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

Abstract

Fix a smooth function $Φ: [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b\le {\Phi }^{\prime }\le b^{-1}$. 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 ${\partial }_t\rho ={\sum }_{k=1}^d(d/d{x}_k)(d/d{W}_k)\Phi (\rho )$. We also derive some properties of the operator ${\sum }_{k=1}^d(d/d{x}_k)(d/d{W}_k)$.

References in corpus (3)

• Quenched invariance principles for random walks with random conductances
• Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances
• Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit

Cited by in corpus (5)

• Hydrodynamic limit of zero range processes among random conductances on the supercritical percolation cluster
• Quenched scaling limits of trap models
• Hydrodynamic Limit for a type of Exclusion Processes with slow bonds in dimension $\ge 2$
• $W$-Sobolev spaces: Theory, Homogenization and Applications
• Equilibrium fluctuations for gradient exclusion processes with conductances in random environments