Exponential ergodicity of infinite system of interating diffusions

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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.