We consider a random walk on Z in a stationary and ergodic random environment, whose states are called types of the vertices of Z. We nd conditions for which the speed of the random walk is positive. In the case of a Markov chain environment with nitely many states, we give an explicit formula for the speed and for the asymptotic proportion of time spent at vertices of a certain type. Using the...

We prove that the process of the most visited site of Sinai's simple random walk in random environment is transient. The rate of escape is characterized via an integral criterion. Our method also applies to a class of recurrent diffusion processes with random potentials. It is interesting to note that the corresponding problem for the usual symmetric Bernoulli walk or for Brownian motion remain...

We study space-time fluctuations around a characteristic line for a onedimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n where n is the ratio of macroscopic and microscopic sc...