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Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

Part of: Probability theory on algebraic and topological structures Ergodic theory Stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  20 March 2018

Nicolas Champagnat  and
Denis Villemonais
Nicolas Champagnat*
Affiliation:
Inria Nancy Grand Est
Denis Villemonais*
Affiliation:
Université de Lorraine
*
* Postal address: Institut Élie Cartan de Lorraine (IECL, UMR CNRS 7502), Université de Lorraine, Campus Scientifique, B.P. 70239, Vandœuvre-lès-Nancy Cedex, F-54506, France.
* Postal address: Institut Élie Cartan de Lorraine (IECL, UMR CNRS 7502), Université de Lorraine, Campus Scientifique, B.P. 70239, Vandœuvre-lès-Nancy Cedex, F-54506, France.
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Abstract

In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

Keywords

One-dimensional diffusionabsorbed processquasi-stationary distributionuniform exponential mixing propertystrict local martingaleone-dimensional process with jumps

MSC classification

Primary: 60J60: Diffusion processes 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 37A25: Ergodicity, mixing, rates of mixing 60B10: Convergence of probability measures 60F99: None of the above, but in this section
Secondary: 60G44: Martingales with continuous parameter 60J75: Jump processes
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 178 - 203
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1] Amir, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. Stoch. Process. Appl. 39, 221–237. Google Scholar
[2] Cattiaux, P. et al. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37, 1926–1969. Google Scholar
[3] Champagnat, N. and Villemonais, D. (2016). Exponential convergence to quasi-stationary distribution and Q-process. Prob. Theory Relat. Fields 164, 243–283. Google Scholar
[4] Champagnat, N. and Villemonais, D. (2016). Population processes with unbounded extinction rate conditioned to non-extinction. Preprint. Available at https://arxiv.org/abs/1611.03010. Google Scholar
[5] Champagnat, N. and Villemonais, D. (2018). Uniform convergence of penalized time-inhomogeneous Markov processes. To appear in ESAIM Prob. Statist. Google Scholar
[6] Champagnat, N. and Villemonais, D. (2017). Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing. ALEA Latin Amer. J. Prob. Math. Statist. 14, 177–199. Google Scholar
[7] Champagnat, N. and Villemonais, D. (2017). Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes. Preprint. Available at https://arxiv.org/abs/1704.01928. Google Scholar
[8] Champagnat, N. and Villemonais, D. (2017). Uniform convergence to the Q-process. Electron. Commun. Prob. 22, 33. Google Scholar
[9] Champagnat, N., Coulibaly-Pasquier, A. and Villemonais, D. (2018). Exponential convergence to quasi-stationary distribution for multi-dimensional diffusion processes. To appear in Séminaire de Probabilités. Google Scholar
[10] Freedman, D. (1983). Brownian Motion and Diffusion, 2nd edn. Springer, New York. Google Scholar
[11] Hening, A. and Kolb, M. (2014). Quasistationary distributions for one-dimensional diffusions with singular boundary points. Preprint. Available at https://arxiv.org/abs/1409.2387. Google Scholar
[12] Itô, K. and McKean, H. P., Jr. (1974). Diffusion Processes and Their Sample Paths, 2nd edn. Springer, Berlin. Google Scholar
[13] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York. Google Scholar
[14] Kolb, M. and Steinsaltz, D. (2012). Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Prob. 40, 162–212. Google Scholar
[15] Kotani, S. (2006). On a condition that one-dimensional diffusion processes are martingales. In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX (Lecture Notes Math. 1874), Springer, Berlin, pp. 149–156. Google Scholar
[16] Le Gall, J.-F. (1983). Applications du temps local aux équations différentielles stochastiques unidimensionnelles. In Seminar on Probability, XVII (Lecture Notes Math. 986), Springer, Berlin, pp. 15–31. Google Scholar
[17] Littin, C. J. (2012). Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular. J. Appl. Prob. 49, 719–730. Google Scholar
[18] Matsumoto, H. (1989). Coalescing stochastic flows on the real line. Osaka J. Math. 26, 139–158. Google Scholar
[19] Méléard, S. and Villemonais, D. (2012). Quasi-stationary distributions and population processes. Prob. Surveys 9, 340–410. Google Scholar
[20] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press. Google Scholar
[21] Miura, Y. (2014). Ultracontractivity for Markov semigroups and quasi-stationary distributions. Stoch. Anal. Appl. 32, 591–601. Google Scholar
[22] Perkowski, N. and Ruf, J. (2012). Conditioned martingales. Electron. Commun. Prob. 17, 48. Google Scholar
[23] Pollett, P. K. (2015). Quasi-stationary distributions: a bibliography. Preprint. Available at http://www.maths.uq.edu.au/~pkp/papers/qsds/. Google Scholar
[24] Protter, P. (2013). A mathematical theory of financial bubbles. In Paris-Princeton Lectures on Mathematical Finance 2013 (Lecture Notes Math. 2081), Springer, Cham, pp. 1–108. Google Scholar
[25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin. Google Scholar
[26] Steinsaltz, D. and Evans, S. N. (2007). Quasistationary distributions for one-dimensional diffusions with killing. Trans. Amer. Math. Soc. 359, 1285–1324. Google Scholar
[27] Van Doorn, E. A. and Pollett, P. K. (2013). Quasi-stationary distributions for discrete-state models. Europ. J. Operat. Res. 230, 1–14. Google Scholar