Polar Sets and Relative Dimension for Generalized Brownian Sheet

Abstract

Let $\ifmmode\expandafter\tilde\else\expandafter\~\fi{W} \overset{\wedge}{=}{\left\{ { \ifmmode\expandafter\tilde\else\expandafter\~\fi{W}{\left( t \right)};t \in R^{N}_{ + } } \right\}}$ be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for ${\left( {t, \ifmmode\expandafter\tilde\else\expandafter\~\fi{W}{\left( t \right)}} \right)}$ are proved. It is also proved that if 2N ≤ αd, then for any compact set $E \subset R^{N}_{ > } ,$

$$ d - \frac{2} {\alpha }{\text{Dim}}E \leqslant \inf {\left\{ {\dim F:F \in {\user1{\mathcal{B}}}{\left( {R^{d} } \right)},P{\left\{ { \ifmmode\expandafter\tilde\else\expandafter\~\fi{W}{\left( E \right)} \cap F \ne \emptyset} \right\}} > 0} \right\}} \leqslant d - \frac{2} {\beta }{\text{Dim}}E, $$

and if 2N >αd, then for any compact set F ⊂ R ^d \ {0},

$$ \frac{\alpha } {2}{\left( {d - {\text{Dim}}F} \right)} \leqslant \inf {\left\{ {\dim E:E \in {\user1{\mathcal{B}}}{\left( {R^{N}_{ > } } \right)},P{\left\{ { \ifmmode\expandafter\tilde\else\expandafter\~\fi{W}{\left( E \right)} \cap F \ne \emptyset} \right\}} > 0} \right\}} \leqslant \frac{\beta } {2}{\left( {d - {\text{Dim}}F} \right),} $$

where ${\user1{\mathcal{B}}}{\left( {R^{d} } \right)}$ and ${\user1{\mathcal{B}}}{\left( {R^{N}_{ > } } \right)}$ denote the Borel σ-algebra in R ^d and in ${R^{N}_{ > } }$ respectively, dim and Dim are Hausdorff dimension and Packing dimension respectively.