Infinite Dimensional Systems of Particles with Interactions Given by Dunkl Operators

Abstract

Firstly we consider a finite dimensional Markov semigroup generated by Dunkl Laplacian with drift terms. For this semigroup we prove gradient bounds involving a symmetrised carre-du-champ operator, and we show that for small coefficients this semigroup has a unique invariant measure which satisfies ergodicity properties. We then extend this analysis to an infinite dimensional model on $(\mathbb {R}^{N})^{\mathbb {Z}^{d}}$ , consisting of interacting finite dimensional models. We construct an associated Markov semigroup for this model using gradient bounds, and finally we study the existence of invariant measures and ergodicity properties.