Since the dawn of stochastic chemical reaction network theory over 50 years ago, there have been many general results about (positive) recurrence, especially in the case of mass-action kinetics. One less-explored area is that of mass-action models whose rate constants, rather than being static, are themselves stochastic. Such models have relevance in applications, since biomolecular systems...
We prove that, for weakly reversible chemical reaction networks with stochastic mass-action kinetics in two species, the associated continuous-time Markov chain is positive recurrent on each closed irreducible communicating class. Equivalently, the process returns to finite sets infinitely often with finite expected return times, and it possesses an invariant probability measure supported on...
The Togashi–Kaneko (TK) model is a prototypical example of an auto- catalytic reaction network exhibiting dramatic switching behavior that is a result of the stochastic dynamics at small volumes. I will present a study of the TK model with additional mutations, using a stochastic averaging principle to make use of the multi-scale feature of its dynamics. I will demonstrate a sensitivity of the...
Chemical reaction networks (CRNs) are commonly analyzed through deterministic or stochastic models that track molecular populations over time. In regimes with large molecule counts, stochastic dynamics are typically approximated by deterministic mass-action kinetics. We present a CRN that defies this expectation: while the deterministic system is unstable, exhibiting finite-time blow-up of...
This set of talks is devoted to the investigation of the mathematical properties of chemical reaction networks described as Markov processes in a finite dimensional state space. The stochastic component of dynamics of the chemical reaction networks of this mini-symposium play a very important role for each of the examples investigated. Several properties of stochastic chemical reaction...
