1Introduction

Consider a family of stochastic reaction–diffusion equation in a weakly nonlinear regime:

with \(\varepsilon \in (0, 1]\), \(T > 0\), initial condition \(\bar{u}_{0, \varepsilon} : (\mathbb{T}/ \varepsilon)^3 \rightarrow \mathbb{R}\)\(\), \(F_{\varepsilon} \in C^9 (\mathbb{R})\) with exponential growth at infinity, \(\alpha > 0\) and \(\mathscr{L} \hspace{.2em} : = (\partial_t - \Delta)\) the heat flow operator and \(\mathbb{T}=\mathbb{R}/ (2 \pi \mathbb{Z})\). Here \(\eta\) denotes a family of centered Gaussian noises on \([0, T / \varepsilon^2] \times (\mathbb{T}/ \varepsilon)^3\) with stationary covariance

such that \(\)\(\tilde{C}^{\varepsilon} (t - s, x, y) = \Sigma (t - s, x - y)\) if \(\operatorname{dist} (x, y) \leqslant 1\) and \(0\) otherwise where \(\Sigma : \mathbb{R} \times \mathbb{R}^3 \rightarrow \mathbb{R}^+\) is a smooth, positive function compactly supported in \([0, 1] \times B_{\mathbb{R}^3} (0, 1)\). We assume also that there exists a compactly supported function \(\psi\) such that \(\psi \ast \psi = \Sigma\) (this is true e.g. when \(\eta\) is obtained by space-time convolution of the white noise with \(\psi\)).

We look for a large scale description of the solution to eq. (1) and we introduce the “mesoscopic” scale variable \(u_{\varepsilon} (t, x) = \varepsilon^{- \beta} u (t / \varepsilon^2, x / \varepsilon)\) where \(\beta > 0\). Substituting \(u_{\varepsilon}\) into (1) we get

In order for the term \(\varepsilon^{- 2 - \beta} \eta (t / \varepsilon^2, x / \varepsilon)\) to converge to a non–trivial random limit we need that \(\beta = 1 / 2\). Indeed the Gaussian field \(\eta_{\varepsilon} (t, x) : = \varepsilon^{- 5 / 2} \eta (t / \varepsilon^2, x / \varepsilon)\) has covariance \(\varepsilon^{- 5} \tilde{C}^{\varepsilon} (t / \varepsilon^2, x / \varepsilon)\) and converges in distribution to the space-time white noise on \(\mathbb{R} \times \mathbb{T}^3\). For large values of \(\alpha\) the non–linearity will be negligible with respect to the additive noise term. Heuristically, we can attempt an expansion of the reaction term around the stationary solution \(Y_{\varepsilon}\) to the linear equation

i.e. \(Y_{\varepsilon} (t, x) = \int_{- \infty}^t \check{P} (t - s, x - y) \eta_{\varepsilon} (s, y) \mathrm{d} s \mathrm{d} y\) with \(\check{P} (t, x) = \frac{1}{(4 \pi t)^{3 / 2}} e^{- \frac{| x |^2}{4 t}} e^{- t} \mathbb{1}_{t \geqslant 0}\).

Let us denote with \(C_{\varepsilon}\) the covariance of \(Y_{\varepsilon}\). We approximate the reaction term as

The Gaussian r.v. \(\varepsilon^{1 / 2} Y_{\varepsilon} (t, x)\) has variance \(\sigma_{\varepsilon}^2 = \varepsilon \mathbb{E} [ (Y_{\varepsilon} (t, x))^2] = \varepsilon \mathbb{E} [ (Y_{\varepsilon} (0, 0))^2] = \varepsilon C_{\varepsilon} (0, 0)\) independent of \((t, x)\). Although \(\sigma_{\varepsilon}^2\) depends on \(\varepsilon\)\(\), it can be bounded from above and below by two positive constants uniformly on \(\varepsilon \in (0, 1]\). We can expand the random variable \(F_{\varepsilon} (\varepsilon^{1 / 2} Y_{\varepsilon} (t, x))\) according to the chaos decomposition relative to \(\varepsilon^{1 / 2} Y_{\varepsilon} (t, x)\) and obtain

where \(H_n (x, \sigma_{\varepsilon}^2)\) are standard Hermite polynomials with variance \(\sigma_{\varepsilon}^2\) and highest-order term normalized to \(1\). Note also that the coefficients \((f_{n, \varepsilon})_{n \geqslant 0}\) do not depend on \((t, x)\) by stationarity of the law of \(\varepsilon^{1 / 2} Y_{\varepsilon} (t, x)\) since they are given by the formula

where \(G\) is a standard Gaussian variable of unit variance.

Let \(X\) be the stationary solution to the equation

with \(\xi\) the space–time white noise on \(\mathbb{R} \times \mathbb{T}^3\) and denote by \(\mathopen{\unicode{x27E6}} X^N \mathclose{\unicode{x27E7}}\) the generalized random fields given by the \(N\)-th Wick power of \(X\), which are well defined as random elements of \(\mathcal{S}' (\mathbb{R} \times \mathbb{T}^3)\) as long as \(N \leqslant 4\). Denote with \(C_X\) the covariance of \(X\). The Gaussian analysis which we set up in this paper shows in particular that if \(\varepsilon^{(n - N) / 2} f_{n, \varepsilon} \rightarrow g_n\) as \(\varepsilon \rightarrow 0\) for every \(0 \leqslant n \leqslant N\), \(N \leqslant 4\), and \((F_{\varepsilon})_{\varepsilon} \subseteq C^{N + 1} (\mathbb{R})\) with exponential growth, then the family of random fields

converges in law in \(\mathcal{S}' (\mathbb{R} \times \mathbb{T}^3)\) as \(\varepsilon \rightarrow 0\) to \(\sum_{n = 0}^N g_n \mathopen{\unicode{x27E6}} X^n \mathclose{\unicode{x27E7}}\).

Consider the smallest \(n\) such that \(f_{n, \varepsilon}\) converges to a finite limit as \(\varepsilon \rightarrow 0\). Since \(H_n (\varepsilon^{1 / 2} Y_{\varepsilon}, \sigma_{\varepsilon}^2) = \varepsilon^{n / 2} \mathopen{\unicode{x27E6}} Y_{\varepsilon}^n \mathclose{\unicode{x27E7}}\), the \(n\)-th term in the expansion (4) is \(f_{n, \varepsilon} \varepsilon^{\alpha + (n - 5) / 2} \mathopen{\unicode{x27E6}} Y_{\varepsilon}^n \mathclose{\unicode{x27E7}}\). Therefore, the equation yields a non-trivial limit only if \(\alpha = (5 - n) / 2\). We are interested mainly in the case \(n = 3 \Rightarrow \alpha = 1\) and \(n = 1 \Rightarrow \alpha = 2\). The case \(\alpha = 2\) gives rise to a Gaussian limit and its analysis its not very difficult. In the following we will concentrate in the analysis of the \(\alpha = 1\) case where the limiting behaviour of the model is the most interesting and given by the \(\Phi^4_3\) family of singular SPDEs. In this case we obtain the family of models

with initial condition \(u_{0, \varepsilon} (\cdot) := \varepsilon^{- \frac{1}{2}} \bar{u}_{0, \varepsilon} (\varepsilon^{- 1} \cdot)\) where \(\bar{u}_{0, \varepsilon}\) is the initial condition of the microscopic model (1).

In order to state our main result, Theorem 2 below, let us introduce some notations and specify our assumptions. Let \(\tilde{F}_{\varepsilon}\) be the centering (up to the third Wiener chaos relative to \(\varepsilon^{1 / 2} Y_{\varepsilon} (t, x)\)) of the function \(F_{\varepsilon}\), i.e.

The decomposition of \(\tilde{F}_{\varepsilon}\) is obviously the same as in (4) except for the fact that we have discarded the orders 0,1,2. \(\)Let \(\tilde{F}_{\varepsilon}^{ (m)}\) be the \(m\)-th derivative of the function \(\tilde{F}_{\varepsilon}\) for \(0 \leqslant m \leqslant 9\) and define the following \(\varepsilon\)–dependent constants:

where \(P_s (x)\) is the heat kernel and \(\int_{s, x}\) denotes integration on \(\mathbb{R}^+ \times \mathbb{T}^3\).

Here \(C_T \mathscr{C} \hspace{.1em}^{- \alpha} (\mathbb{T}^3)\) denotes the space of continuous functions from \([0, T]\) to the Besov space \(\mathscr{C} \hspace{.1em}^{- \alpha} (\mathbb{T}^3) = B^{- \alpha}_{\infty, \infty} (\mathbb{T}^3)\) (see Appendix A.1 for the notation on Besov spaces and paraproducts). Theorem 2 is actually just a corollary of the more precise result Theorem 7, in which we identify the paracontrolled equation satisfied by the limit random field \(u (\lambda)\).

Remark 3. We are interested only in local-in-time convergence of \(u_{\varepsilon}\), as a way to show the potential of our method for controlling stochastic terms with infinite chaos decomposition (developed in Section 26). Nevertheless, we expect it to be possible to obtain global-in-time convergence of the solution with more stringent assumptions on \(F_{\varepsilon}\), although we do not treat this problem here.

Remark 4. As a special case we can take

so that

and

where \(L_{\varepsilon} := 2 \int_{s, x} P_s (x) (C_{Y, \varepsilon} (s, x))^2 .\) Choosing

we obtain \(\lambda_{\varepsilon} \rightarrow (\lambda_0, \lambda_1, \lambda_2, \lambda_3)\). This shows that all the possible limits \(\lambda \in \mathbb{R}^4\) are attainable. In this case (5) takes the form

The name dynamic \(\Phi^4_3\) equation (or stochastic quantisation equation) derives from the fact that the simplest class of models which approximate the limiting random field \(u (\lambda)\) is precisely obtained by choosing a cubic polynomial like in (10) as non-linear term (which is the gradient of a fourth order polinomial playing the role of local potential).

In two dimensions, this model has been subject of various studies since more than thirty years [12, 1, 5]. For the three dimensional case, the kind of convergence results described above are originally due to Hairer [8, 9] and constitute one of the first groundbreaking applications of his theory of regularity structures. Similar results were later obtained by Catellier and Chouk [4] using the paracontrolled approach of Gubinelli, Imkeller and Perkowski [6]. Kupiainen [13] described a third approach using renormalization group ideas.

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

The first result of weak universality for a singular stochastic PDE has been given by Hairer and Quastel [10] in the context the Kardar–Parisi–Zhang equation. Using the machinery developed there Hairer and Wu [11] proved a weak universality result for three dimensional reaction–diffusion equations in the case of Gaussian noise and a polynomial non–linearity, within the context of regularity structures. Weak universality for reaction–diffusion equations driven by non Gaussian noise is analysed in Shen and Wu [22]. Recently, important results concerning the stochastic quantisation equation we obtained by Mourrat and Weber. In particular the convergence to the dynamic \(\Phi^4_2\) model for a class of Markovian dynamics of discrete spin systems [15] and also the global wellposedness of \(\Phi^4_2\) in space and time [16] and in time for \(\Phi^4_3\) [17]. The recent preprint [21] analyzes an hyperbolic version of the stochastic quantisation equation in two dimensions, including the associated universality in the small noise regime.

The present work is the first which considers in detail the weak universality problem in the context of paracontrolled distributions, showing that on the analytic side the apriori estimates can be obtained via standard arguments and that the major difficulty is related to showing the convergence of a finite number of random fields to universal limiting objects. The main novelty of our work is our use of the Malliavin calculus [20, 19] to perform the analysis of these stochastic terms without requiring polynomial non–linearity as in the previous works cited above. In particular we were inspired by the computations in [25] and in general by the use of the Malliavin calculus to establish normal approximations [19]. The main technical results of our paper, Theorem 8 below, is not particularly linked to paracontrolled distributions. A similar analysis is conceivable for the stochastic models in regularity structures. Moreover the same tools can also allow to prove similar non-polynomial weak universality statements for the KPZ along the lines of the present analysis. This is the subject of ongoing work.

Acknowledgments 5. The authors would like to thank the anonymous referee for the detailed and constructive critique which contributed to improve the overall exposition of the results. Support via SFB CRC 1060 is also gratefully acknowledged.

2Analysis of the mesoscopic model

The goal of this section is to obtain a paracontrolled structure for equation (5) analogous to that introduced in [4] for the cubic polynomial case and use it to set up the limiting procedure. Convergence of the stochastic terms and some apriori estimates will be the subject of the following sections. Definitions and a reminder of the basic results of paradifferential calculus needed here can be found in Appendix A.

2.1Paracontrolled structure

Let us start our analysis by centering the reaction term \(F_{\varepsilon} (\varepsilon^{1 / 2} u_{\varepsilon})\) in (5) using decomposition (6) to obtain:

We write \(u_{\varepsilon} = Y_{\varepsilon} + v_{\varepsilon}\) with \(Y_{\varepsilon}\) as in (3), and perform a Taylor expansion of \(\tilde{F}_{\varepsilon} (\varepsilon^{1 / 2} Y_{\varepsilon} + \varepsilon^{1 / 2} v_{\varepsilon})\) around \(\varepsilon^{1 / 2} Y_{\varepsilon}\) up to the third order to get

where \(R_{\varepsilon} (v_{\varepsilon})\) is the remainder of the Taylor series and we used the fact that \(H_2 (\varepsilon^{1 / 2} Y_{\varepsilon}, \sigma_{\varepsilon}^2) = \varepsilon \mathopen{\unicode{x27E6}} Y_{\varepsilon}^2 \mathclose{\unicode{x27E7}}\). Notice that we stopped the Taylor expansion at the first term for which \(\varepsilon\) does not appear anymore with a negative exponent (that is \(\tilde{F}^{(3)}_{\varepsilon} (\varepsilon^{\frac{1}{2}} Y_{\varepsilon})\)). One can then expect the remainder \(R_{\varepsilon} (v_{\varepsilon})\) to converge to zero in some sense. On the other hand, all the other terms except \(\tilde{F}^{(3)}_{\varepsilon} (\varepsilon^{\frac{1}{2}} Y_{\varepsilon})\) and \(R_{\varepsilon} (v_{\varepsilon})\) appear to diverge in the limit \(\varepsilon \rightarrow 0\), but in analogy with well-known renormalization methods for random fields, we try to find a combination of them that can be made to converge in some function space. Define the following random fields:

with \(Y_{\varepsilon}\) stationary solution, while \(Y_{\varepsilon}, Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-2.pdf}}}, \bar{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-3.pdf}}}\) have \(0\) initial condition in \(t = 0\). The last four trees \(\bar{Y}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\) are obtained from the others via the resonant Bony's paraproduct \(\circ\) recalled in Appendix A, and \(\bar{d}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(d_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(d_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \({d'_{\varepsilon}}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(d_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\) are just \(\varepsilon\)-dependent constants whose exact value will matter only in Section 26. Indeed, in the scope of this section we only need the following relation to be verified:

The notation \(\)\(\bar{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) denotes that this tree has finite chaos expansion and can be treated with the well-known techniques of [4] or [18] (we put a bar on \(\bar{Y}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\) just because is the only tree obtained from \(\)\(\bar{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\))

With the definitions (12), equation (11) takes the form

At this point it is worth noting that the trivial case \(\tilde{F}_{\varepsilon} (x) = H_3 (x, \sigma^2_{\varepsilon})\) yields \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} = \mathopen{\unicode{x27E6}} Y_{\varepsilon}^3 \mathclose{\unicode{x27E7}}\), \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} = \mathopen{\unicode{x27E6}} Y^2_{\varepsilon} \mathclose{\unicode{x27E7}}\), \(Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-1.pdf}}} = Y_{\varepsilon}\), \(Y^{\varnothing} = 1\). By comparing these random fields to the ones defined in [4] we can guess that \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-2.pdf}}}, Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-3.pdf}}}, Y^{\varnothing}\) can be controlled respectively in \(\mathscr{C} \hspace{.1em}^{1 / 2 - \kappa}, \mathscr{C} \hspace{.1em}^{- 1 - \kappa}, \mathscr{C} \hspace{.1em}^{- 1 / 2 - \kappa}, \mathscr{C} \hspace{.1em}^{- \kappa}\) \(\forall \kappa > 0\) for any \(F_{\varepsilon}\) satisfying Assumption 1, and carry on the paracontrolled analysis of (14) as if it were the case. Clearly, the paracontrolled structure is robust and does not depend on how the terms \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-2.pdf}}}, Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-3.pdf}}}, Y^{\varnothing}\) are defined as long as they have the desired regularity.

From these observations, we do not expect to be able to control the products \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} v_{\varepsilon}\), \(Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-1.pdf}}} v_{\varepsilon}^2\) and \(Y_{\varepsilon} v_{\varepsilon}\) in eq. (14) uniformly in \(\varepsilon > 0\). In order to proceed with the analysis we make the Ansatz:

and proceed to decompose the ill-defined products using the paracontrolled techniques recalled in Appendix A. We start by writing \(v_{\varepsilon} Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} = v_{\varepsilon} \prec Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-2.pdf}}} + v_{\varepsilon} \succ Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-3.pdf}}} + v_{\varepsilon} \circ Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-4.pdf}}}\). The resonant term, together with Ansatz (15), yields:

with the definition and bounds of \(\overline{\operatorname{com}}_1 (\cdot, \cdot, \cdot)\) given in Lemma 26. Then we define

Moreover we have for \(v_{\varepsilon} Y_{\varepsilon}\):

where we introduced the shorthand \(\varphi_{\varepsilon} := v_{\varepsilon} + Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\). So we let

Finally to analyse the product \(Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-1.pdf}}} v_{\varepsilon}^2\) we write

and consider the products involving only \(Y^{\tau}\) factors: first

and then we define the term \(\)\(Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-1.pdf}}} \diamond (Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-2.pdf}}})^2\) as follows:

so that

We note also that

with \(\operatorname{com}_2 (\cdot, \cdot)\) and \(\operatorname{com}_3 (\cdot, \cdot)\) specified in Lemma 26 of the Appendix. Substituting these renormalized products into (14) and recalling the definition (9) for \(\lambda_{\varepsilon} = (\lambda_{0, \varepsilon}, \lambda_{1, \varepsilon}, \lambda_{2, \varepsilon}, \lambda_{3, \varepsilon})\), we obtain the following equation for \(v^{\natural}_{\varepsilon}\) :

where we can use the constraint (13) to remove the term proportional to \(v_{\varepsilon}\). Summarizing, we obtain the following equation, together with Ansatz (15):

with initial condition \(v_{\varepsilon, 0} = u_{0, \varepsilon} - Y_{\varepsilon} (0)\) and \(U\) given by

The enhanced noise vector \(\mathbb{Y}_{\varepsilon}\) is defined by

for every \(\kappa > 0\), \(T > 0\). We use the notation \(\| \mathbb{Y}_{\varepsilon} \|_{\mathcal{X}_T} = \sum_{\tau} \| \mathbb{Y}^{\tau}_{\varepsilon} \|_{\mathcal{X}^{\tau}}\) for the associated norm where \(Y_{\varepsilon}^{\tau}\) is a generic tree in \(\mathbb{Y}_{\varepsilon}\). The homogeneities \(| \tau | \in \mathbb{R}\) are given by

Note that for every \(\varepsilon > 0\) eq. (16) is equivalent to eq. (5).

Remark 6. The paracontrolled structure we developed in this section is the same as in the work of Catellier and Chouk [4], plus an additive source term (which is \(R_{\varepsilon} (v_{\varepsilon})\) in equation (16)). Therefore, there exists \(T = T \left( \| \mathbb{Y}_{\varepsilon} \|_{\mathcal{X}_T}, \| u_{\varepsilon, 0} \|_{\mathscr{C} \hspace{.1em}^{- 1 / 2 - \kappa}}, | \lambda_{\varepsilon} | \right)\) such that we can define for \(\alpha \in (1 / 2, 2 / 3)\), \(p \in [4, \infty)\)\(\), \(\gamma > \frac{1}{4} + \frac{3}{2} \kappa\) a solution map

so that \(u_{\varepsilon} = \Gamma (u_{\varepsilon, 0}, \mathbb{Y}_{\varepsilon}, \lambda_{\varepsilon}, \mathcal{R})\) with \(u_{\varepsilon} = Y_{\varepsilon} + v_{\varepsilon}\) and \(v_{\varepsilon}\) that solves (16) with the remainder \(R_{\varepsilon} (v_{\varepsilon})\) replaced by \(\mathcal{R}\). The space \(\mathcal{M}^{\gamma}_T L^p (\mathbb{T}^3)\) is specified in Appendix A.1. Indeed, we can use Lemma 20 and Lemma 22 to control \(I\mathcal{R}\) as

for \(\delta > 0\) small enough, and thanks to this bound it is easy to see that the the fixed point procedure of Section 3 of [4] still holds with a fixed additive source term \(\mathcal{R}\). In the same way, the continuity of the solution map \(\Gamma\) follows easily as in Theorem 1.2 of [4].

2.2Identification of the limit

In order to identify interesting limits for equation (5), we introduce \(\forall \lambda = (\lambda_0, \lambda_1, \lambda_2, \lambda_3) \in \mathbb{R}^4\) the enhanced noise \(\mathbb{Y} (\lambda)\) which is constructed from universal noises \(X^{\tau}\) as

where \(X\) is the stationary solution to to the linear equation \(\mathscr{L} \hspace{.2em} X = - X + \xi\) and \(\xi\) is the time-space white noise on \(\mathbb{R} \times \mathbb{T}^3\). We will sometimes use the shorter notation \(\mathbb{Y} (\lambda) = (Y^{\tau} (\lambda))_{\tau}\) for (19).

We define the universal fields \(X^{\tau}\) through their Littlewood-Paley decomposition \(\forall (t, \bar{x}) \in \mathbb{R}^+ \times \mathbb{T}^3\) as:

where as before \(\mathopen{\unicode{x27E6}} \cdot \mathclose{\unicode{x27E7}}\) stands for the Wick product, \(\zeta_i = (x_i, s_i) \in \mathbb{R} \times \mathbb{T}^3\), \(C (\cdot, \cdot)\) is the covariance of \(X\) and \(\mu_{q, \zeta_1, \zeta_2}\) is the measure

with the usual heat kernel \(P_t (x) = \frac{1}{(4 \pi t)^{3 / 2}} e^{- \frac{| x |^2}{4 t}} \mathbb{1}_{t \geqslant 0}\). We commit an abuse of notation by writing \(X (\zeta)\) since \(X\) is actually a distribution in space: the integrals in (20) should obviously be intended as functionals.

Standard computations (see e.g. [4] or [18]) show that, \(\forall \lambda \in \mathbb{R}^4\) and for any \(T > 0\), \(0 < \kappa < \kappa'\)

almost surely.

Using the paracontrolled structure we developed in Section 2.1 we can identify the limiting solution \(u (\lambda)\) introduced in Theorem 2.

Proof. Fix \(T > 0\). Let \(u_{\varepsilon} = Y_{\varepsilon} + v_{\varepsilon}\) be the solution of eq. (5) for fixed \(\varepsilon > 0\), which is seen to be unique in the (\(\varepsilon\)-dependent) time interval \([0, T_{\varepsilon}]\) by a usual fixed-point argument on the original equation (without resorting to the paracontrolled decomposition). Let \(u_{\varepsilon} = \Gamma (u_{\varepsilon, 0}, \mathbb{Y}_{\varepsilon}, \lambda_{\varepsilon}, R_{\varepsilon} (v_{\varepsilon}))\) on \([0, T_{\varepsilon}]\) with \(\Gamma\) defined as in Remark 6 and \(R_{\varepsilon} (v_{\varepsilon})\) seen as an exogenous source term. We know from the a-priori estimations of Section 4.2 that there exists a time \(T_{\star} = T_{\star} \left( \| \mathbb{Y}_{\varepsilon} \|_{\mathcal{X}_T}, \| u_{\varepsilon, 0} \|_{\mathscr{C} \hspace{.1em}^{- 1 / 2 - \kappa}}, | \lambda_{\varepsilon} | \right)\) and a family of events \((\mathcal{E}_{\varepsilon})_{\varepsilon > 0}\) such that \(\mathbb{P} (\mathcal{E}_{\varepsilon}) \rightarrow 1\) for \(\varepsilon \rightarrow 0\) and we can control \(\| v_{\varepsilon} \|_{\mathcal{M}^{1 / 4 + 3 \kappa / 2}_{T_{\star}} L^{\infty}}\). Thus, we can control the \(L^{\infty}\) norm of \(v_{\varepsilon} (t)\) in \([T_{\varepsilon} / 2, T_{\varepsilon}]\) and extend the solution \(v_{\varepsilon}\) on \([0, T_{\star}]\) for every \(\varepsilon\). Denote by \(u_{\varepsilon}^{\star} = \Gamma_{\star} (u_{\varepsilon, 0}, \mathbb{Y}_{\varepsilon}, \lambda_{\varepsilon}, R_{\varepsilon} (v_{\varepsilon}))\) the process \(u_{\varepsilon}\) stopped at time \(T_{\star}\) and \(\Gamma_{\star}\) the corresponding stopped solution map.

Note that \(u (\lambda)\) solves the same equation as \(u_{\varepsilon}^{\star}\)\(\) with \(\mathbb{Y}_{\varepsilon}\) replaced by \(\mathbb{Y} (\lambda)\)\(\), \(u_{\varepsilon, 0}\) replaced by \(u_0\), \(\lambda_{\varepsilon}\) replaced by \(\lambda\) and \(R_{\varepsilon} (v_{\varepsilon}) = 0\). So \(u (\lambda) = u^{\star} = \Gamma_{\star} (u_0, \mathbb{Y} (\lambda), \lambda, 0)\) up to time \(T_{\star}\). Let us introduce the random field \(\bar{u}_{\varepsilon}^{\star} = \Gamma_{\star} (u_{\varepsilon, 0}, \mathbb{Y}_{\varepsilon}, \lambda_{\varepsilon}, 0)\) with \(\bar{v}^{\star}_{\varepsilon} = \bar{u}_{\varepsilon}^{\star} - Y_{\varepsilon}\) that solves the paracontrolled equation (16) but with remainder \(R_{\varepsilon} (v_{\varepsilon}) = 0\).

Consider the \(n\)-tuple of random variables \((u_{\varepsilon, 0}, \mathbb{Y}_{\varepsilon}, u_{\varepsilon}^{\star}, \bar{u}_{\varepsilon}^{\star})\) and let \(\mu_{\varepsilon}\) be its law on \(\mathcal{Z}= \mathscr{C} \hspace{.1em}^{- \alpha} \times \mathcal{X}_T \times \left( C_T \mathscr{C} \hspace{.1em}^{- \alpha} \right)^2\) conditionally on \(\mathcal{E}_{\varepsilon}\). Observe that \(\Gamma_{\star}\) is continuous as discussed in Remark 6, and this gives that \(\forall \delta > 0\), \(\mu_{\varepsilon} \left( \| u_{\varepsilon}^{\star} - \bar{u}_{\varepsilon}^{\star} \|_{C_T \mathscr{C} \hspace{.1em}^{- \alpha}} > \delta \right) \rightarrow 0\) as \(\varepsilon \rightarrow 0\). Indeed \(R_{\varepsilon} (v_{\varepsilon}) \rightarrow 0\) in probability in the space \(\mathcal{M}^{\gamma, p}_{T_{\star}} L^p (\mathbb{T}^3)\) by Lemma 19. This shows that \(\mu_{\varepsilon}\) concentrates on \(\mathscr{C} \hspace{.1em}^{- \alpha} \times \mathcal{X}_T \times \left\{ (z, z) \in \left( C_T \mathscr{C} \hspace{.1em}^{- \alpha} \right)^2 \right\}\). Let \(\mu\) any accumulation point of \((\mu_{\varepsilon})_{\varepsilon}\). Then \(\mu \left( \mathscr{C} \hspace{.1em}^{- \alpha} \times \mathcal{X}_T \times \left\{ (z, z) \in \left( C_T \mathscr{C} \hspace{.1em}^{- \alpha} \right)^2 \right\} \right) = 1\). The apriori estimations of Section 4.2 yield the tightness of \(\mu_{\varepsilon}\) and from the concentration of \(\mu\) on the diagonal we know that there exists a subsequence such that for any test function \(\varphi\),

Moreover, still along subsequences we have that for any bounded continuous function \(\varphi\)

since by Theorem 8 the vector \(\mathbb{Y}_{\varepsilon}\) converges in law to \(\mathbb{Y} (\lambda)\), \(u_{\varepsilon, 0}\) to \(u_0\), and \(\Gamma_{\star}\) is a continuous function as discussed in Remark 6. This shows that

and then by comparing (22) and (23) we can conclude that there exists a subsequence such that

We can identify the limit distribution \(\mu\) by noting that since \(\mathbb{P} (\mathcal{E}_{\varepsilon}) \rightarrow 1\) we have

for any test function \(\psi\). So the first two marginals of \(\mu\) have the law of \((u_0, \mathbb{Y} (\lambda))\) and they are independent since \((u_{\varepsilon, 0}, \mathbb{Y}_{\varepsilon})\) are independent for any \(\varepsilon\). Calling \(\nu\) the law of \((u_0, \mathbb{Y} (\lambda))\) we have that

which implies that \(\mu\) is unique and that the whole family \((\mu_{\varepsilon})_{\varepsilon}\) converges to \(\mu\). We can conclude that \(u^{\star}_{\varepsilon} \rightarrow u^{\star}\) in law with \(u^{\star} = u (\lambda)\) up to the time \(T_{\star} \left( \| \mathbb{Y} (\lambda) \|_{\mathcal{X}_T}, \| u_0 \|_{\mathscr{C} \hspace{.1em}^{- 1 / 2 - \kappa}}, | \lambda | \right)\) since the function \(T^{\star}\) is lower semicontinuous (as obtained from the a-priori estimates).\(\Box\)

3Convergence of the enhanced noise

This is the central section of the paper, in which we present a new method to estimate certain random fields that do not have a finite chaos decomposition, and we apply it to the treatment of the random fields \(\mathbb{Y}_{\varepsilon}\) of (12).

3.1An example of convergence

We choose to give first a complete example (the convergence of the tree \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) to \(\)\(Y^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} (\lambda)\)) in order to put in evidence the main idea in the proof of Theorem 8. Recall its definition (12):

with \(\tilde{F}_{\varepsilon}^{(1)}\) being the first derivative of the centered function \(\tilde{F}_{\varepsilon}\) defined in (6). Since \(\frac{\mathrm{d}}{\mathrm{d} x} H_n (x, \sigma^2_{\varepsilon}) = n\)\(H_{n - 1} (x, \sigma^2_{\varepsilon})\) the Wiener chaos decomposition of \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) reads:

where \(\mathopen{\unicode{x27E6}} \cdot \mathclose{\unicode{x27E7}}\) is the Wiener product and we used the fact that \(\varepsilon^{- \frac{n}{2}} H_n (\varepsilon^{1 / 2} Y_{\varepsilon}, \sigma_{\varepsilon}^2) = \mathopen{\unicode{x27E6}} Y_{\varepsilon}^n \mathclose{\unicode{x27E7}}\). Now one can use hypercontractivity (as done in [10],[11]) to control the \(L^p\) norm of each chaos order by its \(L^2\) norms. However this strategy does not give useful bounds for the infinite series in the second term of (24). Instead, we just observe that

where \(J_i\) is the projection of on the \(i\)-th chaos, and look for a different way to write this remainder. One of the main insights of this paper \(\)is that we can write it as:

where \(\mathrm{D}\), \(\delta\) are the Malliavin derivative and divergence operators, and \(G_{[1]}^{[3]} = (1 - L)^{- 1} (2 - L)^{- 1} (3 - L)^{- 1}\) with \(L\) the Ornstein-Uhlenbeck operator. This is proven in Lemma 37.

To compute the Malliavin derivative of \(\tilde{F}^{(1)}_{\varepsilon} (\varepsilon^{\frac{1}{2}} Y_{\varepsilon})\) we observe that for every \(\varepsilon > 0\) \((t, x) \in \mathbb{R} \times \mathbb{T}^3\) there exists \(h_{(t, x)} \in L^2 (\mathbb{R} \times \mathbb{T}^3)\) such that the Gaussian random variable \(Y_{\varepsilon, (t, x)} := Y_{\varepsilon} (t, x)\)\(\) can be written as

Here \(\xi\) is the Gaussian white noise on \(\mathbb{R} \times \mathbb{T}^3\), which can be seen as a Gaussian Hilbert space \(\langle \xi, h \rangle_{h \in H} = \{ W (h) \}_{h \in H}\) indexed by the Hilbert space \(H := L^2 (\mathbb{R} \times \mathbb{T}^3)\). This is the framework in which we apply the Malliavin calculus results of Appendix 26. Notice that by construction

The function \(h_{(t, x)}\) can actually be written as the space-time convolution

with \(\check{P} (t, x) = \frac{1}{(4 \pi t)^{3 / 2}} e^{- \frac{| x |^2}{4 t}} e^{- t} \mathbb{1}_{t \geqslant 0}\) and \(\psi_{\varepsilon}\) such that \(\eta_{\varepsilon} = \psi_{\varepsilon} \ast \xi\), \(\psi_{\varepsilon} (t, x) = \varepsilon^{- 5 / 2} \psi (\varepsilon^{- 2} t, \varepsilon^{- 1} x)\). We omit the dependence on \(\varepsilon\) of \(h_{(t, x)}\) not to burden the notation.

Going back to the calculations, we obtain from (25) that \(\mathrm{D} \tilde{F}^{(1)}_{\varepsilon} (\varepsilon^{\frac{1}{2}} Y_{\varepsilon}) = \varepsilon^{\frac{1}{2}} \tilde{F}^{(2)}_{\varepsilon} (\varepsilon^{\frac{1}{2}} Y_{\varepsilon}) h .\) \(\)Then (noting that \(\tilde{F}^{(4)}_{\varepsilon} = F^{(4)}_{\varepsilon}\)):

It can be easily seen from (25) that \(Y_{\varepsilon}\) has the same law of a time-space mollification of \(X\) by convolution (with \(X\) defined in Section 2.2). Then the convergence in law of \(f_{3, \varepsilon} \mathopen{\unicode{x27E6}} Y_{\varepsilon}^2 \mathclose{\unicode{x27E7}}\) to \(Y^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} (\lambda)\) can be easily established by standard techniques (see [4] or [18]). We are only left to show that \(\hat{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) in (27) converges to zero in \(C_T \mathscr{C} \hspace{.1em}^{- 1 - \kappa}\).\(\)

It is well known (see Section 3.2 for details) that in order to control the norm of \(\hat{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} (t, \cdot)\) for \(t \in (0, T]\) in the Besov space \(\mathscr{C} \hspace{.1em}^{- \alpha - \kappa}\) \(\forall \kappa > 0\) and in probability, it is enough to have suitable estimates for

for any \(\bar{x} \in \mathbb{T}^3\) since \(\hat{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) is stationary in space. We then proceed to compute:

where \(K_{q, \bar{x}} (x)\) is the kernel associated to the Littlewood-Paley block \(\Delta_q\). Observe that \(\| F^{(4)}_{\varepsilon} (\varepsilon^{1 / 2} Y_{\varepsilon, (t, x)}) \|^p_{L^p} = \int_{\mathbb{R}} | F^{(4)}_{\varepsilon} (z) |^p \gamma (\mathrm{d} z)\) where \(\gamma (\mathrm{d} z)\) is the density of a centered Gaussian with variance \(\sigma^2_{\varepsilon}\). This norm is then finite by the bound (8) of Assumption 1.

Another fundamental idea of this work is that we can “estimate out” the bounded term \(\| F^{(4)}_{\varepsilon} (\varepsilon^{1 / 2} Y_{\varepsilon, (t, x)}) \|_{L^p}\), which has an infinite chaos decomposition, and obtain a standard \(\Phi^4_3\) diagram that can be treated with well understood techniques. Using Lemma 38 and Corollary 42, one has

Then note that we can decompose the norm \(\| u \|_{L^p (\Omega, H^{\otimes 3})} = \| \| u \|^2_{H^{\otimes 3}} \|^{1 / 2}_{L^{p / 2}}\) for \(u \in L^p (\Omega, H^{\otimes 3})\) and since the norm of the Hilbert space \(H^{\otimes 3}\) is given by the scalar product \(\| h^{\otimes 3} \|^2_{H^{\otimes 3}} = \langle h, h \rangle_H^3\) we obtain

The norm containing \(\tilde{F}^{(4)}_{\varepsilon}\) can then be easily estimated using Hölder's inequality as

This yields

which is a standard \(\Phi^4_3\) diagram that can be analysed with the techniques of [8] (recalled in Appendix A.3). We just remark that \(\langle h_{(t, x)}, h_{(t, x')} \rangle = C_{\varepsilon} (0, x - x')\) and the bound \(\varepsilon | C_{\varepsilon} (t, x) | \lesssim 1\) of Lemma 31 yields \(\forall \delta \in (0, 1)\):

and then \(\hat{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} (t, \cdot)\) converges to zero in probability in the space \(\mathscr{C} \hspace{.1em}^{- 1 - \delta / 2}\) \(\forall \delta \in (0, 1)\), as \(\varepsilon \rightarrow 0\). The time regularity of \(\hat{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) needed to obtain the convergence in \(C_T \mathscr{C} \hspace{.1em}^{- 1 - \delta / 2}\) does not need new ideas, and it is done in Section 3.3.

The method shown in this section is valid verbatim for the trees \(Y_{\varepsilon}^{\varnothing}\), \(Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-1.pdf}}}\), \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\), while for the composite trees in (12) (namely \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-2.pdf}}}, \bar{Y}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-3.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-4.pdf}}}\)) that are obtained via paraproducts of simple trees, one has to be able to write the remainder \(\hat{Y}_{\varepsilon}^{\tau}\) as an iterated Skorohod integral \(\delta^n (\ldots)\) in order to exploit the boundedness of this operator. Moreover, these trees require a second renormalization (on top of the Wick ordering) which is not easy to control for infinite chaos decompositions. We deal with both these difficulties introducing the product formula (85), which allows to write products of iterated Skorohod integrals as combinations of iterated Skorohod integrals. The details and calculations for composite trees can be found in Section 3.4.

3.2Main theorem and overview of the proof

The rest of Section 26 is dedicated to the proof of Theorem 8.

From the definition \(\mathbb{Y} (\lambda) = (Y^{\tau} (\lambda))_{\tau}\) of (19) it is clear that we need to prove that \(Y^{\tau}_{\varepsilon} \rightarrow Y^{\tau} (\lambda)\) for every tree \(\tau\). Note that we can write each tree \(Y^{\tau} (\lambda)\) as \(Y^{\tau} (\lambda) = f_{\tau} (\lambda) K^{\tau} (X)\) for a the measurable function \(K^{\tau}\) of the Gaussian process \(X \in C_T \mathscr{C} \hspace{.1em}^{- 1 / 2 - \kappa}\) defined (20), and a suitable deterministic function \(f_{\tau} (\lambda)\) of \(\lambda\). For example, we can write \(Y^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} (\lambda) = \lambda_3 \mathopen{\unicode{x27E6}} X^2 \mathclose{\unicode{x27E7}}\) with \(K^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}} (X) = X^{\resizebox{1em}{!}{\includegraphics{image-2.pdf}}} = \mathopen{\unicode{x27E6}} X^2 \mathclose{\unicode{x27E7}}\) and \(f_{\resizebox{6pt}{4.5pt}{\includegraphics{image-1.pdf}}} (\lambda) = \lambda_3\).

We will show (eqs. (36) and (44)) that every random field \(Y^{\tau}_{\varepsilon}\) defined in (12) can be decomposed with the same functions \(f_{\tau} (\cdot)\) and \(K^{\tau}\) as

where \(\hat{Y}^{\tau}_{\varepsilon}\) are suitable remainder terms. For all \(p \geqslant 2\) it is well-known (see [4],[8]) that the term \(f_{\tau} (\lambda_{\varepsilon}) K^{\tau} (Y_{\varepsilon})\) is uniformly bounded in \(L^p (\Omega ; \mathcal{X}^{\tau})\) (with \(\mathcal{X}^{\tau}\) given by (18)). Thus, we will prove that \(\hat{Y}^{\tau}_{\varepsilon}\) converges to zero in \(L^p (\Omega ; \mathcal{X}^{\tau})\). This can be done by showing that, by Besov embedding, for \(p \in [2, \infty)\) and \(\forall \alpha < | \tau |\) we have

thanks to the stationarity of the process \(Y (t, x)\). In order to prove the bound (29) it suffices to show that \(\forall t \in [0, T]\)

which is one of the key estimation of this paper and will be performed in Sections 3.3 and 3.4.2.

In order to obtain uniform convergence for \(t \in [0, T]\) it suffices to show that \(\forall \sigma \in [0, 1 / 2]\), \(q \geqslant - 1\):

Indeed, by the Garsia-Rodemich-Rumsey inequality we obtain for \(\delta > 0\) small enough and \(p\) large enough

which by Besov embedding yields an estimation on \(\mathbb{E} \left( \| Y_{\varepsilon}^{\tau} \|_{C_T^{\sigma - \kappa / 2} \mathscr{C} \hspace{.1em}^{\alpha - 2 \sigma - \kappa}} \right)\) for \(\kappa > 0\) small enough. This gives us the necessary tightness to claim that \(\mathbb{Y}_{\varepsilon}\) has weak limits along subsequences.

The only thing left after proving (28), (30) and (31) is that for each \(\tau\) we have \(K^{\tau} (Y_{\varepsilon}) \rightarrow K^{\tau} (X)\) in law. However this is clear and already well-known, since by hypothesis we can introduce a space-time convolution regularisation of \(X\) (let's call it \(X_{\varepsilon}\)) which has the same law of \(Y_{\varepsilon}\) for any \(\varepsilon > 0\). This yields immediately the convergence \(Y_{\varepsilon} \rightarrow X\) in law. At this point an approximation argument gives that \(K^{\tau} (Y_{\varepsilon})\) has the same law of \(K^{\tau} (X_{\varepsilon})\). Transposing the regularisation to the kernels of the chaos expansion we can write \(K^{\tau} (X_{\varepsilon}) = K_{\varepsilon}^{\tau} (X)\) and now it is easy to check that \(K_{\varepsilon}^{\tau} (X) \rightarrow K^{\tau} (X)\) in probability (as done systematically in [4],[18]). We can then conclude that \(K^{\tau} (Y_{\varepsilon}) \rightarrow K^{\tau} (X)\) and therefore \(Y^{\tau}_{\varepsilon} \rightarrow Y^{\tau} (\lambda)\) in law for every \(\tau\), since from Assumption 1 we have immediately \(f_{\tau} (\lambda_{\varepsilon}) \rightarrow f_{\tau} (\lambda)\).

Let us give some more details on how to prove the decomposition (28) and the bounds (30) and (31). As seen in Section 3.1 we have \(Y_{\varepsilon, \zeta} = \langle \xi, h_{\zeta} \rangle\) in law for \(\zeta = (t, x) \in \mathbb{R} \times \mathbb{T}^3\) and this gives

We define for \(m \in \mathbb{N}\), \(\zeta \in \mathbb{R} \times \mathbb{T}^3\):

Note that the term \(\Phi^{[m]}\) above is not the \(m\)-th derivative of some function \(\Phi\) (we use the square parenthesis notation to emphasize this fact). It easy to see from (32) that \(\mathrm{D}^k \Phi^{[m]}_{\zeta} = \Phi^{[m + k]}_{\zeta} h_{\zeta}^{\otimes k}\). Therefore, the partial chaos expansion (82) takes a more explicit form when applied to \(\Phi_{\zeta}^{[m]}\)\(\):

with \(G_{[1]}^{[n]}\) defined in (80). Here we used the fact that \(\delta^n (h_{\zeta}^{\otimes n}) = \mathopen{\unicode{x27E6}} Y^n_{\varepsilon, \zeta} \mathclose{\unicode{x27E7}}\) (see Remark 48) and that by the definition of \(\Phi^{[m]}_{\zeta}\) we obtain \(\forall \zeta \in \mathbb{R} \times \mathbb{T}^3\):

with \(\tilde{f}_{n, \varepsilon}\)\(\) the coefficients in the decomposition \(\tilde{F}_{\varepsilon} (\varepsilon^{\frac{1}{2}} Y_{\varepsilon}) := \sum_{n \geqslant 0} \tilde{f}_{n, \varepsilon} H_n (\varepsilon^{\frac{1}{2}} Y_{\varepsilon}, \sigma_{\varepsilon}^2)\). Choosing \(n = 4 - m\) in eq. (33) we obtain

and a remainder with an infinite chaos decomposition strictly greater that \(3 - m\):

This is a key step in the proof of Theorem 8. Indeed, it suffices to substitute (34) into definition (12) to identify the remainder \(\hat{Y}^{\tau}_{\varepsilon}\) in decomposition (28) that has to converge to zero, and see that it always contains the term \(\hat{\Phi}^{[m]}_{\zeta}\). Moreover, the structure (35) of \(\hat{\Phi}^{[m]}_{\zeta}\) makes it possible to bound its \(L^p\) norm and obtain (30), (31) in the same way as done in Section 3.1 for \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\). We will consider separately simple trees (namely \(Y_{\varepsilon}^{\varnothing}, Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-2.pdf}}}, \bar{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-3.pdf}}}, Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-4.pdf}}}\)) which are linear functions of \(\Phi^{[m]}\) in Section 3.3, and composite trees (namely \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-2.pdf}}}, \bar{Y}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-3.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-4.pdf}}}\)) which are quadratic in simple trees and need to be further renormalized in order to converge to some limit as \(\varepsilon \rightarrow 0\). We will show the decomposition (28) for composite trees in Section 3.4.1 and the bounds (30), (31) in Section 3.4.2.

Remark 9. We can easily estimate terms of the form \(\varepsilon^{- (m - 3) / 2} \Phi^{[m]}_{\zeta}\)\(\) for \(3 \leqslant m \leqslant 9\) and every \(p \in [2, \infty)\). We have (as already observed for \(F^{(1)}_{\varepsilon}\)):

where \(\gamma (\mathrm{d} x)\) is the density of a centered Gaussian with variance \(\sigma^2_{\varepsilon}\). The integral is finite by Assumption 1: in particular we only need to assume that the first \(m\) derivatives of \(F_{\varepsilon}\) have exponential growth (actually, it is easy to see that one can require even weaker growth conditions).

3.3Analysis of simple trees

First of all note that the term \(\bar{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) has no remainder, and then it can be shown to converge in law to \(\lambda^{(2)} Y^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) by usual techniques (see [4]). In this section we show the convergence of the trees \(Y_{\varepsilon}^{\varnothing}\), \(Y_{\varepsilon}^{\resizebox{0.5em}{!}{\includegraphics{image-1.pdf}}}\), \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\), \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\). We obtain easily from (34):

with \(\zeta = (s, y)\) and either

where \(K_{q, x} (y)\) is the kernel associated to the Littlewood-Paley block \(\Delta_q\) and \(P_t (x)\) is the heat kernel.

As said before, \(f_{3, \varepsilon} \int_{\zeta} \mathopen{\unicode{x27E6}} Y^{(3 - m)}_{\varepsilon, \zeta} \mathclose{\unicode{x27E7}} \mu_{q, \zeta}\) converges in law in \(L^p\) for every \(2 \leqslant p < + \infty\) to \(\lambda_3 \int_{\zeta} \mathopen{\unicode{x27E6}} Y^{(3 - m)}_{\zeta} \mathclose{\unicode{x27E7}} \mu_{q, \zeta}\) since \(f_{3, \varepsilon} \rightarrow \lambda_3\) by Assumption 1. We can bound the remainder term \(\int_{\zeta} \hat{\Phi}^{[m]}_{\zeta} \mu_{q, \zeta}\) in \(L^p (\Omega)\) using Lemma 38 and the definition of the norm \(\| \cdot \|_{\mathbb{D}^{4 - m, p} (H^{\otimes 4 - m})}\) to obtain:

From Corollary 42 we know that \((j - L)^{- 1}\) and \(\mathrm{D} (j - L)^{- 1}\) are bounded in \(L^p\) for every \(p \in [2, \infty)\) and every \(j \geqslant 1\). Applying repeatedly these estimations we obtain:

Now we can proceed to implement the idea we already described in Section 3.1, i.e. estimating out the term \(\left\| \varepsilon^{- \frac{1}{2}} \Phi^{[4]}_{\zeta} \right\|_{L^p (\Omega)}\) (which is bounded by Remark 9 but with infinite chaos decomposition) and considering the finite-chaos term that is left. We do this by decomposing the \(L^p (\Omega, H^{\otimes 4 - m})\) norm as norms on \(H^{\otimes 4 - m}\) and \(L^{p / 2} (\Omega)\) as follows:

Finally, putting the estimations together and using Hölder's inequality, together with the bound \(\varepsilon | \langle h_{\zeta}, h_{\zeta'} \rangle | = \varepsilon | C_{\varepsilon} (\zeta - \zeta') | \lesssim 1\) of Lemma 31, we obtain for every \(\delta \in (0, 1]\):

Now using Remark 9 (note that to bound \(\varepsilon^{- \frac{1}{2}} \Phi^{[4]}\) we only need to control the first 4 derivatives of \(F_{\varepsilon}\)) and the fact that \(\langle h_{\zeta}, h_{\zeta'} \rangle_H = C_{\varepsilon} (\zeta - \zeta')\) we obtain as a final estimation

From the definition (37) of the measure \(\mu_{q, \zeta}\), the l.h.s of (38) can be estimated in a standard way using Lemma 33 to obtain for every \(x \in \mathbb{T}^3\), \(q > 0\):

Time regularity of trees

We want to show (31). In order to do that, we compute in the same way as before:

We focus on the first term above to obtain that it is bounded by

Now note that

and we can estimate \(\left\| \varepsilon^{- \frac{1}{2}} (\Phi^{[4]}_{t, x} - \Phi^{[4]}_{s, x}) \right\|_{L^p (\Omega)}\) by hypercontractivity and using Lemma 34 as

for any \(\sigma \in [0, 1 / 2]\). The other term can be estimated more easily by

and finally obtain

Which yields estimation (31) by applying Lemma 33 as before. This concludes the treatment of simple trees. Notice that in this section we only needed \(F_{\varepsilon} \in C^5 (\mathbb{R})\) with the first 5 derivatives having exponential growth: indeed we need to take 4 derivatives to bound \(\varepsilon^{1 / 2} \Phi^{[4]}_{\zeta}\) as of Remark 9, plus one more derivative for the time regularity of \(\varepsilon^{1 / 2} \Phi^{[4]}_{\zeta}\).

3.4Analysis of composite trees

In this section we show the decomposition (28) and the bound (30) for the trees \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-2.pdf}}}, \bar{Y}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-3.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-4.pdf}}}\). The time regularity (31) of \(\hat{Y}^{\tau}_{\varepsilon}\) can be obtained with the same technique as in the previous section, assuming that we can control one more derivative of \(F_{\varepsilon}\) than what is needed to prove the boundedness of \(Y^{\tau}_{\varepsilon}\) (thus we will need \(F_{\varepsilon} \in C^9 (\mathbb{R})\) with exponential growth, as discussed in Remark 14). Looking at the definitions in (12) it is clear that we can write the \(q\)-th Littlewood-Paley blocks of \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\), \(\bar{Y}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\) and \(Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\) \(\forall \varepsilon > 0\) as:

for any time-space point \(\bar{\zeta} = (t, \bar{x})\) that we keep fixed throughout this section. In order to keep the notation shorter we defined

which can be thought of as a finite-chaos equivalent of \(\Phi^{[1]}_{\varepsilon}\) (modulo a constant \(f_{2, \varepsilon} / f_{3, \varepsilon}\)) in the same way as \(\bar{Y}_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\) is a finite-chaos equivalent of \(Y_{\varepsilon}^{\resizebox{1em}{!}{\includegraphics{image-1.pdf}}}\)\(\). The measure \(\mu_{q, \zeta_1, \zeta_2}\)on \((\mathbb{R} \times \mathbb{T}^3)^2\) is given by

with \(\zeta_{i :=} (s_i, x_i)\) for \(i = 1, 2\), \(K\) being the kernel associated to the Littlewood-Paley decomposition and \(P\) being the heat kernel. The first step for decomposing (39) is to expand them using the partial chaos expansion (82) to obtain

Like the trees appearing in the \(\Phi^4_3\) model, we expect composite trees to require a further renormalisation, on top of the Wick ordering. We developed (40) to the smallest order that allows us to see the effect of renormalization.

3.4.1Renormalisation of composite trees

In this section we show how to renormalize (39) by estimating terms of the type \(\mathbb{E} [\Phi^{[m]}_{\zeta_1} \Phi^{[n]}_{\zeta_2}]\)\(\) in expansion (40). This poses an additional difficulty, as in principle we would need to compute an infinite number of contractions between \(\Phi^{[m]}_{\zeta_1}\) and \(\Phi^{[n]}_{\zeta_2}\). However, we can again decompose \(\)\(\Phi^{[m]}\) as in (34), and then the product formula (85) ensures that we only need to control a finite number of contractions. This is another important step in the proof and will be carried out in Lemma 12. First we need some preparatory results:

Remark 11. Some caveat on the notation: although we use the same letter for the kernel \(G (\cdot, \cdot)\) and the Green operator \(G_{[m]}^{[n]}\), those two are not related in any possible way. It is always clear which one the notation refers to.

Proof. We have

and by change of variables, exploiting the translation invariance of the problem we obtain:

Using the definition of \(K_q\) we have

Finally we can write

Similar computations holds for the other term, indeed

Substituting the lemma above and (40) in the expressions (39), we can write them as:

with the additional definitions

Now we can characterise the local behaviour of \(\mathbb{E} [\Phi^{[m]}_{\zeta_1} \Phi^{[n]}_{\zeta_2}]\) appearing in the integrals above. Decomposing separately \(\Phi^{[m]}_{\zeta_1}\) and \(\)\(\Phi^{[n]}_{\zeta_2}\) as in (34) we obtain:

where \(\mathbb{E} [\mathopen{\unicode{x27E6}} Y^{3 - m}_{\varepsilon, \zeta_1} \mathclose{\unicode{x27E7}} \mathopen{\unicode{x27E6}} Y^{3 - n}_{\varepsilon, \zeta_2} \mathclose{\unicode{x27E7}}] = (3 - m) ! \delta (3 - m, 3 - n) C_{\varepsilon} (\zeta_1 - \zeta_2)^{3 - n}\) and to bound all other terms we introduce the following result.

Proof. Using formula (83) we decompose

We can bound the term

knowing \(8 - n - i \vee 8 - m - i \leqslant 8\) derivatives of \(F_{\varepsilon}\) (see Remark 9) and using the bound \(\varepsilon | C_{\varepsilon} (\zeta_1 - \zeta_2) | \lesssim 1\) of Lemma 31 with \(| \langle h_{\zeta_1}, h_{\zeta_2} \rangle | = | C_{\varepsilon} (\zeta_1 - \zeta_2) |\) we have:

For the second bound we recall that \(\mathopen{\unicode{x27E6}} Y^m_{\varepsilon, \zeta_1} \mathclose{\unicode{x27E7}} = \delta^m (h_{\zeta_1}^{\otimes m})\) (Remark 48) and compute

Since \(\mathrm{D} h^{\otimes m}_{\zeta_1} = 0\) we obtain \(\mathbb{E} [\mathopen{\unicode{x27E6}} Y^m_{\varepsilon, \zeta_1} \mathclose{\unicode{x27E7}} \hat{\Phi}^{[n]}_{\zeta_2}] = 0\) if \(m < 4 - n\) and

if \(m \geqslant 4 - n\), with

Using Lemma 12 we obtain

and thus \(G (t - s, x) = 18 (f_{3, \varepsilon})^2 \int_{x_1', x_2} \sum_{i \sim j} K_{i, x} (x_1') K_{j, x} (x_2) P_{t - s} (x_1') [C_{\varepsilon} (\zeta_1 - \zeta_2)]^2 + \hat{G} (t - s, x)\) with the remainder term defined by

We have the estimation

Similarly

and

and

We have by Lemma 35 that for all \(\delta \in (0, 1)\) \(\begin{array}{lll} | \hat{G} (t - s, x) | & \lesssim & \varepsilon^{\delta} (| t - s |^{1 / 2} + | x |)^{- 5 - \delta} \end{array} .\) Using estimate (78) together with Lemma 29, we have that for all \(\delta \in (0, 1)\), \(\delta' \in (0, \delta)\) that \(| H (t, x) | \lesssim \varepsilon^{\delta'} (| t - s |^{1 / 2} + | x |)^{- \delta}\). Furthermore, we have

with the remainder term \(\Delta_q \hat{R}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}\) given by

The term

can be shown to converge in law to

in \(C^{\kappa}_T \mathscr{C} \hspace{.1em}^{- 1 / 2 - 2 \kappa}\) with the standard techniques used in the analysis of the \(\Phi^4_3\) model. On the other hand, for all \(\delta > 0\) sufficiently small we have the bounds

which shows that these remainders go to zero in \(\mathscr{C} \hspace{.1em}^{- 1 / 2 - 2 \kappa} \) as \(\varepsilon \rightarrow 0\), since \(\| Y_{\varepsilon} \|_{C^{\kappa}_T \mathscr{C} \hspace{.1em}^{- 1 / 2 - 2 \kappa}}\) is bounded in \(L^p (\Omega)\). Moreover, it is easy to see that \(\| \Delta_q R_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}} - \Delta_q \hat{R}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-2.pdf}}} \|_{L^p (\Omega)} \sim O_{L^{\infty}} (2^{q (1 / 2 + 2 \kappa + 2 \delta)})\). Note that

Here we used the fact that

since \(\int_x K_{i, x} (x_1') K_{j, x} (0) = 0\), where \(| i - j | > 1\). This is readily seen in Fourier space taking into account the support properties of the Littlewood-Paley blocks. Now,

converge to finite constants due to the bounds (41) and (42) and by Lemma 32 \(\int_{s, x} P_s (x) C_{\varepsilon} (s, x)^2 \lesssim | \log \varepsilon | .\)

Finally, from (43) we have

Indeed Lemma 32 again yields \(\varepsilon \int_{s, x} P_s (x) C_{\varepsilon} (s, x)^3 \lesssim 1\).

Thus \(\int_{\zeta_1, \zeta_2} \mathbb{E} [\Phi^{[0]}_{\zeta_1} \Phi^{[1]}_{\zeta_2}] \mu_{q, \zeta_1, \zeta_2}\) gives a diverging constant which depends on all the \((f_{n, \varepsilon})_n\). Making the choice to define the renormalisation constants \(d^{\tau}\) as in eq. (7) we cancel exactly these contributions which are either \((F_{\varepsilon})_{\varepsilon}\) dependent and/or diverging. In particular we verify that we can satisfy the constraint (13).

Finally, noting that \(\delta G_1 \mathrm{D} = (1 - J_0)\) and \(\delta^2 G_{[1]}^{[2]} \mathrm{D}^2 = (1 - J_0 - J_1)\) (as seen in (82)), we can write the trees of (39) as

Let us summarize our results so far. We have shown that \(\Delta_q \hat{R}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}} (\bar{\zeta}) + \Delta_q R_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-2.pdf}}} (\bar{\zeta}) \sim O_{L^{\infty}} (\varepsilon^{\delta} 2^{q (1 / 2 + 2 \kappa + 2 \delta)})\) in \(L^p (\Omega)\) and then these terms converge to \(0\) in the right topology as \(\varepsilon \rightarrow 0\). As already mentioned, the convergence in law of

is easy to establish with standard techniques (as done in [4],[18]) assuming the convergence of \(\lambda_{\varepsilon}\) as in (9) to \(\lambda\). Then, comparing (44) with the canonical trees in (20) we can identify the remainder terms \(\Delta_q \hat{Y}_{\varepsilon}^{\tau}\) that still need to be bounded, that are precisely those in which \(\hat{\Phi}^{[n]}_{\zeta}\) appears. Estimating these terms is the content of next section.

3.4.2Estimation of renormalised composite trees

In this section we prove the bound (30) for composite trees. The difficulty we encounter here is that the remainder \(\hat{Y}^{\tau}_{\varepsilon}\) cannot be written as an iterated Skorohod integral as in Section 3.1, but instead as a product of iterated Skorohod integrals. We will then use the product formula (85) to write the remainder in the desired form. We can write (44) in a much shorter way as:

just substituting again \(\delta G_1 \mathrm{D} = (1 - J_0)\) and \(\delta^2 G_{[1]}^{[2]} \mathrm{D}^2 = (1 - J_0 - J_1)\). In order to treat all trees at the same time, we can write the first terms in the r.h.s. above (modulo a constant that we discard) as:

First notice that by Lemma 38 we have

and from the boundedness of the operator \(\mathrm{D}^{\theta} G_{[1]}^{[r]}\) given by Corollary 42 we obtain:

Computing the \(r\)-th derivative of the integrand we obtain

for \(m + n = 5\) and \(0 \leqslant k + \ell \leqslant 2\). The constraints on \(m, n, k, \ell\) are related to the number of branches in the graphical notation of the trees: each tree has \(m + k - 1\) leaves with height 2 and \(n + \ell - 1\) leaves with height 1, as follows

In (45), the terms which do not contain \(\)\(\hat{\Phi}^{[n]}_{\zeta}\) will generate finite contributions in the limit, as seen in Section 3.4.1 by writing the decomposition (44). We just consider the terms proportional to \(\hat{\Phi}^{[4 - m]}_{\zeta_1} \hat{\Phi}^{[4 - n]}_{\zeta_2}\), because all the other similar terms featuring at least one remainder \(\widehat{\Phi }_{\zeta}^{[m]}\) can be estimated with exactly the same technique, and are easily shown to be vanishing in the appropriate topology.

We can use one of the key observations of this paper, the product formula (85), to rewrite products of Skorohod integrals in the form \(\delta^m (u) \delta^n (v)\) as a sum of iterated Skorohod integrals \(\delta^{\ell} (w) \), which are bounded in \(L^p\) by Lemma 38. We obtain

with \(I = \{ (q, r, i) \in \mathbb{N}^3 : 0 \leqslant q \leqslant m, 0 \leqslant r \leqslant n, 0 \leqslant i \leqslant q \wedge r \}\) and the notation shortcut:

By Remark 39, for every \(n, m \geqslant 1\) and \(\Psi \in \operatorname{Dom} \delta^n\) we can write \(\delta^n (\Psi) h^{\otimes m} = \delta^n (\Psi \otimes h^{\otimes m})\), and therefore

With the term to estimate in this form, we can proceed as in Section 3.1 to estimate separately the terms \(\Theta_{[i]}^{[j]} (\zeta) = \varepsilon^{- \frac{i}{2}} G^{[j]}_{[i]} \Phi_{\zeta}^{[3 + i]}\) in \(L^p (\Omega)\), which are bounded as discussed in Remark 9.

Proof. Thanks to Lemma 38 the integral can be estimated with

with \(V = H^{\otimes m + k + n + \ell - q - r + j}\). We have that \(\| \cdot \|_{L^p (H^{\otimes k + \ell})}^2 = \| \| \cdot \|^2_{H^{\otimes k + \ell}} \|_{L^{p / 2}}^{1 / 2}\) and therefore we can bound each term in the sum above as

Using Hölder's inequality we get the estimate

Now to bound \(\| \langle \mathrm{D}^h \Theta_{[1 + a]}^{[m + a]} (\zeta), \mathrm{D}^h \Theta_{[1 + a]}^{[m + a]} (\zeta') \rangle_{H^{\otimes h}} \|_{L^p}\) (with \(h \leqslant j \leqslant m + n - q - r\) and \(a = r - i\)) we can use the boundedness of the operator \(\mathrm{D}^h G_{[1 + a]}^{[m + a]}\) for \(h \leqslant 2 m\) given by Corollary 42. Consider the two regions \(h \leqslant 2 m\) and \(h > 2 m\). In the first region we just use Corollary 42 to obtain:

If \(h > 2 m\) we first use the bound \(\mathrm{D}^{2 m} G_{[1 + a]}^{[m + a]}\) and then take the remaining \(h - 2 m\) derivatives on \(\varepsilon^{- \frac{1 + a}{2}} \Phi^{[4 + a]}_{\zeta}\):

From Remark 9 we see that this last term is bounded by a constant if \(F \in C^{4 + a + h - 2 m} (\mathbb{R})\) with the first \(4 + a + h - 2 m\) derivatives having an exponential growth, with \(a = r - i\) and \(h \leqslant m + n - q - r\).

Applying the same reasoning to \(\| \langle \mathrm{D}^{j - h} \Theta_{[1 + q - i]}^{[n + q - i]} (\zeta_2), \mathrm{D}^{j - h} \Theta_{[1 + q - i]}^{[n + q - i]} (\zeta'_2) \rangle_{H^{\otimes j - h}} \|_{L^p}\) we conclude that we need to control \(4 + n \vee 4 + m\) derivatives of \(F_{\varepsilon}\) in order to perform the estimates of this Lemma. From the constraints (46) we see that \(4 + n \vee 4 + m \leqslant 8\).\(\Box\)

From Lemma 13 we obtain \(\forall \delta \in [0, 1 / 2)\): \(\varepsilon^{\frac{2 + q + r - 2 i}{2}}\)

Our aim now is to estimate the quantity \(\mathfrak{I}\). The idea is to use the bound \(\varepsilon | \langle h_{\zeta}, h_{\zeta'} \rangle | = \varepsilon C_{\varepsilon} (\zeta - \zeta') \lesssim 1\) of Lemma 31 to cancel strategically some of the covariances \(| \langle h_{\zeta}, h_{\zeta'} \rangle |\). We will consider three regions:

If \(q + r \leqslant 2\) we use the bounds

and then (we suppose \(r < 2\))

to obtain

(If vice-versa \(q < 2\) it suffices to put \(\delta\) on the term \(| \langle h_{\zeta_1}, h_{\zeta_2} \rangle |^{q + \delta}\).) Notice that in this case \(m + k - q > 0.\)

In the case \(q + r = 3\) if \(m + k - q \geqslant 2\) we estimate like before to obtain

Note that \(m + k - q + \delta - 1 > 0\) and \(m + k - q + 2 \delta - 3 > - 1\) here. If \(m + k - q = 1\) we bound

and note that \(m + k - q - 1 / 2 + \delta / 2 > 0\), \(m + k - q - 1 + \delta > 0\), \(n + \ell - r - 2 \geqslant 0\). Finally if \(m + k - q = 0\) we can only have \(m + k = 3, q = 3, r = 0, i = 0\) and thus

If \(\)\(q + r \geqslant 4\) we bound first

(note that \(2 q + 2 r - 2 i + \delta - 4 \geqslant \delta\)) to obtain:

Now in the cases \(m + k = 3, n + \ell = 3\) and \(m + k = 4, n + \ell = 2\) we can just write \(\varepsilon^{6 - q - r - \delta} = \varepsilon^{m + k - q} \varepsilon^{6 - m - k - r - \delta}\) and cancel the corresponding number of covariances to obtain

while for the case \(m + k = 4, n + \ell = 3\) we have either \(\ell \geqslant 1\) or \(k \geqslant 1\) and therefore with one of the following bounds

we obtain the estimates

We can use directly Lemma 36 to obtain a final estimate of (47), (48), (49), (52). For \(\textrm{(\hyperref[e:graph-est-4]{50})}, \textrm{(\hyperref[e:graph-est-5]{51})}\) and (53) notice that the integral over \(\zeta_1, \zeta'_1\) is finite and thus the whole quantity is proportional to \(| \langle h_{\zeta_2}, h_{\zeta_2'} \rangle |^n\). Globally, we have

as needed to prove (30).

Remark 14. Finally, by controlling one more derivative of \(F_{\varepsilon}\) as done in Section 3.3, we can show (31) for \(Y^{\tau} = Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-1.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-2.pdf}}}, \bar{Y}_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-3.pdf}}}, Y_{\varepsilon}^{\resizebox{1.5em}{!}{\includegraphics{image-4.pdf}}}\), thus proving that \(\hat{Y}^{\tau} \rightarrow 0\) in \(C^{\kappa / 2}_T \mathscr{C} \hspace{.1em}^{\alpha - \kappa}\) in probability \(\forall \alpha < | \tau |\). From the proof of Lemma 13 together with this observation, we conclude that we need to control the derivatives of \(F_{\varepsilon}\) up to order \(9\) to be able to show the convergence for composite trees.

4Convergence of the remainder and a-priori bounds

In this section we prove the convergence of the remainder (Lemma 19), as well as some technical results on the norm of the solution, needed in the proof of Theorem 7. In order to prove Lemma 19 we need first to prove Lemma 15, Lemma 16 and Lemma 17 in this order.

4.1Boundedness of the remainder

We show that the remainder \(R_{\varepsilon} (v_{\varepsilon})\) that appears in equation (16) can be controlled by a stochastic term \(M_{\varepsilon, \delta}\) that converges to zero in probability, times a function of the solution \(v_{\varepsilon}\). Let.

for \(p \in [1, \infty)\), \(\delta \in [0, 1]\) and define

Proof. We can write the remainder in two ways:

From assumption (8) on \(F_{\varepsilon}\) we obtain by interpolation of these two expressions, \(\forall \delta \in [0, 1]\), \(\forall t \geqslant 0, x \in \mathbb{T}^3\),

and we estimate, \(\forall \gamma \in [0, 1)\),

We can also verify that \(M_{\varepsilon, \delta} \rightarrow 0\) in probability for every \(\delta > 0\):

Proof. We can use Young's inequality to estimate \(M_{\varepsilon, \delta} (Y_{\varepsilon}, u_{0, \varepsilon})\) for some \(c' > 0\) as

Under Assumptions 1 the term \(\| \varepsilon^{1 / 2} u_{0, \varepsilon} \|_{L^{\infty} (\mathbb{T}^3)}\) is uniformly bounded, so the third term above converges to zero in probability. Note that \(\varepsilon^{1 / 2} Y_{\varepsilon} (t, x)\) and \(P_t \varepsilon^{1 / 2} Y_{\varepsilon} (t = 0)\) are centered Gaussian random variables, and then both \(\mathbb{E} \|e^{c' \varepsilon^{1 / 2} | Y_{\varepsilon} |} \|^p_{L^p [0, T] L^p (\mathbb{T}^3)}\) and \(\mathbb{E} \|e^{c' \varepsilon^{1 / 2} | P_{.} Y_{\varepsilon} (0) |} \|^p_{L^p [0, T] L^p (\mathbb{T}^3)}\) are uniformly bounded in \(\varepsilon > 0\) for every \(p \in [1, \infty)\). This yields the convergence in probability of \(M_{\varepsilon, \delta} (Y_{\varepsilon}, u_{0, \varepsilon})\).\(\Box\)

In order to show that \(\)\(\| R_{\varepsilon} (v_{\varepsilon}) \|_{\mathcal{M}^{\gamma', p}_T L^p} \rightarrow 0\) in probability for \(\gamma' > \frac{1}{4} + \frac{3}{2} \kappa\) as needed in the proof of Theorem 7, we still need to control the norms \(\| v_{\varepsilon} \|^{3 + \delta}_{\mathcal{M}_T^{\gamma / (3 + \delta)} L^{\infty}}\) and \(\|v^{\Box}_{\varepsilon} \|_{C_T L^{\infty}}\) that appear in Lemma 15. This is done in next section.

4.2Apriori bounds on the solution

Proof. We know from Lemma 18 that the bounds above on \(\| v_{\varepsilon} \|_{C_{T_{\star}} \mathscr{C} \hspace{.1em}^{- \frac{1}{2} - \kappa}} + \| v_{\varepsilon} \|_{\mathcal{M}_{T_{\star}}^{\frac{1}{4} + \frac{3 \kappa}{2}} L^{\infty}}\) and \(\)\(\| v_{\varepsilon}^{\Box} \|_{C_{T_{\star}} L^{\infty}}\) hold whenever \(M_{\varepsilon, \delta} \leqslant T_{\star}^{\kappa / 2}\). The event \(\mathcal{E}_{\varepsilon} = \{ M_{\varepsilon, \delta} \leqslant T_{\star}^{\kappa / 2} \}\) has \(\mathbb{P} (\mathcal{E}_{\varepsilon}) \rightarrow 1\) by Lemma 16 and this proves the result.\(\Box\)

The only thing left to prove is Lemma 18, which just a standard application of some well-known bounds on paraproducts, that are recalled in Appendices A.2, and A.2.

First observe that for \(\varepsilon > 0\) a pair \((v_{\varepsilon}, v^{\natural}_{\varepsilon})\) solves the paracontrolled equation (16) if and only if \(v^{\natural}_{\varepsilon} = v^{\flat}_{\varepsilon} + v^{\sharp}_{\varepsilon}\) and \((v_{\varepsilon}, v^{\flat}_{\varepsilon})\) solves:

Here \(U\) is the same as in (16). The initial condition of (56) is given by \(v_{\varepsilon, 0} := u_{0, \varepsilon} - Y_{\varepsilon} (0)\). The a-priori bounds of Lemma 18 come from being able to find closed estimates for (56).

Let us specify now all the notations we are going to use in the rest of this section. We consider the spaces

with the corresponding norms

We refer to Appendix A.1 for the definition of the parabolic spaces \(\mathscr{L} \hspace{.2em}^{\gamma, \alpha}_T\). We let

and \(v^{\Box}_{\varepsilon} (t = 0) = v_{\varepsilon}^{\boxslash} (t = 0) = 0\). We define also the norm

with \(\mathcal{M}^{1 / 4}_T \mathscr{C} \hspace{.1em}^{1 / 2 + 2 \kappa}\) given in Appendix A.1. In order not to get lost in these definitions the reader can keep in mind the following:

• \(v_{\varepsilon}\) is the solution without the linear term;

• \(v^{\sharp}_{\varepsilon}\) is the contribution of the initial condition, which give origin to some explosive norm (near the initial time);

• \(v^{\flat}_{\varepsilon}\) is the regular part of the solution;

• \(v^{\Box}_{\varepsilon}, v_{\varepsilon}^{\boxslash}\) enter in the estimation of the remainder, they are just convenient shortcuts for certain contributions appearing in \(v_{\varepsilon}\).

Proof. Using the well-known Schauder estimates of Lemma 21 (and the fact that \(\| f \|_{\mathscr{L} \hspace{.2em}_T^{\kappa, \alpha}} \lesssim T^{\kappa} \| f \|_{\mathscr{L} \hspace{.2em}_T^{\alpha}}\)) we obtain for \(\kappa, \theta > 0\) small enough

We choose \(\theta > 2 \kappa\) small enough so that

Now

where \(\)we used that \(v^{\Box} (0) = 0\) and as a consequence that \(\|v^{\Box}_{\varepsilon} \|_{C_{_T} L^{\infty}} \leqslant T^{\kappa} \|v^{\Box}_{\varepsilon} \|_{C_{_T}^{\kappa} L^{\infty}} \leqslant T^{\kappa} \| v^{\Box}_{\varepsilon} \|_{\mathcal{V}_T^{\Box}} \) to gain a small power of \(T\). So provided \(T\) is small enough (depending only on \(\mathbb{Y}_{\varepsilon}\)) this yields the following a-priori estimation on \(v_{\varepsilon}^{\Box}\):

Therefore we have an estimation on \(v_{\varepsilon}\):

In order to estimate terms in \(U (\lambda_{\varepsilon}, \mathbb{Y}_{\varepsilon} ; v_{\varepsilon}, v^{\flat}_{\varepsilon} + v^{\sharp}_{\varepsilon})\) we decompose the renormalised products as

We decompose \(U (\lambda_{\varepsilon}, \mathbb{Y}_{\varepsilon} ; v_{\varepsilon}, v^{\flat}_{\varepsilon} + v^{\sharp}_{\varepsilon})\) as

Here \(Q_{\lambda_{\varepsilon}, \mathbb{Y}_{\varepsilon}}\) does not depend from the solution but only on \(\lambda_{\varepsilon}, \mathbb{Y}_{\varepsilon}\) (as the notation suggests) and we have grouped the other terms which we expect to have regularity \(\mathscr{C} \hspace{.1em}^{- 1 / 2 - 2 \kappa}\) in \(Q_{- 1 / 2}\), (and the same for \(Q_0\) and regularity \(\mathscr{C} \hspace{.1em}^{- k}\)). With the same technique we used above for \(v^{\Box}_{\varepsilon}\), we obtain the following estimate on \(v^{\boxslash}_{\varepsilon}\)

and this yields

Then we are ready to bound \(Q_{- 1 / 2}, Q_0, Q_{\lambda_{\varepsilon}, \mathbb{Y}_{\varepsilon}}\) using the standard paraproducts estimations recalled in Appendix A.2:

In order to conclude the estimation of \(\| v^{\flat}_{\varepsilon} \|_{\mathcal{V}^{\flat}_T}\) we have to control \(\| I R_{\varepsilon} (v_{\varepsilon}) \|_{\mathcal{V}^{\flat}_T}\). This is achieved easily by the using the results of Section 4.1. Thanks to Lemma 15 \(\forall \delta \in (0, 1)\), \(\forall \theta > 0\) such that \(\frac{1 - \theta}{3 + \delta} > \frac{1}{4} + \frac{3 \kappa}{2}\) (note that it is possible to choose \(\theta > 2 \kappa\) that satisfies this property as long as \(k\) and \(\delta\) are small enough) we have:

\(\)By Lemma 22 together with (61) we obtain then

Using that

we obtain that \(\exists C' > 0\) such that

with

and

Let \(T_{\star} \in (0, T]\) such that:

Assume that \(M_{\varepsilon, \delta} \leqslant T_{\star}^{\kappa / 2}\). Define a closed interval \([0, S] = \{t \in [0, T_{\star}] : \| v^{\flat}_{{\varepsilon_n} } \|_{\mathcal{V}_t^{\flat}} \leqslant 4 D\} \subseteq [0, T_{\star}]\). This interval is\(\) well defined and non–empty since \(\)\(t \mapsto \| v^{\flat}_{{\varepsilon_n} } \|_{\mathcal{V}_t^{\flat}}\) is continuous and nondecreasing and \(\| v^{\flat}_{{\varepsilon_n} } \|_{\mathcal{V}_0^{\flat}} \leqslant 4 D\)\(\). Let us assume that \(S < T_{\star}\), then we can take \(\epsilon > 0\) small enough such that \(S + \epsilon < T_{\star}\) and by continuity \(\)\(\| v^{\flat}_{\varepsilon} \|_{\mathcal{V}_{S + \epsilon}^{\flat}} \leqslant 5 C\), then

which gives \(\| v^{\flat}_{{\varepsilon_n} } \|_{\mathcal{V}_{S + \epsilon}^{\flat}} \leqslant 4 D\). This implies \(S = T_{\star}\) (by contradiction). From the construction of \(T_{\star}\) it is easy to see that \(T_{\star} \left( \| \mathbb{Y}_{\varepsilon} \|_{\mathcal{X}_T}, \| u_{\varepsilon, 0} \|_{\mathscr{C} \hspace{.1em}^{- 1 / 2 - \kappa}}, | \lambda_{\varepsilon} | \right)\) is lower semicontinuous.\(\Box\)

4.3Convergence of the remainder

It suffices to put together the results obtained in Sections 4.1 and 4.2 to obtain the convergence of \(R_{\varepsilon} (v_{\varepsilon})\):

Proof. From the estimation on \(R_{\varepsilon} (v_{\varepsilon})\) of Lemma 15, together with the fact that \(M_{\varepsilon, \delta} \rightarrow 0\) in probability (Lemma 16) and the bounds on \(\)\(\| v_{\varepsilon} \|_{\mathcal{M}_{T_{\star}}^{\frac{1}{4} + \frac{3 \kappa}{2}} L^{\infty}}\) and \(\| v_{\varepsilon}^{\Box} \|_{C_T L^{\infty}}\) of Lemma 17 we see immediately that \(\)\(\| R_{\varepsilon} (v_{\varepsilon}) \|_{\mathcal{M}^{\gamma, p}_{T_{\star}} L^p} \rightarrow 0\) in probability.\(\Box\)

Appendix AParacontrolled analysis and kernel estimations

In this section we first recall the the basic results of paracontrolled calculus first introduced in [6], without proofs. For more details on Besov spaces, Littlewood–Paley theory, and Bony's paraproduct the reader can refer to the monograph [2]. We then proceed to give some results on the convolution of functions with known singularity and the estimation of finite-chaos Gaussian trees. We refer to Section 10 of [8] and to the nice pedagogic exposition [18] for further details.

A.1Notation

Throughout the paper, we use the notation \(a \lesssim b\) if there exists a constant \(c > 0\), independent of the variables under consideration, such that \(a \leqslant c \cdot b\). If we want to emphasize the dependence of \(c\) on the variable \(x\), then we write \(a (x) \lesssim_x b (x)\). If \(f\) is a map from \(A \subset \mathbb{R}\) to the linear space \(Y\), then we write \(f_{s, t} = f (t) - f (s)\). For \(f \in L^p (\mathbb{T}^d)\) we write \(\|f (x)\|^p_{L^p_x (\mathbb{T}^3)} := \int_{\mathbb{T}^3} |f (x) |^p \mathrm{d} x\).

Given a Banach space \(X\) with norm \(\| \cdot \|_X\) and \(T > 0\), we note \(C_T X = C ([0, T], X)\) for the space of continuous maps from \([0, T]\) to \(X\), equipped with the supremum norm \(\lVert \cdot \rVert_{C_T X}\), and we set \(C X = C (\mathbb{R}_+, X)\). For \(\alpha \in (0, 1)\) we also define \(C^{\alpha}_T X\) as the space of \(\alpha\)-Hölder continuous functions from \([0, T]\) to \(X\), endowed with the seminorm \(\|f\|_{C^{\alpha}_T X} = \sup_{0 \leqslant s < t \leqslant T} \|f (t) - f (s)\|_X / |t - s|^{\alpha}\), and we write \(C^{\alpha}_{\operatorname{loc}} X\) for the space of locally \(\alpha\)-Hölder continuous functions from \(\mathbb{R}_+\) to \(X\). For \(\gamma > 0\), \(p \in [1, \infty)\), we define

The space of distributions on the torus is denoted by \(\mathscr{D} \hspace{.1em}' (\mathbb{T}^3)\) or \(\mathscr{D} \hspace{.1em}'\). The Fourier transform is defined with the normalization

so that the inverse Fourier transform is given by \(\mathscr{F} \hspace{.1em}^{- 1} v (x) = (2 \pi)^{- 1} \sum_k e^{\iota \langle k, x \rangle} v (k)\). Let \((\chi, \rho)\) denote a dyadic partition of unity such that \(\operatorname{supp} (\rho (2^{- i} \cdot)) \cap \operatorname{supp} (\rho (2^{- j} \cdot)) = \emptyset\) for \(|i - j| > 1\). The family of operators \((\Delta_j)_{j \ge - 1}\) will denote the Littlewood-Paley projections associated to this partition of unity, that is \(\Delta_{- 1} u = \mathscr{F} \hspace{.1em}^{- 1} \left( \chi \mathscr{F} \hspace{.1em} u \right)\) and \(\Delta_j = \mathscr{F} \hspace{.1em}^{- 1} \left( \rho (2^{- j} \cdot) \mathscr{F} \hspace{.1em} u \right)\) for \(j \ge 0\). Let \(S_j = \sum_{i < j} \Delta_i\), and \(K_q\)be the kernel of \(\Delta_q\) so that

For the precise definition and properties of the Littlewood-Paley decomposition \(f = \sum_{q \geqslant - 1} \Delta_q f\) in \(\)\(\mathscr{D} \hspace{.1em}' (\mathbb{T}^3)\), see Chapter 2 of [2]. The Hölder-Besov space \(B^{\alpha}_{p, q} (\mathbb{T}^3, \mathbb{R})\) for \(\alpha \in \mathbb{R}\) ,\(p, q \in [1, \infty]\) with \(B^{\alpha}_{p, q} (\mathbb{T}^3, \mathbb{R}) = : \mathscr{C} \hspace{.1em}^a\) is and equipped with the norm

If \(f\) is in \(\mathscr{C} \hspace{.1em}^{\alpha - \varepsilon}\) for all \(\varepsilon > 0\), then we write \(f \in \mathscr{C} \hspace{.1em}^{\alpha -}\). For \(\alpha \in (0, 2)\), we define the space \(\mathscr{L} \hspace{.2em}_T^{\alpha} = C^{\alpha / 2}_T L^{\infty} \cap C_T \mathscr{C} \hspace{.1em}^{\alpha}\), equipped with the norm

The notation is chosen to be reminiscent of

by which we will always denote the heat operator with periodic boundary conditions on \(\mathbb{T}^d\). We also write \(\mathscr{L} \hspace{.2em}^{\alpha} = C^{\alpha / 2}_{\operatorname{loc}} L^{\infty} \cap C \mathscr{C} \hspace{.1em}^{\alpha}\). When working with irregular initial conditions, we will need to consider explosive spaces of parabolic type. For \(\gamma \geqslant 0\), \(\alpha \in (0, 1)\), and \(T > 0\) we define the norm

and the space \(\mathscr{L} \hspace{.2em}^{\gamma, \alpha}_T = \left\{ f : [0, T] \rightarrow \mathbb{R}: \| f \|_{\mathscr{L} \hspace{.2em}^{\gamma, \alpha}_T} < \infty \right\}\). In particular, we have \(\mathscr{L} \hspace{.2em}^{0, \alpha}_T = \mathscr{L} \hspace{.2em}^{\alpha}_T\). We introduce the linear operator \(I : C \left( \mathbb{R}_+, \mathscr{D} \hspace{.1em}' (\mathbb{T}) \right) \rightarrow C \left( \mathbb{R}_+, \mathscr{D} \hspace{.1em}' (\mathbb{T}) \right)\) given by

where \((P_t)_{t \geqslant 0}\) is the heat semigroup with kernel \(P_t (x) = \frac{1}{(4 \pi t)^{3 / 2}} e^{- \frac{| x |^2}{4 t}} \mathbb{I}_{t \geqslant 0}\).

Paraproducts are bilinear operations introduced by Bony [3] in order to linearize a class of non-linear PDE problems. They appear naturally in the analysis of the product of two Besov distributions. In terms of Littlewood–Paley blocks, the product \(fg\) of two distributions \(f\) and \(g\) can be formally decomposed as

where

This decomposition behaves nicely with respect to Littlewood–Paley theory. We call \(f \prec g\) and \(f \succ g\) paraproducts, and \(f \circ g\) the resonant term. We use the notation \(f \preccurlyeq g = f \prec g + f \circ g\). The basic result about these bilinear operations is given by the following estimates, essentially due to Bony [3] and Meyer [14].

When dealing with paraproducts in the context of parabolic equations it would be natural to introduce parabolic Besov spaces and related paraproducts. But to keep a simpler setting, we choose to work with space–time distributions belonging to the scale of spaces \(\left( C_T \mathscr{C} \hspace{.1em}^{\alpha} \right)_{\alpha \in \mathbb{R}}\) for some \(T > 0\). To do so efficiently, we will use a modified paraproduct which introduces some smoothing in the time variable that is tuned to the parabolic scaling. Let therefore \(\varphi \in C^{\infty} (\mathbb{R}, \mathbb{R}_+)\) be nonnegative with compact support contained in \(\mathbb{R}_+\) and with total mass \(1\), and define for all \(i \geqslant - 1\) the operator

We will often apply \(Q_i\) and other operators on \(C \mathscr{C} \hspace{.1em}^{\beta}\) to functions \(f \in C_T \mathscr{C} \hspace{.1em}^{\beta}\) which we then simply extend from \([0, T]\) to \(\mathbb{R}_+\) by considering \(f (\cdot \wedge T)\). With the help of \(Q_i\), we define a modified paraproduct

for \(f, g \in C \left( \mathbb{R}_+, \mathscr{D} \hspace{.1em}' (\mathbb{T}) \right)\). We define the commutators \(\operatorname{com}_1, \overline{\operatorname{com}}_1, \operatorname{com}_2, \operatorname{com}_3\) in Lemma 26. We write \(\check{P} (t, x) = \frac{1}{(4 \pi t)^{3 / 2}} e^{- \frac{| x |^2}{4 t}} e^{- t} \mathbb{1}_{t \geqslant 0}\) for a modified heat kernel which has the same bounds of the usual heat kernel \(P (t, x)\). Let \(Y_{\varepsilon}\) as in (3) and recall that \(C_{\varepsilon}\) is the covariance of \(Y_{\varepsilon}\) , i.e. \(C_{\varepsilon} (t, x) =\mathbb{E} (Y_{\varepsilon} (t, x) Y_{\varepsilon} (0, 0))\). We will sometimes write \(Y_{\varepsilon, \zeta} := Y_{\varepsilon} (t, x)\) for \(\zeta = (t, x) \in \mathbb{R} \times \mathbb{T}^3\). Let \(\sigma_{\varepsilon}^2 = \varepsilon \mathbb{E} [ (Y_{\varepsilon} (0, 0))^2] = \varepsilon C_{\varepsilon} (0, 0)\).

A.2Basic paracontrolled calculus results

First, let us recall some interpolation results on the parabolic time-weighted spaces \(\mathscr{L} \hspace{.2em}^{\gamma, \alpha}_T\):

Schauder estimates

Proofs can be found e.g. in [7]. We need also some well known bounds for the solutions of the heat equation with sources in space–time Lebesgue spaces.

Proof. We only show the second inequality as the first one is easier and obtained with similar techniques. Let \(u = I f\), we have

which allows us to bound \(\| I f \|_{\mathcal{M}^{\gamma}_T \mathscr{C} \hspace{.1em}^{\alpha}}\). In order to estimate \(\| t \mapsto t^{\gamma'} I f \|_{C^{\alpha / 2}_T L^{\infty}}\) we write

We can estimate the first term as

For the third term we have

We obtain then if \(2^{2 i} | t - s | \leqslant 1\)

and if \(2^{2 i} | t - s | > 1\) we just use the trivial estimate

Therefore, for every \(\kappa \in [0, 1]\):

Choosing \(\kappa / q = \alpha / 2\) we obtain the desired estimate.\(\Box\)

Estimates on Bony's paraproducts and commutators

A natural corollary is that the product \(fg\) of two elements \(f \in \mathscr{C} \hspace{.1em}^{\alpha}\) and \(g \in \mathscr{C} \hspace{.1em}^{\beta}\) is well defined as soon as \(\alpha + \beta > 0\), and that it belongs to \(\mathscr{C} \hspace{.1em}^{\gamma}\), where \(\gamma = \min \{\alpha, \beta, \alpha + \beta\}\). We will also need a commutator estimation:

We collect in the following lemma various estimates for the modified paraproduct \(f \prec\!\!\!\prec g\), proofs are again in [7].

We introduce various commutators which allow to control non-linear functions of paraproducts and also the interaction of the paraproducts with the heat kernel.

Proofs can be found in [7].

A.3Estimation of finite-chaos diagrams

In this section we recall a few well-known results on the estimation of finite chaos diagrams. For additional details see Chapter 10 of [8] and [18]. First of all we need to characterize the local behaviour of the heat kernel \(P_t (x)\) and of the covariance \(C_{\varepsilon} (t, x)\) of the Gaussian field \(Y_{\varepsilon}\).

Remark 27. We sometimes use a slightly different version of the heat kernel, namely

in order to have that \(X (t, x) = \int_{- \infty}^t \int_{\mathbb{T}^3} P_{t - s} (x - y) v (s, y) \mathrm{d} s \mathrm{d} y\) is the stationary solution to \(\mathscr{L} \hspace{.2em} X = - X + v\). However, the bounds on the covariance \(C_{\varepsilon}\) remain trivially valid in this setting.

Proof.

In the same way we prove that \(| \partial_t P_t (x) | \lesssim (| t |^{1 / 2} + | x |)^5\), \(| \partial_{x_i} P_t (x) | \lesssim (| t |^{1 / 2} + | x |)^4\) and \(| \partial_{x_i} \partial_{x_j} P_t (x) | \lesssim (| t |^{1 / 2} + | x |)^5\).\(\Box\)

We recall a special case of Lemma 10.14 of [8], which is enough for our purpose. We use the notation \(\unicode{x2980} \zeta \unicode{x2980} := (| t |^{1 / 2} + | x |)\) for \(\zeta = (t, x) \in \mathbb{R} \times \mathbb{T}^3\).

Remark 30. The covariance \(C_{\varepsilon}\) of \(Y_{\varepsilon}\) can be written as \(C_{\varepsilon} = \check{P} \ast \check{C}_{\varepsilon} \ast \check{P} \) with \(\check{C}_{\varepsilon} (t, x) := \mathbb{E} (\eta_{\varepsilon} (t, x) \eta_{\varepsilon} (0, 0))\). Recall from the introduction that \(\check{C}_{\varepsilon} (t, x) = \varepsilon^{- 5} \tilde{C}^{\varepsilon} (\varepsilon^{- 2} t, \varepsilon^{- 1} x)\) where \(\)\(\tilde{C}^{\varepsilon}\) is the covariance of the Gaussian process \(\eta\) defined on \(\mathbb{R} \times (\mathbb{T}/ \varepsilon)^3\), and \(\tilde{C}^{\varepsilon} (t - s, x - y) = \Sigma (t - s, x - y)\) if \(\operatorname{dist} (x, y) \leqslant 1\) and \(0\) otherwise (so that the family of functions \(\tilde{C}^{\varepsilon}\) is bounded uniformly on \(\varepsilon\) by a \(C^{\infty}_c\) function). Then there exists a family of functions \(C_Y^{\varepsilon}\) defined on \(\mathbb{R} \times (\mathbb{T}/ \varepsilon)^3\) such that \(C_{\varepsilon} (t, x) = \varepsilon^{- 1} C_Y^{\varepsilon} (\varepsilon^{- 2} t, \varepsilon^{- 1} x)\) and \(C_Y^{\varepsilon} (t, x) = [\check{P} \ast \check{C}_{\varepsilon} \ast \check{P} ] (t, x)\).

Proof. The first bound is obtained directly from Lemma 28 and Lemma 29. Indeed, since by hypothesis \(\tilde{C}^{\varepsilon}\) has compact support, it is easy to see that \(| \check{C}_{\varepsilon} (t, x) | \lesssim (| t |^{1 / 2} + | x |)^{- 5}\). The second bound is obtained by a simple change of variables in the convolution defining \(C_{\varepsilon}\). \(\Box\)

Proof. From the fact that \(P_{\varepsilon^2 s} (\varepsilon x) = \varepsilon^{- 3} P_s (x)\) together with Remark 30 we obtain

with \(B (0, R) = \{\zeta \in \mathbb{R}^4 : \unicode{x2980} \zeta \unicode{x2980}< R, \zeta \neq 0\}\) a “parabolic” ball centered in the origin. The second estimation is obtained in the same way.\(\Box\)

Proof. The estimation of \(I_m\) is easily obtained by Lemma 31 and a change of variables. For \(\tilde{I}_n\) observe that for every \(q > 0\)

and then we can apply Lemma 29 to obtain the result.\(\Box\)

Proof. It is easy to obtain by interpolation knowing that \(| \partial_t C_{\varepsilon} (t, x) | \lesssim \varepsilon^{- 3}\) from Lemma 31.\(\Box\)

Proof. We will show that

and that

from which we deduce that

Bounding the sum over \(i\) with an integral, we conclude

Let us show (78). We want to estimate

where

When \(t^{1 / 2} \geqslant 2^{- i}, | x |\) we have

When \(2^{- i} \geqslant t^{1 / 2}, | x |\) we estimate simply

When \(| x | \geqslant 2^{- i}, t^{1 / 2}\) we have instead that either \(| x | \geqslant 2 \tau 2^{- i} | y |\) or \(| x | < 2 \tau 2^{- i} | y |\). In the first region \(| x + \tau 2^{- i} y | \geqslant c | x |\) so

while in the second region \(| y | \geqslant 2^i | x | / (2 \tau)\), then \(\)\(| y K_1 (y) | \leqslant | y K_1 (y) |^{1 / 2} f (2^i | x | / (2 \tau))\) where \(f\) is a rapidly decreasing function and in this region

So we conclude that (78) holds. Let us turn to (79). When \(t^{1 / 2} \geqslant 2^{- i}, | x |\) we have

When \(2^{- i} \geqslant t^{1 / 2}, | x |\) we estimate

and finally when \(| x | \geqslant 2^{- i}, t^{1 / 2}\) we have either \(| x | \geqslant 2^{- i + 1} | y |\) or \(| x | < 2^{- i + 1} | y |\). In the first region \(| x + 2^{- i} y | \geqslant c | x |\) so

while in the second \(| y | \geqslant 2^i | x | / 2\), then \(\)\(| K_1 (y) | \leqslant | K_1 (y) |^{1 / 2} f (2^i | x | / 2)\) where \(f\) is another rapidly decreasing function and in this region

concluding our argument.\(\Box\)

Proof. Observe that

where in the first line we used \(\int_y K_{i, x} (y) = 0\) and the fact that \(\int_x K_{i, x} (x_1') K_{j, x} (x_2) = 0\) if \(| i - j | > 1\) and \(\sum_{i, j} K_{i, x} (y) K_{j, x} (x_2) = \delta (x_2 - y) \delta (x_2 - x)\). Now the estimation of the term

with \(\bar{\mu}_{q, \zeta_1, \zeta_2} = K_{q, \bar{x}} (x_2) [P_{t - s_1} (x_2 - x_1) - P_{t - s_1} (\bar{x} - x_1)] \delta (t - s_2) \mathrm{d} \zeta_1 \mathrm{d} \zeta_2\) can be done with Lemma 29 and gives the expected result. The integral

with \(\hat{\mu}_{q, \zeta_1, \zeta_2} = [\sum_{i \sim j} \int_{x, y} [K_{q, \bar{x}} (x) - K_{q, \bar{x}} (x_2)] K_{i, x} (y) K_{j, x} (x_2) P_{t - s_1} (y - x_1)] \delta (t - s_2) \mathrm{d} \zeta_1 \mathrm{d} \zeta_2\) can be estimated by multiple changes of variables. We have \(K_{q, \bar{x}} (x) - K_{q, \bar{x}} (x_2) = 2^{3 q} (x_2 - x) \int_0^1 K' (2^q (x_2 - x) \tau - 2^q (\bar{x} - x_2)) \mathrm{d} \tau\), and by the scaling properties of \(C_{\varepsilon}\) and \(P_{t, y}\), namely \(C_{\varepsilon} (2^{- 2 i} s, 2^{- i} x) \lesssim 2^i C_{\varepsilon} (s, x)\) and \(P_{2^{- 2 i} s} (2^{- i} x) \lesssim 2^{3 i} P_s (x)\) given by Lemma 28 and Lemma 31, we obtain easily the bound on \(\)\(\hat{I}_{k, m, n}\) by rescaling the integral.\(\Box\)

Appendix BSome Malliavin calculus

We recall here some tools from Malliavin calculus that are widely used in the rest of the paper. An introduction to Malliavin calculus and the proofs of some results of this Appendix can be found in [20, 19, 24]. Lemma 43 was inspired by the calculations of [25].

B.1Notation

Let \(\{ W (h) \}_{h \in H}\) be an isonormal Gaussian process indexed by a real separable Hilbert space \(H\). Let \((\Omega, \mathcal{F}, \mathbb{P})\) a probability space with \(\mathcal{F}\) generated by the isonormal Gaussian process \(W\), we note \(L^2 (\Omega) = \bigoplus_{n \geqslant 0} \mathcal{H}_n\) the well-known Wiener chaos decomposition of \(L^2 (\Omega)\). For any real separable Hilbert space \(V\) and \(k \in \mathbb{N}\), let \(\)\(\mathrm{D}^k : L^2 (\Omega ; V) \rightarrow L^2 (\Omega ; H^{\odot k} \otimes V)\) be the Malliavin derivative and \(\delta^k : L^2 (\Omega ; H^{\otimes k} \otimes V) \rightarrow L^2 (\Omega ; V)\) the divergence operator (also called Skorohod integral) defined as the adjoint of \(\mathrm{D}^k\). For \(p \geqslant 1\) we will write \(\mathbb{D}^{k, p} (V) \subset L^p (\Omega ; V)\) for the closure of smooth random variables with respect to the norm

with the notation \(\mathbb{D}^{k, p} := \mathbb{D}^{k, p} (\mathbb{R})\). Let \(\{ P_t \}_{t \in \mathbb{R}^+}\) the Ornstein-Uhlenbeck semigroup and \(L : L^2 (\Omega) \rightarrow L^2 (\Omega)\) its generator (i.e. \(e^{tL} = P_t\)). Following [24] we introduce the Green operator

with the notation

so that \(G_{[j]}^{[j]} = G_j\). To avoid confusion, it is worth stressing that \(G_{[j]}^{[m]}\) is not the \(m\)-th power of the operator \(G_j\) but just a shortcut for \(\prod_{k = j}^m G_k \).

B.2Partial chaos expansion

Let \(\Psi \in L^2 (\Omega)\) which has the Wiener chaos decomposition \(\Psi = \sum_{n \geqslant 0} J_n \Psi\). Then by Proposition 1.2.2 of [20] \(\mathrm{D} J_n \Psi = J_{n - 1} \mathrm{D} \Psi\), and knowing that \(LJ_n \Psi = - nJ_n \Psi\) we obtain the commutation property

for every \(\alpha > 0\), \(j > 0\). The above formula holds also for \(j = 0\) if \(\mathbb{E} (\Psi) = 0\).

The results we have recalled so far let us write an \(n\)th-order Wiener chaos expansion for a random variable in \(\mathbb{D}^{n, 2}\):

Proof. We have for any \(\Psi \in L^2 (\Omega)\), since \(L = - \delta \mathrm{D}\) ([20], Proposition 1.4.3):

where we used (81), and the fact that \((1 - L)^{- 1} \mathrm{D} \Psi \in \operatorname{Dom} \delta\) is obvious by construction. This yields the first order expansion \(\Psi =\mathbb{E} (\Psi) + \delta (1 - L)^{- 1} \mathrm{D} \Psi\). Iterating the expansion up to order \(n\) we obtain (82). It is clear that \(J_0 \mathrm{D}^k \Psi \in \operatorname{Dom} \hspace{1em} \delta^k\) since \(J_0 \mathrm{D}^k \Psi\) is constant with values in \(H^{\otimes k}\). The second equality comes from the fact that \(\delta^k J_0 \mathrm{D}^k \Psi \in \mathcal{H}_k\) \(\forall k \in \mathbb{N}\), indeed \(\forall \Psi' \in \mathbb{D}^{k, 2}\):

In order to obtain \(L^p\) estimations of the remainder term \(\delta^n G_{[1]}^{[n]} \mathrm{D}^n \Psi\) generated by expansion (82), we used the following lemmas:

Remark 39. Let \(V\) be a real separable Hilbert space. For every \(v \in V\) and every \(\Psi \in \mathbb{D}^{q, 2} (H^{\otimes q})\) with \(q \in \mathbb{N}\) we have \(\Psi \otimes v \in \operatorname{Dom} \delta^q\) and

Indeed, notice that for every smooth \(\Psi' \in \mathbb{D}^{q, 2} (V)\) and every smooth \(\Psi \in \mathbb{D}^{q, 2} (H^{\otimes q})\) we have

Now since \(\mathrm{D}^q (\Psi \otimes v) = \mathrm{D}^q \Psi \otimes v\) and \(\Psi \in \mathbb{D}^{q, 2} (H^{\otimes q})\), we have \(\Psi \otimes v \in \mathbb{D}^{q, 2} (H^{\otimes q} \otimes V)\). Lemma 38 yields the bound \(\| \delta^q (\Psi \otimes v) \|_{L^2 (V)} \lesssim \| \Psi \otimes v \|_{\mathbb{D}^{q, 2} (H^{\otimes q} \otimes V)}\) which allows to pass to the limit for \(\Psi'\) and \(\delta^q (\Psi \otimes v)\) in \(L^2 (V)\).

Proof. First notice that we can suppose w.l.o.g. \(\mathbb{E} (\Psi) = 0\) thanks to (81). Therefore we can write \(\mathrm{D} (j - L)^{- \frac{1}{2}}\) as

with \(C = - \sqrt{- L}\). We decompose the second part as \(- C (j - L)^{- 1 / 2} \Psi = \sum_{n = 1}^{\infty} \left( \frac{n}{j + n} \right)^{1 / 2} J_n \Psi = T_{\phi} \Psi\), with \(T_{\phi} \Psi := \sum_{n = 0}^{\infty} \phi (n) J_n \Psi\). We apply Theorem 1.4.2 of [20] to show that \(T_{\phi}\) is bounded in \(L^p\), indeed \(\phi (n) = h (1 / n)\) and \(h (x) = (jx + 1)^{- 1 / 2}\) which is analytic in a neighbourhood of 0. Finally, we can apply Proposition 1.5.2 of [20] to show that \(\mathrm{D} C^{- 1}\) is bounded in \(L^p\), thus concluding the proof.\(\Box\)

The two lemmas above give the following immediate corollary:

The next lemma is one of the most useful tools of this paper. It allows us to write products of decompositions of the type (82) as sums of iterated Skorohod integrals. From now on we will note \(\langle \cdot, \cdot \rangle_{H^{\otimes r}}\) the \(r\)-th contraction, which to avoid inconsistency has to be taken between symmetric tensors. We also note \(h^{\odot n}_{v_1, \ldots, v_n} := h_{v_1} \odot \ldots \odot h_{v_n}\) for \(h_{v_1}, \ldots, h_{v_n} \in H\).

Remark 44. In the special case \(v = g (W (h_v)) h^{\otimes n}_v\) eq. (83) takes the form

which is just a generalization to Skorohod integrals of the multiplication formula for multiple Wiener integrals ([23],[26]). We can write the above formula more explicitly as

Remark 45. Note that one can assume w.l.o.g. the argument of \(\delta^{m + n - q - r}\) in (83) to be symmetric, and this would allow to iterate Lemma 43.

Remark 46. We can give the following intuition for the second formula in Lemma 43. The random variables \(u\) and \(v\) have an infinite chaos decomposition, and following the tree-like notation of [6] or [8] they could be thought of as having an infinite number of leaves which need to be contracted with each other.

It is apparent that the index \(i\) in the second equation denotes contractions between the already existing leaves of the trees \(u, v\). The indexes \(r\) and \(q\) count new leaves in each vertex that are created by the Malliavin derivatives, which are then contracted with other leaves from the other tree. There are then \(m + n - r - q - 2 i\) overall unmatched leaves which are arguments to the iterated Skorokhod integral.

The more intuitive interpretation of the second equation in Lemma 43 is the reason why we gave two distinct expression for the same quantity. Nevertheless, the formula (83) is more practical in the calculations and is more widely used throughout the paper.

Proof. (Lemma 43) Using Cauchy-Schwarz inequality and Lemma 38 we can show that \(\langle D^r \delta^n (v), \delta^j (u) \rangle_{H^{\otimes r}} \in L^2 (\Omega, H^{\otimes m - j - r})\) for every \(0 \leqslant r + j \leqslant m\). \(\)Then we apply Lemma 47 to get:

Using the commutation formula (86) we rewrite the r.h.s. as

We obtain

and using again Lemma 47 we obtain

where we used \(\delta^k (\Psi) h^{\otimes n - r} = \delta^k (\Psi \otimes h^{\otimes n - r})\) for \(\Psi \in \operatorname{Dom} \delta^k\), as seen in Remark 39. Substituting this expression into \(\delta^m (u) \delta^n (v)\) we get

where we set \(A_{m, n, r, i, \ell} := \binom{m}{i} \binom{n}{r} \binom{m - i}{\ell} \binom{r}{i} i!\) and \(J_{m, n} := \{ (r, i, \ell) \in \mathbb{N}^3 : 0 \leqslant r \leqslant n, 0 \leqslant i \leqslant r \wedge n, 0 \leqslant \ell \leqslant m - i \}\).

In order to complete the proof we just have to perform some basic changes of indexes. Taking \(q = \ell + i\) and noting that \(\binom{m}{i} \binom{m - i}{q - i} = \binom{q}{i} \binom{m}{q}\) we have \(A_{m, n, r, i, \ell} = \binom{m}{q} \binom{q}{i} \binom{n}{r} \binom{r}{i} i!\) and this yields (83).

Finally, we perform the change of variables \(q - i \rightarrow q\), \(r - i \rightarrow r\) to get the second formula.\(\Box\)

We give below the results we used to prove Lemma 43.

Remark 48. Note that

where \(\mathopen{\unicode{x27E6}} \cdot \mathclose{\unicode{x27E7}}\) stands for the Wick product. Indeed \(\forall \Psi \in \mathbb{D}^{1, 2}\) we know that \(\mathbb{E} [\delta (h^{\otimes n}) \Psi] =\mathbb{E} [W (h) h^{\otimes n - 1} \Psi]\) using the definition of \(\delta\), and then \(\delta^n (h^{\otimes n}) = \delta^{n - 1} (W (h) h^{\otimes n - 1})\). From Lemma 47 we have, since \(\mathrm{D} W (h) = h\):

which gives by induction \(\delta^n (h^{\otimes n}) = \mathopen{\unicode{x27E6}} W^n (h) \mathclose{\unicode{x27E7}}\).

Proof. If \(j = 0\), \(k = 1\) or \(k = 0, j = 1\) eq. (86) is trivial. Let \(j = k = 1\) and \(u \in \mathbb{D}^{2, 2} (H) \subset \mathbb{D}^{1, 2} (H)\). We can apply Proposition 1.3.2 of [20] to obtain \(\langle \mathrm{D} \delta (u), h \rangle = \langle u, h \rangle + \delta (\langle \mathrm{D} u, h \rangle)\) \(\forall h \in H\). Since by hypothesis \(\mathrm{D} u\) is symmetric we have \(\delta (\langle \mathrm{D} u, h \rangle) = \langle \delta \mathrm{D} u, h \rangle\), and then \(\mathrm{D} \delta (u) = u + \delta \mathrm{D} u\). The proof by induction is easy noticing that \(\mathrm{D} \delta^j = \delta^j \mathrm{D} + j \delta^{j - 1}\).\(\Box\)

Bibliography