Uniqueness of critical Gaussian chaos *

We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.


Introduction
The theory of multiplicative chaos was created by Kahane [20,21] in the 1980's in order to obtain a continuous counterpart of the multiplicative cascades, which were proposed by Mandelbrot in early 1970's as a model for turbulence. During the last 10 years there has been a new wave of interest on multiplicative chaos, due to e.g. its important connections to Stochastic Loewner Evolution [3,29,15], quantum field theories and quantum gravity [18,13,14,24,6,23], models in finance and turbulence [25,Section 5], and the statistical behaviour of the Riemann zeta function over the critical line [16,27].
In Kahane's original theory one considers a sequence of a.s. continuous and centered Gaussian fields X n that can be thought of as approximations of a (possibly distribution valued) Gaussian field X. The fields are defined on some metric measure space (T , λ) appears as a critical value.
In order to give a more concrete view of the chaos we take a closer look at a particularly important example of approximating Gaussian fields in the case where d = 1 and µ is the so-called exactly scale invariant chaos due to Bacry and Muzy [4], [25, p. 331]. Consider the hyperbolic white noise W in the upper half plane R 2 + so that E W (A 1 )W (A 2 ) = m hyp (A 1 ∩ A 2 ) for Borel subsets A 1 , A 2 ∈ R 2 + with compact closure in R 2 + . Above dm hyp = y −2 dx dy denotes the hyperbolic measure in the upper half plane. For every t > 0 consider the set A t (x) := {(x , y ) ∈ R 2 + : y ≥ max(e −t , 2|x − x|) and |x − x| ≤ 1/2} (1.2) and define the field X t on [0, 1] by setting X t (x) := √ 2dW (A t (x)).
Note that the sets A t (x) are horizontal translations of the set A t (0). One then defines the subcritical exactly scale invariant chaos by setting If β = 1, the above limit equals the zero measure almost surely. To construct the exactly scaling chaos measure at criticality β = 1, one has to perform a non-trivial normalization as follows: (1.4) where the limit now exists in probability. The need of a nontrivial normalisation at the critical parameter value in (1.4) has been observed in many analogous situations before, e.g. [8,33]. A convergence result EJP 22 (2017), paper 11. analogous to (1.4) was proven by Aidekon and Shi in the important work [2] in the case of Mandelbrot chaos measures that can be thought of as a discrete analogue of continuous chaos. Independently C. Webb [31] obtained the corresponding result (with convergence in distribution) for the Gaussian cascades ( [2] and [31] considered the total mass, but the convergence of the measures can then be verified without too much work). Finally, Duplantier, Rhodes, Sheffield and Vargas [10,12] established (1.4) for a class of continuous Gaussian chaos measures including the exactly scaling one. We refer to [25,11] for a much more thorough discussion of chaos measures and their applications, as well as for further references on the topic.
An important issue is to understand when the obtained chaos measure is independent of the choice of the approximating fields X n . As mentioned before, Kahane's seminal work contained some results in this direction. Robert and Vargas [26] addressed the uniqueness question in the case of subcritical log-correlated fields (1.1) for convolution approximations X n = φ εn * X. Duplantier's and Sheffield's paper [14] gives uniqueness results for particular approximations of the 2-dimensional GFF. More general results developing the method of [26] are contained in the review [25] due to Rhodes and Vargas, whose conditions are very similar to ours. In [9,19] 1 the method is also applied for a class of convolution approximations of the critical chaos. Another approach is contained in the paper of Shamov [28]. The techniques of the latter paper are based on an interesting new characterisation of chaos measures, which produces strong results but is applicable only in the subcritical range. Finally, in the paper [5] Berestycki provides an elegant and simple treatment of convolution approximations, again in the subcritical regime.
In the present paper we develop a new approach to the uniqueness question, which gives a simple proof of uniqueness in the subcritical regime, but more importantly it also applies to the case of critical chaos. Our idea uses a specifically tailored auxiliary field added to the original field in order to obtain comparability directly from Kahane's convexity inequality, and the choice is made so that in the limit the effect of the auxiliary field vanishes. The approach is outlined before the actual proof in the beginning of Section 3. One obtains a unified result that applies in general to chaos measures obtained via an arbitrary normalization, the only requirement is that the chaos measure is nonatomic almost surely. Therefore, our results apply also to a class of chaos measures that lie between the critical and supercritical ones, which one expects to be useful in the study of finer properties of the critical chaos itself.
Our basic result considers the following situation: Let (X n ) and ( X n ) be two sequences of Hölder-regular Gaussian fields (see Section 2 for the precise definition) on a compact doubling metric space (T , d). Assume that for each n ≥ 1 we have a non-negative Radon reference measure ρ n defined on T . Define the measures dµ n (x) := e Xn(x)− 1 2 E [Xn(x) 2 ] dρ n (x) for all n ≥ 1. The measures µ n are defined analogously by using the fields X n instead.
Theorem 1.1. Let C n (x, y) and C n (x, y) be the covariance functions of the fields X n and X n respectively. Assume that the random measures µ n converge in distribution to an almost surely non-atomic random measure µ on T . Moreover, assume that the covariances C n and C n satisfy the following two conditions: There exists a constant K > 0 such that sup x,y∈T |C n (x, y) − C n (x, y)| ≤ K < ∞ for all n ≥ 1, (1.5) and lim n→∞ sup d(x,y)>δ |C n (x, y) − C n (x, y)| = 0 for every δ > 0.
(1.6) 1 We would like to thank the anonymous referee for pointing out the latter article.

Uniqueness of critical Gaussian chaos
Then the measures µ n converge in distribution to the same random measure µ.

Remark 1.2.
For simplicity we have stated the above theorem and will give the proof in the setting of a compact space T . Similar results are obtained for non-compact T by standard localization. For example assume that T has an exhaustion T = ∞ n=1 K n with compacts K 1 ⊂ K 2 ⊂ · · · ⊂ T , such that every compact K ⊂ T is eventually contained in some K n . Then if the assumptions of Theorem 1.1 are valid for the restrictions to each K n , the claim also holds for T , where now weak convergence is defined using compactly supported test functions.
The proof of the above theorem is contained in Section 3, where it is also noted that one may somewhat loosen the condition (1.5), see Remark 3.6. We refer to Section 2 for precise definitions of convergence in the space of measures and other needed prerequisities.
Section 4 addresses the interesting question when the convergence in Theorem 1.1 can be lifted to convergence in probability (or in L p ). Theorem 4.4 below provides practical conditions for checking this when the convergence is known for some other approximation sequence that has a martingale structure -a condition which is often met in applications.
In Section 5 we discuss consequences for convolution approximations (see Corollaries 5.2 and 5.4). In addition to general results we consider both circular averages and convolution approximations of the Gaussian free field in dimension 2 (Corollary 5.8).
Finally, Section 6 illustrates the use of the results of the previous sections. This is done via taking a closer look at the fundamental critical chaos on the unit circle, obtained from the GFF defined via the Fourier series where the A n , B n are independent standard Gaussians. In [3] the corresponding subcritical Gaussian chaos was constructed using martingale approximates defined via the periodic hyperbolic white noise. We shall consider four different approximations of X: 1. X 1,n is the approximation of X obtained by cutting the periodic hyperbolic white noise construction of X on the level 1/n.
3. X 3,n = φ 1/n * X, where φ is a mollifier function defined on T that satisfies some weak conditions.
4. X 4,n is obtained as the nth partial sum of a vaguelet decomposition of X.
converge as n → ∞ in probability to the same nontrivial random measure µ 1,S 1 on T , which is the fundamental critical measure on T . The convergence actually takes place in L p (Ω) for every 0 < p < 1. The same holds for the vaguelet decomposition X 4,n with the normalization √ n log 2 instead of √ log n.
We refer to Section 6 for the precise definitions of the approximations used above. Theorem 1.3 naturally holds true in the subcritical case if above X j,n is replaced by βX j,n with β ∈ (0, 1), and one removes the factor √ log n. We denote the limit measure by µ β,S 1 . A metric space is doubling if there exists a constant M > 0 such that any ball of radius ε > 0 can be covered with at most M balls of radius ε/2. In this work we shall always consider a doubling compact metric space (T , d). We denote by M + the space of (positive) Radon measures on T . The space M of real-valued Radon measures on T can be given the weak * -topology by interpreting it as the dual of C(T ). We then give M + ⊂ M the subspace topology.
The space M + is metrizable (which is not usually the case for the full space M), for example by using the Kantorovich-Rubinstein metric defined by For a proof see [7,Theorem 8.3.2].
Let P(M + ) denote the space of Radon probability measures on M + . One should note that Borel probability measures and Radon probability measures coincide in this situation, as well as in the case of P(T ), since we are dealing with Polish spaces. Let (Ω, F, P) be a fixed probability space. We call a measurable map µ : Ω → M + a random measure on T . For a given random measure µ the push-forward measure µ * P ∈ P(M + ) is called the distribution of µ and we say that a family of random measures µ n converges in distribution if the measures µ n * P converge weakly in P(M + ) (i.e. when evaluated against bounded continuous functions P(M + ) → R). In order to check the convergence in distribution, it is enough to verify that converges in distribution for every f ∈ C(T ), see e.g. [22,Theorem 16.16].
A stronger form of convergence is the following: We say that a sequence of random measures (µ n ) converges weakly in L p to a random measure µ if for all f ∈ C(T ) the random variable´f (x) dµ n (x) converges in L p (Ω) to´f (x) dµ(x). This obviously implies the convergence µ n → µ in distribution. A (pointwise defined) Gaussian field X on T is a random process indexed by T such that (X(t 1 ), . . . , X(t n )) is a multivariate Gaussian random variable for every t 1 , . . . , t n ∈ T , n ≥ 1. We will assume that all of our Gaussian fields are centered unless otherwise stated. Definition 2.1. A (centered) Gaussian field X on a compact metric space T is Hölderregular if the map (x, y) → E |X(x) − X(y)| 2 is α-Hölder continuous on T × T for some α > 0.
for all f ∈ C(T ). In the case where the measures µ n converge in distribution to a random measure µ : Ω → M + , we call µ a Gaussian multiplicative chaos (GMC) associated with the families X n and ρ n . We call the sequence of measures ρ n a normalizing sequence. In the standard models of subcritical and critical chaos the typical choices are ρ n := λ and ρ n := C √ nλ (or ρ n := C √ log nλ), respectively, where λ stands for the Lebesgue measure. Unless otherwise stated, when comparing the limits of two sequences of random measures (µ n ) and ( µ n ), we will always use the same normalizing sequence (ρ n ) to construct both µ n and µ n .
Lastly we recall the following fundamental convexity inequality due to Kahane [20].

Lemma 2.4.
Assume that X and Y are two Hölder-regular fields such that the covariances satisfy C X (s, t) ≥ C Y (s, t) for all s, t ∈ T . Then for every concave function f : for all ρ ∈ M + .

Convergence and uniqueness: Proof of Theorem 1.1
In this section we prove Theorem 1.1. The simple idea of the proof is as follows: We construct a sequence of auxiliary fields Y ε (see especially Lemma 3.5) that we add on top of the fields X n in order to ensure that the covariance of X n + Y ε dominates the covariance of X n pointwise. The fields Y ε become fully decorrelated as ε → 0, and their construction relies on the non-atomicity of the random measure µ. After these preparations one may finish by a rather standard application of Kahane's convexity inequality (Lemma 2.4). The next two lemmata are almost folklore, but we provide proofs for completeness. Proof. First of all, by the definition of tightness one may easily pick an increasing ∞) . One may choose a concave function h that is majorized by g and satisfies both h(0) = 0 and lim x→∞ h(x) = ∞. Finally, set h(x) := ( h(x)) 1/4 . Condition (3) follows, and (2) is then automatically satisfied by concavity. Since compositions of non-negative concave functions remain concave we obtain (1) as well.
Lemma 3.2. For n ≥ 1 let X n and X n be Hölder-regular Gaussian fields on T with covariance functions C n (x, y) and C n (x, y). Define the random measures µ n and µ n using the fields X n and X n , respectively. Assume that there exists a constant K > 0 such that sup x,y∈T ( C n (x, y) − C n (x, y)) ≤ K < ∞ for all n ≥ 1 and that the family ( µ n ) is tight (in P(M + )). Then also the family (µ n ) is tight.
Proof. By the Banach-Alaoglu theorem it is enough to check that Since lim u→∞ h(u) = ∞, it suffices to verify that sup n≥1 E h(µ(T )) < ∞, where h is the concave function given by Lemma 3.1 for the tight sequence µ n . Pick an independent standard Gaussian G. By our assumption the covariance of the field X n := X n + K 1/2 G dominates that of the field X n , and if the random measure µ n is defined by using the field X n , we obtain by Kahane's concavity inequality E (h(µ n (T ))) 2 ≤ E (h( µ n (T ))) 2 ≤ c for any n ≥ 1 for some constant c > 0 not depending on n.
Since µ n = e K 1/2 G−K/2 µ n the properties (2) and (3) of Lemma 3.1 enable us to estimate for all n ≥ 1 that for some c > 0.
Our proof of Theorem 1.1 is based on the following two lemmas.

Lemma 3.3.
Let (X n ) and ( X n ) be two sequences of Hölder-regular Gaussian fields on T . Assume that there exists a constant K > 0 such that the covariances satisfy sup x,y∈T | C n (x, y) − C n (x, y)| ≤ K < ∞ for all n ≥ 1. Assume also that both of the corresponding sequences of random measures (µ n ) and ( µ n ) converge in distribution to measures µ and µ respectively, and that µ is almost surely non-atomic. Then also µ is almost surely non-atomic.
Proof. Let G be an independent centered Gaussian random variable with variance E G 2 = K. Then the covariance of the field X n + G dominates that of the field X n . Define a field U n (x, y) := X n (x) + X n (y) + 2G on the product space T × T . Its covariance is given by and therefore dominates the covariance of the field V n (x, y) := X n (x) + X n (y) given by and let h be as in Lemma 3.1. Then by Kahane's convexity inequality applied to the fields U n and V n w.r.t. the measure ρ n on the product space T × T we have Above we applied Lemma 3.1 (2) twice. By letting n → ∞ we obtain ) is a constant that only depends on K. Letting ε → 0 lets us conclude that (µ ⊗ µ)(∆) = 0 almost surely, which entails that µ is non-atomic almost surely.

Remark 3.4.
One should note that the above proof is not valid as such if one just assumes that the dominance of the covariance is valid in one direction only. In a sense we perform both a convexity and a concavity argument while deriving the required inequality. We do not know whether this is a limitation of our proof, or whether there exists an example where one-sided bound is not enough.

Lemma 3.5.
Assume that the conditions of Theorem 1.1 hold. Then there exists a collection Y ε (0 < ε < 1) of Hölder-regular Gaussian fields on T such that for a fixed 0 < ε < 1 the covariance of the field X n + Y ε is pointwise larger than the covariance of the field X n for all large enough n. Moreover, there exists a constant C = C(K) depending only on the constant K appearing in (1.5) such that for any λ ∈ M + and ε ∈ (0, 1).
Proof. Fix a sequence of independent standard Gaussian random variables A i , i ≥ 1, such that they are also independent of the fields X n . Let ε > 0 and choose a maximal set of points a 1 , . . . , a n in T such that |a i − a j | ≥ ε/2 for all 1 ≤ i < j ≤ n. Let B i be the ball B(a i , ε). Then the balls B i cover T and we may form a Lipschitz partition of unity p 1 , . . . , p n with respect to these balls. That is, p 1 , . . . , p n are non-negative Lipschitz continuous functions such that whence the covariance of Z ε is given by We may now define the field Y ε (x) = εG + √ KZ ε (x) where G is a standard Gaussian random variable independent of the fields Z ε and X n . The conditions (1.5) and (1.6) together with compactness yield that for all large enough n the covariance of the field X n + Y ε is greater than the covariance of the field X n at every point (x, y) ∈ T × T .
Proof of Theorem 1.1. We will first assume that both sequences (µ n ) and ( µ n ) converge in distribution and show how to get rid of this condition at the end.
Let Y ε be the independent field constructed as in Lemma 3.5. We may assume, towards notational simplicity, that our probability space has the product form Ω = Ω 1 ×Ω 2 , and for (ω 1 , ω 2 ) ∈ Ω one has X n ((ω 1 , ω 2 )) = X n (ω 1 ) and X n ((ω 1 , EJP 22 (2017), paper 11. as n → ∞. In particular we have by Fatou's lemma that Accoding to Lemma 3.5, for almost every ω 1 ∈ Ω 1 we know that in L 2 (Ω 2 ). We next note that for a suitable fixed sequence ε k → 0 this convergence also happens for almost every ω 2 ∈ Ω 2 . By Lemma 3.5 we have the estimate . Now a standard argument verifies the almost sure convergence in The almost sure convergence finally lets us to conclude for all non-negative f ∈ C(T ) and non-negative, bounded, continuous and concave ϕ that Similar inequality also holds with the measures µ and µ switched, so we actually have It is well known that this implies µ ∼ µ.
Let us now finally observe that one can drop the assumption that both families of measures converge. By Lemma 3.2 and Prokhorov's theorem we know that every subsequence µ n k has a further subsequence that converges in distribution to a random measure. Lemma 3.3 ensures that the limit measure of any converging sequence has almost surely no atoms, and hence by the previous part of the proof this limit must equal µ. This implies that the original sequence must converge to µ as well.
Remark 3.6. Our proof of Theorem 1.1 may be modified in a way that allows the conditions (1.5) and (1.6) to be somewhat relaxed. E.g. in the case of subcritical logarithmically correlated fields it is basically enough to have for ε > 0 the inequality for n ≥ n(ε). Analogous results exist also for the critical chaos, but in this case the specific conditions are heavily influenced by the approximation sequence X n one uses. In the previous section convergence was established in distribution, which often suffices, and the main focus was on the uniqueness of the limit. In the present section we establish the convergence also in probability, assuming that this is true for the comparison sequence µ n , which is constructed using approximating sequence ( X n ) that has independent increments. Convergence in probability in the subcritical case was also discussed in [28], and our Theorem 4.4 below can be seen as an alternative way to approach the question.
Here is an outline of our method: We assume that the sequence µ n is defined using linear approximations R n X of the field X (see Definition 4.3), and invoke Lemma 4.1 to prove the convergence in probability by showing that if g is any (random) function that depends only on X 1 , . . . , X k for some fixed k ≥ 1, then we have the convergence in distribution g dµ n → g d µ. To establish the latter convergence, we split the measure µ n where E k,n is a σ(X 1 , . . . , X k )-measurable error resulting from the approximation that goes to 0 as n → ∞. By applying Lemma 4.2 we then conclude that g dµ n converges to where ν k is a random measure independent of X 1 , . . . , X k . Finally, by using the convergence in probability of µ n we can write µ = e X k − 1 2 E [X 2 k ] dη k for a random measure η k , also independent of X 1 , . . . , X k , and Lemma 4.2 tells us that ν k and η k have the same distribution. This lets us conclude that ge Enough speculation, it is time to work.
Assume that the real random variables X, X 1 , X 2 , . . . satisfy: X and X k are F ∞ -measurable, and for any F j measurable set E (with arbitrary j ≥ 1) it holds that Proof. We first verify that (4.1) remains true also if the set E is just F ∞ -measurable. For that end define h j := E (χ E |F j ) and construct an F ∞ -measurable approximation E j := h −1 j ((1/2, 1]). The martingale convergence theorem yields that P(E j ∆E) → 0 as j → ∞. Since the claim holds for each E j , it also follows for the set E by a standard approximation argument.
Let us then establish the stated convergence in probability. Fix ε > 0 and pick M > 0 large enough so that P(|X| > M/2) ≤ ε/2, and such that P(|X| = M ) = 0. Then for overlapping half open intervals I 1 , . . . , I of length less than ε/2 and denote E j := X −1 (I j ) for j = 1, . . . , . In the construction we may assume that 0 is the center point of one of these intervals and P(X = a) = 0 if a is an endpoint of any of the intervals. We fix j and Then the Portmonteau theorem yields that lim k→∞ P(χ Ej X k ∈ I j ) = P(χ Ej X ∈ I j ), or in other words In particular, for large enough k we have that If 0 ∈ I j we obtain in a similar vein that lim k→∞ P(χ Ej X k ∈ (I j ) c ) = P(χ Ej X ∈ (I j ) c ) = 0, or in other words P({X ∈ I j } ∩ {X k ∈ I c j }) → 0, so that we again get that P E j ∩ (|X − EJP 22 (2017), paper 11. X k | > ε ≤ ε 2 for large enough k. By summing the obtained inequalities for j = 1, . . . , and observing that P( k=1 E k ) > 1 − ε/2 we deduce for large enough k the inequality P(|X − X k | > ε) < ε, as desired.
Lemma 4.2. Let X be a Hölder-regular Gaussian field on T that is independent of the random measures µ and ν on T .
(ii) If (µ n ) is a sequence of random measures such that the sequence (e X µ n ) converges in distribution, then also the sequence (µ n ) converges in distribution.
Proof. We will first show that if X is of the simple form N f with N a standard Gaussian random variable and f ∈ C(T ), then the claim holds. To this end let us fix g ∈ C(T ) and consider the function ϕ : R → C defined by Because N is independent of µ and ν, we may write Because the Fourier transform of h is also Gaussian we deduce by taking convolutions that the Fourier transforms u and v coincide as Schwartz distributions. Since u and v are continuous, this implies that u(x) = v(x) for all x. In particular setting x = 0 gives us for all g ∈ C(T ), whence the measures µ and ν have the same distribution.
To deduce the general case, note that we have the Karhunen-Loève decomposition for all g ∈ C(T ), which shows the claim.
The second part of the lemma follows from the first part. Since sup t∈T X(t) < ∞ almost surely, one checks that the sequence (µ n ) inherits the tightness of the sequence (e X µ n ). It is therefore enough to show that any two converging subsequences have the same limit. Indeed, assume that µ kj → µ and µ nj → ν in distribution. Then by independence we have e X µ kj → e X µ and e X µ nj → e X ν, but by assumption the limits are equally distributed and hence also µ and ν have the same distribution. A typical example of a linear regularization process described in the following definition is given by a standard convolution approximation sequence. We denote by C α (T ) the Banach space of α-Hölder continuous functions on T . Definition 4.3. Let (X k ) be a sequence of approximating fields on T . We say that a sequence (R n ) of linear operators R n : α∈(0,1) C α (T ) → C(T ) is a linear regularization process for the sequence (X k ) if the following properties are satisfied: 1. We have lim n→∞ R n f − f ∞ = 0 for all f ∈ α∈(0,1) C α (T ).
2. The limit R n X := lim k→∞ R n X k exists in C(T ) almost surely.  Let R n be some linear regularization process for the sequence X k such that Then also dµ n = e RnX− 1 2 E [(RnX) 2 ] dρ n converges to µ in probability. Proof. Define the filtration F n := σ(X 1 , . . . , X n ). First of all, since e Xn− 1 n ] dρ n converges to µ in probability as n → ∞, we also have To see this, one uses that E [(X n − X k ) 2 ] = E [X 2 n ] − E [X 2 k ] and considers almost surely converging subsequences, if necessary. We denote η k := e −X k + 1 by the independent increments and the definition of R n X. We may thus write Above on the right hand side the term in brackets is negligible as n → ∞. To see this, we note first that e RnX k −X k tends almost surely to the constant function 1 uniformly according to Definition 4.3 (1). Moreover, E [X 2 k − (R n X k ) 2 ] tends to 0 in C(T ), since the field X k takes values in a fixed C γ (T ) for some γ > 0, and by the Banach-Steinhaus theorem sup n≥1 R n C γ (T )→C(T ) < ∞. Namely, whence the dominated convergence theorem applies, since X k C γ (T ) has a super exponential tail by Fernique's theorem. All in all, invoking the assumption on the convergence of µ n we deduce that in distribution as n → ∞.
where the limit ν k may be assumed to be independent of F k . In particular, recalling (4.5) we deduce that e X k − 1 k ] η k . Lemma 4.2 now verifies that ν k ∼ η k . In order to invoke Lemma 4.1, fix any F k measurable bounded random variable g. Then g and X k are independent of X − X k , and we therefore have the distributional convergence where the second last equality followed by independence. Finally, again by the negli- and using (4.4) we see that (4.6) in fact entails the convergence of g dµ n to g d µ in distribution. At this stage Lemma 4.1 applies and the desired claim follows.

Remark 4.6.
In the previous theorem it was crucial that we already have an approximating sequence of fields along which the corresponding chaos converges in probability.
In general if one only assumes convergence in distribution in (4.3), one may not automatically expect that it is possible to lift the convergence to that in probability, even for natural approximating fields. However, for most of the standard constructions of subcritical chaos this problem does not occur, as we have even almost sure convergence in (4.3) due to the martingale convergence theorem.

Convolution approximations
In this section we provide a couple of useful results for dealing with convolution approximations, -scale invariant fields and circular averages of 2-dimensional Gaussian fields. We also note that the results can be applied to a 2-dimensional Gaussian free field in a domain. The next lemma and its corollaries show that any two convolution approximations (with some regularity) applied to log-normal chaos stay close to each other in the sense of Theorem 1.1.
Proof. One can use the mean zero property and decay of ϕ − ψ together with a standard BMO-type estimate [17, Proposition 7.1.5.] to see that for any ε > 0 we have ˆR EJP 22 (2017), paper 11. Corollary 5.2. Let f (x, y) = 2dβ 2 log + 1 |x−y| + g(x, y) be a covariance kernel of a distribution valued field X defined on R d . Here g is a bounded uniformly continuous function. Assume that ϕ and ψ are two locally Hölder continuous convolution kernels in R d that satisfy the conditions of Lemma 5.1. Let (ε n ) be a sequence of positive numbers ε n converging to 0. Then the approximating fields X n := ϕ εn * X and X n := ψ εn * X satisfy the conditions (1.5) and (1.6) of Theorem 1.1.

Remark 5.3. One may easily state localized versions of the above corollary.
Corollary 5.4. Assume that f (x, y) = 2β 2 log + 1 2| sin(π(x−y))| + g(x, y) is the covariance of a (distribution valued) field X on the unit circle. Here g is a bounded continuous function that is 1-periodic in both variables x and y and we have identified the unit circle with R/Z. Assume that ϕ and ψ are two locally Hölder continuous convolution kernels in R that satisfy the conditions of Lemma 5.1, and let (ε n ) be a sequence of positive numbers ε n converging to 0. Then the approximating fields X n := ϕ εn * X and X n := ψ εn * X satisfy the conditions (1.5) and (1.6) of Theorem 1.1.

Remark 5.5.
Above when defining the approximating fields X n we assume that X stands for the corresponding periodized field on R and the fields X n will then automatically be periodic so that they also define fields on the unit circle.
Proof. One easily checks that (x) = 2β 2 log + 1 2| sin(πx)| is in BM O(R). The rest of the proof is analogous to the one of the previous corollary.
The previous result showed that different convolution approximations lead to the same chaos. In turn, in order to show that a single convolution approximation converges to the desired chaos, one may often compare the convolution approximation directly to a martingale approximation field used originally to define the chaos. As an example of this, we show that the convolutions of -scale invariant fields are comparable (in the sense of Theorem 1.1) with the natural approximating fields arising from thescale decomposition. This also extends the convergence of the critical chaos in [12] to convolution approximations. Lemma 5.6. Let k : [0, ∞) → R be a compactly supported and positive definite C 1function with k(0) = 1. Define the -scale invariant field X on R d , whose covariance is (formally) given by Moreover, let ϕ be a convolution kernel satisfying the conditions of Corollary 5.2. Then the approximating fields X n := ϕ e −n * X and the fields X n whose covariance is given by E X n (x) X n (y) =ˆe Proof. One may easily check that the covariance in (5.1) is of the form log + 1 |x − y| + g(x, y), and therefore by Corollary 5.2 it is enough to show the claim for one mollifier ϕ. In particular, we may without generality assume that the support of ϕ is contained in B(0, 1/2) and that ϕ is a symmetric non-negative function. A short calculation shows that we have E X n (0)X n (x) =ˆ∞ 1 (ϕ e −n * ϕ e −n * k(u| · |))(|x|) u du.
Let ψ = ϕ * ϕ. Then the support of ψ is contained in B(0, 1) and ψ e −n = ϕ e −n * ϕ e −n . Thus we get We also have the bound |(ψ e −n * k(u| · |))(x)| ≤ k ∞ˆB for some constant C > 0. Using just the upper bounds of these estimates for all x we get ˆen verifying (1.5). Assume then that δ > 0 is fixed and |x| > δ. Then for large enough n we have that e −n < δ/2 and showing (1.6).
Finally, we state a result for circle averages of 2-dimensional Gaussian fields.
Clearly we can assume that g = 0, since that part of the integral is bounded by a constant and converges uniformly. Moreover, we may assume that |x − y| ≤ 1 2 , since the integral converges uniformly to the right value as n → ∞ when |x − y| ≥ 1 2 . Thus we may write for n large enough so that |x + e −n+is − y − e −n+it | < 1. Now if |x − y| > 2e −n , then by invoking the harmonicity of the logarithm and using the mean value principle twice, we have On the other hand if |x − y| ≤ 2e −n , then we may write where the integrand on the right hand side is bounded from below, and boundedness from above of the whole integral follows since the inner integral contains at most a logarithmic singularity, which is integrable. Thus we have shown that This is enough to show the claim, since it is easy to check that certain convolution kernels ϕ yield approximations with similar covariance structure.
We then very briefly note that the above results can be applied to the 2-dimensional Gaussian free field and its variants. We refer to the paper [12] for the definition of the massless free field (MFF) and a Gaussian free field (GFF) in a bounded domain. Proof. The MFF is of the -scale invariant form, so our result applies directly. In the case of a GFF, we may write X as a smooth perturbation of the MFF (see [12]), whence the claim follows easily. Remark 5.9. We note that Theorem 4.4 often applies for convolution approximations. Especially it can be easily localized and it works for the MFF and GFF, including circular average approximations. The verification of the latter fact is not difficult and we omit it here. EJP 22 (2017), paper 11. Remark 5.10. Convergence of convolution approximations for the critical GFF in the unit circle have also been proven in [19]. The method used there is 'interpolation' of Gaussian fields X 1 and X 2 by the formula √ tX 1 + √ 1 − tX 2 , used already in [26]. It is not immediately clear how far beyond convolution approximations this approach can be extended.

An application (Proof of Theorem 1.3)
The main purpose of this chapter is to prove Theorem 1.3 and explain carefully the approximations mentioned there. For the reader's convenience we try to be fairly detailed, although some parts of the material are certainly well-known to the experts.
We start by defining the approximation X 2,n of the restriction of the free field on the unit circle S 1 := {(x 1 , x 2 ) ∈ R 2 : x 2 1 + x 2 2 = 1}. Following [3] recall that the trace of the Gaussian free field on the unit circle (identified with R/Z) is defined to be the Gaussian where A k , B k and G are independent standard Gaussian random variables. The field X is distribution valued and its covariance (more exactly, the kernel of the covariance operator) can be calculated to be E [X(x)X(y)] = 4 log(2) + 2 log 1 2| sin(π(x − y))| .

(6.2)
A natural approximation of X is then obtained by considering the partial sum of the Fourier series Another way to get hold of this covariance is via the periodic upper half-plane white noise expansion that we define next -recall that the non-periodic hyperbolic white noise W and the hyperbolic area measure m hyp were already defined in the introduction. We define the periodic white noise W per to be W per (A) = W (A mod 1), where A mod 1 = {(x mod 1, y) : (x, y) ∈ A} and we define x mod 1 to be the number x ∈ [− 1 2 , 1 2 ) such that x − x is an integer. Now consider cones of the form It was noted in [3] that the field x → √ 2W per (H(x)) has formally the right covariance (6.2), whence a natural sequence of approximation fields (X 1,n ) is obtained by cutting the white noise at the level 1/n. More precisely we define the truncated cones and define the regular field X 1,n by the formula X 1,n (x) := √ 2W per (H log n (x)). The third approximation fields X 3,n are defined by using a Hölder continuous function ϕ ∈ L 1 (R) that satisfies´ϕ = 1 and possesses the decay for some C, δ > 0. We then set X 3,n := ϕ 1/n * X per , where X per (x) = X(x + 2πZ) is the natural lift of X to a map R → R. This form of convolution is fairly general, and encompasses convolutions against functions ϕ defined on the circle whose support do not contain the point (−1, 0). Example 6.1. Let u be the harmonic extension of X in the unit disc and consider the approximating fields X n (x) = u(r n x) for x ∈ S 1 and for an increasing sequence of radii r n tending to 1. Then X n (x) is obtained from X by taking a convolution against the Poisson kernel ϕ εn on the real axis, where ϕ(x) = 2 1+4π 2 x 2 and ε n = log 1 rn . This kind of approximations might be useful for example in studying fields that have been considered in [24].
The fourth approximation fields X 4,n are defined by using a wavelet ψ : R → R, that is obtained from a multiresolutional analysis, see [32,Definition 2.2]. We further assume that ψ is of bounded variation, so that the distributional derivative ψ is a finite measure.
Finally we require the mild decay with some constants C > 0 and α > 2, and the tail condition We next consider vaguelets that can be thought of as half-integrals of wavelets. Our presentation will be rather succinct -another more detailed account can be found in the article by Tecu [30]. The vaguelet ν : R → R is constructed by setting An easy computation utilizing the decay of ψ and the fact that´ψ = 0 verifies that ν : R → R satisfies |ν(x)| ≤ C (1 + |x|) 1+δ (6.8) for some C, δ > 0. We may then define the periodized functions for all j ≥ 0 and 0 ≤ k ≤ 2 j − 1. It is straightforward to check that the Fourier coefficients of ν j,k satisfy ν j,k (n) = ψ j,k (n) |2πn| when n = 0. The field X 4,n can now be defined by A j,k ν j,k (x), (6.10) where G and A j,k are independent standard Gaussian random variables. To see that this indeed has the right covariance one may first notice that defines a distribution valued field satisfying E Y, u Y, v = u, v for all 1-periodic C ∞ functions u and v. The field X 4,n (x) is essentially the half integral of this field, whose covariance is given by where the lift semigroup I β f for functions f on S 1 is defined by describing its action on the Fourier basis: I β e 2πinx = (2π|n|) −β e 2πinx for any n = 0 and I β 1 = 0. A short calculation shows that the operator I has the right integral kernel 1 π log 1 2| sin(π(x−y))| .
Proof of Theorem 1.3. The road map for the proof (as well as for the rest of the section) is as follows: 1. We first show in Lemma 6.4 below that the chaos measures constructed from the white noise approximations converge weakly in L p by comparing it to the exactly scale invariant field on the unit interval by using Proposition A.2.
2. Next we verify in Lemma 6.5 that the Fourier series approximations give the same result as the white noise approximations. This is done by a direct comparison of their covariances to verify the assumptions of Theorem 1.1.
3. Thirdly we deduce in Lemma 6.7 that convolution approximations also yield the same result by comparing a convolution against a Gaussian kernel to the Fourier series and again using Theorem 1.1.
4. Fourthly we prove in Lemma 6.8 that a vaguelet approximation yields the same result by comparing it against the white noise approximation.
5. Finally, in Lemma 6.9 convergence in probability is established for the Fourier series, convolution and vaguelet approximations by invoking Theorem 4.4.
After the steps (1)-(5) the proof of Theorem 1.3 is complete.
The following lemma gives a quantitative estimate that can be used to compare fields defined using the hyperbolic white noise on H. Proof. Let us first show that for some C > 0. By translation invariance of the covariance it is enough to consider E |W (U s + x) − W (U s )| 2 and we can clearly assume that 0 < x < 1. Obviously the 1-dimensional Lebesgue measure of the set ((U s + x) ∩ {y = a})∆(U s ∩ {y = a}) equals 2 min(f (a), x). Hence we have It follows that the map (x, s) → W (U s +x) is Hölder-regular both in x and s, and therefore also jointly. By Lemma 2.2 the realizations can be chosen to be almost surely continuous in the rectangle [a, b] × [0, 1] which obviously yields the claim.
The claim concerning the approximating fields X 1,n follows from the next lemma by taking into account the definitions (6.3) and (6.4). In the proof we identify the field on the unit circle locally as a perturbation of the exactly scaling field on the unit interval. For the chaos corresponding to the last mentioned field the fundamental result on convergence was proven in [12], and we use this fact as the basis of the proof of the following lemma. defined on the unit circle (which we identify with R/Z) converge weakly in L p (Ω) to a non-trivial measure µ β,S 1 for 0 < p < 1.
Proof. As our starting point we know that the measures defined by on the interval [− 1 2 , 1 2 ] converge weakly in L p (Ω) to a non-trivial measure for 0 < p < 1 under the assumptions we have on β and ρ t . Here A t stands for the cone defined in (1.2) in the introduction. One should keep in mind that we are using the same hyperbolic white noise when defining both W and W per .
and similarly for the limit fields (which clearly exist in the sense of distributions) write .
. We next make sure that Z t (x) is a Hölder regular field, the realizations of which converge almost surely uniformly to the Hölder regular We define the truncated versions of L t , R t and T t by cutting the respective sets at the level e −t as usual, so so f (u) ≤ Cu 3 for some constant C > 0. It follows from Lemma 6.3 that L t (x) and R t (x) converge almost surely uniformly to the fields L(x) and R(x), so Z t (x) converges almost surely uniformly to Z(x) as t → ∞.
Note that E [Z t (x)X t (x)] tends to a finite constant as t → ∞, so the assumptions of Proposition A.2 are satisfied. Therefore the measures on (−δ, δ) converge weakly in L p (Ω) for all 0 < p < 1. Because Y + is a regular field, we may again use Proposition A.2 to conclude that also the measures on (−δ, δ) converge in L p (Ω). By the translation invariance of the field the same holds for any interval of length 2δ. Let I 1 , . . . , I n be intervals of length 2δ that cover the unit circle and let p 1 , . . . , p n ∈ C(S 1 ) be a partition of unity with respect to the cover I k . The on the whole unit circle can be expressed as a sum dµ t (x) = p 1 (x)d µ (1) t (x) + · · · + p 2 (x)d µ (n) t (x). Because each of the summands converges in L p (Ω), we see that also the family of measures µ t converges in L p (Ω). .
Because the function h tn without the bounded term −2 log(cos( π 2 x)) is linear and decreasing on the interval [0, x n ] we know that it is actually 2 log(n) + O(1) on that whole interval. Similarly it is easy to check that for the Fourier series we have f n (x) = 2 log(n) From the above considerations and symmetry it follows that the covariances of the fields X 1,n and X 2,n satisfy the assumptions of Theorem 1.1. This finishes the proof. Remark 6.6. The somewhat delicate considerations in the previous proof are necessary because of the fairly unwieldy behaviour of the Dirichlet kernel.
Proof. It is enough to show the assumptions of Theorem 1.1 for one kernel satisfying the conditions of the lemma because of Corollary 5.4, and because of Lemma 6.5 we can do our comparison against the covariance obtained from the Fourier series construction.
We will make the convenient choice of ϕ(x) = 1 √ 2π e − x 2 2 as our kernel. The covariance of the field ϕ ε * X per is given by Because both of the covariances converge locally uniformly outside the diagonal, we again see that the assumptions of Theorem 1.1 are satisfied.
Our next goal is to prove the convergence in distribution for the vaguelet approximation X 4,n . In the lemma below we recall the definition of the field X 4,n in (6.10). The elementary bounds on vaguelets we use are gathered in Appendix B. Proof. The covariance C n (x, y) of the field X 4,n is given by Let ψ j,k be the periodized wavelets. Then there exists a constant D > 0 such that ψ j,k ∞ ≤ D2 j/2 for all j ≥ 0, 0 ≤ k ≤ 2 j − 1. It follows from Lemma B.1 and Lemma B.3 that when |x − y| ≤ 2 −n , we have |C n (x, x) − C n (x, y)| ≤2π n j=0 2 j −1 k=0 |ν j,k (x)||ν j,k (x) − ν j,k (y)| (6.14) ≤2πC |x − y| n j=0 for some constant E > 0. From Lemma B.3 it also follows that for any ε > 0 the covariances C n (x, y) converge uniformly in the set V ε = {(x, y) : dist(x, y) ≥ ε}. Obviously by definition there is a distributional convergence to the right covariance 4 log 2 + 2 log 1 2| sin(π(x−y))| and this must agree with the uniform limit in V ε . Especially, by invoking again the bound from Lemma B.3 we deduce that |C n (x, x + 2 −n ) − 4 log 2 − 2 log 1 2 sin(π2 −n ) | ≤ 2πB. (6.15) Thus by combining (6.14) and (6.15) the covariance satisfies |C n (x, y) − 2n log 2| ≤ F for all (x, y) ∈ {(x, y) : dist(x, y) ≤ 2 −n } for some constant F > 0. From the known behaviour (see e.g. the end of the proof of Lemma 6.5) of the covariance of the white noise field X 1,n it is now easy to see that the assumptions of Theorem 1.1 are satisfied for the pair (X 4,n ) and (X 1,n ).
Finally we observe that the convergence in lemmas 6.5, 6.7 and 6.8 also takes place weakly in L p . Lemma 6.9. The convergences stated in lemmas 6.5, 6.7 and 6.8 take place in L p for 0 < p < 1 (especially in probability).
Proof. We only prove the claim in the critical case since the subcritical case is similar.
We will use the fields X 1,n as the fields X n in Theorem 4.4. Then according to Lemma 6.4 we have that e Xn− 1 n ] dρ n converges in probability to a measure µ 1,S 1 when dρ n = √ log n dx.
In the case of the Fourier approximation we can define R n in Theorem 4.4 to be the nth partial sum of the Fourier series. That is Recalling Jackson's theorem on the uniform convergence of Fourier series of Hölder continuous functions, it is straightforward to check that R n is a linear regularization process.
In the case of convolutions we take R n to be the convolution against 1 εn ϕ( x εn ), where (ε n ) n≥1 is a sequence of positive numbers tending to 0. The sequence (R n ) obviously satisfies the required conditions. Finally, we sketch the proof for the vaguelet approximations. This time we employ the sequence of operators Because of finiteness of the defining series it is easy to see that (R n ) satisfies the second condition in Definition 4.3. For the first condition we first fix α ∈ (0, 1/2) and observe that R n ν j ,k = ν j ,k as soon as n ≥ j . By the density of vaguelets, in order to verify the first condition it is enough to check that the remainder term tends uniformly to 0 for any f ∈ C α (S 1 ). We begin by noting that d dx = −iHI −1 , where H is the Hilbert transform, which yields for f ∈ C α (S 1 ) ˆ1 0 ψ j,k (y) I −1/2 f (y) = ˆ1 0 d dy ψ j,k (y) HI +1/2 f (y) ≤ C2 −αj , x ∈ [0, 1), since HI +1/2 f (x) ∈ C α+1/2 (S 1 ) by the standard mapping properties of I β , and the Hilbert transform is bounded on any of the C α -spaces. Above, the final estimate was obtained by computing for any g ∈ C α+1/2 (S 1 ) with periodic continuation G to R that ˆ1 The last integral is finite by the assumption (6.6). Together with Lemma B.3 this obviously yields the desired uniform convergence. The proofs of the lemmas 6.5, 6.7 and 6.8 show that the covariances stay at a bounded distance from the covariance of the field X 1,n , and therefore a standard application of Kahane's convexity inequality gives us an L p bound. Combining this with Theorem 4.4 yields the result. As noted in the beginning of this section, having proved all the lemmas above we may conclude the proof of Theorem 1.3. Remark 6.10. In the case of vaguelet approximations we may also rewrite where A i and ν i are the random coefficents and vaguelets appearing in (6.10) ordered in their natural order. The convergence and uniqueness then also holds for the chaos constructed from the fields with the normalizing measure dρ n (x) = √ log n dx.
Remark 6.11. There are many interesting questions that we did not touch in this paper.
For example (this question is due to Vincent Vargas), it is natural to ask whether the convergence or uniqueness of the derivative martingale [10] depends on the approximations used.

A Localization
The Proposition A.2 below is needed in a localization procedure in Lemma 6.4 that is used to carry results from the real line to the unit circle. For its proof we need the following lemma.
Lemma A.1. Assume that µ n is a sequence of random measures that converges to µ weakly in L p (Ω). Let F : Ω → C(T ) be a function valued random variable and assume that there exists q > 0 such that E sup x∈T F (x) α < ∞ for some α > pq p−q . Then´F (x) dµ n (x) tends to´F (x) dµ(x) in L q (Ω).
Proof. It is again enough to show that any subsequence possesses a converging subsequence with the right limit. To simplify notation let us denote by µ n an arbitrary subsequence of the original sequence.
Directly from the definition of the metric in the space M + we see that µ n → µ in probability, meaning that we can pick a subsequence µ nj that converges almost surely. Then the almost sure convergence holds also for the sequence´F (x) dµ nj (x). Finally, for any allowed value of q a standard application of Hölder's inequality shows that E |´F (x) dµ nj (x)| q+ε is uniformly bounded for some ε > 0. This yields uniform integrability and we may conclude.
Proposition A.2. Let (X n ) and (Z n ) be two sequences of (jointly Gaussian) Hölderregular Gaussian fields on T . Assume that the pseudometrics arising in Definition 2.1 can be chosen to have the same Hölder exponent and constant for all the fields Z n . Assume further that there exists a Hölder-regular Gaussian field Z such that Z n converges to Z uniformly almost surely and that E [X n (x)Z n (x)] converges uniformly to some By combining this with the bound (A.2) we see that E |ν n (f ) − c n (f )| q → 0 as ε → 0, uniformly in n. Finally, by Lemma A.1 we have c n (f ) → ν(f ) in L q (Ω). This finishes the proof.

B Estimates for vaguelets
In this appendix we have collected a couple of elementary estimates concerning vaguelets, see (6.9) in Section 6 for the definition of ν j,k .
Lemma B.1. Let f : R → R be a bounded integrable function and let |x − t| dt be its half-integral. Then there exists a constant C > 0 (not depending on f ) such that for all x, y ∈ R we have |F (x) − F (y)| ≤ C f ∞ |x − y|. We can without loss of generality assume that x < y and split the domain of integration to the intervals (−∞, x], [x, x+y 2 ], [ x+y 2 , y] and [y, ∞). On each of the intervals the value of the integral is easily estimated to be less than some constant times |x − y|, which gives the result.
Proof. Without loss of generality we may assume that 0 ≤ x < 1 and let d = dist(x, 0).
Proof. By using Lemma B.2 and the fact that ν j,k (x) = ν j,0 (x − k2 −j ) we have which shows the first claim.