Random Walks in Random Environment
Random walks in random environment constitute the best example to apply the theory developed in Chap. 2. In order to keep notation and results consistent with the previous chapter, we discuss only the time continuous case. By exploiting the translation invariance of the random environment, the position of the random walk is seen as an additive functional of the environment as seen from the particle process. Double-stochastic (or bi-stochastic) random walks are the basic examples where the ergodic stationary measure for the environment process is known explicitly. Random walks with random conductances (so called bond diffusions) are the simplest example and the resulting environment process is reversible. If the condition of a finite cycle decomposition is satisfied the sector condition for the environment process holds. For more general bi-stochastic environments we can prove the central limit theorem in dimension 3 or higher if a mixing condition is satisfied. In dimension 1 the sector condition is always satisfied if the drift has null average.
KeywordsRandom Walk Central Limit Theorem Sector Condition Dirichlet Form Random Environment
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