Existence of equilibrium for Infinite System of Interacting Diffusions

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.