Renormalising SPDEs in regularity structures

Yvain Bruned, A. Chandra, Ilya Chevyrev, M. Hairer

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

The formalism recently introduced in [BHZ16] allows one to assign a regularity
structure, as well as a corresponding “renormalisation group”, to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in [CH16] that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT.
The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general
black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.
Original languageEnglish
Pages (from-to)869–947
Number of pages85
JournalJournal of the European Mathematical Society
Issue number3
Early online date2 Dec 2020
Publication statusE-pub ahead of print - 2 Dec 2020


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