Extinction Rate Of Continuous State Branching Processes In Critical Levy Environments

We study the speed of extinction of continuous state branching processes in a Levy environment, where the associated Levy process oscillates. Assuming that the Levy 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 Levy 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 Levy processes and a result on exponential functionals of Levy processes due to Patie and Savov [28].