# Poincar\'e and log-Sobolev inequalities for mixtures

## TL;DR

This paper investigates how Poincaré and log-Sobolev inequalities behave for mixtures of probability measures, especially Gaussian mixtures, revealing that the constants can vary significantly depending on the mixture ratio and component properties.

## Contribution

It provides bounds on these inequalities for two-component mixtures under certain conditions, extending previous results to multidimensional cases and illustrating complex behaviors.

## Key findings

- Poincaré constant remains bounded in the mixture parameter.
- Log-Sobolev constant can blow up as the mixture ratio approaches 0 or 1.
- Mixture behavior can be more complex than individual components.

## Abstract

This work studies mixtures of probability measures on $\mathbb{R}^n$ and gives bounds on the Poincar\'e and the log-Sobolev constant of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the $\chi^2$-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 Poincar\'e 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 Chafa\"i 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.