Abstract

In this chapter we introduce various functionals such as entropy and free entropy that are defined for suitable probability density functions on Rn. Then we introduce the derivatives of such functionals in the style of the calculus of variations. This leads us to the gradient flows of probability density functions associated with a given functional; thus we recover the famous Fokker–Planck equation and the Ornstein–Uhlenbeck equation. A significant advantage of this approach is that the free analogues of the classical diffusion equations arise from the corresponding free functionals. We also prove logarithmic Sobolev inequalities, and use them to prove convergence to equilibrium of the solutions to gradient flows of suitable energy functionals. Positive curvature is a latent theme in this chapter; for recent progress in metric geometry has recovered analogous results on metric spaces with uniformly positive Ricci curvature, as we mention in the final section.

Variation of functionals and gradient flows

In this chapter we are concerned with evolutions of families of probability distributions under partial differential equations. We use ρ for a probability density function on Rn and impose various smoothness conditions as required. For simplicity, the reader may suppose that ρ is C∞ and of compact support so that various functionals are defined. The fundamental examples of functionals are:

• Shannon's entropy S(ρ) = -∫ ρ(x) log ρ(x) dx;

• Potential energy F(ρ) = ∫ v(x)ρ(x)dx with respect to a potential function v;