Open Access
2023 A unified framework for limit results in chemical reaction networks on multiple time-scales
Timo Enger, Peter Pfaffelhuber
Author Affiliations +
Electron. J. Probab. 28: 1-33 (2023). DOI: 10.1214/22-EJP897


If (XN)N=1,2,... is a sequence of Markov processes which solve the martingale problems for some operators (GN)N=1,2,..., it is a classical task to derive a limit result as N, in particular a weak process limit with limiting operator G. For slow-fast systems XN=(VN,ZN) where VN is slow and ZN is fast, GN consists of two (or more) terms, and we are interested in weak convergence of VN to some Markov process V. In this case, for some fD(G), the domain of G, depending only on v, the limit Gf can sometimes be derived by using some gN0 (depending on v and z), and study convergence of GN(f+gN)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.

Funding Statement

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.


Download Citation

Timo Enger. Peter Pfaffelhuber. "A unified framework for limit results in chemical reaction networks on multiple time-scales." Electron. J. Probab. 28 1 - 33, 2023.


Received: 13 January 2022; Accepted: 26 December 2022; Published: 2023
First available in Project Euclid: 8 February 2023

MathSciNet: MR4546022
zbMATH: 1517.60037
arXiv: 2111:15396
Digital Object Identifier: 10.1214/22-EJP897

Primary: 60F17 , 60J35 , 60J76 , 60K35

Keywords: functional central limit thoerem , Markov jump process , stochastic averaging

Vol.28 • 2023
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