Space-time localisation for the dynamic $\Phi^4_3$ model – Hendrik Weber (University of Bath)

December 13, 2018 @ 9:00 am – 10:00 am
Seminar Room 1
Newton Institute

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>