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

In this paper, we study the Wasserstein gradient flow structure of the porous medium equation restricted to q-Gaussians. The JKO-formulation of the porous medium equation gives a variational functional K_h, which is the sum of the (scaled-) Wasserstein distance and the internal energy, for a time step h. We prove that, for the case of q-Gaussians on the real line, K_h is asymptotically equivalent, in the sense of Γ-convergence as h tends to zero, to a rate-large-deviation-like functional. The result explains why the Wasserstein metric as well as the combination of it with the internal energy play an important role.