Absolute continuity of the invariant measure in Piecewise Deterministic Markov Processes having degenerate jumps

TL;DR

This paper proves the absolute continuity and regularity of the invariant measure in a class of piecewise deterministic Markov processes with degenerate jumps, using a splitting scheme and integration by parts.

Contribution

It introduces a novel splitting scheme based on jump times to establish regularity properties of the invariant measure in degenerate PDMPs.

Findings

Invariant measure is absolutely continuous.

Finer regularity results for one-dimensional marginals.

Method applicable to degenerate jump kernels.

Abstract

We consider piecewise deterministic Markov processes with degenerate transition kernels of the "house-of-cards"-type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the invariant measure of the process. Finally, we obtain finer results on the regularity of the one-dimensional marginals of the invariant measure, using integration by parts with respect to the jump times.