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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  

Generalized random processes are classified by various types of continuity. Representation theorems of a generalized random process on

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{M p} on a set with arbitrary large probability, as well as representations of a correlation operator of a generalized random process on
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{M p} and L r(R), r > 1, are given. Especially, Gaussian generalized random processes are proven to be representable as a sum of derivatives of classical Gaussian processes with appropriate growth rate at infinity. Examples show the essence of all the proposed assumptions. In order to emphasize the differences in the concept of generalized random processes defined by various conditions of continuity, the stochastic differential equation y′(ω; t) = f(ω; t) is considered, where y is a generalized random process having a point value at t = 0 in the sense of Lojasiewicz.

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. F. , The Itô-Clifford integral II, Stochastic differential equations, J. London Math. Soc. 27 (1983), 373–384. MR 84m :46079b Wilde I. F. The Itô-Clifford integral II

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’s formula using discrete Itô’s formula, Studia. Sci. Math. Hungar. , to appear. Ikeda , N. and Watanabe , S., Stochastic differential equations and diffusion processes , Second Ed., North

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46 449 478 El-Nouty, C. , The lower classes of the sub-fractional Brownian motion , in Stochastic Differential Equations and Processes

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