Einstein relation for reversible random walks in random environment on ZZ

The aim of this paper is to consider reversible random walk in a random environment in one dimension and prove the Einstein relation for this model. It says that the derivative at 00 of the effective velocity under an additional local drift equals the diffusivity of the model without drift (Theorem 1.2). Our method here is very simple: we solve the Poisson equation (Pω−I)g=f(Pω−I)g=f and then use the pointwise ergodic theorem in Wiener (1939) [10] to treat the limit of the solutions to obtain the desired result. There are analogous results for Markov processes with discrete space and for diffusions in random environment.