Renormalising SPDEs in regularity structures

  • Yvain Bruned

    University of Edinburgh, UK
  • Ajay Chandra

    Imperial College London, UK
  • Ilya Chevyrev

    University of Edinburgh, UK
  • Martin Hairer

    Imperial College London, UK
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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