Smoothness of the law of manifold-valued Markov processes with jumps

TL;DR

This paper establishes conditions under which the probability law of manifold-valued Markov processes with jumps is smooth, extending results to noncompact manifolds and specific cases like Lie groups.

Contribution

It provides new sufficient conditions for the smoothness of laws of jump-driven processes on manifolds, including noncompact cases, using a localization approach.

Findings

Smoothness conditions are derived for manifold-valued processes with jumps.

Results extend to Lie groups and homogeneous spaces.

Localization technique connects Euclidean and manifold cases.

Abstract

Consider on a manifold the solution $X$ of a stochastic differential equation driven by a L\'evy process without Brownian part. Sufficient conditions for the smoothness of the law of $X_t$ are given, with particular emphasis on noncompact manifolds. The result is deduced from the case of affine spaces by means of a localisation technique. The particular cases of Lie groups and homogeneous spaces are discussed.