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

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