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Piecewise-Deterministic Markov Processes as Limits of Markov Jump Processes

Part of: Limit theorems Physiological, cellular and medical topics Markov processes

Published online by Cambridge University Press:  04 January 2016

Uwe Franz ,
Volkmar Liebscher  and
Stefan Zeiser
Uwe Franz*
Affiliation:
University of Franche-Comté
Volkmar Liebscher*
Affiliation:
Ernst Moritz Arndt University Greifswald
Stefan Zeiser*
Affiliation:
Kinesis Pharma BV
*
∗ Postal address: Faculty of Mathematics of Besançon, University of Franche-Comté, Route de Gray 16, 25 030 Besançon cedex, France. Email address: [email protected]
∗∗ Postal address: Faculty of Mathematics and Sciences, Ernst Moritz Arndt University Greifswald, Walther-Rathenau-Straβe 47, 17487 Greifswald, Germany. Email address: [email protected]
∗∗∗ Postal address: Kinesis Pharma BV, Lage Mosten 29, 4822 NK Breda, The Netherlands. Email address: [email protected]
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Abstract

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A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

Keywords

Piecewise-deterministic Markov processstochastic model of gene regulationlimit theoremSkorokhod space

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92C40: Biochemistry, molecular biology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 729 - 748
Copyright
© Applied Probability Trust 

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