Advertisement

Probabilistic Approach to the Stochastic Burgers Equation

  • Massimiliano Gubinelli
  • Nicolas PerkowskiEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

We review the formulation of the stochastic Burgers equation as a martingale problem. One way of understanding the difficulty in making sense of the equation is to note that it is a stochastic PDE with distributional drift, so we first review how to construct finite-dimensional diffusions with distributional drift. We then present the uniqueness result for the stationary martingale problem of (M. Gubinelli and N. Perkowski, Energy solutions of KPZ are unique. 2015, [18]), but we mainly emphasize the heuristic derivation and also we include a (very simple) extension of (M. Gubinelli and N. Perkowski, Energy solutions of KPZ are unique. 2015, [18]) to a non-stationary regime.

Keywords

Stochastic Burgers equation Martingale problem Diffusions with distributional drift 

References

  1. 1.
    Assing, S.: A pregenerator for burgers equation forced by conservative noise. Commun. Math. Phys. 225(3), 611–632 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bertini, L., Giacomin, G.: Stochastic burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker-Planck-Kolmogorov Equations. American Mathematical Society, USA (2015)Google Scholar
  4. 4.
    Cannizzaro, G., Chouk, K.: Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential. arXiv:1501.04751 (2015)
  5. 5.
    Catellier, R., Gubinelli, M.: Averaging along irregular curves and regularisation of ODEs. Stoch. Process. Appl. 126(8), 2323–2366 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces. Cambridge University Press, UK (2002)Google Scholar
  7. 7.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, UK (2014)Google Scholar
  8. 8.
    Delarue, F., Diel, R.: Rough paths and 1d SDE with a time dependent distributional drift: application to polymers. Probab. Theory Relat. Fields 165(1), 1–63 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diehl, J., Gubinelli, M., Perkowski, N.: The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions. Comm. Math. Phys. 354(2), 549–589 (2017)Google Scholar
  10. 10.
    Flandoli, F.: Random Perturbation of PDEs and Fluid Dynamic Models: Ecole d’été de Probabilités de Saint-Flour XL-2010. Springer Science and Business Media, Berlin (2011)CrossRefGoogle Scholar
  11. 11.
    Flandoli, F., Russo, F., Wolf, J.: Some SDEs with distributional drift. Part I: General calculus. Osaka J. Math. 40(2), 493–542 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Flandoli, F., Russo, F., Wolf, J.: Some SDEs with distributional drift. Part II: Lyons- Zheng structure, Itô’s formula and semimartingale characterization. Random Oper. Stoch. Equ. 12(2), 145–184 (2004)CrossRefGoogle Scholar
  13. 13.
    Flandoli, F., Issoglio, E., Russo, F.: Multidimensional stochastic differential equations with distributional drift. Trans. Am. Math. Soc. 369, 1665–1688 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Funaki, T., Quastel, J.: KPZ equation, its renormalization and invariant measures. Stoch. Partial Differ. Equ. Anal. Comput. 3(2), 159–220 (2015)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gonçalves, P., Jara, M.: Universality of KPZ Equation. arXiv:1003.4478 (2010)
  16. 16.
    Gonçalves, P., Jara, M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212(2), 597–644 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gonçalves, P., Jara, M., Sethuraman, S.: A stochastic Burgers equation from a class of microscopic interactions. Ann. Probab. 43(1), 286–338 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gubinelli, M., Perkowski, N.: Energy solutions of KPZ are unique. J. Amer. Math. Soc. 31(2), 427–471 (2018)Google Scholar
  19. 19.
    Gubinelli, M., Jara, M.: Regularization by noise and stochastic Burgers equations. Stoch. Partial Differ. Equ. Anal. Comput. 1(2), 325–350 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3(6), 1–75 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hairer, M.: Rough stochastic PDEs. Commun. Pure Appl. Math. 64(11), 1547–1585 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, UK (1997)CrossRefGoogle Scholar
  25. 25.
    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer Science and Business Media, Berlin (2013)zbMATHGoogle Scholar
  26. 26.
    Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov processes: Time Symmetry and Martingale Approximation. Springer, Berlin (2012)CrossRefGoogle Scholar
  27. 27.
    Krylov, N.V., Röckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat Fields 131(2), 154–196 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kupiainen, Antti: Renormalization group and stochastic PDEs. Ann. Henri Poincaré 17(3), 497–535 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mathieu, P.: Zero white noise limit through Dirichlet forms, with application to diffusions in a random medium. Probab. Theory Relat Fields 99(4), 549–580 (1994)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mathieu, P.: Spectra, exit times and long time asymptotics in the zero-white-noise limit. Stochastics 55(1–2), 1–20 (1995)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations, vol. 1905. Springer, Berlin (2007)zbMATHGoogle Scholar
  32. 32.
    Russo, F., Vallois, P.: Elements of Stochastic Calculus via Regularization. Séminaire de Probabilités XL, pp. 147–185. Springer, Berlin (2007)zbMATHGoogle Scholar
  33. 33.
    Veretennikov, A.J.: On strong solutions and explicit formulas for solutions of stochastic integral equations. Matematicheskii Sbornik 153(3), 434–452 (1980)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. Ecole d’Eté de Probabilités de Saint Flour XIV-1984, pp. 265–439. Springer, Berlin (1986)Google Scholar
  35. 35.
    Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Math. USSR-Sbornik 22(1), 129–149 (1974)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hausdorff Center for Mathematics & Institute for Applied MathematicsUniversität BonnBonnGermany
  2. 2.Institut für MathematikHumboldt–Universität zu BerlinBerlinGermany

Personalised recommendations