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Article Dans Une Revue Communications in Mathematical Sciences Année : 2008

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

Résumé

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.
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Dates et versions

hal-00196935 , version 1 (13-12-2007)

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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⟩
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