Communications in Mathematical Physics

, Volume 356, Issue 3, pp 673–753 | Cite as

The Dynamic \({\Phi^4_3}\) Model Comes Down from Infinity

  • Jean-Christophe Mourrat
  • Hendrik WeberEmail author
Open Access


We prove an a priori bound for the dynamic \({\Phi^4_3}\) model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows one to construct invariant measures via the Krylov–Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean \({\Phi^4_3}\) field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.



JCM is partially supported by the ANR Grant LSD (ANR-15-CE40-0020-03). HW acknowledges support by an EPSRC First Grant, a Royal Society University Research Fellowship and the Mathematical Sciences Research Institute where part of this work was completed.


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Authors and Affiliations

  1. 1.Ecole normale supérieure de Lyon, CNRSLyonFrance
  2. 2.University of WarwickCoventryUK

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