  Consider a branching random walk on the real line with a random environment
in time (BRWRE). A necessary and sufficient condition for the non-triviality of
the limit of the derivative martingale is formulated. To this end, we
investigate the random walk in time-inhomogeneous random environment (RWRE),
which related the BRWRE by the many-to-one formula. The key step is to figure
out Tanaka's decomposition for the RWRE conditioned to stay non-negative (or
above a line), which is interesting itself as well.

Probability