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 $\lambda\in\mathbb{R}$. 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 \cite{Z} 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 \cite{LD16}.