Extinction behavior and recurrence of n -type Markov branching–immigration processes

In this paper, we consider n -type Markov branching–immigration processes. The uniqueness criterion is ﬁrst established. Then, we construct a related system of diﬀerential equations based on the branching property. Furthermore, the explicit expression of extinction probability and the mean extinction time are successfully obtained in the absorbing case by using the unique solution of the related system of diﬀerential equations and Kolmogorov forward equations. Finally, the recurrence and ergodicity criteria are given if the zero state 0 is not absorbing.


Introduction
Markov branching processes occupy a major niche in the theory and applications of probability.Good general references are Asmussen and Hering [2], Athreya and Jagers [3], Athreya and Ney [4] and Harris [7].Within the branching structure, both stateindependent and state-dependent immigration have been studied.For the former, Sevast'yanov [13] and Vatutin [14] and [15] considered a branching process with stateindependent immigration.Aksland [1] considered a modified birth-death process where the state-independent immigration is imposed.On the other hand, for the latter, Kulkarni and Pakes [8] discussed the total progeny of a branching process with state-dependent immigration.Foster [6] and Pakes [11] considered a discrete-time branching process with immigration at state 0. Yamazato [16] and Pakes and Tavaré [12] investigated the continuoustime version.
Let (Z t : t ≥ 0) denote an n-type Markov branching process (nTMBP) with per capita birth rate and offspring distribution of the type k particle being θ k > 0 and {p (k)  j : j ∈ Z n + } (k = 1, . . ., n), respectively, where Z n + = {j = (j 1 , . . ., j n ) : j 1 , . . ., j n ∈ Z + } with Z + = {0, 1, . ..}.In this paper, we mainly consider a modification (X t : t ≥ 0) of the nTMBP that allows it to be resurrected whenever it hits the zero state and allows immigration when it does not hit the zero state.(X t : t ≥ 0) is called an n-type Markov branching-immigration process (nTMBPI).In order to clearly describe the evolution of (nTMBPI), we adopt the following conventions throughout this paper.
(ii) Let α > 0 and {a j : j ∈ Z n ++ } be a discrete law.When the system is nonempty, then Poisson immigration events with parameter α may occur with random numbers of immigrates according to the law {a j : j ∈ Z n ++ }.Immigration is independent of particles in the system.
(iii) Let β ≥ 0 and {h j : j ∈ Z n ++ } be a discrete law.When the system is empty, then Poisson resurrection events with parameter h may occur with random numbers of immigrates according to the law {h j : j ∈ Z n ++ }.Resurrection, immigration, and particles in the system are independent of each other.
By the above description, (X t : t ≥ 0) is a Markov process satisfying the following conditions: (a) the state space is Z n + ; (b) its generator Q = (q ij : i, j ∈ Z n + ) satisfies otherwise.
(1.1) Remark 1.1 θ k , α, and β are viewed as "branching rate", "immigration rate", and "resurrection rate", respectively.The matrix Li and Chen [9] considered the one-type case.The aim of this paper is to consider the extinction behavior and recurrence property of n-type Markov branching-immigration processes.In contrast to the one-type cases, when a particle of one type in the system splits, the number of particles of different type may change.Therefore, the method used in the one-type case fails and some new approaches should be used in the current situation.In this paper, we find a new method to investigate the extinction behavior and recurrence property of the n-type Markov branching-immigration processes (see, Theorems 3.1 and 3.2).
The structure of this paper is as follows.Regularity and uniqueness criteria together with some preliminary results are first established in Sect. 2. In Sect.3, we concentrate on discussing the extinction behavior of the absorbing nTBIP (i.e., β = 0) and the explicit extinction probability is obtained.In Sect.4, the recurrence criterion is presented in the case β > 0.

Preliminaries and uniqueness
Since Q is determined by the sequences {p (i) j : j ∈ Z n + } (i = 1, . . ., n), {a j : j ∈ Z n ++ }, and {h j : j ∈ Z n ++ }, we define their generating functions as It is obvious that all the generating functions are well defined at least on [0, 1] n .We now investigate the properties of the generating functions {B i (u); i = 1, . . ., n}, α(u), and β(u).Let where u ∈ [0, 1] n and δ ij is the Dirac function.The matrices (B ij (u)) and (g ij (u)) are denoted by B(u) and G(u), respectively.
Definition 2. 1 The system {B i (u) : 1 ≤ i ≤ n} is called singular if there exists an n × n matrix M such that where u denotes the transpose of the vector u.
Definition 2.2 A nonnegative n × n matrix A = (a ij ) is called positively regular if there exists an integer N > 0, such that A N > 0.
+ ) be the Feller minimal Q-function and Q-resolvent, respectively.

Lemma 2.3 For any
where F i (t, u) = j∈Z n + p ij (t)u j , or in the resolvent version ) Proof By the Kolmogorov forward equations, we have that for any i, j ∈ Z n + , Multiplying by u j on both sides of the above equality and summing over j ∈ Z n + we immediately obtain (2.2).Taking a Laplace transform on (2.2) immediately yields (2.3).
Theorem 2.1 Let Q be given in (1.1).Then, there exists exactly one nTMBPI, i.e., the Feller minimal process.
Proof By Lemma 2.4, We only need to consider the case that ρ(1) > 0. For this purpose, we will show that the equations have only trivial solution.Suppose that the contrary is true and let η = (η j : j ∈ Z n + ) be a nontrivial solution of (2.4) corresponding to λ = 1.Then, by (2.4) we have Multiplying by u j on both sides of (2.5) and using some algebra yields that If ρ(1) > 0, then by Lemma 2.2 and the irreducibility of Z n + \ 0 we know that (2.1) has a solution (q 1 , . . ., q n ) ∈ (0, 1) n .Let u = (q 1 , . . ., q n ) in (2.6), we can see that the right-hand side of (2.6) is zero.Therefore, the left-hand side of (2.6) must be zero, which implies that η j = 0 (∀j ∈ Z n + ).The proof is complete.

Extinction
In this section, we shall discuss the extinction property of the absorbing nTMBPI (i.e., β = 0).Let Q denote the absorbing nTBI Q-matrix and P(t) = (p ij (t) : i, j ∈ Z n + ) denote the Feller minimal Q-function.Also, let a i0 = lim t→∞ pi0 (t) be the extinction probability of P(t) starting at state i.In order to discuss the extinction property, we need the following important result, which plays a key role in our discussion.
has the same solution as (3.1).
Before stating our main result in this section, we first provide two useful lemmas.
and thus Moreover, for any i ∈ Z n ++ and u ∈ [0, 1) n , we have Proof By the construction of Q, all the states in Z n ++ are transient.Hence, (3.4) and (3.5) hold.
Here, the last equality follows from the integration by parts.Hence, (x * j : j = 0) is a solution of the equation By Lemma 3.2 in Li and Chen [9], we then have a i0 ≤ x * i (i = 0) since a i0 is the minimal solution of the above equation.(ii) is proved.
Thus, (3.13) is proved.Finally, it is fairly easy to show that the expression in (3.13) is finite if and only if (3.12) holds.

Recurrence Property
In this section we consider the recurrence property of nTMBPI in the case that β = 0 and thus 0 is no longer an absorbing state.We shall assume that the nTBI Q-matrix Q is regular.
It is well known that the nTMBPI is recurrent if and only if the extinction probability of the related absorbing nTMBPI (i.e., β = 0) equals 1.Therefore, by Theorem 3.2 we have the following result.Moreover, if ρ(1) < 0 and n j=1 (I j (1) + R j (1)) < ∞, then the process is exponentially ergodic.
Suppose that ρ(1) ≤ 0 and (4.1) holds.By Chen [5], in order to prove the positive recurrence, we only need to show that the equation then 0 ≤ y j < ∞ (j ∈ Z n + ) and it can be checked that j∈Z n + q ij y j = -1 (i = 0) and j =0 q 0j y j ≤ e Therefore, the nTMBPI is positive recurrent.Conversely, suppose that the process is positive recurrent and thus possesses an equilibrium distribution (π j : j ∈ Z n + ).Letting t → ∞ in (2.2) and using the dominated convergence theorem yields R(s)π 0 + Ĩ(s)  It is easy to see that ρ(1, 1) = 1 -2p.Moreover, the solution of (3.1) is v(u) = u and the smallest nonnegative solution of (2.1) is q 1 = q 2 = min(1, p 1-p ).(i) For the case β = 0, by Theorem 3.1, (ii) For the case β > 0, by Theorem 4.2, the process is positive recurrent if and only if p > 1  2 .