Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 1099–1164 | Cite as

Weak universality for a class of 3d stochastic reaction–diffusion models

  • M. FurlanEmail author
  • M. Gubinelli


We establish the large scale convergence of a class of stochastic weakly nonlinear reaction–diffusion models on a three dimensional periodic domain to the dynamic \(\Phi ^4_3\) model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (in particular \(C^9\) is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their \(L^p\) norms in terms of the graphs of the standard \(\Phi ^4_3\) stochastic terms.


Weak universality Paracontrolled distributions Stochastic quantisation equation Malliavin calculus Partial chaos expansion 

Mathematics Subject Classification

60H15 60H07 



The authors would like to thank the anonymous referee for the detailed and constructive critique which contributed to improve the overall exposition of the results. Support via SFB CRC 1060 is also gratefully acknowledged.


  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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    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)CrossRefzbMATHGoogle Scholar
  4. 4.
    Catellier, Rémi, Chouk, K.: Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation. arXiv:1310.6869 [math-ph] (Oct 2013)
  5. 5.
    Da Prato, G., Debussche, A.: Strong solutions to the stochastic quantization equations. Ann. Probab. 31(4), 1900–1916 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3(6) (2015)Google Scholar
  7. 7.
    Gubinelli, M., Perkowski, N.: KPZ reloaded. Commun. Math. Phys. 349(1), 165–269 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hairer, M.: Regularity structures and the dynamical \(\Phi ^4_3\) model. arXiv:1508.05261 [math-ph] (Aug 2015)
  10. 10.
    Hairer, M., Quastel, J.: A class of growth models rescaling to KPZ. arXiv:1512.07845 [math-ph] (Dec 2015)
  11. 11.
    Hairer, M., Xu, W.: Large scale behaviour of 3d phase coexistence models. arXiv:1601.05138 [math-ph] (Jan 2016)
  12. 12.
    Jona-Lasinio, G., Mitter, P.K.: On the stochastic quantization of field theory. Commun. Math. Phys. (1965–1997) 101(3), 409–436 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kupiainen, A.: Renormalization group and stochastic PDEs. In: Annales Henri Poincaré, pp. 1–39. Springer (2014)Google Scholar
  14. 14.
    Meyer, Y.: Remarques sur un théorème de J.-M. Bony. In: Rendiconti del Circolo Matematico di Palermo. Serie II, pp. 1–20 (1981)Google Scholar
  15. 15.
    Mourrat, J.-C., Weber, H.: Convergence of the two-dimensional dynamic ising-kac model to \(\Phi ^4_2\). Commun. Pure Appl. Math. 70(4), 717–812 (2017)CrossRefzbMATHGoogle Scholar
  16. 16.
    Mourrat, J.-C., Weber, H.: Global well-posedness of the dynamic \(\Phi ^4\) model in the plane. Ann. Probab. 45(4), 2398–2476 (2017). 07MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mourrat, J.-C., Weber, H.: The dynamic \(\Phi ^4_3\) model comes down from infinity. Commun. Math. Phys. 356(3), 673–753 (2017). arXiv:1601.01234 CrossRefzbMATHGoogle Scholar
  18. 18.
    Mourrat J.-C., Weber, H., Xu, W.: Construction of \(\Phi ^4_3\) diagrams for pedestrians. arXiv:1610.08897 [math-ph] (Oct 2016)
  19. 19.
    Nourdin, I., Nualart, D.: Central limit theorems for multiple Skorohod integrals. J. Theor. Probab. 23, 39–64 (2010)CrossRefzbMATHGoogle Scholar
  20. 20.
    Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  21. 21.
    Nualart, D.: The Malliavin Calculus and Related Topics. Probability and Its Applications, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  22. 22.
    Oh, T., Gubinelli, M., Koch, H.: Renormalization of the two-dimensional stochastic nonlinear wave equations. Trans. Am. Math. Soc. (2017, to appear)Google Scholar
  23. 23.
    Shen, H., Xu, W.: Weak universality of dynamical \(\Phi ^4_3\): non-Gaussian noise. arXiv:1601.05724 [math-ph] (Jan 2016)
  24. 24.
    Shigekawa, I.: Derivatives of wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20(2), 263–289 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shigekawa, I.: Stochastic Analysis. American Mathematical Society, Providence (2004)CrossRefzbMATHGoogle Scholar
  26. 26.
    Üstünel, A.S.: A sophisticated proof of the multiplication formula for multiple Wiener integrals. arXiv:1411.4877 [math] (2014)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CEREMADEUniversité Paris DauphineParisFrance
  2. 2.IAM & HCMUniversität BonnBonnGermany

Personalised recommendations