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

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

2008-11-13

arxiv.org/abs/0811.2061v2

Mathematics

Probability

Scientific paper

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.

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