diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlutz" "b/data_all_eng_slimpj/shuffled/split2/finalzzlutz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlutz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe occurrence of spontaneous symmetry breaking in atomic nuclei leads to \nvarious nuclear shapes\nwhich can usually be described by the parametrization of the nuclear surface or the\nnucleon density distribution~\\cite{Bohr1998_Nucl_Structure_1,Ring1980}.\nIn mean-field calculations, the following parametrization\n\\begin{equation}\n \\beta_{\\lambda\\mu} = {4\\pi \\over 3AR^\\lambda} \\langle Q_{\\lambda\\mu} \\rangle,\n \\label{eq:01}\n\\end{equation}\nis usually used, where $Q_{\\lambda\\mu}$ are the mass multipole operators.\nIn Fig.~\\ref{Pic:deformations}, a schematic show is given for some typical nuclear shapes.\nThe majority of observed nuclear shapes is of spheroidal form which\nis usually described by $\\beta_{20}$.\nHigher-order deformations with $\\lambda>2$ such as $\\beta_{30}$\nalso appear in certain atomic mass regions~\\cite{Butler1996_RMP68-349}.\nIn addition, non-axial shapes in atomic nuclei, in particular,\nthe nonaxial-quadrupole (triaxial) deformation $\\beta_{22}$ has been\nstudied both experimentally and\ntheoretically~\\cite{Starosta2001_PRL86-971,Odegard2001_PRL86-5866,Meng2010_JPG37-064025}.\nThe influence of the nonaxial octupole $\\beta_{32}$ deformation\non the low-lying spectra has been also investigated~\\cite{Hamamoto1991_ZPD21-163,Skalski1991_PRC43-140,%\nLi1994_PRC49-R1250,Takami1998_PLB431-242,Yamagami2001_NPA693-579,%\nDudek2002_PRL88-252502,Dudek2006_PRL97-072501,Olbratowski2006_IJMPE15-333,%\nZberecki2006_PRC74-051302R,Dudek2010_JPG37-064032}.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{1.0\\columnwidth}{!}{%\n \\includegraphics{deformations} }\n\\end{center}\n\\caption{\\label{Pic:deformations}(Color online)\nA schematic show of some typical nuclear shapes. \nFrom left to right, the 1st row: (a) Sphere, (b) Prolate spheroid,\n(c) Oblate spheroid, (d) Hexadecapole shape, \nand the second row: (e) Triaxial ellipsoid,\n(f) Reflection symmetric octupole shape, (g) Tetrahedron,\n(h) Reflection asymmetric octupole shape with very large quadrupole deformation\n and large hexadecapole deformation.\nTaken from Ref.~\\cite{Lu2012_PhD}.\n}\n\\end{figure}\n\nIn nuclear fission study,\nvarious shape degrees of freedom play important and different roles\nin the occurrence and in determining the heights of the inner and outer barriers\nin actinide nuclei (in these nuclei double-humped fission barriers usually appear).\nFor example, the inner fission barrier is usually lowered when the triaxial\ndeformation is allowed, while for the outer barrier the reflection asymmetric\n(RA) shape is favored~\\cite{Pashkevich1969_NPA133-400,Moeller1970_PLB31-283,%\nGirod1983_PRC27-2317,Rutz1995_NPA590-680,Abusara2010_PRC82-044303,%\nPrassa2012_PRC86-024317,Prassa2013_PRC88-044324}.\n\nIn order to give a microscopic and self-consistent description of\nthe potential energy surface (PES) with more shape degrees of freedom included,\nmulti-dimensional constrained covariant density functional theories were developed\nrecently~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513}.\nIn these theories, all shape degrees of freedom $\\beta_{\\lambda\\mu}$ \nwith even $\\mu$ are allowed.\nIn this contribution, we present two recent applications of these theories:\nthe PES's of actinide nuclei and\nthe non-axial reflection-asymmetric $\\beta_{32}$ shape in some transfermium\nnuclei.\nIn Section~\\ref{sec:formalism}, the formalism of our multi-dimensional constrained\ncovariant density functional theories will be given briefly.\nThe results and discussions are presented in Section~\\ref{sec:results}.\nFinally we give a summary in Section~\\ref{sec:summary}.\n\n\n\\section{Formalism}\n\\label{sec:formalism}\n\nThe details of the formalism for covariant density functional theories\ncan be found in Refs.~\\cite{Serot1986_ANP16-1,Reinhard1989_RPP52-439,%\nRing1996_PPNP37-193,Vretenar2005_PR409-101,Meng2006_PPNP57-470,Niksic2011_PPNP66-519}.\nThe CDFT functional in our multi-dimensional constrained calculations\ncan be one of the following four forms:\nthe meson exchange or point-coupling nucleon interactions combined with\nthe non-linear or density-dependent couplings~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513,%\nLu2012_EPJWoC38-05003,Lu2013_arXiv1304.6830}.\nHere we show briefly the one corresponding to the non-linear point coupling\n(NL-PC) interactions.\nThe starting point of the relativistic\nNL-PC density functional is the following Lagrangian:\n\\begin{equation}\n \\mathcal{L} = \\bar{\\psi}(i\\gamma_{\\mu}\\partial^{\\mu}-M)\\psi\n -\\mathcal{L}_{{\\rm lin}}\n -\\mathcal{L}_{{\\rm nl}}\n -\\mathcal{L}_{{\\rm der}}\n -\\mathcal{L}_{{\\rm Cou}},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n \\mathcal{L}_{{\\rm lin}} & = & \\frac{1}{2} \\alpha_{S} \\rho_{S}^{2}\n +\\frac{1}{2} \\alpha_{V} \\rho_{V}^{2}\n +\\frac{1}{2} \\alpha_{TS} \\vec{\\rho}_{TS}^{2}\n +\\frac{1}{2} \\alpha_{TV} \\vec{\\rho}_{TV}^{2} ,\n \\nonumber \\\\\n \\mathcal{L}_{{\\rm nl}} & = & \\frac{1}{3} \\beta_{S} \\rho_{S}^{3}\n +\\frac{1}{4} \\gamma_{S}\\rho_{S}^{4}\n +\\frac{1}{4} \\gamma_{V}[\\rho_{V}^{2}]^{2} ,\n \\nonumber \\\\\n \\mathcal{L}_{{\\rm der}} & = & \\frac{1}{2} \\delta_{S}[\\partial_{\\nu}\\rho_{S}]^{2}\n +\\frac{1}{2} \\delta_{V}[\\partial_{\\nu}\\rho_{V}]^{2}\n +\\frac{1}{2} \\delta_{TS}[\\partial_{\\nu}\\vec{\\rho}_{TS}]^{2}\n \\nonumber \\\\\n & & \\mbox{} +\\frac{1}{2} \\delta_{TV}[\\partial_{\\nu}\\vec{\\rho}_{TV}]^{2} ,\n \\nonumber \\\\\n \\mathcal{L}_{{\\rm Cou}} & = & \\frac{1}{4} F^{\\mu\\nu} F_{\\mu\\nu}\n +e\\frac{1-\\tau_{3}}{2} A_{0} \\rho_{V} ,\n\\label{eq:lagrangian}\n\\end{eqnarray}\nare the linear coupling, nonlinear coupling, derivative coupling,\nand the Coulomb part, respectively.\n$M$ is the nucleon mass, $\\alpha_{S}$, $\\alpha_{V}$, $\\alpha_{TS}$,\n$\\alpha_{TV}$, $\\beta_{S}$, $\\gamma_{S}$, $\\gamma_{V}$, $\\delta_{S}$,\n$\\delta_{V}$, $\\delta_{TS}$, and $\\delta_{TV}$ are coupling constants\nfor different channels and $e$ is the electric charge.\n$\\rho_{S}$, $\\vec{\\rho}_{TS}$, $\\rho_{V}$, and $\\vec{\\rho}_{TV}$ are the isoscalar density,\nisovector density, time-like components of isoscalar current, and time-like components of\nisovector current, respectively.\nThe densities and currents are defined as\n\\begin{eqnarray}\n \\rho _{S} = \\bar{\\psi}\\psi , & \\qquad &\n \\vec{\\rho}_{TS} = \\bar{\\psi}\\vec{\\tau}\\psi ,\n \\nonumber \\\\\n \\rho_{V} = \\bar{\\psi} \\gamma^{0} \\psi , & \\qquad &\n \\vec{\\rho}_{TV} = \\bar{\\psi} \\vec{\\tau} \\gamma^{0} \\psi.\n \\label{eq:densities}\n\\end{eqnarray}\nStarting from the above Lagrangian, using the Slater determinants as\ntrial wave functions and neglecting the Fock term as well as the contributions\nto the densities and currents from the negative energy levels, one\ncan derive the equations of motion for the nucleons,\n\\begin{equation}\n \\hat{h}\\psi_{i} = \\left(\\bm{\\alpha}\\cdot\\vec{p}+\\beta(M+S(\\vec{r}))+V(\\vec{r})\\right)\\psi_{i}\n = \\epsilon_{i}\\psi_{i}\n ,\n\\end{equation}\nwhere the potentials $V(\\bm{r})$ and $S(\\bm{r})$ are calculated as\n\\begin{eqnarray}\nS & = & \\alpha_{S}\\rho_{S}+\\beta_{S}\\rho_{S}^{2}+\\gamma_{S}\\rho_{S}^{3}+\\delta_{S}\\triangle\\rho_{S}\\nonumber \\\\\n & & +\\left(\\alpha_{TS}\\rho_{TS}+\\delta_{TS}\\triangle\\rho_{TS}\\right)\\tau_{3} ,\\\\\nV & = & \\alpha_{V}\\rho_{V}+\\gamma_{V}\\rho_{V}^{3}+\\delta_{V}\\triangle\\rho_{V}W\\nonumber \\\\\n & & +\\left(\\alpha_{TV}\\rho_{TV}+\\delta_{TV}\\triangle\\rho_{TV}\\right)\\tau_{3} .\n\\end{eqnarray}\n\nAn axially deformed harmonic oscillator (ADHO) basis is adopted for solving the\nDirac equation~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513,Lu2011_PRC84-014328}.\nThe ADHO basis is defined as the eigen solutions of the Schrodinger\nequation with an ADHO potential~\\cite{Gambhir1990_APNY198-132,Ring1997_CPC105-77},\n\\begin{eqnarray}\n \\left[ -\\frac{\\hbar^{2}}{2M} \\nabla^{2} + V_{B}(z,\\rho) \\right] \\Phi_{\\alpha}(\\bm{r}\\sigma)\n & = &\n E_{\\alpha} \\Phi_{\\alpha}(\\bm{r}\\sigma)\n ,\n \\label{eq:BasSchrodinger-1}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n V_{B}(z,\\rho) = \\frac{1}{2} M ( \\omega_{\\rho}^{2}\\rho^{2} + \\omega_{z}^{2}z^{2} )\n ,\n\\end{equation}\nis the axially deformed HO potential and $\\omega_{z}$ and $\\omega_{\\rho}$\nare the oscillator frequencies along and perpendicular to $z$ axis, respectively.\nThese basis states are also eigen functions of the $z$ component of the\nangular momentum $j_{z}$ with eigen values $K=m_{l}+m_{s}$.\nFor any basis state $\\Phi_{\\alpha}(\\bm{r}\\sigma)$, the time reversal state\nis defined as $\\Phi_{\\bar{\\alpha}}(\\bm{r}\\sigma)=\\mathcal{T}\\Phi_{\\alpha}(\\bm{r}\\sigma)$,\nwhere $\\mathcal{T}=i\\sigma_{y}K$ is the time reversal operator and\n$K$ is the complex conjugation.\nApparently we have $K_{\\bar{\\alpha}}=-K_{\\alpha}$\nand $\\pi_{\\bar{\\alpha}}=\\pi_{\\alpha}$.\nThese basis states form a complete set for expanding any two-component spinors.\nFor a Dirac spinor with four components,\n\\begin{equation}\n \\psi_{i}(\\bm{r}\\sigma) =\n \\left( \\begin{array}{c}\n \\sum_{\\alpha}f_{i}^{\\alpha} \\Phi_{\\alpha}(\\bm{r}\\sigma) \\\\\n \\sum_{\\alpha}g_{i}^{\\alpha} \\Phi_{\\alpha}(\\bm{r}\\sigma)\n \\end{array}\n \\right),\n\\end{equation}\nwhere the sum runs over all the possible combination of the quantum\nnumbers $\\alpha=\\{n_{z},n_{r},m_{l},m_{s}\\}$, and $f_{i}^{\\alpha}$ and\n$g_{i}^{\\alpha}$ are the expansion coefficients.\nIn practical calculations, one should truncate the basis\nin a proper way~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513,Lu2011_PRC84-014328}.\n\nThe nucleus is assumed to be symmetric under the $V_4$ group, that is, all the\npotentials and densities can be expanded as \n\\begin{equation}\n f(\\rho,\\varphi,z) = f_{0}(\\rho,z) \\frac{1}{\\sqrt{2\\pi}}\n+ \\sum_{n=1}^{\\infty} f_{n}(\\rho,z) \\frac{1}{\\sqrt{\\pi}}\\cos(2n\\varphi),\n\\end{equation}\n\nThe PES is obtained by the constrained self-consistent calculation,\n\\begin{equation}\n E^{\\prime} = E_{{\\rm RMF}} +\n \\sum_{\\lambda\\mu} \\frac{1}{2} C_{\\lambda\\mu}Q_{\\lambda\\mu} ,\n\\end{equation}\nwhere the variables $C_{\\lambda\\mu}$'s change their values during the iteration.\n\nBoth the BCS approach and the Bogoliubov transformation are implemented\nin our model to take into account the pairing effects.\nFor convenience, we name the MDC-CDFT with the BCS approach for the pairing \nas the MDC-RMF model and that with the Bogoliubov transformation as the MDC-RHB model.\nMore details of the multi-dimensional constraint covariant density\nfunctional theories can be found in Refs.~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513}.\n\n\n\\section{Results and discussions}\n\\label{sec:results}\n\n\\subsection{PES's of actinides} \n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.9\\columnwidth}{!}{%\n \\includegraphics{Pu240PES1d.eps} }\n\\end{center}\n\\caption{\\label{Pic:PU240-1d}(Color online)\nPotential energy curves of $^{240}$Pu with various self-consistent symmetries imposed.\nThe solid black curve represents the calculated fission path with $V_4$ symmetry imposed:\nthe red dashed curve is that with axial symmetry (AS) imposed,\nthe green dotted curve that with reflection symmetry (RS) imposed,\nthe violet dot-dashed line that with both symmetries (AS \\& RS) imposed.\nThe empirical inner (outer) barrier height $B_\\mathrm{emp}$ is denoted by the grey square (circle).\nThe energy is normalized with respect to the binding energy of the ground state.\nThe parameter set used is PC-PK1.\nTaken from Ref.~\\cite{Lu2012_PRC85-011301R}.\n}\n\\end{figure}\n\nIn Refs.~\\cite{Lu2012_PRC85-011301R,Lu2013_arXiv1304.2513}, one- (1-d),\ntwo- (2-d), and three-dimensional (3-d) constrained calculations were \nperformed for the actinide nucleus $^{240}$Pu.\nThe MDC-RMF model with the parameter set PC-PK1~\\cite{Zhao2010_PRC82-054319} was used.\nIn Fig.~\\ref{Pic:PU240-1d} we show the 1-d potential energy curves \nfrom an oblate shape with $\\beta_{20}$ about $-0.2$ to the fission\nconfiguration with $\\beta_{20}$ beyond 2.0\nwhich are obtained from calculations with different self-consistent symmetries\nimposed: the axial (AS) or triaxial (TS) symmetries combined with\nreflection symmetric (RS) or asymmetric cases.\nThe importance of the triaxial deformation on the inner barrier and\nthat of the octupole deformation on the outer barrier are clearly seen:\nThe triaxial deformation reduces the inner barrier height by more than 2 MeV\nand results in a better agreement with the empirical value~\\cite{Abusara2010_PRC82-044303};\nthe RA shape is favored beyond the fission isomer and lowers very much\nthe outer fission barrier~\\cite{Rutz1995_NPA590-680}.\nBesides these features, it was found for the first time that the outer\nbarrier is also considerably lowered by about 1 MeV when the triaxial\ndeformation is allowed.\nIn addition, a better reproduction of the empirical barrier height can be seen for\nthe outer barrier.\nIt has been stressed that this feature can only be found when the axial and\nreflection symmetries are simultaneously broken~\\cite{Lu2012_PRC85-011301R}.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.95\\columnwidth}{!}{%\n \\includegraphics{Pu240_Inner.eps} }\n\\end{center}\n\\caption{\\label{Pic:PU240_2dI}(Color online)\nPotential energy surfaces of $^{240}$Pu in the $(\\beta_{20},\\beta_{22})$ plane\naround the inner barrier.\nThe energy is normalized with respect to the binding energy of the ground state.\nThe least-energy fission path is represented by a dash-dotted line.\nThe saddle point is denoted by the full star.\nThe contour interval is 0.5 MeV.\n}\n\\end{figure}\n\nTwo-dimensional PES's in the $(\\beta_{20},\\beta_{22})$ plane near the inner and \nouter barriers are shown in Figs.~\\ref{Pic:PU240_2dI} and~\\ref{Pic:PU240_2dO}, respectively.\nStarting from the axially symmetric ground state, the nucleus goes through the \ntriaxial valley to the isometric state. \nThe inner barrier is located at $\\beta_{20} \\approx 0.65$ and $\\beta_{22} \\approx 0.06$. \nThe isomeric state keeps an axially symmetric shape.\nAs $\\beta_{20}$ further increases, the nucleus goes through a triaxial valley\nagain, and then goes fission. \nThe outer barrier is located at $\\beta_{20} \\approx 1.21$, $\\beta_{22} \\approx 0.02$, \nand $\\beta_{30} \\approx 0.37$.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.95\\columnwidth}{!}{%\n \\includegraphics{Pu240_Outer.eps} }\n\\end{center}\n\\caption{\\label{Pic:PU240_2dO}(Color online)\nPotential energy surfaces of $^{240}$Pu in the $(\\beta_{20},\\beta_{22})$ plane\naround the outer barrier.\nThe energy is normalized with respect to the binding energy of the ground state.\nThe least-energy fission path is represented by a dash-dotted line.\nThe saddle point is denoted by the full star.\nThe contour interval is 0.25 MeV.\n}\n\\end{figure}\n\nA systematic study of even-even actinide nuclei has been carried out\nand the results were presented in Ref.~\\cite{Lu2013_arXiv1304.2513}\nwhere we have shown that the triaxial deformation lowers\nthe outer barriers of these actinide nuclei by about $0.5 \\sim 1$ MeV \n(about $10 \\sim 20 \\%$ of the barrier height).\n\n\\subsection{$Y_{32}$-correlations in $N=150$ isotones}\n\nIt has been anticipated that the tetrahedral shape \n($\\beta_{\\lambda\\mu} = 0$, if $\\lambda\\ne3$ and $\\mu\\ne2$) may appear in the ground states of\nsome nuclei with special combinations of the neutron and\nproton numbers~\\cite{Li1994_PRC49-R1250,Dudek2010_JPG37-064032,Dudek2002_PRL88-252502}.\nThe tetrahedral symmetry-driven quantum effects may also lead to a large increase \nof binding energy in superheavy nuclei~\\cite{Chen2013_NPR30-278}. \nHowever, no solid experimental evidence has been found for nuclei with\ntetrahedral shapes.\nOn the other hand, $\\beta_{32}$ deformation may appear together with other shape\ndegrees of freedome, say, $\\beta_2$.\nFor example, it has been proposed that the non-axial\noctupole $Y_{32}$-correlation results in the experimentally observed low-energy $2^-$\nbands in the $N = 150$ isotones~\\cite{Robinson2008_PRC78-034308} and the RASM calculations\nreproduces well the experimental observables of these $2^-$ bands~\\cite{Chen2008_PRC77-061305R}.\n\nIn Ref.~\\cite{Zhao2012_PRC86-057304} the non-axial reflection-asymmetric $\\beta_{32}$\ndeformation in $N=150$ isotones, namely $^{246}$Cm, $^{248}$Cf,\n$^{250}$Fm, and $^{252}$No was investigated using the MDC-RMF model\nwith the parameter set DD-PC1~\\cite{Niksic2008_PRC78-034318}.\nIt was found that \ndue to the interaction between a pair of neutron orbitals,\n$[734]9\/2$ originating from $\\nu j_{15\/2}$ and\n$[622]5\/2$ originating from $\\nu g_{9\/2}$, and\nthat of a pair of proton orbitals,\n$[521]3\/2$ originating from $\\pi f_{7\/2}$ and\n$[633]7\/2$ originating from $\\pi i_{13\/2}$,\nrather strong non-axial octupole $Y_{32}$ effects appear in $^{248}$Cf and\n$^{250}$Fm which are both well deformed with large axial-quadrupole deformations,\n$\\beta_{20} \\approx 0.3$.\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.90\\columnwidth}{!}{%\n \\includegraphics{Cf248_b32_Pair_bin} }\n\\end{center}\n\\caption{\\label{fig:b32} (Color online)\nThe binding energy $E$ (relative to the ground state) for $^{248}$Cf\nas a function of the non-axial octupole deformation parameter $\\beta_{32}$.\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig:b32}, the potential energy curve,\ni.e., the total binding energy as a function of $\\beta_{32}$ was shown\nfor $^{248}$Cf.\nAt each point of the potential energy curve, the energy is minimized\nautomatically with respect to other shape degrees\nof freedom such as $\\beta_{20}$, $\\beta_{22}$, $\\beta_{30}$, and $\\beta_{40}$, etc.\nOne finds in this curve a clear pocket with the depth more than 0.3 MeV.\nSimilar potential energy curve was also obtained for $^{250}$Fm.\nFor $^{246}$Cm and $^{252}$No, only a shallow minimum develops along\nthe $\\beta_{32}$ shape degree of freedom.\nIt was also shown that the evolution of the non-axial octupole $\\beta_{32}$ effect along\nthe $N=150$ isotonic chain is not very sensitive to the form of\nthe energy density functional and the parameter set we used~\\cite{Zhao2012_PRC86-057304}.\n\nBoth the non-axial octupole parameter $\\beta_{32}$ and the energy gain due to\nthe $\\beta_{32}$-distortion reach maximal values at $^{248}$Cf \nin the four nuclei along the $N=150$ isotonic chain~\\cite{Zhao2012_PRC86-057304}.\nThis is consistent with the analysis given in\nRefs.~\\cite{Chen2008_PRC77-061305R,Jolos2011_JPG38-115103} and the experimental observation\nthat in $^{248}$Cf, the $2^-$ state is the lowest among these\nnuclei~\\cite{Robinson2008_PRC78-034308}.\nThese results indicate a strong $Y_{32}$-correlation in these nuclei.\n\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this contribution we present the formalism and some applications of the multi-dimensional\nconstrained relativistic mean field (MDC-RMF) models in which all shape degrees of freedom\n$\\beta_{\\lambda\\mu}$ with even $\\mu$ are allowed.\nThe potential energy surfaces (curves) of actinide nuclei \nand the effect of the triaxiality on the first and second fission barriers \nwere investigated.\nIt is found that besides the octupole deformation, the triaxiality also\nplays an important role upon the second fission barriers.\nThe non-axial reflection-asymmetric $\\beta_{32}$ shape in $N=150$ isotones were studied\nand rather strong non-axial octupole $Y_{32}$ effects have been found in $^{248}$Cf and\n$^{250}$Fm which are both well deformed with large axial-quadrupole deformations,\n$\\beta_{20} \\approx 0.3$.\n\n\n\\begin{ack}\nThis work has been supported by Major State Basic Research Development \nProgram of China (Grant No. 2013CB834400), \nNational Natural Science Foundation of China (Grant Nos. 11121403, 11175252, \n11120101005, 11211120152, and 11275248), \nthe Knowledge Innovation Project of Chinese Academy of Sciences (Grant No. KJCX2-EW-N01). \nThe results described in this paper are obtained on the ScGrid of Supercomputing\nCenter, Computer Network Information Center of Chinese Academy of Sciences.\n\\end{ack}\n\n\n\n\n\\providecommand{\\newblock}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\\label{sec:intro}\n\nIn the past years, our group has extensively used the {\\it Hubble Space Telescope} ({\\it HST}\\,) archive to study star clusters in both Magellanic Clouds \\citep[e.g.][and references therein]{milone2009a, milone2020a}. The exquisite stellar photometry and astrometry provided by {\\it HST}, together with the most advanced techniques for the analysis of astronomical \nimages \\citep[e.g.][]{anderson2008a, sabbi2016a, bellini2017a}, has provided significant advances in understanding Magellanic-Cloud star clusters and their stellar populations.\n\nInspired by the discovery that the color-magnitude diagram (CMD) of the Large Magellanic Cloud (LMC) cluster NGC\\,1806 is not consistent with a single isochrone \\citep{mackey2007a}, we started a series of papers to investigate the so-called extended-main sequence turn off phenomenon (eMSTO), in clusters with ages from about one to 2.3\\,Gyr. \n The main results include the discovery that the eMSTO is a common feature of Magellanic Cloud clusters \\citep{milone2009a}, the early discoveries of split main sequence (MS) in young Magellanic Cloud clusters \\citep{milone2013a, milone2015a, milone2016a, milone2017a}, and the characterization of the multiple populations in young and intermediate-age LMC and Small Magellanic Cloud (SMC) clusters \\citep{milone2018a}. We provided the first direct evidence, based on high-resolution spectra, that the blue and red MS are made up of stellar populations with different rotation rates \\citep{marino2018a} and the color and magnitude of an eMSTO star depend on stellar rotation \\citep{dupree2017a, marino2018a}. \n The high-precision photometry resulting from this project has been instrumental to shed light on the physical mechanisms that are responsible for generating multiple populations in young clusters and has been used both by our team and by other groups to constrain the effect of rotation and stellar mergers on the eMSTO and the split MS \\citep[e.g.][]{bastian2009a, dantona2015a, dantona2017a, wang2022a, cordoni2022a} and the contribution of variable stars on the eMSTO \\citep{salinas2018a}.\n \n Although our main purpose consisted in investigating the eMSTO phenomenon, the resulting photometric and astrometric catalogs have been used for various investigations of stellar astrophysics, including multiple stellar populations in Magellanic Cloud globular clusters \\citep[GCs,][]{lagioia2019a, lagioia2019b, milone2020a, dondoglio2021a}, photometric binaries \\citep[][]{milone2009a, milone2013a}, Be stars \\citep[][]{milone2018a, hastings2021a}, and extinction \\citep{demarchi2020a}.\n \n Driven by these results, we decided to homogeneously analyze all archival images collected with the Ultraviolet and Visual Channel (UVIS) and the Near Infrared Channel (NIR) of Wide Field Camera 3 (WFC3) and with the Wide Field Channel of the Advanced Camera for Surveys (WFC\/ACS) on board {\\it HST}. \n In this work, we present high-precision stellar positions and magnitudes for stars in 101 fields of Magellanic Clouds that include 113 star clusters.\n \n \n The paper is organized as follows. Section \\ref{sec:data} describes the dataset and the methods used for homogeneously reducing the data and presents the CMDs. The methods for correcting the photometry for differential reddening and the differential-reddening maps are discussed in Section \\ref{sec:reddening}, while Section \\ref{sec:centri} is dedicated to the determination of the cluster centers. \n Absolute stellar proper motions are derived in Section \\ref{sec:pms}. Section \\ref{sec:secrets}\n provides some scientific cases that arise from early inspection of our catalogs. Finally, we report in the appendix the serendipitous discovery of one gravitational lens and two stellar clusters.\n \n\\section{Data and data analysis}\\label{sec:data}\nThe dataset used in this paper comprises images collected through the UVIS\/WFC3 and NIR\/WFC3, and WFC\/ACS on board {\\it HST}.\nThe images include 84 known star clusters in the Large Magellanic Cloud (LMC) and 29 clusters in the Small Magellanic Cloud (SMC). These clusters span wide intervals of age and stellar density, from sparse star-forming regions to old and dense GCs. \n The main properties of the available exposures are listed in Table\\,\\ref{tab:data}.\n\nPhotometry and astrometry are obtained from calibrated, flat-fielded WFC3\/NIR ({\\rm \\_flt}) exposures, while in the case of UVIS\/WFC3 and WFC\/ACS data we used the calibrated, flat-fielded exposures corrected for the effects of the poor charge-transfer efficiency (CTE) of the detectors \\citep[{\\rm \\_flc,}][]{anderson2010a}.\nStars are measured by means of distinct approaches\n that work best in different brightness regimes, \n as discussed in the following subsections. \n\n\\subsection{First-pass photometry}\n\nWe accounted for spatial variations of the Point-Spread Function (PSF) by using the grids of library PSFs provided by Jay Anderson for each filter and camera. The PSFs can change from one exposure to another due to focus variations produced by the breathing of {\\it HST}, small guiding inaccuracies, and residual CTE. To derive the optimal PSF we perturbed the library PSFs by using a version of the \\citet{anderson2006a} computer program adapted to UVIS\/WFC3 and WFC\/ACS \\citep[see also][]{bellini2013a}. In a nutshell, we divided each image into a grid containing $n \\times n$ cells, with $n$ ranging from 1 to 5. Bright, isolated, and unsaturated stars within each cell are fitted by the library PSF model, and the residuals of the fit are iteratively used to improve the PSF model itself.\nWe calculated the appropriate PSF model of each star based on its location in the detector by linearly interpolating the four nearest PSFs of the grid \\citep{anderson2000a}.\nThe number of cells in the grid has been fixed with the aim of obtaining the best quality-fit parameters for bright stars and depends on the number of available reference stars used to constrain the PSF perturbation in the cell. \n\n\nThese PSFs are then used to measure the magnitudes and positions of unsaturated stars in each image. Saturated stars in the UVIS\/WFC3 and WFC\/ACS images are measured using the methods by \\citet{gilliland2004a} and \\citet{gilliland2010a}. These authors noted that the total number of electrons of saturated stars in the UVIS\/WFC3 and WFC\/ACS detectors is conserved and this information is preserved in the ${\\rm \\_flt}$ images with gain=2.\nHence, we measured each saturated star in an aperture of 5-pixel radius and added the contiguous saturated pixels that had bled outside this radius \\citep[see][for details]{anderson2008a}. \n\nAll catalogs derived from each filter and camera have been tied to the same photometric zero point, corresponding to the zero point of the deepest exposure in the filter that we used as a reference frame to construct the photometric master frame. To do this, we used the bright, unsaturated stars that are well-fitted by the PSF to calculate the difference between the magnitudes in the master frame and in each exposure. We used the mean of these magnitude differences to transform stars measured in each exposure into this reference frame.\n\nStellar positions are corrected for geometric distortion by using the solutions provided by \\citet{anderson2006b} for WFC\/ACS and \\citet{bellini2009a} and \\citet{bellini2011a} for UVIS\/WFC3.\nThe coordinates of stars in all images of each cluster are transformed into a common reference system based on Gaia Early Data Release 3 (eDR3) catalogs \\citep[][]{gaia2020a}, in such a way that the abscissa and the ordinate are aligned with the West and North direction, respectively. \nWe first de-projected the right ascension and declination into the plane tangential to the center of the main cluster in the field. \nWe assumed for these coordinates a scale factor of 0.04 arcsec per pixel. \nWe first used bright, unsaturated stars that are well-fitted by the PSF to derive the six-parameter linear transformations used to convert the coordinates of all stars in each exposure into this reference frame. \nThen, we derived the $3\\sigma$-clipped average stellar positions to derive a new astrometric catalog, that we used as a master frame to improve the transformations. \n\n\\subsection{Multi-pass photometry}\nThe main outcomes from first-pass photometry, including PSF models, coordinate transformations, photometric zero points, stellar magnitudes, and positions, are used to simultaneously identify and measure all point-like sources in all exposures. \nTo do this, we used the FORTRAN computer program KS2 developed by Jay Anderson \\citep[e.g.\\,][]{sabbi2016a, bellini2017a, nardiello2018a}, which is the evolution of {\\it kitchen sink}, originally written to reduce WFC\/ACS images \\citep{anderson2008a}. \nKS2 exploits various iterations to find and measure stars. It first identifies the brightest and most isolated stars, calculates their fluxes and positions, and subtracts them from the image. In the subsequent iterations, it finds, measures, and subtracts stars that are gradually fainter and closer to neighbor stars. \n We used the stellar positions and magnitudes derived from first-pass photometry to generate appropriate masks for bright stars, including saturated ones. These masks optimize the detection and measurement of faint sources that are close to bright stars. They also minimize the detection of spurious sources that are typically associated with diffraction spikes and other structures of the stellar profile. \n\nThis program adopts three distinct methods to measure stars, each providing optimal photometry for different ranges of stellar luminosity and density. \n\\begin{itemize}\n \\item Method I is optimal for relatively bright stars. It provides accurate measurements of all stars that generate distinct peaks within their local 5$\\times$5-pixel raster after neighbor stars are subtracted. \n Each star is measured by using the PSF model corresponding to its position, while the sky level is estimated from the annulus between 4 and 8 pixels from the center of the star.\n \\item Method II provides the best photometry for faint stars, which do have not enough flux to provide robust fits with the PSF. After subtracting neighbor stars, KS2 performs the aperture photometry of the star in the $5\\times5$ pixel raster. Each pixel is properly weighted to ensure low weight to those pixels contaminated by nearby stars. The sky is calculated as in Method I.\n \\item Method III provides the best photometry in very crowded regions and for faint stars when a large number of exposures are available. It works as Method II, but aperture photometry is calculated over a circle with a radius of 0.75 pixels and the local sky in the annulus between 2 and 4 pixels from the position measured during the finding stage. \n\\end{itemize}\n Stellar fluxes and positions are measured in each exposure separately and then are properly averaged together to derive our best determinations of magnitudes and positions. \n\n Figure\\,\\ref{fig:methods} compares the CMDs of stars in the field of view of Lindsay\\,1 obtained from the three methods. We have chosen this GC as an example because of the deep F275W and F814W photometry available, which comprises 16 and 7 exposures in F275W and F814W, with total integration times of 27,341s and 2,206s, respectively. \n A visual comparison of the top panels reveals that methods\\,II and III are optimal for faint stars as they provide well-defined MSs. The latter method provides slightly better photometry for stars at the bottom of the MS alone, whereas Method\\,II provides the best photometry for the remaining faint MS stars as highlighted in the middle panels, where we show the zoomed CMDs for MS stars with instrumental $-6<$F275W$<-4$ mag\\footnote{Instrumental magnitudes are defined as the $-$2.5 log$_{10}$ of the detected photo-electrons.}. Clearly, the MS plotted in the central panel is much narrower and better defined than that shown in the left and right panels.\n On the contrary, Method I provides the best photometry for stars with bright instrumental magnitudes as demonstrated by the narrow red-giant and sub-giant branch (RGB and SGB) sequences in the bottom-left F555W vs.\\,F555W$-$F814W CMDs.\n For each field, we derived three distinct catalogs from Methods\\,I, II, and III. The science results shown in this paper, which are all focused on bright stars, are based on the photometry derived from Method\\,I.\n\n\\subsection{Photometry calibration}\nPhotometry of each filter and camera has been calibrated to the Vega mag system by computing the aperture correction to the PSF-fit-derived magnitudes and applying to the corrected instrumental magnitude a photometric zero-point.\n To calculate the aperture corrections, we used unsaturated and isolated stars only.\n \nWe measured aperture magnitudes within circular regions of $\\sim$0.4 and 0.5 arcsec radius for UVIS\/WFC3 and WFC\/ACS, respectively. To do this, we used the drizzled and CTE-corrected (${\\rm \\_drc,}$) images, which are normalized to 1s exposure time. Aperture photometry has been calibrated by adding to these instrumental magnitudes the corresponding aperture corrections and the zero points \\citep{bohlin2016a, deustua2017a}. \nFinally, we calculated the 3 $\\sigma$-clipped average of the difference between instrumental PSF magnitudes and calibrated aperture magnitudes for the stars in common. The resulting average values are then added to all the stars to derive calibrated magnitudes. \n\n\\subsection{Quality parameters}\\label{subsec:quality}\nThe computer program KS2 computes for each star various parameters that can be used as diagnostics of the photometric and astrometric quality. For each filter it provides three main quantities:\n\\begin{itemize}\n\n \\item the $RADXS$ parameter is a shape parameter that indicates the amount of flux that exceeds the predictions from the best-fitting PSF \\citep{bedin2008a}. It is defined as $RADXS=(\\sum_{\\rm i,j} pix_{\\rm i,j}-PSF_{\\rm i,j})\/10^{-{\\rm mag}\/2.5}$ where the sum is calculated within an annulus between 1.0 and 2.5 pixels from the center of the star and is normalized to the star's total flux. This quantity is negative when the object is sharper than the PSF (e.g.\\,cosmic rays and PSF artifacts) and it is positive when the object is broader than the PSF (e.g.\\,galaxies). The perfect PSF fit corresponds to $RADXS$=0. \n\n \\item the quality-fit parameter, $qfit$, which is indicative of the goodness of the PSF fit. It is defined as $qfit=\\frac{\\sum_{i,j} pix_{\\rm i,j} PSF_{\\rm i,j}}{\\sqrt{\\sum_{\\rm i,j} pix^{2}_{\\rm i,j} PSF^{2}_{\\rm i,j}}}$, and it is calculated in a 5$\\times$5 pixel area centered on the star, and $pix_{\\rm i,j}$ and $PSF_{\\rm i,j}$ are the values of the pixel and the best-fitting PSF model, respectively, estimated in the pixel (i,j). It ranges from unit, in the case of a perfect fit, to zero.\n \n \\item the root mean scatter of the magnitude determinations, $rms$.\n\\end{itemize}\n\nAs an example, in the top-left panels of Figure \\ref{fig:selezioni} we plot the $RADXS$ and $qfit$ parameters derived from F336W photometry of the star cluster Reticulum as a function of the F336W instrumental magnitude. Top-right panels show the analogous figures but for the F814W filter. The azure lines are drawn by hand with the criteria of separating the bulk of well-measured point-like sources from sources that are poorly fitted by the PSF model.\nThe bottom panels compare the CMD of stars that pass the selection criteria in both filters and the CMD of stars that have been rejected in at least one filter.\nAlthough the magnitude $rms$ is another diagnostic of photometric quality, we prefer not to use it to select the sample of stars with high-quality measurements to avoid excluding variable stars. \n\\begin{figure} \n\\centering\n\\includegraphics[width=8.5cm,trim={9.5cm 0.0cm 9.5cm 0.0cm},clip]{methodsLR.pdf}\n \\caption{Comparison of the instrumental F275W vs.\\,F275W$-$F814W CMDs of stars in the field of view of the star cluster Lindsay\\,1 as derived from Method I (top-left), Method II (top-middle) and Method III (top-right). Well-measured stars are colored black, while stars with poor photometry are plotted with light gray dots. Azure crosses mark stars where the F814W magnitude is derived from saturated images. Middle panels are zoomed-in views of the top-panel CMD around the MS. Bottom panels compare the region of the instrumental F555W vs.\\,F555W$-$F814W CMDs populated by bright stars with F555W$<-9.75$ mag. Calibrated magnitudes and colors are indicated by the top and right axes. } \n \\label{fig:methods} \n\\end{figure} \n\n\n\n\\begin{centering} \n\\begin{figure} \n \\includegraphics[width=9.5cm,trim={0.5cm 5.5cm 1cm 4.0cm},clip]{selezioni.pdf}\n \\caption{\\textit{Top panels}. $RADXS$ and $qfit$ parameters derived from F336W (left) and F814W (right) photometries of stars in the field of view of Reticulum. The azure lines separate well-measured stars (black dots) from poorly measured sources (gray crosses).\n The bottom panels show the instrumental F336W vs.\\,F336W$-$F814W CMD for stars that pass the selection criteria in both filters (left) and for the remaining stars (right).}\n \\label{fig:selezioni} \n\\end{figure} \n\\end{centering} \n\n\n\n\n\\subsection{The color-magnitude diagrams}\\label{subsec:CMDs}\nIn the four panels of Figure\\,\\ref{fig:HESS} we show the $m_{\\rm F336W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ (left) and the $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ (right) Hess diagrams of all the observed fields in the LMC (top) and SMC (bottom).\nClearly, these diagrams reveal the complexity of stellar populations in the Magellanic Clouds, from bright and blue MSs composed of young and metal-rich stars to old and metal-poor stellar populations characterized by blue and faint MSs and faint RGBs.\n\n\n\\begin{figure*} \n\\centering\n \\includegraphics[height=13.5cm,trim={0.0cm 0.0cm 10.0cm 1.0cm},clip]{ALLcmd.pdf}\n\n \\caption{$m_{\\rm F336W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ (left panels) and $m_{\\rm F555W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ (right panels) Hess diagrams for all LMC (top) and SMC stars (bottom).}\n \\label{fig:HESS} \n\\end{figure*} \n\nTo further illustrate the variety of stellar populations and environments contained in the dataset of this paper, we show in Figure\\,\\ref{fig:CMDs} the stacked images and the CMDs of stars in three distinct fields that host stellar populations with different stellar densities, ages, and metallicities. The F475W image and the CMD of stars in the field around the open cluster NGC\\,1966 are plotted in the top panels. This region, which has never been studied with {\\it HST}, hosts a conspicuous population of very young stars\n\n that populate the\n upper MS and the pre-MS.\n The CMD also reveals old-RGB and red-clump stars that likely belong to foreground and background LMC old stellar populations. \n Notably, the region hosts various nebula like NGC\\,1965 and the gas nebula around the Wolf-Rayet star HD-269546 \\citep[ the brightest star visible in the stacked image, ][]{westerlund1964a}, which are visible here in unprecedented detail.\nThe figures in the middle and bottom panels refer to the regions around the intermediate-age cluster NGC\\,2121 (age $\\sim 3$ Gyr) and the dense and old GC NGC\\,2210 (age $\\sim 12$ Gyr), respectively. \n\n\\begin{figure*} \n\\centering\n\\includegraphics[height=7.78cm,trim={0.0cm 0.0cm 0.0cm 0.0cm},clip]{FoVngc1966.pdf}\n\\includegraphics[height=7.78cm,trim={0.0cm 0.0cm 0.0cm 0.0cm},clip]{FoVngc2121.pdf}\n\\includegraphics[height=7.78cm,trim={0.0cm 0.0cm 0.0cm 0.0cm},clip]{FoVngc2210.pdf}\n \\caption{Stacked images and CMDs of stellar fields with different ages and stellar densities. North is up and East to the right.} The top panels show the F475W image and the $m_{\\rm F475W}$ vs.\\,$m_{\\rm F475W}-m_{\\rm F814W}$ CMD of stars in the star-forming region around the very young cluster NGC\\,1966. The middle and bottom panels illustrate the F814W stacked images and the CMDs of the intermediate-age cluster NGC\\,2121 (age $\\sim 3$ Gyr) and the old GC NGC\\,2210 (age $\\sim 12$ Gyr), respectively. \n \\label{fig:CMDs} \n\\end{figure*} \n\nThe CMDs are used to estimate age, distance modulus, (m$-$M$)_{0}$, metallicity, [M\/H], and reddening, E(B$-$V), by using isochrones from the Padova database \\citep{marigo2017a}. \n To minimize the contamination of field stars, we excluded from the analysis the stars at a large distance from the cluster center. Moreover, we statistically subtracted the field stars from the CMD of cluster members by using the method of \\citet{gallart2003a}, in close analogy with what is done in previous papers from our group \\citep[e.g.\\,][]{marino2014a, milone2018a}. In a nutshell, we defined by eye a region that is centered on the cluster and includes the bulk of cluster stars (hereafter cluster field) and a reference field with the same area, and at a large distance from the cluster center, which is mostly composed of field stars. \n We associated with each star in the reference field, the star in the cluster field at the smallest distance in the CMD, where the distance is defined as \\\\\n $\n {\\rm distance}=\\sqrt{(k \\times {\\Delta \\rm color})^{2} + {(\\Delta \\rm magnitude)^{2}} }\n $\\\\\n where $\\Delta$color and $\\Delta$magnitude are the color and magnitude differences, respectively, and $k$ is a factor enhancing the difference in color with respect to the magnitude difference, which is derived as in \\citet[][see their section 3.1]{marino2014a}. These stars are excluded from the comparison with the isochrones.\nThe cluster parameters and the best-fitting isochrones are used in Section\\,\\ref{sec:reddening} to estimate differential reddening maps and to investigate the eMSTO phenomenon (Section\\,\\ref{sec:secrets}). \nTo find the best-fitting isochrone, we used the CMD from the UVIS\/WFC3 and\/or WFC\/ACS photometry providing the widest color baseline, thus maximizing the sensitivity to metallicity. \nHowever, when photometry in optical filters is available, we excluded the UV filters F225W, F275W, F336W, and F343N from the analysis to minimize the effect of multiple populations \\footnote{These UV filters encompass various molecular bands that include carbon, nitrogen, and oxygen.\nTheir fluxes are sensitive to the abundances of these elements, which are not constant within clusters with multiple populations. Hence, the CMDs made with UV filters may lead to less accurate determinations of the cluster parameters than optical CMDs.}. Indeed, these UV filters are sensitive to the potential effects due to stellar populations with different nitrogen and oxygen abundances, \\citep[e.g.][]{marino2008a, milone2020a, dondoglio2021a}, which are typical features of GCs older than $\\sim$2.3 Gyr, and to stellar populations with different rotation rates \\citep[e.g.][]{dantona2015a, milone2016a, li2017a, marino2018a}, which are present in all clusters younger than $\\sim$2 Gyr.\n\nThe observed CMDs are compared with grids of isochrones with different reddening values, distances, metallicities, and ages. The resulting best-fitting parameters are provided in Table\\,\\ref{tab:info} and are estimated as follows.\n\n We first determined the isochrone and the values of reddening and distance modulus that, based on the visual comparison with the CMD, provide the best match with the CMD. Then, we improved the determination of the best-fitting parameters using the following iterative approach. We fixed the values of age, distance, and reddening and better constrained the cluster metallicity by comparing the slopes of the fiducial lines of the observed RGB and the MS of the CMDs, and the slope of the corresponding magnitude intervals of the isochrones. \n\n %\n Then, we assumed the metallicity value corresponding to the minimum difference between the slopes of the observed CMDs and the isochrones to improve the estimates of reddening, age, and distance modulus. To do this, we \n adopted the criteria of obtaining the best match between the isochrone and the observed CMD, {which may change for clusters with different ages.}\n\nThe best-fitting parameters of clusters older than $\\sim 2.3$\\,Gyr were estimated by determining the isochrone that best fits the CMD from the MSTO through the SGB \\citep[e.g.][]{dotter2010a}. Specifically, we calculated the $\\chi^{2}$ values of the distances in the CMD between the fiducial lines of the MSTO and SGB stars and the isochrone. The best-fitting values of age, reddening and distance modulus are derived by means of $\\chi^{2}$ minimization.\nA visual inspection at the CMDs reveals that all clusters between $\\sim 10$ Myr and $\\sim$2.5 Gyr exhibit eMSTOs. Since the eMSTOs challenge their age determinations we provide two age values.\nWe list in column 11 of Table\\,\\ref{tab:info} the age of the isochrone that best fits the lower part of the eMSTO. Clearly, this age value would represent the oldest cluster stars, if the eMSTO is entirely due to age variation. \nAlternatively, if the eMSTO is entirely due to rotation, our age estimate would provide an upper limit to cluster age, as the fast-rotating stars populate the lower part of the eMSTO \\citep[e.g.][]{dupree2017a, marino2018a, marino2018b, kamann2020a}. Hence, we provide in column 12 of Table\\,\\ref{tab:info} the age of the isochrone that best fits the upper part of the eMSTO.\n In the clusters younger than $\\sim 10$\\,Myr, where it is challenging to identify the MSTO, our age determination is largely based on evolved stars. In these young clusters and in the clusters with the eMSTO, the age, distance modulus, and reddening were derived by eye.\n\n To quantify the typical precision of the values of metallicity, age, reddening, and distance modulus inferred by the isochrones we applied the following procedure to four couples of clusters with different ages. The photometry of the clusters of each pair comes from datasets with large differences in the number of images and in the total exposure times. Hence, the range of uncertainties on the fitting parameter inferred from each couple of clusters would comprise the parameters' uncertainties of all studied clusters with similar ages.\n\n We first linearly added to the slopes of the fiducial lines of the RGB and MS stars that were used to constrain the metallicity, the corresponding errors. Hence, we derived the best value of [Fe\/H] that corresponds to the isochrone that provides the best match with the used slope. We repeated the same procedure but by using the slopes of the MS and RGB fiducials after subtracting the errors. \n We consider the semi-difference between the maximum and the minimum [M\/H] value, $\\Delta$[M\/H] as a quantity indicative of the precision of our metallicity estimate.\n\nSimilarly, we shifted each point of the fiducial line of the MS and the SGB to the bright and blue side of the CMD, perpendicular to the isochrone. We indicate the resulting line as blue-shifted fiducial. The shift is applied in such a way that 68.27\\% of the stars on the blue side of the original fiducial line are located on the red side of the blue-shifted fiducial. We applied a similar procedure to derive a red-shifted fiducial line.\nHence, we repeated four times the procedure described above to estimate the values of age, reddening, and distance modulus but by assuming the various combinations of the largest and minimum values of [M\/H] and the blue-shifted and red-shifted fiducials.\nWe consider the semi-differences between the maximum values of age ($\\Delta$age), distance modulus, ($\\Delta$(m$-$M)$_{0}$), and reddening ($\\Delta$E(B$-$V)), as a proxy of the precision of the estimates of the corresponding quantities.\n\nThe results are listed in Table\\,\\ref{tab:errori} for the pairs of clusters of old GCs NGC\\,2005 and NGC\\,1939 (ages of $\\sim 13$\\,Gyr), intermediate age clusters Kron\\,3 and Kron\\,1 (ages of $\\sim$6--7 Gyr). We also investigated the $\\sim$2 Gyr-old clusters NGC\\,1846 and Hodge\\,7 and the young clusters NGC\\,1866 and BSDL\\,1650 (ages of $\\sim$300 Myr).\n\n\n\n\\section{Differential reddening}\\label{sec:reddening}\nTo derive high-resolution reddening maps, we applied to our dataset the method originally developed by \\citet{milone2012a} to correct the ACS\/WFC F606W and F814W magnitudes of Galactic GCs for differential reddening \\citep[see also][]{bellini2017a, jang2022a}. \nThe main difference of the adopted procedure is that the catalogs of several GCs comprise photometry in more than two bands. \nThe main steps of our iterative method, which is illustrated in Figure \\ref{fig:red} for NGC\\,416, can be summarized as follows:\n\\begin{itemize}\n \\item We built the $m_{\\rm F814W}$ vs.\\,$m_{\\rm X}-m_{\\rm F814W}$ diagrams, where X=F275W, F336W, F343N, F438W, F555W, and F814W. Each diagram has been used to gather information on differential reddening from a sample of reference stars.\n Reference stars are selected in the CMD region where the reddening direction defines a wide angle with the cluster fiducial line in such a way that we can easily disentangle the effect on stellar colors and magnitudes due to differential reddening from the shift due to photometric uncertainties.\n As an example, panel a of Figure \\ref{fig:red} highlights in black the selected reference stars of NGC\\,416 in the $m_{\\rm F814W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ CMD.\n \n \\item \n We first derived the reddening direction corresponding to each star as $\\theta={\\rm arctan} \\frac{A_{\\rm X}}{A_{\\rm X}-A_{\\rm F814W}}$, where $A_{\\rm X}$ and $A_{\\rm F814W}$ are the absorption coefficients in the X and F814W bands, respectively. To derive them, we identified the point on the best-fitting isochrone with the same $m_{\\rm X}$ magnitude as the reference star and calculate the $m_{\\rm X}$ and $m_{\\rm F814W}$ magnitude differences with the corresponding point of the isochrone with E(B$-$V)=0 mag. This procedure allows us to account for the dependence of reddening direction from the total amount of reddening and from its spectral type. \n As an example, panel a of Figure\\,\\ref{fig:red} shows the reddening direction associated with the reference star indicated by the red cross.\n \n \\item We translated the CMD into a new reference frame where the origin corresponds to the reference stars as illustrated in panels a and b of Figure \\ref{fig:red}. This CMD is rotated counterclockwise by an angle $\\theta$ so that the abscissa and the ordinate of the new reference frame are parallel and orthogonal, respectively, to the reddening direction. \n \n \\item We generated the fiducial line of MS, SGB, and RGB stars, which we plot as a continuous red line in panel b. To do this, we divided the sample of MS stars into 'ordinate' intervals. For each bin, we calculated the median abscissa associated with the median 'ordinate' of the stars in the bin. The fiducial line has been derived by linearly interpolating these median points.\n \n \\item We calculated the distance of the reference star from the fiducial line along the reddening direction, $\\Delta x'$ as shown in panel b for a reference star of NGC\\,416 that we marked with a large red cross \\footnote{ The differential reddening is responsible for shifting the stars along the reddening line. The amount of such shift, which is proportional to the amount of reddening along the line of sight, depends on the star's position in the field of view. As a consequence, the stars in the different regions of the field are systematically shifted towards larger or lower values of x' with respect to the cluster fiducial line depending on whether they are affected by a larger or smaller amount of reddening with respect to the median cluster reddening (see Figure 5 for an example). On the contrary, photometric errors are responsible for a random scatter along the fiducial line, but such a scatter is essentially not dependent on the reddening direction and the position of the star in the field of view.}.\n \n \\item We calculated the projection of $\\Delta x'$ along the $m_{\\rm X}-m_{\\rm F814W}$ color direction, $\\Delta$ ($m_{\\rm X}-m_{\\rm F814W}$) and plotted this quantity for the available X filters as shown in panel c of Figure \\ref{fig:red}. The observed values of $\\Delta$ ($m_{\\rm X}-m_{\\rm F814W}$) are compared with corresponding quantities derived from the isochrones and corresponding to reddening variations ranging from $\\Delta$ E(B$-$V)=$-$0.3 to 0.3 mag in steps of 0.001 mag. The value of $\\Delta$ E(B$-$V) that provides the minimum $\\chi^{2}$ is assumed as the best differential-reddening estimate associated with the reference star marked with the red cross.\n \n \n %\n \n %\n\\end{itemize}\n\nTo derive the amount of differential reddening associated with each star in the catalog, we selected a sample of N spatially nearby reference stars, (light-blue crosses in Figure\\,\\ref{fig:red}) as shown in panel d. The best determination of differential reddening is provided by the median of the $\\Delta$ E(B$-$V) values of these $N$ neighbors. We excluded the target star from its own differential reddening determination.\n We derived various determinations of differential reddening by assuming different values of $N$, from 35 to 95 in steps of 5 and from 100 to 150 in steps of 10. For each determination, we calculated the pseudo-color distances between the value of $x'$ of the reference stars, corrected for differential reddening, and the fiducial line of Figure\\,\\ref{fig:red}. \n We assumed that our best determination of differential reddening is given by the value of $N$ that provides the minimum value of the r.m.s of these distances. In particular, we used N=75 for NGC\\,416.\n\nAs an example, Figure\\,\\ref{fig:redmap} shows the reddening map in the direction of NGC\\,416 and compares the original CMD to the CMD corrected for differential reddening. A collection of reddening maps for six clusters is provided in Figure \\ref{fig:redmaps}. \n\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=21.0cm,trim={0.5cm 1.2cm 0.0cm 0.0cm},clip]{red12.pdf}\n\n \\caption{This figure illustrates the procedure to estimate the amount of differential reddening associated with the target star represented with the large blue dot. \n Panel a shows the $m_{\\rm F814W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ CMD of all the stars. Reference stars, are located between the two dotted gray lines and are colored black, whereas the neighboring reference stars are marked with light-blue crosses. The gray continuous lines are the abscissa and the ordinate of the rotated reference frame centered on the reference star marked with the large red cross, while the red arrow indicates the reddening direction. Panel b shows the same stars as panel a but in the rotated reference frame. The red continuous line is the fiducial of reference stars and the inset highlights the relative position between one reference star and the fiducial. \n Panel c represents the values of $\\Delta x'$ inferred from different filters (red dots). Gray crosses are the corresponding values derived for $\\Delta$ E(B$-$V) ranging from 0.01 to 0.10 mag in steps of 0.01 mag, while the black crosses provide the best fit to the observations and correspond to $\\Delta$ E(B$-$V)=0.058 mag. Finally, the finding chart zoomed in around the target is illustrated in panel d. See the text for details.\n }\n \\label{fig:red} \n\\end{figure*} \n\n\n\\section{Cluster centers}\\label{sec:centri}\n\nTo determine the coordinates of the center of each star cluster we followed the procedure described in \\citet{cordoni2020b}. In a nutshell, we first selected by eye a sample of probable cluster members based on their location in the CMD and smoothed their stellar spatial distribution with a Gaussian kernel of fixed size. The kernel size has been chosen with the criteria of favoring the overall shape of the cluster, instead of the small-scale structures. \nWe derived five contour lines within 50 arcsec from the cluster center and interpolated each of them with an ellipse by using the algorithm by \\citet{halir1998a}.\nOur best cluster-center determination corresponds to the median value of the centers of the ellipses, while the corresponding uncertainty has been estimated as the dispersion of the center determinations inferred from each ellipse. \n Due to the low number of stars, it was not possible to apply the method above in 13 poorly-populated star clusters, namely BRHT\\,5b, BSDL\\,1650, KMK\\,8827, KMHK\\,1073, KMHK\\,8849, OGLE-CL-LMC390, NGC\\,1749, NGC\\,290, NGC\\,1850A, NGC\\,1858, NGC\\,1938 and NGC\\,1966. For these clusters, we provide raw center determination based on the peak of the histogram distributions of the coordinates of the probable cluster members.\nResults are provided in Table \\ref{tab:info}.\n\n\n\\begin{figure*} \n\\centering\n \\includegraphics[width=16.0cm,trim={3cm 0.0cm 3.0cm 0.0cm},clip]{redmapLR.pdf}\n\n \\caption{\\textit{Left.} Comparison between the original $m_{\\rm F814W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ CMD of NGC\\,416 (top) and the CMD corrected for differential reddening (bottom). \\textit{Right.} Differential-reddening map in the direction of NGC\\,416. The levels of gray are proportional to the reddening variation as indicated on the top-right. The panels on the right show $\\Delta$ E(B$-$V) against the abscissa for stars in eight ordinate intervals. Similarly, the panels on the top represent the reddening variation as a function of the ordinate for stars in eight intervals of X. The field is centered around the center of NGC\\,416 (X,Y=6019,5507) and the X and Y axis are parallel to the right ascension and declination direction, respectively. We adopted a scale of 0.04 arcsec per pixel.}\n \\label{fig:redmap} \n\\end{figure*} \n\n\n\n\\begin{figure*} \n\\centering\n \\includegraphics[width=16.0cm,trim={3cm 0.0cm 3.0cm 0.0cm},clip]{redmaps.pdf}\n\n \\caption{Differential-reddening maps of the regions in front of HW\\,57, NGC\\,416, NGC\\,1751, NGC\\,1850, NGC\\,1852 and NGC\\,1856.}\n \\label{fig:redmaps} \n\\end{figure*} \n\n\\begin{figure} \n\\centering\n\\includegraphics[height=12cm,trim={1.0cm 5.8cm 5.2cm 4.5cm},clip]{PMGAIA.pdf}\n\\caption{This figure illustrates the procedure to identify stars that, based on the position in the CMDs from {\\it HST} photometry and in the proper-motion diagram from GAIA eDR3, are probable members of NGC\\,1806. The probable members are represented with red circles in the CMDs plotted in the top panels and the bottom-left panel and in the proper motion diagram shown in the bottom-right panel. Stars that are not located on the main evolutionary sequences in at least one CMD are represented with blue-starred symbols. The arrow plotted in the proper-motion diagram indicates the mean cluster motion, while the circle is used to select the stars that are not included in the sample of probable cluster members, due to their large proper motions (aqua-starred symbols). See the text for details. }\n \\label{fig:PMGAIA} \n\\end{figure} \n\\begin{figure*} \n\\centering\n\\includegraphics[height=9cm,trim={0.0cm 1cm 0.0cm 1cm},clip]{pmsLR.pdf}\n \\caption{ \\textit{Left.} Coordinates, in degrees, relative to the SMC center of the studied SMC and LMC clusters. The arrows in the top panel are indicative of the absolute proper motions of each cluster, while the bottom panel represents the proper motions of LMC and SMC clusters after subtracting the average motion of the corresponding galaxy. \\textit{Right.} Proper motions of stars brighter than g$_{\\rm BP}=16.0$ mag in the region around the LMC and the SMC (black points). The studied LMC and SMC clusters are plotted in all panels with red and aqua dots, respectively. }\n \\label{fig:PMs} \n\\end{figure*} \n\\section{Proper motions}\\label{sec:pms}\n\nTo estimate the absolute proper motions of the studied clusters, we combined information from {\\it HST} photometry and Gaia eDR3 proper motions. Specifically, for each cluster, we selected by eye stars that, based on their positions in all available CMDs, are probable cluster members. Then, we used the Gaia eDR3 catalog to select stars with magnitude $g_{\\rm BP}<19.0$ mag, which according to the criteria by \\citet{cordoni2018a} have high-quality proper motions. The average proper motion of each cluster has been calculated as the 3-$\\sigma$ clipped average of the proper motions of selected cluster members for which are available both {\\it HST} photometry and Gaia eDR3 high-quality proper motions. \nWe estimated the corresponding uncertainty by following the method of \\citet{vasiliev2019a}, which accounts for systematic errors.\n\n The main steps of the procedure used to derive the absolute proper motion are illustrated in Figure\\,\\ref{fig:PMGAIA} for NGC\\,1806. For this cluster, we have photometry in five photometric bands of UVIS\/WFC3 and WFC\/ACS. We constructed ten CMDs of stars in the FoV of NGC\\,1806 including four $m_{\\rm F814W}$ vs.\\,$m_{\\rm X}-m_{\\rm F814W}$ CMDs, where X=F336W, F343N, F435W, and F555W, three $m_{\\rm F555W}$ vs.\\,$m_{\\rm X}-m_{\\rm F555W}$ CMDs, where X=F336W, F343N, and F435W, two $m_{\\rm F435W}$ vs.\\,$m_{\\rm X}-m_{\\rm F435W}$ CMDs, where X=F336W and F343N, and the $m_{\\rm F343N}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F343N}$ CMD. For each CMD, we selected by eye the stars that, based on their colors and magnitudes, are located on the main cluster evolutionary sequences. As an example, the stars that, based on their positions in all CMDs, likely belong to the RGB, AGB, and red clump of NGC\\,1806 are colored black in the three CMDs of Figure\\,\\ref{fig:PMGAIA}. The colored symbols mark stars with available Gaia eDR3 proper motions in both the CMDs and in the proper-motion diagram. The stars that do not belong to the RGB, AGB, and red clump of NGC\\,1806 in at least one CMD are represented with blue-starred symbols and are not included in the determination of the cluster proper motion. \n We also excluded the selected stars with proper motions that differ from the average cluster motion by more than three times the proper motion dispersion (i.e.\\, the stars outside the black circle shown in the bottom-right panel of Figure\\,\\ref{fig:PMGAIA} represented with aqua-starred symbols). The remaining stars are marked with red open dots.\n\n\n\n\n\n\nResults are provided in Table\\,\\ref{tab:data}. The left panels of Figure\\,\\ref{fig:PMs} show the positions of the studied LMC and SMC clusters relative to the SMC center. In the top-left panel, we associate to each cluster the corresponding proper-motion vector, while in the bottom-left panel we show the proper-motion residuals after subtracting to LMC and SMC clusters the average motion of the corresponding Magellanic Cloud from \\citet{gaia2018a}\\footnote{Although the investigation of the Magellanic Clouds' rotation is beyond our scope, we note that no clear rotation pattern is evident from the bottom-left panel of Figure\\,\\ref{fig:PMs}. This statement, which is based on a visual inspection of this figure, seems to contrast with the evidence of the LMC rotation pattern shown by \\citet{vandermarel2014a, helmi2018a}. We also note that the right panel of Figure\\,\\ref{fig:PMs} highlights the relative motions within the SMC following the pattern of the SMC tidal expansion along the bridge and counter-bridge as detected in previous works \\citep[e.g.][]{zivick2018a, piatti2021a, dias2021a, schmidt2022a}.}. \nThe proper motion diagram is plotted in the right panel of Figure\\,\\ref{fig:PMs} and reveals that, based on proper motions, all clusters are consistent with being either LMC or SMC members. \n\n\\subsection{Proper motions from Gaia eDR3 and {\\it HST} }\nFor thirteen GCs, we take advantage of having more than one epoch observations with appropriate signal-to-noise ratio and temporal baselines to disentangle the internal kinematics of Magellanic Cloud stars and separate cluster members and field stars by using {\\it HST} data alone. Detailed information on the {\\it HST} images available for these clusters are provided in Table\\,\\ref{tab:dataPM}. Relative {\\it HST} motions are then transformed into absolute motions based on Gaia eDR3 proper motions.\n\nTo derive relative proper motions we applied to our dataset the procedure described by \\citet[][]{piotto2012a} and described in the following for NGC\\,1978. \nIn a nutshell, we first identified the distinct groups of images collected at the same epoch through the same filter and camera. We reduced each group of images, separately, as described in Section\\,\\ref{sec:data}, and obtained the corresponding astrometric and photometric catalogs. \n\n The reference frame defined by the first-epoch images collected through the reddest filter is adopted as a master frame. The coordinates of stars in each catalog are transformed into the master frame by means of six-parameter linear transformations \\citep{anderson2006a}. \n To minimize the effect of possible small residual distortions we applied local transformations based on the nearest 70 reference stars. Target stars are never included in the calculation of their own transformations. \n \n The abscissa and the ordinate of each star, expressed in milliarcsec, are plotted against the epoch, expressed in years, as shown in panels a1--a4 of Figure\\,\\ref{fig:MetPM} for two stars in the field of view of NGC\\,1978. For simplicity, in this figure, we show the displacements DX and DY, and the time relative to the stellar position and time at the first epoch. \n These points are finally fitted with a weighted least-squares straight line, whose slope corresponds to the best proper motion estimate. \n \nThe selection of the stars used to derive the transformation is a critical step for accurate proper-motion determination. Hence, we selected bright and unsaturated stars that pass the criteria of selection discussed in Section\\,\\ref{subsec:quality}. We derived proper motions relative to a sample of cluster members that have been selected iteratively. As a consequence, the average relative motion of the cluster is set to zero.\nWe first identified probable cluster stars that, based on all available CMDs, lie on the main evolutionary sequences and used them to derive initial proper motion estimates.\nThen, we iteratively excluded those stars that do not share the same motion as the bulk of cluster members (i.e.\\,stars with proper motions greater than three times the proper-motion dispersion of cluster stars).\nPanels b and c of Figure\\,\\ref{fig:MetPM} mark with black points the probable cluster members that we selected for deriving stellar proper motions in the $m_{F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD and in the $m_{\\rm F814W}$ vs.\\,$DR=\\sqrt{DX^{2}+DY^{2}}$ plane, whereas aqua crosses are probable cluster members. Gray points mark the remaining stars with cluster-like proper motions, i.e.\\, the saturated stars, the faint stars, and the stars that do not lie on the main evolutionary sequences in the CMDs.\n\nTo transform relative proper motions into absolute ones we derived the difference between the relative proper motions derived from {\\it HST} images and the absolute proper motions from Gaia eDR3 for an appropriate sample of stars.\n Specifically, this selected sample includes stars with high-quality relative proper motions (i.e.\\,bright, unsaturated stars that pass the criteria of selection of Section\\,\\ref{subsec:quality}). In addition, the selected sample includes stars that, based on the proper motion uncertainties and on the values of the Renormalized Unit Weight Error (\\texttt{RUWE}), the astrometric\\_gof\\_al\n(\\texttt{As\\_gof\\_al}) parameters of the Gaia eDR3 catalog have accurate Gaia eDR3 absolute proper motions. We refer to papers by \\citet{cordoni2018a, cordoni2020a}, for details on the procedure.\n The sample includes both cluster and field stars, with the exception of a few stars with parallaxes significantly larger than zero.\n\nThe $3\\sigma$-clipped mean differences of the proper motion along each direction ($\\mu_{\\alpha} {\\rm cos}{\\delta}$ and $\\mu_{\\delta}$) are considered as the best estimate of the zero points of the motions and are used to convert relative proper motions into absolute ones.\nAs an example, panels d1 and d2 of Figure\\,\\ref{fig:MetPM} show the histogram distributions of the quantities $\\Delta \\mu_{\\alpha} {\\rm cos} {\\delta}$=$\\mu_{\\alpha} {\\rm cos} \\delta$-DX and $\\Delta \\mu_{\\delta}$=$\\mu_{\\delta}$-DY for stars in the field of view of NGC\\,1978.\n\n\\begin{figure*} \n\\centering\n \\includegraphics[height=9.5cm,trim={2cm 0cm 1.0cm 0cm},clip]{MetPMLR.pdf}\n\\caption{Procedure to estimate absolute proper motions. Panels a1 and a2 show the displacements along the X and Y directions in four epochs of a probable field star (blue triangles) relative to the mean motion of NGC\\,1978. Similarly, panels a3 and a4 show the displacements of a candidate cluster member (red dots). The $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD of stars in the field of view of NGC\\,1978 is plotted in panel b, while panel c shows relative stellar proper motions against $m_{\\rm F814W}$. Aqua crosses are probable field stars selected on the basis of their proper motions. The stars used as references to calculate relative proper motions are colored black, while the remaining stars with cluster-like proper motions are gray. Panels d1 and d2 show the histogram of the difference between our relative proper motions and the absolute proper motions from Gaia eDR3. }\n \\label{fig:MetPM} \n\\end{figure*} \n\nThe proper motion diagrams for NGC\\,1978 stars are plotted in the left panels of Figure\\,\\ref{fig:NGC1978vi} in four distinct magnitude bins. These diagrams can be used to separate the bulk of cluster members (black dots) from probable field stars (red crosses). Here, the red circles that enclose the NGC\\,1978 stars have radii equal to 2.5$\\sigma$, where $\\sigma$ is the average between the $\\sigma-$clipped dispersion values of $\\mu_{\\alpha} {\\rm cos} \\delta$ and $\\mu_{\\delta}$. For illustration purposes, we only mark in red the most-evident field stars with $\\mu_{\\rm \\alpha} \\cos{\\delta} >1.6$ mas\/yr and a distance of more than 0.2 mas\/yr from the average motion of NGC\\,1978, while the remaining stars are colored gray. \n The $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD of probable cluster members and field stars is shown in the right panel of Figure\\,\\ref{fig:NGC1978vi}.\n\\begin{figure} \n\\centering\n \\includegraphics[height=9.0cm,trim={0.1cm 5.cm 0.0cm 3.5cm},clip]{NGC1978pmS.pdf}\n\n \\caption{Proper motion diagrams of stars in the field of view of NGC\\,1978 in four F814W magnitude intervals (left). The $m_{\\rm F814W}$ vs.\\,$m_{\\rm F555W}-m_{\\rm F814W}$ CMD of stars in the left panels is plotted on the right. Stars within the red circles plotted in the left panels are considered probable cluster members and are colored black, whereas the most-evident field stars are represented with red crosses. The remaining stars are colored gray. See the text for details.}\n \\label{fig:NGC1978vi} \n\\end{figure} \n\n\n\\section{A saucerful of secrets}\\label{sec:secrets}\nThe photometry and astrometry of this work are exquisite tools to investigate various astrophysical topics. In this section, we provide further examples of science outcomes that arise from visual inspections of the photometric diagrams and of the proper-motion diagrams. \nSpecifically, in Section\\,\\ref{sec:NEWeMSTO} we report the discovery of eMSTOs in the clusters KMHK\\,361 and NGC\\,265. Section\\,\\ref{sec:AGEemsto} compares the CMDs of LMC clusters younger than $\\sim$2.3 Gyr and investigates the color and magnitude distribution of eMSTO in clusters with different ages. \nGaps and color discontinuities along the MS of NGC\\,1783 are investigated in Section\\,\\ref{sub:zigzag} while Section\\,\\ref{sec:NGC1783} provides evidence of new features along the eMSTO and the upper MS of NGC\\,1783. \nFinally, Section\\,\\ref{sec:pm} is focused on the proper motions of the star clusters and of Magellanic Cloud stellar populations in eleven fields.\n\n\\subsection{Clusters without previous evidence of eMSTO}\\label{sec:NEWeMSTO}\nFigure\\,\\ref{fig:NEWeMSTO} provides evidence that the CMDs of the star clusters KMHK\\,361 (age of 1.35 Gyr) and NGC\\,265 (age of 450 Myr) are not consistent with a single isochrone.\n In this figure, we compare the CMDs of stars in circular fields centered on the cluster (hereafter cluster fields) and in reference fields of the same area. We adopted radii of 20 and 24 arcsec for KMHK\\,361 and NGC\\,265, respectively, enclosing the bulk of cluster stars. To minimize the contamination from cluster stars, the reference fields are as\n far away from the cluster centers as possible, while still being within the FoV.\n\n By assuming a uniform distribution of field stars in the small {\\it HST} field of view, the distribution of stars in the reference-field CMD is indicative of the contamination due to field stars. \n \n Clearly, KMHK\\,361 exhibits an eMSTO, which cannot be explained by field-star contamination alone. \n Similarly, NGC\\,265 shows an intrinsic eMSTO. The upper MS is split in the F435W magnitude interval between $\\sim$21 and 22 mag, with the red MS hosting about two-thirds of MS stars. The two MSs merge around $m_{\\rm F435W} \\sim 22.5$ mag. The comparison between the CMDs of stars in the field and reference fields reveals that the split MS and the eMSTO are not due to field-star contamination.\n \n The visual inspection of the CMDs from our survey suggests that all clusters with ages between $\\sim 0.1$ and $\\sim$2.3 Gyr exhibit the eMSTO (Cordoni et al.\\,in preparation).\n These findings corroborate the evidence that eMSTOs are common features of clusters younger than $\\sim$2.3 Gyr, while split MSs are widespread phenomena among clusters younger than $\\sim$0.8 Gyr \\citep[e.g.][]{milone2009a, milone2022a, niederhofer2015a, goudfrooij2011a, li2017a, correnti2017a}. \n\n\n\n\n\\begin{figure*} \n\\centering\n \\includegraphics[height=6.5cm,trim={0.5cm 5cm 0.2cm 4.5cm},clip]{KMHK361.pdf}\n \\includegraphics[height=6.5cm,trim={0.5cm 5cm 0.2cm 4.5cm},clip]{NGC0265.pdf}\n\n\\caption{CMDs of the clusters KMHK\\,361 and NGC\\,265 without previous evidence of eMSTOs. Stars in the cluster field and reference field of each cluster are represented with black points and aqua crosses, respectively. See text for details.}\n \\label{fig:NEWeMSTO} \n\\end{figure*} \n \n\n\n\n\\subsection{The eMSTO in clusters of different age}\\label{sec:AGEemsto}\n\\begin{centering} \n\\begin{figure*} \n \\includegraphics[height=17cm,trim={0.1cm 5cm 0.0cm 4.5cm},clip]{emstoLMC.pdf}\n\n \\caption{Collection of $M_{\\rm F336W}$ vs.\\,$M_{\\rm F336W}-M_{\\rm F814W}$ CMDs of LMC clusters younger than 2.5 Gyr. All panels have the same scale and are zoomed around the MS, while the insets highlight the MSTO. Clusters are sorted by age.}\n\n\n\n \\label{fig:emstoLMC} \n\\end{figure*} \n\\end{centering} \n\nOur dataset provides a unique opportunity for comparing CMDs of clusters with different ages derived with homogeneous methods. As an example, we take advantage of the collection of $M_{\\rm F336W} vs.\\, M_{\\rm F336W}-M_{\\rm F814W}$ CMDs shown in Figure \\ref{fig:emstoLMC} to investigate how the eMSTO phenomenon changes as a function of cluster age. In this figure, star clusters\n are sorted by age, from $\\sim$10 Myr (NGC\\,1818) to $\\sim$2.5 Gyr (NGC\\,1978). \n The observed magnitudes have been converted into absolute ones by adopting the values of distance modulus and reddening listed in Table\\,\\ref{tab:info}. \n\nA visual inspection of this figure corroborates the previous conclusion that the split MS is visible in all LMC clusters younger than $\\sim$800 Myr (from NGC\\,1818 to NGC\\,1953) and seems to disappear at older ages \\citep{milone2018a}.\nThe color separation between the blue and red MSs approaches its maximum value around the MSTO and decreases towards faint luminosities \\citep{milone2016a}.\nAs pointed out by \\citet{wang2022a}, the gap between the blue and red MSs significantly narrows down around $M_{\\rm F336W}=1.0$ mag, \n which is the luminosity level where the fraction of blue-MS stars approaches its minimum value \\citep{milone2018a}.\nNoticeably, this magnitude value corresponds to an MS mass of $\\sim$2.5 $\\mathcal{M}_{\\odot}$, where the slowly rotating component of MS field stars disappears \\citep{zorec2012a}.\n\nWe also confirm that the eMSTO is a ubiquitous feature of LMC clusters younger than $\\sim$2.3 Gyr. It is visible in all clusters where the turn-off is brighter than the MS bending around $M_{\\rm F336W}=3.0$ mag and disappears in NGC\\,1978 \\citep[e.g.][]{milone2009a, goudfrooij2014a}. \nSince the MS bending is due to a change in the stellar structure, the eMSTO is associated with stars with radiative envelopes alone. In addition, the split MS is visible among stars brighter than the MS bend.\n\nFigure\\,\\ref{fig:emstoLMC} reveals that the color and magnitude distributions of stars across the eMSTO significantly change from one cluster to another. As an example, the Hess diagrams plotted in the top panels (a1, a2, and a3) of Figure \\ref{fig:cumulative} suggest that most TO stars of NGC\\,1868 populate the bright and blue region of the eMSTO, whereas NGC\\,2173 shows higher stellar density on the bottom-red side of its eMSTO. NGC\\,1852 seems to show an intermediate distribution. \n \n To parametrize the stellar distribution of eMSTO stars in the CMDs,\n we adopted the procedure illustrated in Figure \\ref{fig:cumulative}b for NGC\\,1852.\n We defined a new reference frame where the origin, {\\it O}, is set by hand on the bright and blue side of the eMSTO, and the abscissa, {\\it X'}, envelopes the bright part of the eMSTO and points towards the red.\n We derived the red and blue fiducials of the eMSTO in the new reference frame and represented them as red and blue lines in the CMD of Figure \\ref{fig:cumulative}b. To derive the fiducials, we follow the recipe by \\citet{milone2017a}, which is based on the naive estimator \\citep{silverman1986a}. We first divided the eMSTO into a series of bins with fixed pseudo-magnitude, {\\it $\\delta$Y'}. The bins are defined over a grid of points separated by intervals of fixed pseudo-magnitude (s=$\\delta$Y\/3). For each interval, we calculate the 4$^{\\rm th}$ and the 96$^{\\rm th}$ percentile of the {\\it X'} distribution and associated these values with the mean pseudo-magnitude {\\it Y'} of stars in the bin. These values are then linearly interpolated to derive the red and blue boundaries of the eMSTO. These lines are used to calculate the quantity\n \\begin{equation}\n\\Delta_{\\rm X'}=\\frac {X'-X'_{\\rm blue\\,fiducial}} {X'_{\\rm red\\,fiducial}-X'_{\\rm blue\\,fiducial}} \n \\end{equation}\n that is defined in such a way that the stars on the blue and red fiducials have $\\Delta_{X'}$=0 and 1, respectively.\n \n Figure\\,\\ref{fig:cumulative} compares the kernel density (panel c) and the cumulative distributions of $\\Delta_{X'}$ (panel d) for NGC\\,1852 (black), NGC\\,1868 (aqua), and NGC\\,2173 (orange).\n We confirm the visual impression of a predominance of blue eMSTO stars in NGC\\,1868, whereas the $\\Delta_{ X'}$ distribution of NGC\\,2173 is peaked towards the red. NGC\\,1852 has an intermediate distribution.\n\n\n\\begin{centering} \n\\begin{figure} \n \\includegraphics[height=8.0cm,trim={1.0cm 5.5cm 0.0cm 4.5cm},clip]{cumulative.pdf}\n\n \\caption{$m_{\\rm F336W}$ vs.\\,$m_{\\rm F336W}-m_{\\rm F814W}$ Hess diagrams of NGC\\,1868 (a1), NGC\\,1852 (a2), and NGC\\,2173 (a3) zoomed around the eMSTO. \n The blue and red lines are the boundaries of the eMSTOs. Panel b illustrates the scheme to derive the $\\Delta_{X'}$ quantity for eMSTO stars, while the corresponding kernel-density distributions and cumulative distributions are plotted in panels c and d, respectively, for NGC\\,1868 (aqua), NGC\\,1852 (black) and NGC\\,2173 (orange). See the text for details.}\n \\label{fig:cumulative} \n\\end{figure} \n\\end{centering} \n\nTo quantify the $\\Delta_{X'}$ differences among the various clusters, we define two quantities: i) the area, A, below the cumulative curve shown in Figure\\,\\ref{fig:cumulative} and ii) the median value of $\\Delta_{\\rm X'}$, $<\\Delta_{\\rm X'}>$. If the distribution is dominated by blue and bright MSTO stars we would expect large values of A and small values of $<\\Delta_{\\rm X'}>$, while a predominance of faint and red eMSTO stars corresponds to small A and large $<\\Delta_{\\rm X'}>$. Results are shown in Figure\\,\\ref{fig:RelEMSTO}, where we plot both quantities against cluster age. \n LMC clusters\n (red dots in Figure\\,\\ref{fig:RelEMSTO}) exhibit a strong anti-correlation between A and age and a correlation between $<\\Delta_{\\rm X'}>$ and age, as also indicated by the values of the Spearman's rank correlation coefficients of $-$0.91 and 0.92, respectively. \nIntriguingly, the $\\sim$2 Gyr old SMC clusters\n\n NGC\\,411 and NGC\\,416 exhibit larger values of A and smaller values of $<\\Delta_{\\rm X'}>$ than LMC clusters with similar ages, with NGC\\,411 having the largest differences. The small statistical sample of clusters prevents us from reaching a firm conclusion on whether NGC\\,411 is an outlier or SMC and LMC clusters exhibit different trends.\n\nThe multiple populations of young and intermediate-age star clusters share common features that have been instrumental to shed light on the origin of split MSs and eMSTOs \\citep[see][for a recent review]{milone2022a}. \nAs an example, the eMSTO width depends on cluster age. Specifically, if the eMSTO is interpreted as an age spread, the resulting age range is proportional to cluster age \\citep[e.g.][]{niederhofer2015a, cordoni2018a}.\nMoreover, the fractions of stars along the blue and the red MS correlate with stellar mass. The fraction of blue-MS stars varies from $\\sim 40\\%$ among stars with masses of $\\sim 1.5 \\mathcal{M_{\\odot}}$ to $\\sim 15\\%$ among $\\sim 2.5-3.0 \\mathcal{M_{\\odot}}$ stars. It arises again in more massive stars, up to $\\sim 40\\%$ in $\\sim 5.0 \\mathcal{M_{\\odot}}$-stars.\nThe fractions of blue- and red-MS stars do not depend on other properties of the host cluster like the global cluster's mass \\citep[][]{milone2018a}. These results have been instrumental to demonstrate that rotation plays a major role in shaping the eMSTOs and the split MSs of Magellanic-Cloud clusters.\n\nThe evidence that the $\\Delta_{X'}$ distribution of stars along the eMSTO depends on cluster age provides a potential further constraint to the eMSTO phenomenon.\nTo start investigating the physical reasons responsible for the relations shown in Figure \\ref{fig:RelEMSTO}, we used stellar models from the Padova database \\citep{marigo2017a} to simulate a group of CMDs of non-rotating stellar populations with ages of 100, 200, 500, 1,000, 1,250, 1,500 and 2,000 Myrs and internal age spreads. We assumed a flat distribution and maximum width corresponding to the average age variations inferred by \\citet{cordoni2018a} for Magellanic Cloud clusters with the same age. We derived the A and $<\\Delta_{\\rm X'}>$ quantities for each simulated CMD by using the same procedure adopted for real stars and plotted the resulting values against the oldest age of the simulated stellar population (open triangles of Figure\\,\\ref{fig:RelEMSTO}). \n\n Similarly, we simulated another group of CMDs for coeval stellar populations where 33\\% of stars have no rotation, whereas the remaining 67\\% of stars have rotation equal to 0.9 times the breakout value. The simulated diagrams have ages of 100, 150, 500, 800, and 1,250 Myr and are derived by means of Geneva models \\citep{ekstrom2012a, ekstrom2013a, mowlavi2012a, wu2016a}. \n We assumed random viewing-angle distributions and adopted the gravity-darkening model by \\citet{espinosa2011a} and the limb-darkening effect \\citep{claret2000a}. Stellar magnitudes for the available {\\it HST} filters have been derived using the model atmospheres by Castelli \\& Kurucz (2003). \n The resulting A and $<\\Delta_{\\rm X'}>$ quantities are represented with filled diamonds in Figure\\,\\ref{fig:RelEMSTO}. \n For completeness, we used the Geneva models to simulate non-rotating stellar populations with internal age spreads, in close analogy with what we did with the Padova models. Results are represented with filled triangles.\n \n \n Clearly, the A and $<\\Delta_{\\rm X'}>$ quantities inferred from both groups of simulated diagrams provide poor fits to the observations. This fact indicates that internal age variation alone is not responsible for the eMSTO when we assume a flat age distribution for all clusters.\n Similarly, rotation alone is not responsible for the eMSTO when we assume two populations for all clusters: one of non-rotating stars and one of fast rotators with $\\omega=0.9\\omega_{c}$.\n \n It is now widely accepted that the luminosity of eMSTO stars depends on gravity darkening and that its effect is strong for large values of the ratio between the rotational velocity and the critical velocity. Our results could indicate that this ratio increases when stars age on the MS as suggested by \\citet{hastings2020a}.\n \n To properly constrain the contribution of rotation and age variation on the eMSTO, it is mandatory to extend the analysis to simulated diagrams that account for different internal age distributions, different rotation-rate distributions \\citep[e.g.][]{huang2006a, huang2010a, goudfrooij2018a}, and that account for both age variations and stellar populations with different rotation rates. \n \n The interpretation of the eMSTO phenomenon should also account for binary evolution effects \\citep[e.g.][]{wang2022a}. As an example, the stellar models by \\citet{wang2020a} show that the fraction of evolutionary-driven mergers rises for smaller stellar masses, at the expense of the binaries that survive the mass transfer and produce spun-up accretors. An appropriate comparison between the observations illustrated in Figures\\,\\ref{fig:cumulative} and \\ref{fig:RelEMSTO} and the predictions of stellar models that account for binary evolution is mandatory to shed light on the effect of binary evolution on the eMSTO. \n \n\n\n\\begin{figure} \n\\centering \n \\includegraphics[height=5.0cm,trim={0.0cm 5.5cm 0.0cm 12.0cm},clip]{RelEMSTO.pdf}\n\n %\n\n \\caption{Area below the $\\Delta_{\\rm X'}$ cumulative curve, A, (left) and median $\\Delta_{\\rm X'}$ value as a function of cluster age for LMC (red dots) and SMC (aqua dots) clusters with the eMSTO. Open and filled triangles are inferred from simulated CMDs of non-rotating stellar populations with different ages derived from the Padova and Geneva database, respectively. The diamonds correspond to coeval stellar populations with different rotation rates from the Geneva database. \nSee text for details.}\n \\label{fig:RelEMSTO} \n\\end{figure} \n\n\\subsection{A population of UV-dim stars along the eMSTO of NGC\\,1783}\\label{sub:1783emsto}\n \n\\begin{centering} \n\\begin{figure*} \n \\includegraphics[height=9.0cm,trim={0.1cm 4.cm 0.0cm 3.5cm},clip]{NGC1783pm.pdf}\n \\includegraphics[height=9.0cm,trim={0.1cm 4.cm 0.0cm 3.5cm},clip]{NGC1783nuv.pdf}\n\n \\caption{Proper motion diagrams of stars in the field of view of NGC\\,1783 in five F438W magnitude intervals (left). The $m_{\\rm F438W}$ vs.\\,$m_{\\rm F438W}-m_{\\rm F814W}$ CMD for stars in the left panels is plotted on the middle. Stars within the black circles plotted in the proper motion diagrams are considered probable cluster members and are colored black, whereas field stars are represented with aqua crosses. The right panel shows the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD for probable cluster members, while the inset represents the Hess diagram of the CMD region around the upper MS.} \n \\label{fig:NGC1783pm} \n\\end{figure*} \n\\end{centering} \nThe stellar proper motions derived from our dataset allow the partial separation of bright field stars from NGC\\,1783 cluster members, thus providing new insights on its stellar populations. \nThe left panels of Figure\\,\\ref{fig:NGC1783pm} show the proper-motion diagram for stars in the field of view of NGC\\,1783 in five magnitude bins. The black circles are centered on the absolute proper motion of NGC\\,1783, and are used to separate probable cluster members (black points) from field stars (aqua crosses). \n\nThe corresponding $m_{\\rm F438W}$ vs.\\,$m_{\\rm F438W}-m_{\\rm F814W}$ CMD (middle panel) highlights several characteristics of NGC\\,1783 in unprecedented detail. These include the eMSTO \\citep{mackey2008a, milone2009a, goudfrooij2014a} together with a well-populated sequence of MS-MS binaries with large mass ratio \\citep{milone2009a}. The SGB also exhibits intrinsic broadening in color and magnitude, with the majority of stars populating the upper SGB. \nMoreover, the CMD reveals a broad, possibly dual, sequence of stars brighter and bluer than the turn-off.\n This blue sequence, which will be investigated in detail in Section\\,\\ref{sec:NGC1783} was first identified by \\citet{li2016a} who associated it with the young stellar populations within NGC\\,1783. Their result has been challenged by \\citet{cabreraziri2016a} who suggested that the blue sequence is composed of field stars. \n\nHere, we focus on the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD of NGC\\,1783, which is illustrated in the right panel of Figure\\,\\ref{fig:NGC1783pm}. \nAn unexpected feature of this CMD is the sparse cloud of stars on the red side of the eMSTO. These stars, which we dub UV-dim, are marked with red triangles in the left panel of Figure\\,\\ref{fig:NGC1783emsto} where we reproduce the $m_{\\rm F438W}$ vs.\\,$m_{\\rm F275W}-m_{\\rm F438W}$ CMD zoomed around the eMSTO.\nUV-dim stars comprise a small fraction of $\\sim$7\\% of the total number of eMSTO stars with $20.40$, $\\alpha\\in(1-H,\\frac{1}{2})$ and $(F(t), t\\in [0,T] )$ be a stochastic process taking values in the space of \nlinear bounded operators on $L^2(D)$. Assume that\n\\begin{equation*}\n\\sup_{i\\in\\mathbb{N}^+}\\Vert F(s)e_i\\Vert_{\\alpha,1}<+\\infty,\n\\end{equation*}\nwhere for a function $f:[0,T]\\to L^2(D)$,\n\\begin{equation*}\n\\Vert f\\Vert_{\\alpha,1}= \\int_0^T\\left(\\frac{\\Vert f(s)\\Vert_2}{s^\\alpha}+\\int_0^s \\frac{\\Vert f(s)-f(r)\\Vert_2}{(s-r)^{\\alpha+1}}dr \\right) ds.\n\\end{equation*}\nFollowing \\cite{maslonua} we can define a pathwise generalized Stieltjes integral (see also \\cite{zaehle1})\n\\begin{equation*}\n\\int_0^T F(s) W^H(ds):=\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}} \\int_0^T F(s)e_i B_i^H(ds),\n\\end{equation*}\nwhich satisfies the property\n\\begin{align*}\n\\left\\Vert\\int_0^T F(s) W^H(ds)\\right\\Vert_2&\\le \\sup_{i\\in\\mathbb{N}^+}\\Vert F(s)e_i\\Vert_{\\alpha,1}\n\\left(\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}\\Lambda_\\alpha(B_i^H)\\right).\n\\end{align*}\nHere $\\Lambda_\\alpha(B_i^H)$ is a positive random variable defined in terms of a Weyl derivative (see \\cite{maslonua}, Equation (2.4)),\nsatisfying $\\sup_{i\\in\\mathbb{N}^+}E\\left(\\Lambda_\\alpha(B_i^H)\\right)<+\\infty$, as is proved in \\cite{nuarasca}, Lemma 7.5.\nConsequently, if $\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}<+\\infty$, the random variable \n\\begin{equation}\nr_{\\alpha}^{H}:=\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\Lambda_{\\alpha}(B_{i}^{H}). \\label{29}\n\\end{equation}\nis finite, a.s., and then\n\\begin{equation}\n\\label{i}\n\\left\\Vert\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}} \\int_0^T F(s)e_i B_i^H(ds)\\right\\Vert_2\\le \nr_{\\alpha}^H \\sup_{i\\in\\mathbb{N}^+}\\Vert F(s)e_i\\Vert_{\\alpha,1}.\n\\end{equation}\n\\smallskip\n\nNext, we introduce the class of real, parabolic,\ninitial-boundary value problems formally given by\n\\begin{align}\ndu(x,t) & = \\left(\\operatorname*{div}(k(x,t)\\nabla\nu(x,t))+g(u(x,t))\\right) dt+h(u(x,t))W^{H}(x,dt),\\nonumber\\\\\n(x,t) & \\in D\\times \\left( 0,T\\right] ,\\nonumber\\\\\nu(x,0) & =\\varphi(x),\\text{ \\ \\ }x\\in\\overline{D},\\nonumber\\\\\n\\frac{\\partial u(x,t)}{\\partial n(k)} & =0,\\text{ \\ \\ }(x,t)\\in\\partial\nD\\times\\left( 0,T\\right] , \\label{2}\n\\end{align}\nwhere the last relation stands for the conormal derivative of $u$ relative to the matrix-valued field $k$.\n\nIn the next section we shall give a rigorous meaning to such a formal expression and for this, we shall use the \npathwise integral described before.\n\nIn the sequel we write $n(x)$ for the unit outer normal vector at $x\\in$ $\\partial D$ and introduce \nthe following set of assumptions:\n\\begin{description}\n\\item{(C)} The square root $C^{\\frac{1}{2}}$ of the covariance operator is\ntrace-class, that is, we have $\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}<+\\infty$.\n\\item{(${\\rm K}_{\\beta,\\beta^{\\prime}}$)} The entries of $k$ satisfy $k_{i,j}%\n(.)=k_{j,i}(.)$ for all $i,j\\in\\left\\{ 1,...,d\\right\\} $ and there exists a\nconstant $\\beta^{\\prime}\\in\\left( \\frac{1}{2},1\\right] $ such that\n$k_{i,j}\\in\\mathcal{C}^{\\beta,\\beta^{\\prime}}(\\overline{D}\\times\\left[0,T\\right])$\nfor each $i,j$. In addition, we have\n$k_{i,j,x_{l}}:=\\frac{\\partial k_{i,j}}{\\partial x_{l}}\\in\\mathcal{C}\n^{\\beta,\\frac{\\beta}{2}}(\\overline{D}\\times\\left[ 0,T\\right])$\nfor each $i,j,l$ and there exists a constant $\\underline{k}\\in\\mathbb{R}%\n_{\\ast}^{+}$ such that the inequality\n$\n\\left( k(x,t)q,q\\right) _{\\mathbb{R}^{d}}\\geq\\underline{k}\\left| q\\right|\n^{2}\n$\nholds for all $q\\in\\mathbb{R}^{d}$ and all $(x,t)\\in\\overline{D}\\times\\left[\n0,T\\right]$.\\\\\nFinally, we\nhave\n$$\n(x,t)\\mapsto\\sum_{i=1}^{d}k_{i,j}(x,t)n_{i}(x)\\in\\mathcal{C}^{1+\\beta\n,\\frac{1+\\beta}{2}}(\\partial D\\times\\left[ 0,T\\right])\n$$\nfor each $j$ and the conormal vector-field $(x,t)\\mapsto\nn(k)(x,t):=k(x,t)n(x)$ is outward pointing, nowhere tangent to $\\partial D$\nfor every $t$.\n\\item{(L)} The functions $g,h:\\mathbb{R\\mapsto R}$ are Lipschitz continuous.\n\\item{(I)} The initial condition satisfies $\\varphi\\in\\mathcal{C}^{2+\\beta}%\n(\\overline{D})$ and the conormal boundary condition relative to $k$.\n\\end{description}\n\n\n\nFinally, we consider the following assumption which also appears in \\cite{nuavu}: \n\\smallskip\n\n\\noindent (${\\rm H}_\\gamma$) The derivative $h^{\\prime}$ is H\\\"{o}lder continuous with exponent $\\gamma\\in\\left(0,1\\right]$\nand bounded; moreover, the Hurst parameter satisfies $H\\in\\left(\\frac{1}{\\gamma+1},1\\right)$.\n\nNotice that if the derivative $h^{\\prime}$ is Lipschitz continuous this amounts to assuming $H\\in\\left(\\frac{1}{2},1\\right)$.\n\n\nProblem (\\ref{2}) is identical to the\ninitial-boundary value problem investigated in \\cite{nuavu}, up to Hypotheses\n(K$_{\\beta,\\beta^{\\prime}}$) which imply Hypotheses (K)\nof that article. This\nimmediately entails the existence of what is called there a variational\nsolution of type II for (\\ref{2}), henceforth simply coined variational solution. \nWith (K$_{\\beta,\\beta^{\\prime}}$) we have the existence and regularity properties of the Green function\nassociated with the differential operator governing (\\ref{2}). We shall give more details on this in the \nnext section.\n\\medskip\n\nWe organize this article in the following way. In Section \\ref{s2} we first recall\nthe notion of {\\it variational solution} and introduce a\nnotion of {\\it mild solution} for (\\ref{2}) by means of a family of evolution\noperators in $L^{2}(D)$ generated by the corresponding deterministic \nGreen's function. We then proceed by stating our main\nresults concerning the existence, uniqueness, and H\\\"{o}lder regularity of the mild\nsolution along with its indistinguishability from the\nvariational solution when $h$ is an affine function. \nThe section ends with a discussion about the results and methods of their proofs. \nThese are gathered in Section \\ref{s3}.\n\\bigskip\n\n\n\n\\section{Statement and Discussion of the Results}\n\\label{s2}\n\nIn the remaining part of this article we write $H^{1}(D\\times(0,T))$ for the\nisotropic Sobolev space on the cylinder $D\\times(0,T)$, which consists of all\nfunctions $v\\in L^{2}(D\\times(0,T))$ that possess distributional derivatives\n$v_{x_{i}}$, $v_{\\tau}\\in L^{2}(D\\times(0,T))$.\nThe set of all $v\\in H^{1}(D\\times(0,T))$ which do not depend\non the time variable identifies with $H^{1}(D)$, the usual Sobolev\nspace on $D$ whose\nnorm we denote by\n$\\left\\| .\\right\\| _{1,2}$. \n\nFor $0<\\alpha<1$\nwe introduce the Banach space $\\mathcal{B}^{\\alpha,2}(0,T;L^{2}(D))$\nof all Lebesgue-measurable mappings $u:\\left[ 0,T\\right] \\mapsto L^{2}(D)$\nendowed with the norm \n\\begin{equation}\n\\left\\| u\\right\\| _{\\alpha,2,T}^{2}:= \\left(\\sup_{t\\in\\left[0,T\\right]}\\left\\|u(t)\\right\\|_{2}\\right)^2 + \\int_{0}^{T}dt\\left( \\int_{0}\n^{t}d\\tau\\frac{\\left\\| u(t)-u(\\tau)\\right\\| _{2}^{{}}}{(t-\\tau)^{\\alpha+1}\n}\\right) ^{2}<+\\infty. \\label{3}\n\\end{equation}\nNotice that $\\Vert \\cdot\\Vert_{\\alpha,1}\\le c\\left\\| \\cdot \\right\\| _{\\alpha,2,T}$, \nand also that the spaces $\\mathcal{B}^{\\alpha,2}(0,T;L^{2}(D))$ decrease when $\\alpha$ increases. \n\nWe recall the following notion introduced in \\cite{nuavu}, in which the function $x\\mapsto v(x,t)\\in L^{2}(D)$\nis interpreted as the Sobolev trace of $v\\in H^{1}(D\\times(0,T))$ on the\ncorresponding hyperplane. \n\n\\begin{definition}\n\\label{d1}\nFix $H\\in\\left(\\frac{1}{2},1\\right)$ and let $\\alpha\\in\\left(1-H,\\frac{1}{2}\\right)$. We\nassume that conditions (C), (L) are satisfied and that the initial condition $\\varphi$ belongs to $L^2(D)$. In addition we suppose that the symmetric matrix valued function $k$ satisfies\n\\begin{equation*}\n\\underline k |q|^2 \\le (k(x,t)q,q)_{\\mathbb{R}^d}\\le \\overline k |q|^2,\n\\end{equation*}\nfor any $q\\in\\mathbb{R}^d$ and some positive constants $\\underline k$, $\\overline k$\nindependent of $x$ and $t$.\n\nUnder these conditions, the $L^{2}(D)$-valued random field $\\left( \nu_{V}(.,t)\\right) _{t\\in\\left[ 0,T\\right] }$ defined and measurable on\n$\\left( \\Omega,\\mathcal{F},\\mathbb{P}\\right) $ is a \\textit{variational\nsolution }to Problem (\\ref{2}) if\n\n(1) $u_{V}\\in L^{2}(0,T;H^{1}(D))\\cap\\mathcal{B}^{\\alpha,2}%\n(0,T;L^{2}(D))$ a.s., which means that \n\\[\n\\int_{0}^{T}dt\\left\\| u_{V}(.,t)\\right\\| _{1,2}^{2}=\\int_{0}^{T}dt\\left(\n\\left\\| u_{V}(.,t)\\right\\| _{2}^{2}+\\left\\| \\nabla u_{V}(.,t)\\right\\|\n_{2}^{2}\\right) <+\\infty\n\\]\nand $\\left\\| u_{V}\\right\\| _{\\alpha,2,T}<+\\infty$\nhold a.s.\n\n(2) The integral relation\n\\begin{align}\n\\int_{D}dx\\ v(x,t)u_{V}(x,t) & =\\int_{D}dx\\ v(x,0)\\varphi(x)+\\int_{0}^{t}\nd\\tau\\int_{D}dx\\ v_{\\tau}(x,\\tau)u_{V}(x,\\tau)\\nonumber\\\\\n& -\\int_{0}^{t}d\\tau\\int_{D}dx\\left( \\nabla v(x,\\tau),k(x,\\tau)\\nabla\nu_{V}(x,\\tau)\\right) _{\\mathbb{R}^{d}}\\nonumber\\\\\n& +\\int_{0}^{t}d\\tau\\int_{D}dxv(x,\\tau)g(u_{V}(x,\\tau))\\nonumber\\\\\n&+\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{t}\\left(v(.,\\tau),h(u_{V}(.,\\tau))e_{i}\\right) _{2}B_{i}^{H}(d\\tau).\n \\label{4}\n\\end{align}\nholds a.s. for every $v\\in$ $H^{1}(D\\times(0,T))$ and every $t\\in\\left[\n0,T\\right] $.\n\\end{definition}\n\n\nWith the standing hypotheses we easily infer that each term in\n(\\ref{4}) is finite a.s. In particular, Hypothesis (C) and the fact that $h$ is Lipschitz continuous,\nalong with (\\ref{i}) imply the\nabsolute convergence, a.s., of the series of the last term in (\\ref{4}).\n\n\n\n\\bigskip\n\nLet\n$G:\\overline{D}\\times\\left[ 0,T\\right] \\times\\overline{D}\\times\\left[ 0,T\\right]\n\\diagdown\\left\\{ s,t\\in\\left[ 0,T\\right] :s\\geq t\\right\\} \\to\\mathbb{R}$\nbe the parabolic Green's function associated with\nthe principal part of (\\ref{2}). Assume that (K$_{\\beta,\\beta^{\\prime}}$) holds; it is well-known that \n$G$ is a continuous function, twice continuously differentiable in $x$, once continuously\ndifferentiable in $t$. For every $(y,s)\\in D\\times (0,T]$, it is also a classical solution to the linear initial-boundary\nvalue problem\n\\begin{align}\n\\partial_{t}G(x,t;y,s) & =\\operatorname*{div}(k(x,t)\\nabla_{x}\nG(x,t;y,s)),\\text{ \\ \\ }(x,t)\\in D\\times\\left(0,T\\right],\\nonumber\\\\\n\\frac{\\partial G(x,t;y,s)}{\\partial n(k)} & =0,\\text{ \\ \\ \\ }(x,t)\\in\n\\partial D\\times\\left( 0,T\\right], \\label{6}\n\\end{align}\nwith\n\\[\n\\int_{D}dyG(.,s;y,s)\\varphi(y):=\\lim_{t\\searrow s}\\int_{D}dyG(.,t;y,s)\\varphi\n(y)=\\varphi(.),\n\\]\nand satisfies the heat kernel estimates\n\\begin{equation}\n\\left| \\partial_{x}^{\\mu}\\partial_{t}^{\\nu}G(x,t;y,s)\\right| \\leq\nc(t-s)^{-\\frac{d+\\left| \\mu\\right| +2\\nu}{2}}\\exp\\left[ -c\\frac{\\left|\nx-y\\right| _{{}}^{2}}{t-s}\\right] \\label{7}\n\\end{equation}\nfor $\\mu=(\\mu_{1},...,\\mu_{d})\\in\\mathbb{N}^{d}$, $\\nu\\in\\mathbb{N}$ and\n$\\left|\\mu\\right| +2\\nu\\leq 2$, with $\\left| \\mu\\right| =\\sum_{j=1}^{d}\n\\mu_{j}$ (see, for instance, \n\\cite{eidelzhita} or\n\\cite{ladyuralsolo}). In particular, for $\\left| \\mu\\right| =\\nu=0$ we have\n\\begin{equation}\n\\left| G(x,t;y,s)\\right| \\leq c(t-s)^{-\\frac{d}{2}}\\exp\\left[\n-c\\frac{\\left| x-y\\right| _{{}}^{2}}{t-s}\\right]. \\label{8}\n\\end{equation}\nWe shall refer to (\\ref{8}) as the {\\it Gaussian\nproperty} of $G$. \n\nWe can now define the notion of\nmild solution for (\\ref{2}).\n\n\\begin{definition}\n\\label{d2}\nFix $H\\in\\left(\\frac{1}{2},1\\right)$ and let $\\alpha\\in\\left(1-H,\\frac{1}{2}\\right)$. Assume that the hypotheses (C), (K$_{\\beta,\\beta^{\\prime}}$), (L)\nhold and that the initial condition $\\varphi$ is bounded.\n\nUnder these assumptions, the $L^{2}(D)$-valued random field $\\left(\nu_{M}(.,t)\\right) _{t\\in\\left[ 0,T\\right] }$ defined and measurable on\n$\\left( \\Omega,\\mathcal{F},\\mathbb{P}\\right) $ is a \\textit{mild}\n\\textit{solution }to Problem (\\ref{2}) if the following two conditions are satisfied:\n\n(1) $u_{M}\\in L^{2}(0,T;H^{1}(D))\\cap\\mathcal{B}^{\\alpha,2}\n(0,T;L^{2}(D))$ a.s.\n\n(2) The relation\n\\begin{align}\nu_{M}(.,t) & =\\int_{D}dy\\ G(.,t;y,0)\\varphi(y)+\\int_{0}^{t}d\\tau\\int\n_{D}dy\\ G(.,t;y,\\tau)g\\left( u_{M}(y,\\tau)\\right) \\nonumber\\\\\n&+\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{t}\\left( \\int\n_{D}dy\\ G(.,t;y,\\tau)h\\left(u_{M}(y,\\tau)\\right) e_{i}(y)\\right)\nB_{i}^{H}(d\\tau) \\label{9}\n\\end{align}\nholds a.s. for every $t\\in\\left[ 0,T\\right] $ as an equality in $L^{2}(D)$.\n\\end{definition}\n\nWe shall prove in Lemma \\ref{l2} that with the standing assumptions, each term in (\\ref{9}) indeed\ndefines a $L^{2}(D)$-valued stochastic process.\n\\medskip\n\nThe main results of this article are gathered in the next theorem.\n\\bigskip\n\n\\begin{theorem}\n\\label{t1}\nAssume that Hypotheses ($C$), ($K_{\\beta,\\beta^{\\prime}}$), ($L$), ($I$) and ($H_{\\gamma}$) hold; then\nthe following statements are valid:\n\n\\begin{description}\n\\item{(a)} Fix $H\\in\\left(\\frac{1}{\\gamma+1},1\\right)$ and let $\\alpha\\in\\left(1-H,\\frac{\\gamma}{\\gamma+1}\\right)$. Then\nProblem (\\ref{2}) possesses a variational solution\n$u_V$; moreover, every such variational solution is a mild solution $u_M$ to (\\ref{2}).\nMore presicely, for every $t\\in\\left[ 0,T\\right]$, $u_V(.,t)=u_M(.,t)$ a.s. in $L^{2}(D)$.\n\n\n\\item{(b)} Fix $H\\in\\left(\\frac{1}{\\gamma+1}\\vee\\frac{d+1}{d+2},1\\right)$ and then $\\alpha\\in\\left(1-H,\\frac{\\gamma}{\\gamma+1}\\wedge\\frac{1}{d+2}\\right)$. Assume in addition that $h$ is an affine function. Then $u_V$ is the unique variational solution to\n (\\ref{2}), while $u_M$ is its unique mild solution. \n \n\\item{(c)} Let $H$ and $\\alpha$ be as in part (b).\nThen every mild solution $u_{M}$ to Problem (\\ref{2}) \nis H\\\"{o}lder continuous with respect to the time variable. More precisely,\nthere exists a positive, a.s. finite random variable $R_{\\alpha}^{H}$\nsuch that the estimate\n\\begin{equation}\n\\left\\| u_{M}(.,t)-u_{M}(.,s)\\right\\| _{2}\\leq R_{\\alpha}^{H}\n\\left| t-s\\right| ^{\\theta}\\left( 1+\\left\\|u_{M}\\right\\| _{\\alpha,2,T}\\right)\n\\label{h1}\n\\end{equation}\nholds a.s. for all $s,t\\in\\left[ 0,T\\right] $ and every\n$\\theta\\in\\left( 0,\\left(\\frac{1}{2}-\\alpha\\right)\\wedge\\frac{\\beta}{2}\\right)$.\n \\end{description}\n\\end{theorem}\n\n\n\n\n{\\bf Remarks}\n\\begin{enumerate}\n\\item The existence of a mild solution will be proved by reference to\nthe existence of a variational solution. \nThis is in contrast with the method of \\cite{maslonua}, in which the\nauthors prove the existence of mild solutions for a class of\n\\textit{autonomous}, parabolic, fractional stochastic initial-boundary value\nproblems by means of Schauder's fixed point theorem. Their method thus\nrequires the construction of a continuous map operating in a compact and\nconvex set of a suitable functional space. \nIf $h$ is an affine function, the arguments of the proof of Statement (b)\n(see (\\ref{322})) show that a similar approach might be possible for\nour equation.\nTo the best of our knowledge, there exists as yet no such direct way to prove the\nexistence of mild solutions to (\\ref{2}) for a non affine $h$. \n\n\\item If $h$ is not affine, the question of uniqueness remains unsettled. \nIn fact, uniqueness could be proved if we were able to extend the inequality (\\ref{322})\nto any Lipschitz function $h$. This does not seem to be a trivial point, due to the form of the second term in the right-hand side of (\\ref{3}).\n\n\\item If $h$ is an affine function, Theorem \\ref{t1} establishes the complete\nindistinguishability of mild and variational solutions, although we do not\nknow whether this property still holds for a general $h$.\n\n\\item If $h$ is a constant function the setting of the problems and their proofs become much \nsimpler. Indeed, in Definitions \\ref{d1} and \\ref{d2} the space $\\mathcal{B}^{\\alpha,2}(0,T; L^2(D))$\ncan be replaced by the larger one $L^\\infty(0,T; L^2(D))$, consisting of Lebesgue-measurable\nmappings $u:[0,T] \\to L^2(D)$ such that $\\sup_{t\\in[0,T]}\\Vert u(t)\\Vert_2 <\\infty$. This can be checked\nby going through the proofs of \\cite{nuavu} and Lemma \\ref{l2} of Section \\ref{s3}.\nMoreover, the range of values of $\\theta$ in statement (c) of Theorem \\ref{t1} can be extended to the interval $\\left(0,\\frac{\\beta}{2}\\right)$.\nThis can be easily checked by going through the proof of Proposition \\ref{p4}, by checking first that the right-hand side\nof the inequalities (\\ref{40}), (\\ref{41}) can be replaced by $c(t-s)^{\\delta} (s-\\tau)^{-\\delta}$ and \n$c(t-s)^{\\frac{\\delta}{2}} (s-\\tau)^{-\\frac{1}{2}}(\\tau-\\sigma)^{\\frac{1}{2}(1-\\delta)}$, respectively.\n\n\\item \nBy using the \\textit{factorization method}, and under a different set of assumptions on the range of admissible values of $H$ and $\\alpha$,\nwe can obtain a different range of values for the H\\\"older exponent which in general do not provide as good an estimate as (\\ref{h1}) does.\nWe deal with this question in Proposition \\ref{pfact}. \nThe \\textit{factorization method}\nhas been introduced in \\cite{DP-K-Z} and since then\nextensively used for the analysis of the sample paths of solutions to\nparabolic stochastic partial differential equations (see, for instance,\n\\cite{sansovu1}). \n\\end{enumerate}\n\n\\section{Proofs of the Results}\n\\label{s3}\n\nIn what follows we write $c$ for all the irrelevant deterministic constants that occur in\nthe various estimates. \nWe begin by\nrecalling that the uniformly elliptic partial differential operator with\nconormal boundary conditions in the principal part of (\\ref{2}) admits a\nself-adjoint, positive realization $A(t):=-\\operatorname*{div}(k(.,t)\\nabla)$\nin $L^{2}(D)$ on the domain\n\\begin{equation}\n\\mathcal{D}(A(t))=\\left\\{ v\\in H^{2}(D):\\left( \\nabla\nv(x),k(x,t)n(x)\\right) _{\\mathbb{R}^{d}}=0,\\text{ \\ }(x,t)\\in\\partial\nD\\times\\left[ 0,T\\right] \\right\\} \\label{11}\n\\end{equation}\n(see, for instance, \\cite{lions}). An important consequence\nof this property is that the parabolic Green's function $G$ is also, for every\n$(x,t)\\in D\\times\\left( 0,T\\right] $ with $t>s$, a classical solution to the\nlinear boundary value problem\n\\begin{align}\n\\partial_{s}G(x,t;y,s) & =-\\operatorname*{div}(k(y,s)\\nabla_{y}\nG(x,t;y,s)),\\text{ \\ \\ }(y,s)\\in D\\times\\left( 0,T\\right] ,\\nonumber\\\\\n\\frac{\\partial G(x,t;y,s)}{\\partial n(k)} & =0,\\text{ \\ \\ \\ }(y,s)\\in\n\\partial D\\times\\left( 0,T\\right] , \\label{12}\n\\end{align}\ndual to (\\ref{6}) (see, for instance, \\cite{eidelzhita} or \\cite{friedman});\nthis means that along with (\\ref{7}) we also have \n\\begin{equation}\n\\left| \\partial_{y}^{\\mu}\\partial_{s}^{\\nu}G(x,t;y,s)\\right| \\leq\nc(t-s)^{-\\frac{d+\\left| \\mu\\right| +2\\nu}{2}}\\exp\\left[ -c\\frac{\\left|\nx-y\\right| _{{}}^{2}}{t-s}\\right] \\label{13}\n\\end{equation} \nfor $\\left| \\mu\\right| +2\\nu\\leq2$. We now use these facts to prove in the next lemma\nestimates for $G$, which we shall invoke repeatedly in\nthe sequel to analyze various singular integrals. For the sake of clarity we\nlist those inequalities by their chronological order of appearance in the\nproofs below.\n\n\\begin{lemma}\n\\label{l1}\nAssume that Hypothesis ($K_{\\beta,\\beta^{\\prime}}$) holds.\n Then, for all \n$x,y\\in D$ and for every $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$ we have the\nfollowing inequalities.\n\\begin{description}\n\\item{(i)} For all $t, \\tau, \\sigma \\in[0,T]$ with $t>\\tau>\\sigma$ and some $t^*\\in(\\sigma,\\tau)$,\n\\begin{align}\n& \\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| \\nonumber\\\\\n& \\leq c\\left( t-\\tau\\right) ^{-\\delta}(\\tau-\\sigma)^{\\delta}(t-t^{\\ast\n})^{-\\frac{d}{2}}\\exp\\left[ -c\\frac{\\left| x-y\\right| _{{}}^{2}}{t-t^{\\ast\n}}\\right]. \\label{14}\n\\end{align}\n\\item{(ii)} For all $t,s,\\tau\\in[0,T]$ with $t>s>\\tau$ and some $\\tau^*\\in(s,t)$,\n\\begin{align}\n& \\left| G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right| ^{{}}\\nonumber\\\\\n& \\leq c\\left( t-s\\right) ^{\\delta}(s-\\tau)^{-\\delta}(\\tau^{\\ast}\n-\\tau)^{-\\frac{d}{2}}\\exp\\left[ -c\\frac{\\left| x-y\\right| _{{}}^{2}}\n{\\tau^{\\ast}-\\tau}\\right] \\label{15}\n\\end{align}\nand\n\\begin{align}\n& \\left| G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right| ^{\\delta}\\nonumber\\\\\n& \\leq c\\left( t-s\\right) ^{\\delta}(s-\\tau)^{-\\frac{d+2}{2}\\delta+\\frac\n{d}{2}}(\\tau^{\\ast}-\\tau)^{-\\frac{d}{2}}\\exp\\left[ -c\\frac{\\left|\nx-y\\right| _{{}}^{2}}{\\tau^{\\ast}-\\tau}\\right]. \\label{16}\n\\end{align}\n\\item{(iii)} For all $t, s, \\tau, \\sigma\\in[0,T]$ with $t>s>\\tau>\\sigma$, \n\\begin{equation}\n\\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| ^{1-\\delta}\n\\leq c\\left( \\tau-\\sigma\\right) ^{1-\\delta}(s-\\tau)^{-\\frac{d+2}\n{2}(1-\\delta)} \\label{17}\n\\end{equation}\nuniformly in $t$.\n\\end{description}\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }By applying successively (\\ref{8}), the\nmean-value theorem for $G$ and (\\ref{13}) with $\\left|\n\\mu\\right| =0$ and $\\nu=1$ we may write\n\\begin{align*}\n& \\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| \\\\\n& \\leq\\left( \\left| G(x,t;y,\\tau)\\right| +\\left| G(x,t;y,\\sigma)\\right|\n\\right) ^{1-\\delta}\\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| ^{\\delta}\\\\\n& \\leq c\\left( (t-\\tau)^{-\\frac{d}{2}}+(t-\\sigma)^{-\\frac{d}{2}}\\right)\n^{1-\\delta}(\\tau-\\sigma)^{\\delta}\\left| G_{t^{\\ast}}(x,t;y,t^{\\ast})\\right|\n^{\\delta}\\\\\n& \\leq c(t-\\tau)^{-\\frac{d}{2}(1-\\delta)}(t-t^{\\ast})^{-\\frac{d+2}{2}\n\\delta+\\frac{d}{2}}(\\tau-\\sigma)^{\\delta}(t-t^{\\ast})^{-\\frac{d}{2}}\n\\exp\\left[ -c\\frac{\\left| x-y\\right| _{{}}^{2}}{t-t^{\\ast}}\\right] \\\\\n& \\leq c\\left( t-\\tau\\right) ^{-\\delta}(\\tau-\\sigma)^{\\delta}(t-t^{\\ast\n})^{-\\frac{d}{2}}\\exp\\left[ -c\\frac{\\left| x-y\\right| _{{}}^{2}}{t-t^{\\ast\n}}\\right]\n\\end{align*}\nfor some $t^{\\ast}\\in(\\sigma,\\tau)$, since $-\\frac{d+2}{2}\\delta+\\frac{d}{2}<0$\nand $-\\frac{d}{2}(1-\\delta)-\\frac{d+2}{2}\\delta+\\frac{d}{2}=-\\delta$. This\nproves (\\ref{14}). Up to some minor but important changes, the\nremaining inequalities can all be proved in a similar way. \\hfill $\\blacksquare$\n\n\\bigskip\n\nEstimate (\\ref{14}) now allows us to prove that our notion of\nmild solution in Definition \\ref{l2} is indeed well-defined; to this end for arbitrary mappings $\\varphi$ and $u$\ndefined on $D$ and $D\\times [0,T]$, respectively, we introduce\nthe functions $A(\\varphi)$, $B(u)$, $C(u): D\\times\\left[0,T\\right] \\mapsto\\mathbb{R}$ by\n\\begin{align}\nA(\\varphi)(x,t) & :=\\mathit{\\ }\\int_{D}dy\\ G(x,t;y,0)\\varphi\n(y),\\label{19}\\\\\nB(u)(x,t) & :=\\mathit{\\ }\\int_{0}^{t}d\\tau\\int_{D}dy\\ G(x,t;y,\\tau\n)g\\left( u(y,\\tau)\\right) ,\\label{20}\\\\\nC (u)(x,t) & :=\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}\n}\\int_{0}^{t}\\left( \\int_{D}dy\\ G(x,t;y,\\tau)h\\left( u(y,\\tau)\\right)\ne_{i}(y)\\right) B_{i}^{H}(d\\tau), \\label{21}\n\\end{align}\nand prove the following result.\n\n\\begin{lemma}\n\\label{l2}\nThe hypotheses are the same as in Definition \\ref{d2}.\n Then, for every\n $u\\in\\mathcal{B}^{\\alpha,2}(0,T;L^{2}(D))$ we have $A(\\varphi)(.,t) , B(u)(.,t)\\in L^{2}(D)$, and also\n$C(u)(.,t)\\in L^{2}(D)$ a.s., for every $t\\in\\left[0,T\\right]$.\n\\end{lemma}\n\n\\bigskip\n\n\\noindent\\textbf{Proof. }The assertion is evident for $A(\\varphi)(.,t)$, \nsince $\\varphi$ is bounded and (\\ref{8}) holds.\nAs for $B(u)(.,t)$, we infer from the Gaussian property of $G$ that\nthe measure $d\\tau dy\\left| G(x,t;y,\\tau)\\right| $ is finite on $\\left[\n0,T\\right] \\times D$ uniformly in $(x,t)\\in D\\times\\left[ 0,T\\right] $, so\nthat by using successively Schwarz inequality with respect to this measure\nalong with Hypothesis (L) for $g$ we obtain\n\\begin{align*}\n& \\left|B(u)(x,t)\\right| \\leq\\int_{0}^{t}d\\tau\\int_{D}dy\\left|\nG(x,t;y,\\tau)g\\left( u(y,\\tau)\\right) \\right| \\\\\n& \\leq c\\left( \\int_{0}^{t}d\\tau\\int_{D}dy\\left| G(x,t;y,\\tau)\\right|\n\\left( 1+\\left| u(y,\\tau)\\right| ^{2}\\right) \\right) ^{\\frac{1}{2}}\n\\end{align*}\nfor every $x\\in D$. We then get the inequalities\n\\begin{align*}\n& \\left\\|B(u)(.,t)\\right\\| _{2}^{2}=\\int_{D}dx\\left| \\int\n_{0}^{t}d\\tau\\int_{D}dy\\ G(x,t;y,\\tau)g\\left( u(y,\\tau)\\right) \\right| ^{2}\\\\\n& \\leq c\\int_{0}^{t}d\\tau\\int_{D}dy\\left( 1+\\left| u(y,\\tau)\\right|\n^{2}\\right) \n\\leq c\\left( 1+\\int_{0}^{t}d\\tau\\left\\| u(.,\\tau)\\right\\| _{2}\n^{2}\\right) < +\\infty. \n\\end{align*}\n\n It remains to show that $\\left\\|C(u)(.,t)\\right\\| _{2}^{2} <+\\infty$ a.s. for every $t\\in\\left[ 0,T\\right] $. \n\nDefine the functions $f_{i,t}(u):\\left[0,t\\right) \\mapsto L^{2}(D)$ by\n\\begin{equation}\nf_{i,t}(u)(.,\\tau):=\\int_{D}dy\\ G(.,t;y,\\tau)h\\left( u(y,\\tau)\\right)\ne_{i}(y). \\label{22}\n\\end{equation}\nWe shall prove that \n\\begin{equation}\n\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{0}^{t}\nf_{i,t}(u)(.,\\tau)B_{i}^{H}(d\\tau)\\right\\| _{2}\n \\leq c\\ r_{\\alpha}^{H}\\left( 1+\\left\\| u\\right\\|\n_{\\alpha,2,T}\\right), \\label{23}\n\\end{equation}\na.s., where $r_\\alpha^H$ is the a.s. finite and positive random variable\ndefined in (\\ref{29}).\n\nIndeed, by using an argument similar to the one above, since $h$ is Lipschitz\ncontinuous and $\\sup_{i\\in\\mathbb{N}^{+}}\\left\\| e_{i}\\right\\| _{\\infty\n}<+\\infty$, we first obtain\n\\begin{equation}\n\\sup_{i\\in\\mathbb{N}^{+}}\\left\\| f_{i,t}(u)(.,\\tau)\\right\\| _{2}\\leq c\\left( 1+\\left\\|\nu(.,\\tau)\\right\\| _{2}\\right) \\label{24}\n\\end{equation}\nfor every $\\tau\\in\\left[0,t\\right)$. Furthermore,\nfor every $x\\in D$ and all $\\sigma,\\tau\\in\\left[ 0,t\\right) $ with\n$\\tau>\\sigma$ we have\n\\begin{align*}\n&\\left| f_{i,t}(u)(x,\\tau)-f_{i,t}(u)(x,\\sigma)\\right| \n \\leq c\\Big(\\int_{D}dy\\left| G(x,t;y,\\tau)\\right| \\left| u(y,\\tau\n)-u(y,\\sigma)\\right| \\\\\n&\\quad \\quad +\\int_{D}dy\\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| (1+\\left|\nu(y,\\sigma)\\right|)\\Big),\n\\end{align*}\nso that we get successively\n\\begin{align*}\n& \\left| f_{i,t}(u)(x,\\tau)-f_{i,t}(u)(x,\\sigma)\\right| ^{2}\\\\\n& \\leq c\\int_{D}dy\\left| G(x,t;y,\\tau)\\right| \\left| u(y,\\tau\n)-u(y,\\sigma)\\right| ^{2}\\\\\n& +c\\int_{D}dy\\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right| \\left( 1+\\left|\nu(y,\\sigma)\\right| ^{2}\\right) \\\\\n& \\leq c\\int_{D}dy\\left| G(x,t;y,\\tau)\\right| \\left| u(y,\\tau\n)-u(y,\\sigma)\\right| ^{2}\\\\\n& +c\\left( t-\\tau\\right) ^{-\\delta}(\\tau-\\sigma)^{\\delta}\\int\n_{D}dy(t-t^{\\ast})^{-\\frac{d}{2}}\\exp\\left[ -c\\frac{\\left| x-y\\right| _{{}\n}^{2}}{t-t^{\\ast}}\\right] \\left( 1+\\left| u(y,\\sigma)\\right| ^{2}\\right)\n\\end{align*}\nfor some $t^{\\ast}\\in(\\sigma,\\tau)$ and for every $\\delta\\in\\left( \\frac\n{d}{d+2},1\\right)$. This is achieved by using Schwarz inequality with respect to the finite\nmeasures $dy\\left| G(x,t;y,\\tau)\\right| $ and $dy\\left| G(x,t;y,\\tau\n)-G(x,t;y,\\sigma)\\right| $ on $D$, respectively, along with\n(\\ref{14}). We then integrate the preceding estimate with\nrespect to $x\\in D$ and apply the Gaussian property of $G$ to eventually\nobtain\n\\begin{align}\n&\\sup_{i\\in\\mathbb{N}^{+}}\\left\\| f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\right\\|_{2}\\nonumber\\\\\n&\\leq c\\left(\\left\\| u(.,\\tau)-u(.,\\sigma)\\right\\| _{2}\n+\\left( t-\\tau\\right) ^{-\\frac{\\delta}{2}}(\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left( 1+\\left\\| u(.,\\sigma)\\right\\| _{2}\\right)\\right).\\label{25}\n\\end{align}\nTherefore, by applying (\\ref{i}) we have\n\\begin{align}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{0}^{t}\nf_{i,t}(u)(.,\\tau)B_{i}^{H}(d\\tau)\\right\\| _{2}\\nonumber\\\\\n&\\le r_\\alpha^H\\sup_{i\\in\\mathbb{N}^+} \\int_{0}^{t}d\\tau\\left( \\frac{\\left\\|\nf_{i,t}(u)(.,\\tau)\\right\\| _{2}}{\\tau^{\\alpha}}+\\int_{0}^{\\tau}\nd\\sigma\\ \\frac{\\left\\| f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\right\\| _{2}\n}{\\left( \\tau-\\sigma\\right) ^{\\alpha+1}}\\right) \\nonumber\\\\\n&\\le c r_\\alpha^H\\left( 1+\\int_{0}^{t}d\\tau\\ \\frac{\\left\\|\nu(.,\\tau)\\right\\| _{2}^{{}}}{\\tau^{\\alpha}}+\\int_{0}^{t}d\\tau\\int_{0}^{\\tau\n}d\\sigma\\ \\frac{\\left\\| u(.,\\tau)-u(.,\\sigma)\\right\\| _{2}^{{}}}{\\left(\n\\tau-\\sigma\\right) ^{\\alpha+1}}\\right. \\nonumber\\\\\n& +\\left. \\int_{0}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{0}^{\\tau\n}d\\sigma(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left( 1+\\left\\|\nu(.,\\sigma)\\right\\| _{2}\\right) \\right) \\label{26}\n\\end{align}\na.s. \n \nLet us now examine more closely the singular integrals in the\nabove terms. On the one hand, we may write\n\\begin{equation}\n\\int_{0}^{t}d\\tau\\ \\frac{\\left\\|u(.,\\tau)\\right\\|_2}{\\tau^{\\alpha}}\n+\\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma\\ \\frac{\\left\\|u(.,\\tau)-u(.,\\sigma)\\right\\|_2}{\\left(\\tau-\\sigma\\right)^{\\alpha+1}}\n\\leq c\\,\\left\\| u\\right\\|_{\\alpha,2,T}, \\label{27}\n\\end{equation}\nby using Schwarz inequality relative to the measure $d\\tau$ on $(0,t)$ in\nthe last two integrals along with (\\ref{3}). On the other hand, \nin (\\ref{26}) the exponent $\\delta$ can be taken arbitrarly close to $1$; consequently \nour range of values of $\\alpha$ allows the condition $2\\alpha<\\delta$ to be satisfied.\nThus we can\nintegrate the singularities of the time increments in the last line of\n(\\ref{26}) and get the bound\n\\begin{equation}\n\\int_{0}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{0}^{\\tau}d\\sigma\n(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left( 1+\\left\\| u(.,\\sigma)\\right\\| _{2}\\right)\n\\leq c\\left(1+\\sup_{t\\in[0,T]}\\left\\|u(.,t)\\right\\|_2\\right).\n\\label{28}\n\\end{equation}\nTherefore, we can\nsubstitute (\\ref{27}), (\\ref{28}) into (\\ref{26}) to\nobtain (\\ref{23}). \n\n\\hfill $\\blacksquare$\n\n\\bigskip\n\nIn order to relate the notions of variational and mild solution, we recall\nthat the self-adjoint operator $A(t)=-\\operatorname*{div}(k(.,t)\\nabla)$\ndefined on (\\ref{11}) generates the family of evolution operators\n$U(t,s)_{0\\leq s\\leq t\\leq T}$ in $L^{2}(D)$ given by\n\\begin{equation}\nU(t,s)v=\\begin{cases}\nv,&\\text{if}\\ \\ s=t,\\\\\n\\int_{D}dy\\ G(.,t;y,s)v(y), &\\text{if}\\ \\ t>s,\n\\label{30}\n\\end{cases}\n\\end{equation}\nand that each such $U(t,s)$ is itself self-adjoint (see, for instance,\n\\cite{tanabe}), which means that the symmetry property \n\\begin{equation}\nG(x,t;y,s)=G(y,t;x,s) \\label{symmetry}\n\\end{equation}\nholds for every $(x,t;y,s)\\in\\overline{D}\\times\\left[ 0,T\\right]\n\\times\\overline{D}\\times\\left[ 0,T\\right] \\diagdown\\left\\{ s,t\\in\\left[0,T\\right] :s\\geq t\\right\\}$. \n\n\\bigskip\n\n\\noindent{\\bf Proof of Statement (a) of Theorem \\ref{t1}}\n\\medskip\n\nThe existence of a variational solution $u_V$ was proved in Theorem of \\cite{nuavu}.\n\nIn order to prove that every variational solution is mild, we follow the same approach as in Theorem 2 of \\cite{sansovu1}.\nFor the sake of completeness, we sketch the main ideas. \n\nWe shall check that the $L^{2}(D)$-valued stochastic process\n\\begin{align*}\n& u_{V}(.,t)-\\int_{D}dy\\ G(.,t;y,0)\\varphi(y)-\\int_{0}^{t}d\\tau\\int\n_{D}dy\\ G(.,t;y,\\tau)g\\left( u_{V}(y,\\tau)\\right) \\\\\n& -\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}\\int_{0}^{t}\\left(\\int_D dy\\ G(.,t;y,\\tau)h\\left(u_{V}(y,\\tau)e_i(y)\\right)\\right)\nB_i^{H}(d\\tau)\n\\end{align*}\nis a.s. orthogonal for every $t\\in\\left[ 0,T\\right] $ to the dense subspace\n$\\mathcal{C}_{0}^{2}(D)$ consisting of all twice continuously differentiable\nfunctions with compact support in $D$. To this end, for every $v\\in\\mathcal{C}_{0}^{2}(D)$\nand all $s,t\\in\\left[ 0,T\\right]$ with $t\\geq s$\nwe define $v^{t}(.,s):=U(t,s)v$, that is, \n\\begin{equation}\nv^{t}(x,s)=\n\\begin{cases}\nv(x),& \\text{if}\\ \\ s=t,\\\\\n\\int_{D}dyG(y,t;x,s)v(y),& \\text{if}\\ \\ t>s,\n\\end{cases}\n\\label{31}\n\\end{equation}\nfor every $x\\in D$ by taking (\\ref{30}) and (\\ref{symmetry}) into\naccount. It then follows from (\\ref{12}), (\\ref{symmetry}) and Gauss'\ndivergence theorem that $v^{t}\\in H^{1}(D\\times(0,T))$, and that for every $t\\in\\left[ 0,T\\right]$, the relation\n\\begin{equation}\n\\int_{0}^{t}d\\tau\\int_{D}dx\\ v_{\\tau}^{t}(x,\\tau)u_{V}(x,\\tau)=\\int_{0}^{t}\nd\\tau\\int_{D}dx\\left( \\nabla v^{t}(x,\\tau),k(x,\\tau)\\nabla u_{V}\n(x,\\tau)\\right) _{\\mathbb{R}^{d}} \\label{32}\n\\end{equation}\nholds a.s.\nTherefore, we may take (\\ref{31}) as a test function in\n(\\ref{4}), which, as a consequence of (\\ref{32}),\nleads to the relation\n\\begin{align*}\n(v,u_{V}(.,t))_{2} & =(v^{t}(.,0),\\varphi)_{2}+\\int_{0}^{t}d\\tau\n(v^{t}(.,\\tau),g(u_{V}(.,\\tau)))_{2}\\\\\n& +\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{t}\\left(\nv^{t}(.,\\tau),h(u_{V}(.,\\tau))e_{i}\\right) _{2}B_{i}^{\\text{\\textsc{H}}%\n}(d\\tau),\n\\end{align*}\nvalid a.s. for every $t\\in\\left[0,T\\right]$. After some rearrangements, the substitution of\n(\\ref{31}) into the right-hand side of the preceding expression then\nleads to the equality\n\\begin{align*}\n(v,u_{V}(.,t))_{2} & =\\left( v,\\int_{D}dyG(.,t;y,0)\\varphi(y)\\right)_{2}\\\\\n&+\\left( v,\\int_{0}^{t}d\\tau\\int_{D}dyG(.,t;y,\\tau)g\\left( u_{V}\n(y,\\tau)\\right) \\right)_{2}\\\\\n& +\\left( v,\\sum_{i=1}^{+\\infty} \\lambda_i^{\\frac{1}{2}}\\int_{0}^{t}\\left(\\int_D dy\\ G(.,t;y,\\tau)h\\left( u_{V}(y,\\tau)e_i(y)\\right)\\right)\nB_i^{H}(d\\tau)\\right)_{2},\n\\end{align*}\nwhich holds for every $t\\in\\left[ 0,T\\right]$ a.s. and every\n$v\\in\\mathcal{C}_{0}^{2}(D)$, thereby leading to the desired orthogonality\nproperty. \\hfill $\\blacksquare$\n\\bigskip\n\\newpage\n\n\\noindent{\\bf Proof of Statement $(b)$ of Theorem \\ref{t1}}\n\\bigskip\n\n\\noindent Under the standing assumptions, we already know from \\cite{nuavu}\nthat the variational solution is unique. Moreover, we have just proved that every variational solution is also\na mild solution. Hence, it suffices to prove that uniqueness holds\nwithin the class of mild solutions.\nTo this\nend, let us write $u_{M}$ and $\\tilde{u}_{M}$ for any two such solutions\ncorresponding to the same initial condition $\\varphi$; from (\\ref{9}) and\n(\\ref{19})-(\\ref{21}) we have\n\\begin{align}\n& \\left\\Vert u_{M}(.,t)-\\tilde{u}_{M}(.,t)\\right\\Vert _{2}\\nonumber\\\\\n& \\leq\\left\\Vert B(u_{M})(.,t)-B(\\tilde{u}_{M})(.,t)\\right\\Vert\n_{2}+\\left\\Vert C(u_{M})(.,t)-C(\\tilde{u}_{M})(.,t)\\right\\Vert _{2}\n\\label{302}\n\\end{align}\na.s. for every $t\\in\\left[ 0,T\\right] $.\n\nWe proceed by estimating both terms\non the right-hand side of (\\ref{302}). \nSince $g$ is Lipschitz, we have\n\\begin{equation}\n\\left\\| B\\left( u_M\\right) (.,t)-B(\\tilde{u}_M)(.,t)\\right\\| _{2}^{2}\n\\leq c\\int_{0}^{t}d\\tau\\left\\| u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\| _{2}^{2} \\label{303}\n\\end{equation}\na.s. for every $t\\in\\left[ 0,T\\right]$.\n\nIn order to analyze the second term or the right-hand side of (\\ref{302}) we will need the following preliminary result.\n\n\\begin{lemma}\n\\label{l5}\nThe hypotheses are the same as in part (a) of Theorem \\ref{t1} and let\nthe $f_{i,t}(u)$'s be the functions given by (\\ref{22}). Then,\nthe estimate\n\\begin{equation}\n\\sup_{(i,t)\\in\\mathbb{N}^+\\times[0,T]} \\left\\| f_{i,t}(u_M)(.,\\tau)-f_{i,t}(\\tilde{u}_M)(.,\\tau)\\right\\|_2\n\\leq c\\left\\| u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\|_{2}\n\\label{304}\n\\end{equation}\nholds a.s. for every $\\tau\\in\\left[0,t\\right)$.\n\nMoreover, if $h$ is an affine function we\nhave\n\\begin{align}\n& \\sup_{i\\in\\mathbb{N}^+}\\left\\| \\ f_{i,t}(u_M)(.,\\tau)-f_{i,t}(\\tilde{u}_M)(.,\\tau)\n -f_{i,t}(u_M)(.,\\sigma)+f_{i,t}(\\tilde{u}_M)(.,\\sigma)\\right\\|_{2}\\nonumber\\\\\n&\\quad \\leq c(t-\\tau)^{-\\delta}(\\tau-\\sigma)^{\\delta}\\left\\|\nu_M(.,\\sigma)-\\tilde{u}_M(.,\\sigma)\\right\\| _{2}\\nonumber\\\\\n& \\quad +c\\left\\| u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)-u_M(.,\\sigma)\n+\\tilde{u}_M(.,\\sigma)\\right\\| _{2} \\label{305}\n\\end{align}\na.s. for all $t,\\tau,\\sigma\\in\\left[0,T\\right]$ with $t>\\tau>\\sigma$ and every $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$.\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }Up to minor modifications, we can prove (\\ref{304}) as we\nargued in the proof of (\\ref{24}).\n\nFor the proof of (\\ref{305}) we first write\n\\begin{align*}\n&\\left\\| \\ f_{i,t}(u_M)(.,\\tau)-f_{i,t}(\\tilde{u}_M)(.,\\tau)\n -f_{i,t}(u_M)(.,\\sigma)+f_{i,t}(\\tilde{u}_M)(.,\\sigma)\\right\\|_{2}^2\\\\\n &\\quad \\le 2\\left(F^1(i,t,\\tau,\\sigma)+F^2(i,t,\\tau,\\sigma)\\right),\n \\end{align*}\n with\n \\begin{align*}\n& F^1(i,t,\\tau,\\sigma)\\\\\n&= \\int_D dx\\left\\vert\\int_D dy\\ e_i(y)\\left(G(x,t;y,\\tau)-(G(x,t;y,\\sigma)\\right)\\left(u_M(y,\\tau)-\\tilde u_M(y,\\tau)\\right)\\right\\vert^2,\\\\\n&F^2(i,t,\\tau,\\sigma)\\\\\n&=\\int_D dx\\left\\vert\\int_D dy\\ e_i(y) G(x,t;y,\\sigma)\\left(u_M(y,\\tau)-\\tilde u_M(y,\\tau)-u_M(y,\\sigma)+\\tilde u(y,\\sigma)\\right)\\right\\vert^2.\n\\end{align*}\nFrom the Gaussian property of $G$ we clearly see that $F^2(i,t,\\tau,\\sigma)$ is bounded above by the square of the last term of (\\ref{305}). \nMoreover, by applying first (\\ref{14}) and then Schwarz inequality we obtain\n\\begin{equation*}\nF^1(i,t,\\tau,\\sigma)\\le c(t-\\tau)^{-2\\delta}(\\tau-\\sigma)^{2\\delta}\\left\\|u_M(.,\\sigma)-\\tilde{u}_M(.,\\sigma)\\right\\|_{2}^2.\n\\end{equation*}\nHence (\\ref{305}) is proved.\n\n\\hfill $\\blacksquare$\n\\medskip\n\nThe preceding result now leads to the following estimate for the second term\non the right-hand side of (\\ref{302}).\n\n\\begin{lemma}\n\\label{l6}\nThe hypotheses are those of Theorem \\ref{t1} part (b).\nThen we have\n\\begin{align}\n& \\left\\|C(u_M)(.,t)-C(\\tilde{u}_M\n)(.,t)\\right\\| _{2}\\nonumber\\\\\n&\\quad \\leq cr_{\\alpha}^{H}\\left( \\int_{0}^{t}d\\tau\\left(\n\\frac{1}{\\tau^{\\alpha}}+\\frac{1}{\\left( t-\\tau\\right) ^{\\alpha}}\\right)\n\\left\\| u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\| _{2}\\right. \\nonumber\\\\\n&\\quad +\\left. \\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma\\frac{\\left\\| u_M\n(.,\\tau)-\\tilde{u}_M(.,\\tau)-u_M(.,\\sigma)+\\tilde{u}_M(.,\\sigma\n)\\right\\| _{2}}{\\left( \\tau-\\sigma\\right) ^{\\alpha+1}}\\right)\n\\label{306}\n\\end{align}\na.s. for every $t\\in\\left[ 0,T\\right]$.\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }From (\\ref{21}), (\\ref{22}),\nand by using (\\ref{i}), (\\ref{304}), (\\ref{305}), we have\n\\begin{align}\n& \\left\\| C(u_M)(.,t)-C(\\tilde{u}_M\n)(.,t)\\right\\| _{2}\\nonumber\\\\\n& \\leq cr_{\\alpha}^{H}\\left( \\int_{0}^{t}d\\tau\\frac{\\left\\|\nu_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\| _{2}}{\\tau^{\\alpha}}\\right.\n\\nonumber\\\\\n& +\\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma(t-\\tau)^{-\\delta}\n(\\tau-\\sigma)^{\\delta-\\alpha-1}\\left\\| u_M(.,\\sigma\n)-\\tilde{u}_M(.,\\sigma)\\right\\| _{2}\\nonumber\\\\\n& +\\left. \\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma\\frac{\\left\\| u_M\n(.,\\tau)-\\tilde{u}_M(.,\\tau)-u_M(.,\\sigma)+\\tilde{u}_M(.,\\sigma\n)\\right\\| _{2}}{\\left( \\tau-\\sigma\\right) ^{\\alpha+1}}\\right)\n\\label{307}\n\\end{align}\na.s. for every $t\\in\\left[ 0,T\\right]$.\n\nFurthermore, by swapping each\nintegration variable for the other in the second term on the right-hand side\nand by using Fubini's theorem we may write\n\\begin{align*}\n& \\int_{0}^{t}d\\tau\\int_{0}^{\\tau}d\\sigma(t-\\tau)^{-\\delta}\n(\\tau-\\sigma)^{\\delta-\\alpha-1}\\left\\| u_M(.,\\sigma\n)-\\tilde{u}_M(.,\\sigma)\\right\\| _{2}\\\\\n& =\\int_{0}^{t}d\\tau\\left\\| u_M(.,\\tau)-\\tilde{u}_M(.,\\tau)\\right\\|\n_{2}\\int_{\\tau}^{t}d\\sigma(t-\\sigma)^{-\\delta}(\\sigma-\\tau\n)^{\\delta-\\alpha-1}\\\\\n& =c\\int_{0}^{t}d\\tau\\frac{\\left\\| u_M(.,\\tau)-\\tilde{u}_M\n(.,\\tau)\\right\\| _{2}}{\\left( t-\\tau\\right) ^{\\alpha}},\n\\end{align*}\nafter having evaluated the singular integral explicitly in terms of Euler's\nBeta function; this is possible since we can choose $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$ such that $\\alpha<\\delta$. The\nsubstitution of the preceding expression into (\\ref{307}) then proves\n(\\ref{306}). \\hfill $\\blacksquare$\n\\bigskip\n\nIn what follows, we write $R$ for all the irrelevant a.s. finite and positive\nrandom variables that appear in the different estimates, unless we specify these\nvariables otherwise.\nThe preceding inequalities then lead to a\ncrucial estimate for $z_{M}:=u_{M}-\\tilde{u}_{M}$ with respect to the norm in\n$B^{\\alpha,2}(0,t;L^{2}(D))$.\n\\begin{lemma}\n\\label{l30}\nWe assume the same hypotheses as in part (b) of Theorem \\ref{t1}.\nThen we have\n\\begin{align*}\n\\left\\Vert z_{M}\\right\\Vert _2^{2} & \\leq R\\left( \\int_{0}^{t}d\\tau\n\\sup_{\\sigma\\in\\lbrack0,\\tau]}\\left\\Vert z_{M}(.,\\sigma)\\right\\Vert_{2}^{2}\\right. \\nonumber\\\\\n& +\\left. \\int_{0}^{t}d\\tau\\left( \\int_{0}^{\\tau}d\\sigma\\frac{\\left\\Vert\nz_{M}(.,\\tau)-z_{M}(.,\\sigma)\\right\\Vert _{2}}{\\left( \\tau-\\sigma\\right)^{\\alpha+1}}\\right)^{2}\\right) \n\\end{align*}\na.s. for every $t\\in\\left[ 0,T\\right]$.\n\\end{lemma}\n\n\\noindent{\\bf Proof.}\nWe apply Schwarz inequality relative to the measure $d\\tau$ on\n$(0,t)$ to both integrals on the right-hand side of (\\ref{306}). This leads\nto\n\\begin{align}\n& \\Vert C(u_{M})(.,t)-C(\\tilde{u}_{M})(.,t)\\Vert_{2}^{2}\\nonumber\\\\\n& \\leq R\\left( \\int_{0}^{t}d\\tau\\Vert z_{M}(.,\\tau)\\Vert_{2}^{2}+\\int\n_{0}^{t}d\\tau\\left( \\int_{0}^{\\tau}d\\sigma\\frac{\\Vert z_{M}(.,\\tau\n)-z_{M}(.,\\sigma)\\Vert_{2}}{(\\tau-\\sigma)^{\\alpha+1}}\\right) ^{2}\\right)\n\\label{3701}\n\\end{align}\na.s. for every $t\\in\\left[ 0,T\\right] $. This estimate along with (\\ref{302}),\n(\\ref{303}) yields the result.\n\n \\hfill $\\blacksquare$\n\nAs a consequence of the preceding Lemma we obtain\n\\begin{align}\n\\left\\Vert z_{M}\\right\\Vert _{\\alpha,2,t}^{2} & \\leq R\\left( \\int_{0}^{t}d\\tau\n\\sup_{\\sigma\\in\\lbrack0,\\tau]}\\left\\Vert z_{M}(.,\\sigma)\\right\\Vert_{2}^{2}\\right. \\nonumber\\\\\n& +\\left. \\int_{0}^{t}d\\tau\\left( \\int_{0}^{\\tau}d\\sigma\\frac{\\left\\Vert\nz_{M}(.,\\tau)-z_{M}(.,\\sigma)\\right\\Vert _{2}}{\\left( \\tau-\\sigma\\right)^{\\alpha+1}}\\right)^{2}\\right) \\label{308},\n\\end{align}\nby the very definition of the norm $\\left\\Vert \\cdot\\right\\Vert _{\\alpha,2,t}^{2}$. \n\n\\bigskip\n\nWe proceed by analyzing further the second\nterm on the right-hand side of (\\ref{308}), so as to eventually obtain an inequality of Gronwall type\nfor $\\Vert z_M\\Vert_{\\alpha,2,t}^2$. \n\nFirst we introduce some notation.\nFor $0\\le \\tau\\sigma$\n\nOur next goal is to estimate the $L^{2}(D)$-norm of each contribution on\nthe right-hand side of (\\ref{309}). Regarding the first two terms\nwe have the following result.\n\n\\begin{lemma}\n\\label{l7}\nThe hypotheses are the same as in part (a) of Theorem \\ref{t1}; then we\nhave\n\\begin{align}\n& \\left\\| \\int_{\\sigma}^{\\tau}d\\rho\\int_{D}dy\\ G(.,\\tau;y,\\rho)\\left(\ng(u_M(y,\\rho))-g(\\tilde{u}_M(y,\\rho))\\right) \\right\\|\n_{2}\\nonumber\\\\\n& \\leq c\\left( \\tau-\\sigma\\right) ^{\\frac{1}{2}}\\left( \\int_{\\sigma}\n^{\\tau}d\\rho\\left\\| z_M(.,\\rho)\\right\\| _{2}^{2}\\right)\n^{\\frac{1}{2}}\n\\label{310}\n\\end{align}\nand\n\\begin{align}\n& \\left\\| \\int_{0}^{\\sigma}d\\rho\\int_{D}dy\\left( G(.,\\tau;y,\\rho\n)-G(.,\\sigma;y,\\rho)\\right) \\left( g(u_M(y,\\rho\n))-g(\\tilde{u}_M(y,\\rho))\\right) \\right\\| _{2}\\nonumber\\\\\n& \\leq c\\left( \\tau-\\sigma\\right) ^{\\frac{\\delta}{2}}\\left( \\int\n_{0}^{\\sigma}d\\rho\\left( \\sigma-\\rho\\right) ^{-\\delta}\\left\\| z_M\n(.,\\rho)\\right\\| _{2}^{2}\\right) ^{\\frac{1}{2}}\n\\label{311}\n\\end{align}\na.s. for all $\\sigma,\\tau\\in\\left[ 0,t\\right] $ with \n$\\tau>\\sigma$ and every $\\delta\\in\\left( \\frac{d}{d+2},1\\right)$.\n\\end{lemma}\n{\\bf Proof:} The inequality (\\ref{310}) follows by applying Schwarz inequality and using the Gaussian property along with\nassumption (L). As for (\\ref{311}), we first apply Schwarz inequality with respect to the measure on $[0,\\sigma]\\times D$ given by\n$|G(x,\\tau;y,\\rho)-G(x,\\sigma;y,\\rho)|d\\rho\\ dy$ and then (\\ref{15}).\n\n\\hfill $\\blacksquare$\n\n\n\n\nNext, we turn to the analysis of the third term on the right-hand side of\n(\\ref{309}).\n\n\\begin{lemma}\n\\label{l8}\nWith the same hypotheses as in part (b) of Theorem \\ref{t1}, we have\n\\begin{align*}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{\\sigma}^{\\tau\n}\\left( f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M\n)(.,\\rho)\\right) B_{i}^{H}(d\\rho)\\right\\| _{2}\\\\\n& \\leq R\\left( \\int_{\\sigma}^{\\tau}\nd\\rho\\left( \\frac{1}{\\left( \\rho-\\sigma\\right) ^{\\alpha}}+\\frac{1}{\\left(\n\\tau-\\rho\\right) ^{\\alpha}}\\right) \\left\\| z_M(.,\\rho)\\right\\|\n_{2}\\right. \\\\\n& +\\left. \\int_{\\sigma}^{\\tau}d\\rho\\int_{\\sigma}^{\\rho}d\\varsigma\n\\frac{\\left\\| z_M(.,\\rho)-z_M(.,\\varsigma)\\right\\| _{2}\n}{\\left( \\rho-\\varsigma\\right) ^{\\alpha+1}}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[0,t\\right]$ with \n$\\tau>\\sigma$.\n\\end{lemma}\n\\noindent\\textbf{Proof.} \nIn terms of the variables $\\tau,\\rho$ and $\\varsigma$,\ninequalities (\\ref{304}), (\\ref{305}) of Lemma \\ref{l5} now read\n\\begin{equation}\n\\sup_{(i,\\tau)\\in\\mathbb{N}^+\\times [0,T]}\\left\\| \\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}\n(\\tilde{u}_M)(.,\\rho)\\right\\| _{2}\n\\leq c\\left\\| z_M(.,\\rho)\\right\\|_{2} \\label{312}\n\\end{equation}\nand\n\\begin{align}\n& \\sup_{i\\in\\mathbb{N}^+} \\left\\| \\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}\n(\\tilde{u}_M)(.,\\rho)\\ -f_{i,\\tau}(u_M)(.,\\varsigma\n)+f_{i,\\tau}(\\tilde{u}_M)(.,\\varsigma)\\right\\| _{2}^{{}}\\nonumber\\\\\n& \\leq c(\\tau-\\rho)^{-\\delta}(\\rho-\\varsigma)^{\\delta}\n\\left\\| z_M(.,\\varsigma)\\right\\| _{2}\n+c\\left\\| z_M(.,\\rho)-z_M(.,\\varsigma)\\right\\| _{2},\n\\label{313}\n\\end{align}\nrespectively. Thus, by an extended version of (\\ref{i}) for indefinite generalized Stieltjes integrals\n(see Proposition 4.1 in \\cite{nuarasca}),\n\\begin{align*}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{\\sigma}^{\\tau}\n\\left( f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M)(.,\\rho)\\right) B_{i}^{H}(d\\rho)\\right\\| _{2}\\\\\n& \\leq r_\\alpha^H \\sup_{i\\in\\mathbb{N}^+}\\Big\n\\Big(\\int_{\\sigma}^{\\tau}d\\rho\\frac{\\left\\|\\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M%\n)(.,\\rho)\\right\\|_{2}}{\\left( \\rho-\\sigma\\right) ^{\\alpha}}\\\\\n& +\\int_{\\sigma}^{\\tau}d\\rho\\int_{\\sigma}^{\\rho}\\frac{d\\varsigma}{\\left(\\rho-\\varsigma\\right)^{\\alpha+1}}\\\\\n&\\times\\left\\| \\ f_{i,\\tau}(u_M)(.,\\rho)-f_{i,\\tau}(\\tilde{u}_M)(.,\\rho)\\ -f_{i,\\tau}(u_M)(.,\\varsigma)+f_{i,\\tau}(\\tilde{u}_M)(.,\\varsigma)\\right\\|_2\\Big)\\Big) \\\\\n& \\leq R\\left( \\int_{\\sigma}^{\\tau}d\\rho\n\\frac{\\left\\| z_M(.,\\rho)\\right\\| _{2}}{\\left( \\rho\n-\\sigma\\right) ^{\\alpha}}+\\int_{\\sigma}^{\\tau}d\\rho\\left( \\tau-\\rho\\right)\n^{-\\delta}\\int_{\\sigma}^{\\rho}d\\varsigma\\left( \\rho-\\varsigma\n\\right) ^{\\delta-\\alpha-1}\\left\\| z_M(.,\\varsigma\n)\\right\\| _{2}\\right. \\\\\n& +\\left. \\int_{\\sigma}^{\\tau}d\\rho\\int_{\\sigma}^{\\rho}d\\varsigma\n\\frac{\\left\\| z_M(.,\\rho)-z_M(.,\\varsigma)\\right\\|_{2}\n}{\\left( \\rho-\\varsigma\\right) ^{\\alpha+1}}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[ 0,t\\right] $ with $\\tau>\\sigma$ and every\n$\\delta\\in\\left( \\frac{d}{d+2},1\\right)$. But the second term on the\nright-hand side is equal to\n\\begin{equation*}\nc\\int_{\\sigma}^{\\tau}d\\rho\\left( \\tau-\\rho\\right) ^{-\\alpha}\\left\\|\nz_M(.,\\rho)\\right\\| _{2},\n\\end{equation*}\nas can be easily checked by applying Fubini's theorem and by evaluating the\nresulting inner integral in terms of Euler's Beta function. \\hfill $\\blacksquare$\n\\bigskip\n\nAs for the analysis of the fourth term on the right-hand side of\n(\\ref{309}) we need some preparatory results. In particular we shall use the estimate for \ntime increments of the Green function, valid for any $\\delta\\in\\left(\\frac{d}{d+2},1\\right)$: \n\\begin{align}\n& \\left|G(x,t;y,\\tau)-G(x,s;y,\\tau)-G(x,t;y,\\sigma)+G(x,s;y,\\sigma)\\right|\\nonumber\\\\\n& \\leq \\left( \\left| G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|^{\\delta}\n+\\left|G(x,t;y,\\sigma)-G(x,s;y,\\sigma)\\right|^{\\delta}\\right) \\nonumber\\\\\n& \\times\\left( \\left| G(x,t;y,\\tau)-G(x,t;y,\\sigma)\\right|^{1-\\delta}\n+\\left|G(x,s;y,\\tau)-G(x,s;y,\\sigma)\\right|^{1-\\delta}\\right) \\nonumber\\\\\n& \\leq \\left( t-s\\right)^\\delta(s-\\tau)^{-\\frac{d+2}{2}\\delta+\\frac{d}{2}}\n\\left( \\tau-\\sigma\\right) ^{1-\\delta}(s-\\tau)^{-\\frac{d+2}{2}(1-\\delta)}\\nonumber\\\\\n& \\times\\left( (\\tau^{\\ast}-\\tau)^{-\\frac{d}{2}}\n\\exp\\left[ -c\\frac{\\left|x-y\\right|^{2}}{\\tau^{\\ast}-\\tau}\\right] \n+(\\sigma^{\\ast}-\\sigma)^{-\\frac{d}{2}}\\exp\\left[-c\\frac{\\left| x-y\\right|^{2}}{\\sigma^{\\ast}-\\sigma}\\right] \\right) \\nonumber\\\\\n& =c\\left( t-s\\right) ^{\\delta}\\left( s-\\tau\\right) ^{-1}\\left(\n\\tau-\\sigma\\right) ^{1-\\delta}\\nonumber\\\\\n& \\times\\left( (\\tau^{\\ast}-\\tau)^{-\\frac{d}{2}}\\exp\\left[ -c\\frac{\\left|\nx-y\\right| _{{}}^{2}}{\\tau^{\\ast}-\\tau}\\right] +(\\sigma^{\\ast}\n-\\sigma)^{-\\frac{d}{2}}\\exp\\left[ -c\\frac{\\left| x-y\\right|^{2}}{\\sigma^{\\ast}-\\sigma}\\right] \\right), \\label{44}\n\\end{align}\nwith $\\tau^*, \\sigma^* \\in(s,t)$, which follows from (\\ref{16})--(\\ref{17}).\n\n\n\\begin{lemma}\n\\label{l9}\nThe hypotheses are the same as in part (b) of Theorem \\ref{t1} and the\n$f_{i,\\tau,\\sigma}^{\\ast}(u)$'s are the functions given by\n(\\ref{39}). Then, the estimates\n\\begin{equation}\n\\sup_{i\\in\\mathbb{N}^+}\\left\\|\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(u_M)\n(.,\\rho)-\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(\\tilde{u}_M\n)(.,\\rho)\\right\\| _{2}\n\\leq c(\\tau-\\sigma)^{\\frac{\\delta}{2}}(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left\\|z_M(.,\\rho)\\right\\| _{2} \\label{314}\n\\end{equation}\nand\n\\begin{align}\n& \\sup_{i\\in\\mathbb{N}^+}\\left\\|\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(u_M\n)(.,\\rho)-\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(\\tilde{u}_M\n)(.,\\rho)-\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(u_M)(.,\\varsigma\n)+\\mathit{\\ }f_{i,\\tau,\\sigma}^{\\ast}(\\tilde{u}_M)(.,\\varsigma\n)\\right\\| _{2}\\nonumber\\\\\n& \\leq c(\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left( (\\sigma-\\rho)^{-\\frac{1}{2}\n}(\\rho-\\varsigma)^{\\frac{1}{2}(1-\\delta)}\\left\\| z_M(.,\\varsigma\n)\\right\\| _{2}\\right. \\nonumber\\\\\n& +\\left. (\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left\\| z_M\n(.,\\rho)-z_M(.,\\varsigma)\\right\\| _{2}\\right) \\label{315}\n\\end{align}\nhold a.s. for all $\\tau,\\sigma,\\rho,\\varsigma\\in\\left[ 0,T\\right]$ with \n$\\tau>\\sigma>\\rho>\\varsigma$ and every $\\delta\\in\\left( \\frac{d}{d+2},1\\right)$.\n\\end{lemma}\n\\noindent\\textbf{Proof.} It follows from the same type of \narguments as those outlined in the\nproof of Lemma \\ref{l5}. For the proof of (\\ref{314}) the key estimate is (\\ref{15}). For (\\ref{315}), we also apply (\\ref{15}) along with\n(\\ref{44}). \n\n \\hfill $\\blacksquare$\n\\bigskip\n\nThe last relevant $L^{2}(D)$-estimate regarding (\\ref{309}) is then\nthe following.\n\\begin{lemma}\n\\label{l10}\nThe hypotheses are the same as in part (b) of Theorem \\ref{t1}. Then we\nhave\n\\begin{align*}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{0}^{\\sigma\n}\\left( f_{i,\\tau,\\sigma}^{\\ast}(u_M)(.,\\rho)-f_{i,\\tau,\\sigma\n}^{\\ast}(\\tilde{u}_M)(.,\\rho)\\right) B_{i}^{\\text{\\textsc{H}}\n}(d\\rho)\\right\\| _{2}\\\\\n& \\leq R (\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left(\n\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left( \\frac{1}\n{\\rho^{\\alpha}}+\\frac{1}{(\\sigma-\\rho)^{\\alpha}}\\right) \\left\\|\nz_M(.,\\rho)\\right\\| _{2}\\right. \\\\\n& +\\left. \\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\int\n_{0}^{\\rho}d\\varsigma\\frac{\\left\\| z_M(.,\\rho)-z_M\n(.,\\varsigma)\\right\\| _{2}}{\\left( \\rho-\\varsigma\\right) ^{\\alpha+1}\n}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[ 0,t\\right]$ with \n$\\tau>\\sigma$ and every $\\delta\\in\\left( \\frac{d}{d+2},1-2\\alpha\\right)$.\n\\end{lemma}\n\\noindent\\textbf{Proof.} By applying (\\ref{i}), \ntogether with (\\ref{314}), (\\ref{315}), we get\n\\begin{align*}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{0}^{\\sigma\n}\\left( f_{i,\\tau,\\sigma}^{\\ast}(u_M)(.,\\rho)-f_{i,\\tau,\\sigma\n}^{\\ast}(\\tilde{u}_M)(.,\\rho)\\right) B_{i}^{\\text{\\textsc{H}}\n}(d\\rho)\\right\\| _{2}\\\\\n& \\leq R (\\tau-\\sigma)^{\\frac{\\delta}{2}}\\left(\n\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\frac{\\left\\|\nz_M(.,\\rho)\\right\\| _{2}}{\\rho^{\\alpha}}\\right. \\\\\n& +\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{1}{2}}\\int_{0}^{\\rho\n}d\\varsigma(\\rho-\\varsigma)^{\\frac{1}{2}(1-\\delta)-\\alpha-1}\\left\\|\nz_M(.,\\varsigma)\\right\\| _{2}\\\\\n& +\\left. \\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\int\n_{0}^{\\rho}d\\varsigma\\frac{\\left\\| z_M(.,\\rho)-z_M(.,\\varsigma)\n\\right\\| _{2}}{\\left( \\rho-\\varsigma\\right) ^{\\alpha+1}\n}\\right)\n\\end{align*}\na.s. for all $\\sigma,\\tau\\in\\left[ 0,t\\right] $ with $\\tau>\\sigma$. But for every \n$\\delta\\in\\left( \\frac{d}{d+2},1-2\\alpha\\right) $, we have \n\\begin{align*}\n& \\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\frac{1}{2}}\\int_{0}^{\\rho}\nd \\varsigma(\\rho-\\varsigma)^{\\frac{1}{2}(1-\\delta)-\\alpha-1}\\left\\|\nz_M(.,\\varsigma)\\right\\| _{2}\\\\\n& =c\\int_{0}^{\\sigma}d\\rho(\\sigma-\\rho)^{-\\alpha-\\frac{\\delta}{2}}\\left\\|\nz_M(.,\\rho)\\right\\| _{2}.\n\\end{align*}\nThis yields the result.\n\n\\hfill$\\blacksquare$\n\\bigskip\n\nLet us go back to the inequality (\\ref{308}). Owing to (\\ref{309}) and by using the estimates\n(\\ref{310}), (\\ref{311}) together with Lemmas \\ref{l8} and \\ref{l10}, we have\n\\begin{equation}\n\\label{3160}\n\\Vert z_M\\Vert_{\\alpha,2,t}^2 \\le R \\int_0^t d\\tau \\left(\\sup_{\\rho\\in[0,\\tau]}\\Vert z_M(.,\\rho)\\Vert_2^2\n+\\sum_{k=1}^6 \\left[I_k(\\tau)\\right]^2\\right)\n\\end{equation}\na.s., where\n\\begin{align*}\nI_1(\\tau)&= \\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{\\frac{1}{2}+\\alpha}}\n\\left(\\int_\\sigma^\\tau d\\rho \\Vert z_M(.,\\rho)\\Vert_2^2\\right)^{\\frac{1}{2}},\\\\\nI_2(\\tau)&=\\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{-\\frac{\\delta}{2}+\\alpha+1}}\n\\left(\\int_0^\\sigma d\\rho (\\sigma-\\rho)^{-\\delta}\\Vert z_M(.,\\rho)\\Vert_2^2\\right)^{\\frac{1}{2}},\\\\\nI_3(\\tau)&= \\int_0^\\tau \\frac{d\\sigma }{(\\tau-\\sigma)^{\\alpha+1}}\n\\int_\\sigma^\\tau d\\rho\\left(\\frac{1}{(\\rho-\\sigma)^\\alpha}+\\frac{1}{(\\tau-\\rho)^\\alpha}\\right )\\Vert z_M(.,\\rho)\\Vert_2,\\\\\nI_4(\\tau)&=\\int_0^\\tau \\frac{d\\sigma }{(\\tau-\\sigma)^{\\alpha+1}}\n\\left(\\int_\\sigma^\\tau d\\rho \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right),\\\\\nI_5(\\tau)&=\\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{-\\frac{\\delta}{2}+\\alpha+1}}\n\\int_0^\\sigma \\frac{d\\rho}{(\\sigma-\\rho)^{\\frac{\\delta}{2}}}\\left(\\frac{1}{\\rho^\\alpha}+\\frac{1}{(\\sigma-\\rho)^\\alpha}\\right) \\Vert z_M(.,\\rho)\\Vert_2,\\\\\nI_6(\\tau)&=\\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^{-\\frac{\\delta}{2}+\\alpha+1}}\n\\int_0^\\sigma \\frac{d\\rho}{(\\sigma-\\rho)^{\\frac{\\delta}{2}}} \\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}.\\\\\n\\end{align*}\nSet $T_k(t)= \\int_0^t d\\tau\\ \\left[I_k(\\tau)\\right]^2$, $k=1, \\ldots, 6$.\n\nThe function $\\sigma\\mapsto (\\tau-\\sigma)^{-\\frac{1}{2}-\\alpha}$ is integrable on $(0,\\tau)$ for $\\alpha\\in(0,\\frac{1}{2})$. Thus\nwe have \n\\begin{equation} \n\\label{316}\nT_1(t)\\le c \\int_0^t d\\tau \\Vert z_M(.,\\tau)\\Vert_2^2.\n\\end{equation}\n Since we can choose $\\delta>2\\alpha$, we have that $\\sigma\\mapsto (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}$ is integrable on $(0,\\tau)$. Then, applying Schwarz inequality with respect to the measure \n given by $d\\sigma(\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}$, we obtain\n \\begin{align}\n \\label{317}\nT_2(t)&\\le c \\int_0^t d\\tau\\ \\int_0^\\tau d\\sigma\\ (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}\\left(\\int_0^\\sigma d\\rho\\ (\\sigma-\\rho)^{-\\delta}\n\\Vert z_M(.,\\rho)\\Vert_2^2\\right)\\nonumber\\\\\n&\\le c\\int_0^t d\\tau \\left(\\sup_{0\\le \\rho\\le \\tau}\\Vert z_M(\\cdot,\\rho\\Vert_2)^2\\right)\\left(\\int_0^\\tau d\\sigma (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}\\sigma^{1-\\delta}\\right)\\nonumber\\\\\n&\\le c\\int_0^t d\\tau \\left( \\sup_{0\\le \\rho\\le \\tau}\\Vert z_M(\\cdot,\\rho\\Vert_2)^2\\right),\n\\end{align}\nwhere in the last inequality we have used that $\\alpha+\\frac{\\delta}{2}<1$ along with the definition of Euler's Beta function.\n\nBy integrating one obtains\n\\begin{equation*}\n\\int_\\sigma^\\tau d\\rho\\left(\\frac{1}{(\\rho-\\sigma)^\\alpha}+\\frac{1}{(\\tau-\\rho)^\\alpha}\\right )= \\frac{2(\\tau-\\sigma)^{1-\\alpha}}{1-\\alpha}.\n\\end{equation*}\nMoreover, the function $\\sigma\\mapsto (\\tau-\\sigma)^{-2\\alpha}$ is integrable on $(0,\\tau)$. Consequently,\n\\begin{align}\n\\label{318}\nT_3(t)&\\le c \\int_0^t d\\tau \\left(\\sup_{\\rho\\in[0,\\tau]} \\Vert z_M(.,\\rho)\\Vert_2^2\\right) \\int_0^\\tau d\\sigma\\ (\\tau-\\sigma)^{-2\\alpha}\\nonumber\\\\\n &\\le c \\int_0^t d\\tau \\left( \\sup_{\\rho\\in[0,\\tau]} \\Vert z_M(.,\\rho)\\Vert_2^2\\right).\n\\end{align}\nFor any $\\tau\\in (0,t)$, set\n\\begin{equation*}\nI_\\tau= \\int_0^\\tau d\\sigma \\ (\\tau-\\sigma)^{-\\alpha-1+\\frac{\\delta}{2}}\\left(\\int_0^\\sigma d\\rho\\ (\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\left(\\frac{1}{\\rho^\\alpha}\n+\\frac{1}{(\\sigma-\\rho)^\\alpha}\\right)\\right).\n\\end{equation*}\nIt is a simple exercise to check that for $\\alpha+\\frac{\\delta}{2}<1$,\n$\\sup_{\\tau\\in[0,t]} I_\\tau < +\\infty$.\n\nSince\n\\begin{equation*}\nT_5(t) \\le \\int_0^t d\\tau\\ I_\\tau^2 \\left(\\sup_{\\rho\\in[0,\\tau]} \\Vert z_M(.,\\rho)\\Vert_2^2\\right),\n\\end{equation*}\nwe conclude that\n\\begin{equation}\n\\label{319}\nT_5(t) \\le c \\int_0^t d\\tau \\left(\\sup_{\\rho\\in[0,\\tau]} \\Vert z_M(.,\\rho)\\Vert_2^2\\right).\n\\end{equation}\n\n\nFix $\\eta\\in(0,1)$ so that $\\sigma\\mapsto (\\tau-\\sigma)^{-\\eta}$ is integrable on $(0,\\tau)$. Applying Schwarz inequality first with respect to the\nmeasure $d\\sigma (\\tau-\\sigma)^{-\\eta}$, and then with respect to the Lebesgue measure on the interval $(\\sigma,\\tau)$ yields\n\\begin{align*}\nT_4(t)&= \\int_0^t d\\tau \\Big( \\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^\\eta} (\\tau-\\sigma)^{-\\alpha-1+\\eta}\\\\\n&\\quad \\times \\Big(\\int_\\sigma^\\tau d\\rho \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\Big)\\Big)^2\\\\\n&\\le c \\int_0^t d\\tau\\ \\int_0^\\tau \\frac{d\\sigma}{(\\tau-\\sigma)^\\eta} (\\tau-\\sigma)^{-2\\alpha-2+2\\eta}\\\\\n&\\quad \\times \\left(\\int_\\sigma^\\tau d\\rho \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2\\\\\n&\\le c \\int_0^t d\\tau\\ \\int_0^\\tau d\\sigma\\ (\\tau-\\sigma)^{\\eta -2\\alpha-1}\\\\\n&\\quad \\times \\int_\\sigma^\\tau d\\rho \\left( \\int_\\sigma^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2.\n\\end{align*}\nBy choosing $\\eta>2\\alpha$, the function $\\sigma\\mapsto (\\tau-\\sigma)^{\\eta -2\\alpha-1}$ is integrable on $(0,\\tau)$. Thus, from the preceding inequalities\nwe obtain\n\\begin{align}\n\\label{320}\nT_4(t)&\\le c \\int_0^t d\\tau\\ \\int_0^\\tau d\\rho \\left( \\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2\\nonumber\\\\\n&\\le c \\int_0^t d\\tau\\ \\Vert z_M\\Vert_{\\alpha,2,\\tau}^2\\ .\n\\end{align}\nBy Fubini's theorem and evaluations based upon Euler's Beta function, we have\n\\begin{align}\n\\label{321}\nT_6(t)&= \\int_0^t d\\tau \\Big(\\int_0^\\tau d\\rho \\Big( \\int_\\rho^\\tau d\\sigma (\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}(\\sigma-\\rho)^{-\\frac{\\delta}{2}}\\Big)\\nonumber\\\\\n&\\quad\\times \\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\Big)^2\\nonumber\\\\\n&\\le c \\int_0^t d\\tau\\left(\\int_0^\\tau d\\rho \\left(\\int_0^\\rho d\\xi \\frac{\\Vert z_M(.,\\rho)-z_M(.,\\xi)\\Vert_2}{(\\rho-\\xi)^{\\alpha+1}}\\right)^2\\right)\\nonumber\\\\\n&\\le c \\int_0^t d\\tau\\ \\Vert z_M\\Vert_{\\alpha,2,\\tau}^2.\n\\end{align}\n\n\n\n\nFinally, inequalities (\\ref{3160}) to (\\ref{321}) imply\n\\begin{equation}\n\\label{322}\n\\Vert z_M\\Vert_{\\alpha,2,t}^2 \\le R \\int_0^t d\\tau\\ \\Vert z_M\\Vert_{\\alpha,2,\\tau}^2\\ \n\\end{equation}\na.s.\nBy Gronwall's lemma, this clearly implies the uniqueness of the mild solution.\nNow the proof of part (b) of Theorem \\ref{t1} is complete. \\hfill $\\blacksquare$\n\\bigskip\n\n\n\\noindent{\\bf Proof of Statement $(c)$ of Theorem \\ref{t1}}\n\\bigskip\n\nWe investigate\neach of the functions (\\ref{19})--(\\ref{21}) separately.\n\n\\begin{proposition}\n\\label{p2}\nAssume that Hypotheses ($K_{\\beta,\\beta^{\\prime}}$) and ($I$) hold. Then, there exists $c\\in(0,+\\infty)$ \nsuch that the estimate\n\\begin{equation}\n\\left\\|A(\\varphi)(.,t) -A(\\varphi)(.,s)\\right\\| _{2}\\leq\nc\\left| t-s\\right| ^{\\theta^{\\prime}} \\label{33}\n\\end{equation}\n\\textit{holds for all }$s,t\\in\\left[ 0,T\\right] $ and every $\\theta^{\\prime\n}\\in\\left( 0,\\frac{\\beta}{2}\\right] $.\n\\end{proposition}\n\n\\noindent\\textbf{Proof. }Relation (\\ref{19}) defines a classical solution to\n(\\ref{2}) when $g=h=0$, so that the standard regularity theory for\nlinear parabolic equations gives $(x,t)\\mapsto A(\\varphi)\n(x,t)\\in\\mathcal{C}^{\\beta,\\frac{\\beta}{2}}(\\overline{D}\\times\\left[\n0,T\\right] \\mathbb{)}$ (see, for instance, \\cite{eidelzhita}), from which\n(\\ref{33}) follows immediately. \\hfill $\\blacksquare$\n\n\\bigskip\n\nRegarding (\\ref{20}) we have the following result.\n\\begin{proposition}\n\\label{p3}\nAssume that the same hypotheses as in \nTheorem \\ref{t1} (a) hold and let $u_{M}$ be any mild solution to (\\ref{2}).\nThen, there exists $c\\in(0,+\\infty)$ such that the\nestimate\n\\begin{equation}\n\\left\\|B(u_{M})(.,t) - B(u_{M})(.,s)\\right\\| _{2}\\leq\nc\\left| t-s\\right| ^{\\theta^{\\prime\\prime}}\\left(1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right) \\label{34}\n\\end{equation}\nholds a.s. for all $s,t\\in\\left[ 0,T\\right]$ and every\n$\\theta^{\\prime\\prime}\\in\\left( 0,\\frac{1}{2}\\right)$.\n\\end{proposition}\n\n\\noindent\\textbf{Proof. } Without restricting the generality, we may assume that $t>s$.\nWe have\n\\begin{align}\n&B(u_{M})(.,t) - B(u_{M})(.,s)\n=\\int_{s}^{t}d\\tau\\int_{D}dy\\ G(.,t;y,\\tau)g\\left( u_{M}(y,\\tau)\\right)\n\\nonumber\\\\\n&\\quad \\quad +\\int_{0}^{s}d\\tau\\int_{D}dy\\left( G(.,t;y,\\tau)-G(.,s;y,\\tau)\\right)\ng\\left( u_{M}(y,\\tau)\\right), \\label{35}\n\\end{align}\nand remark that in order to keep track of the increment $t-s$ we can estimate\nthe first term on the right-hand side of (\\ref{35}) by using the same\nkind of arguments as we did in the first part of the proof of Lemma \\ref{l2}. For\nevery $x\\in D$ this gives\n\\begin{align*}\n& \\int_{s}^{t}d\\tau\\int_{D}dy\\ \\left| G(x,t;y,\\tau)g\\left( u_{M}\n(y,\\tau)\\right) \\right| \\\\\n& \\leq c(t-s)^{\\frac{1}{2}}\\left( \\int_{s}^{t}d\\tau\\int_{D}dy\\left|\nG(x,t;y,\\tau)\\right| \\left( 1+\\left| u_{M}(y,\\tau)\\right| ^{2}\\right)\n\\right) ^{\\frac{1}{2}},\n\\end{align*}\nso that we eventually obtain\n\\begin{equation}\n\\left\\| \\int_{s}^{t}d\\tau\\int_{D}dy\\ G(.,t;y,\\tau)g\\left( u_{M}(y,\\tau)\\right) \\right\\| _{2}\n\\leq c(t-s)^{\\frac{1}{2}}\\left( 1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right)\n\\label{36}\n\\end{equation}\na.s. for all $s,t\\in\\left[ 0,T\\right] $ with $t>s$. In \\ a similar manner,\nwe can keep track of the increment $t-s$ in the second term on the right-hand\nside of (\\ref{35}) by using (\\ref{15}). We thus have\n\\begin{align}\n& \\left\\| \\int_{0}^{s}d\\tau\\int_{D}dy\\left( G(.,t;y,\\tau)-G(.,s;y,\\tau\n)\\right) g\\left( u_{M}(y,\\tau)\\right) \\right\\| _{2}^{2}\\nonumber\\\\\n& \\leq c\\int_{0}^{s}d\\tau\\int_{D}dy\\int_{D}dx\\left| G(x,t;y,\\tau\n)-G(x,s;y,\\tau)\\right| \\left( 1+\\left| u_{M}(y,\\tau)\\right| ^{2}\\right)\\nonumber \\\\\n& \\leq c(t-s)^{\\delta}\\int_{0}^{s}d\\tau(s-\\tau)^{-\\delta}\\left( 1+\\left\\|\nu_{M}(.,\\tau)\\right\\| _{2}^{2}\\right)\\nonumber \\\\\n& \\leq c(t-s)^{\\delta}\\left( 1+\\sup_{t\\in[0,T]}\\left\\|u_{M}(.,t)\\right\\|_2^{2}\\right)\\label{37}\n\\end{align}\nfor every $\\delta\\in\\left( \\frac{d}{d+2},1\\right) $, \na.s. for all $s,t\\in\\left[ 0,T\\right] $ with $t>s$. This last relation holds\n\\textit{a fortiori} for each $\\delta\\in\\left( 0,1\\right)$, so that\n(\\ref{36}) and (\\ref{37}) indeed prove (\\ref{34}).\n\n \\hfill $\\blacksquare$\n\n\\bigskip\n\nAs for the stochastic term (\\ref{21}), we have the following.\n\n\\begin{proposition}\n\\label{p4}\nAssume the same hypotheses as in \nTheorem \\ref{t1} (c), and let $u_{M}$ be any mild solution to (\\ref{2}).\nThen, there exists $c\\in(0,+\\infty)$ such that the\nestimate\n\\begin{equation}\n\\left\\|C(u_{M})(.,t) - C(u_{M})(.,s)\\right\\| _{2}\\leq\ncr_{\\alpha}^{H}\\left| t-s\\right| ^{\\theta^{\\prime\n\\prime\\prime}}\\left( 1+\\left\\| u_{M}\\right\\|_{\\alpha,2,T}\\right)\n\\label{38}\n\\end{equation}\nholds a.s. for all $s,t\\in\\left[ 0,T\\right]$ and every\n$\\theta^{\\prime\\prime\\prime}\\in\\left( 0,\\frac{1}{2}-\\alpha\\right)$.\n\\end{proposition}\n\nThe proof of Proposition \\ref{p4} is more complicated than that of Proposition \\ref{p3}.\nWe begin with a\npreparatory result whose proof is based on inequalities\n(\\ref{15})--(\\ref{17}).\n\n\n\\begin{lemma}\n\\label{l3}\nWith the same hypotheses as in part (a) of Theorem \\ref{t1}, \n the estimates\n\\begin{equation}\n\\sup_{i\\in \\mathbb{N}^+}\\left\\| f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)\\right\\| _{2}\\leq c\\left(\nt-s\\right) ^{\\frac{\\delta}{2}}\\left( s-\\tau\\right) ^{-\\frac{\\delta}{2}\n}\\left(1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right) \\label{40}\n\\end{equation}\n\\textit{and }\n\\begin{align}\n& \\sup_{i\\in\\mathbb{N}^+}\\left\\| f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)-f_{i,t,s}^{\\ast}(u_{M}\n)(.,\\sigma)\\right\\| _{2}\\nonumber\\\\\n& \\leq c\\left( t-s\\right) ^{\\frac{\\delta}{2}}\\left( s-\\tau\\right)\n^{-\\frac{\\delta}{2}}\\left\\| u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|\n_{2}\\nonumber\\\\\n& +c\\left( t-s\\right) ^{\\frac{\\delta}{2}}\\left( s-\\tau\\right) ^{-\\frac\n{1}{2}}\\left( \\tau-\\sigma\\right) ^{\\frac{1}{2}(1-\\delta)}\\left(1+\\sup_{t\\in[0,T]}\\left\\|u_{M}(.,t)\\right\\|_2\\right) \\label{41}\n\\end{align}\nhold a.s. for every $\\delta\\in\\left( \\frac{d}{d+2},1\\right)$ and for all $\\sigma,\\tau\\in\\left[ 0,s\\right)$ with $\\tau>\\sigma$ in\n(\\ref{41}).\n\\end{lemma}\n\n\\noindent\\textbf{Proof. }The proof of (\\ref{40}) is analogous to that of\n(\\ref{25}) and is thereby omitted. As for (\\ref{41}), by using\n Schwarz inequality relative to the measures $dy\\left|\nG(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|$ and\n\\[\ndy\\left| G(x,t;y,\\tau)-G(x,s;y,\\tau)-G(x,t;y,\\sigma)+G(x,s;y,\\sigma)\\right|\n\\]\non $D$ along with Hypothesis (L) for $h$, we get\n\\begin{align}\n& \\left\\| f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)-f_{i,t,s}^{\\ast}(u_{M}\n)(.,\\sigma)\\right\\| _{2}^{2}\\nonumber\\\\\n& \\leq c\\int_{D}dx\\int_{D}dy\\left| G(x,t;y,\\tau)-G(x,s;y,\\tau)\\right|\n\\left| u_{M}(y,\\tau)-u_{M}(y,\\sigma)\\right| ^{2}\\nonumber\\\\\n& +c\\int_{D}dx\\int_{D}dy\\left| G(x,t;y,\\tau)-G(x,s;y,\\tau)-G(x,t;y,\\sigma\n)+G(x,s;y,\\sigma)\\right|\\nonumber\\\\\n&\\quad\\quad \\times \\left( 1+\\left| u_{M}(y,\\sigma)\\right|\n^{2}\\right) \\nonumber\\\\\n& \\leq c\\left( t-s\\right) ^{\\delta}\\left( s-\\tau\\right) ^{-\\delta\n}\\left\\| u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\| _{2}^{2}\\nonumber\\\\\n&\\leq c\\left( t-s\\right) ^{\\delta}\\left( s-\\tau\\right) ^{-1}\\left(\n\\tau-\\sigma\\right) ^{1-\\delta}\\left(1+\\sup_{t\\in[0,T]}\\left\\|u_{M}(.,t)\\right\\|_2^2\\right), \\label{42}\n\\end{align}\na.s. for all $s,t,\\sigma,\\tau\\in\\left[ 0,T\\right] $ with $t\\geq\ns>\\tau>\\sigma$ and every $\\delta\\in\\left( \\frac{d}{d+2},1\\right) $, as a\nconsequence of (\\ref{15}), (\\ref{44}) and the Gaussian property.\n\n\n\\hfill $\\blacksquare$\n\\bigskip\n\n\\noindent\\textbf{Proof of Proposition \\ref{p4}.} For $t>s$ we write\n\\begin{align}\nC(u_{M})(.,t) - C(u_{M})(.,s) &= \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{s}^{t}f_{i,t}\n(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\nonumber\\\\\n& +\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}^{s}f_{i,t,s}^{\\ast\n}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau). \\label{45}\n\\end{align}\nIn order to estimate the first\nterm on the right-hand side of\n(\\ref{45}), we can start by using inequalities (\\ref{24}) and\n(\\ref{25}) to obtain\n\\begin{align}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{s}^{t}\nf_{i,t}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\| _{2}\n\\nonumber\\\\\n& \\leq cr_{\\alpha}^{H}\\left( \\int_{s}^{t}\\frac{d\\tau}\n{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\frac{\\left\\| u_{M}(.,\\tau)\\right\\|\n_{2}^{{}}}{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\int_{s}^{\\tau}d\\sigma\n\\frac{\\left\\| u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\| _{2}^{{}}}{\\left(\n\\tau-\\sigma\\right) ^{\\alpha+1}}\\right. \\nonumber\\\\\n& +\\left. \\int_{s}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{s}^{\\tau\n}d\\sigma(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left( 1+\\left\\|\nu_{M}(.,\\sigma)\\right\\| _{2}\\right) \\right) \\label{46}\n\\end{align}\na.s. for every $s,t$ $\\in\\left[ 0,T\\right] $ with $t>s$ and each $\\delta\n\\in\\left( \\frac{d}{d+2},1\\right)$.\n\nFurthermore, we have\n\\begin{align}\n& \\int_{s}^{t}\\frac{d\\tau}{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\frac{\\left\\|\nu_{M}(.,\\tau)\\right\\| _{2}^{{}}}{(\\tau-s)^{\\alpha}}+\\int_{s}^{t}d\\tau\\int\n_{s}^{\\tau}d\\sigma\\frac{\\left\\| u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|\n_{2}^{{}}}{\\left( \\tau-\\sigma\\right) ^{\\alpha+1}}\\nonumber\\\\\n& \\leq c\\left( \\left( t-s\\right) ^{1-\\alpha}\\left( 1+\\left\\|\nu_{M}\\right\\| _{\\alpha,2,T}\\right) +(t-s)^{\\frac{1}{2}}\\left\\|\nu_{M}\\right\\| _{\\alpha,2,T}\\right) \\nonumber\\\\\n& \\leq c(t-s)^{\\frac{1}{2}}\\left( 1+\\left\\| u_{M}\\right\\|_{\\alpha,2,T}\\right) \\label{47}\n\\end{align}\nsince $\\alpha<\\frac{1}{2}$. Moreover,\n\\begin{align}\n& \\int_{s}^{t}d\\tau(t-\\tau)^{-\\frac{\\delta}{2}}\\int_{s}^{\\tau}d\\sigma\n(\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left( 1+\\left\\| u_{M}\n(.,\\sigma)\\right\\| _{2}\\right) \\nonumber\\\\\n& \\le c\\ \\left(1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right)\\int_{s}^{t}d\\tau\\ (t-\\tau)^{-\\frac{\\delta}{2}}\\int_s^\\tau d\\sigma\\ \\left(\\tau-\\sigma\\right)\n^{\\frac{\\delta}{2}-\\alpha-1} \\nonumber\\\\\n& \\leq c(t-s)^{1-\\alpha}\\left(1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right), \\label{48}\n\\end{align}\nby virtue of the convergence of the\nintegral, which can be expressed in terms of Euler's Beta function \nsince $\\alpha<\\frac{\\delta}{2}$. The substitution of (\\ref{47}) and (\\ref{48}) into\n(\\ref{46}) then leads to the inequality\n\\begin{equation}\n\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{s}^{t}\nf_{i,t}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\| _{2}\\leq\ncr_{\\alpha}^{H}(t-s)^{\\frac{1}{2}}\\left( 1+\\left\\|\nu_{M}\\right\\|_{\\alpha,2,T}\\right) \\label{hoelder8}\n\\end{equation}\na.s. for every $s,t$ $\\in\\left[ 0,T\\right] $ with $t>s$. \n\nIt remains to\nestimate the second term on the right-hand side of (\\ref{45}). \nFrom (\\ref{i}) with $T$ replaced by $s$, we have\n\\begin{align*}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{0}^{s}\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|\n_{2}\n\\leq r_\\alpha^H\\\\\n&\\times\\sup_{i\\in\\mathbb{N}^+}\\int_{0}^{s}d\\tau\\left( \\frac{\\left\\|\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)\\right\\| _{2}}{\\tau^{\\alpha}}+\\int\n_{0}^{\\tau}d\\sigma\\frac{\\left\\| f_{i,t,s}^{\\ast}(u_{M})(.,\\tau)-f_{i,t,s}\n^{\\ast}(u_{M})(.,\\sigma)\\right\\| _{2}}{\\left( \\tau-\\sigma\\right)\n^{\\alpha+1}}\\right).\n\\end{align*}\n\nBy substituting (\\ref{40}) and (\\ref{41}) we obtain\n\\begin{align}\n& \\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{0}^{s}\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|\n_{2}\\nonumber\\\\\n& \\leq cr_{\\alpha}^{H}(t-s)^{\\frac{\\delta}{2}}\\left(\n\\int_{0}^{s}d\\tau\\left( s-\\tau\\right) ^{-\\frac{\\delta}{2}}\\tau^{-\\alpha\n}\\left(1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right) \\right.\n\\nonumber\\\\\n& +\\int_{0}^{s}d\\tau\\left( s-\\tau\\right) ^{-\\frac{\\delta}{2}}\\int_{0}\n^{\\tau}d\\sigma\\frac{\\left\\| u_{M}(.,\\tau)-u_{M}(.,\\sigma)\\right\\|_{2}\n}{\\left( \\tau-\\sigma\\right) ^{\\alpha+1}}\\nonumber\\\\\n& +\\left. \\int_{0}^{s}d\\tau\\left( s-\\tau\\right) ^{-\\frac{1}{2}}\\int\n_{0}^{\\tau}d\\sigma\\left( \\tau-\\sigma\\right) ^{\\frac{1}{2}\\left(\n1-\\delta\\right) -\\alpha-1}\\left(1+\\sup_{t\\in[0,T]}\\left\\| u_{M}(.,t)\\right\\|_2\\right) \\right) \\nonumber\\\\\n& \\leq c\\ r_{\\alpha}^{H}(t-s)^{\\frac{\\delta}{2}}\\left(1+\\left\\| u_{M}\\right\\|_{\\alpha,2,T}\\right)\\nonumber\\\\\n&\\quad\\times \\left(1+\\int_{0}^{s}d\\tau\\left( s-\\tau\\right) ^{-\\frac{1}{2}}\\int_{0}^{\\tau}d\\sigma\n\\left( \\tau-\\sigma\\right) ^{\\frac{1}{2}\\left( 1-\\delta\\right)-\\alpha-1}\\right),\\label{49}\n\\end{align}\nwhere we have got the last estimate using Schwarz inequality with\nrespect to the measure $d\\tau$ on $\\left( 0,s\\right)$ along with\n(\\ref{3}) in the first two integrals on the right-hand side.\n\nBy imposing the additional restriction $\\delta<1-2\\alpha$, we have\n\\begin{equation*}\n\\int_{0}^{s}d\\tau\\left( s-\\tau\\right) ^{-\\frac{1}{2}}\\int_{0}^{\\tau}d\\sigma\n\\left( \\tau-\\sigma\\right) ^{\\frac{1}{2}\\left( 1-\\delta\\right)-\\alpha-1}\n<+\\infty.\n\\end{equation*}\nThus, we have proved that\n\\begin{equation}\n\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\left\\| \\int_{0}^{s}\nf_{i,t,s}^{\\ast}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\right\\|\n_{2}\\leq cr_{\\alpha}^{H}(t-s)^{\\frac{\\delta}{2}}\\left(\n1+\\left\\| u_{M}\\right\\|_{\\alpha,2,T}\\right) \\label{50}\n\\end{equation}\na.s. for all $s,t$ $\\in\\left[ 0,T\\right] $ with $t>s$ and every $\\delta\n\\in\\left( \\frac{d}{d+2},1-2\\alpha\\right)$. The existence of this restricted\ninterval of values of $\\delta$ is possible by our choice of $\\alpha$. Relations\n(\\ref{45}), (\\ref{hoelder8}) and (\\ref{50}) clearly \nyield (\\ref{38}) with \n$\\theta^{\\prime\\prime\\prime}\n=\\frac{\\delta}{2}\\in\\left( 0,\\frac{1}{2}-\\alpha\\right) $.\n \\hfill $\\blacksquare$\n\n\\bigskip\n\nIt is immediate that Propositions \\ref{p2} to \\ref{p4} imply statement $(c)$ of Theorem \\ref{t1}.\nNotice that $R_\\alpha^H = c(1+r_\\alpha^H)$, with $r_\\alpha^H$ defined in (\\ref{29}).\n\\bigskip\n\nFinally, we give an alternate to the result proved before, as mentioned in Section \\ref{s2}, Remark 5.\n\n\\begin{proposition}\n\\label{pfact}\nThe assumptions are as in Theorem \\ref{t1} part (a). Then,\n\\begin{equation}\n\\Vert C(u_{M})(.,t)-C(u_{M})(.,s)\\Vert_{2}\\leq R |t-s|^{\\theta^{\\prime\\prime\\prime\\prime}}\n\\left(1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right)\n\\label{323}\n\\end{equation}\nholds a.s. for all $s,t\\in\\lbrack0,T]$ and every \n$\\theta^{\\prime\\prime\\prime\\prime}\\in\\left( 0,\\frac{2}{d+2}\\wedge\\frac{1}{2}\\right)$. Consequently, \n\\begin{equation}\n\\label{3231}\n\\Vert u_{M})(.,t)-u_{M})(.,s)\\Vert_2\\le R |t-s|^\\theta \\left(1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right),\n\\end{equation}\na.s. for all $s,t\\in\\lbrack0,T]$ and each $\\theta\\in\\left(0,\\frac{2}{d+2}\\wedge\\frac{\\beta}{2}\\right)$.\n\\end{proposition}\n{\\bf Proof.} \nWe use the factorization method we alluded to in Section \\ref{s2}. For this we express \n$C(u_{M})(.,t)$ in terms of the auxiliary $L^{2}(D)$-valued process\n\\begin{equation*}\nY_{\\varepsilon}(u_{M})(.,t):=\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}%\n\\int_{0}^{t}(t-\\tau)^{-\\varepsilon}f_{i,t}(u_{M})(.,\\tau)B_{i}^H(d\\tau)\n\\end{equation*}\ndefined for every $\\varepsilon\\in\\left( 0,\\frac{1}{2}\\right) $. In fact, by\nrepeated applications of Fubini's theorem and by using the fundamental\nproperty $U(t,\\tau)U(\\tau,\\sigma)=U(t,\\sigma)$ for the evolution operators\ndefined in (\\ref{30}) we obtain\n\\begin{align}\nC(u_{M})(.,t) & =\\sum_{i=1}^{+\\infty}\\lambda_{i}^{\\frac{1}{2}}\\int_{0}%\n^{t}f_{i,t}(u_{M})(.,\\tau)B_{i}^{H}(d\\tau)\\nonumber\\\\\n& =\\frac{\\sin(\\varepsilon\\pi)}{\\pi}\\int_{0}^{t}d\\tau(t-\\tau)^{\\varepsilon\n-1}\\int_{D}dyG(.,t;y,\\tau)Y_{\\varepsilon}(u_{M})(y,\\tau) \\label{factorization}%\n\\end{align}\nfor every $t\\in\\left[ 0,T\\right]$ a.s. \n\nNext we prove that a.s., \n\\begin{equation}\n\\sup_{t\\in\\lbrack0,T]}\\Vert Y_{\\varepsilon}(u_{M})(.,t)\\Vert_{2}\\leq R\\left(\n1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right). \\label{324}\n\\end{equation}\nIndeed, according with (\\ref{i}),\n\\begin{align*}\n&\\Vert Y_{\\varepsilon}(u_{M})(.,t)\\Vert_{2}\\le r_\\alpha^H \\sup_{i\\in\\mathbb{N}^+} \\int_0^t d\\tau (t-\\tau)^{-\\varepsilon}\\\\\n&\\quad\\times\\left\\{\\frac{\\Vert f_{i,t}(u)(.,\\tau)\\Vert_2}{\\tau^\\alpha}+\\int_0^\\tau d\\sigma \\frac{\\Vert f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\Vert_2 }{(\\tau-\\sigma)^{\\alpha+1}}\\right\\}.\n\\end{align*}\nFrom the estimate (\\ref{24}) and the definition of the Beta function, we have\n\\begin{equation*}\n\\int_0^t d\\tau (t-\\tau)^{-\\varepsilon}\\frac{\\Vert f_{i,t}(u)(.,\\tau)\\Vert_2}{\\tau^\\alpha}\\le c\\left(1+\\sup_{\\tau\\in[0,t]} \\Vert u(.,\\tau)\\Vert_2\\right).\n\\end{equation*}\nSince $\\varepsilon\\in(0,\\frac{1}{2})$, applying Schwarz's inequality yields\n\\begin{align*}\n&\\int_0^t d\\tau (t-\\tau)^{-\\varepsilon}\\int_0^\\tau \\frac{\\Vert u(.,\\tau)-u(.,\\sigma)\\Vert_2}{(\\tau-\\sigma)^{\\alpha+1}}\\\\\n&\\quad\\le \\left[\\int_0^t d\\tau\\ (t-\\tau)^{-2\\varepsilon}\\right]^{\\frac{1}{2}} \\left[\\int_0^t d\\tau\\ \\left(\\int_0^\\tau d\\sigma \\frac{\\Vert u(.,\\tau)-u(.,\\sigma)\\Vert_2}{(\\tau-\\sigma)^{\\alpha+1}}\\right)^2\\right]^{\\frac{1}{2}}\\\\\n&\\quad\\le c \\Vert u\\Vert_{\\alpha,2,t}.\n\\end{align*}\nMoreover, for any $\\delta\\in(0,\\frac{1}{2})$ such that $\\frac{\\delta}{2}-\\alpha>0$, we have\n\\begin{equation*}\n\\int_0^t d\\tau (t-\\tau)^{-\\varepsilon-\\frac{\\delta}{2}} \\int_0^\\tau d\\sigma (\\tau-\\sigma)^{\\frac{\\delta}{2}-\\alpha-1}\\left(1+\\Vert u(.,\\sigma)\\Vert_2\\right)\n\\le c\\left(1+\\sup_{\\sigma\\in[0,t]}\\Vert u(.,\\sigma)\\Vert_2\\right).\n\\end{equation*}\nBy virtue of (\\ref{25}) the two above estimates imply\n\\begin{align*}\n\\int_0^t d\\tau\\int_0^\\tau d\\sigma (t-\\tau)^{-\\varepsilon}\\frac{\\Vert f_{i,t}(u)(.,\\tau)-f_{i,t}(u)(.,\\sigma)\\Vert_2 }{(\\tau-\\sigma)^{\\alpha+1}}\n\\le c \\left(1+\\Vert u\\Vert_{\\alpha,2,t}\\right).\n\\end{align*}\nThis ends the proof of (\\ref{324}).\n\\smallskip\n\n\n We can now proceed by estimating the\ntime increments of $C(u_{M})$ using (\\ref{factorization}) and (\\ref{324}) rather than\nwith the expressions of Proposition \\ref{p4}. For this\nwe follow the arguments of the proof of (66) in Proposition 6 of \\cite{sansovu1}\nto see that, by choosing\n $\\theta^{\\prime\\prime\\prime\\prime}\\in\\left(0,\\frac{2}{d+2}\\wedge\\frac{1}{2}\\right)$ with the additional restriction\n$\\varepsilon\\in\\left( \\theta^{\\prime\\prime\\prime\\prime},\\frac{2}{d+2}%\n\\wedge\\frac{1}{2}\\right)$, we obtain\n\\begin{align*}\n& \\left\\Vert C(u_{M})(.,t)-C(u_{M})(.,s)\\right\\Vert _{2}\\\\\n& \\leq c\\left(\\left\\Vert \\int_{s}^{t}d\\tau(t-\\tau)^{\\varepsilon-1}\\int\n_{D}dyG(.,t;y,\\tau)Y_{\\varepsilon}(u_{M})(y,\\tau)\\right\\Vert _{2}\\right. \\\\\n& +\\left. \\left\\Vert \\int_{0}^{s}d\\tau\\int_{D}dy\\left( (t-\\tau\n)^{\\varepsilon-1}G(.,t;y,\\tau)-(s-\\tau)^{\\varepsilon-1}G(.,s;y,\\tau)\\right)\nY_{\\varepsilon}(u_{M})(y,\\tau)\\right\\Vert _{2}\\right) \\\\\n& \\leq R\\left( \\left\\vert t-s\\right\\vert ^{\\varepsilon}+\\left\\vert\nt-s\\right\\vert ^{\\theta^{\\prime\\prime\\prime\\prime}}\\right) (1+\\Vert\nu_{M}\\Vert_{\\alpha,2,T})\n\\leq R |t-s|^{\\theta^{\\prime\\prime\\prime\\prime}}\\left(\n1+\\Vert u_{M}\\Vert_{\\alpha,2,T}\\right),\n\\end{align*}\nproving (\\ref{323}).\n\nFinally, this estimate along with those established in Propositions \\ref{p2} and \\ref{p3} provide (\\ref{3231}) and finish the proof of\nthe proposition.\n\n\\hfill $\\blacksquare$\n\n\n\n\n\\bigskip\n\n\\noindent\\textbf{Acknowledgements.}\nThe research of the first author concerning this paper was completed at the Institute Mittag-Leffler\nin Djursholm. \nThe research of the second author was supported in part by the \nInstitute of Mathematics of the University of Barcelona where this work was begun, and in\npart by the ETH-Institute of Theoretical Physics in Zurich. They would like to\nthank the three institutions for their very kind hospitality.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\label{sec:Intro}\n\nThe global star formation rate (SFR) of large galaxy samples at intermediate redshift has been an area of active study for the past decade \\citep{Steidel99, Hopkins06, Zhu09, Gilbank10}. Each SFR study has chosen a particular star-formation rate indicator (or combination of indicators) from a variety of possible measures. The indicators associated with star formation cover a broad range of wavelengths of the electromagnetic spectrum from x-ray to radio. The most studied measures include UV luminosity, which indicates the amount of massive star formation within the composite stellar population of a galaxy \\citep{Heckman98}, and the luminosity of nebular emission lines such as H$\\alpha$ and [\\ion{O}{2}], which measure the amount of ionizing radiation from massive stars \\citep[hearafter M06]{Kennicutt98, Kewley04, Moustakas06}. Additionally, deep observations from \\emph{Spitzer} have tied the mid-IR continuum emission at $24\\micron$ to compact star formation regions in local galaxies \\citep{Calzetti07, Kennicutt09}.\n\nAll SFR diagnostics essentially measure the energy output of young stars either through direct observation of the UV flux or indirect observation of the ionizing luminosity through nebular emission lines or reprocessed emission from dust. Regardless of the diagnostic used, the SFR indicators have systematic effects which must be removed in order to give an accurate measure of the SFR. The primary physical effect that dominates systematic uncertainty in restframe UV\/optical observations is the amount of dust extinction in the host galaxy. Galaxy dust extinction attenuates the UV continuum or emission line luminosity and biases the indicators to lower SFRs \\citep{Kennicutt98, Kewley02, Kewley04}. Further, the amount of dust extinction within the star-forming galaxy population also correlates with galaxy luminosity \\citep{WH96}, stellar mass \\citep{Brinchmann04, Noeske07b, Elbaz07, Gilbank10}, and star-formation history \\citep{Noeske07a}, presumably due to increased metallicity and dust production from massive stars and mass-loss return to their ISMs. Producing accurate dust-corrected SFRs from UV continuum or nebular emission lines requires at minimum a measure of the dust extinction through diagnostics such as the UV\/optical slope \\citep{Calzetti94, Kong04} or the Balmer decrement (H$\\alpha$\/H$\\beta$) assuming case B recombination for typical galaxy environments \\citep{Calzetti01}. \n\nAdditional systematic uncertainties can also limit the SFR accuracy after dust extinction has been removed. For example, emission line luminosities from [\\ion{O}{2}] and [\\ion{O}{3}] depend on other nebular characteristics such as the chemical abundance and excitation states of the gas \\citep{KD02}. M06 estimates that even after an ideal Balmer-decrement dust correction is applied, [\\ion{O}{2}]-based SFRs have a lower error limit of 32\\% for galaxies with typical metallicities and can have considerably larger error for more extreme abundances. Further, the gas can be excited from sources that are not associated with actual star formation, such as the hard-ionizing background radiation created by AGN, which can be a dominant effect in quiescent galaxies \\citep{Yan06}. \n\nBeyond the UV and optical regime, IR measurements can be used to trace galaxy SFR \\citep{Kennicutt98, Daddi07}; UV photons from massive stars are absorbed by dust within the host galaxy ISM and re-emmited at IR wavelengths. IR-based SFRs are most accurate for heavily attenuated environments such as young starbursts where dust-corrections and measurements of the UV luminosity are difficult. However, IR observations tend to underestimate the dust luminosity in normal spirals where much of the UV radiation is unabsorbed. IR luminosities can also have systematic variations due to differing dust geometries in the host galaxy or contamination from old stellar populations, particularly in red-sequence galaxies \\citep[hereafter S09]{Salim09}. Nevertheless, recent work has calibrated the SFR to IR luminosity in various bands for these dusty galaxies \\citep{Zhu08, Calzetti09, Rieke09} and calibrated the $24\\micron$ luminosity to H$\\alpha$ luminosity to provide a measure of both the SFR and account for typical dust attenuation encountered across a broad range of blue galaxies \\citep{Kennicutt09}. \n\nOne way to mitigate these various systematic effects is to obtain integrated spectral and photometric measurements over a wide wavelength range, providing constraints on the dust extinction and independent cross-checks in SFR. In S09, galaxy template SEDs were fit to measurements spanning FUV to $K$-band wavelengths available from the All-Wavelength Extended Groth Strip International Survey (AEGIS) \\citep{Davis07}. The S09 SFRs were based primarily on measurements of the UV luminosity and are combined with a Baysian analysis of stellar population synthesis models to fit the template SEDs. By simultaneously fitting across the panchromatic data and correcting for various astrophysical effects such as dust extinction, UV upturn in red galaxies, and SFR timescales, the dust-corrected SFRs are robust against large systematic errors that would typically limit SFR measurements at intermediate redshifts across galaxy types. We consider the SED SFRs fit to UV\/optical measurements to be the least biased measure of SFR in red galaxies, particularly when the UV upturn is taken into account \\citep{Han07}. These SED-based SFRs provide a reference for the SFR calibration presented here and will be discussed further in Section~\\ref{sec:Datasets}.\n\nThe goal of this work is to take advantage of these robust SED-fit SFRs and provide a SFR calibration for optically-selected galaxy samples where wide multi-wavelength coverage or emission line data may not be available. Specifically, we wish to develop a suitable calibration using SFR data in the AEGIS field and extend that calibration to $z\\sim1$ galaxies in the DEEP2 Redshift Survey (Davis, 2003) outside the AEGIS field. Because the SFR calibration is based on a large galaxy sample, the final means-tested SFR calibration will not be highly accurate for individual galaxies. However, it will reproduce the global SFR trends seen in the $z\\sim1$ AEGIS sample and allow us to confidently study the galaxy environment at these redshifts with respect to SFR. In \\S\\ref{sec:Datasets} we describe the volume-limited galaxy sample selected from the DEEP2 survey that is matched to the S09 AEGIS SFRs. In \\S\\ref{sec:SFRcorr}, we investigate which combination of DEEP2 measurements, observed [\\ion{O}{2}] luminosity, restframe $B$-band luminosity, or restframe color, that are most correlated with SFR and will therefore provide the best leverage in calibration. Our SFR model for these tests uses a weighted linear combination of observed parameters (not corrected for dust extinction), and we investigate which parameter combination delivers the most accurate SFR calibration for galaxies with both red and blue optical restframe colors (\\S\\ref{sec:MBUBcal} and \\S\\ref{sec:RedBlueCal}). In \\S\\ref{sec:OIISFRs}, we describe two empirical SFR estimators which use local [\\ion{O}{2}] luminosity measurements and correct the SFRs for systematic effects via M$_B$ or stellar mass. We then compare these diagnostics to our best-fit SFR calibration at intermediate redshifts, and we present our conclusions in \\S\\ref{sec:Conclusions}. Throughout this work, we adjust all masses and SFRs to a \\citet{Salpeter55} IMF and assume a $\\Lambda$CDM cosmology ($\\Omega_{M}=0.3$, $\\Omega_{\\Lambda}=0.7$, $h=0.7$). All magnitudes used in this study are in the AB system.\n\n\\section{Datasets}\n\\label{sec:Datasets}\n\nThe main sample for this work is drawn from the DEEP2 survey \\citep{Davis03} where the [\\ion{O}{2}] doublet emission lines have been measured with the DEIMOS spectrograph on Keck \\citep{Faber03}. The DEEP2 survey contains four fields that are separated across the Northern sky for year-round observation, and one of the fields (Field 1) overlaps the AEGIS survey in the Extended Groth Strip (EGS). Within the EGS, DEEP2 targets all galaxies to R$_{AB}<24.1$ and is limited to $z<1.4$, beyond which the [\\ion{O}{2}] emission line falls outside the observed wavelength range (6500-9200\\AA). In the non-EGS fields, targets selected by DEEP2 are limited by design at a lower redshift of $z>0.7$ due to an optical $BRI$ color cut. The total DEEP2 sample with good redshifts ($>95\\%$ confidence, $z$ quality flag $>3$) is 31,656 galaxies. When available, the emission line equivalent widths in DEEP2 are measured using a nonlinear least-squares fit of a Gaussian to the emission line profiles. A line flux is then computed by measuring the continuum in a 20-60\\AA\\ window around the line and calibrating the flux to $K$-corrected $RI$ photometry. This method takes into account both the throughput and slit losses for DEEP2 galaxies of which nearly all have an effective radius less than the 1\" DEIMOS slit width (Weiner et. al, 2012, in preparation; description of method in \\citealp{Weiner07, Zhu09}).\n\nTo build a sample of $z\\sim1$ SFRs, we match the DEEP2 galaxy sample to available panchromatic data from the AEGIS survey. The matched AEGIS sample contains a total of 5345 objects and is a subset of the total available DEEP2 spectra in the EGS field. From this sample, we obtained SFR estimates from template SEDs which are fit to UV, optical, and IR $K$-band photometry in S09. The composite templates are constructed from stellar population synthesis models of \\citet{BC03} augmented with a two-component dust attenuation model of \\citet{CF00} to account for dust associated with various star-formation histories and the intergalactic medium. The templates used in S09 also take into account extreme blue horizontal branch stars which are thought to be responsible for the ``UV upturn\" seen in some early-type galaxies with old stellar populations \\citep[see][for details of the SED models]{Salim07}. For 45\\% of the S09 sample, the SED-fit SFRs have been compared with $24\\micron$ measurements for both red sequence and blue cloud galaxies. The total IR luminosities are inferred from the $24\\micron$ flux densities and galaxy redshifts by fitting to IR SED templates \\citep{DH02}. S09 found that SED SFRs of the blue galaxy population (NUV-R$<3$) were consistent with the measured IR luminosities to 0.3 dex RMS. The SFRs for red galaxies (NUV-R$>5$) were less well matched in IR luminosity due to the increasing contribution of light from old stellar populations in the IR measurements. Because S09 accounts for the latter systematic effect though template fitting, the S09 SFRs based on UV SEDs are a better estimate of the true SFR than the $24\\micron$ measurements in red galaxies. \n\nThe stellar masses for our galaxy sample are computed from the \\citet{Bell03} color-M\/L relations, modified to DEEP2 redshifts through the \\citet{Weiner09} calibration. The color-M\/L calibration is a linear set of equations that require several measured quantities, include M$_{B}$, restframe (U-B) and (B-V) colors, and galaxy redshift. Independently, we confirm that our calculated stellar masses from these relations agree with stellar masses derived from $K$-band measurements \\citep{Bundy06} to 0.3 dex RMS with no systematic offset. As an additional consistency check, we find that the color-M\/L stellar masses agree with those generated by the more sophisticated S09 SED-fitting techniques to within 0.3 dex. We use stellar masses from the M\/L-color calibration rather than stellar masses obtained from the SED fitting because we want the input parameters in our SFR calibration to be easily reproducible and independent from the SED fitting that yields our fiducial SFRs. \n\n\\subsection{Sample selection}\nTo produce a statistical sample volume-limited in M$_{B}$ for both red and blue galaxies, we restrict the matched DEEP2 \/ AEGIS galaxy sample to M$_{B}<-20$ for $0.74-1.0$ and [\\ion{O}{2}] line luminosity limit of log(L[\\ion{O}{2}])$>39.7$ at $z\\sim1$.\\footnote{The SFR $\\psi$ is in units of M$_{\\sun}$ yr$^{-1}$, and L[\\ion{O}{2}] is in ergs s$^{-1}$ throughout this study.}\n\nWhile blue galaxies in our sample generally have well detected L[\\ion{O}{2}] measurements ($>3\\sigma$), a significant fraction of the red galaxies with L[\\ion{O}{2}] measurements have large error due to inherently lower line luminosities. Restricting the L[\\ion{O}{2}] measurements to better than 3$\\sigma$ in DEEP2\nreduces the sample to 80\\% of the volume-limited blue galaxies and 27\\% of red galaxies. For comparison, \\citet{Yan06} found that 35\\% of all volume-limited red galaxies in SDSS had [\\ion{O}{2}] detected at a $3\\sigma$ level in EW. Because we are interested in testing SFR calibrations for a uniform sample with both L[\\ion{O}{2}] and M$_B$, we wish not to limit the completeness in M$_B$, which is well measured for all galaxies, due to the higher L[\\ion{O}{2}] errors. Therefore, we opt to keep all galaxies with measured L[\\ion{O}{2}] in our sample and weight the SFR fits by the measurement errors. \n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/OII-SEDsfr-Selection.pdf}\n\\caption{Comparison of the S09 SFRs for blue galaxies from $0.7420$ in dust correction. Note that the uncorrected SFRs are tightly correlated with observed L[\\ion{O}{2}], indicating that the dust correction introduces systematic uncertainty into the corrected SFR values.}\n\\label{fig:SFRreject}\n\\end{figure}\n\nWe place one final restriction on the sample to remove galaxies that have a large dust-corrected SFRs in the S09 data. Figure~\\ref{fig:SFRreject} shows the dust-corrected and uncorrected S09 SFRs in the matched DEEP2 \/ AEGIS sample as a function of observed [\\ion{O}{2}] luminosity from DEEP2. We find that the uncorrected UV\/optical SED SFRs and observed L[\\ion{O}{2}] are tightly correlated due to similar dust extinction properties in the UV. After the dust correction is applied, a small population (13\\%) of galaxies have SFRs that are $>3\\sigma$ outliers from the mean dust-corrected distribution. These ``dusty\" star-forming galaxies have SFR corrections of more than a factor of 20 and are noted in S09 as primarily intermediate color (green valley) galaxies. Since we are attempting to calibrate SFRs at $z\\sim1$ with limited optical observations, we assume that the dust extinction for these obscured galaxies cannot be accurately measured on an individual galaxy basis (usually done through IR measurements) and remove them from the following analysis. While removing the outliers allows our calibrations to more closely track the ``typical\" galaxy in our sample, this underlying assumption restricts the SFR calibrations in this study to galaxy samples with moderate dust extinction and we acknowledge that our results will not be accurate for galaxies that require extraordinarily large dust corrections. The sample used in our SFR calibration is reduced to a total of 771 galaxies after we remove these galaxies.\n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/SelectionHists.pdf}\n\\caption{Histograms of the selected and matched AEGIS \/ DEEP2 sample from $0.741$ at these redshifts can either be old early-type galaxies with low SFR or heavily-reddened late-type galaxies with a young stellar population and high SFR.\n \n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-2color-MB-O2cal-contour-th.pdf}\n\\caption{The diagnostic M$_{B}$ - (U-B) plane with SFRs calibrated using the M$_{B}$ - L[\\ion{O}{2}] calibration split between red and blue galaxies (dotted line). The solid lines correspond to constant contours of SFR generated from the matched S09 AEGIS galaxy sample with log($\\psi$)=[0.3, 0.6, 0.9, 1.2] M$_{\\sun}$ yr$^{-1}$ and are plotted here for easy reference to the original calibration data. See Section~\\ref{sec:RedBlueCal} for details of the color-split SFR calibration.}\n\\label{fig:2colorSFRcont}\n\\end{figure}\n\nAs developed in Section~\\ref{sec:MBUBcal}, the SFR transition from red to blue galaxies can be \\emph{approximated} by a second-order polynomial fit in M$_{B}$ and (U-B) color that is valid within the range of SFRs and errors considered in our matched DEEP2 \/ AEGIS sample. The left hand plot of Figure~\\ref{fig:SFRcontours} shows the calibrated SFRs for the full DEEP2 sample from $0.74-20$. This unphysical behavior is a consequence of the second-order term in the fit where there are few matched S09 data points to constrain the SFR calibration. \n\\begin{figure*}[tb]\n\\centering\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-MB-UB-UB2cal-contour-th.pdf}}\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-MB-UB-BVcal-contour-th.pdf}}\n\\caption{The diagnostic M$_{B}$ - (U-B) plane with calibrated $0.741$). The S09 data shows that bright star-forming galaxies with heavily-reddened restframe colors do exist in this region. However, because the transition from high to low SFR is a strong function of (U-B) color, the SFRs in this region will not be highly accurate on an individual galaxy basis, but the SFR trend for the entire galaxy population will on average be correct. \n\nWhile the two calibrations used in Figure~\\ref{fig:SFRcontours} are statistically equivalent in comparison to the S09 matched sample, we opt for the linear multicolor calibration to produce better qualitative agreement for faint blue galaxies. Figure~\\ref{fig:SFRfit} shows the SFR produced by the multicolor calibration matches the ``true\" SED SFRs very well, with a residual scatter of $\\sigma_{all}=0.28$ dex for all galaxies with no offset in the mean of the distribution. In fact, the scatter is slightly reduced relative to the residual scatter seen in the original volume-limited calibration generated between $0.741$. The SFR calibration slope differs by 11\\% per decade, perhaps due to additional systematic bias such as an evolution in the [\\ion{O}{2}]\/H$\\alpha$ ratio between $z=1$ and the current epoch. Alternatively, the disagreement could be due to differential luminosity evolution as our assumption of a constant $Q=1.3$ mag\/$z$ corresponds primarily to $L_{*}$ galaxies. \n\nThe bottom panel of Figure~\\ref{fig:SFRdiff} shows that the G10 calibration, modified to correct dust extinction in L[\\ion{O}{2}] through the stellar mass, is an excellent match to our multicolor SFR calibration. The mean of the SFR distributions have a very small offset of 0.01 dex and the slope of the SFRs are in agreement to better than 2\\% per decade. Overall, both the M06 and G10 SFR calibrations seem to be generally consistent with our multicolor SFR calibration, an encouraging result considering the multicolor calibration is based only on broadband optical colors and restframe magnitude. We do not observe a significant systematic bias in the red galaxy SFRs estimated through either L[\\ion{O}{2}]-based SFR calibrations relative to our multicolor SFR calibration.\n\nFinally, we compare the SED-fit SFR contours in the restframe color-magnitude plane to those generated from the M06 and G10 calibrations. As shown in Figure~\\ref{fig:OIISFRcontours}, both L[\\ion{O}{2}]-based calibrations produce similar behavior as the S09 SFR contours in the color-magnitude plane, but the G10 calibration is slightly better at reproducing the color-independent SFR behavior seen in the blue cloud and therefore is in better agreement with our multicolor calibration for blue galaxies (c.f. Figure~\\ref{fig:SFRdiff}). Combined with the previous comparison to the SED-fit SFRs, these results show that our SFR calibration using only restframe optical colors and $B$-band luminosity are at least as accurate as using a local L[\\ion{O}{2}]-based SFR relation and extrapolating it to $z\\sim1$. \n\n\\begin{figure*}[thb]\n\\centering\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-M06cal-contour-th.pdf}}\n\\subfigure{\\includegraphics[width=0.98\\columnwidth]{.\/SEDsfr-G10cal-contour-th.pdf}}\n\\caption{The M$_{B}$ - (U-B) plane with calibrated $0.74