Smooth invariant densities for random switching on the torus

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

doi:10.1088/1361-6544/aaa04f

Smooth invariant densities for random switching on the torus

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

Mathematics

Research output: Contribution to journal › Article › peer-review

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 language	English (US)
Pages (from-to)	1331-1350
Number of pages	20
Journal	Nonlinearity
Volume	31
Issue number	4
DOIs	https://doi.org/10.1088/1361-6544/aaa04f
State	Published - Feb 20 2018

Keywords

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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10.1088/1361-6544/aaa04f

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AU - Hurth, Tobias

AU - Lawley, Sean D.

AU - Mattingly, Jonathan C.

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Y1 - 2018/2/20

N2 - 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.

AB - 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.

KW - integration by parts

KW - invariant densities

KW - piecewise deterministic Markov processes

KW - randomly switched ODEs

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SP - 1331

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JO - Nonlinearity

JF - Nonlinearity

IS - 4

ER -

Smooth invariant densities for random switching on the torus

Abstract

Keywords

ASJC Scopus subject areas

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