Rough solutions for the periodic Korteweg-de Vries equation

We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain and with initial condition in $\FF L^{\alpha,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.

∂ ξ u(t, ξ) 2 = 0, u(0, ξ) = u 0 (ξ), (t, ξ) ∈ R × T where the initial condition u 0 belongs to some Sobolev space H α (T) of the torus T = [−π, π]. In a remarkable series of papers by Bourgain [2,17], Kenig-Ponce-Vega [27,26,25] and later Colliander-Keel-Staffilani-Takaoka-Tao [5] this equation has been proved to possess global solutions starting from initial conditions in H α for any α ≥ −1/2. The existence of solutions in negative Sobolev spaces is possible due to the regularizing effect of the dispersive linear term. This regularization is more effective in the whole line and there global solutions exists for any α ≥ −3/4 [26,5,22]. Other references on the analysis of the KdV equation are [16,24]. More recently Kappeler-Topalov, taking advantage of the complete integrability of this model, extended the global well-posedness in the periodic setting to any α ≥ −1 using the inverse scattering method [23]. Inspired by the theory of rough paths we will look for an alternative approach to the construction of solutions of eq. (1) and in general for dispersive equations with polynomial non-linearities. Our method turns out to have some similarities with Christ's power series approach [3,32] and to allow to consider KdV with initial condition in FL α,p for 1 ≤ p ≤ +∞.
Rough paths (introduced by Lyons in [31,30]) allow the study of differential equations driven by irregular functions. They have been applied to the path-wise study of stochastic differential equations driven by Brownian motion, by fractional Brownian motion of any index H > 1/4 [8] and other stochastic processes [7,13]. Part of the theory has been reformulated in terms of the sub-Riemmannian geometry of certain Carnot groups [11]. In [18] we showed how to reinterpret the work of Lyons in the terms of a cochain complex of finite increments and a related integration theory. The key step is the introduction of a map Λ (called sewing map) which encodes a basic fact of rough path theory. We exploited this point of view to treat stochastic partial differential equations of evolution type [29,20] and to study the initial value problem for a partial differential equation modeling the approximate evolution of random vortex filaments in 3d fluids [1].
We would like to show that the concepts of the theory can be used fruitfully for problems not related to stochastic processes. The periodic Korteweg-de Vries equation is used as a case study for our ideas. The point of view here developed should be applicable also to KdV on the full real line or other dispersive semi-linear equations like the modified KdV or the non-linear Schrödinger equation (indeed the results [3,32] obtained via the power-series method can be understood in terms of rough paths following the lines of the present investigation).
No previous knowledge of rough paths theory is required nor any result on the periodic KdV is used in the following. We took care to make the paper, as much as possible, self-contained. Since the arguments are similar to those used in finite dimensions in [18], the reader can refer to this last paper to gain a wider perspective on the technique and on the stochastic applications of rough paths.
Let us describe the main results of this paper in terms of distributional solutions of KdV: let P N be the Fourier projector on modes k ∈ Z such that |k| ≤ N and N (ϕ)(t, ξ) = ∂ ξ (ϕ(t, ξ) 2 )/2 for smooth functions ϕ. Then Theorem 1 For any 1 ≤ p ≤ ∞, α > α * (p) = max(−1/p, −1/2) and u 0 ∈ FL α,p there exists a T * > 0 and a continuous function u(t) ∈ C([0, T * ], FL α,p ) with u(0) = u 0 for which the distribution N (P N u) converge as N → ∞ to a limit which we denote by N (u) and moreover the distributional equation This solution is the limit of smooth solutions and of some modified Galerkin approximations. There exists a natural space of continuous functions on FL α,p for which the nonlinear term can be defined as a distribution and where uniqueness of solutions holds. We show also how to implement the L 2 (T) conservation law in the rough path approach and obtain in this way global solutions in L 2 (T). Following the lines of the numerical study of SDEs driven by rough paths [9,14,12] we analyze an Euler-like time-discretization of the PDE which converges to the above solution. Finally we also prove an existence and uniqueness result under random perturbation of white-noise type in time of the form where Φ is a bounded linear operator from FL 0,∞ to FL α,p and ∂ t ∂ ξ B a white noise on R × T.
Plan. In Sect. 1 we start by recasting the KdV eq. (1) in its mild form and to perform some manipulation to motivate the finite-increment equation which we will study in Sect. 2 where we prove existence and uniqueness of local solutions and discuss the distributional meaning of these solutions. Then we prove a-priori estimates necessary show that global rough solutions exists for initial conditions in L 2 (T) (Sect. 2.1). In Sect. 2.2 we prove that the rough solutions are limits of suitable Galerkin approximations. Moreover in Sect. 2.3 we introduce a time discretization scheme and prove its convergence. The equation that we study is just one of a stack of finite-increment equations that can be generated starting from KdV. In Sect. 2.4 we derive the next member of this hierarchy. Finally Sect. 3 addresses the problem of the presence of an additive stochastic forcing. We collect some longer and technical proofs in the Appendix.

Notations.
We denote with f : Z → C the Fourier coefficients of a real function f : T → R: f (ξ) = k∈Zf (k)e ikξ and define the space where S ′ (T) is the space of real Schwartz distributions on the torus T and x = (1 + |x| 2 ) 1/2 . We restrict the space to the mean zero functions since this is the natural setting to discuss periodic KdV. Then H α (T)\R = FL α,2 . Sometimes we note F the Fourier transform operator so that Ff =f . Given two Banach spaces V, W , denote with L(V, W ) the Banach space of bounded linear operators from V to W endowed with the operator norm and with L n V = L(V n , V ). If X ∈ L n V we write X(a) = X(a, · · · , a) when the operator is applied to n copies of the same argument a ∈ V . The symbol C in the r.h.s. of estimates denotes a positive constant which can be different from line to line and usually we write A B for A ≤ CB.

Formulation of the problem
Duhamel's perturbation formula applied to eq. (1) gives where U (t) is the Airy group of isometries of any FL α,p given by the solution of the linear part of eq. (1). The group U (t) acts on ϕ as F(U (t)ϕ)(k) = e ik 3 t ϕ(k). Conservation of mass for KdV guarantees that if u 0 (0) = 0 then we will have u(t, 0) = 0 for any t ≥ 0. Using as unknown the twisted where with the convention that k 2 = k − k 1 and where we use the convention that the primed summation over k 1 ∈ Z does not include the terms when k 1 = 0 or k 1 = k. In obtaining this equation we have used the algebraic identity k 3 − k 3 1 − k 3 2 = 3kk 1 k 2 valid for any triple k, k 1 , k 2 such that k = k 1 + k 2 . In the rest of the paper we will concentrate on the study of solution to eq. (3) with the fixed point method.

Power series solutions and generalized integration
Proceeding formally we can expand any solution to (1) into a series involving only the operatorsẊ and the initial condition: This perturbative expansion is naturally indexed by (binary) trees representing the various ways of applyingẊ to itself and of performing time integrations. A possible approach to define a solution to the differential equation is then to prove that each term of the series is well defined and that the sum of the series converges. There is quite a bit of literature on this method for Navier-Stokes like equations [15,28,33,34,21] and recently Christ [3] advocated this approach in the context of nonlinear dispersive equations and used it to solve a modified periodic non-linear Schrödinger equation in FL α,p spaces. See also [32] for another recent paper studying power-series solution for the periodic modified KdV equation using Christ's approach.
The advantage of the power series expansion is that the relevant objects are the operators obtained fromẊ by successive application and time integrations and applied to the inital condition. For example the first term in the expansion involves the the symmetric bilinear operator X ts defined for any s ≤ t as WhileẊ σ is usually unbounded in any FL α,p , the operator X ts will be shown to be bounded in FL α,p for appropriate values of α, p and a similar behavior will hold for higher order operators. The regularizing effect of time integrations is due to the dispersive nature of the linear part combined with absence of resonances in the non-linear term.
On the other end the main disadvantage of the power-series method is that one has to control arbitrary terms of the series and the proof of summability require usually a big analytical and combinatorial effort. Another interesting difference is that in the rough path approach there is available a natural space where uniqueness of solutions holds while in the power-series approach no uniqueness result is available.
Rough path theory allows to bypass the complete power-series expansion exploiting the smallness of the leading terms for small time intervals and using a generalized notion of integration to pass from the approximate solution for an very small time interval to an exact solution for an O(1) interval of time. Instead of writing the series solution from time 0 to time t we write it between two times s ≤ t denoting by δv ts = v t − v s with v t = v(t) (the index notation being more comfortable in the following) so that δv ts = X ts (v s ) + r ts (5) where r ts stands for the rest of the series and will be treated as a negligible remainder term. For this to make sense we need that X ts gives itself a small contribution when |t − s| is small. Our first result give then a quantitative control of the size of X ts as a bounded operator in FL α,p . When 2 ≤ p ≤ +∞ define the set D ⊂ R × R + of pairs (γ, α) by Then Lemma 2 For any couple (γ, α) ∈ D the operator X ts is bounded from (FL α,p ) 2 to FL α,p and |X ts | L 2 F L α,p γ,α |t − s| γ .
The proof of this lemma is postponed to Appendix A. A trivial but important observation is that the family of operators {X ts } s≤t satisfy the algebraic equation as can be easily checked using the definition.

The sewing map
The first term in the r.h.s. of eq. (5) is well understood thanks to Lemma 2, while the r term contains all the difficulty. However, due to the particular structure of eq. (5), the r term must satisfy a simple algebraic equation. Indeed, for any triple s ≤ u ≤ t we have δv ts − δv tu − δv us = 0 and substituting eq. (5) in this relation we get where we used eq. (6) to simplify the r.h.s.. The main observation contained in the work [18] is that sometimes this equation determines r uniquely. To explain the conditions under which we can solve eq. (7) we need some more notation. Given a normed vector space (V, | · |) introduce the vector space C n V ⊂ C([0, T ] n ; V ) such that a ∈ C n V iff a t 1 ,...,tn = 0 when t i = t j for some 1 ≤ i < j ≤ n. We have already introduced the operator δ : Moreover, is useful to introduce another operator δ : C 2 V → C 3 V defined on continuous functions of two parameters on a vector space V as δa tus = a ts − a tu − a us . The two operators satisfy the relation δδf = 0 for any where Imδ is the image of the δ operator. We measure the size of elements in C n V for n = 2, 3 by Hölder-like norms defined in the following way: for f ∈ C 2 V let In the same way, for h ∈ C 3 V , set where the last infimum is taken over all finite sequences {h i ∈ C 3 V } such that h = i h i and for all choices of the numbers ρ i ∈ (0, z). Then · µ is easily seen to be a norm on C 3 V , and we set Eventually, let C 1+ n V = ∪ µ>1 C µ n V for n = 2, 3 and remark that the same kind of norms can be considered on the spaces ZC 3 V , leading to the definition of the spaces ZC µ 3 V and ZC 1+ 3 V . The following proposition is the basic result which allows the solution of equations in the form (7).
Proposition 3 (The sewing map Λ) There exists a unique linear map Λ : This proposition has been first proved in [18]. A simplified proof is contained in [20]. Using the notations just introduced, eq. (7) take the form where, by construction, the r.h.s. belongs to ZC 3 FL α,p . If we can prove that it actually belongs to ZC z 3 FL α,p for some z > 1, Prop. 3 will give us the possibility to state that the unique solution of eq. (10) in C 1+ 2 FL α,p is given by r = Λ[X(δv, v) + X(v, δv)].

Λ equations
Let us come back to our initial problem. Since we aim to work in distributional spaces the rigorous meaning of eq. (3) is a priori not clear (even in a weak sense). By formal manipulations we have been able to recast the initial problem in a finite-difference equation involving the Λ map which reads: where we used an abbreviated notation since all the terms have been already described in detail. Note that in this equation the Λ map has replaced the integral so we will call this kind of equations: Λ-equations. Instead of solving the integral equation we would like to solve, by fixed-point methods, the Λ-equation (11). Afterwards we will show rigorously that solution to such Λ-equations gives generalized solutions to KdV. Unfortunately, for the particular form of our X operator, we are not able to show that this equation has solutions. Recall Lemma 2 where we showed that X belongs to C γ 2 L 2 FL α,p for (γ, α) ∈ D so we must expect that δv ∈ C γ 2 FL α,p since the Λ term will belong (at worst) to C 1+ 2 FL α,p . But then we have X(δv, v) + X(v, δv) ∈ C 2γ 3 FL α,p so that we must require 2γ > 1 in order for this term to be in the domain of Λ. The set D however contains only values of γ ≤ 1/2 so we are not able to study eq. (11) in C γ 1 FL α,p for (γ, α) ∈ D. This difficulty can be overcome by truncating the power-series solution (4) to higher order. By looking at eq. (5) we see that the next step in the expansion will generate the following expression where the trilinear operator X 2 ∈ C 2 L 3 FL α,p is defined by and satisfies the algebraic relation which show that X 2 is related to the previously defined operator X by a "quadratic" relationship (this motivates the abuse of the superscript 2 in the definition of X 2 ). For the regularity of X 2 we have the following result whose proof is again postponed to Appendix A.
Lemma 4 There exists unbounded operatorsX 2 ,X 2 ∈ L 3 FL α,p such that X 2 ts =X 2 ts +X 2 ts , δX 2 = 0 and when α > α * (p) = max(−1/p, −1/2) we have |X ts | L 3 F L α,p α,γ |t−s| and for any couple Note that the full operator X 2 can be controlled only in the smaller region D ′ = D ∩ {α > α * (p)} of the (γ, α) plane. This limitation has some connection with the low regularity ill-posedness discussed in [4] and is related to the periodic setting. To infer the Λ-equation related to eq. (12) we simply apply to it δ operator and we get

Rough solutions of KdV
From eq. (15) we can write down a second Λ-equation (beside eq. (11)) associated to the KdV equation: with To give a welldefined meaning to eq. (16) it will be enough that the argument of Λ belongs actually to its domain. Given an allowed pair (γ, α) ∈ D ′ which fixes the regularity of X and X 2 sufficient (and natural) requirements for v are Under these conditions They are in the domain of Λ if 3γ > 1 i.e. γ > 1/3. This fixes the limiting time regularity for the Λ eq. (16) and it turns out that for any 1 ≤ p ≤ ∞ and α > α * (p) there is a pair (γ, α) ∈ D ′ with γ > 1/3. This will fix the regularity of the initial data that we are able to handle.
To define solutions of eq. (16) let us introduce a suitable space to enforce the all the conditions in eq. (17). For any 0 < η ≤ γ and consider the complete metric space Q η whose elements are triples (y, for any two elements y, z ∈ Q η . With abuse of notations we will denote a triple (y, y ′ , y ♯ ) ∈ Q η using simply its first component. Moreover we denote with y 0 also the constant path in Q η with value (y 0 , 0, 0). The main result of the paper is the following theorem.
Since (γ, α) ∈ D the operators X and X 2 are regular enough so that the proof of the theorem can follow essentially the pattern of a similar result in [18]. A proof is given in Appendix B. Since v ∈ Q γ we have that v ♭ ∈ C 3γ 2 FL α,p with 3γ > 1, then v satisfy the property where Π = {0 = t 0 ≤ t 1 ≤ · · · ≤ t n = t} is a partition of [0, t] and |Π| = max |t i+1 − t i | its size. The limit is in FL α,p . The proof is simple: by eq. (16) and the second term goes to zero as |Π| → 0. Below we will also prove that an Euler scheme related to eq. (18) converges as the size of the partition goes to zero. A nice property of these solutions is the following (already noted by Christ for power series solutions of NLS [3]).

Corollary 6
Let v be the unique solution of the Λ-equation given by Thm. 5 and let u(t) = U (t)v(t).
Let P N be the Fourier projector on modes k such that |k| ≤ N and let N (ϕ)(t, ξ) = ∂ ξ (ϕ(t, ξ) 2 )/2 for smooth functions ϕ. Then in the sense of distributions in C([0, T * ], S ′ (T)) we have convergence of N (P N u) to a limit which we denote by N (u) and moreover the distributional equation Proof. We start by proving that N (P N u) → N (u) distributionally. It is enough to prove for any 0 ≤ s ≤ t ≤ T * the convergence of V t = t 0 U (−r)N (P N u(r))dr in FL α,p since any smooth test function can be approximated in time by step functions. Now using eq. (16) we have where V ♭ is a remainder term, Y ts = X ts (P N × P N ) and From the regularity proofs for X and X 2 the following facts are easy to prove: (i) Y, Y 2 enjoy at least the same regularity of X and X 2 ; (ii) as N → ∞ they are equibounded in C γ L 2 FL α,p and C 2γ L 3 FL α,p respectively; (iii) Y ts → X ts and Y 2 ts → X 2 ts in the strong operator norm for fixed t, s. For V ♭ ts we have then the following equation (15)). Using the sewing map we have that the functions V ♭ are also equibounded in C 3γ This gives the distributional convergence of N (P N u). If we call N (u) the limit we have V t = t 0 U (−r)N (u(r))dr and u(t) = U (t)u(0) + t 0 U (t − r)N (u(r))dr which is the mild form of the differential equation. Some remarks are in order. The function v is γ-Hölder continous in FL α,p , p so that by dominated convergence the function u is only continuous FL α,p without any further regularity. It is not difficult to prove that for (γ, α) in the interior of D the solution v is actually in C γ FL α+ε,p for some small ε > 0. In this case it is clear that u ∈ C ε/3 FL α,p (cfr. the discussion of convergence of Galerkin approximation below and eq. (42)).
Fix p, α and γ such that γ > 1/3 and (γ, α) ∈ D ′ . Then an interesting property of the space Q γ is that for any continuous function z in FL α,p such that y(t) = U (−t)z(t) is in Q γ the distribution N (P N z) converge to a limit N (z). This follows from the proof of the previous corollary. Indeed for general elements y ∈ Q γ the analog of eq. (19) reads and by the convergence of the couple Y, Y 2 to X, X 2 we have δV → δV ∞ in C γ 1 FL α,p where δV ∞ is given by the Λ-equation δV ∞ = (1 − Λδ)[X(y, y) + X 2 (y, y ′ , y ′ )]. We can then define the distribution N (z) by letting t 0 U (−r)N (z(r))dr = V ∞ t and have N (P N z) → N (z) in weak sense.

L 2 conservation law and global solutions
It is well known that the KdV equation formally conserves the L 2 (T) norm of the solution. This conservation law can be used to show existence of global solution when the initial condition is in L 2 (T). Denote with ·, · the L 2 scalar product.
Proof. We will prove that δ v, v = 0. Let us compute explicitly this finite increment: Substituting in this expression the Λ-equation (16) we get Lemma 7 implies that v s , X ts (v s , v s ) = 0 and allows to cancel the X 2 term with the quadratic X term. After the cancellations the increment of the L 2 norm squared is Each term on the r.h.s. of this expression belongs at least to C 3η 2 R and since 3η > 1 this implies that the function t → |v t | 2 L 2 (T) is an Hölder function of index greater than 1 hence it must be constant.

Corollary 9
If v 0 ∈ L 2 there exist a unique global solutions to the Λ-equation (16).
Proof. By Thm. 5 there exists a unique local solution up to a time T * which depends only on |v 0 | L 2 . Since |v T * | L 2 = |v 0 | L 2 we can start from T * and extend uniquely this solution to the interval [0, 2T * ] and then on any interval.
It would be interesting to try to adapt the I-method of Colliander-Steel-Staffilani-Takaoka-Tao [5] to extend the global well-posedness at least in the case p = 2 for any α > α * (2) = −1/2. The handling of correction terms to the conservation law seems however to require some efforts and we prefer to leave this study to a further publication.

Galerkin approximations
Recall that P N is the projection on the Fourier modes |k| ≤ N . In [6] it is proven that the solutions of the approximate KdV equation do not converge even weakly to the flow of the full KdV equation. In the same paper the authors propose a modified finite dimensional scheme and prove its convergence in H α (T) for any α ≥ −1/2.
Here we would like to propose a different scheme inspired by the rough path analysis. By partial series expansion for the twisted variable v (N ) (t) = U (−t)u (N ) (t) is not difficult to show that the unique solution of equation (21) satisfy the Λ-equation where X (N ) = P N X(P N × P N ) and where the trilinear operator X (N ),2 is defined as so that δX (N ),2 (ϕ 1 , ϕ 2 , ϕ 3 ) = 2X (N ) (ϕ 1 , X (N ) (ϕ 2 , ϕ 3 )). These are just multi-linear operators in a finite-dimensional space and to have convergence of the Galerkin approximation it would be enough that both converge in norm to their infinite-dimensional analogs X, X 2 . A decomposition for X (N ),2 analogous to that of X 2 described in Lemma 4 holds X and we will prove in the appendix that Lemma 10 For any pair (γ, α) in the interior of D we have that as N → ∞, Unfortunately it is not difficult to see thatX (N ),2 cannot converge in norm, indeed we have and there is no way to make this converge in norm toX 2 ts =X (∞),2 ts due to the explicit dependence of the cutoff on |k|, |k 2 | which cannot be compensated by the regularity of the test functions or by the 1/k 1 factor. A way to remove this difficulty is to modify the finite dimensional ODE in order to remove this operator in the Λ-equation. This is possible sinceX (N ),2 ts is proportional to t − s so that it admits an obvious differential counterpart. Let use define the trilinear operator Γ (N ) as and note that Then the modified Galerkin scheme is still finite dimensional since Γ (N ) P ×3 N = P N Γ (N ) P ×3 N and is equivalent to the Λ-equation whereX (N ),2 =X (N ),2 +X 2 now do converge in norm to X 2 and satisfy the correct algebraic relations. As a consequence of the Lipschitz continuity of the solution of the Λ-equation (25) w.r.t X, X 2 and v 0 implies the following convergence result.
Corollary 11 Let 1 ≤ p ≤ +∞ and α > α * (p), then for any u 0 ∈ FL α,p as N → ∞ the Galerkin approximations v It is whortwhile to note that this result imply that sup t∈[0,T * ] |u (N ) (t) − u(t)| F L α,p → 0 as N → ∞ while for the finite dimensional scheme devised in [6] the convergence holds only in the sense that sup t∈[0, only for a very low frequency part of the solution. It is interesting to remark that the modified ODE (24) remains an Hamiltonian flow on P N (H −1/2 (T)\R) endowed with the symplectic structure given by Ω(u, v) = 0<|k|≤N u(−k)v(k)/(ik). Its Hamiltonian is given by

A discrete time scheme
The solution described by the Λ-equation (16) can be approximated by a discrete Euler-like scheme defined as follows. For any n > 0 let y n 0 = v 0 and y n i = X i/n,j/n (y n i−1 ) + X 2 i/n,j/n (y n i−1 ) for i ≥ 1. The combination of this scheme with the Galerkin approximation discussed before provide an implementable numerical approximation scheme for the solutions of KdV with low regularity initial conditions. Indeed the next theorem can be combined with Corollary 11 to obtain effective rates of convergence.
Using the triangular array τ k l we rewrite the above expression as a telescopic sum: where r ij = j−1 l=i v ♭ l/n,(l+1)/n . This last term is readily estimated by we want to show that M n N ≤ An 1−3γ uniformly in n for some constant A depending only on the data of the problem: this will imply the statement of the theorem since then |∆ n i | ≤ |q n 0i − p 0i | + |r 0i | + An 1−3γ i n ≤ |r 0i | + An 1−3γ i n ≤ Cn 1−3γ for any i ≤ N and We proceed by induction on ℓ. For ℓ = 1 the statement is clearly true since ∆ n 1 − ∆ n 0 − q n 0,1 + p 0,1 + r 0,1 = 0, moreover for the same reason we have, for all l that ∆ n l+1 − ∆ n l − q n l,l+1 + p l,l+1 + r l,l+1 = 0. Assume then that M n ℓ−1 ≤ A for some ℓ > 0. The basic observation is that when |i − j| ≤ ℓ the sums can be estimated in terms of M n ℓ−1 and various norms of v,X,X 2 much like in the proof of Thm. 5. The bound has the form where C = C(v 0 , X, X 2 ) and where the factor n 1−3γ is due to the previous estimate on r ij . Then since 3γ > 1 and Let m n ℓ = n 3γ−1 M n ℓ , then, for n large enough m n ℓ ≤ F (m n ℓ−1 ) where F is the increasing map which, for ℓ/n small enough has a unique attracting fix-point under iteration starting from 0. In particular the iterations stay bounded and if we set By repeating this argument it is easy to prove that the bound holds for all ℓ ≤ nT , i.e. in the whole existence interval found in Thm. 5.

Remark 13
With a bit more of work it is possible to prove the existence of the solution stated in Thm. 5 using directly the discrete approximation as done by Davie [9] for rough differential equations.

Additive stochastic forcing
As another application of this approach we would like to discuss the presence of an additive random force in the KdV eq. (1): where ∂ t ∂ ξ B a white noise on R×T and where Φ is a linear operator acting on the space variable which is diagonal in Fourier space: Φe k = λ k e k where {e k } k∈Z is the orthonormal basis e k (ξ) = e ikξ / √ 2π and such that λ 0 = 0. In this way the noise does not affect the zero mode. In the rest of this section fix 1 ≤ p ≤ +∞, α > α * (p) and γ ∈ (1/3, 1/2) such that (γ, α) ∈ D ′ and assume that The transformed integral equation (analogous to eq. (3)) associated to (32) is where w t (k) = λ k β k t and {β k · } k∈Z * is a family of complex-valued centered Brownian motions such that β −k = β k and with covariance E[β k t β q s ] = δ k,q (t ∧ s). The relation between the initial noise B and the family {β k · } k∈Z * is given by (34) can be expanded in the same way as we have done before and the first interesting Λ-equation which appears is the following: Here the random operator X w ts : FL α,p → FL α,p is given by and satisfy the equation δX w tus = X tu (ϕ, δw us ). For any couple of integers n, m we have where we used the Gaussian bound By interpolation this gives E |δw ts | r F L α,p |λ| r ℓ α,p |t − s| r/2 for all r ≥ p ≥ 2 which is finite by assumption (33). By the standard Kolmogorov criterion this implies that a.s. δw ∈ C ρ 2 FL α,p for any ρ < 1/2 and a-fortiori δw ∈ C γ 2 FL α,p by choosing ρ ∈ [γ, 1/2). To prove that a sufficiently regular version of the Gaussian stochastic process X w exists we will use a generalization of the classic Garsia-Rodemich-Rumsey lemma which has been proved in [18].

Lemma 15
For any θ > 0 and p ≥ 1, there exists a constant C such that for any R ∈ C 2 V ((V, | · |) some Banach space), we have where U θ,p (R) = The operator X w behaves not worse than X 2 : Lemma 16 Under condition (33) we have X w ∈ C 2γ 2 LFL α,p a.s.. Proof. After an integration by parts, X w can be rewritten as The first term in the r.h.s. belongs to C 2γ 2 LFL α,p path-wise: |X ts (ϕ, δw ts )| F L α,p ≤ |X ts | LF L α,p |δw ts | α,p |ϕ| α,p ≤ X C γ 2 LF L α,p δw C γ 1 F L α,p |ϕ| F L α,p |t − s| 2γ . Let us estimate the random operator I ts : ϕ → I ts (ϕ). Its Fourier kernel is The majorizing kernel Q is the same appearing in the estimates for X so that we already know that Y 2 (Q) F L p < ∞ for all allowed pairs (γ, α). It remains to show that E sup k,k 1 |J ts (k, k 1 , k 2 )| n |t − s| n(1+γ) for arbitrarily large n. It is then enough to bound where the sum is finite for n large enough (depending on γ − γ ′ ). Then choosing p sufficiently large, we have U 2γ+2/p,p (I) < ∞ a.s.. Moreover δI tus (ϕ) = X tu (ϕ, δw us ) so that δI can be bounded path-wise in C 2γ 3 LFL α,p . Then Lemma 15 implies that I 2γ ≤ C(U 2γ+2/p,p (I) + δI 2γ ) < ∞ a.s. ending the proof.
Then, modifying a bit the proof of Thm. 5, is not difficult to prove the following.
Theorem 17 For any 1 ≤ p ≤ +∞ and α > α * (p) eq. (35) has a unique local path-wise solution in C γ FL α,p for any initial condition in FL α,p .
When p = 2 we obtain solutions for noises with values in H α (T) for any α > −1/2. In this way we essentially cover and extend the results of De Bouard-Debussche-Tsutsumi [10]. Their approach consist in modifying Bourgain's method to handle Besov spaces in order to compensate for the insufficient Sobolev time regularity of Brownian motion.

Acknowledgment
I would like to thank A. Debussche which delivered a series of interesting lectures on stochastic dispersive equations during a 2006 semester on Stochastic Analysis at Centro de Giorgi, Pisa. They constituted the motivation for the investigations reported in this note. I'm also greatly indebted with J. Colliander and with an anonymous referee for some remarks which helped me to discover an error in an earlier version of the paper.

A Regularity of some operators
Some elementary results needed in the proofs of this appendix are the subject of the next few lemmas.

A.2 Proof of Lemma 4
Proof. We start the argument as in the proof of Lemma 2. The Fourier transform of X 2 reads we have X 2 ts = Y 3 (Φ 2 ts ). Indeed it is enough to restrict the sums over the set of k i for which kk 1 k 2 k 21 k 22 = 0. The multiplier Φ 2 ts has two different behaviors depending on the quantity h = kk 1 k 2 + k 2 k 21 k 22 being zero or not. We let Φ 2 ts =Φ 2 ts +Φ 2 ts wherȇ since the factorization h = k 2 (k 1 + k 21 )(k − k 21 ) implies that in the expression forΦ 2 ts the only relevant contributions come from the case where k = k 21 and k 1 = −k 22 or k = k 22 and k 1 = −k 21 and thus defining the operatorsX 2 ts = Y 3 (Φ 2 ts ) andX 2 ts = Y 3 (Φ 2 ts ) we have X 2 ts =X 2 ts +X 2 ts with δX 2 tus = 0.
Bound forX 2 . For any 0 ≤ γ ≤ 1/2, Lemma 19 gives So in order to boundX 2 ts in FL α,p with a quantity of order |t − s| 2γ it will be enough to bound separately the multipliers in FL p . The expression A 1 is nicely factorized in its dependence on the couple k 21 , k 22 and the multiplier Θ 1 can be easily bounded using (twice and iteratively) the same arguments as in Lemma 2. We will concentrate on the multiplier Θ 2 which requires different estimates. Since at least one of |k 1 |, |k 21 |, |k 22 | is larger than |k|/4 and similarly one of |k − k 1 |, |k − k 21 |, |k − k 22 | is larger than |k 1 |/2. Using this fact, when 1 + α − γ ≥ 0 and 1 − 2γ ≥ 0, we have where (a, b, c) and (d, e, f ) are two permutations of (1, 21, 22) (depending on k 1 , k 21 , k 22 ) so that we must have {a, b} ∩ {d, e} = ∅. It is clear then that Θ 2 1 so that Y 3 (Θ 2 ) is bounded in L 3 FL α,1 when 1 + α − γ ≥ 0 and 1 − 2γ ≥ 0. The bound in L 3 FL α,∞ will follow by showing that sup k k 1 ,k 21 Θ 2 (k, k 1 , k 21 , k 22 ) is finite. If 1 + α − γ > 1 and 1 − 2γ ≥ 0 we can bound this quantity by When 1 + α − γ ≥ 0 a bound in any L 3 FL α,p for p ∈ [1, +∞] is obtained from a bound of . This last quantity is then finite for any γ < 1/(2p). By interpolation of these various estimates we obtain again that Y 3 (Θ 2 ) is bounded in L 3 FL α,p for any (γ, α) ∈ D. Note that the extra factors in the majorizing sums can be used to show that actually in the interior of the region D the operator X 2 in bounded from FL α,p to FL α+ε,p for some ε > 0.