The Description to the Infinitesimal Operator of the Minimal Q Process
Yang Xiang-qun · 2002
Given a Q matrix, whose opponents are finite. Feller solved the problem about existence of Q process,and constructed one mininal Q 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 (,D()). 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.