The Generalized Branching Process And Its Corresponding Markov Integrater Semigroup

In the study of theories of Markov processes.there traditionally are two methods:the probablistic method and analytical method .Recetly mathematicians have got a series of result using the analytical method.In this paper, we mainly apply some of these conclusions to a specific q-matrix- the generalized branching q-matrix Q . By usingthe theory of semigroups of linear operators. , we discuss the generalized branching q-matrix Q and some properties of its transition function F(t),especially, the property of the generalized branching q-matrix Q in l_∞. We get that the operators Q_l∞ clerived from the generalized branching q-matrix Q and generates Q-integrater semigroup onl_∞, and the operators (?) derived from the generalized branching q-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.Consider the generalized branching q-matrix Q which is a continuous-time Markov chains ou the state space E = Z₊ = {0,1,2,…} and the q-matrix Q = (q_ij,i,j∈E): A generalized Markov branching process(GMBP)[1] X_t is a continuous-time Markov chain on the state space Z⁺ = {0,1,2,…} . where the q-matrix Q = {q_ij;i,j∈Z⁺ } , is given bywith In order to avoid discussing some trivial case , we assume that b₀ > 0 and∑_j=2^∞b_j > 0. It is convenient to introduce the generating function of sequences {b_j;j≥0} as in(3)In chapter two, we discuss the properties of the generalized branching q-matrix Q and its minimal Q-function F(t) . We get under given conditions,matrix Q is stochastically sub-monontone,regular,non-dual and zero-exit in Theorem 2.1.1 , we also get that under given conditions, its minimal Q-function F(t) is unique and honest,non-stochastically monontone and dual.; in Theorem 2.1.2, as following:Theorem 2.1.1 When one of the following three conditions holds trueï¼›where q is the root of the B(s) = 0 with 0 < q < 1 for someε∈(q, 1) , then( 1 ) the generalized branching q-matrix Q is stochastically sub-monontone ;( 2 ) the generalized branching q-matrix Q is regular ;( 3 ) the generalized branching q-matrix Q is zero-exit ;( 4 ) the generalized branching q-matrix Q is non-dual.Theorem 2.1.2 When one of the three conditions Theorem 2.1.1 holds true, then(1) F(t) is unique and honest;(2) F(t) is stochastically non-monontone ;(3) F(t) is dual.In chapter three, we get some properties under which operators Q_l∞, Q_ol1 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 (?) 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:Theorem 3.1.1 When one of the three conditions Theorem 2.1.1 holds true, then(1)λI - Q_l∞ is injective on l_∞, for (?)λ> 0;(2)λI - Q_l∞ is surjective on l_∞, for (?)λ> 0;(3) Q_l∞ is dissipative ;(4) Q_l∞ is closed operater .Theorem 3.1.2 When one of the three conditions Theorem 2.1.1 holds true, then(1) Q_0l1 is dense linear operater on l₁ ;(2) Q_0l1 is dissipative , Q_0l1 is closable operater , (?) is dissipative ;(3)λI - (?) is injective on l₁ , for (?)λ> 0 ;(4)λI - (?) is surjective on l₁, for (?)λ> 0 ,Theorem 3.1.3 When one of the three conditions Theorem 2.1.1 holds true, then(1) Q_c0 is dense linear operater on c₀;(2)Q_c0 is dissipative ;(3) Q_c0 is closable linear operater on c₀ ;(4)λI -Q_co is injective on c₀ , for (?)λ> 0.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 propertiesof transition functions on l_∞. In chapter four, on the basis of Y.R.Li[5],we place restrictions on The generalized branching q-matrix Q , and get the sufficient and necessaryconditions under which the operater Q_l∞ derived from Q generates a once positive contracition integrated semigroup on l_∞and the condition of generating Q-integrater semigroup,also get some porperties of Q-integrater semigroup. we also studing thecondition under which the operater Q_ol1 derived from Q generates a once positive contracition integrated semigroup on l₁ and the condition of generating Q-integrater semigroup. We have the following results:Theorem 4.1.1 The generalized branching q-matrix Q generates a positive once integrated semigroup of contractions T(t) = (T_ij(t);i,j∈Z⁺) on l_∞if and only if one of the three conditions of Theorem 2.1.1 holds true . And (T'_ij(t)) is exactly its Q-function P(t) = (p_ij(t)).Theorem 4.1.2 Contraction integrated semigroup T(t) that is generated by Q_l∞ on l_∞is Q-integrater semigroup, if one of the three conditions of Theorem 2.1.1 holds true .Theorem 4.1.4 (?) generates a positive contraction semigroup S(t) = (S_ij(t); i,j∈E) on l₁,when one of the three conditions of Theorem 2.1.1 holds true ,then S(t) = F(t).In chapter four,we get the the properties of positive once integrated semigroups that generalized branching q-matrix Q generates of contractions on l_∞. on the basis of Y.R.Li[5] .Theorem 5.1.1 The integrated semigroup T(t) generated by generalized branchingq-matrix is stochastically sub-monontone .Theorem 5.1.2 If T(t) is a positive once integrated semigroup of contractions by generalized branching q-matrix Q , then (T'_ij(t)) = P(t) is Feller and so is T(t) , that is ,lim_{i→∞}T_ij(t) = 0 for all j∈Z⁺ and t > 0.