Premium
Space‐Time Localisation for the Dynamic Φ 3 4 Model
Author(s) -
Moinat Augustin,
Weber Hendrik
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21925
Subject(s) - mathematics , compact space , a priori and a posteriori , upper and lower bounds , space (punctuation) , boundary (topology) , invariant (physics) , argument (complex analysis) , realisation , space time , scale (ratio) , scale invariance , set (abstract data type) , boundary value problem , mathematical analysis , pure mathematics , computer science , mathematical physics , statistics , philosophy , biochemistry , chemistry , physics , epistemology , quantum mechanics , chemical engineering , engineering , programming language , operating system
We prove an a priori bound for solutions of the dynamic Φ 3 4 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. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC