Local behaviour of local times of super-brownian motion

Local behaviour of local times of super-Brownian motion Comportement local des temps locaux du super-mouvement

brownien

Mathieu Merle

D.M.A., Ecole normale supérieure, 45, rue d’Ulm, 75005 Paris, France Received 22 February 2005; accepted 14 June 2005

Available online 7 December 2005

Abstract

Forx₀∈R^d\ {0},d3, we study the local behaviour near 0 of the local times(L^y_t)of a super-Brownian motionXinitially inδx₀. More precisely, ifψ (c)is a suitable normalization, our main result implies that the process(ψ (c)(L^x/c_t −L⁰_t), x∈R^d, t0)converges in distribution to a non-degenerate limit asc→ ∞. This allows us to study the local behaviour of the occupation measure ofX, then to recover and to generalise a result of Lee concerning the occupation measure of three-dimensional super- Brownian motion conditioned to hit a distant ball.

©2005 Elsevier SAS. All rights reserved.

Résumé

Pourx₀∈R^d\ {0},d3, on étudie le comportement local au voisinage de 0 des temps locaux(L^y_t)du super-mouvement brownienXde valeur initialeδx₀. Plus précisément, si on noteψ (c)la normalisation adéquate, notre résultat principal implique que le processus(ψ (c)(L^x/c_t −L⁰_t), x∈R^d, t0)converge en loi lorsquec→ ∞vers une limite non dégénérée. Ce résultat nous permettra d’étudier le comportement local de la mesure d’occupation deX, puis de redémontrer et généraliser un résultat de Lee concernant la mesure d’occupation d’un super-mouvement brownien tri-dimensionel conditionné à toucher une boule lointaine.

©2005 Elsevier SAS. All rights reserved.

Keywords:Super-Brownian motion; Local time; Occupation measure

1. Introduction and statement of results 1.1. Introduction

The main goal of this work is to study the local behaviour of the occupation measure of super-Brownian motion in dimensiond3.

E-mail address:merle@dma.ens.fr (M. Merle).

0246-0203/$ – see front matter ©2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2005.06.004

Super-Brownian motion is a model for spatial populations undergoing a continuous branching phenomenon, which arises in a variety of different contexts. It was introduced independently by Watanabe (69) and Dawson (75) as the weak limit of branching particle systems. Connections between the wider class of superprocesses and partial differ- ential equations have helped derive some of the basic properties of super-Brownian motion and have also allowed to prove analytic results using probabilistic arguments. More recently, it has been shown that super-Brownian motion appears in scaling limit of a wide range of lattice systems such as the voter model, the contact process, lattice trees or oriented percolation. Therefore, in a way similar to standard Brownian motion, super-Brownian motion can be thought of as a universal object providing information on the asymptotics of many interacting particle systems or statistical mechanics models.

Local times of superprocesses have been studied by many authors (cf. Sugitani [9], Adler and Lewin [1], Krone [4]).

Our main result Theorem 1 gives precise information about the local behaviour of local times of super-Brownian motion, in dimensiond=2 or 3. Let us be more precise. Letx₀∈R^d\ {0},Xa super-Brownian motion initially in δ_x0, andL^x_t the local time ofX at timet and pointx. As a direct consequence of Theorem 1, ifx∈R^d and if we setψ (c)=√

cwhend=3,ψ (c)=c(ln(c))^−1/2whend=2, we will obtain that(ψ (c)(L^x/c_t −L⁰_t))_t0converges in distribution asc→ ∞to(β_a(x)L0

t)_t0whereβ is a one-dimensional Brownian motion which is independent ofX anda(x)is a constant depending only onx. Theorem 1 in fact gives a more general statement involving finitely many different values ofx. This allows us to study the local behaviour of the occupation measure ofX(Proposition 2).

Our results are related to a recent paper of Lee [5]. Lee considers a super-Brownian motion started atδ_cx0 withc large and conditioned on hitting the unit ball. Using analytic methods, he obtains limit theorems for the occupation measure of this ball by super-Brownian motion (see Proposition 1 below). We will show how to recover and to generalise Lee’s results from our main theorem. To do this, we will need to study the local behaviour of the occupation measure ofXunder its canonical measureNx0 (Proposition 5).

Our results on the local behaviour of local times of super-Brownian motion are analogous to the ones obtained by Yor for local times of standard Brownian motion. LetB be a linear Brownian motion started at the origin and let^x_t denote the local time ofB at levelx and timet. Then Yor [11], proved that(√

c(^x/c_t −⁰_t))_{t0, x0}converges in distribution to a Brownian sheet asc→ ∞. This should be compared with Theorem 1 below.

After introducing the basic notation, Section 1 first summarises known results which motivate our study, mainly found in [5,9], and [7], then states our results. It also provides an outline of the proof of our main result (Theorem 1) about the local behaviour of local times. Section 2 is devoted to the proof of Theorem 1. In Section 3, we apply Theorem 1 to prove a non-conditional equivalent form of Lee’s result (Proposition 2), then we recover and extend Lee’s result (Proposition 1). Section 4 is devoted to the one-dimensional case.

1.2. Notation

Let M_F(R^d) be the space of finite measures on R^d. For μ∈ MF(R^d), f:R^d →R, μ, f is shorthand for

R^df (x)μ(dx).

We denote byB(R^d)the Borelσ-algebra inR^d. The notationB(x, r)stands for the open ball of radiusrcentered atx∈R^d.

We denote byC_b(R^d)the space of bounded continuous functions fromR^dintoR, and byC_b(R^d)₊the space of non-negative functions inC_b(R^d). IfK is a compact subset ofR^d,C_K(R^d)is the subset ofC_b(R^d)consisting of functions supported onK. Ifn∈N, we letC_bⁿ(R^d)be the set of allntimes continuously differentiable functions on R^dwith bounded derivatives of order less thann, andC_b^∞(R^d):=

n0C_bⁿ(R^d).

We denote byp_tthe transition density ofd-dimensional Brownian motion, that is fort0,z∈R^d, p_t(z)=(2π t )^−d/2exp

−|z|² 2t

. Setq_t(x)=_t

0p_s(x)ds, and ifμ∈M_F(R^d), μqt(z)=

R^d

qt(z−y)μ(dy).

We consider a super-Brownian motion (X_t, t0)in the space M_F(R^d). We will writeP_μ for the probability measure under whichX₀=μ. When there is no confusion we will writeP forP_μ,EforE_μ. We denote by(Ft^X)_t0 the right-continuous filtration generated byX.

1.3. Basic properties of super-Brownian motion and its local times

In this section we consider a super-Brownian motionXunderP_μ, forμ∈M_F(R^d).

It is well known (see for example [7], Theorem II.5.9) that for any function φ∈C_b(R^d)₊ we can express E_μ[exp(−_t

0X_s, φds)]in terms of the solution of a partial differential equation:

Eμ

exp

− t 0

Xs, φds =exp

−

v(t, x, φ)μ(dx)

, (1)

where(t, x)→v(t, x, φ)is continuous onR₊×R^dand solves the partial differential equation

∂v(s, x)

∂s =1

2v(s, x)−

v(s, x)2

+φ, (2)

on]0,∞[ ×R^d, with initial valuev(0, x)=0.

(1) is a particular case of theLaplace equationfor the super-Brownian motionX. Note that formula (1) remains valid if φ∈C_K(R^d)is not necessarily non-negative, but only then ift is less than a certain explosion timet^∗>0.

More precisely, for a fixedT >0, (1) will remain valid fortT for any compactly supported function that is bounded from below by a constant depending onT.

As shown in Chapter IV of [6] by the Brownian snake approach (see also Section II.7 of [7]), for anyx ∈R^d, there exists aσ-finite measureNxonC(R₊, M_F(R^d))called the excursion measure of super-Brownian motion such that the law of(X_t)_{t >0}underP_μis the same as the law of(

Xⁱ_t)_{t >0}, where

δ_Xi is a Poisson point process with intensity

Nxdμ(x). This fact is called thecanonical decomposition of super-Brownian motion.

Intuitively, if one thinks of super-Brownian motion as the scaling limit of critical branching random walks as it is introduced in [7], the measureX_t underNx fort >0 represents the contribution to the population of the descendants at time t of one single individual alive at time 0 at the point x. The canonical decomposition expresses that the super-Brownian motion at timetis obtained by superimposing a Poisson number of such contributions.

It is also well known that the processXsolves amartingale problem(see [7], Theorem II.5.1). For anyφ∈C_b²(R^d), X_t, φ = X₀, φ +M_t(φ)+1

2 t 0

X_s, φds, (3)

whereMt(φ)is anFt^X-martingale such thatM0(φ)=0 and the quadratic variation ofM(φ)is M(φ)

t= t 0

Xs, φ² ds.

Sugitani [9] proved that ford3, there exists a random continuous function(t, x)→L^x_t from(0,∞)×R^dinto R₊such that for anyΨ ∈C_b(R^d),

t 0

Xs, Ψds=

R^d

Ψ (x)L^x_t dx. (4)

L^x_t is called thelocal timeofXat pointx∈R^dand timet >0.

TakeL^x₀=0 for everyx∈R^d. The function(L^x_t)_{t0, x∈R}d needs not being continuous at points of the form(0, x), x∈R^d. However Sugitani established ford2 thatL^x_t is continuous in the pair(x, t )on the set of continuity points forμq_t(x). For example, note that this set containsR₊×B(0, r)wheneverμ(B(0, r))=0.

Sugitani also proved that ford=1,L^x_t is always continuous in the pair(x, t )and even differentiable with respect to the space variable at pointsx∈R^d whereμ({x})=0.

Under the assumption that(t, x)→μq_t(x)is continuous in R₊×R^d, Sugitani [9] obtained the existence of exponential moments for local times: for anyT >0, there exists a constantK(μ, T ) >0 such that

E_μ exp

rL^x_t

<∞ (5)

holds for anytT,x∈R^d andr <K(μ, T ). We shall denote byG^t,x_μ :{z∈C, |z|<K(μ, t )} →Cthe function such that for anyz∈C,|z|<K(μ, t ),

Eμ

exp zL^x_t

=exp

G^t,x_μ (z)

. (6)

For convenience we will writeG^t,x_δ

x0 =:G^t,x_x0 .

IfK is a compact set inR^d, and if we only know that(t, x)→μq_t(x)is continuous inR₊×K (for example if μ(K)=0), it is easy to adapt the proof of (5) in [9] to obtain for anyT >0 the existence of constantsK(μ, T , K) >0, C(μ, T , K) >0 such that

E_μ exp

rL^x_t

C(μ, T , K) (7)

holds for anytT,x∈K andr <K(μ, T , K). In this case, this clearly allows us to define the functionsG^t,x_μ for t0, x∈Kin a way such that (6) remains valid forz∈C, |z|<K(μ, t, K).

Let us now discuss thescaling properties of super-Brownian motion. Letc >0, and iff ∈C_b(R^d)₊letf_cbe the functionf_c(x)=f (cx). From (2) we get

v(t, x, f_c)=c²v

c²t, cx, c⁻⁴f .

The scaling properties of super-Brownian motion easily follow from (1) and that observation. Letμ∈M(R^d). For anyA∈B(R^d),t0, let

X^(c)_t (A)=c⁻²X_c2t(cA), μ^(c)(A)=c²μ c⁻¹A

.

Then, the law of the process(X^(c)_t , t0)underP_μ(c)is equal to the law of the process(X_t, t0)underP_μ. Consequently, for anyφ∈C_K(R^d), the law of the process

c⁻⁴

c²t

0

X_u, φdu, t0

underP_μ(c)is the same as the law of the process t

0

X_u, φ_cdu, t0

underP_μ.

We are now in position to state our main result concerning the local behaviour of local times of super-Brownian motion.

1.4. The main result

Theorem 1.Assumed=2or3. Let ψ (c)=

√

c ifd=3,

√c

ln(c) ifd=2.

Letν∈M_F(R^d)such thatν(B(0, ρ))=0for someρ >0, and letXbe a super-Brownian motion inR^d with initial valueν. Ifx₁, . . . , x_kare fixed points inR^d\ {0}, the process

X_t, ψ (c)

L^x_t^1/c−L⁰_t

, . . . , ψ (c)

L^x_t^k/c−L⁰_t

t0

converges asc→ ∞in the sense of weak convergence in the spaceC(R⁺, M(R^d)×R^k)to a limiting process which can be written in the form

X_t, β^x1

L⁰_t, . . . , β^xk

L⁰_t

t0.

Here,(β_t^x)_t0,x∈Rd is a centered Gaussian process independent ofXsuch that cov

β_t^x, βs^y

=a(x, y)(t∧s), anda(x, y)is given by

a(x, y)= ₁

4π²

R³ dz ₁

|z−x|−_|¹_z|

× ₁

|z−y|−_|¹_z|

ifd=3,

1

πx·y ifd=2.

For convenience we will writea(x, x)=:a(x).

Part of the motivation for Theorem 1 came from a recent paper of Lee [5] dealing with asymptotics for the occu- pation measure of super-Brownian motion.

1.5. Lee’s result and extensions

Lee [5] only considers the three-dimensional case, but we will extend his result to the case of dimension 2.

Letd=2 or 3,Xa super-Brownian motion inR^d, with initial valueμ=δcx₀, wherex0∈R^d,x0=0 andc >0.

LetKbe a compact set inR^d, and letφ,ξbe integrable functions onR^dsupported onKsatisfying

Kξ(y)dy=0,

Kφ(y)dy=0. We also defineφ_c,ξ_cthe functions such that for anyx∈R^d, φ_c(x)=φ(cx), ξ_c(x)=ξ(cx).

Note thatφ_c,ξ_care supported onK^(c):=c⁻¹K= {y: cy∈K}.

One may think of the particular example whereK= B(0,1), and where we consider the functions φ⁰=1_B(0,1) , ξ⁰=1_{B(0,1) ∩{x:xd0}}−1_{B(0,1) ∩{x:xd<0}}.

ForT >0, let us introduce the quantities Dφ,T =

T 0

Xt, φdt, Dξ,T = T 0

Xt, ξdt, T 0.

We also set a_ξ:=

K

K

ξ(y)ξ(z)a(y, z)dydz.

The following result, is proved by Lee in [5] (for the cased=3), by analysing the behaviour ofv(t, x0, ac³φ_c+ bc^7/2ξc)asc→ ∞(cf. (2)).

Proposition 1.Letd=2or3,t >0,x₀∈R^d\ {0}and letKbe a compact subset ofR^d. Letφandξ be integrable functions on R^d supported on K satisfying

Kφ(y)dy =0,

Kξ(y)dy =0. Under P_δcx

0(·|X hitsK)the random vector

c^d−4D_φ,c2t, c^d−4ψ (c)D_ξ,c2t

converges in distribution asc→ ∞to a non-degenerate limit(D1, D2). Furthermore, if|a|and|b|are small enough the Fourier transform of the limit is

E

exp(iaD1+ibD2)

=1+2|x₀|² 4−dG^{T ,0}_x

0

ia

K

φ(y)dy−b²a_ξ 2

,

where the functionG^{T ,0}_x0 is determined by(6).

The cased =3 stands as Corollary 1.2 in [5]. The form of the Fourier transform of (D₁, D₂)implies that the conditional law ofD2knowingD1is Gaussian with mean 0 and varianceaξD1(

φ(y)dy)⁻¹.

In the exampleφ=φ⁰,ξ =ξ⁰, Proposition 1 has the following interpretation: conditionally on the event that a super-Brownian motionX started atδcx₀ hits the unit ball, the occupation time of the unit ball up to timec²t (that isD_φ0,c²t)is of orderc^4−d, while the difference between the occupation time of the top half of the unit ball and the occupation time of the bottom half (that isD_ξ0,c²t) is of order√

cwhend=3, andc(ln(c))^1/2whend=2.

In Section 3.2, we will derive Proposition 1 from the following proposition, which is itself a consequence of Theorem 1, as we will see in Section 3.1.

Proposition 2.Letd=2or3. Lett,K,φ,ξ be as in Proposition1, and letν be as in Theorem1. UnderPν, the random vector

c^dD_φc,t, c^dψ (c)D_ξc,t

converges in distribution asc→ ∞to

L⁰_t

K

φ(y)dy, Ut

,

where conditionally onL⁰_t,U_t is Gaussian with variancea_ξL⁰_t.

As we will explain below, the first componentc^dD_φc,t indeed converges almost surely. Also note that, from (6), the Fourier transform of(

φ(y)dyL⁰_t, U_t)is simply E_ν

exp

iaL⁰_t

K

φ(y)dy+ibUt

=exp

G^{T ,0}_ν

ia

K

φ(y)dy−b²a_ξ 2

.

Let us explain how Proposition 2 follows from Theorem 1. The following approach is also valid in the one- dimensional case, and so we consider the general cased3. Let us first give a simple argument showing thatc^dD_φc,t convergesPν-almost surely asc→ ∞. By (4), and our assumption on the support ofφ,

c^dDφ_c,t=c^d

R^d

φ(cx)L^x_t dx=

K

φ(y)L^y/c_t dy.

Since(t, x)→L^x_t is continuous onR₊×B(0, ρ), c^dD_φc,t −→

c→∞L⁰_t

K

φ(y)dy

almost surely underP_ν by dominated convergence. See Theorem 5 in [9] for a related statement ([9] only states the convergence in distribution ofc^dD_φc,t).

Ifφis replaced byξ, the preceding argument is not sufficient. Indeed, since

ξ(x)dx=0, the last step yields c^dD_ξc,t→0 asc→ ∞. Still, we can write

R^d

ξ(y)L^y/c_t dy=

R^d

ξ(y)

L^y/c_t −L⁰_t dy,

which suggests to focus on the convergence ofψ (c)(L^y/c_t −L⁰_t), whereψ (c)is a suitable normalization. This approach is similar to the work of Stroock–Varadhan–Papanicolaou [10] and Yor [11] concerning limit theorems for additive functionals of standard Brownian motion.

The one-dimensional case follows almost immediately from [9] and will be treated briefly in Section 4.Until then we setd=2or3.

1.6. Outline of the proof of Theorem 1

We will first consider the casek=1.We fixx∈R^d\ {0}and a measureνsatisfying the assumption of Theorem 1.

Forα0(α >0 ifd=2)we setg_α(z)=_∞

0 exp(−αt )p_t(z)dt, and fory∈R^d,g_α^y(z)=g_α(z−y). Ifd=3, note that we haveg₀(z)=1/(2π|z|).

The key idea is to use the Tanaka formula for local times of super-Brownian motion in dimension less than 3 (see [2], Theorem 6.1). Let y ∈R^d, d 3, μ∈M_F(R^d), and let α0 ifd =3 or α >0 ifd =2. Under the assumptionμ, gα^y<∞, we havePμ-almost surely,

Xt, g_α^y

= μ, g^y_α

+Mt

g_α^y +α

t 0

Xs, g_α^y

ds−L^y_t, (8)

where M_t(g_α^y)is an Ft^X-martingale which is defined in terms of the martingale measure associated with super- Brownian motion. In particular,M0(gα^y)=0 andMt(gα^y)has quadratic variation

M g^y_α

t= t 0

Xs, g^y_α2

ds.

Ifc is large enough so thatx/c∈B(0, ρ), the conditionsν, g_α⁰<∞,ν, g^x/c_α <∞, will clearly be satisfied.

Thus we can use (8) fory=x/candy=0 to obtain,P_ν-almost surely, L^x/c_t −L⁰_t =

X₀−X_t, g^x/c_α −g_α⁰ +M_t

g_α^x/c

−M_t g⁰_α

+α t 0

X_s, g^x/c_α −g_α⁰

ds. (9)

In what follows we will takeα=0 whend=3. Note that the last term in the right-hand side of the previous formula then vanishes.

In Section 2.1 we will prove the following lemmas:

Lemma 1.Letd=3, T >0. ThenP_ν-almost surely sup

tT

√ c

X₀−X_t, g^x/c₀ −g₀⁰−→

c→∞0.

Lemma 2.Letd=2, T >0. ThenP_ν-almost surely (a) sup

tT

X0−X_t, c

√ln(c)

g_α^x/c−g⁰_α−→

c→∞0, (b) sup

tT

t 0

X_s, c

√ln(c)

g_α^x/c−g⁰_α ds

−→

c→∞0.

From (9), Lemmas 1 and 2 we see that the convergence of (ψ (c)(L^x/c_t −L⁰_t))_t0 follows from that of (ψ (c)Mt(g^x/cα −g_α⁰))t0.

All that remains to do is thus to study the convergence of the martingales M_t^x,c:=

⎧⎨

⎩

√cM_t((g₀^x/c−g⁰₀)) ifd=3,

√c

ln(c)Mt((gα^x/c−g⁰_α)) ifd=2.

Lemma 3.Pν-almost surely, M^x,c

t −→

c→∞a(x)L⁰_t.

This lemma will be proven in Section 2.2.

We then have to discuss the convergence in distribution of the martingaleM_t^x,c. Using the Dubins–Schwarz theorem (see [8], Theorem V.1.6), we can write

M_t^x,c=β^x,c_Mx,ct, (10)

whereβ_t^x,c is a standard Brownian motion. We may and will assume that for suM^x,c∞, β_s^x,c −β_u^x,c = γ_s^x−γ_u^x, where γ^x is a one-dimensional Brownian motion independent ofX. The collection of the laws of the family(X, β^x,c)_c>0is clearly tight.

In Section 2.3 we will prove

Lemma 4.Suppose that along a subsequencec_n ∞we have X, β^x,cn (d)

n−→→∞

X, β^x . Thenβ^xis independent ofX.

From the tightness of the laws of(X, β^x,c)and Lemma 4, it follows that X, β^x,c (d)

n−→→∞

X, β^x

(11) with a Brownian motionβ^xindependent ofX. We know from Lemma 3 thatM^x,ct convergesP_ν-almost surely to a(x)L⁰_t, and the convergence is uniform on every compact time interval by Dini’s theorem. It then follows from (10) and (11) that(Xt, M_t^x,c)t0converges in distribution to(Xt, β^x

a(x)L⁰_t)t0.

The casek=1 of Theorem 1 now follows from this fact and Lemmas 1 and 2. We will then extend this argument to the general case in Section 2.4.

2. Proof of Theorem 1 2.1. Preliminary reduction

In this section we will prove Lemmas 1 and 2. Recall that the pointx=0 is fixed, and that we have also fixed a measureνsatisfying the assumption of Theorem 1: there existsρ >0 such thatν(B(0, ρ))=0.

We start with a preliminary result providing a uniform bound for the measure of small balls. From [2], Corollary 4.8, we know the following

Proposition 3. Letδ >0be fixed. Ifd=2or3then, for anyμ∈M_F(R^d),P_μ-almost surely, lim sup

r0

sup

tδ

sup

y∈R^d

Xt

B(y, r)

Φ(r)⁻¹κμ,

whereΦ(r)=r²(ln(1/r))^4−d andκμis a constant depending onμ.

An easy consequence is the following Corollary 1.Pν-almost surely,

lim sup

r0

sup

t0

sup

y∈B0,ρ/2)

X_t(B(y, r) r⁻²

ln

1 r

d−4

κ_ν.

Proof of Corollary 1. Sinceν(B(0, ρ))=0, it is well known thatP_ν-almost surely there existsn₀(ω)∈Nsuch that supp(Xt)does not intersectB(0,^3ρ₄)for anyt∈ [0,2⁻ⁿ⁰]. Providedrρ/4, we then have

sup

t2^−n0(ω)

sup

y∈B(0,ρ/2)

X_t

B(y, r) r⁻²

ln

1 r

d−4

=0,

and thanks to Proposition 3, forP_ν-almost allω, lim sup

r0

sup

t2^−n0(ω)

sup

y∈R^d

X_t B(y, r)

r⁻²

ln 1

r d−4

κ_ν. Corollary 1 follows. 2

In the following we writeκ₀=κ₀(x),κ₁=κ₁(x), . . .for constants that depend on the pointxwhich is fixed.

2.1.1. The cased=3

For convenience we will use the notation h^x,c(z):=ψ (c)

g₀^x/c(z)−g⁰₀(z)

=

√c 2π

1

|z−x/c|− 1

|z|

, forz=0, x/c.

We will use the following easy estimates onh^x,c: (A1) If|z|^|_2c^x|, then|h^x,c(z)|^√_|z^c_|.

(A2) Letr^2|_c^x|. Then, the maximum of|h^x,c|outside the ballB(0, r)is attained at the point ^rx_|x| and its value is (2π√

c r(_|x^r_|−¹_c))⁻¹.

(A3) If|z| ∧ |z−x/c|^|_2c^x| and|z|^4|_c^x|, then|h^x,c(z)|^2c_π|^√_x^c_|. Proof of Lemma 1. Ifcis sufficiently large andz /∈B(0, ρ), using(A2),

h^x,c(z) 1 2π√

cρ(ρ/|x| −1/c) −→

c→∞0.

ThusX₀, h^x,c = ν, h^x,cclearly goes to 0 asc→ ∞.

Hence, to get Lemma 1, it is enough to prove thatP_ν-almost surely, sup

tT

Xt, h^x,c=sup

tT

R³

h^x,c(z)Xt(dz) −→

c→∞0. (12)

Ift >0 is fixed we know that 0 does not belong to suppX_t,P_ν-almost surely. Thus the same argument as whent=0 givesX_t, h^x,c →0 asc→ ∞. The point is that we need this convergence uniformly intT, and we know that there may exist exceptional timestT such that 0∈suppXt.

It will be convenient to cut the domainR³ of the integral in (12) into different areas where we will be able to estimate the integrand. First of all, ifr >0 is fixed, ifz /∈B(0, r)andcis large enough, using again(A2)we have

|h^x,c(z)|κ₀/√ cso that

R³\B(0,r)

h^x,c(z)X_t(dz) κ₀

√cX_t,1.

Since sup_t0X_t,1isP_ν-almost surely bounded, we get sup

tT

R³\B(0,r)

h^x,c(z)X_t(dz) −→

c→∞0.

Thus, to get (12) it only remains to prove that for somer1>0, sup

tT

B(0,r1)

h^x,c(z)X_t(dz) −→

c→∞0. (13)

From Corollary 1 it follows thatP_ν-almost surely, there existsr₁>0 such that, for anyrr₁, sup

t0

sup

y∈B(0,ρ/2)

X_t B(y, r)

2κνr²ln 1

r

. (14)

Clearly we can assumer₁ρ/4. Letcbe large enough so thatB(0,4|x|/c)⊂B(0, r1). To get rid of the singularities ofh^x,c, we will first deal with the integrals in neighbourhoods of 0 andx/c. First, we have

B(0,^|_2c^x|)

h^x,c(z)X_t(dz)

p1

B(0,_{2p c}^|x|)\B(0, ^|x|

2p+1c)

h^x,c(z)X_t(dz) .

Using(A1)and (14) we see thatPν-almost surely sup

tT

B(0,_{2p c}^|x|)\B(0, ^|x|

2p+1c)

h^x,c(z)X_t(dz) 2κν

2^p+1c√ c

|x|

|x| 2^pc

2

ln 2^pc

|x|

.

It is now clear thatP_ν-almost surely sup

tT

B(0,^|_2c^x|)

h^x,c(z)X_t(dz)

4κν|x| c^1/2

p1

pln(2)+ln(c)−ln(|x|)

2^−pκ₁κ_νc^−1/2ln(c).

Clearly we can obtain an analogous bound for the quantity sup

tT

B(^x_c,^|_2c^x|)

h^x,c(z)X_t(dz) .

Now using(A3)and (14) we get sup

tT

B(0,^4|_c^x|)\(B(0,^|_2c^x|)∪B(^x_c,^|_2c^x|))

h^x,c(z)Xt(dz) 2κν

2c√ c π|x|

4|x| c

2

ln c

4|x|

,

so we finally obtain that sup

tT

B(0,^4|x|_c )

h^x,c(z)X_t(dz)

κ_νκ₂ln(c)

√c . (15)

Let us now consider the integral onB(0, r1)\B(0,^4|_c^x|). LetNbe such that r1

2^N 4|x| c r1

2^N−1.

Note that, for 1pN,r1/2^p2^N−p+1|x|/cwhiler1/2^p−12^N−p+3|x|/c. Once again, using(A2)and (14), we obtain for 1pN

sup

tT

B(0, ^r1

2p−1)\B(0,^r1

2p)

h^x,c(z)X_t(dz)

2κνc^3/2

|x|(2^N−p+1−1)²

2^N−p+3|x| c

2

ln c

2^N−p+3|x|

2κν

2⁶|x| c^1/2 ln

c

|x|

. Therefore,

sup

tT

B(0,r₁)\B(0,^4|_c^x|)

h^x,c(z)X_t(dz)

N×2⁷κν|x| c^1/2 ln

c

|x|

κ_νκ₃(ln(c))² c^1/2 .

The limit (13) now follows from the above and (15), which completes the proof of Lemma 1. 2

2.1.2. The case d=2

Now we considerα >0 and the function h^x,c(z):= c

√ln(c)

g_α^x/c−g_α⁰

(z), forz=0, x/c.

We will need the following estimates onh^x,cwhich shall be proven in Appendix A. Whencis large enough, (B1) Ifz∈B(0,^|_2c^x|),

h^x,c(z)κ₄ c

√ln(c)ln⁺ 1

|z|

. Ifz∈B(^x_c,^|_2c^x|),

h^x,c(z)κ₄ c

√ln(c)ln⁺ 1

|z−x/c|

. (B2) Ifz∈B(0, c^−3/4)\(B(0,^|_2c^x|)∪B(^x_c,^|_2c^x|)),

h^x,c(z)κ₅c ln(c).

(B3) Letr > c^−7/8. Then ifz /∈B(0, r), h^x,c(z)κ6

1 r√

ln(c).

Proof of Lemma 2. We will use a similar method as for proving Lemma 1.

FixT >0. To obtain Lemma 2(a), it is enough to establish thatP_ν-almost surely, sup

tT

X_t, h^x,c−→

c→∞0. (16)

Ifr >0 is fixed andcis sufficiently large we can use(B3)to get sup

tT

R³\B(0,r)

h^x,c(z)X_t(dz)

κ₆ r√

ln(c)sup

tT

X_t,1.

Since sup_t0|Xt,1|is finitePν-almost surely, we have sup

tT

R²\B(0,r)

h^x,c(z)X_t(dz) −→

c→∞0. (17)

Now, from Corollary 1 we know thatP_ν-almost surely, there existsr₂>0 such that for anyrr₂, sup

t0

sup

y∈B(0,ρ/2)

Xt

B(y, r)

2κνr²

ln 1

r 2

. (18)

We can chooser2ρ/4 andclarge enough so thatB(0, c^−3/4)⊂B(0, r2). We will first deal with the neighbourhood of singularities 0 andx/c. Using (18) and the estimate(B1)we get for everyp1,

sup

tT

B(0,_{2p c}^|x|)\B(0, ^|x|

2p+1c)

h^x,c(z)Xt(dz) κ4

2κνc

√ln(c)

ln 2^p+1c

|x|

3|x| 2^pc

2

,

and thus sup

tT

B(0,^|_2c^x|)

h^x,c(z)X_t(dz)

p1

2κ4κ_νc

√ln(c)

ln 2^p+1c

|x|

3|x| 2^pc

2

κ₇(ln(c))^5/2

c −→

c→∞0.