Fractional Fokker–Planck equation

This paper deals with the long time behavior of solutions to a "fractional Fokker-Planck" equation of the form tf = I[f ] + div(xf ) where the operator I stands for a fractional Laplacian. We prove an exponential in time convergence towards equilibrium in new spaces. Indeed, such a result was already obtained in a L 2 space with a weight prescribed by the equilibrium in [I. Gentil and C. Imbert, Asymptot. Anal., 59, 125-138, 2008]. We improve this result obtaining the convergence in a L 1 space with a polynomial weight. To do that, we take advantage of the recent paper [M. P. Gualdani, S. Mischler, and C. Mouhot, http://hal.archives-ouvertes.