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
.
This partially answered a question of Kahane (1993) by showing that the graph of the Wiener
process
(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
is almost
surely
.
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
