On the Bakry-Emery criterion for linear diffusions and weighted porous media equations

Jean Dolbeault (1) , Bruno Nazaret (1) , Giuseppe Savaré (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Dipartimento di matematica F. Casorati

The goal of this paper is to give a non-local sufficient condition for generalized Poincaré inequalities, which extends the well-known Bakry-Emery condition. Such generalized Poincaré inequalities have been introduced by W. Beckner in the gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the $L^2$ norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same strategy to non-linear equations of porous media type. This provides new interpolation inequalities and decay estimates for the solutions of the evolution problem. The criterion is again a non-local condition based on the positivity of the lowest eigenvalue of a Schrödinger operator. In both cases, we relate the Fisher information with its time derivative. Since the resulting criterion is non-local, it is better adapted to potentials with, for instance, a non-quadratic growth at infinity, or to unbounded perturbations of the potential.

Mots clés: Fisher information; Parabolic equations; diffusion; Ornstein-Uhlenbeck operator; porous media; Poincaré inequality; logarithmic Sobolev inequality; convex Sobolev inequality; interpolation; decay rate; entropy; free energy; Fisher information.

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Bruno Nazaret, Giuseppe Savaré. On the Bakry-Emery criterion for linear diffusions and weighted porous media equations. Communications in Mathematical Sciences, 2008, 6, pp.477-494. ⟨hal-00196935⟩