Abstract
We prove an a priori bound for solutions of the dynamic (Formula presented.) equation. This bound provides a control on solutions on a compact spacetime 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 spacetime boundary conditions. We treat the large and smallscale behaviour of solutions with completely different arguments. For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully selfcontained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solutiondependent regularisation procedure. The fact that our bounds do not depend on spacetime boundary conditions makes them useful for the analysis of largescale properties of solutions. They can, for example, be used in a compactness argument to construct solutions on the full space and their invariant measures.
Original language  English 

Pages (fromto)  25192555 
Number of pages  37 
Journal  Communications on Pure and Applied Mathematics 
Volume  73 
Issue number  12 
Early online date  31 Jul 2020 
DOIs  
Publication status  Published  15 Oct 2020 
ASJC Scopus subject areas
 Mathematics(all)
 Applied Mathematics
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Profiles

Hendrik Weber
 Department of Mathematical Sciences  Professor of Probability
 Probability Laboratory at Bath
Person: Research & Teaching