About Hydrodynamic Limit of Some Exclusion Processes via Functional Integration

TL;DR

This paper introduces a novel method using functional integration and PDE analysis to derive hydrodynamic limits of exclusion processes, with potential applications to higher-dimensional and multi-type systems.

Contribution

It proposes a new approach based on variational calculus and generalized measures to analyze hydrodynamic limits of exclusion processes.

Findings

Analysis of ASEP on the torus Z/N/Z

Method extends to higher dimensions and multi-type processes

Derivation of coupled PDEs of Burgers' type

Abstract

This article considers some classes of models dealing with the dynamics of discrete curves subjected to stochastic deformations. It turns out that the problems of interest can be set in terms of interacting exclusion processes, the ultimate goal being to derive hydrodynamic limits after proper scalings. A seemingly new method is proposed, which relies on the analysis of specific partial differential operators, involving variational calculus and functional integration: indeed, the variables are the values of some functions at given points, the number of which tends to become infinite, which requires the construction of \emph{generalized measures}. Starting from a detailed analysis of the \textsc{asep} system on the torus Z/N/Z, we claim that the arguments a priori work in higher dimensions (ABC, multi-type exclusion processes, etc), leading to sytems of coupled partial differential equations of Burgers' type.