Regularity of biased 1D random walks in random environment

We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity lambda is an element of R. For ergodic shift-invariant environments, we show that the limiting velocity v(lambda) is always increasing and that it is everywhere analytic except at most in two points lambda_ and lambda(+). When lambda_ and lambda(+) are distinct, v(lambda) might fail to be continuous. We refine the assumptions in Zeitouni (2004) for having a recentered CLT with diffusivity sigma(2)(lambda) and give explicit conditions for sigma(2)(lambda) to be analytic. For the random conductance model we show that, in contrast with the deterministic case, sigma(2)(lambda) is not monotone on the positive (resp. negative) half-line and that it is not differentiable at lambda = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of Lam and Depauw (2016).