Hopf-algebraic deformations of products and Wick polynomials

TL;DR

This paper introduces a Hopf algebraic framework for understanding cumulant-moment relations and Wick polynomials, generalizing their construction through algebraic deformations and connecting to regularity structures.

Contribution

It develops a novel Hopf algebra deformation approach to define Wick polynomials and cumulant relations, extending classical concepts to a broader algebraic context.

Findings

Wick polynomials can be constructed via Hopf algebra deformations.

The approach unifies cumulant-moment relations within a coalgebra framework.

Connections to regularity structures are established through generalized deformed products.

Abstract

We present an approach to classical definitions and results on cumulant--moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand the construction of Wick polynomials as the result of a Hopf algebra deformation under the action of linear automorphisms induced by multivariate moments associated to an arbitrary family of random variables with moments of all orders. We also generalise the notion of deformed product in order to discuss how these ideas appear in the recent theory of regularity structures.