A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality
Analysis of PDEs
2024-05-28 v1
Abstract
In this paper, we provide a new PDE proof for the celebrated Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection between the Borell--Brascamp--Lieb inequality and properties of diffusion equations of porous medium type pertaining to the large time asymptotics and preservation of a generalized concavity of the solutions. We also recover the equality condition in the special case of the Pr\'ekopa--Leindler inequality by further exploiting known properties of the heat equation including the eventual log-concavity and backward uniqueness of solutions.
Keywords
Cite
@article{arxiv.2405.16721,
title = {A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality},
author = {Kazuhiro Ishige and Qing Liu and Paolo Salani},
journal= {arXiv preprint arXiv:2405.16721},
year = {2024}
}