The Birth-death Process With Uniform Catastrophes And Its Corresponding Generators Of Integrated Semigruop

In the study of theories of Markov processes, there traditionally are two methods: the probabilistic method and the analytical method. The probable method is straightforward, vivid and distinct in expression. As far as the application is concerned, many specialists such as physicists, biologists and chemists are fond of the results expressed by the probable method. However, the results expressed by the analytical method is much easier to combine achievements of modern mathematics. In this paper, we use the analytical method. By using the theory of semigroups of linear operators, we study the birth-death process with Uniform Catastrophes.Consider the birth-death processes with uniform catastrophes which is a continuous-time Markov chains on the state space E = {0,1, 2,…} and the q- matrix Q = (q_ij, i,j∈E):where a > 0, b≥0, d > 0.First, we discuss the properties of birth-death matrix with uniform catastrophes Q and its minimal Q-function F(t) such as monotonicity, duality, FRR and so on, we get:Theorem 2.1.1 (1) Q is dual;(2) Q is monotone;(3) Q is not FRR;(4) Q is zero-exit;(5) Q is regular.Theorem 2.1.2 (1) F(t) is unique and honest;(2) F(t) is stochastically monotone;(3) F(t) is not FRR;(4) F(t) is not strongly monotone; (5) F(t) is not dual;(6) F(t) is strongly ergodic.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_∞. On the basis of Y R.Li[5], we place restrictions on Q, and prove that the operators Q_l∞ derived from the birth-death matrix with uniform catastrophes Q generates a once positive contraction integrated semigroup on l_∞. We get a good result:Theorem 4.1.1 Q_l∞ gnerates once positive contraction integrated semigroup T(t) = (T_ij(t);i,j∈E) on l_∞, therefore T'(t) = F(t).Theorem 4.1.2 We assume T(t) as Theorem 4.1.1, then:(1) T(t) is stochastically monotone;(2) T(t) is not FRR;(3) T(t) is not strongly monotone, and the limitexits.Theorem 4.1.3 Qod gnerates a positive contraction semigroup S(t) = (S_ij(t);i,j∈E) on l₁ and S(t) = F(t).Theorem 4.1.4 We assume S(t) as Theorem 4.1.3, then:(1) S(t) is stochastically monotone;(2) S(t) is not FRR;(3) S(t) is not strongly monotone;(4) S(t) is not dual;(5) S(t) is strongly ergodic.We know that the birth-death process with uniform catastrophes is stochastically monotone, so there exits dual process from Siegmund Theorem. Then we will introduce another Continuous-Time Markov Chains——the dual of birth-death process with uniform catastrophes. Wewill discuss the properties of dual birth-death matrix with uniform catastrophes Q² and its minimal Q—function F²(t); And we will prove the operators derived from dual matrix are generators of positive contraction semigroups on l₁ or c₀. We have the following results:Theorem 5.1.1 (1) Q² is FRR;(2) Q² is dual;(3) Q² is zero-entrance in l₁⁺; (4) Q² is strongly zero-entrance in l₁;(5) Q² is zero-exit;(6) Q² is not monotone. Theorem 5.1.2 (1) F²(t) is FRR;(2) F²(t) is dual;(3) F² (t) is not stochastically monotone;(4) F² (t) is not strongly monotone;(5) F² (t) is strongly egodic.Theorem 6.1.2 (1) Q_l1² gnerates a positive contraction semigroup F²(t) on l₁;(2) gnerates a positive contraction semigroup F²(t) on l₁;(3) gnerates a positive contraction semigroup F²(t) on c₀;(4) Q_c0² gnerates a positive contraction semigroup F²(t) on c₀;(5) gnerates once positive contraction integrated semigroup on 1_∞.