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January, 2022 Infinite-dimensional stochastic differential equations and tail $\sigma$-fields II: the IFC condition
Yosuke KAWAMOTO, Hirofumi OSADA, Hideki TANEMURA
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J. Math. Soc. Japan 74(1): 79-128 (January, 2022). DOI: 10.2969/jmsj/85118511

Abstract

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 “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 it 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.

Funding Statement

The second author is supported in part by JSPS KAKENHI Grant Numbers JP20K20885, JP18H03672, JP16H06338. The third author is supported in part by JSPS KAKENHI Grant Number JP19H01793.

Citation

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Yosuke KAWAMOTO. Hirofumi OSADA. Hideki TANEMURA. "Infinite-dimensional stochastic differential equations and tail $\sigma$-fields II: the IFC condition." J. Math. Soc. Japan 74 (1) 79 - 128, January, 2022. https://doi.org/10.2969/jmsj/85118511

Information

Received: 6 July 2020; Published: January, 2022
First available in Project Euclid: 1 October 2021

Digital Object Identifier: 10.2969/jmsj/85118511

Subjects:
Primary: 82C22
Secondary: 60B20 , 60H10 , 60K35

Keywords: infinite-dimensional stochastic differential equations , Interacting Brownian motions , random matrices

Rights: Copyright ©2022 Mathematical Society of Japan

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Vol.74 • No. 1 • January, 2022
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