An Increment-Type Set-Indexed Markov Property

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.