The Limiting Behavior of a One-Dimensional Random Walk in a Random Medium
The Limiting Behavior of a One-Dimensional Random Walk in a Random Medium
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DOI：
10.1137/1127028
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发表时间：
1983
影响因子：
0.6
通讯作者：
Y. Sinai
中科院分区：
文献类型：
--
作者：
Y. Sinai
我们考虑直线ZI的整数点集上最简单的随机游动，在该直线下，随机移动点分别以概率p（x）和q（x）1 p（x）从x s Z到x +1。在n次转换之后的点的位置由x（n）表示。假设x（0）= 0。所有p（x）的知识决定了随机变量x（n）的概率分布，n> 0。所有与x（n）（n> 0）的行为有关的事件的概率都用P表示。如果概率p（x）本身是某个随机过程的实现，则可以说在随机介质中发生了随机游动。当p（x）形成一个独立随机变量序列时，出现最简单的版本;例如，p（x）21-+ e（x），其中（x）+/-1是一个随机符号，并且符号：（x）对于不同的x是相互独立的，0 <e <1/2。本文主要讨论p（x）独立时的情形。最后指出了某些概括性的结论。与依赖于p（x）的实现的事件有关的概率用P表示。与随机介质中随机游动的常返性和非常返性有关的问题在[1]-[3]中进行了研究。本文研究了x（n）在n->时的行为.基本假设是p（x），q（x）=> const> 0和q（x）Ep log px）O。在这些条件下，与普通的随机游动相反，对于大的n，变量x（n）取log n阶的值。但是如果x（n）是归一化的，即考虑变量x（n）/log n，那么作为n-，x（n）/log 2 n的概率分布变得局部化，即集中在某个点的任意小邻域中，这取决于实现p-{p（x）}。基本结果可以用下面的方式更精确地表述。
We consider the simplest random walk on the set of integer points of the straight line ZI, under which a randomly moving point passes from x s Z to x+ 1 with probabilities p (x) and q (x) 1 p (x), respectively. The position of the point after n transitions is denoted by x (n). It is assumed that x (0)= 0. A knowledge of all the p (x) determines theprobability distribution of the random variables x (n), n> 0. All the probabilities relating to events connected with the behavior of x (n), n> 0, will be denoted belowby P. It is said that a random walk occurs in a random medium if the probabilities p (x) themselves are realizations ofsome randomprocess. The simplest version arises when p (x) forms a sequence of independent random variables; for example, p (x) 21-+ e (x), where (x)+/-1 is a random sign, and the signs:(x) for different x are mutually independent, 0< e< 1/2. In this paper we mainly discuss the situation when the p (x) are independent. Certain generalizations are indicated at the end. The probabilities relating to events depending on the realization of p (x) are denoted by P. Problems connected with the recurrence and non-recurrence properties of a random walk in a random medium were examined in [1]-[3]. In this paper we investigate the behaviour of x (n) as n-->. The basic assumption is that p (x), q (x)=> const> 0 and q (x) Ep log px) O.It will be shown that, under these conditions, in contrast to the ordinary random walk, for large n the variable x (n) takes on values of order log n. But if x (n) is normalized, ie, the variable x (n)/log n is considered, then as n-the probability distribution for x (n)/log2 n becomes localized, ie, concentrated in an arbitrarily small neighborhood of some point depending on the realization p-{p (x)}. The basic result can be formulated more precisely in the following manner.