  The motion of a quantum particle hopping on a simple cubic lattice under the
influence of thermal noise and of a static random potential is expected to be
diffusive, i.e., the particle is expected to exhibit `quantum Brownian motion',
no matter how weak the thermal noise is. This is shown to be true in a model
where the dynamics of the particle is governed by a Lindblad equation for a
one-particle density matrix. The generator appearing in this equation is the
sum of two terms: a Liouvillian corresponding to a random Schr\"odinger
operator and a Lindbladian describing the effect of thermal noise in the
kinetic limit. Under suitable but rather general assumptions on the
Lindbladian, the diffusion constant characterizing the asymptotics of the
motion of the particle is proven to be strictly positive and finite. If the
disorder in the random potential is so large that transport is completely
suppressed in the limit where the thermal noise is turned off, then the
diffusion constant tends to $0$ proportional to the coupling of the particle to
the heat bath.
