Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials

We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.

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

Potential Analysis ◽

10.1007/s11118-020-09872-2 ◽

2020 ◽

Author(s):

Yosuke Kawamoto ◽

Hirofumi Osada ◽

Hideki Tanemura

Keyword(s):

Finite Volume ◽

Stochastic Dynamics ◽

Matrix Theory ◽

Dirichlet Forms ◽

Interaction Potentials ◽

Brownian Motions ◽

Infinite Systems ◽

Point Field ◽

First Main Theorem ◽

Class Interaction

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 $ (\mathcal {E}^{\mathsf {upr}},\mathcal {D}^{\mathsf {upr}})$ ( E u p r , D u p r ) and $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) 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 $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) 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 $ {C_{0}^{3}} $ C 0 3 -class.

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