Renormalising SPDEs in regularity structures

Abstract

The formalism recently introduced in arXiv:1610.08468 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 arXiv:1612.08138 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.

Fixed an error regarding the topology of space of functions of functions with blow-up, see Remark 2.20. The numbering of environments in Section 2, 2.20 later has been shifted by 1, for instance Theorem 2.21 in the published version is now Theorem 2.22 in this arxiv version

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• Geometric stochastic heat equations
• A solution theory for quasilinear singular SPDEs
• The motion of a random string
• Controlling Rough Paths

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