Hydrodynamic limit in a particle system with topological interactions

We study a system of particles in the interval [0, \eps^{ −1}] ∩ Z, \eps^{−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 (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 \eps \sum φ(\eps x) ξ_{\eps^{-2}t} (x) (φ a test function, ξ_{t}(x) the number of particles at site x at time t) concentrates in the limit t → 0 on the deterministic value R \int φ ρ_t, ρ_t interpreted as the limit density at time t. We characterize the limit ρ_t as a weak solution in terms of barriers of a limit free boundary problem.