Continuous-time Dual Processes Of Markov Branching Processes And Their Generalization

In the study of theories of Markov processes, there traditionally are two methods: the probabilistic method and the analytical method. Recently mathematician investigate theories of Markov processes using the analytical method to deal with Markov branching processes (briefly, MBP) and dual generalised Markov branching processes (briefly, DGMBP), and obtain many results. In this paper, we also use the analytical method to discuss the dual Markov branching processes.It is well-known that MBPs occupy a major nicle in the theory and applications of stochastic processes. Indeed, their importance can hardly be overestimated. From Harris(1963), Athreya and Ney(1972), and Asmussen and Hering(1983), we know that an ordinary (one dimension) Markov branching process is a continuous-time Markov chains on the state space E = Z₊ = {0, 1, 2,…} whose development mechanism is governed by the independent property, namely, different particles act independently when give birth (or death). Moreover, by A. Chen[5] and Y. Li[6], we have the q-matrix _bQ of MBP is not always conservative. When the _bQ is conservative, then _bQ is FRR, monotone and regular, also its transition function _bF(t) is FRR and stochastically monotone. However. when _bQ is not conservative, _bQ also has some good properties such as regular, monotone and so on. From Siegmund's theorem, we obtain that _bF(t) must be a dual of some monotone process, i. e, dual Markov branching process(DMBP). In eharpter 2, we'll give the existence and definitions of DMBP. In [8], Y. Li has given the relationship among these definitions. At last in chapter 3 and chapter 4, we give the critia of recurrence, ergodicity and strong ergodicity for DMBP. The results are as follows:Theorem 2. 1. 1 Let Q be the q-matrix of DBMP and F(t) be its minimal Q-function, then(1) Q is stochastically monotone and it is also dual;(2) Q is regular;(3) F(t) is stochastically monotone, and its dual process is the Markov branching process; (4) F(t) satisfies the dual branching property where f_1.0(t) = 1, f_{-1, k}(t) = 0(k≠0).Theorem 2. 1. 2 Assume lim_{k→∞} a_k = 0. The generating function which derived by {a_j-1 - a_j, j≥1} isthen(1) If A＜+∞, the F(t) is FRR.(2) If A = +∞, then F(t) is FRR if and only if, for some and therefor all∈₁∈(q1, 1), where q1 is the smallest root of A(s) = 0.Theorem 3. 1. 1 If -a1/a0≥2, then DMBP is recurrent; and if-a1/a0＜2, then DBMP is transient.Theorem 4. 1. 1 (1) If lim_{k→∞}ak = a＞0, then F(t) is strongly ergodic;(2) If A＜+∞, then F(t) is not strongly ergodic;(3) If A = +∞, then F(t) is strongly ergodic if and only if, for some and therefor all∈₁∈(q₁, 1), such thatHowever, in most realistic situations, the above independence property is unlikely to be appropriate. In particular, if the particles become larger or the particles move with high speed, the particles may interact and, as a result, the birth and death rates may change. In order to model such behaviour, it seems appropriate to replace the branching rates {ib_j-i+1} by the more general tbrm of {i^θb_j-i+1}. Here we generalize the DMBP to DGMBP and obtain the basic properties in chapter 2. Also, the existence and their definitions are given. At last in chapter 3 and chapter 4, we give the critia of recurrence, ergodicity and strong ergodicity for DGMBP. The results are as follows:Theorem 2. 2. 3 For a given GMB transition function _bP(t), ifθ≤1 orθ＞1 and m₁＞0, there always exists another transition function P(t) satisfying whereTheorem 2.2.4 Let P(t) be a transition function on E, then the following two statements are equivalent:(1) ifθ＜1 or 0＞1 and m₁＞0, there exists a GB transition function _bP(t) satisfying(2) the q-matrix Q = P'(O) of P(t) takes the form In this case, the sequences {a_k}, {b_k}, which appeared in Q and _bQ have the following relationship: Theorem 3.1.2 The DGMBP is recurrent if and only if where R_n is defined recursively by R₀ = 1 andTheorem 4.1.2 The DGMBP is ergodic (or positive recurrent) if and only if 0＜γ≤+∞, whereγ: =∑_{k≥1 ak/|a0}. Moreover, the ergodic limits are given bywhere q is the smallest root of the equation B(s) = O,s∈[O, 1],and q < 1 if and only ifγ＞1.Theorem 4.2.3 Let P(t) be a DGMB transition function with q-matrix Q satisfying (2.10-11),(1) if lim_{k-→∞} a_k = a＞0, then P(t) is always strong ergodicity.(2) if limk→∞a_k = 0 and 1≤γ≤+∞, then the following statements are equivalent:(â…°) the DGMBP is strongly ergodic;(â…±) there exists some e with q＜e＜1 such that the integral (â…²) the following integral convergent, i. e.,whereγ=∑_k≥1 a_k/|a₀|, B(s) andξ(with, 0＜ξ＜1) are the same as defined in (2.8).(â…³) for all i≥1, we have...