一维简单随机游动的最爱下穿点

Favorite down-crossing sites of one-dimensional simple random walk

摘要:

随机游动是一类重要马氏过程，有广泛应用. 对于一维简单对称随机游动，一个点在某个时间被称为最爱下穿点，如果在此时刻没有其他点的下穿次数比该点大. 本文研究了最爱下穿点集的个数，证明如下的以概率1成立：4个及4个以上的最爱下穿点只能出现有限次，3个最爱下穿点则会出现无穷多次. 此外，本文还提出了一些相关的开问题.

Abstract:

Random walk is a very important Markov process and has important applications in many fields. For a one-dimensional simple symmetric random walk, a site is called a favorite down-crossing site at a time if its down-crossing local time at the time achieves the maximum among all sites. In this paper, we study the cardinality of the favorite down-crossing site set, and show that with probability 1 there are only finitely many times at which there are at least four favorite down-crossing sites and three favorite down-crossing sites occurs infinitely often. Some related open questions are introduced as well.