Logarithmic Sobolev inequalities: regularizing effect of L\'evy operators and asymptotic convergence in the L\'evy-Fokker-Planck equation

In this paper we study some applications of the L\'evy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the L\'evy-Ornstein-Uhlenbeck process.