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Strict Kantorovich contractions for Markov chains and Euler schemes with general noise

Author

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  • Huang, Lu-Jing
  • Majka, Mateusz B.
  • Wang, Jian

Abstract

We study contractions of Markov chains on general metric spaces with respect to some carefully designed distance-like functions, which are comparable to the total variation and the standard Lp-Wasserstein distances for p≥1. We present explicit lower bounds of the corresponding contraction rates. By employing the refined basic coupling and the coupling by reflection, the results are applied to Markov chains whose transitions include additive stochastic noises that are not necessarily isotropic. This can be useful in the study of Euler schemes for SDEs driven by Lévy noises. In particular, motivated by recent works on the use of heavy tailed processes in Markov Chain Monte Carlo, we show that chains driven by the α-stable noise can have better contraction rates than corresponding chains driven by the Gaussian noise, due to the heavy tails of the α-stable distribution.

Suggested Citation

  • Huang, Lu-Jing & Majka, Mateusz B. & Wang, Jian, 2022. "Strict Kantorovich contractions for Markov chains and Euler schemes with general noise," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 307-341.
    • Handle: RePEc:eee:spapps:v:151:y:2022:i:c:p:307-341
    DOI: 10.1016/j.spa.2022.06.011
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    References listed on IDEAS

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    1. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.
    2. Majka, Mateusz B., 2017. "Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4083-4125.
    3. Liang, Mingjie & Wang, Jian, 2020. "Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3053-3094.
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