Algebraic renormalisation of regularity structures

By Yvain Bruned, Martin Hairer, L. Zambotti. Inventiones Mathematicae, 2016

291 citations recorded by the source corpus.

We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory.

Subfields: Algebra and Number Theory, Finance, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9499588f5ecccf6947d0697b213e047da2f3456c

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Citations in the corpus

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].”
Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity (2018, methodology): “piainen’s RG approach [57, 58], and the approach of Otto, Weber, and coauthors [78, 5, 77]. On the one hand, the theory of regularity structures [41, 27, 42] has since grown into a complete framework [13, 19, 12, 44] which can deal with a large class of parabolic equations (in the socalled subcritic”
A regularity structure for rough volatility (2017, background): “Although this, and negative renormalization, is well understood ([31, 10], also [9] for a rough path perspective, all complete exposition would lead us to far astray from the main topic of this paper.”
HIGH ORDER PARACONTROLLED CALCULUS (2016, background): “Such a kind of equation appears in the study of the random motion of a string on a manifold [19], where α ă 12 in that case ; its study in the setting of regularity structures is the object of Bruned-Hairer-Zambotti’s forthcoming work [3].”
Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras (2019, methodology): “Introduction The purpose of this paper is to introduce the various concepts in order to make connection with the critical notion of multiple pre-Lie algebras that was introduced recently from the remarkable work of Bruned, Hairer and Zambotti [10] on algebraic renormalization of regularity structure”