Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups

We consider stochastic equations in Hilbert spaces with singular drift in the framework of [G. Da Prato, M. Rockner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261-303]. We prove a Harnack inequality (in the sense of [F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417-424]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultra-boundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure 14 (satisfying some mild integrability conditions). Finally, we prove existence of such a measure mu for noncontinuous drifts. (C) 2009 Elsevier Inc. All rights reserved.