The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien . More precisely, for a given realization ω of the environment, we consider the Poisson equation (Pω - I)g = f, and then use the pointwise ergodic theorem in  to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.
"A quenched central limit theorem for reversible random walks in a random environment on Z." J. Appl. Probab. 51 (4) 1051 - 1064, December 2014.