Vol. 11, No. 1, 2018

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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
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.

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