If is a sequence of Markov processes which solve the martingale problems for some operators , it is a classical task to derive a limit result as , in particular a weak process limit with limiting operator G. For slow-fast systems where is slow and is fast, consists of two (or more) terms, and we are interested in weak convergence of to some Markov process V. In this case, for some , the domain of G, depending only on v, the limit can sometimes be derived by using some (depending on v and z), and study convergence of . 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.
Our work is partially funded by the Freiburg Center for Data analysis and Modeling (FDM).
We thank Martin Hutzenthaler for valuable comments on possible improvements of Theorem 2.3.
"A unified framework for limit results in chemical reaction networks on multiple time-scales." Electron. J. Probab. 28 1 - 33, 2023. https://doi.org/10.1214/22-EJP897