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An Increment-Type Set-Indexed Markov Property

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Abstract

We present and study a Markov property, named \(\mathcal C\) -Markov, adapted to processes indexed by a general collection of sets. This new definition fulfils one important expectation for a set-indexed Markov property: there exists a natural generalization of the concept of transition operator which leads to characterization and construction theorems of \(\mathcal C\) -Markov processes. Several usual Markovian notions, including Feller and strong Markov properties, are also developed in this framework. Actually, the \(\mathcal C\) -Markov property turns out to be a natural extension of the two-parameter \(*\) -Markov property to the multiparameter and the set-indexed settings. Moreover, extending a classic result of the real-parameter Markov theory, sample paths of multiparameter \(\mathcal C\) -Feller processes are proved to be almost surely right-continuous. Concepts and results presented in this study are illustrated with various examples.

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Acknowledgments

The author would like to thank his supervisor Erick Herbin and the anonymous referee whose careful proofreadings and useful comments have helped to greatly improve this paper.

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  1. MAS Laboratory, École Centrale Paris, Grande Voie des Vignes, 92295 , Châtenay-Malabry, France

    Paul Balança

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  1. Paul Balança
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Correspondence to Paul Balança.

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This article is part of the Ph.D. thesis prepared by the author under the supervision of Erick Herbin.

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Balança, P. An Increment-Type Set-Indexed Markov Property. J Theor Probab 28, 1271–1310 (2015). https://doi.org/10.1007/s10959-014-0555-y

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