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

TL;DR

This paper investigates the hitting probabilities, Hausdorff dimension, and thermal capacity related to the intersection of the Brownian sheet with sets in space and time, extending previous results for Brownian motion and sheets.

Contribution

It provides a necessary and sufficient condition for intersection probability and characterizes the Hausdorff dimension of intersections using thermal capacity, generalizing earlier work.

Findings

Established a criterion for positive intersection probability.

Determined the Hausdorff dimension of intersection sets.

Extended previous results to more general sets in space and time.

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.