Speed of extinction for continuous-state branching processes in a weakly subcritical Lévy environment

We continue with the systematic study of the speed of extinction of continuous-state branching processes in Levy environments under more general branching mechanisms. Here, we deal with the weakly subcritical regime under the assumption that the branching mechanism is regularly varying. We extend recent results of Li and Xu (2018) and Palau et al. (2016), where it is assumed that the branching mechanism is stable, and complement the recent articles of Bansaye et al. (2021) and Cardona-Tobon and Pardo (2021), where the critical and the strongly and intermediate subcritical cases were treated, respectively. Our methodology combines a path analysis of the branching process together with its Levy environment, fluctuation theory for Levy processes, and the asymptotic behaviour of exponential functionals of Levy processes. Our approach is inspired by the last two previously cited papers, and by Afanasyev et al. (2012), where the analogue was obtained.