Asymptotic behaviour of a Brownian motion on exterior domains
type: Item
instance of: scholarly article
title: Asymptotic behaviour of a Brownian motion on exterior domains
zbMATH Open document ID: 0959.60077
author: P. Collet
author: Servet Martínez
author: Jaime San Martín
publication date: 2000-09-28T00:00:00Z
review text: Let \(p_t(x,y)\) denote the transition density of the Brownian motion of \(\mathbb{R}^d\) killed at the boundary \(\partial D\) of a domain \(D\) having non-polar compact complement. It is shown that, as \(t\to+\infty\), \(p_t(x,y)\) behaves as \({2\over\pi} u(x)u(y) t^{-1}(\log t)^{-2}\) for \(d=2\) and as \((2\pi t)^{-d/2}u(x)u(y)\) for \(d\geq 3\), where \(u(x)\) is the unique positive harmonic function vanishing on \(\partial D\) which is at infinity equivalent to \(\log|x|\) for \(d=2\) and to 1 for \(d\geq 3\). The proof is based on Harnack's inequality and a comparison with the particular case of \(D\) being the complement of the unit ball. \(u(x)\) takes place naturally, since it equals the limit as \(t\to\infty\) of \(\mathbb{P}_x\)(lifetime \(>t\)) for \(d\geq 3\) and of \({1\over 2}\log t \mathbb{P}_x\)(lifetime \(>t\)) for \(d=2\).
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J45
zbMATH DE Number: 1454832
zbMATH Keywords: killed Brownian motion
zbMATH Keywords: transition density
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: harmonic function
MaRDI profile type: Publication
full work available at URL: s004400050251
OpenAlex ID: W2022349036
DOI: 10.1007/S004400050251
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 1454832