  We study the survival probability and the growth rate for branching random
walks in random environment (BRWRE). The particles perform simple symmetric
random walks on the $d$-dimensional integer lattice, while at each time unit,
they split into independent copies according to time-space i.i.d. offspring
distributions. The BRWRE is naturally associated with the directed polymers in
random environment (DPRE), for which the quantity called the free energy is
well studied. We discuss the survival probability (both global and local) for
BRWRE and give a criterion for its positivity in terms of the free energy of
the associated DPRE. We also show that the global growth rate for the number of
particles in BRWRE is given by the free energy of the associated DPRE, though
the local growth rateis given by the directional free energy.
