Renormalising SPDEs in regularity structures
By Yvain Bruned, A. Chandra, Ilya Chevyrev, Martin Hairer. 2017
157 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory
View Paper: https://www.semanticscholar.org/paper/9fcdfbaaf1c6f1b8693a2a19fab5d3c7eab11b23
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In Algebra and Number Theory
Citations in the corpus
- HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “We shall not touch in this work on renormalisation matters, and we invite the reader to read the latest developments of Hairer & co-authors on the subject; see [5, 7, 10].”
- A diagram-free approach to the stochastic estimates in regularity structures (2021, background): “They provide a framework for local well-posedness for a large class of semilinear SPDEs, as worked out in [20, 5, 8, 4].”
- The infinitesimal generator of the stochastic Burgers equation (2018, background): “Although by now there are very sophisticated and powerful black box type tools available in regularity structures that should handle the complicated expansion automatically [2,4,7].”