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arXiv · 1708.01390

Smooth invariant densities for random switching on the torus

Yuri Bakhtin·Tobias Hurth·Sean D. Lawley·Jonathan C. Mattingly
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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.

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Yuri Bakhtin, Tobias Hurth, Sean D. Lawley, Jonathan C. Mattingly. 2018-03-21. Smooth invariant densities for random switching on the torus. https://doi.org/10.1088/1361-6544%2Faaa04f

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