Marco Furlan
email: furlan@ceremade.dauphine.fr

Massimiliano Gubinelli
email: gubinelli@iam.uni-bonn.de

Weak universality for a class of 3d stochastic reactiondiusion models

Keywords: weak universality, paracontrolled distributions, stochastic quantisation equation, Malliavin calculus, partial chaos expansion

Weak universality for a class of 3d stochastic reaction-diffusion models.

where G is a standard Gaussian variable of unit variance. Let X be the stationary solution to the equation L X = ¡X + with the spacetime white noise and denote by JX N K is the generalized random elds given by the N -th Wick power of X which are well dened as random elements of S 0 (R T 3 ) as long as N 6 4. Denote with C X the covariance of X. The Gaussian analysis which we set up in this paper shows in particular the following convergence result. Theorem 1. Fix N 6 4 and assume that " (n¡N )/2 f n;" ! g n for 0 6 n 6 N as " ! 0, that (F " ) " C N +1 (R) and there exists constants c; C > 0 such that sup ";x X k=0 N +1 j@ x k F " (x)j 6 Ce cjxj :

Then the family of random elds F " N : (t; x) 7 ! " ¡N /2 F " (" 1/2 Y " (t; x));

(t; x) 2 R T 3 ;
converges in law in S 0 (R T 3 ) as " ! 0 to

P n=0 N g n JX N K.
Now take the smallest n such that f n;" converges to a nite limit as "! 0. Since H n (" 1/2 Y " ; Y ;" 2 ) = " n/2 JY " n K, the n-th term in the expansion of F " (" 1/2 Y " ) is f n;" " +(n¡5)/2 JY " n K. From Theorem 1 the equation yields a non-trivial limit only if = (5 ¡ n) /2. We are interested mainly in the case n = 3 ) = 1 and n = 1 ) = 2. The case = 2 gives rise to a Gaussian limit and its analysis its not very dicult.

In the following we will concentrate in the analysis of the = 1 case where the limiting behaviour of the model is the most interesting and given by the 3 4 family of singular SPDEs. In this case we obtain the family of models

L u " (t; x) = ¡" ¡ 3 2 F " (" 1 2 u " (t; x)) + " (t; x) (3)
with initial condition u 0;" () := " ¡ 1 2 u 0;" (" ¡1 ) where u 0;" is the initial condition of the microscopic model [START_REF] Albeverio | Stochastic dierential equations in innite dimensions: solutions via Dirichlet forms[END_REF]. Dene for m > 0 and = (t; x) 2 R + T 3 (m) := " (m¡3)/2 F ~"(m) (" 1/2 Y "; ):

where F ~" is the centered function F ~" (x) := F " (x) ¡ f 0;" ¡ f 1;" x ¡ f 2;" H 2 (x; Y ;" 2 ) = X n>3 f n;" H n (x; Y ;" 2 );

and Y ;"

2
is the variance of the centered Gaussian process " 1/2 Y " . Note that H n (" 1/2 Y " (); Y ;" 2 ) = " n/2 JY " n K and denote with f n;" the coecients in the chaos expansion of F " (" 1/2 Y "; ). Dene also various "dependent constants

d " := 1 9 Z s;x P s (x)E[ 0 (1)
(s;x) (1) ]; d ~" := 2 " ¡1/2 f 3;" f 2;" Z s;x P s (x)[C Y ;" (s; x)] 2 ;

d " := 1 6

Z s;x P s (x)E[ 0 (0)
(s;x) (2) ]; d ^" := 1 3

Z s;x P s (x)E[ 0 (0)
(s;x) (1) ];

d " := 2 d " + 3 d " ( 5
)
where P s (x) is the heat kernel and R s;x denotes integration on R + T 3 .

Assumption 2. All along the paper we enforce the following assumptions: a) fu 0;" g "2(0;1] converges to a limit u 0 in C ¡1/2¡ 8 > 0 and is independent of ; b) fu 0;" g "2(0;1] is uniformly bounded in L 1 , i.e. 9C > 0 such that 8" 2 (0; 1] ku 0;" k L 1 ((T/") 3 ) 6 C; c) fF " g "2(0;1] C 9 (R) and there exist constants c; C > 0 such that sup ";x X k=0 9 j@ x k F " (x)j 6 Ce cjxj ; [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] d ) the family of vectors f " g "2(0;1] = ( " (0)

; "

; "

(2)

; "

) "2(0;1] R 4 given by "

(3) = f 3;" " (1) = " ¡1 f 1;" ¡ 3d " " (2) = " ¡1/2 f 2;" " (0) = " ¡3/2 f 0;" ¡ " ¡1/2 f 2;" d " ¡ 3d ~" ¡ 3d ^" [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic nonlinear wave equation[END_REF] has a nite limit = ( (0) ; (1) ; (2) ; (3) ) 2 R 4 as " ! 0.

We can now formulate our main result.

Theorem 3. Under Assumptions 2 the family of random elds fu " g "2(0 ;1] given by the solution to eq. ( 3) converge in law and locally in time to a limiting random eld u() in the space C T C ¡ (T 3 ) for every 1 / 2 < < 1/2 + . The law of u() depends only on the value of and not on the other details of the nonlinearity or on the covariance of the noise term. We call this limit the dynamic 3 4 model with parameter vector 2 R 4 .

In Theorem 3 and in Assumption 2, C T C ¡ (T 3 ) denotes the space of continuous functions from [0; T ] to the Besov space C ¡ (T 3 ) = B 1;1 ¡ (T 3 ) (see Appendix A for details).

Remark 4. In particular we can take F " (x) = (3) H 3 (x; Y ;" 2 ) + " 1/2 (2) H 2 (x; Y ;" 2 ) + " ¡ (1) + " (1) H 1 (x; Y ;" 2 ) + " 3/2 ¡ (0) + " (0) so that f 3;" = (3) ; " ¡1/2 f 2;" = (2) ; " ¡1 f 1;" = (1) + "

(1)

; " ¡3/2 f 0;" = (0) + " (0)

; and

d " = 3( (3) ) 2 L " ; d ~" = (3) (2) L " ; d " = d ^" = 0;
where L " := 2 R s;x P s (x)(C Y ;" (s; x)) 2 : Choosing " (1) := 3d " = 9( (3) ) 2 L " ; " (0) := 3d ~" = 3 (3) (2) L " ;

we obtain " ! ( (0) ; (1) ; (2) ; (3) ). This shows that all the possible limits 2 R 4 are attainable. In this case (3) takes the form L u " = ¡ (3) u " 3 ¡ (2) u " 2 ¡ [ (1) ¡ 3 (3) " ¡1 Y ;" 2 + 9( (3) ) 2 L " ]u " ¡ (0) + (2) Y ;" 2 ¡ 3 (3) (2) L " + " : [START_REF] Gubinelli | KPZ Reloaded[END_REF] When the nonlinearity is given by a cubic polynomial like in [START_REF] Gubinelli | KPZ Reloaded[END_REF] the corresponding limiting model is called dynamic 3 4 equation or stochastic quantisation equation. In two dimensions, this model has been subject of various studies since more than thirty years [START_REF] Jona-Lasinio | On the stochastic quantization of eld theory[END_REF][START_REF] Albeverio | Stochastic dierential equations in innite dimensions: solutions via Dirichlet forms[END_REF][START_REF] Da | Strong solutions to the stochastic quantization equations[END_REF]. For the three dimensional case, the kind of convergence results described above are originally due to Hairer [START_REF] Hairer | A theory of regularity structures[END_REF][START_REF] Hairer | Regularity structures and the dynamical 3 4 model[END_REF] and constitute one of the rst groundbreaking applications of his theory of regularity structures. Similar results were later obtained by Catellier and Chouk [START_REF] Catellier | Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation[END_REF] using the paracontrolled approach of Gubinelli, Imkeller and Perkowski [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. Kupiainen [START_REF] Kupiainen | Renormalization Group and Stochastic PDEs[END_REF] described a third approach using renormalization group ideas.

Weak universality is the observation that the same limiting object describes the large scale behaviour of the solutions of more general equations, in particular that of the many parameters present in a general model, only a nite number of their combinations survive in the limit to describe the limiting object. The adjective weak is related to the fact that in order to control the large scale limit the non-linearity has to be very small in the microscopic scale. This sets up a perturbative regime which is well suited to the analysis via regularity structures or paracontrolled distributions.

The rst result of weak universality for a singular stochastic PDE has been given by Hairer and Quastel [START_REF] Hairer | A class of growth models rescaling to KPZ[END_REF] in the context the KardarParisiZhang equation. Using the machinery developed there Hairer and Wu [START_REF] Hairer | Large scale behaviour of 3d phase coexistence models[END_REF] proved a weak universality result for three dimensional reactiondiusion equations in the case of Gaussian noise and a polynomial nonlinearity, within the context of regularity structures. Weak universality for reactiondiusion equations driven by non Gaussian noise is analysed in Shen and Wu [START_REF] Shen | Weak universality of dynamical $\Phi4_3$: non-Gaussian noise[END_REF]. Recently, important results concerning the stochastic quantisation equation we obtained by Mourrat and Weber. In particular the convergence to the dynamic 2 4 model for a class of Markovian dynamics of discrete spin systems [START_REF] Mourrat | Convergence of the two-dimensional dynamic Ising-Kac model to P h i 2 4[END_REF] and also the global wellposedness of 2 4 in space and time [START_REF] Mourrat | Global well-posedness of the dynamic P h i 4 model in the plane[END_REF] and in time for 3 4 [START_REF] Mourrat | Global well-posedness of the dynamic P h i 3 4 model on the torus[END_REF]. The recent preprint [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic nonlinear wave equation[END_REF] analyzes an hyperbolic version of the stochastic quantisation equation in two dimensions, including the associated universality in the small noise regime.

The present work is the rst which considers in detail the weak universality problem in the context of paracontrolled distributions, showing that on the analytic side the apriori estimates can be obtained via standard arguments and that the major diculty is related to showing the convergence of a nite number of random elds to universal limiting objects. The main point of our analysis is our use of the Malliavin calculus [START_REF] Nualart | The Malliavin calculus and related topics[END_REF][START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF] to perform the analysis of these stochastic terms without requiring polynomial nonlinearity as in the previous works cited above. In particular we were inspired by the computations in [START_REF] Nourdin | Central limit theorems for multiple Skorohod integrals[END_REF] and in general by the use of the Malliavin calculus to establish normal approximations [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF]. The main technical results of our paper, Theorem 10 below, is not particularly linked to paracontrolled distributions. A similar analysis is conceivable for the stochastic models in regularity structures. Moreover the same tools can also allow to prove similar nonpolynomial weak universality statements for the KPZ along the lines of the present analysis. This is the subject of ongoing work.

The paper is structured as follows. Section 1 contains the paracontrolled analysis of eq. ( 3) which will allow to obtain uniform estimates to pass to the limit. Section 2 in the core of the paper, it contains the stochastic analysis based on Malliavin calculus which allows to control the limit of the random elds involved in the paracontrolled description of eq. ( 3) and to identify their limits. All the rest of the paper consists in three appendices which do not contain original material but allow the paper to be self contained. In particular Appendix A collects notations and basic results of paracontrolled calculus. Appendix C collects basic denitions and results from Malliavin calculus which will be needed in the analysis of the stochastic terms. Finally Appendix B contains mostly some technical estimates on kernels needed in the stochastic analysis. The reader not familiar with paracontrolled calculus and/or Malliavin calculus is encouraged to read Appendix A and/or Appendix C before going on. In particular please refer to Section A.1 for the notations and conventions in force all along the paper.

Analysis of the mesoscopic model

In this section we describe the paracontrolled approach [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] to a solution theory for eq. ( 3) along the lines of the CatellierChouk [START_REF] Catellier | Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation[END_REF] analysis. The basic results of paracontrolled calculus we need in this section are included in Appendix A.

The continuity of the solution map for a paracontrolled equation (already established in [START_REF] Catellier | Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation[END_REF] and recalled here) allows us to prove convergence of the solution u " (Theorem 3) by showing the convergence of the enhanced noise Y " and the remainder R " in the appropriate space. Finally, in Theorem 7 we identify the limiting process as the solution of a paracontrolled equation.

Paracontrolled structure

Write u " = Y " + v " , and perform a Taylor expansion of the reaction term F ~"(" 1/2 Y " + " 1/2 v " ) in (3) around " 1/2 Y " up to the third order to get

L u " = " ¡ (0) ¡ (1) v " ¡ 1 2 (2) v " 2 ¡ 1 6 (3) v " 3 ¡ R " (v " ) ¡" ¡3/2 f 0;" ¡ " ¡1 f 1;" (Y " + v " ) ¡ " ¡1/2 f 2;" (JY " 2 K + 2v " Y " + v " 2 ): (9)
with (m) dened in (4) 8 2 R T 3 and R " (v " ) the remainder of the Taylor series.

Dene the following random elds: 3) ; [START_REF] Hairer | Regularity structures and the dynamical 3 4 model[END_REF] with (m) dened in (4), Y " stationary and Y " ; Y " starting from 0 in t = 0. In the scope of this section we can take any set of constants fd " ; d " ; d " ; d ~" ; d ^" g 3 d " that satisfy

L Y " := ¡Y " + " Y ~" := " ¡1/2 f 2;" JY " 2 K; L Y " := (0) ; Y " := Y " Y " ¡ d " ; Y " := 1 3 (1) Y " := Y " Y " ¡ d " Y " ¡ d ^" ; L Y " := Y " ; Y " := Y " Y " ¡ d " ; Y " := 1 6 (2) ; Y ~" := Y ~" Y " ¡ d ~" ; Y " ? := 1 6 (
d " = 2 d " + 3 d " : (11)
Equation ( 9) takes the form

L v " = Y " ¡ Y " ¡ Y ~" ¡ 3Y " v " ¡ 3Y " v " 2 ¡ Y " ? v " 3 ¡" ¡3/2 f 0;" ¡ " ¡1 f 1;" (Y " + v " ) ¡ " ¡1/2 f 2;" (2Y " v " + v " 2 ) ¡ R " (v " ): (12)
In this expression the products Y " v " , Y " v " 2 and Y " v " do not meet the conditions for continuity. In order to continue the analysis we pose the paracontrolled Ansatz

v " = ¡Y " ¡ Y ~" ¡ 3v " Y " + v " ] ; ( 13
)
and proceed to decompose the ill-dened products using the paracontrolled techniques recalled in Appendix A .

We start with

v " Y " = v " Y " + v " Y " + v " Y " :
The resonant term, together with Ansatz (13), yields:

v " Y " = ¡Y " Y " ¡ Y ~" Y " ¡ 3 v " (Y " Y " ) ¡3 com 1 (v " ; Y " ; Y " ) + v " ] Y " :
So we let

Y " ^v" := v " Y " ¡ v " Y " + (3 v " d " + d " Y " + d ^" + d ~" ) = v " Y " ¡ Y ~" ¡ Y " ¡ 3v " Y " + v " ] Y " ¡ 3 com 1 (v " ; Y " ; Y " )
Moreover we have for v " Y " :

v " Y " = ' " Y " ¡ Y " Y " ¡ Y " Y " ¡ Y " Y "
where we introduced the shorthand

' " = v " + Y " . So we let v " Y " := v " Y " + d " = ' " Y " ¡ Y " Y " ¡ Y " Y " ¡ Y "
Finally to analyse the product Y " v " 2 we write

Y " v " 2 = Y " (Y " ) 2 ¡ 2Y " Y " ' " + Y " ' " 2 ;
and consider the products involving only Y factors: rst

Y " Y " = Y " Y " + Y " Y " + Y " + d " =: Y " Y " + d " ;
and then we dene the term Y " (Y " ) 2 as follows:

Y " (Y " ) 2 := Y " (Y " ) 2 ¡ 2d " Y " = Y " (Y " ) 2 + Y " (Y " ) 2 + Y " (Y " Y " ) + 2com 1 (Y " ; Y " ; Y " ) + 2Y " Y " : so that Y " v " 2 := Y " v " 2 + 2d " v " = Y " (Y " ) 2 ¡ 2 (Y " Y " )' " + Y " ' "
Substituting these renormalized products into [START_REF] Hairer | Large scale behaviour of 3d phase coexistence models[END_REF] we obtain the following equation for v " ] :

L v " ] = 3 com 3 (v " ; Y " ) + 3 com 2 (v " ; Y " ) ¡Y " ? v " 3 ¡ 3Y " v " 2 ¡ 3Y " ^v" +Y " ¡ " (2) (2v " Y " + v " 2 ) ¡ " (1) (Y " + v " ) + [9d " + 6d " ¡ 3d " ]v " ¡ " (0) ¡ R " (v " ) = U ( " ; Y " ; v " ; v " ] ) ¡ R " (v " ) (14)
with R " (v " ) the Taylor remainder which appears in [START_REF] Hairer | A theory of regularity structures[END_REF] and " = ( " (0)

; "

; "

; "

) 2 R 4 given by eq. ( 7). We can use the constraint [START_REF] Hairer | A class of growth models rescaling to KPZ[END_REF] to remove the term proportional to v " . The enhanced noise vector Y " is dened by

Y " := (Y " ? ; Y " ; Y " ; Y ~" ; Y " ; Y " ; Y " ; Y ~" ; Y " ) 3 X T := C T C ¡ C T C ¡ 1 2 ¡ (C T C ¡1¡ ) 2 L T 1/2¡ (C T C ¡ ) 3 C T C ¡ 1 2 ¡ ( 15
)
for every > 0. We use the notation kY " k XT = P kY " k X for the associated norm where Y " is a generic tree in Y " . The homogeneities j j 2 R are given by

Y " = Y " ? Y " Y " Y ~" Y " Y " Y " Y ~" Y " j j = 0 ¡1/2 ¡1 ¡1 1/2 0 0 0 ¡1/2
Notice that for every " > 0 eq. ( 14) is equivalent to (3) together with Ansatz (13).

A-priori estimates

In this section we show uniform a-priori estimates for the pair (v " ; v " [ ) which solves the following system of equations 8 > > > < > > > :

v " = ¡Y " ¡ Y ~" ¡ 3v " Y " + v " [ + v " \ L v " [ = U ( " ; Y " ; v " ; v " [ + v " \ ) ¡ R " (v " ) v " [ (0) = Y " (0) + Y ~" (0) + 3v ";0 Y " (0) (16)
with v " \ (t) := P t v ";0 and v ";0 := u 0;" ¡ Y " (0) 2 C ¡1/2¡ . U is given in (14). It is easy to see, by taking v " ] = v " [ + v " \ , that this is equivalent to eq. ( 14) together with Ansatz (13) on v " . We consider the spaces

V T [ := L T 2 \ L T 1/4;1/2+2 \ L T 1/2;1+2 ; V T := L T 1/2;1/2¡ \ L T 1/4+3/2;2 ;
with the corresponding norms

kv " [ k V T [ := kv " [ k L T 2 + kv " [ k L T 1/4;1/2+2 + kv " [ k L T 1/2;1+2 ; ( 17
)
kv " k VT := kv " k L 1/2;1/2¡ + kv " k L 1/4+3/2;2: (18)
Dene 8 2 (0; 1) the quantity

M "; (Y " ; u 0;" ) := k" /2 e c" 1/2 jY"j+c" 1/2 jPv";0j k L p [0;T ]L p (T 3 ) (19)
which will be used to control the remainder R " . The main result of this section is the following lemma.

Lemma 5. There exists a time T ? = T ? (kY " k XT ; ku ";0 k C ¡1/2¡ ; j " j) 2 (0; T ] depending only on kY " k XT , ku ";0 k C ¡1/2¡ and j " j, constants 2 (0; 1) and M "; = M "; (Y " ; u 0;" ) > 0 dened by [START_REF] Mourrat | Construction of P h i 3 4 diagrams for pedestrians[END_REF], and a universal constant C > 0 such that, whenever M "; 6 T ? /2 we have

kv " n [ k V T? [ 6 C(1 + j " n j) (1 + kY " n k X T ) 3 (1 + ku ";0 k C ¡1/2¡) 3 ; kv " k VT ? 6 C ¡ kY " k XT + ku ";0 k C ¡1/2¡ + kv " [ k V T? [ : Proof. Dene v " := v " ¡ v " \ such that v " = ¡Y " ¡ Y ~" ¡ 3(v " + v " \ ) Y " + v " [ ;
and v " (0) = 0. Note also v " := v " + Y " . Using Lemma 16 (and the fact that kf k L T ; . T kf k L T ) we obtain for ; > 0 small enough

kIf k L T ¡+2;2 + kIf k L T 1/4¡+2;1/2+2 + kIf k L T 1/2¡+2;1+2 . T 2 (kf k M T 1¡ C ¡ + kf k M 1/2+2 C ¡1/2¡2): ( 20
)
We choose > 2 small enough so that

L T ¡+3/2;2 \ L T 1/4¡+3/2;1/2+2 \ L T 1/2¡+3/2;1+2 V T [
We dene also the norm

kv " k V T := kv k L T 2 + kv k M T 1/4 C 1/2+2: Now kv " k V T . kY " + Y ~" k V T + kv " k C T L 1(kY " k CTC 1¡ + kY " k CTC ¡1¡) + ¡ kv " \ k CTC ¡1/2¡ + kv " \ k M T 1/4 C ¡ (kY " k CTC 1¡ + kY " k CTC ¡1¡) + kv " [ k V T . kY " k X T + T kv " k V T + kv ";0 k C ¡1/2¡ + kv " [ k V T [
where we used that v (0) = 0 and as a consequence that kv "

k C T L 1 6 T kv " k C T L 1 6 T kv " k V T to gain a small
power of T . So provided T is small enough (depending only on Y " ) this yields the following a-priori estimation on v "

:

kv " k C T L 1 . kv " k V T . kY " k X T + kv ";0 k C ¡1/2¡ + kv " [ k V T [ :
Therefore we have an estimation on v " :

kv " k V T 6 kv " \ k V T + kv k V T . kv ";0 k C ¡1/2¡ + kv " k V T . kY " k X T + kv ";0 k C ¡1/2¡ + kv " [ k V T [ :
In order to estimate terms in U ( " ; Y " ; v " ; v " [ + v " \ ) we decompose the renormalised products as

Y " ^v" = v " Y " ¡ Y ~" ¡ Y " ¡ 3v " Y " + v " [ Y " + v " \ Y " ¡ 3 com 1 (v " ; Y " ; Y " ) v " Y " = ¡Y ~" Y " ¡ 3(v " Y " )Y " + Y " 4 (v " [ + v " \ ) + Y " (v " [ + v " \ ) ¡Y " Y " ¡ Y " Y " ¡ Y " Y " v " 2 = Y " (Y " ) 2 + 2(Y " Y " )(Y ~" + 3v " Y " ) ¡ 2 (Y " Y " ) 4 (v " [ + v " \ ) +2 (Y " Y " ) (v " [ + v " \ ) + Y " 4 (v " + v \ ) 2 + Y " (v " + v \ ) 2 : We have U ( " ; Y " ; v " ; v " [ + v " \ ) = Q ¡1/2 ( " ; Y " ; v 0;" ; v " ; v " [ ) + Q 0 ( " ; Y " ; v 0;" ; v " ; v " [ ) + Q ";Y" with the denitions Q ¡1/2 := ¡3[v " Y " ¡ 3 com 1 (v " ; Y " ; Y " ) + Y " (v " + v \ ) 2 ] ¡6[(Y " Y " )(3v " Y " ) + (Y " Y " ) (v " [ + v " \ )] +2 " (2) (3(v " Y " )Y " ¡ Y " (v " [ + v " \ )) + 3 com 3 (v " ; Y " ) + 3 com 2 (v " ; Y " ) Q 0 := 3[3v " Y " ¡ v " [ Y " ¡ v " \ Y " + 2 (Y " Y " ) 4 (v " [ + v " \ ) ¡ Y " 4 (v " + v \ ) 2 ] ¡Y " ? v " 3 ¡ " (2) [v " 2 + 2Y " 4 (v " [ + v " \ )] Q ";Y" := ¡ 1 ¡ " (1) Y " ¡ " (0) + 3 [Y ~" + Y " ¡ Y " (Y " ) 2 ¡ 2(Y " Y " )Y ~" ] +2 " (2) (Y ~" Y " + Y " Y " + Y " Y " + Y " )
With the same technique we used above for v "

, we obtain the following estimate on v "

kv " k L T 1/2+3/2;1/2+2 + kv " k L T 1/4+; . kY " k X T + kv ";0 k C ¡1/2¡ + kv " [ k V T [
and this yields k(v "

) 2 k L 3/4+5/2;1/2+2 + k(v " ) 2 k L 1/2+2; . (kY " k XT + kv ";0 k C ¡1/2¡ + kv " [ k V T [ ) 2 .
We obtain, using the results of Appendix A

kQ ¡1/2 k M 1/2+2 C ¡1/2¡2 + kQ 0 k M T 1¡ C ¡ . (1 + j " j)(1 + kY " k X T ) 3 (1 + kv ";0 k C ¡1/2¡ + kv " [ k V T [ ) 3 kQ ";Y" k C T C ¡1/2¡ . (1 + j " j) (1 + kY " k XT ) 3
In order to conclude the estimation of kv "

[ k V T [ we have to bound kI R " (v " )k V T [ . By Lemma 6 8; 2 (0; 1)
such that

1 ¡ 3 + > 1 4 + 3 2 we have kt 7 ! t 1¡ I R " (v " )(t; x)k L p [0;T ]L p (T 3 ) . M "; (Y " ; u 0;" )kv " k V T 3+ e c" 1/2 kv " k V . By
Lemma 17 together with (50) we obtain then for these values of and :

k IR " (v " )k V T [ . M "; (Y " ; u 0;" )kv " k V T 3+ e c" 1/2 kv " k V : Using that kPv " [ (0)k V T [ . kv " [ (0)k CTC 1/2¡2 . (1 + kv ";0 k C ¡1/2¡)kY " k X T we obtain that 9C 0 > 0 such that kv "n [ k V T [ 6 C 0 (1 + j "n j) (1 + kY "n k XT ) 3 (1 + kv ";0 k C ¡1/2¡) 3 + C 0 T /2 (1 + j " j)(1 + kY " k XT ) 3 kv " [ k V T [ 3 +C 0 M "; (Y " ; u 0;" )e c" 1/2 ¡ kY"kX T +kv";0k C ¡1/2¡ e c" 1/2 kv " [ k V T [ kv " k VT 3+ 6 D + C M " (Y " ; u 0;" )e c" 1/2 kv " [ k V T [ + C T /2 kv " [ k V T [ 3 + C M "; (Y " ; u 0;" )e c" 1/2 kv " [ k V T [ kv " [ k V T [ 3+ with C := C 0 [(1 + j " j)(1 + kY " k X T ) 3 + e c" 1/2 ¡ kY"kX T +kv";0k C ¡1/2¡ (1 + (kY " k X T + kv ";0 k C ¡1/2¡) 3+ )];
and

D := C 0 (1 + j " n j) (1 + kY " n k X T ) 3 (1 + kv ";0 k C ¡1/2¡) 3 :
Let T ? 2 (0; T ] such that: that S < T ? , then we can take > 0 small enough such that S + < T ? and by continuity kv

C T ? /2 [(5 C) 2 + e c" 1/2 (5C) (5 C) 2+ ] 6 1
" [ k V S + [ 6 5C, then kv "n [ k V S + [ 6 D + C M " (Y " ; u 0;" )e c" 1/2 kv " [ k V S + [ + C (S + ) /2 kv " [ k V S + [ 3 + C M "; (Y " ; u 0;" )e c" 1/2 kv " [ k V S + [ kv " [ k V S + [ 3+ 6 D + C M " (Y " ; u 0;" )e c" 1/2 (5C) + C T ? /2 (5 C) 2 kv " [ k V S + [ + C T ? /2 e c" 1/2 (5C) (5C) 2+ kv " [ k V S + [ 6 2D + 1 2 kv " [ k V S + [ which gives kv "n [ k V S + [ 6 4D
. This implies S = T ? (by contradiction). Lemma 6. For every 2 (0; 1), 2 [0; 1] we have

kt 7 ! t R " (v " ; v " [ ; v " \ )(t; x)k L p [0;T ]L p (T 3 ) . M "; (Y " ; u 0;" )kv " k M /(3+) L 1 3+ e c" 1/2 kv " k C T L 1 with v " := ¡Y " ¡ Y ~" ¡ 3v " Y " + v " [ .
Proof. We can write the remainder in two ways:

R " (v " ) = v " 3 Z 0 1 [F " (3)
("

1 2 Y " + " 1 2 v " ) ¡ F " (3) (" 1 2 Y " )] (1 ¡ ) 2 2! d = " 1 2 v " 4 Z 0 1 F " (4)
("

1 2 Y " + " 1 2 v " ) (1 ¡ ) 3 3! d :
From assumption (6) on F we obtain by interpolation of these two expressions, 8 2

[0; 1], 8t > 0; x 2 T 3 , jR " (v " )(t; x)j . " /2 jv " (t; x)j 3+ e c" 1 2 jY"(t;x)j+c " 1 2 jv " \ (t;x)j+cj" 1 2 v " (t;x)j ;
and we estimate, 8 2 [0; 1),

kt 7 ! t R " (v " )(t; x)k L p [0;T ]L p (T 3 ) . kt 3+ v " (t)k CTL 1 3+ e c" 1/2 kv " k C T L 1 " 2 e c" 1 2 jY"(t;x)j+c " 1 2 jv " \ (t;x)j L p [0;T ]L p (T 3 ) :
1.

Identication of the limit

In order to identify interesting limits for equation ( 3), we introduce the enhanced universal noise X, dened as X = (X ; X ; X ; X ; X ; X );

where X is the stationary solution to to the linear equation L X = ¡X + and is the time-space white noise on R T 3 . We dene

X := JX 3 K; X := JX 2 K; q X := q (X X) = Z 1 ; 2 JX 1 3 KX 2 1;2 ; q X := q (1 ¡ J 0 )(X X ) = Z 1;2 (1 ¡ J 0 )(JX 1 2 KJX 2 2 K) 1;2 ; q X := Z 1 ; 2 (1 ¡ J 1 )(JX 1 3 KJX 2 2 K) 1 ; 2 + 6 Z s;x [ q X(t + s; x ¡ x) ¡ q X(t; x )]P s (x) [C X (s; x)] 2 ; ( 21
)
with X (t = 0) = X (t = 0) = 0. Here as before JK stands for the Wick product, i = (

x i ; s i ) 2 R T 3 , C X (t; x)
is the covariance of X and 1 ; 2 is dened as

1; 2 := [ Z x; y K q;x (x) X i j K i;x (y)K j ;x (x 2 )P t¡s1 (y ¡ x 1 )](t ¡ s 2 )d 1 d 2 :
Standard computations (see e.g. [START_REF] Catellier | Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation[END_REF] or [START_REF] Mourrat | Construction of P h i 3 4 diagrams for pedestrians[END_REF]) show that, for any T > 0, 0 < < 0 ,

X 2 C T C ¡ 1 2 ¡2 0 (C T C ¡1¡2 0 ) 2 C T C 1 2 ¡2 0 (C T C 0¡2 0 ) 2 C T C T ¡ 1 2 ¡2 0 ;
almost surely. Finally, for every = ( (0) ; (1) ; (2)

; (3) ) 2 R 4 we dene Y() := ( (3) ; (3) X ; (3) X ; (2) X ; (3) X ; ( (3) ) 2 X ; ( (3) ) 2 X ; (3) (2) X ; ( (3) ) 2 X ): (22)
Using the paracontrolled structure we developed in the preceding sections and its continuity with respect to Y " , we can state the convergence of the solution of the mesoscopic equation, under the hypothesis that Y " and M "; as dened in ( 19) converge (this is shown in Theorem-10 and Lemma 9).

Theorem 7. Under Assumption 2, the family of random elds u " given by the solutions to eq. ( 3) converges in law and locally in time to a limiting random eld u() in the space C T C ¡ (T 3 ) for every 1 / 2 < < 1/ 2 + . The limiting random eld u() solves the paracontrolled equation

8 > > > > < > > > > : u() = X + v() v() = X ¡ (3) X ¡ (2) X ¡ 3 (3) v() X + v ] () L v ] () = U (; Y(); v(); v ] ()) v ] ()(t = 0) = v 0 + (3) X (t = 0) + (2) X (t = 0) + 3 (3) v ";0 X (t = 0) (23)
with U dened in [START_REF] Kupiainen | Renormalization Group and Stochastic PDEs[END_REF] and v 0 = u 0 ¡ X(t = 0).

Proof. Fix T > 0. Let us denote via ¡ the solution map for ( 16) so that u " = ¡(u ";0 ; Y " ; " ; R " (v " )). Denote by u " = ¡ (u ";0 ; Y " ; " ; R " (v " )) the process u " stopped at the time T ? (kY " k X T ; ku ";0 k C ¡1/2¡; j " j) and ¡ the corresponding solution map. Note that u solves the same equation with Y " replaced by Y(), u ";0 with u 0 , " replaced by and R " = 0. So u() = u () = ¡ (u 0 ; Y(); ; 0) up to time T ? (kY()k X T ; ku 0 k C ¡1/2¡; jj). Let us introduce the random eld u ~" = ¡ (u ";0 ; Y " ; " ; 0) which solves the paracontrolled equation ( 16) but with remainder R " = 0. Consider the n-uple of random variables (u ";0 ; Y " ; u " ; u ~" ) and let " be its law on Z = C ¡ X T (C T C ¡ ) 2 conditionally on E " := fM "; 6 T ? /2 g for 2 (0; 1) xed in Lemma 5. Note that we know that P(E " ) ! 1 from Lemma 9. By the apriori bounds of Lemma 5 we have tightness of the family ( " ) " . By standard arguments it is easy to obtain continuity of the map ¡ and also to observe that for any 0 > 0, " (ku " ¡ u ~" k > 0 ) ! 0 as " ! 0 since M "; ! 0 in probability 8 2 (0; 1). This shows that " concentrates on

C ¡ X T f(z; z) 2 C T C ¡ g. Let any accumulation point of ( " ) " . Then (C ¡ X T f(z; z) 2 C T C ¡ g) = 1.
Moreover along subsequences we have that for any bounded continuous function

E('(u ";0 ; Y " ; u ~" )) = E('(u ";0 ; Y " ; ¡ (u ";0 ; Y " ; " ; 0))) ! E('(u 0 ; Y(); ¡ (u 0 ; Y(); ; 0)))
since by Theorem 10 the vector Y " converges in law to Y() and u ";0 to u 0 and ¡ is a continuous function. We deduce that, still along subsequences, for any test function ', Z Z '(x; y; z; t)d " (x; y; z; t) ! Z Z '(x; y; t; t)d(x; y; z; t) = Z Z '(x; y; ¡ (x; y; ; 0); ¡ (x; y; ; 0))d(x; y; z; t) but we know also that since P(E " ) ! 1 we have

E[ (u ";0 ; Y " )jE " ] = E[ (u ";0 ; Y " )I E " ] P(E " ) ! E[ (u 0 ; Y())];
for any test function . So the rst two marginals of have the law of (u 0 ; Y()) and they are independent since (u ";0 ; Y " ) are independent for any ". Calling the law of (u 0 ; Y()) we have that

Z Z '(x; y; z; t)d(x; y; z; t) = Z C ¡ X T '(x; y; ¡ (x; y; ; 0); ¡ (x; y; ; 0))d(x; y)
which implies that is unique and that the whole family ( " ) " converges to .

Remark 8. In particular this proves Theorem 3.

Convergence of random elds

In this section we prove the convergence of the random elds Y " and M "; . The convergence in probability of M "; is easily obtained as we show in the following lemma.

Lemma 9. Under Assumptions 2 the random variable M "; (Y " ; u 0;" ) dened in [START_REF] Mourrat | Construction of P h i 3 4 diagrams for pedestrians[END_REF] converges to zero in probability for every 2 (0; 1).

Proof. Recalling that v ";0 := u 0;" ¡ Y " (0) we can use Young's inequality estimate M "; (Y " ; u 0;" ) for some c 0 > 0 as

M "; (Y " ; u 0;" ) . " /2 ke c 0 " 1/2 jY"j k L p [0;T ]L p (T 3 ) + " /2 ke c 0 " 1/2 jP.Y"(0)j k L p [0;T ]L p (T 3 ) +" /2 T 1/p e c 0 k" 1/2 u0;"k L 1 :
Under Assumptions 2 the term k" 1/2 u 0;" k L 1 (T 3 ) is uniformly bounded, so the third term above converges to zero almost surely. Note that " 1/2 Y " (t; x) and P t " 1/2 Y " (t = 0) are centered Gaussian random variables, and then both

Eke c 0 " 1/2 jY"j k L p [0;T ]L p (T 3 ) p and Eke c 0 " 1/2 jP.Y"(0)j k L p [0;T ]L p (T 3 )
p are uniformly bounded in " > 0. This yields the convergence in probability of M "; (Y " ; u 0;" ) by Markov inequality.

The central result of this paper is the convergence of the enhanced noises (or trees) Y in law, and their uniform boundedness. Theorem 10. Under Assumptions 2 there exists C > 0 such that for any p 2 [2; 1) we have kY

" k XT < C in L p (P). Moreover, Y " ! Y() 2 X T in law.

Strategy of the proof

The strategy of proof is the following. Denote X = (X ) , Y() = (Y ()) and let K the measurable function of the Gaussian process X 2 C T C ¡1/2¡ such that X = K (X) and Y () = f ()K (X) with f () suitable deterministic functions of . For each we will show that Y " can be decomposed as

Y " = f ( " )K (Y " ) + Y ^" (24)
where Y ^" are suitable remainder terms. For all p > 2 it is well-known (see [START_REF] Catellier | Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation[END_REF], [START_REF] Hairer | A theory of regularity structures[END_REF]) that the term f ( " )K (Y " ) is uniformly bounded in L p (; X ) (with X given by ( 15)), then we will just prove that Y ^" converges to zero in L p (; X ). This con be done by showing that, by Besov embedding, for 1 6 p < +1 and 8 < j j we have

E( Y ^" (t) C ¡3/p p ) . E Y ^" (t) B p; p p 6 X q 2 pq Z T 3 q Y ^" (t; x) L p () p dx 6 C " ! 0 (25)
thanks to the stationarity of the process Y (t; x). For this it suces to show X q

2 pq sup x q Y ^" (t; x) L p () p ! 0 as " ! 0 (26)
In order to conclude uniform convergence for t 2 [0; T ] it suces to show that for 2 [0; 1/2], q > ¡1:

sup x q Y ^" (t; x) ¡ q Y ^" (t; x) L p () p 6 C " jt ¡ sj p 2 ¡(¡2)pq with C " ! 0: (27)
Indeed, by the Garsia-Rodemich-Rumsey inequality we obtain for > 0 small enough and p large enough sup

" E Y ^" C T ¡2/p B p; p ¡2 ¡ p 6 T 2 X q 2 (¡2 ¡)pq sup s<t2[0;T ] sup x q Y ^" (t; x) ¡ q Y ^" (t; x) L p () p jt ¡ sj p 6 C " T 2 X q 2 ¡pq
which by Besov embedding yields an estimation on

E ¡ kY " k C T ¡/2 C ¡2 ¡
for > 0 small enough. This gives us the necessary tightness to claim that Y " has weak limits along subsequences. The only thing left to prove is that for each we have K (Y " ) ! K (X) in law. However this is clear since we can introduce a convolution regularisation of X called X " which has the same law of Y " for any " > 0. At this point an approximation argument gives that K (Y " ) has the same law of K (X " ). Transposing the regularisation to the kernels of the chaos expansion we can write K (X " ) = K " (X) and now it is easy to check that K " (X) ! K (X) in probability (as done e.g. in [START_REF] Catellier | Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation[END_REF], [START_REF] Hairer | A theory of regularity structures[END_REF]). We can then conclude that K (Y " ) ! K (X) in law for any .

Details of the proof

Let us now give the details of the the decomposition (24) and the convergence to zero of the remainder Y ^" in L p (; X ). We need to introduce some notations based on the results of Appendix C. Looking at the denitions of trees listed in [START_REF] Hairer | Regularity structures and the dynamical 3 4 model[END_REF], it is clear that q Y " can be written in the form

(3 ¡ j)! 3! Z (j) ; or (3 ¡ j)!(3 ¡ k)! 3! 3! Z 1;2 1 (j) 2 (k) 1;2 ¡ [renormalisation];
for ; 1 ; 2 2 R T 3 , 0 6 j ; k 6 3 and some measures and 1; 2 . Note that the k-th Malliavin derivative of

(m) , namely D k (m) is (m+k) h k . Then expansion (69) of Appendix C takes a more explicit form 8n > 1: (m) = X k=0 n¡1 E ¡ (m+k) k! JY "; k K + n ¡ Q 1 n (m+n) h n = X k=0 n¡1 " (m+k ¡3)/2 (m + k)! k! f ~m+k;" JY "; k K + n ¡ Q 1 n (m+n) h n (28) with Q n m := Q k=n m (k ¡ L) ¡1 and Y "; := Y " (t; x), = (t; x) 2 R T 3 .
Here we used the fact that n (h

n ) = JY n K
(see Remark 40) and that by the denition of (m) (4) we obtain

8 2 R T 3 , E ¡ (m+k) = " (m+k¡3)/2 (m + k)! f ~m+k;" ;
where f ~n;" is the n-th coecient of the chaos expansion of F ~"(" 1/2 Y " ) relative to " 1/2 Y " (t; x), so that f ~n;" = 0 for n < 3 and f ~n;" = f n;" for n > 3. Choosing n = 4 ¡ m in eq. ( 28) we obtain

(m) = 3! (3 ¡ m)! f 3;" JY "; 3¡m K + 4¡m ¡ Q 1 4¡m (4) h 4¡m = 3! (3 ¡ m)! f 3;" JY "; 3¡m K + ^ (m) ; (29) with ^ (m) := 4¡m (Q 1 4¡m (4) h 4¡m ): (30)
This yields:

(0) = f 3;" JY "; 3 K + 4 (Q 1 4 (4) h 4 ); (1) = 3f 3;" JY "; 2 K + 3 (Q 1 3 (4) h 3 ); (2) = 6f 3;" Y "; + 2 (Q 1 2 (4) h 2 ); (3) = 6f 3;" + (Q 1 (4) h ):
It suces to substitute this decomposition in [START_REF] Hairer | Regularity structures and the dynamical 3 4 model[END_REF] to identify the remainder terms Y ^" for every tree Y " . In the next two sections we will consider separately rst order trees, which are dened a function of (m) and second order trees, which in turn are obtained by multiplying rst order trees (and renormalising). We will show that each of these remainder terms satisfy (26) and ( 27).

Remark 11. Note that for m > 3 we can easily estimate terms of the form " ¡(m¡3)/2 (m) 8 2 R T 3 . We have "

¡ m¡3 2 (m) L p p = F " (m) (" 1/2 Y "; ) L p p = Z R F " (m) (x) p (dx)
where (dx) is the density of a centered Gaussian with variance Y ;" 2 . The integral is nite by Assumption 2.

First-order trees

First of all note that the term Y ~" has no remainder and then it can be shown to converge in law to (2) X by usual techniques. We start with the bound (26) for q Y " ; q Y " ; q Y " , q Y " ?

. We obtain from (29) that

q Y " (t; x ) := (3 ¡ m)! 3! Z (m) = f 3;" Z q Y "; (3¡m) y + (3 ¡ m)! 3! Z ^ (m) = f ( " ) q K (Y " )(t; x ) + q Y ^" (t; x ):
As said before, f 3;" R q Y "; (3¡m) y converges in law in L p for every 2 6 p < +1 to (3) R q X (3¡m) y since f 3;" ! (3) . We can bound the remainder term R ^ (m) in L p () using Lemma 33 and Lemma 36 to obtain

Z ^ (m) L p () = 4¡m Z Q 1 4¡m (4) h 4¡m L p () 6 Q 1 4¡m Z (4) h 4¡m D 4¡m; p . X k=0 4¡m D k Q 1 4¡m Z (4) h 4¡m L p () . Z (4) h 4¡m H 4¡m 2 L p/2 () 1/2 . Z (4) 0 ( 4
)
hh 4¡m ; h 0 4¡m i H 4¡m 0 L p/2 () 1/2 . Z ; 0 (4) 0 (4) L p/2 () jhh ; h 0ij 4¡m j 0j 1/2 . " Z ; 0 " ¡ 1 2 (4)
L p () " ¡ 1 2 0 (4) L p () jhh ; h 0ij 4¡m j 0j 1 2 . " Z ; 0 " ¡ 1 2 (4)
L p () " ¡ 1 2 0 (4) L p () jhh ; h 0ij 3¡m+ j 0j 1 2 ;
for every > 0, where we used the estimation of Lemma 26. Now using Remark 11 and the fact that hh ; h 0i H = C Y ;" ( ¡ 0 ) we obtain as a nal estimation

Z ^ (m) L p () . " 2 Z jC Y ;" ( ¡ 0 )j 3¡m+ j 0j 1/2 : (31)
Remark 12. These last computations are one of the key observations of this paper, exploiting the properties of Malliavin calculus to replace hypercontractivity in the estimation of L p norms with arbitrarily large p without resorting to explicit expansions.

The measure =(s;y) being either [ R x K q;x (x)P t¡s (x ¡ y)]d for q Y "; or K q;x (y)(t ¡ s)d for the other trees, the l.h.s of (31) can be estimated with Lemma 28 to obtain for every x 2 T 3 , q > 0:

q Y ^" (t; x )

L p ()
. "

2 2 ¡ 1¡ 2 q q Y ^" (t; x ) L p () . " 2 2 1+ 2 q q Y ^" (t; x ) L p () . " 2 2 2+ 2 q q Y ^"? (t; x ) L p () . " 2 2 2 q
The time regularity of trees We want to show (27). In order to do that, we compute

Z ¡ ^t;x (m) ¡ ^s;x (m) L p () . 4¡m Z Q 1 4¡m ¡ t;x (4) h t;x 4¡m ¡ s;x (4) h s;x 4¡m L p () . Z ¡ t;x (4) ¡ s;x (4) h s;x 4¡m H 4¡m 2 L p/2 () 1/2 + Z s;x (4) (h t;x 4¡m ¡ h s;x 4¡m ) H 4¡m 2 L p/2 () 1/2 :
We focus on the rst term above to obtain that it is bounded by Now note that

Z ¡ t;x ( 4
" ¡ 1 2 ¡ t;x (4) ¡ s;x (4) = F (4) (" 1 2 Y " (t; x)) ¡ F (4) (" 1 2 Y " (s; x)) = " 1 2 Z 0 1 F (5) [" 1 2 Y " (s; x) + " 1 2 (Y " (t; x) ¡ Y " (s; x))] (Y " (t; x) ¡ Y " (s; x));
and we can estimate "

¡ 1 2 ¡ t;x (4) ¡ s;x (4) L p ()
by hypercontractivity and using Lemma 29 as

. p " 1/2 Z 0 1 F (5) [" 1 2 Y " (s; x) + " 1 2 (Y " (t; x) ¡ Y " (s; x))] L 2p () kY " (t; x) ¡ Y " (s; x)k L 2 () . " 1/2 [C Y ;" (0; 0) ¡ C Y ;" (t ¡ s; 0)]

Second-order trees

In this section we show the decomposition (24) and the bound (26) for the trees Y " ; Y " ; Y ~" ; Y " . The time regularity (27) of Y " can be obtained with the same technique as in the previous section assuming that (F " ) " C 9 (R), and we do not repeat the argument here. Looking at the denitions in [START_REF] Hairer | Regularity structures and the dynamical 3 4 model[END_REF] it is clear that we can write the Littlewood-Paley blocks of Y " , Y " , Y ~" and Y " 8" > 0 as:

q Y " ( ) = 1 6 Z 1;2 1 (0) 2 (2) 1; 2 ¡ d " q (1)( ); q Y " ( ) = 1 9 Z 1;2 1 (1) 2 (1) 1 ; 2 ¡ d " q (1)( ); q Y ~" ( ) = 1 3 Z 1 ; 2 ~1 (1) 2 (1) 1; 2 ¡ d ~" q (1)( ); q Y " ( ) = 1 3 Z 1;2 1 (0) 2 (1) 1; 2 ¡ d " q Y " ( ) ¡ d ^" q (1)( ); (32)
for = (t; x ) where

(m) is dened in (4), ~1 (1)
:= " ¡1/2 f 2;" JY " 2

( 1 )K and the measure 1 ; 2 on (R T 3 ) 2 is given by

1 ; 2 := [ Z x;y K q;x (x) X ij K i;x (y)K j ;x (x 2 )P t¡s 1 (y ¡ x 1 )](t ¡ s 2 )d 1 d 2 with i = (s i ; x i ) i = 1; 2.
The rst step for decomposing (32) is to expand them using the partial chaos expansion (69) to obtain

1 (0) 2 (2) = E 1 (0) 2 (2) + Q 1 D ( 1 (0) 2 (2)
);

1 (1) 2
(1) = E 1

(1)

2 (1) + Q 1 D ( 1 (1) 2 (1)
);

1 (0) 2 (1) = E 1 (0) 2 (1) + J 0 D ¡ 1 (0) 2 (1) + 2 Q 1 2 D 2 ( 1 (0) 2 (1) ) = E 1 (0) 2 (1) + Y " ( 1 )E 1 (1)
2 (1) + Y " ( 2 )E 1 (0) 2 (2) + 2 Q 1 2 D 2 ( 1 (0) 2 (1)
):

(33)

Like the trees appearing in the 3 4 model, we expect second-order trees to require a further renormalisation, on top of the Wick ordering.

Renormalisation of second-order trees

In this section we show how to renormalise (32) by estimating the terms of the type E 1

(m) 2 (n)
in expansion (33). We are going to need the following result:

Lemma 13. We have Z 1; 2 Y " ( 1 )E[ 1 (1) 2 (1) ] 1;2 = Z s;x q Y " (s; x ¡ x) G(t ¡ s; x): and Z 1;2 Y " ( 2 )E[ 1 (0) 2 (2) ] 1; 2 = Z x q Y " (t; x ¡ x)H(t; x);
where

G(t ¡ s; x) := Z x 1 0 ;x2 X i j K i;x (x 1 0 )K j ;x (x 2 )P t¡s (x 1 0 )E 0 (1)
(t¡s;x2)

(1) ;

H(t; x) := Z s;x1;x 1 0 X ij K i;x (x 1 0 )K j ;x (0)P t¡s (x 1 0 ¡ x 1 )E 0 (0) (t¡s;¡x1) (2)
:

Proof. We have Z 1;2 Y " ( 1 )E 1 (1) 2 (1) 1 ; 2 = Z s 1 ;x 1 ;x 2 ;x;x 1 0 K q;x (x) X i j K i;x (x 1 0 )K j ;x (x 2 )P t¡s 1 (x 1 0 ¡ x 1 )Y " (s 1 ; x 1 )E 0 (1) (t¡s 1 ;x 2 ¡x 1 ) (1)
and by change of variables, exploiting the translation invariance of the problem we obtain:

= Z s1;x1;x K q;x (x + x 1 )Y (s 1 ; x 1 ) Z x 1 0 ;x2 X i j K i;x (x 1 0 )K j;x (x 2 )P t¡s1 (x 1 0 )E 0 (1)
(t¡s1;x2)

(1) :

Using the denition of K q we have = Z

s 1 ;x q Y " (s 1 ; x ¡ x) Z x 1 0 ;x 2 X ij K i;x (x 1 0 )K j ;x (x 2 )P t¡s 1 (x 1 0 )E 0 (1) (t¡s 1 ;x 2 )
(1) :

Finally we can write

Z 1 ; 2 Y " ( 1 )E 1 (1) 2 (1) 1;2 = Z s 1 ;x q Y " (s 1 ; x ¡ x) G(t ¡ s 1 ; x):
Similar computations holds for the other term, indeed

Z 1;2 Y " ( 2 )E 1 (0) 2 (2) 1 ; 2 = Z s 1 ;x 1 ;x 2 ;x;x 1 0 K q;x (x) X i j K i;x (x 1 0 )K j ;x (x 2 )P t¡s 1 (x 1 0 ¡ x 1 )Y " (t; x 2 )E 0 (0) (t¡s 1 ;x 2 ¡x 1 ) (2) = Z x 2 K q;x (x + x 2 )Y " (t; x 2 ) Z s 1; x 1 ;x;x 1 0 X i j K i;x (x 1 0 )K j;x (0)P t¡s 1 (x 1 0 ¡ x 1 )E 0 (0) (t¡s 1 ;¡x 1 ) (2) = Z x q Y " (t; x ¡ x) Z s 1; x 1 ;x 1 0 X i j K i;x (x 1 0 )K j ;x (0)P t¡s 1 (x 1 0 ¡ x 1 )E 0 (0) (t¡s 1 ;¡x 1 ) (2)
= Z x q Y " (t; x ¡ x)H(t; x)

Using the lemma above and the partial chaos expansion (33), we can write (32) as:

q Y " ( ) = 1 9 Z 1; 2 Q 1 D ( 1 (1) 2 (1)
) 1;2 + q (1)( ) 1 9

Z s;x G(t ¡ s; x) ¡ d " q Y ~" ( ) = 1 3 Z 1; 2 Q 1 D ( ~1 (1) 2 (1)
) 1 ; 2 + q (1)( ) 1 3

Z s;x G ~(t ¡ s; x) ¡ d ~" q Y " ( ) = 1 6 Z 1 ; 2 Q 1 D ( 1 (0) 2 ( 2
)
) 1;2 + q (1)( ) 1 6 Z x H(t; x) ¡ d " q Y " ( ) = 1 3 Z 1; 2 2 Q 1 2 D 2 ( 1 (0) 2 (1)
) 1; 2 + q (1)( ) 1 3 Z 1; 2 E 1 (0) 2 (1) 1;2 ¡ d ^" + q Y " ( ) 1 3 Z s;x G(t ¡ s; x) + 1 3 Z x H(t; x) ¡ d " + 1 3 q R " ( ) + 1 3 q R " ( )
with the additional denitions

G ~(t ¡ s; x) := Z x 1 ;x 1 0 X i j K i;x (x 1 0 )K j ;x (0)P t¡s 1 (x 1 0 ¡ x 1 )E ~0 (1) (t¡s;¡x1) (1) ; q R " ( ) := Z s;x [ q Y " (s; x ¡ x) ¡ q Y " (t; x )] G(t ¡ s; x); q R " ( ) := Z x [ q Y " (t; x ¡ x) ¡ q Y " (t; x )] H(t; x):
To proceed further in the estimation of these integrals, we need to characterise the local behaviour of

E[ 1 (m) 2 (n)
]. From the decomposition (29) we can write

E 1 (m) 2 (n) = 3! 2 (3 ¡ m)!(3 ¡ n)! (f 3;" ) 2 E[JY ";1 3¡m KJY "; 2 3¡n K] + 3! (3 ¡ m)! f 3;" E JY ";1 3¡m K ^2 (n) + 3! (3 ¡ n)! f 3;" E JY ";2 3¡n K ^1 (m) + E ^1 (m) ^2 (n) ;
where

E[JY ";1 3¡m KJY ";2 3¡n K] = (3 ¡ m)!(3 ¡ m; 3 ¡ n)C Y ;" ( 1 ¡ 2 ) 3¡n
and to bound all other terms we introduce the following result. Lemma 14. Under Assumption 2 we have, for every 0 6 m; n 6 3 and m 6 n:

E ^1 (m) ^2 (n) . X i=0 4¡n " 1+ n ¡m 2 +i jhh 1 ; h 2 ij 4¡m+i . " jhh 1 ; h 2 ij 3¡ m+n 2 + ; 8 2 [0; 1]:
Moreover for every 0 6 m; n 6 3,

E JY "; 1 m K ^2 (n) . " m+n¡3 2 jhh 1 ; h 2 ij m if m > 4 ¡ n; E JY "; 1 m K ^2 (n) = 0 if m < 4 ¡ n:
Proof. Using the integration by parts formula (72) we decompose

E ^1 (m) ^2 (n) = E 4¡m (Q 1 4¡m 1 (4) h 1 4¡m ) 4¡n (Q 1 4¡n 2 (4) h 2 4¡n ) = X i=0 4¡n 4 ¡ m i 4 ¡ n i i! E ¡ Q 5¡n¡i 8¡m¡n¡i 1 (8¡n¡i) Q 5¡m¡i 8¡m¡n¡i 2 (8¡m¡i) hh 1 ; h 2 i 8¡m¡n¡i :
We can bound the term

" m+n 2 +i¡5 E ¡ Q 5¡n¡i 8¡m¡n¡i 1 (8¡n¡i) Q 5¡m¡i 8¡m¡n¡i 2 (8¡m¡i) . " n+i ¡5 2 1 (8¡n¡i) L 2 " m+i¡5 2 2 (8¡m¡i) L 2
(see Remark 11) and therefore, using the bound "jhh 1 ; h

2 ij = "C Y ;" ( 1 ¡ 2 ) . 1, E ^1 (m) ^2 (n) . X i=0 4¡n " 1+ n ¡m 2 +i jhh 1 ; h 2 ij 4¡m+i . " jhh 1 ; h 2 ij 3¡ m+n 2 + :
For the second bound we compute

E h JY "; 1 m K ^2 (n) 2 i = E m (h 1 m ) 4¡n (Q 1 4¡n 2 (4) h 2 4¡n ) = X i=0 m^4¡n m i 4 ¡ n i i! E ¡ D 4¡n¡i (h 1 m ); Q m+1¡i m+4¡n¡i 2 (4+m¡i) h 2 m+4¡n¡i H m+4¡n ¡i : Since Dh 1 m = 0 we obtain E JY "; 1 m K ^2 (n) = 0 if m < 4 ¡ n and E JY "; 1 m K ^2 (n) . " m+n¡3 2 E[" ¡ 3¡m ¡n 2 Q m+n¡3 m 2 (m+n) ]jhh 1 ; h 2 ij m if m > 4 ¡ n, with E[" ¡ 3¡m¡n 2 Q m+n¡3 m 2 (m+n)
] . 1:

Using Lemma 14 we obtain E 1

(1)

2 (1)
= 9 E (f 3;" JY ";1 2 K + ^"; 1 (1) )(f 3;" JY "; 2 2 K + ^2

(1)

) = 18 (f 3;" ) 2 [C Y ;" ( 1 ¡ 2 )] 2 + E ^1 (1)
^2

and thus G(t ¡ s; x) = 18(f 3;"

) 2 R x 1 0 ;x2 P ij K i;x (x 1 0 )K j;x (x 2 )P t¡s (x 1 0 )[C Y ;" ( 1 ¡ 2 )] 2 + G ^(t ¡ s; x) with G ^(t ¡ s; x) := Z x 1 0 ;x2 X ij K i;x (x 1 0 )K j ;x (x 2 )P t¡s (x 1 0 )E ^0 (1) ^(t¡s;x 2 )
(1) :

We have the estimation E ^1

(1)

^2 (1) . " C Y ;" ( 1 ¡ 2 ) 2+ : (34) Similarly E ~1 (1) 2 (1) = 3 " ¡1/2 f 2;" E JY ";1 2 K(f 3;" JY "; 2 2 K + ^2 (1)
) = 6 " ¡1/2 f 2;" f 3;" [C Y ;" ( 1 ¡ 2 )] 2 ; and E 1 (0) 2 (2) = E ¡ f 3;" JY "; 1 3 K + ^1 (0) ¡ 6f 3;" Y "; 2 + ^2 (2)
. jf 3;" j E JY "; 1

3 K ^2 (2) + E ^1 (0) ^2 (2) . " (jf 3;" j + 1)C Y ;" ( 1 ¡ 2 ) 2+ ; (35) and E 1 (0) 2 (1) = E (0) ¡ 3f 3;" JY ", 2 2 K + ^(1) . jf 3;" jE JY "; 1 3 K ^2 (1) + E ^1 (0) ^2 (1)
.

" 1/2 (jf 3;" j + 1)C Y ;" ( 1 ¡ 2 ) 3 : (36)
We have by Lemma 30 that for all 2 (0; 1) jG ^(t ¡ s; x)j . " (jt ¡ sj 1/2 + jxj) ¡5¡ : Using estimate (64) together with Lemma 24, we have that for all 2 (0; 1), 0 2 (0; ) that jH(t; x)j . " 0 (jt

¡ sj 1/2 + jxj) ¡ . Furthermore, letting q R ^" = Z s;x [ q Y " (t; x ¡ x) ¡ q Y " (t; x )] G ^(t ¡ s; x); we have 1 3 q R " = 6(f 3;" ) 2 Z s;x [ q Y " (t + s; x ¡ x) ¡ q Y " (t; x )]P s (x)[C Y ;" (s; x)] 2 + 1 3 q R ^" :
The term

6(f 3;" ) 2 Z s;x [ q Y " (t + s; x ¡ x) ¡ q Y " (t; x )]P s (x)[C Y ;" (s; x)] 2 can be shown to converge in law to 6 R s;x [ q X(t + s; x ¡ x) ¡ q X(t; x )]P s (x) [C X (s; x)] 2 in C T C ¡1/2¡2
with the standard techniques used in the analysis of the 3 4 model. For all > 0 suciently small we have the bounds, q R " L 1

+ q R ^" L 1 6 " kY " k C T C ¡1/2¡22 q(1/2+2+2) Z s;x (jxj + jt ¡ sj 1/2 ) ¡5 . " kY " k C T C ¡1/2¡22 q(1/2+2+2) ;
which shows that these remainders go to zero in C ¡1/2¡2 as " ! 0, since kY

" k C T C ¡1/2¡2 is bounded in L p ().
Moreover, it is easy to see that q R " ¡ q R ^" is bounded in L p (;

C ¡1/2 ). Note that Z s;x G(t ¡ s; x) = Z s;x P s (x)E 0 (1) (s;x) (1) = 18(f 3;" ) 2 Z s;x P s (x)[C Y ;" (s; x)] 2 + Z s;x P s (x)E ^0 (1) ^(s;x) (1) ; Z x H(t; x) = Z s;x P s (x)E 0 (0) (s;x) (2) = Z s;x P s (x)E 0 (0) ^(s;x) (2)
:

Here we used the fact that

Z x X i j K i;x (x 1 0 )K j ;x (0) = Z x X i; j K i;x (x 1 0 )K j;x (0) = (x 1 0 ); since R x K i;x (x 1 0 )K j ;x (0) = 0, where ji ¡ j j > 1.
; 2 E 1 (0) 2 (1) 1;2 = Z s;x P s (x)E 0 (0) (s;x) (1) = O(" ¡1/2 ): Indeed Lemma 27 again yields " R s;x P s (x)C Y ;" (s; x) 3 . 1. Thus R 1;2 E 1 (0) 2 (1)
1;2 gives a diverging constant which depends on all the (f n;" ) n . Making the choice to dene the renormalisation constants d as in eq. ( 5) we cancel exactly these contributions which are either (F " ) " dependent and/or diverging. In particular we verify that we can satisfy the constraint [START_REF] Hairer | A class of growth models rescaling to KPZ[END_REF].

Finally, noting that

Q 1 D = (1 ¡ J 0 ) and 2 Q 1 2 D 2 = (1 ¡ J 0 ¡ J 1 )
we can write the trees of (32) as

q Y " ( ) = (f 3;" ) 2 Z 1; 2 (1 ¡ J 0 )(JY "; 1 2 KJY "; 2 2 K) 1; 2 + f 3;" 3 Z 1;2 Q 1 D ( ^1 (1)
JY "; 2 2 K + JY "; 1 2 K ^2

(1)

) 1; 2 + 1 9 Z 1; 2 Q 1 D ( ^1 (1)

^2

(1)

) 1;2 ; q Y ~" ( ) = " ¡ 1 2 f 2;" f 3;" Z 1;2 (1 ¡ J 0 )(JY "; 1 2 KJY ";2 2 K) 1 ; 2 + 1 3 Z 1;2 Q 1 D ( ~1 (1)
^2

(1)

) 1 ; 2 ; q Y " ( ) = (f 3;" ) 2 Z 1 ; 2 JY ";1 3 KY 2 1;2 + f 3;" 6 Z 1;2 Q 1 D (6 ^1 (0) Y ";2 + JY "; 1 3 K ^2 ( 2
)
) 1;2 + 1 6 Z 1;2 Q 1 D ( ^1 (0) ^2 ( 2
)
) 1;2 ; q Y " ( ) = (f 3;" ) 2 Z 1; 2 (1 ¡ J 1 )(JY ";1 3 KJY "; 2 2 K) 1; 2 + 1 3 q R " ( ) +6(f 3;" ) 2 Z s;x [ q Y " (t + s; x ¡ x) ¡ q Y " (t; x )]P s (x)[C Y ;" (s; x)] 2 + 1 3 q R ^" + + 1 3 Z 1 ; 2 2 Q 1 2 D 2 (3 ^1 (0) JY ";2 2 K + JY "; 1 3 K ^2 (1)
) 1; 2 + 1 3

Z 1 ; 2 2 Q 1 2 D 2 ( ^1 (0)
^2

) 1; 2 ;

(37) with

1 3 q R " ( ) O L 1(2 q(1/2+2+2) ) and 1 3 q R ^" ( ) + 1 3 q R " ( ) O L 1(" 2 q(1/2+2+2)
). Comparing (37) with the canonical trees in [START_REF] Nourdin | Normal Approximations with Malliavin Calculus: From Stein's Method to Universality[END_REF] we can identify the remainder terms q Y ^" that need to converge to zero in order for q Y " to converge to q X .

Estimation of renormalised second-order trees

In this section we show that the remainder terms identied in (37) converge to zero in probability. First notice that we can bound (37) using Lemmas 33,35 and 36 as

n Q 1 n Z 1 ; 2 D k ( 1 (i) 2 (j) ) 1 ; 2 L p . Z 1 ; 2 D k ( 1 (i) 2 (j) ) 1 ; 2 L p :
Therefore, taking the derivatives in (37) we see that it suces to bound in L p (H k+`) the term

1 (4¡m) 2 (4¡n) h 1 k h 2 `= 3!f 3;" (3 ¡ m)! JY 1 m¡1 K + ^1 (4¡m) 3!f 3;" (3 ¡ n)! JY 2 n¡1 K + ^2 (4¡n) h 1 k h 2 `(

38)

for m + n = 5 and 0 6 k + `6 2. This yields some constraints on the number of branches of trees:

q Y " $ m + k = 3; n + `= 3 q Y ~" $ m = 3; k = 0; n = 2; `= 1 q Y " $ m + k = 4; n + `= 2 q Y " $ m + k = 4; n + `= 3: (39)
In (37), the terms proportional to (f 3;" ) 2 will generate nite contributions in the limit. In particular it is easy to see that they converge respectively to

( (3) ) 2 X ; ( (3) ) 2 X ; (3) (2) X ; ( ( 3
) ) 2 X . All other terms will vanish in probability, verifying (26), due to estimates we are going to establish now.

We consider the terms proportional to ^1

(4¡m) ^2 (4¡n)
, all the other similar terms featuring at least one remainder ^ (m) can be estimated with exactly the same technique, and are easily shown to be vanishing in the appropriate topology. One of the key observations of this paper, Lemma 37, allows us to rewrite products of divergencies in the form m (u) n (v) as a sum of divergences `(w), which are then easy to estimate in L p using Lemma 33. We obtain

^1 (4¡m) ^2 (4¡n) = m (Q 1 m 1 (4) h 1 m ) n ¡ Q 1 n 2 (4) h 2 n = X (q;r;i)2I C q;r;i m+n¡q ¡r ¡ D r ¡i Q 1 m 1 (4) h 1 m ; D q ¡i Q 1 n 2 (4) h 2 n H q+r ¡i = X (q;r;i)2I C q;r;i " 1+ r+ q
2 ¡i m+n¡ q ¡r (h 1+r ¡i m+r ¡i ( 1 )h 1 m+r ¡i ; 1+ q ¡i n+ q ¡i ( 2 )h 2 n+ q ¡i i H q +r ¡i)

with I = f(q; r; i) 2 N 3 : 0 6 q 6 m; 0 6 r 6 n; 0 6 i 6 q ^rg and i j () := "

¡ i 2 Q i j (3+i)
:

By Remark 34, for every n; m > 1 and 2 Dom n we can write n ( )h m = n ( h m ), and therefore Z ^1

(4¡m) ^2 (4¡n) h 1 k h 2 ` 1;2 = = X I C q;r;i " 2+ q+r ¡2i 2 m+n¡ q ¡r Z 1+r ¡i m+r ¡i ( 1 ) 1+ q ¡i n+ q ¡i ( 2 ) h 1 m¡q h 2 n¡r h 1 k h 2 `j hh 1 ; h 2 ij q+r ¡i 1; 2 :
The following result allows us to estimate the quantity above in L p (H k+`) .

Lemma 15. Under Assumption 2 we have the bound

m+n¡ q ¡r Z 1+r ¡i m+r ¡i ( 1 ) 1+q ¡i n+ q ¡i ( 2 ) h 1 m¡ q h 2 n¡r h 1 k h 2 `j hh 1 ; h 2 ij q+r ¡i 1;2 L p (H k+`) 2 .
Z jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r jhh 1 ; h 2 ij q+r ¡i jhh 1 0; h 2 0ij q+r ¡i j 1; 2 jj 1 0 ; 2 0j:

Proof. Thanks to Lemma 33 the integral can be estimated with

X j=0;h6 j m+n¡ q ¡r Z D h 1+r ¡i m+r ¡i ( 1 ) D j ¡h 1+ q ¡i n+ q ¡i ( 2 ) h 1 m¡q h 2 n¡r h 1 k h `j hh 1 ; h 2 ij q+r ¡i 1 ; 2 L p (V ) 2 ;
with V = H m+k+n+`¡ q ¡r+ j . We have that kk

L p (H k+`) 2 = kkk H k+2 k L p/2
1/2 and therefore we can bound each term in the sum above as

. ( Z khD h 1+r ¡i m+r ¡i ( 1 )D j ¡h 1+ q ¡i n+ q ¡i ( 2 ); D h 1+r ¡i m+r ¡i ( 1 0 )D j ¡h 1+ q ¡i n+ q ¡i ( 2 0 )i H j k L p/2 jhh 1 ; h 1 0ij m+k ¡q jhh 2 ; h 2 0ij n+`¡r jhh 1 ; h 2 ij q+r ¡i jhh 1 0; h 2 0ij q+r ¡i j 1;2 jj 1 0 ; 2 0j) 1/2
Using Hölder's inequality we get the estimate

khD h 1+r ¡i m+r ¡i ( 1 )D j ¡h 1+q ¡i n+q ¡i ( 2 ); D h 1+r ¡i m+r ¡i ( 1 0 )D j ¡h 1+ q ¡i n+ q ¡i ( 2
0 )i H j k L p/2 . khD h 1+r ¡i m+r ¡i ( 1 ); D h 1+r ¡i m+r ¡i ( 1 0 )i H h k L p khD j ¡h 1+ q ¡i n+ q ¡i ( 2 ); D j ¡h 1+ q ¡i n+ q ¡i ( 2 0 )i H j ¡h k L p
Now to bound terms of the type khD h 1+a m+a (); D h 1+a m+a ( 0 )i H h k L p we consider the cases h 6 m and h > m. In the rst region we use Lemma 36 to estimate

khD h 1+a m+a (); D h 1+a m+a ( 0 )i H h k L p . D h Q 1+a m+a " ¡ 1+a 2 1 (4+a) L 4p (H h ) 2 D h Q 1+a m+a " ¡ 1+a 2 1 0 (4+a) L 4p (H h ) 2 . " ¡ 1+a 2 1 (4+a) L 4p 2 " ¡ 1+a 2 1 0 (4+a) L 4p 2 :
If h > m we rst commute h ¡ m derivatives in the expression D h Q 1+a m+a using formula (68) and then apply Lemma 36 to obtain the bound

D h Q 1+a m+a " ¡ 1+a 2 1 (4+a) L 4p (H h ) 2 . D h¡m " ¡ 1+a 2 1 (4+a) L 4p 2 . " ¡ 1+a 2 1 (4+a) D h ¡m;4p :
Thus, we need to know (N ) up to the order

(4 + r ¡ i + h ¡ m) _ (4 + q ¡ i + j ¡ h ¡ n) 6 (4 + n) _ (4 + m) to perform this estimates.
From Lemma 15 we obtain 8 2 [0; 1/2): "

2+ q+r ¡2i 2 m+n¡q ¡r Z 1+r ¡i m+r ¡i ( 1 ) 1+ q ¡i n+ q ¡i ( 2 ) h 1 m¡ q h 2 n¡r h 1 k h 2 `j hh 1 ; h 2 ij q+r ¡i 1; 2 L p (H k+`)
."

" 2+q+r ¡2i¡ Z jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r jhh 1 ; h 2 ij q+r ¡i jhh 1 0; h 2 0ij q+r ¡i j 1; 2 jj 1 0 ; 2 0j 1 2 :=" 2 (I) 1 2 :
Our aim now is to estimate the quantity I. The idea is to use the bound "jhh ; h 0ij = "C Y ;" ( ¡ 0 ) . 1 to cancel strategically some of the covariances jhh ; h 0ij. We will consider three regions: If q + r 6 2 we use the bounds

" q+r ¡2i jhh 1 ; h 2 ij q+r ¡i jhh 1 0; h 2 0ij q+r ¡i . " 2 jhh 1 ; h 2 ij q jhh 1 0; h 2 0ij r
and then (we suppose r < 2)

" 2¡r ¡ jhh 2 ; h 2 0ij n+`¡r . jhh 2 ; h 2 0ij n+`¡2+
to obtain

I . " r ¡ Z jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡2 jhh 1 ; h 2 ij q jhh 1 0; h 2 0ij r j 1;2 jj 1 0 ; 2 0j . Z jhh 1 ; h 1 0ij m+k ¡q jhh 2 ; h 2 0ij n+`¡2+ jhh 1 ; h 2 ij q j 1;2 jj 1 0 ; 2 0j: (40)
(If vice-versa q < 2 it suces to put on the term jhh 1 ; h 2 ij q+ .) Notice that in this case m + k ¡ q > 0:

In the case q + r = 3 if m + k ¡ q > 2 we estimate like before to obtain

I . " 2¡ Z jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r jhh 1 ; h 2 ij q+r 2 jhh 1 0; h 2 0ij q +r 2 j 1;2 jj 1 0 ; 2 0j . Z jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r jhh 1 ; h 2 ij 1+ j 1;2 jj 1 0 ; 2 0j: (41) Note that m + k ¡ q + ¡ 1 > 0 and m + k ¡ q + 2 ¡ 3 > ¡1 here. If m + k ¡ q = 1 we bound I . Z jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r ¡2 jhh 1 ; h 2 ij 3+ 2 jhh 1 0; h 2 0ij 3+ 2 j 1 ; 2 jj 1 0 ; 2 0j (42)
and note that m

+ k ¡ q ¡ 1/ 2 + / 2 > 0, m + k ¡ q ¡ 1 + > 0, n + `¡ r ¡ 2 > 0.
Finally if m + k ¡ q = 0 we can only have m + k = 3; q = 3; r = 0; i = 0 and thus

I . " 3¡2 Z jhh 2 ; h 2 0ij n+`j hh 1 ; h 2 ij 2¡ jhh 1 0; h 2 0ij 2¡ j 1;2 jj 1 0 ; 2 0j . Z jhh 2 ; h 2 0ij n+`+m+k¡6 jhh 1 ; h 2 ij 2¡ jhh 1 0; h 2 0ij 2¡ j 1; 2 jj 1 0 ; 2 0j (43)
If q + r > 4 we bound rst

" 2q+2r ¡2i+ ¡4 jhh 1 ; h 2 ij q+r ¡i jhh 1 0; h 2 0ij q+r ¡i . jhh 1 ; h 2 ij 2¡ 2 jhh 1 0; h 2 0ij 2¡ 2 (note that 2q + 2r ¡ 2i + ¡ 4 > ) to obtain: I . " 6¡q ¡r ¡ Z jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r jhh 1 ; h 2 ij 2¡ 2 jhh 1 0; h 2 0ij 2¡ 2 j 1; 2 jj 1 0 ; 2 0j
Now in the cases m + k = 3; n + `= 3 and m + k = 4; n + `= 2 we can just write " 6¡ q ¡r ¡ = " m+k¡ q " 6¡m¡k¡r ¡ and cancel the corresponding number of covariances to obtain

I . Z jhh 2 ; h 2 0ij jhh 1 ; h 2 ij 2¡
while for the case m + k = 4; n + `= 3 we have either `> 1 or k > 1 and therefore with one of the following bounds " m+k¡1¡ q " n+`¡r ¡ jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r . jhh 1 ; h 1 0ijjhh 2 ; h 2 0ij " m+k¡ q " n+`¡1¡r ¡ jhh 1 ; h 1 0ij m+k¡ q jhh 2 ; h 2 0ij n+`¡r . jhh 2 ; h 2 0ij 1+

we obtain the estimates

I . Z jhh 2 ; h 2 0ij 1+ jhh 1 ; h 2 ij 2¡ 2 jhh 1 0; h 2 0ij 2¡ 2 j 1; 2 jj 1 0 ; 2 0j (45) I . Z jhh 1 ; h 1 0ijjhh 2 ; h 2 0ij jhh 1 ; h 2 ij 2¡ 2 jhh 1 0; h 2 0ij 2¡ 2 j 1; 2 jj 1 0 ; 2 0j: (46)
We can use directly Lemma 31 to obtain a nal estimate of (40), ( 41), ( 42), (45). For (43); ( 44) and ( 46) notice that the integral over 1 ; 1 0 is nite and thus the whole quantity is proportional to jhh 2 ; h 2 0ij n . Globally, we have

I . 2 (m+k+n+`¡6)q
as needed to prove (26). Lastly, by taking one more derivative of F " as done in Section 2.1, we can show ( 27)

for Y = Y " ; Y " ; Y ~" ; Y " , thus proving that Y ^ ! 0 in C T /2
C ¡ in probability 8 < j j.

If f is in C ¡" for all " > 0, then we write f 2 C ¡ . We let K q the kernel of q so that q f(x ) = R T 3 K x ;q (x)f (x)dx.

A.2 Schauder estimates

For 2 (0; 2), we dene the space

L T = C T /2 L 1 \ C T C , equipped with the norm kf k L T = max kf k C T /2 L 1 ; kf k CTC :
The notation is chosen to be reminiscent of L = @ t ¡ , by which we will always denote the heat operator with periodic boundary conditions on T d . We also write

L = C loc /2
L 1 \ CC . When working with irregular initial conditions, we will need to consider explosive spaces of parabolic type. For > 0, 2 (0; 1), and T > 0 we dene the norm

kf k L T ; = max kt 7 ! t f (t)k C T /2 L 1 ; kf k M T C
and the space L T ; = ff : [0; T ] ! R: kf k L T ; < 1g. In particular, we have L T 0; = L T . We introduce the linear operator I: C(R + ; D 0 (T)) ! C(R + ; D 0 (T)) given by (50) Proofs can be found in [START_REF] Gubinelli | KPZ Reloaded[END_REF]. We need also some well known estimates for the solutions of the heat equation with sources in spacetime Lebesgue spaces.

If(t
Lemma 17. Let 2 R and f 2 L T p B p;1 , then for every 2 [0; 1] we have If 2 C /q C +2(1¡)¡(2¡2+d)/p with kIf k C T /q C +2(1¡)¡(2¡2+d)/p . T kf k L T p B p;1 ; with 1 q + 1 p = 1.
Moreover, for every < 0 < 1 ¡ 1/ p and every < 2 ¡ 5/ p + we have

kIf k L T 0 ; . kv 7 ! v f (v)k L T p B p;1
Proof. We only show the second inequality as the rst one is easier and obtained with similar techniques. Let u = If , we have

t k i u(t)k L 1 6 t 1/q 2 di/p Z 0 1
s ¡q e ¡cq2 2i t(1¡s) ds

1/q Z 0 t s p k i f(s)k L p p d s 1/p . ; q 2 id/p 2 ¡2i/q Z 0 t s p k i f (s)k L p p d
kt 0 i u(t) ¡ s 0 i u(s)k L 1 . Z s t v 0 ¡1 k i u(v)k L 1dv + jt ¡ sj2 i(d+2)/p kv 7 ! v i f (v)k L t;x p + Z s t v 0 i f (v) d v L 1
We can estimate the rst term as

Z s t v 0 ¡1 k i u(v)k L 1dv . 2 i(d+2)/p kv 7 ! v i f (v)k L t;x p Z s t v 0 ¡ ¡1 dv:
For the third term we have

Z s t v i f (v) d v L 1 . Z s t dv 1/q Z s t v p k i f (s)k L 1 p dv 1/p . 2 id/p jt ¡ sj 1/q kv 7 ! v i f (v)k L t;x p
We obtain then if 2 2i jt ¡ sj 6 1

kt 0 i u(t) ¡ s 0 i u(s)k L 1 . 2 id/p jt ¡ sj 1/q kv 7 ! v i f (v)k L t;x p
and if 2 2i jt ¡ sj > 1 we just use the trivial estimate

kt 0 i u(t) ¡ s 0 i u(s)k L 1 . 2 id/p 2 ¡2i/q kv 7 ! v i f (v)k L t;x p . 2 id/p jt ¡ sj 1/q kv 7 ! v i f (v)k L t;x p :
Therefore, for every 2 [0; 1]:

kt 0 i u(t) ¡ s 0 i u(s)k L 1 . 2 ( d+2 p ¡2)i 2 2i/q jt ¡ sj /q kv 7 ! v i f (v)k L t;x p :
Choosing / q = /2 we obtain the desired estimate.

A.3 Bony's paraproduct and some commutators

Paraproducts are bilinear operations introduced by Bony [START_REF] Bony | Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires[END_REF] in order to linearize a class of non-linear PDE problems. They appear naturally in the analysis of the product of two Besov distributions. In terms of Little-woodPaley blocks, the product f g of two distributions f and g can be decomposed as

f g = f g + f g + f g;
where

f g = g f := X j>¡1 X i=¡1 j ¡2 i f j g and f g := X ji¡ j j61 i f j g:
This decomposition behaves nicely with respect to LittlewoodPaley theory. We call f g and f g paraproducts, and f g the resonant term. We use the notation f 4 g = f g + f g. The basic result about these bilinear operations is given by the following estimates, essentially due to Bony [START_REF] Bony | Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires[END_REF] and Meyer [START_REF] Meyer | Remarques sur un théorème de J.-M. Bony[END_REF].

Lemma 18. (Bony's paraproduct estimates)

For any 2 R we have

kf g k C . kf k L 1k g k C ; ( 51
)
and for < 0 furthermore

kf g k C + . ; kf k C kg k C : (52)
For + > 0 we have

kf g k C + . ; kf k C kg k C : (53)
A natural corollary is that the product f g of two elements f 2 C and g 2 C is well dened as soon as + > 0, and that it belongs to C , where = min f; ; + g.

We will also need the several commutator lemmas: When dealing with paraproducts in the context of parabolic equations it would be natural to introduce parabolic Besov spaces and related paraproducts. But to keep a simpler setting, we choose to work with spacetime distributions belonging to the scale of spaces (C T C ) 2R for some T > 0. To do so eciently, we will use a modied paraproduct which introduces some smoothing in the time variable that is tuned to the parabolic scaling. Let therefore ' 2 C 1 (R; R + ) be nonnegative with compact support contained in R + and with total mass 1, and dene for all i > ¡1 the operator

Q i : CC ! CC ; Q i f (t) = Z 0 1 2 ¡2i '(2 2i (t ¡ s))f (s)ds:
We will often apply Q i and other operators on CC to functions f 2 C T C which we then simply extend from [0; T ] to R + by considering f ( ^T ). With the help of Q i , we dene a modied paraproduct

f g := X i (Q i S i¡1 f ) i g
for f ; g 2 C(R + ; D 0 (T)). We collect in the following lemma various estimates for the modied paraproduct, proofs are again in [START_REF] Gubinelli | KPZ Reloaded[END_REF].

Lemma 20.

a) For any 2 R and 2 [0; 1) we have

t kf g(t)k C . kf k M t L 1kg(t)k C ; ( 54
)
for all t > 0, and for < 0 furthermore . Then for all 2 (0; ] we have

t kf g(t)k C + . kf k M t C kg(t)k C : (55
kf k L T . kf (0)k C + T (¡)/2 kf k L T ; kf k L T ; . T (¡)/2 kf k L T ; : (57)
Finally we introduce various commutators which allow to control non-linear functions of paraproducs and also the interaction of the paraproducts with the heat kernel.

Lemma 21.

a) For ; ; 2 R such that + + > 0 and 2 (0; 1) there exists bounded trilinear maps

com 1 ; com 1 : C C C ! C + + ;
such that for smooth f ; g; h they satisfy

com 1 (f ; g; h) = (f g) h ¡ f (g h): (58) com 1 (f ; g; h) = (f g) h ¡ f (g h):
(59) b) Let 2 (0; 2), 2 R, and 2 [0; 1). Then the bilinear maps

com 2 (f ; g): =f g ¡ f g: (60) com 3 (f ; g): =[L ; f ]g := L (f g) ¡ f L g: (61)
have the bounds

t
Z K i (x ¡ y)P t (y)dy Z jK j (x ¡ y)j (jyj + t 1/2 ) dy . X i 2 ¡i (jxj + t 1/2 + 2 ¡i ) 4+ :
Bounding the sum over i with an integral, we conclude Z

0 1 d (jxj + t 1/2 + ) 4+ = 1 (jxj + t 1/2 ) 3+ Z 0 1/(jxj+t 1/2 ) d (1 + ) 4+ . 1 (jxj + t 1/2 ) 3+ :
Let us show (64). We want to estimate

I = Z K i (x ¡ y)P t (y)dy = Z K i (x ¡ y)[P t (y) ¡ P t (x)]dy = Z 0 1 d Z K i (x ¡ y)[P t 0 (x + (y ¡ x))(y ¡ x)]dy jI j . Z 0 1 d Z j(y ¡ x)K i (x ¡ y)jjP t 0 (x + (y ¡ x))jdy . 2 ¡i Z 0 1 d Z jyK 1 (
C Y ;" ( 1 ¡ 2 ) k C Y ;" ( 1 ¡ 2 ) `CY ;" ( 1 ¡ 1 0 ) m C Y ;" ( 2 ¡ 2 0 ) n j 1;2 jj 1 0 ; 2 0j;
with 1 ; 2 for = (t; x ), i = (s i ; x i ) i = 1; 2 dened as

1; 2 := [ Z x; y K q;x (x) X i j K i;x (y)K j ;x (x 2 )P t¡s1 (y ¡ x 1 )](t ¡ s 2 )d 1 d 2 : If `= 0, 0 < m + k ¡ 2 < 5, m + k ¡ 2 2 (¡1 ;
I k;m;n . 2 (k+`+m+n¡4)q : Proof. Observe that 1 ; 2 = [ X i j Z x; y K q;x (x 2 )K i;x (y)K j ;x (x 2 )(P t¡s 1 (y ¡ x 1 ) ¡ P t¡s 1 (x ¡ x 1 ))](t ¡ s 2 )d 1 d 2 +[ X ij Z x;y (K q;x (x) ¡ K q;x (x 2 ))K i;x (y)K j ;x (x 2 )P t¡s 1 (y ¡ x 1 )](t ¡ s 2 )d 1 d 2 = K q;x (x 2 )[P t¡s 1 (x 2 ¡ x 1 ) ¡ P t¡s 1 (x ¡ x 1 )](t ¡ s 2 )d 1 d 2 +[ X ij Z x;y [K q;x (x) ¡ K q;x (x 2 )]K i;x (y)K j ;x (x 2 )P t¡s1 (y ¡ x 1 )](t ¡ s 2 )d 1 d 2 = 1; 2 + ^1;2
where in the rst line we used R y K i;x (y) = 0 and the fact that

R x K i;x (x 1 0 )K j ;x (x 2 ) = 0 if ji ¡ j j > 1 and P i; j K i;x (y)K j ;x (x 2 ) = (x 2 ¡ y)(x 2 ¡ x)
. Now the estimation of the term

I k;m;n := Z 1;2; 1;2 0 C Y ;" ( 1 ¡ 2 ) k C Y ;" ( 1 0 ¡ 2 0 ) `CY ;" ( 1 ¡ 1 0 ) m C Y ;" ( 1 ¡ 2 ) n j 1; 2 jj 1 0 ; 2 0j;
with 1 ; 2 = K q;x (x 2 )[P t¡s1 (x 2 ¡ x 1 ) ¡ P t¡s1 (x ¡ x 1 )](t ¡ s 2 )d 1 d 2 can be done with Lemma 24 and gives the expected result. The integral

I ^k;m;n := Z 1;2 ; 1;2 0 C Y ;" ( 1 ¡ 2 ) k C Y ;" ( 1 0 ¡ 2 0 ) `CY ;" ( 1 ¡ 1 0 ) m C Y ;" ( 1 ¡ 2 ) n j ^1;2 jj ^1 0 ; 2 0j;
with ^1;2 = [ P i j R x;y [K q;x (x) ¡ K q;x (x 2 )]K i;x (y)K j;x (x 2 )P t¡s 1 (y ¡ x 1 )](t ¡ s 2 )d 1 d 2 can be estimated by multiple changes of variables. We have K q;x (x) ¡ K q;x (x 2 ) = 2 3q (x 2 ¡ x) R Proof. First notice that we can suppose w.l.o.g. E( ) = 0 thanks to the commutation (68). Therefore we can write D (j ¡ L) with C = ¡ ¡L p . We decompose the second part as:

0 1 K 0 (2 q (x 2 ¡ x) ¡ 2 q (x ¡ x 2 ))d ,
¡C (j ¡ L) ¡ 1 2 = X n=1 1 n j + n 1 2 J n = T
With T := P n=0 1 (n)J n . We can apply Theorem 1.4.2 of [START_REF] Nualart | The Malliavin calculus and related topics[END_REF] to show that T is bounded in L p , indeed

(n) = h(1 / n) and h(x) = (j x + 1) ¡1/2 which is analytic in a neighbourhood of 0. Finally, we can apply Proposition 1.5.2 of [START_REF] Nualart | The Malliavin calculus and related topics[END_REF] to show that DC ¡1 is bounded in L p , thus concluding the proof.

The following lemma is the most useful tool we used in the paper. It allows us to write products of decompositions of the type (69) as sums of iterated Skorohod integrals.

Lemma 37.

Let

u = F u (W (h u ))h u m , v = F v (W (h v ))h v n with F u ; F v 2 C m+n (R) such that u 2 D m+n;2 (H m ), v 2 D m+n;2 (H n ). Then: m (u) n (v) = X i=0 m^n X q=0 m¡i X r=0 n¡i m q + i q + i i n r + i
r + i i i! m+n¡ q ¡r ¡2i (hD r u; D q vi H q+r+i):

(71)

And also m (u) n (v) = X q=0 m X r=0 n X i=0 q^r m q q i n r r i i! m+n¡ q ¡r (hD r ¡i u; D q ¡i vi H q +r ¡i):

(72)

Proof. We have using Cauchy-Schwarz inequality and Lemma 33 that hD r n (v); j (u)i H r 2 L 2 (; H m¡ j ¡r ) for every 0 6 r + j 6 m. Then we can apply Lemma 39 to obtain m (u) n (v) = X r=0 n n r n¡r (hD r m (u); vi H r):

Then using the commutation formula (73) we rewrite the r.h.s. as

m (u) n (v) = X r=0 n n r X i=0 r^m r i m i i! n¡r (h m¡i (D r ¡i u); v i H r):
We write h m¡i (D r ¡i u); vi H r = h m¡i (D r ¡i u); F v (W (h v ))h v r i H r h v n¡r and verify in the same way as before that h m¡i (D r ¡i u); F v (W (h v ))h v r i H r satises the hypotheses of Lemma 41. We obtain m (u) n (v) = X where we used the fact that k ( )h n¡r = k ( h n¡r ) for 2 Dom k , as seen in Remark 34. Call q = `+ i, then

m (u) n (v) = X i=0 m^n X q=i m X r=i n n r r i m i m ¡ i
q ¡ i i! m+n¡ q ¡r (hD r ¡i u; D q ¡i vi H q+r ¡i);

and noting that m i m ¡ i q ¡ i = (m ¡ i)!m! (q ¡ i)!(m ¡ q)!i!(m ¡ i)! = q!m! i!(q ¡ i)!q!(m ¡ q)! = q i m q ;

we have the nicer symmetric expression

m (u) n (v) = X i=0 m^n X q=i m X r=i n m
q q i n r r i i! m+n¡q ¡r (hD r ¡i u; D q ¡i v i H q+r ¡i):

Finally we perform the change of variables q ¡ i ! q, r ¡ i ! r to get

m (u) n (v) = X i=0 m^n X q=0 m¡i X r=0 n¡i m q + i q + i i n r + i r + i i i! m+n¡ q ¡r ¡2i (hD r u; D q vi H q+r+i):
The second formula is a straightforward change of indexes of the rst one.

Remark 38. Our choice of giving two distinct but closely related formulas in Lemma 37 is due to the fact that the rst formula has a more evident physical meaning. Indeed, vertices u and v (being non-polynomial) have an innite chaos decomposition, which can be represented as having innite legs in a Feynman-like diagram.

It is apparent that the index i in rst equation denotes contractions between the already existing legs of the vertices u; v and that r; q stay for new legs in each vertex created by the Malliavin derivatives which are then contracted with other legs from the other vertex. This leaves m + n ¡ r ¡ q ¡ 2i legs overall uncontracted which are arguments to the iterated Skorokhod integral and would be contracted with other composite vertices in the L p estimates. The second formula however, is more practical in the calculations.

We give below the results we used to prove Lemma 37.

Lemma 39. ([20], Lemma 2.1) Let q > 1, F 2 D q;2 , u 2 Dom( q ) and symmetric. Assume also that 80 6 r + j 6 q hD r F ; j (u)i H r 2 L 2 (; H q ¡r ¡j ). Then 80 6 r 6 q hD r F ; ui r 2 Dom( q ¡r ) and F q (u) = X r=0 q q r q ¡r (hD r F ; ui H r):

Remark 40. Note that

k (h k ) = JW k (h)K
where JK stands for the Wick product. Indeed 8F 2 D 1;2 we know that E[(h n )F ] = E[W (h) h n¡1 ] using the denition of and therefore n (h n ) = n¡1 (W (h)h n¡1 ). We have also n¡1 (W (h)h n¡1 ) = n¡1 (h n¡1 )W (h) ¡ (n ¡ 1)hh; hi n¡2 (h n¡2 ) using Lemma 39, and the result is proved by induction.

Lemma 41. Let `2 N, F 2 D q;2 (H `) , u 2 Dom ( q ) with values in H q+`a nd symmetric. Assume also that h j (u); D r F i H `+r 2 L 2 (; H q ¡r ¡ j ) 80 6 r + j 6 q. Then 80 6 r 6 q hu; D r F i H `+r 2 Dom( q ¡r ) and h q (u); F i H `= X r=0 q q r q ¡r (hu; D r F i H `+r)

Proof. Let q = 1. We have for smooth G 2 D 1;2 , F 2 D q;2 (H `) and u 2 Dom ( q ):

E(hhu; F i H `; DGi H ) = E(hu; DG F i H `+1) = E(h(u); F i H `G) ¡ E(hu; DF i H `+1 G)
where we used the fact that D(G F ) = DG F + G DF for smooth functions. The equality hhu; F i H `; DGi H = hu; DG F i H `+1 holds because u is symmetric.

, g 2 C

 2 ) b) Let ; 2 (0; 2), 2 [0; 1), T > 0, and let f 2 L T ; T C , and L g 2 C T C ¡2 . Then kf g k L T ; . kf k L T ;(k g k C T C + kL g k CTC ¡2): (56) c) Let 2 (0; 2), 2 (0; 1); T > 0, and let f 2 L T

5 )

 5 and k + m + n ¡ 4 2 (0; 5) we have the estimateI k;m;n . 2 (k+m+n¡4)q : If (k + m ¡ 2); (`+ m ¡ 2) 2 (0; 5), k + m + `¡4 2 (0; 5) and k + `+ m + n ¡ 4 2 (0; 5) we have the estimate

¡

1 2

 1 = D(¡C) ¡1 (¡C) (j ¡ L)

  n¡r [h m¡i (D r ¡i u); F v (W (h v ))h v r i H r h v n¡r ] m ¡ i ` n¡r [ m¡i¡`( hD r ¡i u; D `Fv (W (h v ))h v r i H r+`)h v n¡r ] m+n¡r ¡i¡`( hD r ¡i u; D `v i H r+`)

  Which yields estimation (27) by applying Lemma 28 as before.

	and nally obtain
	Z	¡	^t;x (m) ¡ ^s;x (m)	L p ()	. " /2¡2 jt ¡ sj	Z	; 0	jhh s;x ; h s;x 0ij 3¡m++2 j 0j	1 2 :
					1/2
	. " ¡2 jt ¡ sj ;
	for any 2 [0; 1/2]. The other term can be estimated more easily by Z
			"
			; 0
								1
								2 ;

jhh s;x ; h s;x 0ij 2¡m+ jhh t;x ¡ h s;x ; h t;x 0 ¡ h s;x 0ijj 0j 1 2 . " ¡2 jt ¡ sj " Z ; 0 jhh s;x ; h s;x 0ij 3¡m++2 j 0j

  This is readily seen in Fourier space taking into account the support properties of the Littlewood-Paley blocks. Now, Z

					Z
	s;x	P s (x)E	^0 (1)	^(s;x) (1)	;	s;x	P s (x)E	0 (0)	^(s;x) (2)

; converge to nite constants due to the bounds (34) and (35) and by Lemma 27 R s;x P s (x)[C Y ;" (s; x)] 2 . jlog"j: Finally, from (36) we have Z 1

  ) t>0 is the heat semigroup. Standard estimates in exposive spaces that are summarized in the following Lemma.

			Z	t
			) =	P t¡s f (s)ds;
			0
	where (P t Lemma 16. Let 2 (0; 2) and 2 [0; 1). Then
	kIf k L t ks 7 ! P s u 0 k L t ; . kf k M t for all t > 0. If further > ¡, then ( +)/2; . ku 0 k C ¡: C ¡2	(47) (48)
	For all 2 R, 2 [0; 1), and t > 0 we have kIf k M t	C . kf k M t	C ¡2	(49)
	For all 2 (0; 2), 2 [0; 1), " 2 [0; ^2), t > 0 and f 2 L t	; with f (0) = 0 we have
	kf k L t	¡"/2;¡" . kf k L t	;

  s which allows us to bound kIf k M T C . In order to estimate kt 7 ! t 0 If k C T /2 L 1 we write

	1/p

Lemma 19. (Bony's commutator estimate)

  Let > 0, 2 R, and let f ; g 2 C , and h 2 C . Thenkf (g h) ¡ (fg) hk C + . kf k C kg k C khk C :

  Proof. The rst bound is obtained directly from Lemma 22 and Lemma 24. Indeed, since by hypothesis C " has compact support, it is easy to see that C ~"(t; x) . (jtj 1/2 + jxj)¡5 . The second bound is obtained by a simple change of variables in the convolution dening C Y ;" . From the fact that P " 2 s (" x) = " ¡3 P s (x) together with Remark 25 we obtain Z R 4 : 99 < R; = / 0g a parabolic ball centered in the origin. The second estimation is obtained in the same way.

	Moreover, we have
	" jkj+1 jD Lemma 27. We have R s;x P s (x)[C Y ;" (s; x)] 2 . jlog"j and for every n > 3 " n¡2 R	s;x P s (x)[C Y ;" (s; x)] n . 1.
	Proof. RT 3 P s (x)[C Y ;" (s; x)] 2 dsdx .	Z	B(0;" ¡1 ) P s (x)[C Y " (s; x)] 2 dsdx . jlog"j
	with B(0; R) = f 2 Lemma 29. We have for every 2 [0; 1]
			sup
			x2T 3
	Lemma 30. We have for every < 3
	X i j	Z	K i (x ¡ y)P t (y)dy	Z	jK j (x ¡ y)j (jyj + t 1/2 )	dy .	1 (jxj + t 1/2 ) 3+
	Proof. We will show that	Z
	and that from which we deduce that		. 2 ¡i (jxj + t 1/2 + 2 ¡i ) ¡4 ; dy . (jxj + t 1/2 + 2 ¡i ) ¡ ; K i (x ¡ y)P t (y)dy Z jK i (x ¡ y)j (jyj + t 1/2 )	(64) (65)
	X
	i j
		kcom 2 (f ; g)(t)k + . kf k L t	; kg(t)k C ;	t > 0:	(62)

k C Y ;" (t; x)j . 1 Lemma 28. For m 2 (0; 3), n 2 (3; 5), dene for ; 0 2 R T 3

I m := Z jC Y ;" ( ¡ 0 )j m j 0j; I ~n := Z jC Y ;" ( ¡ 0 )j n j ~ ~ 0j with := K q;x (y)(t ¡ s)d and ~ := [ R x K q;x (x)P t¡s (x ¡ y)]d for = (s; y). Then I m . 2 mq and I ~n . 2 (n¡4)q :

Proof. The estimation of I m is easily obtained by Lemma 26 and a change of variables. For I ~n observe that for every q > 0 ~ = [

Z x K q;x (x)(P t¡s (x ¡ y) ¡ P t¡s (x ¡ y))]d

and then we can apply Lemma 24 to obtain the result.

jC Y ;" (t; x) ¡ C Y ;" (0; x)j . " ¡1¡2 jtj Proof. It is easy to obtain by interpolation knowing that j@ t C Y ;" (t; x)j . " ¡3 from Lemma 26.

  When t 1/2 > 2 ¡i ; jxj we havejI j . 2 ¡i t ¡2 . 2 ¡i (jxj + t 1/2 + 2 ¡i ) ¡4 :When 2 ¡i > t 1/2 ; jxj we estimate simply 2i . (jxj + t 1/2 + 2 ¡i ) ¡ ; and nally when jxj > 2 ¡i ; t 1/2 we have either jxj > 2 ¡i+1 jyj or jxj < 2 ¡i+1 jyj. In the rst region jx + 2 ¡i yj > cjxj ¡i yj dy . jxj ¡ . (jxj + t 1/2 + 2 ¡i ) ¡ ; while in the second jyj > 2 i jxj / 2, then jK 1 (y)j 6 jK 1 (y)j 1/2 F (2 i jxj / 2) where F is another rapidly decreasing function and in this region Z jK

	. 2 ¡i Z jI j . 2 ¡i Z 0 0 1 d 1 dF (2 i jxj/(2 )) Z jyK 1 (y)j e ¡cjx+ 2 ¡i y j 2 /t t 2 jP t 0 (z)j = C e ¡jz j 2 /t dy t 4/2 jI j . Z e ¡cjx+ 2 ¡i yj 2 /t t 1/2 z t 2 dy . 2 ¡i Z 0 1 dF (2 i jxj/(2 )) Z e ¡c 0 j 2 ¡i yj 2 /t t 3/2 j 2 ¡i yj dy . 2 ¡i 6 C Z 0 . 2 ¡i jxj 4 Z 0 1 dF (2 i jxj/(2 )) 2 3i jxj 3 3 . 2 ¡i jxj When 2 ¡i > t 1/2 ; jxj we estimate where Z jK i (x ¡ y)j (jyj + t 1/2 ) dy . 2 i Z jK 1 (y)j j2 i x + yj dy . 2 2i sup so Z jK i (x ¡ y)j (jyj + t 1/2 ) dy . Z jK 1 (y)j Z jx + 2 Lemma 31. For m; n 2 (0; 5), k; `2 [0; 2) dene I k;m;n := 1;2; 1;2 0	y)jjP t 0 (x + 2 ¡i y)jdy dF (2 i jxj/(2 )) e ¡cjz j 2 /t t 2 : 1 2 3i 3 jxj Z e ¡c 0 jyj 2 /t t 3/2 dy

Z

jK i (x ¡ y)jP t (y)dy . 2 3i . 2 ¡i (jxj + t 1/2 + 2 ¡i ) ¡4 :

When jxj > 2 ¡i ; t 1/2 we have instead that either jxj > 2 2 ¡i jyj or jxj < 2 2 ¡i jyj. In the rst region jx + 2 ¡i y j > cjxj so

jI j . 2 ¡i Z 0 1 d Z jyK 1 (y)j e ¡c 0 jxj 2 /t t 2 dy . 2 ¡i e ¡c 0 jxj 2 /t t 2 . 2 ¡i jxj ¡4 . 2 ¡ i (jxj + t 1/2 + 2 ¡i ) ¡4 :

while in the second region jyj > 2 i jxj / (2 ), then jy K 1 (y)j 6 jy K 1 (y)j 1/2 F (2 i jxj / (2 )) where F is a rapidly decreasing function and in this region 4 . 2 ¡i (jxj + t 1/2 + 2 ¡i ) ¡4 : So we conclude that (64) holds. Let us turn to (65). When t 1/2 > 2 ¡i ; jxj we have Z jK i (x ¡ y)j (jyj + t 1/2 ) dy . 1 t Z jK i (x ¡ y)jdy . 1 t . (jxj + t 1/2 + 2 ¡i ) ¡ : z Z jK 1 (y)j jz + yj dy . 2 i (x ¡ y)j (jyj + t 1/2 ) dy . F (2 i jxj/2) Z jK 1 (y)j 1/2 j2 ¡i yj dy . 2 i F (2 i jxj/2) . jxj ¡ . (jxj + t 1/2 + 2 ¡i ) ¡ ; concluding our argument.

  and by the scaling properties of C Y ;" and P t; y , namely C Y ;" (2 ¡2i s; 2 ¡i x) . 2 i C Y ;" (s; x) and P 2 ¡2i s (2 ¡i x) . 2 3i P s (x) given by Lemma 22 and Lemma 26, we obtain easily the bound on I ^k;m;n by rescaling the integral. and the result follows applying repeatedly this inequality.Lemma 36. Let j 2 Nnf0g and p > 1. There exists a nite constant c p such that for every 2 L p : D (j ¡ L) (h 1 ); :::; W (h n ) and can be extended by density on L p ).

			¡	1 2	L p (H)	6 c p k k L p
	(where the operator D(j ¡ L)	¡	1 2 is dened on every	polynomial in W

jhh 1 0; h 2 0ij 2¡ 2 j 1; 2 jj 1 0 ; 2 0j(44)

Appendix A Basics of paracontrolled analysis

In this section we recall the notations and the basic results of paracontrolled calculus introduced in [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] without proofs. For more details on Besov spaces, LittlewoodPaley theory, and Bony's paraproduct the reader can refer to the monograph [START_REF] Bahouri | Fourier analysis and nonlinear partial dierential equations[END_REF].

A.1 Notation and conventions.

Throughout the paper, we use the notation a . b if there exists a constant c > 0, independent of the variables under consideration, such that a 6 c b, and we write a ' b if a . b and b . a. If we want to emphasize the dependence of c on the variable x, then we write a(x) . x b(x). For index variables i and j of Littlewood-Paley decompositions (see below) we write i . j if there exists N 2 N, independent of j, such that i 6 j + N , and we write i j if i . j and j . i.

An annulus is a set of the form A = fx 2 R 3 : a 6 jxj 6 bg for some 0 < a < b. A ball is a set of the form B = fx 2 R 3 : jxj 6 bg. If f is a map from A R to the linear space Y , then we write

Given a Banach space X with norm kk X and T > 0, we write C T X = C([0; T ]; X) for the space of continuous maps from [0; T ] to X, equipped with the supremum norm kk CTX , and we set CX = C(R + ; X). For 2 (0; 1) we also dene C T X as the space of -Hölder continuous functions from [0; T ] to X, endowed with the seminorm kf k C T X = sup 06s<t6T kf (t) ¡ f(s)k X / jt ¡ sj , and we write C loc X for the space of locally -Hölder continuous functions from R + to X. For > 0, we dene

The space of distributions on the torus is denoted by D 0 (T 3 ) or D 0 . The Fourier transform is dened with the normalization

so that the inverse Fourier transform is given by F ¡1 v(x) = (2) ¡1 P k e hk;xi v(k). Throughout the paper, (; ) will denote a dyadic partition of unity such that supp((2 ¡i )) \ supp((2 ¡j )) = ; for ji ¡ j j > 1. The family of operators ( j ) j ¡1 will denote the Littlewood-Paley projections associated to this partition of unity, that is ¡1 u = F ¡1 (F u) and j = F ¡1 ((2 ¡j )F u) for j 0. We also use the notation S j = P i< j i . The Hölder-Besov space B 1;1 (T 3 ; R) for 2 R will be denoted by C and equipped with the norm

as well as

Proofs can be found in [START_REF] Gubinelli | KPZ Reloaded[END_REF].

Appendix B Estimation of the kernels

In this section we recall a few well-known results on convolution of functions with known singularities around zero. We remand to Section 10.3 of [START_REF] Hairer | A theory of regularity structures[END_REF] for an extensive treatment of this subject. First of all we need to characterize the local behaviour of the heat kernel P t (x) and of the covariance C Y ;" (t; x) of the Gaussian eld Y " .

Lemma 22. The heat kernel P : R R 3 ! R dened by

Then for every multi-index jkj 6 2 we have:

In this article we use a slightly dierent version of the heat kernel, namely

T 3 P t¡s (x ¡ y)v(s; y)dsdy is the stationary solution to L X = ¡X + v. However, every estimate remains trivially valid in this setting.

Proof.

In the same way we prove that j@ t P t (x)j . (jtj 1/2 + jxj) 5 , j@ xi P t (x)j . (jtj 1/2 + jxj) 4 and j@ xi @ xj P t (x)j . (jtj 1/2 + jxj) 5 .

We recall a special case of Lemma 10.14 of [START_REF] Hairer | A theory of regularity structures[END_REF], which is enough for our purpose. We use the notation 99 := (jtj 1/2 + jxj) for = (t; x) 2 R T 3 . Lemma 24. Let f ; g: R T 3 n f0g ! R smooth, integrable at innity and such that jf ()j . 99 and jg()j . 99 in a ball B = f 2 R T 3 : 99 < 1; = / 0g. Then if ; 2 (¡5; 0) and + + 5 < 0 we have jf g()j . 99 + +5 in a ball centered in the origin.

Moreover, if ; 2 (¡5; 0) and 0 < + + 5 < 1 and for every multi-index jkj 6 2 we have jD k f ()j . 99 ¡jkj and jD k g()j . 99 ¡jkj , then

in a ball centered in the origin.

Remark 25. The covariance C Y ;" of Y " can be written as C Y ;" = P C ~" P with C ~"(t; x) := E( " (t; x) " (0; 0)). Recall from the introduction that C ~"(t; x) = " ¡5 C ~"(" ¡2 t; " ¡1 x) where C ~" is the covariance of the Gaussian process dened on R (T / ") 3 , and C ~"(t ¡ s; x ¡ y) = (t ¡ s; x ¡ y) if dist(x; y) 6 1 and 0 otherwise (so that the family of functions C ~" is bounded uniformly on " by a C c 1 function). Then there exists a family of functions

Appendix C Some Malliavin calculus results

Let D be the Malliavin derivative, the divergence (dened as the adjoint of D) and P t the Ornstein-Uhlenbeck semigroup. We refer to [START_REF] Nualart | The Malliavin calculus and related topics[END_REF] for an extensive discussion on these operators. Call fW (h)g h2H the isonormal Gaussian process indexed by H some real separable Hilbert space. For every 2 L 2 (), P t can be written via the well-known Mehler's formula for = F (W (h)):

where fW 0 (h)g h2H is an independent copy of W . In our case we will consider the Gaussian process Y " indexed by h t;x 8(t; x) 2 R T 3 , with H = L 2 (R T 3 ), with Y " (t; x) dened as in Section 1. By a direct calculation we obtain

This gives a commutation result between the Malliavin derivative D and the generator of the Ornstein-Uhlenbeck semigroup L (dened by P t = e tL , recall that L = ¡D ([22], Proposition 1.4.3)). Indeed, let such that E( ) = 0, then we have for every > 0 and every j > 0:

and the same works for every (not necessarily centered) if j > 0. It is well-known (see [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]) that L acts on square integrable functions as

where J n is the projection of on the n-th Wiener chaos. We can dene (j ¡ L) ¡1 by its action on n-th order chaoses as

The results recalled above allow for the following partial chaos expansion : Lemma 32. Let 2 L 2 (). Then for every n 2 Nnf0g:

Proof. We have for any 2 L 2 ():

where we used (68) and the fact that the Malliavin derivative of a constant is zero. This yields

Iterating this formula up to an order n and using the fact that J 0 (k ¡ L) ¡1 = 1 k J 0 we obtain the result.

Notice that the lemma above implies n Q 1 n D n = (1 ¡ J 0 ::: ¡ J n¡1 ) . In order to have L p estimations of the terms n Q 1 n D n generated by expansion (69), we will need the following lemmas: Lemma 33. ([22], Proposition 1.5.7) Let V be a real separable Hilbert space. For every p > 1 and every q 2 N; k > q and every u 2 D k;p (H q V ) we have k q (u)k D k ¡q; p (V ) . k;p kuk D k; p (H q V ) Remark 34. Using Lemma 33 we can state a simplied version of Lemma 39 in the case where F 2 V is deterministic. Let V be a real separable Hilbert space. For every F 2 V and every u 2 D q;2 (H q ) with q 2 N we have u F 2 Dom q and q (u)F = q (u F ):

We can prove this formula as follows. First notice that for every smooth G 2 D q;2 (V ) and every smooth u 2 D q;2 (H q ) we have

Now since D q (u F ) = D q u F and u 2 D q;2 (H q ) we have that u F 2 D q;2 (H q V ). Lemma 33 yields the bound k q (u F )k L 2 (V ) . ku F k D q;2 (H q V ) which allows to pass to the limit for G and q (u F ) in L 2 (V ).

Lemma 35. For every

Proof. We have

and (0) = 0. Then the operator T satises the hypotheses of Theorem 1.4.2 of [START_REF] Nualart | The Malliavin calculus and related topics[END_REF], with (n) = h(n ¡1 ) and h(x) =

x j x + 1 analytic in a neighbourhood of 0. Therefore

We can pass to the limit thanks to the assumption h j (u); D r F i H `+r 2 L 2 (; H q ¡r ¡j ) and obtain

Now suppose the statement true for q ¡ 1. We have that

= X r=0 q q r q ¡r (hu; D r F i H `+r)

Lemma 42. Let j ; k 2 N, u 2 D j +k;2 (H j ) symmetric and such that all its derivatives are symmetric. We have

Proof. If j = 0, k = 1 or k = 0; j = 1 we have identities. Now let j = k = 1 and u 2 D 2;2 (H). We have that D and since by hypothesis Du is symmetric we have (hDu; hi) = hDu; hi and then D(u) = u + Du. The proof by induction is easy noticing that v is symmetric whenever v is symmetric, and using the fact that D j = j D + j j ¡1 .