Einstein relation for biased random walk on Galton--Watson trees
  We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps. 
Gerard Ben Arous, Yueyun Hu, Stefano Olla, Ofer Zeitouni
math.PR