Abstract：A new structure of the resolvent of two kinds of birth and death processes was given by using RayKnight compactness and run-length measure in boundary processes. Firstly, the different classification of boundary points under R-K compaction theory was discussed. Secondly, it was proved that there exist only trivial solutions for the Kolmogorov equation when the boundary point is natural and outflow, and nontrivial solutions exist in the case of inflow and regularity. At last, the probabilistic construction of the resolvent of the two kinds of birth-death process of boundary points in the case of outflow and regularity was given by introducing the boundary process and the run-length theory. The defects of uncertain probability meaning of analytical method and complex structure of traditional probability method were overcome.

Keyword：birth and death processe;Kolmogorov backward equation;Ray-Knight compactification;resolvent;