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

TL;DR

This paper establishes Harnack inequalities for transition semigroups of singular stochastic equations in Hilbert spaces, leading to regularity results and existence of invariant measures under mild conditions.

Contribution

It proves Harnack inequalities for singular stochastic equations in Hilbert spaces and demonstrates regularizing effects and invariant measure existence.

Findings

Harnack inequality for the transition semigroup

Regularizing and ultraboundedness properties

Existence of invariant measures for non-continuous drifts

Abstract

We consider stochastic equations in Hilbert spaces with singular drift in the framework of [Da Prato, R\"ockner, 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.