Some new examples of Markov processes which enjoy the time-inversion property
type: Item
instance of: scholarly article
title: Some new examples of Markov processes which enjoy the time-inversion property
zbMATH Open document ID: 1087.60058
author: Léonard Gallardo
author: Marc Yor
publication date: 2005-05-03T00:00:00Z
review text: \textit{S. Watanabe} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 115--124 (1975; Zbl 0286.60035)] established the problem of finding time-depending homeomorphisms \(g(t,.)\) for the state spaces \(E\) and \(E'\) of 2 conservative diffusions \((X(t),P)\) and \((X'(t),P')\) in such a way that \((g(t,X(1/t)),P)\) and \((X'(t),P')\) are equivalent. As an important tool to find some answers Watanabe used M. G. Krein's spectral theory of the string. The present authors have modified the time-inversion problem from above to the following one: For a given homogeneous Markov process \((X(t),P.)\) in \(\mathbb{R}^n\) find conditions for the associated Markov semigroup under which the process \((tX(1/t),P.)\) is also a homogeneous Markov process. Using quite different methods some answers are given. The impressive set of very different examples contains, e.g., the examples of Bessel processes considered by Watanabe and several new ones as Dunkl processes and their radial parts.
reviewed by: Wilfried Schenk
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 2164757
zbMATH Keywords: Homogeneous Markov processes
zbMATH Keywords: Semi-stable processes
zbMATH Keywords: Time inversion property
zbMATH Keywords: ``Opérateur carré du champ''
zbMATH Keywords: Bessel processes
zbMATH Keywords: Wishart processes
zbMATH Keywords: Dunkl processes
zbMATH Keywords: Radial Dunkl processes
zbMATH Keywords: Dunkl processes with drift
zbMATH Keywords: Brownian motion in a Weyl chamber
MaRDI profile type: Publication
full work available at URL: s00440-004-0399-y
OpenAlex ID: W2002049454
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DOI: 10.1007/S00440-004-0399-Y
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 2164757