Local bounds for the dynamic $\Phi^4_3$ model

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 very much akin to those presented in 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. 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.