Gaussian Fluctuations for the stochastic Burgers equation in dimension $d\geq 2$

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a $d$-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985]. In both the critical $d=2$ and super-critical $d\geq 3$ cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For $d\ge3$ the scaling adopted is the classical diffusive one, while in $d=2$ it is the weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way.


Introduction
The study of (singular) Stochastic Partial Differential Equations (SPDEs) has known tremendous advances in recent years. Thanks to the groundbreaking theory of regularity structures [Hai14], paracontrolled calculus [GIP15] or renormalization group techniques [Kup14], a local solution theory can be shown to hold for a large family of SPDEs in the so-called sub-critical dimensions (see [Hai14] for the notion of sub-criticality in this context). In contrast, SPDEs in the critical and super-critical dimensions are, from a mathematical viewpoint, still largely unexplored and poorly understood. The aim of the present paper is to establish a framework which allows to rigorously determine the large-scale (Gaussian) fluctuations for a class of SPDEs at and above criticality under (suitable) diffusive scaling.
To present the ideas and the techniques in our analysis, we will focus on a specific example, the Stochastic Burgers Equation (SBE), which, in any dimension, can (formally) be written as ∂ t η = 1 2 ∆η + w · ∇η 2 + div ξ (1.1)

Introduction
The most important result is that of [Yau04], which states that two-dimensional ASEP is logarithmically superdiffusive, meaning that the diffusivity D(t) satisfies D(t) ∼ (log t) 2/3 for t large.1 As mentioned above, our focus is on the critical and super-critical dimensions, i.e. we will be studying (1.1) (in the continuum) in d ≥ 2. In both cases, we will prove that the large-scale behaviour is Gaussian and that the fluctuations evolve as a SHE with renormalized coefficients (see Theorem 1.3), but while for d ≥ 3 the scaling adopted is the classical diffusive scaling, for d = 2 we will consider the so-called weak coupling scaling. This corresponds to tuning down the strength of the nonlinearity (morally, the "norm" of the vector w) at a suitable rate together with the scale parameter, see (1.6). The tuning is chosen so that we can "tame" the growth caused by the non-linear term without destroying its large-scale effects (see e.g. [CD20, CSZ20, Gu20, CES21, CETar] for a similar choice of scaling in the critical dimension for different equations).
Let us mention that the study of large-scale Gaussian fluctuations of space-time random fields with local non-linear random dynamics is not restricted the context of SPDEs but is a classical topic in probability theory and mathematical physics. From the more general point of view of theoretical physics, large-scale Gaussian fluctuations are expected for equilibrium diffusive systems above their critical dimension and this belief is informally explained via (non-rigorous) Renormalization Group (RG) arguments and effective dynamical field theories.
A variety of tools have been used to mathematically justify these heuristics in the static setting, i.e. for the equilibrium measure of local non-linear systems. RG methods have been applied to derive the scaling limit of unbounded spin systems (and other statistical mechanical models) at and above their critical dimension d ≥ 4 in [BBS15, BBS14, BBS19]2. The related problem of triviality of Φ 4 d theories for d > 4 was treated in [Aiz82,Frö82,FFS92] using apriori bounds on correlation functions, an approach which ultimately lead to a proof in the more challenging critical dimension d = 4, see [AD21]. Another class of equilibrium models for which scaling limits have been successfully determined are equilibrium interface models, so called ∇φ models in all dimensions [BY90,NS97,FS97,GOS01], for which the Helffer-Sjöstrand representation of correlations together with homogeneization arguments was exploited. Analogous results have been proven, with very different methods, for discrete ∇φ models in dimension d = 2 (in particular dimer models, both integrable ones [Ken01] and their non-integrable perturbations [GMT20]).
On the other hand, establishing large-scale Gaussian fluctuations for dynamical problems has proven to be challenging and we are not aware of general results in this direction. The major issue seems to stem from the fact that there is currently no framework which allows to naturally identify the law of the limit process. A lot of Introduction attention has been recently devoted to the above mentioned KPZ equation in high dimension, whose expression in its most general form is as (1.1) but with nonlinearity given by ⟨∇η, Q∇η⟩, for Q a d×d deterministic symmetric matrix, and a real valued space-time white noise in place of div ξ. Most of the works focus on the case in which Q is given by the identity matrix and the specific form of the resulting equation plays a crucial role. Indeed, one can use the Cole-Hopf transformation to linearise the SPDE and turn it into a linear multiplicative stochastic heat equation or the associated directed-polymer models. These latter models possess an "explicit" representation (via, e.g., Feynman-Kac formula) and one can then leverage Malliavin calculus tools to derive the scaling limit (see [DGRZ21,GRZ18b] for the multiplicative stochastic heat equation and [GRZ18a, CCM20, LZ22, CNN20] for KPZ). In this context, let us mention the recent striking results on the fluctuations of KPZ at the critical dimension d = 2 in the weak coupling scaling (or the intermediate disorder regime) starting with [CD20, CSZ20, Gu20] (for KPZ) and culminating with the characterisation of the stochastic heat flow [CSZ23]. An alternative approach is that used in [MU18], which is based on re-expressing the SPDE as a functional integral via the Martin-Siggia-Rose formalism (essentially a Girsanov transformation, which again is only possible in view of the specific form of the equation) and then leveraging constructive quantum field theory techniques and RG ideas to control the large scale fluctuations in the regime of small non-linearity.
In the critical dimension d = 2, for Q diagonal and with diagonal entries 1 and −1 (so-called Anisotropic KPZ or AKPZ), progress has been made in [CET23], where it is shown that the equation is logarithmically superdiffusive, and in [CETar], where the large-scale behaviour in the weak coupling scaling is determined. In both articles, the tools adopted are closer to those in the present paper and we will comment more on them below.
Let us also mention recent work in extending the homogeneization methods (originated in the work of Naddaf and Spencer [NS97]) to dynamical problems [AD22,CDS22].
One of the main advantages of the approach taken in the present paper is that it moves away from the usual technique of Cole-Hopf transformation, which is not applicable, towards a set of tools which on the one hand makes a link with interacting particle systems, i.e. via the martingale problem and the associated infinitedimensional generator, and on the other has better chances to apply more generally.

Introduction
Before turning to the core of the proof, in the next section we will precisely state our main result and outline the ideas behind our arguments.

The equation and main result
As written (1.1) is meaningless as the noise is too irregular for the non-linearity to be well-defined. Nonetheless, as we are interested in the large-scale behaviour, we regularise it, so to have a well-defined field at the microscopic level, and our goal then becomes to control its fluctuations while zooming out at the correct scale. We choose the regularisation in such a way to retain a fundamental property of the solution, namely its (formal) invariant measure. This amounts to smoothening the quadratic term via a Fourier cut-off as follows where for a > 0, Π a acts in Fourier space as Π a η(k) = η(k)1 |k|≤a .
(1.3) Remark 1.1 In principle, the results below can be shown to hold using the same techniques of the present paper, even if instead of the Fourier cut-off we used a more general mollifier. That said, to simplify some (minor but annoying) technical points, we will stick to Π 1 and further assume that • for d ≥ 3, 1/ε ∈ N + 1/2 and that the norm in (1.3) is the sup-norm | · | ∞ , • for d = 2, the norm in (1.3) is the Euclidean norm | · | (no further assumption on ε is made).
We refer the reader to Remark 2.6 for some additional (technical) advantages of our choice.
In order to avoid technicalities related to infinite volume issues, we restrict (1.2) to the d-dimensional torus T d ε of side-length 2π/ε. Here, ε is the "scale" parameter which will later be sent to 0. Note that via the diffusive rescaling where now the spatial variable takes values in the d-dimensional torus T d def = T d 1 of side-length 2π, and ξ is a space-time white noise on R + × T d . The coefficient λ ε depends on the dimension and is defined as (1.6) While in the super-critical case d ≥ 3, λ ε is directly determined by (1.4), for d = 2, (1.1) is formally scale invariant under the diffusive scaling (another reason why such dimension is critical) so that one would have λ ε = 1. The choice made in (1.6) corresponds to the so-called weak coupling scaling alluded to earlier and we will comment more on it after we state the main result. Before doing so, let us give the notion of solution for (1.5) we will be working with.
Definition 1. 2 We say that η ε is a stationary solution to (1.5) with coupling constant λ ε if η ε solves (1.5) and its initial condition is η(0, ·) def = µ, for µ a spatial white noise on T d , i.e. a mean zero Gaussian distribution on T d such that where ⟨·, ·⟩ L 2 is the usual scalar product in L 2 (T d ) and L 2 0 (T d ) is the space of zero-average square integrable functions on T d . We will denote by P the law of µ and by E the corresponding expectation.
The reason why the solution η ε of (1.5) started from a spatial white noise is called stationary is that P is its invariant measure and, as we will prove in Lemma 2.1 below, this holds irrespective of the choice of w, ε. Instead, the law P (with corresponding expectation E) of a stationary solution to (1.5) clearly depends on w, ε.
For T > 0, let us denote by C([0, T ], S ′ (T d )) the space of continuous functions with values in the space of tempered distributions S ′ (T d ) and by F the Fourier transform on S ′ (T d ) (see (1.13) for a definition). We are now ready to state the main result of the paper.

Introduction
In the case d = 2, D SHE is explicit and given by the formula |w| being the Euclidean norm of w.
Before proceeding to the proof of the above theorem, let us make a few comments on its statement and on the choice of λ ε in (1.6). Notice that in any dimension d ≥ 2, the constant D SHE appearing in the limiting equation is strictly positive. Hence, despite the presence of the vanishing factor λ ε in front of the nonlinearity, not only the quadratic term does not go to 0 but actually produces a new "Laplacian" and a new noise. Further, this new Laplacian (and noise), in a sense, "feels" the small-scale behaviour of the field η in (1.2) as it depends on the vector w, which in turn describes its microscopic dynamics. A similar phenomenon has already been observed for other critical and super critical SPDEs, even though we are not aware of examples in which a Laplacian of the form above was derived.
We have already pointed out that in d = 2 (1.1) is formally scale invariant. That said, the above mentioned result in [Yau04] for ASEP suggests that also (1.2) is logarithmically superdiffusive and the additional diffusivity can only come from the non-linear term. What the previous theorem shows is that the choice in (1.6) guarantees that the nonlinearity ultimately gives a non-vanishing order 1 contribution. What might appear puzzling is that, by translating the result of Yau to (1.5), the diffusivity D ε (t) of the scaled process grows as | log ε| 2/3 so that one might be led to think that λ ε should be chosen accordingly. While this is not the case, the expression in (1.9) formally implies the result of Yau, as can be seen by taking w = w ε such that |w ε | def = λ −1 ε . Furthermore, a similar scaling has been considered in the critical dimension d = 2 in the context of the KPZ equation [CD20, CSZ20, Gu20, CSZ23], of the Anisotropic KPZ equation [CES21,CETar] and of the stochastic Navier-Stokes equation [CK23]. What is interesting is that, even though these latter examples in principle have different large scale diffusivity, i.e., D(t) ∼ t β for some β > 0 [FT90] for KPZ, D(t) ∼ difficulty is to obtain operator estimates similar to those of Lemma 2.4, which though seem highly non-trivial.
• Prove the analog of Theorem 1.3 (for d = 2) for ASEP on Z 2 in the same limit of weak asymmetry. • In dimension d = 2, defineη ε (x, t) = ε −1 η(x/ε, t/(ε 2 τ (ε))) (to be compared with (1.4)) and find the right correction τ (ε) to the diffusive scaling so that a non-trivial scaling limit as ε → 0 exists. This corresponds to a strong coupling regime. On the basis of [Yau04], the natural guess is τ (ε) = | log ε| 2/3 but the identification of the limit is hard as the regularising properties of the Laplacian vanish.
More challenging will be to move towards models for which the invariant measure is non-Gaussian or non-explicit. One guiding heuristic could be that the limiting Gaussian fluctuations and associated Gaussian stationary distribution could still provide a good setting for the analysis.

Idea of the proof
The idea of the proof finds its roots in the approach detailed in [KLO12] in the context of interacting particle systems. To summarise it and see how it translates to the present context, let us consider the weak formulation of (1.5) started from a spatial white noise µ, which, for a given test function φ, reads where N ε φ stands for the nonlinearity tested against φ and M is the martingale associated to the space-time white noise ξ. Now, upon assuming the sequence {η ε } ε to be tight in the space C([0, T ]S ′ (T d )), we see that all the terms in the above expression converge (at least along subsequences) but we have no information concerning the nonlinearity. For Theorem 1.3 to be true, we need this latter term to produce both a dissipation term of the form of D SHE L w 0 η, where L w 0 def = 1 2 (w · ∇) 2 , and a fluctuation part which should encode the additional noise in (1.8). In other words, the problem is to identify a fluctuation-dissipation relation [LY97a], i.e. to determine D SHE > 0 and W ε such that where L ε is the generator of η ε . Indeed, since by Dynkin's formula, there exists a martingale M ε ( W ε ) such that given (1.11), we could rewrite the nonlinear term in (1.10) as where o(1) is a vanishing summand which contains the boundary terms. The advantage of the above is that we have expressed the nonlinearity in terms of a drift which captures the additional diffusivity and a martingale part which instead econdes the extra noise. At this point, tightness would ensure convergence of the former and we would be left to prove that the sequence M ε ( W ε ) converges to a martingale with the correct quadratic variation.
The (hardest) problem is clearly the derivation of the fluctuation-dissipation relation and this is the point from which our analysis departs from that in [KLO12]. Even though, as we will see, it is enough to determine (1.11) approximately (i.e. that the difference of left and right hand side is small in a suitable sense), in the equation there are two unknowns, D SHE L w 0 which is not apriori given to us, and W ε which, even if we were given the previous, is the solution of an infinite dimensional equation. To separate the two issues, we introduce a suitable truncation which removes the second summand from the right hand side of (1.11), so that we can first solve for W ε and then project back and determine D SHE L w 0 . This is done in d ≥ 3 and d = 2 in very different ways. In the former case, we will control W ε similarly to [LY97a], as we can show it satisfies a graded sector condition in the spirit of [KLO12, Section 2.7.4]. Nonetheless, their functional analytic approach does not apply in our setting and in particular, for the identification of D SHE and L w 0 , we devise a new method which is detailed in Section 2.4. For d = 2 instead, we introduce a novel ansatz which is based on the idea that at large scales the generator of η ε should approximate a modulated version of the generator of (1.8). This ansatz simplifies dramatically the (iterative) analysis performed in [CETar] and is heavily based on the Replacement Lemma in Section 2.1.1.

Organisation of the article
The rest of this work is organised as follows. Below we introduce notations, conventions, function spaces and elements of Wiener space analysis which will be used throughout. Section 2 is the bulk of the paper. In Section 2.1, after recalling basic properties of (1.5) and its generator, we discuss the one of the main technical tools, i.e. the Replacement Lemma for d = 2. In Section 2.2 we determine apriori estimates on the solution to the generator equation and in the next two we identify D SHE and L w 0 , so to obtain a refined version of the fluctuation-dissipation relation (1.11). Section 3 is devoted to the proof of Theorem 1.3. At last, Appendix A contains some technical steps necessary in the proof of the Replacement Lemma.

Notations, function spaces and Wiener space analysis
We let T d be the d-dimensional torus of side length 2π. We denote by {e k } k∈Z d the Fourier basis defined via e k (x) and, for k ∈ Z d is given by the formula (1.13) so that in particular (1.14) Let S(T d ) be the space of smooth functions on T d and S ′ (T d ) the space of real-valued distributions given by the dual of S(T d ). For any η ∈ S ′ (T d ) and k ∈ Z d , we will denote its Fourier transform byη .
(1.15) Note thatη(k) =η(−k). Since the zero modeη(0) of the solution is automatically zero, we will only care aboutη(k) for k ∈ Z d (1.16) In particular, (−∆) 1 2 is an invertible linear bijection on distributions with null 0-th Fourier mode. We denote by H 1 (T d ) the space of mean-zero functions φ such that the norm ∥φ∥ 2 Let (Ω, F, P) be a complete probability space and η be a mean-zero spatial white noise on the d-dimensional torus T d , i.e. η is a Gaussian field with covariance , the space of square-integrable functions with 0 total mass, and ⟨·, ·⟩ L 2 is the usual scalar product in L 2 (T d ). For n ∈ N, let H n be the n-th homogeneous Wiener chaos, i.e. the closed linear subspace of L 2 (Ω) generated by the random variables H n (η(h)), where H n is the n-th Hermite polynomial, and h ∈ H has norm 1. By [Nua06, Theorem 1.1.1], H n and H m are orthogonal whenever m ̸ = n and L 2 (Ω) = n H n . Moreover, there exists a canonical contraction I : n≥0 L 2 (T 2n ) → L 2 (Ω), which restricts to an isomorphism I : ΓL 2 → L 2 (Ω) on the Fock space ΓL 2 := n≥0 ΓL 2 n , where ΓL 2 n denotes the space L 2 sym (T dn ) of functions in L 2 0 (T dn ) which are symmetric with respect to permutation of variables. The restriction I n of I to ΓL 2 n , is itself an isomorphism from ΓL 2 n to H n and, by [Nua06, Theorem 1.1.2], for every F ∈ L 2 (Ω) there exists a family of kernels (f n ) n∈N ∈ ΓL 2 such that F = n≥0 I n (f n ) and where L 2 n def = L 2 (T dn ), and we take the right hand side as the definition of the scalar product on ΓL 2 , i.e. (1.19)

Remark 1.4
In view of the isometry between ΓL 2 and L 2 (Ω), throughout the paper, we will abuse notation and denote with the same symbol operators acting on ΓL 2 and their composition with the isometry I, which is an operator acting on L 2 (Ω).
For F (η) = f (η(φ 1 ), . . . , η(φ n )), f : R n → R smooth and growing at most polynomially at infinity and φ 1 , . . . , φ n ∈ L 2 0 (T d ), we define the Malliavin derivative DF according to (1.20) and, for k ∈ Z d 0 , denote its Fourier transform by (1.21) Notice that, for p ∈ Z, the action of D p on f ∈ ΓL 2 n is F(D p f )(k 1:n−1 ) = nf n (k 1:n−1 , p) , (1.22) so that in particular D p f ∈ ΓL 2 n−1 . At last, we also recall the following integration by parts formula on Wiener space, i.e. (1.23) Throughout the paper, we will write a ≲ b if there exists a constant C > 0 such that a ≤ Cb. We will adopt the previous notations only in case in which the hidden constants do not depend on any quantity which is relevant for the result. When we write ≲ T for some quantity T , it means that the constant C implicit in the bound depends on T .

The equation and its generator
In this section, we first collect a number of preliminary properties of the solution η ε to the regularised Burgers equation (1.5) and its generator (Section 2.1). Then, we carry out a detailed analysis of the generator equation (a.k.a. resolvent equation) and obtain the main estimates we will need in order to determine the large-scale behaviour of η ε .

Multidimensional Burgers generator
As in Section 1.3, let us first write (1.5) in its weak formulation. For φ ∈ S(T d ) and t ≥ 0, it reads where η ε 0 is the initial condition, ξ is a space-time white noise so that In accordance with Remark 1.1, for d = 2 the sup-norm | · | ∞ in (2.3) is replaced by the Euclidean norm | · | instead. From (2.1) follows that the Fourier modesη ε (k), |k| ≥ ε −1 evolve like independent (and ε-independent) Ornstein-Uhlenbeck processes.

Lemma 2.1
For every deterministic initial condition η 0 , the solution t → η ε t of (2.1) exists globally in time and is a strong Markov process. The generator L ε of η ε can be written as L ε = L 0 + A ε and the action of L 0 and A ε on smooth cylinder functions F is given by Moreover, L 0 is symmetric with respect to P while A ε is skew-symmetric. Finally, the law P of the average-zero space white noise is stationary.
Proof. Very similar arguments were provided in a number of references for equations which share features similar to those of (2.1), e.g. [CES21,GJ13] and, more comprehensively, [Gub19], so we will limit ourselves to sketch some of the proofs.
For the global in time existence of (2.1) for fixed ε, we refer the reader to [CES21,Prop. 3.4]  To see that P is stationary, we start by noting that stationarity for the linear equation with w = 0 is well-known (and easy to check), see e.g. [Gub19, Section 2.3], where it is further shown that L 0 is symmetric with respect to P. Hence, by [Ech82], it suffices to check that E[A ε F ] = 0 for every cylinder function F . In fact, this follows immediately if we prove that A ε is skew-symmetric, since, if so, E[A ε F ] = −E[F A ε 1] = 0 as the Malliavin derivative of the constant random variable F = 1 is 0. Now, to prove skew-symmetry for A ε , we apply Gaussian integration by parts (1.23) and get the proof is completed.
Next, we want to determine the action of L 0 and A ε on the Fock space ΓL 2 . To lighten notations, for a set of integers I, we will denote by k I the vector (k i ) i∈I and For any n ∈ N, the action of the operators L 0 , A ε − , A ε + on f ∈ ΓL 2 n is given by and, if n = 1, then A ε − f is identically 0. Moreover, for any i = 1, . . . , d, the mo- Proof. The proof is identical to that of [CES21, Lemma 3.5], to which we refer the interested reader. Regarding the fact that the L 0 , A ε ± commute with the momentum operator, this is immediate to verify.
In order to state the following lemma, we introduce the so-called number operator.
Our analysis will need some preliminary estimates on the asymmetric part of the generator, which corresponds to proving the so-called graded sector condition in [KLO12, Section 2.7.4].

Lemma 2.4
For d ≥ 2 here exist a constant C = C(d) > 0 such that for every ψ ∈ ΓL 2 the following estimate holds for σ ∈ {+, −}. In particular, for σ ∈ {+, −}, it implies Proof. Clearly, once we establish (2.10), (2.11) follows immediately by choosing ψ therein to be (−L 0 ) −1/2 ϱ for ϱ ∈ ΓL 2 . For (2.10), we claim that we only need to prove that for any ψ ∈ ΓL 2 n and ϱ ∈ ΓL 2 n+1 , we have We will show (2.12) at the end. Assuming it holds, we first derive (2.10) for A ε + . The variational characterisation of the (−L 0 ) −1/2 ΓL 2 -norm gives where in the first step we used (2.12) (and C 0 is the universal constant implicit in that inequality) while in the last we chose γ Invoking again the variational formula above, we deduce It remains to prove (2.12). By definition of A ε + , we have Let us look at the sum over k 1 , k 2 . This equals We leave A as it stands and focus on (I). Note that this can be bounded by where we used that for d = 2, λ 2 ε = (log ε −2 ) −1 and As a consequence, for any γ > 0, we have where we used that, by the definition of A and B in (2.13) and (2.14) respectively, we have and similarly for B. Then, (2.12) follows.
Before turning to the analysis of the generator equation, we need some further estimates which are specific to the case d = 2.

The replacement lemma: d = 2
While in dimension d ≥ 3 the inverse Laplacian is integrable close to the origin, and this will be crucial in identifying the limiting diffusivity, in d = 2, this is not the case. To overcome the lack of integrability, we devise an alternative route, which (morally) approximates the inverse of the generator with a linear and diagonal (both in Fourier and in the chaos) operator, given by a non-trivial order 1 perturbation of L 0 .
To be more precise, we need to introduce some notation. For ε > 0, let L ε be the function defined on [1/2, ∞) as and G ε be the operator on ΓL 2 given by which means that for ψ ∈ ΓL 2 n , n ∈ N, the action of G ε on ψ is We are now ready to state the main result of this section, namely, the replacement lemma. Recall the definition of the scalar product ⟨·, ·⟩ on ΓL 2 given in (1.19).

Lemma 2.5 (Replacement Lemma)
There exists a constant C > 0 such that for every ψ 1 , ψ 2 ∈ ΓL 2 we have where G ε is the operator defined according to (2.20) and (2.19) and, for n ∈ N, L w 0 is the operator acting on ψ ∈ ΓL 2 n as

Remark 2.6
As it appears from the proof of Proposition A.1 in the appendix, the form of the function G in (2.18) is dictated by the following fixed point equation .
Now, if we had chosen a different regularisation for the nonlinearity, e.g. the | · | ∞ -distance in (2.3) instead of the Euclidean one or a smooth regularisation instead of the Fourier cut-off in (2.2), then we would have not obtained an explicit G but the statement above would have remained true. Apart from working with an explicit function, the advantage of our choice lies in the fact that it allows us to make an explicit connection with the work of Yau [Yau04] on the two-dimensional ASEP. Indeed, the exponent 2/3 in (2.19) precisely reflects that obtained therein and strongly suggests that, if λ ε were taken to be a constant independent of ε, then the diffusivity would diverge as (log t) 2/3 for t → ∞.
Proof. Notice first that, for any n ∈ N, the operator n into itself, so that, to establish (2.21), it suffices to consider ψ 1 , ψ 2 ∈ ΓL 2 n . Moreover, we can write (2.24) Recalling the form of A ε + in (2.9), we see that the scalar product at the right hand side can be split into a diagonal part, corresponding to making the same choice for the indices i, j in the two occurrences of A ε + , an off-diagonal part of type 1, corresponding to choosing i, j and i ′ , j ′ with one element in common, and off-diagonal part of type 2, corresponding to all remaining choices. (See for instance [CET23, Lemma 3.6], where a similar distinction was made.) To write this in formulas, let .

(2.25)
Then, the scalar product in (2.24) satisfies where diagonal part, given by the first summand, is defined as while off-diagonal parts of type 1 and 2 are respectively given by where, for i = 1, 2, c off i (n) is an explicit positive constant only depending on n and such that c off i (n) = O(n i+1 ).
We are ready to control the left hand side of (2.21). At first, we split all the diagonal parts from the off-diagonal Let us begin by estimating the first summand. By (2.20), (2.22) and (2.26), we see that where P ε is defined as and, to shorten the notations, we wrotẽ By Proposition A.1, we can upper bound the above by n!n so that this term satisfies (2.21).
We are now left to estimate the off diagonal terms. This is done by following mutatis mutandis the same steps as in the proof of [CETar, Lemma 3.4], so we will only point out the necessary changes. For the off-diagonal terms of type 1, we need to control where σ ε n+1 , given in (2.25), is the Fourier multiplier of S ε in ΓL 2 n+1 and, as usual, we have bounded |w · k| ≲ |k|. As in [CETar, Eq. (3.21)], an application of Cauchy-Schwarz shows that (2.32) is upper bounded by (2.33) This is the same expression as in [CETar, Eq. (3.21)], with σ ε n+1 replacing J N there. Since σ ε n+1 satisfies which is the same estimate that J N satisfies, one can argue as in [CETar] and deduce As for the off-diagonal terms of type 2, we proceed similarly, recalling that, this time, c off 2 (n) ≲ n 3 . Namely, applying Cauchy-Schwarz and exploiting once more (2.34), we bound them as

A priori estimates on the truncated generator equation
The goal of this section is to derive a priori estimates on suitable elements of the n-th inhomogeneous Fock space, which will be crucial in characterising the limit ε → 0 of the solution to (1.5). The specific functions we need will depend on the dimension, as for d = 2 we want to take full advantage of the Replacement Lemma 2.5. Let us begin with some definitions. Let n ∈ N, i ≥ 2 and P n i be the projection onto n-th inhomogeneous Fock space with the first i − 1 chaos removed, i.e. onto n j=i ΓL 2 j . The truncated generator L ε i,n , and the corresponding truncated operators A ε i,n , A ε,+ i,n and A ε,+ i,n are respectively defined as For d ≥ 2, let u ε,n be the unique solution of the truncated generator equation which is defined as In the specific case of d = 2, we further considerũ ε which satisfies where L w 0 and G ε are defined according to (2.22) and (2.20) respectively. Note that, even if in (2.37) there is no truncation in the chaos, the equation admits a unique solution. Indeed, it is a triangular equation which can be explicitly solved starting fromũ ε 1 = · · · =ũ ε i−1 = 0, and then inductively setting That said, we will be concerned with the projection ofũ ε onto ⊕ n j=i ΓL 2 j , that we denote byũ ε,n and can be easily seen to solvẽ As we will see in the proof of the main convergence theorem (see Section 3.2), a case which will play a crucial role for us is when the input g above coincides with the action of A ε + on a smooth element of a fixed Fock space. For this, consider test functions φ, ψ ∈ S(T d ) and let f 1 def = φ ∈ ΓL 2 1 and be the symmetric version of φ ⊗ ψ, which lives in ΓL 2 2 . For i = 2, 3, we will study v ε,n (that we distinguish from u ε,n as the right-hand side is fixed A ε + f i−1 ), that is the solution to the equation We are ready to state the main result of this section.
Then, for any k ∈ N, there exists a constant C > 0, depending only on k, m and the dimension d, and ε d (n) > 0 such that for any ε < ε d (n) the following estimate holds where we recall that w ε,n i is the component of w ε,n in the i th chaos. If d ≥ 3, ε d (n) can be taken to be 1.
As a first step in the proof of the previous theorem, we derive weighted estimates on the solution to (2.39) which hold in any dimension d ≥ 2. The argument we exploit follows closely that of [LY97b, Lemma 2.5] and since the proof works for any choice of ψ, we formulate and prove the result in the more general case of (2.36).
where N is the number operator in Definition 2.3.
Proof. As n and ε are fixed throughout the proof, we will write u and u j , j = 1, . . . , n in place of u ε,n and u ε,n j , respectively. Also, by convention, we let u n+1 = 0 = u i−1 = · · · = u 1 . To denote constants which do not depend on ε, n or g we will use C, and such C might change from line to line.
Let us test the j-th component, j ≥ i, with L ε u, which, since by Lemma 2.2 Via a summation by parts, for any a > 0, we have As a consequence, there exists some finite strictly positive constant C = C(a, m, k), which might change from line to line, such that To handle the last sum, we bound where in the second bound we used (2.10) in Lemma 2.4. We now plug the result into (2.43) and rearrange the terms, so that we conclude Choosing a sufficiently large, in such a way that (1 + a j 2k − C √ j ) ≥ 1/2 for every j ≥ 2, (2.42) follows.
In the next two lemmas, we consider the specific case of d = 2. In the first one, we show that, for any g,ũ ε,n defined according to (2.38) can be estimated in terms of g.

Lemma 2.9
For d = 2, let g ∈ m j=i ΓL 2 j , n ∈ N, i ≤ m ≤ n andũ ε,n be given by (2.38). Then, there exists a constant C > 0 independent of n, ε and ε 2 (n) > 0 such that for every ε < ε 2 (n) Proof. At first, we study how the generator L ε acts onũ ε,n . We have where in the last step we used (2.38). Now, A ε +ũ ε,n n ∈ ΓL 2 n+1 , so that, sincẽ u ε,n ∈ ⊕ n j=i ΓL 2 j , the two are orthogonal. Therefore, by testing both sides byũ ε,n , we obtain where in the last step we neglected the negative term at the right hand side, we used (2.11) together with the positivity of −L w 0 G ε and the fact that the operators L 0 , L w 0 and G ε commute to control the second term containing g, and the Replacement Lemma 2.5 to bound the third summand. Now, we choose ε 2 (n) in such a way that Cn 2 λ 2 ε 2 (n) < 1/2. Therefore, (2.46) follows upon rearranging the terms in (2.48).
In the next lemma, we turn our attention toṽ ε,n given in (2.40) and prove that it satisfies the same weighted estimates as those in (2.42). To do so, in view of Proposition 2.8 and the previous Lemma, it suffices to control the difference betweeñ v ε,n and the solution of the truncated generator equation v ε,n . Lemma 2.10 Let d = 2. For n ∈ N and i = 2, 3, letṽ ε,n be given by (2.40) and f 1 , f 2 be as in Theorem 2.7. Then, for any k ∈ N there exists a constant C > 0 and ε 2 (n, k) > 0 such that for every ε < ε 2 (n, k) Let v ε,n be the solution of (2.39) in d = 2. Notice that, trivially, Now, in view of Proposition 2.8, the second summand is bounded by (a constant times) For the first, we claim that for any j there exists C > 0 such that for ε small enough, we have Assuming the claim, we choose in (2.51) j = k and ε 2 (n, k) > 0 such that n k+2 λ ε 2 (n,k) < 1, so that we obtain where the last step follows by (2.10). Hence, (2.49) is proved.
We now prove (2.51). To shorten the notation, set g ♯ We begin by evaluating −L ε onṽ ε,n − v ε,n . To do so, we exploit (2.47) and the fact that, as noted in the proof of Proposition 2.8, since v ε,n solves (2.39), we have This means that , which implies that they are orthogonal to bothṽ ε,n and v ε,n as these belong to ⊕ n j=i ΓL 2 j . Hence, by testing both sides of (2.52) byṽ ε,n − v ε,n , we obtain Let us analyse the two terms at the right hand side separately. For the first, note that the operator L w 0 G ε is negative, so that and in the last bound we used that −L w 0 G ε ≲ −L 0 and that L 0 , L w 0 and G ε commute. Now, the a priori estimates in Proposition 2.8 and Lemma 2.9 allow to upper bound the previous by For the second term in (2.53), we apply the Replacement Lemma 2.5 first and the same a priori estimates as above, thus getting where the constant C changed in the last two lines. Now, by using that for ε < ε 2 (n, k), Cn 2 λ 2 ε < 1, collecting the previous bounds and rearranging the terms, (2.51) follows.
We are now ready for the proof of Theorem 2.7, which is an easy consequence of Lemma 2.4 and, for d ≥ 3, Proposition 2.8, while, for d = 2 Lemmas 2.9 and 2.10.
Proof of Theorem 2.7. Let us first treat the case of d ≥ 3. As noted in the proof of Proposition 2.8, since v ε,n solves (2.39), we have (2.54) By (2.10), for any k ≥ 1, we have Hence, (2.41) follows directly by (2.42) and (2.10).
We now turn to d = 2, in which caseṽ ε,n satisfies (2.40). Notice that, by the recursive definition ofṽ ε,n ,ṽ ε,n 1 = · · · =ṽ ε,n i−1 = 0 and thereforẽ Hence, upon replacing ψ with A ε + f i−1 in (2.47), we see that Now, for the first two terms we use Lemma 2.4 and −L w 0 G ε ≲ −L 0 , so that we can bound them by where the last step follows by Lemma 2.10. For the third term, we note that the ΓL 2 -norm satisfies a variational formulation, i.e.
where in the third step we used the Replacement Lemma 2.5 and in the last, Lemma 2.9. By choosing ε 2 (n, k) > 0 such that λ 2 ε 2 (n,k) n 2+k < 1, (2.41) follows at once.

The diffusivity
The main advantage of Theorem 2.7 is that it allows to replace the nonlinearity with the sum of two terms: one involves the generator of our process, and therefore encodes its fluctuations, while the other consists of A ε − applied to an element in the Fock space of degree i = 2, 3, so that overall it is of degree i − 1. This suggests that this latter term should give us the additional diffusivity appearing in the limit (1.8). The goal of this section is to analyse this object and determine its behaviour.
The analysis for d = 2 and d ≥ 3 will be very different, but the limits we need to establish have the same expression. Let us first state the main theorem and then provide different proofs in the two cases.  Let us begin with the proof the above theorem in d = 2 since, thanks to the replacement lemma, it is way easier.
Proof of Theorem 2.11 in d = 2. Notice first that the recursive definition ofṽ ε,n in (2.40) immediately gives As a consequence, we have For the first term at the right hand side, we use the same variational formulation as in (2.56) so that following the same steps, we deduce where in the last step we used the Replacement Lemma 2.5. For the second, the analysis is identical for f 1 or f 2 , so we only consider the latter. Using the explicit expression of L w 0 , G ε , D and f 2 , we see that Since φ is smooth we can take the limit in ε inside the sum and, provided we take D SHE = G(1), we conclude that the right hand side goes to 0. Hence, the proof of (2.57) is completed.
Concerning (2.58), we note that for any j ∈ N and ψ ∈ ΓL 2 j , Now, the inner sum is clearly finite, and therefore we deduce We apply the previous estimate to the definition ofṽ ε,n , so that we obtain where in the first step, we replaced the truncated operator A ε,+ i,n with A ε + . By the weighted a priori estimate onṽ ε,n in Lemma 2.10, we see that the norm at the right hand side of (2.61) is bounded uniformly in ε. Now, since λ ε goes to 0 as ε → 0, (2.58) follows.
We now turn to the case d ≥ 3. The proof of (2.57) (and (2.58)) requires to be able to pass to limit first in ε and then in n. Since the first limit is the most delicate, we single it out in the statement below, which also includes (2.58), and postpone its proof to the next subsection.

Proposition 2.12
Consider the setting of Theorem 2.11 for d ≥ 3. Then, for every n fixed, there exists a unique constant D n > 0 such that, for i = 2, 3, where D n is the operator given by D n def = D n L w 0 for L w 0 defined according to (2.22). Further, as ε → 0, v ε,n converges to 0 in ΓL 2 .
We now turn to the proof of Theorem 2.11 for d ≥ 3, for which, thanks to Proposition 2.12 we only need to ensure that the limit in n of the constants D n is unique and strictly positive.
Proof of Theorem 2.11 in d ≥ 3. Notice that (2.58) was established in Proposition 2.12, so that we only need to focus on (2.57). For this, in light of (2.62), it suffices to prove that the sequence of operators {D n } n converges to a unique limit, which translates into showing that the sequence of constants {D n } n is Cauchy. The definition of the operator D n ensures that for any Furthermore, the left hand side can be bounded by where, for ε > 0, any n ∈ N and φ = f 1 any smooth function, v ε,n is the solution of the corresponding generator equation (2.39) truncated at level n. To control the last term, we apply (2.10) to get By definition of v ε,n+1 and v ε,n , we deduce that We now test both sides by v ε,n+1 − v ε,n and, using the antisymmetry of A ε , obtain dropped out because of orthogonality of Fock spaces with different indices (recall that v ε,n+1 1 = v ε,n 1 = 0 = v ε,n+1 n+2 = v ε,n n+1 ). Now, the right hand side can be bounded above by We can go back to (2.66) and get where the last step follows by Lemma 2.4. To control the last term, we apply Proposition 2.8, which, for any j > 1, gives At this point, we upper bound the right hand side of (2.64) with the above so that, by using (2.63) and taking the limit in ε, (2.62) ensures that Since f 1 is an arbitrary smooth test function, we can choose it to be such that the norm at the left hand side is non-zero. Consequently, the sequence {D n } n is Cauchy and therefore converges to a unique constant D SHE , so that the proof of (2.57) is concluded.
It remains to show that D SHE is strictly positive. To do so, fix f 1 def = e k for k ∈ Z 2 such that w · k ̸ = 0 and set v ε,n [−k] to be the solution of the truncated generator equation whose input function is f 1 . Notice that Let us consider the first term at the right hand side more carefully. Since A ε − v ε,n 2 [−k] ∈ ΓL 2 1 , we can expand it in the Fourier basis, i.e.
where the last step is a consequence of the translation invariance of the equation, i.e. the commutation between the operators L 0 , A ε with the momentum operator as stated in Lemma 2.2. We can now exploit orthogonality of Fock spaces with different indices and the antisymmetry of A ε , to see that It follows that so that we will only establish a lower bound on the latter norm. We have where in the second equality we used [KLO12, Theorem 4.1], in last step we restricted the supremum to ΓL 2 2 , thus making it smaller, and exploited Lemma 2.2 and the truncation P n 2 in the definition of A ε 2,n in (2.35) to simplify the last term. According to Lemma 2.10, for ψ ∈ ΓL 2 2 , there exists a constant C > 0 such that which implies a further lower bound of the form We are left with estimating ∥(−L 0 ) −1/2 A ε + e k ∥ 2 from below. Thanks to (2.9), we have an explicit expression for the latter, which reads As a consequence, from (2.68) and the previous estimates, we obtain Now, taking the ε → 0 limit at both sides we see that, by (2.62), the second term converges to 0, while the first goes to a strictly positive constant independent of n. Thus, the strict positivity of D SHE follows by passing to the limit in n so that the proof of the theorem is concluded.

Proof of Proposition 2.12
The proof of Proposition 2.12 is the most delicate part of the present paper. Since in d ≥ 3 we have no analogue of the replacement lemma (and we do not even expect it to hold), we need to resort to a different strategy. Our proof is centred around a suitable expansion of the norms of v ε,n (see e.g. (2.80) below) which involves several subsequent applications of the operators A ε + , A ε − interposed by inverses of −L 0 . In order to see why such an expansion is meaningful, it is convenient to write it in terms of operators which are skew-Hermitian and bounded in ΓL 2 (uniformly in ε), and this is the first thing we will do.
Let n ∈ N, n ≥ 2, be fixed throughout the section. For i = 2, 3, let ε > 0 and A ε i,n be given as in (2.35). We introduce the operators T ε i,n and for σ ∈ {+, −}, T ε,σ , T ε,σ i,n , according to Note that, by Lemma 2.4 and in particular (2.11), each of the operators above is bounded in ΓL 2 uniformly in ε (the bound depends on n, but this is irrelevant as n is fixed) and, by Lemma 2.2, it is easy to see that T ε i,n is skew-Hermitian. To have a notation for multiple applications of the operators T ε,σ , σ ∈ {+, −}, we set, for m ≥ 1 and a ≥ 0, Π (n) a,m to be the set of simple random walk paths p = (p 0 , . . . , p a ) consisting of a steps, starting at height 1, i.e. p 0 = 1, ending at height m, i.e. p a = m, that do not reach height n + 1 and that can have height 1 only at the endpoints. Given p ∈ Π (n) a,m we let |p| def = a and T ε p be defined according to In other words, there is a factor T ε,+ (resp. T ε,− ) whenever the path takes a step up (resp. down)3.
We enclose the most technical aspect of the proof of Proposition 2.12 in Proposition 2.13 below. Before giving the proof of the latter, we will show how it implies the former.

Proposition 2.13
For j 1 , j 2 ∈ Z d 0 , let j be either j 1 or j 1:2 , and denote by e j either e j 1 or e j 1:2 def = [e j 1 ⊗ e j 2 ] sym , the symmetric version of e j 1 ⊗ e j 2 , where e j i is the j i -th element of the Fourier basis. In the notations introduced above, for every 3For instance, for a = 4 the path p = (1, 2, 3, 2, 3) ∈ Π (n) 4,3 , n ≥ 4 corresponds to T ε p = T ε,+ T ε,− T ε,+ T ε,+ . a 1 , a 2 ≥ 0 and p r ∈ Π (n) ar,1 , r = 1, 2 (so that in particular a r ∈ 2N) the limit4 exists, with c(p) ∈ R depending only on the path p and such that c(p) = 1 if |p| = 0. Furthermore, for every a r > 0, m r ≥ 2, and p r ∈ Π (n) ar,mr , r = 1, 2, one has Proof of Proposition 2.12 given Proposition 2.13. Let us first introduce a notation thanks to which we can express both the norm in (2.62) and the ΓL 2 -norm of v ε,n . For this, let S and c be, respectively, a linear operator and a constant given by either S = P i−1 A ε − and c = 1 or S = (−L 0 ) 1/2 and c = 0, with P i−1 the orthogonal projector on the (i − 1)-th chaos, ΓL 2 i−1 . Then, the quantity of interest in (2.62) is The rest of the proof will be divided in several steps, the first of which reduces our study to the case where the test functions φ and ψ in the definition of f i−1 are two given elements of the Fourier basis.
Step 1: Reduction to Fourier basis. For this step, we focus on the case i = 3 since i = 2 is easier and can be argued similarly. For j 1 , j 2 ∈ Z d 0 , let v ε,n [−j 1:2 ] be the solution to (2.39) in which we take f 2 to be e j 1:2 = [e j 1 ⊗ e j 2 ] sym . By linearity, we clearly have v ε,n = j 1 ,j 2φ (j 1 )ψ(j 2 )v ε,n [−j 1:2 ]. The definition of the norm in ΓL 2 immediately gives Now, for α ≥ 0, we can apply Cauchy-Schwarz to get ⟨Sv ε,n − cD n−1 f 2 , e −k 1:2 ⟩ 2 = j 1 ,j 2φ (j 1 )ψ(j 2 )⟨Sv ε,n [−j 1:2 ] − cD n−1 e j 1:2 , e −k 1:2 ⟩ 2 ≤ j 1 ,j 2 |j 1:2 | 2α |φ(j 1 )| 2 |ψ(j 2 )| 2 j 1 ,j 2 1 |j 1:2 | 2α ⟨Sv ε,n [−j 1:2 ] − cD n−1 e j 1:2 , e −k 1:2 ⟩ 2 , so that we conclude 4in (2.74), the overbar denotes complex conjugation Now, the norm inside the sum is bounded by (a constant times) |j 1:2 | 2 . Indeed, if S = P 2 A ε − and c = 1, thanks to (2.10), the apriori estimates in Proposition 2.8 and the definition of D n−1 in Proposition 2.12, we get In instead, S = (−L 0 ) 1/2 and c = 0, then As a consequence, upon choosing α big enough, and we are allowed to since f 2 is smooth, by dominated convergence the statement follows if we prove that for all given j 1 , j 2 ∈ Z d 0 , the norm in the sum vanishes. Now, by opening up the squared norm and using the definition of D n−1 we see that this is equivalent to showing that Step 2: Analysis of Sv ε,n [−j]. Recall the definition of the operators T ε i,n and T ε,σ , σ ∈ {+, −} in (2.72). Thanks to (2.39), (2.78) Hence, the scalar product in (2.77) satisfies where in the last step we applied Fubini, which is allowed as the operator e sT ε 2,n has norm at most 1, since the spectrum of T ε i,n is purely imaginary. For the same reason, once we take the ε-limit at both sides we can apply dominated convergence and pass the limit inside the integral in s, i.e.
where in the last step we could exchange the limit with both the summation over b and the scalar product because, for fixed n, s, the norm of sT ε n is uniformly bounded (recall that after the first, in the subsequent steps s can be treated as fixed).
For the norm in (2.76), we can follow the exact same arguments so that we are led to (2.80) The previous arguments guarantee that we are left to determine, for every b 1 , b 2 ≥ 0 and θ ∈ {0, 1}, the limit 2.81) and to do so we will separately consider the case in which S = P i−1 A ε − and c = 1, or S = (−L 0 ) 1 2 and c = 0, starting with the former.
Step 3: The case S = P i−1 A ε − and c = 1. With this choice of S and c we need to prove both (2.76) and (2.77), which respectively correspond to taking θ = 1 and θ = 0 in (2.81). Furthermore, by (2.72), we have so that the scalar product in (2.81) becomes Let us begin with a few remarks. First of all, because of the projection P i−1 and the fact that e j ∈ ΓL 2 i−1 , b 2 must necessarily be even, hence we set a 2 ∈ N \ {0} to be such that b 2 = 2a 2 − 2. The same holds for b 1 if θ = 1 (and we write b 1 = 2a 1 − 2, a 1 ∈ N \ {0}) while if θ = 0, b 1 plays no role and for convenience we will set a 1 ≡ 0. Next, we expand where we define Π 2a,1 to be the set of all walks p which start at time 1 from 1, i.e. p 0 = 1, end at time 2a at 1, i.e. p 2a = 1, have increments of size 1 and are such that p 1 − p 0 = 1 = p 2a−1 − p 2a . If either of the previous conditions were violated the projection onto P i−1 would set the scalar product to 0. Moreover, the last condition on the p's ensures that σ 1 = + and σ 2a = − (so that the T ε,− and T ε,+ in the line above are present). Because of the projector P n i entering the definition of T ε,σ i,n , if a path p r reaches height n + 3 − i or height i − 1 (except at the endpoints), then the corresponding scalar product is annihilated. Hence, we can replace the index set of the sum at the right hand side of (2.82) to Π (n+2−i) 2ar,1 , and get where in the first step we replaced all instances of T ε,σ i,n with T ε,σ , since for p r ∈ Π (n+2−i) 2ar,1 the projection P n i acts as the identity operator, and in the second we used the convention that a 1 = 0 if θ = 0, in which case p 1 has length zero and T ε p 1 = 1. At last, getting back to (2.79), and applying (2.74) in Proposition 2.13, we obtain where the last constant D n+2−i is uniquely defined by the expression before, so that (2.77) follows at once. Similarly, (2.76) holds since, from (2.80) and by (2.74), we get for D n+2−i be given as above. Therefore, the proof of (2.76) is concluded and so is that of (2.62).
Step 4: The case S = (−L 0 ) 1 2 and c = 0. This choice of S turns the scalar product in (2.81) into (2.84) By expanding and arguing as in the previous step, we deduce that where Π br,mr , r = 1, 2, is the set of all walks p r which start at time 1 from 1, i.e. p 0 = 1, end at time b r at m r , i.e. p br = b r , have increments of size 1 and are such that p 1 − p 0 = 1. In the third step, because of the projector P n i in the definition of T ε,σ i,n , we replaced the sum over Π br,mr with that over Π (n+2−i) br,mr and consequently removed the projection. Now, since the scalar product at the right hand side converges to 0 by (2.75), from (2.80) we deduce that ∥v ε,n ∥ 2 converges to 0 so that the proof of Proposition 2.12 is concluded.
We now turn to the proof of Proposition 2.13, for which we will need some notation and a few preliminary results. First of all, the action of the operators A ε + , A ε − of (2.9) on ΓL 2 n can be rewritten as follows: where, for 1 ≤ i < i ′ ≤ n + 1 and 1 ≤ q ≤ n − 1, The advantage of this decomposition is that the operators in (2.86) can be extended to functions which are not necessarily symmetric, i.e. they can be viewed as operators on n L 2 Before detailing their properties, we also introduce operators A ε,δ ± , δ ≥ 0, which will allow us to derive a uniform integrability-type condition. This will be the essential tool to prove the limit in (2.74). For f ∈ L 2 n , 1 ≤ i < i ′ ≤ n + 1, 1 ≤ q ≤ n − 1 and δ ≥ 0, set and and similarly for T ε,δ,− [q]. In the following lemma, we collect estimates on the operators introduced above.
Lemma 2.14 Let ε > 0 and n ∈ N. For every 1 ≤ i < i ′ ≤ n + 1 and n+1 and L 2 n−1 respectively, and commute with the momentum operator defined in Lemma 2.2. Moreover, there exists a constant C > 0 such that for every f ∈ L 2 n , and for every δ ∈ [0, (d − 2)/2) we also have Proof. Commutativity with the momentum operator is immediately checked from the definition of the operators. Concerning (2.89), it follows from (2.90) upon noting that for every f ∈ L 2 with δ = 0, and similarly for T ε,− [q]. Let δ ∈ [0, (d − 2)/2) and f ∈ L 2 n . Then, with a simple change of variables, we have Since the sum above equals (2.91) and the right hand side is bounded as long as δ < (d − 2)/2, (2.90) follows. Concerning T ε,δ,− [q], we apply Cauchy-Schwarz so to obtain and once again the inner sum is bounded uniformly over k q thanks to (2.91), so that the proof of (2.90) is complete.
In the following lemma, we show that the main contribution to the norm of T ε,+ [(i, i ′ )]f and T ε,− [q]f comes from Fourier modes whose size diverges with ε.

Lemma 2.15
In the setting of Lemma 2.14, let γ ∈ (0, 1) and, for f ∈ L 2 n , define the operators wheref > (k 1:n ) def = 1 |k 1 |∧|k 2 |>ε −γf (k 1:n ). Then, there exists a constant C > 0 such that and the same bound holds for Proof. By the definition of the operator A ε + [(i, i ′ )] in (2.86) and that of T ε,+ [(i, i ′ )] in (2.93), we have Hence, where in the last step we simply relabelled all the variables in the sum. Now, the inner sum can be bounded as Before proceeding, we want to rewrite (2.73) in terms of the operators we just introduced. Let a ≥ 0, m ≥ 1, p ∈ Π (n) a,m , j, e j be as above. For f ∈ L 2 i , let κ(f ) = i denote the eigenvalue of the number operator, so that κ def = κ(e j ) = 1 if j = j 1 and κ = 2 if j = j 1:2 . Replacing each of the T ε,+ , T ε,− with the sum of the T ε,+ [(i, i ′ )], T ε,+ [q] (as in (2.85)), we obtain where C m,κ = m−1 j=1 (κ+j) (in particular, C 1,κ = 1), G κ [p] is a set whose elements g = (g s ) s=1,...,a are of the form In particular, the cardinality of G κ [p] is finite and independent of ε.
Thanks to the representation in (2.96), we are ready to state the next lemma which will give (2.75) as a corollary.

Lemma 2.16
In the same setting as Proposition 2.13, let a > 0, m ≥ 2 and p ∈ Π (n) a,m . Then, for every g ∈ G κ [p], we have where κ = κ(e j ) is defined as above, and m κ def = m + κ − 1.
Proof. By the definition of T ε p [g] in (2.96), we have Let γ ∈ (0, 1) and recall the definition of T ε,σ [g] from Lemma 2.15. Thanks to (2.94), we can replace every instance of T ε,σ [g] with T ε,σ [g] since, by (2.89), the T ε,σa [g a ]'s are bounded operators and e j has clearly finite L 2 norm. Therefore we are left to consider where, in the second step, we used the fact that |k 1:mκ | > ε −γ . To see this, notice that, by definition, the new Fourier modes produced by T ε,+ [g] have modulus bigger than ε −γ . Now, the assumption m ≥ 2 ensures that the number of variables on which T ε,σa [g a ] . . . T ε,σ 1 [g 1 ]e j depends (i.e. m κ ) is strictly bigger than that of e j . Therefore, at least one among k 1 , . . . , k mκ must be generated by a T ε,+ [g], which then implies |k 1:mκ | > ε −γ . To conclude, it suffices to observe that by (2.94) and (2.89), the operators T ε,σ [g] are bounded and that e j has finite L 2 -norm, so that the right hand side of (2.98) converges to 0, thus proving (2.97).
To prove (2.74), let a ≥ 0, m ≥ 1, p ∈ Π (n) a,m and g ∈ G κ [p], for κ = 1, 2. We introduce a graphical representation for g ∈ G κ [p], which consists of associating to it a graph as that depicted in Figure 1.
The vertices of the graph are divided in a + 1 columns, labeled from 0 to a. The column with label s contains p s + κ − 1 vertices; the vertices inside each column are labeled with increasing positive integers, starting from the uppermost vertex. The edges only connect vertices in consecutive columns, with each vertex in one column connected to at least one and at most two vertices in the next. column s − 1 is called a branching point and we draw two edges connecting it to the vertices with label i s and i ′ s in column s; if instead g s = q s , then the vertex with label q s in column s is called a merging point and we draw an edge from the vertices with labels 1 and 2 in column s − 1 to q s ; for the other vertices in column s − 1 (i.e. those whose label is not 1 if g s = (i s , i ′ s ), and neither 1 nor 2 if g s = q s ), we inductively draw an edge from the vertex with the lowest label to that of lowest label in column s which is not already connected to another vertex in column s − 1.
One should think of the vertices as carrying a momentum, and of the edges as the relation the momenta on the vertices connected by them satisfy. In particular, when T ε p [g] is applied to e j 1 or5 e j 1 ⊗ e j 2 , the vertex in position 1 of the first column carries momentum j 1 , that in position 2 (if there is one) carries momentum j 2 , a branching point determines the creation of a new momentum, while a merging point, the annihilation of one. Moreover, if x, y, z are vertices then • if x in column s is either a branching or a merging point, and is connected to y, z in column s + 1 or s − 1 respectively, then the sum of the momenta at y, z must coincide with the momentum at x; • if there is an edge between x in column s and y in column s + 1, x is not a branching point and y is not a merging point, then the momenta at x and y coincide.
In the next lemma, we identify a large class of elements g ∈ G 2 [p] for which T ε p [g](e j 1 ⊗ e j 2 ) does not contribute to the limit in (2.74). To state it, let us define 5here we atually mean ej 1 ⊗ ej 2 , not its symmetrized version. Recall that the action of the operators T ε p [g] is well-defined on not necessarily symmetric functions a path π in the graph associated to g as a directed sequence of connected vertices in which π(j) is the label of the vertex π encounters in column j.

Lemma 2.17
Let a ≥ 0, p ∈ Π (n) a,1 and g ∈ G 2 [p]. If in the graph associated to g there exists a path π starting from the second vertex of the 0-th column that contains either a branching or a merging point, then Proof. Let π be a path starting from vertex with label 2 in the column of label 0 and containing either a branching or a merging point. See Fig. 2. Letā ∈ {2, . . . , a} be such that, in the former case, the first branching point π encounters lies in the (ā − 1)-th column, so that in particular π(ā − 1) = 1. In the latter case,ā is such that π(ā) is a merging point, and π(ā − 1) ∈ {1, 2}. Then, invoking (2.89), we upper bound the norm in (2.99) as for M def = 1 + pā −1 . To simplify the notation, set and Ψ ∈ L 2 M . Let us begin by considering the case in which the path reaches a branching point. Then, by definition, σā = pā − pā −1 = + and there exist iā < i ′ā such that gā = (iā, i ′ā ). Let us immediately point out that, by the description of the graph we have given above, the momentum carried by the vertex at the branching point is necessarily j 2 since this is the first branching point on π by construction. Also, the sum of the momenta carried by the vertices at iā, i ′ā must be j 2 . Hence, the quantity we need to control equals where in the last step we renamed the variables (k iā , k i ′ a ) = (ℓ, m) and removed the variable corresponding to j 2 from the outer sum since j 2 is fixed. Now, the inner sum is bounded uniformly in k 1:M −1 (as can be seen by (2.91) with δ = 0) so that we deduce Since by definition (2.100), Ψ = T ε p ′ [g ′ ] for a suitable choice of p ′ and g ′ , the last norm at the right hand side converges to 0 by (2.97).
We now turn to the case in which π reaches a merging point. This time, σā = − and there exists qā such that gā = qā, and π(ā − 1) ∈ {1, 2}. Without loss of generality, assume π(ā − 1) = 1. The norm under study is Now, in the sum above, if ℓ is the momentum carried by the vertex at π(ā − 1), then ℓ must be j 2 . In particular, ℓ = j 2 is fixed and, therefore, so is m since it must coincide with k qā − j 2 (k qā is already summed in the external sum). This means that the sum inside the square at the right hand side of (2.101) has a unique summand. corresponding to g. In particular,ĝ is obtained by simply attaching the path π below g, andg is obtained fromĝ by changing π at the very last step. Note that, when π is removed fromg, the last column contains a single vertex, so the resulting graph is again g. The path π cannot cross the other connected component anywhere else.
Hence, by applying the change of variables k qā → k qā − j 2 and lower bounding the denominator with 1, we deduce that the right hand side of (2.101) is given by Since the norm of Ψ is bounded uniformly in ε, (2.99) follows at once.
Recall that our final goal is to prove Proposition 2.13 and that each T ε p e j has been decomposed as a sum over g ∈ G κ [p]. Thanks to Lemma 2.17, we are left to consider only g ∈ G 1 [p] and those g ∈ G 2 [p] whose associated graph has two connected components, one of which is a path with no branching points. We call the collection of such graphs G 2 [p]; see Fig. 4 for an example. In the next lemma we determine a correspondence between G 1 [p] and G 2 [p] valid in case m = 1.

Lemma 2.18
Let a ≥ 0 be even and p ∈ Π (n) a,1 . Then, for every g ∈ G 1 [p] there exist exactly two elements of G 2 [p], calledĝ andg, such that the graph associated to g can be obtained from that ofĝ andg by removing their connected component corresponding to the single path π.
Proof. The construction of the graphsĝ,g ∈ G 2 [p] given g ∈ G 1 [p] is explained in Figure 3. The fact that there is no other g ′ ∈ G 2 [p] that reduces to g once the path π is removed, follows from the fact that the path π can cross the connected component starting at the vertex 1 of colum labelled 0 only in the step from the second to last to the last column. This can be seen for instance by induction, following the graph from right to left.  Figure 4: A graph corresponding to j = j 1:2 and to p ∈ Π (n) a,m with n ≥ 4, a = 4, m = 3. In this graph, the path π has neither branching nor merging points: the graph is disconnected. The label of the unique vertex of π in column i defines π(i).
(note that in the latter case, e j ̸ = e j if j 1 ̸ = j 2 ). Then, T ε p [g]e j ∈ L 2 mκ , and we write the Fourier transform of its kernel in the form in which we singled out the scaling factor ε d 2 (mκ−1) . We derive an expression for F ε g in terms of ratios of polynomials, whose form and main properties are summarised in the following lemma.
We emphasise that the functions P g , Q g , I g andP g are ε-independent. The only ε-dependence in F ε p comes from the fact that the Riemann sums runs over εZ d 0 .
Proof. The argument for j = j 1 is identical (and simpler) than that for j = j 1:2 , so we will only focus on the latter. The statement is proven via induction on |p| = a ≥ 1. In order to verify condition (iii), we will simultaneously show that Q g further satisfies (iii ′ ) Q g (y 1:M ; x 1:m+1 ) is the product of homogeneous quadratic polynomials Q r , 1 ≤ r ≤ a − 1, each being the sum of |x π(a) | 2 and the squared norms of linear combinations of the other arguments (which are (y 1:M ; x 1:m+1\{π(a)} )). For every 1 ≤ r ≤ a − 1, Q r ̸ = | i̸ =π(a) x i | 2 , |x π(a) | 2 or the sum of the two.
We begin our induction with a = 1. In this case, if p ∈ Π (n) a,m , then necessarily m = 2, M = 0, σ 1 = + and g 1 = (i, i ′ ) for some 1 ≤ i < i ′ ≤ 3. From (2.86) and the definition of T ε,+ [(i, i ′ )], one sees that where the first indicator function comes from the fact that T ε,+ [(i, i ′ )] commutes with the momentum operator (see Lemma 2.14) and that the kernel of e j 1 ⊗ e j 2 at (k 1 , k 2 ) is 1 k 1 =j 1 1 k 2 =j 2 , This expression is of the correct form (2.104), with P g ≡ −4/(2π) d/2 , Q g ≡ 1, I g ≡ J 1 x i ,x i ′ , so that (i), (ii) and (iii ′ ) are clearly satisfied. As for the induction step, we assume the validity of the statement for a given a ≥ 1 and take p ∈ Π (n) a+1,m and g ∈ a,m−σ a+1 and g ′ ∈ G 2 [p ′ ]. Consider first the case σ a+1 = + and let i, i ′ be such that g a+1 = (i, i ′ ). Recalling the definition of T ε,+ [(i, i ′ )] and (2.103), we have By induction, F g ′ has the form in (2.104), and therefore so does F g once we set where now the arguments of P g ′ and I g ′ are (y 1:M ; x i + x i ′ , x 1:m+1\{i,i ′ ,π(a+1)} ) (and the lack of dependence on x π(a+1) follows from the fact that, by construction, π(a + 1) ̸ = i, i ′ ) while the arguments of Q g ′ are (y 1:M ; x i + x i ′ , x 1:m+1\{i,i ′ } ).
If σ a+1 = −, then there exists q such that g a+1 = q. By (2.86), where in this case Again, as F ε g ′ is of the form (2.104) by the induction hypothesis, the same holds for F ε g , where this time we need to set where the omitted arguments of the functions P g ′ and I g ′ are (y 1:M −1 ; y M , x q − y M , x 1:m−1\{q,π(a+1)} ), while those for Q g ′ are (y 1:M −1 ; y M , x q − y M , x 1:m−1\{q} ). Once again, the lack of dependence of P g and I g on x π(a+1) is due to the fact that, by construction, q ̸ = π(a). Now, for both σ a+1 = ±, (2.109) and (2.111) clearly ensure that if P g ′ , Q g ′ and I g ′ satisfy conditions (i) − (ii) so do P g , Q g and I g . The first part of (iii ′ ) is obvious since the variable corresponding x π(a) is simply turned in x π(a+1) and is not affected by the operators A ε,± [g a+1 ] in the definition of T ε,± [g a+1 ]. For the second part of (iii ′ ), note that the first factor in the expression for Q g in (2.109) and (2.111) is different from | i̸ =π(a+1) x i | 2 , |x π(a+1) | 2 or their sum. While this is clear for (2.111) because of the non-trivial y M dependence, for (2.109) this follows by the fact that m + 1 ≥ 4. Indeed, in the base induction step (see (2.106)) corresponding to a = 1, m + 1 = p a + 1 = 3. Now, for a > 1 and σ a = +, by definition of p ∈ Π (n) a,m , p a = m > 2, which implies that the first factor in (2.109) has at least three summands. It remains to show that Q g ′ , evaluated at its new arguments, does not contain a term of the aforementioned form. But this can only happen if a similar term was there in the old variables and this is ruled out by the induction hypothesis. As for property (iii), note that by (iii ′ ), Q g can be written as the product of quadratic polynomials Q r , 1 ≤ r ≤ a − 1, and if Q r depends on at least one of the y i , i ≤ M , then necessarily Q r (y 1:M ; 0) ̸ = 0 for almost all y 1:M . The only potentially problematic case arises when one of the Q r is such that Q r (y 1:M ; x 1:2 ) = |x 1 | 2 , |x 2 | 2 or their sum, but this is ruled out by (iii ′ ) for m = 1.
At last, to verify (2.105), we note that (2.103) and (2.104) give But, since m = 1, the left-most operator in the product T ε As a last result before turning to the proof of Proposition 2.13, in the next lemma we show that (2.105) converges in the limit for ε → 0. Lemma 2.20 In the setting and notation of the previous lemma, let a ≥ 1 be even and p ∈ Π (n) a,1 . Let g be an element of either G 1 [p] or G 2 [p]. Then, there exists a constant c g (p) ∈ R independent of j such that for every j ′ Proof. Since a is even, set a = 2b so that M = b. Our goal is to show that the Riemann-sum in (2.105) converges and that the limit is independent of j. Formally, such limit should be where6 µ is the uniform probability measure on [−1, 1] bd and the integrand is welldefined and finite (except possibly on a zero-measure set) thanks to the properties of Q g stated in Lemma 2.19. Note that if c[g] is given by the expression above, then not only it is independent of j but also (2.114) clearly holds. Hence, it remains to prove that the sum indeed converges to the integral, and that the integral is finite. At first, we want to write the Riemann-sum in (2.105) as an integral with respect to the uniform probability measure on [−1, 1] bd . This is quite standard as it suffices to take a piece-wise constant extension of the polynomialsP g and Q g , but since the quotient displays singularities, we provide the details below.
Recall that we assume 1/ε ∈ N + 1/2. Then, the box [−1, 1] d contains exactly (2/ε) d points of εZ d and it can be written as the union of cubes C(y * ) of side ε, centred at each of the points y * ∈ εZ d ∩ [−1, 1] d . Given y ∈ [−1, 1] d , we let y * (y) denote the (unique, up to Lebesgue-measure zero sets) y * such that y ∈ C(y * ). The sum in (2.105) is then where µ is the uniform probability measure on [−1, 1] ad , P (ε) g is given bỹ 6note that we added the volume factor 2 ad to pass from the Lebesgue measure to the uniform one. The scope of this change of measure will soon become apparent.
using that ∥e j ∥ = 1 or = 1/ √ 2 according to whether j = j 1 or j = j 1:2 , so that (2.120) is proven and the proof of Proposition 2.13 is complete.

Convergence of Burgers to renormalised SHE
Thanks to the results in the previous section, we have all the ingredients we need in order to prove Theorem 1.3. To do so, we will first show that the sequence η ε is tight and then verify that every limit point solves the stationary stochastic heat equation (SHE) where µ is a zero-average space white noise and D SHE > 0 is the constant identified in Theorem 2.11.

Tightness
The proof of tightness follows a by now standard route (see e.g. [GJ13, GT20, CK23, CES21]) which exploits Mitoma's criterion [Mit83], Kolmogorov's continuity theorem and, more importantly, the so-called Itô trick, introduced in [GJ13] and since then exploited in a variety of contexts, see [Gub19] for a pedagogical introduction. For the reader's convenience, we recall below the statement of the latter: Lemma 3.1 (Itô trick) Let d ≥ 2, η ε be the stationary solution to (1.5) with λ ε given as in (1.6). For any p ≥ 2, T > 0 and F ∈ L 2 (Ω) with finite chaos expansion, i.e. F ∈ ⊕ n j=1 H j for some n ∈ N, there exists a constant C = C(p, n) > 0 such that For p = 2, C can be taken independent of n.
We refer the reader to [CES21, Lemma 4.1] for a proof. The n-dependence of C for p > 2 arises when estimating the L p norm in the right-hand side of [CES21, Eq. (4.7)] with the L 2 norm, using Gaussian hypercontractivity. We now move to the proof of tightness.
Proof. By Mitoma's criterion [Mit83], the sequence {η ε } ε is tight in C([0, T ], S ′ (T d )) if and only if {η ε (φ)} ε is tight in C([0, T ], R) for all φ ∈ S(T d ). Therefore, we fix φ ∈ S(T d ) and consider the process {η ε t (φ)} t∈[0,T ] . In view of (2.1), it suffices to show that each of the terms at the right hand side is tight. This is clear for the last, as it is independent of ε, while for the first and second, Lemma 3.1 implies that, for every 0 ≤ t ≤ T and p > 2, we have where in the last step we used that the kernel of N ε φ (η ε ) is A ε + φ. Since the process {η ε t (φ)} t∈[0,T ] is Markov, (3.3) and (3.4) together with Kolmogorov's criterion ensure that the sequence {η ε (φ)} ε is tight in C([0, T ], R), so that the proof of the statement is concluded.

Convergence
In order to identify the limit equation, we will prove that any limit point satisfies the martingale problem associated to (3.1) which we now recall. . Let µ be a zero-average space white noise on T d . We say that a probability measure P on (Ω, G) solves the martingale problem for L eff def = L 0 + D = 1 2 (∆ + D SHE (w · ∇) 2 ) with initial distribution µ, if for all φ ∈ S(T d ), the canonical process η under P is such that is a local martingale, where F 1 (η) def = η(φ) and F 2 (η) def = η(φ) 2 − ∥φ∥ 2 L 2 (T d ) .
As a consequence, Now, all terms at the right hand side are local martingales, which implies that so is the left hand side. Hence, the proof of the statement is complete.
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3. By Proposition 3.2, we know that the sequence {η ε } ε is tight in C([0, T ], S ′ (T d )), hence it converges along subsequences. If we prove that any limit point is a solution of the martingale problem in Definition 3.3 then the statement follows by Theorem 3.4. Let η ∈ C([0, T ], S ′ (T d )) be a limit point. To verify that η satisfies the martingale problem, we need to show that for every given φ ∈ S(T d ) the processes M(F i−1 ), i = 2, 3, defined according to (3.5) with F 1 = η(φ) and F 2 = η(φ) 2 − ∥φ∥ 2 L 2 (T d ) , are local martingales, for which it suffices to prove that for every s ∈ where we introduced a convenient notation for the time increment, i.e. δ s,t f def = f (t) − f (s). Now, by the definition of M(F i−1 ), we deduce that (3.7) where we used that η ε converges in law to η in C([0, T ], S ′ (T d )) together with [CETar, Eq. (5.11)] to approximate the first factor in the expectation with bounded continuous functionals. To analyse this latter term, let us write the equation for η ε in such a way that we can identify the elements which are relevant to the limit. By Dynkin's formula and the weak formulation of (1.5) in (2.1), we have is the martingale whose quadratic variation is and D k is the Malliavin derivative in (1.21). The term which is responsible for creating the new noise and the new Laplacian, is that containing A ε + F i−1 . In order to describe it, notice that the kernel of F 1 in ΓL 2 is f 1 = φ, while that of F 2 is φ ⊗ φ. Then, we consider the random variable W ε,n ∈ L 2 (P), for n ∈ N, whose kernel is w ε,n , which is the solution of the truncated generator equation v ε,n in (2.39) if d ≥ 3 orṽ ε,n given by (2.40) if d = 2. By Dynkin's formula applied to W ε,n , we have where M ε ( W ε,n ) is the martingale whose quadratic variation is the same as (3.9) with F i−1 replaced by W ε,n . We now go back to (3.8) which we rewrite as F i−1 (η ε t )−F i−1 (η ε s )− t s L eff F i−1 (η ε r )dr = δ s,t (M ε · (F i−1 )+M ε · ( W ε,n ))+δ s,t R ε,n (3.11) where R ε,n def = 4 j=1 R ε,n j and the R ε,n j 's are defined as R ε,n 1 (t) def = W ε,n (µ) − W ε,n (η ε t ) , R ε,n 2 (t) We now get back to (3.7), which, in view of (3.10) equals lim n→∞ lim ε→0 E δ s,t (M ε · (F i−1 ) + M ε · ( W ε,n )) + δ s,t R ε,n G(η ε ↾ [0,s] ) .