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Generalized Stochastic Heat Equations

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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 study some properties about the solution of generalized stochastic heat equations driven by a Gaussian noise, white in time and correlated in space, and where the diffusion operator is the inverse of a Riesz potential for any positive fractional parameter. We prove the existence and uniqueness of solution and the Hölder continuity of this solution. In time, Hölder’s parameter does not depend on the fractional parameter. However, in space, Hölder’s parameter has a different behavior depending on the fractional parameter. Finally, we show that the law of the solution is absolutely continuous with respect to Lebesgue’s measure and its density is infinitely differentiable.

Mathematics Subject Classifications 2010:Primary 60H15, 60H30; Secondary 60G60, 60G15, 60H07

Received 10/10/2011; Accepted 12/24/2011; Final 3/12/2012

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References

  1. Ahn, V.V., Angulo, J.M., Ruiz-Medina, M.D.: Possible long-range dependence in fractional randoms fields. J. Stat. Plan. Inference 80, 95–110 (1999)

    Article  Google Scholar 

  2. Angulo, J.M., Ruiz-Medina, M.D., Anh, V.V., Grecksch, W.: Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32, 1077–1099 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Angulo, J.M., Anh, V.V., McVinish, R., Ruiz-Medina, M.D.: Fractional kinetic equations driven by Gaussian or infinitely divisible noise. Adv. Appl. Prob. 37(2), 366–392 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anh, V.V., Leonenko, N.N.: Spectral analysis of fractional kinetic equation with random data. J. Statist. Phys. 104(5–6), 1349–1387 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anh, V.V., Leonenko, N.N.: Renormalization and homogenization of fractional diffusion equations with random data. Prob. Theory Relat. Fields 124(3), 381–408 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boulanba, L., Eddahbi, M., Mellouk, M.: Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density. Osaka J. Math. 47(1), 41–65 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Cabré, X., Sanchón, M.: Semi-stable and extremal solutions of reaction equations involving the p-Laplacian. Commun. Pure Appl. Anal. 6(1), 43–67 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equat. 32(7–9), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dalang, R.C.: Extending the martingale measure stochastic integral with applications to spatially homogeneous spde’s. Electron. J. Probab. 4(6), 29 (electronic) (1999)

    Google Scholar 

  10. Dalang, R.C., Frangos, N.E.: The stochastic wave equation in two spatial dimensions. Ann. Prob. 26(1), 187–212 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  11. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2. Springer, Berlin (1995)

    Google Scholar 

  12. Debbi, L., Dozzi, M.: On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension. Stochastic Proc. Appl. 115, 1764–1781 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dibenedetto, E., Gianazza, U., Vespri, V.: Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electron. Res. Announc. Amer. Math. Soc. 12, 95–99 (electronic) (2006)

    Google Scholar 

  14. Hochberg, K.J.: A signed measure on path space related to Wiener measure. Ann. Prob. 6(3), 433–458

    Google Scholar 

  15. Jourdain, B., Méléard, S., Woyczynski, W.A.: A probabilistic approach for nonlinear equations involving the fractional Laplacian and a singular operator. Potential Anal. 23(1), 55–81 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jourdain, B., Méléard, S., Woyczynski, W.A.: Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws. Bernoulli 11(4), 689–714 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Krylov, V.Y.: Some properties of the distribution corresponding to the equation \(\frac{\partial u} {\partial t} = (-1)^ q+1 \frac{{\partial }^{2}q} {\partial {x}^{2q}}\). Soviet Math. Dokl. 1, 760–763 (1960)

    MathSciNet  MATH  Google Scholar 

  18. Márquez-Carreras, D.: Generalized fractional kinetic equations: another point of view. Adv. Appl. Prob. 41(3), 893–910 (2009)

    Article  MATH  Google Scholar 

  19. Márquez-Carreras, D., Sarrà, M.: Large deviation principle for a stochastic heat equation with spatially correlated noise. Electron. J. Probab. 8(12), 39 (electronic) (2003)

    Google Scholar 

  20. Márquez-Carreras, D., Mellouk, M., Sarrà, M.: On stochastic partial differential equations with spatially correlated noise: smoothness of the law. Stoch. Process. Appl. 93, 269–284 (2001)

    Article  MATH  Google Scholar 

  21. Micu, S., Zuazua, E.: (2006) On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44(6), 1950–1972 (electronic)

    Google Scholar 

  22. Ruiz-Medina, M.D., Angulo, J.M., Anh, V.V.: Scaling limit solution of a fractional Burgers equation. Stoch. Process. Appl. 93(2), 285–300 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives. Gordon and Breach Science Publishers, New York (1987)

    MATH  Google Scholar 

  24. Sanz-Solè, M., Sarrà, M.: Path properties of a class of Gaussian processes with applications to SPDEs. CMS Proceedings 28, 308–316 (2000)

    Google Scholar 

  25. Sanz-Solè, M., Sarrà, M.: Hölder continuity for the stochastic heat equation with spatially correlated noise. Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999). Progress in Probability, vol. 52, pp. 259–268. Birkhäuser, Basel (2002)

    Google Scholar 

  26. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)

    MATH  Google Scholar 

  27. Stein, E.M.: Singular integrals and differentiating properties of Functions. Princeton University Press, Princeton (1970)

    Google Scholar 

  28. Walsh, J.B.: An introduction stochastic partial differential equations. Ecole d’ete de probabilites de Saint Flour XIV 1984, Lecture notes in Mathemathics, vol. 1180, pp. 265–439. Springer, Berlin (1986)

    Google Scholar 

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Acknowledgements

David Márquez-Carreras was partially supported by MTM2009-07203 and 2009SGR-1360.

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Authors and Affiliations

  1. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    David Márquez-Carreras

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  1. David Márquez-Carreras
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Correspondence to David Márquez-Carreras .

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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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Márquez-Carreras, D. (2013). Generalized Stochastic Heat Equations. 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_12

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