A general continuous-state nonlinear branching process
aa r X i v : . [ m a t h . P R ] O c t Submitted to the Annals of Applied Probability
A GENERAL CONTINUOUS-STATE NONLINEARBRANCHING PROCESS
By Pei-Sen Li ¶ , Xu Yang k and Xiaowen Zhou ∗∗ § Renmin University of China ¶ , North Minzu University k and ConcordiaUniversity ∗∗ In this paper we consider the unique nonnegative solution to thefollowing generalized version of the stochastic differential equationfor a continuous-state branching process. X t = x + Z t γ ( X s )d s + Z t Z γ ( X s − )0 W (d s, d u )+ Z t Z ∞ Z γ ( X s − )0 z ˜ N (d s, d z, d u ) , where W (d t, d u ) and ˜ N (d s, d z, d u ) denote a Gaussian white noiseand an independent compensated spectrally positive Poisson ran-dom measure, respectively, and γ , γ and γ are functions on R + with both γ and γ taking nonnegative values. Intuitively, this pro-cess can be identified as a continuous-state branching process withpopulation-size-dependent branching rates and with competition. Us-ing martingale techniques we find rather sharp conditions on extinc-tion, explosion and coming down from infinity behaviors of the pro-cess. Some Foster-Lyapunov type criteria are also developed for sucha process. More explicit results are obtained when γ i , i = 0 , ,
1. Introduction.
Continuous-state branching processes.
Suppose that (Ω , F , F t , P ) isa filtered probability space satisfying the usual hypotheses. Let P x be thelaw of a process started at x , and denote by E x the associated expectation.A continuous-state branching process X = ( X t ) t ≥ is a c`adl`ag [0 , ∞ ]-valued ¶ Supported by NSERC (RGPIN-2016-06704) and NSFC (No. 11771046 and 11571043) k Supported by NSFC (No. 11771018 and No. 11401012), NSF of Ningxia(No. 2018AAC03245) and First-Class Disciplines Foundation of Ningxia (No.NXYLXK2017B09) ∗∗ Supported by NSERC (RGPIN-2016-06704) and NSFC (No. 11731012) § Corresponding author.
MSC 2010 subject classifications:
Primary 60G57; secondary 60G17, 60J80
Keywords and phrases: continuous-state branching process, nonlinear branching, com-petition, extinction, explosion, coming down from infinity, weighted total population,Foster-Lyapunov criterion, stochastic differential equation LI, P.S., YANG, X. AND ZHOU, X. ( F t )-adapted process satisfying the branching property, i.e. for any x, y ≥ t, θ ≥ E x + y (cid:2) e − θX t (cid:3) = E x (cid:2) e − θX t (cid:3) E y (cid:2) e − θX t (cid:3) . Consequently, its Laplace transform is determined by E x (cid:2) e − θX t (cid:3) = e − xu t ( θ ) , where the non-negative function u t ( θ ) solves the differential equation ∂u t ( θ ) ∂t + ψ ( u t ( θ )) = 0with initial value u ( θ ) = θ ≥ ψ ( λ ) = bλ + 12 σ λ + Z ∞ (e − λx − λx ) π (d x )for b ∈ R , σ ≥ σ -finite measure π on (0 , ∞ ) satisfying R ∞ ( z ∧ z ) π (d z ) < ∞ .Via the Lamperti random time change the continuous-state branchingprocess is associated to a spectrally positive L´evy process, which allows manysemi-explicit expressions. In particular, extinction and explosion behaviorsfor continuous-state branching processes were studied by Grey (1974) andKawazu and Watanabe (1971), respectively, and the conditions for extinctionand explosion were expressed in terms of the respective integral tests on thefunction ψ .Bertoin and Le Gall (2006) and Dawson and Li (2006, 2012) noticed thefollowing alternative way of characterizing continuous-state branching pro-cesses through stochastic differential equations (SDEs in short). Let { W (d t, d u ) : t, u ≥ } denote an ( F t )-Gaussian white noise with density measure d t d u on (0 , ∞ ) . In this paper we always write π = 0 for a σ -finite measureon (0 , ∞ ). Let { N (d t, d z, d u ) : t, z, u > } denote an independent ( F t )-Poisson random measure with intensity measure d tπ (d z )d u on (0 , ∞ ) andlet { ˜ N (d t, d z, d u ) : t, z, u > } denote the corresponding compensated mea-sure. Then the continuous-state branching process is a pathwise unique non-negative solution to the following SDE that is called a Dawson-Li SDE inPardoux (2016): X t = x + b Z t X s d s + σ Z t Z X s − W (d s, d u )+ Z t Z ∞ Z X s − z ˜ N (d s, d z, d u ) . (1.2) ONLINEAR BRANCHING PROCESS SDEs similar to (1.2) were studied by Dawson and Li (2006, 2012) and byFu and Li (2010). See also Bertoin and Le Gall (2003, 2005) for related work.We refer to Kyprianou (2006), Li (2011, 2012) and Pardoux (2016) forreviews and literature on continuous-state branching processes.1.2.
Continuous-state branching processes with nonlinear branching.
Mod-els with interactions have gained interests in the study of branching pro-cesses. Athreya and Ney (1972) introduced population-size-dependent Galton-Watson branching processes in which the reproduction mechanism dependson the population size; see also Klebaner (1984) and H¨opfner (1985) forprevious work on population-size-dependent Galton-Watson processes. An-other class of interacting Galton-Watson processes is the so called controlledbranching processes. For a controlled branching process, the reproductionlaw is fixed. But before each branching time the population is regulated bya control function. Previous work on controlled branching processes can befound in Sevast’yanov and Zubkov (1974) and references therein. A discretestate, continuous time branching process with population dependent branch-ing rate can be found in Chen (1997). When the branching rate function isa power function of the population, the extinction probability for such abranching process was obtained in Chen (2002). When the branching rate isa general positive nonlinear function, such a model called nonlinear Markovbranching process was studied in Pakes (2007).The previous work on discrete-state interacting branching processes mo-tivates the study of their continuous-state counterparts. Some population-size-dependent continuous-state branching processes arising as scaling limitsof the corresponding discrete-state branching processes can be found in Li(2006, 2009).In this paper we introduce a class of continuous-state branching processeswhose branching rates depend on their current population sizes. To thisend, we consider a nonnegative solution to the following modification ofSDE (1.2): X t = x + Z t γ ( X s )d s + Z t Z γ ( X s − )0 W (d s, d u )+ Z t Z ∞ Z γ ( X s − )0 z ˜ N (d s, d z, d u ) , (1.3)where γ , γ and γ are Borel functions on R + , and both γ and γ takenonnegative values. The unique nonnegative solution to (1.3) up to the min-imum of its first time of hitting 0 and its explosion time can be treated asa continuous-state nonlinear branching process, where γ i ( x ) /x, i = 1 , LI, P.S., YANG, X. AND ZHOU, X. be interpreted as population-size-dependent branching rates and the driftterm involving γ can be related to either competition or population-size-dependent continuous immigration. We refer to Duhalde et al. (2014) forwork on continuous-state branching processes with immigration. If γ i ( x ) = c i x for c , c ≥ , then the solution to (1.3) reduces to the classical continuous-state branching process and satisfies the branching property (1.1). Observethat the solution X to (1.3) can also be treated as a continuous-state con-trolled branching process.For γ ≡ , γ ( z ) = z and γ satisfying certain conditions, the SDE (1.3)was studied in Pardoux and Wakolbinger (2015) and in Pardoux (2016)where the function γ models an impact of the current population size onthe individuals’ reproduction dynamics. If the interaction is of the type ofcompetition for rare resources, then increasing the population size resultsin a reduction of the individuals’ birth rate and/or increment of the deathrate.For γ ( z ) = γ ( z ) = z and γ ( z ) = θz − γz with positive constants θ and γ , solution to SDE (1.3) can be used to model the density dependence inpopulation dynamics of a large population with competition called logisticbranching process, and it was studied in detail by Lambert (2005). Thequadratic regulatory term has an ecological interpretation as it describesnegative interactions between each pair of individuals in the population.The extinction behavior and the probability distribution of the extinctiontime were considered in Lambert (2005). A similar model with more generalfunction γ was considered in Le et al. (2013) with its first passage timesstudied. The total mass for this model was also studied using the Lampertitransform. Berestycki et al. (2017) gave a genealogical description for theprocess based on interactive pruning of L´evy–trees, and established a Ray–Knight representation result.For γ ( z ) = γ ( z ) ≡
0, the extinction/survival behaviors for process X asthe total mass process of a superprocess with mean field interaction werediscussed in Wang et al. (2017) by a martingale approach. More generally, for γ ( z ) ≡ X are associated to the classification of its boundariesat 0 and ∞ , respectively; see Karlin and Taylor (1981, p. 229).For γ i ( z ) = c i z r with r > c ∈ R and c i ≥ i = 1 ,
2, the solutionto SDE (1.3), called a continuous-state polynomial branching process, wasstudied by Li (2018), where the parameter r describes the degree of inter-action. The polynomial branching process also arises as time-space scalinglimit of discrete-state nonlinear branching processes. Intuitively, functions γ and γ are population-dependent rates for branching events producing small ONLINEAR BRANCHING PROCESS and large amount of children, respectively. By solving the corresponding Kol-mogorov equations, necessary and sufficient conditions in terms of integraltests were obtained for extinction, explosion and coming down from infinity,respectively. Expectations of the extinction time and explosion time werealso discussed in Li (2018), which generalizes those results in Chen (2002)for discrete-state processes to the corresponding continuous-state processes.The nonlinear branching processes considered in this paper generalize thosein Li (2018) by allowing different rates for different branching events.Note that if ˜ N is the compensated measure of a one-sided α -stable randommeasure with α ∈ (1 , π (d z ) = α ( α − − α ) 1 { z> } z − − α d z (1.4)for Gamma function Γ, then on an enlarged probability space, SDE (1.3)can be transformed into the following SDE: X t = x + Z t γ ( X s )d s + Z t p γ ( X s − )d B s + Z t γ ( X s − ) /α Z ∞ u ˜ M (d s, d u ) , (1.5)where { B t : t ≥ } is a Brownian motion and { ˜ M (d t, d u ) : t, u ≥ } is anindependent compensated Poisson random measure with intensity measured tπ (d u ); see Theorem 9.32 in Li (2011) for a similar result. Equation (1.5)has a pathwise unique non-negative strong solution if γ ( z ) = a z + a , γ ( z ) = z r and γ ( z ) = z r for a ∈ R , a ≥ r ∈ [1 / ,
1] and r ∈ ( α − , α ]; see Corollary 4.3 in Li and Mytnik (2011). By Theorem 4.1.2 inLi (2012) one can also convert (1.3) to another SDE: X t = x + Z t γ ( X s )d s + Z t p γ ( X s − )d B s + Z t Z ∞ u ˜ M γ (d s, d u ) , where ˜ M γ (d s, d u ) is an optional compensated Poisson measure with pre-dictable compensator γ ( X s − )d sπ (d u ).Using the Lamperti transform for positive self-similar Markov processes,Berestycki et al. (2015) found the extinction condition of solution to (1.5)for γ ( z ) = θz η f ( z ) , γ ( z ) ≡ , γ ( z ) = z αβ and π (d z ) = α ( α − − α ) 1 { z> } z − − α d z with α ∈ (1 , , θ ≥ , β ∈ [1 − /α, , η = 1 − α (1 − β ) ∈ [0 ,
1) and forcertain nonnegative Lipschitz continuous function f . In particular, for f ≡ LI, P.S., YANG, X. AND ZHOU, X. it is shown that the extinction occurs within finite time with probability onefor 0 ≤ θ < Γ( α ) and with probability 0 for θ ≥ Γ( α ); see Theorems 1.1 and1.4 of Berestycki et al. (2015).We refer to Lambert (2005), Berestycki et al. (2010), Bansaye et al. (2015)and Li (2018) for previous studies of coming down from infinity for a branch-ing process with logistic growth, coalescents, birth and death processes andthe polynomial branching process, respectively.Other than the above mentioned results, we are not aware of any previousresults on hitting probability and coming down from infinity for solutions toSDEs of type (1.3). There is some literature on nonexplosion of solutions togeneral SDE with jumps; see Dong (2016) for a recent result. But we do notfind any systematic discussions on the explosion/nonexplosion dichotomyand the coming down from infinity property of the solutions.The main purpose of this paper is to investigate the extinction, explosionand coming down from infinity behaviors of the continuous-state nonlinearbranching process as solution to (1.3) and specify the associated conditionson functions γ i , i = 0 , , γ i ( x ) as x → γ or large enough fluc-tuations due to γ or γ . Even when the process has a (small) positive driftnear 0, it might still die out because of relative large fluctuations.We are also interested in the relations between the asymptotes of functions γ i ( x ) , i = 0 , , x → ∞ and the explosion and coming down from infinitybehaviors of the nonlinear branching processes as solutions to SDEs (1.3).When γ i , i = 0 , , et al. (2017).Such an approach typically involves understanding how the process exitsfrom consecutive intervals near 0 with the interval lengths decreasing geo-metrically, or consecutive intervals near ∞ with the interval lengths increas-ing geometrically. To this end, we construct the corresponding martingalein each situation. These martingales allow to obtain estimates on both thesequential exit probabilities and sequential exit times via optional stopping,where the lack of negative jumps for process X comes in handy. The desiredresults then follow from Borel-Cantelli type arguments. Although we focuson SDEs of type (1.3), we expect that this approach could also adapted tostudy similar properties of solutions to other SDEs with more general jumpmechanism, and it remains to be checked how sharp the desired results canbe. ONLINEAR BRANCHING PROCESS In addition, we show that the general nonlinear branching processes con-sidered in this paper are closed under a Lamperti type transform, whichallows us to discuss the finiteness of a weighted occupation time until ex-tinction or explosion of the continuous-state nonlinear branching processvia considering the extinction or explosion behaviors of the time changedprocess.We also find Foster-Lyapunov type criteria to show the irreducibility ofthe nonlinear continuous-state branching processes, which is of independentinterest. We refer to Chen (2004) and Meyn and Tweedie (1993) for theFoster-Lyapunov type criteria for explosion and stability of Markov chains.This paper is structured as follows. After introductions in Subsections 1.1and 1.2 on the continuous-state branching processes, Section 2 summarizesthe main results of this paper with an application and examples, where ourresults are compared with the known results. In Section 3 we show thatSDE (1.3) has a unique strong solution up to the first time of reaching 0or explosion given that the functions γ i , i = 0 , , , ∞ ). Section 4 contains Foster-Lyapunov criteria type results that can beused to show the irreducibility of the solution as a Markov process. Proofsof the main results in Section 2 are included in Section 5.
2. Extinction, explosion and coming down from infinity.
Withthe convention inf ∅ := ∞ , for y > τ − y ≡ τ − ( y ) := inf { t > X t < y } , τ + y ≡ τ + ( y ) := inf { t > X t > y } and τ − := inf { t > X t = 0 } . By a solution to SDE (1.3) we mean a c`adl`ag process X = ( X t ) t ≥ satisfying(1.3) up to time τ n := τ − /n ∧ τ + n for each n ≥ X t = lim sup n →∞ X τ n − for t ≥ τ := lim n →∞ τ n . Then both of the boundary points 0 and ∞ areabsorbing for X by definition.Throughout this subsection we assume that SDE (1.3) allows a uniqueweak solution denoted by X := ( X t ) t ≥ , and consequently the process X hasthe strong Markov property. In Theorem 3.1 we are going to show that (1.3)allows a pathwise unique solution if the coefficient functions γ i , i = 0 , , γ γ γ , γ and γ are all locally bounded on [0 , ∞ ).In the following we present our main results on extinction, explosion andcoming down from infinity properties of process X . Most of the proofs aredeferred to Section 5. LI, P.S., YANG, X. AND ZHOU, X.
For a > u >
0, let H a ( u ) := Z ∞ (cid:2) (1 + zu − ) − a − − (1 − a ) zu − (cid:3) π (d z )= a ( a − u − Z ∞ z π (d z ) Z (1 + zu − v ) − − a (1 − v )d v, (2.1)where we use the following form of Taylor’s formula that is often needed inthe proofs of this paper; see e.g. Zorich (2004, p.364) for its proof. Lemma . If function g has a bounded continuous second derivativeon [0 , ∞ ) , then for any y, z > we have g ( y + z ) − g ( y ) − zg ′ ( y ) = z Z g ′′ ( y + zv )(1 − v )d v. Note that for π (d z ) = cz − − α with α ∈ (1 ,
2) and c > H a ( u ) = a ( a − u − α Z ∞ cy − α d y Z (1 + yv ) − − a (1 − v )d v. (2.2)Put G a ( u ) := ( a − γ ( u ) u − − − a ( a − u − γ ( u ) − γ ( u ) H a ( u ) . (2.3)We choose the function G a to be of the particular form in (2.3) so that,by Ito’s formula, the process constructed in Lemma 5.1 can be shown to bea martingale, which is key for the main proofs in Section 5. The martingaleallows to obtain estimates on exits times of the processes X via optionalstopping. The conditions for extinction, explosion and coming down frominfinity for the process X can be identified from the asymptotic behaviors of G a ( u ) for u near 0 or near ∞ . An earlier version of G a can be found in Wang et al. (2017) where it was also used to construct a continuous martingale tostudy the extinction behavior for the interacting super-Brownian motion. Remark . Suppose that π = 0 and u ∈ (0 , c ) for some constant c > .One can see that • If there exists a constant α ∈ (1 , so that sup We first present the two main results on theextinction behaviors for X . Here we only consider the case that the ini-tial value of X is small. In this way we only have to impose conditions onfunction G ( u ) for small positive values of u . These results, combined withFoster-Lyapunov criteria (Lemmas 4.1 and 4.2), can be used to discuss theextinction behaviors for X with arbitrary initial value. Theorem . (i) Suppose that there exist constants a > and r < so that G a ( u ) ≥ − (ln u − ) r for all small enough u > . Then we have P x { τ − < ∞} = 0 for all x > . (ii) Suppose that there exist constants < a < and r > so that G a ( u ) ≥ (ln u − ) r for all small enough u > . Then P x { τ − < ∞} > for all small enough x > . Proof of Theorem 2.3 is deferred to Section 5.The next results concern the first passage probabilities for which we needthe following condition. Condition . (i) For any x and a with x > a > , P x { τ − a < ∞} > . (2.4)(ii) For any x and a with x > a > , P x { τ − a < ∞} = 1 . (2.5)Proof of the next corollary is deferred to Section 5. Corollary . Suppose that the assumption of Theorem 2.3 (ii) holds.Then • P x { τ − < ∞} > for all x > if Condition 2.4 (i) holds; • P x { τ − < ∞} = 1 for all x > if Condition 2.4 (ii) holds. LI, P.S., YANG, X. AND ZHOU, X. For a ≤ b defineΦ( a, b ) := inf y ∈ [ a,b ] γ ( y ) + inf y ∈ [ a,b ] γ ( y )1 { R zπ (d z )= ∞} . We can show that (i) or (ii) of Condition 2.4 hold under certain conditionson γ i , i = 0 , , Proposition . (i) Given x > a > , (2.4) holds if Φ( a, b ) > and sup a ≤ y ≤ b γ ( y ) < ∞ for all b > a . (ii) Given x > a > , suppose that Φ( a, b ) > for all b > a and that γ ( y ) ≤ for all large enough y . Then (2.5) holds. (iii) If γ ( a ) ≤ and Φ( a, b ) > for all b ≥ a > , then for each x > , P x -a.s. X t → as t → ∞ . Further, by the strong Markov propertyeither P x { X t = 0 for all t large enough } = 1 or P x { X t → , but X t > for all t } = 1 , and we say extinguishing occurs in the latter case. Proof of Proposition 2.6 is deferred to Section 4 after Lemma 4.2. Remark . (i ) Combining Proposition 2.6 and Theorem 2.3 (ii) wefind conditions for extinction with probability one and extinguishingwith probability one, respectively. (ii) If γ = γ ≡ , then the process X is the total mass of an interactingsuper-Brownian motion and Theorem 2.3 generalizes Theorems 3.4and 3.5 of Wang et al. (2017). Explosion behaviors. Let τ + ∞ := lim n →∞ τ + n be the explosion time.The solution X to SDE (1.3) explodes at a finite time if τ + ∞ < ∞ . We nowpresent results on the explosion behaviors for X in the following, and again,we only consider the case of large initial values. Theorem . (i) If there exist constants < a < and r < sothat G a ( u ) ≥ − (ln u ) r for all u large enough, then P x { τ + ∞ < ∞} = 0 for all x > . (ii) If there exist a > and r > so that G a ( u ) ≥ (ln u ) r for all u largeenough, then P x { τ + ∞ < ∞} > for all large x . The proof of Theorem 2.8 is deferred to Section 5. ONLINEAR BRANCHING PROCESS Condition . For any x and b with b > x > , P x { τ + b < ∞} > . (2.6)Putting Theorem 2.8 (ii) and the above condition together we reach thefollowing remark. Remark . If Condition 2.9 and the assumption in Theorem 2.8 (ii)hold, then P x { τ + ∞ < ∞} > for all x > . The proof for the next result is deferred to the end of Section 4. Proposition . Given b > x > , if there exists a ∈ (0 , x ) so that inf y ∈ [ a,b ] γ ( y ) + inf y ∈ [ a,b ] γ ( y ) > , then (2.6) holds. Coming down from infinity. We say that the process X comes downfrom infinity if(2.7) lim b →∞ lim x →∞ P x { τ − b < t } = 1 for all t > , and it stays infinite iflim x →∞ P x { τ − b < ∞} = 0 for all b > . We first present equivalent conditions for coming down from infinity. Fromthe proof one can see that they hold for any real-valued Markov processeswith no downward jumps. Proposition . The following statements are equivalent:(i) Process X comes down from infinity.(ii) lim x →∞ E x [ τ − b ] < ∞ for all large b .(iii) (2.8) lim b →∞ lim x →∞ E x [ τ − b ] = 0 . Proof. For the proof that (i) implies (ii), we refer to the proof of Theo-rem 1.11 of Li (2018).Suppose that (ii) holds. Then for any x ′ > b , we have(2.9) lim x →∞ E x [ τ − b ] = lim x →∞ ( E x [ τ − x ′ ] + E x ′ [ τ − b ]) . LI, P.S., YANG, X. AND ZHOU, X. First letting x ′ → ∞ , and then letting b → ∞ in (2.9), we obtain (2.8). (iii)thus holds.(i) follows from (iii) by the Markov inequality. (cid:3) Theorem . (i) If there exist constants a > and r < such that (2.10) G a ( u ) ≥ − (ln u ) r for all u large, then process X stays infinite. (ii) If there exist constants < a < , r > such that (2.11) G a ( u ) ≥ (ln u ) r for all u large enough, then process X comes down from infinity. The proof of Theorem 2.13 is deferred to Section 5. Remark . More recently, for the process X with γ = γ = γ ,the speeds of coming down from infinity are studied in details in Dawson etal. (2018) for the cases that either the function γ i is regularly varying atinfinity or γ i ( x ) = g ( x )e θx for θ > and function g that is regularly varyingat infinity. An application: weighted total population. Let γ be a strictly positivefunction defined on [0 , ∞ ) that is bounded on any bounded interval. In thefollowing, we consider the weighted occupation time, or the weighted totalpopulation of X before explosion, defined as S = Z τ − ∧ τ + ∞ γ ( X s )d s. For t ≥ U t := Z t ∧ τ − ∧ τ + ∞ γ ( X s )d s and V t := inf { s > U s > t } . Define the process ¯ X ≡ { ¯ X t : t ≥ } by ¯ X t := X V t for V t < ∞ and¯ X t := X ∞ := lim sup t →∞ X t for V t = ∞ . Define stopping times ¯ τ − and ¯ τ + ∞ similarly to τ − and τ + ∞ , respectively, with X replaced by ¯ X .We first observe that with the above-mentioned Lamperti type transform,a time changed solution to the generalized Dawson-Li equation (1.3) remainsa solution to another generalized Dawson-Li equation.We leave the proof of the next result to the interested readers. ONLINEAR BRANCHING PROCESS Theorem . For i = 0 , , and y > define ¯ γ i ( y ) := γ i ( y ) /γ ( y ) .Then there exist, on an extended probability space, a Gaussian white noise { W (d s, d u ) : s ≥ , u > } with intensity d s d u and an independent com-pensated Poisson random measure { ˜ N (d s, d z, d u ) : s ≥ , z > , u > } with intensity d sπ (d z )d u so that { ¯ X t : t ≥ } solves the following SDE: ¯ X t = x + Z t ¯ γ ( ¯ X s )d s + Z t Z ¯ γ ( ¯ X s )0 W (d s, d u )+ Z t Z ∞ Z ¯ γ ( ¯ X s − )0 z ˜ N (d s, d z, d u )(2.12) for ≤ t < ¯ τ − ∧ ¯ τ + ∞ . We leave the proof of the next key observation to interested readers. Proposition . We have S = ¯ τ − ∧ ¯ τ + ∞ . Remark . By Proposition 2.16 and Theorem 2.15, the finiteness for S is translated into extinction and explosion behaviors for the time changedprocess ¯ X for which we can apply Theorems 2.3 and 2.8. More details canbe found later in Example 2.23 in Section 2.5. If γ ( x ) = γ ( x ) = γ ( x ) ≡ x and γ satisfies certain interaction condition, then the behaviors for S havebeen studied in Theorems 4.3.1 and 4.3.2 of Le (2014). Processes with power branching rate functions. To obtain more ex-plicit results, in this subsection we only consider processes with power func-tion branching rates, i.e. γ i ( x ) = b i x r i , x > , i = 0 , , , for r , r , r ≥ b ∈ R , b , b ≥ , b + b > 0. In addition, we assume thatthe measure π is defined in (1.4) with 1 < α < 2. Then by (2.2), G a ( u ) =( a − u − b u r − − a ( a − u − b u r − a ( a − u − α b u r c α,a , where c α,a := α ( α − − α ) Z ∞ y − α d y Z (1 + yv ) − − a (1 − v )d v. It is easy to see from properties of the beta function that c α, := α ( α − − α ) Z ∞ y − α d y Z (1 + yv ) − (1 − v )d v LI, P.S., YANG, X. AND ZHOU, X. = α ( α − − α ) Z ∞ x − α (1 + x ) − d x Z v α − (1 − v )d v = α ( α − − α ) × Γ(2 − α )Γ( α )Γ(2) × Γ( α − α + 1) = Γ( α ) . Example . In order to apply Theorems 2.3, 2.8 and 2.13, we onlyneed to compare powers and coefficients of the three terms in the polynomial G a ( u ) for < a < or a > , respectively. To handle the critical case of r = r + 1 or (and) r = r + α − where some terms have the samepower, we further choose the value of a close enough to to obtain thebest possible results. For instance, if both r = r + 1 and r = r + α − hold, for b > b / c α, b , we choose the constant a satisfying < a b ≤ b < 0, then r ≥ b > 0, then r ≥ b > 0, then r ≥ α ;(ii) b > b > 0, then r ≥ ( r + 1) ∧ b > 0, then r ≥ ( r − α ) ∧ α ;(iic) b > b { r = r +1 < } + Γ( α ) b { r = r + α − <α } . In addition, under condition (i), for all x > P x { τ − = ∞ and X t → t → ∞} = 1 , i.e., extinguishing occurs.For extinction with a positive probability we have P x { τ − < ∞} > x > b ≤ ONLINEAR BRANCHING PROCESS (ia) b < r < b > r < b > r < α .(ii) b > b > r < ( r + 1) ∧ b > r < ( r + α − ∧ α ;(iic) b < b { r = r +1 < } + Γ( α ) b { r = r + α − <α } . In addition, P x { τ − < ∞} = 1 for all x > Remark . Note that the above Condition (iic) for P x { τ − = ∞} = 1 and the above Condition (iic) for P x { τ − < ∞} > agree with the corre-sponding results in Berestycki et al. (2015); see the corresponding commentsin Section 1.2. By Theorem 2.8 and Proposition 2.11, we obtain rather sharp conditionsof explosion/non-explosion for the process X in Example 2.18.For non-explosion we have P x { τ + ∞ < ∞} = 0 for all x > b ≤ b > r ≤ b > , r > b > r > r + 1;(iib) b > r > r + α − b < b { r = r +1 } + Γ( α ) b { r = r + α − } . For explosion with a positive probability we have P x { τ + ∞ < ∞} > x > b > , r > b > 0, then r ≤ r + 1;(ii) if b > 0, then r ≤ r + α − b > b { r = r +1 } + Γ( α ) b { r = r + α − } . LI, P.S., YANG, X. AND ZHOU, X. Similarly, by Theorem 2.13 we obtain rather sharp conditions for comingdown from infinity.The process X in Example 2.18 comes down from infinity if one of thefollowing holds(i) b ≤ b < r > b > r > b > r > α .(ii) b > b > r > ( r + 1) ∨ b > r > ( r + α − ∨ α ;(iic) b < b { r = r +1 > } + Γ( α ) b { r = r + α − >α } . Process X in Example 2.18 stays infinite if at least one of the followinghold.(i) b ≤ b < 0, then r ≤ b > 0, then r ≤ b > 0, then r ≤ α .(ii) b > b > 0, then r ≤ ( r + 1) ∨ b > 0, then r ≤ ( r + α − ∨ α ;(iic) b > b { r = r +1 > } + Γ( α ) b { r = r + α − >α } . From the above example we make the following observations. Remark . (i) There is no extinction if the process X has a smallenough negative drift together with small enough fluctuations near .If X has a positive drift, then the requirements on the fluctuations areweaker. Extinction happens with a positive probability if X has eithera large enough negative drift or large enough fluctuations near . Evenif X has a small positive drift near , extinction can still happen witha positive probability if the fluctuations are large enough. ONLINEAR BRANCHING PROCESS (ii) The explosion is caused by a large enough drift associated with the func-tion γ . The fluctuations of the process X associated with the functions γ and γ cannot cause explosion. But large enough fluctuations canprevent the explosion from happening. (iii) A large enough negative drift or large enough fluctuations near infinitycan cause coming down from infinity. Even if the process X has a pos-itive drift, large enough fluctuations can still cause coming down frominfinity. On the other hand, the process X with a moderate negativedrift and moderate fluctuations near infinity stays infinite, and if itallows large fluctuations, with a large enough positive drift it can stillstay infinite. Remark . If b = 0 , then X is a diffusion whose explosion behavioris characterized by Feller’s criterion; see e.g. Corollary 4.4 of Cherny andEngelbert (2005). One can check that the explosion/nonexplosion conditionsin Example 2.18 are consistent with it. Remark . Example 2.18 recovers, for the case with spectrally posi-tive stable L´evy measure specified in (1.4), the integral tests for extinction,explosion and coming down from infinity in Theorems 1.7, 1.9 and 1.11 ofLi (2018), which were proved using a very different approach. Recall that thecontinuous-state polynomial branching process in Li (2018) is the process X with power branching rate functions satisfying r i = r, i = 0 , , . By Example2.18 we have for the continuous-state polynomial branching process, • P x { τ − < ∞} > for all x > , i.e. extinction occurs, if and only if { b =0 } + α { b =0 ,b =0 } > r ; • P x { τ + ∞ < ∞} > for all x > , i.e. explosion occurs, if and only if b > and r > ; • The process X comes down from infinity if and only if b ≤ and { b =0 } + α { b =0 ,b =0 } + 21 { b =0 ,b =0 ,b =0 } < r ; which agree the integral tests in Li (2018). The next example is on the finiteness of the weighted total population S of X introduced in Section 2.4. The next results follow from Remark 2.17,Theorem 2.15 and Example 2.18. Example . Let γ ( x ) = x r for < r < min { r , r , r } in Theorem2.15. Observe that P x { S = ¯ τ − ∧ ¯ τ + ∞ = ∞} = 1 if and only if P x { ¯ τ − = ∞} = 1 LI, P.S., YANG, X. AND ZHOU, X. and P x { ¯ τ + ∞ = ∞} = 1 . The conditions for P x { S < ∞} = P x { ¯ τ − ∧ ¯ τ + ∞ < ∞} = 0 for x > can be found in Example 2.18.Similarly, observe that P x { ¯ τ − ∧ ¯ τ + ∞ < ∞} > if and only if P x { ¯ τ − < ∞} > or P x { ¯ τ + ∞ < ∞} > . Then the conditions for P x { S < ∞} > canalso be found in Example 2.18. 3. Existence and uniqueness of solution. In this section we findconditions on the functions γ i , i = 0 , , X , and consequently X is a Markov process. For thispurpose, we only need the functions γ i , i = 0 , , Theorem . Suppose that the functions γ i , i = 0 , , are locally Lip-schitz; i.e., for each closed interval A ⊂ (0 , ∞ ) , there is a constant c ( A ) > so that for any x, y ∈ A , | γ ( x ) − γ ( y ) | + | γ ( x ) − γ ( y ) | + | γ ( x ) − γ ( y ) | ≤ c ( A ) | x − y | . Then (i) For any initial value X = x ≥ , there exists a pathwise uniquesolution (defined at the beginning of Section 2) to SDE (1.3) . (ii) If in addition, γ is an increasing function, then for any y ≥ x ∈ [0 , ∞ ) and solutions X x := ( X xt ) t ≥ and X y := ( X yt ) t ≥ to SDE (1.3) with X x = x and X y = y , we have P { X yt ≥ X xt for all t ≥ } = 1 . Proof. (i) We prove the result by an approximation argument. For each n ≥ i = 0 , , γ ni ( x ) := γ i ( n ) , n < x < ∞ ,γ i ( x ) , /n ≤ x ≤ n,γ i (1 /n ) , ≤ x < /n. By pp.245–246 in Ikeda and Watanabe (1989), for each n ≥ ξ nt ) t ≥ to ξ nt = x + Z t γ n ( ξ ns )d s + Z t Z γ n ( ξ ns )0 W (d s, d u )+ Z t Z ∞ Z γ n ( ξ ns − )0 z ˜ N (d s, d z, d u ) . (3.1) ONLINEAR BRANCHING PROCESS For m, n ≥ τ nm := inf { t ≥ ξ nt ≥ m or ξ nt ≤ /m } . Then we have ξ nt = ξ mt for t ∈ [0 , τ m ∧ nm ∧ n ) and τ n + in = τ nn , i = 1 , , . . . Clearly,the sequence of stopping times { τ nn } is increasing in n . Let τ := lim n →∞ τ nn .We define the process X := ( X t ) t ≥ by X t = ξ nt for t ∈ [0 , τ nn ) and X t =lim sup n →∞ ξ nτ nn for t ∈ [ τ, ∞ ). Then τ nn := inf { t ≥ X t ≥ n or X t ≤ /n } and X is a solution of (1.3). Since the pathwise uniqueness of the solutionholds for (3.1) in the time interval [0 , τ nn ) for each n ≥ 1, there exists apathwise unique solution to (1.3).(ii) Let ( ξ xn ( t )) t ≥ denote the solution of (3.1) to indicate its dependenceon the initial state. To apply Theorem 2.2 in Dawson and Li (2012), weidentify the notation in Dawson and Li (2012) with that in this paper in thefollowing equations, where the notation on the left hand sides comes fromDawson and Li (2012) and that on the right hand sides is from the presentpaper. E = (0 , ∞ ) , U = (0 , ∞ ) , π (d u ) = d u, g ( x, z, u ) ≡ ,µ (d z, d u ) = π (d z )d u, ˜ N (d s, d z, d u ) = ˜ N (d s, d z, d u )and b ( x ) = b ( x ) = γ n ( x ) , σ ( x, u ) = 1 { u ≤ γ n ( x ) } , g ( x, z, u ) = z { u ≤ γ n ( x ) } . Then conditions (2.a,b,c) in Dawson and Li (2012) are satisfied due to theLipschitz properties of γ ni for i = 0 , , 2. Let l ( x, y, u ) := 1 { u ≤ γ ( x ) } − { u ≤ γ ( y ) } . Since the function γ ( x ) is non-decreasing in x , then for x < y we have I ( x, y ) := Z ∞ d u Z l ( x, y, u ) (1 − t )1 {| l ( x,y,u ) |≤ n } | ( x − y ) + tl ( x, y, u ) | d t = Z γ ( y ) γ ( x ) d u Z − t | ( x − y ) − t | d t ≤ ( γ ( y ) − γ ( x ))(ln( y − x + 1) − ln( y − x )) < ∞ . Similarly, I ( x, y ) < ∞ for all x ≥ y . Then condition (2.d) of Theorem 2.2in Dawson and Li (2012) holds. Now for any y ≥ x ≥ 0, by Theorem 2.2 LI, P.S., YANG, X. AND ZHOU, X. in Dawson and Li (2012) we can show that ξ yn ( t ) ≥ ξ xn ( t ) a.s. for all n and t ≥ 0. Consequently, X yt ≥ X xt a.s. for all t ≥ (cid:3) Throughout the rest of this paper, we always assume that SDE (1.3) hasa unique weak solution which is a Markov process. Remark . The solution to SDE (1.3) also arises as the weak limitin the Skorokhod space D ([0 , ∞ ) , R + ) for a sequence of discrete-state andcontinuous-time Markov chains that can be interpreted as discrete-state branch-ing processes with population dependent branching rates; see Li et al. (2018)for more details. 4. Foster-Lyapunov criteria for extinction and explosion. In thissection, we first present Foster-Lyapunov criteria type results for the process X which generalize a similar result for Markov chains; see Chen (2004, p.84).Let C [0 , ∞ ) be the space of twice continuously differentiable functionson [0 , ∞ ). Define the operator L on C [0 , ∞ ) by Lg ( y ) := γ ( y ) g ′ ( y ) + 12 γ ( y ) g ′′ ( y )+ γ ( y ) Z ∞ ( g ( y + z ) − g ( y ) − zg ′ ( y )) π (d z ) . Lemma . Given a ≥ , let g ∈ C [0 , ∞ ) be a non-negative functionsatisfying the following conditions: (i) sup y ∈ [ a,b ) | Lg ( y ) | < ∞ for all b > a , i.e., Lg is locally bounded on [ a, ∞ ) ; (ii) sup y ∈ [ a, ∞ ) g ( y ) < ∞ ; (iii) g ( a ) > and lim y →∞ g ( y ) = 0 ; (iv) For all b > a , there is a constant d b > so that Lg ( y ) ≥ d b g ( y ) for all y ∈ ( a, b ) .Then for any x > a , we have P x { τ − a < ∞} ≥ g ( x ) /g ( a ) . (4.1) Proof. For any b > x > a , by Itˆo’s formula and conditions (i) and (ii),we have g ( X t ∧ τ − a ∧ τ + b ) = g ( x ) + Z t ∧ τ − a ∧ τ + b Lg ( X s )d s + mart . ONLINEAR BRANCHING PROCESS Taking expectations on both sides, we have E x (cid:2) g ( X t ∧ τ − a ∧ τ + b ) (cid:3) = g ( x ) + Z t E x h Lg ( X s )1 { s<τ − a ∧ τ + b } i d s. By integration by parts, Z ∞ e − d b t E x h Lg ( X t )1 { t<τ − a ∧ τ + b } i d t = Z ∞ e − d b t d E x h g ( X t ∧ τ − a ∧ τ + b ) i = d b Z ∞ e − d b t E x h g ( X t ∧ τ − a ∧ τ + b ) i d t − g ( x ) . Then by (iv), d b Z ∞ e − d b t E x h g ( X t ∧ τ − a ∧ τ + b ) i d t − g ( x ) ≥ d b Z ∞ e − d b t E x h g ( X t )1 { t<τ − a ∧ τ + b } i d t. It follows that g ( x ) ≤ d b Z ∞ e − d b t E x h g ( X τ − a ∧ τ + b )1 { t ≥ τ − a ∧ τ + b } i d t ≤ g ( a ) P x { τ − a < ∞} + sup y ≥ b g ( y ) . Inequality (4.1) thus follows by letting b → ∞ and (iii). (cid:3) The proof for the next lemma is similar to that of Lemma 4.1 and we omitit. Lemma . Given < x < b , suppose there exist constants a ∈ [0 , x ) , d > and a function g ∈ C [0 , ∞ ) satisfying the following conditions: (i) sup y ∈ [ a,b ] | Lg ( y ) | < ∞ ; (ii) sup y ∈ [ a,b ) | g ( y ) | < ∞ ; (iii) g ( a ) = 0 and g ( x ) > ; (iv) Lg ( y ) ≥ dg ( y ) for all y ∈ [ a, b ] .Then we have P x { τ + b < ∞} > . As applications of Lemmas 4.1 and 4.2, we prove Propositions 2.6 and2.11 in this section. LI, P.S., YANG, X. AND ZHOU, X. Proof of Proposition 2.6. (i) Let g ( y ) = e − λy with λ > g satisfies the conditions of Lemma 4.1. Then by Lemma 2.1, we haveuniformly for all a < y < b , Lg ( y ) ≥ λ e − λy n − sup a ≤ z ≤ b ( γ ( z ) ∨ 0) + λ a ≤ z ≤ b γ ( z )+ λ inf a ≤ z ≤ b γ ( z ) Z ∞ z π (d z ) Z e − λzu (1 − u )d u o . (4.2)Observe that λ Z ∞ z π (d z ) Z e − λzu (1 − u )d u ≥ − λ Z ∞ z π (d z ) Z / e − λzu d u ≥ − Z ∞ z (1 − e − λz/ ) π (d z ) ≥ − (1 − e − / ) Z ∞ /λ zπ (d z )converges to R ∞ zπ (d z ) = ∞ as λ → ∞ . It then follows that for each b > a there is a constant d b ( λ ) > Lg ( y ) ≥ d b ( λ )e − λy , a < y < b (4.3)as λ large enough. Thus by Lemma 4.1, for x > a and large enough λ ,(4.4) P x { τ − a < ∞} ≥ e − λ ( x − a ) > , which gives (2.4).(ii) Suppose that there is a constant c > γ ( y ) ≤ y ≥ c .Similar to the argument in (4.2) and (4.3), given any λ > 0, uniformly for c ∨ a < y < b , we have Lg ( y ) ≥ λ γ ( y )e − λy + λ γ ( y )e − λy Z ∞ z π (d z ) Z e − λzu (1 − u )d u ≥ d ′ b ( λ )e − λy for some constant d ′ b ( λ ) > 0. It follows again from Lemma 4.1 that for all λ > P x { τ − l < ∞} ≥ e − λ ( x − l ) > , x > l ≥ c ∨ a. Letting λ → 0, we have(4.5) P x { τ − l < ∞} = 1 , x > l ≥ c ∨ a. It follows from (4.4) that for large enough λ , P x { τ − a < ∞} ≥ e − λ ( x − a ) , x > a. (4.6) ONLINEAR BRANCHING PROCESS For any x > a > t > 0, combining (4.5) and (4.6), by the strongMarkov property, we have P x { τ − a < ∞} = P x { τ − a < t } + Z c ∨ aa P x (cid:8) t ≤ τ − a < ∞ , X t ∈ d z (cid:9) P z { τ − a < ∞} + Z ∞ c ∨ a P x (cid:8) t ≤ τ − a < ∞ , X t ∈ d z (cid:9) P z { τ − a < ∞}≥ P x { τ − a < t } + Z c ∨ aa P x (cid:8) t ≤ τ − a < ∞ , X t ∈ d z (cid:9) P c ∨ a { τ − a < ∞} + Z ∞ c ∨ a P x (cid:8) t ≤ τ − a < ∞ , X t ∈ d z (cid:9) P c ∨ a { τ − a < ∞}≥ P x { τ − a < t } + e − λ ( c ∨ a − a ) (1 − P x { τ − a < t } ) . (4.7)Letting t → ∞ in (4.7), we have P x { τ − a < ∞} = 1. The desired result thenfollows.(iii) For any small enough ε > 0, let A n := n τ − ( ε n +1 ) < ∞ , τ + ε ◦ θ τ − ( ε n +1 ) < τ − ( ε n +2 ) ◦ θ τ − ( ε n +1 ) o , n ≥ . Since γ ( y ) ≤ y ∈ R , then ( X t ) t ≥ is a supermartingale, whichimplies ε n +1 = X τ − ( ε n +1 ) ≥ E ε n +1 h X τ + ε ∧ τ − ( ε n +2 ) i ≥ ε P ε n +1 { τ + ε < τ − ε n +2 } by optional stopping. Thus, P x { A n } ≤ E x h P τ − ( ε n +1 ) (cid:8) τ + ε < τ − ε n +2 (cid:9)i ≤ ε n . It follows from the Borel-Cantelli lemma that P x { A n i.o. } = 0. Therefore,by Proposition 2.6 (ii), we have P x -a.s. X t < ε for all t large enough andthe desired result follows. (cid:3) Proof of Proposition 2.11. Observe that there is a constant b ′ > R b ′ z π (d z ) > 0. Let m := sup y ∈ [ a,b ] | γ ( y ) | < ∞ , m = inf y ∈ [ a,b ] γ ( y ) and m = inf y ∈ [ a,b ] γ ( y ) . Since m ∨ m > 0, there exists a large enough constant c > − cm + 12 c m + 12 c m Z b ′ z π (d z ) ≥ . LI, P.S., YANG, X. AND ZHOU, X. Let g be a convex function, i.e. g ′′ ( y ) ≥ 0, satisfying g ( y ) = e cy − e ca for y ∈ [ a, b + b ′ ] and g ′′ ( y ) = 0 for y > b + b ′ + 1. Then by Lemma 2.1, it is easyto see that Z ∞ ( g ( y + z ) − g ( y ) − zg ′ ( y )) π (d z )= Z b + b ′ +10 ( g ( y + z ) − g ( y ) − zg ′ ( y )) π (d z )+ Z ∞ b + b ′ +1 ( g ( y + z ) − g ( y ) − zg ′ ( y )) π (d z ) ≤ 12 sup y ∈ [ a,b + b ′ +1] g ′′ ( y ) Z b + b ′ +10 z π (d z ) , which implies that condition (i) in Lemma 4.2 is satisfied. Observe that forany y ∈ [ a, b ], we have Z ∞ ( g ( y + z ) − g ( y ) − zg ′ ( y )) π (d z ) ≥ Z b ′ ( g ( y + z ) − g ( y ) − zg ′ ( y )) π (d z )= e cy Z b ′ (e cz − − cz ) π (d z ) ≥ c e cy Z b ′ z π (d z ) . Therefore, for any y ∈ [ a, b ], we have Lg ( y ) = γ ( y ) g ′ ( y ) + 12 γ ( y ) g ′′ ( y )+ γ ( y ) Z ∞ ( g ( y + z ) − g ( y ) − zg ′ ( y )) π (d z ) ≥ γ ( y ) c e cy + c γ ( y )e cy + c γ ( y )e cy Z b ′ z π (d z ) ≥ e cy h − cm + c m + c m Z b ′ z π (d z ) i ≥ e Cy ≥ g ( y ) . Applying Lemma 4.2 yields P x { τ + b < ∞} > (cid:3) 5. Proofs of the main results in Section 2. Recall the definitions of H a and G a in (2.1) and (2.3), respectively. We now present the martingaleswe use to show the main results on extinction, explosion and coming downfrom infinity. It is remarkable that such a martingale is enough to show allthe main results in this paper. Some other forms of martingales can only beused to prove partial results. ONLINEAR BRANCHING PROCESS Lemma . For b > ε > c > let T := τ − c ∧ τ + b . Then the process X − at ∧ T exp n R t ∧ T G a ( X s )d s o is an ( F t ) -martingale and E ε (cid:20) X − aT exp n Z T G a ( X s )d s o(cid:21) ≤ ε − a for a = 1 . Proof. By Itˆo’s formula, we can see that X − at = X − a − Z t G a ( X s ) X − as d s + (1 − a ) Z t Z γ ( X s )0 X − as W (d s, d u )+ Z t Z ∞ Z γ ( X s − )0 (cid:2) ( X s − + z ) − a − X − as − (cid:3) ˜ N (d s, d z, d u ) , and it then follows from the integration by parts formula (see e.g. Protter(2005, p. 68)) that X − at exp (cid:26)Z t G a ( X s )d s (cid:27) = X − a + Z t X − as exp (cid:26)Z s G a ( X u )d u (cid:27) G a ( X s )d s + Z t exp (cid:26)Z s G a ( X u )d u (cid:27) d( X − as )= X − a + local mart.Therefore, t X − at ∧ T exp n Z t ∧ T G a ( X s )d s o (5.1)is a local martingale. By Protter (2005, p. 38), (5.1) is a martingale if E ε (cid:20) sup t ∈ [0 ,δ ] X − at ∧ T exp n Z t ∧ T G a ( X s )d s o(cid:21) < ∞ . (5.2)for each δ > 0. Observe that for 0 ≤ t ≤ δ exp n Z t ∧ T G a ( X s )d s o is uniformly bounded from above by a positive constant. Then (5.2) is obvi-ous for a > 1. In the following we consider the case a < 1. By the Burkholder-Davis-Gundy inequality, we have E ε (cid:20) sup t ∈ [0 ,δ ] (cid:12)(cid:12)(cid:12) Z t ∧ T Z γ ( X s − )0 W (d s, d u ) (cid:12)(cid:12)(cid:12) (cid:21) LI, P.S., YANG, X. AND ZHOU, X. ≤ C E ε (cid:20) Z δ ∧ T d s Z γ ( X s − )0 d u (cid:21) ≤ Cδ sup x ∈ [0 ,b ] γ ( x )(5.3)and E ε (cid:20) sup t ∈ [0 ,δ ] (cid:12)(cid:12)(cid:12) Z t ∧ T Z Z γ ( X s − )0 z ˜ N (d s, d z, d u ) (cid:12)(cid:12)(cid:12) (cid:21) ≤ C E ε (cid:20) Z δ ∧ T d s Z z π (d z ) Z γ ( X s − )0 d u (cid:21) ≤ Cδ Z z π (d z ) sup x ∈ [0 ,b ] γ ( x ) . (5.4)Observe that E ε (cid:20) sup t ∈ [0 ,δ ] (cid:12)(cid:12)(cid:12) Z t ∧ T Z ∞ Z γ ( X s − )0 z ˜ N (d s, d z, d u ) (cid:12)(cid:12)(cid:12)(cid:21) ≤ E ε (cid:20) sup t ∈ [0 ,δ ] (cid:12)(cid:12)(cid:12) Z t ∧ T Z ∞ Z γ ( X s − )0 zN (d s, d z, d u ) (cid:12)(cid:12)(cid:12)(cid:21) + E ε (cid:20) sup t ∈ [0 ,δ ] (cid:12)(cid:12)(cid:12) Z t ∧ T d s Z ∞ zπ (d z ) Z γ ( X s − )0 d u (cid:12)(cid:12)(cid:12)(cid:21) ≤ δ Z ∞ zπ (d z ) sup x ∈ [0 ,b ] γ ( x ) . (5.5)It then follows from (1.3) and (5.3)–(5.5) that E ε [sup t ∈ [0 ,δ ] X t ∧ T ] < ∞ , whichimplies (5.2). Now by Fatou’s lemma, we get E ε (cid:20) X − aT exp n Z T G a ( X s )d s o(cid:21) = E ε (cid:20) lim t →∞ X − at ∧ T exp n Z t ∧ T G a ( X s )d s o(cid:21) ≤ lim t →∞ E ε (cid:20) X − at ∧ T exp n Z t ∧ T G a ( X s )d s o(cid:21) = ε − a , which finishes the proof. (cid:3) Proof of Theorem 2.3. (i) In the present proof for n = 2 , , . . . , let T n := τ − ( ε n ) ∧ τ + b for small enough 0 < ε < b . It follows from Lemma 5.1 that ε − a ≥ E ε h X − aτ − ( ε n ) ∧ τ + ( b ) exp (cid:8) − (ln ε − n ) r ( τ − ( ε n ) ∧ τ + ( b ) (cid:9)i ≥ E ε h X − aτ − ( ε n ) exp (cid:8) − (ln ε − n ) r d n (cid:9) { τ − ( ε n ) <τ + b ∧ d n } i = ε (1 − a ) n exp { ln ε n ( a − / } P ε (cid:8) τ − ( ε n ) < τ + b ∧ d n (cid:9) , ONLINEAR BRANCHING PROCESS where d n := ln ε n ( a − / − (ln ε − n ) r = n ( a − / ε − n r (ln ε − ) r → ∞ as n → ∞ . Then P ε (cid:8) τ − ( ε n ) < τ + b ∧ d n (cid:9) ≤ ε ( a − n − / . By the Borel-Cantelli Lemma we have P ε n τ − ( ε n ) < τ + b ∧ d n i.o. o = 0 . (5.6)Then P ε -a.s., τ − ( ε n ) ≥ τ + b ∧ d n for n large enough.Now if there are infinitely many n so that τ − ( ε n ) ≥ d n , (5.7)then we have τ − = ∞ ; on the other hand, if (5.7) holds for at most finitelymany n , then by (5.6) we have τ + b < τ − ( ε n ) for all n large enough. Com-bining these two cases,(5.8) P ε { τ − = ∞ or τ + b < τ − } = 1 . It follows from the Markov property and lack of negative jumps for X thatif E ε [e − λτ − ; τ − < ∞ ] > λ > 0, then E ε [e − λτ − ; τ − < ∞ ] = E ε [e − λτ − ; τ + b < τ − < ∞ ] ≤ E ε [e − λτ + b ; τ + b < τ − ] E b [e − λτ − ε ; τ − ε < ∞ ] × E ε [e − λτ − ; τ − < ∞ ] < E ε [e − λτ − ; τ − < ∞ ] , where we need (5.8) for the first equation. Therefore, E ε [e − λτ − ; τ − < ∞ ] = 0and consequently, P ε { τ − < ∞} = 0.One can also find similar arguments in the proof of Theorem 4.2.2 in Le(2014) and the proof of Theorem 2.8(2) in Le and Pardoux (2015).(ii) Given 0 < δ < − a , consider the martingale X − at ∧ T exp n Z t ∧ T G a ( X s )d s o LI, P.S., YANG, X. AND ZHOU, X. for T = τ − ( ε δ ) ∧ τ + ( ε − δ ). By Lemma 5.1, ε − a ≥ E ε h X − aτ + ( ε − δ ) exp n Z τ + ( ε − δ )0 G a ( X s )d s o { τ + ( ε − δ ) <τ − ( ε δ ) } i ≥ ε (1 − a )(1 − δ ) P ε (cid:8) τ + ( ε − δ ) < τ − ( ε δ ) (cid:9) . Then(5.9) P ε (cid:8) τ + ( ε − δ ) < τ − ( ε δ ) (cid:9) ≤ ε (1 − a ) δ . Similarly, ε − a ≥ E ε h X − at exp n Z t G a ( X s )d s o { τ + ( ε − δ )= τ − ( ε δ )= ∞} i . Letting t → ∞ we have(5.10) P ε (cid:8) τ + ( ε − δ ) = τ − ( ε δ ) = ∞ (cid:9) = 0 . By Lemma 5.1 again, for t ( ε ) := [ − (1 − δ ) ln ε ] − r we have ε − a ≥ E ε h X − aτ − ( ε δ ) exp n Z τ − ( ε δ )0 G a ( X s )d s o { t ( ε ) <τ − ( ε δ ) <τ + ( ε − δ ) } i ≥ ε (1 − a )(1+ δ ) E ε h e [ − (1 − δ ) ln ε ] r t ( ε ) { t ( ε ) <τ − ( ε δ ) <τ + ( ε − δ ) } i = ε (1 − a )(1+ δ ) ε − (1 − δ ) P ε n t ( ε ) < τ − ( ε δ ) < τ + ( ε − δ ) o . Then P ε n t ( ε ) < τ − ( ε δ ) < τ + ( ε − δ ) o ≤ ε ( a − δ ε − δ = ε a − δ . (5.11)Combining (5.9), (5.10) and (5.11), we have P ε { τ − ( ε δ ) > t ( ε ) } ≤ ε a − δ + ε (1 − a ) δ < ε (1 − a ) δ . By the strong Markov property and lack of negative jumps for process X , P ε n m \ n =0 (cid:8) τ − ( ε (1+ δ ) n ) < ∞ , τ − ( ε (1+ δ ) n +1 ) ◦ θ τ − ( ε (1+ δ ) n ) ≤ t ( ε (1+ δ ) n ) (cid:9)o = m Y n =0 P ε (1+ δ ) n n τ − ( ε (1+ δ ) n +1 ) ≤ t ( ε (1+ δ ) n ) o ≥ m Y n =0 h − ε (1+ δ ) n (1 − a ) δ i ≥ m Y n =0 e − ε (1+ δ ) n (1 − a ) δ ≥ e − ε (1 − a ) δ . ONLINEAR BRANCHING PROCESS Letting m → ∞ we have P ε n ∞ \ n =0 n τ − ( ε (1+ δ ) n ) < ∞ ,τ − ( ε (1+ δ ) n +1 ) ◦ θ τ − ( ε (1+ δ ) n ) ≤ t ( ε (1+ δ ) n ) oo ≥ e − ε (1 − a ) δ . Since under P ε , τ − − = ∞ X n =0 τ − ( ε (1+ δ ) n +1 ) ◦ θ τ − ( ε (1+ δ ) n ) , then P ε n τ − − ≤ ∞ X n =0 t ( ε (1+ δ ) n ) o ≥ e − ε (1 − a ) δ . Notice that for ε n := ε (1+ δ ) n , ∞ X n =1 t ( ε n ) = ∞ X n =1 (cid:2) ( δ − 1) ln ε n (cid:3) − r = ∞ X n =1 (cid:2) (1 + δ ) n ( δ − 1) ln ε (cid:3) − r < ∞ , we thus have P ε (cid:8) τ − − < ∞ (cid:9) ≥ e − ε (1 − a ) δ . By the definition of solution to SDE (1.3) at the beginning of Section 2, wehave P ε (cid:8) τ − = τ − − < ∞ (cid:9) ≥ e − ε (1 − a ) δ , (5.12)which finishes the proof. (cid:3) Proof of Theorem 2.8. (i) In the present proof, for small enough b − and ε satisfying 0 < b < ε − and for n = 2 , , . . . , let T n := τ − ( b ) ∧ τ + ( ε − n ). ByLemma 5.1 we have ε a − ≥ E ε − h X − aτ + ( ε − n ) ∧ τ − ( b ) exp n − (ln ε − n ) r ( τ + ( ε − n ) ∧ τ − ( b )) oi ≥ E ε − h X − aτ + ( ε − n ) exp (cid:8) − (ln ε − n ) r d n (cid:9) { τ + ( ε − n ) <τ − ( b ) ∧ d n } i ≥ ε ( a − n E ε − h exp { ln ε (1 − a ) n/ } { τ + ( ε − n ) <τ − ( b ) ∧ d n } i for b and ε − large enough, where d n := (1 − a ) n ln ε − ε − n ) r = (1 − a ) n − r ε − ) − r → ∞ LI, P.S., YANG, X. AND ZHOU, X. as n → ∞ . Then P ε − n τ + ε − n < τ − ( b ) ∧ d n o ≤ ε (1 − a )( n − / for large enough b and ε − . The desired result of part (i) then follows froman argument similar to that in the proof for Theorem 2.3 (i).(ii) Taking T := τ − ( ε − δ ) ∧ τ + ( ε − − δ ) in Lemma 5.1, we get ε a − ≥ E ε − h X − aτ − ( ε − δ ) exp n Z τ − ( ε − δ )0 G a ( X s )d s o × { τ − ( ε − δ ) <τ + ( ε − − δ ) } i ≥ ε ( a − − δ ) P ε − n τ − ( ε − δ ) < τ + ( ε − − δ ) o . Then(5.13) P ε − n τ − ( ε − (1 − δ ) ) < τ + ( ε − (1+ δ ) ) o ≤ ε ( a − δ . Similarly, ε a − ≥ E ε − h X − an exp n Z n G a ( X s )d s o { τ − ( ε − δ ) ∧ τ + ( ε − − δ ) >n } i ≥ ε (1+ δ )( a − e nε − δ P ε − { τ − ( ε − δ ) ∧ τ + ( ε − − δ ) > n } . Letting n → ∞ we have(5.14) P ε − n τ − ( ε − (1 − δ ) ) = τ + ( ε − (1+ δ ) ) = ∞ o = 0 . Let t ( y ) := (ln y − δ ) − r for y > δ . With T replacedby t ( ε − ) ∧ τ − ( ε − δ ) ∧ τ + ( ε − − δ ), similar to the above argument we get ε a − ≥ E ε − h X − at ( ε − ) exp n Z t ( ε − )0 G a ( X s )d s o × { t ( ε − ) <τ + ( ε − (1+ δ ) ) <τ − ( ε − (1 − δ ) ) } i ≥ ε ( a − δ ) E ε − h e (( δ − 1) ln ε ) r t ( ε − ) { t ( ε − ) <τ + ( ε − (1+ δ ) ) <τ − ( ε − (1 − δ ) ) } i = ε ( a − δ ) E ε − h e ( δ − 1) ln ε { t ( ε − ) <τ + ( ε − (1+ δ ) ) <τ − ( ε − (1 − δ ) ) } i . Then P ε − n t ( ε − ) < τ + ( ε − (1+ δ ) ) < τ − ( ε − (1 − δ ) ) o ≤ ε (1 − a ) δ e − ( δ − 1) ln ε = ε − aδ . (5.15) ONLINEAR BRANCHING PROCESS Combining (5.13), (5.14) and (5.15), we have(5.16) P ε − (cid:8) τ + ( ε − (1+ δ ) ) > t ( ε − ) (cid:9) ≤ ε ( a − δ . Write ˜ τ := 0 and ˜ τ n +1 := ˜ τ + (( X ˜ τ n ∨ δ ) ◦ ˜ τ n for n = 0 , , , . . . withthe convention X ∞ = 0. Notice that X allows possible positive jumps, andunder P ε − for n ≥ X ˜ τ n ≥ ε − (1+ δ ) n if ˜ τ n < ∞ .Observe that under P ε − , if ˜ τ n < ∞ for all n ≥ 1, then ∞ X n =1 t ( X ˜ τ n ) ≤ ∞ X n =1 (cid:2) ln ε − (1+ δ ) n (1 − δ ) (cid:3) − r = ∞ X n =1 (cid:2) (1 + δ ) n ( δ − 1) ln ε (cid:3) − r < ∞ . By the strong Markov property and estimate (5.16), we can show that P ε − { ˜ τ + ∞ < ∞} = P ε − (cid:8) lim n ˜ τ n < ∞ (cid:9) ≥ P ε − (cid:8) ˜ τ n +1 < t ( X ˜ τ n ) for all n ≥ (cid:9) ≥ ∞ Y n =1 h − ε ( a − δ (1+ δ ) n i > . This finishes the proof. (cid:3) Proof of Theorem 2.13. To show part (i), for any constants d > b > u > b , for any 0 < ε < b − , we have P ε − { τ − b < d }≤ P ε − { τ − b < d ∧ τ + ( ε − ) } + ∞ X n =1 P ε − ( τ + ( ε − n ) < τ − b < d, sup ≤ s ≤ τ − b X s ∈ [ ε − n , ε − n +1 ) ) ≤ P ε − { τ − b < d ∧ τ + ( ε − ) } + ∞ X n =1 P ε − (cid:8) τ + ( ε − n ) < τ − b ,τ − b ◦ θ ( τ + ( ε − n )) < d ∧ τ + ( ε − n +1 ) ◦ θ ( τ + ( ε − n )) o ≤ P ε − { τ − b < d ∧ τ + ( ε − ) } + ∞ X n =1 E ε − n { τ + ( ε − n ) <τ − b } P X τ +( ε − n ) n τ − b < d ∧ τ + ( ε − n +1 ) oo . (5.17) LI, P.S., YANG, X. AND ZHOU, X. By Lemma 5.1, for ε − n ≤ x < ε − n +1 , T := τ − b ∧ τ + ( ε − n +1 ) ∧ d and n = 0 , , , . . . , x − a ≥ E x X − aT exp (cid:26)Z T G a ( X s )d s (cid:27) ≥ E x X − aT exp (cid:26) − Z T (ln( X s )) r d s (cid:27) ≥ b − a E x h exp {− d (ln ε − n +1 ) r } ; τ − b < τ + ( ε − n +1 ) ∧ d i = b − a E x h exp {− d (2 n +1 ln ε − ) r } ; τ − b < τ + ( ε − n +1 ) ∧ d i . Then P x { τ − b < τ + ( ε − n +1 ) ∧ d } ≤ b a − exp { (1 − a )2 n ln ε − + d (2 n +1 ln ε − ) r }≤ b a − e (1 − a )2 n − ln ε − = b a − ε ( a − n − . for all small enough ε > 0. It follows from (5.17) and the strong Markovproperty that P ε − { τ − b < d } ≤ b a − ∞ X n =0 ε ( a − n − , which goes to 0 as ε → b > d > X thus stays at infinity.We now proceed to show the part (ii). Write t ( x ) := (1 + δ ) r (ln x ) − r for x > 1. Then by Lemma 5.1, for large x , x − a ≥ E x ( X − aτ + ( x (1+ δ ) ) exp (Z τ + ( x (1+ δ ) )0 G a ( X s )d s ) { τ − ( x (1+ δ ) − ) >τ + ( x δ ) } ) ≥ x (1 − a )(1+ δ ) P x { τ − ( x (1+ δ ) − ) > τ + ( x δ ) } . Then(5.18) P x { τ − ( x (1+ δ ) − ) > τ + ( x δ ) } ≤ x − δ (1 − a ) . ONLINEAR BRANCHING PROCESS By condition (2.11) we also have x − a ≥ E x ( X − aτ − ( x (1+ δ ) − ) exp (Z τ − ( x (1+ δ ) − )0 G a ( X s )d s )) × { t ( x ) <τ − ( x (1+ δ ) − ) <τ + ( x δ ) } ≥ x (1 − a ) / (1+ δ ) E x ( exp (Z t ( x )0 (ln X s ) r d s ) { t ( x ) <τ − ( x (1+ δ ) − ) <τ + ( x δ ) } ) ≥ x (1 − a ) / (1+ δ ) e (1+ δ ) − r (ln x ) r t ( x ) P x { t ( x ) < τ − ( x (1+ δ ) − ) < τ + ( x δ ) } = x (1 − a ) / (1+ δ ) x P x { t ( x ) < τ − ( x (1+ δ ) − ) < τ + ( x δ ) } . It follows that(5.19) P x { t ( x ) < τ − ( x (1+ δ ) − ) < τ + ( x δ ) } ≤ x δ (1 − a )1+ δ − = x − (1+ δa ) / (1+ δ ) . Combining (5.18) and (5.19) we have P x { t ( x ) < τ − ( x (1+ δ ) − ) } ≤ x − (1+ δa ) / (1+ δ ) + x − δ (1 − a ) ≤ x − δ (1 − a ) for small enough δ > 0. Then for b ≡ b ( δ ) large enough, by the strongMarkov property P b (1+ δ ) m n ∩ mn =1 n τ − ( b (1+ δ ) n ) < ∞ , τ − ( b (1+ δ ) n − ) ◦ θ τ − ( b (1+ δ ) n ) ≤ t ( b (1+ δ ) n ) oo = m Y n =1 P b (1+ δ ) n n τ − ( b (1+ δ ) n − ) ≤ t ( b (1+ δ ) n ) o ≥ m Y n =1 (cid:16) − b − δ (1 − a )(1+ δ ) n − (cid:17) ≥ m Y n =1 e − b − δ (1 − a )(1+ δ ) n − = e − P mn =1 b − δ (1 − a )(1+ δ ) n − ≥ e − b − δ (1 − a ) . Let m → ∞ . Thenlim x →∞ P x ( τ − b ≤ ∞ X n =1 t ( b (1+ δ ) n ) = (1 + δ ) r (ln b ) − r ∞ X n =1 (1 + δ ) (1 − r ) n < ∞ ) ≥ e − b − δ (1 − a ) (5.20)for r > 1. Letting b → ∞ in (5.20), we obtain the limit (2.7) and the process X comes down from infinity. (cid:3) LI, P.S., YANG, X. AND ZHOU, X. Acknowledgements. The authors are grateful to an anonymous ref-eree and an associate editor for very helpful comments. Pei-Sen Li thanksConcordia University where part of the work on this paper was carried outduring his visit as a postdoctoral fellow. References. 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Springer-Verlag, Berlin. LI, P.S., YANG, X. AND ZHOU, X. Address of the First and Second authorsE-mail: [email protected]@mail.bnu.edu.cn