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 products 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 invariant 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. (C) 2001 Editions scientifiques et medicales Elsevier SAS.