Elliptic equations for invariant measures on Riemannian manifolds: existence and regularity of solutions

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 global regularity results for such measures. These results a re extended to countable products of finite dimensional manifolds. A new co ncept of weak elliptic equations for measures on infinite dimensional manif olds is introduced. As an application we obtain some a priori estimates for Gibbs measures on countable products of manifolds and prove a new existenc e result for such measures. (C) 2001 Academie des sciences/Editions scienti fiques et medicales Elsevier SAS.