JournalsjfgVol. 1, No. 2pp. 153–176

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
Minkowski dimension of Brownian motion with drift cover
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Abstract

We study fractal properties of the image and graph of Brownian motion in RdR^d with an arbitrary cadlag drift ff. We prove that the Minkowski (box) dimension of both the image and the graph of B+fB+f over A[0,1]A \subset [0,1] are a.s. constants. We then show that for all d1d \geq 1 the Minkowski dimension of (B+f)(A)(B+f)(A) is at least the maximum of the Minkowski dimension of f(A)f(A) and that of B(A)B(A). We also prove analogous results for the graph. For linear Brownian motion, if the drift ff is continuous and A=[0,1]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