## Journal of the Mathematical Society of Japan

### Finite-particle approximations for interacting Brownian particles with logarithmic potentials

#### Abstract

We prove the convergence of $N$-particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). For this proof we present two general theorems on the finite-particle approximations of interacting Brownian motions. In the first general theorem, we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy a uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem. In the second general theorem, we restrict to the case of reversible particle systems and simplify the sufficient condition. We deduce the second theorem from the first. We apply the second general theorem to $\mathrm{Airy}_\beta$ interacting Brownian motion with $\beta=1, 2, 4$, and the Ginibre interacting Brownian motion. The former appears in the soft-edge limit of Gaussian (orthogonal/unitary/symplectic) ensembles in one spatial dimension, and the latter in the bulk limit of Ginibre ensemble in two spatial dimensions, corresponding to a quantum statistical system for which the eigen-value spectra belong to non-Hermitian Gaussian random matrices. The passage from the finite-particle stochastic differential equation (SDE) to the limit ISDE is a sensitive problem because the logarithmic potentials are long range and unbounded at infinity. Indeed, the limit ISDEs are not easily detectable from those of finite dimensions. Our general theorems can be applied straightforwardly to the grand canonical Gibbs measures with Ruelle-class potentials such as Lennard-Jones 6-12 potentials and and Riesz potentials.

#### Note

The first author is supported by Grant-in-Aid for JSPS JSPS Research Fellowships (No. 15J03091). The second author is supported in part by a Grant-in-Aid for Scenic Research (KIBAN-A, No. 24244010; KIBAN-A, No. 16H02149; KIBAN-S, No. 16H06338) from the Japan Society for the Promotion of Science.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 3 (2018), 921-952.

Dates
Received: 23 July 2016
Revised: 29 November 2016
First available in Project Euclid: 18 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1529309020

Digital Object Identifier
doi:10.2969/jmsj/75717571

Mathematical Reviews number (MathSciNet)
MR3830792

Zentralblatt MATH identifier
06966967

#### Citation

KAWAMOTO, Yosuke; OSADA, Hirofumi. Finite-particle approximations for interacting Brownian particles with logarithmic potentials. J. Math. Soc. Japan 70 (2018), no. 3, 921--952. doi:10.2969/jmsj/75717571. https://projecteuclid.org/euclid.jmsj/1529309020

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