the notion of pixel difference between sequential states, which has been successfully used in a 3D domain \\cite{jaderberg2016reinforcement} as a criteria to find the occurrence of an event. In MOL, we use this notion as a similarity between states. While using the mean of the pixel difference in the whole trajectory seems natural, it gets awkward when the environment changes dramatically. This pixel difference $\\delta$ can be viewed as\n\\begin{equation}\n\\delta = \\delta_{agent} + \\delta_{env}\n\\end{equation}\n\nThough we need to take the environment into account, the environment alone may have too great an effect to be a good criterion for sampling. This can result in sampling only the states that have huge environmental changes, even if they are not related to the task. To address this issue, we use the mean of recent pixel differences as a criteria because recent pixel difference will exclude meaningless changes in the current environment.\n\nWhile algorithm \\ref{alg:dissimilar sampling} takes the trajectory as an input which assumes using an off-policy algorithm, we can still do dissimilar sampling when using an on-policy algorithm because we sample the states in sequential order. At each step, we can update the recent history of the states and decide whether the current state should be chosen or not. If the state is chosen, we add the state into the sampled trajectory and give an additional reward to it. When the reward is given, we update the pseudo-count density model with the sampled trajectory and re-initialize the sampled trajectory. Remember, a successful trajectory is defined as a trajectory that ends with the only goal state in the trajectory. This means that there can be several successful trajectories in a single episode. Splitting an episode trajectory into several successful trajectories is reasonable because for each reward, micro-objectives should be counted separately. In other words, if a single state is important in acquiring several goals, we need to count it several times to represent its importance.\n\n\\begin{algorithm}[tb]\n   \\caption{Dissimilar Sampling}\n   \\label{alg:dissimilar sampling}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Input:} A trajectory $L =$ ($s_0, s_1, \u2026 , s_n$), RecentHistorySize = $h$\n   \\STATE {\\bfseries Output:} sampled trajectory $L^*$\n   \\STATE $L^*$ = [$s_0$]\n   \\FOR{$i=1$ {\\bfseries to} $n$}\n   \\STATE $\\delta$ = $||((s_{i-h}:s_i)-(s_{i-h+1}:s_{i+1}))||$\n   \\IF{all($||(L^*[j]-s_i)||$ $\\geq$ $\\delta$) (for $j = 1$ to the length\\\\ of ($L^*$)))}\n   \\STATE Append $s_i$ to $L^*$\n   \\ENDIF\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\nWhen designing an additional reward function, we have considered several requirements.\n\\begin{itemize}\n  \\item {\\bf Convergence}\nIt must have limits as the pseudo-count grows larger. Though we are going to clip the reward, an expanding reward is undesirable because it makes the Q function unstable.\n \\item {\\bf Early stage exploitation}\n It must be significant, even in the early stages of learning, because exploiting micro-objectives is most effective when the learning is more imperfect.\n \\item {\\bf Distinguish micro-objectives and non micro-objectives}\n It must be able to distinguish between micro-objectives and non micro-objectives, which have an importance of nearly 0.\n\\end{itemize}\n\nScaling the reward with a maximum value of (1-$R_{exp}$) for steps leading to the current learning step, will result in giving substantial reward values in the early stages of learning. Also, we clip the reward with $R_{max}$ to limit the maximum additional reward.\n\n\\begin{algorithm}[tb]\n   \\caption{Micro-Objective Learning in Deep Q-Learning}\n   \\label{alg:micro-objective learning}\n\\begin{algorithmic}\n   \\STATE Initialize replay memory $D$ and action-value function $Q$\n   \\STATE Initialize trajectory $L$ and pseudo-count model $M$ for $R_{obj}$\n   \\FOR{episode=$1$ {\\bfseries to} $n$}\n   \\REPEAT\n   \\STATE Select an action $a$ with epsilon-greedy policy\n   \\STATE Do action $a$ in current state $s$\n   \\STATE Update parameters in function $Q$ using replay memory $D$\n   \\STATE Use dissimilar sampling($L+s', h$) to get sampled trajectory $L^*$ where $s'$ is the next state\n   \\IF{$s'$ $\\in$ $L^*$}\n   \\STATE $R$ = $R$ + $R_{obj}(s')$\n   \\STATE Append $s'$ to $L$\n   \\ENDIF\n   \\STATE Insert $(s, a, s', R, terminal)$ in replay memory $D$\n   \\IF{next state is a goal state(external reward $> 0$)}\n   \\STATE Update $M$ with $L$\n   \\STATE Re-initialize trajectory $L$\n   \\ENDIF\n   \\UNTIL{Terminal is True}\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Analysis on Micro-Objective Learning}\nConsider a Markov Decision Process (MDP) which is defined with $(S, A, R, \\rho_0, \\gamma, P)$. $S$ is a set of states, $A$ is a set of possible actions, $R$ is an external reward from the environment, $\\rho_0$ is an initial state distribution, gamma is a discount factor, and $P : S \\times A \\times S \\rightarrow \\mathbb{R}$, is the transition probability distribution.\n\nWe define the importance of a state in a given trajectory.\n\n\\theoremstyle{definition}\n\\begin{definition}{Importance Count}\\\\\nLet there be a successful trajectory $L$ with visited states $s_0$ to $s_n$ in sequential order. We define a {\\bf importance count} $I^L(s_i)$ for every state $s_i$ in a MDP as follows, where $L^*$ is the optimal path from $s_0$ to $s_n$:\n\\end{definition}\n\n\\begin{equation*}\nI^L (s_i) = \n\\begin{cases}\n & 1 \\text{ if } s_i \\in L^*  \\\\ \n & 0 \\text{ otherwise}\n\\end{cases}\n\\end{equation*}\n\nFor example, in the MDP shown in figure \\ref{simple-mdp2}, assume the successful trajectory $L$ has all (state, next state) pairs except for $(s_6, s_7)$. By definition, we get importance of 1 except for the states $s_4$, and $s_6$. These two states did not contribute to reaching the goal state. This definition naturally arises from what humans do when searching for valuable steps in a complex task. Humans try to figure out why a trial was successful by tracing back the cause of the success, which is a similar process to finding the optimal path given a successful trajectory. We use the importance count to define the importance of a state given the policy.\n\n\\begin{definition}{Micro-objective} \\\\\nA micro-objective is a state $s_i$ that has importance $M_{\\pi}(s_i)$ $> 0$ given the current policy $\\pi$. Let $H$ be the set of possible successful trajectories in the given MDP.\n\\begin{equation}\nM_{\\pi}(s_i) = \\sum_{L \\in H} I^L(s_i) \\cdot p_{\\pi}(L)\n\\end{equation}\n, where $p_{\\pi}(L)$ is the probability of following the trajectory $L\n$ when using the policy $\\pi$.\n\\end{definition}\n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{simple_mdp}}\n\\caption{A simple MDP with $s_0$ as an initial state and $s_8$ as a goal state.}\n\\label{simple-mdp2}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nThe state importance should be dependent on the current policy as the value function $V(s)$. For example, in the MDP in figure \\ref{simple-mdp}, if we have a policy of going up, then the important states would be the states from below because in the current policy, getting to the states in the bottom row makes the probability of getting to the goal state higher. However, if we have a policy of going down, the states that are above are more important for reaching the goal state.\n\nConsidering that the definition of importance count is whether or not the state has contributed to achieving the goal, the importance of the micro-objective defines how likely we succeed if we are in that state using the current policy. Because we update the policy at every step, using recent successful trajectories is a reasonable approximation. For convenience, we used all successful trajectories to estimate the importance of micro-objectives.\n\nWe argue that giving an additional reward proportional to the estimated importance accelerates learning to reach the goal. Because of the discount factor $\\gamma$, states near the initial states have small $V(s)$. Also, to update the $V(s)$ of those states, updating $V(s)$ of the states between those states and the goal state is required. However, if we give an additional reward to each state, $V(s)$ will be updated quickly and the requirements mentioned above are eliminated. For example, in the figure \\ref{simple-mdp2} MDP, to update the value of state $s_1$, the values of state $s_k$ $(k \\geq 2)$ need to be updated. However, when giving an additional reward, a direct update is possible with $(s_1, a, s_j)$ $(j = 2, 3, 4)$ in the replay pool $D$.\n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{montezuma-frames}}\n\\caption{The states that have the largest, medium, and smallest $R_{obj}$ in Montezuma's Revenge: First, second, and third row each. For Montezuma's Revenge, we used 0.2 and 0.4 as the criteria. The states with the largest $R_{obj}$ are traditional subgoal states or actual goal states (key, rope, and the door). The states with the medium $R_{obj}$ are not necessary to reach the goal states, but are helpful. The states with the smallest $R_{obj}$ do not have much relationship with the goal states.}\n\\label{montezuma-frames}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nWhile the benefits of giving an additional reward to the appropriate states are obvious, estimating the appropriate states is not. When using first-visit sampling, we can effectively approximate the state importance. Assuming that we are using a substantial number of successful trajectories obtained from the current policy, with an estimated importance count $I^L_{est}$, actual importance count $I^L$, the set of states $S$, and the difference between the estimated importance $M_{est}$ and the actual importance $M$, $L_{M}$,\n\\begin{equation}\n\\label{equation_loss}\nL_{M_{est}} = \\sum_{s_i \\in S}\\sum_{L \\in H} (I^L(s_i)-I^L_{est}(s_i)) \\cdot p_{\\pi}(L)\n\\end{equation}\nAs in equation \\ref{equation_loss}, the loss comes from the difference of importance count, which is caused by states that are not included in the optimal path of a successful trajectory.\n\nTo approximate the loss of importance count $(I^L(s_i)-I^L_{est}(s_i)$, we analyze how the agent is trained. When giving rewards to every state that is visited in a successful trajectory, there can be states that distract the agent from reaching the goal.  With appropriate exploration methods, the count of these states will be low. Also, though the agent may not follow the optimal trajectory, since the estimated importance is as follows,\n\n\\begin{equation}\nM_{est}(s_i) = \\sum_{L \\in H} I^L_{est}(s_i) \\cdot p_{\\pi}(L)\n\\end{equation}\nthe agent is encouraged to follow the successful trajectory that is the most likely in the current policy, resulting in fast convergence of the policy for getting to the goal state. \n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{seaquest-frames}}\n\\caption{The states that have the largest, medium, and smallest $R_{obj}$ in Seaquest: First, second, and third row each. For Seaquest, we used $0.01, 0.1$, and $0.5$ as the criteria. The states with the largest $R_{obj}$ are states that give an external reward. The states with the medium $R_{obj}$ are the states where the submarine is firing its weapon, which is a way to get an external reward. The states with the smallest $R_{obj}$ do not have much relationship with the goal states.}\n\\label{seaquest-frames}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nTherefore, though estimated convergence by first-visit sampling does not converge to the actual importance, $L_{M_{est}}$ converges. Though MOL does not guarantee an optimal policy, it helps for an agent to learn the policy that succeeds in reaching the goal state and this makes it possible to get to the next reward, resulting in a higher average score.\n\n\n\n\\section{Experiments}\nIn the experiments, we focused on 1) analyzing which states are chosen as micro-objectives and which states have large or small importance, and 2) evaluating how much MOL accelerates learning using the discovered micro-objectives. We compared our agent to existing methods in two Atari games: Montezuma's Revenge and Seaquest. Montezuma's Revenge is notorious for its difficult exploration while Seaquest is one of the most dense reward games in Atari.\n\nIn Montezuma's Revenge, we compared a pseudo-count exploration model from \\cite{bellemare2016unifying}, which is the state-of-the-art model in this domain, with and without MOL. This was because Deep Q-learning has a difficult time obtaining successful trajectories which are needed for MOL. We tested our agent in a stochastic ALE setting with a probability to repeat previous actions of 0.25, the same setting as in \\cite{bellemare2016unifying}. \n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{montezuma_result}}\n\\caption{Average Training Episode Score of Montezuma's Revenge for 3 million training frames with three random seeds - Pseudo count exploration model from \\cite{bellemare2016unifying} versus MOL with Pseudo-count exploration model.}\n\\label{montezuma-result}\n\\end{center}\n\\vskip -0.2in\n\\end{figure} \nIn Seaquest, we compared the Double Q-learning with Monte-Carlo return, with and without MOL. Gym environment was used as an emulator\\cite{1606.01540}. Parameters used are set the same as \\cite{van2016deep}.\n\nFor dissimilar sampling, we used recent history size h = 5, and 2,500 as a minimum pixel difference for clear sampling. We reset the emulator every 250,000 frames to lower the memory usage as extremely long episodes take too much memory. Also, we used $\\alpha = 1$ and $R_{max} = 0.9$ for $R_{obj}$.\n\nWe trained the agents for 3 million frames in both games with three different random seeds. In Montezuma's Revenge, after getting the reward of 300 by reaching the door, the rewards following that come from exploring additional rooms. 3 million frames were chosen to see how the agent is trained in diverse rooms. Also, in Seaquest, it is sufficient to observe the effect of MOL.\n\n\\subsection{Analysis on Micro-Objectives}\n\nTo analyze the pseudo-count used for $R_{obj}$, we took one successful trajectory of the agent trained with 3 million frames. We sampled with dissimilar sampling and compared $R_{obj}$ of the sampled frames. Figures \\ref{montezuma-frames} and \\ref{seaquest-frames} show three sample states with the largest, medium, and the smallest $R_{obj}$ for each game when training on 3 million frames.\n\nAs we can see in the figure \\ref{montezuma-frames}, in Montezuma's Revenge, the states where the character is reaching the key, rope, and the door have the largest counts, which are traditional subgoal states. \n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{seaquest_result}}\n\\caption{Average Training Episode Score of Seaquest for 3 million training frames with three random seeds - Double Q-learning with Monte Carlo return(a model from \\cite{bellemare2016unifying} without exploration bonus) versus MOL with Double Q-learning with Monte Carlo return model.}\n\\label{seaquest-result}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nHowever, the states which have medium counts are not traditional subgoal states, but are important in reaching the key or the door. In the first sample, the character is reaching the rope while he does not need to use that path to reach it. Also in the third sample, the character does not need to go left since he can also jump to the left. The states which have the smallest counts appear to have weak relationships with the goal states.\n\nIn Seaquest, the initial game states and goal states have the largest counts as in figure \\ref{seaquest-frames}. In the states with medium counts, the submarine appears to do the \"fire\" action which is needed to get additional points. As in Montezuma's Revenge, the states which have the smallest counts seem to have weak relationships with the goal states.\n\n\n\n\n\\subsection{Accelerating Learning}\nThe learning curve of both games are shown in figure \\ref{montezuma-result} and \\ref{seaquest-result}. We averaged the training episode scores of three experiments. In Montezuma's Revenge, because the reward is sparse and the subgoal state is clear, the gap dramatically increases as expected. After three million frames of training, the average training episode score of the agent with MOL exceeded 100, which means the agents are constantly getting to the door after obtaining the key. Meanwhile, Seaquest is a dense reward game which is rather hard to interpret the subgoal states. However, even in Seaquest, the gap between the baseline and MOL increased as training proceeds. This suggests MOL can be applied to games with unclear subgoal states. After training 3 million frames, using MOL resulted in 120.34\\%, and 18.25\\% increase in Montezuma's Revenge and Seaquest scores, respectively.\n\\begin{table}[t]\n\\label{ratio-table}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lcc}\n\\hline\n\\abovespace\\belowspace\nAlgorithm & Seaquest & Montezuma's\\ Revenge \\\\\n\\hline\n\\abovespace\nBaseline       & 267.10 $\\pm$\\ 11.88 & 51.40 $\\pm$\\ 10.73 \\\\\nWith MOL       & 315.84 $\\pm$\\ 15.01 & 113.26 $\\pm$\\ 40.62\\\\\n\\hline\n\\abovespace\nRatio(\\%) & 18.25 & 120.34 \\\\\n\\hline\n\\belowspace\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\caption{Comparison between the baseline and with MOL after training on 3 million frames.}\n\\end{table}\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{montezumaroom}}\n\\caption{Explored rooms in Montezuma's Revenge after 1.5 million training frames - PSC only explored room 1 and 2 in all 3 experiments with room 0 explored in 2 experiments and room 6 and 7 explored in only 1 experiment. PSC+MOL explored room 0, 1, 2, 6, and 7 in all 3 experiments with room 5 explored in 2 experiments.}\n\\label{montezuma-room}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\nUsing MOL can be viewed as an exploitation of the successful trajectories which might distract exploration. However, the agent can explore better with improved exploitation methods. As in figure \\ref{montezuma-room}, with MOL, the agent normally explores 6 rooms (one experiment fails to explore room 5), before 1.5 million training frames. The fastest room search was exploring 6 rooms in 0.4 million training frames while the baseline searches 6 rooms after 5 million training frames according to \\cite{bellemare2016unifying}. Also, in one experiment, the agent with MOL started to get the external reward of 2,500 before training 1.5 million frames, while the agent needs to collect three items (a key, a door, and a knife) to reach a score of 2,500.\n\n\\section{Conclusion}\nIn this paper, we proposed an autonomous and effective hierarchical reinforcement learning method, Micro-Objective Learning, which accelerates learning by setting micro-objectives with pseudo-counts. Using dissimilar sampling, we avoided counting and giving rewards to similar states, which was critical to discovering precise micro-objectives and learning. Experimental results in Montezuma's Revenge and Seaquest show that micro-objectives embrace the subgoals which were heuristically designed previously and are effective in both sparse and dense reward settings.\n\nIn this work, we have successfully applied the notion of pixel difference for dissimilar sampling. However, for generalization, we have to use higher level features instead of pixel difference. Currently, we need to pre-train the networks to get higher level features, but it does not give sufficiently good features. With additional exploration in unsupervised learning, we could generalize further. In addition, although giving an additional reward directly to an original MDP is effective in accelerating the learning process, it does not guarantee convergence to the optimal policy. Therefore, finding a way to guarantee convergence while still getting the advantages of directly giving an additional reward will be our next step.\n\n\\section*{Acknowledgements} \nThe authors would like to thank Heidi Lynn Tessmer for discussion and helpful comments.\n\\nocite{silver2016mastering}\n\\nocite{sutton1999between}\n\\nocite{konidaris2009skill}\n\\nocite{bellemare2012investigating}\n\\nocite{stolle2002learning}\n\\nocite{stadie2015incentivizing}\n\\nocite{hasselt2010double}\n\\nocite{jaderberg2016reinforcement}\n\\nocite{chiu2011subgoal}\n\\nocite{csimcsek2005identifying}\n\\nocite{goel2003subgoal}\n\\nocite{bakker2004hierarchical}\n\\nocite{oh2015action}\n\\nocite{houthooft2016vime}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOnline learning has received little attention from physics education researchers relative to topics such as conceptual understanding or student discussions in the classroom \\citep{docktor_synthesis_2014}. Physics courses are comparatively rare in online offerings, in part because of the hands-on laboratory courses required by the introductory sequence. However, many instructors are interested in promoting more student discussion in their classes, and web-based forums are a readily available tool for this purpose \\citep{howard_discussion_2015}. Some work in physics has analyzed student discussion posts about homework problems \\citep{kortemeyer_analysis_2006} or in textbook annotation \\citep{miller_analysis_2016}, but more general-purpose forums of the type commonly discussed in distance learning literature are only beginning to be studied \\citep{traxler_coursenetworking_2016,gavrin_connecting_2017}. \n\nForums are included with learning management systems at universities, and are freely available on various stand-alone platforms. Thus, they represent a tool that is available to instructors regardless of their choice of homework system or textbook. \nTo better understand these tools, this paper turns to network methods, which are a natural framework for analyzing the intricate record of transactions produced by discussion forums \\citep{garton_studying_1997}.\nWe consider data from three semesters of an introductory physics course, taking the entire forum transcript (on- or off-topic) as our data. We use network analysis to explore and compare the structure of the discussions between semesters, drawing on the ``map'' of student connectivity that electronic records preserve. Since online environments have not been extensively studied in physics, we will summarize key results and questions of interest from \neducational technology and distance learning. \nThis study applies network analysis in a new context for physics education research and aims to begin building a physics-specific understanding of how students form asynchronous discussion communities that help their learning.\n\n\n\\subsection{Computer-mediated communication}\n\nResearch about how students talk online is typically published \nunder keywords such as computer-mediated communication (CMC) and computer-supported collaborative learning (CSCL). \nNumerous CSCL studies compare online to offline classes in terms of student achievement or satisfaction, and in many cases find that the online environment does at least as well as face-to-face classes \\citep{johnson_synchronous_2006}. \nPotential strengths of asynchronous forums include longer ``think time'' and the ability to easily reference comments from previous weeks, while drawbacks include reluctance to participate and high variability in comment quality \\citep{guzdial_effective_2000,howard_discussion_2015}. The reduced-social-cues nature of text communication can lead to an unpredictable social gestalt in CMC. Researchers have observed both impersonal, highly task-focused environments, and equally strong interpersonal groups where a sense of community can even interfere with ``on-task'' discussions\nif members hesitate to disagree with each other \\citep{walther_computer-mediated_1996}. A review by \\citet{walther_computer-mediated_1996} synthesizes early results to suggest that the speed and quality of community development are shaped by a sense of shared purpose among users, longevity of\nthe group, and outside cues or facilitation.\n\nEducational settings vary in formality from technical, highly-focused project work to free-for-all socializing, producing a range of conversation styles from \nexpository to epistolary \\citep{fahy_epistolary_2002}. \nShared purpose might be expected as a given in course forums, but in practice is often missing, and this is one area where instructor guidance can be very influential \\citep{guzdial_effective_2000,howard_discussion_2015}. \nTo analyze the cognitive level of discussions, many researchers have turned to content analysis. Key results from this area are summarized by \\citet{de_wever_content_2006} in their review of 15 \ncontent analysis schemes for asynchronous discussion groups. They find that content analysis tools vary widely in how clearly they connect learning theory to content codes and how (or if) they report inter-rater reliability measures. Few schemes were used in more than one study, and there is no wide consensus about how to break online conversations into an appropriate ``length scale'' (post, sentence, etc.) for analysis \\citep{rourke_methodological_2001}.\n\nMany researchers instead seek purely quantitative ways to study online talk, including social network analysis. \\citet{garton_studying_1997} argue that social network analysis can effectively describe online interactions with concepts like tie strength, multiplexity (different channels or purposes of communication), or structural roles of nodes in the network. \\citet{wortham_nodal_1999} notes that different network topologies could be productively linked to claims about communities of practice or cognitive apprenticeship. Though network analysis does not speak to the details of messages between students, it can show who talks to whom, the density and frequency of those ties, and how they evolve over time. For instructors trying to build a useful community for an online or online-supplemented course, there are many open questions, some of them first posed decades ago \\citep{rice_electronic_1987,guzdial_effective_2000}: What time scales are appropriate to characterize discussions? What does reciprocity in relationships mean online, where many students might read a post but few respond? How much instructor involvement is needed to promote useful conversation?\n\nIn this study, we include data from the entire semester, to eliminate possible selection effects from only sampling a slice of weeks. The question of reciprocity is taken up again in Section \\ref{sec:meth-nw} where our network model is discussed. We found no obvious link between the instructor's posting frequency and the discussion network that develops, but a future content analysis of the data may better address this question. \nA final caution in generalizing from the CSCL literature is that most results come from fully online courses, and graduate-level courses are overrepresented. \nIt may be possible to draw on the discussion strengths of forums without the isolating effects of a distance course by using a web-based forum to supplement a traditional live class. Studies of this type of forum use are still relatively rare \\citep{guzdial_effective_2000,yang_effects_2003}, especially at the introductory undergraduate level. This adjunct or ``anchored'' mode may be of the most interest to physics educators, whose courses are typically offered face-to-face \nand who increasingly want to build community as part of active learning.\n\n\\subsection{Network analyses of online learning}\n\nIn a recent review of social network analyses in educational technology, \\citet{sie_social_2012} classify study goals as visualization, analysis, simulation, or intervention. Work reviewed here fits in the first two types, and \ncan be grouped into two broad categories: descriptive studies of the structure of student networks in online education, and research connecting students' network positions with performance measures. \nIn the first category, work appearing in distance learning or online education literature has used network methods to understand online community structures (or lack thereof). \nResearchers use network analysis to show power relations in the group or the engagement level of learners \\citep{wortham_nodal_1999,de_laat_investigating_2007}. Other work contrasts between semesters or between student groups within a semester \\citep{reffay_social_2002,aviv_network_2003}, and uses visual displays or clustering analysis to show differences in the community structure. \nThese studies all function as proofs of concept for analyzing online talk via networks, and some suggest best practices for constructing learning environments, but they are primarily exploratory.\nThey also span a range of communication channels, from synchronous text chat to asynchronous forums or email lists. \nOne larger pattern that emerges from this literature review is that \nthe communication medium affects network models---for example, using emails to link the network may produce many one-way but few reciprocal connections.\nWe will return to this issue in Section \\ref{sec:methods}.\n\nA second category of studies chooses one or more markers of course success and tries to link students' network centrality with those outcomes. \\citet{yang_effects_2003} correlated centrality in friendship, advice, and adversarial networks with several components of course grade in an undergraduate business course that used a forum to supplement the face-to-face class. They found that centrality in the advice network was positively correlated with performance in both online and offline class activities. Centrality in the adversarial network (\\textit{e.g.}, ``Which of the following individuals are difficult to keep a good relationship with?'') was negatively correlated with final exam and overall course grade. \\citet{cho_social_2007} collected survey-based networks at the beginning and end of a two-semester online course sequence on aerospace system design. They looked for links between centrality and final grade and between a Willingness-to-Communicate (WTC) construct and network growth. They found that post-course (but not pre-course) degree and closeness centrality were positively correlated with final grade, and that students with higher WTC were more likely to form new ties during the two semesters. \n\nOther approaches use different positive outcomes or look for network characteristics of successful students rather than course-wide correlations. \n\\citet{dawson_study_2008} correlated students' centrality in course forum networks and their sense of course community as measured by Rovai's Classroom Community Scale \\citep{rovai_development_2002}. He found that degree and closeness centrality were positively correlated and betweenness centrality was negatively correlated with greater feelings of classroom community. \nHowever, the data pools 25 courses at undergraduate and graduate levels, different amounts of online integration, and different communication channels, so direct comparisons with these results are difficult. \nIn a second study \\citep{dawson_seeing_2010}, the same author examined student participation in an optional (but encouraged) discussion forum used as a supplement to a large introductory chemistry course. Focusing on the ``ego networks'' (immediate connections, see \\citep{hanneman_introduction_2005}) of individual students in the top and bottom 10\\% of the grade distribution, he found that students in the high-performing group had larger ego networks, and the members of those networks had higher average grades. Additionally, there was a higher percentage of instructor presence in the networks of high-scoring students, who tended to ask a larger number of conceptual questions. Students in the lower-performing group often asked more fact-based questions, which were typically answered by other students, leading to an unintended ``rich get richer'' effect of the higher-performing students receiving a larger share of instructor attention. \n\nThere is thus evidence to support networks' ability to distinguish between at least some types of online dialog structure, and to support links between network centrality and final grade. The latter point has been observed in some physics classrooms \\citep{bruun_talking_2013}, but not previously sought in electronic forums. \nWith some exceptions \\citep{aviv_network_2003}, most of the online network studies either give results for a single course offering or pool multiple courses together. They thus provide interesting cases, but it is unclear how stable their results may be from one semester to the next. Since network analysis requires start-up time for data cleaning and analysis, it is also reasonable to ask if it shows anything not already evident from the participation statistics reported by most forum software. \nBuilding on the literature above, we consider three research questions:\n\\begin{enumerate}\n\\item How do discussion forum networks differ among multiple semesters of an introductory physics course, and can this information be extracted more easily from participation statistics?\n\\item How much are student final grades correlated with their centrality in the discussion forum network?\n\\item Do centrality\/grade correlations, if present, strengthen when reducing the network to a more simplified ``backbone?''\n\\end{enumerate}\nThe third question has not been considered in any prior educational network studies we could find, but emerged from the high density of our discussion networks (Sec.\\ \\ref{sec:results}) and recent work piloting network sparsification in physics education research \\citep{brewe_using_2016}. \n\n\\section{Methods\\label{sec:methods}}\n\nBelow, we describe data collection, the process of building forum networks, comparison of network measures with final course grades, and how we simplified the network using backbone extraction. Further details on the backbone process, including source code, are in the Supplemental Material. \n\n\\subsection{Data collection}\n\nWe adopted the CourseNetworking (CN) platform \\citep{theCN}, which combines a robust forum tool with features typical of learning management systems. CN is a cloud-based platform, accessible either through a web browser or through apps on IOS and Android mobile devices. We selected CN primarily because the interface is ``student-centric,'' that is, student work occupies the majority of the view, and faculty focused tools are secondary. Although it is possible to use CN as a standalone LMS, the instructor coupled it with another system (Canvas) and used CN exclusively as a forum. The CN forum has a look and feel similar to other popular social media, so students pick it up with minimal introduction. The forum supports starting threads as either posts or polls and allows hyperlinks, embedded images and videos, and downloadable files.\nPolls may be structured as multiple choice, ranking, free response, and other formats, allowing students to create and post ``sample questions'' for one another. Students may also post Reflections (comments) beneath Posts and Polls, and rate Posts and Polls using a 1--3 star system. Our network analysis is based on which students post comments on one another's work, as detailed in the next subsection.\n\nOne of us (AG) used the CN forum in three full semesters of a calculus-based introductory mechanics class. The initial enrollment was over 160 students each semester, with the majority of the students being engineering and computer science majors. The institutional context is an urban, public university enrolling approximately 30,000 students. \nIn all three semesters, the university had undergraduate racial\/ethnic demographics of 71--72\\% white, 10\\% African American, 6--7\\% Hispanic\/Latino, other groups (including international students) 4\\% or less. \nThe majority of students commute, and most work part- or full-time in off campus jobs. \n\nThe course is heavily interactive, using Peer Instruction \\citep{mazur_peer_1997} and Just-in-Time Teaching (JiTT) \\citep{novak_just--time_1999} in the lectures, and group problem solving in the recitations. Students received extra credit (maximum 5\\% of the course grade) for use of the forum. (All calculations below involving student grades exclude forum bonus points.)\nFurther course details are described by \\citet{gavrin_students_2016}. In all semesters, CN was introduced on the first day of class with a brief demonstration. In Semesters 1 and 3, the instructor used the CN ``Tasks'' feature to provide an optional weekly discussion topic, which took place in the forum and did not involve extra class time. Finally, in Semester 3, the first-day introduction included mention of a new ability in the software to tag posts with instructor- or user-created ``hashtags.'' In all other respects, the CN implementation was identical across terms.\n \n\n\\subsection{Casting forum data as networks\\label{sec:meth-nw}}\n\nThe forum transcript contains the following data: Content ID (Post, Poll, or Reflection); a unique student identifier code; the date, time, and text of the post; the number of attachments (pictures or ``other''); and the star rating (pre-2016, number of ``likes'') accumulated by the post or comment. \nIn this analysis, the fields for text, number of attachments, and stars or Likes are not used; content ID, student code, date, and time are retained. The transcript also groups all reflections below their parent post or poll, showing a threaded view that corresponds to the student view of the forum. The CN software does track the ``nesting'' level of a reply (whether a student hit the reply button for the original post, or for another reply to that post). In practice, most students did not organize their replies in a multi-layer fashion, using a single reply layer even when the content was clearly a response to another comment. For this analysis, we treat each thread as consisting of a root plus single reply level (Fig.\\ \\ref{fig:nesting}, left). This has consequences for constructing a network---in \nsome other studies using transcript data \\citep{aviv_network_2003,de_laat_investigating_2007}, clear\nnested structure in the electronic logs \nled the authors to draw links only between a poster and the person to whom they were immediately replying. In our data, accurate nesting information is largely unavailable, requiring a different model for drawing connections between participants in a thread. \n\n\\begin{figure}[bthp]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{fig1}\n\\caption{Structure of forum transcript data. The CN data largely shows a post with a single reply layer (left), in contrast to studies where more nested structure is retained and informs the network construction (right). \\label{fig:nesting}}\n\\end{center}\n\\end{figure}\n\nThough it is intuitive that students talking in a forum are interacting with each other somehow, some set of assumptions must be chosen to map the logs into a network object. Prior studies of forum networks have used several different approaches: adding a link between a student commenter and the poster they were directly replying to~\\citep{aviv_network_2003,de_laat_investigating_2007}, surveying students at the beginning and end of the semester \\citep{yang_effects_2003,cho_social_2007}, or unspecified methods \\citep{dawson_study_2008,fahy_patterns_2001}. \nWe used a bipartite network model, often used to model situations where both people (``actors'') and some set of shared activities (\"events\") are of interest \\citep{borgatti_network_1997}. This approach has been used to model scientific collaboration networks and is starting to see use in online education studies \\citep{ouyang_influences_2017,rodriguez_exploring_2011}.\nAfter constructing a bipartite network, the analysis presented here focuses on the actor projection (see Fig.\\ \\ref{fig:bipartite}), which links together all students who posted together in the same discussion thread \\citep{borgatti_analyzing_2011,traxler_coursenetworking_2016}. \nFor the full-semester forum network, this process creates a dense, heavily-interlinked representation of student nodes, including the instructor's place in this web. People who post in many threads in common with each other will be connected by high-weight links (``edges''), while those who post in only one or two threads can only have low-weight links to others. \n\n\\begin{figure}[tbph]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{fig2}\n\\caption{Bipartite network model for transforming forum transcript into a network object. Students are ``actor'' nodes, who post to thread or ``event'' nodes. The actor network projection links together student nodes who posted to the same thread.\\label{fig:bipartite}}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Network measures}\n\nEven a small network quickly becomes unwieldy to describe by naming all actors and listing their connections. (But for very small classes this kind of description can be very illuminating, see \\citep{de_laat_investigating_2007}.)\nStructural measures condense broad features of network objects, and centrality measures quantify the position and importance of a particular node. Basic structure descriptors include the number of nodes ($N$) and edges ($N_E$) and the network density, defined as the ratio of total to possible edges: \n\\begin{equation}\n\\rho = \\frac{N_E}{N(N-1)\/2} \\label{eq:dens}\n\\end{equation}\nfor an undirected network. \nLarger social networks tend to be less dense---mathematically, because the denominator of (\\ref{eq:dens}) scales as $N^2$, and practically because any individual can only sustain relationships with so many other people \\citep{dunbar_neocortex_1992}. In a forum environment, where both rare and frequent interactions are recorded, higher density values may be expected unless some thresholding process is used (see Sec. \\ref{sec:meth-bb}). \n\nUncertainty in the network density can be estimated using boostrap methods~\\citep{snijders_non-parametric_1999}. Using this technique, a new sample of $N$ nodes is drawn from the network and an artificial network is constructed using the connections belonging to those nodes. The density of this artificial network represents a new possible value, and the process is repeated many times, generating a distribution of artificial density values. This distribution can be used to calculate a standard error for the observed statistic. We use the bootstrap method of~\\citet{snijders_non-parametric_1999} with 5000 samples.\n\nCentrality measures describe the position or importance of a node in a network. ``Position'' does not refer to physical location on a network diagram, as plotting algorithms use randomized processes to find reasonable display configurations (for example, minimizing overlap of edges). The number and strength of a node's connections to others, and the extent to which that person is at the core or periphery of the whole network, form the basis of centrality. \nThe most basic measure is degree centrality, which counts the number of edges (connections) attached to a node. In weighted networks, this concept is often expanded to node strength \\citep{hanneman_introduction_2005}, which is the sum of the node's weighted degree. In directed networks such as the reduced backbones described below, directionality of links can be tracked using in- and out-degree or in- and out-strength. All of these values account for only the direct neighbors of a node, but in many networks the \nwider set of the neighbors' connections can also constrain or boost a node's access to information or resources (for example, study group invitations).\n\nA later generation of centrality measures accounts for both the number of neighbors of a node and the importance of each of those neighbors. Their importance, in turn, depends on that of their own neighbors, requiring simultaneous solution over the whole network. Measures of this type can be computed as eigenvalue problems \\citep{bonacich_power_1987}. One of the most popular measures of this type is PageRank, the same base algorithm used by the Google search engine to rank the importance of pages on the internet \\citep{brin_anatomy_1998}. PageRank designates a node as being important if a large number of important pages point to it. It was developed for directed networks (on the internet, linking to another page makes a directed network edge), but can be used in undirected networks as well. PageRank is one of the three centrality measures used below. \n\nThe other two centrality values we will test, Target Entropy (TE) and Hide \\citep{sneppen_hide-and-seek_2005}, have been used in network analyses of classroom interactions between physics students over a university term \\citep{bruun_talking_2013}. Target Entropy is a measure of the diversity of a node's information sources; high TE nodes will have many neighbors who themselves talk to a wide array of other students. Conversely, Hide quantifies how difficult it is to ``find'' a node in the network. High-Hide nodes will have few neighbors, who may themselves be more sparsely connected than average. \n\nFor each semester, we calculate PageRank, Target Entropy, and Hide for all nodes, with PageRank using the \\texttt{igraph} package in R~\\citep{R_software,igraph} and the other two measures using code from \\citet[][Supplemental Material]{bruun_talking_2013}. \nWe then calculate Pearson correlations between each of these three centrality measures and final course grade.\nNetwork centralities inherently violate the assumption of independence that underlies typical correlation calculations. To correct for this issue, permutation tests can be used, where the data set is repeatedly resampled and the correlation re-calculated, typically thousands or tens of thousands of times~\\citep{grunspan_understanding_2014}. The resulting distribution of correlation coefficients gives an estimate of how likely the observed correlation was to occur by chance in a network of the same size and density---in other words, an empirical $p$-value. \nThough network measures are our primary interest, for comparison we also report Pearson correlations between final grade and a student's total contributions to the forum (their combined number of Posts, Polls, and Reflections). \n\n\\subsection{Backbone extraction\\label{sec:meth-bb}}\n\nThe forum networks generated by the process described above are much more dense than typical survey-based networks in a physics class of comparable size~\\citep{brewe_changing_2010,traxler_community_2015,sandt_TNT_2016}. Since they are built from thousands of posts, with content ranging from physics-based conversations to ``post count'' boosting, it seems reasonable that not all interactions are equally important. The most active individuals might be connected by some core structure underlying the ``noisy'' full network, and it is these types of structures that backbone extraction is designed to uncover \\citep{serrano_extracting_2009}. \n\nVarious methods exist for extracting backbones, and for this work we used the Locally Adaptive Network Sparsification (LANS) algorithm of~\\citet{foti_nonparametric_2011}, which has been tested on several real-world dense networks including answer distributions from the Force Concept Inventory~\\citep{brewe_using_2016}.\n LANS is tuned through a parameter $\\alpha$: for each node in the network, all edges below the $(1-\\alpha)$ percentile of edge weight are discarded. Thus, an alpha value of 0.05 would correspond to keeping only the 95th percentile and above of a node's strongest links. For a node with edges of weights 1, 5, and 10, a threshold of $\\alpha=0.05$ would remove all but the weight-10 edge(s). There is no single value of alpha which will suit for all network problems; rather, each analysis should test several values and select one that simplifies to the desired density while still preserving necessary information. Here, we test several values of $\\alpha$ and re-run permutation correlation tests between centrality \n and final grade, investigating whether backbone extraction strengthens the correlations by removing the effect of extraneous low-weight connections.\n\n\\section{Results\\label{sec:results}}\n\n\\subsection{Participation and network statistics}\n\n\\begin{table*}\n\\begin{ruledtabular}\n\\caption{Forum participation and network statistics by semester. Participation includes students enrolled ($N_{class}$), percent who posted in the forum, total number of threads and replies, and average replies per thread and posts per student plus or minus standard deviation. Network statistics include number of nodes ($N$), isolates, average degree ($\\pm SD$), and network density ($\\pm$ boostrapped standard error).\\label{tab:part-nw}}\n\\begin{center}\n\\begin{tabular}{c|cccccc|cccc}\nSemester\t& $N_{class}$\t& Part.\\ (\\%)\t& Threads\t& Replies\t& Replies\/thread\t& Posts\/student\t& $N$\t& Isolates\t& Degree\t& Density \\\\ \\hline\n1 \t& 173\t& 90\t& 936\t& 2376\t& $2.5\\pm\t3.6$\t& $21\\pm16$ & 156\t& 12\t& $53\\pm30$\t& $0.32\\pm0.03$ \\\\\n2\t& 152\t& 86\t& 912\t& 2253\t& $2.5\\pm2.4$\t& $23\\pm24$ & 131\t& 5\t& $29\\pm22$\t& $0.22\\pm0.03$ \\\\\n3\t& 166\t& 87\t& 762\t& 2508\t& $3.3\\pm3.3$\t& $22\\pm22$ & 145\t& 6\t& $42\\pm29$\t& $0.28\\pm0.03$ \n\\end{tabular}\n\\end{center}\n\\end{ruledtabular}\n\\end{table*}\n\nTable \\ref{tab:part-nw} shows summary participation statistics for the forum. Each semester, 85--90\\% of the enrolled students posted at least once. The number of threads was similar between the first two semesters and lower in the third, when the average number of replies per thread increased. We compared thread length and posts per student between semesters using pairwise Wilcoxan tests, which account for the non-normal distribution and presence of outliers in the data. Only Semester 3 had a significantly different ($p < 10^{-5}$) average number of replies per thread. \nThere were no significant differences in the number of posts per student between semesters.\n\nAverage posts per student can mask very different posting patterns, if some semesters have a few high-volume participants and others have a lower but more widespread posting rate.\nFigure \\ref{fig:par} shows the distribution of forum contributions among students. \nTo control for varying class size, the figure shows the density, essentially a smoothed histogram normalized to integrate to 1 for each semester. \nAll three semesters have a peak at low activity (0--15 contributions), a few very active members around 75--100 contributions, and a high-activity ``tail.'' Semester 1 has its largest peak around 25 contributions, while the other two semesters had a less prominent ``shoulder'' there. \n\n\\begin{figure}[ptbh]\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth]{fig3}\n\\caption{Density distribution of forum activity (combined posts, polls, and comments) for class members by semester. The instructor's contribution totals are included and are 94, 182, and 141 by semester.\\label{fig:par}}\n\\end{center}\n\\end{figure}\n\n\nTable \\ref{tab:part-nw} also shows descriptive statistics for the forum discussion networks. Nodes are all students who posted at least once, and isolates are students who only posted one thread, which received no replies (see  student D in Fig.\\ \\ref{fig:bipartite}). Though there are fewer isolates in the second semester compared to the first, the average degree of nodes is lower, as is the network density. Because larger networks will tend to have lower density, \nthe ``natural'' ranking of density values in the three semesters would be (2, 3, 1) for a comparable level of network structure. The observed ranking reverses this. \n\nThe aggregate forum network for the whole semester is too dense to be visually useful without extensive filtering of low weight edges (see Fig.\\ 1 in \\citet{traxler_coursenetworking_2016}). Fig.\\ \\ref{fig:networks} shows the week 7--8 subset of Semesters 1 and 2, \na time of similar activity in the middle of the semester. Each circle shows a student, \nsized by total contributions over the semester and colored by final grade. \nDarker connecting lines indicate higher-weight edges, resulting from more common posting by a pair of students.\n Though total forum activity was similar between the two semesters, the Semester 2 network is less dense and more dominated by a few high-participation, high-grade students during the time shown.\nSemester 3 (not shown) is visually ``between'' the two pictures, with fewer students posting than Semester 1 during this time slice but notably higher density than Semester 2.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=3.3in]{fig4a}\n\\includegraphics[width=3.3in]{fig4b}\n\\caption{Forum networks from weeks 7--8 in Semester 1 (left) and Semester 2 (right). Line opacity is scaled by edge weight, so darker lines indicate more threads in common for a student pair. Nodes are sized by total contributions over the semester and colored by grade (red low, yellow medium, blue high). Withdrawals and instructor or CN staff accounts are white, and the instructor's node is labeled ``I.''\\label{fig:networks}}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Centrality\/grade correlations}\n\nTable \\ref{tab:corr} shows the results of the bootstrap correlations between final grade and centrality in the discussion forum network. In the first semester, PageRank and Target Entropy are positively correlated with final grade and Hide is negatively correlated, all at small effect sizes (using Cohen's suggested thresholds of (0.1, 0.3, 0.5) for size of effect \\citep{cohen_power_1992}). In the second semester, no correlations are significant. The third semester repeats the pattern of semester 1, with the PageRank and Hide correlations now above the threshold for medium effect size. \nThe table also gives the Pearson correlation between total number of forum contributions and final grade for each semester. This correlation is only significant in Semester 3, at a medium effect size.\n\n\\begin{table}[htbp]\n\\begin{ruledtabular}\n\\caption{Correlation coefficients between final grade, the network centrality measures PageRank (PR), Target Entropy (TE), and Hide, and forum participation (total threads$+$comments). Asterisks show the level of statistical significance ($^* p < 0.05$, $^{**} p < 0.01$, and $^{***} p < 0.001$).\\label{tab:corr}}\n\\begin{center}\n\\begin{tabular}{cllll}\nSemester\t& PR\t\t\t& TE \t\t\t& Hide\t\t\t\t& Participation \t\\\\ \\hline\n1 \t\t& 0.18 $^*$\t& 0.29 $^{**}$\t& -0.27 $^{**}$\t\t\t& 0.091 \t\t\\\\\n2\t\t& 0.13 \t\t& 0.17 \t\t& -0.18 \t\t\t\t& 0.12\t\t\\\\\n3\t\t& 0.34 $^{***}$\t& 0.28 $^{**}$\t& -0.31 $^{**}$\t\t& 0.33 $^{***}$\t\t\n\\end{tabular}\n\\end{center}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\subsection{Backbone extraction}\n\nThe goal of backbone extraction is to simplify a network to its essential structure, so high-density forum networks are ideal candidates for this technique. \nFor each semester of data, we calculated the LANS backbone extraction at values of $\\alpha=(0.5, 0.1, 0.05, 0.01)$. Table~\\ref{tab:bbstats} shows the number of edges and the fraction of the original total edge weight remaining~\\citep{serrano_extracting_2009} for each reduction of the three semesters. \n\n\\begin{table}[htbp]\n\\begin{ruledtabular}\n\\caption{Edges ($N_E$) and fraction of total original weight (\\%$W_T$) at each level of backbone extraction; $\\alpha=1$ is the original network.\\label{tab:bbstats}}\n\\begin{center}\n\\begin{tabular}{lrcrcrc}\n\t& \\multicolumn{2}{c}{Semester 1}\t& \\multicolumn{2}{c}{Semester 2}\t& \\multicolumn{2}{c}{Semester 3}\t\\\\ \n$\\alpha$\t& $N_E$\t& \\%$W_T$\t& $N_E$\t& \\%$W_T$\t& $N_E$\t& \\%$W_T$ \\\\ \\hline \n1\t& 7628\t& 1.00\t& 3704\t& 1.00\t& 5858\t& 1.00 \\\\\n0.5\t& 5635\t& 0.88\t& 2476\t& 0.88\t& 4042\t& 0.88 \\\\\n0.1\t& 1173\t& 0.36\t& 572\t& 0.39\t& 1000\t& 0.39 \\\\\n0.05\t& 661\t& 0.24\t& 334\t& 0.26\t& 530\t& 0.25 \\\\\n0.01\t& 194\t& 0.09\t& 186\t& 0.12\t& 221\t& 0.10 \n\\end{tabular}\n\\end{center}\n\\end{ruledtabular}\n\\end{table}\n\nThere are competing criteria for judging a backbone extraction to be appropriate or a value of alpha to be suitably small. One heuristic is that a large portion of the original network weight (the sum of its weighted degree) should remain~\\citep{serrano_extracting_2009}. \nAnother possible metric is to lower $\\alpha$ until the forum network reaches a comparable density or average degree to a classroom survey-based network of similar size~\\citep{traxler_community_2015}. By the first measure, values of $\\alpha=0.05$ or lower may be cause for concern in this data, since they hold only a quarter of the original network weight (a small amount in comparison to the example backbones of\\citet{serrano_extracting_2009}). By the second measure, values of $\\alpha=0.05$ or 0.01 might be most appropriate. \n\nTo resolve this possible contradiction, the ultimate arbiter is what happens to the centrality values of the nodes: their relative distribution and their correlations with students' final grades. For all three semesters, backbone reduction appears to destroy rather than strengthen correlations between network centrality and final grade. \nThe negative Hide\/grade correlation vanishes immediately, \nwith similar though less severe effects on PageRank and Target Entropy (see Supplemental Material for details). In the third semester, there is some suggestion that backbone reduction does not hurt and may even help the PageRank and Target Entropy correlations down to $\\alpha=0.1$. However, the overall effect of the technique is to reduce rather than highlight the useful information.\n\nFigure \\ref{fig:boxplot-PR} shows boxplots of the PageRank scores of nodes for the original ($\\alpha=1$) and reduced networks for Semester 1. These distributions help to explain why correlations with final grade decrease as supposedly ``extraneous'' links are removed. Backbone extraction flattens calculated centrality values for most nodes in the network as $\\alpha$ decreases, with the distribution skewing lower and many nodes eventually occupying the minimum possible PageRank value. \nPlots for the other semesters and the other two centrality measures show a similar effect.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.93\\linewidth]{fig5.eps}\n\\caption{Boxplot of PageRank centrality values for the Semester 1 network backbones. \nThe bottom, middle, and top line of the boxes show first quartile, median, and third quartile. \nThe upper and lower ``whiskers'' extend to the maximum\/minimum values or 1.5 times the inter-quartile range, whichever is larger.\nAs alpha decreases, more node centralities cluster at the minimum value.\\label{fig:boxplot-PR}}\n\\end{center}\n\\end{figure}\n\n\n\\section{Discussion}\n\n\\subsection{Network analysis reveals important differences in forum use between semesters}\n\nOur first research question was: {\\em How do discussion forum networks differ among multiple semesters of an introductory physics course, and can this information be extracted more easily from participation statistics?} From the analyses summarized in Table \\ref{tab:part-nw} and Fig.\\ \\ref{fig:networks}, we find that the forum networks have different densities, average degree, and breadth of participation between semesters. In particular, Semesters 1 and 3 show a higher level of connectivity that is not easily explained by fluctuations in class size or numbers of discussion threads and comments.\nIn contrast, non-network participation statistics show few significant differences between the classes, with only Semester 3 having longer discussion threads (but not more activity overall).\nSome essential structure of discussions is captured by network analysis, beyond that available by participation tracking but without time-consuming content analysis of posts.\n\nOur second research question was: {\\em How much are student final grades correlated with their centrality in the discussion forum network?}\nStudents who are more central in the forum network tend to score higher in the course, but not in every semester---in particular, the higher-density networks are those in which centrality is correlated with grade. \nTarget Entropy and Hide seem to be the most reliable predictors, \nwith PageRank somewhat less consistent. Exploratory analysis shows that in this data set, Target Entropy and Hide are highly correlated, so we focus our discussion below on Target Entropy. \nThis result builds on the findings of question 1: not only do networks better capture the discussion connectivity, but they track a kind of interaction that benefits students in the course.\n\nOur third research question was: {\\em Do centrality\/grade correlations, if present, strengthen when reducing the network to a more simplified ``backbone?''}\nWe predicted that using backbone extraction on the forum networks would clarify correlations between centrality and final grade, by streamlining low-weight links that proliferate in long ``chat'' threads and leaving the most important connections between students. We found that instead, this ``noise'' is part of the signal, and that reducing the forum networks to backbone representations destroys correlations between centrality and grade.\nIt is possible that\na backbone extraction method developed specifically for bipartite networks \\citep{neal_backbone_2014} might improve this result. However, plotting PageRank, Target Entropy, and Hide at successive alpha levels shows that backbone extraction flattens these centrality distributions and pushes more and more nodes to the minimum value. This issue seems likely to recur even with a change in algorithm.\n\n\\subsection{Implications for network research}\n\nOne recommendation that emerges from the literature review of this paper is for researchers to carefully document their choices in using network models to describe online learning. Some past studies have used survey methods to gather network data \\citep{yang_effects_2003,cho_social_2007}, while others draw from electronic logs \\citep{wortham_nodal_1999,reffay_social_2002,aviv_network_2003,de_laat_investigating_2007,dawson_seeing_2010}. Studies in the first category base their approach on earlier social network analysis studies of business organizations, though physics education researchers have tied similar data collection to theoretical frameworks of transformation of participation or communities of practice \\citep{brewe_investigating_2012,bruun_networks_2012}. \n\nStudies that derive their data from electronic logs are more common in the CSCL literature, and this is a promising direction given the growing amount of data that is available from learning management systems.\n\\citet{kortemeyer_analysis_2006} argues that these data open a more natural window onto students' thought processes than think-aloud interviews, where students may be trying to perform to the interviewer's expectations.  \nFor instructors, detecting differences in student participation early in the semester, based on their use of resources like forums, can give early warnings about at-risk students in live or online courses \\citep{dawson_seeing_2010,reffay_social_2002}. \n\nA few studies do not specify how they constructed their networks. \nBoth the data source (survey or logs) and the assumptions made about how to connect the network have consequences for the density and structures that result. In other words, the network model---what constitutes a link between students?---is an interaction model \\citep{freeman_centrality_1978}, which makes a statement about what communication processes the researcher thinks are important to learning in a given environment. Our bipartite model generated far denser networks than survey-based classroom studies, even those drawn from weekly sampling (see \\citep{bruun_talking_2013}, Supplemental Material Fig.\\ 5 for link weight distribution of their densest network). We chose an expansive definition of interaction, and find that centrality in the resulting network is an equally strong predictor of grades as a sparser survey approach. \nOur measured correlations between network centrality and grades are also comparable to those found between annotation quality and exam grade in a physics content analysis study \\citep{miller_analysis_2016}.\nDifferent online learning studies have used a variety of centrality measures, and it is not at all clear that a ``best'' set will emerge. Only by documenting their assumptions can researchers allow for any hope of comparing between or replicating results. \n\n\n\\subsection{Implications for online learning research}\n\nAs outlined above, the range of data sources, network statistics, and outcome measures makes it challenging to check results between CSCL network analyses. However, we can look for alignment in trends or effect sizes of results. \\citet{dawson_seeing_2010} \nfound that high-performing students had more connections and were more likely to be linked to the instructor. High Target Entropy students in our Semesters 1 and 3, who were more likely to do well in the course, would tend to have a large number of connections like the high-scoring students in Dawson's study. Similarly, low Target Entropy---signaling limited sources of information---would generally correspond to student ego networks with only a few connections. \n\nThough the instructor in our data was not intentionally making an anchored forum with the traits recommended by \\citet{guzdial_effective_2000}, the CN interface builds in two of those authors' recommendations: a thread-grouped view with always-visible archives and the ability to choose a post category (through instructor- or user-created ``hashtags''). The authors make a third recommendation of ``anchor'' threads that prompt students with a few key discussion topics and include a link to post their contributions. In Semesters 1 and 3, the instructor created anchor threads via the Tasks feature on CN. Tasks show at the top of the forum page, and were updated once a week in those two semesters. \nThe instructor did not use these weekly tasks in Semester 2, and this change came with (though we cannot say it was the sole cause of) a loss of network connectivity.\n\n\\citet{aviv_network_2003} compared two semesters and found that the level of integration between the forum and class assignments was linked to substantial differences in the amount and cognitive level of discussion by students. Our results match theirs in part: the raw amount of discussion was not necessarily tied to facilitation, but the resulting network between students was more dense and appears to be more educationally useful in the more-structured semesters. \nThe work by Aviv and collaborators is one of a small but growing number of studies that combine network measures with content analysis of posts \\citep{rice_electronic_1987,fahy_epistolary_2002,de_laat_investigating_2007}. Work in physics has shown links between the cognitive level of student comments on homework problems \\citep{kortemeyer_analysis_2006} or textbook annotation \\citep{miller_analysis_2016} with their grades \\citep{kortemeyer_analysis_2006,miller_analysis_2016} or conceptual gains \\citep{miller_analysis_2016}.\nContent analysis of the CN data, currently in progress, will let us look for interplay between the quantitative network structures and qualitative content of discussions. \n\n\\citet{cho_social_2007} and \\citet{yang_effects_2003} found that degree centrality positively correlated with final grade in survey-based classroom networks, though in the first study, the correlation was only marginally significant and a significant correlation instead appeared with closeness centrality. Though their network construction methods were different, the correlations found ($r=0.442$ for \\citep{cho_social_2007}, $r=0.4$ or 0.46 for \\citep{yang_effects_2003}) are similar to the results of this study as well as the correlations with PageRank, Target Entropy, and Hide found by \\citet{bruun_talking_2013}. \n\nThe closest comparison study in physics is \\citet{bruun_talking_2013}, who used weekly surveys to build an aggregate network for an introductory mechanics course. We found that the three centrality measures that emerge as most important in their study---PageRank, Target Entropy, and Hide---are also useful for exploring position\/grade correlations in the forum data. Of these, Target Entropy and Hide seem to show the most consistent signal; these measures originate from a theoretical perspective of quantifying information flow \\citep{bruun_talking_2013,sneppen_hide-and-seek_2005}, which may be especially suited for describing long post chains in forum networks.\n\n\n\\section{Limitations and future work}\n\nLike most CSCL studies \\citep{johnson_synchronous_2006}, this is not a control-group experimental study. One possible reading of our results is that more engaged students tend to participate in the forum, and that high-centrality nodes are merely the ``good'' students (however a reader might define that) who would succeed regardless of the presence of a forum or discussion prompts. Certainly, there is evidence that students who are inclined to talk to others are more likely to benefit from forums \\citep{cho_social_2007}. However, the lack of forum\/grade correlations in Semester 2 suggests that this explanation is incomplete. First, and as a general argument for forum use, even students who are predisposed to talk about class material can benefit from tools for doing so outside of class hours at commuter schools. Furthermore, the differences in Semester 2 show that even a similarly-active forum may not be equally useful. There is no reason to believe that the fraction of engaged, self-starting students was substantially different between our three semesters, but there are significant differences in network structure and in correlations between forum position and grade. Taken together, these points suggest that not only does instructor facilitation matter, but that network analysis can detect this difference even when participation tracking does not. \n\nFinally, a detailed content analysis is beyond the scope of this paper, but spot-checking suggests that the most active threads (which contribute to higher network density) are a mixture of physics-based and social topics. This further weakens the idea that the correlations we found only show the ``best'' students using the forum for strictly studious purposes. The nature of the conversations and community that arise are more complicated than an on\/off-topic dichotomy \\citep{rourke_assessing_1999}, so the \nnext stage of this project will use post text to analyze the discussion differences between semesters and the effect of anchoring by weekly tasks.\nUltimately, content analysis results can be combined with a time-developing picture of the network characteristics \\citep{ouyang_influences_2017} to better understand instructor facilitation, the student discussion culture that emerges, and the benefits that both have for learning in physics forums.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn insurance and risk, a pivotal role is played by the classical {\\it Cram\\'er-Lundberg model} (also known as the {\\it compound Poisson model}). In this model independent and identically distributed claims arrive according to a Poisson process, whereas premiums are earned at a constant rate. This means that if the initial reserve is given by $u\\geqslant 0$, then the reserve level at time $t\\geqslant 0$ is given by \n\\begin{equation}X_t:= u + rt - \\sum_{i=1}^{N_t} L_i,\\label{CP}\\end{equation}\nwith $r>0$ the premium rate, $(N_t)_{t\\geqslant 0}$ a Poisson process with intensity $\\lambda>0$, and $(L_i)_{i\\in{\\mathbb N}}$ a sequence of i.i.d.\\ random variables. The key quantity of interest is the (finite-horizon) ruin probability\n${\\mathbb P}(\\exists s\\in[0,t]: X_s<0)$ and its infinite-horizon counterpart ${\\mathbb P}(\\exists s\\geqslant 0: X_s<0)$. A broad set of techniques has been developed to analyze this quantity, for the Cram\\'er-Lundberg model itself as well as for more advanced variants; we refer to \\cite{AsmussenAlbrecher} for an exhaustive overview. With the random variable $L$ denoting a generic claim, often the {\\it net profit condition} ${\\mathbb E}\\,(X_t-X_0)= rt - {\\mathbb E}N_t\\,{\\mathbb E}L>0$ is imposed. Under this condition, which effectively means that $r> \\lambda\\,{\\mathbb E}L$, it is guaranteed that ruin is rare. A practically relevant objective is to select the initial reserve $u$ such that the (finite or infinite-horizon) ruin probability is below some threshold $\\varepsilon.$\n\nEssentially the same modeling framework can be applied in the context of credit as well. Then the claim arrival process describes the default epochs, the premiums correspond to the interest received from the obligors, and the claims are the corresponding losses. One may wonder, however, whether in this setting the assumption of Poisson arrivals is any realistic: {whereas in the insurance context the number of claims issued can in principle exceed any bound, it is obvious that in the credit context the number of defaults cannot exceed the number of obligors. More concretely, as soon as an obligor goes into default, it effectively leaves the system.} Motivated by this observation, we study in this paper the ruin probability in a transient variant of the Cram\\'er-Lundberg model. We do so by defining for each obligor a random variable (e.g.\\ exponentially distributed) corresponding to the time-to-default, where after the default the obligor can neither cause any new default nor generates any interest anymore. \n\n\\vspace{3mm}\n\n{\\it Model.} We proceed by providing a more formal description of our transient variant of the classical Cram\\'er-Lundberg model. Here we state the main model, which we will generalize in various directions later in the paper.\n\nWe consider a setting in which there are initially $n\\in{\\mathbb N}$ obligors, each of which goes into default after some random amount of time. The corresponding $n$ times-to-default are assumed to be i.i.d.\\ non-negative random variables, characterized by the density $f(\\cdot)$. {In the credit context, risk is quantified over a finite time horizon justifying the use of a model in which clients eventually all go into default.} Let the loss-at-default, per obligor, be distributed as a non-negative random variable $L$, and let these losses be i.i.d., each with Laplace transform $\\ell(\\cdot)$. It is natural to assume that the income per unit of time is proportional to the number of obligors that have not gone into default yet. In other words, the surplus process increases at a rate $ri$ per unit of time, for some $r>0$, when there are $i$ obligors that have not defaulted yet, for $i\\in\\{0,\\ldots,n\\}.$ \nThe company has an initial reserve level $u>0$. Because of the similarity with insurance and risk models, throughout this paper we sometimes refer to losses as claims. \n\nThe primary objective of this paper is to evaluate $p_n(u,t)$, defined as the ruin probability of the company before time $t$, given there are $n$ obligors at time $0$ and that the initial reserve is $u$. Being able to compute $p_n(u,t)$, one can pick $u$ such that this ruin probability remains below an acceptable level $\\varepsilon>0$. In addition, when a new obligor wishes to get a loan, knowledge of $p_n(u,t)$ allows one to decide if (and, if yes, by how much) the initial level should be adjusted. \n\n\n\n\\vspace{3mm}\n\n{\\it Contributions.} For the main model, we provide a procedure by which, for any $n$, the double transform (in space and time, that is) of $p_n(u,t)$ can be determined. More specifically, we develop a recursive relation by which these transforms can be determined. While this means that one can evaluate the finite-horizon ruin probability $p_n(u,t)$ by numerical inversion, we in addition point out how to efficiently estimate this rare-event probability relying on importance sampling simulations; the procedure proposed has provable optimality properties. In addition we provide the logarithmic asymptotics of $p_n(nu,t)$ as $n$ grows large (i.e., in this setting the initial reserve $u$ is scaled by the number of obligors $n$). \n\nBesides the base model, four generalizations are dealt with in this paper.\nOne could argue that the assumption of the times-to-default being independent is not realistic, as in reality defaults tend to cluster. To resolve this issue, in one of the generalizations we allow a {\\it regime switching} mechanism (also frequently referred to as {\\it Markov modulation}) that induces dependence between the obligors. The regime could be thought of as the `state of the economy', wherein every state of the economy the dynamics of the reserve level are described by a specific Cram\\'er-Lundberg model. \nIn a second generalization, we consider a model in which some loss events correspond to defaults (reducing the number of obligors by one) while others do not (leaving the number of obligors unchanged). Another unrealistic feature of the main model is that the obligors are homogeneous: their times-to-default (losses, respectively) stem from the same distribution. To remedy this, we\nalso analyze a model variant corresponding to heterogenous obligors: there are multiple groups, each of them consisting of statistically identical obligors. {This extension offers an important additional flexibility as one can cluster obligors based on the loss distribution, which is often deterministic in the credit context, and consider classes of obligors that do not go into default or have a class-specific income rate.}\nA last extension that we discuss in this paper concerns a model in which between loss events the reserve level behaves as a Brownian motion (rather than as a deterministic drift).\n\n\\vspace{3mm}\n\n{\\it Related literature.} \nStarting from the pioneering papers by Cram\\'er \\cite{C1} and Lundberg \\cite{L1,L2}, focusing on the classical compound Poisson model \\eqref{CP},\na broad range of risk models has been analyzed. Without attempting to provide a complete overview, we proceed by discussing a few important branches; we refer to \\cite{MIK,KGS, TEU} for general accounts of risk theory.\nIn the first place, the assumption of the cumulative claim process being of compound Poisson type has been lifted, thus allowing a compound Poisson claim process perturbed by a diffusion \\cite{G2, G1}, and even a (spectrally one-sided) L\\'evy claim process; see e.g.\\ \\cite[Ch.\\ X and XI]{AsmussenAlbrecher} and \\cite{DM,KYP}. In addition, some models incorporate returns on investment, while in other models the dynamics of the reserve process are level-dependent; see e.g.\\ \\cite[Ch.\\ VIII]{AsmussenAlbrecher} and \\cite{AC, BM}. Finally, there is a substantial body of papers exploring the effect of specific dependence structures; see e.g.\\ \\cite{CKM} and, for an overview, \\cite[Ch.\\ XIII]{AsmussenAlbrecher}. More specifically, the effect of parameter uncertainty can be analyzed through the resampling model recently proposed in \\cite{CDMR}.\n\n\\vspace{3mm}\n\n{\\it Organization.} Section \\ref{S3} provides an explicit analysis, in terms of transforms, for the base model introduced above. A large deviations analysis of the tail probability is presented in Section \\ref{asy}, together with an importance-sampling based simulation approach and a uniform upper bound. The four extensions of the base model are presented by Section \\ref{ext}. {The final section contains a series of numerical experiments.}\n\n\\section{Exact analysis}\\label{S3}\nIn this section we analyze the base model that was described in the introduction. We start by defining the key quantities of this base model, pertaining to the case that each of the obligors has a time-to-default that is exponentially distributed.\nWe then present our analysis yielding a recursion for the double transform of the ruin probability. \n\n\\subsection{Notation and preliminaries}\nPer obligor the rate of going into default is $\\lambda>0$. This means that if there are still $i$ obligors left (i.e., being not in default), the time till the next default is exponentially distributed with mean $(\\lambda i)^{-1}$.\n\nRecall that $p_n(u,t)$ is the probability of ruin before time $t$, starting with $n$ obligors at time $0$, given the initial reserve level is $u$.\nIn our approach we (uniquely) characterize $p_n(u,t)$ through its double transform\n\\[\\psi_n(\\gamma) := \\int_0^\\infty e^{-\\gamma u} \\int_0^\\infty \\vartheta e^{-\\vartheta t} p_n(u,t)\\,{\\rm d}t\\,{\\rm d}u\n=\\int_0^\\infty e^{-\\gamma u}  p_n(u)\\,{\\rm d}u,\\]\nwhere $p_n(u)$ can be interpreted as the probability of ruin before an exponentially distributed clock with mean $\\vartheta^{-1}$ (which is sampled independently from anything else). The case of $t=\\infty$ corresponds with $\\vartheta\\downarrow 0$.\nThe main result of this section is an expression (recursive in $n$) for $\\psi_n(\\gamma)$: we express $\\psi_n(\\cdot)$ in terms of $\\psi_{n-1}(\\cdot)$. Observe that we can equivalently write $p_n(u)$ as ${\\mathbb P}(Z_n\\geqslant u)$, where $Z_n$ is the maximum of the {\\it net} cumulative {loss} process (the net cumulative {claim} process, in the insurance context) over the above-mentioned exponentially distributed amount of time (with mean $\\vartheta^{-1}$, that is). \n\nIn practical settings, one typically has that $r>-\\lambda\\ell'(0)=\\lambda\\,{\\mathbb E}L$, so that at any point in time ruin is rare, in the sense that the expected reserve increases as a function of time; to this end, realize that when there are $i\\in\\{0,\\ldots,n\\}$ obligors left, the `local drift' of the reserve process is $ri + \\lambda i\\,\\ell'(0)>0.$\n\n\\subsection{Analysis}\\label{subsec_analysis}\nIn this subsection we present a recursive scheme to evaluate $\\psi_n(\\gamma)$.\nThe main idea is to condition on the first event, being either the first default (which happens after an exponentially distributed time with mean $(\\lambda n)^{-1})$ or the expiration of the exponential clock (which happens after an exponentially distributed time with mean $\\vartheta^{-1})$. If the former event happens to apply first, then we can still reach ruin, but now with $n-1$ obligors and an adapted initial reserve. If the latter events occurs first, then we won't be facing ruin before the exponential clock expires.\nThese ideas can be translated into mathematical terms as\n\\begin{equation}\n\\label{RECU}p_n(u) = \\int_0^\\infty \\lambda n\\, e^{- (\\lambda n+\\vartheta) t} \\,{\\mathbb P}(Z_{n-1}+L\\geqslant u+rnt)\\,{\\rm d}t;\\end{equation}\nuse that the time till the first event is exponentially distributed with mean $(\\lambda n+\\vartheta)^{-1}$, and that the first event is a default with probability $\\lambda n\/(\\lambda n+\\vartheta)$. \n\nWe proceed by analyzing $\\psi_n(\\gamma)$ using the relation \\eqref{RECU}, with the objective to express it in terms of $\\psi_{n-1}(\\cdot)$. By a change-of-variable $v:=u+rnt$, we obtain\n\\begin{align*}\\psi_n(\\gamma) &=\\int_0^\\infty e^{-\\gamma u}  \\int_0^\\infty \\lambda n\\, e^{- (\\lambda n+\\vartheta) t} \\,{\\mathbb P}(Z_{n-1}+L\\geqslant u+rnt)\\,{\\rm d}t\\,{\\rm d}u\\\\\n&=\\frac{1}{rn} \\int_0^\\infty e^{-\\gamma u}  \\int_u^\\infty \\lambda n\\, e^{- (\\lambda n+\\vartheta) (v-u)\/(rn)}\\, {\\mathbb P}(Z_{n-1}+L\\geqslant v)\\,{\\rm d}v\\,{\\rm d}u.\n\\end{align*}\nThe next step is to swap the order of the integrals, exploiting the fact that the integral over $u$ allows an elementary solution:\n\\begin{align*}\\frac{1}{rn}& \\int_0^\\infty  \\left(\\int_0^v e^{-\\gamma u} e^{ (\\lambda n+\\vartheta) u\/(rn)} \\,{\\rm d}u\\right)\n \\lambda n\\, e^{- (\\lambda n+\\vartheta) v\/(rn)}\\, {\\mathbb P}(Z_{n-1}+L\\geqslant v)\\,{\\rm d}v\\\\\n &=\\frac{\\lambda n}{\\gamma rn -\\lambda n-\\vartheta}\\int_0^\\infty \\big(e^{- (\\lambda n+\\vartheta) v\/(rn)}-e^{-\\gamma v}\\big) {\\mathbb P}(Z_{n-1}+L\\geqslant v)\\,{\\rm d}v.\\end{align*}\nIn the last expression, we see an object that resembles a Laplace transform, but observe that it features a complementary cumulative distribution function rather than a density. Recall however the standard identity\n\\begin{equation}\\label{e1}\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}(X\\geqslant u){\\rm d}u =\\frac{1}{\\gamma}-\\frac{1}{\\gamma}\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}(X\\in{\\rm d}u)= \\frac{1-{\\mathbb E}\\,e^{-\\gamma X}}{\\gamma}.\\end{equation}\nIn addition, using integration by parts, for the non-negative random variable $Z_{n-1}$,\n\\begin{equation}\n\\label{e2}{\\mathbb E}\\,e^{-\\gamma Z_{n-1}} = \\int_0^\\infty e^{-\\gamma x} {\\mathbb P}(Z_{n-1}\\in {\\rm d}x) = 1-\\gamma  \\int_0^\\infty {\\mathbb P}(Z_{n-1}>x)e^{-\\gamma x}{\\rm d}x= 1-\\gamma\\psi_{n-1}(\\gamma).\\end{equation}\nBy the identity (\\ref{e1}), and using the independence between the random variables $Z_{n-1}$ and $L$, we obtain, for any $\\gamma\\geqslant 0$, with $d_n:=(\\lambda n+\\vartheta)\/(rn)$,\n\\begin{align*}\\psi_n(\\gamma) &=\\frac{\\lambda n}{\\gamma rn -\\lambda n-\\vartheta}\\Big(\\frac{rn}{\\lambda n+\\vartheta}\\left(1- {\\mathbb E}\\,e^{-(\\lambda n+\\vartheta)\/(rn)\\,(Z_{n-1}+L)}\\right)\\,-\\frac{1}{\\gamma}\\left(1- {\\mathbb E}\\,e^{-\\gamma\\,(Z_{n-1}+L)}\\right)\\Big)\\\\\n&=\\frac{\\lambda n}{\\lambda n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda n}{\\gamma rn -\\lambda n-\\vartheta}\\left(\\frac{{\\mathbb E}\\,e^{-\\gamma Z_{n-1}}\\ell(\\gamma)}{\\gamma}-\\frac{{\\mathbb E}\\,e^{-d_n Z_{n-1}}\\ell(d_n)}{d_n}\\right),\n\\end{align*}\nwhich, by applying (\\ref{e2}) and a few elementary algebraic steps, equals\n\\[\\frac{\\lambda n}{\\lambda n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda n}{\\lambda n +\\vartheta-\\gamma r n}\\left(B\\left(\\frac{\\lambda n +\\vartheta}{rn},\\psi_{n-1}\\left(\\frac{\\lambda n +\\vartheta}{rn}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right),\\]\nwhere we define\n\\[B(x,y):= \\ell(x)\\left(\\frac{1}{x}-y\\right).\\]\n\nConclude that we have expressed $\\psi_n(\\cdot)$ in terms of $\\psi_{n-1}(\\cdot)$, so that we would obtain a recursion if we would have an explicit expression for $\\psi_0(\\cdot)$. Recall that $\\psi_0(\\cdot)$ corresponds to ruin in the scenario without any obligor left. Obviously $p_0(u,t)\\equiv 0$ for any $u$ and $t$, entailing that $\\psi_0(\\gamma)\\equiv 0$ for any value of $\\gamma$.\nIt means that we can thus recursively compute $\\psi_n(\\gamma)$.\nThe theorem below summarizes the findings so far.\n\n\\begin{theorem} \\label{TH1} For any $\\gamma\\geqslant 0$ and $n\\in{\\mathbb N}$, we have the recursion\n\\[\\psi_{n}(\\gamma)= \\frac{\\lambda n}{\\lambda n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda n}{\\lambda n +\\vartheta-\\gamma r n}\\left(B\\left(\\frac{\\lambda n +\\vartheta}{r n},\\psi_{n-1}\\left(\\frac{\\lambda n +\\vartheta}{rn}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right),\\]\nwhere $\\psi_0(\\gamma)\\equiv 0$.\n\\end{theorem}\n\n\n\\begin{remark}\\label{R1a}{\\em \nInterestingly, one could interpret the departure of an obligor as a time change: \nthe default arrival rate drops from $\\lambda n$ to $\\lambda (n-1)$, and simultaneously the aggregate income per time unit drops from $rn$ to $r(n-1)$. As a consequence, in the infinite-horizon setting ($\\vartheta=0$, that is) the recursion in Theorem \\ref{TH1} greatly simplifies. \n}$\\hfill\\Diamond$\\end{remark}\n\n\\begin{remark}\\label{R1}{\\em \nUpon inspecting the above proof, it is readily checked that it has not been used that the income rate is proportional to the number of obligors present; similarly, it is not crucial that the time till the next default when there are still $i$ obligors is exponential with parameter $\\lambda i$. This effectively means that we can work with an income rate $r_i$ (rather than $ri$) and a default rate $\\lambda_i$ (rather than $\\lambda i$) during times that there are $i$ obligors left. We thus obtain the recursion\n\\[\\psi_{n}(\\gamma)= \\frac{\\lambda_n}{\\lambda_n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda_n}{\\lambda_n +\\vartheta-\\gamma r_n}\\left(B\\left(\\frac{\\lambda_n +\\vartheta}{r_n},\\psi_{n-1}\\left(\\frac{\\lambda_n +\\vartheta}{r_n}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right),\\]\nwhere $\\psi_0(\\gamma)\\equiv 0$. It is also remarked that one can make the loss distribution dependent on the number of obligors in the system, by working with the transform $\\beta_i(\\cdot)$ when there are still $i$ obligors that have not gone into default yet.\n}$\\hfill\\Diamond$\\end{remark}\n\n\\begin{remark}\\label{rem2}{\\em\nAn interesting special case relates to the situation in which $r_n = r$ and $\\lambda_n=\\lambda$, i.e., the conventional Cram\\'er-Lundberg model. Sending $n\\to\\infty$, one should recover the (transient version of the) Pollaczek-Khinchine formula. As an illustration, we show this computation for $\\vartheta=0$, writing $a$ for $\\lambda\/r$ and assuming that $-a\\ell'(0)<1$. We obtain the relation, with the limit of $\\psi_n(\\cdot)$ being denoted by $\\psi(\\cdot)$,\n\\[\\psi(\\gamma) = \\frac{1}{\\gamma}+\\frac{a}{a-\\gamma}\\big(B(a,\\psi(a))-B(\\gamma,\\psi(\\gamma)\\big).\\]\nIt yields, after some elementary algebra, that \n\\[1-\\gamma\\psi(\\gamma) = \\frac{\\gamma}{\\gamma-a+a\\ell(\\gamma)}\\ell(a)(1-a\\psi(a)).\\]\nThe constant $\\ell(a)(1-a\\psi(a))$ can be identified by observing that the left-hand side goes to $1$ as $\\gamma\\downarrow 0$; hence, an application of H\\^opital's rule yields that\n\\[\\ell(a)(1-a\\psi(a)) =\\lim_{\\gamma\\downarrow 0}\\frac{\\gamma-a+a\\ell(\\gamma)}{\\gamma} = 1+a\\ell'(0).\\]\nWe conclude\n\\[\\psi(\\gamma) =\\frac{1}{\\gamma}-\\frac{1+a\\ell'(0)}{\\gamma-a+a\\ell(\\gamma)},\\]\nwhich directly corresponds to the Pollaczek-Khinchine formula \\cite{AsmussenAlbrecher,DM}. Our new results can be thus be seen as a true generalization of the classical results from ruin theory. $\\hfill\\Diamond$\n}\\end{remark}\n\n\\begin{remark}{\\em\nThe recursion featuring in Thm.\\ \\ref{TH1} can be made more explicit when working with its generating function. To demonstrate this, we focus on the case of $\\vartheta=0$, $r_n=rn$, and $\\lambda_n=\\lambda n$. We have, again with $a=\\lambda\/r$,\n\\[\\psi_n(\\gamma) = \\frac{1}{\\gamma} +\\frac{a}{a-\\gamma}\\left(\\ell(a)\\left(\\frac{1}{a}-\\psi_{n-1}(a)\\right)-\\ell(\\gamma)\\left(\\frac{1}{\\gamma}-\\psi_{n-1}(\\gamma)\\right)\\right).\\]\nWe thus obtain that, using that $\\psi_0(\\gamma)=0$, \n\\begin{align*}\n\\Psi(z,\\gamma)&:= \\sum_{n=1}^\\infty z^n\\psi_n(\\gamma)\\\\\n&=\\sum_{n=1}^\\infty z^n\\frac{1}{\\gamma}+z\\frac{a}{a-\\gamma} \\sum_{n=1}^\\infty z^{n-1}\\left(\\ell(a)\\left(\\frac{1}{a}-\\psi_{n-1}(a)\\right)-\\ell(\\gamma)\\left(\\frac{1}{\\gamma}\\psi_{n-1}(\\gamma)\\right)\\right)\\\\\n&=\\frac{z}{1-z}\\frac{1}{\\gamma}+z\\frac{a}{a-\\gamma}\\left(\\ell(a)\\left(\\frac{1}{a{(1-z)}}-\\Psi(z,a)\\right)-\\ell(\\gamma)\\left(\\frac{1}{\\gamma{(1-z)}}-\\Psi(z,\\gamma)\\right)\\right) .\\end{align*} \nWe conclude that \\[\\Psi(z,\\gamma)=\\frac{1}{a-\\gamma-za\\,\\ell(\\gamma)}\\,\\left( \\frac{z}{1-z} \\frac{a-\\gamma}{\\gamma} + za\\, \\ell(a) \\left(\\frac{1}{a{(1-z)}}-\\Psi(z,a)\\right)-\\frac{z}{1-z} \\frac{ a\\,\\ell(\\gamma)}{\\gamma}\\right).\\]\nWe are thus left with determining $\\Psi(z,a)$. \nFor $a$ and $z$ fixed there is a unique positive $\\gamma\\equiv \\gamma(z,a)$ for which the denominator equals 0 (as follows from the fact that $\\nu(\\gamma):=a-\\gamma-za\\,\\ell(\\gamma)$ is concave with $\\nu(0)=a(1-z)>0$ and $\\nu(\\gamma)\\to-\\infty$ as $\\gamma\\to\\infty$). We therefore have that in $\\gamma\\equiv \\gamma(z,a)$ the numerator should equal $0$ as well. \nThis leads to\n\\begin{align*}\\Psi(z,a) &= \\frac{1}{a{(1-z)}}+\\frac{1}{1-z}\\frac{1}{\\gamma(z,a)\\,\\ell(a)}\\left(\\frac{a-\\gamma(z,a)}{a} -\\ell(\\gamma(z,a))\\right)\\\\\n&=\\frac{1}{a{(1-z)}}+{\\frac{a-\\gamma(z,a)-a\\,\\ell(\\gamma(z,a))}{(1-z)\\,a\\gamma(z,a)\\,\\ell(a)}}.\\end{align*}\nCombining the above, we have thus identified \\[\\Psi(z,\\gamma)=\\frac{z}{1-z}\\frac{1}{a-\\gamma-za\\,\\ell(\\gamma)}\\left(\\frac{a-\\gamma -a\\,\\ell(\\gamma)}{\\gamma}-\\frac{a-\\gamma(z,a)-a\\,\\ell(\\gamma(z,a))}{\\gamma(z,a)}\\right).\\] \nBy multiplying with $(1-z)$, we obtain the transform at a geometrically distributed time with success probability $z$. Sending $z\\uparrow 1$, and realizing that $\\gamma(1,a)=0$, we recover the stationary result discussed in Remark \\ref{rem2}. \n $\\hfill\\Diamond$ }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     \n\\end{remark}\n\n\n \n\\section{Asymptotics, efficient simulation, and uniform bound}\\label{asy}\nThe previous section provides us with a way of computing $p_n(u,t).$ Here one should realize that $\\psi_n(\\gamma)$ is a (double) transform, so that numerical Laplace inversion needs to be applied in order to evaluate $p_n(u,t)$. Over the past decades significant progress has been made in the domain of Laplace inversion; see for instance the fast, accurate, and generally applicable algorithms described in\\ \\cite{AW,dI}. If one wishes to avoid numerical inversion, two frequently used alternatives are  (i)~asymptotic techniques, and (ii)~simulated-based estimation. In approach (i), one scales one or more of the model parameters, and aims at finding an explicit expression for the quantity under study (in our case the ruin probability) in the regime that this scaling parameter grows large. Approach (ii) has the intrinsic drawback that, in order to obtain reliable estimates in the domain of small ruin probabilities, many runs are needed. These issues can be remedied by simulating under a suitably chosen alternative measure rather than the actual one, and correcting the simulation output by likelihood ratios; this method is known as importance sampling. \n\nIn this section we present a series of results that help to quantify the ruin probability $p_n(u,t)$ without the need to resort to numerical inversion. Our findings come in three flavors. In the first place we find, for a given $u$ and $t$, the asymptotics of $p_n(nu,t)$ as $n$ grows large; i.e., we scale the initial capital level by the initial number of obligors.\nSecondly, we derive a uniform upper bound on $p_n(u,t)$, comparable to the well-known Lundberg inequality for the conventional Cram\\'er-Lundberg model. Finally, we develop a provably efficient importance-sampling based simulation algorithm. \nImportantly, in this section we can lift the assumption of exponentially distributed time-to-defaults. \n\n\n\\subsection{Notation and preliminaries}\nThroughout this entire section we let the times-to-default $T_1,\\ldots,T_n$ be non-negative i.i.d.\\ random variables, with density $f(\\cdot)$ and distribution function $F(\\cdot)$, distributed as a generic random variable $T$.\nLet $Z_n(t)$ be the net {cumulative loss amount} at time $t\\geqslant 0$, given that at time $0$ there are $n\\in{\\mathbb N}$ obligors present. We denote, for $i=1,\\ldots,n$ and $t\\geqslant 0$, by $W_i(t)$ the net {cumulative loss amount} of the $i$-th obligor at time $t$. By distinguishing between the scenario that obligor $i$ has gone into default at time $t$ and its complement, we can write $W_i(t)$ as\n\\begin{equation}\\label{weetje}W_i(t):=1_{\\{T_i\\leqslant t\\}}L_i-r\\min\\{T_i,t\\}.\\end{equation}\nWe define the moment generating function ${\\mathbb E}\\, e^{\\alpha L}$ of the loss $L$ by $\\bar\\ell(\\alpha):=\\ell(-\\alpha)$. Then, due to fact that the obligors are statistically identical,\n\\[{\\mathbb E}\\,e^{\\alpha Z_n(t)} =\\left({\\mathbb E}\\,e^{\\alpha W_1(t)} \\right)^n.\\]\nIn addition, we can compute the moment generating function of the net {loss} amount of a single obligor at time $t$. By conditioning on the time-to-default, using \\eqref{weetje},\n\\begin{align*}\n\\omega_t(\\alpha):={\\mathbb E}\\,e^{\\alpha W_1(t)}&=\\bar\\ell(\\alpha)\\int_0^t f(s) e^{-r\\alpha s}{\\rm d}s  + \ne^{-r\\alpha t}\\int_t^\\infty f(s) {\\rm d}s\\\\\n&=\\bar\\ell(\\alpha)\\int_0^t f(s) e^{-r\\alpha s}{\\rm d}s  + \ne^{-r\\alpha t} (1-F( t)).\n\\end{align*}\nFor instance, in the special case that the time-to-defaults are exponentially distributed with mean $\\lambda^{-1}$, we have\n\\[\\omega_t(\\alpha)=\\left(1-e^{-(\\lambda+r\\alpha)t}\\right) \\frac{\\lambda}{\\lambda+r\\alpha}\\bar\\ell(\\alpha) + e^{-(\\lambda+r\\alpha)t}.\\]\n\n\\subsection{Large-deviations asymptotics}\\label{subsec_LD}\nThe goal of this subsection is to establish a limit theorem for our ruin probability, given that we start with $n$ obligors and an initial capital reserve level $nu>0$, as $n$ grows large. In other words, we analyze how the probability\n\\begin{equation}\\label{qn}q_n(t):=  p_n(nu,t)={\\mathbb P}\\left(\\exists s\\in[0,t]: Z_n(s) \\geqslant nu\\right)= {\\mathbb P}\\left(\\exists s\\in[0,t]:\\sum_{i=1}^n W_i(s) \\geqslant nu\\right)\\end{equation}\nbehaves as $n\\to\\infty.$\nWe do so under the evident `rarity condition' that, for all $t\\geqslant 0$, ${\\mathbb E}Z_n(t)$ is smaller than $nu$, or, equivalently, \n\\[\\sup_{t\\geqslant 0}\\big({\\mathbb P}(T\\leqslant t) \\,{\\mathbb E}L - r\\,{\\mathbb E}\\min \\{T,t\\} \\big) < u,\\]\nwhere we use that ${\\mathbb E}W_1(t)=\\omega'_t(0) = {\\mathbb P}(T\\leqslant t) \\,{\\mathbb E}L - r\\,{\\mathbb E}\\min \\{T,t\\}.$\nWe start by establishing a lower bound. The underlying principle is that the probability of a union of events is bounded from below by the probability of the most likely event among them. This entails that, for any $s\\in[0,t]$ we have that $q_n(t)\\geqslant \\check q_n(s)$, where\n\\[\\check q_n(s):= {\\mathbb P}\\left(\\sum_{i=1}^n W_i(s) \\geqslant nu\\right).\\]\nDefine the Legendre transform pertaining to $W_1(s)$:\n\\[I(s):=\\sup_{\\alpha}\\left(\\alpha u - \\log \\omega_s(\\alpha)\\right).\\]\nBecause of the rarity condition $\\omega'_s(0) <u$ for all $s\\geqslant 0$, we can restrict ourselves to maximizing over $\\alpha>0$ only;\nwe define $\\alpha^\\star(s):=  \\arg\\sup_{\\alpha}\\left(\\alpha u - \\log \\omega_s(\\alpha)\\right).$\nBy Cram\\'er's theorem \\cite{DZ}, we immediately have that, for any $s\\in[0,t]$,\n\\begin{equation}\\label{LBS}\\liminf_{n\\to\\infty}\\frac{1}{n}\\log q_n(t) \\geqslant\\liminf_{n\\to\\infty}\\frac{1}{n}\\log \\check q_n(s) =\n-\\alpha^\\star(s) u + \\log \\omega_s(\\alpha^\\star(s))=-I(s).\\end{equation}\nWe also define \n\\[t^\\star:= \\arg\\inf_{s\\in[0,t]}I(s),\\]\nwhich has the informal interpretation of the most likely time $Z_n(\\cdot)$ exceeds $nu.$\nFrom the fact that the lower bound \\eqref{LBS} applies for any $s\\in[0,t]$, we thus obtain that\n\\[\\liminf_{n\\to\\infty}\\frac{1}{n}\\log q_n(t) \\geqslant-\\inf_{s\\in[0,t]}I(s) = -I(t^\\star).\\]\n\nWe proceed by proving that $-I(t^\\star)$ is also an upper bound on the decay rate of $q_n(t)$. The first step is to realize that ruin occurs at the default time of one of the $n$ obligors. As a consequence, we can rewrite $q_n$ in terms of the union of $n$ events:\n\\[q_n(t)={\\mathbb P}\\left(\\exists j\\in\\{1,\\ldots,n\\}: T_j\\in[0,t],  \\sum_{i=1}^n W_i(T_j) \\geqslant nu\\right),\\]\ninstead of the union of uncountable many events featuring in the representation (\\ref{qn}).\nBy the union bound, we obtain that this probability $q_n(t)$ is majorized by $n\\hat q_n(t)$, where\n\\[\\hat q_n(t):={\\mathbb P}\\left(T_1\\in[0,t],  \\sum_{i=1}^n W_i(T_1) \\geqslant nu\\right).\\]\nAs $n^{-1} \\log n \\to 0$, it suffices to prove that $\\limsup_{n\\to\\infty} n^{-1} \\log \\hat q_n(t)\\leqslant -I(t^\\star).$ To this end,  by conditioning on $T_1$,\n\\[\\hat q_n(t) = \\int_0^t f(s) \\,{\\mathbb P}\\left( \\sum_{i=2}^{n} W_i(s) +L_1 -rs \\geqslant nu\\right) {\\rm d}s.\\]\nThen observe that the $W_i(T_1)$ are dependent, but once conditioned on $T_1=s$ they have become independent.  The next step is to apply the Markov inequality: for any $\\alpha\\geqslant 0$, with $L_1$ being independent from $W_2(s),\\ldots,W_{n}(s)$,\n\\begin{align*}\n{\\mathbb P}\\left(\n \\sum_{i=2}^{n} W_i(s) +L_1 -rs \\geqslant nu\n \\right)&=\n{\\mathbb P}\\left( \\exp\\left({\\alpha \\sum_{i=2}^{n} W_i(s) +\\alpha L_1}\\right) \\geqslant \\exp({\\alpha (nu+rs)})\\right)  \\\\\n&\\leqslant (w_s(\\alpha))^{n-1}\\bar\\ell(\\alpha) \\,e^{-\\alpha(nu+rs)}\n\\leqslant (w_s(\\alpha))^{n-1}\\bar\\ell(\\alpha) \\,e^{-\\alpha(n-1)u}\n.\\end{align*}\nUpon combining the above, we have thus found that for any $\\alpha(\\cdot)\\geqslant 0$,\n\\[\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\hat q_n(t) \\leqslant \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s) \\,(w_s(\\alpha(s)))^{n-1} \\bar\\ell(\\alpha(s))\\,e^{-\\alpha(s)\\,(n-1)u} {\\rm d}s.\\]\nRecall that, for any $t\\geqslant 0$,  $I(t)=\\alpha^\\star(t) u - \\log \\omega_t(\\alpha^\\star(t)).$\nPlugging in $\\alpha(\\cdot)=\\alpha^\\star(\\cdot),$ we thus obtain, in the second inequality using the definition of $t^\\star$,\n\\begin{align}\\nonumber\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\hat q_n (t)&\\leqslant \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\,e^{-(n-1)I(s)} {\\rm d}s\\\\ \\nonumber\n&\\leqslant \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\,e^{-(n-1)I(t^\\star)} {\\rm d}s\\\\\n&=-I(t^\\star)+ \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\, {\\rm d}s.\\label{laatste}\n\\end{align}\nObserve that we are done if we succeed in proving that the second term in \\eqref{laatste} is $0$, for which it suffices to prove that the integral appearing in this term is finite.\nTo this end, first observe that, with $\\tau(\\alpha):= {\\mathbb E}\\,e^{\\alpha T}$, \n\\[\\lim_{t\\to\\infty} \\omega_t(\\alpha) = \\bar\\ell(\\alpha) \\tau(-r\\alpha)=:\\Delta(\\alpha),\\]\nso that $\\alpha^\\star(\\infty)$ solves $\\Delta'(\\alpha)\/\\Delta(\\alpha) = u$. \n\n\\begin{assumption} \\label{ASS1} The function $\\alpha^\\star(\\cdot)$ is bounded on $[0,t]$.\n\\end{assumption}\n\nUnder Assumption \\ref{ASS1}, we have $\\sup_{s\\in[ 0,t]}\\alpha^\\star(s)\\leqslant M$ for some finite $M$. Note that this holds whenever the function $\\alpha^\\star(\\cdot)$ is continuous, whereas in case $t=\\infty$ we additionally require $\\alpha^\\star(\\infty)<\\infty$. \nWith this assumption in place and using that $\\alpha\\mapsto \\bar\\ell(\\alpha)$ is increasing, we conclude that \n\\[\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\, {\\rm d}s \\leqslant \\bar\\ell(M) \\int_0^t f(s)\\, {\\rm d}s \\leqslant \\bar\\ell(M)<\\infty.\\]\nSummarizing, we have shown\n\\[\\limsup_{n\\to\\infty}\\frac{1}{n}\\log q_n(t) \\leqslant -I(t^\\star).\\]\nWe have arrived at the following result.\n\\begin{theorem} \\label{TH31} As $n\\to\\infty$, under Assumption $\\ref{ASS1}$, \n\\[\\frac{1}{n}\\log q_n(t) \\to -I(t^\\star).\\]\n\\end{theorem}\n\n\\subsection{Efficient simulation}\\label{subsec_is}\nAs the above theorem only provides us with the logarithmic asymptotics of $q_n$, it is inherently imprecise. For instance, if the true asymptotic shape of $q_n$ is $n^\\alpha \\,\\exp({-nI(t^\\star)})$ for some $\\alpha\\in{\\mathbb R}$, or $\\exp(n^\\eta) \\exp({-nI(t^\\star)})$\nfor some $\\eta\\in (0,1)$, the effect of the $\\alpha$ and $\\eta$ is not visible. \nOne can get accurate estimates in an efficient way, however, applying importance sampling.  Below we present an importance sampling algorithm, which we prove to be logarithmically efficient. \n\nThe key idea is that we decompose our rare-event probability $q_n$ into $n$ rare-event probabilities, which we will be dealing with separately. We write\n\\begin{equation}\\label{sum}q_n(t) = \\sum_{j=1}^n q_{nj}(t),\\end{equation}\nwhere \n\\[q_{nj}(t):={\\mathbb P}\\left({\\mathscr F}_j\\right),\\:\\:\\:\\:\n{\\mathscr F}_j:=\n{\\mathscr E}_j\\cap \\bigcap_{i=1}^{j-1} {\\mathscr E}_i^{\\rm c},\\:\\:\\:\\:\n{\\mathscr E}_j:= \\left\\{T_j\\in[0,t],\n \\sum_{i\\not=j} W_i(T_j) +L_j -rT_j \\geqslant nu\n\\right\\};\\]\nthe validity of \\eqref{sum} is due to the events ${\\mathscr F}_j$ being (by construction) disjoint, while the union of the ${\\mathscr E}_j$ equals the union of the ${\\mathscr F}_j$. The problem of efficiently estimating $q_n(t)$ thus reduces to the problem of efficiently estimating each of the $q_{nj}(t)$ (and adding up all the resulting estimates). \n\nFix a $j\\in\\{1,\\ldots,n\\}$ and focus on the estimation of $q_{nj}$. We now define an importance sampling probability measure ${\\mathbb Q}$.\n\\begin{itemize}\n\\item[$\\circ$]\nUnder ${\\mathbb Q}$ the density of $T_j$ remains $f(\\cdot)$. \n\\item[$\\circ$] Conditionally on $T_j=s$, the moment generating function of $L_j$ becomes\n\\[\\bar\\ell^{\\mathbb Q}(\\alpha) = \\frac{\\bar\\ell(\\alpha+\\alpha^\\star(s)}{\\bar\\ell(\\alpha^\\star(s))}.\n\\]\nSampling $L_j$ from ${\\mathbb Q}$ amounts to sampling from an exponentially twisted version of the actual distribution. This is a standard procedure in applied probability; for many frequently used distributions the twisted distribution remains within the same class of distributions, but with different parameters. For instance, the $\\alpha$-twisted version of an exponentially distributed random variable with parameter $\\mu$ corresponds to an exponentially distributed random variable with parameter $\\mu-\\alpha$ (requiring that $\\alpha\\in[0,\\mu)$).\n\\item[$\\circ$] Conditionally on $T_j=s$, the moment generating function of $W_i(s)$ (for $i\\not= j$) becomes\n\\begin{equation}\\label{alp}\\omega_s^{\\mathbb Q}(\\alpha):= \\frac{\\omega_s(\\alpha+\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))}.\\end{equation}\nTo decide whether the event ${\\mathscr F}_j$ applies, we have to sample the default times $T_i$ and (if $T_i<t$) the losses $L_i$, for $i\\not=j$, in accordance with \\eqref{alp}. This can be done as follows. By (\\ref{alp}), the exponentially twisted version of $W_i(s)$ has the moment generating function\n\\begin{align*}\\omega_s^{\\mathbb Q}(\\alpha)=\n\\frac{1}{\\omega_s(\\alpha^\\star(s))}&\\left(\\int_0^s f(v)\\, e^{-(\\alpha+\\alpha^\\star(s))\\, rv} \\bar\\ell(\\alpha+\\alpha^\\star(s)){\\rm d}v \\:+\\right.\\\\ &\\hspace{3mm}\\left.\\int_s^\\infty f(v)\\, e^{-(\\alpha+\\alpha^\\star(s))\\, rs} {\\rm d}v\\right).\\end{align*}\nFrom this identity we observe that the $L_i$ can be sampled from a distribution with moment generating function $\\bar\\ell^{\\mathbb Q}(\\cdot)$, as defined above, whereas the density $f^{\\mathbb Q}(\\cdot)$ of the $T_i$ (for $i\\not=j$) becomes\n\\[f^{\\mathbb Q}(v) =\\frac{1}{\\omega_s(\\alpha^\\star(s))}f(v)\\left( e^{-\\alpha^\\star(s)\\, rv} \\bar\\ell(\\alpha^\\star(s))1_{\\{v\\leqslant s\\}} +  e^{-\\alpha^\\star(s)\\, rs} 1_{\\{v>s\\}}\\right).\\]\n\\end{itemize}\nWe proceed by detailing the importance-sampling based simulation procedure, and establishing its asymptotic efficiency. \nTo this end, we first observe that a generic sample of the likelihood ratio, say ${\\mathscr L}_j$, has the form\n\\[\n{e^{-\\alpha^\\star(T_j)\\,L_j}}\\cdot{\\bar\\ell(\\alpha^\\star(T_j))}\n\\prod_{i\\not=j} \\left({e^{-\\alpha^\\star(T_j)\\,W_i(T_j)}}\\cdot{\\omega_{T_j}(\\alpha^\\star(T_j))}\\right).\\]\nRecall that on the event ${\\mathscr F}_j$ we have $ \\sum_{i\\not= j} W_i(T_j) +L_j -rT_j \\geqslant nu$. As a consequence, on the event ${\\mathscr F}_j$ the likelihood ratio ${\\mathscr L}_j$ is majorized by\n\\begin{align*}e^{-\\alpha^\\star(T_j)(nu + r T_j)}\\cdot{\\bar\\ell(\\alpha^\\star(T_j))}&\\cdot\\big(\\omega_{T_j}(\\alpha^\\star(T_j))\\big)^{n-1}\\\\\n&\\leqslant e^{-\\alpha^\\star(T_j)(n-1)u}\\cdot{\\bar\\ell(\\alpha^\\star(T_j))}\\cdot\\big(\\omega_{T_j}(\\alpha^\\star(T_j))\\big)^{n-1}\\\\\n&=\\bar\\ell(\\alpha^\\star(T_j))\\,e^{-(n-1)\\,I({T_j})} \\leqslant \\bar\\ell(M)\\,e^{-(n-1)\\,I({T_j})}\\\\\n&\\leqslant  \\bar\\ell(M)\\,e^{-(n-1)\\,I({t^\\star})},\n\\end{align*}\nwith $M$ as defined in Section \\ref{subsec_LD} (where we let Assumption \\ref{ASS1} be in force).\nWe thus find that, with ${\\mathscr I}_j$ denoting the indicator function of ${\\mathscr F}_j$, the almost-sure inequality\n${\\mathscr L}_j\\,{\\mathscr I}_j \\leqslant \\bar\\ell(M)\\,e^{-(n-1)\\,I({t^\\star})}$, and therefore\n\\[\\sum_{j=1}^n {\\mathscr L}_j\\,{\\mathscr I}_j \\leqslant n \\,\\bar\\ell(M)\\,e^{-(n-1)\\,I({t^\\star})}.\\]\nEvidently, to obtain an estimator with good precision, we have to repeat the above experiment sufficiently often. Suppose, for each $j\\in\\{1,\\ldots,n\\}$, we perform $N\\in{\\mathbb N}$ independent trials. The corresponding likelihood ratios are denoted by \n${\\mathscr L}_{j,k}$, and the indicator functions are ${\\mathscr I}_{j,k}$, with $j\\in\\{1,\\ldots,n\\}$ and $k\\in\\{1,\\ldots,N\\}$. \nOur estimator thus becomes  \n\\[\\xi_N:=\\frac{1}{N} \\sum_{k=1}^N \\sum_{j=1}^n {\\mathscr L}_{j,k}\\,{\\mathscr I}_{j,k},\\]\nwhich is (by construction) unbiased.\nThe next step is to analyze the performance of this estimator. To this end, we observe in relation to its second moment that\n\\[{\\mathbb E}_{\\mathbb Q}\\left(\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right)^2\\right) \\leqslant n^2 \\,(\\bar\\ell(M))^2\\,e^{-2(n-1)\\,I({t^\\star})},\\]\nwith ${\\mathbb E}_{\\mathbb Q}(\\cdot)$ denoting expectation under ${\\mathbb Q}$.\nWe find the following upper bound for the second moment: \n\\[\\limsup_{n\\to\\infty}\\frac{1}{n}\\log {\\mathbb E}_{\\mathbb Q}\\left(\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right)^2\\right) \\leqslant -2 I(t^\\star).\\]\nBy Theorem \\ref{TH31}, and in addition using that variances are non-negative, we also have the corresponding lower bound:\n\\begin{align*}\n\\liminf_{n\\to\\infty}\\frac{1}{n}\\log {\\mathbb E}_{\\mathbb Q}\\left(\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right)^2\\right)& \\geqslant \\liminf_{n\\to\\infty}\\frac{2}{n}\\log {\\mathbb E}_{\\mathbb Q}\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right) \\\\&= \\liminf_{n\\to\\infty}\\frac{2}{n}\\log q_n(t) = -2 I(t^\\star).\\end{align*}\nThe above bounds lead to the following conclusion, which in practical terms entails that the number of runs needed to obtain an estimate of a given relative precision, grows sub-exponentially in $n$. For the definition of logarithmic efficiency, and related performance notions in rare-event simulation, we refer to \\cite[Ch. VI]{AG}.\n\\begin{theorem} Under Assumption $\\ref{ASS1}$, the estimator $\\xi_N$ is logarithmically efficient as $N\\to\\infty$.\n\\end{theorem}\n\n\\subsection{Uniform bound} \\label{Unif}\nIntrinsic drawbacks of the large-deviations asymptotics is that they only kick in for large $n$, and they provide us with the decay rate only. This motivates the search for a uniform upper bound on the ruin probability $p_n(u,t)$. The result is a Lundberg-type inequality derived along the same lines was done in \\cite[Section XIII.5a]{AsmussenAlbrecher} for the conventional Cram\\'er-Lundberg model in which claims (or losses in the credit context) arrive according to a fixed-intensity Poisson process. We focus on the situation that when there are $n$ obligors the time to the first default is exponentially distributed with mean $\\lambda_n^{-1}$ and the income rate is $r_n.$ Let $\\gamma_n$ be the positive solution for $\\gamma$ in \n\\[\\bar\\ell(\\gamma) \\frac{\\lambda_n}{\\lambda_n+\\gamma r_n}=1.\\]\n\n\\vspace{3mm}\n\n\\begin{theorem}\\label{th_upper}\nSuppose that $\\gamma_n$ is non-increasing in $n$. Then \n\\[p_n(u,t)\\leqslant p_n(u,\\infty)\\leqslant e^{-\\gamma_n u}.\\]\n\\end{theorem}\n\\begin{proof} It is evident that $p_n(u,t)\\leqslant p_n(u,\\infty)$.\nLet $Y_n$ be distributed as $L-r_n\\,T_1$, where $T_1$ is assumed exponentially distributed with mean $\\lambda_n^{-1}$ (independent of $L$).\nConditioning on $Y_n$ immediately yields\n\\[p_n(u,\\infty)=\\mathbb{P}(Y_{n}>u)+\\int_{-\\infty}^u p_{n-1}(u-y,\\infty){\\mathbb P}({Y_{n}}\\in{\\rm d}y).\\]\nWe claim that this implies $p_n(u,\\infty)\\leqslant e^{-\\gamma_n u}$. The proof is by induction. First note that the claim holds true for $n=0$ as $p_0(u,\\infty)=0$ for all $u> 0$. Assuming the inequality holds true for $n-1$,\\begin{align*}\np_{n}(u,\\infty)&\\leqslant\\mathbb{P}(Y_{n}>u)+\\int_{-\\infty}^u e^{-\\gamma_{n-1} (u-y)}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)\\\\\n&\\leqslant\\mathbb{P}(Y_{n}>u)+\\int_{-\\infty}^u e^{-\\gamma_{n} (u-y)}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)\\\\\n&\\leqslant e^{-\\gamma_{n} u}\\int^{\\infty}_u e^{\\gamma_{n} y}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)+\\int_{-\\infty}^u e^{-\\gamma_{n} (u-x)}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)\\\\\n&=e^{-\\gamma_{n} u}\\,{\\mathbb E}\\,e^{\\gamma_{n} Y_n}=e^{-\\gamma_{n} u}\\,\\bar\\ell(\\gamma_n) \\frac{\\lambda_n}{\\lambda_n+\\gamma_n r_n}=e^{-\\gamma_n u},\n\\end{align*}\nwhere in the second inequality it has been used that that $\\gamma_n$ is non-increasing in $n$.\n\\end{proof}\n\n\\begin{remark}{\\em \nIn the special case the default arrival intensity $\\lambda_n$ and the income rates $r_n$ are linear in the number of obligors $n$, it is readily checked that $\\gamma_n$ does not depend on $n$. As a consequence, also the upper bound derived above does not depend on $n$. $\\hfill\\Diamond$\n}\\end{remark}\n\n\n\n\\section{Non-default losses, Markov modulation, \\\\Brownian perturbations, and multiple groups} \\label{ext}\n\nIn this section we consider four important extensions of our base model.\n\\begin{itemize}\n\\item[$\\circ$] In the first extension there are both losses due to defaults (reducing the number of obligors by one) and losses that do not correspond to defaults (leaving the number of obligors unchanged).\n\\item[$\\circ$] Then we consider a model in which the dynamics are affected by a Markovian background process, thus creating dependence between the individual obligors.\n\\item[$\\circ$] We proceed by analyzing a model in which the cumulative process between jumps behaves as a Brownian motion (rather than being linear).\n\\item[$\\circ$] Finally we discuss an extension that allows heterogeneous obligors (by working with multiple groups). \n\\end{itemize}\nNote that, as opposed to the analysis presented in the previous section, in this section we let the default times be exponentially distributed. In principle, the four generalizations introduced above can be combined, but to keep the presentation as transparent as possible we have decided to discuss them separately.\n\n\n\\subsection{Non-default losses} In this subsection we consider the following extension of the model analyzed in Section \\ref{S3} (or, actually, the more general one featured in Remark \\ref{R1}). Next to losses due to defaults (happening at a Poisson rate $\\lambda_n$ with the losses having Laplace transform $\\ell(\\cdot)$ when $n$ obligors are present) there are losses that do {\\it not} correspond to defaults (happening at a Poisson rate $\\lambda^\\circ_n$ with the losses having Laplace transform $\\ell^\\circ(\\cdot)$ when $n$ obligors are present).\n\n\nWe again start our derivations by conditioning on the first event, being the first default, the first loss (not leading to default), or the expiration of the exponential clock. If a default happens first, then we can still reach ruin, but now with $n-1$ obligors and an adapted initial reserve. In case the first event is a loss which does not correspond to a default, then we can still reach ruin with $n$ obligors but an adapted initial reserve. If the exponential clock expires, then we will not be facing ruin.\n\nThis idea can be formalized as follows. With $L^\\circ$ denoting a generic random variable corresponding with a non-default loss, we obtain the relation\n\\begin{align*}p_n(u) &= \\int_0^\\infty e^{- (\\bar \\lambda_n+\\vartheta) t}\\Big(\n\\lambda_n\\, {\\mathbb P}(Z_{n-1}+L\\geqslant u+r_nt)+ \\lambda^\\circ_n\\,  {\\mathbb P}(Z_n+L^\\circ\\geqslant u+r_nt)\\Big){\\rm d}t.\n\\end{align*}\nGoing through the same type of computations as those relied on in Section \\ref{S3}, we end up with a relation between $\\psi_n(\\cdot)$ and $\\psi_{n-1}(\\cdot)$. More specifically, \nfor any $\\gamma\\geqslant 0$, using the notation $\\bar{\\lambda}_n=\\lambda_n+\\lambda_n^\\circ$, we find that\n\\begin{align}\\nonumber\\psi&_{n}(\\gamma)= \\frac{\\bar\\lambda_n}{\\bar\\lambda_n+\\vartheta}\\frac{1}{\\gamma}\\:+\\\\\\nonumber&\\:\\:\\:\\frac{\\lambda_n}{\\bar\\lambda_n+\\vartheta-\\gamma r_n}\\left(B\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n},\\psi_{n-1}\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right)\\:+\\\\&\\:\\:\\:\\frac{\\lambda_n^\\circ}{\\bar\\lambda_n+\\vartheta-\\gamma r_n}\\left(B^\\circ\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n},\\psi_{n}\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n}\\right)\\right)-B^\\circ\\left(\\gamma,\\psi_{n}(\\gamma)\\right)\\right),\\label{psin}\\end{align}\nwhere $B^\\circ(\\cdot,\\cdot)$ is defined as $B(\\cdot,\\cdot)$ but with $\\ell(\\cdot)$ replaced by $\\ell^\\circ(\\cdot)$. \nUnfortunately, this relation between $\\psi_n(\\cdot)$ and $\\psi_{n-1}(\\cdot)$ cannot be directly written in terms of an explicit recursion (as opposed to the model without non-default losses; see Theorem \\ref{TH1}). The $\\psi_n(\\cdot)$, however, can still be found recursively, using the following procedure.\n\nTo this end, we start by defining the (yet unknown) constants\n\\[A_n:=B^\\circ\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n},\\psi_{n}\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n}\\right)\\right).\\]\nThen, using that $\\psi_0(\\cdot)\\equiv 0$, observe that $\\psi_1(\\gamma)$ obeys \n\\begin{align}\\nonumber\\psi_1(\\gamma) = \\:&\\frac{\\bar\\lambda_1}{\\bar\\lambda_1+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda_1}{\\bar\\lambda_1+\\vartheta-\\gamma r_1}\\left(\\frac{r_1}{\\bar\\lambda_1+\\vartheta}\\ell\\left(\\frac{\\bar\\lambda_1+\\vartheta}{r_1}\\right)-\\frac{\\ell(\\gamma)}{\\gamma}\\right)\\:+\\\\\n&\\:\\:\\:\\frac{\\lambda_1^\\circ}{\\bar\\lambda_1+\\vartheta-\\gamma r_1}\\left(B^\\circ\\left(\\frac{\\bar\\lambda_1+\\vartheta}{r_1},\\psi_{1}\\left(\\frac{\\bar\\lambda_1+\\vartheta}{r_1}\\right)\\right)-B^\\circ\\left(\\gamma,\\psi_{1}(\\gamma)\\right)\\right).\\label{psi1_nd}\\end{align}\nWe can rewrite (\\ref{psi1_nd}), for a known function $F(\\cdot)$, as\n\\[\\psi_1(\\gamma) = F(\\gamma) +\\frac{\\lambda_1^\\circ}{\\bar\\lambda_1+\\vartheta-\\gamma r_1}\\left(A_1-\\ell^\\circ(\\gamma)\\left(\\frac{1}{\\gamma}-\\psi_1(\\gamma)\\right)\\right),\\]\nwhich can be rearranged to\n\\[1-\\gamma \\psi_1(\\gamma) = 1 -\\frac{\\gamma F(\\gamma)(\\bar\\lambda_1+\\vartheta-\\gamma r_1)+\\gamma\\lambda_1^\\circ A_1-\\lambda_1^\\circ\\ell^\\circ(\\gamma)}{\\bar\\lambda_1-\\lambda_1^\\circ\\ell^\\circ(\\gamma)+\\vartheta-\\gamma r_1}.\\]\nAs we know that $1-\\gamma \\psi_1(\\gamma)$ is a Laplace transform, its value should be between 0 and 1 for any $\\gamma\\geqslant 0$.\nHence, any zero of the denominator is necessarily also a zero of the numerator. It is standard to verify that the numerator has a single positive zero, say $\\bar\\gamma$. Then it follows that \n\\[A_1 = \\frac{\\ell^\\circ(\\bar\\gamma)}{\\bar{\\gamma}}-F(\\bar\\gamma)\\frac{\\bar\\lambda_1+\\vartheta-\\bar\\gamma r_1}{\\lambda_1^\\circ}.\\]\nNow that we have found $A_1$ and hence $\\psi_1(\\gamma)$, we can identify $A_2$ and $\\psi_2(\\gamma)$ along the same lines:\nwe first express $\\psi_2(\\gamma)$ in terms of $A_2$ using \\eqref{psin}, and then identify $A_2$ using that the zero of the denominator (which we know to equal $\\bar\\lambda_2 -\\lambda_2^\\circ\\ell^\\circ(\\gamma)+\\vartheta-\\gamma r_2$) is a zero of the numerator as well. Continuing this procedure, all $\\psi_n(\\gamma)$ (and constants $A_n$) can be found. \n\n\n\\subsection{Markov modulation}\nIn the models discussed so far the individual obligors are independent. In reality they may be affected by common external factors, to be thought of as the `state of the economy', and hence behave dependently. In this subsection we consider a model in which a particular dependence structure is incorporated, through the mechanism of Markov modulation (also known as regime-switching). \n\nWe start by describing the model. Let $(J(t))_{t\\geqslant 0}$ be an irreducible continuous-time Markov process living on $\\{1,\\ldots,d\\}$. We denote by $q_{jk}\\geqslant 0$ (for $j\\not=k$) the transition rate from state $j$ to state $k$, and $q_j:=-q_{jj}=\\sum_{k\\not=j} q_{jk}$. Let $r_{nj}$ be the rate at which the surplus process increases when there are $n$ obligors and the background process is in state $j$, let $\\lambda_{nj}$ be the corresponding hazard rate of the time to the next default, and let $\\ell_j(\\cdot)$ be the Laplace transform of the loss (with the associated generic random variable being denoted by $L_j$). \n\nLet $T_n$ be the minimum of the time of the first default and the expiration of an exponential clock of rate $\\vartheta$. Denote by\n\\[R(T_n):=\\int_0^{T_n} r_{nJ(t)}{\\rm d}t\\]\nthe increase of the surplus process till $T_n$. We start by analyzing the distribution of $R(T_n)$ through the object\n\\[F_{i,j,n}(x):= {\\mathbb P}_i(R(T_n)\\geqslant x, J(T_n)=j) := {\\mathbb P}(R(T_n)\\geqslant x, J(T_n)=j\\,|\\, J(0)=i).\\]\nUsing the standard `Markovian reasoning', i.e., by distinguishing between all possible events in a (small) time interval of length $\\Delta$ and using the memory-less property, we obtain the relation, as $\\Delta\\downarrow 0$,\n\\[F_{i,j,n}(x) = \\sum_{k\\not = j}F_{i,k,n}(x)\\,q_{kj}\\Delta + F_{i,j,n}(x-r_j\\Delta)\\big(1- (q_j+\\lambda_{nj}+\\vartheta)\\big)+o(\\Delta).\\]\nSubsequently subtracting $F_{i,j,n}(x-r_j\\Delta)$ from both sides, dividing by $\\Delta$ and taking the limit $\\Delta\\downarrow 0$, we end up with a system of linear differential equations:\n\\[F'_{i,j,n}(x) = \\sum_{k=1}^d F_{i,k,n}(x)\\,q_{kj} + F_{i,j,n}(x)\\,(\\lambda_{nj}+\\vartheta).\\]\nFor given $i$ and $n$, this is a system of $d$ coupled linear differential equations, that can be solved in the standard manner; the resulting structure depends on the multiplicities of the eigenvalues. In the sequel we assume that its solution is such that the corresponding density obeys\n\\[{\\mathbb P}_i(R(T_n)\\in {\\rm d} x, J(T_n)=j) = \\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x},\\]\nbut a similar analysis can be done if the terms in the right-hand side of the previous display also involve polynomial factors (as a consequence of the multiplicities of some of the eigenvalues being larger than one). \n\nThe key observation is the identity\n\\begin{align*}{\\mathbb P}_i(Z_n\\geqslant u) &= \\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty {\\mathbb P}_j(Z_{n-1}\\in {\\rm d}z) {\\mathbb P}_i(L_j\\geqslant R(T_n)+u-z, J(T_n)=j)\\\\\n&= \\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\int_0^\\infty {\\mathbb P}_j(Z_{n-1}\\in {\\rm d}z) {\\mathbb P}(L_j\\geqslant x+u-z) \\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x}\\,{\\rm d}x\\\\\n&= \\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty {\\mathbb P}_j(Z_{n-1}+L_j\\geqslant x+u)\\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x}\\,{\\rm d}x\n\\end{align*}\nTherefore, using the by now familiar steps concerning a change-of-variables and swapping the order of integration,\n\\begin{align*}\n\\psi_{ni}(\\gamma)&:=\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}_i(Z_n\\geqslant u)\\,{\\rm d}u\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}_j(Z_{n-1}+L_j\\geqslant x+u)\\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x}\\,{\\rm d}x\\,{\\rm d}u\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\int_u^\\infty e^{-\\gamma u} {\\mathbb P}_j(Z_{n-1}+L_j\\geqslant v)\\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}(v-u)}\\,{\\rm d}v\\,{\\rm d}u\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\sum_{k=1}^d \\xi_{i,j,k,n}\n\\left(\\int_0^v e^{-\\gamma u} e^{\\zeta_{k,n}u}\n\\,{\\rm d}u\\right){\\mathbb P}_j(Z_{n-1}+L_j\\geqslant v) e^{-\\zeta_{k,n}v}\\,{\\rm d}v\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\sum_{k=1}^d \\xi_{i,j,k,n}\n\\frac{e^{-\\zeta_{k,n}v}-e^{-\\gamma v}}{\\gamma- \\zeta_{k,n}}\n{\\mathbb P}_j(Z_{n-1}+L_j\\geqslant v)\\,{\\rm d}v.\n\\end{align*}\nFrom now on we can follow the approach presented in Section \\ref{S3}: the last expression in the previous display can be expressed in terms of $\\psi_{n-1,j}(\\cdot)$, for $j=1,\\ldots,d.$ We thus end up with a vector-valued recursion. As the derivation is fully analogous to the one corresponding to the non-modulated case, we omit the details. \n\n\\subsection{Brownian perturbations}\n\nWe proceed by making the model more realistic by allowing the process to evolve, between defaults, as Brownian motion rather than a deterministic drift. The parameters of this Brownian motion depend on the number of obligors that have not gone in default yet, say with drift coefficient $r_i$ and variance coefficient $\\sigma_i^2$ when there are $i$ obligors left. In this section the time between the $i$-th and $(i+1)$-st default is exponentially distributed with mean $\\lambda_i^{-1}$.\n\n\nConsidering a Brownian motion with parameters $r$ and $\\sigma^2$ over an interval with exponentially distributed length with mean $\\lambda^{-1}$, it is known from Wiener-Hopf theory, that \n\\begin{itemize}\n\\item[$\\circ$]\nthe maximum value $M^+$ achieved is exponentially distributed with the\nparameter \n\\[\\nu^+\\equiv \\nu^+(r,\\sigma^2,\\lambda):= \\frac{\\sqrt{r^2+2\\lambda\\sigma^2}}{\\sigma^2}-\\frac{r}{\\sigma^2}.\\]\n\\item[$\\circ$] the (absolute value of the) amount by which the process goes down after the maximum is achieved until the end of the exponentially distributed interval, say $M^-$, is exponentially distributed with the parameter\n\\[\\nu^-\\equiv \\nu^-(r,\\sigma^2,\\lambda):= \\frac{\\sqrt{r^2+2\\lambda\\sigma^2}}{\\sigma^2}+\\frac{r}{\\sigma^2}.\\]\n\\item[$\\circ$] the random variables $M^+$ and $M^-$ are independent. The rates $\\nu^+$ and $\\nu^-$ are the roots of the equation\n$\\lambda+r\\alpha-\\frac{1}{2}\\alpha^2\\sigma^2=0$. \n\\end{itemize}\nNow define $\\nu_n^\\pm:= \\nu^\\pm(-r_n,\\sigma_n^2,\\lambda_n+\\vartheta)$; note that the first parameter is $-r_n$ rather than $r_n$, as we consider the event of the cumulative claim process exceeding the value $u$ (i.e., the reserve level dropping below $0$). \nAs before, we set up a relation between $\\psi_n(\\cdot)$ and $\\psi_{n-1}(\\cdot)$. Realize that, due to the Brownian term, ruin can occur before the exponential clock (with parameter $\\vartheta$) expires; this happens with probability $ e^{-\\nu_n^+ u}$. \nFollowing the approach we have been using in the case without the Brownian term, we thus obtain the relation\n\\[p_n(u) =  e^{-\\nu_n^+ u}  + I_n(u,\\vartheta),\\]\nwhere\n\\begin{align*}I_n(u,\\vartheta)&:=\\int_0^u \\int_0^\\infty\\nu_n^+ e^{-\\nu_n^+ v}\\nu_n^- e^{-\\nu_n^- w}\n\\frac{\\lambda_n}{\\lambda_n+\\vartheta}\\,{\\mathbb P}(Z_{n-1}+L\\geqslant u-v+w)\\,{\\rm d}w\\,{\\rm d}v\\\\\n&=\\frac{\\lambda_n}{\\lambda_n+\\vartheta}\\int_0^u \\int_{u-v}^\\infty\\nu_n^+ e^{-\\nu_n^+ v}\\nu_n^- e^{-\\nu_n^- (z-u+v)}\\,\n{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}z\\,{\\rm d}v\n.\\end{align*}\n\nThe next step is to evaluate $\\psi_n(\\gamma)$, by multiplying $p_n(u)$ by $e^{-\\gamma u}$ and integrating over $u\\in[0,\\infty).$ We obtain that, interchanging the order of the integrals such that the `easy' integration (over $u$, that is) can be done first,\n\\begin{align*}\\int_0^\\infty &e^{-\\gamma u} I_n(u,\\vartheta){\\rm d}u \\\\&= \\frac{\\lambda_n}{\\lambda_n+\\vartheta}\n\\int_0^\\infty\\int_0^\\infty\\int_v^{z+v} e^{-\\gamma u} \\,\\nu_n^+ e^{-\\nu_n^+ v}\\nu_n^- e^{-\\nu_n^- (z-u+v)}\n\\,{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}u\\,{\\rm d}v\\,{\\rm d}z\\\\\n&=\n \\frac{\\lambda_n}{\\lambda_n+\\vartheta}\n\\int_0^\\infty\\int_0^\\infty {\\nu_n^-}e^{-\\gamma v}\\frac{e^{-\\nu_n^- z}-e^{-\\gamma z}}{\\gamma-\\nu_n^-}\\nu_n^+e^{-\\nu_n^+ v}\n\\,{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}v\\,{\\rm d}z\\\\\n&= \\frac{\\lambda_n}{\\lambda_n+\\vartheta} \\frac{\\nu_n^-\\nu_n^+}{(\\gamma-\\nu_n^-)(\\gamma+\\nu_n^+)} \\int_0^\\infty \n\\big(e^{-\\nu_n^- z}-e^{-\\gamma z}\\big) \\,{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}z.\n\\end{align*}\nPerforming the same steps as in the proof of Theorem \\ref{TH1}, as before relying on the identities \\eqref{e1} and \\eqref{e2} in combination with the independence of $L$ and $Z_{n-1}$, we find\nafter some standard algebra the following result.\n\n\\begin{theorem} \\label{TH41} For any $\\gamma\\geqslant 0$ and $n\\in{\\mathbb N}$, we have the recursion,\n\\begin{align*}\\psi_n(\\gamma)&=\\frac{1}{\\gamma+\\nu_n^+}+\\frac{\\lambda_n}{\\lambda_n+\\vartheta}\n\\frac{1}{\\gamma+\\nu_n^+}{\\frac{\\nu_n^+}{\\gamma}}\\\\\n&\\hspace{20mm}-\\frac{\\lambda_n}{\\lambda_n+\\vartheta} \\frac{\\nu_n^-\\nu_n^+}{(\\gamma-\\nu_n^-)(\\gamma+\\nu_n^+)}\\big( B(\\nu_n^-,\\psi_{n-1}(\\nu_n^-))-B(\\gamma,\\psi_{n-1}(\\gamma))\\big),\n\\end{align*}\nwhere $\\psi_0(\\gamma)\\equiv 0.$\n\\end{theorem}\n\n\\begin{remark}{\\em\nIn Theorem \\ref{TH41} we can simplify\n\\[\\frac{\\lambda_n}{\\lambda_n+\\vartheta} \\frac{\\nu_n^-\\nu_n^+}{(\\gamma-\\nu_n^-)(\\gamma+\\nu_n^+)}=\\frac{\\lambda_n}{\\lambda_n+\\vartheta+r_n\\gamma-\\frac{1}{2}\\gamma^2\\sigma_n^2},\\]\nusing that $\\nu_n^+$ and $\\nu_n^-$ solve\n$(\\lambda_n+\\vartheta)+r_n\\alpha-\\frac{1}{2}\\alpha^2\\sigma_n^2=0$. $\\hfill\\Diamond$\n}\\end{remark}\n\n\\subsection{Multiple groups}\\label{MG}\n\nTo make the model more realistic, one could work with multiple (heterogeneous) groups of obligors. Suppose there are $G\\in\\mathbb{N}$ groups of obligors with initially $n_j$ obligors in group $j\\in\\{1,\\ldots,G\\}$; write ${\\boldsymbol n}=(n_1,\\ldots, n_G).$ We consider the multi-group counterpart of the base model of Section \\ref{S3}:\neach obligor in group $j$ has a time-to-default that is exponentially distributed with rate $\\lambda_j$. The losses at default per obligor in group $j$ are i.i.d.\\ random variables with Laplace transform $\\ell_j(\\cdot)$; in addition these per-group sequences are assumed independent. The income per unit time for this group is $r_j i$ when there are $i\\in\\{1,\\ldots, n_j\\}$ obligors that have not gone into default yet. \n\nThe company's capital reserve is given by the sum of the reserves of the individual groups; its initial level is $u>0$. Let $\\psi_{{\\boldsymbol n}}(\\gamma)$ denote the double transform of the probability of ruin over an exponentially distributed interval (with, as usual, mean $\\vartheta^{-1}$), given there $n^j$ obligors in group $j$ that have not gone into default yet. Then by the same argumentation as before we find, for ${\\boldsymbol n}$ component-wise at least equal to 1, and with ${\\boldsymbol e}_j$ the $j$-th unit vector,\n\n\\begin{align*}\n\\psi_{\\boldsymbol n}(\\gamma)=&\\sum_{j=1}^G\\frac{\\lambda_j n_j}{\\sum_{k=1}^G\\lambda_k n_k+\\vartheta}\\frac{1}{\\gamma}+\\sum_{j=1}^G\\frac{\\lambda_j n_j}{\\sum_{k=1}^G\\lambda_k n_k +\\vartheta-\\gamma r_j n_j}\\\\\n&\\times\\Bigg(B_j\\left(\\frac{\\lambda_j +\\vartheta\/n_j}{r_j},\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_j}\\left(\\frac{\\lambda_j +\\vartheta\/n_j}{r_j}\\right)\\right)-B_j\\left(\\gamma,\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_j}(\\gamma\\right)\\Bigg),\n\\end{align*}\n\\normalsize\nwhere \\[B_j(x,y):= \\ell_j(x)\\left(\\frac{1}{x}-y\\right).\\] We have thus expressed $\\psi_{\\boldsymbol n}(\\gamma)$ as a linear function of $\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_1}(\\gamma)$ up to $\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_G}(\\gamma)$.\nA similar recursive relation be found if some of the entries of ${\\boldsymbol n}$ equal 0. Given that $\\psi_{{\\boldsymbol 0}}(\\gamma)=0$, with ${\\boldsymbol 0}$ denoting the $G$-dimensional all-zeroes vector, we have thus devised a procedure to identify $\\psi_{\\boldsymbol n}(\\gamma)$.\n\n\n\\begin{remark}{\\em \nThe above model extension with multiple classes offers an important additional flexibility. In the first place, one could cluster the obligors in terms of the loss distributions. Per class this loss can even be deterministic; this is a useful property, as in the credit context the losses of some obligors may be a priori known. In addition, we could work with some classes in which the obligors do not go bankrupt and some classes in which they do. Also, one could work with a class-specific income rate.  $\\hfill\\Diamond$\n}\\end{remark}\n\n\n\n\\section{Numerical experiments}\\label{num}\nIn this section we focus on issues concerning the numerical evaluation of the ruin probability. \nIn the first subsection, we specialize to the case that the losses are exponentially distributed, where some of the quantities that feature in the numerical analysis allow closed-form analysis. In the second subsection, we present a couple of illustrative examples. These in particular quantify the effect of the size of the obligor population.\n\n\\subsection{Exponentially distributed losses}\\label{subsec_exp}\nIn Section \\ref{subsec_analysis} the focus was on finding an expression for the double transform $\\psi_n(\\gamma)$,\nwhich can then be inverted numerically. In Section~\\ref{asy} we presented a couple of other approaches: asymptotics, an efficient importance sampling algorithm, and bounds. In this section we present an alternative technique, namely an iterative procedure that directly provides the ruin probabilities $p_n(u,t)$ themselves.  We consider the model variant in which the default rate and the income rate are $\\lambda_i$ and $r_i$, respectively, during time periods in which there are $i$ obligors left. \n\nAs in Section \\ref{subsec_analysis}, the idea is to condition on the first default.\nWe thus obtain, with $W(\\cdot)$ as introduced in Section \\ref{asy}, the following recursive relation:\n\\begin{align}\\nonumber\np_n(u,t)&=\\int_0^t  \\lambda_n e^{- \\lambda_n s} {\\mathbb P}\\left(\\sup_{0\\leqslant v\\leqslant t-s} \\sum_{i=1}^{n-1}W_i(v)+L\\geqslant u+r_ns\\right)\\,{\\rm d}s\\\\ \\nonumber\n&=\\int_0^t \\lambda_n e^{- \\lambda_ns} \\,{\\rm d}s-\\int_0^t  \\lambda_n e^{- \\lambda_n s} {\\mathbb P}\\left(\\sup_{0\\leqslant v\\leqslant t-s} \\sum_{i=1}^{n-1}W_i(v)+L\\leqslant u+r_ns\\right)\\,{\\rm d}s\\\\\n&=1-e^{-\\lambda_n t}-\\int_0^t\\int_0^{u+r_n s}  \\lambda_n e^{- \\lambda_n s}\\left(1-p_{n-1}(u+r_n s-x,t-s)\\right)\\mathbb{P}(L\\in {\\rm d}x)\\,{\\rm d}s. \\label{ECURS}\n\\end{align}\n\nWhen there is only one obligor left, there is only one scenario leading to ruin: default should take place before the exponential clock (with mean $\\vartheta^{-1}$)  expires and the loss should be sufficiently large. In other words, \n\\begin{align}\n\\nonumber p_1(u,t)&= \\int_0^t\\int^\\infty_{u+r_1s}  \\lambda_1 e^{- \\lambda_1 s} \\mathbb{P}(L\\in {\\rm d}x)\\,{\\rm d}s= \\int_0^t \\lambda_1 e^{- \\lambda_1 s} \\mathbb{P}(L\\geqslant u+r_1s)\\,{\\rm d}s\n\\end{align}\nFrom this point on we focus on the case of exponentially distributed claims with mean $\\mu^{-1}$, i.e., ${\\mathbb P}(L\\geqslant x)=e^{-\\mu x}$. We readily obtain\n\\[p_1(u,t)=\\int_0^t \\lambda_1 e^{- \\lambda_1 s} e^{-\\mu (u+r_1s)}\\,{\\rm d}s=\\frac{\\lambda_1 e^{-\\mu u}}{\\lambda_1+\\mu r_1}\\left(1-e^{-(\\lambda_1+\\mu r_1) t}\\right).\\]\nWe can thus obtain $p_2(u,t)$ applying numerical integration to \\eqref{ECURS} with $n=2$. Continuing along these lines,  $p_n(u,t)$ can be numerically evaluated for higher values of $n$. \n\n\\vspace{3mm}\n\nWe now point out how to evaluate the large-deviations asymptotics that were presented in Section~\\ref{subsec_LD}, in the case of exponentially distributed claims.\nThe moment generating function of $W_1(s)$ is for $\\alpha<\\mu$ given by\n\\[\\omega_s(\\alpha)=\\left(1-e^{-(\\lambda+ r\\alpha)s}\\right) \\frac{\\lambda}{\\lambda+r\\alpha}\\frac{\\mu}{\\mu-\\alpha} + e^{-(\\lambda+r\\alpha)s},\\]\nwhereas for $\\alpha\\geqslant \\mu$ the moment generating function is infinite. We continue by computing the mean net loss corresponding to a single obligor (as a function of time):\n\\begin{align*}m(s)&:={\\mathbb E}W_1(s) = \\frac{1}{\\mu}(1-e^{-\\lambda s}) - r\\int_0^s u\\, \\lambda e^{-\\lambda v}{\\rm d}v\n- rs\\int_s^\\infty  \\lambda e^{-\\lambda v}{\\rm d}v\\\\\n&=\\left(\\frac{1}{\\mu}-\\frac{r}{\\lambda}\\right)(1-e^{-\\lambda s}).\n\\end{align*}\n{In the sequel we will assume $u>m(\\infty)$, or equivalently $\\lambda-r\\mu<\\lambda\\mu u$, to make sure the event under consideration is rare.}\n\nThe Legendre transform pertaining to $W_1(s)$ reads\n\\[I(s):=\\sup_{0<\\alpha<\\mu}\\left(\\alpha u - \\log \\omega_s(\\alpha)\\right);\\]\nwe can rule out $\\alpha\\geqslant \\mu$ as  $\\omega_s(\\alpha)=\\infty$ for these $\\alpha$.\nBecause the first-order condition does not allow an explicit solution, one cannot write $I(s)$ in closed form. \nTwo boundary cases can be dealt with explicitly, though. It is first observed that, denoting by $\\omega'_{s,1}(\\alpha)$ the derivative of $\\omega_s(\\alpha)$ with respect to $\\alpha$, and by $\\omega'_{s,2}(\\alpha)$ the derivative of $\\omega_s(\\alpha)$ with respect to $s$,\n\\begin{align}\\nonumber I'(s)& = \\frac{\\rm d}{{\\rm d}s} \\left(\\alpha^\\star(s) u - \\log \\omega_s(\\alpha^\\star(s))\\right)\\\\&=\\frac{{\\rm d}\\alpha^\\star(s)}{{\\rm d}s}\\left(u- \\frac{\\omega'_{s,1}(\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))}\\right)- \\frac{\\omega'_{s,2}(\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))}=- \\frac{\\omega'_{s,2}(\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))},\\label{IS}\\end{align}\nwhere the last equality is due to the definition of $\\alpha^\\star(s)$. By an elementary computation,\n\\begin{equation}\\label{afge}\\omega'_{s,2}(\\alpha)= \\left(\\frac{\\lambda\\mu}{\\mu-\\alpha}-(\\lambda+r\\alpha)\\right)e^{-(\\lambda +r\\alpha)s}=\n\\frac{r\\alpha^2+\\lambda\\alpha -r\\mu\\alpha}{\\mu-\\alpha}\\,e^{-(\\lambda +r\\alpha)s}.\\end{equation}\nWe observe that the Legendre transform $I(s)$ is decreasing in $s$ whenever $\\alpha^*(s)>\\mu-{\\lambda}\/{r}$.\n\\begin{itemize}\n\\item[$\\circ$] For $s=0$, we immediately see that $\\omega_0(\\alpha) = 1$ for all $\\alpha$, so that $\\alpha^\\star(0)=\\mu$ and $I(0) = \\mu u.$ In addition, we obtain by some straightforward algebra that\n\\[I'(0) = - \\lim_{\\alpha\\uparrow \\mu} \\frac{\\omega'_{0,2}(\\alpha)}{\\omega_0(\\alpha)}=-\\infty.\\]\n\\item[$\\circ$] For $s=\\infty$,\n\\[I(s) = \\sup_{0<\\alpha<\\mu} \\kappa(\\alpha),\\:\\:\\:\\:\\kappa(\\alpha):=\\alpha u - \\log (\\lambda\\mu) +\\log(\\lambda+r\\alpha)+\\log(\\mu-\\alpha).\\]\nObserve that $\\kappa(\\cdot)$ is concave, with $\\kappa'(0)>0$ (under the assumption $u>m(\\infty)$) and $\\kappa(\\alpha)\\to-\\infty$ as $\\alpha\\uparrow\\mu.$ In other words, $\\kappa(\\cdot)$ attains a maximum in $(0,\\mu).$\nThe first order condition, determining $\\alpha^\\star(\\infty)$, is\n\\[u=\\frac{1}{\\mu-\\alpha}-\\frac{r}{\\lambda+r\\alpha},\\]\nor equivalently\n\\[ru\\alpha^2 +\\big((\\lambda-r\\mu )u+2r\\big)\\alpha - \\lambda\\mu\\big(u-m(\\infty)\\big)=0.\\]\nAs $\\lambda\\mu (u-m(\\infty))>0$, this equation has a positive and negative root. \nConsequently, $\\alpha^\\star(\\infty)$ is the positive root, i.e.,\n\\[\\alpha^\\star(\\infty) = \\frac{-2r-\\lambda u+r\\mu  u +\\sqrt{4r^2+\\lambda^2 u^2+2r\\lambda\\mu  u^2 + r^2\\mu^2  u^2}}{2ru},\\]\nso that $I(\\infty) = \\kappa(\\alpha^\\star(\\infty)).$\nNext, we want to find the sign  of $I(s)$ in the regime that $s\\to\\infty$. Based on \\eqref{IS} and \\eqref{afge}, this is the sign of \n$-r\\alpha^\\star(\\infty)-\\lambda+r\\mu$. Using the explicit solution of $\\alpha^\\star(\\infty)$, it requires some straightforward calculus to verify that this leads to a negative sign, i.e. $I(s)$ is decreasing in the regime that $s\\to\\infty$, if and only if $ \\lambda- r\\mu>-\\lambda\\mu u$. \n\\end{itemize}\n\n\\subsection{Numerical example}\nFor the numerical results we have used a setup that aligns with the one considered in \\cite{Asmussen1984}. \n\\begin{enumerate}\n\\item[$\\circ$] We consider the case that both the income rates $r_i$ and the default intensity $\\lambda_i$ are linear in the number of obligors $i$ that have not gone into default yet. We let the proportionality constants be $r=  1$ and $\\lambda=0.9$, respectively. In other words, when there are $i$ obligors in the system that have not gone into default yet, the income rate is given by $i$ and the default intensity rate by $0.9\\,i$.\n\\item[$\\circ$] The losses are exponentially distributed with parameter $\\mu=1$.\n\\end{enumerate}\nWith these parameter settings the rarity condition $m(\\infty)<u$ is satisfied for all $u>0$, as we have that $0.9-1=-0.1<0<0.9\\,u$.\n\nFirst, we focus on the evaluation of the large-deviation asymptotics. For $s\\rightarrow \\infty$ we have that the Legendre transform $I(s)$ is decreasing (increasing) if $u>\\frac{1}{9}$ (if $u<\\frac{1}{9}$, respectively). For illustrational purposes we have plotted the functions $\\alpha^\\star(s)$ and $I(s)$ in Figure \\ref{fig1}, as a function of time $s$, for $u=5$ as well as $u=0.1$.\n In the first instance, with $u=5$, the function $I(\\cdot)$ is decreasing, so that the optimal $t^\\star=\\infty$, whereas for $u=0.1$ we see that $I(\\cdot)$ attains a minimal value at $t^\\star=2.3$. \n\n\\begin{figure}\n  \\centering\n \n\\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 2.75, ymax = 4.75,\n            xtick distance = 1,\n            ytick distance = 0.25,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {b}] {\\table};\n                       \\legend{$I(s)$}\n        \\end{axis}\n    \\end{tikzpicture}}\n  \\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0.75, ymax = 1,\n            xtick distance = 1,\n            ytick distance = 0.05,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *,mark size = 0.3pt] table [x = {x}, y = {a}] {\\table};\n         \n                     \\legend{\n                $\\alpha^\\star(s)$}\n        \\end{axis}\n    \\end{tikzpicture}\n}\n\n\n\\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0.0098, ymax = 0.0107,\n            xtick distance = 1,\n            ytick distance = 0.00025,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {d}] {\\table};\n          \n            \\legend{$I(s)$}\n        \\end{axis}\n    \\end{tikzpicture}}\n    \\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.75,\n            xtick distance = 1,\n            ytick distance = 0.25,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {c}] {\\table};\n          \n            \\legend{\n                $\\alpha^\\star(s)$}\n        \\end{axis}\n    \\end{tikzpicture}}\n\n\n  \\caption{The Legendre transform $I(s)$ and the underlying optimal $\\alpha^\\star(s)$ parameter as a function of time $s$ (for $s\\in[0,5]$). In the top panels we took  for $u=5$, whereas in the bottom panels we took $u=0.1$.}\\label{fig1}\n\\end{figure}\n\n\\vspace{3mm}\n\nIn Figure \\ref{fig_Exact_n10} we present, for different values of the initial number of obligors $n$ and $u=5$,  the ruin probabilities as a function of time. This has been done relying on the iterative approach presented of Section \\ref{subsec_exp}. The double integral involved has been evaluated analytically for $n=1,2$ while numerical integration methods have been employed for $n>2$. We do observe that the ruin probability increases in the length of the time interval, as desired. \nThe upper bound (as derived in Section \\ref{Unif}) in this instance is given by 0.6065, and is independent of the number of obligors $n$. As can be observed, this upper bound is rather conservative, in particular  when there are only a few obligors in the system.\n\n\n\\begin{figure}\\resizebox{12.8cm}{9cm}{\n \\pgfplotstableread{Data2.txt}{\\table}\\pgfplotsset{scaled y ticks=false}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 10,\n            ymin = 0, ymax = 0.15,\n            xtick distance = 1,\n            ytick distance = 0.01,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            yticklabel style={\n        \/pgf\/number format\/fixed,\n        \/pgf\/number format\/precision=5\n},\nscaled y ticks=false,\n            legend pos = north west]\n                       \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {a}] {\\table};\n             \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {b}] {\\table};\n              \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {c}] {\\table};\n               \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {d}] {\\table};\n                \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {e}] {\\table};\n                 \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {f}] {\\table};\n                  \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {g}] {\\table};\n                   \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {h}] {\\table};\n                    \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {i}] {\\table};\n                     \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {j}] {\\table};\n          \n           \n        \\end{axis}\n         \\end{tikzpicture}}\n\\caption{Ruin probabilities over time: $p_n(u,t)$ as a function of $t$, for $n=1$ (bottom line) to $n=10$ (top line), with $u=5$.}\n\\label{fig_Exact_n10}\n\\end{figure}\n\nIn a next experiment we study the performance of the importance sampling technique that was presented Section \\ref{subsec_is}. The top panel of Figure \\ref{fig3}  shows, for the initial capital reserve $u$ being equal to $5$, the estimates of the ruin probability as a function of time, obtained by \nsimulation, using our importance sampling algorithm. The values nearly coincide with what is obtained \nby applying the na\\\"{i}ve, direct simulation approach (i.e., without a change of measure); \nfrom Figure \\ref{fig_Exact_n10} we in addition observe that there is a highly accurate match with the values computed using the iterative approach of Section \\ref{subsec_exp}. \nRegarding the  importance sampling simulations  it is noted that we let the events ${\\mathscr E}_j$ correspond to the event where the net cumulative loss process exceeds the initial level $u$ (instead of $nu$), as $u$ in this example corresponds to the {\\it unscaled} initial capital level. The fact that we have used as many as $10^6$ runs guarantees estimates with a high precision. The importance sampling based approach substantially outperforms direct simulation, in that it greatly reduces the variance of the estimator, as can be observed in the bottom panels of Figure \\ref{fig3}.\n\n\n\\begin{figure}\n  \\centering\n\\begin{center}\n\\resizebox{11cm}{7cm}{\n     \\pgfplotstableread{Data3.txt}{\\table}\n    \\begin{tikzpicture}    \n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.045,\n            xtick distance = 1,\n            ytick distance = 0.005,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=1}] {\\table};\n             \\addplot[purple, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=2}] {\\table};\n              \\addplot[black, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=3}] {\\table};\n               \\addplot[red, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=4}] {\\table};\n                      \n                       \\legend{\n                $n=1$,\n                $n=2$,\n                $n=3$,\n                $n=4$}   \n           \n        \\end{axis}\n         \\end{tikzpicture}}\\end{center}\n\n\n\\resizebox{8cm}{4.99cm}{\n     \\pgfplotstableread{Data3.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.175,\n            xtick distance = 1,\n            ytick distance = 0.025,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=1}] {\\table};\n             \\addplot[purple, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=2}] {\\table};\n              \\addplot[black, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=3}] {\\table};\n               \\addplot[red, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=4}] {\\table};\n \n\n \n   \n                      \n                       \\legend{\n                $n=1$,\n                $n=2$,\n                $n=3$,\n                $n=4$}   \n           \n        \\end{axis}\n         \\end{tikzpicture}}\n         \\resizebox{8cm}{4.99cm}{\n     \\pgfplotstableread{Data3.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.175,\n            xtick distance = 1,\n            ytick distance = 0.025,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n     \n       \n         \n            \n                \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=1}] {\\table};\n             \\addplot[purple, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=2}] {\\table};\n              \\addplot[black, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=3}] {\\table};\n               \\addplot[red, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=4}] {\\table};\n                      \n                       \\legend{\n                $n=1$,\n                $n=2$,\n                $n=3$,\n                $n=4$}   \n           \n        \\end{axis}\n         \\end{tikzpicture}}\n\n\\caption{Top panel: ruin probabilities, as simulated by importance sampling: $p_n(u,t)$ as a function of time $t$. Bottom left panel: variance of the estimator under direct simulation as a function of $t$. Bottom right panel: variance of the estimator under importance sampling as a function of $t$. In all experiments we took $u=5.$}\\label{fig3}\n\\end{figure}\n\n\\section{Concluding remarks}\nMotivated by applications in credit risk,  we have analyzed in this paper a transient counterpart of the classical Cram\\'er-Lundberg model. We have presented a broad range of results: exact analysis in terms of transforms, asymptotic analysis including an efficient rare-event simulation algorithm, and four model variants (viz.\\ a setup that also includes non-default losses, one with Markov modulation to make the obligors dependent, one in which the linear drifts are replaced by Brownian motions, and a last one in which there are multiple groups of obligors). \n\nFollow-up research could relate to the next steps to make this model operational. A main challenge concerns dealing with the heterogeneity between the obligors. When there are relatively few groups (with homogeneity within these groups) the approach of Section \\ref{MG} can be relied upon, but when effectively all obligors have a specific time-to-default and loss distribution, an alternative approach needs to be developed. Another topic for future research could concern procedures to on-the-fly adjust the capital level given realizations of the defaults; cf.\\ e.g.\\ the approach proposed in\\cite{DMSW}. \n\n\n\\bibliographystyle{plain}\n{\\small ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPolaritons in semiconductor microcavities are light-matter bosonic quasi-particles formed by strong coupling of cavity photons and intra-cavity excitons ~\\cite{deng2010}. Their excitonic part gives rise to strong interactions essential for fast thermalization and condensation, while their photonic part contributes to their very low effective mass ($~5 \\times 10^{-5}m_{e}$) allowing for high temperature condensation ~\\cite{roomtemperature}. Polariton condensates have been observed both under non-resonant optical excitation~\\cite{BEC2006} and more recently under electrical injection of carriers~\\cite{schneider_2013,solid_2013}. However, polaritons populate a two dimensional plane where a true Bose phase transition is theoretically possible only in the presence of a confining potential~\\cite{berman_theory_2008} and this was first demonstrated with a stress induced trap~\\cite{balili_bose-einstein_2007}. Unlike the weak atom-atom interactions in cold atomic Bose Einstein condensates (BEC), inter-particle interactions in a semiconductor microcavity are strong enough to substantially renormalise polariton self-energy, experimentally observed as a local blue-shift of the polariton spectrum. Variations of the polariton density in the plane of the cavity result in a potential landscape that can be externally controlled through real space modulation of the optical excitation beam. The malleability of the potential landscape can be used to imprint scattering centres~\\cite{sanvitto_all-optical_2011} and devise polariton traps~\\cite{Ring,chiral2014} and gates~\\cite{gao_polariton_2012}. The dynamics of polariton condensates in externally modulated potential landscapes can lead to trapped states, standing polariton waves and phase-locking of remote condensates in non-trivial configurations~\\cite{Ring,manni_spontaneous_2011,opticalSPT2013,rotating_2014,kalevich,ohadi_dissipative_2014}. Extensive control over mesoscopic polariton wavefunctions and their transitions between quantum states, coupled with the extensive propagation of polaritonic flows~\\cite{sanvitto_all-optical_2011,schmutzler_all-optical_2015}, bares applications in quantum control, quantum circuits and on-chip quantum information processing~\\cite{YamamotoReview_2014,qubits_2014}.\n\n\nIn this letter, we investigate the dynamics of pure quantum state transitions of polariton condensates under optical confinement. We utilise a ring-shaped, non-resonant optical excitation scheme to create a size-tunable annular potential trap. Under continuous wave excitation, we study the steady-state regime of trapping and condensate formation. We control the height of the potential trap by tuning the optical excitation density and observe that at coherence threshold, polaritons coalesce preferentially at the uppermost confined energy state that has the largest wavefunction overlap to the exciton reservoir that forms the trap barriers. To confirm that excited state polariton condensates are realised predominantly by polariton confinement in the optically induced potential trap, we study the transient dynamics of the formation mechanism. For this purpose, we change from continuous wave to pulsed excitation, while keeping all other parameters unaltered, and time-resolve the evolution of the spatial polariton state. Under pulsed excitation, the height of the potential barrier is transiently diminishing following the decay of the exciton reservoir. We observe that the mesoscopic polariton condensate switches between states, progressively transforming to the highest available confined energy state. The experimental observations are accurately reproduced using the extended Gross-Pitaevski equation. \n\n\n\\begin{figure*}\n\t\\includegraphics[scale=0.35]{fig1.pdf}\n\t\\centering\n\t\\caption{False color-scale experimental \\textbf{(a)-(e)} and theoretical \\textbf{(f)-(j)} states of polariton condensates.\n\t\\textbf{(a),(f)} $\\Psi_{00}$, \\textbf{(b),(g)} $\\Psi_{11}$, \\textbf{(c),(h)} $\\Psi_{01}$, \n\t\\textbf{(d),(i)} $\\Psi_{02}$, \\textbf{(e),(j)} $\\Psi_{03}$. \n\t$\\epsilon$ denotes the ellipticity of individual configurations.}\n\t\\label{fig:modes}\n\t\\vspace{-0.4cm}\n\\end{figure*}\n\nNon-ground state condensates of spatially-confined polaritons were previously observed in optical defect sites and in pillar microcavities, under Gaussian-shaped non-resonant optical excitation incident to the confinement area ~\\cite{sanvitto2009,maragkou,nardin_2010}. While gain competition in thermodynamic equilibrium has been predicted to give rise to occupation of a single or several excited states~\\cite{eastham2008,portolan2008}, in both cases, excited state condensates were shown to be driven by the dynamics of energy relaxation across the confined energy states resulting in multi-state condensation. In the case of ring-shaped excitation, two characteristically different regimes of polariton condensates have been realised. For ring radii comparable to the thermal de-Broglie wavelength a phase-locked standing-wave condensate co-localised to the excitation area was observed ~\\cite{manni_spontaneous_2011}. For ring radii comparable to the polariton propagation length in the plane of the cavity, the excitation ring acted as a potential barrier and a Gaussian-shaped ground state polariton condensate was realized ~\\cite{Ring}. \nChristofolini and co-workers examined the transition between phase-locked and trapped condensates using multiple-excitation spots and a ring-shaped excitation pattern~\\cite{opticalSPT2013}. Despite earlier work by Manni et al~\\cite{manni_spontaneous_2011}, the authors claimed that for ring-shaped pumps, no phase-locked state is geometrically possible, and that when the spacing between the pumps reduces, the trapped condensate collapses into a Gaussian-shaped ground state. Here, we show that under ring-shaped excitation, the formation of excited state condensates is driven by polariton confinement in the linear potentials and that the presence of non-ground polariton condensates does not necessitate asymmetries in the shape and\/or power distribution of the ring excitation. The dependence of the state selection on the height of the trap's barrier and shape at threshold, provides a robust platform for engineering switches of mesoscopic multi-particle coherent states.\n\n\n\nThe experimental configuration that produces an annular beam of zero angular momenta consists of a double axicon arrangement. A variable telescope is used to control the radii of the excitation beam that we project on the sample. The excitation and detection configuration and the microcavity sample is described in ref.~\\cite{Ring}. The microcavity is held in a cold finger cryostat operating at 6 K. We study the steady-state dynamics under non-resonant excitation at \\unit[752]{nm} using a single mode quasi-continuous wave (CW) laser (2\\% on-off ratio at 10kHz). The microcavity used in these experiments is a high Q factor ($>$ 15000) $5\\lambda\/2$ GaAs\/AlGaAs microcavity with 4 triplets of \\unit[10]{nm} GaAs quantum wells, with a Rabi splitting of \\unit[9]{meV} and a cavity lifetime of \\unit[7]{ps}, as described in ref.~\\cite{tsotsis_lasing_2012}. All experiments were performed for a small negative detuning range of $\\unit[-7]{meV}\\leq$ d $\\leq\\unit[-5]{meV}$.\n\nFigures \\fig{fig:modes}a-e, show the spatial profile of mesoscopic wavefunctions for a range of excitation radii and asymmetries, characterised by the ellipticity and radius of the excitation ring, at the coherence threshold that defines the depth of the trap via the interactions in the reservoir. Theses states resemble the TEM modes of a harmonic oscillator and in what follows we will adapt their symbolism to annotate the state of the polariton wavefunction. For an excitation ring with a radius of $\\sim$\\unit[10]{$\\mu$m} we observe a ground-state polariton condensate (\\Fig{fig:modes}a), as in ref.~\\cite{Ring}, which remains in the ground-state as long as the long axis of the asymmetric excitation does not exceed $\\sim$\\unit[10]{$\\mu$m}. For larger excitation ring radius ($\\sim$\\unit[17]{$\\mu$m}) and similar ellipticity as in \\Fig{fig:modes}a ($\\epsilon=0.22$) at coherence threshold we observe that polaritons coalesce at a higher excited state ($\\psi_{11}$) as shown in \\Fig{fig:modes}b. We note that the symmetry of the excited state wavefunction is robust to small asymmetries in the excitation ring ($0<\\epsilon<0.23$) and the transition from ground to non-ground polariton condensates is predominantly dependent on the radius of the ring. Increasing the ring radius and the asymmetry of the excitation it is possible to observe excited state polariton condensates as shown in \\Fig{fig:modes}c-e. On top of each panel we have annotated the ellipticity of the excitation ring. Interferometric measurements of excited states $\\psi_{01},\\psi_{02},\\psi_{03}$ confirm that these are coherent mesoscopic wavefunctions of extended condensates (\\Fig{int}a-c). \n\n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.45]{fig2.pdf}\n\t\\centering \\vspace{-0.25cm}\n\t\\caption{\\textbf{Interference patterns of trapped polariton condensates: (a)} $\\Psi_{01}$, \\textbf{(b)} $\\Psi_{02}$, \\textbf{(c)} $\\Psi_{03}$. The interference patterns where obtained with a retro-reflector configuration.}\n\t\\vspace{-0.2cm}\n\t\\label{int}\n\\end{figure}\n\n\n\nWe investigate the dependence of the quantum state selectivity on the barrier height by varying the non-resonant excitation density of a geometrically fixed, ring-shaped, asymmetric excitation profile. We use an excitation ring of radius $\\sim$16 $\\mu m$ and $\\epsilon=0.27$ that at coherence threshold produces the $\\Psi_{04}$ polariton state as shown in Fig.\\ref{Ptran}a. By increasing the excitation density above the coherence threshold, while keeping all other parameters the same, we observe the transition from $\\Psi_{04}$ to $\\Psi_{05}$ (Fig. \\ref{Ptran}b). The order of the latter state is clearly revealed in Fig.\\ref{Ptran}c, where we plot the normalised spatial profiles along the white dashed lines of the real space intensity images of Fig.\\ref{Ptran}a,b. Fig.\\ref{Ptran}c shows the presence of an extra node at the higher excitation density indicative of $\\Psi_{05}$. In Fig. \\ref{Ptran}d we plot the energy shift of the condensate in the transition from $\\Psi_{04}$ to $\\Psi_{05}$ with respect to its energy at the coherence threshold ($\\Delta(E_P-E_{P_{th}})$). A sharp increase of the energy shift ($\\sim 45\\mu eV$) is observed in Fig.\\ref{Ptran}d at $P\\sim 1.12 P_{th}$. Within the grey stripe intensity fluctuations of the excitation beam artificially blur the two states. The top panels in Fig.\\ref{Ptran}a,b depict the calculated energy levels for the trap shape and the corresponding probability density of the confined states. In both panels, the red-filled probability density corresponds to the occupied state. It is worth noting here the greater overlap of the probability density of the highest energy level ($\\Psi_{04}$ in \\ref{Ptran}(a) and $\\Psi_{05}$ \\ref{Ptran}(b)) with the reservoir compared to the lowest energy levels. Evidently, with increasing the barrier height a polariton condensate is realised at the next confined energy level as a pure-quantum-state that can be singularly described by the principal quantum number $n$ $(\\Psi_{0,n+1})$.  \n\n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.25]{fig3.pdf}\n\t\\centering \n\t\\caption{\\textbf{Evolution of $\\Psi_{04}$ for increasing excitation density. Bottom panel: (a)}  $\\Psi_{04}$ at P=1.03$P_{th}$. \\textbf{(b)} Subsequent increase of the power results in the appearance of $\\Psi_{05}$. Top panel: Schematic representation of the confined energy states for two different barrier heights. \\textbf{(c)} Profiles of the wavefunction for different excitation densities extracted along the dashed white lines at \\textbf{(a)} and \\textbf{(b)}. \\textbf{(d)} Corresponding energy difference with respect to the energy at coherence threshold for increasing excitation power normalised at the coherence threshold power $P_{th}$. Inset in \\textbf{(d)} shows the spectra of the points denoted by the arrows}\n\t\\label{Ptran}\n\t\\vspace{-0.25cm}\n\\end{figure}\n\nWe explore the robustness of the formation of pure-quantum-states on density fluctuations in the exciton reservoir, by extending our study from the excitation density dependent switching between successive states in the dynamic equilibrium regime, to transitions in the time domain under non-resonant pulsed excitation. We use a ring-shaped non-resonant $200$ femtosecond pulse at \\unit[755]{nm} with $\\sim$\\unit[11]{$\\mu$m} radius of the major axis and $\\epsilon= 0.3$ at $\\sim$1.6$P_{th}$. We record the spatio-temporal dynamics of the emission and observe the formation of the $\\Psi_{01}$ polariton state and its transition to $\\Psi_{00}$~\\cite{supp}. We set the transition point to define the zero time frame for the rest of our analysis. Figure \\ref{Ttran}a shows a snapshot of the $\\Psi_{01}$ state at \\unit[-30]{ps}. At later times, the two lobes of the $\\Psi_{01}$ state appear to move closer together and the condensate rapidly transforms to the ground polariton state ($\\Psi_{00}$) of Fig.\\ref{Ttran}b. The decrease of the density in the  barriers in the time-domain results in a shallower trap in which the $\\Psi_{01}$ state is no longer confined, leading to a polariton condensate at the next available state, here the ground state $\\Psi_{00}$. We spectrally and time resolve the decay of emission at normal incidence with an angular width corresponding to $|k|\\leq 1.4 \\mu$m and observe a sharp energy shift from $\\Psi_{01}$ to $\\Psi_{00}$ as shown in Figure \\ref{Ttran}c. This dynamic transition further illustrates that under optical confinement a polariton condensate spontaneously occurs at a higher confined state as defined by the barrier height of the trap and that the transition to the ground state is hindered solely by the existence of higher energy levels. \n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.29]{fig4.pdf}\n\t\\centering \\vspace{-0.35cm}\n\t\\caption{\\textbf{False colour-scale real space tomographic frames in the time domain. (a)} $\\Psi{01}$ state at -30ps and \\textbf{(b)} subsequent \n\t\ttransition to $\\Psi_{00}$ at 30ps. \\textbf{(c)} Intensity normalised time evolution of the emission energy showing the characteristic energy jump upon the transition threshold.}\n\t\\label{Ttran}\n\t\\vspace{-0.25cm}\n\\end{figure}\n\n\n\\begin{figure*}\n\t\\vspace{-0.1cm}\n\t\\includegraphics[scale=0.29]{fig5.pdf}\n\t\\centering \\vspace{-0.15cm}\n\t\\caption{ Polariton dispersion at -30ps a), 0ps, b) and 30ps c). White dotted lines in a) denote the integrated area from which Fig.\\fig{Ttran}c was extracted. d) Schematic representation of the momentum acquired by polaritons tunnelling outside of the potential trap (potential and dispersion energy not in scale). e) Intensity normalised time evolution of $k_x$ showing the characteristic $\\Delta k_x$ jump of the tunnelling mode.}\n\t\\label{tunel}\n\t\\vspace{-0.25cm}\n\\end{figure*}\n\nThe time resolved dispersion images from which the energy evolution of the system was extracted (Fig.\\fig{Ttran}c) are presented in Fig.\\fig{tunel}a-c.\nThe appearance of the $\\Psi_{01}$ mode is accompanied by a distinct doublet mode in the dispersion (Fig.~\\fig{tunel}a), which corresponds to the counter-propagating components of the standing wave~\\cite{rotating_2014}. As the barrier dynamically decays and the $\\Psi_{00}$ mode is switched on as previously discussed, it quickly overtakes $\\Psi_{01}$ in intensity at $\\sim$\\unit{0}{ps}. The first excited state quickly dissipates after this point with the polariton lifetime and the dispersion is dominated by the emission of the trap ground state. Interestingly, Fig.\\fig{tunel}a-c also reveal distinct satellite modes at the same energy of the confined modes but for greater in plane wave-vector. For quantum states in traps with a finite barrier width, coherent tunnelling modes are a characteristic feature. Moreover, in our system these modes will be accelerated by the potential landscape outside the trap eventually acquiring momentum characteristic of the difference between the energy level in the trap and of the low-density polariton dispersion of the system outside the excitation region (Fig.\\fig{tunel}d). From this description it becomes clear that the tunnelling modes are expected to be at the same energy but with higher momentum, as observed in Fig.\\fig{tunel}a-c. \n\nIntegrating the time-dispersion images over energy, while intensity normalising for every time frame, we compile the time evolution of k$_x$ (Fig.\\fig{tunel}e). This analysis reveals the expected $\\Delta$k$_x$ difference of the tunnelling modes of the two states. Intuitively, the relative (to the trapped state) intensity of the $\\Psi_{01}$ tunnelling mode at the transition is substantial, as the width of the barrier goes to zero at this energy level. In contrast to the tunnelling amplitude of the ground state that is effectively suppressed as the potential width at the $\\Psi_{01}$ energy level is still significant. Nevertheless, following the dynamic dissipation of the barrier, due to the decay of particles as well as draining of the reservoir by the condensate, we observe a continuous increase of the relative intensity of the tunnelling amplitude of the ground state at k$_x\\sim \\unit{1.4}{\\mu m ^{-1}}$. The observation of a strong tunnelling component from the E$_{01}$ energy just before the transition, verifies that the barrier width for this level is indeed minimal and that $\\Psi_{01}$ is close to the rim of the trap barrier, further corroborating our interpretation.\n\n\nThe system can be theoretically modelled with a non-linear Schr\\\"{o}dinger equation, namely the Gross-Pitaevski equation.\nSimulations with the Gross-Pitaevski equation with a potential similar to the one from the experimental measurements in our system \nqualitatively reproduce the states recorded experimentally. Using a potential $V(r)$ that consists of the exciton-exciton interactions in \nthe reservoir, that blue-shift the polariton energy levels, and of the polariton-polariton interactions in the condensate,the Hamiltonian of the system is: \n\\begin{IEEEeqnarray}{l}\nH(r) = T+V(r) \\\\\nV(r)  =  V_{r}(r)+V_{c}(r)  \\\\\n V_{r}(r)=N_{r}U_{ex-ex} f_{r}(r)\n\\end{IEEEeqnarray}\nwhere $N_{r}$ is the density of excitons in the reservoir, $U_{ex-ex}$ the exciton-exciton interaction \nstrength, $f_r(\\mathbf{r})$ is the spatial distribution of the exciton \nreservoir taking into account exciton diffusion beyond the pump spot and $V_c=U_{pol-pol}|\\psi_n(\\mathbf{r})|^2$ where $U_{pol-pol}$ the polariton-polariton interaction strength and $\\psi(\\mathbf{r})$ the condensate wavefunction. In addition to kinetic and potential energy terms in the above Hamiltonian, to account for \npolariton spatial dynamics, a generalization of the extended Gross-Pitaevskii equation is required to include incoherent pumping and \ndecay~\\cite{Wouters2007}. In continuous wave experiments one expects the excitation of a steady state of hot excitons with the spatial profile set by the optical pumping extended by exciton diffusion. One can then make use of the Landau-Ginzburg approach for describing the dynamics of the 2D polariton wavefunction~\\cite{Keeling2008}:\n\\begin{align}\ni\\hbar\\frac{d\\psi(\\mathbf{r},t)}{dt}&=\\left[-\\frac{\\hbar^2\\hat{\\nabla}^2}{2m_P}+\\left(U_{pol-pol}-i\\Gamma_\\mathrm{NL}\\right)|\\psi(\\mathbf{r},t)|^2\\right.\\notag\\\\\n&\\hspace{10mm}\\left.+\\left(U_{pol-ex}+ir\\right)N_rf_r(\\mathbf{r})-\\frac{i\\Gamma}{2}\\right]\\psi(\\mathbf{r},t)\\notag\\\\\n&\\hspace{10mm}+i\\hbar\\mathfrak{R}\\left[\\psi(\\mathbf{r},t)\\right].\\label{eq:GP}\n\\end{align}\nHere $m_P$ is the polariton effective mass and $f_r(\\mathbf{r})$ describes the 2D spatial distribution of $N_r$ excitons. The condensation rate $r$ describes the gain of polaritons in the presence of the exciton reservoir. The polaritons experience both a linear decay $\\Gamma$ and non-linear loss $\\Gamma_{NL}$, which represents the scattering of polaritons out of the condensate when its density is high~\\cite{Keeling2008}. The final term in Eq.~\\ref{eq:GP} represents a phenomenological energy relaxation~\\cite{Wouters2012} in the system, which can play an important role when non-ground state polaritons interact with a potential gradient~\\cite{Wouters2010,Wertz2012,Anton2012}:\n\\begin{equation}\n\\mathfrak{R}[\\psi(x,t)]=-\\lambda N_rf_r(\\mathbf{r})\\left(\\hat{E}_\\mathrm{LP}-\\mu(\\mathbf{r},t)\\right)\\psi(x,t).\\label{eq:relax}\n\\end{equation}\nwhere $\\lambda$ determines the strength of energy relaxation~\\cite{Wouters2012,Wertz2012} and $\\mu(\\mathbf{r},t)$ is a local effective chemical potential that conserves the polariton population~\\cite{Wouters2012}. Kinetic energy relaxation of this form was derived with a variety of methods~\\cite{Solnyshkov2014,Sieberer2014} and offers a simple model for the qualitative description of our experiment. We note however that this model does not distinguish between different mechanisms of energy relaxation, which may have different power dependences~\\cite{Haug2014}.\n\n\n\nFixing $N_rf_r(\\mathbf{r})$ to represent a ring shaped excitation (with slight asymmetry), the numerical solution of Eq.~\\ref{eq:GP} gives the steady state intensity profiles shown in \\Fig{fig:modes}f-j. Different configurations are accessed by varying the spatial distribution ($f_r(\\mathbf{r})$) and population ($N_r$) of hot excitons, as in the experiment~\\cite{TimV}. The simulations support that excited state condensation occurs preferentially at the uppermost confined energy state. \n\n\nAlthough it cannot be explicitly verified that there is no available state in the trap above the condensate energy level, since the polariton potential landscape is not directly measurable, the evidence presented from the steady state switching, the transient study including the dynamic behaviour of the tunnelling components of the system, as well as the theoretical simulations and the calculations for the condensate reservoir overlap~\\cite{supp} strongly supports our interpretation that polaritons condense in the highest available energy state within the optical trap.\n\nIn conclusion, we have investigated the dynamics of polariton condensates under optical confinement and observed that, in contrast to previously reported excited state condensation in defect traps and pillar structures, injection of polaritons from the trap barriers leads to the formation of a pure quantum-confined state with a mesoscopic coherent wavefunction above condensation threshold. This behaviour is in agreement with theoretical expectations for a true Bose condensate that is anticipated to resist multi-mode behaviour~\\cite{combescot_stability_2008,nelsen_dissipationless_2013} in the presence of inter-particle interactions.  Moreover we revealed that the state selectivity of this system strongly depends on the geometric properties of the trap and have demonstrated a highly controllable switching between successive mesoscopic coherent quantum confined states, in the dynamic equilibrium regime and in the time domain. These results highlight the capability of tailoring and manipulating on-chip pure-quantum-states in semiconductor microcavities that can facilitate the implementation of polariton bosonic cascade lasers~\\cite{liew_proposal_2013}. Taking into account that the extensive propagation~\\cite{nelsen_dissipationless_2013} as well as the susceptibility of the polaritonic flow to the potential landscape~\\cite{nguyen_realization_2013} has been widely demonstrated, these results also indicate the potential of engineering confined condensate lattices, coupled by their respective tunnelling amplitudes. Moreover the coupling strength in this architecture can be finely tuned by controlling the barrier height enabling the emergence of applications such as many-body quantum circuitry and quantum simulators. \n\n\n\n\nP. S. acknowledges funding from Greek GSRT programm APOLLO. A. A. acknowledges useful discussions with W. Langbein and S. Portolan.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}