Exit times, moment problems and comparison theorems
type: Item
instance of: scholarly article
title: Exit times, moment problems and comparison theorems
zbMATH Open document ID: 1273.60102
author: Patrick McDonald
published in: Potential Analysis
publication date: 2013-05-13T00:00:00Z
review text: Consider a Riemannian manifold \(M\) with volume measure \(dx\), a domain \(D\subset M\) which has a compact closure and a smooth boundary, the associated Brownian exit time \(r_D\), and \[ Q_D(t):= \int_D \mathbb{P}_x(r_D> t)\,dx. \] It is known that \(Q_D(t)= \sum_{\lambda\in Sp(D)} A^e_\lambda e^{-\lambda t/2}\), where \(Sp(D)\) denotes the Dirichlet Laplacian spectrum of \(D\). The author makes the easy observation that \[ \int_D\mathbb{E}_x(r^n_D)\,dx= n!\;\sum_{\lambda\in Sp(D)} A^2_\lambda\times (2/\lambda)^n, \] for all \(n\in\mathbb{N}\), and as a consequence, that a comparison between the sequences \(\int_D\mathbb{E}_x(r^n_D)\,dx\) for different domains \(D\) induces a comparison between their first Dirichlet eigenvalues, their volumes and their boundary areas.
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 58J65
zbMATH DE Number: 6162459
zbMATH Keywords: spectral geometry
zbMATH Keywords: comparison theorems
zbMATH Keywords: Brownian motion
zbMATH Keywords: exit time
MaRDI profile type: Publication
full work available at URL: s11118-012-9318-5
OpenAlex ID: W2035881335
cites work: Q5508534
cites work: Brownian motion and harnack inequality for Schrödinger operators
cites work: Comparison theorems for exit times
cites work: Q3344779
cites work: Q4821519
cites work: Heat Content, Heat Trace, and Isospectrality
cites work: The spectral geometry of operators of Dirac and Laplace type
cites work: Q3672783
cites work: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
cites work: Torsional rigidity of submanifolds with controlled geometry
cites work: TORSIONAL RIGIDITY OF MINIMAL SUBMANIFOLDS
cites work: Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below
cites work: Isoperimetric conditions, Poisson problems, and diffusions in Riemannian manifolds
cites work: Dirichlet spectrum and heat content
cites work: Q3976730
cites work: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization
cites work: Estimates for the torsion function and Sobolev constants
cites work: Hardy inequality and L p estimates for the torsion function
cites work: Heat content asymptotics of a Riemannian manifold with boundary
DOI: 10.1007/S11118-012-9318-5
description: scientific article; zbMATH DE number 6162459