Existence and regularity of infinitesimally invariant measures, transition functions and time homogeneous Itô-SDEs

Abstract

We show existence of an infinitesimally invariant measure $m$ for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some local integrability order. Subsequently, we derive regularity properties of the corresponding semigroup which is defined in $L^s({\mathbb{R}}^d,m)$, $s\in [1,\infty ]$, including the classical strong Feller property and classical irreducibility. This leads to a transition function of a Hunt process that is explicitly identified as a solution to an SDE. Further properties of this Hunt process, like non-explosion, moment inequalities, recurrence and transience, as well as ergodicity, including invariance and uniqueness of $m$, and uniqueness in law, can then be studied using the derived analytical tools and tools from generalized Dirichlet form theory.

Revised version, slightly modified title

References in corpus (2)

• Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms
• Existence, uniqueness and ergodic properties for time-homogeneous Itô-SDEs with locally integrable drifts and Sobolev diffusion coefficients

Cited by in corpus (2)

• Well-posedness for a class of degenerate Itô-SDEs with fully discontinuous coefficients
• Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space