Space‐Time Localisation for the Dynamic Φ34 Model

• Published in: Communications on pure and applied mathematics Vol. 73; no. 12; pp. 2519 - 2555
• Main Authors: Moinat, Augustin, Weber, Hendrik
• Format: Journal Article
• Language: English
• Published: Melbourne John Wiley & Sons Australia, Ltd 01.12.2020; John Wiley and Sons, Limited
• Subjects: Applications of mathematics; Boundary conditions; Enlargement; Maximum principle; Regularization
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21925

We prove an a priori bound for solutions of the dynamic Φ34 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