Quasi-stationary distributions for Lévy processes
type: Item
instance of: scholarly article
title: Quasi-stationary distributions for Lévy processes
zbMATH Open document ID: 1130.60054
author: Zbigniew Palmowski
author: A. E. Kyprianou
published in: Bernoulli
publication date: 2006-11-06T00:00:00Z
review text: Let \(X=\{ X_{t}:t\geq 0\} \) be a Lévy process defined on a filtered space \(( \Omega ,\mathcal{F},\mathbf{F},P) \) where the filtration \(\mathbf{F}=\{ \mathcal{F}_{t}:t\geq 0\} \) is assumed to satisfy the usual assumptions of right continuity and completion. Consider the probabilities \(\{ P_{x}:x\in \mathbb{R}\} \) such that \(P_{x}( X_{0}=x) =1\) and \(P_{0}=P\). Define the first passage time into the lower half-line \(( -\infty ,0) \) by \(\tau =\inf \{t>0:X_{t}<0\} \).\ The main object of the authors' interest is the existence and characterization of the so-called limiting quasi-stationary distribution (or Yaglom's limit) \(\lim_{t\uparrow \infty}P_{x}( X_{t}\in B| \tau >t) =\mu ( B) \) for \(B\in \mathcal{B}( [0,\infty )) \) where, in particular, \(\mu \) does not depend on the initial state \(x\geq 0\). The authors consider this limit for \(x\geq 0\) in the case of Lévy processes for which \(( -\infty ,0) \) is irregular for \(0\), and for \(x>0\) in the case of Lévy processes for which \(( -\infty ,0) \) is regular for \(0\). The existence and characterization result obtained improves the analogous result for spectrally positive compound Poisson processes with negative drift and jump distribution with rational characteristic function proved (within the context of the \(M/G/1\) queue) by \textit{E. K. Kyprianou} [J. Appl. Prob. 8, 494--507 (1971; Zbl 0234.60121)] as well as the result for Brownian motion with drift proved by \textit{S. Martinez} and \textit{J. San Martin} [J. Appl. Probab. 31, 911--920 (1994; Zbl 0818.60071)], and complements an analogous result for random walks obtained by D. L. Iglehart as well as related results due to R. A. Doney.
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 5070919
zbMATH Keywords: Levy process
zbMATH Keywords: quasi-stationary distribution
zbMATH Keywords: random walk
reviewed by: Marius Iosifescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3150/bj/1155735927
OpenAlex ID: W2078821221
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DOI: 10.3150/BJ/1155735927
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description: scientific article; zbMATH DE number 5070919