Elliptic Stochastic Quantization of Sinh-Gordon QFT

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1 Introduction

One of the first steps in Euclidean Quantum Field Theory (EQFT) is the (more or less explicit) construction of quantum models satisfying one of the equivalent formulations of Osterwalder–Schrader axioms (see, e.g, [28, 61]). In the case of scalar bosonic quantum fields, this problem is equivalent to the definition d of a probability measures ν on the space of tempered distributions ′(R ) formally given by S 1 dν = exp λ V (ϕ(x))dx µ(dϕ), (1) Z − Rd Z where µ is the Gaussian Free Field with mass m > 0, which is the Gaussian measure with covariance

2 1 f, ϕ 2 R2 g, ϕ 2 R2 dµ = f, (m ∆)− g , h iL ( )h iL ( ) h − i Z V : R R is a regular function describing the interaction of the field ϕ, and Z R+ is a normalization constant.→ Important examples, of measures of the form (1), which have been extensively∈ studied in the literature include, are: • 4 4 ϕ2,3 models, where V (ϕ)= ϕ , and d =2, 3 (see, e.g. [18, 20, 46, 57]), • Sine-Gordon model, where V (ϕ) = cos(βϕ), d = 2 and β2 < 8π (see, e.g., [5, 45]), • Høegh-Krohn models, where √8π V (ϕ)= exp(βϕˆ )η(dβˆ), (2) √8π Z− where η is a positive measure supported in √8π, √8π , and d = 2 (see, e.g. [4]). − Since the Gaussian measure µ, in the formal equation (1), is supported on generic distributions of negative regularity, it is not clear how to define the composition of the field ϕ with the nonlinear function V (ϕ). In order to solve this problem we need to exploit renormalization techniques: We approximate the measure µ by a sequence more regular probability law µε (depending on some real parameter ε> 0) such that µε µ as ε 0. We also replace the potential V by a sequence of functions V , depending on the regularization→ → ε µε chosen, for which the limit of the sequence Vε(ϕε) (as ε 0) well defined and nontrivial. A second problem that we face in the definition of measure (1), is that→ since µ is translation-invariant, and thus one cannot expect that the integral λ V (ϕ(x))dx (3) Rd Z to converge when integrated over the whole Rd, even if one is able able to give a precise meaning to the composition V (ϕ). To deal with this difficulty, one can approximate (3) by

λ ρ(x)V (ϕ(x))dx, Rd Z where ρ : Rd [0, 1] is a smooth function decaying sufficiently quickly at infinity, and then send ρ 1. To summarize→ to construct a rigorously meaningful version of the measure (1) we have to consider→ the weak limit 1 lim exp λ ρVε(ϕ) dµε. (4) ρ 1,ε 0 Z − R2 → → ρ,ε Z Many different methods have been used to prove the existence and nontriviality of the limit (4). Many of them analyze the measure in (4) directly, proving tightness, and obtaining the existence of some limit measure, at least up to passing to suitable subsequences.

2 In the recent decade the technique of stochastic quantization (SQ), ﬁrst proposed by Parisi and Wu [56], for studying (1) has become increasingly popular. In this approach one consider the measure (1) as a invariant measure of some suitable stochastic partial diﬀerential equations (SPDEs).An important example, of such SPDEs, is the Langevin dynamics associated with the measure (1), which is described by the equation

2 d ∂ φ(t, x) + (m ∆)φ(t, x)+ λV ′(φ(t, x)) = ξ(t, x), (t, x) R R (5) t − ∈ + × d where ξ is a space time Gaussian white noise on R+ R . The difficulties in solving (5) are analogous to the ones mentioned before for the direct construction× of the measure (1): The expected regularity of the solution φ is not enough to provide a natural sense to the composition with the nonlinearity V ′. Furthermore the noise ξ is expected to grow at infinity, and, thus, one can only expect to solve equation (5) in a weighted space. One of the advantages of SQ methods is that we can apply many SPDEs and PDEs techniques in solving equation (5) which are, in general, not available for the direct construction of the measure (1). 4 T2 The first well posedness result for (5), in the case of Φ2 measure on the torus , were obtained by 4 Da Prato and Debussche in [23] (see also [49] by Mourrat and Weber about Φ2 stochastic quantization SPDE on the the plane R2). The paracontrolled distribution theory of Gubinelli, Imkeller, and Perkowski [30] (see also [21]), as well as Hairer’s regularity structures [33] and Kupiainen’s approach [41], based on renormalization group techniques, allowed to prove well posedness results in the more singular case of 4 T3 Φ3 measure on . Later these methods were improved to obtain a priory bounds on the solutions to (5). These bounds were strong enough to control the invariant measure and thus proof the existence of 4 4 the limit (4) in Φ2 and Φ3 [7, 8, 29, 50]. Another approach using an infinite dimensional stochastic control problem was used by Gubinelli and 3 T3 one of the authors for constructing Φ4 measure on in [14, 15], and by Bringmann in [19] for the Gibbs measure of Hartree hyperbolic equation.

Equation (5) is not the only possibility of pursuing stochastic quantization. In [54, 55] Parisi and Sourlas proposed an elliptic approach to this problem. They conjectured that if one solves the equation

2 (m ∆)φ(z, x)+ λV ′(φ(z, x)) = ξ(z, x) (6) − on (z, x) R2 Rd, and where ξ is a R2 Rd Gaussian white noise, then, for any z R2, the following equivalence∈ should× hold × ∈ 1 φ(z, ) exp( λV (ϕ))dµ (7) · ∼ Z − In the mathematics literature this kind of stochastic quantization was investigated in [40] and proved in [3] for a class of potentials V including convex potentials. Let us remark that, although both the elliptic and parabolic SQ methods so far often fall short of providing enough information on the decay of correlation functions of ν, they are capable of proving pathwise properties even when absolute continuity is lost, since (5) and (6) provide a coupling between the Gaussian Free Field (having law µ) and the interacting ﬁelds (having law ν). Similar results have also been obtained using the Polchinski equation, see, e.g., [17].

The main goal of this paper is to study the equation elliptic stochastic quantization for the Sinh- 2 Gordon model in dimension d = 2 (hereafter called cosh(βϕ)2 model) in the L regime of the charge parameter β, namely when β < √4π. In this case the SQ elliptic equation is formally given by | | (m2 ∆)φ(z, x)+2λ sinh(αφ(z, x)) = ξ(z, x), (z, x) R2 R2 (8) − ∈ × 2 2 2 2 where ξ is a Gaussian white noise on R R , and α =4πβ . The cosh(βϕ)2 model is a particular case × 1 of the Høegh-Krohn model (2) cited before, i.e. when the measure η(dβˆ)= (δ β(dβˆ)+ δβ(dβˆ)) (where 2 − δ β is the Dirac delta with unitary mass in β) and studied in [1, 4, 9] (see also the related model in [10])± using constructive techniques. The Sinh-Gordon± model (in the limit case of zero mass m 0) is an Integrable Quantum Field Theory see, e.g, [51, 62] and has received considerable attention→ in the physics literature [58, 68, 69]. We remark also that a probabilistic approach developed to the Lioville model (V (ϕ) = exp(βϕ),m = 0) developed by Kupianen, Rhodes, Vargas and their coauthors has found

3 remarkable success [32, 42, 43, 44]. As of this moment we are not aware of a probabilistic construction of the Sinh-Gordon in the case m = 0, and giving such a construction appears to be an interesting and challenging problem.

In order to solve equation (8) can be useful to rewrite it introducing a change of variables (introduced 4 by Da Prato and Debussche in [23] in for Φ2 model) in the following way

2 αφ¯ αW αφ¯ αW (m ∆)φ¯(z, x)+ λαe “e ” λαe− “e− ” = 0 (9) − − where φ¯ = φ W and W is the solution to the linear equation ( ∆+ m2)W = ξ. It is possible to give − αW 2 β2 − 2 a precise meaning to the expressions e± (when α = 4π 6 2(4π) ) thanks to Gaussian Multiplicative Chaos (see, e.g., [59]) through the limit

( αW c (α)) α lim e ± ε− ε dx = µ± (dx), (10) ε 0 → 4 where Wε is a smooth (translation invariant) approximation of the Gaussian Free Field (in R ) W , converging almost surely to W as ε 0, and c (α) R is given by the relation → ε ∈ + α2 c (α)= E[W 2], ε 2 ε and thus cε(α) + as ε + . The limit (10) is known to exits in the space of tempered distribution 4 4 → ∞ → ∞ ′(R ) on R . There are two main properties that make the random distribution µα different from, for Sexample, Wick powers : W n : and the Wick complex exponential : eiαW :. The first one is the fact that α µ± is a positive distribution, i.e. a σ-finite measure (this permits to exploit the signed structure of α α the noise). The second property is that µ± exhibits multifractality, i.e. the Besov regularity s of µ± 2 depends on the parameter α and on the integrability p 1, 2(4π) in the following way, for any δ > 0 ∈ α2 we have 2 h 2 α (p 1) δ β (p 1) δ µ B (4π)2 − − (R4)= B (4π) − − (R4). α ∈ p,p,loc p,p,loc These two properties are essential in the study of SPDEs involving the random distribution (10). Al- though, to the best of our knowledge, if we exclude a brief mention in Remark 1.3 and Remark 1.18 in [53] (on T2 and for β2 < 8π 1.37π), there are no studies of singular SPDEs involving a sinh 3+2√2 ≃ nonlinearity, SPDEs driven by a random forcing of the form (10) has been subject of a certain number of works in particular concerning the stochastic quantization of V (ϕ) = exp(βϕ)2 models corresponding to the aforementioned Liouville quantum gravity: see [27] by Garban on the parabolic equation on T2 and S2 for β2 < 8 3 √2 π, [37, 38] by Hoshino, Kawabi, and Kusuoka addressing the parabolic problem on T2 in the whole− subcritical regime β2 < 8π, [53] by Oh, Robert, and Wang about the parabolic and hyperbolic equation for β2 < 4π, [52] by Oh, Robert, Tzvetkov, and Wang about the parabolic equation in the L2 regime β2 6 4π on a general compact manifold, and [2] by Albeverio, Gubinelli and one of the authors on the elliptic quantization on R2 with charged parameter β2 < 4 8 4√3 π (see also [6] by Albeverio, Kawabi, Mihalache and Roeckner on the SQ of exponential and trigonometric− problem using a Dirichlet form approach).

All the previous works use in an essential way the fact that the exponential exp(αψ) is a positive function, in this way the stochastic quantization equation is equivalent to an equation with bounded coefficients. In the case of sinh(αφ) nonlinearity, we cannot directly use the sign of the coefficients and we are forced to deal with an SPDE with unbounded coefficients. For this reason we decide to use a ¯ 4 sort of energy estimate for φ (see Theorem 4.3), inspired by the works on Φ3 model in [7, 8, 29]. This idea permits us to obtaining some a priori estimates for the weighted H1 norm of φ¯ and the integrals αφ¯(x) α α e± µ± (dx) (here is where positivity of the distributions µ± plays a central role). The use of this energy method, and so the choice of H1 norm on φ¯ and not the weaker norms like the Sobolev WR 1,p for p< 2, is the main reason for the limitation of our techniques to the L2 regime for the charged parameter β, i.e. α2 = 4πβ2 6 (4π)2. These a priori estimates allow us to prove the existence and uniqueness of the solution to equation (8) using some regular approximation (for example a Galerkin approximation). Unfortunately Theorem 4.3 is not enough to prove the stochastic quantization for the

4 Sinh-Gordon model, i.e. to rigorously establish the heuristic relation (7). For obtaining this result we need some improved a priori estimates (see Theorem 4.15) obtained by a bootstrap procedure. In order to implement this bootstrap argument a “local” approximation of equation (8) is needed, and the Galerkin or other nonlocal smoothing methods, exploited for studying the exponential models in the articles mentioned before, cannot be used.

For all these reasons, we approximate the SPDE (8) by the following equation on the lattice on (x, z) R2 εZ2: ∈ × α2 2 ε 2 (m ∆R2 ∆ Z2 )φ (x, z)+ λε (4π) sinh(αφ(x, z)) = ξ (x, z), (11) − − ε ε 2 2 where ∆εZ2 is the discrete Laplacian on εZ , ∆R2 is the standard Laplacian on R , and ξε is an ap- 2 propriate white noise having Cameron-Martin space L2(R2 εZ2). The factor εα /4π is the necessary renormalization correction for obtaining the nontrivial limit× (10).

In order to study equation (11), we use Besov spaces on R2 εZ2, which are very similar to the ones developed on εZ2 by Martin and Perkowski in [47] and by Gubinelli× and Hofmanova in [29]. Other works involving the discretization of singular SPDEs are [22, 31, 48, 60, 70] on Td, using an extension operator by interpolation with trigonometric polynomials, and [24, 25, 34, 35] introducing a discrete version of regularity structures. About this framework our main contribution is the following: the ε extension operators ε and used in the current paper differ from the ones employed in [29, 47]. Indeed, in the cited papers,E the extensionE operators, associating to any function on the lattice a distribution on the continuum, are nonlocal, and thus, it is not clear how they behave with respect the composition with the nonlinear coefficients. This problem would cause technical difficulties in the identification of the ε limiting equation for (11). Instead the extension operator , proposed in the present work, is the same as that employed in [57] (see also the more recent [35]), andE it takes values into linear superposition of step ε functions. Although using we loose some regularity, due to the limited smoothness of step functions, it commutes with the nonlinearitiesE and we can prove that, in the ε 0 limit, it is possible to recover the original regularity (see Theorem 2.25 for a precise statement of this→ fact). Let us mention also Theorem 2.18 where we prove a sort of equivalent definition of discrete Besov norm using the discrete differences which could be of interest in the comparison between the discrete Besov methods used in [29, 47] and the ones based on discrete regularity structures of [24, 25, 34, 35].

A ﬁnal contribution of our work is about the convergence of the discrete Gaussian Multiplicative chaos α α µε to its continuum counterpart µ . This convergence problem was already addressed in other works (see, e.g., Section 5.3 in [59] and references there in). The novelty of our result is that the convergence of ε α α (µε ) to µ is not only weakly∗ in the space of positive σ-ﬁnite measures, but also in the Besov spaces E α2 (p 1) ε 2 of the form B− (4π)2 − − (R4) for suitable p 1, 2(4π) , α2 6 (4π)2 and ε > 0 (see Section 3). We p,p,loc ∈ α2 conclude the introduction with the main theoremh proved in the paper.

Theorem 1.1 Equation (8) admits a unique solution for α < 4π. Furthermore we have that φ(x0, ) 2 cosh,β | | · ∈ ′(R ) has probability distribution ν associated with the action S m 1 S(ϕ)= ( ϕ(z) 2 + m2ϕ(z)2)dz + λ : cosh(βϕ(z)) : dz, 2 R2 |∇ | R2 Z Z where β = α (see Deﬁnition 4.1 for the precise deﬁnition of νcosh,β). √4π m

Structure of the Paper. In Section 2 we recall some standard results on Besov spaces on Rd and some of the results proved in [29, 47] on Besov spaces on εZd. We also extend these deﬁnitions to R2 εZ2. Furthermore we introduce diﬀerence spaces on εZd, introduce our extension operator and prove bounds× for it. Finally we discuss regularity of positive distributions multiplied by a density on the lattice, which we will need to to take advantage of the positivity of the Gaussian Multiplicative Chaos. In Section 3 we will prove the estimates on Gaussian Multiplicative Chaos which we require for the a priori estimates 2 2 on (11). To achieve this we will establish estimates on the Green’s function of (m ∆R2 εZ2 ) on − ×

5 R2 εZ2. In Section 4 we prove that a priori estimates on equation (11) which are strong enough that we× can find a convergent subsequence of the solution (after applying the extension operator). The limit will be shown to be in H1(R4) almost surely and to satisfy equation (8). Furthermore we will show that solutions to equation (8) are unique. Finally we will show that the solutions have enough regularity to be restricted to a two dimensional subspace or R2 thus establishing the main theorem. Finally, in Appendix cosh,β A, we prove the the measure νm constructed satisfies the Osterwalder-Schrader axioms, exploiting 4 the methods of [8, 29, 35] for ϕ3 measures. To our knowledge this is the first selfcontained proof of the Osterwalder-Schrader axioms including mass-gap by stochastic quantization, however it relies heavily on the convexity of the renormalized interaction.

Acknowledgements. The authors would like to thank Massimiliano Gubinelli and Sergio Albeverio for the comments and suggestions on an earlier version of the paper. N.B gratefully acknowledges ﬁnan- cial support from ERC Advanced Grant 74148 “Quantum Fields and Probability” and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through CRC 1060. F.C.D. is supported by the DFG under Germany’s Excellence Strategy - GZ770 2047/1, project-id 390685813.

2 The setting

For all the rest of the paper we employ the following notations: If H,K : S R+ are two functions defined in the same set S we say that H . K (or with an abuse of notation that→ H(f) . K(f) (where f S)), if there is some constant C such that for any f S we have H(f) 6 CK(f). We write also H ∈ K (or with an abuse of notation that H(f) K(f) (where∈ f S)) if H . K and K . H. ∼ ∼ ∈ 2.1 Besov spaces on Rd Here we report some standard definitions and properties of Besov spaces on Rd. The main definitions and theorems hold also for the cases of Besov spaces on Td or Rd1 Td2 . × We fix some notations: For any ℓ R we define the (weight) function ρ(d) : Rd R as ∈ ℓ → + 1 ρ(d)(y)= , y Rd. ℓ (1 + y 2)ℓ/2 ∈ | | If p [1, + ) and ℓ R we define ∈ ∞ ∈

p (d) p f Lp(R2) = f(y)ρℓ (y) dy, k k ℓ R2 | | Z where dy is the standard Lebesgue measure on Rd, and

f ∞ R2 = sup f(x)ρ (x) . Lℓ ( ) ℓ k k x R2 | | ∈ p Rd p Rd Rd In the following we write L ( ) for L0( ). We denote by ( ) the Fr´echet space of Schwartz functions d S d and by ′(R ) its strong dual, i.e. the nuclear space of tempered distributions. We denote by : (R ) (Rd) theS Fourier transform on Rd, i.e. F S → S

i(k y) d d (f)(k)= e · f(y)dy, k R ,f (R ) F Rd ∈ ∈ S Z 1 and by − the inverse Fourier transform namely F 1 ˆ 1 i(k y) ˆ Rd ˆ Rd − (f)(y)= d e− · f(k)dk, y , f ( ). F (2π) Rd ∈ ∈ S Z We can extend the Fourier transform on the space of tempered distributions by duality.

6 (d) Rd (d) We consider a smooth dyadic partition of unity (ϕj )j> 1 deﬁned on such that ϕ 1 is supported − − 1 (d) (d) (d) j in a ball around 0 of radius , ϕ is supported in an annulus, ϕ ( ) = ϕ (2− ) for j > 0 and if 2 0 j · 0 · i j > 1 then supp ϕ(d) supp ϕ(d) = . We denote by | − | j ∩ i ∅ (d) 1 (d) K = − (ϕ ). j F j

Remark 2.1 In all the rest of the paper we consider an additional condition on the dyadic partition of unity function, which, although not being completely standard, can always be assumed without loss of (d) (d) generality (see, e.g., Section 3 and Appendix A in [49]): The functions ϕ 1 and ϕ0 (and thus all the (d) − functions ϕj j> 1) have to belong to a Gevrey class of index θ for some θ > 1, i.e. there is a constant { } − C > 0 for which, for any α = (α , , α ) Nd, we have 1 ··· d ∈ 0 α (d) α θ α (d) α θ ∂ ϕ 1 L∞(Rd) 6 C| |(α!) , ∂ ϕ0 L∞(Rd) 6 C| |(α!) , k − k k k where α = αi and α! = αi!. Under the previous conditions we get that there are some constants a>¯ 0 and| | β¯ = 1 for which P θ Q (d) 2 β/¯ 2 (d) 2 β/¯ 2 Rd K 1 (y) . exp( a¯(1 + y ) ), K0 (y) . exp( a¯(1 + y ) ), y . | − | − | | | | − | | ∈ (d) (d) By the deﬁnition of ϕj in terms of ϕ0 we also get, for any j > 0,

¯ K(d)(y) . 2jd exp( a¯(1+22j y 2)β/2), y Rd, | j | − | | ∈ where the constants hidden in the symbol . do not depend on j > 0. We introduce also the following function

(d) (d) (d) j υ (x) := 1 ϕ 1(x)= ϕ0 (2− x), (12) − − > Xj 0 (d) (d) (d) j where we use that ϕj is a partition of unity and thus ϕ 1(x)+ j>0 ϕ0 (2− x)=1. The previous relations imply that − P (d) k k k j k j υ (2− x) := 1 ϕ 1(2− x)= ϕ0(2− − x)= ϕ0(2− − x). (13) − − > > Xj 0 Xj k d d If f ′(R ) we denote by ∆ f (R ) the i-th Littelwood-Paley block of the tempered distribution ∈ S j ∈ S f, namely ∆j f is given by 1 (d) ∆ f = − (ϕ (f)) = K f j F j ·F j ∗ where is the natural product between Schwartz functions and tempered distributions. · R R s Rd Definition 2.2 If p, q [1, + ], s and ℓ we define the Besov space Bp,q,ℓ( ) as the subset of ∈ d ∞ ∈ s∈ d tempered distributions ′(R ) for which f B (R ) if the norm S ∈ p,q,ℓ (j 1)s s p q f B (Rd) := 2 − ∆j 1f L (Rd) j N0 ℓ (N ), k k p,q,ℓ k{ k − k ℓ } ∈ k 0 q is finite. Here q N is the ℓ norm on the space of sequences starting at 0. k·kℓ ( 0) We report here some results useful in what follows.

Proposition 2.3 For any s,ℓ,m R, p, q [1, + ] we have that for any f Bs (Rd) ∈ ∈ ∞ ∈ p,q,ℓ (d) f Bs (Rd) f ρ Bs (Rd) k k p,q,ℓ ∼k · ℓ k p,q,0 m/2 − f Bs (Rd) (1 ∆) (f) Bs m(Rd). k k p,q,ℓ ∼k − k p,q,ℓ

7 Proof The proof can be found in Theorem 6.5 in [66]. ✷

Proposition 2.4 Consider p ,p , q , q [1, + ], s >s and ℓ ,ℓ R such that 1 2 1 2 ∈ ∞ 1 2 1 2 ∈ d d ℓ1 6 ℓ2, s1 > s2 , − p1 − p2

s1 d s2 d d d then B (R ) is continuously embedded in B (R ), and if ℓ1 < ℓ2 and s1 > s2 the p1,q1,ℓ1 p2,q2,ℓ2 p1 p2 embedding is compact. − − Proof The proof can be found in Theorem 6.7 in [66]. ✷

1 1 1 Proposition 2.5 Consider p1,p2,p3, q1, q2, q3 [1, + ] such that + = and q1 = q3 and q2 = . ∈ ∞ p1 p2 p3 ∞ Moreover consider ℓ1,ℓ2,ℓ3 R with ℓ1 + ℓ2 = ℓ3 and consider s1 < 0 s2 0 and s3 = s1 + s2 > 0. ∈ d ≥ d n Finally, consider the bilinear functional Π(f,g)= f g deﬁned on ′(R ) (R ) taking values in (R ). Then there exists a unique continuous extension of Π· as the map S ×S S

Π: Bs1 Bs2 Bs3 p1,q1,ℓ1 × p2,q2,ℓ2 → p3,q3,ℓ3 and we have, for any f,g for which the norms are deﬁned:

Π(f,g) Bs3 . f Bs1 g Bs2 . k k p3,q3,ℓ3 k k p1,q1,ℓ1 k k p2,q2,ℓ2 Proof The proof can be found in [49] in Section 3.3 for Besov spaces with exponential weights. The proof for polynomial weights is similar. ✷

2.2 Besov spaces on lattice

Here we want to introduce a modification of Besov spaces when the base space is a square lattice (see 2 2 [29, 47]). First we consider the set εZ for some ε > 0, i.e. z εZ if z = (εn1,εn2) where n1,n2 Z. We endow εZ2 with the discrete topology and with the measure∈ dz defined, for any function f : εZ2 ∈ R, as → f(z)dz := ε2 f(z). Z2 Zε z εZ2 X∈ We can define also the weighted Lebesgue space Lp(εZ2), where p [1, + ) and ℓ R. with the norm ℓ ∈ ∞ ∈

p p p f Lp = (ρℓ(z)) f(z) dz (14) k k ℓ Z2 | | Zε p Z2 (2) R2 R Z2 where f Lℓ (ε ), where hereafter we write ρℓ(x) := ρℓ (x), x and ℓ . We can define Lℓ∞(ε ) with the∈ norm ∈ ∈ f ∞ Z2 = sup f(z)ρ (z) Lℓ (ε ) ℓ k k z εZ2 | | ∈ 2 2 2 where f L∞(εZ ) if the norm is finite. We define the space (εZ ) = L∞(εZ ) as the space of ∈ ℓ S ℓ<0 ℓ Schwartz functions defined on εZ2, which forms a Fréchet space with the set of seminorms ∞ . Lℓ ℓ<0 2 2 T {k·k } We define also ′(εZ ) as the topological dual of (εZ ) with respect the Fréchet space structure of S S Z2 Z2 T2 T2 (ε ). It is possible to define the Fourier transform ε : (ε ) C∞ 1 , where 1 is the torus of S F S → ε ε 2π T2 π π 2 length ε , i.e. 1 = ε , ε , which has the following form ε − iz h ε(f)(h) := fˆ(h)= e · f(z)dz F Z2 Zε

8 T2 1 T2 where h 1 . In the usual way it is possible to define the inverse Fourier transform − : C∞ 1 ∈ ε F ε → (εZ2) S 1 ˆ 1 iz h ˆ ε− (f)(z) := 2 e− · f(h)dh, F (2π) T2 1 Z ε 1 T2 Z2 from ε− : C∞ 1 (ε ). We can extend the Fourier transform ε, and the inverse Fourier F ε → S F 1 Z2 T2 T2 transform ε− from the space ′(ε ) into ′ 1 (where ′ 1 is the space of distributions, i.e. F S D ε D ε 2 2 2 the topological dual of C∞ T 1 ), and from ′ T 1 into ′(εZ ) respectively. ε D ε S We can extend the concept of weighted Besov spaces Bs (εZ2), where s R, p, q [1, + ], ℓ R, p,q,ℓ ∈ ∈ ∞ ∈ of tempered distributions defined on εZ2. In the following we consider a fixed dyadic partition of unity (2) R2 (ϕj )j> 1 := (ϕj )j> 1 (recall the notation preceding Remark 2.1) defined on having the properties − − required in Section 2.1 (in particular Remark 2.1). For the definition of Besov spaces on the lattice εZ2 N ε T2 with ε =2− , we introduce a suitable partition of unity (ϕj ) 16j6Jε for 1 as follows − ε

ε ϕj (k) if j 1, supp(ϕ ) , . (16) ε − j 6⊂ − ε ε h i 2 If f ′(εZ ), for any 1 6 i 6 J we deﬁne ∈ S − ε 1 ε ε ∆ f = − (ϕ (f)) = K f i Fε i ·Fε i ∗ε ε 1 ε where K = − (ϕ ). i Fε i ε Remark 2.6 It is important to note that when j ¯ 0 and β¯ (0, 1) such that for any z εZ2 ∈ ∈ ¯ Kε(z) . 22j exp( a¯(1+22j z 2)β/2), | j | − | | where the constants hidden in the symbol . do not depend both on j > 1 and 0 <ε 6 1. − ε k N Remark 2.7 In some sense also the function KJε does not depend on ε, when ε =2− for some k 0. ¯ 1 ε 1 1 i(z q) ∈ More precisely consider K = K0 (that is K0 when ε = 1, i.e. K0 (z)= (2π)2 T2 e− · (1 ϕ 1(q))dq), 1 − − we have that R

ε 1 ε K (z) = − 1 ϕ (x) (z) Jε Fε  − j  16j6J − X ε 1 (2) Jε  = − (υ (2− ))(z) Fε · (2) Jε = υ (2− )I 2 (x) (z) F · [2Jε+1 ,2Jε+1]

2Jε (2) Jε I 2 = 2 υ ( ) 1 , 1 (x) (2 z) F · [− 2 2 ] 1 z = 22Jε K¯ (2Jε z)= K¯ , ε2 ε where we used the function υ(2) introduced in (12) and the relation (13).

9 (2) 2 In particular, since υ (x)=1 ϕ 1(x) is in Gevrey class of in θ > 1 as a function on T , we have that there isa> ¯ 0 and 0 < β¯ < 1 such− − that

1 z 2 β/2 Kε (z) . exp a¯ 1+ | | , | Jε | ε2 − ε2 !

k where the constants hidden in the symbol . do not on ε =2− for any k N . ∈ 0 R s Z2 Deﬁnition 2.8 For any p, q [1, + ], s,ℓ we deﬁne the Besov space Bp,q,ℓ(ε ) the Banach space, 2 ∈ s ∞ 2 ∈ 2 subset of ′(εZ ), for which f B (εZ ) if f ′(εZ ) S ∈ p,q,ℓ ∈ S js s 2 p q f B (εZ ) = 2 ∆j f L (εZ2) 16j6Jε ℓ ( 1, ,J ), (17) k k p,p,ℓ k{ k k ℓ }− k {− ··· ε} q where ℓq ( 1, ,J ) is the natural norm of the small ℓ space ℓ ( 1, ,Jε ). k·k {− ··· ε} {− ··· } Many properties of standard Besov spaces can be extended to the case of Besov spaces on the lattice εZ2. We report here some of them which will be useful in what follows (see [29, 47] for a discussion of other properties of Besov spaces on lattices).

Remark 2.9 From now on when the symbols . and appear in expressions involving norms, or other s Z2 ∼ functionals deﬁned on Besov spaces Bp,q,ℓ(ε ) or one of their generalization in Section 2.5, we assume that the constants hidden in the symbols . and can be chosen uniformly with respect to 0 < ε 6 1. Without this assumption the following results are∼ trivial.

p 2 p 2 First we introduce the discrete Laplacian ∆ Z2 : L (εZ ) L (εZ ) deﬁned as ε ℓ → ℓ

f(z + εei)+ f(z εei) 2f(z) ∆ Z2 f(z)= − − ε ε2 j=1,2 X 2 where e ,e is the standard basis of R . It is simple to compute the Fourier transform of ∆ Z2 , indeed { 1 2} ε

iq z f(x + εei)+ f(x εei) 2f(x) iq z Z2 Z2 ε(∆ε f) = ∆ε f(z)e · dz = 2− − e · dz F Z2 Z2 2ε ε j=1,2 ε Z X Z 1 iεqj +iεqj iq z = 2 (e− + e 2)f(z)e · dz ε Z2 − j=1,2 ε X Z 4 εq = sin2 i (f)(q), − ε2 2 Fε j=1,2 X T2 where q = (q1, q2) 1 . Writing ∈ ε

4 2 εqi σ 2 (q)= sin , −εZ ε2 2 j=1,2 X and taking any c > 0, we get that the operator (c ∆εZ2 ) is a self-adjoint, positive and invertible on 2 − ′(εZ ) with the power s R given by S ∈ s 1 s 2 (c ∆εZ ) (f)= ε− c + σ Z2 (q) ε(f)(q) . − F −ε F

R T2 R Lemma 2.10 Consider s,ℓ , p, q [1, + ]. Furthermore assume that σε = σε(q): 1 is ∈ ∈ ∞ ε → a diﬀerentiable function such that there is an m R such that for any β = (β ,β ) N2, for which ∈ 1 2 ∈ 0 β = β1 + β2 6 4, we have | | β 2 (m β )/2 ∂ σ (q) 6 C (1 + q ) −| | | q ε | β | |

10 T2 π π 2 R 2 2 where : 1 2ε , 2ε + deﬁned as q = q1 + q2, form some constant Cβ > 0 independent | · | ε ≃ − → | | 6 on 0 <ε 1. Then the operator p σ 1 s 2 T ε f = − (σ (q)( f)(q)), f B (εZ ) Fε ε Fε ∈ p,q,ℓ s Z2 s m Z2 is well deﬁned from Bp,q,ℓ(ε ) into Bp,q,ℓ− (ε ) and we have

σ ε − T f Bs m(εZ2) . f Bs (εZ2). k k p,q,ℓ k k p,q,ℓ Proof Our goal is to show that if m R then ∈ 1 ε mj ε ε− (ϕj σε εf) Lp(εZ2) . 2 ∆j f Lp(εZ2). kF F k ℓ k k ℓ We write 1 σε − (ϕ σ f)= K ∆ f Fε j εFε j ∗ε j where σε 1 K = − (ϕ σ ). j Fε i ε i j <2 | −X| N σε mj We want to prove that for any l , K L1 (εZ2) 6 C(l)2 , which implies the statement by weighted ∈ k j k −l Young’s inequality. Let us ﬁrst consider the case j = 1. We can choose a smooth function ψ : R2 [0, 1] supported in the ball B(0, 2) and such that ψ(x) = 1− if x 6 1 and, writing → | |

σε 1 i(z q) K 1(z)= 2 e · ψ(q)σε(q)dq, − (2π) T2 1 Z ε since supp( ε(∆ 1f)) B(0, 1), we have F − ⊂

1 σε ε− (σε( ε∆ 1f)(k)) = K 1 ε ∆ 1f. F F − − ∗ − Considering β = (β ,β ), α = (α , α ) N2, from (2.2) we get that 1 2 1 2 ∈ 0

M 2 M σε 1 q i(z q) (1 + z ) K 1(z) = 2 1 ∆T2 e · ψ(q)σε(q)dq | | | − | (2π) T2 − 1 Z 1 ε ! ε

M 1 i(z q) q = 2 e · 1 ∆T2 (ψ(q)σε(q))dq (2π) T2 − 1 Z 1 ε ε

i(z q) α β 6 cα,β e · ∂q ψ(q)(∂q σε(q)) dpd q 6 CM (18) T2 | | α + β =2M Z 1 | | X| | ε q T2 where ∆T2 is the standard Laplacian on 1 with respect to the q = (q1, q2) variables, and cα,β, CM > 0 are 1 ε ε ˜ suitable constants independent on 0 <ε 6 1. Choosing M =2+ ℓ we can deduce that K L1(εZ2) 6 C3 k k ℓ for some constant C˜3 not depending on 0 <ε 6 1. For 0 6 j 6 Jε we proceed similarly after a rescaling argument: We can choose a smooth function ˜ R2 ε ψ : [0, 1] supported in an annulus which is identically 1 with in the support of i 1 <2 ϕi = → | − | i 1 <2 ϕi (when ε 6 1), and writing, | − | P P 1 1 σε i(z q) ˜ i(z q) ˜ Kλ (z)= 2 e · ψ(q)σε(λq)dq = 2 e · ψ(q)σε(λq)dq, (2π) T2 (2π) R2 1 Z λε Z

j ˜ 2 where λ =2 , since supp(ψ) T 1 , we get ⊂ λε

1 σε − (σ (q)( ∆ f)(q)) = λK (λz) ∆ f, Fε ε Fε j λ ∗ε j

11 Then, using the same argument in equation (18), we obtain

2 M σ 1 2 M i(z q) ˜ (1 + z ) Kλ (z) 6 2 (1 + z ) e · ψ(q)σε(λq)dq | | | | (2π) | | R2 Z

1 i(z q) M ˜ 6 2 e · (1 ∆q) (ψ(q)σε(λq))dq (2π) R2 − Z

˜ Since ψ is supported in an annulus B(0, R1) B(0, R2) for some constant radius R1 > R2 > 0 we can 2 (m β )/2 \ estimate (1 + λq ) −| | on B(0, R ) B(0, R ) getting | | 1 \ 2 β 2 (m β )/2 m β β m β m β m m β mj λ| |(1 + λq ) −| | 6 R −| |λ| | λ −| | 6 R −| | λ 6 R −| |2 . | | 2 | | 2 | | 2

(2) ε ε (2) Jε Finally for s = Jε we recall that, setting υ = 1 ϕ 1 we have ϕJ ( )=1 ϕi = υ (2− ). − − ε · − j

Remark 2.11 Consider a positive function σ for which both σ and the 1 satisfy the hypotheses of ε ε σε R R s Z2 Lemma 2.10 for some m and m respectively, then, s,ℓ , p, q [1, + ] and f Bp,q,ℓ(ε ), by ∈ 1 − ∈ ∈ ∞ ∈ using the fact that Tσ = T −1 and applying Lemma 2.10 to both Tσ and T 1 , we get ε ε σ σε ε

s−m s 2 1 s−m Tσε f B (εZ2) . f B (εZ ) = T (Tσε f) . Tσε (f) B (εZ2). k k p,q,ℓ k k p,q,ℓ σε Bs (εZ2) k k p,q,ℓ p,q,ℓ

This implies that

s−m s 2 Tσε f B (εZ2) f B (εZ ). k k p,q,ℓ ∼k k p,q,ℓ Theorem 2.12 For any p, q [1, + ], s,ℓ R we have that ∈ ∞ ∈ s 2 f Bs (εZ2) f ρℓ Bs (εZ2), f B (εZ ) (19) k k p,q,ℓ ∼k · k p,q,0 ∈ p,q,ℓ furthermore for any m R we get ∈ m/2 − (1 ∆εZ2 ) f Bs m(εZ2) f Bs (εZ2) (20) k − k p,q,ℓ ∼k k p,q,ℓ Proof The proof of the equivalence (19) follows the same line of the analogous statement on Rd (see, e.g., Theorem 6.5 of [66]) and it is consequence of the fact that, for any ℓ R, α N2, there are some ∈ ∈ 0 constants Cα,ℓ, Cℓ > 0 for which

∂αρ (x) 6 C ρ (x), x R2 | ℓ | α,ℓ ℓ ∈ and ℓ ρ (x) 6 C ρ (y)(1 + x y )| |. ℓ ℓ ℓ | − |

12 The equivalence (20) follows from Lemma 2.10 and Remark 2.11, by noting that m 2 m/2 σε(q)= ε((1 ∆εZ2 ) )(q)= 1+ σ ∆ (q) 6 (1 + Csing(m) q ) F − − εZ2 | | where 2 C+1 =1, C 1 = , − π and similarly for the derivatives of σε(q) with respect to q. ✷ We now extend the standard estimates on products between a function and a distribution with suitable Besov regularity. More precisely if f (εZ2) and g Bs (εZ2) we can deﬁne the product ∈ S ∈ p,q,ℓ fg = f g = ∆ f∆ g (21) · i j 16i6J , 16j6J − Xε − ε s Z2 which is a well deﬁned element of Bp,q,ℓ(ε ).

1 1 1 Theorem 2.13 Consider s1 6 0,s2 > 0 such that s1 +s2 > 0, p1,p2, q1, q2 [1, + ] , + = 6 1, p1 p2 p3 1 1 1 2 ∈ ∞2 2 and + = 6 1 then the product Π(f,g) = f g deﬁned on ′(εZ ) (εZ ) into ′(εZ ) can be q1 q2 q3 extended in a unique continuous way to a map · S × S S Π: Bs1 (εZ2) Bs2 (εZ2) Bs1 (εZ2). p1,q1,ℓ1 × p2,q2,ℓ2 → p3,q3,ℓ1+ℓ2 Furthermore for any f Bs1 (εZ2) and g Bs2 (εZ2) we have ∈ p1,q1,ℓ1 ∈ p2,q2,ℓ2 Π(f,g) = f g Bs1 (εZ2) . f Bs1 (εZ2) g Bs2 (εZ2). k k k · k p3,q3,ℓ1+ℓ2 k k p1,q1,ℓ1 k k p2,q2,ℓ2

Proof The proof of the present theorem is given in Lemma 4.2 of [47] in the unweighted (i.e. ℓ1 = ℓ2 = 0) and inﬁnite integrability (i.e. p = q = p = q = ) case. The proof in [47], is based on the fact that 1 1 2 2 ∞ j T2 the Fourier transform of the product i R2 >0 real constants), and the Fourier transform of the A ⊂ \ j T2 product if∆jg, where i j < 2, is supported in B(0, 2 R3) 1 , for a suitable constant R3 > 0 (see | − | ∩ ε Lemma A.6 in [47]). Exploiting these observations the proof follows the same lines of the analogous result for Besov spaces on Rd (see, e.g., Chapter 2 of [13] and the proof of Theorem 3.17 of [49]). ✷

Theorem 2.14 Consider p ,p , q , q [1, + ], s >s and ℓ ,ℓ R if 1 2 1 2 ∈ ∞ 1 2 1 2 ∈ d d ℓ1 6 ℓ2, s1 > s2 , − p1 − p2 then Bs1 (εZ2) is continuously embedded in Bs2 (εZ2), and the norm of the embedding is uniformly p1,q1,ℓ1 p2,q2,ℓ2 bounded in 0 <ε 6 1.

si 2 si 2 Proof By Theorem 2.12, we have that Iℓ : B (εZ ) B (εZ ), deﬁned as i pi,qi,ℓi → pi,qi,0

Iℓi (f)= ρℓi f -is an isomorphism. This means that proving that Bs1 (εZ2) ֒ Bs2 (εZ2) continuously is equiv p1,q1,ℓ1 → p2,q2,ℓ2 1 s1 2 s2 2 alent to prove that the map I I− is a continuous embedding of B (εZ ) into B (εZ ). On ℓ2 ℓ1 p1,q1,0 p2,q2,0 the other hand we have that ◦

1 s1 Z2 Iℓ2 I− (g)= ρℓ2 ℓ1 g, g Bp ,q ,0(ε ) (22) ◦ ℓ1 − ∈ 1 1 k Z2 N 1 Since ℓ2 ℓ1 > 0 and so we have that ρℓ2 ℓ1 B , ,0(ε ), for any k 0, by Theorem 2.13 Iℓ2 Iℓ− − − ∈ ∞ ∞ ∈ ◦ 1 s1 Z2 is a continuous operator from Bp1,q1,0(ε ) into itself and with norm 1 − s1 2 s1 2 . k 2 . k 2 Iℓ1 Iℓ (B (εZ ),B (εZ )) ρℓ2 ℓ1 B∞ ∞ (εZ ) ρℓ2 ℓ1 B∞ ∞ (R ) k ◦ 2 kL p1,q1,0 p1,q1,0 k − k , ,0 k − k , ,0 which is uniformly bounded in 0 < ε 6 1. This implies that we can reduce the proof of the present .theorem to the proof of the fact that Bs1 (εZ2) ֒ Bs2 (εZ2) and the inclusion is continuous p1,q1,0 → p2,q2,0 This last statement can be proved following the argument for the analogous statement on Rd (see, e.g., Proposition 2.20 in [13]). ✷

13 2.3 Diﬀerence spaces and Besov spaces on lattice In this subsection we discuss a diﬀerent space of functions with fractional positive regularity on εZ2. More precisely we introduce the following Banach function spaces:

Definition 2.15 Consider 0 6 s 6 1, p (1, + ) and ℓ R we define the (difference) fractional Sobolev space W s,p(εZ2) the Banach space of measurable∈ ∞ functions∈ f : εZ2 R such that ℓ → 1/p s 2 p p f s,p Z2 = f p Z2 + sup k − ε ρ (x) f(x + k) f(x) . (23) Wℓ (ε ) Lℓ (ε ) ℓ k k k k k εZ2,0< k 61 | | | − | ∈ | | x εZ2 ! X∈ 1,p Remark 2.16 In the case s = 1 an equivalent norm on Wℓ is given by the following expression

1/p 2 p ps p f W s,p(εZ2) f Lp(εZ2) + ε (ρℓ(x)) ε− f(x + εei) f(x) (24) k k ℓ ∼k k ℓ  | − |  x εZ2,i=1,2 ∈ X   where e is the standard basis of Z2. Indeed we have that for any k εZ2 of the form k = (k , 0), { i}i=1,2 ∈ 1 ε < k1 6 1 we have

1/p 2 p p p 1 p ε x εZ2,i=1,2(ρℓ(x)) k − f(x + kei) f(x) = k f(k1ei + ) f( ) L (R2) = ∈ | | | − | | 1| k · − · k ℓ 1 p k1 p 6 k 16h6 k1 +1 f(εhei + ) f(ε(h 1)ei + ) L (R2) k ε f(εei + ) f( ) L (R2) P1 ε k · − − · k ℓ ∼ 1 k · − · k ℓ | | 1/p 2 p ps p P ε x εZ2,i=1,2(ρℓ(x)) ε− f(x + εei) f(x) . ∼ ∈ | − | P A similar argument can be done for k = (0, k2) and then, combining these two results, we get the equivalence (24), since the converse inequality is obvious.

Remark 2.17 Let F : R R be a globally Lipschitz function such that F (0) = 0, then it is easy to see that if f W s,p(εZ2) then→ also F (f) W s,p(εZ2) and ∈ ℓ ∈ ℓ

F (f) W s,p(εZ2) . f W s,p(εZ2). k k ℓ k k ℓ 2 2 2 Hereafter we denote by τ : ′(εZ ) ′(εZ ) the translation with respect to the vector z εZ z S → S ∈ 2 2 τ (f)(z′)= f(z + z′), z,z′ εZ ,f ′(εZ ) z ∈ ∈ S Furthermore if k εZ2 we introduce the notation ∈ D f = (τ f f). k k − Finally we write s s ε− (τεe1 f(x) f(x)) εf(x)= s − , ∇ ε− (τεe2 f(x) f(x)) − where e is the standard basis of Z2. We also use the notation f = 1f. { i}i=1,2 ∇ε ∇ε s,p Z2 s δ Z2 We want to establish a relation between the space Wℓ (ε ) and analogous Besov space Bp,p,ℓ± (ε ) (where δ > 0).

R s Z2 Theorem 2.18 Consider 0 6 s 6 1, p, q [1, + ], ℓ , and δ1 > 0 then for any f Bp,q,ℓ(ε ) we have ∈ ∞ ∈ ∈ − s,p f Bs δ1 (εZ2) . f W (εZ2) . f Bs+δ1 (εZ2). (25) k k p,p,ℓ k k ℓ k k p,p,ℓ In order to prove Theorem 2.18 we establish the following lemma.

Lemma 2.19 Consider 0 6 s 6 1, p (1, + ), ℓ R, and δ > 0 then for any f W s,p(εZ2) we have ∈ ∞ ∈ 1 ∈ ℓ

− s,p f Bs δ1 (εZ2) . f W (εZ2). k k p,p,ℓ k k ℓ

14 ε R T1 Proof Letϕ ˜i : [0, 1] be a dyadic partition of unity of 1 as the one introduced in equation (15) → ε T2 ε R for 1 . Furthermore we require that it does not depend on ε forϕ ˜i : [0, 1] is a dyadic partition of ε → unity not depending on ε when i

supp(ϕ ˜ε(q )˜ϕε(q )) supp ϕε , supp(ϕε ) supp ϕ˜ (q )˜ϕε(q ) , i 1 j 2 ⊂  k k ⊂  i 1 j 2  k max(i,j) <2 max(i,j) k <2 | − X | | X− |     ε T2 where ϕk is the dyadic partition of unity of 1 introduced in equation (15). Consider also some constants ε 2 kℓ,j = (kℓ,j,1, kℓ,j,2) εZ forming sequence for which ε 6 kℓ,j 6 1 and there is a constant C1 > 0 independent of 0 <ε∈6 1 for which | | k q sin ℓ,j · > C , (26) 2 1 for any q = (q , q ) supp(ϕ ˜ (q )˜ϕ (q )) for i, j = 1. It is important to note that the inequality (26) 1 2 ∈ j 1 ℓ 2 6 − implies that there is a C2 > 0, independent of ε> 0, for which C k > 2 (27) | ℓ,j | q | | T2 R for q = (q1, q2) supp(ϕ ˜j (q1)˜ϕℓ(q2)). Let σε : 1 be a smooth function deﬁned as ∈ ε →

iki,j q ε ε (e− · 1) σε(q)= ϕ˜j (q1)˜ϕℓ (q2) − s 2s 2 kℓ,j q 16ℓ6J 16j6Jε kℓ,j 1+4 ki,j − sin · − X ε − X | | | | 2 We have that

1 ε ε s 1 ε ε 1 ε ε − (σ ϕ˜ ϕ˜ ( k − D f)) = − (˜ϕ ϕ˜ (f)) − (¯σ ϕ˜ ϕ˜ (f)) Fε ε j ℓ Fε | ℓ,j | kℓ,j Fε i j Fε −Fε ε j ℓ Fε where ε ε 1 σ¯ε(q)= ϕ˜j (q1)˜ϕℓ (q2) . 2s 2 kℓ,j q 16ℓ6J 16j6J 1+4 kℓ,j − sin · − X ε − X ε | | 2 N2 By inequalities (26) and (27), the functions σε(q) andσ ¯ε(q) are built in such a way that for any β 0, such that β 6 4, there are some constants C > 0 (independent of 0 <ε 6 1) for which ∈ | | β β s β β 2s β ∂ σ (q) 6 C (1 + q )− −| |, ∂ σ¯ (q) 6 C (1 + q )− −| |. | q ε | β | | | q ε | β | | By Lemma 2.10, this implies that

1 ε ε s s−δ−δ′ 6 s−δ−δ′ f B (εZ2) Tσε ( ε− (˜ϕj ϕ˜r ε( kr,j − Dkr,j f))) B (εZ2) k k p,p,ℓ k F F | | k p,p,ℓ r,j X 1 ε ε − − ′ + Tσ¯ε ( ε− (˜ϕj ϕ˜r ε(f))) Bs δ δ (εZ2) k F F k p,p,ℓ r,j X 1 ε ε s . −δ−δ′ ε− (˜ϕj ϕ˜r ε( kr,j − Dkr,j f)) B (εZ2) + kF F | | k p,p,ℓ r,j X 1 ε ε − − − + ε− (˜ϕj ϕ˜r ε(f)) B s δ δ(εZ2) kF F k p,p,ℓ r,j X ′ m(δ+δ ) 1 s p . 2− ε− (ϕm ε( kr,j − Dkr,j f)) L (εZ2) kF F | | k ℓ m max(r,j) m <2 X | X− | m(δ+δ′) 1 + 2− ε− (ϕm ε(f)) Lp(εZ2) kF F k ℓ m max(r,j) m <2 X | X− |

− p where we used that g B δ (εZ2) . g L (εZ2) and the constants hidden in the symbol . do not depend k k p,p,ℓ k k ℓ on 0 <ε 6 1. If we choose δ, δ′ > 0 such that δ + δ′ = δ1 we get the thesis. ✷

Remark 2.20 The result of Lemma 2.19 can be extended also to the case of s > 1 in the following way: for any h εZ2, n N, and f Lp(εZ2) deﬁne the operator ∈ ∈ ∈ ℓ n n n Dnf(z)= ( 1)m+1f x + m h h m − 2 − m=0 X n n 1 s,p Z2 when n is even and Dh f = Dh− (Dhf) when n is odd. Then the norm of Wℓ (ε ) when n 1 ~~0, p [1, + ] and δ > 0 ∈ ∞~~

~~− s,p f Bs δ (εZ2) . f W (εZ2) k k p,p,ℓ k k ℓ where the norm W s,p(εZ2) is deﬁned in equation (28). k·k ℓ Lemma 2.21 Consider 0~~

f W s,p(εZ2) . f Bs (εZ2). k k ℓ k k p,p,ℓ s 2 s 2 Proof Consider f B (εZ ) B (εZ ). First we compute τy∆j f ∆j f p Z2 , for ε 6 y 6 1. p,p,ℓ p, ,ℓ Lℓ (ε ) For j = 1 we get ∈ ⊂ ∞ k − k −

τy∆ 1f ∆ 1f Lp(εZ2) = (τy I)K 1 ε ∆if Lp(εZ2) k − − − k ℓ k − − ∗ k ℓ i+1 <2 | X| . τyK 1 K 1 L1(εZ2) ∆if Lp(εZ2) k − − − k ℓ k k ℓ i+1 <2 | X| s . 2− τyK 1 K 1 L1(εZ2) f Bs (εZ2). k − − − k ℓ k k p,∞,ℓ Furthermore we get 1 y τyK 1 K 1 (x)= yK 1 x + t dt, | − − − | ∇ − y Z0 | | which implies that τyK 1 K 1 L1(εZ2) K 1 L1(εZ2) y k − − − k ℓ ≤ k∇ − k ℓ | |

~~16 ε ε s p 2 s Z2 and so τy∆ 1f ∆ 1f L (εZ ) . 2− y f B ∞ (ε ). k − − − k ℓ | |k k p, ,ℓ If 1~~

ε ε sJe ε sJe+Jε τy∆J f ∆J f Lp(εZ2) . 2− y KJ L1(εZ2) f Bs (εZ2) . 2− y f Bs (εZ2). k ε − ε k ℓ | |k∇ ε k ℓ k k p,∞,ℓ | |k k p,∞,ℓ Finally we can use the following trivial inequality

~~ε ε ε js τy∆i f ∆i f Lp(εZ2) 6 2 ∆i f Lp(εZ2) . 2− f Bs (εZ2). k − k ℓ k k ℓ k k p,∞,ℓ Exploiting the previous inequalities we get~~

(1 s)j sj τyf f Lp(εZ2) . f Bs (εZ2) y 2 − + 2− (29) k − k ℓ k k p,∞,ℓ | |  6 6 jXjy jy 0 independent of 0 <ε 6 1. By Lemma 2.10 and the embedding f Lp . f δ Z2 , inequality (32) implies that k k k kBp,p(ε )

~~ε2 ρ (x)p f p . f p ℓ ε B1+δ (εZ2) x R2 |∇ | k k p,p,ℓ Z ∈ z εZ2,i=1,2 ∈ X which in turn gives the statement. ✷~~

~~2.4 Extension operator on lattice~~

In this section we want to study the problem of extending a function (or a distribution) on the lattice Z2 R2 s R2 ε to a function (or distribution) on . Here in the deﬁnition of Besov space Bp,p,ℓ( ) we use the same dyadic partition of unity ϕj j> 1 introduced in Section 2.2. { } − 2 1 I 2 We introduce the reconstruction operator. Let w0(x)= 1 , 1 (x) and w0,ε(x)= ε− w0(ε− x). We (− 2 2 ] deﬁne an operator ε 2 2 : ′(εZ ) ′(R ) E S → S

17 2 ε 2 deﬁned as follows: if f ′(εZ ) we deﬁned the measurable function (f) on R as ∈ S E ε 2 I 2 Z (f)(z)= w0,ε ε f(z)= f(z′) ε , ε (z z′), f ′(ε ) (33) E ∗ ( 2 2 ] − ∈ S z′ εZ2 − X∈ ε In other words, for any f ′(R), the function (f) is constant on the squares ∈ S E ε ε ε ε 2 Q (z′) := + z′ , + z′ + z′ , + z′ R (34) ε −2 1 2 1 × −2 2 2 2 ⊂ i i Z2 ε R2 where z′ = (z1′ ,z2′ ) ε . More precisely for any z Qε(z′) the function (f) on is equal to f(z′) ∈ ∈ 2 E which is the value of f on the intersection Q (z′) εZ . ε ∩ Remark 2.22 Since, for any 0 <ε 6 1,

2 sup ρℓ(x) ρℓ(y), x,y R , (35) x y <ε ∼ ∈ | − | R s Z2 ε p R2 then for any s and p [1, + ] if f Bp,p,ℓ(ε ) then function (f) Lℓ ( ). Furthermore, if s> 0, we have ∈ ∈ ∞ ∈ E ∈ ε (f) Lp(R2) f Lp(εZ2) . f Bs (Z2). kE k ℓ ∼k k ℓ k k p,p,ℓ 1 ε s,p Z2 Theorem 2.23 For any p [1, + ) and 0

1 ε 1,p Z2 p R2 In the case s =1 we have that is continuous from Wℓ (ε ) into Bp,p,ℓ( ) with the operator norm being uniformly bounded in 0 <εE 6 1. s R2 Proof In order to prove the theorem we consider an equivalent formulation of the norm Bp, ,ℓ( ) (see, s R2 ∞ e.g., Theorem 6.9 Chapter 6 of [66]) namely, for any g Bp, ,ℓ( ), ∈ ∞ s g s R2 g p R2 + sup h − τhg g p R2 . (36) Bp,∞,ℓ( ) Lℓ ( ) Lℓ ( ) k k ∼k k 0< h 61,h R2 | | k − k | | ∈ ε ε By Deﬁnition 2.15, using the fact that f Lp(εZ2) (f) Lp(R2) and writing g = (f), we have k k ℓ ∼ kE k ℓ E s g p R2 + sup h − τhg g p R2 . f s,p Z2 . Lℓ ( ) Lℓ ( ) Wℓ (ε ) k k ε< h 61 | | k − k k k | | When h 6 ε the function τ g g is nonzero in a rectangle of area εh, and when it is nonzero we get | | | h − | s s τ g(z′) g(z′) 6 ε f(z) , | h − | |∇ε | 2 1 where z εZ such that z′ z 6 ε. Since h 6 ε and s< we get ∈ | − | | | p 1 s 1 1 h p − s s p s p s s s h − τhg g Lp(R2) . h − h ε − εf Lp(εZ2) . | | εf Lp(εZ2) . εf Lp(εZ2). | | k − k ℓ | | | | k∇ k ℓ ε k∇ k ℓ k∇ k ℓ ✷

Corollary 2.24 For any p [1, + ) and 0 < s 6 1 , the operator ε is continuous from Bs (εZ2) ∈ ∞ p E p,q,ℓ s R2 into Bp, ,ℓ( ). Furthermore the norm ∞ ε sup s Z2 s R2 < + . (Bp,q,ℓ(ε ),Bp,∞,ℓ( )) 0<ε61 kE kL ∞

~~s Z2 s,p Z2 ( Proof The corollary is a simple consequence of the continuous immersion Bp,p,ℓ(ε ) ֒ Wℓ (ε proved in Theorem 2.18. → ✷~~

18 s,p Z2 Theorem 2.25 Consider 0 < s 6 1 and let fε be a sequence of functions in Wℓ (ε ) such that p s,p ε R2 s R2 sup0<ε61 fε W (εZ2) < + , and (fε) f in Lℓ ( ), then f Bp, ,ℓ( ). Furthermore if s = 1 k k ℓ ∞ E → ∈ ∞ 1,p R2 1,p R2 and 1

s f s R2 f p R2 + sup h − τhf f p R2 Bp,∞,ℓ( ) Lℓ ( ) Lℓ ( ) k k ∼ k k h <1 | | k − k | | s f p R2 + sup sup h − τhf f p R2 Lℓ ( ) Lℓ ( ) ∼ k k δ>0 δ< h <1 | | k − k | | s ε ε . f p R2 + sup lim sup sup h − τh (fε) (fε) p Lℓ ( ) Lℓ k k δ>0 ε 0 δ< h <1 | | k E −E k → | | . f Lp(R2) + sup lim sup fε W s,p(εZ2) k k ℓ δ>0 ε 0 k k ℓ → . sup fε W s,p(εZ2) < + . 0<ε61 k k ℓ ∞

˜ p R2 For the case s = 1 we recall the following properties of Sobolev spaces: Deﬁne the operator Dh : Lℓ ( ) p R2 → Lℓ ( ) as τ g g D˜ (g)= h − , g Lp(R2),h R2 h h ∈ ∈ | | then if sup ( D˜hg p R2 ) < C, Lℓ ( ) h 61 k k | | where C R is a suitable constant, then g W s,p(R2) and ∈ + ∈ ℓ

~~g Lp(R2) 6 2C, k∇ k ℓ see, e.g., Theorem 3 in Chapter 5, Section 5.8 of [26] for the proof in unweighted case, the case with weights is similar. On the other hand we have that, if h 6 1, | |~~

Dhf Lp(R2) 6 lim sup εfε Lp(εZ2) 6 sup εfε Lp(εZ2) < + , k k ℓ ε 0 k∇ k ℓ 0<ε61 k∇ k ℓ ∞ → which implies that sup h 61 Dhf Lp(R2) 6 sup0<ε61 εfε Lp(εZ2) and so | | k k ℓ k∇ k ℓ

~~f Lp(R2) 6 2 sup εfε Lp(εZ2). k∇ k ℓ 0<ε61 k∇ k ℓ ✷~~

We prove now a result about the extension operator and Besov spaces with negative regularity s< 0 which, although probably not being optimal, is enough for proving the main results in the current paper.

~~Theorem 2.26 Consider p (1, + ), s 6 0, δ > 0 and ℓ R then the extension operator ε is s Z2 ∈ s ∞δ R2 ∈ E continuous from Bp,p,ℓ(ε ) into Bp,p,ℓ− ( ). Furthermore the norm~~

~~ε − sup (Bs (εZ2),Bs δ (R2)) < + . 0<ε61 kE kL p,p,ℓ p,p,ℓ ∞~~

Before proving Theorem 2.26, we introduce the notion of discretization operator ε from the space p 2 2 D of L (R ), where p [1, + ], into ′(εZ ) deﬁned as ℓ ∈ ∞ S 1 ε(ϕ)(z)= ϕ(x)dx, ϕ Lp(R2),z εZ2 D ε2 ∈ ℓ ∈ ZQε(z) where Qε(z) as deﬁned in equation (34).

19 ε p R2 p Z2 Lemma 2.27 Consider p (1, + ) then the linear operator : Lℓ ( ) Lℓ (ε ) is continuous with a norm uniformly bounded∈ in 0 <∞ ε 6 1, and also, writing qD (1, + ) such→ that 1 + 1 = 1, for any ∈ ∞ p q f Lp(R2) and g Lq(εZ2) we have ∈ ℓ ∈ ℓ

f(x) ε(g)(x)dx = ε(f)(z)g(z)dz. (37) R2 E Z2 D Z Zε s R2 Furthermore for any s> 0, δ > 0 and f Bp, ,ℓ( ) we have ∈ ∞ ε s−δ . s R2 f B (εZ2) f B ∞ ( ). kD k p,p,ℓ k k p, ,ℓ Proof Consider f Lp(εZ2) then, by Jensen inequality, we have ∈ ℓ p ε p 2 1 (f) Lp(εZ2) = ε 2 ρℓ(z)f(x)dx ℓ ε kD k 2 Qε(z) z εZ Z X∈ 2 1 p . ε ρℓ(x)f(x) d x ε2 | | z εZ2 ZQε(z) X∈ p p . ρℓ(x)f(x) dx = f Lp(R2). | | k k ℓ Z Equality (37) follows from the deﬁnition of ε(g). s R2 E N Consider now f Bp, ,ℓ( ), if s δ

n m p Dh f L (εZ2) ′ ℓ f s f p Z2 + sup k k ′ B Lℓ (ε ) m s k k p,∞,ℓ ∼k k Z2 h m=1 h ε ,0< h 61 ∧ X ∈ | | | | m R2 where Dh is the standard extension to of the operator having the same name introduced in (28). Z2 m ε When h ε the operator Dh commutes with the operator in the following sense: for any f Lp(R2) and∈ for even n D ∈ ℓ n m ε n m+1 ε Dh ( (f))(z) = ( 1) τ n m h (f)(z) D m − ( 2 − ) D m=0 X n 1 n m+1 = 2 ( 1) f(x)dx ε m n − Qε(z+( m)h) m=0 Z 2 − Xn 1 n = ( 1)m+1 f(x)dx 2 m n ε − Qε(z+( m)h) m=0 Z 2 − Xn 1 n n = ( 1)m+1 f x m h dx ε2 m − − 2 − m=0 ZQε(z) ε Xm = (D h(f))(z), D − and similarly for n odd. Then, using the result of the ﬁrst part of the proof, we get

n m ε p Dh ( (f)) L (εZ2) ε ′ ε ℓ (f) s ,p = (f) p Z2 + sup k D k′ W (εZ2) Lℓ (ε ) m s kD k ℓ kD k h εZ2,ε< h 61 h ∧ m=1 ∈ | | | | Xn ε m (D f) p Z2 ε h Lℓ (ε ) = (f) p Z2 + sup kD k′ Lℓ (ε ) m s kD k h εZ2,ε< h 61 h ∧ m=1 ∈ | | | | n X m D f p R2 h Lℓ ( ) p k k . f L (R2) + sup m s′ k k ℓ Z2 h m=1 h ε ,ε< h 61 ∧ X ∈ | | | | . f Bs (R2). k k p,∞,ℓ

20 ε − ε s,p ✷ Since, by Lemma 2.19 and Remark 2.20, (f) Bs δ (εZ2) . (f) W (εZ2) the lemma is proved. kD k p,p,ℓ kD k ℓ Proof of Theorem 2.26 Consider f Bs (εZ2) then we have ∈ p,p,ℓ

~~ε ε (f) s−δ R2 = sup (f)(x)g(x)dx . (38) Bp,p,ℓ( ) kE k −s+δ 2 R2 E g B (R ), g −s+δ =1 q,q,−ℓ B (R2) Z ∈ k k q,q,−ℓ~~

~~On the other hand, by Lemma 2.27, we have~~

~~ε(f)(x)g(x)dx = f(z) ε(g)(z)dz. R2 E Z2 D Z Zε Furthermore, since (1 ∆ Z2 ) is self-adjoint, by H¨older inequality, we get − ε f(z) ε(g)(z)dz Z2 D Zε~~

s δ/2 s+δ/2 ε = ρℓ(z)( ∆εZ2 + 1) − (f)(z)ρ ℓ(z)( ∆εZ2 + 1)− ( (g))(z)dz Z2 − − − D Zε s δ/2 s+δ/2 ε 6 ( ∆εZ2 + 1) − (f) Lp(εZ2) ( ∆εZ2 + 1)− ( (g)) Lq (εZ2). k − k ℓ k − D k −ℓ By Theorem 2.12 and Theorem 2.14 we obtain

s δ/2 ( ∆εZ2 + 1) − (f) Lp(εZ2) . f Bs (εZ2) k − k ℓ k k p,p,ℓ s+δ/2 ε ε q − ( ∆ Z2 + 1)− ( (g)) Z2 . (g) s+3δ/4 2 . ε L−ℓ(ε ) B (εZ ) k − D k kD k q,q,−ℓ Thus by Lemma 2.27 we have

~~ε − − − (g) B s+3δ/4(εZ2) . g B s δ (εZ2), kD k p,p,ℓ k k p,p,−ℓ and so by equality (38) the thesis holds. ✷~~

2.5 Besov spaces on R2 εZ2 × In this section we discuss the definition and properties of Besov spaces defined on a four dimensional space where the first two dimensions are continuous and the second two are discrete. More precisely we want to describe the following Besov spaces

Bs1,s2 (R2 εZ2) := Bs1 (R2,Bs2 (εZ2)) (39) p,ℓ1,ℓ2 × p,p,ℓ1 p,p,ℓ2 Bs (R2 εZ2) (40) p,r,ℓ × where p, r [1, + ], ℓ,ℓ ,ℓ R and s ,s R. ∈ ∞ 1 2 ∈ 1 2 ∈ Before giving a precise deﬁnition of the spaces (39) and (40), we introduce some sets of functions on the topological space R2 εZ2 equipped with the natural product topology. If p [1, + ) and × p 2 2 p 2 2 ∈ ∞ ℓ,ℓ1,ℓ2 R we deﬁne the (weighted) Lebesgue space L (R εZ ) and L (R εZ ) as the subset of ∈ ℓ1,ℓ2 × ℓ × (Borel) measurable functions f : R2 εZ2 R for which the norms × → p p f Lp (R2 εZ2) := f(x, z)ρℓ1 (x)ρℓ2 (z) dxdz k k ℓ1,ℓ2 × R2 εZ2 | | Z ×

p (4) p f Lp(R2 εZ2) := f(x, z)ρℓ (x, z) dxdz k k ℓ × R2 εZ2 | | Z × respectively are ﬁnite. First we consider the space of continuous functions C0(R2 εZ2) and also the space of diﬀerentiable function Ck(R2 εZ2), i.e. the subset of C0(R2 εZ2) for which× f Ck(R2 εZ2) if and only if for any α = (α , α ) N×2 with α = α + α 6 k we have× ∂αf C0(R2 ε∈Z2) where× 1 2 ∈ 0 | | 1 2 x ∈ × α α1 α2 ∂x f(x, z)= ∂x1 ∂x2 f(x, z)

21 2 and x = (x1, x2) R . It is important to note that there is a natural identification between the continuous functions C0(R2 ∈ εZ2) and the Fréchet space C0(R2, C0(εZ2)) of continuous functions from R2 into the Fréchet space C0×(εZ2) (which since εZ2 has the discrete topology is equivalent to the set of functions from εZ2 into R2). In a similar way we can identify the space Ck(R2 εZ2) with the Fréchet space Ck(R2, C0(εZ2)) of k differentiable functions from R2 into εZ2. We can also× define

2 2 2 2 2 2 2 2 (R εZ ) := (R ) ˆ (εZ ), ′(R εZ ) := ′(R ) ˆ ′(εZ ), S × S ⊗S S ⊗ S ⊗S 2 2 2 where (R ) and ′(R ) are the space of Schwartz functions and tempered distribution on R respectively, and ˆ Sis the (unique)S topological completion of the tensor product between the nuclear spaces (R2) and ⊗2 2 2 S 2 2 ′(R ) and the Fréchet spaces (εZ ) and ′(εZ ). By standard arguments we can identify (R εZ ) S 2 2 2 S 2 S S × and ′(R εZ ) with (R , (εZ )) (i.e. the space of Schwartz functions taking values in the the S × 2 S S2 2 Fréchet space (εZ )) and ′(R , ′(εZ )) (i.e. the space of tempered distribution taking values in the S S S Z2 R2 Z2 R2 T2 Fréchet space ′(ε )). We can define a Fourier transform from ( ε ) into 1 = S F S × S × ε R2 T2 ( ) ˆC∞ 1 as follows S ⊗ ε i(k x+h z) x z R2 T2 ε(f)(k,h)= e · · f(x, z)dxdz = ( ε (f)), k ,h 1 F R2 εZ2 F F ∈ ∈ ε Z × where f (R2 εZ2) and x is the Fourier transform with respect the continuum variables x R2 and z is∈ the S Fourier× transformF with respect the discrete variables z εZ2. We also consider the inverse∈ Fε ∈ 1 R2 T2 R2 Z2 Fourier transform ε− : 1 ( ε ) as F S × ε → S × 1 ˆ 1 i(k x+h z) ˆ 1,x 1,z R2 Z2 ε− (f)(x, z)= 4 e− · · f(k,h)dkdh = − ( ε− (f)), x ,z ε . F (2π) R2 T2 F F ∈ ∈ 1 Z × ε 1 R2 Z2 In a standard way it is possible to extend ε and ε− on ′( ε ). Using the previous identification R2 Z2 R2 Z2 F F S × x R2 Z2 if f ( ε ) or g ′( ε ) we can define the Schwartz function ∆i f ( ε ) and ∈ S × x ∈ S 2 × 2 ∈ S × tempered distribution g ′(R εZ ) as follows i ∈ S × x 1,x x x 1,x x ∆ f = K f = − (ϕ (f)), ∆ g = K g = − (ϕ (g)) i i∗x F iF i i ∗x F iF 2 where the convolution x is taken only with respect the real variables x R . We use a similar notation ∆z for the Littlewood-Paley∗ blocks with respect the discrete variable z ∈εZ2. j ∈ R R s R2 Z2 Definition 2.28 Consider p, q [1, + ], s and ℓ we define the Besov space Bp,q,ℓ( ε ) 2 2 ∈ ∞ ∈s 2 ∈ 2 × the subspace of ′(R εZ ) for which f B (R εZ ) if and only if S × ∈ p,q,ℓ ×

s(r 1) x z f Bs (R2 εZ2) := 2 − ∆i ∆j f , (41) k k p,q,ℓ ×   16i, 16j6J ,max(i,j)=r 1 p  ε L (R2 εZ2)  − − X − ℓ r N q N × 0 ℓ ( 0) ∈   is ﬁnite.   2 2 Hereafter we write, for any f ′(R εZ ) and r > 1, ∈ S × −

~~1 ε ∆˜ f = − ϕ (k)ϕ (q) (f)(k, q) . (42) r Fε  i j F  16i, 16j6J ,max(i,j)=r − − Xε  ~~

~~Remark 2.29 Consider the functions K˜ ε : R2 εZ2 R, r > 1, deﬁned as r × → −~~

~~ε 1 ε K˜ (x, z) = − ϕ (k)ϕ (q) r Fε  i j  16i, 16j6J ,max(i,j)=r − − Xε  ε  = Ki(x)Kj (z). 16i, 16j6J ,max(i,j)=r − − Xε~~

~~22 ˜ ε When r 6 Jε 1 the functions Kr does not depend on ε (in the sense explained in Remark 2.6 and Remark 2.7) and− we have~~

K˜ ε(x, z)= K (x) K (z)+ K (x) K (z)+ K (x)K (z). r r j  i  r r r 16j6r 1 16i6r 1 − X − − X −   Using the fact that (ϕj )j> 1 is a dyadic partition of unity it is simple to see that − 2r r 2 K 1(2 x)= Kj (x). − 16j6r − X This means that, when r 6 J , there area ¯ R and β¯ (0, 1) such that ε ∈ + ∈ ¯ K˜ ε(x, z) . 24r exp( a¯(1+22r x 2 +22r z 2)β ), | r | − | | | | where the constants hidden in the symbol . do note depend on 0 <ε 6 1 and 1 6 r 6 J 1. − ε − In the case r = Jε, the following equality holds

~~ε ε K˜ (x, z)= KJ (x)+ Ki(x) K (z). Jε ε   Jε 16i6J 1 − X ε−   Thus, by Remark 2.1 and Remark 2.7, we obtain~~

¯ ¯ K˜ ε (x, z) . (22Jε exp( a¯(1+22Jε x 2)β)+24Jε exp( a¯(1+22r x 2 +22r z 2)β)), | Jε | − | | − | | | | where the constants hidden in the symbol . do note depend on 0 <ε 6 1. 2 Finally, when r>Jε, the function K˜r(x, z) = Kr(x) does not depend on z εZ , and, by Remark 2.1, we get ∈ ¯ K˜ ε(x, z) . 22r exp( a¯(1+22r x 2)β) | r | − | | where, as usual, the constants hidden in the symbol . do note depend on 0 <ε 6 1.

Deﬁnition 2.30 Consider p [1, + ], s1,s2 R, ℓ1,ℓ2 R we deﬁne the Besov space of functions from R2 taking values in the Banach∈ ∞ space Bs2 ∈ (εZ2) as∈ the Banach space of f (R2 εZ2) such p,ℓ1,ℓ2 ′ that ∈ S ×

~~s1(j 1) x s1,s2 2 2 s2 2 f B (R εZ ) := 2 − ∆(j 1)(f) B (εZ ) p . (43) k k p,ℓ2,ℓ2 × k − k p,p,ℓ2 L (R2) ( ℓ1 ) N j 0 p N ∈ ℓ ( 0)~~

Remark 2.31 It is important to note that the norm (43) coincides with the one of the Besov space Bs1 of distributions from R2 taking values in the Banach space Bs2 (εZ2). More generally if E is p,p,ℓ1 p,p,ℓ2 R s Rd a Banach space, p, q [1, + ] and s,ℓ it is possible to deﬁne the Besov space Bp,q,ℓ( , E) of the d ∈ ∞ ∈ d d distributions on R taking values in E as the subset of ′(R , E) := ′(R ) ˆ E with the ﬁnite norm S S ⊗ s1(j 1) s p q g B (Rd,E) = 2 − ∆(j 1)(f) E L (R2) j N0 ℓ (N ) k k p,q,ℓ k{ kk − k k ℓ } ∈ k 0 d where g ′(R , E) (see Chapter 2 of [11] for the details). We will use this notion and the notation s R2∈ S Bp,q,ℓ( , E) in what follows.

~~We prove a useful equivalent expression of the norm Bs1,s2 (R2 εZ2). k·k p,ℓ2,ℓ2 ×~~

~~s1,s2 2 2 Theorem 2.32 Consider p [1, + ), s1,s2 R, ℓ1,ℓ2 R if f B (R εZ ) we have that ∈ ∞ ∈ ∈ ∈ p,ℓ1,ℓ2 ×~~

p s1ip s2jp x z p p p f Bs1,s2 (R2 εZ2) = 2 2 ∆i ∆j f(x, z) ρℓ1 (x) ρℓ2 (z) dxdz, k k p,ℓ2,ℓ2 R2 Z2 | | × i,j> 1,j6J ε −X ε Z × s1ip s2jp x z p = 2 2 ∆i ∆j f Lp (R2 εZ2), (44) k k ℓ1,ℓ2 i,j> 1,j6J × −X ε

23 where ∆x∆zf = K x (Kε z f) i j i ∗ j ∗ where x is the convolution with respect the continuum variables x R2 and z is the convolution with respect∗ the discrete variables z εZ2. ∈ ∗ ∈ 2 2 2 2 Proof The proof follows from the fact that we can identify ′(R , ′(εZ )) with ′(R εZ ), from the S S S × s monotone convergence theorem and from the definition of the norm (17) of Besov space B 1 (εZ2). ✷ p,p,ℓ1 We now propose an extension to the Besov spaces on R2 εZ2 of the results of the previous sections. × 2 1 2 Lemma 2.33 Assume that the smooth function σ = σ(k, q): R ε− T R satisfies × → α β 2 2 (m ( α + β ))/2 ∂ ∂ σ(k, q) 6 C(1 + k + q ) − | | | | , | k q | | | | | where α, β N2, α , β 6 4, and m R. Consider s R, ℓ R and p, r [1, + ], then, the operator ∈ 0 | | | | ∈ ∈ ∈ ∈ ∞ σ 1 s 2 2 T f = − (σ(k, q)( f)(k, q)), f B (R εZ ) Fε Fε ∈ p,ℓ × s 2 2 s m 2 2 is well defined from B (R εZ ) into B − (R εZ ) and p,r,ℓ × p,r,ℓ × σ − T f Bs m(εZ2) . f Bs (εZ2). (45) k k p,r,ℓ k k p,r,ℓ

Furthermore if m 6 0 and considering s1,s2 R and θ1,θ2 > 0 such that 0 < θ1 + θ2 6 1 we have that σ s1,s2 2∈ 2 s1 θ1m,s2 θ2m 2 2 T is a well defined operator from B (R εZ ) into B − − (R εZ ) and we have p,ℓ1,ℓ2 × p,ℓ1,ℓ2 × σ − − s ,s T f s1 θ1m,s2 θ2m 2 2 . f 1 2 R2 Z2 . (46) B (R εZ ) Bp,ℓ ,ℓ ( ε ) k k p,ℓ1,ℓ2 × k k 1 2 × Proof The proof follows the same arguments of Lemma 2.10, we report here only the main differences. If we define

Kσε (x, z) λ1,λ2 1 i(z q)+(x k) ˜ ˜ = 4 e · · ψ(k)ψ(q)σε(λ1k, λ2q)dkdq (2π) R2 T2 Z × 1 λ2ε 1 i(z q)+(x k) ˜ ˜ = 4 e · · ψ(k)ψ(q)σε(λ1k, λ2q)dkdq (2π) R4 Z i j Jε 2 where λ1 = 2 , λ2 = 2 6 2 and ψ : R [0, 1] is a suitable function supported in an annulus B (0) B (0) of constant radius R > R > 0→ as in the proof of Lemma 2.10, then we have R1 \ R2 1 2

~~2 2 M σ α1 β1 | | | | (1 + z + x ) Kλ ,λ (x, z) 6 C β + α cα1,α2,β1,β2 λ1 λ2 | | | | | 1 2 | | | | | × α1 + a2 + β1 + β2 =2M | | | | X| | | |~~

~~i(z q)+(x k) α2 ˜ β2 ˜ e · · ∂q ψ(k)∂q ψ(q) R2 × Z 2 2 (m β α )/2 (1 + λ k + λ q ) −| 1|−| 1| dkdq . | 2 | | 2 |~~

~~˜ ˜ ˜ So, since ψ is supported in an annulus of constant radius, we can estimate on the support of ψ(k)ψ(q):~~

α1 β1 2 2 ( α1 + β1 )/2 λ1| |λ|2 |(1 + λ1k + λ2q )− | | | | α1 2 2| α | /2 | β1| 2 2 β /2 = (λ| |(1 + λ k + λ q )−| 1| )(λ| |(1 + λ k + λ q )−| 1| ) 1 | 1 | | 2 | 2 | 1 | | 2 | α1 2 α1 /2 β1 2 β1 /2 6 λ|1 |(1 + λ1k )−| | λ2| |(1 + λ2q )−| | | | α1 β1 | | 6 R1| |R1| |.

~~On the other hand on the support of ψ˜(k)ψ˜(q)~~

~~m m 2 2 m/2 R1 (max(λ1, λ2)) if m 6 0 (1 + λ1k + λ2q ) 6 m m . | | | | (1+2R2) (max(λ1, λ2)) ifm> 0 ~~

~~24 This implies that m 1 Kλ1,λ2 L (R2) . max(λ1, λ2) k k l which, by following the same argument of the proof of Lemma 2.10, implies inequality (45). On the other hand, when m 6 0, we have~~

(1 + λ k 2 + λ q 2)m/2 6 (1 + λ k 2)θ1m/2(1 + λ q 2)θ2m/2 6 R λθ1mλθ2m, | 1 | | 2 | | 1 | | 2 | 1 1 1 and following the previous line of reasoning we can analogously also obtain the second claim. ✷ Hereafter we write ∆R2 εZ2 = ∆R2 + ∆εZ2 × 2 2 where ∆R2 is the standard Laplacian on R acting on the continuum variables x R , and ∆εZ2 is the discrete Laplacian on εZ2 acting on the discrete variables z εZ2. ∈ ∈ R R s R2 Z2 Theorem 2.34 Consider s , ℓ and p, q [1, + ] then for any f Bp,q,ℓ( ε ) and any m R we have ∈ ∈ ∈ ∞ ∈ × ∈ (4) f Bs (R2 εZ2) f ρℓ (x, z) Bs (R2 εZ2) k k p,q,ℓ × ∼k · k p,q,ℓ × m − ( ∆R2 εZ2 + 1) (f) Bs m(R2 εZ2) f Bs (R2 εZ2). k − × k p,q,ℓ × ∼k k p,q,ℓ × s1,s2 2 2 Consider p [1, + ], s1,s2 R and ℓ1,ℓ2 R then, for any f B (R εZ ),and m 6 0, θ1,θ2 > 0 ∈ ∞ ∈ ∈ ∈ p,ℓ1,ℓ2 × and 0 <θ1 + θ2 < 1 we have

s ,s s ,s f B 1 2 (R2 εZ2) f ρℓ1 (x)ρℓ2 (z) B 1 2 (R2 εZ2), k k p,ℓ1,ℓ2 × ∼k · k p,0,0 × m − − s ,s ( ∆R2 εZ2 + 1) (f) s1 θ1m,s2 θ2m 2 . f 1 2 Z2 . B (εZ ) Bp,ℓ ,ℓ (ε ) k − × k p,ℓ1,ℓ2 k k 1 2 Proof The proof is analogous to the one of Theorem 2.12, where we use the expressions of the norm given in Theorem 2.32, and the results of Lemma 2.10 are now replaced by the analogous ones of Lemma 2.33. Indeed we have that

~~m m R2 Z2 σ( ∆ 2 2 +1) = ε(( ∆ ε + 1) ) − R ×εZ F − × m 4 εq = k 2 + sin2 i +1 | | ε2 2  j=1,2 X  2 m  = k + σ 2 (q)+1 | | −εZ We have that~~

2 2 m α m ( k + q + 1) −| | for m > 0 α ∂ σ (k, q) . | | | | 2 m α (47) ( ∆R2×εZ2 +1) 2 2 q −| | − ( k + | | +1 for m< 0 | | π where the constants hidden in the symbol . do not depend on 0 < ε 6 1. Inequality (47) permits the application of Lemma 2.33 in a way analogous of the one of the proof of Theorem 2.12, getting the thesis. ✷

1 1 1 Theorem 2.35 Consider s1 6 0,s2 > 0 such that s1 +s2 > 0, p1,p2, q1, q2 [1, + ] , + = < 1, p1 p2 p3 1 1 1 2 ∈ ∞2 2 and + = < 1 then the product Π(f,g) = f g deﬁned on ′(εZ ) (εZ ) into ′(εZ ) can be q1 q2 q3 extended in a unique continuous way to a map · S × S S

~~Π: Bs1 (R2 εZ2) Bs2 (R2 εZ2) Bs1 (R2 εZ2), p1,q1,ℓ1 × × p2,q2,ℓ2 × → p3,q3,ℓ1+ℓ2 × with the estimate, for any f Bs1 (R2 εZ2) and g Bs2 (R2 εZ2), ∈ p1,q1,ℓ1 × ∈ p2,q2,ℓ2 ×~~

~~f g Bs1 (R2 εZ2) . f Bs1 (R2 εZ2) g Bs2 (R2 εZ2). k · k p3,q3,ℓ1ℓ2 × k k p1,q1,ℓ1 × k k p2,q2,ℓ2 ×~~

25 Proof The proof is similar to the one sketched in the proof of Theorem 2.13. Indeed the support of 2 2 r the Fourier transform of ∆˜ r(f), for f ′(R εZ ) and r > 0, is supported in the set (B(0, 2 R1) B(0, 2rR )) (B(0, 2rR ) B(0, 2rR )) (where∈ S B×(x, R) R2 is the ball of radius R> 0 and center x R2×) 1 \ 2 × 2 ⊂ ∈ for suitable positive constants R1 > R2 > 0. From this observation the proof follows the same line of the one on Rd (see, e.g., Chapter 2 of [13] and the proof of Theorem 3.17 of [49]). ✷

Theorem 2.36 Consider p ,p , q , q [1, + ], s >s and ℓ ,ℓ R if 1 2 1 2 ∈ ∞ 1 2 1 2 ∈ 4 4 ℓ1 6 ℓ2, s1 > s2 , − p1 − p2 then Bs1 (R2 εZ2) is continuously embedded in Bs2 (R2 εZ2), and the norm of the embedding p1,q1,ℓ1 p2,q2,ℓ2 is uniformly bounded× in 0 <ε 6 1. × Furthermore if p ,p [1, + ], s ,s ,s ,s R such that s >s , s >s , and ℓ ,ℓ ,ℓ ,ℓ R if 1 2 ∈ ∞ 1 2 3 4 ∈ 1 3 2 4 1 2 3 4 ∈ 2 2 2 2 ℓ1 6 ℓ3, ℓ2 6 ℓ4, s1 > s3 , s2 > s4 − p1 − p2 − p1 − p2 then Bs1,s2 (R2 εZ2) is continuously embedded in Bs3,s4 (R2 εZ2), and the norm of the embedding p1,ℓ1,ℓ2 p2,ℓ3,ℓ4 is uniformly bounded× in 0 <ε 6 1. × Proof The proof of the ﬁrst part of the theorem can be reduced to the one of Theorem 2.14, using the same of observation in the proof of Theorem 2.35. The second part of the theorem is a special case of Theorem 2.2.4 and Theorem 2.2.5 of [11] about the embedding of Besov spaces formed by distributions taking values in Banach spaces. ✷

~~2.6 Diﬀerence spaces and extension operator on R2 εZ2 ×~~

~~First we introduce the analogous of the diﬀerence spaces introduced in Section 2.3.~~

R s,p R2 Z2 Deﬁnition 2.37 Consider s > 0, p (1, + ) and ℓ we deﬁne the space Wℓ ( ε ) as the linear subspace of Lp(R2 εZ2) whose∈ elements∞ f W s,p∈ (R2 εZ2) are such that the norm× ℓ × ∈ ℓ ×

~~f W s,p(R2 εZ2) := f Lp + f SW s,p(R2 εZ2), k k ℓ × k k ℓ k k ℓ × where SW s,p(R2 εZ2) is the seminorm k·k ℓ ×~~

τhf f p R2 Z2 Lℓ ( ε ) f s,p R2 Z2 = sup k − k × , SWℓ ( ε ) s k k × 0< h 61,h R2 εZ2 h | | ∈ × | | where τhf(y)= f(h + y), is ﬁnite. For the space W s,p(R2 εZ2) we can prove a result analogous to Theorem 2.18. ℓ ×

~~Theorem 2.38 Consider 0 < s 6 1, p (1, + ) and ℓ R then for any δ1 > 0 and any f Bs (R2 εZ2) we have ∈ ∞ ∈ ∈ p,p,ℓ ×~~

− s,p f s δ1 2 2 . f R2 Z2 . f s+δ1 2 2 (48) B (R εZ ) Wℓ ( ε ) B (R εZ ) k k p,p,ℓ × k k × k k p,p,ℓ × Proof We note that for any h R2 εZ2 we have ∈ × i(h (k,q)) (τ f)(k, q)= e · (f). (49) Fε h Fε Using identity (49) and Lemma 2.33, the arguments of Lemma 2.19, Lemma 2.21 and Theorem 2.18 can be easily adapted for proving inequalities (48). ✷

26 ε 2 2 2 2 2 2 We study the following extension operator from ′(R εZ ) into ′(R R ) ′(R ) ˆ ′(R ) defined as E S × S × ≃ S ⊗S ε ε := I ′(R2) E S ⊗E ε 2 2 2 2 where : ′(εZ ) ′(R ) is defined in (33). In particular if f is a measurable function on R εZ ε we canE identifyS (f→) withS a measurable function on R4 and we have × E ε (f)(x, z)= f(x, z′)I ′ (z) E Qε(z ) z′ εZ2 X∈ where, as usual Qε(z′) is defined in equation (34).

~~Theorem 2.39 For any p (1, + ), 0~~

ε s,p R4 s R4 Furthermore if sup0<ε61 fε W (R2 εZ2) < + and (fε) f in ′( ), then f Bp, ,ℓ( ) and ℓ × ∞ k k ∞ E → S 1,p 4 ∈ s R4 s,p R2 Z2 R 1,p f B ∞ ( ) . sup0<ε61 fε W ( ε ), furthermore when s = 1 f Wℓ ( ) and f W (R4) . k k p, ,ℓ k k ℓ × ∈ k k ℓ sup0<ε61 fε W 1,p(R2 εZ2). Finally if s 6 0 and δ > 0 we get k k ℓ × ε − (fε) Bs δ (R4) . fε Bs (R2 εZ2). kE k p,p,ℓ k k p,p,ℓ × Proof The proof is a suitable modiﬁcation of the proof of Theorem 2.23, Theorem 2.25 and Theorem 2.26. ✷ We conclude this section with an useful compactness result about diﬀerences and Besov spaces on R2 εZ2. ×

s2,p 2 2 Theorem 2.40 Consider p [1, + ], 0 < s1 < s2 6 1 and ℓ1 > ℓ2 then the space W (R εZ ) is ∈ ∞ ℓ1 × compactly embedded in W s1,p(R2 εZ2). ℓ2 × Proof We prove the theorem in the case ℓ2 = 0. The general case can be obtained as in exploiting the same argument of the proof of Theorem 2.14. s,p Z2 p Z2 s2,p R2 Z2 For ε > 0, since W0 (ε ) is isomorphic to L (ε ), the space W0 ( ε ) is isomorphic to s2 R2 p Z2 p Z2 p Z2 × Bp, ,0( ,L (ε )). Since L (ε ) is compactly embedded in L ℓ1 (ε ), by Theorem 7.2.3 of [11], ∞ 2 s2 R2 p Z2 the space Bp, ,0( ,L (ε )) is compactly embedded (with respect to the natural embedding) in the ∞ space Bs1 R2,Lp (εZ2) . Using the isomorphism between Lp (εZ2) and Bs1 (εZ2), the space p, , ℓ1 ℓ1 ℓ1 p, , ℓ1 ∞ 2 2 2 ∞ 2 Bs1 R2,Lp (εZ2) is continuously embedded in W s1,p(R2 εZ2). The thesis, thus, follows from p, , ℓ1 ℓ1 ℓ1 ∞ 2 2 × the fact that the composition of a compact embedding a continuous embedding is compact. ✷

s,p Z2 p Z2 Remark 2.41 The isomorphism between W0 (ε ) and L (ε ) (equipped with their natural norm (23) and (14) respectively) is not an isometry, and the norm of this isomorphism diverges to + as ε 0. This is the reason why we cannot use it in the proofs of the previous results. Since compactness∞ is → s,p Z2 p Z2 a topological property and not a metric one we can use the isomorphism between W0 (ε ) and L (ε ) in the proof of Theorem 2.40.

s2 2 2 Corollary 2.42 Consider 0 < s1 < s2 6 1 and ℓ1 > ℓ2 then the space B (R εZ ) is compactly p,p,ℓ1 × embedded in Bs1 (R2 εZ2). p,p,ℓ1 × Proof The corollary is a consequence of Theorem 2.38 and Theorem 2.40. ✷

~~27 2.7 Positive distribution and Besov spaces~~

In this section we want to study positive (tempered) distribution on Rd or R2 εZ2. Namely we 4 2 2 × 4 want to consider elements η or ηε respectively in ′(R ) or ′(R εZ ) such that, for any f (R ) or f (R2 εZ2) positive, i.e. for each y R4 andS (x, z) SR2 ε×Z2 we have f(y),f (x, z) >∈0, S we have ε ∈ S × ∈ ∈ × ε f, η , f , η > 0. h i h ε εi For well known theorems (see, for example, [64]) on positive tempered distribution on Rd, which can be easily extended to the case of R2 εZ2. In particular we can identify every positive tempered distribution on R2 εZ2 with a Radon (σ-ﬁnite)× positive measure (see Theorem 21.1 of [64] for a proof in Rd case). This implies× that considering a positive distribution in Bs (Rd), Bs1,s2 (R2 εZ2) etc. is equivalent p,q,ℓ p,ℓ1,ℓ2 to consider a Radon measure that is also in the previous spaces. ×

~~We introduce the following functions~~

a,β 2i+2j 2i 2 2j 2 β E (x, z)=2 exp a(1+2 x +2 z ) 2 (50) i,j − | | | | where a R , 0 <β< 1, i, j N and 0 6 j 6 J , and ∈ + ∈ ε a,β 2i 2i 2 β E˜ (x)=2 exp a(1+2 x ) 2 (51) i − | | We introduce the following positive functionals

1/p ′ ′ Ms,s (η,a,β) := 2spj+s pi (Ea,β (ρ (x)ρ (z)η(x, z)))pdxdz (52) p,ℓ1,ℓ2,ε i,j ℓ1 ℓ2  R2 Z2 ∗  i N ,j6J ε ∈ X0 ε Z ×  1/p  ′ ′ Ms,s (η,a,β) := 2spj+s pi (Ea,β (ρ (x)ρ (z)η(x, z)))pdxdz (53) p,ℓ1,ℓ2 i,j ℓ1 ℓ2  R4 ∗  i,j N X∈ 0 Z  1/p Ns rsp a,β (4) p p,ℓ,ε(η,a,β) := 2 (Er,r (ρℓ (x, z)η(x, z))) dxdz  R2 Z2 ∗  16r6J ε − X ε Z ×  1/p rsp ˜a,β (4) p + 2 (Er x(ρℓ (x, z)η(x, z))) dxdz  R2 Z2 ∗  > ε rXJε Z ×  1/p  + (E˜a,β (ρ(4)(x, z)η(x, z)))pdxdz (54) Jε z ℓ R2 εZ2 ∗ Z × 1/p Ns rsp a,β (4) p p,ℓ(η,a,β) := 2 (Er,r (ρℓ (x, z)η(x, z))) dxdz (55)  R4 ∗  r> 1 X− Z  

~~Theorem 2.43 For any p [1, + ], s1,s2,s R and ℓ1,ℓ2 R, for any positive Radon measures s1,s2 2 2 ∈ ∞s1,s2 4 ∈ ∈ ηε B (R εZ ) and η B (R ) we have ∈ p,p,ℓ1,ℓ2 × ∈ p,p,ℓ1,ℓ2~~

s1,s2 s1,s2 η s1,s2 2 2 . M (η , a,¯ β¯), η s1,s2 4 . M (η, a,¯ β¯), ε B (R εZ ) p,ℓ1,ℓ2,ε ε B (R ) p,ℓ1,ℓ2 k k p,ℓ1,ℓ2 × k k p,ℓ1,ℓ2 where a¯ R+ and β¯ (0, 1) are the constants appearing in Remark 2.1, Remark 2.6 and Remark 2.7. Furthermore∈ for any s∈ R and η Bs (R2 εZ2) and η Bs (R4) we have ∈ ε ∈ p,p,ℓ × ∈ p,p,ℓ Ns ¯ Ns ¯ ηε Bs (R2 εZ2) . p,ℓ,ε(ηε, a,¯ β), η Bs (R4) . p,ℓ(η, a,¯ β). k k p,p,ℓ × k k p,p,ℓ

28 Proof The thesis is a consequence of the fact that ηε, η are positive measure (thus fε, ηε 6 gε, ηε , 2 2 4 h i h i f, η 6 g, η for any fε,gε (R εZ ) and f,g (R ) such that fε 6 gε and f 6 g) and of Remark 2.1,h Remarki h i 2.6 and Remark∈ 2.7.S × ∈ S ✷

M s1, s2 Theorem 2.44 Consider a positive measure η, suppose that r,ℓ,ε− − (η,a,β)) < + for some s1,s2 > p′ ∞ 0, ℓ > 0 and r [1, + ] and consider g L (dη) for some p′ (1, + ). Then we have that gdη ∈ ∞ ∈ ℓ ∈ ∞ ∈ k1 R2 k2 Z2 1 1 2 si Bp,p,ℓ− ( ,Bp,p,ℓ− (ε )) with p = rq′, where p′ + q′ =1, and ki = p′ + q′ . Furthermore we have

s1, s2 r/p −k ,−k . p′ M− − ¯ gdη B 1 2 (R2 εZ2) g L (R2 εZ2,dη)( r,ℓ,ε (η, q′a,¯ β)) k k p,ℓ,ℓ × k k ℓ × where a¯ R and β¯ (0, 1) are the constants in Remark 2.1, Remark 2.6 and Remark 2.7. ∈ + ∈ Proof Let K ( ) be the Littlewood-Paley function linked with the Littlewood-Paley block ∆ . By our j · j assumptions on Kj (see Remark 2.1, Remark 2.6 and Remark 2.7) there exists somea ¯ R+ and β¯ (0, 1) such that ∈ ∈ ¯ ¯ K (x) . 22j exp( a¯2jβ x β ) | j | − | | for i N , x R2 and uniformly in ε. We have that by H¨older’s inequality ∈ 0 ∈

Ki xKj z(ρℓ′ (x′)ρℓ′ (z′)gdη)(x, z) ∗ ∗ 1 q′ q′ q . p′ ′ ′ g L (dη) R2 εZ2 Ki(x x′) Kj(z z′) ρℓ (x′)ρℓ (z′)dη(x′,z′) ℓ′ k k × | − | | − | 1 R a,¯ β¯ q′ q (56) . p′ ′ ′ g L (dη) (Ei,j (x x′,z z′)) ρℓ (x′)ρℓ (z′)dη(x′,z′) ℓ′ k k − − 1 2j +2i ′ ¯ q p′ Rp′ q a,¯ β . p′ ′ ′ g L (dη)2 Ei,j (x x′,z z′)ρℓ (x′)ρℓ (z′)dη(x′,z′) , k k ℓ′ − − R where we used that

~~′ a,¯ β¯ q′ 2i(q′ 1)+2j(q′ 1) q a,¯ β¯ (E (x x′,z z′)) =2 − − E (x x′,z z′). i,j − − i,j − − On the other hand, by Theorem 2.32, we have~~

p k1pi k2pj gdη − − . 2− − s1, s2 R2 Z2 k kB ′ ′ ( ε ) × p,ℓ ,ℓ × i>0 16j6J X − X ε p Ki x Kj z (ρℓ′ (x)ρℓ′ (z)gdη) dxdz × R2 εZ2 | ∗ ∗ | Z × 2 2 p k1 ′ pi k2 ′ pj − − p − − p . g p′ 2 k kL ′ (dη) × ℓ i>0 16j6J X − X ε ′ ¯ p q a,¯ β q′ (Ei,j (ρℓ(x)ρℓ(z)η(x, z))) dxdz × R2 εZ2 ∗ Z × p s1ri s2rj . g p′ 2− − k kL ′ (dη) × ℓ i>0 16j6J X − X ε ′ q a,¯ β¯ r (Ei,j (ρℓ(x, z)η(x, z))) dxdz × R2 εZ2 ∗ Z × p s1, s2 r . g ′ (M− − (η, q′a,¯ β¯)) . Lp (dη) r,ℓ,ε k k ℓ′ ✷

~~3 Stochastic estimates and regularity of the noise~~

~~In this section we want to study the regularity of the stochastic term in the equation.~~

~~29 3.1 The setting~~

4 2 2 Let ξ be a white noise on R . Using the noise ξ we can define a white noise ξε on R εZ in the following way × ξ (x, z)= ξ(δ I ) ε x ⊗ Qε(z) 2 where Qε(z) is defined in equation (34) δx is the Dirac delta with unitary mass in x R . We can define also the following natural free fields on R4 and R2 εZ2. ∈ × 2 1 W = ( ∆R4 + m )− (ξ), − 2 1 Wε = ( ∆R2 εZ2 + m )− (ξε), − × ¯ 2 1 Wε = ε(( ∆R2 εZ2 + m )− (ξε)). E − × We have to define four Green functions:

: R2 εZ2 R, : R4 R, : R4 R, : R4 R, Gε × → Gε → Gε → G → where ei(x y+k z) 1 e− · · ε(x, z) := 4 2 dydk; G (2π) R2 T2 2 2 2 εk1 2 2 εk2 2 1 y +4ε− sin 2 +4ε− sin 2 + m Z × ε | | 2 εk1 2 εk2 i(x y+ k z) 1 sin 2 sin 2 e− · · ε(x, z) := 4 4 2 2 2 dydk G (2π) R4 ε k k 2 2 2 εk1 2 2 εk2 2 1 2 y +4ε− sin +4ε− sin + m Z | | 2 2 2 εk1 2 εk2 i(x y+k z) 1 sin 2 sin 2 e− · · ε(x, z) := 4 4 2 2 2 2 2 2 dydk G (2π) R4 ε k k ( y + k + m ) Z 1 2 | | | | and e i(x y+k z) 1 e− · · (x, z) := 4 2 2 2 2 dydk. G (2π) R4 ( y + k + m ) Z | | | | Using the previous notation we have that

~~(x x′,z z′)= E[W (x, z)W (x′,z′)], (x x′,z z′)= E[W (x, z)W (x′,z′)], G − − Gε − − ε ε~~

(x, z)= E[W¯ (x, z)W¯ (x′,z′)]. Gε ε ε 3.2 Wick exponential and multiplicative Gaussian chaos Giving a meaning to equation (9) will require to us deﬁne the exponential of W . However W is almost surely not a function, so we require renormalization. We want to deﬁne

2 α α 2 µ =: exp(αW ) :=′′ exp αW E[W ] ′′ − 2 which does not immediately make sense since E[W 2]= . However we can approximate it in the following ∞ way: Let σs be such that T ρ (p) σ2(n)ds = T s (m2 + p 2)2 Z0 | | 4 with ρ C∞(R ) depending smoothly on T and ρ 1 as T . Then deﬁne T ∈ c T → → ∞ T WT = σ(D)dXt Z0 2 4 where Xt is a cylindrical Browning motion in L (R ). Since WT has almost surely compact support in frequency it is a function and we can compute by Ito’s isometry

~~E[W (x)W (y)] = K (x y) T T T −~~

30 1 ρT (n) where we have introduces KT = − (m2+ n 2)2 . From this it follows that F | | W W T → as T at least in law since W is Gaussian. We can then introduce the approximation → ∞ T α2 µα,T = exp αW E[W 2 ] . T − 2 T n 1 Recall that for WT we can also deﬁne the wick powers : WT : inductively by : WT := WT and

n+1 n : WT := WT d : WT : . Z An alternative definition of µα,T would be to write the exponential as a series and Wick order the individual terms. The following lemma shows that these two ways are equivalent and that µα,T converges α α to a well defined limit as T which we will denote by µ . Furthermore define µ− as the limit of → ∞ 2 α,T α 2 µ− = exp αW E[W ] − T − 2 T Lemma 3.1 For any α R,we have that ∈ α2 αn exp αW E[W 2 ] = : (W )n : . T − 2 T n! T X Furthermore, when α2 6 2(4π)2, we get that

n n α n α n lim : WT := : W : T n! n! →∞ X X almost surely. Proof We deﬁne α2 M 1 = exp αW E[W 2 ] T T − 2 T and αn M 2 = : W n : . T n! T 1 2 1 X2 We claim that MT = MT . Obviously M0 = M0 . By the well known calculation for the stochastic exponential 1 1 dMT = αMT dWT , 2 from which it also follows that M is a local martingale. Furthermore we will show below that sup E[ M − ] < T T T B s k k 2,2,l s for some s> 0, so considered as a process on the Hilbert space B2−,2,l MT has bounded quadratic vari- ∞ s ation and converges to some M in B2−,2,l almost surely. On the other hand ∞

~~αn dM 2 = d : W n : T n! T n N ! X∈ αn = d : W n : n! T n N ! X∈ n α n 1 = : W − : dW (n 1)! T T n N ! X∈ − 2 = αMT dWT .~~

31 Now the result follows by standard SDE theory. ✷ We also deﬁne the positive measures on R4 and R2 εZ2 × α2 µα =: exp(αW ) := exp αW E[W 2] , ε ε ε − 2 ε and α2 µ¯α =: exp(αW¯ ) := exp αξ¯ E[W¯ 2] . ε ε ε − 2 ε We recall that, using the fact that commutes with the composition with functions, we get Eε µ¯α(x, y) := (µα)(x, y). ε Eε ε Furthermore we can show in analogy with Lemma 3.1 that αn αn µα = : W n : , µ¯α = : W¯ n : ε n! ε ε n! ε X X We recall a well know fact on the Green function : G

Proposition 3.2 For any m> 0 there is a constant C1 (depending only on m) such that 2 (x, z) 6 log x2 + z2 1 + C . |G | −(4π)2 ∧ 1 p Proof See Section 3, Chapter V of [63]. ✷ More generally, in order to stress the dependence of on the mass m> 0, we use also the notation G i(y x+k z) Gm2 1 e · · (x, z)= 4 2 2 2 2 dydk. (2π) R4 ( y + k + m ) Z | | | |

3.3 Estimates on ε G In this section we prove the following useful estimate on the Green function ε. Our proof differs from proofs of the analogous statement on εZ2 (see, e.g., Section 4.2.2 of [39] or AppendixG A of [16]) and uses 2 2 4 a comparison with the Green’s function on (m ∆R4 ) on R which is well known. − Theorem 3.3 For any m > 0, 0 < ε 6 1 and C > 2 there exists a D R such that, for any (4π)2 C ∈ + x R2 and z εZ2, we have ∈ ∈ 1 ε(x, z) 6 C log + DC (57) |G | + √x2 + z2 ε ∧ where log (x) = log(x) 0. + ∧ Proof We distinguish the cases max( z , x ) > ε and max( z , x ) <ε. Consider first max( z , x ) > ε, and| the| | difference| (x, z|) | | |(x, z). Then we have | | | | Gε − G (x, z) (x, z) |Gε − G | i(y x+k z) i(y x+k z) 1 e · · e · · 6 4 2 2 2 2 2 dkdy (2π) R2 T2 y 2 +4ε 2 sin2 εk1 +4ε 2 sin2 εk2 + m2 − ( y + k + m ) Z × 1 − 2 − 2 | | | | ε | |

~~i(y x+k z) 1 e · · + 4 2 2 2 2 dkdy (2π) R4 T2 ( y + k + m ) Z \ 1 | | | | ε~~

~~= J1 + J2~~

~~32 For J1 we have~~

4 (2π) J1 2 2 2 2 2 2 2 2 εk1 2 2 εk2 2 i(y x+k z) ( y + k + m ) y +4ε− sin 2 +4ε− sin 2 + m = e · · | | | | − | | 2 dkdy R2 T2 2 2 2 εk1 2 2 εk2 2 2 2 2 2 1 y +4ε− sin 2 +4ε− sin 2 +m ( y + k + m ) ! Z × ε | | | | | | i(y x+k z) (A + B)(A B) = e · · 2 2 − dkdy. R2 T2 A B 1 Z × ε

~~i(y x+k z) (A + B)(A B) (A B) e · · 2 2 − dkdy . 2 −2 dkdy. R2 T2 A B R2 T2 A B Z × 1 Z × 1 ε ε~~

2 2 ε k1 If we apply Taylor expansion to ε− sin |2 | we get 2 2 ε k1 2 1 3 4ε− sin | | = k + ((ε k ) ) 2 | 1| ε2 O | 1| and so (A B) ε( k )3 − . | 1| A2B ( y 2 + ε 2 sin2(εk )+ ε 2 sin2(εk )+ m2)( y 2 + k 2 + m2)2 | | − 1 − 2 | | | | ε( k )3 . | 1| ( y 2 + k 2 + m2)3 | | | |ε . ( y 2 + k 2 + m2)3/2 | | | | and since 1 . 1 2 2 2 3/2 ε− R2 T2 ( y + k + m ) 1 Z × ε | | | | we can conclude that

~~i(k,y) x i(k,y) x 1 e · e · J1 = 4 2 2 2 2 2 dkdy 6 K1 (2π) R2 T2 y 2 +4ε 2 sin2 εk1 +4ε 2 sin2 εk2 + m2 − ( y + k + m ) Z × 1 − 2 − 2 | | | | ε | |~~

~~where the constant K1 > 0 is independent of 0 < ε 6 1. Let us now turn to J2. Consider Rε (y, k) = 1 (εm2+ y 2+ k 2)2 | | | |~~

~~1 x z x z i(y ε +k ε ) i(y ε +k ε ) J2 = 4 Rε(y, k)e · · dydk . Rε(y, k)e · · dydk + 4 2 2 (2π) R T R , k1 >1, k2 <1 Z \ Z | | | |~~

~~i(y x +k z ) i(y x +k z ) + Rε(y, k)e · ε · ε dydk + Rε(y, k)e · ε · ε dydk 2 2 R , k2 >1, k1 <1 R , k2 >1, k1 >1 Z | | | | Z | | | |~~

~~= J2,1 + J2,2 + J2,3.~~

~~33 First we focus on J2,1 obtaining~~

~~i(y x +k z ) e · ε · ε J2,1 = 2 2 2 2 2 dydk R2 R2 (εm + y + k1 + k2 + 1) Z × | | | | | |~~

~~1 1~~

~~+ 2 2 2 2 2 2 2 2 2 2 dydk R2 , k2 >1, k1 >1 (εm + y + k1 + k2 ) − (εm + y + k1 + k2 + 1) Z | | | | | | | | | | | | | | | | ~~

+K 2 2 x z 6 Gεm +1 , + K + ε ε 2 2 2 2 2 2((εm + y + k1 + k2 + 1)) + 2 2 2 | |2 | 2 | 2| | 2 2 2 2 dydk 2 R , k2 >1, k1 >1 (εm + y + k1 + k2 + 1) (εm + y + k1 + k2 ) Z | | | | | | | | | | | | | | | | εm 2+1 6 G (1, 0)+ K3 6 K4

x z where we use the fact that max |ε| , |ε| > 1 and K2,K3,K4 > 0 are suitable constants independent of 0 <ε 6 1. In a similar way we can get an estimate J2,2,J2,3 6 K5 for some constant K5 independent of 0 <ε 6 1. Since (x, z) satisﬁes inequality (57) when max( x , z ) > ε the thesis is proved in this case. Consider nowGz = 0 (that means z <ε since z εZ2) and| | |x| <ε, then we have | | ∈ | | 1 1 (x, 0) 6 dk dk ε 2 2 2 2 2 1 2 G (4π)(2π) T2 sin (k ) + sin (k )+ ε m Z 1 2 1 1 1 1 6 2 2 dk1dk2 + 2 2 dk1dk2, (4π)(2π) k 0. Since H 1 as H 0 the theorem is 7,H sin(H) → → proved. ✷

3.4 Estimates on ε G In this section we focus our attention on the the Green functions and . Gε Gε Theorem 3.4 For any m> 0 there is a constant C2 > 0 (not dependinge on 0 <ε 6 1) such that (¯x) (¯x) 6 C . |Gε − Gε | 2 Proof We have that e (¯x) (¯x) |Gε − Gε | 2 εk1 2 εk2 1 sin 2 sin 2 = 4 e 4 2 2 (2π) R4 ε k k Z 1 2

~~ 1 1  2 2 2 2 dk × 4 sin2( εk1 ) 4 sin2( εk2 ) − (m + k )  m2 + 2 + 2 + k2 + k2 | |   ε2 ε2 3 4     ~~

~~34 Integrating with respect to the variables k3, k4 we obtain~~

1 sin2 εk1 sin2 εk2 1 1 K = 2 2 dk. 3 4 2 2  εk εk 2 2  (2π) R2 ε k k 4 sin2( 1 ) 4 sin2( 2 ) − (m + k ) Z 1 2 m2 + 2 + 2  ε2 ε2 | |      We can decompose the integral in

~~3 k π K< . ε 2 | | 2 2 dk . ε . 1. k < π (m + k ) ε Z| i| ε | |~~

~~For K>,1 instead we have~~

1 sin2 k1 sin2 k2 1 1 6 2 2 K>,1 2 2  k k + 2  dk 2 3 4 sin2 1 4 sin2 2 k ε (2π) ki >π k1k2 ( 2 ) ( 2 ) 2 Z| | m2 + + m + |ε2|  ε2 ε2    2 k 2 k   1 sin 1 sin 2 1 . 2 2 2 2 k k dk + ε2 k k 4 sin2 1 4 sin2 2 ki >π 1 2 ( 2 ) ( 2 ) Z| | 1+ m2ε2 + m2ε2 1 sin2 k1 sin2 k2 1 + 2 2 dk 2 2 2 k 2 ε ki >π k1k2 2 Z| | m + |ε2| 1 1 sin2 k1 sin2 k2 . 2 2 dk + 2 2 2 k k ε z z 2 4 sin2 1 4 sin2 2 Z2 1 2 [ π,π] ( 2 ) ( 2 ) z ,z1=0,z2=0 Z − 1+ + ∈ X6 6 m2ε2 m2ε2 1 1 1 k k 2 1 2 2 . + 2 2 2 z 2 sin sin dk 1 ε z z 2 2 2 Z2 1 2 | 2| [ π,π] z ,z1=0,z2=0 ε Z − ∈ X6 6

35 since sin2 k1 sin2 k2 k2k2 k2k2 2 2 . 1 2 . 6 1 2 k k dk 2 2 dk ε 2 2 dk 2 1 2 2 k k π π 2 [ π,π]2 4 sin ( 2 ) 4 sin ( 2 ) [ π,π]2 1 2 , (1 + k + k ) 1+ 2 + 2 [ ε ε ] 1 2 Z − 1+ m2ε2 + m2ε2 Z − ε ε Z − 6 6 1 1 2 . ε k1k2dk . ε . ε (58) 2 2 2 [ π , π ] · ε · ε Z − ε ε and also k k sin2 1 sin2 2 dk . 1. [ 2π,2π]2 2 2 Z − For K>,2 we have

2 εk2 3 1 ε sin 2 k1 K>,2 . εk | εk| dk + π π 2 1 2 2 2 ε k < , k > 4 sin ( 2 ) 4 sin ( 2 ) k 1 ε 2 ε 2 2 Z| | | | k2 m + ε2 + ε2 1+ |ε2| 1 1 + 2 dk2 ε k > π k2 Z| 2| ε 1+ | ε2| 1 1 . ε3 log(ε)+ ε . 1, ε · · ε · and a similar estimate can be proved for K>,3. Since all the previous constants do not depend on 0 <ε 6 1 the theorem is proved. ✷

~~2 a Corollary 3.5 For any m> 0 and C > (4π)2 there is a constant DC independent of 0 <ε 6 1 such that~~

1 a ε(x, z) 6 C log + D |G | + √x2 + z2 ε C ∨ Proof This is a consequence of Theorem 3.4, and the fact that, using Proposition 3.2, we get that 2 (x, z) = (w w )(x, z) 6 log x2 + z2 ε + K, Gε 0,ε ∗ 0,ε ∗ G −(4π)2 ∨ p 1 1 eI 2 ✷ where w0,ε(z)= ε2 1 , 1 (ε− ), for some constant K > 0 independent of 0 <ε 6 1. (− 2 2 ] ·

~~3.5 Stochastic estimates From the estimate of Theorem 3.4 we get the following results.~~

Lemma 3.6 For any a> 0, β (0, 1), C > 2 , i > 0 and 0 6 j 6 J we have ∈ (4π)2 ε 2 1 E a,β n n! R2 εZ2 Ei,j (x, z): Wε(x, z) : dxdz × n (59) R a,β a,β 1 . R4 Z4 E (x, z)E (x′,z′) DC + C log dxdzdx′dz′ ε i,j i,j + √ x x′ 2+ z z′ 2 ε × | − | | − | ∨ R 2 1 E ã,β n n! R2 εZ2 Ei (x): Wε(x, z) : dx × n (60) ã,βR ã,β 1 . R4 E (x)E (x′) DC + C log dxdx′ i i + √ x x′ 2 ε | − | ∨ R 2 1 E Eã,β(z): W (x, z)n : dz n! R2 εZ2 Jε ε × n (61) ã,βR ã,β 1 . R4 E (z)E (z′) DC + C log dxdx′ i i + √ z z′ 2 ε | − | ∨ a,β ã,β R where Ei,j and Ei are defined in equations (50) and equation (51).

36 Proof By deﬁnition and Fubini’s theorem 2 E a,β n Ei,j (x, z): Wε(x, z) : dxdz R2 εZ2 "Z × # a,β a,β E n n = Ei,j (x, z)Ei,j (x′,z′) [: Wε(x, z) :: Wε(x′,z′) :]dxdzdx′dz′. R4 εZ4 Z × Now by Wick’s theorem we have n n n n E[: W (x, z) :: W (x′,z′) :] = n!E[W (x, z)W (x′,z′)] = n![ (x x′,z z′)] . ε ε ε ε Gε − − Thus, then inequality (59) follows from Theorem 3.3. Using the fact that

2 E ã,β n ã,β ã,β E n n Ei (x): Wε(x, z) : dx = Ei (x)Ei (x′) [: Wε(x, z) :: Wε(x′,z) :]dxdx′ R2 R4 εZ4 "Z # Z × 2 a,β n a,β a,β n n E E˜ (z): W (x, z) : dxdz = E˜ (z)E˜ (z′)E[: W (x, z′) :: W (x, z′) :]dzdz′ Jε ε Jε Jε ε ε R2 R4 εZ4 "Z # Z × and again Theorem 3.3, we get the thesis. ✷

Lemma 3.7 Under the same hypothesis of Lemma 3.6, inequalities (59), (60) and (61) hold by replacing W by W¯ and the integration on R2 εZ2 and R4. ε ε × Proof The proof is the similar to the one of Lemma 3.6 where the role of Theorem 3.3 is replaced by Corollary 3.5. ✷

R α2 R Theorem 3.8 Fix α such that (4π)2 < 2, then for any 0 < ε 6 1, a + and 0 <β< 1, p > 2 2 ∈ ∈ 2 such that α (p 1) < 2, any δ > 0 and s ,s > δ such that s + s δ > α (p 1) we have (4π)2 − 1 2 2 1 2 − (4π)2 − δ δ a,β α p a,β α p p[(s1 )i+(s2 )j] E[ E µ (x)] = E[ E µ (0)] 6 C 2 − 2 − 2 (62) | i,j ∗ ε | | i,j ∗ ε | α,a,β δ δ a,β α p a,β α p p[(s1 )i+(s2 )j] E[ E µ¯ (x)] = E[ E µ¯ (0)] 6 C¯ 2 − 2 − 2 (63) | i,j ∗ ε | | i,j ∗ ε | α,a,β where Cα,a,β, C¯α,a,β > 0 is some constants depending only on α, a and β and not on 0 <ε 6 1. Proof We provide a detailed proof only for inequality (62) the proof of (63) being completely analogous where the roles of Theorem 3.3 and Lemma 3.6 are played by Corollary 3.5 and Lemma 3.7. First we consider the case p = 2. We have that 2 E a,β α 2 E a,β α [ Ei,j µε ] = Ei,j (x, z)µε (dx, dz) | ∗ | R2 εZ2 "Z × # + 2 ∞ n E α a,β n = Ei,j (x, z): W (x, z) : dxdz  R2 Z2 n!  n=0 ε ! X Z × +  2n 2 ∞ α E a,β n = 2 Ei,j (x, z): W (x, z) : dxdz (n!) R2 Z2 n=0 " ε # X Z × a,β a,β Lemma 3.6 6 Ei,j (x, z)Ei,j (x′,z′) R4 εZ4 Z × 2 2 1 exp α DC + Cα log dxdzdx′dz′ + 2 2 × x x′ + z z′ ε!! | − | | − | ∨ a,β a,β p 2 . Ei,j (x, z)Ei,j (x′,z′) exp(α DC ) R8 Z (C+κ)α2 1 1+ dxdzdx′dz′ 2 2 ×  x x′ + z z′ ε!  | − | | − | ∨  p  37 where κ is any positive number, the constant in . depends only on a,β and κ, we use the orthogonality of n 1 2 2 0 2 2 the Wick monomials : Wε(x, z) : of diﬀerent degrees, the fact that for any g L (R εZ ) C (R εZ ) we have ∈ × ∩ ×

2 n n E g(x, z): Wε(x, z) : dxdz = n! g(x, z)g(x′,z′)( ε(x x′,z z′)) dxdzdx′dz′, R2 εZ2 R4 εZ4 G − − "Z × # Z × and, in the last two steps, Theorem 3.3. θ 1 θ 2 2 For any 1 <θ< 0 we have x z − 6 x + z which implies that when x , z 6 ε then | | | | | | | | | | | | 1 p 1 1 = θ 1 θ 6 θ 1 θ x 2 + z 2 ε ε ε − x z − | | | | ∨ | | | | and in the same way, when at leastp one of x or z is bigger than ε, we have | | | | 1 1 1 = 6 θ 1 θ . x 2 + z 2 ε x 2 + z 2 x z − | | | | ∨ | | | | | | | | This implies that p p

E[ Ea,β µα 2] . | i,j ∗ ε | 2 2 a,β a,β α DC θ 1 θ (C+κ)α . Ei,j (x, z)Ei,j (x′,z′)e (1 + ( x x′ z z′ − )− )dxdzdx′dz′ R8 | − | | − | Z 2 2 2 (iθ+j(1 θ))(C+κ)α a,β a,β α DC θ 1 θ (C+κ)α . 2 − E0,0 (x, z)E0,0 (x′,z′)e (1 + ( x x′ z z′ − )− )dxdzdx′dz′. R8 | − | | − | Z (64)

2 If we choose C > 2 and κ > 0 small enough such that 2α < (C + κ)α2 < (s + s δ) and if we (4π)2 (4π)2 1 2 − choose 0 <θ< 1 such that δ δ θ(C + κ)α2 ~~2, and consider any γ > 1, then, using hypercontractivity of Gaussian Wick monomials (see Chapter IV, Theorem 1 of [67]), i.e.~~

~~n n p n 2 p E[(: W :) ] 6 p 1 (E[: W : ]) 2 , ε − ε p we have~~

+ n a,β α p 1 ∞ α a,β n p 1 (E[ E µ ]) p 6 | | (E[ E : W : ]) p | i,j ∗ ε | n! | i,j ∗ ε | n=0 X + n ∞ α √p 1 a,β n 2 1 . | | − (E[ E : W : ]) 2 n! | i,j ∗ ε | n=0 X 1 1 + 2 γ2 2 ∞ γ2α2n(p 1)n . − E[ Ea,β : W n : 2] γ2 1 (n!)2 | i,j ∗ ε | − n=0 ! X 1 1 2 2 2 2 γ a,β γα√(p 1) . E E µε − . γ2 1 i,j ∗ " #! −

~~2α2γ2(p 1) If we choose γ > 1 small enough such that s + s δ > − we get the thesis from the ﬁrst part 1 2 − (4π)2 of the proof where p = 2. ✷~~

38 R α2 R Theorem 3.9 Fix α such that (4π)2 < 2, then for any 0 < ε 6 1, a + and 0 <β< 1, p > 2 2 ∈ 2 ∈ such that α (p 1) < 2, for any s> α (p 1) we have (4π)2 − (4π)2 − a,β α p p max(i,j)s E[ E µ ] 6 C′ 2 (65) | i,j ∗ ε | α,a,β

a,β α p pis a,β α p pJεs E[ E˜ xµ ] 6 C′ 2 , E[ E˜ zµ ] 6 C′ 2 (66) | i ∗ ε | α,a,β | Jε ∗ ε | α,a,β a,β α p p max(i,j)s E[ E µ¯ ] 6 C¯′ 2 (67) | i,j ∗ ε | α,a,β a,β α p pis a,β α p pJεs E[ E˜ xµ¯ ] 6 C′ 2 , E[ E˜ zµ¯ ] 6 C′ 2 (68) | i ∗ ε | α,a,β | Jε ∗ ε | α,a,β ¯ where Cα,a,β′ , Cα,a,β′ > 0 are some universal constants depending only on α, a and β. Proof The proof is analogous to the one of Theorem 3.8 where, using the notations of the proof of Theorem 3.8, we exploit the fact that

~~2 2 (C+κ)α 2 (C+κ)α 2 ix 2 + 2 j z 2 − 6 2max(i,j)(C+κ)α x 2 + z 2 − . | − | | − | | | | | p p ✷~~

2 2 Corollary 3.10 For α R, 0 <β< 1, p > 2 such that α (p 1) < 2, for any s > α (p 1) we ∈ (4π)2 − (4π)2 − have E Ms1 ,s2 α p Ms1,s2 α p Ns α p Ns α p sup p,ℓ ,ℓ ,ε(µε ) + p,ℓ ,ℓ (¯µε ) + p,ℓ,ε(µε ,a,β) + p,ℓ(¯µε ,a,β) < , ε | 1 2 | | 1 2 | | | | | ∞ h i provided that l1,l2 > 2, l> 4. Proof This follows from Theorem 3.9 and the deﬁnition of M, N. ✷

α 3.6 Convergence of µ¯ε α p α In this section we want to prove thatµ ¯ε converges (in suitable Besov spaces) in L to µ . We also want to discuss some important consequences that can be deduced from this convergence.

Theorem 3.11 We have that for any f Lr(Rn) (for some 1 2. In particular there is a subsequence ε =2− k 0, such that for any n N we ′ k have : W¯ n : : W n : weakly in Lr (Rn) and almost surely. → ∈ εk → In the following we write

~~εk1 εk2 sin 2 sin 2 Kε(k)= . 4 sin2 εk1 4 sin2 εk2 2 2 ( 2 ) (2 ) 2 2 ε k1k2 m + ε2 + ε2 + k3 + k4 and 1 K(k)= , (m2 + k 2) | |~~

~~Lemma 3.12 We have that K K in L2p(R4) for any 1~~

39 Proof We observe that 1 1 1 K (k) 2p . + + ε 2 p p 3 p | | (1 + k ) 1+ k1 (1 + k 2 + k 2 + k 2) 2 | | | | | 2| | 3| | 4| 1 1 + . (69) p 3 p 1+ k2 (1 + k 2 + k 2 + k 2) 2 | | | 1| | 3| | 4| where the constant in implicit in the symbol . do not depend on ε. Since the function in the right 1 4 hand side of equation (69) is in L (R ) and since Kε K pointwise, we can apply dominate convergence theorem to K K 2pdk obtaining the thesis. → ✷ R4 | ε − | Proof of TheoremR 3.11 Here we follow an argument similar to [61] Theorem V.3. By hypercontractivity it is suﬃcient to prove the convergence in L2(Ω). We have that

E ¯ n n 2 ˆ 2 : Wε :,f : W :,f = f(k1 + + kn) |h i − h i| R4r | ··· | × Z K (k ) K (k ) K(k ) K(k ) 2dk dk ×| ε 1 ··· ε n − 1 ··· n | 1 ··· n r ℓ 1 2 − 2 . fˆ(k1 + + kn) Kε(ka) R4r | ··· | | | × ℓ=1 Z a=1 X n Y K (k ) K(k ) 2 K(k ) 2dk dk (70) ×| ε ℓ − ℓ | | a | 1 ··· n a=Yℓ+1 2 r′ n 2 We use the fact that, by Hausdorﬀ-Young inequality, fˆ L (R ) for some r′ > 1. Since Kε K ′ p Rn nr K 2 | | ∈ p Rn | − | converges to 0 in L ( ) for p = (r′ 1) and ε is uniformly bounded in L ( ), by Young inequality − | | for convolutions, the integrand in (70) converges to 0 in L1, which implies the ﬁrst thesis. Furthermore observe in view of the previous discussion (70) can reformulated as

n n 2 2 E : W¯ :,f : W :,f 6 C o(ε) f r , |h ε i − h i| n k kL where Cn > 0 is some constant depending on n but not on ε. Now take a sequence fm which is dense in the ball B(0, 1) Lr(R2), of center 0 and radius 1, and deﬁne the quantity ⊂ 1 d(ϕ, ψ)= ϕ ψ,f 2 R2 . m2 |h − miL ( )|

X ′ Clearly d(ϕ , ψ) 0 implies ϕ ψ weakly in Lr . On the other hand k → k → 1 Ed : W¯ n :, : W n : 6 C1/2o(ε) 0. ε n m2 → X Extracting a subsequence such that d : W¯ n :, : W n : 0 almost surely implies the statement. ✷ ε → ′ We are now going to prove a theorem about the convergence of the functionals Ms,s (µα,a,β), p,ℓ1,ℓ2,ε ε ′ ′ Ms,s (¯µα,a,β), Ms,s (µα,a,β), Ns (µα,a,β), Ns (¯µα,a,β) and Ns (µα,a,β) deﬁned in equations p,ℓ1,ℓ2 ε p,ℓ1,ℓ2 p,ℓ,ε ε p,ℓ ε p,ℓ (52), (53), (54) and (55).

R α2 R Theorem 3.13 Fix α such that (4π)2 < 2, then for any a + and 0 <β< 1, p > 2 such that 2 ∈ ∈ 2 2 α (p 1) < 2, any δ > 0, any s ,s > δ such that s + s δ > α (p 1) and s> α (p 1), and (4π)2 − 1 2 2 1 2 − (4π)2 − (4π)2 − for any ℓ1,ℓ2 > 2 and ℓ> 4, we have

Ms1,s2 (¯µα,a,β) Ms1,s2 (µα,a,β), Ns (¯µα,a,β) Ns (µα,a,β) p,ℓ1,ℓ2 ε → p,ℓ1,ℓ2 p,ℓ ε → p,ℓ in Lp(Ω). Proof We only prove the assertion for Ms1,s2 (¯µα,a,β), the proof for Ns (¯µα,a,β) is similar. p,ℓ1,ℓ2 ε p,ℓ ε

40 a,β α a,β α 1 p R4 First we want to prove that Ei,j µ¯ε Ei,j µ in L (Ω,Lℓ ( )). First we consider the case p = 2. For any N N, write ∗ → ∗ ∈ N α µ¯α,N = : W¯ n : . ε n! ε n=0 X We have that

E a,b α,N a,b α 2 Ei,j µ¯ε Ei,j µ¯ε L2 (R4) k ∗ − ∗ k ℓ1,ℓ2 +h i ∞ α E ¯ n 2 2 = Ei,j : Wε : L2(R4) [ρℓ1 (x)ρℓ2 (z)] dxdz n! k ∗ k ℓ R4 n=XN+1 Z + ∞ α a,β a,β . Ei,j (x, z)Ei,j (x′,z′) n! R8 n=N+1 Z X n 1 DC + C log dxdzdx′dz′ + 2 2 × x x′ + z z′ ε!! | − | | − | ∨ 0, as N → → ∞ p if we choose ℓ1,ℓ2 > 2. So for any δ > 0 we can choose an N large enough such that E a,b α,N a,b α sup [ Ei,j µ¯ε Ei,j µ¯ε L2(R4)] 6 δ. 0<ε61{ k ∗ − ∗ k ℓ }

~~ε Now from Theorem 3.11 (: W n :) = W¯ : W n : weakly in Lr (R4) P-almost surely. Since E are ε ε ℓ1,ℓ2 i,j smooth, it is not hard toE deduce from the previous→ convergence that~~

~~E W¯ E : W n : i,j ∗ ε → i,j ∗ in L2 (R4), and almost surely, after extracting a suitable subsequence. Since ℓ1,ℓ2~~

~~E a,β ¯ 2 sup Ei,j Wε L2 (R4) < , ε k ∗ k ℓ1,ℓ2 ∞ h i we can deduce by uniform integrability that~~

~~E ¯ n n Ei,j : Wε : Ei,j : W : L2 (R4) 0. k ∗ − ∗ k ℓ1,ℓ2 → h i Summing everything up we get~~

N E a,β α a,β α α E ¯ n ˆn lim Ei,j µ¯ε Ei,j µ L2 (R4) 6 lim : W : : ξ : L2 (R4) +2δ ε 0 k ∗ − ∗ k ℓ1,ℓ2 ε 0 n! k − k ℓ1,ℓ2 → → n=0 h i X = 2δ, since δ > 0 is arbitrary, this gives the statement for p = 2. Now to prove the full statement for p> 2 we proceed similarly: Again from equation (65) we can deduce that for any δ > 0 there exists an N such that that

~~′ E Ms1,s2 α p spj+s pi a,β α p sup p,ℓ,ε (µε ,a,β) 2 (Ei,j (ρℓ(x, z)¯µε (x, z))) dxdz 6 δ. (71) ε  − N R4 ∗  i 0,j6N Z ∈ X  ~~

~~By the ﬁrst step we have that~~

spj+s′pi a,β α p limε 0 i N ,j6N 2 R4 (Ei,j (ρℓ1 (x)ρℓ2 (z)¯µε (x, z))) dxdz → 0 ′ ∈ spj+s pi a,β ∗ α p (72) = i N ,j6N 2 R4 (Ei,j (ρℓ1 (x)ρℓ2 (z)µ (x, z))) dxdz ∈ 0 P R ∗ P R

~~41 Combining (71) and (72) it is not hard to deduce that~~

~~E spj+s′pi a,β α p lim 2 (E (ρℓ1 (x)ρℓ2 (z)¯µε (x, z))) dxdz ε 0 i,j N R4 ∗ → i,j 0 Z X∈~~

~~spj+s′pi a,β α p 2 (Ei,j (ρℓ1 (x)ρℓ2 (z)µ (x, z))) dxdz =0 − N R4 ∗ i,j 0 Z X∈~~

✷

Remark 3.14 A consequence of proof of Theorem 3.11 and the proof of Theorem 3.13 is the following: α2 α2 0 R4 consider (4π)2 < 2 and p > 2 such that (4π)2 (p 1) < 2, then for any continuous function f C ( ) − α ∈ decreasing at infinity faster then ρℓ (for some ℓ > 4) we have that R4 f(x, z)µε (dx, dz) converges to f(x, z)µα(dx, dz) in Lp(Ω). R4 R α R In the following theorem we want to analyze the relationship between µ¯ε (which, we recall, is defined on R4) and it discrete counterpart µα defined on R2 εZ2. ε ×

~~Theorem 3.15 Under the same hypotheses on α, s1, s2, s, ℓ1, etc. of Theorem 3.13 we have~~

E[ Ms1 ,s2 (µα,a,β) Ms1,s2 (¯µα,a,β) p] 0 | p,ℓ1,ℓ2,ε ε − p,ℓ1,ℓ2 ε | → E[ Ns (µα,a,β) Ns (¯µα,a,β) p] 0 | p,ℓ,ε ε − p,ℓ ε | → and so Ms1 ,s2 (µα,a,β) Ms1,s2 (µα,a,β), Ns (µα,a,β) Ns (µα,a,β), p,ℓ1,ℓ2,ε ε → p,ℓ1,ℓ2 p,ℓ,ε ε → p,ℓ in Lp(Ω).

~~Ms1,s2 Ns Ms1,s2 Proof We only prove the statement for p,ℓ,ε , the proof for p,ℓ,ε is analogous. By deﬁnition of p,ℓ,ε we have~~

E Ms1,s2 (µα,a,β)p Ms1,s2 (¯µα,a,β)p | p,ℓ,ε ε − p,ℓ ε | s1pj+s2piE α p α p = 2 ρℓ1 ρℓ2 Ei,j µ¯ε dzdx ρℓ1 ρℓ2 Ei,j µε dxdz R4 | ∗ | − R2 εZ2 | ∗ | i,j Z Z × X

~~E α p α p ρℓ1 ρℓ2 Ei,j µ¯ε dzdx ρℓ1 ρℓ2 Ei,j µε dxdz 0 (73) R4 | ∗ | − R2 εZ2 | ∗ | → Z Z × ~~

~~To prove this observe that~~

~~α p 2 α p ρℓ1 ρℓ2 Ei,j µ¯ε (x, z)dxdz ε ρℓ1 ρℓ2 Ei,j µ¯ε (x, z)dx 0 as ε 0 R4 R2 | ∗ | − 2 | ∗ | → → Z z εZ Z X∈ almost surely since we have that~~

~~α ∞ ρℓ2 δ (R2), Ei,j µ¯ε (ω) (x, z) L (R2,Cδ (R2)) . Ci,j (ω) k kC−ℓ2 k| ∗ | k −ℓ1 −ℓ2 where Ci,j (ω) is an almost surely ﬁnite positive random variable.~~

~~42 Indeed for any x R2 we get ∈~~

~~α p 2 α p ρℓ1 ρℓ2 Ei,j µ¯ε (x, z)dxdz ε ρℓ1 ρℓ2 Ei,j µε (x, z)dx R4 R2 | ∗ | − 2 | ∗ | Z z εZ Z X∈~~

~~α p 2 α p = ρℓ1 ρℓ2 (y) Ei,j µ¯ε (x, y) dy dx ε ρℓ1 ρℓ2 Ei,j µε (x, z)dx R2 | ∗ | − R2 | ∗ | 2 y Qε(z) ! 2 Z z εZ Z ∈ z εZ Z X∈ X∈~~

~~α p α p = ρℓ1 (x) (ρℓ2 (y) Ei,j µ¯ε (x, y)dy ρℓ2 (z) Ei,j µε (x, z)) dy dx R2 | ∗ | − | ∗ | 2 " y Qε(z) # Z z εZ Z ∈ X∈~~

~~α p = ρℓ1 (x) (ρℓ2 (y) ρℓ2 (z) Ei,j µ¯ε (x, y))dy+ R2 − | ∗ | Z2 " y Qε(z) Z z ε Z ∈ X∈~~

~~α p α p + ρℓ2 (z)( Ei,j µ¯ε (x, y) Ei,j µε (x, z))dy dx y Qε(z) | ∗ | − | ∗ | # Z ∈~~

~~α p 6 ρℓ1 (x) (ρℓ2 (y) ρℓ2 (z) Ei,j µ¯ε (x, y)dy)dx + R2 2 y Qε(z) − | ∗ | Z z εZ Z ∈ X∈~~

α p 1 α p 1 + ρℓ1 ρℓ2 ( Ei,j µ¯ε − (x, y)+ Ei,j µε − (x, z)) R2 2 y Qε(z) | ∗ | | ∗ | × z εZ Z Z ∈ X∈ α α ( Ei,j µ¯ (x, y) Ei,j µ¯ (x, z))dydxdz × | ∗ ε | − | ∗ ε |

~~Now the ﬁrst term can be estimated by~~

δ α p ε ρℓ1 δ R2 (ρℓ1 (x)ρℓ2 (y) Ei,j µ¯ (x, y)dy)dx −ℓ ( ) ε k kC 2 R2 y Q (z) | ∗ | Z z εZ2 Z ∈ ε X∈ δ α p = ε ρ δ R2 ρ (x)ρ (y) E µ¯ (z, x)dzdx ℓ1 − ( ) ℓ1 ℓ2 i,j ε k kC ℓ2 R4 | ∗ | Z For the second term we get the bound

α p 1 α p 1 ρℓ2 (z)( Ei,j µ¯ε − (x, y)+ Ei,j µ¯ε − (x, z)) x Q (z) | ∗ | | ∗ | z εZ2 Z ∈ ε X∈ ( E µ¯α (x, y) E µ¯α (x, z))dy | i,j ∗ ε | − | i,j ∗ ε | δ α 2 α p 1 α p 1 6 ε Ei,j µ¯ (ω) (z) ∞ R2 δ R2 ρ (z) ( Ei,j µ¯ − (x, y)+ Ei,j µ¯ − (x, z))dy ε L ( ,C−ℓ ( )) ℓ1 ε ε k| ∗ | k 2 x T (z) | ∗ | | ∗ | z εZ2 Z ∈ X∈ δ 2 α p 1 α p 1 α 6 ε ρ (y)( E µ¯ − (y)+ E µ¯ − (z)) E µ¯ (ω) (z) δ i,j ε i,j ε i,j ε C− R2 | ∗ | | ∗ | k| ∗ | k ℓ Z δ α p 1 α = ε Ei,j µ¯ε −p−1 Ei,j µ¯ε (ω)(z) Cδ k ∗ kL k ∗ k −ℓ and by H¨older inequality

p−1 α p 1 α α p p α p 1/p E[ E µ¯ −− E µ¯ (ω)(z) δ ] 6 E[ E µ¯ − ] E[ E µ¯ (ω)(z) ] . i,j ε Lp 1 i,j ε C− i,j ε Lp 1 i,j ε Cδ k ∗ k k ∗ k ℓ k ∗ k k ∗ k −ℓ ✷

~~3.7 Convergence of the periodic approximation α Z2 In this section we discuss a periodic approximation of the measure µε (with respect the ε variables) that we will use in Section 4.~~

~~43 First we introduce a periodized version of ξε in the following way, for any N N even number, we 2 ∈ denote by [z] N , N the remainder r of the division zi = Nq + r (where 0 6 r~~

~~1 W = ( ∆R2 ∆ Z2 + m)− (ξ ) ε,N − − ε ε,N and then we introduce the positive measure~~

~~+ ∞ α α2 µα = : W n := exp αW E[(W )2] . ε,N n! ε,N ε,N − 2 ε,N n=0 X It is useful also to write~~

i(x y+k z) 1 e− · · ε,N (x, z)= 2 2 2 dy. G N (2π) R2 2 2 2 εk1 2 2 εk2 2 2 y +4ε− sin +4ε− sin + m T π Z Z 2 2 k 1XεN | | ∈ ε ∩ We remark that E[W (x, z)W (0, 0)] = (x, z). ε,N ε,N Gε,N

Lemma 3.16 The function (x, z) is uniformly bounded in N and it converges pointwise to . Gε,N Gε Proof We have that 1 1 ε,N 6 2 2 2 2 2 dy. |G | ε (2π) R2 ( y + m ) Z | | The convergence of ε,N (x, z) follows from the convergence of the Riemann sums to the Riemann integral and the fact that theG integrands deﬁning and are smooth and summable. ✷ Gε,N Gε

Proposition 3.17 For any ε> 0, s< 0, s1 R, s2 < 0, p [2, + ], a R+, 0 <β< 1, and ℓ> 4 we have that ∈ ∈ ∞ ∈ E[ Ms1,s2 (µα ,a,β) Ms1,s2 (µα,a,β) 2] 0 | p,ℓ,ε ε,N − p,ℓ,ε ε | → E[ Ns (µα ,a,β) Ns (¯µα,a,β) 2] 0 | p,ℓ,ε ε,N − p,ℓ,ε ε | → as N + . → ∞ Proof Arguing as in the proof of Theorem 3.15 it is enough to show that

E i,j α i,j α 2 E µε,N E µε Lp (R2 εZ2) 0 k ∗ − ∗ k ℓ1,ℓ2 × → as N . Observe that for any g L1(R2 εZ2) we have → ∞ ∈ × 2 E α g(x, z)dµε,N R2 εZ2 "Z × # α2 (x x′,z z′) = g(x, z)g(x′,z′)e Gε,N − − dxdx′dzdz′ (R2 εZ2)2 Z × 1 2 . exp 2 g L1(R2 εZ2) ε k k ×

~~44 uniformly in N N. Furthermore by binomial formula ∈ 2 E α α g(x, z)dµε,N g(x, z)dµε = R2 εZ2 − R2 εZ2 "Z × Z × #~~

~~2 2 E α E α = g(x, z)dµε,N + g(x, z)dµε R2 εZ2 R2 εZ2 "Z × # "Z × # E α α 2 g(x, z)dµε,N g(x, z)dµε − R2 εZ2 R2 εZ2 Z × Z × A consequence of Wick theorem is that~~

E α α g(x, z)dµε,N g(x, z)dµε R2 εZ2 R2 εZ2 Z × Z × α2E[W (x,z)W (x′,z′)] = g(x, z)g(x′,z′)e ε,N ε dxdx′dzdz′. (R2 εZ2)2 Z × And since 2 2 Wε,N (x, z)Wε(x′,z′) 6 (Wε,N (x, z)) + (Wε(x′,z′)) and clearly W (x, z) W (x, z) ε,N → ε almost surely as N , we have by dominated convergence that → ∞

~~E[W (x, z)W (x′,z′)] (x x′,z z′). ε,N ε → Gε − − So~~

2 2 E α E α lim g(x, z)dµε,N + g(x, z)dµε N R2 εZ2 R2 εZ2 →∞ "Z × # "Z × # E α α lim 2 g(x, z)dµε,N g(x, z)dµε − N R2 εZ2 R2 εZ2 →∞ Z × Z × α2 (x x′,z z′) = 2 g(x, z)g(x′,z′)e Gε − − dxdx′dzdz′ (R2 εZ2)2 Z × α2 (x x′,z z′) 2 g(x, z)g(x′,z′)e Gε − − dxdx′dzdz′ − (R2 εZ2)2 Z × = 0.

This implies that E[ Ei,j µα Ei,j µα 2(x, z)] 0 for x, z R2 εZ2. Now since | ∗ ε,N − ∗ ε | → ∈ × i,j α i,j ∞ 2 2 ∞ 2 2 E µε L (R εZ ) . E L (R εZ ) ρℓ1 (x)ρℓ2 (z)µ M(ρ) k ∗ k ℓ1,ℓ2 × k k × k k we have by dominated convergence that, for any ℓ1,ℓ2 > 2,

~~E Ei,j µα Ei,j µα p(x, z)(ρ (x)ρ (z))pdxdz 0, | ∗ ε,N − ∗ ε | ℓ1 ℓ2 → Z which implies the statement. ✷~~

~~4 Elliptic stochastic quantization of cosh(βϕ)2 model~~

~~4.1 The cosh(βϕ)2 model~~

~~45 In this section we want to recall brieﬂy the deﬁnition and the construction of Euclidean cosh(βϕ)2 model.~~

2 First consider the Gaussian measureν ˜m,ε,N on ′(ε(Z/NZ) ), where m> 0 is the mass of the boson, N N is the size of the spatial cut-off, and ε> 0S is the size of ultra-violet cut-off, describing a random ∈ 2 fieldϕ ˜ ′(ε(Z/NZ) ) with covariance ε,N ∈ S

2 1 2 ϕ˜ (x)˜ϕ (y)˜ν (dϕ) = ( ∆ Z2 + m )− (x y), x,y ε(Z/NZ) , ε,N ε,N m,ε,N − ε ,N − ∈ Z 2 1 2 where ( ∆εZ2,N +m )− (x y) is the Green function of the operator ∆εZ2 +m with periodic boundary − − 2 − condition, and ∆εZ2,N is the discrete Laplacian on (Z/NZ) . There is a map natural continuous map

2 2 i : ′(ε(Z/NZ) ) ′(εZ ) ε,N S → S 2 given by the periodic extension, namely if g ′(ε(Z/NZ) ) we have ∈ S i [g](z)= g([z] ), z εZ2 ε,N ε,N ∈ 2 2 where [z]ε,N ε(Z/NZ) is the equivalent class of the point z εZ . We recall that the extension ∈ 2 2 ∈ operator ε : ′(εZ ) ′(R ) (introduced in Section 2.4) is a continuous map (and so measurable) we then denoteE byS → S νm,ε,N = ( ε iε,N ) (˜νm,ε,N ). E ◦ ∗ 2 The measure ν is a Gaussian measure on ′(R ) whose associated random field is constant in boxes m,ε,N S Z2 1 Tˆ2 1 1 2 of side ε centers on ε and periodic with period N . We use the following notation N = 2N , 2N and we denote − cosh,β λ αϕ(x) αϕ(x) VN (ϕ)= (: e : + : e− :)dx, 2 Tˆ2 Z N αϕ which is well defined νm,ε,N almost surely, and : e± : is defined

~~+ n αϕ ∞ ( α) n : e± := ± : ϕ : n! n=0 X as the Wick exponential with of the random ﬁeld ϕ with respect the measure νm,ε,N . We also write~~

~~V cosh,β (ϕ) Zm,ε,N = e− N νm,ε,N (dϕ). Z~~

cosh,β R2 Deﬁnition 4.1 We say that the measure νm on ′( ) is the measure related with the Euclidean quantum ﬁeld theory having action S 1 S(ϕ)= ( ϕ(z) 2 + m2ϕ(z)2)dz +2λ : cosh(βϕ(z)) : dz, 2 R2 |∇ | R2 Z Z if there are two sequences ε ′ 0 and N + (where ε ′ > 0 and N N) such that n → n → ∞ n n ∈ cosh,β VN (ϕ) cosh,β e− n ′ νm (dϕ)= lim′ lim νm,εn ,Nn (dϕ) , n 0 n + Z ′ → → ∞ m,εn ,Nn !! where the limits are taken in weak sense in the space of probability measures.

~~46 4.2 The equation and its approximations Here we consider the following elliptic (heuristic) SPDE~~

2 ( ∆R4 + m )φ +2α sinh(αφ )= ξ. − − ∞ 4 More precisely we say that a random field taking values in ′(R ) is a solution to equation (4.2), if, 2 1 S writing φ¯ = φ W , where as usual W = ( ∆R4 + m )− (ξ), then the random field solves the following equation − − 2 αφ¯ α αφ¯ α ( ∆R4 + m )φ¯ + αe µ αe− µ− = 0 (74) − − + ( α)n α α α ∞ where µ and µ− are the positive measure introduced in Section 3 and defined by µ± := n=0 ±n! : W n :. P We introduce some approximations to equation (74), using the discretization and the perioditization introduced in Section 3. We define the following equations

¯ ¯ 2 ¯ αφε,N α αφε,N α ( ∆R2 εZ2 + m )φε,N + αe µε,N αe− µε,N− = 0 (75) − × − ¯ ¯ 2 ¯ αφε α αφε α ( ∆R2 εZ2 + m )φε + αe µε αe− µε− =0, (76) − × − where ∆R2 εZ2 (g)(x, z)= ∆R2 g ∆εZ2 g, and − × − − + n + n α ∞ ( α) n α ∞ ( α) n µ± := ± : W :, µ± := ± : W :, ε,N n! ε,N ε n! ε n=0 n=0 X X 2 1 2 1 and where as, usual, Wε,N = ( ∆R2 εZ2 + m )− (ξε,N ) and Wε = ( ∆R2 εZ2 + m )− (ξε) (see Section − × − × 3.7 and Section 3.1 for the deﬁnition of ξε,N and ξε).

~~In rest of of the present section we want to prove the existence and uniqueness of solution to equation (74) using an approximation argument starting from the solutions to equations (75) and (76).~~

~~More precisely we want to prove the following theorem.~~

¯ ¯ 1 R4 ¯ αφ¯ 1 R4 Theorem 4.2 There is a unique solution to equation φ (74) such that φ H ( ) and φe Lℓ ( , dµα) αφ¯ 1 4 α ∈ ∈ and φe¯ − L (R , dµ− ), for any ℓ > ℓ (where ℓ > 0 big enough). ∈ ℓ 0 0 4.3 Some a priori estimates In order to prove the existence for equation (76) we need to proof some a priori estimates for an equation of the form 2 αψ αψ (m ∆R2 ∆εZ2 )ψ + e dη+ e− dη = 0 (77) − − − − where • α,m > 0

• R2 Z2 1+κα R2 Z2 η+, η ( ε ) B2−,2,ℓ ( ε ) where κα is a suitable constant depending on α, and ℓ> 0− big∈M enough.× ∩ × Hereafter we use the notation, if υ : R2 εZ2 R is a function weakly differentiable with respect the first two variable we write × → R2 υ R2 εZ2 υ = ∇ ∇ × ευ ∇ where the first component is a vector in R2 obtained as the gradient with respect to the first two coordinates and the second is the finite difference defined in Section 2.3.

~~47 (4) (4) Theorem 4.3 Suppose that ψ solves (77), and suppose that ρ (x, z)dη+(x, z), ρ (x, z)dη (x, z) < ℓ ℓ − + and ψ H1(R2 εZ2) < + then ∞ k k ℓ × ∞ R R~~

(4) 2 (4) 2 (4) αψ (4) αψ ρℓ R2 εZ2 ψ + ρℓ ψ L2 + ρℓ ψ e dη+ + ρℓ ψ e− dη k ∇ × k k k R2 εZ2 | | R2 εZ2 | | − Z × Z × (4) 2 2 (4) 2 2 . (ρ dη+)(R εZ ) + (ρ dη )(R εZ ) ℓ × ℓ − × (4) Proof Multiplying equation (77) by ρℓ,λψ, where

~~(4) (4) ρℓ,λ(x, z)= ρℓ (λx, λz) where λ> 0, and then taking the integral, we obtain~~

(4) 2 (4) 2 ρℓ,λψ(∆R2 εZ2 ψ)dxdz + m ρℓ,λ(ψ) dxdz − R2 εZ2 × R2 εZ2 Z × Z × (4) αψ (4) αψ + ρℓ,λψe dη+ ρℓ,λψe− dη =0 R2 εZ2 − R2 εZ2 − Z × Z × x x Now since both xe and xe− are functions bounded below by C, for some constant C > 0, we can estimate the last two terms as follows: −

(4) αψ (4) ρℓ,λψe dη+ > C ρℓ,λdη+ R2 εZ2 − R2 εZ2 Z × Z × and analogously (4) αψ (4) ρℓ,λψe− dη > C ρℓ,λdη − R2 εZ2 − − R2 εZ2 − Z × Z × furthermore by integration by parts

(4) (4) ρℓ,λψ(∆R2 εZ2 ψ)dxdz = R2 εZ2 ψ ( R2 εZ2 (ρℓ,λψ))dxdz − R2 εZ2 × R2 εZ2 ∇ × · ∇ × Z × Z × and (4) (4) (4) R2 εZ2 (ρℓ,λψ)= ρℓ,λ, R2 εZ2 ψ +ψ R2 εZ2 ρℓ,λ ∇ × ≪ ∇ × ≫ ∇ × (4) where ρℓ,λ, R2 εZ2 ψ is the vector with components ≪ ∇ × ≫ (4) ρℓ,λ(x, z)∂x1 ψ(x, i) (4) (4)  ρℓ,λ(x, z)∂x2 ψ(x, i)  ρℓ,λ, R2 εZ2 ψ (x, z)= (4) ≪ ∇ × ≫ ε 1ρ (x, z + εe )D ψ  − ℓ,λ 1 εe1   1 (4)   ε− ρ (x, z + εe2)Dεe2 ψ   ℓ,λ    2 2 where e1,e2 is the standard basis of εZ R , and Dh(f)(x, z)= f(x, z + h) f(x, z). For any δ > 0 we can{ choose} ℓ large enough and λ small enough⊂ such that, by computing −

~~2 (ℓ+1) R4 ρ (x, z)= (k + 1)λ(1 + λ (x, z) )− (x, z), ∇ ℓ,λ − k k we get R4 ρ (x) 6 δ ρ (x) . |∇ ℓ,λ | | ℓ,λ | Using the fundamental theorem of calculus, we get~~

~~R2 εZ2 ρℓ,λ(x, z) 6 δ ρℓ,λ(x, z) . |∇ × | | |~~

~~48 a similar estimate holds for the discrete diﬀerence. Then~~

2 2 ρl,λ, R2 εZ2 ψ R2 εZ2 ψ = ρℓ,λ(x, z)( ∂x1 ψ(x, z) + ∂x2 ψ(x, z) )dxdz ≪ ∇ × ≫ ∇ × R2 εZ2 | | | | Z Z × 1 2 + ρℓ,λ(x, z + ε11) ε− Dε11 ψ(x, i) dxdz R2 εZ2 | | Z × 1 2 + ρℓ,λ(x, i + ε12) ε− Dε12 ψ(x, i) dxdz R2 εZ2 | | Z × 1 2 > ρℓ,λ R2 εZ2 ψ dxdz 2 R2 εZ2 |∇ × | Z × and it is not hard to see that

~~ρ, R2 εZ2 ψ 6 R2 εZ2 ρℓ,λ ψ + ρℓ,λ R2 εZ2 ψ | ≪ ∇ × ≫ | |∇ × || | | ||∇ × | and we can estimate~~

~~2 2 R2 εZ2 (ρℓ,λ)ψ R2 εZ2 ψdxdz 6 δ ρℓ,λ(ψ + R2 εZ2 ψ )dxdz R2 εZ2 ∇ × ∇ × R2 εZ2 |∇ × | Z × Z × Choosing δ small enough we can conclude. ✷~~

1+κ 2 Theorem 4.4 Suppose that η , ψ satisfy the hypotheses of Theorem 4.3 and suppose that η B− (R ± ± ∈ 2,2,ℓ × εZ2) where κ (0, 1), then for any β ( 1, 1) we have ∈ ∈ − 2 βαψ 2 βαψ (4) (1 2β)αψ R2 εZ2 (e± ) L2 + ψ e± + ρℓ e± ± dη k∇ × k ℓ/2 | | L2 ± ℓ/2 Z p

(4) 2 2 R Z − . Pβ ρℓ dη ( ǫ ), η B 1+κ (R2 εZ2) R2 εZ2 ± × k ±k 2,2,ℓ/2 × Z × where Pβ is a suitable polynomial depending on β and the constant implicit in the symbol . are independent on ε. Before proving the previous result we prove the following lemma.

~~Lemma 4.5 Consider a measurable function ψ : R2 εZ2 R, and consider γ > 0 then there exists a constant C > 0 independent of ε such that × →~~

γ/2ψ 2 γψ D e± C(D ψ)D e± (78) | εei | ≤ εei εei where e ,e is the standard basis of Z2. { 1 2} Proof We have to show that, for any (x, z) R2 εZ2, we have ∈ × γ (ψ(x,z+εe ) ψ(x,z)) 2 γψ(x,z) e± 2 i − 1 e± γψ(x,z) γ(ψ(x,z+εe ) ψ(x,z)) − (79) 6 Ce± (e± i − 1)(ψ(x, z + εe ) ψ(x, z)). − i − γψ(x,z) Dividing the inequality (79) by e± , (79) is equivalent to

~~2 γ (ψ(x,z+εe ) ψ(x,z)) γ(ψ(x,z+εe ) ψ(x,z)) e± 2 i − 1 6 C(e± i − 1)(ψ(x, z + εe ) ψ(x, z)) − − i − So proving inequality (79) is equivalent to show that~~

~~γ(ψ(x,z+εe ) ψ(x,z)) 1 (e± i − 1)(ψ(x, z + εe ) ψ(x, z)) 6 − i − (80) γ (ψ(x,z+εe ) ψ(x,z)) 2 C e 2 i 1 ± − −~~

~~49 for some C > 0. In order to prove this, if we denote by g = ψ(x, z + εei) ψ(x, z), inequality (80) is equivalent to what follows −~~

γg γ g γ g γ g 1 (e± 1)g e± 2 1 e± 2 +1 g e± 2 +1 γg g 6 = − = γ g = coth g = γ − 2 γ 2 (g) γg C (g) g 2 4 tanh e± 2 1 e± 2 1 e± 1 4 − − − γg γg Now it is not hard to see that limg coth 4 g = and coth 4 g > 0 if g = 0. By L’Hopital rule we get →±∞ ∞ 6 g 4 4 4 lim γg = lim γg = lim 2 γg = g 0 tanh g 0 γ tanh′ g 0 γ 1 tanh γ → 4 → 4 → − 4 γg ✷ Now by continuity of coth 4 gwe can obtain inequality (80) and thus the thesis. 2k 1 1 Proof of Theorem 4.4 The proof can be done by induction on β 6 2−k . For k = 1 (and β = 2 ), αφ¯ using the notations of the proof of Theorem 4.3, we multiply equation (77) by ρ e± , ± ℓ,λ

~~αψ 2 αψ αψ (∆R2 εZ2 ψ)ρℓ,λe± dxdz m ρℓ,λψe± dxdz ρℓ,λe± dη ρℓ,λdη =0 ± × ± ± ± ∓ ∓ Z Z Z Z Now integrating by parts on the ﬁrst term we obtain~~

αψ (∆R2 εZ2 ψ)ρℓ,λe− dxdz × Z αψ = R2 εZ2 ψ R2 εZ2 (ρℓ,λe− )dxdz ∇ × ∇ × Z αψ αψ = R2 εZ2 (ψ) ρℓ,λ, R2 εZ2 e− dxdz + R2 εZ2 (ψ) R2 εZ2 (ρℓ,λ)e− dxdz ∇ × ≪ ∇ × ≫ ∇ × ∇ × Z Z with, similarly at the proof of Theorem 4.3,

αψ(x,z) ρℓ,λ(x, z)∂x1 e− αψ(x,z) αψ ρℓ,λ(x, z)∂x2 e− ρ, R2 εZ2 e− (x, z)=  1 αψ(x,z)  ≪ ∇ × ≫ ρℓ,λ(x, z + εe1)ε− Dεe1 e−  ρ (x, z + εe )ε 1D e αψ(x,z)   ℓ,λ 2 − εe2 −    and computing the scalar product we have

αψ R2 εZ2 ψ ρℓ,λ, R2 εZ2 e− (x, z)= ∇ ×αψ(x,z)≪ ∇ × ≫ αψ(x,z) = ρℓ,λ(x, z)∂x1 e− ∂x1 ψ(x, z)+ ρℓ,λ(x, z)∂x2 e− ∂x2 ψ(x, z) 2 αψ(x,z) +ε− ρℓ,λ(x, z + εe1)Dεe1 e− Dεe1 ψ(x, z)+ 2 αψ(x,z) +ε− ρℓ,λ(x, z + εe2)Dεe2 e− Dεe2 ψ(x, z)= 2 α/2ψ(x,z) 2 2 α/2ψ(x,z) 2 = α ρℓ,λ(x, z)(∂x1 e− ) + α ρℓ,λ(x, z)(∂x2 e− ) + 2 αψ(x,z) +ε− ρℓ,λ(x, z + εe1)Dεe1 e− Dεe1 ψ(x, z)+ 2 αψ(x,z) +ε− ρℓ,λ(x, z + εe2)Dεe2 e− Dεe2 ψ(x, z). In analogous way as in the proof of Theorem 4.3 and the result of Lemma 4.5 ﬁxing δ > 0 we can choose λ small enough such that we have

~~αψ 1 α ψ 2 R2 εZ2 ψ ρℓ,λ, R2 εZ2 e− > ρℓ,λ R2 εZ2 e− 2 . ∇ × ≪ ∇ × ≫ C ∇ ×~~

~~Now we have to estimate~~

αψ αψ αψ R2 εZ2 (ψ)( R2 εZ2 ρℓ,λ)e− dxdz = ∂x1 ρℓ,λ∂x1 ψe− dxdz + ∂x2 ρℓ,λ∂x2 ψe− dxdz ∇ × ∇ × Z Z Z 2 αψ +ε− Dεe1 ρℓ,λDεe1 ψe− dxdz + Z 2 αψ +ε− Dεe2 ρℓ,λDεe2 ψe− dxdz Z

~~50 For the ﬁrst term we apply chain rule to get~~

1 δ 1/2 α αψ αψ/2 αψ/2 2 ψ ∂xi ρℓ,λ∂xi ψe− dxdz = ∂xi ρℓ,λe− ∂xi e− dxdz 6 ρℓ,λ ∂xi e− | | α | | α L2 Z Z 1 1 for i 1, 2 . Now for the 3rd and 4th term we observe that denoting by ε− Dεe1 ρℓ,λ(x, z)= ε − (ρℓ,λ(x, z+ ∈{ } 1 εe1) ρℓ,λ(x, z)) we have that ε− Dεe1 ρℓ,λ 6 δρℓ,λ. Observing− that for any g R ∈ g g/2 g . e− 1 g + e− 1 | | | − || | | − | we get αψ(x,z) (ψ(x, z + εei) ψ(x, z))e− | −αψ(x,z) α(ψ(x,z+εe|i) ψ(x,z)) 6 C ψ(x, z + εei) ψ(x, z) e− (e− − 1) + | −αψ(x,z) || α(ψ(x,z+εe ) ψ(x,z))/2 − | +C e− (e− i − 1) | αψ αψ/2 −αψ/2| 6 CDεei ψDεei e− + Ce− Dεei e−

~~From this we have~~

2 αψ(x,z) ε− Dεei (ρℓ,λ)(x, z) Dεei (ψ)(x, z) e− dxdz 6 2 | αψ || 2 | αψ/2 2 αψ 6 Cδε− ρ (x, z + εe )D ψD e− + ε− Cδ ρ D e− dxdz + δ ρ Ce− dxdz ℓ,λ i R εei εei ℓ,λ| εei | ℓ,λ

R R R αφ¯n By Lemma 4.5, the ﬁrst two terms can be absorbed into the positive term R2 εZ2 ψ ρ, R2 εZ2 e− . For the third term we estimate ∇ × ≪ ∇ × ≫ R αψ αψ C ρ e− dxdz 6 C + ρ ψ e− dxdz ℓ,λ ℓ,λ| | Z Z (4) R2 Z2 (4) R2 Z2 which was estimated in Theorem 4.3 by (ρℓ dη+)( ε ) + (ρℓ dη )( ε ). This proves the 1 × − × theorem for β 2 . ≤ 2k 1 2k+1 1 Suppose that we prove the theorem for β 2−k and consider βk+1 = 2k+1− . Multiplying both 2β αφ¯ ≤ sides by ρe± k+1 and repeating the previous estimates we obtain

~~2 1/2 βk+1αψ 2 1/2 ¯ βk+1αψ (1+2βk)αψ ρℓ R2 εZ2 e± L2 + ρℓ φe± + ρℓe± dη k ∇ × k 2 ± L Z q~~

~~β kαψ +βk αψ . ρe− dη+ + ρe dη . − Z Z On the other hand by~~

~~β αψ β αψ k k − − ρℓe∓ dη e∓ B1 κα+ε(R2 εZ2) η B 1+κα (R2 εZ2), ± ≤k k 2,2,ℓ/2 k ±k 2,2,ℓ/2 Z × × and using induction hypothesis we obtain the thesis. ✷~~

4.4 Existence and uniqueness of solutions to equation (74) In this section we prove the existence and uniqueness of the solution to equation (74). We start with a α α α α results on the boundedness of measures µε,N , µε ,µ ¯ε and µ with respect to ε and N in suitable Besov spaces.

2 α kn Proposition 4.6 Fix α < 4π and 0 <δ< 1 2 , then there are a sequence ε =2− + (for | | − (4π) n → ∞ some k N) and a sequence of N 0 such that for any ℓ > 4 we have n ∈ r → α α α α sup ( µ± −1+δ R2 Z2 , µε± 1+δ R2 Z2 , µ¯ε± −1+δ R4 , µ± −1+δ R4 ) < + εn,Nr B2,2,ℓ ( εn ) n B2,2,ℓ( εn ) n B2,2,ℓ ( ) B2,2,ℓ ( ) εn,Nr k k × k k × k k k k ∞

α α α α α 1+δ 2 almost surely. Furthermore µ± µ± and µ¯± = ε (µ± ) µ± almost surely in B− (R εn,Nr → εn εn E n εn → 2,2,ℓ × Z2 1+δ R4 εn ) and in B2−,2,ℓ ( ) respectively.

51 Ns α Ns α Ns α Proof From Theorem 3.13 and Theorem 3.15 we have that p,ℓ(¯µε ,a,β) p,ℓ(µ ,a,β) and p,ℓ,ε(µε ,a,β) 2 → 2 → Ns (µα,a,β) in Lp(Ω) for any a R , β (0, 1), p > 2, α (p 1) < 2 and s< α (p 1). We can p,ℓ ∈ + ∈ (4π)2 − − (4π)2 − β¯ choose β¯ (0, 1) anda ¯ R+ such that Ki . exp( a¯( x + z ) ) for any i > 1, and a subsequence k∈n ∈ | 1+|δ α − | | | |1+δ α − 1+δ α εn = 2− (where kn N) such that N− (¯µ± , a,¯ β¯) N− (µ± , a,¯ β¯) and N− (µ± , a,¯ β¯) ∈ 2,ℓ εn → 2,ℓ 2,ℓ,εn εn → N 1+δ α ¯ −2,ℓ (µ± , a,¯ β) almost surely. Since, by Theorem 2.43, we have

~~α 1+δ α ¯ µ± 1+δ 2 2 . N− (µ± , a,¯ β) (81) εn B (R ε Z ) 2,ℓ,εn εn k k 2,2,ℓ × n~~

α 1+δ α α 1+δ α −1+δ . N ¯ −1+δ . N ¯ µ¯ε±n B (R4) −2,ℓ (¯µ±εn , a,¯ β), µ± B (R4) −2,ℓ (µ± , a,¯ β), (82) k k 2,2,ℓ k k 2,2,ℓ α 2 1 δ R4 the sequenceµ ¯ε±n is a bounded in L (Ω,B2,−2,ℓ( )), which, by Banach-Alaoglu is weakly precompact. 2 1 δ R4 α α Let κ L (Ω,B2,−2,ℓ( )) be a weak limit of some (convergent) subsequenceµ ¯εn ofµ ¯εn . By Remark 3.14 ∈ k we have that, for any f (R4) and A (where is the σ-algebra of the probability space (Ω, , P)), we have that ∈ S ∈F F F I α I α A(ω) f(x, z)¯µεn (dx, dz) A(ω) f(x, z)µεn (dx, dz) R4 k → R4 Z Z p I α I in L (Ω), which implies that A(ω) R4 f(x, z)µ (dx, dz) = A(ω) R4 f(x, z)κ(dx, dz) for some κ 2 1 δ R4 I 2 1∈δ R4 L (Ω,B − ( )). Since the linear combinations of the elements of the form ( ) 4 f(x, z) (L (Ω,B − ( )))∗, 2,2,ℓ R R A · R · ∈ 2,2,ℓ R4 2 1 δ R4 α for f ( ) and A , is dense in (L (Ω,B2,−2,ℓ( )))∗ we get that κ = µ , andR thus the whole sequence α ∈ S ∈F α α α µ¯εn weakly converges to µ . In a similar way it is possible to prove thatµ ¯ε−n converges to µ− . α α α −1+δ −1+δ −1+δ Inequalities (81) and (82) imply that supεr ( µε±n B (R2 ε Z2), µ¯ε±n B (R4), µ± B (R4)) < k k 2,2,ℓ × n k k 2,2,ℓ k k 2,2,ℓ + . Thus, by the extended Lebesgue dominated convergence theorem (see, i.e. Theorem 2.3.11 of [12]), ∞ E α 2 E α 2 2 1 δ R4 we have that [ µ¯± 1+δ ] converges to [ µ± −1+δ ]. Since L (Ω,B − ( )) is a Hilbert space εn B (R4) B (R4) 2,2,ℓ k k 2,2,ℓ k k 2,2,ℓ α α (more generically a uniformly convex space), and sinceµ ¯ε±n weakly converges to µ± , we have that the α α 2 1 δ R4 sequenceµ ¯ε±n converges to µ± in L (Ω,B2,−2,ℓ( )). By passing to a suitable subsequence we have that α α 1 δ R4 µ¯ε±n converges to µ± almost surely in B2,−2,ℓ( ). This prove the statements of the theorem involving α α µ¯ε±n and µ± . α α By applying an analogous reasoning to µ± and µ , replacing Theorem 3.13 and Theorem 3.15 εn,Nr ε±n by Proposition 3.17, the thesis is proved. ✷

~~(4) 1+δ 4 1 δ 2 2 Remark 4.7 Since ρ (B− (R ))∗, (B − (R εZ ))∗ an important consequence of Proposition ℓ ∈ 2,2,ℓ/2 2,2,ℓ/2 × 4.6, is that~~

(4) α (4) α (4) α sup ρ (x, z)dµ± , ρ (x, z)dµ± , ρ (x, z)d¯µ± < + ℓ εn,Nr ℓ εn ℓ εn εn,Nr R2 ε Z2 R2 ε Z2 R4 ∞ Z × n Z × n Z almost surely. We can now prove the uniqueness to equation (74).

Proposition 4.8 Suppose that φ¯ and ψ¯ are two solution to equation (74) such that, there exists ℓ > 4 α ¯ α ¯ ¯ ¯ ¯ ¯ 2 ψ 2 φ 1 R4 2αφ 2αψ 1 α ¯ ¯ for which φ, ψ,e± ,e± Hℓ ( ) and e± ,e± L2ℓ(dµ± ) almost surely, then φ = ψ almost surely. ∈ ∈

α 1 δ 4 Proof By Proposition 4.6, µ± B − (R ) almost surely for any ℓ > 4. By Young inequality, and ∈ 2,2,ℓ α ψ¯ α φ¯ 1 4 αψ¯ αφ¯ 1 α using the fact that e± 2 ,e± 2 H (R ) and so cosh , cosh L (dµ± ), we get that ∈ ℓ 2 2 ∈ 2ℓ αφ¯ αφ¯ αψ¯ αφ¯ 1 α ψ¯ e± , φ¯ e± , ψ¯ e± , φ¯ e± L (dµ± ). (83) | | | | | | | | ∈ 2ℓ Since φ,¯ ψ¯ are solution to (74), then φ¯ ψ¯ satisﬁes − 2 α α (m ∆)(φ¯ ψ¯)+ α[exp(αφ¯) exp(αψ¯)]dµ α[exp( αφ¯) exp( αψ¯)]dµ− =0. − − − − − − −

52 By the property (83), we can test the previous equation with (φ¯ ψ¯)ρ2ℓ,λ (where ρ2ℓ,λ(x, z)= ρ2ℓ(λx, λz)) ¯ ¯ − 1 R4 and integrating by parts (since we have assume that (φ ψ) in in Hℓ ( )), and choosing λ > 0 small enough, we obtain −

2 ¯ ¯ 2 ¯ ¯ 2 0 = m R4 ρ2ℓ,λ(x, z) φ ψ dxdz + R4 ρ2ℓ,λ(x, z) (φ ψ) dxdz ¯ ¯ | − | ¯ ¯ |∇ − | + R4 (φ ψ) (ρ2ℓ,λ)(x, z) (φ ψ)dxdz R ¯− ¯∇ ∇ −¯ R ¯ α +α R4 (φ ψ)ρ2ℓ,λ(x, z)(exp(αφ) exp(αψ))dµ (dx, dz) (84) R ¯ − ¯ ¯− ¯ α α 4 (φ ψ)ρ (x, z)(exp( αφ) exp( αψ))dµ− (dx, dz) − RR − 2ℓ,λ − − − > 1 ρ (x, z) φ¯ ψ¯ 2dxdz + ρ (x, z) (φ¯ ψ¯) 2dxdz 2 RR4 2ℓ,λ | − | R4 2ℓ,λ |∇ − | where in the last line we haveR used that R

(φ¯ ψ¯)(exp(αφ¯) exp(αψ¯)), (φ¯ ψ¯)(exp( αψ¯) exp( αφ¯)) > 0, − − − − − − due to convexity of exp( αx), exp(αx) and the fact that ρ2ℓ,λ 6 δρ when λ > 0 is small enough, and thus − ∇

2 2 (φ¯ ψ¯) ρ2ℓ,λ (φ¯ ψ¯) 6 δ ρ2ℓ,λ(x, z) φ¯ ψ¯ dxdz + ρ2ℓ,λ(x, z) (φ¯ ψ¯) dxdz R4 | − ∇ ∇ − | R4 | − | R4 |∇ − | Z Z Z Inequality (84) implies the thesis. ✷

Remark 4.9 Using a reasoning similar to the one in the proof of Proposition 4.8, we get that also equations (75) and (76) admit a unique solution. In particular if φ¯ε,N , and φ¯ε are solutions to equation ¯ ¯ 1 R2 Z2 (76) and (75) respectively such that φε,N , φε Hℓ ( ε ), by Theorem 4.3 and Theorem 4.4, we have α φ¯ α φ¯ 1 2 2 2αφ¯ ∈ 1 (4) × α 2αφ¯ 1 (4) α e± 2 ε,N ,e± 2 ε H (R εZ ), e± ε,N L (ρ dµ± ) and e± ε L (ρ dµ± ). This implies ∈ ℓ × ∈ 2ℓ ε,N ∈ 2ℓ ε that φ¯ , and φ¯ are the unique solutions to (76) and (75) in H1(R2 εZ2). ε,N ε ℓ × In order to prove existence for equation (74) we need the next lemma.

¯ 1 R2 Z2 Lemma 4.10 The solution φε,N to equation (75) exists and is unique in Hℓ ( ε ) (where ℓ>ℓ0 for ¯ × ¯ (4) R βαφε,N 1 R2 Z2 (α 2β)φε,N 1 α some ℓ0 +) almost surely. Furthermore e± Hℓ ( ε ) and e± ± L (ρℓ dµε,N± ) for any 0∈6 β < 1. ∈ × ∈ Proof We consider the following equation

~~¯ ¯ 2 ¯ αφε,N,K α αφε,N,K α ( ∆R2 εZ2 + m )φε,N,K + αe fK(x)µε,N αe− fK(x)µε,N− = 0 (85) − × −~~

2 1/2 where f (x) = exp x +1 +1 . Equation (85) admits a unique solution (see Lemma 7 in [3]) K − K 2 2 2 δ 2 2 in L∞(R εZ ) − (R ,L∞(εZ )) for any δ > 0. × ∩C By Lemma 48 in [3] (see also the discussion of Section 6 of [3]) the solution φ¯ε,N,K converges to φ¯ε,N R2 Z2 2 δ R2 Z2 R2 Z2 in Lexp(∞ β)( ε ) Cloc− ( ,L∞(ε )) (where Lexp(∞ β)( ε ) is the space of L∞ functions with × ∩ 2 × 2 an exponential weight, more precisely the norm of L∞ (R εZ ) is exp(β) × 2 2 f ∞ R2 Z2 = sup exp( β( x + z ))f(x, z) , f L∞ (R εZ ), Lexp(β)( ε ) exp(β) k k × (x,z) R2 εZ2 | − | | | | | ∈ × ∈ × 2 2 2 δ 2 2 where β R+) for some β > 0. Furthermore, since φ¯ε,N,K L∞(R εZ ) − (R ,L∞(εZ )), then ¯ ∈ 1 R2 Z2 ∈ × ∩C φε,N,K Hℓ ( ε ) for any ℓ > 4, and so by Theorem 4.3 and Theorem 4.4 and Proposition 4.6 we have that∈ ×

βαφ¯ε,N,K (α 2β)φ¯ε,N,K ¯ 1 1 ± φε,N,K H (R2 εZ2), e± H (R2 εZ2), e± ± L1(ρ(4) dµ α ) . 1 k k ℓ × k k ℓ × k k 2ℓ ε,N where the constants, hidden in the symbol ., do not depend on K. Since φ¯ε,N,K converges strongly to ¯ ¯ ¯ 1 R2 Z2 φε,N , this implies that φε,N,K converges weakly to φε,N in Hℓ ( ε ). By lower semicontinuity of the × ¯ norm H1(R2 εZ2) with respect to the weak convergence we have φε,N H1(R2 εZ2) < + . k·k ℓ × k k ℓ × ∞ The uniqueness of solution to equation (75) is proved in Remark 4.9. ✷

53 Lemma 4.11 For any α < (4π) and εn,Nk > 0 as in the thesis of Proposition 4.6, there exists a (unique) solution to equation| | (76) such that φ¯ H1(R2 ε Z2) almost surely (for any ℓ>ℓ > 0 for εn ∈ ℓ × n 0 some ℓ0 not depending on εn). ¯ 1 R2 Z2 Proof By Lemma 4.10, equation (75) admits a (unique) solution φεn,Nk in Hℓ ( εn ) for ℓ big enough. By Proposition 4.6, Theorem 4.3 and Theorem 4.4 we obtain ×

βαφ¯ε ,N (α 2β)φ¯ε ,N ¯ 1 n k n k φεn,Nk H (R2 ε Z2), e± , e± ± . 1, (86) ℓ n 1 R2 Z2 1 (4) ±α k k × H ( εn ) L (ρ dµε ,N ) ℓ × 2ℓ n k where the constants hidden in the symbol . are independent of Nk. ¯ ¯ βαφεn,Nk This means in particular that the sequences (in Nk) φεn,Nk and e± are weakly precompact in ¯ 1 2 2 1 2 2 βαφε ,N R Z R Z ¯ ± n ko Hℓ ( εn ). Consider κ,κ′ Hℓ ( εn ) such that there is a subsequence φεn,Nko and e × ± ∈ × 1 R2 Z2 converging to κ and κ′ respectively weakly in Hℓ ( εn ). Since, by Theorem 4.3 and Theorem 4.4, ¯ ± × βαφεn,N s,2 2 2 ¯ ± ko ′ R Z φεn,Nko and e converges strongly to κ and κ′ in Wℓ ( εn ), for any ℓ′ >ℓ and 0 ℓ > 4.

2 2 Let K > 0 and consider the function FK : R R deﬁned as FK (x) = x K . The function FK is → ±∧α φ¯ 1 2 2 (globally) Lipschitz and bounded, this implies, by Remark 2.17, that F e 2 ε,N H (R εZ ) K ∈ ℓ × ∩ 2 2 L∞(R εZ ), and × ±α φ¯ε,N (x,z) αφ¯ε,N (x,z) FK e 2 = e±

± α α ¯ ±α φ¯ (x,z) φε ,N (x,z) κ 1 (4) α when e 2 ε,N 6 K. In particular we have that F e 2 n ko , F e 2 L (ρ µ ) and, K K ∈ ℓ εn ± 1 2 2 α φ¯ (x,z) by the convergence of φ¯ to κ in H (R ε Z ), and since F is Lipschitz, F e 2 ε,Nk εn,Nk ℓ × n K K converges to FK (κ) Lebesgue almost everywhere. This means that, by Fatou lemma and using the fact α α that µ± are functions converging to µ Lebesgue almost everywhere, we get εn,Nk ε±n

(4) α 2 κ α 2 2 ρ ′ (x, z)FK e± µ (dx, dz) R εnZ ℓ+ℓ εn × α (4) φ¯ε,N (x,z) α ′ 2 k 6 lim infNk + R2 Z2 ρ (x, z)FK e± µε ,N (dx, dz) o → R∞ εn ℓ+ℓ n ko × ¯ (4) αφε,N (x,z) α 6 lim inf ρ ′ (x, z)e± ko µ (dx, dz) . 1 Nko + R R2 ε Z2 ℓ+ℓ εn,Nk → ∞ × n o Applying again Fatou lemma for KR + → ∞ (4) ακ α 2 2 ρ ′ (x, z)e± µ± (dx, dz)= R εnZ ℓ +ℓ εn ×(4) α 2 κ α = R2 Z2 ρ ′ (x, z) limK + FK e± µε (dx, dz) εn R ℓ +ℓ → ∞ n × (4) α 2 κ α 6 limK + 2 2 ρ ′ (x, z)FK e± µ (dx, dz) . 1 R R εnZ ℓ +ℓ εn → ∞ × R ακ 1 α This implies that e± L (dµ± ). What remain to prove is that κ solves (weakly) equation (76). ∈ εn First, for any σ> 0, by inequality (86), we have that there exists a p> 1 such that

(4) αφ¯ (x,z) p α ρ (x, z)e± ε,N L (µ± (dx, dz) ℓ ∈ ε,N uniformly in N.This implies by Jensen’s inequality that there is a Kσ > 0 for which, for any K>K¯ σ and any N N ∈ (4) αφ¯ (x,z) α ρ (x, z)e± ε,N µ± (dx, dz) < σ. ± ¯ ℓ ε,N αφε,N ¯ Ze >K Consider g (R2 εZ2) by multiplying equation (75) by g, taking the integral and using integration ∈ S ×

~~54 by parts, for any K>K¯ σ, we get~~

2 α φ¯ (x,z) R2 Z2 ¯ 2 εn,N R2 εZ2 [(m ∆ ε )g(x, z)]φεn,Nko dxdz + α R2 εZ2 g(x, z) FK e × − × × × α ¯ α α 2 φε,N (x,z) R µε ,N (dx, dz) FK¯ e− R µ− (dx, dz) 6 n Ko − εn,NKo α ¯ ¯ 2 φεn,N (x,z) αφεn,N (x,z) α . 2 2 g(x, z) F ¯ e e µ (d x, dz) + (87) R εZ K εn,NKo × − α ¯ ¯ α 2 φεn,N (x,z) αφεn,N (x,z) + 2 2 g(x, z) F ¯ e− e− µ− (dx, dz) R εnZ K εn,NK × − o αφ¯ε ,N (x,z) α ±αφ¯ n ∞ R2 Z2 . εn,N ¯ g(x, z)e± µε± ,N (dx, dz) . σ g L ′ ( ε ) R e >K n −ℓ−ℓ | | k k × R α α φ¯ε ,N (x,z) 1 2 2 κ Since φε ,N and F ¯ e± 2 n converge weakly (in H ′ (R εnZ )) to κ and F ¯ e± 2 n ko K ℓ × K α α 1 δ R2 Z2 respectively, and, by Proposition 4.6, µε± ,N converge to µε±n strongly in B2,−2,ℓ( εn ), we can n Ko × take the limit as N + in inequality (87) we get ko → ∞ α 2 2 κ R2 Z2 [(m ∆R2 εZ2 )g(x, z)]κdxdz + α R2 Z2 g(x, z) FK¯ e ε − × ε × × α α κ α × (88) 2 ∞ 2 2 µε (dx, dz) FK¯ e− µε (dx, dz) 6 σ g L ′ (R εZ ) R n − n R k k −ℓ−ℓ ×

~~αφ¯ε(x,z) 1 (4) α Since e± L (ρ ′ dµ± ), by Lebesgue dominate convergence theorem, we can take the limit ∈ ℓ +ℓ εn K¯ + , obtaining → ∞~~

~~2 ακ α ακ α R2 Z2 [(m ∆ εn )g]κdxdz + α g (e µεn (dx, dz) e− µεn (dx, dz)) . σ R2 ε Z2 − × R2 ε Z2 − Z × n Z × n~~

R2 Z2 1 R2 Z2 Since σ is arbitrary and ( εn ) is dense in the dual of Hℓ+ℓ′ ( ε ) we have that κ solves the S × × ¯ equation (76). By the uniqueness of solution to equation (76) obtained in Remark 4.9 κ = φεn the unique solution to equation (76). ✷

¯ Theorem 4.12 Consider εn as in Proposition 4.6 and let φεn be the solution to equation (76) for ε = εn. εn 4 1 4 Then (φ¯ ) converges in ′(R ) to the unique solution φ¯ to equation (74) in H (R ) (for any ℓ > ℓ E εn S ℓ 0 and a suitable ℓ0 > 0). Proof The proof is similar to the one of Lemma 4.11. We report here only a sketch of the proof pointing out the main diﬀerences.

~~By the Lemma 4.11, there is a unique solution to equation (76) and by Theorem 4.3, Theorem 4.4 and Proposition 4.6 we get for β ( 1, 1) ∈ −~~

βαφ¯ε (α 2β)φ¯ε ¯ 1 n n φεn H (R2 ε Z2), e± , e± ± . 1 (89) ℓ n 1 R2 Z2 1 (4) ±α k k × Hℓ ( εn ) L (ρ2ℓ dµεn ) × where the constants hidden in the symbol . are independent on εn. Using the same methods of Lemma 1 ε p 4 nk 4.11 there is a subsequence εn 0 and a κ B (R ) (for any 1 6 p 6 2) such that φ¯ε k → ∈ p,p,ℓ E nk → ε 1 1 υ nk βαφ¯ αβκ p 4 p 4 κ, e± εn e± weakly in B (R ) (and thus strongly in B − (R ) for any υ > 0). E → p,p,ℓ p,p,ℓ βακ 1 R4 Furthermore, by the second part of Theorem 2.39, we also have that κ,e± Hℓ ( ). What remains ακ 1 (4) α ∈ to prove is that e± L (ρ dµ± ) and κ is a solution to equation (74). ∈ ακ 1 (4) α (4) 2 2 (4) First we prove that e± L (ρ dµ± ). Let ρ : R εZ R be the restriction of ρ , since ∈ ℓ,εn × → ℓ (4) k R4 s,p R2 Z2 ρℓ B , , ℓ( ) for any k> 0, it is simple to prove (using the diﬀerence spaces W ℓ ( ε ) and ∈ ∞ ∞ − − × s υ R2 Z2 (4) k R2 Z2 their continuous immersion in Bℓ− ( ε ) for any υ> 0), that ρℓ,ε B , , ℓ( εn ) and also × n ∈ ∞ ∞ − × ε 1 υ n (4) (4) p − R4 that (ρℓ,ε ) converges strongly to ρℓ in Bp,p, ℓ( ) for any υ > 0. E n − εn R2 Z2 R2 Z2 εn Consider operator ∆N : ′( εn ) ′( εn ), deﬁned as ∆N f = f εn ρN where f R2 Z2 εn SR2 T×2 → S × ∗ ∈ ′( εn ) and ρN : 1 [0, 1] is smooth and compactly supported in a ball of radius S × F × en → 2N and εn ρ (ξ) = 1 for ξ 6 N. The operator ∆εn has a regularizing property: for any ∆εn : F N | | N N

55 s 2 2 s′ 2 2 B (R ε Z ) B (R ε Z ) for any s′ > s with a norm depending only on p,ℓ,s and s′ but p,p,ℓ × n → p,p,ℓ × n not on εn. Ns α Furthermore, since by Theorem 3.13 and Theorem 3.15 and by the proof of Proposition 4.6, p,ℓ,εn (µεn ,a,β) Ns (µα,a,β) almost surely, for any υ > 0 there is a random variable N : Ω N, almost surely ﬁnite, → p,ℓ υ → such that for any N¯ > Nυ εn α (1 ∆ )µ± 1−δ 2 2 6 υ. N¯ εn B (R εnZ ) k − k 2,2,ℓ × ε n εn α Furthermore using again the proof of Proposition 4.6, one can prove that (∆N µεn ) converges to 1 υ′ E α p − R4 ∆N µ± in Bp,p,ℓ ( ) for any ℓ > 4 and υ′ > 0. Consider K > 0 and FK as in the proof of Lemma 4.11, then (4) α 2 κ α ρ2ℓ (x, z)FK e± ∆N (µ± )(dx, dz) Z εk (4) α ¯ εk 2 φε α = lim ρ FK e± k (∆N (µε± )(dx, dz)) k + E 2ℓ,εk E k → ∞ Z (4) α ¯ 2 φε α = lim ρ FK e± k ∆N µε± (dx, dz) k + 2ℓ,εk k → ∞ Z α α (4) φ¯ε α (4) φ¯ε α 6 sup ρ FK e± 2 k µ± (dx, dz) + ρ FK e± 2 k (1 ∆N )(µ± )(dx, dz) 2ℓ,εk εk 2ℓ,εk − εk k Z Z α (4) αφ¯ε α φ¯ε . sup ρ e± k µ± (dxdz) + υ sup FK e± 2 k . 2ℓ,εk εk 1 R2 Z2 k k Hℓ ( εnk ) Z × Since υ is arbitrary taking the limit N + we get → ∞

α (4) κ α (4) αφ¯ε α ρ (x, z)FK e± 2 µ± (dx, dz) . sup ρ e± k µ± (dxdz) < + . 2ℓ 2ℓ,εk εk ∞ Z k Z ακ 1 (4) α By Fatou lemma, the limit K + gives that e± L (ρ2ℓ dµ± ). Using a similar strategy it is 2ακ 1→(4) ∞ α ∈ simple to proof that e± L (ρ dµ± ) as required in Proposition 4.8. ∈ 2ℓ For proving that κ is also a (weak) solution to equation (74) using a combination of the technique 4 in the proof of Lemma 4.11 and in the previous part of the proof. Consider g (R ) and let gε ∈ S1 ∈ R2 Z2 R2 Z2 ε p R4 ( ε ) the restriction of g on ε . We have that (gε) converges to Bp,p, ℓ( ) and also Sε × α ε α × E − (g dµ± )= (g )d¯µ± E ε ε E ε ε 2 αφ¯ α αφ¯ α 2 2 ¯ ¯ εn εn 0= gεn , ∆R εZ (φεn ) + m gεn , φεn + α R2 Z2 gεn e dµε α R2 Z2 gεn e− dµε− = × ε n ε n −h i 2 h i × αφ¯ α − × αφ¯ α = ∆R2 Z2 g , φ¯ + m g , φ¯ + α g e εn dµ α g e ε dµ = ε εn εn εn εn R2 εZ2 εn εn R2 εZ2 ε − ε−n −h × i h i R × − R × 2 αφ¯ α αφ¯ α = ∆R2 Z2 g , φ¯ + m g , φ¯ + α g F ¯ e εn dµ α g F ¯ (e ε )dµ ε εn εn εn εn R2 εRZ2 εn K εn R R2 εZ2 ε K − ε−n −h × i h i × − × ¯ ¯ αφεn ¯ α αφεn ¯ α +α αφ¯ε ¯ gε e K R dµ α −αφ¯ε ¯ gε e− R K dµ− e n >K n − εn − e n >K n − εn ε ε ¯ 2 ε ε ¯ R = R4 (∆ R2 εZ2 gε) (φε)dxdzR+ m R4 (gε) (φε)dxdz+ − E ε × E E E ε ε α n (φ¯ ) ε α ε α n (φ¯ ) ε α +α (g )F ¯ e εn d ( µ )+ α (g )F ¯ e εn d ( µ ) R2 εZ2 εn RK E N εn R2 RεZ2 εn K E N εn × E E × E E ¯ ¯ αφεn ¯ α αφεn ¯ α R +α αφ¯ε ¯ gε e K dµ −αRφ¯ε ¯ gε e− K dµ− + e n >K n − εn − e n >K n − εn αφ¯ εn α αφ¯ εn α ¯ εn ¯ εn +α R2 εRZ2 gεn FK e d((1 N )µεn )+Rα R2 εZ2 gεn FK e− d((1 N )µε−n ). × − × − R R The previous equality (together with a reasoning similar to the one of equation (87)) implies that there

~~56 exist Kσ,Nυ such that for K>K¯ σ and N >Nυ we get~~

εk εk 2 εk εk 2 2 ¯ ¯ (∆R εk Z gεk ) (φεk )dxdz + m (gεk ) (φεk )dxdz + − R4 E × E R4 E E Z Z ε ε εk α k (φ¯ ) εk α εk α k (φ¯ ) εn α ¯ εk ¯ εk α (gεk )FK e E d (N µεk ) α (gεk )FK e− E d (N µε−k ) R4 E E − R2 ε Z2 E E Z Z × k ¯ ¯ αφεn ¯ α αφεn ¯ α . α gεn e K dµε gεn e− K dµε− ¯ n − ¯ n eαφεn >K¯ − − e αφεn >K¯ − Z Z ¯ ¯ αφεn εn α αφεn εn α + α gεn FK¯ e d((1 N )µεn )+ α gεn FK¯ e− d((1 N )µε−n ) R2 εZ2 − R2 εZ2 − Z × Z × . 2(σ + υ)(sup g 1 R2 Z2 ). εk B∞ ∞ − ( εk ) k k k , , ℓ ×

~~Taking the limit in order εn + , N + , K¯ + , the previous terms in the last inequality converges to → ∞ → ∞ → ∞~~

~~2 αφ¯ α αφ¯ α (∆R4 g)φ¯dxdz + m gφ¯dxdz + α g(e dµ e− dµ− ) 6 2(σ + υ) gε 1 . k B∞,∞,−ℓ − R4 R4 R4 − k k Z Z Z~~

~~4 By the arbitrarity of σ,υ > 0 and g (R ) we get the thesis. ✷ ∈ S~~

~~4.5 Improved a priori estimates and stochastic quantization~~

In this section we want to prove the (elliptic) stochastic quantization of the cosh(ϕ)2 model. First we 2 2 4 2 recall that, for any x R and f (R ), the tempered distribution δx f ′(R ) (where δx ′(R ) is the Dirac delta with∈ unitary mass∈ S in x) belongs to the Cameron-Martin⊗ ∈ S space associated∈ with S the 2 2 4 Gaussian random distribution W = ( ∆R2 εZ2 + m )(ξ), namely δx f H (R ). This means that for any x R2, it is well deﬁned the Gaussian− × random distribution W (x,⊗) ∈ ∈ · 2 ′(R2) W (x, ),f (R2) := W, δx f L (Ω) (90) S h · iS h ⊗ i ∈ where the pairing W, δx f is understood in the probabilistic sense of Wiener-Skorokhod integral (see, h ⊗ i 2 2 e.g., [67]). Furthermore it is possible to build a version of the map x W (x, ) ′(R ) L (Ω) which 2 2 7→ · ∈ S ⊗ is continuous with respect to the natural topology of ′(R ) L (Ω) (see Lemma 33 of [2]). Hereafter we write W (x, ) for the continuous version of the previousS map⊗ satisfying the relation (90). · In the remainder of this section we prove the following statement.

Theorem 4.13 For any α < 4π and for any x R2 the random tempered distribution φ(x, ) = 2 | | ∈ · φ¯(x, )+ ξˆ(x, ) ′(R ), where φ¯ is the (unique) solution to equation (74) satisfying the hypotheses of Proposition· 4.8,· ∈ is S well deﬁned and distributed as νcosh,β for β = α , where νcosh,β is a probability m √4π m measure associated with the cosh(ϕ)2 model in the sense of Deﬁnition 4.1.

In order to prove Theorem 4.13 we use the approximations φ¯ε,N and φ¯ε satisfying equations (75) and cosh,β ¯ (76) respectively. The ﬁrst link between the measure νm and the approximation φε,N is given by the following lemma.

2 Lemma 4.14 There exists a unique solution φ¯ε,N to equation (75) which is continuous in x R and decay at most exponentially at inﬁnity, furthermore for any x R2 we have ∈ ∈ V cosh,β (ϕ) ε e− N cosh,β (φε,N )(x, ) νm,ε,N (dϕ)= νm,ε,N , (91) E · ∼ Zm,ε,N where φε,N = φ¯ε,N + Wε,N

~~57 Proof The random ﬁeld φ¯ε,N solves equation (75) if and only if φε,N ( ,z) solves the following (ﬁnite) set of elliptic SPDE on R2 ·~~

2 2 2 2 ( ∆R2 + m )φ (x,iε)+ ∂ i V ( φ (x,iε) )= ξ (x,iε) i , (92) − ε,N y ε { ε,N } − ε ∈ −N N N N 2 where V : R × R is defined as ε → 2 2 (yi yi+1 ) (yi yi+1 ) 2 1 2 α cε RN N Vε(y)= − + − +2e− cosh(αyi) , yi 2 2 2 × 2 2 i [ N , N ) 2 ε ε { } ∈ − ∈ i 2 , 2 ∈[−XN N ) 1 E 2 where cε = 2 [Wε ]. Since the function Vε is convex, by Theorem 4 of [3] there is a unique solution to equation (75) which grows at most exponentially at infinity. Furthermore, for any x R2, the law of φ (x, ) satisfies relation (91). ∈ ✷ ε,N · R2 ¯ ¯ Thanks to Lemma 4.14 what remains to prove is that, for any x , φεn,Nk (x, ) φεn (x, ) and εn ¯ ¯ ∈ · → · (φεn )(x, ) φ(x, ) almost surely. The convergence results proved above, namely the convergence of E¯ · ε→n ¯ · ¯ ¯ 1 R2 Z2 1 R4 φεn,Nk and (φεn ) to φεn and φ in Hℓ ( εn ) and Hℓ ( ) respectively, is too weak for proving Theorem 4.13.E We need some a priori estimates× stronger than Theorem 4.3 and Theorem 4.4.

1 τ R2 Z2 Theorem 4.15 Suppose that η Bℓ − ( ε ) (for any τ > 0 and any ℓ>ℓ0) and suppose that ± ∈p × p′ there are r> 1, 1 6 p′ < 3, θ,s > 2 such that, writing q′ = p′ 1 , we have − ( θ+ 2 )q′,( s+ 2 )q′ M − p − p (η , a,¯ β¯) < + r,ℓ,ε ± ∞ (where a¯ > 0 and 0 < β¯ < 1 are the constants appearing in Remarks 2.6 and Remark 2.29). If ψ¯ is the unique solution to equation (77) belonging to H1(R2 εZ2) then ℓ × 2 θ 2 s 2 ψ¯ B − (R ,B− (εZ )) ∈ p,p,ℓ p,p,ℓ and also, for τ > 0 small enough and ℓ> 0 big enough we have,

2 ′ 2 ′ ( θ+ p )q ,( s+ p )q ¯ − − − M − − ¯ ψ B2 θ (R2,B s (εZ2)) . Pθ,α,r,τ η B1 τ (R2 εZ2), r,ℓ,ε (η+, a,¯ β)+ k k p,p,ℓ p,p,ℓ k ±k ℓ × 1 ′ ( θ+ 2 )q′,( s+ 2 )q′ q +M − p − p (η , a,¯ β¯) r,ℓ,ε − where Pθ,α,r,τ is a polynomial independent of 0 <ε 6 1.

αψ 3−δ 1−κ 1−κ Proof By Theorem 4.4 that we have that, for any δ > 0, e± L (dη±) < Pδ( η+ B (R2 εZ2), η B (R2 εZ2)) k k ℓ′ k k 2,2,ℓ × k −k 2,2,ℓ × for a suitable polynomial Pδ (independent on ε). By Theorem 2.44, choosing p = p′, we get that

′ ′ ′ 1/q ¯ ¯ ′ ( θ+ 2 )q ,( s+ 2 )q αψ 1/2 αψ 1/p M − p − p ¯ e± η −θ,−s . e± 3−δ (η , a,¯ β) . ± r,ℓ,ε k ±kB2,2,2ℓ k kLℓ (η ) ± Then, since ¯ 2 1 αψ¯ αψ¯ ψ = (∆R2 εZ2 + m )− (αe dη+ αe− dη ), × − − by Theorem 2.34, the thesis follows. ✷

R α2 2 α2 Remark 4.16 Fix α such that 2 < 1, then there exist 1 ′ , r> 1, 2 > 2 (r 1) ∈ (4π) p (4π) − such that 2(p′ 1) 2 θ − > 0. − − rp′ ′ ′ ′ 2 2(p 1) 2 2(p 1) 1 2(p 1) Indeed writing θ = p′ + δ, we have 2 θ rp−′ =2 p′ δ rp−′ = 1 r p−′ δ which is ′ − − − − − − − 2(p 1) 1 positive if we choose 0 <δ< −′ 1 . p − r 58 R α2 2 Proposition 4.17 Let α , (4π)2 < 1 and consider p′,r > 1 and θ > p′ satisfying the requirement of ∈ ′ p Remark 4.16, and, writing q′ = (p′ 1) and p = rq′, and take − α2 4 sθ,α,r > 2 (r 1)q′ θ + . (4π) − − p′

h Then there is a sequence ε =2− n 0 as n + and N + , such that, for any ℓ > 4, we have n → → ∞ k → ∞ 2 ′ 2 ′ 2 ′ 2 ′ ( θ+ p )q ,( sθ,α,r + p )q α ( θ+ p )q ,( sθ,α,r + p )q α sup M − − (µ± , a,¯ β¯), M − − (µ± , a,¯ β¯) < + r,ℓ,εn εn r,ℓ,εn εn,Nk ∞ εn,Nk almost surely. Proof The proof is similar to the one of Proposition 4.6, thus we omit the details. ✷

Lemma 4.18 Let εn and Nk as in Proposition 4.17, then, for any εn, the sequence (in Nk) of measures νcosh,β on (R2) converges to a unique measure νcosh,β. Furthermore, for any x R2, we have m,εn,Nk ′ m,εn ε S ε ∈ n (φ¯ + W )(x, )= n (φ¯ )(x, )+ W¯ (x, ) νcosh,β. E εn εn · E εn · εn · ∼ m,εn Proof By Theorem 4.15 and Proposition 4.17, if we choose p,sθ,α,r,ℓ,θ such as in Remark 4.16 and

¯ − −s ¯ − −s 6 Proposition 4.17, we have that φεn,Nk 2 θ R2 θ,α,r Z2 , φεn 2 θ R2 θ,α,r Z2 C(ω) (for k kBp,p,ℓ( ,Bp,p,ℓ (εn )) k kBp,p,ℓ( ,Bp,p,ℓ (εn )) some random variable 0 6 C(ω) < + almost surely) uniformly in εn and Nk. This implies, by Theorem 2.26, that∞

~~εn εn (φ¯ ) −s −δ′ , (φ¯ ) −s −δ′ 6 C′(ω) (93) εn,Nk 2−θ R2 θ,α,r R2 εn 2−θ R2 θ,α,r R2 kE kBp,p,ℓ( ,Bp,p,ℓ ( )) kE kBp,p,ℓ( ,Bp,p,ℓ ( ))~~

(for some random variable 0 6 C(ω) < + almost surely and any δ′ > 0) uniformly in εn and Nk. ′′ ¯ ∞ ¯ 1 δ R2 Z2 Since, by the proof of Lemma 4.11, φεn,Nk φεn almost surely in Hℓ − ( εn ), by Theo- εn ¯ εn ¯ 2 R4 → × ε ¯ rem 2.39, (φεn,Nk ) (φεn ) in Lℓ ( ) almost surely. Thus, by the bound (93), (φεn,Nk ) E → E ′ E → εn 2 θ sθ,α,r δ ¯ R2 − − R2 (φεn ) almost surely and (weakly) in Bp,p,ℓ− ( ,Bp,p,ℓ ( )). On the the other hand since 2 E 2 − θ p > 0 (by the choice in Remark 4.16), by Besov embedding theorem (see Theorem 2.2.4 of [11]), − ′ ′ ′ 2 θ sθ,α,r δ sθ,α,r δ sθ,α,r δ R2 − − R2 0 R2 − − R2 0 R2 − − R2 we have Bp,p,ℓ− ( ,Bp,p,ℓ ( )) C ( ,Bp,p,ℓ ( )), where C ( ,Bp,p,ℓ ( )) denotes ⊂ s δ′ R2 − θ,α,r − R2 the Fréchet space of (locally bounded) continuous functions from into Bp,p,ℓ ( ), which implies ′ 2 θ 2 sθ,α,r δ 2 2 2 that δ f (B − (R ,B− − (R )))∗ for any x R . So, we can deduce that, for any x R , x ⊗ ∈ p,p,ℓ p,p,ℓ ∈ ∈ εn ¯ εn ¯ R2 (φεn,Nk )(x, ) (φεn )(x, ) almost surely and (weakly) in ′( ). E · → E · S εn Furthermore, using a proof similar to the one of Theorem 3.11, it is easy to prove that (Wεn,Nk )(x, ) εn R2 E εn · → (Wεn )(x, ) in probability and weakly in ′( ). This means that, since, by Lemma 4.14, (Wεn,Nk )(x, )+ Eεn · S E · (φ¯ )(x, ) νcosh,β , that νcosh,β is tight in (R2) and it converges to a (unique) measure εn,Nk m,εn,Nk m,εn,Nk ′ Ecosh,β · ∼ S εn ν which is the probability law of the random distribution (φ¯ε + Wε )(x, ). ✷ m,εn E n n · Proof of Theorem 4.13 The proof follows the same lines of the proof of Lemma 4.18. More pre- εn ¯ ¯ R4 εn ¯ cisely, by inequality (93), since by Theorem 4.12 (φεn ) φ almost surely in ′( ), (φεn ) ′ 2 θ E sθ,α,r→ δ S E ¯ R2 − − R2 converges to φ almost surely and weakly in Bp,p,ℓ− ( ,Bp,p,ℓ ( )). Furthermore, since δx f ′ ⊗ ∈ 2 θ 2 sθ,α,r δ 2 2 2 εn (B − (R ,B− − (R )))∗ for any x R and any f (R ), the random distribution (φ¯ )(x, ) p,p,ℓ p,p,ℓ ∈ ∈ S E εn · ¯ R2 εn converges to φ(x, ) almost surely and in ′( ). By Theorem 3.11, we have that (Wεn )(x, ) W (x, ) 2 · R2 S cosh,β εn ¯ E · → · in L (Ω) and in ′( ). By Lemma 4.18 νm,εn (φεn + Wεn )(x, ), which, by the previous conver- S cosh,β R2 ∼ E · cosh,β R2 gences, implies that νm,εn is tight in ′( ) and it converges to a measure νm on ′( ) which is ¯S cosh,β S the law of the random field W (x, )+ φ(x, ). The measure νm is unique due to the uniqueness of the law of the free field W and the uniqueness· · of the solution to equation (74). ✷

~~59 A Osterwalder-Schrader Axioms using stochastic quantization~~

In this appendix we will show that the measure we constructed satisfies the Osterwalder-Schrader axioms, using the stochastic quantization equation introduced above. The study of the properties of the measure cosh,β 4 νm presented here are inspired by the analogous stochastic quantization proofs for ϕ3 measure in [8, 29, 35]. We recall them here briefly for the convenience of the reader. Hereafter we define the distributions 2n S ′(R ) as n ∈ S cosh,β Sn(f1 fn)= ϕ, f1 ′, ϕ, fn ′, dνm (ϕ). ⊗···⊗ h iS S ···h iS S Z

Axiom 1 (Regularity) S =1 and there exists a Schwartz semi-norm ,β > 1,K > 0 such that 0 k·ks n S (f ... f ) 6 Kn(n!)β f | n 1 ⊗ ⊗ n | k iks i=1 Y Axiom 2 (Euclidean Invariance) Let the Euclidean group with G = (R,a) R O(d),a Rd acts on functions by ∈ ∈ (Gf)(x)= f(Rx a). − Then S (Gf ... Gf )= S (f ... f ). n 1 ⊗ ⊗ n n 1 ⊗ ⊗ n Rnd Axiom 3 (Reﬂection Positivity) Let + be the set

nd d 1 x R : x = (x , . . . ., x ) and x = (τ ,y )withτ > 0 and y R − ∈ 1 n i i i i i ∈ nd Furthermore deﬁne the reﬂection Θ(x) = Θ((τ,y)) = ( τ,y) and its action on a function f ′(R ) by − ∈ S

~~Θf(x1,...,xn)= f(Θx1, . . . ., Θxn).~~

nd nd Now we require that for all ﬁnite families f (R ) 6 such that supp f R we have { n ∈ S }n M n ⊆ + M S (Θf f ) > 0. i+j i ⊗ j i,j=1 X Axiom 4 (Symmetry) Let π be a permutation of 1, . . . ., n . Then { } S (f ... f )= S (f ... f ) n π(1) ⊗ ⊗ π(n) n 1 ⊗ ⊗ n Axiom 5 (Clustering) For any j, n N, 1 6 j 6 n and a Rd such that a =1 as λ ∈ ∈ | | → ∞

Sn(f1 ... fj fj+1( + λa) ... fn( + λa)) Sj (f1 ... fj)Sn j (fj+1 ... fn). ⊗ ⊗ ⊗ · ⊗ ⊗ · → ⊗ ⊗ − ⊗ ⊗ Rd If furthermore there exists m> 0 such that for any family of fj Cc∞( ) there exists a constant C > 0 such that ∈ m λ S (f f ( + λa)) S (f )S (f ) 6 Ce− | | | 2 1 ⊗ 2 · − 1 1 1 2 | We say that the clustering is exponential.

~~Remark A.1 Exponential clustering implies clustering by Theorem VIII.36 in [61].~~

β2 α2 In what fallows β will always be a (positive) constant such that (4π) = (4π)2 < 1. Furthermore now on we denote by εn 0 and Nr + two sequences satisfying the thesis of Proposition 4.6 and of Proposition 4.17. → → ∞

~~cosh,β Theorem A.2 The measure νm introduce in Deﬁnition 4.1 satisﬁes the Osterwalder-Schrader axioms with exponential clustering.~~

60 Proof Axiom 2 follows from the fact that the solution to Equation (74) is unique, and therefore invariant cosh,β under the Euclidean group, since the equation itself is invariant. Axiom 3 follows from the fact that νm is a weak limit of reﬂection positive measures. Axiom 4 is obvious. It remains to show Axiom 1 and exponential clustering. This is the content of the next two subsections. ✷

~~A.1 Analyticity The aim of this section is to prove the following~~

~~Theorem A.3 There exits ℓ,δ,γ > 0~~

2 cosh,β exp(γ ϕ −2 2 )dν (ϕ) < − B (R ) m B 2δ (R2) k k 2,2,ℓ ∞ Z 2,2,2ℓ We will follow the strategy developed in [35]. The proof is postponed until the end of the subsection, while we ﬁrst collect some assertions needed for it. We introduce the notation

γ 1 ε 2 cosh,β − d˜νN,ε(ϕ)= ε,N,γ exp(γ (ϕ) B 1 (R2))dνm,ε,N (ϕ), (94) Z kE k 2,2,ℓ where ε,N,γ 1/2 ε 2 cosh,β Z = exp(γ (1 ∆R2 )− (ϕ) 2 R2 )dν (dϕ), (95) − L ( ) m,ε,N B δ (R2) k − E k ℓ Z 2,2,ℓ cosh,β the measure νm,ε,N is deﬁned in equation (91), and δ > 0 small enough. We also introduce the operator ε : Lp(R4) Lp(R2 εZ2) deﬁned as D ℓ → ℓ × ε(f) = (I ε)(f), f Lp(R4), D ⊗ D ∈ ℓ where ε is the operator introduced in ThenD we have the following proposition:

~~Proposition A.4 For γ > 0 small enough we have that~~

ν˜γ = Law(W (0, )+ φγ (0, )) N,ε ε,N · ε,N · γ δ 2 2 where φ B− (R εZ ) is the unique solution to the equation ε,N ∈ 2,2,ℓ × γ γ γ 2 ¯ αφN,ε α αφN,ε α ( ∆R2+εZ2 + m )φε,N + αe dµε,N αe− dµε,N− − ε 1 ε − 1 ε γ (96) = γ (ρ (z)(1 ∆R2 )− (W¯ )) + γ (ρ (z)(1 ∆R2 )− ( (φ¯ ))), D ℓ − ,z ε,N D ℓ − ,z E ε,N 2 where ∆R2 is the Laplacian with respect to the second set of variables z R . ,z ∈ Proof This is a consequence of Theorem 4 in [3] in the same way as Lemma 4.14 since for γ small enough 2 2 2 1/2 ε 2 ( ϕ L2(εZ2) + m ϕ L2(εZ2)) γ (1 ∆R2 )− (ϕ) L2(R2) k∇ k ℓ k k ℓ − k − E k ℓ is (strictly) convex. Equation (96) follows from the fact that

~~γ d 1 1/2 ε 2 ε − ε (1 ∆R2 )− (ϕ + εψ) L2(R2) = γ ρℓ(z) 1 ∆R2 (ϕ) ψ(z)dz 2 dεk − E k ℓ εZ2 D − E Z ✷~~

~~γ Proposition A.5 Let φN,ε be the solution to equation (96) deﬁned by Proposition A.4. Then for ℓ large enough and γ small enough we have~~

~~E ¯γ 2 sup [ φ 1 R2 Z2 ] < + . εn,Nr Hℓ ( εn ) Nr,εn k k × ∞~~

61 ε ε Proof Observe that, by the continuity of the operators and (see E D ε n 1 εn ¯γ ¯γ ¯γ 2 γ ρ (x, z) (ρ (z)(1 ∆R2 )− (φ ))φ dxdz . γ φ 2 2 2 ℓ ℓ ,z εn,Nr εn,Nk εn,Nr L (R εZ ) R2 ε Z2 D − E k k ℓ × Z × n and, thus,

ε 1 ε γ γ ρ (x, z) (ρ (z) (1 ∆R2 )− (W ))φ¯ dxdz ℓ ℓ ,z εn,Nr εn,Nr R2 εZ2 D · − E Z × ¯γ 6 γ W −1 2 2 φ 1 R2 Z2 εn,Nr H (R ε Z ) εn,Nr B ( ε ) k k 2,2,ℓ/2 × n k k 2,2,ℓ/2 × 2 ¯γ 2 6 γ( W −1 + φ 1 2 2 ). εn,Nr H (R2 ε Z2) εn,Nr B (R εZ ) k k 2,2,ℓ/2 × n k k 2,2,ℓ/2 × Using this observation we can repeat the proof of Theorem 4.3 to obtain the statement. ✷

γ ε γ γ Proposition A.6 The measures ν¯ = ( )∗(˜ν ), where ν˜ is deﬁned by equation (94), are εn,Nr E εn,Nr εn,Nr 2δ R2 tight on B2−,2,2ℓ( ) (where δ > 0). Proof By Proposition A.4 we have

εn 2 γ −2δ (ϕ) −δ d˜ν B (R2) B (R2) εn,Nr 2,2,2ℓ kE k 2,2,ℓ/2 ε 2 γ . −2δ (ϕ) − d˜ν RB (R2) B δ (R2) εn,Nr 2,2,2ℓ kE k 2,2,ℓ/2 2 δ 2 γ . −2δ ρ (m ∆ Z2 ) ϕ 2 2 d˜ν (97) RB (R2) ℓ/2 εn − L (ε Z ) εn,Nr 2,2,2ℓ k − k n E 2 δ/2 2 . 2 ρℓ(z) [((m ∆ Z2 )− Wε ,N (0,z)) ]dz RεnZ εn n k E −2 δ/2 ¯γ 2 + Z2 ρℓ(z) [((m ∆ε Z2 )− φ (0,z)) ]dz R εn − n εn,Nk The ﬁrst term is uniformly boundedR in εn,Nk by well known properties of the Gaussian free ﬁeld. For the second term we observe that since φ¯γ (0,z) is the unique solution to (96) we have by invariance εn,Nk of the law of W (0, ) and of equation (96) under shifts in the x coordinate, that: εn,Nk · E 2 δ/2 ¯γ 2 2 ρℓ(z) [((m ∆ Z2 )− φ (0,z)) ]dz εnZ εn εn,Nk 1 − E 2 δ/2 ¯γ 2 6 2 2 ρ (z)ρ (x) [((m ∆ Z2 ) φ (x, z)) ]dxdz (98) ρ (x)dx R εZ ℓ ℓ ε − εn,Nk R ℓ × − R 6 C R where in the last line we have used that ρℓ/2(z)ρℓ/2(x) . ρℓ(x, z) and Proposition A.5. This implies 2δ R2 ✷ tightness on B2−,2,2ℓ( ) by compactness of the Besov embedding. It will be helpful to have the following lemma.

Lemma A.7 Let (Ω, ) be a measurable space, let ν be a probability measure defined on Ω, and let S :Ω R be a measurableF function such that eS(x) L1(Ω,ν) then → ∈ eS(x)ν(dx) 6 e S(x)νS(dx) R ZΩ eS(x) where νS is the probability measure defined on Ω, for which νS(dx)= S(x) ν(dx). Ω e ν(dx) R Proof By definition of νS and using the fact that ν is a probability measure we get

S(x) S(y) e ν(dx) e− νS(dy) =1. ZΩ ZΩ S(y) ✷ Applying Jensen inequality to the term Ω e− νS(dy) we get the thesis. Proof of Theorem A.31 Since, by Proposition A.6, the measuresν ¯γ are tight, what remains to R εn,Nr show is that the constants Zε,N,γ deﬁned by equation (95) satisfy

~~sup Zεn,Nr,γ < . εn,Nr ∞~~

~~1this argument is due to M. Gubinelli (personal communication).~~

~~62 γ cosh,β 2 If we apply Lemma A.7 to the measuresν ¯ and ν (which means taking S(ϕ)= ε (ϕ) −2δ , εn,Nr m,εn,Nr n B (R2) kE k 2,2,ℓ we get~~

~~εn,Nr,γ 2 γ Z 6 exp ε (ϕ) −2δ 2 ν (dϕ) . − n B (R ) εn,Nr B 2δ (R2) kE k 2,2,ℓ Z 2,2,ℓ ! On the other hand by inequalities (97) and (98) we have that~~

~~2 γ sup (ϕ) −2δ ν (dϕ) 6 C < + . εn R2 εn,Nr −2δ B2,2,ℓ( ) εn,Nr B (R2) kE k ∞ Z 2,2,ℓ From this we get the thesis. ✷~~

~~A.2 Clustering~~

~~In this section we prove the exponential decay of the two point function S2. More precisely we prove the following theorem.~~

~~Theorem A.8 There exist some constants C,κ > 0 such that for any ξ, ψ H1(R2) such that supp(ψ) supp(ξ) B(0, 1) we have that ∈ ⊂ S~~

cosh,β ζ, ϕ L2(R2) ψ( x), ϕ L2(R2)dν (ϕ) 6 C exp( κ x ) ψ H1(R2) ξ H1(R2). − m B 2δ (R2)h i h ·− i − | | k k k k Z 2,2,2ℓ Note that since cosh(βϕ) is even and the law of the free ﬁeld is invariant with respect to reﬂection we have that

cosh,β cosh,β ζ, ϕ L2(R2)dν (ϕ)= ψ( x), ϕ L2(R2)dν (ϕ)=0. − m − m B 2δ (R2)h i B 2δ (R2)h ·− i Z 2,2,2ℓ Z 2,2,2ℓ Again we postpone the proof until the end of the section. First let us introduce the measure

γζ 1 ε cosh,β d¯ν = exp( γ ζ, ϕ 2 R2 )dν (φ). ε,N Zγζ − hD iL ( ) m,ε,N where ε,N,γζ ε cosh,β Z = exp( γ ζ, ϕ L2(R2))dν (ϕ) − m,ε,N B 2δ (εZ2) − hD i Z 2,2,2ℓ ε 1 Z2 This is a well deﬁned measure since (ζ), L2(εZ2) is continuous on Hloc− (ε ) which contains the cosh,β hD ·i support of νm . Immediately form the deﬁnitions we get the following lemma:

~~Lemma A.9 Under the previous hypotheses we have~~

~~ε ε cosh,β ζ, ϕ L2(εZ2) ψ( x), ϕ L2(εZ2)dν (ϕ)= − m,ε,N B 2δ (εZ2)hD i hD ·− i Z 2,2,2ℓ~~

~~d ε γζ ψ( x), φ L2(εZ2)d¯ν (ϕ) . − ε,N − dγ B 2δ (εZ2)hD ·− i "Z 2,2,2ℓ #γ=0~~

~~Furthermore, if εn and Nr are the sequences in~~

εn εn cosh,β lim ζ, ϕ L2(ε Z2) ψ( x), ϕ L2(ε Z2)dν (ϕ) − n n m,εn,Nr εn 0,Nr B 2δ (ε Z2)hD i hD ·− i → →∞ Z 2,2,2ℓ n cosh,β = ζ, ϕ L2(R2) ψ( x), ϕ L2 (R2)dν (ϕ) − m B 2δ (ε Z2)h i h ·− i Z 2,2,2ℓ n

~~63 Proof For proving the ﬁrst assertion observe that~~

~~d εn γζ ψ( x), ϕ L2(ε Z2)d¯ν (φ) − n εn,Nr dγ B 2δ (εZ2)hD ·− i Z 2,2,2ℓ~~

~~1 εn d εn cosh,β = ψ( x), ϕ L2(ε Z2) [exp( γ ζ, ϕ L2(ε Z2))]γ=0dν (ϕ) εn,Nr,0 − n n m,εn,Nr Z B 2δ (ε Z2)hD ·− i dγ − hD i Z 2,2,2ℓ n~~

~~d 1 εn cosh,β + ψ( x), ϕ L2 (ε Z2)dν (ϕ) εn,Nr,γζ − n m,εn,Nr dγ Z B 2δ (ε Z2)hD ·− i γ=0 Z 2,2,2ℓ n~~

~~εn ε cosh,β = ψ( x), ϕ L2(ε Z2) ζ, ϕ L2(ε Z2)dν (ϕ), − n n m,εn,Nr − B 2δ (ε Z2)hD ·− i hD i Z 2,2,2ℓ n~~

εn p 2 2 where we used the fact exp( γ ζ, ϕ L (εnZ )) is, by Theorem A.3 and its proof, uniformly in L and so we can exchange the derivative− hD operationi with the integral, and in the last line we exploit the property that ε γζ ψ( x), ϕ L2(ε Z2)d¯ν (ϕ)=0, − n B 2δ (ε Z2)hD ·− i Z 2,2,2ℓ n 1 and that Zεn,Nr,γζ is diﬀerentiable in γ. For the second assertion it is enough to observe that

~~ε ε cosh,β ζ, φ L2(ε Z2) ψ( x), φ L2 (ε Z2)dν (ϕ) − n n m,εn,Nr B 2δ (ε Z2)hD i hD ·− i Z 2,2,2ℓ n~~

εn εn cosh,β = ζ, ϕ L2(R2) ψ( x), ϕ L2(R2)dν (ϕ) − m,εn,Nr B 2δ (ε Z2)h E i h ·− E i Z 2,2,2ℓ n ε cosh,β cosh,β 2δ 2 and we have established in Theorem 4.13 that lim n (ν )= ν weakly on B− (R ), εn 0,Nr , m,εn,Nr m 2,2,2ℓ for any δ > 0. The weak convergence together with→ the bound→∞ E

~~εn εn~~

½ −1/2 2 2 R2 2 R2 (1 ∆ Z2 ) (ϕ) >N ζ, ϕ L ( ) ψ( x), ϕ L ( ) ε L2(ε Z2) {kk − k ℓ n k }h E i h ·− E i 1/2 2 6 exp( γN/2) exp(γ (1 ∆εZ2 )− (ϕ) L2(ε Z2)) − k − k ℓ n proves the thesis. ✷

Proposition A.10 For γ > 0 small enough we have that ν¯γ = Law(W (0, )+ φ¯γζ (0, )) N,ε ε,N · ε,N · γζ δ 2 2 where φ¯ B− (R εZ ) is the unique solution to the equation ε,N ∈ 2,2,l × γζ γζ 2 γζ αφ¯ α αφ¯ α ( ∆R2 Z2 + m )φ¯ + αe ε,N µ αe− ε,N µ− − +ε ε,N ε,N − ε,N (99) = γ εζ D where εζ : R2 εZ2 R is deﬁned by D × → εζ(x, z)= εζ(z). D D Proof The proof is analogous to the one of Proposition A.4. ✷ Let us introduce the notation

~~2 2 2 1/2 n 2 1/2 w(x, z): R εZ (1 + κ(1 + x ) )− exp(κ(1 + z ) ). × → | | | |~~

¯γζ Proposition A.11 Let φε,N be the solution to equation (99). Then provided κ is small enough there exists a constant C > 0 such that 1/2 γζ 0 2 E w(x, z)(φ¯ φ¯ ) dxdz 6 Cγ ζ 2 R2 εn ,Nr εn,Nr L ( ) R2 ε Z2 − k k Z × n

64 Proof By the deﬁnitions of φ¯γζ , φ¯0 we have εn,Nr εn,Nr 2 ¯γζ ¯0 ( ∆R2+ε Z2 + m )(φ φ ) − n εn,Nr − εn,Nr +αc (sinh(α(W + φ¯γζ )) sinh(α(W + φ¯0 ))) ε,α εn ,Nr εn,Nr εn,Nr εn,Nr ε − = n ζ. D where 2 α E[W 2 ] cε,α = e− 2 εn,Nr . Testing the previous equation with the function w(x, z)(φ¯γζ φ¯0 ) we get εn,Nr − εn,Nr 2 γζ 0 γζ 0 w(x, z)( ∆R2 Z2 + m )(φ¯ φ¯ )(φ¯ φ¯ )dxdz+ εn εn,Nr εn,Nr εn,Nr εn,Nr R2 ε Z2 − × − − Z × n +α 2w(x, z)c (sinh(α(W + φ¯γζ )) sinh(α(W + φ¯0 ))) ε,α εn ,Nr εn,Nr εn,Nr εn,Nr R2 ε Z2 − × Z × n (φ¯γζ φ¯0 )dxdz = γ w(x, z) εζ(φγζ φ0 )dxdz. εn,Nr εn,Nr N,ε N,ε × − R2 ε Z2 D − Z × n Now since cosh is convex

~~γζ 0 γζ 0 α[sinh(α(Wε ,N + φ¯ )) sinh(α(Wε ,N + φ¯ ))](φ¯ φ¯ ) > 0. n r εn,Nr − n r εn,Nr εn,Nr − εn,Nr So~~

2 γζ 0 γζ 0 w(x, z)( ∆R2 Z2 + m )(φ¯ φ¯ )(φ¯ φ¯ )dxdz 6 εn εn,Nr εn,Nr εn,Nr εn,Nr R2 ε Z2 − × − − Z × n γζ 0 w(x, z) εn ζ(φ¯ φ¯ )dxdy. εn,Nr εn,Nr R2 ε Z2 D − Z × n On the other hand

2 γζ 0 γζ 0 w(x, z)( ∆R2 Z2 + m )(φ¯ φ¯ )(φ¯ φ¯ )dxdz εn εn ,Nr εn,Nr εn,Nr εn,Nr R2 ε Z2 − × − − Z × n = m2 w(x, z)(φ¯γζ φ¯0 )2dxdz + εn ,Nr εn,Nr R2 ε Z2 − Z × n γζ 0 2 + w(x, z)( R2 Z2 (φ¯ φ¯ )) dxdz + εn εn,Nr εn,Nr R2 ε Z2 ∇ × − Z × n γζ 0 γζ 0 + R2 Z2 (φ¯ φ¯ )(φ¯ φ¯ )( R2 Z2 w)dxdz ε εn,Nr εn,Nr εn ,Nr εn,Nr ε R2 ε Z2 ∇ × − − ∇ × Z × n It is easy to prove that R2 ε Z2 w 6 κ w |∇ × n | | | so for εn and κ small enough. This and Cauchy-Schwartz inequality give us

2 γζ 0 γζ 0 w(x, z)( ∆R2 Z2 + m )(φ¯ φ¯ )(φ¯ φ¯ )dxdz εn εn ,Nr εn,Nr εn,Nr εn,Nr R2 ε Z2 − × − − Z × n 1 > m2 w(x, z)(φ¯γζ φ¯0 )2dxdz εn,Nr εn,Nr 2 R2 ε Z2 − Z × n 1 γζ 0 2 + w(x, z)( R2 Z2 (φ¯ φ¯ )) dxdz εn εn,Nr εn,Nr 2 R2 ε Z2 ∇ × − Z × n So putting everything together we get

2 γζ 0 2 γζ 0 2 w(x, z)(m (φ¯ φ¯ ) + ( R2 Z2 (φ¯ φ¯ )) )dxdz εn,Nr εn,Nr εn εn,Nr εn,Nr R2 ε Z2 − ∇ × − Z × n γζ 0 6 2 w(x, z) εn ζ(φ¯ φ¯ )dxdz. εn,Nr εn,Nr R2 ε Z2 D − Z × n

~~65 Now since by Young’s inequality~~

γζ 0 2 w(x, z) εn ζ(φ¯ φ¯ )dxdz 6 εn,Nr εn,Nr R2 ε Z2 D − Z × n 2 γζ 0 2 w(x, z) εn ζ dxdz + w(x, z)(φ¯ φ¯ ) dxdy εn,Nr εn,Nr R2 ε Z2 D R2 ε Z2 − Z × n Z × n and since 2 εn 2 w(x, z) ζ dxdz 6 C ζ L2(R2), R2 ε Z2 D k k Z × n for some constant C > 0 independent on ζ (due to the fact that we have assumed ζ to be supported in B(0, 1)) we can conclude. ✷ Proof of Theorem A.8 By Lemma A.9 we have that

εn εn cosh,β sup −2δ ζ, ϕ 2 Z2 ψ( x), ϕ 2 Z2 dν (ϕ) εn,Nk B (ε Z2) L (εn ) L (εn ) m,εn,Nr 2,2,2ℓ n hD i hD ·− i εn γζ cosh,β 6 sup sup −2δ ψ( x), ϕ 2 Z2 (d¯ν (ϕ) dν (ϕ)) εn,Nk Rγ>0 B (ε Z2) L (εn ) m,εn,Nr m,εn,Nr 2,2,2ℓ n hD ·− i − R On the other hand, by the invariance of the law of φ¯γζ (x, ) and φ¯0 (x, ) with respect to the εn,Nr εn,Nr translation of x coordinates, · ·

εn γζ cosh,β ψ( y), ϕ 2 Z2 (d¯ν (ϕ) dν (ϕ)) L (εn ) m,εn,Nr m,εn,Nr −2δ 2 B (εnZ )hD ·− i − Z 2,2,2ℓ ε γζ 0 = E[ n ψ( y), (φ¯ (0, ) φ¯ (0, )) ] | hD ·− εn,Nr · − εn,Nr · i | = ( εn ψ(z y))(E[φ¯γζ (0,z) φ¯0 (0,z)])dz εn ,Nr εn,Nr Z2 D − − Zεn n ε γζ 0 = C (1 + x )− n ψ(x, z y) (E[φ¯ (x, z) φ¯ (x,z)])dxdz εn,Nr εn,Nr R2 ε Z2 | | D − − Z × n γζ 0 6 C exp( κ y ) w(x, z) εn ψ(x, z y) (E[φ¯ (x, z) φ¯ (x,z)])dxdz εn,Nr εn,Nr − | | R2 ε Z2 D − − Z × n where again 2 1/2 n 2 1/2 w(x, z)=(1+ κ(1 + x ) )− exp(κ(1 + z ) ). | | | | and we have used that ψ is supported in B(0, 1). From this we can conclude by Proposition A.11 and Cauchy-Schwartz inequality. ✷

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