发表机构
Center for Applied Mathematics and KL-AAGDM, Tianjin University; Tianjin University; School of Mathematics, Harbin Institute of Technology; Beijing Research Institute, Harbin Institute of Technology（天津大学应用数学中心与KL-AAGDM; 天津大学; 哈尔滨工业大学数学学院; 哈尔滨工业大学北京研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对可约随机线性反应网络，基于谱与结构分类，明确其在两类距离下的长时间行为，证明网络非爆炸，给出稳定、临界、发散 regimes 的收敛特性，并通过耦合方法得到指数收敛时的最优收敛率。
AI 中文摘要
近年来，随机反应网络的指数遍历性已受到大量关注[SIAM J. Appl. Dyn. Syst. 24, 1668-1710 (2025)]。本文针对可约随机线性反应网络，在$L^1$-Wasserstein距离与全变差距离下，对其长时间行为进行结构分类。该分类由一阶流入矩阵$A$的最大特征值$\lambda_{\max}$、零阶流入向量$b$相对于$A$左零空间的位置，以及网络限制于持续物种的守恒律结构决定。我们首先证明，所有随机线性反应网络均非爆炸。在稳定 regime $\lambda_{\max}<0$中，过程指数快速收敛至唯一平稳分布；在临界 regime $\lambda_{\max}=0$中，当$A$的零特征值为半单且$b$正交于$A$的左零空间时，指数收敛当且仅当受限网络满足正则性条件；若这两个谱条件成立但正则性条件不满足，则仅在全变差下收敛且非指数收敛。在发散 regime 中，不存在闭的不可约正常返类包含内部状态，且若存在平稳分布，其支撑必在边界上。此外，通过耦合方法，我们得到了指数收敛情形下的最优收敛率。
英文摘要
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 $L^1$-Wasserstein and total variation distances. The classification is determined by the maximal eigenvalue $λ_{\max}$ 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 $λ_{\max}<0$, 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.
