Bernoulli

  • Bernoulli
  • Volume 21, Number 1 (2015), 505-536.

Exponential ergodicity for Markov processes with random switching

Bertrand Cloez and Martin Hairer

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Abstract

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.

Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 505-536.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597080

Digital Object Identifier
doi:10.3150/13-BEJ577

Mathematical Reviews number (MathSciNet)
MR3322329

Zentralblatt MATH identifier
1330.60094

Keywords
ergodicity exponential mixing piecewise deterministic Markov process switching Wasserstein distance

Citation

Cloez, Bertrand; Hairer, Martin. Exponential ergodicity for Markov processes with random switching. Bernoulli 21 (2015), no. 1, 505--536. doi:10.3150/13-BEJ577. https://projecteuclid.org/euclid.bj/1426597080


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