最小Q过程的无穷小算子的刻划

The Description to the Infinitesimal Operator of the Minimal Q Process

给定一矩阵Q ,其元素均有限 .Feller解决了Q过程存在性问题 ,且构造了一个最小Q过程 f(t) .设Q过程P(t)的Lapalace变换即预解算子为Ψ(λ) ,P(t)所生成的无穷小算子为A ,由文 [2 ]可知P(t) ,Ψ(λ) ,A三者一一对应 ,且已知Q过程P(t) ,可决定A ,Ψ(λ) .而对于给定的矩阵 Q如何求出P(t) ,Ψ(λ) ,或A的问题 ,实际上是可列马尔可夫过程的一个核心问题 :即Q过程的构造问题 .文献 [1]对此课题作了深入研究 ,其结果是以Q过程P(t)所对应的预解算子Ψ(λ)表述的 .由Ψ(λ)的Lapalace反变换可决定P(t) ,然而自然要问 :对于Ψ(λ) ,其对应的A怎样刻划呢 ?特别地 ,记最小Q过程为Φ(λ) ,对应的无穷小算子为 A ,我们的首要问题是如何刻划最小Q过程的无穷小算子 ( A ,D( A) ) .本文对此问题作了一些基本工作 .当 Q矩阵零流出和单流出时 ,分别求出了最小Q过程Φ(λ)所对应的无穷小算子 .

Given a Q matrix, whose opponents are finite. Feller solved the problem about existence of Q process, and constructed one minimal kQ process-f(t). Let Ψ(λ) denote the lapalace transform of Q process P(t), i. e. its the resolvent operator, and A denote the infinitesimal operator generated by P(t). Knocon from ref. [2], there are one-one correspondences among P(t), Ψ(λ) and A, also A and Ψ(λ) can be determined when Q process- P(t) is given. How to find P(t), Ψ(λ) or A when Q matrix is given in fact is the key problem in denumerable Markov processes i. e. construction problem of Q process. In ref. [1], this problem was studied further, and corresponding results were stated in style of the resolvent operator Ψ(λ) corresponding to Q process-P(t). By the anti-Lapalace transform of Ψ(λ), P(t) is determined, of course we will surpose the question that how to describe the corresponding A for Ψ(λ)? Especially, if we let Φ(λ) denote the mininal Q process, and A denote the corresponding infinitestmal operator, then the primary problem is how to describe the infinitesimal operator (A, D(A)). We make some basic work on this problem. When the Q matrix is in zero exit case and in single exit case the infinitesimal operatores for the minimal Q process- Φ(λ) are described respectively. The main result is described in the domain independent of λ.