The growing tree-valued process

...

Figure 3.1: The pruning process, starting from explosion time A defined in (3.32).

Consider the ascension time (or explosion time):

A=inf©

θ∈Θ^ψ, σθ< ∞ª

, (3.32)

where we use the convention ^inf; =θ∞. The following Theorem gives the distribution of the ascension time A and the distribution of the tree at this random time. Recall that θ¯=ψ⁻¹(ψ(θ))is defined in (3.10).

Theorem 3.24([AD12a]). Letψbe a critical branching mechanism satisfying Assumptions 1 and 2.

1. For allθ∈Θ^ψ, we haveN^ψ[A>θ]=θ¯−θ.

2. Ifθ∞<θ<0, underN^ψ, we have, for any non-negative measurable functional^F, N^ψ[F(T_A+θ⁰,θ⁰≥0)|A=θ]=ψ⁰( ¯θ)N^ψh

F(T_θ⁰,θ⁰≥0)σ0e^−ψ(θ)σ0i . 3. For allθ∈Θ^ψ, we haveN^ψ[σA< +∞|A=θ]=1.

In other words, at the ascension time, the tree can be seen as a size-biased critical Lévy tree. A precise description of TA is given in [AD12a]. Notice that in the setting of [AD12a], there is no need of Assumption 2.

3.3 The growing tree-valued process

Special Markov Property of pruning

In [ADV10], the authors prove a formula describing the structure of a Lévy tree, conditionally on theθ-pruned tree obtained from it in the (sub)critical case. We will give a general version of this result. From the measure of marks, M in (3.30), we define a measure of increasing marks by:

M^↑(d x,dθ⁰)=X

i∈I^↑

δ(xi,θi)(d x,dθ⁰), (3.33)

with

I^↑=n

i∈I^ske∪I^nod;M(‚;,xiƒ ×[0,θi])=1o .

The atoms^(xi,θi)forⁱ∈I^↑correspond to marks such that there are no marks ofM on‚;,x_iƒ with aθ-component smaller thanθi. In the case of multipleθj for a given nodexi∈Br_∞(T), we only keep the smallest one. In the caseΠ=0, the measureM^↑ describes the jumps of a record process on the tree, see [AD11] for further work in this direction. Theθ-pruned tree can alternatively be defined usingM^↑instead ofM as forθ≥0:

Λθ(T,M)=n

x∈T, M^↑(‚;,x‚×[0,θ])=0o . We set:

I_θ^↑=n

i∈I^↑,x_i∈Lf(Λ_θ(T,M))o

=n

i∈I^↑,θi<θ and M^↑(‚;,x_i‚×[0,θ])=0o and fori∈I_θ^↑:

Tⁱ=T\T^;,xi={x∈T, xi∈ ‚;,xƒ},

whereT^y,x is the connected component ofT\{x}containing ^y. For ⁱ∈I_θ^↑,Tⁱ is a real tree, and we will consider^xi as its root. The metric and mass measure onTⁱ are the restriction of the metric and mass measure ofT onTⁱ. By construction, we have:

T =Λθ(T,M)~_i∈I^↑

θ(Tⁱ,xi). (3.34)

Now we can state the general special Markov property.

Theorem 3.25(Special Markov Property). Letψbe a branching mechanism satisfying Assump-tions 1 and 2. Letθ>0. Conditionally onΛθ(T,M), the point measure:

M_θ^↑(d x,dT⁰,dθ⁰)=X

i∈I_θ^↑

δ(xi,Tⁱ,θi)(d x,dT⁰,dθ⁰)

underP^ψr₀ (or underN^ψ) is a Poisson point measure onΛθ(T,M)×T×(0,θ]with intensity:

m^Λθ(T,M)(d x)³

2βN^ψ[dT⁰]+ Z

(0,+∞)Π(d r)re^−θ0rP^ψr(dT⁰)´

1(0,θ](θ⁰)dθ⁰. (3.35) Proof. It is not difficult to adapt the proof of the special Markov property in [ADV10] to get Theorem 3.25 in the (sub)critical case by taking into account the pruning times θi and the w-tree setting; and we omit this proof which can be found in [ADH12b]. We prove how to extend the result to the super-critical Lévy trees using the Girsanov transform of Definition 3.18.

Assume thatψis super-critical. For^a>0, we shall writeΛθ,a(T,M)=πa(Λθ(T,M))for short. According to (3.34) and the definition of super-critical Lévy trees, we have that for any a>0, the truncated treeπa(T)can be written as:

πa(T)=Λ_θ,a(T,M)~_i∈I^↑

θ, H_xi≤a

³πa−H_xi(Tⁱ),x_i´

3.3. The growing tree-valued process and we have to prove that^P_i∈I^↑

θδ(xi,Tⁱ,θi)(d x,dT⁰,dθ⁰)is conditionally onΛθ(T,M)a Pois-son point measure with intensity (3.35). Since ^a is arbitrary, it is enough to prove that the point measureMa, defined by

Ma(d x,dT⁰,dθ⁰)= X non-negative measurable functional onT. Let

B=N^ψ£

F(Λθ,a(T,M)) exp(− 〈Ma,Φ〉)¤ .

Thanks to Girsanov formula (3.22) and the special Markov property for critical branching mechanisms, we get:

Thanks to the Girsanov formula and (3.29), we get:

and thanks to (3.7), we get:

G(h,x,θ)= Using (3.39) with^F replaced by^{F R} gives:

N^ψh

exp(−〈Ma,Φ〉)F(Λθ,a(T,M))i

=B=N^ψ£

F(Λθ,a(T,M))R(Λθ,a(T,M))¤ . This implies thatMa is, conditionally onΛθ,a(T,M), a Poisson point measure with intensity (3.36). This ends the proof.

3.3. The growing tree-valued process An explicit construction of the growing process

In this section, we will construct the growth process using a family of Poisson point measures.

Letψ be a branching mechanism satisfying Assumptions 1 and 2. Letθ∈Θ^ψ. According to (3.20) and (3.7), we have:

N^ψθ[T ∈ •]=2βN^ψθ[T ∈ •]+ Z

(0,+∞)Π(d r)re^−θrP^ψr^θ(T ∈ •). (3.40) LetT⁽⁰⁾∈Twith root;. For^q∈Θ^ψand ^q≤θ, we set:

T⁽⁰⁾_{q =T}⁽⁰⁾ and ^m(0)q =m^T(0).

We define the w-trees grafted on T⁽⁰⁾ by recursion on their generation. We suppose that all the random point measures used for the next construction are defined on T under a probability measure^QT(0)(dω).

Suppose that we have constructed the family⁽⁽T^(k)_q ,m⁽ⁿ⁾_q ), 0≤k≤n,q∈Θ^ψ∩(−∞,θ)). We write

T⁽ⁿ⁾₌ ^G

q∈Θ^ψ,q≤θ

T⁽ⁿ⁾_{q .}

We define the (ⁿ+1)-th generation as follows. Conditionally on all trees from generations smaller thanⁿ,(T^(k)_{q , 0≤}k≤n, q∈Θ^ψ∩(−∞,θ)), let

N_θⁿ⁺¹(d x,dT,d q)= X

j∈J⁽ⁿ⁺¹⁾

δ(xj,T^j,θj)(d x,dT,d q) be a Poisson point measure onT⁽ⁿ⁾_×T×Θ^ψ with intensity:

µⁿ⁺¹_θ (d x,dT,d q)=m⁽ⁿ⁾_q (d x)N^ψq[dT]1{q≤θ}d q.

For^q∈Θ^ψ and^q≤θ, we set

J⁽ⁿ⁺¹⁾_q =©

j∈J⁽ⁿ⁺¹⁾, q<θj

ª

and we define the treeT⁽ⁿ_q⁺¹⁾and the mass measurem⁽ⁿ_q ⁺¹⁾ by:

T⁽ⁿ_q⁺¹⁾₌T⁽ⁿ⁾_q ~_j∈Jq⁽ⁿ⁺¹⁾⁽T ^j,xj) and ^m(nq ⁺¹⁾= X

j∈J⁽ⁿ_q⁺¹⁾

m^Tj(d x).

Notice that by construction, (T⁽ⁿ⁾_q ,n∈N) is a non-decreasing sequence of trees. We set T_q the completion of∪n∈NT⁽ⁿ⁾_q , which is a real tree with root ;and obvious metric^dTq, and we define a mass measure onT_q by^mTq=P

n∈Nm⁽ⁿ⁾_q .

For ^q∈Θ^ψ and ^q <θ, we consider Fq the σ-field generated by T⁽⁰⁾ and the sequence of random point measures ⁽¹{q⁰∈[q,θ]}N_θ⁽ⁿ⁾(d x,dT,d q⁰),n∈N). We setNθ=P

n∈NN_θⁿ. The backward random point process q7→1_{q≤q⁰}Nθ(d x,dT,d q⁰) is by construction adapted to the backward filtration⁽Fq,q∈Θ^ψ∩(−∞,θ]).

The proof of the following result is postponed to Section3.3.

Theorem 3.26. Let ψbe a branching mechanism satisfying Assumptions 1 and 2. UnderQ^ψθ:=

N^ψθ[dT⁽⁰⁾]Q^T(0)(dω), the process

³³

T_q,d^Tq,;,m^T¯q´

,q∈Θ^ψ∩(−∞,θ]´

is aT-valued backward Markov process with respect to the backward filtration F^θ=(Fq,q ∈ Θ^ψ∩(−∞,θ]). It is distributed as⁽⁽Tq,m^Tq),q∈Θ^ψ∩(−∞,θ])underN^ψ.

Notice the Theorem in particular entails that⁽T_q,d^Tq,;,m^T¯q)is a w-tree for all^q. We shall use the following lemma.

Lemma 3.27. Let ψ be a branching mechanism satisfying Assumptions 1 and 2. Let K be a measurable non-negative process (as a function of^q) defined on^R+×T×Twhich is predictable with respect to the backward filtrationF^θ. We have:

Q^ψθ This means that the predictable compensator ofNθ is given by:

µθ(d x,dT,d q)=m^Tq(d x)N^ψq[dT]1_{q∈Θψ,q≤θ}d q.

Notice this construction does not fit in the usual framework of random point measures as the support at time q of the predictable compensator is the (predictable backward in time) random setT_q×T×Θ^ψ.

Proof. Based on the recursive construction, we have:

Q^ψθ Now, by construction, we have that:

T_q=T⁽ⁿ⁾_q ~_j∈J⁽ⁿ⁾_q ^{( ˜}Tj,xj)

3.3. The growing tree-valued process

It can be noticed that the mapq7→T_q is non-decreasing càdlàg (backwards in time) and that we have, for ^j∈ ∪n∈NJ⁽ⁿ⁾, ^xj∈T_θj: T_θj−=T_θj~⁽T ^j,xj). In particular, we can recover the random measureNθ from the jumps of the process ⁽T_q,q∈Θ^ψ∩(−∞,θ]). This and the natural compatibility relation ofN_θ with respect toθgives the next Corollary.

Corollary 3.28. Letψbe a branching mechanism satisfying Assumptions 1 and 2. Let⁽Tθ,θ∈Θ^ψ) be defined underN^ψ. Let

N =X

j∈J

δ(xj,T^j,θj)

be the random point measure defined as follows:

• The set^{θj;j∈J}is the set of jumping times of the process⁽Tθ,θ∈Θ^ψ): for ^j∈J,Tθj−6=Tθj.

with respect to the backward left-continuous filtrationF=(Fθ,θ∈Θ^ψ)defined by:

F_θ=σ((xj,T ^j,θj);θ≤θj)=σ(Tq−;θ≤q).

More precisely, for any non-negative predictable process^K with respect to the backward filtrationF, we have:

N^ψ

·Z

N(d x,dT,d q)K³

q,Tq,Tq−

´¸

=N^ψ

·Z

µ(d x,d T,d q)K³

q,Tq,Tq~^(T,x)´

¸ . (3.41) Remark 5. Notice that Assumption 2 is assumed only for technical measurability condition, see Remark2. We conjecture that this results holds also if Assumption 2 is not in force.

As a consequence, thanks to property 3 of Theorem 3.24, we get, with the convention sup; =θ∞, that:

A=sup{θj,j∈J andσ^j= +∞} with σj=m^Tj(T ^j).

Proof of Theorem3.26

By construction, it is clear that the process ⁽T_q,q ∈Θ^ψ∩(−∞,θ]) is a backward Markov process with respect to the backward filtration (Fq,q ∈Θ^ψ∩(−∞,θ]). By construction this process is càglàd in backward time. Since the process ⁽Tq,q ∈Θ^ψ) is a forward càdlàg Markov process, it is enough to check that forθ0∈Θ^ψ, such thatθ0<θ, the two dimensional marginals(T_θ0,T_θ)and(Tθ0,Tθ) have the same distribution.

Replacing ψby ψθ0, we can assume that θ0=0 and 0<θ. We shall decompose the big treeT0 conditionally on the small tree Tθ by iteration. This decomposition is similar to the one which appears in [AD07] or [Voi10] for the fragmentation of the (sub)critical Lévy tree, but roughly speaking the fragmentation is here frozen but for the fragment containing the root.

We set T⁽⁰⁾=Tθ and ^m˜(0)=m^Tθ, so that ⁽T⁽⁰⁾,m⁽⁰⁾) and ⁽T⁽⁰⁾, ˜m⁽⁰⁾) have the same distribution. Recall notationM^↑from (3.33) as well as (3.34):T0=T⁽⁰⁾~_i∈I^↑,1

θ (Tⁱ,xi), where we write^I_θ^↑,1=I_θ^↑ and whereP¹=P

i∈I_θ^↑,1δ(xi,Tⁱ,θi) is, conditionally onT⁽⁰⁾, a Poisson point measure with intensity:

ν¹(d x,dT⁰,d q)=m˜⁽⁰⁾(d x)³

2βN^ψ[dT⁰]+ Z

(0,+∞)Π(d r)re^−qrP^ψr(dT⁰)´

1(0,θ](q)d q.

Fori∈I_θ^↑,1, we define the sub-tree ofTⁱ: T˜ⁱ=

n

x∈Tⁱ;M^↑(ƒxi,x‚×[0,θi])=0o .

Since Tⁱ is distributed according to N^ψ (or to P^ψr_i for some ^ri >0), using the property of Poisson point measures, we have that conditionally onT⁰andθi, the treeT^˜i is distributed as

3.3. The growing tree-valued process Λθi(T,M)underN^ψ(or underP^ψr_i) that is the distribution ofT^˜i isN^ψθi[dT](orP^ψr_i^θi(dT)), thanks to the special Markov property. Furthermore we haveTⁱ=T˜ⁱ~_i0∈I_θ,i^↑,2(Tⁱ⁰,xi⁰)where

X

i⁰∈I_θ,i^↑,2

δ_(xi0,Tⁱ⁰_,θi0)

is, conditionally onT⁽⁰⁾ andT^˜i a Poisson point measure on T^˜i×T×(0,θ]with intensity:

m^T˜i(d x)³

2βN^ψ(dT⁰)+ Z

(0,+∞)Π(d r)re^−qrP^ψr(dT⁰)´

1_[0,θi)(q)d q.

Thus we deduce, using again the special Markov property, that:

N˜_θ¹(d x,dT,d q)= X

i∈I^↑,1

δ(xi, ˜Tⁱ,θi)(d x,dT,d q)

is conditionally onT⁰ a Poisson point measure onT⁽⁰⁾×T×Θ^ψ with intensity:

µ˜¹(d x,dT,d q)=m˜⁽⁰⁾_q (d x)N^ψq[dT]1[0,θ)(q)d q, withm^˜(0)_q (d x)=m˜⁽⁰⁾(d x). We setT⁽¹⁾=T⁽⁰⁾~_i∈I^↑,1

θ ( ˜Tⁱ,xi)for the first generation tree and forq∈[0,θ]:

m˜⁽¹⁾_q (d x)= X

i∈I_θ^↑,1

m^T˜i(d x)1[0,θi)(q).

See Figure 3.2 for a simplified representation. We get that (T⁽¹⁾_θ , (m⁽¹⁾_q ,q ∈[0,θ]),T⁽⁰⁾,m^T(0)) and⁽T⁽¹⁾, ( ˜m⁽¹⁾_q ,q∈[0,θ]),T⁽⁰⁾, ˜m⁽⁰⁾)have the same distribution.

Furthermore, by collecting all the trees grafted onT⁽¹⁾, we get that T =T⁽¹⁾~_i0∈I_θ^↑,2(Tⁱ⁰,x_i0),

where ^I_θ^↑,2= ∪_i∈I^↑,1

θ I_θ,i^↑,2 and where

P²= X

i⁰∈I_θ^↑,2

δ_(xi0,Tⁱ⁰_,θi0)

is, conditionally on (T⁽¹⁾, ( ˜m⁽¹⁾_q ,q∈[0,θ]),T⁽⁰⁾, ˜m⁽⁰⁾) a Poisson point measure onT⁽¹⁾×T×

(0,θ]with intensity:

ν²(d x,dT,d q)=m˜⁽¹⁾_q (d x)³

2βN^ψ(dT⁰)+ Z

(0,+∞)Π(d r)re^−qrP^ψr(dT⁰)´

1_[0,θ](q)d q.

Notice that:

T⁽¹⁾={x∈T0;M^↑(‚;,x‚×[0,θ])≤1} and ^m˜(1)_θ ^{(d x)}+m˜⁽⁰⁾(d x)=1_T⁽¹⁾(x)m^T0(d x). (3.42)

Figure 3.2: The tree T0, T⁽⁰⁾, and a tree Tⁱ and its sub-tree T^˜i belonging to the first generation treeT⁽¹⁾\T⁽⁰⁾.

Then we can iterate this construction, and by taking increasing limits we obtain that the pair⁽⁽∪n∈NT⁽ⁿ⁾_θ ,P

n∈Nm⁽ⁿ⁾_θ ),T₀)has the same distribution as⁽T⁰,T⁽⁰⁾), where:

T⁰=n

x∈T0;M^↑(‚;,x‚×[0,θ])< +∞o

and m^˜0(d x)=1_T⁰(x)m^T0(d x).

To conclude, we need to check first that the completion ofT⁰isT0, or asT0 is complete that the closure ofT⁰ as a subset ofT0 is exactlyT0 and then thatm^T0(T^0c)=0.

Notice thatM^↑has less marks thanM. Then Proposition 1.2 in [AD07] in the case when β=0or an elementary adaptation of it in the general framework of [Voi10], gives there is no loss of mass in the fragmentation process. This implies that, ifψis (sub)critical, then:

m^T0({x∈T0;M(‚;,x‚×[0,θ])= ∞}=0. (3.43) Then, if ψis super-critical, by considering the restriction of T0 up to level a, πa(T0), and using a Girsanov transformation from Definition3.18 withθ=θ^∗ and (3.43), we deduce that (3.43) holds forπa(T0). Since^a is arbitrary, we deduce by monotone convergence that (3.43)

3.4. Application to overshooting

Dans le document THÈSE présentée pour obtenir (Page 89-99)