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

Abstract

We consider stochastic equations in Hilbert spaces with singular drift in the framework of [Da Prato, Röckner, PTRF 2002]. We prove a Harnack inequality (in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure $\mu$ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure $\mu$ for non-continuous drifts.

References in corpus (1)

• Harnack inequality and applications for stochastic generalized porous media equations

Cited by in corpus (4)

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• Strong uniqueness for SDEs in Hilbert spaces with nonregular drift
• Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises