Abstract
In this paper, we introduce a random environment for the exclusion process in Zd obtained by assigning a maximal occupancy to each site. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we derive a quenched hydrodynamic limit in path space by strengthening the mild solution approach initiated in Nagy (2002) and Faggionato (2007). To this purpose, we prove, employing the technology developed for the random conductance model, a homogenization result in the form of an arbitrary starting point quenched invariance principle for a single particle in the same environment, which is a result of independent interest. The self-duality property of the partial exclusion process allows us to transfer this homogenization result to the particle system and, then, apply the tightness criterion in Redig et al. (2020).
Highlights
In recent years there has been extensive study of the scaling limit of random walks in both static and dynamic random environment
The macroscopic effects of the environment may be studied through scaling limits such as hydrodynamic limits, fluctuations and large deviations around the hydrodynamic limit, as well as via the study of non-equilibrium behavior of systems coupled to reservoirs which, in random environment, is still a challenge
We introduce a random environment for the exclusion process in Zd obtained by assigning a maximal occupancy αx ∈ N to each site x ∈ Zd and we study its hydrodynamic limit
Summary
In recent years there has been extensive study of the scaling limit of random walks in both static and dynamic random environment. In such systems, the macroscopic equation can be guessed from the behavior of the expectation of the local particle density which, in turn, amounts to understand the scaling behavior of a single “dual” particle This intuitive “transference principle” from the scaling limit of one random walker to the macroscopic equation has to be made rigorous. Given a realization α of the random environment, the partial (simple) exclusion process in the environment α, abbreviated by SEP(α), is the Markov process on X α whose generator acts on bounded cylindrical functions φ : X α → R, i.e., functions which depend only on a finite number of occupation variables, as follows (all throughout the paper, |·| will always denote the Euclidean norm): Lαφ(η) =.
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