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+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\nThe discovery of quantum vacuum nonlinearities\n\\cite{Heisenberg:1935qt,Weisskopf,Schwinger:1951nm} under controlled\nlaboratory conditions using real photons or macroscopic\nelectromagnetic fields is a major goal of contemporary strong-field\nphysics. Many proposals rely on a pump-probe scheme, where a\nwell-controlled, say optical, photon beam probes a region of space\nthat is exposed to a strong field (``pump''). A typical example is given\nby schemes intended to verify vacuum birefringence\n\\cite{Toll:1952,Baier,BialynickaBirula:1970vy,Adler:1971wn} that can\nbe searched for using macroscopic magnetic fields\n\\cite{Cantatore:2008zz,Berceau:2011zz} or with the aid of\nhigh-intensity lasers \\cite{Heinzl:2006xc}, see e.g.,\n\\cite{Dittrich:2000zu,Marklund:2008gj,Dunne:2008kc,DiPiazza:2011tq} for reviews.\n\nAs these setups require techniques such as high-purity ellipsometry\n\\cite{Cantatore:2008zz,Marx:2011} to separate the (small) signal from\na typically huge background, a recent proposal has focused on a\nquantum-reflection scheme that facilitates a built-in noise\nsuppression \\cite{Gies:2013yxa}. In this scheme, incident\nprobe photons propagate towards a spatially localized field\ninhomogeneity (``pump''), as, e.g., generated in the focal spots of a\nhigh-intensity laser system.  Even though the inhomogeneity acts similar to\nan attractive potential, probe photons can be scattered backwards due to\nquantum reflection.  Looking for reflected photons in the field free\nregion, this scenario inherently allows for a clear geometric\nseparation between signal and background.  First estimates of the\nnumber of reflected photons attainable in present and near future\nlaser facilities look promising. Figure~\\ref{fig:QRef} depicts a\ntypical Feynman diagram contributing to the effect.\n\nAs quantum reflection crucially relies on the presence of an\ninhomogeneous pump field, it belongs to a general class of\nquantum-induced interference effects\n\\cite{King:2013am,Tommasini:2010fb,Hatsagortsyan:2011} with the\nparticular property of optimizing the signal-to-noise ratio.\n\nThe pump-probe scheme is typically also reflected by the theoretical\ndescription, in which the nonlinearities are kept for the pump-probe\ninteraction, but the equations are linearized with respect to the\nprobe propagation. In the present work, we rely again on an optical\npump-probe setup which however requires a nonlinear treatment of the\nprobe-field. The idea is to look for laser photon merging in the\npresence of an electromagnetic field inhomogeneity.  This effect\nresembles the standard nonlinear optical process of {\\it second\n  harmonic generation} (SHG) -- or in general {\\it high harmonic\n  generation} -- with the nonlinear crystal replaced by the quantum\nvacuum subject to strong electromagnetic fields. Higher harmonic\ngeneration in an electromagnetized vacuum has been discussed on the\nlevel of the Heisenberg-Euler action in\n\\cite{BialynickaBirula:1981,Kaplan:2000aa,Valluri:2003sp,Fedotov:2006ii},\nsee also the discussion in \\cite{Marklund:2006my}, or using the\nconstant-field polarization tensor in \\cite{DiPiazza:2005jc}. Laser\nphoton merging in proton-laser collisions have been investigated in\ndetail in \\cite{DiPiazza:2007cu,DiPiazza:2009cq}, where a promising\nscenario has been proposed for a discovery of the merging phenomenon\nthat involves a nowadays conventional optical high-intensity laser at\na high-energy proton collider. A related effect is called four-wave\nmixing for which also a concrete experimental proposal has been\nexplored in \\cite{Lundin:2006aa,King:2012aw}.  The same underlying\nquantum vacuum nonlinearity could even be used to radiate photons from\nthe focal spot of a single focused laser beam (``vacuum emission'') as\nproposed in \\cite{Monden:2011}. More generally, frequency mixing induced\nby quantum vacuum effects has even been suggested as a sensitive probe\nto search for new hypothetical particles \\cite{Dobrich:2010hi}.\n\n\\begin{figure}\n\\includegraphics[width=0.57\\textwidth]{QRef} \n\\caption{Typical Feynman diagram contributing to the effect of quantum reflection \\cite{Gies:2013yxa}. For field strengths of the inhomogeneity well below the critical field strength (cf. main text), the leading contribution arises from diagrams with two couplings to the field inhomogeneity. As there is no energy transfer from static fields, the frequencies of the incident and outgoing photons match.}\n\\label{fig:QRef}\n\\end{figure}\n\nIn the present work, we concentrate on an ``all-optical'' parameter\nregime realizable with high-intensity lasers. As the signal is\nexpected to be very small, we again consider specifically the\nkinematics of the reflection process for an appropriate\nsignal-to-noise reduction.  As in \\cite{Gies:2013yxa}, we limit\nourselves to the study of time-independent field inhomogeneities, such\nthat there is no energy transfer from the field inhomogeneity. Depending on the spatial field\ninhomogeneity, the propagation direction of the merged photons can\ndiffer from that of the incident probe photons.  For the specific\nreflecting kinematic situation, the merged photons can even\npropagate -- somewhat counter-intuitively -- into the backward\ndirection.  For a straightforward comparison of the signals resulting\nfrom quantum reflection \\cite{Gies:2013yxa} and the photon merging\nscenario of this work, we focus on a one-dimensional magnetic field\ninhomogeneity. As is shown by an explicit calculation below, our\nfindings confirm the expectation that the merging process for the\nreflective scenario is dominated by the quantum reflection process for\nthe all-optical parameter regime. Nevertheless, due to a different\npolarization and frequency dependence, filtering techniques might\nallow for a discovery of the merging process in this set up as well.\n\nLet us briefly outline the theoretical framework of our study,\ntailored to an all-optical scenario.\nOptical lasers operate at frequencies $\\omega\\sim{\\cal O}({\\rm eV})$\nmuch smaller than the electron mass $m\\approx511\\,{\\rm keV}$,\nconstituting a typical scale associated with quantum effects in\nquantum electrodynamics (QED), such that $\\frac{\\omega}{m}\\ll1$.\nMoreover, the maximum field strengths attainable with present and near\nfuture laser facilities are small in comparison to the {\\it critical\n  field strength} $E_{\\rm cr}\\equiv\\frac{m^2}{e}$\n\\cite{Heisenberg:1935qt}, i.e., $\\{\\frac{e{\\mathfrak\n    E}}{m^2},\\frac{eB}{m^2}\\}\\ll1$, with $\\mathfrak{E}$ denoting the\nelectric field strength of the probe laser and $B$ the peak magnetic field\nstrength of the spatially localized inhomogeneity.\nHence, for a given number $2n$, $n\\in\\mathbb{N}$, of probe laser\nphotons of frequency $\\omega$ (wavelength\n$\\lambda=\\frac{2\\pi}{\\omega}$), the dominant merging process into a\nsingle photon of frequency $2n\\omega$ is expected to arise from an\ninteraction of the type depicted in Fig.~\\ref{fig:merging_cartoon},\nexhibiting a single coupling to the (magnetic) field\ninhomogeneity. Higher order couplings to the field inhomogeneity are\nstrongly suppressed due to the fact that $\\frac{eB}{m^2}\\ll1$.\nFurry's theorem (charge conjugation symmetry of QED) dictates the\ninteraction to vanish for any odd number of couplings to the\nelectron-positron loop, which justifies that we have \ntailored the merging process to $2n$ laser photons. \nThe dominant contribution in the weak-field limit is expected to arise from\nthe merging of two laser photons, described by Feynman diagrams with\nfour legs (cf. Fig.~\\ref{fig:merging_cartoon}).\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{merging_cartoon} \n\\caption{Cartoon of the photon merging process. In the presence of a stationary but\n  spatially inhomogeneous electromagnetic field $2n$ laser photons of frequency\n  $\\omega$ can merge into a single photon of frequency $2n\\omega$.\n  Depending on the spatial field inhomogeneity, the propagation\n  direction of the merged photons can differ from that of the incident\n  probe photons. In curly braces we introduce our notation for the corresponding fields\/polarizations and four-momenta; cf. also Eqs.~\\eqref{eq:background}, \\eqref{eq:Ak} and \\eqref{eq:M4}, as well as Fig.~\\ref{fig:perspective}.}\n\\label{fig:merging_cartoon}\n\\end{figure}\n\nA sketch of the geometry of the reflective scenario\nof the merging process\nto be investigated in this paper can be found in  Fig.~\\ref{fig:perspective}.\nHere we already summarize the notation to be introduced and discussed below.\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{QS-beschr-crop} \n\\caption{Schematic depiction of the two-photon merging process. Incident probe photons (wave vector $\\vec{\\kappa}$, energy $\\kappa^0=|\\vec{\\kappa}|=\\omega$) hit a one-dimensional field inhomogeneity $\\vec{B}({\\rm x})=B({\\rm x})\\vec{e}_{\\rm z}$ of width $w$ under an angle of $\\theta$. Due to nonlinear effective couplings between electromagnetic fields mediated by virtual charged particle fluctuations, the field inhomogeneity can impact incident probe photons to merge and form an outgoing photon (wave-vector $\\vec{k}_f$) of twice the energy of the incident probe photons, i.e., $k_f^0=|\\vec{k}_f|=2\\omega$. \nMost notably, the inhomogeneity can affect the outgoing merged photons to reverse their momentum component along $\\vec{e}_{\\rm x}$ with respect to the incident probe photons.\nThe vectors $\\vec{a}_1$, $\\vec{a}_2$ and $\\vec{\\epsilon}^{\\,(1)}$, $\\vec{\\epsilon}^{\\,(2)}$ span the polarization degrees of freedom of the incident and outgoing photons, respectively. For the depiction we specialized to $\\varphi=\\varphi'=0$ (cf. the main text).}\n\\label{fig:perspective}\n\\end{figure}\n\nThe leading quantum reflection process in the perturbative regime also\narises from four leg diagrams (cf. Fig.~\\ref{fig:QRef}).  While\nquantum reflection necessitates at least two couplings to the field\ninhomogeneity, photon merging just needs a single coupling to the\ninhomogeneity. Conversely, quantum reflection can be considered as a\ntwo-photon (one incident, one outgoing) process, whereas photon\nmerging involves at least three photons (two incident, one\noutgoing). From this observation, it can already be anticipated\nthat the dependence of the observables on the various parameters\nwill differ between the two processes.\n\nNotably, the merging process in Fig.~\\ref{fig:merging_cartoon} can be\nevaluated straightforwardly, owing to the fact that the photon\npolarization tensor is explicitly known for generic monochromatic\nplane wave backgrounds \\cite{Baier:1975ff,Becker:1974en}.\nInterpreting the plane wave background in terms of incident probe\nphotons of frequency $\\omega$, the two open legs of the polarization\ntensor can be identified with the field inhomogeneity and the outgoing\nmerged photon, respectively.  The polarization of the incident photons\ncan be controlled by adjusting the polarization of the monochromatic\nplane wave background.\n\nOur paper is organized as follows: In Sec.~\\ref{sec:calculation} we\nexplain in detail the various steps needed to evaluate the photon\nmerging process.  A crucial technical step is to find a\ncontrolled approximation to the photon polarization tensor in a plane\nwave background, facilitating an analytical treatment of the photon\nmerging process.  Such an approximation, especially suited to the\nparameters of an all-optical experimental scenario, is derived in\nSec.~\\ref{subsec:Pi}.  Section~\\ref{seq:Ex+Res} is devoted to the\ndiscussion of explicit examples and results. It contains a thorough\ncomparison of the effects of laser photon merging and quantum reflection. \nWe end with conclusions and an outlook in Sec.~\\ref{seq:Con+Out}.\n\n\n\\section{Calculation} \\label{sec:calculation}\n\n\\subsection{Photon polarization tensor in plane wave field} \\label{subsec:Pi}\n\nWe briefly recall and summarize the basic structure of the photon\npolarization tensor in a generic, elliptically polarized monochromatic\nplane wave background \\cite{Baier:1975ff,Becker:1974en}. The latter is parametrized by the following gauge potential in Coulomb-Weyl gauge\n\\begin{equation}\n{\\cal A}_\\mu(x)={\\mathfrak a}_{1}a_{1\\mu}\\cos(\\kappa x)+{\\mathfrak a}_{2}a_{2\\mu}\\sin(\\kappa x), \\label{eq:background}\n\\end{equation}\nwith $\\mathcal{A}_0=0$, $\\kappa^2=0$ and\n$a_1\\kappa=a_2\\kappa=a_1a_2=0$. Moreover, we will use\nthe frequency $\\omega\\equiv\\kappa^0$.  The four-vectors $a_{i\\mu}$ with\n$i\\in\\{1,2\\}$ are normalized to unity, i.e., $a_i^2=1$, and the field\namplitude is encoded in the coefficients ${\\mathfrak a}_i\\geq0$.  Our\nmetric convention is $g_{\\mu \\nu}=\\mathrm{diag}(-1,+1,+1,+1)$, and we use $c=\\hbar=1$.\n\nFor the normalized plane wave field strength in momentum space,\nwe introduce $f^{\\mu\\nu}_i=\\kappa^\\mu a_i^\\nu-\\kappa^\\nu a_i^\\mu$. \nIn the following, we will frequently use the shorthand notation $(kf_i)^\\mu=k_\\nu f^{\\nu\\mu}_i$. \n\nIn momentum space the photon polarization tensor mediates between two\nfour-momenta $k_1$ and $k_2$.  Since the wave~\\eqref{eq:background} is\ncharacterized by the single four-momentum $\\kappa$ and a change in the\nincident momentum is determined by an interaction with the wave, the\nkinematics are such that $k_2=k_1+C\\kappa$, with scalar constant $C$\n\\cite{Baier:1975ff}.  Correspondingly, $\\kappa k_2=\\kappa\nk_1\\equiv\\kappa k$ and also $(k_1f_i)^\\mu=(k_2f_i)^\\mu=(kf_i)^\\mu$.\n\nFollowing \\cite{Baier:1975ff}, the associated photon\npolarization tensor can then be compactly represented as\n\\begin{equation}\n \\Pi^{\\mu\\nu}(k_1,k_2)=c_1\\Lambda_1^\\mu\\Lambda_2^\\nu+ c_2\\Lambda_2^\\mu\\Lambda_1^\\nu+c_3\\Lambda_1^\\mu\\Lambda_1^\\nu+c_4\\Lambda_2^\\mu\\Lambda_2^\\nu+c_5\\Lambda_3^\\mu\\Lambda_4^\\nu, \\label{eq:PIstructure}\n\\end{equation}\nwith scalar coefficients $c_j(k_1,k_2)$, $j\\in\\{1,\\ldots,5\\}$.\nThe tensor structure is encoded in products of the normalized four vectors\n\\begin{multline}\n \\Lambda_i^\\mu=\\frac{(kf_i)^\\mu}{(\\kappa k)}=a_i^\\mu-\\frac{(ka_i)}{(\\kappa k)}\\kappa^\\mu \\quad \\text{for} \\quad i\\in\\{1,2\\},\\\\\n \\Lambda_3^\\mu=\\frac{\\kappa^\\mu k_1^2-k_1^\\mu(\\kappa k)}{(\\kappa k)\\sqrt{-k_1^2}}, \\quad \\Lambda_4^\\mu=\\frac{\\kappa^\\mu k_2^2-k_2^\\mu(\\kappa k)}{(\\kappa k)\\sqrt{-k_2^2}}, \\label{eq:Lambdas}\n\\end{multline}\nfulfilling\n$\\Lambda_1^2=\\Lambda_2^2=\\Lambda_3^2=\\Lambda_4^2=1$. This tensor\nstructure guarantees that $\\Pi^{\\mu\\nu} (k_1,k_2)$ satisfies the\nWard identities $k_{1,\\mu}\\Pi^{\\mu\\nu} (k_1,k_2)=\\Pi^{\\mu\\nu}(k_1,k_2)k_{2,\\nu}=0$.\n\nApart from a trivial overall factor of $\\alpha=e^2\/(4\\pi)$,\nthe coefficients $c_j$ depend on the kinematic variables $k_1$, $k_2$\nand $\\kappa$ as well as the electron mass $m$, and account for\nthe entire field strength dependence. The latter dependence is most\nconveniently expressed in terms of the two invariant intensity\nparameters $\\xi_i=\\frac{e\\mathfrak{a}_i}{m}$ with $i\\in\\{1,2\\}$.\nIn Coulomb-Weyl gauge, the amplitude $\\mathfrak{a}_i$ is intimately related to the amplitude\nof the associated electric field $\\mathfrak{E}_i$ via\n$\\mathfrak{a}_i=\\frac{\\mathfrak{E}_i}{\\omega}$, such that -- in terms\nof parameters directly accessible in the lab -- we have\n$\\xi_i=\\frac{e\\mathfrak{E}_i}{m\\omega}$.\n\nIn consequence of Furry's theorem, the field dependence can be encoded in\n$\\xi_1^2$, $\\xi_2^2$ and $\\xi_1\\xi_2$, i.e., combinations even in the\ncharge $e$, only.  It is moreover helpful to introduce the\ndimensionless parameter $\\lambda=-\\frac{\\kappa k}{2m^2}$,\nparametrizing the relative momenta of the involved photons. \nIn summary, the relevant dimensionless parameters for the\noff-shell  polarization tensor in a plane wave field are given by\n\\begin{equation}\n\\xi_i=\\frac{e\\mathfrak{E}_i}{m\\omega}, \\quad \\lambda=-\\frac{\\kappa k}{2m^2}, \\quad \\frac{k_1 k_2}{4m^2}, \n\\end{equation}\nwhere the last parameter characterizes the relative momenta of the in- and outgoing photon legs.\n\nIn the following, we are only interested in a situation with actual interactions\nwith the plane wave field~\\eqref{eq:background} and thus omit the zero\nfield contribution in \\Eqref{eq:PIstructure}.\\footnote{More precisely,\n  the coefficients $c_j$ provided in the following correspond to the\n  quantity $\\Pi^{\\mu\\nu}(A)-\\Pi^{\\mu\\nu}(A=0)$.  The zero field term\n  can be included straightforwardly, noting that\n  $g^{\\mu\\nu}-\\frac{k_1^\\mu\n    k_1^\\nu}{k_1^2}=\\Lambda_1^\\mu\\Lambda_1^\\nu+\\Lambda_2^\\mu\\Lambda_2^\\nu+\\Lambda_3^\\mu\\Lambda_3^\\nu$. \\label{ftnt}}\n\nThe coefficients $c_j$ generically decompose into an {\\it elastic}\npart characterized by zero momentum exchange with the wave and an {\\it\n  inelastic} part with finite momentum exchange.  The latter part is\nmade up of an infinite number $l\\in\\mathbb{Z}\\setminus\\{0\\}$ of\ncontributions with momentum transfer $2\\kappa l$ to be associated with\nthe absorption\/release of $2l$ laser photons.  Correspondingly, we\nwrite\n\\begin{equation}\n c_j=i(2\\pi)^4m^2\\frac{\\alpha}{\\pi}\\Bigl[\\delta(k_1-k_2)G_j^0+\\sum_{l\\in\\mathbb{Z}\\setminus\\{0\\}}\\delta(k_1-k_2-2l\\kappa)G_j^l\\Bigr], \\label{eq:c_j}\n\\end{equation}\nwhere the dimensionless coefficients $G_j^l(k_1,k_2)$, with\n$l\\in\\mathbb{Z}$, are most conveniently represented in terms of double\nparameter integrals that cannot be tackled analytically in a\nstraightforward way.  One of the integrals is over a proper-time type\nparameter $\\rho\\in [0,\\infty[$, and the other one over an additional parameter $\\nu\\in [-1,1]$\nrelated to the momentum routing in the loop.\n\nIn order to state them most compactly, it is convenient to define\n\\begin{multline}\n A=\\frac{1}{2}\\Bigl(1-\\frac{\\sin^2\\rho}{\\rho^2}\\Bigr),\\quad A_0=\\frac{1}{2}\\rho(\\partial_\\rho A),\\quad A_1=A+2A_0, \\\\\n z=\\frac{2(\\xi_1^2-\\xi_2^2)}{|\\lambda|(1-\\nu^2)}\\rho A_0,\\quad  y=\\frac{2(\\xi_1^2+\\xi_2^2)}{|\\lambda|(1-\\nu^2)}\\rho A. \\label{eq:defs}\n\\end{multline}\nTaking these definitions into account, the explicit expressions for $G_j^l$ read\n\\begin{equation}\n G_j^l=\\int_{-1}^1{\\rm d}\\nu\\int_0^\\infty\\frac{{\\rm d}\\rho}{\\rho}\\,{\\rm e}^{-i\\phi_0\\rho}\\,g_j^l\\,{\\rm e}^{-iy}, \\label{eq:Gs}\n\\end{equation}\nwhere\n\\begin{equation}\n \\phi_0=\\frac{2}{|\\lambda|(1-\\nu^2)}\\Bigl[1-i\\epsilon+\\frac{k_1k_2}{4m^2}(1-\\nu^2)\\Bigr],\n\\end{equation}\nwith $\\epsilon\\to0^+$, and\n\\begin{align}\n g_1^l&=\\xi_1\\xi_2\\Bigl(2\\,{\\rm sign}(\\lambda)\\frac{1+\\nu^2}{1-\\nu^2}\\,\\rho A_0-A_1\\frac{l}{z}\\Bigr)i^lJ_l(z), \\nonumber\\\\\n g_2^l&=g_1^l\\left(A_0\\to-A_0,z\\to z,A_1\\to A_1\\right), \\nonumber\\\\\n g_3^l&=\\Bigl(\\xi_1^2A_1-\\frac{\\xi_1^2-\\xi_2^2}{1-\\nu^2}\\sin^2\\rho\\Bigr)i^{l}\\bigl(J_l(z)-iJ_l'(z)\\bigr)+\\xi_1^2\\frac{1+\\nu^2}{1-\\nu^2}\\sin^2\\rho\\, i^lJ_l(z) \\nonumber\\\\\n      &\\quad +\\frac{1}{4}\\Bigl(\\frac{k_1k_2}{m^2}-\\frac{i|\\lambda|(1-\\nu^2)}{\\rho}\\Bigr)i^l\\bigl(J_l(z)-\\delta_{l0}\\,{\\rm e}^{iy} \\bigr), \\nonumber\\\\\n g_4^l&=g_3^l\\left(\\xi_1^2\\leftrightarrow\\xi_2^2\\right)(-1)^l, \\nonumber\\\\\n g_5^l&=-\\frac{\\sqrt{k_1^2k_2^2}}{4m^2}(1-\\nu^2)i^l\\bigl(J_l(z)-\\delta_{l0}\\,{\\rm e}^{iy} \\bigr), \\label{eq:gs}\n\\end{align}\nfor $l\\in\\mathbb{Z}$. Here, $J_l(z)$ denotes the Bessel function of\nthe first kind, and $\\delta_{ll'}$ is the Kronecker delta.\nEquations~\\eqref{eq:PIstructure}-\\eqref{eq:gs} constitute the full\nexpression of the photon polarization tensor in a generic plane wave\nbackground of type~\\eqref{eq:background}\n\\cite{Baier:1975ff,Becker:1974en}; see \\cite{Meuren:2013oya} for\na more recent derivation and an alternative representation.\nNoteworthily, whenever one of the momenta $k_1$ and $k_2$ is on the\nlight cone, i.e., either $k_1^2=0$ or $k_2^2=0$, the coefficients\n$G_5^l$ vanish for all $l\\in\\mathbb{Z}$, such that $c_5=0$. Except for\nthe zero field contribution (cf. footnote~\\ref{ftnt}), the tensor\nstructure of the photon polarization tensor {under these conditions\n  can be written entirely in terms of the four-vectors $\\Lambda_1^\\mu$\n  and $\\Lambda_2^\\mu$.\n\nFor completeness, note that for a circularly polarized plane wave\nbackground, corresponding to the choice of $\\xi_1=\\xi_2$, we have\n$z=0$.  Hence, taking into account that $J_l(z)\\sim z^{|l|}$\n[cf. \\Eqref{eq:Jseries} below], the only nonvanishing\ncontributions~\\eqref{eq:gs} are those with $l\\in\\{0,\\pm1\\}$,\ncorresponding to the possibility of an elastic interaction and an\ninteraction involving the emission\/absorption of just two photons from\nthe circularly polarized wave.  The physical reason for this is that a\ncircularly polarized wave has definite chirality, such that\ntransitions are only possible without a change in the chirality of the\nincident photon $(l=0)$ or with a reversal of its chirality ($l=\\pm1$)\n\\cite{Baier:1975ff,Becker:1974en}.\n\nAs the expressions are rather cumbersome, we subsequently aim at an\napproximation particularly suited for all-optical experiments.  Our\nstrategy to achieve this relies on series expansions of the expression\n$g_j^l\\,{\\rm e}^{-iy}$ in the integrand of \\Eqref{eq:Gs}, such that\nboth integrals can be performed explicitly and handy approximations\nfor the polarization tensor are obtained. Similar expansion\nstrategies have recently also led to new analytical insights into\nthe well-known polarization tensor for constant fields \\cite{Karbstein:2013ufa}.\n\nFor this purpose it is particularly helpful to note that $A$ and $A_0$\nhave the following infinite series representations\n[cf. \\Eqref{eq:defs}],\n\\begin{equation}\n A=\\frac{\\rho^2}{6}\\sum_{n=0}^\\infty A^{(2n)}\\rho^{2n},\\quad  A_0=\\frac{\\rho^2}{6}\\sum_{n=0}^\\infty (1+n)A^{(2n)}\\rho^{2n}, \\label{eq:As_series}\n\\end{equation}\nwith $A^{(2n)}=\\frac{3}{2}\\frac{(2i)^{2n+4}}{(2n+4)!}$; our definitions are such that $A^{(0)}=1$.\n\nThe above series representations suggest to define\n\\begin{equation}\n \\zeta^\\pm\\equiv\\frac{(\\xi_1^2\\pm\\xi_2^2)\\rho^3}{3|\\lambda|(1-\\nu^2)}\n\\end{equation}\nand to rewrite the quantities $y$ and $z$ as follows\n\\begin{equation}\n  y=\\zeta^+\\sum_{n=0}^\\infty A^{(2n)}\\rho^{2n},\\quad z=\\zeta^-\\sum_{n=0}^\\infty (1+n)A^{(2n)}\\rho^{2n}. \\label{eq:yzseries}\n\\end{equation}\n\nAnother important ingredient in our approach is the series\nrepresentation of $J_l(z)$, which, for $l\\in\\mathbb{Z}$, reads\n(cf. formulae 8.404 and 8.440 of \\cite{Gradshteyn})\n\\begin{equation}\n J_{l}(z)=\\sum_{j=0}^{\\infty}\\frac{(-1)^j[{\\rm sign}(l)]^l}{j!\\,(|l|+j)!}\\left(\\frac{z}{2}\\right)^{|l|+2j}\\quad {\\rm for}\\quad |{\\rm arg}(z)|<\\pi\\,, \\label{eq:Jseries}\n\\end{equation}\nwhere $[{\\rm sign}(l)]^l=1$ for $l=0$ is implicitly understood.\nInserting \\Eqref{eq:yzseries} into \\Eqref{eq:Jseries}, all the Bessel\nfunctions occurring in \\Eqref{eq:gs} can be expanded in powers of $\\zeta^-$ and $\\rho^2$.\nAnalogously, factors of ${\\rm e}^{-iy}$ can be expanded in powers of\n$\\zeta^+$ and $\\rho^2$.\n\nIn the following, let us\nassume that $|\\frac{k_1k_2}{4m^2}|<1$, which is well compatible with\nan all-optical experimental scenario.  Building on this assumption,\nand resorting to the identity $\\int_0^\\infty \\frac{{\\rm\n    d}\\rho}{\\rho}\\,{\\rm\n  e}^{-i\\phi_0\\rho}\\,\\rho^{l+1}=l!\\bigl(\\frac{-i}{\\phi_0}\\bigr)^{l+1}$\nfor $l\\in\\mathbb{N}_0$, we obtain\n\\begin{equation}\n \\int_0^\\infty \\frac{{\\rm d}\\rho}{\\rho}\\,{\\rm e}^{-i\\phi_0\\rho}\\,\\rho^{l+1}=l!\\left(-\\frac{i}{2}|\\lambda|(1-\\nu^2)\\right)^{l+1}\\sum_{n=0}^\\infty\\binom{n+l}{n}\\left(-\\frac{k_1k_2}{4m^2}(1-\\nu^2)\\right)^n  .\n\\end{equation}\nHaving implemented the above expansions, the polarization tensor can formally be written in terms of multiple infinite sums.\nNoteworthily, all $\\nu$ integrals are of the following type\n\\begin{align}\n \\int_{-1}^1{\\rm d}\\nu\\,(1-\\nu^2)^n&=\\frac{2^{2n+1}(n!)^2}{(2n+1)!}, \\nonumber\\\\\n \\int_{-1}^1{\\rm d}\\nu\\,(1+\\nu^2)(1-\\nu^2)^n&=\\left(1+\\frac{1}{2n+3}\\right)\\int_{-1}^1{\\rm d}\\nu\\,(1-\\nu^2)^n,\n\\end{align}\nwith $n\\in\\mathbb{N}_0$, and can straightforwardly be performed explicitly for each contribution.\n\nThus, with the collective notation $\\xi^2\\in\\{\\xi_1^2,\\xi_2^2,\\xi_1\\xi_2\\}$ a generic\ncontribution to the photon polarization tensor reads\n\\begin{multline}\n \\int_{-1}^1{\\rm d}\\nu\\int_0^\\infty \\frac{{\\rm d}\\rho}{\\rho}\\,{\\rm e}^{-i\\phi_0\\rho}\\Bigl(\\frac{\\xi^2\\rho^{2}}{6}\\Bigr)^s\\rho^{l}(\\zeta^+)^n(\\zeta^-)^j\n \\left\\{\n \\begin{array}{c}\n  1 \\\\\n  1-\\nu^2 \\\\\n  \\frac{1}{1-\\nu^2}\\\\\n  \\frac{1+\\nu^2}{1-\\nu^2}\n \\end{array}\n \\right\\} \\\\\n=\\Bigl(-\\frac{2\\xi^2\\lambda^2}{3}\\Bigr)^{s}\\Bigl(i\\frac{2(\\xi_1^2+\\xi_2^2)\\lambda^2}{3}\\Bigr)^n\\Bigl(i\\frac{2(\\xi_1^2-\\xi_2^2)\\lambda^2}{3}\\Bigr)^j\\bigl(-2i|\\lambda|\\bigr)^{l}\\\\\n\\times c(n,j,s,l)\n \\left\\{\n \\begin{array}{c}\n  1 \\\\\n  1-\\frac{1}{4(n+j+s)+2l+3} \\\\\n  1+\\frac{1}{2}\\frac{1}{2(n+j+s)+l}\\\\\n  1+\\frac{1}{2(n+j+s)+l}\n \\end{array}\n \\right\\} \n\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^2)\\Bigr) , \\label{eq:genblock}\n\\end{multline}\nwith integers $\\{l,n,j\\}\\in\\mathbb{N}_0$ and $s\\in\\{0,1\\}$, fulfilling\n$l+n+j+s>0$. The components in the columns in braces exhaust all possible\ntypes of occurring $\\nu$ integrands. The explicit\nexpression for the numeric coefficient in \\Eqref{eq:genblock} is\n\\begin{equation}\n c(n,j,s,l)=\\frac{2[3(n+j)+2s+l-1]!\\{[2(n+j+s)+l]!\\}^2}{[4(n+j+s)+2l+1]!}.\n\\end{equation}\nBoth integrations can be carried out, and \\Eqref{eq:genblock} provides\nus with the full numeric prefactor for given integers $l$, $n$, $j$\nand $s$ at leading order in a double expansion in\n$|\\frac{k_1k_2}{4m^2}|\\ll1$ and $|\\lambda|\\ll1$, both\ncorresponding to a soft-photon limit.\nMost importantly, the parameters} $\\xi_i$ never come alone but\nalways appear in combination with a factor of $\\lambda$. This\nimplies that any perturbative expansion of the photon polarization\ntensor in plane wave backgrounds which is superficially in powers of\n$\\xi^2$ in fact amounts to an expansion in the combined parameter\n$\\xi^2\\lambda^2$. This is of substantial practical relevance, as\noptical high-intensity lasers are entering the regime $\\xi\\gg1$.\nStill the present expansion remains valid as long as\n$\\xi^2\\lambda^2\\ll 1$ which is typically well satisfied for\ncontemporary optical high-intensity lasers.\nFirst indications of a larger validity regime of the naive ``small-$\\xi$'' expansion had\nalready been observed in \\cite{DiPiazza:2009cq}. Our all-order series expansion of the\npolarization tensor now clarifies the systematics of the underlying\nphysical parameter regimes.\n\nCorrespondingly, the photon polarization tensor can be organized in\nterms of an expansion in the dimensionless quantities\n$\\frac{k_1k_2}{4m^2}$, $\\lambda$ and $\\xi^2\\lambda^2$.  In particular,\nthe leading contributions to \\Eqref{eq:c_j} are of ${\\cal\n  O}(\\xi^2\\lambda^2)$ and read\n\\begin{align}\n G_1^0&=-G_2^0=\\frac{32}{315}\\xi_1\\xi_2\\lambda^2i\\lambda\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr), \\nonumber\\\\\n G_3^0&=-\\frac{2}{45}\\left(4\\xi_1^2\\lambda^2+7\\xi_2^2\\lambda^2\\right)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr),  \\nonumber\\\\\n G_4^0&=G_3^0\\left(\\xi_1^2\\leftrightarrow\\xi_2^2\\right), \\nonumber\\\\\n G_5^0&=-\\frac{8}{105}\\frac{\\sqrt{k_1^2k_2^2}}{4m^2}(\\xi_1^2\\lambda^2+\\xi_2^2\\lambda^2)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr),  \\label{eq:G_j^0}\\\\\n\\intertext{and}\n G_1^{\\pm 1}&=G_2^{\\pm 1}=\\pm\\frac{i}{15}\\,\\xi_1\\xi_2\\lambda^2\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr), \\nonumber\\\\\n G_3^{\\pm 1}&=\\frac{1}{45}\\left(4\\xi_1^2\\lambda^2-7\\xi_2^2\\lambda^2\\right)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr),  \\nonumber\\\\\n G_4^{\\pm 1}&=-G_3^{\\pm 1}\\left(\\xi_1^2\\leftrightarrow\\xi_2^2\\right), \\nonumber\\\\\n G_5^{\\pm 1}&=\\frac{4}{105}\\frac{\\sqrt{k_1^2k_2^2}}{4m^2}(\\xi_1^2\\lambda^2-\\xi_2^2\\lambda^2)\\Bigl(1+{\\cal O}(\\tfrac{k_1k_2}{4m^2})+{\\cal O}(\\lambda^{2})\\Bigr), \\label{eq:G_j^1}\n\\end{align}\nwhereas the leading contributions to $G^l_j$ with $|l|\\geq2$ scale as\n$\\sim(\\xi^2\\lambda^2)^{|l|}$ and thus are at least of ${\\cal\n  O}((\\xi^2\\lambda^2)^2)$.\nPlugging these terms into Eqs.~\\eqref{eq:PIstructure}-\\eqref{eq:c_j},\nwe obtain a compact approximation to the photon polarization tensor\nfor a generic, elliptically polarized plane wave background in the\nparameter regime where\n$\\{\\xi^2\\lambda^2,|\\lambda|,|\\frac{k_1k_2}{4m^2}|\\}\\ll1$.  The above\nfindings imply that the infinite sum in \\Eqref{eq:c_j} at ${\\cal\n  O}(\\xi^2\\lambda^2)$ receives contributions only for $l=\\pm1$.\nHence, the persistent inelastic interactions can be associated with\nthe absorption\/release of just two laser photons.\n\nAs a particular example, we consider\nthe special case of an incoming on-shell photon with $k_1^\\mu=\\omega_1(1,\\vec{k}_1\/|\\vec{k}_1|)$, fulfilling\n$k_1^2=0$. In this case, the parameter $\\lambda$ can be written as\n$\\lambda\\to\\frac{\\omega\\omega_1}{2m^2}\\bigl(1-\\cos\\varangle(\\vec{\\kappa},\\vec{k}_1)\\bigr)$,\nsuch that\n\\begin{equation}\n \\lambda^2\\xi^2 \\quad\\to\\quad \\Bigl(\\frac{e\\mathfrak{E}}{m^2}\\Bigr)^2\\frac{\\omega_1^2}{4m^2}\\bigl(1-\\cos\\varangle(\\vec{\\kappa},\\vec{k}_1)\\bigr)^2,\n\\end{equation}\nwhere we employed the shorthand notation\n$\\mathfrak{E}^2\\in\\{\\mathfrak{E}^2_1,\\mathfrak{E}^2_2,\\mathfrak{E}_1\\mathfrak{E}_2\\}$. Obviously,\nthe dependence on the frequency $\\omega$ of the plane wave background\ndrops out and the combination $\\lambda^2\\xi^2$ becomes $\\omega$\nindependent.  Correspondingly, the photon polarization tensor at ${\\cal O}(\\xi^2\\lambda^2)$ in the\nlimit $\\omega\\to0$ is obtained straightforwardly in this case:\nIt is given by \\Eqref{eq:PIstructure} with $c_5=0$ [see the remarks\n  below \\Eqref{eq:gs}], and the projectors~\\eqref{eq:Lambdas} and\nother coefficients~\\eqref{eq:c_j} specialized to $\\omega=0$.\nObviously, it only features an elastic contribution and its\ncoefficients [cf. \\Eqref{eq:c_j}] are given by\n\\begin{equation}\n c_j\\quad\\to\\quad i(2\\pi)^4m^2\\frac{\\alpha}{\\pi}\\delta(k_1-k_2)\\tilde G_j, \\label{eq:c_jCrossed}\n\\end{equation}\nwith $\\tilde G_j\\equiv\\bigl[G_j^0+G_j^{+1}+G_j^{-1}\\bigr]\\big|_{\\omega=0}$ and $j\\in\\{1,\\ldots,4\\}$. \nInserting the explicit expressions from Eqs.~\\eqref{eq:G_j^0} and \\eqref{eq:G_j^1} into \\Eqref{eq:c_jCrossed}, we obtain $\\tilde G_1=\\tilde G_2=0$ as well as $\\tilde G_3=-\\frac{28}{45}\\xi_2^2\\lambda^2$ and $\\tilde G_4=-\\frac{16}{45}\\xi_2^2\\lambda^2$.\nAs expected the dependence on $\\xi_1$ completely drops out and the polarization tensor in this limit eventually depends only on the single field strength $\\mathfrak{E}_2$. Recall that the electromagnetic field components follow by differentiations of the four-vector potential~\\eqref{eq:background}, which explains why the electric field $\\mathfrak{E}_2$, persists even though it comes along with a factor of $\\sin(\\kappa x)$ in \\Eqref{eq:background}.\nPutting everything together, we finally obtain\n\\begin{equation}\n \\Pi^{\\mu\\nu}(k_1,k_2) \\ \\ \\to\\ \\ -i(2\\pi)^4\\delta(k_1-k_2)\\frac{\\alpha}{\\pi}\\,\\omega_1^2\\bigl(1-\\cos\\varangle(\\vec{\\kappa},\\vec{k}_1)\\bigr)^2\\Bigl(\\frac{e\\mathfrak{E}_2}{m^2}\\Bigr)^2\\biggl[ \\frac{7}{45}\\Lambda_1^\\mu\\Lambda_1^\\nu+\\frac{4}{45}\\Lambda_2^\\mu\\Lambda_2^\\nu\\biggr]. \\label{eq:PIstructureCrossed}\n\\end{equation}\nThis reproduces the photon polarization tensor for constant crossed fields at ${\\cal\n  O}\\bigl((\\frac{e\\mathfrak{E}}{m^2})^2\\bigr)$ and on-the-light-cone\ndynamics \\cite{narozhnyi:1968,ritus:1972}.\n\n\n\\subsection{Laser photon merging} \\label{subsec:photonmerging}\n\nFor a given laser photon polarization, i.e., a particular\nchoice of the monochromatic plane wave\nbackground~\\eqref{eq:background}, the photon merging amplitude\ndepends on both the explicit expression for the field inhomogeneity\nand the polarization state $\\epsilon_\\mu^{*(p)}(k)$ of the outgoing\nphoton, with $p$ labeling the two transverse photon polarizations.  It\nis given by \\cite{Yakovlev:1967}\n\\begin{equation}\n {\\cal M}^{(p)}(k)=\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{2k^0}}\\int\\frac{{\\rm d}^4q}{(2\\pi)^4}\\,\\Pi^{\\mu\\nu}(k,q)A_\\nu(q)\\,, \\label{eq:M}\n\\end{equation}\nwhere $A_\\nu(q)=\\int_{x}\\,{\\rm e}^{-ixq}A_\\nu(x)$ is the Fourier\ntransform of the gauge field representing the inhomogeneous electromagnetic field \nin position space; the star symbol $^*$ denotes complex\nconjugation.  The explicit expression for $k^\\mu=(k^0,\\vec{k})$\ndepends of course on the specific merging process to be\nconsidered. For the merging of $2n$ laser photons of frequency\n$\\omega$ in a static field, momentum conservation and the fact that\nthe outgoing photon is real and propagates on the light cone imply\nthat $k^0=|\\vec{k}|=2n\\omega$.  Moreover, given this condition, the\ncoefficient $c_5$ in \\Eqref{eq:PIstructure} vanishes [cf. below\n  \\Eqref{eq:gs}], such that the tensor structure of\n$\\Pi^{\\mu\\nu}(q,k)$ can be expressed solely in terms of\n$\\Lambda_1^\\mu$ and $\\Lambda_2^\\mu$.\n\nAs outlined in detail above, in this article we limit ourselves to the\nstudy of the merging process in a static magnetic field.  We consider\nfield inhomogeneities of the form $\\vec{B}(x)=B(x)\\vec{e}_B$, such\nthat the direction of the magnetic field $\\vec{e}_B$ is fixed globally\nand only its amplitude is varied.  More specifically, we set\n$\\vec{e}_B=\\vec{e}_{\\rm z}$ and focus on a one dimensional spatial\ninhomogeneity in $\\rm x$ direction, i.e., $B(x)\\to B({\\rm x})$, such\nthat $\\vec{\\nabla}B({\\rm x})\\sim\\vec{e}_{\\rm x}$.  The wave vector of\nthe laser photons is assumed to be $\\vec{\\kappa}={\\kappa}_{\\rm\n  x}\\vec{e}_{\\rm x}+{\\kappa}_{\\rm y}\\vec{e}_{\\rm y}$, i.e., the\nincident laser photons do not have a momentum component parallel to\nthe magnetic field (cf. Fig~\\ref{fig:perspective}). Even if they\nhad, such a component would not be affected due to translational\ninvariance along the ${\\rm z}$ direction.\n\nUtilizing $\\kappa^2=0$ it is convenient to introduce the angle parameter\n$\\theta\\in[0\\ldots\\frac{\\pi}{2}]$ and write\n$\\kappa^\\mu=\\omega(1,\\cos\\theta,\\sin\\theta,0)$ with $\\omega>0$.\nCorrespondingly, the orthogonality relations\n$a_1\\kappa=a_2\\kappa=a_1a_2=0$ imply that the parametrization of the\northonormal vectors $a_1^\\mu$ and $a_2^\\mu$ just requires one\nadditional angle parameter which we denote by\n$\\varphi\\in[0\\ldots2\\pi)$.  We write\n\\begin{align}\n a_1^\\mu&=(0,-\\sin\\theta\\cos\\varphi,\\cos\\theta\\cos\\varphi,-\\sin\\varphi), \\nonumber\\\\\n a_2^\\mu&=(0,-\\sin\\theta\\sin\\varphi,\\cos\\theta\\sin\\varphi,\\cos\\varphi), \\label{eq:a_12}\n\\end{align}\ni.e., our conventions are such that the spatial components of\n$\\kappa^\\mu$, $a_1^\\mu$ and $a_2^\\mu$ form a right-handed trihedron\n(cf. Fig~\\ref{fig:perspective}).  The choice of $\\theta$ fixes the\npropagation direction $\\vec{\\kappa}$ of the incident photons relative\nto the inhomogeneity, while $\\varphi$ controls the orientation of the\nvectors $\\vec a_1$ and $\\vec a_2$ spanning the spatial subspace\ntransverse to $\\vec{\\kappa}$.\n\nA convenient choice for the four-vector potential giving rise to a\nmagnetic field of the desired type is\n\\begin{equation}\n A^\\mu(x)=A({\\rm x})e^\\mu_{\\rm y}, \\quad\\text{with}\\quad A({\\rm x})=\\int^{\\rm x}{\\rm d}{\\rm x}'\\,B({\\rm x}'), \\label{eq:Ax}\n\\end{equation}\nwhere we have defined $e^\\mu_{\\rm y}\\equiv(0,\\vec{e}_{\\rm y})$.\nThe lower limit of the integral is left unspecified as it does\nnot have any observable consequences and thus can be chosen\narbitrarily.  Finally, a Fourier transform of \\Eqref{eq:Ax} yields the\nmomentum space representation of the four-vector potential as needed\nin \\Eqref{eq:M},\n\\begin{equation}\n A^\\mu(q)=(2\\pi)^3\\delta(q_0)\\delta(q_{\\rm y})\\delta(q_{\\rm z})A(q_{\\rm x})e^\\mu_{\\rm y}, \\quad\\text{with}\\quad A(q_{\\rm x})=\\int_{-\\infty}^\\infty{\\rm d}{\\rm x}\\,{\\rm e}^{-i{\\rm x}q_{\\rm x}}\\,A({\\rm x}) . \\label{eq:Ak}\n\\end{equation}\n\nPlugging this expression into \\Eqref{eq:M} and introducing $\\bar q^\\mu\\equiv(0,q_{\\rm x}\\vec{e}_{\\rm x})$, the photon merging amplitude can be simplified significantly and reads\n\\begin{equation}\n {\\cal M}^{(p)}(k)=\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{2k^0}}\\int\\frac{{\\rm d}q_{\\rm x}}{2\\pi}\\,\\Pi^{\\mu2}(k,\\bar q)\\,A(q_{\\rm x})\\,. \\label{eq:M1}\n\\end{equation}\nSubstituting $k_2\\to\\bar q$ into the expressions for $\\Lambda_1^\\mu$ and $\\Lambda_2^\\mu$ in\n\\Eqref{eq:Lambdas} we obtain together with \\Eqref{eq:a_12}\n\\begin{align}\n \\Lambda_1^\\mu&=\\bigl(\\tan\\theta\\cos\\varphi,0,\\tfrac{\\cos\\varphi}{\\cos\\theta},-\\sin\\varphi\\bigr), \\nonumber\\\\\n \\Lambda_2^\\mu&=\\bigl(\\tan\\theta\\sin\\varphi,0,\\tfrac{\\sin\\varphi}{\\cos\\theta},\\cos\\varphi\\bigr). \\label{eq:lambda12}\n\\end{align}\nAnalogously to \\Eqref{eq:c_j}, we write \n\\begin{equation}\n \\Pi^{\\mu2}(k,\\bar q)=(2\\pi)^4\\sum_{l\\in\\mathbb{Z}}\\delta(k-\\bar q-2l\\kappa)\\Pi^{\\mu2}_l(k,\\bar q), \\label{eq:Pi2nu}\n\\end{equation}\nwhere the explicit representation\n\\begin{equation}\n \\Pi^{\\mu2}_l= im^2\\frac{\\alpha}{\\pi}\\frac{1}{\\cos\\theta}\\Bigl[\\Lambda_1^\\mu\\,(G_1^l\\sin\\varphi+G_3^l\\cos\\varphi)+\\Lambda_2^\\mu\\,(G_2^l\\cos\\varphi+G_4^l\\sin\\varphi)\\Bigr]\n\\end{equation}\nmakes use of \\Eqref{eq:lambda12}.  Using \\Eqref{eq:Pi2nu} in\n\\Eqref{eq:M1}, the residual integration over $q_{\\rm x}$ can be\nperformed and we obtain\n\\begin{equation}\n {\\cal M}^{(p)}(k)=(2\\pi)^3 \\delta(k_{\\rm z})\\sum_{l\\in\\mathbb{Z}}\\delta(k^0-2l\\omega)\\delta(k_{\\rm y}-2l\\omega\\sin\\theta)\\,\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{2k^0}}\n\\Pi^{\\mu2}_l(k,\\tilde k)A(\\tilde k_{\\rm x})\\,, \\label{eq:M2}\n\\end{equation}\nwith $\\tilde k^\\mu\\equiv(0,(k_{\\rm x}-2l\\omega\\cos\\theta)\\vec{e}_{\\rm x})$.\n\nTaking into account the fact that the outgoing photon has positive\nenergy ($k^0>0$) and propagates on the light cone ($k_\\mu k^\\mu=0$),\nand also because of the $\\delta$ functions for the ${\\rm y}$ and ${\\rm z}$\nmomentum components, we identify $k^0\\equiv2l\\omega$ and\nrewrite the $\\delta$ function implementing energy conservation in\n\\Eqref{eq:M2} as follows,\n\\begin{equation}\n \\delta(k^0-2l\\omega)\\ \\to\\ \\delta_{l0}\\,\\delta(k_{\\rm x})+ \\Theta(l+0^+)\\,\\frac{1}{\\cos\\theta}\\Bigl[\\delta(k_{\\rm x}-2l\\omega\\cos\\theta)+\\delta(k_{\\rm x}+2l\\omega\\cos\\theta)\\Bigr]\\,, \\label{eq:delta}\n\\end{equation}\nwhere $\\Theta(.)$ is the Heaviside function.\nCorrespondingly, we have\n\\begin{multline}\n {\\cal M}^{(p)}(k)=(2\\pi)^3 \\delta(k_{\\rm z})\\sum_{l=1}^{\\infty}\\frac{1}{\\cos\\theta}\\Bigl[\\delta(k_{\\rm x}-2l\\omega\\cos\\theta)+\\delta(k_{\\rm x}+2l\\omega\\cos\\theta)\\Bigr] \\\\ \\times\\delta(k_{\\rm y}-2l\\omega\\sin\\theta)\\,\\frac{\\epsilon_\\mu^{*(p)}(k)}{\\sqrt{4l\\omega}}\\,\\Pi^{\\mu2}_l(k,\\tilde k)A(\\tilde k_{\\rm x})\\,, \\label{eq:M3}\n\\end{multline}\nwith $k^\\mu=(2l\\omega,k_{\\rm x},k_{\\rm y},0)$, where we have\nmade use of the fact that the $l=0$ contribution vanishes: it scales\n$\\sim\\delta(\\vec{k})\\,\\frac{\\Pi^{\\mu2}_l(k,\\tilde\n  k)}{\\sqrt{4l\\omega}}\\sim\\delta(\\vec{k})\\,l^{3\/2}\\to0$ [cf. also\n  \\Eqref{eq:expansionparameters->} below].\n\nWhen adapted to the particular kinematics in \\Eqref{eq:M3} (cf. the arguments of the photon polarization tensor), the dimensionless parameters $\\frac{k_1k_2}{4m^2}$, $\\lambda$ and $\\xi^2\\lambda^2$\ngoverning the expansion of the photon polarization tensor performed in Sec.~\\ref{subsec:Pi} all vanish for the contribution $\\sim\\delta(k_{\\rm x}-2l\\omega\\cos\\theta)$.\nFor the contribution $\\sim\\delta(k_{\\rm x}+2l\\omega\\cos\\theta)$ they are non-zero and read\n\\begin{align}\n \\frac{k_1k_2}{4m^2}\\equiv \\frac{k \\tilde{k}}{4m^2} &\\quad\\to\\quad\\frac{1}{2}\\left(\\frac{2l\\omega \\cos\\theta}{m}\\right)^2, \\nonumber\\\\\n \\lambda&\\quad\\to\\quad\\left(\\frac{2l\\omega \\cos\\theta}{m}\\right)\\frac{\\omega\\cos\\theta}{m}, \\nonumber\\\\\n \\xi^2\\lambda^2&\\quad\\to\\quad\\left(\\frac{e\\mathfrak{E}}{m^2}\\right)^2\\left(\\frac{2l\\omega \\cos\\theta}{m}\\right)^2\\cos^2\\theta. \\label{eq:expansionparameters->}\n\\end{align}\nNeglecting higher-order contributions of ${\\cal O}(\\frac{k_1k_2}{4m^2})\\sim{\\cal O}(\\lambda)\\sim{\\cal O}(\\frac{\\omega^2}{m^2})$, our result will of course be fully governed by the remaining parameters $\\xi_1^2\\lambda^2$, $\\xi_2^2\\lambda^2$ and $\\xi_1\\xi_2\\lambda^2$.\n\nAs a result, the number of merged photons with four wave-vector $k_f^\\mu$\nand polarization $p$ according to Fermi's golden rule is given by \n\\begin{equation}\n {\\cal N}^{(p)}(k_f)=\\int\\frac{d^3k}{(2\\pi)^3}\\,\\bigl|{\\cal M}^{(p)}(k)\\bigr|^2= TL_{\\rm y}L_{\\rm z}\\sum_{l=1}^{\\infty}\\frac{\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_l(k_f,\\tilde k_f)A(\\tilde k_{f,{\\rm x}})\\bigr|^2}{4l\\omega\\cos\\theta}\\,, \\label{eq:M4}\n\\end{equation}\nwith $k^\\mu_f=2l\\omega(1,-\\cos\\theta,\\sin\\theta,0)$, i.e., the\noutgoing photon of energy $2l\\omega$ propagates in\n$(-\\cos\\theta,\\sin\\theta,0)$ direction.  Moreover, $\\tilde\nk^\\mu_f=-4l\\omega\\cos\\theta(0,\\vec{e}_{\\rm x})$ encodes the momentum\ntransfer from the field inhomogeneity, $T$ is the interaction time and\n$L_{\\rm y}L_{\\rm z}$ is the interaction area transverse to the\ninhomogeneity.  The total number of merged photons is\n\\begin{equation}\n {\\cal N}(k_f)=\\sum_{p}{\\cal N}^{(p)}(k_f).  \\label{eq:N}\n\\end{equation}\nObviously the dominant contribution is due to the merging of just two\nlaser photons, $l=1$, as higher photon processes are suppressed\nby at least a factor of $\\xi^2\\lambda^2$.  Correspondingly,\n\\Eqref{eq:M3} can be written as\n\\begin{equation}\n {\\cal N}^{(p)}(k_f)= TL_{\\rm y}L_{\\rm z}\\frac{\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_1(k_f,\\tilde k_f)A(\\tilde k_{f,{\\rm x}})\\bigr|^2}{4\\omega\\cos\\theta}\\bigl(1+{\\cal O}(\\tfrac{e^2{\\mathfrak E}^2}{m^4}\\tfrac{\\omega^2}{m^2})\\bigr)\\,. \\label{eq:M4a}\n\\end{equation}\nWe emphasize that the terms written out explicitly in \\Eqref{eq:M4a} account for\nthe entire two-photon merging process.  We approximate the infinite\nsum in \\Eqref{eq:M4} by its contribution for $l=1$, and thereby\nneglect merging processes of $2l$ laser photons with $l>1$.\n\nEmploying the substitutions $\\varphi\\to\\varphi'$ and $\\theta\\to\\pi-\\theta$ in \\Eqref{eq:a_12}, we introduce the following two vectors\n\\begin{align}\n \\epsilon^{(1)\\mu}(k_f)&=(0,-\\sin\\theta\\cos\\varphi',-\\cos\\theta\\cos\\varphi',-\\sin\\varphi'), \\nonumber\\\\\n \\epsilon^{(2)\\mu}(k_f)&=(0,-\\sin\\theta\\sin\\varphi',-\\cos\\theta\\sin\\varphi',\\cos\\varphi'), \\label{eq:epsilons}\n\\end{align}\nwith $\\varphi'\\in[0\\ldots2\\pi)$ fixed, to span the subspace transverse to the wave-vector $\\vec{k}_f$ of the merged photon.\nThe two polarization degrees of freedom of the outgoing photon are then conveniently expressed in terms of the vectors $\\epsilon^{(p)\\mu}(k_f)$, with $p\\in\\{1,2\\}$, representing linear polarization states in the particular basis characterized by a particular choice of $\\varphi'$.\nPolarizations other than linear can be obtained through linear combinations of the vectors~\\eqref{eq:epsilons}.\n\nWe are now in a position to provide the explicit expressions of\nthe polarization tensor in \\Eqref{eq:M4} contracted with a given\npolarization vector of the outgoing photon, which read\n\\begin{multline}\n \\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_l\n= im^2\\frac{\\alpha}{\\pi}\\frac{1}{2\\cos\\theta}\\Bigl[\\sin\\varphi'\\,(G_1^l-G_2^l)-\\sin(\\varphi'+2\\varphi)(G_1^l+G_2^l)  \\\\\n-\\cos(\\varphi'+2\\varphi)\\,(G_3^l-G_4^l) -\\cos\\varphi'\\,(G_3^l+G_4^l)\\Bigr], \\label{eq:epsilonPi}\n\\end{multline}\nand\n\\begin{equation}\n \\epsilon_\\mu^{*(2)}(k_f)\\Pi^{\\mu2}_l=\\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_l\\big|_{\\varphi'\\,\\to\\,\\varphi'-\\frac{\\pi}{2}}\\,. \\label{eq:epsilonPi2}\n\\end{equation}\nIntroducing the dimensionless field strengths $\\varepsilon_i\\equiv\\frac{e\\mathfrak{E}_i}{m^2}$ with $i\\in\\{1,2\\}$, \nin particular the $l=1$ contribution to \\Eqref{eq:epsilonPi} can be written as\n\\begin{multline}\n \\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_{1}\n=i(\\omega \\cos\\theta)^2\\frac{\\alpha}{\\pi}\\frac{2}{15}\\cos\\theta\\Bigl[-2i\\sin(\\varphi'+2\\varphi)\\,\\varepsilon_1\\varepsilon_2  \\\\\n+\\cos(\\varphi'+2\\varphi)\\,(\\varepsilon_1^2+\\varepsilon_2^2) -\\frac{11}{3}\\cos\\varphi'\\,(\\varepsilon_1^2-\\varepsilon_2^2)\\Bigr]\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr), \\label{eq:epsilonPi_l=1}\n\\end{multline}\nwhere we have made use of Eqs.~\\eqref{eq:G_j^1} and\n\\eqref{eq:expansionparameters->}.\n\nIf $A(\\tilde k_{f,{\\rm x}})$ is either purely real or imaginary\nvalued, which is true for the field inhomogeneities symmetric in\n  $\\rm x$ to be considered below, the modulus squared can be split\nand \\Eqref{eq:M4a} be represented as follows,\n\\begin{equation}\n {\\cal N}^{(p)}(k_f)= TL_{\\rm y}L_{\\rm\n   z}\\frac{\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_1(k_f,\\tilde\n   k_f)|^2\\,|A(\\tilde k_{f,{\\rm x}})\\bigr|^2}{4\\omega\\cos\\theta}\\bigl(1+{\\cal \n   O}(\\tfrac{e^2{\\mathfrak\n     E}^2}{m^4}\\tfrac{\\omega^2}{m^2})\\bigr)\\,. \\label{eq:M4b}\n\\end{equation}\n\nThe modulus squared of \\Eqref{eq:epsilonPi_l=1} is obtained straightforwardly and reads\n\\begin{multline}\n \\bigl|\\epsilon_\\mu^{*(1)}(k_f)\\Pi^{\\mu2}_1\\bigr|^2\n=(\\omega \\cos\\theta)^4\\frac{\\alpha^2}{\\pi^2}\\frac{4}{225}\\cos^2\\theta\\Bigl\\{4(\\varepsilon_1\\varepsilon_2)^2\n-\\frac{22}{3}\\cos\\varphi'\\cos(\\varphi'+2\\varphi)\\,(\\varepsilon_1^4-\\varepsilon_2^4) \\\\\n+\\Bigl[\\frac{121}{9}\\cos^2\\varphi'+\\cos^2(\\varphi'+2\\varphi)\\Bigr](\\varepsilon_1^2-\\varepsilon_2^2)^2\\Bigr\\}\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr),\n\\label{eq:modpisquared_1}\n\\end{multline}\nwhile the analogous expression for the other polarization mode ($p=2$) follows from \\Eqref{eq:epsilonPi2}.\n\nAiming at the total number of merged photons in the polarization basis characterized by a particular choice of $\\varphi'$, we have to add the moduli squared corresponding to the two different polarization states [cf. Eqs.~\\eqref{eq:N} and \\eqref{eq:M4b}].\nThis results in\n\\begin{multline}\n \\sum_{p=1}^2\\bigl|\\epsilon_\\mu^{*(p)}(k_f)\\Pi^{\\mu2}_1\\bigr|^2\n=(\\omega \\cos\\theta)^4\\frac{\\alpha^2}{\\pi^2}\\frac{8}{225}\\cos^2\\theta \\\\\n\\times\\Bigl[4(\\varepsilon_1\\varepsilon_2)^2\n-\\frac{11}{3}\\cos(2\\varphi)\\,(\\varepsilon_1^4-\\varepsilon_2^4)\n+\\frac{65}{9}(\\varepsilon_1^2-\\varepsilon_2^2)^2\\Bigr]\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr),\n\\label{eq:modpisquared}\n\\end{multline}\nwhich is completely independent of the choice of $\\varphi'$, as it should.\nNoteworthily, in case of circularly polarized incident\nlaser photons for which $\\xi_1=\\xi_2$ and thus\n$\\varepsilon_1=\\varepsilon_2$, the contributions for both polarization\nmodes individually become independent of $\\varphi$ and $\\varphi'$;\ncf. \\Eqref{eq:modpisquared_1}. Equation \\eqref{eq:modpisquared}\nupon insertion into \\Eqref{eq:N} and accounting for the prefactors\ndisplayed in \\Eqref{eq:M4a} represent a central result of this work.\n\nSubsequently, we assume the probe laser to deliver incident laser\npulses of duration $\\tau$, entering under an angle $\\theta$ and\nfeaturing a circular transverse beam profile.  The longitudinal\nevolution of the probe laser pulses follows the envelope of a Gaussian\nbeam, with beam waist right at the intersection with the field\ninhomogeneity.  We denote the transverse cross-section area at the\nbeam waist by $\\sigma$.  Correspondingly, the transversal area $L_{\\rm\n  y}L_{\\rm z}$ can be identified with the intersection area of such a\nbeam profile with the ${\\rm y}$--${\\rm z}$ plane, i.e., $L_{\\rm y}L_{\\rm\n  z}=\\frac{\\sigma}{\\cos\\theta}$ (cf. Fig.~\\ref{fig:yzSchnitt}).\nAssuming that the magnetic field inhomogeneity is long-lived as compared to the\npulse duration $\\tau$ of the probe laser, it is reasonable to consider\n$\\tau$ as a measure of the interaction time $T$, and set $T=\\tau$.\nHence, we can make use of the following substitution,\n\\begin{equation}\n TL_{\\rm y}L_{\\rm z} \\quad \\to \\quad \\frac{\\sigma\\tau}{\\cos\\theta}\\,. \\label{eq:subst}\n\\end{equation}\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.67\\textwidth]{Gaussbeam-beschr-crop} \n\\caption{Sketch of the envelope of a Gaussian beam intersecting the ${\\rm y}$--${\\rm z}$ plane in\n the vicinity of its waist under an angle of $\\theta$ (cf. also\n Fig.~\\ref{fig:perspective}).  Given that the transverse cross-section\n area of the Gaussian beam at the beam waist is a circle of area\n $\\sigma$, the intersection area is an ellipse with area $\\frac{\\sigma}{\\cos\\theta}$.}\n\\label{fig:yzSchnitt}\n\\end{figure}\n\n\n\\section{Results and Discussion} \\label{seq:Ex+Res}\n\nLet us now consider explicit examples of localized magnetic field inhomogeneities which can be tackled analytically.\nWe limit ourselves to two elementary shapes, characterized by just two parameters, namely an amplitude $B$ and a typical extension $w$.\nFor a Lorentz profile characterized by its full width at half maximum (FWHM),\n\\begin{equation}\n B({\\rm x})=\\frac{B}{1+(\\frac{2{\\rm x}}{w})^2}, \\label{eq:Lorentz}\n\\end{equation}\nthe associated gauge field in position space can be determined by \\Eqref{eq:Ax}. We obtain\n\\begin{equation}\n A({\\rm x})=\\frac{Bw}{2}\\arctan\\!\\left(\\frac{2{\\rm x}}{w}\\right),\n\\end{equation}\nand Fourier transforming to momentum space via \\Eqref{eq:Ak},\n\\begin{equation}\n A(q_{\\rm x})=-i\\frac{\\pi Bw}{2q_{\\rm x}}\\,{\\rm e}^{-\\frac{|q_{\\rm x}|w}{2}}. \\label{eq:inh1}\n\\end{equation}\n\n\nAnalogously, for a Gaussian type inhomogeneity characterized by its full width at $1\/{\\rm e}$ of its maximum,\n\\begin{equation}\n B({\\rm x})=B\\,{\\rm e}^{-\\left(\\frac{2{\\rm x}}{w}\\right)^2}, \\label{eq:Gauss}\n\\end{equation}\nwe obtain\n\\begin{equation}\n A({\\rm x})=\\frac{\\sqrt{\\pi}Bw}{4}\\,{\\rm erf}\\!\\left(\\frac{2{\\rm x}}{w}\\right),\n\\end{equation}\nwhere ${\\rm erf}(.)$ denotes the error function, and finally\n\\begin{equation}\n A(q_{\\rm x})=-i\\frac{\\sqrt{\\pi} Bw}{2q_{\\rm x}}\\,{\\rm e}^{-\\left(\\frac{q_{\\rm x}w}{4}\\right)^2}. \\label{eq:inh2}\n\\end{equation}\nEquations~\\eqref{eq:inh1} and \\eqref{eq:inh2} share an overall prefactor $\\sim(-i\\frac{\\sqrt{\\pi} Bw}{2q_{\\rm x}})$, but differ in the exponential decay.\nFor the Lorentz profile the decay is linear in $|q_{\\rm x}|w$, while for the Gaussian inhomogeneity it is quadratic in this dimensionless parameter.\n\nIt is now straightforward to derive the number of merged laser photons, \\Eqref{eq:M4b}, for these inhomogeneities.\nThe number of outgoing merged laser photons with polarization\n$p=1$ and energy $2\\omega$ reads [cf. Eqs.~\\eqref{eq:M4b} and\n  \\eqref{eq:subst}]\n\\begin{multline}\n {\\cal N}^{(1)}(k_f)\n= w\\sigma\\tau(eB)^2(\\omega w)\\frac{\\alpha\\cos^2\\theta}{57600\\,\\pi}\\left\\{\\begin{array}{c}\n            {\\rm e}^{-(4\\omega\\cos\\theta) w} \\\\\n            \\frac{1}{\\pi}\\,{\\rm e}^{-\\frac{1}{8}(4\\omega\\cos\\theta)^2w^2}\n        \\end{array}\\right\\} \\\\\n        \\times\\Bigl\\{4(\\varepsilon_1\\varepsilon_2)^2\n-\\frac{22}{3}\\cos\\varphi'\\cos(\\varphi'+2\\varphi)\\,(\\varepsilon_1^4-\\varepsilon_2^4) \\\\\n+\\Bigl[\\frac{121}{9}\\cos^2\\varphi'+\\cos^2(\\varphi'+2\\varphi)\\Bigr](\\varepsilon_1^2-\\varepsilon_2^2)^2\\Bigr\\}\\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr),\n \\label{eq:Np1}\n\\end{multline}\nwhere the upper line in braces is the result for the Lorentz~\\eqref{eq:Lorentz}\nand the lower line that for the Gaussian~\\eqref{eq:Gauss} profile, and\n\\begin{equation}\n {\\cal N}^{(2)}(k_f)={\\cal N}^{(1)}(k_f)\\big|_{\\varphi'\\,\\to\\,\\varphi'-\\frac{\\pi}{2}}\\,. \\label{eq:Np2}\n\\end{equation}\nAs the results for the Lorentz and Gaussian inhomogeneities -- apart\nfrom the different exponential behavior -- are of very similar\nstructure, we find it convenient to adopt the two-component notation\nemployed in \\Eqref{eq:Np1} in the remainder of this paper.\nEquation \\eqref{eq:Np1} exhibits several characteristic\ndependencies on the involved parameters: as to be expected from the\nunderlying Feynman diagram, the leading order effect is proportional\nto the square of the plane wave intensity, i.e., $\\sim\n\\mathfrak{E}^4$, and to the square of the magnetic field $\\sim B^2$.\nIn particular the latter dependence represents a comparatively\nstrong increase of the effect with an enhancement of the peak\nmagnetic background field. Other typical nonlinear phenomena such\nas photon scattering off a magnetic field $\\sim B^4$ or photon splitting\n$\\sim B^6$ are more strongly suppressed since the $B$ field scale is\nmeasured in terms of the electron mass scale. On the other hand, the\ninhomogeneous field has to provide the necessary momentum transfer\n$\\sim 4\\omega\\cos\\theta$, and the effect is exponentially damped with $\\sim(4\\omega\\cos\\theta)w$.\n\nIn this respect, it is instructive to compare these expressions\nwith the number of photons experiencing quantum reflection\n\\cite{Gies:2013yxa} for the very same conditions, i.e., for incident\nphotons of the same energy, angle of incidence and polarization, and\nexactly the same field inhomogeneities as in Eqs.~\\eqref{eq:Lorentz}\nand \\eqref{eq:Gauss}.\n\nFor completeness, we note that in Ref.~\\cite{Gies:2013yxa}, the field inhomogeneity was not accounted for exactly in the sense that the photon polarization tensor was evaluated {\\it a priori} in the presence of the magnetic field inhomogeneity,\nbut rather the inhomogeneity was built in {\\it a posteriori} by resorting to the result for a constant magnetic background field and using the constant-field expressions locally.\nAs argued in detail in~\\cite{Gies:2013yxa}, such an approach is\njustifiable for inhomogeneities whose typical scale of variation $w$\nis much larger than the Compton wavelength $\\lambda_c$ of the charged\nvirtual particles, i.e., $w\\gg \\lambda_c$.  Particularly in quantum\nelectrodynamics (QED), where the virtual particles are electrons,\n$\\lambda_c\\approx2\\cdot10^{-6}{\\rm eV}^{-1}\\approx3.9\\cdot10^{-13}{\\rm\n  m}$, many field inhomogeneities available in the laboratory can be\ndealt with along these lines.\n\nReference~\\cite{Gies:2013yxa} identifies two situations for which the\ncalculations become particularly simple, corresponding to\nspecial alignments of the incident photons' wave vector $\\vec{k}$ and\npolarization plane, the magnetic field $\\vec{B}$, and the direction of\nthe inhomogeneity $\\vec{\\nabla}B$.\nThe one reconcilable with incident photons of four wave-vector\n$\\kappa^\\mu=\\omega(1,\\cos\\theta,\\sin\\theta,0)$ and\n$\\vec{B}\\sim\\vec{e}_{\\rm z}$ is that with polarization vector in the\nplane spanned by $\\vec{\\kappa}$ and $\\vec{B}$, labeled by $\\parallel$\nin~\\cite{Gies:2013yxa}.  To bring the $\\parallel$ case\nof~\\cite{Gies:2013yxa} and the merging scenario discussed here into\nfull kinematic agreement, we specialize the quantum reflection formulae to\n$\\varangle(\\vec{\\kappa},\\vec{B})=\\frac{\\pi}{2}$ and set\n$\\varphi=\\varphi'=0$, $\\varepsilon_1=0$ and\n$\\varepsilon_2=\\frac{e\\mathfrak{E}}{m^2}$ in Eqs.~\\eqref{eq:Np1} and\n\\eqref{eq:Np2}, i.e., we specialize to incident laser photons\npolarized linearly along ${\\rm z}$, and look for induced outgoing\nphotons in the same polarization basis.  Incidentally, it can be shown\nstraightforwardly that the polarization direction is conserved under\nthese circumstances for quantum reflection (cf. \\cite{Gies:2013yxa}),\ni.e., the quantum reflected photons are still polarized along ${\\rm z}$,\nwhile for laser photon merging the induced outgoing photons are\npolarized differently, namely their polarization vector lies in the\n${\\rm x}$--${\\rm y}$ plane [cf. \\Eqref{eq:Np-linpol1} below].\n\nThe number of quantum reflected photons ${\\cal N}_{\\rm Qref}$ is\nobtained by multiplying the number of incident probe photons $N_{\\rm\n  probe}$ with the adequate reflection coefficient, given in Eqs.~(27)\nand (29) of \\cite{Gies:2013yxa}.\nIn order to allow for a more direct comparison with the merging result, we first rewrite $N_{\\rm probe}$: The number of incident photons per pulse amounts to the ratio of the pulse energy of the probe laser $\\cal E$ and its frequency $\\omega$, i.e., $N_{\\rm probe}=\\frac{\\cal E}{\\omega}$.\nThe intensity $I_{\\rm probe}$ at the focal spot, which is related to the electric field strength in the focus via $I_{\\rm probe}={\\mathfrak E}^2$, is determined by $I_{\\rm probe}=\\frac{\\cal E}{\\sigma\\tau}$.\nHence, the number of probe photons can be expressed as $N_{\\rm probe}=\\frac{{\\mathfrak E}^2\\sigma\\tau}{\\omega}$, and -- neglecting corrections of ${\\cal O}\\bigl((\\tfrac{eB}{m^2})^6\\bigr)$ -- we finally obtain\n\\begin{equation}\n{\\cal N}_{\\rm Qref}\n= w\\sigma\\tau \\frac{49\\,\\alpha}{129600\\pi}\n\\biggl(\\frac{eB}{m^2}\\biggr)^4(e{\\mathfrak E})^2(\\omega w)\\frac{1}{\\cos^2\\theta}\n\\left\\{\\begin{array}{c}\n        \\frac{1}{4}(1+\\omega w\\cos\\theta)^2\\,{\\rm e}^{-2\\omega w\\cos\\theta}\\\\\n        \\frac{1}{2\\pi}\\,{\\rm e}^{-\\frac{1}{4}(\\omega w\\cos\\theta)^2} \n       \\end{array}\\right\\} . \\label{eq:Np-Qref}\n\\end{equation}\nFor the merging process, Eqs.~\\eqref{eq:Np1} and \\eqref{eq:Np2}, the same choice of parameters results in\n\\begin{equation}\n{\\mathcal N}^{(1)}\n= w\\sigma\\tau\\frac{49\\,\\alpha}{129600\\,\\pi}\\biggl(\\frac{e\\mathfrak{E}}{m^2}\\biggr)^4(eB)^2(\\omega w)  \\cos^2\\theta \\left\\{\\begin{array}{c}\n            {\\rm e}^{-4\\omega w\\cos\\theta} \\\\\n            \\frac{1}{\\pi}\\,{\\rm e}^{-2(\\omega w\\cos\\theta)^2}\n        \\end{array}\\right\\}\n \\Bigl(1+{\\cal O}(\\tfrac{\\omega^2}{m^2})\\Bigr) , \\label{eq:Np-linpol1}\n\\end{equation}\nwhile ${\\mathcal N}^{(2)}=0$, such that ${\\mathcal N}_{\\rm\n  merg}\\equiv{\\mathcal N}^{(1)}$. Both results exhibit an exponential\nsuppression with exponent $\\sim w\\omega\\cos\\theta=w\\kappa_{\\rm\n  x}$, with $\\kappa_{\\rm x}$ being the momentum component of the\nincident probe photons in the direction of the inhomogeneity\n[cf. above \\Eqref{eq:a_12}]. The suppression is more pronounced for the merging\nprocess.  This can also be understood intuitively by recalling that\nthe momentum transfer from the inhomogeneity is $|2\\kappa_{\\rm x}|$\nfor the process of quantum reflection (cf. \\cite{Gies:2013yxa}), while\nit is twice as large, namely $|4\\kappa_{\\rm x}|$, for the merging of\ntwo laser photons.\n\nAnother important point to notice is that in \\Eqref{eq:Np-Qref} the\ntransition to large incidence angles $\\theta\\lesssim\\pi\/2$ provides a\nconvenient handle to damp the exponential suppression while at the\nsame time increasing the overall prefactor, which scales inversely\nwith $\\cos^2\\theta$.  Conversely, in \\Eqref{eq:Np-linpol1} an\nanalogous increase of the angle of incidence to $\\theta\\lesssim\\pi\/2$\ndiminishes the overall prefactor $\\sim\\cos^2\\theta$.\nThe ratio of Eqs.~\\eqref{eq:Np-linpol1} and \\eqref{eq:Np-Qref} can be derived straightforwardly, and reads\n\\begin{equation}\n\\frac{\\mathcal{N}_{\\rm merg}}{{\\cal N}_{\\rm Qref}}\n\\approx 4\\,\n\\biggl(\\frac{\\mathfrak{E}}{B}\\,\\cos^2\\theta\\biggr)^2 \n\\left\\{\\begin{array}{c}\n        \\frac{1}{(1+\\omega w\\cos\\theta)^{2}}\\,{\\rm e}^{-2\\omega w\\cos\\theta} \\\\\n        \\frac{1}{2}\\,{\\rm e}^{-\\frac{7}{4}(\\omega w\\cos\\theta)^2}\n       \\end{array}\\right\\} . \\label{eq:ratio}\n\\end{equation}\nIt is governed by just two dimensionless quantities, namely the product $\\omega w \\cos\\theta$, measuring the width $w$ of the inhomogeneity in units of the inverse of the momentum component of the incident photons in $\\vec{\\nabla}B$ direction,\nand $\\mathfrak{E}\/B\\,\\cos^2\\theta$, i.e., the ratio of the field strength of the probe relative to that of the pump, augmented by an extra factor of $\\cos^2\\theta$.\n\nIt is now natural to ask for the conditions which have to be met such that photon merging dominates quantum reflection, i.e., $\\mathcal{N}_{\\rm merg}\\geq{\\cal N}_{\\rm Qref}$.\nInserting this condition into \\Eqref{eq:ratio}, we obtain\n\\begin{equation}\n\\frac{\\mathfrak{E}}{B}\\,\\cos^2\\theta\n\\geq \\frac{1}{2}\n\\left\\{\\begin{array}{c}\n        |1+\\omega w\\cos\\theta|\\,{\\rm e}^{\\omega w\\cos\\theta} \\\\\n        \\sqrt{2} \\,{\\rm e}^{\\frac{7}{8}(\\omega w\\cos\\theta)^2}\n       \\end{array}\\right\\}\\geq \\frac{1}{2}\n\\left\\{\\begin{array}{c}\n        1 \\\\\n        \\sqrt{2}\n       \\end{array}\\right\\} , \\label{eq:ratio2}\n\\end{equation}\nwhere we made use of the fact that the expression on the right-hand\nside of the first inequality is bounded from below by its value for\n$\\omega w\\cos\\theta=0$. \nThe latter condition tells us that for the particular set-up\nconsidered here, the yields for photon merging can dominate those for\nquantum reflection only if the quantity $(\\mathfrak{E}\/B)\\cos^2\\theta$ is larger\nthan the numerical bounds given on the rightmost side of\n\\Eqref{eq:ratio2}.\n\nIn Fig.~\\ref{fig:ratio}, we exemplarily set $\\mathfrak{E}=B$\nwhich is a natural choice if all fields are provided by a\nhigh-intensity laser system. We\ninvestigate the implications of the first inequality in\n\\Eqref{eq:ratio2} as a function of $\\theta$ and $\\omega w$.\nObviously, for this choice of the field strengths laser photon merging\ncan only dominate quantum reflection if\n$\\cos\\theta\\geq\\frac{1}{\\sqrt{2}}$ $\\leftrightarrow$\n$\\theta\\leq45^\\circ$ ($\\cos\\theta\\geq 2^{-1\/4}$ $\\leftrightarrow$\n$\\theta\\leq32.7^\\circ$) for a Gaussian (Lorentzian) inhomogeneity.\nQualitatively speaking, the merging process tends to dominate for\nsmall angles of incidence $\\theta$ and small values of $\\omega w$.\nEquation~\\eqref{eq:ratio} implies that this region (in the\n$\\theta$--$\\omega w$ plane) can be enlarged by increasing the ratio of\n$\\mathfrak{E}\/B$.\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{140514merging_ratios_E=B_omegawofc} \n\\caption{Choosing $\\mathfrak{E}=B$ as an example,\nwe depict the regimes where photon merging dominates quantum\nreflection and vice versa based on \\Eqref{eq:ratio2}.  Photon\nmerging dominates quantum reflection in the regime in the lower left\nbounded by the blue (solid) and red (dotted) lines for Gauss and\nLorentz type inhomogeneities, respectively.}\n\\label{fig:ratio}\n\\end{figure}\n\nSo far we only focused on the relative\nimportance of the two effects, but did not\nprovide absolute quantitative estimates.  Most obviously, as\nboth effects are suppressed by powers of $\\frac{eB}{m^2}$ and\n$\\frac{e\\mathfrak{E}}{m^2}$ [cf. Eqs.~\\eqref{eq:Np-Qref} and\n  \\eqref{eq:Np-linpol1}], in order to increase them it is preferable\nto enlarge the field strengths as much as possible.\n\nBefore providing some explicit quantitative estimates, let us briefly\ndiscuss the generic features of \\Eqref{eq:Np-linpol1} and confront it\nwith \\Eqref{eq:Np-Qref}.\nConsider the first derivative of the number of merged photons~\\eqref{eq:Np-linpol1} with respect to $w\\cos\\theta$,\n\\begin{equation}\n\\frac{d{\\mathcal N}_{\\rm merg}}{d(w\\cos\\theta)}\n\\approx  \\frac{2\\,{\\mathcal N}_{\\rm merg}}{w\\cos\\theta}\n\\left\\{\\begin{array}{c}\n            1-2\\omega w\\cos\\theta \\\\\n            1-2(\\omega w\\cos\\theta)^2\n        \\end{array}\\right\\}\\stackrel{!}{=}0 \\quad \\to\\quad\n w\\cos\\theta = \\frac{1}{2\\omega}\\left\\{\\begin{array}{c}\n        1\\\\\n        \\sqrt{2}\n       \\end{array}\\right\\}.\n \\label{eq:N_diff}\n\\end{equation}\nTaking into account the sign of the second derivative, we find that \nthe number of outgoing merged photons has a\nmaximum as a function of $w\\cos\\theta$ for the above values and reads \n\\begin{equation}\n{\\mathcal N}_{\\rm merg}\\big|_{\\rm max}\n\\approx \\frac{\\sigma\\tau}{\\omega}\\frac{49\\,\\alpha}{129600\\,\\pi}\\biggl(\\frac{e\\mathfrak{E}}{m^2}\\biggr)^4(eB)^2\\,\\frac{1}{4}\\left\\{\\begin{array}{c}\n            {\\rm e}^{-2} \\\\\n            \\frac{2}{\\pi}\\,{\\rm e}^{-1}\n        \\end{array}\\right\\} . \\label{eq:Nmax}\n\\end{equation}\n\nHence, keeping $w$ fixed, \nthe number of merged photons increases monotonically as a\nfunction of $\\theta$ from its value for $\\theta=0$ until it reaches\na maximum at $\\theta = \\arccos(\\frac{1}{2\\omega w})$ in case of the\nGaussian, and $\\theta = \\arccos(\\frac{1}{\\sqrt{2}\\omega w})$ for the\nLorentz type inhomogeneity. Increasing $\\theta$ even further, it\ndecreases rapidly until it reaches ${\\mathcal N}_{\\rm merg}=0$ at\n$\\theta=90^\\circ$.\n\nConversely, for fixed $\\omega$ the number of quantum reflected photons~\\eqref{eq:Np-Qref} exhibits a monotonic increase throughout the interval from $\\theta=0$ to $\\theta=90^\\circ$. \nActually, ${\\cal N}_{\\rm Qref}$ even diverges for $\\theta\\to90^\\circ$ due to the cosine squared term in its denominator, an unphysical feature which can be attributed to the unphysical limit of an infinitely long interaction of the probe photons and the inhomogeneity at ``grazing incidence'' $\\theta\\to90^\\circ$.\n\nFinally, we provide some rough estimates on the numbers of merged and\nquantum reflected photons attainable in an all optical pump--probe\nexperiment based on high-intensity lasers.  Even though we have just\nfocused on a one-dimensional field inhomogeneity, as in\n\\cite{Gies:2013yxa} we exemplarily adopt the design parameters of the\ntwo high-intensity laser systems to become available in Jena \\cite{Jena}:\nJETI~200 \\cite{JETI200} ($\\lambda=800{\\rm nm}\\approx4.06{\\rm\n  eV}^{-1}$, ${\\cal E}=4{\\rm J}\\approx2.50\\cdot10^{19}{\\rm eV}$,\n$\\tau=20{\\rm fs}\\approx30.4{\\rm eV}^{-1}$) as probe, and POLARIS\n\\cite{POLARIS} ($\\lambda_{\\rm pump}=1030{\\rm nm}\\approx5.22{\\rm\n  eV}^{-1}$, ${\\cal E}_{\\rm pump}=150{\\rm\n  J}\\approx9.36\\cdot10^{20}{\\rm eV}$, $\\tau_{\\rm pump}=150{\\rm\n  fs}\\approx228{\\rm eV}^{-1}$) as pump.\nThis is meant to give a first order of magnitude estimate of the\nnumber of induced outgoing photons.  Let us \nemphasize that it is certainly a rather crude approximation to adopt\nthe formula derived for a stationary, one-dimensional magnetic field\ninhomogeneity of Gaussian type~\\eqref{eq:Gauss} to mimic the field\ninhomogeneity as generated in the focal spot of a high-intensity\nlaser.  Such an approximation ignores the {\\it longitudinal modulation\n  and evolution} of the pump laser pulse.  A more rigorous and refined\ntreatment in the context of an all optical pump--probe experiment\nwould require us to account also for the temporal structure and\nevolution of field inhomogeneities. Fully accounting for pulse shape\ndependencies has become a subject of increasing importance in\nstrong-field phenomenology with high-intensity lasers. Progress has\nalready been made, for instance, for the case of vacuum birefringence\n\\cite{DiPiazza:2006pr,Dinu:2014tsa}.\n\nIn generic high-intensity laser experiments the focal spot area cannot\nbe chosen at will, but is limited by diffraction.\nAssuming Gaussian beams, the effective focus area is conventionally\ndefined to contain $86\\%$ of the beam energy ($1\/e^2$ criterion for\nthe intensity).  The minimum value of the beam diameter in the focus\nis given by twice the laser wavelength multiplied with $f^\\#$, the\nso-called $f$-number, defined as the ratio of the focal length and the\ndiameter of the focusing aperture \\cite{Siegman}; $f$-numbers as low\nas $f^\\#=1$ can be realized experimentally.  Thus, assuming both probe\nand pump lasers to be focused down to the diffraction limit, the\nattainable field strengths are of the order of\n\\begin{equation}\n \\mathfrak{E}^2=I_{\\rm probe}\\approx \\frac{0.86\\,{\\cal E}}{\\tau\\,\\sigma}\\,, \\quad B^2=2I_{\\rm pump}\\approx2\\,\\frac{0.86\\,{\\cal E}_{\\rm pump}}{\\tau_{\\rm pump}\\,\\sigma_{\\rm pump}}\\,,\n\\label{eq:EBpump}\n\\end{equation}\nwith $\\sigma\\approx\\pi\\lambda^2$ and $\\sigma_{\\rm pump}\\approx\\pi\\lambda_{\\rm pump}^2$. The additional factor of two in the definition of $B$ accounts for the fact that, focusing on a purely magnetic field inhomogeneity, the entire laser intensity is considered to be available in terms of a magnetic field, as could, e.g., be realized by superimposing two counter propagating laser beams.\n\nIn the most straightforward experimental setting to imagine, the pump laser beam propagates along the ${\\rm y}$ axis, while its transversal profile,\nparametrized by the coordinate ${\\rm x}$, evolves along the well-defined envelope of a Gaussian beam, and in the vicinity of the beam waist is to be understood as constituting the Gaussian field inhomogeneity~\\eqref{eq:Gauss} of width $w\\approx2\\lambda_{\\rm pump}$.\n\nFor beams focused down to the diffraction limit, the Rayleigh length \nis given by the wavelength of the beam multiplied with a factor of\n$\\pi$ \\cite{Siegman}, i.e., for the pump, $z_{\\rm R}=\\pi\\lambda_{\\rm\n  pump}$.  Hence, over distances of the order of several wave lengths\n$\\lambda_{\\rm pump}$ about the beam waist, the beam diameter remains\napproximately constant along $\\vec{e}_{\\rm y}$ and an experimental\nsetting resembling Fig.~\\ref{fig:perspective} is conceivable.\n\nIn Fig.~\\ref{fig:qualitative}, we plot the number of induced outgoing photons for both effects as a\nfunction of $\\theta$. The respective results are obtained\nstraightforwardly by plugging the design parameters of the Jena\nhigh-intensity laser systems JETI~200 and Polaris given above into\n\\Eqref{eq:EBpump} and the lower components of Eqs. \\eqref{eq:Np-Qref},\n\\eqref{eq:Np-linpol1} and \\eqref{eq:Nmax}. \n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.8\\textwidth]{140514merging_ratios_E=B_omegawofc_PolarisJETI} \n\\caption{Number of induced outgoing photons per shot ${\\cal N}_{\\rm\n    merg}$ due to the effects of laser photon merging and quantum\n  reflection as a function of $\\theta$, adopting the design parameters of the\n  Jena high-intensity laser systems, JETI~200 and Polaris (cf. main\n  text).  The horizontal dashed line shows where the number of induced\n  outgoing photons per shot becomes one.  For quantum reflection this\n  is the case for $\\theta\\geq78^\\circ$ \\cite{Gies:2013yxa}.\n  Conversely, the number of outgoing merged photons reaches a maximum\n  at $\\theta\\approx87^\\circ$ and stays below ${\\cal N}_{\\rm\n    merg}\\big|_{\\rm max}\\approx1.3\\cdot10^{-4}$ throughout the\n  interval $0\\leq\\theta\\leq90^\\circ$; cf. \\Eqref{eq:Nmax} for the\n  Gaussian inhomogeneity and the discussion below.  For completeness,\n  we note that ${\\cal N}_{\\rm\n    Qref}\\bigr|_{\\theta=0}\\approx3\\cdot10^{-29}$ while ${\\cal N}_{\\rm\n    merg}\\bigr|_{\\theta=0}\\approx3\\cdot10^{-228}$ .}\n\\label{fig:qualitative}\n\\end{figure}\n\nObviously, for this particular all-optical experimental setup the\nphoton merging process is substantially suppressed in comparison with\nquantum reflection.  As detailed below \\Eqref{eq:Np-linpol1}, the\ndifferences observed in Fig.~\\ref{fig:qualitative} can be attributed\nto the different scaling of Eqs. \\eqref{eq:Np-Qref} and\n\\eqref{eq:Np-linpol1} with $\\cos^2\\theta$.  While quantum reflection\nreceives an overall enhancement with $\\sim\\frac{1}{\\cos^2\\theta}$ for\nlarge angles of incidence $\\theta\\lesssim90^\\circ$, photon merging\nbecomes maximal if the condition~\\eqref{eq:Nmax} is met (for the\nJETI~200 -- Polaris setup this is the case for an angle of\n$\\theta\\approx 87^\\circ$, wherefore ${\\cal N}_{\\rm merg}\\big|_{\\rm\n  max}\\approx1.3\\cdot10^{-3}$) and dies off to zero for\n$\\theta\\to90^\\circ$.\n\nIn practice, an all-optical setup designed to benefit from the\ngeometric noise reduction will work at a reflection angle near or\nsomewhat above $\\theta \\simeq 80^\\circ$. For parameters similar to\nthe ones studied here, photon merging then is clearly a negligible\nbackground to the quantum reflection signal. Nevertheless, because\nof its different polarization and frequency dependence, appropriate\nfiltering techniques could still render photon merging detectable in the long run.\n\n\n\n\\section{Conclusions and Outlook} \\label{seq:Con+Out}\n\nIn this paper we have studied laser photon merging in the presence of\na one dimensional, stationary magnetic field inhomogeneity.  We have\nin particular confronted the number of outgoing merged photons with\nthe number of quantum reflected photons for the same conditions and\ndiscussed in detail the similarities and differences of the two\neffects.  Sticking to the design parameters of the high-intensity\nlaser facilities to be available in Jena, consisting of a petawatt and\na terawatt class laser system, we have provided a first rough estimate\nof the number of merged photons to be potentially attainable in an\nall-optical experiment. Our results confirm that the quantum\nreflection signal is a most promising candidate for the discovery of\nquantum vacuum nonlinearities under controlled laboratory conditions\nwith high-intensity lasers. In particular, it dominates photon\nmerging in a wide parameter range.\n\nThe expression for the photon merging number is determined most\nstraightforwardly from\nthe photon polarization tensor in a plane wave background.  Actually,\nthe main difficulty in determining the number of outgoing merged laser\nphotons is the problem of finding a convenient and controllable\nexpansion of the photon polarization tensor, allowing us to represent\nour results in concise expressions.  This has led us to adopt a novel\nexpansion strategy to obtain analytical insights into the photon\npolarization in plane wave backgrounds. We believe that this\nrepresentation will also be useful in many other strong field physics\nquestions beyond the merging process.\n\nOf course, a natural extension of our present study in the future\nwould be the investigation of the photon merging process in more\ngeneric, time-dependent inhomogeneities.  Such a study is necessary to\nallow for definitive answers about the the numbers of outgoing merged\nphotons attainable in the focal spot of high-intensity lasers, taking\ninto account the full longitudinal evolution of the pump laser pulse.\n\n\n\\section*{Acknowledgments}\n\nWe are particularly indebted to Maria~Reuter for creating\nFigs.~\\ref{fig:perspective} and \\ref{fig:yzSchnitt}.  FK is\ngrateful to Matt~Zepf for many interesting and enlightening discussions.\nHG acknowledges support by the DFG under grants Gi 328\/5-2 (Heisenberg\nprogram) and SFB-TR18. RS acknowledges support by the Ministry of Education\nand Science of the Republic of Kazakhstan.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nControl charts such as the Shewhart chart \\citep{Shewhart1931ECo} and\nthe cumulative sum (CUSUM) chart \\citep{Page1954CIS} have been\nvaluable tools in many areas, including reliability\n\\citep{OConnor2002Pre,Xie2002Sec}, medicine\n\\citep{Carey2003Ihw,Lawson2005Sas,Woodall2006Tuo} and finance\n\\citep{Frisen2008FS}.  See \\cite{Stoumbos2000SoS} and the special\nissues of ``Sequential Analysis'' (2007, Volume 26, Issues 2,3) for an\noverview.  Often, heterogeneity between observations is accounted for\nby using risk-adjusted charts based on fitted regression models\n\\citep{Grigg2004oor,Horvath2004Mci,Gandy2010ram}.\n\nA common convention in monitoring based on control charts is to assume\nthe probability distribution of in-control data to be known.  In\npractice this usually means that the distribution is estimated based\non a sample of in-control data and the estimation error is ignored.\nExamples of this are\n\\cite{Steiner2000Msp,Grigg2004oor,Bottle2008Iin,Biswas2008rCi,Fouladirad2008Otu,Sego2009Rmo,Gandy2010ram}.\n\nHowever, the estimation error has a profound effect on the performance\nof control charts. This has been mentioned at several places in the\nliterature, e.g.\\ \\cite{jones2004rld,Albers2004Esc,jensen2006epe,Stoumbos2000SoS,Champ2007PoM}.\n\nTo illustrate the effect of estimation, we consider a CUSUM chart\n\\citep{Page1954CIS} with normal observations and estimated in-control\nmean. We observe a stream of independent random variables\n$X_1,X_2,\\ldots$ which in control have an $ N(\\mu,1)$ distribution and\nout of control have an $N(\\mu+\\Delta,1)$ distribution, where $\\Delta>0$\nis the shift in the mean.  The chart\nswitches from the in-control state to the out-of-control state at an\nunknown time $\\kappa$.  The unknown in-control mean $\\mu$ is estimated\nby the average $\\hat\\mu$ of $n$ past in-control observations\n$X_{-n},\\dots,X_{-1}$ (this is often called phase 1 of the monitoring;\nthe running of the chart is called phase 2). We consider the  CUSUM chart\n$$S_t=\\max(0, S_{t-1}+X_t-\\hat \\mu - \\Delta\/2), \\quad S_0=0 $$\nwith hitting time $\\tau=\\inf\\{t>0: S_t\\geq c\\}$ for some threshold\n$c>0$.\n\n\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.95\\linewidth]{simpaper\/estimerr_guaranteed_CUSUM.pdf}\n  \\caption{In-control distribution of ARL=$\\E(\\tau|\\hat\\mu)$ for\n    CUSUMs for standard normally distributed data.  The mean $\\hat\n    \\mu$ used in the monitoring is estimated based on $n$ past\n    observations.  The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%,\n    75\\%, 90\\% and 97.5\\% quantiles.The top part of the plot shows\n    the situation when estimation error is ignored. In the middle part  the\n    threshold has been chosen to give an unconditional ARL of 100\n    (averaging out the parameter estimation). In the bottom part the threshold is\n    adjusted to guarantee  with 90\\% probability an in-control ARL of at least 100.  }\n  \\label{fig:estimerr}\n\\end{figure}\n\nThe in-control average run length,\n$\\ARL=\\E(\\tau|\\hat\\mu,\\kappa=\\infty)$, depends on $\\hat \\mu$ and is\nthus a random quantity. The top part of the plot in\nFigure~\\ref{fig:estimerr} shows boxplots of its distributions with\nthreshold $c=2.84$, $\\Delta=1$ and various numbers of past\nobservations. If  $\\hat \\mu=\\mu$, i.e.\\ $\\mu$ was know,  this\nwould give an in-control $\\ARL$ of 100. The estimation error is having a substantial effect on\nthe attained $\\ARL$ even for  large samples such as\n$n=1000$.\nFor further illustrations of the impact of estimation error see\n\\cite{jones2004rld} for CUSUM charts and  \\cite{Albers2004Esc} for\nShewhart charts.\n\n\nSo far, no general approach for taking the estimation error into\naccount has been developed, but there are many special constructions\nfor specific situations.  For instance, for some charts so called\nself-starting charts\n\\citep{Hawkins1987SCC,Hawkins1998csc,Sullivan2002SCC}, maximum\nlikelihood surveillance statistics\n to eliminate parameters\n\\cite[e.g.][]{Frisen2009}, correction\nfactors for thresholds\n\\citep{Albers2004Esc,jones2002statistical},\nmodified thresholds \\citep{Zhang2011TSX} and threshold\nfunctions \\citep{Horvath2004Mci,Aue2006Cpm} have been\ndeveloped. Various bootstrap schemes for specific situations have also\nbeen suggested, see for instance\n\\cite{Kirch2008BSC,Chatterjee2009Dcs,Capizzi2009Bdo,Huskova10Bsc}.\nFurther, some nonparametric charts which account for the estimation\nerror in past data have been proposed, see \\cite{Chakraborti2007Ncc}\nand references therein.   Recently some modified charts for monitoring\nvariance in the normal distribution with estimated parameters have\nbeen suggested by \\cite{Maravelakis2009AEC}  and\n\\cite{Castagliola2011ACC}.\n\nWhen addressing estimation error, the above methods mainly  focus\non the performance of the charts averaged over both the estimation of\nthe in-control state as well as  running  the chart once.\nIn the middle part of Figure~\\ref{fig:estimerr}, the threshold has\nbeen chosen such that,  averaged over both the estimation of\nthe in-control state as well as  running  the chart once, the average\nrun length is $100$ (this results in a different threshold\nfor each $n$). It turns out that only a small change in the threshold\nis needed and that the distribution of the conditional $\\ARL=\\E(\\tau|\\hat\n\\mu)$ is only changed slightly. This bias correction\nfor the $\\ARL$ actually goes in the wrong direction in the sense that\nit implies more short $\\ARL$s. This is due to the $\\ARL$ being\nsubstantially influenced by the right tail of the run length\ndistribution, see the discussion in Section 2 of \\cite{Albers2006SAC}.\n\n\n\nHowever, usually, after the chart parameters are estimated, the\nchart is run for some time without any reestimation of the in-control\nstate even if the chart signals. Moreover, in some situations, several\ncharts are run based on the same estimated parameters.\nIn these situations the ARL conditional on the estimated in-control\nstate is more relevant than the unconditional ARL. In the middle and\nupper part of Figure \\ref{fig:estimerr}, one sees that the conditional\nARL can be much lower than 100, meaning that both the unadjusted\nthreshold and the\n threshold adjusted for bias in the unconditional ARL\nlead, with a substantial probability, to  charts\nthat have a considerably decreased time until false alarms.\n\n\nTo overcome these problems we will look at the performance of the\nchart conditional on the estimated in-control distribution, averaging\nonly over different runs of the chart. This will lead to the\nconstruction of charts that with high probability have an in-control\ndistribution with desired properties conditional on the observed past\ndata, thus reducing the situations in which there are many false\nalarms due to estimation error.\n\n\nThe bottom part of Figure \\ref{fig:estimerr}  shows the distribution of the in control\nARL when   the threshold for each set of past data is adjusted to\nguarantee an in-control ARL of at least 100 with probability 90\\%. The adjustment is\ncalculated using a bootstrap procedure explained later in the paper.\nThe adjustment succeeds to avoid the too low ARLs with the\nprescribed probability, and we will see later that the cost in a\nhigher out-of-control ARL is modest.  Using hitting probabilities instead of ARL as\ncriterion leads to similar results.\n\n\\xx{talk about adjustments}\n\n\nOur approach is similar in spirit to the exceedance probability\nconcept developed by Albers and Kallenberg for various types of\nShewhart \\citep{Albers2004AEC,Albers2005Ncf,Albers2005EPF} and\nnegative binomial charts \\citep{Albers2009CUM,Albers2010Toc}. They\ncalculate approximate adjusted thresholds such that there is only a\nsmall prescribed probability that some performance measure, for\ninstance an ARL, will be a certain amount below or above a specified\ntarget.\n\n\nThe main difference between their approach and what we present is that\nour approach applies far more widely, to many different types of\ncharts and without having to derive specific approximation formulas in\neach setting. If we apply a nonparametric bootstrap, the\nproposed procedure will be robust against model misspecification.  In\naddition to that, our approach allows not only to adjust the threshold\nbut also to give a confidence interval for the in-control performance\nof a chart for a fixed threshold.  Lastly, even though not strongly\nadvocated in this paper, the bootstrap procedure we propose can also\nbe used to do a bias correction for the unconditional performance of\nthe chart, as in the middle part of Figure \\ref{fig:estimerr}.\n\n\n\nNext, we describe our approach more formally.\nSuppose we want to use a monitoring scheme and that the in-control\ndistribution $P$ of the observations is unknown, but that\nbased on past in-control behaviour we have an estimate $\\hat P$ of the\nin-control distribution.  Let $q$ denote the in-control property of\nthe chart we want to compute, such as the $\\ARL$, the false alarm\nprobability or the threshold needed for a certain $\\ARL$ or false alarm\nprobability. In the above example we were interested to find a\nthreshold such that the in-control ARL is 100.\n\nGenerally, $q$ may depend on both the true in-control distribution $P$\nand on estimated parameters of this distribution which for many charts\nare needed to run the chart. We denote these parameters by $\\hat\n\\xi=\\xi(\\hat P)$.\nIn the above CUSUM chart example $\\hat \\xi = \\hat\n\\mu$.  We are interested in $q(P;\\hat\\xi)$, that\nis the in-control performance of the chart conditional on the\nestimated parameter. In the above CUSUM example,  $q(P;\\hat\\xi)$ is the threshold\nneeded to give an $\\ARL$ of 100 if the observations are from the true\nin-control distribution  $P$ and the\nestimated parameter $\\hat \\mu$ is used. As $P$ is not observed $q(P;\\hat\n\\xi)$ is not observable.  As mentioned above, many papers pretend\nthat the estimated in-control distribution $\\hat P$ equals the true\nin-control distribution $P$ and thus use $ q(\\hat P;\\hat \\xi)$.\nOur suggestion is to use bootstrapping of past data to construct an\napproximate one-sided confidence intervals for $q(P;\\hat \\xi)$. From\nthis we get a  guaranteed conditional performance of the control\nscheme.\n\n\n\n\nIn Section~\\ref{sec:monitorhomobs} we present the general idea in the\nsetting with homogeneous observations, and discuss this for Shewhart\nand CUSUM charts.  The main theoretical results are presented in\nSection~\\ref{sec:gentheor}, with most of the proofs given in the\nAppendix.  Section \\ref{sec:simulsingle} contains simulations\nillustrating the performance of charts for homogeneous observations.\nIn Section~\\ref{sec:regmod} extensions to charts based on regression\nand survival analysis models are presented. Some concluding comments\nare given in Section~\\ref{sec:conclusion}.  The suggested methods are\nimplemented in a flexible R-package, that will be made available on the\nComprehensive R Archive Network (CRAN).\n\n\n\n\\section{Monitoring homogeneous observations}\n\\label{sec:monitorhomobs}\n\\subsection{General idea}\n\\label{subsec:generalidea}\n\n\nSuppose that in control we have independent observations\n$X_1,X_2,\\dots$ following an unknown distribution $P$.  We want to use\nsome monitoring scheme\/control chart that detects when $X_{i}$ is no\nlonger coming from $P$. The particular examples we discuss in\nthis paper are Shewhart and CUSUM charts, but the methodology we\nsuggest applies more widely.\n\nTo run the charts, one often needs certain parameters $\\xi$. For\nexample, in the CUSUM control chart of the introduction, we\nneeded $\n\\xi= \\mu$, the assumed in-control mean.  These parameters will usually\nbe estimated.\n\nLet $\\tau$ denote the time at which the chart signals a change. As $\\tau$ may\ndepend on $\\xi$, we sometimes write $\\tau(\\xi)$.  The\ncharts we consider use a threshold $c$, which determines how quickly\nthe chart signals (larger $c$ lead to a later signal).\n\nThe performance of such a control chart with the in-control\ndistribution $P$ and the parameters $ \\xi$ can, for example, be\nexpressed as one of the following.\n\\begin{itemize}%\n\\setlength{\\itemsep}{0pt}%\n\\setlength{\\parskip}{0pt}%\n\\item $\\ARL(P;\\xi)=\\E(\\tau( \\xi))$, where $\\E$ is the expectation with respect to $P$.\n\\item ${\\hit}(P;\\xi)=\\Prob(\\tau( \\xi)\\leq T)$ for\n  some finite $T>0$, where $\\Prob$ is the probability measure under which $X_1,X_2,\\dots\\sim P$. This is the false alarm probability in $T$ time\n  units.\\xx{do we want to distinguish between $\\Prob$ and $P$?}\n\\item $c_{\\ARL}(P;\\xi)=\\inf\\{c>0:\\ARL(P;\\xi)\\geq\\gamma\\}$ for some\n  $\\gamma>0$. Assuming appropriate continuity, this is the threshold\n  needed to give an in-control average run length of $\\gamma$.\n\\item $c_{\\hit}(P;\\xi)=\\inf\\{c>0:\\hit(P;\\xi)\\leq\\beta\\}$ for\n  some $0<\\beta<1$. This is the threshold needed\n  to give a false alarm probability of $\\beta$.\n\\end{itemize}\nThe latter two quantities are very important in practice, as they are\nneeded to decide which threshold to use to run a chart.  In the\nnotation we have suppressed the dependence of the quantities on $c$,\n$T$, $\\gamma$, $\\beta$ and $\\Delta$.\n\nIn the following, $q$ will denote one of $\\ARL$, $\\hit$, $c_{\\ARL}$\nor $c_{\\hit}$, or simple transformations such as $\\log(\\ARL)$,\n$\\logit(\\hit)$, $\\log(c_{\\ARL})$ and $\\log(c_{\\hit})$, where\n$\\logit(x)=\\log\\left(\\frac{x}{1-x}\\right)$.\n\nThe true in-control distribution $P$ and the parameters $\\xi=\\xi(P)$\nneeded to run the chart are usually estimated.  We assume that we have\npast in-control observations $X_{-n},\\dots,X_{-1}$ (independent of $X_1,X_2,\\dots$), which\nwe use to  estimate the in-control distribution $P$ parametrically or\nnon-parametrically. We denote this estimate by $\\hat P$.  The estimate\nof $\\xi$ will be denoted by $\\hat \\xi=\\xi(\\hat P)$. For example, in\nthe CUSUM control chart of the introduction, $\\hat \\xi=\\hat \\mu$ is\nthe estimated in-control mean.\n\nThe observed performance of the chart will depend on the true\nin-control distribution $P$ as well as on the estimated parameters\n$\\hat \\xi$ that are used to run the chart.  Thus we are interested in\n$q(P;\\hat\\xi)$, the performance of the control chart\n\\emph{conditional} on $\\hat \\xi$.  This is an unknown quantity as $P$\nis not known.  Based on the estimator $q(\\hat P;\\hat\\xi)$, we\nconstruct a one-sided confidence interval for this quantity to\nguarantee, with high probability, a certain performance for the\nchart. We choose to call the interval a  confidence interval,\neven though the quantity $q(P;\\hat\\xi)$ is random.\n\nWe suggest the following for guaranteeing an upper bound on $q$ (which\nis relevant for  $q=\\hit$, $q=c_{\\ARL}$ or $q=c_{\\hit}$).  For $\\alpha\\in (0,1)$,\nlet $p_\\alpha$ be a constant such that\n$$\n\\Prob(q(\\hat P;\\hat\\xi) - q(P;\\hat\\xi) >p_\\alpha)=1-\\alpha,\n$$\nassuming that such a $p_{\\alpha}$ exists.\nHence,\n$$\n\\Prob(q(P;\\hat\\xi)< q(\\hat P;\\hat\\xi)-p_{\\alpha})= 1-\\alpha.\n$$\nThus $(-\\infty,q(\\hat P;\\hat\\xi)-p_{\\alpha})$ could be considered an\nexact lower one-sided confidence interval of $q(P;\\hat\\xi)$.\n\\xx{Or,\n tolerance interval?}\n\\cc{This is not really a confidence interval in the classical sense -\n $q( P;\\hat\\xi)$ is an\n  unobserved random quantity... and not just a fixed parameter.\nThe article  Weerahandi (1993, JASA) ``Generalized Confidence Intervals'' might be a useful reference. }\n\nOf course, $p_{\\alpha}$ is unknown.  We suggest to obtain an\napproximation of $p_{\\alpha}$ via bootstrapping.  In the following,\n$\\hat P^\\ast$ denotes a parametric or non-parametric bootstrap\nreplicate of the estimated in-control distribution $\\hat P$.\nWe can approximate $p_\\alpha$ by $p^\\ast_\\alpha$ such that\n$$\\Prob(q(\\hat P^\\ast;\\hat\\xi^\\ast)-q(\\hat P;\\hat\\xi^\\ast)> p^\\ast_\\alpha|\\hat P)=1-\\alpha.$$\n\\cc{alternatively we could use $\\hat P(q(\\hat P^\\ast;\\hat\\xi^\\ast)-q(\\hat P;\\hat\\xi^\\ast)> p^\\ast_\\alpha)=1-\\alpha.$\n}\nThus\n\\begin{equation}\n  \\label{eq:onesidedapproxconfint}\n (-\\infty,q(\\hat P;\\hat\\xi)-p^\\ast_\\alpha)\n\\end{equation}\nis a one-sided (approximate) confidence interval for $q(P;\\hat\\xi)$.\nIn this paper, we will use the following generic algorithm to\nimplement the bootstrap.\n\\begin{algorithm}[Bootstrap]\n  \\label{alg:Bootstrap}\n\\hspace*{2mm}\\\\[-7mm]\n\\begin{enumerate}%\n\\setlength{\\itemsep}{0pt}%\n\\setlength{\\parskip}{0pt}%\n\\item From the past data $X_{-n},\\dots,X_{-1}$, estimate\n  $\\hat P$ and  $\\hat\\xi$.\n\\item Generate bootstrap samples $X^{\\ast}_{-n},\\dots,X^{\\ast}_{-1}$\n  from $\\hat P$.  Compute the corresponding estimate $\\hat P^{\\ast}$\n  and $\\hat \\xi^{\\ast}$. Repeat $B$ times to get $\\hat\n  P^{\\ast}_1,\\dots,\\hat P^{\\ast}_B$ and $\\hat \\xi^{\\ast}_1,\\dots,\\hat\n  \\xi^{\\ast}_B$.\n\\item Let $p_{\\alpha}^{\\ast}$ be the $1-\\alpha$ empirical quantile of\n  $q(\\hat P^{\\ast}_b;\\hat\\xi^{\\ast}_b)-q(\\hat P;\\hat\\xi^{\\ast}_b)$, $b=1,\\dots,B$.\n\\end{enumerate}\n\\end{algorithm}\n\n\n\n\nFor guaranteeing a lower bound on $q$, which is for example relevant\nfor $q=\\ARL$, a similar upper one-sided confidence interval can be\nconstructed.\n\n\n\n\nIn a practical situation, the focus would be on deciding which\nthreshold to use for the control chart to obtain desired in-control\nproperties. We suggest to use either $q=c_{\\ARL}$ or $q=c_{\\hit}$, or\nlog transforms of these, and\nthen run the chart with the adjusted threshold\n\\begin{equation}\n  \\label{eq:adjThreshold}\nq(\\hat P;\\hat \\xi)-p^{\\ast}_{\\alpha}.\n\\end{equation}\nThis will guarantee that in (approximately) $1-\\alpha$ of the\napplications of this method, the control chart actually has the\ndesired in-control properties.\n\n\n\\cc{The following are some comments which are probably not quite relevant to practice.\nIn some application of control charts the chart parameters $\\xi$ are\nnot estimated but determined according to certain specifications the\nprocess should meet. A typical example would be industrial\napplications like monitoring of properties of mass produced units\nwhere there are precise specification of physical properties of the\nunits which should be monitored. Then $\\xi$ may be determined\naccording to these specifications, and the point of the monitoring is\nto detect deviations from the specifications.  However, the full\nin-control distribution $P$ would usually still be unknown and our\napproach would still apply for constructing confidence intervals for\n$q(P;\\xi_s)$ where $\\xi_s$ denotes a specified $\\xi$.\n\n\nWould this in practice be relevant? Or would one also specify $P$,\n  or at least parts of $P$ like the  mean?  Could using our approach\n  here e.g.\\ lead to picking a far too   large $c$ to get a guaranteed\n  ARL if $P$ actually is far off from   where it ``should be''?\n}\n\n\n\n\\subsection{Specific charts}\n\n\\subsubsection{Shewhart charts}\n\\label{subsubsec:Shewhart}\n\nThe one-sided Shewhart chart \\citep{Shewhart1931ECo} signals at\n$$\n\\tau=\\inf\\{t \\in \\{1,2,\\dots\\}: f(X_t,\\xi)>c\\}\n$$\nfor some threshold $c$, where  $f$ is some function, $X_t$ is the\nobservation at time $t$ and $\\xi$ are\nsome parameters.\n $X_t$ can be a single\nmeasurement or e.g.\\ the average, range or standard deviation of a\nspecified number of measurements, or some other statistic like a\nproportion.\nIt is common to use a Shewhart chart with a threshold of\nthe mean plus 3 times the standard deviation, in\nthis case one would use $c=3$ and   $f(x,\\xi)=\\frac{x-\\xi_1}{\\xi_2}$\nwith $\\xi_1$ being the mean and $\\xi_2$ being the standard deviation.\n For two-sided charts one could just use  $f(x,\\xi)=\\frac{|x-\\xi_1|}{\\xi_2}$.\n\nConditionally on fixed parameters  $ \\xi$, the stopping time $\\tau$ follows a\ngeometric distribution with parameter\n$p=p(c;P,\\xi)=\\Prob(f(X_t,\\xi)>c)$.\nThen the performance measures mentioned in the previous section simplify to\n\\begin{align*}\n\\ARL(P;\\xi)=&\\frac{1}{p(c;P,\\xi)},& \\hit(P;\\xi)=&1-(1-p(c;P,\\xi))^T,\\\\\n c_{\\ARL}(P;\\xi)=&p^{-1}\\left( \\frac{1}{\\gamma}\n   ;P,\\xi\\right)\n\\;\\;\\;\\;\\;\\;\n\\text{ and}&\n c_{\\hit}(P;\\xi)=&p^{-1}\\left( 1-(1-\n    \\beta)^{\\frac{1}{T}};P,\\xi\\right),\n\\end{align*}\n where $p^{-1}(\\cdot;P,\\xi)$ is the inverse of $p(\\cdot;P,\\xi)$.\n\\cc{To get the formula for $c_{\\hit}$ set\n    $\\beta=\\hit$ in the second item and solve for $c$.}\n\n\nSuppose that the in-control distribution comes from a parametric\nfamily $P_{\\theta}, \\theta\\in \\Theta$.  Furthermore, suppose that we\nhave some way of computing an estimate $\\hat\\theta$ of $\\theta$ based on the\nsample.\nThen we can use  Algorithm \\ref{alg:Bootstrap} with $\\hat P=P_{\\hat \\theta}$ to compute a\nconfidence interval as given by (\\ref{eq:onesidedapproxconfint}).\n\nShewhart charts depend heavily on the tail behaviour of the\ndistribution of the observations. This is particularly problematic\nwhen the sample size is small and we use non-parametric methods or a\nsimple non-parametric bootstrap. We thus primarily suggest to use a\nparametric bootstrap for Shewhart charts.\n\n\n\\begin{remark}\nIn certain cases the parametric bootstrap will actually be exact when\n$B \\to \\infty$. This happens when the distribution of\n$q(P_{\\hat\\theta};\\hat\\xi) - q(P_{\\theta};\\hat\\xi)$ under $P_{\\theta}$\ndoes not depend on $\\theta$. In particular, this implies that\n$q(P_{\\hat\\theta^\\ast};\\hat\\xi^\\ast) - q(P_{\\hat\\theta};\\hat\\xi^\\ast)$\nhas the same distribution and $p^\\ast_\\alpha\\to p_\\alpha$ as $B\\to\n\\infty$.\n\n As an example, consider the case when\n $f(x,\\xi)=\\frac{x-\\xi_1}{\\xi_2}$ and $X_t$ follows an\n $N(\\xi_1,\\xi_2^2)$ distribution and $q$ is any of the performance\n measures described above. We use $\\theta=\\xi$ and as estimator $\\hat\n \\xi_1$ we use the sample mean and as estimator $\\hat\\xi_2$ we use the\n sample standard deviation. Then\n\\begin{align*}\np(c;P_{\\!\\xi},\\hat\\xi)\n=\\Prob_{\\!\\xi}\\!\\left(\\frac{X_t-\\hat\\xi_1}{\\hat\\xi_2}> c\\right)\n=1-\\Phi\\left(\\frac{c\\hat\\xi_2+\\hat\\xi_1-\\xi_1}{\\xi_2}\\right),\n\\end{align*}\nwhere $\\Phi$ is the cdf of the standard normal distribution, and\nunder $P_\\xi$,\n$$\n\\frac{c\\hat\\xi_2+\\hat\\xi_1-\\xi_1}{\\xi_2}=c\\frac{\\hat\\xi_2}{\\xi_2}+\\frac{\\hat\\xi_1-\\xi_1}{\\xi_2}\\sim\\frac{c}{\\sqrt{n-1}}\\sqrt{W}+\\frac{1}{\\sqrt{n}}Z,\n$$\nwhere $W\\sim\\chi_{n-1}^2$ and $Z\\sim N(0,1)$ are independent. Thus the\ndistribution of $p(c;P_\\xi,\\hat\\xi)$, and hence $q(P_{\\xi};\\hat\\xi)$,\nis completely known.  As $\np(c;P_{\\!\\hat\\xi},\\hat\\xi)=\\Prob_{\\!\\hat\\xi}\\left(\\frac{X_t-\\hat\\xi_1}{\\hat\\xi_2}>\n  c\\right) =1-\\Phi(c)$, and thus $q(P_{\\!\\hat\\xi};\\hat\\xi)$, is not\nrandom, the distribution of $q(P_{\\!\\hat\\xi};\\hat\\xi) -\nq(P_{\\xi};\\hat\\xi)$ also does not depend on any unknown\nparameters. Thus the parametric bootstrap is exact in this example.\n\\end{remark}\n\n\n\\subsubsection{CUSUM charts}\n\\label{subsubsec:CUSUM}\nThis section considers the one-sided CUSUM chart \\citep{Page1954CIS}.\nThe classical CUSUM chart was designed to detect a shift of size\n$\\Delta>0$ in the mean of normally distributed observations. Let $\\mu$ and $\\sigma$\ndenote, respectively, the in-control mean and standard\ndeviation. A CUSUM chart can be defined by\n\\begin{equation}\n  \\label{eq:discrCUSUM_meanshift}\nS_t=\\max(0, S_{t-1}+(X_t-\\mu - \\Delta\/2)\/\\sigma), \\quad S_0=0\n\\end{equation}\nwith hitting time $\\tau=\\inf\\{t>0: S_t\\geq c\\}$ for some threshold\n$c>0$.\n\nAlternatively, we could drop the scaling and not divide by the\n standard deviation $\\sigma$ in\n(\\ref{eq:discrCUSUM_meanshift}). See Chapter 1.4 in\n\\cite{Hawkins1998csc} for a discussion on scaled versus unscaled\nCUSUMs.\n\n\nMore generally, to accommodate observations with general in-control distribution with\ndensity $f_0$ and general out-of-control distribution with density $f_{1}$, it\nis optimal in a certain sense \\citep{Moustakides1986OST} to modify the\nCUSUM chart by replacing $(X_t-\\mu - \\Delta\/2)\/\\sigma$ by the log\nlikelihood ratio $\\log(f_1(X_t,\\theta)\/f_0(X_t, \\theta))$ such\nthat the CUSUM chart is\n\\begin{equation}\n  \\label{eq:discrCUSUM_loglikelihood}\nS_t=\\max(0, S_{t-1}+\\log(f_1(X_t,\\theta)\/f_0(X_t, \\theta))), \\quad S_0=0.\n\\end{equation}\n\nLet $\\xi$ denote either $(\\mu,\\sigma)$ in\n(\\ref{eq:discrCUSUM_meanshift}) or $\\theta$ in\n(\\ref{eq:discrCUSUM_loglikelihood}).  Usually, $\\xi$ needs to be\nestimated from past data, and we can then use Algorithm\n\\ref{alg:Bootstrap} to compute a confidence interval\n(\\ref{eq:onesidedapproxconfint}) for the performance measure\n$q(P;\\hat\\xi)$. For (\\ref{eq:discrCUSUM_loglikelihood}) it is most\nnatural to use a parametric bootstrap with $\\hat P=P_{\\hat \\theta}$,\nwhile for (\\ref{eq:discrCUSUM_meanshift}) we can use either a\nparametric or a nonparametric bootstrap. In the latter case we let\n$\\hat P$ be the empirical distribution of $X_{-n},\\dots,X_{-1}$, i.e.\nin Algorithm \\ref{alg:Bootstrap}, $X^{\\ast}_{-n},\\dots,X^{\\ast}_{-1}$\nare sampled with replacement from $X_{-n},\\dots,X_{-1}$.\n\n\n\\begin{remark}\nSimilar as for Shewhart charts, this parametric bootstrap is exact\nwhen the distribution of\n$q(P_{\\hat\\theta};\\hat\\xi)-q(P_{\\theta};\\hat\\xi)$ does not have any\nunknown parameters.  This is, for instance, the case if we use\n(\\ref{eq:discrCUSUM_loglikelihood}) for an exponential distribution\nwith the out-of-control distribution specified as an exponential\ndistribution with mean $\\Delta\\lambda$, where $\\lambda$ is the\nin-control mean. Another example of this is when we have normally\ndistributed data and use a CUSUM with the increments\n$(X_t-\\hat\\mu)\/\\hat\\sigma-\\Delta\/2$.\n\\end{remark}\n\n\n\n\n\n\\section{General theory}\n\\label{sec:gentheor}\n\nIn this section, we show that asymptotically, as the number of past\nobservations $n$ increases, our procedure works.  An established way\nof showing asymptotic properties of bootstrap procedures is via a\nfunctional delta method \\citep{Vaart1996WCa,Kosorok2008ItE}. Whilst we\nwill follow a similar route, our problem does not fit directly into\nthe standard framework, because the quantity of interest, $q(P,\\hat\n\\xi)$, contains the random variable $\\hat \\xi$.\nWe present the setup and\nthe main result in Section \\ref{sec:th:main}, followed by examples\n(Section \\ref{sec:th:examples}).\n\\cc{The asymptotic development in this section only show that things\n  do not go badly wrong as $n\\to \\infty$. They only establish\n  consistency of the correction\/confidence intervals.  However, the need to use\n  these confidence intervals disappears as $n$ increases.}\n\n\n\\subsection{Main theorem}\n\\label{sec:th:main}\nLet $D_q$ be the set in which  $P$ and its\nestimator $\\hat P$ lie, i.e.\\ a set describing the potential\nprobability distribution of our observations. This could be a subset\nof $\\mathbb{R}^d$ for parametric distributions, the set of cumulative\ndistribution functions for non-parametric situations, or the set of\njoint distributions of covariates and observations.  We assume that\n$D_q$ is a subset of a complete normed vector space $D$. \\cc{Do we\n  want to \/need to assume that $D$ is complete (every Cauchy sequence\n  converges)?  This should not be a problem as $R^k$ and\n  $l_{\\infty}(\\mathbb{R})$ are complete metric spaces. } Let $\\Xi$ be a\nnon-empty topological space containing the potential parameters $\\xi$\nused for running the chart. In our examples, we will let $\\Xi\\subset\n\\mathbb{R}^d$ be an open set.\n\nWe assume that $\\hat P^{\\ast}=\\hat P^{\\ast}(\\hat P, W_n)$ is a\nbootstrapped version of $\\hat P$ based both on the observed data $\\hat P$ and\non an independent random vector $W_n$.  For example, when resampling\nwith replacement then $W_n$ is a weight vector of length $n$,\nmultinomially distributed, that determines how often a given\nobservation is resampled.  In a parametric bootstrap, $W_n$ is the\nvector of random variables needed to generate observations from the\nestimated parametric distribution.\n\n\nIn the main theorem we will need that the mapping $q:D_q\\times\n\\Xi\\to\\mathbb{R}$, which returns the property of the chart we are interested\nin, satisfies the following extension of Hadamard differentiability.\nFor the usual definition of Hadamard differentiability see e.g.\\\n\\citep[Section 20.2]{Vaart1998AS}. The extension essentially consists in\nrequiring Hadamard differentiability in the first component when the second\ncomponent is converging.\n\\begin{definition}\n\\label{def:haddiffamily}\nLet $D,E$ be metric spaces, let $D_f\\subset D$ and let $\\Xi$ be a\nnon-empty topological space. \\cc{We need at least to be able to speak\n  about convergence in $\\Xi$.}  The family of functions\n$\\{f(\\cdot;\\xi):D_f \\to E: \\xi\\in \\Xi\\}$ is called \\emph{Hadamard\n  differentiable at $\\theta\\in D_f$ around $\\xi \\in \\Xi$ tangentially\n  to $D_0\\subset D$}  if there exists a continuous linear map \\cc{this is a requirement that also appears in the original definition and which we may be using in our proofs; however, we never prove for our derivatives that they are continuous and linear}\n$f'(\\theta;\\xi):D_0\\to E$ such that\n$$\n\\frac{f(\\theta+t_nh_n;\\xi_n)-f(\\theta;\\xi_n)}{t_n}\\to\nf'(\\theta;\\xi)(h)\\quad(n\\to\\infty)\n$$\nfor all sequences $(\\xi_n)\\subset \\Xi$, $(t_n)\\subset \\mathbb{R}$, $(h_n)\\subset D$\nthat satisfy $\\theta+t_nh_n\\in D_f \\,\\forall n$ and $\\xi_n\\to \\xi$, $t_n\\to 0$, $h_n\\to h\\in D_0$ as $n\\to \\infty$.\n\\end{definition}\n\n\n\nIn the following theorem we understand convergence in distribution, denoted by $\\leadsto$, as defined\nin \\citet[Def 1.3.3]{Vaart1996WCa} or in \\citet[p.108]{Kosorok2008ItE}.\n\\cc{ Let\n  $(\\Omega_n, {\\cal A}_n, P_n)$ be a sequence of probability spaces,\n  let $(\\Omega, \\cal A, P)$ be a further probability space, let $D$ be\n  a metric space and let $X_n:\\Omega_n\\to D$ be a sequence of maps and\n  let $X:\\Omega\\to D$ be a Borel measurable map. Then $X_n\\leadsto X$\n  if $\\E^{\\ast}f(X_n) \\to \\E f(X)$ for all continuous, bounded $f:D\\to\n  \\mathbb{R}$.  }\n\\cc{Outer expectation is defined in \\cite{Vaart1996WCa} and\n  in \\cite{Kosorok2008ItE}, essentially $\\E^{\\ast}X = \\inf \\{\\E Y:\n  Y\\geq X, Y \\text{ measurable}\\}$.}\n\\begin{theorem}\n\\label{th:main}\n  Let $q:D_q\\times \\Xi\\to \\mathbb{R}$ be a mapping, let $P\\in D_q$ and let\n  $\\xi:D_q\\to \\Xi$ be a continuous function.\n  Suppose that the following conditions are satisfied.\n\\begin{itemize}%\n\\setlength{\\itemsep}{0pt}%\n\\setlength{\\parskip}{0pt}%\n\\item[a)] $q$ is Hadamard differentiable at $P$ around $\\xi$ tangentially to $D_0$ for some $D_0\\subset D$.\n\\item[b)] $\\hat P$ is a sequence of random elements in $D_q$ such that\n$\n  \\sqrt{n}(\\hat P-P)\\leadsto Z\n$ as $n\\to \\infty$\nwhere $Z$ is some tight random element in $D_0$.\n\\item[c)]\n$\\sqrt{n}(\\hat P^{\\ast}-\\hat P)\\condweakconv{\\hat P} Z$ as $n\\to\\infty$\nwhere $\\condweakconv{\\hat P}$ denotes weak convergence conditionally on $\\hat\nP$ in probability as defined in \\citet[p.19]{Kosorok2008ItE}.  \\cc{\n  i.e. $\\sup_{h\\in \\text{BL}_1}|E_Wh(\\hat X_n) - E\n  h(X)|\\stackrel{P}{\\to}0$ and $E_Wh(\\hat X_n)^{\\ast}-E_Wh(\\hat\n  X_n)_{\\ast}\\stackrel{P}{\\to}0$ for all $f\\in \\text{BL}_1$ where the\n  subscript $W$ denotes conditional expectation over the weights given\n  the remaining data.  }\n\\item[d)] The cumulative distribution function  of  $q'(P;\\xi)Z$ is continuous.\n\\item[e)] Outer-almost surely, the map $W_n\\mapsto h(\\hat P^{\\ast}(\\hat P, W_n))$ is measurable for each $n$ and for every continuous bounded function $h:D_q\\to \\mathbb{R}$.\n\\item[f)] $q(\\hat P; \\hat \\xi)-q(P;\\hat \\xi)$ and  $p_\\alpha^{\\ast}$ are  random variables, i.e.\\ measurable,\nwhere $\\hat \\xi =\\xi(\\hat P)$ and $p^{\\ast}_{\\alpha}=\\inf\\{t\\in \\mathbb{R}:\n\\hat \\Prob(q(\\hat P^\\ast;\\hat\\xi^\\ast)-q(\\hat P;\\hat\\xi^\\ast)\\leq t)\\geq\n\\alpha\\}$.\n\\end{itemize}\nThen\n\\begin{equation*}\n\\Prob(q(P;\\hat\\xi)\\in (-\\infty,  q(\\hat P;\\hat\\xi)-p^{\\ast}_{\\alpha}))\\to  1-\\alpha \\quad (n\\to \\infty).\n\\end{equation*}\n\\end{theorem}\nA similar result  holds for upper confidence intervals.\n\nThe proof is in Appendix \\ref{sec:proof}.  The theorem essentially is\nan extension of the delta-method.  Condition a) ensures the necessary\ndifferentiability.  Conditions b) and c) are standard assumptions for\nthe functional delta method; b) for the ordinary delta method and c)\nfor the bootstrap version of it. Condition d) ensures that, after\nusing an extension of the delta-method, the resulting confidence\ninterval will have the correct asymptotic coverage probability.\nCondition e) is a technical measurability condition, which will be\nsatisfied in our examples. Condition f) is a measurability condition,\nwhich should usually be satisfied.\n\n\n\\subsection{Examples}\n\\label{sec:th:examples}\nThe following sections give examples in which Theorem \\ref{th:main}\napplies.  We consider hitting probabilities ($q=\\hit$) and thresholds to\nobtain certain hitting probabilities ($q=c_{\\hit}$).\n\nThese examples are\nmeant to be illustrative rather than exhaustive. For example, other\nparametric setups could be considered along similar lines to Section\n\\ref{sec:cusum-charts-with}. Furthermore, other performance measures such as\n$\\log(c_{\\hit})$ or $\\logit(\\hit)$ would essentially require application of chain rules\nto show differentiability.\n\n\\subsubsection{Simple nonparametric setup for CUSUM charts}\n\\label{sec:theor:ex:CUSUM:nonpar}\nWe show how the above theorem applies to the CUSUM chart described in\n(\\ref{eq:discrCUSUM_meanshift}) when using a non-parametric bootstrap\nversion of Algorithm \\ref{alg:Bootstrap}.\n\nLet $D=l_{\\infty}(\\mathbb{R})$ be the set of bounded functions $\\mathbb{R}\\to\\mathbb{R}$\nequipped with the sup-norm $\\|x\\|=\\sup_{t\\in \\mathbb{R}}|x_t|$. \\cc{This is a\n  Banach space, i.e. a complete normed vector space}\nLet $D_q\\subset D$ be the set of cumulative distribution functions on\n$\\mathbb{R}$ with finite second moment.\nThe parameters needed to run the chart are the mean and the standard deviation of the in-control observations, thus we may choose\n $\\Xi=\\mathbb{R}\\times(0,\\infty)$ and  $\\xi:D_q\\to \\Xi, P\\mapsto (\\int x P(dx),\n\\int x^2 P(dx)-(\\int x P(dx))^2)$.\n\n\nAs quantities $q$ of interest we are considering hitting probabilities\n($q=\\hit$) and thresholds ($q=c_{\\hit}$) needed to achieve a certain hitting\nprobability.  The probability $\\hit:D_q\\times \\Xi\\to \\mathbb{R}$ of hitting a\nthreshold $c>0$ up to step $T>0$ can be written as\n$\\hit(P;\\xi)=\\Prob(m(Y) \\geq c)$, where\n$m(Y)=\\max_{i=1,\\dots,T}R_i(Y)$ is the maximum value of the chart up\nto time $T$,\n$R_i(Y)=\\sum_{j=1}^iY_j-\\min_{0\\leq k\\leq i}\\sum_{j=1}^kY_j$ is the\nvalue of the CUSUM chart at time $i$, $Y=(Y_1,\\dots,Y_T)$,\n$Y_t=\\frac{X_t-\\xi_1-\\Delta\/2}{\\xi_2}$ and $X_1,\\dots,X_T \\sim P$ are the\nindependent observations.  The threshold needed to achieve a certain hitting\nprobability $\\beta \\in (0,1)$ is $c_{\\hit}: D_q\\times \\Xi\\to \\mathbb{R}$,\n$c_{\\hit}(P;\\xi)=\\inf\\{c>0:\\hit(P;\\xi)\\leq \\beta\\}$.\n\nThe setup for the nonparametric bootstrap is as follows. $W_{n}$ is an\n$n$-variate multinomially distributed random vector with probabilities\n$1\/n$ and $n$ trials. The resampled distribution is $\\hat\nP^{\\ast}=\\frac{1}{n}\\sum_{j=1}^nW_{nj}\\delta_{X_{-j}}$, where $\\delta_x$\ndenotes the Dirac measure at $x$.\n\n\nThe following lemma shows\ncondition a) of  Theorem \\ref{th:main},\nthe Hadamard differentiability of $\\hit$ and $c_{\\hit}$.\n\\begin{lemma}\n\\label{le:HaddiffCUSUMhit}\nFor every $P\\in D_q$, and every $\\xi \\in \\mathbb{R}\\times(0,\\infty)$, the\nfunction $\\hit$ is Hadamard differentiable at $P$ around $\\xi$\ntangentially to $D_0=\\{H:\\mathbb{R}\\to \\mathbb{R}: H\\text{ continuous}, \\lim_{t\\to\n  \\infty}H(t)=\\lim_{t\\to-\\infty}H(t)=0\\}$.  If, in addition, $P$ has a\ncontinuous bounded positive derivative $f$ with $f(x)\\to 0$ as $x\\to\n\\pm \\infty$, then $c_{\\hit}$ is also Hadamard differentiable at $P$\naround $\\xi$ tangentially to $D_0$.\n\\end{lemma}\nThe proof  is in Appendix \\ref{sec:haddifhitprobex}, with preparatory results in\nAppendix \\ref{sec:chain-rule} - \\ref{sec:diff-hitt-prob}.\n\nConditions b) and c) of Theorem \\ref{th:main} follow directly from empirical process theory,\nsee e.g.\\  \\cite[p.17,Theorems 2.6 and 2.7]{Kosorok2008ItE}.\n\\cc{To see conditions b) and c) of  Theorem \\ref{th:main}, we can argue as follows.\n  In the language of empirical process theory, consider ${\\cal\n    F}=\\{\\mathbb{R}\\to \\mathbb{R}, x\\mapsto 1_{(-\\infty,a]}(x):a\\in \\mathbb{R}\\}$ and let\n  $l_{\\infty}({\\cal F})$ be the set of all bounded function ${\\cal\n    F}\\to\\mathbb{R}$.  As $\\cal F$ can be identified with $\\mathbb{R}$, we can\n  idenfity $l_{\\infty}({\\cal F})$ with $l_{\\infty}(\\mathbb{R})$, the set of\n  bounded functions $\\mathbb{R}\\to \\mathbb{R}$.  By \\cite[p.17]{Kosorok2008ItE}, $\\cal F$ is\n  Donsker, i.e.\\ if $X_1,\\dots,X_n\\sim P$ independently, and letting\n  $P_n=\\frac{1}{n}\\sum_{i=1}^n\\delta_{X_i}$ be the corresponding\n  empirical measure, then $G_n=\\sqrt{n}(P_n-P)\\leadsto G$ in\n  $l_{\\infty}(\\cal F)$ (or equivalently in $l_{\\infty}(\\mathbb{R})$) for some\n  $G$.\n\n  thus $G_n$ is considered a random element in $l_{\\infty}({\\cal\n      F})$ (or equivalently $l_{\\infty}(\\mathbb{R})$), via\n    $\\sqrt{n}(P_n-P)(1_{(-\\infty,a]})=\\sqrt{n}(P_n((-\\infty,a])-P((-\\infty,a]))$\n\n  Now, Theorems 2.6 and 2.7 of \\cite{Kosorok2008ItE} give conditional convergence\n  results for the nonparametric bootstrap, i.e. they show that\n$\\hat G_n\\condweakconv{\\hat P} G$ in $l_{\\infty}({\\cal F})$ and that the sequence $\\hat G_n$ is asymptotically measurable.\nSufficient conditions for c)  are e.g.\\ given in Theorems 3.6.1 and 3.6.2 on p.347 of \\cite{Vaart1996WCa}\n}\nCondition e)  is satisfied as well, see  bottom of p.189  and after Theorem 10.4 (p.184) of \\cite{Kosorok2008ItE}.\n\nVerifying condition d) in full is  outside the scope of the present paper.\nA starting point could be the fact that by\nthe Donsker theorem, $Z\\sim G\\circ P$, where $G$ is a Brownian bridge.\n\\cc{We would need to consider the derivative in Lemma\n  \\ref{le:diffhitprob} and in Lemma \\ref{le:Haddiffinversemap}.}\n\n\n\n\\subsubsection{CUSUM charts with normally distributed observations}\n\\label{sec:cusum-charts-with}\nIn this section, we consider a similar setup to the monitoring based\non (\\ref{eq:discrCUSUM_meanshift}) considered in the previous\nsubsection with the difference that we now use parametric assumptions.\nMore specifically,  the observations $X_i$ follow a normal\ndistribution with unknown mean $\\mu$ and variance $\\sigma^2$. We will\nuse this both for computing the properties of the chart as well as in\nthe bootstrap, which will be a parametric bootstrap version of\nAlgorithm \\ref{alg:Bootstrap}.\n\n\nThe distribution of the observations can be identified with its\nparameters which we estimate by $\\hat P = (\\hat \\mu, \\hat \\sigma^2)$,\nwhere $\\hat \\mu=\\frac{1}{n}\\sum_{i=1}^nX_{-i}$ and $\\hat\n\\sigma^2=\\frac{1}{n-1}\\sum_{i=1}^n(X_{-i}-\\hat \\mu)^2$.  The set of\npotential parameters is $D_q=\\mathbb{R}\\times(0,\\infty)$ which is a subset of\nthe Euclidean space $D=\\mathbb{R}^2$. The parameters needed to run the chart\n(\\ref{eq:discrCUSUM_meanshift}) are just the same as the one needed to\nupdate the distribution, thus $\\Xi=D_q$ and $\\xi:D_q\\to \\Xi,\n(\\mu,\\sigma)\\mapsto (\\mu,\\sigma)$ is just the identity.\n\nAs before, we are interested in hitting probabilities within the first\n$T$ steps.  Using the function $\\hit$ defined in the previous\nsubsection, we can write the hitting probability in this parametric\nsetup as $\\hit^N:D_q\\times \\Xi\\to\\mathbb{R}$, $(\\mu,\\sigma;\\xi)\\mapsto\\hit(\n\\Phi_{\\mu,\\sigma^2};\\xi)$, where $\\Phi_{\\mu,\\sigma^2}$ is the cdf of\nthe normal distribution with mean $\\mu$ and variance $\\sigma^{2}$ and\nthe superscript $N$ stands for normal distribution.  Furthermore,\nusing $c_{\\hit}$ from the previous subsection, the threshold needed to\nachieve a given hitting probability is $c_{\\hit}^N:D_q\\times\\Xi\\to\\mathbb{R}$,\n$(\\mu,\\sigma;\\xi)\\mapsto c_{\\hit}(\\Phi_{\\mu,\\sigma^2};\\xi)$.\n\nThe resampling is a parametric resampling. To put this in the framework of the main theorem, we let $W_n=(W_{n1},\\dots,W_{nn})$, where\n$W_{n1},\\dots,W_{nn}\\sim N(0,1)$ are independent. The\nresampled parameters are then  $\\hat\n\\mu^{\\ast}_n=\\frac{1}{n}\\sum_{i=1}^nX_{ni}^{\\ast}$ and $\\hat \\sigma^{\\ast\n  2}_n=\\frac{1}{n-1}\\sum_{i=1}^n(X^{\\ast}_{ni}-\\hat \\mu^{\\ast}_n)^2$\nwhere $ X^{\\ast}_{ni}=\\hat P_2W_{ni}+\\hat P_1$.\n\nThe following lemma shows that condition a) of  Theorem \\ref{th:main} is  satisfied.\n\\begin{lemma}\n\\label{le:HaddiffCUSUMhitNORMAL}\nFor every $\\theta\\in \\mathbb{R}\\times (0,\\infty)$ and every  $\\xi \\in \\mathbb{R}\\times(0,\\infty)$,\nthe functions $\\hit^N$ and $c_{\\hit}^{N}$ are Hadamard differentiable at $\\theta$ around $\\xi$.\n\\end{lemma}\nThe proof can be found in Appendix \\ref{sec:haddifhitprobex}, using again the preparatory results of\nAppendix \\ref{sec:chain-rule} - \\ref{sec:diff-hitt-prob}.\n\nConcerning the other conditions of Theorem \\ref{th:main}: Condition b)\ncan be shown using standard asymptotic theory, e.g.\\ maximum likelihood\ntheory, which will yield that $Z$ is normally distributed. \\cc{could\n  argue via the $(\\hat \\mu, (n-1)\/n\\hat \\sigma^2)$ being the MLE}\nCondition c) is essentially the requirement that the parametric\nbootstrap of normally distributed data is working.  \\cc{This should be\n  easy to shown by arguing conditionally on the estimators. There may\n  be something in vdVaart, asymptotic statistics - but he is just\n  using nonparametric resampling.}  As $Z$ is a normally distributed\nvector, condition d) holds unless $q'$ equals 0.  Condition e) is\nsatisfied, as the mapping $W_n\\mapsto\\hat P^{\\ast}(\\hat P, W_n) =\n(\\hat\\mu_n^{\\ast}, \\hat\\sigma_n^{\\ast 2})$ is continuous and hence\nmeasurable.\n\n\n\n\n\n\n\\subsubsection{Setup for Shewhart charts}\nFor Shewhart charts, the same setup as in the previous two sections\ncan be used, the only difference is the choice of $q$. Conditions b),\nc) and e) are as in the previous two sections.  We conjecture that it is possible to show the  Hadamard\ndifferentiability  more directly, as the properties are\navailable in closed form, see Section \\ref{subsubsec:Shewhart}.\n\n\n\\cc{\nWith $G=1-p$,\n $\\hit(G)=(c\\mapsto 1-G(c)^T)$,  $\\hit'(G)(H)=(c\\mapsto -G(c)^{T-1}H(c))$,\n\n\n $\\frac{\\partial}{\\partial c} \\hit(G)(c)=-T G(c)^{T-1}g(c)$\n\n $\\ARL(G)=(c\\mapsto\\frac{1}{1-G(c)})$, $\\ARL'(G)(H)=(c\\mapsto\\frac{1}{(1-G(c))^2}H(c))$ \\cc{see \\cite[Lemma 3.9.25]{Vaart1996WCa}}\n $\\frac{\\partial}{\\partial c}\\ARL(G)(c)=\\frac{1}{(1-G(c))^2}g(c)$\n}\n\n\n\n\\section{Simulations for homogeneous observations}\n\\label{sec:simulsingle}\n\n\nWe now illustrate our approach by some simulations using\nCUSUM charts. The simulations were done in R \\citep{R}.\n\n\nWe use two past sample sizes, $n=50$ and\n$n=500$. The in-control distribution of $X_t$ is $N(0,1)$ and\nwe use 1000 replications and $B=1000$ bootstrap\nreplications. We employ both the parametric bootstrap and the\nnonparametric bootstrap mentioned in the previous sections.  For the\nparametric bootstrap we used the sample mean and sample standard\ndeviation of $X_{-n},\\dots,X_{-1}$ as estimates for the mean and the standard deviation of\nthe observations.\n\nFor the performance measures $\\ARL$, $\\log(\\ARL)$, $\\hit$ and\n$\\logit(\\hit)$ we use a threshold\nof $c=3$. For $c_{\\ARL}$ we calibrate to  an $\\ARL$ of $100$\nin control and for $c_{\\hit}$ we calibrate to a false alarm probability of\n$5\\%$ in 100 steps.\n\n\n\nWe use the CUSUM chart (\\ref{eq:discrCUSUM_meanshift}) with $\\Delta=1$\nand $\\mu$ and $\\sigma$ estimated from the past data. To compute\nproperties such as $\\ARL$ or hitting probabilities, we  use a\nMarkov chain approximation (with 75 grid points), similar to the one\nsuggested in \\cite{BROOK1972atp}\\cc{there is a precise description of\n  a grid that is being used in that paper - I think we are using\n  something similar but most likely not completely identical}. \\cc{We\nchecked that this gave very good  approximations.}\n\n\n\n\n\n\\subsection{Coverage probabilities}\n\\label{subsec:simulcoverage}\n\n\nTable \\ref{tab:covprob_simnormal} contains coverage probabilities of\nnominal 90\\% confidence intervals. These are the one-sided lower confidence\nintervals given by (\\ref{eq:onesidedapproxconfint}), except for\n$q=\\ARL$ and $\\log(ARL)$ where the corresponding upper interval is used.\n\n\n\\begin{table}\n    \\caption{Coverage probabilities of nominal 90\\% confidence intervals for CUSUM charts.\n  \\label{tab:covprob_simnormal}}\n\\begin{center}\n\\parbox{0.58\\textwidth}{\n    \\input{simpaper\/tablepaper_stdNormal_CenterScale.tex}\\\\\n      The standard deviation of the results is roughly 0.01.\n }\n    \\end{center}\n\\end{table}\n\n\nIn the parametric case, for $n=50$, the coverage probabilities are\nsomewhat off for untransformed versions, in particular for $q=\\ARL$.\nUsing $\\log$ or $\\logit$ transformations seems to improve the coverage\nprobabilities considerably.  In the parametric case, for $n=500$, all\ncoverage probabilities seem to be fine, except for $q=\\ARL$, which\nalthough  shows some marked improvement compare to $n=50$.  In the\nnonparametric case, a similar picture emerges, but the coverage\nprobabilities are a bit worse than in the parametric case.\n\n\n\\begin{remark}\n\\label{rem:scalingdoesnotmatter}\n  For $q= \\log(c_{\\ARL})$ and $q= \\log(c_{\\hit})$ the division by\n  $\\hat\\sigma$ in\n  (\\ref{eq:discrCUSUM_meanshift}) could be skipped without making a\n  difference to the coverage probabilities.  Indeed, the division by\n  $\\hat\\sigma$ just scales the chart (and the resulting threshold) by\n  a multiplicative factor, which is turned into an additive factor by\n  $\\log$ and which then cancels out in our adjustment.\n\\end{remark}\n\n\n\n\n\\subsection{The benefit of an adjusted threshold}\nIn this section, we consider both the in- and out-of-control\nperformance of CUSUM charts when adjusting the threshold $c$ to give a\nguaranteed in-control $\\ARL$ of 100.  Setting the threshold is, in our\nopinion, the most important practical application of our method.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{simpaper\/adjusted_unadjusted_ARL_CenterScale_boxplot.pdf}\n  \\caption{Distribution of the conditional $\\ARL$ for CUSUMs in a\n    normal distribution setup. Thresholds are calibrated to an\n    in-control $\\ARL$ of 100.  The adjusted thresholds have a\n    guarantee of 90\\%.  A log transform is used in the calibration.\n    The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%, 75\\%, 90\\% and\n    97.5\\% quantiles.  The white boxplots are in-control, the gray\n    boxplots out-of-control.}\n  \\label{fig:adjusted_unadjusted_ARL}\n\\end{figure}\n\n\nFigure \\ref{fig:adjusted_unadjusted_ARL} shows average run lengths\nfor both  the unadjusted threshold $c(\\hat P;\\hat \\mu, \\hat \\sigma)$ and the\nadjusted threshold $\\exp(\\log( c(\\hat P;\\hat \\mu, \\hat \\sigma))-p^{\\ast}_{0.1})$, where $p^{\\ast}_{0.1}$ is computed via the parametric\nbootstrap using $q=\\log(c_{\\ARL})$.  Thus, with  90\\%  probability, the  adjusted threshold should\nlead to an $\\ARL$ that is above 100. In this and in all  following\nsimulations,\nthe out-of-control ARL refers\nto the  situation where the chart is  out-of-control from the\nbeginning, i.e.\\ from\ntime 0 onwards.\n\nFor the unadjusted threshold, the desired in-control average run\nlength is only reached in roughly half the cases.  More importantly,\nfor $n=50$, the probability of having an in-control $\\ARL$ of below $50$\nis greater than 20\\%.\n\n\nWith the adjusted threshold we should get an average run length of at\nleast 100 in 90\\% of the cases.  This is achieved.  The\nout-of-control $\\ARL$ using the adjusted thresholds increases only\nslightly compared to the unadjusted version.\n\n\nSimilarly to Remark \\ref{rem:scalingdoesnotmatter}, removing the\nscaling by $\\hat \\sigma$ in (\\ref{eq:discrCUSUM_meanshift}) would not\nchange the results of this section.\n\n\n\\subsection{Nonparametric bootstrap - advantages and disadvantages}\n\nIn this section, we compare the parametric and the non-parametric\nbootstrap.  We consider CUSUM charts that are calibrated to an\nin-control average run length of 100 assuming a normal distribution.\nWe use the adjusted threshold $\\exp(\\log( c_{\\ARL}(\\hat P;\\hat\n\\mu, \\hat \\sigma))-p^{\\ast}_{0.1})$.\n\nFigure \\ref{fig:par_nonpar_ARL} shows the distribution of $\\ARL$ for\n$n=50$ and $n=500$ for both the parametric bootstrap that assumes a normal\ndistribution of the updates and the nonparametric bootstrap. We consider both a\ncorrectly specified model where $X_t\\sim N(0,1)$ as well as two\nmisspecified models where $X_t\\sim \\text{Exponential}(1)$ and $\\sqrt{20}X_t\\sim\n\\chi^2_{10}$ (all of the $X_t$ have variance 1).  We show both the in- as well as the\nout-of-control performance of the charts.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{simpaper\/par_nonpar_ARL_CenterScale_boxplot.pdf}\n  \\caption{Effects of misspecification. Thresholds are calibrated to\n    an in-control ARL of 100 and adjusted to the estimation error with\n    a guarantee of 90\\%.  A log transform is used in the\n    calibration. The white boxplots are in-control, the gray\n    boxplots are  out-of-control. The boxplots show\n    the 2.5\\%, 10\\%, 25\\%, 50\\%, 75\\%, 90\\% and 97.5\\% quantiles.}\n  \\label{fig:par_nonpar_ARL}\n\\end{figure}\n\nIn the correctly specified model ($X_t\\sim N(0,1)$), the performance\nof the parametric and the non-parametric chart seems to be almost\nidentical. The only difference is a slightly worse in-control\nperformance for the non-parametric chart for $n=50$.\n\nIn the misspecified model with $X_t\\sim \\text{Exponential}(1)$, the\nparametric chart does not have the desired in-control\nprobabilities. The non-parametric chart seems to be doing well, in\nparticular for $n=500$.  We have a similar results in the other\nmisspecified model, with $\\sqrt{20}X_t\\sim \\chi^2_{10}$.\n\n\n\n\n\n\\section{Regression models}\n\\label{sec:regmod}\n\nIn many monitoring situations, the units being monitored are heterogeneous,\nfor instance when monitoring patients at hospitals or bank customers.\nTo make sensible monitoring systems in such situations, the explainable\npart of the heterogeneity should be accounted for by relevant\nregression models. The resulting charts are often called risk\nadjusted, and an overview of some such charts can be  found in \\cite{Grigg2004oor}.\n\nTo run risk adjusted charts, the regression model needs to be estimated\nbased on past data, and this estimation needs to be accounted for.  Our\napproach for setting up charts with a guaranteed performance applies\nalso to risk adjusted charts,\nand we will in particular look at linear, logistic and survival\nmodels.\n\n\\subsection{Linear models}\n\\label{subsec:linmod}\n\nSuppose we have independent observations $(Y_1,X_1),$  $(Y_2,X_2)$, $\\ldots$,\nwhere $Y_i$ is a response of interest and $X_i$ is a corresponding\nvector of covariates, with the first component usually equal to 1.\nLet $P$ denote the joint distribution of $(Y_i,X_i)$ and suppose that\nin control $\\E(Y_i|X_i)=X_i\\xi$.  From some observation\n$\\kappa$ there is a shift in the mean response to\n$\\E(Y_i|X_i)=\\Delta+X_i\\xi$ for $i=\\kappa,\\kappa+1,\\dots$.\n\nMonitoring schemes for detecting changes in regression models can\nnaturally be based on residuals of the model, see for instance\n\\cite{Brown1975TfT} and \\cite{Horvath2004Mci}.\n We can, for instance,\ndefine a CUSUM to monitor changes in the conditional mean of $Y$ by\n$$S_t=\\max(0, S_{t-1}+Y_t-X_t \\xi - \\Delta\/2), \\quad S_0=0, $$\nwith hitting time $\\tau=\\inf\\{t>0: S_t\\geq c\\}$ for some threshold\n$c>0$.\nIn a similiar manner we could also set up charts for\nmonitoring changes in other components of  $\\xi$.\n\nThe parameter vector $\\xi$ is estimated from past in\ncontrol data, e.g.\\ by the standard least squares estimator.  We\nsuggest to use a nonparametric version of the general Algorithm\n\\ref{alg:Bootstrap} with $\\hat P$ being the\nempirical distribution putting weight $1\/n$ on each of the past\nobservations $(Y_{-n},X_{-n}),\\dots,(Y_{-1},X_{-1})$. Resampling is\nthen equivalent to resampling\n$(Y^{\\ast}_{-n},X^{\\ast}_{-n}),\\dots,(Y^{\\ast}_{-1},X^{\\ast}_{-1})$ by\ndrawing with replacement from $\\hat P$.\n\nThe suggested method should  work  even if the linear model is misspecified,\ni.e.\\  $\\E(Y_i|X_i)=X_i\\xi$ does not necessarily hold. The nonparametric\nbootstrap should take  this into account.\n\n\nAn analogous approach can be used for Shewhart charts. In settings\nwhere it is reasonable to consider the covariate vector to be\nnon-random one could alternatively use bootstrapping of residuals, see\nfor example \\cite{Freedman1981BRM}.\n\n\n\\subsubsection{Theoretical considerations}\n\nObtaining precise results is more demanding than in the examples\nwithout covariates in Section~\\ref{sec:th:examples}. We only\ngive an idea of the setup that might be used.\n\n\nThe set of distributions of the observations $D_q$ can be chosen as the\nset of cdfs on $\\mathbb{R}^{d+1}$ with finite second moments, where $d$ is the dimension of the covariate. The first cdf corresponds to the responses, the others to the covariates. $D_q$ is contained\nin the vector space $D=l_{\\infty}(\\mathbb{R}^{d+1})$, the set of bounded functions $\\mathbb{R}^{d+1}\\to \\mathbb{R}$.\nThe parameters needed to run the chart are the regression coefficients contained in the set  $\\Xi=\\mathbb{R}^d$.\nThese parameters are obtained from the distribution of the observations via  $\\xi:D_q\\to \\Xi$, $F\\mapsto\n(E(X^TX))^{-1}E(X Y)$ where $(Y,X)\\sim F$ where $X$ is considered to\nbe a row vector.\n\n\nWe conjecture that the conditions of Theorem \\ref{th:main} are broadly\nsatisfied if the cdf  of $Y-X\\xi$ is differentiable and if for\nthe property $q$ we use  hitting probabilities or thresholds to\nachieve a given hitting probability.  In particular, it should be\npossible to show Hadamard differentiability similarly to Lemma\n\\ref{le:HaddiffCUSUMhit}: write $q$ as concatenation of two functions\nand use the chain rule in Lemma \\ref{le:chainrule}.  The first mapping\nreturns the distribution of the updates of the chart depending on\n$F\\in D_q$ and $\\xi\\in \\Xi$ via $(F;\\xi)\\mapsto {\\cal L}(Y-X\\xi-\n\\Delta)$, where ${\\cal L}$ denotes the law of a random variable.  The\nsecond takes the distribution of the updates and returns the property of\ninterests. The differentiability of the second map has been shown in\nLemmas \\ref{le:Haddiffinversemap} and  \\ref{le:diffhitprob}.\n\n\n\\subsubsection{Simulations}\n\\label{example:CUSUMLinReg}\n\nWe illustrate the performance of the bootstrapping scheme using a\nCUSUM and the linear in-control model\n$Y=X_{1}+X_{2}+X_3+\\epsilon$. Let $\\epsilon\\sim N(0,1)$, $X_{1}\\sim\n\\text{Bernoulli}(0.4)$, $X_2\\sim U(0,1)$ and $X_3\\sim N(0,1)$, where\n$X_1,X_2,X_3$ and $\\epsilon$ are all independent. The out-of-control\nmodel is $Y=1+X_{1}+X_{2}+X_3+\\epsilon$,\ni.e. $\\Delta=1$. Figure~\\ref{fig:regression_ARL} shows the distribution\nof the attained ARL for CUSUMs with thresholds calibrated to give an\nin control ARL of 100. We see that the behaviour of the adjusted\nversus unadjusted thresholds are very similar to what we observed for\nthe simpler model in Figure~\\ref{fig:adjusted_unadjusted_ARL}. The\ncoverage probabilities obtained for this regression model, not\nreported here, are also very similar to the covarage probabilities\nreported in Table~\\ref{tab:covprob_simnormal}, though with a tendency\nto be slightly worse.\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{simpaper\/regression_ARL_boxplot.pdf}\n  \\caption{Distribution of the conditional $\\ARL$ for CUSUMs in a\n    linear regression setup. Thresholds are calibrated to an\n    in-control $\\ARL$ of 100. A log transform is used in the\n    calibration.  The adjusted thresholds have a guarantee of\n    90\\%. The white boxplots are in control, the gray\n    out-of-control. The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%,\n    75\\%, 90\\% and 97.5\\% quantiles.}\n  \\label{fig:regression_ARL}\n\\end{figure}\n\n\n\n\n\n\\subsection{Logistic regression}\n\\label{subsec:logreg}\n\nControl charts, in particular CUSUM charts, based on logistic\nregression models are popular for modelling of binary\noutcomes in medical contexts. See e.g.\\ \\cite{Lie1993nsp}, \\cite{Steiner2000Msp},\n\\cite{Grigg2004oor} and \\cite{Woodall2006Tuo}.\n\nSuppose we have independent observations $(Y_1,X_1),(Y_2,X_2),\\ldots,$\nwhere $Y_i$ is a binary response variable and $X_i$ is a corresponding\nvector of covariates. Further, suppose that in control the log odds\nratio is $\\logit(\\Prob(Y_i=1|X_i))=X_i\\xi$, and that from some observation\n$\\kappa$ there is a shift in the log odds ratio to\n$\\logit(\\Prob(Y_i=1|X_i))=\\Delta+X_i\\xi$ for $i=\\kappa,\\kappa+1,\\dots$\n\nA CUSUM to monitor changes in the odds ratio can be defined by \\citep{Steiner2000Msp}\n$$S_t=\\max(0, S_{t-1}+R_t), \\quad S_0=0, $$\nwhere $R_t$ is the log likelihood ratio between the in-control and out-of-control model for observation $t$. More precisely\n$$\n\\exp(R_t)=\\frac{\\exp(\\Delta+X_t\\xi)^{Y_t}\/(1+\\exp(\\Delta+X_t\\xi))}{\\exp(X_t\\xi)^{Y_t}\/(1+\\exp(X_t\\xi))}\n=\\exp(Y_t\\Delta)\\frac{1+\\exp(X_t\\xi)}{1+\\exp(\\Delta+X_t\\xi)}.\n$$\n\nThe parameter vector $\\xi$ is estimated from past in-control\ndata by e.g.\\  the standard maximum likelihood estimator. The same\nnonparametric bootstrap approach as described for the linear model in\nSection~\\ref{subsec:linmod} can now be applied to this CUSUM based on\nthis logistic regression model. Moreover, this approach would also\napply to control charts based on other generalized linear models, for\ninstance Poisson regression models for monitoring count data. The only\namendment needed is to replace $R_t$ by the relevant log likelihood\nratio.\n\n\n\nWe have run simulations, not reported here, based on the same covariate\nspecifications as in Section~\\ref{example:CUSUMLinReg}.  The results\nare similar to the results for the linear model of\nSection~\\ref{example:CUSUMLinReg}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Survival analysis models}\n\nRecently, risk adjusted control charts based on survival models have\nstarted to appear, see\n\\cite{Biswas2008rCi,Sego2009Rmo,Steiner2009ras,Gandy2010ram}. In none\nof these papers any adjustment for estimation error is done, but\n\\cite{Sego2009Rmo} are  illustrating, by simulations, the impact of\nestimation error on the attained average run length  for the\naccelerated failure time model based CUSUM studied in their paper.\n\n\nIn the following, we provide a brief simulation example of our\nadjustment in a survival setup where we use the methods described in\n\\cite{Gandy2010ram}.\n\nWe observe the survival of individuals over a fixed time interval of\nlength $n$ (we will use $n=100$ and $n=500$).  Individuals arrive at\ntimes $B_i$ (in our simulation according to a Poisson process with\nrate $1$), and survive for $T_i$ time units. Individuals may arrive\nbefore the observation interval, as long as $B_i+T_i$ is after the\nstart of the observation interval.  Right-censoring, at $C_i$ time\nunits after arrival, is taking place after a maximum follow-up time of\n$t=60$ time units or after the individuals leave the observation\ninterval.  In the simulation, the true hazard rate of $T_{i}$ is\n$h_i(t)= 0.1\\exp( X_{1i}+X_{2i})$, where $X_{1i}\\sim\n\\text{Bernoulli}(0.4)$ and $X_{2i}\\sim N(0,1)$ are covariates.\n\nBased on the observed data we fit a Cox proportional hazard model\nwith $X_{1i}$ and $X_{2i}$ as covariates and nonparametric baseline,\ngiving estimates $\\hat \\beta$ for the covariate effects and $\\hat\n\\Lambda_0(t)$ for the the integrated baseline.\n\nWe use the CUSUM chart described in\n\\cite{Gandy2010ram} against a proportional alternative with\n$\\rho=1.25$.  The parameters\nneeded to run the chart are $\\xi=(\\beta, \\Lambda_0)$ estimated by\n$\\hat\\xi=(\\hat \\beta, \\hat \\Lambda_0)$.\nTo be precise, the chart signals at time\n$\\tau=\\inf\\{t>0:S(t)\\geq c\\}$, where\n$S(t)=R(t)-\\inf_{s\\leq t}R(s)$,\n$\n                              R (t ) = \\log(\\rho ) N (t ) - (\\rho - 1)\n\\Lambda(t),\n$\n $N (t )$ is the number of events until time $t$ and $\\Lambda(t ) =\n\\sum_{i} \\exp( \\beta_1X_{i1}+ \\beta_2X_{2i})\n\\Lambda_0(\\min((t-B_i)^{+},T_i,C_i))$.\n\n\nWe are interested in finding a threshold that gives a desired\nhitting probability, i.e. we use $q=c_{\\hit}$.  We compute\n$c_{\\hit}(P,\\xi)$ via simulations (simulate new data from $P$ and run\nthe chart with $\\xi$). We estimate the threshold needed to get a 10\\%\nfalse alarm probability in $n$ time units in control, by the 90\\% quantile of 500\nsimulations of the maximum of the chart.\n\nTo resample, we resample individuals with replacement. We use 500\nbootstrap samples.  Figure \\ref{fig:hitprobsurvanal} shows the\ndistribution of the resulting hitting probabilities based on 500\nsimulated observation intervals.\n\n\n\n\\begin{figure}\n \\centering\n\\includegraphics[width=\\linewidth]{simpaper\/adjusted_unadjusted_coxhitprob_boxplot.pdf}\n\\caption{Distribution of the conditional hitting probability for\n  survival analysis CUSUMs. Thresholds are calibrated to an in-control\n  hitting probability of 0.1.  The adjusted thresholds have a\n  guarantee of 90\\%.  The white boxplots are in control, the gray\n  out-of-control. The boxplots show the 2.5\\%, 10\\%, 25\\%, 50\\%, 75\\%,\n  90\\% and 97.5\\% quantiles. \\label{fig:hitprobsurvanal}}\n\\end{figure}\n\n\nIn control, without the adjustment, the desired false\nalarm probability of 0.1 is only reached in roughly 60\\% of the cases.  The\nbootstrap correction seems to work fine, leading to a false alarm probability\nof at most 10\\% in roughly 90\\% of the cases.  As expected, increasing the\nlength of the fitting period and the length of time the chart is run\nfrom $n=100$ to $n=500$ results in higher out-of-control hitting probabilities.\n\n\n\nIf the length of the fitting period and the deployment period of the chart differ then\na somewhat more complicated resampling procedure needs to be used.\nFor example, one could\n resample arrival times and survival times\/covariates separately.\nThe former could be done by assuming a Poisson process as arrival time and the\nlatter either by resampling with replacement or by sampling from an estimated\nCox model and an estimated censoring distribution.\n\n\n\n\n\n\\cc{\n              In the survival analysis case with a proportional alternative, the chart is  based on\n              $$\n              R(t) = \\log(\\rho ) N (t )- (\\rho - 1)\\hat \\Lambda (t ),\n              $$\n              After the time transformation of $N$ to the standard Poisson process $\\tilde N$ this becomes\n              \\begin{align*}\n                \\tilde R(t) = R(\\Lambda^{-1}(t))=\\log(\\rho)\\tilde\n                N(t)-(\\rho-1)\\hat\\Lambda(\\Lambda^{-1}(t))\n              \\end{align*}\n              Thus the nice Markov-approximation will not work $\\hat\n              \\Lambda$ and $\\Lambda$ will not have independent\n              increments. Therefore we needed to simulate.\n}\n\n\n\n\n\n\n\n\n\n\\section{Conclusions and discussion}\n\\label{sec:conclusion}\n\n\nWe have presented a general approach for handling estimation error in\ncontrol charts with estimated parameters and unknown in-control\ndistributions. Our suggestion is, by bootstrap methods, to tune the\nmonitoring scheme to guarantee, with high probability, a certain\nconditional in-control performance (conditional on the estimated\nin-control distribution). If we apply a nonparametric bootstrap, the\napproach is  robust against model specification error.\n\nIn our opinion, focusing on a guaranteed conditional in-control  performance is\ngenerally more relevant than focusing on some average\nperformance, as an estimated chart usually is run for some time without\nindependent reestimation. Our approach can also easily be adapted to\nmake for instance bias adjustments. Bias adjustments, in\ncontrast to guaranteed performance, tend to\nbe substantially influenced by tail behaviour for heavy\ntailed distributions which for instance the average run length has.\nThis implies that the bias adjustments need not be useful in the\nmajority of cases as the main effect of the adjustment is to adjust\nthe tail behaviour.\n\n\nWe have in particular demonstrated our approach for various variants\nof Shewhart and CUSUM charts, but the general approach will\napply to other charts as well. The method is generally\nrelevant when the in-control distribution is unknown and the conditions of\nTheorem~\\ref{th:main} hold. We conjecture that this will be the case\nfor many of the most commonly used control charts.\n\\cc{for instance be the case for charts like  EWMA charts \\citep{Roberts1959CCT}, general\nlikelihood ratio based charts \\cite{Frisen1991Ops,Frisen2003SSO},  the\nSets method \\citep{Chen1978SSC,Grigg2004ARA}}\nNumerous extensions of control charts to other settings exist, for example\nto other  regression\nmodels, to  autocorrelated data, to multivariate data.\nWe do  conjecture that our approach will also apply  in\nmany of these settings.\n\n\n\n\n\n\n\n\\small\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFew-shot learning is a research challenge that assesses a model's ability to quickly adapt to new tasks or new environments.\nThis has been the leading area where researchers apply meta-learning algorithms - where a strategy that learns to learn quickly is likely to be the most promising.\nHowever, it was recently shown by Tian et al. \\cite{Tian2020} that a model with a good embedding is able to match and beat many modern sophisticated meta-learning algorithms on a number of few-shot learning benchmarks.\nIn addition, there seems to be growing evidence that this is a real phenomena \\cite{Chen2019, Chen, Dhillon2019, Huang2019}.\nFurthermore, analysis of the representations learned by Model Agnostic Meta-Learning (MAML) \\cite{maml} (on few-shot learning tasks) revealed that MAML mainly works by learning a feature that is re-usable for many tasks \\cite{Raghu} -- what we are calling a good embedding in this paper.\n\nThese discoveries reveal a lack of understanding on when and why meta-learning algorithms work and are the main motivation for this work. \nIn particular, our contributions are:\n\n\\begin{enumerate}\n    \\item We show that it is possible to define a synthetic task that results in lower degree of feature re-use, thus suggesting that \n   current few-shot learning benchmarks might not have the properties needed for the success of meta-learning algorithms;\n   \\item Meta-overfitting occurs when the number of classes (or concepts) are finite, and the issue disappears once the tasks have an unbounded number of concepts;\n   \\item More adaptation for MAML does not necessarily result in representations that change significantly or even perform better at meta-test time.\n\\end{enumerate}\n\n\\section{Unified Framework for Studying Meta-Learning and Absolute Performance}\\label{metric_ml}\n\nWe propose that future work on meta-learning should not only report absolute performance, but also quantify and report the degree of meta-learning.\nWe hypothesize this is important because previous work \\cite{Tian2020} has observed that supervised learning (while only fine-tuning the final layer) is sufficient to solve current meta-learning benchmarks.\nThis might give the potentially false impression that current trends in meta-learning are irrelevant.\nTo avoid that, we hypothesize that measuring the degree of meta-learning we provide in this section will provide a step forward in explaining those important observations.  \n\nIn this work, we make an important first step by emphasizing the analysis done by \\cite{Raghu}, by defining the degree of meta-learning as the normalized degree of change in the representation of a neural network $nn_{\\theta}$ after using meta-adaptation $A$:\n\\begin{equation}\\label{eq_ml}\n    ML(nn_{\\theta}) = \\mathrm{Diff}( nn_{\\theta},  A(nn_{\\theta}) ).\n\\end{equation}\nIn this work we set $ML(nn_{\\theta})$ to be distance based Canonical Correlation Analysis (dCCA) \\cite{Morcos}.\nNote that dCCA is simply 1 minus CCA to switch the similarity based metric to a difference based metric between 0 and 1.\n\n\\section{Benchmarks that Require Meta-Learning}\n\n\\subsection{Background}\n\n\\textbf{Model-Agnostic Meta-Learning (MAML).}\nThe MAML algorithm \\cite{maml} attempts to meta-learn an initialization of parameters for a neural network that is primed for quick gradient descent adaptation. \nIt consists of two main optimization loops: 1) an outer loop used to prime the parameters for fast adaptation, and 2) an inner loop that does the fast adaptation.\nDuring meta-testing, only the inner loop is used to adapt the representation learned by the outer loop.\n\n\\textbf{Feature re-use.} \nIn the context of MAML, this term usually means that the inner loop provides little adaptation during meta-testing, when solving an unseen task.\nIn particular, Raghu et al. \\cite{Raghu} showed that MAML has little representation change as measured with CCA and CKA after adaptation, during meta-testing on the MiniImageNet few-shot learning benchmark.\n\n\\subsection{Motivation for Our Work}\n\nThe analysis by Raghu et al. \\cite{Raghu} showing that MAML works mainly by feature re-use is the main motivation for our work.\nHowever, we argue that their conclusion is highly dependent on the data set (or benchmark) used.\nThis motivates us to construct a different benchmark and show that by {\\em only} constructing a different benchmark, we can exhibit lower degrees of feature re-use in a statistically significant way.\nTherefore, our goal will be to show a lower degree of feature re-use than them.\nIn particular, their work \\cite{Raghu} showed that the representation layer of a neural network trained with MAML had a dCCA of $0.1 \\pm 0.02$ \\cite{Raghu}.\n{\\em Therefore, our concrete goal will be to show that the dCCA on our task is greater than $0.12$}.\nIf this is achieved, it is good evidence that this new benchmark benefits from meta-learning and can be detected at a higher degree than previous work \\cite{Raghu} in a statistically significant way.\nThis is our main result of this section and is discussed in detail in Section \\ref{main_result}.\n\n\\subsection{Synthetic Task that Requires Meta-learning}\n\n\\subsubsection{Overview and Goal}\\label{goals}\nThe main idea is to sample functions to be approximated, such that the final layer needs little or no adaptation, but the feature layers require a large amount of adaptation.\nThis type of task would forcibly require that the meta-learner learns a representation that requires the feature layers to change to achieve good meta-test performance (i.e., it cannot rely solely on feature re-use).\nTherefore, to perform well, not only would it be good to adapt the representation layers, but additionally performance is likely to be obtained from a (meta-learned) initialization that is primed to change flexibly.\nIn summary, our goal will be to construct synthetic tasks such achieving high meta-test performance and detectable meta-learning - as discussed in section \\ref{metric_ml} - one needs to go beyond feature sharing.\n\n\\subsubsection{Definition}\\label{def}\n\nIn this section, we describe a family of benchmarks that exhibits detectable meta-learning and requires more than a re-usable representation layer to be solved.\nWe propose a set of regression functions specified as a fully connected neural network (FCNN), such that the magnitude of parameters of the representation are larger than the head.\nIn particular, we sample the parameters of the representation layer from a Gaussian with a larger standard deviation, compared to the parameter sampling of the head.\nWe define the representation layer to be the first $L-1$ layers, and the head to be the final layer.\n\nNext, we describe the process to sample one function (regression task\n) from a Gaussian distribution.\nWe have two pairs of benchmark parameters $[(\\mu^{(1)}, \\sigma^{(1)}), (\\mu^{(2)}, \\sigma^{(2)})]$:\n$(\\mu^{(1)}, \\sigma^{(1)})$ to sample the parameters for the representation layer, \nand $(\\mu^{(2)}, \\sigma^{(2)})$ to sample the parameters for the final layer.\nThen each regression task $f^{(t)}$ (with index $t$) is sampled as follows:\n\\begin{itemize}\n    \\item Sample the representation parameters $w^{(l)} \\sim N(\\mu^{(1)}, \\sigma^{(1)})$ for each layer $l \\in [L-1]$ in the representation layers\n    \\item Sample the final layer parameters $w^{(L)} \\sim N(\\mu^{(2)}, \\sigma^{(2)})$\n\\end{itemize}\n\nThe idea is that for some $c \\in \\mathbb R$ we have $\\sigma^{(1)} > c \\cdot  \\sigma^{(2)}$ such that the variance in tasks is due to the representation layers, and therefore adapting the representation layers is necessary.\nFor all our experiments $\\sigma^{(2)}=1.0$.\nAn example task can be seen in Figure \\ref{fun_reg}.\nDuring meta-training, points are uniformly sampled from $[-1,1]$, and the standard support set and query set are constructed by computing $f^{(t)}_{w}(x)$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.46\\linewidth]{figs\/fun_reg.png}\n\\caption{An example regression task constructed as described in Section \\ref{def}. Addressing such tasks requires high degree of meta-learning.}\n\\label{fun_reg}\n\\end{figure}\n\n\\subsubsection{Results on Benchmarks that Require Meta-Learning}\\label{main_result}\n\nIn this section, we show a higher degree of meta-learning and a lower degree of feature re-use from an initialization trained with MAML on the benchmarks described in Section \\ref{def}. \nIn particular, we show this in Figure \\ref{best_relu_vs_std} because the dCCA value exhibited is much larger than $0.12$ of previous work \\cite{Raghu}.\n{\\em Most importantly, the results are statistically significant, because the error bars do not intersect with the red dotted line with (worst case) dCCA value of $0.12$. }\nThe red dotted line is the top error band of previous work - i.e. the mean plus the standard deviation.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]{figs\/best_relu_vs_std.png}\n\\caption{\nShows the of lack of feature re-use and a higher degree of meta-learning, as the standard deviation of the representation layer $\\sigma^{(1)}$  for generating regression.\nThe x-axis is the standard deviation (std) of the parameter $\\sigma^{(1)}$ for generating the tasks for the data sets. \nThe models used for each point in the plot are models selected from early stopping (using the meta-validation MSE loss) when meta-trained with MAML.\nThe models are the same architecture as the target function (4 layers fully connected neural network) with ReLU activation function. \nWe also show the meta-validation loss vs the standard deviation of the task.\nThe dCCA was computed by from the average and standard deviation over the representation layers, in this case the first three layers.\nThe average is across different runs using the same meta-learned initialization.\nThe red dotted line shows the value of $0.12$ that our models have to be statistically significant.\nThe only difference of this figure with respect to figure \\ref{relu_metaoverfitted} is that we selected a model with the best validation here and in the figure \\ref{relu_metaoverfitted} we selected the model in last step.\n}\n\\label{best_relu_vs_std}\n\\end{figure}\n\nNote that a dCCA higher than $0.12$ was observed across all of our experiments in over sixteen different benchmarks. \nIn particular, this happened even in models that had meta-overfitted, e.g., see Figure \\ref{relu_metaoverfitted}.\nThis is strongly suggestive that the benchmarks we defined in Section \\ref{def} require meta-learning, since they do not solely rely on feature re-use to be solved.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]\n                   {figs\/relu_metaoverfitted.png}\n\\caption{\nThis figure supports the main result of the paper because a higher degree of meta-learning and a lack of feature re-use are present -- even in models that are meta-overfitted.\nA meta-overfitted model can be easily obtained in our experiments by selecting a model at the final iteration.\nThe x-axis is the standard deviation (std) of the parameter $\\sigma^{(1)}$ for generating the tasks for the data sets.\nThe red dotted line shows the value of $0.12$ that our models have to be above for statistically significant results that support our claims.\nThe only difference of this figure with respect to figure \\ref{best_relu_vs_std} is that we selected a model in last step (after trough, and it had meta-overfitted) while in figure \\ref{best_relu_vs_std} we select the model with lowest meta-validation loss.\n}\n\\label{relu_metaoverfitted}\n\\end{figure}\n\n\\section{Meta-Overfitting} \\label{meta_overfitting}\n\nIn this section, we show how being armed with the additional metric discussed in Section \\ref{metric_ml}, we are able to identify an increasing gap between the meta-test and meta-train losses\/accuracy -- a term we refer to as \\textit{meta-overfitting}.\nIn particular, this phenomenon is observed when we meta-train models with MAML, and becomes more pronounced as the number of iterations increases.\nWe attribute this to the adaptation, because this increase in the meta-generalization gap is observed in conjunction with the low degree of feature re-use (as discussed in Section \\ref{main_result}), and is most noticeable in our synthetic benchmarks compared to MiniImagent \\cite{Raghu}. \nNote that the dCCA of the models was much larger in our synthetic benchmarks than in MiniImagent.\nIn addition, we show that if the number of regression tasks (in this case functions) is not fixed, then the meta-overfitting issue is no longer observed\n\n\\subsection{Finite Number of Tasks}\\label{finite_metaoverfit}\n\nWhen the number of regression tasks (functions) is finite ($200$ in our experiments), we consistently observe meta-overfitting.\nWe show this in Figure \\ref{meta_overfit1} by increasing the meta-generalization gap (i.e. an increase in the difference between the meta-train and the meta-validation losses).\nThis is consistently observed in over $30$ experiments with a finite number of regression tasks.\n\nFurthermore, meta-overfitting is also observed in a few-shot image recognition benchmark.\nThis is shown in Figure \\ref{overfit_mini} with MiniImagent.\nWith a PyTorch ResNet-18 model, one can observe a meta-generalization gap of about $30\\%$.\nWith a state-of-the-art ResNet-12 \\cite{Tian2020}, the meta-generalization gap is instead about $20\\%$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.55\\linewidth]{figs\/meta_overfit1.png}\n\\caption{\nShows meta-overfitting when the number of tasks (functions) is finite at $200$ regression tasks because the meta-validation loss increases as the meta-train loss decreases.\nIn particular, the dCCA for these models was $0.36 \\pm 0.12$ corresponding to $\\sigma^{(1)}=1.0$. \nThe plot is the learning curve for a 4-layered fully connected neural network trained with MAML \\cite{maml} using episodic meta-learning.\nNote that we use a (large) meta-batch size of $75$ to decrease the noise during training in the figure.\nThe main difference of this figure with figure \\ref{no_overfit} is that in this one has a finite set of tasks using our synthetic benchmark while the other has an infinite set of tasks using the sinusoidal tasks suggested in \\cite{maml}.\n}\n\\label{meta_overfit1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.67\\linewidth]{figs\/resnet18_metaoverfitting_horizontal.png}\n\\caption{ \nShows that meta-overfitting is a real phenomenon in MiniImagent.\nWe interpret this due to the peak in the meta-validation accuracy followed by a decline as the number of iterations increases.\nImportantly, the meta-train accuracy continues to increase as it converges.\nThe model trained is an out-of-the-box PyTorch ResNet-18.\nNote that the higher noise of the meta-validation accuracy is due to having a meta-batch size of $2$ to speed up experiments.\nWe smoothed the meta-validation curve with a TensorBoard smoothing weight of $0.8$.\nWe consistently saw that increases in meta-batch size lead to decreases in noise in the learning curves, but we didn't re-run these experiments since it can take up to a week to reproduce an episodic meta-learning run - even on a Quadro RTX 6000.\n}\n\\label{overfit_mini}\n\\end{figure}\n\n\\subsection{Infinite Number of Tasks}\n\nIt is interesting to highlight that meta-overfitting was not observed when the number of regression tasks is unbounded, as shown in Figure \\ref{no_overfit}.\nThis evidence suggests that, when the number of tasks is unbounded but sampled from a related set of tasks, meta-learning algorithms can leverage their power to adapt without meta-overfitting.\n\nTo measure the amount of meta-learning and the lack of feature re-use, we compute the dCCA value of the model as in Section \\ref{main_result} and observe a value of  $0.44 \\pm 0.11$.\nThis also implies that the degree of meta-learning is higher when the number of tasks is unbounded. \n\nThe main contribution is that this evidence suggests {\\it we need to rethink how we define the few-shot learning benchmarks for meta-learning}. \nWe hypothesize this is true because changing the property - like the number of concepts available to the learning - changes the behaviors of meta-learning algorithms. \nIn particular, MAML stops meta-overfitting.\nThis suggests to practitioners that MAML is a good algorithm for online or lifelong learning benchmarks - rather than deploying it to benchmarks with a fixed number of concepts.\nOverall, our evidence suggests that, as a research community, we are applying meta-learning algorithms to the wrong data sets and benchmarks.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.55\\linewidth]{figs\/no_overfit.png}\n\\caption{\nShows that meta-overfitting does not occur and perfect meta-generalization occurs when the number of tasks (functions) is unbounded when training with MAML.\nIn other words, the meta-train and meta-validation error are indistinguishable and decrease together as the meta-iterations increases.\nThe main difference of this figure with figure \\ref{meta_overfit1} is that in this one has a finite set of tasks using our synthetic benchmark while the other has an infinite set of tasks using the sinusoidal tasks suggested in \\cite{maml}.\n}\n\\label{no_overfit}\n\\end{figure}\n\n\\section{Effects of More Meta-Adaptation}\n\nIn this section, we show that increasing the number of inner steps for MAML during adaptation does not necessarily change the representation further as measured with dCCA (as in Equation \\ref{eq_ml}).\nIn addition, the meta-validation performance also does not change.\n\nTo show this, we obtain a single neural network meta-trained with MAML using a dataset as described in Section \\ref{def}.\nThen we plot how the representation changes and how the meta-validation error changes as a function of the inner steps.\nWe show this in Figures \\ref{ml_loss_vs_inner_steps_sigmoid_best} and \\ref{ml_loss_vs_inner_steps_relu_best}.\nWe observe that the MAML neural networks are robust to meta-overfitting with respect to the inner steps of its inner adaptation rule.\n\nNote that this is different from what was observed in Section \\ref{finite_metaoverfit}, because that section shows it as a function of the meta iterations (what is sometimes called outer iterations).\nIn addition, it is important to emphasize that the representation change in the plots is above the $0.12$ compared to previous work \\cite{Raghu}, supporting the main results of section \\ref{main_result}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]{figs\/ml_loss_vs_inner_steps_sigmoid_best.png}\n\\caption{\nShows 1) the lack of representation change and b) meta-validation change as the number of inner steps increases.\n1 is shown by the relative flatness of the blue and orange lines in the upper plot.\nSimilarly, 2 is shown by the flatness of the green line in the lower plot.\nIn particular, notice that we exponentially increase the inner steps from 1 to 2 to 32.\nThe models used are 4 layered FCNN trained with MAML with 1 inner step and 0.1 inner learning rate, selected using early stopping using the meta-validation set with the Sigmoid activation function.\nThe only difference of this figure with figure \\ref{ml_loss_vs_inner_steps_relu_best} is that this figure uses a sigmoid activation and the other one uses a ReLU.\nNote that this is the model used for figure \\ref{meta_overfit1}.\nNote the dCCA value remains above 0.12 suggesting lower degree of feature re-use.\n}\n\\label{ml_loss_vs_inner_steps_sigmoid_best}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\linewidth]{figs\/ml_loss_vs_inner_steps_relu_best.png}\n\\caption{\nShows 1) the lack of representation change and b) meta-validation change as the number of inner steps increases.\n1 is shown by the relative flatness of the blue and orange lines in the upper plot.\nSimilarly, 2 is shown by the flatness of the green line in the lower figure.\nWe want to emphasize that we exponentially increase the inner steps from 1 to 2 to 32.\nThe models used are 4 layered FCNN trained with MAML with 1 inner step and 0.1 inner learning rate, selected using early stopping using the meta-validation set with the ReLU activation function.\nThe only difference of this figure with figure \\ref{ml_loss_vs_inner_steps_sigmoid_best} is that this figure uses a ReLU activation and the other one uses a sigmoid.\nNote the dCCA value remains above 0.12 suggesting lower degree of feature re-use.\n}\n\\label{ml_loss_vs_inner_steps_relu_best}\n\\end{figure}\n\n\\section{Related Work}\\label{related_section}\n\nOh et al. \\cite{boil} show that one can encourage models to use less feature re-use purely algorithmically by setting the inner learning rate to zero for the final layer. \nThey showed BOIL outperforms MAML in both traditional few-shot learning (e.g. meta-trained on MiniImagent then meta-tested on MiniImagent) and cross-domain few-shot learning (meta-trained on MiniImagent then meta-tested on tiered ImageNet). \nIn particular, their cross-domain few-shot learning is similar in spirit to the synthetic task we propose in section \\ref{def}.\nHowever, note that we show that even MAML - an algorithm that has been shown to work by feature re-use \\cite{Raghu, boil} - can exhibit large representation changes if it is trained solely on a task that requires large feature changes.\nConcisely, we encourage rapid learning by only changing the task, while Oh et al. \\cite{boil} encourage it by changing the algorithm itself.\n\nGuo et al.'s \\cite{bscd_fsl} work is similar to ours in that they focus on defining a benchmark more appropriate for meta-learning and transfer learning. \nThey propose that meta-learning should be done in a fashion where the distribution of tasks sampled changes considerably when moving from meta-training to meta-evaluation.\nOur work is different in that we emphasize that the meta-training tasks themselves need to have diversity to be able to encourage meta-learning.\nAlthough Guo et al.'s \\cite{bscd_fsl} meta-evaluation procedure is excellent, we hypothesize - based on our results - that their benchmark won't have enough diversity to encourage large representation changes during meta-training.\nHowever, we conjecture that combining our ideas and theirs is a promising step for creating a better benchmark for meta-learning.\n\n\nSimilar work by Triantafillou et al. \\cite{Triantafillou2019} attempt to improve benchmarks by merging more data sets, but we hypothesize their data sets are not diverse enough to achieve this.\nIn terms of methods, our work is most similar to Raghu et al. \\cite{Raghu}, but they lack an analysis of the role of the tasks in explaining their observations.\nThere is also other work \\cite{Chen2019, Chen, Dhillon2019, Huang2019} that shows that a good representation is sufficient to achieve a high meta-accuracy on modern few-shot learning tasks e.g. MiniImagent, tiered-Imagenet, Cifar FS, FC100, Omniglot, \\cite{Tian2020}, which we hope to analyze in the future.\nWe conjecture in is imperative that a definition of meta-learning is developed and explicitly connected to the general intelligence.\nChollet \\cite{Chollet2019} takes this direction, but to our understanding the proposed definition is mostly focused on program synthesis. \nWe also hope that in the future a metric for AI safety is ubiquitously reported as suggested in Miranda et al. \\cite{foundationsmetalearning}.\n\n\\section{Discussion}\n\nIt is exciting evidence that by only changing the few shot learning benchmark, one can consistently show higher degrees of representation changes as measured by two different metrics. \nWe hypothesize this is the case because the meta-learning system has to be trained explicitly with a task that demands it to learn to adapt.\n\nAn important  discussion point is the lack of an authoritative definition for measuring meta-learning in our work and in the general literature.\nIn particular, in our work, we decided to not report any results with CKA.\nWe decided this because Ding et al. \\cite{Ding} showed that it's possible to remove up to 97\\% of the principal components of the weights of a layer until CKA starts to detect it.\nThus, we used dCCA which doesn't have the problem.\nIt instead has a higher variance, but it's easier to address this with experiment repetition sand error bars (which we did).\nHowever, we hypothesize it would be interesting to use and extend Orthogonal Procrustes as suggest by \\cite{Ding} in future work.\n\nThe most obvious gap in our work is a thorough analysis with a real world vision data set.\nWe hope to repeat our work with the hinted extension in section \\ref{related_section} benchmarks as suggested in \\cite{bscd_fsl, Triantafillou2019}. \n\nIn addition, Figures \\ref{ml_loss_vs_inner_steps_sigmoid_best}, \\ref{ml_loss_vs_inner_steps_relu_best} shows that as the number of inner steps increases, the dCCA does not increase.\nThis is somewhat surprising given the meta-overfitting results observed in section \\ref{meta_overfitting} and further experiments would be valuable.\n\n\n\n\\begin{ack}\nWe'd like to thank Intel for providing our team with access to their Academic Cluster Environment (ACE). \nTheir computational resources and support from their staff were essential to the successful completion of our project.\nIn addition, this work utilized resources supported by the National Science Foundation's Major Research Instrumentation program, grant 1725729, as well as the University of Illinois at Urbana-Champaign \\cite{Kindratenko2020}.\nWe'd like to acknowledge the work and authors of Anatome, TorchMeta and higher \\cite{anatome, torchmeta, higher} for making their code available and answering ours questions in their project's GitHub repository.\nWe'd like to acknowledge the weights and biases (wandb) framework for powerful tracking of experiments \\cite{wandb}.\nWe acknowledge the feedback from Open Philanthropy on the proposal on the foundations of meta-learning \\cite{foundationsmetalearning} that inspired this work.\nWe acknowledge the anonymous reviewers from NeurIPS for the valuable feedback for this work.\n\\end{ack}\n\n\n\n\\medskip\n\n\n\\printbibliography\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nChange detection obtains ground feature change information by comparing images from different periods.\nRemote sensing images have become common data for detecting changes in the surface due to their high spatial coverage and high time resolution \\cite{Hussain.2013}. At the same time, the increased spatial resolution of remote sensing images can provide more details of ground objects.\nIn-depth study of urban change is essential to promote sustainable urbanization \\cite{Huang.2017}. Therefore, UCD has become a research hotspot. \nUrban areas often have a wide variety of objects and strong regional heterogeneity. Ground objects, even in the same class, may have very different geometric shapes, and local features. In order to better analyze urbanization, different requirements are also put forward for the usage data.\n\\textbf{(1) Higher spatial resolution.}\n Higher spatial resolution images can provide more information to distinguish features between different images to obtain a clear boundary of change.\n\\textbf{(2) Richer prior information on land cover.} Knowing the prior information about land cover can detect the direction of change and analyze land cover changes.\n\\textbf{(3) Longer time series images.} \nThe changes in ground objects are time-dependent, and a more continuous sequence of images can realize time series analysis to monitor urban changes.\n\nWe collected public UCD datasets (Table~\\ref{tab:dataset}) and found that they have some limitations: \n\\textbf{(1) Lack of high spatial resolution images.} The image spatial resolutions of OSCD \\cite{Daudt.2018}, ZY3 \\cite{Zhang.2020}, and SZTAKI AirChange \\cite{Benedek.2009} are 10 m, 5.8 m, and 1.5 m respectively. Although the resolution is gradually increasing, it still cannot meet the requirements of UCD, especially for buildings.\n\\textbf{(2) Lack of semantic annotation.} ABCD \\cite{Fujita.2017}, LEVIR-CD \\cite{Chen.2020}, WHU Building \\cite{Ji.2019b} only label building-related changes, and Season-varying \\cite{Lebedev.2018} directly labeled land cover related changes, all of them are lack of semantic changes. It is difficult to perform multi-class change detection to obtain fine change directions. Although HRSCD \\cite{CayeDaudt.2019} provided the direction of changes, its labeling accuracy is only\n80\\% to 85\\%. In addition, the ground objects of the urban area are classified into five categories,\nwhich is relatively rough and difficult to reflect the changes of typical objects in urban areas.\n\\textbf{(3) Lack of long-range multi-temporal images.} The above-mentioned public datasets only contain bi-temporal images of the same area. Therefore, it is difficult to obtain satisfactory refined detection results for UCD.\n\n\\begin{table*}[!t]\n  \\centering\n  \\caption{ Open datasets in remote sensing change detection } \\label{tab:dataset}\n   \n   \n    \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{cccccccc}\n      \\toprule\n      {Dataset}& Resolution (m) & \\#Images & Image size (Pixel)  & Years & Change &Interesting object& Classes \\\\\n      \\midrule\n\n      OSCD \\cite{Daudt.2018}  & 10 & 24 & $600\\times600$  & 2 & Binary change & Land cover&- \\\\\n    \n      ZY3 \\cite{Zhang.2020}  & 5.8 & 1 & $1154\\times740$  & 2  & Binary change & Land cover&- \\\\\n\n       SZTAKI AirChange \\cite{Benedek.2009}  & 1.5 & 13 & $952\\times640$  & 2 & Binary change & Land cover&-\\\\\n  \n     \n       AICD \\cite{Bourdis.2011}  & 0.5 & 1000 & $800\\times600$  & 2 & Binary change & Land cover&-\\\\\n      \n       {ABCD\\cite{Fujita.2017}}  & {0.4} & 8506\/8444 & $160\\times160$\/$120\\times120$  &{2} &Binary change & {Building} &-\\\\\n\n        LEVIR-CD \\cite{Chen.2020}  & 0.5 & 637 & $1024\\times1024$  & 2 & Binary change & Building&-\\\\\n\n       WHU Building \\cite{Ji.2019b}& 0.2 & 1 & $32207\\times15354$  & 2  & Binary change & Building&-\\\\\n\n      Season-varying \\cite{Lebedev.2018}&0.03-1&16000& $256\\times256$&2&Binary change&Land cover&-\\\\\n      HRSCD \\cite{CayeDaudt.2019}  & 0.5 & 291 & $10000\\times10000$  & 2 & Semantic change & Land cover&5 \\\\\n      Hi-UCD (ours)  & 0.1 & 1293 & $1024\\times1024$  & 3 & Semantic change & Land cover & 9 \\\\\n      \\bottomrule\n    \\end{tabular}}\n\n\\end{table*}\n\nTo solve these problems, we introduce a large-scale semantic annotated ultra-high resolution UCD dataset named Hi-UCD.\nOur dataset uses aerial images with a spatial resolution of 0.1m to clearly show the spatial details of ground objects and capture small changes in them.\nHi-UCD obtains fine semantic changes of objects by labeling the land cover classes of images in different periods. We have selected 9 land cover classes including natural and artificial objects to achieve full coverage of urban ground objects.\nIn addition, Hi-UCD contains images of three time phases in the same area, which is conducive to studying the temporal correlation of changes in ground features.\nOverall, Hi-UCD is a large-scale, multi-temporal, ultra-high resolution urban semantic change detection data set, which can realize comprehensive detection and analysis of urban changes.\nTo verify the validity of Hi-UCD, we selecte the classic method in the binary and multi-class change detection task to conduct the experiments, finally provide a benchmark.\n\n\n\n\n\\section{Hi-UCD Dataset}\n\nHi-UCD focuses on urban changes and uses ultra-high resolution images to construct multi-temporal semantic changes to achieve refined change detection.\nThe study area of Hi-UCD is a part of Tallinn, the capital of Estonia, with an area of 30 $km^2$. The Estonian Land Board provides aerial images\\footnote{ Orthophoto, Land Board 2020.} taken by Leica ADS100-SH100 in 2017, 2018, and 2019, with topographic database\\footnote{Estonian Topographic Database, Land Board 2020.} for the area. Hi-UCD obtained semantic changes by annotating the land cover classes in different periods. We have considered topographic documents and changes in ground objects to\nselect 9 land cover classes to achieve complete coverage of ground objects in Estonia.\n Finally, we cut each year's images into patches with a size of $1024\\times1024$, and filter out patches with change pixels more than 200 to form the Hi-UCD dataset. There are 359 image pairs in 2017-2018, 386 pairs in 2018-2019, and 548 pairs in 2017-2019, including images, semantic maps and change maps at different times.\n\\begin{figure}[!ht]\n  \\centering\n  \\includegraphics[scale=0.4,trim=0 140 150 0,clip]{figure\/classess.pdf}\n  \\caption{Examples of Hi-UCD dataset for images, semantic maps and change maps.}\\label{fig:example}\n\\end{figure}\n In Figure \\ref*{fig:example}, examples of the Hi-UCD dataset are given. Compared with other public datasets, its characteristics are as follows:\n\n \\begin{enumerate}\n   \\item \\textbf{Ultra-high spatial resolution.}  \n   Hi-UCD uses aerial images with a spatial resolution of 0.1m, which is the highest resolution in public data. In these images, the geometric shape of the ground objects is clear, and the boundary is obvious, which provides rich spatial texture information. Therefore, it is conducive to detecting local changes of ground objects and realizing refined change detection.\n\n  \\item \\textbf{Multi-temporal images.} Hi-UCD contains the images of the three years from 2017 to 2019, and gives the semantic annotation and change mask every two years (2017-2018, 2018-2019, 2017-2019). Changes are highly time-dependent, and multi-temporal data can provide temporal features, which helps researchers to conduct long-term serial studies and improve the temporal precision of UCD. In addition, the images of different years have undergone orthorectification without registration errors. At the same time, they were taken in the same season, which greatly reduced the influence of seasonal changes in vegetation.\n\n  \\item \\textbf{Semantic annotation.} Considering typical urban objects and change-related objects, we developed Hi-UCD semantic annotation categories.\n   There are 9 types of objects, including natural objects (water, grassland, woodland, bare land), artificial objects (Building, greenhouse, road, bridge), and others (change-related), basically include all types of urban land cover in Estonia. The above categories are mapped with the shapefile layer in the Estonian Topographic Database (ETD). Due to the inconsistency of the vector boundaries between different years in ETD, the buildings are based on the vector of each year, and the other classes are based on the vector of 2019. Through visual interpretation, we check the topographic shapefiles and modify them. Meanwhile, we compare images of different years to determine the relevant objects of the change and add the category \"other\". Finally, the binary and multi-classes change masks generated through the semantic annotation results.\n  \\end{enumerate}\nBecause of these characteristics, Hi-UCD is full of challenges: (1) the increase in spatial resolution has aggravated the shadows and occlusions in the image.\n(2) The changes in uninteresting ground objects such as cars will cause serious background noise during change detection. (3) High-rise buildings are tilted and geometrically mismatched due to different shooting angles at different times. \n(4) The number of category transitions caused by changes is much greater than the number of semantic categories, which increases the difficulty of multi-classes change detection task. In summary, Hi-UCD is far more diverse, comprehensive, and challenging.  \n\n\\section{Benchmark}\n\n\nIn order to establish a fair benchmark,  we evaluated the classic methods of binary and multi-class change detection\nunder a unified experimental setting and data division conditions.\n\n\\textbf{Methods} After decades of development, change detection methods have evolved from pixel-based direct comparison to data-driven deep learning methods \\cite{Hussain.2013,Tewkesbury.2015,Shi.2020}. \nWe chose different methods according to the different detection task. \nFor binary change detection, these methods are the commonly used as comparison methods, including traditional methods (change vector analysis (CVA) \\cite{Malila.1980}, multivariate alteration detection (MAD) \\cite{Nielsen.1998}, the regularized iteratively reweighted MAD (IRMAD) \\cite{Nielsen.2007}), and deep learning methods (FC\\_EF \\cite{CayeDaudt.2018}, FC\\_Siam\\_diff \\cite{CayeDaudt.2018}, FC\\_Siam\\_diff \\cite{CayeDaudt.2018}, FC\\_Res\\_EF \\cite{CayeDaudt.2019}).\nFor multi-class change detection, the commonly used method is the post-classification comparison. After classifying images of different time phases through a classifier, like support vector machines [18], random forest [19], convolutional neural network [20] are compared to obtain change information. Considering the complexity of the objects in Hi-UCD, we only chose the classic semantic segmentation deep learning networks in computer vision ( Deeplab v3 \\cite{chen2017rethinking.2017}, Deeplab v3+ \\cite{Chen_2018_ECCV}, PSPNet \\cite{Zhao_2017_CVPR} ) and  remote sensing ( FarSeg \\cite{Zheng.2020b} ) for classification to obtain multi-class changes.\n\n\\textbf{Settings} We used 300 pairs of images in 2017 and 2018 for training, the remaining 59 pairs as the validation set, and 386 pairs of images in 2018 and 2019 for testing.\nIn the traditional method, the clustering method proposed in \\cite{Celik.2009} was used to obtain the change mask. For all deep learning methods, the learning rate was 0.01 and use a polynomial decay with a decay factor of 0.9. The batch size was 4 and trained on a single GPU.  The stochastic gradient descent (SGD) was used for optimization with weight decay of 0.0001 and a momentum of 0.9. For data augmentation, horizontal and vertical flip, rotation of 90 degrees and random cut ( $size=(512,512)$) were adopted during training. \nIn binary change detection, the number of iterations is 10k, and the loss function is the binary cross-entropy and dice loss. While in classification,  we used the cross-entropy loss function with 20k iterations. The backbone used for classification methods was ResNet-50, which was pre-trained on ImageNet \\cite{Deng.2009}. \n\n\\textbf{Metrics}\n We used overall accuracy (OA), kappa coefficient to evaluate the overall performance of the change detection results. For binary change detection methods, we used intersection over union (IoU) to only evaluate the ability to detect changes. In addition, We added mean intersection over union (mIoU) to evaluate algorithm performance in classification and multi-class change detection. The parameters and number of operations measured\n by multiply-adds (MAdd) calculated by a tensor with a size of $1\\times C\\times256\\times256 (C=3,6)$ are given to show deep learning model complexity.\n The accuracy evaluation results of different methods are shown in Table~\\ref{tab:result}.\n\n \\textbf{Analysis} In Table~\\ref{tab:firsttable},  most binary change detection methods can effectively detect unchanged ground objects, the IoU of change does not exceed 50\\%.\n IRMAD performed the best with kappa 8\\% higher than the other traditional methods. Deep learning methods are significantly higher than traditional methods in all metrics, which fully reflects the potential of deep learning in change detection. Traditional methods cannot distinguish false changes caused by shadows, occlusions, and uninteresting objects, while deep learning methods rely on their powerful learning capabilities to effectively remove background noise.\n Among them, the FC\\_Siam\\_diff method is slightly better than FC\\_Siam\\_conc in all metrics. FC\\_EF improves IoU of change by nearly 7\\%. After adding the residual module, FC\\_EF\\_Res increased by nearly 5\\% and has the smallest parameters.  \n In Table~\\ref{tab:secondtable}, the metrics of all methods for land cover classification in 2018 are higher than in 2019. Because the change of the ground objects leads to differences in the distribution of ground features at different times. \n Through post-classification to get the results of multi-class change detection, there are many false alarms at the boundary of the ground objects in multi-class change results. In Table~\\ref{tab:firsttable}, although Deeplab v3 obtains the best accuracy in multi-class change detection, it also has the highest computational complexity. \n Besides, the accuracy of the change highly depends on the accuracy of the classification,\nHow to obtain reliable multi-class change detection results in urban areas is still a problem that needs research. \n\n\n\n\\begin{table} [tb]\n  \\caption{The quantitative evaluation of the baseline methods for Hi-UCD}\n  \\label{tab:result}\n  \\centering\n  \\subtable[ Change detection accuracy ]{\n    \\resizebox{0.48\\linewidth}{!}{\n    \\begin{tabular}{ccccccccc}\n      \\toprule\n      {Binary change detection} &\n      \\#Params (M)&\n      MAdds (B)&\n       OA (\\%) &\n       Kappa (\\%) & \n       IoU (\\%) &\n     \n     \n     \n        \\\\\n       \\midrule\n      CVA \\cite{Malila.1980} &-&-&40.79\t&\t3.98\t&\t11.51\t\t\\\\\n       MAD \\cite{Nielsen.1998}&-&-\t&\t88.95\t&\t3.64\t&\t3.41\t\\\\\n       IRMAD \\cite{Nielsen.2007}&-&-\t&\t88.08\t&\t11.78\t&\t9.38\t\t\\\\  \n       FC\\_Siam\\_conc \\cite{CayeDaudt.2018}  &1.546&5.8& 91.25 \t&\t44.16\t&\t32.26 \t \t \t\\\\\n       FC\\_Siam\\_diff \\cite{CayeDaudt.2018}&1.35&4.67 &91.74 \t&\t47.67  \t&\t35.19 \t \t \t\\\\\n       FC\\_Siam\\_EF \\cite{CayeDaudt.2018}&1.35&3.54 &91.50 \t&\t54.92 \t\t&\t42.51 \t \t \t \t\\\\\n        Siam\\_Res\\_EF \\cite{CayeDaudt.2019}& 1.104&1.98&93.05 \t&\t60.67 \t&\t47.62  \t \t \\\\\n       \\midrule\n       { Multi-class change detection} &\n       \\#Params (M)&\n       MAdds (B)&\n         OA (\\%) &\n         Kappa (\\%) & \n         mIoU (\\%) \\\\\n         \\midrule\n   \n       Deeplab v3 \\cite{chen2017rethinking.2017}&39.046&80.58& 76.82\t&\t29.48\t&\t17.51\\\\\n       Deeplab v3+ \\cite{Chen_2018_ECCV}&39.897&26.34&75.83\t&\t28.31\t&\t15.65\\\\\n       PSPNet \\cite{Zhao_2017_CVPR}&46.588&88.60\t& 76.17\t&\t28.49\t&\t14.29\\\\\n       FarSeg \\cite{Zheng.2020b}&33.881&28.68& 73.58\t&\t25.68\t&\t13.34\\\\\n        \\bottomrule\t\n         \\end{tabular}}\n         \\label{tab:firsttable}\n  }\n  \\hfill\n  \\subtable[Land cover accuracy ]{  \n    \\resizebox{0.48\\linewidth}{!}{      \n    \\begin{tabular}{ccccccc}\n      \\toprule\n       Methods&\n       \\#Params (M)&\n       MAdds (B)&\n       Year&\n        OA (\\%) &\n        Kappa (\\%) & \n        mIoU (\\%) \n        \\\\\n        \\midrule\n        \\multirow{2}{*}{Deeplab v3 \\cite{chen2017rethinking.2017}}\n        &\\multirow{2}{*}{39.046}&\\multirow{2}{*}{40.29}&2018\t&\t87.59\t& 83.94 &\t72.39 \\\\\n        &&&2019&77.19&69.74&42.55\\\\\n        \\midrule\n        \\multirow{2}{*}{Deeplab v3+ \\cite{Chen_2018_ECCV}} &\\multirow{2}{*}{39.897 }&\\multirow{2}{*}{13.17}& 2018\t&\t86.28\t& 82.22&\t67.24 \\\\\n        &&&2019&76.23&68.45&37.98\\\\\n        \\midrule\n        \\multirow{2}{*}{PSPNet \\cite{Zhao_2017_CVPR}} &\\multirow{2}{*}{46.588 }&\\multirow{2}{*}{44.30} &2018\t&\t86.50\t&82.50&69.88 \\\\\n        &&&2019&74.50&65.79&37.37\\\\\n        \\midrule\n        \\multirow{2}{*}{FarSeg \\cite{Zheng.2020b}} &\\multirow{2}{*}{33.881 }&\\multirow{2}{*}{14.34 }& 2018\t&\t86.78\t&82.88 &69.98 \\\\\n        &&&2019&75.58&64.83&36.02\\\\\n        \\bottomrule\n         \\end{tabular}}\n         \\label{tab:secondtable}\n  }\n  \\end{table}\n\n\n\n\n\\section{Conclusion}\nIn this article, we introduce a new multi-temporal ultra-high-resolution aerial image UCD dataset, which has rich semantic annotations to detect more details of urban change.\nAt the same time, we have established a benchmark for UCD in binary and multi-class change detection tasks.\nIn the next work, we will continue to expand the scale of the dataset and provide different large-area test sets to verify the generalization and migration of the algorithm better. We hope the release of Hi-UCD will promote the development of UCD.\n\n\n\\medskip\n\n\\ack\nThis work was supported by National Natural Science Foundation of China under Grant Nos.41771385, 41801267 and 42071350. The authors would like to thank the Estonian Land Board for acquiring and providing the data used in this study.\n\\small\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction to the problem}\n\\noindent \nConsider $u(x,t)$ as a function describing the temperature at the time $t$ of a point $x$ in an infinite isolated rod, being hence a solution of the heat equation. As usual, it is assumed that heat has spread from hotter zones to colder ones. Now, if one considers an ancient non-negative solution, the diffusive process has already gone on for an infinite amount of time, and it is reasonable to question if $u(x,t)$ has become constant. This fact, stated in this way, is generally false, as shown by the following examples:\n\\begin{equation}\\label{examples}\n    u_1(x,t)= e^{x_N+t}, \\quad u_2(x,t)=e^{-t} \\sin(x_1), \\quad x \\in {\\mathbb R}^N.\n\\end{equation}\nThe two functions above are {\\it eternal} solutions of the heat equation, i.e. solutions in ${\\mathbb R}^N \\times {\\mathbb R}$. We call {\\it ancient} solutions those solutions that solve the parabolic equation in $ {\\mathbb R}^N \\times (-\\infty, T)$ for some time $T \\in {\\mathbb R}$. In line with the literature, we call {\\it Liouville property} any rigidity condition that ensures the triviality of solutions. It is clear from $u_1$ that a sign condition is not enough to confirm our suspect, while the sign-changing solution $u_2$ shows that boundedness at a fixed time is not enough. Although Appel \\cite{Appel} already proved in 1892 that an ancient solution to the heat equation which is two-sided bounded (as for instance $0 \\leq u(x,t) \\leq M$) is constant, the first optimal parabolic Liouville theorem for ancient solutions was found in 1952 by Hirschman (see \\cite{Hirschman}, Bear \\cite{Bear} and Widder \\cite{Widder}, \\cite{WidderBook} for the case $N=1$), stating that a non-negative ancient solution to the heat equation is constant if one adds the assumption that, for a time $t_o<T$,\n\\begin{equation}\\label{log}\n    \\lim_{r \\uparrow + \\infty} \\frac{\\log(\\sup_{|x|<r}u(x,t_o))}{r} \\leq 0, \\qquad \\text{that is,} \\qquad u(x,t_o) \\leq e^{o(|x|)} \\quad \\text{as} \\quad |x|\\rightarrow +\\infty.\n\\end{equation} This result was sharp in the sense that any function of the kind\n\\begin{equation*}\\label{count-Hirsch}\n    u(x,t)= e^{a^2t} \\cosh{ax}\n\\end{equation*} shows that if \\eqref{log} above is violated then $u(x,t)$ is not necessarily constant; but \\eqref{log} is just a condition on the space variables. Sub-exponential optimal growth conditions have been generalized to different metric contexts; see for instance \\cite{Sunra} and references therein. Not much later, in 1958, Friedman gave a condition on the behavior of non-negative ancient solutions to more general second order parabolic equations as\n\\begin{equation}\\label{seond-order}\n    \\partial_t u(x,t) = \\sum_{i,j=1}^N a_{i,j}\\frac{\\partial^2 u(x,t)}{\\partial x_i \\partial x_j} + \\sum_{i=1}^N b_i \\frac{\\partial_i u(x,t)}{x_i} +c u(x,t),  \\quad \\text{in} \\quad {\\mathbb R}^N \\times {\\mathbb R}_+,\n\\end{equation} being $b_i,c$ real numbers and $\\{a_{i,j}\\}_{i,j}$ a positive matrix. Now the assumption concerns infinite past times as\n\\begin{equation} \\label{fried}\n\\lim_{t\\downarrow - \\infty} \\frac{\\log(u(0,t))}{t}= c+ \\gamma, \\qquad \\gamma>0,\\quad  c+\\gamma \\ge 0. \\end{equation} \\noindent See \\cite{Friedman} for the result and \\cite{Eidelman} for the earlier case of systems. Furthermore, conditions guaranteeing the stabilization of the solution to a constant were studied for a fixed space variable (see \\cite{Eidelman-Kamin-Tedeev} and its references for an account). This short preamble is just to highlight that different assumptions, mainly on the second bound, may be requested to solutions of these parabolic equations in order to ensure Liouville property; it is therefore an incomplete list. The literature on these rigidity results is wide, so we refer the reader to the book \\cite{QS-libro} and the survey \\cite{Kogoj} for a more complete account.\\vskip0.1cm\n\\noindent \nThe heat equation can be regarded as a special case of the anisotropic $p$-Laplacian equation\n\n\\noindent \\begin{equation} \\label{EQ}\n    \\partial_t u= \\sum_{i=1}^N \\partial_i (|\\partial_i u|^{p_i-2} \\partial_i u),  \n   \n\\end{equation} \\noindent\nwhen $p_i\\equiv 2$ for all $i=1,\\dots,N$. When $2<p_i< \\bar{p}(1+1\/N)$, this equation describes the effect of competing diffusions along coordinate axes in finite speed of propagation (see \\cite{Ant-Sh}, \\cite{Mosconi}, \\cite{Ruzicka} for an introduction to the parabolic problem). In one spatial dimension, the equation \\eqref{EQ} is a one-dimensional $p$-Laplacian, which has a very interesting change of behavior from the degenerate case ($p>2$) to the singular one ($1<p<2$). Roughly speaking, the solutions to $p$-Laplacian singular equations behave more like solutions to elliptic equations and only one bound is enough to infer that they are constant (see \\cite{DGV-Liouville} for more details). On the other hand, in the degenerate case two bounds are required to infer a Liouville property; see Section \\ref{LiouvilleSection} below for a counterexample. In this paper we show some Liouville properties for non-negative solutions to \\eqref{EQ} for a range of $p_i$s which is degenerate and allows a finite speed of propagation. In many physical circumstances, this is a more reasonable assumption than the sudden infinite expansion of the support of solutions to the heat equation.\\vskip0.2cm \n\\noindent The theory of regularity for solutions to \\eqref{EQ}, even if much investigated, is still incomplete and fragmented (see, e.g., \\cite{MingRadu} and \\cite{Mar-survey} for an account on the elliptic case). The Liouville properties that we are about to describe are entailed by recent Harnack estimates, obtained with an approach of expansion of positivity. This has been shown relying on the behavior of abstract fundamental solutions in \\cite{Ciani-Mosconi-Vespri}. Here we start from the aforementioned Harnack inequality (see Section \\ref{Preliminaries}, estimate \\eqref{Harnack}), which is formulated in an intrinsic geometry (\\ref{anisocubes}) reflecting the natural scaling of the equation, and study some rigidity connections between local and global behavior of solutions.\\newline\nSimilarly to the Liouville property inferred by Hirschman, we will prove that it is sufficient to have a one-sided bound (say, from below) and an estimate from the other side (say, from above), just for a fixed time. If these conditions are met, solutions are forced to be constant (Theorem \\ref{Liouville1}). This clearly implies that a solution which is bounded both from above and below is bounded; on the other hand, it is unreasonable to expect that just a one-sided bound suffices (see the example in Section \\ref{LiouvilleSection}) for our range of $p_i$s. As a known fact, we comment that a precise decay on oscillation given by H\\\"older continuity estimates is enough as Liouville property. This decay is usually easier to show than a complete Harnack inequality, and as such deserves its own attention: already in the range $\\bar{p}(1+1\/N)\\leq p_i<\\bar{p}(1+2\/N)$, although continuity is expected by the regularizing properties of diffusion, no Harnack estimate may be available, because the competition among the diffusions is too strong (see for instance \\cite{AS}). It would be of interest to compare this behavior to the one in porous materials by the sole control on the oscillation, for example as in \\cite{Eurica}.\\newline\nOn a similar track to Friedman's result, we prove that, fixed any spatial point and assuming that the solution is bounded at infinity in time, again it is forced to be constant (Theorem \\ref{Liouville2}). Finally we state a Harnack inequality that frees the time variable to be intrinsic (Theorem \\ref{WeakHarnack}, see \\cite{DB} for the isotropic counterpart): being the Harnack estimate not anymore intrinsic in time, the estimate is more suitable for an application to rigidity. Moreover, this turns out to be useful to determine the optimal growth on the initial data when $|x|\\rightarrow \\infty $ for the solvability of the Cauchy problem for \\eqref{EQ} (see for instance \\cite{DB-He}). Clearly, this implies that the domain where the equation is solved must be, in turn,\n`compatible' with the anisotropy of the diffusion: this is certainly the case for ancient solutions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Structure of the paper}\nSection \\ref{Preliminaries} is devoted to set up the functional framework and to recall some known properties of the solutions, as the existence of fundamental solutions, comparison principles and the Harnack inequality. In Section \\ref{LiouvilleSection} we prove the Liouville-type results, while Section \\ref{WHSection} concerns an alternative formulation of the Harnack inequality, which turns out to be locally equivalent to the known one. Finally we gather in the Appendix a detailed proof of the H\\\"older continuity of solutions.\n\n\\section*{Notations}\n\\begin{itemize}\n\\item[-] We define the following function of $p_i$s, called {\\it harmonic mean}: $\\,\n\\bar{p}=N(\\sum_{i=1}^N1\/p_i)^{-1}$.\\newline We suppose that $ 1<p_1\\leq p_2 \\leq \\ldots \\leq p_N$ are ordered, as well as $\\bar{p}<N$.\n\\vskip0.1cm \\noindent \n    \n\\item[-] For any $\\rho, \\theta>0$ and $x\\in{\\mathbb R}^N$, we denote by $K_{\\rho}(x) \\subset {\\mathbb R}^N$ the cube of side $2\\rho$ centered at $x$.\n\\noindent Let $x_o+\\mathcal{K}_{\\rho}(\\theta)$ stand for the anisotropic cube of radius $\\rho$, ``magnitude'' $\\theta$, and center $x_o$, i.e.,\n\\begin{equation*}\\label{anisocubes}\nx_o+\\mathcal{K}_{\\rho}(\\theta)= \\prod_{i=1}^N\\bigg{\\{}|x-x_{o,i}|<\\theta^{{(p_i-\\bar{p})}\/{p_i}}\\rho^{{\\bar{p}}\/{p_i}}\\bigg{\\}}.\n\\end{equation*}\nIf either $\\theta=\\rho$ or $p_i=p$ for all $i=1,\\ldots,N$, then $x_o+\\mathcal{K}_{\\rho}(\\theta)=K_{\\rho}(x_o)$.\n\n\\item[-] For any $\\rho, \\theta,C >0$ and $(x_o,t_o) \\in {\\mathbb R}^{N+1}$, we consider the following anisotropic cylinders:\n\\begin{equation*}\\label{cylinders}\n\\begin{cases}\n\\text{centered: }(x_o,t_o)+\\mathcal{Q}_{\\rho}(\\theta,C)=\n(x_o+\\mathcal{K}_{\\rho}(\\theta) )\\times (t_o-\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}},t_o+\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}});\\\\\n\\text{forward: }(x_o,t_o)+\\mathcal{Q}^+_{\\rho}(\\theta,C)= (x_o+\\mathcal{K}_{\\rho}(\\theta) )\\times [t_o,t_o+\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}});\\\\\n\\text{backward: }(x_o,t_o)+\\mathcal{Q}^-_{\\rho}(\\theta,C)=\n(x_o+\\mathcal{K}_{\\rho}(\\theta) )\\times (t_o-\\theta^{2-\\bar{p}}(C\\rho)^{\\bar{p}},t_o].\n\\end{cases}\n\\end{equation*} \\noindent We omit the index $C$ when the constant is clear from the context. \\vskip0.1cm \\noindent \n\\item[-] For $\\Omega \\subset \\subset {\\mathbb R}^N$, i.e., $\\Omega$ open and bounded set in ${\\mathbb R}^N$, we denote with $\\Omega_T= \\Omega \\times [-T,T]$, $T>0$, the parabolic domain, and for $s\\in {\\mathbb R}$ with $S_s= {\\mathbb R}^N \\times (-\\infty, s)$ the space strip.\\vskip0.1cm \\noindent \n\\item[-] We adopt the nowadays classic convention that constants may change from line to line.\n\\end{itemize}\n\n\n\n\n\n\\section*{Acknowledgements}\n\\noindent \nWe are grateful to S.A. Marano and V. Vespri for encouraging us toward this project. We wish to thank professor S. Mosconi for his precious suggestions and Emanuele Macca for a numerical insight about Barenblatt-type solutions. Moreover we are indebted with Eurica Henriques for the careful reading of the manuscript and for pointing out an early mistake about H\\\"older continuity of solutions. Finally, S. Ciani is supported by the department of Mathematics of the Technical University of Darmstadt, and U. Guarnotta is supported by: (i) PRIN 2017 `Nonlinear Differential \nProblems via Variational, Topological and Set-valued Methods' (Grant \nNo. 2017AYM8XW) of MIUR; (ii) GNAMPA-INdAM Project \nCUP$\\underline{\\phantom{x}}$E55F22000270001; (iii) grant `PIACERI \n20-22 Linea 3' of the University of Catania. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries and Tools of the Trade} \\label{Preliminaries}\n\\noindent \nWe begin with the definition of local weak solution. For $\\Omega \\subseteq {\\mathbb R}^N$ open rectangular domain and $T>0$, the Banach spaces \n\\[W^{1,{\\bf{p}}}_{loc}(\\Omega):= \\{ u \\in W^{1,1}_{loc}(\\Omega) |\\,  \\partial_i u \\in L^{p_i}_{loc}(\\Omega) \\}, \\]\n\\[ L^{{\\bf{p}}}_{loc}(0,T;W^{1,{\\bf{p}}}_{loc}(\\Omega)):= \\{u \\in W^{1,1}_{loc}(0,T;L^1_{loc}(\\Omega))|\\, \\partial_i u \\in L^{p_i}_{loc}(0,T;L^{p_i}_{loc}(\\Omega))   \\}, \\]\nare called anisotropic spaces (see, for instance, \\cite{Ant-Sh}). When $\\bar{p}>N$ and $\\partial \\Omega$ is regular enough, the space $W^{1,{\\bf{p}}}(\\Omega)$ is embedded in the space of H\\\"older continuous functions \\cite{VenTuan}. A function \\[ u \\in C^0_{loc}(0,T; L^2_{loc}({\\mathbb R}^N)) \\cap L^{\\bf{p}}_{loc}(0,T;W^{1,{\\bf{p}}}_{loc}({\\mathbb R}^N))\\] is called a {\\it local weak solution} of \\eqref{EQ} in $S_T$ if, for all $0<t_1<t_2<T$ and any $\\varphi \\in C^{\\infty}_{loc}(0,T;C_o^{\\infty}({\\mathbb R}^N))$, it holds\n\\begin{equation} \\label{anisotropic-localweaksolution}\n\\int u \\varphi \\, dx \\bigg|_{t_1}^{t_2}+ \\int_{t_1}^{t_2} \\int (-u \\, \\varphi_t + \\sum_{i=1}^N \\, |\\partial_i u|^{p_i-2} \\partial_i u \\, \\partial_i \\varphi) \\, dx dt=0,\n\\end{equation} \\noindent where the integral is assumed to be taken in ${\\mathbb R}^N$ when no otherwise specified. When $\\Omega\\subset {\\mathbb R}^N$ is bounded we still call local weak solution to \\eqref{EQ} in $\\Omega_T$ a function satisfying the above requirements with ${\\mathbb R}^N$ replaced by $\\Omega$.\nHereafter we suppose that\n\\begin{equation}\\label{pi}\n    2<p_1\\leq p_N<\\bar{p}(1+1\/N),\\qquad \\bar{p}<N.\n\\end{equation}\nMoreover, throughout the paper, the constants $C_i$, $i=1,2,3$, are the ones of Theorem \\ref{Harnack-Inequality} below.\\newline\nNow we recall the main tools for our proofs, namely, the Harnack inequality, the existence of an abstract Barenblatt-type solution, and the comparison principle.\n\\begin{theorem} \\label{Harnack-Inequality}\nLet $u\\ge 0$ be a  local weak solution to \\eqref{EQ} in $\\Omega \\times [-T,  T]$ and let \\eqref{pi} be valid. Suppose that $u(x_o,t_o)>0$ for a Lebesgue point $(x_o,t_o) \\in \\Omega_T$ for $u$. Then there exist $C_{1}\\ge 0, C_3\\ge C_2\\ge 1$, depending only on $N$ and the $p_{i}$s, such that, letting $\\theta=u(x_o,t_o)\/C_1$, it holds\n\\begin{equation}\\label{Harnack}\n  \\frac{1}{C_{3}}\\sup_{x_o+\\mathcal{K}_{\\rho}(\\theta)}u(\\,\\cdot\\, , t_o - \\theta^{2-\\bar p}\\, (C_{2}\\, \\rho)^{\\bar p} )\\le  u(x_o,t_o) \\le C_{3} \\inf_{x_o+\\mathcal{K}_{\\rho}(\\theta)} u(\\,\\cdot\\, ,   t_o + \\theta^{2-\\bar{p}}\\, (C_{2}\\, \\rho)^{\\bar{p}})\n    \\end{equation}\n    whenever \n    \\begin{equation} \\label{side-condition}\n    \\theta^{2-\\bar p}\\, (C_{3}\\,  \\rho)^{\\bar p}<T-|t_o| \\qquad \\text{and} \\qquad x_o+\\mathcal{K}_{C_{3}\\, \\rho}(\\theta)\\subseteq \\Omega.\\end{equation}\n\\end{theorem}\n\n\n\\noindent The assumption $u(x_o,t_o)>0$ is defined by a suitable limit process, as customary. Semi-continuity clarifies this definition, as long as a theoretical maximum principle is in force (see \\cite{CianiGuarnotta}, \\cite{Mosconi}, \\cite{Liao} for an account). \nWe stress this definition because Theorem \\ref{Harnack-Inequality} has been proven in \\cite{Ciani-Mosconi-Vespri} without the assumption of H\\\"older continuity of solutions, that can be shown (see Section \\ref{Appendix}) to be a sole consequence of \\eqref{Harnack}. This important property has been faced several times in the past, with imprecise proofs or an unclear geometric setting. For this reason, and in order to explain the main adversities that anisotropic diffusion obliges us to face, we include in Section \\ref{Appendix} the proof of local H\\\"older continuity of solutions to \\eqref{EQ}, which follows the Moser's ideas \\cite{Moser} through an appropriate anisotropic intrinsic geometry. Taking for granted the continuity of solutions, in what follows we will refer directly to the pointwise values of solutions.\\vskip0.1cm \\noindent Secondly, let us comment Theorem \\ref{Harnack-Inequality} from a global point of view: if we pick a point $(x_o,t_o) \\in \\Omega_T$ where $u$ is positive, it is possible to `detect' the sets where the pointwise controls \\eqref{Harnack} hold true. This is the core of the next proposition.\n\n\\begin{proposition}\\label{paraboloids}\nSuppose the assumptions of Theorem \\ref{Harnack-Inequality} to be satisfied for $(x_0,t_0) \\in \\Omega\\times[-T,T]$. Then\n\\begin{equation} \\label{estimate-paraboloid}\n\\inf_{\\mathcal{P}_\\theta^+(x_0,t_0)} u \\ge u(x_0,t_0)\/C_3 \\qquad \\text{and}\\qquad \\sup_{\\mathcal{P}_\\theta^-(x_0,t_0)}u \\leq C_3 u(x_0,t_0),\\end{equation}\nwhere, setting $\\theta= u(x_o,t_o)\/C_1$, the paraboloids $\\mathcal{P}^+_{\\theta}(x_o,t_o)$ and $\\mathcal{P}^-_{\\theta}(x_o,t_o)$ are defined by\n\\[\n\\mathcal{P}_\\theta^+(x_0,t_0)= \\bigg{\\{}(x,t) \\in \\Omega_T:\\, \\,  C_2^{\\bar{p}} |x_i-x_{0,i}|^{p_i}\\theta^{2-p_i}\\leq  (t-t_0)\\leq C_2^{\\bar{p}}\\varrho^{\\bar{p}}\\theta^{2-\\bar{p}}, \\, \\,   \\forall i=1,..N \\bigg{\\}},\n\\]\n\\[\n\\mathcal{P}_\\theta^-(x_0,t_0)= \\bigg{\\{}(x,t) \\in \\Omega_T:\\,\\,    -C_2^{\\bar{p}}\\varrho^{\\bar{p}}\\theta^{2-\\bar{p}}\\leq (t-t_0) \\leq -C_2^{\\bar{p}}|x_i-x_{0,i}|^{p_i} {\\theta}^{2-p_i},  \\, \\,   \\forall i=1,..N \\bigg{\\}},\n\\] \n\\noindent with $\\varrho$ depending on $u$, $\\Omega_T$, and $(x_0,t_0)$ according to the following expression (see Figure \\ref{FigA}): \n\\begin{equation}\\label{rho+}\n    \\varrho^{\\bar{p}}= C_3^{-\\bar{p}} \\bigg( \\frac{u(x_0,t_0)}{C_1} \\bigg)^{\\bar{p}-2} \\min_{i=1,\\dots,N} \\bigg{\\{}(T-|t_0|), \\, \\bigg( \\frac{\\mathrm{dist}(x_0, \\partial \\Omega)}{2}\\bigg)^{p_i} \\bigg( \\frac{u(x_0,t_0)}{C_1} \\bigg)^{2-p_i} \\bigg{\\}}.\n\\end{equation}\n\\end{proposition}\n\\noindent It is remarkable that estimate \\eqref{Harnack} is prescribed on a {\\it{space}} configuration depending on the solution, in contrast to what happens with $p$-Laplacian type equations. This is due to the natural scaling of the equation (see \\cite{CianiGuarnotta} and the end of Section \\ref{WHSection}), because the expansion of positivity of solutions is readily checked via comparison with the following family of Barenblatt-type solutions.\n\n\n\\begin{theorem}\n\\label{Barenblatt}\nSet $\\lambda=N(\\bar{p}-2)+\\bar{p}$ and let \\eqref{pi} be satisfied. For each $\\sigma >0$ there exists $\\tilde{\\eta}>0$ and a local weak solution $\\mathcal{B}_{\\sigma}(x, t)$ to \\eqref{EQ} \nwith the following properties, valid for any $t\\in(0,T)$:\n\\begin{enumerate}\n\\item $\\displaystyle{\\|\\mathcal{B}_{\\sigma}(\\cdot, t)\\|_{\\infty}=\\sigma \\,  t^{-\\alpha}}$,\n\\item\n$\\displaystyle{{\\rm supp}(\\mathcal{B}_{\\sigma}(\\cdot, t))\\subseteq \\prod_{i=1}^N \\big{\\{} |x_i|\\le    \\sigma^{(p_i-2)\/p_i}\\, t^{\\alpha_i} \\big{\\}}}$, $\\qquad \\qquad$  $\\alpha=N\/\\lambda$, $\\alpha_i=(1+2\\alpha)\/p_i-\\alpha$, \n\\item\n$\\displaystyle{\\{\\mathcal{B}_{\\sigma}(\\cdot, t)\\ge \\eta\\,  \\sigma \\, t^{-\\alpha}\\}\\supseteq \\prod_{i=1}^N \\big{\\{} |x_i|\\le \\eta\\, \\sigma^{(p_{i}-2)\/p_{i}}\\, t^{\\alpha_i} \\big{\\}}=:\\mathcal{P}_t}$.\n\\end{enumerate}\n\\end{theorem}\n\\noindent The existence of a Barenblatt Fundamental solution $\\mathcal{B}$ is a consequence of the finite speed of propagation of solutions to \\eqref{EQ} combined with a particular correspondence of the Cauchy problems associated to \\eqref{EQ} and to an anisotropic Fokker-Planck equation. On the other hand, the properties of $\\mathcal{B}$ stated above stem from comparison techniques and the invariance of the equation \\eqref{EQ} under scaling, which entitles $\\mathcal{B}$ to be a self-similar solution. We refer to \\cite{Ciani-Mosconi-Vespri} for the proofs of these facts; see also \\cite{CSV}, \\cite{Vazquez} for the singular case.\n\\begin{proposition} \\label{local-comparison}\nLet $\\Omega \\subset {\\mathbb R}^N$ be a bounded open set and $u,v$ be weak local solutions to the equation \\eqref{EQ} in $\\Omega_T$, satisfying $u(x,t) \\ge v(x,t)$ in the parabolic boundary of $\\Omega_T$. Then $u \\ge v$ in $\\Omega_T$.\n\\end{proposition} \\noindent \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Liouville-type results}\\label{LiouvilleSection}\n\\noindent In their origins, Liouville properties were discovered for harmonic functions. Indeed, for solutions to $\\Delta u =0$ in ${\\mathbb R}^N$, Liouville properties occur when $u$ is just one-sided bounded, or also when it grows sublinearly at infinity. \nThe two classic examples in \\eqref{examples} follow respectively from an application of Harnack's inequality and from gradient estimates. Here we observe that gradient bounds of logarithmic type are unknown for solutions to the stationary counterpart of \\eqref{EQ} and seem very difficult to obtain, chiefly because of the lack of homogeneity of the operator. On the other hand, for parabolic equations a one-side bound is not sufficient to imply that solutions are constant, as we remarked. The Liouville property is false, stated as it is, also for non-negative solutions to degenerate $p$-Laplacian equations (i.e., for $p>2$). Indeed, the one-parameter family of non-negative functions \n\\[\n{\\mathbb R} \\times {\\mathbb R}  \\ni (x,t)\\rightarrow u(x,t;c)= c^{{1}\/({p-2})} \\bigg(\\frac{p-2}{p-1}  \\bigg)^{{(p-1)}\/{(p-2)}} (1-x+ct)_+^{{(p-1)}\/{(p-2)}}\n\\] is a family of non-negative, non-constant weak solutions to $u_t=\\Delta_{p}u$ in ${\\mathbb R}^2$. This naturally provides a counterexample also in case of equation \\eqref{EQ} in one spatial dimension. Similarly, the anisotropic driving example we have in mind is\n\\[{\\mathbb R}^N \\times {\\mathbb R} \\ni (x,t) \\rightarrow u(x,t;c)= \\bigg(1-ct + \\sum_{i=1}^N {(\\alpha_i\/p_i') |x_i|^{p_i'}}\\bigg)_+,\\]\nfor $\\alpha_i >0$ such that $\\sum_{i=1}^N |\\alpha_i|^{p_i-1}\\alpha_i=c$ and being $p_i'$ the H\\\"older conjugate of $p_i$ for all $i=1,\\dots, N$. On the other hand, if a lower bound is coupled with a specific upper bound at some time level, Liouville property is true, as the following result shows.\n\n\\begin{theorem}\\label{Liouville1}\nLet $T\\in {\\mathbb R}$, $S_T={\\mathbb R}^N \\times (-\\infty, T)$ and $u$ be a solution to \\eqref{EQ} which is bounded below in $S_T$. Assume moreover that, for some $s<T$, one has \n\\begin{equation}\\label{rs-bound}\n    \\sup_{{\\mathbb R}^N} u(\\cdot,s)=M_s <\\infty.\n\\end{equation} Then $u$ is constant in $S_s= {\\mathbb R}^N \\times (-\\infty, s)$.\n\\end{theorem} \n\\begin{corollary}\n\\label{LiouvilleCor}\nLet $T\\in {\\mathbb R}$, $S_T={\\mathbb R}^N \\times (-\\infty, T)$, and $u$ be a solution to \\eqref{EQ}. If $u$ is bounded from above and below in $S_T$, then it is constant.\n\\end{corollary}\n\n\\begin{proof}[Proof of Theorem \\ref{Liouville1}]\nLet $u$ be a solution to \\eqref{EQ} bounded from below in $S_T$. We define\n\\[\nm:=\\inf_{S_T} u.\n\\]\nWe prove the following fact, which is interesting in its own:\n\\begin{equation}\\label{perse}\n    \\lim_{t\\rightarrow -\\infty} u(x,t)= \\inf_{S_T} u \\quad \\mbox{for any} \\quad x \\in {\\mathbb R}^N.\n\\end{equation}\\noindent To this aim, fix any $x\\in{\\mathbb R}^N$ and $\\varepsilon>0$. Notice that there exists a point $(y_\\varepsilon, s_\\varepsilon) \\in S_T$ such that $u(y_{\\varepsilon},s_{\\varepsilon})-m\\leq \\varepsilon\/C_3$. Set $\\theta_{\\varepsilon}= (u(y_{\\varepsilon},s_{\\varepsilon})-m)\/C_1$. Exploiting the left-hand side of \\eqref{Harnack}, written for the solution $u-m$, we obtain the estimate\n\\begin{equation}\\label{infimum}\nm \\leq u(y,s) \\leq m+\\varepsilon, \\quad \\text{for all} \\quad (y,s) \\in \\mathcal{P}_{\\theta_\\varepsilon}^-(y_{\\varepsilon},s_{\\varepsilon}).\n\\end{equation}\nConsider the half line $R:=\\{x\\}\\times(-\\infty,T)$. Observe that\n\\[\nR \\cap \\mathcal{P}_{\\theta_\\varepsilon}^-(y_{\\varepsilon},s_{\\varepsilon}) = \\{x\\} \\times (-\\infty,t_{\\varepsilon,x}), \\quad \\mbox{being} \\quad t_{\\varepsilon,x} := s_\\varepsilon-C_2^{\\bar{p}}(2-|x_i-y_{\\varepsilon,i}|)^{p_i}\\theta^{2-p_i}.\n\\]\nAccording to \\eqref{infimum}, this shows that\n\\begin{equation*}\nm \\leq u(x,s) \\leq m+\\varepsilon, \\quad \\text{for all} \\quad s < t_{\\varepsilon,x}.\n\\end{equation*}\nAccordingly, \\eqref{perse} is proved, by arbitrariness of $x$ and $\\varepsilon$. \nA similar argument shows that\n\\begin{equation}\\label{perse2}\n    \\sup_{S_T} u <\\infty \\quad \\Rightarrow \\quad \\lim_{t\\rightarrow -\\infty} u(x,t) = \\sup_{S_T} u \\quad \\forall x \\in {\\mathbb R}^N.\n\\end{equation}\nEventually this implies that any $u$ solution to \\eqref{EQ} which is bounded from both above and below in the whole $S_T$ is necessarily constant. Indeed, by \\eqref{perse} and \\eqref{perse2} we have $\\sup_{S_T} u= \\inf_{S_T} u$. This argument proves Corollary \\ref{LiouvilleCor}.\n\\vskip0.2cm \\noindent \nIn order to conclude the proof of Theorem \\ref{Liouville1}, we use the assumption that there exists $\\bar{s} \\in (-\\infty,\\, T)$ such that $u(\\cdot, \\bar{s})$ is bounded from above in the whole ${\\mathbb R}^N$ by a suitable $M_s\\in {\\mathbb R}$. Indeed, letting $\\theta_x=(u(x,\\bar{s})-m)\/C_1$ for any $x\\in{\\mathbb R}^N$ and using the intrinsic backward Harnack inequality for $u-m$ again, we get the uniform bound \n\\[\nu(y,s) \\leq C_3 u(x,\\bar{s}) \\leq C_3 M_{\\bar{s}}, \\quad \\text{for all} \\quad x \\in {\\mathbb R}^N \\quad \\mbox{and} \\quad (y,s) \\in \\mathcal{P}_{\\theta_{x}}^-(x,\\bar{s}).\n\\] Reasoning as above, with $\\mathcal{P}_{\\theta_x}^-(x,\\bar{s})$ instead of $\\mathcal{P}_{\\theta_{\\varepsilon}}^- (y_{\\varepsilon},\\, s_{\\varepsilon})$, besides recalling that $u$ bounded from both above and below in $\\mathcal{P}_{\\theta_x}^-(x,\\bar{s})$ uniformly in $x \\in {\\mathbb R}^N$, we conclude that $u$ is constant in $S_{\\bar{s}}$.\n\\end{proof} \n\n\\noindent As a general principle, the bigger the set where the equation is solved the stronger the rigidity: for solutions of \\eqref{EQ} in ${\\mathbb R}^N \\times {\\mathbb R}$, it suffices to check their asymptotic (in time) two-side boundedness at a single point $y \\in {\\mathbb R}^N$ to infer that they are constant, as shown by the next theorem. \n\\begin{theorem}\\label{Liouville2}\nSuppose \\eqref{pi} to be satisfied. Let $u$ be a local weak solution to \\eqref{EQ} in ${\\mathbb R}^N \\times {\\mathbb R}$ which is bounded from below. If, in addition, there exists $y \\in {\\mathbb R}^N$ and a sequence $\\{ s_n\\} \\subset {\\mathbb R}$, $s_n\\to+\\infty$, such that $\\{u(y, s_n)\\}$ is bounded, then $u$ is constant.\n\\end{theorem}\\noindent \n\\begin{remark}\nWe explicitly point out the following straightforward consequence of Theorem \\ref{Liouville2}. Let $u$ be a local weak solution to \\eqref{EQ} in ${\\mathbb R}^N \\times {\\mathbb R}$ which is bounded from below. Suppose that, for some $y \\in {\\mathbb R}^N$, one has\n\\begin{equation}\\label{Liouville2HP}\n    \\liminf_{t\\rightarrow +\\infty} u(y,t)=\\alpha \\in {\\mathbb R}.\n\\end{equation}\\noindent Then $u$ is constant.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{Liouville2}]\nLet $m:=\\inf u$ and consider $\\tilde{u}:=u+m+C_1$, which is a solution to \\eqref{EQ}. By assumption, there exist $M \\in {\\mathbb R}$ and $\\{s_n\\}\\subset{\\mathbb R}$ such that $s_n \\to +\\infty$ and\n\\[\n\\tilde{u}(y,s_n)<M \\quad  \\quad \\forall n \\in {\\mathbb N}.\n\\]\nLet us fix arbitrarily $\\bar{s}\\in {\\mathbb R}$ and let $\\bar{n} \\in {\\mathbb N}$ be big enough such that $s_n >\\bar{s}$ for all $n \\ge \\bar{n}$. Then, for all $n\\ge\\bar{n}$, we set $\\theta_n:=\\tilde{u}(y,s_n)\/C_1$ and define a sequence of radii $\\{\\rho_n\\}$ through\n\\[\ns_n-\\theta_n^{2-\\bar{p}} (C_2 \\rho_n)^{\\bar{p}} =\\bar{s}, \\quad \\quad \\mbox{that is,} \\quad \\quad \\rho_n= \\bigg[\\theta_n^{\\bar{p}-2}\\frac{(s_n-\\bar{s})}{C_2^{\\bar{p}}} \\bigg]^{1\/\\bar{p}}.\n\\] We want to apply the Harnack inequality to deduce an upper bound for $\\tilde{u}(\\cdot,\\bar{s})$ in the whole ${\\mathbb R}^N$; so we need to check that the intrinsic anisotropic cubes $\\mathcal{K}_{\\rho_n}(\\theta_n)$ expand as $s_n\\to+\\infty$. An explicit computation yields\n\\[\n\\mathcal{K}_{\\rho_n}(\\theta_n)= \\prod_{i=1}^N \\bigg{\\{}|x_i|<\\theta_n^{{(p_i-2)}\/{p_i}}  \\left(\\frac{s_n-\\bar{s}}{C_2^{\\bar{p}}}\\right)^{{1}\/{p_i}} \\bigg{\\}}\\quad \\xrightarrow[n \\to \\infty]{} \\quad {\\mathbb R}^N,\n\\] since $1\\le\\theta_n\\le M\/C_1$ and $\\{s_n\\}$ diverges. By the intrinsic backward Harnack inequality \\eqref{Harnack} we have\n\\[\n\\sup_{y+\\mathcal{K}_{\\rho_n}(\\theta_n)} \\tilde{u} \\bigg(\\, \\,  \\cdot\\, \\, ,\\,  s_n-\\theta_n^{2-\\bar{p}} (C_2\\rho_n)^{\\bar{p}} \\bigg)\\leq C_3 \\,\\tilde{u}(y,s_n) \\leq C_3 M \\quad \\forall n\\ge \\bar{n}.\n\\] Thus, recalling the definition of $\\{\\rho_n\\}$, we get the uniform estimate \n\\[\n\\sup_{y+\\mathcal{K}_{\\rho_n}(\\theta_n)} \\tilde{u}(\\cdot, \\bar{s} ) \\leq C_3 M \\quad \\forall n\\ge \\bar{n},\n\\]\nwhence, letting $n\\to\\infty$,\n\\[\n\\sup_{{\\mathbb R}^N} \\tilde{u}(\\cdot, \\bar{s}) \\leq C_3 M.\n\\]\nNow we are in the position to apply Theorem \\ref{Liouville1}. The proof is concluded by arbitrariness of $\\bar{s}\\in{\\mathbb R}$, after transforming $\\tilde{u}$ back to $u$.\n\\end{proof} \n\n\n\\noindent Finally, we show that the oscillation estimates \\eqref{control} constitute a Liouville property for non-negative ancient solutions. This allows us to get rid of the range of $p_i$s of finite speed of propagation in Theorem \\ref{Harnack-Inequality}, at the price of assuming a suitable oscillation decay. \n\n\n\\begin{theorem} \\label{Cutilisci}\nLet $u$ be a bounded function in $S_T$ satisfying the oscillation estimates \\eqref{control}. Then $u$ is constant.\n\\end{theorem}\n\n\n\\begin{proof} The proof is an adaptation of an early idea already present in \\cite{Glagoleva1} (see also \\cite{Landis}); here the adversity is the intrinsic geometry, which in the case of global boundedness turns out to be simpler. The natural geometry will be dictated by $\\omega_o= 2 ||u||_{\\infty, S_T}$. Indeed, let $A,B \\in S_T$ be two points such that $u(A) \\ne u(B)$ and call $Z_T= \\{T\\} \\times {\\mathbb R}^N$. Setting\n\n\\[d= \\max \\{\\mathrm{dist}(A, Z_T),\\,  \\mathrm{dist}(B, Z_T),\\, \\mathrm{dist}(A,B) \\},\\]\nwe choose a radius $R_o>0$ big enough to enclose $A$ and $B$ inside an intrinsic backward cylinder $\\tilde{\\mathcal{Q}}_0:=\\mathcal{Q}_{R_o}^-(\\omega_o\/C_1,C_2)$,\nso that $R_o$ satisfies \n\\begin{equation*}\n    \\begin{cases}\n  (\\omega_o\/C_1)^{p_i-\\bar{p}} R_o^{{\\bar{p}}} >d^{p_i},\\quad i =1,..,N,\\\\\n({\\omega_o}\/{C_1})^{2-\\bar{p}} (C_2 R_o)^{\\bar{p}} >d.\n    \\end{cases\n\\end{equation*}\nLet us set $\\delta=4C_3\/(1+4C_3) \\in (0,1)$ and define $\\varepsilon= \\delta^{({\\bar{p}-2})\/{\\bar{p}}}\/A \\in (0,1)$, being $A=4^{p_N}$ as in Proposition \\ref{birra} below. Suppose, without loss of generality, that $T=0$, and construct the sequence of expanding backward cylinders $\\tilde{\\mathcal{Q}}_n$ centered in $(y,s)=(0,0)$\nas\n\\[\n\\tilde{\\mathcal{Q}}_n= \\prod_{i=1}^N \\bigg{\\{}|x_i|<[\\omega_o\/(C_1\\delta^n)]^{{(p_i-\\bar{p})}\/{p_i}} (R_o\/\\varepsilon)^{{\\bar{p}}\/{p_i}}  \\bigg{\\}} \\times [(\\omega_o\/(C_1 \\delta^{n})]^{2-\\bar{p}} [(C_2 R_o)\/\\varepsilon) ]^{\\bar{p}} ,\\, 0\\bigg].\n\\] Using the oscillation inequalities \\eqref{control}, we arrive at the contradiction \n\\[\n    2 ||u||_{\\infty, S_T} \\ge \\operatornamewithlimits{osc}_{\\tilde{\\mathcal{Q}}_0} u \\ge (\\operatornamewithlimits{osc}_{\\tilde{\\mathcal{Q}}_n} u)\/ \\delta^n \\ge |u(A)-u(B)|\/ \\delta^{n} \n\\] for large $n \\in {\\mathbb N}$.\n\\end{proof}\n\\noindent Theorem \\ref{Cutilisci} can be formulated without the assumption that $u$ is a solution of any equation. Indeed, this general principle goes far beyond equation \\eqref{EQ} and is a key argument to prove rigidity results in a very general class of equations (see, e.g., \\cite[Prop. 18.4]{DBGV-mono}). Its importance enters into play when a Harnack inequality ceases to hold true.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Time-extrinsic Harnack inequality} \\label{WHSection}\n In this section we show how it is possible to free the Harnack inequality from its intrinsic geometry in time. More specifically, we give a formulation of the Harnack inequality allowing the solution to be evaluated at any time level, independently of the anisotropic geometry, provided there is enough room for the anisotropic evolution inside $\\Omega_T$. Unlike the isotropic case, here it looks more difficult to get rid of the intrinsic geometry along the space variables. The proof of the next theorem exploits a comparison with the abstract Barenblatt fundamental solution $\\mathcal{B}$ of Theorem \\ref{Barenblatt} to control the positivity.\n\n\n\\begin{theorem}\n\\label{WeakHarnack} Let $u \\ge 0$ be a local weak solution to \\eqref{EQ} in $\\Omega_T$, and assume \\eqref{pi}. Then there exist $ \\tilde{\\eta} >0$ and $\\gamma>1$, depending only on $N$ and $p_i$s, such that for all $(x_o,t_o)\\in \\Omega_T$ and $\\rho, \\tilde{\\theta}>0$ fulfilling the condition\n\\begin{equation}\\label{domain}\n(x_o, t_o+ \\tilde{\\theta})+\\mathcal{Q}_{C_3 \\rho}(u(x_o,t_o)\/C_1,C_2) \\subseteq \\Omega_T\n\\end{equation} we have\n\\begin{equation}\\label{WH}\n    u(x_o,t_o)\\leq \\gamma \\bigg{\\{} \\bigg(\\frac{\\rho^{\\bar{p}}}{\\tilde{\\theta}}  \\bigg)^{{1}\/{(\\bar{p}-2)}}+ \\bigg( \\frac{\\tilde{\\theta}}{\\rho^{\\bar{p}}} \\bigg)^{N\/\\bar{p}} \\bigg[\\inf_{x_o+K_{\\tilde{\\eta}\\rho}(\\tilde{\\eta} u(x_o,t_o)\/C_1)} u( \\cdot,\\, t_o+\\tilde{\\theta})  \\bigg]^{\\lambda\/\\bar{p}}\\bigg{\\}},\n\\end{equation} \\noindent where $C_1,C_3>1$ come from Theorem \\ref{Harnack-Inequality} while $\\lambda,\\tilde{\\eta}>0$ stem from Theorem \\ref{Barenblatt}. \\end{theorem}\n\\begin{proof}\nLet $\\rho, \\tilde{\\theta} >0$ be such that \\eqref{domain} holds true. Set\n\\begin{equation} \\label{t*}\nt^*:= \\bigg( \\frac{C_1}{u(x_o,t_o)} \\bigg)^{\\bar{p}-2} (C_2 {\\rho})^{\\bar{p}}.\n\\end{equation} We can suppose $t^*<\\tilde{\\theta}\/2$; otherwise we get $u(x_o,t_o) \\leq \\gamma ({\\rho}^{\\bar{p}}\/\\tilde{\\theta})^{1\/(\\bar{p}-2)}$ for a suitable $\\gamma= \\gamma (C_1, C_2,\\bar{p})$, and \\eqref{WH} is valid. Observe that $t^*<\\tilde{\\theta}\/2$ and \\eqref{domain} imply\n\\[\nt_0+\\bigg( \\frac{C_1}{u(x_o,t_o)}\\bigg)^{\\bar{p}-2} (C_2 {\\rho})^{\\bar{p}} <t_0+ \\tilde{\\theta}\/2 < T \\quad \\mbox{and} \\quad x_o + \\mathcal{K}_{C_3 {\\rho}}(u(x_o,t_o)\/C_1) \\subset \\Omega.\n\\] Hence the forward Harnack inequality \\eqref{Harnack} furnishes\n\\[\nu(x_o,t_o) \\leq C_3 u(x,\\, t_o+t^*) \\quad \\quad \\forall \\, x \\in x_o + \\mathcal{K}_{{\\rho}}(u(x_o,t_o)\/C_1).\n\\] This initial value can be considered for a comparison with the Barenblatt solution  $\\mathcal{B}_{\\sigma}(x-x_o,t-s)$ centered at $(x_o,s)$, being $s,\\sigma>0$ to be chosen such that $\\mathcal{B}_{\\sigma}(x-x_o,t_o+t^*-s)$ lies below $u$ in $x_0+\\mathcal{K}_{\\rho}(u(x_o,t_o)\/C_1)$. These requirements can be written as\n\\begin{equation}\\label{supportami}\n\\begin{cases}\n\\operatorname{supp}{\\mathcal{B}_{\\sigma}}(\\cdot-x_o, t_o+t^*-s) \\subseteq x_o+ \\mathcal{K}_{\\rho}(u(x_o,t_o)\/C_1),\\\\\n\\|\\mathcal{B}_{\\sigma}(\\cdot-x_o, t_o+t^*-s)\\|_\\infty \\leq u(x_o,t_o)\/C_3.\n\\end{cases}\n\\end{equation}\nAccording to Theorem \\ref{Barenblatt}, conditions in \\eqref{supportami} are fulfilled as long as\n\\begin{equation}\\label{supportami2}\n\\begin{cases}\n \\sigma^{(p_i-2)\/p_i} (t_o+t^*-s)^{\\alpha_i} \\leq \\rho^{\\bar{p}\/p_i} (u(x_o,t_o)\/C_1)^{(p_i-\\bar{p})\/p_i},\\\\\n \\sigma (t_o+t^*-s)^{-\\alpha} \\leq u(x_o,t_o)\/C_3.\n\\end{cases}\n\\end{equation}\n\\noindent Inequalities in \\eqref{supportami2} are in turn ensured by choosing\n\\[\n\\sigma= (t_o+t^*-s)^{N\/\\lambda} u(x_o,t_o)\/C_3 \\quad \\mbox{and} \\quad s= t_o+t^*- \\bigg( \\frac{\\rho^{\\bar{p}}}{u(x_o,t_o)^{\\bar{p}-2}} \\bigg) \\gamma_1,\\] \nwhere $\\gamma_1=\\min \\{(C_3^{p_i-2})\/(C_1^{p_i-\\bar{p}})\\, |\\, i=1,\\dots,N\\}$. Therefore the comparison principle, applied at the time $t_o+\\tilde{\\theta}>t_o+t^*$, gives\n\\begin{equation}\\label{comparison} \\begin{aligned}\nu(x, t_o+\\tilde{\\theta}) &\\ge \\tilde{\\eta} \\sigma |t_o+t^*-(t_o+\\tilde{\\theta})|^{-\\alpha} = \\tilde{\\eta} \\bigg( \\frac{u(x_o,t_o)}{C_3} \\bigg) (t_o+t^*-s)^{N\/\\lambda} (\\tilde{\\theta}-t^*)^{-N\/\\lambda}\\\\\n&\\ge \\tilde{\\eta} \\bigg( \\frac{u(x_o,t_o)}{C_3} \\bigg) \\bigg(  \\frac{\\gamma_1  \\rho^{\\bar{p}}}{u(x_o,t_o)^{\\bar{p}-2}} \\bigg)^{N\/\\lambda} \\tilde{\\theta}^{-N\/\\lambda} \\ge \\gamma u(x_o,t_o)^{\\bar{p}\/\\lambda} \\bigg( \\frac{\\rho^{\\bar{p}}}{\\tilde{\\theta}} \\bigg)^{N\/\\lambda},\n\\end{aligned} \\end{equation} being $\\gamma= \\gamma(\\gamma_1,\\tilde{\\eta})$, for every $x$ in the set of positivity\n\\begin{equation*}\n\\begin{aligned}\n\\mathcal{P}_{t_o+\\tilde{\\theta}-s}(x_o)\\supseteq \\mathcal{P}_{t_o+t^*-s}(x_o) &= \\prod_{i=1}^N\\{|x_i-x_{o,i}|\\leq\\tilde{\\eta} \\rho^{\\bar{p}\/p_i} (u(x_o,t_o)\/C_1)^{(p_i-\\bar{p})\/p_i} \\}\\\\\n&=x_o+ \\mathcal{K}_{\\tilde{\\eta}\\rho}(\\tilde{\\eta}u(x_o,t_o)\/C_1),\n\\end{aligned}\n\\end{equation*}\nwith $\\tilde{\\eta}$ depending only on the data $N, p_i$. Passing \\eqref{comparison} to the infimum on $x_o+ \\mathcal{K}_{\\tilde{\\eta}\\rho}(\\tilde{\\eta}u(x_o,t_o)\/C_1)$ concludes the proof.\n\\end{proof}\n\n\n\n\n\n\\begin{remark}\nIn Theorem \\ref{WeakHarnack} the lower bound $u(x_o,t_o)>0$ is not required; moreover, $\\tilde{\\theta}>0$ is arbitrarily chosen between those numbers that preserve the inclusion of the intrinsic cylinder translated to time $\\tilde{\\theta}$ into $\\Omega_T$. Henceforth, when the equation is solved in ${\\mathbb R}^{N+1}$, the proof furnishes inequality \\eqref{WH} without the first term on the right, and one can infer similar Liouville properties as Theorem \\ref{Liouville2}. Actually, Theorems \\ref{Harnack-Inequality} and \\ref{WeakHarnack} are equivalent for small radii.\\vskip0.2cm \n\n\\noindent Indeed, we proved that Theorem \\ref{Harnack-Inequality} implies Theorem \\ref{WeakHarnack}. Now we show that the converse statement can be obtained by a simple choice of $\\tilde{\\theta}$. Indeed, let us pick\n\\[ \\tilde{\\theta}= \\frac{(2\\gamma)^{\\bar{p}-2}\\rho^{\\bar{p}}}{u(x_o,t_o)^{\\bar{p}-2}},\n\\] and suppose that $(x_o,t_o+\\tilde{\\theta}) + \\mathcal{Q}_{C_3\\rho} (u(x_o,t_o)\/C_1)\\subset \\Omega_T$. The weak Harnack inequality \\eqref{WH} leads to\n\\[\nu(x_o,t_o) \\leq \\gamma \\bigg{\\{} \\frac{u(x_o,t_o)}{2\\gamma}+ \\bigg( \\frac{2\\gamma}{u(x_o,t_o)} \\bigg)^{{N(\\bar{p}-2)}\/{\\bar{p}}}\\bigg[ \\inf_{x_o+ \\mathcal{K}_{\\tilde{\\eta} \\rho}(\\tilde{\\eta} u(x_o,t_o)\/C_1)} u(\\cdot, \\, t_o+ \\bigg( \\frac{u(x_o,t_o)}{2\\gamma} \\bigg)^{2-\\bar{p}} \\rho^{\\bar{p}}) \\bigg]^{{\\lambda}\/{\\bar{p}}}  \\bigg{\\}},\n\\]\nwhence\n\\[\nu(x_o,t_o) \\leq \\tilde{C_3} \\inf_{x_o+ \\mathcal{K}_{\\tilde{\\rho}}(M)} u(\\cdot, \\, t_o+ \\tilde{C_2} M^{2-\\bar{p}} \\tilde{\\rho}^{\\bar{p}}), \\quad M= u(x_o,t_o)\/\\tilde{C}_1, \\] for all $\\tilde{\\rho} \\leq \\tilde{\\eta} \\rho$ and with positive constants \n\\[\\tilde{C_1}= C_1\/\\tilde{\\eta}, \\quad \\quad  \\tilde{C_2}=\\frac{(2\\gamma\/\\tilde{C}_1)^{\\bar{p}-2}}{\\tilde{\\eta}^2}, \\quad \\quad \\tilde{C_3}= 2 \\gamma.\n\\] \\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix: H\\\"older Continuity of solutions}\n\\label{Appendix}\n\n\n\\begin{theorem}\\label{HC}\nUnder condition \\eqref{pi}, any local weak solution $u$ to \\eqref{EQ} is locally H\\\"older continuous. More precisely, there exist $\\gamma>1$ and $\\chi \\in (0,1)$, depending only upon $p_i,N$, with the following property: for each compact set $K \\subset \\subset \\Omega_T$ there exist a set $\\Lambda$ and $\\omega_o=\\omega_o(K, \\|u\\|_{\\infty,K})$ such that $K \\subset \\Lambda \\subseteq \\Omega_T$ and, for every $(x,t)$, $(y,s) \\in K$,\n\\begin{equation} \\label{HContinuity}\n    |u(x,t)-u(y,s)| \\leq \\gamma \\omega_o \\bigg(\\frac{\\sum_{i=1}^N |x_i-y_i|^{{p_i}\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-p_i)}\/{\\bar{p}}}+ |t-s|^{1\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-2)}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda) } \\bigg)^{\\chi},\n\\end{equation}\\noindent with\n\\begin{equation} \\label{pi-dist}\n\\begin{aligned}\n&{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda):=\\inf \\{ {\\bf{p}}_x, {\\bf{p}}_t \\}, \\quad \\text{being}\\\\\n& {\\bf{p}}_x=\\inf \\bigg{\\{} \n|x_i-y_i|^{{p_i}\/{\\bar{p}}}(\\omega_o\/C_1)^{{(\\bar{p}-p_i)}\/{\\bar{p}}}\\, : \\, (x,t) \\in K, (y,s) \\in \\partial \\Lambda,\\, i=1,..,N\\bigg{\\}},\\\\\n& {\\bf{p}}_t=\\inf \\bigg{\\{} \n|t-s|^{{1}\/{\\bar{p}}}(\\omega_o\/C_1)^{{(\\bar{p}-2)}\/{\\bar{p}}}\\, : \\, (x,t) \\in K, (y,s) \\in \\partial \\Lambda\\bigg{\\}}.\n\\end{aligned}\n\\end{equation} \\noindent Furthermore, if $u$ is bounded in $\\Omega_T$ then \\eqref{HContinuity} holds with $\\Lambda= \\Omega_T$.\n\\end{theorem} \\noindent We prove Theorem \\ref{HC} in four steps, without assuming that $u$ is globally bounded.\n\\begin{proof}\nLet us fix a compact set $K \\subset \\subset \\Omega_T$ and two points $(y,s), (x,t) \\in K$. \\vskip0.2cm \n\\noindent {\\small{STEP 1-{\\it A global bound for the solution in $K$.}}}\n\\vskip0.2cm \\noindent Set $g(k)=\\sum_{i=1}^{N}k^{ p_{i}-2}$ and $h(k)=\\left(\\sum_{i=1}^{N}k^{p_{i}-\\bar p_{2}}\\right)^{-1}$. We use the estimates in \\cite[Lemma 4.2]{Mosconi}: under the exponent range \\eqref{pi}, subsolutions to \\eqref{EQ} satisfy the estimate \n\\begin{equation}\n\\label{supest}\n\\|u_{+}\\|_{L^{\\infty}(Q_{\\lambda\/2, M})}\\leq g^{-1}(1\/M)+ h^{-1}\\left(C\\Big(M\\, \\bint\\kern-0.15cm\\bint_{Q_{\\lambda, M}} u_{+}^{\\bar p_{2}}\\, dx\\Big)^{{\\bar p}\/{(N+\\bar p)}}\\right)\n\\end{equation} in the anisotropic cylinders \n\\begin{equation} \\label{anisocylinder}\nQ_{\\lambda, M}= \\prod_{i=1}^{N}\\left[-\\lambda^{{1}\/{p_{i}}}, \\lambda^{{1}\/{p_{i}}}\\right]\\times [-M\\, \\lambda, 0],\\quad \\quad  M, \\lambda>0.\n\\end{equation}\n\\vskip0.2cm \\noindent By compactness of $K$, we find $(x_i,t_i) \\in K$ and $\\lambda_i, M_i \\in \\mathbb{R}_+$, $i=1,\\dots,m$, for $m\\in{\\mathbb N}$, such that \n\\begin{equation*}\n    K \\subset \\Lambda:=\\bigcup_{j=1}^m \\{(x_j,t_j)+Q_{\\lambda_j,M_j}\\}\\subseteq \\bigcup_{j=1}^m \\{(x_j,t_j)+Q_{2\\lambda_j,M_j} \\}\\subseteq \\Omega_T,\n\\end{equation*} \\noindent being $Q_{\\lambda,M}$ as in \\eqref{anisocylinder}. \n\\noindent According to \\eqref{supest}, for each anisotropic cylinder  $\\hat{Q}_{\\lambda_j,M_j}=(x_j,t_j)+Q_{\\lambda_j,M_j}$, $j=1,\\dots,m$, we deduce the estimate \n\\begin{equation*} \\begin{aligned}\\label{A}\n\\| u\\|_{L^{\\infty}(\\hat{Q}_{\\lambda_j, M_j})} &\n\\leq g^{-1} (1\/\\min_{j} M_j)+ h^{-1} \\bigg( C \\max_{j=1,\\dots,m}  \\bigg( M_j \\Xint{\\raise4pt\\hbox to7pt{\\hrulefill}} \\Xint{\\raise4pt\\hbox to7pt{\\hrulefill}}_{\\hat{Q}_{2\\lambda_j,M_j}} |u|^{\\bar{p}_2}\\,  dxdt\\bigg)^{{\\bar{p}}\/{(N+\\bar{p})}} \\bigg)=: \\mathcal{I},\n\\end{aligned}\n\\end{equation*} \\noindent because $h$,$g$, are monotone increasing.\n\\noindent Finally, we define $\\omega_o=\\omega_o(K)$ as \\begin{equation} \\label{0}\n\\omega_o:= 2 \\mathcal{I}. \\end{equation} \\noindent  Accordingly,\n\\[\nK \\subset \\bigcup_{j=1}^m \\hat{Q}_{\\lambda_j, M_j}(x_j,t_j)= \\Lambda \\qquad \\mbox{and} \\qquad 2 \\|u\\|_{L^{\\infty}(K)} \\leq \\omega_o.\n\\]\n\\vskip0.2cm \\noindent {\\small{STEP 2-{\\it Accommodation of degeneracy and alternatives.}}}\n\\vskip0.2cm \\noindent\nRecalling \\eqref{pi-dist}\nwe define $R:= [{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda)]\/(2C_3)$. Now, by definition of $R$, the intrinsic cylinder centered at $(y,s)\\in K$ and constructed with $R$ and $\\omega_o$ is contained inside  $\\Lambda$, that is,\n\\[\n(y,s)+ \\mathcal{Q}_{R}(\\omega_o\/C_1,C_2) \\subseteq \\Lambda.\n\\]\n\n\n\\noindent Now consider any other point $(x,t) \\in K$. We reduce the study of the oscillation only in $(y,s) + Q_R(\\omega_o\/C_1)$, having elsewhere the H\\\"older continuity of $u$. Indeed, if $|s-t| \\ge (\\omega_o\/C_1)^{2-\\bar{p}}(C_2 R)^{\\bar{p}}$, we have\n\\[\n|u(y,s)-u(x,t)|\\leq |u(y,s)|+|u(x,t)|\\leq \\omega_o \\leq 2C_3 \\omega_o \\bigg( \\frac{(\\omega_o\/C_1)^{{(\\bar{p}-2)}\/{\\bar{p}}}|s-t|^{{1}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda)} \\bigg) \\] by definition of $R$.\nSimilarly, if $|y_i-x_i| \\ge (\\omega_o\/C_1)^{{(p_i-\\bar{p})}\/{\\bar{p}}} R^{{\\bar{p}}\/{p_i}}$ for some $i \\in \\{1,\\dots,N\\}$, the same conclusion follows from\n\\[\n|u(y,s)-u(x,t)|\\leq |u(y,s)|+|u(x,t)|\\leq \\omega_o \\leq 2C_3 \\omega_o \\bigg( \\frac{(\\omega_o\/C_1)^{{(\\bar{p}-p_i)}\/{\\bar{p}}}|y_i-x_i|^{{p_i}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda)} \\bigg).\n\\]\nHence we can assume that\n\\begin{equation}\\label{exclusion}\n   |s-t|<(\\omega_0\/C_1)^{2-\\bar{p}} (C_2 R)^{\\bar{p}} \\quad \\text{and}  \\quad  |y_i-x_i|< (\\omega_o\/C_1)^{{(p_i-\\bar{p})}\/{p_i}} R^{{\\bar{p}}\/{p_i}}  \\quad \\forall i=1,\\dots, N, \n\\end{equation}\nthat is, \n\\[(x,t) \\in (y,s)+ \\mathcal{Q}_R(\\omega_o\/C_1,C_2). \\]\nWe take the cylinder $\\mathcal{Q}_0:=(y,s)+\\mathcal{Q}_R^-(\\omega_o\/C_1,C_2)$ as the first element of a net $\\{\\mathcal{Q}_n\\}_n$ of cylinders shrinking to the center $(y,s)$. This net will be constructed to control uniformly the oscillation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vskip0.2cm \\noindent \n\n\n\\noindent {\\small{STEP 3- { \\it Controlled reduction of oscillation }}} \n\\begin{proposition}[Reduction of oscillation in shrinking cylinders] \\label{birra} Let the hypothesis of Theorem \\ref{HC} be satisfied and assume also \\eqref{exclusion}. Then, setting \n\\begin{equation*}\n    \\begin{cases}\n    \\omega_0= \\omega_o(K),\\\\\n    \\omega_n=\\delta \\omega_{n-1}, \\, n\\ge 1,\n    \\end{cases} \n    \\begin{cases}\n    \\theta_n= \\omega_n\/C_1, \\, n\\ge 0,\\\\\n    \\rho_0=R,\\\\\n    \\rho_n= \\varepsilon \\rho_{n-1}, \\, n\\ge 1,\\\\\n    \\end{cases}\n    \\begin{cases}\n    \\delta=4C_3\/(1+4C_3),\\\\\n    \\varepsilon=\\delta^{{(\\bar{p}-2)}\/{\\bar{p}}}\/A, \\\\\n    A=4^{p_N}, \\end{cases}\n\\end{equation*} \\noindent we have both the inclusions\n\\[\n\\mathcal{Q}_{n}\\subset \\mathcal{Q}_{n-1}, \\quad \\text{with} \\quad \\mathcal{Q}_n= (y,s)+ \\mathcal{Q}_{\\rho_n}^-(\\theta_n)= \\prod_{i=1}^N \\bigg{\\{}|y_i-x_i|<\\theta_n^{{(p_i-\\bar{p})}\/{p_i}}\\rho_n^{{\\bar{p}}\/{p_i}}  \\bigg{\\}} \\times \\bigg(s-\\theta_n^{2-\\bar{p}} (C_2\\rho_n)^{\\bar{p}} ,\\, s\\bigg],\n\\] and the inequalities\n\\begin{equation}\\label{control}\n    \\operatornamewithlimits{osc}_{\\mathcal{Q}_n} u \\leq \\omega_n = \\delta^n \\omega_o.\n\\end{equation}\n\\end{proposition}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Proposition \\ref{birra}]\\noindent First of all, we prove that $\\mathcal{Q}_{n} \\subset \\mathcal{Q}_{n-1}$ for all $n\\in{\\mathbb N}$. By direct computation,\n\\begin{equation*} \\begin{aligned}\n\\theta_{n}^{2-\\bar{p}} (C_2\\rho_{n})^{\\bar{p}} = \\bigg(\\frac{\\delta \\omega_{n-1}}{C_1}\\bigg)^{2-\\bar{p}}\\bigg((C_2\\rho_{n-1}\/A)^{\\bar{p}}\\delta^{\\bar{p}-2}\\bigg)= \\theta_{n-1}^{2-\\bar{p}} (C_2\\rho_{n-1}\/A)^{\\bar{p}}.\n\\end{aligned} \\end{equation*}  For each $i\\in \\{1,..,N\\}$, since $p_i>2$ and $\\delta \\in (0,1)$, it holds\n\\[\n\\theta_{n}^{p_i-\\bar{p}} \\rho_{n}^{{\\bar{p}}} = \\delta^{p_i-2} \\theta_{n-1}^{{p_i-\\bar{p}}} ( {\\rho_{n-1}}\/{A} )^{{\\bar{p}}} \\leq \\theta_{n-1}^{{p_i-\\bar{p}}} ( {\\rho_{n-1}}\/{A} )^{{\\bar{p}}}.\n\\] This computation shows a little more, by allowing indeed $\\mathcal{Q}_{n}\\subset (y,s)+\\mathcal{Q}_{\\rho_{n-1}\/A}^-(\\theta_{n-1})\\subset \\mathcal{Q}_{n-1}$.\n\\noindent Now we prove \\eqref{control} by induction. The base step holds true: indeed, the accommodation of degeneracy (see Step 2 above) entails $\\mathcal{Q}_0\\subset\\Lambda$, so that the bound produced in Step 1 yields\n\\[\n\\operatornamewithlimits{osc}_{\\mathcal{Q}_0} u \\leq \\operatornamewithlimits{osc}_{\\Lambda} u \\leq 2\\|u\\|_{L^\\infty(\\Lambda)} \\leq \\omega_o.\n\\]\nWe assume now that the statement \\eqref{control} is true until step $n$ and we show it for $n+1$. This will determine the number $A$. More precisely, we assume that $\\operatornamewithlimits{osc}_{\\mathcal{Q}_n}u \\leq \\omega_n$ and, by contradiction, that $\\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}} u > \\omega_{n+1}$. We set\n\\[\nM_{n}= \\sup_{\\mathcal{Q}_{n}} u, \\qquad m_{n}= \\inf_{\\mathcal{Q}_{n}}u, \\qquad P_{n}=(y,\\, s-\\theta_{n}^{2-\\bar{p}}(C_2\\rho_{n})^{\\bar{p}}).\n\\] Now we observe that one of the following two inequalities must hold:\n\\[\nM_{n}-u(P_{n}) > \\omega_{n+1}\/4 \\quad \\text{or} \\qquad u(P_{n})-m_{n} > \\omega_{n+1}\/4.\n\\] Indeed, if both alternatives are violated, then by adding the opposite inequalities we obtain $\\operatornamewithlimits{osc}_{\\mathcal{Q}_{n}} u  \\leq \\omega_{n+1}\/2< \\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}}$, generating an absurd because $\\mathcal{Q}_{n+1} \\subseteq \\mathcal{Q}_n$. Let us suppose $M_{n}-u(P_{n}) \\ge  \\omega_{n+1}\/4$, the other case being similar. In particular we have the double bound\n\\begin{equation}\n    \\label{doublebound}\n    \\omega_{n+1}\/4\\leq M_n - u(P_n) \\leq \\omega_n.\n\\end{equation}\nLet us set $\\hat{\\theta}_n=(M_n-u(P_n))\/C_1$. We work in the half-paraboloid $\\mathcal{P}^+_n=\\mathcal{P}^+_{\\hat{\\theta}_n}(P_n)$ for times restricted to the ones of $\\mathcal{Q}_n$.\n\\noindent\nWe notice that the starting time of $P_n^+$ is the same of $\\mathcal{Q}_n$ (see \\ref{FigA}). To show that $\\mathcal{P}_n^+\\subset\\mathcal{Q}_n\\subset\\Omega_T$, we control the space variables. By definition of $\\omega_n$, we obtain the following estimate:\n\\begin{equation}\\label{spatial-est}\n|x_i-y_i|^{p_i}<  \\bigg(\\frac{M_n-u(P_n)}{C_1}\\bigg)^{p_i-2}\\rho_n^{\\bar{p}}\\bigg(\\frac{\\omega_n}{C_1}\\bigg)^{2-\\bar{p}}=\\bigg(\\frac{M_n-u(P_n)}{C_1}\\bigg)^{p_i-2} \\bigg(\\frac{R}{A^n}\\bigg)^{\\bar{p}} \\bigg(\\frac{\\omega_o}{C_1} \\bigg)^{2-\\bar{p}},\n\\end{equation}\nfor all $x\\in\\pi_x(\\mathcal{P}_n^+)$, being $\\pi_x$ is the projection on the space variables. \\\\\n\\noindent\nNow we show that, after a certain time $\\bar{t}$, the whole cylinder $\\mathcal{Q}_{n+1}$ is contained in the paraboloid $ \\mathcal{P}^+_n$; see Figure \\ref{FigA} for a representation. For times $t >s-(\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}}$, we denote by $\\mathcal{P}^+_n(t)$ the time-section of $\\mathcal{P}^+_n$ at time $t$:\n\\[ \\mathcal{P}^+_n(t)= \\bigg{\\{} x \\in {\\mathbb R}^N: \\, \\,    |x_i-y_i|^{p_i}< C_2^{-\\bar{p}} [(M_n-u(P_n))\/C_1]^{p_i-2}(t-s+ (\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}})  \\bigg{\\}}.\\]\n\\noindent Let us set \\begin{equation} \\label{t} \\bar{t}=s-(\\omega_{n+1}\/C_1)^{2-\\bar{p}}(C_2 \\rho_{n+1})^{\\bar{p}},\\end{equation}\nand let us prove that at time $\\bar{t}$ we have the inclusion $\\pi_{x}(\\mathcal{Q}_{n+1})\\subset \\mathcal{P}^+_n(\\bar{t})$. This reduces to show that\n\\[\\rho_{n+1}^{\\bar{p}} (\\omega_{n+1}\/C_1)^{p_i-\\bar{p}} \\leq (A^{\\bar{p}}-1)  [(M_n-u(P_n)\/C_1)]^{p_i-2} \\rho_{n+1}^{\\bar{p}} \n (\\omega_{n+1}\/C_1)^{2-\\bar{p}},\\]\n that is,\n\\[\\omega_{n+1}^{p_i-2} \\leq (A^{\\bar{p}}-1) (M_n-u(P_n))^{p_i-2}.\\]\nAccording to \\eqref{doublebound}, this inequality holds true if we choose $A$ such that $4^{p_N-2}<A^{\\bar{p}}-1$, as for instance $A= 4^{p_N}$. \\vskip0.1cm \\noindent Therefore, by the Harnack inequality \\eqref{Harnack} and \\eqref{doublebound}, we can estimate the infimum of $M_{n}-u$ in $\\mathcal{Q}_{n+1}$ as\n\\begin{equation} \\label{bound}\n \\inf_{\\mathcal{Q}_{n+1}} (M_{n}-u) \\ge \\frac{M_{n}-u(P_{n})}{C_3} \\ge\\, \\,  \\omega_{n+1}\/(4C_3),\n\\end{equation}\\noindent again referring to Figure \\ref{FigA}. Thus\n\\[\nM_{n} \\ge \\sup_{\\mathcal{Q}_{n+1}} u+ \\omega_{n+1}\/(4C_3).  \n\\]\nAdding $-\\inf_{\\mathcal{Q}_n} u \\ge - \\inf_{\\mathcal{Q}_{n+1}}u$ to both sides, besides using $\\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}}u >\\omega_{n+1}$, we get \n\\[\\omega_n \\ge M_n - \\inf_{\\mathcal{Q}_n} u \\ge  \\sup_{\\mathcal{Q}_{n+1}} u+ \\omega_{n+1}\/(4C_3)  - \\inf_{\\mathcal{Q}_{n+1}} u= \\operatornamewithlimits{osc}_{\\mathcal{Q}_{n+1}}u+\\omega_{n+1}\/(4C_3)> \\bigg(1+\\frac{1}{4C_3}  \\bigg) \\omega_{n+1}\\, .\n\\]This leads to a contradiction by definition of $\\delta$, since\n\\[\n\\omega_n > \\bigg(1+\\frac{1}{4C_3}  \\bigg) \\delta \\omega_{n}=  \\bigg(\\frac{4C_3}{1+4C_3}\\bigg) \\bigg(1+\\frac{1}{4C_3} \\bigg) \\omega_n = \\omega_n.\n\\] \n\n\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.25]\n\n\\draw[thick,->] (25,0) -- (25,6) node[anchor=north west] {\\small{$x \\in {\\mathbb R}^N$}};\n\\draw[thick,->] (25,0) -- (31,0) node[anchor=south west] {\\small{$t \\in {\\mathbb R}$}};\n\\draw (20,0) rectangle (-6,6);\n\n\n\\draw (20,0)  rectangle (6,4);\n\n\n\n\\draw (20,0) rectangle (-6,-6);\n\n\n\\draw (20,0) rectangle (6,-4);\n\n\\draw (18, 2) node{$\\mathcal{Q}_{n+1}$};\n\\draw (-4, 4) node{$\\mathcal{Q}_{n}$};\n\\draw (4, 4.6) node{\\textcolor{red}{$\\mathcal{P}^+_n$}};\n\\draw (-7, 0) node{$P_n$};\n\\draw (21, 0) node{$s$};\n\n\n\\draw (5.6,-0.65) node{$\\bar{t}$};\n\n\n\n\n\\draw[red]  (20,5.6) parabola (-6,0);\n\n\n\n\n\\draw[red]  (20,-5.6) parabola (-6,0);\n\n\n\n\n\\end{tikzpicture}\n\\caption{{\\small Scheme of the proof of \\eqref{bound}. The anisotropic paraboloid $\\mathcal{P}^+_n$ (in red), that is centered in $P_n=(\\,y,\\, s-(\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}})$, evolves in a time $(\\omega_n\/C_1)^{2-\\bar{p}}(C_2\\rho_n)^{\\bar{p}}$ to cover completely $\\mathcal{Q}_{n+1}$.}}\n \\label{FigA}\n\\end{figure}\n    \n\n\n\n\\end{proof}\n\\noindent \n{\\small STEP 4.{\\it Conclusion of the proof of Theorem \\ref{HC}}}\n\\vskip0.2cm \n\n\\noindent If we consider a point $(x,t) \\in (y,s) +\\mathcal{Q}_R^-(\\omega_o\/C_1)$, let $n \\in \\mathbb{N}$ be the last number such that we have $(x,t)\\in \\mathcal{Q}_n$, so that $(x,t) \\not\\in \\mathcal{Q}_{n+1}$. From the first condition and \\eqref{control} we have \n\n\\[|u(x,t)-u(y,s)| \\leq \\operatornamewithlimits{osc}_{\\mathcal{Q}_n} u\\leq \\delta^n \\omega_o.\\]\nThe rest of the job is standard and consists in determining from condition $(x,t) \\not\\in \\mathcal{Q}_{n+1}$ an upper bound for $\\delta^n$. For the sake of simplicity, we just show the case $x \\not\\in y+\\mathcal{K}_{\\rho_{n+1}}$.\n\n\\noindent \nLet $\\beta>0$ be such that $\\delta^{{(\\bar{p}-2)}\/{\\bar{p}}}\/A=\\delta^{\\beta}$. By assumption, there must be an index $i \\in \\{1,\\dots, N\\}$ such that\n\\begin{equation*}\\label{i}\n\\begin{aligned}\n|x_i-y_i|^{p_i}> \\rho_{n+1}^{{\\bar{p}}} (\\omega_{n+1}\/C_1)^{{p_i-\\bar{p}}} = \\gamma(A)(\\delta^n)^{p_i-2}R^{\\bar{p}}(\\omega_o\/C_1)^{p_i-\\bar{p}} \\geq \\gamma(A)(\\delta^n)^{[{\\bar{p}(\\beta-1)+p_i}]}R^{\\bar{p}}(\\omega_o\/C_1)^{p_i-\\bar{p}}\n\\end{aligned}\n\\end{equation*}\n\\noindent that gives us, for $\\chi_i=\\bar{p}\/(\\bar{p}(\\beta-1)+p_i)$, the following estimate of $\\delta^n$:\n \\begin{equation*} \\begin{aligned}\n \\delta^n \\leq&\n \\gamma  \\bigg( \\frac{ |x_i-y_i|^{{p_i}\/{\\bar{p}}}  (\\omega_o\/C_1)^{{(\\bar{p}-p_i)}\/{\\bar{p}}}}{R} \\bigg)^{{\\bar{p}}\/[{{\\bar{p}(\\beta-1)+p_i}}]} \\\\\n &\\leq \\gamma \\bigg(\\frac{\\sum_{i=1}^N |x_i-y_i|^{{p_i}\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-p_i)}\/{\\bar{p}}}+ |t-s|^{{1}\/{\\bar{p}}}\\omega_o^{{(\\bar{p}-2)}\/{\\bar{p}}}}{{\\bf{p}}\\text{-dist}(K,\\partial \\Lambda) } \\bigg)^{\\chi_i}.\\end{aligned} \\end{equation*}\n \\noindent\n From $A>4>\\delta^{-1-2\/\\bar{p}}$ we infer $\\beta>2$, whence $\\chi_i\\in(0,1)$. A similar estimate follows from the case where times are not contained, with $\\chi_t= \\bar{p}\/(\\bar{p}(\\beta-1)+2)$. Therefore, recalling that $p_N>2$, we choose the H\\\"older exponent\n\\begin{equation} \\label{alfa}\n    \\chi = \\min \\{\\chi_i, \\chi_t, \\quad i=1,\\dots,N \\}=\\frac{\\bar{p}}{\\bar{p}(\\beta-1)+p_N}.\n\\end{equation}\n \\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\small\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}