The universal equivariance properties of exotic aromatic B-series

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.

26 pages

References in corpus (5)

• The structure group for quasi-linear equations via universal enveloping algebras
• The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
• The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators
• Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces
• Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations

Cited by in corpus (4)

• The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
• The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators
• Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations
• Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration