Images of the Brownian Sheet
| dc.creator | Khoshnevisan, Davar | |
| dc.creator | Xiao, Yimin | |
| dc.date | 2004-09-25 | |
| dc.date.accessioned | 2026-07-07T05:12:34Z | |
| dc.date.available | 2026-07-07T05:12:34Z | |
| dc.description | An N-parameter Brownian sheet in R^d maps a non-random compact set F in R^N_+ to the random compact set B(F) in \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 (1977), J.-P. Kahane (1985a; 1985b), and one of the present authors (1999). (2) If the Hausdorff dimension of F is strictly greater than (d/2), then with probability one, we can find a finite number of points ζ_1,...,ζ_m such that for any rotation matrix θthat leaves F in B(θF), one of the ζ_i's is interior to B(θF). In particular, B(F) has interior-points a.s. This verifies a conjecture of T. S. Mountford (1989). This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of ``sectorial local-non-determinism (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). | |
| dc.description | 27 pages, submitted for publication | |
| dc.identifier | https://arxiv.org/abs/math/0409491 | |
| dc.identifier | http://arxiv.org/abs/math/0409491 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72622 | |
| dc.subject | Probability | |
| dc.subject | 60G15; 60G17; 28A80 | |
| dc.title | Images of the Brownian Sheet | |
| dc.type | text |