The extremal process of super-Brownian motion: a probabilistic approach via skeletons

Recently Ren et al. [Stoch. Proc. Appl., 137 (2021)] have proved that the extremal process of the super-Brownian motion converges in distribution in the limit of large times. Their techniques rely heavily on the study of the convergence of solutions to the Kolmogorov-Petrovsky-Piscounov equation along the lines of [M. Bramson, Mem. Amer. Math. Soc., 44 (1983)]. In this paper we take a different approach. Our approach is based on the skeleton decomposition of super-Brownian motion. The skeleton may be interpreted as immortal particles that determine the large time behaviour of the process. We exploit this fact and carry asymptotic properties from the skeleton over to the super-Brownian motion. Some new results concerning the probabilistic representations of the limiting process are obtained, which cannot be directly obtained through the results of [Y.-X. Ren et al., Stoch. Proc. Appl., 137 (2021)]. Apart from the results, our approach offers insights into the driving force behind the limiting process for super-Brownian motions.


Introduction 1.Super-Brownian motion and the skeleton space
Throughout this paper, we use ":=" as a way of definition.Suppose M F (R) is the space of finite measures on R equipped with the topology of weak convergence.The set of finite and compactly supported measures on R is denoted by M c (R).We use B b (R) (respectively, B + (R)) to denote the space of bounded (respectively, nonnegative) Borel functions on R. The space of continuous (and compactly supported) functions on R will be denoted as C(R) (and C c (R) resp.).We use the notation f, µ := R f (x)µ(dx) and µ := 1, µ .The main process of interest in this paper is an M F (R)-valued Markov process X = {X t : t ≥ 0} with evolution depending on two quantities P t and ψ.Here P t is the semigroup of the standard Brownian motion {((B t ) t≥0 , Π x ) : x ∈ R} and ψ is the so-called branching mechanism, which takes the form ψ(λ) = −αλ + βλ 2 + (0,∞) e −λy − 1 + λy π(dy) for λ ≥ 0, (1.1) with α > 0, β ≥ 0 and π(dy) being a measure concentrated on (0, ∞) such that (0,+∞) y ∧ y 2 π(dy) < +∞.
The distribution of X is denoted by P µ if it is started at µ ∈ M F (R) at t = 0.With abuse of notation, we also use P µ to denote the expectation with respect to P µ .X is called a (supercritical) (P t , ψ)-superprocess or super-Brownian motion with branching mechanism ψ if X is an M F (R)-valued process such that for any µ ∈ M F (R), f ∈ B + b (R) and t ≥ 0, P µ e − f,Xt = e − u f (t,•),µ , ( where u f (t, x) := − log P δx e − f,Xt (1.3) is the unique nonnegative locally bounded solution to the following integral equation: P s (ψ(u f (t − s, •))) (x)ds for any x ∈ R and t ≥ 0.
We note that u f (t, x) is also a solution to the partial differential equation with initial condition u(0, x) = f (x).Moreover, if f is a nonnegative bounded continuous function on R, lim t→0 u f (t, x) = f (x), for x ∈ R. We will refer to (1.4) as the Kolmogorov-Petrovsky-Piscounov (K-P-P) equation.The existence of such a process X is established by [10].Moreover, such a super-Brownian motion X has a Hunt realization in M F (R) (see, for example, [17,Theorem 5.12]) such that t → f, X t is almost surely right continuous for any bounded continuous functions f .We shall always work with this version.The (P t , ψ)-superprocess can be constructed as the high density limit of a sequence of branching Markov processes.Another link between superprocesses and branching Markov processes is provided by the socalled skeleton decomposition, which is developed by [5,9,11,16].The skeleton decomposition provides a pathwise description of a superprocesses in terms of immigrations along a branching Markov process called the skeleton.The following condition is fundamental for the skeleton construction.
If P µ denotes the measure P µ,ν with ν replaced by a Poisson random measure with intensity λ * µ(dx), then X := X * + I; P µ is Markovian and has the same distribution as (X; P µ ).Moreover, under P µ , given X t , the measure Z t is a Poisson random measure with intensity λ * X t (dx).
Since ( X; P µ ) is equal in distribution to the (P t , ψ)-superprocess (X; P µ ), we may work on this skeleton space whenever it is convenient.For notational simplification, we will abuse the notation and denote X by X.We will refer to (Z t ) t≥0 as the skeleton branching Brownian motion (skeleton BBM) of X.Since the distributions of X * (resp.I) under P µ,ν do not depend on ν (resp.µ), we sometimes write P µ,• (resp.P •,ν ) for P µ,ν .
The aim of this article is to study the asymptotic behavior of the super-Brownian motion.Since the solutions to the K-P-P equation fully capture the space-time behavior of the super-Brownian motion, it is natural to use this relationship to investigate the large time behaviour for super-Brownian motions.K-P-P equation has been studied extensively using both analytic and probabilistic methods (see, for example, [7,8,14,15]).Among these papers, the seminal work of Bramson [7] established the convergence of u(t, x + m(t)) to travelling waves for some centering term m(t).We use u ∈ Z t to denote a particle of the skeleton BBM which is alive at time t and z u (t) for its spatial location at t. Based on the results of [7], [1,2] proved that for a branching Brownian motion, the extremal process, namely the random point measure converges in distribution to a limiting process as t → +∞, and gave an explicit construction of the limiting process.In Section 1.2.2 we will review some of the facts concerning on the convergence of the solutions to the K-P-P equation with applications to branching Brownian motions.It is to be noted that Bramson [7] considers the solutions to (1.4) with initial condition taking values in [0, 1], and thus cannot be applied directly to super-Brownian motions.A key step is to establish the convergence of solutions to (1.4) for more general initial conditions.Methodologically, the convergence result established in [7] relies mainly on approximations of the solutions by Feynman-Kac formula.Bramson's reasoning can be applied, with modifications, to solutions of the K-P-P equation with initial conditions taking values in [0, ∞).This method has been adopted by Ren et al. in their recent work [20].In this paper we shall offer a different approach.We appeal to the skeleton techniques for superprocesses.Intuitively, the super-Brownian motion may be interpreted as a cloud of subcritical diffusive mass immigrating off a supercritical branching Brownian motion, the skeleton, which governs the large time behaviour of the process.We exploit this fact and carry the long time behaviour from the skeleton over to the super-Brownian motion.Our work is partly inspired by [9,11], where the skeleton techniques have been used successfully to establish the laws of large numbers for superprocesses.Apart from the result itself, our approach provides structural insights into the driving force behind the limiting process for the super-Brownian motions.

1.2
The extremal process of the skeleton and facts

Derivative martingales for the skeleton
In this and the next two subsections we assume (A1) and (1.6) hold.Additional conditions used are stated explicitly.Recall that u ∈ Z t and z u (t) denote, respectively, a particle of the skeleton BBM which is alive at time t and its spatial location at t. Define for t ≥ 0, It is known that ((∂M t ) t≥0 , P •,ν ) is a signed martingale for every compactly supported finite point measure ν on R, which is referred to as the derivative martingale of the skeleton BBM (Z t ) t≥0 .This martingale is deeply related to the travelling wave solutions to the K-P-P equation and plays an important role in the limit theory of the skeleton BBM.[14] has proved that the martingale ((∂M t ) t≥0 , P •,δ0 ) has an almost sure nonnegative limit, and later [22] established the sufficient and necessary condition for the limit to be non-degenerate.We give the statement below which reproduces the same results in the setting of skeleton space.
Proof.By decomposing ∂M t into contributions derived from the population at time s ∈ [0, t), one has Here u ≺ v means that u is an ancestor of v.We use z Then we have In particular by setting s = 0, we have It is easy to see that for u ∈ Z 0 , ∂M (u) t and M (u) t are independent copies of (∂M t , P •,δ0 ) and (M t , P •,δ0 ), respectively.By [14] one has P •,δ0 (lim t→+∞ ∂M t exists and is nonnegative) = 1 and P •,δ0 (lim t→+∞ M t = 0) = 1.We also note that Z 0 is a Poisson random measure with compactly supported intensity µ.So each of the sums in the right hand side of (1.9) contains finite terms almost surely.These facts together with (1.9) imply that the limit ∂M ∞ = lim t→+∞ ∂M t exists and is nonnegative P µ -a.s. for every µ ∈ M c (R).

The extremal process of the skeleton
Let M(R) be the space of all Radon measures on R equipped with the vague topology.A sequence {Ξ n : n ≥ 1} of random Radon measures on R is said to converge in distribution to Ξ if and only if for all φ ∈ C + c (R), the random variables φ, Ξ n converges in distribution to φ, Ξ .For x ∈ R and a function f on R, we define the shift operator T x by T x f (y) := f (x + y) for all y ∈ R. For µ ∈ M(R), we use µ + x and sometimes T x µ to denote the measure induced by Given (1.6), (1.5) can be written as with initial condition u(0, x) = f (x).Then (t, x) Recall the definition of u f (t, x) for f ∈ B + b (R) given in (1.3).We note that u f (t, x) is the unique nonnegative solution to (1.4) with initial condition f .In particular, if where (Z t ) t≥0 is the skeleton BBM.Let max Z t := max{z u (t) : u ∈ Z t } be the maximal displacement of the skeleton BBM.In particular, P •,δx (max Z t > 0) is a solution to (1.4) with initial condition 1 (0,+∞) (x).Bramson [7] studied the asymptotic behavior of the solution to the K-P-P equation (1.4) with initial condition u(0, x) taking values in [0, 1].Actually in [7], the nonlinear function −ψ can be any function on [0, 1] satisfying that Bramson [7] shows in particular that if ψ also satisfies that for some ρ > 0, then when the initial condition u(0, x) satisfies a certain integrability condition, it holds that where and w(x) is a travelling wave solution with speed √ 2, that is, w(x) is the unique (up to translations) solution to the ordinary differential equation with 1 − w(x) being a distribution function on R. The integral representation of w(x) is established in [18] (see also, [14,22]): When ∂M ∞ is nondegenerate, one has for some constant C > 0.Moreover, it holds that Later [2] recovered the above representation of the form (1.18) and provided an expression for the constant C as a function of the initial condition.As a result [2] established the convergence in distribution of the extremal process of the branching Brownian motion.To apply [2,7]'s results directly to the skeleton BBM, we assume the following condition holds.
Define  x) for some g ∈ C + c (R). Taking φ = 1 − e −g for some g ∈ C + c (R), one can rewrite u φ (t, x) as 1 − P •,δx e − g,Zt , and the above result yields that P •,δ0 e − g,Zt−m(t) → P •,δ0 [exp{−C(φ)∂M ∞ }] as t → +∞.(1.20)This implies the convergence in distribution of the extremal process of a branching Brownian motion.To be more specific, [2] proved that, the extremal process of a branching Brownian motion defined by dz and decoration law △ Z , where and △ Z is a random point measure supported on (−∞, 0], with an atom at 0, which satisfies that We denote proved the same result by a totally different approach almost at the same time as [2].
The limiting extremal process E Z ∞ can be constructed as follows.Given ∂M ∞ , let {e i : i ≥ 1} be the atoms of a Poisson point process on R with intensity c * ∂M ∞ √ 2e − √ 2z dz and {△ Z i : j ≥ 1} be a sequence of i.i.d.point measures with the same law as △ Z , then Recall that max Z t = max{z u (t) : u ∈ Z t } is the maximal displacement of the skeleton BBM.Then is a solution to (1.4) with initial condition 1 (0,+∞) (x).Proposition 1.3 yields that for any x ∈ R, This implies that under the assumptions of Proposition 1.3, the maximal displacement of the skeleton BBM centered by m(t) converges in distribution to a randomly shifted Gumbel distribution.In fact, Proposition 1.3 implies the joint convergence in distribution of (E Z t , max E Z t ), see, for example, [4, Lemma 4.4].

Relation between the limits of the derivative martingales of super-Brownian motion and its skeleton
Define for t ≥ 0, By [15] for every µ ∈ M c (R), ((∂W t ) t≥0 , P µ ) is a martingale which is usually called the derivative martingale of the super-Brownian motion.Obviously (∂W t ) t≥0 is the counterpart of (∂M t ) t≥0 in the setting of superprocess.Interest of this martingale is stimulated by its close connection with the travelling wave solutions to the K-P-P equation (see, for example, [15] and the references therein).Note that ((∂W t ) t≥0 , P µ ) is a signed martingale which does not necessarily converge almost surely.To study its convergence, [15] imposed the following condition on ψ: The for all C > 0 and x ∈ R. We observe that for every x ∈ R, the process ((X t ) t≥0 , Using this and the branching property of superprocesses, one has for any λ > 0, Hence by (1.24) and (1.11) one gets P µ e −λ∂W∞ = P µ e −λ∂M∞ for all λ > 0 and so (∂W ∞ , P µ ) d = (∂M ∞ , P µ ).

Statement of main results
In what follows and for the remainder of the paper we assume (A1), (A2) and (1.6) hold.Additional conditions used are stated explicitly. Define This result is obtained independently in [20,Proposition 1.3(1)].In fact, the constant C(φ) given in (1.27) is the same as the one given in [20,Proposition 1.3].This is because our u φ (t, x) is the solution to (1.4) with initial condition u(0, x) = φ(x), while U φ (t, x) defined in [20] is the solution to (1.4) with initial condition u(0, x) = φ(−x).It holds that U φ (t, x) = u φ (t, −x).Comparing the definitions of C(φ) in (1.27) and the one in [20, Proposition 1.3], we see that they are the same.We also note that the "locally uniform convergence" is slightly stronger than [20, Proposition 1.3(1)], where the convergence of (1.28) is established for each fixed x ∈ R.
(ii) Let max X t denote the supremum of the support of X t , i.e., max Here we take the convention that inf ∅ = +∞.Unlike for the skeleton BBM, Theorem 1.6 does not imply the growth order of max X t is m(t).In fact, the asymptotic behavior of max X t depends heavily on the branching mechanism ψ(λ) and it may grow much faster than m(t).We give such examples in Remark 2.12.
Theorem 1.6 yields the existence of the limiting process of the extremal process of super-Brownian motion.
Theorem 1.8.For t ≥ 0, set Then for every x ∈ R, the process In the above statement and for the remainder of this paper, when we talk about the distributional limit, we do not specify the probability space where the limit is defined, just use P to denote the probability measure, and E to denote the corresponding expectation.We remark that the distribution of (E ∞ , E Z ∞ ) depends on x since it is the distributional limit of ((E t , E Z t ) t≥0 , P δx ).We will not remark this dependence in similar situation later in this paper.
In the following proposition, we establish a dichotomy on the finiteness of the supremum of the support for the limiting process.Proposition 1.9.Suppose x ∈ R and E ∞ is the limit of ((E t ) t≥0 , P δx ) in distribution.Let max E ∞ be the supremum of the support of E ∞ .Then max E ∞ is a.s.finite if and only if We also obtain some new results on the probabilistic representations for the limiting process.For u ∈ Z t , denote by I (u) s the immigration at time t + s that occurred along the subtree of the skeleton rooted at u with location z u (t).Lemma 3.4 below shows that under P δx , for every s > 0, conditioned on {max Moreover, the limit △ I,s , △ Z does not depend on x.
dy and decoration law △ Z , here c * = C(1 (0,+∞) ).Let {d i : i ≥ 1} be the atoms of E Z ∞ , and for every s > 0, let {△ s i : i ≥ 1} be an independent sequence of i.i.d.random measures with the same law as (I s − √ 2s, P •,δ0 ), then (1.30) The limit in (1.30) can not be put into the summation.In fact, for each i ≥ 1, △ s i converges in distribution to the null measure as s → +∞, see Remark 3.7 below.The following result gives an alternative description of E ∞ .
Proposition 1.11.Suppose the assumptions of Theorem 1.10 hold.Given (∂M ∞ , P δx ), let {e i : i ≥ 1} be the atoms of a Poisson point process with intensity c * ∂M ∞ √ 2e − √ 2x dx, and for every s > 0, let {△ I,s i : i ≥ 1} be an independent sequence of i.i.d random measures with the same law as △ I,s , then T ei △ I,s i .
For every s > 0, i≥1 T ei △ I,s i is a Poisson random measure with exponential intensity, in which each atom is decorated by an independent copy of an auxiliary measure.However, their distributional limit E ∞ may not inherit such a structure.This is revealed by the following theorem.
Theorem 1.12.Suppose x ∈ R and E ∞ is the limit of ((E t ) t≥0 , P δx ) in distribution.There exist a constant ι ≥ 0 and a measure Λ on M(R) \ {0} satisfying that The constant ι may not be 0 in general.The argument of Remark 3.11 shows that ι = 0 if the following condition holds: 13 below further shows that Λ(dµ) = c0 c * P △ X ∈ dµ , where c0 is a constant given by (2.39) with φ = 0, and △ X is the limit of X t − max X t conditioned on {max X t − √ 2t > 0}.In fact, it is proved in Lemma 3.13 that under (A3), conditioned on {max X t − √ 2t > 0}, the random measures (X t − max X t , Z t − max X t ) converges, as t → +∞, in distribution to a limit ( △ X , △ Z ), and that given △ X , △ Z is a Poisson random measure with intensity △ X .Theorem 1.13.Assume in addition that (A3) holds.Suppose x ∈ R and (E ∞ , E Z ∞ ) is the limit of ((E t , E Z t ) t≥0 , P δx ) in distribution.Let c0 be given by (2.39) with φ = 0. Given (∂M ∞ , P δx ), let {ẽ i : i ≥ 1} be the atoms of a Poisson point process with intensity c0 ∂M ∞ √ 2e − √ 2y dy and {( △ X i , △ Z i ) : i ≥ 1} be an independent sequence of i.i.d.random measures with the same law as ( △ X , △ Z ).Then we have Then we have, for λ > 1, < E e −E∞(0,∞) = P δx e −C(1 (0,+∞) )∂M∞ .
2 Convergence of the extremal process of super-Brownian motion 2.1 Proof of Proposition 1.3 Recall that ((B t ) t≥0 , Π x ) a standard Brownian motion starting at x, and that u f (t, x) is the unique nonnegative solution to (1.4) with initial condition f .
(4) Fix s ≥ 0 and y ∈ R. Let v(t, x) := u f (t + s, x + y) for all t ≥ 0 and x ∈ R. It is easy to verify that v is the unique nonnegative solution to (1.4) with initial condition v(0, x) = u f (s, x + y) = T y u f (s, •)(x).Hence we get v(t, x) = u Tyu f (s,•) (t, x).In particular by setting s = 0 we get that u f (t, x + y) = u Tyu f (0,•) (t, x) = u Tyf (t, x).
Our proof of Proposition 1.3 follows from two main steps: the first step is to establish the convergence for the Laplace functionals of the skeleton BBM which are truncated by a certain cutoff, and the second step is to show the convergence continues to hold when the cutoff is lifted.We need the following lemmas, which are refinements of [2, Proposition 4.4 and Lemma 4.9].In fact, [2] proves the same results for [0, 1]-valued functions with support bounded on the left.We extend their results to all functions of H 1 .Though the idea of our proofs is similar to [2], we give the details here for the reader's convenience.exists and is finite.Moreover, the limit exists and is finite, and for every x ∈ R, dy We get by (2.1) and (2.2) that We may rewrite Ψ(r, t, x + √ 2t) as follows: where G(z) := (1 − e −z ) /z.Using the fact that G(z) ∈ [0, 1] for all z > 0 and G(z) ∼ 1 as z → 0, we get by the bounded convergence theorem that Consequently by letting t → +∞ in (2.3), we have Then by letting r → +∞, we have This implies that the limit lim r→+∞ C r (φ) exists and is finite, and is equal to Therefore we complete the proof.
Proof.We note that for every δ ∈ R, Thus by Lemma 2.1(3), we have ) Proof of Proposition 1.3:The first part of this proposition follows from Lemma 2.2.We only need to show the second part.For c > 0 and x ∈ R, let By the uniqueness (up to translations) of the travelling wave solution, one has w c (x) = w 1 (x − ln c/ √ 2) for all x ∈ R. We need to show that (2.9) For δ ≥ 0, by (2.8) we have that and

.11)
We note that and Then by the continuity of w 1 we get from (2.12) and (2.13) that On the other hand, by Lemma 2.4 we have for δ ≥ 0, as t → +∞.Hence we get (2.9) by letting first t → +∞ and then δ → +∞ in both (2.10) and (2.11).

Proof of Theorem 1.6
First we introduce notation to refer to the different parts of the skeleton decomposition which will be used later in the computation.For t ≥ 0, let F t denote the σ-filed generated by Z, X * and I up to time t.Denote by I * ,t s the immigration at time t + s that occurred along the skeleton before time t.For u ∈ Z t , denote by I (u) s the immigration at time t + s that occurred along the subtree of the skeleton rooted at u with location z u (t).We have It is known (see, e.g., [9]) that given F t , (X * s+t + I * ,t s ) s≥0 is equal in distribution to ((X * s ) s≥0 ; P Xt ) and I (u) := (I (u) s ) s≥0 is equal in distribution to (I; P •,δ zu(t) ).Moreover, the processes {I (u) : u ∈ Z t } are mutually independent and are independent of t , we have Using the fact that (Z 0 , P δx ) is a Poisson random measure with intensity δ x (dy), one has (2.16) In this section we will make extensive use of (2.16), mostly when we deal with u f (t, x − √ 2t) for large t, in which case, u * f (t, x − √ 2t) becomes easy to handle.Recall the definition of φ δ given in (2.4).The following lemma gives an upperbound for the constant C(φ 0 ) which will be used later.
The last inequality is from the fact that 1 − e −x ≤ x for all x ≥ 0. This together with (2.20) yields that for t ≥ 3r and Hence we complete the proof.
Lemma 2.7.Suppose {φ s (x) : s ≥ 0} is a sequence of functions in H 1 .If φ s (x) → 0 as s → +∞ for all x ∈ R, and Proof.Suppose α(s) ≥ 0 for all s > 0. (The explicit value of α(s) will be given later.)By Lemma 2.1(3), one has So it suffices to prove that for all x ∈ R. By Corollary 2.3 and Lemma 2.6, we have We recall the definition of H given in (1.25).
Proof.It follows by Jensen's inequality that Thus by (2.2) we have The assumption that φ ∈ H implies that The following lemma extends the result of Lemma 2.2 to all functions of H.
Lemma 2.10.Suppose φ ∈ H. Then for any r > 0, exists and is finite.The limit exists and is finite.Moreover, for every x ∈ R, Proof.Without loss of generality we assume φ ∈ H \ H 1 .Let M := φ ∞ and φ 1 = φ/M .Then The finiteness of C r (φ) is immediate since by Lemma 2.1(2) u φ (r, −y − √ 2r) ≤ M u φ1 (r, −y − √ 2r) for all r ≥ 0 and y ∈ R and so C r (φ) ≤ M C r (φ 1 ) < +∞.Since we get by Lemma 2.1(3)(4) that This implies that ) ∈ H 1 for s large enough, we get by (2.27) and Proposition 1.3 that Since the C r (φ) ≤ M C r (φ 1 ) for all r > 0 and the latter is bounded in r, (2.28) implies that the limit C(φ) = lim r→+∞ C r (φ) exists and is finite, and satisfies that On the other hand, it follows from Lemma 2.2 and (2.26) that Letting s → +∞, we get by (2.29) and Lemma 2.9 that Corollary 2.11.For f, f 1 , f 2 ∈ H and M ≥ 1, Proof.This result follows directly from Lemma 2.10 and Lemma 2.1.
Proof of Theorem 1.6: We first suppose that φ ∈ H 1 .Then by Proposition 1.3 we have By (1.11) we have
Remark 2.12.Recall that max X t denotes the supremum of the support of X t .Unlike for the skeleton BBM, Theorem 1.6 does not imply the growth order of max X t is m(t).We make a short discussion here.
By (2.37) and the monotone convergence theorem, one has for t, r ≥ 0 and x ∈ R, exists and is finite, and for all x ∈ R, P δ0 e − φ,Xt−m(t)−x ; max X t − m(t) ≤ x → P δ0 e − C(φ)∂M∞e − √ 2x as t → +∞.
3 Probabilistic representation of the limiting process

Laws of decorations
For the proofs of Theorem 1.10 and Proposition 1.11 we need to show the existence of the limit for (Z t − max Z t , u∈Zt I (u) . This is completed by the following lemmas.
Lemma 3.1.For any f, g ∈ B + b (R), x, z ∈ R and t, y ≥ 0, we have On the other hand, by Fubini's theorem and Lemma 3.1 we have , where the limit is independent of x and z, and Y is an exponential random variable with mean 1/ √ 2.Moreover, we have for any where c * = C(1 (0,+∞) ).
Proof.In view of (3.4) we have for any x, z ∈ R and y ≥ 0, The final equality follows from Lemma 2.2.This implies that, conditioned on {max Z t − √ 2t − z > 0}, max Z t − √ 2t − z converges in distribution to an exponentially distributed random variable with mean 1/ √ 2. Suppose f, h, g are functions satisfying our assumptions.Recall that F t is the σ-filed generated by Z, X * and I up to time t, and given F t , I (u) s d = (I s , P •,δ zu(t) ) for u ∈ Z t .We have On the other hand by Corollary 2.11 we have for any δ ∈ R Using (3.13) and the fact that lim λ1→0 C(λ 1 f ) = 0 we get by (3.14) that lim λ1,λ2,λ3→0 This implies that lim λ1,λ2,λ3→0 and that for every y ∈ R, lim λ1,λ2,λ3,λ4→0 In view of (3.15) and (3.16), one can use the bounded convergence theorem to show that the right hand side of (3.12) converges to 1 as λ i → 0, i = 1, 2, 3, 4. Hence we complete the proof.
Proof.The first conclusion is a direct result of Lemma 3.3 and [2, Lemma 4.13].We only need to show the independence.Suppose f, g, h ∈ C + c (R) and y ≥ 0. We have This yields the independence.
Proof of Theorem 1.10:It follows from Theorem 1.8 and Lemma 3.6 that for all g ∈ B + (R) with 1 − e −g ∈ H, The above equations hold in particular for all g ∈ C + c (R).This implies that E Z ∞ is a decorated Poisson point process with intensity c * ∂M ∞ √ 2e − √ 2y dy and decoration law △ Z .We have for all The final equality follows from (3.19).Since lim s→+∞ C(1 )) = C(φ), we get by the above equality that lim for all φ ∈ C + c (R). Hence we prove (1.30).
Remark 3.7.We claim that for for each i ≥ 1, △ s i converges in distribution to the null measure as s → +∞.This is because, by (2.16)  Hence we complete the proof.

Proof of Theorem 1.12
We prove Theorem 1.12 in this section.First we relate C(φ) to the Laplace functional of a certain random Radon measure.Then we observe that this random measure is infinitely divisible and thus get an expression for C(φ) which leads to the probabilistic interpretation presented in Theorem 1.12.Our observation on C(φ) is inspired by the work of [19].
This implies that E e − g, E Z ∞ | E X ∞ , Y = e − 1−e −g , E X ∞ P-a.s.So we prove the second conclusion of this lemma.Lemma 3.13.Suppose x, z ∈ R.Under P δx , conditioned on {max X t − √ 2t − z > 0}, the random elements (X t −max X t , Z t −max X t , max X t − √ 2t−z) converges, as t → +∞, in distribution to a limit ( △ X , △ Z , Y ) := ( E X ∞ −Y, E Z ∞ −Y, Y ), where the limit is independent of x and z, and ( △ X , △ Z ) is independent of Y .Moreover, given △ X , △ Z is a Poisson random measure with intensity △ X .
Proof.The first conclusion follows from Lemma 3.12 (in place of Lemma 3.3 ) in the same way as Lemma 3.4.We only need to show the second conclusion.
14.It is easy to see that condition (A3) implies(1.23).So by Proposition 1.5 one can replace ∂M ∞ by ∂W ∞ in the statement of Theorem 1.13.△ Z ).The two theorems give two interpretations of E Z ∞ as a decorated Poisson point process.Though the two interpretations are equal in law, they have different intensities and then different decoration laws.To see this, we only need to prove that c * < c0 .Using P δx (∂M ∞ > 0) > 0 and Theorem 1.10, one has that Define m H 1/2 (t) := sup{x ∈ R : v H (t, x) ≥ 1/2} for t ≥ 0. By [7, Proposition 8.1 and Proposition 8.2], there are constants C H ≥ v H (s, y) for all s ≥ 0 and y ∈ R, it follows by the convexity of ψ that k(s, y) ≤ k H (s, y) for all s ≥ 0 and y ∈ R.