Regularity of the stationary density for systems with fast random switching

Michel Benaïm; Oliver Tough

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arXiv · 2212.03632

Regularity of the stationary density for systems with fast random switching

Michel Benaïm·Oliver Tough

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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.

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Michel Benaïm, Oliver Tough. 2022-12-07. Regularity of the stationary density for systems with fast random switching. https://arxiv.org/abs/2212.03632

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