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Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

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  • Liang, Mingjie
  • Wang, Jian

Abstract

We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to α-stable-like processes. By assuming that the SDE has a unique strong solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the L1-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which has been open for a long time in this area.

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  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3053-3094
    DOI: 10.1016/j.spa.2019.09.001
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    References listed on IDEAS

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    1. Bass, Richard F. & Burdzy, Krzysztof & Chen, Zhen-Qing, 2004. "Stochastic differential equations driven by stable processes for which pathwise uniqueness fails," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 1-15, May.
    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. Mátyás Barczy & Zenghu Li & Gyula Pap, 2015. "Yamada-Watanabe Results for Stochastic Differential Equations with Jumps," International Journal of Stochastic Analysis, Hindawi, vol. 2015, pages 1-23, January.
    4. Wang, Linlin & Xie, Longjie & Zhang, Xicheng, 2015. "Derivative formulae for SDEs driven by multiplicative α-stable-like processes," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 867-885.
    5. 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.
    6. Kulik, Alexey M., 2009. "Exponential ergodicity of the solutions to SDE's with a jump noise," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 602-632, February.
    7. Masuda, Hiroki, 2007. "Ergodicity and exponential [beta]-mixing bounds for multidimensional diffusions with jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 35-56, January.
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    Cited by:

    1. Kulczycki, Tadeusz & Ryznar, Michał, 2020. "Semigroup properties of solutions of SDEs driven by Lévy processes with independent coordinates," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7185-7217.
    2. Bao, Jianhai & Wang, Jian, 2022. "Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 114-142.
    3. 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.
    4. Bao, Jianhai & Fang, Rongjuan & Wang, Jian, 2024. "Exponential ergodicity of Lévy driven Langevin dynamics with singular potentials," Stochastic Processes and their Applications, Elsevier, vol. 172(C).

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