Elliptic equations for invariant measures on finite and infinite dimensional manifolds

We obtain sufficient conditions in terms of Lyapunov functions for the exis tence of invariant measures for diffusions on finite-dimensional manifolds and prove some regularity results for such measures. These results are exte nded to countable products of finite-dimensional manifolds. We introduce an d study a new concept of weak elliptic equations for measures on infinite-d imensional manifolds. Then we apply our results to Gibbs distributions in t he case where the single spin spaces are Riemannian manifolds. In particula r, 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 a pply our techniques to prove absolute continuity of invariant measures on t he infinite-dimensional torus, improving a recent result of A.F. Ramirez. F urthermore, we obtain a new result concerning the question whether invarian t measures are Gibbsian. (C) 2001 Editions scientifiques et medicales Elsev ier SAS.