A novel approach to hydrodynamics for long-range generalized exclusion

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.

Figures (2)

• Figure 1: Illustration of the bulk dynamics of the long-range exclusion process ($\kappa = 1$) on $\mathbb Z$.
• Figure 2: Illustration of the reservoir dynamics for the long-range exclusion process ($\kappa = 1$) with density $\alpha\in [0,1]$ to the left and $\beta\in [0,1]$ to the right.

Theorems & Definitions (27)

• remark 1: A sufficient condition
• definition 1
• remark 2
• theorem 1
• lemma 1: Tightness
• lemma 2
• lemma 3
• lemma 4
• proof : Proof of Theorem \ref{['theo:hydrodynamics']}
• proof : Proof of Lemma \ref{['lem:tight']}