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Contractive Markov Systems
Author(s) -
Werner Ivan
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704006088
Subject(s) - mathematics , markov chain , aperiodic graph , discrete mathematics , probability measure , bounded function , markov kernel , partition (number theory) , measure (data warehouse) , markov process , variable order markov model , combinatorics , markov model , pure mathematics , computer science , mathematical analysis , statistics , database
Certain discrete‐time Markov processes on locally compact metric spaces which arise from graph‐directed constructions of fractal sets with place‐dependent probabilities are studied. Such systems naturally extend finite Markov chains and inherit some of their properties. It is shown that the Markov operator defined by such a system has a unique invariant probability measure in the irreducible case and an attractive probability measure in the aperiodic case if the vertex sets form an open partition of the state space, the restrictions of the probability functions on their vertex sets are Dini‐continuous and bounded away from zero, and the system satisfies a condition of contractiveness on average.