Strong approximations of three-dimensional Wiener sausages

Endre CSÁKI^{1} and Yueyun HU^{2}

Abstract. In this paper we prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]’s fine L^{2}-norm estimates between the Wiener sausage and the Brownian intersection local times.

Keywords. Wiener sausage, intersection local times, strong approximation.

AMS Classification 2000. 60F15.

1 Introduction

Let {B_{t}, t ≥ 0} be a d-dimensional Brownian motion and let K ⊂ ℝ^{d} be a compact set. The Wiener sausages associated to (B_{t}) and K are the compact sets

S^{K}(0, t) =^{def} ⋃_{0≤s≤t}(B_{s} + K), t ≥ 0.

Denote by m(dx) the Lebesgue measure. The volume of Wiener sausage m(S^{K}(0, t)) is a very rich topic of researches, we only mention some recent references ([3], [7], [8]), see their lists of references for related studies.

We restrict our attention to the three dimensional case (d = 3). Denote by C(K) the Newtonian electrostatic capacity of the compact K, see [6] pp. 250. Concerning the asymptotic of m(S^{K}(0, t)) when t → ∞, Kesten, Spitzer and Whitman ([6] pp. 252, Problem 4), Spitzer [17], and Le Gall [11] proved respectively the following results (1.1), (1.2) and (1.3):

1A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, P.O.B. 127, Budapest, H–1364, Hungary. E-mail: csaki@renyi.hu. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886 and T 043037

2Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR–7599), Université Paris VI, 4 Place Jussieu, F–75252 Paris cedex 05, France. E-mail: hu@proba.jussieu.fr

Theorem A When t → ∞,

\frac{1}{t}m(S^{K}(0, t)) ⟶^{a.s.} C(K),  (1.1)
𝔼(m(S^{K}(0, t))) = C(K)t + \frac{4}{(2π)^{3/2}} C(K)^{2} \sqrt{t} + o(\sqrt{t}),  (1.2)
\frac{m(S^{K}(0, t)) − C(K)t}{\sqrt{t log t}} ⟶^{(d)} \frac{C(K)^{2}}{π\sqrt{2}} N ,  (1.3)

where N denotes a centered real-valued Gaussian variable, of variance 1, and C(K) denotes Newtonian capacity.

This paper aims at establishing a strong approximation of the process {m(S^{K}(0, t)), t ≥ 0}:

Theorem 1.1 Let l ≥ 1 and K_{1}, ..., K_{l} be l compact sets in ℝ^{3}. There exists a one dimensional Brownian motion (β(t), t ≥ 0) such that almost surely, for all 1 ≤ i ≤ l,

m(S^{K_{i}}(0, t)) = C(K_{i})t + \frac{C(K_{i})^{2}}{π\sqrt{2}} β(t log t) + o(t^{1/2} (log t)^{1/4+δ}),

for any positive constant δ > 0.

The above strong approximation is in agreement with Bass and Kumagai [2] where they deal with the range of a three-dimensional (and two-dimensional) random walk. Note also that the Brownian motion β can be chosen simultaneously the same for all the l compact sets (K_{i})_{1≤i≤l}, a fact already pointed out by Le Gall [11].

Theorem 1.1 implies the convergence in terms of process of the normalized Wiener sausage to Brownian motion. The error term o(t^{1/2} (log t)^{1/4+δ}) being sufficiently small, in addition to weak convergence, we can deduce functional almost sure central limit theorem and laws of the iterated logarithm.

Corollary 1.2 Let K ⊂ ℝ^{3} be a compact set such that C(K) > 0. Put

M_{t}(x) = \frac{π\sqrt{2}}{C(K)^{2}\sqrt{t log t}} (m(S^{K}(0, xt)) − C(K)xt) , 0 ≤ x ≤ 1.

We have

L (M_{t}(·)) ⟶^{w} L(β), t → ∞
\frac{1}{log T} ∫_{1}^{T} \frac{1}{t}δ (M_{t}(·)) dt ⟶^{w} L(β), a.s., T → ∞
lim sup_{t→∞} \frac{1}{\sqrt{t log t log log t}} (m(S^{K}(0, t)) − C(K)t) = \frac{C(K)^{2}}{π} ,  a.s.
lim inf_{t→∞} (\frac{log log t}{t log t})^{1/2} sup_{0≤s≤t} |m(S^{K}(0, s)) − C(K)s| = \frac{C(K)^{2}}{4} ,  a.s.,

where ⟶^{w} denotes weak convergence in C[0, 1], L(·) denotes the law of the process in bracket and δ(f(x)) stands for a point mass at f ∈ C[0, 1].

The proof of Theorem 1.1 relies on two steps:

(i) Establish the strong approximation of the Wiener sausage by an intersection local time;

(ii) Use a Tanaka-Rosen-Yor ([14], [20]) type formula and approximate the intersection local time by a Brownian motion.

This paper is organized as follows: In Section 2, we give a general estimate on the rate of growth of a process through its L^{p}-norms; Section 3 is devoted to the intersection local times, in particular to show the above step (i); the step (ii) is done in Section 4. Finally, we prove Theorem 1.1 in Section 5.

Throughout this paper, ∥ξ∥_{p} denotes the L^{p}-norm of a random variable ξ, while c_{p}, c′_{p}, c′′_{p} denote some positive constants, depending on p > 0, whose exact values may vary from one paragraph to another.

2 Rate of growth

We shall estimate the rate of growth of some process through its L^{p}-norms. The following result on the modulus of continuity, due to Barlow and Yor [1], is obtained as a consequence of Garsia-Rodemich-Rumsey lemma:

Lemma 2.1 ([1], (3.b)) Let r > 0. Assume that (ξ_{t}, 0 < t < r) is a real-valued continuous process such that for some constants a > 0, b > 1 and κ > 0,

𝔼 |ξ_{t} − ξ_{s}|^{a} ≤ κ |t − s|^{b}, ∀ 0 < s ≤ t < r.

Then for any 0 < γ < b − 1, there exists some constant c(a, b, γ) > 0 such that

𝔼 ( sup_{0<s≠t<r} \frac{|ξ_{t} − ξ_{s}|^{a}}{|t − s|^{γ}} ) ≤ c(a, b, γ) κ r^{b−γ}.

Proposition 2.2 Let (X_{t}, t > 0) be a real-valued process. Let 0 < a ≤ 1, b_{2} ≥ b_{1} ≥ 0 be three constants. Assume that for any p > 1, there exists some constant c_{p} > 0 such that

∥X_{t}∥_{p} ≤ c_{p} t^{a} (log t)^{b_{1}}, ∀ t > t_{0},  (2.1)
∥X_{t} − X_{s}∥_{p} ≤ c_{p} (t − s)^{a} (log t)^{b_{2}}, t_{0} ≤ s ≤ t − 1,  (2.2)
∥ sup_{t≤u≤t+1} |X_{u} − X_{t}|∥_{p} ≤ c_{p} t^{a/2}, ∀ t > t_{0},  (2.3)

for some constant t_{0} > 1. Then for any ϵ > 0,

X_{t} = o (t^{a}(log t)^{b_{1}+ϵ}) , a.s., t → ∞.

The power a/2 in (2.3) can be replaced by any positive constant smaller than a. Note that in (2.2), we only require for the increments X_{t} − X_{s} when t − s > 1, the local fluctuations are controlled by (2.3).

Proof: Let p > 4/a and n > t_{0}. By (2.3),

ℙ( max_{t_{0}≤j≤n} sup_{j≤t≤j+1} |X_{t} − X_{j}| > n^{a}) ≤ ∑_{j=[t_{0}]}^{n} ℙ( sup_{j≤t≤j+1} |X_{t} − X_{j}| > n^{a})

≤ ∑_{j=[t_{0}]}^{n} n^{−ap} c_{p}^{p} (1 + n)^{ap/2}
≤ c′ n^{−ap/2+1},

whose sum on n converges. Applying Borel-Cantelli’s lemma shows that

max_{0≤j≤n} sup_{j≤t≤j+1} |X_{t} − X_{j}| = O(n^{a}), a.s., n → ∞.

Then it remains to prove that

X_{n} = o(n^{a} (log n)^{b_{1}+ϵ}), a.s., n → ∞.

Let 0 < ϵ < a/4 and fix any constant η ∈ (0, 1/2) satisfying ηb_{2} ≤ a(1 − η)/2. Let p > 8/(ηϵ). Consider large j and define n_{j} = [e^{j^{η}}]. Since ϵ > 0 is arbitrary, it suffices to prove that

∑_{j} ℙ( max_{n_{j}≤k≤n_{j+1}} |X_{k}| ≥ n_{j}^{a} (log n_{j})^{b_{1}+ϵ}) < ∞.  (2.4)

To this end, we have by triangle inequality

ℙ( max_{n_{j}≤k≤n_{j+1}} |X_{k}| ≥ n_{j}^{a} (log n_{j})^{b_{1}+ϵ})

≤ ℙ(|X_{n_{j}}| ≥ \frac{1}{2}n_{j}^{a} (log n_{j})^{b_{1}+ϵ}) + ℙ( max_{n_{j}≤k≤n_{j+1}} |X_{k} − X_{n_{j}}| ≥ \frac{1}{2}n_{j}^{a} (log n_{j})^{b_{1}+ϵ})
=^{def} I_{1} + I_{2}.

Using the L^{p}-norm of X_{n_{j}},

I_{1} ≤ (\frac{1}{2}n_{j}^{a} (log n_{j})^{b_{1}+ϵ})^{−p} c_{p}^{p} n_{j}^{ap} (log n_{j})^{pb_{1}}
≤ c′_{p} j^{−pϵη} ≤ j^{−8}.

We shall apply Lemma 2.1 to estimate I_{2}. Consider a continuous process ξ such that ξ_{k} = X_{k+n_{j}} − X_{n_{j}} for 0 ≤ k ≤ n_{j+1} − n_{j}, and ξ_{·} is linear on each interval [k, k + 1]. From (2.2), we easily deduce that for any 0 < u, v < n_{j+1} − n_{j},

𝔼|ξ_{u} − ξ_{v}|^{p} ≤ \{ \begin{matrix}c_{p}^{p} |v − u|^{ap} (log n_{j+1})^{pb_{2}}, & |u − v| > 2; \\ c′_{p} |v − u|^{p} (log n_{j+1})^{pb_{2}}, & |u − v| < 1. \end{matrix}

Since a ≤ 1, there exists some constant c_{a,p} > 0 such that for any 0 < u, v < n_{j+1} − n_{j},

𝔼|ξ_{u} − ξ_{v}|^{p} ≤ c_{a,p} (log n_{j+1})^{b_{2}p} |u − v|^{ap}.

Applying Lemma 2.1 with γ = p(a − ϵ/2) < ap − 1 gives that for some constant c(ϵ, p) > 0

𝔼( sup_{0≤u≠v<n_{j+1}−n_{j}} \frac{|ξ_{v} − ξ_{u}|^{p}}{|v − u|^{p(a−ϵ/2)}} ) ≤ c(ϵ, p) (n_{j+1} − n_{j})^{pϵ/2}(log n_{j+1})^{b_{2}p}.

It follows from Chebyshev’s inequality that

I_{2} ≤ ℙ( sup_{0<k≤n_{j+1}−n_{j}} \frac{|ξ_{k}|^{p}}{k^{p(a−ϵ/2)}} > 2^{−p} \frac{n_{j}^{ap}}{(n_{j+1} − n_{j})^{p(a−ϵ/2)}} (log n_{j})^{p(b_{1}+ϵ)})
≤ c_{ϵ,p}2^{p} (\frac{n_{j+1} − n_{j}}{n_{j}} )^{ap} (log n_{j+1})^{pb_{2}} (log n_{j})^{−p(b_{1}+ϵ)}
≤ c′ j^{−(1−η)pa−pη(b_{1}+ϵ)+pηb_{2}}
≤ c′ j^{−ap(1−η)/2} ≤ j^{−2},

by our choice of p and η. Then the sum of I_{1} and I_{2} over j converges and it proves (2.4) hence the Proposition. □

When the estimate (2.2) on the increments holds for all t > s, we may get rid of the condition on the local fluctuations (2.3), as stated in the following result:

Proposition 2.3 Let (X_{t}, t > 0) be a real-valued process. Let 0 < a ≤ 1, b_{2} ≥ b_{1} ≥ 0 be three constants. Assume that for any p > 1, there exists some constant c_{p} > 0 such that

∥X_{t}∥_{p} ≤ c_{p} t^{a} (log t)^{b_{1}}, ∀ t ≥ t_{0},
∥X_{t} − X_{s}∥_{p} ≤ c_{p} (t − s)^{a} (log t)^{b_{2}}, t_{0} ≤ s ≤ t,

for some t_{0} > 1. Then for any ϵ > 0,

X_{t} = o (t^{a}(log t)^{b_{1}+ϵ}) , a.s., t → ∞.

Proof: The proof goes in the same way as in Proposition 2.2. Let us only mention the main difference. Let ϵ, η, p, n_{j} be the same as in Proposition 2.2. To control the probability

ℙ( sup_{n_{j}≤t≤n_{j+1}} |X_{t} − X_{n_{j+1}}| ≥ \frac{1}{2}n_{j}^{a} (log n_{j})^{b_{1}+ϵ}),

we define ξ_{u} := X_{u(n_{j+1}−t_{0})+t_{0}}, 0 ≤ u ≤ 1. Then for any 0 < u, v < 1,

𝔼|ξ_{u} − ξ_{v}|^{p} ≤ c_{p}^{p} n_{j+1}^{ap} (log n_{j+1})^{b_{2}p} |u − v|^{ap}.

Applying Lemma 2.1 with γ = p(a − ϵ/2) < ap − 1 gives that for some constant c(ϵ, p) > 0

𝔼( sup_{0<s≠t<n_{j+1}} \frac{|X_{t} − X_{s}|^{p}}{|t − s|^{p(a−ϵ/2)}} ) ≤ c(ϵ, p) n_{j+1}^{pϵ/2}(log n_{j+1})^{b_{2}p}.

Thus the proposition follows in the same way. □

3 Intersection local times

Denote by {α(x, F), x ∈ ℝ^{3}, F bounded measurable F ⊂ ℝ_{+}^{2}} the family of intersection local times of B. For any F, almost surely α(x, F) is the density of the measure ∫_{F} dsduδ_{(B_{s}−B_{u})}(dx), which means that for any test function f : ℝ^{3} → ℝ_{+},

∫_{(s,u)∈F} dsdu f(B_{s} − B_{u}) = ∫_{ℝ^{3}} dxf(x)α(x, F).  (3.1)

We refer to Rosen [13], Le Gall [9] for the construction of the family {α(x, F)}. We shall be particularly interested in two families of subsets F:

A_{t} = {(s, u) : 0 ≤ s ≤ u − 1 ≤ t − 1},  t > 1.
F_{δ} = {(s, u) : 0 ≤ s ≤ u − δ ≤ 1 − δ},  0 < δ < 1.

Let us introduce the following notation: For 0 ≤ a < b, S_{ϵ}^{K}(a, b) =^{def} ⋃_{a≤t≤b}(B_{t} + ϵK) and S_{ϵ}^{K} = ⋃_{0≤s≤1}(B_{s} + ϵK) when there is no risk of confusion. Then we have the following scaling property: For t > 1,

(m(S^{K}(0, t)), α(0, A_{t})) =^{law} (t^{3/2}m(S_{1/\sqrt{t}}^{K}), t^{1/2}α(0, F_{1/t})).

Write

{ξ} = ξ − 𝔼(ξ).

The following L^{p}-norm estimate is essentially due to Le Gall who proved the case p = 2 in [11], Theorem 3.1. His arguments can be adopted to deal with the L^{p} case.

Proposition 3.1 Fix a compact set K ⊂ ℝ^{3}. For any p ≥ 2, there exists some constant c_{p} > 0 depending on p and K such that for all 0 < ϵ < 1/2,

∥\frac{1}{ϵ^{2}} \{m(S_{ϵ}^{K})\} − C(K)^{2}\{α(0, F_{ϵ^{2}})\}∥_{p} ≤ c_{p}.

Consequently, for any t > 1,

∥\{m(S^{K}(0, t))\} − C(K)^{2}\{α(0, A_{t})\}∥_{p} ≤ c_{p} \sqrt{t}.

We shall make use of the following Rosenthal inequality, see Petrov [12], Theorem 2.9:

Lemma 3.2 For any n centered, independent and identically distributed random variables X_{1}, ...X_{n} there exists a constant c_{p} > 0 depending only on p ≥ 2 such that

∥∑_{i=1}^{n} X_{i}∥_{p} ≤ \{ \begin{matrix}c_{p} \sqrt{n} ∥X_{1}∥_{p}, \\ c_{p} (\sqrt{n} ∥X_{1}∥_{2} + n^{1/p} ∥X_{1}∥_{p}). \end{matrix}  (3.2)

Proof of Proposition 3.1: The proof of the case p > 2 is almost the same as the case p = 2 given by Le Gall ([11], pp. 1006). In fact, let n be an integer such that 1/4 < ϵ2^{n/2} ≤ 1/2. Then

using the notation S_{ϵ}^{K}(a, b) =^{def} ⋃_{a≤t≤b}(B_{t} + ϵK), I_{−}^{(k,q)} = (\frac{2q−2}{2^{k}}, \frac{2q−1}{2^{k}}) and I_{+}^{(k,q)} = (\frac{2q−1}{2^{k}}, \frac{2q}{2^{k}}) for

1 ≤ q ≤ 2^{k−1},

\{m(S_{ϵ}^{K})\} = ∑_{k=1}^{2^{n}} \{m (S_{ϵ}^{K} (\frac{k − 1}{2^{n}}, \frac{k}{2^{n}} ))\} − ∑_{k=1}^{n} ∑_{q=1}^{2^{k−1}} \{m(S_{ϵ}^{K}(I_{−}^{(k,q)}) ∩ S_{ϵ}^{K}(I_{+}^{(k,q)}))\}.

Therefore by the independent increment property of Brownian motion, we can apply (3.2) and get

∥\frac{1}{ϵ^{2}} ∑_{k=1}^{2^{n}} \{m (S_{ϵ}^{K} (\frac{k − 1}{2^{n}}, \frac{k}{2^{n}} ))\}∥_{p} ≤ c_{p}ϵ^{−2} 2^{n/2} ∥\{m (S_{ϵ}^{K} (0, \frac{1}{2^{n}} ))\}∥_{p}
≤ c_{p}ϵ^{−2}2^{n} sup_{1/4<r≤1/2} ∥{S_{r}^{K}(0, 1)}∥_{p}
≤ c′_{p},  (3.3)

where the second inequality follows from the scaling property: m(S_{ϵ}^{K}(0, \frac{1}{2^{n}})) =^{law} 2^{−3n/2} m(S_{r}^{K}(0, 1)) with r = ϵ2^{−n/2} ∈ (\frac{1}{4}, \frac{1}{2}]. On the other hand,

∥\frac{1}{ϵ^{2}} ∑_{k=1}^{n} ∑_{q=1}^{2^{k−1}} \{m(S_{ϵ}^{K}(I_{−}^{(k,q)}) ∩ S_{ϵ}^{K}(I_{+}^{(k,q)}))\} − C(K)^{2}\{α(0, F_{2^{−n}})\}∥_{p}

≤ ∑_{k=1}^{n} ∥\frac{1}{ϵ^{2}} ∑_{q=1}^{2^{k−1}} \{m(S_{ϵ}^{K}(I_{−}^{(k,q)}) ∩ S_{ϵ}^{K}(I_{+}^{(k,q)}))\} − C(K)^{2}\{α(I_{−}^{(k,q)} × I_{+}^{(k,q)})\}∥_{p}

≤ c_{p} ∑_{k=1}^{n} (2^{(k−1)/2}∥ξ(ϵ, k)∥_{2} + 2^{(k−1)/p}∥ξ(ϵ, k)∥_{p}),

by applying the second inequality in (3.2) and where

ξ(ϵ, k) = \frac{1}{ϵ^{2}} \{m(S_{ϵ}^{K}(0, 2^{−k}) ∩ S_{ϵ}^{K}(2^{−k}, 2^{−k+1}))\} − C(K)^{2}\{α((0, 2^{−k}) × (2^{−k}, 2^{−k+1}))\}

Le Gall ([11], pp.1006) has already shown that

∑_{k=1}^{n} 2^{(k−1)/2}∥ξ(ϵ, k)∥_{2} = O(1).  (3.4)

We shall prove that

∑_{k=1}^{n} 2^{(k−1)/p}∥ξ(ϵ, k)∥_{p} = O(1),

which together with (3.3) and (3.4) implies the Proposition.

To this end, by scaling,

∑_{k=1}^{n} 2^{(k−1)/p}∥ξ(ϵ, k)∥_{p} = 2^{−1/p} ∑_{k=1}^{n} 2^{−( \frac{1}{2} − \frac{1}{p})k} ∥r_{k}^{−2}\{m(S_{r_{k}}^{K} ∩ S̃_{r_{k}}^{K})\} − C(K)^{2}\{L_{2}([0, 1]^{2})\}∥_{p}

≤ 2^{−1/p+1} ∑_{k=1}^{n} 2^{−( \frac{1}{2} − \frac{1}{p})k} (∥r_{k}^{−2}m(S_{r_{k}}^{K} ∩ S̃_{r_{k}}^{K})∥_{p} + C(K)^{2} ∥L_{2}([0, 1]^{2})∥_{p}) ,

where r_{k} = ϵ2^{k/2} < 1, S_{r_{k}}^{K} = S_{r_{k}}^{K}(0, 1) and L_{2}([0, 1]^{2}) denotes the intersection local times of two independent Brownian motion B and B̃ over [0, 1]^{2}. Using Le Gall ([10], Corollary 3.2), we have that for any compact K, ∥L_{2}([0, 1]^{2})∥_{p} < ∞ and

sup_{0<r<1} ∥r^{−2}m(S_{r}^{K} ∩ S̃_{r}^{K})∥_{p} < ∞.

It follows that

∑_{k=1}^{n} 2^{(k−1)/p}∥ξ(ϵ, k)∥_{p} ≤ c_{p} ∑_{k=1}^{n} 2^{−( \frac{1}{2} − \frac{1}{p})k} = O(1),

completing the proof. □

The main technical estimate in this section is the following L^{p}-norm:

Lemma 3.3 Let p ≥ 1. Denote by D the unit ball of ℝ^{3}. We have for all t > 1,

||m(S^{D}(0, t) ∩ S̃^{D}(0, ∞))||_{p} ≤ c_{p} \sqrt{t},

where c_{p} > 0 only depending on p and S̃^{D} denotes the Wiener sausage associated with another Brownian motion independent of B.

We may choose c_{p} = c^{p} in the above lemma for some numerical constant c > 1.

Proof of Lemma 3.3: By scaling

m(S^{D}(0, t) ∩ S̃^{D}(0, ∞)) =^{law} ϵ^{−3} m(S_{ϵ}^{D}(0, 1) ∩ S̃_{ϵ}^{D}(0, ∞)),

with ϵ = t^{−1/2}. It is equivalent to prove that for any integer p ≥ 1,

𝔼(m(S_{ϵ}^{D}(0, 1) ∩ S̃_{ϵ}^{D}(0, ∞)))^{p} ≤ c′_{p} ϵ^{2p},  0 < ϵ < 1.  (3.5)

Denote by T_{ϵ}^{K}(x) the first entry time into x − ϵK:

T_{ϵ}^{K}(x) = inf{t ≥ 0 : B_{t} ∈ x − ϵK},

with convention that inf ∅ = ∞. Define

ψ(r) = \frac{1}{r}1_{(r<1)} + e^{−r^{2}/16}, r > 0.

Recall the following estimate, see Le Gall [11], Lemma 3.2:

ℙ(T_{ϵ}^{D}(x) ≤ 1) ≤ c ψ(|x|),  (3.6)
ℙ(T_{ϵ}^{D}(x) < ∞) = \frac{ϵ}{|x|} ∧ 1.  (3.7)

Note that

Υ_{p}(ϵ) =^{def} 𝔼(m(S_{ϵ}^{D}(0, 1) ∩ S̃_{ϵ}^{D}(0, ∞)))^{p}
= ∫ dx_{1}...dx_{p} ℙ( max_{1≤i≤p} T_{ϵ}^{D}(x_{i}) ≤ 1) ℙ( max_{1≤i≤p} T_{ϵ}^{D}(x_{i}) < ∞)
= ∫_{∃i≠j:|x_{i}−x_{j}|≤2ϵ} + ∫_{∀i≠j:|x_{i}−x_{j}|>2ϵ},  (3.8)

with respect to the obvious density. By symmetry, we have

∫∃i≠j:|x_{i}−x_{j}|≤2ϵ ≤ \frac{p(p − 1)}{2} ∫ dx_{1}...dx_{p−1} ∫_{|x_{p}−x_{p−1}|≤2ϵ} dx_{p}
≤ c (2ϵ)^{3} \frac{p(p − 1)}{2} ∫ dx_{1}...dx_{p−1} ℙ( max_{1≤i≤p−1} T_{ϵ}^{D}(x_{i}) ≤ 1) ℙ( max_{1≤i≤p−1} T_{ϵ}^{D}(x_{i}) < ∞)
≤ 4cp^{2}ϵ^{3} Υ_{p−1}(ϵ).  (3.9)

Let S_{p} be the set of permutations on [1, ..., p]. By the strong Markov property, we have

ℙ( max_{1≤i≤p} T_{ϵ}^{D}(x_{i}) < ∞)

≤ ∑_{σ∈S_{p}} ℙ(T_{ϵ}^{D}(x_{σ(1)}) ≤ ... ≤ T_{ϵ}^{D}(x_{σ(p)}) < ∞)

≤ ∑_{σ∈S_{p}} ℙ(T_{ϵ}^{D}(x_{σ(1)}) ≤ ... ≤ T_{ϵ}^{D}(x_{σ(p−1)}) < ∞) sup_{|y−x_{σ(p−1)}|≤ϵ} ℙ_{z}(T_{ϵ}^{D}(x_{σ(p)}) < ∞)

≤ ∑_{σ∈S_{p}} ∏_{i=1}^{p} (\frac{ϵ}{(|x_{σ(i)} − x_{σ(i−1)}| − ϵ)^{+}} ∧ 1),

by means of (3.7), with convention that σ(0) = 0 and x_{0} = 0. Similarly, we get from (3.6) that

ℙ( max_{1≤i≤p} T_{ϵ}^{D}(x_{i}) ≤ 1) ≤ (c_{p})^{p} ϵ^{p} ∑_{σ∈S_{p}} ∏_{i=1}^{p} ψ(|x_{σ(i)} − x_{σ(i−1)}| − ϵ)^{+}).

Plugging these into (3.8), we obtain that

∫_{∀i≠j:|x_{i}−x_{j}|>2ϵ} dx_{1}...dx_{p} ℙ( max_{1≤i≤p} T_{ϵ}^{D}(x_{i}) ≤ 1) ℙ( max_{1≤i≤p} T_{ϵ}^{D}(x_{i}) < ∞) ≤ c^{p} p! ϵ^{2p} J_{p}^{∗},

with

J_{p}^{∗} =^{def} max_{σ∈S_{p}} J_{p}(σ) =^{def} max_{σ∈S_{p}} ∫ dx_{1}...dx_{p} ∏_{i=1}^{p} \frac{ψ(|x_{σ(i)} − x_{σ(i−1)}|)}{|x_{i} − x_{i−1}|}).

We are going to prove that there exists some constant c > 0 such that

J_{p}^{∗} ≤ c^{p} 2^{3p^{2}},  (3.10)

which in view of (3.8) and (3.9) implies (3.5) and completes the proof.

To show (3.10), we firstly remark that the function y(∈ ℝ^{3}) → ∫_{ℝ^{3}} \frac{ψ(|x|)}{|x−y|}dx is continuous on the whole ℝ^{3} and goes to 0 when |y| → ∞, hence

sup_{y∈ℝ^{3}} ∫_{ℝ^{3}} \frac{ψ(|x|)}{|x − y|}dx = c < ∞.  (3.11)

This implies that for any y, z ∈ ℝ^{3},

∫ dx\frac{ψ(|x|) ψ(|x − y|)}{|x − z|} = ∫_{|x−y|≤|y|/2} + ∫_{|x−y|>|y|/2}
≤ ψ(\frac{|y|}{2}) ∫_{|x−y|≤|y|/2} \frac{ψ(|x − y|)}{|x − z|}dx + ψ(\frac{|y|}{2}) ∫_{|x−y|>|y|/2} \frac{ψ(|x|)}{|x − z|}dx
≤ 2 c ψ(\frac{|y|}{2}),  (3.12)

where in the first inequality, we make use of the monotonicity of ψ(·) and the fact that |x| ≥ |y|/2 on \{|x − y| ≤ |y|/2\}. We prove (3.10) by recurrence on p. When p = 1, J_{1}^{∗} = ∫ \frac{ψ(|x|)}{|x|}dx < ∞. Let p ≥ 2 and consider σ ∈ S_{p}. If σ(p) = p, we have

J_{p}(σ) = ∫ dx_{1}...dx_{p−1} ∏_{i=1}^{p−1} \frac{ψ(|x_{σ(i)} − x_{σ(i−1)}|)}{|x_{i} − x_{i−1}|}) ∫ dx_{p} \frac{ψ(|x_{p} − x_{σ_{p−1}}|)}{|x_{p} − x_{σ_{p−1}}|}
≤ c J_{p−1}(σ) ≤ c J_{p−1}^{∗},

by (3.11). If σ(p) ≠ p, we denote by j = j(σ, p) ≤ p − 1 such that σ(j) = p. We have

J_{p}(σ) = ∫ \frac{dx_{1}...dx_{p−1}}{|x_{1}|...|x_{p−1} − x_{p−2}|} ∏_{i=1,i≠j−1,i≠j}^{p} ψ(|x_{σ(i)} − x_{σ(i−1)}|) ∫ dx_{p} \frac{ψ(|x_{σ_{j}} − x_{σ_{j−1}}|)ψ(|x_{σ_{j+1}} − x_{σ_{j}}|)}{|x_{p} − x_{p−1}|}

≤ 2c ∫ \frac{dx_{1}...dx_{p−1}}{|x_{1}|...|x_{p−1} − x_{p−2}|} ∏_{i=1,i≠j−1,i≠j}^{p} ψ(|x_{σ(i)} − x_{σ(i−1)}|) ψ(\frac{|x_{σ_{j+1}} − x_{σ_{j−1}}|}{2})

≤ 2c ∫ dx_{1}...dx_{p−1} ∏_{i=1}^{p−1} \frac{ψ(|x_{σ̃(i)} − x_{σ̃(i−1)}|/2)}{|x_{i} − x_{i−1}|}

where we used (3.12) for the first inequality and the monotonicity of ψ for the second inequality, the new permutation σ̃ ∈ S_{p−1} is defined by: σ̃(k) = σ(k) for k < j and σ̃(k) = σ(k + 1) for j ≤ k ≤ p − 1. By the change of variable x = 2x̃, we get

J_{p}(σ) ≤ 2c 2^{3(p−1)} J_{p−1}(σ̃) ≤ 2^{3p−2} c J_{p−1}^{∗}.

Hence J_{p}^{∗} ≤ 2^{3p−2} c J_{p−1}^{∗} ≤ 2^{3p^{2}}c^{p}, yielding that for all p ≥ 1 and ϵ < 1/2,

𝔼(m(S_{ϵ}^{D}(0, 1) ∩ S̃_{ϵ}^{D}(0, ∞))^{p}) ≤ ϵ^{2p} c 2^{4p^{2}}.
□

Corollary 3.4 Fix the compact set K. For any p > 1, there exists some constant c_{p} > 0 such that

∥\{m(S^{K}(0, t))\} − \{m(S^{K}(0, s))\}∥_{p} ≤ c_{p} \sqrt{(t − s) log t}, 0 < s < t − 1;  (3.13)
∥ sup_{t≤u≤t+1} |m(S^{K}(0, u)) − m(S^{K}(0, t))|∥_{p} ≤ c_{p}.  (3.14)

Proof: Consider firstly the case t − s > 1. Write F \G := F ∩ G^{c}. Define B̃_{u} := B_{u+s} − B_{s}, u ≥ 0 and S̃^{K} the associated Wiener sausage. We have

{m(S^{K}(0, t))} − {m(S^{K}(0, s))} = {m(S^{K}(s, t)\S^{K}(0, s))}
=^{law} {m(S̃^{K}(0, t − s)\S^{K}(0, s))}
= {m(S̃^{K}(0, t − s))} − {m(S̃^{K}(0, t − s) ∩ S^{K}(0, s))}.

Firstly, the convergence in law of (1.3) holds in fact in L^{p} for any p > 1. This together with (1.2) shows that

∥\{m(S̃^{K}(0, t − s))\}∥_{p} ≤ c_{p} \sqrt{(t − s) log(3 + t − s)}, ∀ 0 < s < t − 1.

Secondly, we have

∥{m(S̃^{K}(0, t − s) ∩ S^{K}(0, s))}∥_{p} ≤ 2∥m(S̃^{K}(0, t − s) ∩ S^{K}(0, s))∥_{p}
≤ 2∥m(S̃^{K}(0, t − s) ∩ S^{K}(0, ∞))∥_{p}
≤ c′_{p} \sqrt{t − s},

by Lemma 3.3. Then (3.13) is proven.

For t ≤ u ≤ t + 1, we have

∥ sup_{t≤u≤t+1} |m(S^{K}(0, u)) − m(S^{K}(0, t))|∥_{p} = ∥{m(S^{K}(t, u)\S^{K}(0, t))}∥_{p}
≤ 2∥m(S^{K}(t, t + 1)\S^{K}(0, t))∥_{p}
= 2∥m(S̃^{K}(0, 1)\S^{K}(0, t))∥_{p}
≤ 2∥m(S^{K}(0, 1))∥_{p}
≤ c_{p},

which completes the proof. □

4 Increments of intersection local times

We quote a Tanaka-Rosen-Yor type formula for the intersection local time (see [16], [19], [20]):

Lemma 4.1 ([20], (3.c)) For any t ≥ 1, we have

α(0, A_{t}) = −\frac{1}{2π} ∫_{1}^{t} dB_{u} ∫_{0}^{u−1} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} − \frac{1}{2π} ∫_{1}^{t−1} ds (\frac{1}{|B_{t} − B_{s}|} − \frac{1}{|B_{1+s} − B_{s}|}) .

We shall estimate the L^{p} norm of several quantities related to the intersection local times α(0, A_{t}). Let us collect some preliminary results in the following lemma:

Lemma 4.2 Let t ≥ 27 and p ≥ 2. There exists some constant c_{p} > 0,

∥∫_{1}^{t} ds\frac{B_{s}}{|B_{s}|^{3}} ∥_{p} ≤ c_{p} \sqrt{log t}  (4.1)
∥∫_{1}^{t} ds\frac{B_{s}}{|B_{s}|^{3}} ∥_{2} = \sqrt{2 log t} + O(1),  (4.2)
∥∫_{0}^{t−1} \frac{ds}{|B_{t} − B_{s}|}∥_{p} ≤ c_{p}\sqrt{t},  (4.3)
∥\{∫_{0}^{t} \frac{ds}{|B_{1+s} − B_{s}|}\}∥_{p} ≤ c_{p}\sqrt{t}.  (4.4)

Proof: We shall constantly use the Burkholder-Davis-Gundy inequality for martingales: For any p > 0, there exist some constants c_{p} > 0 and c′_{p} > 0 such that for any continuous martingale (M_{t}), we have

c_{p} (∥⟨M⟩_{t}∥_{p/2})^{1/2} ≤ ∥ sup_{0≤s≤t} |M_{s}|∥_{p} ≤ c′_{p} (∥⟨M⟩_{t}∥_{p/2})^{1/2} .  (4.5)

We define R_{t} =^{def} |B_{t}|, t ≥ 0 which is the radial part of the three dimensional Brownian motion B. It is well-known ([15], Chap. 12) that R is a three-dimensional Bessel process and admits the following canonical decomposition:

R_{t} = γ_{t} + \frac{1}{2} ∫_{0}^{t} \frac{ds}{R_{s}} , t ≥ 0,

for some one-dimensional Brownian motion γ.

Applying Itô’s formula to \frac{x}{|x|} gives that

∫_{1}^{t} ds\frac{B_{s}}{|B_{s}|^{3}} = − (\frac{B_{t}}{|B_{t}|} − \frac{B_{1}}{|B_{1}|}) + ∫_{1}^{t} \frac{dB_{s}}{|B_{s}|} − ∫_{1}^{t} \frac{B_{s}}{|B_{s}|} \frac{B_{s} ∘ dB_{s}}{|B_{s}|^{2}}
=^{def} − (\frac{B_{t}}{|B_{t}|} − \frac{B_{1}}{|B_{1}|}) + M_{t},  u > 1.  (4.6)

Using the Burkholder-Davis-Gundy inequality (4.5) for the martingale M = (M_{u}^{(i)}), 1 ≤ i ≤ 3 implies that

∥M_{t}∥_{p} ≤ c_{p} ∑_{i=1}^{3} ∥⟨M^{(i)}⟩_{t}∥_{p/2}^{1/2}

= c_{p} ∑_{i=1}^{3} ∥∫_{1}^{t} ds (\frac{1}{|B_{s}|^{2}} − \frac{(B_{s}^{(i)})^{2}}{|B_{s}|^{4}} )∥_{p/2}^{1/2}

≤ 6c_{p}∥∫_{1}^{t} \frac{ds}{R_{s}^{2}} ∥_{p/2}^{1/2}
≤ c′_{p}\sqrt{log t},

where the last inequality is a consequence of the fact that

\frac{1}{log t} ∫_{1}^{t} \frac{ds}{R_{s}^{2}} → 1,  a.s. and in L^{p}.  (4.7)

Thus (4.1) is proven. Observe that

𝔼|M_{t}|^{2} = ∑_{i=1}^{3} 𝔼⟨M^{(i)}⟩_{t} = ∑_{i=1}^{3} 𝔼 ∫_{1}^{t} ds (\frac{1}{|B_{s}|^{2}} − \frac{(B_{s}^{(i)})^{2}}{|B_{s}|^{4}} ) = 2 log t, t > 1,

which together with (4.6) yields (4.2). It follows from Brownian time-reversal that

𝔼 (∫_{0}^{t} \frac{ds}{|B_{t} − B_{s}|})^{p} = 𝔼 (∫_{0}^{t} \frac{ds}{R_{s}} )^{p}
= 2^{p}𝔼 (R_{t} − γ_{t})^{p}
≤ 4^{p} (𝔼(R_{t})^{p} + 𝔼|γ_{t}|^{p})
= c′_{p}t^{p/2},

by the scaling property. This yields (4.3). To obtain (4.4), we write

∫_{0}^{t} \frac{ds}{|B_{1+s} − B_{s}|} = ∑_{j=0}^{[t/2]} ∫_{2j}^{2j+1} \frac{ds}{|B_{1+s} − B_{s}|} + ∑_{j=0}^{[t/2]} ∫_{2j+1}^{2j+2} \frac{ds}{|B_{1+s} − B_{s}|} − ∫_{t}^{2[t/2]+2} \frac{ds}{|B_{1+s} − B_{s}|}.

In each of two sums on j, we have independent and identically distributed variables with common law that of ∫_{0}^{1} \frac{ds}{|B_{1+s}−B_{s}|} ∈ L^{p}; hence by Rosenthal’s inequality (3.2), the two sums centered by their expectations have a L^{p} norm bounded by O(\sqrt{t}). The L^{p}-norm of the third term itself is less than ∥∫_{t}^{t+2} \frac{ds}{|B_{1+s}−B_{s}|} ∥_{p} = ∥∫_{0}^{2} \frac{ds}{|B_{1+s}−B_{s}|} ∥_{p} is bounded on t. Then (4.4) is proven and completes the proof

of Lemma 4.2. □

Lemma 4.3 For any ϵ > 0, almost surely,

\{∫_{1}^{t−1} \frac{ds}{|B_{1+s} − B_{s}|}\} = o (\sqrt{t} (log t)^{δ}) ,  a.s., t → ∞,  (4.8)
∫_{1}^{t−1} \frac{ds}{|B_{t} − B_{s}|} = o (\sqrt{t} (log t)^{δ}) ,  a.s., t → ∞,  (4.9)

Proof: The proof is based on an application of Propositions 2.2 and 2.3. We shall prove that for any t > s > 2,

∥∫_{0}^{t−1} du\frac{1}{|B_{t} − B_{u}|} − ∫_{0}^{s−1} du\frac{1}{|B_{s} − B_{u}|}∥_{p} ≤ c_{p}\sqrt{(t − s) log t}.  (4.10)

Therefore, by virtue of (4.3) and (4.10), Proposition 2.3 implies (4.8). The (4.9) can be proven similarly.

To show (4.10), we have

∥∫_{0}^{t−1} du\frac{1}{|B_{t} − B_{u}|} − ∫_{0}^{s−1} du\frac{1}{|B_{s} − B_{u}|}∥_{p}
≤ ∥∫_{s−1}^{t−1} du\frac{1}{|B_{t} − B_{u}|}∥_{p} + ∥∫_{0}^{s−1} du (\frac{1}{|B_{t} − B_{u}|} − \frac{1}{|B_{s} − B_{u}|})∥_{p}
≤ c_{p}\sqrt{t − s} + ∥∫_{s}^{t} dB_{r} ∫_{0}^{s−1} du\frac{B_{r} − B_{u}}{|B_{r} − B_{u}|^{3}} ∥_{p},

by using respectively (4.3) to obtain the first term and the Itô’s formula for the second term. Now, by using (4.5), the L^{p} norm in the above inequality is smaller than

c_{p}∥∫_{s}^{t} dr |∫_{0}^{s−1} du\frac{B_{r} − B_{u}}{|B_{r} − B_{u}|^{3}} |^{2}∥_{p/2}^{1/2} ≤ c_{p} (∫_{s}^{t} dr∥∫_{1}^{r} \frac{du}{R_{u}^{2}} ∥_{p})^{1/2} ≤ c′_{p}\sqrt{(t − s) log t}.

□

We can now prove our main estimate in this section:

Lemma 4.4 For 1 ≤ s < t and t > 2,

∥∫_{1}^{t} du |∫_{0}^{u−1} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} |^{2} − 2t log t∥_{p} ≤ c_{p} t\sqrt{log t log log t},  (4.11)

∥∫_{s}^{t} du |∫_{0}^{u−1} dv \frac{B_{u} − B_{v}}{|B_{u} − B_{v}|^{3}} |^{2}∥_{p} ≤ c_{p}(t − s) log t,  (4.12)

Proof: Notice that by Brownian time-reversal and (4.1),

∥∫_{0}^{u−1} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} ∥_{p} = ∥∫_{1}^{u} ds\frac{B_{s}}{|B_{s}|^{3}} ∥_{p} ≤ c_{p} \sqrt{log u}  u ≥ 2,  (4.13)

which yields (4.12). In view of (4.2), (4.11) is equivalent to prove that

∥∫_{1}^{t} du\{|∫_{0}^{u−1} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} |^{2}\}∥_{p} ≤ c_{p}t\sqrt{log t log log t}.

Define

Y_{u}(a, b) = ∫_{a}^{b} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} , 0 < a < b < u.

Then by scaling property, ∥∫_{1}^{t} du\{|∫_{0}^{u−1} ds\frac{B_{u}−B_{s}}{|B_{u}−B_{s}|^{3}} |^{2}\}∥_{p} = t ∥∫ 1_{\frac{1}{t}} du\{|Y_{u}(0, u − \frac{1}{t})|^{2}\}∥_{p}. Then

it suffices to show that for 0 < ϵ < 1/27,

∥∫_{ϵ}^{1} du\{|Y_{u}(0, u − ϵ)|^{2}\}∥_{p} ≤ c_{p}\sqrt{log \frac{1}{ϵ} log log \frac{1}{ϵ}}.  (4.14)

To this end, we remark that for any 0 < a < b < u,

∥Y_{u}(a, b)∥_{p} = ∥∫_{u−b}^{u−a} ds\frac{B_{s}}{|B_{s}|^{3}} ∥_{p}
≤ 2 + ∥M_{u−a} − M_{u−b}∥_{p}
≤ 2 + c_{p}∥∫_{u−b}^{u−a} \frac{ds}{R_{s}^{2}} ∥_{p/2}^{1/2}
≤ 2 + c_{p}\sqrt{log \frac{u − a}{u − b}},

by means of (4.7). Choose δ = \frac{1}{n} ∼ (log \frac{1}{ϵ})^{−θ} for some constant θ > 2. We have

∫_{ϵ}^{1} du{|Y_{u}(0, u − ϵ)|^{2}} = ∫_{ϵ}^{δ} du{|Y_{u}(0, u − ϵ)|^{2}} + ∫_{δ}^{1} du{|Y_{u}(0, u − ϵ)|^{2}}.

Observe that

∥∫_{ϵ}^{δ} du{|Y_{u}(0, u − ϵ)|^{2}}∥_{p} ≤ 2∥∫_{ϵ}^{δ} du|Y_{u}(0, u − ϵ)|^{2}∥_{p}
≤ 2 ∫_{ϵ}^{δ} du∥|Y_{u}(0, u − ϵ)|^{2}∥_{p}
≤ c_{p} ∫_{ϵ}^{δ} du(1 + log(u/ϵ))
≤ c_{p}δ log(1/ϵ) = c_{p}(log(1/ϵ))^{1−θ}.  (4.15)

We claim that

∥∫_{δ}^{1} du\{|Y_{u}(0, u − ϵ)|^{2}\} − ∫_{δ}^{1} du\{|Y_{u}(u − δ, u − ϵ)|^{2}\}∥_{p} ≤ c_{p}\sqrt{log \frac{1}{ϵ} log log \frac{1}{ϵ}}.  (4.16)

In fact, by Cauchy-Schwarz’ inequality,

∥∫_{0}^{1} dsf(u)g(u)∥_{p} ≤ (∥∫_{0}^{1} du|f(u)|^{2}∥_{p})^{1/2} (∥∫_{0}^{1} du|g(u)|^{2}∥_{p})^{1/2} .

Then by writing Y_{u}(0, u − ϵ) = Y_{u}(0, u − δ) + Y_{u}(u − δ, u − ϵ), we have

∥∫_{δ}^{1} du{|Y_{u}(0, u − ϵ)|^{2}} − ∫_{δ}^{1} du{|Y_{u}(u − δ, u − ϵ)|^{2}}∥_{p}
≤ 2∥∫_{δ}^{1} du |Y_{u}(0, u − δ)|^{2}∥_{p} + 2∥∫_{δ}^{1} du |Y_{u}(0, u − δ)|^{2}∥_{p}^{1/2} ∥∫_{δ}^{1} du |Y_{u}(u − δ, u − ϵ)|^{2}∥_{p}^{1/2}.

But, ∥∫_{δ}^{1} du |Y_{u}(0, u − δ)|^{2}∥_{p} ≤ c_{p} ∫_{δ}^{1} du(1 + log(1/δ)) ≤ c_{p} log 1/δ ≤ c′p log log(1/ϵ), and similarly ∥∫_{δ}^{1} du |Y_{u}(u − δ, u − ϵ)|^{2}∥_{p} ≤ c_{p} ∫_{δ}^{1} du(1 + log δ/ϵ) ≤ c′_{p} log(1/ϵ). Hence (4.16) is obtained.

To bound the L^{p} norm of ∫_{δ}^{1} du{|Y_{u}(u − δ, u − ϵ)|^{2}}, we cut the interval [δ, 1] into n = 1/δ parts:

∫_{δ}^{1} du{|Y_{u}(u − δ, u − ϵ)|^{2}} = ∑_{j≤n−1:j odd} ∫_{jδ}^{(j+1)δ} + ∑_{j≤n−1:j even} ∫_{jδ}^{(j+1)δ} .

Remark that the sum on odd j ≤ n − 1 is a sum of iid variables whose common law is that of

∫_{δ}^{2δ} \{|Y_{u}(u − δ, u − ϵ)|^{2}\} =^{law} δ ∫_{1}^{2} du\{|∫_{u−1}^{u−ϵ/δ} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} |^{2}\}.

Applying (3.2), we obtain that

∥ ∑_{j≤n−1:j odd} ∫_{jδ}^{(j+1)δ} |Y_{u}(u − δ, u − ϵ)|^{2}∥_{p} ≤ c_{p} n^{1/2} δ ∥∫_{1}^{2} du\{|∫_{u−1}^{u−ϵ/δ} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} |^{2}\}∥_{p}
≤ c′_{p} n^{1/2} δ ∫_{1}^{2} du(1 + log δ/ϵ)
≤ c′_{p}δ^{1/2} log(1/ϵ)
= c′_{p}(log 1/ϵ)^{1−θ/2}.

The same holds for the sum on even j. Since θ > 2, we get ∥∫_{δ}^{1} du{|Y_{u}(u − δ, u − ϵ)|^{2}}∥_{p} = O(1) which in view of (4.15) and (4.16) implies (4.14), as desired. □

5 Proof of Theorem 1.1

Denote by

N_{t} = −\frac{1}{2π} ∫_{1}^{t} dB_{u} ∫_{0}^{u−1} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} , t > 1.

We claim that

X_{t} =^{def} \{m(S^{K}(0, t))\} − C(K)^{2}N_{t} = o (\sqrt{t} (log t)^{δ}) ,  a.s., t → ∞.  (5.1)

In fact, by Proposition 3.1, (4.3) and (4.4), we have

∥X_{t}∥_{p} ≤ c_{p}\sqrt{t}.  (5.2)

On the other hand, applying (4.5) to N_{t}, we deduce from Lemma 4.4 and Corollary 3.4 that

∥X_{t} − X_{s}∥_{p} ≤ c_{p}\sqrt{(t − s) log t},  t > s > 27,
∥ sup_{t≤u≤t+1} |X_{u} − X_{t}|∥_{p} ≤ c_{p}\sqrt{log t},  t > 27,

which together with (5.2) allows us to apply Proposition 2.2, and shows (5.1).

We can also apply Proposition 2.3 in view of Lemma 4.4, and we obtain that

∫_{1}^{t} du |∫_{0}^{u−1} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} |^{2} − 2t log t = o (t(log t)^{\frac{1}{2} +δ}) ,  a.s., t → ∞.  (5.3)

By Dubins-Schwarz’s representation, there exists some one-dimensional Brownian motion β such that

N_{t} = \frac{1}{π\sqrt{2}}β (\frac{1}{2} ∫_{1}^{t} du |∫_{0}^{u−1} ds\frac{B_{u} − B_{s}}{|B_{u} − B_{s}|^{3}} |^{2})
= \frac{1}{π\sqrt{2}}β(t log t) + o (t^{1/2}(log t)^{\frac{1}{4} +δ}) ,  a.s., t → ∞,  (5.4)

by using (5.3) and the Brownian increments (cf. [4], Theorem 1.2.1): For a non-decreasing function 0 < a_{t} ≤ t such that t/a_{t} ↑, we have

lim sup_{t→∞} \frac{1}{\sqrt{2a_{t}(log(t/a_{t}) + log log t)}} sup_{0≤s≤t−a_{t}} sup_{0≤v≤a_{t}} |β̃(s + v) − β̃(s)| = 1, a.s.

Theorem 1.1 follows by assembling (5.1) and (5.4). □

Acknowledgements. Cooperation between the authors was supported by the joint French-Hun-garian Intergovernmental Grant ”Balaton” no. F-39/00.

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[17] Spitzer, F.: Electrostatic capacity, heat flow, and Brownian motion. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964) 110–121.

[18] Weinryb, S. and Yor, M.: Théorème central limite pour l’intersection de deux saucisses de Wiener indépendantes. Probab. Theory Related Fields 97 (1993), no. 3, 383–401.

[19] Yor, M.: Compléments aux formules de Tanaka-Rosen. Séminaire de Probabilités XIX 1983/84, pp. 332–349, Lecture Notes in Math., 1123, Springer, Berlin, 1985.

[20] Yor, M.: Renormalisation et convergence en loi pour les temps locaux d’intersection du mouvement brownien dans ℝ^{3}. In the same reference as [19] pp. 350–365.