一类随机环境中的广生灭过程

Generalized Birth and Death Process in A Random Environment

设1={0,1,2,……},I上广生灭Q—矩阵Q(θ)=(qij(θ))(gij,(θ)=0,j=i>1■的元素均是某概率空间(Γ,/A,M)上的随机变量。本文构造了一个概率空间(Γ×IR^([0,∞)),/A×/F,P),以及此空间上的随机过程X={X(θ,t);θ∈Γ,t([0,∞)};当θ∈Γ固定时,X(θ)={X(θ,t);t∈[0,∞)}是广生灭Q(θ)过程。称随机过程X为随机环境中的广生灭过程。对一类随机环境,本文讨论了随机环境中广生灭过程的“常返”性、“遍历”性以及平均到达时间。作为特殊情况,包括了随机环境中生灭过程、随机环境中单生全灭过程。

Let I={0,1,2,……},and the generalized brith and death Q-matrix Q=(q_(ij);i,j∈I)(q_(ij)=0,j-i>1) Now,suppose the members of the Q be random variables on a certain probability space(Γ,|A,M). In this paper,a probability space(Γ×I_R~■·J,|A×|F,P)is constructed on which a stochastic process x=(χ(θ,t);θ∈Γ, t∈[0,∞))is defined,When θ∈Γ is fixed,the x=(χ(θ,t); t∈[0,∞))is the generalized birth and death Q(θ)process.The stochastic process x is called generalized birth and death process in random enviroment. For a random environment,the“recurrence”property, “ergodic”property and mean reaching times of the x are studied in this paper.Furthermore,birth and death process and one brith and all death process in the random environment are dealt with as the special cases of the x.