{"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/the-expected-signature-of-stopped-brownian","title":"Expected signature of stopped Brownian motion on $d$-dimensional $C^{2, α}$-domains has finite radius of convergence everywhere: $2\\leq d \\leq 8$","arxiv_id":"2011.07917","date":"2020-11-16","proceeding":null,"authors":["Siran Li","Hao Ni"],"abstract":"A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $\\Omega$, it is only known that the radius of convergence for the expected signature on sufficiently regular $\\Omega$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $\\Omega$ is the $2$-dimensional unit disc ([1]). In this paper, we prove that on any bounded $C^{2,\\alpha}$-domain $\\Omega \\subset \\mathbb{R}^d$ with $2\\leq d \\leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a \"domain-averaging hyperbolic development\" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.","url_abs":"https://arxiv.org/abs/2011.07917v4","url_pdf":"https://arxiv.org/pdf/2011.07917v4.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":"the-expected-signature-of-stopped-brownian","repo_url":"https://github.com/hello0630/esigstoppedbm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}