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Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems

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  • Wu, Liming

Abstract

A classical damping Hamiltonian system perturbed by a random force is considered. The locally uniform large deviation principle of Donsker and Varadhan is established for its occupation empirical measures for large time, under the condition, roughly speaking, that the force driven by the potential grows infinitely at infinity. Under the weaker condition that this force remains greater than some positive constant at infinity, we show that the system converges to its equilibrium measure with exponential rate, and obeys moreover the moderate deviation principle. Those results are obtained by constructing appropriate Lyapunov test functions, and are based on some results about large and moderate deviations and exponential convergence for general strong-Feller Markov processes. Moreover, these conditions on the potential are shown to be sharp.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:91:y:2001:i:2:p:205-238
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    1. Albeverio, Sergio & Kolokoltsov, Vassily N., 1997. "The rate of escape for some Gaussian processes and the scattering theory for their small perturbations," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 139-159, May.
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    Cited by:

    1. Dexheimer, Niklas & Strauch, Claudia, 2022. "Estimating the characteristics of stochastic damping Hamiltonian systems from continuous observations," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 321-362.
    2. Gourcy, Mathieu, 2007. "A large deviation principle for 2D stochastic Navier-Stokes equation," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 904-927, July.
    3. Comte, Fabienne & Prieur, Clémentine & Samson, Adeline, 2017. "Adaptive estimation for stochastic damping Hamiltonian systems under partial observation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3689-3718.
    4. P. Cattiaux & José R. León & C. Prieur, 2015. "Recursive estimation for stochastic damping hamiltonian systems," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 401-424, September.
    5. Liming Wu, 2009. "A Φ-Entropy Contraction Inequality for Gaussian Vectors," Journal of Theoretical Probability, Springer, vol. 22(4), pages 983-991, December.
    6. Anna Melnykova, 2020. "Parametric inference for hypoelliptic ergodic diffusions with full observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 595-635, October.
    7. Douc, Randal & Fort, Gersende & Guillin, Arnaud, 2009. "Subgeometric rates of convergence of f-ergodic strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 897-923, March.
    8. Xi, Fubao, 2009. "Asymptotic properties of jump-diffusion processes with state-dependent switching," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2198-2221, July.
    9. Zhang, Xicheng, 2010. "Stochastic flows and Bismut formulas for stochastic Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1929-1949, September.
    10. Ankit Kumar & Manil T. Mohan, 2023. "Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation," Journal of Theoretical Probability, Springer, vol. 36(1), pages 661-709, March.
    11. Guillin, A. & Liptser, R., 2005. "MDP for integral functionals of fast and slow processes with averaging," Stochastic Processes and their Applications, Elsevier, vol. 115(7), pages 1187-1207, July.
    12. Xi, Fubao & Yin, George, 2013. "The strong Feller property of switching jump-diffusion processes," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 761-767.
    13. Quentin Clairon & Adeline Samson, 2022. "Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations," Computational Statistics, Springer, vol. 37(5), pages 2471-2491, November.
    14. 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.
    15. Hu, Shulan & Wu, Liming, 2011. "Large deviations for random dynamical systems and applications to hidden Markov models," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 61-90, January.
    16. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    17. Xi, Fubao & Yin, G., 2010. "Asymptotic properties of nonlinear autoregressive Markov processes with state-dependent switching," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1378-1389, July.
    18. 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.
    19. 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.
    20. Hu, Shulan & Wang, Ran, 2020. "Asymptotics of stochastic Burgers equation with jumps," Statistics & Probability Letters, Elsevier, vol. 162(C).
    21. Wang, Ran & Xu, Lihu, 2018. "Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1772-1796.
    22. Guillin, Arnaud, 2001. "Moderate deviations of inhomogeneous functionals of Markov processes and application to averaging," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 287-313, April.
    23. Kontoyiannis, I. & Meyn, S.P., 2017. "Approximating a diffusion by a finite-state hidden Markov model," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2482-2507.
    24. Kulik, Alexey M., 2011. "Asymptotic and spectral properties of exponentially [phi]-ergodic Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1044-1075, May.
    25. Cattiaux, Patrick & León, José R. & Prieur, Clémentine, 2014. "Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1236-1260.

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