PUB - Publikationen an der Universität Bielefeld

In this paper we prove new results on the regularity (i.e., smoothness) of measures mu solving the equation L*mu=0 for operators of type L=Delta+B .del on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where Delta=Delta(H) is the Gross-Laplacian, (E, H, y) is an abstract Wiener space and B=-id(E)+v where v takes values in the Cameron-Martin space H. Using Gross' logarithmic Sobolev-inequality in an essential way we show that mu is always absolutely continuous w.r.t. the Gaussian measure y and that the square root of the density is in the Malliavin test function space of order 1 in L(2)(y). Furthermore, we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for L*mu=0. These include results on the ''inverse problem'', i.e., we give conditions ensuring that B is the (vector) logarithmic derivative of a measure. We also prove necessary and suf ficient conditions for mu to be symmetrizing (i.e., L is symmetric on L(2)(mu)). Finally, a substantial part of this work is devoted to the uniqueness of symmetrizing measures for L. We characterize the cases, where we have uniqueness, by the irreducibility of the associated (classical) Dirichlet forms. (C) 1995 Academic Press, Inc.