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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 realvalued 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 driftdiffusion processes and combine it with the classical work of Kahane on Brownian images.
Original language  English 

Pages (fromto)  115132 
Number of pages  18 
Journal  Analysis & PDE 
Volume  11 
Issue number  1 
Early online date  17 Sept 2017 
DOIs  
Publication status  Published  2018 
Keywords
 Brownian motion
 Wiener process
 Itô calculus
 Itô driftdiffusion process
 Fourier transform
 Fourier dimension
 Salem set
 Graph
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 1 Finished

Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
1/09/16 → 30/06/18
Project: Fellowship