Steady States, Fluctuation–Dissipation Theorems and Homogenization for Reversible Diffusions in a Random Environment

Abstract

Prolongating our previous paper on the Einstein relation, we study the motion of a particle diffusing in a random reversible environment when subject to a small external forcing. In order to describe the long time behavior of the particle, we introduce the notions of steady state and weak steady state. We establish the continuity of weak steady states for an ergodic and uniformly elliptic environment. When the environment has finite range of dependence, we prove the existence of the steady state and weak steady state and compute its derivative at a vanishing force. Thus we obtain a complete ‘fluctuation–dissipation Theorem’ in this context as well as the continuity of the effective variance.