# A note on Fisher Information hypocoercive decay for the linear Boltzmann equation

## TL;DR

This paper proves that for the linear Boltzmann equation with a near-quadratic confining potential, the Fisher Information and relative entropy decay exponentially fast to zero from smooth initial data.

## Contribution

It establishes exponential decay of Fisher Information for the linear Boltzmann equation in a non-compact setting with near-quadratic potential, extending hypocoercivity results.

## Key findings

- Fisher Information decays exponentially fast to zero.
- Relative entropy also converges exponentially.
- Results hold for smooth initial data in a non-compact setting.

## Abstract

This note deals with the linear Boltzmann equation in the non-compact setting with a confining potential which is close to quadratic. We prove that in this case, starting from a smooth initial datum, the Fisher Information (hence, the relative entropy) with respect to the stationary state converges exponentially fast to zero.