Einstein relation and steady states for the random conductance model

We consider random walk among i.i.d., uniformly elliptic conductances on Z^d, and prove the Einstein relation (see Theorem 1). It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle. These invariant measures are often called steady states. The Einstein relation follows at least for d = 3, from an expansion of the steady states as a function of the bias (see Theorem 2), which can be considered our main result. This expansion is proved for d ≥ 3. In contrast to Guo [Ann. Probab. 44 (2016) 324-359], we need not only convergence of the steady states, but an estimate on the rate of convergence (see Theorem 4).