Dimension of polar sets for Brownian sheet (Q1418272)

Let \(W = \{W(t), t \in R_+^N\}\) be the Brownian sheet in \(R^d\). For given compact sets \(E \subset R_>^N=(0, \infty)^N\) and \(F \subset R^d \backslash \{0\}\), the authors prove a sufficient condition for \(P\big\{ W(E) \cap F \neq \emptyset\big\} =0\). Using this condition, together with a condition of \textit{Z. Chen} [J. Math., Wuhan Univ. 17, 373--378 (1997; Zbl 0933.60035)], the authors show that: (i)\ If \(2N \leq d\), then for every compact set \(E \subset (0, \infty)^N\), \[ \inf\Big\{\dim F: \;F \in {\mathcal B}(R^d),\;P\big\{ W(E) \cap F \neq \emptyset\big\} >0\Big\} = d - 2 \text{Dim }E. \] (ii)\ If \(2N > d\), then for every compact set \(F \subset R^d \backslash \{0\}\), \[ \inf\Big\{\dim E: \;E \in {\mathcal B}(R_>^N),\;P\big\{ W(E) \cap F \neq \emptyset\big\} >0\Big\} = (d - \text{Dim }F)/2. \] In the above, \(\dim\) and \(\text{Dim}\) denote Hausdorff and packing dimension, respectively. The corresponding results for fractional Brownian motion have been established by the reviewer [Stochastics Stochastics Rep. 66, 121-151 (1999; Zbl 0928.60027)].