SOME PROPERTIES OF THE RANGE OF SUPER-BROWNIAN MOTION

JEAN-FRANÇOIS DELMAS

Abstract. We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the ε-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of X_{t} is capacity-equivalent to [0, 1]^{2} in ℝ^{d}, d ≥ 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1]^{4} in ℝ^{d}, d ≥ 5.

Introduction

Super-Brownian motion, denoted here by X = (X_{t}, t ≥ 0), is a measure-valued process in ℝ^{d}. It can be obtained as a limit of branching Brownian particle systems. We refer to Dynkin [8] for such an approximation in a more general setting. Another way to study super-Brownian motion, is to use the path-valued process, called the Brownian snake, which was introduced by Le Gall [9, 12]. Furthermore this approach allows us to study also the integrated super-Brownian excursion (ISE). This process appears naturally when one consider the limit of rescaled lattice trees in high dimension (see Derbez and Slade [4, 3]). For every bounded Borel set A ⊂ ℝ^{d}, we denote by A^{ε} = {x ∈ ℝ^{d}; d(x, A) ≤ ε} and by |A| the Lebesgue measure of the set A. Recently Tribe [19] (see also Perkins [16]) proved a convergence result for the volume of the ε-neighborhood of the support at time t > 0, supp X_{t}, of super-Brownian motion in dimension d ≥ 3. More precisely, Tribe showed that the quantity ε^{2−d} |(supp X_{t})^{ε} ∩ A| converges a.s. to a deterministic constant times ∫ 1_{A}(x)X_{t}(dx). Using results of Le Gall [11] on hitting probabilities for the Brownian snake, we give a similar result for the range of the Brownian snake. We then derive an analogous result (theorem 2.1) for the range of super-Brownian motion after time t > 0, R_{t}(X) defined as the closure of ∪_{s≥t}supp X_{s}. More precisely, we show that there exists a positive constant C_{0} depending only on d such that for every Borel set A ⊂ ℝ^{d}, d ≥ 4, for every t > 0, we have a.s.

lim_{ε→0} φ_{d}(ε) |R_{t}(X)^{ε} ∩ A| = C_{0} ∫_{t}^{∞} ds ∫ 1_{A}(z)X_{s}(dz),

where φ_{4}(ε) = log(1/ε) and φ_{d}(ε) = ε^{4−d} if d ≥ 5. We also give a similar result for the support of ISE (corollary 2.4).

1991 Mathematics Subject Classification. 60G57, 60J80.

Key words and phrases. Superprocesses, integrated super-Brownian excursion, measure valued process, Brownian snake, hitting probabilities, capacity-equivalence.

The research was done at the École Nationale des Ponts et Chaussées and at MSRI, supported by NSF grant DMS-9701755.

Pemantle and Peres [14] defined the notion of capacity-equivalence for two random Borel sets, and later Pemantle and al. [15] showed that the range of Brownian motion in ℝ^{d}, d ≥ 3, is capacity-equivalent to [0, 1]^{2}. As an application of the previous results , we show (proposition 4.3) that a.s. on {X_{t} ≠ 0}, the set supp X_{t} ⊂ ℝ^{d}, d ≥ 3, is capacity-equivalent to [0, 1]^{2}, and that a.s. the range R_{t}(X) ⊂ ℝ^{d} and the support of ISE for d ≥ 5 are capacity-equivalent to [0, 1]^{4}.

Let us now describe more precisely the contents of the following sections. In section 1, we recall the definition of the path-valued process W = (W_{s}, s ≥ 0) called the Brownian snake. We denote by ζ_{s} the lifetime of the path W_{s}. We recall the links between the Brownian snake, super-Brownian motion and ISE.

In section 1.3, we introduce the main tools concerning the Brownian snake. In particular, we consider T_{(x,ε)} the hitting time for the Brownian snake of B̄(x, ε), the closed ball with center x and radius ε:

T_{(x,ε)} = inf {s ≥ 0; ∃t ∈ [0, ζ_{s}], W_{s}(t) ∈ B̄(x, ε)} .

The function u_{ε}(x) = ℕ_{0} [T_{(x,ε)} < ∞], where ℕ_{0} is the excursion measure of the Brownian snake away from the trivial path 0, is the maximal nonnegative solution of Δu = 4u^{2} on ℝ^{d}\B(0, ε) (see also Dynkin [7]). The study of |R(W)^{ε} ∩ A| = ∫_{A} dx 1_{{T_{(x,ε)}<∞}}, where R(W) is the range of the Brownian snake, relies on the explicit law of the first hitting path (W_{T_{(x,ε)}}, ζ_{T_{(x,ε)}}) under the excursion measure. This law has been computed by Le Gall [11, 13]. It is closely related to the law of the process (x_{t}^{ε}, 0 ≤ t ≤ τ^{ε}), defined as the unique strong solution of

dx_{t}^{ε} = dβ_{t} + \frac{∇u_{ε}(x_{t}^{ε} − x)}{u_{ε}(x_{t}^{ε} − x)}dt, for 0 ≤ t ≤ τ^{ε},

where β is a Brownian motion in ℝ^{d} started at β_{0} = 0 and τ^{ε} = inf {t ≥ 0; |x_{t}^{ε} − x| = ε}.

In section 2, we state the main result on the convergence of the volume of the ε-neighbor-hood of R_{t}(X). The method of the proof is completely different from the one used by Tribe in [19]. It is derived from the convergence of the volume of the ε-neighborhood of the range of the Brownian snake in L^{2}(ℕ_{0}) (proposition 2.3).

Section 3 is devoted to the proof of the latter convergence. The proof of the L^{2}(ℕ_{0}) convergence is somewhat technical because we need a precise rate of convergence. The derivation of this estimate relies heavily on the explicit law of (W_{T_{(x,ε)}}, ζ_{T_{(x,ε)}}) under ℕ_{0}. It also depends on precise information on the behavior of the function u_{1} at infinity. In particular we give the asymptotic expansion of u_{1} at infinity in the appendix.

In section 4 we prove the results on capacity-equivalence for the support and the range of super-Brownian motion and for the support of ISE. Let f : [0, ∞) → [0, ∞] be a decreasing function. We define the energy of a Radon measure ν on ℝ^{d} with respect to the kernel f by: I_{f}(ν) = ∫∫ f(|x − y|)ν(dx)ν(dy), and the capacity of a set Λ ⊂ ℝ^{d} by cap_{f}(Λ) = [inf_{ν(Λ)=1} I_{f}(ν)]^{−1}. Following the terminology introduced in [14], we say that two sets Λ_{1} and Λ_{2} are capacity-equivalent if there exist two positive constants c and C such that for every kernel f, we have

c cap_{f}(Λ_{1}) ≤ cap_{f}(Λ_{2}) ≤ C cap_{f}(Λ_{1}) .

Proposition 4.3 states that a.s. the set supp X_{t} ⊂ ℝ^{d}, d ≥ 3, is capacity-equivalent to [0, 1]^{2}, and that a.s. the range R_{t}(X) ⊂ ℝ^{d}, as well as the support of ISE for d ≥ 5 are capacity-equivalent to [0, 1]^{4}. The proof follows the method of [15].

1. Preliminaries on the Brownian snake and super-Brownian motion

We first introduce some notation. We denote by (M_{f}, M_{f}) the space of all finite measures on ℝ^{d}, endowed with the topology of weak convergence. We denote by B_{b+}(ℝ^{p}), respectively B_{b+}(ℝ^{+} × ℝ^{p}), the set of all real bounded nonnegative measurable functions defined on ℝ^{p}, respectively on ℝ^{+} × ℝ^{p}. We also denote by B(ℝ^{p}) the Borel σ-field on ℝ^{p}. For A ∈ B(ℝ^{p}), let Cl(A) = Ā be the closure of A. For every measure ν ∈ M_{f}, and f ∈ B_{b+}(ℝ^{d}), we shall write ∫ f(y)ν(dy) = (ν, f). We also denote by supp ν the closed support of the measure ν. If S is a Polish space, we denote by C(I, S) the set of all continuous functions from I ⊂ ℝ into S.

1.1. The Brownian snake. We recall some facts about the Brownian snake, a path-valued Markov process introduced by Le Gall [9, 12]. A stopped path is a continuous function w : [0, ζ] → ℝ^{d}, where ζ = ζ_{(w)} is called the lifetime of the path. We shall denote by ŵ the end point w(ζ). Let W be the space of all stopped paths in ℝ^{d}. When equipped with the metric

d(w, w′) = |ζ_{(w)} − ζ_{(w′)}| + sup_{s≥0} |w(s ∧ ζ_{(w)}) − w′(s ∧ ζ_{(w′)})|,

the space W is a Polish space.

Let w ∈ W and a, b ≥ 0, such that a ≤ b ∧ ζ_{(w)}. There exists a unique probability measure on W denoted by Q_{a,b}^{w}(dw′) such that:

(i) ζ_{(w′)} = b, Q_{a,b}^{w}(dw′)-a.s. (ii) w′(t) = w(t) for every 0 ≤ t ≤ a, Q_{a,b}^{w}(dw′)-a.s. (iii) The law of (w′(t + a), 0 ≤ t ≤ b − a) under Q_{a,b}^{w}(dw′) is the law of Brownian motion in

ℝ^{d} started at w(a) and stopped at time b − a. We shall also consider Q_{a,b}^{w}(dw′) as a probability on the space C([0, b], ℝ^{d}). We set W_{x} = {w ∈ W; w(0) = x} for x ∈ ℝ^{d}. Let w ∈ W_{x}. We restate theorem 1.1 from [9]:

Theorem 1.1 (Le Gall). There exists a continuous strong Markov process with values in W_{x}, W = (W_{s}, s ≥ 0), whose law is characterized by the following two properties.

(i) The lifetime process ζ = (ζ_{s} = ζ_{(W_{s})}, s ≥ 0) is a reflecting Brownian motion in ℝ^{+}. (ii) Conditionally given (ζ_{s}, s ≥ 0), the process (W_{s}, s ≥ 0) is a time-inhomogeneous contin-

uous Markov process, whose transition kernel between times s and s′ ≥ s is

P_{s,s′}(w, dw′) = Q_{m(s,s′),ζ_{s′}}^{w} (dw′),

where m(s, s′) := inf_{r∈[s,s′]} ζ_{r}.

From now on we shall consider the canonical realization of the process W defined on the space C(ℝ^{+}, W_{x}). The law of W started at w is denoted by E_{w}. We will use the following consequence of (ii): outside a E_{w}-negligible set, for every s′ > s, one has W_{s}(t) = W_{s′}(t) for every t ∈ [0, m(s, s′)]. We shall write E_{w}^{∗} for the law of the process W killed when its lifetime reaches zero. The distribution of W under E_{w}^{∗} can be characterized as in theorem 1.1, except that its lifetime process is distributed as a linear Brownian motion killed at its first hitting time of {0}. The state space for (W, E_{w}^{∗}) is the space W_{x}^{∗} = W_{x} ∪ ∂, where ∂ is a cemetery point. The trivial path x such that ζ_{(x)} = 0, x(0) = x is clearly a regular point for the process (W, E_{w}). Following [2] chapter 3, we can consider the excursion measure, ℕ_{x}, outside {x}. The distribution of W under ℕ_{x} can be characterized as in theorem 1.1, except that now the lifetime process ζ is distributed according to Itô measure of positive excursions of linear Brownian motion. We normalize ℕ_{x} so that, for every ε > 0,

ℕ_{x} [sup_{s≥0} ζ_{s} > ε] = \frac{1}{2ε} .

The Brownian snake enjoys a scaling property: if λ > 0, the law of the process W_{s}^{(λ)}(t) = λ^{−1}W_{λ^{4}s}(λ^{2}t) under ℕ_{x} is λ^{−2}ℕ_{λ^{−1}x}.

We recall the strong Markov property for the snake under ℕ_{x} (see [12]). Let T be a stopping time of the natural filtration F^{W} of the process W. Assume T > 0 ℕ_{x}-a.e., and let F, H nonnegative measurable functionals on C(ℝ^{+}, W_{x}^{∗}) such that F is F_{T}^{W} measurable. Then if θ denotes the usual shift operator, we have

ℕ_{x} [T < ∞; F · H ∘ θ_{T}] = ℕ_{x} [T < ∞; F · E_{W_{T}}^{∗} [H]] .

Let σ = inf {s > 0; ζ_{s} = 0} denote the duration of the excursion of ζ under ℕ_{x}. The range R = R(W) of W is defined under ℕ_{x} by

R = {W_{s}(t); 0 ≤ t ≤ ζ_{s}, 0 ≤ s ≤ σ} = {Ŵ_{s}; 0 ≤ s ≤ σ} .

For every nonnegative measurable function F on W_{x}^{∗}, we have

ℕ_{x} [∫_{0}^{σ} F (W_{s}, ζ_{s})ds] = ∫_{0}^{∞} 𝔼_{x} [F (β_{[0,t]}, t)] dt,

where β_{[0,t]} is under ℙ_{x} the restriction to [0, t] of a Brownian motion in ℝ^{d} started at β_{0} = x. Now consider under ℕ_{x} the continuous version (l_{s}^{t}, t > 0, s ≥ 0) of the local time of ζ at level t and time s. We define a measure valued process Y on ℝ^{d} by setting for every t > 0, for every φ ∈ B_{b+}(ℝ^{d}),

(Y_{t}, φ) = ∫_{0}^{σ} dl_{s}^{t} φ(Ŵ_{s}).

We shall sometimes write Y_{t}(W) to recall that Y_{t} is a function of the Brownian snake. From the joint continuity of the local time and the continuity of the map s ↦ Ŵ_{s}, we get that ℕ_{x}-a.e., the process Y is continuous on (0, ∞) for the Prohorov distance on M_{f}. Let φ ∈ B_{b+}(ℝ^{d}). We define on ℝ^{+} × ℝ^{d} the function v(t, x) = ℕ_{x} [1 − exp −(Y_{t}, φ)], if t > 0, and v(0, x) = φ(x). We will write v(t) for the function v(t, ·). We recall that the function v is the unique nonnegative measurable solution of the integral functional equation

v(t) + 2 ∫_{0}^{t} ds P_{s} [v(t − s)^{2}] = J(t) t ≥ 0, (1)

where J(t, x) = P_{t}[φ](x), and (P_{t}, t ≥ 0) is the Brownian semi-group in ℝ^{d}. A few other remarks on the solution of (1) are presented in section 5.1 below.

1.2. Super-Brownian motion and ISE. Let us now recall the definition of super-Brownian motion and its connection with the Brownian snake. The second part of the next theorem is lemma 4.1 from [6]. Let ν ∈ M_{f}.

Theorem 1.2. There exists a continuous strong Markov process X = (X_{s}, s ≥ 0) defined on the canonical space C(ℝ^{+}, M_{f}), whose law is characterized by the two following properties under ℙ_{ν}^{X}.

(i) X_{0} = ν, ℙ_{ν}^{X}-a.s. (ii) For every φ ∈ B_{b+}(ℝ^{d}), t ≥ s > 0, we have

𝔼_{ν}^{X} [exp [−(X_{t}, φ)] | σ(X_{u}, 0 ≤ u ≤ s)] = exp [−(X_{s}, v(t − s))],

where the function v is the unique nonnegative solution of (1) with J(t) = P_{t}[φ].

Furthermore, for every integer m ≥ 1, t_{m} > ⋯ > t_{1} ≥ 0, φ_{1}, …, φ_{m} ∈ B_{b+}(ℝ^{d}), we have

𝔼_{ν}^{X} [exp [− ∑_{{i;t_{i}≤t}}(X_{t−t_{i}}, φ_{i})]] = exp [−(ν, v(t))], (2)

where v is the unique nonnegative solution to the integral equation (1) with right-hand side J(t) = ∑_{{i;t_{i}≤t}} P_{t−t_{i}}[φ_{i}].

Theorem 1.3 (Le Gall [9, 12]). Let ∑_{i∈I} δ_{W^{i}} be a Poisson measure on C(ℝ^{+}, W) with intensity ∫ ν(dx)ℕ_{x}[·], then the process Z defined by Z_{0} = ν and Z_{t} = ∑_{i∈I} Y_{t}(W^{i}) if t > 0, is distributed according to ℙ_{ν}^{X}.

We deduce from the normalization of ℕ_{x} that ℕ_{x} [Y_{t} ≠ 0] = 1/2t < ∞. This implies that for every t > 0, there is only a finite number of indices i ∈ I such that the process (Y_{s}(W^{i}), s ≥ t) is nonzero.

We now recall the connection between ISE and Brownian snake. There exists a unique collection (ℕ_{0}^{(r)}, r > 0) of probability measure on C(ℝ^{+}, W_{0}^{∗}) such that:

1. For every r > 0, ℕ_{0}^{(r)}[σ = r] = 1.

2. For every λ > 0, r > 0, F, nonnegative measurable functional on C(ℝ^{+}, W_{0}^{∗}),

ℕ_{0}^{(r)} [F (W^{(λ)})] = ℕ_{0}^{(λ^{−4}r)} [F (W)] .

3. For every nonnegative measurable functional F on C(ℝ^{+}, W_{0}^{∗}),

ℕ_{0}[F] = \frac{1}{\sqrt{2π}} ∫_{0}^{∞} dr r^{−3/2}ℕ_{0}^{(r)}[F]. (3)

The measurability of the mapping r ↦ ℕ_{0}^{(r)}[F] follows from the scaling property 2. Under ℕ_{0}^{(1)}, the distribution of W is characterized as in theorem 1.1, except that the lifetime process is distributed according to the normalized Itô measure. The law of the ISE is the law of the continuous tree associated to \sqrt{2}W, under ℕ_{0}^{(1)} (see corollary 4 in [10] and [1]). In particular the law of the support of ISE is the law of \sqrt{2}R under ℕ_{0}^{(1)}, where we set λA = \{x; λ^{−1}x ∈ A\}.

1.3. Hitting probabilities for the Brownian snake. We now recall a few results from [11]. Let w ∈ W ∪ C(ℝ^{+}, ℝ^{d}), we introduce the first hitting time of A ∈ B(ℝ^{d}):

τ_{A}(w) = inf {t ≥ 0; w(t) ∈ A} ,

with the usual convention inf ∅ = ∞. We omit w when there is no risk of confusion. Consider the Brownian snake W, and set

T_{(y,ε)} = inf {s ≥ 0; ∃t ∈ [0, ζ_{s}], W_{s}(t) ∈ B̄(y, ε)} ,

where B(y, ε) is the open ball in ℝ^{d} centered at y with radius ε > 0, and B̄(y, ε) its closure. We know from [12] that the function defined on ℝ^{d}\B̄(0, ε),

u_{ε}(y) := ℕ_{0} [T_{(y,ε)} < ∞] = ℕ_{0} [R ∩ B̄(y, ε) ≠ ∅] = ℕ_{−y} [R ∩ B̄(0, ε) ≠ ∅] ,

is the maximal nonnegative solution on ℝ^{d}\B̄(0, ε) of

Δu = 4u^{2}.

This result was first proved in a more general setting by Dynkin [7] in terms of superprocesses. The function u_{ε} is strictly positive on ℝ^{d}\B̄(0, ε). For every y_{0} ∈ ∂B(0, ε), we have

lim_{y∈B̄(0,ε)^{c};y→y_{0}} u_{ε}(y) = ∞.

Scaling and symmetry arguments show that for every y ∈ ℝ^{d}\B̄(0, ε),

u_{ε}(y) = ε^{−2}u_{1} (\frac{|y|}{ε}) , (4)

where the function u_{1}(r), r ∈ (1, ∞) is the maximal nonnegative solution on (1, ∞) of

u′′_{1}(r) + \frac{d − 1}{r}u′_{1}(r) = 4u_{1}^{2}(r).

It is easy to see that the function u_{1} is decreasing. In section 5.2 we give the asymptotic expansion of u_{1} at infinity.

We give the following result on the probability of the event {T_{(y,ε)} < ∞} (see lemma 2.1 of [11]). Assume x_{0} ∉ B̄(y, ε). Then ℕ_{x_{0}}-a.e. for every T ≥ 0, we have

(5)  E_{W_{T}}^{∗} [T_{(y,ε)} < ∞] = 2 ∫_{0}^{ζ_{T} ∧τ_{B(y,ε)}(W_{T})} dt u_{ε}(W_{T} (t) − y)) e^{[−2 ∫_{0}^{t} u_{ε}(W_{T} (s)−y)ds]}

= 1 − exp [−2 ∫_{0}^{ζ_{T} ∧τ_{B(y,ε)}(W_{T})} u_{ε}(W_{T} (s) − y)ds].

Let x_{0}, x ∈ ℝ^{d}. We will now describe the law of W_{T_{(x,ε)}} under ℕ_{x_{0}}[· | T_{(x,ε)} < ∞]. First of all we denote by β a Brownian motion in ℝ^{d} started at x_{0} under ℙ_{x_{0}}. Assume x_{0} ∉ B̄(x, ε). Corollary 2.3 from [11] ensures that there exists ℙ_{x_{0}}-a.s. a unique continuous process x^{ε} = (x_{t}^{ε}, 0 ≤ t ≤ τ^{ε}) taking values in ℝ^{d} such that for every η ∈ (0, |x − x_{0}| −ε), for every t ≤ τ_{η}^{ε} = inf {s ≥ 0; |x_{s}^{ε} − x| ≤ ε + η},

x_{t}^{ε} = β_{t} + ∫_{0}^{t} \frac{∇u_{ε}(x_{s}^{ε} − x)}{u_{ε}(x_{s}^{ε} − x)}ds,

furthermore, ℙ_{x_{0}}-a.s. τ^{ε} = lim_{η→0} τ_{η}^{ε} < ∞ and |x_{τ^{ε}}^{ε} − x| = ε. We also recall that thanks to Girsanov’s theorem, we have for every nonnegative measurable function F on C([0, t], ℝ^{d})

𝔼_{x_{0}} [τ^{ε} > t; F (x_{[0,t]}^{ε})]

= 𝔼_{x_{0}} [τ_{B(x,ε)}(β) > t; F (β_{[0,t]}) \frac{u_{ε}(β_{t} − x)}{u_{ε}(x_{0} − x)} exp [−2 ∫_{0}^{t} u_{ε}(β_{s} − x)ds]] ,

where x_{[0,t]}^{ε} and β_{[0,t]} are the restriction of x^{ε} and β to [0, t]. The law of x^{ε} under ℙ_{x_{0}} can be interpreted as a probability measure on W_{x_{0}}^{∗}. Consider the closed set

A = {w ∈ W_{x_{0}}^{∗}; τ_{B̄(x,ε)}(w) < ∞} .

It has been proved in [11] that its capacitary measure with respect to the Brownian snake with initial point x_{0} is exactly u_{ε}(x_{0} − x) times the law of x^{ε} under ℙ_{x_{0}}. It is not hard to check however that the normalized capacitary measure can be interpreted as the hitting distribution under ℕ_{x_{0}} (cf [13], this is proved in a way similar to the classical interpretation of the capacitary measure as a last exit distribution, see e.g. Port and Stone [17]). Thus we deduce that for every nonnegative measurable function F on W_{x_{0}}^{∗}, we have

ℕ_{x_{0}} [T_{(x,ε)} < ∞; F (W_{T_{(x,ε)}}, ζ_{T_{(x,ε)}})] = u_{ε}(x_{0} − x)𝔼_{x_{0}} [F (x^{ε}, τ^{ε})] .

Hence for every t ≥ 0, and every nonnegative measurable function F on C([0, t], ℝ^{d}), we have

(6) ℕ_{x_{0}} [T_{(x,ε)} < ∞; ζ_{T_{(x,ε)}} > t; F ((W_{T_{(x,ε)}}(s), s ∈ [0, t]))]

= 𝔼_{x_{0}} [τ_{B(x,ε)} > t; F (β_{[0,t]}) u_{ε}(β_{t} − x) exp [−2 ∫_{0}^{t} u_{ε}(β_{s} − x)ds]] .

Finally we shall use the following inequality, that can be derived from the Feynman-Kac formula (use the fact that u_{ε} solves Δu = 4u_{ε}u)

u_{ε}(x) ≥ 2𝔼_{0} [∫_{0}^{τ_{B(x,ε)}} dt u_{ε}(β_{t} − x)^{2} exp [−4 ∫_{0}^{t} u_{ε}(β_{s} − x)ds]] . (7)

There is in fact equality in (7) (see the remark on page 293 of [11]).

2. A property of the range of super-Brownian motion

For A ∈ B(ℝ^{d}), ε > 0, we set A^{ε} := {x ∈ ℝ^{d}; d(x, A) ≤ ε}, with d(x, A) = inf {|x − y|; y ∈ A}. We will write |A| for the Lebesgue measure of A. We also set

C_{0} = a_{0}2π^{d/2}Γ([d − 2]/2)^{−1},

where the constant a_{0} is defined in lemma 5.1 (see also the remark below the lemma). We set R_{t}(X) = Cl (⋃_{s≥t} supp X_{s}). Let φ_{d}(ε) = ε^{4−d} if d ≥ 5 and φ_{4}(ε) = log(1/ε) for ε > 0.

Theorem 2.1. Let ν ∈ M_{f}. For every Borel set A ⊂ ℝ^{d}, d ≥ 4, for every t > 0, ℙ_{ν}^{X}-a.s.

lim_{ε→0} φ_{d}(ε) |R_{t}(X)^{ε} ∩ A| = C_{0} ∫_{t}^{∞} ds (X_{s}, 1_{A}). (8)

If there exists ρ < 4 such that lim_{ε→0} ε^{ρ−d} |(supp ν)^{ε}| = 0 then (8) holds with t = 0.

Let K a compact subset of ℝ^{d}. We consider the measure ϕ(K) defined by ϕ(K)(A) = |K ∩ A|. Since the set R_{t}(X) is compact for t > 0, the theorem implies that a.s. the sequence of measures (φ_{d}(ε)ϕ(R_{t}(X)^{ε}), ε > 0) converges weakly to C_{0} ∫_{t}^{∞} ds (X_{s}, 1_{A}).

Let us recall the main theorem of [19] (see also [16]).

Theorem 2.2 (Tribe). Let A a bounded Borel set in ℝ^{d}, d ≥ 3. Fix t > 0 and ν ∈ M_{f}. Then there exists a positive constant α_{0} depending only on d such that

lim_{ε→0} ε^{2−d} |(supp X_{t})^{ε} ∩ A| = α_{0}(X_{t}, 1_{A}),

where the convergence holds ℙ_{ν}^{X}-a.s. and in L^{2}(ℙ_{ν}^{X}).

We shall deduce theorem 2.1 from the next proposition on the range of the Brownian snake, whose proof will be given in the next section. For θ ∈ (0, 1), we set h_{d,θ}(ε) = ε^{1−θ} if d ≥ 5 and h_{4,θ}(ε) = log(1/ε)^{−1/θ} for ε ∈ (0, 1). For short we will write h_{d} for h_{d,θ}.

Proposition 2.3. Let d ≥ 4. For every θ ∈ (0, 1/d) and every R_{0} > 0, there exists a constant κ = κ(θ) > 0 and ε_{0} > 0 such that for every ε ∈ (0, ε_{0}], for every x_{0} with |x_{0}| ≤ R_{0}, and every Borel set A ⊂ B̄(0, R_{0}), we have

|ℕ_{x_{0}} [φ_{d}(ε) |R(W)^{ε} ∩ A ∩ B̄(x_{0}, h_{d}(ε))^{c}| −C_{0} ∫_{0}^{∞} ds (Y_{s}, 1_{A})]| ≤ h_{d}(ε)^{κ/2},

and

ℕ_{x_{0}} [[φ_{d}(ε) |R(W)^{ε} ∩ A ∩ B̄(x_{0}, h_{d}(ε))^{c}| −C_{0} ∫_{0}^{∞} ds (Y_{s}, 1_{A})]^{2}] ≤ h_{d}(ε)^{κ}.

Remark. We have trivially B(x_{0}, ε) ⊂ R(W)^{ε}, ℕ_{x_{0}}-a.e. Since ℕ_{x_{0}} is an infinite measure, ℕ_{x_{0}} [|R(W)^{ε} ∩ B(x_{0}, δ)|] = ∞ for every ε, δ > 0. This is the reason why we consider A ∩ B̄(x_{0}, h_{d}(ε))^{c} rather than A in the previous proposition.

We first give a consequence of this proposition.

Corollary 2.4. Let d ≥ 4. For every Borel set A ⊂ ℝ^{d}, ℕ_{x_{0}}-a.e., we have

lim_{ε→0} φ_{d}(ε) |R(W)^{ε} ∩ A| = C_{0} ∫_{0}^{∞} ds (Y_{s}, 1_{A}).

The results holds ℕ_{0}^{(1)}-a.s. if |∂A| = 0.

Proof of corollary 2.4. Since ℕ_{x_{0}}-a.e. the range R(W) is bounded, we only need to consider a bounded Borel set A. Let κ > 0 be fixed as in proposition 2.3. Let ε_{n} such that h_{d}(ε_{n}) = n^{−2/κ} for n ≥ 1. Using the Borel-Cantelli lemma and the second upper bound of proposition 2.3, we get that the sequence (φ_{d}(ε_{n}) |R(W)^{ε_{n}} ∩ A|, n ≥ 1) converges ℕ_{x_{0}}-a.e. to C_{0} ∫_{0}^{∞} ds (Y_{s}, 1_{A}). But for ε′ ≤ ε, since R(W)^{ε′} ⊂ R(W)^{ε}, we have

φ_{d}(ε′) |R(W)^{ε′} ∩ A| ≤ φ_{d}(ε) |R(W)^{ε} ∩ A| φ_{d}(ε′)/φ_{d}(ε).

A monotonicity argument using the fact that φ_{d}(ε_{n+1})/φ_{d}(ε_{n}) converges to 1, completes the proof of the first part.

The above result implies that ℕ_{0}-a.e. the sequence of measures (φ_{d}(ε)ϕ(R(W)^{ε}), ε > 0) converges weakly to C_{0} ∫_{0}^{∞} ds Y_{s}. Using (3) we see this convergence also holds dr-a.e. ℕ_{0}^{(r)}- a.s. By the scaling property the Brownian snake and the family (ℕ_{0}^{(r)}, r > 0), we get this convergence holds ℕ_{0}^{(1)}-a.s. Thus we have for every Borel set A ⊂ ℝ^{d}, ℕ_{0}^{(1)}-a.s.

C_{0} ∫_{0}^{∞} ds (Y_{s}, 1_{Int(A)}) ≤ lim inf_{ε→0} φ_{d}(ε) |R(W)^{ε} ∩ A|

≤ lim sup_{ε→0} φ_{d}(ε) |R(W)^{ε} ∩ A| ≤ C_{0} ∫_{0}^{∞} ds (Y_{s}, 1Ā),

where Int(A) denotes the interior of A. To prove the second part of the corollary we just need to check that if |∂A| = 0 then ∫_{0}^{∞} ds (Y_{s}, 1_{Int(A)}) = ∫_{0}^{∞} ds (Y_{s}, 1Ā). It is enough to prove that |A| = 0 implies ∫_{0}^{∞} ds (Y_{s}, 1_{A}) = 0 ℕ_{0}^{(1)}-a.s. Conditioning by the lifetime process, we get

ℕ_{0}^{(1)} [∫_{0}^{∞} ds (Y_{s}, 1_{A})] = ℕ_{0}^{(1)} [∫_{0}^{1} dt 1_{A}(Ŵ_{t})] = ∫_{0}^{1} dt ℕ_{0}^{(1)} [P_{ζ_{t}}[1_{A}](0)] .

This is equal to zero if |A| = 0. This ends the proof of the second part of the corollary. □

As a byproduct of the proof we get that ℕ_{x_{0}}-a.e. and ℕ_{0}^{(1)}-a.s. the sequence of measures (φ_{d}(ε)ϕ(R(W)^{ε}), ε > 0) converges weakly to C_{0} ∫_{0}^{∞} ds Y_{s}.

We first state some straightforward consequences of (4) and lemma 5.1. We say that ε_{0} > 0 satisfies the condition (C) if ε_{0}^{−θ} ≥ 4/3 if d ≥ 5 or log(1/ε_{0}) ≥ 4 log(2/θ)/θ if d = 4. For θ ∈ (0, 1/4) this implies that for ε ∈ (0, ε_{0}), h_{4}(ε)/ε ≥ 4/3 and that

(9) log(log(1/ε))/[θ log(1/ε)] ≤ 1/2.

For d ≥ 4, θ ∈ (0, 1/d), there exists a constant b_{1} such that for every ε satisfying (C), x ∉ B(0, h_{d}(ε)) we have

(10)  u_{ε}(x) ≤ b_{0}φ_{d}(ε)^{−1} |x|^{2−d},
(11)  u_{ε}(x) ≤ φ_{d}(ε)^{−1} |x|^{2−d} [a_{0} + b_{1}h_{d}(ε)^{θ/2}] .

For |x| > ε, we have

u_{ε}(x) ≥ a_{0}φ_{d}(ε)^{−1} |x|^{2−d} if d ≥ 5, (12)

(13) u_{ε}(x) ≥ a_{0}φ_{4}(ε)^{−1} |x|^{−2} [1 + log(2 |x|)/ log(1/ε)]^{−1} if d = 4.

We will also often use the following inequality for ε satisfying (C): φ_{d}(ε)h_{d}(ε)^{d} ≤ h_{d}(ε)^{3}. Proof of theorem 2.1. Recall that for every t > 0, ℙ_{ν}^{X} a.s. the set R_{t}(X) is bounded. Thus we only need to consider a bounded Borel set A. Thanks to the Markov property of X at time t and theorem 2.2 it is clearly enough to prove the second part of theorem 2.1. Let ν ∈ M_{f} and ρ < 4 such that lim_{ε→0} ε^{ρ−d} |(supp ν)^{ε}| = 0. For short we write a.s. for ℙ_{ν}^{X}-a.s.

First step. Recall we can write for every t > 0, X_{t} = ∑_{i∈I} Y_{t}(W^{i}), where ∑_{i∈I} δ_{W^{i}} is a Poisson measure on C(ℝ^{+}, W) with intensity measure ∫ ν(dx)ℕ_{x}[·]. We let x_{0}^{i} denote the starting point of the Brownian snake W^{i} (i.e. x_{0}^{i} = W_{0}^{i}(0)). Notice that a.s. for every i ∈ I, x_{0}^{i} ∈ supp ν, which is bounded thanks to the hypothesis on supp ν. Fix θ ∈ (0, 1/d) such that d − ρ ≥ (d − 4)/(1 − θ) (and θ < 4 − ρ if d = 4). Fix R_{0} such that supp ν ⊂ B(0, R_{0}). Let κ and ε_{0} < 1 fixed as in proposition 2.3. We notice that for every bounded Borel set A ⊂ B(0, R_{0}),

φ_{d}(ε) |R_{0}(X)^{ε} ∩ A| ≤ ∑_{i∈I} V_{ε}(W^{i}) + φ_{d}(ε) |A ∩ (supp ν)^{h_{d}(ε)}|,

where

V_{ε}(W^{i}) = φ_{d}(ε) |R(W^{i})^{ε} ∩ A ∩ B̄(x_{0}^{i}, h_{d}(ε))^{c}| .

We set V_{0}(W^{i}) = C_{0} ∫_{0}^{∞} ds (Y_{s}(W^{i}), 1_{A}). We use the second moment formula for a Poisson measure to get:

𝔼_{ν}^{X} [[∑_{i∈I} V_{ε}(W^{i}) − ∑_{i∈I} V_{0}(W^{i})]^{2}] = ∫ ν(dx)ℕ_{x} [[V_{ε}(W) − V_{0}(W)]^{2}]

+ [∫ ν(dx)ℕ_{x} [V_{ε}(W) − V_{0}(W)]]^{2} .

We deduce from proposition 2.3 that for every ε ∈ (0, ε_{0}],

𝔼_{ν}^{X} [[∑_{i∈I} V_{ε}(W^{i}) − ∑_{i∈I} V_{0}(W^{i})]^{2}] ≤ [(ν, 1) + (ν, 1)^{2}]h_{d}(ε)^{κ}.

Notice the hypothesis on supp ν and θ imply that lim_{ε→0} φ_{d}(ε) |(supp ν)^{h_{d}(ε)}| = 0. Arguments similar to those used in the first part of the proof of corollary 2.4 show then a.s.

lim_{ε→0} ∑_{i∈I} V_{ε}(W^{i}) = ∑_{i∈I} V_{0}(W^{i}).

Notice we have ∑_{i∈I} V_{0}(W^{i}) = C_{0} ∫_{0}^{∞} ds (X_{s}, 1_{A}). Using the above remark on supp ν, we deduce that a.s.

lim sup_{ε→0} φ_{d}(ε) |R_{0}(X)^{ε} ∩ A| ≤ C_{0} ∫_{0}^{∞} ds (X_{s}, 1_{A}).

Second step. To get a lower bound, consider an increasing sequence (E_{p}, p ≥ 1) of measurable subsets of E = C(ℝ^{+}, W) such that ⋃_{p≥1} E_{p} = E and ∫ ν(dx)ℕ_{x}[E_{p}] = α_{p} < ∞. (For instance we can take E_{p} = {W; sup_{s≥0} ζ_{s} ≥ 1/p}.) Then a.s. the set I_{p} = {i ∈ I; W^{i} ∈ E_{p}} is finite. We have

φ_{d}(ε) |R_{0}(X)^{ε} ∩ A| ≥ ∑_{i∈I_{p}} V_{ε}(W^{i}) − ∑_{(i,j)∈I_{p}^{2}; i≠j} U_{ε}(W^{i}, W^{j}),

where

U_{ε}(W^{i}, W^{j}) = φ_{d}(ε) |R(W^{i})^{ε} ∩ R(W^{j})^{ε} ∩ A ∩ B̄(x_{0}^{i}, h_{d}(ε))^{c} ∩ B̄(x_{0}^{j}, h_{d}(ε))^{c}|
= φ_{d}(ε) ∫_{A∩B̄(x_{0}^{i},h_{d}(ε))^{c}∩B̄(x_{0}^{j},h_{d}(ε))^{c}} dy 1_{{T_{(y,ε)}(W^{i})<∞}}1_{{T_{(y,ε)}(W^{j})<∞}}.

Arguments similar to those of the first step show that a.s.

lim_{ε→0} ∑_{i∈I_{p}} V_{ε}(W^{i}) = ∑_{i∈I_{p}} V_{0}(W^{i}) = ∑_{i∈I_{p}} C_{0} ∫_{0}^{∞} ds (Y_{s}(W^{i}), 1_{A}).

Now conditionally on the cardinal of I_{p}, the Brownian snakes (W^{i}, i ∈ I_{p}) are independent and have the same law: μ_{p} = α_{p}^{−1} ∫ ν(dx)ℕ_{x}[· ∩ E_{p}]. For two independent Brownian snakes (W, W′) under μ_{p} ⊗ μ_{p}, we get using (10), that for ε satisfying (C),

μ_{p} ⊗ μ_{p}[U_{ε}(W, W′)] ≤ α_{p}^{−2} ∫∫ ν(dx_{0})ν(dx′_{0})ℕ_{x_{0}} ⊗ ℕ_{x′_{0}}[U_{ε}(W, W′)]

≤ φ_{d}(ε)α_{p}^{−2} ∫∫ ν(dx_{0})ν(dx′_{0}) ∫_{A∩B̄(x_{0},h_{d}(ε))^{c}∩B̄(x′_{0},h_{d}(ε))^{c}} dy

[b_{0}φ_{d}(ε)^{−1} |y − x_{0}|^{2−d}] [b_{0}φ_{d}(ε)^{−1} |y − x′_{0}|^{2−d}]

≤ φ_{d}(ε)^{−1}α_{p}^{−2}(ν, 1)^{2}b_{0}^{2} sup_{x_{0}∈ℝ^{d}} ∫_{B̄(0,R_{0})\B̄(x_{0},h_{d}(ε))} dy |y − x_{0}|^{4−2d}

≤ cφ_{d}(ε)^{−1}h_{d}(ε)^{4−d} if d ≥ 5

≤ cφ_{4}(ε)^{−1} log(log(1/ε)) if d = 4

≤ ch_{d}(ε)^{θ/2} if d ≥ 4,

where the constant c is independent of ε and A. Using the Borel-Cantelli lemma for the sequence (h_{d}(ε_{n}) = n^{−4/θ}, n ≥ 1), and a monotonicity argument, we get that μ_{p} ⊗ μ_{p}-a.s. lim_{ε→0} U_{ε}(W, W′) = 0. Then since the cardinal of I_{p} is a.s. finite, we get that for every integer p ≥ 1, a.s.,

lim_{ε→0} ∑_{(i,j)∈I_{p}^{2}; i≠j} U_{ε}(W^{i}, W^{j}) = 0.

We deduce that for every integer p ≥ 1, a.s.

lim inf_{ε→0} φ_{d}(ε) |R_{0}(X)^{ε} ∩ A| ≥ ∑_{i∈I_{p}} C_{0} ∫_{0}^{∞} ds (Y_{s}(W^{i}), 1_{A}).

We get the lower bound by letting p → ∞. This and the upper bound of the first step ends the proof of the theorem. □

3. Proof of proposition 2.3

We shall use many times in the sequel the fact that ∫_{0}^{∞} ds (Y_{s}, 1_{A}) = ∫_{0}^{σ} ds 1_{A}(Ŵ_{s}) ℕ_{x_{0}}- a.e. We assume d ≥ 4. We recall easy equalities, which can readily be deduced from the results of section 5.1. For every A ∈ B(ℝ^{d}), we have

(14)  ℕ_{x} [∫_{0}^{σ} ds 1_{A}(Ŵ_{s})] = ∫_{A} dy G(x, y),

where G is the Green kernel in ℝ^{d}: G(x, y) = 2^{−1}π^{−d/2}Γ([d − 2]/2) |x − y|^{2−d}, and

(15)  ℕ_{x} [[∫_{0}^{σ} ds 1_{A}(Ŵ_{s})]^{2}] = 4 ∫ dy G(x, y) [∫_{A} dz G(y, z)]^{2} .

We can also compute the first moment under E_{w}^{∗}. For every A ∈ B(ℝ^{d}), w ∈ W, we have with ζ = ζ_{(w)},

E_{w}^{∗} [∫_{0}^{σ} ds 1_{A}(Ŵ_{s})] = 2 ∫_{0}^{ζ} dt ℕ_{w(t)} [∫_{0}^{σ} ds 1_{A}(Ŵ_{s})] = 2 ∫_{0}^{ζ} dt ∫_{A} dy G(w(t), y). (16)

Thanks to the space invariance of the law of the Brownian snake, we shall only consider the case x_{0} = 0 and A ⊂ B̄(0, R_{0}), for R_{0} fixed. We fix θ ∈ (0, 1/d) and R_{0} > 1. Let ε′_{0} > 0 satisfying (C). We consider ε ∈ (0, ε′_{0}). In this section, we denote by c, c_{1}, c_{2}, … positive constants whose values depend only on d, θ and R_{0}. The value of c may vary from line to line. For short we shall write A_{ε} = A ∩ B̄(0, h_{d}(ε))^{c} (not to be confused with A^{ε}) and R for R(W).

We first consider the case d ≥ 5. Notice that

ℕ_{0} [|R^{ε} ∩ A_{ε}|] = ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞] = ∫_{A_{ε}} dx u_{ε}(x).

Thus we deduce from (12) and (11), that for ε ∈ (0, ε′_{0}),

a_{0}ε^{d−4} ∫_{A} dx |x|^{2−d} −a_{0}ε^{d−4} ∫_{B(0,ε^{1−θ})} dx |x|^{2−d}

≤ ℕ_{0} [|R^{ε} ∩ A_{ε}|] ≤ ε^{d−4}[a_{0} + b_{1}h_{d}(ε)^{θ/2}] ∫_{A} dx |x|^{2−d} .

Therefore using also (14), we have

|ℕ_{x_{0}} [ε^{4−d} |R(W)^{ε} ∩ A_{ε}| −C_{0} ∫_{0}^{∞} ds (Y_{s}, 1_{A})]| ≤ ch_{d}(ε)^{θ/2}.

Thus we get the first bound of proposition 2.3 (take κ < θ and ε_{0} small enough). The proof is similar for d = 4 (use (29) instead of (28) and the fact that |x| is bounded by R_{0}).

Now we will prove the second bound. To this end we have to find an upper bound on I = ℕ_{0} [|R^{ε} ∩ A_{ε}|^{2}] and a lower bound on J = ℕ_{0} [|R^{ε} ∩ A_{ε}| ∫_{0}^{σ} ds 1_{A}(Ŵ_{s})].

3.1. An upper bound on I. The term I can also be written

I = ∫∫ _{A_{ε}×A_{ε}} dx dy ℕ_{0} [T_{(x,ε)} < ∞; T_{(y,ε)} < ∞] .

Consider the above integral as the sum of the integral over |x − y| ≤ 2h_{d}(ε) (denoted by I_{1}) and the one over |x − y| > 2h_{d}(ε) (denoted by I_{2}). Using (10) we easily obtain an upper bound on I_{1}:

I_{1} ≤ |B(0, 2h_{d}(ε))| ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞]
≤ ch_{d}(ε)^{d} ∫_{A_{ε}} dx φ_{d}(ε)^{−1}b_{0} |x|^{2−d} ≤ c_{1}φ_{d}(ε)^{−2}h_{d}(ε)^{3}.

Notice the event {T_{(x,ε)} < ∞; T_{(y,ε)} < ∞} is a subset of

{T_{(x,ε)} < ∞; T_{(y,ε)} ∘ θ_{T_{(x,ε)}} < ∞} ∪ {T_{(y,ε)} < ∞; T_{(x,ε)} ∘ θ_{T_{(y,ε)}} < ∞} ,

where θ_{t} is the usual shift operator. By symmetry, we get

I_{2} ≤ 2 ∫∫ _{A_{ε}×A_{ε}} dx dy 1_{{|x−y|>2h_{d}(ε)}}ℕ_{0} [T_{(x,ε)} < ∞; T_{(y,ε)} ∘ θ_{T_{(x,ε)}} < ∞] . (17)

Using the strong Markov property of the Brownian snake under ℕ_{0} at the stopping time T_{(x,ε)}

and (5), we see that the quantity ℕ_{0} [T_{(x,ε)} < ∞; T_{(y,ε)} ∘ θ_{T_{(x,ε)}} < ∞] is equal to

ℕ_{0} [T_{(x,ε)} < ∞; 2 ∫_{0}^{ζ_{T_{(x,ε)}}∧τ_{B(y,ε)}(W_{T (x,ε)})} dt u_{ε} (W_{T_{(x,ε)}}(t) − y) e^{[−2 ∫_{0}^{t} u_{ε}(W_{T_{(x,ε)}}(s)−y)ds]}] .

Finally the law of the stopped path W_{T_{(x,ε)}} under ℕ_{0} is given by (6). Thus the previous expression is equal to

2 ∫_{0}^{∞}dt 𝔼_{0} [τ_{B(x,ε)} > t; τ_{B(y,ε)} > t; u_{ε}(β_{t} − x)u_{ε}(β_{t} − y) e^{[−2 ∫_{0}^{t} ds [u_{ε}(β_{s}−x)+u_{ε}(β_{s}−y)]]}] .

We substitute this last expression for ℕ_{0} [T_{(x,ε)} < ∞; T_{(y,ε)} ∘ θ_{T_{(x,ε)}} < ∞] in (17), and then decompose the right-hand side of (17) in three terms by considering the integral in dxdy on the sets |β_{t} − x| ∧ |β_{t} − y| > h_{d}(ε) (integral I_{21}), |β_{t} − x| ≤ h_{d}(ε) (integral I_{22}), and |β_{t} − y| ≤ h_{d}(ε) (integral I_{23}) (recall |x − y| > 2h_{d}(ε)).

An upper bound on I_{21}. We shall need the following notation:

I_{0} = 4a_{0}^{2} ∫ dz G(0, z) [∫_{A} dx |z − x|^{2−d}]^{2} .

We use (11) to bound I_{21} above by: for ε ∈ (0, ε′_{0}),

4 ∫∫ _{A_{ε}×A_{ε}} dx dy 1_{{|x−y|>2h_{d}(ε)}} ∫_{0}^{∞} dt 𝔼_{0}[ |β_{t} − x| > h_{d}(ε); |β_{t} − y| > h_{d}(ε);
φ_{d}(ε)^{−2} |β_{t} − x|^{2−d} |β_{t} − y|^{2−d} (a_{0} + b_{1}h_{d}(ε)^{θ/2})^{2} ]
≤ 4φ_{d}(ε)^{−2} [a_{0}^{2} + ch_{d}(ε)^{θ/2}] ∫∫ _{A×A} dx dy ∫ dz G(0, z) |z − x|^{2−d} |z − y|^{2−d}
≤ φ_{d}(ε)^{−2}I_{0} + c_{2}φ_{d}(ε)^{−2}h_{d}(ε)^{θ/2}.

An upper bound on I_{22} and I_{23}. By symmetry we have I_{22} = I_{23}. Before getting an upper bound on I_{22}, notice that |β_{t} − x| ≤ h_{d}(ε) and |x − y| > 2h_{d}(ε) imply |β_{t} − y| > h_{d}(ε). Furthermore thanks to (10), we get

∫_{A_{ε}} dy 1_{{|β_{t}−y|>h_{d}(ε)}}u_{ε}(β_{t} − y) e^{−2 ∫_{0}^{t} u_{ε}(β_{s}−y)ds} ≤ ∫_{A} dy [b_{0}φ_{d}(ε)^{−1} |β_{t} − y|^{2−d}]
≤ b_{0}φ_{d}(ε)^{−1} ∫_{B(0,R_{0})} dy |y|^{2−d} = c_{3}φ_{d}(ε)^{−1}.

Thus the sum I_{22} + I_{23} is bounded above by

8c_{3}φ_{d}(ε)^{−1} ∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0} [τ_{B(x,ε)} > t; 1_{{|β_{t}−x|≤h_{d}(ε)}}u_{ε}(β_{t} − x) e^{−2 ∫_{0}^{t} u_{ε}(β_{s}−x)ds}] .

Using the Cauchy-Schwarz inequality and formula (7), we get

I_{22} + I_{23} ≤ 8c_{3}φ_{d}(ε)^{−1} [∫_{A_{ε}} dx ∫_{0}^{∞} dt ℙ_{0} [|β_{t} − x| ≤ h_{d}(ε)]]^{1/2}

× [∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0} [τ_{B(x,ε)} > t; u_{ε}(β_{t} − x)^{2} e^{−4 ∫_{0}^{t} u_{ε}(β_{s}−x)ds}]]^{1/2}

≤ 8c_{3}φ_{d}(ε)^{−1} [∫_{A} dx ∫ dz G(0, z)1_{{|z−x|≤h_{d}(ε)}}]^{1/2} [∫_{A_{ε}} dx 2^{−1}u_{ε}(x)]^{1/2} .

Then thanks to (10), we get I_{22} + I_{23} ≤ c_{4}φ_{d}(ε)^{−3/2}h_{d}(ε)^{d/2} ≤ c_{4}φ_{d}(ε)^{−2}h_{d}(ε)^{3/2}.

Conclusion on the upper bound on I. By combining the previous results, we get for d ≥ 4

I ≤ c_{1}φ_{d}(ε)^{−2}h_{d}(ε)^{3} + φ_{d}(ε)^{−2}I_{0} + c_{2}φ_{d}(ε)^{−2}h_{d}(ε)^{θ/2} + c_{4}φ_{d}(ε)^{−2}h_{d}(ε)^{3/2}.

Thus we get φ_{d}(ε)^{2}I ≤ I_{0} + c_{5}h_{d}(ε)^{θ/2}.

3.2. A lower bound on J. We shall need the last hitting time of B̄(x, ε) under ℕ_{0} for the Brownian snake:

L_{(x,ε)} = sup {s ≥ 0; ∃t ∈ [0, ζ_{s}], W_{s}(t) ∈ B̄(x, ε)} .

We then get

J = ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{0}^{L_{(x,ε)}} ds 1_{A}(Ŵ_{s})]

+ ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{T_{(x,ε)}}^{σ} ds 1_{A}(Ŵ_{s})]

− ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{T_{(x,ε)}}^{L_{(x,ε)}} ds 1_{A}(Ŵ_{s})] .

The time-reversal invariance property of the Itô measure and the characterization of the excursion measure ℕ_{x} readily imply that the latter itself enjoys the same invariance property. Thus the first two terms of the right-hand side are equal. We shall denote their sum by J_{1}. Let J_{2} denote the third term.

A lower bound on J_{1}. Let us use the strong Markov property of the Brownian snake at time T_{(x,ε)}, then (16) and (6), to get

J_{1} = 2 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; 2 ∫_{0}^{ζ_{T_{(x,ε)}}} dt ∫_{A} dy G (W_{T_{(x,ε)}}(t), y)]
= 4 ∫_{A_{ε}} dx ∫_{A} dy ∫_{0}^{∞} dt 𝔼_{0} [τ_{B(x,ε)} > t; G(β_{t}, y)u_{ε}(β_{t} − x) e^{−2 ∫_{0}^{t} u_{ε}(β_{s}−x)ds}] .

Fatou’s lemma gives that lim inf_{ε→0} φ_{d}(ε)J_{1} ≥ J_{0}, where

J_{0} = 4a_{0} ∫∫ _{A×A} dxdy ∫ dz G(0, z)G(z, y) |z − x|^{2−d} .

Unfortunately, we need an estimate on the rate of convergence. This requires some technical calculations. Notice that on {τ_{B(x,h_{d}(ε))}(β) > t}, inequalities (12), (13) and (10) imply

a_{0}φ_{d}(ε)^{−1}F_{d}(β_{t} − x) |β_{t} − x|^{2−d} ≤ u_{ε}(β_{t} − x) ≤ b_{0}φ_{d}(ε)^{−1} |β_{t} − x|^{2−d},

where F_{d}(z) = 1 if d ≥ 5 and F_{4}(z) = [1 + log(2 |z|)/ log(1/ε)]^{−1}. For short we write Γ_{t} = 2b_{0}φ_{d}(ε)^{−1} ∫_{0}^{t} |β_{s} − x|^{2−d} ds. Then φ_{d}(ε)J_{1} is bounded below by

J′_{1} = 4a_{0} ∫_{A_{ε}}dx ∫_{A}dy ∫_{0}^{∞}dt 𝔼_{0} [τ_{B(x,h_{d}(ε))} > t; G(β_{t}, y) |β_{t} − x|^{2−d} F_{d}(β_{t} − x) e^{−Γ_{t}}] .

In order to obtain an upper bound on |J′_{1} − J_{0}|, we have to find an upper bound on

∫∫ _{A×A} dxdy ∫_{0}^{∞} dt 𝔼_{0} [G(β_{t}, y) |β_{t} − x|^{2−d} [1 − 1_{A_{ε}}(x)1_{{τ_{B(x,h_{d}(ε))}>t}}F_{d}(β_{t} − x) e^{−Γ_{t}}]] .

Thus we shall decompose 1−1_{A_{ε}}(x)1{τ_{B(x,h_{d}(ε))} > t}F_{d}(β_{t} −x) e^{−Γ_{t}} into a sum of four terms:

[1 − 1_{A_{ε}}(x)] + 1_{A_{ε}}(x) [1 − 1_{{τ_{B(x,h_{d}(ε))}>t}}]

+ 1_{A_{ε}}(x)1_{{τ_{B(x,h_{d}(ε))}>t}} [1 − F_{d}(β_{t} − x)] + 1_{A_{ε}}(x)1_{{τ_{B(x,h_{d}(ε))}>t}}F_{d}(β_{t} − x) [1 − e^{−Γ_{t}}] .

We denote by J_{11}, J_{12}, J_{13} and J_{14} the corresponding integrals. The integral

J_{11} = ∫_{A\A_{ε}} dx ∫_{A} dy ∫_{0}^{∞} dt 𝔼_{0} [G(β_{t}, y) |β_{t} − x|^{2−d}]

is easily bounded above by

∫_{B(0,h_{d}(ε))} dx ∫_{B(0,R_{0})} dy ∫ dz G(0, z)G(z, y) |z − x|^{2−d} ≤ c_{6}h_{d}(ε)^{2}.

We bound J_{12} by applying the strong Markov property of Brownian motion at time τ_{B(x,h_{d}(ε))},

J_{12} = ∫_{A_{ε}} dx ∫_{A} dy ∫_{0}^{∞} dt 𝔼_{0} [τ_{B(x,h_{d}(ε))} ≤ t; G(β_{t}, y) |β_{t} − x|^{2−d}]
≤ ∫_{A_{ε}} dx ∫_{A} dy 𝔼_{0} [τ_{B(x,h_{d}(ε))} < ∞; ∫ dz G(β_{τ_{B(x,h_{d}(ε))}}, z)G(z, y) |z − x|^{2−d}] .

An easy calculation shows that there exists a constant c_{7} such that for every (x, x′) ∈ B(0, 2R_{0}) × B(0, 2R_{0}), |x − x′| ≤ 1/2,

∫_{B(0,R_{0})} dy ∫ dz G(x′, z)G(z, y) |z − x|^{2−d} ≤ c_{7}φ_{d}(|x′ − x|)

Furthermore we have for every r ∈ (0, 1),

∫_{B(0,R_{0})} dx ℙ_{0} [τ_{B(x,r)} < ∞] = ∫_{B(0,R_{0})} dx [(\frac{r}{|x|})^{d−2} ∧ 1] ≤ cr^{d−2}. (18)

We deduce from the previous remarks that if d ≥ 5,

J_{12} ≤ ch_{d}(ε)^{4−d} ∫_{A_{ε}} dx ℙ_{0} [τ_{B(x,h_{d}(ε))} < ∞] ≤ ch_{d}(ε)^{4−d+d−2} = ch_{d}(ε)^{2},

and if d = 4, J_{12} ≤ c log(1/h_{d}(ε))h_{d}(ε)^{2}. Thus we get that for d ≥ 4, J_{12} ≤ c_{8}h_{d}(ε)^{3/2}. If d ≥ 5 then J_{13} = 0. For d = 4 thanks to (9) we have for |z| ≥ h_{4}(ε), |1 − F_{4}(z)| ≤ 2 |log(2 |z|)| / log(1/ε). We deduce that

J_{13} ≤ log(1/ε)^{−1} ∫∫ _{A×A}dxdy ∫_{0}^{∞}dt 𝔼_{0} [τ_{B(x,h_{d}(ε))} > t; G(β_{t}, y)2 |log(2 |β_{t} − x|)| |β_{t} − x|^{−2}]
≤ c log(1/ε)^{−1} ∫∫ _{A×A} dxdy ∫ dz G(0, z) |log(2 |z − x|)| |z − x|^{−2} G(z, y)
≤ c log(1/ε)^{−1} ≤ c_{9}h_{d}(ε)^{θ}.

Notice first that thanks to (9), F_{d}(z) ≤ 2 for |z| ≥ h_{d}(ε). We have, using the Markov property for Brownian motion at time s,

J_{14} ≤ 2 ∫∫ _{A×A} dxdy ∫_{0}^{∞} dt

𝔼_{0}[τ_{B(x,h_{d}(ε))} > t; G(β_{t}, y) |β_{t} − x|^{2−d} 2b_{0}φ_{d}(ε)^{−1} ∫_{0}^{t} |β_{s} − x|^{2−d} ds]

≤ cφ_{d}(ε)^{−1} ∫∫ _{A×A} dxdy ∫_{0}^{∞} ds ∫_{0}^{∞} dt
𝔼_{0} [|β_{s} − x|^{2−d} 𝔼_{β_{s}} [|β_{t} − x| > h_{d}(ε); G(β_{t}, y) |β_{t} − x|^{2−d}]]
≤ cφ_{d}(ε)^{−1}M(d, h_{d}(ε)),

where

M(d, ε) = ∫∫ _{B(0,R_{0})^{2}} dxdy ∫∫ dzdz′ G(0, z) |z − x|^{2−d} G(z, z′)G(z′, y) |z′ − x|^{2−d} 1_{|z′−x|>ε}.

An easy computation shows there exists a constant c such that for ε ∈ (0, 1],

(19) M(d, ε) ≤ \{ \begin{matrix}c & if d ∈ \{4, 5\}, \\ c + c log(1/ε) & if d = 6, \\ cε^{6−d} & if d ≥ 7. \end{matrix}

Thus we easily deduce that J_{14} ≤ c_{10}h_{d}(ε)^{θ}.

We have ε^{4−d}J_{1} ≥ J_{0} − 4a_{0}(J_{11} + J_{12} + J_{13} + J_{14}). Putting together the previous results, we get for d ≥ 4,

ε^{4−d}J_{1} ≥ J_{0} − 4a_{0}[c_{6}h_{d}(ε)^{2} + c_{8}h_{d}(ε)^{3/2} + c_{9}h_{d}(ε)^{θ} + c_{10}h_{d}(ε)^{θ}] ≥ J_{0} − c_{11}h_{d}(ε)^{θ}.

An upper bound on J_{2}. We will first recall the decomposition of the Brownian snake under E_{w}^{∗} (see theorem 2.5 in [12]). We denote by (α_{i}, β_{i}), i ∈ I, the excursion intervals of ζ above its minimum process (i.e. of the process (ζ_{t} − inf_{s∈[0,t]} ζ_{s}) above 0) before σ under E_{w}^{∗}. For i ∈ I the paths W_{s}, s ∈ [α_{i}, β_{i}] coincide over [0, ζ_{α_{i}}]. For every i ∈ I, and s ≥ 0 we set W_{s}^{i}(t) = W_{(α_{i}+s)∧β_{i}}(t + ζ_{α_{i}}), t ∈ [0, ζ_{s}^{i}] with ζ_{s}^{i} = ζ_{(α_{i}+s)∧β_{i}} − ζ_{α_{i}}. Then W_{s}^{i} is a stopped path (W_{s}^{i} ∈ W) with initial point W_{(α_{i}+s)∧β_{i}}(ζ_{α_{i}}) = Ŵ_{α_{i}} = w(ζ_{α_{i}}).

Proposition 1 (Le Gall). The random measure ∑_{i∈I} δ_{(ζ_{α_{i}},W^{i})} is under E_{w}^{∗} a Poisson point measure on [0, ζ_{(w)}] × C(ℝ^{+}, W) with intensity 2dt ℕ_{w(t)}[·].

The process ∑_{i∈I,ζ_{α_{i}}≤t} δ_{W^{i}} for t ∈ [0, ζ_{w}] is a Poisson point process with inhomogeneous intensity. We will now describe the law under E_{W_{T_{(x,ε)}}}^{∗} of the first excursion (ζ_{α_{i_{0}}} , W^{i_{0}}) which

hits the ball B̄(x, ε), that is, with evident notation, the first excursion for which T_{(x,ε)}(W^{i}) is finite. We first notice that under ℕ_{0}[· | T_{(x,ε)} < ∞], E_{W_{T_{(x,ε)}}}^{∗} -a.s. there are such excursions.

Indeed we have thanks to lemma 2.1 of [11] that ℕ_{0}[. | T_{(x,ε)} < ∞]-a.s.

E_{W_{T_{(x,ε)}}}^{∗} [∃i ∈ I, T_{(x,ε)}(W^{i}) < ∞] = 1 − exp −2 ∫_{0}^{ζ_{T_{(x,ε)}}} dt u_{ε}(W_{T_{(x,ε)}}(t) − x) = 1.

Since the integral ∫_{0}^{r} dt u_{ε}(W_{T_{(x,ε)}}(t) − x) is finite for r < ζ_{T_{(x,ε)}}, we deduce there exists a unique first excursion i_{0} which hits B̄(x, ε). Classical arguments on Poisson point process implies that the law of (ζ_{α_{i_{0}}} , W^{i_{0}}) is 21_{[0,ζ_{T_{(x,ε)}})}(t)dt ℕ_{W_{T_{(x,ε)}}(t)}[T_{(x,ε)} < ∞, ·]. We introduce

the random time M_{(x,ε)} = inf {s > T_{(x,ε)}; ζ_{s} = m(T_{(x,ε)}, L_{(x,ε)})}. It is clear from the definition of the excursion i_{0} that α_{i_{0}} = M_{(x,ε)} under E_{W_{T_{(x,ε)}}}^{∗}. We will now express J_{2} using the

excursion i_{0}. We have

J_{2} = 2 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{M_{(x,ε)}}^{L_{(x,ε)}} ds 1_{A}(Ŵ_{s})]

= 2 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; E_{W_{T_{(x,ε)}}}^{∗} [∫_{M_{(x,ε)}}^{L_{(x,ε)}} ds 1_{A}(Ŵ_{s})]]

= 2 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; E_{W_{T_{(x,ε)}}}^{∗} [∫_{α_{i_{0}}}^{L_{(x,ε)}(W^{i_{0}})} ds 1_{A}(Ŵ_{s}^{i_{0}})]]

= 4 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{0}^{ζ_{T_{(x,ε)}}} dt ℕ_{W_{T_{(x,ε)}}(t)} [T_{(x,ε)} < ∞; ∫_{0}^{L_{(x,ε)}} ds 1_{A}(Ŵ_{s})]] .

We used the time reversal property of the Brownian snake for the first equality, then the strong Markov property and at last the definition of the excursion i_{0} and its law. We will distinguish according to {t ≥ τ_{B(x,h_{d}(ε))}} (integral J_{21}) and {t < τ_{B(x,h_{d}(ε))}} (integral J_{22}). Notice that since x ∈ A_{ε} we have τ_{B(x,h_{d}(ε))}(W_{T_{(x,ε)}}) < ζ_{T_{(x,ε)}} ℕ_{0}-a.e.

We now bound J_{21} using (14).

J_{21} = 4 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{τ_{B(x,h_{d}(ε))}}^{ζ_{T_{(x,ε)}}} dt ℕ_{W_{T_{(x,ε)}}(t)} [T_{(x,ε)} < ∞; ∫_{0}^{L_{(x,ε)}} ds 1_{A}(Ŵ_{s})]]

≤ 4 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{τ_{B(x,h_{d}(ε))}}^{ζ_{T_{(x,ε)}}} dt ℕ_{W_{T_{(x,ε)}}(t)} [∫_{0}^{σ} ds 1_{A}(Ŵ_{s})]]

= 4 ∫_{A_{ε}} dx ℕ_{0} [T_{(x,ε)} < ∞; ∫_{τ_{B(x,h_{d}(ε))}}^{ζ_{T_{(x,ε)}}} dt ∫_{A} dy G(W_{T_{(x,ε)}}(t), y)] .

Now we use (6), the Cauchy-Schwarz inequality and (7) to get

J_{21} ≤ 4 ∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0} [τ_{B(x,ε)} > t ≥ τ_{B(x,h_{d}(ε))}; ∫_{A} dy G(β_{t}, y)u_{ε}(β_{t} − x) e^{−2 ∫_{0}^{t} u_{ε}(β_{r}−x)dr}]

≤ 4 [2^{−1} ∫_{A_{ε}} dx u_{ε}(x)]^{1/2} [∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0} [t ≥ τ_{B(x,h_{d}(ε))}; (∫_{A} dy G(β_{t}, y))^{2}]]^{1/2}

≤ cφ_{d}(ε)^{−1/2} [∫_{A_{ε}} dx ℙ_{0} [τ_{B(x,h_{d}(ε))} < ∞] sup_{x′∈B(0,2R_{0})} ∫ dz G(z, x′) (∫_{A} dy G(z, y))^{2}]^{1/2}

≤ cφ_{d}(ε)^{−1/2}h_{d}(ε)^{(d−2)/2}.

We used the strong Markov property at time τ_{B(x,h_{d}(ε))} and (18) for the last two inequalities. This implies that J_{21} ≤ c_{12}φ_{d}(ε)^{−1}h_{d}(ε)^{1/2}.

Using the time reversal property of the Brownian snake, the strong Markov property at time T_{(x,ε)} and (16) we get

J_{22} = 4 ∫_{A_{ε}} dx ℕ_{0}[T_{(x,ε)} < ∞; ∫_{0}^{τ_{B(x,h_{d}(ε))}} dt ℕ_{W_{T_{(x,ε)}}(t)} [T_{(x,ε)} < ∞; ∫_{T_{(x,ε)}}^{σ} ds 1_{A}(Ŵ_{s})] ]

= 8 ∫_{A_{ε}} dx ℕ_{0}[T_{(x,ε)} < ∞; ∫_{0}^{τ_{B(x,h_{d}(ε))}} dt

ℕ_{W_{T_{(x,ε)}}(t)} [T_{(x,ε)} < ∞; ∫_{0}^{ζ_{T_{(x,ε)}}} ds ∫_{A} dy G(W_{T_{(x,ε)}}(s), y)] ].

We will distinguish according to {s ≥ τ_{B(x,h_{d}(ε))}} (integral J_{23}) and {s < τ_{B(x,h_{d}(ε))}} (integral J_{24}). We now bound J_{23}. We have

J_{23} = 8 ∫_{A_{ε}} dx ℕ_{0}[T_{(x,ε)} < ∞; ∫_{0}^{τ_{B(x,h_{d}(ε))}} dt
ℕ_{W_{T_{(x,ε)}}(t)} [T_{(x,ε)} < ∞; ∫_{τ_{B(x,h_{d}(ε))}}^{ζ_{T_{(x,ε)}}} ds ∫_{A} dy G(W_{T_{(x,ε)}}(s), y)] ]
= 8 ∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0}[τ_{B(x,h_{d}(ε))} > t; u_{ε}(β_{t} − x) e^{−2 ∫_{0}^{t} u_{ε}(β_{r}−x)dr} ∫_{0}^{∞} ds
𝔼_{β_{t}} [τ_{B(x,ε)} > s ≥ τ_{B(x,h_{d}(ε))}; ∫_{A} dy G(β_{s}, y)u_{ε}(β_{s} − x) e^{−2 ∫_{0}^{s} u_{ε}(β_{v}−x)dv}] ]
≤ cφ_{d}(ε)^{−1} ∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0}[τ_{B(x,h_{d}(ε))} > t; |β_{t} − x|^{2−d}
[∫_{0}^{∞} ds 𝔼_{β_{t}} [τ_{B(x,ε)} > s; u_{ε}(β_{s} − x)^{2} e^{−4 ∫_{0}^{s} u_{ε}(β_{v}−x)dv}]]^{1/2}
[∫_{0}^{∞} ds 𝔼_{β_{t}} [s ≥ τ_{B(x,h_{d}(ε))}; (∫_{A} dy G(β_{s}, y))^{2}]]^{1/2} ]
≤ cφ_{d}(ε)^{−1} ∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0}[τ_{B(x,h_{d}(ε))} > t; |β_{t} − x|^{2−d} [2^{−1}u_{ε}(β_{t} − x)]^{1/2}
[𝔼_{β_{t}} [τ_{B(x,h_{d}(ε))} < ∞; 𝔼_{β_{τ_{B(x,h_{d}(ε))}}} [∫_{0}^{∞} ds (∫_{A} dy G(β_{s}, y))^{2}]]]^{1/2} ]

≤ cφ_{d}(ε)^{−3/2} ∫_{A_{ε}} dx ∫_{0}^{∞} dt 𝔼_{0} [τ_{B(x,h_{d}(ε))} > t; |β_{t} − x|^{(6−3d)/2} ℙ_{β_{t}} [τ_{B(x,h_{d}(ε))} < ∞]^{1/2}]

[ sup_{x′∈B(0,2R_{0})} ∫ dz′G(x′, z′) (∫_{A} dy G(z′, y))^{2}]^{1/2}
≤ cφ_{d}(ε)^{−3/2} ∫_{A_{ε}} dx ∫_{|z−x|≥h_{d}(ε)} dz G(0, z) |z − x|^{(6−3d)/2} h_{d}(ε)^{(d−2)/2} |z − x|^{(2−d)/2} .

We used (6) twice for the second equality, (10) and Cauchy-Schwarz inequality for the first inequality, (7) and the strong Markov property at time τ_{B(x,h_{d}(ε))} for the second and (18) for the last. We easily deduce that J_{23} ≤ c_{13}φ_{d}(ε)^{−1}h_{d}(ε).

For J_{24} we have using (6) twice and (10) twice,

J_{24} = 8 ∫_{A_{ε}} dx ℕ_{0}[T_{(x,ε)} < ∞; ∫_{0}^{τ_{B(x,h_{d}(ε))}} dt

ℕ_{W_{T_{(x,ε)}}(t)} [T_{(x,ε)} < ∞; ∫_{0}^{τ_{B(x,h_{d}(ε))}} ds ∫_{A} dy G(W_{T_{(x,ε)}}(s), y)] ]

= 8 ∫_{A_{ε}} dx ∫_{0}^{∞} dt ∫_{0}^{∞} ds 𝔼_{0}[τ_{B(x,h_{d}(ε))} > t; u_{ε}(β_{t} − x) e^{−2 ∫_{0}^{t} u_{ε}(β_{r}−x)dr}

𝔼_{β_{t}} [τ_{B(x,h_{d}(ε))} > s; ∫_{A} dy G(β_{s}, y)u_{ε}(β_{s} − x) e^{−2 ∫_{0}^{s} u_{ε}(β_{v}−x)dv}] ]

≤ cφ_{d}(ε)^{−2} ∫_{A_{ε}} dx ∫_{0}^{∞} dt ∫_{0}^{∞} ds 𝔼_{0}[τ_{B(x,h_{d}(ε))} > t; |β_{t} − x|^{2−d}

𝔼_{β_{t}} [τ_{B(x,h_{d}(ε))} > s; ∫_{A} dy G(β_{s}, y) |β_{s} − x|^{2−d}] ]
≤ cφ_{d}(ε)^{−2}M(d, h_{d}(ε)).

Using (19) we get J_{24} ≤ c_{14}φ_{d}(ε)^{−1}h_{d}(ε)^{θ}. As a conclusion we get

J_{2} ≤ c_{12}φ_{d}(ε)^{−1}h_{d}(ε)^{1/2} + c_{13}φ_{d}(ε)^{−1}h_{d}(ε) + c_{14}φ_{d}(ε)^{−1}h_{d}(ε)^{θ}.

Conclusion on the lower bound on J.

By combining the previous results, we get for d ≥ 4,

φ_{d}(ε)J ≥ J_{0} − c_{11}h_{d}(ε)^{θ} − φ_{d}(ε)J_{2} ≥ J_{0} − c_{15}h_{d}(ε)^{θ}.

3.3. End of the proof of proposition 2.3. We deduce from formula (15), that

J_{0} = C_{0}ℕ_{0} [[∫_{0}^{σ} 1_{A}(Ŵ_{s})ds]^{2}] , and I_{0} = C_{0}^{2}ℕ_{0} [[∫_{0}^{σ} 1_{A}(Ŵ_{s})ds]^{2}] .

Thus we get from section 3.1 and 3.2 that for ε small enough

ℕ_{0} [[φ_{d}(ε) |R^{ε} ∩ A_{ε}| −C_{0} ∫_{0}^{σ} ds 1_{A}(Ŵ_{s})]^{2}] ≤ c_{5}h_{d}(ε)^{θ/2} + 2c_{15}h_{d}(ε)^{θ}.

Take κ < θ/2 and ε_{0} small to get the second upper bound of proposition 2.3. □

4. Capacity equivalence for the support and the range of X

Let f : (0, ∞) → [0, ∞) be a decreasing function. We put f(0) = lim_{r↓0} f(r) ∈ [0, ∞]. We define the energy of a Radon measure ν on ℝ^{d} with respect to the kernel f by: I_{f}(ν) = ∫∫ f(|x − y|)ν(dx)ν(dy), and the capacity of a set Λ ∈ B(ℝ^{d}) by

cap_{f}(Λ) = [ inf_{ν(Λ)=1} I_{f}(ν)]^{−1} .

Following [14], we say that two sets Λ_{1} and Λ_{2} are capacity-equivalent if there exist two positive constants c and C such that for every kernel f, we have

c cap_{f}(Λ_{1}) ≤ cap_{f}(Λ_{2}) ≤ C cap_{f}(Λ_{1}) .

The next lemma is an immediate consequence of the remarks in [15] p.385.

Lemma 4.1. Let Λ ⊂ ℝ^{d} be a bounded Borel set. Suppose there exist two positive constants c′ and γ such that

lim_{ε→0} ε^{γ−d} |Λ^{ε}| = c′.

Then there exists a constant C such that for every kernel f, we have

cap_{f}(Λ) ≤ C [∫_{0}^{1} f(r)r^{γ−1}dr]^{−1} .

For every measure μ ∈ M_{f}, we set

S_{ε}(μ) = ∫∫ μ(dx)μ(dy) p(ε^{2}, x − y),

where p is the Brownian transition density in ℝ^{d}: p(t, x) = (2πt)^{−d/2} e^{− |x|^{2} /2t}, (t, x) ∈ (0, ∞) × ℝ^{d}. The next lemma is also an immediate consequence of [15] (p.387).

Lemma 4.2. Let Λ ⊂ ℝ^{d} be a bounded Borel set. Suppose there exist two positive constants c′ and γ and a measure μ ∈ M_{f} such that μ(Λ^{c}) = 0 and

lim_{ε→0} ε^{d−γ}S_{ε}(μ) = c′.

Then there exists a constant c such that for every kernel f, we have

c [∫_{0}^{1} f(r)r^{γ−1}dr]^{−1} ≤ cap_{f}(Λ) .

For example, for every integer p ≤ d, we can consider the cube [0, 1]^{p} as a subset of ℝ^{d}, and then we obviously have

lim_{ε→0} ε^{p−d} |([0, 1]^{p})^{ε}| = 2π^{(d−p)/2}/Γ((d − p)/2),

and if μ is Lebesgue measure on [0, 1]^{p},

lim_{ε→0} ε^{d−p}S_{ε}(μ) = (2π)^{(p−d)/2}.

Thus we deduce from lemma 4.1 and 4.2 that there exist two positive constants c′_{p}, C′_{p}, such that for every kernel f,

c′_{p} [∫_{0}^{1} f(r)r^{p−1}dr]^{−1} ≤ cap_{f}([0, 1]^{p}) ≤ C′_{p} [∫_{0}^{1} f(r)r^{p−1}dr]^{−1} . (20)

We shall prove the following result on super-Brownian motion and ISE.

Proposition 4.3. (i) Assume d ≥ 3. Let t > 0, ν ∈ M_{f}. ℙ_{ν}^{X}-a.s. on {X_{t} ≠ 0}, the set supp X_{t} is capacity-equivalent to [0, 1]^{2}.

(ii) Assume d ≥ 5. Let t > 0, ν ∈ M_{f}. ℙ_{ν}^{X}-a.s. on {X_{t} ≠ 0}, the set R_{t}(X) is capacityequivalent to [0, 1]^{4}. Furthermore, if there exists a positive number ρ < 4 such that lim_{ε→0} ε^{ρ−d} |(supp ν)^{ε}| = 0, then ℙ_{ν}^{X}-a.s. the set R_{0}(X) is capacity-equivalent to [0, 1]^{4}.

(iii) Assume d ≥ 5. The set R_{t}(W) is capacity-equivalent to [0, 1]^{4} ℕ_{0}^{(1)}-a.s.

Proof of proposition 4.3 (i). Let d ≥ 3. It is well-known that for t > 0, ℙ_{ν}^{X}-a.s. the set supp X_{t} is bounded. Thus, thanks to theorem 2.2, ℙ_{ν}^{X}-a.s., we have

lim_{ε→0} ε^{2−d} |(supp X_{t})^{ε}| = α_{0}(X_{t}, 1).

Now apply lemma 4.1 to Λ = supp X_{t}, with γ = 2 and take p = 2 in (20). We get that ℙ_{ν}^{X}-a.s., on {X_{t} ≠ 0}, there exists a (random) constant C_{1} > 0, such that for every kernel f,

cap_{f}(supp X_{t}) ≤ C_{1} cap_{f}([0, 1]^{2}) .

For the second part of (i), we use lemma 4.4 below. Recall notation Y_{t} from section 1.1.

Lemma 4.4. Fix t > 0 and x ∈ ℝ^{d}, d ≥ 3. Then we have

lim_{ε→0} ε^{d−2}(2π)^{d/2}S_{ε}(Y_{t}) = \frac{4}{d − 2}(Y_{t}, 1),

where the convergence holds ℕ_{x}-a.e. and in L^{2}(ℕ_{x}).

Let us explain how the proof is completed using lemma 4.4. Thanks to lemma 4.2, the above lemma and (20) imply that ℕ_{x}-a.e. on {Y_{t} ≠ 0}, there exists a positive constant c_{1} such that for every kernel f,

cap_{f}(supp Y_{t}) ≥ c_{1} cap_{f}([0, 1]^{2}) .

Now remember that for t > 0, under ℙ_{ν}^{X}, we can write X_{t} = ∑_{i∈I} Y_{t}(W^{i}), where ∑_{i∈I} δ_{W^{i}} is a Poisson measure on C(ℝ^{+}, W) with intensity ∫ ν(dx) ℕ_{x}[·]. On {X_{t} ≠ 0}, there exists i_{0} such that Y_{t}(W^{i_{0}}) ≠ 0. Then we have supp Y_{t}(W^{i_{0}}) ⊂ supp X_{t}. Thus the previous lemma entails that there exists a.s. a positive constant c_{1}(W^{i_{0}}) such that for every kernel f,

cap_{f}(supp X_{t}) ≥ cap_{f}(supp Y_{t}(W^{i_{0}})) ≥ c_{1}(W^{i_{0}}) cap_{f}([0, 1]^{2})

This completes the proof of (i). □

Proof of proposition 4.3 (ii). Let d ≥ 5. We argue as in the proof of (i) using theorem 2.1 instead of theorem 2.2 and the following lemma instead of lemma 4.4.

Lemma 4.5. Fix t ≥ 0 and x ∈ ℝ^{d}, d ≥ 5. Then we have for every T > t ≥ 0,

lim_{ε→0} ε^{d−4}(2π)^{d/2}S_{ε} (∫_{t}^{T} ds Y_{s}) = \frac{16}{(d − 2)(d − 4)} ∫_{t}^{T} ds (Y_{s}, 1),

where the convergence holds ℕ_{x}-a.e. and in L^{2}(ℕ_{x}).

□

Proof of proposition 4.3 (iii). Let d ≥ 5. For the first part we argue as in the proof of (i) using the second part of corollary 2.4 instead of theorem 2.2. Notice that thanks to (3) and the scaling property of the family (ℕ_{0}^{(r)}, r > 0), the convergence in lemma 4.5 also holds ℕ_{0}^{(1)}-a.s. The second part of (iii) is then a direct consequence of lemma 4.2 (with μ = ∫_{0}^{T} ds Y_{s} and γ = 4) and (20) (with p = 4). □

The proofs of lemma 4.4 and lemma 4.5 are very similar. We shall only prove the latter. The former uses the same techniques in a simpler way. Proof of lemma 4.5. We first want to show the convergence in L^{2}(ℕ_{x}). Fix T > t ≥ 0. By standard monotone class arguments, we deduce from the results of section 5.1 an explicit expression for

ℕ_{x} [∫_{0}^{T} ⋯ ∫_{0}^{T} ds_{1} … ds_{4} ∫ ⋯ ∫ Y_{s_{1}}(dx_{1}) … Y_{s_{4}}(dx_{4})g(s_{1}, …, s_{4}, x_{1}, …, x_{4})] ,

where g is any measurable positive function on (ℝ^{+})^{4} × (ℝ^{d})^{4}. Specializing to the case g(s_{1}, …, s_{4}, x_{1}, …, x_{4}) = ∏_{i=1}^{4} 1_{[t,T]}(s_{i})p(ε^{2}, x_{1} − x_{2})p(ε^{2}, x_{3} − x_{4}), we get

ℕ_{x} [S_{ε} (∫_{t}^{T} ds Y_{s})^{2}]

= \frac{1}{3} 4!2^{3} ∫_{0}^{T} ds ∫ dy p(s, x − y)\{4 ∫_{(t−s)_{+}}^{T −s} ds_{1} ∫ dy_{1} p(s_{1}, y − y_{1}) ∫_{0}^{T −s} ds_{2}

∫ dy_{2} p(s_{2}, y − y_{2}) ∫_{(t−s−s_{2})_{+}}^{T −s−s_{2}} ds_{3} ∫ dy_{3} p(s_{3}, y_{2} − y_{3}) ∫_{0}^{T −s−s_{2}} ds_{4}

∫ dy_{4} p(s_{4}, y_{2} − y_{4}) ∫_{(t−s−s_{2}−s_{4})_{+}}^{T −s−s_{2}−s_{4}} ds_{5} ∫ dy_{5} p(s_{5}, y_{4} − y_{5})

∫_{(t−s−s_{2}−s_{4})_{+}}^{T −s−s_{2}−s_{4}} ds_{6} ∫ dy_{6} p(s_{6}, y_{4} − y_{6})

[p(ε^{2}, y_{1} − y_{3})p(ε^{2}, y_{5} − y_{6}) + p(ε^{2}, y_{1} − y_{5})p(ε^{2}, y_{3} − y_{6})

+ p(ε^{2}, y_{1} − y_{6})p(ε^{2}, y_{3} − y_{5})]

+ ∫_{0}^{T −s} ds_{7} ∫ dy_{7} p(s_{7}, y − y_{7}) ∫_{(t−s−s_{7})_{+}}^{T −s−s_{7}} ds_{8} ∫ dy_{8} p(s_{8}, y_{7} − y_{8})

∫_{(t−s−s_{7})_{+}}^{T −s−s_{7}} ds_{9} ∫ dy_{9} p(s_{9}, y_{7} − y_{9})

∫_{0}^{T −s} ds_{10} ∫ dy_{10} p(s_{10}, y − y_{10}) ∫_{(t−s−s_{10})_{+}}^{T −s−s_{10}} ds_{11} ∫ dy_{11} p(s_{11}, y_{10} − y_{11})

∫_{(t−s−s_{10})_{+}}^{T −s−s_{10}} ds_{12} ∫ dy_{12} p(s_{12}, y_{10} − y_{12})

[p(ε^{2}, y_{8} − y_{9})p(ε^{2}, y_{11} − y_{12}) + p(ε^{2}, y_{8} − y_{11})p(ε^{2}, y_{9} − y_{12})

+ p(ε^{2}, y_{8} − y_{12})p(ε^{2}, y_{9} − y_{11})]} We write J_{1}, J_{2}, J_{3}, J_{4}, J_{5}, and J_{6}, respectively for the integrals corresponding to the integrands p(ε^{2}, y_{1} − y_{3})p(ε^{2}, y_{5} − y_{6}), p(ε^{2}, y_{1} − y_{5})p(ε^{2}, y_{3} − y_{6}), p(ε^{2}, y_{1} − y_{6})p(ε^{2}, y_{3} − y_{5}), p(ε^{2}, y_{8} − y_{9})p(ε^{2}, y_{11} − y_{12}), p(ε^{2}, y_{8} − y_{11})p(ε^{2}, y_{9} − y_{12}), and p(ε^{2}, y_{8} − y_{12})p(ε^{2}, y_{9} − y_{11}) respectively. As we shall see the integral J_{4} gives the main contribution. Before proceeding to the calculations, we give three useful bounds: for every positive real numbers s, ε^{2} < 2^{−1}(T^{−1} ∧ T), we have for d ≥ 5

(21)  ∫_{0}^{T} (ε^{2} + s + r)^{−d/2} dr ≤ \frac{2}{d − 2} (ε^{2} + s)^{1−d/2} ,
(22)  ∫_{0}^{T} (ε^{2} + s + r)^{1−d/2} dr ≤ \frac{2}{d − 4} (ε^{2} + s)^{2−d/2} ,

(23)  ∫_{0}^{T} (ε^{2} + r)^{2−d/2} dr ≤ H_{T} (ε) := \begin{cases}2(d − 6)^{−1}ε^{6−d} & if d ≥ 7, \\ 4 ln ε^{−1} & if d = 6, \\ \sqrt{6T} & if d = 5. \end{cases}

From now on, we assume that ε^{2} < 2^{−1}(T^{−1}∧T) and also ε^{2} ln ε^{−1} < T if d = 6. Let us derive an upper bound on J_{1}. By repeated applications of the Chapman-Kolmogorov identities, we get

J_{1} ≤ 2^{8} ∫_{0}^{T} ⋯ ∫_{0}^{T} ds … ds_{6} ∫ dy p(s, x − y) ∫ dy_{1} p(s_{1}, y − y_{1})
∫ dy_{2} p(s_{2}, y − y_{2}) ∫ dy_{3} p(s_{3}, y_{2} − y_{3}) ∫ dy_{4} p(s_{4}, y_{2} − y_{4})
∫ dy_{5} p(s_{5}, y_{4} − y_{5}) ∫ dy_{6} p(s_{6}, y_{4} − y_{6})p(ε^{2}, y_{1} − y_{3})p(ε^{2}, y_{5} − y_{6})
≤ 2^{8} ∫_{0}^{T} ⋯ ∫_{0}^{T} ds … ds_{6} p(ε^{2} + s_{1} + s_{2} + s_{3}, 0)p(ε^{2} + s_{5} + s_{6}, 0).

We can apply (21), (22) and (23) to get:

J_{1} ≤ \frac{2^{8}}{(2π)^{d}} T ∫_{0}^{T} ds_{1} \frac{4}{(d − 2)(d − 4)}(ε^{2} + s_{1})^{2−d/2} \frac{4}{(d − 2)(d − 4)}T ε^{4−d}
≤ c_{1}T^{2}ε^{4−d}H_{T} (ε),

where the constant c_{1} depends only on d. We can use the same method for J_{2}:

J_{2} ≤ 2^{8} ∫_{0}^{T} ⋯ ∫_{0}^{T} ds … ds_{6} ∫ dy p(s, x − y) ∫ dy_{1} p(s_{1}, y − y_{1})
∫ dy_{2} p(s_{2}, y − y_{2}) ∫ dy_{3} p(s_{3}, y_{2} − y_{3}) ∫ dy_{4} p(s_{4}, y_{2} − y_{4})
∫ dy_{5} p(s_{5}, y_{4} − y_{5}) ∫ dy_{6} p(s_{6}, y_{4} − y_{6})p(ε^{2}, y_{1} − y_{5})p(ε^{2}, y_{3} − y_{6})
≤ 2^{8} ∫_{0}^{T} ⋯ ∫_{0}^{T} ds … ds_{6} ∫ dz p(s_{4}, z)p(ε^{2} + s_{1} + s_{2} + s_{5}, z)p(ε^{2} + s_{3} + s_{6}, z),

where we made the change of variables z = y_{2} −y_{4}. Since p(ε^{2} +s_{3} +s_{6}, z) ≤ p(ε^{2} +s_{3} +s_{6}, 0) and p(ε^{2} + s_{1} + s_{2} + s_{5}, z) ≤ p(ε^{2} + s_{1} + s_{2} + s_{5}, 0) , we can argue as for J_{1} to get:

J_{2} ≤ 2^{8} ∫_{0}^{T} ⋯ ∫_{0}^{T} ds … ds_{6} p(ε^{2} + s_{1} + s_{2} + s_{5}, 0)p(ε^{2} + s_{3} + s_{6}, 0).
≤ c_{1}T^{2}ε^{4−d}H_{T} (ε).

By symmetry, we get J_{2} = J_{3}. We want now to find an upper bound on J_{4}. Using (21), (22) and (23) we get:

J_{4} = 2^{6} ∫_{0}^{T} ds ∫ dy p(s, x − y)[ ∫_{0}^{T −s} ds_{7} ∫_{(t−s−s_{7})_{+}}^{T −s−s_{7}} ds_{8} ∫_{(t−s−s_{7})_{+}}^{T −s−s_{7}} ds_{9}

∫ dy_{7} p(s_{7}, y − y_{7}) ∫ dy_{8} p(s_{8}, y_{7} − y_{8}) ∫ dy_{9} p(s_{9}, y_{7} − y_{9})p(ε^{2}, y_{8} − y_{9})]^{2}

= 2^{6} ∫_{0}^{T} ds[ ∫_{0}^{T −s} ds_{7} ∫_{(t−s−s_{7})_{+}}^{T −s−s_{7}} ds_{8} ∫_{(t−s−s_{7})_{+}}^{T −s−s_{7}} ds_{9} p(ε^{2} + s_{8} + s_{9}, 0)]^{2}

≤ 2^{6}(2π)^{−d} ∫_{0}^{T} ds[ ∫_{0}^{T −s} ds_{7} \frac{4}{(d − 2)(d − 4)} [ε^{2} + 2(t − s − s_{7})_{+}]^{2−d/2} ]^{2}

= \frac{2^{10}}{(2π)^{d} [(d − 2)(d − 4)]^{2}}

∫_{0}^{T} ds[ε^{4−d}[(T − s) − (t − s)_{+}] + ∫_{0}^{(t−s)_{+}} ds_{7} [ε^{2} + 2(t − s − s_{7})_{+}]^{2−d/2} ]^{2}

≤ \frac{2^{10}}{(2π)^{d} [(d − 2)(d − 4)]^{2}} ∫_{0}^{T} ds[ε^{4−d}[(T − s) ∧ (T − t)] + 2^{−1}H_{2T} (ε)]^{2}

≤ \frac{2^{10}}{(2π)^{d}} [\frac{ε^{4−d}}{(d − 2)(d − 4)}]^{2} [\frac{(T − t)^{3}}{3} + (T − t)^{2}t] + c_{2}T^{2}ε^{4−d}H_{T} (ε),

where the constant c_{2} depends only on d. We now compute an upper bound on J_{5}:

J_{5} ≤ 2^{6} ∫_{0}^{T} ⋯ ∫_{0}^{T} ds … ds_{12} ∫ dy p(s, x − y) ∫ dy_{7} p(s_{7}, y − y_{7})
∫ dy_{8} p(s_{8}, y_{7} − y_{8}) ∫ dy_{9} p(s_{9}, y_{7} − y_{9}) ∫ dy_{10} p(s_{10}, y − y_{10})

∫ dy_{11} p(s_{11}, y_{10} − y_{11}) ∫ dy_{12} p(s_{12}, y_{10} − y_{12})p(ε^{2}, y_{8} − y_{11})p(ε^{2}, y_{9} − y_{12})

≤ 2^{6} ∫_{0}^{T} ⋯ ∫_{0}^{T} ds … ds_{12} ∫ dz p(s_{7} + s_{10}, z)p(ε^{2} + s_{8} + s_{11}, z)p(ε^{2} + s_{9} + s_{12}, z),

where we made the change of variables z = y_{10}−y_{7}. Since p(ε^{2}+s_{9}+s_{12}, z) ≤ p(ε^{2}+s_{9}+s_{12}, 0), and p(ε^{2} + s_{7} + s_{8} + s_{10} + s_{11}, 0) ≤ p(ε^{2} + s_{7} + s_{8} + s_{10}, 0), we can argue as for J_{1}, and get:

J_{5} ≤ c_{1}T^{2}ε^{4−d}H_{T} (ε).

By symmetry we get J_{6} = J_{5}. Combining the previous bounds leads to

ℕ_{x} [S_{ε} (∫_{0}^{t} ds Y_{s})^{2}] ≤ \frac{2^{10}}{(2π)^{d}} [\frac{ε^{4−d}}{(d − 2)(d − 4)}]^{2} [\frac{(T − t)^{3}}{3} + (T − t)^{2}t] + c_{3}T^{2}ε^{4−d}H_{T} (ε),

where the constant c_{3} depends only on d.

We shall now find a lower bound for ℕ_{x} [S_{ε}(∫_{t}^{T} ds Y_{s}) ∫_{t}^{T} ds (Y_{s}, 1)]. Using similar arguments as in the beginning of the proof, we get

I :=ℕ_{x} [S_{ε} (∫_{t}^{T} ds Y_{s}) ∫_{t}^{T} ds (Y_{s}, 1)]
= \frac{1}{3} 3!2^{3} ∫_{0}^{T} ds ∫ dy p(s, x − y) ∫_{(t−s)_{+}}^{T −s} ds_{1} ∫ dy_{1} p(s_{1}, y − y_{1})

∫_{0}^{T −s} ds_{2} ∫ dy_{2} p(s_{2}, y − y_{2}) ∫_{(t−s−s_{2})_{+}}^{T −s−s_{2}} ds_{3} ∫ dy_{3} p(s_{3}, y_{2} − y_{3}) ∫_{(t−s−s_{2})_{+}}^{T −s−s_{2}} ds_{4}

∫ dy_{4} p(s_{4}, y_{2} − y_{4}) [p(ε^{2}, y_{1} − y_{3}) + p(ε^{2}, y_{1} − y_{4}) + p(ε^{2}, y_{3} − y_{4})] .

Since we are looking for a lower bound, we restrict our attention to the term p(ε^{2}, y_{3} − y_{4}). We get

I ≥ 2^{4} ∫_{0}^{T} ds ∫_{(t−s)_{+}}^{T −s} ds_{1} ∫_{0}^{T −s} ds_{2} ∫_{(t−s−s_{2})_{+}}^{T −s−s_{2}} ds_{3} ∫_{(t−s−s_{2})_{+}}^{T −s−s_{2}} ds_{4} p(ε^{2} + s_{3} + s_{4}, 0)

≥ \frac{2^{4}}{(2π)^{d/2}} \frac{4}{(d − 2)(d − 4)} ∫_{0}^{T} ds [(T − s) ∧ (T − t)]

∫_{0}^{T −s} ds_{2} [(ε^{2} + 2(t − s − s_{2})_{+})^{2−d/2} − 2 (ε^{2} + (T − s − s_{2}))^{2−d/2}]

≥ \frac{2^{6}}{(2π)^{d/2}} \frac{1}{(d − 2)(d − 4)} ∫_{0}^{T} ds [(T − s) ∧ (T − t)] [ε^{4−d}(T − s − (t − s)_{+}) − 2H_{T} (ε)]

≥ \frac{2^{6}}{(2π)^{d/2}} \frac{ε^{4−d}}{(d − 2)(d − 4)} [\frac{(T − t)^{3}}{3} + (T − t)^{2}t] − c_{4}T^{2}H_{T} (ε),

where c_{4} depends only on d. Finally we deduce from section 5.1, with φ(s) = 1_{[0,T −t]}(s), that

ℕ_{x} [[∫_{t}^{T} ds (Y_{s}, 1)]^{2}] = 4 [\frac{(T − t)^{3}}{3} + (T − t)^{2}t] .

Combining the previous results, we get for ε small enough

ℕ_{x} [[ε^{d−4}(2π)^{d/2}S_{ε} (∫_{t}^{T} ds Y_{s}) − \frac{2^{4}}{(d − 2)(d − 4)} ∫_{t}^{T} ds (Y_{s}, 1)]^{2}] ≤ c_{5}T^{2}ε^{d−4}H_{T} (ε)
≤ c_{6}T^{2}ε,

where c_{6} depends only on d. This gives the convergence in L^{2}(ℕ_{x}). Now S_{ε} (∫_{t}^{T} ds Y_{s}) is monotone decreasing in ε (cf lemma 5.3 in [15]). The ℕ_{x}-a.e. convergence then follows from the previous estimate by an application of the Borel-Cantelli lemma and monotonicity arguments. □

5. Appendix

5.1. Formula for moments of the Brownian snake. For the reader’s convenience, we recall some explicit formulas for moments of the Brownian snake. These formulas are wellknown, at least in the context of superprocesses (see e.g. Dynkin [5]). We can compute the Laplace functional of ∫_{0}^{t} ds(Y_{s}, φ(s)) for φ ∈ B_{b+}(ℝ^{+} × ℝ^{d}). To this end start from the finite dimensional Laplace functional (2) with t_{i} = i/m, φ_{i} = \frac{1}{m} φ(i/m) for a nonnegative continuous function φ with compact support on ℝ^{+} × ℝ^{d}. Thanks to the continuity of the process X, by a suitable passage to the limit, we get for ν ∈ M_{f}

𝔼_{ν}^{X} [exp [− ∫_{0}^{t}(X_{t−s}, φ(s))ds]] = exp [−(ν, v(t))],

where v is a nonnegative solution of (1) with right-hand side J(t, x) = ∫_{0}^{t} ds P_{t−s}[φ(s)](x). This can be extended by monotone class arguments to any φ ∈ B_{b+}(ℝ^{+}×ℝ^{d}). The uniqueness of the solution is easily established using arguments similar to the classical Gronwall lemma. Then we get v(t, x) = ℕ_{x} [1 − exp [− ∫_{0}^{t} ds (Y_{t−s}, φ(s))]], thanks to theorem 1.3.

Now we introduce an auxiliary power series. Let us consider the analytic function f(λ) = 1 − \sqrt{1 − λ} for |λ| < 1. It is easy to check that for |λ| < 1, we have

f(λ) = ∑_{n=1}^{∞} γ_{n}λ^{n},

where the sequence (γ_{n}, n ≥ 1) is defined by γ_{1} = 1/2 and the recurrence

γ_{n} = \frac{1}{2} ∑_{k=1}^{n−1} γ_{k}γ_{n−k} for n ≥ 2

(use the fact that f solves 2f(λ) = f(λ)^{2} + λ). Now let T > 0 and J a nonnegative measurable function on ℝ^{+} × ℝ^{d}, such that M_{T} = sup_{[0,T]×ℝ^{d}} J(t, x) < ∞. We define the family of measurable functions (h_{n}, n ≥ 1) on ℝ^{+} × ℝ^{d}, by the initial condition

h_{1}(t) = J(t),

and the recurrence

(24)  h_{n}(t) = 2 ∑_{k=1}^{n−1} ∫_{0}^{t} ds P_{s} [h_{k}(t − s)h_{n−k}(t − s)] for n ≥ 2.

We clearly have for every n ≥ 1,

sup_{[0,T]×ℝ^{d}} |h_{n}| ≤ [4T]^{n−1}[2M_{T}]^{n}γ_{n}.

Thus the power series w(λ, t) = ∑(−1)^{n+1}λ^{n}h_{n}(t) is normally convergent on [0, T] × ℝ^{d} for |λ| < [8T M_{T}]^{−1}. And it clearly solves the integral equation on [0, T] × ℝ^{d}

w(t) + 2 ∫_{0}^{t} ds P_{s} [w(t − s)^{2}] = λJ(t). (25)

To get the uniqueness of the solution to the previous integral equation, use arguments similar to Gronwall’s lemma. Finally we can compute the moments for the process Y under ℕ_{x}. Indeed, let φ ∈ B_{b+}(ℝ^{+} × ℝ^{d}). We have shown that for λ > 0, the function v_{λ}(t, x) = ℕ_{x} [1 − exp −λ ∫_{0}^{t}(Y_{t−s}, φ(s))ds] is the unique solution to (25) on ℝ^{+} × ℝ^{d} with J(t, x) =

∫_{0}^{t} ds P_{s}[φ(t − s)](x). Thus for λ ≥ 0 small enough, we have v_{λ}(t) = w(λ, t). Then from the series expansion for w(λ, t), we get for every integer n ≥ 1

ℕ_{x} [(∫_{0}^{t} ds (Y_{t−s}, φ(s)))^{n}] = n!h_{n}(t, x),

where the functions h_{n} are defined by h_{1}(t) = ∫_{0}^{t} ds P_{s}[φ(t − s)], and the recurrence (24). In the same way it can be shown that for every φ ∈ B_{b+}(ℝ^{d}), for every t ≥ 0, n ≥ 1,

ℕ_{x} [(Y_{t}, φ)^{n}] = n!h_{n}(t, x),

where the functions are defined by h_{1}(t) = P_{t}[φ], and the recurrence (24).

5.2. Some properties of the function u_{1}. We consider the function u_{1}, which is the maximal solution on (1, ∞) of the non linear differential equation

u′′(r) + \frac{d − 1}{r}u′(r) = 4u(r)^{2}.

Lemma 5.1. There exist positive constants a_{0}, b_{0} and b′_{1}, depending only on d, such that

lim_{r→∞} r^{d−2}u_{1}(r) = a_{0} if d ≥ 5, lim_{r→∞} r^{2} log(r) u_{1}(r) = a_{0} = 1/2 if d = 4;

furthermore for every r > 1,

(26)  u_{1}(r) ≥ a_{0}r^{2−d} if d ≥ 5, u_{1}(r) ≥ a_{0}r^{−2} log(2r)^{−1} if d = 4;

and for every r ≥ 4/3,

(27)  u_{1}(r) ≤ b_{0}r^{2−d} if d ≥ 5, u_{1}(r) ≤ b_{0}[2r^{2} log(r)]^{−1} if d = 4;
(28)  u_{1}(r) ≤ a_{0}r^{2−d} + b′_{1}r^{6−2d} if d ≥ 5,
(29)  u_{1}(r) ≤ a_{0}r^{−2} log(r)^{−1} + b′_{1}r^{−2} log(r)^{−2} log(log(r)) if d = 4.

We will prove this lemma by giving the asymptotic expansion of u_{1} at ∞. For d ≥ 5, we will see the constant a_{0} can be expressed as the radius of convergence of a series. We introduce the auxiliary function

z(t) = 4(d − 2)^{−2(d−1)/(d−2)}t u_{1} [(\frac{t}{d − 2})^{1/(d−2)}] , for t > d − 2.

This function is solution on (d − 2, ∞) of

y′′(t) = t^{−δ−2}y(t)^{2}, (30)

where δ = \frac{d − 4}{d − 2}. We deduce from [18] p.132 that the function z(t) (i.e. r^{d−2}u_{1}(r)) is decreasing. Proof in the case d ≥ 5. We deduce from theorem 1.1 of [18] that the limit q = lim_{t→∞} z(t) exists and is positive. Hence by integrating (30) twice from t to ∞, we get for t > d − 2,

z(t) − q = ∫_{t}^{∞}(r − t)r^{−δ−2}z(r)^{2}dr. (31)

Now consider the sequence (q_{n}, n ≥ 0) defined by q_{0} = 1 and the recurrence

q_{n} = \frac{1}{nδ(nδ + 1)} ∑_{k=0}^{n−1} q_{k}q_{n−k−1}, for n ≥ 1.

Clearly we have for every n ≥ 0, q_{n} ≤ 2 [4/δ]^{n} γ_{n+1}, where the sequence (γ_{n}, n ≥ 1) has been introduced in section 5.1. The power series ∑ q_{n}q^{n+1}t^{−δn} is convergent and even C^{∞} as a function of t at least for t > t_{1} = [4q/δ]^{1/δ}. This power series also solves (31) for t > t_{1}. The same arguments as in the proof of the Gronwall lemma show that equation (31) possesses a unique solution bounded in a neighborhood of infinity. Thus the function z and the power series agree for t > t_{1}. Since the function z is analytic on (d − 2, ∞) and since q and the coefficients q_{n} are positive, we deduce that the radius of convergence of the series ∑ q_{n}s^{n} is q(d − 2)^{−δ} and that for t > d − 2

z(t) = ∑_{n=0}^{∞} q_{n}q^{n+1}t^{−nδ}.

Thus we get with obvious notation for r > 1,

u_{1}(r) = 4^{−1}(d − 2)^{d/(d−2)}r^{2−d} ∑_{n=0}^{∞} q_{n}q^{n+1}(d − 2)^{−n(d−4)/(d−2)}r^{−n(d−4)}

= r^{2−d} ∑_{n=0}^{∞} a_{n}r^{−n(d−4)}.

Since the function r^{d−2}u_{1}(r) is decreasing we get (26). Since the real numbers (a_{n}, n ≥ 0) are positive, (27) and (28) follow easily. Notice that 4(d − 2)^{−2}a_{0} is the radius of convergence of the series ∑ q_{n}s^{n}. □

Proof in the case d = 4. We write f(t) ∼ g(t) at 0+ when the real function f and g are positive or negative on I = (0, 0+ε) for some ε > 0 and lim_{t∈I,t→0} f(t)/g(t) = 1. We also write f(t) ∼ g(t) at ∞ when f(1/t) ∼ g(1/t) at 0+. We know from [18] p.133 that z(t) ∼ log(t)^{−1} at ∞. We deduce from (30) that z′(t) is negative on (2, ∞) and z′′(t) ∼ [t log(t)]^{−2} at ∞. By integration, we get z′(t) ∼ t^{−1} log(t)^{−2} at ∞. We now consider the function w(s) = z(e^{s}) which solves w′′ − w′ = w^{2} on (log 2, ∞). Notice that the function w is positive decreasing and w′ is negative. We also have w(s) ∼ s^{−1}, w′(s) ∼ −s^{−2} and w′′(s) = o(s^{−2}) at ∞. Thus the function defined on (0, ∞) by

p(w(s)) = w′(s), for s ∈ (log 2, ∞),

is well defined and even of class C^{1}, and p′(w(s)) = w′′(s)/w′(s). Thus the function p can be extended as a C^{1} function on [0, ∞) by setting p(0) = 0 and p′(0) = 0. Furthermore it solves

p(w)p′(w) − p(w) = w^{2} on [0, ∞).

We also have p(w) ∼ −w^{2} at 0+. We consider the sequence (ρ_{n}, n ≥ 2) defined by ρ_{2} = 1 and the recurrence

ρ_{n} = ∑_{k=2}^{n−1} kρ_{k}ρ_{n−k+1}, for n ≥ 3.

The radius of convergence of the series ∑(−1)^{n+1}ρ_{n}w^{n} is 0, nevertheless we will prove it is the asymptotic expansion of p at 0+. We set H_{n}(w) = ∑_{k=2}^{n}(−1)^{k+1}ρ_{k}w^{k} for n ≥ 2. We now prove by recurrence that p(w) = H_{n}(w) + h_{n}(w), where h_{n}(w) = o(w^{n}) at 0+. This is true for n = 2. Let us assume it is true at stage n. Let g_{n,α}(w) = (1 − α)(−1)^{n}ρ_{n+1}w^{n+1} − h_{n}(w). We easily have

g′_{n,α}(w)p(w) + g_{n,α}(w)[H′_{n}(w) − 1] = α(−1)^{n}ρ_{n+1}w^{n+1} + o(w^{n+1}),
= (−1)^{n+1}ρ_{n+2}w^{n+2} + o(w^{n+2}), if α = 0.

Let us assume n is even. For α = 0, the above right hand side is negative on (0, ε], for ε small enough. Since p is negative and [H′_{n}(w) − 1] < 0 on [0, ε], for ε small, we see that g_{n,0}(w) < 0 implies g′_{n,0}(w) ≥ 0. As g_{n,0}(0) = 0, we get by contradiction that g_{n,0} ≥ 0 on [0, ε]. This implies h_{n}(w) ≤ ρ_{n+1}w^{n+1}. Similar arguments for α > 0 implies that g_{n,α} ≤ 0 on [0, ε_{α}] for ε_{α} > 0 small enough. Since this holds for any α > 0 and since h_{n}(w) ≤ ρ_{n+1}w^{n+1} for w small enough, we deduce that h_{n+1}(w) = h_{n}(w) − ρ_{n+1}w^{n+1} = o(w^{n+1}). If n is odd the proof is similar.

From the definition of p, we then have w′(s) = H_{n}(w(s)) + O(w(s)^{n+1}) at ∞. For n = 3 this gives w′(s) = −w(s)^{2} + 2w(s)^{3} + O(w(s)^{4}) at ∞. Since w(s) ∼ s^{−1} at +∞, we deduce by integration that

\frac{1}{w(s)} −2 log w(s) + O(1) = s at infinity.

Standard arguments yields w(s) = s^{−1} + 2s^{−2} log(s) + O(s^{−2}) at infinity. Thus we have

u_{1}(r) = \frac{1}{r^{2}} [\frac{1}{2 log(r)} +\frac{log(log(r))}{4 log(r)^{2}} + O (log(r)^{−2})] at + ∞.

Notice the previous calculation can be continued to give an asymptotic expansion of u_{1} at infinity. The inequalities (27) and (29) follow easily. We will now prove that for every r > 1, u_{1}(r) ≥ [2r^{2} log(2r)]^{−1}. We consider the function w(r) = u_{1}(r)−[2r^{2} log(2r)]^{−1}. The function w is positive at least over (1, 1 + η) ∩ (η^{−1}, ∞) for η small. Let us assume that w achieves its minimum at r_{0} and that w(r_{0}) ≤ 0. Then we have r_{0} ∈ [1 + η, η^{−1}], w′(r_{0}) = 0 and w′′(r_{0}) ≥ 0. An easy computation gives

w′′(r) = 4w(r) [u_{1}(r) + \frac{1}{2r^{2} log(2r)}] − \frac{3}{r}w′(r) − \frac{1}{2r^{4}(log(2r))^{3}} .

Evaluation at r = r_{0} implies that w′′(r_{0}) < 0. This contradicts the assumption. Hence w is positive, that is we get (26) for d = 4. □

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