Fractional Fokker-Planck equation

Abstract

This paper deals with the long time behavior of solutions to a "fractional Fokker-Planck" equation of the form ${\partial }_{t}f=I[f]+\text{div}(xf)$ 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 \cite{GI}. 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 \cite{GMM} in which an abstract theory of enlargement of the functional space of the semigroup decay is developed.

17 pages

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