Preliminaries Stopping times Organization of the paper

Let ˜ F ∞ be the σ-field generated by ˜ S = ρ C t , m C t , t ≥ 0. Recall that P ∗ µ,Π dS denotes the law of the marked exploration process S started at µ, Π ∈ S and stopped when ρ reaches 0. For ℓ ∈ 0, +∞, we will write P ∗ ℓ for P ∗ ℓδ ,0 . If Q is a measure on S and ϕ is a non-negative measurable function defined on the measurable space R + × S × S, we denote by Q[ ϕu, ω, ·] = Z S ϕu, ω, S QdS . In other words, the integration concerns only the third component of the function ϕ. We can now state the Special Markov Property. Theorem 3.2 Special Markov property. Let ϕ be a non-negative measurable function defined on R + × M f R + × S. Then, we have P-a.s. E  exp − X i ∈I ϕA α i , ρ α i − , S i ˜ F ∞   = exp ‚ − Z ∞ du α 1 N ” 1 − e −ϕu,µ,· — |µ= ˜ ρ u Œ exp − Z ∞ du Z 0,∞ π 1 dℓ € 1 − E ∗ ℓ [e −ϕu,µ,· ] |µ= ˜ ρ u Š . 34 In other words, the law under P of the excursion process X i ∈I δ A αi , ρ αi − , S i du dµ dS , given ˜ F ∞ , is the law of a Poisson point measure with intensity 1 {u≥0} du δ ˜ ρ u dµ α 1 Nd S + Z 0,∞ π 1 dℓP ∗ ℓ dS . Informally speaking, this Theorem gives the distribution of the marked exploration process “above” the pruned CRT. The end of this section is now devoted to its proof. Let us first remark that, if lim λ→+∞ φ 1 λ +∞, we have α 1 = 0 and π 1 is a finite measure. Hence, there is no marks on the skeleton and the number of marks on the nodes is finite on every bounded interval of time. The proof of Theorem 3.2 in that case is easy and left to the reader. For the rest of this Section, we assume that lim λ→+∞ φ 1 λ = +∞.

3.1 Preliminaries

Fix t 0 and η 0. For S = S s = ρ s , m s , s ≥ 0, we set B = {σS = +∞} ∪ {T η S = +∞} ∪ {L η S = −∞} where σS = inf{s 0; ρ s = 0}, T η S = inf{s ≥ η; 〈ρ s , 1 〉 ≥ η} and L η S = sup{s ∈ [0, σS ]; 〈η s , 1 〉 ≥ η}, with the convention inf ; = +∞ and sup ; = −∞. We consider non-negative bounded functions ϕ satisfying the assumptions of Theorem 3.2 and these four conditions: h 1 ϕu, µ, S = 0 for any u ≥ t. 1446 h 2 u 7→ ϕu, µ, S is uniformly Lipschitz with a constant that does not depend on µ and S . h 3 ϕu, µ, S = 0 on B; and if S ∈ B c then ϕu, µ, S depends on S only through S u , u ∈ [T η , L η ]. h 4 The function µ 7→ ϕu, µ, S is continuous with respect to the distance Dµ, µ ′ + |〈µ, 1〉 − 〈µ ′ , 1 〉| on M f R + , where D is a distance on M f R + which defines the topology of weak convergence. Lemma 3.3. Let ϕ satisfies h 1 − h 3 and let w be defined on 0, ∞ × [0, ∞ × M f R + by w ℓ, u, µ = E ∗ ℓ [e −ϕu,µ,· ]. Then, for N − a.e. µ ∈ M f R + , the function ℓ, u 7→ wℓ, u, µ is uniformly continuous on 0, ∞ × [0, ∞. Proof. Let u 0 and ℓ ′ ℓ. If we set τ ℓ = inf{t ≥ 0, ρ t {0} = ℓ} we have, by the strong Markov property at time τ ℓ and assumption h 3 , that E ∗ ℓ ′ ” e −ϕu,µ,· — = E ∗ ℓ ′ h 1 {T η τ ℓ } E ∗ ℓ ” e −ϕu,µ,· —i + E ∗ ℓ ′ h e −ϕu,µ,· 1 {T η ≤τ ℓ } i . Therefore, wℓ ′ , u, µ − wℓ, u, µ| ≤ E ∗ ℓ ′ h 1 {T η ≤τ ℓ } E ∗ ℓ ” e −ϕu,µ,· —i + E ∗ ℓ ′ h e −ϕu,µ,· 1 {T η ≤τ ℓ } i ≤ 2P ∗ ℓ ′ T η ≤ τ ℓ = 2P ∗ ℓ ′ −ℓ T η +∞. Using Lemma 1.6, for ℓ ′ − ℓ η, we get |wℓ ′ , u, µ − wℓ, u, µ| ≤ 2 € 1 − e −ℓ ′ −ℓN[T η ∞] Š . Since N[T η ∞] ∞, we then deduce there exists a finite constant c η s.t. for all function ϕ satisfying h 3 , |wℓ ′ , u, µ − wℓ, u, µ| ≤ c η |ℓ ′ − ℓ|. 35 The absolute continuity with respect to u is a direct consequence of assumptions h 1 − h 2 .

3.2 Stopping times

Let Rd t, du be a Poisson point measure on R 2 + defined on S, F independent of F ∞ with inten- sity the Lebesgue measure. We denote by G t the σ-field generated by R· ∩ [0, t] × R + . For every ǫ 0, the process R ǫ t := R[0, t] × [0, 1ǫ] is a Poisson process with intensity 1ǫ. We denote by e ǫ k , k ≥ 1 the time intervals between the jumps of R ǫ t , t ≥ 0. The random variables e ǫ k , k ≥ 1 are i.i.d. exponential random variables with mean ǫ, and are independent of F ∞ . They define a mesh of R + which is finer and finer as ǫ decreases to 0. 1447 For ǫ 0, we consider T ǫ = 0, M ǫ = 0 and for k ≥ 0, M ǫ k+ 1 = inf{i M ǫ k ; m T ǫ k + P i j=M ǫ k +1 e ǫ j 6= 0}, S ǫ k+ 1 = T ǫ k + M ǫ k+ 1 X j=M ǫ k +1 e ǫ j , T ǫ k+ 1 = inf{s S ǫ k+ 1 ; m s = 0}, 36 with the convention inf ; = +∞. For every t ≥ 0, we set τ ǫ t = Z t ds 1 S k ≥1 [T ǫ k ,S ǫ k+ 1 s and F e t = σF t ∪ G τ ǫ t . 37 Notice that T ǫ k and S ǫ k are F e -stopping times. Now we introduce a notation for the process defined above the marks on the intervals ” S ǫ k , T ǫ k — . We set, for a ≥ 0, ¯ H a the level of the first mark, ρ − a the restriction of ρ a strictly below it and ρ + a the restriction of ρ a above it: ¯ H a = sup{t 0, m a [0, t] = 0}, ρ − a = ρ a · ∩ [0, ¯ H a 38 and ρ + a is defined by ρ a = [ρ − a , ρ + a ], that is for any f ∈ B + R + , 〈ρ + a , f 〉 = Z [ ¯ H a , ∞ f r − ¯ H a ρ a d r. 39 For k ≥ 1 and ǫ 0 fixed, we define S k , ǫ = € ρ k , ǫ , m k , ǫ Š in the following way: for s 0 and f ∈ B + R + ρ k , ǫ s = ρ + S ǫ k +s∧T ǫ k , 〈m a k , ǫ s , f 〉 = Z ¯ H Sǫ k ,+ ∞ f r − ¯ H S ǫ k m a S ǫ k +s∧T ǫ k d r, with a ∈ {nod, ske}, and m k , ǫ s = m nod k , ǫ s , m ske k , ǫ s . Notice that ρ k , ǫ s {0} = ρ S ǫ k { ¯ H S ǫ k }. For k ≥ 1, we consider the σ-field F ǫ,k generated by the family of processes S T ǫ ℓ +s∧S ǫ ℓ+1 − , s ℓ∈{0,...,k−1} . Notice that for k ∈ N ∗ F ǫ,k ⊂ F e S ǫ k . 40

3.3 Approximation of the functional