> <\body> <\hide-preamble> |->> > > > > > > > > > > > > > > >>>> > \; <\doc-data|>>|> \; Hausdorff Center for Mathematics & Institute for Applied Mathematics University of Bonn, Germany >>|> \; > measure on a bounded domain. Our starting point is the interpretation of its Laplace transform as the value function of a stochastic optimal control problem along the flow of a scale regularization parameter. Once small scale singularities have been renormalized by the standard counterterms, >-convergence allows to extend the variational characterization to the unregularized model.>|>-convergence.>> The > Gibbs measure on the -dimensional torus =\=\=/L\|)>|)>> is the probability measure > obtained as the weak limit for \> of the family |)>0>> given by\ <\equation> \\|)>= |)>|]>|>\\|)>, where <\equation*> V|)>\\>|)>|\|>-a|)>|\|>-b|)>\\,\e|)>>\\|)> Here \0> is a fixed constant, > is the Laplacian on >, > is the centered Gaussian measure with covariance |)>>, > is a normalization factor, ,b> given constants and =\\\> with > some appropriate smooth and compactly supported cutoff function such that \\> as \>. The measures > and > are realized as probability measures on |)>>, the space of tempered distributions on >. They are supported on the Hölder\UBesov space /2-\>|)>> for all small \0>. The existence of the limit > is conditioned on the choice of a suitable sequence of ,b|)>0>>. The constant > is not necessary, but is useful to decouple the behavior of the numerator from that of the denominator in eq.. \; The aim of this paper is to give a proof of convergence using a variational formula for the partition function > and for the generating function of the measure >. As a byproduct we obtain also a variational description for the generating function of the limiting measure > via >-convergence of the variational problem. Let us remark that, to our knowledge, it is the first time that such explicit description of the unregulated > measure is available. \; \ Our work can be seen as an alternative realization of Wilson's and Polchinski's continuous renormalization group (RG) method. This method has been made rigorous by Brydges, Slade et al. and as such witnesses a lot of progress and successes. The key idea is the nonperturbative study of a certain infinite dimensional Hamilton\UJacobi\UBellman equation describing the effective, scale dependent, action of the theory. Here we avoid the analysis involved by the direct study of the PDE by going to the equivalent stochastic control formulation, well established and understood in finite dimensions. The time parameter of the evolution corresponds to an increasing amount of small scale fluctuations of the Euclidean field and our main tool is a variational representation formula, introduced by Boué and Dupuis, for the logarithm of the partition function interpreted as the value function of the control problem. See also the related papers of Üstünel and Zhang where extensions and further results on the variational formula are obtained. The variational formula has been used by Lehec to prove some Gaussian functional inequalities, following the work of Borell. In this representation we can avoid the analysis of an infinite dimensional second order operator and concentrate more on pathwise properties of the Euclidean interacting fields. We are able to leverage techniques developed for singular SPDEs, in particular the paracontrolled calculus developed in, to perform the renormalization of various non-linear quantities and show uniform bounds in the \> limit.\ \; Define the normalized free energy > for the cutoff > measure, as <\equation> \\-|\|>>log |)>>exp|\|>f|)>-V|)>|]>\\|)> where C|)>;\|)>> is a given function. The main result of the paper is the following <\theorem> Let and take a small \0>. There exist renormalization constants ,b> (which depend polynomially on >) such that the limit <\equation*> \\lim\>\, exists for every C>;\|)>> with linear growth. Moreover the functional > has the variational form <\equation*> \=inf\>>\>+Z>|)>+\>+\|Z>|\<\|\|\>>>+|l|\<\|\|\>>|)>\\|)>>|]> where <\itemize-dot> > denotes expectations on the Wiener space of a cylindrical Brownian motion |)>0>> on |)>> with law >; > a collection of polynomial functions of the Brownian motion |)>0>> comprising a Gaussian process |)>0>> such that >|)>=Law>|)>>; >> is the space of predictable processes (wrt. the Brownian filtration) in ;H>|)>>; ,l|)>0>> are explicit (non-random) functions of \>> and >; >> a nice polynomial (non-random) functional of ,u|)>>, independent of .\ See Section and in particular Theorem for precise definitions of the various objects and a more detailed statement of this result. With respect to the notations in Lemma, observe that <\equation*> f>+Z>|)>+\>=\>,Z,K|)>, where > is another functional of ,u|)>>. \; Theorem implies directly the convergence of |)>> to a limit measure > on |)>>. Taking in the linear dual of >> it also gives the following formula for the Laplace transform of >: <\equation> |)>>exp|)>|)>\\|)>=exp|\|>|\|>|)>-\|)>|)>. \; To our knowledge this is the first such explicit description (i.e. without making reference of the limiting procedure). The difficulty is linked to the conjectured singularity of the > measure with respect to the reference Gaussian measure. Another possible approach to an explicit description goes via integration by parts (IBP) formulas, see for an early proof and a discussion of this approach. More recently gives a self\Ucontained proof of the IBP formula for any accumulation point of the > in the full space. However is still not clear how to use these formulas directly to obtain uniqueness of the measure and/or other properties (either on the torus or on the more difficult situation of the full space). Therefore, while our approach here is limited to the finite volume situation, it could be used to prove additional results, like large deviations or weak universality very much like for SPDEs, see e.g.. \; The parameter , which determines the size of the spatial domain =\>, will be kept fixed all along the paper and we will not attempt here to obtain the infinite volume limit \>. For this reason we will avoid to explicitly show the dependence of > with >. However some care will be taken to obtain estimates uniform in the volume |\|>>.\ \; An easy consequence of the estimates needed to establish the main theorem is the following corollary (well known in the literature, see e.g.): <\corollary> There exists functions |)>,E|)>> not depending on |\|>>, such that\ <\equation*> lim\0+>>|)>|\>=0, and, for any \0>, <\equation*> E|)>\\\E|)>. A similar statement for will be sketched below in order to introduce some of the ideas on which the proof is based.\ \; The construction of the > measure in finite volume is basic problem of constructive quantum field theory to which many works have been devoted, especially in the case. It is not our aim to provide here a comprehensive review of this literature. As far as the case is concerned, let us just mention some of the results that, to different extent, prove the existence of the limit as the ultraviolet (small scale) regularization is removed. After the early work on Glimm and Jaffe, in part performed in the Hamiltonian formalism, all the subsequent research has been formulated in the Euclidean setting: i.e. as the problem of construction and study of the probability measures > on a space of distributions. Feldman, Park, Benfatto et al., Magnén and Seneor and finally Brydges et al. obtained the main results we are aware of. Recent advances in the analysis of singular SPDEs put forward by the invention of regularity structures by M.Hairer and related approaches or even RG\Uinspired ones, have allowed to pursue the stochastic quantization program to a point where now can be used to prove directly the existence of the finite volume > measure in two different ways. Uniqueness by these methods requires additional efforts but seems at reach. Some results on the existence of the infinite volume measure and dynamics have been obtained recently. For an overview of the status of the constructive program wrt. the analysis of the > models the reader can consult the introduction to and \; This paper is organized as follows. In Section we set up our main tool, the Boué\UDupuis variational formula of Theorem. Then, as a warmup exercise, we use the formula to show bounds and existence of the > measure in Section. We then pass to the more involved situation of three dimensions in Section where we introduce the renormalized variational problem. In Section we establish uniform bounds for this new problem and in Section we prove Theorem. Section and Section are concerned with some details of the analytic and probabilistic estimates needed throughout the paper. Appendix gather background material on functional spaces, paraproducts and related functional analytic background material. <\acknowledgments*> The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program SRQ: Scaling limits, Rough paths, Quantum field theory during which part of the work on this paper was undertaken. This work was supported by the German DFG via CRC 1060 and by EPSRC Grant Number EP/R014604/1. <\convention*> Let us fix some notations and objects. Let |a|\>=|)>>. Denote with |)>> the space of Schwartz functions on > and with |)>> the dual space of tempered distributions. The notation > or f> stands for the space Fourier transform of . \ In order to easily keep track of the volume dependence of various objects we normalize the Lebesgue measure on > to have unit mass. We denote the normalized integral and measure by <\equation*> f\|\|>>>f ,x=|\|>>\x where |\|>> is the volume of >. Norms in all the related functional spaces (Lebesgue, Sobolev and Besov spaces) are understood similarly normalized unless stated otherwise. The various constants appearing in the estimates will be understood uniform in |\|>>, unless otherwise stated. The constant \0> represents a small positive number which can be different from line to line. The reader is referred to the Appendix for an overview of the functional spaces and the additional notations used in the paper. We begin by constructing a probability space > endowed with a process |)>|]>>> belonging to |]>,/2-\>|)>|)>> and such that >|)>=Law>|)>> for all 0> and >>|)>=\>. \ Fix \-d/2> and let \C;H>|)>>, |)>0>> the canonical process on > and > the Borel >\Ualgebra of >. On ,|)>> consider the probability measure > which makes the canonical process a cylindrical Brownian motion in |)>>. In the following > without any qualifiers will denote expectations wrt. > and >> will denote expectations wrt. some other measure >. On the measure space ,,\|)>> there exists a collection |)>\|)>>> of complex (2-dimensional) Brownian motions, such that |\>=B>, ,B> independent for \n> and =|\|>\|)>>e|n,\|\>>B>. Note that has a.s. trajectories in ,>|)>|)>> for any \0> by standard arguments. Fix some \C>,\|)>> such that =1>. Let \\> and <\equation*> \\|\>\|)>=\\|)>/t, where |\>> is the partial derivative of > with respect to . Consider the process |)>0>> defined by\ <\equation> Y\|\|>>\|)>>||n|\>>e|n,\|\>>dB,t\0. It is a centered Gaussian process with covariance\ <\eqnarray*> |Y,\|\>|Y,\|\>|]>>|||\|>>\|)>>\||n|\>>dB |^>||m|\>>dB |^>|\>|]>>>||||\|>>\|)>>>||n|\>>|^>|^>|\>,>>>> for any ,\\|)>> and 0>, by Fubini theorem and Ito isometry. By dominated convergence \>>|Y,\|\>|Y,\|\>|]>=|\|>\|)>>|n|\>|^>|^>|\>> for any \,\\L|)>>. Note that up to any finite time the r.v. > has a bounded spectral support and the stopped process =YT>> for any fixed 0>, is in ,W|)>|)>> for any \>. Furthermore |)>> only depends on a finite subset of the Brownian motions |)>>.\ We will usually write |)>> for the Fourier multiplier operator with symbol . With this convention we can compactly denote\ <\equation> Y=J\X,t\0, where \|D|\>\>. We observe that > has a distribution given by the pushforward |)>|)>>\> of > through >. We write the measure > in in terms of expectations over > as <\equation> g|)>\\|)>=|)> e|)>>|]>|>, for any bounded measurable |)>\\>.\ For fixed the polynomial appearing in the expression for |)>> is bounded below (since \0>) and > is well defined and also bounded away from zero (this follows easily from Jensen's inequality). However as \> we tend to loose both these properties due to the fact that we will be obliged to take \+\> to renormalize the non\Ulinear terms. To obtain uniform upper and lower bounds we need a more detailed analysis and we proceed as follows. Denote by > the space of progressively measurable processes which are >\Ualmost surely in \L\\|)>>. We say that an element of > is a . Below we will need also drifts belonging to >\L;H>|)>|)>> for some \\>, we denote the corresponding space with >>. Consider the measure > on ,|)>> whose Radon\UNykodim derivative wrt. > is given by <\equation*> \|\\>=|)>>|>. Since > depends on finitely many Brownian motions |)>> then it is well known that any >\Uabsolutely continuous probability can be expressed via Girsanov transform. In particular, by the Brownian martingale representation theorem there exists a drift \\> such that <\equation*> \|\\>=exp>u\X-|\|>|2>>|u|\<\|\|\>>>\s|)>, (recall that we normalized the |)>> norm) and the entropy of > wrt.> is given by <\equation*> H\|\|)>=\>\|\\>|]>=|\|>|2>\>>|u|\<\|\|\>>>\s|]>. Here equality holds also if one of the two quantities is >. By Girsanov theorem, the canonical process is a semimartingale under > with decomposition <\equation*> X=+u\s,t\0, where |)>> is a cylindrical >\UBrownian motion in |)>>. Under > the process |)>> has the semimartingale decomposition =W+U> with <\equation*> W\J\X,U=I|)>, where for any drift \> we define <\equation*> I\Jv\s. The integral in the density can be restricted to > since =0> if T>. Now <\equation> -log =-log |)>>\|\\>|)>|]>=V|)>+>u\X-|\|>|2>>|u|\<\|\|\>>\s, and taking expectation of wrt > we get <\equation> -log =\>+I|)>|)>+|\|>|2>>|u|\<\|\|\>>\s|]>. For any \> define the measure > by <\equation*> \|\\>=exp>v\X-|\|>|2>>|v|\<\|\|\>>\s|)>. Denote with \\> the set of drifts \> for which |)>=1>, in particular \\>. By Jensen's inequality and Girsanov transformation we have <\equation*> -log =-log \>|)>>|]>=-log \|)>->v\X+|\|>|2>>|v|\<\|\|\>>\s>|]> <\equation*> \\|)>+>v\X-|\|>|2>>|v|\<\|\|\>>\s|]>, \ for all \>, where \\>>. We conclude that <\equation> -log \\+I|)>+|\|>|2>>|v|\<\|\|\>>\s|]>, where +I> and >|)>=Law>>. The bound is saturated when >. We record this result in the following lemma which is a precursor of our main tool to obtain bounds on the partition function and related objects. <\lemma> The following variational formula for the free energy holds: <\equation*> \=-|\|>>log \ |)>>|]>=min\> \|\|>>V+I|)>+>|v|\<\|\|\>>>\s|]>. where \|\|>f+V>. This formula is nice and easy to prove but somewhat inconvenient for certain manipulations since the space > is indirectly defined and the reference measure > depends on the drift . A more straightforward formula has been found by Boué\UDupuis which involves the fixed canonical measure > and a general adapted drift \>. This formula will be our main tool in the following. <\theorem> The Boué\UDupuis (BD) variational formula for the free energy holds: <\equation*> \=-|\|>>log \ |)>>|]>=inf\> \|\|>>V+I|)>+>|v|\<\|\|\>>>\s|]>. where the expectation is taken wrt to the measure > on >. <\proof> The original proof can be found in Boué\UDupuis for functionals bounded above. In our setting the formula can be proved using the result of Üstünel by observing that |)>> is a functional, according to his definitions. Namely, for some 1> such that we have <\equation*> \|)>|\|>|]>+\|)>>|]>\+\. <\remark> Some observations on these variational formulas. <\enumerate-alpha> They originates directly from the variational formula for the free energy of a statistical mechanical systems: > playing the role of the internal energy and the quadratic term playing the role of the entropy. The infimum could not be attained in Theorem while it is attained in Lemma. The drift generated by absolutely continuous perturbations of the Wiener measure has been introduced and studied by Föllmer.\ They are a non\UMarkovian and infinite dimensional extension of the well known stochastic control problem representation of the Hamilton\UJacobi\UBellman equation in finite dimensions. The BD formula is easier to use than the formula in Lemma since the probability do not depend on the drift . Going from one formulation to the other requires proving that certain SDEs with functional drift admits strong solutions and that one is able to approximate unbounded functionals > by bounded ones. See Üstünel and Lehec for a streamlined proof of the BD formula and for applications of the formula to functional inequalities on Gaussian measures. For example, from this formula is not difficult to prove integrability of functionals which are Lipshitz in the Cameron\UMartin directions. The next lemma provides a deterministic regularity result for > which will be useful below. It says that the drift generates shifts of the Gaussian free field in directions which belong to > uniformly in the scale parameter up to >. The space > is the Cameron\UMartin space of the free field. <\lemma> Let \\>. For any L|)>,H>|)>> we have <\equation*> supt\T>|I|\<\|\|\>>+1>>+sups\t\T>|I-I|\<\|\|\>>+1>>|1\>\|v|\<\|\|\>>>>\r. <\proof> Using the fact that |)>> is diagonal in Fourier space, and denoting with |)>\>> the basis of trigonometric polynomials, we have <\equation*> |\|)>v\s|\<\|\|\>>>>>|||\|>>|k|\>>|\|)>e,v|\>\s|\|>>>||>||\|>>|k|\>>|\|)>e,e|\>|\|>\s|)>|e,v|\>|\|>\s|)>>>||>||v|\<\|\|\>>>>\s*sup|e,\|)>e|\>\s>>||>||v|\<\|\|\>>>>\s sup|e,\|)>e|\>\|v|\<\|\|\>>>>\s.>>>>> On the other hand > is a smooth Fourier multiplier and using Proposition we have the estimate |\|)>f|\<\|\|\>>>>\|f|\<\|\|\>>>>/|s|\>> uniformly in 0>, therefore, for all r\t\T>, we have <\equation*> |\|)>v\s|\<\|\|\>>>>>|>||\|)>v|\<\|\|\>>>>\s|)>\|\|)>v|\<\|\|\>>>>\s>>||>||v|\<\|\|\>>>>\s|)>.>>>>> We conclude that <\equation*> | I-I|\<\|\|\>>+1>>\|\|)>v\s|\<\|\|\>>>>\|]>|v|\<\|\|\>>>>\s. \; <\notation> In the estimates below the symbol |)>> will denote a generic positive deterministic quantity, not depending on |\|>> and such that |)>/\\0> as \0>. Moreover the symbol > will denote a generic random variable measurable wrt. |)>>|)>> and belonging to |)>> uniformly in and |\|>>. > As a warm up we consider here the case setting for simplicity. From Theorem we see that the relevant quantity to bound is of the form <\equation> F\\|\|>>V+I|)>+|u|\<\|\|\>>>|]>, for \>. Let =I> and write =Y> as a mnemonic of the fact that under > the process is a martingale. From now on we leave implicit the integration variable over the spatial domain >. Choosing <\equation> a=6\|]>,b=3\|]>, we have <\equation*> V+Z|)>=\|W|\>+4\|W|\>Z+6\|W|\>Z+4\WZ+\Z, where\ <\equation*> |W|\>>|>|-6\|]> W+3\|]>,>>||W|\>>|>|-3\|]>W,>>||W|\>>|>|-\|]>,>>>>> denote the Wick powers of the Gaussian r.v. >. These polynomials, when seen as stochastic processes in , are >\Umartingales wrt. the filtration of |)>>. In particular they have an expression as iterated stochastic integrals wrt. the Brownian motions |)>> introduced in eq.. Using Theorem with we readily have an upper bound for the free energy: <\equation*> -|\|>>log \\\|W|\>|]>=0. For a lower bound we need to estimate from below the average under > of the variational expression\ <\equation*> \|W|\>+4\|W|\>Z+6\|W|\>Z+4\WZ+\Z+|u|\<\|\|\>>>. The strategy we adopt is to bound , and for a generic drift , the contributions <\equation*> \\|W|\>Z|\>>+|W|\>Z|\>\>+WZ|\>\\> , in term of quantities involving only the Wick powers of which we can control in expectation and the last two \ <\equation*> |u|\<\|\|\>>>+\Z. Any residual positive contribution depending on can be dropped in the lower bound making the dependence on the drift disappear. To control term > we see that by duality and Young's inequality, for any \0>, <\equation> |W|\>Z|\|>\4\||W|\>|\<\|\|\>>>|Z|\<\|\|\>>>\C,d|)>\||W|\>|\<\|\|\>>>+\|u|\<\|\|\>>>\s. For the term \> the following fractional Leibniz rule is of help: <\proposition> Let p\\> and ,p,p,p\1> such that >+>=>+>=>. Then for every \0> there exists a constant C such that\ <\equation*> ||\|\>|\<\|\|\>>>\||\|\>>f|\<\|\|\>>>> ||\|\>>g|\<\|\|\>>>>+||\|\>>g|\<\|\|\>>>> ||\|\>>f|\<\|\|\>>>>. <\proof> See. Using Proposition we get, for any \0>, <\equation> |W|\>Z |\|>>|>|||W|\>|\<\|\|\>>,5>>|Z|\<\|\|\>>,>>>>||>|||W|\>|\<\|\|\>>,5>>|Z|\<\|\|\>>,2>>|Z|\<\|\|\>>>>>>||>|||W|\>|\<\|\|\>>,5>>|Z|\<\|\|\>>>|Z|\<\|\|\>>>>>||>|\|2\>||W|\>|\<\|\|\>>,5>>+|4>|Z|\<\|\|\>>>+\|4>|Z|\<\|\|\>>>.>>>>> \; In order to bound the term \\> we observe the following: <\lemma> Let W,p>> for every p\\> and \0>. We claim that for any \0> there exists a constant ,d|)>>, and an exponent \> such that for any W>\ <\equation*> \f g|\|>\E|)>|f|\<\|\|\>>,p>*>+\|g|\<\|\|\>>,2>>+\|g|\<\|\|\>>*>|)>. <\proof> By duality f g|\|>\|f|\<\|\|\>>,p>>|g|\<\|\|\>>,p>>>. Applying again Proposition and Proposition , we get <\equation*> |g|\<\|\|\>>,14/13>>>|>|||\|\>>g|\<\|\|\>>>>\||\|\>g|\<\|\|\>>>|g|\<\|\|\>>>>>||>||g|\<\|\|\>>>|g|\<\|\|\>>>.>>>>> So <\equation*> f g|\|>>|>||f|\<\|\|\>>,14>>|g|\<\|\|\>>>|g|\<\|\|\>>>>>||>||f|\<\|\|\>>,14>>+\|g|\<\|\|\>>>+\|g|\<\|\|\>>>|)>.>>>>> \; <\remark> For the case it would have been enough to estimate |g|\<\|\|\>>,p>>>. The stronger estimate will be useful below for since there we will only have \W,p>> for any large . Then <\equation> WZ|\|>\C E|)>|W|\<\|\|\>>,p>*>+\*|Z|\<\|\|\>>,2>>+\|Z|\<\|\|\>>*>|)>. Using eqs., and we obtain, for > small enough, <\equation> |\|>\Q+\|u|\<\|\|\>>>+\Z|]>, where <\equation*> Q=O|)>||W|\>|\<\|\|\>>>+||W|\>|\<\|\|\>>,5>>+|W|\<\|\|\>>,p>*>|]>. Therefore <\equation*> F\-\|]>+|)>|u|\<\|\|\>>>+\Z|]>\-\|]>. This last average do not depends anymore on the drift and we are only left to show that\ <\equation*> sup \|]>\\. However, it is well known that the Wick powers of the two dimensional Gaussian free field are distributions belonging to ,W,b>|)>> for any 1> and 1> and hypercontractivity plus an easy argument gives the uniform boundedness of the above averages, see e.g.. We have established: <\theorem> For any \0> we have <\equation*> sup|\|>>|\|>\O|)>, where the constant in the r.h.s. is independent of >.\ <\remark> Observe that the argument above remains valid upon replacing > with p> with 1>. This implies that |)>>> is in all the > spaces wrt. the measure > uniformly in and for any 1>. > In three dimensions the strategy we used in two dimensions fails. Indeed here the Wick products are less regular: |W|\>\>> uniformly in for any small \0> and |W|\>> does not even converge to a well-defined random distribution. This implies that there is no straightforward approach to control the terms\ <\equation> |W|\>Z,|W|\>Z, like we did in Section. The only apriori estimate on the regularity of =I> is in >, coming from Lemma and the quadratic term in the variational functional >. It is also well known that in three dimensions there are further divergences beyond the Wick ordering which have to be subtracted in order for the limiting measure to be non-trivial. For these reasons we introduce in the energy > further scale dependent renormalization constants ,\> \ beyond Wick ordering to have <\equation> |\|>>V|)>=f|)>+|Y|\>-\\|Y|\>-\|)>. Repeating the computation from Section we arrive at\ <\equation> >||+Z|)>+\\Z+|2>\Z+4\W Z|]>>>|||\WZ+\\Z+\\|]>+\Z+|u|\<\|\|\>>>|]>.>>>>> where we introduced the convenient notations <\equation*> \\4|W|\>,\\12|W|\>,t\0, and we recall that is a fixed function belonging to >;\|)>> with linear growth.\ This form of the functional is not very useful in the limit \> since some of the terms, taken individually, are not expected to behave well. We will perform a change of variables in the variational functional in order to obtain some explicit cancellations which will leave well behaved quantities of . The main drawback is that the functional will have a less compact and canonical form. Some care has to be taken in order for the resulting quantities to be still controlled by the coercive terms. We will need some regularization which will make compatible Fourier cutoffs with > estimates. To introduce such a regularization fix a smooth function :\\\> such that |)>=1> if \1/4> and |)>=0> if \1/2>. Then \ <\equation> |)>\|)>>|| for t>,>>>||)>>|| for \c t> 0>.>>>>>> By the Mihlin-Hörmander theorem we deduce that the operator =\|)>> is bounded on > for any p\\>, see Proposition. In the following, for any C|]>,|)>|)>> we define >\\f> then \ <\equation*> Z>=\Z=\|\|\>\|)>u \s=\|\|\>\|)>u \s=\Z. In this way we have |Z>|\<\|\|\>>>\|Z|\<\|\|\>>>> for all T>. The renormalized functional will depend on some specific renormalized combinations of the martingales |\|\>|)>>. Therefore it will be also convenient to introduce a collective notation for all the stochastic objects appearing in the functionals and specify the topologies in which they are expected to be well behaved. Let <\equation*> \\,\,\|3|\>>,\\1>,\>,\|2|\>\|2|\>>|)>, with \W>, <\equation*> \|3|\>>\J\,\>\J\|3|\>>\s,\\1>\\\\>, <\equation*> \>\\\\>+2\\,\|2|\>\|2|\>>\\|)>\\|)>+2|\>. We not not need to include >> since it can be obtained as a function of |3|\>>> thanks to the bound <\equation*> |\>-\>|\<\|\|\>>>>\|J\|3|\>>|\<\|\|\>>>>\r\|J\|3|\>>|\<\|\|\>>>>\r|]> <\equation*> \|\|3|\>>|\<\|\|\>>>>r||r|\>>>|]>\sup>|\|3|\>>|\<\|\|\>>>>, valid for all s\t\T> which shows that the deterministic map |3|\>>\\>> is continuous from |]>,>|)>> to |]>,>|)>>. The pathwise regularity of all the other stochastic objects follows from the next Lemma, provided the function > is chosen appropriately. <\lemma> There exists a function \C,\|)>> such that <\equation> |\|>+|t|\>|\>|\|>\log|t|\>,t\0. and such that the vector > is almost surely in > where > is the Banach space <\equation*> \=C|]>,\|)>\|3|\>>\L,>|)>,\|2|\>\|2|\>>\L,>|)>|}> with <\equation*> \=\>\>\>\>\>\>\>, and equipped with the norm\ <\equation*> |\|\<\|\|\>>>\|\|\<\|\|\>>|]>,\|)>>+|\|3|\>>|\<\|\|\>>,>|)>>+|\|2|\>\|2|\>>|\<\|\|\>>,>|)>>. The norm |\|\<\|\|\>>>> belongs to all > spaces. Moreover the averages of the Besov norms >> of the components of > of regularity > are uniformly bounded in the volume |\|>> if \>. The proof is based on the observation that one can choose > in such a way that every component >> of the vector > is such that \>|)>0>> for -1> and \> is a martingale wrt. the Brownian filtration. The reader can find details for this statement in the proof of Lemma below, in particular eq. and give the stochastic integral representations of the most difficult terms which require renormalization by >. The quadratic variation of these martingales can be controlled uniformly up to >. As a consequence, there exists a version of \>|)>0>> which is continuous in |]>>. Since each Littlewood\UPaley block is a smooth function of is not difficult from this to deduce that \\\>|)>\C|]>\\;\|)>>. By Burkholder\UDavid\UGundy (BDG) estimates and > estimates on the the quadratic variations one can conclude that |\|\<\|\|\>>>> is in all > spaces provided we can control the appropriate moments of the norm of > uniformly in . This is achieved in Lemma for the more difficult resonant products where we show also the claimed uniformly of the finite Besov norms >>. > For convenience of the reader we summarize the probabilistic estimates in Table. ||||||>>|>>||3|\>>>>|>>>|\1>>>|>>>||2|\>\|2|\>>>>>|>>|>>|\L>>|>>|>>|>>|>\>>>>>>|Regularities of the various stochastic objects, the domain of the time variable is understood to be |]>>. Estimates in these norms holds a.s. and in |)>> for all 1> (see Lemma).> <\remark> The requirement that |3|\>>\L> will be used in Section to establish equicoercivity and to relax the variational problem to a suitable space of measures.\ We are now ready to perform a change of variables which renormalizes the variational functional. <\lemma> Define \\>, \C|]>,H>|)>>, \C|]>,H>|)>> such that <\equation> \u+\\T>\|3|\>>+\\T>J\Z>|)>,>>|\I,K\I,w\-\\T>J\Z>|)>+l,>>>>>t\0, Then the functional > defined in eq. has the form <\eqnarray*> >||,Z,K|)>+\|)>+|l|\<\|\|\>>>|]>,>>>> where <\equation*> \,Z,K|)>\f+Z|)>+\>, <\eqnarray*> >>|>||2>\,K,K|)>+|2>\K|)>K-\\\>|)>K>>|>>|>|\-Z>|)>|)>K>>|>>|>|\>>|)>K\t>>|>>|>|WK-12\W\>K+12\W>|)>K>>|>>|>|\Z>-Z>|)>-\\-Z>|)>-2\0>\Z>>>\t>>|>>|>|\>K-|2>\|2|\>\|2|\>>>|)>\t-|2>\,\,Z>,Z>|)>\t>>>> where > and > are linear forms defined in the Appendix (and recalled in the proof below) and we have chosen <\equation> >|>||2>\|3|\>>|)>\t+|2>\\>|)>>>|||\\W\>-4\\W>|)>.>>>>> <\proof> We are going to absorb the mixed terms via the quadratic cost function. To do so we develop them along the flow of the scale parameter via Ito formula. For the first we have <\equation*> \\Z=\\>\t+martingale, and we can cancel the first term on the r.h.s. by making the change of variables <\equation> w\u+\\T>\|3|\>>,t\0, into the cost functional to get <\equation*> \\Z+>|u|\<\|\|\>>>\s=-|2>|3|\>>|)>\t+>|w|\<\|\|\>>>\s+martingale, where we used that > is self-adjoint. The divergent term \Z> has been replaced with a divergent but purely stochastic term |3|\>>|)>\t> which does not affect anymore the variational problem and can be explicitly removed by adding its average to >. As a consequence, we are not able to control |)>> in > anymore and we should rely on the relation and on a control over the > norm of |)>> coming from the residual quadratic term |w|\<\|\|\>>>>. From we have the relation <\equation*> Z=-\\>+K, which can be used to expand the second mixed divergent term in as <\equation> |2>\Z=|2>\>|)>-\\\>K+|2>\K. Again, the first term on the r.h.s. a purely stochastic object and will give a contribution independent of the drift absorbed in >. We are still not done since this operation has left two new divergent terms on the r.h.s. of eq.: the > regularity of > is not enough to control the products with > which has regularity >>, a bit below . In order to proceed further we will isolate the divergent parts of these products via a paraproduct decomposition (see Appendix for details) and expand <\eqnarray*> \\>K+|2>\K>||\Z|)>K-\\\>|)>K>>|||\\>|)>K+|2>\K|)>K>>||||2>\K|)>K-\K|)>K|)>.>>>> The first two terms will require renormalizations which we put in place in Step3 below. All the other terms will be well behaved and we collect them in >>. In particular we observe that the last one can be rewritten as\ <\equation*> |2>\K|)>K-\K|)>K|)>=-|2>\,K,K|)> using the trilinear form > defined in Proposition.\ As we anticipated, the resonant term \\>> needs renormalization. In the expression of > in we have the counterterm \WZ> available, which we put in use now by writing <\equation*> -\\\>|)>K-2\\WZ=-\\\>+2\W|)>|\>>>K+2\\W\>. The first contribution is collected in >> and the expectation of the second will contribute to >.\ As far as the term \Z|)>K> is concerned, we want to absorb it into |w|\<\|\|\>>\s> like we did with the linear term in Step2. Before we can do this we must be sure that, after applying Ito's formula, it will be still possible to use Z> to control some of the growth of this term. Indeed the quadratic dependence in > (via >) cannot be fully taken care of by the quadratic cost |w|\<\|\|\>>\s>.\ We decompose <\equation*> \\Z|)>K=\\Z>|)>K+\\-Z>|)>|)>K and using the fact that the functions -Z>> and -K>> are spectrally supported outside of a ball or radius we will be able to show that the second term is nice enough as \> to not require further analysis and we collect it in >>. For the first we apply Ito's formula to decompose it along the flow of scales as <\equation*> \\Z>|)>K=\\Z>|)>|\>\t+\\>>|)>K\t+. The second term will be fine and we collect it in >>. We are left with the singular term \Z>|)>|\>\t>. Using eq. and expanding in the residual quadratic cost function obtained in Step1, we compute\ <\equation*> \\Z>|)>|\>\t+>|w|\<\|\|\>>>\t=-|2>\Z>|)>|)>\t+>|l|\<\|\|\>>>\t <\equation> =-|2>\Z>|)>|)>\Z>|)>|)>\t+|l|\<\|\|\>>> To renormalize the first term on the r.h.s. we observe that the remaining couterterm can be rewritten as <\equation> -\\Z=-\\>|)>-\\Z>-Z>|)>-\\-Z>|)>. Differentiating in the first term in the r.h.s. of eq. we get\ <\equation> -\\>|)>=-\|\>>|)>\t-2\0>\Z>>>\t. The last term in eq. and the last two contributions in are collected in >>. The first contribution in eq. has the right form to be used as a counterterm for the resonant product in. Using the commutator > introduced in Proposition we have <\equation*> -|2>\Z>|)>|)>\Z>|)>|)>+2|\>>|)>|]>\t <\equation*> =-|2>\|)>\\|)>+2|\>|]>|\>|2|\>\|2|\>>>>|)>\t-|2>\,\,Z>,Z>|)>\t and collect both terms in >>.\ We are now left with the cubic term which we rewrite as <\equation*> 4\WZ=-4\W>|)>+12\W>|)>K-12\W\>K+4\WK. The average of the first term is collected in > while all the remaining terms in >>. At last we have established the claimed decomposition since the residual cost functional, from eq. has the form |l|\<\|\|\>>>>. The aim of this section is to give upper and lower bounds on > uniformly on and |\|>>. In particular we will prove the bounds of Corollary taking the explicit dependence on the coupling constant > into account. <\lemma> There exists a finite constant , which does not depend on |\|>>, such that <\equation*> sup|\|>\C. <\proof> Observe that from Lemma and Section we have that\ <\equation*> \,Z,K|)>\Q+\|Z|\<\|\|\>>>+>|l|\<\|\|\>>>\t|)>, which immediately gives \ <\equation> -\|]>\-\|]>+|)>\|Z|\<\|\|\>>>+>|l|\<\|\|\>>>\t|)>\\. On the other hand for any suitable drift >\\> we get the bound\ <\equation> \\\|]>+|)>\|I>|)>|\<\|\|\>>>+>|l>|)>|\<\|\|\>>>\t|)>, where <\equation> l>|)>=>+\\T>J+\\>|)>|)>>|)>. Therefore it remains to produce an appropriate drift >> for which the r.h.s. in eq. is finite (and so uniformly in |\|>> and of |)>>). \; One possible strategy is to try and choose >> such that >|)>=0>, however this fails since estimates on this via Gronwall's inequality would rely on the Hölder norm of > for which we do not have uniform control in the volume. In order to overcome this problem we decompose > and use weighted estimates similarly as done in. \; Consider the decomposition <\equation*> \=>\+>\, where the random field >\> is constructed as follows. Let > be smooth function, positive and supported on > and such that \\\>\-m|)>=1>. Denote by =\-m|)>>. Let |~>> be a smooth function supported in >. Denote by N>f> the Fourier multiplier operator |~>> and similarly N>f=|~>|)>>. Set \|\\|\<\|\|\>>|)>>>>>>> and let <\equation*> >\\\\\>\L>\|)>, and <\equation*> >\\\\\>\L>\|)>. (with slight abuse of notation we drop the time dependence of the operators >>, >>). Observe that the laws of both >\> and >\> are translation invariant w.r.t to translations by \\\>. By , Theorem 2.4.7 and Bernstein inequality\ <\eqnarray*> |>\|\<\|\|\>>>>>>|>||L>\|)>|\<\|\|\>>>>>>>||>||\\|\<\|\|\>>>>>|\\|\<\|\|\>>>>>\1>>>> Furthermore for a weight > (see Appendix for precisions on the weighted spaces |)>>, >|)>> and >|)>> used below): <\equation> |>\|\<\|\|\>>>>|)>>>|>||\>\|\<\|\|\>>>>|)>>>>||>||\\|\<\|\|\>>>>>|)>|\\|\<\|\|\>>>|)>>>>||>| \*|\\|\<\|\|\>>>>>|)>|\\|\<\|\|\>>>|)>>>>||>| |\\|\<\|\|\>>>>|)>>|)>>|\\|\<\|\|\>>>>|)>>>>||>||\|\<\|\|\>>>>|)>>,>>>>> where we used the possibility to compare weighted and unweighted norms once localized via >. We now let >> be the solution to the (integral) equation <\equation> >=-\\T>|3|\>>+J*>\\\>|)>|)>|]>,t\0, which can be solved globally. For \1/2> and 1>, we have, for >, <\equation*> |I>|)>|\<\|\|\>>>|)>>\\|J\|\<\|\|\>>>|)>>+\|J>\\\>|)>|)>|\<\|\|\>>>|)>>|]>\s <\equation*> \\s||s|\>>>|J\|\<\|\|\>>>|)>>+\s||s|\>/2>>|>\|\<\|\|\>>>/2>>|I>|)>|\<\|\|\>>>|)>>. \; Therefore Gronwall's lemma implies that, for >: <\equation> |I>|)>|\<\|\|\>>>|)>>>|>|s||s|\>>>|J\|\<\|\|\>>>|)>>|)>exp|>\|\<\|\|\>>>>>\s||s|\>>>|)>>>||>|s||s|\>>>|J\|\<\|\|\>>>|)>>|)>>>||>||\|3|\>>|\<\|\|\>>,B>|)>|)>>.>>>>> Taking =>||\|>>> and using Besov embedding we deduce from: <\equation*> sup\|I>|)>|\<\|\|\>>>\\\>s||s|\>>>|J\|\<\|\|\>>>>|)>\\. Now computing >|)>> from eq. and, we obtain <\equation*> l>|)>=\\T>J >\\\>|)>|)>,t\0. It now remains to prove that |l>|)>|\<\|\|\>>>|]>\O|)>> uniformly in 0>. Note that, for >, <\equation> ||J >\\\>|)>|)>|\<\|\|\>>|)>>>|>||s|\>/2>>|>\\\>|)>|)>|\<\|\|\>>/2>|)>>>>||>||s|\>/2>>|>\|\<\|\|\>>>/2>|)>>|I>|)>|\<\|\|\>>/2>|)>>.>>>>> We know that the distribution of >> is invariant under translation by \\\>. Recalling that \\\>\-m|)>=1> and letting > be a polynomial weight with sufficient decay and such that \\>, we have <\eqnarray*> |l>|)>|\<\|\|\>>>|]>>||\|s\\T>J>\\\>|)>|)>|\<\|\|\>>>|]>>>||>|\\\>\|s\\-m|)>J>\\\>|)>|)>|\<\|\|\>>>|]>>>|>|>||\|>\|s\\T>\J>\\\>|)>|)>|\<\|\|\>>>|]>>>|\\>)>>|>|\s \|J>\\\>|)>|)>|\<\|\|\>>|)>>|]>>>|)>>|>|s||s|\>>>\|>\|\<\|\|\>>>/2>|)>>|I>|)>|\<\|\|\>>/2>|)>>|]>>>||>|s||s|\>>>\|>\|\<\|\|\>>>/2>|)>>+\|I>|)>|\<\|\|\>>/2>|)>>|]>>>|,)>>|>|>s||s|\>>>|\|\<\|\|\>>>/2>|)>>+\\|\|3|\>>|\<\|\|\>>,B>|)>|)>>|]>>>||>|.>>>> This last quantities are bounded since standard arguments allow to bound, for sufficiently large, <\equation*> |\|\<\|\|\>>>/2>|)>>|]>\\|\|\<\|\|\>>>/2>|)>>\\|\|\<\|\|\>>>|)>> <\equation*> =-1>2/2|)>p>>\x\\\|\|>\-1>2/2|)>p>\\|\|>\1 uniformly in 0>. Similarly, we have <\equation*> |\|\<\|\|\>>|B|)>|)>>|]>\\|\|\<\|\|\>>|)>>=\|\|>, Now\ <\equation*> supT>\|\|>\supT>|\|>|)>\T and <\equation*> |\|3|\>>|\<\|\|\>>,B>|)>|)>>\>|J\|\<\|\|\>>>|)>> <\eqnarray*> |\|3|\>>|\<\|\|\>>,B>|)>|)>>>|>|>|J\|\<\|\|\>>>|)>>\s>>||>|>|||D|\>>\|\<\|\|\>>>|)>>\s>>||>|>|\\|\<\|\|\>>>>\s>>||>|>|t|\>>|t|\>|\|\<\|\|\>>>|)>\s>>>> This concludes the proof of the lemma. \; <\remark> The decomposition of the noise is similar to the one given in but differs in the fact that we choose the frequency cutoff dependent on the size of the noise instead of the point, to preserve translation invariance. The price to pay is that the decomposition is nonlinear, however this does not present any inconvenience in our context. > In this section we establish the >-convergence of the variational functional obtained in Lemma as \>. >-convergence is a notion of convergence introduced by DeGiorgi which is well suited for the study of variational problems. The book is a nice introduction to >-convergence in the context of the calculus of variations. For the convenience of the reader we recall here the basic definitions and results.\ \ <\definition> Let > be a topological space and let :\\,\|]>>. We say that the sequence of functionals |)>> >-converges to iff\ <\enumerate-roman> For every sequence \x> in >\ <\equation*> F\liminf\> F|)>; For every point there exists a sequence \x> (called a recovery sequence) such that\ <\equation*> F\limsup\> F|)>. <\definition> A sequence of functionals :\\,\|]>> is called equicoercive if there exists a compact set \\> such that for all \>\ <\equation*> inf\>F=inf>F. A fundamental consequence of >-convergence is the convergence of minima. <\theorem> If |)>> >-converges to and |)>> is equicoercive, then admits a minimum and\ <\equation*> min> F=lim\> inf> F. For a proof see.\ \; In this section we allow all constants to depend on the volume |\|>>: this is not critical since, at this point, the aim is to obtain explicit formulas at fixed >.\ \; We denote <\equation*> \,p>\L|)>;W,p>|)>,\\\, and by ,p>> the reflexive Banach space ,p>> endowed with the weak topology. We will write >> for ,2>>, and > for > and let \\,3>>, this space will be useful as it gives sufficient control over :\ <\lemma> For > small enough Z> is a compact map \C|]>,L|)>>. <\proof> By definition of we have for any \\1/8-\/2>, <\eqnarray*> |Z>-Z>|\<\|\|\>>>>>|||>>Ju\s|\<\|\|\>>>>\>>|||D|\>>u|\<\|\|\>>>>\s>>||>|>>|s|\>>>||D|\>>u|\<\|\|\>>>>\s>>||>|>>|s|\>>>||D|\>>u|\<\|\|\>>,3>>\s>>||>|>>|s|\>>>\s*|u|\<\|\|\>>,3>>\s\|u|\<\|\|\>>>>>|s|\>>>\s,>>>> where we have used a Sobolev embedding in the second to last line. Since\ <\equation*> lim\t>>>|s|\>>>\s=0,>|s|\>>>\s\\, for any \|]>>, we can conclude by the Rellich\UKondrachov embedding theorem and the Arzela\UAscoli theorem, that bounded sets in > are mapped to compact sets in |]>,L|)>>, proving the claim. We will need the following lemma, which establishes pointwise convergence for the functional > defined in Lemma. In the sequel, by an abuse of notation, we will denote both a generic element of > and the canonical random variable on > by <\equation*> \=,\,\|3|\>>,\\1>,\|2|\>\|2|\>>,\>|)> <\lemma> Define >=l>,u|)>\\> such that <\equation> l>\u+\\|3|\>>+\J\Z>|)>,t\0. For any sequence ,\|)>> such that \u> in >, =l,u|)>\l=l>,u|)>> in > and <\equation*> |>||,\,\|3|\>>,\\1>,\|2|\>\|2|\>>,\>|)>>>|>||>|>||,\,\|3|\>>,\\1>,\>,\|2|\>\|2|\>>|)>>>>>> in > we have\ <\equation*> lim\>\,Z|)>,K|)>|)>=\>,Z,K|)>, where >> is defined by\ <\equation*> \>,Z,K|)>\f>+Z>|)>+\>>,Z,K|)>, with >>,Z,K|)>=\>>> given by <\eqnarray*> >>>|>||2>>,K>,K>|)>+|2>>\K>|)>K>-\>\\>>|)>K>,>>|>>>||>|>>>|>|>\>>|)>K\t,>>|>>>|>|\>K>-12\>\>>|)>K>+12\\>>>|)>K>,>>|>>>|>|>0>\Z>>>\t,>>|>>>|>|\>>K>-\>\|2|\>\|2|\>>>|)>\t+|2>>,\,Z>,Z>|)>\t,>>>> where, with abuse of notation, we set <\equation> >\>>>|>|>\\>>+\>\\>>+\>\1>,>>|>>>|)>>|>|>>>\\>>|)>+2\>\1>\>>+2>>,\>>,\>|)>>>|||>\>>\\>>|)>+2\>\>>\\>>|)>,>>>>> and where ,\,\> are the trilinear forms defined in Proposition, Proposition and Proposition respectively. <\proof> Lemma implies that for any \u> in > we have |)>\Z> in |]>,L|)>> and by the convergence of \l> in > we have also |)>\K> in |]>,H>|)>>. The products \>> and >|)>> can be decomposed using paraproducts and, after replacing the resonant products by the corresponding stochastic objects in >, we obtain the finite analogs of the expressions in eq.. After this preprocessing, it is easy to see by continuity that we have \>\\>\>>> and >|)>\\>>>|)>> in >>. For >> and >> and the first term of >> the statement follows from uniform bounds for ,Z|)>,K|)>|)>> on \C|]>,H>|)>\C|]>,H>|)>> and multilinearity. For >> and the first two terms of >> convergence to follows from the bounds established in Lemma and Lemma. For >>, the last term of >> and the last two terms of >> we can again use uniform bounds and multilinearity as well as dominated convergence, thanks to Proposition. Going back to our particular setting recall that from Lemma we learned that\ <\equation*> \=inf\>F, with <\equation*> F=\,Z,K|)>+\|Z|\<\|\|\>>>+|l|\<\|\|\>>>|]>, where ,Z,K> are functions of according to eq.. This form of the functional is appropriate to analyze the limit \> and obtain the main result of the paper, stated precisely in the following theorem which is a simple restatement of the basic consequence of Theorem below.\ <\theorem> We have <\equation*> lim\>\=\\inf\>F>, where <\equation*> F>=\>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|]>, and where >> and >> are defined in Lemma. In order to use >-convergence, we need to properly modify the variational setting in order to guarantee enough compactness and continuity uniformly as \>. \; The analytic estimates contained in Section below allow to infer that there exists a small \>, and a finite constant \0> uniformly bounded in such that <\equation> -Q+|)>\|Z|\<\|\|\>>>+>>|l|\<\|\|\>>>|]>\F, and <\equation> F\Q+|)>\|Z|\<\|\|\>>>+>>|l|\<\|\|\>>>|]>. As long as is finite, the original potential > is bounded below so in particular we have <\equation> -C+\>>|u|\<\|\|\>>>|]>\F. From this we conclude that we can relax the optimization problem and ask that \>, once we have established Lemma where > is the space of predictable processes in >: <\equation*> \=inf\>F. The reason of this relaxation lies in the fact that the cost terms |l|\<\|\|\>>>,and |Z|\<\|\|\>>>> control the > norm of uniformly in , modulo constants depending only on |\|\<\|\|\>>>> which are bounded in average uniformly in . Note that eq. implies that for any sequence |)>> such that |)>> remains bounded we must have that also <\equation> sup \|l|)>|\<\|\|\>>>|]>\\. To prove >-convergence we need to find a space with a topology which, on the one hand is strong enough to enable to prove the >-liminf inequality, and on the other hand allows to obtain enough compactness from >. Almost sure convergence on \\> would allow for the former but is to strong for the latter. For this reason we need a setting based on convergence in law as precised in the following definition. <\definition> Denote by ,u|)>> be the canonical variables on \\> and consider the space of probability measures <\equation*> \\\\|)> > =Law>,u|)>> for some \> with >|u|\<\|\|\>>>|]>\\>> |}>. Equip > with the following topology: \\> iff <\enumerate-alpha> > converges to > weakly on \\>, \>|u|\<\|\|\>>>|]>\\>. Denote by |\>> the closure of > in the space of probability measures > on \\> such that >|u|\<\|\|\>>>|]>\\> . Condition (b) allows to exclude pathological points in |\>> and makes possible Lemma below. Then\ <\equation> \=inf\\>>|)>, where <\equation*> >|)>\\>,Z,K|)>+\|Z|\<\|\|\>>>+|l|\<\|\|\>>>|]> and where >> denotes the expectation on \\> wrt. the probability measure >. Our first aim will be to prove that the family >|)>> is indeed equicoercive on |\>>.\ <\lemma> Let |)>> be a family of measures on \\> such that \>|u|\<\|\|\>>>|]>\\>. Then |)>> is tight on \\>, in particular there exists a subsequence converging in |\>>. <\proof> Observe that |)>> on > is tight since > is a separable metric space, so for any \0>, we can find a compact set >\> such that \>|)>\\|)>\\/2>. Now let >\>\B\\\>, for some large to be chosen later. Then >> is a compact subset of \\> and\ <\eqnarray*> >,u|)>\>|]>>|>|+*\>|u|\<\|\|\>>>|]>.>>>> Choosing sup2\>|u|\<\|\|\>>>|]>/\> gives tightness. <\lemma> There exists a constant , depending only on > and >, such that <\equation*> \>|u|\<\|\|\>>>|]>\C+2\\>|Z|\<\|\|\>>>|]>+\>|l|\<\|\|\>>>|]>. <\proof> We use |l|\<\|\|\>>>\|l|\<\|\|\>>>> in the bound <\equation*> |>|u|\<\|\|\>>>|]>>|>|<\cell> \\>|\|3|\>>|\<\|\|\>>>|]>+\\>|s\J\\Z|)>|\<\|\|\>>>|]>+\>|l|\<\|\|\>>>|]> >||>|<\cell> \\>|\|3|\>>|\<\|\|\>>>|]>+\\>>|\|\<\|\|\>>>>>||s|\>>>|Z|\<\|\|\>>>\s|]> >|||<\cell> +\>|l|\<\|\|\>>>|]> >||>|<\cell> \\>|\|3|\>>|\<\|\|\>>>|]>+|2>\>>|\|\<\|\|\>>>>>||s|\>>>\s|]>+2\\>|Z|\<\|\|\>>>|]> >|||<\cell> +\>|l|\<\|\|\>>>|]>. >>>> \; <\corollary> The family >|)>> is equicoercive on |\>>. <\proof> Define for some 0> large enough\ <\equation*> =:\\>|Z|\<\|\|\>>>|]>+\>|l|\<\|\|\>>>|]>\K|}> From eq. we have <\equation*> \\>|Z|\<\|\|\>>>|]>+\>|l|\<\|\|\>>>|]>\C+>|)>, so\ <\equation*> inf\>>|)>\K-C on the other hand from it follows that\ <\equation*> supinf\|\>>>|)>\\. So for large enough\ <\equation*> inf\|\>>>|)>=inf\>>|)> And by \ Lemma and Lemma > is a compact set.\ To be able to use the equicoercivity we will need to show that we can extend the infimum in to |\>>. For this we will first need some properties of the space |\>>. In particular we will need to show that measures with sufficiently high moments are dense in |\>> in a way which behaves well with respect to >>. For this we introduce some approximations which will be useful in the sequel.\ <\definition> Let \>, \>, and >|)>\0>> be a smooth Dirac sequence on > and >|)>\0>> be another smooth Dirac sequence compactly supported on \\>. Denote by >> the convolution only wrt the space variable, and by > the space-time convolution. Define the following approximations of the identity:\ <\equation*> >|)>>|>|>\>,>>|>|)>>|>|t> u\\>=et>u\>\> \s.>>>>> Denote by\ <\equation*> \inf0|u|\<\|\|\>>,3>>\s\N|}>, and\ <\equation*> |)>\u|}>>. Observe the following properties of these maps: <\itemize> >> is a continuous map \\> and \\>; >> is a continuous map \\>; > is continuous as a map \B\\>; if is a predictable process then >>, >>, > will also be predictable. Furthermore we have the bounds\ <\equation*> |>|\<\|\|\>>>,|>|\<\|\|\>>>,||\<\|\|\>>>\|u|\<\|\|\>>>. uniformly in ,N>, and for every \>, <\equation*> lim\0>|>-u|\<\|\|\>>>=lim\0>|>-u|\<\|\|\>>>=lim\>|-u|\<\|\|\>>>=0. With abuse of notation, for \\\|)>> and \\>, we let <\equation*> f>\=>\=Law>,f|)>. <\remark> Let us briefly comment on the uses of these approximation in the sequel. >> will be used when one wants to obtain a sequence of weakly convergent measures on \\> or \\> from a sequence of measures weakly convergent on \\>. >> will be used when one wants to obtain a measure on \\> from one on \\>, while preserving the estimates on the moments of > since >\>|)>=Z\>\>>. <\lemma> Let \|\>>. Then there exist |)>> in > such that \\> on \\> (now with the norm topology) and \>|u|\<\|\|\>>>|]>\\>. <\proof> By definition of |\>> of there exists |~>\\> weakly on \ \\>. Then >|)>>|~>\>|)>>\> on \\> as \>, and since >|)>>\\\> weakly on \\> as \0>, we obtain the statement by taking a diagonal sequence. <\lemma> Let \\> on \\>, such that \>|u|\<\|\|\>>>|]>>\\. Then\ <\enumerate> for every Lipschitz function f on >, >|u|)>|\>|]>\\>|u|)>|\>|]>>; for every Lipschitz function f on |]>,L|)>> we have >|)>|]>\\>|)>|]>>. <\proof> \ Let be a Lipschitz function on > with Lipschitz constant . Let \C,\|)>> be supported on > with =1> on >, and =\>. Then f*\|u|\<\|\|\>>>|)>> is bounded,\ <\equation*> lim\>\> \|u|\<\|\|\>>>|)>|]>=\> \|u|\<\|\|\>>>|)>|]>, and \ \ <\eqnarray*> >\|u|\<\|\|\>>>|)>|]>-\>|]>>||>\|u|\<\|\|\>>>|)>-f|)>|u|\<\|\|\>>>\N|}>>|\>>>||>|>|u|\<\|\|\>>> |u|\<\|\|\>>>\N|}>>|]>>>||>|>|u|\<\|\|\>>> |]>\|u|\<\|\|\>>>\N|)>>>||>|\>|u|\<\|\|\>>> |]>.>>>> Using that \>|u|\<\|\|\>>> |]>\\> we have <\eqnarray*> \>\>|]>-\>|]>>|>|\>\> \|u|\<\|\|\>>>|)>|]>-\> \|u|\<\|\|\>>>|)>|]>|\|>>>|||> \|u|\<\|\|\>>>|)>|]>-\>|]>|\|>>>|||> \|u|\<\|\|\>>>|)>|]>-\>|]>|\|> >>||>| sup\>|u|\<\|\|\>>> |]>\N,>>>> and sending \> gives the statement. The second statement follows from the first and Lemma. From the definition of > we have that <\equation*> \=inf\\>>|)>, where <\equation*> >|)>\\>,Z,K|)>+\|Z|\<\|\|\>>>+|l|\<\|\|\>>>|]>. The space > is not necessarily closed w.r.t weak convergence of measures. To be able to use an argument involving equicoercivity we need to show that we can pass to the closure |\>> of > in the infimum. To do this we first need to prove that we can approximate measures in |\>> by measures with bounded support in the second marginal which are still in |\>>. This is the content of the following <\lemma> Let \|\>> such that >|Z|\<\|\|\>>>|]>+E>|u|\<\|\|\>>>|]>\\>. For any 0> there exists \|\>> such that |u|\<\|\|\>>>\L>, >-almost surely, \\> weakly on \ \\> as \>,\ <\equation*> \>|Z|\<\|\|\>>>|]>\\>|Z|\<\|\|\>>>|]>,\>|u|\<\|\|\>>>|]>\\>|u|\<\|\|\>>>|]>. Furthermore for any > there exists |)>\> such that |u|\<\|\|\>>>\L>, >-almost surely and \\> weakly on \\>. <\proof> \; First let us show how to approximate > with |~>> which are defined such that |Z|\<\|\|\>>>\L>, |~>> almost surely. As \|\>>, there exists |)>\> such that \\> on \\> and \>|u|\<\|\|\>>>|]>\\>. Since \> there exist |)>> adapted such that =Law,u|)>>. Define \\Ju\t\||]>=J\\||]>\t>. Then > is a martingale with continuous paths in |)>>. Define the stopping time =inf\|||\<\|\|\>>>\L|}>> where the inf is equal to if the set is empty. Observe that >=J\\|>|]>\t=Z|)>> with \\\|>|]>> adapted, by optional sampling, and almost surely |>|\<\|\|\>>>\L>. Now set |~>\Law>,u|)>>.\ (Tightness) The next goal is to show that for fixed , we can select a suitable convergent subsequence from |~>|)>>. For this we first show that |~>|)>> is tight on \\>. From the definition of > we have that \>|u|\<\|\|\>>>|]>\\>, and by construction \ <\equation*> sup \|~>>|u|\<\|\|\>>>|]>\sup\>|\\|>|]>|\<\|\|\>>>|]>\sup \>|u|\<\|\|\>>>|]>=sup \>|u|\<\|\|\>>>|]>\\, which gives tightness according to . We can then select a subsequence which converges on >.\ (Bounds) Let |~>> be the limit of the sequence constructed in Step 1.1. In this step we prove bounds on the relevant moments of |~>>. Let ,f> be sequences of functions on > which are Lipschitz, convex and monotone for every , while for every \> <\eqnarray*> f\x,>||\>f=x,>>|f\x,>||\>f=x.>>>> Then |u|\<\|\|\>>>|)>> is a lower-semi continuous positive function on > so by the Portmanteau lemma we have\ <\equation*> \|~>>|u|\<\|\|\>>>|)>|]>\liminf\>\|~>>|u|\<\|\|\>>>|)>|]>, and since it is also Lipschitz continuous and convex we have\ <\eqnarray*> \> \|~>>|u|\<\|\|\>>>|)>|]>>||\> \>||\>|\>|]>|\<\|\|\>>>|)>|]>>>||>|\> \>|u|\<\|\|\>>>|)>|]>=\>|u|\<\|\|\>>>|)>|]>.>>>> Therefore <\eqnarray*> |~>>|u|\<\|\|\>>>|]>>||\>\|~>>|u|\<\|\|\>>>|)>|]>>>||>|\>\>|u|\<\|\|\>>>|)>|]>=\>|u|\<\|\|\>>>|]>.>>>> Proceeding similarly for , we see that |Z|\<\|\|\>>>|)>> is a continuous function on > bounded below, and Lipschitz-continuous and convex on > so we again can estimate <\equation*> \|~>>|Z|\<\|\|\>>>|)>|]>=lim\>\|~>>|Z|\<\|\|\>>>|)>|]>, <\eqnarray*> |~>>|Z|\<\|\|\>>>|)>|]>>||\>\|~>>|Z|\<\|\|\>>>|)>|]>>>|||\>\>|\|)>>|\|>|]>|\<\|\|\>>>|)>|]>>>||>|\>\>||Z|)>|]>|\<\|\|\>>>|)>|]>=\>||Z|)>|]>|\<\|\|\>>>|)>|]>.>>>> and, taking \>, obtain\ <\equation*> \|~>>|Z|\<\|\|\>>>|]>\\>|Z|\<\|\|\>>>|]>. \; (Weak convergence) Now we prove weak convergence of |~>> to > on \\>. Let \\\\> be bounded and continuous. By dominated convergence and continuity of , >\|~>>,>|)>|]>=\|~>>,u|)>|]>>. Using furthermore that ,u|)>\f,>|)>> is continuous on \\> and Lemma in the 5th line below, we can estimate\ <\eqnarray*> ||\>>,u|)>|]>-\|~>>,u|)>|]>|\|>>>|||\>lim\0>\>\>,>|)>|)>|]>-\|~>>,>|)>|)>|]>|\|>>>|||\>lim\0>\>\>,>|)>|)>-f,\>|)>>|\>|]>|)>|]>|\|>>>|||\>lim\0>\>\>,>|)>|)>-f,\>|)>>|\>|]>|)>\\|}>>|]>|\|>>>||>|\>lim\0>\>\>,>|)>|)>-f,\>|)>>|\>|]>|)>|u|\<\|\|\>>>\c L|}>>|]>|\|>>>||>||f|\<\|\|\>>> lim\>sup|u|\<\|\|\>>>|]>|L>=0.>>>> \; In this step we improve the approximation to have bounded support. Let \\> be the subsequence selected in Step 1.1. Recall that =Law,u|)>> with adapted >. Define \\|)>\|]>>, and similarly to Step 1, \inf0 \|||\<\|\|\>>>\L|}>>, set \\\>|]>>. Then |u|\<\|\|\>>>\N> uniformly in and >-almost surely, so =Law,u|)>> is tight on \\> \ and we can select a weakly convergent subsequence. Denote the limit by >. Now we follow the strategy from Step 1. (Bounds) Let > be defined like in Step 1.2. Then again we have\ <\eqnarray*> \>\>|u|\<\|\|\>>>|)>|]>>||\> \>||\>>|\>|]>|\<\|\|\>>>|)>|]>>>||>|\>\>|>|\<\|\|\>>>|)>|]>>>>>|||>||\<\|\|\>>>|)>|]>\\>|u|\<\|\|\>>>|)>|]>.>>>> It follows that\ <\eqnarray*> >|u|\<\|\|\>>>|]>>||\>\|~>>|u|\<\|\|\>>>|)>|]>>>||>|\>liminf\> \>|u|\<\|\|\>>>|)>|]>>>||>|\>\>|u|\<\|\|\>>>|)>|]>=\>|u|\<\|\|\>>>|]>.>>>> \; (Weak convergence) Now we prove that \|~>> weakly on >. Let \\\\> be bounded and continuous. By dominated convergence and continuity of , >\|~>>,>|)>|]>=\|~>>,u|)>|]>>, and furthermore since,>|)>> is continuous on \\> we have <\eqnarray*> ||\>|~>>,u|)>|]>-\>,u|)>|]>|\|>>>|||\>lim\0>\>\>,>|)>|]>-\|~>>,>|)>|]>|\|>>>|||\>lim\0>\>\>,\>|)>>|\>|]>|)>-f,\>>|)>>|\>|]>|)>|]>|\|>>>|||\>lim\0> \>\>,\>|)>>|\>|]>|)>-f,\>>|)>>|\>|]>|)>|)>\\|}>>|]>|\|>>>||>|\>sup>\>,\>|)>>|\>|]>|)>-f,\>>|)>>|\>|]>|)>|)>||u|\<\|\|\>>>|}>\ N|}>>|]>|\|>>>||>|\\>\f\|)> lim\>sup|u|\<\|\|\>>>|]>|N>>>|||>>> We now put everything together. Since all > are supported on the set |Z|\<\|\|\>>>\L|}>>, weak convergence and Lemma imply\ <\equation*> lim\>\>|Z|\<\|\|\>>>|]>=\|~>>|Z|\<\|\|\>>>|]>. By the Portmanteau lemma, <\equation> liminf\> \>|u|\<\|\|\>>>|]>\\|~>>|u|\<\|\|\>>>|]>, and\ <\equation*> liminf\> \|~>>|u|\<\|\|\>>>|]>\\>|u|\<\|\|\>>>|]> which together with Step1.2 imply \>\|~>>|u|\<\|\|\>>>|]>=\>|u|\<\|\|\>>>|]>>, and by the same argument \>\|~>>|Z|\<\|\|\>>>|]>=\>|Z|\<\|\|\>>>|]>>. For any \0> we can choose a |~>> such that\ <\equation*> |~>>|Z|\<\|\|\>>>|]>-\>|Z|\<\|\|\>>>|]>|\|>+|~>>|u|\<\|\|\>>>|]>-\>|u|\<\|\|\>>>|]>|\|>\\. Since then by <\equation*> \>|u|\<\|\|\>>>|]>\liminf\>\>|u|\<\|\|\>>>|]>\\>|u|\<\|\|\>>>|]>-\ we can choose large enough so that\ <\equation*> >|Z|\<\|\|\>>>|]>-\>|Z|\<\|\|\>>>|]>|\|>+>|u|\<\|\|\>>>|]>-\>|u|\<\|\|\>>>|]>|\|>\\ which implies the statement of the theorem.\ <\lemma> If \> we have <\equation*> \=inf\|\>>>|)>. <\proof> To prove the claim it is enough to show that for any \|\>>, for any \0>, there exists a sequence \> such that \>>|)>\>|)>+\>. W.l.o.g we can assume that >|)>\\>. Observe that, as long as \> we can also express <\equation*> >|)>=\>|\|>>V+Z|)>+>>|u|\<\|\|\>>>|]>, and deduce that >|u|\<\|\|\>>>\\> since > is bounded below at fixed . By Lemma there exists a sequence |)>\|\>> , such that >, |u|\<\|\|\>>>\L> almost surely, \\> on \\> and <\equation*> \>|Z|\<\|\|\>>>|]>\\>|Z|\<\|\|\>>>|]>,\>|u|\<\|\|\>>>|]>\\>|u|\<\|\|\>>>|]>. First we have to improve the regularity of > to get convergence on \\> but without affecting our control on the moments of >, so let >\>|)>>\> and >\>|)>>\>. Then\ <\equation*> \>>|Z|\<\|\|\>>>|]>\E>>|Z|\<\|\|\>>>|]>,\>>|u|\<\|\|\>>>|]>\E>>|u|\<\|\|\>>>|]>, and >\>>> on \\>. By continuity of >> and the bound, >>|)>\>>|)>> as \> and >>|)>\>|)>> as \0>. In particular we can find and > such that >>|)>->|)>|\|>\\/2>. By Lemma there exists a sequence |)>> such that each measure > is supported on \B> and \\> weakly on \\>. Setting ,\>\>|)>>>|)>>\> and ,\>\>|)>>>|)>>\> we have ,\>\\,\>> on \\> with norm topology. Then, for some \C,\|)>,\=1 on B> supported on > and for any \>, +Z|)>\|\|\<\|\|\>>>/N|)>,|u|\<\|\|\>>>> are continuous bounded functions on the common support of ,\>> and\ <\eqnarray*> ||\>,\>>|\|>>V+Z|)>+>>|u|\<\|\|\>>>|]>|\>->>||||-\,\>>|\|>>V+Z|)>+>>|u|\<\|\|\>>>|]>|\|>>>||>|\>,\>>|\|>>\|\|\<\|\|\>>>/N|)>V+Z|)>+>>|u|\<\|\|\>>>|]>-|\>>>||||-\,\>>|\|>>\|\|\<\|\|\>>>/N|)>V+Z|)>+>>|u|\<\|\|\>>>|]>|\|>>>|||,\>>|\|>>|)>|\|\<\|\|\>>>/N|)>V+Z|)>|]>|\|>>>|||,\>>|\|>>|)>|\|\<\|\|\>>>/N|)>V+Z|)>|]>|\|>>>||>|,\>>|\|\<\|\|\>>>\N|}>>V+Z|)>|]>|\|>>>|||,\>>|\|\<\|\|\>>>\N|}>>V+Z|)>|]>|\|>>>|)>>|>|,\>|\|\<\|\|\>>>\N|)>|)>\,\>>|\|\<\|\|\>>>+L|]>|\|>>>||>|,\>>|\|\<\|\|\>>>|]>|N>\,\>>|\|\<\|\|\>>>+L|]>>>|\>)>>|>|>>> and by dominated convergence (since ,\>> is supported on \B>) we can find a > such that >,\>|)>->>|)>|\|>\\/2> which proves the statement. \ The proof of Lemma does not apply when >. However also in this case the functional <\equation> >>|)>\\>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|]>, has a well defined meaning, so we can investigate the relation between the two variational problems (on > and on |\>>) when the cutoff is removed. An additional difficulty derives from the fact that approximating the drift we might destroy the regularity of >>, since now >> needs to be more regular than , contrary to the finite case. To resolve this problem we need to be able to smooth out the remainder without destroying the bound on >. To do so smoothing >> directly, and constructing a corresponding new will not work, since >> by itself does not give enough control on and >. However we are still able to prove the following lemma by regularizing an \Paugmented\Q version of >>. <\lemma> There exists a family of continuous functions >>>:\\\>, which are also continuous \\>, such that for any |]>>, <\eqnarray*> |>|\<\|\|\>>>>|>||\|\<\|\|\>>>+|u|\<\|\|\>>>,>>||Z>|)>|\<\|\|\>>>>|>||\|\<\|\|\>>>+|Z|\<\|\|\>>>,>>||l>>,u|)>|)>|\<\|\|\>>>>|>>||\|\<\|\|\>>>|)>+|Z>|\<\|\|\>>>+|u|\<\|\|\>>>,>>>> and |l>>,u|)>|)>|\<\|\|\>>>> depends continuously on ,u|)>\\\>. Furthermore\ <\equation*> >,u|)>\u \>>, and if >\\> <\equation*> l>>,u|)>|)>\l> >> \\0. <\proof> \ \ Let =>\+>\> be the decomposition introduced in Section, and observe that for any 0> we can easily modify it to ensure that |>\|\<\|\|\>>>>>\c >, almost surely for any \|\>> and for any p\\>, >|>\|\<\|\|\>>>>>|]>\C> where depends on |\|>,\,c,p>. Now set =-\J>\\Z>|)>+l>>. Then satisfies\ <\equation*> u=-\\|3|\>>-\J>\\Z>|)>+. From this equation we can see that, like in Section, <\equation*> |u|\<\|\|\>>>\\|\|3|\>>|\<\|\|\>>>+\>|s|\>>>|>\|\<\|\|\>>>>>\s|u|\<\|\|\>>>+||\<\|\|\>>>, and choosing, small enough we get\ <\equation> |u|\<\|\|\>>>\\|\|3|\>>|\<\|\|\>>>+||\<\|\|\>>>. And similarly we observe that\ <\equation*> Z=-\\>-\J>\\Z>|)>\s+Z|)>, so again with small enough and since >=\Z> for T>: <\equation> |Z|\<\|\|\>>>\\|\>|\<\|\|\>>>+|Z|)>|\<\|\|\>>>. Conversely, it is not hard to see that we have the inequalities\ <\eqnarray*> |Z|)>|\<\|\|\>>>>|>||\>|\<\|\|\>>>+|Z|\<\|\|\>>>,>>>> and\ <\equation> ||\<\|\|\>>>\\|\|3|\>>|\<\|\|\>>>+|u|\<\|\|\>>>. Clearly the map ,u|)>\,|)>> is continuous as a map \ \\\\> and using Lemma also as a map \\\\\> , and the inverse is clearly continuous \\\\\>. We now show that it is also continuous as a map \\\\\>. Assume that |)>\l> weakly, since then |l|)>|\<\|\|\>>>> bounded, this implies by that also |u|\<\|\|\>>>> is bounded, and so we can select a weakly convergent subsequence, converging to >>. Then >> solves the equation\ <\equation*> u>=-\\|3|\>>-\J>\\Z>>|)>|)>+, (which can be seen for example by testing with some \>>) which implies that >=u> (e.g. by Gronwall). Now define >> to be the solution to the equation\ <\equation*> rem>=-\\|3|\>>-\J>\\Z>>|)>|)>+>|)>. Then by the properties discussed above rem>> is continuous in both the weak and the norm topology, we also have from and that\ <\eqnarray*> |rem>|\<\|\|\>>>>|>||\|3|\>>|\<\|\|\>>>+|u|\<\|\|\>>>,>>>> from we have\ <\equation*> |Z >|)>|\<\|\|\>>>\\|\>|\<\|\|\>>>+|Z|\<\|\|\>>>, and by definition of >>\ <\eqnarray*> |>|)>|\<\|\|\>>>>|||>|)>|\<\|\|\>>>>>||>>||\|3|\>>|\<\|\|\>>>+|u|\<\|\|\>>>>>>> Now observe that\ <\eqnarray*> |l>>|)>|\<\|\|\>>>>|>||s\\J>\\Z>>|)>|)>|\<\|\|\>>>+|>|)>|\<\|\|\>>>>>||>>||s|\>>>|>\|\<\|\|\>>>>>|Z>>|)>|\<\|\|\>>>\s+\|\|3|\>>|\<\|\|\>>>+|u|\<\|\|\>>>>>||>||\|\<\|\|\>>>|)>+|Z>>|)>|\<\|\|\>>>+|u|\<\|\|\>>>>>||>||\|\<\|\|\>>>|)>+|Z>|\<\|\|\>>>+|u|\<\|\|\>>>.>>>> Observing that also |\J>\\Z>>|)>|)>|\<\|\|\>>>> depends continuously on ,u|)>> (both in the weak and strong topology on >) gives the statement. <\lemma> For any \|\>> such that >>|)>\\> there exists a sequence of measures \|\>> such that <\enumerate-roman> For any \>, <\equation> \>|u|\<\|\|\>>>|]>+\>|l>|\<\|\|\>>>|]>\\, \\> weakly on \\> and >>|)>\Law>>|)>> weakly on >, <\equation*> lim\>>>|)>=>>|)>, For any > there exists a sequence \> such that\ <\equation> sup>|u|\<\|\|\>>>|]>+\>|l>|\<\|\|\>>>|]>|)>\\, \\> weakly on \\> and >>|)>\Law>>|)>> weakly on >. <\proof> By Lemma there exists a sequence >\\> weakly on \\> such that\ <\equation*> \>>|Z|\<\|\|\>>>|]>\\>|Z|\<\|\|\>>>|]>,\>>|u|\<\|\|\>>>|]>\\>|u|\<\|\|\>>>|]>, and >> is supported on \B|)>\\\>. Now set >>\>|)>>\>>. Then >>\\>\>|)>>\> on \\> and by the bounds from Lemma also >>>|Z|\<\|\|\>>>|]>\\>>|Z|\<\|\|\>>>|]>> and >>>|l>|\<\|\|\>>>|]>\\>>|l>|\<\|\|\>>>|]>>. The bounds from Lemma imply also >>|Z|\<\|\|\>>>|]>\\>|Z|\<\|\|\>>>|]>>, >>|l>|\<\|\|\>>>|]>\\>|l>|\<\|\|\>>>|]>>, and furthermore\ <\eqnarray*> >>>|u|\<\|\|\>>>|]>>|>|>>|\|\<\|\|\>>>+|u|\<\|\|\>>>|)>>>||>|>>|\|\<\|\|\>>>|)>+,>>>> and similarly\ <\eqnarray*> >>>|l>|\<\|\|\>>>|]>>|>>|>>|\|\<\|\|\>>>+|u|\<\|\|\>>>|)>>>||>|>>|\|\<\|\|\>>>|)>+,>>>> and by continuity of >>> and we are also able to deduce that we can find > small enough and > large enough depending on > such that >>>|)>->>|)>|\|>\1/2L> and >>>|)>->>>|)>|\|>\1/2L>. Choosing =\>>> we obtain the first three points of the Lemma. For the fourth point recall that from Lemma we have sequences >\\>> weakly on \\>, and >\>, which have support in \B|)>> and since >> is continuous on \\> setting >>\>|)>>\>> we obtain the desired sequence.\ <\lemma> If > we have <\equation*> inf\\>>>|)>=inf\|\>>>>|)>. <\proof> One can now proceed very similarly to the proof of Lemma. Let \|\>> such that >>|)>\\>. By Lemma, for any \|\>>, there exists a > such that >>|)>->>|)>|\|>\1/L>, and a sequence |)>> such that \>, \\> weakly on \\>, and such that is satisfied. Define ,\>\Law,rem>>|)>|)>>, and observe that now ,\>\\,\>> on \\>, ,\>>,l>|)>\Law,\>>,l>|)>> on \\>, and that we have ,\>>|u|\<\|\|\>>>|]>+\,\>>|l>|\<\|\|\>>>|]>|)>\\>. Then for some \C,\|)>,\=1 on B> supported on >, for any \>, the function\ <\eqnarray*> |||\|\<\|\|\>>>+|u|\<\|\|\>>>+|l>|\<\|\|\>>>|N>|)>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|)>>>||||~>,u|)>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|)>>>>> is bounded and continuous on \\>, and so by weak convergence\ <\eqnarray*> ||\>>>,\>|)>->>,\>|)>|\|>>>||>|\>,\>>|~>,u|)>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|)>|]>-|\>>>||||-\,\>>|~>,u|)>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|)>|]>|\|>>>|||\,\>>|~>,u|)>|)>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|)>|\|>|]>>>|||,\>>|~>,u|)>|)>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|)>|\|>|]>>>||>|\,\>>|\|\<\|\|\>>>+|u|\<\|\|\>>>+|l>|\<\|\|\>>>\N|}>>>,Z,K|)>+\|Z>|\<\|\|\>>>+|l>|\<\|\|\>>>|\|>|]>>>||>|,\>|\|\<\|\|\>>>+|u|\<\|\|\>>>+|l>|\<\|\|\>>>\N|)>\,\>>|\|\<\|\|\>>>+|u|\<\|\|\>>>+|l>|\<\|\|\>>>|]>|)>>>||>|\,\>>|\|\<\|\|\>>>+|u|\<\|\|\>>>+|l>|\<\|\|\>>>|]>\,\>>|\|\<\|\|\>>>+|u|\<\|\|\>>>+|l>|\<\|\|\>>>|]>|)>>>||>|\>>>>>> As we can find ,\> such that >>,\>|)>->>|)>|\|>\1/L> we can conclude.\ Finally we can state the key result of this section. <\theorem> The family >|)>> >\Uconverges to >>> on |\>>. Therefore <\equation*> lim \=lim inf\|\>>>|)>=inf\|\>>>>|)>=\. <\proof> In order to establish >-convergence consider a sequence \\> in |\>>. We need to prove that \> >|)>\>>|)>>. It is enough to prove this statement for a subsequence, the full statement follows from the fact that every sequence has a subsequence satisfying the inequality. Take a subsequence (not relabeled) such that <\equation> sup >|)>\\. If there is no such subsequence there is nothing to prove. Otherwise tightness for the subsequence follows like in the proof of equicoercivity. Then invoking the Skorokhod representation theorem of we can extract a subsequence (again, not relabeled) and find random variables |~>,|)>> and |~>,|)>> on some probability space |~>,|~>|)>> such that |~>>|~>,|)>=\>, |~>>|~>,|)>=\> and almost surely |~>\|~>> in >, \> in >. Note that \l|~>,|)>\l\l>|~>,u|)>> in > and using we deduce that the almost sure convergence \l> in >, maybe modulo taking another subsequence, again not relabeled. Note that, by our analytic estimates (which hold pointwise on the probability space) we have <\equation*> \|~>,Z|)>,K|)>|)>+\|Z|)>|\<\|\|\>>>+>>|l|)>|\<\|\|\>>>+H|~>|)>\0, for some |~>|)>> random variable |~>|)>> such that |~>>|~>|)>|]>=\|)>|]>>. Fatou's lemma and Lemma then give <\equation*> liminf\> >|)>=liminf\> \|~>>|~>,Z|)>,K|)>|)>+\|Z|)>|\<\|\|\>>>+>>|l|\<\|\|\>>>|]> <\equation*> =liminf\> \|~>>|~>,Z|)>,K|)>|)>+\|Z|)>|\<\|\|\>>>+>>|l|\<\|\|\>>>+H|~>|)>|]>-\|)>|]> <\equation*> \ \|~>> liminf\>|~>,Z|)>,K|)>|)>+\|Z|)>|\<\|\|\>>>+>>|l|\<\|\|\>>>+H|~>|)>|]>-\|)>|]> <\equation*> \\|~>> >|~>,Z|)>,K|)>|)>+|Z>|)>|\<\|\|\>>>+>>|l>|)>|\<\|\|\>>>|]>=>>|)>, which is the >-liminf inequality. Now all that remains is constructing a recovery sequence, for this we can again assume w.l.o.g that >>|)>\\>. From Lemma there is > such that >>|)>->>|)>|\|>\> and is satisfied. Then choosing =Law>,T|}>>u|)>> we obtain that T|}>>u|)>=T|}>>l>>, so |lT|}>>u|)>|\<\|\|\>>>\|l>|\<\|\|\>>>>, and |ZT|}>>u|)>|\<\|\|\>>>=|Z|\<\|\|\>>\|u|\<\|\|\>>>>, which is integrable by . By dominated convergence and Lemma we obtain \>>|)>=>>|)>>. Extracting a suitable diagonal sequence gives the recovery sequence. > In this section we collect a series of analytic estimate which together allow to establish the pointwise bounds and and the continuity required for Lemma. First of all note that <\equation> |K|\<\|\|\>>>>>|>||t|\>>>|\|\<\|\|\>>>>\s |Z|\<\|\|\>>>+|l|\<\|\|\>>>\s>>||>||t|\>>>|\|\<\|\|\>>>>\s |)>+\|Z|\<\|\|\>>>+|l|\<\|\|\>>>\s,>>>>> which implies that quadratic functions of the norm |K|\<\|\|\>>>>> with small coefficient can always be controlled, uniformly in |]>>, by the coercive term\ <\equation*> \Z+>|l|\<\|\|\>>>\s. <\lemma> For any small \0> there exists \0> such that <\equation*> >|\|>\C,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>. <\proof> By Proposition, <\equation> \K|)>K-\K|)>K|\|>>>|\\|\|\<\|\|\>>>> |K|\<\|\|\>>>\\|\|\<\|\|\>>>> |K|\<\|\|\>>>>>|\\|\|\<\|\|\>>>> |K|\<\|\|\>>>|K|\<\|\|\>>>>>|\\|\|\<\|\|\>>>>+|K|\<\|\|\>>>+\|K|\<\|\|\>>>.>>>>> By Proposition, <\eqnarray*> \K|)>K|\|>>|>||\|\<\|\|\>>>> |K|\<\|\|\>>>>>>> which is estimated in the same way and finally, <\eqnarray*> \\>|)>K|\|>>|>||\|\<\|\|\>>/2>> |\>|\<\|\|\>>/2>>>|K|\<\|\|\>>>>>>||>||)> \|\|\<\|\|\>>/2>> |\>|\<\|\|\>>/2>>>|)>+\|K|\<\|\|\>>>>.>>>> <\lemma> For any small \0> there exists \0> such that <\eqnarray*> >|\|>>|>|>,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>|)>>>>> <\proof> Using the spectral support properties of the various terms we observe that <\equation*> |\|\<\|\|\>>>>\>|\|\<\|\|\>>>>T>, and <\equation*> T>|Z-Z>|\<\|\|\>>>\|Z-Z>|\<\|\|\>>>>\|Z-Z>|\<\|\|\>>|1/2-\>>>>|Z-Z>|\<\|\|\>>|1/2-\>>> <\equation*> \|Z|\<\|\|\>>|1/2-\>>>>|Z|\<\|\|\>>|1/2-\>>>, where we used also interpolation and the > bound |Z>|\<\|\|\>>>\|Z|\<\|\|\>>>>. We recall also that <\equation> Z=K+\\>. Therefore we estimate as follows \; <\eqnarray*> \-Z>|)> |)>K>||\-K>|)> |)>K+\\>-\,\>|)> |)>K>>>> For the second term we can estimate\ <\eqnarray*> \>-\,\>|)> |)>K>|>||W|\<\|\|\>>>>>|\>-\,\>|\<\|\|\>>>|K|\<\|\|\>>>>>>||>|T>|W|\<\|\|\>>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>>>>> for the first term we get <\eqnarray*> \-K>|)> |)>K>|>||W|\<\|\|\>>>>>|K|\<\|\|\>>>|K|\<\|\|\>>>>>>||>||W|\<\|\|\>>>>>TT>|K|\<\|\|\>>>>>>||>|T>|W|\<\|\|\>>>>>|K|\<\|\|\>>>>,>>>> which we can again estimate like in Lemma . <\lemma> For any small \0> there exists \0> such that <\eqnarray*> >|\|>>|>|,\|)>E|)>Q+\supt\T>|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>.>>>> <\proof> First note that |\>|)>=|\|\>/t|)>|\>|\|\>/t|)>>. In particular >> is spectrally supported in an annulus with inner radius and outer radius . Then for any \> <\equation*> |>>|\<\|\|\>>>>=||\>|\|\>|t>|)>|\|\>|t>Z|\<\|\|\>>>>\||\>|\|\>|t>|)>|\|\>>|t>>Z|\<\|\|\>>>>\|Z|\<\|\|\>>>||t|\>>>. By Proposition, for any \0> there exists \0> such that <\equation*> \>>|)>K \t|\|>\\|\|\<\|\|\>>>>>|>>|\<\|\|\>>>|K|\<\|\|\>>>>\t>>|\\|\|\<\|\|\>>>>>|Z|\<\|\|\>>>>|K|\<\|\|\>>>>t||t|\>>>>>|\\|Z|\<\|\|\>>>>supt\T>|K|\<\|\|\>>>>|\|\<\|\|\>>>>>t||t|\>>>>>|\\|Z|\<\|\|\>>>>|Z|\<\|\|\>>>supt\T>|K|\<\|\|\>>>>|\|\<\|\|\>>>>>t||t|\>>>>>|\\|Z|\<\|\|\>>>supt\T>|K|\<\|\|\>>>>|\|\<\|\|\>>>>>t||t|\>>>>>|+\|Z|\<\|\|\>>>supt\T>|K|\<\|\|\>>>>|\>|\<\|\|\>>>>|\|\<\|\|\>>>>>t||t|\>>>>>>>> and again\ <\equation*> |Z|\<\|\|\>>>supt\T>|K|\<\|\|\>>>>|\|\<\|\|\>>>>>t||t|\>>>>>|\C\|\|\<\|\|\>>>>>t||t|\>>>+\supt\T>|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>.>>>>> While\ <\eqnarray*> |||Z|\<\|\|\>>>supt\T>|K|\<\|\|\>>>>|\>|\<\|\|\>>>>|\|\<\|\|\>>>>>t||t|\>>>>>||>||\|\<\|\|\>>>>>t||t|\>>>|\>|\<\|\|\>>>>+supt\T>|K|\<\|\|\>>>>+\|Z|\<\|\|\>>>.>>>> <\lemma> For any small \0> there exists \0> such that <\equation*> >|\|>\C,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>. <\proof> \ Using Lemma we establish that\ <\equation*> WK|\|>\E|)>|W|\<\|\|\>>,p>>+\|K|\<\|\|\>>>>+\|K|\<\|\|\>>>|)>. Next, we can write,\ <\equation*> \W>|)>K|\|>\\W>\\>|)>K|\|>+\|W|\<\|\|\>>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>. which can be easily estimated by Young's inequality. Decomposing <\equation*> W>\\>|)>=W\>\\>|)>+W\>\\>|)>+W\>\\>|)>. We can estimate the first two terms by\ <\equation*> \W\>\\>|)>K|\|>\\|W|\<\|\|\>>>>>|\>|\<\|\|\>>>|K|\<\|\|\>>>>, and <\equation*> \W\>\\>|)>K|\|>\\|W|\<\|\|\>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>. Young's inequality gives then the appropriate result. For the final term we use Proposition to get <\equation*> W\>\\>|)>K|\|>>>|\\\>\>K|\|>+\|W|\<\|\|\>>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>>>|\\|\>|\<\|\|\>>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>+\|W|\<\|\|\>>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>>>|\\C,\|)>|\>|\<\|\|\>>>>>|\>|\<\|\|\>>>>+|W|\<\|\|\>>>>>|\>|\<\|\|\>>>>|]>+\|K|\<\|\|\>>>>.>>>>> For the last term we estimate <\eqnarray*> \>|)>K|\|>>|>||W\>|\<\|\|\>>>>> |K|\<\|\|\>>>>,>>>> which can be estimated like in Lemma after we observe that <\equation*> |W\>|\<\|\|\>>>>>\|W\\>|\<\|\|\>>>>>+|W\\>|\<\|\|\>>>>>+|W\\>|\<\|\|\>>>>> <\equation*> \|W|\<\|\|\>>>>>|\>|\<\|\|\>>>>+|\>|\<\|\|\>>>>> and use Lemma to bound >>. <\lemma> Assume that\ <\equation*> sup|\|>||T|\>>+|\|>*\t||t|\>>\\. Then for any small \0> there exists \0> such that <\equation*> >|\|>\C>E|)>|\|>||T|\>>+|\|>*\t||t|\>>|]>+\|Z|\<\|\|\>>>>+\\|Z|\<\|\|\>>>. <\proof> We can estimate <\equation*> \Z>-Z>|)>|\|>\\|\|>|Z>|\<\|\|\>>>|Z-Z>|\<\|\|\>>>\\|\|>||T|\>>|Z>|\<\|\|\>>>|Z-Z>|\<\|\|\>>>, and\ <\equation*> \-Z>|)>|\|>\\|\|>|Z-Z>|\<\|\|\>>>\\|\|>||T|\>>|Z>-Z|\<\|\|\>>>|Z-Z>|\<\|\|\>>>. These bounds imply that both of them remain bounded provided > does not grow too fast in which is indeed insured by Lemma. For the last term we can apply the estimate <\equation*> \Z>>>\t|\|>\\>|\|>|Z>|\<\|\|\>>> |>>|\<\|\|\>>>\t\\|Z|\<\|\|\>>> |Z|\<\|\|\>>>|\|>*\t||t|\>>. Again, after we have fixed > below, we will see this to be bounded. Collecting these bounds we get <\equation*> >|\|>\C>\|\|>||T|\>>+|\|>*\t||t|\>>|]>+\\|Z|\<\|\|\>>>+\|Z|\<\|\|\>>>>. <\lemma> For any small \0> there exists a \0> such that <\eqnarray*> >|\|>>|>|,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>.>>>> <\proof> We start by observing that <\equation*> \\\>+2\W|)>K|\|>\\|\>|\<\|\|\>>,2>>|K|\<\|\|\>>,2>>. and using Lemma and eq. we have this term under control. Next split <\equation*> |2>\\Z>|)>|)>+2|\>>|)>|]>\t|\|> <\equation*> \|2>\\Z>|)>|)>-\\J\|)>>|)>\t|\|>+\\|2|\>\|2|\>>>|)>\t|\|> and note that |t|\>J> satisfies the assumptions of Proposition with . Therefore\ <\equation*> |J\Z>|)>-\|)>\Z>|\<\|\|\>>>>\|t|\>|\|\<\|\|\>>>>>|Z> |\<\|\|\>>>>, and by Proposition, <\equation*> |J\Z>|)>|\<\|\|\>>>>+|J\Z>|)>|\<\|\|\>>>>\ |t|\>>|\|\<\|\|\>>>>>|Z> |\<\|\|\>>>. and by binomial formula\ <\equation*> |2>\Z>|)>|)>\t-|2>\|)>\Z>|)>\t|\|> <\equation*> \\ supT> |Z> |\<\|\|\>>>|Z> |\<\|\|\>>>>|]>|\|\<\|\|\>>>>>t||t|\>>>. <\equation*> \\ supT> |Z> |\<\|\|\>>>|Z> |\<\|\|\>>>>|]>|\|\<\|\|\>>>>>t||t|\>>> Which can be easily estimated by Young's inequality. From Proposition and Proposition <\equation*> |2>\\Z>|)>|)>-|2>\\Z>|)>\J\ Z>|\|>\\|J\|\<\|\|\>>>>> |Z>|\<\|\|\>>>>> |Z>|\<\|\|\>>>. and by interpolation <\equation*> \\|J\|\<\|\|\>>>>> |Z> |\<\|\|\>>>|Z> |\<\|\|\>>>> The integrability of this term in time follows from the inequality\ <\equation*> |J\|\<\|\|\>>>>>\|t|\>>|\|\<\|\|\>>>>>. To prove this last bound, recall that J> is a Fourier multiplier with symbol <\equation*> |k|\>|^>|k|\>/t|)>|k|\>/t|)>=|k|\>\|k|\>/t|)>, where > is a smooth function supported in an annulus of radius . Using this observation and applying Proposition gives the estimate. Applying Proposition and Proposition we get <\equation*> \|\\Z>|)>\J\-\\J\|)>>|)>|\<\|\|\>>>>\\|J\|\<\|\|\>>>>> |Z>|\<\|\|\>>>>>. and after using duality and interpolation we obtain\ <\equation*> |2>\\Z>|)>|)>-\\J\|)>>|)>\t|\|> <\equation*> \\ supT> |Z> |\<\|\|\>>>|Z> |\<\|\|\>>>>|]>|\|\<\|\|\>>>>>t||t|\>>> <\equation*> \ \ supT>|Z> |\<\|\|\>>>>+\|Z|\<\|\|\>>>|)>+C,\|)>\|\|\<\|\|\>>>>>t||t|\>>>|)> <\equation*> \ \ |Z|\<\|\|\>>>>+\|Z|\<\|\|\>>>|)>+C,\|)>\|\|\<\|\|\>>>>>t||t|\>>>. Finally we have <\equation*> \\|2|\>\|2|\>>>|)>\t|\|>\\|\|2|\>\|2|\>>|\<\|\|\>>>\t|]>|Z|\<\|\|\>>>>|Z|\<\|\|\>>> <\equation*> \C|)>\|\|2|\>\|2|\>>|\<\|\|\>>>\t|]>+\\|Z|\<\|\|\>>>+\|Z|\<\|\|\>>>>. Using eq. to control |Z|\<\|\|\>>>>> in terms of > we obtain the claim. > In this section we close our argument proving the following lemmas which give uniform estimates as \> of the stochastic terms appearing in our analytic estimates.\ <\lemma> For any \0> and any 1,r\\,q\|]>>, there exists a constant ,p,q|)>> which does not depend on > such that\ <\equation> sup\|W\W>|\<\|\|\>>>>|]>\C,p,q|)>. Moreover there exists a function \C,\|)>> such that for any \0> and any 1>, <\equation> sup \|\\>-2\W|)>|\<\|\|\>>>>|]>\C,p,q|)>, <\equation> \>|J\\J\-2|\>|\<\|\|\>>>>\t|)>|]>\C,p,q|)>. <\equation*> sup \|J\\J\-2|\>|\<\|\|\>>>>|]>\C,p,q|)> and <\equation> |\|>+|t|\>|\>|\|>\1+log|t|\>,t\0. Furthermore > is independent of >. By Besov embedding Hölder norms of these objects are also uniformly bounded in (but not uniformly in >). <\proof> We will concentrate in proving the bounds on the renormalized terms in eq. and and leave to the reader to fill the details for the easier term in eq.. Recall the representation of =Y> in terms of the family of Brownian motions |)>> in eq.. Wick's products of the Gaussian field > can be represented as iterated stochastic integrals wrt. |)>>. In particular, if we let w=|k|\>\\B>, we have <\equation*> \=12|W|\>=24,k>e+k|)>*\x>>\w>|)>\w>|)>, <\equation*> \>=24,k,k>e>*\x>>>>>|)>||k>|\>>\u|)>\w>|)>\w>|)>\w>|)>, where >\k+k+k>. Now products of iterated integrals can be decomposed in sums of iterated integrals and we get <\equation> >|)>>||\\>-2\W|)>>>|||,\,k>>G>5>|)>\w>|)>\\w>|)>>>|||,\,k>>G>3>|)>\w>|)>\\w>|)>>>|||>>G>|)>\w>|)>,>>>>> where \s\\\s\T|}>\> and where the deterministic kernels are given by <\eqnarray*> >5>|)>>|>||)>K5|)>>|)>e5|)>>|)>*\x>\Sh>j>\>>|||K\|)>>|)>K\\|)>>|)>>>\\|)>>|)>||k\\|)>>|\>>\u|)>,>>|>3>|)>>|>||)>K3|)>>|)>e3|)>>|)>*\x>\Sh>j>\r||p|\>>\>>|||K>+q|)>K\|)>>-q|)>>\r>\|)>>-p|)>||k\|)>>-p|\>>\u|)>,>>|>|)>>|>||)>K|)>e*\x>j>,p>\r\r>|)>||p|\>>>|)>||p|\>>\>>|||K+p|)>K-p-p|)>\r\s>-p-p|)>||k-p-p|\>>\u|)>,>>|>|)>>|>|>|)>-2\K|)> e*\x>,>>>> where > is the set of permutations of ,k+l|}>> keeping the orders \\\\> and \\\\> and where, for any symbol , we denote with expression of the form n>> the vector ,\,z|)>>. Estimation of \\>|)>> reduces then to estimate each of the three iterated integrals using BDG inequalities to get, for any 2>, <\equation*> I=,\,k>>G>5>|)>\w>|)>\\w>|)>|\|>|]>|}> <\equation*> \\,\,k>>G>5>|)>\w>|)>\\w>|)>|\|>|]> <\equation*> \,\,k>>>5>|)>|\|>>|)>||k|\>>\>|)>||k|\>>\s\\s. The kernel >5>|)>> being a symmetric function of its argument, we can simplify this expression into an integral over >: <\equation*> \; <\equation*> I\,\,k>>>5>|)>|\|>>|)>||k|\>>\>|)>||k|\>>\s\\s. Under the measure >|)>||k|\>>\s>, we have <\equation*> >>\\|)>>|)>||k\\|)>>|\>>\u|\|>\|k>|\>>. Therefore with some standard estimates we can reduce us to consider <\equation*> I\,\,k>>5|)>>|)>||k|\>>>\k>>>|)>||k|\>>\>|)>||k|\>>\s\\s <\equation*> \,\,k>>5|)>>|)>||k|\>>>\k>>>|)>||k|\>>\>|)>||k|\>>\s\\s <\equation*> \,\,k>5|)>>|)>||k|\>>>\k>>|k|\>>\|k|\>> <\equation*> \,p>\p>K+p|)>,\,k>|k|\>>>=p,k>=p>|k|\>>\|k|\>> <\equation*> \,p>\p>K+p|)>|p|\>>|p|\>>\,r>K|p|\>>|p+r|\>>\K|r|\>>\2. Now by similar reasoning we also have <\equation*> >3>|)>|\|>\\Sh>3|)>>|)>|\|>j>\r>+p|)>K\|)>>-p|)>|\|>||p|\>|k>+p|\>> <\equation*> \\Sh>3|)>>|)>|\|>||k>|\>> so <\equation*> I=,\,k>>G>5>|)>\y>|)>\\y>|)>|\|>|]>|}> <\equation*> \,\,k>>\Sh>3|)>>|)>|\|>||k>|\>>|\|>>|)>||k|\>>\>|)>||k|\>>\s\\s <\equation*> \,\,k>3|)>>|)>|\|>||k|\>|k|\>|k|\>>\||r|\>>\2. Finally, we note that the same strategy cannot be applied to the first chaos, since the kernel >> cannot be uniformly bounded. We let <\equation*> ,k|)>>|>||)>j>,q>\r\r>|)>||q|\>>>|)>||q|\>>\>>|||K+q|)>K-q-q|)>\r\s>-q-q|)>||k-q-q|\>>\u|)>,>>>>> so <\equation*> G>|)>=K|)>e*\x>,k|)>-2\|]>. Observe that <\equation*> >||\2|)>,q>\r\r>|)>||q|\>>>|)>||q|\>>\>>|||\r>+q|)>||q+q|\>>\u j>K+q|)>K-q|)>.>>>>> We choose > as\ <\equation> \=A=\2|)>,q>\u\r\r>|)>||q|\>>>|)>||q|\>>+q|)>||q+q|\>> where we used the fact that for all \> we have j>KK=1>, since f\g=f g>. Note that, as claimed, <\equation*> |\|>\,q>|\|>,|\|>,+q|\|>\T>||q|\>|q|\>|q+q|\>>\1+log|T|\>. Now <\equation*> A,k|)>-2\=\6|)>,q>\r\r>|)>||q|\>>>|)>||q|\>>j>K+q|)>\ <\equation*> \-q-q|)>\r\r>-q-q|)>||k-q-q|\>>\u-K+q|)>\r>+q|)>||q+q|\>>\u|)> so when +q|\|>\|\|>> the quantity in round brackets can be estimated by |\|>|q+q|\>> while when +q|\|>\|\|>> it is estimated by |q+q|\>> so we have <\equation*> ,k|)>-\|\|>\,q>|q|\>>|q|\>>|q+q|\>>+q|\|>\|\|>>++q|\|>\|\|>>|\|>||q+q|\>>|)> <\equation*> \1+log|k|\>. And then with this choice of > the kernel >> stays uniformly bounded as \> and satisfies <\equation*> >|)>|\|>\K|)>log|k|\>. From this we easily deduce that <\equation*> I=>>G>5>|)>\y>|)>|\|>|]>|}>\q 2,q\-1. All together these estimates imply that <\equation*> \|\\>|\<\|\|\>>>\|)>,q\-1. Standard argument allows to deduce eq.. The analysis of the other renormalized product proceeds similarly. Let <\equation*> V\\|2|\>\|2|\>>=J\\J\-2|\>,t\0. First note that by definition of Besov spaces we have <\equation*> \>|V|\<\|\|\>>-d/r>>\t|)>|]> <\equation*> \\>2+d/r|)>>|\V|\<\|\|\>>>|)>\t|)>|]>. By Minkowski's integral inequality this is bounded by <\equation*> \>\t2+d/r|)>>|\V|\<\|\|\>>>|)>|]>|}>|)>. When p> Jensen's inequality and Fubini's theorem give <\equation*> \>\t2+d/r|)>>>x||\|>> \V|\|>|]>|}>|)>. Finally hypercontractivity and stationarity allow to reduce this to bound <\equation*> \>\t2+d/r|)>> V|\|>|]>|)>|}>|)>. Letting =\V|\|>|]>> we have <\equation*> \>|\|2|\>\|2|\>>|\<\|\|\>>-d/r>>\t|)>|]>\>\t2+d/r|)>> |)>|}>|)>. Now we decompose the random field |2|\>\|2|\>>|)>> into homogeneous stochastic integral as above and obtain <\equation> |2|\>\|2|\>>|)>>||,\,k>>G|2|\>\|2|\>>4>|)>\w>|)>\\w>|)>>>|||,k>>G|2|\>\|2|\>>|)>\w>|)>\w>|)>>>|||J2>>>>>> with <\eqnarray> |2|\>\|2|\>>4>|)>>|||)>K4|)>>|)>e4|)>>|)>*\x>\>>|||\Sh>j>K\|)>>|)>K\|)>>|)>\|)>>|)>||k\|)>>|\>>\|)>>|)>||k\|)>>|\>>>>||2|\>\|2|\>>|)>>|||)>K>|)>e>|)>*\x>\Sh>j>\>>|||\r||q|\>>K>+q|)>K>-q|)>>+q|)>||k>+q|\>>>-q|)>||k>-q|\>>|)>>>||2|\>\|2|\>>>|||)> j>,q>\r\r\>>|||>|)>||q|\>>>|)>||q|\>>K+q|)>K-q|)>+q|)>||q+q|\>>>>||||\>.>>>> Using our choice of > in eq. we have that <\equation*> |\>=\2|)>,q>\r\r>|)>||q|\>>>|)>||q|\>>+q|)>||q+q|\>>\u, which implies also that\ <\equation*> G|2|\>\|2|\>>=0,|\>|\|>\|t|\>||t|\>>. as claimed. We pass now to estimate the other two chaoses. The technique is the same we used above. Consider first <\equation*> I\\,\,k>>G|2|\>\|2|\>>4>|)>\w>|)>\\w>|)>|\|>|]> <\equation*> \,\,k>>|2|\>\|2|\>>4>|)>|\|>>|)>||k|\>>\>|)>||k|\>>\s\\s <\equation*> \,\,k>>|2|\>\|2|\>>4>|)>|\|>>|)>||k|\>>\>|)>||k|\>>\s\\s <\equation*> \,\,k>K4|)>>|)>>>|)>||k>|\>>>|)>||k>|\>>>|)>||k|\>>\>|)>||k|\>>\s\\s <\equation*> \,\,k>K4|)>>|)>>|)>||k>|\>>>|)>||k>|\>>|k|\>>\|k|\>> <\equation*> \\t>||t|\>>,\,k>4|)>>|)>||k|\>|k|\>|k|\>|k|\>>\\t>||t|\>>2 where we used that |\|>\tt>>. Now taking +d/r\0> we have <\equation*> >\t2+d/r|)>> |)>|}>\>\t\t> -d/r|)>>||t|\>>|}> <\equation*> \>t||t|\>+d/r>>\1. Taking into account that |\|>,|\|>\t> we can estimate \; <\equation*> |2|\>\|2|\>>|)>|\|>\>|)>|\|>\t>||p|\>>+p|)>||k+p|\>>-p|)>||k-p|\>>|)>\>|)>|\|>|t|\>, from which we deduce that <\equation*> I\\,k>>G|2|\>\|2|\>>|)>\w>|)>\w>|)>|\|>|]> <\equation*> \,k>>|2|\>\|2|\>>|)>|\|>>|)>||k|\>>>|)>||k|\>>\s\s <\equation*> \|t|\>,k>>|)>|\|>>>|)>||k|\>>>|)>||k|\>>\s\s <\equation*> \|t|\>,k>>|)>|\|>\t>||k|\>>\t>||k|\>>\|t|\>2\t>, and then, as for >, we have \; <\equation*> >\t2+d/r|)>> |)>|)>\>t||t|\>>2-d/r|)>>\t>|)>\1, as claimed. From these estimates standard arguments give eq.. <\lemma> <\equation*> \|\|\<\|\|\>>>|]>\T \ This implies that |3|\>>\C|]>,B>|)>\L,B>|)>> for any \> uniformly in the volume and |3|\>>\C|]>,>|)>\L,>|)>> Estimating the cube\ <\equation*> \=12|W|\>=24,k,k>e+k+k|)>*\x>>\w>|)>\w>|)>\w>|)> We get for any , by space homogeneity, <\equation*> \|\|\<\|\|\>>>|]>=\,k,k>>>\w>|)>\w>|)>\w>|)>|\|>|]> <\eqnarray*> |\|\<\|\|\>>>|]>>||,k,k>>>\w>|)>\w>|)>\w>|)>|\|>|]>>>||>|,k,k>>>\w>|)>\w>|)>\w>|)>|\|>|]>>>|||,k,k>>>>|)>||k|\>>\>|)>||k|\>>\s\\s>>>> Now\ <\eqnarray*> ||,k,k>>>>|)>||k|\>>\>|)>||k|\>>\s\\s>>||>|,k,k>>|)>||k|\>>\>|)>||k|\>>\s\\s>>|||||k|\>>\s|)>>>||>|>>>> Now the second properties follow by the fact that > is supported in an annulus of radius , so\ <\equation*> |\|3|\>>|\<\|\|\>>>>=|||D|\>>\|\<\|\|\>>>>\|\\|\<\|\|\>>>>\|t|\>>|t|\>|\|\<\|\|\>>> and Hoelder estimates follow by Besov-embedding. <\bibliography|bib|tm-plain|qft-drift> <\bib-list|50> S.AlbeverioS.Kusuoka. The invariant measure and the flow associated to the $\\Phi \B4_3$-quantum field model. Nov 2017. Sergio Albeverio, Song LiangBoguslav Zegarlinski. Remark on the integration by parts formula for the >-quantum field model. , 9(1):149\U154, 2006. H.Bahouri, J.-Y.CheminR.Danchin. . Springer, jan 2011. R.Bauerschmidt, D.C.BrydgesG.Slade. A renormalisation group method. III. Perturbative analysis. , mar 2014. ArXiv: 1403.7252. G.Benfatto, M.Cassandro, G.Gallavotti, F.Nicoló, E.Olivieri, E.PresuttiE.Scacciatelli. Ultraviolet stability in Euclidean scalar field theories. , 71(2):95\U130, jun 1980. C.Borell. Diffusion equations and geometric inequalities. , 12(1):49\U71, 2000. M.BouéP.Dupuis. A variational representation for certain functionals of Brownian motion. , 26(4):1641\U1659, oct 1998. A.Braides. . Oxford Lecture Series in Mathematics and Its Applications, 22. Clarendon Press, 1, 2002. Haïm BrezisPetru Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. , 35(5):1355\U1376, 2018. D.C.Brydges. Lectures on the renormalisation group. , 16, 7\U93. Amer. Math. Soc., Providence, RI, 2009. D.C.Brydges, R.FernándezC.Ecublens. . 1993. D.C.Brydges, J.FröhlichA.D.Sokal. A new proof of the existence and nontriviality of the continuum > and > quantum field theories. , 91(2):141\U186, 1983. D.C.BrydgesT.Kennedy. Mayer expansions and the Hamilton-Jacobi equation. , 48(1-2):19\U49, jul 1987. D.C.BrydgesG.Slade. A renormalisation group method. I. Gaussian integration and normed algebras. , mar 2014. ArXiv: 1403.7244. D.C.BrydgesG.Slade. A renormalisation group method. II. Approximation by local polynomials. , mar 2014. ArXiv: 1403.7253. D.C.BrydgesG.Slade. A renormalisation group method. IV. Stability analysis. , mar 2014. D.C.BrydgesG.Slade. A renormalisation group method. V. A single renormalisation group step. , mar 2014. ArXiv: 1403.7256. D.Brydges, J.DimockT.R.Hurd. The short distance behavior of >. , 172(1):143\U186, 1995. R.CatellierK.Chouk. Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation. , oct 2013. ArXiv: 1310.6869. A.Dal Maso. >-Convergence>. Birkhäuser Boston, 1993. J.Feldman. The *\> field theory in a finite volume. , 37:93\U120, 1974. W.H.FlemingH.M.Soner. . Springer, 2nd, nov 2005. H.Föllmer. An entropy approach to the time reversal of diffusion processes. ProfM.MetivierProfE.Pardoux, , 69Lecture Notes in Control and Information Sciences, 156\U163. Springer Berlin Heidelberg, 1985. M.FurlanM.Gubinelli. Weak universality for a class of 3d stochastic reaction\Udiffusion models. , may 2018. J.Glimm. Boson fields with the > interaction in three dimensions. , 10(1):1\U47, 1968. J.GlimmA.Jaffe. Positivity of the > Hamiltonian. , 21(7):327\U376, jan 1973. M.GubinelliM.Hofmanová. Global solutions to elliptic and parabolic > models in Euclidean space. , apr 2018. ArXiv: 1804.11253. M.Gubinelli, P.ImkellerN.Perkowski. Paracontrolled distributions and singular PDEs. , 3:0, 2015. Massimiliano GubinelliMartina Hofmanova. A PDE construction of the Euclidean > quantum field theory. Oct 2018. Massimiliano Gubinelli, BarisEvren UgurcanImmanuel Zachhuber. Semilinear evolution equations for the anderson hamiltonian in two and three dimensions. 2018. A.GulisashviliM.A.Kon. Exact Smoothing Properties of Schrödinger Semigroups. , 118(6):1215\U1248, 1996. M.Hairer. A theory of regularity structures. , 198(2):269\U504, mar 2014. Martin HairerWeijun Xu. Large scale behaviour of 3d phase coexistence models. , jan 2016. ArXiv: 1601.05138. Martin HairerWeijun Xu. Large-scale limit of interface fluctuation models. , feb 2018. ArXiv: 1802.08192. A.Jakubowski. The Almost Sure Skorokhod Representation for Subsequences in Nonmetric Spaces. , (42(1)):167\U174, 1998. S.Janson. . Cambridge University Press, jun 1997. A.Kupiainen. Renormalization Group and Stochastic PDEs. , 17(3):497\U535, mar 2016. J.Lehec. Representation formula for the entropy and functional inequalities. , 49(3):885\U899, 2013. J.MagnenR.Sénéor. The infinite volume limit of the > model. , 24(2):95\U159, 1976. J.-C.MourratH.Weber. The dynamic > model comes down from infinity. , jan 2016. Jean-Christophe MourratHendrik Weber. Global well-posedness of the dynamic > model in the plane. 45:0, 01 2015. Jean-Christophe Mourrat, Hendrik WeberWeijun Xu. Construction of > diagrams for pedestrians. , oct 2016. ArXiv: 1610.08897. F.OttoH.Weber. Quasilinear SPDEs via rough paths. , may 2016. ArXiv: 1605.09744. Y.M.Park. The *\> Euclidean quantum field theory in a periodic box. , 16(11):2183\U2188, 1975. J.Polchinski. Renormalization and effective lagrangians. , 231(2):269\U295, jan 1984. D.RevuzM.Yor. . Springer, 3rd, dec 2004. Hans Triebel. . Springer, 1992. A.S.Üstünel. Variational calculation of Laplace transforms via entropy on Wiener space and applications. , 267(8):3058\U3083, 2014. K.G.Wilson. The renormalization group and critical phenomena. , 55(3):583\U600, 1983. X.Zhang. A variational representation for random functionals on abstract Wiener spaces. , 49(3):475\U490, 2009. \; \; \; In this section we will recall some well known results about Besov spaces, embeddings, Fourier multipliers and paraproducts. The reader can find full details and proofs in and for weighted spaces in . First recall the definition of Littlewood\UPaley blocks. Let ,\> be smooth radial functions \\> such that\ <\itemize> \B>, \B\B>; \,\\1>, |)>+0>\\|)>=1> for any \\>; \|)>\supp \\|)>=\> if \1>. Introduce the notations =\>, =\\|)>> for 0>. For any |)>> we define the operators f\\|)>|)>>, -1>. <\definition> We say a function :\\\> of the form =|x|\>>> for \0> is a weight. <\definition> \ For a weight > let\ <\equation*> |f|\<\|\|\>>|)>>=>|\|>\ \x|)> and by |)>> the space of functions for which this norm is finite. For function defined on a torus in > we consider their periodic extensions on >. <\definition> Let \,p,q\|]>>, and > be a weight. For a Schwarz distribution |)>> define the norm\ <\equation*> |f|\<\|\|\>>|)>>=||\f|\<\|\|\>>|)>>|)>-1>|\<\|\|\>>>. Then the space >(r is the set of functions in |)>> such that this norm is finite. We denote =B|\|>|)>> the normalized Besov space and =B> the Sobolev spaces, and by =B,\>> the (unweighted) Hoelder spaces. <\definition> Let \> and > be a weight. Then we denote by\ <\equation*> |f|\<\|\|\>>|)>>=||\\f|\<\|\|\>>>>|)>-1>|\<\|\|\>>>> and by |)>> the space of Schwarz distributions such that this norm is finite.\ <\proposition> Let \0>.We have for any ,q\|]>,q\q> <\equation*> |f|\<\|\|\>>>>\|f|\<\|\|\>>>>\|f|\<\|\|\>>>>> Furthermore, if we denote by > the normalized fractional Sobolev spaces then for any |]>> <\equation*> |f|\<\|\|\>>>\|f|\<\|\|\>>,p>>\|f|\<\|\|\>>>>> <\proposition> For any ,s\\> such that \s>, any |]>> the Besov space >>> is compactly embedded into >>. <\definition> Let >|)>>. We define the paraproducts and resonant product <\equation*> f\g=g\f\i-1>\f\g,f\g\\1>\f\g. Then <\equation*> f*g=f\g+f\g+f\g. <\proposition> Let >|)>>. We define the paraproducts and resonant product by <\equation*> f\g=g\f\i-1>\f\g,f\g\\1>\f\g. Then <\equation*> f*g=f\g+f\g+f\g. Moreover for any weight >, \0,\\\> and ,p\|]>>, >+>=> we have the estimates\ <\equation*> |f\g|\<\|\|\>>+\>|)>>>|>||f|\<\|\|\>>>,\>|)>>|g|\<\|\|\>>>,q>|)>>,>>>>> and for any ,\\\> such that +\\0> the estimates\ <\equation*> |f\g|\<\|\|\>>+\>|)>>>|>||f|\<\|\|\>>>,\>|)>>|g|\<\|\|\>>>,q>|)>>,>>>>> For a proof see Theorem3.17 and Remark3.18 in. <\proposition> For any weight ,\\0,\\\> we have\ <\equation*> |f\g|\<\|\|\>>+\>|)>>>|>||f|\<\|\|\>>>|)>>|g|\<\|\|\>>>|)>>,>>>>> The proof is an easy modification of the proof of Theorem 3.17 in. <\proposition> Let \> ,\\\> such that +\\0>, +\+\\0> and ,p,p,p\|]>> such that >+>+>=>. Then there exists a bounded trilinear form > such that for any \0>, <\equation*> |\|\<\|\|\>>,\>+\+\>>\|g|\<\|\|\>>,\>>>|f|\<\|\|\>>,\>>>|h|\<\|\|\>>,\>>>, and when > we have <\equation*> \=g|)>\h-gh|)>. <\proof> The proof is a slight modification of the one given in Lemma 2.97 from and interpolation implies that |\f g-\|\<\|\|\>>>\2>|f|\<\|\|\>>,p>>|g|\<\|\|\>>>>>. This in turn gives after some algebraic computations(see )that\ <\equation*> \g|)>=f|)>\g+R with |R|\<\|\|\>>>\2+\|)>>|f|\<\|\|\>>,\>>>|g|\<\|\|\>>,\>>>>. Now to prove the statement of the proposition observe that for smooth we have\ <\equation*> \=-1>\1>\\g|)>\h-\g\f\h Now observe that the term \g> has Fourier transform outside of B> for some Ball independent of , so choosing large enough we can rewrite the sum as\ <\equation*> \=-1>\1>i+N>f\g\h+Rg|)>|)>-\g\f\h <\equation*> -1>\1>i+N>Rg|)>\h-i+N>\g\f\h Now we estimate the norm of the two terms separately. First note that for fixed \ <\equation*> -1>\1>i+N>Rg|)> has a Fourier transform supported in B>. By Lemma 2.69 from it is enough to get an estimate on |2+\+\|)>j>-1>\1>i+N>Rg|)>\h|\<\|\|\>>>> to estimate it in ,\>+\+\>>, so by Hölder inequality <\eqnarray*> |\1>R-1>\g|)>\h|\<\|\|\>>>>|>|\1>2+\|)>>2>|g|\<\|\|\>>,\>>>|f|\<\|\|\>>,q>>>|h|\<\|\|\>>,q>>>>>||>|+\+\|)>>|g|\<\|\|\>>,\>>>|f|\<\|\|\>>,q>>>|h|\<\|\|\>>,q>>>>>>> For the second term observe that for fixed the Fourier transform of\ <\equation*> -1>\1>i+N>\g\f\h is supported in B>. Now we can estimate again by Hölder inequality\ <\eqnarray*> |>||-1>\1>i+N>\g\f\h|\<\|\|\>>>>>||>|k>-1>2+\|)>k>i+N>|g|\<\|\|\>>,\>>>|f|\<\|\|\>>,\>>>|h|\<\|\|\>>,\>>>>>||>|+\+\|)>>|g|\<\|\|\>>,\>>>|f|\<\|\|\>>,q>>>|h|\<\|\|\>>,q>>>>>>> <\equation*> \; <\proposition> Assume \>, ,\\\> such that +\\0>, and +\+\=0> , >+>+>=1> and >+>=1>. Then there exists a bounded trilinear form > for which <\equation*> |\|>\|f|\<\|\|\>>,\>>>|g|\<\|\|\>>,q>>>|h|\<\|\|\>>,q>>> and <\equation*> \=|\|>>>g|)>h-h|)>g|]> for smooth functions. <\proof> This is modification of the proof of Lemma A.6 in . Repeating an algebraic computation given in in the proof of Lemma A.6, we get that for smooth we have\ <\equation*> \=k-1,\L>-k,1\\L>|)>\\g,\h\f\ Then we estimate\ <\eqnarray*> |\|>>|>|k-1,\L>-k,1\\L>|)>\\g,\h\f\>>||>|k,j\k>\g,\h\f\|\|>>>||>|k,j\k>|\f|\<\|\|\>>>>|\g|\<\|\|\>>>>|\h|\<\|\|\>>>>>>||>| k>|\f|\<\|\|\>>>>|)>k,j\k>2+\|)>k>|\g|\<\|\|\>>>>|\h|\<\|\|\>>>>>>||>||f|\<\|\|\>>,\>>>|g|\<\|\|\>>,q>>>|h|\<\|\|\>>,q>>>>>>> <\proposition> There exists a family |)>0>> of bounded multilinear forms on >\>\H>\H>> such that for smooth ,\,g>,g>> it holds <\equation*> ,\\g>,g>|)>=\g>|)>J\g>|)>-\\J\|)>g>g>|]>, and\ <\equation*> ,\,g>,g>|)>|\|>\|t|\>>>|\|\<\|\|\>>>>|\|\<\|\|\>>>>|g>|\<\|\|\>>>>|g>|\<\|\|\>>>>, for some \0>. Note that |t|\>J> satisfies the assumptions of Proposition and with , therefore using also Proposition <\equation*> |J\g>|)>-J\\g>|\<\|\|\>>-\>>\|t|\>|\|\<\|\|\>>>>|g>|\<\|\|\>>>> and therefore <\eqnarray*> ||\g>|)>-\\g>|)>|]>J\g>|)>|\|>>>||>||J\g>|)>-J\\g>|\<\|\|\>>-\>>|J\g>|)>|\<\|\|\>>+\>>>>||>||t|\>|\|\<\|\|\>>>>|g>|\<\|\|\>>>>|t|\>>|\|\<\|\|\>>>>|g>|\<\|\|\>>>>>>>> and by symmetry also\ <\eqnarray*> ||\g>|)>J\g>|)>-\\g>|)>\\g>|)>|]>|\|>>>||>||t|\>>|\|\<\|\|\>>>>|g>|\<\|\|\>>>>|\|\<\|\|\>>>>|g>|\<\|\|\>>>>>>>> \ Furthermore from Proposition and for sufficiently small ,\> <\eqnarray*> ||\\g>|)>\\g>|)>-\\g>|)>\J\|)>g>|\|>>>||>||J\|\<\|\|\>>-\>> |g>|\<\|\|\>>>> |J\|\<\|\|\>>>>|g>|\<\|\|\>>>>>>||>||t|\>>|\|\<\|\|\>>>> |g>|\<\|\|\>>>> |\|\<\|\|\>>>>|g>|\<\|\|\>>>>>>>> and applying Proposition-\ <\eqnarray*> |||\>\g>|)>\J\-\\J\|)>>|)>|\<\|\|\>>>>>>||>||J\|\<\|\|\>>-\>> |g>|\<\|\|\>>>> |J\|\<\|\|\>>>>>>||>||t|\>>|\|\<\|\|\>>>> |g>|\<\|\|\>>>> |\|\<\|\|\>>>>>>>> and putting things together gives the estimate.\ \; > <\definition> A smooth function > is said to be an > multiplier if for every multiindex > there exists a constant >> such that\ <\equation> >|\\>>f|)>|\|>\>|\|>|)>|\|>>,\\\. We say that a family > is a uniformly > multiplier if is satisfied for every with >> independent of .\ <\proposition> Let > be an > multiplier, \>, |]>>, and B|)>>, then\ <\equation*> |\|)>f|\<\|\|\>>>\|f|\<\|\|\>>>. Furthermore the constant depends only on and the constants >> in . For a proof see Lemma 2.78. <\proposition> Assume 0>, \,\\\>. Let > be an > multiplier and ,p\|]>>, >+>=>, B,\>>>, B,\>>>. Then for any \0>. <\equation*> |\|)>g|)>-|)>f\g|)>|\<\|\|\>>+\-m-\>>\|f|\<\|\|\>>,\>>>|g|\<\|\|\>>,\>>>. The constant depends only on ,\,\> and the constants in . For a proof see Lemma 2.99. <\proposition> Let > ,p> and ,s> be such that =|p>+|p>> and s+|)>s> and assume that W,p>\W,p>>. Then\ <\equation*> |f|\<\|\|\>>>\|f|\<\|\|\>>>,p>>|f|\<\|\|\>>>,p>>. For a proof see . \; <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> wilson_renormalization_1983 polchinski_renormalization_1984 brydges_functional_1993 brydges_short_1995 brydges_lectures_2009 brydges_renormalisation_2014_3 brydges_renormalisation_2014_2 bauerschmidt_renormalisation_2014 brydges_renormalisation_2014_1 brydges_renormalisation_2014 brydges_short_1995 brydges_mayer_1987 fleming_controlled_2005 boue_variational_1998 ustunel_variational_2014 zhang_variational_2009 lehec_representation_2013 borell_diffusion_2000 gubinelli_paracontrolled_2015 albeverio_remark_2006 gubinelli_pde_2018 hairer_large_2016 hairer_large-scale_2018 furlan_weak_2018 benfatto_ultraviolet_1980 glimm_boson_1968 glimm_positivity_1973 feldman_lambda_1974 park_lambda_1975 benfatto_ultraviolet_1980 magnen_infinite_1976 brydges_new_1983 hairer_theory_2014 gubinelli_paracontrolled_2015 catellier_paracontrolled_2013 otto_quasilinear_2016 kupiainen_renormalization_2016 mourrat_dynamic_2016 albeverio_invariant_2017 gubinelli_pde_2018 gubinelli_global_2018 albeverio_invariant_2017 revuz_continuous_2004 follmer_entropy_1985 boue_variational_1998 boue_variational_1998 ustunel_variational_2014 follmer_entropy_1985 fleming_controlled_2005 ustunel_variational_2014 lehec_representation_2013 janson_gaussian_1997 janson_gaussian_1997 gulisashvili_exact_1996 mourrat_construction_2016 gubinelli_global_2018 Triebel_1992 gubinelli_global_2018 braides_gamma_convergence_2002 DalMaso1993 jakubowski bahouri_fourier_2011 gubinelli_paracontrolled_2015 gubinelli_global_2018 mourrat_plane_2015 mourrat_plane_2015 mourrat_plane_2015 gubinelli_paracontrolled_2015 bahouri_fourier_2011 gubinelli_paracontrolled_2015 bahouri_fourier_2011 gubinelli_hamiltonian_2018 gubinelli_hamiltonian_2018 bahouri_fourier_2011 bahouri_fourier_2011 Brezis_2017 <\associate|table> |1>||Regularities of the various stochastic objects, the domain of the time variable is understood to be ||]>>. Estimates in these norms holds a.s. and in |L|)>> for all |p\1> (see Lemma |->|-0.3em|>|0em||0em|>>).>|> <\associate|toc> |math-font-series||1.Introduction > |.>>>>|> |math-font-series||2.A stochastic control problem> |.>>>>|> |math-font-series||3.Two dimensions> |.>>>>|> |math-font-series||4.Three dimensions> |.>>>>|> |math-font-series||5.Bounds> |.>>>>|> |math-font-series||6.Gamma convergence> |.>>>>|> |math-font-series||7.Analytic estimates> |.>>>>|> |math-font-series||8.Stochastic estimates> |.>>>>|> |math-font-series||Bibliography> |.>>>>|> |math-font-series||Appendix A.Besov spaces and paraproducts> |.>>>>|>