A note on the applications of Wick products and Feynman diagrams in the study of singular partial differential equations

Abstract

The study of singular partial differential equations has seen rapid significant developments very recently. In particular, the works of Hairer (2013, 2014), and Gubinelli et al. (2015) paved the way for others to follow, providing blueprints for further study. Yet, many manuscripts in this field consist of extensive applications of techniques from physics, specifically quantum field theory, such as Wick products which are best explained in terms of Feynman diagrams. The purpose of this short note is to describe how the necessity of Wick products comes about, their applications using Feynman diagrams, as well as the utility of Gaussian hypercontractivity theorem. We also conclude with a description of an open problem that seems to be very mathematically challenging and physically meaningful. The author’s intention is to make this note as accessible as possible to a wide audience by providing sufficient details, as well as relatively self-contained by including all relevant results which are necessary for our discussions.