Title: Geometric Properties of the Brownian Sheet
Speaker: Professor Yimin Xiao
Speaker Info: Department of Statistics and Probability, Michigan State University
Brief Description:
Special Note:
Abstract:
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\title{Geometric Properties of the Brownian Sheet}
\author{\bf Yimin Xiao \\ Department of Statistics and Probability\\ Michigan State University \\ East Lansing, MI 48823 \\ {\tt E-mail: xiao@stt.msu.edu.}}
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\begin{abstract}
The $N$-parameter Brownian sheet $B= \{B(t), t \in {\bf R}_+^N\}$ in ${\bf R}^d$ is a multiparameter extension of Brownian motion and is one of the most important Gaussian random fields. This talk is concerned with geometric properties of the random set $B(F)$, where $F \subseteq {\bf R}^N_+$ is a non-random Borel set. We prove the following results on the image-set $B(F)$: \\ (1) It has positive $d$-dimensional Lebesgue measure if and only if $F$ has positive $\frac d 2$-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P.\@ Kahane (1985a, 1995b) and D. Khoshnevisan (1999).\\ % (2) If $\dim F > \frac d 2$, then with probability one, $B(F)$ has interior-points a.s. This verifies a conjecture of T. S. Mountford (1989).\\ % (3) If $\dim F \le d/2$, then $B(F)$ is almost surely a Salem set.
The proofs of these results rely on two ideas: To prove {(1)}, we introduce and analyze a family of bridged sheets. Items {(2)} and (3) are proved by developing a notion of ``sectorial local-nondeterminism (LND).'' Both ideas may be of independent interest.
This talk is based on joint work with Davar Khoshnevisan. \end{abstract}
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