  This article studies a Fokker-Planck type equation of fractional diffusion
with conservative drift $\partial$f/$\partial$t = $\Delta$^($\alpha$/2) f +
div(Ef), where $\Delta$^($\alpha$/2) denotes the fractional Laplacian and E is
a confining force field. The main interest of the present paper is that it
applies to a wide variety of force fields, with a few local regularity and a
polynomial growth at infinity. We first prove the existence and uniqueness of a
solution in weighted Lebesgue spaces depending on E under the form of a
strongly continuous semigroup. We also prove the existence and uniqueness of a
stationary state, by using an appropriate splitting of the fractional Laplacian
and by proving a weak and strong maximum principle. We then study the rate of
convergence to equilibrium of the solution. The semigroup has a property of
regularization in fractional Sobolev spaces, as well as a gain of integrability
and positivity which we use to obtain polynomial or exponential convergence to
equilibrium in weighted Lebesgue spaces. Mathematics Subject Classification
(2000): 47D06 One-parameter semi-groups and linear evolution equations [See
also 34G10, 34K30], 35P15 Estimation of eigenvalues, upper and lower bounds
[See also 35P05, 45C05, 47A10], 35B40 Partial differential equations,
Asymptotic behavior of solutions [see also 45C05, 45K05, 35410],
