Continuous state branching processes in random environment: The Brownian case

We consider continuous-state branching processes that are perturbed by a Brownian motion. These processes are constructed as the unique strong solution of a stochastic differential equation. The long-term behaviours are studied. In the stable case, the extinction and explosion probabilities are given explicitly. We find three regimes for the asymptotic behaviour of the explosion probability and five regimes for the asymptotic behaviour of the extinction probability. In the supercritical regime, the process conditioned on eventual extinction has three regimes for the asymptotic behaviour of the extinction probability. Finally, the process conditioned on non-extinction and the process with immigration are given.