  For the Vlasov-Poisson equation with random uncertain initial data, we prove
that the Landau damping solution given by the deterministic counterpart
(Caglioti and Maffei, {\it J. Stat. Phys.}, 92:301-323, 1998) depends smoothly
on the random variable if the time asymptotic profile does, under the
smoothness and smallness assumptions similar to the deterministic case. The
main idea is to generalize the deterministic contraction argument to more
complicated function spaces to estimate derivatives in space, velocity and
random variables. This result suggests that the random space regularity can
persist in long-time even in time-reversible nonlinear kinetic equations.
