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Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts

Author

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  • Wujun Lv

    (Donghua University)

  • Xing Huang

    (Tianjin University)

Abstract

The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where the drift is assumed to be Hölder–Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack inequality is derived by the method of coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case. An example is presented in the final part of the paper.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:2:d:10.1007_s10959-020-00989-z
    DOI: 10.1007/s10959-020-00989-z
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    References listed on IDEAS

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    1. Freidlin, Mark & Weber, Matthias, 2001. "On random perturbations of Hamiltonian systems with many degrees of freedom," Stochastic Processes and their Applications, Elsevier, vol. 94(2), pages 199-239, August.
    2. G. Prato & F. Flandoli & E. Priola & M. Röckner, 2015. "Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1571-1600, December.
    3. 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.
    4. Zhang, Xicheng, 2005. "Strong solutions of SDES with singular drift and Sobolev diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1805-1818, November.
    5. Chetan D. Pahlajani, 2015. "Stochastic Averaging for a Hamiltonian System with Skew Random Perturbations," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1165-1226, September.
    6. Bao, Jianhai & Wang, Feng-Yu & Yuan, Chenggui, 2015. "Hypercontractivity for functional stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3636-3656.
    7. Wang, Feng-Yu & Yuan, Chenggui, 2011. "Harnack inequalities for functional SDEs with multiplicative noise and applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2692-2710, November.
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