Nonlinear Branching Processes with Immigration

Transcription

1 Acta Mathematica Sinica, English Series Aug., 217, Vol. 33, o. 8, pp Published online: April 24, 217 DOI: 1.17/s Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 217 onlinear Branching Processes with Immigration Pei-Sen LI School of Mathematical Sciences, Beijing ormal University, Beijing 1875, P. R. China peisenli@mail.bnu.edu.cn Abstract The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established. Keywords onlinear branching process, immigration, stochastic integral equation, regularity, recurrence, ergodicity, strong ergodicity MR21 Subject Classification 6J27, 6J8 1 Introduction Markov branching processes are models for the evolution of populations of particles. Those processes constitute one of the most important subclasses of continuous-time Markov chains. Standard references on those processes are [9] and [2]. The basic property of an ordinary linear branching process is that different particles act independently when giving birth or death. In most realistic situations, however, this property is unlikely to be appropriate. In particular, when the number of particles becomes large or the particles move with high speed, the particles may interact and, as a result, the birth and death rates can either increase or decrease. Those considerations have motivated the study of nonlinear branching processes. On the other hand, a branching process describes a population evolving randomly in an isolated environment. A useful and realistic modification of the model is the addition of new particles from outside sources. This consideration has provided the stimulation for the study of branching models with immigration and/or resurrection. Let {r i : i } be a sequence of nonnegative constants with r =and{b i : i } a discrete probability distribution on := {, 1,...} with b 1 =. A continuous-time Markov chain is called a nonlinear branching process if it has density matrix R =r ij givenby r i b j i1, j i 1, i 1, r i, j = i 1, r ij = 1.1 r i b, j = i 1, i 1,, otherwise. A typical special case is where r i = αi θ for α andθ>, which reduces to the ordinary linear branching process when r i = αi. Let γ andlet{a i : i } be another discrete Received October 8, 216, revised December 22, 216, accepted January 17, 217 Supported by SFC Grant os , and

2 122 Li P. S. probability distribution on satisfying a =. A continuous-time Markov chain is called a nonlinear branching process with resurrection if its density matrix is given by r i b j i1, j i 1, i 1, r i, j = i 1, r i b, j = i 1, i 1, ρ ij = 1.2 γa j, j > i =, γ, j = i =,, otherwise. Here the resurrection means that at each time when the process gets extinct, some immigrants come into the population at rate γ according to the distribution {a i }. By a nonlinear branching process with immigration we mean a Markov chain with density matrix Q =q ij givenby r i b j i1 γa j i, j i 1, i, r i γ, j = i, q ij = 1.3 r i b, j = i 1, i 1,, otherwise. In this model, the immigrants come at rate γ according to the distribution {a i } independently of the inner population. The purpose of this paper is to investigate the construction and basic properties of the nonlinear branching process with immigration defined by 1.3. Let m = ja j, M = jb j, j= which represent the birth mean and immigration mean of the process, respectively. Moreover, we introduce the functions F s = a i s i, As =γ1 F s, Gs = i= j= b i s i, Bs =Gs s, s [, 1]. i= Let q be the smaller root of the equation Gs =s in [, 1]. We sometimes denote r i by ri for notational convenience. Suppose that Ω, F, F t,p is a probability space satisfying the usual hypotheses. Denote mi =b i and ni =a i for each i. Let {pt} and {qt} be F t -Poisson point processes with characteristic measures dumdz andγndz, respectively. We assume {pt} and {qt} are independent of each other. Let p ds, du, dz and q ds, dz be the Poisson random measures associated with {pt} and {qt}, respectively. Given an -valued F -measurable random variable X, let us consider the stochastic integral equation rxs X t = X z 1 p ds, du, dz z q ds, dz. 1.4

3 onlinear Branching Processes 123 Let ζ = lim k τ k, where τ k =inf{t :X t k}. The above equation only makes sense for t<ζ.we call ζ the explosion time of {X t } and make the convention X t = for t ζ. We say the solution is non-explosive if ζ =. As a special case of 1.4 we also consider the equation rxs X t = X z 1 p ds, du, dz. 1.5 We now state the main results of the paper. Theorem 1.1 There exists a pathwise unique solution to 1.4. Moreover, if the solution to 1.5 is non-explosive, then so is the solution to 1.4. Theorem 1.2 Let {X t } be the solution to 1.4 and let Q ij t =P X t = j X = i. Then Q ij t solves the Kolmogorov forward equation of Q. Theorem 1.3 The solution to 1.4 is the minimal process of Q and the solution to 1.5 is the minimal process of R. Theorem 1.4 The density matrix R is regular if and only if Q is regular. Theorem If M 1, then Q is regular. 2 Suppose that i=1 r 1 i <. ThenQ is regular if and only if M 1. 3 Suppose that 1 <M and r i = αi θ for α> and θ>. Then Q is regular if and only if for some ε q, 1, we have 1 ln 1 θ 1 ds =. ε Bs s In the following three theorems, we assume γr i b > for every i 1, so the matrix Q is irreducible. Theorem Suppose that m<, M < 1 and lim i r i =. Then the nonlinear branching process with immigration is recurrence. 2 Suppose that r i is increasing and there exist constants α>and > such that r i /i α holds for each i>. Then the nonlinear branching process with immigration is recurrent if M 1 and [ 1 y ] J := αby exp Ax αbx dx dy =. 3 Suppose that M>1. Then the nonlinear branching process with immigration is transient. 4 Suppose that r i is increasing and there exist constants α>and > such that r i /i α holds for each i>. Then the nonlinear branching process with immigration is transient if M 1 and [ 1 y ] J := αby exp Ax αbx dx dy <. Theorem If m<, M 1, r i is increasing and i=1 r 1 i <, then the nonlinear branching process with immigration is ergodic.

4 124 Li P. S. 2 Suppose that r i = αi θ for α> and θ 1. Then the recurrent nonlinear branching process with immigration is ergodic if and only if As ln 1 θ 1 ds <. 1.6 αbs s 3 If m<,m < 1 and lim inf i r i /i >, then the nonlinear branching process with immigration is exponentially ergodic. Theorem If m<, M<1, r i is increasing and i=1 r 1 i <, then the process is strongly ergodic. 2 Suppose that r i = αi θ for α> and θ>1. Then the nonlinear branching process with immigration is strongly ergodic if and only if 1 ln 1 θ 1 ds <. 1.7 αbs s 3 If i=1 r 1 i =, then the nonlinear branching process with immigration is not strongly ergodic. The nonlinear branching process with resurrection defined above was introduced by [8], who studied the problems of uniqueness, recurrence and ergodicity of the process. The model has attracted the attention of a number of authors. In particular, [16] gave criteria for strong ergodicity of the process. [4] and [14] established some criteria for their regularity and uniqueness. [3] studied some interesting differential-integral equations associated with a special class of nonlinear branching processes and gave some characterizations of their mean extinction times. [5] established a Harris regularity criterion for such processes. The existence and uniqueness of linear branching processes with instantaneous resurrection were studied in [6]. However, most of the study of models with immigration have been focused on linear branching structures. The branching process with immigration was studied in [11], who gave a characterization of the onedimensional marginal distributions of the process starting from zero. An ergodicity criterion for the process was given in [12, 15] established some recurrence criteria for linear branching processes with immigration and resurrection. The first three theorems above give constructions of nonlinear branching processes with and without immigration. These provide convenient formulations of the processes. In particular, the result of Theorem 1.4 is derived as an immediate consequence of 1.4 and 1.5. We hope the equations can also be useful in some other similar situations. The proof of Theorem 1.5 is based on Theorem 1.4 and the results of [8] and [5]. The study of recurrence of the immigration model is more delicate since the problem cannot be reduced to the extinction problem of the original nonlinear branching process as in the case of a resurrection model. Theorem 1.6 was proved by using the results of the minimal nonnegative solutions as developed in [7] and comparing the process with some linear branching processes which was studied by [12]. The proofs of the ergodicities in Theorems 1.7 and 1.8 are based on comparisons of the process with some suitably designed birth-death process and estimates of the mean extinction time.

5 onlinear Branching Processes Stochastic Integral Equations Stochastic integral equations with jumps have been playing increasingly important roles in the study of Markov processes. In this section, we give a construction of the solution to 1.4 and prove the solution is a minimal nonlinear branching process with immigration. This result is then used to study the regularity of the density matrix Q. We refer to [1] for the general theory of stochastic equations with jumps. Proposition 2.1 The pathwise uniqueness of solutions holds for the equation 1.4. Proof Let {X t } and {X t} be any two solutions of equation 1.4 with X = X. By passing to the conditional probability P F, we may and do assume X = X is deterministic. Let τ m =inf{t :X t m}, τ m =inf{t :X t m} and σ m = τ m τ m. It is sufficient to show that τ m = τ m = σ m and X t = X t for all t σ m m =1, 2,... Then σm X t σm X t σ m = z 1[1 {<u rxs } m1 1 {<u rx s }] p ds, du, dz, where m = {, 1, 2,...,m}. Taking the expectation, we get E[ X t σm X t σ m ] { σm } E z 1[1 {<u rxs } 1 {<u rx s }] p ds, du, dz m1 { σm } E z 1 1 {<u rxs } 1 {<u rx s } dsdumdz m1 { σm } M m1 1E rx s rx s ds M m1 1 E[ rx s σm rx s σ m ]ds, where M m := m zmdz. By taking m X, we have X s X s m for <s σ m. Denote d m =sup{ ri rj/i j : i j, i, j m}. Then we have E[ X t σm X t σ m ] M m1 1d m E[ X s σm X s σ m ]ds. 2.1 Since X s σm and X s σ m only have countably many discontinuous points, we can also use X s σm and X s σ m instead of X s σm and X s σ m in the right hand side of 2.1. Using Gronwall s inequality we have E[ X s σm X s σ m ] =. Thus we can conclude that X t = X t for all t [,σ m a.s. This clearly implies that τ m = τ m = σ m a.s. and the pathwise uniqueness of solutions of 1.4 is proven. Theorem 2.2 For any -valued F -measurable random variable X, there is a pathwise unique solution to 1.5. Proof Without loss of generality, we assume X is deterministic. Let D 1 = {s : ps,rx ] }. Since rx E[ p,t],rx ] ] = ds du mdz =trx <,

6 126 Li P. S. the set D 1 is discrete in,. Let σ 1 be the minimal element in D 1 and pσ 1 =u 1,z 1. Then set X, t [,σ 1, X t = X z 1 1, t = σ 1. The process {X t :<t σ 1 } is clearly the solution of 1.5. Set D 2 = {s : ps σ 1 [,rxσ 1 ] }, σ 2 be the minimal element in D 2 and pσ 1 σ 2 =u 2,z 2. Define {X t : σ 1 <t σ 1 σ 2 } by xσ 1, t σ 1,σ 1 σ 2, X t = xσ 1 z 2 1, t = σ 1 σ 2. It is easy to see that {X t :<t σ 1 σ 2 } is the unique solution of 1.5. Continuing this process successively, we get a process {X t : t<τ}, whereτ = i=1 σ i. ext, we show τ = ζ := lim k τ k, where τ k =inf{t :X t k}. Clearly, for each n wehavex t < for t [, n i= σ i]. Then n i= σ i <ζholds for each n, and so τ ζ. On the other hand, since [ τm E rxs ] p ds, du, dz t max rk <, k m the process {X t } has finitely many jumps before t τ m, therefore t τ m <τ,since t and m 1 can be arbitrary, we get ζ τ. Then we have τ = ζ. Hence X t is determined in the time interval [,ζ; the uniqueness is clear from Proposition 2.1. Proof of Theorem 1.1 Let {Xt } denote the solution to 1.5. Let {v k : k =1, 2,...} be the set of jump times of the Poisson process t q ds, dz. We have clearly v k as k. For t<v 1 set X t = Xt. Suppose that X t has been defined for t<v k and let ξ = X vk z q ds, dz. {v k } Hereandinthesequelwemaketheconvention =. By the assumption there is also asolution{xt k } to rxs X t = ξ z 1 p v k ds, du, dz. Let η k be the explosion time of {Xt k }. Ifv k η k >v k1, we define X t = Xt v k k for v k t<v k1. If v k η k v k1,wesetx t = Xt v k k for v k t<v k η k and X t = for v k η k t<v k1. By induction that defines a process {X t }, which is clearly the pathwise unique solution to 1.4. Obviously, if the solution of 1.5 is non-explosive for each deterministic initial state X = i, we have η k = for all k, andso{x t } is non-explosive. Proof of Theorem 1.2 Let Ñpds, du, dz = p ds, du, dz dsdumdz andñqds, dz = Ñ q ds, du, dz dsdundz. For any bounded function f on, wehave τm rxs fx t τm =fx [fx s z 1 fx s ] p ds, du, dz

7 onlinear Branching Processes 127 where τm [fx s z fx s ] q ds, dz τm rxs = fx [fx s z 1 fx s ]dsdumdz τm [fx s z fx s ]γdsndzm t f, 2.2 M t f := τm rxs τm [fx s z 1 fx s ]Ñpds, du, dz [fx s z fx s ]Ñqds, dz is a martingale. Since X s X s for at most countably many s, we can also use X s instead of X s in the right hand side of 2.2. In particular, for f =1 {j},wehave τm 1 {Xt τ m =j} =1 {X =j} b k rx s [1 {Xs k 1=j} 1 {Xs =j}]ds k= τm γa k [1 {Xs k=j} 1 {Xs =j}]ds M t 1 {j}. Write E i = E X = i fori. Taking the expectation in both sides of the above equation and letting m,weget ζ E i 1 {Xt ζ =j} =E i 1 {X =j} b k E i rx s [1 {Xs k 1=j} 1 {Xs =j}]ds k= ζ γa k E i [1 {Xs k=j} 1 {Xs =j}]ds. Obviously, here we can remove the truncation ζ and obtain Q ij t =δ ij j = δ ij j b k [r j k1 Q i,j k1 s r j Q ij s]ds k= j1 j 1 γa k [Q i,j k s Q ij s]ds Q ik sr k b j k1 Q ij sr j ds Q ik sγa j k γq ij s ds. k= Differentiating both sides we get j1 j 1 Q ijt = Q ik tr k b j k1 Q ij tr j Q ik tγa j k γq ij t = Q ik tq kj. k= k=

8 128 Li P. S. This is just the Kolmogorov forward equation of Q. Proof of Theorem 1.3 By Theorem 1.1, the solution {X t } to 1.4 is a time homogeneous Markov process with state space := {, 1, 2,..., }. Suppose that σ 1 and z 1 are given in the proof of Theorem 2.2. Let qv 1 =y 1. By the properties of Poisson point process, we can see that P σ 1 >t=e rxt,pz 1 = i =m{i} =b i,pv 1 >t=e γt,py 1 = i =n{i} =a i and σ 1, z 1, v 1,y 1 are mutually independent. Write P i = P X = i fori. Let ξ t =max{n m : n i= σ i, m i= v i t}. Obviously, we have P i [X t = j, ξ t =]=δ ij. By the Markov property of {X t }, otice that P i {X t = j, ξ t = m 1} = P i {1 {σ1 v 1 <t}p Xσ1 v 1 [X t σ1 v 1 = j, ξ t σ1 v 1 = m]} = P i {1 {σ1 <t}1 {v1 σ 1 }P Xσ1 [X t σ1 = j, ξ t σ1 = m]} P i {1 {v1 <t}1 {v1 <σ 1 }P Xv1 [X t v1 = j, ξ t v1 = m]} { } = P i r i e rit s e γt s P Xt s [X s = j, ξ s = m]ds { } P i γe γt s e rit s P Xt s [X s = j, ξ s = m]ds { = P i r i e r iγt s P i { = k i γe r iγt s k=i 1 k=i1 e r iγt s q ik P k [X s = j, ξ s = m]ds. P i [X t = j] = } P z 1 = k i 1P k [X s = j, ξ s = m]ds } P y 1 = k ip k [X s = j, ξ s = m]ds P i [X t = j, ξ t = m]. m= From the theory of Markov chains we know P ij t :=P i [X t = j] is the minimal solution to the Kolmogorov equation of the density matrix Q, see Chen [7, p. 78]. Then {X t } is the minimal process of the density matrix Q. Proof of Theorem 1.4 Suppose that R is regular. Then the minimal solution of its Kolmogorov backward equation is honest, i.e., the minimal process of R is non-explosive. Applying Theorems 1.1 and 1.3 we know the minimal process of Q is non-explosive. Thus Q is regular. Conversely, suppose that R is not regular. Then by [1, Theorem 2.7 3], there exists a non-trivial solution u i to ui r ik u k, u i 1. 2γ r i k i Since r ik q ik, we see u i is also a solution to u i q ik u k. γ q i k i Using [1, Theorem 2.7 3] again, we see Q is not regular.

9 onlinear Branching Processes 129 Proof of Theorem 1.5 Theorem 2.3 of [5]. By Theorem 1.4, we derive the results from Theorem 1.2 of [8] and 3 Recurrence Proof of Theorem Under the assumption, there exists a constant 1 such that r i γm/1 M holds for each i. Takex i = i for i. For i, wehave q ij x j = r i b i 1 r i b j1 γa j i j j= =r i γi r i M 1 γm r i γi = q ii x i. Let π ij be the embedded chain of q ij. The above calculations imply that x i is a finite solution of π ij x j x i, i. j= Then Q is recurrent by Theorem 4.24 in [7]. 2 Suppose that M 1andJ =. We shall prove the process is recurrent by comparison arguments. Let Q = q ij be the density matrix defined by αib j i1 γa j i, j i 1, αi γ, j = i, q ij = αib, j = i 1,, otherwise, which corresponds to a linear branching process with immigration. It was proved in [12] that this process is recurrent. ext, we define the density matrix Q =qij by r i b j i1 γa j i r i /αi, j i 1, r i γr i /αi, j = i, qij = r i b, j = i 1, q ij, i <,, otherwise. Let π ij andπij denote the embedded chains of q ij andqij, respectively. It is easy to see that π ij = πij for i and j. Then Q is also recurrent. For l i>,wehave q ij = r i b qlj. Moreover, we have and j=k j=k j=i j=i q ij = qlj =, k i 1 j=k j=k q ij qij qlj, k l 1. j=k

10 13 Li P. S. Then Q and Q are stochastically comparable, so we can construct a Q-process X t anda Q -process Xt on some probability space in such a way that X = X and X t Xt for all t ; see Example 5.51 in [7]. ow the recurrence of X t follows from that of Xt. 3 Since M>1, there exists an s, 1 such that Bs <, i.e. i= b is i 1 < 1. Take H = {} and x i =1 s i. For i 1, we have π ik x k = π i,i 1 x i 1 π i,ik x ik k= = r ib r i γ x i 1 = 1 [ r i b 1 s i 1 r i γ [ =1 si r i r i γ k= r i b k1 γa k x ik r i γ γa k 1 s ik b k s k 1 γ ] a k s k 1 s i = x i. ] r i b k1 1 s ik Then the process is transient by Theorem 8..2 in [13]. 4 Since the proof is similar to that of 2, we omit it. 4 Mean Extinction Time In this section, we assume r i = αi θ for α>andθ 1. Let X t be a realization of the nonlinear branching process with immigration. Its jump times are given successively by τ = and τ n =inf{t : t>τ n 1,X t X τn 1 }. We also define σ k =inf{t τ 1 : X t = k}. Inorderto prove the criterion for the ergodicity of X t, let us consider the absorbing process X t := X t σ. The density matrix of this process is given by q ij, i, q ij =, i =. For this process, we define τ =, τ n =inf{t : t> τ n 1, Xt X τ n 1 } and σ k =inf{t τ 1 : Xt = k}. It is easy to see that E i σ = E i σ. 4.1 Let p ij t and φ ij λ denote the transition function and the resolvent of X t, respectively. Lemma 4.1 For any i and s [, 1, we have p ijts j = αbs p ij tj θ s j 1 As p ij ts j, t, 4.2 and j= j= λ φ ij λs j s i = αbs φ ij λj θ s j 1 As φ ij λs j, λ >. 4.3 Proof From the Kolmogorov forward equation of the transition function we obtain that j 1 p ij t = p ik tr k b j k1 γa j k p ij tr j γ p i,j1 tr j1 b.

11 onlinear Branching Processes 131 Multiplying s j on both sides of the above equality and then summing over j, wehave j 1 j 1 p ijts j = p ik tr k b j k1 s j γ p ik ta j k s j j= j= j= j= p i,j1 tr j1 s j b p ij tr j s j γ p ij ts j, Then we can interchange the order of summation to see j 1 p ik tr k s k 1 and It follows that j= k= p ik tr k b j k1 s j = k l γ j= j 1 j= k= p ik ta j k s j = γ k l p ik ts k j=k1 j=k1 j= b j k1 s j k1 a j k s j k. p ijts j = p ij tr j s j 1 αbs p ij ts j As. That proves 4.2 and 4.3 is just the Laplace transform of 4.2. Lemma 4.2 For any i, k 1, we have p ik tdt < and lim t p ik t =.Furthermore, for i 1 and s [, 1, we have p ik tdt s k <. 4.4 Proof Fixing an i 1, we can use the Kolmogorov forward equation to see which means that p i t =b α p i1 udu, j= p i1 tdt b 1 α 1 <. Suppose that p ik tdt < for k j. By the Kolmogorov forward equations, we can see for j 1, j 1 p ij t δ ij = αk θ b j k1 γa j k p ik udu αj θ γ p ij udu Letting t,wehave αj 1 θ b p ij1 udu. p ij1 tdt <. Then p ik tdt < by induction. Since the limit lim t p ik t always exists, we see lim t p ik t = immediately.

12 132 Li P. S. We next tend to prove 4.4. Since M 1, we have Bs > forafixeds [, 1. Then there exists a k 1sothatkαBs sas >. Using 4.2, we have p ijus j = αbs p ij uj θ s j 1 As p ij us j j= αbs j=k1 [kαbs sas] p ij uj θ s j 1 As j=k1 p ij us j p ij us j 1 As k p ij us j. 4.5 Let A =max s [,1] As and B =max s [,1] αbs. Thenforeachs [, 1, p ij usj du B p ij uj θ s j 1 du A p ij us j du j= t B j θ s j 1 t A s j <. Then we use Fubini s theorem to see p ijus j du = j= j= Integrating both sides of 4.5, p ij ts j s i [kαbs sas] j= As p ijus j du. j=k1 k p ij udu s j. Letting t and using the fact that p ik tdt <, we have p ij udu s j 1 <, j=k1 p ij udu s j 1 which implies 4.4. Proposition 4.3 Suppose that the nonlinear branching process with immigration is recurrent and 1.6 holds. Then for i 1 we have E i σ 1 1 y i ln 1 θ 1 [ 1 Ay dy exp ln 1 θ 1 ] dy Γθ αby y Γθ αby y 4.6 and E i σ 1 y i ln 1 θ 1 dy. 4.7 αby y Proof Multiplying 4.3 by lns/y θ 1, dividing by αbs and integrating both sides, we have s φ ij j θ y j 1 ln s θ 1 s λ Ay φ ij λy j y i λ φ i λ dy = ln s θ 1 dy. y αby y

13 onlinear Branching Processes 133 Letting y = se x j in the left hand side of the above equation, we get s φ ij j θ y j 1 ln s θ 1 dy = φ ij λs j x θ 1 e x dx =Γθ φ ij λs j. y Using the above two equations, we obtain φ ij λs j = 1 s λ Ay Γθ For i 1, λ>ands [, 1], let ote that Then, by 4.8, By Lemma 4.2, λ φ i λ = ψ i λ, s = ψ i λ, s 1 Γθ 1 Γθ s φ ij λy j y i λ φ i λ αby φ ij λs j. e t p i t λ 1 y i αby lim λ φ ij λ = lim λ λ It follows that, for s [, 1], Denote By 4.4, we have dt ln 1 y θ 1 dy λ Ayψ i λ, y αby lim λψ iλ, s lim λ λ λ C i := 1 Γθ 1 Γθ 1 Γθ ψ i,s= 1 y i αby 1 y αby Ay αby for each s<1. Letting λ in 4.9, we have ψ i,s C i 1 Γθ s e t p ij t λ e t dt =1. ln s y θ 1 dy. 4.8 ln 1 y θ 1 dy. 4.9 dt =. φ ij λ =. 4.1 ln 1 y θ 1 dy ln 1 y θ 1 dy ln 1 y θ 1 dy <. p ik tdt s k < Ayψ i,y αby ln 1 y θ 1 dy.

14 134 Li P. S. Using the Gronwall s inequality, we have [ s Ay ψ i,s C i exp ln 1 θ 1 ] dy Γθ αby y Letting s 1, we see lim ψ i,s = lim p ij ts j dt = 1 p i tdt = E i σ. s 1 s 1 Hence 4.6 follows from 4.1 and Similarly, by 4.8, we have ψ i λ, s 1 s Γθ Letting λ and then letting s 1, we obtain 4.7. λ φ i λ y i ln s θ 1 dy. αby y 5 Ergodicity and Strong Ergodicity One of the main steps to prove Theorems 1.7 and 1.8 is to compare our nonlinear branching process with immigration with a suitably designed birth-death process, which we now introduce. A similar birth-death process was used by [8] in her study of the regularity of the nonlinear branching process with resurrection. Let L = M b 1= kb k1 and let ˆX t be a birth-death process with birth rate d i = r i L γm and death rate c i = r i b. We denote the density matrix of ˆX t byˆq ij. Let T := inf{t : ˆXt =}. Lemma Suppose that m<, M<1, r i is increasing and i=1 r 1 i <. Then the birth-death process ˆX t is strongly ergodic. 2 Suppose that m<, M 1, r i is non-decreasing and i=1 r 1 i <. Then the birth-death process ˆX t is ergodic. Proof 1 It is easy to check that the birth-death process is regular. Fix an ε> satisfying L ε<b. Then there exists an such that d i γ i L ε foreachi>.let 1 n d k d n S =. 5.1 c n1 c k c n1 n=1 It is obvious that n=1 c 1 n1 <. otice that for each n>,wehave n d k d n d k d max c k c n1 1 k c k c ρ n n d 1 d n c 1 c n1 n d k d ρ n k1 max, 1 k c k c c n1 d k d n c k c n1 where ρ = b 1 L ε < 1. Then S<. By Corollary 2.4 of [16], we conclude that ˆX t is strongly ergodic.

15 onlinear Branching Processes Since L b,wehave d d n 1 R := d c 1 c n n=1 n=1 c 1 γmc 2 γm c n 1 γm c 1 c 2 c n. Taking logarithm on the right-hand side, we get n 1 c1 γmc 2 γm c n 1 γm ln = ln 1 γm 1 ln. c 1 c 2 c n r i b c n Since lim i r i =, wehaveln 1 γm r i b γm r i b as i. Then there exists a constant C such that for sufficiently large n, c1 γmc 2 γm c n 1 γm 1 1 ln C ln, c 1 c 2 c n r i c n and hence i=1 c 1 γmc 2 γm c n 1 γm c 1 c 2 c n for another constant T. That implies R<. By Theorem 4.55 in [7] the birth-death processisergodic. Lemma 5.2 If the nonlinear branching process with immigration has a stationary distribution μ =μ j, then the generating function fs := j= μ js j satisfies the following equation : Proof s Γθfs =Γθμ Ay αby i=1 T c n ln s y θ 1 fydy, s [, 1]. 5.2 The stationary distribution μ j satisfiesμq =. In view of 1.3, we have j 1 j1 μ j γ αj θ = μ i γa j i μ i αi θ b j i i= Multiplying s j on both sides of the above equality and then summing over j, wehave j 1 j1 γ μ j s j αs μ j j θ s j 1 = γ μ i a j i s j α μ i i θ b j i1 s j. 5.4 Interchanging the order of summation, l.h.s. of 5.4 = γ μ i s i i= α i= i= j=i1 μ i i θ s i 1 i=1 i= a j i s j i αμ 1 b j=i 1 i=1 b j i1 s j i1 = γ μ i s i F s αμ 1 b α μ i i θ s i 1 Gs. 5.5 Letting j = in 5.3, we see μ γ = αμ 1 b. Therefore, from 5.4 it follows that μ j j θ s j 1 = fsas αbs. i=1

16 136 Li P. S. Multiplying the above equation by ln s y θ 1 and integrating the both sides, we have s Letting y = se x j,weget μ j j θ y j 1 ln s θ 1 s Ay dy = ln s θ 1 fydy. 5.6 y αby y l.h.s. of 5.6 = = θ 1 μ j j θ se x j j 1 x s x j j e j dx μ j s j x θ 1 e x dx =Γθ[fs μ ]. Then fs is a solution to the differential equation 5.2. Proof of Theorem By Lemma 5.1, the birth-death process ˆX t isergodic. Thusby Theorem 4.45 in [7], the equation u =, d i u i1 u i c i u i 1 u i 1=, i 5.7 has a finite nonnegative solution u i. By Remark 2.5 of [16], we have i 1 1 d k1 d j u =, u i =. 5.8 c k1 c k1 c j1 k= It is apparent that u i u i1. Moreover, we have u i1 u i = 1 c i1 j=i1 j=k1 d i1 d j c i1 c j1, u i u i 1 = 1 c i j=i d i d j c i c j1. Since d i1 /c i1 <d i /c i and 1/c i1 < 1/c i, it is not hard to show that u i1 u i is non-increasing in i. Coming back to the matrix Q, fori 1, q ij u j = q ij u j u i j= j= = c i u i 1 u i r i γa k b k1 k u il u il 1 l=1 k u il u il 1 l=1 c i u i 1 u i r i and q j u j = q j u j u 1 = kb k1 u i1 u i γ ka k u i1 u i = c i u i 1 u i r i L γmu i1 u i = c i u i 1 u i d i u i1 u i = q j i=1 j u i u i 1 < q j ju 1 γmu 1 <. 5.1

17 onlinear Branching Processes 137 Then u i is a nonnegative bounded solution to the following equation q j u j <, q ij u j 1, i 1. By Theorem 4.45 in [7], we know the process is positive recurrent. 2 Suppose that the process is ergodic. Then letting s = 1 in 5.2, we get j= > Γθ μ j μ jy j Ay ln 1 θ 1 μ Ay dy ln 1 θ 1 dy. αby y αby y j= Since μ >, we have 1.6. Conversely, suppose that 1.6 holds. By the strong Markov property, we have E σ =E τ 1 E [E Xτ1 σ ] = 1 q i E i σ = 1 q q γ a i E i σ. Using 4.6 we have E σ 1 γ 1 γγθ [ i=1 Ay ln 1 θ 1 ] [ 1 Ay dy exp ln 1 θ 1 ] dy. αby y Γθ αby y By 1.6, the right-hand side is finite. Thus the process is ergodic. 3 By the assumption, there exists C>such that r i C b Γi for large enough i. Therefore, jq ij = i kr i b k1 i kγa k i 1r i b γ r i i j= m r i b Γ m Ci. Applying Corollary 4.49 in [7], we know the process is exponentially ergodic. Proof of Theorem Using Lemma 5.1, we see the birth-death process ˆX t is strongly ergodic. Let u i := E i T fori. Applying Theorem 4.44 and Lemma 4.48 in [7], we find that u i is a bounded non-negative solution to equation 5.7. By 5.9 and 5.1, u i isalso a non-negative bounded solution to the following equation q j u j <, q ij u j 1, i 1. By Theorem 4.45 in [7], we know the process is strongly ergodic. 2 Suppose that 1.7 holds. Then j= Ay ln 1 θ 1 dy γ αby y Letting i in 4.6, we get sup E i σ 1 i Γθ [ 1 αby i=1 1 ln 1 θ 1 dy <. αby y ln 1 θ 1 ] [ 1 dy exp y Γθ Ay ln 1 θ 1 ] dy <. αby y Then by Theorem 4.44 in [7], the process is strongly ergodic. Conversely, suppose that X t is strongly ergodic. By Theorem 4.44 in [7] and 4.7, we know 1.7 holds.

18 138 Li P. S. 3 By the strong Markov property, for i 1, we have E i σ = i E kσ k 1. otice that E k σ k 1 E k [time spent at k until the next jump] = 1 r k γ. Thus E i σ i r k γ 1. By the assumption i=1 r 1 i =, wehavesup i E i σ =. Applying Theorem 4.44 in [7], we know the process is not strongly ergodic. Acknowledgements The author would like to thank Professors Mu-Fa Chen, Yong-Hua Mao and Yu-Hui Zhang for their advice and encouragement. I am grateful to the two referees for pointing out a number of typos in the first version of the paper. References [1] Anderson, W. J.: Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer, ew York, 1991 [2] Athreya, K. B., ey, P. E.: Branching Processes, Springer, Berlin, 1972 [3] Chen, A. Y.: Ergodicity and stability of generalised Markov branching processes with resurrection. J. Appl. Probab., 39, [4] Chen, A. Y., Li, J. P., Ramesh,. I.: Uniqueness and extinction of weighted Markov branching processes. Methodol. Comput. Appl. Prob., 7, [5] Chen, A. Y., Li, J. P., Ramesh,. I.: General Harris regularity criterion for non-linear Markov branching process. Statist. Probab. Letters., 76, [6] Chen, A. Y., Renshaw, E.: Markov branching processes with instantaneous immigration. Probab. Theory Relat. Fields, 87, [7] Chen, M. F.: From Markov Chains to on-equilibrium Particle Systems, Second edition, World Scientific, Singapore, 24 [8] Chen, R. R.: An extended class of time-continuous branching processes. J. Appl. Probab., 34, [9] Harris, T. E.: The Theory of Branching Processes, Springer, Berlin, 1963 [1] Ikeda,., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, Second edition, orth- Holland/Kodasha, Amsterdam/Tokyo, 1989 [11] Karlin, S., Taylor, H. M.: A First Course in Stochastic Processes. Second edition, Academic Press, ew York, 1975 [12] Li, J. P., Chen, A. Y.: Markov branching processes with immigration and resurrection. Markov Processes Relat. Fields, 12, [13] Meyn, S. P., Tweedie, R. L.: Markov Chains and Stochastic Stability, Second edition, Cambridge Univ. Press, Cambridge, 29 [14] Pakes, A. G.: Extinction and explosion of nonlinear Markov branching processes. J. Austr. Math. Soc., 82, [15] Yang, Y. S.: On branching processes allowing immigration. J. Appl. Probab., 9, [16] Zhang, Y. H.: Strong-ergodicity for single-birth processes. J. Appl. Probab., 38,