Elliptic equations for invariant measures on finite and infinite dimensional manifolds

We obtain sufficient conditions in terms of Lyapunov functions for the existence of invariant measures for diffusions on finite dimensional manifolds and prove some regularity results for such measures. These results are extended to countable prod-ucts of finite dimensional manifolds. We introduce and study a new concept of weak elliptic equations for measures on infinite dimensional manifolds. Then we apply our results to Gibbs distributions in the case where the single spin spaces are Riemannian manifolds. In particular, we obtain some a priori estimates for such Gibbs distributions and prove a general existence result applicable to a wide class of models. We also apply our techniques to prove absolute continuity of in-variant measures on the infinite dimensional torus, improving a recent result of A. F. Ramirez. Furthermore, we obtain a new result concerning the question whether invariant measures are Gibbsian