Large deviations for Markov jump processes with uniformly diminishing rates

TL;DR

This paper establishes a large deviation principle for jump Markov processes with rates that diminish uniformly, extending the applicability of LDPs to systems like chemical reaction networks with vanishing reaction rates.

Contribution

It proves an LDP for jump Markov processes with slowly vanishing rates and relaxes previous assumptions, broadening the scope of applications such as chemical kinetics.

Findings

LDP holds even when jump rates vanish uniformly in the state space.

The assumptions on decay rates are shown to be optimal.

Application to chemical reaction networks with mass action kinetics.

Abstract

We prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further discuss the optimality of our assumptions on the decay of the jump rates. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of mass action kinetics.