Abstract
We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain Ω, with both fast and slow boundary. For the random walks on Ω dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on Ω 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 Ω, and analyze their stationary nonequilibrium 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.
Funding Statement
The first author gratefully acknowledges funding by the Austrian Science Fund (FWF) grant F65, by the European Research Council (ERC, grant agreement No 716117, awarded to Prof. Dr. Jan Maas). He also gratefully acknowledges funding of his current position by the Austrian Science Fund (FWF) grant ESPRIT 208.
The second author gratefully acknowledges funding by the Hausdorff Center for Mathematics at the University of Bonn. Part of this work was completed while this author was a member of the Institute of Science and Technology Austria. He gratefully acknowledges funding of his position at that time by the Austrian Science Fund (FWF) grants F65 and W1245.
The third author gratefully acknowledges funding by the Lise Meitner fellowship, Austrian Science Fund (FWF): M3211. Part of this work was completed while funded by the European Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie grant agreement No. 754411.
Acknowledgments
The authors are very grateful to Antonio Agresti for many useful conversations about boundary value problems in Lipschitz domains. The first named author wishes to thank Kazuhiro Kuwae for a useful discussion about the paper [53]. He is also grateful to Alessandra Faggionato, Lorenzo Bertini, and Giada Basile for fruitful conversations on the subject of this paper. The third named author is grateful to Patrícia Gonçalves for some fruitful conversations on an earlier draft of this paper, and to Claudio Landim for kindly pointing out the reference [54]. The authors wish to express their gratitude to three anonymous reviewers for their careful reading and very helpful suggestions.
Citation
Lorenzo Dello Schiavo. Lorenzo Portinale. Federico Sau. "Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains." Ann. Appl. Probab. 34 (2) 1789 - 1845, April 2024. https://doi.org/10.1214/23-AAP2007
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