B-series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-series and S-series. Precisely, we prove the relationship between the Grossman-Larson algebras over exotic and grafted forests and the corresponding duals to the Connes-Kreimer coalgebras and use it to study the natural composition laws on exotic S-series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge-Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

翻译：B-级数及其推广是数值积分器分析的有力工具。为研究遍历随机微分方程不变测度采样的阶条件，引入了名为奇异芳香B-级数的推广形式。通过引入新的对称归一化系数，我们分析了与奇异B-级数和S-级数相关的代数结构。具体而言，我们证明了奇异森林与嫁接森林上的Grossman-Larson代数及其对应的Connes-Kreimer余代数对偶之间的关系，并利用这一关系研究奇异S-级数的自然复合律。将这一代数框架应用于一类随机龙格-库塔方法阶条件的推导，我们提出了一种乘法性质，该性质确保某些阶条件自动得到满足。