Hydrodynamic limit in a particle system with topological interactions

We study a system of particles in the interval $[0,\epsilon^{-1}] \cap \mathbb Z$, $\epsilon^{-1}$ a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously new particles are injected at site 0 at rate $j\epsilon$ ($j>0$) and removed at same rate from the rightmost occupied site. The removal mechanism is therefore of topological rather than metric nature. The determination of the rightmost occupied site requires a knowledge of the entire configuration and prevents from using correlation functions techniques. We prove using stochastic inequalities that the system has a hydrodynamic limit, namely that under suitable assumptions on the initial configurations, the law of the density fields $\epsilon \sum \phi(\epsilon x) \xi_{\epsilon^{-2}t}(x)$ ($\phi$ a test function, $\xi_t(x)$ the number of particles at site $x$ at time $t$) concentrates in the limit $\epsilon\to 0$ on the deterministic value $\int \phi \rho_t$, $\rho_t$ interpreted as the limit density at time $t$. We characterize the limit $\rho_t$ as a weak solution in terms of barriers of a limit free boundary problem.