Properly Posed Problem Of Solution And Asymptotic Solution Of Kolmogorov Backward Differential Equation

In this thesis, the solution of Kolmogorov backward differential equations in birth and death process theory has been proved to be well-posedness by using the theories and methods of linear operator Co semigroup in functional analysis. And the existence of superior eigenvalue of the coefficient matrix of the equations has been studied by using the theories of positive operator and conjugate operator.This thesis consists of four chapters. Main parts of the thesis are chapter 3 and chapter 4. Detail contents are as follows. In chapter 1 we state the history of birth and death process theory, and expound the organization of the paper and the problem that needs to solve. In chapter 2 we arrange some concepts, theorems and results which will be used. In chapter 3, firstly we establish the operator equation. Secondly by using the theories and methods in functional analysis, and study the properties of birth and death Q-matrix. Thirdly it is proved that a birth and death Q-matrix generates a linear operator Co semigroup under certain conditions, that is to say, the birth and death Q-matrix is the infinitesimal generator of the Co semigroup. By which the existence, uniqueness and stability of solution of Kolmogorov backward differential equations is proved. Finally we prove the semigroup is a positive Co semigroup. Besides, we apply Co semigroup theory to queue. In chapter 4 it is proved that the coefficient matrix of Kolmogorov backward differential equations has unique nonnegative proper function. We prove the correspondence eigenvalue is real, discrete and larger than any other eigenvalues' real. By which we can conclude that the eigenvalue is the superior eigenvalue of the Q-matrix. And we give an asymptotic representation of solution of the equation.