Anderson–Kim positive recurrence conjecture for weakly reversible stochastic mass‑action systems

Prove that every 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 paper studies non-explosivity and recurrence properties of continuous-time Markov chains arising from stochastic mass-action systems. A central open direction in the field is a conjecture of Anderson and Kim asserting positive recurrence for all weakly reversible stochastic mass-action systems on each closed communicating class.

The authors survey partial progress and related results (e.g., complex-balanced cases and low-dimensional settings) and contribute new non-explosivity results that reduce certain instances of the conjecture to establishing the existence of stationary distributions.