Smooth invariant densities for random switching on the torus

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

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1331-1350
Number of pages20
Issue number4
StatePublished - Feb 20 2018


  • integration by parts
  • invariant densities
  • piecewise deterministic Markov processes
  • randomly switched ODEs

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


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