On the spectral function of the Poisson-Voronoi cells

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ON THE SPECTRAL FUNCTION OF THE POISSON–VORONOI CELLS

André GOLDMAN^∗, Pierre CALKA

Université Claude Bernard Lyon 1, LaPCS, bât. B, domaine de Gerland, 50, avenue Tony-Garnier, 69366 Lyon cedex 07, France

Received 26 March 2002, accepted 26 February 2003

ABSTRACT. – Denote by ϕ(t)=

n1e⁻^λⁿ^t, t >0, the spectral function related to the Dirichlet Laplacian for the typical cellCof a standard Poisson–Voronoi tessellation inR^d, d2.

We show that the expectation Eϕ(t),t >0, is a functional of the convex hull of a standardd- dimensional Brownian bridge. This enables us to study the asymptotic behaviour of Eϕ(t), when t →0⁺,+∞. In particular, we prove that the law of the first eigenvalueλ1 ofC satisfies the asymptotic relation ln P{λ1t} ∼ −2^dωdj_(d^d₋_2)/2·t⁻^d/2whent→0⁺, whereωd andj(d−2)/2

are respectively the Lebesgue measure of the unit ball in R^d and the first zero of the Bessel functionJ_(d₋_2)/2.

2003 Éditions scientifiques et médicales Elsevier SAS

MSC: primary 35P15, 60D05, 60J65; secondary 52A22, 60G55

Keywords: Brownian bridge; Convex hull; Eigenvalues; Empirical distributions;

Johnson–Mehl tessellations; Laplace operator; Palm distribution; Spectral function; Stochastic geometry; Voronoi tessellations

RÉSUMÉ. – On considèreϕ(t)=

n1e⁻^λⁿ^t, t >0, la fonction spectrale du Laplacien avec condition de Dirichlet au bord de la cellule typique C d’une mosaïque de Poisson–Voronoi standard deR^d, d2. Nous montrons que l’espérance Eϕ(t),t >0, est une fonctionnelle de l’enveloppe convexe d’un pont browniend-dimensionnel standard. Cela nous permet d’étudier le comportement asymptotique de Eϕ(t), quandt→0⁺,+∞.En particulier, nous prouvons que la loi de la première valeur propreλ₁deCsatisfait la relation ln P{λ₁t} ∼ −2^dω_dj_(d^d₋_2)/2·t⁻^d/2 quandt→0⁺, oùω_d etj_(d₋_2)/2sont respectivement la mesure de Lebesgue de la boule-unité deR^det le premier zéro de la fonction de BesselJ_(d₋_2)/2.

2003 Éditions scientifiques et médicales Elsevier SAS

∗Corresponding author.

E-mail addresses: andre.goldman@univ-lyon1.fr (A. Goldman), pierre.calka@univ-lyon1.fr (P. Calka).

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0. Introduction and statement of the main results

Consider= {xn; n1}a homogeneous Poisson point process inR^d, d 2, with thed-dimensional Lebesgue measureVd for intensity measure. The set of cells

C(x)=y∈R^d; y−xy−x, x∈, x∈,

(which are almost surely bounded polyhedra) is the well known Poisson–Voronoi tessellation of R^d. Introduced by Meijering [17] and Gilbert [8] as a model of crystal aggregates, it provides now models for many natural phenomena as thermal conductivity [16], telecommunications [2], astrophysics [28] and ecology [22]. An extensive list of the areas in which the tessellation has been used can be found in Stoyan et al. [25] and Okabe et al. [21].

In order to describe the statistical properties of the tessellation, the notion of typical cellCin the Palm sense is commonly used [19]. Consider the spaceKof convex compact sets of R^d endowed with the usual Hausdorff metric. Let us fix an arbitrary Borel set B ⊂R^d such that 0< Vd(B) <+∞. The typical cell C is defined by means of the identity [19]:

Eh(C)= 1

Vd(B)E

x∈B∩

hC(x)−x,

whereh:K→Rruns throughout the space of bounded measurable functions.

Consider now the cell

C(0)=y∈R^d; yy−x, x∈

obtained when the origin is added to the point process . It is well known [19] that C(0)andCare equal in law. On the other hand, the typical cell can also be characterized by means of the empirical distributions. Indeed, let Vd,R be the set of all cells C(x), x ∈, included in the ball B(R) centered at the origin and of radius R >0. Let us defineNR=#Vd,R and fixh:K→Ran arbitrary bounded measurable function. Then (see [11]):

Eh(C)= lim

R→+∞

1 NR

C(x)∈Vd,R

hC(x)−x, a.s.

Explicit formulas for the distributions of the main geometric characteristics of the typical cell of the Poisson–Voronoi tessellation have been recently obtained [3,4].

Nevertheless their expressions are intricate and in general, we do not have any precise idea about the asymptotic behaviour of these distributions.

In this paper, we are interested in the properties of the fundamental frequencies of the typical cellC. More precisely let us consider 0< λ1< λ2· · ·λn· · ·the (random) eigenvalues of the Laplacian under Dirichlet boundary conditions onCand denote by

ϕ(t)=

n1

e⁻^λⁿ^t, t >0,

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the associated spectral function. Let us remark that the spectral function of deterministic sets has been largely studied in connection with their geometrical structure [7,14,29,30].

Besides, let

W (t)∈R^d, t∈ [0,1], W (0)≡W (1)≡0,

be a d-dimensional Brownian bridge on the interval [0,1] independent of the point process. Let us denote by

W=W (t); 0t1⊂R^d (1)

the associated path and byW its closed convex hull.

For a setD⊂R^d, we define B(D;x)=

y∈D

By,y−x, B(D;x)=

y∈D

By,y−x,

whereB(y, r)(resp. B(y, R)= {x∈R^d; x−y< R}) is the closed (resp. open) ball centered at y and of radius R >0. Vd(D, x) will denote the d-dimensional Lebesgue measure of the setB(D;x), i.e.,

Vd(D, x)=Vd B(D;x).

The notation P (resp. P) is used for the probability associated to the point process (resp. to the Brownian bridgeW). Likewise the expectations E and E refer respectively toandW.

We show that the expectation Eϕ(t), t >0, is a functional of the Lebesgue measure of the setsB(W;x), x∈R^d.

THEOREM 1. – Ford2, Eϕ(t)= 1

(2π )^d/2

E exp−(2t)^d/2Vd W, xdx

= 1 (4π t)^d/2

E exp−Vd

√

2tW, x dx, t >0. (2) Formula (2) is the key point for the proof of our main result providing the logarithmic asymptotic behaviour of the distribution function (and of the Laplace transform) of the square of the fundamental frequency (e.g., the first eigenvalue λ1 of the Dirichlet laplacian) of the Poisson–Voronoi typical cell.

THEOREM 2. – Denote byµ1the first eigenvalue of the largest random ball centered at the origin and included inC(0). We have

tlim→∞t⁻^d/(d⁺²⁾ln E e⁻^{t λ}¹= lim

t→∞t⁻^d/(d⁺²⁾ln Eϕ(t)= lim

t→+∞t⁻^d/(d⁺²⁾ln E e⁻^{t µ}¹

= −2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²₋_2)/2 d

d/(d+2)

, (3)

(4)

and

lim

t→0⁺t^d/2ln P{λ1t} = lim

t→0⁺t^d/2ln P{µ1t} = −2^dωdj_(d^d₋_2)/2, (4) whereωd=σd/d andj(d−2)/2are respectively the Lebesgue measure of the unit-ball of R^d and the first positive zero of the Bessel functionJ(d−2)/2.

This non-trivial result is strongly connected with the Donsker–Varadhan theorem about the volume of the Wiener sausage [6] providing the logarithmic equivalent, when t→ +∞, of the survival probability P{√

2tW⊂ [x∈B(x, ε)]^c}of a Brownian path until timet >0 in a random medium of Poissonian obstacles which are the balls centered atx∈and of fixed radiusε >0. More precisely, we will show as an intermediary result of our proof that this last logarithmic equivalent is precisely the same as the logarithmic equivalent of the probability P{√

2tW⊂2·C(0)}that the Brownian bridge stays until timet >0 in the homothetic 2·C(0)of the Poisson–Voronoi typical cell.

Besides, Theorem 2 shows that the asymptotic behaviour of the first eigenvalue of the typical cell C(0)is the same as the first eigenvalue of the largest ball included inC(0).

This result is substantially close to the work of A.S. Sznitman who proved that the large deviations of the first eigenvalue of the complementary of a Poisson cloud of obstacles (e.g., balls) in a fixed box are controlled by the largest ball free of obstacles (see [27], p. 182).

We can also exploit Theorem 1 by investigating the asymptotic estimation of Eϕ(t) when t → 0⁺. Let us recall that H. Weyl proved in 1911 [30] that for a bounded domain U ⊂R^d with a piecewise-smooth boundary, the spectral function ϕU satisfies the asymptotic relation

ϕU(t) ∼

t→0⁺

Vd(U ) (4π t)^d/2.

For a bounded polygonal convex domain in R², it was shown (see [14,23,29]) that whent→0⁺,

ϕU(t)=V2(U )

4π t − V1(U ) 4√

4π t + 1 24

π α⁻¹(U )−N0(U )+2+Oe⁻^c/t, (5)

where

(i) V1(U )is the perimeter ofU;

(ii) N0(U )is the number of vertices ofU;

(iii) α⁻¹(U ) = ^Ni=⁰1^{(U )}1/αi is the harmonic mean of the inside-facing angles α₁, . . . , αN₀(U )at the vertices ofU;

(iv) c >0 is a positive constant independent oft >0.

For a bounded convex non-degenerate polyhedron U⊂R^d, d 3, Fedosov [7] proved that we have whent→0⁺,

ϕU(t)= Vd(U )

(4π t)^d/2− Vd−1(U ) 4(4π t)^(d⁻^1)/2

+ 1

8(4π t)^(d⁻^2)/2

Nd−2(U ) i=1

1 3

ωi

π − π ωi

Vd−2(Fi)+O 1

t^(d⁻^3)/2

, (6)

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where

(i) Vd−1(U )is the(d−1)-dimensional measure of the boundary ofU; (ii) Fi, i=1, . . . , Nd−2(U ), are the(d−2)-dimensional faces ofU;

(iii) ωi is the magnitude of the dihedral angle at the faceFi, 1iNd−2(U ).

Returning now to the spectral function of the typical cellC, we prove that for allk1 we have the asymptotic relation (whent→0⁺)

Eϕ(t)= 1 (4π t)^d/2

_k₋₁

i=0

cd,it^i/2+Ot^k/2

, (7)

where the coefficientscd,iare expressed in terms of the covariances

c(u1, . . . , uj)=E(Mu₁· · ·Mu_j), u1, . . . , uj∈S^d⁻¹, 1j i,

and Mu =sup₀_s₁(u ·W (s)), u ∈ S^d⁻¹, is the projection of the d-dimensional Brownian path W on the half-lineR₊u(see Remark 3).

The calculation of the coefficients c2,i, i0 (resp. cd,i, i=0,1), shows that they have the same geometrical meaning (up to the expectation) as in deterministic formulas (5) and (6).

This paper is structured as follows. The first section is devoted to some useful preliminaries on the convex hull of the Brownian bridge. We then prove in the second section Theorem 1 and we provide some easy consequences of it. The following two sections are focused respectively on the proof of Theorem 2 about the asymptotic behaviour of Eϕ(t), t→ +∞, and on the asymptotic estimation of Eϕ(t), t →0⁺. We finally enunciate some concluding remarks.

The principal results of this paper were announced in [12].

1. Preliminaries on the convex hull of thed-dimensional Brownian bridge We begin with the following elementary facts:

PROPOSITION 1. – Consider a bounded closed setC⊂R^dand denote byCits closed convex hull. Furthermore letD⊂C be a countable, dense subset ofC. We have

B(C;x)=B C;x, (8)

Vd(D, x)=Vd(C, x)=Vd

B(D;x), x∈R^d. (9) Proof of (8). – Letu∈S^d⁻¹be a unit vector such that

(x+R+u)∩B C;x= ∅.

It can be easily seen that there exist a pointz∈(x+R+u)and a support hyperplaneHu

ofC perpendicular tousuch that:

(i) (x+R+u)∩B(C;x)=xz, wherexz= {λx+(1−λ)z, 0λ1}is the closed segment with bounding pointsx, z;

(ii) y−z = y−x, ∀y∈Hu∩∂C.

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Moreover it is known ([24], Corollary 18.3.1) that the intersectionHu∩∂Cmust contain at least one pointy∈C. Therefore

(x+R+u)∩B C;x⊂B(C;x) ∀u∈S^d⁻¹, which implies (8). ✷

Proof of (9). – Fixy∈B(D;x). An elementary geometrical argument shows (the set Dbeing bounded) that

λx+(1−λ)y; 0< λ1⊂B(D;x).

Integrating then in spherical coordinates (withxas center) we obtain the equality Vd(D, x)=Vd

B(D;x).

Combining this with the obvious inclusions

B(D;x)⊂B(C;x)⊂B(C;x)⊂B(D;x), x∈R^d, we obtain the result. ✷

Our next task is to give a useful estimate of the difference Vd(C, x)−ωdx^d=Vd

B(C;x)\Bx

for setsC⊂R^d containing the origin. It follows from (8) that we may suppose that the setCis convex. Let us introduce a few notations. Fixx∈R^d\ {0}and define:

(i) H= {y∈R^d; (y−x)·x=0}the polar hyperplane of the pointx;

(ii) H⁺= {y∈R^d;(y−x)·x0}the half-space associated withHand containing the origin;

(iii) S⁺=S^d⁻¹∩(H⁺−x), S⁻=S^d⁻¹\S⁺;

(iv) H0,u the hyperplane perpendicular to the vector u∈S^d⁻¹ and containing the origin;

(v) Huthe support hyperplane ofCperpendicular touand included in the half-space H_0,u⁺ = {y∈R^d; y·u0};

(vi) A(C, x)= {u∈S⁻; Hu∩(x+R+u)= ∅}(notice thatHu∩(x+R+u)= ∅for allu∈S⁺);

(vii) m(x, u) = d(x, Hu) the distance between x and Hu, ρ(x, u) = |x · u| = d(x, H0,u) the distance between x and H0,u, h(u)=d(0, Hu) the distance betweenHuandH0,u.

Remark 1. – For allx∈R^d\ {0}, we have







m(x, u)=h(u)+ρ(x, u) ifu∈S⁺, m(x, u)= |h(u)−ρ(x, u)| ifu∈S⁻, A(C, x)= {u∈S⁻; h(u)−ρ(x, u)0}.

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PROPOSITION 2. – Forx∈R^d, we have:

Vd(C, x)−ωdx^d=2^d d

d j=1

d j S⁺

h(u)^jρ(x, u)^d⁻^jdνd(u)

+2^d d

S⁻

h(u)−ρ(x, u)∨0^ddνd(u), (11)

and in particular,

Vd(C,0)=2^d d

S^d⁻¹

h(u)^ddνd(u). (12)

Proof. – Foru∈S^d⁻¹andx∈R^d\ {0}, three possibilities occur:

Case 1.u /∈S⁺∪A(C, x)and consequently

(x+R+u)∩B(C;x)=(x+R+u)∩Bx= {x}. Case 2.u∈S⁺which implies that

(x+R+u)∩B(C;x)=xz, (x+R+u)∩Bx=xz, with

x−z =2m(x, u), x−z =2ρ(x, u).

Case 3.u∈A(C, x)which implies that

(x+R₊u)∩B(C;x)=xz, (x+R₊u)∩Bx= {x}, with

x−z =2m(x, u).

Then integration in spherical coordinates (withx as center) gives that Vd(C, x)−ωdx^d

=

S⁺

2m(x,u)

2ρ(x,u)

r^d⁻¹dr

dνd(u)+

A(C,x)

2m(x,u)

0

r^d⁻¹dr

dνd(u)

= 2^d

d S⁺

m(x, u)^d−ρ(x, u)^ddνd(u)+

A(C,x)

m(x, u)^ddνd(u)

. (13) From (10) we get that

m(x, u)^d−ρ(x, u)^d= d j=1

d j

h(u)^jρ(x, u)^d⁻^j, u∈S⁺, and

(8)

h(u)−ρ(x, u)∨0=

m(x, u) foru∈A(C, x), 0 foru∈S⁻\A(C, x).

Substituting these expressions in (13) we find the final result (11).

To prove (12), it suffices to notice that for allu∈S^d⁻¹, we have R+u∩B(C;0)=0z

with

z =2h(u),

so integrating in spherical coordinates we obtain the result. ✷

Suppose now thatC is a random convex set containing the origin and invariant (in law) by rotations with the origin as center. Thus the random variables h(u), u∈S^d⁻¹, are equal in law and we obtain:

PROPOSITION 3. – Suppose thatCsatisfies the above conditions and that E{h(u)^d}

<∞, u∈S^d⁻¹. Fixingu0∈S^d⁻¹, we have then EVd(C, x)−ωdx^d

= d j=1

Id,jx^d⁻^jEh(u0)^j+2^d d

S−

Eh(u0)−ρ(x, u)∨0^ddνd(u), where

Id,j=2^d d σd−1

d j

¹

0

t^d⁻^j1−t²^(d⁻^3)/2dt

=2^d d

d j

π^(d⁻^1)/20((d+1−j )/2)

0(d−j/2) , 1jd.

Proof. – Taking the expectation in (11) the result follows from the direct evaluation of the integrals

S⁺

ρ(x, u)^d⁻^jdνd(u)= x^d⁻^jId,j. ✷

Remark 2. – Under the conditions stated in Proposition 3 we obtain that EVd(εC, x)−ωdx^d ∼

ε→0Id,1εx^d⁻¹Eh(u0).

In particular, in dimension d =2, it follows from the Cauchy formula giving the perimeter of a convex set that

EV2(εC, x)−πx² ∼

ε→0ε4x

π EV1(C), whereV1(C)denotes the perimeter of the convex setC.

Choose now for C the closed convex hull W⊂ R^d, of the sample path of the d-dimensional Brownian bridge on the interval [0,1]. Recall that W is invariant by

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rotations with the origin as center. Hence the random variablesh(u), u∈S^d⁻¹, defined above coincide in law with the maximum M0 of the 1-dimensional Brownian bridge.

The law ofM0is explicitely known, namely [26]

P{M0u} =e⁻^2u². (14) Hence all the moments ofM0are finite, and we have

EM02k= 1

2 k

k!, EM02k+1=(2k+1)!

8^kk!

√π 2√

2, k∈N. In particular,

EVd

εW, x −ωdx^d ∼

ε→0Id,1εx^d⁻¹

√π 2√

2. Now, denote by

M=sup

y∈W

y = sup

0s1

W (s) the maximum of the radial part of the Brownian bridge

W (s)=W1(s), . . . , Wd(s), 0s1.

The components Wi(s), 0s 1, i =1, . . . , d, are independent one-dimensional Brownian bridges. Hence

P{Ms}dP

sup

0s1

W₁(s) s

√d

2dP

M₀ s

√d

=2de⁻^2s²^/d, s0, (15) and

EM^k<+∞, ∀k0. (16)

As a consequence we deduce the following result:

PROPOSITION 4. – For allk∈N^∗there exists a constant 0< ck<+∞such that EVd

εW, x −ωdx^d^kck

kd i=k

εⁱx^kd⁻ⁱ,

ε >0, x∈R^d.

Proof. – From Proposition 2 we have Vd

εW , x −ωdx^d 2^d

d d−1

j=1

d j

(εM)^j

S⁺

ρ(x, u)^d⁻^jdνd(u)+ωd(εM)^d

,

which by (16) implies the result. ✷

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2. Proof of Theorem 1 and consequences Proof of Theorem 1. – Consider the spectral function

ϕ(t)=

n1

e⁻^{t λ}ⁿ, t >0, of the typical cellC.

Let us recall first (see [9,26]) that the spectral functionϕU(t), t >0, of any bounded domainU⊂R^d can be expressed in term of the Brownian bridge under the form:

ϕU(t)= 1 (4π t)^d/2

U

Px+√

2tW⊂Udx. (17)

Applying the formula above to the domain

C(0)=y∈R^d; yy−x, x∈^law=C,

and taking the expectation we obtain (by Fubini theorem) Eϕ(t)= 1

(4π t)^d/2E

Px+√

2tW⊂C(0)dx Observe that

−x+√

2tW⊂C(0) ⇐⇒ ∩−x+B√

2tW;x= ∅. Therefore applying the property of the Poisson point processwe obtain

P−x+√

2tW⊂C(0)=exp−Vd

√

2tW, x , and consequently

Eϕ(t)= 1 (4π t)^d/2

E exp−Vd

√

2tW, x dx.

The obvious identity Vd

√

2tW, x=(2t)^d/2Vd

W, x

√2t

, t >0, x∈R^d, and an elementary change of variable provide the result. ✷

In particular, Theorem 1 provides an infinite expansion of Eϕ(t)ford=2, which is valid for everyt >0.

THEOREM 3. – Ford=2 there existst0>0 such that:

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Eϕ(t)= 1 4π t

n0

(−1)ⁿ n!

EV2

√

2tW, x−πx²ⁿe⁻^π^x²dx, (18) for all 0< t < t0, the series being absolutely convergent.

Proof. – Regarding (2) it suffices to prove that there existst0>0 such that

E expV2

2t0W, x −2πx²dx <+∞.

Using the obvious inclusion B√

2tW;x⊂B2√

2t M+ x, we obtain the inequality

V2

√

2tW, xπx +2√

2t M²3π

2 x²+24tπ M². (19) Hence,

E expV2

√

2tW, x −2πx²dx

E exp

V2

√

2tW, x −π 2x²

dxE exp24tπ M²

=

+∞

0

e^sP

24π M²s t

ds+11+4 +∞

0

e^se⁻^{s/(24π t )}ds, by using (15). Thus it suffices to taket0<1/(24π ). ✷

In the two-dimensional case d = 2, formula (2) provides straightforwardly an identification, in term of Brownian bridge, of the expectation of the distribution function of the eigenvalues

N (t)=

n1

1_{λnt}, t >0.

Indeed on one hand we have Eϕ(t)=t

+∞

0

e⁻^{t s}E

n1

1_{λns}ds, t >0.

On the other hand an elementary computation yields 1

2π

e⁻^{2t V}²⁽^W,x) dx= t 2π

+∞

0

e⁻^{t s} P2V2W, xsdx

ds.

So by injectivity of the Laplace transform and Theorem 1 we obtain

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THEOREM 4. – In dimension d =2, the expectation of the distribution function N (s), s >0, is of the form

EN (t)= 1 2π

P2V2W, xtdx.

3. Proof of Theorem 2

The main object in this paragraph is to obtain the logarithmic equivalent of the Laplace transform of the distribution of the square of the fundamental frequency (that means the first eigenvalueλ1of the Dirichlet–Laplacian) of the Poisson–Voronoi typical cell. Using a Tauberian argument, this result leads us to the asymptotic evaluation whent→0⁺of the logarithm of the distribution function P{λ1t}.

Proof of (3). – Let us first notice that

e⁻^{t µ}¹ e⁻^{t λ}¹ ϕ(t), t >0.

Consequently, it suffices to prove that

t→+∞lim t⁻^d/(d⁺²⁾ln E e⁻^{t µ}¹= −2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²₋_2)/2 d

d/(d+2)

(20) and

lim sup

t→+∞ t⁻^d/(d⁺²⁾ln Eϕ(t)−2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²₋_2)/2 d

d/(d+2)

. (21) In order to obtain the asymptotic result (20), let us remark that the random radiusRmof the largest disc centered at the origin and included inC(0)has the distribution

P{Rmr} =e⁻²^d^ω^d^r^d, r0. (22) Thus using the fact that the first eigenvalue of a disc of radius r >0 is equal to j_(d²₋_2)/2/r², we deduce that

E e⁻^{t µ}¹=

+∞

0

exp

−tj_(d²₋_2)/2

r² −2^dωdr^d

d2^dωdr^d⁻¹dr

=t^d/(d⁺²⁾ +∞

0

exp

−t^d/(d⁺²⁾

j_(d²₋_2)/2

r² −2^dωdr^d

d2^dωdr^d⁻¹dr.

We then get from Laplace method that

t→+∞lim t⁻^d/(d⁺²⁾ln E e⁻^{t µ}¹= −min

r>0

j_(d²₋_2)/2

r² +2^dωdr^d

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= −2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²_−2)/2 d

d/(d+2)

,

which proves the result (20).

The proof of (21) is far more delicate. Using Theorem 1, we first have Eϕ(t)= 1

(2π )^d/2

E e⁻^{(2t )}^d/2^V^d⁽^{W ,x)} dx

= 1 (2π )^d/2

{xt}

E e⁻^{(2t )}^d/2^V^d⁽^{W ,x)} +

{x>t|}

E e⁻^{(2t )}^d/2^V^d⁽^{W ,x)} dx

.

The obvious following inequality

Vd W , xωdx^d, x∈R^d, then implies that

{x>t|}

E e⁻^{(2t )}^d/2^V^d⁽^{W ,x)} dx

{x>t}

e⁻^{(2t )}^d/2^ω^d^x^ddx

=(2t)⁻^d/2e⁻²^d/2^ω^d^t^3d/2, t >0.

Therefore

t→+∞lim t⁻^d/(d⁺²⁾ln

{x>t}

E e⁻^{(2t )}^d/2^V^d⁽^{W ,x)} dx= −∞. Consequently in order to obtain (21), it suffices to prove that

lim sup

t→+∞ t⁻^d/(d⁺²⁾

{xt}

E e⁻^{(2t )}^d/2^V^d⁽^{W ,x)} −2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²₋_2)/2 d

d/(d+2)

. (23)

The key estimation which enables us to derive the above asymptotic behaviour is contained in the following lemma.

LEMMA 1. – For everyt >0, andx∈R^d,

E e⁻^{(2t )}^d/2^V^d⁽^W,x) E e⁻^{(2t )}^d/2^V^d⁽^W,0) , (24) or equivalently

EPx+√

2tW⊂C(0)EP√

2tW⊂C(0). (25)

This last result will be proved at the end of the section. Let us notice that Lemma 1 implies the inequality

{xt}

E e⁻^{(2t )}^d/2^V^d⁽^{W ,x)} dxωdt^dE e⁻^{(2t )}^d/2^V^d⁽^{W ,0)}

=ωdt^dEP√

2tW⊂C(0), t >0.

(14)

Therefore in order to prove (23), it suffices to have lim sup

t→+∞ t⁻^d/(d⁺²⁾ln EP√

2tW⊂C(0) −2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²₋_2)/2 d

d/(d+2)

. (26)

This last result is a consequence of the Donsker–Varadhan theorem about the volume of the Wiener sausage [6].

More precisely, let us denote byLthe distance from the origin to its nearest neighbor.

The distribution ofLis given by the equality P{Lt} =P

Rm t

2

=e⁻^ω^d^t^d, t >0. (27) Consequently,

EP√

2tW⊂C(0)=

+∞

0

EP√

2tW⊂C(0)|L=sdωds^d⁻¹e⁻^ω^d^s^dds. (28) Moreover, it is obvious from the definition of the Voronoi cell C(0) that for every c∈ [1,2),

C(0)⊂

x∈

B 1

c ·x, 1

c −1 2

·L c

. (29)

This leads us to compare for the Brownian bridge the probability to stay in the cellC(0) with the probability to avoid Poissonian obstacles. For every fixedε >0 andc∈ [1,2), we obtain

+∞

2ε

EP√

2tW⊂C(0)|L=sdωds^d⁻¹e⁻^ω^d^s^dds

+∞

2ε

EP √

2tW⊂

x∈

B 1

c·x, 1

c −1 2

ε

c L=s

dωds^d⁻¹e⁻^ω^d^s^dds

EP

√ 2tW⊂

(c·x)∈

B

x, 1

c−1 2

ε

c

=E exp

−c^dVd y∈√

2tW

By,c⁻¹−0.5ε.

It is well known that the estimation of the Wiener sausage provided by Donsker and Varadhan [6] is still relevant for the path of a Brownian bridge. Consequently, we obtain that

lim sup

t→+∞ t⁻^d/(d⁺²⁾ln +∞

2ε

EP√

2tW⊂C(0)|L=sdωds^d⁻¹e⁻^ω^d^s^dds

(15)

−2^d/(d⁺²⁾(c^dωd)^2/(d⁺²⁾ d+2

2

j_(d²_−2)/2 d

d/(d+2)

. (30)

Taking the limit in (30) whenc→2⁻, we get lim sup

t→+∞ t⁻^d/(d⁺²⁾ln +∞

2ε

EP√

2tW⊂C(0)|L=sdωds^d⁻¹e⁻^ω^d^s^dds −2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²₋_2)/2 d

d/(d+2)

. (31)

Besides, let us notice that we have the following inequality for everys < s∈R+: EP√

2tW⊂C(0)|L=sEP√

2tW⊂C(0)|L=s. (32) Indeed, considering a uniformly distributed pointX0on the sphereS^d⁻¹, the equality in law

(|L=s)^law=∩B(0, s)^c∪ {s·X₀}, is clearly satisfied. We then deduce from (32) that

2ε 0

EP√

2tW⊂C(0)|L=sdωds^d⁻¹e⁻^ω^d^s^ddsEP√

2tW⊂C(0)|L=2ε

+∞

2ε

EP√

2tW⊂C(0)|L=sdωds^d⁻¹e⁻^ω^d^s^dds.

Using (31), we then obtain that lim sup

t→+∞ t⁻^d/(d⁺²⁾ln 2ε

0

EP√

2tW⊂C(0)|L=sdωds^d⁻¹e⁻^ω^d^s^dds −2^3d/(d⁺²⁾ωd2/(d+2)

d+2 2

j_(d²₋_2)/2 d

d/(d+2)

. (33)

Combining the equality (28) with (30) and (33), we get the estimation (26), which completes the proof of the convergence (3).

Let us now focus on the proof of Lemma 1.

Proof of Lemma 1. – Let us fixx∈R^d. Using the identity E e⁻^{(2t )}^d/2^V^d⁽^W,x) =(2t)^d/2

∞ 0

PVd W, xue⁻^{(2t )}^d/2^udu, t >0, we obtain that it suffices to show the following inequality.

PVd W, xuPVd W,0u, u0. (34)