On the Fourier analytic structure of the Brownian graph

TL;DR

This paper proves that the Fourier dimension of Brownian motion graphs is almost surely 1, using Ito calculus to analyze Fourier transforms and extend previous bounds on Fourier dimensions of function graphs.

Contribution

It introduces a novel method based on Ito calculus to determine the Fourier dimension of Brownian graphs, establishing it as almost surely 1.

Findings

Fourier dimension of Brownian graph is almost surely 1

New method using Ito calculus for Fourier transform estimates

Extends previous bounds on Fourier dimensions of function graphs

Abstract

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.