Probability Theory and Related Fields

, Volume 155, Issue 1–2, pp 71–126 | Cite as

Rough Burgers-like equations with multiplicative noise

  • Martin Hairer
  • Hendrik Weber


We construct solutions to vector valued Burgers type equations perturbed by a multiplicative space–time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.

Mathematics Subject Classification (2010)

60H15 35R60 


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  1. 1.
    Albeverio S., Röckner M.: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Relat. Fields 89(3), 347–386 (1991)CrossRefzbMATHGoogle Scholar
  2. 2.
    Benth F.E., Deck T., Potthoff J.: A white noise approach to a class of non-linear stochastic heat equations. J. Funct. Anal. 146(2), 382–415 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertini L., Cancrini N., Jona-Lasinio G.: The stochastic Burgers equation. Commun. Math. Phys. 165(2), 211–232 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chan T.: Scaling limits of Wick ordered KPZ equation. Commun. Math. Phys. 209(3), 671–690 (2000)zbMATHGoogle Scholar
  6. 6.
    Da Prato G., Debussche A.: Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196(1), 180–210 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Da Prato G., Debussche A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Da Prato G., Debussche A., Temam R.: Stochastic Burgers’ equation. Nonlinear Differential Equations Appl. 1(4), 389–402 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Da Prato G., Zabczyk J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  10. 10.
    Friz P., Victoir N.: Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46(2), 369–413 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Friz P.K., Victoir N.B.: Multidimensional stochastic processes as rough paths. Cambridge Studies in Advanced Mathematics, vol. 120. Theory and applications. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  12. 12.
    Garsia, A.M., Rodemich, E., Rumsey, H. Jr.: A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20, 565–578 (1970/1971)Google Scholar
  13. 13.
    Gubinelli M.: Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gubinelli M., Tindel S.: Rough evolution equations. Ann. Probab. 38(1), 1–75 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gyöngy I.: Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process. Appl. 73(2), 271–299 (1998)CrossRefzbMATHGoogle Scholar
  16. 16.
    Hairer M.: Rough stochastic PDEs. Commun. Pure Appl. Math. 64, 1547–1585 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hairer, M.: Singular perturbations to semilinear stochastic heat equations. Probab. Theory Relat. Fields (2011)Google Scholar
  18. 18.
    Hairer, M., Maas, J.: A spatial version of the Itô-Stratonovich correction. Ann. Probab. (2011, to appear)Google Scholar
  19. 19.
    Hairer M., Stuart A.M., Voss J.: Analysis of SPDEs arising in path sampling. II. The nonlinear case. Ann. Appl. Probab. 17(5-6), 1657–1706 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hairer, M., Voss, J.: Approximations to the stochastic Burgers equation. J. Nonl. Sci. (2011)Google Scholar
  21. 21.
    Jona-Lasinio G., Mitter P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101(3), 409–436 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Karatzas I., Shreve S.E.: Brownian motion and stochastic calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)Google Scholar
  23. 23.
    Kardar M., Parisi G., Zhang Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lions P.-L., Souganidis P.E.: Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 326(9), 1085–1092 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lyons, T., Qian, Z.: System control and rough paths. Oxford Mathematical Monographs. Oxford University Press/Oxford Science Publications, Oxford (2002)Google Scholar
  26. 26.
    Lyons T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2), 215–310 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lyons, T.J., Caruana, M., Lévy, T.: Differential equations driven by rough paths. In: Lecture Notes in Mathematics, vol. 1908. Springer, Berlin, 2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the Summer School by Jean PicardGoogle Scholar
  28. 28.
    Nualart D., Răşcanu A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53(1), 55–81 (2002)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Nualart D., Rozovskii B.: Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise. J. Funct. Anal. 149(1), 200–225 (1997)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Teichmann J.: Another approach to some rough and stochastic partial differential equations. Stoch. Dyn. 11(2–3), 535–550 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of WarwickCoventryUK

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