TY - JOUR

T1 - Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions

AU - Kawamoto, Yosuke

AU - Osada, Hirofumi

AU - Tanemura, Hideki

N1 - Funding Information:
Y.K. is supported by a Grant-in-Aid for for Scientific Research (Grant No.15J03091) from the Japan Society for the Promotion of Science. H.O. is supported in part by a Grant-in-Aid for Scientific Research (S), No.16H06338; Grant-in-Aid for Challenging Exploratory Research No.16K13764 from the Japan Society for the Promotion of Science. H.T. is supported in part by a Grant-in-Aid for Scientific Research (C), No.15K04010; Scientific Research (B), No.19H01793 from the Japan Society for the Promotion of Science.
Publisher Copyright:
© 2020, The Author(s).

PY - 2021/12

Y1 - 2021/12

N2 - The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms (Eupr, Dupr) and (Elwr, Dlwr) on L2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium state μ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by (Elwr, Dlwr) is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle’s class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle’s class interaction potentials of C03-class.

AB - The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms (Eupr, Dupr) and (Elwr, Dlwr) on L2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium state μ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by (Elwr, Dlwr) is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle’s class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle’s class interaction potentials of C03-class.

KW - Infinite-dimensional stochastic differential equations

KW - Infinitely many particle systems

KW - Interacting Brownian motions

KW - Random matrices

KW - Uniqueness of Dirichlet forms

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U2 - 10.1007/s11118-020-09872-2

DO - 10.1007/s11118-020-09872-2

M3 - Article

AN - SCOPUS:85090762096

SN - 0926-2601

VL - 55

SP - 639

EP - 676

JO - Potential Analysis

JF - Potential Analysis

IS - 4

ER -