On Monotonicity and Propagation of Order Properties
For researchers in dynamical systems and stochastic processes, this work provides a theoretical bridge between deterministic monotonicity and stochastic order propagation, though the results are largely theoretical and incremental.
The paper establishes a link between monotonicity in deterministic dynamical systems and propagation of order by Markov processes, deriving infinitesimal characterizations for increasing and increasing convex orders. It shows that increasing order equals standard monotonicity, while increasing convex order corresponds to monotone systems with convex vector fields, with applications to diffusion processes and biological systems.
In this paper, a link between monotonicity of deterministic dynamical systems and propagation of order by Markov processes is established. The order propagation has received considerable attention in the literature, however, this notion is still not fully understood. The main contribution of this paper is a study of the order propagation in the deterministic setting, which potentially can provide new techniques for analysis in the stochastic one. We take a close look at the 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 increasing order is equivalent to the standard monotonicity, while the class of systems propagating the increasing convex order is equivalent 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.