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arXiv · 2305.10993

The universal equivariance properties of exotic aromatic B-series

Adrien Laurent·Hans Munthe-Kaas
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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.

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Adrien Laurent, Hans Munthe-Kaas. 2023-05-18. The universal equivariance properties of exotic aromatic B-series. https://doi.org/10.1007/s10208-024-09668-5

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