  A stochastic discrete drift-diffusion model is proposed to account for the
effects of shot noise in weakly coupled, highly doped semiconductor
superlattices. Their current-voltage characteristics consist of a number stable
multistable branches corresponding to electric field profiles displaying two
domains separated by a domain wall. If the initial state corresponds to a
voltage on the middle of a stable branch and a sudden voltage is switched so
that the final voltage corresponds to the next branch, the domains relocate
after a certain delay time. Shot noise causes the distribution of delay times
to change from a Gaussian to a first passage time distribution as the final
voltage approaches that of the end of the first current branch. These results
agree qualitatively with experiments by Rogozia {\it et al} (Phys. Rev. B {\bf
64}, 041308(R) (2001)).
