Images of the Brownian Sheet

D. Khoshnevisan and Yimin Xiao

Abstract. An \(N\)-parameter Brownian sheet in \({\bf R}^d\) maps a nonrandom compact set \(F\) in \({\bf R}^N_+\) to the random compact set \(B(F)\) in \({\bf R}^d\). We prove two results on the image-set \(B(F)\):

(1) It has positive \(d\)-dimensional Lebesgue measure if and only if \(F\) has positive \((d/2)\)-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1979), J.-F. Kahane (1985a, 1985b), and one of the present authors (Khoshnevisan 1999).

(2) If \(\dim_H F > (d/2)\), then with probability one, we can find a finite number of points \(\zeta_1,\ldots,\zeta_m\in{\bf R}^d\) such that for any rotation matrix \(\theta\) that leaves \(F\) in \({\bf R}^N_+\), one of the \(\zeta_i\)'s is interior to \(B(\theta F)\). A simple consequence of this latter fact is that \(B(F)\) has interior-points, and this verifies a conjecture of Mountford (1989).

This paper contains two novel ideas: On one hand, to prove (1), we introduce and analyse a family of bridged sheets. On the other hand, (2) is proved by developing a notion of "sectorial local-nondeterminism (LND)." Both ideas may be of independent interest. We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).

Keywords. Brownian sheet, image, Bessel--Riesz capacity, Hausdorff dimension, interior-point

AMS Classification (2000). 60G15, 60G17, 28A80.

Support. Research supported in part by grants from the U.S. National Science Foundation.

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: September 13, 2004
© 2004 - Davar Khoshnevisan and Yimin Xiao