Hydrodynamic limit of gradient exclusion processes with conductances on $\bb Z^d$

Abstract

Fix a smooth function $\Phi : [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b \le \Phi'\le b^{-1}$. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes in $\bb Z^d$, with conductances given by special class of functions $W$, is described by the weak solutions of the non-linear parabolic partial differential equation $\partial_t \rho = \sum_{k=1}^d (d/dx_k)(d/dW_k)\Phi(\rho)$. We also derive some properties of the operator $\sum^d_{k=1}(d/dx_k)(d/dW_k)$.