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 cadlag drift f. We prove that the Minkowski (box) dimension of both the image and the graph of B + f over A subset of [0, 1] are a.s. constants. We then show that for all d >= 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.