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Regularity of invariant measures on finite and infinite dimensional spaces and applications. (English) Zbl 0840.60069
Summary: 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.

MSC:
 60J45 Probabilistic potential theory 31C25 Dirichlet forms
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