Minkowski Dimension of Brownian Motion With Drift

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.