Stochastic Dynamics of Discrete Curves and Exclusion Processes. Part 1: Hydrodynamic Limit of the ASEP System

Abstract

This report is the foreword of a series dedicated to stochastic deformations of curves. Problems are set in terms of exclusion processes, the ultimate goal being to derive hydrodynamic limits for these systems after proper scalings. In this study, solely the basic \textsc{asep} system on the torus is analyzed. The usual sequence of empirical measures, converges in probability to a deterministic measure, which is the unique weak solution of a Cauchy problem. The method presents some new features, letting hope for extensions to higher dimension. It relies on the analysis of a family of parabolic differential operators, involving variational calculus. Namely, the variables are the values of functions at given points, their number being possibly infinite.