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2018 | OriginalPaper | Chapter

An Analyst’s Take on the BPHZ Theorem

Author : Martin Hairer

Published in: Computation and Combinatorics in Dynamics, Stochastics and Control

Publisher: Springer International Publishing

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Abstract

We provide a self-contained formulation of the BPHZ theorem in the Euclidean context, which yields a systematic procedure to “renormalise” otherwise divergent integrals appearing in generalised convolutions of functions with a singularity of prescribed order at their origin. We hope that the formulation given in this article will appeal to an analytically minded audience and that it will help to clarify to what extent such renormalisations are arbitrary (or not). In particular, we do not assume any background whatsoever in quantum field theory and we stay away from any discussion of the physical context in which such problems typically arise.

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previous chapter Heavy Tailed Random Matrices: How They Differ from the GOE, and Open Problems

next chapter Parabolic Anderson Model with Rough Dependence in Space

It would have looked more natural to impose the stronger condition \(\deg _\infty \mathfrak {t}^{(k)} = \deg _\infty \mathfrak {t} - |k|\) as before. One may further think that in this case one would be able to extend Theorem 4.3 to all diagrams Γ, not just those in \(\mathcal {H}_+\). This is wrong in general, although we expect it to be true after performing a suitable form of positive renormalisation as in [2, 3]. This is not performed here, and as a consequence we are unable to take advantage of the additional large-scale cancellations that the stronger condition \(\deg _\infty \mathfrak {t}^{(k)} = \deg _\infty \mathfrak {t} - |k|\) would offer.

I.e. T _i is such that if u ∈ T _i and v ≤ u, then v ∈ T _i.

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Title: An Analyst’s Take on the BPHZ Theorem
Author: Martin Hairer
Publisher: Springer International Publishing
Book: Computation and Combinatorics in Dynamics, Stochastics and Control
Print ISBN: 978-3-030-01592-3

Electronic ISBN: 978-3-030-01593-0

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-030-01593-0_16

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