The Excursion Theory Of Birth And Death Process

In this paper,we present the local time ,inverse local time ,Poisson point process , excursion, excursion measure of birth and death process when it move away from ∞. Building the excursion theory of birth and death process,and using which we investigate the sample path of birth and death process when oo regular and instantaneous. We introduce the concept as t tends to 0+,i.e. initial variable.Using the Markov property of excursion under excursion measure and the entrance law of secondary transition function (p_ij(t))_i,j∈E,We obtain the relevant results when the initial variable e(0) taking value in state space. In conclusion,we obtain the representation in excursion measure of matrix resolvent.