We consider the weakly asymmetric exclusion process on a bounded interval with particle reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. We prove the associated dynamical large deviations principle.
"Dynamical large deviations for the boundary driven weakly asymmetric exclusion process." Ann. Probab. 37 (6) 2357 - 2403, November 2009. https://doi.org/10.1214/09-AOP472