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Composition and substitution of Regularity Structures B-series

Yvain Bruned

TL;DR

The paper develops Regularity Structures B-series as a rigorous extension of Butcher-type tree expansions to singular SPDEs. It introduces two coupled series, $B_-$ for the solution and $B_+$ for the nonlinear right-hand side, indexed by decorated trees, and proves composition and substitution rules via deformed products $\star_2$ and $\star_1$ that arise from a co-interaction of BCK and EC coproducts. This algebraic framework naturally encodes renormalisation through maps $M_{\beta}$ and re-centering operators $\Pi_z$, linking the analytic renormalisation of iterated integrals to precise algebraic operations on decorated trees. The results provide a coherent, diagrammatic approach to describe, renormalise, and analyze the solutions of singular SPDEs within the Regularity Structures setting, offering a bridge between numerical-analytic B-series techniques and stochastic PDE renormalisation.

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.

Composition and substitution of Regularity Structures B-series

TL;DR

The paper develops Regularity Structures B-series as a rigorous extension of Butcher-type tree expansions to singular SPDEs. It introduces two coupled series, for the solution and for the nonlinear right-hand side, indexed by decorated trees, and proves composition and substitution rules via deformed products and that arise from a co-interaction of BCK and EC coproducts. This algebraic framework naturally encodes renormalisation through maps and re-centering operators , linking the analytic renormalisation of iterated integrals to precise algebraic operations on decorated trees. The results provide a coherent, diagrammatic approach to describe, renormalise, and analyze the solutions of singular SPDEs within the Regularity Structures setting, offering a bridge between numerical-analytic B-series techniques and stochastic PDE renormalisation.

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.
Paper Structure (4 sections, 10 theorems, 127 equations)

This paper contains 4 sections, 10 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. B-series in Numerical Analysis
  3. Algebraic structures on decorated trees
  4. Regularity Structures B-series

Key Result

Theorem 1.1

One has where the product $\star_2$ (resp. $\star_1$) is a deformation of $\star_{\text{\tiny{BCK}}}$ ( $\star_{\text{\tiny{EC}}}$) in the sense of BM22.

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 3.1
  • Proposition 3.2
  • ...and 12 more