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

✍ Scribed by S. Albeverio; V. Bogachev; M. Röckner

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 556 KB
• Volume: 52
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199903)52:3<325::aid-cpa2>3.0.co;2-v

✦ Synopsis

We prove uniqueness of "invariant measures," i.e., solutions to the equation L * µ = 0 where L = ∆ + B • ∇ on R n with B satisfying some mild integrability conditions and µ being a probability measure on R n . 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 L 1 (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 L 1 (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 L 2 (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.