A unified framework for limit results in chemical reaction networks on multiple time-scales

Abstract

If ${(X^N)}_{N=1,2,\mathrm{...}}$ is a sequence of Markov processes which solve the martingale problems for some operators ${(G^N)}_{N=1,2,\mathrm{...}}$, it is a classical task to derive a limit result as $N\to \mathrm{\infty }$, in particular a weak process limit with limiting operator G. For slow-fast systems $X^N=(V^N,Z^N)$ where $V^N$ is slow and $Z^N$ is fast, $G^N$ consists of two (or more) terms, and we are interested in weak convergence of $V^N$ to some Markov process V. In this case, for some $f\in \mathcal{D}(G)$, the domain of G, depending only on v, the limit $Gf$ can sometimes be derived by using some $g_N\to 0$ (depending on v and z), and study convergence of $G^N(f+g_N)\to Gf$. We develop this method further in order to obtain functional Laws of Large Numbers (LLNs) and Central Limit Theorems (CLTs). We then apply our general result to various examples from Chemical Reaction Network theory. We show that we can rederive most limits previously obtained, but also provide new results in the case when the fast-subsystem is first order. In particular, we allow that fast species to be consumed faster than they are produced, and we derive a CLT for Hill dynamics with coefficient 2.