On smoothing properties of transition semigroups associated to a class of SDEs with jumps

TL;DR

This paper investigates the smoothing effects of transition semigroups linked to a class of jump-driven SDEs, establishing conditions under which these semigroups are strong Feller and map $L_p$ spaces to continuous functions.

Contribution

It introduces new probabilistic and analytic conditions ensuring smoothing properties for nonlocal semigroups of jump SDEs with non-Lipschitz drifts.

Findings

Semigroups are strong Feller under certain conditions.

Transition semigroups map $L_p$ to continuous bounded functions.

Regularizing properties depend on negative moments of the subordinator.

Abstract

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) driven by additive pure-jump L\'evy noise. In particular, we assume that the L\'evy process driving the SDE is the sum of a subordinated Wiener process and of an independent arbitrary L\'evy process, that the drift coefficient is continuous (but not necessarily Lipschitz continuous) and grows not faster than a polynomial, and that the SDE admits a Feller weak solution. By a combination of probabilistic and analytic methods, we provide sufficient conditions for the Markovian semigroup associated to the SDE to be strong Feller and to map $L_p$ to continuous bounded functions. A key intermediate step is the study of regularizing properties of the transition semigroup associated to the subordinated Wiener process in terms of negative moments of the subordinator.