On the heat kernel and the Dirichlet form of Liouville Brownian Motion

In \cite{GRV}, a Feller process called Liouville Brownian motion on $\R^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field $e^{\gamma X}$ and is the right diffusion process to consider regarding 2d-Liouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially \cite{fuku} and the techniques introduced in \cite{GRV}. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a non-trivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in \cite{GRV} was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. One possible approach was to use the theory developed for example in \cite{stollmann,sturm1,sturm2}, whose aim is to capture the"geometry"of the underlying space out of the Dirichlet form of a process living on that space. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide an intrinsic metric which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.

A Index of notations 27 B Reinforced Kolmogorov's continuity criterion 28 1 Introduction This paper is concerned with the study of a Feller process, called the Liouville Brownian motion, that has been introduced in [12] to have further insight into the geometry of 2d-Liouville quantum gravity.More precisely, one major mathematical problem in (critical) 2d-Liouville quantum gravity is to construct a random metric on a two dimensional Riemannian manifold D, say a domain of R 2 (or the sphere) equipped with the Euclidean metric dz 2 , which takes on the form where X is a (massive) Gaussian Free Field (GFF) on the manifold D and γ ∈ [0, 2[ is a coupling constant (see [19,6,8,9,13,20] for further details and insights in Liouville quantum gravity).If it exists, this metric should generate several geometric objects: instead of listing them all, let us just say that each object that can be associated to a smooth Riemannian geometry raises an equivalent question in 2d-Liouville quantum gravity.Mathematical difficulties originate from the short scale logarithmically divergent behaviour of the correlation function of the GFF X.So, for each object that one wishes to define, one has to apply a renormalization procedure.For instance, one can define the volume form associated to this metric.The theory of renormalization for measures formally corresponding to the exponential of Gaussian fields with logarithmic correlations first appeared in the beautiful paper [16] under the name of Gaussian multiplicative chaos and applies to the GFF.Thereafter, convolution techniques were developed in [10,23,24].This allows to make sense of measures formally defined by: where dz stands for the volume form (Lebesgue measure) on D (to be exhaustive, one should integrate against h(z) dz where h is a deterministic function involving the conformal radius at z but this term does not play an important role for our concerns).This approach made possible in [10,22] (see also [1,2]) a rigorous measure based interpretation of the Knizhnik-Polyakov-Zamolodchikov formula (KPZ for short, see the original paper [19]).
In [12], the authors defined the Liouville Brownian motion.It can be thought of as the diffusion process associated to the metric (1.1) and is formally the solution of the stochastic differential equation: where B is a standard Brownian motion living on D. Furthermore, they proved that this Markov process is Feller and generates a strongly continuous semigroup (P X t ) t 0 , which is symmetric in L 2 (D, M).In particular, the Liouville Brownian motion preserves the Liouville measure.They also noticed that one can attach to the Liouville semigroup (P X t ) t 0 a Dirichlet form by the formula: .4) with domain F , which is defined as the set of functions f ∈ L 2 (R 2 , M) for which the above limit exists and is finite.This expression is rather non explicit.The purpose of this paper is to pursue the stochastic analysis of 2d-Liouville quantum gravity initiated in [12].We denote by H 1 (D, dx) the standard Sobolev space: and by H 1 loc (D, dx) the functions which are locally in H 1 (D, dx).First, we will make explicit the Liouville Dirichlet form (1.4), relying on techniques developed in [11,16], more precisely traces of Dirichlet forms and potential theory: Theorem 1.1.For γ ∈ [0, 2[, the Liouville Dirichlet form (Σ, F ) takes on the following explicit form: its domain is and for all functions f, g ∈ F : Furthermore, it is strongly local and regular.
Let us stress here that understanding rigorously the above Theorem is not obvious since the Liouville measure M and the Lebesgue measure dx are singular.The domain F is composed of the functions u ∈ L 2 (D, M) such that there exists a func- It is a consequence of the general theory developped in [11] (see chapter 6) and of the tools developped in [12] that the definition of (Σ, F ) actually makes sense: indeed, if f, g in H 1 loc (D, dx) are such that f (x) = g(x) for M(dx)-almost every x then ∇f (x) = ∇g(x) for dx-almost every x.
Then we perform an analysis of the Liouville resolvent family (R X λ ) λ 0 : We will prove that this family possesses strong regularizing properties.In particular, our two main theorems concerning the resolvent family are: Theorem 1.2.Almost surely in X, for γ ∈ [0, 2[, the resolvent operator R X λ is strong Feller in the sense that it maps the bounded measurable functions into the set of continuous bounded functions.
There is an exponent α ∈ (0, 1) (depending only on γ), such that, almost surely in X, for all λ > 0 the Liouville resolvent is locally α-Hölder.More precisely, for each R and λ 0 > 0, we can find a random constant C R,λ 0 , which is P X -almost surely finite, such that for all λ ∈]0, λ 0 ] and for all continuous function f : D → R vanishing at infinity, ∀x, y ∈ B(0, R): As a consequence, we obtain the existence of the massive Liouville Green functions, which are nothing but the densities of the resolvent operator with respect to the Liouville measure (see Theorem 3.10).
For symmetric semigroups, Fukushima and al. [11] proved the highly non-trivial Theorem (see their Theorems 4.1.2and 4.2.4) which states that absolute continuity of the resolvent family is equivalent to absolute continuity of the semigroup.As such, this allows us to obtain the following theorem on the existence of a heat-kernel: Theorem 1.4.Liouville heat kernel.The Liouville semigroup (P X t ) t>0 is absolutely continuous with respect to the Liouville measure.There exists a family (p X t (•, •)) t 0 , called the Liouville heat kernel, of jointly measurable functions such that: and such that: 1) (positivity) for all t > 0 and for all x ∈ D, for M(dy)-almost every y ∈ D, p X t (x, y) 0, 2) (symmetry) for all t > 0 and for every x, y ∈ D: These properties have interesting consequences regarding the stochastic structure of 2d-Liouville quantum gravity.For instance, the Liouville Brownian motion spends Lebesgue almost all the times in the set of points supporting the Liouville measure, nowadays called the thick points of the field X and first introduced by Kahane in the case of log-correlated Gaussian fields [16] like Free Fields (see also [15]).Furthermore, for a given time t, the Liouville Brownian motion is almost surely located on the thick points of X.We will also define the Liouville Green function to investigate the ergodic properties of the Liouville Brownian motion, which turns out to be irreducible and recurrent.
Finally, let us end this introduction by a discussion on the Liouville Dirichlet form as well as its possible relevance to the construction of the Liouville metric.Over the last 20 years, a rich theory has been developed whose aim is to capture the "geometry" of the underlying space out of the Dirichlet form of a process living on that space.See for example [25,26,27].This geometric aspect of Dirichlet forms can be interpreted in a sense as an extension of Riemannian geometry applicable to non differential structures.Among the recent progresses of Dirichlet forms has emerged the notion of intrinsic metric associated to a strongly local regular Dirichlet form [4,5,7,25,26,27,28]. It is natural to wonder if this theory is well suited to this problem of constructing the Liouville metric.More precisely, the intrinsic metric is defined by This metric is actually a metric in the wide sense, meaning that it can possibly take values d X (x, y) = 0 or d X (x, y) = +∞ for some x = y.Let us point out that, when the field X is smooth enough (and therefore not a free field), the distance (1.5) coincides with the Riemannian distance generated by the metric tensor e γX(z) dz 2 .Generally speaking, the point is to prove that the topology associated to this metric is Euclidean, in which case d X is a proper distance and (D, d X ) is a length space (see [25,Theorem 5.2]).Unfortunately, in the context of 2d-Liouville quantum gravity, we prove that this intrinsic metric turns out to be 0. Anyway, this fact is also interesting as it sheds some new light on the mechanisms involved in the renormalization of the Liouville metric (if exists).
The reader may find a list of notations used throughout the paper in Section A.

Notations
We stick to the notations of [12] (see the section "Background"), where the basic tools needed to define 2d-Liouville quantum gravity are described.In particular, a description of the construction of Free Fields and their cutoff regularization are given.Once regularized with a proper cutoff, the Free Field becomes smooth enough so that we can define an associated Riemannian manifold together with the standard associated objects.The mathematical purpose of this paper is to study what become (part of) these objects when the cutoff is removed.

Liouville Dirichlet form
The purpose of this section is to give an explicit description of the Liouville Dirichlet form, namely the Dirichlet form of the Liouville Brownian motion, by combining [11] and the results in [12].The first part of this section is devoted to recalling a few material about Dirichlet form in order to facilitate the reading of this paper.Then we identify the Dirichlet form and, finally, we discuss some questions naturally raised by the construction of the Dirichlet form.Among them: "Can we construct the Liouville Brownian motion from the only use of [11]?"

Background on positive continuous additive functionals and Revuz measures
In this subsection, to facilitate the reading of our results, we first summarize the content of section 5 in [11] applied to the standard Brownian (Ω, (B t ) t 0 , (F t ) t 0 , (P x ) x∈D ) in D which is of course reversible for the canonical volume form dx of D. We suppose that the space Ω is equipped with the standard shifts (θ t ) t 0 on the trajectory.One may then consider the classical notion of capacity associated to the Brownian motion.In this context, we have the following definitions: The capacity of a Borel measurable set K is then defined as: The set K is said polar when Cap(K) = 0.
Definition 2.2 (Revuz measure).A Revuz measure µ is a Radon measure on D which does not charge the polar sets.

Definition 2.3 (PCAF).
A positive continuous additive functional (PCAF) (A t ) t 0 is a F t -adapted continuous functional with values in [0, ∞] that satisfies for all ω ∈ Λ: where Λ is defined in the following way: there exists a polar set N (for the standard Brownian motion) such that for all x ∈ D \ N, P x (Λ) = 1 and θ t (Λ) ⊂ Λ for all t 0.
In particular, a PCAF is defined for all starting points x ∈ R 2 except possibly on a polar set for the standard Brownian motion.One can also work with a PCAF starting from all points, that is when the set N in the above definition can be chosen to be empty.In that case, the PCAF is said in the strict sense.
Finally, we conclude with the following definition on the support of a PCAF: Definition 2.4 (support of a PCAF).Let (A t ) t 0 be a PCAF with associated polar set N. The support of (A t ) t 0 is defined by: From section 5 in [11], there is a one to one correspondence between Revuz measures and PCAFs.In the next subsection, we will identify the Liouville measure M as the Revuz measure associated to the increasing functional F constructed in [12].Let us first check that the measure M is a Revuz measure, i.e. it does not charge polar sets: Lemma 2.5.Almost surely in X, the Liouville measure M does not charge the polar sets of the (standard) Brownian motion.
Proof.Let A be a bounded polar set and let R > 0 be such that A ⊂ B(0, R).From [21] (see also [17]), it suffices to prove that the mapping x → G R (x, y)M(dy) is bounded, where G R stands for the Green function of the Brownian motion killed upon touching ∂B(0, R).Recall that the Green function over B(0, R) takes on the form for some bounded function g over B(0, R), where d stands for the usual Riemannian distance on D. The result thus follows from Corollary 2.21 in [12] where it is proved that the Liouville measure uniformly integrates the ln over compact sets.

The Revuz measure associated to Liouville Brownian motion
In this subsection, we identify the measure M as the Revuz measure associated to the functional F introduced in [12].This functional F is defined almost surely in X for all x ∈ D by where B is a standard Brownian motion on D. By setting We claim: Lemma 2.6.Almost surely in X, F is a PCAF in the strict sense whose support is the whole domain D. Also, the Revuz measure of F is the Liouville measure M.
Proof.The fact that F is a PCAF in the strict sense whose support is the whole domain D is a direct consequence of [12] as summarized in (2.1).
The Revuz measure µ associated to F is the unique measure on D that does not charge polar sets and such that: for all continuous compactly supported function f .Here B x stands for the law of a Brownian motion starting from x.Let us denote by p t (x, y) the standard heat kernel on D. To identify the measure µ it suffices to compute its values on the set of continuous functions with compact support.For such a function, we have: Then The proof is complete.

Construction of the Liouville Dirichlet form (Σ, F )
In this subsection, we want to apply Theorem 6.2.1 in [11].Recall that the Liouville Brownian motion is defined in [12] as a continuous Markov process defined for all starting points x by the relation: We know that M is the Revuz measure associated to F .Hence, we can now straightforwardly apply the abstract framework of Theorem 6.2.1 in [11] to get the following expression for the Dirichlet form associated to Liouville Brownian motion: Theorem 2.7.The Liouville Dirichlet form (Σ, F ) takes on the following explicit form on L 2 (D, M): with domain Furthermore, it is strongly local and regular.
In fact, for any PCAF (A t ) t 0 associated to Brownian motion, one can define the Dirichlet form associated to the Hunt process B A −1 t .In this general case, theorem 6.2.1 in [11] gives an expression to the Dirichlet form which is non explicit and involves an abstract projection construction involving the support Y of the PCAF.Nonetheless, there is one case where the Dirichlet form takes on the simple form (2.2): when the support Y is the whole space D (recall that this constitutes a large part of the work [12]).
Remark 2.8.This result may appear surprising for non specialists of Dirichlet forms.Let us forget for a while Liouville quantum gravity and assume that the measure M is a smooth measure, meaning that it has a density w.r.t. the Lebesgue measure bounded from above and away from 0. Then we obviously have L 2 (D, dx) = L 2 (D, M).In that case, the domain and expression of the time changed Dirichlet form coincide with those of the Dirichlet form of the standard Brownian motion on D. So, a natural question is: "How do we differentiate the Markov process associated to this time changed Dirichlet form from the standard Brownian motion?".The answer is hidden in the fact that a Dirichlet form uniquely determines a Markovian semi-group provided that you fix a reference measure with respect to which you impose the semi-group to be symmetric.In the case of the standard Brownian motion, the reference measure is the Lebesgue measure dx whereas the reference measure is M in the case of the time changed Brownian motion.

Discussion about the construction of the Dirichlet form and the associated Hunt process
A natural question regarding the theory of Dirichlet forms is: "Can one construct directly the Liouville Brownian motion via the theory of Dirichlet forms without using the results in [12]?".Since the Liouville measure is a Revuz measure, it uniquely defines a PCAF (A t ) t .This PCAF may be used to change the time of a reference Dirichlet form, here that of the standard Brownian motion on D. The time changed Dirichlet form constructed in [11, Theorem 6.2.1] corresponds to that of a Hunt process where B is a standard Brownian motion and A −1 t is the inverse of the PCAF (A t ) t .Nevertheless, we stress that identifying this Hunt process explicitly is not obvious without using the tools developed in [12].Moreover, this abstract construction of H t rigorously defines a Hunt process living in the space D \ N where N is a polar set.To our knowledge, there is no general theory on Dirichlet forms which enables to get rid of this polar set, hence constructing a PCAF in the strict sense and a Hunt process starting from all points of D. In conclusion, without using the tools developped in [12], one can construct the Liouville Brownian motion in a non explicit way living in D \ N and for starting points in D \ N where N is a polar set (depending on the randomness of X); in this context, one can not start the process from one given fixed point x ∈ D or define a Feller process in the strict sense.Even if this was the case, in order to identify the corresponding Dirichlet form by the simple formula (2.2), one must show that the PCAF has full support (which is also part of the work done in [12]).
Let us mention that a measurable Riemannian structure associated to strongly local regular Dirichlet forms is built in [14].In [18], harmonic functions and Harnack inequalities for trace processes (i.e.associated to time changed Dirichlet forms) are studied.In particular, it is proved that harmonic functions for the Liouville Brownian motion are harmonic for the Euclidean Brownian motion and that harmonic functions for the Liouville Brownian motion satisfy scale invariant Harnack inequalities.Actually, there are many powerful tools that can be associated with a Dirichlet forms and listing them exhaustively is far beyond the scope of this paper.

Liouville Heat Kernel and Liouville Green Functions
The Liouville Brownian motion generates a Feller semi-group (P X t ) t , which can be extended to a strongly continuous semigroup on L p (D, M) for 1 p < +∞ and is reversible with respect to the Liouville measure M (see [12]).Recall that Proposition 3.1 ( [12]).For γ < 2, almost surely in X, the n-regularized semigroup (P n t ) t converges towards the Liouville semi-group (P X t ) t in the sense that for all function f ∈ C b (D): The main purpose of this section is to prove the existence (almost surely in X) of a heat-kernel p t (x, y) for this Feller semi-group (P X t ) t 0 .Our strategy for establishing the existence of the heat-kernel will be first to prove that the resolvent associated to our Liouville Brownian motion is (a.s. in X) absolutely continuous w.r.t the Liouville measure M. See Theorem 3.10.In general, the absolute continuity of the resolvent is far from implying the absolute continuity of the semi-group (think for example of the process defined on the circle by X x t = e i(x+t) ).Nevertheless, as stated in the introduction, the symmetry of the Liouville semi-group w.r.t. the Liouville measure M allows us to apply a deep theorem of Fukushima and al. [11] to conclude: see Theorem 3.16.Finally we deduce some corollaries along this section such as the fact that the Liouville Brownian motion a.s.spends most of his time in the thick points of the field X, the construction of the Liouville Green function or the study of the ergodic properties of the Liouville Brownian motion.

Analysis of the Liouville resolvent
One may also consider the resolvent family (R X λ ) λ>0 associated to the semigroup (P X t ) t .In a standard way, the resolvent operator reads: Furthermore, the resolvent family (R X λ ) λ>0 is self-adjoint in L 2 (D, M) and extends to a strongly continuous resolvent family on the L p (D, M) spaces for 1 p < +∞.This results from the properties of the semi-group.From Proposition 3.1, it is straightforward to deduce: Proposition 3.2.For γ < 2, almost surely in X, the n-regularized resolvent family (R n λ ) λ converges towards the Liouville resolvent (R X λ ) λ in the sense that for all function f ∈ C b (D): Also, it is possible to get an explicit expression for the resolvent operator: Proposition 3.3.For γ < 2, almost surely in X, the resolvent operator takes on the following form for all measurable bounded function f on D: Proof.Given a measurable bounded function f on D, we have: which completes the proof.
Theorem 3.4.For γ < 2, almost surely in X, the resolvent operator R X λ is strong Feller, i.e. maps the measurable bounded functions into the set of continuous bounded functions.
Proof.Let us consider a bounded measurable function f and let us prove that R X λ f (x) is a continuous function of x.To this purpose, write for some arbitrary ǫ > 0: We are going to prove that the family of functions (N ǫ ) ǫ uniformly converges towards 0 on the compact subsets as ǫ → 0 (obviously, if D is compact, we will prove uniform convergence on D) and that the functions R X,ǫ λ f are continuous.First we focus on (N ǫ ) ǫ and write the obvious inequality: From [12], we now that the latter quantity converges to 0 as ǫ to 0. Let us now prove the continuity of R X,ǫ λ f .By the Markov property of the Brownian motion, we get: Now we consider two points x and y in D and realize the coupling of (B x , F (x, •)) and (B y , F (y, •)) explained in [12].Recall that this coupling lemma allows us to construct a Brownian motion B x starting from x and a Brownian motion B y starting from y in such a way that they coincide after some random stopping time τ x,y .Let us denote by P B the law of the couple (B x , B y ) and E B the corresponding expectation.We obtain: y,ǫ) .Concerning the first quantity, observe that it is different from 0 only if the two Brownian motions have not coupled before time ǫ, in which case we use the rough This latter quantity converges towards 0 as |x − y| → 0. The second quantity is treated with the same idea: =E B e −λF (x,ǫ) − e −λF (y,ǫ) ; ǫ τ x,y + E B e −λF (x,ǫ) − e −λF (y,ǫ) ; ǫ > τ x,y 2P(ǫ τ x,y ) + E B e −λF (x,τ x,y )−λF (x,]τ x,y ,ǫ]) − e −λF (y,τ x,y )−λF (y,]τ x,y ,ǫ]) ; ǫ > τ x,y .
By gathering the above considerations, we have proved that x → R X,ǫ λ f (x) is continuous over D. Since the family (R X,ǫ λ f ) ǫ uniformly converges towards R X λ f on the compact sets as ǫ → 0, we deduce that R X λ f is continuous.Now we focus on another aspect of the regularizing properties of the resolvent family: Theorem 3.5.Assume D = R 2 and γ ∈ [0, 2[.There is an exponent α ∈ (0, 1) (depending only on γ), such that, almost surely in X, for all λ > 0 the Liouville resolvent is locally α-Hölder.More precisely, for each R and λ 0 > 0, we can find a random constant C R,λ 0 , which is P X -almost surely finite such that, for all λ ∈]0, λ 0 ] and for all continuous function f : R 2 → R vanishing at infinity: Proof.Fix λ > 0. Let f : R 2 → R be a bounded Borelian function.Let us prove that x → R X t f (x) is locally Hölder.Without loss of generality, we may assume that f ∞ 1.To this purpose, let us work inside a ball centered at 0 with fixed radius, say 1. Inside this ball, we consider two different points x, y.From this two points, we start two independent Brownian motions B x and B y , and couple them in the usual fashion to produce two new Brownian motions, call them still B x and B y , that coincide after some stopping time τ x,y .By applying the strong Markov property, we get: First we focus on N x,y : Let us now treat the term R X,x,y λ f .By the strong Markov property of the Brownian motion, we get: Therefore we have: ) − e −λF (y,τ x,y ) .
In the above inequalities, we have used the facts that B x τ x,y = B y τ x,y and R X λ f ∞ λ −1 .It is readily seen that this quantity can be estimated by: E B e −λF (x,τ x,y ) − e −λF (y,τ x,y ) E B min 1, λ|F (x, τ x,y ) − F (y, τ x,y )| E B min 1, λF (x, τ x,y ) + λF (y, τ x,y ) ( Therefore, we can take the q-th power (q 1) and use the Jensen inequality to get Observe that this bound holds for all λ and all functions f with f ∞ 1.So, let us choose a countable family (g n ) n of functions in C 0 (R d ) dense for the topology of uniform convergence over compact sets and set f n = g n / g n ∞ .By gathering (3.4)+(3.6),we get We claim: Lemma 3.6.For all x, y ∈ B(0, 1) and all χ ∈]0, 1 2 [, ǫ > 0, p ∈]0, 1[ and q 1 such that pq > 1, we have for some constant C χ,p,q which only depends on χ, p, q and ∀q 0, We postpone the proof of this lemma and come back to the proof of Theorem 3.5.We deduce that for all x, y ∈ B(0, 1), χ ∈]0, 1  2 [, ǫ > 0, p ∈]0, 1[ and q 1 such that pq > 1, we have for some constant C χ,p,q,λ 0 which only depends on χ, p, q, λ 0 .Now we fix χ ∈]0, 1 2 [.Then we choose δ > 0 such that 1 + δ < min(2, 4 γ 2 ).Since ξ(1 + δ) > 1, we can choose ǫ > 0 such that (2 − ǫ)ξ(1 + δ) > 2. Then we choose q > 1 large enough so as to make ǫχq > 2. Then we choose p ∈]0, 1[ such that pq = 1 + δ.We get for some β > 2 only depending on γ ∈ [0, 2[.From Theorem B.1, we deduce that for some α > 0 (only depending on γ) and some positive P X -almost surely finite random variable C independent of n and λ ∈]0, λ 0 ] ∩ Q: Observe that this relation is then necessarily true for all λ ∈]0, λ 0 ] because of the continuity of the resolvent with respect to the parameter λ.Now consider a function It is plain to deduce from the uniform convergence of (g n k ) k towards f (and therefore the uniform convergence of R X λ g n towards R X λ f ) and (3.7) that: The proof is over.
Proof of Lemma 3.6.Let us consider R > 0 such that R|x − y| 2 1.We have The last inequality results from the fact that 0 < p < 1.Therefore, for any χ ∈]0, 1 2 [ the last inequality resulting from the fact that the law of the random variable τ x,y |x− y| −2 is independent from x, y and possesses moments of order χ for all χ ∈]0, 1 2 [.By taking the q-th power and integrating with respect to E X , we get: Now we take R = |x − y| −ǫ and use (see [12]): Proposition 3.7.If γ 2 < 4 and x ∈ R 2 , the mapping F (x, •) possesses moments of order 0 q < min(2, 4/γ 2 ).Furthermore, if F admits moments of order q 1 then, for all s ∈ [0, 1] and t ∈ [0, T ]: where C q > 0 (independent of x, T ) and Thus we get: and we prove the Lemma.When D is compact, i.e. when D = T 2 or S 2 , we get: There is an exponent α ∈ (0, 1) (depending only on γ), such that, almost surely in X, for all λ > 0 the Liouville resolvent is α-Hölder.More precisely, for each λ 0 > 0, we can find a random constant C λ 0 , which is P X -almost surely finite such that, for all λ ∈]0, λ 0 ] and for all continuous function f : D → R, ∀x, y ∈ D: Corollary 3.9.For each λ > 0, the resolvent operator R X λ : ) is compact for the topology of convergence over compact sets.In the case of the sphere S 2 (or the torus T 2 ) equipped with a GFF X, the resolvent operator Proof.This is just a consequence of Theorems 3.5 or 3.8.Theorem 3.4 has the following consequences on the structure of the resolvent family: Theorem 3.10.(massive Liouville Green kernels).The resolvent family (R X λ ) λ>0 is absolutely continuous with respect to the Liouville measure.Therefore there exists a family (r X λ (•, •)) λ , called the family of massive Liouville Green kernels, of jointly measurable functions such that: and such that: 1) (strict-positivity) for all λ > 0 and for all x ∈ D, for M(dy)-almost every y ∈ D, 2) (symmetry) for all λ > 0 and for every x, y ∈ D: 3) (resolvent identity) for all λ, µ > 0, for all x, y ∈ D, 4) (λ-excessive) for every y, e −λt P X t (r λ (•, y))(x) r λ (x, y) for M-almost every x and for all t > 0.
Proof.It suffices to prove that, almost surely in X, Since the Liouville semigroup is invariant under the Liouville measure, we have for all bounded Borelian set A Therefore, M(A) = 0 implies that for M-almost every x ∈ D: R X λ 1 A (x) = 0. Since M has full support, we thus have at hand a dense subset We apply Theorem [11,Lemma 4.2.4] to prove the existence of the massive Liouville Green kernels.It remains to prove item 1.Consider a Borel set K and x ∈ D such that R X λ 1 K (x) = 0. Then P B a.s., we have We deduce, by using the Markov property: Since the transition probability of the standard Brownian motion are strictly positive, we deduce that the mapping x → R X λ 1 K (x) vanishes over a set with full Lebesgue measure.Furthermore, it is continuous by Theorem 3.4.Thus we have R X λ 1 K = 0 identically.Finally, we get: thus showing that the resolvent density is positive M almost surely.

Recurrence and ergodicity
As prescribed in [11, section 1.5], let us define the Green function for f ∈ L 1 (D, M) by We further denote g D the standard Green kernel on D.
Following [11], we say that the semi-group (P X t ) t , which is symmetric w.r.t. the measure M, is irreducible if any Proof.Irreducibility is a straightforward consequence of Theorem 3.10.Let us establish recurrence.We carry out the proof in the case of D = R 2 .The reader can easily adapt the proof to D = S 2 , T 2 .We first observe that We have used the fact that F (x, t) almost surely converges towards +∞ as t → ∞ (see [12]).Now observe that Let us assume that f has compact support.The above quantity diverges as t → ∞ like C ln t.To see this, we first compensate the divergence at infinity as follows: 2πr dr f (u) M(du) By passing to the limit as t → ∞, we get In fact, the above computations need some further explanations.The Green kernel appears through the relation To prove that this log term does not affect the convergence of the integral (3.9), we have to use Corollary 2.21 in [12].Obviously, we are thus left with two options.Either Remark 3.12.It may be worth saying here that the above proof works in dimension 2 (or 1) only.It is based on the recurrence of the standard 2-dimensional Brownian motion.This is one point were techniques related to the Liouville Brownian motion differ according to the dimension.Let us also point out that the integral formula for the Liouville Green function should be convenient to study its regularizing property.For instance, it is almost obvious to see that Gf is continuous when f is bounded.
We can now apply [11,Theorem 4. 2. Assume further that f is locally uniformly bounded.Then, P x -almost surely Remark 3.14.Observe that the above theorem entails uniqueness of the invariant probability measure for the Liouville Brownian motion in the case of the sphere or the torus.With additional efforts, one could also establish in this way uniqueness in the case of the whole plane up to multiplicative constants.
From [11, lemma 4.8.3],we deduce that there exists a positive constant c and a nearly Borel measurable finely closed set B ⊂ D such that M(B) < +∞ and r 1 (x, y) > c for all x, y ∈ B. Recall that r 1 stands for the massive Liouville Green kernel (see Theorem 3.10).Then [11, for some constant C and all functions f ∈ F (i.e. in the domain of the Liouville Dirichlet form).

Liouville heat kernel
In this subsection, we investigate the existence of probability densities of the Liouville semi-group with respect to the Liouville measure.
Theorem 3.16.Liouville heat kernel.For γ ∈ [0, 2[, the Liouville semigroup (P X t ) t>0 is absolutely continuous with respect to the Liouville measure.There exists a family (p X t (•, •)) t 0 , called the Liouville heat kernel, of jointly measurable functions such that: and such that: 1. (positivity) for all t > 0 and for all x ∈ D, for M(dy)-almost every y ∈ D, p X t (x, y) 0, 2. (symmetry) for all t > 0 and for every x, y ∈ D: Let us stress that a complete analysis of the Liouville heat kernel is still missing.From Theorem 3.10, we know that for all x ∈ D and M-almost every y ∈ D, p t (x, y) > 0 for some t belonging to a measurable set with positive Lebesgue measure.Yet, one may expect that p t (x, y) > 0 for all t.Furthermore, one may also expect that p t (x, y) is a continuous function of (t, x, y).Finally, another interesting question is to characterize the function p t as the unique minimal solution of the Liouville heat equation, as it is standard in stochastic analysis.We end this section by collecting some consequences of the above analysis on the behavior of B t with respect to the Liouville measure M.

The Liouville Brownian motion spends most of his time
in the thick points of X In this section, it is convenient to make explicit the dependence of γ of the Liouville measure, i.e. we write M γ instead of M. Following Kahane [16], let us introduce the γ-thick points of X: where the series (c n ) n 1 was introduced in [12], i.e.
It is a well known fact that the Borel set K γ gives full mass to the measure M γ , i.e.M γ (K c γ ) = 0: this was proved in Kahane's seminal work [16] using the socalled Peyrière measure.The sets K γ appear also frequently in the general context of multifractal formalism for multifractal measures (log-Poisson, discrete cascades, etc...).The terminology "thick points" is not due to Kahane but appears in [15] for example.This implies that, if γ, γ ′ ∈ [0, 2[ are such that γ = γ ′ , the measures M γ and M γ ′ are singular with respect to each other.In particular, the measure M γ for γ ∈]0, 2[ is singular with respect to the Lebesgue measure (which corresponds to γ = 0).Let us also stress that K γ should be distinguished from the support of M γ which is D.
As a consequence of Theorem 3.10, we obtain the following result where λ is the Lebesgue measure: Corollary 3.17.For γ ∈ [0, 2[, the Liouville Brownian motion spends Lebesguealmost all its time in the γ-thick points of X for all starting points x: a.s. in X, ∀x ∈ D, a.s.under P B x , λ{t 0; B x t ∈ K c γ } = 0.
• A weaker form of this result is proved in [3]: the author proves that and for only one starting point.
• This result may also be recovered from the invariance of the measure M proved in [12], but in that case only for M(dx)-almost all starting points (which is thus also slightly weaker than our Corollary).
If one now relies on Theorem 3.16 instead, one obtains the following Corollary (note that it different from the above one, not stronger, nor weaker).
Corollary 3.19.For γ ∈ [0, 2[, almost surely in X, for all t > 0 P B x a.s., B x t ∈ K γ .Observe that the above corollary was already known when the initial law of the Liouville Brownian motion is the Liouville measure [12].Replacing the starting law by the Dirac mass at x (a.s. in X for all x) is a much stronger statement.As a strongly local regular Dirichlet form, (Σ, F ) can be written as where Γ is a positive semidefinite, symmetric bilinear form on F with values in the signed Radon measures on R 2 (the so-called energy measure).Denoting by P t (x, dy) the transition probabilities of the semi-group, the energy measure can be defined by the formula for every f ∈ F ∩ L ∞ (D, M) and every φ ∈ F ∩ C c (D).The energy measure is local, satisifes the Leibniz rule as well as the chain rule [11].Let us denote by F loc = {f ∈ L 2 loc (D, M); Γ(f, f ) is a Radon measure}.The energy measure defines in an intrinsic way a pseudo metric d X on D by [4,5,7,28].The condition Γ(f, f ) M means that the energy measure Γ(f, f ) is absolutely continuous w.r.t to M with Radon-Nikodym derivative d dM Γ(f, f ) 1.In general, d X may be degenerate d X (x, y) = 0 or d X (x, y) = +∞ for some x = y.

Why it vanishes in the setting of Liouville quantum gravity
Here we provide a rigorous proof in the next subsection followed by a more heuristical explanation by considering the intrinsic metric associated to the n-regularized Dirichlet forms (Σ n , F n ) obtained by using X n .Proof.For f ∈ F loc ∩ C(D), the energy measure characterized by (see [11, (3.2.14) and Th.6.2.1]):

A proof that the intrinsic metric vanishes
It is worth mentioning here that the above formula implicitly implies that the energy measure of f ∈ L 2 (D, M) does not depend on the choice of the element f ∈ H 1 loc (D, dx) such that f = f M-almost everywhere.Routine computations on differentiation then entails that Since M and the Lebesgue measure are singular with respect to each other (see above), the condition Γ(f, f ) M entails that ∇f = 0.In particular, f is constant.

A heuristical justification by looking at the n-regularized forms
It is tempting to write in a loose sense that Now, it is easy to check that the intrinsic metric d n associated to the Dirichlet form with domain is exactly the Riemannian metric with metric tensor given by We have the following result, which is in some sense folklore within the community but to our knowledge is not written done anywhere: converges towards the trivial metric, meaning that for all x, y ∈ D, a.s. in X: Proof.By definition, Obviously, this distance is bounded from above by the weight of the segment joining x to y, i.e.

Conclusion
Roughly speaking, one may say that the geometric aspect of Dirichlet forms at the level of constructing a metric is not as powerful as one might hope looking at its degree of generality.If the machinery seems to be efficient when the underlying space is not too far from a smooth Riemannian geometry, it does overcome the issue of renormalization.The reader may object that it is not clear that the Liouville metric exists and this could be an explanation to the fact that the intrinsic metric of Dirichlet forms vanishes: we stress that this objection is not relevant since it does not even work in dimension 1 though the metric is perfectly explicit.
A Index of notations

B Reinforced Kolmogorov's continuity criterion
In this section, we prove the following result: Theorem B.1.Assume that (f n ) n is a sequence of random functions defined on the same probability space (Ω, F , P) such that for some q, β > 0 and for all x, y ∈ B(0, R) ⊂ R d : For all α ∈]0, β q [, we can find a modification of f n for each n (still denoted by f n ) and a random constant C, which is P-almost surely finite such that: ∀n, ∀x, y ∈ B(0, R), |f n (x) − f n (y)| C|x − y| α .
Proof.For simplicity, we carry out the proof in dimension 1 and we assume that x, y belong to the set [0, 1].Let us consider α ∈]0, β/q[.We get: For each n, the mapping t → f n (t, ω) is therefore α-Hölder on D ∩ [0, 1] so that it can be extended to the whole [0, 1] while remaining α-Hölder with the same Hölder constant.Since f n is continuous in probability for each n, this extension is a modification of f n for all n.

Definition 2 . 1 (
Capacity and polar set).The capacity of an open set O ⊂ D is defined by

Theorem 3 .
11. (Liouville Green function) The Liouville semi-group is irreducible and recurrent.The Liouville Green function, denoted by G X D , is given by

R 2 f 2 f
(u) M(du) = 0 and f = 0 M almost surely, which entails Gf = 0 M-almost surely, or R (u) M(du) > 0 leading to Gf (x) = +∞.Put in other words, the Liouville semi-group is recurrent.The exact expression of the Liouville Green function results from the above considerations.

1 t t 0 f
7.3] to get Theorem 3.13.Let us denote by P M the law of the Liouville Brownian motion with initial distribution M. Let f ∈ L 1 (D, M) be a Borel measurable function.1.It holds P M -almost surely that lim t→∞ (B r ) dr = 1 M(D) D f (x) M(dx).
theorem 4.8.2]yields Theorem 3.15.(Poincaré type inequality) There exists a strictly positive bounded function g ∈ L 1 (D, M) such that g 1 B and:

D
. (semigroup property) for all s, t 0, for all x, y ∈ D, p X t+s (x, y) = D p X t (x, z)p X s (z, y) M(dz).Proof.All the properties result from Theorem 3.10 and [11, Theorems 4.1.2and 4.2.4].

4
Degenerescence of the intrinsic metric associated to the Liouville Dirichlet form 4.1 Background on the geometric theory of Dirichlet forms and extension of Riemannian geometry

2 n
(s)] ds where ds stands for the standard arc length on D. The arc length restricted to the segment [x, y] is a Radon measure in the class R + 1 of [16].Therefore, for γ ∈ [0, 2[, the limit C x,y = lim n→∞ [x,y]

• X :
Gaussian Free Field, • M (or M γ ): Liouville measure, • (B x t ) t : a standard Brownian motion starting from x, • (Σ, F ): Dirichlet-form, • (R X λ ) λ 0 : Liouville resolvent operator, • B b (D): space of bounded measurable functions on D, • C(D): space of continuous functions on D, • C b (D): space of bounded continuous functions on D, • C 0 (D): space of continuous functions on D vanishing at infinity, • C c (D): space of continuous functions on D with compact support, • L p (D, µ): Borel measurable functions on D with µ-integrable p-th power, • H 1 (D, dx): standard Sobolev space, • H 1 loc (D, dx): functions which are locally in H 1 (D, dx).