Poincaré and Log–Sobolev Inequalities for Mixtures

Abstract

This work studies mixtures of probability measures on $\mathbb{R}^n$ and gives bounds on the Poincaré and the log-Sobolev constant of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the $χ^2$ -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter whereas the log-Sobolev may blow up as the mixture ratio goes to $0$ or $1$ . This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.