Monotonicity and complete monotonicity for continuous-time Markov chains

TL;DR

This paper explores the concepts of monotonicity and complete monotonicity in continuous-time Markov chains, revealing their differences and conditions under which they coincide, especially in relation to discrete-time cases.

Contribution

It provides a detailed analysis of monotonicity notions in continuous-time Markov chains and identifies conditions where these notions align or differ from discrete-time counterparts.

Findings

Monotonicity and complete monotonicity are not equivalent in continuous-time Markov chains.

In some partially ordered sets, these notions coincide in continuous time but not in discrete time.

The paper characterizes when these properties are equivalent or distinct in various settings.

Abstract

We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.