Redig, Frank; Saada, Ellen; Sau, FedericoIST Austria
We consider the symmetric simple exclusion process in Zd with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic behavior of the density field. While the hydrodynamic limit at fixed macroscopic times is obtained via a generalization to the time-inhomogeneous context of the strategy introduced in , in order to prove tightness for the sequence of empirical density fields we develop a new criterion based on the notion of uniform conditional stochastic continuity, following . In conclusion, we show that uniform elliptic dynamic conductances provide an example of environments in which the so-called arbitrary starting point invariance principle may be derived from the invariance principle of a single particle starting from the origin. Therefore, our hydrodynamics result applies to the examples of quenched environments considered in, e.g., , ,  in combination with the hypothesis of uniform ellipticity.
Electronic Journal of Probability
We warmly thank S.R.S. Varadhan for many enlightening discussions at an early stage of this work. We are indebted to Francesca Collet for fruitful discussions and constant support all throughout this work. We thank Simone Floreani and Alberto Chiarini for helpful conversations on the final part of this paper as well as both referees for their careful reading and for raising relevant issues on some weak points contained in a previous version of this manuscript; we believe this helped us to improve it. Part of this work was done during the authors’ stay at the Institut Henri Poincaré (UMS 5208 CNRS-Sorbonne Université) – Centre Emile Borel during the trimester Stochastic Dynamics Out of Equilibrium. The authors thank this institution for hospitality and support (through LabEx CARMIN, ANR-10-LABX-59-01). F.S. thanks laboratoire MAP5 of Université de Paris, and E.S. thanks Delft University, for financial support and hospitality. F.S. acknowledges NWO for financial support via the TOP1 grant 613.001.552 as well as funding from the European Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie grant agreement No. 754411. This research has been conducted within the FP2M federation (CNRS FR 2036).
Redig F, Saada E, Sau F. Symmetric simple exclusion process in dynamic environment: Hydrodynamics. Electronic Journal of Probability. 2020;25. doi:10.1214/20-EJP536
Redig, F., Saada, E., & Sau, F. (2020). Symmetric simple exclusion process in dynamic environment: Hydrodynamics. Electronic Journal of Probability. Project Euclid. https://doi.org/10.1214/20-EJP536
Redig, Frank, Ellen Saada, and Federico Sau. “Symmetric Simple Exclusion Process in Dynamic Environment: Hydrodynamics.” Electronic Journal of Probability. Project Euclid, 2020. https://doi.org/10.1214/20-EJP536.
F. Redig, E. Saada, and F. Sau, “Symmetric simple exclusion process in dynamic environment: Hydrodynamics,” Electronic Journal of Probability, vol. 25. Project Euclid, 2020.
Redig F, Saada E, Sau F. 2020. Symmetric simple exclusion process in dynamic environment: Hydrodynamics. Electronic Journal of Probability. 25, 138.
Redig, Frank, et al. “Symmetric Simple Exclusion Process in Dynamic Environment: Hydrodynamics.” Electronic Journal of Probability, vol. 25, 138, Project Euclid, 2020, doi:10.1214/20-EJP536.