  We study well-posedness and long time behavior of the nonlinear
Vlasov-Poisson- Fokker-Planck system with an external confining potential. The
system describes the time evolution of particles (e.g.$\,\,$in a plasma)
undergoing diffusion, friction and Coulomb interaction. We prove existence and
uniqueness of mild solutions in a weighted Sobolev space. Moreover, we prove
that the solutions converge to the global equilibrium exponentially. Our
results hold for a wide class of external potentials and the estimates on the
rate of convergence are explicit and constructive. The technique is based on
the construction of Lyapunov functionals, new short and long time estimates for
the linearized system, and fixed point arguments.
