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Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet

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1.	Title	Title of document	Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet
2.	Creator	Author's name, affiliation, country	Yimin Xiao; Michigan State University
2.	Creator	Author's name, affiliation, country	Davar Khoshnevisan; The University of Utah
2.	Creator	Author's name, affiliation, country	Dongsheng Wu; Michigan State University
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Brownian sheet, sectorial local nondeterminism, image, Salem sets, multiple points, Hausdorff dimension, packing dimension.
3.	Subject	Subject classification	60G15, 60G17, 28A80
4.	Description	Abstract	We prove the following results about the images and multiple points of an $N$-parameter, $d$-dimensional Brownian sheet $B =\{B(t)\}_{t \in R_+^N}$: (1) If $\text{dim}_H F \leq d/2$, then $B(F)$ is almost surely a Salem set. (2) If $N \leq d/2$, then with probability one $\text{dim}_H B(F) = 2 \text{dim} F$ for all Borel sets of $R_+^N$, where "$\text{dim}_H$" could be everywhere replaced by the ``Hausdorff,'' ``packing,'' ``upper Minkowski,'' or ``lower Minkowski dimension.'' (3) Let $M_k$ be the set of $k$-multiple points of $B$. If $N \leq d/2$ and $ Nk > (k-1)d/2$, then $\text{dim}_H M_k = \text{dim}_p M_k = 2 Nk - (k-1)d$, a.s. The Hausdorff dimension aspect of (2) was proved earlier; see Mountford (1989) and Lin (1999). The latter references use two different methods; ours of (2) are more elementary, and reminiscent of the earlier arguments of Monrad and Pitt (1987) that were designed for studying fractional Brownian motion. If $N>d/2$ then (2) fails to hold. In that case, we establish uniform-dimensional properties for the $(N,1)$-Brownian sheet that extend the results of Kaufman (1989) for 1-dimensional Brownian motion. Our innovation is in our use of the sectorial local nondeterminism of the Brownian sheet (Khoshnevisan and Xiao, 2004).
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Research of D. Khoshnevisan and Y. Xiao is supported in part by the NSF grant DMS-0404729. The research of D. Wu is supported in part by the NSF grant DMS-0417869.
7.	Date	(YYYY-MM-DD)	2006-09-19
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/353
10.	Identifier	Digital Object Identifier	10.1214/EJP.v11-353
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 11
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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