Regularity of invariant measures: The case of non-constant diffusion part

Bogachev VI, Krylov N, Röckner M. Regularity of invariant measures: The case of non-constant diffusion part. Journal of Functional Analysis. 1996;138(1):223-242.We prove regularity (i.e., smoothness) of measures mu on R(d) satisfying the equation L*mu = 0 where L is an operator of type Lu = tr(Au-'') + B . del u. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely mu-square integrable. We also treat a class of corresponding infinite dimensional cases where R(d) is replaced by a locally convex topological vector space X. In this cases mu is proved to be absolutely continuous w.r.t. a Gaussian measure on X and the square root of the Radon-Nikodym density belongs to the Malliavin test function space D-2,D-1. (C) 1996 Academic Press. Inc