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

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SRQW03 - Scaling limits & SPDEs: recent developments and future directions

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. &nbsp; <span><br>We treat the&nbsp; large and small scale behaviour of solutions with completely different arguments. </span> For small scales we use bounds 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. &nbsp; <br>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&nbsp; in a compactness argument to construct solutions on the full space and their invariant measures. &nbsp; <br><span><br>Joint work with A. Moinat.</span>

This talk is part of the Isaac Newton Institute Seminar Series series.

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