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

A study of Landau damping with random initial inputs