关键词: Brownian Sheet, polar set, Hausdorff dimension, packing dimension

Abstract:

Let $W\hat{=}\{W(t);t\in
R^N_+\}$ be the $d$-dimensional $N$-parameter Brownian Sheet.
Sufficient conditions for a compact set $F\subset R^d\setminus
\{0\}$ to be a polar set for $W$ are proved. It is also proved
that if $2N\leq d$, then for any compact set $ E\subset R^N_>$,
$$
\inf\{{\rm dim} F:F\in {\cal B}(R^d), P\{W(E)\cap F\not= \phi\}>0\}
=d-2{\rm Dim}E,
$$
and if $2N>d$, then for any compact set $F\subset R^d\setminus \{0\}$,
$$
\inf\{{\rm dim}E:E\in {\cal B}(R^N_>), P\{W(E)\cap F\not= \phi\}>0\}
=\frac{d}{2}-\frac{{\rm Dim}F}{2},
$$
where ${\cal B}(R^d)$ and ${\cal B}(R^N_>)$ denote the Borel
$\sigma$-algebra in $R^d$ and  $R^N_>$ respectively, and
dim and Dim are Hausdorff dimension and Packing dimension
respectively.

Key words: Brownian Sheet, polar set, Hausdorff dimension, packing dimension