Bimolecular case of the positive recurrence problem for weakly reversible stochastic mass‑action systems

Prove that every bimolecular weakly reversible stochastic mass‑action system, modeled as a continuous‑time Markov chain under stochastic mass‑action kinetics, is positive recurrent on each closed communicating class.

Background

The general Anderson–Kim conjecture predicts positive recurrence for all weakly reversible systems. Substantial progress exists for several subclasses, yet the bimolecular case remains unresolved.

The authors establish that all bimolecular weakly reversible systems are non-explosive, thereby reducing positive recurrence to proving the existence of stationary distributions, but they emphasize the remaining gap for the full positive recurrence claim in the bimolecular setting.

References

In this paper, even though we are still unable to prove every bimolecular weakly reversible stochastic mass-action system is positive recurrent, we indeed show that every such system is non-explosive (Corollary~\ref{cor:endotactic-non-explosive}(iv)).

— Non-explosivity of endotactic stochastic reaction systems  (2409.05340 - Xu, 2024) in Section 1, Introduction