  We consider the diffusion approximation of branching processes in random
environment (BPREs). This diffusion approximation is similar to and
mathematically more tractable than BPREs. We obtain the exact asymptotic
behavior of the survival probability. As in the case of BPREs, there is a phase
transition in the subcritical regime due to different survival opportunities.
In addition, we characterize the process conditioned to never go extinct and
establish a backbone construction. In the strongly subcritical regime, mean
offspring numbers are increased but still subcritical in the process
conditioned to never go extinct. Here survival is solely due to an immortal
individual, whose offspring are the ancestors of additional families. In the
weakly subcritical regime, the mean offspring number is supercritical in the
process conditioned to never go extinct. Thus this process survives with
positive probability even if there was no immortal individual.
