Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 1 (2018), 115-132.

On the Fourier analytic structure of the Brownian graph

Jonathan M. Fraser and Tuomas Sahlsten

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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 1. This partially answered a question of Kahane (1993) by showing that the graph of the Wiener process Wt (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 Wt is almost surely 1. 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.

Article information

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

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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]

Brownian motion Wiener process Itô calculus Itô drift-diffusion process Fourier transform Fourier dimension Salem set graph


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.

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