Some simple but challenging Markov processes

Malrieu, Florent

doi:10.5802/afst.1468

Cited by 37 publications

(43 citation statements)

References 73 publications

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“…Many authors have studied systems with random switchings (or, switching systems), and they are known independently under various titles: hybrid systems [YZ10], piecewise deterministic Markov processes (PDMP) (e.g. [Dav93], [Mal15]), random evolutions, see [Her03] for the history of the subject and extensive bibliography. Much of this work was inpired by [Kac74] (a reprint of an article published in 1956) where the first probabilistic representation of a second-order hyperbolic equation was obtained.…”

Section: Introductionmentioning

confidence: 99%

Smooth invariant densities for random switching on the torus

et al. 2018

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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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Section: Introductionmentioning

confidence: 99%

Smooth invariant densities for random switching on the torus

et al. 2018

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“…Interestingly, they were introduced and rediscovered many times by different groups of researchers. Here we give just a few references to works of some of these groups: [Kac74], [Dav93], [Her03], [Ani08], [FGRC09], [YZ10], [BLBMZ12], [BH12], [LMR15], [Mal15].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium

Bakhtin

Gaál

2019

Stoch. Dyn.

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We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus this setting is in the universality class of the unstable equilibrium exit under small white-noise perturbations.

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“…PDMPs were first introduced by Davis [18] in an even more general setting. For instance, PDMPs may involve jumps not only on the collection of vector fields but also on R d [19,39]. The class of PDMPs considered in this article is also known under the names of hybrid systems [50] and random evolutions [28], [21,Chapter 12].…”

Section: Introductionmentioning

confidence: 99%

Random switching near bifurcations

Hurth

Kuehn

2019

Stoch. Dyn.

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The interplay between bifurcations and random switching processes of vector fields is studied. More precisely, we provide a classification of piecewise deterministic Markov processes arising from stochastic switching dynamics near fold, Hopf, transcritical and pitchfork bifurcations. We prove the existence of invariant measures for different switching rates. We also study, when the invariant measures are unique, when multiple measures occur, when measures have smooth densities, and under which conditions finite-time blow-up occurs. We demonstrate the applicability of our results for three nonlinear models arising in applications.

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Some simple but challenging Markov processes

Cited by 37 publications

References 73 publications

Smooth invariant densities for random switching on the torus

Smooth invariant densities for random switching on the torus

Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium

Random switching near bifurcations

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