REGULARITY OF INVARIANT-MEASURES ON FINITE AND INFINITE-DIMENSIONAL SPACES AND APPLICATIONS

In this paper we prove new results on the regularity (i.e., smoothness ) of measures mu solving the equation Lmu=0 for operators of type L=D elta+B .del on finite and infinite dimensional state spaces E. In part icular, 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 meas ure 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 applica tions to infinite dimensional stochastic differential equations and pr ove some new existence results for Lmu=0. These include results on th e ''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 symm etric on L(2)(mu)). Finally, a substantial part of this work is devote d to the uniqueness of symmetrizing measures for L. We characterize th e cases, where we have uniqueness, by the irreducibility of the associ ated (classical) Dirichlet forms. (C) 1995 Academic Press, Inc.