Algebraic deformation for (S)PDEs

Abstract

We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide a new construction of the algebraic objects at play in regularity structures in the works by Bruned, Hairer and Zambotti (2019) and by Bruned and Schratz (2022) for deriving a general scheme for dispersive PDEs at low regularity. This construction also explains how the algebraic structure by Bruned et al. (2019) cited above can be viewed as a deformation of the Butcher–Connes–Kreimer and the extraction-contraction Hopf algebras. We start by deforming various pre-Lie products via a Taylor deformation and then we apply the Guin–Oudom procedure which gives us an associative product whose adjoint can be compared with known coproducts. This work reveals that pre-Lie products and their deformation can be a central object in the study of (S)PDEs.