Abstract
Studying the subexponential convergence towards equilibrium of a strong Markov process, we exhibit an intermediate Lyapunov condition equivalent to the control of some moment of a hitting time. This provides a link, similar (although more intricate) to the one existing in the exponential case, between the coupling method and the approach based on the existence of a Lyapunov function for the generator, in the context of the subexponential rates found by Fort-Roberts (2005), Douc-Fort-Guillin (2009) and Hairer (2016).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
O.E. Barndorff-Nielsen, N. Shephard, Non-gaussian ornstein-uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat Methodol. 63(2), 167–241 (2001). https://doi.org/10.1111/1467-9868.00282
A. Bernou, N. Fournier, A coupling approach for the convergence to equilibrium for a collisionless gas. Ann. Appl. Probab. (2021, In press)
P. Cattiaux, A. Guillin, Hitting times, functional inequalities, lyapunov conditions and uniform ergodicity. J. Funct. Anal. 272(6), 2361–2391 (2017). https://doi.org/10.1016/j.jfa.2016.10.003
P. Cattiaux, N. Gozlan, A. Guillin, C. Roberto, Functional inequalities for heavy tailed distributions and application to isoperimetry. Electron. J. Probab. 15, 346–385 (2010). https://doi.org/10.1214/EJP.v15-754
M. Davis, Markov Models and Optimization, 1 edn. (Routledge, England, 2018). https://doi.org/10.1201/9780203748039
R. Douc, G. Fort, A. Guillin, Subgeometric rates of convergence of F-Ergodic strong Markov processes. Stochastic Processes Appl. 119(3), 897– 923 (2009)
R. Douc, G. Fort, E. Moulines, P. Soulier, Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14(3), 1353–1377 (2004). https://doi.org/10.1214/105051604000000323
D. Down, S.P. Meyn, R.L. Tweedie, Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4), 1671–1691 (1995)
G. Fort, G.O. Roberts, Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15(2), 1565–1589 (2005)
M. Hairer, Convergence of Markov Processes (2016). http://www.hairer.org/notes/Convergence.pdf. Lecture notes
J. Jacod, A.N. Shiryaev, Limit theorems for Stochastic processes, in Grundlehren Der Mathematischen Wissenschaften, vol. 288. (Springer, Berlin, 1987). https://doi.org/10.1007/978-3-662-02514-7
O. Kallenberg, Foundations of modern probability, in Probability and Its Applications. (Springer, New York, 2002). https://doi.org/10.1007/978-1-4757-4015-8
H.W. Kuo, T.P. Liu, L.C. Tsai, Free molecular flow with boundary effect. Commun. Math. Phys. 318(2), 375–409 (2013). https://doi.org/10.1007/s00220-013-1662-9
R.B. Lund, S.P. Meyn, R.L. Tweedie, Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Probab. 6(1) (1996). https://doi.org/10.1214/aoap/1034968072
S.P. Meyn, R.L. Tweedie, Stability of Markovian processes I: criteria for discrete-time chains. Adv. Appl. Probab. 24(3), 542–574 (1992). https://doi.org/10.2307/1427479
S.P. Meyn, R.L. Tweedie, A survey of Foster-Lyapunov techniques for general state space Markov processes, in Proceedings of the Workshop on Stochastic Stability and Stochastic Stabilization (1993)
S.P. Meyn, R.L. Tweedie, Stability of Markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Probab. 25(3), 487–517 (1993). https://doi.org/10.2307/1427521
S.P. Meyn, R.L. Tweedie, Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25(3), 518–548 (1993). https://doi.org/10.2307/1427522
E. Nummelin, General Irreducible Markov Chains and Non-Negative Operators, 1st edn. (Cambridge University, Cambridge, 1984). https://doi.org/10.1017/CBO9780511526237
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, in Grundlehren Der Mathematischen Wissenschaften, vol. 293. (Springer, Berlin, 1991). https://doi.org/10.1007/978-3-662-21726-9
H. Thorisson, The Queue GI/G/1: finite moments of the cycle variables and uniform rates of convergence. Stochastic Processes Appl. 19(1), 85–99 (1985). https://doi.org/10.1016/0304-4149(85)90041-9
P. Tuominen, R.L. Tweedie, Subgeometric rates of convergence of f-Ergodic Markov chains. Adv. Appl. Probab. 26(3), 775–798 (1994). https://doi.org/10.2307/1427820
Acknowledgements
I would like to acknowledge Nicolas Fournier (LPSM, Sorbonne Université) for all the fruitful discussions and advices he offered me while preparing this note. The author warmly thanks the anonymous referee for their suggestions and remarks. This work was supported by grants from Région Île de France.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bernou, A. (2022). On Subexponential Convergence to Equilibrium of Markov Processes. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités LI. Lecture Notes in Mathematics(), vol 2301. Springer, Cham. https://doi.org/10.1007/978-3-030-96409-2_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-96409-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-96408-5
Online ISBN: 978-3-030-96409-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)