  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).
