Exotic aromatic B-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 orthogonal-equivariance. 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 classify with stronger equivariance properties the main subsets of the exotic aromatic B-series, in particular the exotic B-series. 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.

翻译：异域芳香B-级数最初是为计算$\mathbb{R}^d$空间及流形上遍历随机微分方程高阶数值积分中的阶条件而引入的。本文证明异域芳香B-级数满足一种普遍几何性质，即它们由局部性和正交等变性所刻画。这一刻画证实了异域芳香B-级数是一种基础几何对象，其与芳香B-级数及B-级数共享类似等变性性质，因此自然推广了后两者。此外，我们基于更强的等变性性质对异域芳香B-级数的主要子集进行了分类，特别包括异域B-级数。在分析过程中，我们提出了异域芳香树与对偶向量场的广义定义，并探讨了退化性对分类的影响。