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.
Yimin Xiao
Department of Statistics and Probability
A-413 Wells Hall
Michigan State University
East Lansing, MI 48824, U.S.A.

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