一维简单随机游动的最爱下穿点  

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

作　　者：郝晨旭 胡泽春[1] 马婷 宋仁明 HAO Chen-Xu;HU Ze-Chun;MA Ting;SONG Ren-Ming(Schoolof Mathematics,Sichuan University,Chengdu 610064,China;Department of Mathematics,University of Illinois,Urbana IL 61801,USA)

机构地区：[1]四川大学数学学院,成都610064 [2]美国伊利诺伊大学香槟分校数学系,厄巴纳IL61801

出　　处：《四川大学学报（自然科学版）》2023年第5期45-52,共8页Journal of Sichuan University(Natural Science Edition)

基　　金：国家自然科学基金(12171335,12101429,12071011,11931004,11871184);西蒙斯基金(429343);四川大学理科发展专项项目(2020SCUNL201)。

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(S_(n)),点x在时间n被称为最爱下穿点,如果在时刻n没有其它点的下穿次数比点x的下穿次数大.本文研究了最爱下穿点集的个数,证明如下的命题以概率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.Random walk is a very important Markov process and has important applications in many fields.For a one-dimensional simple symmetric random walk(S_(n)),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.

关 键 词：随机游动 最爱点 最爱边 最爱下穿点 局部时 

分 类 号：O211.5[理学—概率论与数理统计]