关键词: 分枝过程, 随机环境中的分枝随机游动, 依赖于代的分枝随机游动

Abstract:

Suppose {Z_n;n=0,1,2,…} is a branching random walk in the random environment, and ξ ={ξ₀,ξ₁,ξ₂, …} is the environment process. Let Z(n,x) be the number of the nth generation located in the interval (-∞, x], f_ξn(s)=∑^∞_j=0 p_ξn(j)s^j be the generating function of the distribution of the particle in the nth generation, and m_ξn=f_ξn^' (1). We show that under the specific conditions, there exists a sequence of random variables {t_n}, so that Z(n,t_n)(∏_i=0^n-1m_ξi)^-1 converges in L². For branching random walks in varying environments, we have similar results.

Key words: branching process, branching random walks in random environments, branching random walks in varying environments