Symmetric simple exclusion process in dynamic environment: hydrodynamics

Abstract

We consider the symmetric simple exclusion process in 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.

Collisions of Random Walks in Dynamic Random Environments

“…We will refer to such environments as stationary, strongly reversible Markovian environments; see Section 2 for detailed definitions. This class includes many of the most natural and interesting examples of dynamic RCMs appearing in the literature, including dynamical percolation [26,34,[34][35][36], the simple symmetric exclusion process [7,38,39], and dynamic RCMs in which the conductances evolve according to an SDE such as those arising in the Helffer-Sjöstrand representation of gradient fields, see e.g. [17,25].…”

Hydrodynamics for the partial exclusion process in random environment

“…While for this latter step we follow closely the mild solution approach initiated in [33] and further developed in e.g. [16], [19], [35], to get the desired homogenization result we employ, via a suitable random time change, several concepts and results developed in the context of the random conductance model.…”

“…[28] and [14]) do not apply in this framework. In [35], in which the hydrodynamic limit of the simple exclusion process in presence of dynamic random conductances is studied, a criterion for relative compactness, based on the notion of uniform stochastic continuity, has been presented. This criterion applies to the simple symmetric exclusion process in both static and dynamic random conductances and, in this paper, we apply it also to our context of partial exclusion.…”

“…[25], [24], [22], [18]). However, in presence of self-duality as for the symmetric simple exclusion process, the works [33], [16], [15], [17] and [35] delineate a rigorous strategy to prove the hydrodynamic limit in random environment as a consequence of an arbitrary starting point quenched invariance principle of the corresponding random walk.…”

“…Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we prove the quenched hydrodynamic limit. To this purpose, we exploit the self-duality property of the interacting particle system to transfer, via a mild solution representation of the empirical density fields and the tightness criterion developed in [35], a homogenization result concerning random walks in the same environment with arbitrary starting points to the particle system.…”