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We consider the weakly asymmetric exclusion process on a bounded intervalwith particles reservoirs at the endpoints. The hydrodynamic limit for theempirical density, obtained in the diffusive scaling, is given by the viscousBurgers equation with Dirichlet boundary conditions. In the case in which thebulk asymmetry is in the same direction as the drift due to the boundaryreservoirs, we prove that the quasi-potential can be expressed in terms of thesolution to a one-dimensional boundary value problem which has been introducedby Enaud and Derrida \cite{de}. We consider the strong asymmetric limit of thequasi-potential and recover the functional derived by Derrida, Lebowitz, andSpeer \cite{DLS3} for the asymmetric exclusion process.

Strong asymmetric limit of the quasi-potential of the boundary driven weakly asymmetric exclusion process