The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes
type: Item
instance of: scholarly article
title: The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes
zbMATH Open document ID: 1004.60046
DOI: 10.1023/A:1016377720516
author: Jean Bertoin
author: Marc Yor
published in: Potential Analysis
publication date: 2002-08-28T00:00:00Z
review text: A Markov process \(X_t\) on \((0,\infty)\) with law \(\mathbb P^{x}\), \(x>0\), is called semi-stable (or self-similar) with index \(1/\alpha\), if \(\text{law}(kX_{k^{-\alpha}t}, t\geq 0) = \mathbb P^{kx}\), \(k,x>0\). Lamperti established a one-to-one correspondence between semi-stable Markov processes \(X_t\) and Lévy processes \(\xi_t\) which involves time-changes using an exponential functional of the Lévy process. The present paper investigates the limiting behaviour of \(\mathbb P^x\) as \(x\downarrow 0\). This question comes up quite naturally when one wants to characterize the class of all Markov processes which can arise in limit theorems [cf. \textit{J. Lamperti}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 22, 205-225 (1972; Zbl 0274.60052)], but it is also linked, via the scaling property, to the long-time asymptotics of \(X_t\): \(\lim_{x\to 0} \mathbb P^x \circ X_1^{-1} = \mathbb P^0\circ X^{-1}_1\) weakly if and only if weak-\(\lim_{t\to\infty} \mathbb P^y\circ(t^{-1/\alpha}X_t)^{-1! } = \mathbb P^0\circ X_1^{-1}\). The main result of the paper shows that \(\mathbb P^x \to \mathbb P^0\) as \(x\downarrow 0\) in the sense of finite-dimensional distributions whenever the Lévy process \(\xi_t\) is not of lattice type and has strictly positive means. In this case, the semigroup associated with \(X_t\) is Feller, i.e., acts on the space of continuous functions on \((0,\infty)\) which vanish at infinity. This result extends a recent paper by \textit{J. Bertoin} and \textit{M.-E. Caballero} [Bernoulli 8, 195-205 (2002; Zbl 1002.60032)] where the Lévy process was assumed to be a subordinator. The proof of the theorem is based on the renewal theorem and duality arguments which are well-known for the Lévy process \(\xi_t\) and which are extended to the semi-stable Markov process \(X_t\).
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 60G18
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 1790689
zbMATH Keywords: semi-stable Markov process
zbMATH Keywords: entrance distribution
zbMATH Keywords: Lévy process
zbMATH Keywords: exponential functional
zbMATH Keywords: scaling property
zbMATH Keywords: time-reversal
zbMATH Keywords: Feller semigroup
zbMATH Keywords: renewal theorem
zbMATH Keywords: resolvent
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: a:1016377720516
OpenAlex ID: W1523738930
Recommended article: Entrance from \(0+\) for increasing semi-stable Markov processes
Recommended article: Recurrent extensions of self-similar Markov processes and Cramér's condition
Recommended article: Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes
Recommended article: Evolution equations for Lévy stable processes
Recommended article: Q3979061
Recommended article: Expansion of transition distributions of Lévy processes in small time
Recommended article: Semi-stable processes and their extensions
Recommended article: Conditioned stable Lévy processes and the Lamperti representation
Recommended article: Criteria for recurrence and transience of semistable processes
description: scientific article; zbMATH DE number 1790689