Probabilistic approach for comparing first eigenvalues
type: Item
instance of: scholarly article
title: Probabilistic approach for comparing first eigenvalues
zbMATH Open document ID: 0564.60071
author: Cristina Betz
author: Henryk Gzyl
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77253
review text: A probabilistic method is used to compare the first eigenvalues of the problems \[ G\psi +\lambda \psi =0,\quad G\psi -q\psi +\lambda \psi =0, \] \[ G\psi -q\psi +\lambda \rho \psi =0\quad in\quad D,\quad \psi |_{\partial D=0}, \] when G is the generator of a certain class of Hunt processes \((X_ t)\). By considering subordinated and time changed processes obtained from the process \((X_ t)\) and characterizing first eigenvalues as \(\lambda_ 1=\sup \{\lambda:\sup_{x\in D}E^ x[e^{\lambda T}]<\infty \}\), the comparison of first eigenvalues can be reduced to the comparison of stopping times. The results generalize classical ones for the Laplace operator. The paper also contains a section with various examples.
Mathematics Subject Classification ID: 60J40
zbMATH DE Number: 3899894
zbMATH Keywords: time change
zbMATH Keywords: first eigenvalues
zbMATH Keywords: Hunt processes
zbMATH Keywords: stopping times
zbMATH Keywords: Laplace operator
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3899894