Ratio limit theorems for a Brownian motion killed at the boundary of a Benedicks domain
type: Item
instance of: scholarly article
title: Ratio limit theorems for a Brownian motion killed at the boundary of a Benedicks domain
zbMATH Open document ID: 0965.60076
author: P. Collet
author: Servet Martínez
author: Jaime San Martín
published in: The Annals of Probability
publication date: 2001-08-02T00:00:00Z
review text: Let be \(n\geq 0\) and consider \(E\), a closed proper subset of \(\mathbb{R}^{n-1}\). Also let \({\mathcal P}_E\) be the conc of positive harmonic functions in \({\mathcal D}=\mathbb{R}^n\setminus E\times \{0\}\) vanishing at \(E\). A result of \textit{M. Benedicks} [Ark. Math. 18, 53-72 (1980; Zbl 0455.31009)] states that either all functions in \({\mathcal P}_E\) are proportional or \({\mathcal P}_E\) is generated by two linearly independent, minimal positive harmonic functions. The authors characterize for an integral test both cases and remark that in both cases there is a unique, up to a multiplicative constant, positive harmonic and symmetric function. The authors show ratio limit theorems for the associated heat kernlel. When the hole is compact, the Martin boundary is two-dimensional. Sharp estimates on the life time probabilities are also obtained and the various constants appearing in the theory are identified in probabilistic terms.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 1462547
zbMATH Keywords: Brownian motion
zbMATH Keywords: heat kernel
zbMATH Keywords: ratio limit theorems
zbMATH Keywords: Benedicks domain
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677443
OpenAlex ID: W1539397934
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DOI: 10.1214/AOP/1022677443
description: scientific article; zbMATH DE number 1462547