gubinelli@iam.uni-bonn.de

1.Introduction

We show how to renormalise a class of general quasilinear equations of which one of the simplest examples is the following parabolic SPDE:

with \(a : \mathbb{R} \rightarrow [\lambda, 1]\) for \(\lambda > 0\) a uniformly bounded \(C^3\) diffusion coefficient, and \(\| a^{(k)} \|_{L^{\infty}} \leqslant 1\) for \(k = 0, \ldots, 3\). We assume that \(\xi \in \mathscr{C} \hspace{.1em} ^{\alpha - 2} (\mathbb{T}^2)\) with \(2 / 3 < \alpha < 1\) where \(\mathscr{C} \hspace{.1em}^{\alpha} (\mathbb{T}^2)\) is the Besov space \(B^{\alpha}_{\infty, \infty} (\mathbb{T}^2)\). This would apply for example to the space white noise on \(\mathbb{T}^2\). In this case we only expect that \(u (t, \cdot) \in \mathscr{C} \hspace{.1em}^{\alpha} (\mathbb{T}^2)\) and the term \(a (u (t, \cdot)) \Delta u (t, \cdot)\) is not well defined when \(2 \alpha - 2 < 0\). Eq. (1) is a quasilinear generalisation of the two–dimensional periodic parabolic Anderson model (PAM).

Let us remark from the beginning that the framework developed in this paper allows to deal with a class of equations of the form

where \(a_1, a_2\) are sufficiently smooth non-degenerate coefficients and \(\xi (z, t, x)\) is a Gaussian process with covariance

with \(F\) a smooth function and \(Q\) a distribution of parabolic regularity \(\rho > - 4 / 3\). This includes as a special case the space white noise discussed before, but we could consider a time white noise with a regular dependence on the space variable, or some noise which is mildly irregular in space and time.

Moreover, the scalar character of the equation or of the non-linear diffusion coefficient will not play any specific role and we could consider vector–valued equations with general diffusion coefficients, provided the template problem (6) below is uniformly parabolic.

For the sake of clarity and simplicity we will discuss mainly the basic example (1) since this contains already most of the technical difficulties. The fact that one can handle models as general as (2) can be seen as a direct byproduct of the techniques we will introduce below. Let us state one simple result that can be obtained via the theory developed in this paper:

In order to devise a suitable formulation of eq. (1) and obtain a theory with \(u \in \mathscr{C} \hspace{.1em}^{\alpha}\) we decompose the non-linear diffusion term in the l.h.s. with the help of Bony's paraproduct [31] and write

with

where \(\prec, \succ\) are standard paraproducts and \(\circ\) denotes the resonant product (see Appendix A for the definitions). In (4) the l.h.s. is always well defined, irrespectively of the regularity of the function \(u\), and the problem becomes that of controlling the resonant product \(a (u) \circ \Delta u\) appearing in the r.h.s. . A key point of the analysis put forward below is that this term can be expected to be of regularity \(2 \alpha - 2 > \alpha - 2\), better than the leading term \(\xi \in \mathscr{C} \hspace{.1em}^{\alpha - 2}\).

Our approach can be described as follows. For an equation of the form

we consider at first a parametric “template” problem with constant coefficients \(\eta_1, \eta_2\)

where now \(\eta = (\eta_1, \eta_2)\)\(\) is a fixed vector in \(\mathbb{R}^2\). A nonlinear paraproduct \(\Pi_{\prec\!\!\!\prec}\) will allow us to modulate the parametric solution \(\vartheta (\eta)\) with the quantity \(a (u) = (a_1 (u), a_2 (u))\) in order to capture the most irregular part of the solution \(u\) itself. As a consequence, the paracontrolled Ansatz

will define a regular remainder term \(u^{\sharp}\) which solves a standard PDE. With this decomposition the resonant products appearing in eq. (4) can be estimated along the lines of the standard paracontrolled arguments introduced in [15], and all the arguments introduced there can be extended in a straightforward manner to the quasilinear setting.

Recently Otto and Weber [33] and Bailleul, Debussche and Hofmanová [7] investigated quasilinear SPDEs in the context of pathwise methods and in a range of regularities compatible with the ones we will consider in this paper.

• In [33] the authors obtained a local well-posedness result for equations of the form

  \(\displaystyle \partial_t u (t, x) - a (t, x) \partial_x^2 u (t, x) = f (u (t, x)) \xi (t, x), \qquad (t, x) \in \mathbb{T}^2\)

  where both space and time variables take values in a one dimensional periodic domain and their noise can be white in time but colored in space, essentially behaving like a distribution of parabolic regularity in \((- 4 / 3, 1)\). In order to do that they introduce a specific notion of modelled function and related estimates.

• Bailleul, Debussche and Hofmanová in [7] obtain local well-posedness for the generalised parabolic Anderson model equation

  \(\displaystyle \partial_t u (t, x) - a (u (t, x)) \Delta u (t, x) = g (u (t, x)) \xi (x) \qquad t \geqslant 0, x \in \mathbb{T}^2 .\)

  (7)

  The authors obtain the same result as the one presented in Section 6 of our work, without the machinery of nonlinear paraproducts introduced here, but using only the basic tools of paracontrolled analysis and some clever transformations.

Let us comment a bit on relations of our results with those contained in these two papers.

• The parametric controlled Ansatz of Otto and Weber [33] is the main source of inspiration for the present work. We think that it is a very deep and fundamental observation which is quite orthogonal to the development of an alternative theory for singular SPDEs which is the main aim of their work. Our paper shows that this idea survives outside their specific methodology, in particular that Safanov's approach to Schauder estimates is not necessary to handle quasilinear equations, and a relatively straightfoward extension of the paracontrolled approach is sufficient to encompass in a compact manner the results on quasilinear equations contained in their paper.

• While the approach of Bailleul, Debussche and Hofmanová [7] is more straightforward than ours, it has fundamental limitations. In particular we remark that the generalization of (7) to matrix-valued diffusion coefficients \((a_{i j})_{i, j}\), namely to equations of the form

  \(\displaystyle \partial_t u (t, x) - a_{i j} (u (t, x)) \frac{\partial^2}{\partial x_i \partial x_j} u (t, x) = g (u (t, x)) \xi, \qquad t \geqslant 0, x \in \mathbb{T}^2, \)

  (8)

  is out of reach of the techniques used in [7], while can be treated directly in our framework and by Otto and Weber's approach.

• Another interesting observation we make in the present work is that the parametric Ansatz can pervade without problems all the coefficients of the equation. In particular we can allow very general noise terms whose law itself depends on the solutions of the equation. This is one of the main novelties of this paper, not present in other works on singular SPDEs. See eq. (2) above and Section 7 below for a detailed explanation of this case.

• Let us mention that Otto and Weber's parametric Ansatz has predecessors in the theory of rough paths and in standard stochastic analysis. Non–linear versions of rough paths have been considered by one of the authors in order to study the Korteweg–de Vries equation [14]. Non–linear Young integrals were used by one of the authors in joint work with Catellier [12] to study the the regularising properties of sample paths of stochastic processes. See also the related work of Hu and Le [24] on differential systems in Hölder media. Relevant to this discussion of non-linear variants of rough paths is the work of Bailleul on rough flows [8] and their application to homogeneisation [6]. By looking at the composition \(f (g (x))\) as the action of the distribution \(\delta_{g (x)}\) on the function \(f\), non–linear constructions can be linearised at the price of working in infinite–dimensional spaces: this is the approach chosen by Kelly and Melbourne to avoid non–linear generalisations of rough path theory in their study of homogeneisation of fast–slow system with random initial conditions [26]. It is also worth mentioning Kunita's theory of semimartingale vector fields [27], which occupies a place in stochastic analysis quite similar to that occupied by these recent developments in the landscape of rough paths/paracontrolled distributions theories. A non–linear generalisation of the classic bilinear paraproducts already appeared in the notion of paracomposition introduced by Alinhac [10].

Usefulness of paraproducts in the analysis of non-linear PDEs is by now well established: see for example the seminal paper of Meyer [31], the early review of Bony [10], the recent books of Alinhac and Gérard [1] and Bahouri, Chemin, and Danchin [3]. Let us mention also the interesting paper of Hörmander [25] where paradifferential operators allow to bypass the Nash–Moser fixpoint theorem in some applications where the loss of regularity prevents straightforward use of standard Banach fixpoint theorem. The main observation in that paper is that, with the aid of paradifferential operators, it is possible to identify a “corrected” problem for which standard Banach fixpoint applies.

Paracontrolled calculus for singular SPDEs has beed introduced by Gubinelli, Imkeller and Perkowski [15] (see also the lecture notes [17]) and used to study various equations like the KPZ equation [16], the dynamic \(\Phi^4_3\) model [11] in \(d = 3\) and its global well–posedness [32], the spectrum of the continuous Anderson Hamiltonian in \(d = 2\) [2]. By using heat–semigroup techniques paracontrolled calculus has been extended to the manifold context by Bailleul and Bernicot [4].

Paracontrolled calculus is currently limited to “first order” computations. This limitation is also ubiquitous in the present work. Even if, in practice, this is not a big issue, and the calculus is still able to deal with a large class of problems, it makes the paracontrolled approach less appealing for a general theory of singular SPDEs. Let us remark that recently Bailleul and Bernicot [5] developed an higher order version of the paracontrolled calculus. However, apart from these recent development, whose impact is still to be assessed, the most general theory for singular SPDEs has been developed by Hairer [19, 20, 13] under the name of regularity structures theory. Regularity structures are a vast generalisation of Lyons' rough paths [28, 30, 29] which give effective tools to describe non-linear operations acting on certain spaces of distributions, their renormalization by subtraction of local singularities and their use to solve singular SPDEs. Regularity structures have been successfully applied to all the models mentioned so far [19, 18], to other models like the Sine–Gordon model [22] (which however can also be handled via paracontrolled techniques) and to study weak universality conjectures [21, 23]. In their current instantiation it does not seem possible to solve quasilinear SPDEs via regularity structures. The results of the present paper hint to the fact that a non-linear version of regularity structures is conceivable, at least in principle. Indeed one can imagine models depending on additional parameters and modelled distributions acting as evaluations of the parametric models at certain space–time dependent values of the parameters. It would be interesting to pursue further this intuition.

The structure of the paper is the following. In Section 2 we introduce our basic tools: the non-linear paraproduct decomposition and some related commutation lemmas. In Section 3 we introduce the paracontrolled Ansatz which allows to transform the singular problem (1) into a well–behaved PDE. In Section 4 we discuss a-priori estimates, the uniqueness of the solution of the transformed PDE and its continuity w.r.t. the random data and the initial condition. We also introduce the algebraic structure which allows to renormalise the model. Section 5 deals with the renormalization of the stochastic data and the construction of the enhanced noise associated to white noise. Section 6 deals with the extension of the results to the more general equation (7). In Section 7 we develop a paracontrolled structure for equation 2 and define its renormalised enhanced noise. Finally, in Appendix A we review some reference material on Besov spaces and linear paraproducts, and prove some technical lemmas.

Notations.

Moreover for convenience we denote \(\mathscr{C} \hspace{.1em}_T^{\alpha} := C_T \mathscr{C} \hspace{.1em}^{\alpha}\). We will avoid to write explicitly the time span \(T\) whenever this does not cause ambiguities. We will need also spaces for functions of \((\eta, t, x)\) where \(\eta\) is an additional parameter in \([\lambda, 1]\) for \(\lambda \in (0, 1)\) which we denote \(C_{\eta}^k V\) with the norm

where \(V\) is a Banach space of functions on \([0, T] \times \mathbb{T}^2\): in our case \(V = \mathscr{C} \hspace{.1em}_T^{\alpha}\) or \(V = \mathscr{L} \hspace{.2em}^{\alpha}_T\).

We will denote by \(K_{i, x} (y) = 2^{2 i} K (2^i (x - y))\) the kernel of the Littlewood–Paley operator \(\Delta_i\) and \(Q_{i, t} (s) = Q_i (t - s) = 2^{2 i} Q (2^{2 i} (t - s))\) a smoothing kernel at scale \(2^{2 i}\) in the time direction where \(Q\) is a smooth, positive function with compact support in \(\mathbb{R}_+\) and mass \(1\). We introduce also the shortcut \(P_{i, x} = K_{< i - 1, x} = \sum_{j < i - 1} K_{i, x}\). Another notation shortcut widely used in this article is to write \(\int_{x, y}\) for integrals on \(\mathbb{T}^2\) or \(\mathbb{R}\) with respect to the measures \(\mathrm{d} x\) and \(\mathrm{d} y\) without specifying the integration bounds, whenever this does not create ambiguity. Finally, we will note \(\delta f^{t x}_{s y} = f (t, x) - f (s, y)\) and \(\delta_{\tau} f_{s y}^{t x} = f (s, y) + \tau (f (t, x) - f (s, y))\) for \(\tau \in [0, 1]\).

2.Nonlinear paraproducts

In this section we introduce nonlinear paraproduct and some related results that will be used in Section 3 to analyse equation (1).

Let \(g : [0, T] \times \mathbb{T}^2 \rightarrow \mathbb{R}\), and \(h : \mathbb{R} \times [0, T] \times \mathbb{T}^2 \rightarrow \mathbb{R}\) be smooth functions. We can decompose the composition \(h (g (\cdot), \cdot)\) via nonlinear paraproducts as follows. Define

This gives a map

that can be uniquely extended to

thanks to the following bounds:

Proof. Using the fact that

and

we obtain the bounds on \(\Pi_{\prec} (g, h)\), \(\Pi_{\succ} (g, h)\), \(\Pi_{\circ} (g, h)\) and \(\Pi_{\prec} (g_1, h) - \Pi_{\prec} (g_2, h)\). We proceed to estimate the term \(\Pi_{\succ} (g_1, h) - \Pi_{\succ} (g_2, h)\). We will use the following notation for brevity:

Then

With the same reasoning we can bound the norm of \(\Pi_{\circ} (g_1, h) - \Pi_{\circ} (g_2, h)\).\(\Box\)

We will need the following time-smoothed nonlinear paraproduct

with \(Q \in C^{\infty}_c (\mathbb{R})\) a kernel with total mass \(1\) as specified in the introduction, and \(Q_{i, t} (s) := 2^{2 i} Q (2^i (t - s))\). In (15) we use the convention that a continuous function \(t \mapsto g (t)\) on \(\mathbb{R}_+\) is extended to \(\mathbb{R}\) by defining \(g (t) = g (0)\) for \(t \leqslant 0\). This preserves the Hölder norms of index in \([0, 1]\). The modified nonlinear paraproduct enjoys similar bounds to the regular one.

Proof. The norm \(\| \Pi_{\prec\!\!\!\prec} (g, h) \|_{C_T \mathscr{C} \hspace{.1em}^{\gamma}}\) can be treated in the same way as in Lemma 2. We estimate \(\| \Pi_{\prec\!\!\!\prec} (g, h) \|_{C_T^{\gamma / 2} \mathscr{C} \hspace{.1em}^0}\) as follows:

The second inequality can be obtained easily with the same techniques used so far.\(\Box\)

Remark 4. Using the Fourier support properties of the kernel \(P_{q, x} (\cdot)\) it is easy to see that it has mass \(1\), i.e. \(\int_y P_{q, x} (y) = 1\) \(\forall x \in \mathbb{T}^2, q \in \mathbb{N}\). Then the paraproducts (11) and (15) can be written as follows for a constant smooth function \(g (t, x) = \bar{g}\), \(\forall (t, x) \in \mathbb{R} \times \mathbb{T}^2\):

2.1.Nonlinear commutator.

The next technical ingredient is a commutator lemma between the non-linear paraproduct of (15) and the standard resonant product. Since it will be needed below to analyse a term of the form \(\Pi_{\prec\!\!\!\prec} (g, h) \circ \Delta \Pi_{\prec\!\!\!\prec} (g, h)\), we will specialise our discussion to this specific structure. Notice that in the following the various space–time operators act pointwise in the parameter \(\eta\), in the sense that, for example:

Proof. Let \(A (t, z) := \Pi_{\prec\!\!\!\prec} (g, h) \circ \Delta \Pi_{\prec\!\!\!\prec} (g, h) (t, z)\) . We can approximate \(A (t, z)\) with its value for a fixed \(g = g (t, z)\), to obtain

with the remainders (16) and (17). To estimate the first term notice that

and that, by Remark 4

This yields

We proceed then to estimate (16) and (17). We obtain

Using Lemma 3 we have

Lemma 6 gives

and thus (16) is bounded by \(2^{- (2 \gamma + \rho - 2 - \varepsilon) q}\)\(\| g \|_{\mathscr{L} \hspace{.2em}^{\rho}_T} \| h \|^2_{\mathscr{C} \hspace{.1em}^{\gamma}_{_T}}\).

We can easily bound (17) in the same way, and this proves the first inequality. For the second inequality, Lemma 3 yields

and using the second inequality of Lemma 6 we obtain the desired bound.

The extension of \(\Lambda\) to \(\mathscr{L} \hspace{.2em}^{\rho}_T \times C_{\eta}^2 \mathscr{L} \hspace{.2em}_T^{\gamma}\) is standard (see e.g. the proof of the commutator lemma in [15], Lemma 2.4). \(\)\(\Box\)

Proof.

where we used the notation \(\delta g^{s y}_{t z} = g (s, y) - g (t, z)\), \(\delta_{\tau} g^{s y}_{t z} = g (t, z) + \tau (g (s, y) - g (t, z))\) and Lemma 22. This proves the first bound.

The second inequality can be obtained in the same way with the techniques already used here and in Lemma 2.\(\Box\)

2.2.Approximate paradifferential problem.

In this section we construct an approximate solution to the equation

with data \(f \in \mathscr{C} \hspace{.1em}^{\gamma - 2}\) and \(g \in \mathscr{L} \hspace{.2em}^{\rho}_T\), for some fixed \(\rho, \gamma \in (0, 1)\). The idea is to obtain it via a certain class of paradifferential operators. We introduce the operator \(\mathscr{L} \hspace{.2em}\) acting on functions of \((\eta, t, x)\) by

We will also use the notation \(\mathscr{L} \hspace{.2em}_{\eta_1} := \partial_t - \eta_1 \Delta\). \(\)

Observe that if \(u\) does not depend on \(\eta\) we can define

and from definition (11) with \(h = \mathscr{L} \hspace{.2em} u\) we obtain \(\Pi_{\prec} \left( g, \mathscr{L} \hspace{.2em} \right) u = \partial_t u - g \prec \Delta u\).

We can describe the commutation between the differential operator \(\mathscr{L} \hspace{.2em}\) and the paraproduct \(\Pi_{\prec\!\!\!\prec} (g, \cdot)\) via the following estimate:

Proof. We start considering \(g \in C^{\infty} ([0, T], \mathbb{T}^2)\) and \(U \in C_{\eta}^2 C^{\infty} ([0, T], \mathbb{T}^2)\), and prove \(\textrm{(\hyperref[e:quasidef-psi]{22})}\) in this setting. Notice that \(\Pi_{\prec\!\!\!\prec} \left( g (t, y), \mathscr{L} \hspace{.2em}_{g (t, y)} U \right) = \mathscr{L} \hspace{.2em} U (g (t, y))\). As a consequence, we can estimate

with the commutators

We have

with the definition

and

with the definition

Indeed:

This shows that \(\textrm{(\hyperref[e:quasidef-psi]{22})}\) holds for smooth functions.

With the techniques used in Lemma 6 we can estimate

By the spectral support properties of the commutators we have that

and

This yields

We have so far proved (21) and then (22) follows by continuity. The local Lipshitz dependence on \(g\) can be obtained via similar computations.\(\Box\)

Remark 8. If \(f\) does not depend on \(\eta\) we consider the parametric problem

which is solved by

Remark that

We have, thanks to the well-known Schauder estimates of Lemma 21 (since \(\eta \geqslant \lambda\)):

We define then

and observe that \(u (t, x)\) is an approximate solution of equation (18), indeed

and the estimation in Lemma 7 together with the bound (27) yield immediately the following inequality:

3.Paracontrolled Ansatz

In order to give a meaning to the PDE in (4) with initial condition \(u_0 \in \mathscr{C} \hspace{.1em}^{\alpha}\), our initial goal will be to get informations on solutions \(\theta = \theta (g)\) of the equation

for a fixed \(g \in \mathscr{C} \hspace{.1em}^{\alpha}_T\), \(2 / 3 < \alpha < 1\), \(g \in [\lambda, 1]\). Using the results of Section 2.2, we consider to this effect the parametric problem

for \(\eta \in [\lambda, 1]\). We will consider the stationary solution of this problem which has the form

and in order for (30) to be well defined we impose that the noise \(\xi\) has zero mean on \(\mathbb{T}^2\) (this is a simplifying assumption which can be easily removed, e.g. at the price of adding a linear term to the equation). We can control (30) by bounding its Littlewood-Paley blocks with a Bernstein lemma for distributions with compactly supported Fourier transform ([3], Lemma 2.1) to obtain:

We define now for every \(t \in [0, T]\)

Thanks to Lemma 3 we have the bound \(\| \theta \|_{\mathscr{L} \hspace{.2em}^{\alpha}_{_T}} \lesssim \| \vartheta \|_{C_{\eta} \mathscr{L} \hspace{.2em}^{\alpha}_{_T}} \lesssim \| \xi \|_{\mathscr{C} \hspace{.1em}^{\alpha - 2}}\) . We observe that this definition together with Lemma 7 gives

with \(\| \Psi (a (u), \vartheta) \|_{\mathscr{C} \hspace{.1em}^{2 \alpha - 2 - \varepsilon}_T} \lesssim \| a (u) \|^2_{\mathscr{L} \hspace{.2em}^{\alpha}_T} \| \xi \|_{\mathscr{C} \hspace{.1em}^{\alpha - 2}_T}\) . We expect then \(\Psi (a (u), \vartheta)\) to be bounded in \(\mathscr{C} \hspace{.1em}^{2 \alpha - 2 - \varepsilon}_T\) for any \(\varepsilon > 0\). At this point let us introduce the Ansatz

Remark 9. Notice that we are not making any assumption on the existence of such \(u\), which is the subject of Section 4. Our aim here is to find the equation that a couple \((u, u^{\sharp}) \in \mathscr{C} \hspace{.1em}_T^{\alpha} \times \mathscr{C} \hspace{.1em}_T^{2 \alpha}\) verifying (32) must solve, in order for \(u\) to solve (4).

Observe that

It follows that \(u^{\sharp}\) must solve

with \(\Phi (u) = a (u) \circ \Delta u + a (u) \succ \Delta u\), and if we can make sense of the resonant term \(a (u) \circ \Delta u\), it is reasonable to expect \(u^{\sharp} (t, \cdot) \in \mathscr{C} \hspace{.1em}^{2 \alpha}\) \(\forall t \in (0, T]\). Indeed, take \(U^{\sharp} := U_Q\) to be the solution of

for some \(Q = Q (u^{\sharp})\) to be determined and \(\eta \in [\lambda, 1]\). Using again Lemma 7 as shown in Remark 8 we have

For \(\eta \in [\lambda, 1] \)we define \(\mathscr{P} \hspace{.1em}_t u_0^{\sharp} (\eta) := e^{\eta \Delta t} u_0^{\sharp}\) so that \(\mathscr{L} \hspace{.2em} \left( \mathscr{P} \hspace{.1em}_t u_0^{\sharp} \right) = 0\), with \(\mathscr{L} \hspace{.2em}\) as in (19).

We set

Taking

we obtain that \(U^{\sharp}\) solves equation (34) if and only if \(u^{\sharp}\) solves equation (33). As we will see, \(Q (u^{\sharp}) (t)\) belongs to \(\mathscr{C} \hspace{.1em}^{2 \alpha - 2}\) \(\forall t \in (0, T]\) but not uniformly as \(t \rightarrow 0\). However it belongs to \(\mathscr{C} \hspace{.1em}^{\alpha - 2}\) uniformly as \(t \rightarrow 0\).

It remains to control the resonant term \(a (u) \circ \Delta u\) appearing in \(\Phi (u)\). We have

By paralinearization (see Theorem 25) \(a (u) = a' (u) \prec u + R_a (u)\) with \(\| R_a (u) \|_{\mathscr{C} \hspace{.1em}^{2 \alpha}_T} \lesssim 1 + \| u \|^2_{\mathscr{C} \hspace{.1em}^{\alpha}_T}\), and then

In order to use the commutator lemma (Lemma 26) we can estimate \(a' (u)\), recalling that \(\alpha \in (0, 1)\), as

and write

Then, Ansatz (32) gives

Summarizing, we have:

Thanks to the nonlinear commutator (Lemma 5), we can decompose the resonant term \(\theta \circ \Delta \theta\) to obtain

and \(\Lambda (a (u), \vartheta) \in \mathscr{C} \hspace{.1em}^{3 \alpha - 2 - \varepsilon}_T\) if \(u \in \mathscr{L} \hspace{.2em}^{\alpha}_T\). Here we defined

Finally, recalling the decomposition of \(u^{\sharp}\) in two terms (35) we obtain

where

Thanks to Lemma 2 the terms \(a' (u) \Pi_{\diamondsuit} (a (u), \Theta_2)\) and \(\Phi_1 (u)\) can be estimated in \(\mathscr{C} \hspace{.1em}^{2 \alpha - 2}_T\), provided \(\Theta_2 \in C^2_{\eta} \mathscr{C} \hspace{.1em}^{2 \alpha - 2}_T\) (see Section 5). On the other hand the term \(\Phi_2 (u) (t)\) can be estimated in \(\mathscr{C} \hspace{.1em}^{2 \alpha - 2}\) only for strictly positive times \(t > 0\) due to the lack of regularity of the initial condition \(u_0^{\sharp}\) which a priori lives only in \(\mathscr{C} \hspace{.1em}^{\alpha}\).

Note moreover that the specific form of \(\Phi\) allows to deduce that if we replace \(\Theta_2\) by \(\tilde{\Theta}_2 = \Theta_2 - H\) with \(H \in C_{\eta}^2 \mathscr{C} \hspace{.1em}^{2 \alpha - 2}_T\) then this is equivalent to consider an equation for \(u\) of the form

Let us resume this long discussion in the following theorem:

We call the elements in \(\mathcal{X}^{\alpha}\) enhanced noises. In the next section we will exploit the space \(\mathcal{X}^{\alpha}\) for \(2 / 3 < \alpha < 1\) to solve equations (38) and (32).

4.Local wellposedness

The main result of this section is the local well–posedness for equations (32) and (38) when \((\vartheta, \Theta_2) \in \mathcal{X}^{\alpha}\) and \(u_0 \in \mathscr{C} \hspace{.1em}^{\alpha}\) for \(2 / 3 < \alpha < 1\). This yields a unique solution to (37), thanks to Theorem 10.

Remark 13. The proof is based on a Picard fixed point argument. In order to have a contraction map on a small time interval \([0, T]\), we carry on our analysis of \(\)\(U^{\sharp}\) in the space \(C_{\eta}^2 \mathscr{L} \hspace{.2em}^{2 \delta}_T \supset C_{\eta}^2 \mathscr{L} \hspace{.2em}^{2 \alpha}_T\) and make use of the estimates of Lemma 21 to obtain a factor \(T^{\varepsilon}\) for some (small) \(\varepsilon > 0\).

Proof. (Theorem 12) Let \(\mathcal{G}_T = \mathscr{L} \hspace{.2em}^{\alpha}_T \times C_{\eta}^2 \mathscr{L} \hspace{.2em}^{2 \delta}_T\). We introduce the map

by

and

We will establish that this map is a contraction in the space \(\mathcal{G}_T .\)

First, we have to show that there exists a ball \(B \subset \mathcal{G}_T\) such that \(\Gamma (B) \subseteq B\). We have the bound \(\left\| \mathscr{P} \hspace{.1em} u_0^{\sharp} \right\|_{C^1_{\eta} \mathscr{L} \hspace{.2em}^{\alpha}_T} \lesssim \| u^{\sharp}_0 \|_{\mathscr{C} \hspace{.1em}^{\alpha}_T}\) . It is easy to obtain, using the estimates of Section 2 and Lemma 21:

for some \(\kappa > 0\).

By the assumption that \((\vartheta, \Theta_2) \in \mathcal{X}^{\alpha}\) we deduce that there exists \(M > 0\) such that \(\| \Theta_2 \|_{C_{\eta}^2 \mathscr{C} \hspace{.1em}_T^{2 \alpha - 2}} \leqslant M\). We have

To bound the term \(\Phi_2 (u)\) we observe that \(\left\| \mathscr{P} \hspace{.1em}_t u^{\sharp}_0 \right\|_{C_{\eta}^2 \mathscr{C} \hspace{.1em}^{2 \alpha}} \lesssim t^{- \frac{\alpha}{2}} \| u^{\sharp}_0 \|_{\mathscr{C} \hspace{.1em}^{\alpha}}\) thanks to Lemma 21. This gives

and then \(\Gamma_{U^{\sharp}} (u, U^{\sharp})\) is bounded in \(C^2_{\eta} \mathscr{L} \hspace{.2em}_T^{2 \delta}\) for \(T\) small enough. We have also

and these bounds show that \(\Gamma (B) \subseteq B\). The contractivity of \(\Gamma_{U^{\sharp}} (u, U^{\sharp})\) can be obtained in the same way. Now consider \(\Gamma_u (u, U^{\sharp})\): we have

while for the other terms in \(\Gamma_u (u_1, U^{\sharp}_1) - \Gamma_u (u_2, U^{\sharp}_2)\) we remark that

Then \(\forall 0 < \varepsilon < \alpha\), using Lemma 3 and Lemma 22:

With the same reasoning \(\)we estimate

and then \(\Gamma\) is a contraction for small times.

The uniqueness of the solution \((u, U^{\sharp}) \in \mathscr{L} \hspace{.2em}^{\alpha}_T \times C_{\eta}^2 \mathscr{L} \hspace{.2em}^{2 \delta}_T\) and the Lipshitz continuity of the localized solution map \(\Sigma_{\tau}\) can be proved along the same lines via standard arguments.\(\Box\)

5.Renormalization

At this point we want to construct an enhanced noise \(\Xi\) associated to the white noise \(\xi\). Already in the standard setting of the generalised PAM model with constant diffusion matrix, the construction of the enhancement requires a renormalization since the resonant product \(\vartheta \circ \Delta \vartheta\) is not well defined.

Let \(\psi \in \mathcal{S} (\mathbb{T}^2)\) be a cutoff function and let \(\psi_{\varepsilon} (x) = \varepsilon^{- 2} \psi (x / \varepsilon)\). Then define a regularised noise by \(\xi_{\varepsilon} = \psi_{\varepsilon} \ast \xi\) and let \(\vartheta_{\varepsilon} = (- \eta \Delta)^{- 1} \xi_{\varepsilon}\). Notice that

where

as \(\varepsilon \rightarrow 0\). Subtracting the diverging quantity \(H_{\varepsilon}\) to \(\vartheta_{\varepsilon} \circ \Delta \vartheta_{\varepsilon}\) and then taking the limit as \(\varepsilon \rightarrow 0\) delivers a finite result.

Proof. The proof is a mild modification of the proof for PAM [15]. In order to establish the required \(C^2_{\eta} \mathscr{C} \hspace{.1em}^{2 \alpha - 2}_T\) regularity for \(\Xi_2\) we follow the computations for the case where the diffusion coefficient is constant. We only have to discuss the additional regularity in the parameter \(\eta\). In order to do so observe that

where \(\mathopen{\unicode{x27E6}} \mathclose{\unicode{x27E7}}\) denotes the Wick product with respect to the Gaussian structure of \(\xi\). Then we have

and

Now the computations relative to the regularities of these additional stochastic objects are equivalent to those for the term \(\Xi_{2, \varepsilon}\) where one or two instances of \(\vartheta_{\varepsilon} (\eta)\) are replaced by Gaussian fields of similar regularities of the form \(\partial_{\eta} \vartheta_{\varepsilon} (\eta)\) and \(\partial_{\eta}^2 \vartheta_{\varepsilon} (\eta)\). A direct inspection of the proof contained in [15] allows us to deduce that we have almost sure \(\mathscr{C} \hspace{.1em}^{2 \alpha - 2}\) regularity for these terms and also for random fields \(\partial_{\eta}^n {\Xi_{2, \varepsilon}} \) for any finite \(n\). This allows also to deduce that the random field is almost surely smooth in the parameter \(\eta\). Similar computations allow to prove continuity in \(\varepsilon\) for \(\varepsilon > 0\). The rest of the proof is standard.\(\Box\)

In conclusion we see that in order to be able to use this convergence result we need to modify our approximate PDE and consider instead

which gives the renormalised equation (3).

Our well–posedness results for the paracontrolled formulation of this equation together with the convergence result of Theorem 14 allow to deduce that \(u_{\varepsilon} \rightarrow u\) in \(\mathscr{C} \hspace{.1em}^{\delta}_T\) for any \(2 / 3 < \delta < \alpha < 1\) and that the limiting process \(u\) satisfies a modified version of eq. (1), namely

where \(a (u) \diamond \Delta u\) denotes a renormalized diffusion term given by

6.Nonlinear source terms

Let us start by discussing the presence of a \(u\) dependent r.h.s. in eq. (1). We want to solve

where \(a_1\) is a non-linear diffusion coefficient as before and \(a_2 : \mathbb{R} \rightarrow \mathbb{R}\) is another bounded function with sufficiently many bounded derivatives. We rewrite this equation as

where now \(a (u) = (a_1 (u), a_2 (u))\) is a vector valued non-linearity. Since we don't need \(u\) to depend on any parameter \(\eta = (\eta_1, \eta_2)\), we have defined \(\mathscr{L} \hspace{.2em}\) as

and used the identity \(\Pi_{\prec} \left( a (u), \mathscr{L} \hspace{.2em} \right) u = (\partial_t - a_1 (u) \prec \Delta) u\), similarly to what we have done in (20).

Notice that the non–linear paraproduct can be extended trivially to the vector valued case in such a way that, for example,

As before we make the Ansatz

where now \(\vartheta\) solves

for \(\eta = (\eta_1, \eta_2) \in [\lambda, 1] \times [- L, L]\) where \(L\) is a large but fixed constant. The bounded domain is important to be able to have uniform estimates and reuse the estimates proved above in the simple situation of \(\eta_2 = 1\). The solution of this equation is

Observe that

and recall that by Lemma 7

Now

with \(\Xi (\eta) (t, x) = \eta_2 \xi (x)\) and then

In conclusion

and the equation for \(u^{\sharp}\) reads

where now all the terms on the r.h.s. can be considered remainder terms. Let us just remark that the commutation term \(a_2 (u) \prec \xi - a_2 (u) \prec\!\!\!\prec \xi\) can be handled easily via Lemma 27. Of course, the first two terms in the equation above require to be treated as resonant terms. Note that, modulo terms of order \(\mathscr{C} \hspace{.1em}^{3 \alpha - 2}_T\) (or \(\mathcal{E}^{\alpha / 2} \mathscr{C} \hspace{.1em}^{2 \alpha - 2}\) as defined in Lemma 21) the terms \(a_1 (u) \circ \Delta u + a_2 (u) \circ \xi\) are equivalent to

and that by computations similar to those of the previous sections one can prove that

so the resonant terms are comparable to the sum of the two terms

which require renormalization of the form

and the convergence follows with the same arguments of Section 5.

We remark that the structure of the second renormalisation term, which is due to the r.h.s. in the equation, is the same of that found by Bailleul, Debussche and Hofmanová in [7].

Remark 15. Our approach works straightforwardly for equation (8), namely

with \(a : \mathbb{R} \rightarrow M_2 (\mathbb{R})\) such that \(\sum_{i, j} a (u)_{i j} x_i x_j \geqslant C | x |^2\) \(\forall x \in \mathbb{R}^2\) for \(C > 0\) and \(\partial^2_{i j} := \frac{\partial^2}{\partial x_i \partial x_j}\).

To see that, let \(a (u) := (a_{i j} (u), g (u)) \in \mathbb{R}^5\) and \(\eta = (\eta_{i, j}, \eta_g) \in \mathbb{R}^5\). Let \(\mathscr{L} \hspace{.2em} (\eta) := \partial_t - \eta_{i j} \partial^2_{i j}\) and \(\Xi (\eta) := \eta_g \xi\) with the uniform ellipticity condition \(\sum_{i, j} \eta_{i j} x_i x_j \geqslant C | x |^2\) \(\forall x \in \mathbb{R}^2\). It is easy to verify that Lemma 7 and Lemma 5 hold within this setting, just considering nonlinear paraproducts for functions depending on 5 parameters. We have then:

with \(\vartheta (\eta)\) stationary solution of \(\mathscr{L} \hspace{.2em} \vartheta (\eta) = \Xi (\eta)\), \(\mathscr{P} \hspace{.1em}_t u_0^{\sharp} := e^{\eta_{i j} \partial^2_{i j} t} u_0^{\sharp}\) and \(U^{\sharp} (\eta)\) which solves

with \(Q (u, U^{\sharp}) \in \mathscr{C} \hspace{.1em}^{2 \alpha - 2 - \varepsilon}\), \(\Theta_1 \in C^2_{\eta} C^2_{\eta'} \mathscr{C} \hspace{.1em}^{2 \alpha - 2} = \vartheta (\eta) \circ \eta'_{i j} \partial^2_{i j} \vartheta (\eta)\), \(\Theta_2 (\eta, \eta') = \vartheta (\eta) \circ \eta'_g \xi\) and \(U^{\sharp} (t = 0) = 0\). Note that we can write \(\vartheta\) as

From the uniform ellipticity condition we have \(\| \vartheta \|_{C_{\eta}^k \mathscr{C} \hspace{.1em}^{\alpha}} \lesssim \| \xi \|_{\mathscr{C} \hspace{.1em}^{\alpha}}\), and Schauder estimates analogous to those of Lemma 21 hold as well.

Now consider the renormalization. We have

The convergence of \(\Theta_1^{\varepsilon} - H_1^{\varepsilon}\), \(\Theta_2^{\varepsilon} - H_2^{\varepsilon}\) in \(C_{(\eta, \eta')}^k \mathscr{C} \hspace{.1em}^{2 \alpha - 2} (\mathbb{T}^2)\) can be obtained with the techniques used in [15], Section 5.2.

7.Full generality

Within the framework of the present work we are actually able to treat equations of the form

where \(\xi (\eta_2, x)\) is a Gaussian process with covariance

where \(F\) is a smooth covariance function. Let as before \(2 / 3 < \alpha < 1\). In this case we can take as a parametric equation

whose solution \(\vartheta\) is a Gaussian process, smooth with respect to the variable \(\eta = (\eta_1, \eta_2)\) which we assume taking value in a compact subset of \(\mathbb{R}^2\) for which \(\eta_1 \geqslant \lambda > 0\) with fixed \(\lambda\). Letting \(a (u) = (a_1 (u), a_2 (u))\) we can rewrite the l.h.s. of eq. (41) in the form

and the r.h.s. as

where \(\Xi (\eta, x) = \xi (\eta_2, x)\). Now we perform the paraproduct decomposition to get

We have introduced here the parametric differential operator \(\mathfrak{D} (\eta) := \eta_1 \Delta\) for \(\eta = (\eta_1, \eta_2)\).

Let \(\mathscr{P} \hspace{.1em}_t (\eta) := e^{t \eta_1 \Delta}\) as before, and invoke the paracontrolled Ansatz in the usual form

Using that

and observing that we can take \(\mathscr{L} \hspace{.2em} \vartheta = \Xi\) and that \(\mathscr{L} \hspace{.2em} \mathscr{P} \hspace{.1em} u^{\sharp}_0 = 0\) to get

where

which is solved by \(U^{\sharp}\) satisfying

Indeed \(\Pi_{\prec} (a (u), F (u, U^{\sharp})) = F (u, U^{\sharp})\), since \(F (u, U^{\sharp})\) does not depend on the additional parameter. Remark that the term \(\Pi_{\prec} (a (u), \Xi) - \Pi_{\prec\!\!\!\prec} (a (u), \Xi)\), which does not appear in the simpler case, can be treated with Lemma 28.

It remains now to discuss the handling of the resonant products under the paracontrolled assumption, namely \(\Pi_{\circ} (a (u), \Xi)\) and \(\Pi_{\circ} (a (u), \mathfrak{D}u)\). Next lemma is a paralinearization result adapted to our non-linear context.

Proof.

and observe that the first term is equal to \(u \circ \Pi_{\prec} ((a (u), a' (u)), \mathcal{D}Z)\) while the second term can be easily estimated in \(\mathscr{C} \hspace{.1em}_T^{\gamma + 2 \rho}\). \(\Box\)

Using this result and Lemma 28 we can expand

and similarly, noting that

where \((\mathcal{D}\mathfrak{D}) (\eta, \eta') = \eta_1' \Delta\), we have

Finally the equation for \(U^{\sharp}\) reads

This can be solved essentially as we did in the simpler context. We see that the general enhancement has the form

which of course will require renormalization like we did before. In particular

where we used that \(\eta_1 \Delta \vartheta (\eta) = - \xi (\eta_2, \cdot)\). Now observe that

and that

with \(\partial_1 F\) denoting the derivative with respect to the first entry.

In the end the renormalised enhanced noise is obtained as the limit in \(\mathcal{X}^{\alpha}\) of \((\xi_{\varepsilon}, \Xi_{2, \varepsilon})\) where

with

We remark that if we take \(F (\eta_2, \tilde{\eta}_2) = \eta_2 \tilde{\eta}_2\) we obtain again the situation treated in Section 6. Indeed in this case

which coincides with (40).

Remark 17. Consider the more general equation (2), where the noise depends explicitly on time, e.g. with a covariance

with \(F\) a smooth function and \(Q\) a distribution of parabolic regularity \(\rho > - 4 / 3\). First note that the coefficient \(a_1 (u) \in [\lambda, 1]\) in front of the time derivative can be eliminated trivially by dividing.

In order to handle the time dependence of the noise, the framework of this paper will still apply, provided we consider space–time paraproducts instead of paraproducts which act only on the space variable. This can be done exactly following the lines of the paper [15], where time paraproducts are employed in the paracontrolled approach to solutions to SDE driven by gaussian signals.

The constraint of regularity \(\rho > - 4 / 3\) does allow to treat a noise which is white in time and smooth in space, but not a space–time white noise. It is well known that the first order paracontrolled approach on which the present paper is based does not allow to treat this kind of irregular signals in full generality.

Appendix A.Besov spaces and linear paraproducts

In this Appendix we collect some classical results from harmonic analysis needed in the paper. For a gentle introduction to Littlewood-Paley theory and Besov spaces see the recent monograph [3], where most of our results are taken from. There the case of tempered distributions on \(\mathbb{R}^d\) is considered. The Schauder estimates for the heat semigroup (Lemma 21) are classical and can be found in [15, 17].

Fix \(d \in \mathbb{N}\) and denote by \(\mathbb{T}^d = (\mathbb{R}/ (2 \pi \mathbb{Z}))^d\) the \(d\)-dimensional torus. We focus here on distributions and SPDEs on the torus, but everything in this Appendix applies mutatis-mutandis on the full space \(\mathbb{R}^d\), see [15]. The space of distributions \(\mathscr{D} \hspace{.1em}' = \mathscr{D} \hspace{.1em}' (\mathbb{T}^d)\) is defined as the set of linear maps \(f\) from \(C^{\infty} = C^{\infty} (\mathbb{T}^d, \mathbb{C})\) to \(\mathbb{C}\), such that there exist \(k \in \mathbb{N}\) and \(C > 0\) with

for all \(\varphi \in C^{\infty}\). In particular, the Fourier transform \(\mathscr{F} \hspace{.1em} f : \mathbb{Z}^d \rightarrow \mathbb{C}\), \(\mathscr{F} \hspace{.1em} f (k) = \langle f, e^{- i k \cdot} \rangle\), is defined for all \(f \in \mathscr{D} \hspace{.1em}'\), and it satisfies \(\left| \mathscr{F} \hspace{.1em} f (k) \right| \leqslant | P (k) |\) for a suitable polynomial \(P\). We will also write \(\hat{f} (k) = \mathscr{F} \hspace{.1em} f (k)\). Conversely, if \((g (k))_{k \in \mathbb{Z}^d}\) is at most of polynomial growth, then its inverse Fourier transform

defines a distribution, and we have \(\mathscr{F} \hspace{.1em}^{- 1} \mathscr{F} \hspace{.1em} f = f\) as well as \(\mathscr{F} \hspace{.1em} \mathscr{F} \hspace{.1em}^{- 1} g = g\). To see this, it suffices to note that the Fourier transform of \(\varphi \in C^{\infty}\) decays faster than any rational function (we say that it is of rapid decay). Indeed, for \(\mu \in \mathbb{N}^d_0\) we have \(| k^{\mu} \hat{g} (k) | = \left| \mathscr{F} \hspace{.1em} (\partial^{\mu} g) (k) \right| \leqslant \| \partial^{\mu} g \|_{L^1 (\mathbb{T}^d)}\) for all \(k \in \mathbb{Z}^d\). As a consequence we get the Parseval formula \(\left\langle f, \overline{\varphi} \right\rangle = (2 \pi)^{- d} \sum_k \hat{f} (k) \overline{\hat{\varphi} (k)}\) for \(f \in \mathscr{D} \hspace{.1em}'\) and \(\varphi \in C^{\infty}\).

Linear maps on \(\mathscr{D} \hspace{.1em}'\) can be defined by duality: if \(A : C^{\infty} \rightarrow C^{\infty}\) is such that for all \(k \in \mathbb{N}\) there exists \(n \in \mathbb{N}\) and \(C > 0\) with \(\sup_{| \mu | \leqslant k} \| \partial^{\mu} (A \varphi) \|_{L^{\infty}} \leqslant C \sup_{| \mu | \leqslant n} \| \partial^{\mu} \varphi \|\), then we set \(\langle^t A f, \varphi \rangle = \langle f, A \varphi \rangle\). Differential operators are defined by \(\langle \partial^{\mu} f, \varphi \rangle = (- 1)^{| \mu |} \langle f, \partial^{\mu} \varphi \rangle\). If \(\varphi : \mathbb{Z}^d \rightarrow \mathbb{C}\) grows at most polynomially, then it defines a Fourier multiplier

which gives us a distribution \(\varphi (\mathrm{D}) f \in \mathscr{D} \hspace{.1em}'\) for every \(f \in \mathscr{D} \hspace{.1em}'\).

Littlewood-Paley blocks give a decomposition of any distribution on \(\mathscr{D} \hspace{.1em}'\) into an infinite series of smooth functions.

Dyadic partitions of unity exist, as shown in [3]. The reason for considering smooth partitions rather than indicator functions is that indicator functions do not have good Fourier properties. We fix a dyadic partition of unity \((\chi, \rho)\) and define the dyadic blocks \(\)

We also use the notation

and notice that

with \(K_{j, x} (y) = 2^{dj} K (2^j (x - y))\), \(P_{j, x} (y) = \sum_{i < j - 1} K_{i, x} (y)\), \(K\) radial with zero mean.

Every dyadic block has a compactly supported Fourier transform and therefore it belongs to \(C^{\infty}\). It is easy to see that \(f = \sum_{j \geqslant - 1} \Delta_j f = \lim_{j \rightarrow \infty} S_j f\) for all \(f \in \mathscr{D} \hspace{.1em}'\).

For \(\alpha \in \mathbb{R}\), the Hölder-Besov space \(\mathscr{C} \hspace{.1em}^{\alpha}\) is given by \(\mathscr{C} \hspace{.1em}^{\alpha} = B^{\alpha}_{\infty, \infty} (\mathbb{T}^d, \mathbb{R})\), where for \(p, q \in [1, \infty]\) we define

with the usual interpretation as \(\ell^{\infty}\) norm in case \(q = \infty\). Then \(B^{\alpha}_{p, q}\) is a Banach space and while the norm \(\lVert \cdot \rVert_{B^{\alpha}_{p, q}}\) depends on \((\chi, \rho)\), the space \(B^{\alpha}_{p, q}\) does not, and any other dyadic partition of unity corresponds to an equivalent norm.

If \(\alpha \in (0, \infty) \setminus \mathbb{N}\), then \(\mathscr{C} \hspace{.1em}^{\alpha}\) is the space of \(\lfloor \alpha \rfloor\) times differentiable functions whose partial derivatives of order \(\lfloor \alpha \rfloor\) are (\(\alpha - \lfloor \alpha \rfloor\))-Hölder continuous (see page 99 of [3]). Note however, that for \(k \in \mathbb{N}\) the space \(\mathscr{C} \hspace{.1em}^k\) is strictly larger than \(C^k\), the space of \(k\) times continuously differentiable functions.

The following lemma gives useful characterisation of Besov regularity for functions that can be decomposed into pieces which are localized in Fourier space.

The Bernstein inequalities of the next lemma are extremely useful when dealing with functions with compactly supported Fourier transform.

We recall some standard heat kernel estimations (see Chapter 2 of [3] and [15, 17]).

We need the following interpolation lemma:

Proof.

and choosing \(2^{- n - 1} \leqslant | t - s |^{1 / 2} \leqslant 2^{- n}\) we obtain

and this yields the result.\(\Box\)

Terms of the type \(\| a (u (t, x)) \|_{\mathscr{C} \hspace{.1em}^0}\) with \(a : \mathbb{R} \rightarrow \mathbb{R}\) cannot be estimated directly with their Hölder norm. In the following lemma we prove some bounds used in Section 4.

Proof. The bound on \(\| a (u) \|_{\mathscr{C} \hspace{.1em}_T^{\alpha}}\) is trivial. We estimate \(\| a (u_t) - a (u_s) \|_{\mathscr{C} \hspace{.1em}^0}\) as

If \(k > - 1\) we have

and then the first term above becomes

The second term is

The third term can be estimated as the first one when \(k > - 1\). Otherwise we just bound it as

For the three terms together we have the bound

With the same technique we obtain

and this gives the second estimate. The third one can be obtained easily.\(\Box\)

A.1.Bony's paraproduct.

In terms of Littlewood–Paley blocks, the product of two smooth functions \(fg\) can be decomposed as

Where the paraproducts \(f \prec g\) and \(f \succ g\) and the resonant product \(f \circ g\) are defined as

We will often use the shortcuts \(\sum_{i \sim j}\) for \(\sum_{| i - j | \leqslant 1}\) and \(\sum_{i \lesssim j}\) for \(\sum_{i < j - 1}\). Of course, the decomposition depends on the dyadic partition of unity used to define the blocks \(\Delta_j\), and also on the particular choice of the pairs \((i, j)\) in the diagonal part. The choice of taking all \((i, j)\) with \(|i - j| \leqslant 1\) into the diagonal part corresponds to property 2 in our definition of dyadic partitions of unity.

Bony's crucial observation [9, 31] is that the paraproduct \(f \prec g\) (and thus \(f \succ g\)) is always a well-defined distribution. Heuristically, \(f \prec g\) behaves at large frequencies like \(g\) (and thus retains the same regularity), and \(f\) provides only a frequency modulation of \(g\). The only difficulty in constructing \(fg\) for arbitrary distributions lies in handling the resonant product \(f \circ g\). The basic result about these bilinear operations is given by the following estimates.

Bony proved also a basic paralinearisation result, soon after improved by Meyer. We give here a particular version suited to our purposes.

The additional key ingredient at the core of the paracontrolled approach is the following commutation result proved in [15], Lemma 2.4:

We will need the following two lemmas to compare standard and time-smoothed paraproducts. The first one has essentially the same proof as [15], Lemma 5.1.

The second lemma has a standard proof.

Bibliography