@article{EJP353,
author = {Yimin Xiao and Davar Khoshnevisan and Dongsheng Wu},
title = {Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {11},
year = {2006},
keywords = {Brownian sheet, sectorial local nondeterminism, image, Salem sets, multiple points, Hausdorff dimension, packing dimension.},
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).},
pages = {no. 32, 817-843},
issn = {1083-6489},
doi = {10.1214/EJP.v11-353},
url = {http://ejp.ejpecp.org/article/view/353}}