Einstein relation and steady states for the random conductance model

We consider the random conductance model: random walk among iid, uniformly elliptic conductnace on the d-dimensional lattice. We state,and explain, the Einstein relation for this model: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 dimensions three and larger, from an expansion of the steady states as a function of the bias.

The talk is gase on joint work with Jan Nagel and Xiaoqin Guo