Diffusion in planar Liouville quantum gravity

www.imstat.org/aihp 2015, Vol. 51, No. 3, 947–964

DOI:10.1214/14-AIHP605

© Association des Publications de l’Institut Henri Poincaré, 2015

Diffusion in planar Liouville quantum gravity

Nathanaël Berestycki

Statistical Laboratory, University of Cambridge, Cambridge, UK.

Received 20 February 2013; revised 17 December 2013; accepted 22 January 2014

Abstract. We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domainD of the complex plane for which the Riemannian metric tensor at a pointz∈D is given by exp(γ h(z)), appropriately normalised. Herehis an instance of the Gaussian free field onDandγ∈(0,2)is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian free field, and show that it spends Lebesgue-almost all its time in the set ofγ-thick points, almost surely. Similar but deeper results have been independently and simultaneously proved by Garban, Rhodes and Vargas.

Résumé. Nous construisons une diffusion naturelle associée ê la géométrie aléatoire de la gravité quantique de Liouville. For- mellement, il s’agît d’un mouvement Brownien dans un domaineDdu plan complexe, muni d’un tenseur de Riemann donné par exp(γ h(z)), correctement renomalisé. Icihest une réalisation du champ libre Gaussien surD, etγ∈ ]0,2[est un paramètre. Il est montré que ce processus est presque sûrement continu et possède certains propriétés d’invariance conforme. Une borne sur la di- mension de Hausdorff des instants passés dans les points épais du champ libre Gaussien est obtenue, qui montre que cette diffusion passe Lebesue-presque tout son temps dans les pointsγ-épais, presque sûrement. Des résultats semblables mais plus profonds ont été indépendemment et simultanément obtenus par Garban, Rhodes et Vargas.

MSC:60K37; 60J67; 60J65

Keywords:Brownian motion; Gaussian free field; Liouville quantum gravity; Thick points

1. Introduction

This paper is motivated by a recent series of works on planar Liouville quantum gravity and the so-called KPZ relation (for Knizhnik, Polyakov and Zamolodchikov). The KPZ relation describes a way to relate geometric quantities associated with Euclidean models of statistical physics to their formulation in a setup governed by a certainrandom geometry, the so-called Liouville (critical) quantum gravity. This is a problem which has a long and distinguished history and for which we refer the interested reader to the recent breakthrough paper of Duplantier and Sheffield [4]

and the excellent survey article by Garban [5].

A central problem in this area is the construction of a natural random metric in the plane, enjoying properties of conformal invariance, such that a KPZ relation holds. By this we mean that given a setAin the plane, the Hausdorff dimensions ofAendowed either with the Euclidean metric or the random (quantum) metric are related by a determin- istic transformation. Given these requirements, it is reasonably natural to look for or postulate that the local metric at a pointzcan be written in the form (up to normalising factor) exp(γ h(z)), wherehis a Gaussian free field andγis a parameter. Unfortunately,his not a function but a random distribution, and the exponential of a distribution is not in general well-defined.

1Research supported in part by EPSRC grants EP/GO55068/1 and EP/I03372X/1.

While making sense of this notion of random metric is still wide open, Duplantier and Sheffield, in the paper mentioned above, were able to construct a random measure, called the quantum gravity measure, which intuitively speaking corresponds to the volume measure of the metric. Remarkably, using this measure, they were able to de- fine suitable notions of scaling dimensions for a setAand show that a KPZ relation holds, where the deterministic transformation involves a quadratic polynomial.

The purpose of this paper is to show that a natural notion ofdiffusionalso makes sense in this context. Roughly speaking, one can summarise the main result by saying that, while we still do not know how to measure the distance between two pointszandw, it is possible to say how long it would take a Brownian motion to go fromztow. The key idea is to note that, using conformal invariance of Brownian motion in two dimensions, it suffices to parametrise the Brownian motion correctly.

Note. As I was preparing this paper,I learnt that Garban,Rhode and Vargas were working on a similar problem[6].

Their results are of a similar nature,though are more precise in some aspects.

1.1. Looking for the right object

What follows is an informal discussion which is aimed to explain where the definition comes from. Bylocal metric ρ(z)at a pointz∈D, we mean that small segments of Euclidean lengthεare in the Riemannian metric considered to have distanceρ(z)εat the first order whenε→0.

LetU, Dbe two proper simply connected domains, and letf:U→Dbe a conformal isomorphism. We think of Uas being a (wild) domain endowed with the random geometry, andDa nice domain such as the unit disc, in which we read this geometry. If(W_t, t≥0)is a standard Brownian motion in U (i.e., stopped upon leavingU), then we simply wish to describe how(W_t, t≥0)is parametrized byD. To do this, it suffices to considerX_t=f (W_t). By Itô’s formula,

Z_t=f (W_t)=Bt

0|f(Ws)|²ds; (1)

and henceZis a time-change of a Brownian motion(Bt, t≥0)inD. This cannot directly be used as a definition as the time-change still involvesW and we only want to define the processZ in terms ofB and the metricρ(z)inD derived from mapping the metric inUviaf. Clearly,ρ(z)is simply equal to 1/|f(w)|²= |g(z)|², whereg=f⁻¹ (see Figure1).

The reader can then easily check that setting Z_t=B

μ⁻_t ¹

; whereμ_t= t

0

g(B_s)²ds, (2) andμ⁻_t ¹:=inf{s >0:μs> t}, gives the same process as (1). The advantage of this way of writingZis that it involves only the standard Brownian motion(B_t, t≥0)in the nice domainD and the local metricρ(z)at any pointz∈D, which we assume to be given.

U

D z w

ε |f(w)|²ε

f

Fig. 1. Local metric and conformal map.

1.2. Statements

We will thus use (2) as our definition. Fix a proper connected domainD⊂C, and lethbe an instance of the Gaussian free field in D. (We use the Duplantier–Sheffield normalisation of the Green function.) Formally, h is a centered Gaussian process indexed by the Sobolev spaceH₀¹(D), which is the completion ofC_K^∞(D)with respect to the scalar product

(f, g)_∇= 1 2π

D

(∇f )·(∇g).

Thenhis a centered Gaussian process such that if(h, f )_∇is the value of the field at the functionf∈H₀¹(D), then Cov

(h, f )_∇, (h, g)_∇

=(f, g)_∇ by definition.

Forz∈Dandεsufficiently small, we leth_ε(z)be the well-defined average ofhover a circle of radiusεaboutz.

(We refer the reader to [13] for a proof that this is indeed well-defined and other general facts about the Gaussian free field.) Then we define a process(Zε(t), t≥0)as in (2). That is, letz∈U and let(Bt, t≥0)be a planar Brownian motion such thatZ0=zalmost surely. We put

Zε(t)=B μ⁻_ε¹(t)

, whereμε(t)= _t∧T

0

e^{γ hε(Bs)}ε^γ2/2ds, andμ⁻_ε¹(t)=inf{s≥0: μ_ε(s) > t},T =inf{t≥0:B_s∈/D}.

Definition 1.1. The Liouville diffusion,if it exists,is the limit asε→0of the processZ_ε.

Obviously, sinceBdoes not depend onε, the issue of convergence of the processZ_εreduces to that of thequantum clock process(με(t), t≤T ).

Theorem 1.2. Assume0≤γ <2.Then(Z_ε(t), t≥0)converges almost surely along the dyadic sequenceε=2^−kas k→ ∞to a random process(Z(t), t≥0)which is almost surely continuous up to the hitting time of∂D.

Remark 1.3. It can be seen from the proof thatZdoes not stay stuck anywhere almost surely,in the sense thatμ⁻¹(t) is strictly increasing.

We now address conformal invariance properties. LetD,D˜ be two simply connected domains and letφ:D→ ˜D be a conformal isomorphism (a bijective conformal map with conformal inverse).

Theorem 1.4. LetQ=^γ₂+_γ² andψ=φ⁻¹.Then we can write φ(B_μ−1(t))= ˜B_μ˜−1

ψ (t),

whereB˜ is a Brownian motion inD,˜

˜

μ_ψ(t)=lim

ε→0

_t

0

ε^γ2/2e^{γ[ ˜hε+Qlog|ψ|](B˜s)}ds andh˜is the Gaussian free field inD.˜

In other words, mapping the Liouville diffusion Z(t )by the transformation φ, one obtains the corresponding Liouville diffusion in D, except that the Gaussian free field˜ h˜ inD˜ has been replaced byh˜+Qlog|ψ|. (This is similar to Proposition 2.1 in Duplantier and Sheffield [4].)

Finally, it is of interest to quantify how much time the Brownian motion spends at points for which the value of the fieldhis unusually big. Consider thethick pointsof the Gaussian free field: forα >0, let

Tα⁻:= {z∈D: lim infε→0 hε(z) log(1/ε)≤α},

Tα⁺:= {z∈D: lim sup_{ε→0log(1/ε)}^hε(z) ≥α}. (3)

Hu, Miller and Peres [7, Theorem 1.2] proved that the Hausdorff dimension ofTα is a.s.(2−α²/2)∨0. (Note that these authors were working with a slightly different normalisation of the Gaussian free field which differs by a factor of√

2π, and ourαis√

2ain the notations of that paper.) Theorem 1.5. Let0< γ <2and letα > γ.Then almost surely,

dim

t:Z(t )∈T_α⁺

≤ 2−α²/2

2−αγ+γ²/2. (4)

The same result holds whenα < γ andTα⁻replaced withTα⁺. This upper bound is enough to deduce the following result:

Corollary 1.6. With probability one, Leb

t: Z(t ) /∈Tγ⁼

=0, where

Tα⁼=

z∈D: lim

ε→0

h_ε(z) log(1/ε)=α

.

By contrast, using the methods of Benjamini and Schramm [2] (see also Rhode and Vargas [12]) it is possible to show the following analogue of the KPZ relation.

Proposition 1.7. FixA⊂Da nonrandom Borel set and letd0=dim(A)wheredimrefers to the(Euclidean)Haus- droff dimension.Then almost surely,

dim{t: Z_t∈A} =d, (5)

wheredsolves the equationd0+d²γ²/2−d(2+γ²/2)=0.

Recall that in the case whereA=T_α^±, as mentionned above, the Hausdorff dimension is(2−α²/2)∨0. Neverthe- less, the formula in (5) does not match that from Theorem1.5. This is of course becauseT_α^±depends very strongly on the Gaussian free field. The difficulty in Theorem1.5is thus essentially to understand the effect on the clock process of coming near a thick point, and hence to disentangle the separate effects linked on the one hand to the trajectory of a standard Brownian motion and on the other hand to the frequency of those thick points.

2. Proof of Theorem1.2: Convergence

For the rest of the paper, with a slight abuse of notation, we call (B_t, t≥0)a Brownian motion stopped at time T :=Tr=inf{t >0: dist(Bt, ∂D)≤r}. Herer >0 is a small arbitrary number. We will still call

Zε(t)=B μ⁻_ε¹(t)

, whereμε(t)= t∧T

0

e^{γ hε(Bs)}ε^γ2/2ds, andμ⁻_ε¹(t)=inf{s≥0:με(s) > t},T =Tr.

In this section we prove that the clock processμ_ε(t)converges asε→0 to a limit (which might still be degenerate).

By Proposition 3.2 in [4], Var

h_ε(z)

= −logε+log

R(z;D)

for allεsuch thatB(z, ε)⊂D, whereR(z;D)is the conformal radius ofzinD. That is,R(z;D)=1/|φ(z)|, where φ:D→Dis a conformal map such thatφ(z)=0. Note that fort≤T =T_r, there are two nonrandom constantsc1, c2

such that

c₁≤R(B_t;D)≤c₂.

We will denoteh¯ε(z)=γ hε(z)+(γ²/2)logε.

Proof. We start by pointing out a potential source of confusion. Note that for eachfixedz∈D, the sequenceα_ε(z)= e^{γ hε(z)−(γ2/2)Varhε(z)} (viewed as a function of ε) forms a nonnegative martingale, in its own filtration. Hence by Fubini’s theorem,

E t

0

α_ε(B_s)ds

=t.

Nevertheless, the above doesnotimply thatα_ε(t)is a martingale as a function ofε: this is because the martingale property ofα_ε(z)ceases to hold when the filtration contains all the information about(h_ε(w), w∈D).

Nevertheless, the random variablesμ_ε(t)converge asε→0 almost surely to a limit. We now prove this statement.

In fact we only prove this along the subsequenceε=2^−k, k≥1. With an abuse of notation we writeμ_kforμ₂₋k and h_k for h₂₋k. Then it suffices to prove that|μ_k−μ_k+1| ≤Cr^k for somer <1 andC <∞, almost surely. Assume without loss of generality thatt=1 and lets∈ [0,1]. LetS_k^s= [0,1] ∩ {s+2^−2kZ}, and let

Xk(s)= 1 2^2k

t∈S_k^s

e^h¯k(Bt)

and

Y_k(s)= 1 2^2k

t∈S_k^s

e^h¯k+1(Bt).

Note thatμk=1

0Xk(s)dsandμk+1=1

0Yk(s)dsso it suffices to prove that

Xk(s)−Yk(s)≤Cr^k (6) for someC, r <1 uniformly ins∈ [0,1]. As in [4], we start with the caseγ <√

2 where an easy second moment argument suffices. LetE˜(·)=E(·|σ (Bs, s≤t )). Then note that

E˜X_k(s)−Yk(s)²

= 1 2^4k

t,t∈S^s_k

E˜

e^h¯k(Bt)−e^h¯k+1(Bt)

e^h¯k(Bt)−e^h¯k+1(Bt) .

Let t, t∈S_k^s and assume that |B_t −B_t|>2^−k. Then conditionally onh_k(B_t)andh_k(B_t), the random variables h_k+1(B_t)andh_k+1(B_t)are independent Gaussian random variables with meanh_k(B_t)(resp.h_k(B_t)) and variance log 2. Thus, in that case,

E˜

e^h¯k(Bt)−e^h¯k+1(Bt)

e^h¯k(Bt)−e^h¯k+1(Bt)

|hk(Bt), hk(B_t)

= ˜E

e^h¯k(Bt)−e^h¯k+1(Bt)|hk(Bt), hk(B_t)

× ˜E

e^h¯k(Bt)−e^h¯k+1(Bt)|hk(Bt), hk(B_t) .

Now, observe thath¯_k+1(B_t)= ¯h_k(B_t)+γ X−(γ²/2)log 2 where X is a centred Gaussian random variable with variance log 2, which is independent fromB, h˜ _k(B_t), h_k(B_t). Hence

E˜

e^h¯k(Bt)−e^h¯k+1(Bt)|h_k(B_t), h_k(B_t)

=e^h¯k(Bt)

1−e^{−(γ2/2)log 2}E˜ e^{γ X}

=0.

Of course the same also holds once we uncondition onhk(Bt), hk(B_t). It follows by Cauchy–Schwarz’s inequality that

E˜X_k(s)−Y_k(s)²

= 1 2^4k

t,t∈S_k^s

1_{|Bt−Bt|≤2}−kE˜

e^h¯k(Bt)−e^h¯k+1(Bt)

e^h¯k(Bt)−e^h¯k+1(Bt)

≤ 1 2^4k

t,t∈S_k^s

1_{|Bt−Bt|≤2}−k

E˜

e^h¯k(Bt)−e^h¯k+1(Bt)2E˜

e^h¯k(Bt)−e^h¯k+1(Bt)2

= C 2^4k

t,t∈S_k^s

1_{|Bt−Bt|≤2}−kE

e^h¯k(z)−e^h¯k+1(z)2

, (7)

wherezis any point inDsuch that dist(z, ∂D)≥r. To compute the expectation in the sum, we condition onhk(z) and get, lettingε=2^−k,

E

e^h¯k(z)−e^h¯k+1(z)2

≤E e^2¯hk(z)

=ε^γ2E

e^{2γ hk(z)}

≤Cε^γ2−4γ2/2=Cε^−γ2. Therefore,

E˜X_k(s)−Y_k(s)²

≤Cε⁴ε^−γ2

t,t∈S_k^s

1_{|Bt−Bt|≤2}−k. (8)

We will need the following lemma on two-dimensional Brownian motion:

Lemma 2.1. There existsC=C(ω)depending only on the realisation ofB,such that,uniformly ins∈ [0,1]:

t,t∈S_k^s

1_{|Bt−Bt|≤2}−k ≤C2^2kk⁴,

almost surely.

Proof. Key to the proof will be a result of Dembo, Peres, Rosen and Zeitouni [3]. Let μ denote the occupation measure of Brownian motion at time 1. Then Theorem 1.2 of [3] states that

δlim→0sup

x∈R²

μ(D(x, δ))

δ²(log(1/δ))² =2 (9)

almost surely. In particular, there existsM(ω)such thatμ(D(x, δ))≤M(ω)δ²(log(1/δ))². LetAdenote the event where there is somet∈ [0,1]such that

t∈S_k^t

1_{|Bt−Bt|≤2}−k > Ck⁴,

whereC=C(ω)is chosen suitably. By Lévy’s modulus of continuity theorem, we know that sup

|s−t|≤2^−2k

|B_s−B_t| ≤δ:=c2^−k√ k

almost surely for some universalc >0 and for allksufficiently large. On the eventA, we can thus findx=B_twhere μ(D(x, δ))≥Ck⁴2^−2k for ksufficiently large. But note that δ²(log(1/δ))²≤c2^−2kk³ for some nonrandomc >0.

Hence choosingC(ω)=cM(ω)we see from (9) thatP(A)=0, as desired.

Plugging the estimate of Lemma2.1into (8), we get E˜X_k(s)−Y_k(s)²

≤Cε^4−γ2−2(log 1/ε)⁴, which proves (6) at least ifγ <√

2.

To prove (6) in the general case (γ <2), we introduce the set S˜_k^s=

t∈S_k^s: h_ε(B_t) <−αlog

ε/R(B_t;D) ,

whereα > γ is a fixed parameter which will be chosen close enough toγlater on, andR(z;D)denotes the conformal radius at the pointz∈D. We letT_k^s=S_k^s\ ˜S_k^s. Then we have

Xk(s)= 1 2^2k

t∈T_k^s

e^h¯k(Bt)+ 1 2^2k

t∈ ˜S_k^s

e^h¯k(Bt).

It is easy to show that the first is negligible. IfQ˜ denotes the law of the exponential tilting ofP˜ by e^{γ hk(Bt)}, i.e., dQ˜

dP˜(ω)= e^{γ hk(Bt)} E˜(e^{γ hk(Bt)}), then lettingσ²= −log(ε/R(Bt;D)),

Q˜

h_k(B_t)∈dx

= e^−x2/(2σ2)e^{γ x}dx/√ 2πσ² e^−x2/(2σ2)e^{γ x}dx/√

2πσ² =e^{−(x−m)2/(2σ2)} dx

√2πσ²

, (10)

wherem=γ σ², and hence the law ofh_K(B_t)underQ˜ isN(m, σ²). Thus E˜

1t∈T_k^se^h¯k(Bt)

=ε^γ2/2Q˜

h_k(B_t) > ασ²

× ˜E

e^{γ hk(Bt)}

≤ε^γ2/2exp

−1

2(α−γ )²σ²

R(B_t;D)^γ2/2,

where the bound above is obtained by using standard bounds on the normal tail distribution. This decays exponentially fast withkuniformly int≤T.

Likewise, by conditioning onh¯_k(B_t), we get that E˜ 1

2^2k

t∈T_k^s

e^h¯k+1(Bt)≤CE˜ 1 2^2k

t∈T_k^s

e^h¯k(Bt),

whereC <∞depends only onγ, and thus this tends to 0 exponentially fast.

Define nowX˜_k(s)=₂¹2k

t∈ ˜S_k^se^h¯k(Bt) andY˜_k(s)=₂¹2k

t∈ ˜S^s_ke^h¯k+1(Bt). We wish to boundE((X˜_k(s)− ˜Y_k(s))²).

Applying the same reasoning as in (7) shows that EX˜_k(s)− ˜Y_k(s)2

≤ C 2^4k

t,t∈ ˜S^s_k

1_|Bt−B

t|≤2^−k

E

e^2¯hk(Bt) E

e^2¯hk(Bt) .

Now whent∈ ˜S_k^s, using the same reasoning as in (10) but with tilting proportional to e^{2γ hk(Bt)}instead E

1_{t∈ ˜S}s

ke^2¯hk(Bt)

=ε^γ2Q

X < ασ² E

e^{2γ hk(Bt)} ,

whereX∼N(2γ σ², σ²). We may if we wish assume thatα <2γ, so Q

X < ασ²

≤exp

−1

2(2γ−α)²σ²

≤Cexp 1

2(2γ−α)²logε

.

Thus using Lemma2.1again, EX˜k(s)− ˜Yk(s)2

≤Cε⁴×

ε⁻²log(1/ε)⁴

×ε^{γ2+(2γ−α)2/2}ε^−4γ2/2

≤C(log 1/ε)⁴ε^{2+(2γ−α)2/2−γ2}.

Choosingαarbitrarily close toγ we find that the exponent ofεis arbitrarily close to 2−γ²/2 which is positive since γ <2. Thus we can findαclose enough toγ such that the exponent is positive, in which case (6) follows.

As discussed at the beginning of the section, this implies almost sure convergence ofμ_ε(t)to a limitμ(t )(which

might still be identically zero at this stage).

3. Proof of Theorem1.2: Nondegeneracy Letr >0 and letT_r=inf{t≥0: dist(Bt, ∂U )≤r}.

We will first show that Pz

εlim→0

_Tr

0

e^{γ hε(Bs)}ε^γ2/2ds >0

>0.

It suffices to show that the integral is bounded inL^q for someq >1. Our strategy is inspired by work of Bacry and Muzy [1] on multifractal random measures. Since the proof can appear a bit convoluted, we start by explaining what lies behind it. Essentially, theqth moment of the integral can be understood as the sum of two terms: one diagonal term which gives the sum of the local contribution of the field at each point, and a cross-diagonal term which evaluates how these various bits interact with one another. Consider a squareSin the domain and such thatz∈S. The strategy will be to slice the square into many squares of sidelength 2^−m, wheremwill be a large but finite number. The key part of the estimate is to show that the sum of the contributions inside each smaller square is small. To achieve this, we use a scaling argument, as the Gaussian free field in a small square can be thought of as a general ‘background’

height plus an independent Gaussian free field in the square.

Without loss of generality, we will assume thatz∈S=(0,1)²the unit square, andDcontains the squareS, where Sis the square centered onSwhose sidelength is 3 (i.e.,S=(−1,2)²). Then it will suffice to check that

_τ

0

e^{γ hε(Bs)}ε^γ2/2dsis bounded inL^q,

whereτ =inf{t≥0: B_t∈/S}. We will in fact show the slightly stronger statement that _T

0

e^{γ hε(Bs)}ε^γ2/21_{B_s∈S}dsis bounded inL^q, (11)

whereT =inf{t≥0: B_t∈/S}.

3.1. Auxiliary field

Fix a bounded continuous functionφ:[0,∞)→ [0,∞)which is a bounded positive definite function and such that φ(x)=0 ifx≥1. For instance, we chooseφ(x)=

(1− |x|)₊, see [10] and the discussion in Example 2.3 in [11].

Define an auxiliary centered Gaussian random field(X_ε(x))_x∈Rd by specifying its covariance c_ε(x, y):=E

X_ε(x)X_ε(y)

=log₊ 8

|x−y| ∨ε

+φ

|y−x| ε

.

Define also the normalized field to be X¯_ε(x)=γ X_ε(x)−(γ²/2)σε, with σ_ε=c_ε(0,0)=log(8/ε)+1, so that E(e^X¯ε(x))=1. Because we have assumed thatS⊂D, it is easy to check that the covariance structure ofX¯_ε and γ h_εare very close: more precisely, there are constantsaandb, independent ofε, such that

c_ε(x, y)−a≤E

γ h_ε(x)γ h_ε(y)

≤c_ε(x, y)+b (12)

for all x, y∈S. Condition for a moment on the trajectory of the Brownian path (B_s, s≤T ), and let E˜ denote the corresponding conditional expectation. Define a measureμ_ε to be the (random) Borel measure on[0, T]whose density with respect to Lebesgue measure is e^{γ hε(Bs)}ε^γ2/21_{B_s∈S}, s∈ [0, T]. LetI = {s∈ [0, T]: B_s∈S}. Note that conditionally givenB, the process(hε(Bs))s∈I is a centered Gaussian process indexed byI with covariance function η_ε(B_s, B_t), whereη_ε(x, y)is the covariance function of the (unconditional) Gaussian field(h_ε(x))_x∈D. By Theorem 2 of Kahane [8], we deduce from the right-hand side of (12) that

E˜

μ_ε(0, T )q

≤ ˜E _T

0

e^{Yε(s)−(1/2)E˜(Yε(s))2}1_{Bs∈S}ds q

, (13)

where fors∈I,Y_ε(s)is a Gaussian centered field with covariancec_ε(B_s, B_t)+b. ThusY_ε(s)may be realized as Y_ε(s)= ¯X_ε(B_s)+W, whereW is a fixed independent centered normal random variable of varianceb. We deduce that

E˜

μ_ε(0, T )q

≤ ˜E

e^qW−qb/2E˜ _T

0

e^X¯ε(Bs)1_{Bs∈S}ds q

=e^q(q−1)b/2E˜ _T

0

e^X¯ε(Bs)1_{Bs∈S}ds q

.

Taking expectations, E

μ_ε(0, T )q

≤e^q(q−1)b/2E T

0

e^X¯ε(Bs)1_{Bs∈S}ds q

. (14)

Reasoning similarly with the left-hand side of (12) gives us E

μ_ε(0, T )q

≥e^{−q(q−1)a/2}E T

0

e^X¯ε(Bs)1_{Bs∈S}ds q

. (15)

The crucial observation aboutX_ε(and the reason why we introduce it) is that it enjoys an exact scaling relation, as follows:

Lemma 3.1. Forλ <1, X_λε(λx)

x∈B(0,4)=d

Ω_λ+X_ε(x)

x∈B(0,4),

whereΩ_λis an independent centered Gaussian random variable with variancelog(1/λ).

Proof. One easily checks that for allx, y∈R²such thatx−y ≤8,c_λε(λx, λy)=log(1/λ)+c_ε(x, y).

3.2. Moments of orderq >1

Therefore, fix 1< q <2 and consider forz∈S= [0,1]²the unit square, f_ε(z)=Ez

T 0

e^X¯ε(Bs)1_{Bs∈S}ds q

, (16)

where as before,T =inf{t≥0:B_t∈/S}, andS= [−1,2]²⊂B(0,4). We letM_ε=sup_z∈Sf_ε(z). Our goal will be to show that

M_εis uniformly bounded inεfor some choice ofq >1. (17)

The strategy for the proof of (17) will be the following. We fixm≥1, which we will choose suitably large (but fixed) at some point. We split the squareS into a checkerboard pattern of squaresS_i, each of which has sidelength 2^−m. By Minkowski’s inequality, it suffices to show that

i∈I

_T

0

e^X¯ε(Bs)1_{Bs∈Si}ds

is uniformly bounded inL^q, where(S_i)_i∈I is a maximal subset of squares such that|z−w|>2^−mforz∈S_i, w∈S_j andi=j ∈I. In words, we have retained “every other subsquare” inI. Note that there are at most|I| ≤4^m such subsquares.

Sinceq <2, the functionx→x^q/2is concave. Being equal to 0 atx=0, it is therefore subadditive. Hence letting d_i=_T

0 e^X¯ε(Bs)1_{Bs∈Si}ds, Ez

i∈I

_T

0

e^X¯ε(Bs)1_{Bs∈Si}ds q

≤Ez

i∈I

d_i^q/2 2

=

i∈I

Ez

d_i^q

+

i=j

Ez

d_i^q/2d_j^q/2 .

We treat separately the diagonal terms and the nondiagonal ones. We start by the diagonal terms.

Lemma 3.2. There existsCindependent ofm, qandεsuch that

i∈I

Ez

d_i^q

≤Cm2^{m(ζ (q)+2)}Mε2^m, where

ζ (q):=q²γ² 2 −q

2+γ²

2

. (18)

Proof. LetU_i=inf{t≥0:B_t∈S_i}and letT_i=inf{t≥U_i:B_s∈/S_i}, whereS_iis the square centered onS_icontaining the 8 adjacent dyadic squares of same size asS_i. LetN_i denote the number of times that the path of the Brownian motion returns toS_iafter having touched the boundary ofS_i, beforeT.

Then applying the Markov property at each such return, we get maxz∈S Ez

d_i^q

≤Cmax

w∈S_iEw

β_i^q

Ez(N_i), (19)

where β_i=

_Ti

0

e^X¯ε(Bs)1_{Bs∈S_i}ds.

(Note that by the strong Markov property, (19) holds even when z /∈Si, as before the first hitting time ofSi the contribution to the integral indi is zero.) Now, a simple martingale argument shows that for some constantC >0,

Ez(Ni)≤Clog 2^m

, uniformly inz∈Sandi∈I.

We now use the scaling properties of bothBandXε(i.e., Lemma3.1) to estimate the diagonal terms. Letλ=1/2^m, and write 2^mBs = ˜B_s22m, whereB˜ is a planar Brownian motion starting fromw˜ =2^mw. LetS˜i=2^mSi,S˜_i=2^mS_i,