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Existence and uniqueness of fixed points for Markov operators andMarkov processes

Published online by Cambridge University Press:  01 May 1998

O Hernández-Lerma  and
JB Lasserre
O Hernández-Lerma
Affiliation:
Departamento de Matemáticas, CINVESTAV-IPN, Apartado Postal 14-740, México D.F. 07000, Mexico. E-mail: ohernand@math.cinvestav.mx
JB Lasserre
Affiliation:
LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse Cédex, France. E-mail: lasserre@laas.fr
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Abstract

This paper concerns a Markov operator $T$ on a space $L_1$, and a Markov process $P$ which defines a Markov operator on a space $M$ of finite signed measures. For $T$, the paper presents necessary and sufficient conditions for: \begin{enumerate}\item [(a)] the existence of invariant probability densities (IPDs)\item [(b)] the existence of strictly positive IPDs, and\item [(c)] the existence and uniqueness of IPDs.\end{enumerate} Similar results on invariant probability measures for $P$ are presented. The basic approach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitable Banach space, and then obtain necessary and sufficient conditions for this equation to have a solution.

1991 Mathematics Subject Classification: 60J05, 47B65, 47N30.

Keywords

Markov operatorsMarkov processesfixed pointsinvariant probability measuresinvariant probability densitieslinear equations in Banach space
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 76 , Issue 3 , May 1998 , pp. 711 - 736
Copyright
London Mathematical Society 1998

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