Abstract
We prove an a priori bound for solutions of the dynamic 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. We stress immediately that our proof is fully self‐contained, 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 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. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
