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

We prove uniqueness of "invariant measures," i.e., solutions to the equation L*mu = 0 where L = Delta + B . del on R-n with B satisfying some mild integrability conditions and mu 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 L-1(mu) generates a strongly continuous semigroup having mu as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L-1(mu) 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*mu = 0 implies that L is symmetric on L-2(mu) Or L*mu = 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. (C) 1999 John Wiley & Sons, Inc.