Exponential ergodicity of infinite system of interating diffusions

TL;DR

This paper introduces a new probabilistic approach to establish exponential ergodicity for infinite interacting diffusion processes on unbounded lattices, including cases with subelliptic generators like the Heisenberg group.

Contribution

It develops a novel method for proving exponential ergodicity in infinite dimensions using weak solutions via martingale problems, applicable to subelliptic operators.

Findings

Proves exponential convergence in the uniform norm for certain infinite-dimensional diffusions.

Extends ergodicity results to processes with subelliptic generators such as those from the Heisenberg group.

Provides a framework for analyzing a broader class of interacting diffusions on unbounded lattices.

Abstract

We develop and implement new probabilistic strategy for proving exponential ergodicity for interacting diffusion processes on unbounded lattice. The concept of the solution used is rather weak as we construct the process in infinite dimension as a solution to suitable infinite dimensional martingale problem. However the techniques allow us to consider cases where the generator of the particle is subelliptic operator. In addition we prove exponential convergence in the uniform norm, which appears to be new in this context. As a model case we present situation, where the operator arises from Heisenberg group. In the last section we mention some further examples that can be handled using our methods.