Volume 25, 2021
|Page(s)||346 - 375|
|Published online||27 July 2021|
Extinction rate of continuous state branching processes in critical Lévy environments
CMAP, École Polytechnique, Route de Saclay,
Palaiseau Cedex, France.
2 CIMAT A.C. Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico.
3 Université Clermont Auvergne, Irstea, UR LISC, Centre de Clermont-Ferrand, 9 avenue Blaise Pascal CS 20085, 63178 Aubière, France.
4 Complex Systems Institute of Paris Ile-de-France, 113 rue Nationale, Paris, France.
* Corresponding author: email@example.com
Accepted: 29 June 2021
We study the speed of extinction of continuous state branching processes in a Lévy environment, where the associated Lévy process oscillates. Assuming that the Lévy process satisfies Spitzer’s condition, we extend recent results where the associated branching mechanism is stable. The study relies on the path analysis of the branching process together with its Lévy environment, when the latter is conditioned to have a non-negative running infimum. For that purpose, we combine the approach developed in Afanasyev et al. , for the discrete setting and i.i.d. environments, with fluctuation theory of Lévy processes and a result on exponential functionals of Lévy processes due to Patie and Savov .
Mathematics Subject Classification: 60J80 / 60G51 / 60H10 / 60K37
Key words: Continuous state branching processes / Lévy processes conditioned to stay positive / random environment / Spitzer’s condition / extinction / long time behaviour
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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