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

Abstract

We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain $\Omega$, with both fast and slow boundary. For the random walks on $\Omega$ dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on $\Omega$ with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions; the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on $\Omega$, and analyze their stationary non-equilibrium fluctuations. All scaling limit results for SEP/SIP concern finite-dimensional distribution convergence only, as our duality techniques do not require to establish tightness for the fields associated to the particle systems.

56 pages, 3 figures; final version

References in corpus (6)

• Stationary and Nonequilibrium Fluctuations in Boundary Driven Exclusion Processes
• Discrete approximations to reflected Brownian motion
• Exact solution of an integrable non-equilibrium particle system
• Sharp Convergence to Equilibrium for the SSEP with Reservoirs
• Thermodynamics of nonequilibrium driven diffusive systems in mild contact with boundary reservoirs
• Hydrodynamic limit of Exclusion Processes with slow boundaries on hypercubes

Cited by in corpus (2)

• Tiny fluctuations of the averaging process around its degenerate steady state
• Spectral gap of the symmetric inclusion process