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Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises

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  • Bao, Jianhai
  • Wang, Jian

Abstract

We establish exponential ergodicity for the stochastic Hamiltonian system (Xt,Vt)t≥0 on R2d with Lévy noises dXt=(aXt+bVt)dt,dVt=U(Xt,Vt)dt+dLt,where a≥0, b>0, U:R2d→Rd and (Lt)t≥0 is an Rd-valued pure jump Lévy process. The approach is based on a new refined basic coupling for Lévy processes and a Lyapunov function for stochastic Hamiltonian systems. In particular, we can handle the case that U(x,v)=−v−∇U0(x) with double well potential U0 which is super-linear growth at infinity such as U0(x)=c1(1+|x|2)l−c2|x|2 with l>1 or U0(x)=c1e(1+|x|2)l−c2|x|2 with l>0 for any c1,c2>0, and also deal with the case that the Lévy measure ν of (Lt)t≥0 is degenerate in the sense that ν(dz)≥c|z|d+θ01{00 and θ0∈(0,2), where z1 is the first component of the vector z∈Rd.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:146:y:2022:i:c:p:114-142
    DOI: 10.1016/j.spa.2021.12.014
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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. Komorowski, Tomasz & Walczuk, Anna, 2012. "Central limit theorem for Markov processes with spectral gap in the Wasserstein metric," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2155-2184.
    3. 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.
    4. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    5. 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.
    6. Wu, Liming, 2001. "Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 205-238, February.
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