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

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 Prékopa–Leindler inequality by further exploiting known properties of the heat equation including the eventual log-concavity and backward uniqueness of solutions.