Natural densities of Markov transition probabilities
type: Item
instance of: scholarly article
title: Natural densities of Markov transition probabilities
zbMATH Open document ID: 0581.60057
DOI: 10.1007/BF01845990
author: Rainer Wittmann
publication date: 1986-01-01T00:00:00Z
review text: Let \((P_ t)\), \((P^*\!_ t)\) be two measurable submarkovian semigroups on a measurable space E which are absolutely continuous and in duality with respect to a \(\sigma\)-finite measure \(\mu\). Then it is shown that there exists a unique measurable function \(p: (0,\infty)\times E\times E\to {\bar {\mathbb{R}}}_+\) satisfying \[ (i)\quad P_ tf(x)=\int p(t,x,y)f(y)\mu (dy),\quad P^*\!_ tf(x)=\int p(t,y,x)f(y)\mu (dy)\quad (f\in E_+,\quad x\in E) \] \[ (ii)\quad p(s+t,x,y)=\int p(s,x,z)p(t,z,y)\mu (dz),\quad s,t>0,\quad x,y\in E. \] In particular, for a symmetric semigroup, there exists a unique symmetric density satisfying (i), (ii). A more general result for inhomogeneous transition probabilities is also given.
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 3930065
zbMATH Keywords: submarkovian semigroups
zbMATH Keywords: symmetric density
zbMATH Keywords: inhomogeneous transition probabilities
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3930065