Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$

[2502.09032] Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$ Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2502.09032 (math) [Submitted on 13 Feb 2025 ( v1 ), last revised 28 Aug 2026 (this version, v4)] Title: Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$ Authors: Chun-Kit Lai , Cheuk Yin Lee View PDF HTML (experimental) Abstract: We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than $1/2$ is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Faà di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric $\alpha$-stable process has Fourier dimension 1 almost surely when $\alpha \in [1,2]$ and is a Salem set when $\alpha = 1$. Subjects: Probability (math.PR) ; Classical Analysis and ODEs (math.CA) Cite as: arXiv:2502.09032 [math.PR] (or arXiv:2502.09032v4 [math.PR] for this version) https://doi.org/10.48550/arXiv.2502.09032 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Cheuk Yin Lee [ view email ] [v1] Thu, 13 Feb 2025 07:37:52 UTC (37 KB) [v2] Thu, 4 Sep 2025 05:27:07 UTC (39 KB) [v3] Wed, 20 May 2026 08:35:02 UTC (40 KB) [v4] Fri, 28 Aug 2026 07:16:26 UTC (41 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2025-02 Change to browse by: math math.CA 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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