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June, 1978 The Uniform Dimension of the Level Sets of a Brownian Sheet
Robert J. Adler
Ann. Probab. 6(3): 509-515 (June, 1978). DOI: 10.1214/aop/1176995535


Let $W_N(\mathbf{t})$ denote the $N$-parameter Brownian sheet (Wiener process) taking values in $R^1$. For $0 < T \leqq 1$, set $\Delta(T) = \{\mathbf{t} \in R^N: 0 < t_i \leqq T, i = 1, \cdots, N\}$ and let $E(x, T) = \{\mathbf{t} \in \Delta(T): W_N(\mathbf{t}) = x\}$, the set of $\mathbf{t}$ where the process is at the level $x$. Then we show that, with probability one, the Hausdorff dimension of $E(x, T)$ equals $N - \frac{1}{2}$ for all $0 < T \leqq 1$ and every $x$ in the interior of the range of $W_N(\mathbf{t}, \mathbf{t} \in \Delta(T)$. This provides an answer to a question raised earlier by Pyke.


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Robert J. Adler. "The Uniform Dimension of the Level Sets of a Brownian Sheet." Ann. Probab. 6 (3) 509 - 515, June, 1978.


Published: June, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0378.60028
MathSciNet: MR490818
Digital Object Identifier: 10.1214/aop/1176995535

Primary: 60G17
Secondary: 60J55 , 60J65

Keywords: Brownian sheet , Hausdorff dimension , Level sets , Local time

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 3 • June, 1978
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