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We obtain the order conditions for sampling the invariant measure for a class of Runge\u2013Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.<\/jats:p>","DOI":"10.1007\/s10208-021-09495-y","type":"journal-article","created":{"date-parts":[[2021,6,7]],"date-time":"2021-06-07T22:04:17Z","timestamp":1623103457000},"page":"649-695","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds"],"prefix":"10.1007","volume":"22","author":[{"given":"Adrien","family":"Laurent","sequence":"first","affiliation":[]},{"given":"Gilles","family":"Vilmart","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,7]]},"reference":[{"issue":"3","key":"9495_CR1","doi-asserted-by":"publisher","first-page":"A1800","DOI":"10.1137\/110846609","volume":"34","author":"A Abdulle","year":"2012","unstructured":"A.\u00a0Abdulle, D.\u00a0Cohen, G.\u00a0Vilmart, and K.\u00a0C. 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