Strong Feller properties for degenerate SDEs with jumps

TL;DR

This paper proves the strong Feller property for a class of degenerate SDEs with jumps under Hörmander conditions, extending existing criteria and providing an example involving coupled oscillators.

Contribution

It extends criteria for law convergence of Wiener functionals to establish strong Feller properties for SDEs driven by subordinate Brownian motion with unbounded drifts.

Findings

Established strong Feller property under Hörmander conditions.

Extended criteria for convergence of Wiener functionals.

Verified example with coupled oscillators.

Abstract

Under full H\"ormander's conditions, we prove the strong Feller property of the semigroup determined by an SDE driven by additive subordinate Brownian motion, where the drift is allowed to be arbitrarily growth. For this, we extend a criterion due to Malicet-Poly \cite{Ma-Po} and Bally-Caramellino \cite{Ba-Ca} about the convergence of the laws of Wiener functionals in total variations. Moreover, the example of a chain of coupled oscillators is verified.