In probability theory, a piecewise deterministic markov process pdmp is a process whose behaviour is governed by random jumps at points in time, but whose evolution is deterministically governed by an ordinary differential equation between those times. Existence of an invariant probability measure for a stochastic process. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the. We present a short introduction into the framework of piecewise deterministic markov processes. Our analysis is based on known results on stability of continuoustime markov processes 4, 5, and properties of piecewise deterministic markov processes pdmps 6, 7. Stability of piecewisedeterministic markov processes ieee xplore. Dufour abstractin this paper we consider the long run average continuous control problem of piecewisedeterministic markov processes pdps for short. One can find the definition and many properties of these processes in the founding book of davis 1993. Piecewisedeterministic markov processes pdmps were in troduced by davis 3 as a general family of nondiffusion stochastic models suitable for formulating.
Pdf a general class of nondiffusion stochastic models is introduced. Piecewise deterministic markov processes springerlink. Piecewise deterministic markov processes for continuoustime. We first state tractable stability conditions for two typical frameworks motivated by. Qualitative properties of certain piecewise deterministic.
We derive a sufficient condition on the coefficients of the model to ensure the exponential ergodicity of the process under two different assumptions on the jumps. Piecewise deterministic markov processes and dynamic reliability. Control and communication in engineering, springer, new york. The main goal of this paper is to establish some equivalence results on stability, recurrence, and ergodicity between a piecewise deterministic markov process pdmp xt and an embedded. With markovian systems, convergence is most likely in a distributional. Stability and ergodicity of piecewise deterministic markov processes article pdf available in proceedings of the ieee conference on decision and control 47. The piecewise deterministic markov processes denoted pdmps were. Abstract of kinetic limits of piecewise deterministic markov processes and grain boundary coarsening by joe klobusicky, ph. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. Lyapunov stability criteria for such processes involve solving partial integrodifferential. Averaging for a fully coupled piecewisedeterministic markov.
The main goal of this paper is to establish some equivalence results on stability, recurrence, and ergodicity between a piecewise deterministic markov process pdmp x t and an embedded discretetime markov chain thetan generated by a markov kernel g that can be explicitly characterized in terms of the three local characteristics of the pdmp, leading to tractable criterion. We study a class of piecewise deterministic markov processes with state space rd. Stability conditions for a piecewise deterministic markov. Stability and ergodicity of piecewise deterministic markov processes o. Piecewise deterministic markov process archive ouverte hal.
In this paper, we study a form of stability for a general family of nondiffusion markov processes known in the literature as piecewisedeterministic markov process pdmp. Kinetic limits of piecewise deterministic markov processes. Markov processes with jumps, such as piecewisedeterministic markov processes, offer a signi. Simulation of genetic networks modelled by piecewise. Thinning and multilevel monte carlo methods for piecewise deterministic markov processes with an application to a stochastic morrislecar model part of.
In the present paper we study the stability of a threshold continuostime model that belongs to the class of piecewise deterministic markov processes. Average continuous control of piecewise deterministic markov processes o. Introduction the development of cell cycle models to account for the statistical prop. In this paper, we study a form of stability for a general family of nondiffusion markov processes known in the literature as piecewise deterministic markov process pdmp. The modeling is a key step in order to study the properties of the involved physical process. Oct 01, 2012 read linearization techniques for controlled piecewise deterministic markov processes. Stability of piecewise deterministic markovian metapopulation.
Some relations between pdmp without boundary and point processes are developed in the book of jacobsen. Approximation methods for piecewise deterministic markov. Piecewise markov deterministic processes, stochastic semigroups, par. Advances in applied probability latest issue cambridge core. Piecewise deterministic markov processes, applications in biology. Stability of fluid queueing systems with parallel servers and. Recently, interest has grown for their use to sample from a target distribution 4, 23, 5. Stability of piecewise deterministic markovian metapopulation processes on networks. Piecewise deterministic markov processes pdmp, similarly to diffusion processes, form an important class of markov processes, which are used to model random dynamical systems in numerous fields see e. Random dynamical systems with jumps and with a function type. On the stability of planar randomly switched systems.
We give an informal introduction to piecewise deterministic markov processes, covering the aspects relevant to these new monte carlo algorithms, with a view to making the development of new continuoustime monte carlo more accessible. Application to zubovs method, applied mathematics and optimization on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The term stability is not commonly used in the markov chain literature. In this paper, we study a form of stability for a general family of nondiffusion markov processes known in the literature as piecewisedeterministic markov process. This chapter contains the basic theory for piecewise deterministic markov processes, whether homogeneous or not, based exclusively on the theory of marked point processes from the previous chapters and presented through the device of viewing a pdmp as a process adapted to the filtration generated by an rcm. Piecewise deterministic markov processes pdmp appear in many areas, such as engineering, operations research, biology, economics, etc. Stability conditions for a piecewise deterministic markov process. Stability of piecewisedeterministic markov processes siam. Piecewise deterministic markov processes in biological models. Average continuous control of piecewise deterministic markov. Numerical methods for piecewise deterministic markov. The authors propose piecewise deterministic markov processes as an alternative approach to model gene regulatory networks. Stability and ergodicity of piecewise deterministic markov. We illustrate the abstract mathematical setting with a series of examples related to dispersal of biological systems, cell cycle models, gene expression, physiologically structured populations, as well as neural activity.
Davist imperial college, london read before the royal statistical society at a meeting organized by the research section on wednesday, may 2nd 1984, professor j. Piecewise deterministic markov processes and their invariant. Dufour abstract the main goal of this paper is to establish some equivalence results on stability, recurrence between a piecewise deterministic markov process pdmp for short fx tg and an embedded discretetime markov chain f n g generated by. Request pdf stability and ergodicity of piecewise deterministic markov processes the main goal of this paper is to establish some equivalence results on stability, recurrence, and ergodicity. Vincent lemaire, michele thieullen, nicolas thomas. Lyapunov stability criteria for such processes involve solving.
Probabilistic methods, simulation and stochastic differential equations. Meaningful results on the stability of jackson networks have been derived that are deeply rooted on considerations about the graph structure 34. In las04, stability and ergodicity via meyntweedie arguments are. Stability and ergodicity of piecewise deterministic markov processes. Stability of piecewisedeterministic markov processes. There is, to our knowledge, little literature on general stability criteria for r nvalued piecewise deterministic markov processes.
A hybrid simulation algorithm is presented and discussed, and several standard regulatory modules are analysed by numerical means. The the continuous component evolves according to a smooth vector. In dynamical systems literature, it is commonly used to mean asymptotic stability, i. We will henceforth call these piecewise deterministic processes or pdps. A general class of nondiffusion stochastic models by m. By stability here we mean the existence of an invariant probability measure for the pdmp. Control, optimisation and calculus of variations doi.