Long-time behavior of reducible stochastic linear reaction networks: a spectral and structural classification

Abstract: Exponential ergodicity of stochastic reaction networks has attracted considerable attention in recent years [SIAM J. Appl. Dyn. Syst. 24, 1668-1710 (2025)]. Here, we provide a structural classification of the long-time behavior of reducible stochastic linear reaction networks under the $L<sup>1$-Wasserstein and total variation distances. The classification is determined by the maximal eigenvalue $λ<em>{\max}$ of the first-order influx matrix $A$, the position of the zero-order influx vector $b$ relative to the left nullspace of $A$, and the conservation-law structure of the network restricted to the persistent species. We first prove that every stochastic linear reaction network is non-explosive. In the stable regime $λ</em>{\max}&lt;0$, the process converges exponentially fast to a unique stationary distribution. In the critical regime $λ_{\max}=0$, when the zero eigenvalue of $A$ is semisimple and $b$ is orthogonal to the left nullspace of $A$, exponential convergence occurs if and only if a regularity condition on the restricted network is satisfied. If these two spectral conditions hold but the regularity condition fails, convergence occurs only in total variation and is non-exponential. In the divergent regimes, no closed irreducible positive recurrent class can contain an interior state, and any stationary distribution, if it exists, must be supported on the boundary. Moreover, by employing a coupling method, we obtain the optimal convergence rate in the exponentially convergent cases.