Existence of invariant measures for diffusion processes on a Wiener space

Abstract

In this paper, we discuss existence of finite invariant measures for diffusion processes with unbounded drift on an abstract Wiener space (B,H,μ). The processes we treat are the ones which have generators formally expressed as — (I/2)D*σD + (6, D ), where D stands for the ^-derivative in the sense of Malliavin calculus, D* the adjoint operator of D with respect to the Wiener measure μ, σ a diffusion coefficient, being a function on B taking values in the space of positive symmetric operators on H, and b a drift coefficient, being an ^-valued function on B. When σ = identity, a given generator has rigorous meaning in L sense under a mild condition on 6, and existence of the associated diffusion and invariant measures was proved by Shigekawa [17] in the case that b is bounded. Partially generalized cases were treated by Vintschger [21] and Zhang [22]. We extend these results to the case where σ is not constant and b is not necessarily bounded but has only some kind of exponential integrability. The diffusion process is constructed by using the theory of Dirichlet forms and the Girsanov transformation, so measures charging no set of zero capacity are allowed to be initial measures. In this paper, however, we restrict our attention to absolutely continuous measures with respect to μ, as in the papers mentioned above. That is, our formulation is as follows: Let {Tt} be the associated Markovian semigroup on L°°(μ). Is there any non-zero p £ L (μ) such that