Continuous-time Dual Weighted Markov Branching Processes

Over the years, with many mathematicians' study, the theory of Markov processes have become more and more complete. In the long-term course of the study, scientists have gotten not only a great deal of useful results but also varied research methods. This paper defines a new class of Markov processes - dual weighted Markov branching process (referred to as the dual weighted branch process), and focus on discussing some basic properties of this process with the analytical method.As we all know, the theory of branching processes and its application in stochasticprocesses play an important role. Known from Harris (1963), Athreya and Ney (1972), Asmusse and Hering (1983) , an ordinary (one-dimensional) Markov branching process is a continuous-time Markov chain in the state space Z₊ , whose development mechanism is governed by the independent property, called branching property, that is particles act independently when given birth or death. However, in most realistic situations, the above independence nature is not so applicable. Especially in the actual operation, birth and death interact each other. This is why people are always with great interest and effort to study more general branching models. In particular, the literature [2] defined the weighted branching processes, which is a branch of generalized branching processes.This article defines a now branching process-dual weighted branching process based on theories of weighted branching one. Subsequently, we study such process's characterization, as existence, uniqueness, regularity Feller, recurrence and strong er-godicity. The important results as follows:Definition 2.1.1 A q-matrix Q = (q_ij;i,j∈E) is called a dual weighted Markov branching q-matrix(henceforth referred to as a dual WB-q-matrix) ifwhere 0≤a_k+1≤a_k <∞(k≥1), lim_{i→∞}a_i = 0, d > 0, and 0 =ω₀ <ω_j≤ω_j+1(j≥0)Definition 2.1.2 A dual weighted Maokov branching processes(D-WMBPs) is a E-valued continuous time Markov chain whose transition funtion P(t) = (P_ij(t);i,j∈E) satisfies the Kolmogorov forward equationswhere Q is a dual WB-q-matrix as in (1).Theorem 3.1.2 Define m := (?)k(a_k-a_k+1 ) and generating function B(s) =d - (a₁ + d)s +(?) (a_k -a_k+10s^k+1 by {a_j - a_j+1;j≥1}. Let q is the minimal root of B(s) = 0 in [0,l],then(1) If (?) =∞, Q is regular;(2) If (?) <∞, m > d and (?), Q is regular;(3) If (?)<∞, m < d and (?) < 1, Q is regular,where (?) = lim_{n→∞}, sup(?)Theorem 3.2.2 Let F(t) be the minimal Q-function of dual weighted branching q-matrix, then(1) If a_∞= 0:(i) (?)=∞and m <∞,F(t) is dual;(ii)(?)<∞,F(t) is dual if and only if m≤d.(2)If a_∞> 0, F(t) is not dual.Theorem 3.3.1 Let F(t) be the minimal Q-function of dual weighted branching q-matrix, then F(t) is the only one if(1)∑∞_n=1(?) =∞and m <∞; or(2) (?)<∞. m≤d and (?)< 1. Theorem 4.1.1 If dual weighted branching q-matrix is regular, then dual weighted branching function is recurrence if and only ifwhere H_n is defined as H₀ = 1 andTheorem 4.1.2 If (?) =∞d∞> 0 . F(t) is strong ergodicity, ifTheorem 5.1.3 Let Q be the dual weughted branching q-matrix, F(t) is strong ergodcity if a = 0 and...