本文研究无限树上随机环境中的随机游动。Lyons和Pemantle [11]给出了一个精确的常返/瞬变判据。本文主要研究正则树上随机环境中常返随机游动$(X\_n)$的几乎几乎必然渐近性态，它与Mandelbrot [13]的乘法级联密切相关.我们证明了，在一些一般的假设下，分布的环境，存在一个新的指数$ν\in (0, {1\over 2}]$，使$\max\_{0\le i \le n} |X\_i|$的行为渐近像$n^ν$。$ν$的值是根据环境的分布明确制定的。
英文摘要：
We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk $(X\_n)$ in random environment on a regular tree, which is closely related to Mandelbrot [13]'s multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent $ν\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$ behaves asymptotically like $n^ν$. The value of $ν$ is explicitly formulated in terms of the distribution of the environment.
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