Communications in Mathematical Physics

, Volume 349, Issue 1, pp 165–269 | Cite as

KPZ Reloaded

  • Massimiliano GubinelliEmail author
  • Nicolas Perkowski


We analyze the one-dimensional periodic Kardar–Parisi–Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from deriving a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between the KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also show that there is a natural approximation scheme for the nonlinearity in the stochastic Burgers equation. Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation. Moreover, we study Sasamoto–Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation that does not rely on the Cole–Hopf transform.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AC15.
    Allez, R., Chouk, K.: The continuous Anderson hamiltonian in dimension two. arXiv:1511.02718 (2015)
  2. ACQ11.
    Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64(4), 466–537 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  3. AKQ11.
    Alberts T., Khanin K., Quastel J.: The intermediate disorder regime for directed polymers in dimension 1 + 1. Ann. Probab. 42(3), 1212–1256 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  4. Ass02.
    Assing S.: A pregenerator for Burgers equation forced by conservative noise. Commun. Math. Phys. 225(3), 611–632 (2002)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  5. Ass13.
    Assing S.: A rigorous equation for the Cole–Hopf solution of the conservative KPZ equation. Stoch. Partial Differ. Equ. Anal. Comput. 1(2), 365–388 (2013)zbMATHMathSciNetGoogle Scholar
  6. BB16.
    Bailleul I., Bernicot F.: Heat semigroup and singular PDEs. J. Funct. Anal. 270(9), 3344–3452 (2016)zbMATHMathSciNetCrossRefGoogle Scholar
  7. BBF15.
    Bailleul, I., Bernicot, F., Frey, D.: Higher order paracontrolled calculus and 3d-PAM equation. arXiv:1506.08773 (2015)
  8. BCD11.
    Bahouri H., Chemin J.-Y., Danchin R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, New York (2011)zbMATHCrossRefGoogle Scholar
  9. BCK14.
    Bakhtin Y., Cator E., Khanin K.: Space–time stationary solutions for the Burgers equation. J. Am. Math. Soc. 27(1), 193–238 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  10. BD98.
    Boué M., Dupuis P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641–1659 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  11. BG97.
    Bertini L., Giacomin G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183(3), 571–607 (1997)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  12. Bon81.
    Bony J.-M.: Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér. (4) 14, 209–246 (1981)zbMATHGoogle Scholar
  13. CC13.
    Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation. arXiv:1310.6869 (2013)
  14. CC15.
    Cannizzaro, G., Chouk, K.: Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential. arXiv:1501.04751 (2015)
  15. CF14.
    Chouk, K., Friz, P.: Support theorem for a singular semilinear stochastic partial differential equation. arXiv:1409.4250 (2014)
  16. CFG15.
    Cannizzaro, G., Friz, P., Gassiat, P.: Malliavin calculus for regularity structures: the case of gPAM. arXiv:1511.08888 (2015)
  17. CK14.
    Chen, L., Kim, K.: On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. arXiv:1410.0604 (2014)
  18. CO14.
    Catuogno P., Olivera C.: Strong solution of the stochastic Burgers equation. Appl. Anal. 93(3), 646–652 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  19. Cor12.
    Corwin I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001, 76 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  20. CT15.
    Corwin, I., Tsai, L.-C.: KPZ equation limit of higher-spin exclusion processes. arXiv:1505.04158 (2015)
  21. DD16.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  22. DPZ14.
    Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions, pp. 152. Cambridge University Press, Cambridge (2014)zbMATHCrossRefGoogle Scholar
  23. DT13.
    Dembo A., Tsai L.-C.: Weakly asymmetric non-simple exclusion process and the Kardar–Parisi–Zhang equation. Commun. Math. Phys. 341(1), 219–261 (2013)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  24. Ech82.
    Echeverría P.: A criterion for invariant measures of Markov processes. Z. Wahrsch. Verw. Gebiete. 61(1), 1–16 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  25. FH14.
    Friz, P.K., Hairer, M.: A course on rough paths. Universitext, Springer, Cham (2014) (with an introduction to regularity structures)Google Scholar
  26. FNS77.
    Forster D., Nelson David R., Stephen Michael J.: Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A. 16(2), 732–749 (1977)ADSMathSciNetCrossRefGoogle Scholar
  27. FQ14.
    Funaki T., Quastel J.: KPZ equation, its renormalization and invariant measures. Stoch. Partial Differ. Equ. Anal. Comput. 3(2), 159–220 (2014)zbMATHMathSciNetGoogle Scholar
  28. FS13.
    Friz P., Shekhar A.: Doob–Meyer for rough paths. Bull. Inst. Math. Acad. Sin. (N.S.). 8(1), 73–84 (2013)zbMATHMathSciNetGoogle Scholar
  29. Fur14.
    Furlan, M.: Stochastic Navier–Stokes equation in 3 dimensions. Master’s thesis (2014) (supervised by Massimiliano Gubinelli)Google Scholar
  30. Gär88.
    Gärtner J.: Convergence towards Burgers’ equation and propagation of chaos for weakly asymmetric exclusion processes. Stoch. Process. Appl. 27(2), 233–260 (1988)zbMATHMathSciNetGoogle Scholar
  31. Gär88.
    Gubinelli M.: Controlling rough paths. J. Funct. Anal. 1, 86–140 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  32. GIP16.
    Gubinelli, M., Imkeller, P., Perkowski, N.: A Fourier analytic approach to pathwise stochastic integration. Electron. J. Probab. 21(2) (2016)Google Scholar
  33. GIP16.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi. 3(6) (2015)Google Scholar
  34. GJ13.
    Gubinelli M., Jara M.: Regularization by noise and stochastic Burgers equations. Stoch. Partial Differ. Equ. Anal. Comput. 1(2), 325–350 (2013)zbMATHMathSciNetGoogle Scholar
  35. GJ14.
    Gonçalves P., Jara M.: Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212(2), 597–644 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  36. GLT06.
    Gubinelli M., Lejay A., Tindel S.: Young integrals and SPDEs. Potential Anal. 25(4), 307–326 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  37. GP15a.
    Gubinelli M., Perkowski N.: Lectures on singular stochastic PDEs. Ensaios Math. 29, 1–89 (2015)zbMATHMathSciNetGoogle Scholar
  38. GP15b.
    Gubinelli, M., Perkowski, N.: Energy solutions of KPZ are unique. arXiv:1508.07764 (2015)
  39. Hai11.
    Hairer M.: Rough stochastic PDEs. Commun. Pure Appl. Math. 64(11), 1547–1585 (2011)zbMATHMathSciNetGoogle Scholar
  40. Hai13.
    Hairer M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  41. Hai14.
    Hairer M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  42. HL15.
    Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space. preprint arXiv:1504.07162 (2015)
  43. HM15.
    Hairer, M., Matetski, K.: Discretisations of rough stochastic PDEs. arXiv:1511.06937 (2015)
  44. HM12.
    Hairer M., Maas J.: A spatial version of the Itô–Stratonovich correction. Ann. Probab. 40(4), 1675–1714 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  45. HMW14.
    Hairer M., Maas J., Weber H.: Approximating rough stochastic PDEs. Commun. Pure Appl. Math. 67(5), 776–870 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  46. HP13.
    Hairer M., Pillai Natesh S.: Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Ann. Probab. 41(4), 2544–2598 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  47. HQ15.
    Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ. arXiv:1512.07845 (2015)
  48. HS15.
    Hairer, M., Shen, H.: A central limit theorem for the KPZ equation. arXiv:1507.01237 (2015)
  49. KPZ86.
    Kardar M., Parisi G., Zhang Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)ADSzbMATHCrossRefGoogle Scholar
  50. KS88.
    Karatzas I., Shreve Steven E.: Brownian Motion and Stochastic Calculus. Springer, New York (1988)zbMATHCrossRefGoogle Scholar
  51. KS91.
    Krug, J., Spohn, H.: Kinetic Roughening of Growing Surfaces, vol. 1(99), p. 1. C. Godreche, Cambridge University Press, Cambridge (1991)Google Scholar
  52. Kup14.
    Kupiainen, A.: Renormalization group and stochastic PDEs. Ann. Henri Poincaré (Springer), 1–39 (2014)Google Scholar
  53. LS98.
    Lam C.-H., Shin Franklin G.: Improved discretization of the Kardar–Parisi–Zhang equation. Phys. Rev. E 58(5), 5592–5595 (1998)ADSCrossRefGoogle Scholar
  54. Mey81.
    Meyer, Y.: Remarques sur un théorème de J.-M. Bony. Rend. Circ. Mate. Palermo. Ser. II, 1–20 (1981)Google Scholar
  55. MF14.
    Moreno F., Gregorio R.: On the (strict) positivity of solutions of the stochastic heat equation. Ann. Probab. 42(4), 1635–1643 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  56. Mue91.
    Mueller C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37(4), 225–245 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  57. MW14.
    Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic Ising–Kac model to \({\phi^{4}_{2}}\). arXiv:1410.1179 (2014)
  58. Per14.
    Perkowski, N.: Studies of robustness in stochastic analysis and mathematical finance. Ph.D. thesis (2014)Google Scholar
  59. PT16.
    Prömel David J., Trabs M.: Rough differential equations driven by signals in Besov spaces. J. Differ. Equ. 260(6), 5202–5249 (2016)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  60. QS15.
    Quastel J., Spohn H.: The one-dimensional KPZ equation and its universality class. J. Stat. Phys. 160(4), 965–984 (2015)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  61. Qua14.
    Quastel, J.: The Kardar–Parisi–Zhang equation and universality class. In: XVIIth International Congress on Mathematical Physics, pp. 113–133 (2014)Google Scholar
  62. SS09.
    Sasamoto T., Spohn H.: Superdiffusivity of the 1D lattice Kardar–Parisi–Zhang equation. J. Stat. Phys. 137(5–6), 917–935 (2009)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  63. SS10.
    Sasamoto T., Spohn H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B. 834(3), 523–542 (2010)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  64. Üst14.
    Üstünel A.S.: Variational calculation of Laplace transforms via entropy on Wiener space and applications. J. Funct. Anal. 267(8), 3058–3083 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  65. WKMS00.
    Weinan E., Khanin K., Mazel A., Sinai Ya.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. (2) 151(3), 877–960 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  66. ZK65.
    Zabusky Norman J., Kruskal Martin D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15(6), 240 (1965)ADSzbMATHCrossRefGoogle Scholar
  67. ZZ14.
    Zhu, R., Zhu, X.: Approximating three-dimensional Navier–Stokes equations driven by space–time white noise. arXiv:1409.4864 (2014)
  68. ZZ15a.
    Zhu R., Zhu X.: Three-dimensional Navier–Stokes equations driven by space–time white noise. J. Differ. Equ. 259(9), 4443–4508 (2015)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  69. ZZ15b.
    Zhu, R., Zhu, X.: Lattice approximation to the dynamical \({\Phi_{3}^{4}}\) model. arXiv:1508.05613 (2015)

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.CEREMADE and UMR 7534 CNRS, Institut Universitaire de FranceUniversité Paris-DauphineParisFrance
  2. 2.Hausdorff Center for Mathematics, Institute for Applied MathematicsUniversity of BonnBonnGermany
  3. 3.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations