Multiscale Piecewise Deterministic Markov Process in Infinite Dimension: Central Limit Theorem and Langevin Approximation

TL;DR

This paper analyzes the fluctuations of infinite-dimensional slow-fast PDMPs, proving a central limit theorem and exploring Langevin approximations, motivated by stochastic Hodgkin-Huxley models of nerve fibers.

Contribution

It extends previous work by providing a detailed mathematical analysis of fluctuations and Langevin approximations for infinite-dimensional PDMPs, including general results for Hilbert space valued systems.

Findings

Proved a central limit theorem for infinite-dimensional PDMPs.

Developed Langevin approximations for the fluctuations.

Applied results to stochastic Hodgkin-Huxley models.

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.