Regularity of invariant measures on finite and infinite dimensional spaces and applications.

*(English)*Zbl 0840.60069Summary: We prove new results on the regularity (i.e., smoothness) of measures \(\mu\) solving the equation \(L^* \mu = 0\) for operators of type \(L = \Delta + B \cdot \nabla\) on finite- and infinite-dimensional state spaces \(E\). In particular, we settle a conjecture of I. Shigekawa [Osaka J. Math. 24, 37-59 (1987; Zbl 0636.60080)] in the situation where \(\Delta = \Delta_H\) is the Gross-Laplacian, \((E,H, \gamma)\) is an abstract Wiener space and \(B = - \text{id}_E + v\) 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 \(\gamma\) and that the square root of the density is in the Malliavin test function space of order 1 in \(L^2 (\gamma)\). 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 sufficient 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.