Stationarity, time--reversal and fluctuation theory for a class of piecewise deterministic Markov processes

TL;DR

This paper analyzes the properties of a class of piecewise deterministic Markov processes, focusing on stationarity, reversibility, fluctuations, and symmetries, and explores their asymptotic deterministic behavior as jump frequency increases.

Contribution

It investigates the asymptotic behavior and fluctuation structure of PDMPs, extending non-Markovian results to this class and examining symmetry relations beyond time-reversal.

Findings

As jump frequency increases, the process becomes asymptotically deterministic.

Fluctuation structures are characterized, extending previous non-Markovian results.

A Gallavotti--Cohen--type symmetry relation with a different involution is identified.

Abstract

We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states $(x, \s)\in \O\times \G$, $\O$ being a region in $\bbR^d$ or the $d$--dimensional torus, $\G$ being a finite set. The continuous variable $x$ follows a piecewise deterministic dynamics, the discrete variable $\s$ evolves by a stochastic jump dynamics and the two resulting evolutions are fully--coupled. We study stationarity, reversibility and time--reversal symmetries of the process. Increasing the frequency of the $\s$--jumps, we show that the system behaves asymptotically as deterministic and we investigate the structure of fluctuations (i.e. deviations from the asymptotic behavior), recovering in a non Markovian frame results obtained by Bertini et al. \cite{BDGJL1, BDGJL2, BDGJL3, BDGJL4}, in the context of Markovian stochastic interacting particle systems. Finally, we discuss a Gallavotti--Cohen--type symmetry relation with involution map different from time--reversal. For several examples the above results are recovered by explicit computations.