# Minkowski dimension of Brownian motion with drift

### Philippe H. A. Charmoy

University of Oxford, UK### Yuval Peres

Microsoft Research, Redmond, USA### Perla Sousi

University of Cambridge, UK

## Abstract

We study fractal properties of the image and graph of Brownian motion in $R^d$ with an arbitrary cadlag drift $f$. We prove that the Minkowski (box) dimension of both the image and the graph of $B+f$ over $A \subset [0,1]$ 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.

## Cite this article

Philippe H. A. Charmoy, Yuval Peres, Perla Sousi, Minkowski dimension of Brownian motion with drift. J. Fractal Geom. 1 (2014), no. 2, pp. 153–176

DOI 10.4171/JFG/4