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Malliavin Calculus for Fractional Heat Equation

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Malliavin Calculus and Stochastic Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 34))

Abstract

In this article, we give some existence and smoothness results for the law of the solution to a stochastic heat equation driven by a finite dimensional fractional Brownian motion with Hurst parameter H > 1 ∕ 2. Our results rely on recent tools of Young integration for convolutional integrals combined with stochastic analysis methods for the study of laws of random variables defined on a Wiener space.

MSC Subject Classifications 2000: Primary 60H35; Secondary 60H07, 60H10, 65C30.

Received 9/2/2011; Accepted 6/1/2012; Final 6/26/2012

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References

  1. Adams, R.A.: Sobolev spaces. Pure and Applied Mathematics, vol. 65. Academic [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)

    Google Scholar 

  2. Baudoin, F., Hairer, M.: A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Relat. Fields 139(3–4), 373–395 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caruana, M., Friz, P.: Partial differential equations driven by rough paths. J. Differ. Equations 247(1), 140–173 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Caruana, M., Friz, P.K., Oberhauser, H.: A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(1), 27–46 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deya, A.: Numerical schemes for rough parabolic equations. Appl. Math. Optim. 65(2), 253–292 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deya, A.: A discrete approach to Rough Parabolic Equations. Electron. J. Probab. 16, 1489–1518 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Deya, A., Gubinelli, M., Tindel, S.: Non-linear rough heat equations. Probab. Theory Relat. Fields 153(1–2), 97–147 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Friz, P., Oberhauser, H.: Rough path stability of SPDEs arising in non-linear filtering. ArXiv preprint (2010)

    Google Scholar 

  9. Gubinelli, M., Lejay, A., Tindel, S.: Young integrals and SPDEs. Potential Anal. 25(4), 307–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gubinelli, M., Tindel, S.: Rough evolution equations. Ann. Probab. 38(1), 1–75 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202(1), 277–305 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nualart, D.: The Malliavin calculus and related topics. Probability and its Applications (New York), 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  13. Nualart, D., Saussereau, B.: Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stoch. Process. Appl. 119(2), 391–409 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    Google Scholar 

  15. Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. de Gruyter Series in Nonlinear Analysis and Applications, vol. 3. Walter de Gruyter, Berlin (1996)

    Google Scholar 

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Acknowledgements

S. Tindel is partially supported by the (French) ANR grant ECRU.

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Authors and Affiliations

  1. Institut Élie Cartan Nancy, Université de Nancy 1, 239, 54506, Vandœuvre-lès-Nancy Cedex, France

    Aurélien Deya & Samy Tindel

Authors
  1. Aurélien Deya
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  2. Samy Tindel
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Corresponding author

Correspondence to Aurélien Deya .

Editor information

Editors and Affiliations

  1. , Department of Statistics, Purdue University, North University Street 150, West Lafayette, 47907, Indiana, USA

    Frederi Viens

  2. , Department of Mathematics, University of Kansas, Jayhawk Blvd 1460, Lawrence, 66045, Kansas, USA

    Jin Feng

  3. Dept. Mathematics, University of Kansas, Snow Hall 405, Lawrence, 66045-2142, Kansas, USA

    Yaozhong Hu

  4. , Department of Economics and Business, University Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona, 08005, Spain

    Eulalia Nualart 

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Deya, A., Tindel, S. (2013). Malliavin Calculus for Fractional Heat Equation. In: Viens, F., Feng, J., Hu, Y., Nualart , E. (eds) Malliavin Calculus and Stochastic Analysis. Springer Proceedings in Mathematics & Statistics, vol 34. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5906-4_16

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