Regularity of the stationary density for systems with fast random switching

TL;DR

This paper investigates the regularity of stationary densities in systems with fast random switching between vector fields, establishing conditions under which the density is smooth, unbounded, or lower semi-continuous.

Contribution

It provides H"ormander-type conditions that guarantee smoothness of the stationary density depending on the switching rates, using a novel probabilistic approach with stopping times.

Findings

Stationary density is $C^k$ for fast switching rates.

Density becomes unbounded with slow switching.

Lower semi-continuity holds regardless of switching rates.

Abstract

We consider the piecewise-deterministic Markov process obtained by randomly switching between the flows generated by a finite set of smooth vector fields on a compact set. We obtain H\"ormander-type conditions on the vector fields guaranteeing that the stationary density is: $C^k$ whenever the jump rates are sufficiently fast, for any $k<\infty$; unbounded whenever the jump rates are sufficiently slow and lower semi-continuous regardless of the jump rates. Our proofs are probabilistic, relying on a novel application of stopping times.