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Stochastic flows and Bismut formulas for stochastic Hamiltonian systems

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  • Zhang, Xicheng

Abstract

We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is obtained. We also use Zvonkin's transformation to derive a stochastic flow of C1-diffeomorphisms for non-degenerate SDEs with Hölder continuous drifts. Next, we prove a Bismut type formula for certain degenerate SDEs. Lastly, we apply our results to stochastic Hamiltonian systems, which in particular covers the following stochastic nonlinear oscillator equation where has a bounded first order derivative, and is a one dimensional Brownian white noise.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:10:p:1929-1949
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    References listed on IDEAS

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    1. Carmona, Philippe, 2007. "Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1076-1092, August.
    2. Zhang, Xicheng, 2005. "Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 435-448, March.
    3. 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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    Cited by:

    1. Uda, Kenneth, 2021. "Averaging principle for stochastic differential equations in the random periodic regime," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 1-36.
    2. Xu, Jie & Wen, Jiaping & Mu, Jianyong & Liu, Jicheng, 2019. "Stochastic flows of SDEs with non-Lipschitz coefficients and singular time," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 118-127.
    3. Zhang, Xicheng, 2013. "Derivative formulas and gradient estimates for SDEs driven by α-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1213-1228.
    4. Feng-Yu Wang, 2014. "Derivative Formula and Gradient Estimates for Gruschin Type Semigroups," Journal of Theoretical Probability, Springer, vol. 27(1), pages 80-95, March.
    5. Xiliang Fan, 2019. "Derivative Formulas and Applications for Degenerate Stochastic Differential Equations with Fractional Noises," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1360-1381, September.
    6. Yan, Litan & Yin, Xiuwei, 2018. "Bismut formula for a stochastic heat equation with fractional noise," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 165-172.
    7. Jianhai Bao & Xing Huang & Chenggui Yuan, 2019. "Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts," Journal of Theoretical Probability, Springer, vol. 32(2), pages 848-871, June.
    8. Bao, Jianhai & Wang, Feng-Yu & Yuan, Chenggui, 2019. "Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4576-4596.
    9. Wujun Lv & Xing Huang, 2021. "Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts," Journal of Theoretical Probability, Springer, vol. 34(2), pages 827-851, June.

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