Smooth invariant densities for random switching on the torus

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

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)1331-1350
Number of pages20
JournalNonlinearity
Volume31
Issue number4
DOIs
StatePublished - Feb 20 2018

Fingerprint

Torus
Invariant
Vector Field
Integration by Parts Formula
poisson process
Random Dynamical Systems
Malliavin Calculus
Switched Systems
calculus
Poisson process
dynamical systems
Periodic Orbits
Dynamical systems
Orbits
orbits

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

Smooth invariant densities for random switching on the torus. / Bakhtin, Yuri; Hurth, Tobias; Lawley, Sean D.; Mattingly, Jonathan C.

In: Nonlinearity, Vol. 31, No. 4, 20.02.2018, p. 1331-1350.

Research output: Contribution to journalArticle

Bakhtin, Y, Hurth, T, Lawley, SD & Mattingly, JC 2018, 'Smooth invariant densities for random switching on the torus', Nonlinearity, vol. 31, no. 4, pp. 1331-1350. https://doi.org/10.1088/1361-6544/aaa04f
Bakhtin, Yuri ; Hurth, Tobias ; Lawley, Sean D. ; Mattingly, Jonathan C. / Smooth invariant densities for random switching on the torus. In: Nonlinearity. 2018 ; Vol. 31, No. 4. pp. 1331-1350.
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