一类分段确定马氏过程的平稳性、时反性和涨落理论
Stationarity, time--reversal and fluctuation theory for a class of piecewise deterministic Markov processes
摘要：
本文考虑一类随机动力系统，称为分段确定马尔可夫过程，其状态为$(x, \s)\in Ø\times \G$，$Ø$是$\bbR^d$或$d$-维环面中的一个区域，$\G$是一个有限集.连续变量$x$服从分段确定性动力学，离散变量$\s$服从随机跳跃动力学，两个演化过程是完全耦合的.我们研究了该过程的平稳性、可逆性和时间反演对称性。增加频率的$\s$-跳跃，我们表明，该系统的行为渐近确定性和我们调查的波动结构（即偏离渐近行为），恢复在一个非马尔可夫框架的结果得到贝尔蒂尼等人。\cite{BDGJL1, BDGJL2, BDGJL3, BDGJL4}，在马尔可夫随机相互作用粒子系统的背景下。最后讨论了一种不同于时间反演的对合映射的Gallavotti-Cohen型对称关系。对于几个例子，上述结果通过显式计算恢复。
英文摘要：
We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states $(x, \s)\in Ø\times \G$, $Ø$ 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.
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