Hopf-algebraic deformations of products and Wick polynomials

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.

Revised and improved Section 9. 29 pages

Cited by in corpus (5)

• The geometry of the space of branched Rough Paths
• Signature Cumulants, Ordered Partitions, and Independence of Stochastic Processes
• Algebraic groups in non-commutative probability theory revisited
• Perturbation theory for the ${\Phi }_3^4$ measure, revisited with Hopf algebras
• Wick polynomials in non-commutative probability