Hydrodynamics for symmetric exclusion in contact with reservoirs

Abstract

We consider the symmetric exclusion process with jumps given by a symmetric, translation invariant, transition probability $p(\cdot)$. The process is put in contact with stochastic reservoirs whose strength is tuned by a parameter $\theta\in\mathbb R$. Depending on the value of the parameter $\theta$ and the range of the transition probability $p(\cdot)$ we obtain the hydrodynamical behavior of the system. The type of hydrodynamic equation depends on whether the underlying probability $p(\cdot)$ has finite or infinite variance and the type of boundary condition depends on the strength of the stochastic reservoirs, that is, it depends on the value of $\theta$. More precisely, when $p(\cdot)$ has finite variance we obtain either a reaction or reaction-diffusion equation with Dirichlet boundary conditions or the heat equation with different types of boundary conditions (of Dirichlet, Robin or Neumann type). When $p(\cdot)$ has infinite variance we obtain a fractional reaction-diffusion equation given by the regional fractional Laplacian with Dirichlet boundary conditions but for a particular strength of the reservoirs.