Regularity of Invariant Measures: The Case of Non-constant Diffusion Part

We prove regularity (i.e., smoothness) of measuresμon Rdsatisfying the equationL*μ=0 whereLis an operator of typeLu=tr(Au″)+B·∇u. HereAis a Lipschitz continuous, uniformly elliptic matrix-valued map andBis merelyμ-square integrable. We also treat a class of corresponding infinite dimensional cases where Rdis replaced by a locally convex topological vector spaceX. In this casesμis proved to be absolutely continuous w.r.t. a Gaussian measure onXand the square root of the Radon–Nikodym density belongs to the Malliavin test function space D2, 1.