  In a previous article (\textit{Int. Math. Res. Not.} 2014, 2730--2745) T.
Orponen and the authors proved that the Fourier dimension of the graph of any
real-valued function on $\mathbb{R}$ is bounded above by $1$. This partially
answered a question of Kahane ('93) 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 Ito calculus to estimate Fourier transforms by reformulating the question in
the language of Ito drift-diffusion processes and combine it with the classical
work of Kahane on Brownian images.
