On monotonicity and propagation of order properties

In this paper, a link between monotonicity of deterministic dynamical systems and propagation of order by Markov processes is established. Order propagation has received a considerable attention in the literature, however, this notion is yet to be fully understood. The main contribution of this paper is a study of order propagation in the deterministic setting, which potentially can provide new techniques for analysis in the stochastic one. We take a close look at propagation of the so-called increasing and increasing convex orders. Infinitesimal characterisations of these orders are derived, which resemble the well-known Kamke conditions for monotonicity. It is shown that propagation of the increasing order is equivalent to classical monotonicity, while the class of systems propagating the increasing convex order is equal to the class of monotone systems with convex vector fields. The paper is concluded by deriving a novel result on order propagating diffusion processes and an application of this result to biological processes.