Existence and Uniqueness of Invariant Measures: An Approach via Sectorial Forms

Abstract.

We prove the existence and uniqueness for invariant measures of strongly continuous semigroups on L ² (X;μ ) , where X is a (possibly infinite-dimensional) space. Our approach is purely analytic based on the theory of sectorial forms. The generators covered are, e.g., small perturbations (in the sense of sectorial forms) of operators generating hypercontractive semigroups. An essential ingredient of the proofs is a new result on compact embeddings of weighted Sobolev spaces H ^1,2 (ρ . . . dx) on \bf R ^d (resp. a Riemannian manifold) into L ² (ρ dx) . Probabilistic consequences are also briefly discussed.