  We study the kinetic Fokker-Planck equation perturbed by a stochastic Vlasov
force term. When the noise intensity is not too large, we solve the Cauchy
Problem in a class of well-localized (in velocity) functions. We also show
that, when the noise intensity is sufficiently small, the system with
prescribed mass admits a unique invariant measure which is exponentially
mixing. The proof uses hypocoercive decay estimates and hypoelliptic gains of
regularity. At last we also exhibit an explicit example showing that some
restriction on the noise intensity is indeed required.
