时间随机环境中一维紧邻随机游动的常返性质

设环境q={q(n)}∞0是取值于0,1]上一列独立同分布的随机变量列,且Eq(0)=p;{Sn}∞0是随机环境q中取整数值随机游动,S0=0,且满足:对任意的整数xi(i≥0),x,y,P(Sn+1=y|S1=x1,…,Sn-1=xn-1,Sn=x,q)={q(n),y=x+1,1-q(n),y=x-1,0,其他.我们证明了:p>1/2时,Sn→+∞,a.e.,n→∞;p<1/2时,Sn→-∞,a.e.,n→∞;p=1/2时,-∞=(lim infSn)/(n→+∞)<(lim supSn)/(n→+∞)=+∞,a.e.,n→∞.

The Recurrence of One-dimensional Nearest-neighbour Random Walks in a Time Random Environments

Suppose that random environment q={q(n)}∞0, is i.i.d. and valued on 0,1], RWRE {Sn}∞0,S0=0, integers xi (i≥0),x,y, satisfies P(Sn+1=y|S1=x1,…,Sn-1=xn-1,Sn=x,q)={q(n),y=x+1,1-q(n),y=x-1,0,,others. We prove: If p>1/2,Sn→+∞,a.e., n→∞; If p<1/2,Sn→-∞,a.e., n→∞; If p=1/2,-∞=(lim infSn)/(n→+∞)<(lim supSn)/(n→+∞)=+∞,a.e., n→∞.