Existence of equilibrium for Infinite System of Interacting Diffusions

TL;DR

This paper introduces a probabilistic approach to establish the existence of equilibrium states for infinite systems of interacting diffusions, including cases with degenerate elliptic operators like those from the Heisenberg group.

Contribution

It develops a new probabilistic framework for analyzing long-term behavior of infinite interacting diffusions, accommodating degenerate operators and unbounded lattices.

Findings

Existence of equilibrium states for infinite interacting diffusions.

Applicable to degenerate elliptic operators, including Heisenberg group cases.

Framework extends to various complex models.

Abstract

We develop and implement new probabilistic strategy for proving basic results about long time behaviour for interacting diffusion processes on unbounded lattice. The concept of the solution used is rather weak as we construct the process as a solution to suitable infinite dimensional martingale problem. However the techniques allow us to consider cases where the generator of the particle is degenerate elliptic operator. As a model example 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.