Hausdorff and Fourier dimension of graph of continuous additive processes

TL;DR

This paper investigates the Hausdorff and Fourier dimensions of graphs of continuous additive processes, revealing their dependence on local regularity and providing explicit calculations for self-similar cases, including the Cantor staircase.

Contribution

It establishes the relationship between the dimensions of the graph and the local Hölder indices of the process, and computes these dimensions explicitly for self-similar functions.

Findings

Hausdorff dimension is 3/2 when V is bi-Lipschitz.

Fourier dimension is positive if V has points with positive lower Hölder regularity.

Hausdorff dimension for the Cantor staircase graph is 1 + (1/2) * (log 2 / log 3).

Abstract

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive additive processes with zero mean. Such processes can be represented as $X_t = B_{V(t)}$ where $B$ is Brownian motion and $V$ is a continuous increasing function. We show that these dimensions depend on the local uniform H\"{o}lder indices. In particular, if $V$ is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if $V$ admits at least one point with positive lower H\"{o}lder regularity. It is also possible to estimate the Hausdorff dimension of the graph through the $L^q$ spectrum of $V$. We will show that if $V$ is generated by a self-similar measure on ${\mathbb R}^{1}$ with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its $L^q$ spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is $1+\frac12\cdot\frac{\log 2}{\log 3}$. Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely.