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We derive a formula for the quasi-potential of one-dimensional symmetricexclusion process in weak contact with reservoirs. The interaction with theboundary is so weak that, in the diffusive scale, the density profile evolvesas the one of the exclusion process with reflecting boundary conditions. Inorder to observe an evolution of the total mass, the process has to be observedin a longer time-scale, in which the density profile becomes immediatelyconstant.

Dynamic and static large deviations of a one dimensional SSEP in weak contact with reservoirs