Renormalising SPDEs in regularity structures

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 e...

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Veröffentlicht in:Journal of the European Mathematical Society : JEMS 2021-01, Vol.23 (3), p.869-947
Hauptverfasser: Bruned, Yvain, Chandra, Ajay, Chevyrev, Ilya, Hairer, Martin
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:1435-9855
1435-9863
DOI:10.4171/jems/1025