Space-time localisation for the dynamic $\Phi^4_3$ model

TL;DR

This paper establishes a boundary-condition independent a priori bound for solutions of the dynamic ^3 model, combining regularity structures and PDE techniques to analyze solutions across scales.

Contribution

It introduces a novel, self-contained approach to bound solutions of the ^3 model that works uniformly across scales without boundary condition dependence.

Findings

Provides a scale-dependent bound for the ^3 equation solutions.

Connects regularity structures with PDE maximum principle methods.

Enables analysis of large-scale properties and invariant measures.

Abstract

We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ 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.