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We study a rescaling of the zero-range process with homogenous jump rates$g(k)=k^\alpha$ with arbitrary $\alpha\ge 1$. With a simultaneous rescaling ofspace, time and particle size, the rescaled empirical density is shown toconverge to the porous medium equation with homogeneity $\alpha$. We furtheridentify the dynamical large deviations from the porous medium equation, usingpathwise discretised regularity estimates to prove a version of thesuperexponential estimate in the spirit of the Aubin-Lions-Simons lemma.Finally, we use the large deviation principle to give an expression of theporous medium equation as the gradient flow of the Boltzmann entropy withrespect to a tailor-made Wasserstein-type distance.

A Rescaled Zero-Range Process for the Porous Medium Equation: Hydrodynamic Limit, Large Deviations and Gradient Flow