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Solutions to the Linear Stochastic Heat and Wave Equation

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Analysis of Variations for Self-similar Processes

Part of the book series: Probability and Its Applications ((PIA))

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Abstract

The solutions to certain stochastic partial differential equations with linear Gaussian noise constitute interesting examples of self-similar processes. In this chapter we analyze these classes of self-similar processes. We focus on the solution to the linear heat and wave equation driven by a Gaussian noise which behaves as a Brownian motion or fractional Brownian motion with respect to the time variable and is white or colored with respect to the space variable. We consider various aspects of these self-similar processes. In particular we present the conditions for the existence of the solution, the sharp regularity of their trajectories, we study the law of the solution to the linear heat equation and its connection with the bifractional Brownian motion.

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References

  1. R. Balan, C.A. Tudor, The stochastic heat equation with fractional-colored noise: existence of the solution. Latin Am. J. Probab. Math. Stat. 4, 57–87 (2008)

    MathSciNet  MATH  Google Scholar 

  2. R.M. Balan, C.A. Tudor, The stochastic wave equation with fractional noise: a random field approach. Stoch. Process. Appl. 120, 2468–2494 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Bayraktar, V. Poor, R. Sircar, Estimating the fractal dimension of the SP 500 index using wavelet analysis. Int. J. Theor. Appl. Finance 7, 615–643 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems. Electron. Commun. Probab. 12, 161–172 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Bourguin, C.A. Tudor, On the law of the solution to a stochastic heat equation with fractional noise in time. Preprint (2011)

    Google Scholar 

  6. D. del Castillo-Negrete, B.A. Carreras, V.E. Lynch, Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach. Phys. Rev. Lett. 91 (2003)

    Google Scholar 

  7. J. Clarke De la Cerda, C.A. Tudor, Hitting times for the stochastic wave equation with fractional-colored noise. Rev. Mat. Iberoam. (2012, to appear)

    Google Scholar 

  8. R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDE’s. Electron. J. Probab. 4, 1–29 (1999). Erratum in Electr. J. Probab. 6 (2001) 5 pp.

    Article  MathSciNet  Google Scholar 

  9. R.C. Dalang, N. Frangos, The stochastic wave equation in two spatial dimensions. Ann. Probab. 26, 187–212 (1998)

    Article  MathSciNet  Google Scholar 

  10. R.C. Dalang, C. Mueller, Some non-linear SPDE’s that are second order in time. Electron. J. Probab. 8, 1–21 (2003)

    Article  MathSciNet  Google Scholar 

  11. R.C. Dalang, M. Sanz-Solé, Regularity of the sample paths of a class of second order SPDE’s. J. Funct. Anal. 227, 304–337 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three. Mem. Am. Math. Soc. 199(931), 1–70 (2009)

    Google Scholar 

  13. R.C. Dalang, M. Sanz-Solé, Criteria for hitting probabilities with applications to systems of stochastic wave equations. Bernoulli 16(4), 1343–1368 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Denk, D. Meintrup, S. Schaffer, Modeling, simulation and optimization of integrated circuits. Int. Ser. Numer. Math. 146, 251–267 (2004)

    Google Scholar 

  15. K. Dzhaparidze, H. van Zanten, A series expansion of fractional Brownian motion. Probab. Theory Relat. Fields 130, 39–55 (2004)

    Article  MATH  Google Scholar 

  16. M. Gubinelli, A. Lejay, S. Tindel, Young integrals and SPDE’s. Potential Anal. 25, 307–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. S.C. Kou, X. Sunney, Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett. 93(18) (2004)

    Google Scholar 

  18. J.A. Leon, S. Tindel, Ito’s formula for linear fractional PDEs. Stoch. Int. J. Probab. Stoch. Process. 80(5), 427–450 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Maejima, C.A. Tudor, Wiener integrals and a Non-central limit theorem for Hermite processes. Stoch. Anal. Appl. 25(5), 1043–1056 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Maslovski, D. Nualart, Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202, 277–305 (2003)

    Article  MathSciNet  Google Scholar 

  21. A. Millet, M. Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27, 803–844 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Nualart, Malliavin Calculus and Related Topics, 2nd edn. (Springer, New York, 2006)

    MATH  Google Scholar 

  23. D. Nualart, P.-A. Vuillermont, Variational solutions for partial differential equations driven by fractional a noise. J. Funct. Anal. 232, 390–454 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Ouahhabi, C.A. Tudor, Additive functionals of the solution to fractional stochastic heat equation. Preprint. J. Fourier Anal. Appl. (2012). doi:10.1007/s00041-013-9272-7

    Google Scholar 

  25. S. Peszat, J. Zabczyk, Nonlinear stochastic wave and heat equations. Probab. Theory Relat. Fields 116, 421–443 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Quer-Sardanyons, S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stoch. Process. Appl. 117, 1448–1472 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Sanz-Solé, Malliavin Calculus with Applications to Stochastic Partial Differential Equations (EPFL Press, Lausanne, 2005)

    Book  MATH  Google Scholar 

  28. J. Swanson, Variations of the solution to a stochastic heat equation. Ann. Probab. 35(6), 2122–2159 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Tindel, C.A. Tudor, F. Viens, Stochastic evolution equations with fractional Brownian motion. Probab. Theory Relat. Fields 127, 186–204 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Treves, Basic Linear Partial Differential Equations (Academic Press, New York, 1975)

    MATH  Google Scholar 

  31. J.B. Walsh, An introduction to stochastic partial differential equations, in Ecole d’Eté de Probabilités de Saint-Flour XIV. Lecture Notes in Math., vol. 1180 (Springer, Berlin, 1986), pp. 265–439

    Google Scholar 

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Tudor, C.A. (2013). Solutions to the Linear Stochastic Heat and Wave Equation. In: Analysis of Variations for Self-similar Processes. Probability and Its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-00936-0_2

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