Composition and substitution of Regularity Structures B-series

TL;DR

This paper introduces Regularity Structures B-series for singular SPDEs, defining composition and substitution operations via algebraic structures, which enhances the understanding and solution methods for these complex equations.

Contribution

It develops a novel algebraic framework for Regularity Structures B-series, linking them with products and Hopf algebras to improve analysis of singular SPDEs.

Findings

Defines composition and substitution of B-series for SPDEs

Rewrites operations via products and Hopf algebras

Provides a new perspective on solving singular SPDEs

Abstract

In this work, we introduce Regularity Structures B-series which are used for describing solutions of singular stochastic partial differential equations (SPDEs). We define composition and substitutions of these B-series and as in the context of B-series for ordinary differential equations, these operations can be rewritten via products and Hopf algebras which have been used for building up renormalised models. These models provide a suitable topology for solving singular SPDEs. This new construction sheds a new light on these products and open interesting perspectives for the study of singular SPDEs in connection with B-series.