2019/04/25 by Anderson, David F., Schnoerr, David, Yuan, Chaojie
#FOS: Biological sciences #FOS: Mathematics #Molecular Networks (q-bio.MN) #Probability (math.PR) #Quantitative Methods (q-bio.QM)
paper · doi:10.48550/arxiv.1904.11583
It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that the initial distribution of a stochastically modeled reaction network is a product of Poissons, under what conditions will the distribution remain a product of Poissons for all time? By drawing inspiration from Crispin Gardiner's "Poisson representation" for the solution to the chemical master equation, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Interestingly, the condition is a dynamical "complex-balancing" for only those complexes that have multiplicity greater than or equal to two (i.e. the higher order complexes that yield non-linear terms to the dynamics). We term this new condition the "dynamical and restricted complex balance" condition (DR for short).