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Exponential ergodicity of Lévy driven Langevin dynamics with singular potentials

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  • Bao, Jianhai
  • Fang, Rongjuan
  • Wang, Jian

Abstract

In this paper, we address exponential ergodicity for Lévy driven Langevin dynamics with singular potentials, which can be used to model the time evolution of a molecular system consisting of N particles moving in Rd and subject to discontinuous stochastic forces. In particular, our results are applicable to the singular setups concerned with not only the Lennard-Jones-like interaction potentials but also the Coulomb potentials. In addition to Harris’ theorem, the approach is based on novel constructions of proper Lyapunov functions (which are completely different from the setting for Langevin dynamics driven by Brownian motions), on invoking the Hörmander theorem for non-local operators and on solving the issue on an approximate controllability of the associated deterministic system as well as on exploiting the time-change idea.

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  • 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).
    • Handle: RePEc:eee:spapps:v:172:y:2024:i:c:s0304414924000474
    DOI: 10.1016/j.spa.2024.104341
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    References listed on IDEAS

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    1. Zimo Hao & Xuhui Peng & Xicheng Zhang, 2021. "Hörmander’s Hypoelliptic Theorem for Nonlocal Operators," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1870-1916, December.
    2. 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.
    3. 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.
    4. 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.
    5. 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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