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Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

Part of: Equations and systems with randomness Markov processes Limit theorems

Published online by Cambridge University Press:  10 January 2022

Amarjit Budhiraja ,
Nicolas Fraiman Open the ORCID record for Nicolas Fraiman [Opens in a new window]  and
Adam Waterbury
Amarjit Budhiraja*
Affiliation:
University of North Carolina at Chapel Hill
Nicolas Fraiman*
Affiliation:
University of North Carolina at Chapel Hill
Adam Waterbury*
Affiliation:
University of North Carolina at Chapel Hill
*
*Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA.
*Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA.
*Postal address: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC 27599, USA.
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Abstract

We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems.

In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

Keywords

Quasi-stationary distributionsuniform large deviation principlesrandom perturbationslong-time behavior

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 34F05: Equations and systems with randomness
Secondary: 60F10: Large deviations 92D25: Population dynamics (general)
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 1 , March 2022 , pp. 64 - 110
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Buckley, F. and Pollet, P. (2010). Limit theorems for discrete-time metapopulation models. Prob. Surv. 7, 53–83.10.1214/10-PS158CrossRefGoogle Scholar
Champagnat, N. and Villemonais, D. (2017). General criteria for the study of quasi-stationarity. Preprint. Available at https://arxiv.org/abs/1712.08092.Google Scholar
Dupuis, P. and Ellis, R. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York.10.1002/9781118165904CrossRefGoogle Scholar
Faure, M. and Schreiber, S. (2014). Quasi-stationary distributions for randomly perturbed dynamical systems. Ann. Appl. Prob. 24, 553–598.10.1214/13-AAP923CrossRefGoogle Scholar
Ferrari, P., Kesten, A., MartÍnez, H. and Picco, P. (1995). Existence of quasi–stationary distributions. A renewal dynamical approach. Ann. Prob. 23, 501–521.10.1214/aop/1176988277CrossRefGoogle Scholar
Freidlin, M. and Wentzell, A. (1984). Random Perturbations of Dynamical Systems. Springer, New York.10.1007/978-1-4684-0176-9CrossRefGoogle Scholar
Gosselin, F. (1996). The expected extinction time of a population within a system of interacting biological populations. Bull. Math. Biol. 58, 555–568.Google Scholar
Gosselin, F. (2001). Asymptotic behavior of absorbing Markov chains conditional on non-absorption for applications in conservation biology. Ann. Appl. Prob. 11, 261–284.10.1214/aoap/998926993CrossRefGoogle Scholar
Gyllenberg, M. and Silvestrov, D. (1994). Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol. 33, 35–70.10.1007/BF00160173CrossRefGoogle Scholar
Hasselblatt, B. and Katok, A. (1995). Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press.Google Scholar
Högnäs, G. (1997). On the quasi-stationary distribution of a stochastic Ricker model. Stoch. Process. Appl. 70, 243–263.10.1016/S0304-4149(97)00064-1CrossRefGoogle Scholar
Kifer, Y. (1988). Random Perturbations of Dynamical Systems. BirkhÄuser, Basel.10.1007/978-1-4615-8181-9CrossRefGoogle Scholar
Marmet, B. (2013). Quasi-stationary distributions for stochastic approximation algorithms with constant step size. Preprint. Available at https://arxiv.org/abs/1303.7081.Google Scholar
Méléard, S. and Villemonais, D. (2012). Quasi-stationary distributions and population processes. Prob. Surveys 9, 340–410.10.1214/11-PS191CrossRefGoogle Scholar
Pollett, P. (1996). Modelling the long-term behaviour of evanescent ecological systems. Ecol. Modelling 86, 135–139.10.1016/0304-3800(95)00040-2CrossRefGoogle Scholar
Pollett, P. (2001). Quasi-stationarity in populations that are subject to large-scale mortality or emigration. Environm. Internat. 27, 231–236.10.1016/S0160-4120(01)00092-7CrossRefGoogle Scholar
Salins, M., Budhiraja, A. and Dupuis, P. (2017). Uniform large deviation principles for Banach space valued stochastic differential equations. To appear in Trans. Amer. Math. Soc. Google Scholar
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683–700.10.2307/1427670CrossRefGoogle Scholar
Van Doorn, E. A. and Pollett, P. K. (2009). Quasi-stationary distributions for reducible absorbing Markov chains in discrete time. Markov Process. Relat. Fields 15, 191–204.Google Scholar