Extinction rate of continuous state branching processes in critical Lévy environments
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 and the existence of some exponential moments, 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. \cite{Afanasyev2005}, for the discrete setting and i.i.d. environments, with fluctuation theory of Lévy processes and a remarkable result on exponential functionals of Lévy processes under Spitzer's condition due to Patie and Savov \cite{patie2016bernstein}.