  We consider a randomly perturbed Korteweg-de Vries equation. The perturbation
is a random potential depending both on space and time, with a white noise
behavior in time, and a regular, but stationary behavior in space. We
investigate the dynamics of the soliton of the KdV equation in the presence of
this random perturbation, assuming that the amplitude of the perturbation is
small. We estimate precisely the exit time of the perturbed solution from a
neighborhood of the modulated soliton, and we obtain the modulation equations
for the soliton parameters. We moreover prove a central limit theorem for the
dispersive part of the solution, and investigate the asymptotic behavior in
time of the limit process.
