本文对随机游动的常返性的判定进行了比较深入的研究.主要介绍了如何用逃逸概率来判定随机游动的常返性.逃逸概率即随机游动的质点在到达a之前先到达b的概率,利用其调和性可与电网络上的电压建立联系,从而可以通过电网中相关理论来计算逃逸概率. 文章从一维随机游动入手,先将常返性的判定问题转化为了逃逸概率的计算问题,再利用逃逸概率的调和性将其与电压建立联系,并逐步拓展到图上的的随机游动,从而得出结论——随机游动的常返性与其对应的电网从原点到无穷处的有效电阻有关,若有效电阻为∞,则该随机游动是常返的,否则为暂留的.最后利用这一结论,通过Rayleigh’s短路断路法证明了二、三维上的简单随机游动的常返性与暂留性.

In this thesis, we will make a deep research into the proof for the recurrence and transience of Random walks. The main purpose of this paper is to introduce how to use the escape probability to prove the recurrence and transience of Random walks. The escape probability is the probability that the Random walks reaches position b before it reaches position a. For the escape probability is a harmonic function, we can establish the connection between the escape probability and the voltage in electric network. Then we can calculate the escape probability by using the relevant theory in electric network. Starting from the Random walks in one dimension, we firstly transform the problem of proving its recurrence into the calculation of the escape probability. Then, by using the harmonicity of the escape probability, we establish the connection between the escape probability and the voltage. Gradually, we extend it to the Random walks on graph. The conclusion is that the recurrence of Random walks is related to the effective resistance to infinity of corresponding electric network. Finally, we successfully prove the recurrence and transience of simple random walk in two as well as three dimensions by using the conclusion above and the Rayleigh’s short-cut method.