Motivated by the work of Boeinghoff and Huznethaler on branching diffusions on random environment, we introduce continuous state branching processes in a Brownian random environment. A process in such class is defined as a strong solution of an stochastic differential equation and satisfies a quenched branching property. Special attention is given to the case when the branching mechanism has finite mean, where we can provide more explicit results on long- term behaviour and we can include an immigration term.
Restricting our attention to the stable case, we show that there two different behaviour depending on the scaling index . If , then the process can explode at finite time with positive probability and three different regimes appear. For the case , we show that four regimes arise for the speed of extinction, similarly to the discrete case. The precise asymptotics for the speed of extinction allow us to introduce the so called Q-process or process condition to be never extinct. This is a joint work with Sandra Palau.
Continuous state branching processes in a Brownian random environment