The Monotonicity, Dulity And Feller Property Of Weighted Markov Branching Processes

In the study of theories of Markov processes,there are two methods traditionaly:the probabilistic method and the analystical method.Recently,mathematician investigate theories of Markov processes using the analytical method to deal with Markov branchingprocesses and obtain many results.In this paper ,we also use the analytical method to investigate one of the most important processes which called the weighted Markov branching processes. The basic property which govens the evolution of Markov branchingprocesses is branching property.This is that different particles act independently when giving birth or death. But in most realistic situations,particles usually interact with each other,so many authors have great intrest to generalize the branching processes to more general branching models.A particularly interesting class of one generalized-branchingprocesses was considered in Chen(1997),this paper will further consider the branching processes which called the weighed Markov branching processes.Consider the weighed Markov branching processes which is a continuous-time Markov chains on the state space E =Z⁺ = {0,1,2,…} and the q-matrix Q = (q_ij,i,j∈E):Where: b_j≥0(j≥1), d > 0,ω_j > 0(j≥1) , 0 < b = (?) b_j < +∞The regularity and uniqueness criteria was obtained in [2],This article will discuss the duelity ,monotonicity and Feller property.The main results are as follows: Theorem 4.1.2 For a WB -q-matrix Q ,the minimal Q-function F(t) is monotone if and only if(1)ω_i(b-(?)b_k)≤ω_i+1(b-(?)b_k),i≥1,j≥1,where(?)b_k= 0 and(2)one of the following cases holds(a) d≥m_b(b) db<+∞and (?) =+∞(c ) m_b = +∞and (?) =+∞Theorem 4.1.3 For a WB -q-matrix Q,the minimal Q-function F(t) is dual (of some monotone one) if and only if(1) (?) and(2)one of the following cases holds(a) (?)=+∞(b) (?)<+∞,db≤+∞(C) (?)<+∞,∑_n=1^+∞R_n=+∞where R₀= 1;R_n = (?)+(?), (n≥1Ï„_n =(?)b_jTheorem 4.1.4 For a WB - q-matrix Q,let F(t) is the minimal Q-function,we have the following results :(1)If db< +∞, then F(t) is always Feller.(2)If d≥m_b, F(t) is Feller if and only if (?)R_n=+∞where R₀ = 1;R_n =(?)+(?)R_k-1(n≥1);Ï„_n= (?)b_jRemark If (?) = +∞,then F(t) is Feller.For general case,check (?)R_n=+∞may not simple.Since the sequence {R_n;n≥1} is given recursively.So we get a much better sufficient conditions as follows:Theorem 5.1.1 For a WB - q-matrix Q Jet q be the minimal solution of U(s) in [0,l],then(l)If (?), then Q is zero-entrance.(2)If(?), then Q is nonzero-entrance.(3)If (?) exits,and then ifω< (?), Q is zero-entrance; ifω> (?), Q isnonzero-entrance. Corollary5.1.2 For a WB -q-matrix Q satisfyingω_n≤ω_n+1 ,(n≥1), so (?) = w exists,q is the minimal solution of U(s) in [0,1] satisfying 0 < q < 1.(1) Suppose d≥m_b(a)Ifω< 1, then F(t) is dual, Feller and monotone.(b)Ifω> 1, then F(t) is monotone, but neither dual nor Feller.(2) Suppose d < m_b < +∞(a)Ifω<(?),then F(t) is dual and Feller, furthermore if (?) = +∞holds,then F(t) is monotone.(b)Ifω> (?), furthermoreif (?) < +∞, then F(t) is dual and Feller,if (?)= +∞, then F(t) is monotone.(3) Suppose m_b= +∞(a)Ifω< (?), F(t) is dual and Feller; Furthermore if (?) = +∞then F(t)is monotone.(b)If w > (?), furthermoreif either (?)< +∞or (?)=+∞and∑_n=1^+∞(?)<+∞then F(t) is dual and Feller,if (?)=+∞then F(t) is monotone.