A Scaling Approach to Stochastic Chemical Reaction Networks

Abstract: We investigate the asymptotic properties of Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action. Their transition rates exhibit a polynomial dependence on the state variable, with possible discontinuities of the dynamics along the boundary of the state space. We investigate the scaling properties of these networks when the norm of the initial state is converging to infinity and the reaction rates are fixed. This scaling approach is used to have insight on the time evolution of these networks when they start from a ``large'' initial state. The main difference with the scalings of the literature is that it does not change neither the graph structure of the CRN, nor its reaction rates. Several simple and interesting examples of CRNs are investigated with this scaling approach, including the detailed analysis of a CRN with several unsual asymptotic properties in the last section. We also show that a stability criterion due to Filonov for positive recurrence of Markov processes may simplify significantly the stability analysis of these networks.