TY - JOUR

T1 - Infinite-dimensional stochastic differential equations and tail σ-fields II

T2 - the IFC condition

AU - Kawamoto, Yosuke

AU - Osada, Hirofumi

AU - Tanemura, Hideki

N1 - Funding Information:
2020 Mathematics Subject Classification. Primary 82C22; Secondary 60K35, 60H10, 60B20. Key Words and Phrases. interacting Brownian motions, infinite-dimensional stochastic differential equations, random matrices. 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.
Publisher Copyright:
©2022 The Mathematical Society of Japan

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Infinite-dimensional stochastic differential equations

KW - Interacting Brownian motions

KW - Random matrices

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U2 - 10.2969/jmsj/85118511

DO - 10.2969/jmsj/85118511

M3 - Article

AN - SCOPUS:85124297160

SN - 0025-5645

VL - 74

SP - 79

EP - 128

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

IS - 1

ER -