The Uniform Dimension of the Level Sets of a Brownian Sheet

Abstract

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.