Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales

We investigate a class of stochastic chemical reaction networks with n≥1 chemical species S₁, …, Sₙ, and whose complexes are only of the form kᵢSᵢ, i=1,…, n, where (kᵢ) are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter N. A natural hierarchy of fast processes, a subset of the coordinates of (Xᵢ(t)), is determined by the values of the mapping i↦kᵢ. We show that the scaled vector of coordinates i such that kᵢ=1 and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as N gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.