Qualitative properties of certain piecewise deterministic Markov processes

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

doi:10.48550/arxiv.1204.4143

Cited by 8 publications

(13 citation statements)

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“…In the following, we propose a numerical scheme in order to compute numerically solutions of equation (1). A result of uniqueness is essential and needs an adapted test function space.…”

Section: Uniquenessmentioning

confidence: 99%

“…Indeed the method permits to view the global behavior of the density, whereas the classical approach with particles and Monte-Carlo simulations only describes the trajectories. With our method we can see how the stochastic jumps interact with the flow, which leads sometimes to some unexpected behaviors (see [1]). Moreover the method is robust with large values of δt, which permits to compute the asymptotic stationary states of a PDMP.…”

Section: Numerical Test and Conclusionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical methods for piecewise deterministic Markov processes with boundary

Goudenège¹

2018

Preprint

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In this paper is described the general aspect of a numerical method for piecewise deterministic Markov processes with boundary. Under very natural hypotheses, a crucial result about uniqueness of solution of a generalized Kolmogorov equation with respect to a test function space is proved. Next we prove the existence and uniqueness of a positive solution to the finite volume scheme without result about convergence. Finally different models of transmission control protocol window-size processes are simulated to illustrate the efficiency of the numerical method for describing the evolution of the density of a piecewise deterministic Markov process with boundary. Obviously some technical aspects have been skipped for reader convenience but the full theory will be exposed in a forthcoming paper in collaboration with C.

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“…In the following, we propose a numerical scheme in order to compute numerically solutions of equation (1). A result of uniqueness is essential and needs an adapted test function space.…”

Section: Uniquenessmentioning

confidence: 99%

Section: Numerical Test and Conclusionmentioning

confidence: 99%

Numerical methods for piecewise deterministic Markov processes with boundary

Goudenège¹

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…We start by proving an accessibility result for (2) in Proposition 4 below, which holds without any assumption on the existence of densities and relies on some stability properties of the underlying deterministic system (3). We refer the reader to [4] for similar ideas in the framework of Piecewise Deterministic Markov Processes.…”

Section: Positivity Of Densities For Models (2)mentioning

confidence: 99%

Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems

Höpfner¹,

Löcherbach²,

Thieullen³

2017

Bernoulli

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In this paper we study the existence of densities for strongly degenerate stochastic differential equations (SDEs) whose coefficients depend on time and are not globally Lipschitz. In these models neither local ellipticity nor the strong Hörmander condition is satisfied. In this general setting we show that continuous transition densities indeed exist in all neighborhoods of points where the weak Hörmander condition is satisfied. We also exhibit regions where these densities remain positive.We then apply these results to stochastic Hodgkin-Huxley models with periodic input as a first step towards the study of ergodicity properties of such systems in the sense of [27]- [28].

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“…These two results are different in nature since the first one relies on stability properties of the underlying deterministic system (5) while the second one does not. For the proof of the first result (Proposition 5) we refer the reader to [4] for similar ideas in the framework of Piecewise Deterministic Markov Processes.…”

Section: Positivity Of Densitiesmentioning

confidence: 99%

Transition densities for strongly degenerate time inhomogeneous random models

Höpfner¹,

Löcherbach²,

Thieullen³

2013

Preprint

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In this paper we study the existence of densities for strongly degenerate stochastic differential equations whose coefficients depend on time and are not globally Lipschitz. In these models neither local ellipticity nor the strong Hörmander condition is satisfied. In this general setting we show that continuous transition densities indeed exist in all neighborhoods of points where the weak Hörmander condition is satisfied. We also exhibit regions where these densities remain positive.We then apply these results to stochastic Hodgkin-Huxley models as a first step towards the study of ergodicity properties of such systems.

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Qualitative properties of certain piecewise deterministic Markov processes

Cited by 8 publications

References 0 publications

Numerical methods for piecewise deterministic Markov processes with boundary

Numerical methods for piecewise deterministic Markov processes with boundary

Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems

Transition densities for strongly degenerate time inhomogeneous random models

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