Realizable monotonicity for continuous-time Markov processes

TL;DR

This paper investigates the concepts of stochastic and realizable monotonicity in continuous-time Markov chains on finite partially ordered sets, revealing conditions under which these notions coincide or differ from discrete-time cases.

Contribution

It formalizes and compares stochastic and realizable monotonicity in continuous-time Markov processes, highlighting differences from discrete-time scenarios and identifying conditions for their equivalence.

Findings

Stochastic and realizable monotonicity are not equivalent in continuous-time Markov chains.

There exist specific partially ordered sets where these notions coincide in continuous-time but not in discrete-time.

The paper provides a formal framework for analyzing monotonicity properties in continuous-time Markov processes.

Abstract

We formalize and analyze the notions of stochastic monotonicity and realizable mono-tonicity 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 stochastic monotonicity and realizable monotonicity coincide in continuous-time but not in discrete-time.