  In this paper we study the Fokker-Planck operator with potential V(x), and
analyze some kind of conditions imposed on the potential to ensure the validity
of global hypoelliptic estimates. As a consequence, we obtain the compactness
of resolvent of the Fokker-Planck operator if either the Witten Laplacian on
0-forms has a compact resolvent or some additional assumption on the behavior
of the potential at infinity is fulfilled. This work improves the previous
results of H\'erau-Nier and Helffer-Nier, by obtaining a better global
hypoelliptic estimate under weaker assumptions on the potential.
