aa r X i v : . [ m a t h . P R ] F e b Liouville Brownian motion at criticality
R´emi Rhodes ∗ Vincent Vargas † Abstract
In this paper, we construct the Brownian motion of Liouville Quantum Gravity with centralcharge c = 1 (more precisely we restrict to the corresponding free field theory). Liouvillequantum gravity with c = 1 corresponds to two-dimensional string theory and is the conjecturalscaling limit of large planar maps weighted with a O ( n = 2) loop model or a Q = 4-state Pottsmodel embedded in a two dimensional surface in a conformal manner.Following [28], we start by constructing the critical LBM from one fixed point x ∈ R (or x ∈ S ), which amounts to changing the speed of a standard planar Brownian motion dependingon the local behaviour of the critical Liouville measure M ′ ( dx ) = − X ( x ) e X ( x ) dx (where X is a Gaussian Free Field, say on S ). Extending this construction simultaneously to all pointsin R requires a fine analysis of the potential properties of the measure M ′ . This allows usto construct a strong Markov process with continuous sample paths living on the support of M ′ , namely a dense set of Hausdorff dimension 0. We finally construct the associated Liouvillesemigroup, resolvent, Green function, heat kernel and Dirichlet form.In passing, we extend to quite a general setting the construction of the critical Gaussianmultiplicative chaos that was initiated in [21, 22] and also establish new capacity estimates forthe critical Gaussian multiplicative chaos. Key words or phrases:
Gaussian multiplicative chaos, critical Liouville quantum gravity, Brownian motion, heatkernel, potential theory.
MSC 2000 subject classifications: 60J65, 81T40, 60J55, 60J60, 60J80, 60J70, 60K40
Contents ∗ Universit´e Paris-Dauphine, Ceremade, F-75016 Paris, France. Partially supported by grant ANR-11-JCJCCHAMU † Ecole Normale Sup´erieure, DMA, 45 rue d’Ulm, 75005 Paris, France. Partially supported by grant ANR-11-JCJCCHAMU Critical LBM starting from one fixed point 12 F ′ on the whole of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Definition and properties of the critical LBM . . . . . . . . . . . . . . . . . . . . . 42 Liouville Quantum Field Theory is a two dimensional conformal field theory which plays an im-portant part in two dimensional models of Euclidean quantum gravity. Euclidean quantum gravityis an attempt to quantize general relativity based on Feynman’s functional integral and on theEinstein-Hilbert action principle. More precisely, one couples a Conformal Field Theory (CFT)with central charge c to gravity. A famous example is the coupling of c free scalar matter fields togravity, leading to an interpretation of such a specific theory of 2 d -Liouville Quantum Gravity asa bosonic string theory in c dimensions [52].It is shown in [52, 39, 15]) that the coupling of the CFT with gravity can be factorized as atensor product: the random metric governing the geometry of the space that the CFT lives onis independent of the CFT and roughly takes on the form [52, 39, 15] (we consider an Euclideanbackground metric for simplicity): g ( x ) = e γX ( x ) dx , (1.1)where the fluctuations of the field X are governed by the Liouville action and the parameter γ isrelated to the central charge of the CFT via the famous result in [39] (for c γ = 1 √ √ − c − √ − c ) . (1.2)Therefore γ ∈ [0 , d -Liouville quantum gravity. For these reasons and though it may be an interesting field in itsown right, Liouville quantum field theory (governing the metric in 2 d -Liouville quantum gravity)is an important object in theoretical physics.Here we will restrict to the case when the the cosmological constant in the Liouville action isset to 0, turning the field X in (1.1) into a Free Field, with appropriate boundary conditions. Inthe subcritical case γ < , c <
1, the geometry of the metric tensor (1.1) (with X a free field) ismathematically investigated in [23, 28, 29, 55, 57] and the famous KPZ scaling relations [39] are2igorously proved in [4, 23, 56] in a geometrical framework (see also [16] for a non rigorous heatkernel derivation). This paper focuses on the coupling of a CFT with a c = 1 central charge togravity or equivalently 2 d string theory: in this case, one has γ = 2 by relation (1.2). For an excellentreview on 2 d string theory, we refer to Klebanov’s lecture notes [37]. As expressed by Klebanov in[37]: ”Two-dimensional string theory is the kind of toy model which possesses a remarkably simplestructure but at the same time incorporates some of the physics of string theories embedded inhigher dimensions”. Among the γ γ = 2 probably possesses the richeststructure, inherited from its specific status of phase transition. For instance, the construction ofthe volume form, denoted by M ′ , associated to the metric tensor (1.1) is investigated in [21, 22]where it is proved that it takes on the unusual form: M ′ ( dx ) = − X ( x ) e X ( x ) dx, (1.3)which also coincides with a proper renormalization of e X ( x ) dx . The reader may also consult [44]for a construction of non trivial conformally invariant gravitationally dressed vertex operators, theso-called tachyonic fields , and [37] for more physics insights. In this paper, we will complete thispicture by constructing the Brownian motion (called critical Liouville Brownian motion, criticalLBM for short), semi-group, resolvent, Dirichlet form, Green function and heat kernel of the metrictensor (1.1) with γ = 2.We further point out that Liouville quantum gravity is conjecturally related to randomly tri-angulated random surfaces (see [18] for precise conjectures) weighted by discrete critical statisticalphysics models. For c = 1, these models include one-dimensional matrix models (also called “ma-trix quantum mechanics” (MQM)) [9, 30, 31, 33, 34, 36, 38, 50, 51, 61], the so-called O ( n ) loopmodel on a random planar lattice for n = 2 [40, 41, 42, 43], and the Q -state Potts model on arandom lattice for Q = 4 [8, 13, 25]. The critical LBM is therefore conjectured to be the scalinglimit of random walks on large planar maps weighted with a O ( n = 2) loop model or a Q = 4-state Potts model, which are embedded in a two dimensional surface in a conformal manner asexplained in [18]. For an introduction to the above mentioned 2 d -statistical models, see, e.g., [49].We further mention [12, 17] for recent advances on this topic in the context of pure gravity, i.e.with no coupling with a CFT.To complete this overview, we point out that the notions of diffusions or heat kernel are atthe core of physics literature about Liouville quantum gravity (see [2, 3, 10, 11, 14, 16, 62] forinstance): for more on this, see the subsection on the associated distance. Basically, our approach of the metric tensor e X ( x ) dx (where X is a Free Field) relies on theconstruction of the associated Brownian motion B , called the critical LBM. Standard results of 2 d -Riemannian geometry tell us that the law of this Brownian motion is a time change of a standardplanar Brownian motion B (starting from 0): B xt = x + B hB x i t (1.4)where the quadratic variations hB x i are formally given by: hB x i t = F ′ ( x, t ) − , F ′ ( x, t ) = Z t e X ( x + B r ) dr. (1.5)3ut in other words, we should integrate the weight e X ( x ) along the paths of the Brownian motion x + B to construct a mapping t F ′ ( x, t ). The inverse of this mapping corresponds to the quadraticvariations of B x . Of course, because of the irregularity of the field X , giving sense to (1.5) is notstraightforward and one has to apply a renormalization procedure: one has to apply a cutoff tothe field X (a procedure that smoothes up the field X ) and pass to the limit as the cutoff isremoved. The procedure is rather standard in this context. Roughly speaking, one introduces anapproximating field X ǫ where the parameter ǫ stands for the extent to which one has regularizedthe field X (we have X ǫ → X as ǫ goes to 0). One then defines F ǫ ( x, t ) = Z t e X ǫ ( x + B r ) dr (1.6)and one looks for a suitable deterministic renormalization a ( ǫ ) such that the family a ( ǫ ) F ǫ ( x, t )converges towards a non trivial object as ǫ →
0. In the subcritical case γ <
2, the situation israther well understood as the family a ( ǫ ) roughly corresponds to a ( ǫ ) ≃ exp( − γ E [ X ǫ ( x ) ])(the dependence of the point x is usually fictive and may easily get rid of) in such a way that a ( ǫ ) Z t e γX ǫ ( x + B r ) dr (1.7)converges towards a non trivial limit. We will call this renormalization procedure standard: it hasbeen successfully applied to construct random measures of the form e γX ( x ) dx [35, 58, 59] (thereader may consult [55] for an overview on Gaussian multiplicative chaos theory). Though thechoice of the cutoff does not affect the nature of the limiting object, a proper choice of the cutoffturns the expression (1.7) into a martingale. This is convenient to handle the convergence of thisobject.At criticality ( γ = 2), the situation is conceptually more involved. It is known [21] that thestandard renormalization procedure of the volume form yields a trivial object. Logarithmic cor-rections in the choice of the family a ( ǫ ) are necessary (see [21, 22]) and the limiting measure thatwe get is the same as that corresponding to a metric tensor of the form − X ( x ) e X ( x ) . Observethat it is not straightforward to see at first sight that such metric tensors coincide, or even arepositive. This subtlety is at the origin of some misunderstandings in the physics literature wherethe two forms of the tachyon field e X ( x ) and − X ( x ) e X ( x ) appear without making perfectly clearthat they coincide.The case of the LBM at criticality obeys this rule too. We will prove that non standard loga-rithmic corrections are necessary to make the change of time F ǫ converge and they produce thesame limiting change of times as that corresponding to a metric tensor of the form − X ( x ) e X ( x ) .This summarizes the almost sure convergence of F ǫ for one given fixed point x ∈ R . Yet, if onewishes to define a proper Markov process, one has to go one step further and establish that, almostsurely, F ǫ ( x, · ) converges simultaneously for all possible starting points x ∈ R : the place of the”almost sure” is important and gives rise to difficulties that are conceptually far different from theconstruction from one given fixed point. In [28], it is noticed that this simultaneous convergence ispossible as soon as the volume form M γ ( dx ) associated to the metric tensor (1.1) is regular enoughso as to make the mapping x Z R ln + | x − y | M γ ( dx )4ontinuous. When γ <
2, multifractal analysis shows that the measure M γ possesses a power lawdecay of the size of balls and this is enough to ensure the continuity of this mapping. In thecritical case γ = 2, the situation is more complicated because the measure M ′ is rather wild: forinstance the Hausdorff dimension of its support is zero [5]. Furthermore, the decay of the size ofballs investigated in [5] shows that continuity and even finiteness of the mapping x Z R ln + | x − y | M ′ ( dx ) (1.8)is unlikely to hold on the whole of R . Yet we will show that we can have a rather satisfactorycontrol of the size of balls for all x belonging to a set of full M ′ -measure, call it S : ∀ x ∈ S, sup r ∈ ]0 , M ′ ( B ( x, r ))( − ln r ) p < + ∞ for some p large enough. In particular this estimate shows that the expression (1.8) is finite forevery point x ∈ S . Also, this estimate answers a question raised in [5] (a similar estimate wasproved by the same authors [6] in the related case of the discrete multiplicative cascades). Wewill thus construct the change of time F ′ ( x, · ) simultaneously for all x ∈ S . What happens on thecomplement of S does not matter that much since it is a set with null M ′ measure. Yet we willextend the change of time F ′ ( x, · ) to the whole of R .Once F ′ is constructed, potential theory [26] tells us that the LBM at criticality is a strongMarkov process, which preserves the critical measure M ′ . We will then define the semigroup,resolvent, Green function and heat kernel associated to the LBM at criticality.We stress that, once all the pieces of the puzzle are glued together, this LBM at criticalityappears as a rather weird mathematical object. It is a Markov process with continuous samplepaths living on a very thin set, which is dense in R and has Hausdorff dimension 0. And in spiteof this rather wild structure, the LBM at criticality is regular enough to possess a (weak form of)heat kernel. Beyond the possible applications in physics, we feel that the study of such an objectis a fundamental and challenging mathematical problem, which is far from being settled in thispaper. An important question related to this work is the existence of a distance associated to the metrictensor (1.1) for γ ∈ [0 , γ < F ( x, t ) = R t e γX ( x + B r ) dr the associated additive functional along the Brownian paths. We know that forall t > p Xt ( x, y ). Many papers in the physics literature haveargued that the Liouville heat kernel should have the following representation which is classical inthe context of Riemannian geometry p Xt ( x, y ) ∼ e − d ( x,y ) /t t (1.9)where d ( x, y ) is the associated distance, i.e. the ”Riemannian distance” defined by (1.1). From therepresentation (1.9), physicists [2, 3, 62] have derived many fractal and geometrical properties of5iouville quantum gravity. In particular, the paper [62] established an intriguing formula for thedimension d H of Liouville quantum gravity which can be defined by the heuristic M γ ( { y ; d ( x, y ) r } ) ∼ r d H where M γ is the associated volume form. Note that the meaning of the above definition is notobvious since M γ is a multifractal random measure.Along the same lines, a recent physics paper [16] establishes an interesting heat kernel derivationof the KPZ formula. The idea behind the paper is that, if relation (1.9) holds, then one can extractthe metric from the heat kernel by using the Mellin-Barnes transform given by Z ∞ t s − p Xt ( x, y ) dt. Indeed, a standard computation gives the following equivalent for s ∈ ]0 , Z ∞ t s − e − d ( x,y ) /t t dt ∼ d ( x,y ) → C s d ( x, y ) − s ) (1.10)where C s is some positive constant. Though it is not clear at all that such a relation holds rigorously,it gives at least a way of defining a notion of capacity dimension for which a KPZ relation hasbeen heuristically derived in [16]. Thanks to the relation (1.10), the authors claim that this yieldsa geometrical version of the KPZ equation which does not rely on the Euclidean metric. Recallthat the rigorous geometrical derivations of KPZ in [23, 56] rely on the measure M γ and implyworking with Euclidean balls . Remark 1.1.
In fact, after the present work and based on results in the field of fractal diffusions,it is argued in [46] that one should replace the heuristic (1.9) by the following heuristic p Xt ( x, y ) ∼ e − (cid:18) d ( x,y ) dHt (cid:19) dH − t where d H is the dimension of Liouville quantum gravity. Hence, we should also rather get therelation Z ∞ t s − p Xt ( x, y ) dt ∼ d ( x,y ) → C s d ( x, y ) d H (1 − s ) . Acknowledgements
The authors wish to thank Antti Kupiainen, Miika Nikula and Christian Webb for many veryinteresting discussions which have helped a lot in understanding the specificity of the criticalcase and Fran¸cois David who always takes the time to answer their questions with patience andkindness.
In this section, we draw up the framework to construct the Liouville Brownian motion at criticalityon the whole plane R . Other geometries are possible and discussed at the end of the paper. At the time of publishing this manuscript, this heat kernel based KPZ formula has been rigorously proved in [7]. .1 Notations In what follows, we will consider Brownian motions B or ¯ B on R (or other geometries) independentof the underlying Free Field. We will denote by E Y or P Y expectations and probability with respectto a field Y . For instance, E X or P X (resp. E B or P B ) stand for expectation and probability withrespect to the log-correlated field X (resp. the Brownian motion B ). For d >
1, we consider thespace C ( R + , R d ) of continuous functions from R + into R d equipped with the topology of uniformconvergence over compact subsets of R + . In this section we introduce the log-correlated Gaussian fields X on R that we will work withthroughout this paper. One may consider other geometries as well, like the sphere S or the torus T (in which case an adaptation of the setup and proofs is straightforward). We will representthem via a white noise decomposition.We consider a family of centered Gaussian processes (( X ǫ ( x )) x ∈ R ) ǫ> with covariance structuregiven, for 1 > ǫ > ǫ ′ >
0, by: K ǫ ( x, y ) = E [ X ǫ ( x ) X ǫ ′ ( y )] = Z ǫ k ( u, x, y ) u du (2.1)for some family ( k ( u, · , · )) u > of covariance kernels satisfying: A.1 k is nonnegative, continuous. A.2 k is locally Lipschitz on the diagonal, i.e. ∀ R > ∃ C R > ∀| x | R , ∀ u > ∀ y ∈ R | k ( u, x, x ) − k ( u, x, y ) | C R u | x − y | A.3 k satisfies the integrability condition: for each compact set S ,sup x ∈ S,y ∈ R Z ∞ | x − y | k ( u, x, y ) u du < + ∞ . A.4 the mapping H ǫ ( x ) = R ǫ k ( u,x,x ) u du − ln ǫ converges pointwise as ǫ → K , sup x,y ∈ K sup ǫ ∈ ]0 , | H ǫ ( x ) − H ǫ ( y ) || x − y | < + ∞ . A.5 for each compact set K , there exists a constant C K > k ( u, x, y ) > k ( u, x, x )(1 − C K u / | x − y | / ) + for all x ∈ K and y ∈ R . for all u > x ∈ K and y ∈ R .Such a construction of Gaussian processes is carried out in [1, 54] in the translation invariant case.Furthermore, [A.2] implies the following relation that we will use throughout the paper: for eachcompact set S , there exists a constant c S > k ) such that for all y ∈ S , ǫ ∈ (0 ,
1] and w ∈ B ( y, ǫ ), we haveln 1 ǫ − c S K ǫ ( y, w ) ln 1 ǫ + c S . (2.2)We denote by F ǫ the sigma algebra generated by { X u ( x ); ǫ u, x ∈ R } .7 .3 Examples We explain first a Fourier white noise decomposition of log-correlated translation invariant fieldsas this description appears rather naturally in physics. Consider a nonnegative even function ϕ defined on R such that lim | u |→∞ | u | ϕ ( u ) = 1. We consider the kernel K ( x, y ) = Z R e i h u,x − y i ϕ ( u ) du. (2.3)We consider the following cut-off approximations K ǫ ( x, y ) = 12 π Z B (0 ,ǫ − ) e i h u,x − y i ϕ ( u ) du. (2.4)The kernel K can be seen as the prototype of kernels of log-type in dimension 2. It has obviouscounterparts in any dimension. The cut-off approximation is quite natural, rather usual in physics(sometimes called the ultraviolet cut-off) and has well known analogues on compact manifolds (interms of series expansion along eigenvalues of the Laplacian for instance). If X has covariancegiven by (2.3), then X has the following representation X ( x ) = Z R e i h u.x i p ϕ ( u )( W ( du ) + iW ( du ))where W ( du ) and W ( du ) are independent Gaussian distributions. The distributions W ( du ) and W ( du ) are functions of the field X (since they are the real and imaginary parts of the Fouriertransform of X ). The law of W ( du ) is W ( du )+ W ( − du ) where W is a standard white noise and thelaw of W ( du ) is f W ( du ) − f W ( − du ) where f W is also standard white noise. One can then considerthe following family with covariance (2.4) and which fits into our framework as it corresponds toadding independent fields X ǫ ( x ) = Z B (0 ,ǫ − ) e i h u.x i p ϕ ( u )( W ( du ) + iW ( du ))Notice also that the approximations X ǫ are functions of the original field X since W ( du ) and W ( du ) are functions of the field X .Notice that K ǫ can be rewritten as ( S stands for the unit sphere and ds for the uniformprobability measure on S ) K ǫ ( x, y ) = Z ǫ k ( r, x, y ) r dr, with k ( r, x, y ) = r Z S ϕ ( rs ) cos( r h x − y, s i ) ds. Let us simplify a bit the discussion by assuming that ϕ is isotropic. In that case, it is plain tocheck that assumptions [A.1-5] are satisfied (in the slightly extended context of integration over[0 , ǫ ] instead of [1 , ǫ ] but this is harmless as K ( x, y ) is here a very regular Gaussian kernel). Example 1. Massive Free Field (MFF).
The whole plane MFF is a centered Gaussian distri-bution with covariance kernel given by the Green function of the operator π ( m − △ ) − on R ,i.e. by: ∀ x, y ∈ R , G m ( x, y ) = Z ∞ e − m u − | x − y | u du u . (2.5)8 he real m > is called the mass. This kernel is of σ -positive type in the sense of Kahane [35]since we integrate a continuous function of positive type with respect to a positive measure. It isfurthermore a star-scale invariant kernel (see [1, 54]): it can be rewritten as G m ( x, y ) = Z + ∞ k m ( u ( x − y )) u du. (2.6) for some continuous covariance kernel k m ( z ) = R ∞ e − m v | z | − v dv and therefore satisfies theassumptions [A.1-5].One may also choose the Fourier white noise (2.4) decomposition with ϕ ( u ) = | u | + m or thesemigroup covariance structure E [ X ǫ ( x ) X ǫ ′ ( y )] = π Z ∞ max( ǫ,ǫ ′ ) p ( u, x, y ) e − m u du, which also satisfies assumptions [A.1-5] (modulo a change of variable u = 1 /v in the aboveexpression: see [22, section D]). Example 2. Gaussian Free Field (GFF).
Consider a bounded open domain D of R . Formally,a GFF on D is a Gaussian distribution with covariance kernel given by the Green function ofthe Laplacian on D with prescribed boundary conditions. We describe here the case of Dirichletboundary conditions. The Green function is then given by the formula: G D ( x, y ) = π Z ∞ p D ( t, x, y ) dt (2.7) where p D is the (sub-Markovian) semi-group of a Brownian motion B killed upon touching theboundary of D , namely for a Borel set A ⊂ D Z A p D ( t, x, y ) dy = P x ( B t ∈ A, T D > t ) with T D = inf { t > , B t D } . The most direct way to construct a cut-off family of the GFF on D is then to consider a white noise W distributed on D × R + and to define: X ( x ) = √ π Z D × R + p D ( s , x, y ) W ( dy, ds ) . One can check that E [ X ( x ) X ( x ′ )] = π R ∞ p D ( s, x, x ′ ) ds = G D ( x, x ′ ) . The corresponding cut-offapproximations are given by: X ǫ ( x ) = √ π Z D × [ ǫ , ∞ [ p D ( s , x, y ) W ( dy, ds ) . They have the following covariance structure E [ X ǫ ( x ) X ǫ ′ ( y )] = π Z ∞ max( ǫ,ǫ ′ ) p D ( u, x, y ) du, (2.8) which also satisfies assumptions [A.1-5] (on every subdomain of D and modulo a change of variable u = 1 /v in the above expression: see also [22, section D]). For some technical reasons, we will sometimes also consider either of the following assumptions:
A.6 k ( v, x, y ) = 0 for | x − y | > Dv − (1 + 2 ln v ) α for some constants D, α > A.6’ ( k ( v, x, y )) v is the family of kernels presented in examples 1 or 2.9 .4 Regularized Riemannian geometry We would like to consider a Riemannian metric tensor on R (using conventional notations inRiemannian geometry) of the type e X ǫ ( x ) dx , where dx stands for the standard Euclidean metricon R . Yet, as we will see, such an object has no suitable limit as ǫ goes to 0. So, for futurerenormalization purposes, we rather consider: g ǫ ( x ) dx = √− ln ǫ ǫ e X ǫ ( x ) dx . The Riemannian volume on the manifold ( R , g ǫ ) is given by: M ǫ ( dx ) = √− ln ǫ ǫ e X ǫ ( x ) dx, (2.9)where dx stands for the Lebesgue measure on R , and will be called ǫ -regularized critical measure.The study of the limit of the random measures ( M ǫ ( dx )) ǫ is carried out in [21, 22] in a less generalcontext. It is based on the study of the limit of the family M ′ ǫ ( dx ) defined by: M ′ ǫ ( dx ) := (2 ln 1 ǫ − X ǫ ( x )) e X ǫ ( x ) − ǫ dx. We will extend the results in [21] and prove
Theorem 2.1.
Almost surely, the (locally signed) random measures ( M ′ ǫ ( dx )) ǫ> converge as ǫ → towards a positive random measure M ′ ( dx ) in the sense of weak convergence of measures. Thislimiting measure has full support and is atomless. Concerning the Seneta-Heyde norming, we have
Theorem 2.2.
Assume [A.1-5] and either [A.6] or [A.6’]. We have the convergence in probabilityin the sense of weak convergence of measures: M ǫ ( dx ) → r π M ′ ( dx ) , as ǫ → . The proof of Theorem 2.2 is carried out in [22, section D] (in fact, it is assumed in [22] thatthe family of kernels ( k ( v, x, y )) v is translation invariant but adapting the proof is straightforwardand thus left to the reader).Beyond its conceptual importance, the Seneta-Heyde norming is crucial to establish, via Ka-hane’s convexity inequalities [35], the study of moments carried out in [21, 22] and obtain Proposition 2.3.
Assume [A.1-5] and either [A.6] or [A.6’]. For each bounded Borel set A and q ∈ ] − ∞ , , the random variable M ′ ( A ) possesses moments of order q . Furthermore, if A has nontrivial Lebesgue measure and x ∈ R : E [ M ′ ( λA + x ) q ] ≃ C ( q, x ) λ ξ M ′ ( q ) where ξ M ′ is the power law spectrum of M ′ : ∀ q < , ξ M ′ ( q ) = 4 q − q . M ′ is establishedin [5]. We stress here that we pursue the discussion at a heuristic level since the result in [5] isnot general enough to apply in our context (to be precise, it is valid for a well chosen family ofkernels ( k ( v, x, y )) v in dimension 1 in order to get nice scaling relations for the associated measure M ′ ). Anyway, we expect this result to be true in greater generality and we will not use it in thispaper: we are more interested in its conceptual significance. So, by analogy with [5], the measure M ′ is expected to possess the following modulus of continuity of ”square root of log” type: for all γ < /
2, there exists a random variable C almost surely finite such that ∀ ball B ⊂ B (0 , , M ′ ( B ) C (ln(1 + | B | − )) − γ . (2.10)Furthermore, the Hausdorff dimension of the carrier of M ′ is 0. By analogy with the results thatone gets in the context of multiplicative cascades [6], one also expects that the above theorem 2.10cannot be improved. In particular, the measure M ′ does not possess a modulus of continuity betterthan a log unlike the subcritical situation explored in [28], where this property turned out to becrucial for the construction of the LBM as a whole Markov process. This remark is at the origin ofthe further complications arising in our paper (the critical case) in comparison with [28, 29] (thesubcritical case). The main concern of this paper will be the Brownian motion associated with the metric tensor g ǫ : following standard formulas of Riemannian geometry, one can associate to the Riemannianmanifold ( R , g ǫ ) a Brownian motion B ǫ : Definition 2.4 ( ǫ -regularized critical Liouville Brownian motion, LBM ǫ for short) . For any fixed ǫ > , we define the following diffusion on R . For any x ∈ R , B ǫ,xt = x + Z t ( − ln ǫ ) − / ǫ − e − X ǫ ( B ǫ,xu ) d ¯ B u . (2.11) where ¯ B t is a standard two-dimensional Brownian motion. We stress that the fact that there is no drift term in the definition of the Brownian motion istypical from a scalar metric tensor in dimension 2. By using the Dambis-Schwarz Theorem, onecan define the law of the LBM ǫ as Definition 2.5.
For any ǫ > fixed and x ∈ R , B ǫ,xt = x + B hB ǫ,x i t , (2.12) where ( B r ) r > is a two-dimensional Brownian motion independent of the field X and the quadraticvariation hB ǫ,x i of B ǫ,x is defined as hB ǫ,x i t := inf { s > − ln ǫ ) / ǫ Z s e X ǫ ( x + B u ) du > t } . (2.13)It will thus be useful to define the following quantity on R × R + : F ǫ ( x, s ) = ǫ Z s e X ǫ ( x + B u ) du, (2.14)11n such a way that the process hB ǫ,x i is entirely characterized by:( − ln ǫ ) / F ǫ ( x, hB ǫ,x i t ) = t. (2.15)Several standard facts can be deduced from the smoothness of X ǫ . For each fixed ǫ >
0, theLBM ǫ a.s. induces a Feller diffusion on R , and thus a semi-group ( P ǫt ) t > , which is symmetricw.r.t the volume form M ǫ .We will be mostly interested in establishing the convergence in law of the LBM ǫ as ǫ → ǫ thus boils down to establishing the convergenceof its quadratic variations hB ǫ,x i . The first section is devoted to the convergence of the ǫ -regularized LBM when starting from onefixed point, say x ∈ R . As in the case of the convergence of measures [21, 22], the critical situationhere is technically more complicated than in the subcritical case [28], though conceptually similar.The first crucial step of the construction consists in establishing the convergence towards 0 of thefamily of functions ( F ǫ ( x, · )) ǫ and then in computing the first order expansion of the maximum ofthe field X ǫ along the Brownian path up to time t , and more precisely to prove thatmax s ∈ [0 ,t ] X ǫ ( x + B s ) − ǫ → −∞ , as ǫ → . (3.1)This is mainly the content of subsection 3.2, after some preliminary lemmas in subsection 3.1.Then our strategy will mainly be to adapt the ideas related to convergence of critical measures[21, 22].We further stress that, in the case of measures (see [21]), the content of subsection 3.2 isestablished thanks to comparison with multiplicative cascades measures and Kahane’s convexityinequalities. In our context, no equivalent result has been established in the context of multiplicativecascades in such a way that we have to carry out a direct proof. Let us consider a standard Brownian motion B on the plane R starting at some given point x ∈ R . Let us consider the occupation measure µ t of the Brownian motion up to time t > ∀ ǫ ∈ ]0 , , h ( ǫ ) = ln 1 ǫ ln ln ln 1 ǫ . (3.2)The following result is proved in [45] Theorem 3.1.
There exists a deterministic constant c > such that P B -almost surely, the set E = { z ∈ R ; lim sup ǫ → µ t ( B ( z, ǫ )) ǫ h ( ǫ ) = c } has full µ t -measure. We will need an extra elementary result about the structure of Brownian paths:12 emma 3.2.
For every p > , we have almost surely: Z R × R | x − y | ln (cid:0) | x − y | + 2 (cid:1) p µ t ( dx ) µ t ( dy ) < + ∞ . Proof of Lemma 3.2.
We have: E B h Z R × R | x − y | ln (cid:0) | x − y | + 2 (cid:1) p µ t ( dx ) µ t ( dy ) i = E B h Z [0 ,t ] | B r − B s | ln (cid:0) | B r − B s | + 2 (cid:1) p drds i = Z [0 ,t ] E B h | r − s || B | ln (cid:0) | r − s | / | B | + 2 (cid:1) p i drds. Let us compute for a t the quantity E B h a | B | ln (cid:0) a / | B | +2 (cid:1) p i . By using the density of theGaussian law, we get: E B h a | B | ln (cid:0) a / | B | + 2 (cid:1) p i = 12 π Z R a | u | ln (cid:0) a / | u | + 2 (cid:1) p e − | u | du Z ∞ ar ln (cid:0) a / r + 2 (cid:1) p e − r dr a Z ∞ u ln (cid:0) u + 2 (cid:1) p e − u a du a Z a / u ln (cid:0) u + 2 (cid:1) p e − u a du + 1 a Z ta / u ln (cid:0) u + 2 (cid:1) p e − u a du + 1 a Z ∞ t u ln (cid:0) u + 2 (cid:1) p e − u a du a Z a / u ln p (cid:0) u (cid:1) du + 1 a Z t u ln (cid:0) u + 2 (cid:1) p e − a / du + 2 t ln p e − t a p a ( p −
1) ln p − a + Ca e − a / + 2 t ln p e − t a . Therefore E B h Z R Z R | x − y | ln (cid:0) | x − y | + 2 (cid:1) p µ t ( dx ) µ t ( dy ) i C Z t Z t (cid:16) | r − s | ln p − | r − s | + 1 | r − s | e − | r − s | / + e − t | r − s | (cid:17) drds. As this latter quantity is obviously finite, the proof is complete.
To begin with, we claim:
Proposition 3.3.
For all x ∈ R , almost surely in X , the family of random mapping t F ǫ ( x, t ) converges to in the space C ( R + , R + ) as ǫ → .Proof. Fix x ∈ R . Observe first that F ǫ ( x, t ) = ǫ Z t e X ǫ ( x + B r ) dr = Z t e g ǫ ( x + B r ) e X ǫ ( x + B r ) − E X [ X ǫ ( x + B r ) ] dr g ǫ ( u ) = 2 E X [ X ǫ ( u ) ] − ǫ = Z /ǫ k ( u, x, x ) − u du. By assumption [A.4], the function g ǫ converges uniformly over the compact subsets of R . Fur-thermore, for each t >
0, the set { x + B s ; s ∈ [0 , t ] } is a compact set and g ǫ converges uniformlyover this compact set. So, even if it means considering R t e X ǫ ( x + B r ) − E X [ X ǫ ( x + B r ) ] dr instead of F ǫ ( x, t ), we may assume that E X [ X ǫ ( x + B r ) ] = ln ǫ , in which case F ǫ ( x, t ) is a martingale withrespect to the filtration F ǫ = σ { X r ( x ); ǫ r, x ∈ R } . As this martingale is nonnegative, it con-verges almost surely. We just have to prove that the limit is 0. To this purpose, we use a lemmain [24]. Translated into our context, it reads: Lemma 3.4.
The almost sure convergence of the family ( F ǫ ( x, t )) ǫ towards as ǫ → is equivalentto the fact that lim sup ǫ → F ǫ ( x, t ) = + ∞ under the probability measure defined by: Q |F ǫ = t − F ǫ ( x, t ) P X . The main idea of what follows is to prove that, under Q , F ǫ ( x, t ) is stochastically boundedfrom below by the exponential of a Brownian motion so that lim sup ǫ → F ǫ ( x, t ) = + ∞ and weapply Lemma 3.4 to conclude .To carry out this argument, let us define a new probability measure Θ ǫ on B ( R ) ⊗ F ǫ byΘ ǫ |B ( R ) ⊗F ǫ = t − e X ǫ ( y ) − ǫ P X ( dω ) µ t ( dy ) , (3.3)where µ t stands for the occupation measure of the Brownian motion B x starting from x . We denoteby E Θ ǫ the corresponding expectation. In fact, since the above definition defines a pre-measureon the ring B ( R ) ⊗ S ǫ F ǫ , one can define a measure Θ on B ( R ) ⊗ F by using Caratheodory’sextension theorem. We recover the relation Θ |B ( A ) ⊗F ǫ = Θ ǫ . Similarly, we construct the probabilitymeasure Q on F = σ (cid:0) S ǫ F ǫ (cid:1) by setting: Q |F ǫ = t − F ǫ ( x, t ) d P X , which is nothing but the marginal law of ( ω, y ) ω with respect to Θ ǫ . We state a few elementaryproperties below. The conditional law of y given F ǫ is given by:Θ ǫ ( dy |F ǫ ) = e X ǫ ( y ) − ǫ F ǫ ( x, t ) µ t ( dy ) . If Y is a B ( R ) ⊗ F ǫ -measurable random variable then it has the following conditional expectationgiven F ǫ : E Θ ǫ [ Y |F ǫ ] = Z R Y ( y, ω ) e X ǫ ( y ) − ǫ F ǫ ( x, t ) µ t ( dy ) . Now we turn to the proof of Proposition 3.3 while keeping in mind this preliminary background.Let us observe that it is enough to prove that the set { lim sup ǫ → F ǫ ( x, t ) = + ∞} has probability 1 conditionally to y under Θ to deduce that it satisfies Q (cid:0) { lim sup ǫ → F ǫ ( x, t ) = + ∞} (cid:1) = 1 .
14o we have to compute the law of F ǫ ( x, t ) under Θ( ·| y ). Recall the definition of h in (3.2). Wehave: F ǫ ( x, t ) = Z R e X ǫ ( u ) − ǫ µ t ( du ) > Z R e X ǫ ( u ) − ǫ B ( y,ǫ ) µ t ( du ) . Let us now write X ǫ ( u ) = λ ǫ ( u, y ) X ǫ ( y ) + Z ǫ ( u, y )where λ ǫ ( u, y ) = K ǫ ( u,y )ln ǫ and Z ǫ ( u, y ) = X ǫ ( u ) − λ ǫ ( u, y ) X ǫ ( y ). Observe that the process ( Z ǫ ( u, y )) u ∈ R is independent of X ǫ ( y ). Therefore F ǫ ( x, t ) > Z R e X ǫ ( u ) − ǫ B ( y,ǫ ) µ t ( du )= e X ǫ ( y ) − ǫ Z R e Z ǫ ( u,y )+2( λ ǫ ( u,y ) − X ǫ ( y ) B ( y,ǫ ) µ t ( du )= e X ǫ ( y ) − ǫ +ln h ( ǫ ) × inf u ∈ B ( y,ǫ ) e λ ǫ ( u,y ) − X ǫ ( y ) × ǫ h ( ǫ ) Z R e Z ǫ ( u,y ) B ( y,ǫ ) µ t ( du ) . Let us define a ǫ ( y ) = Z R B ( y,ǫ ) µ t ( du ) . With the help of the Jensen inequality, we deduce F ǫ ( x, t ) > e X ǫ ( y ) − ǫ +ln h ( ǫ ) inf u ∈ B ( y,ǫ ) e λ ǫ ( u,y ) − X ǫ ( y ) a ǫ ( y ) ǫ h ( ǫ ) exp (cid:16) a ǫ ( y ) Z B ( y,ǫ ) Z ǫ ( u, y ) µ t ( du ) (cid:17) . Let us set Y ǫ = a ǫ ( y ) − Z R Z ǫ ( u, y ) B ( y,ǫ ) µ t ( du ) . Finally, for all
R >
0, we use the independence of Y ǫ and X ǫ ( y ) to getΘ (cid:16) { lim sup ǫ → F ǫ ( x, t ) = + ∞}| y (cid:17) (3.4) > Θ (cid:16) { lim sup ǫ → e X ǫ ( y ) − ǫ +ln h ( ǫ ) × inf u ∈ B ( y,ǫ ) e λ ǫ ( u,y ) − X ǫ ( y ) × a ǫ ( y ) ǫ h ( ǫ ) × exp( − R ) = + ∞}| y (cid:17) × Θ( Y ǫ > − R | y ) . (3.5)Now we analyze the behaviour of each term in the above expression.First, notice that Θ( Y ǫ > − R | y ) = P X ( Y ǫ > − R ) and that Y ǫ is a centered Gaussian randomvariable under P X with variance a ǫ ( y ) − Z B ( y,ǫ ) × B ( y,ǫ ) E X (cid:2) ( X ǫ ( u ) − λ ǫ ( u, y ) X ǫ ( y ))( X ǫ ( u ′ ) − λ ǫ ( u ′ , y ) X ǫ ( y )) (cid:3) µ t ( du ) µ t ( du ′ ) . This quantity may be easily evaluated with assumption [A.2] and proved to be less than someconstant C , which does not depend on ǫ and y ∈ { x + B s ; s ∈ [0 , t ] } . ThereforeΘ( Y ǫ > − R | y ) > − ρ ( R ) (3.6)15or some nonnegative function ρ that goes to 0 as R → ∞ .Second, from Theorem 3.1, there exists a constant c such that P B -almost surely, the set E = { z ∈ R ; lim sup ǫ → a ǫ ( z ) ǫ h ( ǫ ) = c } has full µ t -measure. Since E has full µ t measure, even if it means extracting a random subsequence(only depending on B ), we may assume thatlim ǫ → a ǫ ( y ) ǫ h ( ǫ ) = c. (3.7)Third, under Θ( ·| y ), the process X ǫ ( y ) − ǫ is a Brownian motion, call it ¯ B , in logarithmictime, i.e. ¯ B ln ǫ = X ǫ ( y ) − ǫ . We further stress that ¯ B is independent from B , and thus from the random sequence ( a ǫ ( y )) ǫ .From the law of the iterated logarithm, we deduce that Θ( ·| y )-almost surelylim sup ǫ → e X ǫ ( y ) − ǫ +ln h ( ǫ ) = + ∞ . (3.8)Fourth, using assumption [A.2], it is readily seen that there exists some constant C such thatfor all ǫ ∈ (0 , y ∈ { x + B s , s ∈ [0 , t ] } and all u ∈ B ( y, ǫ )2 | λ ǫ ( u, y ) − | C ( − ln ǫ ) − . Since the process X ǫ ( y ) − ǫ is a Brownian motion in logarithmic time under Θ( ·| y ), we deducelim inf ǫ → inf u ∈ B ( y,ǫ ) e λ ǫ ( u,y ) − X ǫ ( y ) > . (3.9)By gathering (3.7)+(3.8)+(3.9), we deduce that, under Θ( ·| y ):lim sup ǫ → e X ǫ ( y ) − ǫ +ln h ( ǫ ) × inf u ∈ B ( y,ǫ ) e λ ǫ ( u,y ) − X ǫ ( y ) × a ǫ ( y ) ǫ h ( ǫ ) × exp( − R ) = + ∞ . (3.10)By plugging (3.6)+(3.10) into (3.5), we get for all R > (cid:16) { lim sup ǫ → F ǫ ( x, t ) = + ∞}| y (cid:17) > − ρ ( R ) . By choosing R arbitrarily large, we complete the proof of Proposition 3.3 with the help of Lemma3.4. Proposition 3.5.
Almost surely in X , for all x ∈ R , P B x almost surely, for all t > we have: sup ǫ> sup s ∈ [0 ,t ] X ǫ ( B xs ) − ǫ + a ln ln 1 ǫ < + ∞ (3.11) for all a ∈ ]0 , [ . roof. Assume that the kernel k ( u, x, y ) in (2.1) is given by k ( u ( x − y )) for some continuouscovariance kernel k with k (0) = 1. It is then proved in [21] that, for each fixed a ∈ ]0 , [, thereexists a sequence ( C n ) n of P X -almost surely finite random variables such that,sup ǫ> sup x ∈ B (0 ,n ) X ǫ ( x ) − ǫ + a ln ln 1 ǫ < C n . (3.12)Now for each fixed x , the Brownian motion B x has P x -almost surely continuous sample paths.Therefore, P x -almost surely, for any t >
0, we can find n such that ∀ s ∈ [0 , t ], B xs ∈ B (0 , n ). Thusthe claim follows from (3.12) in this specific case.One must make some extra effort to extend this result to the more general situation of as-sumption [A]. It will be convenient to set t = ln ǫ for ǫ ∈ ]0 , x ∈ R , we consider themapping t ϕ x ( t ) = E [ X e − t ( x ) ] = Z e t k ( u, x, x ) u du. It is continuous and strictly increasing and we denote by ϕ − x ( t ) the inverse mapping. Let us thenconsider the mapping t T x ( t ) defined by ϕ x ( T x ( t )) = t . We consider the Gaussian process Y t ( x ) = X e − Tx ( t ) ( x ), which has constant variance t . We have E [ Y t ( x ) Y t ( y )] = Z e Tx ( t ) ∧ Ty ( t ) k ( u, x, y ) u du > Z e Tx ( t ) k ( u, x, y ) u du = Z e t k ( e ϕ − x (ln v ) , x, y ) k ( e ϕ − x (ln v ) , x, x ) dvv . By assumption [A.5], for each compact set K , we can find a constant C K such that k ( u, x, y ) k ( u, x, x ) > (1 − C K u / | x − y | / ) + for all x ∈ K and y ∈ R . We deduce, for all x ∈ K E [ Y t ( x ) Y t ( y )] > Z e t (1 − C K e ϕ − x (ln v ) / | x − y | / ) + v dv. By assumption [A.4], one can check that the mapping x e ϕ − x (ln v ) v converges uniformly on K towards a bounded strictly positive function. So even if it means changing the constant C K , wehave E [ Y t ( x ) Y t ( y )] > Z e t (1 − C K | x − y | / ) + v dv. (3.13)for all x ∈ K and y ∈ R . From [21], this latter covariance kernel satisfies the estimate (3.11).From [35], the above comparison between covariance kernels (3.13) with equal variance entails thatthe result also holds for the process Y t . It is then plain to conclude by noticing that the function x T x ( t ) t converges uniformly as t → ∞ over the compact sets towards a strictly positive limit(this results from [A.4]). 17 .3 Limit of the derivative PCAF Inspired by the construction of measures at criticality [21, 22], it seems reasonable to think thatthe change of times F ǫ , when suitable renormalized, should converge towards a random change oftimes that coincides with the limit of the following process F ′ ,ǫ ( x, t ) := Z t (cid:0) ǫ − X ǫ ( x + B u ) (cid:1) e X ǫ ( x + B u ) − ǫ du. Establishing the convergence of the above martingale is the main purpose of this section. Observethat, P B almost surely, the family ( F ′ ,ǫ ( x, t )) ǫ is almost martingale for each t > ǫ by E X [ X ǫ ( x ) ]). Nevertheless, it is not nonnegative and not uniformly integrable. It istherefore not obvious that such a family almost surely converges towards a (non trivial) positivelimiting random variable. The following theorem is the main result of this section: Theorem 3.6.
Assume [A.1-5] and fix x ∈ R . For each t > , the family ( F ′ ,ǫ ( x, t )) ǫ convergesalmost surely in X and in B as ǫ → towards a positive random variable denoted by F ′ ( x, t ) , suchthat F ′ ( x, t ) > almost surely. Furthermore, almost surely in X and in B , the (non necessarilypositive) random mapping t F ′ ,ǫ ( x, t ) converges as ǫ → in the space C ( R + , R + ) towards astrictly increasing continuous random mapping t F ′ ( x, t ) . Throughout this section, we will assume that assumptions [A.1-5] are in force. The observa-tion made in the beginning of the proof of Proposition 3.3 remains valid here and, without lossof generality, we may assume that ln ǫ = E X [ X ǫ ( x ) ]. In this way, the family ( F ′ ,ǫ ( x, t )) ǫ is amartingale. Actually, Proposition 3.5 tells us that it is a positive martingale for t large enough.Therefore it converges almost surely towards a limit F ′ ( x, t ). But it is not uniformly integrable sothat there are several complications involved in establishing non-triviality of the limit. We haveto introduce some further tools to study the convergence. We will introduce a family of auxiliary”truncated” martingales, called below F ′ ,ǫβ ( x, t ), which are reasonably close to F ′ ,ǫ ( x, t ) while beingsquare integrable. This will be enough to get the non triviality of F ′ ( x, t ).Given t > z, x ∈ R and β >
0, we introduce the random variables f βǫ ( z ) = (cid:0) ǫ − X ǫ ( z ) + β (cid:1) { τ βz <ǫ } e X ǫ ( z ) − ǫ (3.14) F ′ ,ǫβ ( x, t ) = Z t (cid:0) ǫ − X ǫ ( B xu ) + β (cid:1) { τ βBxu <ǫ } e X ǫ ( B xu ) − ǫ du = Z t f βǫ ( B xu ) du e F ′ ,ǫβ ( x, t ) = Z t (cid:0) ǫ − X ǫ ( B xu ) (cid:1) { τ βBxu <ǫ } e X ǫ ( B xu ) − ǫ du, where, for each u ∈ [0 , t ], τ βu is the ( F ǫ ) ǫ -stopping time defined by τ βz = sup { r , X r ( z ) − r > β } . In what follows, we will first investigate the convergence of ( F ′ ,ǫβ ( x, t )) ǫ ∈ ]0 , to deduce first theconvergence of ( e F ′ ,ǫβ ( x, t )) ǫ ∈ ]0 , and then the convergence of ( F ′ ,ǫ ( x, t )) ǫ ∈ ]0 , . We claim: Proposition 3.7.
We fix x ∈ R and t > . Almost surely in B , the process ( F ′ ,ǫβ ( x, t )) ǫ ∈ ]0 , is a continuous positive F ǫ -martingale and thus converges almost surely in X and B towards anonnegative random variable denoted by F ′ β ( x, t ) . Proposition 3.8.
We fix x ∈ R and t > . Almost surely in B , the martingale ( F ′ ,ǫβ ( x, t )) ǫ> isuniformly integrable.Proof. Let us first state the following lemma, which will serve in the forthcoming computations.
Lemma 3.9.
Let us denote µ t the occupation measure of the Brownian motion B xt . P B x -almostsurely, for all δ > , there are a compact set K and some constant L such that µ t ( K c ) δ and forall y ∈ K sup r µ t ( B ( y, r )) r g ( r ) L with g ( r ) = ln (cid:0) r + 2 (cid:1) . Proof.
It is a direct consequence of Lemma 3.2.According to this lemma, for each fixed δ >
0, we are given a compact K = K δ satisfying theabove conditions. We denote by µ Kt the measure µ Kt ( dy ) = K ( y ) µ t ( dy ) and F ′ ,ǫβ,K ( x, t ) = Z t (cid:0) ǫ − X ǫ ( y ) + β (cid:1) { τ βy <ǫ } e X ǫ ( y ) − ǫ K ( y ) µ t ( dy ) . To prove Proposition 3.8, it suffices to prove that the family ( F ′ ,ǫβ,K ( x, t )) ǫ is uniformly integrable.Indeed, if true, we have for each R > K c is the complement of K in R ) P X ( F ′ ,ǫβ ( x, t ) { F ′ ,ǫβ ( x,t ) > R } ) P X (2 F ′ ,ǫβ,K ( x, t ) { F ′ ,ǫβ,K ( x,t ) > R } ) + P X (2 F ′ ,ǫβ,K c ( x, t ) { F ′ ,ǫβ,Kc ( x,t ) > R } ) P X (2 F ′ ,ǫβ,K ( x, t ) { F ′ ,ǫβ,K ( x,t ) > R } ) + 2 E X [ F ′ ,ǫβ,K c ( x, t )] . We deduce lim sup R →∞ P X ( F ′ ,ǫβ ( x, t ) { F ′ ,ǫβ ( x,t ) > R } ) E X [ F ′ ,ǫβ,K c ( x, t )] = 2 βδ. By choosing δ arbitrarily small, we prove the uniform integrability of ( F ′ ,ǫβ ( x, t )) ǫ> .So, we just have to focus on the uniform integrability of the family ( F ′ ,ǫβ,K ( x, t )) ǫ> . We introducethe annulus C ( y, ǫ,
1) = B ( y, \ B ( y, ǫ ) for ǫ ∈ (0 , E X [ F ′ ,ǫβ,K ( x, t ) ] = Z R Z R E X [ f βǫ ( y ) f βǫ ( w )] µ Kt ( dy ) µ Kt ( dw )= Z R Z B ( y,ǫ ) E X [ f βǫ ( y ) f βǫ ( w )] µ Kt ( dy ) µ Kt ( dw )+ Z R Z C ( y,ǫ, E X [ f βǫ ( y ) f βǫ ( w )] µ Kt ( dyx ) µ Kt ( dw )+ Z R Z B ( y, c E X [ f βǫ ( y ) f βǫ ( w )] µ Kt ( dy ) µ Kt ( dw ) def = Π ǫ + Π ǫ + Π ǫ . (3.15)It is not difficult to see that Π ǫ Ct for some constant C independent of ǫ . The main terms areΠ ǫ and Π ǫ . We begin with Π ǫ . 19 emark 3.10. Before going further into details, let us just heuristically explain how to completethe proof. On the ball B ( y, ǫ ) , the process X ǫ ( w ) is very close to X ǫ ( y ) . Therefore, with a goodapproximation, we can replace X ǫ ( w ) by X ǫ ( y ) and get: Π ǫ C Z R Z B ( y,ǫ ) E X h (1 + ( X ǫ ( y )) ) e X ǫ ( y )+2 ln ǫ ( β − X ǫ ( y )) { sup s ∈ [ ǫ, X s ( y ) β } ) i µ Kt ( dy ) µ Kt ( dw ) . Let us define a new probability measure on F ǫ by P β ( A ) = E X [ A ( β − X ǫ ( y )) { sup s ∈ [ ǫ, X s ( y ) β } )] and recall that, under P β , the process ( β − X s ) ǫ s has the law of ( β ln s ) ǫ s where ( β u ) u isa -dimensional Bessel process starting from β . Hence Π ǫ C Z R Z B ( y,ǫ ) E β h (1 + ( β ln ǫ ) ) e − β ln 1 ǫ +2 ln ǫ i µ Kt ( B ( y, ǫ )) µ Kt ( dy ) ≃ C Z R (1 + ln 1 ǫ ) e − q ln ǫ +2 ln ǫ µ Kt ( B ( y, ǫ )) µ Kt ( dy ) and this latter quantity goes to as ǫ → since µ Kt ( B ( y, ǫ )) Lǫ g ( ǫ ) . Similar ideas will allow usto treat Π ǫ . Nevertheless, details are a bit more tedious. Let us now try to make rigorous the above remark. Observe that necessarily K s ( y, w ) ln s .Let us define the functions h , h and ¯ h by: h ( s ) = ln 1 s − K s ( y, w ) = h ( s ) , ¯ h ( s ) = K s ( y, w ) . (3.16)By considering 3 independent Brownian motions B , B , ¯ B , we further define P y,ws = B h ( s ) , P w,ys = B h ( s ) , Z s = ¯ B ¯ h ( s ) . (3.17)Observe that the process ( X s ( y ) , X s ( w )) s has the same law as the process ( P y,ws + Z s , P w,ys + Z s ) s . Now we compute Π ǫ and then use a Girsanov transform:Π ǫ = Z R Z B ( y,ǫ ) E X [( β − P w,yǫ − Z ǫ − ǫ ) { sup r ∈ [ ǫ, P w,yr + Z r − r β } ( β − P y,wǫ − Z ǫ − ǫ ) × . . .. . . { sup r ∈ [ ǫ, P y,wr + Z r − r β } e P y,wǫ +4 Z ǫ +2 P w,yǫ − ǫ ] µ Kt ( dy ) µ Kt ( dw )= Z R Z B ( y,ǫ ) E X [( β − P w,yǫ − Z ǫ ) { sup r ∈ [ ǫ, P w,yr + Z r β } ( β − P y,wǫ − Z ǫ ) × . . .. . . { sup r ∈ [ ǫ, P y,wr + Z r β } e Z ǫ − ǫ +4 K ǫ ( y − w ) ] µ Kt ( dy ) µ Kt ( dw ) . Let us set: β y,wǫ = β − min s ∈ [ ǫ, P y,ws . Because of assumption [A.2], we havesup ǫ ∈ (0 , sup w ∈ B ( y,ǫ ) sup s ∈ ] ǫ, h ( s ) + h ( s ) c c > k . Therefore we have:Π ǫ C Z R Z B ( y,ǫ ) E X h(cid:16) Z ǫ ) (cid:17) e Z ǫ +2 ln ǫ ( β y,wǫ − Z ǫ ) { sup r ∈ [ ǫ, Z r β y,wǫ } i µ Kt ( dy ) µ Kt ( dw ) . Let us define a new (random) probability measure on F ǫ by P β,y,w ( A ) = 1 β y,wǫ E Z [ A ( β y,wǫ − Z ǫ ( y )) { sup s ∈ [ ǫ, Z s ( y ) β y,wǫ } ) | β y,wǫ ]with associated expectation denoted by E β,y,w . Recall that, under P β,y,w , the process ( β y,wǫ − Z s ) ǫ s has the law of ( β K s ( y,w ) ) ǫ s where ( β u ) u is a 3-dimensional Bessel process startingfrom β y,wǫ . Hence:Π ǫ C Z R Z B ( y,ǫ ) E X h β y,wǫ E β,y,w h(cid:16) β K ǫ ( y,w ) ) (cid:17) e − β Kǫ ( y,w ) +2 ln ǫ ii µ Kt ( dy ) µ Kt ( dw ) . Let us compute the quantity E β,y,w h(cid:16) β K ǫ ( y,w ) ) (cid:17) e − β Kǫ ( y,w ) i . To this purpose, we use the fact that the law of a 3 d -Bessel process starting from β y,wǫ is givenby p ( B t − β y,wǫ ) + ( B t ) + ( B t ) where B , B , B are three independent Brownian motions.Therefore, by using (2.2) when necessary, we get E β,y,w h(cid:16) β K ǫ ( y,w ) ) (cid:17) e − β Kǫ ( y,w ) i = Z R (1 + ( u − β y,wǫ ) + v + w ) e − √ ( u − β y,wǫ ) + v + w e − u v w Kǫ ( y,w ) dudvdw (2 πK ǫ ( y, w )) / C (1 + ( β y,wǫ ) ) e β y,wǫ Z R (1 + u + v + w ) e − √ u + v + w e − u v w Kǫ ( y,w ) dudvdw (2 πK ǫ ( y, w )) / C (1 + ( β y,wǫ ) ) e β y,wǫ Z ∞ (1 + r ) e − r e − r
22 ln 1 ǫ r dr (ln ǫ ) / , for some constant C >
0, which may have changed along lines. Let us set ∀ a > , H ( a ) = Z ∞ (1 + r ) e − r e − r a r dra / . It is plain to check that H ( a ) C (max(1 , a )) − / for some positive constant C . Hence E β,y,w h(cid:16) β K ǫ ( y,w ) ) (cid:17) e − β Kǫ ( y,w ) i C (1 + ( β y,wǫ ) ) e β y,wǫ H (ln 1 ǫ ) . We deduce: Π ǫ C (ln 1 ǫ ) − / Z R Z B ( y,ǫ ) E X h β y,wǫ (cid:0) β y,wǫ ) (cid:1) e β y,wǫ i µ Kt ( dy ) µ Kt ( dw ) CH (ln 1 ǫ ) ǫ − Z R µ Kt ( B ( y, ǫ )) µ Kt ( dy ) . E X h Cβ y,wǫ (1 + ( β y,wǫ ) ) e β y,wǫ i is finite and does not depend on y, w because h ( s ) is bounded on R + independently of s, y, w . Because of Lemma 3.9, this latter quantitygoes to 0 as ǫ → P B x -almost surely, and so does Π ǫ .Now we treat Π ǫ . We will follow similar arguments as for Π ǫ , though different behaviors areinvolved. Indeed, in this case, we have to face the possible long range correlations of the kernel k .So we adapt the decomposition of the couple ( X s ( y ) , X s ( w )) s ∈ ]0 , into Wiener integrals as follows.Let us consider a smooth function ϕ with compact support in the ball B (0 , ϕ ϕ = 1 over a neighborhood of 0. Let us define the functions h , h , ¯ h and b h by: h ( s ) = ln 1 s − K s ( y, w ) = h ( s ) , ¯ h ( s ) = Z s k ( u, y, w ) ϕ ( u ( y − w )) u du, b h ( s ) = Z s k ( u, y, w ) (cid:0) − ϕ ( u ( y − w )) (cid:1) u du. By considering 4 independent Brownian motions B , B , ¯ B, b B , we further define P y,ws = B h ( s ) , P w,ys = B h ( s ) , Z s = ¯ B ¯ h ( s ) , b Z s = b B b h ( s ) . (3.18)An elementary computation of covariance shows that the process ( X s ( y ) , X s ( w )) s has thesame law as the process ( P y,ws + Z s + b Z s , P w,ys + Z s + b Z s ) s . The process Z s encodes theshort-scale correlations of the two Brownian motions ( X s ( y )) s and ( X s ( w )) s , and is the processthat will rule the behaviour of the expectation E X [ f βǫ ( y ) f βǫ ( w )]. The remaining terms are justnegligible perturbations that we will have to get rid of in the forthcoming computations.We make first a few elementary remarks. Observe that ¯ h ( s ) = ¯ h ( | y − w | ) for all s > | y − w | insuch a way that Z ǫ = Z | y − w | for ǫ | y − w | . We also set D := sup { b h ( s ); s ∈ ]0 , y, w ∈ R } < + ∞ . We will often use the elementary relation: ∀ a > , ∀ x ∈ R , ( β − a − x ) { x β − a } ( β − x ) { x β } . (3.19)Now we begin the computations related to Π ǫ . So we consider w ∈ C ( y, ǫ,
1) and we have bythe Girsanov transform and (3.19): E X [ f βǫ ( w ) f βǫ ( y )]= E X h ( β − P y,wǫ − Z ǫ − b Z ǫ + 2 ln 1 ǫ ) { sup u ∈ [ ǫ, P y,wu + Z u + b Z u − u β } e P y,wǫ +2 Z ǫ +2 b Z ǫ − ǫ . . . · · · × ( β − P w,yǫ − Z ǫ − b Z ǫ + 2 ln 1 ǫ ) { sup u ∈ [ ǫ, P w,yu + Z u + b Z u − u β } e P w,yǫ +2 Z ǫ +2 b Z ǫ − ǫ i e D E X h ( β − P y,wǫ − Z ǫ − b Z ǫ + 2 ln 1 ǫ ) { sup u ∈ [ ǫ, P y,wu + Z u + b Z u − u β } e P y,wǫ +2 Z ǫ − ǫ . . . · · · × ( β − P w,yǫ − Z ǫ − b Z ǫ + 2 ln 1 ǫ ) { sup u ∈ [ ǫ, P w,yu + Z u + b Z u − u β } e P w,yǫ +2 Z ǫ − ǫ i e D E X h ( β − min s ∈ ]0 , b Z s − P y,wǫ − Z ǫ + 2 ln 1 ǫ ) { sup u ∈ [ ǫ, P y,wu + Z u − u β − min s ∈ ]0 , b Z s } e P y,wǫ +2 Z ǫ − ǫ . . . · · · × ( β − min s ∈ ]0 , b Z s − P w,yǫ − Z ǫ + 2 ln 1 ǫ ) { sup u ∈ [ ǫ, P w,yu + Z u − u β − min s ∈ ]0 , b Z s } e P w,yǫ +2 Z ǫ − ǫ i . ǫ replacedby | y − w | . First observe that Z ǫ = Z | y − w | for ǫ | y − w | . Second, from assumption [A.3], we havesup y ∈ K,w ∈ R Z ∞ | y − w | k ( u, y, w ) u du = C K < + ∞ . Therefore we can deduce ∀ u | y − w | , (cid:12)(cid:12) ln | y − w | u − ( h ( u ) − h ( | y − w | )) (cid:12)(cid:12) c for some constant c independent of everything that matters. This means that the quadratic varia-tions of the martingale ( P w,yu − P w,y | y − w | ) u | y − w | can be identified with ln | y − w | ǫ up to some constant c independent of y, w, ǫ . We further stress that both martingales P y,w and P w,y are independent.Therefore, by conditioning with respect to F | y − w | , the integrand in the above expectation essen-tially reduces to the product of two independent martingales (recall that, if X t = R t f ( r ) dB r is aWiener integral, then ( β + 2 E [ X t ] − X t ) { sup s ∈ [0 ,t ] X s − E [ X t ] β } e X t − E [ X t ] is a martingale). Byapplying the stopping time theorem and by setting b β = β − min s ∈ ]0 , b Z s + c , we getΠ ǫ C Z R Z C ( y,ǫ, E X [( b β − P w,y | y − w | − Z | y − w | − | y − w | ) { sup r ∈ [ | y − w | , P w,yr + Z r − r b β } . . . ( b β − P y,w | y − w | − Z | y − w | − | y − w | ) { sup r ∈ [ | y − w | , P y,wr + Z r − r b β } . . . e P y,w | y − w | +4 Z | y − w | +2 P w,y | y − w | − | y − w | ] µ Kt ( dy ) µ Kt ( dw ) . Recall that sup y,w ∈ K,s ∈ ] | y − w | , E [( P y,ws ) + ( P w,ys ) ] c ′ and some constant c ′ > k by assumption [A.2]. Indeed, for s ∈ [ | y − w | , h ( s ) = ln 1 s − Z s k ( u ( y − w )) u du = Z s − k ( u ( y − w )) u du Z s C K u | y − w | u du C K , where C K is the Lipschitz constant given by assumption [A.2]. So, if we use the Girsanov transformand even if it means changing the value of b β by adding C K , we getΠ ǫ C Z R Z C ( y,ǫ, E X [( b β − P w,y | y − w | − Z | y − w | ) { sup r ∈ [ | y − w | , P w,yr + Z r b β } ( b β − P y,w | y − w | − Z | y − w | ) . . . { sup r ∈ [ | y − w | , P y,wr + Z r b β } e Z | y − w | − | y − w | +4 K | y − w | ( y − w ) ] µ Kt ( dy ) µ Kt ( dw ) . Let us then define β y,w = b β − min s ∈ [ | y − w | , P y,w | y − w | . We deduce:Π ǫ C Z R Z C ( y,ǫ, E X h(cid:16) Z | y − w | ) (cid:17) e Z | y − w | +2 ln | y − w | ( β y,w − Z | y − w | ) × . . . · · · × { sup r ∈ [ | y − w | , Z r β y,w } i µ Kt ( dy ) µ Kt ( dw ) . F | y − w | by P β,y,w ( A ) = 1 β y,w E Z [ A ( β y,w − Z | y − w | ( y )) { sup s ∈ [ | y − w | , Z s ( y ) β y,w } ) | β y,w ]and recall that, under P β,y,w , the process ( β y,w − Z s ) | y − w | s has the law of ( β K s ( y − w ) ) | y − w | s where ( β u ) u is a 3-dimensional Bessel process starting from β y,w . HenceΠ ǫ C Z R Z C ( y,ǫ, E X h β y,w E β,y,w h(cid:16) β K | y − w | ( y − w ) ) (cid:17) e − β K | y − w | ( y − w ) +2 ln | y − w | ii µ Kt ( dy ) µ Kt ( dw ) . From (2.2), we have ln 1 | y − w | − C K K | y − w | ( y − w ) ln 1 | y − w | + C K . If we proceed along the same lines as we did previously, we get: E β,y,w h(cid:16) β K | y − w | ( y − w ) ) (cid:17) e − β K | y − w | ( y − w ) i C (1 + ( β y,w ) ) e β y,w Z ∞ (1 + r ) e − r e − r
22 ln 1 | y − w | r dr (ln | y − w | ) / C (1 + ( β y,w ) ) e β y,w H (ln 1 | y − w | ) . (3.20)Therefore, thanks to Lemma 3.9 (or Lemma 3.2),sup ǫ ∈ ]0 , Π ǫ C Z R Z B ( y, H (ln | y − w | ) | y − w | µ Kt ( dw ) µ Kt ( dy ) < + ∞ . The proof of Proposition 3.8 is complete.Before proceeding with the proof of Theorem 3.6, let us first state a few corollaries of theprevious computations. For β > ǫ ∈ ]0 , M βǫ ( dx ) = f βǫ ( x ) dx. Following [21], the family ( M βǫ ) ǫ ∈ ]0 , almost surely converges as ǫ → M β ( dx ) and M β ( dx ) = M ′ ( dx )on compact sets for β (random) large enough. The following corollary proves Theorem 2.1 andtherefore considerably generalizes the results in [21]. Corollary 3.11.
Assume [A.1-5]. Consider the random measure M βǫ ( dx ) = f βǫ ( x ) dx. Then for p < / : E X h Z B (0 , Z B (0 , ln p | y − w | M β ( dy ) M β ( dw ) i < + ∞ . Therefore, the measure M β (and consequently M ′ ) is diffuse. roof. If we just replace the occupation measure of the Brownian motion by the Lebesgue measurealong the lines of the proof of Proposition 3.8, we get: E X h Z B (0 , Z B (0 , ln p | y − w | M β ( dy ) M β ( dw ) i C Z B (0 , Z B (0 , | y − w | ln p | y − w | H (ln 1 | y − w | ) dydw. This latter quantity is finite for p < /
2. The finiteness of such an integral implies that, almostsurely, the measure M β cannot give mass to singletons.This result is closely related, though weaker, than (2.10). Yet, the setup of our proof is quitegeneral whereas (2.10) has been proved in [5] for a specific one-dimensional measure that exhibitsnice scaling relations. Nonetheless, (2.10) is expected to hold in greater generality but we do notknow to which extent the proofs in [5] extend to more general situations. In the same spirit, weclaim: Corollary 3.12.
Fix t > and p < / . For each δ > , there is a compact set K such that µ t ( K c ) δ and E X h Z t Z t ln p + | B xr − B xs | K ( B xr ) K ( B xs ) F ′ β ( x, dr ) F ′ β ( x, ds ) i < + ∞ . In particular, almost surely in X and in B x , the random mapping r F ′ β ( x, r ) does not possessdiscontinuity point on [0 , t ] .Proof. For each δ >
0, we use once again Lemma 3.9 to find a compact set K and some constant L such that µ t ( K c ) δ and for all y ∈ K sup r µ t ( B ( y, r )) r g ( r ) L, with g ( r ) = ln (cid:0) r (cid:1) . (3.21)We denote by µ Kt the measure µ Kt ( dy ) = K ( y ) µ t ( dy ) . Once again, the computations made in proposition 3.8 show that E X h Z t Z t ln p + | B xr − B xs | K ( B xr ) K ( B xs ) F ′ β ( x, dr ) F ′ β ( x, ds ) i C Z t Z t {| B xr − B xs | } | B xr − B xs | ln p | B xr − B xs | H (cid:16) ln 1 | B xr − B xs | (cid:17) K ( B xr ) K ( B xs ) drds + C Z t Z t {| B xr − B xs | > } ln p + | B xr − B xs | K ( B xr ) K ( B xs ) drds C Z t Z t {| u − v | } | u − v | ln p | u − v | H (cid:16) ln 1 | u − v | (cid:17) µ Kt ( du ) µ Kt ( dv ) . Because of (3.21), the above integrals are finite for p < / s F ′ β ( x, [0 , s ]) since the Brownianmotion has continuous sample paths. Indeed, the discontinuity points of this mapping corresponds25o the set A of atoms of the measure F ′ β ( x, ds ), which are countable. For each n ∈ N ∗ , let usdenote by A n the set of atoms in [0 , t ] of this measure that are of size strictly greater than R t K c /n ( B xs ) F ′ β ( x, dr ). For s ∈ A n , we necessarily have K c /n ( B xs ) = 0. Also, we necessarilyhave K /n ( B xs ) = 0 otherwise the integral R t R t ln p | B xr − B xs | K /n ( B xr ) K /n ( B xs ) F ′ β ( x, dr ) F ′ β ( x, ds )would be infinite. We deduce that A n is empty for all n ∈ N ∗ , meaning that there is no atom ofsize greater than R t K c /n ( B xs ) F ′ β ( x, dr ) for all n . Since E x h Z t K c /n ( B xs ) F ′ β ( x, dr ) i = µ t ( K c /n ) /n → n → ∞ , we deduce that the quantity R t K c /n ( B xs ) F ′ β ( x, dr ) converges to 0 in probability as n → ∞ . Wecomplete the proof.We are now in position to handle the proof of Theorem 3.6. Proof of Theorem 3.6.
We first observe that the martingale ( F ′ ,ǫβ ( x, t )) ǫ> possesses almost surelythe same limit as the process ( e F ′ ,ǫβ ( x, t )) ǫ> because | F ′ ,ǫβ ( x, t ) − e F ′ ,ǫβ ( x, t ) | = β Z t { τ βBxu <ǫ } e X ǫ ( B xu ) − ǫ du βF ǫ ( x, t ) (3.22)and the last quantity converges almost surely towards 0 (see Proposition 3.3). Using Corollary 3.5,we have almost surely in X and in B :sup ǫ> max s ∈ [0 ,t ] X ǫ ( B xs ) − ǫ < + ∞ , which obviously implies ∀ ǫ > , F ′ ,ǫ ( t, x ) = e F ′ ,ǫβ ( x, t )for β (random) large enough. We deduce that, almost surely in X and in B , the family ( F ′ ,ǫ ( x, t )) ǫ> converges towards a positive random variable.It is plain to deduce the random measures ( F ′ ,ǫβ ( x, dt )) ǫ converges in the sense of weak con-vergence of measures towards a random measure F β ( x, dt ). To prove convergence in C ( R + , R + ),we just have to prove that the mapping t F β ( t, x ) is continuous. This property is proved inCorollary 3.12. From (3.22) and Proposition 3.3 again, we deduce that the family of random map-pings ( t F ′ ,ǫ ( x, t )) ǫ almost surely converges as ǫ → C ( R + , R + ) towards a nonnegativenondecreasing mapping t F ′ ( x, t ).Let us prove that, almost surely in X and in B , the mapping t F ′ ( x, t ) is strictly increasing.We first write the relation, for ǫ ′ < ǫ , F ′ ,ǫ ′ β ( x, dr ) =(2 ln 1 ǫ − X ǫ ( B xr ) + β ) { τ βBxr <ǫ ′ } e X ǫ ′ ( B xr ) − ǫ ′ dr (3.23)+ (2 ln ǫǫ ′ − X ǫ ′ ( B xr ) + X ǫ ( B xr ) + β ) { τ βBxr <ǫ ′ } e X ǫ ′ ( B xr ) − ǫ ′ dr. By using the same arguments as throughout this section, we pass to the limit in this relation as ǫ ′ → β → ∞ to get F ′ ( x, dr ) = e X ǫ ( B xr ) − ǫ F ′ ǫ ( x, dr ) (3.24)26here F ′ ǫ ( x, dr ) is almost surely defined as F ′ ǫ ( x, dr ) = lim β →∞ lim ǫ ′ → F ′ ,ǫ ′ β,ǫ ( x, dr )and F ′ ,ǫ ′ β,ǫ ( x, dr ) is given by(2 ln ǫǫ ′ − X ǫ ′ ( B xr ) + X ǫ ( B xr ) + β ) { τ βǫ,Bxr <ǫ ′ } e X ǫ ′ − X ǫ )( B xr ) − ǫǫ ′ dr where τ βǫ,z = sup { u X uǫ ( z ) − X ǫ ( z ) − u > β − X ǫ ( z ) + 2 ln 1 ǫ } . Let us stress that we have used the fact that the measure(2 ln 1 ǫ − X ǫ ( B xr ) + β ) { τ βBxr <ǫ ′ } e X ǫ ′ ( B xr ) − ǫ ′ dr goes to 0 (it is absolutely continuous w.r.t. to F ǫ ′ ( x, dr )) when passing to the limit in (3.23) as ǫ ′ →
0. From (3.24), it is plain to deduce that, almost surely in B , the event { F ′ ( x, [ s, t ]) = 0 } (with s < t ) belongs to the asymptotic sigma-algebra generated by the field { ( X ǫ ( x )) x ; ǫ > } .Therefore it has probability 0 or 1 by the 0 − B , P X ( F ′ ( x, [ s, t ]) = 0) = 0 for any s < t .By considering a countable family of intervals [ s n , t n ] generating the Borel sigma field on R + , wededuce that, almost surely in X and in B , the mapping t F ′ ( x, t ) is strictly increasing. Here we explain the Seneta-Heyde norming for the change of times F ǫ . Some technical constraintsprevents us from claiming that it holds under the only assumptions [A.1-5]. So it is important tostress here that the Seneta-Heyde renormalization is not necessary to construct the critical LBM.It just illustrates that the derivative construction of the change of times F ′ ( x, t ) also correspondsto a proper renormalization of F ǫ . Theorem 3.13.
Assume [A.1-5] and either [A.6] or [A.6’]. Then the conclusions of Theorem 2.2hold and almost surely in B , we have the following convergence in P X -probability as ǫ → r ln 1 ǫ F ǫ ( x, t ) → r π F ′ ( x, t ) . Proof.
The proof is a rather elementary adaptation of the proof in [22, section D]. Just reproducethe proof in [22, section D] while replacing the Lebesgue measure by the occupation measure ofthe Brownian motion and use Lemma 3.9 when necessary. Details are left to the reader.
Remark 3.14.
In the case of example (2) , it is necessary to consider the occupation measure ofthe Brownian motion killed upon touching the boundary of the domain D . .5 The LBM does not get stuck In this subsection, we make sure that the LBM does not get stuck in some area of the state space R . Typically, this situation may happen over areas where the field X takes large values, thereforehaving as consequence to slow down the LBM. Mathematically, this can be formulated as follows:check that the mapping t F ′ ( x, t ) tends to ∞ as t → ∞ . Theorem 3.15.
Assume [A.1-5] and fix x ∈ R . Almost surely in X and in B , lim t →∞ F ′ ( x, t ) = + ∞ . (3.25) Proof.
It suffices to reproduce the techniques of [28, subsection 2.5].
Remark 3.16.
Of course, this statement does not hold in the case of a bounded planar domainwhen one considers a Brownian motion killed upon touching the boundary of D . In this case, theLiouville Brownian motion (see below) will run until touching the boundary of D . We are now in position to define the critical LBM when starting from one fixed point. Indeed, oncethe change of times F ′ has been constructed, the strategy is the same as in [28, subsection 2.10]. Definition 3.17.
Assume [A.1-5]. The critical Liouville Brownian motion is defined by: ∀ t > , B xt = B x hB x i t , and ∀ t > , p /π F ′ ( x, hB x i t ) = t. (3.26) As such, the mapping t
7→ hB x i t is defined on R + , continuous and strictly increasing. Theorem 3.18.
Assume [A.1-5] and either [A.6] or [A.6’]. Fix x ∈ R . Almost surely in B andin P X -probability, the family ( B, hB ǫ,x i , B ǫ,x ) ǫ converges in the space C ( R + , R ) × C ( R + , R + ) × C ( R + , R ) equipped with the supremum norm on compact sets towards the triple ( B, hB x i , B x ) . Remark 3.19.
One may wonder whether the process introduced in Definition 2.11 also convergesin probability. Actually, the argument carried out in [28, section 2.12] remains true here: almostsurely in X , the couple of processes ( ¯ B, B ǫ ) ǫ in Definition 2.11 converges in law towards a couple ( ¯ B, B ) where ¯ B and B are independent. Since B ǫ is measurable w.r.t. ¯ B , this shows that the process B ǫ does not converge in probability. This justifies our approach of studying the convergence viathe Dambis-Schwarz representation theorem: it leads to studying a process (definition 2.5) thatconverges in probability. Remark 4.1.
In this whole section, we assume that assumptions [A.1-5] are in force. Sometimes,we make a statement to relate our results to the metric tensor g ǫ . For such a connection to bemade, the Seneta-Heyde norming is needed and thus assumption [A.6] or [A.6’] are required. Toavoid confusion, this will be explicitly mentioned.
28n this section, we will investigate the critical LBM as a Markov process, meaning that we aimat constructing almost surely in X the critical LBM starting from every point. In the previoussection, the guiding line was similar to [28] besides technical difficulties. From now on, the differencewill be conceptual too: in the subcritical situation, the issue of constructing the LBM starting fromevery point is possible because the mapping x Z R ln + | x − y | M γ ( dy )is a continuous function of x , where M γ stands for the subcritical measure with parameter γ < M ′ is of the type M ′ ( B ( x, r )) C p ln(1 + r − )and cannot be improved as proved in [6] in the context of multiplicative cascades. The mapping x Z R ln + | x − y | M ′ ( dy ) (4.1)is thus certainly not continuous and may even take infinite values. The ideas of [28] need be renewedto face the issues of criticality.On the other hand, the theory of Dirichlet forms [26, 53] (or potential theory) tells us thatone can construct a Positive Continuous Additive Functional (PCAF for short) associated to M ′ provided that the mapping (4.1) does not take too many infinite values. More precisely, M ′ isrequired not to give mass to polar sets of the Brownian motion. The problem of the theory ofDirichlet forms is that one can guarantee the existence of the PCAF but it cannot be identifiedand all the information that we get by explicitly constructing the PCAF F ′ is lost in this approach.Also, while being extremely powerful in the description of the Dirichlet form of the LBM, the theoryof Dirichlet forms gives much weaker results than the coupling approach developed in [28, 29]concerning the qualitative/quantitative properties of F ′ . We thus definitely need to gather both ofthese approaches.Our strategy will be to identify a large set of points of finiteness of the mapping (4.1) in orderto construct a perfectly identified PCAF on the whole space via coupling arguments. Then we willprove that M ′ does not charge polar sets in order to identify our PCAF with that of the theory ofDirichlet forms in the sense of the Revuz correspondence. Once this gap is bridged, we can applythe full machinery of [26] to get a lot of further information about F ′ : mainly, a full description ofthe Dirichlet form associated to the critical LBM. To facilitate the reading of our results, we summarize here some basic notions of potential theoryapplied to the standard Brownian (Ω , ( B t ) t > , ( F t ) t > , ( P x ) x ∈ R ) in R seen as a Markov process,which is of course reversible for the canonical volume form dx of R . These notions can be foundwith further details in [26, 53]. One may then consider the classical notion of capacity associatedto the Brownian motion. In this context, we have the following definition:29 efinition 4.2 (Capacity and polar set) . The capacity of an open set O ⊂ R is defined by Cap( O ) = inf { Z D | f ( x ) | dx + Z D |∇ f ( x ) | dx ; f ∈ H ( R , dx ) , f > over O } . The capacity of a Borel measurable set K is then defined as: Cap( K ) = inf O open ,K ⊂ O Cap( O ) . The set K is said polar when Cap( K ) = 0 . Definition 4.3 (Revuz measure) . A Revuz measure µ is a Radon measure on R which does notcharge the polar sets. Then we introduce the notion of PCAF that we will use in the following (see [26, 53]):
Definition 4.4 (PCAF) . A Positive Continuous Additive Functional ( A ( x, t, B )) t > ,x ∈ R /N ofthe Brownian motion (with B = 0 ) on R is defined by:-a polar set N (for the standard Brownian motion),- for each x ∈ R /N and t > , A ( x, t, B ) is F t -adapted and continuous, with values in [0 , ∞ ] and A ( x,
0) = 0 ,- almost surely, A ( x, t + s, B ) − A ( x, t, B ) = A ( x + B t , s, B t + · − B t ) , s, t > . In particular, a PCAF is defined for all starting points x ∈ R except possibly on a polar setfor the standard Brownian motion. One can also work with a PCAF starting from all points, thatis when the set N in the above definition can be chosen to be empty. In that case, the PCAF issaid in the strict sense .Finally, we conclude with the following definition on the support of a PCAF: Definition 4.5 (support of a PCAF) . Let ( A ( x, t, B )) t > ,x ∈ R /N be a PCAF with associated polarset N . The support of ( A ( x, t, B )) t > ,x ∈ R /N is defined by: e Y = n x ∈ R \ N : P ( R ( x ) = 0) = 1 o , where R ( x ) = inf { t > A ( x, t, B ) > } . From section 5 in [26], there is a one to one correspondence between Revuz measures µ andPCAFs ( A ( x, t, B )) t > ,x ∈ R /N under the Revuz correspondence: for any t > f, h : E h.dx h Z t f ( B r ) dA r i = Z t Z R f ( x ) P r h ( x ) µ ( dx ) dr, (4.2)where ( P r ) r > stands for the semigroup associated to the planar Brownian motion on R .30 .2 Capacity properties of the critical measure The purpose of this section is to establish some preliminary results in order to apply the theoryof Dirichlet forms. In particular, we will establish that the critical measure does not charge polarsets. To this purpose, we will need some fine pathwise properties of Bessel processes. So we firstrecall a result from [47]:
Theorem 4.6.
Let X be a d -Bessel process on R + starting from x > with respect to the law P x .1. Suppose that φ ↑ ∞ such that R ∞ φ ( t ) t e − φ ( t ) dt < + ∞ . Then P x (cid:16) X t > √ tφ ( t ) i.o. as t ↑ + ∞ (cid:17) = 0 .
2. Suppose that ψ ↓ such that R ∞ ψ ( t ) t dt < + ∞ . Then P x (cid:16) X t < √ tψ ( t ) i.o. as t ↑ + ∞ (cid:17) = 0 . Recall that we denote by M β the measure M β ( dx ) = lim ǫ → f βǫ ( x ) dx . The main purpose ofthis subsection is to prove the following result: Theorem 4.7. • Let us consider two functions φ, ψ as in the above theorem and define Y ǫ ( x ) = 2 ln 1 ǫ − X ǫ ( x ) . We introduce the set E = { x ∈ R ; lim sup ǫ → Y ǫ ( x ) q ln ǫ φ (ln ǫ ) and lim inf ǫ → Y ǫ ( x ) q ln ǫ ψ (ln ǫ ) > } . Then M β gives full mass to E , i.e. M β ( E c ) = 0 . • Furthermore, for each ball B , for each δ > , there is a compact set K ⊂ B such that M β ( B ∩ K c ) δ and for all p > Z K Z B (cid:16) ln 1 | x − y | (cid:17) p M β ( dx ) M β ( dy ) < + ∞ . Proof.
Let us consider a non empty ball B . We introduce the Peyri`ere probability measure Q β onΩ × B : Z f ( x, ω ) dQ β = 1 β | B | E X [ Z B f ( x, ω ) M β ( dx )] . Let us consider the random process t ¯ X t = X e − t . The main idea is that under Q β , the process( β − ¯ X t + 2 t ) t > has the law very close to a 3 d -Bessel process ( β t ) t starting from β . The first claimthen follows from Theorem 4.6. We just have to precise the notion of ”very close”: some negligibleterms appear because of the difference between t and E X [ ¯ X t ( x ) ] and we have to quantify them.We consider the measure (convergence is established the same way as for M β )¯ M β ( dx ) = lim ǫ → (2 E X [ ¯ X t ( x ) ] − X t ( x ) + β ) { sup u ∈ [0 ,t ] ¯ X u − E X [ ¯ X u ( x ) ] β } e X t − E X [ ¯ X t ( x ) ] dx D = sup t > sup x ∈ B | E X [ ¯ X t ( x ) ] − t | < + ∞ . We set H ( x ) = lim t →∞ E X [ ¯ X t ( x ) ] − t (see assumption [A.4]). Observe that M β ( dx ) e H ( x ) ¯ M β +2 D ( dx ) . Let us set ˆ β = β + 2 D . Therefore, we consider the probability measure ¯ Q ˆ β on Ω × B : Z f ( x, ω ) d ¯ Q ˆ β = 1ˆ β | B | E X [ Z B f ( x, ω ) ¯ M ˆ β ( dx )]and Q β is absolutely continuous with respect to ¯ Q ˆ β . Under ¯ Q ˆ β (following arguments already seen),the process ( ˆ β − ¯ X t +2 E X [ ¯ X t ( x ) ]) t > has the law of (Bess T t ( x ) ) t where (Bess t ) t a 3 d -Bessel processstarting from ˆ β and T t ( x ) = K e − t ( x, x ). The first claim then follows from Theorem 4.6, the factthat sup x ∈ B,t > | K e − t ( x, x ) − t | < + ∞ and the absolute continuity of Q β w.r.t. ¯ Q ˆ β .Now we prove the second statement, which is more technical. To simplify things a bit, weassume that E [ X t ( x ) ] = t . If this not the case, we can apply the same strategy as above, namelyconsidering ¯ M β +2 D instead of M β . We consider now a couple of functions φ ( t ) = (1 + t ) χ and ψ ( t ) = (1 + t ) − χ for some small positive parameter χ close to 0. They satisfy the assumptions ofTheorem 4.6. Then we consider the random compact sets for R > t ∈ [0 , + ∞ ], K R,t = (cid:8) x ∈ B ; sup u ∈ [0 ,t ] β − ¯ X u ( x ) + 2 u (1 + u ) / χ R (cid:9) K R,t = (cid:8) x ∈ B ; inf u ∈ [0 ,t ] β − ¯ X u ( x ) + 2 u (1 + u ) / − χ > R (cid:9) K R,t = K R,t ∩ K R,t . We will write K R for K R, ∞ . From Theorem 4.6, we have lim R →∞ Q β ( K R ) = 1. Therefore, we havelim R →∞ M β ( K cR ∩ B ) = 0 P X almost surely.Let us denote E Q expectation with respect to the probability measure Q β . Without loss ofgenerality, we assume that β | B | = 1: this avoids to repeatedly write the renormalization constantappearing in the definition of Q β . To prove the result, we compute E Q [ K R ( x ) M β ( B ( x, e − t ))] = lim ǫ → E [ Z B K R ( x ) M β ( B ( x, e − t )) M β ( dx )]= lim ǫ → E [ Z B Z B ( x,e − t ) K R ( x ) f βǫ ( w ) dwM β ( dx )] lim ǫ → Z B Z B ( x,e − t ) E [ K R, − ln | x − w | ( x ) f βǫ ( w ) f βǫ ( x )] dwdx. Now we can argue as in the proof of Proposition 3.8 (treatment of Π ǫ ) to see that for ǫ | x − w | E [ K R, − ln | x − w | ( x ) f βǫ ( w ) f βǫ ( x )] C E [ K R, − ln | x − w | ( x ) f ˆ β | x − w | ( w ) f ˆ β | w − x | ( x )]for some irrelevant constant C and ˆ β = β − min u > ln | x − w | ˆ Z u where the process ˆ Z is independentof the sigma algebra F | x − w | and ˆ Z s = B ˆ h ( s ) − B ˆ h ( − ln | x − w | ) for some Brownian motion B andˆ h ( s ) = R e s k ( u,x,w )(1 − ϕ ( u ( x − w ))) u du . 32 emark 4.8. The above technical part is due to the presence of possible long range correlations.Though they do not affect qualitatively our final estimates, getting rid of them may appear technical.The reader who wishes to skip this technical part may instead consider the compact case: k ( u, x, v ) =0 if u | x − w | > . In that case, we just have E [ K R, − ln | x − w | ( x ) f βǫ ( w ) f βǫ ( x )] = E [ K R, − ln | x − w | ( x ) f β | x − w | ( w ) f β | w − x | ( x )] because of the stopping time theorem and the fact that both martingales ( f βǫ ( w ) − f β | x − w | ( w )) ǫ | x − w | and ( f βǫ ( x ) − f β | x − w | ( x )) ǫ | x − w | are independent. We are thus left with computing E [ K R, − ln | x − w | ( x ) f ˆ β | x − w | ( w ) f ˆ β | w − x | ( x )]. To this purpose, we needthe following lemma, the proof of which is straightforward as a simple computation of covariance.Thus, details are left to the reader. Lemma 4.9.
We consider w = x such that | x − w | . The law of the couple ( ¯ X t ( w ) , ¯ X t ( x )) t > can be decomposed as: ( ¯ X t ( w ) , ¯ X t ( x )) t > = ( Z x,wt + P x,wt , P x,wt + D x,wt ) t > where the process Z x,w , P x,w , D x,w are independent centered Gaussian and, for some independentBrownian motions B , B independent of ¯ X : P x,wt = Z t g ′ x,w ( u ) d ¯ X u ( x ) , Z x,wt = Z t (1 − g ′ x,w ( u ) ) / dB u , D x,wt = Z t (1 − g ′ x,w ( u ) ) / dB u . with g x,w ( u ) = R e u k ( y ( x − w )) y dy . Moreover sup t ln | x − w | E [( Z x,wt ) + ( D x,wt ) ] C, (4.3) for some constant C independent of x, w such that | x − w | . Setting s = ln | x − w | , we use this lemma to get E [ K R, − ln | x − w | ( x ) f ˆ β | w − x | ( w ) f ˆ β | w − x | ( x )] E h K R,s ( x )( ˆ β − P x,ws − Z x,ws − s ) + e P x,ws + Z x,ws ) − s ( ˆ β − P x,ws − D x,ws − s ) + × . . . { sup u ∈ [0 ,s P x,wu + D x,wu − u ˆ β } e P x,ws + D x,ws ) − s i . We get rid of the process Z x,w by using first the Girsanov transform with the correspondingexponential term, and then by estimating the remaining terms containing Z x,w with the help of(4.3). We get for some constant C that may vary along lines but does not depend on x, w : E [ K R,s ( x ) f ˆ β | w − x | ( w ) f β | w − x | ( x )] C E h K R,s ( x )(1 + ˆ β )(1 + | P x,ws − E [( P x,ws ) ] | ) e P x,ws − E [( P x,ws ) ] ( β − P x,ws − D x,ws − s ) + × . . . · · · × { sup u ∈ [0 ,s P x,wu + D x,wu − u ˆ β } e P x,ws + D x,ws ) − s i .
33e use the Girsanov transform again to make the term e P x,ws + D x,ws ) − | x − w | disappear: E [ K R,s ( x ) f ˆ β | w − x | ( w ) f ˆ β | w − x | ( x )] C E h K R,s ( x )(1 + ˆ β )(1 + | P x,ws | ) e P x,ws +2 E [( P x,ws ) ] ( ˆ β − P x,ws − D x,ws ) + { sup u ∈ [0 ,s P x,wu + D x,wu ˆ β } i . Now we write P x,ws = P x,ws + D x,ws − D x,ws and use (4.3) to see that we can replace E [( P x,ws ) ] by s even if it means modifying the constant C (which still does not depend on x, w ): E [ K R,s ( x ) f ˆ β | w − x | ( w ) f ˆ β | w − x | ( x )] C E h K R ( x )(1 + ˆ β )(1 + | P x,ws + D x,ws | + | D x,ws | ) e P x,ws + D x,ws )+2 s e − D x,ws × . . . ( ˆ β − P x,ws − D x,ws ) + { sup u ∈ [0 ,s P x,wu + D x,wu ˆ β } i . Now we use the fact that for x ∈ K R,s :sup u − ln | x − w | β − P x,wu − D x,wu (1 + u ) / χ R and inf u − ln | x − w | β − P x,wu − D x,wu (1 + u ) / − χ > R , to get E [ K R,s ( x ) f ˆ β | w − x | ( w ) f ˆ β | w − x | ( x )] C E h K R,s ( x )(1 + ˆ β ) (cid:0) R (1 + s ) / χ + | D x,ws | (cid:1) e − R − (1+ s ) / − χ +2 s e − D x,ws × . . . ( ˆ β − P x,ws − D x,ws ) + { sup u ∈ [0 ,s P x,wu + D x,wu ˆ β } i C | x − w | e − R − s / − χ (1 + Rs / χ ) × . . . · · · × E h (1 + ˆ β )(1 + | D x,ws | ) e − D x,ws ( β − P x,ws − min u ∈ [0 ,s ] D x,wu ) + { sup u ∈ [0 ,s P x,wu β − min u ∈ [0 ,s D x,wu } i C | x − w | e − R − s / − χ (1 + Rs / χ ) E h (1 + ˆ β )(1 + | D x,ws | ) e − D x,ws ( ˆ β − min u ∈ [0 ,s ] D x,wu ) i . Because of (4.3), the last expectation is finite and bounded independently of x, w . Indeed, ˆ β ,( D x,wu ) u s are independent Wiener integrals with bounded variance (independently of x, w ).Therefore we can find α > | x − w | E [ e α (min u s D x,wu ) + e α ˆ β ] < + ∞ . With theseestimates, it is plain to see that the above expectation is finite. To sum up, we have proved E Q [ K R ( x ) M β ( B ( x, e − t ))] Z B ( x,e − t ) C | x − w | e − R − (ln | x − w | ) / − χ (1 + R (ln 1 | x − w | ) / χ ) dw = C Z e − t ρ − e − R − (ln ρ ) / − χ (1 + R (ln 1 ρ ) / χ ) dρ = C Z ∞ t e − R − y / − χ (1 + Ry / χ ) dy. E h Z K R Z B (cid:16) ln 1 | x − y | (cid:17) p M β ( dx ) M β ( dy ) i = E Q h K R ( x ) Z B (cid:16) ln 1 | x − y | (cid:17) p M β ( dy ) i ∞ X n =1 E Q h K R ( x ) Z B ∩{ − n − < | x − y | − n } (cid:16) ln 1 | x − y | (cid:17) p M β ( dy ) i ∞ X n =1 ( n + 1) p ln p E Q h K R ( x ) M β ( B ( x, − n )) i C ∞ X n =1 ( n + 1) p Z ∞ n ln 2 e − R − y / − χ (1 + Ry / χ ) dy. This last series is easily seen to be finite. The proof of Theorem 4.7 is complete.It is then a routine trick to deduce
Corollary 4.10.
For each ball B , for all δ > , there is a compact set K δ ⊂ B such that M β ( B ∩ K cδ ) δ and for all p > , for all x ∈ K δ sup x ∈O open ⊂ B, diam( O ) M β ( O ) (cid:0) − ln diam( O ) (cid:1) p < + ∞ . Corollary 4.11.
1. For each ball B , for each δ > , there is a compact set K ⊂ B such that M ′ ( B ∩ K c ) δ and for all p > Z K Z B (cid:16) ln 1 | x − y | (cid:17) p M ′ ( dx ) M ′ ( dy ) < + ∞ .
2. For each ball B , for all δ > , there is a compact set K δ ⊂ B such that M ′ ( B ∩ K cδ ) δ andfor all p > , for all x ∈ K δ sup x ∈O open ⊂ B, diam( O ) M ′ ( O ) (cid:0) − ln diam( O ) (cid:1) p < + ∞ .
3. Almost surely in X , the Liouville measure M ′ does not charge the polar sets of the (standard)Brownian motion.Proof. Both statements results from the fact that M β coincides with M ′ over bounded sets for β (random) large enough.Though we will not need such a strong statement in the following , we state the optimal modulusof continuity bound that we can get with our methods35 orollary 4.12. For each ball B , for all χ ∈ ]0 , / and for all δ > , there is a compact set K δ ⊂ B such that M ′ ( B ∩ K cδ ) δ and for all x ∈ K δ sup x ∈O open ⊂ B, diam( O ) M ′ ( O ) exp (cid:0) ( − ln diam( O )) − χ (cid:1) < + ∞ . Remark 4.13.
Observe that the question of the capacity properties of the measure M ′ , i.e. Corol-lary 4.11, was initially raised in [5] (see at the end of the first section). F ′ on the whole of R In this subsection, we have two main objectives: to construct the PCAF F ′ ( x, · ) on the wholeof R and prove the convergence of ( F ′ ,ǫβ ( x, · )) ǫ towards F ′ ( x, · ). The main difficulty here is thefollowing: in section 3.3 we have proved the almost sure (in X and B ) convergence of ( F ′ ,ǫβ ( x, · )) ǫ towards F ’ when the starting point x is fixed. Of course, we can deduce that almost surely in X ,for a countable collection of given starting points, this convergence holds. The main difficulty is toprove that this convergence holds for all possible starting points and this definitely requires somefurther arguments.For q = ( q , q ) ∈ Z , we denote by C q the cube [ q , q + 1] × [ q , q + 1]. We fix p > q ∈ Z and δ >
0, we denote by K q δ,L the compact set K q δ,L = K δ ∩ (cid:8) x ∈ C q ; sup x ∈O open , diam( O ) M β ( O ) (cid:0) − ln diam( O ) (cid:1) p L (cid:9) , (4.4)where K δ is the compact set given by Corollary 4.10 applied with B = C q . Then we set K q δ = [ L> K q δ,L , K δ,L = [ q ∈ Z K q δ,L and S = [ δ> , q ∈ Z ,L> K q δ,L . (4.5)From Corollary 4.10, we have M β (( K q δ ) c ∩ C q ) δ . Therefore, M β (( S δ> K q δ ) c ∩ C q ) = 0 for each q ∈ Z and thus M β ( S c ) = 0 . (4.6)We consider a Brownian motion B starting from 0 and define a Brownian motion starting from x by B x = x + B for each point x ∈ R . Following Section 3, we may assume that ( F ′ ,ǫβ ( x, · )) ǫ converges almost surely in X and B in C ( R + ; R + ) towards F ′ β ( x, · ) for each rational points x ∈ Q .For each δ, L > x ∈ Q , we define the adapted continuous random mapping F ′ ,δ,Lβ ( x, t ) = Z t K δ,L ( B xr ) F ′ β ( x, dr ) . (4.7) Proposition 4.14.
Almost surely in X , for each δ, L > and x ∈ R , there exists a B x -adaptedcontinuous random mapping, still denoted by F ′ ,δ,Lβ ( x, · ) such that for all sequence of rational points ( x n ) n converging towards x , the sequence ( F ′ ,δ,Lβ ( x n , · )) n converges in P B -probability in C ( R + , R + ) towards F ′ ,δ,Lβ ( x, · ) . roof. Let us fix x ∈ R . We consider a sequence ( x n ) n of rational points converging towards x . We first establish the convergence in law under P B of the sequence ( B x n , F ′ ,δ,Lβ ( x n , · )) n in C ( R + , R ) × C ( R + , R + ). To this purpose, the main idea is an adaptation of [28, section 2.9] withminor modifications. Yet, we outline the proof because we will play with this argument throughoutthis section. For all 0 < s < t , we write F ′ ,δ,Lβ ( x, ] s, t ]) for F ′ ,δ,Lβ ( x, t ) − F ′ ,δ,Lβ ( x, s ). The proof relieson two arguments: a coupling argument and the estimate:sup y ∈ B (0 ,R ) ∩ Q E B [ F ′ ,δ,Lβ ( y, t )] → , as t → . (4.8)We begin with explaining (4.8). For all y ∈ Q , we have: E B [ F ′ ,δ,Lβ ( y, t )] = Z R Z t p r ( y, z ) dr K δ,L ( z ) M β ( dz ) . Furthermore we have for each p > r ∈ ]0 , / sup x ∈ R ( − ln r ) p M β ( B ( x, r ) ∩ K δ,L ) p L. (4.9)Indeed, observe first that sup r ∈ ]0 , / sup x ∈ K δ,L ( − ln r ) p M β ( B ( x, r ) ∩ K δ,L ) L by definition. Toextend this formula to x K δ,L , take r / B ( x, r ) ∩ K δ,L is empty in which case M β ( B ( x, r ) ∩ K δ,L ) = 0 or we can find y ∈ B ( x, r ) ∩ K δ,L and then M β ( B ( x, r ) ∩ K δ,L ) M β ( B ( y, r ) ∩ K δ,L ) L ( − ln 2 r ) − p p L ( − ln r ) − p .We deduce that for all R > y ∈ B (0 ,R ) Z R Z t p r ( y, z ) dr K δ,L ( z ) M β ( dz ) → , as t → . (4.10)Indeed, the decay of the size of balls induced by (4.9), used with p >
2, is enough to overcomethe ln-singularity produced by the heat kernel integral: R t p r ( y, z ) dr (to be exhaustive, one shouldadapt the argument in [28, section 2.7] but this is harmless). Hence (4.8).Let us now prove that the family ( F ′ ,δ,Lβ ( x n , · )) n is tight in C ( R + ; R + ). This consists in checkingthat for all T, η >
0: lim δ → lim sup n →∞ P B (cid:0) sup s,t T | t − s | δ | F ′ ,δ,Lβ ( x n , ] s, t ]) | > η (cid:1) = 0 . (4.11)The control of this supremum uses two arguments: a control of the involved quantities when s, t are closed to 0 via (4.8) and a control of this supremum via a coupling argument when s, t arefar enough from 0. To quantify the proximity to 0, we introduce a parameter θ >
0. We have for δ < θ : P B (cid:0) sup s,t T | t − s | δ | F ′ ,δ,Lβ ( x n , ] s, t ]) | > η (cid:1) P B (cid:0) F ′ ,δ,Lβ ( x n , θ ) > η/ (cid:1) + P B (cid:0) sup θ s,t T | t − s | δ | F ′ ,δ,Lβ ( x n , ] s, t ]) | > η/ (cid:1) . (4.12)37o establish (4.11), it is thus enough to prove that the lim sup θ → lim sup δ → lim sup n →∞ of eachterm in the right-hand side of the above expression vanishes. The first term is easily treated withthe help of the Markov inequality and (4.8) so that we now focus on the second term. To thispurpose, we recall the following coupling lemma: Lemma 4.15.
Fix x ∈ R and let us start a Brownian motion B x from x . Let us consider anotherindependent Brownian motion B ′ starting from and denote by B y , for a rational y ∈ Q , theBrownian motion B y = y + B ′ . Let us denote by τ x,y the first time at which the first componentsof B x and B y coincide: τ x,y = inf { u > B ,xu = B ,yu } and by τ x,y the first time at which the second components coincide after τ x,y : τ x,y = inf { u > τ x,y ; B ,xu = B ,yu } The random process B x,y defined by B x,yt = ( B ,xt , B ,xt ) if t τ x,y ( B ,yt , B ,xt ) if τ x,y < t τ x,y ( B ,yt , B ,yt ) if τ x,y < t. is a new Brownian motion on R starting from x , and coincides with B y for all times t > τ y .Furthermore, if y − x → , we have for all η > : P ( τ x,y > η ) → . We choose y ∈ Q . We can consider the couple ( F ′ ,δ,L,x n β ( y, · ) , F ′ ,δ,Lβ ( x n , · )) where F ′ ,δ,Lβ ( x n , · ) isthe same as that considered throughout this section and F ′ ,δ,L,x n β ( y, · ) is constructed as F ′ ,δ,Lβ ( y, · )but we have used the Brownian motion ¯ B y,x n of Lemma 4.15 instead of the Brownian motion B y .The important point to understand is that this couple does not have the same law as the couple( F ′ ,δ,Lβ ( y, · ) , F ′ ,δ,Lβ ( x n , · )) but it has the same 1-marginal. Furthermore we have F ′ ,δ,L,x n β ( y, ] s, t ]) = F ′ ,δ,Lβ ( x n , ] s, t ]) for τ y,x n s < t . We deduce: P B (cid:0) sup θ s,t T | t − s | δ | F ′ ,δ,Lβ ( x n , ] s, t ]) | > η/ (cid:1) P B (cid:0) sup θ s,t T | t − s | δ | F ′ ,δ,Lβ ( y, ] s, t ]) | > η/ (cid:1) + P ( τ y,x n > θ ) . It is then obvious to get:lim sup δ → lim sup n →∞ P B (cid:0) sup θ s,t T | t − s | δ | F ′ ,δ,Lβ ( x n , ] s, t ]) | > η/ (cid:1) P ( τ y,x > θ ) . (4.13)Since the choice of y was arbitrary, we can now choose y arbitrarily close to x to make this latterterm as close to 0 as we please for a fixed θ . Hence (4.11) and the family ( F ′ ,δ,Lβ ( x n , · )) n is tight in C ( R + , R + ). 38e can also use the coupling argument to prove that there is only one possible limit in lawfor all subsequences ( x n ) n such that x n → x as n → ∞ , thus showing the convergence in law in C ( R + , R + ) of the family ( F ′ ,δ,Lβ ( x n , · )) n . Here we have only dealt with the convergence of the family( F ′ ,δ,Lβ ( x n , · )) n but it is straightforward to adapt the argument to the family ( B x n , F ′ ,δ,Lβ ( x n , · )) n .Now, we come to the convergence in P B -probability of ( F ′ ,δ,Lβ ( x n , · )) n . We fix t >
0, we considerthe mapping ( x, y ) ∈ Q × Q E B (cid:2) ( F ′ ,δ,Lβ ( x, t ) − F ′ ,δ,Lβ ( y, t )) (cid:3) . We can expand the square and, following the ideas in [28, section 2.7], use (4.9) to control theln-singularities to see that the above mapping extends to a continuous function on R × R , whichvanishes on the diagonal { ( x, x ); x ∈ R } . Therefore, if ( x n ) n is a sequence in Q converging towards x , the sequence ( F ′ ,δ,Lβ ( x n , t )) n converges in L under P B . We can thus extract a subsequence( x φ ( n ) ) n such that for all t ∈ Q ∩ R + , the sequence ( F ′ ,δ,Lβ ( x φ ( n ) , t )) n converges P B -almost surely.Also, we have seen that this subsequence converges in law in C ( R + , R + ) towards a continuousrandom mapping. The Dini theorem implies that ( F ′ ,δ,Lβ ( x φ ( n ) , · )) n converges P B -almost surely in C ( R + , R + ) towards a limit denoted by F ′ ,δ,Lβ ( x, · ).We are now in position to construct F ′ β on the whole of R . Theorem 4.16.
Almost surely in X , for all x ∈ R , the random measure F ′ ,δ,Lβ ( x, dt ) converges P B -almost surely as δ → and L → ∞ in the sense of weak convergence of measures towards arandom measure denoted by F ′ β ( x, dt ) . Furthermore:1. for x ∈ Q , F ′ β ( x, · ) coincides with the limit of the family ( F ′ ,ǫβ ( x, · )) ǫ as ǫ → defined insubsection 3.3.2. for x ∈ S ∪ Q , convergence of the mapping t F ′ ,δ,Lβ ( x, t ) towards t F ′ β ( x, t ) holds P B -almost surely in C ( R + , R + ) as δ → , L → ∞ .3. for x S , for all s > , convergence of the mapping t F ′ ,δ,Lβ ( x, ] s, t ]) towards t F ′ β ( x, ] s, t ]) holds P B -almost surely in C ([ s, + ∞ [ , R + ) .Proof. Observe that K δ,L ⊂ K δ ′ ,L ′ if δ ′ δ and L ′ > L . Therefore, for all x ∈ R and for all t > F ′ ,δ,Lβ ( x, t ) F ′ ,δ ′ ,L ′ β ( x, t ) if δ ′ δ . We can thus define the almost sure limit: F ′ , ∞ β ( x, t ) = lim δ → ,L →∞ F ′ ,δ,Lβ ( x, t ) . Actually, this also implies the weak convergence as δ → L → ∞ of the measure F ′ ,δ,Lβ ( x, dt )towards a random measure, still denoted by F ′ , ∞ β ( x, dt ).For x ∈ Q , let us identify F ′ , ∞ β ( x, · ) with the limit F ′ β ( x, · ) of the family ( F ′ ,ǫβ ( x, · )) ǫ as ǫ → F ′ ,δ,Lβ ( x, t ) = Z t K δ,L ( B xr ) F ′ β ( x, dr ) F ′ β ( x, t ) ,
39n such a way that F ′ , ∞ β ( x, t ) F ′ β ( x, t ). Second, by the dominated convergence theorem we get E B (cid:2) | F ′ β ( x, t ) − F ′ , ∞ β ( x, t ) | (cid:3) = lim δ → ,L →∞ E B (cid:2) Z t K cδ,L ( B xr ) F ′ β ( x, dr ) (cid:3) = lim δ → ,L →∞ Z R Z t p r ( x, y ) dr K cδ,L ( y ) M β ( dy )= Z R Z t p r ( x, y ) dr S c ( y ) M β ( dy )=0because M β ( S c ) = 0. Now that we have identified F ′ β ( x, · ) with F ′ , ∞ β ( x, · ) on x ∈ Q , we skip thedistinction made with the superscript ∞ and write F ′ β for the limit of F ′ ,δ,Lβ .For x ∈ Q , the continuity of the mapping t F ′ β ( x, t ) together with the Dini theorem impliesthe P B -almost sure convergence of F ′ ,δ,Lβ ( x, · ) towards F ′ β ( x, · ) in C ( R + , R + ). Let us now completethe proof of items 2 and 3. We fix s >
0. The coupling argument established in the proof ofProposition 4.14 shows that the mapping t F ′ β ( x, ] s, t ]) is continuous on [ s, + ∞ [ (it coincidesin law with t F ′ β ( y, ] s, t ]) with y ∈ Q as soon as B x and B y are coupled before time s , whichhappens with probability arbitrarily close to 1 provided that y is close enough to x ). The Dinitheorem again implies item 3. Let us stress that it is not clear that we can take s = 0 because weneed to control the decay of balls at x ∈ S c and this decay may happen to be very bad on S c .To prove item 2, we also use the Dini theorem but we further need to prove that the mapping t F ′ β ( x, t ) is continuous at t = 0 with F ′ β ( x,
0) = 0. This can be done by computing E B (cid:2) F ′ β ( x, t ) (cid:3) = Z R Z t p r ( x, y ) dr S ( y ) M β ( dy ) . Following [28], we observe that the mapping y R t p r ( x, y ) dr possesses a logarithmic singularityat y = x . Furthermore, for x ∈ S , we havesup x ∈O open ⊂ B, diam( O ) M ′ ( O ) (cid:0) − ln diam( O ) (cid:1) p < + ∞ . Therefore Z R Z p r ( x, y ) dr S ( y ) M β ( dy ) < + ∞ in such a way that the dominated convergence theorem implies E B (cid:2) F ′ β ( x, t ) (cid:3) = Z R Z t p r ( x, y ) dr S ( y ) M β ( dy ) → , as t → . To sum up, we have proved that the mapping t F ′ β ( x, t ) is continuous at t = 0 with F ′ β ( x,
0) = 0for x ∈ S . We complete the proof of item 2 with the help of the Dini theorem. Remark 4.17.
It is important here to stress that the above Proposition shows that for all x , wehave defined a mapping F ′ β ( x, · ) , which is P B -almost surely continuous with continuous sample aths and satisfies a variant of the additivity property of a PCAF, i.e. for all s, t > we havealmost surely F ′ β ( x, t + s ) = F ′ β ( x, t ) + ¯ F ′ β ( x, s ) , where, conditionally to F t , the variable ¯ F ′ β ( x, s ) is distributed as F ′ β ( x + B t , s ) under P x + B t (measureof a Brownian motion starting from x + B t ). Also, we have not so far proved that F ′ β is a PCAFbecause it is defined for all x P B -almost surely whereas we need to define it P B -almost surely forall x . Yet, we will see that this problem for the construction of a proper PCAF is not too serious. We claim:
Theorem 4.18.
Almost surely in X , we define F ′ ( x, t ) = lim β →∞ F ′ β ( x, ]0 , t ]) where convergence holds P B -almost surely in C ( R + , R + ) . F ′ is some form of PCAF in the strictsense in R : it is defined for all starting points and satisfies the following variant of the additivityproperty F ′ β ( x, t + s ) = F ′ β ( x, t ) + ¯ F ′ β ( x, s ) , s, t > where, conditionally to F t , the variable ¯ F ′ β ( x, s ) is distributed as F ′ β ( x + B t , s ) under P x + B t (measureof a Brownian motion starting from x + B t ).Furthermore,1. for all x ∈ R , F ′ is continuous, strictly increasing and goes to ∞ as t → ∞ .2. F ′ coincides outside a set of zero capacity with a PCAF of Revuz measure M ′ .Proof. For each x ∈ R , for each t >
0, the mapping β F ′ β ( x, ]0 , t ]) is increasing. We can thusdefine F ′ ( x, t ) = lim β →∞ F ′ β ( x, ]0 , t ]). Furthermore, for each ball B containing x , F ′ ( x, t ) coincideswith F ′ β ( x, ]0 , t ]) for t < τ B ( x ) = inf { u > B xu B } and β (random) large enough (more preciselyfor β large enough to make sup x ∈ B sup ǫ ∈ ]0 , X ǫ ( x ) − ǫ < β , see Proposition 3.5). It is obviousto check that F ′ satisfies the additivity (4.14).Now we prove item 1. This results from the coupling argument detailed in Proposition 4.14 as F ′ ( x, · ) is strictly increasing and goes to ∞ as t → ∞ for x ∈ Q (see also [28, Proposition 2.24]).Finally, we prove item 2, more precisely we establish the relation (4.2) for M ′ and F ′ . Theconstruction of F ′ entails that, for any 0 < s < t E B x h Z ts f ( B xr ) F ′ ( x, dr ) i = Z ts Z R f ( y ) p r ( x, y ) M ′ ( dy ) dr. Therefore, for any nonnegative Borel functions f, h ( P r stands for the semigroup associated to theplanar Brownian motion): Z R h ( x ) E B x h Z ts f ( B xr ) F ′ ( x, dr ) i dx = Z ts Z R Z R f ( y ) p r ( x, y ) M ′ ( dy ) h ( x ) dx dr = Z ts Z R f ( y ) P r h ( y ) M ′ ( dy ) dr.
41t suffices to let s → A associated to M ′ because M ′ is a smooth measure in the sense of [26] thanks to Corollary 4.11. Now we have at our disposala PCAF A with Revuz measure M ′ and an ”almost” PCAF F ′ . The reader may check that theuniqueness part of [26, Theorem 5.1.4] can be reproduced to prove that F ′ and A coincide for x outside a set of capacity 0 (just observe that this proof does not use the fact that the set wherethe PCAF is defined does not depend on x ). Definition 4.19. (critical Liouville Brownian motion).
Almost surely in X , for all x ∈ R ,the law of the LBM at criticality, starting from x , is defined by: B xt = x + B hB x i t where hB x i is defined by p /πF ′ ( x, hB x i t ) = t. We stress that B x is a local martingale. Proposition 4.20.
The critical LBM is a strong Markov process with continuous sample paths.Proof.
Strong Markov property results from [26, sect. 6]. Continuity of sample paths results fromthe fact that F ′ is strictly increasing. Theorem 4.21.
Assume further [A.6] or [A.6’]. Almost surely in X , for all x ∈ S , the ǫ -regularizedBrownian motion ( B ǫ,x ) ǫ defined by Definition 2.5 converges in law in the space C ( R + , R ) equippedwith the supremum norm on compact sets towards B x .Proof. This is just a consequence of Theorems 4.16 and 4.18 as explained in [28].From [26, Th. 6.2.1], we claim:
Theorem 4.22. (Dirichlet form).
The critical Liouville Dirichlet form (Σ , F ) takes on thefollowing explicit form on L ( R , M ′ ) : Σ( f, g ) = 12 Z R ∇ f ( x ) · ∇ g ( x ) dx (4.15) with domain F = n f ∈ L ( R , M ′ ) ∩ H loc ( R , dx ); ∇ f ∈ L ( R , dx ) o , Furthermore, it is strongly local and regular.
Let us denote by P Xt (for t >
0) the mapping f ∈ C b ( R ) (cid:0) x ∈ R P Xt f ( x ) = E B [ f ( B xt )] (cid:1) . (4.16)Similarly we define P ǫ as the semigroup generated by the Markov process B ǫ . From [26, sect. 6],we claim: Theorem 4.23. (Semigroup).
The linear operator P Xt , restricted to C c ( R ) , extends to a linearcontraction on L p ( R , M ′ ) for all p < ∞ , still denoted P Xt . Furthermore: ( P X ) t > is a Markovian strongly continuous semigroup on L p ( R , M ′ ) for p < ∞ . • Assume further [A.6] or [A.6’]. Almost surely in X , the ǫ -regularized semigroup ( P ǫ ) ǫ con-verges pointwise for x ∈ S towards the critical Liouville semigroup. More precisely, for allbounded continuous function f , we have: ∀ x ∈ S, ∀ t > , lim ǫ → P ǫt f ( x ) = P Xt f ( x ) . • P X is self-adjoint in L ( R , M ′ ) . • the measure M ′ is invariant for P Xt . The critical Liouville Laplacian ∆ X is defined as the generator of the critical Liouville semi-group times the usual extra factor 2. The critical Liouville Laplacian corresponds to an operatorwhich can formally be written as ∆ X = X − ( x ) e − X ( x ) ∆and can be thought of as the Laplace-Beltrami operator of 2 d -Liouville quantum gravity at criti-cality (of course when X is a free field).One may also consider the resolvent family ( R Xλ ) λ> associated to the semigroup ( P X t ) t . In astandard way, the resolvent operator reads: ∀ f ∈ C b ( R ) , ∀ x ∈ R , R Xλ f ( x ) = Z ∞ e − λt P Xt f ( x ) dt. (4.17)Furthermore, the resolvent family ( R Xλ ) λ> extends to L p ( R , M ′ ) for 1 p < + ∞ , is stronglycontinuous for 1 p < + ∞ and is self-adjoint in L ( R , M ′ ). This results from the properties ofthe semi-group. As a consequence of Theorem 4.23, it is straightforward to see that: Proposition 4.24.
Assume further [A.6] or [A.6’]. Almost surely in X , the ǫ -regularized resolventfamily ( R ǫλ ) λ converges towards the critical Liouville resolvent ( R Xλ ) λ in the sense that for allfunction f ∈ C b ( R ) : ∀ x ∈ S, lim ǫ → R ǫλ f ( x ) = R Xλ f ( x ) . Also and similarly to [26, 29], it is possible to get an explicit expression for the resolventoperator:
Proposition 4.25.
Almost surely in X , the resolvent operator takes on the following form for allmeasurable bounded function f on R : ∀ x ∈ R , R Xλ f ( x ) = p /π E B (cid:2) Z ∞ e − λ √ /πF ′ ( x,t ) f ( B xt ) F ′ ( x, dt ) (cid:3) . The main purpose of this section is to prove the following structure result on the resolventfamily:
Theorem 4.26. (massive Liouville Green kernels at criticality) . For every x ∈ R , theresolvent family ( R Xλ ) λ> is absolutely continuous with respect to the critical Liouville measure. herefore there exists a family ( r Xλ ( · , · )) λ , called the family of massive critical Liouville Greenkernels, of jointly measurable functions such that: ∀ x ∈ R , ∀ f ∈ B b ( R ) , R Xλ f ( x ) = p /π Z R f ( y ) r Xλ ( x, y ) M ′ ( dy ) and such that:1) (strict-positivity) for all λ > and x ∈ R , M ′ ( dy ) a.s., r Xλ ( x, y ) > ,
2) (symmetry) for all λ > and x, y ∈ R : r Xλ ( x, y ) = r Xλ ( y, x ) ,
3) (resolvent identity) for all λ, µ > , for all x, y ∈ R , r Xµ ( x, y ) − r Xλ ( x, y ) = ( λ − µ ) p /π Z R r Xλ ( x, z ) r Xµ ( z, y ) M ′ ( dz ) .
4) ( λ -excessive) for every y : e − λt P Xt ( r λ ( · , y ))( x ) r λ ( x, y ) for M ′ -almost every x and for all t > .Proof. We have to show absolute continuity of the resolvent for x ∈ R . Though inspired by [28, 29],we have to adapt the proof because we do not have ”uniform convergence” of the PCAF towards0 as t →
0. In particular, it is not clear that the resolvent be strong Feller. For δ >
0, we definefor f ∈ B b ( R ) R Xλ,δ f ( x ) = p /π E B (cid:2) Z ∞ δ e − λ √ /πF ′ ( x, ] δ,t ]) f ( B xt ) F ′ ( x, dt ) (cid:3) , where F ′ ( x, dr ) stands for the random measure associated to the increasing function t F ′ ( x, t ).Once again, the coupling argument of Proposition 4.14, it is plain to see that the mapping x ∈ R R Xλ,δ f ( x )is continuous. Now we claim Lemma 4.27.
For every x ∈ R , δ > and all nonnegative bounded Borelian function f , we have R Xλ f ( x ) = 0 = ⇒ R Xλ,δ f ( x ) = 0 . Lemma 4.28.
For every x ∈ R and all nonnegative bounded Borelian function f , we have ∀ δ > , R Xλ,δ f ( x ) = 0 = ⇒ R Xλ f ( x ) = 0 . We postpone the proofs of the above two lemmas. If A is a measurable set such that M ′ ( A ) = 0then by invariance of M ′ for the resolvent family, we deduce that R λ A ( x ) = 0 for M ′ -almost every x ∈ R . Since S has full M ′ -measure and because M ′ has full support, we deduce that R Xλ f ( x ) = 0for x belonging to a dense subset of R . From Lemma 4.27, R Xλ,δ f ( x ) = 0 for x belonging to adense subset of R . Continuity of R Xλ,δ f entails that this function identically vanishes on R forevery δ >
0. With the help of Lemma 4.28, we deduce that R λ A ( x ) = 0 for x ∈ R . Therefore, forall x ∈ R , R λ A ( x ) = 0, thus showing that the measure A R Xλ A ( x ) is absolutely continuouswith respect to M ′ . 44 roof of Lemmas 4.27 and 4.28. For x ∈ R and all bounded nonnegative Borelian function f , wehave (see Proposition 4.25) R Xλ f ( x ) = p /π E B x (cid:2) Z ∞ e − λ √ /πF ′ ( x,t ) f ( B xt ) F ′ ( x, dt ) (cid:3) = p /π E B x (cid:2) Z δ e − λ √ /πF ′ ( x,t ) f ( B xt ) F ′ ( x, dt ) (cid:3) + p /π E B x (cid:2) e − λF ′ ( x,δ ) Z ∞ δ e − λ √ /πF ′ ( x, ] δ,t ]) f ( B xt ) F ′ ( x, dt ) (cid:3) . (4.18)Therefore R Xλ f ( x ) > p /π E B x (cid:2) e − λF ′ ( x,δ ) Z ∞ δ e − λ √ /πF ′ ( x, ] δ,t ]) f ( B xt ) F ′ ( x, dt ) (cid:3) . For x ∈ R , we have e − λF ′ ( x,δ ) > P B x -almost surely. The proof of Lemma 4.27 follows.Furthermore, for x ∈ R , we have p /π E B x (cid:2) Z δ e − λ √ /πF ′ ( x,t ) f ( B xt ) F ′ ( x, dt ) (cid:3) λ − k f k ∞ E B x (cid:2) − e − λ √ /πF ′ ( x,δ ) (cid:3) . Since F ′ ( x, δ ) converges in law towards 0 as δ →
0, we deduce thatlim δ → p /π E B x (cid:2) Z δ e − λ √ /πF ′ ( x,t ) f ( B xt ) F ′ ( x, dt ) (cid:3) = 0 . From (4.18), we deduce: R Xλ f ( x ) R Xλ,δ f ( x ) + p /π E B x (cid:2) Z δ e − λ √ /πF ′ ( x,t ) f ( B xt ) F ′ ( x, dt ) (cid:3) . The proof of Lemma 4.28 follows.As prescribed in [26, section 1.5], let us define the Green function for f ∈ L ( D, M ′ ) by Gf ( x ) = lim t →∞ Z t P Xr f ( x ) dr. We further denote g the standard Green kernel on R . Following [26], we say that the semi-group( P Xt ) t , which is symmetric w.r.t. the measure M ′ , is irreducible if any P Xt -invariant set B satisfies M ′ ( B ) = 0 or M ′ ( B c ) = 0. We say that ( P Xt ) is recurrent if, for any f ∈ L ( D, M ′ ), we have Gf ( x ) = 0 or Gf ( x ) = + ∞ M ′ -almost surely. Theorem 4.29. (Liouville Green function at criticality).
The critical Liouville semi-groupis irreducible and recurrent.The critical Liouville Green function, denoted by G X , is given for every x ∈ S by G X f ( x ) = p /π Z R π ln 1 | x − y | f ( y ) M ′ ( dy ) for all functions f ∈ L ( R , M ′ ) such that Z R f ( y ) M ′ ( dy ) = 0 . roof. Irreducibility is a straightforward consequence of Theorem 4.26 and the remaining part ofthe statement is a straightforward adaptation of [28, 29] for x ∈ S .We investigate now the existence of probability densities of the critical Liouville semi-groupwith respect to the critical Liouville measure. Theorem 4.30. (Critical Liouville heat kernel).
The critical Liouville semigroup ( P X t ) t > is absolutely continuous with respect to the critical Liouville measure. There exists a family ofnonnegative functions ( p Xt ( · , · )) t > , which we call the critical Liouville heat kernel, such that: ∀ x ∈ R , dt a.s. , ∀ f ∈ B b ( R ) , P Xt f ( x ) = p /π Z R f ( y ) p Xt ( x, y ) M ′ ( dy ) . Proof.
From Theorems 4.26 and [26, Theorems 4.1.2 and 4.2.4], the Liouville semi-group is abso-lutely continuous with respect to the Liouville measure.
Remark 4.31.
Though we call the family ( p Xt ( · , · )) t > heat kernel, we are not in position toestablish most of the regularity properties expected from a heat kernel. Furthermore, a weak formof the notion of spectral dimension is obtained in [57], which is . We do not know how to adaptthe argument in the critical case because we cannot prove the continuity of the mapping ( x, y ) p ( t, x, y ) dt . Let us consider the set E defined in Theorem 4.7. Recall that M ′ ( E c ) = 0. As a consequenceof Theorem 4.26, we obtain the following result where λ is the Lebesgue measure: Corollary 4.32.
Almost surely in X , for all starting points x ∈ R , the critical LBM spendsLebesgue-almost all the time in the set E :a.s. in X, ∀ x ∈ R , a.s. under P B x , λ { t > B xt ∈ E c } = 0 . If one applies Theorem 4.30 instead, one obtains the similar but different result:
Corollary 4.33.
Almost surely in X , for all t > P B x a.s. , B xt ∈ E. So far, we constructed in detail the LBM on (subdomains of) R . This construction may beadapted to other geometries like the sphere S or torus T (equipped with a standard GaussianFree Field (GFF) with vanishing average for instance). Actually, our techniques can be adapted toother 2-dimensional Riemannian manifolds with a scalar metric tensor. The main reason is thata Riemannian manifold is locally isometric to R . We will not detail the proofs since the wholemachinery works essentially the same as in the plane: it suffices to have at our disposal a white noisedecomposition of the underlying Gaussian distribution and to adapt properly our assumptions.We rather give here further details in the case of the GFF on planar domains as the associatedBrownian motion possesses important conformal invariance properties. We consider a boundedplanar domain D . The Liouville Brownian motion on D is defined as follows: • consider a white noise cut-off approximation ( X ǫ ) ǫ of the GFF on D with Dirichlet boundary46onditions: see equation (2.8). • first define the time change as the limit F ′ ( x, t ) = lim ǫ → ǫ Z t (2 E [ X ǫ ( x + B u ) ] − X ǫ ( x + B u )) e X ǫ ( x + B u ) du for all t < τ Dx where B is a standard planar Brownian motion and τ Dx its first exit time out of D .This limit turns out to be the same as Z t C ( x + B u , D ) (2 E [ X ( x + B u ) ] − X ( x + B u )) e X ( x + B u ) − E [ X ( x + B u ) ] du where C ( x, D ) is the conformal radius at x in the domain D (see [44] or [23] in a different context). • extend this construction to all possible starting points as in section 4. Define the exit time ofthis LBM out of D by ˆ τ Dx = p /πF ′ ( x, τ Dx ) and then define the Liouville Brownian motion as inDefinition 4.19 for all time t < ˆ τ Dx . • Observe that this Liouville Brownian motion is invariant under conformal reparametrization. Thismeans that for all conformal map ψ : D ′ → D the process ψ − ( B x ) has the law of the LiouvilleBrownian motion on D ′ where in the construction of F ′ we use the standard reparametrizationrule of Liouville field theory X → X ◦ ψ + Q ln | ψ ′ | where Q = γ + γ for a subdomain of C (or Q = γ for a GFF on the sphere or the torus with vanishing mean). References [1] Allez R., Rhodes R., Vargas V.: Lognormal ⋆ -scale invariant random measures, ProbabilityTheory and Related Fields , April 2013, Volume 155, Issue 3-4, pp 751-788, arXiv:1102.1895v1.[2] Ambjørn J., Boulatov D., Nielsen J.L., Rolf J., Watabiki Y.: The spectral dimension of 2 D Quantum Gravity.
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