A novel approach to hydrodynamics for long-range generalized exclusion

TL;DR

This paper introduces a new framework for analyzing long-range exclusion processes with infinite-mean jumps, deriving a superballistic scaling limit governed by a non-local transport equation.

Contribution

It develops a novel approach to the hydrodynamic limit for long-range exclusion processes without requiring knowledge of stationary states.

Findings

Superballistic scaling limit established

Limit evolution described by a non-local transport equation

Method relies solely on algebraic structure of the generator

Abstract

We consider a class of generalized long-range exclusion processes evolving either on $\mathbb Z$ or on a finite lattice with an open boundary. The jump rates are given in terms of a general kernel depending on both the departure and destination sites, and it is such that the particle displacement has an infinite expectation, but some tail bounds are satisfied. We study the superballisitic scaling limit of the particle density and prove that its space-time evolution is concentrated on the set of weak solutions to a non-local transport equation. Since the stationary states of the dynamics are unknown, we develop a new approach to such a limit relying only on the algebraic structure of the Markovian generator.