The Generalized Birth-death Catastrophes Process And Its Corresponding Markov Integrater Semigroup

In the study of theories of Markov processes,mathematicians have got a series of perfect and universal conclusions.In this paper, we mainly apply some of these conclusions to a specific q-matrix-the generalized birth-death catastrophes matrix Q. By using the theory of semigroups of linear operators, firstly, we discuss the generalized birthdeath catastrophes matrix Q and some properties of its transition function F(t) .especially,the property of the generalized birth-death catastrophes matrix in l_∞. Secondly, we prove that the operators Q_l∞ derived from the generalized birth-death catastrophesmatrix Q generates Q-integrater semigroup on l_∞, and the operators Q_ol1^- derived from the generalized birth-death catastrophes matrix Q generates a once positive contraction integrated semigroup on l₁, then discuss some properties of Q-integrater semigroup and once positive contraction integrated semigroup. Finally, we get the duai q-matrix Q^* of Q,and discussion about some properties of Q^* and the minimal Q^*-function.Consider the generalized birth-death catastrophes matrix which is a continuoustime Markov chains on the state space E=Z₊={0,1,2.···} and the q-matrix In chapter two, we discuss the properties of matrix Q and its minimal Q-function F(t) . We get the sufficient and necessary conditions under which matrix Q is monotone,dual and zero-exit in Theorem 2.1.1 , we also get the sufficient and necessary conditions under which its minimal Q-function F(t) is monotone , and the conditions of its duality and Feller in Theorem 2.1.2, we get :In chapter three, we get some properties under which operators Q_l∞,(?) and Q_c0 derived from the generalized birth-death catastrophes matrix Q on l_∞,l₁ and c₀ respectively. We get the conditions under whichλI-Q_l∞ is injective and surjective on l_∞, also get the conditions under which Q_l∞ is dissipative and closed operater inTheorem 3.1.1. We get the conditions under whichλI-(?) is injective and surjectiveon l₁, also get the conditions under which Q_0l1 is dissipative in Theorem 3.1.2. We study that Q_c0 is dissipative and closable linear operater on c₀ in Theorem 3.1.3. we get : Y.R.Li[5] got that there is a one-to-one relationship between transition functions and the positive once integrated semigroups of contractions on l_∞by studing the properties of transition functions on l_∞. In chapter four, on the basis of Y.R.Li[5].wc place restrictions on the generalized birth-death catastrophes matrix Q,and get the sufficient and necessary conditions under which the operater Q_l∞ derived from Q generates a once positive contraction integrated semigroup on l_∞and the condition of generating Q-integrater semigroup,also get some porperties of Q-integrater semigroup. wealso studing the condition under which the operater Q_ol1 derived from Q generates a once positive contraction integrated semigroup on l_∞and the condition of generating Q-integrater semigroupa,also get some porperties . We have the following resuits: Under some conditions , we know that the minimal Q-function F(t) of the generalized birth-death catastrophes matrix Q is stochastically monotone, so there exits dual process from Siegmund Theorem. In charper five, we will introduce Continuous-Time Markon Chains- the dual of the generalized birth-death catastrophes process. We get the dual q-matrix Q^* of the generalized birth-death catastrophes matrix Q, and discuss the properties of Q^* and its minimal Q-function F^*(t). We have the following resuits:In charper six , we study some properties under which operators Q_l1^* derived from dual q-matrix Q^* of the generalized birth-death catastrophes matrix Q , and describe positive contraction semigroup which is generated by operator Q_l1^* on l₁. We have the following resuits: Theorem 6.1.2 Q_l1^* generates a positive contraction semigroup F^*(t) on l₁ when...