August 16, 2025Open Access

Regularity of the stationary density for systems with fast random switching

Key Points

The stationary density is bounded when jump rates are sufficiently fast, and unbounded when they are slow.
Key metrics establish that stationary density is lower semicontinuous for all jump rate conditions.
Probabilistic approaches employ novel use of stopping times to achieve results on stationary density.
Hörmander-type conditions on vector fields are critical for density characterization in compact vector field sets.

Abstract

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 Hörmander-type conditions on the vector fields guaranteeing that the stationary density is: C k whenever the jump rates are sufficiently fast, for any k ∞ ; unbounded whenever the jump rates are sufficiently slow and lower semicontinuous regardless of the jump rates. Our proofs are probabilistic, relying on a novel application of stopping times.

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