Communications in Mathematical Physics

, Volume 358, Issue 2, pp 521–588 | Cite as

Space–Time Discrete KPZ Equation

  • G. Cannizzaro
  • K. Matetski
Open Access


We study a general family of space–time discretizations of the KPZ equation and show that they converge to its solution. The approach we follow makes use of basic elements of the theory of regularity structures (Hairer in Invent Math 198(2):269–504, 2014) as well as its discrete counterpart (Hairer and Matetski in Discretizations of rough stochastic PDEs, 2015. arXiv:1511.06937). Since the discretization is in both space and time and we allow non-standard discretization for the product, the methods mentioned above have to be suitably modified in order to accommodate the structure of the models under study.


  1. BCD11.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier analysis and nonlinear partial differential equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2011).
  2. BG97.
    Bertini, L., Giacomin,G.: StochasticBurgers andKPZ equations fromparticle systems.Commun. Math. Phys. 183(3), 571–607 (1997).
  3. BGS16.
    Blondel, O., Gonçalves, P.C., Simon, M.: Convergence to the stochastic Burgers equation from a degenerate microscopic dynamics. Electron. J. Probab. 21, Paper No. 69, 25 (2016).
  4. BS08.
    Brenner, S.C., Scott, L.R.: The mathematical theory of the element methods, vol. 15 of Texts in Applied Mathematics. Springer, New York, third ed., (2008)
  5. CS16.
    Corwin, I., Shen, H.: Open ASEP in the weakly asymmetric regime (2016). arXiv:1610.04931
  6. CST16.
    Corwin, I., Shen, H., Tsai, L.-C.: ASEP(q,j) converges to the KPZ equation (2016). arXiv:1602.01908
  7. CT17.
    Corwin, I., Tsai, L.-C.: KPZ equation limit of higher-spin exclusion processes. Ann. Probab. 45(3), 1771–1798 (2017).
  8. CW15.
    Chandra, A., Weber, H.: Stochastic PDEs, regularity structures, and interacting particle systems (2015). arXiv:1508.03616
  9. Dau92.
    Daubechies, I.: Ten lectures on wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992).
  10. DGP17.
    Diehl, J., Gubinelli, M., Perkowski, N.: The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions. Commun. Math. Phys. 354(2), 549–589 (2017).
  11. DT16.
    Dembo,A., Tsai, L.-C.:Weakly asymmetric non-simple exclusion process and the Kardar-Parisi-Zhang equation. Commun. Math. Phys. 341(1), 219–261 (2016).
  12. FGS16.
    Franco T., Gonçalves P., Simon M.: Crossover to the stochastic Burgers equation for the WASEP with a slow bond. Commun. Math. Phys. 346(3), 801–838 (2016) ADSMathSciNetCrossRefMATHGoogle Scholar
  13. FH14.
    Friz, P.K., Hairer, M.: A course on rough paths. Universitext. Springer, Cham, With an introduction to regularity structures (2014).
  14. FH17.
    Funaki T., Hoshino M.: A coupled KPZ equation, its two types of approximations and existence of global solutions. J. Funct. Anal. 273(3), 1165–1204 (2017) MathSciNetCrossRefMATHGoogle Scholar
  15. GA88.
    Gärtner J.: Convergence towards Burgers’ equation and propagation of chaos for weakly asymmetric exclusion processes. Stoch. Process. Appl. 27(2), 233–260 (1988) MathSciNetMATHGoogle Scholar
  16. GIP15.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6, 75. (2015).
  17. GJ10.
    Gonçalves, P., Jara, M.: Universality of KPZ equation (2010). arXiv:1003.4478
  18. GJ13.
    Gubinelli M., Jara M.: Regularization by noise and stochastic Burgers equations. Stoch. Partial Differ. Equ. Anal. Comput. 1(2), 325–350 (2013) MathSciNetMATHGoogle Scholar
  19. GJ14.
    Gonçalves P., Jara M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212(2), 597–644 (2014) MathSciNetCrossRefMATHGoogle Scholar
  20. GJ16.
    Gonçalves P., Gonçalves P.: Stochastic Burgers equation from long range exclusion interactions. Stoch. Process. Appl. 127(12), 4029–4052 (2017) MathSciNetCrossRefMATHGoogle Scholar
  21. GJS17.
    Gonçalves P., Jara M., Simon M.: Second order Boltzmann-Gibbs principle for polynomial functions and applications. J. Stat. Phys. 166(1), 90–113 (2017) ADSMathSciNetCrossRefMATHGoogle Scholar
  22. GP15.
    Gubinelli, M., Perkowski, N.: Energy solutions of KPZ are unique. J. Am. Math. Soc.
  23. GP16.
    Gubinelli, M., Perkowski, N.: The Hairer–Quastel universality result in equilibrium (2016). arXiv:1602.02428
  24. GP17.
    Gubinelli M., Perkowski N.: KPZ reloaded. Commun. Math. Phys. 349(1), 165–269 (2017) ADSMathSciNetCrossRefMATHGoogle Scholar
  25. Gub04.
    Gubinelli M.: Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004) MathSciNetCrossRefMATHGoogle Scholar
  26. Hai13.
    Hairer M.: Solving the KPZ equation. Ann. Math. (2) 178(2), 559–664 (2013) arXiv:1109.6811 MathSciNetCrossRefMATHGoogle Scholar
  27. Hai14.
    Hairer M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014) arXiv:1303.5113 ADSMathSciNetCrossRefMATHGoogle Scholar
  28. Hai15.
    Hairer M.: Introduction to regularity structures. Braz. J. Probab. Stat. 29(2), 175–210 (2015) MathSciNetCrossRefMATHGoogle Scholar
  29. HM15.
    Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs (2015). arXiv:1511.06937
  30. Hos16.
    Hoshino, M.: Paracontrolled calculus and Funaki-Quastel approximation for the KPZ equation (2016). arXiv:1605.02624
  31. HQ15.
    Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ (2015). arXiv:1512.07845
  32. HS15.
    Hairer M., Shen H.: A central limit theorem for the KPZ equation. Ann. Probab. 45(6B), 4167–4221 (2017) MathSciNetCrossRefMATHGoogle Scholar
  33. Kal02.
    Kallenberg, O.: Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, second ed., (2002)Google Scholar
  34. KPZ86.
    Kardar M., Parisi G., Zhang Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)ADSCrossRefMATHGoogle Scholar
  35. Lab17.
    Labbé C.: Weakly asymmetric bridges and the KPZ equation. Commun. Math. Phys. 353(3), 1261–1298 (2017) ADSMathSciNetCrossRefMATHGoogle Scholar
  36. Lyo98.
    Lyons T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2), 215–310 (1998) MathSciNetCrossRefMATHGoogle Scholar
  37. Mey92.
    Meyer, Y.: Wavelets and operators, vol. 37 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, Translated from the 1990 French original by D. H. Salinger (1992)Google Scholar
  38. Mue91.
    Mueller C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37(4), 225–245 (1991)MathSciNetCrossRefMATHGoogle Scholar
  39. Nel73.
    Nelson E.: The free Markoff field. J. Funct. Anal. 12, 211–227 (1973) MathSciNetCrossRefMATHGoogle Scholar
  40. Nua06.
    Nualart, D.: The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second ed. (2006)Google Scholar
  41. SS09.
    Sasamoto T., Spohn H.: Superdiffusivity of the 1D lattice Kardar-Parisi-Zhang equation. J. Stat. Phys. 137(5-6), 917–935 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. Wal86.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV—1984, vol. 1180 of Lecture Notes in Math., 265–439. Springer, Berlin, (1986).
  43. ZK65.
    Zabusky N.J., Kruskal M.D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965) ADSCrossRefMATHGoogle Scholar
  44. ZZ14.
    Zhu, R., Zhu, X.: Approximating three-dimensional Navier–Stokes equations driven by space–time white noise (2014). arXiv:1409.4864
  45. ZZ15.
    Zhu, R., Zhu, X.: Lattice approximation to the dynamical \({\phi_3^4}\) model (2015). arXiv:1508.05613

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK
  2. 2.University of TorontoTorontoCanada

Personalised recommendations