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

Presented by: 
Hendrik Weber University of Bath
Date: 
Thursday 13th December 2018 - 09:00 to 10:00
Venue: 
INI Seminar Room 1
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. 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.  

Joint work with A. Moinat.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons