Positive recurrence of every weakly reversible stochastic reaction network
Take a finite set of reactions $y \to y'$ between complexes $y, y' \in \mathbb{N}_0^d$, with rate constants $\kappa_{y \to y'} > 0$, and assume the network is weakly reversible: every directed edge of the graph of complexes lies on a directed cycle. Its stochastic mass-action model is the continuous-time Markov chain on $\mathbb{N}_0^d$ in which reaction $y \to y'$ takes the state $x$ to $x + y' - y$ at rate $$\kappa_{y \to y'} \prod_{i=1}^{d} (x_i)_{y_i}, \qquad (m)_j = \frac{m!}{(m-j)!} \ \text{for } m \ge j, \quad (m)_j = 0 \ \text{otherwise}.$$ Anderson, Craciun and Kurtz showed that if the network also has deficiency zero, the model has a product-form Poisson stationary distribution on each closed irreducible subset of the state space, and Anderson, Cappelletti, Koyama and Kurtz showed that such complex-balanced models are nonexplosive; together these give positive recurrence. Anderson and Kim proved that every state is positive recurrent for binary weakly reversible networks with a single linkage class once the reactions $\varnothing \to S$ and $S \to \varnothing$ are added for every species $S$. Anderson, Cappelletti and Kim proved positive recurrence for binary weakly reversible networks with a single linkage class in which, for each species, some complex is a multiple of that species.
Problem. Must the minimal continuous-time chain be nonexplosive, and every state in each closed communicating class be positive recurrent? For a class with more than one state, positive recurrence means a finite expected return time after leaving the starting state; an absorbing state forms a closed class on its own and counts as positive recurrent.
Source: Anderson and Kim (2018), Section 6, the Positive Recurrence Conjecture. Stated by AIM — Open Applied Problems with nonexplosion added and with positive recurrence asked in closed communicating classes.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, chemical reaction networks, continuous-time Markov chains, positive recurrence
References
@article{anderson2018,
author = {Anderson, David F. and Kim, Jinsu},
title = {Some network conditions for positive recurrence of stochastically modeled reaction networks},
journal = {SIAM J. Appl. Math.},
volume = {78},
number = {5},
year = {2018},
pages = {2692--2713},
doi = {10.1137/17M1161427},
eprint = {1710.11263},
archivePrefix = {arXiv},
}
@article{anderson2010,
author = {Anderson, David F. and Craciun, Gheorghe and Kurtz, Thomas G.},
title = {Product-form stationary distributions for deficiency zero chemical reaction networks},
journal = {Bull. Math. Biol.},
volume = {72},
number = {8},
year = {2010},
pages = {1947--1970},
doi = {10.1007/s11538-010-9517-4},
}
@article{anderson2020,
author = {Anderson, David F. and Cappelletti, Daniele and Kim, Jinsu},
title = {Stochastically modeled weakly reversible reaction networks with a single linkage class},
journal = {J. Appl. Probab.},
volume = {57},
number = {3},
year = {2020},
pages = {792--810},
doi = {10.1017/jpr.2020.28},
eprint = {1904.08967},
archivePrefix = {arXiv},
}
@article{anderson2018nonexplosivity,
author = {Anderson, David F. and Cappelletti, Daniele and Koyama, Masanori and Kurtz, Thomas G.},
title = {Non-explosivity of stochastically modeled reaction networks that are complex balanced},
journal = {Bull. Math. Biol.},
volume = {80},
number = {10},
year = {2018},
pages = {2561--2579},
doi = {10.1007/s11538-018-0473-8},
eprint = {1708.09356},
archivePrefix = {arXiv},
}
@misc{aim-reaction-network-positive-recurrence,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Positive recurrence of every weakly reversible stochastic reaction network}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/272-reaction-network-positive-recurrence.md},
note = {Entry 272 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00466, "Positive recurrence of every weakly reversible stochastic reaction network", clopen.solutions.