A Markov property for set-indexed processes

TL;DR

This paper introduces a Markov property for set-indexed processes, enabling the development of a transition system theory and a generator that fully characterizes the process's distribution.

Contribution

It proposes a new Markov property for set-indexed processes and develops a transition system and generator framework for their construction and characterization.

Findings

All processes with independent increments satisfy this Markov property.

The set-indexed generator fully characterizes the process distribution.

A transition system theory is developed for these processes.

Abstract

We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.