An increment type set-indexed Markov property

Abstract

In this article is introduced and studied a set-indexed Markov property named C-Markov. This new definition fulfils one important expectation for a Markov property: there exists a natural set-indexed generalization of the concept of transition operator which leads to characterization and construction theorems for C-Markov processes. Several other usual Markovian notions, including Feller and strong Markov properties, can also be developed in this framework. Actually, the C-Markov property turns out to be a natural extension of the two-parameter \ast-Markov property to the multiparameter and as well the set-indexed settings. Moreover, generalizing a classic result of the real-parameter Markov theory, sample paths of multiparameter C-Feller processes are proved to be almost surely right-continuous. Concepts and results introduced in this study are illustrated with various examples.