Fractional Fokker-Planck equation

Isabelle Tristani (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Fractional Laplacian; Fokker-Planck equation; spectral gap; exponential rate of convergence; long-time asymptotic

Domaines: Equations aux dérivées partielles [math.AP]

Isabelle Tristani. Fractional Fokker-Planck equation. 2014. ⟨hal-00914059v2⟩