Elliptic equations for invariant measures on finite and infinite dimensional manifolds (Q5932261)

First, the authors obtain some sufficient conditions in terms of Lyapunov function for the existence of invariant measures for diffusions on finite-dimensional manifolds and prove regularity results for the measures. Then those constructions are extended to countable products of finite-dimensional manifolds. A new concept of weak elliptic equations for measures is introduced and investigated. As applications, the authors prove: some a priori estimates for Gibbs distributions (when a single spin space is a Riemannian manifold), general existence results for such measures (applicable to a wide class of models), absolute continuity of invariant measures on infinite-dimensional torus, etc.