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The universal equivariance properties of exotic aromatic B-series

Adrien Laurent, Hans Munthe-Kaas

TL;DR

This work establishes that exotic aromatic $B$-series are not merely calculational devices but universal geometric objects characterized by locality and orthogonal-equivariance, extending the classical $B$-series and aromatic $B$-series. By developing an invariant-tensor framework and a generalized exotic-aromatic-tree formalism, the authors show that local, orthogonal-equivariant maps admit Taylor expansions as exotic aromatic $B$-series and identify the precise conditions under which subfamilies (stolonic, exotic, and connected exotic aromatic $B$-series) arise via categorical equivariance. A detailed construction using dual vector fields and invariant-tensor transfers yields a bijective correspondence between trees and invariant tensors under dimension-dependent limits, enabling a rigorous decomposition of modified vector fields in stochastic ergodic settings. Degeneracies, such as gradient-vector fields, are analyzed to refine the classification and reveal equivalence relations among trees, linking the theory to practical ergodic SDE integrators and suggesting broad applicability to manifolds and symplectic settings. Overall, the paper unifies deterministic and stochastic geometric integration under a universal, representation-theoretic framework and paves the way for future extensions to more general B-series variants and equivariance structures.

Abstract

The exotic aromatic Butcher series were originally introduced for the calculation of order conditions for the high order numerical integration of ergodic stochastic differential equations in $\mathbb{R}^d$ and on manifolds. We prove in this paper that exotic aromatic B-series satisfy a universal geometric property, namely that they are characterised by locality and equivariance with respect to orthogonal changes of coordinates. This characterisation confirms that exotic aromatic B-series are a fundamental geometric object that naturally generalises aromatic B-series and B-series, as they share similar equivariance properties. In addition, we provide a classification of the main subsets of the exotic aromatic B-series, in particular the exotic B-series, using different equivariance properties. Along the analysis, we present a generalised definition of exotic aromatic trees, dual vector fields, and we explore the impact of degeneracies on the classification.

The universal equivariance properties of exotic aromatic B-series

TL;DR

This work establishes that exotic aromatic -series are not merely calculational devices but universal geometric objects characterized by locality and orthogonal-equivariance, extending the classical -series and aromatic -series. By developing an invariant-tensor framework and a generalized exotic-aromatic-tree formalism, the authors show that local, orthogonal-equivariant maps admit Taylor expansions as exotic aromatic -series and identify the precise conditions under which subfamilies (stolonic, exotic, and connected exotic aromatic -series) arise via categorical equivariance. A detailed construction using dual vector fields and invariant-tensor transfers yields a bijective correspondence between trees and invariant tensors under dimension-dependent limits, enabling a rigorous decomposition of modified vector fields in stochastic ergodic settings. Degeneracies, such as gradient-vector fields, are analyzed to refine the classification and reveal equivalence relations among trees, linking the theory to practical ergodic SDE integrators and suggesting broad applicability to manifolds and symplectic settings. Overall, the paper unifies deterministic and stochastic geometric integration under a universal, representation-theoretic framework and paves the way for future extensions to more general B-series variants and equivariance structures.

Abstract

The exotic aromatic Butcher series were originally introduced for the calculation of order conditions for the high order numerical integration of ergodic stochastic differential equations in and on manifolds. We prove in this paper that exotic aromatic B-series satisfy a universal geometric property, namely that they are characterised by locality and equivariance with respect to orthogonal changes of coordinates. This characterisation confirms that exotic aromatic B-series are a fundamental geometric object that naturally generalises aromatic B-series and B-series, as they share similar equivariance properties. In addition, we provide a classification of the main subsets of the exotic aromatic B-series, in particular the exotic B-series, using different equivariance properties. Along the analysis, we present a generalised definition of exotic aromatic trees, dual vector fields, and we explore the impact of degeneracies on the classification.
Paper Structure (15 sections, 14 theorems, 74 equations, 5 tables)

This paper contains 15 sections, 14 theorems, 74 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Locality and equivariance
  4. Exotic aromatic trees
  5. Characterisation of exotic aromatic B-series
  6. Geometric characterisation of exotic aromatic B-series
  7. Invariant tensor spaces and transfer of geometric properties
  8. Correspondence between exotic aromatic trees and invariant tensors
  9. Characterisation of exotic aromatic B-series
  10. Classification of exotic aromatic B-series
  11. Dual vector fields
  12. Categorical characterisations
  13. Impact of degeneracies on the classification
  14. Conclusion and future works
  15. Exotic aromatic trees of order 3

Key Result

Proposition 2.6

Let $\varphi=(\varphi_d\colon \mathfrak{X}(\mathbb{R}\xspace^d)\rightarrow\mathfrak{X}(\mathbb{R}\xspace^d))_d$ be a sequence of smooth maps. If $\varphi$ is left-orthogonal-equivariant, then $\varphi$ is local, orthogonal-equivariant, and trivially decoupling. If $\varphi$ is right-orthogonal-equiv

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: McLachlan16bsm
  • Definition 2.7
  • Example
  • Example
  • Lemma 2.8
  • ...and 34 more