Regularity Properties for a System of Interacting Bessel Processes

TL;DR

This paper investigates the regularity and boundary behavior of a particle system with Coulomb interactions on a simplex, establishing the Feller property, semi-martingale conditions, and heat kernel estimates for both attractive and repulsive regimes.

Contribution

It proves the Feller property for the system with singular drift and boundary reflection, and characterizes when the process is a semi-martingale, providing new insights into interacting Bessel processes.

Findings

Feller property established for both attraction and repulsion cases.

Process is a Euclidean semi-martingale only under repulsion.

Exponential heat kernel gradient estimates obtained in the repulsive regime.

Abstract

We study the regularity of a diffusion on a simplex with singular drift and reflecting boundary condition which describes a finite system of particles on an interval with Coulomb interaction and reflection between nearest neighbors. As our main result we establish the Feller property for the process in both cases of repulsion and attraction. In particular the system can be started from any initial state, including multiple point configurations. Moreover we show that the process is a Euclidean semi-martingale if and only if the interaction is repulsive. Hence, contrary to classical results about reflecting Brownian motion in smooth domains, in the attractive regime a construction via a system of Skorokhod SDEs is impossible. Finally, we establish exponential heat kernel gradient estimates in the repulsive regime. The main proof for the attractive case is based on potential theory in Sobolev spaceswith Muckenhoupt weights.