Asymptotic equivalence of the discrete variational functional and a rate-large-deviation-like functional in the Wasserstein gradient flow of the porous medium equation

TL;DR

This paper demonstrates that in the Wasserstein gradient flow of the porous medium equation, the JKO-discretized functional asymptotically matches a large deviation-like functional for q-Gaussians, clarifying the roles of the Wasserstein metric and Tsallis-entropy.

Contribution

It establishes the asymptotic equivalence between the JKO-discretized functional and a large deviation-like functional for q-Gaussians, revealing underlying connections in the porous medium equation.

Findings

Asymptotic equivalence between functionals for q-Gaussians

Role of Wasserstein metric and Tsallis-entropy clarified

Provides theoretical insight into gradient flow structure

Abstract

In this paper, we study the Wasserstein gradient flow structure of the porous medium equation. We prove that, for the case of $q$-Gaussians on the real line, the functional derived by the JKO-discretization scheme is asymptotically equivalent to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the Tsallis-entropy play an important role.