Existence of equilibrium for infinite system of interacting diffusions

We develop and implement new probabilistic strategy for proving basic results about long-time behavior 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 the situation where the operator arises from Heisenberg group. In the last section, we provide further examples that can be handled using our methods.