Phi-entropy inequalities for diffusion semigroups

François Bolley (1) , Ivan Gentil (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We obtain and study new $\Phi$-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-Plank type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The $\Gamma_2$ criterion of D.~Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.

Mots clés: Functional inequalities; logarithmic Sobolev inequality; Poincaré inequality; $\Phi$-entropies; Bakry-Emery criterion; diffusion semigroups; Fokker-Planck equation

Domaines: Probabilités [math.PR]

François Bolley, Ivan Gentil. Phi-entropy inequalities for diffusion semigroups. 2008. ⟨hal-00344163v2⟩