Itô excursion theory for self-similar Markov processes
type: Item
instance of: scholarly article
title: Itô excursion theory for self-similar Markov processes
zbMATH Open document ID: 0810.60067
published in: The Annals of Probability
publication date: 1995-04-09T00:00:00Z
review text: A standard Markov process on \(([0, \infty), {\mathcal B} [0, \infty))\) with transition function \(P_ t (x,A)\) is considered. It is assumed that the process is \(\alpha\)-selfsimilar, i.e. for some \(\alpha > 0\), \(P_{at} (a^ \alpha x, a^ \alpha A) = P_ t (x,A)\), and that the process is killed when hitting 0. The existence of an \(\alpha\)-selfsimilar extension of the process to \([0, \infty)\) is examined. It is proved that under certain conditions the extension exists and ``either leaves 0 continuously (a.s) or (a.s.) jumps from 0 to \((0, \infty)\) according to the `jumping in' measure \(\eta (dx) = dx/x^{\beta + 1}\), \(\beta > 0\).'' Applications of the result to the diffusion process and to the ``reflecting barrier process'' introduced by S. Watanabe are given.
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J60
zbMATH DE Number: 681374
zbMATH Keywords: selfsimilar Markov process
zbMATH Keywords: entrance law
zbMATH Keywords: diffusion
zbMATH Keywords: reflecting barrier process
zbMATH Keywords: diffusion process
author: Juha Vuolle-Apiala
reviewed by: Eugene Kozarovitsky
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988721
OpenAlex ID: W2010484264
DOI: 10.1214/AOP/1176988721
description: scientific article; zbMATH DE number 681374