We prove regularity (i.e., smoothness) of measures μ on Rd satisfying the equation L*μ = 0 where L is an operator of type Lu = tr(Au″) + B · ∇u. Here A is a Lipschitz continuous, uniformly elliptic matrix-valued map and B is merely μ-square integrable. We also treat a class of corresponding infinite dimensional cases where ℝd is replaced by a locally convex topological vector space X. In this cases μ 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 double-struck D sign 2, 1.
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We thank Sergio Albeverio for fruitful discussions on the contents of this paper. Financial support of the Sonderforschungsbereiche 256 (Bonn) and 343 (Bielefeld), EC-Science Project SC1*CT92-0784, the International Science Foundation (Grant M 38000), and the Russian Foundation of Fundamental Research (Grant, 94-01-01556) is gratefully acknowledged.