Extinction rate of continuous state branching processes in critical Lévy environments

¹ CMAP, École Polytechnique, Route de Saclay, 91128 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: jcpardo@cimat.mx

Received: 24 March 2020
Accepted: 29 June 2021

Abstract

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. [2], 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 [28].