Abstract
We consider the symmetric simple exclusion process in $\mathbb {Z}^{d}$ 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 [41], 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 [50]. 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., [1], [3], [6] in combination with the hypothesis of uniform ellipticity.
Highlights
Dynamic random environments are natural quantities to be inserted in probabilistic models in order to make them more realistic
Our contribution is to carry out this connection between single particle behavior and diffusive hydrodynamic limit in the context of dynamic environment for a nearest-neighbor particle system, namely the symmetric simple exclusion process (SSEP) in a quenched dynamic bond disorder, for which we show that a suitable form of self-duality remains valid
For the symmetric simple exclusion process in a quenched dynamic bond disorder in Zd (whose generator is described in (2.3) below), we show that a form of self-duality still holds and allows us to write the occupation variables of the particle system in terms of positions of suitable time-inhomogeneous backward random walks evolving in the same environment
Summary
Dynamic random environments are natural quantities to be inserted in probabilistic models in order to make them more realistic. In the latter instance, we rely on two main assumptions for this tightness criterion to be effective: a quenched invariance principle for forward random walks and a uniform bound on the maximal number of particles per site. Even other types of quenched disorder are suited for this tightness criterion as proved in [23], in which the environment is designated by assigning a (uniformly bounded) maximal occupancy αx ∈ N to each site In other words, this criterion applies to all particle systems for which a self-duality property and a uniform bound on the maximal number of particles per site hold in combination with the validity of the arbitrary starting point invariance principle. We conclude the paper with the complete proof of our new tightness criterion used (Appendix B, more precisely Theorem B.4 in combination with Theorem B.2) and the study of a non-trivial space-time inhomogeneous scenario in which the invariance principle for the random walk starting from the origin yields an analogous invariance principle for all random walks starting from all macroscopic points and times (Appendix C, see Section 3.1)
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