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Intertwined Markov processes: the extended Chapman–Kolmogorov equation

Part of: Abstract harmonic analysis Partial differential operators Markov processes Groups and semigroups of linear operators, their generalizations and applications Functional-differential and differential-difference equations

Published online by Cambridge University Press:  04 September 2017

Wha-Suck Lee  and
Niko Sauer
Wha-Suck Lee
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield 0028, Pretoria, South Africa (whasuck.lee@up.ac.za; niko.sauer@up.ac.za)
Niko Sauer
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield 0028, Pretoria, South Africa (whasuck.lee@up.ac.za; niko.sauer@up.ac.za)
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Abstract

Uniformization transforms a pseudo-Poisson process with unequal intensities (leaving rates) into one with uniform intensity. Self-transitions is the price to pay. Two intensities arise when one considers an absorbing barrier of a Markov process as a body in its own right: a pair of Markov processes intertwined by an extended Chapman–Kolmogorov equation naturally arises. We show that Sauer's two-state space empathy theory handles such intertwined processes. The price of self-transitions is also avoided.

Keywords

Chapman–Kolmogorov equationintertwined Markov processesempathyimplicitB-evolution equationpseudo-Poisson processFokker–Planck equations

MSC classification

Secondary: 47D07: Markov semigroups and applications to diffusion processes 47F99: None of the above, but in this section 60J27: Continuous-time Markov processes on discrete state spaces 60J99: None of the above, but in this section 60J28: Applications of continuous-time Markov processes on discrete state spaces 60J35: Transition functions, generators and resolvents 34K32: Implicit equations 34K30: Equations in abstract spaces 47D03: Groups and semigroups of linear operators 47D99: None of the above, but in this section 43A05: Measures on groups and semigroups, etc.
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 1 , February 2018 , pp. 123 - 131
Copyright
Copyright © Royal Society of Edinburgh 2018 

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