An application of the backbone decomposition to supercritical super-Brownian motion with a barrier
aa r X i v : . [ m a t h . P R ] F e b An application of the backbonedecomposition to supercriticalsuper-Brownian motion with a barrier
A. Kyprianou ∗ A. Murillo-Salas † and J. L. P´erez ∗ , ‡ November 7, 2018
Abstract
We analyse the behaviour of supercritical super-Brownian motion with a barrier throughthe pathwise backbone embedding of Berestycki et al. [3]. In particular, by consideringexisting results for branching Brownian motion due to Harris et al. [15] and Maillard[20], we obtain, with relative ease, conclusions regarding the growth in the right mostpoint in the support, analytical properties of the associated one-sided FKPP waveequation as well as the distribution of mass on the exit measure associated with thebarrier.
Key words and phrases : Super-Brownian motion, backbone decomposition, killedsuper-Brownian motion.MSC 2010 subject classifications: 60J68, 35C07.
Suppose that X = { X t : t ≥ } is a (one-dimensional) super-diffusion with motion corre-sponding to that of a Brownian motion with drift − ρ ∈ R , stopped at zero, and branchingmechanism ψ taking the form ψ ( λ ) = − αλ + βλ + Z (0 , ∞ ) ( e − λx − λx )Π(d x ) , (1) ∗ Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA27AY, UK. † Departamento de Matem´aticas, Universidad de Guanajuato, Jalisco s/n, Mineral deValenciana, Guanajuato, Gto. C.P. 36240, M´exico. ‡ Department of Statistics, ITAM, Rio Hondo 1, Tizapan 1 San Angel, 01000 M´exico, D.F. λ ≥ α = − ψ ′ (0 + ) ∈ (0 , ∞ ), β ≥ , ∞ )which satisfies R (0 , ∞ ) ( x ∧ x )Π(d x ) < ∞ . We also insist that β > ≡
0. The existenceof this class of superprocesses is guaranteed by [5, 6, 8].Let M F ( I ) be the space of finite measures on I ⊆ R and note that X is a M F [0 , ∞ )-valued Markov process under P µ for each µ ∈ M F [0 , ∞ ), where P µ is law of X with initialconfiguration µ . One may think of P µ as a law on cadlag path-space X := D ([0 , ∞ ) ×M F [0 , ∞ )). Henceforth we shall use standard inner product notation, for f ∈ C + b [0 , ∞ ) and µ ∈ M F [0 , ∞ ), h f, µ i = Z R f ( x ) µ (d x ) . Accordingly we shall write || µ || = h , µ i .Recall that the total mass of the process X is a continuous-state branching process withbranching mechanism ψ . Since there is no interaction between spatial motion and branch-ing we can characterise our ψ -superdiffusion as supercritical on account of the assumption α = − ψ ′ (0 + ) >
0. Such processes may exhibit explosive behaviour, however, under theconditions assumed above, X remains finite at all positive times. We insist moreover that ψ ( ∞ ) = ∞ which means that with positive probability the event lim t ↑∞ || X t || = 0 will oc-cur. Equivalently this means that the total mass process does not have monotone increasingpaths; see for example the summary in Chapter 10 of Kyprianou [17]. The probability of theevent E := { lim t ↑∞ || X t || = 0 } is described in terms of the largest root, say λ ∗ , of the equation ψ ( λ ) = 0. It is known (cf.Chapter 8 of [17]) that ψ is strictly convex with ψ (0) = 0 and hence since ψ ( ∞ ) = ∞ and ψ ′ (0 + ) < , ∞ ), one of which is always 0. For µ ∈ M F [0 , ∞ ) we have P µ (lim t ↑∞ || X t || = 0) = e − λ ∗ || µ || . (2)It is a straightforward exercise (cf. Lemma 2 of [3] or Theorem 2.6 of [25]) to show that thelaw of X under P µ conditioned on E is that of another super-diffusion with the same motioncomponent as X , but with a new branching mechanism which is given by ψ ∗ ( λ ) = ψ ( λ + λ ∗ )for λ ≥
0. Said another way, the aforesaid super-diffusion has semigroup characterised bythe non-linear equation (6) with the quantity ψ replaced by ψ ∗ . We denote its law by P ∗ µ .In this article we shall also assume that Z ∞ qR ξλ ∗ ψ ( u )d u d ξ < ∞ . (3)The condition (3) implies in particular that R ∞ /ψ ( ξ )d ξ < ∞ (cf. [25]) which in turnguarantees that the event E agrees with the event of extinction , namely { ζ X < ∞} where ζ X = inf { t > || X t || = 0 } . Note that (3) cannot be satisfied for branching mechanisms which belong to bounded varia-tion spectrally positive L´evy processes. 2n this paper our objective is to show the robustness of a recent pathwise backbonedecomposition, described in detail in the next section, as a mechanism for transferring resultsfrom branching diffusions directly into the setting of superprocesses. We shall do this bydemonstrating how two related fundamental results for branching Brownian motion with akilling barrier induce the same results for a ψ -super-Brownian motion with killing at theorigin. The latter, which we shall denote by X + = { X + t : t ≥ } , can be defined on the sameprobability space as X by simply taking X + t ( · ) = X t ( · ∩ (0 , ∞ )) . (4)For f ∈ C + b (0 , ∞ ), µ ∈ M F (0 , ∞ ), x > t ≥ − log E µ ( e −h f,X t i ) = Z (0 , ∞ ) u f ( x, t ) µ (d x ) , t ≥ , (5)describes the semi-group of X , where u f is the unique positive solution to u f ( x, t ) = E − ρx [ f ( B t ∧ τ )] − E − ρx (cid:20)Z t ∧ τ ψ ( u f ( B s , t − s ))d s (cid:21) x, t ≥ . (6)Here, E − ρx is expectation with respect to P − ρx , under which { B t : t ≥ } is a Brownianmotion with drift − ρ , issued from x ≥ τ = inf { t > B t < } . The reader is referredto Theorem 1.1 of Dynkin [4], Proposition 2.3 of Fitzsimmons [14] and Proposition 2.2 ofWatanabe [27] for further details; see also Dynkin [6, 8] for a general overview.Our first result, based on the branching particle analogue in [15], shows that the classicalgrowth of the right most point in the support and its intimate relation with non-negativestationary solutions to (6) can also be seen in the superprocess context. Specifically, wemean solutions of the form u ( x, t ) = Φ( x ), which necessarily solveΦ( x ) = E − ρx [Φ( B t ∧ τ )] − E − ρx (cid:20)Z t ∧ τ ψ (Φ( B s ))d s (cid:21) , x ≥ . If we additionally suppose, for technical reasons which are soon to become apparent, thatΦ monotonically connects zero the origin to λ ∗ at + ∞ , then it is a straightforward exerciseusing classical Feynman-Kac representation of solutions to ODEs to show that Φ solves thedifferential equation12 Φ ′′ ( x ) − ρ Φ ′ ( x ) − ψ (Φ( x )) = 0 on x > ∞ ) = λ ∗ . (7)In that case we call Φ a wave solution to (7). Theorem 1.1 (Strong law for the support)
Define R Xt := inf { y > X t ( y, ∞ ) = 0 } = inf { y > X + t ( y, ∞ ) = 0 } (8) and denote the extinction time of X + by ζ X + = inf { t > || X + t || = 0 } . i) Assume that −∞ < ρ < √ α . Then, for all x > , lim t →∞ R Xt t = √ α − ρ on { ζ X + = ∞} , (9) P δ x -almost surely and Φ( x ) := − log P δ x ( ζ X + < ∞ ) , for all x > . (10) is the unique wave solution to (7).(ii) For all ρ ≥ √ α there exists no monotone wave solution to (7) and P δ x ( ζ X + < ∞ ) = 1 for all x > . Remark 1.2
Whilst Theorem 1.1 offers results on the existence and uniqueness of solutionsto (7) we do not claim that these are necessarily new. Indeed one may extract the same orsimilar results using the methods in, for example Kamataka [16], Uchyama [26] and Pinsky[22]. See also the discussion in Remark 3.1 below.Our second result looks at the distribution of mass that is absorbed at the origin, when ρ takes the critical value √ α , in the spirit of recent results of Addario-Berry and Broutin[1], A¨ıd´ekon et al. [2] and Maillard [20]. In order to describe this result we need to introducethe concept of Dynkin’s exit measures.For each x ∈ R , suppose we defined the superprocess Y = { Y t : t ≥ } under Q δ x tohave the same branching mechanism as ( X, P δ x ) however, the underlying motion associatedwith Y is that of a Brownian motion with drift − ρ (i.e. no stopping at 0). The existenceof ( Y, Q δ x ) is justified through the same means as for ( X, P δ x ). In principle it is possible toconstruct these two processes on the same probability space, however, this is unnecessary forour purposes. For each z, t ≥
0, define the space-time domain D t − z = { ( x, u ) ∈ R × [0 , ∞ ) : u < t, x > − z } . According to Dynkin’s theory of exit measures outlined in Section 7 of[7] and Section 1 of [9], it is possible to describe the mass in the superprocess Y as itfirst exits the domain D t − z . In particular, according to the characterisation for branchingMarkov exit measures given in Section 1.1 of [9], the random measure Y D t − z is supported on ∂D t − z = ( {− z } × [0 , t )) ∪ ([ − z, ∞ ) × { t } ) and is characterised by the Laplace functional Q δ x ( e −h f,Y Dt − z i ) = e − u zf ( x,t ) , where x ≥ − z , f ∈ C b ([ − z, ∞ ) × [0 , ∞ )) and u zf ( x, t ) uniquely solves, amongst non-negativesolutions, (cf. Theorem 6.1 of [7]) the equation u zf ( x, t ) = E − ρx [ f ( B t ∧ τ − z , t ∧ τ − z )] − E − ρx (cid:20)Z t ∧ τ − z ψ ( u zf ( B u , t − u ))d u (cid:21) x ≥ − z, t ≥ , (11)where τ − z = inf { t > B t < − z } . Intuitively speaking, one should think of Y D t − z as theanalogue of the atomic measure supported on ∂D t − z which describes the collection of particles4nd their space-time position in a branching Brownian motion with drift − ρ who are first intheir genealogical line of descent to exit the space-time domain ( − z, ∞ ) × [0 , t ).In the case that ρ ≥ √ α , it was shown in Theorem 3.1 of [18] that the the limitingrandom measure Y D − z = lim t ↑∞ Y D t − z (which exists almost surely by monotonicity) is almostsurely finite and has total mass which satisfies Q δ x ( e − θ || Y D − z || ) = e − v θ ( x + z ) , for θ ≥ x ≥ − z , where 12 v ′′ θ ( x ) − ρv ′ θ ( x ) − ψ ( v θ ( x )) = 0 , with v θ (0) = θ . In particular, { v θ ( x ) : x ≥ } is the semigroup of a continuous-statebranching process with branching mechanism which satisfies ψ D ( λ ) = Ψ ′ (Ψ − ( λ )) , for λ ∈ [0 , λ ∗ ], where Ψ is the unique monotone solution to the wave equation12 Ψ ′′ ( x ) + ρ Ψ ′ ( x ) − ψ (Ψ( x )) = 0 on R with Ψ( −∞ ) = λ ∗ and Ψ(+ ∞ ) = 0 . (12)Indeed, it was shown in Theorem 3.1 of [18] that || Y D || := {|| Y D − z || : z ≥ } is a continuous-state branching process with growth rate ρ + p ρ − α .We are now ready to state our second main result, based on the branching Brownianmotion analogue in [20], which in particular focuses on the case that the underlying motionhas a critical speed √ α . Theorem 1.3 (Absorbed mass at criticality)
Set ρ = √ α . Assume that for some ε > , Z [1 , ∞ ) x (log x ) ε Π(d x ) < ∞ . (13) Then for each z, x > we have Q δ x ( || Y D − z || > t ) ∼ √ α ( x + z ) e ( x + z ) √ α t (log t ) . as t ↑ ∞ . Note that in terms of our earlier notation, we see that, X + t under P δ x has the same law as Y D t | (0 , ∞ ) ×{ t } under Q δ x . Whilst Theorem 1.1 therefore concerns the spatial evolution of thesupport of the measure Y D t away from the origin for speeds ρ > √ α , by contrast Theorem1.3 above addresses the distribution of mass accumulated at the origin by the same measure,at the critical speed √ α .The remainder of this paper is structured as follows. In the next section we give a briefoverview of the backbone decomposition for X , noting that similar decompositions also holdfor a number of other processes used in this article. In Section 3 we prove Theorem 1.1 andin Section 4 we prove Theorem 1.3. 5 The backbone decomposition and Poissonisation
As alluded to above, our results are largely driven by the backbone decomposition, re-cently described in the pathwise sense by [3] for conservative processes. Note that backbonedecompositions have been known in the earlier and more analytical setting of semigroupdecompositions through the work of [12] and [11] as well as in the pathwise setting in thework of [23, 24].To describe the backbone decomposition in detail, consider the process { Λ Xt : t ≥ } which has the following pathwise construction. First sample from a branching particle dif-fusion with branching generator F ( r ) = q X n ≥ p n r n − r ! = 1 λ ∗ ψ ( λ ∗ (1 − r )) , r ∈ [0 , , (14)and particle motion which is that of a Brownian motion with drift − ρ , stopped at the origin.Note that in the above generator, we have that q is the rate at which individuals reproduceand { p n : n ≥ } is the offspring distribution. With the particular branching generator givenby (14), q = ψ ′ ( λ ∗ ), p = p = 0, and for n ≥ p n := p n [0 , ∞ ) where for y ≥
0, we definedthe measure p n ( · ) on { , , , . . . } × [0 , ∞ ) by p n (d y ) = 1 λ ∗ ψ ′ ( λ ∗ ) (cid:26) β ( λ ∗ ) δ (d y ) { n =2 } + ( λ ∗ ) n y n n ! e − λ ∗ y Π(d y ) (cid:27) . If we denote the aforesaid branching particle diffusion by Z X = { Z Xt : t ≥ } then weshall also insist that the configuration of particles in space at time zero, Z , is given by anindependent Poisson random measure with intensity λ ∗ µ . Next, dress the branches of thespatial tree that describes the trajectory of Z X in such a way that a particle at the space-timeposition ( ξ, t ) ∈ [0 , ∞ ) has an independent X -valued trajectory grafted on to it with rate2 β d N ∗ ξ + Z ∞ ye − λ ∗ y Π(d y )d P ∗ ξδ y . Here the measure N ∗ ξ is the excursion measure (cf. [9, 19, 10]) on the space X which satisfies N ∗ x (1 − e −h f,X t i ) = u ∗ f ( x, t ) , for x, t ≥ f ∈ C + b [0 , ∞ ), where u ∗ f ( x, t ) is the unique solution to (6) with the branchingmechanism ψ replaced by ψ ∗ . Moreover, on the event that an individual in Z X dies andbranches into n ≥ ξ ∈ [0 , ∞ ), with probability p n (d y ) P ∗ yδ ξ , anadditional independent X -valued trajectory is grafted on to the space-time branching point.The quantity Λ Xt is now understood to be the total dressed mass present at time t togetherwith the mass present at time t of an independent copy of ( X, P ∗ µ ) issued at time zero. Wedenote the law of (Λ X , Z X ) by P µ .The backbone decomposition is now summarised by the following theorem lifted fromBerestycki et al. [3]. 6 heorem 2.1 For any µ ∈ M F ( R d ) , the process (Λ X , P µ ) is Markovian and has the samelaw as ( X, P µ ) . Moreover, for each t ≥ , the law of Z Xt given Λ Xt is that of a Poissonrandom measure with intensity measure λ ∗ Λ Xt . Not much changes in the above account when we replace the role of X by the superprocess Y or indeed the continuous-state branching process || Y D || . Specifically, for the case of Y ,the motion of the backbone, Z Y , is that of a Brownian motion with drift − ρ and ψ remainsthe same. For the case of || Y D || , we may consider the motion process to be that of a particlewhich remains fixed at a point and the branching mechanism ψ is replaced by ψ D . Proof of (i): Using obvious notation in light of (4), and referring to the discussion followingTheorem 1.3, we necessarily have that R Xt is equal in law to inf { y > Y D t | (0 , ∞ ) ×{ t } ( y, ∞ ) =0 } , and the latter is Q δ x -almost surely bounded above by R Yt . It is known from Corollary3.2 of [18] that, under (3), for any ρ ≤ √ α ,lim t →∞ R Yt t = √ α − ρ Q δ x -almost surely on the survival set of Y . It follows that, under the same assumptions,lim sup t →∞ R Xt t ≤ √ α − ρ on { ζ X + = ∞} , (15) P δ x -almost surely.For the lower bound, note that the backbone decomposition allows us to deduce straightaway that, again using obvious notation, on { ζ Λ X + = ∞} , R Z X t ≤ R Λ X t holds P δ x -almostsurely for each x >
0. The restriction of the process Z X to (0 , ∞ ) can be formally identifiedas a branching Brownian motion with killing at the origin. In [15] it was shown that a dyadicbranching Brownian motion with drift − ρ and killing at the origin which branches at rate q has the property that the right most particle speed is equal to √ q − ρ on survival. In factcareful inspection of their proof shows that it is straightforward to replace dyadic branchingby a random number of offspring with mean m ∈ (1 , ∞ ). In that case the right most speedis equal to p q ( m − − ρ . Note that for the process Z X , we easily compute from (14) that q ( m −
1) = α . We now have, thatlim inf t →∞ R Λ X t t ≥ lim t →∞ R Z X t t = √ α − ρ on { ζ Z X + = ∞} , (16) P δ x -almost surely. Let us temporarily assume however that { ζ Z X + < ∞} agrees with theevent { ζ Λ X + < ∞} under P δ x . Theorem 2.1 now allows us to conclude from (16) thatlim inf t →∞ R X t ≥ √ α − ρ on { ζ X + = ∞} P δ x -almost surely. 7o complete the proof of part (i) we must therefore show that { ζ Z X + < ∞} agrees with theevent { ζ Λ X + < ∞} under P δ x and that their common probabilities give the unique solution to(7). To this end, first note that the backbone decomposition, and in particular the Poissonembedding of Z X in Λ X , gives us that { ζ Λ X + < ∞} ⊆ { ζ Z X + < ∞} . Next note that thebackbone decomposition also tells us that { Z X (0 , ∞ ) = 0 } ⊆ { ζ Λ X + < ∞} . If we define themonotone increasing function Φ : [0 , ∞ ) → [0 , ∞ ) by e − Φ( x ) = P δ x ( ζ X + < ∞ ) = P δ x ( ζ Λ X + < ∞ ) , so that in particular Φ(0) = 0, then the previous observations tell us that for x > e − λ ∗ ≤ e − Φ( x ) ≤ P δ x ( ζ Z X + < ∞ ) < . Note that the final inequality above is strict as all initial particles in Z X may hit the stoppingboundary before branching with positive probability. It is a straightforward exercise to show,using the Markov branching property and the fact that Φ(0) = 0, that Φ respects the relation e − Φ( x ) = E δ x ( P X t ( ζ X + < ∞ )) = E δ x ( e −h Φ ,X + t i )= E δ x ( e −h Φ ,X t i ) for all x, t ≥ . (17)Inspecting the semi-group evolution equation (6) for X with data f = Φ and taking accountof the fact that its unique solution given by (5), we see that Φ solves the differential equationin (7).To show that Φ(+ ∞ ) = λ ∗ , note that the law of h Φ , X + t i under P δ x is equal to that of h Φ( x + · ) , Y D t − x | ( − x, ∞ ) ×{ t } i under Q δ . Thanks to the monotonicity of Φ( x ) and Y D t − x | ( − x, ∞ ) ×{ t } in x and the fact that 0 < Φ( x ) ≤ λ ∗ we have, with the help of dominated convergence, e − Φ(+ ∞ ) = lim x ↑∞ Q δ ( e −h Φ( x + · ) ,Y Dt − x | ( − x, ∞ ) ×{ t } i ) = Q δ ( e − Φ( ∞ ) || Y t || ) . (18)On account of the fact that the process {|| Y t || : t ≥ } is a continuous-state branching processwith branching mechanism ψ the equality in (18) together with the fact that Φ(+ ∞ ) ∈ (0 , λ ∗ ]forces us to deduce that Φ(+ ∞ ) = λ ∗ .Now suppose that φ solves (7). The backbone decomposition tells us that for all t ≥ Z t ( · ) given Λ Xt ( · ) is a Poisson random field with intensity measure λ ∗ Λ Xt ( · ). Hence, E δ x h e h log(1 − φ/λ ∗ ) ,Z Xt i i = E δ x E (cid:20) e h log(1 − φ/λ ∗ ) ,Z Xt i (cid:12)(cid:12)(cid:12)(cid:12) Λ Xt (cid:21) = E δ x (cid:20) exp (cid:26) − Z (cid:0) − e log(1 − φ ( y ) /λ ∗ ) (cid:1) λ ∗ Λ Xt (d y ) (cid:27)(cid:21) = E δ x h e −h φ, Λ Xt i i = e − φ ( x ) . Recalling that φ is monotone with φ (0+) = 0 and φ (+ ∞ ) = λ ∗ , and hence that − log(1 − φ/λ ∗ ) ∈ [0 , ∞ ) so that h− log(1 − φ/λ ∗ ) , Z Xt i ≥ − log(1 − φ ( R Z X t )) ,
8t follows with the help of the known asymptotics of R Z X t , eg (16), thatlim sup t →∞ E δ x h e h log(1 − φ/λ ∗ ) ,Z Xt i { ζ ZX + = ∞} i ≤ lim sup t →∞ E δ x h e log(1 − φ ( R ZXt ) /λ ∗ ) { ζ ZX + = ∞} i = 0 . Subsequently e − φ ( x ) = P δ x ( ζ Z X + < ∞ ) + lim t →∞ E δ x h e −h− log(1 − φ/λ ∗ ) ,Z Xt i { ζ ZX + = ∞} i = P δ x ( ζ Z X + < ∞ ) . (19)We conclude from (19) thatΦ( x ) = − log P δ x ( ζ Λ X + < ∞ ) = − log P δ x ( ζ Z X + < ∞ )is the unique monotone solution to (7). Moreover, since { ζ Λ X + < ∞} ⊆ { ζ Z X + < ∞} , we maynow also deduce that { ζ Λ X + < ∞} = { ζ Z X + < ∞} , P δ x -almost surely, which completes theproof of part (i) of the Theorem.Proof of (ii): Suppose now that ρ ≥ √ α . The estimate R Xt ≤ R Yt used in (15) now tellsus that P δ x ( ζ X + < ∞ ) = 1 and hence, because of the backbone decomposition, it also tellsus that P δ x ( ζ Λ X + < ∞ ) = 1. As noted earlier, the Poisson embedding of Z X in Λ X gives usthat { ζ Λ X + < ∞} ⊆ { ζ Z X + < ∞} and hence it follows that P δ x ( ζ Z X + < ∞ ) = 1. Suppose nowthat a monotone wave solution, Φ, to (7) exists. Then the computation in (19) forces us toconclude that Φ ≡ (cid:3) Remark 3.1
Whilst Theorem 1.1 offers results on the existence and uniqueness of solutionsto (7), Proposition 2 of Pinsky [22] and Theorem 1 of Harris et al. [15] also offer the rate ofdecay of monotone solutions at + ∞ to the wave equation12 Ψ ′′ ( x ) − ρ Ψ ′ ( x ) + F (Ψ( x )) on x > ∞ ) = 0 , (20)for ρ < √ q where F ( s ) = q ( s − s ) and q >
0. A straightforward inspection of the proofin Theorem 1 of Harris et al. [15] shows that in fact their result on the decay of Ψ holdsfor more general functions F taking, for example, the form F ( s ) = q ( P ∞ n =2 s n p n − s ) for s ∈ [0 , q > { p n : n ≥ } is a probability distribution satisfying F ′ (1) < ∞ .Specifically, most of the arguments in [15] do not require a dyadic offspring distribution suchas is imposed there, however, in Section 6 one must take care with the exponential term inthe martingale defined in (14). In their terminology, the integrand in the exponential term, β ( f ( Y s ) − G ( Y s ) where G ( s ) = F ( s ) /s . Thereafter, the necessaryadjustments, which pertain largely to bounds, are relatively obvious. In that case their resultreads as follows. For all ρ < p F ′ (1)lim x ↑∞ e − ( ρ − √ ρ +2 q ) x Ψ( x ) = k ρ for some k ρ ∈ (0 , ∞ ).Note that when F is given by (14) it is straightforward to check that Ψ solves (20) if andonly if λ ∗ (1 − Ψ) solves (7). It follows immediately that when ρ < √ α lim x ↑∞ exp n − (cid:16) ρ − p ρ + 2 ψ ′ ( λ ∗ ) (cid:17) x o (1 − Φ( x ) /λ ∗ ) = k ρ . Proof of Theorem 1.3
As alluded to above, our objective is to embed an existing result for branching Brownianmotion with absorption at the origin into the superprocess setting with the help of thebackbone decomposition. For all x ∈ R we shall denote by Q δ x the law of the backbonedecomposition, ( Z Y , Λ Y ) of Y . The existing result in question is due to Maillard [20] andwhen paraphrased in terms of the backbone process Z Y for Y , states that, under the conditionthat P n ≥ n (log n ) ε p n < ∞ , for some ε >
0, and ρ = √ α it follows that for all x ≥ − z ,Q δ x ( || Z YD − z || > n ||| Z Y || = 1) ∼ √ α ( x + z ) e √ α ( x + z ) n (log n ) (21)as N ∋ n ↑ ∞ . Here we understand Z YD − z to mean the atomic valued measure, supportedon {− z } × [0 , ∞ ) which describes the space-time position of particles in the branchingdiffusion Z Y who are first in their line of descent to exit the domain D ∞− z . The process || Z YD || := {|| Z YD − z || : z ≥ } is known to be a continuous time Galton-Watson (cf. Lemma 3.1and Proposition 3.2 in [20] or Proposition 3 in [21]), which, like its continuous-state spaceanalogue || Y D || , has growth rate √ α . Maillard’s result follows by first establishing that F z ( s ) := Q δ ( s || Z YD − z || ||| Z Y || = 1) (22)satisfies F ′′ z (1 − s ) ∼ √ αze z √ α s (log(1 /s )) , as s ↓ , (23)and then applying a classical Tauberian theorem.The strategy for our proof of Theorem 1.3 will be to first show that the moment condition P n ≥ n (log n ) ǫ p n < ∞ is implied by (13). Thereafter, we shall appeal to an analyticalidentity that arises through the Poissonisation property of the backbone decomposition,thereby allowing us to convert the asymptotic (23) into an appropriate asymptotic whichleads, again through an application of a Tauberian theorem, to the conclusion of Theorem1.3. We start with the following lemma. Lemma 4.1 If R [1 , ∞ ) x (log x ) ε Π(d x ) < ∞ for some ε > , then P n ≥ n (log n ) ε p n < ∞ . Proof:
Appealing to the definition of { p n : n ≥ } it suffices to prove that Z ∞ X n ≥ n (log n ) ε ( λ ∗ x ) n n ! e − λ ∗ x Π(d x ) < ∞ . (24)To this end, define the following function f ( x ) = (log(1 + x )) ε , then it is easy to see that f ′′ ( x ) = − (2 + ε ) (log(1 + x )) ε (1 + x ) (log(1 + x ) − (1 + ε )) . N ∈ N such that log(1 + N ) > (1 + ε ) and subsequently that f ′′ ( x ) < x ≥ N . This implies that f is concave in ( N , ∞ ). Hence, using Jensen’s inequality X n ≥ N +1 n (log n ) ε ( λ ∗ x ) n n ! e − λ ∗ x (25)= λ ∗ x X n ≥ N (log( n + 1)) ε ( λ ∗ x ) n n ! e − λ ∗ x ≤ ( λ ∗ x ) X n ≥ N e − λ ∗ x ( λ ∗ x ) n n ! ! log P n ≥ N n ( λ ∗ x ) n n ! e − λ ∗ x P n ≥ N ( λ ∗ x ) n n ! e − λ ∗ x + 1 !! ε ≤ ( λ ∗ x ) log P n ≥ N n ( λ ∗ x ) n n ! P n ≥ N ( λ ∗ x ) n n ! + 1 !! ε . (26)On the other hand we have thatlim x →∞ log (cid:18) P n ≥ N n ( λ ∗ x ) nn ! P n ≥ N λ ∗ x ) nn ! + 1 (cid:19) log x = 1 . So we can find
K > x > K log P n ≥ N n ( λ ∗ x ) n n ! P n ≥ N ( λ ∗ x ) n n ! + 1 ! < x. Using (26), this implies that Z [ K, ∞ ) X n ≥ N +1 n (log n ) ε ( λ ∗ x ) n n ! e − λ ∗ x Π(d x ) < ε λ ∗ Z [ K, ∞ ) x (log x ) ε Π(d x ) < ∞ . On the other hand, by choosing N large enough, we also have that for all n ≥ N + 1,(log n ) ε < C ( n −
1) for some
C >
0. Hence Z (0 ,K ) X n ≥ N +1 n (log n ) ε ( λ ∗ x ) n n ! e − λ ∗ x Π(d x ) ≤ ( λ ∗ K ) X n ≥ N +1 Z (0 , ∞ ) ( λ ∗ x ) n − ( n − e − λ ∗ x Π(d x ) ≤ ( λ ∗ K ) X n ≥ p n < ∞ For the first terms of (24) we have Z (0 , ∞ ) N X n =2 n (log n ) ε ( λ ∗ x ) n n ! e − λ ∗ x Π( dx ) = N X n =2 n (log n ) ε ( λ ∗ ) n n ! Z (0 , ∞ ) x n e − λ ∗ x Π( dx ) < ∞ , which follows from the fact that each term of the sum is finite. This completes the proof. (cid:3) emark 4.2 It is not difficult to show that the converse of the statement in Lemma 4.1 isalso true, however we leave it as an exercise for the reader.Let us now turn to the proof of Theorem 1.3. We approach the proof here on in twosteps. The first step is to show that the process || Z YD || under Q δ x , for which distributionalproperties are known thanks to (21), has the same branching generator as the continuous-time Galton-Watson process Z ′ := { Z ′ z : z ≥ } , where the latter is the backbone embeddedin the continuous-state branching process || Y D || . Thanks to the backbone decomposition of || Y D || , say ( Z ′ , Λ ′ ), and the easily seen fact that ψ D ( λ ∗ ) = 0, we have that the law of Z ′ z given Λ ′ z is that of a Poisson random variable with parameter λ ∗ Λ ′ z . This Poissonisationresult will allow us to feed the known distributional asymptotic for Z ′ z (equiv. || Z YD − z || ) intothe required result for Λ ′ z (equiv. || Y D − z || ). Step 1:
We start by recalling from Maillard [20], Section 3, that || Z YD || has branchinggenerator given by Θ ′ (Θ − ( s )) , s ∈ [0 ,
1] where Θ is the unique monotone solution to thewave equation12 Θ ′′ ( x ) − √ α Θ ′ ( x ) + F (Θ( x )) = 0 on R with Θ( −∞ ) = 1 and Θ(+ ∞ ) = 0 , (27)and F is the branching generator of the backbone Z Y given in (14). Again appealing to (14)but for the backbone decomposition ( Z ′ , Λ ′ ) of || Y D || and the fact that ψ D ( λ ∗ ) = 0, we knowthat Z ′ has branching generator given by F D ( s ) = 1 λ ∗ ψ D ( λ ∗ (1 − s )) , for s ∈ [0 , . Our objective is thus to show that F D ( s ) = Θ ′ (Θ − ( s )) for all s ∈ [0 , ψ D ( λ ) = Ψ ′ (Ψ − ( λ )) for λ ∈ [0 , λ ∗ ] where Ψ solves (12). It isa straightforward exercise to check that Θ( x ) = 1 − Ψ( − x ) /λ ∗ . Indeed Θ(+ ∞ ) = 0 andΘ( −∞ ) = 1 and Θ solves (27) on account of the fact that Ψ solves (12). Moreover, onereadily confirms that 1 λ ∗ Ψ ′ (Ψ − ( λ ∗ (1 − s ))) = Θ ′ (Θ − ( s )) for s ∈ [0 , , This implies in turn that the required equality, F D ( s ) = Θ ′ (Θ − ( s )), holds and in particularthat || Z YD || and Z ′ have the same branching generator. Step 2:
Recall that ( Z ′ , Λ ′ ) is the backbone decomposition of {|| Y D − z || : z ≥ } and denotethe law of former by Q x when the latter has law Q δ x . Appealing to spatial homogeneity, wemay henceforth proceed without loss of generality by assuming that x = 0.It follows from the conclusion of Step 1 and the Poissonisation property of the backbonedecomposition that for z ≥ s ∈ [0 , δ ( s || Z YD − z || ) = Q ( s Z ′ z ) = Q (cid:16) Q ( s Z ′ z | Λ ′ z ) (cid:17) = Q ( e − λ ∗ Λ ′ z (1 − s ) ) = Q δ ( e − λ ∗ || Y D − z || (1 − s ) ) . (28)12ow using the fact that, under Q δ , || Z YD || = || Z Y || is a Poisson random variable withintensity λ ∗ we haveQ δ h s || Z YD − z || i = ∞ X k =0 e − λ ∗ ( λ ∗ ) k k ! F z ( s ) k = exp {− λ ∗ (1 − F z ( s )) } , (29)where F z ( s ) was defined in (22). If we set w z ( s ) = Q δ ( e − s || Y D − z || )then (28) and (29) tell us that w z ( λ ∗ s ) = exp {− λ ∗ (1 − F z (1 − s )) } . Taking second derivatives on both sides of the last equality gives us( λ ∗ ) w ′′ z ( λ ∗ s ) = ( λ ∗ F ′′ z (1 − s ) + ( λ ∗ F ′ z (1 − s )) ) exp {− λ ∗ (1 − F z (1 − s )) } . (30)Recalling (23) and noting that F ′ z (1 − s ) ∼ e z √ α , as s ↓ , which holds on account of the fact that || Z YD || is a continuous-time Galton-Watson processwith growth rate √ α , we have from (30) that w ′′ z ( λ ∗ s ) ∼ √ αze z √ α λ ∗ s (log(1 /s )) , as s ↓ . (31)Taking u = λ ∗ s and using (31) we obtain w ′′ z ( u ) ∼ √ αze z √ α u (log λ ∗ + log(1 /u )) ∼ √ αze z √ α u (log(1 /u )) , as u ↓ . (32)Denote by U z (d y ) the measure in (0 , ∞ ) defined by the relation w z ( s ) = Z [0 , ∞ ) e − sy U z (d y ) . In other words U z (d y ) = Q δ ( || Y D − z || ∈ dy) for y ≥
0. And let us take ˜ U z (d y ) = y U z (d y )on [0 , ∞ ), then it is easy to see that w ′′ z ( s ) = Z [0 , ∞ ) e − sy ˜ U z (d y ) . Then using Theorem 2 XIII.5 in [13] we have using (32), that˜ U z ( t ) ∼ t √ αze z √ α (log t ) as t → ∞ . U z ( t, ∞ ) = Z ( t, ∞ ) y − ˜ U z (d y ) ∼ √ αze z √ α (cid:18) Z ∞ t y (log y ) d y − t (log t ) (cid:19) as t → ∞ . But by Theorem 1 VIII.9 in [13] the integral on the right hand side above is equivalent to1 /t (log t ) . This implies that Q δ ( || Y D − z || > t ) = U z ( t, ∞ ) ∼ √ αze z √ α t (log t ) as t → ∞ , which proves the result. (cid:3) Remark 4.3
Maillard [20] gives further results in the case that ρ > √ α for the asymptoticbehaviour of Q δ x ( || Y D − z || > t ) as t ↑ ∞ . Again using ideas of Poisson embedding through thebackbone, analogous asymptotics can be transferred from the case of branching Brownianmotion to super-Brownian motion. Acknowledgements
All three authors would like to thank and anonymous referee for their comments and remarkson an earlier draft of this paper which lead to its improvement. AEK acknowledges finan-cial support from the Santander Research Grant Fund, JLP acknowledges financial supportfrom CONACyT-MEXICO grant number 150645, AMS acknowledges financial support fromCONACyT-MEXICO grant number 129076.
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