Space-Time Localisation for the Dynamic Φ34 Model

Augustin Moinat, Hendrik Weber

Research output: Contribution to journalArticle

22 Citations (SciVal)


We prove an a priori bound for solutions of the dynamic (Formula presented.) equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions. We treat the large- and small-scale behaviour of solutions with completely different arguments. For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure. The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large-scale properties of solutions. They can, for example, be used in a compactness argument to construct solutions on the full space and their invariant measures.

Original languageEnglish
Pages (from-to)2519-2555
Number of pages37
JournalCommunications on Pure and Applied Mathematics
Issue number12
Early online date31 Jul 2020
Publication statusPublished - 15 Oct 2020

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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