  We study theoretically the contribution of quantum effects to the exciton
diffusion coefficient in atomically thin crystals. It is related to the weak
localization caused by the interference of excitonic wavefunctions on the
trajectories with closed loops. Due to a weak inelasticity of the
exciton-phonon interaction the effect is present even if the excitons are
scattered by long-wavelength acoustic phonons. We consider exciton interaction
with longitudinal acoustic phonons with linear dispersion and with flexural
phonons with quadratic dispersion. We identify the regimes where the weak
localization effect can be particularly pronounced. We also briefly address the
role of free charge carriers in the exciton quantum transport and, within the
self-consistent theory of localization, the weak localization effects beyond
the lowest order.
