An increment type set-indexed Markov property

TL;DR

This paper introduces the C-Markov property for set-indexed processes, extending classical Markov concepts to multiparameter and set-indexed frameworks, with characterization theorems and path regularity results.

Contribution

It defines the C-Markov property, develops related transition operators, and extends classical Markovian notions to set-indexed and multiparameter processes.

Findings

C-Markov property generalizes classical Markov properties

Existence of transition operators for C-Markov processes

Sample paths of C-Feller processes are right-continuous

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.