On the Bakry-Emery criterion for linear diffusions and weighted porous media equations
Savaré, Giuseppe; Nazaret, Bruno; Dolbeault, Jean (2008), On the Bakry-Emery criterion for linear diffusions and weighted porous media equations, Communications in Mathematical Sciences, 6, 2, p. 477-494
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00196935/en/
Journal nameCommunications in Mathematical Sciences
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Abstract (EN)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.
Subjects / KeywordsOrnstein-Uhlenbeck operator; diffusion; Parabolic equations; porous media; Poincaré inequality; logarithmic Sobolev inequality; convex Sobolev inequality; interpolation; decay rate; entropy; free energy; Fisher information
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