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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\u2013Larson algebras over exotic and grafted forests and the corresponding duals to the Connes\u2013Kreimer 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\u2013Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.<\/jats:p>","DOI":"10.1007\/s10208-023-09638-3","type":"journal-article","created":{"date-parts":[[2024,1,19]],"date-time":"2024-01-19T18:01:31Z","timestamp":1705687291000},"page":"271-301","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Exotic B-Series and S-Series: Algebraic Structures and Order Conditions for Invariant Measure Sampling"],"prefix":"10.1007","volume":"25","author":[{"given":"Eugen","family":"Bronasco","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,1,19]]},"reference":[{"issue":"4","key":"9638_CR1","doi-asserted-by":"publisher","first-page":"1600","DOI":"10.1137\/130935616","volume":"52","author":"A Abdulle","year":"2014","unstructured":"A.\u00a0Abdulle, G.\u00a0Vilmart, and K.\u00a0C. 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