Journal of Statistical Physics

, Volume 171, Issue 4, pp 543–598 | Cite as

The Scaling Limit of the KPZ Equation in Space Dimension 3 and Higher

  • Jacques Magnen
  • Jérémie Unterberger


We study in the present article the Kardar–Parisi–Zhang (KPZ) equation
$$\begin{aligned} \partial _t h(t,x)=\nu \Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta (t,x), \qquad (t,x)\in \mathbb {R}_+\times \mathbb {R}^d \end{aligned}$$
in \(d\ge 3\) dimensions in the perturbative regime, i.e. for \(\lambda >0\) small enough and a smooth, bounded, integrable initial condition \(h_0=h(t=0,\cdot )\). The forcing term \(\eta \) in the right-hand side is a regularized space-time white noise. The exponential of h—its so-called Cole-Hopf transform—is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilson’s renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer and Magnen (Commun Math Phys 162(1):85–121, 1994). Standard large deviation estimates for \(\eta \) make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution h may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards–Wilkinson model (\(\lambda =0\)) with renormalized coefficients \(\nu _{eff}=\nu +O(\lambda ^2),D_{eff}=D+O(\lambda ^2)\).


KPZ equation Cole-Hopf transformation Directed polymer Constructive field theory Renormalization Cluster expansion 

Mathematics Subject Classification

35B50 35B51 35D40 35K55 35R60 35Q82 60H15 81T08 81T16 81T18 82C41 



We wish to thank H. Spohn, F. Toninelli and the referee for numerous discussions, suggestions and corrections, which have hopefully contributed in particular to the readability of the paper. J. Unterberger acknowledges the support of the ANR, via the ANR project ANR-16-CE40-0020-01.


  1. 1.
    Abdesselam, A., Rivasseau, V.: Trees, Forests and Jungles: A Botanical Garden for Cluster Expansions. Lecture Notes in Physics, vol. 446. Springer, Berlin (1995)zbMATHGoogle Scholar
  2. 2.
    Altland, A., Simons, B.: Condensed Matter Field Theory. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy on the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. Math. 64, 466–537 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses 10, Société mathématique de France (2000)Google Scholar
  5. 5.
    Bailleul, I., Bernicot. F.: Higher order paracontrolled calculus. arXiv:1609.06966
  6. 6.
    Barabasi, A.-L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bolthausen, E.: A note on the diffusion of directed polymers in random environment. Commun. Math. Phys. 123(4), 529–534 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bricmont, J., Kupiainen, A.: Random walks in asymmetric random environments. Commun. Math. Phys. 142(2), 345–420 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bricmont, J., Kupiainen, A., Lin, G.: Renormalization group and asymptotics of solutions of nonlinear parabolic equations. Commun. Pure Appl. Math. 47, 893–922 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bricmont, J., Gawedzki, K., Kupiainen, A.: KAM theorem and quantum field theory. Commun. Math. Phys. 201, 699–727 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bruned, M., Hairer, M., Zambotti, L.: Algebraic renormalisation of regularity structures. arXiv:1610.08468
  13. 13.
    Cannizzarro, G., Friz, P., Gassiat, P.: Malliavin calculus for regularity structures: the case of gPAM. arXiv:1511.08888
  14. 14.
    Cardy, J.: Field theory and nonequilibrium statistical mechanics, cours de 3ème cycle de la Suisse Romande, semestre d’été de l’année académique 1998-1999 (in English)Google Scholar
  15. 15.
    Carmona, P., Hu, Y.: On the partition function of a directed polymer in a random environment. Probab. Theory Relat. Fields 124, 431–457 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Catellier, R., Chouk, K.: Paracontrolled distributions and the 3-dimensional stochastic quantization equation. arXiv:1310.6869 (2013)
  17. 17.
    Chandra, A., Hairer, M.: An analytic BPHZ for regularity structures. arXiv:1612.08138
  18. 18.
    Chouk, K., Allez, R.: The continuous Anderson hamiltonian in dimension two. arXiv:1511.02718
  19. 19.
    Comets, F., Yoshida, N.: Directed polymers in random environment are diffusive at weak disorder. Ann. Prob. 34, 1746–1770 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 113001 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26, 1493–1517 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Construction and Borel summability of infrared \(\Phi ^4_4\) by a phase space expansion. Commun. Math. Phys. 109, 437–480 (1987)ADSCrossRefGoogle Scholar
  23. 23.
    Fröhlich, J., Spencer, T.: The Kosterlitz-Thouless transition in two-dimensional abelian spin systems and the Coulomb gas. Commun. Math. Phys. 81, 527–602 (1981)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Gallavotti, G., Nicolò, K.: Renormalization theory in 4-dimensional scalar fields. Commun. Math. Phys. 100, 545–590 and 101, 247–282 (1985)Google Scholar
  25. 25.
    Gawedzki, K., Kupiainen, A.: Massless lattice \(\phi ^4_4\) theory: rigorous control of a renormalizable asymptotically free model. Commun. Math. Phys. 99(2), 197–252 (1985)ADSCrossRefGoogle Scholar
  26. 26.
    Glimm, J., Jaffe, A.: Positivity of the \(\varphi ^{4}_{3}\) Hamiltonian. Fortschr. Phys. 21, 327–376 (1973)CrossRefGoogle Scholar
  27. 27.
    Glimm, J., Jaffe, A.: Quantum Physics, A Functional Point of View. Springer, Berlin (1987)zbMATHGoogle Scholar
  28. 28.
    Gu, Y., Ryzhik, L., Zeitouni, O.: The Edwards-Wilkinson Limit of the Random Heat Equation in Dimensions Three and Higher. arXiv:1710.00344
  29. 29.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi. 3 (2015)Google Scholar
  30. 30.
    Hairer, M., Labbé, C.: Multiplicative stochastic heat equations on the whole space. arXiv:1504.07162
  31. 31.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Iagolnitzer, D., Magnen, J.: Polymers in a weak random potential in dimension four: rigorous renormalization group analysis. Commun. Math. Phys. 162(1), 85–121 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Imbrie, J.Z., Spencer, T.: Diffusion of directed polymer in a random environment. J. Stat. Phys. 52, 609–626 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jona-Lasinio, G., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. 101, 401 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Jona-Lasinio, G., Mitter, P.K.: Large deviation estimates in the stochastic quantization of \(\phi ^4_2\). Commun. Math. Phys. 130, 111 (1990)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Jona-Lasinio, G., Sénéor, R.: Study of stochastic differential equations by constructive methods I. J. Stat. Phys. 83, 1109 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer, Berlin (1991)zbMATHGoogle Scholar
  40. 40.
    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    Kupiainen, A., Marcozzi, M.: Renormalization of generalized KPZ equation. arXiv:1604.08712
  42. 42.
    Kupiainen, A.: Renormalization group and stochastic PDEs. Ann. Henri Poincaré 17(3), 497–535 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Le Bellac, M.: Quantum and Statistical Field Theory. Oxford Science Publications, Oxford (1991)Google Scholar
  44. 44.
    Magnen, J., Unterberger, J.: From constructive theory to fractional stochastic calculus. (I) An introduction. Ann. Henri Poincaré 12, 1199–1226 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Magnen, J., Unterberger, J.: From constructive theory to fractional stochastic calculus. (II) The rough path for \(\frac{1}{6}<\alpha <\frac{1}{4}\). Ann. Henri Poincaré 13, 209–270 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Mastropietro, V.: Non-perturbative Renormalization. World Scientific, Singapore (2008)CrossRefzbMATHGoogle Scholar
  47. 47.
    Mourrat, J.-C., Weber, H.: Global well-posedness of the dynamic \(\Phi ^4_3\)-model on the torus. arXiv:1601.01234
  48. 48.
    Nelson, E.: A quartic interaction in two dimensions. In: Goodman, R., Segal, I. (eds.) Mathematical Theory of Elementary Particles. MIT Press, Cambridge (1966)Google Scholar
  49. 49.
    Nelson, E.: Derivation of the Schrödinger Equation from Newtonian Mechanics. Phys. Rev. 150(4), 1079 (1966)ADSCrossRefGoogle Scholar
  50. 50.
    Øksendal, B.: Stochastic Differential Equations. An Introduction with Applications. Springer, Berlin (2000)zbMATHGoogle Scholar
  51. 51.
    Parisi, G., Wu, Y.-S.: Perturbation theory without gauge fixing. Sci. Sin. 24, 483 (1981)MathSciNetGoogle Scholar
  52. 52.
    Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton Series in Physics. Princeton University Press, Princeton (1991)CrossRefGoogle Scholar
  54. 54.
    Salmhofer, M.: Renormalization: An Introduction. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  55. 55.
    Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834, 523–542 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Unterberger, J.: Generalized PDE estimates for KPZ equations through Hamilton-Jacobi-Bellman formalism. arXiv:1312.5293
  58. 58.
    Unterberger, J.: Mode d’emploi de la théorie constructive des champs bosoniques. Avec une application aux chemins rugueux, Confluentes Mathematici 4 (1) (2012) (in French with an English lexicon)Google Scholar
  59. 59.
    Unterberger, J.: PDE estimates for multi-dimensional KPZ equation. (1) PDE estimates. arXiv:1307.1980
  60. 60.
    Velo, G., Wightman, A.S.: Constructive quantum field theory. In: Velo, G., Wightman, A. (eds.) Proceedings of the 1973 Erice Summer School. Lecture Notes in Physics 25, Springer, New York (1973)Google Scholar
  61. 61.
    Wilson, K.G.: Renormalization group and critical phenomena. I. Renormalization group and the kadanoff scaling picture. Phys. Rev. B 4, 3174–3184 (1971)ADSCrossRefzbMATHGoogle Scholar
  62. 62.
    Wilson, K.G., Kogut, J.: The renormalization group and the \( \varepsilon \) expansion. Phys. Rep. 12, 75–200 (1974)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire associé au CNRS UMR 7644, Centre de Physique ThéoriqueEcole PolytechniquePalaiseau CedexFrance
  2. 2.Laboratoire associé au CNRS UMR 7502, Institut Elie CartanUniversité de LorraineVandœuvre-lès-Nancy CedexFrance

Personalised recommendations