Quantitative rates in the irregular critical regime

Establish quantitative subexponential convergence rates in total variation for reducible stochastic linear reaction networks in the irregular critical regime, and identify structural criteria that determine those rates.

Background

The irregular critical regime is defined by λmax⁡=0\lambda_{\max}=0, a semisimple zero eigenvalue, orthogonality of the zero-order influx vector to the left nullspace, and failure of the regularity condition. The paper proves convergence in total variation but shows that convergence is non-exponential; it also proves failure of convergence in the L1L^1-Wasserstein distance.

The authors do not provide a quantitative subexponential rate for total-variation convergence. They explicitly identify both rate estimation and the structural classification of those rates as unresolved problems.

References

Several important questions remain open. First, in the irregular critical regime, it would be desirable to establish quantitative subexponential convergence rates in total variation and to identify structural criteria that determine these rates.

— Long-time behavior of reducible stochastic linear reaction networks: a spectral and structural classification  (2610.11443 - Wu et al., 8 Oct 2026) in Section 6, Conclusions and discussion

Second, in the divergent regimes, a more refined analysis of the boundary dynamics could characterize the probabilities of escape and absorption, classify the closed classes contained in the boundary, and determine the conditions under which the time-dependent distribution converges to a boundary-supported limiting distribution.

— Long-time behavior of reducible stochastic linear reaction networks: a spectral and structural classification  (2610.11443 - Wu et al., 8 Oct 2026) in Section 6, Conclusions and discussion