Space-time discrete KPZ equation

We study a general family of space-time discretizations of the KPZ equation and show that they converge to its solution. The approach we follow makes use of basic elements of the theory of regularity structures [M. Hairer, A theory of regularity structures, Invent. Math. 2014] as well as its discrete counterpart [M. Hairer, K. Matetski, Discretizations of rough stochastic PDEs, 2015]. Since the discretization is in both space and time and we allow non-standard discretization for the product, the methods mentioned above have to be suitably modified in order to accommodate the structure of the models under study.


Introduction
The celebrated KPZ equation is a singular Stochastic Partial Differential Equation (SPDE) which was introduced by three physicists, Kardar, Parisi and Zhang to which it owes its name [KPZ86], as a description of the fluctuations of a randomly growing surface. It is formally given by the expression in which h is the height function (the surface mentioned above) and ξ is a space-time white noise, i.e. a Gaussian random field on L 2 (R 2 ) whose covariance satisfies E ξ, ϕ ξ, ψ = ϕ, ψ L 2 , for ϕ, ψ ∈ L 2 (R 2 ). It is well-known that the realizations of ξ, seen as a random distribution, belong almost surely to a Hölder-type space of distributions of regularity α < − 3 2 (see below for the exact definition of these spaces). The regularising properties of the heat operator ∂ t − ∆ then imply that the expected regularity of h cannot be better than α + 2 < 1 2 which in turn makes it unclear what meaning to attribute to the square of its derivative. That said, the KPZ equation is presumed (and in certain cases, shown) to be a universal object and, by now, people have developed various methods in order to make sense of it. The first significant work in this direction is due to Bertini and Giacomin [BG97]. They defined the solution to (1.1) as the logarithm of the solution to the linear multiplicative stochastic heat equation (known to be strictly positive [Mue91]), which, in turn, can be treated thanks to classical tools from stochastic analysis (see for example [Wal86]). They not only showed well-posedness but also proved that a suitably rescaled particle system (the Weakly Asymmetric Simple Exclusion Process, briefly WASEP) converges to it, thus confirming that their notion of solution is the physically correct one. Nevertheless, their approach is based on a non-linear transformation (the logarithm mentioned above), the so-called Cole-Hopf transform, that is in general difficult to implement at the discrete level (although it has been successfully done in various but specific situations, see [DT16,CT15,CST16,Lab16,CS16]).
More recently, in [GJ10,GJ14], Gonçalves and Jara introduced the notion of energy solution, which can be seen as a martingale problem-type formulation of the KPZ equation, and proved existence of such solution as well as the fact that all the limit points of a significant class of suitably rescaled interacting particle systems (among which WASEP and zero-range processes) are indeed energy solutions. The problem of uniqueness has been settled for a slight reformulation of this notion, given in [GJ13], by Gubinelli and Perkowski in [GP15b] and from then on a number of results confirming the claim of universality has been obtained, e.g. [GS15,FGS15,BGS16,GJ16,GP16,DGP16]. The main drawback of their method lies in the fact that, in order to apply it, it is necessary to know the invariant measure for the system explicitly and have a good control over it already at the discrete level.
Almost contemporarily, a different way of making sense of (1.1) has been established. Thanks to the theory of rough paths first [Hai13] (introduced by Lyons in [Lyo98]) and paracontrolled calculus [GP15a] (first in [GIP15]), regularity structures [Hai14] then, it is now possible to give a pathwise meaning to the KPZ equation which has the significant advantage of dealing with the equation directly (no need of the Cole-Hopf transform) and the invariant measure plays no role whatsoever. Such techniques are extremely stable and general enough to be applicable to a huge number of singular SPDEs. The price to pay though is that, in this context, one has to work with strong topologies and the solution is identified in spaces constrained by certain non-trivial algebraic relations that are not always easily identifiable. Nevertheless, they have been proved to be very powerful and a significant body of literature is now available, [HQ15,HS15,GP15a,FH16,Hos16].
The aim of the present paper is to show that a family of space-time discretizations of the KPZ equation converges to its solution once the discretization is removed. For reasons that will be clarified later, we will work with the Stochastic Burgers Equation (SBE) obtained by formally taking the derivative of h in (1.1), i.e. setting u def = ∂ x h so that u solves where u 0 is the initial condition and ξ a space-time white noise. We will assume throughout periodic boundary conditions, so that the space variable ranges over the one-dimensional torus T def = R/Z. Let ε > 0, ξ be as above and define the discrete noise as , where, given z = (t, x),z = (t,x) ∈ R 2 , |z −z| s def = |t −t| 1/2 ∨ |x −x| is the parabolic norm on R 2 . Then, the family of space-time discretizations we have in mind is obtained by a forward explicit scheme, which, at scale ε reads D t,ε 2 u ε (z) = ∆ ε u ε (z) + D x,ε B ε (u ε , u ε )(z) + D x,ε ξ ε (z) , u ε (0, ·) = u ε 0 (·) , (1.4) where z = (t, x) belongs to the space-time grid (ε 2 N) × T ε , u ε 0 is the initial condition, {ξ ε (z)} z is as in (1.3),D t,ε 2 is the forward difference (i.e.D t,ε 2 f (t) = ε −2 (f (t + ε 2 ) − f (t))) and ∆ ε , D x,ε , B ε are, respectively, the discrete laplacian, spatial derivative and product whose precise definition will be given in (3.17). Only using the most basic elements of the theory of regularity structures [Hai14] and their discrete counterpart (see [HM15]), we can prove the following theorem, which represents the main result of this work. Theorem 1.1. Make Assumptions 3.12, 3.13, 3.14 on the discrete operators ∆ ε , D x,ε , B ε and fix η > −1. Let ξ be a space-time white noise on the probability space (Ω, F , P) and {ξ ε (z)} z , z ∈ (ε 2 Z) × T ε , be given by (1.3). For {u ε 0 } ε , a sequence of random functions on T ε independent of ξ ε , let u ε be the solution to (1.4) with respect to ξ ε , starting at u ε 0 . If there exists u 0 ∈ C η such that lim ε→0 u ε 0 , u 0 (ε) in probability (the norm · (ε) C η being introduced below), then for every α < − 1 2 there exists a stopping time T ∞ and a sequence of constants C ε converging to a finite constant C such that for all T < T ∞ lim ε→0 u ε , u u(0, ·) = u 0 (·) (1.5) and the constant C, introduced above, depends on the specific definition of the discrete laplacian, derivative and product.
The presence of a finite constant in (1.5) does not come as a surprise. If it is true in general that specific features of the discrete model can influence the shape of the limiting equation, it is not clear what is the physical meaning of the transport term in this context. The constant C, in turn, depends on the definition of the discrete operatorsD t,ε 2 , ∆ ε , D x,ε and B ε . We point out that a similar phenomenon has already been observed in [GP15a] when taking discretizations of the type in (1.4) but involving only the space variable.
Remark 1.2. The value of the constant C is specified in (3.21). Properly choosingD t,ε 2 , D x,ε , ∆ ε and B ε , it might happen that C can be taken to be 0 (for example, under the assumptions of Theorem 1.1, if D x,ε is antisymmetric). Nevertheless, in general this is not the case. Moreover, we underline that the discretization in time plays a non-trivial role in the sense that "our" C is different from the one in [GP15a,Theorem 8.2] even in situations in which one might think it could be the same (e.g. B ε being the point-wise product or the Sasamoto-Spohn model).
Before proceeding to the proof of the previous theorem let us point out some of the key aspects of the present work.
• To prove convergence of the discrete systems introduced above, we first have to clarify what we mean by solution for (1.2) and, to do so, we solve SBE with a slightly new methodology which is applicable to a number of different singular SPDEs (the dimension plays no role whatsoever). In order to make sense of the nonlinearity in space, (and only for that!) we will exploit the most basic elements of the theory of Regularity Structures (read: the reconstruction theorem) while the Schauder estimates that one needs in order to close the fixed point are obtained by regarding the heat kernel as a rescaled test function, with time as the scaling parameter (see Section 4.1 for more on this). In particular, Lemma 4.1 and Proposition 4.4, that represent the core of the argument, are valid for more general space-time kernels, namely any for which one can show a sufficiently fast decay at infinity (for the purpose of this paper, we will consider as an example the off-the-grid extension of the space-time discrete heat kernel). This means that, on one side we can avoid the complicated construction carried out in [Hai14,Sections 5], on the other it leaves the possibility to treat time as a separate quantity, and this, in certain situations, can come at hand. The overall approach seems to us more direct (and similar in spirit to [GIP15]) even if definitely not as systematic as [Hai14].
• The fact that we are discretizing the product through the operator B ε will play an important role in our analysis.
For two functions f and g on T ε , B ε is a twisted product given by the average of the product of f and g evaluated at different points (for the definition, see (3.17c)). It includes the usual pointwise product, general Zabusky-type discretizations (see [KZ65]) and many others (in [GP15a], the authors refer to them as discretizations of Sasamoto-Spohn type). As we pointed out before, in order to make sense of the ill-posed terms in the equation one has to identify a suitable subspace of the space of Hölder functions/distributions constrained by specific algebraic relations, depending on the structure of the equation itself. This twisted product naturally requires to modify such relations and all the constructions/definitions behind them in a non-trivial way. This is the first time in which this is done in the discrete setting in the context of regularity structures and hints at the way in which such a procedure might be carried out for actual interacting particle systems.
• In general, one cannot expect the product of gaussians to be invariant for the system in (1.4) for any choice of the discrete operators, and the discrete Cole-Hopf transform (known also as Gärtner transform [Gar88]) of u ε does not seem to solve any suitable discrete version of the stochastic heat equation, so that it is not clear how one could apply these methods in order to obtain a result as Theorem 1.1 in its full generality.
• The advantage of taking space-time discretizations lies also in the fact that such a scheme is in principle directly numerically implementable while when the discretization is only in space then another discretization step is needed.
In this paper though we do not obtain explicit rates of convergence, leaving the question for future investigations.
At last let us discuss previous works and related literature. Although discretizations in both space and time in the context of singular SPDEs and these new pathwise techniques have never been considered before, we point out that the overall strategy is similar to situations in which the discretization is only in space. In this case, we mention in particular [GP15b] where the authors, among the various, show convergence for a class of spatially discrete models to the KPZ equation, imposing assumptions on the discrete laplacian, derivative and product very similar to the ones of the present paper. Their approach is based though on paracontrolled calculus (see also [ZZ14,ZZ15] for the space discretization of stochastic Navier-Stokes and Φ 4 3 equations with similar techniques), which represents a significant difference with respect to our work. On the other hand, more closely related to ours is [HM15], in which it is given a general framework to deal with space discretizations of singular SPDEs. Nevertheless, the discretization of the product there considered is only the pointwise one and it follows more closely [Hai14].
Plan of the paper. In Section 2, we recall the ingredients of the theory of Regularity Structures and its grid counterpart, we will need in order to make sense of the nonlinearity. In Section 3, we explain the ideas behind our approach and the procedure we follow, precisely introducing all the quantities of interest and stating all the main results. Section 4 contains the proof of the discrete and continuous Schauder estimates as well as the fixed point argument and Section 5 is devoted to the stochastic estimates and the definition of the discrete and continuous stochastic processes that appear in the description of our solution. At last, in Section 6, we prove Theorem 1.1.

Spaces and conventions
In this section we want to introduce and recall the definition of the function spaces we will be using throughout the rest of the paper. We begin with some useful conventions that will make the notations lighter.
Fix T > 0. Let t,t ∈ (0, T ], x,x ∈ R and set z = (t, x) andz = (t,x). Then, for a function ζ : (0, T ] × R → R, we define the increment operators in space, time and space-time respectively as (1.6) Given t,t, x,x ∈ R, let us introduce the following quantities where in the last we set z = (t, x) andz = (t,x) and | · | s is the parabolic norm on R 2 . For α ∈ (0, 1), η ∈ R, we say that a function ζ : (0, T ] × R → R is α-Hölder in space with explosion rate η, briefly ζ ∈ C α η,T , if the quantity is finite. We say instead that the function is parabolic α-Hölder continuous, i.e. ζ ∈ C α,s η,T , if In case ζ is (parabolic) α-Hölder continuous but does not exhibit any blow-up as t approaches 0, we will simply write ζ ∈ C α T (or C α,s T ). The norm on the latter space is the same as before but with η = 0 at the first summand of (1.7), and η = α in the second of both (1.7) and (1.8).
For a point z ∈ (0, T ] × R, we also introduce the following quantity where the generic points z,z are given by z = (t, x) andz = (t,x). We will use the previous seminorm also for functions on ((0, T ] × R) 2 . If R(·, ·) is such a function, its · η,α;T,z seminorm has the same definition as above but the increment δ z,· ζ is replaced by R(z, ·). Moreover, it is immediate to verify that ζ ∈ C α,s η,T is equivalent to where again the generic point z is (t, x). Let S(R) be the space of Schwartz functions and S ′ (R) its dual, i.e. the space of tempered distributions. In order to measure the regularity of functions on (0, T ] with values in S ′ (R), we introduce the spaces C α η,T and C α,s η,T , for α < 0 and η ∈ R. C α η,T contains functions of time, ζ : (0, T ] → S ′ (R), such that, for every t ∈ (0, T ], ζ(t, ·) belongs to the dual of C r , r > −⌊α⌋ and where B r 0 is the space of C r compactly supported functions whose C r -norm is bounded by 1 and ϕ λ x is the rescaled version of ϕ ∈ B r 0 centered at x ∈ R, i.e. ϕ λ x (y) = λ −1 ϕ(λ −1 (y − x)). In case η = 0 we simply write C α T , meaning with this the space C((0, T ], C α (R)). On the other hand, ζ ∈ C α,s T if ζ belongs to the dual of C r,s ∞ and where B r,s 0 is the space of C r,s ∞ compactly supported functions whose C r,s ∞ -norm is bounded by 1 and ϕ λ z is the rescaled version of ϕ ∈ B r,s 0 centered at z ∈ R 2 , i.e. ϕ λ z (z) = λ −3 ϕ(λ −2 (t − t), λ −1 (x − x)), where z = (t, x) andz = (t,x). Sometimes we will work with families of distributions ζ z ∈ S ′ (R) parametrised by z ∈ [−T, T ] × R. For α < 0 we denote by L α T the space of such distributions, belonging to the dual space of C r 0 for an integer r > −⌊α⌋, and equipped with the seminorm

Space-time discrete norms
We here define the analog of the norms introduced in the previous paragraph but for functions defined on a grid. Let T > 0, N ∈ N and set ε def = 2 −N . The discrete grids we will be working with are Given t,t > 0, we define |t| ε def = |t| 1/2 0 ∨ ε and |t,t| ε def = |t| ε ∧ |t| ε . Now, for α ∈ (0, 1), η ∈ R and a function ζ : Λ s ε,T → R, we introduce the discrete analog of (1.7) and (1.8) as (1.14) By analogy to what has been done above, we define ζ (ε) η,α;T,z according to (1.9). For α < 0, we set where ·, · ε indicates the discrete pairing, i.e. f, g ε = ε x∈Λε f (x)g(x) and all the quantities above are the same as in the continuous case, and ζ where, in this case, ·, · ε indicates the parabolic discrete pairing, i.e. f, g ε = ε 3 . For discrete families of functions on the grid parametrized by z ∈ Λ s ε we write At last, for α < 0 and η ≤ 0, when comparing a discrete map, ζ ε on Λ s ε,T (or Λ ε ) with a map ζ ∈ C α η,T (or in C α ), with a slight abuse of notation, we will use the following norm (and, correspondingly, when ζ and ζ ε do not depend on time) where, inside the absolute value, ·, · ε is the discrete pairing introduced above, while the second is the usual evaluation of a distribution on a test function.

Notation.
In the rest of the paper, to indicate that a sequence of maps ζ ε parametrized by ε satisfies ζ ε (ε) G < ∞ uniformly in ε, where G is any of the spaces defined above, we will write ζ ε ∈ G ε .

Elements of regularity structures
The aim of this section is twofold. On one side we want to recall the elements of the theory of regularity structures we need in order to make sense of the ill-posed products appearing in the equation we want to treat. On the other, we will see how to modify these notions to be able to consider functions and "objects" defined on the space-time dyadic grid. No claim of completeness is made and the interested reader is addressed to [Hai14,HM15] for the first part (for a thorough introduction see also [CW15,FH14,Hai15]), and to [HM15] for the second.

Basic definitions and inhomogeneous models
We begin by defining what a regularity structure is.
Definition 2.1. A regularity structure T is a triplet (A, T , G), in which A is a locally finite bounded from below set of "homogeneities" such that 0 ∈ A, T = α∈A T α , the model space, is a graded vector space and each of the T α is a finite dimensional Banach space whose norm will be denoted by · , G, the structure group, is a set of linear transformations on T , such that for every Γ ∈ G, α ∈ A and τ ∈ T α one has Γτ − τ ∈ β<α T β . Furthermore, we assume that A ⊂ (−1, 1) and T 0 is generated by the basis vector 1, which represents the abstract unit. For τ ∈ T α , we will denote by |τ | = α the homogeneity of τ .
Remark 2.2. Given a regularity structure T and an element τ ∈ T , we will indicate by Q α τ its canonical projection onto T α , and define τ α def = Q α τ , i.e. the T α -norm of the component of τ in T α . We also write Q <α , Q ≥α etc. for the projection onto T <α def = β<α T β , etc.
In practice, a regularity structure is nothing but a list of symbols constrained by specific algebraic requirements. In order to attribute them a meaning, M. Hairer introduces the notion of model that we here recall in the variation given in [HM15,Def. 2.4].
Definition 2.3. Let T = (A, T , G) be a regularity structure. An (inhomogeneous) model Z = (Π, Γ, Σ) for T consists of three collection of maps: {Γ t } t∈R is such that Γ t : R × R → G and for any x, y, z ∈ R and t ∈ R, where Id is the identity map; {Σ x } x∈R is such that Σ x : R × R → G and for all x ∈ R and s, t, r ∈ R, is linear and for all x, y ∈ R and t ∈ R, Moreover, for any γ > 0 and every T > 0, there exists a constant C γ > 0 for which the bounds hold uniformly over all λ ∈ (0, 1], ϕ ∈ B 1 0 , |x − y| ≤ 1, t ∈ [−T, T ], τ ∈ T with |τ | < γ, and m ∈ A such that m < |τ |. Remark 2.4. For a model Z = (Π, Γ, Σ), we denote by Π γ;T , Γ γ;T and Σ γ;T the smallest constant C γ such that the bounds on Π, Γ and Σ in (2.2) hold and set |||Z||| γ;T def = Π γ;T + Γ γ;T + Σ γ;T . We also define the family of semidistances between two models as Even though, in general, Z −Z is not a model, notice that the expressions on the right hand side still makes sense.
Loosely speaking, for every x ∈ R the map Π t x assigns to every symbol a distribution that vanishes (or explodes) at the right order around x (here, "right" is understood in the sense of homogeneities) while the families of Γ t 's and Σ x 's guarantee that these expansions are consistently sewed together in space and time respectively.
We can now define a space of functions taking values in a regularity structure and representing the model dependent counterpart of the space of Hölder functions (compare (2.3) below with (1.8)). Let T = (A, T , G) be a regularity structure and Z = (Π, Γ, Σ) a model on T . Given γ, η ∈ R, we say that a function H : where the first summand of the right hand side of (2.3) is given by (2.4) Functions belonging to these spaces are called modelled distributions. In order to study their continuity properties with respect to the underlying model, we will need to be able to compare modelled distributions belonging to the space D γ,η T , but based on different models. Let Z = (Π, Γ, Σ),Z = (Π,Γ,Σ) be two models on T , and H ∈ D γ,η T (Z),H ∈ D γ,η T (Z) two modelled distributions, then a natural notion of distance between them can be obtained via replacing in the second summand of (2.3). We denote the result by |||H;H||| γ,η;T , this notation being due to the fact that, as a distance, |||·; ·||| γ,η;T is not a function of H −H.
A modelled distribution can be seen as the generalized abstract Taylor expansion of a given function, in which the elements appearing in its expression are not only abstract polynomials but also other "jets" belonging to the regularity structure and whose meaning is encoded in the model. That said, we want to be able to associate a concrete function/distribution to a modelled distribution and this can be done via the reconstruction operator [Hai14,Thm. 3.10]. The following is an adaptation of the aforementioned result, better suited to deal with inhomogeneous models, and can be found in [HM15, Thm. 2.11].
Theorem 2.5. Let T = (A, T , G) be a regularity structure with α def = min A < 0 and Z = (Π, Γ, Σ) be a model. Then, for every η ∈ R, γ > 0 and T > 0, there exists a unique family of linear operators R t : D γ,η T (Z) → C α (R), parametrised by t ∈ (0, T ], such that the bound is the associated family of operators, then the bound holds uniformly over H ∈ D γ,η T (Z),H ∈ D γ,η T (Z) and the same parameters as above. The map R introduced in Theorem 2.5, is the so-called reconstruction operator. We will always postulate in what follows that R t = 0, for t ≤ 0.
Notice that the statement above does not only guarantee that we can uniquely associate a distribution to a modelled distribution in D γ for γ > 0, but it also tells us what such a distribution "looks like" in a neighborhood of every point. Indeed, the bound (2.5) gives a good control over its reconstruction since the image through the model of the modelled distribution is fully explicit (after all, Π is a linear map on the abstract "jets" in the regularity structure). Moreover, in case a model Z is composed of smooth functions, then, for any modelled distribution H ∈ D γ,η , γ > 0, RH is a continuous function satisfying for all (t, x) ∈ (0, T ] × R the identity As a first concrete application of the previous theorem, we want to recall a proposition stated and proved in [Hai14,Prop. 4.14]. It shows how it is possible to obtain, in this context, a classical result of harmonic analysis concerning the product of two Hölder functions/distributions. We formulate it here for functions/distributions Hölder in space with a blow-up in time as t approaches 0 (see (1.7)).
Proposition 2.6. Let α 1 , α 2 , η 1 , η 2 ∈ R\N and T > 0. Set α def = α 1 ∧α 2 and η def = (η 1 +α 2 ∧0)∧(η 2 +α 1 ∧0)∧(η 1 +η 2 ). Then, the map (f, g) → f g extends to a continuous bilinear map from C α1 η1,T × C α2 η2,T to C α η if and only if α 1 + α 2 > 0 and in this case there exists a constant C > 0 such that As we will see in the upcoming sections, what presented so far is everything we need from the whole theory of regularity structures, in order to be able to solve the equation (1.2). Before delving into the details and understand how to put this into practice, we want to introduce the discrete counterpart of the objects previously defined.

Discrete models and modelled distributions
Since we aim at showing the convergence of the discrete equation to the continuous one exploiting the built-in stability of these techniques, on one side we have to introduce a suitable space-time discrete notion of models, modelled distribution and reconstruction operator while on the other we need to verify that the latter satisfy properties similar to the ones enjoyed by their continuous versions.
We will work on the dyadic grids (1.12) with ε = 2 −N . While the notion of regularity structure clearly does not need to be adapted, we begin with the following.
Let γ, η ∈ R, T = (A, T , G) be a regularity structure and Z ε = (Π ε , Γ ε , Σ ε ) be a discrete model. For a function H ε on Λ s ε,T with values in T <γ , we define where, in both cases l ∈ A. To indicate that a sequence of such maps H ε parametrised by ε satisfies |||H ε ||| (ε) γ,η;T < ∞ uniformly in ε, we will write H ε ∈ D γ,η ε,T (Z ε ) and call such functions discrete modelled distributions. We are now ready to give the following definition.
Definition 2.8. Let γ, η ∈ R, T = (A, T , G) be a regularity structure and Z ε = (Π ε , Γ ε , Σ ε ) be a discrete model. For a discrete modelled distribution H ε ∈ D γ,η ε,T (Z ε ), we define for all (t, x) ∈ Λ s ε,T the discrete reconstruction map R ε by Notice that, while in the continuous case, for a smooth model relation (2.6) is a consequence of the reconstruction theorem, here we are defining the discrete reconstruction operator according to (2.9) because, as we will see, such definition is well-suited for our purposes. Nevertheless, in the discrete setting, we have in principle much more freedom in the choice of the discrete reconstruction operator, because, after all, we only have to specify its value on a finite number of points. In any case, no matter how we define it, what one needs is to obtain an analog of (2.5) which is uniform in ε and this is what the next theorem shows. Theorem 2.9 (Thm. 4.5, [HM15]). Let γ, η ∈ R, T = (A, T , G) be a regularity structure with α def = min A < 0 and Z ε = (Π ε , Γ ε , Σ ε ) be a discrete model. Then the bound holds for all H ε ∈ D γ,η ε,T (Z ε ), locally uniformly over (t, x) ∈ Λ s ε,T , ϕ ∈ B 1 0 and uniformly in λ ∈ [ε, 1], and the hidden constant does not depend on ε. If furthermoreZ ε = (Π ε ,Γ ε ,Σ ε ) is another model andR ε t : D γ,η ε,T (Z) → C α (R) is again defined according to Definition 2.8, then the bound holds uniformly over H ε ∈ D γ,η ε,T (Z ε ),H ε ∈ D γ,η ε,T (Z ε ) and the same parameters as above.

Analysis of the continuous and discrete equations
As remarked in the introduction, the difficulty in making sense of the KPZ equation comes from the fact that the spacetime white noise, as a random distribution, is extremely singular and consequently the expected regularity of the solution does not allow to classically define the non-linear term appearing on the right hand side of (1.1). Instead of the latter, we will focus on the SBE given by where ξ is the space-time white noise on the one-dimensional torus and u 0 the initial condition. We will mainly work with the mild formulation of the previous, so let P : R 2 → R be the Green's function of ∂ t − ∆, which, for t > 0, is given by and is 0 for t ≤ 0. We mollify the noise ξ, i.e. we consider ξ ε def = ξ * ̺ ε , where ̺ is a symmetric compactly supported smooth function integrating to 1 and ̺ ε (t, x) def = ε −3 ̺(ε −2 t, ε −1 x) its rescaled version, and write where P ′ def = ∂ x P , the first summand on the right hand side is a purely spatial convolution and * denotes the convolution in space-time.
The idea developed in [Hai13,GP15a] (and [Hai14]) is then to split the analysis of the equation (1.2) in two distinct modules. At first one infers from (3.1) a minimal set of processes, the controlling processes X, built from ξ, and postulates that there is a way, not only to properly define them but also to prove that they satisfy certain regularity requirements. Then, one identifies a suitable subspace of the space of distributions, depending on such processes, for which it is possible to make sense of the ill-posed operations and formulates a fixed point map that is continuous in these data. At last, one exploits stochastic calculus techniques to show that it is indeed possible to construct X starting from a space-time white noise.
In this section we want to accomplish two goals. First, we will carefully carry out the program outlined above in the context of (1.2), introducing all the quantities of interest. Then we will present the family of discrete models we want to deal with, explaining the procedure to follow in order to prove their convergence to the solution of (1.2), and precisely state all the main results, postponing their proofs to the subsequent sections.

The controlling processes
For reasons that will soon be clarified, we need to split the heat kernel P in a "singular" and a "regular" part. To do so, we consider a smooth compactly supported function χ : R 2 → [0, 1] such that χ(z) = 1 for z s ≤ 1 2 and χ(z) = 0 for z s > 1 (of course the choice of 1 is completely arbitrary). Then we set, for z ∈ R 2 so that P = K +K. Clearly K is compactly supported, smooth except at 0 and coincides with P in a neighborhood of the origin, whileK is even, smooth and, when dealing with convolutions ofK with spatially periodic distributions on the time interval [−1, 1], we can assumeK to be compactly supported (see [Hai14,Lem. 7.7] for the precise statement and the proof of this claim). As for P , we will indicate by K ′ (resp.K ′ ) the spatial derivative of K (resp.K).
Let us try to heuristically understand how to identify what the above mentioned controlling processes in our case are. Recall that we aim at solving (1.2) in a suitable function space of Hölder type and, looking at (3.1), it is reasonable to expect that the main contribution to the regularity of its solution, u, will come from the singular part of the stochastic convolution P ′ * ξ, i.e. K ′ * ξ, since what is left, no matter how (ir-)regular ξ is, is smooth. Hence, if in terms of regularity we can presume that u "looks like" K ′ * ξ at small scales, briefly, u ∼ K ′ * ξ, then the ill-posed product at the right hand side should satisfy u 2 ∼ (K ′ * ξ) 2 . At this point, we can further detail the expansion of u beyond the first term. Indeed, referring again to (3.1), we see that if u 2 ∼ (K ′ * ξ) 2 then u ∼ K ′ * ξ + K ′ * (K ′ * ξ) 2 and so on.
The point here is that it suffices to pursue such an iteration only finitely many times. Let us be more precise. Replace the noise ξ by a smooth function η and, given X (η) = K ′ * η and two non-zero constants a, b ∈ R, we define the following as well as At last, we introduce, for t ∈ R and x, y ∈ R, the remainders where we omitted the dependence on η not to clutter the presentation.
Remark 3.1. The reason why we introduced certain constants in the expression of some of the processes above is that, when η is the space-time white noise, the products there appearing are ill-posed. To be more precise, in the limit as ε goes to 0, these processes would diverge to ∞ and the only way to prevent them from blowing up is through a suitable renormalization procedure, consisting in surgically remove the divergences. As we will see in the following sections, in our context, this simply amounts to subtract the 0-chaos component from the Wiener-chaos expansion of the stochastic process at hand.
Remark 3.2. In principle, one would expect the presence of a renormalisation constant in the definition of X and X as well, but these constants are killed by the convolution with the spatial derivative of the heat kernel and therefore we can forget about them. This is the advantage of working with SBE instead of KPZ.
As pointed out in the introduction, we want to show not only that we can construct these objects in the case in which η is the space-time white noise but we also need to prove that they have certain space-time regularities. Recall that, as a random distribution, the realizations of the space-time white noise belong almost surely to C α s for any α < − 3 2 (see for example the proof of [Hai14,Prop. 9.5]). Starting from this, one can guess what is the expected regularity of the terms in (3.3) by applying the following rules: 1. by usual Schauder estimates, the convolution with the heat kernel P (and, of course, with its singular part K) increases the regularity by 2, and consequently the one with P ′ (resp. K ′ ) by 1; 2. for β, γ ∈ R, the product of two distributions in C β and C γ respectively belongs to C δ where δ = min{β, γ, β + γ} (see [BCD11, Thms. 2.82, 2.85, Prop. 2.71] or simply Proposition 2.6); 3. since we are subtracting the ill-posed part of the product (i.e. the "diagonal"), the regularity of the terms in (3.3c) will be given by the sum of the regularities of the factors.
Even if the second property analytically holds provided that β + γ > 0, we will simply postulate that this is still the case (and this is the point in which the renormalization will enter the game, see Remark 3.1).
In order to lighten the notation, and following the properties 1.-3. stated above, we assign numbers α τ and βτ , representing the regularities of X τ and Rτ respectively, to each label We do it in a recurrent way: • we fix a parameter α ; • we set α τ = i α τi ∧ 0 + 1, in case the forest {τ i } is obtained by removing the root and the adjacent edges from a tree τ ; • we set α τ = α τ1 ∧ 0 + α τ2 ∧ 0 and β τ = α τ1 + α τ2 , for a label τ = τ 1 τ 2 , with τ i being a tree.
With these notations at hand we are ready to give the following definition.
where the elements appearing in the previous are given by (3.3) (and we hid the dependence on the constants a, b for the processes X , X to lighten the notation). Then we define the space X of controlling processes as where cl W {·} denotes the closure of the set in brackets with respect to the topology of W and We denote by X a generic element of this space and, if η ∈ C α −1,s coincides with the first component of X ∈ X , we say that X is an enhancement (or lift) of η. Table 1: Summary of the values of the α τ and βτ for τ ∈ L andτ ∈ L R respectively. Given α ∈ R, α − stands for α − ε for any ε > 0 small enough.
The following proposition, whose proof is provided in Section 5.2, states that it is indeed possible to enhance the space-time white noise.
Proposition 3.4. Let ξ be a space-time white noise on a probability space (Ω, F , P), ̺ a symmetric compactly supported smooth function integrating to 1 and Then, there exists a sequence of constants C ε and C ε and a controlling process At last, even if the specific expression of these constants depends on the choice of the mollifier, their behaviour as ε → 0 does not, i.e. C ε ∼ ε −1 and C ε can be chosen to be 0.

The ill-posed product, regularity structure and a notion of solution
Thanks to Proposition 3.4, we know how to construct a number of stochastic processes, closely related to the equation (1.2), starting from a space-time white noise ξ. Building on the ideas sketched at the beginning of the previous section (see also [Hai13,GP15a]), let ξ ε be given as before and u ε be the solution of (1.2) with ξ ε replacing ξ. We want to proceed to a Wild expansion of u ε around the solution of the linearized equation, or more precisely, around the singular part of it. Indeed, we expect that, upon subtracting sufficiently many irregular terms, what remains has better regularity properties allowing, on one side, to make sense of the ill-posed product (via the theory of regularity structures) and, on the other, to define a map that can be proved to have a unique fixed point in a suitable space.
To be more concrete, let X(ξ ε ) ∈ X be an enhancement of ξ ε according to Definition 3.3 and we define v ε as where u ε is as above (i.e. the solution to (1.2) driven by the mollified noise ξ ε ). Then, v ε solves the following stochastic PDE (written in its mild formulation), with u 0 as initial condition, , the convolution involving only the space-variable, and whereK is the smooth part of the heat kernel introduced in (3.2).
We now want to pass to the limit as ε goes to 0. Since Proposition 3.4 will take care of the terms belonging to X(ξ ε ), we can focus on the others. In the ε-limit, the regularity of v ε , α ⋆ cannot be better than the one of X ε , which is α < 1 2 . Assuming also α ⋆ > −α and thanks to the bounds on α , one easily sees that all the summands in (3.5) and (3.6) are analytically well-defined (all the products fall into the scope of Proposition 2.6) apart from v ε X ε , for which we will need the theory of regularity structures.
To take a glimpse at the idea behind our approach (and, more generally, the rough path/regularity structures approach), we notice that if we look at the small increments of the mild solution to the equation (3.5), then, at least formally, they should behave as the ones of the process X ε . More precisely, it is reasonable to expect the relation to hold, where v ′ ε = 4X ε + 2v ε , and R ε has "better" regularity than X ε . This formal relation hints at the way in which the regularity structure should be defined in order to be able to encapsulate a suitable description of the process v ε and consequently to make sense of the ill-posed term in (3.5) in the limit.
To this purpose we now define a regularity structure (A, T , G) such that each controlling process X ∈ X gives raise to a model on it. Let α τ , βτ be the parameters introduced in the previous section. Then we set where the Banach spaces T α , T β , T 0 and T α are copies of R with the unit vectors , , 1 and respectively.
Let X ∈ X , then we introduce the model Z = (Π, Γ, Σ) given by Π t x 1 = 1 as well as For any points s, t, x, y ∈ R, the maps Γ t and Σ x are defined as Γ t xy 1 = 1, Σ st x 1 = 1 and It follows immediately from the definition of the space X that the maps (Π, Γ, Σ) have all the algebraic and analytic properties required in Definition 2.3. Furthermore, one can define a structure group G in such a way that the operators Γ t xy and Σ st x belong to G, see [Hai14,HM15]. Motivated by the expansion (3.11), we can define the modelled distributions, which describe v and vX at the abstract level. More precisely, given two functions v, v ′ : which corresponds to the product between the modelled distributions V and t (x) def = 1 . The idea is now to define the product vX in (3.5) as R V , where R is the reconstruction operator associated to the model Z = (Π, Γ, Σ). Indeed, provided that V belongs to a space D γ,η T (Z) for some γ > 0, Theorem 2.5 guarantees that this quantity is indeed welldefined.
In order to verify this latter statement and define a fixed point map, we need to introduce a suitable function space. To this end, we fix another global parameter α ⋆ > 0 which, as before, represents the regularity of the solution v. Now, given and we denote by H η,γ T the completion of such smooth triplets under this seminorm. Then we define the space to which the solution will belong. Definition 3.5. Let α ⋆ > 0, γ > −α and η ∈ (−1, 0). Let X be the space of controlling processes as in Definition 3.3. Then we define the space of controlled processes X η,γ and V ∈ H η,γ T . Moreover, we define the set of all processes controlled by X ∈ X as endowed with the norms introduced above.
It is now immediate to verify that, given any controlling process in X ∈ X and any process V controlled by X, there is a unique way to define the ill-posed product as the reconstruction of the modelled distribution given in (3.9).
is an enhancement of η and η is a smooth function, then the following equality holds Remark 3.7. The reason why in the actual equation the constant b will not appear is that, thanks to Proposition 3.4, we know that C ε can be chosen to be 0 in case η is a smooth and symmetric approximation of the space-time white noise.
Proof. There is very little to prove. Indeed, the fact that V ∈ D γ,η T (Z) follows by Notice that the function (0, T ] × R ∋ (t, x) → is a modelled distribution in Dγ ,γ for everyγ > 0 whose element of minimum homogeneity is . Therefore, [Hai14, Prop. 6.12] implies that V ∈ D γ+α ,η+α T (Z) (alternatively one could prove this by direct computation). By assumption, γ + α > 0 hence, Theorem 2.5 guarantees that, for every t ∈ (0, T ], R t (V ) is unique and that the map assigning to every couple (X, V ) in X η,γ T the reconstruction of V is locally Lipschitz continuous. At last, the equality (3.12) is another consequence of the aforementioned theorem. Indeed, for η smooth thanks to (2.6), we have and the last term coincides with the right hand side of (3.12) because of the definition of the model since the second summand is equal to v ′ (t, x)b.
At this point we are ready to formulate the map that we will prove to admit a unique fixed point in the space of controlled processes. Let the map M be defined on , the reconstruction operator R corresponds to the model introduced above for the controlling process X ∈ X , and F v and Q are as in (3.6).
Remark 3.8. Notice that, in case the controlling process X ε def = X(η ε , a ε , b ε ) depends on a non-trivial b ε (that, in principle, might even diverge), then, thanks to Proposition 3.6 the equation for u ε would read (3.14) which corresponds to the renormalized equation.
The following theorem states the existence and uniqueness result for the limit of equation (1.2).
Theorem 3.9. Let α ⋆ ∈ −α , α , γ ∈ α , α + α ⋆ ∧ α and η ∈ (−1, 0). Then for every u 0 ∈ C η and every X ∈ X , Remark 3.10. It was shown in [GP15a], that the solution of SBE exists at all times, i.e. T ∞ = ∞ and an analogous argument would work also in this context, but since we will not prove it explicitly here we refrain from stating the previous theorem so to include this observation.
Remark 3.11. In the setting of Remark 3.8, in case the sequence b ε diverges, what the previous Theorem is stating is that, provided X ε converges to a well-defined controlling process then so does the sequence u ε , even if it is not clear what specific equation the limit would solve. Indeed such equation could formally be written as where b might also be +∞.
Before proving the previous theorem, we want to introduce the family of discrete systems we have in mind, state the main assumptions on the quantities mentioned in (1.4) and see how to translate in the discrete setting what we have done so far.

The discrete setting
As already mentioned in the introduction, we now want to deal with a family of space-time discrete systems and prove that each of its members converges to the solution of SBE given in Theorem 3.9. Such systems are defined on a spacetime grid that mimics the parabolic nature of the equation we are working with. To be more precise, we fix N ∈ N and ε def = 2 −N , define the grids as in (1.12) and consider the discrete equations (1.4). In order to define the discrete operators introduced above, we will need three signed measures, ν, π and µ, the first two on R and the latter on R 2 , satisfying the following assumptions.
Assumption 3.12. ν is a purely atomic signed symmetric measure on R, supported on Z ∩ B(0, R ν ), for some radius and its Fourier transform, given byν(ξ) def = R e −2πiξy ν(dy), vanishes only on Z.
Assumption 3.13. π is a purely atomic signed measure on R, supported on Assumption 3.14. µ is a purely atomic signed measure on R 2 , supported on Then, to describe the action of ∆ ε , D x,ε and B ε on functions on the grid Λ ε , we write ν ε (dy) = ν(ε −1 dy), π ε (dy) = π(ε −1 dy) and µ ε (dy 1 , dy 2 ) = µ(ε −1 dy 1 , ε −1 dy 2 ) for the rescaled versions of ν, π and µ respectively, so that, for any ϕ , ψ ∈ ℓ ∞ (Λ ε ) and x ∈ Λ ε , we can define is the total variation of ν. In order to write the mild formulation of (1.4) and build the discrete counterpart of the controlling processes given in Section 3.1, we introduce the operator P ε , defined as the Green's function of the space-time discrete heat operator. More explicitly, P ε is the unique solution of where z ∈ Λ s ε , δ 0,· is the Kronecker delta function and we impose P ε t = 0 for all t ≤ 0. The mild formulation of (1.4) then becomes , and P ε u ε 0 is a discrete spatial convolution. Moreover, proceeding as in (3.2), we define K ε andK ε in such a way that P ε = K ε +K ε and K ε is compactly supported in a ball centered at 0 and coincides with P ε in a neighborhood of the origin, whileK ε is even and can be assumed to be compactly supported (see Lemma 5.9 for the details).
We are now ready to build the discrete controlling processes, whose definition is analogous to (3.3) but accommodates the structure of (3.19). Let η ε be a function on Λ s ε , then, upon setting X , ε (η ε ) = D x,ε K ε * η ε and given two non-zero constants a, b ∈ R, we have (3.20a) as well as Furthermore, for t ∈ Λ ε 2 and x, y ∈ Λ ε , we set where we omitted the dependence of the previous terms on η ε not to clutter the presentation.
We are now ready for the following definition and the subsequent proposition, representing the discrete version of Definition 3.3 and Proposition 3.4. Definition 3.15 (Discrete Controlling Processes). Let α ∈ − 3 5 , − 1 2 and α τ , βτ , for τ ∈ L andτ ∈ L R , be the same as in Definition 3.3 (see also Table 1). For η ε , a function on Λ s ε , and two non-zero constants a, b ∈ R, we set where the elements appearing in the previous are given by (3.20) (and we hid the dependence on the constants a, b for the processes X ,ε , X ,ε to lighten the notation). Then we define the space X ε of discrete controlling processes as the set of families X ε (η ε , a, b), parametrized by ε, such that X ε (η ε , a, b) ∈ W ε , where We denote by X ε a generic element of this space and, if η ε coincides with the first component of X ε ∈ X ε , we say that X ε is an enhancement (or lift) of η ε . Proposition 3.16. Let {ξ ε (z)} z∈Λ s ε be a family of i.i.d. normal random variables with mean 0 and variance ε −3 on a probability space (Ω, F , P). Let ν, π and µ be three signed measures satisfying Assumptions 3.12, 3.13, 3.14 andν,π,μ be their Fourier transforms. Then there exists a sequence of constants 1 As pointed out in the introduction, we here recall that we say that a family of functions f ε on the grid, parametrized by ε, belongs to (for example) uniformly in ε, where C , ε behaves asymptotically as ε −1 and C , ε can be chosen to be independent of ε. More precisely, they are given by def =π(k)/k and both the integrals are finite. Furthermore, let ̺ be a symmetric compactly supported smooth function integrating to 1 and, forε ≥ ε, ̺ε(t, x) ε be the sequence of constants introduced above associated to ξ ε ε . Then, there exists θ > 0 such that, for all p ≥ 1 one has uniformly in ε: At this point we can focus on the analytical aspects of the discrete equation (3.19) and we will follow once again the same procedure described in the previous paragraph. Let X ε ∈ X ε be an enhancement of ξ ε according to Definition 3.15 and set , the discrete convolution involving only the space-variable, and The term that we have to describe through our regularity structure is B ε v ε , X , ε , i.e. a twisted product in whose expression v ε and X , ε are evaluated at two different points. If we look at the expression (3.22), it seems reasonable to expect that v ε admits an expansion of the form R ε a suitable remainder and, to shorten the notations, we set z 1 = z + ε(0, y 1 ) and z 1 =z + ε(0, y 1 ). It is important to keep in mind this expansion in the definition of the abstract setting, and in the next section we will see how to adapt, in this discrete situation, the construction carried out in the continuous case.

The regularity structure, discrete models and modelled distributions
As we pointed out at the end of the previous section, we want to keep track, at the abstract level, of the fact that the product we are considering is twisted. To do so, we introduce a slightly different regularity structure. Let α τ , βτ , for τ ∈ L andτ ∈ L R , be given as in Definition 3.15 (see also the Table 1), then we define (A D , T D , G D ) (where D stands for "discrete"), as and consequently the maps Γ ε are given by and Σ ε are equal to In the following lemma, whose proof is rather immediate, we show that, for every ε > 0, Z ε defined above is indeed a discrete model.
be the regularity structure given in (3.25) and Z ε = (Π ε , Γ ε , Σ ε ) be defined according to (3.26). Then, Z ε is a discrete model for T D in the sense of Definition 2.7.
Proof. The validity of the analytic properties come from the fact that Z ε is built on X ε , so that for Γ ε and Σ ε there is nothing to verify. Recall that Π ε,t x (·) is given by R , ε (x; t,) for which the correct bound holds by assumption. For the other notice that, given λ ∈ and, by a simple change of variable, Concerning the algebraic properties, they can be verified applying the very definitions of the maps Π ε , Γ ε and Σ ε .
At this point, we are ready to define the discrete modelled distribution we will use in order to describe v ε and B ε (v ε , X , ε ). While for the first we want to recover the controlled structure given in (3.24), for the second we would like an expansion better adapted to our twisted product. Let X ε ∈ X ε and Z ε be defined as in (3.26). Given two functions, v ε and v ′, ε , on Λ s ε,T , we define (3.28) As before, we now would like to replace the twisted product between v ε and X , ε appearing in (3.22) by the discrete recontruction of the modelled distribution V ε , so to obtain a quantity which is uniformly well-defined as ε goes to 0.
Let us fix α ⋆ > 0, γ > α and η ∈ R. Given a triplet of functions and we say that a family of triplets V ε , parametrized by ε belongs to H η,γ ε,T if V ε (ε) η,γ;T < ∞ uniformly in ε. The discrete controlling processes can then be defined as follows.
The next proposition shows that the twisted product defined above, behaves sufficiently similarly to the usual one.
Proposition 3.19. Let α ⋆ > 0, γ > −α and η ∈ (−1, 0). Let X ε ∈ X ε , T D = (A D , T D , G D ) be the regularity structure given in (3.25) and Z ε be defined according to (3.26). Let V ε = (v ε , v ′, ε , R ε ) ∈ H γ,η ε,X ε ,T and V ε , (V ) ε be given by the expressions in (3.27) and (3.28), respectively. Then, V ε ∈ D γ,η ε,T (Z ε ), (V ) ε ∈ D γ+α ,η+α ε,T (Z ε ) and, for all a, b), for a, b ∈ R, is an enhancement of η ε and η ε is a function on Λ s ε , then the following equality holds for all (t, x) ∈ Λ s ε,T . Proof. It is immediate to see, by the definition of the model Z ε , that V ε ∈ D γ,η ε,T (Z), so we focus on (V ) ε . Each of the summands in (2.3) can be easily shown to be bounded uniformly in ε, apart from two terms, whose argument is given by They can be treated exploiting the same techniques, so we will consider only the first. Let x, y ∈ Λ ε , such that x = y, and t ∈ Λ ε 2 ,T . Notice that where, to shorten the notations, we set x 1 = x + y 1 ,x 1 = y + y 1 and z 1 = (t, x 1 ),z 1 = (t,x 1 ). It follows that where we applied the assumptions on v ′, ε , R ε , X , ε and µ, and the facts that γ > α , ε ≤ |x − y|. Moreover, the constants hidden in the previous chain of inequalities are clearly uniform in ε. Hence, we can conclude that (V ) ε ∈ D γ+α ,η+α ε,T (Z ε ) and, by Definition 2.8 and Lemma 3.17, we also have which in turn completes the proof.
Thanks to the previous proposition, we are ready to define the map M ε , which represents the discrete counterpart of (3.13). More precisely, let M ε be the map on C η,ε × X η,γ,ε where, to shorten the notations, we set z 1 = z + ε(0, y 1 ) andz 1 =z + ε(0, y 1 ) and the modelled distribution (V ) ε is defined in (3.28), the reconstruction operator R ε corresponds to the model introduced above for the controlling process X ε ∈ X ε , and F (ε) v ε and Q (ε) are as in (3.23). The following theorem represents the discrete version of Theorem 3.9.
Theorem 3.20. Under the assumptions of Theorem 3.9, let X ε ∈ X ε be a sequence of controlling processes u ε 0 ∈ C η,ε be a sequence of periodic functions on Λ ε . Then there exists T d ∞ ∈ (0, +∞] such that for all T < T d ∞ the sequence of solution maps S ε T that assigns to (u ε 0 , X ε ) the solution u ε of (1.4) is locally Lipschitz continuous uniformly in ε with respect to the discrete norms. Remark 3.21. As in Remark 3.8, if the discrete controlling process X ε def = X(η ε , a ε , b ε ) depends on a non-trivial b ε (which is going to be the case in the setting of Proposition 3.16), then, by the equality 3.31 the equation for u ε would read which is the discrete renormalized equation. Nevertheless, similarly to in Remark 3.11, the previous Theorem states that, provided that the sequence X ε is uniformly bounded (in ε) in the space of discrete controlling processes then so is u ε .

Continuous and discrete Schauder estimates and fixed points
The aim of this section is twofold. We will begin by proving the Schauder estimates needed to treat the convolution with the heat kernel and see how they can be put into practice so to build a solution for (1.2). Then we will translate this construction in the discrete setting and obtain uniform bounds on the discrete solution maps.

Continuous convolutions
As already mentioned in the introduction, our approach is based on the ability of treating the convolution with the heat kernel as a testing against a recentered and rescaled test function, where the time variable plays the role of the scaling parameter. After all, the convolution is nothing but a "centering" around a specific point, as t tends to 0 the heat kernel converges weakly to the Dirac delta function and its L 1 norm is constant in t. The missing ingredient is the compactness of the support which is however not essential. Indeed, for t > 0, the heat kernel is a Schwarz function and the following direct generalization of [GIP15, Lem. 6.3] shows that this is more enough to guarantee the validity of what claimed above (actually we prove that it suffices that the function decays sufficiently fast at ∞).
Remark 4.2. The reason why we take into account maps x → ζ x , where ζ x depends on a base point x, is that we want to be able to replace ζ with the image of a model for a certain regularity structure or the term appearing on the left hand side of the reconstruction bound (2.5). In other words, ζ x will represent a local generalized Taylor expansion around the point x.
The next proposition, joint with the previous lemma represent, as we will shortly see, the Schauder estimates we need.
Remark 4.6. In the previous proposition, the assumptions (4.4), in a sense, require the possibility to trade the regularity of the distribution ζ z , for fixed z ∈ (0, T ] × T with the one of the map (0, T ] × T ∋ z → ζ z . As we will see in the following lemma, this is indeed the case for both the model and the reconstruction operator. Proof. Let z = (t, x), w = (s, y) ∈ (0, T ] × T, then, for a function ϕ ∈ S(R) and λ ∈ (0, 1], the triangular inequality and (4.4) immediately give under the assumption that s ≤ t. We apply the previous to bound the first summand at the left hand side of (4.5) and obtain where the last passage holds because ρ > −2, n < β + 2 and θ > 0. For the second term in (4.5), we can separately deal with space and time increments.So, takez = (t,x) = (t, x) = z and the quantity of interest is (4.7) Now, in case t − s < |x −x| 2 we treat each of the two summands separately. For t − s < t 2 , the second summand at the integrand can be bounded as before (since it equals P (n) * ζ z (z)), thus giving a contribution of where we restricted the integral to the set of s for which t−s < |x−x| 2 and used the facts s ∈ ( t 2 , t] and 0 < θ 1 < β+2−n. For the other we apply (4.6) to the integrand, and obtain two summands, the first of which corresponds to (4.8) and can therefore be analogously bounded while the second is where the last bound holds since s ∈ ( t 2 , t] and 0 < θ 1 < α − n + 2. If instead |x −x| 2 > t − s ≥ t 2 , we follow the same procedure outlined above to bound the integrands, but now the integrals in (4.8) and (4.9) are over s < t 2 , thus respectively giving a contribution of order since ρ > −2, α + 2 > n and β ≥ α. We can now turn to the case t − s ≥ |x −x| 2 . In this situation, we apply Taylor's theorem to the spatial increment of the heat kernel, so that where we hid the dependence of the integrand on ν inx =x(ν) = x + ν(x − x) and the integral in s is taken over t − s ≥ |x −x| 2 . For t − s ≤ t 2 , we apply (4.6) and bound the previous by where we implicitly exploited that, for any ν ∈ [0, 1], |x −x| 2 ≤ |x −x| 2 ≤ t − s and β ≥ α, and the last inequalities are a consequence of s ∈ ( t 2 , t] and n ≥ β + 1 > β + 1 − θ 1 . If instead, t 2 ≤ t − s (≤ t), then the quantity to bound is the same as the first term in the last chain, but this time the integral is over s ∈ [0, t 2 ], so that since n ≥ β + 1 and ρ > −2. At last we need to investigate the time regularity of the convolution of ζ · with the heat kernel. To do so, we take t,t ∈ (0, T ] such that |t −t| ≤ |t,t| 2 0 , we assume, without loss of generality, t <t and set z = (t, x) and z = (t, x). Then, we notice that the time increment of the quantity at study can be written in the following way (4.11) For the first summand, we proceed as usual, i.e. we apply (4.6) to the integrand and bound the resulting integral by which holds since for s ∈ [t,t], s − t ≤t − t and θ 1 ∈ (0, α − n + 2). The analysis of the second summand is similar to the one we carried out for the spatial increments, with some special arrangements that we will now point out. For t − s <t − t, we treat each term separately. Thanks to (4.6), we have where in the first line we exploited the factst − s > t − s and β ≥ α. If t − s < t 2 , the integral to bound is the same as in (4.8) but with a different domain of integration, namely those s such that t − s <t − t, and the bound is analogous. If instead t 2 ≤ t − s <t − t, one proceeds as in the first term of (4.10). In case t − s ≥t − t, we apply Taylor's formula and the equality ∂ t P t = ∆P t , so that the integrand of the second summand in (4.11) becomes where we made use of (4.6) and the fact that, trivially, q − s ≥ t − s for all q ∈ [t,t]. If t − s < t 2 , we integrate the previous over s such that t − s ≥t − t and get As a straightforward corollary of the previous proposition combined with Lemma 4.1, we have the Schauder estimate for Hölder distributions.
Proof. The proof is immediate. Indeed, since ζ does not depend on the base point, (4.4b) becomes trivially true (the left hand side is simply 0) while the validity of (4.4a) is guaranteed by Lemma 4.1 and the assumption ζ ∈ C β ρ,T .
The following is instead a significant application of Proposition 4.4, in which we show how to apply it in the context of the Reconstruction Theorem 2.5.
Remark 4.9. The reason why we focus on (4.13) and not, as might seem more natural, on the whole left hand side of (2.5) will be clarified in the upcoming section. The point is that we only care about the ill-posed part of the product that, in our (and, also, more general) context, will be the one coming from the element in the regularity structure with lowest homogeneity.
Proof. We begin with (4.14). In order to establish the connection with Proposition (4.4), we define, for z = (t, x) ∈ (0, T ] × T and s ∈ (0, T ], ζ z (s, ·) as the right hand side of (4.13). We want to verify that (4.4) hold for z → ζ z . Let us begin by showing (4.1) since it will imply (4.4a). Let ϕ be a function in B r 0 (R) and λ ∈ (0, 1]. Notice that where Q ≥β is the projection onto l≥β T l (recall that the components of H in T l for l > γ are 0 by definition) and the bound on the first term follows by (2.5) while the second by the properties of the model and the definition of modelled distribution. Let x =x,z = (t,x) and z = (t, x). Then where we made use of the algebraic properties of the model and of the easily verifiable equality . At this point, we exploit the analytical properties of the model and the definition of modelled distributions to bound the first by since z andz live in a compact and γ ≥ β, and the second by Then, (4.15) and the previous allow to apply Lemma 4.1, which in turn implies (4.4a). Consider now t =t such that |t − t| ≤ |t,t| 2 0 and let this timez = (t, x) = (t, x) = z. Then, we have which presents essentially the same terms as (4.16). Thus, following the same procedure as above but exploiting the properties of the map Σ (instead of the ones of Γ), one immediately obtains Concerning (4.4b), notice at first that the analytical and algebraic properties of a model and Lemma 4.1 guarantee that for any τ ∈ T α and ψ ∈ S(R) we have Π t x τ, ψ λ x λ α τ where the proportionality constant hidden in the previous bound depends only on the norm of ψ. Now, since in (4.16) and (4.17), we showed that the increment ζz(t, ·) − ζ z (t, ·) can be written as Π t x τ for some τ ∈ T α , the validity of (4.4b) follows at once.
The last part of the statement can be obtained applying the same scheme as before but exploiting the local Lipschitz continuity of the reconstruction operator, Theorem 2.5.
Before proceeding, we state the following lemma, whose proof is provided in [Hai14,Lem. 7.5] or can be easily obtained using Lemma 4.1 and some of the tools exploited in Proposition 4.4.

The fixed point argument: proof of Theorem 3.9
Thanks to the results in the previous section, we now have all the necessary ingredients to suitably bound the map M introduced in (3.13) and that we here recall. For u 0 ∈ C η and X ∈ X we set, with a slight abuse of notation, M(·) and in (4.18b) we unwrapped relation (3.11), making use of the definition ofṽ ′ , X and (4.13). We are now ready to state and prove the next proposition which represents the core of the fixed point argument.
In the proof of the previous we will need the following lemma.
Proof. The proof proceeds along the same lines of Lemma 4.8. We will focus at first on (4.1). Let ϕ ∈ B r 0 (R), λ ∈ (0, 1] and z = (t, x) ∈ (0, T ] × T, then On the other hand, forz = (t,x), withx = x we have where the last equality is a consequence of the fact that the realization of through the model does not depend on the base point. Now, if |x −x| > |t| 0 , we bound each summand of the previous separately using (4.22). Otherwise, we get Hence, the assumption (4.1) is matched and consequently (4.4a) (with ρ = η − α ⋆ and β = α ). Moreover notice that, taking t =t such that |t − t| ≤ |t,t| 2 0 andz = (t, x) = (t, x) = z, we have Proceeding as above, separately taking into account the cases |t −t| ≤ |t,t| 2 0 and |t −t| > |t,t| 2 0 ,one obtains the same bound.
But now, also the validity of (4.4b) (with α = β = α ) can be obtained exploiting the same argument presented at the end of the proof of Proposition 4.4, so that, upon choosing θ 1 = α + 1 − α ⋆ and θ 2 = 0 we are done.
Proof of Proposition 4.11. In order to prove the statement, we need to bound each of the norms appearing in definition (3.10). As we will see, this will essentially amount to verify that the assumptions of Proposition 4.4 are satisfied.
We begin with the C α⋆,s η,T ofṽ, the latter being defined in (4.18a). The bound on the initial condition follows immediately from Lemma 4.10. Concerning the terms of the form X τ , for τ ∈ { , }, the bound follows by the fact that they belong to X. Let us write, F (i) v , i = 1, 2, for the first and second summand in the expression of F v , as given in (4.19). Then by Proposition 2.6, they satisfy for every t ∈ (0, T ] and σ ∈ (0, η + 1), so that thanks to Lemma 4.1, upon choosing for the first θ 1 1 = α + 1 − α ⋆ , θ 1 2 = 0 and for the second θ 2 where, by assumption, the exponents of T are strictly positive. Since the functionK is smooth and compactly supported, the convolution withK ′ is a continuous operation and the required bound on the term Q in (4.18a) follows straightforwardly. We can now deal with the term containing the reconstruction operator. Let β ∈ [α , β ) to be fixed later. By choosing H = V , we see that for z = (t, x) ∈ (0, T ] × T, we have Now, while the second is taken care by Lemma 4.12, since by Proposition 3.6 we know that V ∈ D γ+α ,η+α T and γ + α > 0, η > γ + 2, η + α > −2, we can directly apply Lemma 4.8 to the first summand with θ 1 = β + 1 − α ⋆ , θ 2 = γ − α ⋆ and β such that β + 1 > γ so that its C α⋆,s η,T -norm is bounded by T β+1−γ 2 . For the Gubinelli's derivativeṽ ′ , defined in (4.18a), we immediately get for some θ > 0, as soon as α < γ < α + α ⋆ .
At last we look at the seminorm involving R, whose expression is given in (4.18b). The increment of the terms containing initial condition can be dealt with as before. The bound on R z follows by definition, while for δ z,z P ′ * J V β,z , Lemma 4.8 and in particular (4.14), along with the observations made before, gives v , i = 1, 2, we make use once more of (4.23) but with while the increments of Q are again bounded straightforwardly. The last summand is such that δ z,zX is smooth and |ṽ ′ (z)| |z| The inequality (4.20b) can be shown following essentially the same steps exploited so far, so that the proof is complete.
Thanks to the previous proposition, we have now all the elements to prove Theorem 3.9.
Proof of Theorem 3.9. It follows from (4.20a) that for fixed u 0 ∈ C η , X ∈ X and a sufficiently small T > 0 there exists a ball in H η,γ X,T which is left invariant by M. Furthermore, (4.20b) and the Banach fixed point theorem imply that M admits a unique fixed point in this ball. The uniqueness of this fixed point on whole H η,γ X,T follows from a simple argument as in [Hai14,Thm. 4.8]. We can now restart the procedure at time T and so and so on, constructing in this way the maximal solution to the fixed point problem. The solution to (1.2) is then built from (3.4).
Concerning the local Lipschitzianity of the map, using the same procedure exploited in the proof of the previous proposition and the local Lipschitz-continuity of all the operations we performed (along with a procedure analog to the proof of [Gub04, Prop. 8]), it is not difficult to see that, given X,X ∈ X and u 0 ,v 0 ∈ C η such that for a fixed R > 0, and letting V ∈ H η,γ X,T andV ∈ H η,γ X,T be the respective fixed point of the of equation starting at u 0 (resp.v 0 ) and controlled by X (resp.X) determined above, then the following inequality holds and, invoking again (3.4), the proof of Theorem 3.9 is concluded.

Discrete heat kernel and Schauder estimates
In this section we want to show that it is possible to perform the same operations and obtain the same bounds as in the continuous case but in the discrete setting. As before, we fix N ∈ N, set ε def = 2 −N and define the grids as in (1.12), and T ε def = T ∩ Λ ε . Let us begin with the discrete counterpart of Lemma 4.1.
Remark 4.14. The condition imposed on the family ψ λ is the same mentioned in Remark 4.3, but we are allowing λ to take value in a discrete set.
Proof. The proof is identical to that of Lemma 4.1, with the only difference that we need to use the points from the grid.
In order to be able to follow the same steps of Section 4.1, we need to show that the convolution with the discrete heat kernel can be regarded as the testing against the recentered family ψ λ introduced above, in which time will play the role of λ. Recall that the space-time discrete heat kernel, P ε , is the unique solution of (3.18). From the previous equation, we immediately deduce that for all (t, x) ∈ Λ s ε one has In order to derive suitable bounds on the kernel just defined we need at first to build a regular extension of P ε off the grid Λ s ε . While the time component can simply be dealt with by taking t ∈ R and the integer part of tε −2 at the exponent, for the space one we briefly recall the construction done in [HM15, Sec. 5.1]. Letφ(x) def = sin(πx) πx andφ be a smooth function compactly supported in a ball of radius 1 4 around the origin and integrating to 1. Then, let ψ be the function whose Fourier transform is Fφ * φ. Notice that ψ belongs to S(R), coincides with the Kronecker's function on Z and F ψ is supported on {ζ : |ζ| ≤ 3 4 }. Indicating as usual its rescaled version as ψ ε , i.e. ψ ε (x) def = ε −1 ψ(x/ε), we obviously have the identity (4.24) But now we can defineP ε according to the right hand side of the previous but for (t, x) varying in R 2 . Notice that, by , where * ε denotes the space-time discrete convolution. From now on, with a slight abuse of notation, we will indicate the extensionP ε by P ε , because no confusion can arise.

Remark 4.15.
Let D x,ε be the discrete derivative as in (3.17b), j ∈ N and assume that Assumption 3.13 is satisfied. The same extension procedure that lead to the definition of P ε can be performed on D j x,ε P ε without any modification and, from now on, we will denote by D j x,ε P ε such extensions.
The following lemma provides some bounds on this kernel and its discrete derivative uniform in ε, which will allow to apply Lemma 4.13 and use it in the same way we exploited its continuous counterpart.
Lemma 4.16. Let N ∈ N, ε = 2 −N and Assumptions 3.12 and 3.13 be in place. Let ∆ ε , D x,ε , be respectively the discrete Laplacian and derivative introduced in (3.17a) and (3.17b), and P ε the extension to R 2 of the kernel defined in (4.24).
Then, for any c > 0 fixed, and j, k and m ∈ N such that m < ⌊c⌋ ∨ ⌊c −1 ⌋, there exists a constant C > 0, depending only on j, k and m, such that the following bound holds uniformly over z = (t, x) ∈ R 2 such that z s ≥ cε, where |t| ε = |t| 1/2 ∨ ε. In case z s < cε, upon taking m = 0, (4.25) is still valid.
Proof. The proof of the statement follows the same lines as the one of [HM15, Lem. 5.3] but we need to adapt it to the present situation. Let j ∈ N and, for (t, x) ∈ R 2 and ε > 0, set At first, we aim at obtaining bounds for F ε uniform in t, x and ε. To this purpose, consider its spatial Fourier transform, which is given by (4.26) Take for now c > 1 and m ∈ N such that m < ⌊c⌋. Let us begin with the case t ≥ cε 2 and we first study the situation in which ξ ∈ R is such that ε|t| −1 ε |ξ| ≤ 3 4 . We want to control the m-th derivative of (4.26) and, thanks to the generalized Leibniz rule, we can separately treat each of the factors there appearing. Now, by construction,ψ is a C ∞ -function with compact support hence for any m ∈ N, where the last passage comes from the fact that, since |t| ≥ cε 2 , ε|t| −1 ε 1. For the second factor, we point out that, by Assumption 3.13, π has compact support, hence, for l ∈ N, l ≥ 1, we get where once again we are using ε|t| −1 ε 1. On the other hand for l = 0, thanks to Assumption 3.13, in particular (3.16), and Taylor's formula, we have (4.28) In conclusion this implies that π(−ε|t| −1 ε ·)/(ε|t| −1 ε ) converges to the identity in C ∞ , therefore its j-th power is uniformly bounded by |ξ| j . We now come to the third and last factor, which is the one that guarantees the polynomial decay. Let m ∈ N be such that m < ⌊c⌋. By Faà di Bruno formula, we can write the m-th derivative of the abovementioned term as withT ε,m defined by where the sum runs over (n 1 , · · · , n m ) ∈ N m such that l ln l = m,c is a constant depending on the m-tuple andν, n def = l n l ≤ m and we exploited the fact that, since t ≥ cε 2 , tε −2 = (ε|t| −1 ε ) 2 . It is not difficult to see thatT is uniformly bounded in ξ, t and ε. Indeed, by the very definition of ν and the fact that ν is symmetric by Assumption 3.12, 1 + ν(·)/(2ν) is bounded away from 0 (actually it is greater or equal to 1/2) hence the first factor does not create any problem. While the second is obvious, for the third we proceed as for the second factor of (4.26) but exploiting assumption Assumption 3.12 instead of Assumption 3.13. To be more precise, for l ∈ N, l ≥ 2, the bound is identical to (4.27), and we have for l = 0, 1, we follow (4.28) and use Assumption 3.12, in particular (3.15), and Taylor's formula, so that We can now focus on the main part of the proof, i.e. the one concerning the first term on the right hand side of (4.29). Let us rewrite it as where in the passage from the first to the second line we applied Taylor's formula and introduced the function f (x) def = −x −2ν (x), while to understand in the inequality right after, it suffices to recall that t|t| −2 ε = 1 since |t| > cε 2 and that, as pointed out above, ν(·)/(2ν) ≥ − 1 2 . To understand the reason why also the last bound holds, notice that, by Assumption 3.12, there exists a constant c f > 0 such that f (x) is bounded from below by c f for all |x| ≤ 3 4 . Summarizing what obtained so far, we have ∂ m ξ F ε t (ξ) |ξ| m+j e −2c f ξ 2 1 + |ξ| −k , (4.30) for any k ∈ N, ξ such that ε|t| −1 ε |ξ| ≤ 3 4 and |t| ≥ cε 2 . The very same bound holds trivially also for ε|t| −1 ε |ξ| > 3 4 since, by construction, ψ ≡ 0. At last, in case |t| < cε 2 , (4.26) becomes F ε t (ξ) =ψ(ξ) ( π(−ξ)) j (1 + ν(ξ)/(2ν) ⌊tε −2 ⌋ and the second factor can be bounded by the total variation of the measure π while the last is trivially bounded by a constant. Ultimately, the polynomial decay of the m-th derivative of F ε comes from ψ, and (4.30) holds also in case |t| < cε 2 .
At this point we exploit the continuity properties of the Fourier transform, that guarantee that F ε is C ∞ (since its Fourier transform decays faster than any polynomial) and that its derivatives decay faster than any polynomial of degree less or equal to ⌊c⌋ (since, vice versa, its Fourier transform is m times continuously differentiable). In other words we have, for any k, m ∈ N such that m < ⌊c⌋, there exists a constant C, depending only on k and m, for which (1 + |x|) −m uniformly in z = (t, x) ∈ R 2 and ε > 0. The bound (4.25) is now a direct consequence of the previous since it suffices to change variables (x → |t| −1 ε |x|). For z s < cε, notice that the second factor in the last inequality is bounded by a constant.
At last we remark that, for c < 1, we need to distinguish another case, namely |t| ∈ [cε 2 , c −1 ε 2 ). While for |t| ≥ c −1 ε 2 and |t| < cε 2 we proceed as before, the just mentioned situation can be dealt with as in the latter case.
For the time increments instead, if one looks at (the discrete version of) (4.10), the first summand can be treated as before, while for the second notice that, fort > t ∈ Λ ε 2 ,T , t − s < cε 2 ≤t − t we have which in turn concludes the proof of (4.34).
And again, as a straightforward corollary of the previous proposition combined with Lemma 4.13, we have the following discrete Schauder estimate.
Proof. The proof is identical to the one of Corollary 4.7, but instead of Lemma 4.1 and Proposition 4.4 one has to use Lemma 4.13 and Proposition 4.17.
In the following we show how to apply Proposition 4.17 in the context of the Reconstruction Theorem 2.9.
Lemma 4.19. Let T = (A, T , G) be a regularity structure according to Definition 2.1, α def = min A, Z ε = (Π ε , Γ ε , Σ ε ) a discrete periodic model on T and R ε the corresponding reconstruction operator. Let γ, η, β and n be as in the statement of Lemma 4.8 and T ∈ (0, 1]. For z = (t, x) ∈ Λ ε 2 ,T × T ε and H ε ∈ D η,γ ε,T (Z ε ), define the distribution where Q <β is the projection onto T <β (see Remark 2.2). Then, for θ 1 , θ 2 , θ andβ as in the same Lemma mentioned above, the following bound holds uniformly in ε, where the suprema run over z = (t, x) ∈ Λ ε 2 ,T × T ε . If furthermoreZ ε = (Π ε ,Γ ε , Σ ε ) is another discrete model,R ε t the associated family of discrete reconstruction operators andH ε ∈ D γ,η ε,T (Z), then (4.35) still holds upon replacing J ε H ε by J H −J εH ε and with the constant C proportional to Proof. Also in this case the proof is completely analogous to the one of Lemma 4.8, but using the discrete counterpart of the lemmas and propositions mentioned in the proof of the latter.
At last, we recall the analogous of Lemma 4.10 in the discrete setting whose proof can be easily deduced by Lemma 4.13 and some of the tools exploited in Proposition 4.17.

Uniform bounds on the discrete solution map
We now want to bound, uniformly in ε, the map M ε defined in (3.32) and that we now recall. Let u ε 0 be a periodic function on Λ ε and X ε ∈ X ε a discrete controlling process according to Definition 3.15. We set M ε (·) and to shorten the notation in (4.37b) we set z 1 = z + ε(0, y 1 ), z 2 = z + ε(0, y 2 ) andz 2 =z + ε(0, y 2 ). The equation for the remainderR ε was obtained, as for (4.18), by unwrapping the relation (3.30), exploiting the definitions ofṽ ′ ε , X ,ε and (4.35) but in this case an extra term appears which is due to the fact that the product we are dealing with is twisted. Anyway this term does not create any trouble since it is sufficiently regular and allows us to conclude that bounds similar to the ones of Proposition 4.11 hold also in the discrete setting.
Proposition 4.21. Let α ⋆ , γ and η be as in the statement of Proposition 4.11. Let u ε 0 ∈ C η,ε be a family of periodic functions parametrized by ε and X ε ∈ X ε . Then there exists θ > 0 such that the map M ε , defined in (4.37), satisfies, for all V ε ,V ε ∈ H η,γ ε,X ε ,T the following bounds uniformly in ε, where the second bound holds provided that V ε (ε) η,γ;T , V ε (ε) η,γ;T ≤ M < ∞, M independent of ε, and the hidden constant depends only on M .
The following lemma represents the discrete counterpart of Lemma 4.12 and will play its same role in the proof of the previous proposition. Since the proof is identical to the one already given we omit it.
Proof of Proposition 4.21. As in the case of Proposition 4.11, the proof consists in bounding each of the norms in Definition 3.29, which in turn boils down to verify that for each summand appearing in the equations forṼ ε the assumptions of Proposition 4.17 are satisfied. The terms in (4.37) which have an expression analogous to the corresponding ones in (4.18) can be treated in the same way. What remains to obtain is a bound on the extra element in (4.37b) which however is immediate. Indeed, forgetting about the term that depends only on discrete controlling processes, we have which is the required bound since, by assumption γ < α ⋆ + α , and this concludes the proof.
We conclude this section with the proof of Theorem 3.20.
Proof of Theorem 3.20. Notice that in the discrete context, the problem is not existence and uniqueness of solution for fixed ε since one can explicitly construct the solution at all times. The point is to obtain a bound, uniform in ε of its · (ε) η,γ;T norm. By assumption, X ε ∈ X ε and u ε 0 ∈ C η, ε , hence the sequences {X ε } ε and {u ε 0 } ε are such that M cp η,γ;T ≤ M for all ε > 0 Then, (4.39) tells us that there exists C > 0 such that where the passage from the first to the second line holds upon taking M def = 2C(M 0 +(1+M cp ) 2 ) while in the last inequality we are imposing ε < (1 + M ) −2/θ . Hence, for T = (1 + M ) −θ =C(M 0 + (1 + M cp ) 2 ) −θ and ε < (1 + M ) −2/θ , M ε leaves the ball of radius M in H η,γ ε,X ε ;T invariant, uniformly in ε, and for smaller T and ε, thanks to (4.40) we conclude that M ε is there a contraction. This in turn implies that the unique fixed point (in the whole space H η,γ ε,X ε ;T , invoking again the same argument as [Hai13,Thm. 4.8]) V ε is such that its · (ε) η,γ;T is uniformly bounded in ε by 2C(M 0 + (1 + M cp ) 2 ). By iterating the previous argument, we get the maximal time for which the sequence of fixed points V ε is uniformly bounded. The proof of the local Lipschitz continuity can then be obtained following the scheme suggested in the proof of Theorem 3.9.

Convergence of controlling processes
The focus of this section is on the controlling processes we introduced in Sections 3.1 and 3.3. We aim at showing that it is indeed possible to consistently enhance a space-time white noise ξ to a controlling process and that, if the family of rescaled normal random variables {ξ ε (z)} z∈Λ s ε converges to ξ in a suitable topology, then so does the discrete controlling process associated to it.

A convergence criterion for random distributions
Let us begin by recalling some basic facts about Wiener-chaos decomposition, on which our subsequent analysis is based. Let (Ω, F , P) be a probability space and ξ be a space-time white noise on it, by which we mean a Gaussian random field whose covariance structure is given by It is well-known (see, for example, [Nua06, Ch. 1]) that L 2 (Ω, P) = n∈N H n , H n being defined as the closure of the linear subspace of L 2 (Ω, P) generated by the random variables {H n (ξ(ϕ)) : ϕ ∈ H}, where H n is the Hermite polynomial of degree n ([Nua06, Thm. 1.1.1]).
It is also possible to show that, for every n ∈ N, there exists a bijective isometry, I n (which we will refer to as Wiener-Itô isometry), between the space H ⊗sn of symmetric functions in L 2 (R n × T n ) and H n . Composing the projection that assigns to every element in H ⊗n its symmetrization belonging to H ⊗sn , we obtain a map (that we will still denote by I n ) such that for every ϕ ∈ H ⊗n and I n (ϕ) is called Wiener-Itô integral of order n. We will say that a random variable Y ∈ L 2 (Ω, P) belongs to the n-th homogeneous Wiener chaos if Y ∈ H n , while if Y ∈ k≤n H k we will say that it belongs to the n-the inhomogeneous Wiener chaos.
Let ξ ε def = ξ * ̺ ε , where ̺ a symmetric compactly supported smooth function integrating to 1 and ̺ ε (t, x) def = ε −3 ̺(ε −2 t, ε −1 x). At this point, it should be clear that, upon taking η = ξ ε , each of the stochastic objects introduced in (3.3) lies in a finite inhomogeneous Wiener chaos, whose order is given by the number of occurrences of ξ in their expression (e.g. X is in the first, X in the second and so on).
In order to show that there exist suitable constants for which X(ξ ε , C ε , C ε ) converges to a well-defined object, independent of ̺, in a suitable topology, we will decompose each of its terms in its Wiener-chaos expansion and separately bound its chaos components. From the discussion above, one can deduce that such components will be of the form for a test function ϕ, where I k is the Wiener integral of k-th order with respect to ξ, the function W ε takes values in H ⊗k and is such that W ε (z 1 )W ε (z 2 ) ∈ L 1 ((R 2 ) ⊗k ), for every z 1 = z 2 ∈ R 2 . Moreover, we define its limit, Y , in the same way, but via some H ⊗k -valued function W, such that W(z 1 )W(z 2 ), W(z 1 )W ε (z 2 ) ∈ L 1 ((R 2 ) ⊗k ), for every z 1 = z 2 ∈ R 2 . By (5.1), the covariance functions, given by for every z 1 = z 2 , where δW ε def = W ε − W and the scalar product is taken in H ⊗k , will play an extremely important role in our analysis. In all the cases we will consider these functions can be written as K(z 1 , z 2 ) = K(z 1 − z 2 ), where K is a kernel that can have a singularity at the origin and is smooth away from it. It is this singularity that determines the regularity and convergence properties of the distributions Y and Y ε . The following definition provides a quantitative description of such functions.
Definition 5.1. We say that a smooth compactly supported function K : R 2 \ {0} → R has a singularity of order ζ ∈ R, if for all multiindices k = (k 0 , k 1 ) ∈ N 2 , such that |k| s def = 2|k 0 | + |k 1 | ≤ ζ, the function D k K is defined on R 2 , and there exists a constant C > 0 such that the bound holds uniformly over z = 0. Here, for a point z = (t, x), we use the norm z s = |t| 1/2 ∨ |x|.
Remark 5.2. It is not difficult to see that, if the function K has singularity of order ζ / ∈ N in the sense of Definition 5.1, then the function is well-defined, for z = 0 and for all multiindices k, and satisfies the bound The following proposition provides a criterion to show the convergence of random variables Y ε as in (5.2) to Y .
Proposition 5.3. In the setting described above, let K and δK ε be compactly supported functions on R 2 defined in (5.3) for distributions Y and Y ε of the form (5.2). Then we have the following results 1. if K is singular of order ζ ∈ (−3, −1], then the process Y almost surely belongs to the space C α,s 1 , for any α < ζ 2 . If furthermore the function δK ε has a singularity of order ζ − 2θ, for some θ > 0, and satisfies the estimates (5.4), with the proportionality constants of order ε 2θ , then, for any p ≥ 1 and α < ζ 2 , the following bound holds 2. If K is singular of order ζ ∈ (−1, ∞)\N. Then the process Y almost surely belongs to the space C β/2 ([−1, 1], C α−β ), for any α < ζ 2 and any β ∈ (0, α) if ζ > 0 and any β ∈ 0, ζ+1 2 if ζ < 0. Moreover, if the function δK ε has singularity of order ζ − 2θ, for some θ ∈ 0, ζ+1 2 , and satisfies the estimates (5.4), with the proportionality constants of order ε 2θ , then, for any p ≥ 1 and α < ζ 2 , the following bound holds The proof of the previous definition is based on the following technical lemma. For any two time points s, t ∈ R, we define the operator δ 2 s,t K(x) def = ǫ1,ǫ2∈{0,1} which is finite if α < ζ 2 and p is large enough, where we indicated with C α.s K the space of C α,s functions on K,K is the 1-fattening of K and the proportionality constant depends on p. Here, we have used the fact that the set Ψ ⋆ is finite. In case ζ < 0, this is exactly the first claim of Proposition 5.3.
For ζ > 0, the estimate (5.11) immediately implies the required result of the second claim of the above mentioned proposition (in fact, we have proved a stronger result, that X belongs to the parabolic Hölder space). If instead ζ ∈ (−1, 0), then a special argument is required. To be able to evaluate Y at a fixed time point, we need the bound to hold uniformly over δ, λ ∈ (0, 1], for all functions ϕ δ,λ t,x (s, y) The expectation at the left hand side of the previous is equal to Then, we can simply bound z 1 − z 2 ζ s ≤ |x 1 − x 2 | ζ (recall that ζ < 0), where as before we treat space and time variables separately, i.e. z i = (t i , x i ). Therefore, we can then bound the integral (5.13) by where we have estimated the integral in the same way as in (5.10), under the assumption that ζ > −1. Letting now δ in (5.12) go to 0 and using the dominated convergence theorem we obtain for any time point t ∈ R. Using this bound one can proceed in the same way as in (5.11), but now exploiting the wavelet expansion only in the spatial variable, to obtain for α < ζ 2 , At last, we investigate the regularity of Y in time. Let us denote for brevity δ s,t W(x) def = W(t, x) − W(s, x). Then we can write δ s,t W(x 1 ), δ s,t W(x 2 ) = δ 2 s,t K(x 1 − x 2 ) , where we used the operator δ 2 s,t defined in (5.9). Furthermore, it follows from Lemma 5.4 that for any β ∈ [0, 2]. Taking a function ψ n x from the wavelet basis on R and using the previous bound we derive the estimates where the integral is estimated as in (5.11) with the condition ζ − 2β > −1. From this estimate we obtain in the same way as above for any α < ζ 2 . Now one can apply the Kolmogorov continuity criterion for Banach-valued random variables [Kal02], to prove the existence of a modification of Y which belongs to C β/2 ([−1, 1], C α−β ), for any β < ζ+1 2 . The required estimates on the approximating processes Y ε can be obtained by following the same steps, recalling that the proportionality constants depend on ε.
In the following corollary we deal with the special case in which a random variable Y can be written as K * Ȳ , where K is a singular kernel according to Definition 5.1 andȲ is a random variable of the form (5.2).
Proof. The random distribution K * Y is of the form (5.2) with the respective function as in (5.3) given bŷ where K is as in Proposition 5.3. It follows immediately from [Hai14,Lem. 10.14] thatK has singularity of order ζ + 2ξ + 6 > −1. The claim now follows from point 2 of Proposition 5.3.
We conclude this section with the following lemma which will come at hand when dealing with the remainder terms introduced in (3.3c).
Lemma 5.6. Let us be given a function R ∋ x → R x such that R x is of the form (5.2) with the respective function K x , defined in (5.3), and, for some ζ ∈ (−1, 0), satisfies (5.14) uniformly over the variables z i = (t, x i ), and where the sum runs over finitely many values of δ ∈ [0, 1 + ζ). Let furthermore R x have a strictly positive Hölder regularity in time variable. Then R belongs almost surely to the space C([−1, 1], L β (R)) for any β < ζ 2 . Proof. Similarly to (5.11), but using the wavelet expansion only in the spatial variable, we can prove that the bounds hold for all p ≥ 1, s, t ∈ [−1, 1] and β < ζ 2 , and for some γ > 0. The claim then follows from the Kolmogorov continuity criterion for Banach-valued random variables [Kal02].

Enhancing the space-time white noise
In the setting of the previous section, let ξ be a periodic space-time white noise and ξ ε def = ξ * ̺ ε , where ̺ a symmetric compactly supported smooth function integrating to 1 and ̺ ε (t, x) def = ε −3 ̺(ε −2 t, ε −1 x). The core of this section is the proof of Proposition 3.4. In order to represent the chaos components of the stochastic processes in X(ξ ε ) and X(ξ), we will use the expression in (5.2) and we will denote their covariance by (5.3). Wick's lemma [Nua06, Prop. 1.1.3] will allow us to work with the product of multiple Wiener-Itô integrals, while we will apply Proposition 5.3 to prove convergence of the smooth controlling processes defined in Section 3.1. Notice that, this latter proposition tells us that it suffices to determine the order of the singular kernel appearing in (5.3), which in turn are given by products, convolutions and derivatives of the kernel K, that is the singular part of the heat kernel, as we saw at the beginning of Section 3.1. To deal with these operations, we will extensively use the results of [Hai14,Sec. 10.3], together with the fact that K and K ε def = K * ρ ε have singularities of order −1 in the sense of Definition 5.1 and that, for every θ ∈ [0, 1], K − K ε has a singularity of order −1 − θ with the constant C in (5.4) proportional to ε θ .
Proof of Proposition 3.4. [Hai14,Lem. 10.2] shows that ξ belongs to C − 3 2 −κ,s a.s. for every κ > 0 and that, for all p ≥ 1, ξ ε converges to ξ in L p (Ω, C − 3 2 −κ,s ) as ε → 0. The process X ε can be written as X ε = ∂ x Y ε , where Y ε belongs to the first Wiener chaos, and whose kernel is W ε (z; y) = K ε (z − y). Analogously, the limiting process X is given by X = ∂ x Y , and Y is defined via the kernel W (z; y) = K(z − y). Then it follows from [Hai14,Lem. 10.14] that the covariance functions K and δK ε , which are defined in (5.3) for W ε and W , satisfy the assumptions of Proposition 5.3 with the parameters ζ = 1 and θ < 1 2 . This implies that Y ∈ C β/2 ([−1, 1], C α−β (R)), for any α ∈ (0, 1 2 ) and β ∈ (0, α). The regularity of X is then C β/2 ([−1, 1], C α−1−β (R)), and the estimate (5.27) holds for β < 1 2 − θ. It follows from this result and Corollary 5.5 that the distribution X = K ′ ⋆ X belongs to C β/2 ([−1, 1], C α−β (R)), for any α ∈ (0, 1 2 ) and β ∈ (0, α). The respective bound (5.27) follows trivially. To lighten the notation, in what follows we will represent our kernels via diagrams, similarly to what was done in the proof of [Hai14,Thm. 10.22]. By z y and z y we will denote the kernels K(z − y) and K ε (z − y) respectively, while the difference (K ε − K)(z − y) by z y . Furthermore, we will use a double-headed arrow for the partial derivative in space of a function, e.g. z y will be ∂ x K(z − y). Each node in such diagrams corresponds to a point in R 2 , and if a variable is integrated out, we will draw the corresponding node in gray. Now, we can turn to the process X ε . Let us begin by defining its 0-th chaos component, which will be the constant we need to remove in order to obtain a well-defined object in the limit. By shift-invariance, this is given by and it is not difficult to see that C ε ∼ ε −1 . Then upon subtracting C ε in (3.3a), we conclude that X ε is defined by the kernel W ε (z) = z .
and, correspondingly, the limiting process X is defined by the kernel W which is obtained by replacing dashed arrows by solid ones in the above diagram. The covariance function K is then given by which implies, using [Hai14,Lem. 10.14], that it has a singularity of order ζ = −2. Applying now Proposition 5.3 we conclude that X belongs to C α,s 1 for α < −1. Writing now the kernel δW ε = W ε − W as and treating each term in this sum separately as before, we conclude that the estimate (5.27) holds for X and X ε with θ < 1 2 and α < −1 − θ.
belong to the 4th and 2nd homogeneous Wiener chaoses respectively and we omitted the 0-th chaos component since the spatial derivative will anyway kill it. The kernels defining X ε,4 and X ε,2 are given by Using [Hai14,Lem. 10.14] we conclude that the orders of the singularities of the corresponding covariance functions are ζ = −κ for every κ > 0. Proposition 5.3 now yields X ∈ C β/2 ([−1, 1], C α−β (R)) with α < 0 and β ∈ (0, 1 2 ). Convergence of the approximate processes we show in the usual way. Now, we turn to analysis of the remainders defined in (3.3c). Since these definitions are continuous with respect to the controlling processes, the identities (3.3c) will automatically hold in the limit ε → 0. It remains only to show that the processes belong to the required spaces.
In order to show that X is "controlled" by X , we write where the processes R 2,x and R 0,x belong to the 2nd and 0th homogeneous Wiener chaoses respectively and are characterised by the kernels .
We have already established above that the functions K and K have singularities of orders ζ 1 = 1 − κ and ζ 2 = −1 respectively, which implies, subtracting to each K , K , and applying [Hai14,Lem. 10.14] the bound The same result holds for R 0,x , because it is deterministic and the kernel W 0,x satisfies |W 0,x (t, y)| |y − x| −κ for every κ > 0. It follows furthermore from the previous results that R has a strictly positive Hölder regularity in time, which yields from Lemma 5.6 that R ∈ C([−1, 1], L β (R)) almost surely for any β < ζ1+ζ2 2 . Finally, we turn to the limiting process of R ε . Notice that, by definition R ε = P ′ * R ε , where formally R (x; t, y) = X (t, y) − X (t, x) X (t, y). For the latter we write R (x; t, y) = R 4,x (t, y) + R 2,x (t, y) , where the process R i,x (t, y) belongs to the ith Wiener chaos and is of the form (5.2). Notice that we neglected the 0-th chaos component because it will be anyway killed by the spatial derivative. Acting as before, it is not difficult to show that the respective functions (5.3) satisfy the bound (5.14) with ζ < 0. Applying now Lemma 5.6 we conclude that R ∈ C([−1, 1], L β (R)) almost surely for any β < 0. As a consequence, setting ζ z (s, ·) def = R z (s, ·), we have uniformly over z = (t, x), ∈ (0, T ] × T, λ ∈ (0, 1] and ϕ ∈ B r 0 (R). Moreover, ifz = (t,x) is another point in (0, T ] × T, λ ∈ (0, 1], then uniformly over z = (t, x),z = (t,x) ∈ (0, T ] × T such that |t −t| ≤ |t,t| 0 , λ ∈ (0, 1] and ϕ ∈ B r 0 (R). Now, notice that, in the expression for the increments, the only term that is tested against ϕ λ x is X , for which it is straightforward to verify that the assumptions of Lemma 4.1 are matched so that the previous holds also for ϕ Schwartz. In conclusion, upon choosing α = α , β = β and ρ = 0 in Proposition 4.4 (as well as θ 1 = θ 2 = 0), the proof of the proposition is concluded.

Discrete singular kernels
As we saw in the previous section, in order to obtain suitable bounds for the stochastic processes under study the crucial notion we needed is that of kernels with a singularity at 0 of prescribed order. In the discrete setting, Definition 5.1 becomes Definition 5.7. Let {K ε } ε be a sequence of functions on Λ s ε supported in a ball around the origin. We say that it is of order ζ ∈ R if for some m ∈ N the quantity is bounded uniformly in ε. In the previous, k = (k 0 , k 1 ) ∈ N 2 , |k| s def = 2k 0 + k 1 , z s,ε = z s ∨ ε andD k ε is the forward discrete derivative in space and time, i.e.D k Remark 5.8. Notice that in (5.17), we are not taking a general discrete derivative operator as in Lemma 4.16 but we made the specific choice ofD. Even if not crucial, it is a convention that will turn out to be convenient when proving what it means to multiply or convolve singular kernels. Moreover, this does not affect in any way the definition of the stochastic objects under study.
Before stating a number of lemmas representing the discrete counterpart of those in [Hai14,Sec. 10.3], let us show that the discrete heat kernel (and its discrete derivatives), as given in (4.24) can indeed be decomposed as the sum of a singular and a "regular" part.
Lemma 5.9. For any j ∈ N, the kernel D j x,ε P ε can be written as D j x,ε P ε = D j x,ε K ε + D j x,εK ε , in such a way that the identity is satisfied for all z ∈ Λ s ε and all spatially periodic functions u on Λ + ε 2 × Λ ε (here Λ + ε 2 def = Λ ε 2 ∩ R + ). In the previous decomposition, D j x,ε K ε is a singular kernel of order −1 − j according to Definition 5.7 andK ε is compactly supported, non-anticipative and such that, for any r > 0, K ε (ε) C r,s is bounded uniformly in ε.
Proof. The statement above is similar in spirit to Lemma 5.5 in [Hai14] and Lemma 5.4 in [HM15], but, in a sense, we only need to localize our discrete heat kernel around the origin as we did in (3.2). Indeed, it suffices to consider a smooth compactly supported function χ : R 2 → [0, 1] such that χ(z) = 1 for z s ≤ 1 and χ(z) = 0 for z s > 3 2 (of course the choice of 3 2 is completely arbitrary). Then we set, for z ∈ Λ s ε D j x,ε K ε (z) = χ(z)D j x,ε P ε (z) and D j x,εK At this point the decomposition in the statement is obvious and it remains to prove that indeed K ε andK ε satisfy the correct properties. Now,K ε is clearly non-anticipative and such that, for any r > 0, K ε (ε) C r,s because so is P ε outside of a ball centered at the origin. In order to make it compactly supported, we recall the procedure outlined in Lemma 7.7 of [Hai14]. Assume for simplicity that u has period 1. Fix a function ̺ : R → [0, 1], compactly supported in a ball of radius C centered at 0 such that k∈Z ̺(x + k) = 1 for every x ∈ Λ ε . Then, one can straightforwardly verify that does the job. Concerning K ε we have to prove that there exists a c > 0 such that the quantity in (5.17) is bounded upon taking ζ = −1 − j. Let k = (k 0 , k 1 ) ∈ N 2 and recall that, for z s ≤ 1,D k ε D j x,ε K ε (z) coincides withD k ε D j x,ε P ε (z) so that we can focus on the latter. The proof proceeds along the same steps as the one of Lemma 4.16 hence we will simply point out the main steps. Fix c > 0 such that ⌊c −1 ⌋ > 1 + j + |k| s and, for (t, We want to obtain uniform bounds on its Fourier transform, which is given by where, to streamline the notation, we have setξ def = ε|t| −1 ε ξ, and π d def = δ 1 (dx) − δ 0 (dx), which obviously satisfies Assumption 3.13. Notice that the simplifications that lead to the expression above were possible only because t ∈ Λ ε 2 .
Since all the quantities in the previous expression were already considered in the proof of the aforementioned Lemma, we reach the same conclusions, namely, for any m ∈ N, m < ⌊c −1 ⌋, there exists C > 0 such that uniformly over z = (t, x) ∈ R 2 such that z s ≥ cε, and ε > 0, where |t| ε = |t| 1 2 , for |t| ≥ cε 2 , and is equal to ε otherwise. If instead z s < cε the previous bound holds with m = 0. To see how to get (5.17), we proceed as follows. In case z s < cε, there is nothing to prove. Otherwise, recall that z s = |t| 1 2 ∨ |x|, hence if |t| 1 2 ≥ |x| we choose m = 0, while if |x| > |t| 1 2 we take m = 1 + j + |k| s . In both cases the right-hand side of (5.18) becomes z −1−j−|k|s s which is exactly what requested, hence the proof is complete.
Remark 5.10. It might seem that in the previous proof we showed a slightly different result, namely that there exists a c > 0 such that, for ζ def = −1 − j and k ∈ N 2 if z s < cε.
In the analysis of the stochastic terms, we will run into the convolution of two discrete singular kernels whose order is less or equal than −|s|, hence falling out of the range of applicability of the previous lemma. Nevertheless, we will need to obtain suitable bounds for these objects and, under specific assumptions, this is still possible. Let us begin with the following definition, which represents the discrete counterpart of [Hai14,Def. 10.15].
Definition 5.12. Let −|s| − 1 < ζ ≤ −|s| and K ε be a function on Λ s ε of order ζ. We define the discrete renormalized distribution R ε K ε corresponding to K ε as for every smooth compactly supported test function ψ, where the pairing ·, · ε is a Riemann sum approximation of the usual L 2 scalar product.
The proof of the following lemma is analogous to [Hai14,Lem. 10.16].
Proof of Proposition 5.16. Thanks to the characterization of the discrete Hölder spaces through the discrete wavelets given in Proposition A.2 and the lemma above, it is immediate to see that in order to prove the result it suffices to follow the same steps and use the same arguments as in the proof of Proposition 5.3, replacing the integrals with the discrete pairing and exploiting the analogous bounds.
Proof. Analogous to the one of Corollary 5.5, applying though Lemma 5.11 and Proposition 5.16 in place of [Hai14,Lem. 10.14] and Proposition 5.3.
We conclude this section with the following lemma, whose proof is identical to that of Lemma 5.6, which allows to bound the stochastic processes defined in (3.20c).
Lemma 5.19. Let us be given a family of functions (parametrized by ε) Λ ε ∋ x → R ε x such that R ε x is of the form (5.25) with the respective function K ε x , defined in (5.26), and, for some ζ ∈ (−1, 0), satisfies uniformly over the variables z i = (t, x i ), and where the sum runs over finitely many values of δ ∈ [0, 1 + ζ). Let furthermore R ε x have a strictly positive (discrete) Hölder regularity in time. Then R ε belongs almost surely to the space C ε ([−1, 1], L β,ε ) for any β < ζ 2 .
Now that we derived the discrete version of all the results we needed in the proof of Proposition 3.4, we are ready for the following.
Proof of Proposition 3.16. Let {ξ ε (z)} z∈Λ s ε be a family of i.i.d. normal random variables with mean 0 and variance ε −3 and set, forε ≥ ε, ξ ε ε def = ξ ε * ε ̺ε, where ̺ is a symmetric compactly supported smooth function integrating to 1 and ̺ε(t, x) def =ε −3 ̺(ε −2 t,ε −1 x) its rescaled version. Notice that the definition of the discrete stochastic processes in X ε (ξ ε ε ) (or in X ε (ξ ε )) can be obtained by that of the ones in X(ξ ε ) (or X(ξ)) simply by replacing every occurrence of K with K ε , K ε with K ε ε def = K ε * ε ̺ε, ∂ x with D x,ε and the pointwise product with B ε . Moreover, all the estimates involving these quantities, as well as those we exploited in the proof of Proposition 3.4 are available in this discrete setting (Section 5.3, Proposition 5.16 and Corollary 5.18) and are uniform in ε. This ensures that the validity of Proposition 3.16 can be shown by retracing the steps followed in the proof of the above mentioned proposition.
Now, arguing as before, apart from the first, all the other factors at the right hand side are clearly bounded uniformly k ∈ [− 1 2 , 1 2 ]. For the first instead it suffices to notice that g, by Assumption 3.13 is differentiable and such that g(0) = 0, which in particular implies that g(k)/k is a bounded continuous function. Therefore we conclude that C , ε = 1 2 − 1 2 g(k) k g(−k) 2 4ν 2 (2ν +ν(k)) 2 f (k) 2 (4ν +ν(k)) 2μ (−k, k)dk , (5.32) which is finite and independent of ε.

Convergence of discrete solutions
In this section we want to prove the main result of this paper, namely Theorem 1.1, which states that, upon choosing the sequence of constants C ε as in Proposition 3.16, the sequence of solutions to our family of discrete models converge to the solution of the renormalized SBE.
Proof of Theorem 1.1. Given the stability results obtained in Sections 4 and 5, in order to prove the statement we will exploit a diagonal argument similar to the one, used in analogous contexts, in [HM15, HQ15, HS15] and many others. To be more specific, let ψ be a smooth, symmetric compactly supported function on R 2 which integrates to 1 and, for somē ε ≥ ε, define ξε def = ξ * ψε, where ψε(t, x) =ε −3 ψ(t/ε 2 , x/ε). Let uε be the unique smooth solution to ∂ t uε = ∆uε + ∂ x (uε) 2 + C∂ x uε + ∂ x ξε , uε(0, ·) = u 0 (·) , (6.1) where C is the constant in (3.21), and define the family of controlling processes X(ξε; C) def = X(ξε, Cε , Cε ) as in the proof of Proposition 3.4, where Cε is as in (5.15) and we impose Cε def = C. Let X(ξ; C) be the enhancement of white noise determined in the just mentioned proposition, in which though we subtract to the term X the constant C. Thanks to Theorem 3.9, we know that the solution map S, assigning to any (u 0 , X) ∈ C η × X the solution to 6.1 is jointly locally Lipschitz continuous, which in particular implies that if T ∞ is the explosion time for u def = S(u 0 , X(ξ; C)) then, for all T < T ∞ , we have E u; uε p C α η,T E [ X(ξ; C); X(ξε; C) p X ] and, by Proposition 3.4, for all p ≥ 1, the right hand side converges to 0 uniformly in ε. In order to discretize the noise ξε, forε ≥ ε, we set ξ ε ε (z) def = ψε * ε ξ ε (z) , for z ∈ Λ ε 2 ,T × T ε , where we recall that ξ ε (z) def = ε −3 ξ, ½ |·−z|s≤ε/2 . Let u ε ε and u ε be the unique solutions to (1.4) both with initial condition u ε 0 but with respect to the noise ξ ε ε and ξ ε , respectively. Let X ε (ξ ε ε , C ε ε ) def = X(ξ ε ε , C , ε ε , C , ε ε ) and X ε (ξ ε , C ε ) def = X ε (ξ ε , C , ε , C , ε ) be the discrete enhancement of ξ ε ε and ξ ε respectively, where, in each case, the constants are as in (5.31) and (5.32) respectively, and C ε converges to C as ε goes to 0. Arguing as before but applying Theorem 3.20 and the (uniform) local Lipschitz continuity of the discrete solution map S ε , we have that, for all T < where T d ∞ is the stopping time determined in Theorem 3.20, and, by Proposition 3.16, for all p ≥ 1, the latter converges to 0 uniformly in ε. Hence, we obtain, for all T < T d ∞ ∧ T ∞ , and taking the limit ε → 0 and thenε → 0, the above discussion guarantees that the first and the last summand at the right hand side converge to 0. For the second summand instead, it suffices to notice that, forε fixed, uε is the solution of a parabolic semilinear PDE driven by the smooth noise ξε (hence, smooth itself) therefore the convergence of its discretization is ensured by the convergence of the discrete noise (see [BS08] and references therein). But now, by our definitions we have that ξε = ψε * ξ and ξ ε ε = ψ ε ε * ξ , where we set ψ ε ε (z) def = ε −3 ψε, ½ |·−z|s≤ε/2 . It is easy to see that |D k (ψε(z) − ψ ε ε (z))| εε −|s|−|k|s−1 , for all k ∈ N 2 and uniformly in z ∈ Λ s ε , which implies the required convergence.

A Elements of wavelet analysis and its discrete counterpart
In this appendix we want to present the basic elements of wavelet analysis we need in the definition of the stochastic terms appearing in the description of the solution to our equation and its discrete version. We make no claim of exhaustiveness and the interested reader is addressed to the relevant literature, namely [Dau92], [Mey92]. In the discrete case, we recall the construction in [HM15] and explicitly state and prove the results necessary in our context.
A multiresolution analysis of L 2 (R) (see [Mey92, Def. 1, Sec. 2]) is a sequence {V n } n∈Z of closed subsets of L 2 (R) such that • V n ⊂ V n+1 for all n ∈ Z, the intersection of all V n 's is {0} and their union dense in L 2 (R) • for all n ∈ Z, f ∈ V 0 if and only if f (2 n ·) ∈ V n • there exists a function ϕ ∈ L 2 (R) (called father wavelet) such that {ϕ(· − k)} k∈Z is an orthonormal basis of V 0 .
We now let Λ n def = 2 −n Z and define ϕ n x = 2 n/2 ϕ(2 n (· − x)), the rescaled version of the father wavelet that preserves its L 2 -norm, then it is immediate to see that, for every n ∈ Z, the family {ϕ n x } x∈Λn forms an orthonormal basis of V n . It is even possible to understand whether a given function ϕ can be used as father wavelet for a multiresolution analysis. Indeed, if ϕ ∈ L 2 (R) is such that 1. ϕ , ϕ k = 0 for all k ∈ Z (ϕ k (·) def = ϕ(· − k)), 2. there exists a sequence {a k } k∈Z ∈ ℓ 2 (Z) for which ϕ(·) = k a k ϕ k (2·) 3. for some r ≥ 0, ϕ is a compactly supported C r function integrating to 1, then defining V n , for n ∈ Z, as the closure of span{ϕ n x } x∈Λn , we have that {V n } n∈Z forms a multiresolution analysis of L 2 (R). In the following proposition we collect (without proof) other important results concerning wavelets.
Proposition A.1. For every r ≥ 0 there exists a function ϕ ∈ L 2 (R) such that properties 1,2 and 3 stated above hold and, in 2, the number of non-zero a k 's is finite. Moreover, take n ∈ Z and let W n be the orthogonal complement of V n in V n+1 . Then there exists a finite set of coefficients {b k } k∈K , K ⊂ Z, such that, upon setting ψ(x) = k∈K b k ϕ(2x − k), the family {ψ n x } x∈Λn forms an orthonormal basis of W n , where ψ n x = 2 n/2 ψ(2 n (· − x)). As a consequence, for any n ∈ Z, is an orthonormal basis of the whole L 2 (R). At last, ψ is such that, for any k ∈ N, k ≤ r R ψ(x)x k dx = 0 .

(A.2)
We will refer to the function ψ introduced above as "mother wavelet".
Proof. The proof is immediate, notice that by the definition of the discrete Fourier transform in (B.1) and thanks to a simple change of variables, we have that the right hand side of (B.3) equals ε 2 R x,y∈Λε f (x + εy 1 )g(y + εy 2 ) 1 2ε − 1 2ε e −2πik(x−y) dkµ(dy 1 , dy 2 ) from which the result follows at once.