Infinite-dimensional stochastic differential equations and tail $ σ$-fields II: the IFC condition
Yosuke Kawamoto, Hirofumi Osada, Hideki Tanemura
备注：
This paper is a continuation of "Infinite-dimensional stochastic differential equations and tail $σ$-fields", which published in Probability Theory and Related Fields, https://doi.org/10.1007/s00440-020-00981-y. This paper will be published in Journal of the Mathematical Society of Japan
发表信息：
J. Math. Soc. Japan 74 (1), 79-128, (January, 2022)
In a previous report, the second and third authors gave general theorems for unique strong solutions of infinite-dimensional stochastic differential equations (ISDEs) describing the dynamics of infinitely many interacting Brownian particles. One of the critical assumptions is the \lq\lq IFC" condition. The IFC condition requires that, for a given weak solution, the scheme consisting of the finite-dimensional stochastic differential equations (SDEs) related to the ISDEs exists. Furthermore, the IFC condition implies that each finite-dimensional SDE has unique strong solutions. Unlike other assumptions, the IFC condition is challenging to verify, and so the previous report only verified solution for solutions given by quasi-regular Dirichlet forms. In the present paper, we provide a sufficient condition for the IFC requirement in more general situations. In particular, we prove the IFC condition without assuming the quasi-regularity or symmetry of the associated Dirichlet forms. As an application of the theoretical formulation, the results derived in this paper are used to prove the uniqueness of Dirichlet forms and the dynamical universality of random matrices.
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Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility
KAIST（韩国科学技术院）；Huawei Technologies（华为技术有限公司）；Fordham University（福特汉姆大学）；Kim Jaechul Graduate School of AI, KAIST（韩国科学技术院金在哲人工智能研究生院）；Institute for Basic Science (IBS)（基础科学研究院）；University of Oxford（牛津大学）
