Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling

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.