  We calculate the effects of random density fluctuations on two-neutrino
flavor transformations (${\nu}_{\tau(\mu)} \rightleftharpoons {\nu}_e$) in the
post-core-bounce supernova environment. In particular, we follow numerically
the flavor evolution of neutrino states propagating through a stochastic field
of density fluctuations. We examine the approach to neutrino flavor
depolarization, and study the effects of this phenomenon in both the early
shock reheating epoch and the later r-process nucleosynthesis epoch. Our
results suggest that significant fluctuation-induced neutrino flavor
depolarization effects occur in these environments only when the zero-order
(without density fluctautions) evolution of the neutrino states includes
adiabatic propagation through resonances (mass level crossings). In the shock
reheating epoch, depolarization effects from fluctuations with amplitudes
larger than 0.05% of the local matter density can cause an increase in the
heating rate of the material behind the shock compared to the case with no
neutrino flavor transformation - but this corresponds to a significant decrease
in this quantity relative to the case with adiabatic neutrino transformation.
If r-process nucleosynthesis is to occur during the late stages of supernova
evolution, then the requirement of neutron-rich conditions excludes a region of
neutrino mass-squared difference and vacuum mixing angle ($\delta m^2,\
\sin^22\theta$) parameter space for neutrino flavor transformation. We find
that in the presence of stochastic fluctuations, this excluded region is not
significantly altered even for random fluctuations with an amplitude of 1% of
the local matter density.
