Displacement convexity for the entropy in semidiscrete nonlinear Fokker-Planck equations

TL;DR

This paper demonstrates the displacement convexity of a nonstandard entropy in finite state spaces for nonlinear Fokker-Planck equations using a gradient flow approach and introduces a new mean function for the Onsager operator.

Contribution

It introduces a novel mean function to establish displacement convexity of entropy in semidiscrete nonlinear Fokker-Planck equations.

Findings

Explicit computation of the convexity constant λ.

Application of a new mean function in the gradient flow framework.

Establishment of displacement λ-convexity in finite state spaces.

Abstract

The displacement $\lambda$-convexity of a nonstandard entropy with respect to a nonlocal transportation metric in finite state spaces is shown using a gradient flow approach. The constant $\lambda$ is computed explicitly in terms of a priori estimates of the solution to a finite-difference approximation of a nonlinear Fokker-Planck equation. The key idea is to employ a new mean function, which defines the Onsager operator in the gradient flow formulation.