# Asymptotic behavior of density in the boundary-driven exclusion process on the Sierpinski gasket

## TL;DR

This paper studies the macroscopic and fluctuation behaviors of the exclusion process on the Sierpinski gasket with boundary effects, deriving heat equations with various boundary conditions and analyzing the resulting Ornstein-Uhlenbeck processes.

## Contribution

It establishes the hydrodynamic limit and fluctuation results for the exclusion process on the Sierpinski gasket with boundary-driven dynamics, including novel boundary condition cases.

## Key findings

- Derivation of the heat equation with Dirichlet, Neumann, and Robin boundary conditions.
- Identification of Ornstein-Uhlenbeck processes governing fluctuations.
- Analysis of boundary effects on particle density evolution.

## Abstract

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.