No abstract

Fix a functionwhere d ≥ 1, and each function W k : R → R is strictly increasing, right continuous with left limits. We prove the equilibrium fluctuations for exclusion processes with conductances, induced by W , in random environments, when the system starts from an equilibrium measure. The asymptotic behavior of the empirical distribution is governed by the unique solution of a stochastic differential equation taking values in a certain nuclear Fréchet space.

Fix strictly increasing right continuous functions with left limits and periodic increments,. Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Existence and uniqueness results for W -generalized elliptic equations, and uniqueness results for W -generalized parabolic equations are also established. Finally, an application of this theory to stochastic homogenization is presented.

This contribution aims at presenting and generalizing a recent work of Hernández, Jara and Valentim [10]. We consider the weakly asymmetric version of the so-called discrete Atlas model, which has been introduced in [10]. Precisely, we look at some equilibrium fluctuation field of a weakly asymmetric zero-range process which evolves on a discrete half-line, with a source of particles at the origin. We prove that its macroscopic evolution is governed by a stochastic heat equation with Neumann or Robin boundary conditions, depending on the range of the parameters of the model. 1 The continuous Atlas model is given by a semi-infinite system of independent Brownian motions on R, see for instance [5,11], and also [10] for more details. 2 We refer to [13] for a review on zero-range processes.