Consider a nearest-neighbor random walk with certain asymptotically zero drift on the positive half line.Let M be the maximum of an excursion starting from 1 and ending at 0.We study the distribution of M and characterize its asymptotics,which is quite different from those of simple random walks.

We study a birth and death process {Nt}t≥0 in i.i.d,random environment,for which at each discontinuity,one particle might be born or at most L particles might be dead.Along with investigating the existence and the recurrence criterion,we also study the law of large numbers of {Nt}.We show that the first passage time can be written as a functional of an L-type branching process in random environment and a sequence of independent and exponentially distributed random variables.Consequently,an explicit velocity of the law of large numbers can be given.

考虑随机环境中相关随机游动.对暂留但具有零速度的情形,证明了游动趋向无穷的速度不但是次线性的,而且比ns'还慢,其中s'∈(s,∞),s是某个取值于(0,1)的常数.该结果刻画了游动的慢速度性质.