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

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 L2 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.

Keywords

parabolic equations, diffusion, Ornstein-Uhlenbeck operator, porous media, Poincaré inequality, logarithmic Sobolev inequality, convex Sobolev inequality, interpolation, decay rate, entropy, free energy, Fisher information

2010 Mathematics Subject Classification

35B40, 35J10, 35K20, 35K55, 35K65, 39B62

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Published 1 January 2008