Time-periodic solutions of the Vlasov-Poisson-Fokker-Planck system

In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-amplitude solution with the same period. The proof follows by the Serrin's method on the basis of the exponential time-decay property of the linearized system in the case of the constant background profile.