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We consider a class of stochastic dynamical systems, called piecewisedeterministic 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 finiteset. The continuous variable $x$ follows a piecewise deterministic dynamics,the discrete variable $\s$ evolves by a stochastic jump dynamics and the tworesulting evolutions are fully--coupled. We study stationarity, reversibilityand time--reversal symmetries of the process. Increasing the frequency of the$\s$--jumps, we show that the system behaves asymptotically as deterministicand we investigate the structure of fluctuations (i.e. deviations from theasymptotic behavior), recovering in a non Markovian frame results obtained byBertini et al. \cite{BDGJL1, BDGJL2, BDGJL3, BDGJL4}, in the context ofMarkovian stochastic interacting particle systems. Finally, we discuss aGallavotti--Cohen--type symmetry relation with involution map different fromtime--reversal. For several examples the above results are recovered byexplicit computations.

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