Preprint: Brownian Motion and Thermal Capacity

Davar Khoshnevisan and Yimin Xiao

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 (1978). 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.

Keywords. Brownian motion, thermal capacity, Euclidean and space-time Hausdorff dimension.

AMS Classification (2000) Primary: 60J65, 60G17; Secondary: 28A78, 28A80, 60G15, 60J45.

Support. Research supported in part by the the NSF grant DMS-1006903 (D.K.).

Pre/E-Prints. This paper is available in

 Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu Yimin Xiao Department of Statistics and Probability A-413 Wells Hall Michigan State University East Lansing, MI 48824, U.S.A. xiao@stt.msu.edu

Last Update: October 23, 2013
© 2011 - Davar Khoshnevisan & Yimn Xiao