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

TL;DR

This paper introduces a non-local criterion extending the Bakry-Emery condition to establish generalized Poincaré inequalities and decay estimates for linear and nonlinear diffusion equations, improving existing results.

Contribution

It develops a non-local sufficient condition based on eigenvalues of a Schrödinger operator, applicable to potentials with non-quadratic growth and unbounded perturbations.

Findings

Provides new interpolation inequalities for porous media equations

Establishes decay estimates for solutions of nonlinear diffusions

Extends Bakry-Emery criterion to non-local settings

Abstract

The goal of this paper is to give a non-local sufficient condition for generalized Poincar\'e inequalities, which extends the well-known Bakry-Emery condition. Such generalized Poincar\'e 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 Schr\"odinger 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.