# Infinite-dimensional stochastic differential equations for Coulomb random point fields

## TL;DR

This paper constructs and analyzes infinite-dimensional stochastic differential equations for Coulomb particle systems, proving existence, uniqueness, and connecting finite and infinite particle dynamics.

## Contribution

It introduces a new stochastic analysis method for Coulomb systems, extending beyond determinantal point fields, and provides rigorous construction of Coulomb interacting Brownian motions.

## Key findings

- Constructed strong solutions for Coulomb ISDEs in all dimensions d ≥ 2.
- Proved pathwise uniqueness and reversibility of the infinite-particle dynamics.
- Connected finite-particle systems with infinite-particle limits through approximation schemes.

## Abstract

We study the infinite-dimensional stochastic differential equations (ISDEs) of infinite-particle systems associated with Coulomb random point fields. The stochastic dynamics described by these ISDEs are referred to as Coulomb interacting Brownian motions. In all spatial dimensions $ d \ge 2 $ and for all inverse temperatures $ \beta > 0 $, we construct the Coulomb interacting Brownian motions. We prove that the ISDEs admit strong solutions and that pathwise uniqueness holds. The resulting labeled dynamics form an $ \RdN $-valued diffusion, possibly without an invariant measure, while the corresponding unlabeled process is a reversible diffusion with respect to the underlying Coulomb random point field. Moreover, we identify the infinite-particle stochastic dynamics as the limit in path space of finite-particle systems driven by stochastic differential equations. This identification is achieved through two approximation schemes: finite-domain systems with reflecting boundary conditions and $ N $-particle systems. Although the $ N $-particle approximation is more fundamental, its justification relies crucially on the finite-domain approximation together with the uniqueness of solutions to the ISDEs. Previously, only the case $ d = 2 $ and $ \beta = 2 $, known as the Ginibre interacting Brownian motion, was understood through random matrix theory and determinantal random point fields. Extending this result beyond the determinantal setting has remained a major difficulty. We introduce a new, conceptually clear method based on stochastic analysis of infinite-particle systems with long-range interactions that yields a rigorous construction of Coulomb interacting Brownian motions. A key ingredient is an explicit computation of the logarithmic derivatives of Coulomb random point fields.