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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• The dynamical sine-Gordon model in the full subcritical regime
• The structure group for quasi-linear equations via universal enveloping algebras
• Algebraic renormalisation of regularity structures
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• Stochastic quantisation of Yang-Mills
• A diagram-free approach to the stochastic estimates in regularity structures
• Fluctuations around a homogenised semilinear random PDE
• Algebraic deformation for (S)PDEs
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• Paracontrolled calculus and regularity structures
• Some recent progress in singular stochastic PDEs
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• Renormalisation from non-geometric to geometric rough paths
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• An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures
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• Nondivergence form quasilinear heat equations driven by space-time white noise
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• Renormalised singular stochastic PDEs
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• Lecture notes on Malliavin calculus in regularity structures
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• Local well-posedness of subcritical non-linear heat equations with Gaussian initial data
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• Random models for singular SPDEs
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