<\body> <\hide-preamble> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> > > > > > > > > > > > >>>> \; \; > 3d stochastic reaction\Udiffusion models>|Furlan>|<\author-affiliation> CEREMADE Université Paris Dauphine, France |>>|Gubinelli>|<\author-affiliation> IAM & HCM Universität Bonn, Germany |<\author-email> gubinelli@iam.uni-bonn.de >>> > model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (> is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their > norms in terms of the graphs of the standard > stochastic terms. >|> \; Consider a family of stochastic reaction\Udiffusion equation in a : <\equation> u=-\>*F>|)>+\\|]>\/\|)> with \>, initial condition >>:/\|)>\\>, >\C|)>> with exponential growth at infinity, \0> and :=-\|)>> the heat flow operator and =\/\|)>>. Here > is a centered Gaussian noise with stationary covariance <\equation*> \\|)>=> such that >=\> if \1> and otherwhise where :\\\\\> is a smooth, positive function compactly supported in \B>>.\ We look for a large scale description of the solution to eq. and we introduce the \Pmesoscopic\Q scale variable >=\>u,x/\|)>> where \0>. Substituting >> into we get <\equation> u>=-\-2-\>*F>>*u>|)>+\>*\*>,>|)>. In order for the term >*\,x/\|)>> to converge to a non\Utrivial random limit we need that =1/2>. Indeed the Gaussian field >:=\\,x/\|)>> has covariance >=\>,x/\|)>> and converges in distribution to the space-time white noise on \\>. For large values of > the non\Ulinearity will be negligible with respect to the additive noise term. Heuristically, we can attempt an expansion of the reaction term around the stationary solution >> to the linear equation <\equation*> Y>=-Y>+\>. Let us denote with >> the covariance of >>. We approximate the reaction term as\ <\equation*> \-5/2>F>*u>|)>\\-5/2>F>*Y>|)>. The Gaussian r.v. *Y>> has variance >=\\>|)>|]>=\\>|)>|]>=\C>> independent of >. Therefore we can expand the r.v. >*Y>|)>> according to the chaos decomposition relative to *Y>> and obtain <\equation*> F>*Y>|)>=0>f>* H*Y>,\>|)>, where >|)>> are standard Hermite polynomials with variance >>. Note that also the coefficients >|)>0>> do not depend on > by stationarity of the law of *Y>> since they are given by the formula <\equation*> f>=n!*\>*Y>|)>H*Y>,\>|)>|]>=n!*\>>G|)>H>G,\>|)>|]> where is a standard Gaussian variable of unit variance. Let be the stationary solution to the equation <\equation*> X=-X+\ with > the space\Utime white noise and denote by |X|\>> is the generalized random fields given by the -th Wick power of which are well defined as random elements of \\|)>> as long as 4>. Denote with > the covariance of . The Gaussian analysis which we set up in this paper shows in particular the following convergence result. <\theorem> Fix 4> and assume that /2>f>\g> for n\N> as \0>, that >|)>>\C|)>> and there exists constants 0> such that <\equation*> sup,x>F>|\|>\C e>. Then the family of random fields <\equation*> \>:\\F>*Y>|)> converges in law in \\|)>> as \0> to g|X|\>>. Now take the smallest such that >> converges to a finite limit as \0>. Since *Y>,\>|)>=\|Y>|\>>, the -th term in the expansion of >*Y>|)>> is >*\+/2>|Y>|\>>. From Theorem the equation yields a non-trivial limit only if =/2>. We are interested mainly in the case \=1> and \=2>. The case =2> gives rise to a Gaussian limit and its analysis its not very difficult. 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 > family of singular SPDEs. In this case we obtain the family of models <\equation> u>=-\>*F>>*u>|)>+\> with initial condition >|)>\\>>>\|)>> where >>> is the initial condition of the microscopic model. \ \ Define for 0> and =\\\\> <\equation> \>>\\/2>>>Y,\>|)>. where >> is the centered function <\equation*> > \F>-f>-f>*x-f>H>|)>=3>f>*H>|)> and >> is the variance of the centered Gaussian process Y>>. Note that Y>|)>,\>|)>=\|Y>|\>> and denote with >> the coefficients in the chaos expansion of >Y,\>|)>>. Define also various >\Udependent constants <\equation> |||>>>|>|P\>\>>|]>,>|>|>>>|>|f>f> P>|]>,>>|>>>|>|P\>\>>|]>,>||>>>|>|P\>\>>|]>,>>|>>>|>|>>+3 >>>||||>>>> where > is the heat kernel.\ <\axiom> All along the paper we enforce the following assumptions: <\enumerate-alpha> >|)>>> of eq. converges to a limit > in >> and is independent of >; >>|)>>> is uniformly bounded in >>, i.e. C\0> such that \\> |>>|\<\|\|\>>>/\|)>|)>>\C>; >|)>>\C|)>> and there exists constants 0> such that <\equation> sup,x>F>|\|>\C e>, the vector >=>>,\>>,\>>,\>>|)>\\> given by <\equation> >>>||>>||>>>||f>-3>>>>|>>>||f>>||>>>||f>-\f> >>-3>>-3>>>>>>> has a finite limit =>,\>,\>,\>|)>\\> as \0>.\ We can now formulate our main result. <\theorem> Under Assumptions the random field >> given by the solution to eq. converges in law and locally in time to a limiting random field |)>> in the space >|)>> for every \\2/3>. The law of |)>> 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 > model with parameter vector \\>. \ <\remark> In particular we can take <\equation*> >>||>H>|)>+\\>H>|)>+\>+\>>|)>H>|)>+\>+\>>|)>>>>>> so that <\equation*> f>=\>,\f>=\>,\f>=\>+\>>,\f>=\>+\>>, and <\equation*> >>=3>|)>L>,>>=\>\> L>,>>=>>=0, where >\2P>|)>.> Choosing <\equation*> \>>\3>>=9>|)>L>,\>>\3>>=3\>\> L>, we obtain >\>,\>,\>,\>|)>>. This shows that all the possible limits \\> are attainable. In this case takes the form <\equation> u>=-\>u>-\>u>->-3\>\\>+9>|)>L>|]>u>-\>+\>\>-3\>\> L>+\> . When the nonlinearity is given by a cubic polynomial like in the corresponding limiting model is called dynamic > equation or stochastic quantisation equation. In that case the kind of convergence results described above are originally due to Hairer and constitute one of the first groundbreaking applications of his theory of regularity structures. Similar results were later obtained by Catellier and Chouk using the paracontrolled approach of Gubinelli, Imkeller and Perkowski. Kupiainen described a third approach using renormalization group ideas.\ 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 finite number of their combinations survive in the limit to describe the limiting object. The adjective \Pweak\Q 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 first result of weak universality for a singular stochastic PDE has been given by Hairer and Quastel in the context the Kardar\UParisi\UZhang equation. Using the machinery developed there Hairer and Wu proved a weak universality result for three dimensional reaction\Udiffusion equations in the case of Gaussian noise and a polynomial non\Ulinearity, within the context of regularity structures. Weak universality for reaction\Udiffusion equations driven by non Gaussian noise is analysed in Shen and Wu. Important results concerning the stochastic quantisation equation we obtained by Mourrat and Weber. In particular the convergence to the dynamic > model for a class of Markovian dynamics of discrete spin systems and also the global wellposedness of > in space and time and in time for >.\ The present work is the first 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 all the work is related to showing the convergence of a finite number of random fields to universal limiting objects. The main point of our analysis is our use of the Malliavin calculus to perform the analysis of these stochastic terms without requiring polynomial non\Ulinearity as in the previous works cited above. In particular we were inspired by the computations in and in general by the use of the Malliavin calculus to establish normal approximations. The main technical results of this part of our paper is not particularly linked to paracontrolled distributions. A similar analysis could be possible for the stochastic models in regularity structures. Moreover the same tools can also allow to prove similar non-polynomial weak universality statements for the KPZ along the lines of the present analysis. This is the subject of ongoing work. \; Analysis of the mesoscopic model> In this section we describe the paracontrolled approach to a solution theory for eq. along the lines of the Catellier\UChouk analysis. The fundamental results of paracontrolled calculus we use in this section are listed in Appendix.\ The continuity of the solution map for a paracontrolled equation (already established in and recalled here) allows us to prove convergence of the solution >> (Theorem) by showing the convergence of the enhanced noise >> and the remainder >> in the appropriate space. Finally, in Theorem we identify the limiting process as the solution of a paracontrolled equation. Write >=Y>+v>>, and perform a Taylor expansion of the reaction term >Y>+\v>|)>> in around Y>> up to the third order to get <\equation> u>>||>-\>-\>v>-\>v>-\>v>-R>>|)>>>|||f>-\f>>+v>|)>-\f>|Y>|\>+2 v> Y>+v>|)>.>>>>> with >>> defined in \\\\\> and >>|)>> the remainder of the Taylor series. Define the following random fields: <\equation> Y>>|>|>>|>|>>>|>|f>*|Y>|\>>>|>>>|>|>,>||>>>|>|>>\>>->>>>|>>>|>|\>>||>>>|>|>>\>>->>Y>->>>>|>>>|>|>>,>||>>>|>|>>\>>->>>>|>>>|>|\>>||>>>|>|>>\>>->>>>|>>>|>|\>>||||>>>> with >> defined in and >,>>,>>> stationary solutions. In the scope of this section we can take any constants >>> that satisfy <\equation> >>>||>>+3 >>.>>>>> Equation takes the form <\equation> v>>||>->>->>-3>>v>-3>>v>->>v>>>|||f>-\f>*>+v>|)>-\f>*>*v>+v>|)>-R>>|)>.>>>>> In this expression the products >>*v>>, >>*v>> and >*v>> do not meet the conditions for continuity. In order to continue the analysis we pose the paracontrolled Ansatz <\equation> v>=->>->>-3v>\>>+v>>, and proceed to decompose the ill-defined products using the paracontrolled techniques recalled in Appendix . We start with <\equation*> v>>>=v>\>>+v>\>>+v>\>>. The resonant term, together with Ansatz, yields: <\eqnarray*> >\>>>||>>\>>- >>\>>-3*v>*>>\>>|)>>>|||>>,>>,>>|)>+v>>\>>.>>>> So we let <\equation*> >>|^>v>>|>|>>>-v>\>>+>*>>+>>Y>+>>+>>|)>>>|||>\>>- >>->>-3 v>*>>+v>>\>>-3*>>,>>,>>|)>>>>>> Moreover we have for >Y>>: <\equation*> >*Y>>||>Y>->>\Y>->>\Y>->>\Y>>>>>> where we introduced the shorthand >=v>+>>>. So we let\ <\equation*> v>\Y>\v>*Y>+>>=\>Y>->>\Y>->>\Y>->> Finally to analyse the product >>*v>> we write <\equation*> >>*v>=>>>>|)>-2>>>>\>+>>\>, and consider the products involving only >> factors: first <\equation*> >>>>=>>\>>+>>\>>+>>+>>=:>>\>>+>>, and then we define the term >>\>>|)>> as follows: <\eqnarray*> >>\>>|)>>|>|>>>>|)>-2>>>>>>|||>>\>>|)>+>>\>>|)>+>>\>>\>>|)>+2com>>,>>,>>|)>+2>>>>.>>>> so that <\equation*> >>\v>\>>v>+2>>v>=>>\>>|)>-2*>>\>>|)>\>+>>\> We note also that <\equation*> v>=->>->>+v>>-3v>\>>-3com>,>>|)>-3*com>,>>|)>. \; Substituting these renormalized products into we obtain the following equation for >>> : <\eqnarray*> v>>>||>,>>|)>+3*com>,>>|)>>>|||>>v>-3>>\ v>-3>>|^> v>>>|||>-\>>> \Y>+v>|)>>>|||>> >+v>|)>+>>+6>>-3>>|]>v>-\>>-R>>|)>>>|||>,\>;v>,v>>|)>-R>>|)>>>>> with >>|)>> the Taylor remainder which appears in and >=>>,\>>,\>>,\>>|)>\\> given by eq. and where we can use the constraint to remove the term proportional to >>. The enhanced noise vector >> is defined by <\eqnarray*> >>|>|>>,>>,>>,>>,>>,>>,>>,>>,>>|)>>>|>>||>|>|>|>\C-\>\>|)>\>\>|)>\C-\>>>>> for every \0>. We use the notation |\>|\<\|\|\>>>=>|\>>|\<\|\|\>>>>> for the associated norm where >>> is a generic tree in >>. The homogeneities |\|>\\> are given by <\equation*> |||||||||||||||||||||>>>||>>>|>>>|>>>|>>>|>>>|>>>|>>>|>>>|>>>>||\|>>|||>|||>||||>>>>> Notice that for every \0> eq. is equivalent to together with Ansatz. In this section we show uniform a-priori estimates for the pair >,v>>|)>> which solves the following system of equations <\equation> |||>>||>>->>-3v>\>>+v>>+v>>>>|v>>>||>,\>;v>,v>>+v>>|)>-R>>|)>>>|>>>||>>+>>+3v,0>\>>>>>>>|\> with >>\P*v,0>> and ,0>\u>-Y>\>. is given in . It is easy to see, by taking >>=v>>+v>>>, that this is equivalent to eq. together with Ansatz on >>. We consider the spaces\ <\equation*> \>\>\>\>,\\>\/2,2\>, with the corresponding norms <\eqnarray*> |v>>|\<\|\|\>>>>>|>||v>>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>,>>>> <\eqnarray*> |v>|\<\|\|\>>>>|>||v>|\<\|\|\>>>>+|v>|\<\|\|\>>/2,2\>>. >>>> Define also the quantity <\equation> M>>,u>|)>\|\/2>e*>|\|>+c \>*v,0>|\|>>|\<\|\|\>>L|)>>. which will be used to control the remainder >>. The main result of this section is the following lemma. <\lemma> There exists a time >=T>|\>|\<\|\|\>>>,|u,0>|\<\|\|\>>>>,>|\|>|)>\> depending only on |\>|\<\|\|\>>>>, |u,0>|\<\|\|\>>>>> and >|\|>>, a constant >=M>>,u>|)>\0> defined by, and a universal constant 0> such that, whenever >\T>/2>> we have <\eqnarray*> |v>>|\<\|\|\>>>>>>>|>|>|\|>|)>*|\>|\<\|\|\>>>|)>|u,0>|\<\|\|\>>>>|)>,>>||v>|\<\|\|\>>>>>>|>||\>|\<\|\|\>>>+|u,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>>>|)>.>>>> <\proof> Define >>\v>-v>>> such that <\equation*> |>>>||>>->>-3>>+v>>|)>\>>+v>>,>>>>> and >>=0>. Note also >>\v>>+>>>. Using Lemma (and the fact that |f|\<\|\|\>>,\>>\T>|f|\<\|\|\>>>>>) we obtain for ,\\0> small enough <\equation> |I f|\<\|\|\>>+2\,2\>>+|I f|\<\|\|\>>+2\,1/2+2\>>+|I f|\<\|\|\>>+2\,1+2\>>\T|2>>|f|\<\|\|\>>>>>+|f|\<\|\|\>>>>>|)>. We choose \2\> small enough so that <\equation*> +3\/2,2\>\+3\/2,1/2+2\>\+3\/2,1+2\>\\> We define also the norm <\eqnarray*> |v>>|\<\|\|\>>>>>|>||v>|\<\|\|\>>>>+|v>|\<\|\|\>>>>.>>>> Now <\eqnarray*> |v>>|\<\|\|\>>>>>|>||>>+>>|\<\|\|\>>>>+|v>>|\<\|\|\>>>L>>|>>|\<\|\|\>>>>+|>>|\<\|\|\>>>>|)>>>||||v>>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>|)>|>>|\<\|\|\>>>>+|>>|\<\|\|\>>>>|)>+|v>>|\<\|\|\>>>>>>||>||\>|\<\|\|\>>>+T>|v>>|\<\|\|\>>>>+|v,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>>>>> where we used that >=0> and as a consequence that |v>>|\<\|\|\>>>L>>\T>|v>>|\<\|\|\>>>>L>>\T>|v>>|\<\|\|\>>>>> to gain a small power of . So provided is small enough (depending only on >>) this yields the following a-priori estimation on >>>: <\equation*> |v>>|\<\|\|\>>L>>\|v>>|\<\|\|\>>>>\|\>|\<\|\|\>>>+|v,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>. Therefore we have an estimation on >>: <\equation*> |v>|\<\|\|\>>>\|v>>|\<\|\|\>>>+|v>|\<\|\|\>>>\|v,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>\|\>|\<\|\|\>>>+|v,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>. In order to estimate terms in >,\>;v>,v>>+v>>|)>> we decompose the renormalised products as\ <\equation*> |||>>|^>v>>||>\>>- >>->>-3 v>*>>+v>>\>>+v>>\>>-3*>>,>>,>>|)>>>|>\Y>>||>>Y>-3>\>>|)>Y>+Y>\>>+v>>|)>+Y>\>>+v>>|)>>>|||>>\Y>->>\Y>->>>>|>>\v>>||>>\>>|)>+2>>\>>|)>>>+3v>\>>|)>-2*>>\>>|)>\>>+v>>|)>>>|||>>\>>|)>\>>+v>>|)>+>>\>>+v>|)>+>>\>>+v>|)>.>>>>> \; We have >,\>;v>,v>>+v>>|)>=Q>,\>,v>,v>,v>>|)>+Q>,\>,v>,v>,v>>|)>+Q>,\>>> with the definitions <\eqnarray*> >|>|>\>>-3*>>,>>,>>|)>+>>\>>+v>|)>|]>>>|||>>\>>|)>>\>>|)>+>>\>>|)>\>>+v>>|)>|]>>>|||>>>\>>|)>Y>-Y>\>>+v>>|)>|)>+3*com>,>>|)>+3*com>,>>|)>>>|>|>|>*>>-v>>\>>-v>>\>>+2*>>\>>|)>\>>+v>>|)>->>\>>+v>|)>|]>>>|||>>v>-\>>>+2Y>\>>+v>>|)>|]>>>|>,\>>>|>|>>|)>Y>-\>>+3 >>+>>->>\>>|)>-2>>\>>|)>>>|]>>>|||>>>>Y>+>>\Y>+>>\Y>+>>|)>>>>> With the same technique we used above for >>>, we obtain the following estimate on >>> <\equation*> |v>>|\<\|\|\>>/2,1/2+2\>>+|v>>|\<\|\|\>>,\>>\|\>|\<\|\|\>>>+|v,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>> and this yields |>>|)>|\<\|\|\>>/2,1/2+2\>>+|>>|)>|\<\|\|\>>,\>>\|\>|\<\|\|\>>>+|v,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>|)>>.\ We obtain, using the results of Appendix <\equation*> |||Q|\<\|\|\>>>>>+|Q|\<\|\|\>>>>>>|>|>|\|>|)>|\>|\<\|\|\>>>|)>|v,0>|\<\|\|\>>>>+|v>>|\<\|\|\>>>>|)>>>||Q>,\>>|\<\|\|\>>>>>|>|>|\|>|)>*|\>|\<\|\|\>>>|)>>>>>> \; In order to conclude the estimation of |v>>|\<\|\|\>>>>> we have to bound |I R>>|)>|\<\|\|\>>>>>. By Lemma \\>, \\0> such that |3+\>\+|2>> we have |t\t>I R>>|)>|\<\|\|\>>L|)>>\M>>,u>|)>|v>|\<\|\|\>>>>e|*v>>|\<\|\|\>>>>>>. By Lemma together with we obtain then\ <\eqnarray*> | I R>>|)>|\<\|\|\>>>>>|>|>>,u>|)>|v>|\<\|\|\>>>>e|*v>>|\<\|\|\>>>>>.>>>> Noticing that |P v>>|\<\|\|\>>>>\|v>>|\<\|\|\>>>>\|v,0>|\<\|\|\>>>>|)>|\>|\<\|\|\>>>> we obtain that C\0> such that <\eqnarray*> |v>>|\<\|\|\>>>>>|>|>|\|>|)>*|\>|\<\|\|\>>>|)>|v,0>|\<\|\|\>>>>|)>+C*T/2>>|\|>|)>|\>|\<\|\|\>>>|)>|v>>|\<\|\|\>>>>>>|||*M>>,u>|)>e|\>|\<\|\|\>>>+|v,0>|\<\|\|\>>>>|)>>e|v>>|\<\|\|\>>>>>|v>|\<\|\|\>>>>>>||>|>>,u>|)>e|v>>|\<\|\|\>>>>>+C*T/2>|v>>|\<\|\|\>>>>+C*M>>,u>|)>e|v>>|\<\|\|\>>>>>|v>>|\<\|\|\>>>>>>>>> with C>|\|>|)>|\>|\<\|\|\>>>|)>+e|\>|\<\|\|\>>>+|v,0>|\<\|\|\>>>>|)>>|\>|\<\|\|\>>>+|v,0>|\<\|\|\>>>>|)>>|)>|]>> and C>|\|>|)>*|\>|\<\|\|\>>>|)>|v,0>|\<\|\|\>>>>|)>>.\ Let >\> such that:\ <\equation*> C*T>/2>+e>>|]>\ and <\equation*> C*T>/2>e>\D. Assume that >\T>/2>>. Define a closed interval =>|]>:|v>>|\<\|\|\>>>>\4D|}>\>|]>>. This interval is well defined and non\Uempty since |v>>|\<\|\|\>>>>> is continuous and nondecreasing and |v>>|\<\|\|\>>>>\4D>. Let us assume that T>>, then we can take \0> small enough such that \T>> and by continuity |v>>|\<\|\|\>>>>>\5C>, then <\eqnarray*> |v>>|\<\|\|\>>>>>>|>|>>,u>|)>e|v>>|\<\|\|\>>>>>>+C*|)>/2>|v>>|\<\|\|\>>>>>+C*M>>,u>|)>e|v>>|\<\|\|\>>>>>>|v>>|\<\|\|\>>>>>>>>||>|>>,u>|)>e>+C*T>/2>|v>>|\<\|\|\>>>>>+C*T>/2>e>>|v>>|\<\|\|\>>>>>>>||>||v>>|\<\|\|\>>>>>>>>> which gives |v>>|\<\|\|\>>>>>\4D>. This implies >> (by contradiction). <\lemma> For every \>, \> we have\ <\equation*> |t\t>R>>,v>>,v>>|)>|\<\|\|\>>L|)>>\M>>,u>|)>|v>|\<\|\|\>>>/|)>>L>>e|*v>>|\<\|\|\>>L>>> with >>\->>->>-3v>\>>+v>>>. <\proof> We can write the remainder in two ways: \; <\equation*> R>>|)>=v>>>>*Y>+\*\>v>|)>-F>>>*Y>|)>|]>*|)>|2!>\\=\>v>F>>>*Y>+\*\>v>|)>*|)>|3!>\\. From assumption on we obtain by interpolation of these two expressions, \\>, t\0,x\\>, <\equation*> >>|)>|\|>\\/2>>|\|>>e>*>|\|>+c \>>>|\|>+c>*v>>|\|>>, and we estimate, \\>, <\equation*> |t\t>R>>|)>|\<\|\|\>>L|)>>\|t|3+\>> v>|\<\|\|\>>L>>>e|*v>>|\<\|\|\>>L>>>|\|2>>e>*>|\|>+c \>>>|\|>>|\<\|\|\>>L|)>>. In order to identify interesting limits for equation, we introduce the enhanced universal noise >, defined as <\equation*> \=,,,,|)>, where is the stationary solution to to the linear equation X=-X+\> and > is the time-space white noise on \\>. We define <\equation*> \; <\equation> >|>||X|\>,>>|>|>||X|\>,>>|>|>|\X|)>=,\> |X>|\>X>*\,\>,>>|>|>||)>\|)>=,\> |)>|X>|\>|X>|\>|)>*\,\>,>>|>|>|,\> |)>|X>|\>|X>|\>|)>*\,\>+6 ,x>X>-x|)>-\X>|)>|]>P |]>,\>>>>> where as before |\|\>> stands for the Wick product, =,s|)>\\\\>, > is the covariance of and ,\>> is defined as <\equation*> \,\>\K>>j>KK|)>P>|)>|]>\|)>\\\\. Standard computations (see e.g. or) show that, for any 0>, \\\> <\equation*> \\C>-2\>\>>|)>\C>-2\>\>>|)>\C>-2\>, almost surely. Finally, for every =>,\>,\>,\>|)>\\> we define\ <\equation> \|)>\>,\>X,\>,\>,\>,>|)>,>|)>,\>\>,>|)>|)>. \; Using the paracontrolled structure we developed in the preceding sections and its continuity with respect to >>, we can state the convergence of the solution of the mesoscopic equation, under the hypothesis that >> and >> converge (this will is shown in Theorem- and Lemma). <\theorem> The family of random fields >> given by the solutions to eq. converges in law and locally in time to a limiting random field |)>> in the space >|)>> for every \\2/3>. The limiting random field |)>> solves the paracontrolled equation <\equation> |||||)>>|||)>>>||)>>||>-\>-3\>v|)>\+v>|)>>>|v>|)>>||,\|)>;v|)>,v>|)>|)>>>|>|)>>||+\>+\>+3\>v,0>\>>>>>|\> with defined in and =u-X>. <\proof> Fix 0>. Let us denote via > the solution map for so that u>=\,0>,\>,\>,R>>|)>|)>>. Denote by >>=\>,0>,\>,\>,R>>|)>|)>> the process >> stopped at the time >|\>|\<\|\|\>>>,|u,0>|\<\|\|\>>>>,>|\|>|)>> and >> the corresponding solution map. Note that solves the same equation with >> replaced by |)>>, ,0>> with >, >> replaced by > and >=0>. So |)>=u>|)>=\>,\|)>,\,0|)>> up to time >|\|)>|\<\|\|\>>>,|u|\<\|\|\>>>>,|\|>|)>>. Let us introduce the random field >>=\>,0>,\>,\>,0|)>> which solves the paracontrolled equation but with remainder >=0>. Consider the -uple of random variables ,0>,\>,u>>,>>|)>> and let >> be its law on =>\\\>|)>> conditionally on >\>\T>/2>|}>>. Note that we know that >|)>\1> from Lemma. By the apriori bounds of Lemma 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>, >|u>>->>|\<\|\|\>>\\|)>\0> as \0> since >\0> in probability. This shows that >> concentrates on >\\\\C>|}>>. Let > any accumulation point of >|)>>>. Then >\\\\C>|}>|)>=1>. Moreover along subsequences we have that for any bounded continuous function <\equation*> \,0>,\>,>>|)>|)>=\,0>,\>,\>,0>,\>,\>,0|)>|)>|)>\\,\|)>,\>,\|)>,\,0|)>|)>|)> since by Theorem the vector >> converges in law to |)>> and ,0>> to > and >> is a continuous function. We deduce that, still along subsequences, for any test function >, <\equation*> >\\\>\>\\\=>\>,0|)>,\>,0|)>|)>\\ but we know also that since >|)>\1> we have <\equation*> \,0>,\>|)>\|\>|]>=,0>,\>|)>\>>|]>|\>|)>>\\,\|)>|)>|]>, for any test function >. So the first two marginals of > have the law of ,\|)>|)>> and they are independent since ,0>,\>|)>> are independent for any >. Calling > the law of ,\|)>|)>> we have that <\equation*> >\\\=>\\>\>,0|)>,\>,0|)>|)>\\ which implies that > is unique and that the whole family >|)>>> converges to >. Convergence of random fields> In this section we prove the convergence of the random fields >> and >>. The convergence in probability of >> is easily obtained as we show in the following lemma. <\lemma> Under Assumptions the random variable >>,u>|)>> defined in converges to zero in probability.\ <\proof> Recalling that ,0>\u>-Y>> we can use Young's inequality estimate >>,u>|)>> for some \0> as <\eqnarray*> >>,u>|)>>|>|/2>|e*\*>|\|>>|\<\|\|\>>L|)>>+\/2>|e*\*>Y>|\|>>|\<\|\|\>>L|)>>>>|||/2>Te*|\*u>|\<\|\|\>>>>>.>>>> Under Assumptions the term |\u>|\<\|\|\>>>|)>>> is uniformly bounded, so the third term above converges to zero almost surely. Note that Y>> and \Y>> are centered Gaussian random variables, and then both |e*\*>|\|>>|\<\|\|\>>L|)>>> and |e*\*>Y>|\|>>|\<\|\|\>>L|)>>> are uniformly bounded in \0>. This yields the convergence in probability of >>,u>|)>> by Markov inequality. The central result of this paper is the convergence of the enhanced noises (or ) >> in law, and their uniform boundedness. <\theorem> Under Assumptions there exists 0> such that for any |)>> we have |\>|\<\|\|\>>>\C> in |)>>. Moreover, >\\|)>\\> in law. The strategy of proof is the following. Denote =>|)>>>, |)>=>|)>|)>>> and let >> the measurable function of the Gaussian process C>> such that >=K>> and >|)>=f>|)>K>>with >|)>> suitable deterministic functions of >. For each > we will show that >>> can be decomposed as <\equation> Y>>=f>>|)>K>>|)>+>> where >>> are suitable remainder terms. For all 2> it is well-known (see ,) that the term >>|)>K>>|)>> is uniformly bounded in ;\>|)>> (with >> given by ), then we will just prove that >>> converges to zero in ;\>|)>>. This con be done by showing that, by Besov embedding, for p\+\> and \\|\|>> we have <\equation> \|>>|\<\|\|\>>-3/p>>|)>\ \|>>|\<\|\|\>>>>|)>\2**p*q>> |\>>|\<\|\|\>>|)>>\x\C>\0 thanks to the stationarity of the process >. For this it suffices to show <\equation> 2**p*q> sup |\>>|\<\|\|\>>|)>>\0as \\0 In order to conclude uniform convergence for > it suffices to show that for \>, -1>: <\equation> \ sup |\>>-\>>|\<\|\|\>>|)>>\C>*p>2**-2\|)>p*q>with C>\0. Indeed, by the Garsia-Rodemich-Rumsey inequality we obtain for \0> small enough and large enough <\eqnarray*> > \|>>|\<\|\|\>>-2/p>B-2\-\>>|)>>|>|2-2\-\|)>*p*q>**supt\>*sup|\>>-\>>|\<\|\|\>>|)>>|*p>>>>||>|>T2*p*q>>>>> which by Besov embedding yields an estimation on |Y>>|\<\|\|\>>-\/2>-2\-\>>|)>> for \0> small enough. This gives us the necessary tightness to claim that >> has weak limits along subsequences. The only thing left to prove is that for each > we have >>|)>\K>> in law. However this is clear since we can introduce a convolution regularisation of called >> which has the same law of >> for any \0>. At this point an approximation argument gives that >>|)>> has the same law of >>|)>>. Transposing the regularisation to the kernels of the chaos expansion we can write >>|)>=K>>> and now it is easy to check that \ >>\K>> in probability (as done e.g. in,). We can then conclude that >>|)>\K>> in law for any >.\ Let us now give the details of the \ the decomposition and the convergence to zero of the remainder >>> in ;\>|)>>. We need to introduce some notations based on the results of Appendix. \ Looking at the definitions of trees listed in , it is clear that Y>> can be written in the form <\equation*> !|3!>>\>>*\>or!!|3!*3!>,\>\>>\>>*\,\>- for ,\,\\\\\>, j,k\3> and some measures >> and ,\>>. Note that the -th Malliavin derivative of >>>, namely \>>> is >>h>k>>. Then expansion of Appendix takes a more explicit form n\1>: <\equation> >>>||>>|)>|k!>|Y,\>|\>+\\>>h>n>|)>>>|||\/2>!|k!>>|Y,\>|\>+\\>>h>n>|)>>>>>> with \> and ,\>\Y>>, =\\\\>. Here we used the fact that >n>|)>=|Y>|\>> (see Remark) and that by the definition of >>> we obtain \\\\\> <\equation*> \>>|)>=\/2>*!*> where >> is the -th coefficient of the chaos expansion of >Y>|)>> relative to *Y>>, so that >=0> for 3> and >=f>> for 3>. Choosing in eq. we obtain\ <\equation> >>>||!>f>*|Y,\>|\>+\\>>h>4-m>|)>>>|||!>f>*|Y,\>|\>+|^>>>>>>>> with\ <\equation> |^>>>\\\>>h>4-m>|)>. This yields: <\equation*> >>>||>|Y,\>|\>+\\>>h>4>|)>,>>|>>>||>|Y,\>|\>+\\>>h>3>|)>,>>|>>>||>Y,\>+\\>>h>2>|)>,>>|>>>||>+\\>>h>|)>.>>>>> It suffices to substitute this decomposition in to identify the remainder terms >>> for every tree >>>. In the next two sections we will consider separately , which are defined a function of >>> and , which in turn are obtained by multiplying first order trees (and renormalising). We will show that each of these remainder terms satisfy and . <\remark> Notice that for 3> we can easily estimate terms of the form /2>\>>> \\\\\>. We have <\equation*> |\>\>>|\<\|\|\>>>=|F>>Y,\>|)>|\<\|\|\>>>=>>>|\|>\x|)> where x|)>> is the density of a centered Gaussian with variance >>. The integral is finite by Assumption. First of all note that the term >>> has no remainder and then it can be shown to converge in law to >> by usual techniques. We start with the bound for >>,\>>,\>>>,>>>. We obtain from that <\eqnarray*> Y>>>|)>>|>|!|3!>>\>>\>>>|||>>|Y>,\>|\>\>+!|3!>>|^>>>\>>>|||>>|)>\K>>|)>>|)>+\>>>|)>.>>>> As said before, >>|Y>,\>|\>\>> converges in law in > for every p\+\> to >>|X>>|\>\>> since >\\>>. We can bound the remainder term >|^>>>\>> in |)>> using Lemma and Lemma to obtain <\eqnarray*> |>|^>>>\>|\<\|\|\>>|)>>>|||\>Q*\>>h4-m>>\>|\<\|\|\>>|)>>\|Q>\>>h4-m>>\>|\<\|\|\>>>>>||>||*\Q>*\>>h4-m>>\>|\<\|\|\>>|)>>\||>\>>h>4-m>\>|\<\|\|\>>4-m>>|\<\|\|\>>|)>>>>||>||>\>>\>>|h>4-m>,h>4-m>|\>4-m>>*\>*\>*|\<\|\|\>>|)>>>>||>|,\>|\>>\>>|\<\|\|\>>|)>>*|h>,h>|\>|\|> >*\>|\|>|]>>>||>|,\>|\>\>>|\<\|\|\>>|)>>|\>\>>|\<\|\|\>>|)>>|h>,h>|\>|\|>>\>|\|>|]>>>>||>|>,\>|\>\>>|\<\|\|\>>|)>>|\>\>>|\<\|\|\>>|)>>|h>,h>|\>|\|>>>\>|\|>|]>>,>>>> for every \0>, where we used the estimation of Lemma. \ Now using Remark and the fact that |h>,h>|\>=C>-\|)>> we obtain as a final estimation <\equation> |>|^>>>\>|\<\|\|\>>|)>>\\|2>>>-\|)>|\|>> >*\>|\|>|]>. <\remark> 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 > norms with arbitrarily large without resorting to explicit expansions. The measure =>> being either K>>P|]>\\> for Y,|\>>>> or >>\\\> for the other trees, the l.h.s of can be estimated with Lemma to obtain for every >\\>, 0>: <\equation*> |\>>>|)>|\<\|\|\>>|)>>>|>||2>>2|2>q>>|||\>>>|)>|\<\|\|\>>|)>>>|>||2>>2|2>q>>>||\>>>|)>|\<\|\|\>>|)>>>|>||2>>2|2>q>>|||\>>>|)>|\<\|\|\>>|)>>>|>||2>>2|2>q>>>>>> We want to show. In order to do that, we compute <\eqnarray*> |>|^>>-|^>>|)>\>|\<\|\|\>>|)>>>|>||\>Q*>h4-m>-\>h4-m>|)>\>|\<\|\|\>>|)>>>>||>|||>>-\>|)>h4-m>\>|\<\|\|\>>4-m>>|\<\|\|\>>|)>>>>|||||>\>4-m>-h4-m>|)>\>|\<\|\|\>>4-m>>|\<\|\|\>>|)>>.>>>> We focus on the first term above to obtain that it is bounded by <\eqnarray*> |||>>-\>|)>>>-\>>|)>|h4-m>,h>4-m>|\>4-m>>*\>*\>*|\<\|\|\>>|)>>>>||>|,\>|>-\>|)>>>-\>>|)>|\<\|\|\>>|)>>*|h,h>|\>|\|> >*\>|\|>|]>>>||>|,\>|\>-\>|)>>>-\>>|)>|\<\|\|\>>|)>>|h,h>|\>|\|>>\>|\|>|]>>>>||>|>,\>|\>-\>|)>>>-\>>|)>|\<\|\|\>>|)>>|h,h>|\>|\|>>>\>|\|>|]>>>>>> now notice that <\eqnarray*> >>-\>|)>>||>>Y>|)>-F>>Y>|)>>>|||>F>>Y>+\\>>-Y>|)>|]>*>-Y>|)>>>>> and we can estimate |\>>-\>|)>|\<\|\|\>>|)>>> by hypercontractivity and using Lemma as \; <\eqnarray*> |>||F>>Y>+\\>>-Y>|)>|]>|\<\|\|\>>|)>>*|Y>-Y>|\<\|\|\>>|)>>>>||>|>-C>|]> >>||>|>>,>>>> for any \>. The other term can be estimated more easily by <\eqnarray*> ||>,\>|h,h>|\>|\|>>|h-h,h>-h>|\>|\|>>\>|\|>|]>>>>||>|>>>,\>|h,h>|\>|\|>+2\>>\>|\|>|]>>,>>>> and finally obtain <\equation*> |>|^>>-|^>>|)>\>|\<\|\|\>>|)>>\\/2-2\>>,\>|h,h>|\>|\|>+2\>>\>|\|>|]>>. Which yields estimation by applying Lemma as before. In this section we show the decomposition and the bound for the trees >>,>>,>>,>>>. The time regularity of >>> can be obtained with the same technique as in the previous section assuming that >|)>>\C|)>>, and we do not repeat the argument here. Looking at the definitions in it is clear that we can write the Littlewood-Paley blocks of >>>, >>>, >>> and >>> \\0> as:\ <\equation> >>|\>|)>>||,\>\>>*\>>*\,\>->>\|\>|)>,>>|>>|\>|)>>||,\>\>>*\>>*\,\>->>\|\>|)>,>>|>>|\>|)>>||,\>|~>>>*\>>*\,\>->>\|\>|)>,>>|>>|\>|)>>||,\>\>>*\>>*\,\>->>\Y>|\>|)>->>\|\>|)>,>>>>> for |\>=>|)>> where >>> is defined in , |~>>>\\f>|Y>|)>|\>> and the measure ,\>> on \\|)>> is given by <\equation*> \,\>\K>>j>KK|)>P>|)>|]>\|)>\\\\ with =,x|)>> . The first step for decomposing is to expand them using the partial chaos expansion to obtain <\equation> >>\>>>||>>\>>|]>+\Q\ >>\>>|)>,>>|>>\>>>||>>\>>|]>+\Q\ >>\>>|)>,>>|>>\>>>||>>\>>|]>+\\>>\>>|)>|]>+\Q\>>\>>|)>>>|||>>\>>|]>+Y>|)>\>>\>>|]>+Y>|)>\>>\>>|]>+\Q\>>\>>|)>.>>>>> Like the trees appearing in the > model, we expect second-order trees to require a further renormalisation, on top of the Wick ordering. In this section we show how to renormalise by estimating the terms of the type >>\>>|]>> in expansion . We are going to need the following result: <\lemma> We have <\equation*> ,\>Y>|)>\>>\>>|]>\,\>=\Y>>-x|)> G. and <\equation*> ,\>Y>|)>\>>\>>|]>\,\>=\Y>>-x|)>H, where <\eqnarray*> >|>|,x>*j>K|)>K|)>P|)>\>\>|)>>|]>,>>|>|>|,x>*j>K|)>KP-x|)>\>\>|)>>|]>.>>>> <\proof> We have <\eqnarray*> ||,\>Y>|)>\>>\>>|]>\,\>>>|||,x,x,x,x>K>>j>K|)>K|)>P>-x|)>Y>,x|)>\>\>,x-x|)>>|]>>>>> and by change of variables, exploiting the translation invariance of the problem we obtain: <\equation*> =,x,x>K>>|)>Y,x|)>,x>*j>K|)>K|)>P>|)>\>\>,x|)>>|]>. Using the definition of > we have <\equation*> =,x>\Y>,>-x|)>,x>*j>K|)>K|)>P>|)>\>\>,x|)>>|]>. Finally we can write <\equation*> ,\>Y>|)>\>>\>>|]>\,\>=,x>\Y>,>-x|)> G,x|)>. Similar computations holds for the other term, indeed <\eqnarray*> ||,\>Y>|)>\>>\>>|]>\,\>>>|||,x,x,x,x>K>>j>K|)>K|)>P>-x|)>Y>|)>\>\>,x-x|)>>|]>>>|||>K>>|)>Y>|)>x,x,x>*j>K|)>KP>-x|)>\>\>,-x|)>>|]>>>|||\Y>>-x|)>x,x>*j>K|)>KP>-x|)>\>\>,-x|)>>|]>>>|||\Y>>-x|)>H>>>> Using the lemma above and the partial chaos expansion, we can write as: <\eqnarray*> >>|\>|)>>||,\>\Q\ >>\>>|)>*\,\>+\|\>|)> G->>|]>>>|>>|\>|)>>||,\>\Q\ |~>>>\>>|)>*\,\>+\|\>|)> ->>|]>>>|>>|\>|)>>||,\>\Q\ >>\>>|)>*\,\>+\|\>|)> H->>|]>>>|>>|\>|)>>||,\>\Q\>>\>>|)>\,\>+\|\>|)>,\>\>>*\>>|]>*\,\>->>|]>>>|||Y>|\>|)> G+H->>|]>>>|||\>>|\>|)>+\>>|\>|)>>>>> with the additional definitions <\eqnarray*> >|>|,x>j>K|)>KP>-x|)>\|~>>\>|)>>|]>,>>|>>|\>|)>>|>|,x>Y>>-x|)>-\Y>>|)>|]> G,x|)>,>>|>>|\>|)>>|>|Y>>-x|)>-\Y>>|)>|]> H.>>>> To proceed further in the estimation of these integrals, we need to characterise the local behaviour of >>\>>|]>>. From the decomposition we can write <\eqnarray*> >>\>>|]>>|||!!>>|)>*\|Y,\>|\>|Y,\>|\>|]>+!>f>*\|Y,\>|\>|^>>>|]>>>|||!>f>*\|Y,\>|\>|^>>>|]>+\|^>>>|^>>>|]>,>>>> where |Y,\>|\>|Y,\>|\>|]>=!\C>-\|)>> and to bound all other terms we introduce the following result. <\lemma> Under Assumption we have, for every m,n\3> and n>:\ <\equation*> |^>>>|^>>>|]>|\|>\\+i>|h>,h>|\>|\|>\\>|h>,h>|\>|\|>+\>,\\\. Moreover for every m,n\3>, <\equation*> |Y,\>|\>|^>>>|]>|\|>>|>|>|h>,h>|\>|\|>>||4-n,>>||Y,\>|\>|^>>>|]>>||||4-n.>>>>> <\proof> Using the integration by parts formula we decompose <\eqnarray*> |^>>>|^>>>|]>>||\>>h>4-m>|)>\\>>h>4-n>|)>|]>>>|||i!*\\>>*Q\>>|)>|h>,h>|\>.>>>> We can bound the term <\equation*> \+i-5>\\>>*Q\>>|)>\|\>\>>|\<\|\|\>>>|\>\>>|\<\|\|\>>> (see Remark) and therefore, using the bound |h>,h>|\>|\|>=\C>-\|)>\1>, <\equation*> |^>>>|^>>>|]>|\|>\\+i>|h>,h>|\>|\|>\\>|h>,h>|\>|\|>+\>. For the second bound we compute\ <\eqnarray*> |Y,\>|\>|^>>>>|]>>||>m>|)>\\>>h>4-n>|)>|]>>>|||4-n>i!*\|\>m>|)>,Q\>>h>m+4-n-i>|\>m+4-n-i>>|)>.>>>> Since hm>>=0> we obtain |Y,\>|\>|^>>>|]>=0> if 4-n > and <\eqnarray*> |Y,\>|\>|^>>>|]>|\|>>|>|>*\>Q\>>|]>|h>,h>|\>|\|>>>>> if 4-n>, with <\equation*> \>Q\>>|]>\1. Using Lemma we obtain <\eqnarray*> >>\>>|]>>||>|Y,\>|\>+|^>>,\>|)>>|Y,\>|\>+|^>>>|)>|]>=18*>|)> >-\|)>|]>+\|^>>>|^>>>|]>>>>> and thus =18>|)>,x>*j>K|)>K|)>P|)>>-\|)>|]>+> with\ <\equation*> \,x>*j>K|)>K|)>P|)>\|^>>|^>>|)>>|]>. We have the estimation <\equation> |^>>>|^>>>|]>|\|>\\>C>-\|)>>. Similarly <\eqnarray*> |~>>>\>>|]>>||f>*\|Y,\>|\>>|Y,\>|\>+|^>>>|)>|]>=6*\f>f>>-\|)>|]>,>>>> and <\equation> >>\>>|]>|\|>>||>|Y,\>|\>+|^>>>|)>>Y,\>+|^>>>|)>|]>|\|>\>|\|>|Y,\>|\>|^>>>|]>|\|>+|^>>>|^>>>|]>|\|>>>||>|>>|\|>+1|)>C>-\|)>>,>>>>> and <\equation> >>\>>|]>|\|>>||>>|Y\\>|\>+|^>>|)>|]>|\|>\>|\|>\|Y,\>|\>|^>>>|]>+\|^>>>|^>>>|]>>>||>|>|\|>+1|)>C>-\|)>.>>>>> We have by Lemma that for all \> |\|>>|>|>+|)>>>>>>>. > Using estimate together with Lemma, we have that for all \>, \|)>> that |\|>\\>+|)>>>. Furthermore, letting <\equation*> \>>=,x>Y>>-x|)>-\Y>>|)>|]> ,x|)>, we have <\equation*> \>>=6>|)> ,x>Y>>-x|)>-\Y>>|)>|]>P>|]>+>>. The term <\equation*> 6>|)> ,x>Y>>-x|)>-\Y>>|)>|]>P>|]> \ can be shown to converge in law to ,x>X>-x|)>-\X>|)>|]>P |]>> in >>> with the standard techniques used in the analysis of the > model. For all \0> sufficiently small we have the bounds,\ <\eqnarray*> |\>>|\<\|\|\>>>>+|\>>|\<\|\|\>>>>>|>|>|Y>|\<\|\|\>>>>>2+2\|)>>+|)>-5>>>||>|>|Y>|\<\|\|\>>>>>2+2\|)>>,>>>> which shows that these remainders go to zero in >> as \0>, since |Y>|\<\|\|\>>>>>> is bounded in |)>>. Moreover, it is easy to see that >>->>> is bounded in ,|)>>. Note that <\equation*> G=P\>\>>|]>=18>|)> P>|]>+P\|^>>|^>>>|]>, <\equation*> H=P\>\>>|]>=P\>|^>>>|]>. Here we used the fact that <\equation*> j>K|)>K=K|)>K=\|)>, since K|)>K=0>, where \1>. This is readily seen in Fourier space taking into account the support properties of the Littlewood-Paley blocks. Now, <\equation*> P\|^>>|^>>>|]>,P\>|^>>>|]>, converge to finite constants due to the bounds and and by Lemma P>|]>\|\|>.> Finally, from we have <\equation*> ,\>\>>*\>>|]>*\,\>=P\>\>>|]>=O|)>. Indeed Lemma again yields PC>\1>. Thus ,\>\>>*\>>|]>*\,\>> gives a diverging constant which depends on all the >|)>>. Making the choice to define the renormalisation constants >> as in eq. we cancel exactly these contributions which are either >|)>>> dependent and/or diverging. In particular we verify that we can satisfy the constraint.\ Finally, noting that Q\=|)>> and Q\=-J|)>> we can write the trees of as <\equation> >>|\>|)>>||>|)>,\> |)>|Y,\>|\>|Y,\>|\>|)>*\,\>\>>|||>|3>,\>\Q\ |^>>>|Y,\>|\>+|Y,\>|\>|^>>>|)>*\,\>+\,\>\Q\ |^>>>|^>>>|)>*\,\> ,>>|>>|\>|)>>||>f>*f>,\> |)>|Y,\>|\>|Y,\>|\>|)>*\,\>+\,\>\Q\ |~>>>|^>>>|)>*\,\> ,>>|>>|\>|)>>||>|)>,\>|Y,\>|\>Y>*\,\>>>|||>|6>,\>\Q\ |^>>>Y,\>+|Y,\>|\>|^>>>|)>*\,\>+,\>\Q\ |^>>>|^>>>|)>*\,\> ,>>|>>|\>|)>>||>|)>,\> |)>|Y,\>|\>|Y,\>|\>|)>*\,\>+\>>|\>|)>>>|||>|)> ,x>Y>>-x|)>-\Y>>|)>|]>P>|]>+>>+>>|||,\>\Q\|^>>>|Y,\>|\>+|Y,\>|\>|^>>>|)>*\,\>+,\>\Q\|^>>>|^>>>|)>*\,\> ,>>>>> with \>>|\>|)>\O>>+2\|)>>|)>> and \>>|\>|)>+\>>|\>|)>\O>>>*2+2\|)>>|)>>.\ Comparing with the canonical trees in we can identify the remainder terms >>> that need to converge to zero in order for Y>>> to converge to X>>. In this section we show that the remainder terms identified in converge to zero in probability. First notice that we can bound> using Lemmas, and as <\equation*> |\Q,\>\ >>\>>|)>*\,\>|\<\|\|\>>>\|,\>\ >>\>>|)>*\,\>|\<\|\|\>>>. Therefore, taking the derivatives in > we see that it suffices to bound in k+\>|)>> the term <\equation> \>>\>>hk>>\h\>>=>|!>*|Y>|\>+|^>>>|]>>|!>*|Y>|\>+|^>>>|]>hk>>\h>\> for and k+\\2>. This yields some constraints on the number of branches of trees: <\equation> >>>|>|n+\=3>>|>>>|>|=1>>|>>>|>|n+\=2>>|>>>|>|m+k=4,n+\=3.>>>>> In , the terms proportional to >|)>> will generate finite contributions in the limit. In particular it is easy to see that they converge respectively to >|)>,>|)>,\>\>,>|)>>. All other terms will vanish in probability, verifying, due to estimates we are going to establish now.\ We consider the terms proportional to |^>>>|^>>>>, all the other similar terms featuring at least one remainder |^>>>> 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, allows us to rewrite products of divergencies in the form \> as a sum of divergences >>, which are then easy to estimate in > using Lemma. We obtain <\eqnarray*> |^>>>|^>>>>||\>>hm>>|)>\\>>h>n>|)>>>|||\I>C>*\|\Q\>>h>m>,\Q\>>h>n>|\>q+r-i>>|)>>>|||\I>C>*\-i>*\|\|)>h>m+r-i>,\|)>hn+q-i>>|\>q+r-i>>|)>>>>> with \\:0\q\m,0\r\n,0\i\q\r|}>> and <\equation*> \|)>\\>*Q \>>. By Remark, for every 1> and \Dom \> we can write |)>hm>=\\hm>|)>>, and therefore <\equation*> |^>>>|^>>>hk>>\h\>>*\,\>= <\equation*> =C>*\>*\*\|)>\|)>*h>m-q>\hn-r>>\hk>>\h\>>|h>,h>|\>|\|>\,\>. The following result allows us to estimate the quantity above in >k+\>|)>>. <\lemma> Under Assumption we have the bound\ <\equation*> |\*\|)>\|)>*h>m-q>\hn-r>>\hk>>\h\>>|h>,h>|\>|\|>\,\>|\<\|\|\>>k+\>|)>> <\equation*> \|h>,h>|\>|\|>|h>,h>|\>|\|>-r>|h>,h>|\>|\|>|h>,h>|\>|\|>,\>|\|>,\>|\|>. <\proof> Thanks to Lemma the integral can be estimated with <\equation*> j>|\\|)>*\\|)>*h>m-q>\hn-r>>\hk>>\h\>|h>,h>|\>|\|>\,\>|\<\|\|\>>>, with m+k+n+\-q-r+j>>. We have that |\|\<\|\|\>>k+\>|)>>=||\|\<\|\|\>>k+\>>|\<\|\|\>>>> and therefore we can bound each term in the sum above as <\eqnarray*> |>|||\\|)>\\|)>,\\|)>\\|)>|\>j>>|\<\|\|\>>>|\>\>>|||||h>,h>|\>|\|>|h>,h>|\>|\|>-r>|h>,h>|\>|\|>|h>,h>|\>|\|>,\>|\|>,\>|\|>|)>>>>> Using Hölder's inequality we get the estimate <\eqnarray*> ||||\\|)>\\|)>,\\|)>\\|)>|\>j>>|\<\|\|\>>>>>||>|||\\|)>,\\|)>|\>h>>|\<\|\|\>>>||\\|)>,\\|)>|\>j-h>>|\<\|\|\>>>>>>> Now to bound terms of the type ||\\|)>,\\|)>|\>h>>|\<\|\|\>>>> we consider the cases m> and m>. In the first region we use Lemma to estimate <\eqnarray*> ||\\|)>,\\|)>|\>h>>|\<\|\|\>>>>|>||\Q*\>\>>|\<\|\|\>>h>|)>>|\Q*\>\>>|\<\|\|\>>h>|)>>>>||>||\>\>>|\<\|\|\>>>|\>\>>|\<\|\|\>>>.>>>> If m> we first commute derivatives in the expression Q> using formula and then apply Lemma to obtain the bound <\equation*> |\Q*\>\>>|\<\|\|\>>h>|)>>\|\\>\>>|\<\|\|\>>>\|\>\>>|\<\|\|\>>>. Thus, we need to know >>> up to the order \\\> to perform this estimates. From Lemma we obtain \\>: <\equation*> |\>*\*\|)>\|)>*h>m-q>\hn-r>>\hk>>\h\>>|h>,h>|\>|\|>\,\>|\<\|\|\>>k+\>|)>> <\equation*> \\>>|h>,h>|\>|\|>|h>,h>|\>|\|>-r>|h>,h>|\>|\|>|h>,h>|\>|\|>,\>|\|>,\>|\|>|)>> <\equation*> \\|2>>|)>>. Our aim now is to estimate the quantity >. The idea is to use the bound |h>,h>|\>|\|>=\C>-\|)>\1> to cancel strategically some of the covariances |h>,h>|\>|\|>>. We will consider three regions:\ If 2> we use the bounds <\equation*> \|h>,h>|\>|\|>|h>,h>|\>|\|>\\|h>,h>|\>|\|>|h>,h>|\>|\|> and then (we suppose 2>) <\equation*> \>|h>,h>|\>|\|>-r>\|h>,h>|\>|\|>-2+\> to obtain\ <\eqnarray*> >|>|>|h>,h>|\>|\|>|h>,h>|\>|\|>-2>|h>,h>|\>|\|>|h>,h>|\>|\|>,\>|\|>,\>|\|>>>||>||h>,h>|\>|\|>|h>,h>|\>|\|>-2+\>|h>,h>|\>|\|>,\>|\|>,\>|\|>.>>>> (If vice-versa 2> it suffices to put > on the term |h>,h>|\>|\|>>>.) Notice that in this case 0.>\ In the case if 2> we estimate like before to obtain <\eqnarray*> >|>|>|h>,h>|\>|\|>|h>,h>|\>|\|>-r>|h>,h>|\>|\|>>|h>,h>|\>|\|>>,\>|\|>,\>|\|>>>||>||h>,h>|\>|\|>|h>,h>|\>|\|>-r>|h>,h>|\>|\|>>,\>|\|>,\>|\|>.>>>> Note that -1\0> and -3\-1> here. If we bound\ <\eqnarray*> >|>||h>,h>|\>|\|>|h>,h>|\>|\|>-r-2>|h>,h>|\>|\|>|2>>|h>,h>|\>|\|>|2>>,\>|\|>,\>|\|>>>>> and note that /2\0>, \0>, -r-2\0>. Finally if we can only have and thus <\eqnarray*> >|>|>|h>,h>|\>|\|>>|h>,h>|\>|\|>>|h>,h>|\>|\|>>,\>|\|>,\>|\|>>>||>||h>,h>|\>|\|>+m+k-6>|h>,h>|\>|\|>>|h>,h>|\>|\|>>,\>|\|>,\>|\|>>>>> \; If 4> we bound first <\equation*> \-4>|h>,h>|\>|\|>|h>,h>|\>|\|>\|h>,h>|\>|\|>|2>>|h>,h>|\>|\|>|2>> \ (note that -4\\>) to obtain: <\equation*> \\\>|h>,h>|\>|\|>|h>,h>|\>|\|>-r>|h>,h>|\>|\|>|2>>|h>,h>|\>|\|>|2>>,\>|\|>,\>|\|> Now in the cases =3> and =2> \ we can just write >=\\>> and cancel the corresponding number of covariances to obtain <\eqnarray*> >|>||h>,h>|\>|\|>>|h>,h>|\>|\|>|2>>|h>,h>|\>|\|>|2>>,\>|\|>,\>|\|>>>>> while for the case =3> \ we have either \1> or 1> and therefore with one of the following bounds <\eqnarray*> \-r-\>|h>,h>|\>|\|>|h>,h>|\>|\|>-r>>|>||h>,h>|\>|\|>|h>,h>|\>|\|>>>>|\-1-r-\>|h>,h>|\>|\|>|h>,h>|\>|\|>-r>>|>||h>,h>|\>|\|>>>>>> we obtain the estimates <\eqnarray*> >|>||h>,h>|\>|\|>>|h>,h>|\>|\|>|2>>|h>,h>|\>|\|>|2>>,\>|\|>,\>|\|>>>|>|>||h>,h>|\>|\|>|h>,h>|\>|\|>>|h>,h>|\>|\|>|2>>|h>,h>|\>|\|>|2>>,\>|\|>,\>|\|>.>>>> We can use directly Lemma to obtain a final estimate of , , , . For ,> and notice that the integral over ,\> is finite and thus the whole quantity is proportional to |h>,h>|\>|\|>>. Globally, we have <\equation*> \\2-6|)>q> as needed to prove. Lastly, by taking one more derivative of >> as done in Section, we can show for >=>>,>>,>>,>>>, thus proving that >\0> in /2>-\>> in probability \\|\|>>. In this section we recall the notations and the basic results of paracontrolled calculus introduced in without proofs. For more details on Besov spaces, Littlewood\UPaley theory, and Bony's paraproduct the reader can refer to the nice recent monograph. Throughout the paper, we use the notation b> if there exists a constant 0>, independent of the variables under consideration, such that c\b>, and we write b> if b> and a>. If we want to emphasize the dependence of on the variable , then we write \b>. For index variables and of Littlewood-Paley decompositions (see below) we write j> if there exists >, independent of , such that j+N>, and we write j> if j> and i>. An is a set of the form =:a\\b|}>> for some a\b>. A is a set of the form =:\b|}>>. If is a map from \> to the linear space , then we write =f-f>. For L|)>> we write |f|\<\|\|\>>|)>>\>|\|>\x>. Given a Banach space with norm |\|\<\|\|\>>> and 0>, we write *X=C,X|)>> for the space of continuous maps from > to , equipped with the supremum norm |\|\>*X>>, and we set ,X|)>>. For \> we also define >X> as the space of >-Hölder continuous functions from > to , endowed with the seminorm |f|\<\|\|\>>>*X>=sups\t\T>|f-f|\<\|\|\>>/>>, and we write >X> for the space of locally >-Hölder continuous functions from > to . For \0>, we define <\equation*> \>X=,X|)> : |v|\<\|\|\>>>X>=|t\t>v|\<\|\|\>>X>\\|}>. The space of distributions on the torus is denoted by |)>> or >. The Fourier transform is defined with the normalization <\equation*> u==>e|k,x|\>>u\x,k\\, so that the inverse Fourier transform is given by v=|)>e|k,x|\>>v>. Throughout the paper, ,\|)>> will denote a dyadic partition of unity such that \|)>|)>\supp\|)>|)>=\> for \1>. \ The family of operators |)>-1>> will denote the Littlewood-Paley projections associated to this partition of unity, that is u=u|)>> and =\\|)>u|)>> for 0>. We also use the notation =j>\>. The Hölder-Besov space >,\>,\|)>> for \\> will be denoted by >> and equipped with the norm <\equation*> |f|\>>=|f|\>>,\>>=sup-1>>|\f|\<\|\|\>>>|)>>|)>. If is in -\>> for all \0>, then we write ->>. We let > the kernel of > so that f>|)>=>K>,q>f\x>. For \>, we define the space >=C/2>L>\C>>, equipped with the norm <\equation*> |f|\<\|\|\>>>>=max|f|\<\|\|\>>/2>L>>,|f|\<\|\|\>>>>|}>. The notation is chosen to be reminiscent of =\-\>, by which we will always denote the heat operator with periodic boundary conditions on >. We also write >=C/2>L>\C>>. When working with irregular initial conditions, we will need to consider explosive spaces of parabolic type. For \0>, \>, and 0> we define the norm <\equation*> |f|\<\|\|\>>,\>>=max|t\t>f|\<\|\|\>>/2>L>>,|f|\<\|\|\>>>>>|}> and the space ,\>=\\ : |f|\<\|\|\>>,\>>\\|}>>. In particular, we have >=>>. We introduce the linear operator ,|)>|)>\C,|)>|)>> given by <\equation*> I f=Pf\s, where |)>0>> is the heat semigroup. Standard estimates in exposive spaces that are summarized in the following Lemma. <\lemma> Let \> and \>. Then <\equation> |I f|\<\|\|\>>,\>>\|f|\<\|\|\>>>-2>> for all 0>. If further \-\>, then <\equation> |s\Pu|\<\|\|\>>+\|)>/2,\>>\|u|\<\|\|\>>>>. For all \\>, \>, and 0> we have <\equation> |I f|\<\|\|\>>>>>\|f|\<\|\|\>>>-2>> For all \>, \>, \\2\|)>>, 0> and ,\>> with =0> we have <\equation> |f|\<\|\|\>>-\/2,\-\>>\|f|\<\|\|\>>,\>> Proofs can be found in. We need also some well known estimates for the solutions of the heat equation with sources in space\Utime Lebesgue spaces. <\lemma> Let \\> and L*B>>>, then for every \> we have C/q>*+2*|)>-+d|)>/p>> with <\equation*> |I f|\<\|\|\>>/q>*+2*|)>-+d|)>/p>>\|f|\<\|\|\>>*B>>>, with +=1>. Moreover, for every \\\1-1/p> and every \2-5/p+\> \ we have\ <\equation*> |I f|\<\|\|\>>,\>>\|v\v>f|\<\|\|\>>B>>> <\proof> We only show the second inequality as the first one is easier and obtained with similar techniques. Let , we have <\eqnarray*> >|\*u|\<\|\|\>>>>>|>|*2sq>*e*t*>*\s|]>s*p>|\*f|\<\|\|\>>>*\*s|]>>>||,q>>|2*s*p>|\*f|\<\|\|\>>>*\*s|]>>>>> which allows us to bound |I f|\<\|\|\>>>>>>. In order to estimate |t\t>I f|\<\|\|\>>/2>L>>> we write\ <\eqnarray*> |t>\*u-s>\*u|\<\|\|\>>>>>|>|v-1>|\u|\<\|\|\>>>>\v+2/p>|v\v>\f|\<\|\|\>>>>>||||v>\*f*\*v|\<\|\|\>>>>>>>> We can estimate the first term as <\equation*> v-1>|\u|\<\|\|\>>>>\v\2/p>|v\v>\f|\<\|\|\>>>v-\-1>\v. For the third term we have\ <\eqnarray*> |v>\*f*\*v|\<\|\|\>>>>>|>|\v|]>v*p>|\*f|\<\|\|\>>>>*\v|]>>>||>|**|v\v>\f|\<\|\|\>>>>>>> We obtain then if \1> <\equation*> |t>\*u-s>\*u|\<\|\|\>>>>\2|v\v>\f|\<\|\|\>>> and if \1> we just use the trivial estimate <\equation*> |t>\*u-s>\*u|\<\|\|\>>>>\22*|v\v>\f|\<\|\|\>>>\2|v\v>\f|\<\|\|\>>>. Therefore, for every \>: <\equation*> |t>\*u-s>\*u|\<\|\|\>>>>\2-2|)>*i>*2*i*/q>*/q>*|v\v>\f|\<\|\|\>>>. Choosing /q=\/2> we obtain the desired estimate. Paraproducts are bilinear operations introduced by Bony 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 Littlewood\UPaley blocks, the product of two distributions and can be decomposed as <\equation*> f*g=f\g+f\g+f\g, where <\equation*> f\g=g\f\-1>\f\gf\g\\1>\f\g. This decomposition behaves nicely with respect to Littlewood\UPaley theory. We call g> and g> , and g> the term. We use the notation g=f\g+f\g>. The basic result about these bilinear operations is given by the following estimates, essentially due to Bony and Meyer. <\lemma> For any \> we have <\equation> |f\g|\<\|\|\>>>>\>|f|\<\|\|\>>>>|g|\<\|\|\>>>>, and for \0> furthermore <\equation> |f\g|\<\|\|\>>+\>>\,\>|f|\<\|\|\>>>>|g|\<\|\|\>>>>. For +\\0> we have <\equation> |f\g|\<\|\|\>>+\>>\,\>|f|\<\|\|\>>>>|g|\<\|\|\>>>>. A natural corollary is that the product of two elements >> and >> is well defined as soon as +\\0>, and that it belongs to >>, where =min ,\,\+\|}>>. \; We will also need the several commutator lemmas: <\lemma> Let \0>, \\>, and let >>, and >>. Then <\equation*> |f\h|)>-\h|\<\|\|\>>+\>>\|f|\<\|\|\>>>>|g|\<\|\|\>>>>|h|\<\|\|\>>>>. 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 space\Utime distributions belonging to the scale of spaces >|)>\\>> for some 0>. To do so efficiently, we will use a modified paraproduct which introduces some smoothing in the time variable that is tuned to the parabolic scaling. Let therefore \C>,\|)>> be nonnegative with compact support contained in > and with total mass , and define for all -1> the operator <\equation*> Q:C>\C>,Qf=>2\|)>f\s. We will often apply > and other operators on >> to functions C>> which we then simply extend from > to > by considering \T|)>>. With the help of >, we define a modified paraproduct <\equation*> f\g\Sf|)>\g for C,|)>|)>>. We collect in the following lemma various estimates for the modified paraproduct, proofs are again in. <\lemma> \; <\enumerate-alpha> For any \> and \> we have <\equation> t>|f\g|\<\|\|\>>>>\|f|\<\|\|\>>>L>>|g|\<\|\|\>>>>, for all 0>, and for \0> furthermore <\equation> t>|f\g|\<\|\|\>>+\>>\|f|\<\|\|\>>>>>|g|\<\|\|\>>>>. Let ,\\>, \>, 0>, and let ,\>>, C>>, and g\C-2>>. Then <\equation> |f\g|\<\|\|\>>,\>>\|f|\<\|\|\>>,\>>|g|\<\|\|\>>>>+|g|\<\|\|\>>-2>>|)>. Let \>, \,> 0>, and let >>. Then for all \|]>> we have <\equation> |||f|\<\|\|\>>>>>|>||f|\<\|\|\>>>>+T-\|)>/2>|f|\<\|\|\>>>>,>>||f|\<\|\|\>>,\>>>|>|-\|)>/2>|f|\<\|\|\>>,\>>.>>>>> 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> <\enumerate-alpha> For ,\,\\\> such that +\+\\0> and \> there exists bounded trilinear maps\ <\equation*> com,>:>\>\>\+\+\>, such that for smooth they satisfy <\equation> com=g|)>\h-f*h|)>. <\equation> >=g|)>\h-f*h|)>. Let \>, \\>, and \>. Then the bilinear maps <\equation> com:=f\g-f\g. <\equation> com:=,f\|]>g\g|)>-f\g. have the bounds <\equation> t>|com|\<\|\|\>>+\>\|f|\<\|\|\>>,\>>|g|\<\|\|\>>>>,t\0. as well as <\equation> t>|com|\<\|\|\>>+\-2>\|f|\<\|\|\>>,\>>|g|\<\|\|\>>>>,t\0. Proofs can be found in.\ 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 Section10.3 of for an extensive treatment of this subject. First of all we need to characterize the local behaviour of the heat kernel > and of the covariance >> of the Gaussian field >>. <\lemma> The heat kernel \\\\> defined by =t|)>>e|4t>>*\0>> has the bound <\equation*> |)>|\|>\+|)>. Let \> a multi-index with =2k+k+\+k>. Then for every multi-index \2> we have:\ <\equation*> P|\|>\+|)>>. <\remark> In this article we use a slightly different version of the heat kernel, namely\ <\equation*> P=t|)>>e|4t>>*e*\0> in order to have that =>>Pv\s\y> is the stationary solution to X=-X+v>. However, every estimate \ remains trivially valid in this setting. <\proof> \; <\equation*> |\|>+|)>\|)>|]>e|4>>=|\|>|)>e|\|>|4>>\+\ In the same way we prove that P|\|>\+|)>>, >P|\|>\+|)>> and >\>P|\|>\+|)>>. We recall a special case of Lemma10.14 of , which is enough for our purpose. We use the notation |\|\>\+|)>> for =\\\\>. <\lemma> Let \\\\> smooth, integrable at infinity and such that |)>|\|>\|\|\>>> and |)>|\|>\|\|\>>> in a ball \ \\\:|\|\>\1,\\0|}>>. Then if ,\\> and +\+5\0> we have <\equation*> g|)>|\|>\|\|\>+\+5> in a ball centered in the origin. Moreover, if ,\\> and \+\+5\1> and for every multi-index \2> we have f|)>|\|>\|\|\>->> and g|)>|\|>\|\|\>->>, then <\equation*> g|)>-f\g|\|>\|\|\>+\+5> in a ball centered in the origin. <\remark> The covariance >> of >> can be written as >=P\>\P> with >\\>\>|)>>. Recall from the introduction that >=\>t,\x|)>> where >> is the covariance of the Gaussian process > defined on \/\|)>>, and >=\> if \1> and otherwhise (so that the family of functions >> is bounded uniformly on > by a >> function). \ Then there exists a family of functions >> defined on \/\|)>> such that >=\C>t,\x|)>> and >=>\P|]>>. <\lemma> The covariance >> has the bound, for every multi-index \2>: <\equation*> C>|\|>\+|)>>. Moreover, we have\ <\equation*> \+1>C>|\|>\1 <\proof> The first bound is obtaned directly from Lemma and Lemma. Indeed, since by hypothesis >> has compact support, it is easy to see that >|\|>\+|)>>. The second bound is obtained by a simple change of variables in the convolution defining >>.\ <\lemma> We have P>|]>\|\|>> and for every 3> P>|]>\1>. <\proof> From the fact that s>*x|)>=\P> together with Remark we obtain\ <\equation*> \\>P>|]>\s\x\|)>>P>|]>\s\x\|\|> with =\\:|\|\>\R,\\0|}>> a \Pparabolic\Q ball centered in the origin. The second estimation is obtained in the same way. <\lemma> For >, >, define for ,\\\\\> <\equation*> I\>-\|)>|\|> >*\>|\|>,\>-\|)>|\|> |~>>*|~>>|\|> with >\K>>\\\> and |~>>\K>>P|]>d\> for =>. Then <\equation*> I\2and\2q>. <\proof> The estimation of > is easily obtained by Lemma and a change of variables. For > observe that for every 0> <\equation*> |~>>=K>>-P>-y|)>|)>|]>d\ and then we can apply Lemma to obtain the result. <\lemma> We have for every \> <\equation*> sup\>>-C>|\|>\\>> <\proof> It is easy to obtain by interpolation knowing that C>|\|>\\> from Lemma. <\lemma> We have for every \3>\ <\equation*> j>KP\y|\|>*|\|>|+t|)>>>\y\+t|)>>> <\proof> We will show that\ <\equation> KP\y|\|>\2+t+2|)>, and that <\equation> |\|>|+t|)>>>\y\+t+2|)>>, from which we deduce that <\equation*> j>KP\y|\|>*|\|>|+t|)>>>\y\|+t+2|)>>>. Bounding the sum over with an integral, we conclude <\equation*> \|\>|+t+\|)>>>=+t|)>>>+t|)>>\||)>>>\+t|)>>>. Let us show. We want to estimate <\eqnarray*> ||KP\y=K-P|]>\y>>|||\\K|)>|]>\y>>|>|>|\\K|\|>|)>|\|>\y\2\\|\|>2y|)>|\|>\y>>||>|\\|\|>2y|\|>/t>|t>\y>>>> where <\equation*> |\|>=/t>|t>>|\|>\C/t>|t>. When \2,> we have <\equation*> \2*t\2+t+2|)>. When \ \t,> we estimate simply\ <\equation*> \|\|>P\y\2\2+t+2|)>. When \2,t> we have instead that either \2\2> or \2\2>. In the first region 2y|\|>\c> so <\equation*> \2\\|\|>/t>|t>\y\2/t>|t>\2\2+t+2|)>. while in the second region \2/|)>>, then |\|>\|\|>F/|)>|)>> where is a rapidly decreasing function and in this region <\eqnarray*> >|>|\\ F/|)>|)>2y|\|>/t>|t>\y>>||>|\\ F/|)>|)>2y|\|>/t>|t2y|\|>>\y\2\\ F/|)>|)>|\>/t>|t>\y>>||>||>\\ F/|)>|)>|\>\|>\2+t+2|)>.>>>> So we conclude that holds. Let us turn to. When \2,> we have <\equation*> |\|>|+t|)>>>\y\>>|\|>\y\>>\+t+2|)>>. When \ \t,> we estimate\ <\equation*> |\|>|+t|)>>>\y\2i>|\|>|x+y|\|>>>\y\2 sup|\|>|>>\y\2\+t+2|)>>, and finally when \2,t> we have either \ 2> or \ 2>. In the first region y|\|>\c> so <\equation*> |\|>|+t|)>>>\y\|\|>|y|\|>>>\y\>\+t+2|)>>, while in the second \2/2>, then |\|>\|\|>F/2|)>> where is another rapidly decreasing function and in this region <\equation*> |\|>|+t|)>>>\y\F/2|)>|\|>|y|\|>>>\y\2i>F/2|)>\>\+t+2|)>>, concluding our argument. <\lemma> For >, \> define <\equation*> I\,\>C>-\|)>*C>-\|)>>*C>-\|)>*C>-\|)>,\>|\|>,\>|\|>, with ,\>> for |\>=>|)>>, =,x|)>> defined as <\equation*> \,\>\K>>j>KK|)>P>|)>|]>\|)>\\\\. If =0>, m+k-2\5>, > and > we have the estimate <\equation*> I\2q>. If ,+m-2|)>\>, -4\> and +m+n-4\> we have the estimate <\equation*> I\2+m+n-4|)>q>. <\proof> Observe that <\eqnarray*> ,\>>||j>K>>|)>KK|)>>|)>-P>>-x|)>|)>|]>\|)>\\\\>>|||j>>>-K>>|)>|)>KK|)>P>|)>|]>\|)>\\\\>>|||>>|)>>-x|)>-P>>-x|)>|]>\|)>\\\\>>|||j>>>-K>>|)>|]>KK|)>P>|)>|]>\|)>\\\\>>||||\>,\>+|^>,\>>>>> where in the first line we used K=0> and the fact that K|)>K|)>=0> if \1> and KK|)>=\-y|)>\-x|)>>. Now the estimation of the term <\equation*> >\,\>C>-\|)>*C>-\|)>>*C>-\|)>*C>-\|)>|\>,\>|\|>|\>,\>|\|>, with |\>,\>=K>>|)>>-x|)>-P>>-x|)>|]>\|)>\\\\> can be done with Lemma and gives the expected result. The integral <\equation*> \,\>C>-\|)>*C>-\|)>>*C>-\|)>*C>-\|)>|^>,\>|\|>|^>,\>|\|>, with |^>,\>=j>>>-K>>|)>|]>KK|)>P>|)>|]>\|)>\\\\> can be estimated by multiple changes of variables. We have >>-K>>|)>=2-x|)>K-x|)>\-2>-x|)>|)>\\>, and by the scaling properties of >> and >, namely >s,2x|)>\2C>> and s>x|)>\2P> given by Lemma and Lemma, we obtain easily the bound on > by rescaling the integral. Let > be the Malliavin derivative, > the divergence (defined as the adjoint of >) and > the Ornstein-Uhlenbeck semigroup. We refer to for an extensive discussion on these operators. Call |}>H>> the isonormal Gaussian process indexed by some real separable Hilbert space. For every \L|)>>, > can be written via the well-known Mehler's formula for =F|)>>: <\equation> P\=PF|)>=\>>W+|)>W|)>|]> where |}>H>> is an independent copy of . In our case we will consider the Gaussian process >> indexed by > \\\\>, with \\|)>>, with >> defined as in Section. By a direct calculation we obtain <\equation> \ \ P\=e*\W+|)>*W|)>|]>h=eP \ \ \. This gives a commutation result between the Malliavin derivative > and the generator of the Ornstein-Uhlenbeck semigroup (defined by =e>, recall that \> (, Proposition1.4.3)). Indeed, let > such that |)>=0>, then we have for every \0> and every 0>: <\equation> \ >\=\ |)>>>t-1>*ee\ \t=|)>>>t-1>*et>P \ \=>\ \ \ and the same works for every > (not necessarily centered) if 0>. It is well-known (see ) that acts on square integrable functions > as <\equation*> L \=->n*J\ where \> is the projection of > on the n-th Wiener chaos. We can define > by its action on -th order chaoses as J\=J \> n\0,j\1>. The results recalled above allow for the following partial chaos expansion : <\lemma> Let \L|)>>. Then for every \\\>: <\equation> \=*\JD\+\Q\\ with =>. <\proof> We have for any \L|)>>: <\equation*> \-J\=L L-J\|)>=-\ \ \ L-J\|)>=\*\ \ \\ where we used and the fact that the Malliavin derivative of a constant is zero. This yields <\equation> \=\|)>+\*\ \ \ Iterating this formula up to an order and using the fact that =J> we obtain the result. Notice that the lemma above implies Q\\=\-J|)>\>. In order to have > estimations of the terms QD\> generated by expansion , we will need the following lemmas:\ <\lemma> , Proposition1.5.7>Let be a real separable Hilbert space. For every 1> and every \,k\q> and every \\V|)>> we have\ <\equation*> |\|\<\|\|\>>>\|u|\<\|\|\>>\V|)>> <\remark> Using Lemma we can state a simplified version of Lemma in the case where V> is deterministic. Let be a real separable Hilbert space. For every V> and every \q>|)>> with \> we have F\Dom \> and\ <\equation*> \F=\F|)>. We can prove this formula as follows. First notice that for every smooth \> and every smooth \q>|)>> we have <\equation*> \|\F|)>,G|\>|)>=\|u\F,\ G|\>q>\V>|)>=\|\F, G|\>|)>. Now since F|)>=\u\F> and \q>|)>> we have that F\\q>\V|)>>. Lemma yields the bound |\F|)>|\<\|\|\>>>\|u\F|\<\|\|\>>q>\V|)>>> which allows to pass to the limit for and F|)>> in >. <\lemma> For every \\\>, the operator => is bounded in > for every p\\>.\ <\proof> We have \= \=>J\=\|)>+T>\> with >\=>\J\> and =0>. Then the operator >> satisfies the hypotheses of Theorem1.4.2 of , with =h|)>> and => analytic in a neighbourhood of 0. Therefore\ <\equation*> |Q\|\<\|\|\>>>\\|)>+|T>\|\<\|\|\>>>\|\|\<\|\|\>>> and the result follows applying repeatedly this inequality. <\lemma> Let \\\> and 1>. There exists a finite constant > such that for every \L>: <\equation*> |\*> \|\<\|\|\>>>\c|\|\<\|\|\>>> (where the operator >> is defined on every > polynomial in |)>,\,W|)>> and can be extended by density on >). <\proof> First notice that we can suppose w.l.o.g. |)>=0> thanks to the commutation . Therefore we can write *>> as <\equation*> \>=\ > \ with >. We decompose the second part as: <\equation*> -C >\=>|)>>J\=T>\ With >\\>\J\>. We can apply Theorem1.4.2 of to show that >> is bounded in >, indeed =h> and => which is analytic in a neighbourhood of 0. Finally, we can apply Proposition1.5.2 of to show that > is bounded in >, 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 as sums of iterated Skorohod integrals.\ <\lemma> \ Let |)>|)>hm>>, |)>|)>hn>> with ,F\C|)>> such that \m>|)>>, \n>|)>>. Then: <\equation> \\=n>i!*\|\u,\v|\>q+r+i>>|)>. And also\ <\equation> \\=r>i!*\|\u,\v|\>q+r-i>>|)>. <\proof> We have using Cauchy-Schwarz inequality and Lemma that |D\,\|\>r>>\L,Hm-j-r>|)>> for every r+j\m>. Then we can apply Lemma to obtain <\equation*> \\=\|\ \,v|\>r>>|)>. Then using the commutation formula we rewrite the r.h.s. as\ <\equation*> \\=m>i!*\|\u|)>,v|\>r>>|)>. We write |\u|)>,v|\>r>>=|\u|)>,F|)>|)>hr>|\>r>>*hn-r>> and verify in the same way as before that |\u|)>,F|)>|)>hr>|\>r>>> satisfies the hypotheses of Lemma. We obtain <\eqnarray*> \>||m>i!*\|\u|)>,F|)>|)>hr>|\>r>>*hn-r>|]>>>|||m>i!=0>>\>|\u,\>F|)>|)>hr>|\>r+\>>|)>hn-r>|]>>>|||m>=0>>i!\>|\u,\>v|\>r+\>>|)>>>>> where we used the fact that |)>hn-r>=\\hn-r>|)>> for \Dom \>, as seen in Remark. Call +i>, then\ <\equation*> \\=n>i!\|\u,\v|\>q+r-i>>|)>, and noting that <\equation*> \; <\equation*> =!m!|!!i!!>=!q!!>=, we have the nicer symmetric expression <\equation*> \\=n>i!*\|\u,\v|\>q+r-i>>|)>. <\equation*> \; Finally we perform the change of variables q>, r> to get <\equation*> \\=n>i!*\|\u,\v|\>q+r+i>>|)>. The second formula is a straightforward change of indexes of the first one. <\remark> Our choice of giving two distinct but closely related formulas in Lemma is due to the fact that the first formula has a more evident \Pphysical meaning\Q. Indeed, vertices and (being non-polynomial) have an infinite chaos decomposition, which can be represented as having infinite \Plegs\Q in a Feynman-like diagram. It is apparent that the index in first equation denotes contractions between the already existing legs of the vertices and that 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 legs overall uncontracted which are arguments to the iterated Skorokhod integral and would be contracted with other composite vertices in the > estimates. The second formula however, is more practical in the calculations. We give below the results we used to prove Lemma. <\lemma> , Lemma2.1>Let 1>, \>, Dom|)>> and symmetric. Assume also that 0\r+j\q> |DF,\|\>r>>\L,Hq-r-j>|)>>. Then 0\r\q> |DF,u|\>\Dom|)>> and <\equation*> F*\=\|\F,u|\>r>>|)>. <\remark> Note that <\equation*> \k>|)>=|W|\> where |\|\>> stands for the Wick product. Indeed F\\> we know that n>|)>F|]>=\*hn-1>|]>> using the definition of > and therefore n>|)>=\hn-1>|)>>. We have also hn-1>|)>=\n-1>|)>W-|h,h|\>\n-2>|)>> using Lemma, and the result is proved by induction.\ <\lemma> Let \\>, \\>|)>>, Dom |)>> with values in q+\>> and symmetric. Assume also that |\,DF|\>\+r>>\L,Hq-r-j>|)>> 0\r+j\q>. Then 0\r\q> |u,DF|\>\+r>>\Dom|)>> and <\equation*> |\,F|\>\>>=\|u,\F|\>\+r>>|)> <\proof> Let . We have for smooth \>, \\>|)>> and Dom |)>>: <\eqnarray*> ||u,F|\>\>>,\ G|\>|)>>|||u,\ G\F|\>\+1>>|)>>>||||\,F|\>\>>*G|)>-\|u,\ F|\>\+1>>*G|)>>>>> where we used the fact that =\G\F+G*\F> for smooth functions. The equality ||u,F|\>\>>,\ G|\>=|u,\ G\F|\>\+1>>> holds because is symmetric.\ We can pass to the limit thanks to the assumption |\,DF|\>\+r>>\L,Hq-r-j>|)>> and obtain <\equation*> \|u,F|\>\>>|)>=|\,F|\>\>>-|u,\F|\>\+1>>. Now suppose the statement true for . We have that \ <\eqnarray*> |\,F|\>\>>>||\|\,\F|\>\+r>>|)>>>|||\|u,DF|\>\+r>>|)>+\|u,\ F|\>\+r+1>>|)>>>|||\|u,\F|\>\+r>>|)>>>>> <\lemma> Let \>, \j>|)>> symmetric and such that all its derivatives are symmetric. We have <\equation> \\=j>i!*\ u|)> <\proof> If , or we have identities. Now let and \>. We have that \Dom|)>> (, Proposition1.3.1), and since u\\2>|)>> it is easy to see that h\H> |\ u,h|\>\\> by computing its norm. Then \ we can apply Proposition1.3.2 of to obtain h\H> <\equation*> |\\,h|\>=|u,h|\>+\|\u,h|\>|)> and since by hypothesis u> is symmetric we have |\u,h|\>|)>=|\\u,h|\>> and then \=u+\\u>. The proof by induction is easy noticing that v> is symmetric whenever is symmetric, and using the fact that \=\\+j\>. <\bibliography|bib|tm-plain|biblio> <\bib-list|19> Hajer Bahouri, Jean-Yves CheminRaphael Danchin. . Springer, 2011. Jean-Michel Bony. Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires. , 14:209\U246, 1981. Rémi CatellierKhalil Chouk. Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation. , oct 2013. ArXiv: 1310.6869. Massimiliano Gubinelli, Peter ImkellerNicolas Perkowski. Paracontrolled distributions and singular PDEs. , 3(6), 2015. Massimiliano GubinelliNicolas Perkowski. KPZ Reloaded. , 349(1):165\U269, jan 2017. Martin Hairer. A theory of regularity structures. , 198(2):269\U504, 2014. Martin Hairer. Regularity structures and the dynamical > model. , aug 2015. ArXiv: 1508.05261. Martin HairerJeremy Quastel. A class of growth models rescaling to KPZ. , dec 2015. ArXiv: 1512.07845. Martin HairerWeijun Xu. Large scale behaviour of 3d phase coexistence models. , jan 2016. ArXiv: 1601.05138. Antti Kupiainen. Renormalization Group and Stochastic PDEs. , 1\U39. Springer, 2014. Yves Meyer. Remarques sur un théorème de J.-M. Bony. , 1\U20. 1981. Jean-Christophe MourratHendrik Weber. Convergence of the two-dimensional dynamic Ising-Kac model to >. , oct 2014. ArXiv: 1410.1179. Jean-Christophe MourratHendrik Weber. Global well-posedness of the dynamic > model in the plane. , 2015. Jean-Christophe MourratHendrik Weber. Global well-posedness of the dynamic > model on the torus. , jan 2016. ArXiv: 1601.01234. Jean-Christophe Mourrat, Hendrik WeberWeijun Xu. Construction of > diagrams for pedestrians. , oct 2016. ArXiv: 1610.08897. Ivan NourdinDavid Nualart. Central limit theorems for multiple Skorohod integrals. , Jul 2007. ArXiv: 0707.3448. Ivan NourdinGiovanni Peccati. . Cambridge University Press, May 2012. David Nualart. . Probability and its applications. Springer, 2nd ed, 2006. Hao ShenWeijun Xu. Weak universality of dynamical $\\Phi\B4_3$: non-Gaussian noise. , jan 2016. ArXiv: 1601.05724. \; \; <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> hairer_theory_2014 hairer_regularity_2015 catellier_paracontrolled_2013 gubinelli_paracontrolled_2012 kupiainen_renormalization_2014 hairer_class_2015 hairer_large_2016 shen_weak_2016 mourrat_convergence_2014 mourrat_global_2015 mourrat_global_2016 nualart_2006 nourdin_peccati_2012 nourdin_nualart_2007 nourdin_peccati_2012 gubinelli_paracontrolled_2012 catellier_paracontrolled_2013 catellier_paracontrolled_2013 catellier_paracontrolled_2013 mourrat_construction_2016 catellier_paracontrolled_2013 hairer_theory_2014 catellier_paracontrolled_2013 hairer_theory_2014 hairer_theory_2014 gubinelli_paracontrolled_2012 Bahouri2011 gubinelli_kpz_2017 Bony1981 Bony1981 Meyer1981 gubinelli_kpz_2017 gubinelli_kpz_2017 hairer_theory_2014 hairer_theory_2014 hairer_theory_2014 nualart_2006 nualart_2006 nualart_2006 nualart_2006 nualart_2006 nualart_2006 nualart_2006 nourdin_nualart_2007 nualart_2006 nualart_2006 <\associate|toc> |math-font-series||1Analysis of the mesoscopic model> |.>>>>|> |1.1Paracontrolled structure |.>>>>|> > |1.2A-priori estimates |.>>>>|> > |1.3Identification of the limit |.>>>>|> > |math-font-series||2Convergence of random fields> |.>>>>|> |Strategy of the proof |.>>>>|> > |Details of the proof |.>>>>|> > |2.1First-order trees |.>>>>|> > |The time regularity of trees |.>>>>|> > |2.2Second-order trees |.>>>>|> > |2.2.1Renormalisation of second-order trees |.>>>>|> > |2.2.2Estimation of renormalised second-order trees |.>>>>|> > |math-font-series||Appendix ABasics of paracontrolled analysis> |.>>>>|> |A.1Notation and conventions. |.>>>>|> > |A.2Schauder estimates |.>>>>|> > |A.3Bony's paraproduct and some commutators |.>>>>|> > |math-font-series||Appendix BEstimation of the kernels> |.>>>>|> |math-font-series||Appendix CSome Malliavin calculus results> |.>>>>|> |math-font-series||Bibliography> |.>>>>|>