Vol. 11, No. 1, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
Editors' interests
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author index
To appear
Other MSP journals
On the Fourier analytic structure of the Brownian graph

Jonathan M. Fraser and Tuomas Sahlsten

Vol. 11 (2018), No. 1, 115–132

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.

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
Received: 28 March 2016
Revised: 19 July 2017
Accepted: 5 September 2017
Published: 17 September 2017
Jonathan M. Fraser
School of Mathematics and Statistics
University of St Andrews
St Andrews
United Kingdom
Tuomas Sahlsten
School of Mathematics
University of Manchester
United Kingdom