Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$

TL;DR

This paper proves that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than 1/2 is almost surely 1, extending previous results and confirming parts of a conjecture.

Contribution

It introduces a combinatorial integration by parts formula to compute Fourier transform moments, advancing the understanding of fractal dimensions of stochastic process graphs.

Findings

Fourier dimension of fractional Brownian motion graph with H > 1/2 is almost surely 1

Graph of symmetric alpha-stable process has Fourier dimension 1 for alpha in [1,2]

Graph of alpha-stable process is a Salem set when alpha=1

Abstract

We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than $1/2$ is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Fa\`a di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric $\alpha$-stable process has Fourier dimension 1 almost surely when $\alpha \in [1,2]$ and is a Salem set when $\alpha = 1$.