Infinite dimensional systems of particles with interactions given by Dunkl operators

TL;DR

This paper develops a framework for infinite-dimensional particle systems with interactions modeled by Dunkl operators, establishing invariant measures and ergodic properties through gradient bounds.

Contribution

It introduces a novel infinite-dimensional semigroup based on Dunkl operators and proves the existence of invariant measures and ergodicity for small interaction coefficients.

Findings

Existence of invariant measures for the semigroup

Extension from finite to infinite dimensions

Ergodicity properties established

Abstract

Firstly we consider a finite dimensional Markov semigroup generated by Dunkl laplacian with drift terms. Using gradient bounds we show that for small coefficients this semigroup has an invariant measure. We then extend this analysis to an infinite dimensional semigroup on $(\mathbb{R}^N)^{\mathbb{Z}^d}$ which we construct using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.