  Supercritical branching processes in constant environment conditioned on
eventual extinction are known to be subcritical branching processes. The case
of random environment is more subtle. A supercritical branching diffusion in
random environment (BDRE) conditioned on eventual extinction of the population
is not a BDRE. However the quenched law of the population size of a
supercritical BDRE conditioned on eventual extinction is equal to the quenched
law of the population size of a subcritical BDRE. As a consequence,
supercritical BDREs have a phase transition which is similar to a well-known
phase transition of subcritical branching processes in random environment.
