Probability Theory and Related Fields

, Volume 133, Issue 4, pp 437–463 | Cite as

Lp estimates for the uniform norm of solutions of quasilinear SPDE's

Article

Abstract

In this paper we prove Lp estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.

Mathematics Subject Classifications (2000)

60H15 60G46 35R60 

Key words or phrases

Stochastic partial differential equation Itô's formula Maximum principle Moser's iteration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aronson, D.G., Serrin, J.: Local behavior of solutions of quasi-linear parabolic equations. Arch. Ration. Mech. Anal. 25, 81–122 (1967)Google Scholar
  2. 2.
    Bally, V., Matoussi, A.: Weak solutions for SPDE's and Backward Doubly SDE's. J. Theoret. Probab. 14, 125–164 (2001)CrossRefGoogle Scholar
  3. 3.
    Bally, V., Pardoux, E., Stoica, L.: Backward stochastic equation associated to a symmetric Markov process. Potential Analysis 22, 17–60 (2005)Google Scholar
  4. 4.
    Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space, Kluwer, 1991, (1993)Google Scholar
  5. 5.
    Da Prato, G., Zabczyk, J.: Stochastic equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1992Google Scholar
  6. 6.
    Denis, L.: Solutions of SPDE considered as Dirichlet Processes. Bernoulli Journal of Probability, 10 (5), 783–827, (2004)Google Scholar
  7. 7.
    Denis, L., Stoica, I.L.: A general analytical result for non-linear s.p.d.e.'s and applications. Electronic Journal of Probability 9, 674–709 (2004)Google Scholar
  8. 8.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter studies in Math, 1994Google Scholar
  9. 9.
    Gyöngy, I., Rovira, C.: On LP-solutions of semilinear stochastic partial differental equations. Stochastic Processes and their Applications 90, 83–108 (2000)CrossRefGoogle Scholar
  10. 10.
    Krylov, N.V.: An analytic approach to SPDEs. Stochastic Partial Differential Equations: Six Perspectives, AMS Mathematical surveys an Monographs 64, 185–242 (1999)Google Scholar
  11. 11.
    Kunita, H.: Generalized Solutions of a Stochastic Partial Differential Equation. J. Theoret. Probab 7, 279–308 (1994)CrossRefGoogle Scholar
  12. 12.
    Matoussi, A., Scheutzow, M.: Semilinear Stochastic PDE's with nonlinear noise and Backward Doubly SDE's. J. Theoret. Probab. 15, 1–39 (2002)CrossRefGoogle Scholar
  13. 13.
    Mikulevicius, R., Rozovskii, B.L.: A Note on Krylov's Lp -Theory for Systems of SPDES. Electronic Journal of Probability 6 (12), 1–35 (2001)Google Scholar
  14. 14.
    Moser, J.: On Harnack's theorem for elliptic differential equation. Communications on Pures and applied Mathematics 4, 577–591 (1961)Google Scholar
  15. 15.
    Nualart, D., Pardoux, E.: White noise driven quasilinear SPDEs with reflectio. Probab. Theory Relat. Fields 93, 77–89 (1992)CrossRefGoogle Scholar
  16. 16.
    Pardoux, E.: Stochastic partial differential equations and filtering of diffusion process. Stochastics 3, 127–167 (1979)Google Scholar
  17. 17.
    Pardoux, E., Peng, S.: Backward doubly SDE's and systems of quasilinear SPDEs. Probab. Theory Relat. Field 98, 209–227 (1994)CrossRefGoogle Scholar
  18. 18.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, third edition, 1999Google Scholar
  19. 19.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, volume 2, Itô Calculus, 2000Google Scholar
  20. 20.
    Rozovskii, B.L.: Stochastic Evolution Systems, Kluver, Dordrecht- Boston- London, 1990Google Scholar
  21. 21.
    Walsh, J.B.: An introduction to stochastic partial differential equations. Ecole d'Eté de St-Flour XIV, 1984, Lect. Notes in Math 1180, Springer Verlag, 1986Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Département de Mathématiques, Equipe “Analyse et Probabilités”Université d'Evry-Val-d'EssonneEVRY CedexFrance
  2. 2.Département de Mathématiques, Equipe “Statistique et Processus”Université du MaineLE MANS Cedex 9France
  3. 3.Faculty of MathematicsUniversity of BucharestBucharestRomania

Personalised recommendations