A PDE construction of the Euclidean $\Phi^4_3$ quantum field theory

We present a self-contained construction of the Euclidean $\Phi^4$ quantum field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and side length $M$. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow \infty$. Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder--Schrader axioms for a nontrivial Euclidean QFT apart from rotation invariance and clustering. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson--Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a \emph{distribution} on the space of Euclidean fields. Our argument applies to arbitrary positive coupling constant and also to multicomponent models with $O(N)$ symmetry.


Introduction
From the point of view of probability theory, one of the major achievements of the constructive quantum field theory (CQFT) program [VW73, Sim74, GJ87, Riv91, BSZ92, Jaf00, Jaf08, Sum12] which flourished in the 70s and 80s can be summarized in the existence of a "wonderful new mathematical object" (as Gelfand once put it [Jaf08]): Theorem 1.1 There exists a one parameter family (ν λ ) λ>0 of measures on S ′ (R 3 ) that are non-Gaussian, Euclidean invariant and reflection positive.
A measure µ on the space S ′ (R 3 ) of Schwartz distributions on R 3 is Euclidean invariant (EI) if it is invariant under the rigid motions of R 3 . Denote by Ψ µ (f ) := S ′ (R 3 ) e iϕ(f ) µ(dϕ) the characteristic function of µ. We say that µ is reflection positive (RP) if the matrix (Ψ µ (f i − θf j )) i,j is positive semidefinite for any finite choice of Schwartz functions (f i ) i ⊆ S(R 3 ) with supp(f i ) ⊆ {(x 1 , x 2 , x 3 ) ∈ R 3 : x 1 > 0} and where θf i (x 1 , x 2 , x 3 ) = f i (−x 1 , x 2 , x 3 ) is the reflection with respect to the x 1 = 0 plane. Reflection positivity is a property whose crucial importance for probability theory and mathematical physics [Bis09,Jaf18] and representation theory [NO18,JT18] has been one of the byproducts of the constructive effort.
Surprisingly, a measure which satisfies all these three properties has been quite difficult to find. Euclidean invariance and reflection positivity conspire against each other. Models which easily satisfy one property hardly satisfy the other if they are not Gaussian (see e.g. [AY02,AY09]). In the two dimensional setting the existence of the analogous object has been one of the early successes of CQFT [Sim74,GJ87,BSZ92], while it is likely that in four and more dimensions such an object cannot exist [FFS92].
Theorem 1.1 (provided some additional technical properties are satisfied) implies the existence of a relativistic quantum field theory in the Minkowski space-time R 1+2 which satisfies the Wightman axioms [Wig76] (a minimal set of axioms capturing the essence of the combination of quantum mechanics and special relativity). The translation from the commutative probabilistic setting (Euclidean QFT) to the non-commutative Minkowski QFT setting is operated by a set of axioms introduced by Osterwalder-Schrader [OS73,OS75] for the correlation functions of the measure ν λ (called Schwinger functions or Euclidean correlation functions) which shall satisfy: a regularity axiom (OS0), an Euclidean invariance axiom (OS1), a reflection positivity axiom (OS2), a symmetry axiom (OS3) and a cluster property (OS4).
The standard approach to construction of measures which satisfy EI, RP and are non-Gaussian is to perturb in a non-linear way a Gaussian measure via a Gibbs-type density which is ill-defined due to small scale (ultraviolet, in CQFT parlance) singularities as well as to large scale ones (infrared). One is then led to introduce a cut-offs in order to tame the singularities and regularize the measure (see e.g. our choice in (1.1) below). Such a regularization typically spoils EI or RP (or both) and has to be subsequently removed by a more or less elaborate limiting procedure, whose main duty is to reestablish the simultaneous validity of both properties. This additionally requires, especially in three dimensions, to remove certain diverging quantities, a process called renormalization.
The original proof of the OS axioms, along with additional properties of the family of measures (ν λ ) λ which are called Φ 4 3 measures, is scattered in a series of works covering almost a decade. Glimm [Gli68] first proved the existence of the Hamiltonian (with an infrared regularization) in the Minkowski setting. Then Glimm and Jaffe [GJ73] introduced the phase cell expansion of the regularized Schwinger functions, which revealed itself a powerful and robust tool (albeit complex to digest) in order to handle the local singularities of Euclidean quantum fields and to prove the ultraviolet stability in finite volume. The proof of existence of the infinite volume limit and the verification of Osterwalder-Schrader axioms [OS73,OS75] was then completed by Feldman and Osterwalder for λ small [FO76] using cluster expansion methods, finally the work of Seiler and Simon [SS76] allowed to extend the existence result to all λ > 0 (this is claimed in [GJ87] even though we could not find a clear statement in Seiler and Simon's paper). Equations of motion for the quantum fields were established by Feldman and Raczka [FR77].
Since this first, complete, construction, there have been several other attempts to simplify (both technically and/or conceptually) the arguments and the Φ 4 3 measure has been since considered a test bed for various CQFT techniques. There exists at least six methods of the proof: the original phase cell method of Glimm and Jaffe extended by Feldman and Osterwalder [FO76], Magnen and Seneor [MS76] and Park [Par77] (among others), the probabilistic approach of Benfatto, Cassandro, Gallavotti, Nicoló, Olivieri, Presutti and Schiacciatelli [BCG+78], the block average method of Bałaban [Bał83] (reviewed by Dimock in [Dim13a,Dim13b,Dim14]), the wavelet method of Battle-Federbush [Bat99], the skeleton inequalities method of Brydges, Fröhlich, Sokal [BFS83], the work of Watanabe on rotation invariance [Wat89] via the renormalization group method of Gawędzki and Kupiainen [GK86], and more recently the renormalization group method of Brydges, Dimock and Hurd [BDH95].
It should be said that, apart from the Glimm-Jaffe-Feldman-Osterwalder result, none of the additional constructions seems to be as complete and to verify explicitly all the OS axioms. As Jaffe [Jaf08] remarks: "Not only should one give a transparent proof of the dimension d = 3 construction, but as explained to me by Gelfand [private communication], one should make it sufficiently attractive that probabilists will take cognizance of the existence of a wonderful mathematical object." In our opinion, among all these (incomplete) methods, the simplest and the most "attractive" one seems to be that of skeleton inequalities proposed by Sokal [Sok82] and Brydges, Fröhlich, Sokal [BFS83], which however fails to prove rotational invariance (thus not covering completely Theorem 1.1) and to give information for large λ.
In the present paper we put forward a simple, self-contained, construction of the Φ 4 3 measure based on methods from PDE theory as well as on recent advances in the field of singular SPDEs. We can show invariance under translation, reflection positivity, the regularity axiom of Osterwalder-Schrader and the non-Gaussianity of the measure, thus going a long way (albeit not fully reaching the goal) to a complete proof of Theorem 1.1 and of its consequences for QFT. Our proof applies to all values of the coupling parameter λ > 0 as well as to natural extensions to N -dimensional vectorial variants of the model. Furthermore, we establish an integration by parts formula which leads to the hierarchy of the Dyson-Schwinger equations for the Schwinger functions of the measure.
Our methods are innovative and very different from all the known constructions we enumerated above. In particular, we do not rely on any of the standard tools like cluster expansion or correlation inequalities or skeleton inequalities, and therefore our approach brings a new perspective to this extensively investigated classical problem, with respect to the removal of both ultraviolet and infrared regularizations.
The key idea is to use a dynamical description of the approximate measure which relies on an additional random source term which is Gaussian, in the spirit of the stochastic quantization approach introduced by Nelson [Nel66,Nel67] and Parisi and Wu [PW81] (with a precursor in a technical report of Symanzik [Sym64]).
The concept stochastic quantization refers to the introduction of a reversible stochastic dynamics which has the target measure as the invariant measure, here in particular the Φ 4 d measure in d dimensions. The rigorous study of the stochastic quantization for the two dimensional version of the Φ 4 theory has been first initiated by Jona-Lasinio and Mitter [JLM85] in finite volume and by Borkar, Chari and Mitter [BCM88] in infinite volume. A natural d = 2 local dynamics has been subsequently constructed by Albeverio and Röckner [AR91] using Dirichlet forms in infinite dimensions. Later on, Da Prato and Debussche [DPD03] have shown for the first time the existence of strong solutions to the stochastic dynamics in finite volume. Da Prato and Debussche have introduced an innovative use of a mixture of probabilistic and PDE techniques and constitute a landmark in the development of PDE techniques to study stochastic analysis problems. Similar methods have been used by McKean [McK95b,McK95a] and Bourgain [Bou96] in the context of random data deterministic PDEs. Mourrat and Weber [MW17b] have subsequently shown the existence and uniqueness of the stochastic dynamics globally in space and time. For the d = 1 dimensional variant, which is substantially simpler and does not require renormalization, global existence and uniqueness have been established by Iwata [Iwa87].
In the three dimensional setting the progress has been significantly slower due to the more severe nature of the singularities of solutions to the stochastic quantization equation. Only very recently, there has been substantial progress due to the invention of regularity structures theory by Hairer [Hai14] and paracontrolled distributions by Gubinelli, Imkeller, Perkowski [GIP15]. These theories greatly extend the pathwise approach of Da Prato and Debussche via insights coming from Lyons' rough path theory [Lyo98,LQ02,LCL07] and in particular the concept of controlled paths [Gub04,FH14]. With these new ideas it became possible to solve certain analytically ill-posed stochastic PDEs, including the stochastic quantization equation for the Φ 4 3 measure and the Kardar-Parisi-Zhang equation. The first results were limited to finite volume: local-in-time well-posedness has been established by Hairer [Hai14] and Catellier, Chouk [CC18]. Kupiainen [Kup16] introduced a method based on the renormalization group ideas of [GK86].
Long-time behavior has been studied by Mourrat, Weber [MW17a], Hairer, Mattingly [HM18b] and a lattice approximation in finite volume has been given by Hairer and Matetski [HM18a] and by Zhu and Zhu [ZZ18]. Global in space and time solutions have been first constructed by Gubinelli and Hofmanová in [GH18]. Local bounds on solutions, independent on boundary conditions, and stretched exponential integrability have been recently proven by Moinat and Weber [MW18].
However, all these advances are still falling short to give a complete proof of the existence of the Φ 4 3 measure on the full space and of its properties. Indeed they, including essentially all of the two dimensional results, are principally aimed at studying the dynamics with an a priori knowledge of the existence and the properties of the invariant measure. For example Hairer and Matetski [HM18a] use a discretization of a finite periodic domain to prove that the limiting dynamics leaves the finite volume Φ 4 3 measure invariant using the a priori knowledge of its convergence from the paper of Brydges et al. [BFS83]. Studying the dynamics, especially globally in space and time is still a very complex problem which has siblings in the ever growing literature on invariant measures for deterministic PDEs starting with the work of Lebowitz, Rose and Speer [LRS88,LRS89], Bourgain [Bou94,Bou96], Burq and Tzvetkov [BT08b,BT08a,Tzv16] and with many following works (see e.g. [CO12,CK12,NPS13,Cha14,BOP15]) which we cannot exhaustively review here.
The first work proposing a constructive use of the dynamics is, to our knowledge, the work of Albeverio and Kusuoka [AK17], who proved tightness of certain approximations in a finite volume. Inspired by this result, our aim here is to show how these recent ideas connecting probability with PDE theory can be streamlined and extended to recover a complete, self-contained and simple, proof of existence of the Φ 4 3 measure on the full space. In the same spirit see also the work of Hairer and Iberti [HI18] on the tightness of the 2d Ising-Kac model. Soon after Hairer's seminal paper [Hai14], Jaffe [Jaf14] analyzed the stochastic quantization from the point of view of reflection positivity and constructive QFT and concluded that one has to necessarily take the infinite time limit to satisfy RP. Even with global solution at hand a proof of RP from dynamics seems nontrivial and actually the only robust tool we are aware of to prove RP is to start from finite volume lattice Gibbs measures for which RP follows from the spatial Markov property.
For this reason, the starting point of our analysis is a family (ν M,ε ) M,ε of Gibbs measures on the periodic lattice Λ M,ε = (ε(Z/M Z)) 3 with mesh size ε and side length M , given by (1.1) where ∇ ε denotes the discrete gradient and a M,ε , b M,ε are suitable renormalization constants, m 2 ∈ R is called the mass and λ > 0 the coupling constant of the model. Our goal is to let ε → 0 and M → ∞ in order to recover both full translation invariance and reflection positivity which for ν M,ε is well known to hold. To this end, we prove that the family (ν M,ε ) M,ε is tight once embedded in the space of probability measures on S ′ (R 3 ). The removal of the regularization parameters ε, M requires a precise tuning of the renormalization constants (a M,ε , b M,ε ) M,ε .
An SPDE is used to derive bounds which are strong enough to prove the tightness of the family (ν M,ε ) M,ε . To be more precise, we study a lattice approximation of the (renormalized) stochastic quantization equation where ξ is a space-time white noise on R 3 . The lattice dynamics is a system of stochastic differential equation which is globally well-posed and has ν M,ε as its unique invariant measure. We can therefore consider its stationary solution ϕ M,ε having at each time the law ν M,ε . We introduce a suitable decomposition together with an energy method in the framework of weighted Besov spaces. This allows us, on the one hand, to track down and renormalize the short scale singularities present in the model as ε → 0, and on the other hand, to control the growth of the solutions as M → ∞. As a result we obtain uniform bounds which allow to pass to the limit in the weak topology of probability measures. The details of the renormalized energy method rely on recent developments in the analysis of singular PDEs. In order to make the paper accessible to a wide audience with some PDE background we implement renormalization using the paracontrolled calculus of [GIP15] which is based on Bony's paradifferential operators [Bon81,Mey81,BCD11]. We also rely on some tools from the paracontrolled analysis in weigthed Besov spaces which we developed in [GH18] and on the results of Martin and Perkowski [MP17] on Besov spaces on the lattice.
The method we use here is novel and differs from the approach of [GH18] in that we are initially less concerned with the continuum dynamics itself. We do not try to obtain estimates for strong solutions and rely instead on certain cancellations in the energy estimate that permit to significantly simplify the proof. The resulting bounds are sufficient to provide a rather clear picture of any limit measure as well as some of its physical properties. In contrast, in [GH18] we provided a detailed control of the dynamics (1.2) (in stationary or non-stationary situations) at the price of a more involved analysis. Section 4.2 of the present paper could in principle be replaced by the corresponding analysis of [GH18]. However the adaptation of that analysis to the lattice setting (without which we do not know how to prove RP) would still require further preparatory work that constitutes a large fraction of the present paper. Similarly, the recent results of Moinat and Weber [MW18] (which appeared after we completed a first version of this paper) can be conceivably used to replace a part of Section 4. Our choice of an alternative approach is mostly motivated by the desire to provide a self-contained, elementary (to the extent possible) and accessible argument.
Our main result is the following.
Theorem 1.2 There exists a choice of the sequence (a M,ε , b M,ε ) M,ε such that for any λ > 0 and m 2 ∈ R, the family of measures (ν M,ε ) M,ε (properly extended to S ′ (R 3 )) is tight. Every accumulation point ν is translation invariant, reflection positive and non-Gaussian. In addition, for every small κ > 0 there exists σ > 0, β > 0 and υ = O(κ) > 0 such that 2. The Dyson-Schwinger equations were first derived by Feldman and Raczka [FR77] using the results of Glimm, Jaffe, Feldman and Osterwalder.
3. As already noted by Albeverio, Liang and Zegarlinski [ALZ06] on the formal level, the integration by parts formula gives rise to a cubic term which cannot be interpreted as a random variable under the Φ 4 3 measure. Therefore, the crucial question that remained unsolved until now is how to make sense of this critical term as a well-defined probabilistic object. In the present paper, we obtain fine estimates on the approximate stochastic quantization equation and construct a coupling of the stationary solution to the continuum Φ 4 3 dynamics and the Gaussian free field. This leads to a detailed description of the renormalized cubic term as a genuine random space-time distribution. Moreover, we approximate this term in the spirit of the operator product expansion. 4. To the best of our knowledge, our work provides the first rigorous proof of a general integration by parts formula with an exact formula for the renormalized cubic term. In addition, the method applies to arbitrary values of the coupling constant λ 0 if m 2 > 0 and λ > 0 if m 2 0 and we state the precise dependence of our estimates on λ. In particular, we show that our energy bounds are uniform over λ in every bounded subset of [0, ∞) provided m 2 > 0 (see Remark 4.6).
5. By essentially the same arguments, we are able to treat the vector version of the model, where the scalar field ϕ : R 3 → R is replaced by a vector valued one ϕ : R 3 → R N for some N ∈ N and the measures ν M,ε are given by a similar expression as (1.1), where the norm |ϕ| is understood as the Euclidean norm in R N .
To conclude this introductory part, let us compare our result with other constructions of the Φ 4 3 field theory. The most straightforward and simplest available proof has been given by Brydges, Fröhlich and Sokal [BFS83] using skeleton and correlation inequalities. All the other methods we cited above employ technically involved machineries and various kinds of expansions (they are however able to obtain very strong information about the model in the weakly-coupled regime, i.e. when λ is small). Compared to the existing methods, ours bears similarity in conceptual simplicity to that of [BFS83], with some advantages and some disadvantages. Both works construct the continuum Φ 4 3 theory as a subsequence limit of lattice theories and the rotational invariance remains unproven. The main difference is that [BFS83] relies on correlation inequalities, which, on the one hand, restricts the applicability to weak couplings and only models with N = (0, )1, 2 components (note that the N = 0 models have a meaning only in their formalism but not in ours), but, on the other hand, allow to establish bounds on the decay of correlation functions, which we do not have. However, our results hold for every value of λ > 0 and m 2 ∈ R while the results in [BFS83] works only in the so-called "single phase region", which essentially corresponds to small λ > 0 or m 2 > 0 large.
Our work is intended as a first step in the direction of using PDE methods in the study of Euclidean QFTs and large scale properties of statistical mechanical models. Another related attempt is the variational approach developed in [BG18] for the finite volume Φ 4 3 measure. As far as the present paper is concerned the main open problems is to establish rotational invariance and give more information on the limiting measures, in particular establish uniqueness for small λ. It is not clear how to deduce anything about correlations from the dynamics but it seems to be a very interesting and challenging problem.
Plan. The paper is organized as follows. Section 2 gives a summary of notation used throughout the paper, Section 3 presents the main ideas of our strategy and Section 4, Section 5 and Section 6 are devoted to the main results. First, in Section 4 we construct the Euclidean quantum field theory as a limit of the approximate Gibbs measures ν M,ε . To this end, we introduce the lattice dynamics together with its decomposition. The main energy estimate is established in Theorem 4.5 and consequently the desired tightness as well as moment bounds are proven in Theorem 4.9. In Section 4.4 we establish finite stretched exponential moments. Consequently, in Section 5 we verify the translation invariance and reflection positivity, the regularity axiom and nontriviality of any limit measure. Section 6 is devoted to the integration by parts formula and the Dyson-Schwinger equations. Finally, in Appendix A we collect a number of technical results needed in the main body of the paper.
Acknowledgement. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Scaling limits, rough paths, quantum field theory when work on this paper was undertaken. In particular, we are grateful to Sergio Albeverio, David Brydges, Jürg Fröhlich, Stefan Hollands, Seiichiro Kusuoka and Pronob Mitter for stimulating discussions. This work was supported by EPSRC Grant Number EP/R014604/1. M. G. is partially supported by the German Research Foundation (DFG) via CRC 1060.

Notation
Within this paper we are concerned with the Φ 4 3 model in discrete as well as continuous setting. In particular, we denote by Λ ε = (εZ) d for ε = 2 −N , N ∈ N 0 , the rescaled lattice Z d and by For notational simplicity, we use the convention that the case ε = 0 always refers to the continuous setting. For instance, we denote by Λ 0 the full space Λ 0 = R d and by Λ M,0 the continuous torus Λ M,0 = T d M . With the slight abuse of notation, the parameter ε is always taken either of the form ε = 2 −N for some N ∈ N 0 , N N 0 , for certain N 0 ∈ N 0 that will be chosen as a consequence of Lemma A.9 below, or ε = 0. Various proofs below will be formulated generally for ε ∈ A := {0, 2 −N ; N ∈ N 0 , N N 0 } and it is understood that the case ε = 0 or alternatively N = ∞ refers to the continuous setting. All the proportionality constants, unless explicitly signalled, will be independent of M, ε, λ, m 2 . We will track the explicit dependence on λ as far as possible and signal when the constant depends on the value of m 2 > 0.
For f ∈ ℓ 1 (Λ ε ) and g ∈ L 1 (Λ ε ), respectively, we define the Fourier and the inverse Fourier transform as where k ∈ (ε −1 T) d =:Λ ε and x ∈ Λ ε . These definitions can be extended to discrete Schwartz distributions in a natural way, we refer to [MP17] for more details. In general, we do not specify on which lattice the Fourier transform is taken as it will be clear from the context. Consider a smooth dyadic partition of unity (ϕ j ) j −1 such that ϕ −1 is supported in a ball around 0 of radius 1 2 , ϕ 0 is supported in an annulus, ϕ j (·) = ϕ 0 (2 −j ·) for j 0 and if |i − j| > 1 then supp ϕ i ∩ supp ϕ j = ∅. For the definition of Besov spaces on the lattice Λ ε for ε = 2 −N , we introduce a suitable periodic partition of unity onΛ ε as follows where x ∈Λ ε and the parameter J ∈ N 0 , whose precise value will be chosen below independently on ε ∈ A, satisfies 0 N − J J ε := inf{j : supp ϕ j ∩ ∂(ε −1 T) d = ∅} → ∞ as ε → 0. We note that by construction there exists ℓ ∈ Z independent of ε = 2 −N such that J ε = N − ℓ. Then (2.1) yields a periodic partition of unity onΛ ε . The reason for choosing the upper index as N − J and not the maximal choice J ε will become clear in Lemma A.9 below, where it allows us to define suitable localization operators needed for our analysis. The choice of parameters N 0 and J is related in the following way: A given partition unity (ϕ j ) j −1 determines the parameters J ε in the form J ε = N − ℓ for some ℓ ∈ Z. By the condition N − J J ε we obtain the first lower bound on J. Then Lemma A.9 yields a (possibly larger) value of J which is fixed throughout the paper. Finally, the condition 0 N − J implies the necessary lower bound N 0 for N , or alternatively the upper bound for ε = 2 −N 2 −N 0 and defines the set A. We stress that once the parameters J, N 0 are chosen, they remain fixed throughout the paper.
Remark that according to our convention, (ϕ 0 j ) j −1 denotes the original partition of unity (ϕ j ) j −1 on R d , which can be also read from (2.1) using the fact that for ε = 0 we have J ε = ∞. Now we may define the Littlewood-Paley blocks for distributions on Λ ε by which leads us to the definition of weighted Besov spaces. Throughout the paper, ρ denotes a polynomial weight of the form for some ν 0 and h > 0. The constant h will be fixed below in Lemma 4.4 in order to produce a small bound for certain terms. Such weights satisfy the admissibility condition ρ(x)/ρ(y) ρ −1 (x − y) for all x, y ∈ R d . For α ∈ R, p, q ∈ [1, ∞] and ε ∈ [0, 1] we define the weighted Besov spaces on Λ ε by the norm where L p,ε for ε ∈ A \ {0} stands for the L p space on Λ ε given by the norm (with the usual modification if p = ∞). Analogously, we may define the weighted Besov spaces for explosive polynomial weights of the form ρ −1 . Note that if ε = 0 then B α,ε p,q (ρ) is the classical weighted Besov space B α p,q (ρ). In the sequel, we also employ the following notations C α,ε (ρ) := B α,ε ∞,∞ (ρ), H α,ε (ρ) := B α,ε 2,2 (ρ).
In Lemma A.1 we show that one can pull the weight inside the Littlewood-Paley blocks in the definition of the weighted Besov spaces. Namely, under suitable assumptions on the weight that are satisfied by polynomial weights we have f B α,ε p,q (ρ) ∼ ρf B α,ε p,q in the sense of equivalence of norms, uniformly in ε. We define the duality product on Λ ε by and Lemma A.2 shows that B −α,ε p ′ ,q ′ (ρ −1 ) is included in the topological dual of B α,ε p,q (ρ) for conjugate exponents p, p ′ and q, q ′ .
We employ the tools from paracontrolled calculus as introduced in [GIP15], the reader is also referred to [BCD11] for further details. We shall freely use the decomposition f g = f ≺ g + f • g + f ≻ g, where f ≻ g = g ≻ f and f • g, respectively, stands for the paraproduct of f and g and the corresponding resonant term, defined in terms of Littlewood-Paley decomposition. More precisely, for f, g ∈ S ′ (Λ ε ) we let We also employ the notations f g : For notational simplicity, we do not stress the dependence of the paraproduct and the resonant term on ε in the sequel. These paraproducts satisfy the usual estimates uniformly in ε, see e.g. [MP17], Lemma 4.2, which can be naturally extended to general B α,ε p,q (ρ) Besov spaces as in [MW17b], Theorem 3.17.
Throughout the paper we assume that m 2 > 0 and we only discuss in Remark 4.6 how to treat the case of m 2 0. In addition, we are only concerned with the 3 dimensional setting and let d = 3. We denote by ∆ ε the discrete Laplacian on Λ ε given by where (e i ) i=1,...,d is the canonical basis of R d . It can be checked by a direct computation that the integration by parts formula holds for the discrete gradient We let Q ε := m 2 − ∆ ε , L ε := ∂ t + Q ε and we write L for the continuum analogue of L ε . We let L −1 ε to be the inverse of L ε on Λ ε such that L −1

Overview of the strategy
With the goals and notations being set, let us now outline the main steps of our strategy.
Lattice dynamics. For fixed parameters ε ∈ A, M > 0, we consider a stationary solution ϕ M,ε to the discrete stochastic quantization equation whose law at every time t 0 is given by the Gibbs measure (1.1). Here ξ M,ε is a discrete approximation of a space-time white noise ξ on R d constructed as follows: Let ξ M denote its periodization on T d M given by is a test function, and define the corresponding spatial discretization by Then (3.1) is a finite-dimensional SDE in a gradient form and it has a unique invariant measure ν M,ε given by (1.1).
Recall that due to the irregularity of the space-time white noise in dimension 3, a solution to the limit problem (1.2) can only exist as a distribution. Consequently, since products of distributions are generally not well-defined it is necessary to make sense of the cubic term. This forces us to introduce a mass renormalization via constants a M,ε , b M,ε 0 in (3.1) which shall be suitably chosen in order to compensate the ultraviolet divergencies. In other words, the additional linear term shall introduce the correct counterterms needed to renormalize the cubic power and to derive estimates uniform in both parameters M, ε. To this end, a M,ε shall diverge linearly whereas b M,ε logarithmically and these are of course the same divergencies as those appearing in the other approaches, see e.g. Chapter 23 in [GJ87].
Energy method in a nutshell. Our aim is to apply the so-called energy method, which is one of the very basic approaches in the PDE theory. It relies on testing the equation by the solution itself and estimating all the terms. To explain the main idea, consider a toy model driven by a sufficiently regular forcing f such that the solution is smooth and there are no difficulties in defining the cube. Testing the equation by u and integrating the Laplace term by parts leads to 1 2 ∂ t u 2 L 2 + m 2 u 2 L 2 + ∇u 2 L 2 + λ u 4 L 4 = f, u . Now, there are several possibilities to estimate the right hand side using duality and Young's inequality, namely, . This way, the dependence on u on the right hand side can be absorbed into the good terms on the left hand side by choosing δ ∈ (0, 1). If in addition u was stationary hence in particular t → E u(t) 2 L 2 is constant, then we obtain To summarize, using the dynamics we are able to obtain moment bounds for the invariant measure that depend only on the forcing f . Moreover, we also see the behavior of the estimates with respect to the coupling constant λ. Nevertheless, even though using the L 4 -norm of u introduces a blow up for λ → 0, the right hand side f in our energy estimate below will always contain certain power of λ in order to cancel this blow up and to obtain bounds that are uniform as λ → 0.
Decomposition and estimates. Since the forcing ξ on the right hand side of (1.2) does not possess sufficient regularity, the energy method cannot be applied directly. Following the usual approach within the field of singular SPDEs, we shall find a suitable decomposition of the solution ϕ M,ε , isolating parts of different regularity. In particular, since the equation is subcritical in the sense of Hairer [Hai14] (or superrenormalizable in the language of quantum field theory), we expect the nonlinear equation (1.2) to be a perturbation of the linear problem L X = ξ. This singles out the most irregular part of the limit field ϕ. Hence on the approximate level we set ϕ M,ε = X M,ε + η M,ε where X M,ε is a stationary solution to and the remainder η M,ε is expected to be more regular.
To see if it is indeed the case we plug our decomposition into (3.1) to obtain Here X 2 M,ε and X 3 M,ε denote the second and third Wick power of the Gaussian random variable X M,ε defined by where a M,ε := E[X 2 M,ε (t)] is independent of t due to stationarity. It can be shown by direct computations that appeared already in a number of works (see [CC18], [Hai14], [Hai15], [MWX16]) that X 2 M,ε is bounded uniformly in M, ε as a continuous stochastic process with values in the weighted Besov space C −1−κ,ε (ρ σ ) for every κ, σ > 0, whereas X 3 M,ε can only be constructed as a space-time distribution. In addition, they converge to the Wick power X 2 and X 3 of X. In other words, the linearly growing renormalization constant a M,ε gives counterterms needed for the Wick ordering.
Note that X is a continuous stochastic process with values in C −1/2−κ,ε (ρ σ ) for every κ, σ > 0. This limits the regularity that can be obtained for the approximations X M,ε uniformly in M, ε. Hence the most irregular term in (3.3) is the third Wick power and by Schauder estimates we expect η M,ε to be 2 degrees of regularity better. Namely, we expect uniform bounds for η M,ε in C 1/2−κ (ρ σ ) which indeed verifies our presumption that η M,ε is more regular than ϕ M,ε . However, the above decomposition introduced new products in (3.3) that are not well-defined under the above discussed uniform bounds. In particular, both η M,ε X 2 M,ε and η 2 M,ε X M,ε do not meet the condition that the sum of their regularities is strictly positive, which is a convenient sufficient condition for a product of two distributions to be analytically well-defined.
Therefore, we need to continue with the decomposition in the same spirit in order to cancel the most irregular term in (3.3), namely, X 3 M,ε . The usual way, which can be found basically in all the available works on the stochastic quantization (see e.g. in [CC18], [GH18], [Hai14], [Hai15], [MW17a]) is therefore to define X M,ε as the stationary solution to leading to the decomposition ϕ M,ε = X M,ε − λX M,ε + ζ M,ε . Writing down the dynamics for ζ M,ε we observe that the most irregular term is the paraproduct X 2 M,ε ≻ X M,ε which can be bounded uniformly in C −1−κ (ρ σ ) and hence this is not yet sufficient for the energy method outlined above. Indeed, the expected (uniform) regularity of ζ M,ε is C 1−κ,ε (ρ σ ). However, we point out that not much is missing.
In order to overcome this issue, we proceed differently than the above cited works and let Y M,ε be a solution to where U ε > is the localization operator defined in Section A.2. With a suitable choice of the constant L = L(λ, M, ε) determining U ε > (cf. Lemma A.12, Lemma 4.1) we are able to construct the unique solution to this problem via Banach's fixed point theorem. Consequently, we find our decomposition ϕ M,ε = X M,ε + Y M,ε + φ M,ε together with the dynamics for the remainder The first term on the right hand side is the most irregular contribution, the second term is not controlled uniformly in M, ε, the third term is needed for the renormalization and Ξ M,ε contains various terms that are more regular and in principle not problematic or that can be constructed as stochastic objects using the remaining counterterm −3λ 2 b M,ε (X M,ε + Y M,ε ). The advantage of this decomposition with φ M,ε as opposed to the usual approach leading to ζ M,ε above is that together with X 3 M,ε we cancelled also the second most irregular contribution (U ε > X 2 M,ε ) ≻ Y M,ε , which is too irregular to be controlled as a forcing f using the energy method. The same difficulty of course comes with X 2 M,ε ≻ φ M,ε in (3.7), however, since it depends on the solution φ M,ε we are able to control it using a paracontrolled ansatz. To explain this, let us also turn our attention to the resonant product X 2 M,ε • φ M,ε which poses problems as well. When applying the energy method to (3.7), these two terms appear in the form where we included a polynomial weight ρ as in (2.2). The key observation is that the presence of the duality product permits to show that these two terms approximately coincide, in the sense that their difference denoted by is controlled by the expected uniform bounds. This is proven generally in Lemma A.13. As a consequence, we obtain Finally, since the last term on the left hand side as well as the first term on the right hand side are diverging, the idea is to couple them by the following paracontrolled ansatz. We define and expect that the sum of the two terms on the right hand side is more regular than each of them separately. In other words, ψ M,ε is (uniformly) more regular than φ M,ε . Indeed, with this ansatz we may complete the square and obtain where the right hand side, given in Lemma 4.2, can be controlled by the norms on the left hand side, in the spirit of the energy method discussed above. These considerations lead to our first main result proved as Theorem 4.5 below. In what follows, Q ρ (X M,ε ) denotes a polynomial in the ρ-weighted norms of the involved stochastic objects, the precise definition can be found in Section 4.1.
Tightness. In order to proceed to the proof of the existence of the Euclidean Φ 4 3 field theory, we shall employ the extension operator E ε from Section A.4 which permits to extend discrete distributions to the full space R 3 . An additional twist originates in the fact that by construction the process Y M,ε given by (3.6) is not stationary and consequently also φ M,ε fails to be stationary. Therefore the energy argument as explained above does not apply as it stands and we shall go back to the stationary decomposition ϕ M,ε = X M,ε − λX M,ε + ζ M,ε , while using the result of Theorem 3.1 in order to estimate ζ M,ε . Consequently, we deduce tightness of the family of the joint laws of (ϕ M,ε , X M,ε , X M,ε ) evaluated at any fixed time t 0, proven in Theorem 4.9 below.
To this end, we denote by (ϕ, X, X ) a canonical representative of the random variables under consideration and let ζ := ϕ − X + λX . Theorem 3.2 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). Then the family of joint laws of (E ε ϕ M,ε , E ε X M,ε , E ε X M,ε ), ε ∈ A, M > 0, evaluated at an arbitrary time t 0 is tight. Moreover, any limit measure µ satisfies for all p ∈ [1, ∞) Osterwalder-Schrader axioms. The projection of a limit measure µ onto the first component is the candidate Φ 4 3 measure and we denote it by ν. Based on Theorem 3.2 we are able to show that ν is translation invariant and reflection positive, establishing (partly) OS1 and OS2, see Section 5.2 and Section 5.3. In addition, we prove that the measure is nontrivial, i.e. non-Gaussian. To this end, we make use of the decomposition ϕ = X − λX + ζ together with the moment bounds from Theorem 3.2. Since X is Gaussian whereas X is not, the idea is to use the regularity of ζ to conclude that it cannot compensate X which is less regular. In particular, we show that the connected 4-point function is nonzero, see Section 5.4.
It remains to discuss a stretched exponential integrability of ϕ, leading to the distribution property OS0 shown in Section 5.1. More precisely, we show the following result which can be found in Proposition 4.11. Proposition 3.3 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). For every κ ∈ (0, 1) In order to obtain this bound we revisit the bounds from Theorem 3.1 and track the precise dependence of the polynomial Q ρ (X M,ε ) on the right hand side of the estimate on the quantity X M,ε which will be defined through (4.3), (4.4), (4.5) below taking into account the number of copies of X appearing in each stochastic object. However, the estimates in Theorem 3.1 are not optimal and consequently the power of X M,ε in Theorem 3.1 is too large. To optimize we introduce a large momentum cut-off X 3 M,ε given by a parameter K > 0 and let which allows for refined bounds on Y M,ε , yielding optimal powers of X M,ε .
Integration by parts formula. The uniform energy estimates from Theorem 3.2 and Proposition 3.3 are enough to obtain tightness of the approximate measures and to show that any accumulation point satisfies the distribution property, translation invariance, reflection positivity and nontriviality. However, they do not provide sufficient regularity in order to identify the continuum dynamics or to establish the hierarchy of Dyson-Schwinger equations providing relations of various n-point correlation functions. This can be seen easily since neither the res- is well-defined in the limit. Another and even more severe difficulty lies in the fact that the third Wick power X 3 only exists as a space-time distribution and is not a well-defined random variable under the Φ 4 3 measure, cf. [ALZ06]. To overcome the first issue, we introduce a new paracontrolled ansatz χ M,ε := φ M,ε + 3λX M,ε ≻ φ M,ε and show that χ M,ε possesses enough regularity uniformly in M, ε in order to pass to the limit in the resonant product X 2 M,ε • χ M,ε . Namely, we establish uniform bounds . This not only allows to give meaning to the critical resonant product in the continuum, but it also leads to a uniform time regularity of the processes ϕ M,ε . We obtain the following result proved below as Theorem 6.2. Theorem 3.4 Let β ∈ (0, 1/4) and σ ∈ (0, 1). Then it holds true that for all p ∈ [1, ∞) and ). This additional time regularity is then used in order to treat the second issue raised above and to construct a renormalized cubic term ϕ 3 . More precisely, we derive an explicit formula for ϕ 3 including X 3 as a space-time distribution, where time indeed means the fictitious stochastic time variable introduced by the stochastic quantization, nonexistent under the Φ 4 3 measure. In order to control X 3 we re-introduce the stochastic time and use stationarity together with the above mentioned time regularity. Finally, we derive an integration by parts formula leading to the hierarchy of Dyson-Schwinger equations connecting the correlation functions. The precise result proved in Theorem 6.7 reads as follows.
where for a smooth h : and ϕ 3 is given by an explicit formula, namely, (6.6).
In addition, we are able to characterize J ν (F ) in the spirit of the operator product expansion, see Lemma 6.5.

Construction of the Euclidean Φ field theory
This section is devoted to our main result. More precisely, we consider (3.1) which is a discrete approximation of (1.2) posed on a periodic lattice Λ M,ε . For every ε ∈ (0, 1) and M > 0 (3.1) possesses a unique invariant measure that is the Gibbs measure ν M,ε given by (1.1). We derive new estimates on stationary solutions sampled from these measures which hold true uniformly in ε and M . As a consequence, we obtain tightness of the invariant measures while sending both the mesh size as well as the volume to their respective limits, i.e. ε → 0, M → ∞.

Stochastic terms
Recall that the stochastic objects X M,ε , X 2 M,ε , X 3 M,ε and X M,ε were already defined in (3.2), (3.4) and (3.5). As the next step we provide further details and construct additional stochastic objects needed in the sequel. All the distributions on Λ M,ε are extended periodically to the full lattice Λ ε . Then X M,ε which is a stationary solution to (3.5 where P ε t denotes the semigroup generated by L ε on Λ ε . Then it holds for every κ, σ > 0 and some β > 0 small uniformly in M, ε thanks to the presence of the weight. For details and further references see e.g. Section 3 in [GH18]. Here and in the sequel, T ∈ (0, ∞) denotes an arbitrary finite time horizon and C T and C β/2 T are shortcut notations for C([0, T ]) and C β/2 ([0, T ]), respectively. Throughout our analysis, we fix κ, β > 0 in the above estimate such that β 3κ. This condition will be needed for the control of a parabolic commutator in Lemma 4.4 below. On the other hand, the parameter σ > 0 varies from line to line and can be arbitrarily small.
If U ε > is a localizer defined for some given constant L > 0 according to Lemma A.12, we let Y M,ε be the solution of (3.6) hence Note that this is an equation for Y M,ε , which also implies that Y M,ε is not a polynomial of the Gaussian noise. However, as shown in the following lemma, Y M,ε can be constructed as a fixed point provided L is large enough.
, where the proportionality constant is independent of M, ε.
Proof We define a fixed point map for some L > 0 to be chosen below. Then it holds in view of the Schauder estimates from Lemma 3.4 in [MP17], the paraproduct estimates as well as Lemma A.12 that δ.
In particular, we have that which will be used later in order to estimate the complementary operator U ε by Lemma A.12. Note that L(λ, M, ε) a priori depends on M, ε. However, due to the uniform bound on valid for some γ ∈ (0, 1), we may use compactness to deduce that for every fixed λ > 0 there exists a subsequence (not relabeled) such that L(λ, M, ε) → L 0 (λ). This will also allow to identify the limit of the localized term below in Section 6. Next, it holds Therefore we deduce that K leaves balls in C T C 1/2−κ,ε (ρ σ ) invariant and is a contraction on Hence there exists a unique fixed point Y M,ε and the first bound follows. Next, we use the Schauder estimates (see Lemma 3.9 in [MP17]) to bound the time regularity as follows According to this result, we remark that Y M,ε itself is not a polynomial in the noise terms, but with our choice of localization it allows for a polynomial bound of its norm. As the next step, we introduce further stochastic objects needed below. Namely, Note that we do not include X M,ε in X M,ε since it can be controlled by X 2 M,ε using Schauder estimates. In order to have a precise control of the number of copies of X appearing in each stochastic term we define X M,ε as the smallest number bigger than 1 and all the quantities 3) , Note that it is bounded uniformly with respect to M, ε. Besides, if we do not need to be precise about the exact powers, we denote by Q ρ (X M,ε ) a generic polynomial in the above norms of the noise terms X M,ε , whose coefficients depend on ρ but are independent of M, ε, λ, and change from line to line.

Decomposition and uniform estimates
With the above stochastic objects at hand, we let ϕ M,ε be a stationary solution to (3.1) on Λ M,ε having at each time t 0 the law ν M,ε . We consider its decomposition ϕ M,ε = X M,ε +Y M,ε +φ M,ε and deduce that φ M,ε satisfies Our next goal is to derive energy estimates for (4.6) which hold true uniformly in both parameters M, ε. To this end, we recall that all the distributions above were extended periodically to the full lattice Λ ε . Consequently, apart from the stochastic objects, the renormalization constants and the initial conditions, all the operations in (4.6) are independent of M . Therefore, for notational simplicity, we fix the parameter M and omit the dependence on M throughout the rest of this subsection. The following series of lemmas serves as a preparation for our main energy estimate established in Theorem 4.5. Here, we make use of the approximate duality operator D ρ 4 ,ε as well as the commutators C ε ,C ε andC ε introduced Section A.3.

Lemma 4.2 It holds
Proof Noting that (4.6) is of the form L ε φ ε + λφ 3 ε = U ε , we may test this equation by ρ 4 φ ε to deduce We use the fact that (f ≻) is an approximate adjoint to (f •) according to Lemma A.13 to rewrite the resonant term as and use the definition of ψ in (4.8) to rewrite Φ ρ,ε as . For the first term we write Next, we use again Lemma A.13 to simplify the quadratic term as As the next step, we justify the definition of the resonant product appearing in Ψ ρ 4 ,ε and show that it is given by Z ε from the statement of the lemma. To this end, let and recall the definition of Y M,ε (4.1). Hence by Lemma A.14 which is the desired formula. In this formulation we clearly see the structure of the renormalization and the appropriate combinations of resonant products and the counterterms. ✷ As the next step, we estimate the new stochastic terms appearing in Lemma 4.2. Here and in the sequel, ϑ = O(κ) > 0 denotes a generic small constant which changes from line to line.

Lemma 4.3 It holds true
Proof By definition of Z ε and the discussion in Section 4.1, Lemma 4.1, Lemma A.14, Lemma A.12 and (4.2) we have (since the choice of exponent σ > 0 of the weight corresponding to the stochastic objects is arbitrary, σ changes from line to line in the sequel) (1 + λ + λ| log t| + λ 2 ) X ε 7+ϑ and the first claim follows since σ > 0 was chosen arbitrarily. Next, we recall (4.1) and the fact that X ε = X ε • X ε can be constructed without any renormalization in C T C −κ,ε (ρ σ ). As a consequence, the resonant term reads where the for the second term we have (since U ε > is a contraction) that For the two paraproducts we obtain directly We proceed similarly for the remaining term, which is quadratic in Y ε . We have Accordingly, (4.14) and for the paraproducts This gives the second bound from the statement of the lemma. ✷ Let us now proceed with our main energy estimate. In view of Lemma 4.2, our goal is to control the terms in Θ ρ 4 ,ε + Ψ ρ 4 ,ε by quantities of the from where δ > 0 is a small constant which can change from line to line. Indeed, with such a bound in hand it will be possible to absorb the norms of φ ε , ψ ε from the right hand side of (4.7) into the left hand side and a bound for φ ε , ψ ε in terms of the noise terms will follow. Lemma 4.4 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). Then it holds Since the weight ρ is polynomial and vanishes at infinity, we may assume without loss of generality that 0 < ρ 1 and consequently ρ α ρ β whenever α β 0. We also observe that due to the integrability of the weight it holds (see Lemma A.6) Even though it is not necessary for the present proof, we keep track of the precise power of the quantity X ε in each of the estimates. This will be used in Section 4.4 below to establish the stretched exponential integrability of the fields. We recall that ϑ = O(κ) > 0 denotes a generic small constant which changes from line to line.
In view of Lemma 4.2 we shall bound each term on the right hand side of (4.7). We have This term can be absorbed provided C ρ = ρ −4 [∇ ε , ρ 4 ] L ∞,ε is sufficiently small, such that C δ C 2 ρ m 2 , which can be obtained by choosing h > 0 small enough (depending only on m 2 and δ) in the definition (2.2) of the weight ρ. Next, and we estimate explicitly for another constant C ρ depending only on the weight ρ, which can be taken smaller than m 2 by choosing h > 0 small, and consequently Using Lemma A.2, Lemma A.7, interpolation from Lemma A.3 with for θ = 1−4κ 1−2κ and Young's inequality we obtain . Recall that since σ is chosen small, we have the interpolation inequality (see Lemma A.3) where θ = 1/2−3κ 1−2κ . Similar interpolation inequalities will also be employed below. Then, in view of Lemma A.13 and Young's inequality, we have where we further estimate by Schauder and paraproduct estimates ,ε and hence we deduce by interpolation with θ = 1−6κ 1−2κ and embedding that . Due to Lemma A.14 and interpolation with θ = 1−5κ 1−2κ , we obtain . Then we use the paraproduct estimates, the embedding C 1/2−κ,ε (ρ σ ) ⊂ H 1/2−2κ,ε (ρ 2−σ/2 ) (which holds due to the integrability of ρ 4ι for some ι ∈ (0, 1) and the fact that σ can be chosen small), together with Lemma 4.1 and interpolation to deduce for θ = 1/2−5κ 1−2κ that Here we employ Lemma A.7 and interpolation to obtain for θ = 1/2−4κ and similarly for the other two terms, where we also use Lemma 4.3 and the embedding H 1−2κ,ε (ρ 2 ) ⊂ Next, we obtain Then, by (4.2) and finally for θ = 1/2−4κ 1−2κ The proof is complete. ✷ Now we have all in hand to establish our main energy estimate.
Theorem 4.5 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). There exists a constant α = α(m 2 ) ∈ (0, 1) such that for θ = 1/2−4κ Proof As a consequence of (4.8), we have according to Lemma A.5, Lemma A.4, Lemma A.1 Therefore, according to Lemma 4.4 we obtain that . Choosing δ > 0 sufficiently small (depending on m 2 and the implicit constant C from Lemma A.5) allows to absorb the norms of φ ε , ψ ε from the right hand side into the left hand side and the claim follows. ✷

Remark 4.6
We point out that the requirement of a strictly positive mass m 2 > 0 is to some extent superfluous for our approach.
To be more precise, if m 2 0 then we may rewrite the mollified stochastic quantization equation as )ϕ ε and the same decomposition as above introduces an additional term on the right hand side of (4.7). This can be controlled by where we write C δ,λ −1 to stress that the constant is not uniform over small λ. As a consequence, we obtain an analogue of Theorem 4.5 but the uniformity for small λ is not valid anymore.
Corollary 4.7 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). Then for all p ∈ [1, ∞) and θ = 1/2−4κ Proof Based on (4.21) we obtain . The L 4 -norm on the left hand side can be estimated from below by the L 2 -norm, whereas on the right hand side we use Young's inequality to deduce L 2,ε . Hence we may absorb the second term from the right hand side into the left hand side. ✷

Tightness of the invariant measures
Recall that ϕ M,ε is a stationary solution to (3.1) having at time t 0 law given by the Gibbs measure ν M,ε . Moreover, we have the decomposition ϕ M,ε = X M,ε + Y M,ε + φ M,ε , where X M,ε is stationary as well. By our construction, all equations are solved on a common probability space, say (Ω, F, P), and we denote by E the corresponding expected value.
Theorem 4.8 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). Then for every p ∈ [1, ∞) it holds sup Proof Let us show the first claim. Due to stationarity of In order to estimate the right hand side, we employ Theorem 4.5 together with Lemma 4.1 to deduce E ρ 2 (ϕ M,ε (0) − X M,ε (0)) 2 Finally, taking τ > 0 large enough, we may absorb the second term from the right hand side into the left hand side to deduce Observing that the right hand side is bounded uniformly in M, ε, completes the proof of the first claim. Now, we show the second claim for p ∈ [2, ∞). The case p ∈ [1, 2) then follows easily from the bound for p = 2. Using stationarity as above we have Due to Corollary 4.7 applied to p−1 and the fact that for any σ > 0 it holds τ 0 | log s| 2p/(2+θ) ds C p,σ τ 1+σ for all τ 1, we deduce L 2,ε . Plugging this back into (4.24) and using Young's inequality we obtain and choosing δ > 0 small enough, we may absorb the second term on the right hand side into the left hand side and the claim follows ✷ The above result directly implies the desired tightness of the approximate Gibbs measures ν M,ε . To formulate this precisely we make use of the extension operators E ε for distributions on Λ ε constructed in Section A.4. We recall that on the approximate level the stationary process ϕ M,ε admits the decomposition ϕ M,ε = X M,ε + Y M,ε + φ M,ε , where X M,ε is stationary and Y M,ε is given by (4.1) with X M,ε being also stationary. Accordingly, letting we obtain ϕ M,ε = X M,ε − λX M,ε + ζ M,ε , where all the summands are stationary.
Consequently, by Lemma A.15 the same bounds hold for the corresponding extended distributions and hence the family joint laws of (E ε ϕ M,ε , E ε X M,ε , E ε X M,ε ) at any time t 0 is tight. Therefore up to a subsequence we may pass to the limit as ε → 0, M → ∞ and the uniform moment bounds are preserved for every limit point. ✷ The marginal of µ corresponding to ϕ is the desired Φ 4 3 measure, which we denote by ν. According to the above result, ν is obtained as a limit (up to a subsequence) of the continuum extensions of the Gibbs measures ν M,ε given by (1.1) as ε → 0, M → ∞.

Stretched exponential integrability
The goal of this section is to establish better probabilistic properties of the Φ 4 3 measure. Namely, we show that ρ 2 ϕ M,ε 1−υ H −1/2−2κ,ε is uniformly (in M, ε) exponentially integrable for every υ = O(κ) > 0, hence we recover the same stretched exponential moment bound for any limit measure ν. To this end, we revisit the energy estimate in Section 4.2 and take a particular care to optimize the power of the quantity X M,ε appearing in the estimates. Recall that it can be shown that uniformly in M, ε for a small parameter β > 0 (see [MW18]). Accordingly, it turns out that the polynomial Q ρ (X M,ε ) on the right hand side of the bound in Lemma 4.4 shall not contain higher powers of X M,ε than 8 + O(κ). In the proof of Lemma 4. 4 we already see what the problematic terms are. In order to allow for a refined treatment of these terms, we introduce an additional large momentum cut-off and modify the definition of Y M,ε from (3.6), leading to better uniform estimates and consequently to the desired stretched exponential integrability. More precisely, let K > 0 and take a compactly supported, smooth function v : R → R + such that v L 1 = 1. We define where the convolution is in the time variable and v K (t) := 2 K v(2 K t). With standard arguments one can prove that sup is exponentially integrable for a small parameter and therefore we can modify the definition of X M,ε to obtain while still keeping the validity of (4.25). Moreover, we let X 3 M,ε > := X 3 M,ε − X 3
Next, we redefine Y M,ε to solve . The estimates of Lemma 4.1 are still valid with obvious modifications. In addition, we obtain and by interpolation it follows for a ∈ [0, 1/2 − κ] that From now on we avoid, as usual, to specify explicitly the dependence on M since it does not play any role in the estimates. The energy equality (4.7) in Lemma 4.2 now reads where and Θ ρ 4 ,ε , Ψ ρ 4 ,ε where defined in Lemma 4.2. Our goal is to bound the right hand side of (4.28) with no more than a factor X M,ε 8+ϑ for some ϑ = O(κ). In view of the estimates within the proof of Lemma 4.4 we observe that the bounds (4.15), (4.16), (4.17), (4.18), (4.19) and (4.20) need to be improved. Lemma 4.10 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). Then there is ϑ = O(κ) > 0 such that Proof Let us begin with a new bound for the term with X ε Y 2 ε appearing in (4.15). For the resonant term we get from the interpolation estimate (4.27) that the bound (4.14) can be updated as where we used that, due to the presence of the localizer (see (4.2)), we can bound giving an improved bound for the paracontrolled term which reads as follows Consequently, for θ = 1−4κ For the paraproducts we have for θ = 1/2−4κ Let us now consider the term with X ε Y ε always in (4.15). In view of (4.11), (4.12), (4.13) we shall modify the bound of the resonant product which using the decomposition (4.10) together with (4.11) and the bound (4.29). We obtain and consequently, for θ = 1−4κ 1−2κ , For the paraproducts we have for θ = 1/2−4κ With the improved bound for Y , (4.16), (4.17), (4.18) can be updated as follows Now, let us update the bound (4.19) as Next, we shall improve the bound (4.20). Here we need to use a different modification for each term appearing in ρ 4 φ ε , λ 2 Z ε ε as defined in (4.9). For θ = 1/2−4κ 1−2κ we bound Next, we have and the resonant term is bounded as Next, for θ = 1−5κ 1−2κ , At last, we have This concludes the estimation of ρ 4 φ ε , λ 2 Z ε ε giving us Finally, we arrive to the additional term introduced by the localization. Using (4.26) we obtain where we also see that the power 8 + ϑ is optimal for this decomposition. ✷ Let φ ε := (1 + ρ 2 φ ε 2 L 2,ε ) 1/2 and ϕ ε * := (1 + ρ 2 ϕ ε 2 H −1/2−2κ,ε ) 1/2 . With Lemma 4.10 in hand we can proceed to the proof of the stretched exponential integrability.
Proof We apply (4.28) and Lemma 4.10 to obtain where by writing C δ,λ −1 we point out that the constant is not uniform over small λ. Therefore by absorbing the constant term C δ,λ −1 in X ε 8+ϑ we have Now we can have two situations at any given time, either X ε 2 ς tρφ ε 1−υ L 4,ε or X ε 2 > ς tρφ ε 1−υ L 4,ε for some fixed and small ς > 0. In the first case the right hand side of (4.30) is bounded by and we can choose υ = υ(κ) so that (4 + ϑ/2)(1 − υ) = 4 and by taking ς small (depending on δ, λ through C δ,λ −1 ) we can absorb this term into the left hand side since for t ∈ (0, 1) it will be bounded by provided ρ is chosen to be of sufficient decay, and therefore we simply bound the right hand side of (4.30) by The first claim is proven. It remains to prove the bound for ϕ ε . By Hölder's inequality, we have and we observe that Y ε (1) 1−υ 1 + X ε 2 so the first term on the right hand side is integrable uniformly in ε by (4.25). On the other hand, using Lemma 4.11 we have and therefore E[e 2β φε(1) 1−υ ] E[1 + e (2β/C) Xε 2 ].
We conclude that

The Osterwalder-Schrader axioms and nontriviality
The goal of this section is to establish several important properties of any limit measure ν obtained in the previous section. Osterwalder and Schrader [OS73,OS75] introduced the following axioms for a family (S n ∈ S ′ (R 3 ) ⊗n ) n∈N 0 .
OS0 (Distribution property) It holds S 0 = 1. There is a Schwartz norm · s on S ′ (R 3 + ) and β > 0 such that for all n ∈ N and f 1 , . . . , f n ∈ S(R 3 + ) it holds where (a, R).f n (x) = f n (a + Rx) and where O (3) is the orthogonal group of R 3 .
The reconstruction theorem of [OS75] asserts that functions (S n ) n∈N 0 which satisfy OS0-3 are the Euclidean Green's functions (or Schwinger functions) of a uniquely determined Wightman theory (maybe lacking the cluster property). The reader is referred to [GJ87] for a detailed exposition of the Euclidean approach to QFT.
For any measure µ on S ′ (R 3 ) we define S µ n ∈ (S ′ (R 3 )) ⊗n as In this case OS3 is trivially satisfied. Along this section we will prove that, for any accumulation point ν, the functions (S ν n ) n satisfy additionally OS0, OS2 and OS1 with the exception of invariance with respect to SO(3) (but including reflections) and moreover that it is not a Gaussian measure.

Distribution property
Here we are concerned with proving the bound (5.1) for correlation functions of ν.
Proposition 5.1 There exists β > 1 and K > 0 such that any limit measure ν constructed via the procedure in Section 4 satisfies: for all n ∈ N and all f 1 , . . . , f n ∈ H 1/2+2κ (ρ −2 ) we have In particular, it satisfies OS0.

Translation invariance
Proposition 5.2 Any limit measure ν constructed via the procedure in Section 4 is translation invariant.
Proof By their definition in (1.1), the approximate measures ν M,ε are translation invariant under lattice shifts. That is, for h ε ∈ Λ ε it holds T hε ν M,ε = ν M,ε . In other words, the processes ϕ M,ε and T hε ϕ M,ε coincide in law. In addition, since the translation T hε commutes with the extension operator E ε , it follows that E ε ϕ M,ε and T hε E ε ϕ M,ε coincide in law. Now we recall that the limiting measure ν was obtained as a weak limit of the laws of E ε ϕ M,ε on H −1/2−2κ (ρ 2+γ ).
If h ∈ R d is given, there exists a sequence h ε ∈ Λ ε such that h ε → h. Let κ ∈ (0, 1) be small and arbitrary. Then we have for F ∈ C 0,1 b (H −1/2−3κ (ρ 2+γ )) that where in the third inequality we used the regularity of F and Theorem 4.8 as follows If F ∈ C b (H −1/2−3κ (ρ 2+γ )), then by approximation and dominated convergence theorem we also get T h ν(F ) = ν(F ), which completes the proof. ✷

Reflection positivity
As the next step we recover the reflection positivity of the measure ν. We fix an index i ∈ {1, 2, 3} and establish reflection positivity of ν with respect to the reflection given by the hyperplane To this end, we denote R 3 +,δ = {x ∈ R 3 ; x i > δ} and define the space of functionals F depending on fields restricted to R 3 +,δ by and let H + = H +,0 . For a function f : R 3 → R we define its reflection and extend it to F ∈ H + by θF (ϕ(f 1 ), . . . , ϕ(f k )) := F (ϕ(θf 1 ), . . . , ϕ(θf k )). Hence for F ∈ H +,δ the reflection θF depends on ϕ evaluated at Proposition 5.3 Any limit measure ν constructed via the procedure in Section 4 is reflection positive with respect to all reflections θ = θ i , i ∈ {1, 2, 3}. In particular, it satisfies OS2.
Proof We recall that our Euclidean quantum field theory ν was obtained as a limit of (suitable continuum extensions of) the measures ν M,ε given by (1.1). It is known that for every ε, M the measures ν M,ε reflection positive (on Λ M,ε ), see [GJ87]. Therefore, we obtain Next, we observe that since the function w in the definition of the extension operator E ε was chosen radially symmetric, the reflection and the extension operator commute. Moreover, if F ∈ H +,δ then F • E ε ∈ H + when ε is small enough (depending on δ) and therefore due to the reflection positivity of ν M,ε , for all F ∈ H +,δ we have Using the support properties of ν we can approximate any F ∈ H + by functions in H +,δ and therefore obtain the first claim. Let us now show that (5.2) holds. Note that, thanks to the exponential integrability satisfied by ν any polynomial of the form G = n∈N 0 ϕ ⊗n (f n ) for sequences (f n ∈ S C (R 3n < )) n∈N 0 with finitely many nonzero elements, belongs to L 2 (ν). In particular it can be approximated in L 2 (ν) by a sequence (F n ) n of cylinder functions in H + . Therefore E ν [θGG] = lim n→∞ E ν [θF n F n ] 0 and we conclude that

Nontriviality
This section is devoted to the proof of nontriviality, that is, non-Gaussianity.
Theorem 5.4 If λ > 0 then any limit measure ν constructed via the procedure in Section 4 is non-Gaussian.
Proof In order to show that the limiting measure ν is non-Gaussian, it is sufficient to prove that the connected four-point function is nonzero, see [BFS83]. In other words, we shall prove that the distribution U ν 4 (x 1 , . . . , is nonzero. To this end, we recall that in Theorem 4.9 we obtained a limit measure µ which is the joint law of (ϕ, X, X ) and that ν is the marginal corresponding to the first component. Let K i = F −1 ϕ i be a Littlewood-Paley projector and consider the connected four-point function U ν 4 convolved with (K i , K i , K i , K i ) and evaluated at (x 1 , . . . , x 4 ) = (0, . . . , 0), that is, where L is a quadrilinear form. Since under the limit µ we have the decomposition ϕ = X − λX + ζ, we may write L(ϕ, ϕ, ϕ, ϕ) = L(X, X, X, X) − 4λL(X, X, X, X ) + R (5.3) where R contains terms which are at least bilinear in X or linear in ζ. Due to Gaussianity of X, the first term on the right hand side of (5.3) vanishes. Our goal is to show that the second term behaves like 2 i whereas the terms in R are more regular, namely, bounded by 2 i(1/2+κ) . In other words, R cannot compensate 4λL(X, X, X, X ) and as a consequence L(ϕ, ϕ, ϕ, ϕ) = 0 if λ > 0. Let us begin with L(X, X, X, X ). To this end, we denote k [123] = k 1 + k 2 + k 3 and recall that where · denotes Wick's product. Hence denoting H := [4m 2 + |k [123] | 2 + |k 1 | 2 + |k 2 | 2 + |k 3 | 2 ] we obtain L(X, X, X, Let us now estimate various terms in R. The terms containing only combinations of X, X can be estimated directly whereas for terms where ζ appears it is necessary to use stationarity due to the limited integrability in space. For instance, and similarly for the other terms without ζ which are collectively of order 2 i4κ (λ 2 + λ 4 ). For the remaining terms, we fix a weight ρ as above and use stationarity. In addition, we shall be careful about having the necessary integrability. For instance, for the most irregular term we have and we bound this quantity as 2,2 (ρ 2 ) ]) 1/2 2 i(1/2+5κ) (λ + λ 7/2 ).
where we used Theorem 4.9. Next, Proceeding similarly for the other terms we finally obtain the bound Therefore for a fixed λ > 0 there exists a sufficiently large i such that and the proof is complete. ✷

Integration by parts formula and Dyson-Schwinger equations
The goal of this section is twofold. First, we introduce a new paracontrolled ansatz, which allows to prove higher regularity and in particular to give meaning to the critical resonant product in the continuum. Second, the higher regularity is used in order to improve the tightness and to construct a renormalized cubic term ϕ 3 . Finally, we derive an integration by parts formula together with the Dyson-Schwinger equations and we identify the continuum dynamics.

Improved tightness
In this section we establish higher order regularity and a better tightness which is needed in order to define the resonant product X 2 • φ in the continuum limit. Recall that the equation (4.6) satisfied by φ M,ε has the form we obtain by the commutator lemma, Lemma A.14, can be rewritten as (4.9) and controlled due to Lemma 4.3, where we also estimated X M,ε Y M,ε and X M,ε Y 2 M,ε , we deduce Consequently, the equation satisfied by χ M,ε reads and can be controlled as in the proof of Lemma A.14.
Next, we state a regularity result for χ M,ε , proof of which is postponed to Appendix A.6. While it is in principle possible to keep track of the exact dependence of the bounds on λ we do not pursue it any further since there seems to be no interesting application of such bounds. Nevertheless, it can be checked that the bounds in this section remain uniform over λ belonging to any bounded subset of [0, ∞).
Proposition 6.1 Let ρ be a weight such that ρ ι ∈ L 4,0 for some ι ∈ (0, 1). Let φ M,ε be a solution to (6.1) and let χ M,ε be given by (6.2). Then We apply this result in order to deduce tightness of the sequence (ϕ M,ε ) M,ε as time-dependent stochastic processes. In other words, in contrast to Theorem 4.8, where we only proved tightness for a fixed time t 0, it is necessary to establish uniform time regularity of (ϕ M,ε ) M,ε . To this end, we recall the decompositions Theorem 6.2 Let β ∈ (0, 1/4). Then it holds true that for all p ∈ [1, ∞) and τ ∈ (0, T ) where L ∞ τ,T H −1/2−2κ,ε (ρ 2 ) = L ∞ (τ, T ; H −1/2−2κ,ε (ρ 2 )). Proof Let us begin with the first bound. According to Proposition 6.1 and Theorem 4.8 we obtain that In addition, the computations in the proof of Proposition 6.1 imply that also E L ε χ M,ε is bounded uniformly in M, ε. As a consequence, we deduce that is also bounded uniformly in M, ε. Next, we apply a similar approach to derive uniform time regularity of φ M,ε . To this end, we study the right hand side of (6.1). Observe that due to the energy estimate from Theorem 4.5 and the bound from Proposition 6.1 together with Theorem 4.8 the following are bounded uniformly , whereas all the other terms on the right hand side of (6.1) are uniformly bounded in better function spaces. Hence we deduce that is bounded uniformly in M, ε. Now we have all in hand to derive a uniform time regularity of ζ M,ε . Using Schauder estimates together with (6.4) it holds that is bounded uniformly in M, ε.
Proof According to Theorem 6.31 in [Tri06] we have the compact embedding and consequently since α < β the embedding is compact, see e.g. Theorem 5.1 [Amm00]. Hence the desired tightness of E ε ϕ M,ε follows from Theorem 6.2 and Lemma A.15. The tightness of E ε X M,ε follows from the usual arguments and does not pose any problems. ✷ As a consequence, we may extract a converging subsequence of the joint laws of the processes loc X . Letμ denote any limit point. We denote by (ϕ, X) the canonical processes on W α,1 loc B −1−4κ loc X and let µ be the law of the pair (ϕ, X) underμ (or the projection ofμ to the first two components). Observe that there exists a measurable map Ψ : (ϕ, X) → (ϕ, X) such thatμ = µ • Ψ −1 . Therefore we can represent expectations underμ as expectations under µ with the understanding that the elements of X are constructed canonically from X via Ψ. Furthermore, Y, φ, ζ, χ are defined analogously as on the approximate level as measurable functions of the pair (ϕ, X). In particular, the limit localizer U > is determined by the constant L 0 obtained in Lemma 4.1. Consequently, all the above uniform estimates are preserved for the limiting measure and the convergence of the corresponding lattice approximations to Y, φ, ζ, χ follows. In addition, the limiting process ϕ is stationary in the following distributional sense: for all f ∈ C ∞ c (R + ) and all τ > 0, the laws of coincide. Based on the time regularity of ϕ it can be shown that this implies that the laws of ϕ(t) and ϕ(t + τ ) coincide for all τ > 0 and a.e. t ∈ [0, ∞). The projection of µ on ϕ(t) taken from this set of full measure is the measure ν as obtained in Theorem 4.9.

Integration by parts formula
The goal of his section is to derive an integration by parts formula for the Φ 4 3 measure on the full space. To this end, we begin with the corresponding integration by parts formula on the approximate level, that is, for the measures ν M,ε and pass to the limit.
Let F be a cylinder functional on S ′ (R 3 ), that is, F (ϕ) = Φ(ϕ(f 1 ), . . . , ϕ(f n )) for some Φ : R n → R and f 1 , . . . , f n ∈ S(R 3 ). Let DF (ϕ) denote the L 2 -gradient of F . Then it holds for fields ϕ ε defined on Λ ε where x ∈ Λ ε and w ε is the kernel involved in the definition of the extension operator E ε from Section A.4. By integration by parts it follows that (6.5) According to Theorem 4.9, we can already pass to the limit on the left hand side as well as in the second term on the right hand side of (6.5). Namely, we obtain for any accumulation point ν and any (relabeled) subsequence (ν M,ε • (E ε ) −1 ) M,ε converging to ν that the following convergences hold in the sense of distributions in the variable The remainder of this section is devoted to the passage to the limit in (6.5), leading to the integration by parts formula for the limiting measure in Theorem 6.7 below. In particular, it is necessary to find a way to control the convergence of the cubic term and to interpret the limit under the Φ 4 3 measure. Let us denote ϕ 3 M,ε (y) := ϕ(y) 3 + (−3a M,ε + 3λb M,ε )ϕ(y). We shall analyze carefully the distributions J M,ε (F ) ∈ S ′ (Λ ε ) given by in order to determine the limit of E ε J M,ε (F ) (as a distribution in x ∈ R 3 ) as (M, ε) → (∞, 0). Unfortunately, even for the Gaussian case when λ = 0 one cannot give a well-defined meaning to the random variable ϕ 3 under the measure ν. Additive renormalization is not enough to cure this problem since it is easy to see that the variance of the putative Wick renormalized limiting field is infinite. In the best of the cases one can hope that the renormalized cube ϕ 3 makes sense once integrated against smooth cylinder functions F (ϕ). Otherwise stated, one could try to prove that (J M,ε ) M,ε converges as a linear functional on cylinder test functions over S ′ (R d ).
To this end, we work with the stationary solution ϕ M,ε and introduce the additional notation ϕ 3 M,ε (t, y) := ϕ M,ε (t, y) 3 + (−3a M,ε + 3λb M,ε )ϕ M,ε (t, y). As the next step, we employ the decomposition in order to find a decomposition that can be controlled by our estimates. We rewrite Next, we use the paraproducts and paracontrolled ansatz to control the various resonant products. For the renormalized resonant product 3 X 2 M,ε • (−λX M,ε + ζ M,ε ) + 3λb M,ε ϕ M,ε we first recall that Therefore using the definition of Z M,ε in (4.9) we have The remaining resonant product that requires a decomposition can be treated as where we used the notation f g = f ≺ g + f • g.
These decompositions and our estimates show that the products are all are controlled in the space L 1 (0, T, B −1−3κ,ε 1,1 (ρ 4+σ )). The term X 3 M,ε requires some care since it cannot be defined as a function of t. Indeed, standard computations show that E ε X 3 M,ε → X 3 in W −κ,∞ T C −3/2−κ,ε (ρ σ ), namely, it requires just a mild regularization in time to be well defined and it is the only one among the contributions to ϕ 3 M,ε which has negative time regularity. In particular, we may write ϕ 3 is uniformly bounded in M, ε. The dependence of the function H ε on ε comes from the corresponding dependence of the paraproducts as well as the resonant product on ε. Now, let h : R → R be a smooth test function with supp h ⊂ [τ, T ] for some 0 < τ < T < ∞ and such that R h(t)dt = 1. Then by stationarity we can rewrite the Littlewood-Paley blocks As a consequence of Corollary 6.3 and the discussion afterwards we extract a subsequence converging in law and using the uniform bounds we may pass to the limit and conclude Here ϕ 3 is expressed (as ϕ 3 M,ε before) as a measurable function of (ϕ, X) given by (6.6) where we used the notation f ✶ g = f ≺ g + f ≻ g and ζ, φ, Y are defined as starting from (ϕ, X) = Ψ(ϕ, X) as the operator C is the continuum analog of the commutator C ε defined in (A.8), the localizer U > is given by the constant L 0 from Lemma 4.1 and B(·) (appearing also in the limit Z, cf. (4.9)) is the uniform Remark that our uniform bounds remain valid for the limiting measure µ. As a consequence we obtain the following result.
Lemma 6.4 Let F : S ′ (R 3 ) → R be a cylinder function such that for some n ∈ N. Let µ be an accumulation point of the sequence of laws of (E ε ϕ M,ε , E ε X M,ε ). Then it holds (along a subsequence) that for any function h as above. Moreover, we have the estimate where the implicit constant depends on µ, h but not on F .
Proof For any cylinder function F satisfying the assumptions and since supp h ∈ [τ, T ] we have the following estimate for arbitrary conjugate exponents p, p ′ ∈ (1, ∞) Since for arbitrary conjugate exponents q, q ′ ∈ (1, ∞) it holds (ρ 4+σ ) ) 1/q , we obtain due to Theorem 4.8 that where α = 1 + κ − 1/(pq). Finally, choosing p, q ∈ (1, ∞) sufficiently small and κ ∈ (0, 1) appropriately, we may apply the Sobolev embedding W β,1 T ⊂ W α,pq T together with the uniform bound from Theorem 6.2 (which remains valid in the limit) to deduce To show the second bound in the statement of the lemma, we use the fact that supp h ⊂ [τ, T ] for some 0 < τ < T < ∞ to estimate (ρ 4+σ ) ) 1/2 C F , where the last inequality follows from Theorem 6.2 and the bounds in the proof of Proposition 6.1. ✷ Heuristically we can think of J µ (F ) as given by However, as we have seen above, this expression is purely formal since ϕ 3 is only a space-time distribution with respect to µ and therefore ϕ 3 (0) is not a well defined random variable. One has to consider F → J µ (F ) as a linear functional on cylinder functions taking values in S ′ (R 3 ) and satisfying the above properties. Lemma 6.4 presents a concrete probabilistic representation based on the stationary stochastic quantization dynamics of the Φ 4 3 measure.
Alternatively, the distribution J µ (F ) can be characterized in terms of ϕ(0) without using the dynamics, in particular, in the spirit of the operator product expansion as follows.
Lemma 6.5 Let F be a cylinder function as in Lemma 6.4 and ν the first marginal of µ. Then there exists a sequence of constants (c N ) N ∈N tending to ∞ as N → ∞ such that Then by stationarity of ϕ under µ we have for a function h satisfying the above properties At this point is not difficult to proceed as above and find suitable constants (c N ) N ∈N which deliver the appropriate renormalizations so that and therefore, using the control of the moments, prove that ✷ Remark 6.6 By the previous lemma it is now clear that J µ does not depends on µ but only on its first marginal ν. So in the following we will write J ν := J µ to stress this fact.
Using these informations we can pass to the limit in the approximate integration by parts formula (6.5) and obtain an integration by parts formula for the Φ 4 3 measure in the full space. This is the main result of this section.
In particular, this allow to express the (space-homogeneous) two-point function S ν 2 (x − y) := E ν [ϕ(x)ϕ(y)] of ν as the solution to where the right hand side includes the four point function S ν 4 (x 1 , . . . , Finally, we observe that the above arguments also allow us to pass to the limit in the stochastic quantization equation and to identify the continuum dynamics. To be more precise, we use Skorokhod's representation theorem to obtain a new probability space together with (not relabeled) processes (ϕ M,ε , X M,ε ) defined on some probability space and converging in the appropriate topology determined above to some (ϕ, X). We deduce the following result.
Corollary 6.9 The couple (ϕ, X) solves the continuum stochastic quantization equation where ξ = L X and ϕ 3 is given by (6.6).

A Technical results
In this section we present auxiliary results needed in the main body of the paper.

A.1 Besov spaces
First, we cover various properties of the discrete weighted Besov spaces such as an equivalent formulation of the norms, duality, interpolation, embeddings, bounds for powers of functions and a weighted Young's inequality.
Proof We write ρf = ρ ≺ f + ρ f and estimate by paraproduct estimates f B α,ε p,q (ρ) , which implies one inequality. For the converse one, we write f = ρ −1 ≺ (ρf ) + ρ −1 (ρf ), and estimate . ✷ Lemma A.2 Let α ∈ R, p, p ′ , q, q ′ ∈ [1, ∞] such that p, p ′ and q, q ′ are conjugate exponents. Let ρ be a weight as in Lemma A.1. Then with a proportionality constant independent of ε. Consequently, Proof In view of Lemma A.1 it is sufficient to consider the unweighted case. Let f ∈ B α,ε p,q and g ∈ B −α,ε p ′ ,q ′ . Then by Parseval's theorem and Hölder's inequality we have Then it holds The proof is a consequence of Hölder's inequality. Let us show the claim for p, p 0 , p 1 , q, q 0 , q 1 ∈ [1, ∞) and ε ∈ A \ {0}. If some of the exponents p, p 0 , p 1 , q, q 0 , q 1 are infinite or we are in the continuous setting, the proof follows by obvious modifications. We write and apply Hölder's inequality to the conjugate exponents p 0 θp and p 1 (1−θ)p to obtain and by Hölder's inequality to the conjugate exponents q 0 θq and q 1 We note that by our construction of the Littlewood-Paley projectors on Λ ε , in each of the cases j = −1, j ∈ {0, . . . , N − J − 1} and j = N − J, there exists an L 1 -kernel K such that the Littlewood-Paley block ∆ ε j f is given by a convolution with 2 jd K(2 j ·). See Lemma A.2 in [MP17] for more details. For notational simplicity we omit the dependence of K on the three cases above.
Lemma A.4 Let ε ∈ A and let β > 0. Then it holds and the proportional constants do not depend on ε.
Proof Due to Lemma A.1 together with Parseval's equality we directly obtain the first claim. Consequently, by Young's inequality together with the fact that ρ(y) ρ(x) ρ −1 (x−y) (for a universal proportionality constant that depends only on ρ) we have that ✷ Lemma A.5 Let κ ∈ (0, 1), p ∈ [1, ∞] and let ρ be a polynomial weight , where the proportionality constant does not depend on ε.
Proof Let j 0. Let K j = K j,ε = F −1 ϕ ε j and denoteK j =K j,ε = i∼j K i,ε . Then it holds that ∆ ε j f =K j * ∆ ε j f and we writē For the second term it holds by translation invariance of ∇ ε hence by Young inequality The kernel V j,ℓ := (Id −∆ ε ) −1 ∇ * ε,ℓK j is given by and it is possible to check that (using that ε2 j 1) uniformly in j whereÃ is an annulus centered at the origin. Therefore and from this is easy to deduce that V j,ℓ L 1,ε (ρ −1 ) 2 −j uniformly in j and ε. A similar computation applies to the first term in (A.1) to obtain and the proof is complete. ✷ Lemma A.6 Let ε ∈ A and let ι > 0. Let ρ be a weight such that ρ ι ∈ L 4,0 . Then where the proportionality constant does not depend on ε.
Proof By Hölder's inequality and since for |x − y| 1 the quotient ρ(x) ρ(y) is uniformly bounded above and below, it follows from Lemma A.3 [MP17] that where the proportional constant only depends on ρ. ✷ Lemma A.7 Let α > 0. Let ρ 1 , ρ 2 be weights. Then for every β > 0 it holds true , where the proportionality constants do not depend on ε.
Proof Due to the paraproduct estimates and the embeddings of Besov spaces, we have for every For the cubic term, we write and estimate each term separately. The second and the third term can be estimated directly by . For the remaining term, we have where by the paraproduct estimates and Lemma A.4 which completes the proof. ✷ Lemma A.8 Let ρ be a polynomial weight. Let p, q, r ∈ [1, ∞] be such that 1 r + 1 = 1 p + 1 q . Then where * ε denotes the convolution on Λ ε and the proportionality constants are independent of ε.
Proof We observe that for a polynomial weight of the form ρ(x) = x −ν for some ν 0, it holds that ρ(y) ρ(x)ρ −1 (x − y). Accordingly, hence the claim follows by (unweighted) Young's inequality. For the second bound, we write and apply Hölder's inequality with exponents r, rp r−p , rq Finally, taking the rth power and integrating completes the proof. ✷

A.2 Localizers
As the next step, we introduce another equivalent formulation of the weighted Besov spaces B α,ε ∞,∞ (ρ) in terms of suitable point evaluation of the Littlewood-Paley decomposition. First, for J ∈ N 0 such that N − J J ε , α ∈ R and ε ∈ A we define the Besov space b α,ε ∞,∞ (ρ) of sequences λ = (λ j,m ) −1 j N −J,m∈Z d by the norm Note that we do not stress the dependence of b α,ε ∞,∞ (ρ) on the parameter J as in the sequel we only consider one fixed J for all ε ∈ A given by Lemma A.9 below. The next result shows the desired equivalence.
Lemma A.9 Let α ∈ R, ε ∈ A and let ρ be a weight. There exists J ∈ N 0 (independent of ε) with the following property: f ∈ B α,ε ∞,∞ (ρ) if and only if it is represented by where the proportionality constants do not depend on ε. In particular, given f ∈ B α,ε ∞,∞ (ρ) the coefficients λ are defined by and given λ ∈ b α,ε ∞,∞ (ρ) the distribution f is recovered via the formula where F 2 −j−J Z d denotes the Fourier transform on the lattice 2 −j−J Z d .
Lemma A.13 Let ε ∈ A. Let α, β, γ ∈ R be such that α, γ > 0, β + γ < 0 and α + β + γ > 0 and let ρ 1 , ρ 2 , ρ 3 be weights and let ρ = ρ 1 ρ 2 ρ 3 . There exists a bounded trilinear operator where the proportionality constant is independent of ε, and for smooth functions we have where C ε was defined above. Hence the desired formula holds for smooth functions. By (A.9) and the paraproduct estimates we have , and the right hand side is estimated by which completes the proof. ✷ Next, we show several commutator estimates. To this end, ∆ ε denotes the discrete Laplacian on Λ ε and we define the corresponding elliptic and parabolic operators by Q ε := m 2 − ∆ ε and L ε := ∂ t + Q ε , where m 2 > 0.

A.4 Extension operators
In order to construct the Euclidean quantum field theory as a limit of lattice approximations, we need a suitable extension operator that allows to extend distributions defined on the lattice Λ ε to the full space R d . To this end, we fix a smooth, compactly supported and radially symmetric nonnegative function w ∈ C ∞ c (R d ) such that supp w ⊂ B 1/2 where B 1/2 ⊂ R d is the ball centered at 0 with radius 1/2 and R d w(x)dx = 1. Let w ε (·) := ε −d w(ε −1 ·) and define the extension operator E ε by where by * ε we denote the convolution on the lattice Λ ε .
Proof Within the proof we denote * the convolution on R d whereas * ε stands for the convolution on Λ ε . Let K j = F −1 R d ϕ j and K ε j = F −1 ϕ ε j . First, we observe that for j < N − J we have Consequently, For i < N −J we obtain by Young's inequality for convolutions, Lemma A.8 and the construction of w ε , uniformly in ε, that ∆ ε j f L p,ε (ρ) .
If i N − J then we write letK i = j∼i K j and Hence by Lemma A.8 Now we estimate the first term on the right hand side (using the fact that the weight ρ −1 increases with |x|) as follows y))∆ ℓ w(y)dy ρ −1 (x)dx.

A.5 A Schauder estimate
In this section we establish a suitable Schauder-type estimate needed in Section A.6.
Lemma A.16 Let ρ be a weight and let P ε t = e t(∆ε−m 2 ) denote the semigroup generated by ∆ ε − m 2 . Then there exists c > 0 uniform in ε such that for all −1 j N − J it holds true where the proportionality constant does not depend on ε and t 0.
Proof Applying the Littlewood-Paley projectors we obtain Hence according to Lemma A.16 there exists c > 0 such that for −1 j N − J and uniformly in T > 0 and ε v L 1 Finally, we proceed with the proof of the proof of Proposition 6.1.