Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

Abstract

In [20], the authors addressed the question of the averaging of a slow-fast Piecewise Deterministic Markov Process (PDMP) in infinite dimension. In the present paper, we carry on and complete this work by the mathematical analysis of the fluctuation of the slow-fast system around the averaged limit. A central limit theorem is derived and the associated Langevin approximation is considered. The motivation of this work is a stochastic Hodgkin-Huxley model which describes the propagation of an action potential along the nerve fiber. We study this PDMP in detail and provide more general results for a class of Hilbert space valued PDMP.

Spatio-temporal averaging for a class of hybrid systems and application to conductance-based neuron models

“…To emphasize the role of ε and N in the process, we write from now on (X ε,N , Y ε,N ) for the process satisfying system (20-21) with initial conditions (X ε,N 0 , Y ε,N 0 ) ∈ H × E N (which may be random). Concerning averaging when N is held fixed, such a system have been studied in [9,10]. For a law of large number when N goes to infinity but with ε held fixed, see [19,18].…”

Spatio-temporal hybrid (PDMP) models: Central limit theorem and Langevin approximation for global fluctuations. Application to electrophysiology

“…(5. 16) Note, that in the equations (5.15) and (5.16) we omitted the time-dependence of the derivatives in the right hand sides. Thus the systems (5.15) and (5.16) decouple in two closed systems of differential equations, the first takes values in the space X * × E * and the second is operatorvalued.…”

“…In recent studies, PDMPs proved particularly useful for modelling in neuroscience and physiology under bottom-up ( [3,8,26,23]) or multi-scale approaches ( [15,16,24]). Indeed mathematical models for biological real-life processes are constructed on largely varying temporal and spatial scales.…”

Optimal stopping for measure-valued piecewise deterministic Markov processes

“…They belong to the family of hybrid processes with a discrete component called mode or regime interacting with a Euclidean component. PDMPs can model a wide area of phenomena from insurance and queuing problems [10], finance [1], reliability [11] to neuroscience [14,19], population dynamics [2,7] and many other fields. In this paper we are especially interested in population dynamics applications.…”

“…In this paper we are interested in extending the definition of PDMPs to measure-valued state spaces. Infinite dimensional PDMPs have already been introduced in [4] (see also [14,19]). In those papers, PDMPs take values in a separable Hilbert space and model spatio-temporal phenomena occurring on neuronal membranes.…”