Comptes Rendus
no. 9
Probability theory
A note on the quasi-ergodic distribution of one-dimensional diffusions
[Une note sur la distribution quasi ergodique des diffusions en dimension 1]

Présenté par : Jean-François Le Gall

Guoman He1
1 School of Mathematics and Statistics, Hunan University of Commerce, Changsha, Hunan 410205, PR China
Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 967-972.

Nous étudions la quasi-ergodicité des diffusions unidimensionnelles sur ]0,∞[, où 0 est une frontière de sortie et ∞ une frontière d'entrée. Notre but est d'améliorer des résultats obtenus par He and Zhang (2016) [3]. Ainsi, nous retrouvons les résultats principaux de ce texte sous des hypothèses moins restrictives.

In this note, we study quasi-ergodicity for one-dimensional diffusions on (0,∞), where 0 is an exit boundary and +∞ is an entrance boundary. Our main aim is to improve some results obtained by He and Zhang (2016) [3]. In simple terms, the same main results of the above paper are obtained with more relaxed conditions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.07.009
Guoman He 1

1 School of Mathematics and Statistics, Hunan University of Commerce, Changsha, Hunan 410205, PR China 
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Guoman He. A note on the quasi-ergodic distribution of one-dimensional diffusions. Comptes Rendus. Mathématique, Volume 356 (2018) no. 9, pp. 967-972. doi : 10.1016/j.crma.2018.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.07.009/

[1] L.A. Breyer; G.O. Roberts A quasi-ergodic theorem for evanescent processes, Stoch. Process. Appl., Volume 84 (1999), pp. 177-186

[2] P. Cattiaux; P. Collet; A. Lambert; S. Martínez; S. Méléard; J. San Martín Quasi-stationary distributions and diffusion models in population dynamics, Ann. Probab., Volume 37 (2009), pp. 1926-1969

[3] G. He; H. Zhang On quasi-ergodic distribution for one-dimensional diffusions, Stat. Probab. Lett., Volume 110 (2016), pp. 175-180

[4] N. Ikeda; S. Watanabe Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, vol. 24, North-Holland, Amsterdam, 1989

[5] S. Karlin; H.M. Taylor A Second Course in Stochastic Processes, Academic Press, New York, 1981

[6] J. Littin Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular, J. Appl. Probab., Volume 49 (2012), pp. 719-730

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