Convexity Seminar

---

The role of the Boltzmann entropy and Fisher information in diffusive PDE

Speaker: Nestor Guillen, Texas State University, Courant

Location: Warren Weaver Hall 1302

Date: Tuesday, February 10, 2026, 11 a.m.

Synopsis:

The Boltzmann entropy and the Fisher information are fundamental objects arising in statistics, however they have also become an essential tool in the analysis of nonlinear parabolic PDE. This tradition goes back over half a century starting with Linnik's proof of the central limit theorem and McKean's work on the 1-D model for a Boltzmann gas, and continues to this day in the ruling out of blow up for kinetic equations. In this elementary talk I will survey some of this history, the basic  properties of the functionals (notably their convexity and various symmetries), and provide a simple explanation for their monotonicity in time for various equations -- including the Fokker-Planck equation and nonlocal parabolic equations. If time allows I will discuss how these observations are in a sense analogues to traditional inequalities for mixed volumes.