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作者：Hutchcroft, Tom
作者单位：California Institute of Technology
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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：Jiao, Yong; Osekowski, Adam; Wu, Lian
作者单位：Central South University; University of Warsaw
摘要：We introduce the notion of differential subordination in the context of noncommutative continuous-time martingales and show that for 2 < infinity this domination enforces the corresponding L-p bound between the two processes. As applications, we obtain best-order L-p inequalities for the second-order Riesz transforms on a class of group von Neumann algebras equipped with a conditionally negative length function. The proof relies on a novel stochastic representation of these second-order operat...
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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：Husson, Jonathan; Ko, Justin
作者单位：University of Michigan System; University of Michigan; Ecole Normale Superieure de Lyon (ENS de LYON)
摘要：We consider the asymptotics of rank k spherical integrals when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = o(N)$$\end{document}. We prove that the sublinear rank spherical integrals are approximately the products of rank 1 spherical integrals. Our formulas extend the results for rank k spherical integrals p...
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作者：Fang, Xiao; Liu, Song-Hao; Shao, Qi-Man; Zhao, Yi-Kun
作者单位：Chinese University of Hong Kong; Dalian University of Technology; Southern University of Science & Technology; Southern University of Science & Technology
摘要：The question of whether the central limit theorem (CLT) holds for the total number of edges in exponential random graph models (ERGMs) in the subcritical region of parameters has remained an open problem. In this paper, we establish the CLT. As a result of our proof, we also derive a convergence rate for the CLT, an explicit formula for the asymptotic variance, and the CLT for general subgraph counts. To establish our main result, we develop Stein's method for the normal approximation of gener...
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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...
