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.

• The paper establishes necessary and sufficient conditions using thermal capacity for the Brownian sheet to intersect compact sets with positive probability.
• It generalizes classical Brownian motion results to the multidimensional case, linking hitting probabilities with the Hausdorff dimensions of the intersection sets.
• The rigorous framework provided offers key insights for analyzing intersection geometries in stochastic processes and potential applications in hyperbolic SPDEs.

Hitting Probabilities, Thermal Capacity, and Hausdorff Dimension Results for the Brownian Sheet

The paper by Cheuk Yin Lee and Yimin Xiao explores the intersection properties of the Brownian sheet, particularly focusing on its hitting probabilities, thermal capacity, and Hausdorff dimension. The (N,d)-Brownian sheet $W = \{W(t) : t \in \mathbb{R}^N\}$ is studied concerning the intersection with a compact set $F \subset \mathbb{R}^d$ through its image $W(E)$ for compact subsets $E \subset (0, 0)^N$.

The authors extend previously known results about hitting probabilities for the Brownian motion to the (N,d)-dimensional Brownian sheet. Such results were initially tackled by Watson and Doob for Brownian motion and were further developed for special cases of the Brownian sheet by Khoshnevisan and Shi. In this work, Cheuk Yin Lee and Yimin Xiao generalize these specialized results into a more comprehensive framework applicable to arbitrary compact $E$ and $F$.

The paper addresses two essential queries: determining the necessary and sufficient conditions for the intersection $W(E) \cap F \neq \emptyset$ with positive probability and establishing the Hausdorff dimensions of the intersection sets under such conditions. The authors effectively utilize the concepts of thermal capacity and energy, natural extensions of earlier work for $N = 1$, to provide answers to these questions.

One of the significant findings is the explicit conditions under which the Brownian sheet, characterized by the absence of stationary increments like Brownian motion, intersects $F$ with positive probability. The necessary and sufficient condition for this is expressed in terms of the thermal capacity, $C_2(E \times F) > 0$, which correlates with the parabolic Hausdorff dimension of the Cartesian product $E \times F$.

The results further enrich the understanding of the dimension of the intersections by proving that the essential supremum of the Hausdorff dimension of $W(E) \cap F$ can be deduced using the thermal capacity. An explicit duality exists between hitting probabilities and the geometrical construct of the intersection set's Hausdorff dimension, presented through Theorems 1.3 and 1.4.

In essence, this work provides a rigorous mathematical formulation for analyzing the intersection properties of multidimensional Gaussian fields with specific sets. It proposes methods applicable to other random fields, such as solutions to hyperbolic SPDEs, extending the utility of Brownian sheet analysis beyond classical applications. The formulations presented herein offer a richer conceptual toolkit for understanding the interaction of stochastic processes with complex geometrical structures.

Overall, the paper indicates promising directions for further research, particularly in multidimensional random fields and potential applications in probability theory and spatial statistics. As researchers continue to explore applications involving multi-parameter processes, the techniques from this study could aid in the exploration of spatial phenomena in higher-dimensional contexts.