On uniqueness of invariant measures for finite- and infinite-dimensional diffusions

Abstract

We prove uniqueness of "invariant measures," i.e., solutions to the equation L* μ, = 0 where L = Δ + B · ∇ on ℝn with B satisfying some mild integrability conditions and μ being a probability measure on ℝn. This solves an open problem posed by S. R. S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L1 (μ) generates a strongly continuous semigroup having μ as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L1 (μ) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the "symmetric case") in particular is studied and conditions are identified ensuring that L* μ = 0 implies that L is symmetric on L2(μ) or L* μ = 0 has a unique solution. We also prove infinite-dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in infinite dimensions. © 1999 John Wiley & Sons, Inc.