Phi-entropy inequalities for diffusion semigroups

AbstractWe obtain and study new Φ-entropy inequalities for diffusion semigroups, with Poincaré or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker–Planck type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The Γ2 criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms