We study fractal properties of the image and graph of Brownian motion in with an arbitrary cadlag drift . We prove that the Minkowski (box) dimension of both the image and the graph of over are a.s. constants. We then show that for all the Minkowski dimension of is at least the maximum of the Minkowski dimension of and that of . We also prove analogous results for the graph. For linear Brownian motion, if the drift is continuous and , then the corresponding inequality for the graph is actually an equality.
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Philippe H. A. Charmoy, Yuval Peres, Perla Sousi, Minkowski dimension of Brownian motion with drift. J. Fractal Geom. 1 (2014), no. 2, pp. 153–176DOI 10.4171/JFG/4