Preprint: 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. 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