Infinite-dimensional stochastic differential equations related to Bessel random point fields

TL;DR

This paper solves complex infinite-dimensional stochastic differential equations modeling interacting particles, revealing their equilibrium states as Bessel random point fields and establishing their quasi-Gibbsian nature.

Contribution

It provides a novel solution to ISDEs with Coulomb interactions and characterizes the equilibrium states as Bessel random point fields.

Findings

Equilibrium states are Bessel random point fields.

The ISDEs are solved using logarithmic derivatives.

The random point fields are shown to be quasi-Gibbsian.

Abstract

We solve the infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles in $ \mathbb{R}^+$ interacting through the two-dimensional Coulomb potential. The equilibrium states of the associated unlabeled stochastic dynamics are Bessel random point fields. To solve these ISDEs, we calculate the logarithmic derivatives, and we prove that the random point fields are quasi-Gibbsian.