  We consider Fokker-Planck equations in the whole Euclidean space, driven by
Levy processes, under the action of confining drifts, as in the classical
Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or
sub-exponential decay rates, as time goes to infinity, of zero average
solutions, under some diffusivity condition on the Levy process, which includes
the fractional Laplace operator as a model example. Our approach relies on the
long time oscillation estimates of the adjoint problem and applies to (the
possible superposition of) both local and nonlocal diffusions, as well as to
strongly or weakly confining drifts. Our results extend, with a unifying
perspective, many previous works based on different analytic or probabilistic
methods, with several interesting connections. On one hand, we make a link
between the (nonlinear) PDE methods used for the long time behavior of
Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations;
on another hand, we give a purely analytical approach towards some oscillation
decay estimates which were obtained so far only with probabilistic coupling
methods.
