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Existence and Uniqueness of Fixed Points for Markov Operators and Markov Processes
Author(s) -
HernándezLerma O,
Lasserre JB
Publication year - 1998
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611598000380
Subject(s) - mathematics , markov chain , uniqueness , markov process , fixed point , markov kernel , banach space , invariant (physics) , operator (biology) , pure mathematics , discrete mathematics , markov model , variable order markov model , mathematical analysis , statistics , biochemistry , repressor , transcription factor , mathematical physics , gene , chemistry
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: a the existence of invariant probability densities (IPDs) b the existence of strictly positive IPDs, and c the existence and uniqueness of IPDs. 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.