The Properties of Generalized Collision Branching Processes

We consider basic properties regarding uniqueness, extinction, and explosivity for the Generalized Collision Branching Processes (GCBP). Firstly, we investigate some important properties of the generating functions for GCB q -matrix in detail. Then for any given GCB q -matrix, we prove that there always exists exactly one GCBP. Next, we devote to the study of extinction behavior and hitting times. Some elegant and important results regarding extinction probabilities, the mean extinction times, and the conditional mean extinction times are presented. Moreover, the explosivity is also investigated and an explicit expression for mean explosion time is established.


Introduction
In this paper, we mainly consider extinction and explosivity for the Generalized Collision Branching Processes (GCBP). e particles in the system that evolves can be described as follows. Collisions between particles occur at random, and whenever m particles collide, they are removed and replaced by j "offsprings" with probability p j (j ≥ 0), independently of other collisions. In any small time interval (t, t + Δt), there is a positive probability θΔt + o(Δt) that a collision occurs, and the chance of 2 or more collisions occurring in that time interval is o(Δt).
Assume that there are i particles present at time t and all interactions are equally likely. en, there will be j particles with probability i m θp j− i+m Δt + o(Δt) after time Δt. In this paper, we take X(t) be the number of particles present at time t and therefore X(t) to be a continuous-time Markov chain with nonzero transition rates q ij � i m b j− i+m ,j ≥ i − m, i≥ m, where b m �− θ(1− p m ) and b j � θp j for j≠ m.
is leads us to the following formal definition.

Definition 1.
A q-matrix Q � (q ij ; i, j ∈ Z + ) is called a generalized collision branching q-matrix (henceforth referred to as a GCB q-matrix) if it takes the following form: together with b k > 0(k � 0, 1, . . ., m − 1). e conditions b 0 > 0 and ∞ j�m+1 b j > 0 are essential, while condition b k > 0(k � 1, . . ., m − 1) is imposed for convenience; all our conclusions hold true with some minor and obvious adjustments if this latter condition is removed.
Guided by this fact, we formally define this generalized collision branching process as follows.

Definition 2.
A generalized collision branching process (henceforth referred to simply as a GCBP) is a continuous-time Markov chain, taking values in Z + , whose transition function P(t) � (p ij (t); i, j ∈ Z + ) satisfies the forward equation where Q is a GCB q-matrix as defined in (1) and (2). In order to avoid discussing some trivial cases, we shall assume that Z + is an irreducible class for our q-matrix Q as well as for the corresponding Feller minimal Q-function throughout this paper excepting where we consider the absorbing case.
e more general jump rates will be discussed in subsequence papers. e structure of this paper is as follows. Some preliminary results are obtained in Section 2. In Section 3, we show that there always exists exactly one GCBP for a given GCB q-matrix Q. And then the extinction behavior and hitting times are considered in the Section 4, where some elegant and important results regarding extinction probabilities and mean extinction times and explosion times are obtained.

Preliminaries
In order to investigate properties of GCBPs, we introduce the generating function B(s) of the sequence {b k ; k ≥ 0} in (1) and (2) as e function plays an extremely important role in the following discussion. It is easy to see that It is clear that B′(1) > − ∞. Moreover, the number of solutions to equation B(s) � 0 in s ∈ [0, 1) is determined by the sign of B′(1), and we will give the simple results in the following. However, their proofs are obvious and thus omitted in this paper.

Lemma 2. Suppose that Q is a GCB q-matrix as defined in
or equivalently where Proof. By the Kolmogorov forward equation (3), for any i, j ≥ 0, Multiplying s j on both sides of the above equality and summing over j ∈ Z + , we immediately obtain (6). Finally, (7) is the Laplace transform of (6).

Uniqueness
In this section, we mainly consider the uniqueness of GCBPs.
be the Feller minimal Q-function and Q-resolvent, respectively, where Q is a GCB q-matrix. en for any i ≥ m and |s| < 1, we have Proof. It is easily seen that all states G � {m, m + 1, . . ., } are transient, and thus, (i) follows. is simple fact can also be easily obtained analytically. Indeed, by Kolmogorov forward equation, we have which implies that We now prove (9). Firstly, we know that the Feller minimal Q-resolvent can be obtained by the following (Laplace transform version) forward integral iteration: and that ϕ (n) ij (λ) ↑ ϕ ij (λ) as n ⟶ ∞ for all i, j ∈ E. Now, we consider our GCB q-matrix Q on Z + , and we still denote ϕ (n) ij (λ), i, j ∈ Z + to be the corresponding Feller minimal resolvent. Firstly, we claim that for any n ≥ 0, For j < m, (15) is trivially true, so we assume j ≥ m. We use mathematically induction on n to prove the conclusion. Obviously, it is true for n � 0.
Next, by (14), we can easily get that Define Applying the notation, (16) can be rewritten as By (14), It follows from the above two expressions that and so (15) follows from the induction principle. Also, letting s ↑ 1 in (20) yields that However, it is easily seen that and thus, by (18), we have It follows from the Dominated Convergence eorem and (20) yields that for 0 < s < 1, Letting n ⟶ ∞ in (18) and applying the above equality leads to the conclusion that for 0 < s < 1, However, for all 0 < 1 − ε ≤ s < 1, we may find an ε > 0 such that B(s) ≠ 0. us, Mathematical Problems in Engineering Applying the Monotone Convergence eorem and Dominated Convergence eorem yields It easy to see that the above equality holds for all 0 < s < 1. us, (12) yields from (25). Moreover, (10) is the Laplace transform of (12), which implies that (10) holds for almost all t ≥ 0. Furthermore, note that the left-hand side of (11) is a continuous function of t > 0; thus, (10) holds for all t ≥ 0.
Proof. Firstly, we suppose that B′(1) ≤ 0 and let P(t) � {p ij (t), i, j ≥ 0} be the minimal Q-transition function. Substituting (1) into (3) gives It easily yields that for 0 ≤ s < 1, the right-hand side being strictly positive for s ∈ (0, 1) follows from the Lemma 1. Moreover, it is easy to dictate that for all t ≥ 0, where q i : � − q ii � i m b m < ∞. erefore, the series ∞ j�0 p ij ′ (t)s j converges uniformly on [0, ∞) for every s ∈ [0, 1), and since the derivatives p ij ′ (t) are all continuous, the derivative of ∞ j�0 p ij (t)s j exists and equals ∞ j�0 p ij (t)s j . us, we may integrate (29) to obtain Letting s ↑ 1 in (31) yields ∞ j�0 p ij (t) ≥ 1, which implies that the equality holds for all i ≥ 0. erefore, the minimal Qtransition function is honest, and hence, Q is regular.
Conversely, by the eorem 3.6 of Li and Chen [9], it is easy to obtain the conclusion since ∞ k� m 1/ k m < ∞. e proof is complete. By eorem 1, we can see that if B′(1) ≤ 0, then the GCBP is regular. In the sequel, we will prove that for any given GCB q-matrix Q, there always exists exactly one Q-process satisfying the Kolmogorov forward equation (3).
Proof. It follows from eorem 1, we only need to consider the case 0 < B′(1) ≤ +∞. In order to prove the uniqueness of the GCBP, we will verify Reuter's condition, i.e., we need to prove that the equation has only the trivial solution, and then cover all λ > 0. Let Y � (y i ; i ≥ 0) be a nontrivial solution corresponding to λ � 1, then y 0 > 0 and by (32), It is clear that the nontriviality of the solution η implies that ∞ j�0 η j is well defined for all s ∈ [0, 1] since (34) holds, which implies that because it follows from the root test, these series have the same radius of convergence. Applying Fubini's theorem together with (33) and (36) yields that

Extinction and Explosion
From the previous section, we have obtained that the GCBP is uniquely determined by its q-matrix, so we will examine some of its properties in this section. Let {X(t), t ≥ 0} be the unique GCBP, and denote P(t) � {p ij (t), i, j ≥ 0} be its transition function. Define the extinction times τ k for k � 0, 1, . . ., m − 1 as and denote the corresponding extinction probabilities by 4 Mathematical Problems in Engineering and the overall extinction probability by a k � P(τ < ∞ | X(0) � i) � m− 1 k�0 a ik . Also let E i (·) denote the expectation conditional on X(0) � i.
We now prove (42). It follows from Lemma 1 that we have q < 1 since 0 < B′(1) ≤ ∞. Putting s � q in (11) and noting that B(q) � 0, we discover that ∞ j�0 p ij ′ (t)q j � 0 for any t > 0, implying that ∞ j�0 t 0 p ij ′ (u)du · q j � 0. us, for any t > 0, Letting t ⟶ ∞, we have Noting that all of the limits exist, we may apply the Dominated Convergence eorem in the last term on the left-hand side to obtain (42) since q < 1.
By eorem 3, we know that the process is absorbed with probability less than 1 if 0 < B′(1) ≤ +∞. Our next result establishes that the process must explode if absorption does not occur in such cases.

Theorem 4. For the Feller minimal GCBP,
(1 − y) m− 1 a i0 + a i1 y + · · · + a im− 1 y m− 1 − y i B(y) dy. (45) Proof. It follows from (10), for all s ∈ [0, 1), we have i.e., e apparent singularity at s � q on the left-hand side is removable, because the series on the right-hand side certainly converges for all s ∈ [0, 1). Moreover, the left-hand side is continuous and strictly positive (indeed increasing) on this interval. erefore, integrating (48) with respect to s iteration m times and applying Fubini's theorem yields that for any s ∈ [0, 1), Letting s ↑ 1 in (49), we can get that the equality (49) also holds for s � 1, and en the proof is complete if (46) holds since Lemma 4. Let (p ij (t), i, j ∈ Z + ) and (ϕ ij (λ), i, j ∈ Z + ) be the Feller minimal Q-function and Q-resolvent where Q is a GCB q-matrix.
(i) For any i, k ≥ m, and hence, considering the integrand is nonnegative, we obtain that Proof. By (10), we have Letting t ⟶ ∞ in the equality (55) for s ∈ (− 1, 1), applying the Dominated Convergence eorem on the lefthand side and the Monotone Convergence eorem on the right-hand side, we obtain (53) by the uniqueness of the Taylor expansion. Furthermore, (53) implies (54) is trivial, and hence, the proof is complete.
Proof. It is easily seen from eroem 3 and Lemma 1 that if 0 < B′(1) ≤ ∞, then m− 1 k�0 a ik < 1 which implies E i (τ) � +∞, so let us assume that B′(1) ≤ 0. For these latter cases, it follows from (55) and applying the Monotone Convergence eorem yields us, the proof is complete.
It is easily seen that E i (τ k ) � +∞(i ≥ m, k � 0, 1, . . ., m − 1) when the extinction is not certain. Under these circumstances, it is natural to consider the conditional expected extinction times, given by where μ ik (k ≤ m − 1) satisfy the linear equations Proof. First we consider the case 0 < B′(1) ≤ +∞, and thus, 0 < q 0 < 1, and |q i | < 1 for j � 1, . . ., m − 1, applying the eorem 3 together with k�0 p ik (t)q k � q i yields the expression On integrating (60) and using Noting that |q j | < 1 for j � 1, . . ., m − 1, letting t ⟶ ∞ and applying the monotone convergence theorem yields On the other hand, by the definition of τ, , and then all of the conclusions follow since |q i | < 1 for j � 1, . . ., m − 1.
From now on, we will consider the explosion probabilities and expected explosion times. By eorem 1, we only need to consider the case that 0 < B′(1) ≤ ∞. Denote τ ∞ be the explosion time and let a i∞ � P(τ ∞ | X(0) � i) be the probability of explosion starting in state i. Since we are aiming at the minimal process, p i∞ (t): � 1 − ∞ j�0 p ij (t) � P(τ ∞ ≤ t | X(0) � i) is the probability of explosion by time t starting in state i, and p i∞ (t) ⟶ a i∞ as t ⟶ ∞.
Finally, we consider the time spent in each state over the lifetime of the process. Let T k be the total time spent in state k(k ≥ m) and let μ ik � E i (T k )(i ≥ m). en, is quantity was evaluated in (29). We have therefore the following result.
Theorem 8. All of μ ik (i ≥ m, k ≥ m) are finite and given by

Data Availability
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Conflicts of Interest
e authors declare that they have no conflicts of interest.