Qualitative properties of certain piecewise deterministic Markov processes
Qualitative properties of certain piecewise deterministic Markov processes
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DOI：
10.1214/14-aihp619
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发表时间：
2015-08-01
影响因子：
1.5
通讯作者：
Zitt, Pierre-Andre
中科院分区：
文献类型：
--
作者：
Benaim, Michel;Le Borgne, Stephane;Zitt, Pierre-Andre
研究了一类状态空间为R-dxE的分段确定马尔可夫过程，其中E是有限集.连续分量根据在离散坐标的跳跃时间处切换的平滑向量场而演变。跳跃率可能取决于整个过程的位置。工作的一般假设下，该过程保持在一个紧凑的集合，我们详细说明了一个可能的建设的过程，并描述其支持，在微分包含的解决方案集。我们建立的过程中的长期行为的结果，在一定的一组可访问的点，这是强有力的支持不变的措施。在Hormander型括号条件下，我们证明了存在唯一不变测度，且过程按全变差收敛到平衡点。最后，我们给出的例子中，括号条件不成立，可能有一个或多个不变的措施，这取决于流之间的跳跃率。
We study a class of piecewise deterministic Markov processes with state space R-d x E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hormander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.