Gradient flow structures for discrete porous medium equations

TL;DR

This paper establishes a gradient flow framework for discrete porous medium equations on finite sets, linking them to entropy functionals and non-local transportation metrics, and explores their geometric properties.

Contribution

It introduces a novel gradient flow structure for discrete porous medium equations, extending Otto's Wasserstein gradient flow concept to discrete settings.

Findings

Discrete porous medium equations are gradient flows of entropy functionals.

Counterexample shows lack of geodesic convexity in some cases.

Discussion of convergence to Wasserstein metric via Gromov-Hausdorff limits.

Abstract

We consider discrete porous medium equations of the form \partial_t \rho_t = \Delta \phi(\rho_t), where \Delta is the generator of a reversible continuous time Markov chain on a finite set X, and \phi is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in R^n discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.