APPLICATIONS TO SPECTRAL DIMENSIONS
SEBASTIAN ANDRES AND NAOTAKA KAJINO
ABSTRACT. TheLiouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion pro- cess evolving in a planar random geometry induced by theLiouville measureMγ, formally written asMγ(dz) = eγX(z)−γ2E[X(z)2]/2dz,γ ∈ (0,2), for a (massive) Gaussian free fieldX. It is anMγ-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive func- tional with Revuz measureMγ.
In this paper we provide a detailed analysis of the heat kernelpt(x, y)of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian up- per bound of the formpt(x, y) ≤C1t−1log(t−1) exp`
−C2((|x−y|β∧1)/t)β−11 ´ for t ∈ (0,12]for each β > 12(γ+ 2)2, and an on-diagonal lower bound of the formpt(x, x) ≥ C3t−1`
log(t−1)´−η
fort ∈ (0, tη(x)], withtη(x) ∈ (0,12]heavily dependent onx, for eachη >18forMγ-almost everyx. As applications, we deduce that the pointwise spectral dimension equals2Mγ-a.e. and that the global spectral dimension is also2.
1. INTRODUCTION
One of the main mathematical issues in the theory of two-dimensional Liouville quantum gravity is to construct a random geometry on a two-dimensional manifold (sayR2 equipped with the Euclidian metricdx2) which can be formally described by a Riemannian metric tensor of the form
eγX(x)dx2, (1.1)
where X is a massive Gaussian free field on R2 defined on a probability space (Ω,A,P) and γ ∈ (0,2) is a parameter. The study of Liouville quantum grav- ity is mainly motivated by the so-called KPZ-formula (for Knizhnik, Polyakov and Zamolodchikov), which relates some geometric quantities in a number of models in statistical physics to their formulation in a setup governed by this random geometry.
In this context, by the KPZ relation the parameterγ can be expressed in terms of
Date: October 2, 2015.
2010 Mathematics Subject Classification. Primary: 60J35, 60J55, 60J60, 60K37; Secondary:
31C25, 60J45, 60G15.
Key words and phrases. Liouville quantum gravity, Gaussian multiplicative chaos, Liouville Brown- ian motion, heat kernel, spectral dimension.
S.A. was partially supported by the CRC 1060 “The Mathematics of Emergent Effects”, Bonn.
N.K. was partially supported by JSPS KAKENHI Grant Number 26287017.
1
a certain physical constant called the central charge of the underlying model. We refer to [12] and to the survey article [15] for more details on this topic.
However, to give rigorous sense to the expression (1.1) is a highly non-trivial problem. Namely, as the correlation function of the Gaussian free fieldX exhibits short scale logarithmically divergent behaviour, the field X is not a function but only a random distribution. In other words, the underlying geometry is too rough to make sense in the classical Riemannian framework, so some regularisation is required. While it is not clear how to execute a regularisation procedure on the level of the metric, the method performs well enough to construct the associated volume form. More precisely, using the theory of Gaussian multiplicative chaos established by Kahane in [20] (see also [25]), by a certain cutoff procedure one can define the associated volume measureMγ for γ ∈ (0,2), called the Liouville measure. It can be interpreted as being given by
Mγ(A) =
∫
A
eγX(z)−
γ2
2 E[X(z)2]dz,
but this expression forMγ is only very formal, forMγ is known to be singular with respect to the Lebesgue measure by a result [20, (141)] by Kahane (see also [25, Theorems 4.1 and 4.2]). Recently, in [17] Garban, Rhodes and Vargas have con- structed the natural diffusion processB= (Bt)t≥0associated with (1.1), which they call theLiouville Brownian motion (LBM). Similar results have been simultaneously obtained in a weaker form also by Berestycki [4]. On a formal level,Bis the solution of the SDE
dBt=e−
γ
2X(Bt)+γ2
4 E[X(Bt)2]dB¯t,
whereB¯ = ( ¯Bt)t≥0is a standard Brownian motion onR2independent ofX. In view of the Dambis-Dubins-Schwarz theorem this SDE representation suggests defining the LBMBas a time change of another planar Brownian motionB = (Bt)t≥0. This has been rigorously carried out in [17], and then by general theory the LBM turns out to be symmetric with respect to the Liouville measureMγ. In the companion paper [18] Garban, Rhodes and Vargas also identified the Dirichlet form associated withBand they showed that the transition semigroup is absolutely continuous with respect toMγ, meaning that the Liouville heat kernelpt(x, y)exists. Moreover, they observed that the intrinsic metricdB generated by that Dirichlet form is identically zero, which indicates that
limt↓0 tlogpt(x, y) =−dB(x, y)2
2 = 0, x, y∈R2,
and therefore some non-Gaussian heat kernel behaviour is expected. This degen- eracy of the intrinsic metric is known to occur typically for diffusions on fractals, whose heat kernels indeed satisfy the so-called sub-Gaussian estimates; see e.g. the survey articles [3,23] and references therein.
In this paper we continue the analysis of the Liouville heat kernel, which has been initiated simultaneously and independently in [24]. As our first main results we obtain the continuity of the heat kernel and a rough upper bound on it.
Theorem 1.1. Let γ ∈ (0,2). Then P-a.s. the following hold: A (unique) jointly continuous versionp = pt(x, y) : (0,∞)×R2×R2 → [0,∞) of the Liouville heat kernel exists and is(0,∞)-valued, and in particular the Liouville Brownian motionB is irreducible. Moreover, the associated transition semigroup(Pt)t>0 defined by
Ptf(x) :=Ex[f(Bt)] =
∫
R2pt(x, y)f(y)Mγ(dy), x∈R2,
is strong Feller, i.e.Ptf is continuous for any bounded Borel measurablef :R2 →R.
Theorem 1.2. Letγ ∈ (0,2). ThenP-a.s., for anyβ > 12(γ+ 2)2 and any bounded U ⊂R2 there exist random constantsCi =Ci(X, γ, U, β)>0,i= 1,2, such that
pt(x, y) =pt(y, x)≤C1t−1log(t−1) exp (
−C2
(|x−y|β∧1 t
) 1
β−1
)
(1.2) for allt∈(0,12],x∈R2 andy∈U, where| · |denotes the Euclidean norm onR2.
Since β > 12(γ+ 2)2 >2, the off-diagonal partexp(
−C2((|x−y|β∧1)/t)β−11 ) of the bound (1.2) indicates that the process diffuses slower than the two-dimensional Brownian motion, which is why such a bound is calledsub-Gaussian. We do not expect that the lower bound 12(γ + 2)2 for the exponentβ is best possible. Unfor- tunately, Theorem1.2alone does not even exclude the possibility thatβ could be taken arbitrarily close to2, which in the case of the two-dimensional torus has been in fact disproved in a recent result [24, Theorem 5.1] by Maillard, Rhodes, Vargas and Zeitouni showing thatβ satisfying (1.2) for smalltmust be at least2 +γ2/4.
In this sense the Liouville heat kernel does behave anomalously, which is natural to expect from the degeneracy of the intrinsic metric associated with the LBM.
From the conformal invariance of the planar Brownian motionB it is natural to expect that the LBMBas a time change ofB admits two-dimensional behaviour, as was observed by physicists in [1] and in a weak form proved in [26] (see Remark1.5 below). The on-diagonal partt−1log(t−1) in (1.2) shows a sharp upper bound in this spirit except for a logarithmic correction, and we will also prove the following on-diagonal lower bound valid for Mγ-a.e. x ∈ R2, which matches (1.2) besides another logarithmic correction.
Theorem 1.3. Letγ ∈(0,2). ThenP-a.s., for Mγ-a.e.x ∈R2, for anyη > 18there exist random constantsC3 = C3(X, γ,|x|, η) > 0 andt0(x) = t0(X, γ, η, x) ∈ (0,12] such that
pt(x, x)≥C3t−1(
log(t−1))−η
, ∀t∈(0, t0(x)]. (1.3) Combining the on-diagonal estimates in Theorems 1.2and1.3, we can immedi- ately identify the pointwise spectral dimension as2.
Corollary 1.4. Letγ ∈(0,2). Then P-a.s., forMγ-a.e.x∈R2, limt↓0
2 logpt(x, x)
−logt = 2.
Essentially from Theorems 1.2and 1.3 we shall further deduce that the global spectral dimension, that is the growth order of the Dirichlet eigenvalues of the generator on bounded open sets, is also2; see Subsection6.2for details.
Remark 1.5. In [26, Theorem 3.6] the following result on the spectral dimension has been proved:P-a.s., for anyα >0and for allx∈R2,
ylim→x
∫ ∞
0
e−λttαpt(x, y)dt <∞, ∀λ >0, (1.4) and
ylim→x
∫ ∞
0
e−λtpt(x, y)dt=∞, ∀λ >0. (1.5) In [26] the left hand sides were interpreted as the integrals intof the on-diagonal heat kernel pt(x, x), which was needed due to the lack of the knowledge of the continuity of pt(x, y). By Theorem 1.1 this interpretation can be made rigorous now, and moreover, (1.4) follows immediately from Theorem 1.2. On the other hand, (1.5) is actually an easy consequence of the Dirichlet form theory. Indeed, by [14, Exercises 2.2.2 and 4.2.2] ∫∞
0 e−λtpt(x, x)dt is equal to the reciprocal of theλ-order capacity of the singleton{x}with respect to the LBM, and this capacity is zero by [14, Lemma 6.2.4 (i)] and the fact that the same holds for the planar Brownian motion.
The proofs of our main results above are mainly based on the moment estimates for the Liouville measureMγ by [20,27] and those for the exit times of the LBM Bfrom balls by [17], together with the general fact from time change theory that the Green operator of the LBM has exactly the same integral kernel as that of the planar Brownian motion (see (2.6) below). To turn those moment estimates into P-almost sure statements, we need some Borel-Cantelli arguments that cannot pro- vide us with uniform control on various random constants over unbounded sets.
For this reason we can expect the estimate (1.2) to hold onlylocallyuniformly, so that in Theorem 1.2 we cannot drop the dependence of the constants C1, C2 on U or the cutoff of |x−y|at 1 in the exponential. Also to remove the logarithmic corrections in (1.2) and (1.3) and the restriction toMγ-a.e. points in Theorem1.3 and Corollary1.4one would need to have good uniform control on the ratios of the Mγ-measures of concentric balls with different radii. However, we cannot hope for such control in view of [5, Remark A.2], where it is claimed that
lim sup
r↓0
sup
x∈B(0,1)
Mγ(
B(x,2r)) Mγ(
B(x, r))1−η =∞
for anyη ∈(
−∞,4+γγ22
) P-a.s., withB(x, R) :={y ∈ R2 : |x−y|< R}for x∈R2 andR >0.
The LBM can also be constructed on other domains like the torus, the sphere or planar domainsD ⊂ R2 equipped with a log-correlated Gaussian field like the (massive or massless) Gaussian free field (cf. [17, Section 2.9]). In fact, Theo- rem1.1has been simultaneously and independently obtained in [24] for the LBM on the torus, where thanks to the boundedness of the space one can utilise the eigenfunction expansion of the heat kernel to prove its continuity and the strong Feller property of the semigroup. On the other hand, in our case ofR2the Liouville heat kernelpt(x, y) does not admit such an eigenfunction expansion and the proof of its continuity and the strong Feller property requires some additional arguments.
Therefore, although the proofs of our results should directly transfer to the other domains mentioned above, we have decided to work on the planeR2 in this paper for the sake of simplicity and in order to stress that our methods also apply to the case of unbounded domains.
In [24] Maillard, Rhodes, Vargas and Zeitouni have also obtained upper and lower estimates of the Liouville heat kernel on the torus. Their heat kernel upper bound in [24, Theorem 4.2] involves an on-diagonal part of the formCt−(1+δ) for any δ > 0 and an off-diagonal part of the form exp(
−C(|x−y|β/t)β−11)
for any β > β0(γ), where β0(γ) is a constant larger than our lower bound 12(γ + 2)2 on the exponent β and satisfies limγ↑2β0(γ) = ∞. Thus Theorem 1.2 gives a better estimate, and we prove it by self-contained, purely analytic arguments while the proof in [24] relies on (1.4), whose proof in [26] is technically involved. Concern- ing lower bounds, an on-diagonal lower bound as in Theorem1.3is not treated in [24]. On the other hand, their off-diagonal lower bound [24, Theorem 5.1], which implies the boundβ ≥2 +γ2/4for any such exponentβ as in (1.2) (in the case of the torus) as mentioned above after Theorem1.2, is not covered by our results.
The rest of the paper is organised as follows. In Section2we recall the construc- tion of the LBM in [17] and introduce the precise setup. In Section 3 we prove preliminary estimates on the volume decay of the Liouville measure and on the exit times from balls needed in the proofs. In Section4we show that the resolvent op- erators of the LBM killed upon exiting an open set have the strong Feller property, which is needed in Section 5 to prove Theorems 1.1 and 1.2. In Subsection 5.1 we show the continuity of the Dirichlet heat kernel associated with the killed LBM on a bounded open set by using its eigenfunction expansion, and in Subsection5.2 we then deduce the continuity of the heat kernel and the strong Feller property on unbounded open sets, as well as Theorem1.2, using a recent result in [19]. Finally, in Section 6 we show the on-diagonal lower bound in Theorem 1.3 and thereby identify the pointwise and global spectral dimensions as2.
Throughout the paper, we write C for random positive constants depending on the realisation of the fieldX, which may change on each appearance, whereas the
numbered random positive constantsCi will be kept the same. Analogously, non- random positive constants will be denoted by c or ci, respectively. The symbols
⊂ and ⊃ for set inclusion allow the case of the equality. We denote by | · | the Euclidean norm onR2 and byB(x, R) := {y ∈R2 : |x−y|< R},x ∈R2,R > 0, open Euclidean balls inR2and for abbreviation we setB(R) :=B(0, R). Lastly, for non-emptyU ⊂R2andf :U →Rwe writekfk∞:=kfk∞,U := supx∈U|f(x)|.
2. LIOUVILLE BROWNIAN MOTION
2.1. Massive Gaussian free field and Liouville measure. Consider a massive Gaussian free fieldX on the whole plane R2, i.e. a Gaussian Hilbert space asso- ciated with the Sobolev spaceH1mdefined as the closure ofCc∞(R2)with respect to the inner product
hf, gim :=m2
∫
R2f(x)g(x)dx+
∫
R2∇f(x)· ∇g(x)dx,
where m > 0 is a parameter called the mass. More precisely, (hX, fim)f∈H1
m is a family of Gaussian random variables on a probability space(Ω,A,P) with mean0 and covariance
E[
hX, fimhX, gim
]= 2πhf, gim.
In other words, the covariance function ofXis given by the massive Green function g(m)associated with the operatorm2−∆, which can be written as
g(m)(x, y) =
∫ ∞
0
1 2ue−m
2
2 u−|x−2uy|2 du=
∫ ∞
1
k(m)(
u(x−y))
u du (2.1)
with
k(m)(z) := 1 2
∫ ∞
0
e−m
2
2v|z|2−v2 dv.
Following [17] we now introduce an n-regularised version ofX. To that aim let (an)n≥0 ⊂ R be an unbounded strictly increasing sequence with a0 = 1 and let (Yn)n≥1 be a family of independentcontinuous Gaussian fields on R2 defined also on(Ω,A,P)with mean0and covariance
E[
Yn(x)Yn(y)]
=
∫ an
an−1
k(m)(
u(x−y))
u du=:g(m)n (x, y); (2.2) here, suchYn can be constructed by applying a version [22, Problem 2.2.9] of the Kolmogorov-ˇCentsov continuity theorem to a Gaussian field onR2with mean0and covariancegn(m), which in turn exists by the Kolmogorov extension theorem (see e.g.
[11, Theorems 12.1.2 and 12.1.3]) since(
gn(m)(x, y))
x,y∈Ξis a non-negative definite real symmetric matrix for any finiteΞ⊂R2. Then for eachn≥1, then-regularised fieldXnis defined as
Xn(x) :=
∑n k=1
Yk(x), x∈R2,
and the associated random Radon measureMn=Mγ,n onR2 is given by Mn(dx) := exp
(
γXn(x)−γ22E[
Xn(x)2])
dx (2.3)
with a parameter γ ≥ 0. By the classical theory of Gaussian multiplicative chaos established in Kahane’s seminal work [20] (see also [25]) we have the following:
P-a.s. the family (Mn)n≥1 converges vaguely on R2 to a random Radon measure M =Mγ called theLiouville measure, whose law is uniquely determined byγ and the covariance functiong(m)ofX, andM has full supportP-a.s. forγ ∈[0,2)and is identically zeroP-a.s. forγ ≥2. Throughout the rest of this paper, we assume that γ ∈ (0,2)is fixed and we will drop it from our notation, although the quantities defined through the Liouville measureM =Mγwill certainly depend onγ.
2.2. Definition of Liouville Brownian Motion. The Liouville Brownian motion has been constructed by Garban, Rhodes and Vargas in [17] as the canonical diffu- sion process under the geometry induced by the measureM. More precisely, they have constructed a positive continuous additive functionalF ={Ft}t≥0 of the pla- nar Brownian motion B naturally associated with the measure M and they have defined the LBM asBt=BF−1
t . In this subsection we briefly recall the construction.
LetΩ0 := C([0,∞),R2), letB = (Bt)t≥0 be the coordinate process onΩ0 and set G∞0 := σ(Bs;s <∞) andGt0 :=σ(Bs;s≤t),t≥0. Let{Px}x∈R2 be the family of probability measures on(Ω0,G∞0 ) such that for eachx∈R2,B = (Bt)t≥0 underPx
is a two-dimensional Brownian motion starting atx. We denote by{Gt}t∈[0,∞] the minimum completed admissible filtration forBwith respect to{Px}x∈R2 as defined e.g. in [14, Section A.2]. Moreover, let{θt}t≥0be the family of shift mappings onΩ0, i.e.Bt+s=Bt◦θs,s, t≥0. Finally, we writeqt(x, y) := (2πt)−1exp(
−|x−y|2/(2t)) , t >0,x, y∈R2, for the heat kernel associated withB.
Definition 2.1. i) A[−∞,∞]-valued stochastic processA = (At)t≥0 on(Ω0,G∞)is called apositive continuous additive functional (PCAF)ofB in the strict sense, ifAt
isGt-measurable for every t≥0 and if there exists a setΛ ∈ G∞, called a defining setforA, such that
a) for allx∈R2,Px[Λ] = 1, b) for allt≥0,θt(Λ)⊂Λ,
c) for all ω ∈ Λ, [0,∞) 3 t 7→ At(ω) is a [0,∞)-valued continuous function withA0(ω) = 0and
At+s(ω) =At(ω) +As◦θt(ω), ∀s, t≥0.
ii) Two such functionals A1 andA2 are calledequivalentifPx[A1t = A2t] = 1for allt > 0, x ∈ R2, or equivalently, there exists Λ ∈ G∞ which is a defining set for bothA1 andA2 such thatA1t(ω) =A2t(ω)for allt≥0,ω ∈Λ. Equivalent PCAFs in the strict sense will always be identified hereafter.
iii) For any suchA, a Borel measureµAonR2 satisfying
∫
R2f(y)µA(dy) = lim
t↓0
1 t
∫
R2Ex [∫ t
0
f(Bs)dAs ]
dx
for any non-negative Borel functionf :R2 → [0,∞]is called the Revuz measureof A, which exists uniquely by general theory (see e.g. [6, Theorem A.3.5]).
For everyn∈Nlet nowFtn: Ω×Ω0→[0,∞)be defined as Ftn:=
∫ t
0
exp (
γXn(Bs)−γ22E[
Xn(Bs)2])
ds, t≥0, (2.4)
which is strictly increasing int. Note that for everynthe functionalFn = (Ftn)t≥0
considered as a process defined on(Ω0,G0∞)is a PCAF ofB in the strict sense with defining setΩ0 and Revuz measureMn.
Theorem 2.2([17, Theorem 2.7]). P-a.s. the following hold:
i) There exists a unique PCAFF in the strict sense whose Revuz measure isM. ii) For allx∈R2,Px-a.s.,F is strictly increasing and satisfieslimt→∞Ft=∞. iii) For allx∈R2,Fnconverges toF inPx-probability in the spaceC([0,∞),R)
equipped with the topology of uniform convergence on compact sets.
The process(B,{Px}x∈R2),P-a.s. defined byBt:=BF−1
t ,t≥0, is called the (massive) Liouville Brownian motion (LBM).
Thanks to Theorem 2.2, we can apply the general theory of time changes of Markov processes to have the following properties of the LBM: First, it is a recur- rent diffusion on R2 by [14, Theorems A.2.12 and 6.2.3]. Furthermore by [14, Theorem 6.2.1 (i)] (see also [17, Theorem 2.18]), the LBM isM-symmetric, i.e. its transition semigroup(Pt)t>0 given by
Pt(x, A) :=Ex[Bt∈A]
fort∈(0,∞),x∈R2 and a Borel setA⊂R2, satisfies
∫
R2Ptf·g dM =
∫
R2f·Ptg dM
for all Borel measurable functionsf, g : R2 → [0,∞]. Here the Borel measurabil- ity ofPt(·, A) can be deduced from [17, Corollary 2.20] (or from Proposition 2.4 below).
Remark2.3. [17, Corollary 2.20] states that(Pt)t>0 is a Feller semigroup, meaning that Pt preserves the space of bounded continuous functions. Note that this is different from the notion of a Feller semigroup as for instance in [6, 14], i.e. a strongly continuous Markovian semigroup on the space of continuous functions vanishing at infinity. It is not known whether (Pt)t>0 is a Feller semigroup in the latter sense.
It is natural to expect that the LBM can be constructed in such a way that it depends measurably on the randomness of the fieldX. However, this measurability does not seem obvious from the construction in [17], since there the existence of the PCAFF has been deduced from some general theory on the Revuz correspondence forP-a.e. fixed realisation ofM. To overcome this issue, in the following proposition we show forP-a.e. environment the pathwise convergence ofFn towardsF in an appropriate{Px}x∈R2-a.s. sense which also ensures the measurability ofFtandBt
with respect to the product σ-field A ⊗ G∞0 for all t ≥ 0. The proof is given in AppendixA.
Proposition 2.4. There exists a setΛ∈ A ⊗ G∞0 such that the following hold:
i) For P-a.e. ω ∈ Ω, Px[Λω] = 1 for any x ∈ R2, where Λω := {ω0 ∈ Ω0 : (ω, ω0)∈Λ}.
ii) For every(ω, ω0)∈Λthe following limits exist inRfor all0< s≤t:
Fs,t(ω, ω0) := lim
n→∞
(Ftn(ω, ω0)−Fsn(ω, ω0)) , Ft(ω, ω0) := lim
u↓0Fu,t(ω, ω0).
Moreover, withF0(ω, ω0) := 0,[0,∞)3t7→ Ft(ω, ω0)∈[0,∞)is continuous, strictly increasing and satisfieslimt→∞Ft(ω, ω0) =∞.
iii) Lett≥0and setFt:=tonΛc. ThenFtisA ⊗ G∞0 -measurable.
iv) For P-a.e.ω ∈ Ω, the process(
Ft(ω,·))
t≥0 is a PCAF ofB in the strict sense with defining setΛω.
The previous proposition implies easily thatF indeed has the Revuz measureM. More strongly, we have the following proposition valid for any starting pointx∈R2 P-a.s., which we prove in AppendixBin a slightly more general setting for later use.
Proposition 2.5. P-a.s., for all x ∈ R2 and all Borel measurable functions η : [0,∞)→[0,∞]andf :R2 →[0,∞],
Ex
[∫ ∞ 0
η(t)f(Bt)dFt
]
=
∫ ∞
0
∫
R2η(t)f(y)qt(x, y)M(dy)dt, and in particular, for anyt >0,
∫
R2f(y)M(dy) = 1 t
∫
R2Ex [∫ t
0
f(Bs)dFs ]
dx.
2.3. The Liouville Dirichlet form. By virtue of Propositions 2.4and 2.5, we can apply the general theory of Dirichlet forms to obtain an explicit description of the Dirichlet form associated with the LBM, as it has been done in [17,18].
Denote by H1(R2)the standard Sobolev space, that is
H1(R2) ={f ∈L2(R2, dx) : ∇f ∈L2(R2, dx)}, on which we define the form
E(f, g) = 1 2
∫
R2∇f · ∇g dx. (2.5)
Recall that(E, H1(R2))is the Dirichlet form of the planar Brownian motionB. By He1(R2) we denote the extended Dirichlet space, that is the set of dx-equivalence classes of Borel measurable functions f on R2 such that limn→∞fn = f ∈ Rdx- a.e. for some(fn)n≥1 ⊂H1(R2)satisfying limk,l→∞E(fk−fl, fk−fl) = 0. By [6, Theorem 2.2.13] we have the following identification ofHe1(R2):
He1(R2) ={f ∈L2loc(R2, dx) :∇f ∈L2(R2, dx)}. Thecapacityof a setA⊂R2 is defined by
Cap(A) = inf
B⊂R2open A⊂B
inf
f∈H1(R2) f|B≥1dx-a.e.
{E(f, f) +
∫
R2f2dx }
.
A setA⊂R2 is calledpolarifCap(A) = 0. We call a functionf quasi-continuous if for anyε >0there exists an openU ⊂R2 withCap(U)< εsuch thatf|R2\U is real- valued and continuous. By [14, Theorem 2.1.7] any f ∈ He1(R2) admits a quasi- continuousdx-versionfe, which is unique up to polar sets by [14, Lemma 2.1.4].
Then, as the Liouville measureM is a Radon measure onR2and does not charge polar sets by [17, Theorem 2.2] (or by Propositions2.4,2.5and [6, Theorem 4.1.1 (i)]), the Dirichlet form(E,F)of the LBM B is a strongly local regular symmetric Dirichlet form on L2(R2, M) which takes on the following explicit form by [14, Theorem 6.2.1]: The domain is given by
F={
u∈L2(R2, M) : u=f Me -a.e. for somef ∈He1(R2)} , which can be identified with{
f ∈He1(R2) : fe∈L2(R2, M)}
by [14, Lemma 6.2.1], and forf, g∈ F the formE(f, g)is given by (2.5).
2.4. The killed Liouville Brownian motion. Let U be a non-empty open subset of R2 and let U ∪ {∂U} be its one-point compactification. We denote by TU :=
inf{s ≥ 0 : Bs 6∈ U} the exit time of the Brownian motion B from U and by τU := inf{s≥0 : Bs6∈U}that of the LBMB, whereinf∅:=∞. Since by definition Bt =BF−1
t ,t ≥ 0, and F is a homeomorphism on[0,∞), we haveτU = FTU. Let nowBU = (BtU)t≥0 andBU = (BtU)t≥0 denote the Brownian motion and the LBM, respectively, killed upon exitingU. That is, they are diffusions onU defined by
BUt :=
{Bt ift < TU,
∂U ift≥TU, BtU :=
{Bt ift < τU,
∂U ift≥τU.
Then fort, λ ∈ (0,∞), the semigroup operatorPtU and the resolvent operatorRUλ associated with the killed LBMBU are expressed as, for each Borel functionf :U → [−∞,∞]and with the conventionf(∂U) := 0,
PtUf(x) :=Ex[
f(BtU)]
and RUλf(x) :=Ex [∫ τU
0
e−λtf(Bt)dt ]
, x∈R2,
provided the integrals exist. IfU is bounded, as a time change ofBUthe killed LBM BU has the same integral kernel for its Green operatorGU asBU, namely for any non-negative Borel functionf :U →[0,∞]andx∈R2,
GUf(x) :=Ex
[∫ τU
0
f(Bt)dt ]
=Ex
[∫ TU
0
f(Bt)dFt
]
=
∫
U
gU(x, y)f(y)M(dy) (2.6) (cf. PropositionB.1). HeregU denotes the Euclidean Green kernel given by
gU(x, y) =
∫ ∞
0
qtU(x, y)dt, x, y∈R2, (2.7) for the heat kernel qtU(x, y) of BU: qU = qtU(x, y) : (0,∞)×U ×U → [0,∞) is the jointly continuous function such thatPx[BUt ∈ dy] = qUt (x, y)dyfor t > 0 and x∈ U, and we setqtU(x, y) := 0fort > 0and(x, y) ∈(U ×U)c. Finally, we recall (see e.g. [14, Example 1.5.1]) that the Green functiongB(x0,R)over a ballB(x0, R) is of the form
gB(x0,R)(x, y) = 1
π log 1
|x−y|+ Ψx0,R(x, y), x, y∈B(x0, R), (2.8) for some continuous functionΨx0,R :B(x0, R)×B(x0, R)→R.
3. PRELIMINARY ESTIMATES
3.1. Volume decay estimates. For our analysis of the Liouville heat kernel some good control on the volume of small balls under the Liouville measure is needed. An upper estimate has already been established in [17], and we provide a similar lower bound in the next lemma. The argument is based on some bounds on the negative moments of the measure of small balls. Such bounds have been proved in [27] in the case where the limiting random measure is obtained through approximation of the covariance kernel of the Gaussian free field by convolution. Since it is not clear to the authors whether the cutoff procedure producing the approximating measures Mnis covered by the results in [27], we give a comparison argument in LemmaC.1.
In the rest of this section, we writeξ(q) := (2 +˜ γ22)q+γ22q2 forq >0.
Lemma 3.1. Letα1:= 12(γ+ 2)2 andα2:= 12(2−γ)2. ThenP-a.s., for anyε >0and anyR≥1there existCi =Ci(X, γ, R, ε)>0,i= 4,5, such that
C4rα1+ε≤M(
B(x, r))
≤C5rα2−ε, ∀x∈B(R), r ∈(0,1]. (3.1) Proof. By the monotonicity of (3.1) in εand R it suffices to show (3.1) P-a.s. for eachεandR. The upper bound is proved in [17, Theorem 2.2]. We show the lower bound in the same manner. Letq := 2/γ, so that α1 = (2 + ˜ξ(q))/q. Letε >0 and R≥1be fixed, and forn≥1we setrn:= 2−nRand
ΞR,n:={(k
2nR,2lnR)
: k, l∈Z, |k|,|l| ≤2n}
⊂[−R, R]2. (3.2)
Then for eachn≥1, by ˇCebyˇsev’s inequality and LemmaC.1, P[
x∈minΞR,n
M(
B(x, rn))
≤2−n(α1+ε) ]
=P[
xmax∈ΞR,n
M(
B(x, rn))−q
≥2n(α1+ε)q
]≤ ∑
x∈ΞR,n
P[ M(
B(x, rn))−q
≥2n(α1+ε)q ]
≤2−n(α1+ε)q ∑
x∈ΞR,n
E[ M(
B(x, rn))−q]
≤2−n(α1+ε)q22n+3c2nξ(q)˜ =c2−εqn
for somec=c(γ, R)>0. Thus∑∞
n=1P[
minx∈ΞR,nM(
B(x, rn))
≤2−n(α1+ε)]
<∞, so that by the Borel-Cantelli lemmaP-a.s. for someC =C(X, γ, R, ε)>0we have that M(
B(x, rn))
≥ C2−n(α1+ε) for all n ≥ 1 and all x ∈ ΞR,n. Since for every y ∈ B(R) and r ∈ (0,1] we have B(y, r) ⊃ B(x, rn) for some x ∈ ΞR,n with n satisfying 14r ≤rn< 12r, the claim follows.
3.2. Exit time estimates. In this subsection we provide some lower estimates on the exit times from balls which are needed in the proof of Theorems1.1 and1.2.
More precisely, we establish estimates on the tail behaviour at zero of these exit times by showing certainP-a.s. local uniform bounds on their negative moments.
Let{ϑt}t≥0 denote the family of shift mappings for the LBMB, which is defined byϑt(ω0) :=θF−1
t (ω0)(ω0)fort≥0andω0 ∈Ω0and satisfiesFs+t−1 =Fs−1+Ft−1◦ϑs and henceBt+s=Bt◦ϑsfors, t≥0onΛω by virtue ofFt+s=Ft+Fs◦θt,s, t≥0 (cf. [6, Subsection A.3.2]).
Proposition 3.2. Letq > 0. ThenP-a.s., for anyκ >2 + ˜ξ(q)and anyR ≥1there exists a random constantC6=C6(X, γ, R, q, κ)>0such that
Ex[
τB(x,r)−q ]
≤C6r−κ, ∀x∈B(R), r∈(0,1], (3.3) Proof. Since (3.3) is weaker for larger κ and smaller R, it suffices to show (3.3) P-a.s. for eachκandR. First we note that, lettingn→ ∞in [17, Proposition 2.12]
by using [17, Lemma 2.8] (see also TheoremA.1below) and Fatou’s lemma, we get EEx[
τB(x,r)−q ]
≤cr−ξ(q)˜ , ∀x∈R2, r∈(0,1], (3.4) for some c = c(γ, q) > 0. As in the proof of Lemma 3.1 above let rn := 2−nR andΞR,n be defined as in (3.2) for any n ≥ 1. In the sequel we write Eµ for the expectation operator associated with the lawPµ of a Brownian motion with initial distributionµ. Letx∈R2 and letµx,rn :=Px[BτB(x,rn) ∈ ·]be the distribution of the LBM upon exitingB(x, rn). For anyz ∈ ∂B(x, rn), sinceB(z, rn) ⊂ B(x,2rn) and henceτB(z,rn)≤τB(x,2rn), by using (3.4) we get
EEz[ τB(x,2r−q
n)
]≤EEz[ τB(z,r−q
n)
]≤crn−ξ(q)˜ ,
provided n is large enough so that rn ≤ 1. By Fubini’s theorem, the µx,rn(dz)- integral of this inequality becomes
EEµx,rn[ τB(x,2r−q
n)
]≤crn−ξ(q)˜ .
