  The effective dynamics of solitons for the generalized nonlinear
Schr\"odinger equation in a random potential is rigorously studied. It is shown
that when the external potential varies slowly in space compared to the size of
the soliton, the dynamics of the center of the soliton is almost surely
described by Hamilton's equations for a classical particle in the random
potential, plus error terms due to radiation damping. Furthermore, a limit
theorem for the dynamics of the center of mass of the soliton in the
weak-coupling and space-adiabatic limit is proven in two and higher dimensions:
Under certain mixing hypotheses for the potential, the momentum of the center
of mass of the soliton converges in law to a diffusion process on a sphere of
constant momentum. Moreover, in three and higher dimensions, the trajectory of
the center of mass of the soliton converges to a spatial Brownian motion.
