On the stability of planar randomly switched systems

Benaı̈m, Michel; Borgne, Stéphane Le; Malrieu, Florent; Zitt, Pierre-André

doi:10.1214/13-aap924

Cited by 38 publications

(52 citation statements)

References 8 publications

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“…The transitions from extinction of species y to extinction of species x when the jump rate parameter t increases is reminiscent of the transition occurring with linear systems analyzed in [6] and [26]. The process {X t , Y t , I t } restricted to M y 0 is defined by Y t = 0 and the one dimensional dynamicsẊ = α It X(1 − a It X)…”

Section: Remarkmentioning

confidence: 99%

Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder”

Benaı̈m¹,

Lobry²

2016

Ann. Appl. Probab.

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110

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Section: Remarkmentioning

confidence: 99%

Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder”

Benaı̈m¹,

Lobry²

2016

Ann. Appl. Probab.

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110

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“…To simplify matters we suppose that the F i are C 1 and that there is a compact set K ⊂ R d that is left invariant by all flows (nasty things may occur if this is not the case, see e.g. [BLBMZ12a]). We also suppose that we are given n 2 nonnegative functions λ ij : R d → R (the jump rates), such that λ ii (y) = 0, and for any given y, (λ ij (y)) i,j is irreducible.…”

Section: The Markov Switching Modelmentioning

confidence: 99%

Piecewise deterministic Markov process — recent results

et al. 2014

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Abstract.We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, better results can be proved, especially as regards long time behaviour. We continue our review with an infinite dimensional example of neuronal activity. From the statistical point of view, these models provide specific challenges: we illustrate this point with the example of the estimation of the distribution of the inter-jumping times. We conclude with a short overview on numerical methods used for simulating PDMPs. General introductionThe piecewise deterministic Markov processes (denoted PDMPs) were first introduced in the literature by Davis ( [Dav84,Dav93]). Already at this time, the theory of diffusions had such powerful tools as the theory of Itō calculus and stochastic differential equations at its disposal. Davis's goal was to endow the PDMP with rather general tools. The main reason for that was to provide a general framework, since up to then only very particular cases had been dealt with, which turned out not to be easily generalizable.PDMPs form a family of càdlàg Markov processes involving a deterministic motion punctuated by random jumps. The motion of the PDMP {X(t)} t≥0 depends on three local characteristics, namely the jump rate λ, the flow φ and the transition measure Q according to which the location of the process at the jump time is chosen. The process starts from x and follows the flow φ(x, t) until the first jump time T 1 which occurs either spontaneously in a Poisson-like fashion with rate λ (φ(x, t)) or when the flow φ(x, t) hits the boundary of the state-space. In both cases, the location of the process at the jump time T 1 , denoted by Z 1 = X(T 1 ), is selected by the transition measure Q(φ(x, T 1 ), ·) and the motion restarts from this new point as before. This fully describes a piecewise continuous trajectory for {X(t)} with jump times {T k } and post jump locations {Z k }, and which evolves according to the flow φ between two jumps. Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx

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“…A detailed bifurcation analysis of (21) and the underlying stochastic system will be included in a forthcoming publication. Such a blowup is reminiscent of stochastically switched linear ODEs that blowup despite switching between only stable systems [3,24].…”

Section: Pde Examplesmentioning

confidence: 99%

Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives

Lawley¹

2016

SIAM J. Appl. Dyn. Syst.

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Abstract. Driven by diverse applications, several recent models impose randomly switching boundary conditions on either a PDE or SDE. The purpose of this paper is to provide tools for calculating statistics of these models and to establish a connection between these two perspectives on diffusion in a random environment. Under general conditions, we prove that the moments of a solution to a randomly switching PDE satisfy a hierarchy of BVPs with lower order moments coupling to higher order moments at the boundaries. Further, we prove that joint exit statistics for a set of particles following a randomly switching SDE satisfy a corresponding hierarchy of BVPs. In particular, the M -th moment of a solution to a switching PDE corresponds to exit statistics for M particles following a switching SDE. We note that though the particles are non-interacting, they are nonetheless correlated because they all follow the same switching SDE. Finally, we give several examples of how our theorems reveal the sometimes surprising dynamics of these systems.keywords. random PDE, stochastic hybrid system, piecewise deterministic Markov process, randomly switching boundary, switched dynamical system.

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On the stability of planar randomly switched systems

Cited by 38 publications

References 8 publications

Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder”

Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder”

Piecewise deterministic Markov process — recent results

Boundary Value Problems for Statistics of Diffusion in a Randomly Switching Environment: PDE and SDE Perspectives

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