Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures

In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the $\Phi^4_3$-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential.


Introduction to the series
In this two-paper series, we study the renormalized wave equation with a Hartree nonlinearity and random initial data given by (a) #´B 2 tt u´u`∆u " :pV˚u 2 qu: pt, xq P RˆT 3 , u| t"0 " φ 0 , B t u| t"0 " φ 1 .
Here, T def " R{2πZ is the torus and the interaction potential V : T 3 Ñ R is of the form V pxq " c β |x|´p 3´βq for all small x P T 3 , where 0 ă β ă 3, satisfies V pxq Á 1 for all x P T 3 , is even, and is smooth away from the origin. The nonlinearity : pV˚u 2 qu : is a renormalization of pV˚u 2 qu (see Definition 2.6 below).
The nonlinear wave equation (a) is a prototypical example of a Hamiltonian partial differential equation. The formal Hamiltonian is given by :pV˚u 2 qpt, xqupt, xq 2 : dx, where L 2 x " L 2 x pT 3 q. Based on the Hamiltonian structure, we expect the formal Gibbs measure µ b given by dµ b pφ 0 , φ 1 q " Z´1 expp´Hpφ 0 , φ 1 qq dφ 0 dφ 1 (b) to be invariant under the flow of (a), where Z is a normalization constant. The first part of this series focuses on the rigorous construction and properties of µ b . With a primary focus on the related Φ 4 d -models, similar constructions have been studied in constructive quantum field theory. Recently, this area of research has been revitalized through advances in singular stochastic partial differential equations. The main difficulties come from the quartic interaction :pV˚u 2 qu 2 : in the Hamiltonian. In fact, without the interactions, one obtains the Gaussian free field dg b pφ 0 , φ 1 q " Z´1 0 exp´´1 2 }φ 0 } 2 x¯d φ 1 , which can be constructed through elementary arguments. Using our representation of the Gibbs measure µ b , we also prove that µ b and g b are mutually singular for 0 ă β ă 1{2. In the second part of this series, we study the dynamics of (a) with random initial data drawn from the Gibbs measure µ b . Due to the low spatial regularity, the local theory requires a mix of techniques from dispersive equations, harmonic analysis, and probability theory. More specifically, we rely on ideas from the para-controlled calculus of Gubinelli, Imkeller, and Perkowski [GIP15]. The heart of this series, however, lies in the global theory. Our main contribution is a new form of Bourgain's globalization argument [Bou94], which addresses the singularity of the Gibbs measure and its consequences. We now state an qualitative version our main theorem, which combines our measure-theoretic and dynamical results. For the quantitative version, we refer the reader to Theorem 1.1 below and Theorem 1.3 in the second part of this series. We recall that the parameter 0 ă β ă 3 determines the regularity of the interaction potential V .

Introduction
In the first paper of this series, we rigorously construct and study the formal Gibbs measure µ b from (b) above. Since the Hamiltonian Hrφ 0 , φ 1 s splits into a sum of functions in φ 0 and φ 1 , we can rewrite (b) as dµ b pφ 0 , φ 1 q "Z´1 0 exp´´1 4 ż T 3 :pV˚φ 2 0 qφ 2 0 : dx´1 The construction and properties of the second factor are elementary (as will be explained below), and we now focus on the first factor. As a result, we are interested in the rigorous construction of a measure µ which is formally given by :pV˚φ 2 qφ 2 : dx´1 2 }φ} 2 L 2 pT 3 q´1 2 }∇φ} 2 L 2 pT 3 q¯d φ.
Our Gibbs measure µ is closely related to the Φ 4 d -models, which replace the three-dimensional torus T 3 by the more general d-dimensional torus T d and replace the integrand :pV˚φ 2 qφ 2 : by the renormalized quartic power :φ 4 :. Thus, the Φ 4 d -model is formally given by Aside from their connection to Hamiltonian PDEs, such as nonlinear wave and Schrödinger equations, the Φ 4 d -models are of independent interest in quantum field theory (cf. [Fol08]). In most rigorous constructions of measures such µ or the Φ 4 d -models, the first step consists of a regularization. For instance, one may insert a frequency-truncation in the nonlinearity or replace the continuous spatial domain by a discrete lattice. In a second step, one then proves the convergence of the regularized measures as the regularization is removed, either by direct estimates or compactness arguments.
With a particular focus on Φ 4 d -models, the question of convergence of the regularized measures has been extensively studied over several decades. The first proof of convergence was a major success of the constructive field theory program, which thrived during the 1970s and 1980s. We refer the reader to the excellent introduction of [GH19] for a detailed overview and the original works [BCG`78, BFS83, FO76, GJ87, MS76,Par77,Sim74,Wat89]. In the 1990s, Bourgain [Bou94,Bou97,Bou96] revisted the Φ 4 d -model in dimension d " 1, 2 using tools from harmonic analysis and introduced these problems into the dispersive PDE community. Bourgain's works [Bou94,Bou97,Bou96] also contain important dynamical insights, which will be utilized in the second part of this series. Based on the concept of stochastic quantization, which was introduced by Nelson [Nel66,Nel67] and Parisi-Wu [PW81], the construction and properties of the Φ 4 d -models have also been studied over the last twenty years in the stochastic PDE community. The main idea behind stochastic quantization is that the Φ 4 d -measure is formally invariant under the stochastic nonlinear heat equation (1.3) B t u`u´∆u "´:u 3 :`?2ξ pt, xq P RˆT d , where ξ is space-time white noise. After prescribing simple initial data, such as up0q " 0, one hopes to obtain the Φ 4 d -measure as the limit of the law of uptq as t Ñ 8. In spatial dimensions d " 1, 2, this approach was carried out by Iwata [Iwa87] and Da Prato-Debussche [DPD03], respectively. In spatial dimension d " 3, however, (1.3) is highly singular and the local well-posedness theory of (1.3) is beyond classical methods in stochastic partial differential equations. In groundbreaking work [Hai14], Hairer introduced regularity structures, which provide a detailed description of the local dynamics of (1.3). Alternatively, the local well-posedness of (1.3) was also obtained by Catellier and Chouk in [CC18], which is based on the para-controlled calculus of Gubinelli, Imkeller, and Perkowski [GIP15]. In order to construct the Φ 4 3 -model using (1.3), however, local control over the solution is not sufficient, and one needs a global well-posedness theory. The global theory has been addressed very recently in [AK17, GH19,HM18,MW17], which combine regularity structures or para-controlled calculus with further PDE arguments, such as the energy method. Using similar tools, Barashkov and Gubinelli [BG18,BG20] recently developed a variational approach to the Φ 4 3 -model, which does not directly rely on the stochastic heat equation (1.3). Their work forms the basis of this paper and will be discussed in more detail below.
After this broad overview of the relevant literature, we now begin a more detailed discussion of the previous methods. Throughout this discussion we encourage the reader to think of the nonlinear wave equation as a Hamiltonian system of ordinary differential equations in Fourier space. We begin with the elementary construction of the Gaussian free field. Then, we discuss the construction of the Φ 4 1 and Φ 4 2 -models using harmonic analysis, similar as in Bourgain's works [Bou94,Bou96], and the construction of the Φ 4 3 -model using the variational approach of Barashkov and Gubinelli [BG18].
Given a function φ : T d Ñ R, its Fourier expansion is given by Due to the real-valuedness of φ, the sequence p p φpnqq nPZ d satisfies the symmetry condition p φpnq " p φp´nq. In order to respect this symmetry, we let Λ Ď Z d be such that Z d " t0u where Ţ denotes the disjoint union. For n P Λ, we denote by d p φpnq the Lebesgue measure on C, and for n " 0, we denote by d p φp0q the Lebesgue measure on R. We can then formally identify the d-dimensional Gaussian free field (1.5) dg d pφq " Z´1 exp´´1 2 }φ} 2 L 2 pT d q´1 2 }∇φ} 2 L 2 pT d q¯d φ as the push-forward under the Fourier transform of ( where xny 2 " 1`|n| 2 . While (1.5) is entirely formal, the right-hand side of (1.6) is a welldefined product measure. Under the measure in (1.6), p φp0q is a standard real-valued Gaussian and p p φpnqq nPΛ is a sequence of independent complex Gaussians satisfying E| p φpnq| 2 " xny´2. Turning this formal discussion around, we let pΩ, F, Pq be an ambient probability space containing a sequence of independent complex-valued standard Gaussians pg n q nPΛ and a standard real-valued Gaussian g 0 . Then, we can rigorously define the Gaussian free field g d by (1.7) dg d pφq "´ÿ nPZ d g n xny e ixn,xy¯# P, where the subscript # denotes the push-forward. Using the representation (1.7), we see that a typical sample of g d almost surely lies in H s x pT d q for all s ă 1´d{2 but not in H 1´d{2 x pT d q.
Using either Sobolev embedding or Khintchine's inequality, we obtain g 1 -almost surely that 0 ă }φ} L 4 pTq ă 8. This implies that the density dΦ 4 1 {dg 1 is well-defined, almost surely positive, and lies in L q pg 1 q for all 1 ď q ď 8. In particular, the Φ 4 1 -model is absolutely continuous with respect to the Gaussian free field g 1 . We emphasize that the potential energy in (1.8) does not require a renormalization. Furthermore, we can define truncated Φ 4 1 -models by where N is a dyadic integer and P ďN a Littlewood-Paley projection. As was shown in [Bou94], direct estimates yield the convergence of dΦ 4 1;N {dg 1 in L q pg 1 q for all 1 ď q ă 8 and hence Φ 4 1;N converges to Φ 4 1 in total variation as N tends to infinity. In two spatial dimensions, however, we encounter a new difficulty. Since g 1 -almost surely }φ} L 2 " 8, the potential energy }φ} 4 L 4 is almost surely infinite. As a result, the potential energy requires a renormalization. A direct calculation using the definition of P ďN in (1.14) below yields We then replace the monomial pP ďN φq 4 by the Hermite polynomial :pP ďN φq 4 :" pP ďN φq 4´6 σ 2 N pP ďN φq 2`3 σ 4 N . This leads to the truncated Φ 4 2 -model given by :pP ďN φq 4 : pxq dx¯dg 2 pφq.
After this renormalization, one can show (cf. [OT18]) that the densities dΦ 4 2;N {dg 2 converge in L q pg 2 q for all 1 ď q ă 8 and we can define Φ 4 2 as the limit (in total-variation) of Φ 4 2;N as N Ñ 8. As in one spatial dimension, the Φ 4 2 -model is absolutely continuous with respect to the Gaussian free field g 2 . Using similar tools as for the Φ 4 2 -model, Bourgain [Bou97] constructed the Gibbs measure µ for the Hamiltonian with a Hartree interaction for β ą 2, which corresponds to a relatively smooth interaction potential V . The key point of this paragraph is that the Φ 4 1 -model, the Φ 4 2 -model, and the Gibbs measure µ for a smooth interaction potential can be constructed through "hard" analysis. As a result, one obtains strong modes of convergence and absolute continuity with respect to Gaussian free field.
The construction of the Φ 4 3 -model, however, is much more complicated. As will be described below, several of the "hard" conclusions, such as convergence in total-variation or absolute continuity with respect to the Gaussian free field, are either unavailable or fail. As a result, we have to (partially) replace hard estimates by softer compactness arguments. We now give a short overview of the variational approach in [BG18,BG20], which forms the basis of this paper. In order to use techniques from stochastic control theory, we introduce a family of Gaussian processes pW t pxqq tě0 on an ambient probability space pΩ, F, Pq satisfying Law P pW 8 q " g 3 , which will be defined in Section 2.1. We view t as a stochastic time-variable which serves as a regularization parameter. Using this terminology, we obtain a truncated Φ 4 3 -model by setting dΦ 4 3;T pφq " pW 8 q #`d Φ 4 3;T pφq5 and dΦ 4 3;T pφq " Z´1 T exp`´1 4 We emphasize already that the Φ 4 3;T -measure does not correspond to a truncated Hamiltonian, which will be discussed in full detail in Section 2.1. In order to construct the Φ 4 3 -model, the main step is to prove the tightmess of the Φ 4 3;T -measures in T . Using Prokhorov's theorem, this implies the weak convergence of a subsequence of Φ 4 3;T and we can define the Φ 4 3 -measure as the weak limit. To prove tightness, Barashkov and Gubinelli obtain uniform bounds in T on the Laplace transform The main ingredients for the uniform bounds are the Boué-Dupuis formula (Theorem 2.1) and the para-controlled calculus of Gubinelli, Imkeller, and Perkowski [GIP15], which has also been used in the stochastic quantization approach to the Φ 4 3 -model (cf. [GH19]). While the variational approach yields the existence of the Φ 4 3 -measure, it only yields limited information regarding its properties. In spatial dimensions d " 1, 2, the Φ 4 d -model is absolutely continuous with respect to the Gaussian free field g d , and hence the samples of Φ 4 3 for many purposes behave like a random Fourier series with independent coefficients. This is an essential ingredient in almost all invariance arguments for random dispersive equations (see e.g. [Bou97,Bou96,DNY19,NORBS12]). Unfortunately, the Φ 4 3 -measure is singular with respect to the Gaussian free field g 3 . This fact seems to be part of the folklore in mathematical physics, but it is surprisingly difficult to find a detailed reference. In an unpublished note available to the author [Hai], Martin Hairer proved the singularity using the stochastic quantization approach and regularity structures. Using the Girsanov-transformation, Barashkov and Gubinelli [BG20] constructed a reference measure ν 4 3 for the Φ 4 3 -model, which serves a similar purpose as the Gaussian free field for Φ 4 1 and Φ 4 2 . The samples of ν 4 3 are given by an explicit Gaussian chaos of finite order and Φ 4 3 is absolutely continuous with respect to ν 4 3 . Furthermore, Barashkov and Gubinelli proved that the reference measure ν 4 3 and the Gaussian free field g 3 are mutually singular, which yields a self-contained proof of the singularity of Φ 4 3 with respect to the Gaussian free field g 3 .
1.1. Main results and methods. In the following, we simply write g " g 3 for the threedimensional Gaussian free field. Let N ě 1 be a dyadic integer and define the renormalized potential energy by The coupling constant λ ą 0 is introduced for illustrative purposes, but the reader may simply set λ " 1 as in all previous discussions. The renormalization constants a N , b N , and c λ N are as in Definition 2.8 and Proposition 3.2 and the renormalization multiplier M N is as in Definition 2.8. We emphasize that the renormalization in (1.9) goes beyond the usual Wick-ordering, which is only based on the mass }P ďN φ} 2 L 2 . The additional renormalization is contained in the renormalization constant c λ N , which is related to the mutual singularity of µ b and g (for 0 ă β ă 1{2). The truncated and renormalized Hamiltonian H N is given by where we omit the dependence on λ ą 0 from our notation. We emphasize that only the quartic term contains a frequency-truncation and renormalization, whereas the quadratic terms remain 6 unchanged. As described in the beginning of the introduction, we focus on the first factor of the truncated Gibbs measure µ b N , which is given by Before we state our main result, we recall the assumptions on the interaction potential V : T 3 Ñ R from the introduction to the series. In these assumptions, 0 ă β ă 3 is a parameter.
Assumptions A. We assume that the interaction potential V satisfies (1) V pxq " c β |x|´p 3´βq for some c β ą 0 and all x P T 3 satisfying }x} ď 1{10, V is smooth away from the origin.
We now state the conclusions of this paper which will be needed in the second paper of this series. A more comprehensive version of our results will then be stated in Theorem 1.3, Theorem 1.4, and Theorem 1.5 below. The additional results may be useful in further applications, such as invariant measures for a Schrödinger equation with a Hartree nonlinearity.
Theorem 1.1 (The Gibbs measure). Let κ ą 0 be a fixed positive parameter, let 0 ă β ă 3 be a parameter, and let the interaction potential V be as in the Assumptions A. Then, a subsequence of the truncated Gibbs measures pµ N q N ě1 converges weakly to a probability measure µ 8 on C´1 {2´κ x pT 3 q. If in addition 0 ă β ă 1{2, the limiting measure µ 8 and the Gaussian free field g are mutually singular. Furthermore, there exists a sequence of reference measures pν N q N ě1 on C´1 {2´κ x pT 3 q and an ambient probability space pΩ, F, Pq satisfying the following properties: (1) (Absolute continuity and L q -bounds) The truncated Gibbs measures µ N are absolutely continuous with respect to the reference measures ν N . More quantitatively, there exists a parameter q ą 1 and a constant C ě 1, depending only on β, such that (2) (Representation of ν N ) Let γ " minp1{2`β, 1q. There exists a large integer k " kpβq and two random functions G, R N : pΩ, Fq Ñ C´1 {2´κ x pT 3 q satisfying for all p ě 2 that ν N " Law P`G`RN˘, g " Law P`G˘, and }R N } L p Remark 1.2. After the completion of this series, the author learned of independent work by Oh, Okamoto, and Tolomeo [OOT20], which discusses the focusing and defocusing three-dimensional (stochastic) nonlinear wave equation with a Hartree nonlinearity. In the focusing case, the authors provide a complete picture of the construction and properties of the focusing Gibbs measures, which distinguishes the three regimes β ą 2, β " 2, and β ă 2 (cf. [OOT20]). In the defocusing case, the authors construct the Gibbs measures for β ą 0 and prove the singularity for 0 ă β ď 1{2, which includes the endpoint β " 1{2. In addition, [OOT20] shows the uniqueness of µ 8 , which is not proven in this paper. The reference measures are also briefly discussed in [OOT20, Appendix C], but only play a minor role in their analysis. The L q -bound in Theorem 1.1, which will be essential in the second part of this series, is not proven in [OOT20]. The dynamical results in [OOT20], however, are restricted to β ě 2 and β ą 1 in the focusing and defocusing case, respectively. In particular, the singular regime 0 ă β ă 1{2 in the defocusing case is not covered, which is the main object of this series.
We remark that Theorem 1.1 makes no statement about the uniqueness of µ 8 , which will not pose any problems in the second part of the series. While Theorem 1.1 only yields the weak convergence of a subsequence, we believe that the full sequence converges weakly. For the Φ 4 3 -model, this was proven using Γ-convergence in [BG18]. In [BG18], Barashkov and Gubinelli also obtain a variational description of the limiting measure, which does not rely on the limiting procedure. Due to the already extensive length of this series, however, we do not pursue the question of uniqueness here. In addition to the singular regime 0 ă β ă 1{2, the most interesting cases in Theorem 1.1 are the Newtonian potential |x|´2 (corresponding to β " 1) and the Coulomb potential |x|´1 (corresponding to β " 2). As mentioned earlier in the introduction, Bourgain [Bou97] proved a version of Theorem 1.1 in the limited range β ą 2, which corresponds to a relatively smooth interaction potential.
We now split the main theorem (Theorem 1.1) into three parts: ‚ the tightness of the truncated Gibbs measures µ N , ‚ the construction and properties of the reference measures ν N , ‚ the mutual singularity of the Gibbs measure and the Gaussian free field.
Theorem 1.3 (Tightness). The truncated Gibbs measures pµ N q N ě1 are tight on C´1 {2´κ x pT 3 q. In particular, a subsequence of pµ N q N ě1 weakly converges to a limiting measure µ 8 .
The overall strategy of the proof of Theorem 1.3 is the same as in the variational approach of Barashkov and Gubinelli [BG18]. In comparison with [BG18], the terms in this paper often have a more complicated algebraic structure but obey better analytical estimates. As any reader familiar with regularity structures or para-controlled calculus may certify, the algebraic structure of most stochastic objects is already quite complicated, so this trade-off is not always favorable. In addition, the non-locality of the nonlinearity requires different analytical estimates and we mention the two most important examples: (i) The coercive term }f } 4 L 4 in the variational problem for the Φ 4 3 -model is replaced by the potential energy ż T 3 pV˚f 2 qf 2 dx.
We emphasize that the coercive term in the variational problem does not contain a renormalization, which is a result of the binomial formula in Lemma 2.11. In order to use the potential energy in our estimates, we rely on a fractional derivative estimate of Visan [Vis07,(5.17)]. (ii) In the variational problem, we encounter mixed terms of the form ż where pW t q tě0 is the Gaussian process from the introduction. Based on the literature on random dispersive equations [Bou97, Bou96, DNY19, DNY20, GKO18], it is tempting to bound this mixed term through Fourier-analytic and random matrix techniques. We instead develop a simpler and elegant physical-space approach. The next theorem gives a more detailed description of the reference measures in Theorem 1.1. For notational simplicity, we allow the truncation parameter N to take the value 8.
Theorem 1.4 (Reference measures). There exists a family of reference measures pν N q 1ďN ď8 and an ambient probability space pΩ, F, Pq satisfying the following properties: (1) Absolute continuity and L q -bounds: The truncated Gibbs measures µ N are absolutely continuous with respect to the reference measures ν N . More quantitatively, there exists a parameter q ą 1 and a constant C ě 1, depending only on β, such that (2) Representation of ν N : We have that Here, n " npβq is a large integer and the linear, cubic, and n-th order Gaussian chaoses are explicitly given by where we refer the reader to Section 2.1 and Definition 2.6 for the definitions of J N s and the renormalizations.
We emphasize that the representation of ν N in Theorem 1.4 is much more detailed than stated in Theorem 1.1. This additional information is not required in our proof of global well-posedness and invariance in the second part of the series. However, we believe that the more detailed representation way be relevant for the Schrödinger equation with a Hartree nonlinearity. The reason lies in lowˆlowˆhigh-interactions, which are more difficult in Schrödinger equations than in wave equations. In the last two years, we have seen new and intricate methods dealing with these interactions [Bri18,DNY19,DNY20], but all of these papers heavily rely on the independence of the Fourier coefficients. In fact, overcoming this obstruction is mentioned as an open problem in [DNY20, Section 9.1]. The proof of Theorem 1.4 is based on the Girsanov-approach of Barashkov and Gubinelli [BG20]. As mentioned earlier, however, we cannot use the same approximate Gibbs measures as in [BG20], since they do not correspond to a frequency-truncated Hamiltonian. In the second part of the series, the frequency-truncated Hamiltonians are an essential ingredient in the proof of global wellposedness and invariance. This difference will be discussed in detail in Section 2.1. For now, we simply mention that there is a trade-off between desirable properties from a PDE or probabilistic perspective.
Our last theorem describes the relationship between the Gibbs measure µ 8 and the Gaussian free field g. Theorem 1.5 (Singularity). If 0 ă β ă 1{2, then the Gibbs measure µ 8 and the Gaussian free field g are mutually singular. If β ą 1{2, then the Gibbs measure is absolutely continuous with respect to the Gaussian free field g.
Theorem 1.5 determines the exact threshold between absolute continuity and singularity of µ 8 with respect to g. As mentioned in Remark 1.2, the singularity at the endpoint β " 1{2 has been obtained in independent work by Oh, Okamoto, and Tolomeo [OOT20]. The absolute continuity for β ą 1{2 already follows from the variational estimates in our construction of µ 8 . The main step is the mutual singularity of µ 8 and g for 0 ă β ă 1{2. We provide an explicit event witnessing this singularity, which is based on the behaviour of the frequency-truncated potential energy ż  1.2. Overview. To orient the reader, let us review the rest of this paper. In Section 2.1, we introduce the stochastic control perspective and recall the Boué-Dupuis formula. In Section 2.2, we estimate several stochastic objects, such as the renormalized nonlinearity :pV˚W 2 t qW t :. Our main tools will be Itô's formula and Gaussian hypercontractivity. In Section 3, we prove the tightness of the truncated Gibbs measures µ N and construct the limiting measure µ 8 . Using the Laplace transform and the Boué-Dupuis formula, the proof of tightness reduces to estimates for a variational problem, which occupy most of this section. In Section 4, we first construct the reference measures ν N and then examine their properties. The main ingredients are Girsanov's transformation and our earlier variational estimates. Finally, in Section 5, we prove the singularity of the Gibbs measure µ 8 with respect to the Gaussian free field g for all 0 ă β ă 1{2.
1.3. Notation. In the rest of the paper, we use def " instead of :" for definitions. The reason is that the colon in :" may be confused with our notation for renormalized powers in Definition 2.6 below. With a slight abuse of notation, we write dx for the normalized Lebesgue measure on T 3 . That is, we implicitly normalize ż T 3 1 dx " 1.
We define the Fourier transform of a function f : For any k P N and n 1 , . . . , n k P Z 3 , we define (1.12) n 12...k def " k ÿ j"1 n j .
In particular, it holds that t Þ Ñ ρ t pξq is non-decreasing. In order to break up the frequency truncation, we also set This continuous approach instead of the usual discrete decomposition will be essential in the stochastic control approach (Section 2.1). Nevertheless, we will sometimes use the usual dyadic Littlewood-Paley operators. For any dyadic N ě 1, we define P ďN by (1.14) { P ďN f pnq " ρ N pnq p f pnq.
We further set The corresponding Fourier multipliers are denoted by (1.15) χ 1 pnq " ρ 1 pnq and For any s P R, the C s x pT 3 q-norm is defined as We then define the corresponding space C s x pT 3 q by Due to the additional constraint as N Ñ 8, the space C s x pT 3 q is separable. This allows us to later use Prokhorov's theorem for families of measures on C s x pT 3 q. We also define (1.18) Similar as above, the additional restriction as t Ñ 8 makes C 0 t C s x pr0, 8sˆT 3 q separable. As a measure of tightness in C 0 t C s x pr0, 8sˆT 3 q, we define for any 0 ă α ă 1 and η ą 0 the norm For 1 ď r ď 8, we also define the Sobolev space W s,r x pT 3 q as the completion of C 8 x . We hope that the subscript x prevents any confusion with the stochastic objects in Section 2.2.

Stochastic objects
In this section, we introduce the stochastic control framework and describe several stochastic objects. While the reader with a background in singular SPDE and advanced stochastic calculus can think of this section as standard, much of this section may be new to a reader with a primary background in dispersive PDE. As a result, we include full details for most standard arguments but encourage the expert to skip the proofs.
2.1. Stochastic control perspective. We let pB n t q nPZ 3 zt0u be a sequence of standard complex Brownian motions such that B´n t " B n t and B n t , B m t are independent for n ‰˘m. We let B 0 t be a standard real-valued Brownian motion independent of pB n t q nPZ 3 zt0u . Furthermore, we let B t p¨q be the Gaussian process with Fourier coefficients pB n t q nPZ 3 , i.e., For every t ě 0, the Gaussian process formally satisfies ErB t pxqB t pyqs " t¨δpx´yq and hence B t p¨q is a scalar multiple of spatial white noise. We also let pF t q tě0 be the filtration corresponding to the family of Gaussian processes pB n t q tě0 . For future use, we denote the ambient probability space by pΩ, F, Pq. The Gaussian free field g, however, has covariance p1´∆q´1. To this end, we now introduce the Gaussian process W t pxq. For σ t pxq as in (1.13) and any n P Z 3 , we define We note that W n t is a complex Gaussian random variable with variance ρ 2 t pnq{xny 2 . We finally set It is easy to see for any κ ą 0 that W P C 0 t C´1 {2´κ x pr0, 8sˆT 3 q almost surely. With a slight abuse of notation, we write dPpW q for the integration with respect to the law of W under P, i.e., we omit the push-forward by W , and we write W for the canonical process on Comparing W t and B t , we have changed the covariance from t Id to ρ t p∇q 2 pI´∆q´1. For any fixed T ě 0, we have that (2.4) Law P pW T q " Law P pρ T p∇qW 8 q.
We already emphasize, however, that the processes t Þ Ñ W t and t Þ Ñ ρ t p∇qW 8 have different laws, since only the first process has independent increments. This difference will be important in the definition of r µ T below. To simplify the notation, we also introduce the Fourier multiplier J t , which is defined by Using this notation, we can represent the Gaussian process W t through the stochastic integral In a similar spirit, we define for any u : r0, 8qˆT 3 Ñ R the integral I t rus by We now recall the Boué-Dupuis formula [BD98], where our formulation closely follows [BG18,BG20]. We let H a be the space of F t -progressively measurable processes u : Ωˆr0, 8qˆT 3 Ñ R which P-almost surely belong to L 2 t,x pr0, 8qˆT 3 q.
Theorem 2.1 (Boué-Dupuis formula). Let 0 ă T ă 8, let F : C t pr0, T s, C 8 x pT 3 qq Ñ R be a Borel measurable fuction, and let 1 ă p, q ă 8. Assume that where we regard W as an element of C t pr0, T s, C 8 x pT 3 qq. Then, Remark 2.2. The optimization problem in (2.8) and, more generally, the change of perspective from W 8 to the whole process t Þ Ñ W t , is reminiscent of stochastic control theory. Due to the frequency projection in the definition of J t , we have that W t , I t rus P C t pr0, T s, C 8 x pT 3 qq. In our arguments below, the smoothness can be used to verify (2.7) through soft methods. Of course, a soft method cannot yield uniform bounds in T , and this will be one of the main goals of this section.
In the introduction, we discussed the Gibbs measure µ N corresponding to the truncated dynamics induced by H N defined in (1.10). In the spirit of the stochastic control approach, we now change our notation and use the parameter T to denote the truncation. Since the law of W 8 under P is the same as the Gaussian free field g and P ďT " ρ T p∇q, we obtain that (2.9) dµ T pφq " 1 Z T,λ exp´´:V T,λ pρ T p∇qφq:¯d`pW 8 q # P˘pφq.

12
The renormalized potential energy V T,λ is as in (3.2). We view µ T as a measure on the space C´1 {2´κ x pT 3 q for any fixed κ ą 0. In order to utilize the Boué-Dupuis formula, we lift µ T to a measure on The content of the next lemma explains the relationship between r µ T and µ T .
Lemma 2.4. The Gibbs measure µ T is the pushfoward of r µ T under W 8 , i.e., Due to its central importance to the rest of the paper, we prove this basic identity.
In [BG18,BG20], Barashkov and Gubinelli work with the lifted measure (2.12) ds µ T pW q " 1 Z T,λ exp`´:V T,λ pW T q:˘dPpW q. While W T and ρ T p∇qW 8 have the same distribution, the measures r µ T and s µ T do not coincide. Since this is an important difference between this paper and the earlier works [BG18, BG20], let us explain our motivation for working with r µ T instead of s µ T . From a probabilistic stand-point, the measure s µ T has better properties than r µ T . This is related to the independent increments of the process t Þ Ñ W t and we provide further comments in Remark 4.8 below. From a PDE perspective, however, s µ T behaves much worse than r µ T . For the proof of global well-posedness and invariance in the second part of this series, it is essential that µ T " pW 8 q # r µ T is invariant under the Hamiltonian flow of (1.10). In contrast, the author is not aware of an explicit expression for the pushforward of s µ T under W 8 . In particular, pW 8 q # s µ T is not directly related to µ T and not necessarily invariant under the Hamiltonian flow of H N . Alternatively, we could work with the pushfoward of s µ T under W T . A similar calculation as in the proof of Lemma 2.4 shows that pW T q # s µ T " pρ T p∇qq # µ T . Unfortunately, pρ T p∇qq # µ T also does not seem to be invariant under a truncation of the nonlinear wave equation. To summarize, while the measure s µ T has useful probabilistic properties, it lacks a direct relationship to the truncated dynamics and is ill-suited for our globalization and invariance arguments. Since we rely on ρ T p∇qW 8 in the definition of r µ T , the Gaussian process t Þ Ñ ρ T p∇qW t will play an important role in the rest of this paper. As a result, we now deal with both values T and t simultaneously. In most arguments, T will remain fixed while we use Itô's formula and martingale properties in t. To simplify the notation, we now write Since this will be convenient below, we also define (2.14) Furthermore, we define the integral operator I T t by 2.2. Stochastic objects and renormalization. We now proceed with the construction and renormalization of several stochastic objects. Similar constructions are standard in the probability theory literature and a comprehensive and well-written introduction can be found in [GP18,MWX17,OT18]. In order to make this section accessible to readers with a primary background in dispersive PDEs, however, we include full details. In a similar spirit, we follow a hands-on approach and mainly rely on Itô calculus. In Lemma 2.20, however, this approach becomes computationally infeasible and we also use multiple stochastic integrals (see [Nua06] or Section A.2).
Lemma 2.5. Let S N be the symmetric group on t1, . . . , N u and let W T ,n t be as in (2.13). Then, we have for all n 1 , n 2 , n 3 , n 4 P Z 3 that The integrals in (2.16)-(2.19) are iterated Itô integrals. This lemma is related to the product formula for multiple stochastic integrals, see e.g. [Nua06, Proposition 1.1.3].
Proof. The first equation (2.16) follows from the definition of the Itô derivative dW n t . The second equation (2.17) follows from Itô's product formula. Indeed, we have that The third equation (2.18) follows from Itô's formula and the second equation (2.17). Using Itô's formula, we have that The easiest way to keep track of the pre-factors throughout the proof is to compare the number of terms of each type and the cardinality of the symmetric group. In the formula above, we have three terms of each type and the cardinality #S 3 " 6, so we need the pre-factor 1{2. By inserting the second equation (2.17) and our expression for the cross-variation, we obtain For the second equality, we also used the permutation invariance of any sum over π P S 3 . This completes the proof of the third equation (2.18). We now prove the fourth and final equation (2.19). The argument differs from the proof of the third equation only in its notational complexity. Using Itô's formula and the third equation (2.18), we obtain that The total contribution of the second summand iś This completes the proof of the fourth equation (2.19).
Definition 2.6 (Renormalization). We define the renormalization constants a T t , b T t P R and the multiplier M T t : L 2 pT 3 q Ñ L 2 pT 3 q by Using this notation, we set Remark 2.7. As is clear from the definition, the renormalized powers in (2.20), (2.21), and (2.22) depend on the regularization parameter t. This dependence will always be clear from context and we thus do not reflect it in our notation.
Definition 2.8 (Renormalization of the dynamics). For any N ě 1, we define Throughout most of the paper, we will only work with the renormalization constants from Definition 2.6, which contain two finite parameters. The renormalization constants in Definition 2.8 will be more important in the second part of this series.
Proposition 2.9 (Stochastic integral representation of renormalized powers). With n 12 , n 123 , and n 1234 defined as in (1.12), we have that Furthermore, it holds that Remark 2.10. The "lower-order" terms in (2.6) were chosen precisely to obtain the result in Proposition 2.9. The renormalized powers of W T t can be represented solely using iterated stochastic integrals, which have many desirable properties.
Proposition 2.9 essentially follows from Lemma 2.5, Definition 2.6, and a tedious calculation. For the sake of completeness, however, we provide full details.
Proof. We first prove (2.24). Using (2.17), we have that By subtracting a T t from both sides and symmetrizing, this leads to the desired identity. We now turn to the proof of (2.25). From (2.18), we obtain that After symmetrizing and comparing with Definition 2.6, this leads to the desired identity. Next, we prove the identity (2.26). Using (2.19), we have that It remains to simplify (2.28) and (2.29). Regarding (2.28), we have that After symmetrizing, this completes the proof of (2.26).
Finally, it remains to prove (2.27). Since V is real-valued and even, we have that p V pnq " p V pnq " p V p´nq. As long as n 1234 " 0, this implies Using (2.30), (2.27) follows after inserting (2.25) and (2.26) into the two sides of the identity.
Like the Hermite polynomials, the generalized and renormalized powers in Definition 2.6 satisfy a binomial formula.
Lemma 2.11 (Binomial formula). For any f P H 1 pT 3 q, we have the binomial formulas Remark 2.12. Overall, the terms in (2.32) obey better analytical estimates than their counterparts for the Φ 4 3 -model in [BG20]. However, their algebraic structure is more complicated. The most challenging term is ż which requires a delicate random matrix estimate (Section 3.3).
Proof of Lemma 2.11: This follows from Definition 2.6 and the classical binomial formula. For the quartic binomial formula (2.32), we also used the self-adjointness of the convolution with V and the multiplier M T t . While this is not reflected in our notation, it is clear from Definition 2.6 that the multiplier M T t depends linearly on the interaction potential V . In the proof of the random matrix estimate (Proposition 3.7), we will need to further decompose M T t , both with respect to the interaction potential V and dyadic frequency blocks. We introduce the notation corresponding to this decomposition in the next definition.
Definition 2.13. We let M T t rV ; N 1 , N 2 s be the Fourier multiplier corresponding to the symbol In the next definition, we define our last renormalization of a stochastic object.
Definition 2.14. We define the correlation function on T 3 by We further define Here, τ y denotes the translation operator τ y f pxq " f px´yq.
The next lemma relates the multiplier and correlation function from Definition 2.13 and Definition 2.14, respectively.
is even, the symbol in (2.33) is the convolution of p V with (2.37). Thus, the inverse Fourier transform is given by´ÿ In Lemma 2.5, Proposition 2.9, Lemma 2.11, and Lemma 2.15, we have dealt with the algebraic structure of stochastic objects. We now move from algebraic aspects towards analytic estimates.
In the following lemmas, we show that several stochastic objects are well-defined and study their regularities.
Lemma 2.16 (Stochastic objects I). For every p ě 1, ǫ ą 0, and every 0 ă γ ă minpβ, 1q, we have that Furthermore, as t Ñ 8 and/or T Ñ 8, the stochastic objects : pW T t q 2 :, V˚: pW T t q 2 :, and : pV˚pW T t q 2 qW T t : converge in their respective spaces indicated by (2.38)-(2.40). Remark 2.17. The statement and proof of Lemma 2.16 are standard and the respective regularities can be deduced by simple "power-counting". Nevertheless, we present the proof to familiarize the reader with our set-up and as a warm-up for Lemma 2.20 below.
Proof. The first step in the proof of (2.38)-(2.40) is a reduction to an estimate in L 2 pΩˆT 3 q using Gaussian hypercontractivity. We provide the full details of this step for (2.38), but will omit similar details in the remaining estimates (2.39)-(2.40). Let N ě 1 and let q " qpǫq ě 1 be sufficiently large. By using Hölder's inequality in ω P Ω, it suffices to prove the estimates for p ě q. Using Bernstein's inequality and Minkowski's integral inequality, we obtain By Gaussian hypercontractivity (Lemma A.1), we obtain that Since the distribution of : pW T t q 2 : is translation invariant, the function x Þ Ñ } : pW T t q 2 : } L 2 ω pΩq is constant. We can then replace L q x pT 3 q by L 2 x pT 3 q and obtain In order to prove (2.38), it therefore remains to show uniformly in T, t ě 0 that Using Proposition 2.9, the orthogonality of the iterated stochastic integrals, and Itô's isometry, we have that n,n 1 ,n 2 PZ 3 n 1`n2 "n 1 xny 2`2ǫ xn 1 y 2 xn 2 y 2 ρ T t pn 1 q 2 ρ T t pn 2 q 2 À ÿ n 1 ,n 2 PZ 3 1 xn 1`n2 y 2`2ǫ xn 1 y 2 xn 2 y 2 À 1.
This completes the proof of (2.38). The estimate (2.39) can be deduced from the smoothing properties of V or by repeating the exact same argument. It remains to prove (2.40), which can be reduced using hypercontractivity (and the room in γ) to the estimate } :pV˚pW T t q 2 qW T t : } 2 L 2 ω H´3 2`γ x À 1.
Using Proposition 2.9, the orthogonality of the iterated stochastic integrals, and Itô's isometry, we have that Corollary 2.18. For every 0 ă γ ă minp1, βq and any n P Z 3 , we can control the Fourier coefficients of :pV˚pW T t q 2 qW T t : by Proof. Arguing as in the proof of Lemma 2.16, it suffices to prove that (2.43) ÿ n 1 ,n 2 ,n 3 PZ 3 : n 123 "n 1 xn 12 y 2β xn 1 y 2 xn 2 y 2 xn 3 y 2 À 1 xny 2γ .
Lemma 2.19 (Stochastic objects II). For any sufficiently small δ ą 0 and any N 1 , N 2 ě 1, it holds that Proof. Arguing as in the proof of (2.38) in Lemma 2.16, we have that It only remains to move the supremum in y P T 3 into the expectation. From a crude estimate, we have for all y, y 1 P T 3 that By Kolmogorov's continuity theorem (cf. [Str11, Theorem 4.3.2]), we obtain for any 0 ă α ă 1 that Combining this with (2.45) leads to the desired estimate.
The next lemma is similar to Lemma 2.16, but is concerned with more complicated stochastic objects. In order to shorten the argument, we will no longer use Itô's formula to express products of stochastic integrals. Instead, we will utilize the product formula for multiple stochastic integrals from [Nua06, Proposition 1. We emphasize that W T,r3s t contains the interaction potential V even though this is not reflected in our notation.

22
Lemma 2.20 (Stochastic objects III). For every p ě 1, ǫ ą 0, and every 0 ă γ ă minpβ, 1 2 q, we have that Remark 2.21. The analog of pV˚: pW T t q 2 :qW T,r3s t for the Φ 4 3 -model in [BG18] requires a further logarithmic renormalization. In our case, however, the additional smoothing from the interaction potential V eliminates the responsible logarithmic divergence.
Proof. We first prove (2.47), which is (by far) the easiest estimate. As in the proof of Lemma 2.16, we can use Gaussian hypercontractivity (Lemma A.1) to reduce (2.48) to the estimate The rest of the argument follows from Corollary 2.18 and a deterministic estimate. More precisely, it follows from }σ T s } L 2 s " 1 that For a small δ ą 0, we obtain from Corollary 2.18 (with γ replaced by γ`δ) that We now turn to the proof of (2.48). Using the same reductions based on Gaussian hypercontractivity as before, it suffices to prove that We first rewrite pV˚: pW T t q 2 :qpxqW T,r3s t pxq as a product of multiple stochastic integrals instead of iterated stochastic integrals. This allows us to use the product formula from Lemma A.4, which leads to a (relatively) simple expression. To simplify the notation below, we define the symmetrization of p V pn 1`n2 q by p V S pn 1 , n 2 , n 3 q " 1 6 ÿ πPS 3 p V pn πp1q`nπp2q q.
We now turn to the proof of (2.49). This stochastic object has a more complicated algebraic structure than the stochastic object in (2.48), but a similar analytic behaviour. From the definition of M T t , we obtain that Using the variable names m 1 , m 4 , m 5 P Z 3 instead of m 1 , m 2 , m 3 P Z 3 is convenient once we insert an expression for W T,r3s t . A minor modification of the derivation of (2.52) shows that where the symmetric function f p¨; t, m 1 q is given by f pt 1 , n 1 , t 2 , n 3 , t 3 , n 3 ; t, m 1 q " 1tn 123 " m 1 u 1 xn 123 y 2 p V S pn 1 , n 2 , n 3 q´ż t maxpt 1 ,t 2 ,t 3 q σ T s pn 123 q 2 ds¯1t0 ď t 1 , t 2 , t 3 ď tu.
In the construction of the drift measure (Section 4), we need a renormalization of px∇y´1 {2 W T t q n . The term x∇y´1 {2 W T t has regularity 0´and hence the n-th power is almost defined. While we could use iterated stochastic integrals to define the renormalized power, it is notationally convenient to use an equivalent definition through Hermite polynomials. This definition is also closer to the earlier literature in dispersive PDE. We recall that the Hermite polynomials tH n px, σ 2 qu ně0 are defined through the generating function e tx´1 2 σ 2 t 2 " 8 ÿ n"0 t n n! H n px, σ 2 q.
Definition 2.22. We define the renormalized n-th power by We list two basic properties of the renormalized power in the next lemma.
Since the proof is standard, we omit the details. For similar arguments, we refer the reader to [OT18].

Construction of the Gibbs measure
The goal of this section is to prove Theorem 1.3. The main ingredient is the Boué-Dupuis formula, which yields a variational formulation of the Laplace transform of r µ T . Our argument follows earlier work of Barashkov and Gubinelli [BG18], but the convolution inside the nonlinearity requires additional ingredients (see Section 3.2 and Section 3.3).
3.1. The variational problem, uniform bounds, and their consequences. Due to the singularity of the Gibbs measure for 0 ă β ă 1{2, which is the main statement in Theorem 1.5, the construction will require one final renormalization. We recall that λ ą 0 denotes the coupling constant in the nonlinearity and we let c T,λ be a real-valued constant which remains to be chosen.
For the rest of this section, we let ϕ : pr0, 8sˆRq Ñ R be a functional with at most linear growth. We denote the (non-renormalized) potential energy by V px´yqf pyq 2 f pxq 2 dx dy.
We denote the renormalized version of Vpf q by (3.2) :V T,λ pf q: where : pV˚f 2 qf 2 : is as in Definition 2.6. To further simplify the notation, we denote for any u : r0, 8qˆT 3 Ñ R the space-time L 2 -norm by x pT 3 q dt. With this notation, we can now state the main estimate of this section.
Here, Q T pW, ϕ, λq satisfies for all p ě 1 the estimate ErQ T pW, ϕ, λq p s À p 1, where the implicit constant is uniform in T ě 1.
The argument of ϕ in (3.4) is not regularized, that is, we are working with W instead of W T . This is important to obtain control over µ T , which is the push-foward of r µ T under W 8 .
Remark 3.2. This is a close analog of [BG18, Theorem 1]. Due to the smoothing effect of the interaction potential V , however, the shifted drift l T rus is simpler. In contrast to the Φ 4 3 -model, the difference l T puq´u does not depend on u. As is evident from the proof, we have that (3.7) Ψ T,ϕ λ pW, Irusq " ϕpW`Irusq`Ψ T,0 λ pW, Irusq. This observation will only be needed in Proposition 3.3 below.
We first record the following proposition, which is a direct consequence of Proposition 3.1 and the Boué-Dupuis formula.
Proposition 3.3. The measures r µ T satisfy the following properties: (i) The normalization constants Z T,λ satisfy Z T,λ " λ 1, i.e., they are bounded away from zero and infinity uniformly in T . (ii) If the functional ϕ : C 0 t C´1 {2´κ x pr0, 8sˆT 3 q Ñ R has at most linear growth, then Proof of Proposition 3.3: We first prove i. From the definition of µ T , we have that Using the Boué-Dupuis formula and Proposition 3.1, we have that From (3.6), we directly obtain that (3.8)´logpZ T,λ q ě´C λ .
By choosing u t def "´λJ T t :pV˚pW T t q 2 qW T t :, which is equivalent to requiring l T t rus " 0 and implies I T t rus " W T,r3s t , we obtain from Lemma 2.20 that (3.9)´logpZ T,λ q À λ 1`E P " VpλW T,r3s t q ı À λ 1.
By combining (3.8) and (3.9), we obtain that Z T,λ " λ 1. We now turn to ii, which controls the Laplace transform of r µ T . Using the Boué-Dupuis formula and Proposition 3.1, we obtain that The first summand logpZ T,λ q has already been controlled. The second summand can be controlled using exactly the same estimates. We finally prove iii. Let α, η ą 0 be sufficiently small depending on κ. Since the embedding x is compact (see (1.19) for the definition), it suffices to estimate the Laplace transform evaluated at While this is not a functional on C 0 t C´1 2´κ x , we can proceed using a minor modification of the previous estimates. Using Proposition 3.1 and (3.7), it suffices to prove Proof of Theorem 1.3: The tightness is included in Proposition 3.3. The existence of a subsequential weak limit follows from Prokhorov's theorem.
We also record the following consequence of the proof of Proposition 3.1, which will play an important role in Section 5. The proof of this result will be postponed until Section 3.4.
Proposition 3.1 is the most challenging part in the construction of the measure and the proof will be distributed over the remainder of this subsection.
3.2. Visan's estimate and the cubic terms. In the variational problem, the potential energy VpI T 8 rusq appears with a favorable sign. This is crucial to control the terms in :V T,λ pW T 8`I T 8 rusq: which are cubic in I T 8 rus and hence cannot be controlled by the quadratic terms }u} 2 L 2 or }l T puq} 2 L 2 . In the Φ 4 3 -model, the potential energy term }I T 8 rus} 4 L 4 is both stronger and easier to handle. While we cannot change the strength of VpI T 8 rusq, Lemma 3.5 solves the algebraic difficulties.
Due to the assumed lower-bound of V , we first note that V px´yqf pyq 2 f pxq 2 dx dy " Vpf q.
Unfortunately, the square of f is inside the integral operator x∇y´β 2 , which makes it difficult to use this estimate. The next lemma yields a much more useful lower bound on Vpf q.
Lemma 3.5 (Visan's estimate). Let 0 ă β ă 3 and f P C 8 pT 3 q. Then, it holds that (3.14) }x∇y´β 4 f } 4 L 4 x pT 3 q À Vpf q. This estimate is a minor modification of [Vis07, (5.17)] and we omit the details. We now turn to the primary application of Visan's estimate in this work.
Lemma 3.6 (Cubic estimate). For any small δ ą 0 and any 1`2δ 2 ă θ ď 1, it holds that Proof. We use a Littlewood-Paley decomposition to write We first estimate the contribution for N 3 Á M . We have that ÿ Due to (3.13), this contribution is acceptable. Next, we estimate the contribution of M À N 3 . We further decompose Then, the total contribution can be bounded using Hölder's inequality and Fourier support considerations by In the last line, it is simplest to first perform the sum in N 2 , then in N 1 , and finally in M .

3.3.
A random matrix estimate and the quadratic terms. In the proof of Proposition 3.1, we will encounter expressions such as This term no longer involves an explicit stochastic object, such as : pW T t q 2 : pxq, at a single point x P T 3 . By expanding the convolution, we can capture stochastic cancellations in terms of two spatial variables x P T 3 and y P T 3 , which has already been done in Lemma 2.19. The most natural way to capture stochastic cancellations in (3.16), however, is through random operator bounds. This is the object of the next lemma.
Proposition 3.7 (Random matrix estimate). Let γ ą maxp1´β, 1{2q and let 1 ď r ď 8. We define Then, we have for all 1 ď p ă 8 that Remark 3.8. Aside from Fourier support considerations, the proof below mainly proceeds in physical space. If r " 2, an alternative approach is to view Op T t pγ, 2q as the operator norm of a random matrix acting on the Fourier coefficients. Using a non-trivial amount of combinatorics, one can then bound Op T t pγ, 2q using the moment method (see also [DNY20, Proposition 2.8]). This alternative approach is closer to the methods in the literature on random dispersive equations but more complicated. The estimate for r ‰ 2, which is not needed in this paper, is useful in the study of the stochastic heat equation with Hartree nonlinearity.
Proof. Since this will be important in the proof, we now indicate the dependence of the multiplier on the interaction potential by writing M T t rV s. We use a Littlewood-Paley decomposition of W T t , f 1 , and f 2 . We then have that ż To control this sum, we first define a frequency-localized version of Op T t pγ, rq by Op T t pr; We emphasize the change from W γ,r x pT 3 q to L r x pT 3 q, which simplifies the notation below. By proving the estimate for a slightly smaller γ, (3.17) reduces to By using Lemma 2.16 and Lemma 2.19, it suffices to prove for a small δ ą 0 that By interpolation, we can further reduce to r " 1 or r " 8. Using the self-adjointness of the convolution with V and the multiplier M T t rV ; N 1 , N 2 s, it suffices to take r " 1. We now separate the cases N 1 " N 2 and N 1  N 2 .
Case 1: N 1  N 2 . This is the easier (but slightly tedious) case and it does not contain any probabilistic resonances. We note that M T t rV ; N 1 , N 2 s " 0 and hence we only need to control the convolution term. From Fourier support considerations, we also see that this term vanishes unless maxpK 1 , K 2 q Á maxpN 1 , N 2 q. While our conditions on f 1 and f 2 are not completely symmetric and we already used the self-adjointness to restrict to r " 1, we only treat the case K 1 Á K 2 . Since our proof only relies on Hölder's inequality and Young's inequality, the case K 1 À K 2 can be treated similarly. We now estimatěˇˇˇż We now split the last sum into the cases L ! N 2 and N 2 À L À K 1 . If L ! N 2 , we only obtain a non-zero contribution when N 2 " K 2 . Thus, the corresponding contribution is bounded by In the last line, we also used N 1 À K 1 and γ ą 1{2. If L Á N 2 , we simply estimate provided that γ ą maxp1´β, 1{2q. This completes the estimate in Case 1, i.e., N 1  N 2 .
Case 2: N 1 " N 2 . This is the more difficult case. Guided by the uncertainty principle, we decompose the interaction potential by writing V " P !N 1 V`P ÁN 1 V . Using the linearity of the multiplier M T t rV ; N 1 , N 2 s in V , we decompose ż We now split the proof into two subcases corresponding to the contributions of P !N 1 V and P ÁN 1 V .
Case 2.a: N 1 " N 2 , contribution of P !N 1 V . Similar as in Case 1, we do not rely on any cancellation between the convolution term and its renormalization. As a result, we estimates both terms separately. We first estimate the convolution term. Due to the convolution with P !N 1 V , we only obtain a non-zero contribution if N 1 " K 1 . Using N 1 " N 2 in the second inequality below, we obtain thaťˇˇż Second, we turn to the multiplier term. From the definition of M T t rP !N 1 V ; N 1 , N 2 s (see Definition 2.13), we see that the corresponding symbol is supported on frequencies |n| " N 1 . As a result, we only obtain a non-zero contribution if K 1 " K 2 " N 1 . Using Lemma 2.15, Hölder's inequality, and Young's inequality, we obtaiňˇˇż This completes the estimate of the contribution from P !N 1 V .
Case 2.b: N 1 " N 2 , contribution of P "N 1 V . The estimate for this case relies on the cancellation between the convolution and multiplier term, i.e., the renormalization. One important ingredient lies in the estimate }P "N 1 V } L 1 x À N´β 1 , which yields an important gain.
Using the translation operator τ y , we rewrite the convolution term as Using Lemma 2.15, we obtain that ż By recalling Definition 2.14 and combining both identities, we obtain that ż Using that :pτ y P N 1 W T t qP N 2 W T t :pxq is supported on frequencies À N 1 , we obtain thaťˇˇˇż This completes the estimate of the contribution from P "N 1 V and hence the proof of the proposition.
3.4. Proof of Proposition 3.1 and Corollary 3.4. In this subsection, we reap the benefits of our previous work and prove the main results of this section.
Proof of Proposition 3.1: In this proof, we treat Q T " Q T pW, ϕ, λq like an implicit constant and omit the dependence on W, ϕ, and λ. In particular, its precise definition may change throughout the proof.
From the quartic binomial formula (Lemma 2.11), it follows that ϕpW`Ipuqq`:V T,λ pW T We have grouped the terms according to their importance and their degree in I T 8 rus. The first line consists of the main terms, whereas the second and third line consist of less important terms of increasing degree in I T 8 rus. We will split them further in (3.22)-(3.25) below and introduce notation for the individual terms. Since :pV˚pW T 8 q 2 qW T 8 : has regularity´3 2`β´a nd I T 8 rus has regularity 1, the term λ ż T 3 :pV˚pW T 8 q 2 qW T 8 : I T 8 rus dx is potentially unbounded as T Ñ 8. As in [BG18], we absorb it into the quadratic term 1 2 }u} 2 L 2 . To this end, we want to remove the integral in I T 8 rus and obtain an expression in the drift u. From Itô's formula, it holds that I T t rus dp:pV˚pW T t q 2 qW T t :q.
The second term is a martingale (in the upper limit of integration) and therefore has expectation equal to zero. Together with the self-adjointness of J t , it follows that where l T rus is as in (3.5). To simplify the notation, we write With W T,r3s t as in (2.46), it follows that (3.21) I T t rws " I T t rus`λW T,r3s t .
Since E 0 does not depend on w, we can define The behavior of c T,λ as T Ñ 8 is irrelevant for the rest of the proof. However, it determines whether the Gibbs measure is singular or absolutely continuous with respect to the Gaussian free field (see Section 5). From the estimates (B.3) and (B.4), it is easy to see that Thus, it suffices to bound the terms in E 1 , E 2 , and E 3 pointwise by We treat the individual summands separately.
Contribution of E 1 : For the first summand in E 1 , the linear growth of ϕ, Sobolev embedding, and Lemma imply that For the second summand in (3.27) above, we used a minor modification of (2.47). For the second summand in E 1 , we have from Lemma 2.20 for any 0 ă γ ă minpβ, 1 2 q that λˇˇż T 3 pV˚:pW T 8 q 2 :qW T,r3s 8 I T 8 rws dxˇˇÀ λ}pV˚:pW T 8 q 2 :qW T,r3s For the third summand in E 1 , we have from lemma 2.20 that Contribution of E 2 : For the first summand in E 2 , the random matrix estimate (Proposition 3.7) implies for every 0 ă γ ă minpβ, 1 2 q that The second summand in E 2 can easily be controlled using Lemma 2.16. Contribution of E 3 : We estimate the first summand in E 3 by In the second factor, we bound the contribution of pV˚I T 8 rws 2 qI T 8 rws using Lemma 3.6. In contrast, the terms containing at least one factor of W T,r3s t can be controlled using Lemma 2.20, (B.3) and (B.4). This leads to The second summand in E 3 can be controlled using the same (or simpler) arguments.
Based on the proof of Proposition 3.1, we can also determine the behaviour as T Ñ 8 of the renormalization constants c T,λ . In particular, we obtain a short proof of Corollary 3.4.
Proof of Corollary 3.4: We let β ą 1{2 and choose any 1{2 ă γ ă minpβ, 1q. Using the definition of c T,λ in (3.26), it remains to control the expectation of E 0 , which is defined in (3.22). We treat the four terms in E 0 separately. The first term has zero expectation by Proposition 2.9. For the second term, we obtain from Corollary 2.18 that For the third term, we obtain from Lemma 2.16 and Lemma 2.20 thaťˇˇˇE P " ż T 3 pV˚:pW T 8 q 2 :qpW T,r3s For the fourth term, we obtain from Lemma 2.20 and the random matrix estimate (Proposition 3.7) thaťˇˇˇE This completes the argument.

The reference and drift measures
In this section, we prove Theorem 1.4, which contains information regarding the reference measures.
In this paper, we will use the reference measure ν 8 to prove the singularity of the Gibbs measure (Theorem 1.5). In the second part of this series, the reference measures will play an essential role in the probabilistic local well-posedness theory.
As in previous sections, we replace the truncation parameter N by T . Due to its central importance, let us provide an informal description of the terms in the representation of ν T . The first summand follows the distribution of the Gaussian free field, which has independent Fourier coefficients and regularity´1{2´. The second summand is a cubic Gaussian chaos with regularity 1{2`β´. Finally, the third summand is a Gaussian chaos of order n with regularity 5{2´. The statement of Theorem 1.4 is concerned with measures on C´1 {2´κ x pT 3 q. At this point, it should not be surprising to the reader that the proof mostly uses the lifted measures r µ T and r µ 8 . We will construct a reference measure Q u T for r µ T , and the reference measure ν T will be given by the pushforward of Q u T under W 8 . Since the main tool in the construction of Q u T is Girsanov's theorem, we call Q u T the drift measure. This section is a modification of the arguments in Barashkov and Gubinelli's paper [BG20]. Since l T rus in Proposition 3.1 is simpler than in the Φ 4 3 -model, however, we obtain slightly stronger results. For instance, we prove L q -bounds for the density D T in (4.23), whereas the analogous density in [BG20] only satisfies "local" L q -bounds. 4.1. Construction of the drift measure. We define the forcing term u t "´λJ t´: pV˚pW t´It rusq 2 qpW t´It rusq:¯`J t x∇y´1 2 :´x∇y´1 2`W t´It rus˘¯n : .
Using the binomial formulas (Lemma 2.11 and Lemma 2.23), we see that the integral equation has smooth coefficients on every compact subset of r0, 8qˆT 3 . As a result, it can be solved locally in time using standard ODE-theory. Due to the polynomial nonlinearity, however, we will need to rule out finite-time blowup. To this end, we introduce the blow-up time T exp ru T s P p0, 8s, which we will later show to be infinite almost surely with respect to both P and Q u T . The reason is that the highest-degree term in (4.2), which is given by´J T t x∇y´1 {2 px∇y´1 {2 I T t ru T sq n , is defocusing. We also introduce the stopping time .
From the integral equation, it is clear that u T t p¨q is supported in frequency space on the finite set tn P Z 3 : }n} À xtyu. As a result, the L 2 t L 2 x -norm can be used as a blow-up criterion and the solution u T t exists for all times t ď τ T,N , i.e., T exp ru T s ą τ T,N . We then define the truncated solution by (4.5) From the definition of τ T,N , it follows that x ds ď N. Thus, u T,N satisfies Novikov's condition and we can define the shifted probability measure Q u T,N by Here, the L 2 x -pairing in the integral We emphasize that the stochastic integral ş 8 0 ş T 3 u T,N s dB s only depends on the Brownian process B through the Gaussian process W . This is important in order to view Q u T,N as a measure on pr0, 8sˆT 3 q without changing the expression for the density. To make this direct dependence on W clear, we note that u T and hence τ T,N are functions of W T , and hence W , directly from their definition. By using the definition of u T , the self-adjointness of J T t , and dW T s " J T s dB s , we obtain that The expression on the right-hand side clearly is a function of W T and hence W . With a slight abuse of notation, we will keep writing the integral with respect to dB s , since it is more compact.
By Girsanov's theorem, the process To avoid confusion, let us remark on a technical detail. In the definition (4.8), the drift u T,N s is supported on frequencies |n| À xT y. The right-hand side of (4.8), however, does not contain a further frequency projection. In particular, W and hence W u T,N contain arbitrarily high frequencies.
This is related to the definition of the truncated Gibbs measure µ T , where the density only depends on frequencies À xT y, but whose samples contain arbitrarily high frequencies. Put differently, we regularize the interaction but not the samples themselves. To make notational matters even worse, while W u T,N contains all frequencies, we will often work with ρ T p∇qW u T,N , which only contains frequencies À xT y. Similar as in Section 2.1, we define the truncated process W T,u T,N t by (4.10) Due to the integral equation (4.2), we have that (4.11) u T,N t " 1tt ď τ T,N u "´λ J T t´: pV˚pW T,u T,N t q 2 qW T,u T,N t :¯`J T t x∇y´1 2 :`x∇y´1 2 W T,u T,N t˘n : ı .
We intend to use Q u T,N (and the limit as N Ñ 8) as a reference measure for r µ T . Due to (4.9), the law of W T,u T,N t under Q u T,N does not depend on N . In our estimates of u T,N t through the integral equation, it is therefore natural to view W T,u T,N t as given. Under this perspective, the right-hand side of (4.11) no longer depends on u T and yields an explicit expression for u T . For comparison, the corresponding equation in the Φ 4 3 -model (cf. [BG20,(14)]) is a linear integral equation. We now start to estimate the drift u T .
Lemma 4.1. For all 1 ď M ď N , all S ě 0, and all 0 ă γ ă minp1, βq, it holds that In particular, it holds that Proof. We recall from the definition of the drift measure that Law Q u T,N pW u T,N q " Law P pW q and Law Q u T,N pW T,u T,N q " Law P pW T q As a result, we obtain that For the first summand, we obtain from the definition of J T s and Lemma 2.16 that For the second summand, we obtain from Lemma 2.23 that This yields the desired estimate.

41
The proof of Lemma 4.2 is easier than its counterpart [BG20,(16)] in the Φ 4 3 -model, which requires a Gronwall argument. The second estimate (4.14) is needed for technical reasons related to tightness, and we encourage the reader to ignore it.
Proof. The argument is similar to the proof of Lemma 4.1. From the definition of u T,M and u T,N , we have that (4.15) u T,M s " 1ts ď τ T,M uu T,N s . Thus, we obtain that Using the integral equation (4.2) again, we obtain that Using that Law Q u T,N pW u T,N q " Law P pW q, we obtain from Lemma 2.20 and Lemma 2.23 that This completes the proof of the first estimate. The second estimate (4.14) follows from a minor modification of the proof. To simplify the notation, we set Apsq def " }J s J T s :pV˚pW T,u T,N s q 2 qW T,u T,N s : } L 8 x`} J s J T s x∇y´1 2 :`x∇y´1 2 W T,u T,N s˘n : } L 8 x For any K ě 1, we have from a similar argument as in (4.17) that sup 0ďt 1 ďt : Proceeding as in the first estimate, this implies that The desired estimate of the C α,η t C 0 x -norm then follows by summing over dyadic scales and using a telescoping series if the times are not comparable.
In Lemma 4.1 and Lemma 4.2, we controlled the process u T with respect to the measures Q u T,N . Unfortunately, the proof of Proposition 4.4 below also requires the absence of finite-time blowup for u T with respect P. This is the subject of the next lemma.
Lemma 4.3. For any T ě 1, it holds that T exp ru T s " 8 P-almost surely.
The proof of the analogue for the Φ 4 3 -model (cf. [BG20, Lemma 5]) extends verbatim to our situation and we omit the minor modifications. To ease the reader's mind, let us briefly explain why the same argument applies here. In most of this section, the most important term in the integral equation (4.2) is the first summand. It has the lowest regularity and is closely tied to the interactions in the Hamiltonian. The result of Lemma 4.3, however, is essentially a soft statement. If we fix a time S ě 1 and only want to rule out T exp ru T s ď S, the low regularity is inessential and only leads to a loss in powers of S. The main term is then given by the (auxiliarly) second summand, which is defocusing and exactly the same as in the Φ 4 3 -model. The next proposition eliminates the stopping time from our drift measures.
Proposition 4.4. The family of measures pQ u T,N q T,N ě0 is tight on C 0 t C´1 {2´κ x pr0, 8sˆT 3 q. For any fixed T ě 0, the sequence of measures pQ u T,N q N ě0 weakly converges to a measure Q u T as N Ñ 8. For any S ě 0, the limiting measure Q u T satisfies Our argument differs from the proof of [BG20, Lemma 7], which is the analog for the Φ 4 3 -model. The argument in [BG20] relies on Kolmogorov's extension theorem, whereas we rely on tightness and Prokhorov's theorem. This is important in the proof of Corollary 4.5 below, since the measures Q u T are not (completely) consistent. We also believe that this clarifies the mode of convergence. Before we begin with the proof, we state the following corollary.
Corollary 4.5. The measures Q u T weakly convergence to a measure Q u 8 on C 0 t C´1 {2´κ x pr0, 8sˆT 3 q as T Ñ 8. For any S ě 0, it holds that where u s is as in (4.2) with W T t and I T t replaced by W t and I t . Proof of Proposition 4.4: We first prove that the family of measures pQ u T,N q T,N ě0 , viewed as measures for W , are tight on C 0 t C´1 {2´κ x pr0, 8sˆT 3 q. From (4.8), we have that (4.20) W " W u T,N`I ru T,N s.
Since the law of W u T,N under Q u T,N agrees with the law of W under P, an application of Kolmogorov's continuity theorem (cf. [Str11, Theorem 4.3.2]) yields for any p ě 1, 0 ă α ă 1 2 , and 0 ă η ă κ{2 that Together with Lemma 4.2, this implies Since the embedding C α,η t C´p is compact, this implies the tightness of the family of measures pQ u T,N q T,N ě0 .
By Prokhorov's theorem, a subsequence of pQ u T,N q N weakly converges to a measure Q u T . Once we proved (4.18), this can be upgraded to weak convergence of the full sequence, since (4.18) uniquely identifies the limit. With a slight abuse of notation, we therefore ignore this distinction between a subsequence and the full sequence. Let S ě 0 and let f : pr0, SsˆT 3 q Ñ R be continuous, bounded, and nonnegative. We write f pW q for f pW | r0,Ss q. Using the weak convergence of Q u T,N to Q u T , we have that T,N r1tτ T,N ě Suf pW qs`E Q u T,N r1tτ T,N ă Suf pW qs¯. Using Lemma 4.2, the second term is controlled by which converges to zero as N Ñ 8. Together with the definition of Q u T,N and the martingale property of the Girsanov density, this implies Using monotone convergence and Lemma 4.3, we obtain Proof of Corollary 4.5: Due to Proposition 4.4, the family of measures pQ u T q T ě0 is tight. By Prokhorov's theorem, it follows that a subsequence weakly converges to a measure Q u 8 . Once (4.19) is proven, it uniquely identifies the limit Q u 8 . With a slight abuse of notation, we therefore assume as before that the whole sequence Q u T converges weakly to Q u 8 . Since W T t " W t and I T t " I t for all 0 ď t ď T {4 (by our choice of ρ), it follows from the integral equation (4.2) that u T s " u s for all 0 ď s ď T {4. Using (4.18), it follows for all S ď T {4 that The corresponding identity (4.19) for Q u 8 then follows by taking T Ñ 8. Corollary 4.6. For any T ě 1, S ě 1, and any 0 ă γ ă minpβ, 1{2q, the measure Q u T satisfies the two estimates The corollary directly follows from Lemma 4.1, Lemma 4.2, and Proposition 4.4.

4.2.
Absolutely continuity with respect to the drift measure. We recall the definition of the measure r µ T from (2.10), which states that (4.22) dr µ T dP " 1 Z T,λ exp´´:V T,λ pW T 8 q:¯.
Using Proposition 4.4, we obtain that Since dB t " dB u T t`u T t dt, we also obtain that (4.24) Proposition 4.7 (L q -bounds). If n P N in the definition of u T is odd and sufficiently large, there exists a q ą 1 such that Remark 4.8. We point out two important differences between Proposition 4.7 and the corresponding result for the Φ 4 3 -model in [BG20, Lemma 9]. The first difference is a consequence of working with r µ T instead of s µ T as described in Section 2.1. Barashkov and Gubinelli define and bound the density D T with respect to the same measure Q u 8 for all T ě 1. In contrast, our density is defined with respect to Q u T and we make no statements about the behaviour of D T with respect to Q u S for any S ‰ T . Since the increments of T Þ Ñ ρ T p∇qW 8 are not independent, such a statement would be especially difficult if S and T are close. The second difference is a result of the smoothing effect of the interaction potential V . While the Hartree-nonlinearity allows us to prove the full L q -bound (4.25), the corresponding result in the Φ 4 3 -model requires the localizing factor expp´}W 8 } n The rest of this subsection is dedicated to the proof of the L q -bounds (Proposition 4.7). Since we intend to apply the Boué-Dupuis formula to bound the density D T in L q pQ u T q, we first study the effect of shifts in B u T on the integral equation (4.2). For any w P H a , we define . We also define h T,w " w`u T,w . We further decompose r T,w s " r r T,w s`J T s x∇y´1 2 :px∇y´1 2 pW T,u T s`I T s rwsqq n : . We encourage the reader to continue to ignore this detail on first reading. Before we begin the main argument, we prove the following auxiliary lemma. Lemma 4.9 (Estimate of r r T,w t ). Let ǫ, δ ą 0 be small absolute constants and let n ě npδ, βq be sufficiently large. Then, we have for all t ě 0 that (4.27) xty 1`δ }r r T,w t } 2 L 2 x À n,δ,β,λ C ǫ Q t pW T,u T q`ǫ´}I T t rws} n`1 Remark 4.10. We emphasize that the implicit constant does not depend on ǫ. In the application of Lemma 4.9, we will choose ǫ ą 0 sufficiently small depending on δ, n, β, λ.
Proof. In the following argument, the implicit constants are allowed to depend on n, δ, β, and λ but not on ǫ. We estimate the five terms in r r T,w Proof of Proposition 4.7: The proof splits into two steps.
Step 1: Formulation as a variational problem. In order to prove the desired estimate (4.25), it suffices to obtain a lower bound on´log E Q u T rD q T s. Using the Boué-Dupuis formula, we obtaiń is a martingale, its expectation vanishes. We now insert the formula u T,w " h T,w´w into the formula above, and obtain that q´:V T,λ pW T,u T

8`I
T 8 rh T,w sq:`1 2 q´:V T,λ pW T,u T

8`I
T 8 rh T,w sq:`1 2 Since we want to obtain a lower bound, the most dangerous term in the expression above iś q´1 2 ş 8 0 }w t } 2 L 2 dt. Using our previous information about the variational problem (Proposition 3.1 and Proposition 3.3) and the nonnegativity of VpI T 8 rh T,w sq, we obtain that Recalling the definition of l T t ph T,w q from Proposition 3.1, we obtain that " pr T,w t`w t q. Together with our previous estimate, this leads tó By choosing q sufficiently close to one, it only remains to establish This bound is proven via a Gronwall-type argument.
Step 2: Gronwall-type argument. This step crucially relies on the smoother term in the definition of the drift (4.2). We essentially follow the proof of [BG20, Lemma 11]. As in [BG20], we introduce the auxiliary process (4.30) Aux s pW T,u T , wq " n ÿ i"0ˆn i˙x ∇y´1 2 J T s´: px∇y´1 2 W T,u T s q i : px∇y´1 2 I T s rwsq n´i¯.
With this notation, r T,w " r r T,w`A uxpW T,u T , wq. We then expand (4.31) w 2 s " 2pw s`r T,w s q 2´4 w s r T,w s´2 pr T,w s q 2´w2 s " 2pw s`r T,w s q 2´4 w s r r T,w s´2 pr w s q 2´w2 s´4 Aux s pW T,u T , wq. Using Itô's integration by parts formula, we have for all s ď t that Due to the martingale property, the second summand has zero expectation. After setting (4.32) Aux t pW T,u T , wq def " n ÿ i"0 1 n`1´iˆn i˙ż T 3 :px∇y´1 2 W T,u T t q i : px∇y´1 2 I T t rwsq n`1´i dx, we obtain that We perform the Gronwall-type argument based on the quantity Φptq, which is defined by By [BG20, Lemma 12] and (4.33), we have that From Lemma 4.9, we obtain for ǫ, δ ą 0 that Φpsq.
By choosing ǫ ą 0 sufficiently small depending on δ, this implies the desired estimate.
4.3. The reference measure. Using our construction of the drift measures Q u T , we now provide a short proof of Theorem 1.4. As in the rest of this section, we use the truncation parameter T .
Proof of Theorem 1.4: For any 1 ď T ď 8, we define the reference measure ν T as By using the L q -bound (Proposition 4.7), we have that for all Borel sets This proves the first part of Theorem 1.4. Regarding the representation of ν T , which forms the second part of Theorem 1.4, we have that This completes the proof.

Singularity
In this section, we prove Theorem 1.5. The majority of this section deals with the singularity for 0 ă β ă 1{2. The absolute continuity for β ą 1{2 will be deduced from Corollary 3.4 and requires no new ingredients. Theorem 1.5 is important for the motivation of this series of papers, since we provide the first proof of invariance for a Gibbs measure which is singular with respect to the corresponding Gaussian free field. The methods of this section, however, will not be used in the rest of this two-paper series. We prove the singularity of the Gibbs measure µ 8 and the Gaussian free field g through the explicit event in Proposition 5.1.
Here, g is the Gaussian free field, µ 8 is the Gibbs measure, and φ P C´1 2´κ x pT 3 q denotes the random element.
Remark 5.2. In the statement of the proposition, the reader may wish to replace φ by W 8 , g by P, and µ 8 by r µ 8 . We choose the notation φ to emphasize that this is a property of g and µ 8 only and does not rely on the stochastic control perspective. Of course, the stochastic control perspective is heavily used in the proof.
To simplify the notation, we define :pV˚pW S s q 2 qpW S s q 2 : . We note that the dependence on the interaction potential V is not reflected in this notation. We first study the behaviour of the integral of W S,4 8 with respect to P. This is the easier part of the proof and the statement (5.1) follows from the following lemma. Proof. Let k ě 1 remain to be chosen. We define the auxiliary function (5.16) We will now show that which implies the desired result. We could switch from pW u , Q u 8 q to pW, Pq, which we have done several times above. Since the A S,j in (5.8)-(5.10) are defined in terms of W u , however, we decided not to change the measure. We define A j s similar as in (5.8)-(5.10), but with J S s replaced by J s , I S s replaced by I s , and W S,u replaced by W u . Since all our estimates for A S,j were uniform in S ě 1, they also hold for A j .
Using the Boué-Dupuis formula (Theorem 2.1) and the cubic binomial formula, we have that The main term is given by (5.18). By Lemma 5.5, we see that (5.18) converges to infinity as S Ñ 8. Thus, it remains to obtain a lower bound on the variational problem in (5.19)-(5.21). The terms in (5.19) are coercive and help with the lower bound. In contrast, the terms in (5.20) and (5.21) are viewed as errors and will be estimated in absolute value. Regarding (5.19), we briefly note that In the estimates below, we will often use that A S,j s rvs " 0 for all s " S. We begin with the first term in (5.20). We have thaťˇˇ1 x`} I s rvs} 2 In the last line, we also Lemma B.4. Since S Ñ 8, this contribution can be absorbed in the coercive term (5.22). The estimate of the second summand in (5.20) is exactly the same.
Regarding the error terms in (5.21), we have thaťˇˇ1 x`} A j s rvs} 2 L 2 x¯d x ds.
The right-hand side can now be controlled using the same (or simpler) estimates as for the second summand in (5.22). This completes the proof.
Essentially the same estimates as in the previous proof can also be used to control the minor terms in (5.6) and (5.7). We record them in the following lemma.
Next, we prove (5.26). Using Itô's isometry and (5.29), we have that Finally, we turn to (5.28), which is the most regular term. We first recall the algebraic identity J S s W S,3 s " J S W S,u,3 s`ř where the remainder RpW u , uq contains the terms from (5.6) and (5.7) with an additional S´1`2 β`δ . By Lemma 5.8, there exists a deterministic sequence S m such that the first summand in (5.32) converges to´8 almost surely with respect to Q u 8 . Since 0 ă β ă 1{2, we have that 1´2β ą max´1 2´β , 1´3β, 1 2´2 β, 0¯.
Using Lemma 5.9, this implies that the remainder R S pW u , uq converges to zero in L 1 pQ u 8 q. By passing to a subsequence if necessary, we can assume that R Sm pW u , uq converges to zero almost surely with respect to Q u 8 . Using (5.32), this implies that lim mÑ8 1 S 1´2β´δ m ż T 3 W Sm,4 8 dx "´8 Q u 8 -a.s.
Equipped with Corollary 3.4 and Proposition 5.1, we now provide a short proof of Theorem 1.5.
Proof of Theorem 1.5: If 0 ă β ă 1{2, then the mutual singularity of the Gibbs measure µ 8 and the Gaussian free field g directly follows from Proposition 5.1. If β ą 1{2, we claim that for all p ě 1 that (5.33) dµ T dg P L p pgq with uniform bounds in T ě 1. Since a subsequence of µ T converges weakly to µ 8 , this implies the absolute continuity µ 8 ! g. In order to prove the claim, we recall that µ T " pW 8 q # r µ T and g " pW 8 q # P. Thus, it suffices to bound the density dr µ T {dP in L p pPq for all p ě 1. From the definition of r µ T (Definition 2.3) and the definition of the renormalized potential energy in (3.2), we have that dr µ T dP¯p " 1 Z T,λ˘p exp´´p :V T,λ pW T 8 q:" 1 Z T,λ˘p exp´´λ p 4 ż T 3 :pV˚pW T 8 q 2 qpW T 8 q 2 : dx´pc T,λ" Z T pλ Z T,λ˘p exppc T pλ´p c T,λ q¨1 Z T,pλ exp´´:V T,pλ pW T 8 q:¯.
The first two factors are uniformly bounded in T by Proposition 3.3 and Corollary 3.4. The last factor is uniformly bounded in L 1 pPq for all T ě 1 since we only replaced the coupling constant λ by pλ. This completes the proof of the claim (5.33).