Fractional Fokker–Planck equation

This paper deals with the long time behavior of solutions to a “fractional Fokker–Planck” equation of the form $\partial_t f = I[f] + \mathrm{div}(xf)$ 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.fr/ccsd-0049578, 2013] in which an abstract theory of enlargement of the functional space of the semigroup decay is developed.

Keywords

fractional Laplacian, Fokker–Planck equation, spectral gap, exponential rate of convergence, long-time asymptotic

2010 Mathematics Subject Classification

35B40, 35Q84, 47G20

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Published 22 April 2015