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We consider the open symmetric exclusion (SEP) and inclusion (SIP) processeson a bounded Lipschitz domain $\Omega$, with both fast and slow boundary. Forthe random walks on $\Omega$ dual to SEP/SIP we establish: afunctional-CLT-type convergence to the Brownian motion on $\Omega$ with eitherNeumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality)boundary conditions; the discrete-to-continuum convergence of the correspondingharmonic profiles. As a consequence, we rigorously derive the hydrodynamic andhydrostatic limits for SEP/SIP on $\Omega$, and analyze their stationarynon-equilibrium fluctuations. All scaling limit results for SEP/SIP concernfinite-dimensional distribution convergence only, as our duality techniques donot require to establish tightness for the fields associated to the particlesystems.

Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains