On the Fourier analytic structure of the Brownian graph

Fraser, Jonathan M.; Sahlsten, Tuomas

doi:10.2140/apde.2018.11.115

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On the Fourier analytic structure of the Brownian graph

Jonathan M. Fraser and Tuomas Sahlsten

Vol. 11 (2018), No. 1, 115–132

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 $ℝ$ is bounded above by $1$ . This partially answered a question of Kahane (1993) by showing that the graph of the Wiener process $W_{t}$ (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 $W_{t}$ 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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 42B10, 60H30

Secondary: 11K16, 60J65, 28A80

Milestones

Received: 28 March 2016

Revised: 19 July 2017

Accepted: 5 September 2017

Published: 17 September 2017

Authors

Jonathan M. Fraser
School of Mathematics and Statistics University of St Andrews St Andrews United Kingdom

Tuomas Sahlsten
School of Mathematics University of Manchester Manchester United Kingdom

Vol. 11, No. 1, 2018