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

Cheuk Yin Lee, Yimin Xiao

TL;DR

This work extends hitting-probability and fractal-dimension results from Brownian motion to the Brownian sheet by introducing the $\gamma$-thermal capacity $C_\gamma(E\times F)$. It proves a sharp necessary-and-sufficient criterion: $\mathbb{P}(W(E)\cap F)>0$ if and only if $C_0(E\times F)>0$, and characterizes the essential supremum of the Hausdorff dimension of $W(E)\cap F$ as $\|{\rm dim}_{\rm H}(W(E)\cap F)\|_\infty = \sup\{\gamma>0: C_\gamma(E\times F)>0\}$. To analyze the dimension, the paper employs a codimension argument with an independent additive $\alpha$-stable process, linking hitting properties to parabolic capacity and establishing a parabolic Frostman-type theorem, particularly when $d\ge 2N$ where ${\rm dim}_{\rm H}(W(E)\cap F) = {\rm dim}_{\rm H}(E\times F; \rho) - d$. The methodology combines pinned Brownian sheets, parabolic geometry, and multiparameter martingale inequalities to illuminate the intersection structure of Gaussian random fields and related SPDEs.

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.

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

TL;DR

This work extends hitting-probability and fractal-dimension results from Brownian motion to the Brownian sheet by introducing the -thermal capacity . It proves a sharp necessary-and-sufficient criterion: if and only if , and characterizes the essential supremum of the Hausdorff dimension of as . To analyze the dimension, the paper employs a codimension argument with an independent additive -stable process, linking hitting properties to parabolic capacity and establishing a parabolic Frostman-type theorem, particularly when where . The methodology combines pinned Brownian sheets, parabolic geometry, and multiparameter martingale inequalities to illuminate the intersection structure of Gaussian random fields and related SPDEs.

Abstract

Let be an -Brownian sheet and let and be compact sets. We prove a necessary and sufficient condition for to intersect with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set in terms of the thermal capacity of . 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 is an interval.
Paper Structure (5 sections, 15 theorems, 197 equations)

This paper contains 5 sections, 15 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Thermal capacity
  3. Proof of Theorem \ref{['T:hit']}
  4. Intersection with the range of an additive stable process
  5. Hausdorff dimension of $W(E)\cap F$ and proofs of Theorems \ref{['T:dim:cap']} and \ref{['T:dim:dim-d']}

Key Result

Theorem 1.2

Fix $0<a<b<\infty$. Then, there exists $K=K(N,d,a,b)\in (1,\infty)$ such that for all compact sets $E \subset (a, b)^N$ and $F \subset \{x \in {\mathbb R}^d : \|x\| < b\}$, where $F$ has Lebesgue measure zero, we have As a consequence, ${\mathbb P}(W(E) \cap F \ne \varnothing) > 0$ if and only if $C_0(E \times F) > 0$, and we have

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 19 more