  We couple two-dimensional hydrodynamics to realistic one-dimensional
multigroup flux-limited diffusion neutrino transport to investigate
protoneutron star convection in core collapse supernovae, and more
specifically, the interplay between its development and neutrino transport, for
both 15 and 25 solar mass models. In the presence of neutrino transport,
protoneutron star convection velocities are too small relative to bulk inflow
velocities to result in any significant convective transport of entropy and
leptons. A simple analytical model supports our numerical results, indicating
that the inclusion of neutrino transport reduces the entropy-driven
(lepton-driven) convection growth rates and asymptotic velocities by a factor
of 3 (50) at the neutrinosphere and a factor 250 (1000) at a density of 10^{12}
g/cm^{3}, for both our 15 and 25 solar mass models. Moreover, when transport is
included, the initial postbounce entropy gradient is smoothed out by neutrino
diffusion, whereas the initial lepton gradient is maintained by electron
capture and neutrino escape near the neutrinosphere. Despite the maintenance of
the lepton gradient, protoneutron star convection does not develop over the 100
ms duration typical of all our simulations, except in the instance where
``low-test'' initial conditions are used, which are generated by core collapse
and bounce simulations that neglect neutrino-electron scattering and ion-ion
screening corrections to neutrino- nucleus elastic scattering.
