The universal equivariance properties of exotic aromatic B-series

TL;DR

This paper proves that exotic aromatic B-series are characterized by locality and orthogonal equivariance, establishing them as fundamental geometric objects that generalize traditional B-series and aromatic B-series.

Contribution

It demonstrates the universal geometric property of exotic aromatic B-series and classifies their main subsets based on equivariance properties.

Findings

Exotic aromatic B-series are characterized by locality and orthogonal equivariance.

They generalize aromatic and B-series as fundamental geometric objects.

Classification of exotic B-series subsets based on equivariance.

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.