An Analyst’s Take on the BPHZ Theorem

  • Martin Hairer
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


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.



The author would like to thank Ajay Chandra and Philipp Schönbauer for several useful discussions during the preparation of this article. Financial support through ERC consolidator grant 615897 and a Leverhulme leadership award is gratefully acknowledged.


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Authors and Affiliations

  1. 1.Mathematics InstituteImperial CollegeLondonUK

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