Minkowski dimension of Brownian motion with drift

TL;DR

This paper investigates the fractal Minkowski dimensions of Brownian motion with arbitrary cadlag drifts, establishing that these dimensions are almost surely constant and relate to the dimensions of the drift and Brownian motion itself.

Contribution

It proves that the Minkowski dimensions of the image and graph of Brownian motion with drift are almost surely constant and establishes bounds relating these dimensions to the drift and Brownian motion.

Findings

Minkowski dimensions are almost surely constant for Brownian motion with drift.

The dimension of the image of B+f is at least the maximum of the dimensions of B and f.

For linear Brownian motion with continuous drift, the dimension of the graph equals the maximum of the dimensions of B and f.

Abstract

We study fractal properties of the image and the graph of Brownian motion in $\R^d$ with an arbitrary c{\`a}dl{\`a}g drift $f$. We prove that the Minkowski (box) dimension of both the image and the graph of $B+f$ over $A\subseteq [0,1]$ are a.s.\ constants. We then show that for all $d\geq 1$ the Minkowski dimension of $(B+f)(A)$ is at least the maximum of the Minkowski dimension of $f(A)$ and that of $B(A)$. We also prove analogous results for the graph. For linear Brownian motion, if the drift $f$ is continuous and $A=[0,1]$, then the corresponding inequality for the graph is actually an equality.