A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality

TL;DR

This paper introduces a novel PDE-based proof for the Borell--Brascamp--Lieb inequality, linking it to diffusion equations and their properties, and recovers equality conditions in special cases.

Contribution

It provides a new PDE approach to the Borell--Brascamp--Lieb inequality, connecting it with diffusion equations and generalized concavity preservation.

Findings

Established a PDE proof connecting Borell--Brascamp--Lieb to diffusion equations

Revealed the relationship between inequality properties and large time asymptotics

Recovered equality conditions for Prékopa--Leindler case using heat equation properties

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.