We consider random walks in a balanced random environment in Z d , d ≥ 2. We first prove an invariance principle (for d ≥ 2) and the transience of the random walks when d ≥ 3 (recurrence when d = 2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.

We prove a quenched central limit theorem for balanced random walks in time-dependent ergodic random environments which is not necessarily nearest-neighbor. We assume that the environment satisfies appropriate ergodicity and ellipticity conditions. The proof is based on the use of a maximum principle for parabolic difference operators. * Electronic address: deuschel@math.tu-berlin.de † Electronic address:

We consider random walk among iid, 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 [11], we need not only convergence of the steady states, but an estimate on the rate of convergence (see Theorem 4).

In this article, we consider the speed of the random walks in a (uniformly elliptic and i.i.d.) random environment (RWRE) under perturbation. We obtain the derivative of the speed of the RWRE w.r.t. the perturbation, under the assumption that one of the following holds: (i) the environment is balanced and the perturbation satisfies a Kalikow-type ballisticity condition, (ii) the environment satisfies Sznitman's ballisticity condition. This is a generalized version of the Einstein relation for RWRE. Our argument is based on a modification of Lebowitz-Rost's argument developed in [Stochastic Process. Appl. 54 (1994) 183-196] and a new regeneration structure for the perturbed balanced environment.Comment: Published at http://dx.doi.org/10.1214/14-AOP975 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We prove a quenched local central limit theorem for continuous-time random walks in $${\mathbb {Z}}^d, d\ge 2$$ Z d , d ≥ 2 , in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green’s function.