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⊆[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.