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. 
Ivan Gentil (CEREMADE), Cyril Imbert (CEREMADE)