{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/ergodic-sdes-on-submanifolds-and-related","title":"Ergodic SDEs on submanifolds and related numerical sampling schemes","arxiv_id":"1702.08064","date":"2017-02-26","proceeding":null,"authors":["Wei Zhang"],"abstract":"In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure {\\mu} on the level set of a smooth function $\\xi: \\mathbb{R}^d\\rightarrow \\mathbb{R}^k$, $1\\le k < d$. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure is {\\mu}. Motivated by the previous work of Ciccotti, Leli\\`evre, and Vanden-Eijnden [11], as well as the work of Leli\\`evre, Rousset, and Stoltz [33], in this paper we construct a family of ergodic diffusion processes on the level set of $\\xi$ whose invariant measures coincide with the given one. For the conditional measure, in particular, we show that the corresponding SDEs of the constructed ergodic processes have relatively simple forms, and, moreover, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn't require computing the second derivatives of $\\xi$ and the error estimates of its long time sampling efficiency are obtained.","url_abs":"https://arxiv.org/abs/1702.08064v5","url_pdf":"https://arxiv.org/pdf/1702.08064v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"ergodic-sdes-on-submanifolds-and-related","repo_url":"https://github.com/zwpku/sampling-on-levelset","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=1702.08064","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1702.08064"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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