The Range of Levy's $N$-Parameter Brownian Motion in $d$-Space

Abstract

Let $B^{(N,d)}$ be Levy's $N$-parameter Brownian motion in $d$-space. It is shown that almost surely $B^{(N,d)}$ doubles the Hausdorff dimension of every Borel set in the parameter space when $d \geqslant 2N$. The dimension of the range of $B$ is also determined in this case.