Exponential ergodicity for Markov processes with random switching

By Bertrand Cloez, Martin Hairer. 2013

158 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/207a5af8d758d83d2068614a1ed7244327b972c0

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Citations in the corpus

Quantitative contraction rates for Markov chains on general state spaces (2018, background): “Meanwhile there have been numerous extensions and applications of their result [25, 6, 3, 11].”
Ergodicity of regime-switching diffusions in Wasserstein distances (2014, methodology): “The cost function used in [8] to define the Wasserstein distance is bounded.”
Criteria for transience and recurrence of regime-switching diffusion processes * (2014, methodology): “The cost function used to define the Wasserstein distance in [4] is bounded.”
Qualitative properties of certain piecewise deterministic Markov processes (2012, background): “As a consequence, it is difficult to establish, in our general framework, sufficient conditions for the stability of the process under study without the invariance assumption (1); results in this direction may however be found in the recent [8].”
Approximation of Invariant Measures for Regime-switching Diffusions (2014, background): “So far, the works on RSDPs have included ergodicity [3, 9, 22, 21, 26], stability [15, 21, 27, 31], recurrence and transience [18, 19, 20, 31], invariant densities [1, 2], hypoellipticity [1, 4], and so forth.”