Renormalising SPDEs in regularity structures
Yvain Bruned
University of Edinburgh, UKAjay Chandra
Imperial College London, UKIlya Chevyrev
University of Edinburgh, UKMartin Hairer
Imperial College London, UK
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Abstract
The formalism recently introduced in [BHZ19] 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 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 on 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 non-linear SPDEs.
Cite this article
Yvain Bruned, Ajay Chandra, Ilya Chevyrev, Martin Hairer, Renormalising SPDEs in regularity structures. J. Eur. Math. Soc. 23 (2021), no. 3, pp. 869–947
DOI 10.4171/JEMS/1025