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Algebraic renormalisation of regularity structures

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1610.08468v3.pdfWorking paper1.08 MBAdobe PDFView/Open
Title: Algebraic renormalisation of regularity structures
Authors: Bruned, Y
Hairer, M
Zambotti, L
Item Type: Working Paper
Abstract: 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.
Issue Date: 16-Jan-2018
URI: http://hdl.handle.net/10044/1/64260
Publisher: arXiv
Copyright Statement: © 2018 The Author(s).
Keywords: math.RA
math.AP
math.PR
Notes: 104 pages. The new version has many explanations, examples and images added
Publication Status: Published
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences
Mathematics