关于含瞬时态Q过程的几点注记

Some Notes on Q-Processes in Case Where Some States Are Intaneous

设E为可列集,Q=(q_(ij):i,j∈E是E×E上的Q矩阵,我们并不要求Q全稳定.考虑如下问题,取△∈E,问Q满足什么条件下,存在E上的非负行向量e和E上的非负向量f,使Q=(q_(ij):i,j∈EU{△})为Q-矩阵.其中q△△=-∞,qi△=f_i,q_(△j)=e_j,q_(ij)=q(i),(i,j∈E),本文给出了上述问题的完整解答.同时,我们给出了上述结果在含瞬时态Q过程方面的三个应用.

Let E be a Countable Set, Q=(q_(ij): i,j∈E)be a Q-matrix on E×E,Q needn't be totally stable.The following Problem is discussed. Let △∈E, what conditions does Q Satisfy that there exist a non-negative row vector e on E and a non-negative column vector f on E such that Q=(q_(ij);i,j∈EU{△}is a Q-matrix? Where q _(△△) =-∞ q_(△j)=e_j,q_(i△)=f_i,q_(ij)(i,j∈E) In this paper, a Complete answer to this problem is given, and three of its applications on Q-processes in case where some states are instantaneoas.are also provided.