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作者：Benard, Timothee; Breuillard, Emmanuel
作者单位：Centre National de la Recherche Scientifique (CNRS); University of Oxford
摘要：We establish the (non-lattice) local limit theorem for products of i.i.d. random variables on an arbitrary simply connected nilpotent Lie group G, where the variables are allowed to be non-centered. Our result also improves on the known centered case by proving uniformity for two-sided moderate deviations and allowing measures with a moment of order 2(dimG)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \us...
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作者：[Anonymous]
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作者：Gess, Benjamin; Gvalani, Rishabh S.; Konarovskyi, Vitalii
作者单位：Technical University of Berlin; Max Planck Society; Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Hamburg; National Academy of Sciences Ukraine; Institute of Mathematics of NASU
摘要：The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks ...
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作者：Chen, Hong-Bin; Xia, Jiaming
作者单位：New York University; University of Pennsylvania
摘要：We give a meaning to the Hamilton-Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness. Originally defined on the set of monotone probability measures, these equations can be interpreted, via an isometry, to be defined on an infinite-dimensional closed convex cone with an empty interior in a Hilbert space. We prove the comparison principle, and the convergence of finite-dimensional approximations furnishing the existen...
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作者：Hong, Wei; Li, Shihu; Liu, Wei
作者单位：Jiangsu Normal University
摘要：In this paper, we provide a general framework for investigating McKean-Vlasov stochastic partial differential equations. We first show the existence of weak solutions by combining the localizing approximation, Faedo-Galerkin technique, compactness method and the Jakubowski version of the Skorokhod representation theorem. Then under certain locally monotone condition we further investigate the existence and uniqueness of (probabilistically) strong solutions. The applications of the main results...
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作者：De Vecchi, Francesco; Fresta, Luca; Gordina, Maria; Gubinelli, Massimiliano
作者单位：University of Pavia; Roma Tre University; University of Connecticut; University of Oxford
摘要：We introduce a theory of non-commutative L-p spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Ito-Grassmann processes, Girsanov's formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables including a Grassmann analog of the Phi(2)(4) Euclidean QFT in a ...
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作者：Ding, Jian; Sun, Nike
作者单位：Peking University; Massachusetts Institute of Technology (MIT)
摘要：We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube {-1,+1}N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{-1,+1\}<^>N$$\end{document} intersected by M random half-spaces. The perceptron's capacity is the largest integer MN\documentclass[12pt]{minimal} \usepackage{am...
