Stationarity, time--reversal and fluctuation theory for a class of piecewise deterministic Markov processes
Abstract
We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states (x, )∈ × , being a region in d or the d--dimensional torus, being a finite set. The continuous variable x follows a piecewise deterministic dynamics, the discrete variable 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 --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. 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.