Analysis & PDE
- Anal. PDE
- Volume 11, Number 1 (2018), 115-132.
On the Fourier analytic structure of the Brownian graph
In a previous article (Int. Math. Res. Not. 2014:10 (2014), 2730–2745) T. Orponen and the authors proved that the Fourier dimension of the graph of any real-valued function on is bounded above by . This partially answered a question of Kahane (1993) by showing that the graph of the Wiener process (Brownian motion) is almost surely not a Salem set. In this article we complement this result by showing that the Fourier dimension of the graph of is almost surely . In the proof we introduce a method based on Itô calculus to estimate Fourier transforms by reformulating the question in the language of Itô drift-diffusion processes and combine it with the classical work of Kahane on Brownian images.
Anal. PDE, Volume 11, Number 1 (2018), 115-132.
Received: 28 March 2016
Revised: 19 July 2017
Accepted: 5 September 2017
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63] 60J65: Brownian motion [See also 58J65] 28A80: Fractals [See also 37Fxx]
Fraser, Jonathan M.; Sahlsten, Tuomas. On the Fourier analytic structure of the Brownian graph. Anal. PDE 11 (2018), no. 1, 115--132. doi:10.2140/apde.2018.11.115. https://projecteuclid.org/euclid.apde/1513774490