Coupling and Strong Feller for Jump Processes on Banach Spaces

TL;DR

This paper studies the coupling and strong Feller properties of Markov semigroups for linear SDEs driven by Lévy processes on Banach spaces, emphasizing the role of drift in infinite dimensions and providing gradient estimates and convergence results.

Contribution

It introduces new conditions for coupling and strong Feller properties in infinite-dimensional Lévy-driven SDEs, highlighting the importance of drift for quasi-invariance.

Findings

Establishes coupling and strong Feller properties under lower bound conditions.

Provides gradient estimates and exponential convergence results.

Illustrates main results with models on Wiener and Hilbert spaces.

Abstract

By using lower bound conditions of the L\'evy measure w.r.t. a nice reference measure, the coupling and strong Feller properties are investigated for the Markov semigroup associated with a class of linear SDEs driven by (non-cylindrical) L\'evy processes on a Banach space. Unlike in the finite-dimensional case where these properties have also been confirmed for L\'evy processes without drift, in the infinite-dimensional setting the appearance of a drift term is essential to ensure the quasi-invariance of the process by shifting the initial data. Gradient estimates and exponential convergence are also investigated. The main results are illustrated by specific models on the Wiener space and separable Hilbert spaces.