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 L1-Wasserstein and total variation distances. The classification is determined by the maximal eigenvalue λmax 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 λmax<0, the process converges exponentially fast to a unique stationary distribution. In the critical regime λmax=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.
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BibTeX
@misc{wu2026longtime,
title = {{Long-time behavior of reducible stochastic linear reaction networks: a spectral and structural classification}},
author = {Bingjie Wu and Hao Kang and Chen Jia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11443},
eprint = {2610.11443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11443},
}
RIS
TY - UNPB
TI - Long-time behavior of reducible stochastic linear reaction networks: a spectral and structural classification
AU - Bingjie Wu
AU - Hao Kang
AU - Chen Jia
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11443
UR - https://arxiv.org/abs/2610.11443
AB - 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.
ER -