> <\body> <\hide-preamble> |->> > > > > > > > > > > > > > > >>>> > \; |> \; IAM University of Bonn, Germany >>||<\author-affiliation> IAM & HCM University of Bonn, Germany >>> > Euclidean field theory on a periodic domain. The key ingredient is a variational formula for the partition function involving a stochastic control problem along the flow of a scale regularization parameter. Techniques originating from the analysis of singular SPDEs give uniform bounds allowing to prove the boundedness of the variational functional as the cutoff is removed. The control problem reveals itself as a rigorous and efficient version of Wilson's continous renormalization group.>|> The > Euclidean quantum field theory on the dimensional torus =\=/\|)>|)>> is the probability measure > obtained as the weak limit for \> of the family of measures |)>> given by\ <\equation> \\|)>= exp>|)>-a \|)>|]>\\|]>\\|)>. Here > is the Laplacian on >, > is the centered Gaussian measure with covariance |)>>, > is a normalization factor, > a given constant and =\\\> with > some appropriate smooth and compactly supported cutoff function such that \\> as \>. The measures ,\> are realized as probability measures on |)>>, the space of distributions on >. It is well known that they are supported on the Sobolev space /2-\,p>|)>> for all small \0> and 2>. The existence of the limit > is conditioned on the choice of an appropriate sequence of |)>>. \; The aim of this paper is to present a proof of convergence using stochastic calculus along the direction given by the continuous scale cutoff together with a variational formula for the partition function > and for the generating function of the measure >. Our work can be seen as an attempt to pursue 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 take the stand of trying to avoid the heavy 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. This variational formula has been used by Lehec to reprove some Gaussian functional inequalities. 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.\ \; The main result of the paper is the following bound. <\theorem> Let . There exist renormalization constants ,\>, which depend polynomially on >, and functions |)>,E|)>> not depending on >, such that\ <\equation*> lim\0>|)>|\>=0, and for any \0> <\equation*> E|)>\-|\|>>logexp>|)>-\ \|)>-\|]>\\|]>\\|)>\E|)>. A similar statement for will be proven below in order to introduce some of the ideas on which the proof is based. The method of proof give also boundedness of the moment generating function of > uniformly in allowing to prove convergence in law, at least along subsequences |)>>, and some additional informations on the pathwise behaviour of the interacting fields. We do not pursue further these developments since it is also clear from the methods we introduce that the full convergence of the sequence of measures |)>> and of the corresponding variational problems can be obtained using arguments of >\Uconvergence which we leave to a further publication.\ \; The construction of the > measure in finite volume is basic problem of constructive quantum field theory to which much literature has been devoted. Is not our aim to provide here a comprehensive review of this literature.The case is much simpler than and a large amount of work has been devoted to this situation. 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 fundational work on Glimm and Jaffe which was partly 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 quantisation program to a point where now can be used to prove directly existence of the finite volume > measure in two different ways. Some results on the existince of the infinite volume dynamics have been obtained recently. For a recent overview of the status of the constructive program wrt. the analysis of the > models the reader can consult the introduction to. \; The 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 existence of the > measure in Section. We pass then to the more involved situation in three dimensions in Section and give the proof of Theorem. In Section we collect the proof of renormalizability of the stochastic terms and related probabilistic estimates. Finally the appendix contains background material on functional spaces and paraproducts and also some auxiliary results. The reader is invited to skim over the appendix in order to be introduced to the notations used in the paper.\ <\acknowledgments*> The authors are partially supported by the German DFG via CRC 1060. We begin by constructing a probability space > endowed with a process C,W/2-\,p>|)>|)>> such that >>|)>=\> and >|)>=Law>|)>>.\ \; Let us fix some notations and objects. Let |a|\>=|)>>. Denote with |)>> the space of Schwartz functions on > and with |)>> the dual space of 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 Lebesgue measure on > to have unit mass. We denote the normalized integral and measure by <\equation*> f\|\|>>>f ,x=|\|>>\x. Norms in all the related functional spaces (Lebesgue, Sobolev and Besov spaces) are understood similarly normalized unless \ stated otherwise. The various contants appearing in the estimates will be understood independent of >\ \ Fix \-d/2> and let \C;H>|)>>, |)>> 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 ,|)>>. On the measure space ,,\|)>> there exists a collection |)>\>> of complex (2-dimensional) Brownian motions, such that |\>=B>, \ ,B> independent for \n> and =\>ei|n,\|\>>B>. Note that has a.s. trajectories in ,W,p>|)>|)>> for any \0> and 2> 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 > defined by\ <\equation> Y\\>||n|\>>ei|n,\|\>>dB. It is a centered Gaussian process with covariance\ <\eqnarray*> |Y,\|\>|Y,\|\>|]>>||\>\>\||n|\>>dB |^>||m|\>>dB |^>|\>|]>>>|||\>>||n|\>>|^>|^>|\>,>>>> for any ,\\|)>>, by Fubini's theorem and Ito isometry. Then for any \,\\L|)>> we have \>>|Y,\|\>|Y,\|\>|]>=\>|n|\>|^>|^>|\>>, by dominated convergence.\ Note that up to any finite time the r.v. >Y> has finite 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 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> f|)>\\|)>=|)> e|)>>|]>|>, for any bounded measurable |)>\\> where <\equation*> V|)>\>|)>|\|>-a |)>|\|>-b|)>\\,\\|)>>|]>. The sequence |)>> will be chosen appropriately below in order to decouple the behaviour of the numerator from that of the denominator. The first problem we address is that of giving upper and lower bounds to the quantity > appearing in. Note that, for fixed , the polynomial appearing in the expression for |)>> is bounded below if \0>, in this case > 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 since we will be obliged to take \+\>. In order to obtain uniform upper and lower bounds we proceed as follows. Denote by > the space of progressively measurable processes which are >\Ualmost surely in \\|)>>. We say that an element of > is a . 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|)>, 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, where |)>> is a cylindrical >\UBrownian motion in |)>>. This implies that, under > the process |)>> has the semimartingale decomposition =W+U> with <\equation*> W\J\X,U=I|)>, where for any drift \> we define \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|]>. This will be our key formula to get an upper bound on >. 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>>. This 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 pressure > holds: <\equation*> -|\|>>log =-|\|>>log \ |)>>|]>=min\> \|\|>>V+I|)>+>|v|\<\|\|\>>>\s|]>. This formula is nice and easy to prove but somewhat incovenient for certain manipulations since the space > is indirectly defined and the reference measure > depends on the drift . A more straighforward formula has been found by Boué\UDupuis which involves a the fixed canonical measure > and a general adapted drift \>. This formula will be our main tool in the following. <\theorem> The Boué\UDupuis variational formula for the pressure > holds: <\equation*> -|\|>>log =-|\|>>log \ |)>>|]>=inf\> \|\|>>V+I|)>+|\|>|2>>|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 this variational formulas. <\enumerate-alpha> They originates directly from the variational formula for the free energy of a statistical mechanics systems: > playing the role of the internal energy and the quadratic term playing the role of the entropy. Note that 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.\ The Boué\UDupuis formula in Theorem is easier to use than the formula in Lemma since the probability do not depend on the drift . However 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. This requires the notion of functional, introduce by Üstünel. The Boué\UDupuis formula is a non\UMarkovian and infinite dimensional extension of the well known stochastic control problem representation of the Hamilton\UJacobi\UBellman equation in finite dimensions. See Üstünel and Lehec for a streamlined proof of the Boué\UDupuis 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. The space > is the Cameron\UMartin space of the free field. <\lemma> For any L\\|)>> we have <\equation*> supt\T>| I|\<\|\|\>>>\|v|\<\|\|\>>>\s. <\proof> Using the fact that |)>> is diagonal in Fourier space, and denoting with |)>\>> the basis of trigonometric polynomials, we have\ <\equation*> |\|)>v\s,\|)>v\s|\>=|\|)>e,v|\>\s|\|> <\equation*> \|\|)>e,e|\>|\|>\s|)>|e,v|\>|\|>\s|)>\|\|>|v|\<\|\|\>>>\s*sup|e,\|)>e|\>\s <\equation*> \|\|>|v|\<\|\|\>>>\s sup|e,\|)>e|\>\|\|>|v|\<\|\|\>>>\s. Therefore <\equation*> | I|\<\|\|\>>>\||\|\> I|\<\|\|\>>>=|\|>>|\|)>v\s,\|)>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 .\ As a warm up we consider here the case . From Theorem we see that the relevant quantity to bound is of the form <\equation> \\\|\|>>V+I|)>+>|u|\<\|\|\>>,\x|)>>\s|]>, for \>. Let =I> and let =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+>6\WZ+>\Z, where\ <\equation*> |W|\>|)>\W|)>-6\|)>|]> W|)>+3\|)>|]>, <\equation*> |W|\>|)>\W|)>-3\|)>|]>W|)>, <\equation*> |W|\>|)>\W|)>-\|)>|]>, denote the Wick powers of the Gaussian r.v. |)>>. The reader can check the well known property of these polynomials when seen as stochastic processes in , namely that they 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 pressure: <\equation*> -|\|>>log \\\|W|\>|]>=0. As for an effective 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|\<\|\|\>>>\s. 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|\<\|\|\>>>\t+\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\1> such that >+>=>. Then for every such p,p> and 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 , we get <\eqnarray*> |g|\<\|\|\>>,p>>>|>|||\|\>>g|\<\|\|\>>>>\||\|\>>g|\<\|\|\>>>>|g|\<\|\|\>>>>>||>|||\|\>>g|\<\|\|\>>>|g|\<\|\|\>>>\|g|\<\|\|\>>|1-\>>>>|g|\<\|\|\>>>|1-\>>|g|\<\|\|\>>>>>||>||g|\<\|\|\>>|1-\>>>>|g|\<\|\|\>>>|1-\>>.>>>> So <\eqnarray*> f g|\|>>|>||4|)>>>|f|\<\|\|\>>,p>>|g|\<\|\|\>>|1-\>>>>\|4|)>>>|g|\<\|\|\>>>|1-\>>>>||>|||)>|)>>>|f|\<\|\|\>>>>,p>>+\|g|\<\|\|\>>>>+\|g|\<\|\|\>>>|)>.>>>> If > is chosen small enough, is sufficiently large and > is chosen suffiently close to . <\remark> For the case it would have been enough to estimate |g|\<\|\|\>>,p>>>. The stronger estimate will be useful for since there we will only have \W,p>>. Then <\equation> WZ|\|>\C E|)>|W|\<\|\|\>>,p>*>+\*|Z|\<\|\|\>>,2>>+\|Z|\<\|\|\>>*>|)>, and using eqs., and together we obtain for > small enough <\equation> |\|>\O|)>||W|\>|\<\|\|\>>>+||W|\>|\<\|\|\>>,5>>+|W|\<\|\|\>>,p>*>|]>+\\. and therefore <\equation*> \\-C\||W|\>|\<\|\|\>>>+||W|\>|\<\|\|\>>,5>>+|W|\<\|\|\>>,p>*>|]>. Using eq., eq. and Lemma in eq. we get <\eqnarray*> >|>|||W|\>|\<\|\|\>>>+||W|\>|\<\|\|\>>,5>>+|W|\<\|\|\>>,p>*>|]>+\|]>>>||>|||W|\>|\<\|\|\>>>+||W|\>|\<\|\|\>>,5>>+|W|\<\|\|\>>,p>*>|]>.>>>> This last average do not depends anymore on the drift and we are only left to show that\ <\equation*> sup \||W|\>|\<\|\|\>>>+||W|\>|\<\|\|\>>,5>>+|W|\<\|\|\>>,p>>|]>\\. 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. Therefore we have established that <\theorem> For any \0> we have <\equation*> sup|\|>>|\|>\O|)>, where the constant in the r.h.s. is independent of >.\ <\remark> Note this also implies that |)>>> is in all the > spaces wrt. the measure > uniformly in and for any 1> since the above argument remains valid upon replacing > with p>. <\remark> With slightly different ways to estimate the various terms it would be possible to show that the bound in Theorem is actually |)>> as in the case.\ We would like now to pursue a similar strategy when where the regularity of the Wick products is less pleasant. In particular we only have |W|\>\>>, |W|\>\>> uniformly in for any small \0>. This implies that there is no straighforward way to control the terms\ <\equation> |W|\>Z,|W|\>Z, like we did in Section. Indeed 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 further scale dependent renormalization constants ,\> into the energy > beyond Wick ordering and let <\equation*> |\|>>V|)>=|Y|\>-\|Y|\>-\|)>. Repeating the computation from Section we arrive at\ <\eqnarray*> ||\|>>>||\Z+|2>\Z+4\W Z|]>>>|||WZ+\Z|]>+\Z+|u|\<\|\|\>>>\s|]>,>>>> where we introduced the convenient notations <\equation*> \\4|W|\>,\\12|W|\>. The rest of this section will be concerned with lower and upper bounds of this quantity leading to the proof of Theorem. We are going to use the quadratic term |u|\<\|\|\>>>\s> to absorb the mixed terms. In order to do so we need to represent them as quantities flowing along the scale parameter . This will allow to identify suitable counterterms. By Ito's formula\ <\equation*> \\Z=\\>\t+, where we used that |)>> is a martingale and that |)>> is of bounded variation. We now introduce a new drift via the formula <\equation> u=-\J\+w, we have <\equation> Z=-\J\\t+Jw\t=-\\>+K, where <\equation*> K\I,\>\J\\s. Note that |\>=-\J\+Jw>. It holds <\equation*> \\Z+|u|\<\|\|\>>>\s=-\\|)>\t+|w|\<\|\|\>>>\s+martingale. Here we have eliminated the mixed divergent term \Z> and replaced it with a divergent but purely stochastic term \|)>\t> which does not affect anymore the variational problem and can be explicitly removed by adding its average to >. This implies in turn that 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|\<\|\|\>>> \s>. <\remark> \ This lack of uniform regularity of the full drift is a sign of the fact that the limiting measure > is expected not to be absolutely continuous wrt. to the Gaussian free field. We do not make this connection rigorous in this paper and leave the discussion to a further publication. \ Taking into account the relation we write <\equation*> |2>\Z=|2>\>|)>-\\\>K+|2>\K. The first term is already purely stochastic and will give a contribution independent of which can again be absorbed in >. \ In order to proceed we will isolate the divergent parts of the product via paracontrolled calculus (see Appendix for details). This is the content of the following\ <\lemma> For any small \0> there exists \0> such that <\eqnarray*> ||\\>K+|2>\K-\\Z |)>K+\\\>|)>K|\|>>>|||C,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>.>>>> <\proof> We compute <\eqnarray*> \\>K+|2>\K>||\Z|)>K-\\\>|)>K>>|||\\>|)>K+|2>\K|)>K>>||||2>\K|)>K-\K|)>K|)>.>>>> By Proposition, <\eqnarray*> \K|)>K-\K|)>K|\|>>|>||\|\<\|\|\>>>> |K|\<\|\|\>>>>>>||>||2|)>>>|\|\<\|\|\>>>> |K|\<\|\|\>>|1-\>|)>>>>\|2|)>>>|K|\<\|\|\>>|1-\>|)>>>>>||>||1/2-3\>>>|\|\<\|\|\>>|2|)>>|)>|)>>>>+\|K|\<\|\|\>>>>>>|||\|K|\<\|\|\>>>.>>>> Furthermore by Proposition, <\eqnarray*> \K|)>K|\|>>|>||\|\<\|\|\>>>> |K|\<\|\|\>>>>>>||>||1/2+\>>|\|\<\|\|\>>|2|)>>|)>|)>>>>+\|K|\<\|\|\>>>>+\\|K|\<\|\|\>>>,>>>> and\ <\eqnarray*> \\>|)>K|\|>>|>||\|\<\|\|\>>/2>> |\>|\<\|\|\>>/2>>>|K|\<\|\|\>>>>>>||>||)> \|\|\<\|\|\>>/2>> |\>|\<\|\|\>>/2>>>|)>+\|K|\<\|\|\>>>>.>>>> The resonant term \\>> will behave well enough after suitable renormalization, as we will see later. As far as the term \Z |)>K> in concerned, we want to absorb this into |w|\<\|\|\>>\s> like we did with the linear term. Before we can do this we must make sure that after applying Ito's formula it still possible to use Z> to control some of the growth of this term since we already have quadratic (critical) dependence on |)>> which cannot be fully taken care of by |w|\<\|\|\>>\s>. Choose :\\\> to be a smooth function such =1> if 1/4> and =0> if 1/2>. Set =\|\|\>/t|)>>. Then \ <\eqnarray*> \|)>>|| for t>,>>>||)>>|| for \c t> 0>.>>>>> By the Mihlin-Hörmander theorem we deduce that the operator > is bounded on > for any p\\>. Let >\\Z> and >\\K>, then \ <\equation*> Z>=\Z=\|\|\>\|)>u \s=\|\|\>\|)>u \s=\Z, so in particular we have |Z>|\<\|\|\>>>\|Z|\<\|\|\>>>> for all T>. On the other hand the functions -Z>> and -K>> are spectrally supported outside of a ball or radius and <\equation*> \\Z|)>K=\\Z>|)>K+\\-Z>|)>|)>K. We estimate the second part with the following: <\lemma> For any small \0> there exists \0> such that <\eqnarray*> \-Z>|)>|)>K|\|>>|>|,\|)>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 observe also that <\equation> |Z|\<\|\|\>>>>\|K|\<\|\|\>>>>+\|\>|\<\|\|\>>>> Therefore we estimate as follows <\equation*> ||\-Z>|)> |)>K|\|>>>||>||\|\<\|\|\>>>>|Z-Z>|\<\|\|\>>>|K|\<\|\|\>>>>>>||>||\|\<\|\|\>>>>|K|\<\|\|\>>|1/2-\>>>>|Z|\<\|\|\>>|1/2-\>>>>>||||1/2-\>>|\|\<\|\|\>>>>|\>|\<\|\|\>>|1/2-\>>>>|K|\<\|\|\>>>>|Z|\<\|\|\>>|1/2-\>>>>>||>||4|)>>>|\|\<\|\|\>>>>|K|\<\|\|\>>|1/2-\>>>>\|4|)>>>|Z|\<\|\|\>>|1/2-\>>>>>||||4|)>>>|\|\<\|\|\>>>>|\>|\<\|\|\>>|1/2-\>>>>|K|\<\|\|\>>>>\|4|)>>>|Z|\<\|\|\>>|1/2-\>>>>>||>|,\|)>\|1/2-3\>>|\|\<\|\|\>>|)>|1/2-3\>>>>+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>>>|||,\|)>\|1/2+\>>|\|\<\|\|\>>|)>|1/2+\>>>>|\>|\<\|\|\>>|1/2+\>>>>+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>>>>>> \; It remains to absorb the remaining term. Applying Ito's formula we obtain\ <\equation*> \\Z>|)>K=\\Z>|)>|\>\t+\\>>|)>K\t+. The second term in the r.h.s. is harmless thanks to the following Lemma <\lemma> For any small \0> there exists \0> such that <\eqnarray*> \>>|)>K\t|\|>>|>|,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>.>>>> <\proof> First note that |\>=|n|\>/t|)>|\>|n|\>/t|)>>. In particular >> is spectrally supported in an annulus with inner radius and outer radius .\ <\equation*> |>>|\<\|\|\>>>=\>|\>|n|\>|t>|)>|n|\>|t>|\|>\\>|\>|n|\>|t>|)>|n|\>|t>|n|\>>|t>>|\|>\|Z|\<\|\|\>>>>|t>>. By Proposition <\equation*> \>>|)>K \t|\|>>|>||\|\<\|\|\>>>>|>>|\<\|\|\>>>|K|\<\|\|\>>>>\t>>||>||\|\<\|\|\>>>>|Z|\<\|\|\>>>>|K|\<\|\|\>>>>t|t>>>>||>||Z|\<\|\|\>>>>supt\T>|K|\<\|\|\>>>>|\|\<\|\|\>>>>t|t>>>>||>||Z|\<\|\|\>>>>>|Z|\<\|\|\>>>>supt\T>|K|\<\|\|\>>>>|\|\<\|\|\>>>>t|t>>>>||>||Z|\<\|\|\>>>>t\T>|K|\<\|\|\>>>>|)>>|\|\<\|\|\>>>>t|t>>|)>>>|||>|Z|\<\|\|\>>>>t\T>|K|\<\|\|\>>>>|)>\>>|||\|\>|\<\|\|\>>>>>|\|\<\|\|\>>>>t|t>>|)>>>||>|\|Z|\<\|\|\>>>+\t\T>|K|\<\|\|\>>>>|)>>>|||,\|)>\|1-8\>>|\|\<\|\|\>>>>t|t>>|)>>>||||1+8\>>|\>|\<\|\|\>>>>>|\|\<\|\|\>>>>t|t>>|)>>>|||\|Z|\<\|\|\>>>+\t\T>|K|\<\|\|\>>>>|)>>>>>> \; So we are left with the singular term \Z>|)>|\>\t>. In order to handle it we need to introduce a further decomposition of the drift via a second paracontrolled Ansatz <\equation> w=*-\J\Z>|)>+l>=-\J\Z>|)>+>, where \I>. Now\ <\equation> |K|\<\|\|\>>>>>|>|>>|\|\<\|\|\>>>>\s |Z|\<\|\|\>>>+|l|\<\|\|\>>>\s>>||>|>>|\|\<\|\|\>>>>\s |)>+\|Z|\<\|\|\>>>+|l|\<\|\|\>>>\s,>>>>> which implies that quadractic functions of the norm |K|\<\|\|\>>>>> with a small coeffient can always be controlled, uniformly in >, by the positive term\ <\equation*> \Z+|l|\<\|\|\>>>\s. Using eq. we compute\ \; <\equation*> \\Z>|)>|\>\t+|w|\<\|\|\>>>\t=-|2>\Z>|)>|)>\t+|l|\<\|\|\>>>\t. We now see how we can renormalize the remaining term with the following Lemma. <\lemma> For any small \0> there exists a \0> such that <\eqnarray*> |2>\\Z>|)>|)>-\\J\|)>>|)>|\|>>|>|,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>.>>>> <\proof> To prove this first note that J> satisfies the assumptions of Proposition with . So\ <\equation*> |J\Z>|)>-\|)>\Z>|\<\|\|\>>>>\|t|\>|\|\<\|\|\>>>>|Z> |\<\|\|\>>>>, and by Proposition, <\equation*> |J\Z>|)>|\<\|\|\>>>>\ |t|\>>|\|\<\|\|\>>>>|Z> |\<\|\|\>>>. Summarizing these considerations we have\ <\equation*> |2>\Z>|)>|)>-\Z>|)>|)>\|)>\Z>|)>|]>\t|\|> <\equation*> \\ supT> |Z> |\<\|\|\>>>|Z> |\<\|\|\>>>>|]>|\|\<\|\|\>>>>t||t|\>>>. Again applying Proposition and Proposition, <\equation*> |2>\Z>|)>|)>\t-|2>\|)>\Z> |)>\t|\|> <\equation*> \\ supT> |Z> |\<\|\|\>>>|Z> |\<\|\|\>>>>|]>|\|\<\|\|\>>>>t||t|\>>>. From Proposition we have\ <\equation*> |2>\\Z>|)>|)>-|2>\\Z>|)>\J\ Z>|\|>\\|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. Finally, applying Proposition we get <\equation*> \|\\Z>|)>\J\-\\J\|)>>|)>|\<\|\|\>>>\\|J\|\<\|\|\>>>> |Z>|\<\|\|\>>\>>. Summarizing we have\ <\equation*> |2>\\Z>|)>|)>-\\J\|)>>|)>|\|> <\equation*> \\ supT> |Z> |\<\|\|\>>>|Z> |\<\|\|\>>>>|]>|\|\<\|\|\>>>>t||t|\>>> <\equation*> \ \ supT>|Z> |\<\|\|\>>>>+\|Z|\<\|\|\>>>|)>+C,\|)>\|\|\<\|\|\>>>>t||t|\>>>|)> <\equation*> \ \ |Z|\<\|\|\>>>>+\|Z|\<\|\|\>>>|)>+C,\|)>\|\|\<\|\|\>>>>t||t|\>>>. and using eq. to control |Z|\<\|\|\>>>>> in terms of > we obtain the claim. We are now left to estimate the cubic term which we rewrite as <\equation*> \WZ=\W>|)>+3\W>|)>K+3\W>|)>K+\WK. Using Lemma we establish that\ <\equation*> \WK|\|>\E|)>|W|\<\|\|\>>,p>>+\|K|\<\|\|\>>>>+\|K|\<\|\|\>>>|)>. <\lemma> For any small \0> there exists \0> such that <\equation*> W>|)>K|\|>+W\>K|\|>\C,\|)>E|)>Q+\|K|\<\|\|\>>>>+\\|Z|\<\|\|\>>>. <\proof> \ We can write, by Proposition, <\equation*> 3\W>|)>K|\|>\6\W>\\>|)>K|\|>+C\|W|\<\|\|\>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>. Decompose <\equation*> W>\\>|)>=W\>\\>|)>+W\>\\>|)>+W\>\\>|)>. We can estimate the first two terms by\ <\equation*> 6\W\>\\>|)>K|\|>\\|W|\<\|\|\>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>, and <\equation*> 6\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|\|>+C\|W|\<\|\|\>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>>>||>||\>|\<\|\|\>>>>|W\\>|\<\|\|\>>>>|K|\<\|\|\>>>>+\|W|\<\|\|\>>>>|\>|\<\|\|\>>>>|K|\<\|\|\>>>>.>>||>|C|)>|\>|\<\|\|\>>>>|W\\>|\<\|\|\>>>>+|W|\<\|\|\>>>>|\>|\<\|\|\>>>>|)>+\|K|\<\|\|\>>>>>>>>> For the last term we estimate <\eqnarray*> \>|)>K|\|>>|>||W\>|\<\|\|\>>>>|K|\<\|\|\>>>>,>>||>|\|2-2\>>>|W\>|\<\|\|\>>>>\|2-2\>>>|K|\<\|\|\>>|1-\>|)>>>|K|\<\|\|\>>|1-\>|)>>>>>||>||1/2-6\>>>|W\>|\<\|\|\>>|1/2-6\>>>>+\|K|\<\|\|\>>>+|K|\<\|\|\>>>>>>>> and we observe that <\equation*> |W\>|\<\|\|\>>>>\|W\\>|\<\|\|\>>>>+|W\\>|\<\|\|\>>>>+|W\\>|\<\|\|\>>>> <\equation*> \|W|\<\|\|\>>>>|\>|\<\|\|\>>>>+|W\\>|\<\|\|\>>>> which allows to conclude using Lemma to bound \\>>. Let us summarize the argument so far. Starting from the expression <\equation*> \\4\|W|\>Z+6\|W|\>Z+4\WZ-WZ+\Z|)>+\Z+|u|\<\|\|\>>>\s, we performed a first change of variables to cancel the term linear in > and get <\equation*> \=|2>\Z+4\WZ-WZ+\Z|)>+\ Z+|w|\<\|\|\>>>\s+\ where the dots stands for terms which are under control via the positive terms appearing in the expression. Next, we expanded the term quadratic in > to get <\equation*> \=\\\>K+|2>\K+4\WZ-WZ+\Z|)>+\ Z+|w|\<\|\|\>>>\s+\ Then rewrote it using commutators <\equation*> \=\\\>|)>K+\\Z|)>K+4\WZ <\equation*> -WZ+\Z|)>+\ Z+|w|\<\|\|\>>>\s+\ Applied an Ito formula to the second term (after a preliminary modification) to get <\equation*> \=\\\>|)>K+\\Z>|)>|\>\t+4\WZ <\equation*> -WZ+\Z|)>+\ Z+|w|\<\|\|\>>>\s+\ Introduced a second paracontrolled Ansatz to cancel the resulting contributions and obtained <\equation*> \=\\\>|)>K-|2>\Z>|)>|)>\t+4\WZ <\equation*> -WZ+\Z|)>+\ Z+|l|\<\|\|\>>>\t+\ Next we applied a commutator lemma to the second term to clarify its renormalizability and expanded the counterterms to get <\equation*> \=\\\>|)>K-|2>\\J\|)>>|)>\t+4\WZ <\equation*> -WK+\>|)>|)>+\ Z+|l|\<\|\|\>>>\t+\ It now remains to handle the terms <\equation> \\\>|)>K-|2>\\J\ |)>>|)>\t, which so far are divergent and need to be renormalized. Recall also that we still have the counterterms we introduced in the beginning, namely\ <\equation*> -2\WZ-\Z. To handle the first term in eq. we consider <\equation*> \\\>|)>K-2\WZ=\\\>-2\\W|)>K+2\\W\> and observe that the average of the second term in the r.h.s. can be absorbed in > while for the first we have <\equation*> \\\>-2\\W|)>K|\|>\\|\\\>-2\\W|\<\|\|\>>>>|K|\<\|\|\>>>>. Using Lemma and eq. we have this term under control. On the other hand, to renormalize the second term in eq., we observe that <\equation> \Z=\>|)>+\Z>-Z>|)>+\-Z>|)>. For the second and third term in eq. in this last expression we have the estimates\ <\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. Deriving in the first term in the r.h.s. of eq. we get\ <\equation*> \>|)>=|\>>|)>\t+20>\Z>>>\t. For the second term in this expression 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. We conclude that <\eqnarray*> |2>\\J\ |)>>|)>\t-\Z>|||2>\\J\-2\|\>|)>>|)>\t+R,>>>> with <\equation*> |\|>\|\|>||T|\>>+|\|>*\t||t|\>>|]>|Z|\<\|\|\>>>+|Z|\<\|\|\>>>>|)>\O|)>|Z|\<\|\|\>>>+|Z|\<\|\|\>>>>|)> which is therefore under control. Note that we have used that |)>> is of order >. Finally we have <\equation*> \\J\-2\|\>|)>>|)>\t|\|>\\|J\\J\-2\|\>|\<\|\|\>>>>\t|]> sup>|>|)>|\<\|\|\>>>> <\equation*> \\|\J\\J\-2\|\>|\<\|\|\>>>>\t|]>|Z|\<\|\|\>>>> which can be controlled again using Lemma. At this point we are in the situation where all the terms which are not positive but depending on the drift can be bounded by <\equation*> E|)>Q+\ Z+|l|\<\|\|\>>>\t|)> with a small constant >. Therefore <\equation*> \\-E|)>Q+|)> Z+|l|\<\|\|\>>>\t|)> and taking the expectation we obtain the claimed lower bound in Theorem. For the upper bound we note that the bounds we established give <\equation*> \\E|)>Q+|)> Z+|l|\<\|\|\>>>\t|)> for all admissible drift \>. Therefore it is enough to choose a convenient drift for which <\equation> sup\ Z+|l|\<\|\|\>>>\t|)>\\. Choosing equal to zero, like in the two dimensional case, will not work since then we would not get a useful lower bound due to the presence of the diverging constant .> Instead we need to use a drift corresponding to the choice in eq., more precisely it would have to satify the equation\ <\equation> Z=-\\>->J\Z>|)>\s, for fixed > large enough and T>. It will then be sufficient to find such that <\equation> v=-\J\-\\T>J\\I|)> for all 0>. The reason for introducing > is that we can only solve and control the moments of the solution of eq. for large values of >. We proceed via a fixpoint argument. For any \> consider the affine map\ <\equation*> v\F=-\J\-\\T>J\\I|)>. Now fix some \-1/2>, and \0> and introduce the norm <\equation*> |v|\>\>es>|v|\<\|\|\>>>>\s. Then observe that <\equation*> |F|\<\|\|\>>>>\\|J\|\<\|\|\>>>>+\|J\\I|)>|\<\|\|\>>>> <\equation*> \\|s|\>|)>/2>|\|\<\|\|\>>-1-\>>+\|s|\>|)>/2>|\|\<\|\|\>>-1-\>>|I|\<\|\|\>>> <\equation*> \C\|s|\>|)>/2>|\|\<\|\|\>>-1-\>>+C\\T>|s|\>|)>/2>|\|\<\|\|\>>-1-\>>|v|\<\|\|\>>,L|)>>, where we used the same computation as in Lemma , the fact that > is a bounded operator on >> and the bound on the operator norm of |)>> to get the |s|\>|)>/2>> factor. Therefore we can estimate\ <\equation*> |F|\>=>es>|F|\<\|\|\>>>>\s <\equation*> \C\>|\|\<\|\|\>>-1-\>>s>\s||s|\>>>+C\>|v|\<\|\|\>>>\r>\T>|\|\<\|\|\>>-1-\>>s>\s||s|\>>> <\equation*> \C\>|\|\<\|\|\>>-1-\>>s>\s||s|\>>> <\equation*> \; <\equation*> +C\>>\T>|\|\<\|\|\>>-1-\>>s||s|\>>>|)>>es>\s|)>|v|\<\|\|\>>>\r <\equation*> \C\>|\|\<\|\|\>>-1-\>>s>\s||s|\>>>+||)>>>>|\|\<\|\|\>>-1-\>>s||s|\>>>|)>>er>|v|\<\|\|\>>>\r . <\equation*> \; Choosing > large enough so that <\equation*> ||)>>>>|\|\<\|\|\>>-1-\>>s||s|\>>>|)>\, we have <\equation*> |F|\>\C\>|\|\<\|\|\>>-1-\>>s>\s||s|\>>>+|v|\>. Similarly <\equation*> |F-F|)>|\>\|v-v|\>. These bounds allows to apply a Picard iteration and obtain a drify \> such that > and satisfying the bound <\equation*> |v|\>\2C\>|\|\<\|\|\>>-1-\>>s>\s||s|\>>>. In order to use this drift in the variational argument we need to control in expectation the positive term in eq.. Now\ <\equation*> l=v+\J\+\J\\I|)>=\\T>J\\I|)>,s\0. Using that I=\I>> for all s\T> we have <\equation*> |l|\<\|\|\>>>\s\\>|J\\I>|)>|\<\|\|\>>>\s\\|I>|\<\|\|\>>>>|s|\>>|\|\<\|\|\>>>>\s <\equation*> \\|\\>>|\<\|\|\>>>+\>|s|\>>|\|\<\|\|\>>>>\s|)>, since >=-\\>>>. The expectation of this quantity is clearly finite. In order to control |)>> uniformly in we observe that for T>, <\eqnarray*> |Z|\<\|\|\>>>>>|>||J\|\<\|\|\>>>>\s+\>|J\Z> |)>|\<\|\|\>>>>\s>>||>||s|\>>>||\|\>\|\<\|\|\>>>>\s+\>|s|\>>>||\|\>\|\<\|\|\>>>> |Z|\<\|\|\>>>>\s.>>>> Using Gronwall's lemma, taking expectation of the -th moment and apply Cauchy-Schwarz inequality we obtain\ <\eqnarray*> |||Z|\<\|\|\>>>>|]>>>||>|>||\|\>\|\<\|\|\>>>>s||s|\>>>|)>|]>\>>||\|\>\|\<\|\|\>>>>s||s|\>>>|)>|]>>>||>|\>||\|\>\|\<\|\|\>>>>s||s|\>>>|]>\>>s|s/2>>exp||T|\>/2>>||\|\>\|\<\|\|\>>>>|)>|]>>>||>|\>|\|\<\|\|\>>>>s||s|\>>>|]>*supT> \||T|\>/2>>||\|\>\|\<\|\|\>>>>|)>|]>,>>>> where the second line was obtained by applying Jensen's inequality on the finite measure space ,\|)>,|s|\>/2>\s|)>>. The first factor is known to be finite and for the second we have the following lemma which allows to conclude that the average is finite uniformly in and prove the lower bound for Theorem. <\lemma> If > is large enough only depending on > we have <\equation*> supT> \||T|\>/2>>|\|\<\|\|\>>>>|)>|]>\\. <\proof> It is well known that random variables in the second Gaussian chaos are exponentially integrable for small coefficients. However in order to give a complete proof and show some other uses of the variational formula we give a self contained proof in our particular case. Fix 0>. If we apply Theorem with the bounded random variable =2*p*|T|\>/2>|\|\<\|\|\>>>>\N> we deduce that\ <\equation*> log \||T|\>/2>> |\|\<\|\|\>>>>\N|)>|]> <\equation*> =sup\> \||T|\>/2>>|\+24WU+U|\<\|\|\>>>>\N-|\|>|2>|u|\<\|\|\>>>\t|]>. <\equation*> \sup\> \||T|\>/2>>|\+24WU+U|\<\|\|\>>>>-|\|>|2>|u|\<\|\|\>>>\t|]>. where =I>. By Morrey\<#2019\>s inequality and Sobolev embedding, for large enough , <\equation*> |U|\<\|\|\>>>>\||\|\>U|\<\|\|\>>>\|U|\<\|\|\>>>\|U|\<\|\|\>>>\|U|\<\|\|\>>>\|u|\<\|\|\>>>\t. In order to handle the mixed term we decompose U=W\U+W\U> and estimate each term by Besov's embedding and paraproduct bounds as <\equation*> |W\U|\<\|\|\>>>>\|W\U|\<\|\|\>>>>\|W|\<\|\|\>>>>|U|\<\|\|\>>>, and <\equation*> |W\U|\<\|\|\>>>>\|W|\<\|\|\>>>>|U|\<\|\|\>>>\|W|\<\|\|\>>>>|U|\<\|\|\>>>. By Young's inequality\ <\equation*> |WU|\<\|\|\>>>>\|W|\<\|\|\>>>>+|u|\<\|\|\>>>\t. Therefore, for large enough \0> (depending only on >) <\eqnarray*> ||||T|\>/2>>|+24WU+U|)>|\<\|\|\>>>>-|\|>|2>|u|\<\|\|\>>>\t|]>>>||>|||T|\>/2>>|\|\<\|\|\>>>>+p\|2 |T|\>/2>>|W|\<\|\|\>>>>+||T|\>/2>>-|\|>|2>|]>|u|\<\|\|\>>>\t|]>>>||>|||T|\>/2>>||\|\>\|\<\|\|\>>>>+p\|2 |T|\>/2>>|W|\<\|\|\>>>>|]>>>>> which is bounded uniformly in T>. This allows us to conclude. 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>\ <\equation> sup\|W\W>|\<\|\|\>>>>|]>\\. Moreover there exists a function \C,\|)>> such that for any \0> and any 1> <\equation> sup \|\\>-2\\W|)>|\<\|\|\>>>>|]>\\, <\equation> \>|J\\J\-2\|\>|\<\|\|\>>>>\t|)>|]>\\. and <\equation> |\|>+|t|\>|\>|\|>\\log|t|\>,t\0. <\proof> We will concentrate in proving the bounds on the renomalized terms in eq. and and leave to the readed 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>|)> <\equation*> +,\,k>>G>3>|)>\w>|)>\\w>|)>+>>>|)>\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 have also <\equation*> >3>|)>|\|>\\Sh>3|)>>|)>|\|>j>\r||p|\>>>+p|)>K\|)>>-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|\>>\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*> >|)>=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 now 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|\>>\\log|T|\>. Now <\equation*> A,k|)>-2\\=\6|)>,q>\r\r>|)>||q|\>>>|)>||q|\>>\ <\equation*> \j>K+q|)>-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|\>>|)>\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=>>>5>|)>\y>|)>|\|>|]>|}>\q 2,q\-1. All together these estimates imply that <\equation*> \|\\\>-2\\W|)>|\<\|\|\>>>\|)>,q\-1. Standard argument allows to deduce eq.. The analysis of the other renormalized product proceeds similarly. Let \J\\J\-2\|\>>. First note that by Besov embedding we have <\equation*> \>|V|\<\|\|\>>>>\t|)>|]>\\>|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*> \>|J\\J\-2\|\>|\<\|\|\>>>>\t|)>|]>\>\t2+d/r|)>> |)>|}>|)>. Now we decompose the random field \\J\-2\|\>|)>> into homogeneous stochastic integral as above. And obtain <\equation*> \J\\J\-2\|\>|)>=,\,k>>GJ2>4>|)>\w>|)>\\w>|)> <\equation*> +,k>>GJ2>|)>\w>|)>\w>|)>+GJ2> <\eqnarray> J2>4>|)>>|||)>K4|)>>|)>e4|)>>|)>*\x>\>>|||\Sh>j>K\|)>>|)>K\|)>>|)>\|)>>|)>||k\|)>>|\>>\|)>>|)>||k\|)>>|\>>>>|J2>|)>>|||)>K>|)>e>|)>*\x>\Sh>j>\>>|||\r||q|\>>K>+q|)>K>-q|)>>+q|)>||k>+q|\>>>-q|)>||k>-q|\>>|)>>>|J2>>|||)>\ 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*> GJ2>=0 and <\equation*> |\>|\|>\\|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>>GJ2>4>|)>\w>|)>\\w>|)>|\|>|]> <\equation*> \,\,k>>J2>4>|)>|\|>>|)>||k|\>>\>|)>||k|\>>\s\\s <\equation*> \,\,k>>J2>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>K4|)>>|)>|k|\>>|k|\>>|k|\>>|k|\>>\\t>||t|\>>2 where we used that |\|>\t\t>>. 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*> J2>|)>|\|>\>|)>|\|>\t>||p|\>>+p|)>||k+p|\>>-p|)>||k-p|\>>|)>\>|)>|\|>|t|\>, from which we deduce that <\equation*> I\\,k>>GJ2>|)>\w>|)>\w>|)>|\|>|]> <\equation*> \,k>>J2>|)>|\|>>|)>||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.. 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. 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> Let \,p,q\|]>>. For a Schwarz distribution |)>> define the norm\ <\equation*> |f|\<\|\|\>>>=||\f|\<\|\|\>>>|)>-1>|\<\|\|\>>> Then the space > is the closure of Schwarz distributions under this norm. We denote >=B,\>>> the Besov\UHölder space and >=B>> the Sobolev spaces. \ <\proposition> Let p\p\\,1\q\q\\ B,q>> is continuously included in ,q>>->|)>>>. <\proposition> For any ,s\\> such that \s>, any |]>> the Besov space >>> is compactly embedded into >>. <\definition> Let >|)>>. We define the paraproducs <\equation*> f\g=i-1>\f\g,f\g=i+1>\f\g=g\f. Moreover we introduce the resonant product <\equation*> f\g=\1>\f\g. Then <\equation*> f*g=f\g+f\g+f\g. <\proposition> Let \0,\\\>. For |)>> we have the estimates\ <\eqnarray*> |f\g|\<\|\|\>>-\>>>|>||f|\<\|\|\>>>>|g|\<\|\|\>>>,>>||f\g|\<\|\|\>>>>>|>||f|\<\|\|\>>>>|g|\<\|\|\>>>>,>>||f\g|\<\|\|\>>-\>>>|>||f|\<\|\|\>>>>|g|\<\|\|\>>>>,>>||f\g|\<\|\|\>>>>>|>||f|\<\|\|\>>>>|g|\<\|\|\>>>>.>>>> Let ,\\\> such that +\\0>. Then\ <\eqnarray*> |f\g|\<\|\|\>>+\>>>|>||f|\<\|\|\>>>>|g|\<\|\|\>>>>,>>||f\g|\<\|\|\>>+\>>>|>||f|\<\|\|\>>>>|g|\<\|\|\>>>>.>>>> By density the paraproduct and resonant product also extend to bilnear operators on the respective spaces.\ <\proposition> Let \> ,\\\> such that +\\0>, +\+\\0>. Then for >, and for any \0>, <\equation*> |g|)>\h-gh|)>|\<\|\|\>>+\+\-\>>\|f|\<\|\|\>>>>|h|\<\|\|\>>>>|g|\<\|\|\>>>>, <\equation*> |g|)>\h-gh|)>|\<\|\|\>>+\+\>>\|f|\<\|\|\>>>>|h|\<\|\|\>>>>|g|\<\|\|\>>>>. <\proposition> Assume >,g\H>,h\H>> and +\+\=0>. Then\ <\equation*> |\|>>>g|)>h-h|)>g|]>\|f|\<\|\|\>>>>|g|\<\|\|\>>>>|h|\<\|\|\>>>>. <\remark> Proposition is not proven in the above references but is quite easy and the reader can fill out a proof. <\proposition> Let C,\|)>>, and assume that \> then there exists a map : >\>> such that for any >|)>> <\equation*> F=f\F+R. <\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 . <\proposition> Assume 0>, \,\\\>. Let > be an > multiplier, >>, H>>. Then for any \0>. <\equation*> |\|)>g|)>-|)>f\g|)>|\<\|\|\>>+\-m-\>>\|f|\<\|\|\>>>>|g|\<\|\|\>>>> Again the constant depends only on ,\,\> and the constants in . <\bibliography|bib|tm-plain|qft-drift> <\bib-list|36> Sergio AlbeverioSeiichiro Kusuoka. The invariant measure and the flow associated to the $\\Phi \B4_3$-quantum field model. Nov 2017. Hajer Bahouri, Jean-Yves CheminRaphaël Danchin. . Springer, jan 2011. Roland Bauerschmidt, DavidC.BrydgesGordon 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. Michelle BouéPaul Dupuis. A variational representation for certain functionals of Brownian motion. , 26(4):1641\U1659, oct 1998. D.C.BrydgesT.Kennedy. Mayer expansions and the Hamilton-Jacobi equation. , 48(1-2):19\U49, jul 1987. D.Brydges, J.DimockT.R.Hurd. The short distance behavior of $(\\phi\B4)_3$. , 172(1):143\U186, 1995. DavidC.Brydges. Lectures on the renormalisation group. , 16, 7\U93. Amer. Math. Soc., Providence, RI, 2009. DavidC.Brydges, Roberto FernándezCh- Ecublens. . 1993. DavidC.Brydges, Jürg FröhlichAlanD.Sokal. A new proof of the existence and nontriviality of the continuum $\\varphi \B4_2$ and $\\varphi \B4_3$ quantum field theories. , 91(2):141\U186, 1983. DavidC.BrydgesGordon Slade. A renormalisation group method. I. Gaussian integration and normed algebras. , mar 2014. ArXiv: 1403.7244. DavidC.BrydgesGordon Slade. A renormalisation group method. II. Approximation by local polynomials. , mar 2014. ArXiv: 1403.7253. DavidC.BrydgesGordon Slade. A renormalisation group method. IV. Stability analysis. , mar 2014. ArXiv: 1403.7255. DavidC.BrydgesGordon Slade. A renormalisation group method. V. A single renormalisation group step. , mar 2014. ArXiv: 1403.7256. Rémi CatellierKhalil Chouk. Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation. , oct 2013. ArXiv: 1310.6869. Joel Feldman. The $\\lambda \\varphi \B4_3$ field theory in a finite volume. , 37:93\U120, 1974. WendellH.FlemingHalilMete 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. James Glimm. Boson fields with the $:\\Phi\B{4}:$ interaction in three dimensions. , 10(1):1\U47, 1968. James GlimmArthur Jaffe. Positivity of the \\<#2009\>34 Hamiltonian. , 21(7):327\U376, jan 1973. Massimiliano GubinelliMartina Hofmanová. Global solutions to elliptic and parabolic $\\Phi\B4$ models in Euclidean space. , apr 2018. ArXiv: 1804.11253. Massimiliano Gubinelli, Peter ImkellerNicolas Perkowski. Paracontrolled distributions and singular PDEs. , 3:0, 2015. Archil GulisashviliMarkA.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. Svante Janson. . Cambridge University Press, jun 1997. Antti Kupiainen. Renormalization Group and Stochastic PDEs. , 17(3):497\U535, mar 2016. Joseph Lehec. Representation formula for the entropy and functional inequalities. , 49(3):885\U899, 2013. J.MagnenR.Sénéor. The infinite volume limit of the $\\phi \B4_3$ model. , 24(2):95\U159, 1976. Jean-Christophe MourratHendrik Weber. The dynamic $\\Phi\B4_3$ model comes down from infinity. , jan 2016. ArXiv: 1601.01234. Felix OttoHendrik Weber. Quasilinear SPDEs via rough paths. , may 2016. ArXiv: 1605.09744. YongMoon Park. The $\\lambda \\phi\B4_3$ Euclidean quantum field theory in a periodic box. , 16(11):2183\U2188, 1975. Joseph Polchinski. Renormalization and effective lagrangians. , 231(2):269\U295, jan 1984. Daniel RevuzMarc Yor. . Springer, 3rd, dec 2004. AliSüleyman Üstünel. Variational calculation of Laplace transforms via entropy on Wiener space and applications. , 267(8):3058\U3083, oct 2014. KennethG.Wilson. The renormalization group and critical phenomena. , 55(3):583\U600, 1983. Xicheng Zhang. A variational representation for random functionals on abstract Wiener spaces. , 49(3):475\U490, 2009. <\equation*> \; \; \; <\equation*> \; \; <\initial> <\collection> <\attachments> <\collection> <\associate|bib-bibliography> <\db-entry|+gSjxQIQnAm5UEi|article|wilson_renormalization_1983> <|db-entry> > > <\db-entry|+gSjxQIQnAm5UEh|article|polchinski_renormalization_1984> <|db-entry> > > <\db-entry|+gSjxQIQnAm5UEY|book|brydges_functional_1993> <|db-entry> Roberto Ch- > <\db-entry|+gSjxQIQnAm5UEe|article|brydges_short_1995> <|db-entry> J. T. R. > > )$.pdf:/Users/mgubi/Zotero/storage/SWGIBMTI/Brydges et al. - 1995 - The short distance behavior of $(phi)$.pdf:application/pdf> <\db-entry|+gSjxQIQnAm5UEf|incollection|brydges_lectures_2009> <|db-entry> > <\db-entry|+gSjxQIQnAm5UEd|article|brydges_renormalisation_2014_3> <|db-entry> Gordon > > <\db-entry|+gSjxQIQnAm5UEb|article|brydges_renormalisation_2014_2> <|db-entry> Gordon > > <\db-entry|+gSjxQIQnAm5UEc|article|bauerschmidt_renormalisation_2014> <|db-entry> David C. Gordon > > <\db-entry|+gSjxQIQnAm5UEa|article|brydges_renormalisation_2014_1> <|db-entry> Gordon > > <\db-entry|+gSjxQIQnAm5UEZ|article|brydges_renormalisation_2014> <|db-entry> Gordon > > <\db-entry|+gSjxQIQnAm5UEg|article|brydges_mayer_1987> <|db-entry> T. > > =4\ threshold and another derivation of some earlier results of Göpfert and Mack.> <\db-entry|+TmD8H98ZsgGOs3|book|fleming_controlled_2005> <|db-entry> Halil Mete > <\db-entry|+TmD8H98ZsgGOs0|article|boue_variational_1998> <|db-entry> Paul > > logEe\f(W)=infvE1/2\10\vs\2ds+f(W+\\0vsds)\log\Ee\f(W)=infvE1/2\01\vs\2ds+f(W+\0\vsds)-\\log Ee\B{-f(W)} = \\inf_v E{1/2 \\int_0\B1 \\parallel v_s \\parallel\B2 ds + f(W + \\int_0\B{\\cdot} v_s ds)} holds, where WWW is a standard ddd-dimensional Brownian motion, fff is any bounded measurable function that maps C([0,1]:\d)C([0,1]:Rd)C([0, 1]: \\mathbb{R}\Bd) into \R\\mathbb{R} and the infimum is over all processes vvv that are progressively measurable with respect to the augmentation of the filtration generated by WWW. An application is made to a problem concerned with large deviations, and an extension to unbounded functions is given.> <\db-entry|+CUVfDoZVCp96nY|article|ustunel_variational_2014> <|db-entry> > > ) be the classical Wiener space where H is the Cameron\UMartin space which consists of the primitives of the elements of L 2 ( [ 0 , 1 ] , d t ) \ R d . We denote by L a 2 ( \ , H ) the equivalence classes w.r.t. d t \ d \ whose Lebesgue densities s \ u ( s , w ) are almost surely adapted to the canonical Brownian filtration. If f is a Wiener functional s.t. 1 E [ e \ f ] e \ f d \ is of finite relative entropy w.r.t. \, we prove that J \ = inf \ ( E \ [ f \ U + 1 2 \| u \| H 2 ] : u \ L a 2 ( \ , H ) ) \ \ log \ E \ [ e \ f ] = inf \ ( \ W f d \ + H ( \ \| \ ) : \ \ P ( W ) ) where P ( W ) is the set of probability measures on ( W , B ( W ) ) and H ( \ \| \ ) is the relative entropy of \ w.r.t. \. We call f a tamed functional if the inequality above can be replaced with equality. We characterize the class of tamed functionals, which is much larger than the set of essentially bounded Wiener functionals. We show that for a tamed functional the minimization problem of l.h.s. has a solution u 0 if and only if U 0 = I W + u 0 is almost surely invertible and d U 0 \ d \ = e \ f E \ [ e \ f ] and then u 0 is unique. To do this, we prove the theorem which says that the relative entropy of U 0 \ is equal to the energy of u 0 if and only if it has a \-a.s. left inverse. We use these results to prove the strong existence of the solutions of stochastic differential equations with singular (functional) drifts and also to prove the non-existence of strong solutions of some stochastic differential equations.> <\db-entry|+CUVfDoZVCp96nX|article|zhang_variational_2009> <|db-entry> > > <\db-entry|+TmD8H98ZsgGOs1|article|lehec_representation_2013> <|db-entry> > > <\db-entry|+gSjxQIQnAm5UEI|article|gubinelli_paracontrolled_2015> <|db-entry> Peter Nicolas > > <\db-entry|+gSjxQIQnAm5UEM|article|glimm_boson_1968> <|db-entry> > > $ interaction in .pdf:/Users/mgubi/Zotero/storage/HIVCW679/Glimm - 1968 - Boson fields with the $Phi $ interaction in .pdf:application/pdf;Snapshot:/Users/mgubi/Zotero/storage/V5NGD252/1103840981.html:text/html> <\db-entry|+gSjxQIQnAm5UEN|article|glimm_positivity_1973> <|db-entry> Arthur > \<#2009\>34 Hamiltonian> 1973 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim> > \<#2009\>34 Hamiltonian is bounded from below by a constant proportional to the volume.> 34 Hamiltonian.pdf:/Users/mgubi/Zotero/storage/EJTRN8ZR/Glimm and Jaffe - 1973 - Positivity of the \ 34 Hamiltonian.pdf:application/pdf;Snapshot:/Users/mgubi/Zotero/storage/VWJZEPZ7/abstract.html:text/html> <\db-entry|+gSjxQIQnAm5UEO|article|feldman_lambda_1974> <|db-entry> > > $ field theory in a finit.pdf:/Users/mgubi/Zotero/storage/9DH7MMMI/Feldman - 1974 - The $lambda varphi $ field theory in a finit.pdf:application/pdf;MathSciNet Snapshot:/Users/mgubi/Zotero/storage/4WTCBRXA/publdoc.html:text/html> <\db-entry|+gSjxQIQnAm5UEP|article|park_lambda_1975> <|db-entry> > > $ Euclidean quantum field the.pdf:/Users/mgubi/Zotero/storage/FYCAY49S/Park - 1975 - The $lambda phi$ Euclidean quantum field the.pdf:application/pdf> <\db-entry|+gSjxQIQnAm5UEQ|article|benfatto_ultraviolet_1980> <|db-entry> M. G. F. E. E. E. > > <\db-entry|+gSjxQIQnAm5UER|article|magnen_infinite_1976> <|db-entry> R. > > <\db-entry|+gSjxQIQnAm5UES|article|brydges_new_1983> <|db-entry> Jürg Alan D. > > <\db-entry|+gSjxQIQnAm5UET|article|hairer_theory_2014> <|db-entry> > > 43\34\\Phi \B4_3 Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of 333-dimensional ferromagnets near their critical temperature.> <\db-entry|+gSjxQIQnAm5UEU|article|catellier_paracontrolled_2013> <|db-entry> Khalil > > houk013aracontrolled Distributions and the 3-dimensional Stochastic Quantization.pdf:/Users/mgubi/Zotero/storage/VE6W9JD9/Catellierhouk013aracontrolled Distributions and the 3-dimensional Stochastic Quantization.pdf:application/pdf> <\db-entry|+gSjxQIQnAm5UEW|article|otto_quasilinear_2016> <|db-entry> Hendrik > > \\frac{2}{3}$ on the parabolic H\\"older scale. The example we have in mind is a random forcing $f$ and our assumptions allow, for example, for an $f$ which is white in the time variable $x_2$ and only mildly coloured in the space variable $x_1$; any spatial covariance operator $(1 + \|\\partial_1\|)\B{-\\lambda_1 }$ with $\\lambda_1 \\frac13$ is admissible. On the deterministic side we obtain a $C\B\\alpha$-estimate for $u$, assuming that we control products of the form $v\\partial_1\B2v$ and $vf$ with $v$ solving the constant-coefficient equation $\\partial_2 v-a_0\\partial_1\B2v=f$. As a consequence, we obtain existence, uniqueness and stability with respect to $(f, vf, v \\partial_1\B2v)$ of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing $f$ using stochastic arguments. For this we extend the treatment of the singular product $\\sigma(u)f$ via a space-time version of Gubinelli's notion of controlled rough paths to the product $a(u)\\partial_1\B2u$, which has the same degree of singularity but is more nonlinear since the solution $u$ appears in both factors. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.> <\db-entry|+gSjxQIQnAm5UEV|article|kupiainen_renormalization_2016> <|db-entry> > > <\db-entry|+gSjxQIQnAm5UEX|article|mourrat_dynamic_2016> <|db-entry> Hendrik > > <\db-entry|+gSjxQIQnAm5UEL|article|albeverio_invariant_2017> <|db-entry> Seiichiro > > <\db-entry|+VZu1232XNGWCev|article|gubinelli_global_2018> <|db-entry> Martina > > <\db-entry|+TmD8H98ZsgGOrx|book|revuz_continuous_2004> <|db-entry> Marc > <\db-entry|+TmD8H98ZsgGOs2|incollection|follmer_entropy_1985> <|db-entry> > Prof E. > 1985 Springer-Verlag> > <\db-entry|+TmD8H98ZsgGOrw|book|janson_gaussian_1997> <|db-entry> > <\db-entry|+fi7I2RVCpoE0mt|article|gulisashvili_exact_1996> <|db-entry> Mark A. > > <\db-entry|+gSjxQIQnAm5UEJ|book|bahouri_fourier_2011> <|db-entry> Jean-Yves Raphaël > It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.> <\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 gubinelli_paracontrolled_2015 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_global_2018 albeverio_invariant_2017 revuz_continuous_2004 follmer_entropy_1985 boue_variational_1998 boue_variational_1998 ustunel_variational_2014 follmer_entropy_1985 ustunel_variational_2014 fleming_controlled_2005 ustunel_variational_2014 lehec_representation_2013 janson_gaussian_1997 janson_gaussian_1997 gulisashvili_exact_1996 bahouri_fourier_2011 gubinelli_paracontrolled_2015 <\associate|toc> |math-font-series||1Introduction > |.>>>>|> |math-font-series||2The RG flow as a stochastic control problem> |.>>>>|> |math-font-series||3Two dimensions> |.>>>>|> |math-font-series||4Three dimensions> |.>>>>|> |4.1Lower bound |.>>>>|> > |4.1.1Bound for the linear term |.>>>>|> > |4.1.2Bound for the quadratic term |.>>>>|> > |4.1.3Bound for the cubic term |.>>>>|> > |4.1.4Renormalization |.>>>>|> > |4.2Upper bound |.>>>>|> > |math-font-series||5Estimates of the stochastic terms> |.>>>>|> |math-font-series||Appendix ABesov spaces and paraproducts> |.>>>>|> |math-font-series||Bibliography> |.>>>>|>