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

We give an a priori bound for solutions of the dynamic ${\mathrm{\Phi }}_3^4$ 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 will show the different techniques used for the small and large scale bounds. For small scales we use techniques derived from Hairer’s theory of regularity structures.. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.