1.Introduction

The \(\Phi^4_d\) Gibbs measure on the \(d\)-dimensional torus \(\Lambda = \Lambda_L =\mathbb{T}_L^d = (\mathbb{R}/ (2 \pi L\mathbb{Z}))^d\) is the probability measure \(\nu\) obtained as the weak limit for \(T \rightarrow \infty\) of the family \((\nu_T)_{T > 0}\) given by

where

Here \(\lambda \geqslant 0\) is a fixed constant, \(\Delta\) is the Laplacian on \(\Lambda\), \(\vartheta\) is the centered Gaussian measure with covariance \((1 - \Delta)^{- 1}\), \(\mathscr{Z}_T\) is a normalization factor, \(a_T, b_T\) given constants and \(\phi_T = \rho_T \ast \phi\) with \(\rho_T\) some appropriate smooth and compactly supported cutoff function such that \(\rho_T \rightarrow \delta\) as \(T \rightarrow \infty\). The measures \(\vartheta\) and \(\nu_T\) are realized as probability measures on \(\mathscr{S}' (\Lambda)\), the space of tempered distributions on \(\Lambda\). They are supported on the Hölder–Besov space \(\mathscr{C}^{(2 - d) / 2 - \kappa} (\Lambda)\) for all small \(\kappa > 0\). The existence of the limit \(\nu\) is conditioned on the choice of a suitable sequence of renormalization constants \((a_T, b_T)_{T > 0}\). The constant \(b_T\) is not necessary, but is useful to decouple the behavior of the numerator from that of the denominator in eq. (1).

The aim of this paper is to give a proof of convergence using a variational formula for the partition function \(\mathscr{Z}_T\) and for the generating function of the measure \(\nu_T\). As a byproduct we obtain also a variational description for the generating function of the limiting measure \(\nu\) via \(\Gamma\)-convergence of the variational problem. Let us remark that, to our knowledge, it is the first time that such explicit description of the unregulated \(\Phi^4_3\) measure is available.

Our work can be seen as an alternative realization of Wilson's [49] and Polchinski's [45] continuous renormalization group (RG) method. This method has been made rigorous by Brydges, Slade et al. [11, 18, 10] and as such witnesses a lot of progress and successes [14, 15, 4, 16, 17, 18]. The key idea is the nonperturbative study of a certain infinite dimensional Hamilton–Jacobi–Bellman equation [13] 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 [22]. 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 [7], for the logarithm of the partition function interpreted as the value function of the control problem. See also the related papers of Üstünel [48] and Zhang [50] where extensions and further results on the variational formula are obtained. The variational formula has been used by Lehec [38] to prove some Gaussian functional inequalities, following the work of Borell [6]. 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 [28], to perform the renormalization of various non-linear quantities and show uniform bounds in the \(T \rightarrow \infty\) limit.

Define the normalized free energy \(\mathcal{W}_T\) for the cutoff \(\Phi^4_3\) measure, as

where \(f \in C (\mathscr{S}' (\Lambda) ; \mathbb{R})\) is a given function. The main result of the paper is the following

See Section 4 and in particular Theorem 23 for precise definitions of the various objects and a more detailed statement of this result. With respect to the notations in Lemma 22, observe that

where \(K (u)\) is another functional of \((\mathbb{W}, u)\).

Theorem 1 implies directly the convergence of \((\nu_T)_T\) to a limit measure \(\nu\) on \(\mathscr{S}' (\Lambda)\). Taking \(f\) in the linear dual of \(\mathscr{C}^{- 1 / 2 - \kappa}\) it also gives the following formula for the Laplace transform of \(\nu\):

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 \(\Phi^4_3\) measure with respect to the reference Gaussian measure. Another possible approach to an explicit description goes via integration by parts (IBP) formulas, see [2] for an early proof and a discussion of this approach. More recently [29] gives a self–contained proof of the IBP formula for any accumulation point of the \(\Phi^4_3\) 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. [33, 34, 24].

The parameter \(L\), which determines the size of the spatial domain \(\Lambda = \Lambda_L\), will be kept fixed all along the paper and we will not attempt here to obtain the infinite volume limit \(L \rightarrow \infty\). For this reason we will avoid to explicitly show the dependence of \(\mathcal{W}_T\) with \(\Lambda\). However some care will be taken to obtain estimates uniform in the volume \(| \Lambda |\).

An easy consequence of the estimates needed to establish the main theorem is the following corollary (well known in the literature, see e.g. [5]):

A similar statement for \(d = 2\) will be sketched below in order to introduce some of the ideas on which the \(d = 3\) proof is based.

The construction of the \(\Phi^4_{2, 3}\) measure in finite volume is basic problem of constructive quantum field theory to which many works have been devoted, especially in the \(d = 2\) case. It is not our aim to provide here a comprehensive review of this literature. As far as the \(d = 3\) 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 [25, 26], 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 \(\nu\) on a space of distributions. Feldman [21], Park [44], Benfatto et al. [5], Magnén and Seneor [39] and finally Brydges et al. [12] 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 [32] and related approaches [28, 19, 43] or even RG–inspired ones [37], 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 \(\Phi^4_3\) measure in two different ways [40, 1]. Uniqueness by these methods requires additional efforts but seems at reach. Some results on the existence of the infinite volume measure [29] and dynamics [27] have been obtained recently. For an overview of the status of the constructive program wrt. the analysis of the \(\Phi^4_{2, 3}\) models the reader can consult the introduction to [1] and [29]

This paper is organized as follows. In Section 2 we set up our main tool, the Boué–Dupuis variational formula of Theorem 4. Then, as a warmup exercise, we use the formula to show bounds and existence of the \(\Phi^4_2\) measure in Section 3. We then pass to the more involved situation of three dimensions in Section 4 where we introduce the renormalized variational problem. In Section 5 we establish uniform bounds for this new problem and in Section 6 we prove Theorem 1. Section 7 and Section 8 are concerned with some details of the analytic and probabilistic estimates needed throughout the paper. Appendix A 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 \(\langle a \rangle = (1 + a^2)^{1 / 2}\). Denote with \(\mathscr{S} (\Lambda)\) the space of Schwartz functions on \(\Lambda\) and with \(\mathscr{S}' (\Lambda)\) the dual space of tempered distributions. The notation \(\hat{f}\) or \(\mathscr{F} f\) stands for the space Fourier transform of \(f\). In order to easily keep track of the volume dependence of various objects we normalize the Lebesgue measure on \(\Lambda\) to have unit mass. We denote the normalized integral and measure by

where \(| \Lambda |\) is the volume of \(\Lambda\). 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 \(| \Lambda |\), unless otherwise stated. The constant \(\kappa > 0\) 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.

2.A stochastic control problem

We begin by constructing a probability space \(\mathbb{P}\) endowed with a process \((Y_t)_{t \in [0, \infty]}\) belonging to \(C ([0, \infty], \mathscr{C}^{(2 - d) / 2 - \kappa} (\Lambda))\) and such that \(\operatorname{Law}_{\vartheta} (\phi_T) =\operatorname{Law}_{\mathbb{P}} (Y_T)\) for all \(T \geqslant 0\) and \(\operatorname{Law}_{\mathbb{P}} (Y_{\infty}) = \vartheta\).

Fix \(\alpha < - d / 2\) and let \(\Omega := C (\mathbb{R}_+ ; H^{- \alpha})\), \((X_t)_{t \geqslant 0}\) the canonical process on \(\Omega\) and \(\mathscr{B}\) the Borel \(\sigma\)–algebra of \(\Omega\). On \((\Omega, \mathscr{B})\) consider the probability measure \(\mathbb{P}\) which makes the canonical process \(X\) a cylindrical Brownian motion in \(L^2 (\Lambda)\). In the following \(\mathbb{E}\) without any qualifiers will denote expectations wrt. \(\mathbb{P}\) and \(\mathbb{E}_{\mathbb{Q}}\) will denote expectations wrt. some other measure \(\mathbb{Q}\). On the measure space \((\Omega, \mathscr{B}, \mathbb{P})\) there exists a collection \((B_t^n)_{n \in (L^{- 1} \mathbb{Z})^d}\) of complex (2-dimensional) Brownian motions, such that \(\overline{B^n_t} = B^{- n}_t\), \(B^n_t, B^m_t\) independent for \(m \neq \pm n\) and \(X_t = | \Lambda |^{- 1 / 2} \sum_{n \in (L^{- 1} \mathbb{Z})^d} e^{i \langle n, \cdot \rangle} B^n_t\). Note that \(X\) has a.s. trajectories in \(C (\mathbb{R}_+, \mathscr{C}^{- d / 2 - \varepsilon} (\Lambda))\) for any \(\varepsilon > 0\) by standard arguments.

Fix some \(\rho \in C_c^{\infty} (\mathbb{R}_+, \mathbb{R}_+)\) such that \(\rho (0) = 1\). Let \(\rho_t (x) := \rho (x / t)\) and

where \(\dot{\rho}_t\) is the partial derivative of \(\rho_t\) with respect to \(t\). Consider the process \((Y_t)_{t \geqslant 0}\) defined by

It is a centered Gaussian process with covariance

for any \(\varphi, \psi \in \mathscr{S} (\Lambda)\) and \(t, s \geqslant 0\), by Fubini theorem and Ito isometry. By dominated convergence \(\lim_{t \rightarrow \infty}\)\(\mathbb{E} [\langle Y_t, \varphi \rangle \langle Y_t, \psi \rangle] = | \Lambda |^{- 1} \sum_{n \in (L^{- 1} \mathbb{Z})^d} \langle n \rangle^{- 2} \hat{\varphi} (n) \overline{\hat{\psi} (n)}\) for any \(\varphi , \psi \in L^2 (\Lambda)\).

Note that up to any finite time \(T\) the r.v. \(Y_T\) has a bounded spectral support and the stopped process \(Y^T_t = Y_{t \wedge T}\) for any fixed \(T > 0\), is in \(C (\mathbb{R}_+, W^{k, 2} (\Lambda))\) for any \(k \in \mathbb{N}\). Furthermore \((Y^T_t)_t\) only depends on a finite subset of the Brownian motions \((B^n)_n\).

We will usually write \(g (\mathrm{D})\) for the Fourier multiplier operator with symbol \(g\). With this convention we can compactly denote

where \(J_s := \langle D \rangle^{- 1} \sigma_s (D)\). We observe that \(Y_t\) has a distribution given by the pushforward \((\rho_t (\mathrm{D}))_{\ast} \vartheta\) of \(\vartheta\) through \(\rho_t (D)\). We write the measure \(\nu_T\) in (1) in terms of expectations over \(\mathbb{P}\) as

for any bounded measurable \(g : \mathscr{S}' (\Lambda) \rightarrow \mathbb{R}\).

For fixed \(T\) the polynomial appearing in the expression for \(V_T (Y_T)\) is bounded below (since \(\lambda > 0\)) and \(\mathscr{Z}_T\) is well defined and also bounded away from zero (this follows easily from Jensen's inequality). However as \(T \rightarrow \infty\) we tend to loose both these properties due to the fact that we will be obliged to take \(a_T \rightarrow + \infty\) to renormalize the non–linear terms. To obtain uniform upper and lower bounds we need a more detailed analysis and we proceed as follows.

Denote by \(\mathbb{H}_a\) the space of progressively measurable processes which are \(\mathbb{P}\)–almost surely in \(\mathcal{H} := L^2 (\mathbb{R}_+ \times \Lambda)\). We say that an element \(v\) of \(\mathbb{H}_a\) is a drift. Below we will need also drifts belonging to \(\mathcal{H}^{\alpha} := L^2 (\mathbb{R}_+ ; H^{\alpha} (\Lambda))\) for some \(\alpha \in \mathbb{R}\), we denote the corresponding space with \(\mathbb{H}_a^{\alpha}\). Consider the measure \(\mathbb{Q}_T\) on \((\Omega, \mathscr{B})\) whose Radon–Nykodim derivative wrt. \(\mathbb{P}\) is given by

Since \(Y_T\) depends on finitely many Brownian motions \((B^n)_n\) then it is well known [46, 23] that any \(\mathbb{P}\)–absolutely continuous probability can be expressed via Girsanov transform. In particular, by the Brownian martingale representation theorem there exists a drift \(u^T \in \mathbb{H}_a\) such that

(recall that we normalized the \(L^2 (\Lambda)\) norm) and the entropy of \(\mathbb{Q}_T\) wrt. \(\mathbb{P}\) is given by

Here equality holds also if one of the two quantities is \(+ \infty\). By Girsanov theorem, the canonical process \(X\) is a semimartingale under \(\mathbb{Q}_T\) with decomposition

where \((\tilde{X}_t)_t\) is a cylindrical \(\mathbb{Q}_T\)–Brownian motion in \(L^2 (\Lambda)\). Under \(\mathbb{Q}_T\) the process \((Y_t)_t\) has the semimartingale decomposition \(Y_t = W_t + U_t\) with

where for any drift \(v \in \mathbb{H}_a\) we define

The integral in the density can be restricted to \([0, T]\) since \(u_t^T = 0\) if \(t > T\). Now

and taking expectation of (7) wrt \(\mathbb{Q}_T\) we get

For any \(v \in \mathbb{H}_a\) define the measure \(\mathbb{Q}^v\) by

Denote with \(\mathbb{H}_c \subseteq \mathbb{H}_a\) the set of drifts \(v \in \mathbb{H}_a\) for which \(\mathbb{Q}^v (\Omega) = 1\), in particular \(u^T \in \mathbb{H}_c\). By Jensen's inequality and Girsanov transformation we have

for all \(v \in \mathbb{H}_c\), where \(\mathbb{E}^v := \mathbb{E}_{\mathbb{Q}^v}\). We conclude that

where \(Y = W^v_T + I_T (v)\) and \(\operatorname{Law}_{\mathbb{Q}^v} (W^v) =\operatorname{Law}_{\mathbb{P}} (Y)\). The bound is saturated when \(v = u^T\). 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.

This formula is nice and easy to prove but somewhat inconvenient for certain manipulations since the space \(\mathbb{H}_c\) is indirectly defined and the reference measure \(\mathbb{E}^v\) depends on the drift \(v\). A more straightforward formula has been found by Boué–Dupuis [7] which involves the fixed canonical measure \(\mathbb{P}\) and a general adapted drift \(u \in \mathbb{H}_a\). This formula will be our main tool in the following.

Proof. The original proof can be found in Boué–Dupuis [7] for functionals bounded above. In our setting the formula can be proved using the result of Üstünel [48] by observing that \(V_T^f (Y_T)\) is a tame functional, according to his definitions. Namely, for some \(p, q \geqslant 1\) such that \(1 / p + 1 / q = 1\) we have

Remark 5. Some observations on these variational formulas.

The next lemma provides a deterministic regularity result for \(I (v)\) which will be useful below. It says that the drift \(v\) generates shifts of the Gaussian free field in directions which belong to \(H^1\) uniformly in the scale parameter up to \(\infty\). The space \(H^1\) is the Cameron–Martin space of the free field [36].

Proof. Using the fact that \(\sigma_s (\mathrm{D})\) is diagonal in Fourier space, and denoting with \((e_k)_{k \in \mathbb{Z}^d}\) the basis of trigonometric polynomials, we have

On the other hand \(\sigma_s (D)\) is a smooth Fourier multiplier and using Proposition 59 we have the estimate \(\| \sigma_s (\mathrm{D}) f \|_{H^{\alpha}} \lesssim \| f \|_{H^{\alpha}} / \langle s \rangle^{1 / 2}\) uniformly in \(s \geqslant 0\), therefore, for all \(0 \leqslant r \leqslant t \leqslant T\), we have

We conclude that

\(\Box\)

3.Two dimensions

As a warm up we consider here the case \(d = 2\) setting \(f = 0\) for simplicity. From Theorem 4 we see that the relevant quantity to bound is of the form

for \(u \in \mathbb{H}_a\). Let \(Z_t = I_t (u)\) and write \(W_t = Y_t\) as a mnemonic of the fact that under \(\mathbb{P}\) the process \(W\) is a martingale. From now on we leave implicit the integration variable over the spatial domain \(\Lambda\). Choosing

we have

where

denote the Wick powers of the Gaussian r.v. \(W_T\) [36]. These polynomials, when seen as stochastic processes in \(T\), are \(\mathbb{P}\)–martingales wrt. the filtration of \((W_t)_t\). In particular they have an expression as iterated stochastic integrals wrt. the Brownian motions \((B^n_t)_{t, n}\) introduced in eq. (4). Using Theorem 4 with \(u = 0\) we readily have an upper bound for the free energy:

For a lower bound we need to estimate from below the average under \(\mathbb{P}\) of the variational expression

The strategy we adopt is to bound pathwise, and for a generic drift \(u\), the contributions

in term of quantities involving only the Wick powers of \(W\) which we can control in expectation and the last two positive terms

Any residual positive contribution depending on \(u\) can be dropped in the lower bound making the dependence on the drift disappear. To control term \(\mathrm{I}\) we see that by duality and Young's inequality, for any \(\delta > 0\),

For the term \(\mathrm{I} \mathrm{I}\) the following fractional Leibniz rule is of help:

Proof. See [31].\(\Box\)

Using Proposition 8 we get, for any \(\delta > 0\),

In order to bound the term \(\mathrm{I} \mathrm{I} \mathrm{I}\) we observe the following:

Proof. By duality \(\)\(\left| f g^3 \right| \leqslant \| f \|_{W^{- 1 / 2 - \varepsilon, p}} \| g^3 \|_{{W^{1 / 2 + \varepsilon, p'}} }\). Applying again Proposition 8 and Proposition 61, we get

So

\(\Box\)

Remark 10. For the \(d = 2\) case it would have been enough to estimate \(\| g^3 \|_{W^{\varepsilon, p}}\). The stronger estimate will be useful below for \(d = 3\) since there we will only have \(W_T \in W^{- 1 / 2 - \varepsilon, p}\) for any large \(p\).

Then

Using eqs. (12), (13) and (14) we obtain, for \(\delta\) small enough,

where

Therefore

This last average do not depends anymore on the drift and we are only left to show that

However, it is well known that the Wick powers of the two dimensional Gaussian free field are distributions belonging to \(L^a (\Omega, W^{- \varepsilon, b})\) for any \(a \geqslant 1\) and \(b \geqslant 1\) and hypercontractivity plus an easy argument gives the uniform boundedness of the above averages, see e.g. [42]. We have established:

Remark 12. Observe that the argument above remains valid upon replacing \(\lambda\) with \(\lambda p\) with \(p \geqslant 1\). This implies that \(e^{- V_T (Y_T)}\) is in all the \(L^p\) spaces wrt. the measure \(\mathbb{P}\) uniformly in \(T\) and for any \(p \geqslant 1\).

4.Three dimensions

In three dimensions the strategy we used in two dimensions fails. Indeed here the Wick products are less regular: \(\mathopen{\unicode{x27E6}} W_T^2 \mathclose{\unicode{x27E7}} \in \mathscr{C}^{- 1 - \kappa}\) uniformly in \(T\) for any small \(\kappa > 0\) and \(\mathopen{\unicode{x27E6}} W_T^3 \mathclose{\unicode{x27E7}}\) does not even converge to a well-defined random distribution. This implies that there is no straightforward approach to control the terms

like we did in Section 3. The only apriori estimate on the regularity of \(Z_T = I_T (u)\) is in \(H^1\), coming from Lemma 6 and the quadratic term in the variational functional \(F_T (u)\). 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 \(V_T\) further scale dependent renormalization constants \(\gamma_T, \delta_T\) beyond Wick ordering to have

Repeating the computation from Section 3 we arrive at

where we introduced the convenient notations

and we recall that \(f\) is a fixed function belonging to \(C (\mathscr{C}^{- 1 / 2 - \kappa} ; \mathbb{R})\) with linear growth.

This form of the functional is not very useful in the limit \(T \rightarrow \infty\) 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 \(T\). 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 \(L^4\) estimates. To introduce such a regularization fix a smooth function \(\theta : \mathbb{R}_+ \rightarrow \mathbb{R}_+\) such that \(\theta (\xi) = 1\) if \(\xi \leqslant 1 / 4\) and \(\theta (\xi) = 0\) if \(\xi \geqslant 1 / 2\). Then

By the Mihlin-Hörmander theorem we deduce that the operator \(\theta_t = \theta_t (\mathrm{D})\) is bounded on \(L^p\) for any \(1 < p < \infty\), see Proposition 59. In the following, for any \(f \in C ([0, \infty], \mathscr{S}' (\Lambda))\) we define \(f^{\flat}_t := \theta_t f_t\) then

In this way we have \(\| Z_t^{\flat} \|_{L^p} \lesssim \| Z_T \|_{L^p}\) for all \(t \leqslant T\). The renormalized functional will depend on some specific renormalized combinations of the martingales \((\mathopen{\unicode{x27E6}} \mathbb{W}^k_t \mathclose{\unicode{x27E7}})_{t, k}\). 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

with \(\mathbb{W}^1 := W\),

We not not need to include \(\mathbb{W}^{[3]}\) since it can be obtained as a function of \(\mathbb{W}^{\langle 3 \rangle}\) thanks to the bound

valid for all \(0 \leqslant s \leqslant t \leqslant T\) which shows that the deterministic map \(\mathbb{W}^{\langle 3 \rangle} \mapsto \mathbb{W}^{[3]}\) is continuous from \(C ([0, \infty], \mathscr{C}^{- 1 / 2 - \kappa})\) to \(\)\(C^{1 / 2} ([0, \infty], \mathscr{C}^{1 / 2 - 2 \kappa})\). The pathwise regularity of all the other stochastic objects follows from the next Lemma, provided the function \(\gamma\) is chosen appropriately.

Proof. The proof is based on the observation that one can choose \(\gamma\) in such a way that every component \(\mathbb{W}^{(i)}\) of the vector \(\mathbb{W}\) is such that \((\Delta_q \mathbb{W}_t^{(i)} (x))_{t \geqslant 0}\) for \(q \geqslant - 1\) and \(x \in \Lambda\) is a martingale wrt. the Brownian filtration. The reader can find details for this statement in the proof of Lemma 44 below, in particular eq. (59) and (61) give the stochastic integral representations of the most difficult terms which require renormalization by \(\gamma\). The quadratic variation of these martingales can be controlled uniformly up to \(t = \infty\). As a consequence, there exists a version of \((\Delta_q \mathbb{W}_t^{(i)} (x))_{t \geqslant 0}\) which is continuous in \(t \in [0, \infty]\). Since each Littlewood–Paley block is a smooth function of \(x\) is not difficult from this to deduce that \(\)\(((t, x) \mapsto \Delta_q \mathbb{W}_t^{(i)} (x)) \in C ([0, \infty] \times \Lambda ; \mathbb{R})\). By Burkholder–David–Gundy (BDG) estimates and \(L^p\) estimates on the the quadratic variations one can conclude that \(\| \mathbb{W} \|_{\mathfrak{S}}\) is in all \(L^p\) spaces provided we can control the appropriate moments of the norm of \(\mathbb{W}_T\) uniformly in \(T\). This is achieved in Lemma 44 for the more difficult resonant products where we show also the claimed uniformly of the finite \(r\) Besov norms \(B^{\alpha}_{p, r}\).\(\Box\)

For convenience of the reader we summarize the probabilistic estimates in Table 1.

Remark 14. The requirement that \(\mathbb{W}^{\langle 3 \rangle} \in L^2 \mathscr{C}^{- 1 / 2 -}\) will be used in Section 6 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.

Proof. Step 1. We are going to absorb the mixed terms (16) 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

and we can cancel the first term on the r.h.s. by making the change of variables

into the cost functional to get

where we used that \(J_t\) is self-adjoint. The divergent term \(\mathbb{W}_T^3 Z_T\) has been replaced with a divergent but purely stochastic term \(\int_0^T (\mathbb{W}_t^{\langle 3 \rangle})^2 \mathrm{d} t\) which does not affect anymore the variational problem and can be explicitly removed by adding its average to \(\delta_T\). As a consequence, we are not able to control \((Z_t)_t\) in \(H^1\) anymore and we should rely on the relation (21) and on a control over the \(H^1\) norm of \((K_t)_t\) coming from the residual quadratic term \(\| w \|^2_{\mathcal{H}}\).

Step 2. From (23) we have the relation

which can be used to expand the second mixed divergent term in (16) as

Again, the first term on the r.h.s. a purely stochastic object and will give a contribution independent of the drift \(u\) absorbed in \(\delta_T\). We are still not done since this operation has left two new divergent terms on the r.h.s. of eq. (24): the \(H^1\) regularity of \(K_T\) is not enough to control the products with \(\mathbb{W}^2\) which has regularity \(\mathscr{C}^{- 1 - \kappa}\), a bit below \(- 1\). In order to proceed further we will isolate the divergent parts of these products via a paraproduct decomposition (see Appendix A for details) and expand \(\)

The first two terms will require renormalizations which we put in place in Step 3 below. All the other terms will be well behaved and we collect them in \(\Upsilon^{(1)}_T\). In particular we observe that the last one can be rewritten as

using the trilinear form \(\mathfrak{K}_2\) defined in Proposition 56.

Step 3. As we anticipated, the resonant term \(\mathbb{W}^2_T \circ \mathbb{W}_T^{[3]}\) needs renormalization. In the expression of \(F_T\) in (18) we have the counterterm \(- 2 \lambda^2 \gamma_T W_T Z_T\) available, which we put in use now by writing

The first contribution is collected in \(\Upsilon^{(6)}_T\) and the expectation of the second will contribute to \(\delta_T\).

As far as the term \(\lambda (\mathbb{W}^2_T \succ Z_T) K_T\) is concerned, we want to absorb it into \(\int \| w_s \|^2 \mathrm{d} s\) like we did with the linear term in Step 2. Before we can do this we must be sure that, after applying Ito's formula, it will be still possible to use \(Z^4_T\) to control some of the growth of this term. Indeed the quadratic dependence in \(K_T\) (via \(Z_T\)) cannot be fully taken care of by the quadratic cost \(\int \| w_s \|^2 \mathrm{d} s\).

We decompose

and using the fact that the functions \(Z_T - Z_T^{\flat}\) and \(K_T - K_T^{\flat}\) are spectrally supported outside of a ball or radius \(c T\) we will be able to show that the second term is nice enough as \(T \rightarrow \infty\) to not require further analysis and we collect it in \(\Upsilon^{(2)}_T\). For the first we apply Ito's formula to decompose it along the flow of scales as

The second term will be fine and we collect it in \(\Upsilon^{(3)}_T\).

Step 4. We are left with the singular term \(\int_0^T (\mathbb{W}^2_t \succ Z^{\flat}_t) \dot{K_t} \mathrm{d} t\). Using eq. (21) and expanding \(w\) in the residual quadratic cost function obtained in Step 1, we compute

To renormalize the first term on the r.h.s. we observe that the remaining couterterm can be rewritten as

Differentiating in \(T\) the first term in the r.h.s. of eq. (26) we get

The last term in eq. (26) and the last two contributions in (27) are collected in \(\Upsilon^{(5)}_T\). The first contribution in eq. (26) has the right form to be used as a counterterm for the resonant product in (25). Using the commutator \(\mathfrak{K}_{3, t}\) introduced in Proposition 57 we have

and collect both terms in \(\Upsilon^{(6)}_T\).

Step 5. We are now left with the cubic term which we rewrite as

The average of the first term is collected in \(\delta_T\) while all the remaining terms in \(\Upsilon^{(4)}_T\). At last we have established the claimed decomposition since the residual cost functional, from eq. (25) has the form \(\| l \|^2_{\mathcal{H}}\).\(\Box\)

5.Bounds

The aim of this section is to give upper and lower bounds on \(\mathcal{W}_T (f)\) uniformly on \(T\) and \(| \Lambda |\). In particular we will prove the bounds of Corollary 2 taking the explicit dependence on the coupling constant \(\lambda\) into account.

Proof. Observe that from Lemma 15 and Section 7 we have that

which immediately gives

On the other hand for any suitable drift \(\check{u} \in \mathbb{H}_a\) we get the bound

where

Therefore it remains to produce an appropriate drift \(\check{u}\) for which the r.h.s. in eq. (29) is finite (and so uniformly in \(| \Lambda |\) and of \(o (\lambda^3)\)).

One possible strategy is to try and choose \(\check{u}\) such that \(l (\check{u}) = 0\), however this fails since estimates on this \(u\) via Gronwall's inequality would rely on the Hölder norm of \(\mathbb{W}_t^2\) for which we do not have uniform control in the volume. In order to overcome this problem we decompose \(\mathbb{W}^2\) and use weighted estimates similarly as done in [27].

Consider the decomposition

where the random field \(\mathcal{U}_{\geqslant} \mathbb{W}_s^2\) is constructed as follows. Let \(\varphi\) be smooth function, positive and supported on \([- 2, 2]^3\) and such that \(\sum_{m \in \Lambda \cap \mathbb{Z}^d} \varphi^2 (\bullet - m) = 1\). Denote by \(\varphi_m = \varphi (\bullet - m)\). Let \(\tilde{\chi}\) be a smooth function supported in \(B (0, 1)\). Denote by \(\mathcal{X}_{> N} f\) the Fourier multiplier operator \(\mathcal{F}^{- 1} \tilde{\chi} (k / N) \hat{f}\) and similarly \(\mathcal{X}_{\leqslant N} f = \mathcal{F}^{- 1} (1 - \tilde{\chi} (k / N)) \hat{f}\). Set \(L_m (s) := (1 + \| \varphi_m \mathbb{W}_s^2 \|)_{{\mathcal{\mathscr{C}}^{- 1 - \delta}} }^{\frac{1}{2 \delta}}\) and let

and

(with slight abuse of notation we drop the time dependence of the operators \(\mathcal{U}_{\leqslant}\), \(\mathcal{U}_{>}\)).

Observe that the laws of both \(\)\(\mathcal{U}_{>} \mathbb{W}_s^2\) and \(\mathcal{U}_{\leqslant} \mathbb{W}_s^2\) are translation invariant w.r.t to translations by \(m \in \Lambda \cap \mathbb{Z}^d\). By [47], Theorem 2.4.7 and Bernstein inequality

Furthermore for a weight \(\rho\) (see Appendix A for precisions on the weighted spaces \(L^p (\rho)\), \(\mathscr{C}^{\alpha} (\rho)\) and \(B^{\alpha}_{p, q} (\rho)\) used below):

where we used the possibility to compare weighted and unweighted norms once localized via \(\varphi_m\). We now let \(\check{u}\) be the solution to the (integral) equation

which can be solved globally. For \(3 \delta < 1 / 2\) and \(p \geqslant 1\), we have, for \(t \in [0, T]\),

Therefore Gronwall's lemma implies that, for \(t \in [0, T]\):

Taking \(\rho = \frac{\mathbb{1}_{\Lambda}}{| \Lambda |}\) and using Besov embedding we deduce from (33):

Now computing \(l^T (\check{u})\) from eq. (30) and (32), we obtain

It now remains to prove that \(\mathbb{E} [\| l^T (\check{u}) \|^2_{\mathcal{H}}] \lesssim O (\lambda^3)\) uniformly in \(T > 0\). Note that, for \(s \in [0, T]\),

We know that the distribution of \(\check{u}\) is invariant under translation by \(m \in \Lambda \cap \mathbb{Z}^d\). Recalling that \(\sum_{m \in \Lambda \cap \mathbb{Z}^d} \varphi^2 (\bullet - m) = 1\) and letting \(\rho\) be a polynomial weight with sufficient decay and such that \(\rho^3 \geqslant \varphi^2\), we have

This last quantities are bounded since standard arguments allow to bound, for \(p\) sufficiently large,

uniformly in \(s \geqslant 0\). Similarly, we have

Now

and

This concludes the proof of the lemma.

\(\Box\)

Remark 17. The decomposition of the noise is similar to the one given in [27] 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.

6.Gamma convergence

In this section we establish the \(\Gamma\)-convergence of the variational functional obtained in Lemma 15 as \(T \rightarrow \infty\). \(\Gamma\)-convergence is a notion of convergence introduced by De Giorgi which is well suited for the study of variational problems. The book [8] is a nice introduction to \(\Gamma\)-convergence in the context of the calculus of variations. For the convenience of the reader we recall here the basic definitions and results.

A fundamental consequence of \(\Gamma\)-convergence is the convergence of minima.

For a proof see [20].

In this section we allow all constants to depend on the volume \(| \Lambda |\): this is not critical since, at this point, the aim is to obtain explicit formulas at fixed \(\Lambda\).

We denote \(\)

and by \(\mathcal{H}_w^{\alpha, p}\) the reflexive Banach space \(\mathcal{H}^{\alpha, p}\) endowed with the weak topology. We will write \(\mathcal{H}^{\alpha}\) for \(\mathcal{H}^{\alpha, 2}\), and \(\mathcal{H}\) for \(\mathcal{H}^0\) and let \(\mathcal{L} := \mathcal{H}^{- 1 / 2 - \kappa, 3}\), this space will be useful as it gives sufficient control over \(Z\):

Proof. By definition of \(Z\) we have for any \(0 < \varepsilon < 1 / 8 - \kappa / 2\),

where we have used a Sobolev embedding in the second to last line. Since

for any \(t_2 \in [0, \infty]\), we can conclude by the Rellich–Kondrachov embedding theorem and the Arzela–Ascoli theorem, that bounded sets in \(\mathcal{L}\) are mapped to compact sets in \(C ( [0, \infty], L^4)\), proving the claim.\(\Box\)

We will need the following lemma, which establishes pointwise convergence for the functional \(\Phi_T\) defined in Lemma 15. In the sequel, by an abuse of notation, we will denote both a generic element of \(\mathfrak{S}\) and the canonical random variable on \(\mathfrak{S}\) by

Proof. Lemma 21 implies that for any \(u^T \rightarrow u\) in \(\)\(\mathcal{L}_w\) we have \(Z (u^T) \rightarrow Z (u)\) in \(C ([0, \infty], L^4)\) and by the convergence of \(l^T \rightarrow l\) in \(\mathcal{H}_w\) we have also \(K (u^T) \rightarrow K (u)\) in \(C ([0, \infty], H^{1 - \kappa})\). The products \(\mathbb{X}^{T, 1}_T \mathbb{X}_T^{T, [3]}\) and \(\mathbb{X}^{T, 1}_T (\mathbb{X}_T^{T, [3]})^2\) can be decomposed using paraproducts and, after replacing the resonant products by the corresponding stochastic objects in \(\mathbb{X}^T\), we obtain the finite \(T\) analogs of the expressions in eq. (36). After this preprocessing, it is easy to see by continuity that we have \(\mathbb{X}^{T, 1}_T \mathbb{X}_T^{T, [3]} \rightarrow \mathbb{X}_{\infty}^1 \mathbb{X}_{\infty}^{[3]}\) and \(\mathbb{X}^{T, 1}_T (\mathbb{X}_T^{T, [3]})^2 \rightarrow \mathbb{X}^1_{\infty} (\mathbb{X}_{\infty}^{[3]})^2\) in \(\mathscr{C}^{1 / 2 - \kappa}\). For \(\Upsilon^{(1)}\) and \(\Upsilon^{(4)}\) and the first term of \(\Upsilon^{(6)}\) the statement follows from uniform bounds for \((\mathbb{X}^T, Z (u^T), K (u^T))\) on \(\mathfrak{S} \times C ([0, \infty], H^{1 / 2 - \kappa}) \times C ([0, \infty], H^{1 - \kappa})\) and multilinearity. For \(\Upsilon^{(2)}\) and the first two terms of \(\)\(\Upsilon^{(5)}\) convergence to \(0\) follows from the bounds established in Lemma 39 and Lemma 42. For \(\Upsilon^{(3)}\), the last term of \(\Upsilon^{(5)}\) and the last two terms of \(\Upsilon^{(6)}\) we can again use uniform bounds and multilinearity as well as dominated convergence, thanks to Proposition 57.\(\Box\)

Going back to our particular setting recall that from Lemma 15 we learned that

with

where \(l^T (u), Z (u), K (u)\) are functions of \(u\) according to eq. (21). This form of the functional is appropriate to analyze the limit \(T \rightarrow \infty\) 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 37 below.

In order to use \(\Gamma\)-convergence, we need to properly modify the variational setting in order to guarantee enough compactness and continuity uniformly as \(T \rightarrow \infty\).

The analytic estimates contained in Section 7 below allow to infer that there exists a small \(\delta \in (0, 1)\), and a finite constant \(Q_T > 0\) uniformly bounded in \(T\) such that

and

As long as \(T\) is finite, the original potential \(V_T\) is bounded below so in particular we have

From this we conclude that we can relax the optimization problem and ask that \(u \in \mathbb{L}_a\), once we have established Lemma 26 where \(\mathbb{L}_a\) is the space of predictable processes in \(\mathcal{L}\):

The reason of this relaxation lies in the fact that the cost terms \(\| l^T (u) \|_{\mathcal{H}}, \operatorname{and} \| Z_T \|_{L^4}\) control the \(\mathcal{L}\) norm of \(u\) uniformly in \(T\), modulo constants depending only on \(\| \mathbb{W} \|_{\mathfrak{S}}\) which are bounded in average uniformly in \(T\).

Note that eq. (37) implies that for any sequence \((u^T)_T\) such that \(F_T (u^T)\) remains bounded we must have that also

To prove \(\Gamma\)-convergence we need to find a space with a topology which, on the one hand is strong enough to enable to prove the \(\Gamma\)-liminf inequality, and on the other hand allows to obtain enough compactness from \(F_T\). Almost sure convergence on \(\mathfrak{S} \times \mathcal{L}\) 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.

Condition (b) allows to exclude pathological points in \(\overline{\mathcal{X}}\) and makes possible Lemma 32 below. Then

where

and where \(\mathbb{E}_{\mu}\) denotes the expectation on \(\mathfrak{S} \times \mathcal{L}\) wrt. the probability measure \(\mu\). Our first aim will be to prove that the family \((\breve{F}_T)_T\) is indeed equicoercive on \(\overline{\mathcal{X}}\).

Proof. Observe that \(\operatorname{Law} (\mathbb{W})\) on \(\mathfrak{S}\) is tight since \(\mathfrak{S}\) is a separable metric space, so for any \(\varepsilon > 0\), we can find a compact set \(\mathcal{K}^1_{\varepsilon} \subset \mathfrak{S}\) such that \(\mu ((\mathfrak{S} \setminus \mathcal{K}_{\varepsilon}) \times \mathcal{L}) < \varepsilon / 2\). Now let \(\mathcal{K}_{\varepsilon}^2 := \mathcal{K}^1_{\varepsilon} \times B (0, C) \subset \mathfrak{S} \times \mathcal{L}\), for some large \(C\) to be chosen later. Then \(\mathcal{K}_{\varepsilon}^2\) is a compact subset of \(\mathfrak{S} \times \mathcal{L}_w\) and

Choosing \(C > \sup_T 2\mathbb{E}_{\mu^T} [\| u \|^2_{\mathcal{L}}] / \varepsilon\) gives tightness.\(\Box\)

Proof. We use \(\| l^T (u) \|_{\mathcal{L}} \lesssim \| l^T (u) \|_{\mathcal{H}}\) in the bound

\(\Box\)

Proof. Define for some \(K > 0\) large enough

From eq. (37) we have

so

on the other hand from (38) it follows that

So for \(K\) large enough

And by Lemma 26 and Lemma 25 \(\mathcal{K}\) is a compact set. \(\Box\)

To be able to use the equicoercivity we will need to show that we can extend the infimum in (41) to \(\bar{\mathcal{X}}\). For this we will first need some properties of the space \(\bar{\mathcal{X}}\). In particular we will need to show that measures with sufficiently high moments are dense in \(\overline{\mathcal{X}}\) in a way which behaves well with respect to \(\breve{F}_T\). For this we introduce some approximations which will be useful in the sequel.

Remark 29. Let us briefly comment on the uses of these approximation in the sequel. \(\text{reg}_{t : x, \varepsilon}\) will be used when one wants to obtain a sequence of weakly convergent measures on \(\mathfrak{S} \times \mathcal{H}\) or \(\mathfrak{S} \times \mathcal{L}\) from a sequence of measures weakly convergent on \(\mathfrak{S} \times \mathcal{L}_w\). \(\text{reg}_{x, \varepsilon}\) will be used when one wants to obtain a measure on \(\mathfrak{S} \times \mathcal{H}\) from one on \(\mathfrak{S} \times \mathcal{L}\), while preserving the estimates on the moments of \(Z (u)\) since \(Z (u \ast_{\Lambda} \eta_{\varepsilon}) = Z (u) \ast_{\Lambda} \eta_{\varepsilon}\).

Proof. By definition of \(\overline{\mathcal{X}}\) of there exists \(\tilde{\mu}_n \rightarrow \mu\) weakly on \(\mathfrak{S} \times \mathcal{L}_w\). Then \((\text{reg}_{t : x, \varepsilon})_{\ast} \tilde{\mu}_n \rightarrow (\text{reg}_{t : x, \varepsilon})_{\ast} \mu\) on \(\mathfrak{S} \times \mathcal{L}\) as \(n \rightarrow \infty\), and since \((\text{reg}_{t : x, \varepsilon})_{\ast} \mu \rightarrow \mu\) weakly on \(\mathfrak{S} \times \mathcal{L}\) as \(\varepsilon \rightarrow 0\), we obtain the statement by taking a diagonal sequence.\(\Box\)

Proof. Let \(f\) be a Lipschitz function on \(\mathcal{L}\) with Lipschitz constant \(L\). Let \(\eta \in C (\mathbb{R}, \mathbb{R})\) be supported on \(B (0, 2)\) with \(\eta = 1\) on \(B (0, 1)\), and \(\eta_N (x) = \eta (x / N)\). Then \(u \mapsto f (u) \eta_N (\| u \|_{\mathcal{L}})\) is bounded,

and

Using that \(\sup_n \mathbb{E}_{\mu_n} [\| u \|^2_{\mathcal{L}} ] < \infty\) we have

and sending \(N \rightarrow \infty\) gives the statement. The second statement follows from the first and Lemma 21.\(\Box\)

From the definition of \(\mathcal{X}\) we have that

where

The space \(\mathcal{X}\) 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 \(\overline{\mathcal{X}}\) of \(\mathcal{X}\) in the infimum. To do this we first need to prove that we can approximate measures in \(\overline{\mathcal{X}}\) by measures with bounded support in the second marginal which are still in \(\overline{\mathcal{X}}\). This is the content of the following

Proof.

Step 1 First let us show how to approximate \(\mu\) with \(\tilde{\mu}_L\) which are defined such that \(\|Z_T (u) \|_{L^4} \leqslant L\), \(\tilde{\mu}_L\) almost surely. As \(\mu \in \overline{\mathcal{X}}\), there exists \((\mu_n)_n \subset \mathcal{X}\) such that \(\mu_n \rightarrow \mu\) on \(\mathfrak{S} \times \mathcal{L}\) and \(\sup_n \mathbb{E}_{\mu_n} [\| u \|^2_{\mathcal{L}}] < \infty\). Since \(\mu_n \in \mathcal{X}\) there exist \((u^n)_n\) adapted such that \(\mu_n =\operatorname{Law} (\mathbb{W}, u^n)\). Define \(\tilde{Z}^n_s := \mathbb{E} \left[ \int^T_0 J_t u^n_t \mathrm{d} t| \mathcal{F}_s \right] = \int^T_0 J_t \mathbb{E} [u^n_t | \mathcal{F}_s] \mathrm{d} t\). Then \(\tilde{Z}\) is a martingale with continuous paths in \(L^4 (\Lambda)\). Define the stopping time \(T_{L, n} = \inf \{ t \in [0, T] |\| \tilde{Z}^n_t \|_{L^4} \geqslant L \}\) where the inf is equal to \(T\) if the set is empty. Observe that \(\tilde{Z}_{T_{L, n}} = \int^T_0 J_t \mathbb{E} [u^n_t | \mathcal{F}_{T_{L, n}}] \mathrm{d} t = Z_T (u^{L, n})\) with \(u^{L, n}_t := \mathbb{E} [u^n_t | \mathcal{F}_{T_{L, n}}]\) adapted, by optional sampling, and almost surely \(\| \tilde{Z}_{T_L} \|_{L^4} \leqslant L\). Now set \(\tilde{\mu}_{L, n} := \operatorname{Law}_{\mathbb{P}} (\mathbb{W}, u^{L, n})\).

Step 1.1 (Tightness) The next goal is to show that for fixed \(L\), we can select a suitable convergent subsequence from \((\tilde{\mu}_{L, n})_n\). For this we first show that \((\tilde{\mu}_{L, n})_n\) is tight on \(\mathfrak{S} \times \mathcal{L}_w\). From the definition of \(\mathcal{X}\) we have that \(\sup_n \mathbb{E}_{\mu_n} \left[ \left\| u \right\|^2_{\mathcal{L}} \right] < \infty\), and by construction

which gives tightness according to Lemma 25. We can then select a subsequence which converges on \(\mathcal{L}_w\).

Step 1.2 (Bounds) Let \(\tilde{\mu}_L\) be the limit of the sequence constructed in Step 1.1. In this step we prove bounds on the relevant moments of \(\tilde{\mu}_L\). Let \(f^M_1, f^M_2\) be sequences of functions on \(\mathbb{R}\) which are Lipschitz, convex and monotone for every \(N\), while for every \(x \in \mathbb{R}\)

Then \(f_1^M (\| u \|_{\mathcal{L}})\) is a lower-semi continuous positive function on \(\mathcal{L}_w\) so by the Portmanteau lemma we have

and since it is also Lipschitz continuous and convex we have

Therefore

Proceeding similarly for \(Z\), we see that \(f^N_2 (\| Z_T \|_{L^4})\) is a continuous function on \(L^4\) bounded below, and Lipschitz-continuous and convex on \(L^4\) so we again can estimate

and, taking \(N \rightarrow \infty\), obtain

Step 1.3 (Weak convergence) Now we prove weak convergence of \(\tilde{\mu}_L\) to \(\mu\) on \(\mathfrak{S} \times \mathcal{L}\). Let \(f : \mathfrak{S} \times \mathcal{L} \rightarrow \mathbb{R}\) be bounded and continuous. By dominated convergence and continuity of \(f\), \(\lim_{\varepsilon} \mathbb{E}_{\tilde{\mu}_L} [f (\mathbb{X}, \text{reg}_{t : x, \varepsilon} (u))] =\mathbb{E}_{\tilde{\mu}_L} [f (\mathbb{X}, u)]\). Using furthermore that \((\mathbb{X}, u) \mapsto f (\mathbb{X}, \text{reg}_{t : x, \varepsilon} (u))\) is continuous on \(\mathfrak{S} \times \mathcal{L}_w\) and Lemma 21 in the 5th line below, we can estimate

Step 2 In this step we improve the approximation to have bounded support. Let \(\mu_n \rightarrow \mu\) be the subsequence selected in Step 1.1. Recall that \(\mu_n =\operatorname{Law} (\mathbb{W}, u^n)\) with adapted \(u^n\). Define \(\tilde{Z}_t^{n, N} := \mathbb{E} [Z_T (\text{cut}_N (u)) \mathrel{|} \mathcal{F}_t]\), and similarly to Step 1, \(T_{n, L, N} := \inf \{t \geqslant 0 |\| \tilde{Z}^{n, N}_t \|_{L^4} \geqslant L\}\), set \(u^{n, N, L} := \mathbb{E} [\text{cut}_N (u) \mathrel{|} \mathcal{F}_{T_{n, L, N}}]\). Then \(\| u^{n, N, L} \|_{\mathcal{L}} \leqslant N\) uniformly in \(n\) and \(\mathbb{P}\)-almost surely, so \(\mu_{n, L, N} =\operatorname{Law} (\mathbb{W}, u^{n, N, L})\) is tight on \(\mathfrak{S} \times \mathcal{L}_w\) and we can select a weakly convergent subsequence. Denote the limit by \(\mu_{L, N}\). Now we follow the strategy from Step 1.

Step 2.1 (Bounds) Let \(f_1^M\) be defined like in Step 1.2. Then again we have

It follows that

Step 2.1 (Weak convergence) Now we prove that \(\mu_{L, N} \rightarrow \tilde{\mu}_L\) weakly on \(\mathcal{L}\). Let \(f : \mathfrak{S} \times \mathcal{L} \rightarrow \mathbb{R}\) be bounded and continuous. By dominated convergence and continuity of \(f\), \(\lim_{\varepsilon} \mathbb{E}_{\tilde{\mu}_L} [f (\mathbb{X}, \text{reg}_{t : x, \varepsilon} (u))] =\mathbb{E}_{\tilde{\mu}_L} [f (\mathbb{X}, u)]\), and furthermore since\(f (\mathbb{X}, \text{reg}_{t : x, \varepsilon} (u))\) is continuous on \(\mathfrak{S} \times \mathcal{L}_w\) we have

Step 3. We now put everything together. Since all \(\mu_{L, N}\) are supported on the set \(\{ u : \| Z_T (u) \|_{L^4} \leqslant L \}\), weak convergence and Lemma 21 imply

By the Portmanteau lemma,

and

which together with Step 1.2 imply \(\lim_{L \rightarrow \infty} \mathbb{E}_{\tilde{\mu}_L} [\| u \|_{\mathcal{L}}^2] =\mathbb{E}_{\mu} [\| u \|_{\mathcal{L}}^2]\), and by the same argument \(\lim_{L \rightarrow \infty} \mathbb{E}_{\tilde{\mu}_L} [\| Z_T (u) \|^4_{L^4}] =\mathbb{E}_{\mu} [\| Z_T (u) \|^4_{L^4}]\). For any \(\delta > 0\) we can choose a \(\tilde{\mu}_L\) such that

Since then by (42)

we can choose \(N\) large enough so that

which implies the statement of the theorem. \(\Box\)

Proof. To prove the claim it is enough to show that for any \(\mu \in \overline{\mathcal{X}}\), for any \(\alpha > 0\), there exists a sequence \(\mu_n \in \mathcal{X}\) such that \(\limsup_{n \rightarrow \infty} \breve{F}_T (\mu_n) \leqslant \breve{F}_T (\mu) + \alpha\). W.l.o.g we can assume that \(\breve{F}_T (\mu) < \infty\). Observe that, as long as \(T < \infty\) we can also express

and deduce that \(\mathbb{E}_{\mu} \| u \|_{\mathcal{H}}^2 < \infty\) since \(V_T\) is bounded below at fixed \(T\). By Lemma 32 there exists a sequence \((\mu_L)_L \subset \overline{\mathcal{X}}\) , such that \(\mu_L\), \(\| u \|_{\mathcal{L}} \leqslant L\) almost surely, \(\mu_L \rightarrow \mu\) on \(\mathfrak{S} \times \mathcal{L}\) and

First we have to improve the regularity of \(\mu_L\) to get convergence on \(\mathfrak{S} \times \mathcal{H}_w\) but without affecting our control on the moments of \(Z_T\), so let \(\mu^{\varepsilon}_L := (\text{reg}_{x, \varepsilon})_{\ast} \mu_L\) and \(\mu^{\varepsilon} := (\text{reg}_{x, \varepsilon})_{\ast} \mu\). Then

and \(\mu^{\varepsilon}_L \rightarrow\)\(\mu^{\varepsilon}\) on \(\mathfrak{S} \times \mathcal{H}\). By continuity of \(\breve{F}_T\) and the bound (38), \(\breve{F}_T (\mu^{\varepsilon}_L) \rightarrow \breve{F}_T (\mu^{\varepsilon})\) as \(L \rightarrow \infty\) and \(\breve{F}_T (\mu^{\varepsilon}) \rightarrow \breve{F}_T (\mu)\) as \(\varepsilon \rightarrow 0\). In particular we can find \(L\) and \(\varepsilon\) such that \(| \breve{F}_T (\mu^{\varepsilon}_L) - \breve{F}_T (\mu) | < \alpha / 2\). By Lemma 32 there exists a sequence \((\mu_{n, L})_{n, L}\) such that each measure \(\mu_{n, L}\) is supported on \(\mathfrak{S} \times B (0, L)\) and \(\mu_{n, L} \rightarrow \mu_L\) weakly on \(\mathfrak{S} \times \mathcal{H}_w\). Setting \(\mu^{\varepsilon, \delta}_{n, L} := (\text{reg}_{t ; x, \delta})_{\ast} (\text{reg}_{x, \varepsilon})_{\ast} \mu_{n, L}\) and \(\mu^{\varepsilon, \delta}_L := (\text{reg}_{t ; x, \delta})_{\ast} (\text{reg}_{x, \varepsilon})_{\ast} \mu_L\) we have \(\mu^{\varepsilon, \delta}_{n, L} \rightarrow \mu^{\varepsilon, \delta}_L\) on \(\mathfrak{S} \times \mathcal{H}\) with norm topology. Then, for some \(\chi \in C (\mathbb{R}, \mathbb{R}), \chi = 1\operatorname{on}B (0, 1)\) supported on \(B (0, 2)\) and for any \(N \in \mathbb{N}\), \(V_T (\mathbb{X}^1_T + Z_T (u)) \chi (\| \mathbb{X} \|_{\mathfrak{S}} / N), \| u \|^2_{\mathcal{H}}\) are continuous bounded functions on the common support of \(\mu^{\varepsilon, \delta}_{n, L}\) and

and by dominated convergence (since \(\mu^{\varepsilon, \delta}_L\) is supported on \(\mathfrak{S} \times B (0, L)\)) we can find a \(\delta\) such that \(| \breve{F}_T (\mu^{\varepsilon, \delta}_L) - \breve{F}_T (\mu^{\varepsilon}_L) | < \alpha / 2\) which proves the statement. \(\Box\)

The proof of Lemma 33 does not apply when \(T = \infty\). However also in this case the functional

has a well defined meaning, so we can investigate the relation between the two variational problems (on \(\mathcal{X}\) and on \(\bar{\mathcal{X}}\)) when the cutoff is removed. An additional difficulty derives from the fact that approximating the drift \(u\) we might destroy the regularity of \(l^{\infty} (u)\), since now \(l^{\infty} (u)\) needs to be more regular than \(u\), contrary to the finite \(T\) case. To resolve this problem we need to be able to smooth out the remainder without destroying the bound on \(Z_T (u)\). To do so smoothing \(l^{\infty} (u)\) directly, and constructing a corresponding new \(u\) will not work, since \(l^{\infty} (u)\) by itself does not give enough control on \(u\) and \(Z (u)\). However we are still able to prove the following lemma by regularizing an “augmented” version of \(l^{\infty} (u)\).

Proof. Let \(\mathbb{X}^2 = \mathcal{U}_{\leqslant} \mathbb{X}^2 + \mathcal{U}_{>} \mathbb{X}^2\) be the decomposition introduced in Section 5, and observe that for any \(c > 0\) we can easily modify it to ensure that \(\| \mathcal{U}_{>} \mathbb{X}^2 \|_{\mathcal{\mathscr{C}}^{- 1 - \kappa}} < c\), almost surely for any \(\mu \in \overline{\mathcal{X}}\) and for any \(1 \leqslant p < \infty\), \(\mathbb{E}_{\mu} [\| \mathcal{U}_{\leqslant} \mathbb{X}^2 \|^p_{\mathcal{\mathscr{C}}^{- 1 + \kappa}}] \leqslant C\) where \(C\) depends on \(| \Lambda |, \kappa, c, p\). Now set \(\tilde{l}_t (u) = - \lambda J_s (\mathcal{U}_{\leqslant} \mathbb{X}_t^2 \succ Z_t^{\flat} (u)) + l^{\infty} (u)\). Then \(u\) satisfies

From this equation we can see that, like in Section 5,

and choosing, \(c\) small enough we get

And similarly we observe that

so again with \(c\) small enough and since \(Z^{\flat}_s = \theta_s Z_T\) for \(s \leqslant T\):

Conversely, it is not hard to see that we have the inequalities

and

Clearly the map \((\mathbb{X}, u) \mapsto (\mathbb{X}, \tilde{l} (u))\) is continuous as a map \(\mathfrak{S} \times \mathcal{L} \rightarrow \mathcal{L}\) and using Lemma 21 also as a map \(\mathfrak{S} \times \mathcal{L}_w \rightarrow \mathfrak{S} \times \mathcal{L}_w\) , and the inverse is clearly continuous \(\mathfrak{S} \times \mathcal{L} \rightarrow \mathfrak{S} \times \mathcal{L}\). We now show that it is also continuous as a map \(\mathfrak{S} \times \mathcal{L}_w \rightarrow \mathfrak{S} \times \mathcal{L}_w\). Assume that \(\tilde{l} (u^n) \rightarrow l (u)\) weakly, since then \(\| l (u^n) \|_{\mathcal{L}}\) bounded, this implies by (44) that also \(\| u^n \|_{\mathcal{L}}\) is bounded, and so we can select a weakly convergent subsequence, converging to \(u^{\star}\). Then \(u^{\star}\) solves the equation

(which can be seen for example by testing with some \(h \in \mathcal{L}^{\ast}\)) which implies that \(u^{\star} = u\) (e.g. by Gronwall). Now define \(\operatorname{rem}_{\varepsilon} (u)\) to be the solution to the equation

Then by the properties discussed above \(u \mapsto \operatorname{rem}_{\varepsilon} (u)\) is continuous in both the weak and the norm topology, we also have from (44) and (47) that

from (45) we have

and by definition of \(\operatorname{rem}_{\varepsilon} (u)\)

Now observe that

Observing that also \(\| \lambda J_s (\mathcal{U}_{\leqslant} \mathbb{X}_t^2 \succ Z_t^{\flat} (\operatorname{rem}_{\varepsilon} (u))) \|_{\mathcal{H}}\) depends continuously on \((\mathbb{X}, u)\) (both in the weak and strong topology on \(\mathcal{L}\)) gives the statement.\(\Box\)

Proof. By Lemma 32 there exists a sequence \(\mu_{\tilde{L}} \rightarrow \mu\) weakly on \(\mathfrak{S} \times \mathcal{L}\) such that

and \(\mu_{\tilde{L}}\) is supported on \(\mathfrak{S} \times B (0, \tilde{L}) \subset \mathfrak{S} \times \mathcal{L}\). Now set \(\mu^{\varepsilon}_{\tilde{L}} := (\operatorname{rem}_{\varepsilon})_{\ast} \mu_{\tilde{L}}\). Then \(\mu^{\varepsilon}_{\tilde{L}} \rightarrow \mu^{\varepsilon} := (\operatorname{rem}_{\varepsilon})_{\ast} \mu\) on \(\mathfrak{S} \times \mathcal{L}\) and by the bounds from Lemma 34 also \(\mathbb{E}_{\mu^{\varepsilon}_{\tilde{L}}} [\| Z_T (u) \|^4_{L^4}] \rightarrow \mathbb{E}_{\mu^{\varepsilon} } [\| Z_T (u) \|^4_{L^4}]\) and \(\mathbb{E}_{\mu^{\varepsilon}_{\tilde{L}}} \left[ \left\| l^{\infty} (u) \right\|^2_{\mathcal{H}} \right] \rightarrow \mathbb{E}_{\mu^{\varepsilon}} \left[ \left\| l^{\infty} (u) \right\|^2_{\mathcal{H}} \right]\). The bounds from Lemma 34 imply also \(\mathbb{E}_{\mu^{\varepsilon}} [\| Z_T (u) \|^4_{L^4}] \rightarrow \mathbb{E}_{\mu } [\| Z_T (u) \|^4_{L^4}]\), \(\mathbb{E}_{\mu^{\varepsilon}} \left[ \left\| l^{\infty} (u) \right\|^2_{\mathcal{H}} \right] \rightarrow \mathbb{E}_{\mu} \left[ \left\| l^{\infty} (u) \right\|^2_{\mathcal{H}} \right]\), and furthermore

and similarly

and by continuity of \(\breve{F}_{\infty}\) and (38) we are also able to deduce that we can find \(\varepsilon\) small enough and \(\tilde{L}\) large enough depending on \(\varepsilon\) such that \(| \breve{F}_{\infty} (\mu^{\varepsilon}) - \breve{F}_{\infty} (\mu) | < 1 / 2 L\) and \(| \breve{F}_{\infty} (\mu_L^{\varepsilon}) - \breve{F}_{\infty} (\mu^{\varepsilon}) | < 1 / 2 L\). Choosing \(\mu_L = \mu^{\varepsilon}_{\tilde{L}}\) we obtain the first three points of the Lemma. For the fourth point recall that from Lemma 32 we have sequences \(\mu_{n, \tilde{L}} \rightarrow \mu_{\tilde{L}}\) weakly on \(\mathfrak{S} \times \mathcal{L}_w\), and \(\mu_{n, \tilde{L}} \in \mathcal{X}\), which have support in \(\mathfrak{S} \times B (0, \tilde{L})\) and since \(\operatorname{rem}_{\varepsilon}\) is continuous on \(\mathfrak{S} \times \mathcal{L}_w\) setting \(\mu^{\varepsilon}_{n, \tilde{L}} := (\text{reg}_{\varepsilon})_{\ast} \mu_{n, \tilde{L}}\) we obtain the desired sequence. \(\Box\)

Proof. One can now proceed very similarly to the proof of Lemma 33. Let \(\mu \in \overline{\mathcal{X}}\) such that \(\breve{F}_{\infty} (\mu) < \infty\). By Lemma 35, for any \(L, \mu \in \bar{\mathcal{X}}\), there exists a \(\mu_L\) such that \(| \breve{F}_{\infty} (\mu) - \breve{F}_{\infty} (\mu_L) | < 1 / L\), and a sequence \((\mu_{n, L})_n\) such that \(\mu_{n, L} \in \mathcal{X}\), \(\mu_{n, L} \rightarrow \mu_L\) weakly on \(\mathfrak{S} \times \mathcal{L}_w\), and such that (50) is satisfied. Define \(\mu^{\varepsilon, \delta}_{n, L} := \operatorname{Law} (\mathbb{X}, \operatorname{rem}_{\varepsilon} (\text{reg}_{t : x, \varepsilon} (u)))\), and observe that now \(\mu^{\varepsilon, \delta}_{n, L} \rightarrow \mu^{\varepsilon, \delta}_L\) on \(\mathfrak{S} \times \mathcal{L}\), \(\operatorname{Law}_{\mu^{\varepsilon, \delta}_{n, L}} (\mathbb{X}, l^{\infty} (u)) \rightarrow \operatorname{Law}_{\mu^{\varepsilon, \delta}_L} (\mathbb{X}, l^{\infty} (u))\) on \(\mathfrak{S} \times \mathcal{H}\), and that we have \(\sup_n (\mathbb{E}_{\mu^{\varepsilon, \delta}_{n, L}} [\| u \|^p_{\mathcal{L}}] +\mathbb{E}_{\mu^{\varepsilon, \delta}_{n, L}} [\| l^{\infty} (u) \|^p_{\mathcal{H}}]) < \infty\). Then for some \(\chi \in C (\mathbb{R}, \mathbb{R}), \chi = 1\operatorname{on}B (0, 1)\) supported on \(B (0, 2)\), for any \(N \in \mathbb{N}\), the function

is bounded and continuous on \(\mathfrak{S} \times \mathcal{L}\), and so by weak convergence

As we can find \(\varepsilon, \delta\) such that \(| \breve{F}_{\infty} (\mu^{\varepsilon, \delta}_L) - \breve{F}_{\infty} (\mu_L) | < 1 / L\) we can conclude. \(\Box\)

Finally we can state the key result of this section.

Proof. In order to establish \(\Gamma\)-convergence consider a sequence \(\mu^T \rightarrow \mu\) in \(\overline{\mathcal{X}}\). We need to prove that \(\liminf_{T \rightarrow \infty} \breve{F}_T (\mu^T) \geqslant \breve{F}_{\infty} (\mu)\). 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

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 [35] we can extract a subsequence (again, not relabeled) and find random variables \((\tilde{\mathbb{X}}^T, \tilde{u}^T)_T\) and \((\tilde{\mathbb{X}}, \tilde{u})\) on some probability space \((\tilde{\Omega}, \tilde{\mathbb{P}})\) such that \(\operatorname{Law}_{\tilde{\mathbb{P}}} (\tilde{\mathbb{X}}^T, \tilde{u}^T) = \mu^T\), \(\operatorname{Law}_{\tilde{\mathbb{P}}} (\tilde{\mathbb{X}}, \tilde{u}) = \mu\) and almost surely \(\tilde{\mathbb{X}}^T \rightarrow \tilde{\mathbb{X}}\) in \(\mathfrak{S}\), \(\tilde{u}^T \rightarrow \tilde{u}\) in \(\mathcal{L}_w\). Note that \(\tilde{l}^T := l^T (\tilde{\mathbb{X}}^T, \tilde{u}^T) \rightarrow l := l^{\infty} (\tilde{\mathbb{X}}, u)\) in \(\mathcal{L}_w\) and using (51) we deduce that the almost sure convergence \(l^T \rightarrow l\) in \(\mathcal{H}_w \), maybe modulo taking another subsequence, again not relabeled. Note that, by our analytic estimates (which hold pointwise on the probability space) we have

for some \(L^1 (\tilde{\mathbb{P}})\) random variable \(H (\tilde{\mathbb{X}}^T)\) such that \(\mathbb{E}_{\tilde{\mathbb{P}}} [H (\tilde{\mathbb{X}}^T)] =\mathbb{E} [H (\mathbb{W})]\). Fatou's lemma and Lemma 22 then give

which is the \(\Gamma\)-liminf inequality. Now all that remains is constructing a recovery sequence, for this we can again assume w.l.o.g that \(\breve{F}_{\infty} (\mu) < \infty\). From Lemma 35 there is \(\mu_L\) such that \(| \breve{F}_{\infty} (\mu) - \breve{F}_{\infty} (\mu_L) | < \frac{1}{L}\) and (49) is satisfied. Then choosing \(\mu_L^T =\operatorname{Law}_{\mu_L} (\mathbb{X}, \mathbb{1}_{\{ t \leqslant T \}} u_t) \) we obtain that \(l^T (\mathbb{1}_{\{ t \leqslant T \}} u_t) = \mathbb{1}_{\{ t \leqslant T \}} l^{\infty} (u)\), so \(\| l^T (\mathbb{1}_{\{ t \leqslant T \}} u_t) \|_{\mathcal{H}} \leqslant \| l^{\infty} (u) \|_{\mathcal{H}}\), and \(\| Z_T (\mathbb{1}_{\{ t \leqslant T \}} u_t) \|^4_{L^4} = \| Z_T (u) \|^4 \leqslant \| u \|^4_{\mathcal{L}}\), which is integrable by \(\)(49). By dominated convergence and Lemma 22 we obtain \(\lim_{T \rightarrow \infty} \breve{F}_T (\mu^T_L) = \breve{F}_{\infty} (\mu_L)\). Extracting a suitable diagonal sequence gives the recovery sequence.\(\Box\)

7.Analytic estimates

In this section we collect a series of analytic estimate which together allow to establish the pointwise bounds (37) and (38) and the continuity required for Lemma 22. First of all note that

which implies that quadratic functions of the norm \(\| K_t \|_{H^{1 - \kappa}}\) with small coefficient can always be controlled, uniformly in \([0, \infty]\), by the coercive term

Proof. By Proposition 56,

By Proposition 53,

which is estimated in the same way and finally,

Proof. Using the spectral support properties of the various terms we observe that

and

where we used also interpolation and the \(L^2\) bound \(\| Z^{\flat}_T \|_{L^2} \lesssim \| Z_T \|_{L^2}\). We recall also that

Therefore we estimate as follows

For the second term we can estimate

for the first term we get

which we can again estimate like in Lemma 38.\(\Box\)

Proof. First note that \(\)\(\dot{\theta}_t (\mathrm{D}) = (\langle \mathrm{D} \rangle / t^2) \dot{\theta} (\langle \mathrm{D} \rangle / t)\). In particular \(Z^{\flat}_t\) is spectrally supported in an annulus with inner radius \(t / 4\) and outer radius \(t / 2\). Then for any \(\theta \in [0, 1]\)

By Proposition 53, for any \(\varepsilon > 0\) there exists \(\delta > 0\) such that

and again

While

Proof. Using Lemma 9 we establish that

Next, we can write,

which can be easily estimated by Young's inequality. Decomposing

We can estimate the first two terms by

and

Young's inequality gives then the appropriate result. For the final term we use Proposition 55 to get

For the last term we estimate

which can be estimated like in Lemma 38 after we observe that

and use Lemma 44 to bound \(\mathbb{W}_T^{1 \circ [3]}\).\(\Box\)

Proof. We can estimate

and

These bounds imply that both of them remain bounded provided \(\gamma_T\) does not grow too fast in \(T\) which is indeed insured by Lemma 44. For the last term we can apply the estimate

Again, after we have fixed \(\gamma_t\) below, we will see this to be bounded. Collecting these bounds we get

Proof. We start by observing that

and using Lemma 44 and eq. (52) we have this term under control. Next split

and note that \(\langle t \rangle^{1 / 2} J_t\) satisfies the assumptions of Proposition 60 with \(m = - 1\). Therefore

and by Proposition 59,

and by binomial formula

Which can be easily estimated by Young's inequality. From Proposition 56 and Proposition 50

and by interpolation

The integrability of this term in time follows from the inequality

To prove this last bound, recall that \(t^{1 / 2} J_t\) is a Fourier multiplier with symbol

where \(\eta\) is a smooth function supported in an annulus of radius \(1\). Using this observation and applying Proposition 59 gives the estimate. Applying Proposition 55 and Proposition 50 we get

and after using duality and interpolation we obtain

Finally we have

Using eq. (54) to control \(\| Z_T \|_{H^{1 / 2 - \delta}}\) in terms of \(K_T\) we obtain the claim.\(\Box\)

8.Stochastic estimates

In this section we close our argument proving the following lemmas which give uniform estimates as \(T \rightarrow \infty\) of the stochastic terms appearing in our analytic estimates.

Proof. We will concentrate in proving the bounds on the renormalized terms in eq. (56) and (57) and leave to the reader to fill the details for the easier term in eq. (55). Recall the representation of \(W_t = Y_t\) in terms of the family of Brownian motions \((B_t^n)_{t, n}\) in eq. (4). Wick's products of the Gaussian field \(W_T\) can be represented as iterated stochastic integrals wrt. \((B_t^n)_{t, n}\). In particular, if we let \(\mathrm{d} w_s (k) = \langle k \rangle^{- 1} \sigma_s (k) \mathrm{d} B_s^k\), we have

where \(k_{(123)} := k_1 + k_2 + k_3\). Now products of iterated integrals can be decomposed in sums of iterated integrals and we get

where \(A^n_T := \{ 0 \leqslant s_1 < \cdots < s_n \leqslant T \} \subseteq [0, T]^n\) and where the deterministic kernels are given by

where \(\operatorname{Sh} (k, l)\) is the set of permutations of \(\{ 1, \ldots, k + l \}\) keeping the orders \(\sigma (1) < \cdots < \sigma (k)\) and \(\sigma (k + 1) < \cdots < \sigma (k + l)\) and where, for any symbol \(z\), we denote with expression of the form \(z_{1 \cdots n}\) the vector \((z_1, \ldots, z_n)\). Estimation of \(\Delta_q (\mathbb{W}^2_T \circ \mathbb{W}^{[3]}_T) (x)\) reduces then to estimate each of the three iterated integrals using BDG inequalities to get, for any \(p \geqslant 2\),

The kernel \(G^{2 \diamond [3]}_{0, q} ((s, k)_{1 \cdots 5})\) being a symmetric function of its argument, we can simplify this expression into an integral over \([0, T]^5\):

Under the measure \(\frac{\sigma_{s_5} (k_5)^2}{\langle k_5 \rangle^2} \mathrm{d} s_5\), we have

Therefore with some standard estimates we can reduce us to consider

Now by similar reasoning we also have

so

Finally, we note that the same strategy cannot be applied to the first chaos, since the kernel \(G^{2 \diamond [3]}_{2, q}\) cannot be uniformly bounded. We let

so

Observe that

We choose \(\gamma_T\) as

where we used the fact that for all \(q \in \mathbb{R}^d\) we have \(\sum_{i \sim j} K_i (q) K_j (q) = 1\), since \(\int f \circ g = \int f g\). Note that, as claimed,

Now

so when \(| q_1 + q_2 | \gg | k_1 |\) the quantity in round brackets can be estimated by \(| k_1 | \langle q_1 + q_2 \rangle^{- 4}\) while when \(| q_1 + q_2 | \lesssim | k_1 |\) it is estimated by \(\langle q_1 + q_2 \rangle^{- 2}\) so we have

And then with this choice of \(\gamma_T\) the kernel \(\tilde{G}^{2 \circ [3]}_{2, q}\) stays uniformly bounded as \(T \rightarrow \infty\) and satisfies

From this we easily deduce that

All together these estimates imply that

Standard argument allows to deduce eq. (56). The analysis of the other renormalized product proceeds similarly. Let

First note that by definition of Besov spaces we have

By Minkowski's integral inequality this is bounded by

When \(r \geqslant p\) Jensen's inequality and Fubini's theorem give

Finally hypercontractivity and stationarity allow to reduce this to bound

Letting \(I_q (t) =\mathbb{E} [| \Delta_q V (t) (0) | ^2]\) we have

Now we decompose the random field \(\Delta_q (\mathbb{W}^{\langle 2 \rangle \diamond \langle 2 \rangle}_t) (x)\) into homogeneous stochastic integral as above and obtain

with

Using our choice of \(\gamma_T\) in eq. (60) we have that

which implies also that

as claimed. We pass now to estimate the other two chaoses. The technique is the same we used above. Consider first

where we used that \(| \sigma_t (x) | \lesssim t^{- 1 / 2} \mathbb{1}_{x \sim t}\). Now taking \(\varepsilon + d / r > 0\) we have

Taking into account that \(| k_1 |, | k_2 | \lesssim t\) we can estimate

from which we deduce that

and then, as for \(I_{0, q}\), we have

as claimed. From these estimates standard arguments give eq. (57).\(\Box\)

Estimating the cube

We get for any \(p\), by space homogeneity,

Now

Now the second properties follow by the fact that \(\sigma_t\) is supported in an annulus of radius \(t\), so

and Hoelder estimates follow by Besov-embedding.

Bibliography

Appendix A.Besov spaces and paraproducts

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 [3, 28] and for weighted spaces in [27, 41]. First recall the definition of Littlewood–Paley blocks. Let \(\chi, \varphi\) be smooth radial functions \(\mathbb{R}^d \rightarrow \mathbb{R}\) such that

• \(\operatorname{supp} \chi \subseteq B (0, R)\), \(\operatorname{supp} \varphi \subseteq B (0, 2 R) \setminus B (0, R)\);

• \(0 \leqslant \chi, \varphi \leqslant 1\), \(\chi (\xi) + \sum_{j \geq 0} \varphi (2^{- j} \xi) = 1\) for any \(\xi \in \mathbb{R}^d\);

• \(\operatorname{supp} \varphi (2^{- j} \cdot) \cap \operatorname{supp} \varphi (2^{- i} \cdot) = \varnothing\) if \(| i - j | > 1\).

Introduce the notations \(\varphi_{- 1} = \chi\), \(\varphi_j = \varphi (2^{- j} \cdot)\) for \(j \geqslant 0\). For any \(f \in \mathscr{S}' (\Lambda)\) we define the operators \(\Delta_j f := \mathcal{F}^{- 1} \varphi_j (\xi) \hat{f} (\xi)\), \(j \geqslant - 1\).

For a proof see Theorem 3.17 and Remark 3.18 in [41].

The proof is an easy modification of the proof of Theorem 3.17 in [41].

Proof. The proof is a slight modification of the one given in [28] Lemma 2.97 from [3] and interpolation implies that \(\| \Delta_j f g - \Delta_j (f g) \|_{L^p} \leq 2^{- j \alpha} \| f \|_{W^{\alpha, p_1}} \| g \|_{L^{p_2}}\). This in turn gives after some algebraic computations(see [28])that

with \(\| R_j (f, g) \|_{L^p} \lesssim 2^{- j (\alpha + \beta)} \| f \|_{B_{p_1, \infty}^{\alpha}} {\| g \|_{B_{p_2, \infty}^{\beta}}} \). Now to prove the statement of the proposition observe that for smooth \(f, g, h\) we have

Now observe that the term \(f \succ \Delta_k g\) has Fourier transform outside of \(2^k B\) for some Ball \(B\) independent of \(k\), so choosing \(N\) large enough we can rewrite the sum as

Now we estimate the norm of the two terms separately. First note that for fixed \(j\)

has a Fourier transform supported in \(2^j B\). By Lemma 2.69 from [3] it is enough to get an estimate on \(\) \(\sup_k \left\| {2^{(\alpha + \beta + \gamma) j}} \sum_{j \geqslant - 1} \sum_{| i - j | \leqslant 1} \mathbb{1}_{k \leqslant i + N} R_j (f, \Delta_k g) \Delta_i h \right\|_{L^p}\) to estimate it in \(B_{p , \infty}^{\alpha + \beta + \gamma}\), so by Hölder inequality

For the second term observe that for fixed \(k\) the Fourier transform of

is supported in \(2^k B\). Now we can estimate again by Hölder inequality

Proof. This is modification of the proof of Lemma A.6 in [30]. Repeating an algebraic computation given in [30] in the proof of Lemma A.6, we get that for smooth \(f, g, h\) we have

Then we estimate

For a proof see [3] Lemma 2.78.

For a proof see [3] Lemma 2.99.

For a proof see [9].