Subexponential behaviour of the Dirichlet heat kernel
type: Item
instance of: scholarly article
title: Subexponential behaviour of the Dirichlet heat kernel
zbMATH Open document ID: 1031.35048
DOI: 10.1016/S0022-1236(02)00056-3
published in: Journal of Functional Analysis
publication date: 2003-04-28T00:00:00Z
review text: Let \(D\) be an open and connected set in \(\mathbb{R}^m\), and let \(p_D(x,y; t)\), \(x\in D\), \(y\in D\), \(t> 0\) denote the Dirichlet heat kernel associated to the parabolic operator \(-\Delta_D+ {\partial\over\partial t}\), where \(-\Delta_D\) is the Dirichlet Laplacian for \(D\). Let (\(B(s)\), \(s\geq 0\); \(P_x\), \(x\in\mathbb{R}^m\)) be a Brownian motion associated to \(-\Delta+{\partial\over\partial t}\), and let \(T_D\) be the first exit time from \(D\). In this paper a relation between the decay properties of \(p_D(x,y; t)\) and \(P_x[T_D> t]\) for large time is established, under assumption that \(P_x[T_D> t]\) decays subexponentially. Variational lower bounds for \(p_D(x,x; t)\) and \(P_x[T_D> 1]\) are given, too.
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1901762
zbMATH Keywords: Brownian motion
zbMATH Keywords: decay properties
author: M. van den Berg
reviewed by: A. Cichocka
MaRDI profile type: Publication
cites work: The first exit time of planar Brownian motion from the interior of a parabola
cites work: Brownian motion in cones
cites work: Gaussian bounds for the Dirichlet heat kernel
cites work: A Gaussian Lower Bound for the Dirichlet Heat Kernel
cites work: Asymptotic behaviour of a Brownian motion on exterior domains
cites work: Q3342857
cites work: The first exit time of Brownian motion form a parabolic domain
cites work: Pointwise estimates and quasilinear parabolic equations
cites work: Q5562752
description: scientific article; zbMATH DE number 1901762