Phi-entropy inequalities for diffusion semigroups

We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups, with Poincare 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.

[1]  Djalil CHAFAÏ.  Entropies, convexity, and functional inequalities , 2002 .

[2]  Giuseppe Savaré,et al.  On the Bakry-Emery criterion for linear diffusions and weighted porous media equations , 2007, Communications in Mathematical Sciences.

[3]  C. Villani,et al.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , 2003 .

[4]  M. Ledoux.  On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions , 1992 .

[5]  Djalil CHAFAÏ,et al.  Sur les in'egalit'es de Sobolev logarithmiques , 2000 .

[6]  F. Malrieu,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR INHOMOGENEOUS MARKOV SEMIGROUPS , 2006, math/0602548.

[7]  R. Latala,et al.  Between Sobolev and Poincaré , 2000, math/0003043.

[8]  Djalil CHAFAÏ.  Entropies, convexity, and functional inequalities : On Phi-entropies and Phi-Sobolev inequalities , 2004 .

[9]  D. Bakry.  L'hypercontractivité et son utilisation en théorie des semigroupes , 1994 .

[10]  W. Beckner.  A generalized Poincaré inequality for Gaussian measures , 1989 .

[11]  L. Gross.  Logarithmic Sobolev inequalities and contractivity properties of semigroups , 1993 .

[12]  Giuseppe Toscani,et al.  ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS , 2001 .

[13]  Dominique Bakry,et al.  Functional Inequalities for Markov semigroups , 2009 .

[14]  M. Ledoux.  The geometry of Markov diffusion generators , 1998 .

[15]  B. Helffer.  Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics , 2002 .

[16]  Qiangchang Ju,et al.  Large-time behavior of non-symmetric Fokker-Planck type equations , 2008 .

[17]  Anton Arnold,et al.  Refined convex Sobolev inequalities , 2005 .

[18]  Emanuele Caglioti,et al.  A Non-Maxwellian Steady Distribution for One-Dimensional Granular Media , 1998 .

[19]  Djalil CHAFAÏ.  BINOMIAL-POISSON ENTROPIC INEQUALITIES AND THE M/M/∞ QUEUE , 2006 .