Brownian motion and thermal capacity

Abstract

Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset (0,\infty )$ and $F\subset {\mathbf{R}}^d$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when $d\ge 2$, our formula can be described in terms of the Hausdorff dimension of $E\times F$, where $E\times F$ is viewed as a subspace of space time.

Published in at http://dx.doi.org/10.1214/14-AOP910 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Levy processes: Capacity and Hausdorff dimension