Exponential ergodicity for Markov processes with random switching
Exponential ergodicity for Markov processes with random switching
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DOI：
10.3150/13-bej577
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发表时间：
2013-03
期刊：
影响因子：
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通讯作者：
B. Cloez;Martin Hairer
中科院分区：
文献类型：
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作者：
B. Cloez;Martin Hairer
我们研究了一个马尔可夫过程的两个组成部分：第一个组成部分的演变，根据一个众多的底层马尔可夫动力学，与选择的动态变化的跳跃时间的第二个组成部分。第二个分量是离散的，其跳跃率可能取决于整个过程的位置。根据规律性假设的跳跃率和Wasserstein收缩条件的基本动力学，我们提供了一个具体的标准收敛到平衡的Wasserstein距离。证明是基于一个耦合参数和弱形式的哈里斯定理。特别是，我们得到指数遍历的情况下，不验证任何hypoellipticity假设，但不一致收缩。在适当的正则化假设下，我们还获得了总变差距离的界。给出了一些例子来说明我们的结果，包括一类分段确定的马尔可夫过程。
We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of the Harris theorem. In particular, we obtain exponential ergodicity in situations which do not verify any hypoellipticity assumption, but are not uniformly contracting either. We also obtain a bound in total variation distance under a suitable regularising assumption. Some examples are given to illustrate our result, including a class of piecewise deterministic Markov processes.