  I study a model for a massive one-dimensional particle in a singular periodic
potential that is receiving kicks from a gas. The model is described by a
Lindblad equation in which the Hamiltonian is a Schr\"odinger operator with a
periodic $\delta$-potential and the noise has a frictionless form arising in a
Brownian limit. I prove that an emergent Markov process in a semi-classical
limit governs the momentum distribution in the extended-zone scheme. The main
result is a central limit theorem for a time integral of the momentum process,
which is closely related to the particle's position. When normalized by
$t^{5/4}$, the integral process converges to a time-changed Brownian motion
whose rate depends on the momentum process. The scaling $t^{5/4}$ contrasts
with $t^{3/2}$, which would be expected for the case of a smooth periodic
potential or for a comparable classical process. The difference is a wave
effect driven by Bragg reflections that occur when the particle's momentum is
kicked near the half-spaced reciprocal lattice.
