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Reflected SPDEs Driven by Fractional Noises

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Abstract

In this paper, we study the stochastic partial differential equation with two reflecting smooth walls h1 and h2, driven by a fractional noise, which is fractional in time and white in space. The large deviation principle for the law of the solution to this equation, will be established through developing a classical method. Furthermore, we obtain the Hölder continuity of the solution.

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References

  1. Biagini, F., Hu, Y.Z., Øksendal, B., Zhang, T.S. Stochastic Calculus for Fractional Brownian Motion and Applications. Springer-Verlag, London, 2008

    Book  MATH  Google Scholar 

  2. Chenal, F., Millet, A. Uniform large deviations for parabolic SPDEs and applications. Stoch. Proc. Appl., 72: 161–186 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cerrai, S., Roeckner, M. Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Probab., 32(1): 1100–1139 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Dalang, R., Mueller, C., Zambotti, L. Hitting properties of parabolic SPDE’s with reflection. Ann. Probab., 34(4): 1423–1450 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Donati-Martin, C., Pardoux, E. White noise driven SPDEs with reflection. Probab. Theory Relat. Fields, 95: 1–24 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dalang R., Zhang, T.S. H¨older continuity of solutions of SPDEs with reflection. Commun. Math. Stat., 1: 133–142 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jiang, Y.M., Shi, K.H., Wang, Y.J. Large deviation principle for the forth-order stochastic heat equations with fractional noises. Acta Math. Sinica., 26(1): 89–106 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mémin, J., Mishura, Y., Valkeila, E. Inequalities for moments of Wiener integral with respect to a fractional Brownian motion. Stat. Probab. Letter, 51: 197–206 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Naulart, D., Pardoux, E. White noise driven quasilinear SPDEs with reflection. Probab. Theory Relat. Fields, 93: 77–89 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Nualart, E., Viens, F. The fractional stochastic heat equation on the circle: Time regularity and potential theory. Stoch. Proc. Appl., 119(5): 1505–1540 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Otobe, Y. Stochastic partial differential equations with two reflecting walls. J. Math. Sci. Univ. Tokyo, 13: 139–144 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Øksendal, B., Zhang, T.S. Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stoch. Stoch. Rep., 71(3–4): 141–163 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sower, R.B. Large deviations for a reaction-diffusion equation with non-Gaussian perturbations. Ann. Probab., 20(1): 504–537 (1992)

    Article  MathSciNet  Google Scholar 

  14. Walsh, J. An introduction to stochastic partial differential equations. Lecture Notes in Math., Vol. 1180, Springer, Berlin, Heidelberg, 1986

  15. Xie, B. Gradient estimate and log-Harnack inequality for reflected SPDEs driven by multiplicative noises. arXiv.1805.01169v1 (2018)

    Google Scholar 

  16. Xu, T.G., Zhang, T.S. White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Stoch. Proc. Appl., 119(10): 3453–3470 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang, J., Zhai, J.L. SPDEs with two reflecting walls and two singular drifts. J. Math. Anal. Appl., 438: 991–1009 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, J., Zhang, T.S. Existence and uniqueness of invariant measures for SPDEs with two reflecting walls. J. Theoret. Probab., 27(3): 863–877 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zambotti, L. A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge. J. Funct. Anal., 180: 195–209 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, T.S. White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities. Potential Anal., 33(2): 137–151 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, T.S. Large deviations for invariant measures of SPDEs with two reflecting walls. Stoch. Proc. Appl., 122(10): 3425–3444 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, T.S., Yang, J. White noise driven SPDEs with two reflecting walls. Inf. Dim. Anal. Quant. Probab. Rel. Top., 14(4): 647–659 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are very grateful to the editors and the anonymous referees for their careful reading of the paper and helpful suggestions.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Juan Yang & Qing Zhou

Authors
  1. Juan Yang
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  2. Qing Zhou
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Corresponding author

Correspondence to Qing Zhou.

Additional information

This paper is supported by the National Natural Science Foundation of China (Nos. 11871010, 11871116, 11971040) and by the Fundamental Research Funds for the Central Universities (No. 2019XD-A11).

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Yang, J., Zhou, Q. Reflected SPDEs Driven by Fractional Noises. Acta Math. Appl. Sin. Engl. Ser. 36, 347–360 (2020). https://doi.org/10.1007/s10255-020-0938-z

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  • DOI: https://doi.org/10.1007/s10255-020-0938-z

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