Hitting probabilities, thermal capacity, and Hausdorff dimension results for the Brownian sheet

Abstract

Let $W=\{W(t):t\in {\mathbb{R}}_{+}^N\}$ be an $(N,d)$-Brownian sheet and let $E\subset (0,\infty {)}^N$ and $F\subset {\mathbb{R}}^d$ be compact sets. We prove a necessary and sufficient condition for $W(E)$ to intersect $F$ with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set $W(E)\cap F$ in terms of the thermal capacity of $E\times F$. This extends the previous results of Khoshnevisan and Xiao (2015) for the Brownian motion and Khoshnevisan and Shi (1999) for the Brownian sheet in the special case when $E\subset (0,\infty {)}^N$ is an interval.