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

Abstract

Abstract. 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.Key words. parabolic equations, diffusion, Ornstein-Uhlenbeck operator, porous media, Poincaré inequality, logarithmic Sobolev inequality, convex Sobolev inequality, interpolation, decay rate, entropy, free energy, Fisher information AMS subject classifications. 35B40; 35K55; 39B62; 35J10; 35K20; 35K65The Bakry-Emery method [9] has been extremely successful in establishing logarithmic Sobolev inequalities [31]. It is also known to apply very well to the proof of Poincaré inequalities and inequalities which interpolate between Poincaré and logarithmic Sobolev inequalities [11,6,2,21], the so-called "generalized Poincaré inequalities." The first paper on such inequalities has been written by W. Beckner in [11] in the case of a Gaussian measure. In [6], the inequalities have been slightly generalized by taking into account general x-dependent diffusions and by considering "convex entropies" based on a convex function ψ satisfying the additional admissibility condition:The proof is based on the entropy-entropy production method and the Bakry-Emery condition, [9]. The result has been slightly improved in [5], thus providing "refined inequalities" in the case ψ(s) = (s p − 1 − p(s − 1))/(p − 1), p ∈ (1,2). Other considerations on "generalized Poincaré inequalities" can be found in [2,35,21,3].The Bakry-Emery condition is a sufficient, local condition, which relies on a uniform strict log-concavity of the measure. Using perturbation techniques, it is possible to relax such a strict assumption to some extent, see [32,21]. When Poincaré and logarithmic Sobolev inequalities are known to hold simultaneously, further interpolation inequalities can also be established: see for instance [35,15,10,3] BAKRY-EMERY CRITERION FOR DIFFUSIONSIn case of generalized Poincaré inequalities, many techniques which are available for Poincaré inequalities and spectral gap approaches can be adapted, and more flexib...

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