Smooth invariant densities for random switching on the torus

Abstract

We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.

19 pages

References in corpus (2)

• Quantitative ergodicity for some switched dynamical systems
• Integration by parts formula for locally smooth laws and applications to sensitivity computations

Cited by in corpus (5)

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