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Elementary Pathwise Methods for Nonlinear Parabolic and Transport Type Stochastic Partial Differential Equations with Fractal Noise

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Modern Stochastics and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 90))

Abstract

We survey some of our recent results on existence, uniqueness, and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.

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References

  1. Ayache, A., Leger, S., Pontier, M.: Drap brownien fractionnaire. Potential Anal. 17, 31–43 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berens, H., Butzer, P.L., Westphal, U.: Representation of fractional powers of infinitesimal generators of semigroups. Bull. Am. Math. Soc. 74, 191–196 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bertini, L., Cancrini, N., Jona-Lasinio, G.: The stochastic Burgers equation. Comm. Math. Phys. 165, 211–232 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974)

    Book  MATH  Google Scholar 

  5. Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré 23, 245–287 (1987)

    MATH  MathSciNet  Google Scholar 

  6. Coulhon, T.: Ultracontractivity and Nash type inequalitites. J. Funct. Anal. 141, 510–539 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers equation. Nonlinear Diff. Equat. Appl. 1 (4), 389–402 (1994)

    Article  MATH  Google Scholar 

  8. Da Prato, G., Gatarek, D.: Stochastic Burgers equation with correlated noise. Stoch. Stoch. Rep. 52, 29–41 (1995)

    Article  MATH  Google Scholar 

  9. Da Prato, G., Zabzcyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  10. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  11. Friz, P.K., Victoir, N.: Multidimesnional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Gyöngy, I., Nualart, D.: On the stochastic Burgers equation in the real line. Ann. Probab. 27(2), 782–802 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)

    Book  Google Scholar 

  15. Hinz, M.: Burgers system with a fractional Brownian random force. Stochastics 83(1), 67–106 (2011)

    MATH  MathSciNet  Google Scholar 

  16. Hinz, M., Zähle, M.: Gradient type noises I - partial and hybrid integrals. Complex Var. Elliptic Equat. 54, 561–583 (2009)

    Article  MATH  Google Scholar 

  17. Hinz, M., Zähle, M.: Gradient type noises II - systems of partial differential equations. J. Funct. Anal. 256, 3192–3235 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hinz, M., Zähle, M.: Semigroups, potential spaces and applications to (S)PDE. Potential Anal. 36, 483–515 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations: A Modelling, White Noise Functional Approach. Birkhäuser, Boston (1996)

    Book  Google Scholar 

  20. Hu, Y., Nualart, D.: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields. 143, 285–328 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hu, J., Zähle, M.: Potential spaces on fractals. Studia Math. 170, 259–281 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hu, J., Zähle, M.: Generalized Bessel and Riesz potential spaces on metric measure spaces. Potential Anal. 30, 315–340 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Issoglio, E.: Transport equations with fractal noise: existence uniqueness and regularity of the solution. J. Anal. Appl. 32(1), 37–53 (2013)

    MATH  MathSciNet  Google Scholar 

  24. Klingenhöfer, F., Zähle, M.: Ordinary differential equations with fractal noise. Proc. Am. Math. Soc. 127, 1021–1028 (1999)

    Article  MATH  Google Scholar 

  25. Lyons, T.J.: Differential equations driven by rough signals I: an extension of an inequality by L.C. Young. Math. Res. Lett. 1, 451–464 (1994)

    Google Scholar 

  26. Lyons, T.J.: Differential equations driven by rough signals II. Rev. Iberoam. 14 (2), 215–310 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Nualart, D.: The Malliavin Calculus and Related Topics. Springer, New York (1995)

    Book  MATH  Google Scholar 

  29. Nualart, D., Rascanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53(1), 55–81 (2002)

    MATH  MathSciNet  Google Scholar 

  30. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  31. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1998)

    Google Scholar 

  32. Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, New York (2007)

    Google Scholar 

  33. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Non-linear Partial Differential Equations. De Gruyter, Berlin (1996)

    Book  Google Scholar 

  34. Russo, F., Trutnau, G.: Some parabolic PDE whose drift is an irregular random noise in space. Ann. Probab. 35(6), 2213–2262 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Russo, F., Vallois, P.: Forward, backward and symmetric stochastic integration. Probab. Theory Relat. Fields 97, 403–421 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  36. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  37. Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Annals of Mathematics Studies. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  39. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. J.A. Barth, Heidelberg (1995)

    MATH  Google Scholar 

  40. Varopoulos, N.: Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63, 240–260 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  41. Walsh, J.B.: An introduction to stochastic partial differential equations. École d’été de Probabilités de Saint-Flour, XIV-1984. Lecture Notes in Mathematics, vol. 1180. Springer, New York (1986)

    Google Scholar 

  42. Yosida, K.: Functional Analysis. Springer, New York (1980)

    Book  MATH  Google Scholar 

  43. Young, L.C.: An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)

    Article  MathSciNet  Google Scholar 

  44. Zähle, M.: Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Relat. Fields 111, (1998)

    Google Scholar 

  45. Zähle, M.: Integration with respect to fractal functions and stochastic calculus II. Math. Nachr. 225, 145–183 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  46. Zähle, M.: Forward integrals and stochastic differential equations. In: Dalang, R.C., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications III. Progress in Probability, pp. 293–302. Birkhäuser, Basel (2002)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Bielefeld University, Postfach 100131, 33501, Bielefeld, Germany

    Michael Hinz

  2. Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK

    Elena Issoglio

  3. Institute of Mathematics, Friedrich Schiller University, Jena, Jena, D-07737, Germany

    Martina Zähle

Authors
  1. Michael Hinz
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  2. Elena Issoglio
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  3. Martina Zähle
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Corresponding author

Correspondence to Martina Zähle .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Volodymyr Korolyuk

  2. Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, Compiegne Cedex, France

    Nikolaos Limnios

  3. Dept. of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Yuliya Mishura

  4. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Lyudmyla Sakhno

  5. Dept.of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    Georgiy Shevchenko

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Hinz, M., Issoglio, E., Zähle, M. (2014). Elementary Pathwise Methods for Nonlinear Parabolic and Transport Type Stochastic Partial Differential Equations with Fractal Noise. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_8

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