Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions
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Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions Yosuke Kawamoto1 · Hirofumi Osada2
· Hideki Tanemura3
Received: 4 December 2017 / Accepted: 6 August 2020 / © The Author(s) 2020
Abstract 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 (E upr , D upr ) and (E lwr , D lwr ) 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 (E lwr , D lwr ) 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. Keywords Uniqueness of Dirichlet forms · Interacting Brownian motions · Random matrices · Infinite-dimensional stochastic differential equations · Infinitely many particle systems Mathematics Subject Classification (2010) 60J45 · 60K35 · 82B21 · 60B20 · 60J60 · 60H10 Hirofumi Osada
[email protected] Yosuke Kawamoto [email protected] Hideki Tanemura [email protected] 1
Fukuoka Dental College, Fukuoka 814-0193, Japan
2
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan
3
Department of Mathematics, Keio University, Kohoku-ku, Yokohama 223-8522, Japan
Y. Kawamoto et al.
1 Introduction An infinite system of interacting Brownian motions in Rd can be represented by an (Rd )N valued stochastic process X = (X i )i∈N [10, 11, 14, 18]. This process is realized using several probabilistic constructs such as stochastic differential equation, Dirichlet form theory, and martingale problems. Among them, the second author constructed processes in a general setting using the technique of Dirichlet forms [14, 18]. Specifically, the Dirichlet form introduced, (E upr , D upr ) is obtained by the smallest μ μ extension of the bilinear form (E , D◦ ) on L2 (S, μ) with domain D◦ defined by E (f, g) = D[f, g](s) μ(d s), S
∞
D[f, g](s) =
1 ∇si fˇ · ∇si g, ˇ 2 i=1
D◦μ
= {f ∈ D◦ ∩ L2 (S, μ) ; E (f, f ) < ∞},
(1.1)
where D◦ is the set of all local smooth functions on the (unlabeled) configuration space S introduced in Eq. 2.1, fˇ is a
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