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2018 On the Fourier analytic structure of the Brownian graph
Jonathan M. Fraser, Tuomas Sahlsten
Anal. PDE 11(1): 115-132 (2018). DOI: 10.2140/apde.2018.11.115

Abstract

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.

Citation

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Jonathan M. Fraser. Tuomas Sahlsten. "On the Fourier analytic structure of the Brownian graph." Anal. PDE 11 (1) 115 - 132, 2018. https://doi.org/10.2140/apde.2018.11.115

Information

Received: 28 March 2016; Revised: 19 July 2017; Accepted: 5 September 2017; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 06789260
MathSciNet: MR3707292
Digital Object Identifier: 10.2140/apde.2018.11.115

Subjects:
Primary: 42B10, 60H30
Secondary: 11K16, 28A80, 60J65

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.11 • No. 1 • 2018
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