  We study stationary distributions in the context of stochastic reaction
networks. In particular, we are interested in complex balanced reaction
networks and reduction of such networks by assuming a set of species (called
non-interacting species) are degraded fast (and therefore essentially absent in
the network), implying some reaction rates are large compared to others.
Technically, we assume these reaction rates are scaled by a common parameter
$N$ and let $N\to\infty$. The limiting stationary distribution as $N\to\infty$
is compared to the stationary distribution of the reduced reaction network
obtained by algebraic elimination of the non-interacting species. In general,
the limiting stationary distribution might differ from the stationary
distribution of the reduced reaction network. We identify various sufficient
conditions for when these two distributions are the same, including when the
reaction network is detailed balanced and when the set of non-interacting
species consists of intermediate species. In the latter case, the limiting
stationary distribution essentially retains the form of the complex balanced
distribution. This finding is particularly surprising given that the reduced
reaction network might be non-weakly reversible and exhibit unconventional
kinetics.
