Journal of the European Mathematical Society· 2020Q1
Renormalising SPDEs in regularity structures
- 130citations
- Q1SCImago
- 2020year
Short summary
A new formalism constructs a renormalization group action on singular stochastic PDEs, showing BPHZ-lifted solutions match modified classical PDE solutions.
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Key points
- Constructs a renormalization group action on singular stochastic PDEs within a regularity structure framework.
- Shows that BPHZ-lifted solutions of SPDEs coincide with classical solutions of a modified PDE.
- Yields a general local existence and stability theorem for a wide class of singular non-linear SPDEs.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Journal of the European Mathematical Society, 2020 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics