Sharp Wasserstein Convergence Rates for Empirical Path Laws of Itô Processes

[2608.07879] Sharp functional quantization and empirical Wasserstein rates for Itô processes Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2608.07879 (math) [Submitted on 8 Aug 2026 ( v1 ), last revised 28 Sep 2026 (this version, v3)] Title: Sharp functional quantization and empirical Wasserstein rates for Itô processes Authors: Xihao He , Fengyi Yuan View PDF HTML (experimental) Abstract: We establish functional quantization and empirical Wasserstein rates for continuous Itô processes under the supremum norm. We assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $\rho$-moment for some $\rho>1$. Under this assumption, the $n$-point $L^q$ quantization error is at most $C(\log n)^{-1/2}$ for every $1\leq q<\rho$. The known Brownian lower bound shows that the exponent $1/2$ is optimal over this class. Our proof combines an adaptive dyadic time partition with localization according to the size of the integrands. A general transfer principle yields the sharp mean rate $(\log N)^{-1/2}$ for the $p$-Wasserstein distance between the empirical law of $N$ independent copies and their common path law, together with nonasymptotic deviation bounds, whenever $1\leq p<\rho$. Applications include empirical path-law estimates for path-dependent SDEs and a path-space limit-theory estimate for path-dependent McKean--Vlasov interacting particle systems with common noise. Subjects: Probability (math.PR) MSC classes: 60B10, 49Q22, 60H10, 60K35 Cite as: arXiv:2608.07879 [math.PR] (or arXiv:2608.07879v3 [math.PR] for this version) https://doi.org/10.48550/arXiv.2608.07879 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xihao He [ view email ] [v1] Sat, 8 Aug 2026 03:10:57 UTC (28 KB) [v2] Tue, 25 Aug 2026 03:16:29 UTC (33 KB) [v3] Mon, 28 Sep 2026 00:15:50 UTC (38 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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