A Rescaled Zero-Range Process Converging to the Porous Medium Equation: Large Deviations & Application to Gradient Flow

We consider a particle process on the discrete torus, given by a rescaling of the particle sizes and time in the zero-range model with superlinear jump rate $g(k)=k^\alpha, \alpha\in [1,\infty)$, as well as the usual parabolic rescaling of space and time. Under a relationship between the two scalings, we derive a dynamical large deviation principle, with the same rate function as found for conservative SPDEs by Fehrman and Gess. In this context, we are able to avoid the usual superexponential estimate in the hydrodynamic limit, and the same role is played by stochastic estimates at the level of the particle system and an argument in the spirit of Aubin-Lions-Simons. We finally exhibit a gradient flow structure for the PME through the properties of the large deviation rate function.