Formal differential equations and renormalization

Abstract

The study of solutions of differential equations (analytic or formal) can often be reduced to a conjugacy problem, namely the conjugation of a given equation to a much simpler one, using identity-tangent diffeomorphisms.

On the one hand, following Écalle’s work (with a different terminology), such diffeomorphisms are given by characters on a given Hopf algebra (here a shuffle Hopf algebra). On the other hand, for some equations, the obstacles in the formal conjugacy are reflected in the fact that the associated characters appear to be ill-defined.

The analogy with the need for a renormalization scheme (dimensional regularization, Birkhoff decomposition) in quantum field theory becomes obvious for such equations and deliver a wide range of toy models. We discuss here the case of a simple class of differential equations where a renormalization scheme yields meaningful results.