Gradient flow structures for discrete porous medium equations

Abstract

We consider discrete porous medium equations of the form \partial_t ρ_t = Δϕ(ρ_t), where Δis the generator of a reversible continuous time Markov chain on a finite set X, and ϕis 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.

19 pages