Images of the Brownian Sheet

**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. 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: September 13, 2004*

©
2004 - Davar Khoshnevisan and Yimin Xiao