Xan Duhalde
email: xan.duhalde@upmc.fr

Thomas Duquesne
email: thomas.duquesne@upmc.fr

EXACT PACKING MEASURE OF THE RANGE OF ψ-SUPER BROWNIAN MOTIONS

Keywords: AMS 2000 subject classifications: Primary 60G57, 60J80. Secondary 28A78 Super-Brownian motion, general branching mechanism, Lévy snake, total range, occupation measure, exact packing measure

We consider super processes whose spatial motion is the d-dimensional Brownian motion and whose branching mechanism ψ is critical or subcritical; such processes are called ψ-super Brownian motions. If d > 2γ/(γ -1), where γ ∈ (1, 2] is the lower index of ψ at ∞, then the total range of the ψ-super Brownian motion has an exact packing measure whose gauge function is g(r) = (log log 1/r)/ϕ -1 ((1/r log log 1/r) 2 ), where ϕ = ψ ′ • ψ -1 . More precisely, we show that the occupation measure of the ψ-super Brownian motion is the g-packing measure restricted to its total range, up to a deterministic multiplicative constant only depending on d and ψ. This generalizes the main result of [10] that treats the quadratic branching case. For a wide class of ψ, the constant 2γ/(γ -1) is shown to be equal to the packing dimension of the total range.

1 Introduction.

The main result of this paper provides an exact packing gauge function for the total range of super processes whose spatial motion is the d-dimensional Brownian motion and whose branching mechanism ψ is critical or subcritical. We call such super processes ψ-super Brownian motions (or ψ-SBM, for short). This generalizes the main result of [START_REF] Duquesne | The packing measure of the range of super-Brownian motion[END_REF] that concerns the Dawson-Watanabe super process corresponding to the quadratic branching mechanism ψ(λ) = λ 2 .

Before stating precisely our main results, let us briefly recall previous works related to the fine geometric properties of super processes. Most of these results concern the Dawson-Watanabe super process (Z t ) t≥0 . Dawson and Hochberg [START_REF] Dawson | The carrying dimension of a stochastic measure diffusion[END_REF] have proved that a.s. for all t > 0, the Hausdorff dimension of the topological support of Z t is equal to 2 ∧ d. In [START_REF] Dawson | Historical processes[END_REF], Dawson and Perkins prove that in supercritical dimensions d ≥ 3, the Dawson-Watanabe super process Z t is a.s. equal to the h 1 -Hausdorff measure restricted to the topological support of Z t , where h 1 (r) = r 2 log log 1/r (see also Perkins [34,[START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF] for a close result holding a.s. for all times t). By use of the Brownian snake, Le Gall and Perkins [START_REF] Le Gall | The Hausdorff measure of the support of two-dimensional super-Brownian motion[END_REF] prove a similar result in the critical dimension d = 2 with the gauge function h 2 (r) = r 2 log 1/r log log log 1/r. In [START_REF] Le Gall | The packing measure of the support of super-Brownian motion[END_REF], Le Let us introduce precisely the main results of our paper. We first fix a branching mechanism ψ that is critical or subcritical: namely, ψ : R + → R + is the Laplace exponent of a spectrally positive Lévy process that is of the following Lévy-Khintchine form:

∀λ ∈ R + , ψ(λ) = αλ + βλ 2 + (0,∞) (e -λr -1 + λr) π(dr) , (1)
where α, β ∈ R + , and π is the Lévy measure that satisfies (0,∞) (r ∧ r 2 ) π(dr) < ∞. The branching mechanism ψ is the main parameter that governs the law of the processes that are considered in this paper. We introduce two exponents that compare ψ with power functions at infinity:

γ = sup c ∈ R + : lim λ→∞ ψ(λ)λ -c = ∞ , η = inf{c ∈ R + : lim λ→∞ ψ(λ)λ -c = 0 . (2)
The lower index γ and the upper index η have been introduced by Blumenthal and Getoor [START_REF] Blumenthal | Sample functions of stochastic processes with stationary independent increments[END_REF]: they appear in the fractal dimensions and the regularity of the processes that we consider. The statements of the paper also involve a third exponent:

δ = sup c ∈ R + : ∃ C ∈ (0, ∞) such that Cψ(µ)µ -c ≤ ψ(λ)λ -c , 1 ≤ µ ≤ λ (3)
that has been introduced in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF]. It is easy to check that 1 ≤ δ ≤ γ ≤ η ≤ 2. If ψ is regularly varying at ∞, all these exponents coincide. In general, they are however distinct and we mention that there exist branching mechanisms ψ of the form [START_REF] Bertoin | Lévy processes[END_REF] such that δ = 1 < γ = η (see Lemma 2.3 and Lemma 2.4 in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] for more details). In our paper we shall often assume that δ > 1 which is a mild regularity assumption on ψ (see Comment 1.4 below).

The space R d stands for the usual d-dimensional Euclidian space. We denote by M f (R d ) the space of finite Borel measures equipped with the topology of weak convergence. For all µ ∈ M f (R d ) and for all Borel measurable functions f : R d → R + , we use the following notation:

f, µ = R d f (x) µ(dx) and µ = µ R d .
Then, µ is the total mass of µ. We shall also denote by Supp(µ) the topological support of µ that is the smallest closed subset supporting µ.

Unless the contrary is explicitly mentioned, all the random variables that we consider are defined on the same measurable space (Ω, F). We first introduce a R d -valued continuous process ξ = (ξ t ) t≥0 ; for all x ∈ R d , we let P x be a probability measure on (Ω, F) such that ξ under P x is distributed as a standard d-dimensional Brownian motion starting from x. We also introduce Z = (Z t ) t∈R + that is a M f (R d )valued càglàd process defined on (Ω, F), and for all µ ∈ M f (R d ), we let P µ be a probability measure on (Ω, F) such that Z under P µ is distributed as super Brownian motion with branching mechanism ψ. Namely, under P µ , Z is a Markov process whose transitions are characterized as follows: for all bounded Borel measurable functions f : R d → R + and for all s, t ∈ R + ,

P µ -a.s. E µ exp(-Z t+s , f ) Z s = exp(-Z s , v t ), (4)
where the function (v t (x)) t∈R + ,x∈R d is the unique nonnegative solution of the integral equation

v t (x) + E x t 0 ψ v t-s (ξ s ) ds = E x [f (ξ t )] , x ∈ R d , t ∈ [0, ∞).
Dawson-Watanabe super processes correspond, up to scaling in time and space, to the branching mechanism ψ(λ) = λ 2 . Super diffusions with general branching mechanisms of the form [START_REF] Bertoin | Lévy processes[END_REF] have been introduced by Dynkin [START_REF] Dynkin | A probabilistic approach to one class of nonlinear differential equations[END_REF]; for a detailed account on super processes, we refer to the books of Dynkin [START_REF] Dynkin | superdiffusions and partial differential equations[END_REF][START_REF] Dynkin | Superdiffusions and positive solutions of nonlinear partial differential equations[END_REF], Le Gall [START_REF] Le Gall | Spatial branching processes, random snakes and partial differential equations[END_REF], Perkins [START_REF] Bolthausen | Lectures from the 29th Summer School on Probability Theory held in Saint-Flour[END_REF], Etheridge [START_REF] Etheridge | An introduction to superprocesses[END_REF] and Li [START_REF] Li | Measure-valued branching Markov processes[END_REF]. We easily check that, under P µ , the process ( Z t ) t≥0 of the total mass of the ψ-SBM is a continuous states branching process with branching mechanism ψ. Continuous states branching processes have been introduced by Jirina [START_REF] Ji Řina | Stochastic branching processes with continuous state space[END_REF] and Lamperti [START_REF] Lamperti | Continuous state branching processes[END_REF][START_REF] Lamperti | The limit of a sequence of branching processes[END_REF], and further studied by Bingham [START_REF] Bingham | Continuous branching processes and spectral positivity[END_REF]. The assumption δ > 1, implies γ > 1, which easily entails ∞ dλ/ψ(λ) < ∞ that is called the Grey condition. Under this condition, standard results on continuous states branching processes (see Bingham [START_REF] Bingham | Continuous branching processes and spectral positivity[END_REF]) imply that Z is absorbed in 0 in finite time: namely, P µ (∃t ∈ R + : Z t = 0) = 1. Thus the following definition makes sense:

M = ∞ 0 Z t dt (5)
and M is therefore a random finite Borel measure on R d : it is the occupation measure of the ψ-SBM Z.

We also define the total range of Z by

R = ε>0 t≥ε Supp(Z t ) , (6)
where for any subset B in R d , B stands for its closure. We recall here a result due to Sheu [START_REF] Sheu | Asymptotic behavior of superprocesses[END_REF] that gives a condition on ψ for R to be bounded:

P µ -a.s. R is bounded ⇐= = = =⇒ ∞ 1 db b 1 ψ(a)da
< ∞ and Supp(µ) is compact. [START_REF] Dawson | Super-Brownian motion: path properties and hitting probabilities[END_REF] See also Hesse and Kyprianou [START_REF] Hesse | The mass of super-Brownian motion upon exiting balls and Sheu's compact support condition[END_REF] for a simple probabilistic proof. Note that if γ > 1, then (7) holds true.

We next denote by dim H and dim p respectively the Hausdorff and the packing dimensions on R d . We also denote by dim and dim the lower and the upper box dimensions. We refer to Falconer [START_REF] Falconer | Fractal geometry[END_REF] for precise definitions. We next recall Theorem 6.3 [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] that asserts that for all µ ∈ M f (R d ) distinct from the null measure, the following holds true.

If γ > 1, then P µ -a.s.

dim

H (R) = d ∧ 2η η -1 . (8)
If furthermore Supp(µ) is compact, then R is bounded (by [START_REF] Dawson | Super-Brownian motion: path properties and hitting probabilities[END_REF]) and Theorem 6.3 [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] also asserts that P µ -a.s. dim (R) = d ∧ 2η η-1 . As already mentioned [START_REF] Dawson | Historical processes[END_REF] generalizes the work of Delmas [START_REF] Delmas | Some properties of the range of the super-Brownian motion[END_REF] that treats SBMs whose branching mechanism is stable. Note that Assumption γ > 1 is not completely satisfactory for dim H (R) only depends on d and η (see Proposition 5.7 [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] and the discussion in Section 5.3 of this article). The first result of our paper computes the packing dimension of R under more restrictive assumptions.

Theorem 1.1 Let µ ∈ M f (R d ) be distinct from the null measure. Let ψ be of the form [START_REF] Bertoin | Lévy processes[END_REF]. Let R be the total range of the ψ-SBM with initial measure µ, as defined in [START_REF] Dawson | The carrying dimension of a stochastic measure diffusion[END_REF]. Assume that δ > 1 and that d > 2δ δ-1 . Then,

P µ -a.s. dim p (R) = 2γ γ -1 . ( 9
)
If furthermore Supp(µ) is compact, then P µ -a.s. dim(R) = 2γ γ-1 .
Comment 1.1 Theorem 5.5 [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] shows that if γ > 1, the Hausdorff and the packing dimensions of the ψ-Lévy tree are resp. η η-1 and γ γ-1 . Thus, for all sufficiently large d, (8) and ( 9) imply that the Hausdorff and the packing dimensions of R are twice that of the ψ-Lévy tree. This can be informally explained by the fact that R appears as the range of the ψ-Lévy snake that can be viewed as a Gaussian process indexed by the ψ-Lévy tree that is ( 1 2 -ε)-Hölder regular for any ε ∈ (0, 1/2) (see Lemma 6.4 [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] and see also [START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF] in Section 2.4 of the present paper for more details).

Comment 1.2

The above mentioned Hölder-regularity of the Lévy snake studied in [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] also entails that if Supp(µ) is compact and if γ > 1, then dim(R) ≤ d ∧ 2γ γ-1 . Since dim p (A) ≤ dim(A) for any bounded A ⊂ R d (see e.g. [START_REF] Falconer | Fractal geometry[END_REF]), an easy argument entails that for all non-null finite measure µ, if γ > 1, then P µ -a.s. dim p (R) ≤ d ∧ 2γ γ-1 . This, combined with (8) entails that

if η = γ > 1, then P µ -a.s. dim H (R) = dim p (R) = d ∧ 2γ γ -1 , (10)
with the same equality for the lower and the upper box dimensions if Supp(µ) is compact. The equality γ = η holds true for instance if ψ is regularly varying at ∞. Therefore, the novelty of Theorem 1.1 only concerns the cases where γ = η.

Comment 1.3 In Theorem 1.1, note that Assumption δ > 1 is not optimal since the value of dim p (R) only depends on γ. Our arguments fail to prove [START_REF] Delmas | Some properties of the range of the super-Brownian motion[END_REF]

when d ∈ ( 2γ γ-1 , 2δ δ-1 ). We conjecture that if γ > 1, then P µ -a.s. dim p (R) = d ∧ 2γ γ-1 . Let us set ϕ = ψ ′ • ψ -1 .
The main properties of that increasing function are stated in Section 2.2. Here, we just notice that the reciprocal function of ϕ, that is denoted by ϕ -1 , is defined from [α, ∞) to [0, ∞). Then, we set

g(r) = log log 1 r ϕ -1 ( 1 r log log 1 r ) 2 , r ∈ (0, r 0 ) (11)
where r 0 = min(α -1/2 , e -e ), with the convention α -1/2 = ∞ if α = 0. We check (see Section 2.2) that g is a continuous increasing function such that lim 0+ g = 0.

We next denote by P g the g-packing measure on R d , whose definition is recalled in Section 2.1. The following theorem is the main result of the paper. Theorem 1.2 Let µ ∈ M f (R d ) be distinct from the null measure. Let ψ be of the form [START_REF] Bertoin | Lévy processes[END_REF]. Let Z be a ψ-SBM starting from µ; let R be its total range, as defined by [START_REF] Dawson | The carrying dimension of a stochastic measure diffusion[END_REF], and let M be its occupation measure, as defined by [START_REF] Bolthausen | Lectures from the 29th Summer School on Probability Theory held in Saint-Flour[END_REF]. Let g be defined by [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF]. Assume that

δ > 1 and d > 2γ γ -1 .
Then, there exists a positive constant κ d,ψ that only depends on d and ψ such that

P µ -a.s. M = κ d,ψ P g ( • ∩ R) . ( 12
)
Comment 1.4 Unlike Theorem 1.1, we think that the assumptions of Theorem 1.2 are optimal in the following sense. Indeed, since d ∧ 2γ γ-1 is thought to be the packing dimension of R, (12) probably does not hold true when d ≤ 2γ γ-1 . Moreover, arguing as in Lemma 2.3 [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF], we easily check that δ > 1 if and only if g satisfies a doubling condition (see (13) Section 2.1). Although, it is possible to define packing measures with respect to irregular gauge functions (see Edgar [START_REF] Edgar | Centered densities and fractal measures[END_REF]), the doubling condition is the minimal assumption on a gauge function which implies that the corresponding packing measure has nice properties (regularity, comparison lemmas). In this sense, Assumption δ > 1 is required to stay within the framework of standard geometric measures.

Comment 1.5 In the stable cases where ψ(λ) = λ γ with γ ∈ (1, 2], then

∀r ∈ (0, ∞), g(r) = r 2γ γ-1 log log 1/r -γ+1 γ-1 .
If γ = 2, then g(r) = r 4 (log log 1/r) -3 and we recover the result from [START_REF] Duquesne | The packing measure of the range of super-Brownian motion[END_REF]. Moreover, note that in the stable cases, Theorem 1.1 is a straightforward consequence of Theorem 1.2. However, when γ = η, it turns out that Theorem 1.1 cannot be simply derived from Theorem 1.2: indeed, one important ingredient of the proof of Theorem 1.2 consists in computing the lower local density of M with respect to g; this lower limit is achieved along a deterministic sequence of radii whose images by g are hard to compare with a power function.

The paper is organised as follows. In Section 2, we recall definitions and basic results. Section 2.1 is devoted to packing measures and to two comparison lemmas that are standard technical tools used to compute packing measures. Section 2.2 gather elementary facts on the power exponents δ, γ and η associated with the branching mechanism ψ. In Section 2.3 and in Section 2.4, we recall the definitions of -and various results on -the ψ-height process, the corresponding ψ-Lévy tree and the associated ψ-Lévy snake. In Section 3, we prove estimates on a specific subordinator (Sections 3.1 and Section 3.3) and on functionals of the snake involving the hitting time of a given ball (Section 3.2 and Section 3.4). Section 4 is devoted to the proof of the two main theorems: we prove Theorem 1.2 first and Theorem 1.1 next.

2 Notations, definitions and preliminary results.

Packing measures.

In this section, we briefly recall basic results on packing measures on the Euclidian space R d that have been introduced by Taylor and Tricot in [START_REF] Taylor | Packing measure, and its evaluation for a Brownian path[END_REF].

A gauge function is an increasing continuous function g : (0, r 0 ) → (0, ∞), where r 0 ∈ (0, ∞), such that lim 0+ g = 0 and that satisfies a doubling condition: namely, there exists C ∈ (0, ∞) such that ∀r ∈ (0, r 0 /2), g(2r) ≤ Cg(r) .

(

) 13
Let B ⊂ R d and let ε ∈ (0, ∞). Recall that a (closed) ε-packing of B is a finite collection of pairwise disjoint closed ball (B(x m , r m )) 1≤m≤n whose centers x m belong to B and whose radii r m are not greater than ε. We then set

P (ε) g (B) = sup 1≤m≤n g(r m ) ; B(x m , r m ) 1≤m≤n , ε-packing of B (14)
and

P * g (B) = lim ε→0+ P (ε) g (B) ∈ [0, ∞] , (15)
that is called the g-packing pre-measure of B. The g-packing measure of B is then given by

P g (B) = inf n≥0 P * g (B n ) ; B ⊂ n≥0 B n . (16)
Remark 2.1 The definition [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] of

P (ε)
g that we adopt here is slightly different from the one introduced by Taylor and Tricot [START_REF] Taylor | Packing measure, and its evaluation for a Brownian path[END_REF] who take the infimum of n m=1 g(2r m ) over ε-packings with open balls. However, since g is increasing, continuous and satisfies a doubling condition [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], the resulting measure is quite close to Taylor and Tricot's definition: the difference is irrelevant to our purpose and their main results on packing measures immediately apply to the g-packing measures defined by [START_REF] Duquesne | The Hausdorff measure of stable trees[END_REF].

Next recall from Lemma 5.1 [START_REF] Taylor | Packing measure, and its evaluation for a Brownian path[END_REF] that P g is a Borel-regular outer measure. Moreover, it is obvious from the definition that for any subset B ⊂ R d ,

P g (B) ≤ P * g (B) . (17)
Next, if B is a P g -measurable set such that P g (B) < ∞, then for any ε > 0, there exists a closed subset

F ε ⊂ B such that P g (B) ≤ P g (F ε ) + ε . (18)
We recall here Theorem 5.4 [START_REF] Taylor | Packing measure, and its evaluation for a Brownian path[END_REF] that is a standard comparison result for packing measures.

Theorem 2.1 (Theorem 5.4 [START_REF] Taylor | Packing measure, and its evaluation for a Brownian path[END_REF]) Let µ be a finite Borel measure on R d . Let B be a Borel subset of R d . Let g be a gauge function satisfying a doubling condition [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] with a constant C > 0. Then, the following holds true.

(i) If lim inf r→0 µ(B(x,r)) g(r)

> 1 for any x ∈ B, then µ(B) ≥ P g (B).

(ii) If lim inf r→0 µ(B(x,r)) g(r)

< 1 for any x ∈ B, then µ(B) ≤ C 2 P g (B).

We also recall the following specific result due to Edgar in [START_REF] Edgar | Centered densities and fractal measures[END_REF] We finally recall the definition of the packing dimension: let α ∈ (0, ∞); we simply write P α instead of P g when g(r) = r α , r ∈ (0, ∞). Let B ⊂ R d . Then, the packing dimension of B, denoted by dim p (B) is the unique real number in [0, d] such that

P α (B) = ∞ if α < dim p (B) and P α (B) = 0 if α > dim p (B) . (19)
2.2 Exponents.

In this section we briefly recall from [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] several results relating power exponents associated with ψ to properties of the gauge function g introduced in [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF]. Recall that the branching mechanism ψ has the Lévy-Khintchine form [START_REF] Bertoin | Lévy processes[END_REF]. It is well-known that ψ ′ is the Laplace exponent of a subordinator, just like ψ -1 , the reciprocal of ψ. Thus, ϕ = ψ ′ • ψ -1 is also the Laplace exponent of a subordinator. Note that ψ ′ (0) = α. As already mentioned, the reciprocal function of ϕ, denoted by ϕ -1 , is then defined from [α, ∞) to [0, ∞). We also introduce the function ψ(λ) = ψ(λ)/λ that easily shown to be also the Laplace exponent of subordinator. Next observe that 1/ϕ is the derivative of ψ -1 and recall that ψ is convex and that ψ ′ , ψ, ψ -1 and ϕ are concave. In particular, this implies ψ(2λ) ≤ 2 ψ(λ) and the following

ψ(2λ) ≤ 4ψ(λ), ψ(λ) ≤ ψ ′ (λ) ≤ ψ(2λ) -ψ(λ) λ ≤ 4 ψ(λ) and λ ψ -1 (λ) ≤ ϕ(λ) ≤ 4λ ψ -1 (λ) . (20)
Let φ : [0, ∞) → [0, ∞) be a continuous increasing function. We agree on sup ∅ = 0 and inf ∅ = ∞ and we define the following exponents that compare φ with power functions at infinity:

γ φ = sup c ∈ R + : lim λ→∞ φ(λ)λ -c = ∞ , η φ = inf{c ∈ R + : lim λ→∞ φ(λ)λ -c = 0 . (21)
Then, γ φ (resp. η φ ) is the lower exponent (resp. the upper) of φ at ∞. We also introduce the following exponent

δ φ = sup c ∈ R + : ∃ C ∈ (0, ∞) such that Cφ(µ)µ -c ≤ φ(λ)λ -c , 1 ≤ µ ≤ λ (22)
that plays a important role for the regularity of the gauge function. Thus by ( 2) and (3)

γ = γ ψ , η = η ψ , δ = δ ψ .
It is easy to check that 1 ≤ δ ≤ γ ≤ η ≤ 2. If ψ is regularly varying at ∞, all these exponents coincide.

In general, they are however distinct and we mention that there exist branching mechanisms ψ of the form (1) such that δ = 1 < γ = η: see Lemma 2.3 and Lemma 2.4 in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] for more detail. As a direct consequence of [START_REF] Etheridge | An introduction to superprocesses[END_REF] we have

δ ψ = δ ψ ′ = δ -1, γ ψ = γ ψ ′ = γ -1 and η ψ = η ψ ′ = η -1.
Moreover, we get δ ϕ = (δ -1)/δ, γ ϕ = (γ -1)/γ and η ϕ = (η -1)/η. Recall from [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF] the definition of the function g. The arguments of the proof of Lemma 2.3 (i) in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] can be immediately adapted to prove that g : (0, r 0 ) → (0, ∞) is is an increasing continuous function such that lim 0+ g = 0 and such that it satisfies the following.

(i) The function g satisfies the doubling condition (13) iff δ > 1.

(ii) If ψ is regularly varying at ∞ with exponent c > 1, then δ = γ = η = c and g is regularly varying at ∞ with exponent c/(c -1).

We shall further need the following bound that is a consequence of (20).

Lemma 2.3 Let g the gauge function defined by [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF]. Let c ∈ (0, ∞). Then there exists r(c) ∈ (0, r 0 ) that only depends on c such that ∀r ∈ (0, r(c)), g(r) ψ ′-1 (c/r 2 ) ≤ 4r 2 .

Proof. Take r(c) ∈ (0, r 0 ) such that log log 1/r(c) ≥ 1∨ √ c. Thus,

r ∈ (0, r(c)), ψ ′-1 (cr -2 ) ≤ ψ ′-1 ((r -1 log log r -1 ) 2 ) . Recall that ϕ -1 = ψ • ψ ′-1
. By comparing ψ and ψ ′ thanks to [START_REF] Etheridge | An introduction to superprocesses[END_REF], we get for all r ∈ (0, r(c))

g(r) ψ ′-1 (cr -2 ) = ψ ′-1 (cr -2 ) log log 1 r ψ(ψ ′-1 (( 1 r log log 1 r ) 2 ) ≤ ψ ′-1 (( 1 r log log 1 r ) 2 ) log log 1 r ψ(ψ ′-1 (( 1 r log log 1 r ) 2 ) ≤ log log 1 r ψ ψ ′-1 (( 1 r log log 1 r ) 2 ) ≤ 4r 2 log log 1 r ≤ 4r 2 ,
which is the desired result.

Height process and Lévy trees.

In this section we recall the definition of the height process that encodes Lévy trees. The Lévy trees are the scaling limit of Galton-Watson trees and they are the genealogy of super-Brownian motion.

The height process. Recall that ψ stands for a branching mechanism of the form [START_REF] Bertoin | Lévy processes[END_REF]. We always assume that γ > 1. It is convenient to work on the canonical space D([0, ∞), R) of càdlàg paths equipped with Skorohod topology and the corresponding Borel sigma-field. We denote by X = (X t ) t≥0 the canonical process and by P the distribution of the spectrally positive Lévy processes starting from 0 whose Laplace exponent is ψ. Namely,

∀t, λ ∈ [0, ∞), E exp(-λX t ) = exp(tψ(λ)) .
Note that the specific form (1) of ψ implies that X t is integrable and that E[X t ] = -αt. This easily entails that X does not drift to ∞. Conversely, if a spectrally positive Lévy process does not drift to ∞, then its Laplace exponent is necessarily of the form [START_REF] Bertoin | Lévy processes[END_REF]. We shall assume that γ > 1 which easily implies that either β > 0 (and γ = 2) or (0,1) rπ(dr) = ∞. It entails that P-a.s. X has unbounded variation paths: see Bertoin [START_REF] Bertoin | Lévy processes[END_REF], Chapter VII, Corollary 5 (iii).

Note that γ > 1 entails ∞ dλ/ψ(λ) < ∞ and, as shown by Le Gall and Le Jan [START_REF] Le Gall | Branching processes in Lévy processes: the exploration process[END_REF] (see also [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF],

Chapter 1), there exists a continuous process H = (H t ) t≥0 such that for any t ∈ [0, ∞), the following limit holds true in P-probability:

H t = lim ε→0 1 ε t 0 1 {I s t <Xs<I s t +ε} ds, (23)
where I s t stands for inf s≤r≤t X r . The process H = (H t ) t≥0 is called the ψ-height process; it turns out to encode the genealogy of super-Brownian motion with branching mechanism ψ as explained below. We shall need the following result that is proved in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF].

Lemma 2.4 ([13] Theorem 1.4.4) Assume that γ > 1. Then for every c ∈ (0, γ-1 γ ), H is P-a.s. locally c-Hölder continuous.

Excursions of the height process. We denote by I is the infimum process of X:

∀t ∈ R + , I t = inf 0≤r≤t X r .
When ψ is of the form ψ(λ) = βλ 2 , X is distributed as a Brownian motion and [START_REF] Hesse | The mass of super-Brownian motion upon exiting balls and Sheu's compact support condition[END_REF] easily implies that H is proportional to X -I, which is distributed as a reflected Brownian motion. In the general cases, H is neither a Markov process nor a martingale. However it is possible to develop an excursion theory for H as follows.

Since X has unbounded variation sample paths, basic results of fluctuation theory (see Bertoin [START_REF] Bertoin | Lévy processes[END_REF], Sections VI.1 and VII.1) entail that X -I is a strong Markov process in [0, ∞) and that 0 is regular for (0, ∞) and recurrent with respect to this Markov process. Moreover, -I is a local time at 0 for X -I (see Bertoin [START_REF] Bertoin | Lévy processes[END_REF], Theorem VII.1). We denote by N the corresponding excursion measure of X -I above 0. More precisely, we denote by (a j , b j ), j ∈ I, the excursion intervals of X -I above 0 and we define the corresponding excursions by

X j = X (a j + •)∧b j -I a j , j ∈ I. Then, j∈I δ (-Ia j ,X j ) is a Poisson point measure on [0, ∞)×D([0, ∞), R) with intensity dx N (dX).
Next observe that under P, the value of H t only depends on the excursion of X -I straddling time t and we easily check that

j∈I (a j , b j ) = {t ≥ 0 : H t > 0} .
This allows to define the height process under N as a certain measurable function H(X) of X. We denote by C(R + , R) the space of the continuous functions from [0, ∞) to R equipped with the topology of the uniform convergence on every compact subsets of [0, ∞); by convenience, we shall slightly abuse notation by denoting by H = (H t ) t≥0 the canonical process on C(R + , R) and by denoting by N (dH) the "distribution" of the height process H(X) associated with X under the excursion measure N (dX). Then we derive from the previous results the following Poisson decomposition of the height process H(X) associated with X under P: for any j ∈ I, set H j = H (a j + •)∧b j ; then, under P, the point measure

j∈I δ (-Ia j ,H j ) (24)
is distributed as a Poisson point measure on [0, ∞) × C(R + , R) with intensity dx N (dH). For more details, we refer to [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Chapter 1.

We denote by σ the duration of X under its excursion measure N (with an obvious definition). It is easy to check that H and X under N have the same duration and that the following holds true.

N -a.e. σ < ∞ , H 0 = H σ = 0 and H t > 0 ⇐⇒ t ∈ (0, σ) .
Basic results of fluctuation theory (see Bertoin [START_REF] Bertoin | Lévy processes[END_REF], Chapter VII) also entail:

∀λ ∈ (0, ∞) N 1-e -λσ = ψ -1 (λ). ( 25
)
Local times of the height process. We recall from [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Chapter 1, Section 1.3, the following result: there exists a jointly measurable process (L a s ) a,s∈[0,∞) such that P-a.s. for any a ∈ [0, ∞), s → L a s is continuous, non-decreasing and such that

∀ t, a ≥ 0, lim ε→0 E sup 0≤s≤t 1 ε s 0 dr 1 {a<Hr ≤a+ε} -L a s = 0 . ( 26
)
The process (L a s ) s∈[0,∞) is called the a-local time of H. Recall that I stands for the infimum process of X. One can show (see [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF], Lemma 1.3.2) that for fixed t, L 0 t = -I t . Moreover, one can observe that the support of the random Stieltjes measure dL a

• is contained in the closed set {t ≥ 0 : H t = a}. A general version of the Ray-Knight theorem for H asserts the following. For any x ≥ 0, set T x = inf{t ≥ 0 : X t = -x}; then, the process (L a Tx ; a ≥ 0) is distributed as a continuous-states branching process CSBP with branching mechanism ψ and initial state x (see Le Gall and Le Jan [START_REF] Le Gall | Branching processes in Lévy processes: the exploration process[END_REF], Theorem 4.2, and also [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Theorem 1.4.1).

It is possible to define the local times of H under the excursion measure N as follows. For any b > 0, let us set v(b) = N (supt∈[0,σ] H t > b). Since H is continuous, the Poisson decomposition [START_REF] Ji Řina | Stochastic branching processes with continuous state space[END_REF] implies that v(b) < ∞, for any b > 0. It is moreover clear that v is non-increasing and that lim ∞ v = 0. Then, for every a ∈ (0, ∞), we define a continuous increasing process (L a t ) t∈[0,∞) , such that for every b ∈ (0, ∞) and for any t ∈ [0, ∞), one has

lim ε→0 N 1 {sup H>b} sup 0≤s≤t∧σ 1 ε s 0 dr 1 {a-ε<Hr≤a} -L a s = 0. (27)
We refer to [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] Section 1.3 for more details. Note that N -a.e. L a t = L a σ for all t ≥ σ. The process (L a t ) t∈[0,∞) is the a-local time of the excursion of the height process.

Lévy trees. We briefly explain how the height process H under its excursion measure N can be viewed as the contour process of a tree called the Lévy tree. Recall that σ is the duration of H under N . For any

s, t ∈ [0, σ], we set d(s, t) = H t + H s -2 inf u∈[s∧t,s∨t] H u . (28)
The quantity d(s, t) represents the distance between the points corresponding to s and t in the Lévy tree. Indeed d is obviously symmetric in s and t and we easily check that d satisfies the triangle inequality. Two real numbers s, t ∈ [0, σ] correspond to the same point in the Lévy tree iff d(s, t) = 0, which is denoted by s ∼ t. Observe that ∼ is an equivalence relation. The Lévy tree is then given by the quotient set

T = [0, σ]/ ∼ .
Then, d induces a true metric on T that we keep denoting d. Denote by p : [0, σ] → T the canonical projection. Since H is continuous N -a.e., so is p, which implies that (T , d) is a random compact connected metric space. More specifically, (T , d) is N -a.e a compact real-tree, namely a compact metric space such that any two points are connected by a unique self-avoiding path, that turns out to be geodesic: see [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] for more details on Lévy trees viewed as real-trees.

The mass measure of the Lévy tree T , denoted by m, is the pushforward measure of the Lebesgue measure ℓ on [0, σ] via the canonical projection p. Namely, N -a.e. for all Borel measurable function f :

T → R + , m, f = σ 0 ℓ(dt) f (p(t)) .
One can show that N -a.e. the mass measure is diffuse; obviously its topological support is T and m(T ) = σ. We refer to [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] for more details.

Let r ∈ (0, ∞)

and let t ∈ [0, σ]. Let B(p(t), r
) denote the open ball in (T , d) with center p(t) and radius r. Then the mass measure of B(p(t), r) in (T , d) is then given by

a(t, r) := m (B(p(t), r)) = σ 0 ℓ(ds) 1 {d(s,t)≤r} . ( 29
)
We shall need the following result on the lower density of m at typical points that is proved in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF], Theorem 1.2. To that end, we set

∀r ∈ (0, α ∧ e -e ), k(r) := log log 1 r ϕ -1 ( 1 r log log 1 r ) . ( 30
)
Lemma 2.5 ([12] Theorem 1.2) Let ψ be a branching mechanism of the form [START_REF] Bertoin | Lévy processes[END_REF]. Let σ the the duration of the height process H under its excursion measure N . Let k be as in [START_REF] Le Gall | Branching processes in Lévy processes: the exploration process[END_REF]. Assume that δ > 1. Then, there exists a constant c 1 ∈ (0, ∞) that only depends on ψ such that

N -a.e. for ℓ-almost all t ∈ [0, σ] lim inf r→0 a(t, r) k(r) ≥ c 1 ,
where a(t, r) is defined by ( 29) for all r ∈ (0, ∞) and for all t ∈ [0, σ].

The exploration process. As already mentioned, the height process is not a Markov process. To explore in a Markovian way the Lévy tree, Le Gall and Le Jan in [START_REF] Le Gall | Branching processes in Lévy processes: the exploration process[END_REF] introduce a measure valued process ρ = (ρ t ) t≥0 that is called the exploration process whose definition is the following. Denote by M f (R + ) the set of finite measures on [0, ∞) equipped with the total variation distance. Recall that X under P is a spectrally positive Lévy process starting from 0 whose Laplace exponent is ψ that satisfies γ > 1. We denote by F X t the sigma-field generated by X •∧t augmented with the P-negligible events. Recall that for all s, t ∈ [0, ∞) such that s ≤ t, I s t stands for inf u∈[s,t] X u . Then, for all t ∈ [0, ∞), the following definition makes sense under P or N :

ρ t (dr) = β1 [0,Ht] (r) dr + 0<s≤t X s-<I s t (I s t -X s-) δ Hs (dr). (31)
Note that the M f (R + )-valued process ρ is (F X t ) t≥0 -adapted. The height process H can be deduced from ρ as follows: for any µ ∈ M f (R + ), we denote by Supp(µ) its topological support and we define

H(µ) = sup(Supp(µ)) ,
that is possibly infinite. We can show that

P-a.s. (or N -a.e.) ∀t ∈ [0, ∞), Supp(ρ t ) = [0, H t ] .
As proved in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], the exploration process ρ admits a càdlàg modification under P and N . By Proposition 1.2.3 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], under N , ρ is a càdlàg strong Markov process with respect to (F X t+ ) t≥0 .

2.4

The Lévy Snake.

The ψ-Lévy snake is a generalization of Le Gall's Brownian snake that greatly facilitates the study of super processes: it provides a Markovian parametrisation of the genealogy and the spatial positions of the underlying continuous population that gives rise to the super process. We recall from [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Chapter 4, the following definition of the ψ-Lévy snake. To that end, recall that ξ = (ξ t ) t≥0 is a R d -valued continuous process defined on (Ω, F) and that for any x ∈ R d , P x is a probability measure on (Ω, F) such that under P x , ξ is distributed as a standard d-dimensional Brownian motion starting from x.

Snake with a deterministic Hölder-regular lifetime process. We denote W the set of continuous stopped paths, namely the set of pairs (w, ζ w ) where

ζ w ∈ [0, ∞) and w : [0, ζ] → R d is a continuous function.
Here ζ w is the lifetime of w. We shall slightly abuse notation by simply denoting w instead of (w, ζ) in the sequel. The set W is equipped with the metric d defined for w, w ′ ∈ W by :

d(w, w ′ ) = |ζ w -ζ w ′ | + sup t≥0 w(t∧ζ w )-w ′ (t∧ζ w ′ ) .
Here • stands for the Euclidian norm on R d . It can be shown that (W, d) is a Polish space.

To define the finite dimensional marginal distributions of the snake, we first need to introduce its transition kernels. Let w ∈ W, let a ∈ [0, ζ w ] and let b ∈ [a, ∞). We plainly define a law R a,b (w, dw ′ ) on W by requiring the following.

(i) R a,b (w, dw ′ )-a.s. w ′ (t) = w(t), ∀t ∈ [0, a]. (ii) R a,b (w, dw ′ )-a.s. ζ w ′ = b.
(iii) The law of (w ′ (a+t)) 0≤t≤b-a under R a,b (w, dw ′ ) is the law of (ξ t ) 0≤t≤b-a under P w(a) .

In particular, R 0,b (w, dw ′ ) is the law of (ξ t ) 0≤t≤b under P w(0) . We denote by (W s ) s≥0 the canonical process on the space W R + of the W-valued functions on R + equipped with the product sigma-field. We next fix x ∈ R d . We slightly abuse notation by denoting x the stopped path with null lifetime starting from x (and therefore ending at x). Let h ∈ C(R + , R + ) such that h(0) = 0. We call h the lifetime process. For all s, s ′ ∈ R + such that s ′ ≥ s, we use the notation b h (s, s ′ ) = inf s≤r≤s ′ h(r). From the definition of the laws R a,b and Kolmogorov extension theorem there is a unique probability measure

Q h x on W R + such that for all 0 = s 0 < s 1 < • • • < s n , Q h x W s 0 ∈ A 0 , . . . , W sn ∈ A n (32) = 1 A 0 (x) A 1 ו••×An R b h (s 0 ,s 1 ),h(s 1 ) (w 0 , dw 1 ) . . . R b h (s n-1 ,sn),h(sn) (w n-1 , dw n ). Note that (h, x) → Q h x is measurable and for all t ∈ R + , Q h x -a.s. ζ Wt = h(t).
We next discuss the regularity of the process W under Q h x . To that end, we assume the following.

The lifetime process h is locally Hölder continuous with exponent r ∈ (0, 1].

Fix p ∈ (1, ∞) and t 0 ∈ (0, ∞). The last inequality of the proof of Proposition 4.4.1 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] p. 120, entails that there exists a constant C that only depends on t 0 , on p and on the Hölder constant of h on [0,

t 0 ], such that ∀s, t ∈ [0, t 0 ], Q h x d(W s , W t ) p ≤ C |t-s| pr/2 . ( 33
)
If p > 2/r, then the Kolmogorov continuity criterion applies and asserts that there exists a continuous modification of the process W . We slightly abuse notation by keeping notation Q h x for law on C(R + , W) of such a modification; likewise, we also keep denoting by (W t ) t≥0 the canonical process on C(R + , W). We then call Q h

x the law of the snake with lifetime process h starting from x. Working on

C(R + , W), we see that Q h x -a.s. for all t ∈ R + , ζ Wt = h(t).
We then set

W t = W t (h(t)) . ( 34
)
The process W is called the snake's endpoint process that is Q h x -a.s. continuous. From (32), we easily get that under Q h

x , the endpoint process is Gaussian whose covariance is characterized by

∀s, t ∈ R + , Q h x W t -W s 2 = h(t) + h(s) -2 inf s∧t≤u≤s∨t h(u) . (35)
Moreover since (33) holds for any p > 1, the Kolmogorov criterion implies that for any q ∈ (0, r/2), W is Q h x -a.s. locally q-Hölder continuous.

The definition of Lévy snake. The Lévy snake is the snake whose lifetime process is the height process H introduced in Section 2.3. Recall that we assume that γ > 1 and recall from Lemma 2.4 that H is Pa.s. (or N -a.e.) Hölder regular and that the previous construction of the snake applies. We then define the excursion measure of the ψ-Brownian snake starting from x ∈ R d by

N x = C(R + ,R) N (H ∈ dh) Q h x . ( 36
)
Then H is the lifetime process of W . Namely N x -a.e. for all t ∈ [0, σ], ζ Wt = H t and thus, W t = W t (H t ). Moreover, under N x , the conditional law of W given H is Q H x : we refer to [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Chapter 4, for more details. Lemma 2.4 and the results discussed right after [START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF] entail the following lemma.

Lemma 2.6 Assume that γ > 1. Then, for any q ∈ (0, γ-1 2γ ), W is N x -a.e. locally q-Hölder.

The range of the endpoint process W is a connected compact subset of R d and we use the following notation

R W = W t ; t ∈ [0, σ] . (37)
Recall that for any a ∈ (0, ∞), (L a s ) s≥0 stands for the local time of H at level a. We then denote by Z a (W ) the random measure on R d defined by

Z a (W ), f = σ 0 dL a s f ( W s ),
for any Borel measurable f : R d → R + . We also set Z 0 (W ) = 0, the null measure. Recall that M f (R d ) stands for the space of finite Borel measures on R d equipped with the topology of weak convergence. We can proves that under N x , the M f (R d )-valued process (Z a ) a≥0 has a càdlàg modification that is denoted in the same way by convenience. The occupation measure of the snake M W is then defined by

M W , f = ∞ 0 Z a (W ), f da = σ 0 f W s ds , (38)
for any Borel measurable f : R d → R + .

We then recall Theorem 4.2.1 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] that connects the ψ-Lévy snake to super Brownian motions.

Theorem 2.7 ([13] Theorem 4.2.1) We keep notation from above. Assume that γ > 1.

Let µ ∈ M f (R d ). Let i∈J δ (x i ,W i )
be a Poisson point measure on R d ×W with intensity µ(dx)N x (dW ). For every a ∈ (0, ∞) set

Z a = i∈J Z a (W i ) .
and also set Z 0 = µ. Then, the process (Z a ) a≥0 is a ψ-super Brownian motion starting from µ (as defined in the introduction section). Moreover, if R and M are defined in terms of Z by ( 6) and ( 5), then,

R ∪ {x j ; j ∈ J } = j∈J R W j and M = j∈J M W j . (39)
The last point [START_REF] Taylor | Packing measure, and its evaluation for a Brownian path[END_REF] is not part of Theorem 4.2.1 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] but it is an easy consequence of that result. To simplify notation, when there is no risk of confusion we shall simply write

Z a := Z a (W ), R := R W and M := M W .
Consequences of Markov property. As the height process, the ψ-Lévy snake (W t ) t≥0 defined above is not a Markov process. However W := (ρ t , W t ) t≥0 is a strong Markov process under N x : see Theorem 4.1.2 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] for more details. Instead of fully discussing the Markov property of W , we only state here the various results we need, that are consequences of the strong Markov property.

Denote by (F W t ) t≥0 the filtration generated by (W t ) t≥0 . As a consequence of the strong Markov property for W (see [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], Theorem 4.1.2) and a specific decomposition of the snake into excursions above the infimum of its lifetime proved in Lemma 4.2.4 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], we get the following result that is used in Section 3.4.

N 0 1 {0<T <∞} Y exp - σ T G( W s )ds = N 0 1 {0<T <∞} Y exp - [0,H T ] ρ T (dh) N W T (h) 1-e -σ 0 G( Ws)ds
.

We shall apply the strong Markov property of W at specific hitting times of the snake. More precisely, let us introduce several notations. Let r ∈ (0, ∞). We define the first hitting time of W of the closed ball

B(0, r) in R d by τ r := inf t ∈ R + : W t ∈ B(0, r) , (40)
with the convention inf ∅ = ∞. We also introduce the following function

∀x ∈ B(0, r) c , u r (x) := N x (τ r < ∞) . ( 41
)
Since t → W t is continuous, we also get

∀x ∈ B(0, r) c , u r (x) = N x (R ∩ B(0, r) = ∅) . (42)
From [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] p. 121 and p. 131, we know that u r (x) ∈ (0, ∞), for all r ∈ (0, ∞) and all x ∈ B(0, r) c , and that u r is moreover radial. We then denote by u r the function from (r, ∞) to (0, ∞) such that

∀x ∈ B(0, r) c , u r ( x ) = u r (x) .
Let r ′ ∈ (0, ∞). For all stopped path w ∈ W, we next set

T r ′ (w) = inf s ∈ [0, ζ w ] : w(s) ∈ B(0, r ′ ) , (43)
with the convention inf ∅ = ∞. We then define a function ̟ : R 2 + → R + by

∀λ 1 , λ 2 ∈ R + , ̟(λ 1 , λ 2 ) = (ψ(λ 1 ) -ψ(λ 2 )) /(λ 1 -λ 2 ) if λ 1 = λ 2 , ψ ′ (λ 1 ) if λ 1 = λ 2 . (44)
Recall that ξ = (ξ t ) t≥0 is a R d -valued continuous process defined on (Ω, F) and that for any x ∈ R d , P x is a probability measure on (Ω, F) such that under P x , ξ is distributed as a standard d-dimensional Brownian motion starting from x.

The following proposition is a specific application of Theorem 4.6.2 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] that we use in the proof of Lemma 3.15 in Section 3.4. Proposition 2.9 ([13] Theorem 4.6.2) Let x ∈ R d . Let r, r ′ ∈ (0, ∞) be such that r ′ > r and x ∈ B(0, r ′ ) c . We keep the previous notation. Let F, G : W → R + be Borel-measurable. Then,

N x 1 {τr <∞} F (W τr (s)) 0≤s≤T r ′ (Wτ r ) exp - [0,T r ′ (Wτ r )] ρ τr (dh) G (W τr (s)) 0≤s≤h = u r (r ′ ) E x 1 {T r ′ (ξ)<∞} F (ξ s ) 0≤s≤T r ′ (ξ) exp - [0,T r ′ (ξ)] dh ̟ u r (ξ h ), G((ξ s ) 0≤s≤h )
.

Palm formula We introduce the following notation

∀λ ∈ R + , ψ * (λ) = ψ(λ) -αλ , (45)
that is clearly the Laplace exponent of a spectrally Lévy process. We then fix x ∈ R d . Again, recall that ξ = (ξ t ) t≥0 under P x is distributed as a standard d-dimensional Brownian motion starting from x. Let U = (U a ) a≥0 be a subordinator defined on (Ω, F, P x ) that is assumed to be independent of ξ and whose Laplace exponent is

∀λ ∈ R + , ψ * (λ) := ψ(λ) λ -α .
For any a ∈ R + , we denote by R a (db) the random measure 1 [0,a] (b)dU b . We first recall from [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], formula (106) p. 105, that for any measurable function F : M f (R + )×W → R + , the following holds true:

N x σ 0 ds F (ρ s , W s ) = ∞ 0 da e -αa E x F (R a , (ξ s ) 0≤s≤a ) . (46)
We next provide a Palm decomposition for the occupation measure M of the snake that is used to estimate its lower local density at "typical" points. To that end we need to introduce the following auxiliary random variables. Let (V t ) t≥0 be a subordinator defined on (Ω, F, P 0 ) that is independent of ξ and whose Laplace exponent is ψ * ′ (λ) := ψ ′ (λ)-α. We then introduce the following point measure on

[0, ∞)×C(R + , W): N * (dt dW ) = j∈J * δ (t j ,W j ) , (47)
such that under P 0 and conditionally given (ξ, V ), N * is distributed as a Poisson point process with intensity dV t N ξt (dW ). For all j ∈ J * , we denote by M j the occupation measure of the snake W j . Then for all a ∈ R + , we define the following random measure on R d ;

M * a = j∈J * 1 [0,a] (t j )M j . (48)
Note that M * a is P 0 -a.s. a random finite Borel measure on R d . Informally M * a is the sum of the the occupation measure of the snakes grafted at a rate given by V on the spatial spine ξ between time 0 and a. As a by-product of Formula (113) p.113 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], we get the following Palm decomposition of M under N 0 . Proposition 2.10 ([13] (113)) Let F : R d ×M f (R d ) → R + be measurable. Then,

N 0 R d M(dy) F (y, M) = ∞ 0 da e -αa E 0 [F (ξ a , M * a )] . (49)
We shall mostly use the Palm formula in this way: for any measurable functional G :

D(R + , R) → R + , we get N 0 R d M(dy) G (M(B(y, r))) r≥0 = ∞ 0 da e -αa E 0 G (M * a (B(0, r))) r≥0 . ( 50
)
3 Estimates.

3.1 Tail of a subordinator.

Recall from (45) that ψ * (λ) = ψ(λ)αλ, that is the Laplace transform of a spectrally positive Lévy process. Therefore, ψ * ′ is the Laplace exponent of a subordinator that is conservative for ψ * ′ (0) = 0. By subordination, ψ * ′ • ψ -1 is also the Laplace exponent of a conservative subordinator. The main idea of the proof of Theorem 4.1 consists in comparing the mass of a typical ball with a subordinator whose Laplace exponent is ψ * ′ • ψ -1 . To that end, we first need the following result.

Lemma 3.1 Assume δ > 1. Recall from [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF] the definition of the gauge function g : (0, r 0 ) → (0, ∞). Let (S r ) r∈[0,∞) be a subordinator defined on the auxiliary probability space (Ω, F, P 0 ). We assume that the Laplace exponent of S is ψ * ′ • ψ -1 . Let ρ n ∈ (0, r 0 ), n ∈ N, be such that

ρ n+1 ≤ e -n ρ n and sup n≥0 n -2 log 1/ρ n < ∞ . (51)
Then, n≥0 P 0 S ρn ≤ g(4ρ n ) = ∞. Moreover, we get

P 0 -a.s. lim sup n→∞ S ρ n+1 g(4ρ n ) < ∞ .
Proof. To simplify notation, we set Φ = ψ * ′ • ψ -1 . Thus, E 0 [exp(-λS r )] = exp(-rΦ(r)). Denote by Φ -1 the reciprocal function of Φ. For any r ∈ (0, e -1 ), we set

g * (r) = log log 1 r Φ -1 ( 1 r log log 1 r
)

.

An easy computation implies that Φ -1 (y) = ϕ -1 (y 2 + α). Since α ∈ [0, ∞), we easily get g * (r) ≤ g(r), r ∈ (0, r 0 ). For any n ∈ N, we then set λ n = Φ -1 ((4ρ n ) -1 log log 1/4ρ n )). Then, observe that

λ n g(4ρ n ) ≥ λ n g * (4ρ n ) = log log(1/4ρ n ) .
Next note that for all a, x ∈ [0, ∞), (1e -a )1 {0≤x≤a} ≥ e -xe -a , which easily entails

P 0 S ρn ≤ g(4ρ n ) ≥ exp(-ρ n Φ(λ n )) -exp(-λ n g(4ρ n )) 1 -exp(-λ n g(4ρ n )) ∼ n→∞ (log 1/(4ρ n )) -1 4 .
By the second assumption in (51), n≥0 (log 1/(4ρ n )) -1 4 = ∞, which proves the first point of the lemma. Let us prove the second point. By a standard Markov inequality, we get

P 0 S ρ n+1 ≥ g * (4ρ n ) ≤ 1 -exp(-ρ n+1 Φ(λ n )) 1 -exp(-λ n g * (4ρ n )) ≤ ρ n+1 Φ(λ n ) 1 -exp(-λ n g * (4ρ n ))
.

First observe that 1-exp(-λ n g * (4ρ n )) = 1-(log 1/4ρ n ) -1 -→ 1, as n → ∞. By (51), there exists a constant c 2 ∈ (0, ∞) such that

ρ n+1 Φ(λ n ) = ρ n+1 4ρ n log log 1/(4ρ n ) ≤ c 2 e -n log n , Thus, n≥0 P 0 S ρ n+1 ≥ g(4ρ n ) ≤ n≥0 P 0 S ρ n+1 ≥ g * (4ρ n ) < ∞,
which completes the proof by the Borel-Cantelli lemma.

Estimates on hitting probabilities.

As already mentioned in [START_REF] Dawson | Super-Brownian motion: path properties and hitting probabilities[END_REF], the total range R of a ψ-super Brownian motion is bounded if the starting measure µ has compact support and if

∞ 1 db b 1 ψ(a)da < ∞. (52)
Observe that if δ > 1, then γ > 1 and (52) holds true, which allows to define the following function

∀v ∈ (0, ∞) , I(v) = ∞ v db b v ψ(a) da = ∞ 0 db b 0 ψ(v + a) da . ( 53
)
This function is clearly decreasing and continuous and it plays a role in the proof of an upper bound of the hitting probabilities of the ψ-Lévy snake. We first need the following elementary lemma.

Lemma 3.2 Assume δ > 1. Then, there exists c 3 ∈ (0, ∞) that only depends on ψ such that for all v ∈ (1, ∞) and all r ∈ (0, ∞) satisfying r ≤ I(v), we have

v ≤ ψ ′-1 4c 3 r -2 .
Proof. By an elementary change of variable, we get

I(v) = ∞ 1 √ v db b 1 ψ (va) da . Fix c ∈ (1, δ). By the definition (3) of δ, there exists C ∈ (0, ∞) such that ψ(va) ≥ Cψ(v)a c , for any a, v ∈ (1, ∞). Let v ∈ (1, ∞) and r ∈ (0, ∞) be such that r ≤ I(v). Then, r ≤ I(v) ≤ C -1/2 (v/ψ(v)) 1/2 ∞ 1 db b 1 a c da =: c 3 v/ψ(v) 1/2 ,
which implies the desired result since ψ ′ (v) ≤ 4ψ(v)/v by [START_REF] Etheridge | An introduction to superprocesses[END_REF].

Recall from [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] the definition of the range R of the snake. Recall from (36) the notation N x for the excursion measure of the snake starting from x. Let r ∈ (0, ∞). Recall that B(0, r) stands for the open ball in R d with radius r and center 0. Then, we set for all x ∈ B(0, r) v r (x) = N x (R ∩ B(0, r) c = ∅) .

(54)

From [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] p. 121 and p. 131, we know that ∀x ∈ B(0, r), v r (x) ∈ (0, ∞) and lim

x →r- v r (x) = ∞ .
Moreover, v r is C 2 on B(0, r) and it satisfies 1 2 ∆v r = ψ(v r ). As an easy consequence of Brownian motion isotropy, v r is a radial function: namely, v r (x) only depends on x (and r). Therefore, one can derive estimates on v r by studying the associated ordinary differential equation corresponding to the radial function, as done by Keller in [START_REF] Keller | On solutions of ∆u = f (u)[END_REF], p. 507 inequality [START_REF] Keller | On solutions of ∆u = f (u)[END_REF], who proves the following:

∀r ∈ (0, ∞), 2 √ d r ≤ I(v r (0)) ≤ 2r . ( 55
)
We use this bound as follows. For any r ∈ (0, ∞) and any x ∈ B(0, r) c , recall that

u r (x) = N x (R ∩ B(0, r) = ∅) . (56)
From [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] p. 121 and p. 131, we know that u r (x) ∈ (0, ∞) for all x ∈ B(0, r) c , that u r is radial, lim

x →∞ u r (x) = 0 and lim

x →r+ u r (x) = ∞ . (57)
Recall that we denote by u r the radial function yielded by u r , namely:

∀x ∈ B(0, r) c , u r ( x ) = u r (x) . (58)
Moreover, u r is C 2 in B(0, r) c and it satisfies

1 2 ∆u r = ψ(u r ) in B(0, r) c . ( 59
)
We shall use several times the following upper bound of u r .

Lemma 3.3 Assume that δ > 1 and that d ≥ 3. Let ̺ ∈ (0, ∞). Then there exist r 1 , c 4 ∈ (0, ∞), that only depend on ψ, d and ̺, such that

∀r ∈ (0, r 1 ), ∀x ∈ B 0, (1+̺)r c , u r (x) ≤ (1+̺)r/ x d-2 ψ ′-1 (c 4 r -2 ) .
Proof. Let y ∈ R d be such that y = (1 + ̺)r. First note that

N y R ∩ B(0, r) = ∅ ≤ N y (R ∩ B(y, ̺r) c = ∅) .
By translation invariance of Brownian motion, the right member of the previous inequality does not depend on y and we get u r (y) ≤ v ̺r (0), where v ̺r is defined by (54). For any x ∈ B(0, (1+̺)r) c , we next set w(x) = u r (x)v ̺r (0)((1+̺)r/ x ) d-2 that is clearly subharmonic. The previous upper bound implies that w(y) ≤ 0, if y = (1+̺)r. By (57), lim x →∞ w(x) = 0 and by the maximum principle, we get that w ≤ 0 on B(0, (1+̺)r) c . Namely,

∀r ∈ (0, ∞), ∀x ∈ B 0, (1 + ̺)r c , u r (x) ≤ v ̺r (0) (1+̺)r/ x d-2 . (60)
Since δ > 1, (52) is satisfied and the function I given by ( 53) is well-defined; we easily check that I(v) → 0 iff v → ∞. Then, (55) implies that lim r→0 v ̺r (0) = ∞, so we can find r 1 ∈ (0, ∞) such that for all r ∈ (0, r 1 ), v ̺r (0) ≥ 1 and by the left member of (55) we also have 2r̺/ √ d ≤ I(v ̺r (0)). Thus, Lemma 3.2 applies and asserts that v ̺r (0) ≤ ψ ′-1 (c 4 r -2 ), where c 4 := c 3 d̺ -2 , which completes the proof thanks to (60).

We use the previous lemma to get an upper bound of the expectation of a specific additive functional of the Brownian motion that involves u r . More precisely, for any r ∈ (0, ∞), we define

q r = ψ ′-1 (c 4 r -2 ) . (61)
Recall that c 4 is the constant appearing in Lemma 3.3. Note that for any 0 < r < (c 4 /ψ ′ (1)) 1/2 , we have q r ≥ 1. We then define ∀r ∈ 0, (c 4 /ψ ′ (1))

1 2 , J(r) = r 2 q 2 d-2 r qr 1 ψ ′ (v)v -d d-2 dv . ( 62
)
Recall that ξ = (ξ t ) t≥0 stands for a standard Brownian motion starting from 0 that is defined on the auxiliary probability space (Ω, F, P 0 ). We next prove the following lemma.

Lemma 3.4 Assume that δ > 1 and that d ≥ 3. Let a ∈ (0, ∞). Then, there exist c 5 , c 6 , r 2 ∈ (0, ∞) that only depend on ψ, d and a, such that

∀r ∈ (0, r 2 ) E 0 2a 0 ds 1 { ξs ≥2r} ψ ′ (u r (ξ s )) ≤ c 5 + c 6 J(r) .
Proof. To simplify notation, we denote by L the expectation in the left member of the previous inequality.

Recall from (58) the notation u r for the radial function yielded by u r . By Fubini and easy changes of variable, we have the following.

L = B(0,2r) c dx 2a 0 ds (2πs) -d/2 e -x 2 /2s ψ ′ u r ( x ) = c 7 ∞ 2r dy y d-1 ψ ′ u r (y) 2a 0 ds s -d/2 e -y 2 /2s = c 8 ∞ 2r dy yf (y)ψ ′ u r (y) ,
where for any y ∈ (0, ∞), we have set f (y) = ∞ y 2 /(4a) du u d/2-2 e -u and where c 7 and c 8 are constants that only depend on d. Since we assume that d ≥ 3, f (0) is well-defined and finite, and it is easy to check that ∞ 0 yf (y)dy < ∞. We next use Lemma 3.3 with ̺ = 1 to get for all r ∈ (0, r 1 ) and all y ∈ (2r, ∞) that u r (y) ≤ (2r/y) d-2 q r . We then set α r = 2rq

1/(d-2) r
. Thus, u r (y) ≤ (α r /y) d-2 , for all r ∈ (0, r 1 ) and all y ∈ (2r, ∞). We next set

r 2 := r 1 ∧ (c 4 /ψ ′ (1)) 1/2 .
Observe that for any r ∈ (0, r 2 ), q r ≥ 1, which implies that α r ≥ 2r. Next, observe that for all r ∈ (0, r 2 ) and all y ∈ (α r , ∞), (α r /y) d-2 ≤ 1. Thus, ψ ′ ( u r (y)) ≤ ψ ′ (1). It implies

L ≤ c 8 ψ ′ (1)
c 8 f (0) αr 2r yψ ′ (α r /y) d-2 dy = 1 d-2 c 8 f (0)α 2 r (αr /2r) d-2 1 ψ ′ (v)v -d d-2 dv = c 6 J(r) ,
where we have set c 6 := 4 d-2 c 8 f (0). Then, the desired result follows from (63).

When d is greater than 2δ δ-1 , the function J is bounded for all small values of r as proved in the following lemma. Lemma 3.5 Assume that δ > 1 and that d > 2δ δ-1 . Then, there exists a constant c 9 ∈ (0, ∞) that depends on d and ψ such that J(r) ≤ c 9 for all r ∈ (0, r 2 ).

Proof. Observe that 2 d-2 < δ -1. Recall that δ ψ ′ = δ -1. Let us fix u ∈ ( 2 d-2 , δ -1)
. By the definition (3) of the exponent δ ψ ′ , there exists K ∈ (0, ∞) depending on ψ and u such that

∀ 1 ≤ λ ≤ µ, ψ ′ (λ) ≤ K ψ ′ (µ)(λ/µ) u . Recall from (61) that ψ ′ (q r ) = c 4 r -2
, where c 4 is the constant appearing in Lemma 3.3.

Then we get the following.

J(r) = r 2 q 2 d-2 r qr 1 ψ ′ (v)v -d d-2 dv ≤ r 2 q 2 d-2 r ψ ′ (q r ) qr 1 (v/q r ) u v -d d-2 dv ≤ c 4 q 2 d-2 -u r qr 1 v u-2 d-2 -1 dv ≤ c 4 q 2 d-2 -u r qr 0 v u-2 d-2 -1 dv = c 4 u-2 d-2
, which implies the desired result with

c 9 := c 4 u-2 d-2 . When d ∈ ( 2γ γ-1 , 2δ δ-1 ],
we are only able to prove that lim inf r→0 J(r) < ∞. More precisely, we prove the following lemma. Lemma 3.6 Assume that γ > 1 and that d > 2γ γ-1 . Recall that c 4 appears in Lemma 3.3. Then, there exists a decreasing function

θ ∈ (ψ ′ (1), ∞) -→ r θ ∈ 0 , (c 4 /ψ ′ (1)) 1 2
such that lim θ→∞ r θ = 0 and such that there exists c 10 ∈ (0, ∞), that only depends on d and ψ, and that satisfies ∀θ ∈ (ψ

′ (1), ∞), J(r θ ) ≤ c 10 . (64)
Moreover, for any θ ′ , θ ∈ (ψ ′ (1), ∞) such that θ ′ ≥ θ, we also have

r θ ′ /r θ ≤ (θ/θ ′ ) 1/2 and r θ ≥ c 11 θ -c 12 , ( 65
)
where c 11 , c 12 ∈ (0, ∞) only depend on d and ψ.

Proof. Note that 2 d-2 < γ -1 = γ ψ ′ . Let us fix c ∈ ( 2 d-2 , γ ψ ′ )
. Thus, λ -c ψ ′ (λ) → ∞ as λ → ∞, which allows to define the following for any θ ∈ (ψ ′ (1), ∞):

r θ = c 4 /ψ ′ (λ θ ) 1 2
where

λ θ = inf λ ∈ [1, ∞) : λ -c ψ ′ (λ) = θ . ( 66
)
Note that if θ > ψ ′ (1), then r θ < (c 4 /ψ ′ (1))

Thus, θ → θr 2 θ decreases, which proves the first inequality in (65). To prove the second inequality, we fix ε ∈ (0, ∞) such that c + ε < γ ψ ′ . By definition of γ ψ ′ , there exists

K ∈ (0, ∞) such that λ -c ψ ′ (λ) ≥ Kλ ε , for any λ ∈ [1, ∞). It entails that θ = λ -c θ ψ ′ (λ θ ) ≥ Kλ ε θ . Thus, r θ = c 4 /ψ ′ (λ θ ) 1 2 = c 4 /(θλ c θ ) 1 2 ≥ c 4 K c ε 1 2 θ -1 2 (1+ c ε ) ,
which implies the desired result with

c 11 = c 4 K c ε 1 2 and c 12 = 1 2 (1 + c ε ).
By combing the previous lemmas we obtain the following result.

Lemma 3.7 Assume that δ > 1 and that d > 2γ γ-1 . Then d ≥ 4. Let a ∈ (0, ∞). For any n ∈ N, set ρ n = r e n 2 , where (r θ ) θ∈[ψ ′ (1),∞) is defined as in Lemma 3.6. Then, the sequence (ρ n ) n≥0 satisfies (51) in Lemma 3.1. Moreover, there exists a constant c 14 ∈ (0, ∞), that only depends on d, ψ and a, such that for all sufficiently large n ∈ N,

E 0 2a 0 ds1 { ξs ≥2ρn} ψ ′ (u ρn (ξ s )) ≤ c 14 .
Proof. Note that γ ≤ 2, which easily entails that d ≥ 4. By (65) in Lemma 3.6, which completes the proof of the lemma.

ρ n+1 /ρ n ≤ e n 2 /e (n+1) 2 1/2 = e -n-

3.3

The spine and the associated subordinator.

Recall from Section 2.4, the Palm formula for the occupation measure of the snake. Recall that ξ = (ξ t ) t≥0 is d-dimensional Brownian motion starting from 0 that is defined on (Ω, F, P 0 ). Recall that (V t ) t≥0 is a subordinator defined on (Ω, F, P 0 ) that is independent of ξ and whose Laplace exponent is ψ * ′ (λ) = ψ ′ (λ)α. Recall from (47) that under P 0 , conditionally given (ξ, V ), N * (dtdW ) = j∈J * δ (t j ,W j ) is a a Poisson point process on [0, ∞)×C(R + , W) with intensity dV t N ξt (dW ). Then recall from (48) that for all a ∈ R + , we have set M * a = j∈J * 1 [0,a] (t j )M j where for all j ∈ J * , M j stands for the occupation measure of the snake W j as defined in [START_REF] Sheu | Asymptotic behavior of superprocesses[END_REF].

For any a ∈ R + , we then introduce

T a := M * a , 1 = j∈J * 1 [0,a] (t j )σ j , (67)
where σ j is the total duration of the excursion of the snake W j . By construction of the snake excursion measure,

N x 1-e -λσ = N 1-e -λσ = ψ -1 (λ) . ( 68
)
We shall assume throughout the paper that d ≥ 4. We then introduce the following two last exit times: for all r ∈ R + , we set

ϑ(r) = sup{t ≥ 0 : ξ t ≤ r} (69) γ(r) = sup t ≥ 0 : (ξ (1) t ) 2 + (ξ (2) t ) 2 + (ξ (3) t ) 2 ≤ r (70)
where (ξ

(i)
t ) t≥0 stands for the i-th coordinate of ξ. We then recall the two basic facts on the processes γ and ϑ.

(a) The increments of (ϑ(r)) r≥0 are independents. Moreover,( ξ ϑ(r)+t ) t≥0 is independent of the two processes (ϑ(r ′ )) 0≤r ′ ≤r and ( ξ t∧ϑ(r) ) t≥0 .

(b) The process (γ(r)) r≥0 has independent and stationary increments: it is a subordinator with Laplace exponent λ -→ √ 2λ.

The first point is proved in Getoor [START_REF] Getoor | The Brownian escape process[END_REF]. The second is a celebrated result of Pitman [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF].

Before stating our lemma, we introduce the following random variables:

∀ t ≥ s ≥ 0, N r (s, t) = # j ∈ J * : s < t j < t and R j ∩ B(0, r) = ∅ , (71)
that counts the snakes that are grafted on the spatial spine ξ between times s and t, and that hit the ball B(0, r).

Lemma 3.8 Assume that d ≥ 4. We keep the previous notation. Then, the following holds true.

(i) For all real numbers r > r ′ > ρ > ρ ′ > 0 and all a ∈ (0, ∞), the random variables T ϑ(2ρ) -T ϑ(2ρ ′ ) , T ϑ(2r) -T ϑ(2r ′ ) and N r (ϑ(2r), a+ϑ(2r))

are independent.

(ii) The process (T γ(r) ) r≥0 is a subordinator with Laplace exponent 2 ψ * ′ •ψ -1 .

Remark 3.1 We take the opportunity to mention that it is stated incorrectly in [START_REF] Duquesne | The packing measure of the range of super-Brownian motion[END_REF] (60), that T γ(2ρ) -T γ(2ρ ′ ) , T γ(2r) -T γ(2r ′ ) and N (ϑ(2r), a) are independent. More precisely, the statement (49) in [START_REF] Duquesne | The packing measure of the range of super-Brownian motion[END_REF] is incorrect. We provide here a correct statement and a correct proof.

Proof. Let us prove (i). Let λ, µ, θ ∈ R + . To simplify notation we set

Y = exp -λ(T ϑ(2r) -T ϑ(2r ′ ) ) -µ(T ϑ(2ρ) -T ϑ(2ρ ′ ) )-θN r (ϑ(2r), a+ϑ(2r)) .
Then recall (68) and recall from (58) notation u r ( x ) = N x (R ∩ B(0, r) = ∅). Then, basic results on Poisson processes imply

E 0 Y (ξ, V ) = exp -ψ -1 (λ)(V ϑ(2r) -V ϑ(2r ′ ) )-ψ -1 (µ)(V ϑ(2ρ) -V ϑ(2ρ ′ ) )-(1-e -θ ) (0,a) dV t u r ( ξ ϑ(2r)+t ) .
Since V is a subordinator with Laplace exponent ψ * ′ , we then get

E 0 Y ξ = exp -(ϑ(2r) -ϑ(2r ′ ))ψ * ′ (ψ -1 (λ)) -(ϑ(2ρ)-ϑ(2ρ ′ ))ψ * ′ (ψ -1 (µ)) - a 0 dt ψ * ′ (1-e -θ ) u r ( ξ ϑ(2r)+t ) .
The above mentioned property (a) for last exit times then implies that

ϑ(2r) -ϑ(2r ′ ), ϑ(2ρ)-ϑ(2ρ ′ ) and a 0 dt ψ * ′ (1-e -θ ) u r ( ξ ϑ(2r)+t )
are independent which easily implies (i).

The second point is proved in a similar way: let 0 = r 0 < r 1 < . . . < r n and let λ 1 , . . . λ n ∈ R + . We set

Y ′ = exp - 1≤k≤n λ k T γ(r k ) -T γ(r k-1 )
Thus,

E 0 Y ′ = E 0 exp - 1≤k≤n ψ -1 (λ k ) V γ(r k ) -V γ(r k-1 ) = E 0 exp - 1≤k≤n ψ * ′ (ψ -1 (λ k )) γ(r k )-γ(r k-1 ) = exp - 1≤k≤n (r k -r k-1 ) 2ψ * ′ (ψ -1 (λ k )) .
Indeed, the first equality comes from basic results on Poisson point measures, the second equality comes from the fact that V is a subordinator with Laplace exponent ψ * ′ and the last equality is a consequence of the above mentioned Property (b) of the last exit times γ(r). This completes the proof of (ii).

Recall from (71) the notation N r (s, t) that counts the snakes that are grafted on the spatial spine ξ between times s and t, and that hit the ball B(0, r). We state the following lemma that actually means, in some sense, that in supercritical dimension there is no snake W j grafted far away that hit B(0, r). Lemma 3.9 Assume that γ > 1 and that d > 2γ γ-1 . Then, for all t > s > 0,

∀ t > s > 0, lim r→0 P 0 (N r (s, t) = 0) = 1 .
Proof. Recall from (56) the definition of u r and from (58) that of ũr . By the definition (71), conditionally given (ξ, V ), N r (s, t) is a Poisson random variable with parameter t s dV w u r (ξ w ). Thus

P 0 (N r (s, t) = 0) = E 0 exp - t s dw ψ * ′ (u r (ξ w )) . (72)
Next, note that d > 2γ γ-1 implies that d-2 > 2 γ-1 . Then, there exists b ∈ (0, 1) and a ∈ (0, γ -1) such that

(d -2)(1 -b) > 2 a . (73)
By the definition (69) of the last exit time process ϑ,

E 0 exp - t s dw ψ * ′ (u r (ξ w )) ≥ exp -tψ * ′ (ũ r (r b )) P 0 s > ϑ(r b ) . (74)
Clearly lim r→0 P 0 (s > ϑ(r b )) = 1. Thus, we only have to choose b such that lim r→0 ũr (r b ) = 0. To that end, we apply Lemma 3.3 with ̺ = 1: for all r ∈ (0,

r 1 ∧ 2 -1 1-b ), we get ũr (r b ) ≤ (2r/r b ) d-2 ψ ′-1 (c 4 r -2 ) = 2 d-2 r (d-2)(1-b) ψ ′-1 (c 4 r -2 ). (75)
Since a ∈ (0, γ -1) and since γ ψ ′ = γ -1, the definition (21) of γ ψ ′ entails that for all sufficiently large λ, ψ ′ (λ) ≥ λ a and thus ψ ′-1 (λ) ≤ λ 1/a . Then (75) implies that there exists a constant c ∈ (0, ∞) such that u r (r b ) ≤ c r (d-2)(1-b)-2/a for all sufficiently small r. By (73), this entails lim r→0 ũr (r b ) = 0, which completes the proof by ( 72) and (74).

We next prove a similar estimate for N r (ϑ(2r), a+ϑ(2r)).

Lemma 3.10 Assume that δ > 1 and that d > 2δ δ-1 . Then, there exist two constants c 15 , r 3 ∈ (0, ∞) that only depends on d, ψ and a, such that ∀r ∈ (0, r 3 ) P 0 N r (ϑ(2r), a+ϑ(2r)) = 0 ≥ c 15 .

Proof. Recall from (56) the definition of u r and from (58) that of ũr . By the definition (71), conditionally given (ξ, V ), N r (ϑ(2r), a+ϑ(2r)) is a Poisson random variable with parameter a+ϑ(2r) ϑ(2r) dV t u r (ξ t ). Thus

P 0 N r (ϑ(2r), a+ϑ(2r)) = 0 = E 0 exp - a+ϑ(2r) ϑ(2r) dt ψ * ′ (u r (ξ t )) . (76)
Next note that on {ϑ(2r) ≤ a}, a + ϑ(2r) ≤ 2a and that t ≥ ϑ(2r) implies that ξ t ≥ 2r. Thus, by (76)

P 0 N r (ϑ(2r), a+ϑ(2r)) = 0 ≥ E 0 1 {ϑ(2r)≤a} exp - 2a 0 dt 1 { ξt ≥2r} ψ * ′ (u r (ξ t )) ≥ E 0 exp - 2a 0 dt 1 { ξt ≥2r} ψ * ′ (u r (ξ t )) -P 0 (ϑ(2r) > a) ≥ exp -E 0 2a 0 dt 1 { ξt ≥2r} ψ * ′ (u r (ξ t )) -P 0 (ϑ(2r) > a), (77)
where we use Jensen inequality in the last line. By Lemma 3.4 and Lemma 3.5, for any r ∈ (0, r 2 ),

exp -E 0 2a 0 dt 1 { ξt ≥2r} ψ * ′ (u r (ξ t )) ≥ exp(-c 5 -c 6 c 9 ) .
Since P 0 (ϑ(2r) > a) → 0 as r → 0, there exists r 3 ∈ (0, r 2 ) such that P 0 (ϑ(2r 3 ) > a) ≤ 1 2 e -c 5 -c 6 c 9 . Thus by (77) and the previous inequality, for any r ∈ (0, r 3 ),

P 0 N r (ϑ(2r), a+ϑ(2r)) = 0 ≥ 1 2 exp(-c 5 -c 6 c 9 ) =: c 15 ,
which completes the proof of lemma.

Under the less restrictive condition d > 2γ

γ-1 , we get a similar lower but only for the family of radii (ρ n ) n≥1 introduced in Lemma 3.7. Lemma 3.11 Assume that δ > 1 and that d > 2γ γ-1 . Let (ρ n ) n≥1 be the sequence introduced in Lemma 3.7. Then, there exists a constant c 16 ∈ (0, ∞), that only depend on d, ψ and a, such that

∀n ≥ 1 P 0 (N r (ϑ(2ρ n ), a+ϑ(2ρ n )) = 0) ≥ c 16 .
Proof. The lower bound (77) applies for r = ρ n . Then, Lemma 3.7, entails that for all sufficiently large n,

exp -E 0 2a 0 dt 1 { ξt ≥2ρn} ψ * ′ (u ρn (ξ t )) ≥ exp(-c 14 ) ,
and we completes the proof arguing as in the proof of Lemma 3.10 with c 16 := 1 2 e -c 14 .

We then shall need the following result that is used in the proof of Lemma 4.4.

Lemma 3.12 Assume that δ > 1 and that d ≥ 4. Let (ρ n ) n≥1 be the sequence of radii introduced in Lemma 3.7. Then, (i) :

n≥1 P 0 T γ(2ρn) ≤ g(8ρ n ) = ∞ and (ii) : P 0 -a.s. lim sup n→∞ T γ(2ρ n+1 ) g(8ρ n ) < ∞ .
Proof. Lemma 3.8 shows that T γ(•) is a subordinator with Laplace exponent 2 ψ * ′ •ψ -1 . Then, Lemma 3.7 asserts that the sequence (ρ n ) n≥0 satisfies the conditions (51) in Lemma 3.1, which immediately entails (i) and (ii).

We end this section with the following result that is close to the previous one and that is used in the proof of Theorem 1.1. Lemma 3.13 Assume that γ > 1. Let u ∈ (0, 2γ γ-1 ). Then, there exists a decreasing sequence (s n ) n≥1 that tends to 0, that only depends on ψ and u, and that satisfies the following.

(i) :

n≥1 P 0 T γ(2sn) ≤ s u n = ∞ and (ii) : P 0 -a.s. lim sup n→∞ T γ(2s n+1 ) s u n < ∞ . Proof. Let u ′ ∈ (u, 2γ γ-1 ). We set ϕ * = ψ * ′ • ψ -1 , where ψ * (λ) = ψ(λ) -α. Recall from (21) that γ ϕ = γ ϕ * = γ-1
γ . We fix a ∈ (0, γ-1 γ ). By the definition (21) of γ ϕ * , there exists

λ 0 ∈ (0, ∞) such that ϕ * (λ) ≥ λ a for any λ ∈ [λ 0 , ∞). Next observe that 2/u ′ > γ-1 γ = γ ϕ * .
As an easy consequence of definition (21) of γ ϕ * , we get lim inf λ→∞ ϕ * (λ)λ -2/u ′ = 0. Consequently, there exists an increasing sequence

(λ n ) n≥0 such that ∀n ∈ N, 2 n ≤ λ n and λ a n ≤ ϕ * (λ n ) ≤ λ 2/u ′ n . (78)
We next fix ε ∈ (0, ∞) such that (1 + ε)u/u ′ < 1, which is possible since u ′ > u. Then, we set

∀n ∈ N, s n = ϕ * (λ n ) -1+ε 2 . (79)
Since, by Lemma 3.8, (T γ(r) ) r≥0 is a subordinator with Laplace exponent √ 2ϕ * , a Markov inequality entails

P 0 T γ(2sn) > s u n ≤ 1 -exp -2 √ 2s n ϕ * (λ n ) 1 -exp(-λ n s u n )) ≤ 2 √ 2 s n ϕ * (λ n ) 1 -exp(-λ n s u n ) . (80)
By (79) and the last inequality of (78), we get

λ n s u n = λ n ϕ * (λ n ) -(1+ε)u/2 ≥ λ 1-(1+ε)u/u ′ n ---→ n→∞ ∞. (81)
Moreover, the first two inequalities in (78) and (79) imply

s n ϕ * (λ n ) = ϕ * (λ n ) -ε/2 ≤ λ -aε/2 n ≤ 2 -naε/2 . ( 82
)
Then ( 80), ( 81) and (82) imply that there exists c ∈ (0, ∞) such that

P 0 T γ(2sn) > s u n ≤ c2 -naε/2 .
It immediately implies (i) and (ii) follows from P 0 T γ(2s n+1 ) > s u n ≤ P 0 T γ(2sn) > s u n and from the Borel Cantelli lemma.

Estimates for bad points.

Recall from [START_REF] Pitman | One-dimensional Brownian motion and the three-dimensional Bessel process[END_REF] the definition of the excursion measure N 0 of the ψ-Lévy snake W . Recall that the lifetime process of W is the height process H. Namely, N 0 -a.e. for all t ∈ R + , ζ Wt = H t . Therefore, the duration σ of W under N 0 is the duration of the excursion of H. Recall that W is the endpoint process of the snake. Recall from [START_REF] Sheu | Asymptotic behavior of superprocesses[END_REF] that M stands the occupation measure of the snake, namely the random measure on R d that is the image via W of the Lebesgue measure on [0, σ]. Recall from [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] the definition of R, the range of the snake.

Let λ, r ∈ (0, ∞). Note that M = σ, therefore we get N 0 1-e -λM(B(0,r)) ≤ N 0 1-e -λσ = N 1-e -λσ = ψ -1 (λ),

where the last to equalities comes from [START_REF] Keller | On solutions of ∆u = f (u)[END_REF] and (68). The next lemma states a lower bound of the same kind.

Lemma 3.14 There exists c 17 ∈ (0, 1), that only on d, such that ∀µ, r ∈ (0, ∞),

r 2 µ ψ -1 (µ) ≥ 1 = = =⇒ N 0 1-e -µM(B(0,r)) ≥ c 17 ψ -1 (µ). (84)
Proof. To simplify, set q(µ, r) := N 0 1-e -µM(B(0,r)) . Note that M(B(0, r)) = σ 0 dt 1 { Wt <r} . An easy argument combined with Fubini first entails the following:

q(µ, r) = µ ∞ 0 dt N 0 1 {t≤σ ; Wt <r} e -µ σ t ds 1 { Ws <r} . ( 85
)
We next apply Lemma 2.8 at the (deterministic) time t (recall that this lemma is a specific form of the Markov property for W = (ρ, W )). Then we get for any t ∈ R + :

N 0 1 {t≤σ ; Wt <r} e -µ σ t ds 1 { Ws <r} = N 0 1 {t≤σ ; Wt <r} exp - [0,Ht] ρ t (dh) N Wt(h) 1 -e -µM(B(0,r)) . (86)
From (83), we get that for all t, h ≥ 0, N Wt(h) 1e -µM(B(0,r)) ≤ ψ -1 (µ). Then,

N 0 1 {t≤σ ; Wt <r} exp - [0,Ht] ρ t (dh) N Wt(h) 1 -e -µM(B(0,r)) ≥ N 0 1 {t≤σ ; Wt <r} e -ψ -1 (µ) [0,H t ] ρt(dh) (87)
Then by ( 85), ( 86), (87) and Fubini we get

q(µ, r) ≥ µN 0 σ 0 dt 1 { Wt <r} e -ψ -1 (µ) [0,H t ] ρt(dh) . ( 88
)
We next apply (46) to the right member of the previous inequality: to that end, recall that U = (U a ) a≥0 stands for a subordinator defined on (Ω, F, P 0 ) that is independent of ξ and whose Laplace exponent is ψ * (λ) = ψ(λ)/λ-α. Then, (46) to the right member of (88) entails the following:

µN 0 σ 0 dt 1 { Wt <r} e -ψ -1 (µ) [0,H t ] ρt(dh) = µ ∞ 0 da e -αa E 0 1 { ξa <r} e -ψ -1 (µ)Ua = µ ∞ 0 da e -aµ/ψ -1 (µ) P 0 ( ξ a < r ) ,
since ξ and U are independent. Thus, (88) and a simple change of variable using the scaling property of Brownian motion, entail q(µ, r) ≥ ψ -1 (µ) Before stating the next lemma, we recall a result about the first exit time from a ball for a ddimensional Brownian motion. First set

χ d,r := inf t ∈ R + : ξ t = r . (90)
In dimension 1, one get

∀r, λ ∈ R + , E 0 exp(-λχ 1,r ) = cosh(r √ 2λ) -1 ≤ 2 exp -r √ 2λ . (91)
Indeed note that t → exp( √ 2λξ tλt) is a martingale (see e.g. Revuz and Yor [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Chapter II, (3.7)). In dimension d, observe that P 0 -a.s. χ d,r ≥ min Proof. We fix µ ∈ (0, ∞) and x ∈ R d \B(0, 2r). To simplify notation we set p(x, µ, r) = N x 1 {R∩B(0,r) =∅} e -µM(B(0,2r)) .

Recall from (40) the definition of τ r that is the hitting time in B(0, r) of the snake W . First recall that {R ∩ B(0, r) = ∅} = {τ r < ∞} and observe that

We next apply Proposition 2.8 to the right member of (93) and to the stopping time τ r ; thus we get we get

p(x, µ, r) ≤ N x 1 {τr <∞} exp - [0,Hτ r ] ρ τr (dh) N Wτ r (h) 1 -e -µM(B(0,2r)) . (94)
Let w ∈ W be a continuous stopped path starting from x ∈ B(0, 2r); we denote by ζ w its lifetime. We define T 1 (w), T 2 (w) and T 3 (w) by

T 1 (w) = inf t ∈ [0, ζ w ] : w(t) ≤ 3r/2 T 2 (w) = inf t ∈ [0, ζ w -T 1 (w)] : w(t + T 1 (w))-w(T 1 (w)) > r/4 T 3 (w) = inf t ∈ [0, ζ w ] : w(t) ≤ 5r/4 (95)
with the convention that inf ∅ = ∞. Observe that if T 3 (w) < ∞, then T 1 (w) + T 2 (w) ≤ T 3 (w). Moreover, since x ∈ B(0, 2r) c , N x -a.e. on the event {τ r < ∞}, we have T 1 (W τr ) + T 2 (W τr ) ≤ T 3 (W τr ) < τ r < ∞ and for any t ∈ [T 1 (W τr ), T 1 (W τ ) + T 2 (W τr )], the following inequality holds true:

N Wτ r (t)
1e -µM(B(0,2r)) ≥ N Wτ r (t) 1e -µM(B(Wτ r (t),r/4)) = N 0 1e -µM(B(0,r/4)) =: Λ µ,r , the last equality being a consequence of the invariance of the snake under translation; here Λ µ,r only depends on µ and r. To simplify notation we set T 1 = T 1 (W τr ) and T 2 = T 2 (W τr ). An elementary inequality combined with (94) entails

p(x, µ, r) ≤ N x 1 {τr <∞} exp - [T 1 ,T 1 +T 2 ]
ρ τr (dh) N Wτ r (h) 1e -µM(B(0,2r))

≤ N x 1 {τr <∞} exp -Λ µ,r

[T 1 ,T 1 +T 2 ]
ρ τr (dh) .

(96) Lemma 3.16 Assume they δ > 1 and that d ≥ 3. Recall that c 18 is the constant appearing in Lemma 3.15 and that c 4 is the constant appearing in Lemma 3.3. There exists c 20 , c 21 , r 4 , κ 0 ∈ (0, ∞), that only depend d and ψ, such that for all r ∈ (0, r 4 ), for all κ ∈ (0, κ 0 ) and for all x ∈ B(0, 2r) c ,

N x R ∩ B(0, r) = ∅ ; M(B(0, 2r)) ≤ κg(r) ≤ c 18 ψ ′-1 c 4 r -2 (r/ x ) d-2 (log 1/r) -c 20 κ -c 21 .
Proof. Let κ, µ ∈ (0, ∞). Let r ∈ (0, r 1 ) where r 1 is the constant appearing in Lemma 3.3. Assume that such that r 2 µ ψ -1 (µ) ≥ 16. A Markov inequality, combined with Lemma 3.15 entails :

N x R ∩ B(0, r) = ∅; M (B(0, 2r)) ≤ κg(r) ≤ e κµg(r) N x 1 {R∩B(0,r) =∅} e -µM(B(0,2r)) ≤ c 18 (r/ x ) d-2 ψ ′-1 c 4 r -2 exp F (µ, r, κ) , (101)
where we have set F (µ, r, κ) := κµg(r)-c 19 r ϕ(µ) .

For any q ∈ [1, ∞) and any r ∈ (0, r 0 ), we set

µ r,q = ϕ -1 q( 1 r log log 1 r ) 2 .
We first get an estimate for µ q,r . To that end fix a such that 1/a ∈ (0, δ ϕ ). By the definition ( 22) of δ ϕ , there exists C ∈ (0, ∞) such that for any p, λ ∈ [1, ∞), Cp 1/a ϕ(λ) ≤ ϕ(pλ). Thus, for any z ≥ ϕ(1) and any q ≥ C, we get ϕ -1 (qz) ≤ (q/C) a ϕ -1 (z). This easily entails that there exists r 5 ∈ (0, r 0 ∧ r 1 ) and

c 22 ∈ (0, ∞) such that ∀r ∈ (0, r 5 ), ∀q ∈ [1, ∞), µ r,1 ≤ µ r,q ≤ c 22 q a µ r,1 . (102)
Recall that ψ(λ) = ψ(λ)/λ. We next observe that for all r ∈ (0, e -1 ) and all q ∈ [1, ∞), (20) implies

r 2 µ r,q ψ -1 (µ r,q ) = r 2 ψ(ψ -1 (µ r,q )) ≥ 1 4 r 2 ψ ′ (ψ -1 (µ r,q )) ≥ 1 4 r 2 ψ ′ (ψ -1 (µ r,1 )) = 1 4 (log log 1/r) 2 .
Then we set r 4 = exp(-e 8 ) ∧ r 5 and we get

∀r ∈ (0, r 4 ), ∀q ∈ [1, ∞), r 2 µ r,q ψ -1 (µ r,q ) ≥ 16 . (103)
Let r ∈ (0, r 4 ), q ∈ [1, ∞) and κ ∈ (0, ∞). Observe that g(r) = (log log 1/r)/µ r,1 and that r ϕ(µ r,q ) = √ q log log 1/r. Thus, we get

F (µ r,q , r, κ) = κ µ r,q µ r,1 -c 19 √ q log log 1 r ≤ √ q c 22 κq a-1 2 -c 19 log log 1 r .
Since δ ϕ ≤ 1, we get a > 1 and thus a-1 2 > 1 2 . We then set κ 0 = c 19 /(2c 22 ) and for all κ ∈ (0, κ 0 ) we also set q κ = (κ 0 /κ)

1 a-1 2 . Then q κ ≥ 1 and √ q κ c 22 κq a-1 2 κ -c 19 = - 1 2 c 19 (κ 0 /κ) 1 2a-1 .
We then set c 20 = 1 2 c 19 κ and we complete the proof thanks to (103) that allows to apply (101) for any r ∈ (0, r 4 ).

By the Borel Cantelli Lemma, P 0 -a.s. for all sufficiently large n, N rn (s, a) = 0. Since r → N r (s, a) is non-decreasing, we get that P 0 -a.s. for all sufficiently small r, N r (s, a) = 0. Consequently, ∀s ∈ (0, a), P 0 -a.s. lim inf

r→0+ M * a (B(0, r)) g(r) = lim inf r→0+ M * s (B(0, r)) g(r) . (106)
Let G s be the sigma-field generated by 1 [0,s] (t) N * (dt dW ) and completed by the P 0 -negligible sets. Using properties of Poisson random measures and the Blumenthal zero-one law for ξ, we easily check that G 0+ := s>0 G s is P 0 -trivial: namely, for all A ∈ G 0+ , either P 0 (A) = 0 or P 0 (A) = 1. Then observe that (106) implies that the random variable lim inf r→0 M * a (B(0, r))/g(r) is G 0+ -measurable. It is therefore P 0 -a.s. equal to a deterministic constant κ d,ψ ∈ [0, ∞] that does not depends on a.

M(B(x, r)) g(r) = κ d,ψ ∈ [0, ∞] . (107)
Now, we need to prove that 0 < κ d,ψ < ∞, which is done in two steps.

Lemma 4.4 Assume that δ > 1 and that d > 2γ γ-1 . Then, κ d,ψ < ∞. Proof. Fix a ∈ (0, ∞) and recall from (67) the definition of T a that is the sum of the durations of the snakes that are grafted on the spine (ξ s ) 0≤s≤a . Recall from (71) the definition of N r (s, t). Observe that if ϑ(2r) ≥ a, then M * a (B(0, r)) ≤ T a ≤ T ϑ(2r) . Next not that if ϑ(2r) < a and if N r (ϑ(2r), a + ϑ(2r)) = 0, then, M * a (B(0, r)) ≤ T ϑ(2r) . Therefore, P 0 -a.s. on {N r (ϑ(2r), a+ϑ(2r)) = 0}, M * a (B(0, r)) ≤ T ϑ(2r) .

Then, for all r > r ′ ≥ 0, and all A > 0, P 0 -a.s. on {N r (ϑ(2r), a+ϑ(2r

)) = 0}∩{T ϑ(2r) -T ϑ(2r ′ ) ≤ A}, M * a (B(0, r)) ≤ A + T ϑ(2r ′ ) . (108)
Recall from (69) the definition of γ(r) and ϑ(r) and observe that ϑ(r) ≤ γ(r), which implies P 0 -a.s. ∀r ∈ (0, ∞), T ϑ(r) ≤ T γ(r) .

Thus, by (108), for all r > r ′ ≥ 0, and all A > 0,

P 0 -a.s. on {N r (ϑ(2r), a+ϑ(2r)) = 0}∩{T ϑ(2r) -T ϑ(2r ′ ) ≤ A}, M * a (B(0, r)) ≤ A + T γ(2r ′ ) , (109)
which allows us to bound M * a (B(0, r)) by the subordinator T γ(•) studied in section 3.3. Recall Lemma 3.7, where the sequence (ρ n ) n≥1 is introduced; recall Lemma 3.11 and Lemma 3.12 that provide a control respectively on P 0 (N ρn (ϑ(2ρ n ), a + ϑ(2ρ n )) = 0) and on T γ(2ρn) . For any n ≥ 1, we next introduce the following random variable:

Y n = 1 {Nρ n (ϑ(2ρn),a+ϑ(2ρn))=0}∩{T ϑ(2ρn ) -T ϑ(2ρ n+1 ) ≤g(8ρn)} . (110)
By (109), we get

∀n ≥ 1, P 0 -a.s. on {Y n = 1}, M * a (B(0, ρ n )) ≤ g(8ρ n ) + T γ(2ρ n+1 ) .
We then define the following event:

E = n≥1 Y n = ∞ .
We claim that P 0 (E) > 0.

Then the proof of the lemma is completed as follows: the previous arguments first entail

P 0 -a.s. on E, lim inf n→∞ M * a (B(0, ρ n )) g(8ρ n ) ≤ 1 + lim sup n→∞ T γ(2ρ n+1 ) g(8ρ n ) . (112)
Recall that the assumption δ > 1 implies that g satisfies a C-doubling condition [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], which entails g(8r) ≤ C 3 g(r). Then by (112),

P 0 -a.s. on E, κ d,ψ = lim inf r→0+ M * a (B(0, r)) g(r) ≤ C 3 1 + lim sup n→∞ T γ(2ρ n+1 ) g(8ρ n ) . (113)
By Lemma 3.12 (ii) the right member of ( 113) is P 0 -a.s. finite, which completes the proof of the lemma. It only remain to prove our claim (111).

Proof of (111). We use a second moment method. By the independence property of Lemma 3.8 (i), we first get

E 0 [Y n ] = P 0 (N ρn (ϑ(2ρ n ), a+ϑ(2ρ n )) = 0) P 0 T ϑ(2ρn) -T ϑ(2ρ n+1 ) ≤ g(8ρ n ) .
Then, the lower bound of Lemma 3.11 and the fact that

T ϑ(r) ≤ T γ(r) entail E 0 [Y n ] ≥ c 16 P 0 T ϑ(2ρn) -T ϑ(2ρ n+1 ) ≤ g(8ρ n ) (114) ≥ c 16 P 0 T ϑ(2ρn) ≤ g(8ρ n ) ≥ c 16 P 0 T γ(2ρn) ≤ g(8ρ n ) .
So by Lemma 3.12 (i), we get

n≥1 E 0 [Y n ] = ∞ . (115)
Besides, for n > m ≥ 1, by the independence property of Lemma 3.8 (i)

E 0 [Y n Y m ] ≤ P 0 T ϑ(2ρn) -T ϑ(2ρ n+1 ) ≤ g(8ρ n ); T ϑ(2ρm) -T ϑ(2ρ m+1 ) ≤ g(8ρ m ); N ρm (ϑ(2ρ m ), a+ϑ(2ρ m )) = 0 ≤ P 0 T ϑ(2ρn) -T ϑ(2ρ n+1 ) ≤ g(8ρ n ) E 0 [Y m ] ≤ 1 c 16 E 0 [Y n ] E 0 [Y m ] , (116)
where the last inequality follows from (114). Therefore, if we denote L n = 1≤k≤n Y k , then (115) and (116) entail

lim sup n→∞ E[L 2 n ] E[L n ] 2 < ∞.
and the desired result (111) follows from the Kochen Stone Lemma.

The following lemma completes the proof of Theorem 4.1. Its proof relies on a density result on the ψ-Lévy tree that is proved in [START_REF] Duquesne | The exact packing measure of Lévy trees[END_REF] and that is recalled here as Lemma 2.5. Proof. We work with W under N 0 . The proof consists in lifting to W the estimate of Lemma 2.5 by using the fact that conditionally given H, W is a Gaussian process (see [START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF] in Section 2.4). More precisely, recall that the height process H is the lifetime process of the snake W and recall from (36) the definition of N 0 that shows that conditionally given H, the law of W is Q H 0 (see Section 2.4). Recall from (28) the following notation d for the pseudo-distance in the Lévy tree:

∀s, t ∈ [0, σ], d(s, t) = H t + H s -2 inf u∈[s∧t,s∨t] H u . Let r, R ∈ (0, ∞) and let t ∈ [0, σ]. We set a(t, r) = σ 0 1 {d H (s,t)≤r} ds and b(t, r, R) = σ 0 1 {d H (s,t)≤r}∩{ Ws-Wt ≥R} ds .
The quantity a(t, r) has been already introduced in Lemma 2.5. First note that

∀t ∈ [0, σ], a(t, r) ≤ b(t, r, R) + M(B( W t , R)) . (117)
By [START_REF] Perkins | The Hausdorff measure of the closed support of super-Brownian motion[END_REF], we then get The definition of δ ϕ ∈ (0, 1) easily implies that there exists a constant c 24 ∈ (0, ∞) such that for all n sufficiently large c 24 k(r n ) ≥ g(R n ). Then, Lemma 2.5 entails that N 0 -a.e. for almost all t ∈ [0, σ], lim inf

N (dH)-a.e. ∀t ∈ [0, σ] , Q H 0 [b(t, r, R)] ≤ a(t, r) R d \B(0,R/ √ r) (2π) -d/2 e -x 2 /2 dx . (118)
n→∞ M(B( W t , 2 -n )) g(2 -n ) ≥ c 1 c 24 .
An easy argument involving the doubling property (13) for g completes the proof thanks to (107).

4.2 Proof of Theorem 4.2.

We first introduce a specific decomposition of R d into dyadic cubes. We adopt the following notation: we denote by ⌊ •⌋ the integer part application and we write log 2 for the logarithm in base 2; we fix d > 2γ It is easy to check the following properties. For any r < (2d) -1 , we set n(r) = ⌊log 2 (r -1 (1 + 2 -p ) √ d)⌋, so that the following inequalities hold:

1 2 (1 + 2 -p ) √ d 2 -n(r) < r ≤ (1 + 2 -p ) √ d 2 -n(r) . (120)
Next, for any x = (x 1 , . . . , x d ) ∈ R d and for j ∈ {1, . . . , d}, we set y j = 2 -n(r)-p ⌊x j 2 n(r)+p + 1 2 ⌋ .

Therefore, y = (y 1 , . . . , y d ) ∈ D n(r) and we easily check the following:

• Prop(3) The point x belongs to D • n(r) (y) and D n(r) (y) ⊂ B(x, r).

where c 29 , c 30 , c 31 ∈ (0, ∞) only depend on d and ψ. Therefore, N 0 ( n≥N U n (A)) < ∞, which easily implies the lemma.

We next prove the following lemma. We use the doubling condition [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] to choose c 32 such that c 32 g((1 + 2 -p ) √ d2 -n(r) ) ≤ κ 2 g(2 -n(r) ) for all sufficiently small r ∈ (0, 1). Thus, we get Therefore P g (B A ) < ∞. Suppose now that P g (B A ) > 0. Then, as a consequence of ( 18), there exists a closed subset F , with F ⊂ B A , such that P g (F ) > 0. Since F is closed then F = ε>0 F (ε) ; since F is a subset of B A and since W ∈ W A (where (104) holds true), we get M(F ) ≤ M(B A ) = 0 and by (127) applied to B = F , we obtain P g (F ) = 0, which rises a contradiction. Thus, we have proved that N 0 -a.e.P g (B A ) = 0, which easily entails the lemma by letting A go to ∞, since P g ({0}) = 0.

We now complete the proof of Theorem

Fix x ∈ R d and recall Lemma 2.6 that asserts that for any q ∈ (0, γ-1 2γ ), N x -a.e. ( W s ) s∈[0,σ] is q-Hölder continuous. As already mentioned in Comment 1.2, it easily implies the following:

N x -a.e. dim(R) ≤ 2γ γ -1 , ( 129
)
where R is the range of the Lévy snake as defined [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]. We next prove the following lemma. Proof. Let us first assume that x / ∈ K and set k := inf y∈K xy > 0. For any ε ∈ (0, k/2), denote by n ε the minimal number of balls with radius ε that are necessary to cover K, and denote by B(x ε 1 , ε),

Lemma 2 . 8 (

 28 [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] Theorem 4.1.2 and Lemma 4.2.4) Let T be a (F W t+ ) t≥0 -stopping time. Let Y be a nonnegative F W T + -measurable random variable. Let G : R d → R + be Borel measurable. Then,

∞

  αr yf (y) dy + c 8 αr 2r yf (y)ψ ′ (α r /y) d-2 dy ≤ c 5 + c 8 f (0) αr 2r yψ ′ (α r /y) d-2 dy , (63) where c 5 := c 8 ψ ′ (1) ∞ 0 yf (y)dy. By using the change of variable v = (α r /y) d-2 we get

1 2 ≤ds1{

 2 e -n and n -2 log 1/ρ n ≤ c 12n -2 log c 11 , which proves that (ρ n ) n≥0 satisfies (51) in Lemma 3.1. Moreover, for all n ∈ N such that ρ n ∈ (0, r 2 ), Lemma 3.4 and (64) in Lemma 3ξs ≥2ρn} ψ ′ (u ρn (ξ s )) ≤ c 5 + c 6 J(r e n 2 ) ≤ c 5 + c 6 c 10 =: c 14 ,

∞ 0 P

 0 0 ξ c ≤ r µ/ψ -1 (µ) e -c dc . (89) If r 2 µ/ψ -1 (µ) ≥ 1, then (89) implies (84) with c 17 := ∞ 0 P 0 ( ξ c ≤ 1) e -c dc. Clearly, c 17 only depends on d and c 17 ∈ (0, 1).

t

  stands for the j-th coordinate of ξ t . The previous inequality combined with (91) then entails∀λ, r ∈ R + , E 0 exp(-λχ d,r (ξ)) ≤ 2d exp -r 2λ/d . (92)Lemma 3.15 Assume that δ > 1 and that d ≥ 3. Recall that r 1 and c 4 are the constants appearing in Lemma 3.3. Then, there exist c 18 , c 19 ∈ (0, ∞), that only depend on d and ψ, such that ∀µ ∈ (0, ∞), ∀r ∈ (0, r 1 ) such that r 2 µ ψ -1 (µ) ≥ 16, ∀x ∈ R d \B(0, 2r), N x 1 {R∩B(0,r) =∅} e -µM(B(0,2r)) ≤ c 18 (r/ x ) d-2 ψ ′-1 c 4 r -2 exp -c 19 r ϕ(µ) .

N 1 {µ σ τr 1 {

 11 x -a.e. on the event {R ∩ B(0, r) = ∅}, σ τr Ws <2r} ds ≤ M(B(0, 2r)).Thus, we get p(x, µ, r) ≤ N x 1 {τr <∞} e -Ws <2r} ds .

F

  (µ r,qκ , r, κ) ≤ -c 20 κ -c 21 log log 1 r

Remark 4 . 1

 41 We point out that Lemma 4.3 holds true for all gauge function g. By (50), the previous lemma entails that N 0 -a.e. for M-almost all x, lim inf

  r→0+

Lemma 4 . 5

 45 Assume that δ > 1 and that d > 2γ γ-1 . Then κ d,ψ > 0.

For any integer n ≥ 2 ,N 0 1 rϕ -1 1 r log log 1 r.

 2011 we next set R n = 2 -n and r n = 1 4 R 2 n (log log 1/R n ) -1 . By elementary computation,∀n ≥ 2, R d \B(0,Rn/ √ rn) (2π) -d/2 e -x 2 /2 dx ≤ c 23 n -3/2 ,where c 23 ∈ (0, ∞) only depends on d (note that the power 3/2 is not optimal). By (118), we getN (dH)-a.e. ∀t ∈ [0, σ] , Q H 0   n≥2 b(t, r n , R n ) a(t, r n )   < ∞ . Thus, N (dH)-a.e. for all t ∈ [0, σ], Q H 0 (lim sup n→∞ b(t, r n , R n )/a(t, r n ) > 0) = 0. Then, by Fubini, N (dH)-a.e. Q H -a.e. for ℓ-almost all t ∈ [0, σ], lim n→∞ b(t, r n , R n ) a(t, r n ) = 0(119)(ℓ stands here for the Lebesgue measure on the real line). Recall from[START_REF] Le Gall | Branching processes in Lévy processes: the exploration process[END_REF], the notation k(r):∀r ∈ (0, α ∧ e -e ), k(r) := log log Then, (119) combined with (117), entails N 0 -a.e. for ℓ almost all t ∈ [0, σ], lim inf n→∞ a(t, r n ) k(r n ) ≤ lim inf n→∞ M(B( W t , R n )) k(r n ) .

[ y j - 1 2 2 1 2 2 [ y j - 1 2 2 1 2 2

 12121212 To simplify notation, we set D n = 2 -n-p Z d , for any n ≥ 0. For any y = (y 1 , . . . , y d ) in D n , we also set D n (y) = d j=1 -n ; y j + -n ) and D • n (y) = d j=1 -n-p ; y j + -n-p ).

• Prop( 1 ) 1 2 2

 112 If y, y ′ are distinct points in D n , then D • n (y) ∩ D • n (y ′ ) = ∅.• Prop(2) Let y ∈ D n . Then, we haveD • n (y) ⊂ B(y , -n-p √ d ) ⊂ B(y , 2 -n-p √ d ) ⊂ D n (y) .

Lemma 4 . 7

 47 Assume that δ > 1 and that d > 2γ γ-1 . Then, N 0 -a.e. P gx ∈ R : lim infr→0+ g(r) -1 M(B(x, r)) = κ d,ψ = 0 ,(125)where κ d,ψ is the constant appearing in Theorem 4.1.Proof. We fix A > 100. By Theorem 4.1 and Lemma 4.6 there exists a Borel subset W A of W such that N 0 (W\W A ) = 0 and such that on W A , (104) and (123) hold true. We shall work deterministically onW ∈ W A . Let B be any Borel subset of {x ∈ R d : 1/A ≤ x ≤ A}. Let ε ∈ (0, ∞) and let B(x 1 , r 1 ), . . . B(x k , r k ) be any closed ε-packing of B ∩ R.Namely, the later balls are disjoint, their centres belong to B ∩ R and their radii are smaller than ε. Let c 32 ∈ (0, ∞) be a constant to be specified later. First observe thatk i=1 g(r i ) = k i=1 g(r i )1 {M(B(x i ,r i )>c 32 g(r i )} + k i=1 g(r i )1 {M(B(x i ,r i )≤c 32 g(r i )} ≤ c -1 32 M B (ε) + k i=1 g(r i )1 {M(B(x i ,r i )≤c 32 g(r i )} ,(126)where we have set B (ε) = {x ∈ R d : dist(x, B) ≤ ε}. Next, fix 1 ≤ i ≤ k; recall notation n(r i ) from (120) and denote by y i the point of D n(r i ) corresponding to x i such that Prop(3) holds true. Therefore, by (120), we have M(B(x i , r i ) ) ≤ c 32 g(r i ) and x i ∈ B ∩ R =⇒ M(D n(r i ) (y i )) ≤ c 32 g((1 + 2 -p ) √ d2 -n(r i ) ) and R ∩ D • n(r i ) (y i ) = ∅.

  i )1 {M(B(x i ,r i )≤c 32 g(r i )} ≤ n : 2 -n ≤c 33 ε U n (A) , where c 33 = 2((1 + 2 -p ) √ d) -1 .Since W belongs to W A where (123) holds, this inequality combined with (126) implies the following.P g (B ∩ R) ≤ P * g (B ∩ R) ≤ c -1 32 M ε>0 B (ε) .(127)We next applies (127) with B = B A given byB A = x ∈ R : 1/A ≤ x ≤ Aand lim inf r→0+ g(r) -1 M(B(x, r)) = κ d,ψ .

4 . 2 :

 42 by Theorem 4.1 and Lemma 4.7 there exists a Borel subset W * of W such that N 0 (W\W * ) = 0 and such that (104) and (125) hold true on W * . We fix W ∈ W * and we setGood = x ∈ R : lim inf r→0+ g(r) -1 M(B(x, r)) = κ d,ψand Bad = R\Good.Let B be any Borel subset of R d . By (104) and (125), we haveM(B ∩ Bad) = P g (B ∩ R ∩ Bad) = 0 .Then, we apply Lemma 2.2 to Good ∩ B and we getM(B ∩ Good) = κ d,ψ P g (B ∩ R ∩ Good) .Therefore, on W * , for Borel subset B of R d , M(B) = κ d,ψ P g (B ∩ R), which completes the proof of Theorem 4.2.

4. 3

 3 Proof of theorem 1.2 We derive Theorem 1.2 from Theorem 4.2. To that end, we first need an upper bound of the upper boxcounting dimension of R under N x . Let us briefly recall the definition of the box-counting dimensions of a bounded subset K ⊂ R d : let ε ∈ (0, ∞) and let n ε (K) stands for the minimal number of open balls of radius ε that are necessary to cover A. Then, dim(K) = lim inf ε→0 log n ε (K) log 1/ε and dim(K) = lim sup ε→0 log n ε (K) log 1/ε .

Lemma 4 . 8

 48 Assume that γ > 1 and that d > 2γ γ-1 . Let x ∈ R d . For any compact subset K such that dim(K) ≤ 2γγ-1 , we have N x -a.e. M(K) = 0.

  , Corollary 5.10.

	Lemma 2.2 (Corollary 5.10 [19]) Let µ be a finite Borel measure on R d . Let κ ∈ (0, ∞) and let B be a Borel subset of R d such that
	∀x ∈ B, lim inf r→0+	µ(B(x, r)) g(r)	= κ .
	Then, µ(B) = κP g (B).
	Remark 2.2 The main purpose of Edgar's article [19] is to deal with fractal measures in metric spaces
	with respect to possibly irregular gauge functions. Corollary 5.10 [19] (stated here as Lemma 2.2) holds
	true in this general setting if µ satisfies the so called Strong Vitali Property (see [19] p.43 for a definition
	and a discussion of this topic). A result due to Besicovitch [2] ensures that any finite measure on R d enjoys
	the Strong Vitali Property. Therefore, Lemma 2.2 is an immediate consequence of Edgar's Corollary 5.10
	[19].

Recall from (95) the definition of T 3 . Recall from (44) the definition of the function ̟. We next apply Proposition 2.9 with r ′ = 5 4 r to the right member of (96). Then we get

ρ τr (dh)

= ũr (5r/4) E x 1 {T 3 (ξ)<∞} exp -

dt ̟ u r (ξ t ), Λ µ,r 1 [T 1 (ξ),T 1 (ξ)+T 2 (ξ)] (t)

≤ ũr (5r/4) E x 1 {T 1 (ξ)<∞} exp -

Here recall that ξ under P x is distributed as a standard d-dimensional Brownian motion starting from x.

The convexity of ψ provides the following lower bounds for ̟ (u r (ξ t ), Λ µ,r ):

• if u r (ξ t ) ≥ Λ µ,r , then ϕ (u r (ξ t ), Λ µ,r ) ≥ ψ ′ (Λ µ,r );

• if u r (ξ t ) < Λ µ,r , then ̟ (u r (ξ t ), Λ µ,r ) ≥ ψ(Λ µ,r )/Λ µ,r ≥ 1 4 ψ ′ (Λ µ,r ), by [START_REF] Etheridge | An introduction to superprocesses[END_REF] for the last inequality. These inequalities combined with (96) and (97), entail p(x, µ, r) ≤ ũr (5r/4) E x 1 {T 1 (ξ)<∞} exp -

We now assume that

Recall that Λ µ,r = N 0 [1-e -µM(B(0,r/4) ]. By Lemma 3.14, Λ µ,r ≥ c 17 ψ -1 (µ). We next use the concavity of ψ ′ and the fact that c 17 ∈ (0, 1), to get

Recall that r 1 and c 4 are the constants appearing in Lemma 3.3. We assume that r ∈ (0, r 1 ). Then, Lemma 3.3 with ̺ = 1/4 implies that ũr (5r/4) ≤ ψ ′-1 (c 4 r -2 ). Thus, by (98) and (99) we get

Recall from (95) the definition of T 1 (ξ) and T 2 (ξ). By the Markov property at time T 1 (ξ), we get

. Moreover by (92), we get

Then we set c 18 = 2d (3/2) d-2 and c 19 = c 17 /(32d) and we get

which implies the desired result by (100).

Recall from [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF] the definition of the gauge function g:

4 Proof of the results.

Recall from [START_REF] Sheu | Asymptotic behavior of superprocesses[END_REF] the definition of the total occupation measure of the snake M. We prove in Section 4.1 the following results on the lower density of M.

Theorem 4.1 Let ψ be a branching mechanism the form [START_REF] Bertoin | Lévy processes[END_REF]. Let (W t ) t≥0 be the associated snake. Let g be defined by [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF]. Assume that δ > 1 and that d > 2γ γ-1 . Then, there exists a constant κ d,ψ ∈ (0, ∞), that only depends on d and ψ, such that N 0 -a.e. for M-almost all x, lim inf

This result is then used to prove the following theorem in Section 4.2.

Theorem 4.2 Let ψ be a branching mechanism the form (1). Let (W t ) t≥0 be the associated snake. Let g be defined by [START_REF] Duquesne | Packing and Hausdorff measures of stable trees[END_REF]. Assume that δ > 1 and that d > 2γ γ-1 . Then, there exists a constant κ d,ψ ∈ (0, ∞), that only depends on d and ψ, such that N 0 -a.e. for any Borel set B, M(B) = κ d,ψ P g (B ∩ R ) .

Proof of theorem 4.1.

Recall from Section 2.4, the Palm formula for the occupation measure of the snake. To that end, recall that ξ = (ξ t ) t≥0 is a continuous process defined on the auxiliary measurable space (Ω, F) and recall that P 0 is a probability measure on (Ω, F) under which ξ is distributed as a standard d-dimensional Brownian motion starting from the origin 0. Recall that (V t ) t≥0 be a subordinator defined on (Ω, F, P 0 ) that is independent of ξ and whose Laplace exponent is ψ * ′ (λ) = ψ ′ (λ)-α. Recall from (47) that under P 0 , conditionally given (ξ, V ), N * (dtdW ) = j∈J * δ (t j ,W j ) is a Poisson point process on [0, ∞) × C(R + , W) with intensity dV t N ξt (dW ). Recall from (48) that for all a ∈ R + , we have set M * a = j∈J * 1 [0,a] (t j )M j where for all j ∈ J * , M j stands for the occupation measure of the snake W j as defined in [START_REF] Sheu | Asymptotic behavior of superprocesses[END_REF]. Also recall from (71) the definition of the following random variables:

that counts the snakes that are grafted on the spatial spine ξ between times s and t, and that hit the ball B(0, r). We first prove the following lemma that is a consequence of the Blumenthal 0-1 law.

Proof. Let a ∈ (0, ∞). Let s ∈ (0, a). Observe that if N r (s, a) = 0, then M * a (B(0, r)) = M * s (B(0, r)). By Lemma 3.9, there exists a deterministic sequence (r n ) n≥0 decreasing to 0 such that n≥0 P 0 (N rn (s, a) = 0) < ∞ .

We work under N 0 . Recall Lemma 3.16: we fix

(121)

Since we assume δ > 1, g satisfies the doubling condition [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], which implies that there exists κ 2 ∈ (0, ∞), that only depends on d, ψ and κ 1 , such that for all sufficiently large n,

We then fix A > 100 and for any n such that 2 -n ≤ 1/(2A), we set

We first prove the following lemma.

Lemma 4.6 Assume that δ > 1 (so that (122) holds) and that d ≥ 3. Then, for all A > 100,

Proof. We fix n such that 2 -n ≤ 1/(2A) and we fix y ∈ D n such that 1/A ≤ y ≤ A. By Prop( 2) and (122), we get

the last equality being an immediate consequence of the invraince of the snake by translation. We next apply Lemma 3.16 with x = -y and r = 1 2 2 -n-p √ d and κ = κ 1 (that satisfies (121); thus, there exists c 25 , c 26 ∈ (0, ∞), that only depends on d and ψ, such that

. By Lemma 2.3 and the doubling property (13) of g, there exists c 27 ∈ (0, ∞), that only depends on d and ψ, such that for all sufficiently large n,

which entails the following:

) where c 28 = c 25 c 27 . Elementary arguments entail the following inequalities:

..., B(x ε nε , ε) such balls. Then, (46) combined with standard estimates of d-dimensional Green function entail the following inequalities.

where c 34 , c 35 ∈ (0, ∞) only depend on d. Since d > 2γ γ-1 ≥ dim(K), the previous inequality implies that N x (M(K)) = 0 as ε → 0.

Let us now consider the general case: for any r > 0, the previous case applies to the compact set K ′ = K\B(x, r) and we get

which implies the desired result as r → 0 since M is diffuse.

The end of the proof of Theorem 1.2 follows an argument due to Le Gall in [START_REF] Le Gall | The Hausdorff measure of the range of super-Brownian motion[END_REF] pp. 312-313. Theorem 4.2 and Lemma 4.8 imply that for any compact set K such that dim(K) ≤ 2γ γ-1 , and for any

Recall the connection (39) in Theorem 2.7 between R, M and the excursions W j , j ∈ J , of the Brownian snake. An easy argument on Poisson point processes combined with (129) and (130) implies that almost surely P g (R W j ∩ R W i ) = 0 for any i = j in J . Then, (39) entails

Theorem 2.7 and (39) thus imply

which is the desired result.

Dimension of the range of the ψ-SBM.

We now prove Theorem 1.1. To that end, recall that ξ = (ξ t ) t≥0 is a continuous process defined on the auxiliary measurable space (Ω, F) and recall that P 0 is a probability measure on (Ω, F) under which ξ is distributed as a standard d-dimensional Brownian motion starting from the origin 0. Recall that (V t ) t≥0 be a subordinator defined on (Ω, F, P 0 ) that is independent of ξ and whose Laplace exponent is ψ * ′ (λ) = ψ ′ (λ)-α. Recall from (47) that under P 0 , conditionally given (ξ, V ), N * (dtdW ) = j∈J * δ (t j ,W j ) is a Poisson point process on [0, ∞)×C(R + , W) with intensity dV t N ξt (dW ). Then recall from (48) that for all a ∈ R + , we have set M * a = j∈J * 1 [0,a] (t j )M j where for all j ∈ J * , M j stands for the occupation measure of the snake W j as defined in [START_REF] Sheu | Asymptotic behavior of superprocesses[END_REF]. Also recall from (71) the definition of the following random variables: ∀ t ≥ s ≥ 0 N r (s, t) = # j ∈ J * : s < t j < t and R j ∩ B(0, r) = ∅ , that counts the snakes that are grafted on the spatial spine ξ between times s and t, and that hit the ball B(0, r).

Lemma 4.9 Assume that δ > 1 and that d > 2δ δ-1 . Then, for all a ∈ (0, ∞), and for all u ∈ (0, 2γ γ-1 ),

Proof. The present proof is very similar to that of Lemma 3.10. We detail only the main steps. Recall from Lemma 3.8 the definition of the processes (T ϑ(r) ) r≥0 and (T γ(r) ) r≥0 . Recall from (71) the definition of N r (s, t). Recall from Lemma 3.13 the definition of the sequence (s n ) n≥0 . Then, for all n ≥ 0, we set

Reasoning as in the proof of (109), we get

Thus, if we show that

we get (131), by use of Lemma 3.13 (ii).

The inequality (134) is obtained using Kochen Stone Lemma, as in the proof of (111). Indeed, under the assumption d > 2δ

δ-1 , Lemma 3.10 entails that for all n ∈ N, P 0 (N sn (ϑ(2s n ), a + ϑ(2s n )) = 0) ≥ c 15 > 0; we then argue exactly as in the proof of Lemma 4.4 to obtain (134). We leave the details to the reader.

Proof of Theorem 1.1. Assume that δ > 1 and that d > 2δ δ-1 . Recall from (37) the definition of R, the range of the Lévy snake. By Theorem 2.7 and by spatial invariance of the snake, it is sufficient to prove N 0 -a.e. dim p (R) = dim(R) = 2γ γ -1 .

Recall that for every bounded subset K ⊂ R d , dim p (A) ≤ dim(A) (see e.g. Falconer [START_REF] Falconer | Fractal geometry[END_REF]). By (129), it then only remains to prove

Lemma 4.9 comnied with (50) implies that for all u ∈ (0, 2γ γ-1 ), N 0 -a.e. for M-almost all x, lim inf r→0+ r -u M(B(x, r)) < ∞,

which implies that N 0 -a.e. dim p (R) ≥ u by the comparison results stated here as Theorem 2.1. This entails (135) and the proof of Theorem 1.1 is completed.