We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N −β , with β ∈ [0, ∞). We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β. If β ∈ [0, 1), the hydrodynamic limit is given by the usual heat equation. If β = 1, it is given by a parabolic equation involving an operator d dx d dW , where W is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If β ∈ (1, ∞), it is given by the heat equation with Neumann's boundary conditions, meaning no passage through the slow bonds in the continuum. 2000 Mathematics Subject Classification. 60K35,26A24,35K55.

For a heat equation with Robin's boundary conditions which depends on a parameter α > 0, we prove that its unique weak solution ρ α converges, when α goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann's boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of ρ α obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.

We present the correct space of test functions for the Ornstein-Uhlenbeck processes defined in [2]. Under these new spaces, an invariance with respect to a second order operator is shown, granting the existence and uniqueness of those processes. Moreover, we detail how to prove some properties of the semigroups, which are required in the proof of uniqueness.

Fix a strictly increasing right continuous with left limits function W : R → R and a smooth function Φ :We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes, with conductances given by W , is described by the weak solutions of the non-linear differential equation ∂tρ = (d/dx)(d/dW )Φ(ρ). We derive some properties of the operator (d/dx)(d/dW ) and prove uniqueness of weak solutions of the previous non-linear differential equation.2000 Mathematics Subject Classification. 60K35, 26A24, 35K55, 82C44.

ABSTRACT. We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter ρ ∈ (0,1). The rate of passage of particles to the right (resp. left) is 1 2 + a 2n γ (resp. 1 2 − a 2n γ ) except at the bond of vertices {−1,0} where the rate to the right (resp. left) is given bywe show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if γ > 1 2 , while for γ = 1 2 it is an energy solution of the stochastic Burgers equation. For γ ≥ β = 1, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For γ ≥ β > 1, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.