Critical Brownian sheet does not have double points
type: Item
instance of: scholarly article
title: Critical Brownian sheet does not have double points
zbMATH Open document ID: 1269.60053
DOI: 10.1214/11-AOP665
author: Robert C. Dalang
author: D. Khoshnevisan
author: Eulalia Nualart
author: Y. Xiao
author: Dongsheng Wu
published in: The Annals of Probability
publication date: 2012-08-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.0235
full work available at URL: https://projecteuclid.org/euclid.aop/1341401150
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 6067457
zbMATH Keywords: Brownian sheet
zbMATH Keywords: multiple points
zbMATH Keywords: capacity
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: pinned sheet
zbMATH Keywords: conditional potential theory
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
arXiv ID: 1009.0235
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OpenAlex ID: W3098882617
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review text: Let NEWLINE\[NEWLINET = \{ ( s , t ) ; \in ( 0 , \infty ) ^{2 N } , s _{i } \neq t _{i } , i = 1 , \dotsc , N \},NEWLINE\]NEWLINE let \(B\) be a \(N\)-parameter \(\mathbb{R}^{d }\)-Brownian sheet, \(\emptyset \neq A \subset \mathbb{R}^{d }\), \(W _{1 }\), \(W _{2 }\) other, independent, \(N\)-parameter \(\mathbb{R}^{d }\) Brownian sheets. The main result states that NEWLINE\[NEWLINE\operatorname{P}(\text{exists } ( u _{1 } , u _{2 } ) \in T , B ( u _{1 } ) = B ( u _{2 } ) \in A ) > 0NEWLINE\]NEWLINE is equivalent to NEWLINE\[NEWLINE\operatorname{P} (\text{exists}\; (u_{1},u_{2})\in T , W _{1 } ( u _{1 } ) = W _{2 } ( u _{2 } ) \in A ) > 0.NEWLINE\]NEWLINE A tool used are ``pinned sheets'' \(B _{s } ( t ) = B(t)-E ( B ( t ) ; B ( s ) )\).NEWLINENEWLINE This yields the following corollaries.NEWLINENEWLINE1. If NEWLINE\[NEWLINEM _{2 } = \{ x : \text{exists } s \neq t \text{ in }( 0 , \infty )^{N }\text{ with } B ( s ) = B(t)=x\} NEWLINE\]NEWLINE (the set of double points of \(B\)), then \(\operatorname{P} ( M _{2 } \cap A \neq \emptyset ) > 0\) holds for all \(A \neq \emptyset\) if \(d<2N\), is equivalent to \(\operatorname{Cap}_{2 ( d - 2 N ) } ( A ) > 0\) if \(d > 2 N\) and, if \(d = 2 N\), is equivalent to the existence of a probability \(\mu\) with compact support \(\subset A\) and NEWLINE\[NEWLINE\int \int |\log _{+ } ( | x-y | ^{- 1 } ) | ^{2 } d \mu ( x ) d \mu ( y ) < \infty.NEWLINE\]NEWLINE \(\operatorname{Cap}_{\beta } ( A ) ^{- 1 }\) is defined as NEWLINE\[NEWLINE\inf \int \int k _{\beta } ( x-y ) d \mu ( x ) d \mu ( y )NEWLINE\]NEWLINE over all probabilities \(\mu\) with compact support \(\subset A\), \(k _{\beta } ( x ) = \| x \| ^{- \beta }\) for \(\beta > 0\), \(\log _{+ } ( \| x \| ^{- 1 } )\) for \(\beta = 0\), \(1\) for \(\beta < 0\).NEWLINENEWLINE2. \(B\) has double points if and only if \(d < 4 N\). \(M _{2 }\) has a.s. positive Lebesgue measure if and only if \(d < 2 N\).NEWLINENEWLINE 3. Define \(s < _{\pi } t\) for \(s , t \in \mathbb{R}^{N }\), …
description: scientific article; zbMATH DE number 6067457