Hydrodynamic behavior of long-range symmetric exclusion with a slow barrier: superdiffusive regime

TL;DR

This paper studies the hydrodynamic limits of a long-range symmetric exclusion process with a slow barrier, revealing different superdiffusive regimes characterized by fractional Laplacians and boundary conditions, depending on the barrier's parameters.

Contribution

It provides a detailed analysis of superdiffusive hydrodynamic behavior with fractional PDEs for long-range exclusion processes with a slow barrier, highlighting regime differences based on barrier parameters.

Findings

Different PDE regimes depending on barrier parameters.

Fractional Laplacian with boundary conditions describes the hydrodynamics.

Distinct behaviors for $eta=0$ and $eta>0$ regimes.

Abstract

We analyse the hydrodynamical behavior of the long jumps symmetric exclusion process in the presence of a slow barrier. The jump rates are given by a symmetric transition probability $p(\cdot)$ with infinite variance. When jumps occur from $\mathbb{Z}_{-}^{*}$ to $\mathbb N$ the rates are slowed down by a factor $\alpha n^{-\beta}$ (with $\alpha>0$ and $\beta\geq 0$). We obtain several partial differential equations given in terms of the regional fractional Laplacian on $\mathbb R^*$ and with different boundary conditions. Surprisingly, in opposition to the diffusive regime, we get different regimes depending on whether $\alpha=1$ (all bonds with the same rate) or $\alpha\neq 1$.