Minimal order for absence of stationary distributions in infinite closed classes among endotactic stochastic mass‑action systems

Determine the minimal reaction order k for which there exists an endotactic stochastic mass‑action system of order k that has an infinite closed communicating class admitting no stationary distribution (no finite stationary measure).

Background

The authors note that every stochastic mass-action system possesses a stationary measure on each closed communicating class, but that measure need not be finite, so a stationary distribution may fail to exist.

A strongly endotactic example of order 7 is known to be null recurrent, yet the smallest order at which an endotactic system in an infinite closed class fails to have any stationary distribution is not known.

References

In , a 7th order null recurrent strongly endotactic stochastic mass-action system was constructed. Note that every stochastic mass-action system has a stationary measure in every closed communicating class . To the best knowledge of the author, it seems unknown what is the minimal order for an endotactic stochastic mass-action system to admit no stationary distribution (i.e., no finite stationary measure) in an infinite closed communicating class.

— Non-explosivity of endotactic stochastic reaction systems (2409.05340 - Xu, 9 Sep 2024) in Remark following Corollary 3.6 (Section 3, Non-explosivity/Recurrence)