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The Smoluchowski–Kramers limits of stochastic differential equations with irregular coefficients

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  • Xie, Longjie
  • Yang, Li

Abstract

In this paper, we study the small mass limit of the dynamic of stochastic system with state dependent friction and diffusion. We prove the strong convergence of the original system under weak assumptions on the coefficients, where the limiting equation admits an additional noise-induced drift term. Furthermore, the rate of convergence is also obtained. Our results even provide non-trivial improvement of the previous limit theorems for the classical Langevin equation.

Suggested Citation

  • Xie, Longjie & Yang, Li, 2022. "The Smoluchowski–Kramers limits of stochastic differential equations with irregular coefficients," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 91-115.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:91-115
    DOI: 10.1016/j.spa.2022.04.016
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    References listed on IDEAS

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    1. Freidlin, Mark & Hu, Wenqing & Wentzell, Alexander, 2013. "Small mass asymptotic for the motion with vanishing friction," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 45-75.
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    3. Birrell, Jeremiah & Wehr, Jan, 2018. "Homogenization of dissipative, noisy, Hamiltonian dynamics," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2367-2403.
    4. Song, Renming & Xie, Longjie, 2020. "Well-posedness and long time behavior of singular Langevin stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1879-1896.
    5. 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.
    6. Xia, Pengcheng & Xie, Longjie & Zhang, Xicheng & Zhao, Guohuan, 2020. "Lq(Lp)-theory of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5188-5211.
    7. 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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