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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
Article
  • 113 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

60H15 60H07 

Notes

Acknowledgements

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.

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Copyright information

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

Authors and Affiliations

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

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