Construction and Skorohod representation of a fractional K-rough path

We go ahead with the study initiated in [3] about a heat-equation model with non-linear perturbation driven by a space-time fractional noise. Using general results from Hairer's theory of regularity structures, the analysis reduces to the construction of a so-called K-rough path (above the noise), a notion we introduce here as a compromise between regularity structures formalism and rough paths theory. The exhibition of such a K-rough path at order three allows us to cover the whole roughness domain that extends up to the standard space-time white noise situation. We also provide a representation of this abstract K-rough path in terms of Skorohod stochastic integrals.


Introduction
The aim of this paper is to go a few steps further into the analysis of the SPDE model introduced in [3], namely the equation where F : R×R → R is a quite general vector fields, Ψ is a continuous function, and ∂t∂xB stands for a space-time fractional noise. To be more specific, B is here a fractional sheet with Hurst indexes H1, H2, that is a centered Gaussian field with covariance function given by the formula At this point, let us recall that the whole difficulty raised by this equation (at least when H1 = 1 2 ) lies in the fact that B is not a martingale process, which rules out the possibility to study this model within the classical SPDE framework of [6] or [17]. It is then natural to turn to pathwise methods, and in fact, this equation provides us with an interesting example to test the flexibility of the theory of regularity structures -RS in the sequelrecently introduced by M. Hairer in [10]. The machinery has already proved to be a very powerful tool to study stochastic parabolic dynamics, as a flourishing literature can easily testify. To mention but a few applications, we can quote for instance [1,2,12,13,14].
In this context, our objective behind the study of (1.1) is actually manifold: • RS theory is essentially built upon a sophisticated extension of concepts from Lyons' rough paths (RP) theory. However, based on the introductory paper [10], the fundamental analogies between RP and RS theories may not be obvious to a non-initiated reader. Therefore, and in the continuity of [3], we here propose to somehow go one step back into the formulation, by highlighting the role of an object whose definition and properties look very much like those a classical rough path: the so-called K-rough path (see Definition 2.7). Of course, as the RS-expert reader will soon realize, restricting the analysis to this sole concept of a K-rough path reduces the possible scope of application of RS results, in comparison with the general abstract formalism settled in [10]. In brief, K-rough paths are specifically designed for the dynamics of (1.1), and their definition must be reshaped when turning to other models such as KPZ or Φ 4 3 equations. This being said, we think, or at least we hope, that this more straightforward presentation may help the reader to catch the very "rough-path" essence of RS theory, on a non-trivial SPDE example.
• Still focusing on the notion of a K-rough path, the analysis will give us the opportunity to recall how the technical tools used to study rough paths (Hölder topologies, Garsia-Rodemich-Rumsey lemma,...) extend to the parabolic framework. We will also see that the renormalization procedures, one of the main achievements of RS theory, can be very conveniently expressed in this setting.
• In [3], the above ideas were implemented for a second-order analysis, which in fact corresponds to the situation where 2H1 + H2 > 5 3 . We will here go one step further and consider the study of the model up to third order, which covers the case 2H1 + H2 > 3 2 . This extension will therefore give us the opportunity to go deeper into the exploration of the concept of a K-rough path. What also makes this work important to us is that it makes the link with the classical space-time white noise situation, for which H1 = H2 = 1 2 (and accordingly 2H1 + H2 = 3 2 ). In brief, thanks to the subsequent results, we are now able to cover the whole domain that extends up to -but does not include -the standard space-time white noise (see the comments of Figure 2.2 for more details). It is worth mentioning that in the (very) particular martingale situation where H1 = H2 = 1 2 , the RS machinery has been implemented by M. Hairer and E. Pardoux in [13], leading the authors to a Wong-Zakaï-type property similar to our forthcoming Corollary 2.20.
• The exhibition of a third-order K-rough path above the fractional noise turns out to be a quite technical task. The construction differs from those in the white-noise situation, due to some fractional kernel to be dragged throughout the computations. Our analysis relies on Fourier techniques inherited from the harmonizable (or Fourier) representation of the fractional sheet, that is the formula for some appropriate constant cH 1 ,H 2 > 0 and where F(W ) stands for the Fourier transform of a space-time white noise in R 2 . Our main result, namely the existence of a K-rough path above ∂t∂xB, can then be seen as a parabolic version of the results of Coutin and Qian about fractional rough paths (see [5]). Just as in the latter reference, we will also be able to provide a decomposition of our fractional K-rough path in terms of Skorohod integrals, to be compared with the formulas in [5,Theorem 4]. We consider these chaos-decomposition formulas another substantial improvement with respect to the study in [3]: we are here able to describe our abstract K-rough path in terms of some "pre-existing" stochastic tools. In the -very -particular situation of a white-in-time noise, the decomposition reduces to the sole "Itô" K-rough path, an identification that can then be transposed on the level of the equation itself (see the last statement in Corollary 2.20).
One of the main ideas in both RP and RS theories can be summed up as follows -in a loose manner, of course: in order to interpret and solve the noisy differential equation under consideration (standard differential equations for RP theory, parabolic equations for RS theory), and therefore give a sense to the implicitly-defined solution, we only need to study a finite number of objects that are explicitly defined in terms of the noise only. In other words, all the successive operations involved in the equation, i.e., composition with a smooth vector fields, multiplication with the noise, integration and even the fixed-point argument, nicely combine around these few explicit objects, called the rough path in RP theory and the K-rough path in our setting.
In this paper about Equation (1.1), and for the sake of conciseness, we will not come back to the description of the sophisticated machinery that associates a K-rough path (or a "model" along Hairer's terminology) with a solution of the equation. The details of this sophisticated deterministic procedure can be found in [10,Sections 4 to 7], as well as in [3, Section 2] for a shorter version applying specifically to the dynamics of (1.1).
Thus, in what follows, we will only stick to the problem of constructing a K-rough path above the fractional noise. A natural way to initiate this construction is to start from the so-called canonical K-rough path associated with a given smooth approximation B n of the rough fractional noise B. This object has to be seen as the parabolic counterpart of the iterated integrals -or "canonical rough path" -of RP theory: just as its one-parameter model, the canonical K-rough is indeed derived from a Taylor expansion of the classical equation associated with the smooth path B n . A specific description of this object, that we will denote by B n in the sequel, will be given in Definition 2.6.
Showing the convergence of B n would then immediately provide us with a K-rough path above ∂t∂xB, as desired -again, the situation can for instance be compared with the RP example treated in [5]. Unfortunately, as soon as 2H1 + H2 ≤ 2, such a convergence happens to fail, which forces us to turn to renormalization tricks and exploit the flexibility of the definition of a K-rough path, as illustrated by Lemma 2.12. This divergence phenomenon and the need for an appropriate renormalization were already observed at second order in [3], and can be compared with the well-known divergence properties of KPZ or parabolic Anderson models. Couterbalancing the explosion will prove to be an intricate task at third order, with correction terms inspired by the chaos expansion of the canonical K-rough path.
Throughout the study, we will consider the approximation B n of B given by the formula: where Dn := [−2 2n , 2 2n ] × [−2 n , 2 n ], and cH 1 ,H 2 , F(W ) are defined just as in (1.2). Thanks to the isometry properties satisfied by W (or F(W )), such an approximation readily yields explicit and manageable formulas (in terms of the fractional kernel) when computating related moments. Another advantage of the representation is that the Malliavin calculus with respect to B n (one of the keys of our analysis) can easily be connected with the standard Malliavin calculus for W , or F(W ), as we will see it in Section 3. This being said, we are pretty sure that the consideration of a mollifying procedure (just as in [10]) would lead to very similar constructions and results. Consider indeed a mollifying sequence ρn(s, x) := 2 3n ρ(2 2n s, 2 n x) on R 2 and denote temporarily Then using the representation (1.2) of the fractional sheet and applying Fubbini theorem -at least formallyallow us to write which points out the strong similarity with the above approximation In the same vein, there is no doubt to us that the subsequent constructions could easily be extended to a wider class of fractional noises, provided one can exhibit appropriate bounds on the Fourier transform of their covariance function (this will indeed be the quantity at the core of the computations regarding the noise part). For the sake of conciseness, we have stuck to the prototype fractional-sheet example though.
The paper is organized as follows. In Section 2, we introduce the notion of a K-rough path, which corresponds to the central object of our analysis and -hopefully -offers a clear link between the formalisms of RP and RS theories. We shall also state a few basic properties satisfied by K-rough paths, together with analytical tools suited to these objects. This will put us in a position to state our main result, namely the existence of a K-rough path above the fractional sheet, as well as its important consequences on Equation (1.1). Section 3 and 4 are then devoted to the details of this construction. Section 3 actually consists in a Malliavin-chaos expansion of the components of the (renormalized) canonical K-rough path associated with the smooth approximation B n , while Section 4 focuses on the extension of these formulas above the rough process B, by means of technical moments controls. Finally, in Appendix A, we have collected a few useful (deterministic) estimates related to the interactions between K-rough paths and the heat kernel, at first and second orders.

K-rough paths and main results
The general machinery of RS theory, as introduced in [10], leans on a combination of a high number of sophisticated objects, gathered under the names of models and regularity structures. However, when specializing the analysis to Equation (1.1) and focusing on the essential information within those models/regularity structures, a relatively simple object naturally arises. We will call a K-rough path for its similarity with a classical RP. Just as with classical RP, the definition of a K-rough path highly depends on the roughness of the driver -here, the almost sure roughness ofḂ := ∂t∂xB -, seen as a distribution. In order to quantify this roughness, we will use the (local) Besov-type topologies introduced in [10], as detailled below.
Notations. We will denote by D ′ (R 2 ) the set of general distributions and by D ′ m (R 2 ) (m ≥ 0) the dual of the space C m (R 2 ) of m-times differentiable and compactly-supported test-functions. Let us also consider the usual parabolic scaling s := (2, 1) and the related balls for every x = (x0, x1) ∈ R 2 and R > 0. Given ϕ : R 2 → R, x = (x0, x1) ∈ R 2 and ℓ ≥ 0, we will denote by ϕ ℓ x the x-centered and 2 −ℓ -scaled version of ϕ, that is x1)) , for all y = (y0, y1) ∈ R 2 . Finally, we will denote by B the set of smooth functions on R 2 with compact support included in Bs(x, R) and derivatives uniformly bounded by 1 up to order 4. Definition 2.1. For every α < 0 and every set K ⊂ R 2 , we say that a distribution X ∈ D ′ (R 2 ) belongs to C α (K) if it belongs to D ′ 4 (R 2 ) and if the quantity X α;K := sup x∈K,ϕ∈B,ℓ≥0 is finite. In the sequel, we denote by C α c (R 2 ) the set of distributions X ∈ D ′ (R 2 ) such that X ∈ C α (K) for every compact set K.

Definition 2.2.
For every α < 0 and every set K ⊂ R 2 , we say that a map X : R 2 → D ′ (R 2 ) belongs to C C C α (K) if for every x ∈ R 2 , Xx belongs to D ′ 4 (R 2 ) and if the quantity is finite. We denote by C C C α c (R 2 ) the set of maps X : R 2 → D ′ (R 2 ) such that X ∈ C C C α (K) for every compact set K.
As we already mentionned it in the introduction, the definition of the canonical K-rough above a smooth approximation B n (and by extension the definition of a K-rough path above a rough function B) is derived from the space-time expansion of Equation (1.1), that can also be written as where G stands for the usual heat kernel on R and we have set X := ∂t∂xB n . Therefore, it is not a surprise that the description of the key elements of the dynamics, which together will form the canonical K-rough path, should appeal to the heat kernel. As we are only dealing with local behaviours, it actually suffices to focus on the kernel around its singularity, that is around 0, which gives birth to the following definition, more suited to the topology under consideration: We call a localized heat kernel any function K : R 2 \{(0, 0)} → R satisfying the following conditions: (ii) There exists a smooth function K0 : Again: a localized heat kernel K is nothing but a local representation, around 0, of the heat kernel G (see [10,Lemma 5.5] for more details). The following bound on the Fourier transform (denoted by F) of such a localized heat kernel will be extensively used in the computations of Section 4. Its proof is an immediate consequence of the decomposition (2.2), as the reader can easily check it: Lemma 2.5. Let K be a localized heat kernel, in the sense of Definition 2.4. Then for all a, b ∈ [0, 1) satisfying a + b < 1, there exists a constant c K,a,b such that for all non-zero ξ, η ∈ R, Finally, space-time expansion of (2.1) will naturally lead us to consider space-time expansions of the heat kernel (or its localized version). Let us label those quantities for further use: given a localized heat kernel K and for all x = (x0, x1), y = (y0, y1), z = (z0, z1) ∈ R 2 , we set We are now ready to introduce our central notion of a K-rough path above a given deterministic distribution X ∈ C α c (R 2 ), at least for α < 0 large enough. This definition already appeared in [3] for α ∈ (− 4 3 , 0), which morally corresponds to an expansion of the equation up to second order -with the consideration of some K-Lévy area term. We would like to go one step further here and handle the situation where α ∈ (− 3 2 , 4 3 ], which forces us to introduce third-order elements in the analysis. Thus, from now on and for the rest of Section 2.1, we fix such a coefficient Let us start with the description of the canonical K-rough path in this case: Let us insist one more time on the fact that this definition is motivated by the space-time Taylor expansion of the standard PDE (2.1). Then, just as in RP theory, it turns out that this notion of a canonical K-rough path -and the whole integration machinery built upon it -can be lifted at some more abstract level above a rougher distribution X, which gives rise to the following general definition: Definition 2.7 (K-rough path). Given a localized heat kernel K and a distribution X ∈ C α c (R 2 ), we call a K-rough path (of order α) above X any 4-uplet such that X 1 = X and the following "K-Chen" relations hold: for every x = (x0, x1), y = (y0, y1) ∈ R 2 , and For a fixed localized heat kernel K and given two K-rough paths X, Y (above possibly different distributions X, Y ), we denote, for every compact set K ⊂ R 2 ,

9)
and X α;K := X; 0 α;K . In the sequel, we will denote by EK,α the set of K-rough paths of order α.
Remark 2.8. As its name suggests it, the canonical K-rough path above a continuous function X is a particular example of K-rough path. Relations (2.7) and (2.8) can indeed be easily checked from the explicit formulas in (2.5)-(2.6).
Remark 2.9. With expansion (2.2) in mind and for every n ≥ 0, let us denote by K n the smooth compactlysupported function obtained as the finite sum K n (s, x) := 0≤ℓ≤n 2 3ℓ K0(2 2ℓ s, 2 ℓ x). Then, in (2.7) and (2.8), and although K itself is not a smooth test-function (due to its singularity at 0), the coefficient , is well defined as the limit of the sequence X i x K (i),n x,y (i ∈ {1, 2}), resp.
, where K (i),n is derived from K (i) by replacing each occurence of K with K n . The proof of this assertion can be easily deduced from the properties in Appendix A, which also contains a few regularity results related to these quantities. Remark 2.10. Relations (2.7) and (2.8) can legitimately be considered as a parabolic analog of the classical Chen's relations for a third-order rough path x = (x 1 = δx, x 2 , x 3 ) (see [15] for a detailled definition). To emphasize this analogy, introduce the dual pathx = ( . With these notations, the classical Chen's relations can also be written as which makes the similarity with (2.7)-(2.8) obvious. Obseerve however that the higher complexity of this twoparameter setting forces us to consider more sophisticated structures exhibiting two third-order components (instead of one for RP theory). Also, the implicit presence of the (regularizing) heat kernel in the definition of a K-rough path echoes in a natural way on the choice of the roughness assumptions, that is on the choice of the successive combinations α, 2α + 2, 3α + 4, as Formulas (2.5)-(2.6) and the regularity properties of Appendix A should convince the reader.
Remark 2.11. The set EK,α can easily be turned into a complete metric space by considering the distance where X; Y α;K is defined by (2.9) and we have set R k := [−k, k] 2 . The completeness of (EK,α, dα) can indeed be shown along the arguments of the proof of [3, Proposition 3.1].
The following basic property, the proof of which is immediate, illustrates the flexibility of the definition of a Krough path. It shows in particular that, provided it exists, a K-rough path above a given distribution X ∈ C α c (R 2 ) is not unique at all. Here again, the idea is strongly reminiscent of the well-known non-uniqueness property of classical rough paths. In the sequel, we shall rely on this "renormalization" trick to overcome the diverging issue raised by the canonical K-rough path above the approximation B n . Lemma 2.12 (Renormalization). Fix a localized heat kernel K and a path X ∈ C α c (R 2 ). Given a K-rough path X above X and a constant c ∈ R, consider the path defined along the following formulas: . Then X is a K-rough path above X as well.
In RP theory, the so-called Garsia-Rodemich-Rumsey Lemma and its extensions (see e.g. [9, Section 6]) provide a very efficient tool to study the roughness of the processes under consideration, and therefore represent one of the keys of the RP analysis. Using sophisticated wavelets arguments, M. Hairer succeeded in the exhibition of similar tools in the multiparameter setting. The following statement offers a possible simple way to account for these results. In particular, it should be clear to the reader that the central condition in this statement is directly related to (not to say that it perfectly fits) the structure of the K-Chen relations from Definition 2.7 (compare (2.10) with (2.7)-(2.8)). Let us also recall that the notation B has been introduced at the beginning of this section.

map with increments of the form
) and θ i : R 2 × R 2 → R is such that for every compact set K ⊂ R 2 and every x, y ∈ K, one has Proof. It is a straightforward generalization of the arguments in the proof of [3, Lemma 3.2].
As we evoke it from the beginning, what makes this K-rough-path structure so interesting (beyond its clear analogy with the classical rough-path structure) is the fact that it can be readily injected into the machinery of [10] so as to deduce numerous striking results about the equation driven by X, that is The following general statement sums up these important consequences. At this point, let us recall that we have fixed a parameter α ∈ (− 3 2 , − 4 3 ] for the whole Section 2.1, and that the map Renorm has been defined through Lemma 2.12, while the distance dα has been introduced in Remark 2.11.

Proposition 2.14 (Solution map). Fix an arbitrary time horizon T > 0 as well as a vector field
Then there exists a localized heat kernel K and a "solution" map such that the following properties are satisfied:

Based on the above properties and setting
Proof. The four properties (i)-(iv) are derived from a careful examination of the analysis carried out in [10] (see also [3] for a detailled version of this analysis in the special case given by the dynamics of Equation (1.1)). The only ingredient to specify here is how the K-rough-path structure can be related to the regularity structure/model terminology used in [10]. A part of this connection has already been exhibited in [3, Section 2.1] for α ∈ (− 4 3 , −1]. Taking up the notations (A, T ) of the latter reference and following the general procedure described in [10,Section 8], the regularity structure (Ā,T ) to be considered in our situation is given bȳ T2α+4 := Span{I(ΞI(Ξ))} , T3α+4 := Span{ΞI(ΞI(Ξ)), ΞI(Ξ) 2 } . Then, given a K-rough path X above a distribution X ∈ C α c (R 2 ), we define the two maps Γ X : R 2 × R 2 → L(T ) and Π X : as follows. For any β ∈ A and τ ∈ T β , we define both Γ X xy (τ ) and Π X x (τ ) (x, y ∈ R 2 ) just as in [3, Section 2.1]. Besides, we set, for all x,y ) , and Γ X xy (I(ΞI(Ξ))) := I(ΞI(Ξ)) − axy Y1 − bxy I(Ξ) − cxy 1 , axy . In (2.13), the notation Y1 refers to the abstract symbol for the first-order monomial in the x1-variable (for instance, , where the notation ⋆ refers to the product of the regularity structure (see [3, Section 2.1]).
Using the properties contained in the very definition of a K-rough path, one can now check that for all x, y ∈ R 2 , the identity Π X y = Π X x • Γ X xy holds true, while for every compact set K ⊂ R 2 , it holds, with the notations of [10, for some compact setK ⊃ K.
This leads us to the desired conclusion: the above-described pair (Π X , Γ X ) does define a model for the regularity structureT , in the sense of [10,Definition 2.17]. The construction has actually been designed in such a way that if X is a smooth path and X stands for the canonical (α, K)-rough path above X (along Definition 2.6), then the associated model (Π X , Γ X ) coincides with the canonical model described in [10,Section 8.2].
Based on these observations, the properties (i)-(iv) of our statement are now mere consequences of general results from [10] (see also [3, Section 2] for a shorter presentation applying specifically to Equation (2.12)). For the sake of conciseness, we will not return to the details of this sophisticated analysis.
2.2. Main results. Now endowed with the above preliminary -deterministic -material, we can turn to the detailled presentation of our main result, namely the exhibition of a K-rough path above the stochastic fractional sheet B. This construction will be based on a chaos decomposition of the canonical K-rough path B n , a strategy that will naturally lead us to the consideration of trace-type terms. The procedure will more specifically involve a series of operators L, Θ (i) , L i,X that we propose to introduce right now. Let us recall that, throughout the paper, we denote by F the usual Fourier transformation, that is First, for every (H1, H2) such that 2H1 + H2 > 3 2 , every x = (x0, x1) ∈ R 2 and every smooth compactlysupported function ψ on R 2 , we set (2.14) where cH 1 ,H 2 refers to the constant in the representation (1.2) of the fractional sheet. Then, for every x,y+z (z) , and finally Remark 2.15. Checking the well-posedness of the above operators is one of the objectives behind the computations of Section 4 and Appendix A. Therefore, we refer the reader to these sections for a detailled analysis of L, Θ (i) and L i,X . In fact, the treatment of these trace operators (which are specific to the fractional situation, as emphasized by Proposition 2.19) will prove to be one of the most technical parts of our study.
Theorem 2.16 (Existence of a fractional K-rough path). Fix (H1, H2) ∈ (0, 1) 2 and α < 0 such that Moreover, the following identification formulas hold true for the components of B: for every x ∈ R 2 and every smooth compactly-supported function ψ, one has almost surely x,. x,.
Remark 2.18. By following the arguments of the proof of [3,Proposition 3.12], it is easy to show that the quantity c n H 1 ,H 2 defined by (2.15) asymptotically behaves as In the very particular situation of a white-in-time noise, that is when H1 = 1 2 -and the whole Itô integration theory becomes available -, the symmetry in representation (1.2), combined with the vanishing properties of any (localized) heat kernel, offers a drastic simplification of both the analysis and the results: In the setting of Theorem 2.16, assume that H1 = 1 2 and H2 > 1 2 , that is B is a white-in-time noise with fractional spatial regularity of order H2 > 1 2 . Then, with the above notations, one has, for every test-function ψ with support included in the set {x ∈ R 2 : x0 ≥ 0} and every x ∈ R 2 , Proof. Since H1 = 1 2 , we can lean on the basic isometry properties of the Fourier transform to assert that It now suffices to observe that due to the vanishing assumption on K and the support condition on ψ, the latter integral is necessarily equal to zero. The same combination of arguments (isometry property and support condition) can also be used to show that L 1,Ḃ The identification with the Itô K-rough path immediately follows from Formulas (2.16) to (2.19) and the fact that Skorohod integrals are known to coincide with Itô integrals in this situation (see for instance [8]).
We can finally combine the above construction with the general results of [10] about Equation (1.1), as we summed them up through Proposition 2.14: Then for every 0 < T1 < T0 and as n tends to infinity, the sequence Y n of classical solutions of the (renormalized) equation  Proof. The first assertion is a straightforward application of Proposition 2.14 and Theorem 2.16. When H1 = 1 2 , the identification of the solution with Itô's solution is then a consequence of the identification result of Proposition 2.19 (on the level of the K-rough path): the details of this lifting procedure can be found in [13, Section 6].
The following basic picture illustrates the domain covered by the combination of the above results with the results of [3], regarding the pair (H1, H2). Based on Proposition 2.3, the successive stages for the global roughness α of the noise turn into successive slices for the combination 2H1 + H2. The black, resp. red, slice corresponds to the first-order, resp. second-order, situation where α ∈ (−1, 0), resp. α ∈ (− 4 3 , −1], and was treated in [3]. The blue slice corresponds to the setting of the present paper, with α ∈ (− 3 2 , − 4 3 ]. Its border extends up to the standard space-time white-noise situation (H1 = H2 = 1 2 ), as we pointed it out earlier. The rest of the paper is now devoted to the proof of Theorem 2.16. Therefore, from now on and until the end, we fix, on a complete probability space (Ω, F, P), a fractional sheet B of Hurst index (H1, H2) ∈ (0, 1) 2 satisfying 3 2 < 2H1 + H2 ≤ 2 , and with representation (1.2) with respect to some space-time white noise W . Also, we consider the smooth approximation B n defined by (1.3) and, for some fixed localized heat kernel K, we denote by B n the canonical K-rough path associated with B n , in the sense of Definition 2.6. Finally, we denote by B n := ( B 1,n , B 2,n , B 3.1,n , B 3 3 2 ) are the continuation of the first and second-order results of [3] for the more restrictive case where 2H1 + H2 > 5 3 . In particular, the study of the first and second-order components of the renormalized canonical K-rough path (i. e., B 1,n and B 2,n ) can be done along the very same arguments and estimates as in [3], which already provides us with the following preliminary statement:  ( B 1 , B 2 x,y · B 1 , and for all n, k, p ≥ 1,  [3], the results of (2.25) are only stated under the assumption that 5 3 < 2H1 + H2 ≤ 2 (the bounds in (2.24) are actually true for every (H1, H2) ∈ (0, 1) 2 ). However, a close examination of the technical details in the latter reference would show to the -patient -reader that the computations remain valid for 2H1 + H2 ∈ ( 3 2 , 5 3 ] as well. In any case, the forthcoming proof of Formula (2.17) for B 2 would easily allow us to recover (2.25) (see for instance (4.1)), and these bounds at first and second-order orders turn out to be elementary to obtain in comparison with the subsequent third-order estimates.

Chaos decomposition of the fractional canonical K-rough path
Before we go further, note that, for the sake of clarity in the subsequent computations, and in opposition with the formulation of Section 2, we will henceforth go back to more standard notations regarding time, resp. space, variables, and denote them by s, t, u, resp. x, y, z. Now, in the Gaussian setting under consideration, our strategy to prove the convergence of B n is based on the following natural (albeit technical) two-step procedure: (i) For fixed n, and using Malliavin-calculus tools, expand the components of B n as a sum of Skorohod integrals with respect to B n ; (ii) Show the convergence, as n tends to infinity and for an appropriate topology, of each of the summands in these chaos-type decompositions.
The present section 3 is devoted to the proof of Step (i), and therefore, some preliminary material on Malliavin calculus must be introduced. In fact, for the two-parameter processes we shall consider in the sequel, an exhaustive presentation of this material can be found in [4, Sections 5 and 6], and accordingly we will not return to the definition of the classical objects therein introduced, namely the Hilbert space HZ , the Malliavin derivative D Z and the Skorohod integral δ Z associated with any centered Gaussian field {Z(s, x); s, x ∈ R} defined on a complete probability space (Ω, F, P).
However, what must be underlined in this situation (i.e., when working with B or B n ) is that thanks to the representation (1.2), resp. (1.3), there exists a close link between the Malliavin calculus with respect to B, resp. B n , and the Malliavin calculus with respect to W or W := F(W ). To elaborate on these relations, let us introduce the family of operators Qα 1 ,α 2 (α1, α2 ∈ (0, 1)), resp. Q n α 1 ,α 2 , defined for every measurable, compactly-supported function ϕ and every ξ, η ∈ R as where cα 1 ,α 2 is the same constant as in the representation (1.2). We can then rely on the following identities: for every test-function ϕ, every functional F = F (B), resp. F n = F n (B n ), smooth enough (in the sense of Malliavin calculus) and every HB-valued, resp. HBn -valued, random variable u in an appropriate domain, it holds that (3.2) Here again, we refer the reader to [4] (and more specifically to [4, Lemma 6.1]) for a proof of these identities, as well as for further details regarding the specific assumptions on F , F n and u. Let us also recall the following general product rule satisfied by the Skorohod integral, for either Z := B or Z := B n (see [16]): Identities (3.1) and (3.2) point out the important role played by the operators Qα 1 ,α 2 , Q n α 1 ,α 2 in this setting. The following related estimates will thus prove to be fundamental in the sequel: Let α1, α2 ∈ (0, 1). For every smooth compactly-supported function ϕ : R 2 → R, it holds that 5) where ϕ α 1 ,α 2 is defined along the following formulas: (3.6) • If α1 ∈ (0, 1 2 ) and α2 ∈ ( 1 2 , 0), then • If α1, α2 ∈ (0, 1 2 ), then As for (3.5), it can easily be derived from the same Sobolev-embedding arguments as in [3,Lemma 4.4].
We are now in a position to prove the desired decomposition formulas for the components of B n (for each fixed n), namely: Proposition 3.2. For every smooth compactly-supported function ψ : R 2 → R and every (s, x) ∈ R 2 , one has, in L 2 (Ω), and

11)
where, following the notations of Section 2.2, we have set, for every X ∈ C α c (R 2 ), and L 3,n,X (s,x) (ψ) := c 2 Proof. Let us first focus on the most intricate identity, namely (3.10). To this end, we set and consider two sequences of partitions ti = t ℓ i := i ℓ , yj = y m j := j m (ℓ, m ∈ N, i, j ∈ Z). For all fixed i, j, ℓ, m, we know by (3.3) that, writing 1 ij for 1 [ Now, since we can assert that From this expression, and with the help of (A.1) and (2.24), we can easily justify that as ℓ, m → ∞, At this point, observe that which immediately entails that Going back to (3.12) and observing in addition that B 2,n (s,x) K (2) (s,x),(t,y) = B 2,n (s,x) K (2) (s,x),(t,y) , it remains us to prove that as ℓ, m → ∞, one has Using the basic properties of the Skorohod integral, the proof of the first convergence actually reduces to showing that in L 2 (Ω; HBn ) .
To this end, we can first invoke (3.2) and (3.4) to assert that Going back to the definition of V n (s,x) , the conclusion now follows from the combination of (A.2)-(A.5)-(A.7) and (2.25). As for the second convergence statement in (3.13), observe that Just as above, we can now conclude by using (A.2)-(A.5)-(A.7) and (2.25), together with the fact that E |Ḃ n (t, y)| 4 1/4 E |Ḃ n (t, y)| 2 1/2 This achieves the proof of (3.10).
The -less sophisticated -identities (3.9) and (3.11) can then be shown along the very same arguments, and therefore we leave their proofs to the reader as an exercise.

Convergence of the decomposition
We turn here to the second -and final -step of the strategy sketched out at the beginning of the previous section. Thus, in brief, our aim now is to prove the convergence of B n by showing the convergence of each of the summands in the decompositions (3.9)-(3.11). Identities (2.17)-(2.19) will then be obtained as immediate consequences of this extension.
Throughout the section, and just as in Lemma 2.21, we will use the notation R k := [−k, k] 2 , for k ≥ 0. Besides, we recall that given ψ : R 2 → R, (s, x) ∈ R 2 and ℓ ≥ 0, we denote by ψ ℓ (s,x) the rescaled function For a clear statement of our result, let us introduce the processes B 2 , B 3.1 and B 3.2 defined by the right-hand sides of (2.17), (2.18) and (2.19), that is The result of the main technical step of our analysis now reads as follows (we recall that the set B of testfunctions has been introduced at the beginning of Section 2): Proposition 4.1. For every α ∈ (− 3 2 , −3 + 2H1 + H2), there exists ε > 0 such that for all n, ℓ ≥ 0, k ≥ 1, (s, x) ∈ R k and ψ ∈ B, one has where the proportional constants are independent of (n, k, ℓ) and ψ.
Before we turn to the proof of the technical estimates behind Proposition 4.1, let us see how the latter result can be used in order to derive our main Theorem 2.16.
As far as the identities (2.17)-(2.19) are concerned, it now suffices to write, for every n, k and x ∈ R k , and use (4.4) again while letting n tend to infinity. The same argument obviously holds for B 3.1 and B 3.2 as well, which achieves the proof of the theorem.
The rest of the section is devoted to the proof of (4.1)-(4.3). Based on decompositions (3.9)-(3.11), the strategy reduces to controlling the convergence of each summand in these formulas. To this end, our arguments will strongly rely on the general deterministic bounds collected in Appendix A, and therefore we are rather confident about the fact that the subsequent computations could easily be extended to a more general class of fractional noises.
where the proportional constants are independent of (n, k, ℓ) and ψ.
Proof. The argument is very similar as the one in the proof of Proposition 4.2, and therefore we will only focus on the main ideas. In fact, by setting (s,x),(t,y) , observe that if for instance H1 > 1 2 and H2 < 1 2 , one has by (3.7) (s,x),(s+2 −2ℓ u,x+2 −ℓ y) . Besides, thanks to (A.1), we know that With these estimates in mind, we can easily mimic the proof of Proposition 4.2 and conclude that, for ε > 0 small enough, where we have used (2.24) to derive the last inequality.

Then the estimation of both
where the proportional constant is independent of (n, k, ℓ) and ψ. As a consequence of (4.27) and (2.24), one has in particular Proof. It is based on the same strategy as the proof of Proposition 4.4, that is an appropriate Cauchy-Schwartz argument. As above, let us start with the inequality with We have here used the basic identity (F(D (0,1) K))(ξ, η) = c η F(K)(ξ, η).
Estimation of I ℓ (s,x) . Just as in the proof of Proposition 4.4, write I ℓ (s,x) as Then note that given our assumptions on (H1, H2), we can easily find λ1, λ2, α0 satisfying the following conditions: At this point, observe that 4H1 − 2 − λ1 < 1 and 4H2 − 4 − λ2 < 1. Moreover, with (4.32)-(4.33) in mind, it is easy to exhibit a, b ∈ [0, 1) such that a + b < 1 and Consequently, we can apply Lemma 2.5 and assert that As for the second integral in (4.34), we have, by Lemma 3.1 (and more precisely by (3.7)), Using (A.1), we can here rely on the following bounds: Combined with fact that λ2 > −1 > −2(α + 2), these estimates allow us to conclude that Going back to (4.34), we have thus proved the desired estimate, namely Estimation of II n,ℓ (s,x) . The argument is the same as in the proof of Proposition 4.4: we first bound |II n,ℓ (s,x) | in a similar way as in (4.26) and then use the same Cauchy-Schwartz bound (4.34) (by replacing Hi with Hi − ε, for ε > 0 small enough, and X − Y with X). This leads us to the expected estimate, namely 4.5. Convergence in the first chaos: case of L 3,n,Ḃ n . Proposition 4.6. For every α ∈ (− 3 2 , −3 + 2H1 + H2), there exists ε > 0 such that for all X, Y ∈ C α c (R 2 ), n, ℓ ≥ 0, k ≥ 1, (s, x) ∈ R k and ψ ∈ B, one has where the proportional constant is independent of (n, k, ℓ) and ψ. As a consequence of (4.36) and (2.24), one has in particular Proof. Let us set X ∆ := X − Y . One has (s,x),(t,y) (u, z) , (t,y),(u,z) K (2) (s,x),(t,y) (u, z) .