While backward error analysis does not generalise straightforwardly to the strong and weak approximation of stochastic differential equations, it extends for the sampling of ergodic dynamics. The calculation of the modified equation relies on tedious calculations and there is no expression of the modified vector field, in opposition to the deterministic setting. We uncover in this paper the Hopf algebra structures associated to the laws of composition and substitution of exotic aromatic S-series, relying on the new idea of clumping. We use these algebraic structures to provide the algebraic foundations of stochastic numerical analysis with S-series, as well as an explicit expression of the modified vector field as an exotic aromatic B-series.

翻译：尽管后向误差分析无法直接推广至随机微分方程的强逼近与弱逼近，但其可延伸至遍历动力系统的采样过程。修正方程的计算依赖于繁琐的推导，且与确定性情形不同，修正向量场缺乏显式表达式。本文基于"聚类"这一新思想，揭示了与异域芳香S-级数的复合律及代换律相关联的Hopf代数结构。利用这些代数结构，我们为基于S-级数的随机数值分析奠定了代数基础，并给出了修正向量场作为异域芳香B-级数的显式表达式。