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2006 Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet
Yimin Xiao, Davar Khoshnevisan, Dongsheng Wu
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Electron. J. Probab. 11: 817-843 (2006). DOI: 10.1214/EJP.v11-353

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).

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Yimin Xiao. Davar Khoshnevisan. Dongsheng Wu. "Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet." Electron. J. Probab. 11 817 - 843, 2006. https://doi.org/10.1214/EJP.v11-353

Information

Accepted: 19 September 2006; Published: 2006
First available in Project Euclid: 31 May 2016

zbMATH: 1111.60020
MathSciNet: MR2261054
Digital Object Identifier: 10.1214/EJP.v11-353

Subjects:
Primary: 60G15
Secondary: 28A80 , 60G17

Keywords: Brownian sheet , Hausdorff dimension , image , multiple points , Packing dimension , Salem sets , Sectorial local nondeterminism

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