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White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles

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  • Xu, Tiange
  • Zhang, Tusheng

Abstract

In the first part of this paper, we prove the uniqueness of the solutions of SPDEs with reflection, which was left open in the paper [C. Donati-Martin, E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields 95 (1993) 1-24]. We also obtain the existence of the solution for more general coefficients depending on the past with a much shorter proof. In the second part of the paper, we establish a large deviation principle for SPDEs with reflection. The weak convergence approach is proven to be very efficient on this occasion.

Suggested Citation

  • Xu, Tiange & Zhang, Tusheng, 2009. "White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3453-3470, October.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3453-3470
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    Citations

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    Cited by:

    1. Ben Hambly & Jasdeep Kalsi & James Newbury, 2018. "Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models," Papers 1808.07107, arXiv.org, revised Jun 2019.
    2. Li, Ruinan & Li, Yumeng, 2020. "Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space–time white noise," Statistics & Probability Letters, Elsevier, vol. 161(C).
    3. Juan Yang & Tusheng Zhang, 2014. "Existence and Uniqueness of Invariant Measures for SPDEs with Two Reflecting Walls," Journal of Theoretical Probability, Springer, vol. 27(3), pages 863-877, September.
    4. Salins, M., 2021. "Systems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial data," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 159-194.
    5. Yang, Xue & Zhang, Jing, 2019. "The obstacle problem for quasilinear stochastic PDEs with degenerate operator," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3055-3079.
    6. Liu, Ruoyang & Tang, Shanjian, 2024. "The obstacle problem for stochastic porous media equations," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    7. Li, Ruinan & Zhang, Beibei, 2024. "A transportation inequality for reflected SPDEs on infinite spatial domain," Statistics & Probability Letters, Elsevier, vol. 206(C).
    8. Matoussi, Anis & Sabbagh, Wissal & Zhang, Tusheng, 2017. "Backward doubly SDEs and semilinear stochastic PDEs in a convex domain," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2781-2815.
    9. Zhang, Tusheng, 2012. "Large deviations for invariant measures of SPDEs with two reflecting walls," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3425-3444.
    10. Zhang, Tusheng, 2011. "Systems of stochastic partial differential equations with reflection: Existence and uniqueness," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1356-1372, June.
    11. Yang, Xue & Zhang, Qi & Zhang, Tusheng, 2020. "Reflected backward stochastic partial differential equations in a convex domain," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6038-6063.
    12. Yang, Xue, 2019. "Reflected backward stochastic partial differential equations with jumps in a convex domain," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 126-136.
    13. Jasdeep Kalsi, 2020. "Existence of Invariant Measures for Reflected Stochastic Partial Differential Equations," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1755-1767, September.
    14. Hambly, Ben & Kalsi, Jasdeep, 2020. "Stefan problems for reflected SPDEs driven by space–time white noise," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 924-961.

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