Stochastic Chemical Reaction Networks with Discontinuous Limits and AIMD processes

TL;DR

This paper investigates stochastic chemical reaction networks exhibiting discrete-induced transitions, demonstrating their convergence to AIMD processes under certain conditions, and explores the implications of these discontinuous limits in CRNs.

Contribution

The study provides a rigorous analysis of DIT phenomena in CRNs, establishing convergence to AIMD processes and connecting discrete transitions to continuous deterministic limits.

Findings

CRNs can exhibit DIT properties leading to AIMD limit processes

Scaled CRN processes converge to jump processes or deterministic functions

Discrete blowups cause asymptotic discontinuous behavior in CRNs

Abstract

In this paper we study a class of stochastic chemical reaction networks (CRNs) for which chemical species are created by a sequence of chain reactions. We prove that under some convenient conditions on the initial state, some of these networks exhibit a discrete-induced transitions (DIT) property: isolated, random, events have a direct impact on the macroscopic state of the process. If this phenomenon has already been noticed in several CRNs, in auto-catalytic networks in the literature of physics in particular, there are up to now few rigorous studies in this domain. A scaling analysis of several cases of such CRNs with several classes of initial states is achieved. The DIT property is investigated for the case of a CRN with four nodes. We show that on the normal timescale and for a subset of (large) initial states and for convenient Skorohod topologies, the scaled process converges in distribution to a Markov process with jumps, an Additive Increase/Multiplicative Decrease (AIMD) process. This asymptotically discontinuous limiting behavior is a consequence of a DIT property due to random, local, blowups of jumps occurring during small time intervals. With an explicit representation of invariant measures of AIMD processes and time-change arguments, we show that, with a speed-up of the timescale, the scaled process is converging in distribution to a continuous deterministic function. The DIT analyzed in this paper is connected to a simple chain reaction between three chemical species and is therefore likely to be a quite generic phenomenon for a large class of CRNs.