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Abstract  

We give a new proof of the central limit theorem for one dimensional symmetric random walk in random environment. The proof is quite elementary and natural. We show the convergence of the generators and from this we conclude the convergence of the process. We also investigate the hydrodynamic limit (HDL) of one dimensional symmetric simple exclusion in random environment and prove stochastic convergence of the scaled density field. The macroscopic behaviour of this field is given by a linear heat equation. The diffusion coefficient is the same as that of the corresponding random walk.

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Abstract  

We introduce the directed-edge-reinforced random walk and prove that the process is equivalent to a random walk in random environment. Using Oseledec"s multiplicative ergodic theorem, we obtain recurrence and transience criteria for random walks in random environment on graphs with a certain linear structure and apply them to directed-edge-reinforced random walks.

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1974 Révész, P. , Random walk in random and non-random environments . Second edition , World Scientific Publishing Co. Pte. Ltd

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, Acta Math, Acad, Sci, Hung , 11 ( 1960 ), 137 – 162 . MR 22 #12599 [5] Révész , P. , Random walk in random and non-random

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non-random environments , World Scientić Publishing Co., Inc. (Teaneck, NJ., 1990). MR 92c :60096 Révész P. Random walk in random and non-random

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] R évész , P. , Random walk in random and non-random environments , World Scientific, Singapore, 1990 . [9] R ogers , L. C. G. and S tapleton , E. J

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] R évész , P. , Random, Walk in Random, and Non, Random, Environments , World Scientific , Singapore ( 1990 ). MR 92c:60096 [10] R evuz , D. and Y or

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. , Random, walk in random and non-random, environments , World Scientific Publishing Co. , Teaneck, NJ , 1990 . MR 92c:60096 [13] T alagrand , М. , Lower classes

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