Extinction rate of continuous state branching processes in critical Lévy random environments

We determine the speed of extinction of continuous state branching processes (CSBPs) in an oscillating Lévy random environment satisfying the so-called Spitzer's condition. The study relies on the path analysis of the process together with its environment, conditionally on the survival event, following the approach for discrete time branching processes in random environment of Afanasyev et al. [1]. In particular, we characterize CSBPs in a random environment conditioned to stay positive as a solution of a stochastic differential equation and determine its long time behavior.
This is a joint work with Vincent Bansaye (CMAP, Ecole Polytechnique) and Juan Carlos Pardo (CIMAT).