A consistency-stability approach to scaling limits of zero-range processes

TL;DR

This paper introduces a straightforward quantitative method to analyze the hydrodynamic limit of symmetric zero-range processes, providing convergence rates in Monge-Kantorovich distance and relative entropy, applicable in one and two dimensions.

Contribution

It presents a novel, simplified approach for studying hydrodynamic limits that avoids traditional block estimates, using modulated Monge-Kantorovich distances and microscopic stability.

Findings

Convergence rate estimates in Monge-Kantorovich distance.

Uniform convergence in time for the diffusive scaling.

Applicable to zero-range processes in 1D and 2D.

Abstract

We propose a simple quantitative method for studying the hydrodynamic limit of interacting particle systems on lattices. It is applied to the diffusive scaling of the symmetric Zero-Range Process (in dimensions one and two). The rate of convergence is estimated in a Monge-Kantorovich distance asymptotic to the L^1 stability estimate of Kruzkhov, as well as in relative entropy; and it is uniform in time. The method avoids the use of the so-called ``block estimates''. It is based on a modulated Monge-Kantorovich distance estimate and microscopic stability properties.