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

Electronic Journal of Probability 2026 Open access

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Authors: Cheuk Yin Lee, Yimin Xiao
DOI: 10.1214/26-ejp1483
Cited by: 0
References: 25

Abstract

Let W={W(t):t∈R+N} be an (N,d)-Brownian sheet and let E⊂(0,∞)N and F⊂Rd 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)∩F in terms of the thermal capacity of E×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⊂(0,∞)N is an interval.

Fields of Study & Subjects

Stochastic processes and statistical mechanics Mathematical Dynamics and Fractals Geometric Analysis and Curvature Flows

Publication details

Journal / source
Electronic Journal of Probability
Volume
31
Issue
none
Pages
1–31
Publication date
1 January 2026
Language
EN