On Merging of Stochastic Flow of Semi-Markov Dynamics

TL;DR

This paper studies the merging behavior of semi-Markov processes within a stochastic flow framework, providing explicit formulas for meeting probabilities and conditions for almost sure merging.

Contribution

It introduces a family of stochastic flows for semi-Markov laws and derives conditions under which processes merge with probability one.

Findings

Explicit expressions for meeting and merging probabilities.

Sufficient conditions for almost sure merging of processes.

Framework applicable to various semi-Markov models.

Abstract

Given a semi-Markov law, using an additional parameter, we consider a family of stochastic flows corresponding to that law. Then we suitably select a particular flow, for which we obtain expressions of the meeting and merging probabilities of a pair of semi-Markov processes, solving the same equation but having two different initial conditions. A set of sufficient conditions are also obtained under which any two solutions of the flow eventually merge with probability one.