Odalric-Ambrym Maillard
email: odalric.maillard@inria.fr

Rémi Munos
email: remi.munos@inria.fr

Brownian Motions and Scrambled Wavelets for Least-Squares Regression

We consider ordinary (non penalized) least-squares regression where the regression function is chosen in a randomly generated sub-space G P ⊂ S of finite dimension P , where S is a function space of infinite dimension, e.g. L 2 ([0, 1] d ). G P is defined as the span of P random features that are linear combinations of the basis functions of S weighted by random Gaussian i.i.d. coefficients. We characterize the so-called kernel space K ⊂ S of the resulting Gaussian process and derive approximation error bounds of order O(||f || 2 K log(P )/P ) for functions f ∈ K approximated in G P . We apply this result to derive excess risk bounds for the least-squares estimate in various spaces. For illustration, we consider regression using the so-called scrambled wavelets (i.e. random linear combinations of wavelets of L 2 ([0, 1] d )) and derive an excess risk rate O(||f

which is arbitrarily close to the minimax optimal rate (up to a logarithmic factor) for target functions

a Sobolev space of smoothness order s > d/2. We describe an efficient implementation using lazy expansions with numerical complexity Õ(2 d N 3/2 log N +N 5/2 ), where d is the dimension of the input data and N is the number of data.

Introduction

We consider ordinary least-squares regression using randomly generated feature spaces. Let us first describe the general regression problem: we observe data D N = ({x n , y n } n≤N ) (with x n ∈ X , y n ∈ R), assumed to be independently and identically distributed (i.i.d.) from some distribution P , where x n ∼ P X and

y n = f * (x n ) + η n (x n ),
where f * is some (unknown) target function such that ||f * || ∞ ≤ L and η n is a centered, independent noise of variance bounded by σ 2 . We assume that L and σ are known. Now, for a given class of functions F, and f ∈ F , we define the empirical 2 -error

L N (f ) def = 1 N N k=1 [y n -f (x n )] 2 ,
and the generalization error

L(f ) def = E (X,Y )∼P [(Y -f (X)) 2 ].
The goal is to return a regression function f ∈ F with lowest possible generalization error L( f ). The excess risk L( f ) -L(f * ) measures the closeness to optimality.

In this paper we consider infinite dimensional spaces F that are generated by a denumerable family of functions {ϕ i } i≥1 , called initial features. We assume that f * ∈ F.

Since F is an infinite dimensional space, the empirical risk minimizer in F is certainly subject to overfitting. Traditional methods to circumvent this problem have considered penalization, i.e. one searches for a function in F which minimizes the empirical error plus a penalty term, for example

f = arg min f ∈F L N (f ) + λ||f || p p , for p = 1 or 2.
where λ is a parameter and usual choices for the norm are 2 (ridge-regression [START_REF] Tikhonov | Solution of incorrectly formulated problems and the regularization method[END_REF]) and 1 (LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the Lasso[END_REF]).

In this paper we follow an alternative approach introduced in [START_REF] Maillard | Compressed Least-Squares Regression[END_REF], called Compressed Least Squares Regression, which considers generating randomly a subspace G P (of finite dimension P ) of F, and then returning the empirical risk minimizer in G P , i.e. arg min g∈G P L N (g). Their work considered the case when F is of finite dimension. Here we consider the case of infinite dimension and provide a characterization of the resulting approximation spaces for which learning is possible.

Regression with random subspaces: Let us briefly recall the method described in [START_REF] Maillard | Compressed Least-Squares Regression[END_REF]. The random subspace G P is generated by constructing a set of P random features (ψ p ) 1≤p≤P defined as linear combinations of the initial features {ϕ i } 1≤i≤F (in their work, they assumed that F has a finite dimension F ) weighted by some random coefficients:

ψ p (x) def = F i=1 A p,i ϕ i (x), for 1 ≤ p ≤ P,
where the coefficient A p,i are drawn i.i.d. from a centered distribution with variance 1/P . Here we explicitly choose a Gaussian distribution N (0, 1/P ). We write A the random P × F matrix with elements (A p,i ).

Then G P ⊂ F is defined as the vector space spanned by those features, i.e.

G P def = {g β (x) def = P p=1 β p ψ p (x), β ∈ R P }.
Now, the least-squares estimate g b β ∈ G P is the function in G P with minimal empirical error, i.e. g b β = arg min

g β ∈G P L N (g β ), (1)
and is such that β = Ψ † Y ∈ R P , where Ψ is the N × P -matrix composed of the elements: Ψ n,p def = Ψ p (x n ), and Ψ † is the Moore-Penrose pseudo-inverse of Ψ1 . The final prediction function g(x) is the truncation (to the threshold ±L) of g b β , i.e. g(x)

def = T L [g b β (x)],
where

T L (u) def = u if |u| ≤ L, L sign(u) otherwise.
The result of [START_REF] Maillard | Compressed Least-Squares Regression[END_REF] states that with probability at least 1 -δ (on the choice of the random matrix A, thus on G P ), the generalization error of g is bounded as

L( g) -inf f ∈F L(f ) = O P log(N ) N + ||α * || 2 sup x∈X ||ϕ(x)|| 2 log(N/δ) P , (2)
where α * is the vector coefficient of the target function f * ∈ F (i.e. f * = Our motivation The result (2) says that if the term ||α * || sup x ||ϕ(x)|| is small, then the least-squares estimate in the random subspace G P has low excess risk. The questions we wish to address now are: Can we characterize the spaces for which this is the case? And, since the final bound does not involve the dimension F , can we handle the case when F is of infinite dimension? This paper gives a positive answer to these questions.

An initial motivation for this work lies in the following simple property of the random features (ψ p ) 1≤p≤P :

Lemma 1 Provided that the initial features (ϕ i ) 1≤i≤F are continuous, each random feature ψ p is a Gaussian process indexed by the space X with covariance structure given by 1 P ϕ(x), ϕ(x ) (where ϕ(x) ∈ R F is the vector with components (ϕ i (x)) 1≤i≤F ).

Proof: By definition, ψ p (x) = A p ϕ(x) where A p is the p th row of the random matrix A, which contains i.i.d. N (0, 1 P ) entries. We may rewrite ψ p (x) = ψ x (A p ) to highlight the role of the random variable A p . Thus E A p (ψ x (A p )) = 0, and by definition:

Cov Ap (ψ x , ψ x ) = E Ap (ψ x (A p )ψ x (A p )) = 1 P F i=1 ϕ i (x)ϕ i (x ) = 1 P ϕ(x), ϕ(x )
The continuity of the initial features (ϕ i ) guarantees that there exists a continuous version of the process ψ p which is thus a Gaussian process.

Then it is natural to ask what happens when

F → ∞. Since ψ p (x) = F i=1 A p,i ϕ i (x)
, this means we want to understand and give a meaning to the object W (x) = ∞ i=1 ξ i ϕ i (x) (where we drop the index p for simplicity and introduce the standard normal i.i.d. variables (ξ i ) i ). We will use the theory of Gaussian random functions in Section 2.3 (see [START_REF] Lifshits | Gaussian random functions[END_REF][START_REF] Janson | Gaussian Hilbert spaces[END_REF]) for that purpose. But let us start with a motivating example.

Motivating Example 1: Brownian motions indexed by [0, 1]. Consider the following simple example where the initial features are multiscaled hat functions on the space X = [0, 1]. The mother hat function is

Λ(x) = xI [0,1/2[ + (1 -x)I [1/2,1[
, and the rescaled hat functions are Λ j,l (x) = 2 -j/2 Λ(2 j x -l) for any scale j ≥ 1 and translation index 0 ≤ l ≤ 2 j -1. We also write Λ 0 (x) = x and Λ 1 (x) = 1. This defines a basis of C 0 ([0, 1]) (introduced by Faber in 1910, and known as the Schauder basis, see [START_REF] Jaffard | Décompositions en ondelettes[END_REF] for an interesting overview).

Those functions are indexed by the scale j and translation index l, but all functions may be equivalently indexed by a unique index i ≥ 1.

We will see in Section 2.2 that the random features ψ p (x), defined as linear combinations of those hat functions weighted by Gaussian i.i.d. random numbers, are Brownian motions. In Section 3 we will show that such Brownian motions are indeed good for regression in the sense that if the target function f * belongs to the Sobolev space H 1 ([0, 1]) of order 1 (space of functions which have a weak derivative in L 2 ([0, 1])), then the least-squares estimate in G P (defined by P Brownian motions) has an excess risk of O( In Section 2, we study the random objects (random features, random subspace) considered in this setting, and precisely characterize a special space of functions related to these objects called the kernel space. We make use of two theories: (1) Gaussian Random Function Theory (that studies precise properties of objects like Brownian bridges, Strassen balls, etc.) that allows for good flexibility in the choice of the basis functions of the target space, and (2) Approximation Theory, and more precisely, Multi-Resolution Analysis (MRA) (which deals for instance with wavelets and Besov spaces).

In Section 3, we apply this analysis to provide generalization bounds which extend the results of [START_REF] Maillard | Compressed Least-Squares Regression[END_REF] to the case when F is of infinite dimension. The analysis shows that in Example 2, the excess risk of the least-squares estimate built with scrambled wavelets is arbitrarily close to the minimax rates on the associated kernel space (which is a Sobolev space).

Finally, in Section 5, we describe an efficient numerical implementation using a multiresolution tree structure that generates the expansion of the random features (ψ p ) only at the data points (lazy implementation). The resulting algorithm has numerical cost O(2 d N 3/2 log N ) (where d is the dimension of X ) for building the linear system, and O(N 5/2 ) for solving it.

Elements of theory about Gaussian objects

We now give an interpretation of the random features in terms of random processes and analyze the corresponding limit object when the dimension of the initial feature space F is infinite.

In this Section we will introduce the notion of a Gaussian object W (Section 2.1), define its kernel space K (Section 2.2), and its expansion (Section 2.3).

Gaussian objects

Let S be a vector space and S its dual. We write (•, •) its duality product.

Definition 2 (Gaussian objects) A random W ∈ S is called a Gaussian object if for all ν ∈ S , we have that (ν, W ) is a Gaussian (real-valued) variable. Now we call

• a ∈ S an expectation of W if ∀ν ∈ S , E(ν, W ) = (ν, a). • K : S → S a covariance operator of W if ∀ν, ν ∈ S , Cov((ν, W )(ν , W )) = (ν, Kν ).
When a and K exists, we write W ∼ N (a, K).

Example 1: Consider the case where S = C([0, 1]) is the space of continuous real-valued functions of the unit interval. Then S is the set of signed measures and we can define (ν, f ) = [0,1] f dν. Then the Brownian motion indexed by [0, 1] is a Gaussian object W ∈ C([0, 1]) with a ≡ 0 and K defined by (Kν)(t) = [0,1] min(s, t)ν(ds).

Definition of the kernel space

Given a Gaussian centered object W , one may naturally define a space K ⊂ S called the kernel space of N (0, K). It is built by first enriching S with all measurable linear functionals (w.r.t. W ), and then taking the dual of its closure. We now define it precisely by introducing the canonical injection I of the continuous linear functionals into the space of measurable linear functionals, and its adjoint I. We refer the interested reader to [START_REF] Lifshits | Gaussian random functions[END_REF] or [START_REF] Janson | Gaussian Hilbert spaces[END_REF] for refinements.

For any ν ∈ S , we have (ν, Kν) = E(ν, W ) 2 < ∞. Thus (ν, •) ∈ L 2 (S, N (0, K)) which is the space of square integrable functionals under measure N (0, K), i.e. {z : S → R, E W ∼N (0,K) |z(W )| 2 < ∞}. Now define the injection I : S → L 2 (S, N (0, K)) by I (ν) = (ν, •). Then the space of measurable linear functionals S N = I (S ) is the closure of the image of S by I (in the L 2 sense). It is a Hilbert space with inner product inherited from L 2 (S, N (0, K)), i.e. z 1 , z 2 S N = E(z 1 (W )z 2 (W )) (where z can be written as z = lim n (ν n , •) with ν n ∈ S ).

Provided that I is continuous (see Section 2.4 for practical conditions ensuring when this is the case) we define the adjoint I : S N → S of I , by duality: For any µ ∈ S , (µ,

Iz) = I µ, z S N = E W ((µ, W )z(W )), from which we deduce that (Iz)(x) = E W (W (x)z(W )).
Now the kernel space of N (0, K) is defined as

K def = I(I (S )) ⊂ S.
Equivalent construction of the kernel space. The kernel space can be built alternatively based on a separable Hilbert space H as follows:

Lemma 3 [START_REF] Lifshits | Gaussian random functions[END_REF] Let J : H → S be an injective linear mapping such that K = JJ , where J is the adjoint operator of J. Then the kernel space of

N (0, K) is K = J(H), endowed with inner product Jh 1 , Jh 2 H def = h 1 , h 2 H .
Example 1 (continued) In the case of the Brownian motions already considered, one may build K by choosing the Hilbert space H = L 2 ([0, 1]) and the mapping J : H → S defined by (Jh

)(t) = [0,t] h(s)ds, which satisfies (J ν)(t) = ν([t, 1]) and K = JJ . Thus, the kernel space K is J(L 2 ([0, 1])) = {k ∈ H 1 ([0, 1]); k(0) = 0},
) (Kν)(t) = [0,1] d Π d i=1 min(s i , t i )ν(ds). The Hilbert space is H = L 2 ([0, 1] d ) and we choose J to be the volume integral (Jh)(t) = [0,t] h(s)ds, which implies that K = JJ . Thus K = J(L 2 ([0, 1] d )) is the so-called Cameron-Martin space [8], endowed with the norm ||f || K = || ∂ d f ∂x1...∂x d || L 2 ([0,1] d ) .
One may interpret this space as the set of functions which have a d-th order crossed (weak) derivative

∂ d f ∂x 1 ...∂x d in L 2 ([0, 1] d )
, vanishing on the "left" boundary (edges containing 0) of the unit d-dimensional cube.

Note that in dimension d > 1, this space differs from the Sobolev space H 1 .

Extension of Example 1 to smooth functions One may extend the previous example to the space

S = C p ([0, 1] d ) of p-times continuously differentiable functions, by choosing H = L 2 ([0, 1] d ) and J : H → S defined by (Jh)(s 0 ) = [0,s0] [0,s1] ... [0,sp] h(u)duds p ...ds 1 .
Then the covariance operator defined by:

(Kν)(t 0 ) = [0,t 0 ],...,×[0,t p ] (J ν)(u)dudt p ...dt 1 , with (J ν)(u) = [0,1] d I 0≤u≤sp≤...≤s0 ν(ds 0 )ds 1 ...ds p , corresponds to a Gaussian random object W ∼ N (0, K) whose kernel space is K = J(L 2 ([0, 1] d )).
Indeed, S is the set of signed measures on the unit cube. Thus, for ν ∈ S and h ∈ H, we have

(J ν, f ) = [0,1] d (Jh)(s 0 )ν(ds 0 ), from which we deduce K = JJ .
Note that when p = 0, we recover the previous case of Brownian sheets.

Expansion of a Gaussian object:

Let (ϕ i ) i be an orthonormal basis of K (for the inner product •, • K ). From Lemma 3, in the case of the alternative construction via the mapping J and the Hilbert space H, one can build such an orthonormal basis with the functions ϕ i = Jh i where (h i ) i is an orthonormal basis of H. We now define the expansion of a Gaussian object W (see [START_REF] Lifshits | Gaussian random functions[END_REF]):

Lemma 4 Let {ξ i i.i.d. ∼ N (0, 1)} i≥1 . Then ∞ i=1 ξ i ϕ i is a Gaussian object, written W , with law N (0, K). It is called an expansion of W ∼ N (0, K).
Example 1 (continued) To build an expansion for the Brownian motions, we use the Haar basis of

L 2 ([0, 1]). It is defined by h j,l (x) = 2 j/2 h(2 j x -l), where h(x) = I [0,1/2[ -I [1/2,1[ , together with h 0 (x) = I [0,1] (x).
We deduce that a basis of the kernel space is obtained by the integrals of those functions (since Jh(t) = t 0 h(s)ds), i.e. which are the hat functions defined in the introduction: Λ j,l = Jh j,l , and Λ 0 = Jh 0 .

Note that the rescaling factor inside Λ j,l naturally appears as 2 -j/2 , and not 2 j/2 as usually defined in wavelet-like transformations.

In the sequel, we only consider the case of an orthogonal basis since this corresponds to the Examples 1 and 2 (described below), but note that orthogonality is actually not required (dictionaries of functions could be handled as well).

Expansion of the kernel space

We now give a characterization of the functions of the kernel space in terms of its basis. When S is a Hilbert space with orthogonal basis (h i ) i (thus ϕ i = h i defines a basis of K), the condition that I is continuous (see Section 2.1) can be guaranteed by the condition that i ||h i || 2 is finite. In such case, we have [START_REF] Lifshits | Gaussian random functions[END_REF]:

K = {f def = i α i ϕ i , i α 2 i < ∞}. ( 3
)
In the general case when S is not a Hilbert space, and we use the construction (via Lemma 3) of K via a mapping J : H → S, where H is a Hilbert space with orthonormal basis (h i ) i , the continuity of I is ensured provided that i ||ϕ i || 2 < ∞ where ϕ i = Jh i , in which case we also recover [START_REF] Bungartz | Sparse grids[END_REF].

Note that we write f = i≥1 α i h i , where α i are real coefficients, with the usual meaning that lim F ||f -F i=1 α i h i || = 0 in the least-squares sense. When (h i ) i is an orthogonal basis, the existence of such decomposition corresponds to the separability of the space, otherwise, it corresponds to (h i ) i being a Riesz basis.

Thus the kernel space K may be seen from two equivalent points of view: either as a set of functions that are expectations of some random process, or as a set of functions that are random linear combinations of the initial features. We will use both points of view to derive our results.

Example 2: Scrambled wavelets We now introduce a second example built from a family of orthogonal wavelets ( φε,j,l ) ∈ C q ([0, 1] d ) (where ε is a multi-index, j is a scale index, l a multi-index) with at least q > d/2 vanishing moments. For s ∈ (d/2, q), we define the so-called scrambled wavelets W = ε,j,l ξ ε,j,l ϕ ε,j,l , where ξ ε,j,l i.i.d.

∼ N (0, 1) and ϕ ε,j,l = 2 -js φε,j,l || φε,j,l || 2 are the rescaled wavelets. The term "scrambled wavelets" refers to the disorderly construction of this multi-resolution random process built from wavelets.

Lemma 5 The Kernel space of the law of the scrambled wavelets is the Sobolev space

H s ([0, 1] d ). Proof: The condition s > d/2 guarantees that ε,j,l ||ϕ ε,j,l || 2 = (2 d -1) j 2 -2js 2 jd < ∞,
so that the scrambled wavelets are well defined and the corresponding mapping I is continuous.

When the wavelet functions are C q ([0, 1] d ) with at least q > s vanishing moments, it is known that the homogeneous Besov space B s,β,γ ([0, 1] d ) admits the following characterization (independent of the choice of the wavelets [START_REF] Frazier | Decomposition of besov spaces[END_REF][START_REF] Bourdaud | Ondelettes et espaces de besov[END_REF]

): B s,β,γ = {f ; ||f || s,β,γ < ∞} where ||f || s,β,γ def = ε j 2 j(s+d/2-d/β) ( 2 j -1 l1•••l d =0 | f, φε,j,l | β ) 1/β γ 1/γ
When the family of wavelets satisfies this condition, we say that it is adapted to the space H s ([0, 1] d ) for s < q. One example of such wavelets is given by Daubechies wavelets in dimension 1. For instance, for s = 1, the Daubechies 3 wavelets with 3 > 1 vanishing moments are Lipschitz of order 1, 08 > s, i.e. C 1 ([0, 1]). For s = 2, we can consider Daubechies 10 wavelets with 10 vanishing moments (see [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]). The extension to dimension d is easy.

Thus, any f ∈ B s,β,β writes f = ε,j,l α ε,j,l ϕ ε,j,l with α ε,j,l = (f, ϕ ε,j,l ) and ϕ ε,h,l = φε,h,l || φε,h,l || . Let us consider α ε,j,l = 2 j(s+d/2-d/β) (f, φε,j,l ). Then we deduce that ||α|| β = ||f || s,β,β .

Thus if we introduce the rescaled wavelet functions

ϕ ε,j,l = 2 -j(s+1/2-1/β) φε,j,l || φε,j,l || 2 , any f ∈ B s,β,β writes f = ε,j,l α ε,j,l ϕ ε,j,l with ||α|| β < ∞. When β = 2, B s,β,β is H s ([0, 1] d ). Thus we deduce H s ([0, 1] d ) = {f = ε,j,l α ε,j,l ϕ ε,j,l ; ||α|| < ∞},
which is also the definition of the kernel space of W = ε,j,l ξ ε,j,l ϕ ε,j,l . This example explains how to handle Sobolev spaces H s ([0, 1] d ) of any order s > d/2.

Note that this result extends similarly to inhomogeneous Sobolev spaces H s 2 , where s ∈ R d is a multiindex, via tensorisation of one dimensional Sobolev spaces (see [START_REF] Sickel | Tensor products of sobolev-besov spaces and applications to approximation from the hyperbolic cross[END_REF]). In this case, if s i > 1/2 for all 1 ≤ i ≤ d, and ( φj,l ) j,l is a wavelet basis of H s 2 ([0, 1] d ) (adapted to this space), then one can check that the kernel space of the law of Brownian (inhomogeneous) wavelets ϕ j,l = 2 -

P d i=1 j i s i φj,l || φj,l || 2 is H s 2 ([0, 1] d ).
3

Back to regression

Now we wish to understand when a target function f * can be well approximated by functions in random subspaces. In this section we show that this is exactly when f * belongs to the kernel space K.

When kernel spaces are approximation spaces

We now interpret the functions of the random subspaces G P as empirical estimates of functions of the kernel space K of W . This provides conditions on the coefficients of the linear combinations to guarantee the convergence of the random functions, when the dimension P of the random subspaces increases.

Let us consider P i.i.d. realizations of a Gaussian object W with law N (0, K), i.e. W 1 , ..., W P ∼ N (0, K), together with their expansion on an orthonormal basis (ϕ i ) i≥1 of the kernel space K. Thus

W p = i≥1 ξ p,i ϕ i where ξ p,i i.i.d.
∼ N (0, 1). Lemma 6 G 0 P is the set of (functional) unbiased empirical estimates of functions in K w.r.t. the law of W (i.e. for f α ∈ K, for all x ∈ X , Eg Aα (x) = f α (x) and lim P →∞ g Aα (x) = f α (x), almost surely).

We define the random features as ψ

p def = i≥1 A p,i ϕ i = W p √ P ,
Proof: By definition of I, any function f α of the kernel space K may be written

f α (•) = E(W (•)z(W ))
for some z ∈ S N (see Section 2.2) that satisfies α i = z(ϕ i ). Thus, if we consider the empirical estimates g P (•) = 1 P P p=1 z(W p )W p (•) and introduce β p = z(W p ) √ P , then g P = P p=1 β p ψ p . Now, by linearity of z, we have β = Aα thus g P = g Aα . Since f α = i z(ϕ i )ϕ i , we deduce that g P ∈ G 0 p . Thus, for each f α ∈ K, E( g P ) P = f α and the Law of Large Numbers guarantees that for any x ∈ X , the sequence ( g P (x)) P converges a.s. to f α (x) when P → ∞.

Conversely, let (α i ) i≥1 be a sequence such that ||α|| < ∞. Then the sequence (g Aα ) P converges since z(W )W (•) ∈ L 1 (S, N (0, K)), where the linear functional z is defined by z(ϕ i ) = α i . Indeed, this measurability property is ensured by definition of S N whenever

i z 2 (ϕ i ) < ∞, i.e. ||α|| < ∞. Its limit is Eg Aα (•) = E[ 1 P P p=1 i ξ p,i α i j ξ p,j ϕ j (•)] = i α i ϕ i (•) which is in K.
Note that this extends the case of finitely many linear combinations to the case of infinitely many linear combinations (the transformation β = Aα generalizes the case of random matrices A that appears in [START_REF] Maillard | Compressed Least-Squares Regression[END_REF]).

One can go one step further and characterize the variance of the empirical estimates g Aα ∈ G 0 P . This provides a nice interpretation (discussed in Examples 1 and 2 after the lemma) of the product ||α * || 2 sup x ||ϕ(x)|| 2 that appears in the excess risk bounds of [START_REF] Maillard | Compressed Least-Squares Regression[END_REF] (see e.g. equation (2) of Section 1).

Lemma 7 The variance of the empirical estimate g

Aα ∈ G 0 P of any function f α = i α i ϕ i ∈ K , is, at any x ∈ X , V(g Aα (x)) = 1 P (f 2 α (x) + ||f α || 2 K ||ϕ(x)|| 2 ) ≤ 2 P ||f α || 2 K ||ϕ(x)|| 2 .
Proof: By definition, V(

g P (x)) = 1 P (E[(W (x)z(W )) 2 ] -f 2 (x)), where z(ϕ i ) = α i . Thus, it is sufficient to consider the term: E[(W (x)z(W )) 2 ] = E[( i ξ i ϕ i (x) j ξ j z(ϕ j )) 2 ] = E[( i j ξ i ξ j (γ i,j )) 2 ]
, with γ i,j = ϕ i (x)z(ϕ j ). From the fact that ξ i are i.i.d. N (0, 1) variables (thus E(ξ 4 i ) = 3), we deduce 2 , from which we conclude by using Cauchy-Schwarz for deriving the inequality.

E[(W (x)z(W )) 2 ] = i k =i γ i,i γ k,k + i j =i γ 2 i,j + i j =i γ i,j γ j,i + i 3γ 2 i,i = 2 i j z(k j )k j (•)z(ϕ i )ϕ i (•) + i j z(k j ) 2 ϕ 2 i (•), = 2f 2 α (x) + ||f α || 2 K ||ϕ(x)||
Example 1 (continued) When one considers Brownian sheets for regression with a target function f * = i α * i ϕ i that lies in the Cameron-Martin space K (defined previously), then the term

||α * || 2 sup x∈X ||ϕ(x)|| 2 that appears in (2) is bounded by 2 -d ||f * || 2
K . Indeed:

• Since the Haar basis (h i ) i is an orthonormal basis of L 2 ([0, 1] d ), we have ||f * || 2 K = || ∂ d f * ∂x 1 ...∂x d || 2 L 2 ([0,1] d )
which by definition is also

i (α * i ) 2 ||h i || 2 = ||α * || 2 . Thus ||α * || 2 = ||f * || 2 K .
• Remember that the functions (ϕ i ) i are the hat functions Λ j,l . The mother hat function Λ satisfies ||Λ|| ∞ ≤ 1/2. In dimension d, we consider the tensor product ϕ j,l of one-dimensional hat functions (thus j and l are multi-indices). Since the support of Λ is [0, 1], then for any x ∈ [0, 1] d , for all j there exists at most one l(x

) = l = (l 1 , ..., l d ) such that ϕ j,l (x) = Π d i=1 Λ j i ,l i (x i ) = 0. Thus ||ϕ(x)|| 2 = j,l ϕ 2 j,l (x) = j (Π d i=1 Λ j i ,l i (x i )) 2 ≤ j (2 - P d i=1 ji/2 2 -d ) 2 = 2 -2d (1-2 -1 ) d = 2 -d . Thus sup x∈X ||ϕ(x)|| 2 ≤ 2 -d .
Example 2 (continued) We now consider regression with scrambled wavelets. Assume that the mother wavelet φ has compact support [0, 1] d and is bounded by λ, and assume that the target function

f * = i α * i ϕ i lies in the Sobolev space H s ([0, 1] d ) with s > d/2. Then ||α * || 2 ||ϕ(•)|| 2 ≤ λ 2d (2 d -1) 1-2 -2(s-d/2) ||f * || 2 H s . Indeed: • We have ||α * || 2 = ||f * || 2 Bs,2,2 = ||f * || 2 H s = ||f * || 2 K (see Example 2 of Section 2.4).
• By definition, the rescaled wavelet are ϕ ε,j,l (x) = 2 -js φε,j,l (x) = 2 -js 2 jd/2 φε (2 j x -l), where φε (x) = Π d i=1 φε i (x). Thus for all x ∈ [0, 1] d , by the assumption on the support on φ, ||ϕ(x)|| 2 = ε j (2 -js 2 jd/2 φε (2 j x i -l i )) 2 , and by definition of λ, this is bounded by

ε j (2 -j(s-d/2) λ d ) 2 ≤ (2 d -1) λ 2d 1-2 -2(s-d/2) whenever s > d/2. Thus ||ϕ(x)|| 2 ≤ λ 2d (2 d -1) 1-2 -2(s-d/2) .
Note that in the case of inhomogeneous Sobolev spaces, we would have instead:

||α * || 2 ||ϕ(•)|| 2 ≤ λ 2d Π d i=1 (1-2 -2(s i -1/2) ) ||f * || 2 H s 2 .

Approximation error with random subspaces

We now provide an approximation result that generalizes Theorem 1 of [START_REF] Maillard | Compressed Least-Squares Regression[END_REF] when one considers the approximation of functions f ∈ K by the random subspaces G P .

From the results of the previous paragraph, one knows that whenever f α ∈ K, for any x ∈ X , the empirical estimate g Aα (x) concentrates around f α (x). The next result relies on the additional property that g Aα also concentrates around f α in || • || P -norm (the full proof makes use of Johnson-Lindenstrauss Lemma and is provided in Appendix A).

G 0 P = {g Aα ∈ G P , f α ∈ K}. For illustration, in Example 2, S = L 2 ([0, 1] d ), K = H s ([0, 1] d )
, the initial feature (ϕ i ) i≥1 are the rescaled wavelets, and the random features are the scrambled wavelets.

Let f α = i α i ϕ i ∈ K. Write g * the projection of f α onto G P , i.e. g * = arg min g∈G P ||f α -g|| P , and ḡ * = T L g * its truncation at the threshold L ≥ ||f α || ∞ . Notice that due to the randomness of the features (ψ p ) 1≤p≤P of G P , the space G P is also random, and so is ḡ * . The following result provides bounds for the approximation error ||f α -ḡ * || P both in expectation and in high probability.

Theorem 8 Whenever P ≥ 15 log P (i.e. P ≥ 20) we have:

E G P ||f α -ḡ * || 2 P ≤ 8 log P P (1 + √ 2)||α|| 2 sup x∈X ||ϕ(x)|| 2 + L 2 .
For any η > 0, whenever P/η ≥ 15 log(P/η), we have with probability 1 -η (w.r.t. the choice of the random subspace G P ),

||f α -ḡ * || 2 P ≤ 8 log P/η P (1 + √ 2)||α|| 2 sup x∈X ||ϕ(x)|| 2 + L 2 .

Excess risk bounds

We now return to the initial regression problem and derive excess risk bounds for the two specific examples.

The idea is to combine the approximation error bound from Theorem 8 with usual estimation error bounds for linear spaces (see e.g. [START_REF] Györfi | A distribution-free theory of nonparametric regression[END_REF]).

Let us consider a target function f * = i α * i ϕ i ∈ K. Remember that our prediction function g is the truncation

g def = T L [g b β ]
of the (ordinary) least-squares estimate g b β (empirical risk minimizer in the random space G P ) defined by [START_REF] Barron | Approximation and learning by greedy algorithms[END_REF].

Note that since we consider an infinite number of initial features, the method considered here is not directly implementable. However, we will address in Section 5 practical considerations concerning the additional approximation error and the numerical complexity of an algorithmic implementation.

We now provide upper bounds (both in expectation and in high probability) on the excess risk for the least-squares estimate using random subspaces.

Theorem 9 Whenever P ≥ 15 log(P ), we have the following bound in expectation (w.r.t. all sources of randomness, i.e. input data, noise, and random features):

E G P ,X,Y ||f * -g|| 2 P ≤ c max(σ 2 , L 2 ) (log N + 1)P N + 8 log P P 8(1 + √ 2)||α * || 2 sup x∈X ||ϕ(x)|| 2 + L 2 , ( 4
)
where c is a universal constant (see [START_REF] Györfi | A distribution-free theory of nonparametric regression[END_REF]). Now, for any η > 0, whenever P/η ≥ 15 log(P/η), we have the following bound in high probability (w.r.t. the choice of the random features): (and a similar bound holds in high probability). Indeed, this is a direct consequence of the previous Theorem and the results of Section 3.1 (which says that the kernel space of the law of the Brownian sheet is the Cameron-Martin space and that ||α * || = ||f * || K ).

E X,Y ||f * -g|| 2 P ≤ c max(σ 2 , L
Example 2: Regression with scrambled wavelets. Now consider a target function f * ∈ H s ([0, 1] d ), with s > d/2. Similarly, ordinary least-squares performed on the random subspaces spanned by P scrambled wavelets (as defined previously) has an excess risk (as a direct consequence of Theorem 9 and the results of previous section):

E||f * -g|| 2 P = O log N N P + log P P ||f * || 2 H s .
In both examples, by choosing P of order √ N ||f * || K , one deduces the excess risk

E||f * -g|| 2 P = O ||f * || K log N √ N . ( 6
)

Discussion

Minimax optimality Note that although the rate Õ(N -1/2 ) deduced from Theorem 9, Equation ( 6), does not depend on the dimension d of the input data X , it does not contradict the known minimax lower bounds, which are Ω(N -2s/(2s+d) ) for functions defined over [0, 1] d that possess s-degrees of smoothness (e.g. that are s-times differentiable), see e.g. Chapter 3 of [START_REF] Györfi | A distribution-free theory of nonparametric regression[END_REF]. Indeed, the kernel space is composed of functions whose order of smoothness may depend on d. For illustration, in Example 2, the kernel space is the Sobolev space

H s ([0, 1] d ) with s > d/2. Thus 2s/(2s + d) > 1/2.
Notice that if one considers wavelets with q vanishing moments, where q > d/2, then one may choose s (such that q > s > d/2) arbitrarily close to d/2, and deduce that the excess risk rate Õ(N -1/2 ) deduced from Theorem 9 is arbitrarily close to the minimax lower rate. Thus regression using scrambled wavelets is minimax optimal (up to logarithmic factors). Now, about Example 1, we are not aware of minimax lower bounds for Cameron-Martin spaces, thus we do not know whether regression using Brownian sheets (Example 1) is minimax optimal or not.

Links with RKHS Theory There are strong links between the kernel space of Gaussian objects and Reproducing Kernel Hilbert Spaces (RKHS). We now remind two properties that illustrate those links:

• There is a bijection between Carleman operators and the set of RKHSs [START_REF] Canu | Functional learning through kernel[END_REF][START_REF] Saitoh | Theory of reproducing Kernels and its applications[END_REF]. A Carleman operator is a linear injective mapping J : H → S (where H is a Hilbert space) such that J(h)(t) = Γ t (s)h(s)ds where (Γ t ) t is a collection of functions of H. From Lemma 3, a Carleman operator defines the kernel space of the law of the Gaussian object with covariance operator K = JJ . This kernel space is also a RKHS with a kernel k(s, t) = Γ s , Γ t . Now, conversely, from any kernel k, one can construct a Carleman operator J generating the RKHS [START_REF] Canu | Functional learning through kernel[END_REF].

• Expansion of a Mercer kernel. The expansion of a Mercer kernel k (i.e. when X is compact Haussdoff and k is a continuous kernel) is given by k(x, y) = ∞ i=1 λ i e i (x)e i (y), where (λ i ) i and (e i ) i are the eigenvalues and eigenfunctions of the integral operator

L k : L 2 (X , µ) → L 2 (X , µ) defined by (L k (f ))(x) = X k(x, y)f (y)dµ(y). The associated RKHS is K = {f = i α i ϕ i ; i α 2 i < ∞}, where ϕ i =
√ λ i e i , endowed with the inner product f α , f β = α, β l2 . This space is thus also the kernel space of the Gaussian object

W = i ξ i ϕ i .
The expansion of a Mercer kernel gives an explicit construction of the functions of the RKHS. However it may not be straightforward to compute the eigenvalues and eigenfunctions of the integral operator L k and thus the basis functions ϕ i in the general case.

The approach described in this paper (expansion of a Gaussian object via a linear injective mapping) enables to choose explicitly the initial basis functions, and build the corresponding kernel space. For example one may choose a multiresolution basis of a given Hilbert space (which is interesting for numerical implementations as described later), which is not obvious from the Mercer expansion. We believe that this main difference is of practical interest.

Note also that an alternative construction of a RKHS from a Gaussian object is given by [START_REF] Lifshits | Gaussian random functions[END_REF]: provided S contains Dirac measures, the kernel space K of a Gaussian measure N (0, K) on the space S is the RKHS with kernel k induced by K, i.e. defined as k(s, t) = (δ s , Kδ t ) (for example the Cameron-Martin space considered in Example 1 is a RKHS).

Related works

In [START_REF] Rahimi | Uniform approximation of functions with random bases[END_REF][START_REF] Rahimi | Random features for large-scale kernel machines[END_REF], the authors consider, for a given parameterized function Φ : X × Θ → R bounded by 1, and a probability measure µ over Θ, the space F of functions f

(x) = Θ α(θ)Φ(x, θ)dθ such that ||f || µ = sup θ | α(θ) µ(θ) | < ∞.
They show that this is a dense subset of the RKHS with kernel k(x, y) = Θ µ(θ)Φ(x, θ)Φ(y, θ)dθ, and that if f ∈ F, then with high probability over

(θ p ) p≤P i.i.d ∼ µ, there exist coefficients (c p ) p≤P such that f (x) = P p=1 c p Φ(x, θ p ) satisfies || f -f || 2 2 ≤ O( ||f || µ √ P ).
The method is analogous to the construction of the empirical estimates g Aα ∈ G P of function f α ∈ K in our setting (see Lemma 6). Indeed we may formally identify Φ(x, θ p ) with ψ p (x) = i A p,i ϕ i (x), θ p with the sequence (A p,i ) i , and the law µ with the law of this infinite sequence. However, note that the condition sup x,θ Φ(x, θ) ≤ 1 does not hold in our setting and the fact that Θ is a set of infinite sequences makes the identification tedious without the Gaussian random functions theory. Anyway, we believe that this link provides a better mutual understanding of both approaches (i.e. [START_REF] Rahimi | Uniform approximation of functions with random bases[END_REF] and the present paper).

In the work [START_REF] Barron | Approximation and learning by greedy algorithms[END_REF], the authors provide excess risk bounds for greedy algorithms (i.e. in a non-linear approximation setting). The bounds derived in their Theorem 3.1 is similar to the result stated in our Theorem 9. The main difference is that their bound makes use of the l 1 norm of the coefficients α * instead of the l 2 norm in our setting. It would be interesting to further investigate whether this difference is a consequence of the non-linear aspect of their approximation or if it results from the different assumptions made about the approximation spaces, in terms of rate of decrease of the coefficients.

Efficient implementation using a lazy multi-resolution expansion

In practice, in order to build the least-squares estimate, one needs to compute the values of the random features (ψ p ) p at the data points (x n ) n , i.e. the matrix Ψ = (ψ p (x n )) p≤P,n≤N .

Due to finite memory and precision of computers, numerical implementations can only handle a finite number F of initial features (ϕ i ) 1≤i≤F . In [START_REF] Maillard | Compressed Least-Squares Regression[END_REF] it was mentioned that the computation of Ψ, which makes use of the random matrix A = (A p,i ) p≤P,i≤F , has a complexity O(F P N ). However, thanks to our analysis, one can define the random features (ψ p ) p≤P as an expansion of Gaussian objects built from a multi-resolution basis (ϕ j,l ) j,l = (ϕ i ) i≤F of the underlying kernel space K. This implies that, provided that these functions have compact support (such as the hat functions or the Daubechie wavelets), we can significantly speed up the computation of the matrix Ψ by using a tree-based lazy expansion, i.e. where the expansion is built only when needed for the evaluation at the points (x n ) n .

Consider Example 2 (wavelets). In dimension 1, using a wavelet dyadic-tree of depth H (i.e. F = 2 H+1 ), the numerical cost for computing Ψ is O(HP N ) (using one tree per random feature). Now, in dimension d the classical extension of one-dimensional wavelets uses a family of 2 d -1 wavelets, thus requires 2 d -1 trees each one having 2 dH nodes. While the resulting number of initial features F is of order 2 d(H+1) , thanks to the lazy evaluation (notice that one never computes all the initial features), one needs to expand at most one path of length H per training point, and the resulting complexity to compute Ψ is O(2 d HP N ).

Note that one may alternatively use the so-called sparse-grids instead of wavelet trees, which have been introduced by Griebel and Zenger (see [START_REF] Zenger | Sparse grids[END_REF][START_REF] Bungartz | Sparse grids[END_REF]). The main result is that one can reduce significantly the total number of features to F = O(2 H H d ) (while preserving a good approximation for sufficiently smooth functions). Similar lazy evaluation techniques can be applied to sparse-grids. Now, using a finite F introduces an additional approximation (squared) error term in the final excess risk bounds or order O(F -2s d ) for a wavelet basis adapted to H s ([0, 1] d ). This additional error (due to the numerical approximation) can be made arbitrarily small, e.g. o(N -1/2 ), whenever H ≥ log N d . Thus, using P = O( √ N ) random features, we deduce that the complexity of building the matrix Ψ is O(2 d N 3/2 log N ) and solving the least squares system has a numerical cost O(P N 2 ) = O(N 5/2 ).

Conclusion and future works

We analyzed least-squares regression using sub-spaces G P that are generated by P random linear combinations of infinitely many initial features. We showed that the kernel space K of the related Gaussian object provides an exact characterization of the set of target functions f * for which this random regression works. We illustrated those results on two examples for which the kernel space is a known functional space. We derived a general approximation error result from which we deduced excess risk bounds of order O( log N N P + log P P ||f * || 2 K ).

Example 2 shows that least-squares regression with scrambled wavelets provides rates that are arbitrarily close to minimax optimality. For Example 1, the relevant space of functions is the Cameron-Martin space, for which we are not aware of minimax lower bounds in dimension d > 1. The functions in that space need only to possess one d-th order derivative in L 2 , thus this space is certainly larger than a Sobolev space of order d.

A limitation of Example 2 is that, so far, the scrambled wavelets are unable to consider Sobolev spaces of smoothness s ≤ d/2. Possible directions for handling such spaces may involve developing a similar analysis for Besov spaces with β < 2 and fractional Brownian motions, which is the object of future works.

A nice property of using multiscale objects like Brownian sheets and scrambled wavelets is the possibility to design efficient numerical implementations. We described a method for computing the regression function with complexity O(N 5/2 + 2 d N 3/2 log N ).

A Proof of Theorem 8

Proof: In order to prove that g Aα concentrates around f α in || • || P -norm, we sample auxiliary states

X 1 , • • • , X J i.i.d.
∼ P , and prove that on an event of high probability, we simultaneously have that:

• ||g Aα -f α || 2 P is close to the empirical estimate 1 J J j=1 |g Aα (X j ) -f α (X j )| 2 (thanks to Chernoff- Hoeffding inequality),
• each g Aα (X j ) concentrates around f α (X j ) when P is large (thanks to a version of Johnson-Lindenstrauss Lemma which approximately preserves the inner-products).

We make explicit the corresponding probability spaces. Consider the probability space defined over the product sample space Ω 1 × Ω 2 , where Ω 1 consists of all the possible realizations of J states X 1 , . . . , X J drawn i.i.d. from P X , and Ω 2 is the set of all possible realizations of the random elements (A p,i ) 1≤p≤P,i≥1 (which defines the random feature space G P ).

For any δ > 0, set ε 2 def = 8 P log(4J/δ). Thus for P ≥ 15 log(4J/δ) we have ε ≤ 3/4 thus ε 2 /4 -ε 3 /6 ≥ ε 2 /8 and P ≥ log(4J/δ) ε 2 /4-ε 3 /6 . We may thus apply a version of Johnson-Lindenstrauss (JL) Lemma for inner-products, deduced from JL Lemma for norms by polarization inequality (see Proposition 1 of [START_REF] Maillard | Compressed Least-Squares Regression[END_REF]). This lemma is stated for vectors of finite dimension F , and can be extended in our setting to the case of infinite converging sequences. Indeed, it applies to the two truncated sequences α F = (α 1 , . . . , α F ) and ϕ F = (ϕ 1 (X j (ω 1 )), . . . , ϕ 1 (X j (ω 1 )) F ) for any finite F , and the extension follows due to the converging properties of these two sequences w.r.t. the random elements (A p,i ) 1≤p≤P,i≥1 and the measurability of the limit objects. Thus for any ω 1 ∈ Ω 1 (which defines X 1 (ω 1 ), . . . , X J (ω 1 )), there exists an event Ω 2 (ω 1 ) ⊂ Ω 2 of probability P(Ω 2 (ω 1 )) ≥ 1 -δ such that for all ω 2 ∈ Ω 2 (ω 1 ), for all 1 ≤ j ≤ J,

|A(ω 2 )α • A(ω 2 )ϕ(X j (ω 1 )) -α • ϕ(X j (ω 1 ))| ≤ ε||α|| ||ϕ(X j (ω 1 ))|| ≤ ε||α|| sup x∈X ||ϕ(x)||. ( 7
) Let Ω = {{ω 1 } × Ω 2 (ω 1 ), ω 1 ∈ Ω 1 }. Obviously, P(Ω) = Ω1 P(Ω 2 (ω 1 ))dω 1 ≥ 1 -δ. Now for each ω 2 ∈ Ω 2 , define Ω 1 (ω 2 ) = {ω 1 ∈ Ω 1 , {ω 1 } × {ω 2 } ∈ Ω} ⊂ Ω 1 .
Let δ > 0. By Chernoff-Hoeffding's inequality, we have that for any ω 2 ∈ Ω 2 (which defines A(ω 2 ) and the random subspace G P (ω 2 ))), there exists an event Ω 1 (ω 2 ) ⊂ Ω 1 (ω 2 ) with probability P(Ω 1 (ω 2 )|Ω 1 (ω 2 )) ≥ 1 -δ such that for all ω 1 ∈ Ω 1 (ω 2 ), we have Thus by setting δ = δ = (log P )/P and J = (log P )/4, we deduce that whenever P ≥ 15 log P we have the result in expectation (4).

E X∼P X |A(ω 2 )α • A(ω 2 )ϕ(X) -α • ϕ(X)| 2 ≤ 1 J J j=1 |A(ω 2 )α • A(ω 2 )ϕ(X j (ω 1 )) -α • ϕ(X j (ω 1 ))| 2 +
Similarly, for the result in high probability, we deduce that with probability The result [START_REF] Frazier | Decomposition of besov spaces[END_REF] follows by setting δ = δ = η/P log(P/η) and J = (1/4) log(P/η).

F i=1 α * i ϕ i ) and ||u|| 2 def = F i=1 u 2 i

 2 . The bound shows a usual decomposition in terms of estimation error (first term) and approximation error w.r.t. G P (second term). Now, the best tradeoff, obtained for P of order ||α * || sup x ||ϕ(x)|| √ N , provides an excess risk of O(||α * || sup x ||ϕ(x)|| log N/δ √ N ).

  the Sobolev space of order 1 with functions being equal to 0 on the left boundary. Now, for the extension to dimension d, we consider the space S = C([0, 1] d ) and the covariance operator of the Brownian sheet (Brownian motion in dimension d

where A p,i def = 1 √

 1 P ξ p,i . Using abbreviated notations, we write ψ p (•) = A p ϕ(•). Similarly, for any β ∈ R P , and (α i ) i≥1 ∈ l 2 (N) we write β = Aα to mean β p = i≥1 A p,i α i , for p = 1 . . . P .Define G P as the span of the random features (ψ p ) 1≤p≤P , and G 0 P as the set of functions g Aα ∈ G P such that f α ∈ K, i.e.: G 0 P def = {g β = P p=1 β p ψ p ; s.t. β = Aα, and ||α|| < ∞}.

Figure 1 :

 1 Figure1: Schematic view of the kernel space K, the space G P spanned by the random features (ψ p ) 1≤p≤P , andG 0 P = {g Aα ∈ G P , f α ∈ K}. For illustration, in Example 2, S = L 2 ([0, 1] d ), K = H s ([0, 1] d ), the initial feature (ϕ i ) i≥1 are the rescaled wavelets, and the random features are the scrambled wavelets.

  Our contributionIn this paper, we analyze least-squares regression with random subspaces and illustrate our analysis using two examples: Brownian motions (Example 1) and what we call scrambled wavelets (Example 2), defined as linear combination of wavelets weighted by random Gaussian coefficients.

	P log N N	+||f || 2 H 1	log P P ).
	Now choosing P of order ||f || H 1

√ N , we deduce the excess risk O(||f || H 1 log N √ N ).

  Example 1: Regression with Brownian sheets. Consider a target function f * that lies in the Cameron-Martin space K (equipped with the norm||f || K def = || ∂ d f ∂x1...∂x d || 2 L 2 ). Then ordinary least-squares performed on the random subspaces spanned by P Brownian sheets has an expected excess risk

	2 ) 2)||α E||f * -g|| 2 (log N + 1)P N + 8 log P/η P 8(1 + √ P = O log N N P + log P P ||f * || 2 K ,

* || 2 sup x∈X ||ϕ(x)|| 2 + L 2 . (5)

  ε||α|| 2 supBound in expectation Notice that trivially |f (x) -ḡ(ω 2 )(x)| ≤ |f (x) -g(ω 2 )(x)|. Thus[START_REF] Janson | Gaussian Hilbert spaces[END_REF] impliesE ω 2 ||f -ḡ * ω 2 || 2Now from the definition of ε, by setting δ = δ = (log P )/P and J = (log P )/4, we deduce that whenever P ≥ 15 log P we have:Bound in high probability Define Λ def = {ω 2 , P(Ω 1 (ω 2 )) ≥ 1 -γ} ⊂ Ω 2 . We have P(Ω ) = Ω2 P(Ω 1 (ω 2 ))dω 2 ≤ Ω2 I P(Ω 1 (ω 2 ))≥1-γ dω 2 + (1 -γ) Ω2 I P(Ω 1 (ω 2 ))<1-γ dω 2 ≤ P(Λ) + (1 -γ)(1 -P(Λ)). Now, since P(Ω ) ≥ 1 -δ -δ , we have P(Λ) ≥ 1 -δ+δ γ .We deduce that for η = δ+δ γ > 0, the event Λ has a probability P(Λ) ≥ 1 -η, and for all ω 2 ∈ Λ, from[START_REF] Janson | Gaussian Hilbert spaces[END_REF]. Now, by setting δ = δ = η/P log(P/η) and J = (1/4) log(P/η), we deduce that whenever P/η ≥ 15 log(P/η) we have with probability 1 -η (w.r.t. the random choice of A), Like in the proof of Theorem 8 in Appendix A, we consider the probability space defined over the sample space Ω 1 ×Ω 2 where Ω 1 consists of the possible realizations of J states X 1 , . . . , X J drawn i.i.d. from P X , and Ω 2 is the set of all possible realizations of the random features (ψ p ) p . Note that X 1 , . . . , X J are auxiliary random variable that should not be confused with the random training data (x n , y n ) n≤N . Denote by E X,Y the expectation w.r.t. the input data and the noise.

	||f -ḡ * ω2 || 2 P	≤	Ω 1 (ω 2 )	||f -g * ω2 || 2 P dω 1 +	(Ω 1 (ω 2 )) c	||f -ḡ * ω2 || 2 P dω 1
		≤ ε 2 ||α|| 2 sup x∈X	||ϕ(x)|| 2 1 +	log(1/δ ) 2J	+ (2L) 2 δ + δ η
	||f -ḡ * || 2 P ≤	8 log P/η P	(1 +	√ 2)||α|| 2 sup
							x∈X	||ϕ(x)|| 2 1 +	log(1/δ ) 2J	(8)
							N
		+	64 log(4J/δ) P	||α|| 2 sup x∈X	||ϕ(x)|| 2 1 +	log(1/δ ) 2J	+ (2L)
	P	≤		Ω	||f -g * ω 2 || 2 P dω 1 dω 2 +	(Ω ) c	||f -ḡ * ω 2 || 2 P dω 1 dω 2
		≤ ε 2 ||α|| 2 sup x∈X	||ϕ(x)|| 2 1 +	log(1/δ ) 2J	+ (2L) 2 (δ + δ )
	E ω 2 ||f -ḡ * ω2 || 2 P ≤	8 log P P	(1 +	√ 2)||α|| 2 sup

x∈X ||ϕ(x)|| 2 log(1/δ ) 2J .

We deduce that under the event Ω 1 (ω 2 ), inf

g∈G P (ω2) ||f α -g|| 2 P ≤ ||f α -g A(ω2)α || 2 P = E X∼P X |A(ω 2 )α • A(ω 2 )ϕ(X) -α • ϕ(X)| 2 ≤ ε 2 ||α|| 2 sup Let us define Ω = {Ω 1 (ω 2 ) × {ω 2 }, ω 2 ∈ Ω 2 } ⊂ Ω. Note that P(Ω ) = Ω2 P(Ω 1 (ω 2 ))dω 2 = Ω2 P(Ω 1 (ω 2 ))|Ω 1 (ω 2 ))P(Ω 1 (ω 2 ))dω 2 ≥ (1 -δ ) Ω2 P(Ω 1 (ω 2 ))dω 2 = (1 -δ )P(Ω) ≥ 1 -(δ + δ ) x∈X ||ϕ(x)|| 2 + L 2 . x∈X ||ϕ(x)|| 2 + L 2 .

B Proof of Theorem 9

Proof:

Since P(Ω ) ≥ 1 -(δ + δ ) and ||f * -g|| ∞ ≤ 2L, we have E ω 1 ,ω 2 ,X,Y ||f * -g(ω 2 )|| 2 P ≤ Ω E X,Y ||f * -g(ω 2 )|| 2 P + (2L) 2 (δ + δ ).

(9)

For any ω 2 ∈ Ω 2 , Theorem 11.3 of

[START_REF] Györfi | A distribution-free theory of nonparametric regression[END_REF]

says that

E X,Y ||f * -g(ω 2 )|| 2 P ≤ c max(σ

2 , L 2 ) (log N + 1)P N + 8 inf g∈G P (ω 2 ) ||f * -g|| 2 P , which, combined with (8) and applied in (9) gives that whenever P ≥ 15 log(4J/δ), E ω1,ω2,X,Y ||f * -g(ω 2 )|| 2 P ≤ c max(σ 2 , L 2 ) (log N + 1)P 2 (δ + δ ).

  1 -η, E X,Y ||f * -g(ω 2 )|| 2

	P	≤ c max(σ 2 , L 2 )	(log N + 1)P N
		+	64 log(4J/δ) P	||α|| 2 sup x∈X	||ϕ(x)|| 2 1 +	log(1/δ ) 2J	+ (2L) 2 δ + δ η	.

In the full rank case when N ≥ P , Ψ † = (Ψ T Ψ) -1 Ψ T


Maxime Hauray

Anne Nouri

WELL-POSEDNESS OF A DIFFUSIVE GYROKINETIC MODEL

We study a finite Larmor radius model used to describe the ions distributions in the core of a tokamak plasma, that consist in a gyro-kinetic transport equation, coupled with an electro-neutrality equation. Since the last equation do not provide enough regularity on the electric potential, we introduce a simple linear collision operator adapted to the finite Larmor radius approximation. Next we study the twodimensional dynamics in the direction perpendicular to the magnetic field and prove thanks to the smoothing effects of the collisions and of the gyro-average the global existence of solutions, as well as short time uniqueness and stability.

1 Introduction.

The model studied in that article describes the density of ions in the core of a tokamak plasma. In such highly magnetized plasma, the charged particles have a very fast motion of gyration around the magnetic lines, called the Larmor gyration. A good approximation is then to consider that the particles are uniformly distributed on gyro-circle, parametrized by their gyro-center, and Larmor radius r L (that is proportionnal to the speed of rotation u, and in our article, we will forget the physical constant and write r L = u). The models obtained in that new variables are kinetic in the direction parallel to the magnetic field lines, and fluids (precisely a superposition of fluid models) in the perpendicular direction. For rigorous derivation of such models and more complete discussion on its validity, we refer to FreSon [START_REF] Frénod | The finite Larmor radius approximation[END_REF] and our previous work GheHauNou09 [START_REF] Ghendrih | Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution[END_REF], in which the derivation is perform from a Vlasov equation in the limit of large magnetic field.

Such gyro-kinetic models are usually closed by an electro-neutrality equation, that as usual provide very few regularity for the eletric field, so that the well-posedness of gyro-kinetic models is, at least at our knowledge unknown. In this article, we add a "gyro-averaged" collision operator to the model and study the dynamics in the directions perpendicular to the field only.

Let us now describe our precise model. The ion distribution function f (t, x, u) in gyro-coordinates depends on the time t, the gyro-center position x ∈ T2 and the velocity of the fast Larmor rotation u ∈ R + (which is also proportional to the Larmor radius). The electric potential Φ depends only on (t, x). They satisfy the following system of equation on Ω = T

2 × R + ∂f ∂t + (J 0 u ∇ x Φ) ⊥ • ∇ x f = βu∂ u f + 2βf + ν ∆ x f + 1 u ∂ u (u∂ u f ) (1.1) eq:gyroFP2D (Φ -Φ * x H T )(t, x) = T (ρ(t, x) -1) (1.
2) eq:elecneut ρ(t, x) = (J 0 w f (t, x, w)2πwdw) (1.3) eq:defrho f (0, x, v) = f i (x, v) ∀(x, u) ∈ Ω (1.4) eq:indata where β and ν are two positive constant, ρ is the density in physical space, T is the ion temperature,

J 0 u h(x g ) = 1 2π 2π 0 h(x g + ue iϕc ) dϕ c , (1.5
) eq:J0

is the well known zero-order Bessel operator Watson [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF] and That model without the Fokker-Planck operator (ν = β = 0) was studied in our previous work

H T (x) = e -|x| 2 4T 2π 3

GheHauNou09

[4] -to which we refer for an heuristical derivation of the electro-neutrality equation ( eq:elecneutr eq:elecneutr 1.2) -and is used by physicists for simulation, by instance in Gysela code Gysela06 [START_REF] Grandgirard | GYSELA, a full-f global gyrokinetic Semi-Lagrangian code for ITG turbulence simulations[END_REF]. Here we just mention that ( eq:elecneutr eq:elecneutr 1.2) is obtained in a close to equilibrium setting, with an adiabatic hypothesis on the distribution of the electrons n e = n 0 e-eΦ Te ≈ n 0 1 + eΦ Te , and an hypothesis of adiabatic response of the ions on the gyro-circle which gives rise to the Φ * H T term. As usual in quasi-neutral equation, we have no good a priori estimates on the regularity of E = -∇Φ.

Remark that even if that equation ( eq:gyroFP2D eq:gyroFP2D

1.1) is derived from a Vlasov model (A rigourous derivation of a more general 3D model is performed for fixed field E in section sect:dif sect:dif 2), it is of "fluid" nature. In fact there is no transport in the remaining of the velocity variable u, and the position of the gyro-center is transported by the eletric drift (J 0 u E) ⊥ . So that the equation is similar to the 2D Navier-Stokes equation written in vorticity. More precisely, we have a family of fluid model depending on a parameter u, which are coupled thanks to diffusion in the u variable, and by the closure used for E described below.

Moreover, we will prove in the following that thanks to gyro-average J 0 u , equation has the same regularity than the NS2D equation in vorticity. In fact, the force field J 0 u ∇ x Φ belongs naturally to H 1 if f ∈ L 2 with some weight. That is why we obtain the same result that are known about the NS2D equation : global existence and short time uniqueness and stability. However, our model present an additional difficulties which is the lose of regularity for small u. In fact, for small value of u the H 1 bound (in x only) of J 0 u ∇ x Φ explodes.

To state our reuslts properly, we will need the following definitions and notations :

• In the sequel, the letter C will design a numerical constant, that may change form line to line. Unless it is mentioned, such constants are independent of everything.

• L 2 u (Ω) = L 2 (Ω, udxdu)
is the space of square integrable functions with respect to the measure udxdu. • We shall use various norm on T 2 or on Ω. To avoid confusion, we will use the following convention. All the norm performed on the whole Ω will have their weight with respect to u as additional indice. By instance

• 2πu , • H 1 2πu(1+u 2 )
. All the norms without any indices are norm on T 2 only.

• For any weight function k : R + → R + , the norm • 2,m is defined for any function f on Ω by

f 2,k = F (•, u) 2 k(u) du 1 2
• The most usefull weights will be m(u) = 2πu(1 + u 2 ) and m(u) = 1 + u 2 .

• We change a litlle bit the duality used to define distributions in the following definition Definition 1.1. Using distributions with the weight u means that duality is performed as

f, g u = f g dx g dv || udu .
This definition may seem a little artificial because the simple definition of derivative with respect to u, is not valid. Instead,

∂ u f, g u = -f, ∂ u g u - f u , g u .
However, this weight respects the underlying physics (u is in fact the 1D norm of a 2D velocity variable) and has many advantages. For instance the operator (1/u)∂ u (u∂ u ) is self-adjoint with this weight.

Our precise result are the following. We prove global existence under the hypothesis f i 2,m < +∞.

:existence Theorem 1.1. Let f i satisfy f i 2,m < +∞. Then there exists at least one weak solution

f ∈ L ∞ (R + , L 2 u (Ω))∩ L 2 (R + , H 1 u (Ω)) to (
eq:gyroFP2D eq:gyroFP2D

1.1)-(

eq:elecneutr eq:elecneutr 1.2) with initial condition f i , which also satisfies for any t > 0

f (t) 2 2,u + ν t 0 (∇ x , ∂ u )f 2 2,u ds ≤ f i 2,u ,
and all the a priori estimates of the previous section (Lemma lem:u-moment lem:u-moment

3.5,

lem:disp lem:disp

3.6,

lem:grad_xf lem:grad_xf

3.7) if their initial hypothesis are satisfied.

And we prove short time uniqueness and stability under the additional hypothesis

∇ x f 2,m < +∞. uniqueness Theorem 1.2. Let f i satisfy f i 2,m + ∇ x f 2,m < +∞ .
Then the positive time τ ⋆ defined in Lemma lem:grad_xf lem:grad_xf 3.7 is such that the weak solution to ( eq:gyroFP2D eq:gyroFP2D

1.1)-(

eq:indata eq:indata 1.4) , defined in Theorem thm:existence thm:existence

1.1, is unique on [0, τ ⋆ ] .
Moreover, that solution is stable on that interval of time in the following sense. Assume that (f n ) n∈N is a family of solutions given by theorem thm:existence thm:existence

1.1 with initial conditions f n i satisfying lim n→+∞ f n i -f i 2,m = 0 , and sup n∈N f n i L 2 m (L 4 ) < +∞ . Then lim n→+∞ sup t∈[0,τ * ] f n (t) -f (t) 2,m = 0 .
This local result has some more consequence when relating it to the bound on ν

T 0 ∇ x f 2 2,m dt ≤ f i 2 2,m + C(T ) f i 2 2
satisfied by any solution in the sense of thm:existence thm:existence 1.1. The last bound implies that ∇ x f 2,m is almost surely finite. The local result implies more : that the norm of gradient may blow up only on a closed and negligeable set, of 4/5-capicity zero..... In the next section the diffusive operator of ( eq:gyroFP2D eq:gyroFP2D

1.1) is rigourously derived from a linear Vlasov-Fokker-Planck equation in the limit of large magnetic field. In the third section, some useful lemmas are established, proving regularizing properties of the gyro-average, global preservation of some weighted norm of f , the short time preservation of the u(1 + u 2 )-moment of ∇ x f by the system ( eq:gyroFP2D eq:gyroFP2D

1.1)-(

eq:elecneutr eq:elecneutr 1.2)), and controlling the electric potential by the physical density. This allows to prove the global existence (Theorem thm:existence thm:existence 1.1) of solutions to the Cauchy problem in the fourth section and their short time uniqueness and stability (Theorem thm:uniqueness thm:uniqueness 1.2) in the fifth section. Finally some useful properties of the first Bessel function J 0 are proven in the appendix.

2 Derivation of the gyro-Fokker-Planck operator sect:dif

In that section, we rigorously justify the form of the Fokker-Planck appearing in the right-hand side of ( eq:gyroFP2D eq:gyroFP2D

1.1). The usual collision operator for plasmas is the nonlinear Landau operator originally introduced by Landau Landau [START_REF] Landau | The transport equation in the case of coulom interactions[END_REF]. Because of its complexity, simplified collision operators have been introduced. An important physical litterature exists on the subject, also in the gyro-kinetic case (See Brizard04 [START_REF] Brizard | A guiding-center fokker-planck collision operator for nonuniform magnetic fields[END_REF] and the references therein). In this paper we choose the simplest possible operator possible, namely a linear Fokker-Planck operator. The reasons of this choice are :

-Its simplicity will allow to focus on the other difficulties of the model, -The fact that physicists studying gyro-kinetic models for the core of the plasma mainly assume that the dynamics stays close to equilibrium, in which case a linear approximation of the collision operator is relevant.

-The aim of the paper is not a precise description of collisions. In fact, even if they exists in tokamaks, being needed to produce energy, their effect is small compared to the turbulent transport. However, we are interested by their regularizing effect, since the electro-neutrality equation ( eq:elecneutr eq:elecneutr 1.2) do not provide enough regularity to get a well-posed problem. This is a major difference to the Poisson equation setting.

We start from a simple model for a 3D plasma, i.e. a linear Vlasov-Fokker-Planck equation with an external electric field, an external uniform magnetic field and linear collision and drift terms, and obtain in the limit of large magnetic field a 3D (in position) equation analog to ( eq:gyroFP2D eq:gyroFP2D 1.1). In particular, we show that a usual linear Fokker-Planck term on the speed variables turns into an equation with diffusion terms both in space and Larmor radius variables in the limit.

Precisely, for any small parameter ǫ > 0 we study the distribution f ǫ (t, x, v) of ions submitted to an exterior electric field E(t, x) (independent of ǫ) and an uniform magnetic field B ǫ = (1/ǫ, 0, 0). We also model collisions (with similar particles and the others species) by a simple linear Fokker-Planck operator. To avoid any problem with possible boundary collisions, which are really hard to take into account in gyrokinetic theory, we assume that (x, v) ∈ T 3 × R 3 , where T 3 is the 3D torus. When the scale length of all the parameters are well chosen (in particular the length scale in the direction perpendicular to the magnetic field should be chosen of order ǫ times the length scale in the parallel direction, we refer to our previous work GheHauNou09 [START_REF] Ghendrih | Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution[END_REF] for more details on the scaling), the Vlasov equation f ǫ satisfies is

∂f ∂t + v ∂ x f + E • ∇ v f + 1 ǫ (v ⊥ • ∇ x ⊥ f + v ⊥ • ∇ v ⊥ f ) = div v (βvf ǫ ) + ν∆ v f ǫ , (2.1
) eq:vlares where β, ν are two positive parameters, the subscript (resp. ⊥) denotes the projection on the direction parallel (resp. on the plane perpendicular) to B, and the superscript ⊥ denotes the projection on the plane perpendicular to B composed with the rotation of angle π/2. In others

words if v = (v 1 , v 2 , v 3 ), v ⊥ = (v 1 , v 2 , 0), v || = (0, 0, v 3 ), v ⊥ = (-v 2 , v 1 , 0) .
The next results require the additional notation,

J0 u g(x g , ρ L , v ) = 1 2π 2π 0 g(x g + ue iϕc , ρ L e i(ϕc-π 2 ) + v e ) dϕ c , (2.2
) eq:J0tilde

which is a gyro-average performed in phase space, that will be used as an initial layer to adapt the initial condition to the fast Larmor gyration.

thm:FPgyro Theorem 2.1. Let E ∈ L ∞ t (L 2
) and f ǫ be a family of solutions to equation ( eq:vlares eq:vlares

2.1) with initial condition

f i ∈ L 2 satisfying sup t f ǫ (t) 2 ≤ f i 2 .
Then the family fǫ defined by

fǫ (t, x g , v) = f (t, x g + v ⊥ , v)
admits a subsequence that converges in the sense of distributions towards a function f depending only on (t, x g , u = |v|, v || ) and solution to

∂ t f + v ∂ x f + J 0 u E ∂ v f +(J 0 u E) ⊥ • ∇ xg f = β(v || ∂ v || f + u∂ u f + 3 f ) + ν ∆ xg ⊥ f + 1 u ∂ u (u∂ u f ) , (2.
3) eq:FPlim in the sense of distributions with the weight u, with the initial condition J0 u (f 0 ).

Remark 2.1. The reason for the change of variables is that the 1/ǫ-term in equation ( eq:vlares eq:vlares 2.1) induces a very fast rotation in the perpendicular direction both in the x and v variables,

v(t) = v 0 e it/ǫ , x(t) = x 0 + v 0⊥ + v 0 e i(t/ǫ-π/2) .
But in the gyro-coordinates this fast rotation is simply described by a rotation in v,

v(t) = v 0 e it/ǫ , x g (t) = x 0 g . Remark 2.2.
The final diffusion appears in all dimensions except the x g || one. It does not mean that there is no regularization in that direction. Indeed, the models have diffusion in v || , which after some time regularize in the x g || direction. This mechanism is well known for the Fokker-Planck equation (see for instance bouchut [START_REF] Bouchut | Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system[END_REF]). However, we are not able to prove this phenomena in the non-linear setting because the electric field of the model lacks regularity. This is the reason why we will only study the 2D model.

Proof of Theorem

thm:FPgyro thm:FPgyro 2.1. We proved in a previous work

GheHauNou09 [4] that, provided f 0 ∈ L 2 and E ∈ L 1 t (W 1,2
x ), a subsequence of f ǫ solutions of ( eq:vlares eq:vlares 2.1) without the collision term converges towards a solution of ( eq:FPlim eq:FPlim 2.3) without the collision term. In order to simplify the presentation, we will neglect the electric field and the parallel translation terms. To obtain the result in full generality, the only thing to do is to add the argument given in our previous work to the one given below. For the same reason, we shall also not treat the problem of initial conditions.

So consider the above Vlasov Fokker-Planck equation without electric force field and parallel translation,

∂ t f + 1 ǫ (v ⊥ • ∇ x ⊥ f + v ⊥ • ∇ v ⊥ f ) = div v (βvf ) + ν∆ v f . (2.

4) eq:vlaFP

The first step is to use the change of variables (x, v) → (

x g = x + v ⊥ , v). Since ∇ v f = ∇ v f -∇ ⊥ xg f , ∆ v f = ∆ v f + ∆ xg ⊥ f -2∇ v • ∇ ⊥ xg f , ∇ v • (vf ) = v • ∇ v f + 3 f -v • ∇ ⊥ xg f ,

equation (

eq:vlaFP eq:vlaFP 2.4) becomes

∂ t fǫ + 1 ǫ v ⊥ • ∇ v fǫ = -β v • ∇ v fǫ + 3 fǫ -v • ∇ ⊥ xg fǫ + ν ∆ v fǫ + ∆ xg fǫ -2∇ v • ∇ xg fǫ . (2.5
)

By hypothesis fǫ is bounded in L ∞ t (L 2
x;v ). Therefore, at least a subsequence of ( fǫ ) converges weakly to some f ∈ L ∞ t (L 2 ). Passing to the limit in ( eq:vlaFP eq:vlaFP 2.4), it holds that

v ⊥ • ∇ v f = 0 ,
since all the other terms are bounded. For v = (ue iϕ , v || ) where ϕ is the gyro-phase, the previous equality means that f is independent of the gyro-phase (in the sense of distribution and thus as a L 2 function). Equation ( eq:vlaFP eq:vlaFP 2.4) tested against a smooth function g independent of the gyro-phase writes

fǫ ∂ t g -β(v • ∇ v g -v • ∇ ⊥ xg g) -ν(∆ v g + ∆ xg ⊥ g -2∇ ⊥ xg • ∇ v g) dx g dv = 0 .
(2.6) eq:FP2

We may also pass to the limit when ǫ tends to zero in this equation and obtain that the same equality holds for fǫ replaced by f , considered as a function defined on

T 3 × R 3 . For the change of variable v = (ue iϕ , v || ), ∇ v g = (e iϕ ∂ u g + ie iϕ ∂ ϕ g, ∂ v || ).
Hence, for any function g independent on the gyrophase ϕ, it holds that

∆ vg g = ∂ 2 v || g + 1 u ∂ u (u∂ u g) , (∇ ⊥ xg • ∇ vg )g = ∇ ⊥ xg • (e iϕ ∂ u g) = e iϕ • ∇ ⊥ xg ∂ u g, v • ∇ v g = v || ∂ v || g + u∂ u g .
The other terms appearing in ( eq:FP2 eq:FP2 2.6) remain unchanged. Then,

f ∂ t g -β(v || ∂ v || g + u∂ u g -ue iϕ • ∇ ⊥ xg g) -ν(∂ 2 v || g + 1 u ∂ u (u∂ u g) + ∆ xg ⊥ g -2e iϕ • ∇ ⊥ xg ∂ u g) dx g dv || 2πududϕ = 0 .
(2.7) eq:FPuphi Since f is independent of ϕ, performing the integration in ϕ first makes the term containing ϕ vanish. So the function f of the five variables (x g , u, v || ) satisfies

f ∂ t g -β(v || ∂ v || g + u∂ u g) -ν(∂ 2 v || g + 1 u ∂ u (u∂ u g) + ∆ xg ⊥ g) dx g dv || udu = 0 .
(2.8) eq:FPuphi2

It exactly means that f satisfies the equation

∂ t f = β(v || ∂ v || f + u∂ u f + 3 f ) + ν ∂ 2 v || f + ∆ xg ⊥ f + 1 u ∂ u (u∂ u f ) , (2.9)
in the sense of distributions with weight u. It is the equation ( eq:FPlim eq:FPlim 2.3) without parallel transport nor electric field.

If we look at solutions of this equation invariant by translation in the direction of B, we exactly get the 2D-model announced in the introduction. In fact, if f is a solution of ( eq:FPlim eq:FPlim

2.3), then f (t, x, u) = f (t, x, u, v || ) dv ||
is a solution of ( eq:gyroFP2D eq:gyroFP2D

1.1). Such an assumption on f is reinforced by experiments and numerical simulations, where it is observed that the distribution of ions is quite homogeneous in x .

Some useful lemmas ct:apriori

We prove here some a priori estimates useful for the proof of our theorem. In order to simplify the proof of some of the following Lemmas, we sometimes uses the following formulation of ( eq:gyroFP2D eq:gyroFP2D

1.1) with the genuine two-dimensional velocity variable. Denote by f (t, x, u) = f (t, x, | u|), u ∈ R 2 . It is solution (in the sense of distribution with usual duality) of the following equation with 4D in space and velocity variables

∂ t f -∇ ⊥ x (J 0 | u| Φ) • ∇ x f = ν(∆ x f + ∆ u f ) + β(2 f + u • ∇ u f ). (3.1) eq:VFP4D
Heuristically, radial in u solution of equation ( eq:VFP4D eq:VFP4D

3.1) is a solution of ( eq:gyroFP2D eq:gyroFP2D

1.1). We can state for instance a precise Lemma in the case where φ is fixed and smooth. lem:3Deq4D Lemma 3.1. For a fixed smooth potential Φ, f is the unique solution of ( eq:gyroFP2D eq:gyroFP2D

1.1) with initial condition f i if and only if f is the unique solution of ( eq:VFP4D eq:VFP4D

3.1) with intial condition fi .

Proof of the Lemma lem:3Deq4D lem:3Deq4D

3.1 : The proof relies on the uniqueness of the solution to ( eq:VFP4D eq:VFP4D

3.1) (See

Ladyzen [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF] and the conservation of the radial symmetry of the solution.

3.1 Regularizing properties of the gyro-average.

In this section, some regularizing property of the gyro-average operato are proven. They are based on the fact that Ĵ0 ∼ k -1 2 for large k (the precise bound are proved in App:A App:A A), which implies that J 0 maps H s onto H s+ 1 2 . It is important since the formula ( eq:elecneutr eq:elecneutr 1.2) giving the gyro-averaged potential in terms of the distribution f involves two gyro-averages, and thus a gain of one derivative for the gyro-averaged potential w.r.t. f . However, the regularizing properties of J 0 u are bad for small u, which raises difficulties. The first lemma of this section gives the regularity of the gyro-averaged potential in term of the potential Φ. The second one gives the regularity of the density ρ in terms of the distribution f . We will need the two following definitions before stating it. Definition 3.2. Let f be a measurable function defined on Ω. Denote by

f L 2 m (H s ) = f (•, u) 2 H s m(u) dw 1 2
the norm with the weight m(u) = 2πu(1 + u 2 ).

For any U > 0, let F be a measurable function defined on

Ω U = T 2 × [0, U ]. Denote by F H 1 U = T 2 U 0 |f | 2 + |∇ x f | 2 + |∂ u f | 2 2πu du 1 2
.

The lemmas stating the regularity of Φ and ρ are the following.

lem:phireg Lemma 3.2. For any s ∈ R, u > 0 and Φ with 0-mean, it holds that

i) J 0 u Φ H s ≤ Φ H s , ii) J 0 u Φ H s+ 1 2 ≤ 2 1 4 √ u Φ H s , iii) ∂ u J 0 u Φ H s ≤ 1 √ u Φ H s+ 1 2 .
As a consequence, for any U > 0,

iv) J 0 u Φ H 1 U ≤ 4 √ U Φ H 1 2
.

lem:rhoreg Lemma 3.3. For any s > 0, if f 2πu dxdu = 1 and ρ is defined by ( eq:defrho eq:defrho

1.3), then ρ -1 H s+ 1 2 ≤ 2 1 4 π f L 2 m (H s ) . (3.2)
Proof of Lemma lem:phireg lem:phireg

and

lem:rhoreg lem:rhoreg

3.3. Denote by Φ(k) the k Fourier coefficient of Φ. Then J 0 u Φ 2 H s = ∞ k=1 |J 0 u (k)| 2 |Φ(k)| 2 ≤ ∞ k=1 |Φ(k)| 2 = Φ 2 H s ,
using the bound Ĵ0 ∞ ≤ 1 proved in Lemma lem:boundJ lem:boundJ A.1. It is the inequality i). For the second inequality, remark that 1 A.1 in

+ |k| 2 1 + w 2 |k| 2 = 1 w 2 1 + 1 |k| 2 1 + 1 w 2 |k| 2 ≤ 2 w 2 , k ∈ Z * , (3.3
J 0 u Φ 2 H s+ 1 2 = k =0 | Ĵ 0 u Φ(k, u)| 2 (1 + |k| 2 ) s+ 1 2 = k =0 | Φ(k)| 2 | Ĵ0 (|k|u)| 2 (1 + |k| 2 ) s+ 1 2 ≤ k =0 | Φ(k)| 2 (1 + |k| 2 ) s 1 + |k| 2 1 + |k| 2 u 2 ≤ √ 2 u Φ 2 H s .
For the third estimate of Lemma lem:phireg lem:phireg 3.2, remark that

∂ u Ĵ 0 u Φ (k) = ∂ u Ĵ0 (|k|u) Φ(k) = |k| Φ(k) Ĵ0 ′ (|k|u)
and use the bound iii) of Lemma lem:boundJ lem:boundJ

A.1 to get

|(∂ u Ĵ 0 u Φ)(k)| ≤ |k| u | Φ(k)| .
From this, we obtain

∂ u (J 0 u Φ) H s ≤ 1 √ u Φ H s+ 1 2 .
The point iv) uses the previous inequalities. First remark that the norm H 1 U is also equal to

F H 1 U = U 0 ∂ u F (•, u) 2 L 2 + F (•, u) 2 H 1 2πu du 1 2
.

Using this formulation and ii)-iii) leads to

J 0 u Φ 2 H 1 U = U 0 ∂ u J 0 Φ 2 L 2 + J 0 Φ 2 H 1 2πu du ≤ 2π Φ 2 H 1 2 U 0 1 + √ 2 u u du ≤ 16U Φ 2 H 1 2
, which gives the desired result and ends the proof of Lemma

|ρ(k)| ≤ 2π | f |(k, w)w (1 + w 2 |k| 2 ) 1/4 dw.

It follows from (

eq:calc eq:calc 3.3) that for k = 0,

(1 + |k| 2 ) 2s+1 4 |ρ(k)| ≤ 2 5 4 π ∞ 0 | f |(k, w)(1 + |k| 2 ) s 2 √ w dw ≤ 2 5 4 π ∞ 0 | f | 2 (k, w)(1 + |k| 2 ) s w(1 + w 2 ) dw 1/2 ∞ 0 dw (1 + w 2 ) 1/2 = 2 1 4 π ∞ 0 | f | 2 (k, w)(1 + |k| 2 ) s 2πw(1 + w 2 ) dw 1/2 .
Hence, since ρ(0) = T 2 ρ(x) dx = 1 by mass conservation,

ρ -1 H s+ 1 2 ≤ 2 1 4 π k =0 ∞ 0 | f (k, w)| 2 (1 + |k| 2 ) s 2 2πw(1 + w 2 ) dw ≤ 2 1 4 π ∞ 0 f (w) 2 H s 2πw(1 + w 2 ) dw 1/2
, and Lemma lem:rhoreg lem:rhoreg 3.3 is proved.

Control of the potential by the density.

Denote by L T the operator that maps any function Φ on T 2 with zero mean to 1 T (Φ -Φ * x H T ) and by H s 0 (T d ) the space of H s functions with zero mean. This section is devoted to a proof of the boundedness of

L -1 from H s 0 (T d ) onto H s 0 (T d ).
Recall that in a Fourier setting (See the Appendix of

|1 -Ĥ(k)| ≥ |k| 2 T 4 1 -e -1 |k| 2 T , ∀ k ∈ Z 2 \ {(0, 0)}.
As a consequence, the operator L -1 T maps any H s 0 , s ∈ R, into itself with norm

L -1 H s 0 ≤ c T := 4 1 -e -1 T . (3.4)
Remark 3.3. Lemma lem:boundL lem:boundL 3.4 shows that L -1 is bounded for small T , and of order T for large T , the physical case of interest. The boundedness of the spatial domain is essential. When defined on the whole space R 2 rather than on the torus, the operator L -1 is not bounded. Its norm explodes in the low frequency range.

Proof of the Lemma lem:boundL lem:boundL 3.4 Two bounds on J 0 (l) are used, namely one of the bounds of Lemma lem:boundJ lem:boundJ A.1 for l ≥ 1 and the following bound given by the Taylor expansion of J 0 near 0 for l ≤ 1,

0 ≤ (J 0 (l)) 2 ≤ 1 - l 2 4 , if 0 ≤ l ≤ 1 .
Consequently,

| ĤT (k)| ≤ 2 T 1 |k| 0 1 - (|k|u) 2 4 e -u 2 /T u du + √ 2 |k|T ∞ 1 |k| e -u 2 T du ≤ 2 w 0 1 - x 2 4w 2 e -x 2 x dx + √ 2w ∞ w e -x 2 dx ≤ 1 - 3 4 e -w 2 - 1 4w 2 (1 -e -w 2 ) + √ 2w ∞ w e -x 2 dx,
where w = (|k| √ T ) -1 . Now, using the bounds 2 -1 2 < 3 4 and

w ∞ w e -x 2 dx ≤ ∞ w xe -x 2 dx = e -w 2 2 , it holds that 1 -| ĤT (k)| ≥ 1 4w 2 (1 -e -w 2 ).
This is the first claim of lemma lem:boundL lem:boundL

3.4

The function of w in the right-hand side of the previous inequality on | Ĥ(k)| is decreasing and goes from 1 4 at 0 to 0 at +∞. Consequently its minimal value are obtained for large w i.e. for small |k|, namely |k| = 1. Precisely ,

1 -sup k =0 | ĤT (k)| ≥ T 4 1 -e -1 T .
Since the Fourier representation of L -1 T is the multiplication by T (1 -Ĥ(k)) -1 we obtain that in any H s 0 , s ∈ R,

L -1 T H s 0 = sup k =0 T |1 -Ĥ(k)| ≤ 4
1 -e -1 T , which is the desired result.

Propagation of L 2

m and L 2 m (L 4 ) norms of f .

sec:moment

The two following lemmas will be useful in the sequel.

m:u-moment Lemma 3.5. Assume that f i 2 2,u < +∞. Then, any solution of ( eq:gyroFP2D eq:gyroFP2D

1.1) and ( eq:indata eq:indata

1.4), for regular potential φ, satisfies ∀ t > 0, f (t) 2,u ≤ e βt f i 2,u .
Assume moreover that f i 2,m < +∞. Then any solution f satisfies

f (t) 2 2,m ≤ f i 2 2,m + (2ν + β) e 2βt -1 β f i 2 2,2πu . (3.5) eq:u-moment
with the convention that e 2βt -1

β = 2t if β = 0. lem:disp Lemma 3.6. Assume f i L 2 m (L 4 )
< +∞ and f is a solution of ( eq:gyroFP2D eq:gyroFP2D

1.1) with initial condition f i with a regular potential φ. Then f satisfies

f (t) L 2 m (L 4 ) ≤ e (β+2ν)t f i L 2 m (L 4 )
(3.6) eq:disp Remark 3.4. A more careful analysis will show that

f (t) 2 L 2 m (L 4 ) ≤ f i 2 L 2 m (L 4 ) + (2ν + β) e 2βt -1 β f i 2 L 2 2πu (L 4 ) ,
but the simple estimate of Lemma lem:disp lem:disp 3.6 will be sufficient.

Proof of Lemma

lem:u-moment lem:u-moment

3.5] Multiply equation (

eq:VFP4D eq:VFP4D

3.1) by f . Using the notations

u = | u|, g(t, u) = 1 2 f (t, •, u) 2 2 , (3.7)
and integrating in the x variable leads to

∂ t g -ν∆ u g = -(∇ x , ∇ u ) f (t, •, u) 2 2 + β(4g + u • ∇ u g). (3.8)
Multiply the previous equation by k( u), where k is a smooth function on R 2 with compact support and integrate in the velocity variable u leads to

∂ t g(t, u)k( u) d u + (∇ x , ∇ u ) f (t, •, u) 2 2 k( u) d u = (ν∆ u k( u) + 4βk(u) -β div(k( u) u))g(t, u) d u.
By approximation, this is still true for functions k with unbounded supports. For k( u) = 1,

∂ t e -2βt g(t, u) d u = -e -2βt (∇ x , ∇ u ) f (t, •, u) 2 2 d u ≤ 0.
Coming back to the 1D original quantities, it means that

f (t) 2,u ≤ e βt f i 2,u .
(3.9) eq:calc2

For k( u) = m(u), then ∆k = 4 and

4 m(u) -div( m(u) u) = 2 m(u) -m′ (u)u = 2 .
Therefore,

g(t, u) m(u) d u ≤ g(0, u) m(u) d u + 2(2ν + β) t 0 g(s, u) d u ds
Or in other words

f (t) 2 2,m ≤ f i 2 2,m + 2(2ν + β) t 0 f (s) 2 2,u ds .
Using the bound of equation ( eq:calc2 eq:calc2

3.9), we get

f (t) 2 2,m ≤ f i 2 2,m + (2ν + β) e 2βt -1 β f i 2 2,u .
with the convention that e 2βt -1

β = 2t if β = 0.
Proof of Lemma lem:disp lem:disp 3.6 In order to simplify the presentation, we will first performed calculation without justifying every integration by parts and division. But once we obtain an a-priori result, we will explain the small adaptation needed to make it rigorous. First, we denote

α(t, u) = | f (t, x, u)| 4 dx = f 4 4 , γ(t, u) = | f | 2 |∇ u f | 2 dx
Multiply equation ( eq:gyroFP2D eq:gyroFP2D

1.1) by 3 sign(f )|f | 3 and integrating with respect to x leads to

∂ t α = -12ν f 2 |(∇ x , ∇ u )f | 2 dx + ν∆ u α + 8βα + β u • ∇ u α .
(3.10) eq:alpha

Hence, dividing by √ α

∂ t √ α = ∂ t α 2 √ α ≤ -6ν γ √ α + ν∆ u α 2 √ α + 4β √ α + β u • ∇ u α 2 √ α
Now, we multiply by m(u), integrate with respect to u

∂ t √ α m(u)d u ≤ -6ν γ √ α m(u) d u + ν 2 ∆ u α √ α m(u)d u + 4β √ α m(u) d u + β u • ∇ u α 2 √ α m(u) d u
With the help of some integration by parts, we get that

u • ∇ u α 2 √ α m(u) d u = -2 √ α( m(u) + u 2 ) d u ∆ u α √ α m(u)d u = -2 u • ∇ u α √ α d u + |∇ u α| 2 2α 3 2 m(u) d u = 8 √ α d u + |∇ u α| 2 2α 3 2 m(u) d u .
Thanks to that, the previous inequality simplify in

∂ t √ α m(u)d u ≤ -6ν γ √ α m(u) d u + ν 4 |∇ u α| 2 α 3 2 m(u) d u + 2(β + 2ν) √ α d u
Next we can estimate |∇ u α| in terms of γ. In fact by Hölder inequality

∇ u α = ∇ u f 4 dx = 4 f 3 ∇ u f dx |∇ u α| 2 ≤ 16 f 4 dx f 2 |∇ u f | 2 dx = 16αγ
So that the second term in the right hand side of the previous inequality is controlled up to a constant to the first one. We precisely get

∂ t √ α m(u)d u ≤ -2ν γ √ α m(u) d u + 2(β + 2ν) √ α d u (3.11) eq:ineqdiff
From which we conclude easily.

In the previous calculation, we have not justified all the integrations by part. To make the argument rigorous, a possibility is to choose a smooth function

ξ 1 from R + into [0, ] such that ξ(u) = 1 if u ∈ [0, 1] and ξ(u) = 0 if u ∈ [1,

+∞), and define for all

U > 0 ξ U (u) = ξ u U . Remark that |U ξ ′ U | ∞ ≤ |ξ ′ | ∞ and |U 2 ξ ′′ U | ∞ ≤ |ξ ′′ | ∞ .
Then, we performed the previous calculation with the weight mU = mξ U , we obtain an inequality very similar to ( eq:ineqdiff eq:ineqdiff 3.11)

∂ t √ α mU (u)d u ≤ -2ν γ √ α mU (u) d u + 2(β + 2ν) + C U 2 √ α d u (3.12) eq:diffappr from which we get f (t) L 2 mU (L 4 ) ≤ e (β+2ν+ C U 2 )t f i L 2 mU (L 4 )
, which give the desired result letting U going to infinity.

The other point not rigorously justified is the division by √ α that may be zero. However, since we have a diffusion equation, it may be proved that for t > 0, α > 0 everywhere. Or we can use a family of smooth approximation of √ •. Or we can say that α + ǫ satisfy ( eq:alpha eq:alpha 3.10) with a additional term that has the good sign, so that it will satisfy ( eq:diffapprox eq:diffapprox 3.12), and we will obtain the desired inequality letting ǫ going to zero and then U going to infinity. It is well justified since the maximum principle applies there so that any solution with non-negative initial condition remains non-negative.

Short time estimate of the

m-moment of ∇ x f . nt-grad_xf
The following lemma provide is central in the proof of the stability and uniqueness of the solution for short time.

em:grad_xf Lemma 3.7. Assume that f is a solution of the system ( eq:gyroFP2D eq:gyroFP2D

1.1)-(

eq:indata eq:indata

1.4) satisfying initially ∇ x f i 2,m < +∞.
Then there exists a constant C * and a time τ * depending on (T, ν, ∇ x f i 2,m ), such that

∇ x f i 2 2,m + ν 2 τ * 0 (∇ x , ∂ u )∇ x f 2 2,m dt ≤ C *
We also mention that the result is true if the definition of Φ in ( eq:elecneutr eq:elecneutr

∂ t ∇ x f -∇ ⊥ x (J 0 u Φ)∇ 2 x,x f = β(2∇ x f + u • ∇ u ( f )) + ν∆ x, u (∇ x f ) -∇ x (∇ ⊥ x (J 0 u Φ)∇ x f .
If we now multiply by t ∇ x f on the left and integrate in x, the function h defined by h(t, u) = 1 2

|∇ x f | 2 dx satisfies, ∂ t g(u) = β(4g(u) + u • ∇ u g(u))ν∆ u g(u) -ν ∇ x, u ∇ x f 2 2 - t ∇ x f ∇ x (∇ ⊥ x J 0 u Φ) ∇ x f dx. ( 3

.13) eq:gevol

We may also multiply this equation by m(u) = (1 + u 2 ) and integrate it in u. We obtain after that

1 2 ∂ t ∇ x f 2 2, m + ν ∇ x, u ∇ x f 2 2, m = (4ν + 2β) ∇ x f 2 2,u 0 - t ∇ x f ∇ x (∇ ⊥ x J 0 u Φ) ∇ x f m(u) dx d u. (3.14) eq:gevol2
To go on, we need to understand a little better the matrix

M (t, x, u) = ∇ x (∇ ⊥ x J 0 u Φ). First remember that Φ = L -1
T (ρ -1), and then remark that from there definition, J 0 and L -1 commute with derivation in x. So that our term may be rewritten

M = J 0 u L -1 (∇ x (∇ ⊥ x ρ)).
Using the bound of the Lemma lem:rhoreg lem:rhoreg

and

lem:boundL lem:boundL

we obtain that

∀u > 0 , M (t, u) H 1 = J 0 u L -1 (∇ x (∇ ⊥ x ρ)) H 1 ≤ 2 1 4 c T √ u ρ -1 H 5 2 ≤ Cc T √ u f L 2 (H 2 m ) .
Moreover, the H 2 m -norm of f appears in the right-hand side of ( eq:gevol eq:gevol 3.13). So that we may use it to control M . With a control on the H 1 norm of M , we do not get an infinite bound on M , like

M ∞ ≤ C f H 2 .
In that case, we will be able to use a classical tool to conclude. But this is almost true, we are in a critical case (d = 2 and p = 2) for the Sobolev imbeddings, but we still know that the square of M is exponentially integrable. Precisely, since M is of average 0, we have for all u > 0 the following Trüdinger inequality

x e M 2 6 M 2 H 1 dx ≤ 2
We refer to Moser [START_REF] Moser | A sharp form of an inequality by Trudinger[END_REF] for a proof of that result. To estimate t ∇ f M ∇ f (1 + u 2 ) dxdu, we first perform the integral in x. For this, we apply the inequality

ab ≤ e a -b + b ln b , ∀ a, b > 0, to (a, b) = |M| 6 M 2 H 1 2 , |∇ f | 2 ∇ f 2 2
. The previous inequality comes from the Legendre transform of e a , as the Young inequality, so that our application will be a log-exp analog of the Hölder inequalities. We obtain

M ∇ f 2 = 6 M H 1 ∇ f 2   |M | 6 M 2 H 1 2 |∇ f | 2 ∇ f 2 2 2 dx   1 2 ≤ Cc T √ u f L 2 m(H 2 ) ∇ f 2 2 -1 + |∇ f | 2 ∇ f 2 2 ln |∇ f | 2 ∇ f 2 2 dx 1 2 , ≤ Cc T √ u f L 2 m(H 2 ) ∇ f 2 1 + ln ∇ x f 4 4 ∇ x f 4 2 1/2 , ≤ Cc T √ u f L 2 m(H 2 ) ∇ f 2 1 + ln c s ∇ 2 x,x f 2 2 ∇ x f 2 2 1 2
, where we have used the Jensen inequality -precisely, for a function g of integral 1, g ln g ≤ ln( g 2 ) -in the last but one line, and the Sobolev imbedding from H 1 into L 4 with constant c s in the last one. The constant C may change from line to line. Using this and Jensen inequality in the previous equation, we get

| t ∇ f M ∇ f | m(u) dxdu ≤ M ∇ f 2 ∇ f 2 m(u) du , ≤ Cc T | f L 2 m(H 2 ) ∇ f 2 2 1 + ln cs ∇ 2 x,x f 2 2 ∇x f 2 2 1 2 m(u) √ u du , ≤ Cc T f L 2 m(H 2 ) ∇f 2, m u ∇ f 2, m 1 + m(u) ∇ f 2 2 ∇ f 2 2, m ln c s ∇ 2 x,x f 2 2 ∇ x f 2 2 du 1 2 , ≤ Cc T f L 2 m(H 2 ) ∇f 2, m u ∇ f 2, m 1 + ln cs ∇ 2 x,x f 2 2, m ∇x f 2 2, m 1 2
Remark that the fraction inside the logarithm is always greater than 1 so that the square root is well defined.

In order to get a bound on ∇ x f 2, m u , we use that

∇ x f 2 2, 1 u = 2 h(u) d u |u| = 2 h(u) div u u |u| d u = -∇ u (|∇ x f | 2 ) • u |u| dxd u ≤ 2 ∇ 2 u f 2 ∇ x f 2 , (3.15) so that ∇ x f 2 2, m u = ∇ x f 2 2,u + ∇ x f 2 2, 1 u ≤ 2 ∇ x f 2, m + ∇ u ∇ x f 2, m ∇ x f 2, m Therefore, 1 2 ∂ t ∇ x f 2, m + ν ∇ x, u ∇ x f 2 2, m ≤ (4ν + 2β) ∇ x f 2 2,u 0 + Cc T ∇ 2 x,x f 2, m ∇ f 3 2 2, m ∇ x f 1 2 2, m + ∇ u ∇ x f 1 2 2, m 1 + ln c s ∇ 2 x,x f 2 2, m ∇ x f 2 2, m 1 2 .
(3.16) eq:gevol3

We next use the inequality 1 + ln(x) ≤ x ǫ ǫ , valid for any ǫ ∈ (0, 1), x > 0, and get We will use the Young inequalities xy ≤ x p p + y q q , where 1 p + 1 q = 1. For the first term, with p = 2 1+ǫ and then q = 2 1-ǫ , and for the second with p = 4 3+2ǫ and then q = 4 1-2ǫ . We obtain the following bounds

∂ t ∇ x f 2, m + ν ∇ x, u ∇ x f 2 2, m ≤ (4ν + 2β) ∇ x f 2 2,u 0 + . . . C(ǫ) ∇ x, u ∇ x f 1+ǫ 2, m ∇ x f 3 2 -ǫ 2, m ∇ x f 1 2 2, m + ∇ x, u ∇ x f 1 2 2, m .
b 1+ǫ a 2-ǫ ≤ 1 + ǫ 2 b 2 + 1 -ǫ 2 a 4+ 2ǫ 1-ǫ , and b 3 2 +ǫ a 3 2 -ǫ ≤ 3 + 2ǫ 4 b 2 + 1 -2ǫ 4 a 6+ 8ǫ 1-2ǫ , valid for ǫ < 1 2 .
Taking into account the two constants ν and C(ǫ) we get for ǫ < 1 2

1 2 ∂ t ∇ x f 2 2, m + ν 2 ∇ x, u ∇ x f 2 2, m ≤ (4ν + 2β) ∇ x f 2 2,m + . . . C(ǫ) 2 1-ǫ ν -1+ǫ 1-ǫ ∇ x f 4+ 2ǫ 1-ǫ 2, m + C(ǫ) 4 1-2ǫ ν -3+2ǫ 1-2ǫ ∇ x f 6+ 8ǫ 1-2ǫ 2, m
.

where C(ǫ) has only change from a numerical constant. With the notation h = ∇ x f 2 2, m, it gives

1 2 ∂ t h ≤ (4ν + 2β)h + C(ǫ) 2 1-ǫ ν -1+ǫ 1-ǫ h 2+ ǫ 1-ǫ + C(ǫ) 4 1-2ǫ ν -3+2ǫ 1-2ǫ h 3+ 4ǫ 1-2ǫ .
That is a differential inequality with a growth faster than linear, that give a solution that may explode in a finite time. The time of explosion τ * may be bounded below by something depending only on ν, ǫ, T and h(0) = ∇ x fi 2,m . Since ǫ may be choosen arbitrairy between 0 and 1 2 , τ * depends only on ν,β, T and the initial value h 0 .

Using that in the inequality ( eq:gevol3 eq:gevol3

3.16), and turning back to the original u variable, we obtain the existence of a constant

C 3 (T, ν, ∇ x f i m ) such that ∇ x f i 2 2,m + ν 2 τ * 0 (∇ x , ∂ u )∇ x f 2 2,m dt ≤ C 3 (T, ν, ∇ x f i 2,m )
A bound by below for τ * . In order to simplify the next two paragraphs, we assume that β ≤ ν. A careful analysis of the term in the right hand side of the previous equation shows that if ǫ is choosen small enough so that ν ≤ C(ǫ), then for h ≤ h := ν C(ǫ) 2 the dominant term is 4νh and for h ≥ h the dominant term is h 6+ 8ǫ 1-2ǫ . Then, if h(0) ≥ h, the explosion time is given by the equation

∂ t h ≤ C(ǫ) 2 1-2ǫ ν -3+2ǫ 1-2ǫ h 3+ 8ǫ 1-2ǫ .
For that equation, we get that the explosion time is larger than

τ * = C(ǫ) -4 1-2ǫ ν 3+2ǫ 1-2ǫ h(0) -2 1-2ǫ
In the case h(0) ≤ h, then for the early time, the equation may be rewritten

∂ t h ≤ 12νh ,
till h(0) = h. It take a time greater than T 1 = 1 12ν ln h h(0) . And after that time, the explosion time is given by the later calculation, and due to simplification it comes C ν . Finally, we get an explosion time

τ * = 1 12ν ln h h(0) + C ν
Best choice for ǫ. It is quite difficult to optimize that quantity in ǫ. But, as the condition on ǫ are 0 < ǫ < 1 2 and ν ≤ C(ǫ) = C cT ǫ , we can choose

ǫ = min 1 8 , C c T ν With that choice, we get i) If ν ≤ 8Cc T , τ * =    Cν 13 3 h -8 3 0 if h 0 ≥ C ν cT 2 C ν 1 + ln Cν 2 c 2 T h0 else ii) If ν ≥ 8Cc T , τ * = Cν 13 3 h -8 3 0 if h 0 ≥ 1 C ν (1 -ln h 0 )
else It is then clear that the value of τ * depends only on ν, h 0 and the temperature T .

The case of physical interest is the first one. Since in core of tokamaks, we have a large temperature T which implies a constant c T large, and a small colisionnality, in other words a small ν.

4 Existence of solutions.

:existence

In this section, we prove the existence theorem thm:existence thm:existence 1.1. The proof will use the following notation and a preliminary lemma. A priori estimates of the Lemma lem:u-moment lem:u-moment 3.5 on the solution (f, Φ) to eq:gyroFP2D eq:gyroFP2D

1.1-eq:elecneutr eq:elecneutr 1.2 on [0, T ] lead to the definition of the set K of functions f such that

f (t) 2,m ≤ √ M , a.a.t ∈ 0, T,
where

M = f i 2 2,m + (2ν + β) e 2βt -1 β f i 2 2,2πu
For each n > 0, we also introduce an approximation of the potential Φ n defined for any

f ∈ L 2 m by Φ n (t, x) := |k|≤n;k =0 e ik•x 1 1 -Ĥ(k) 2πJ 0 w fn (t, k, w)wdw -1 , (4.1)
lem:approx Lemma 4.1. For any n ∈ N * and any T > 0, there is a unique f n in K ∩ L 2 (0, T ; H 1 u (Ω)) solution to ( eq:gyroFP2D eq:gyroFP2D

1.1) with the potential Φ replaced by Φ n = Φ n (f n ) and initial condition f i . That solution satisfy all the a priori estimate of the previous section.

Proof of Lemma

lem:approx lem:approx 4.1 Let S be the map defined on K by S(f ) = g, where g is the solution in K ∩ L 2 (0, T ; H 1 u (Ω)) to ( eq:gyroFP2D eq:gyroFP2D

1.1) with the potential Φ n (f ) and initial condition f . The existence and uniqueness of S(F ) follows from

Lions-Magenes

[8] Thm 4.1 p 257, since ∇Φ n is bounded in L ∞ (0, T ; H 3 (T 2 )) by c n M for some constant c n . Then S maps K into K. Moreover, S is a contraction in L ∞ (0, T ; L 2 u (Ω)) for T small enough. Indeed, let g 1 = S(f 1 ) (resp. g 2 = S(f 2 )). By estimates very similar to the one performed in Lemma lem:rhoreg lem:rhoreg

it holds that

∀t ≥ 0, (Φ n (f 1 ) -Φ n (f 2 ))(t, •) L ∞ (T 2 ) ≤ cn (f 1 -f 2 )(t, •) 2,m ,
for some constant cn . Substracting the equation satisfied by g 2 from the equation satisfied by g 1 and integrating over Ω leads to

e 2(2ν+β)t 2 d dt e -2(2ν+β)t g 1 -g 2 2 2,m ≤ -ν (∇ x , ∂ u )(g 1 -g 2 ) 2 2,m -g 2 ∇ ⊥ (J 0 u (Φ n (f 2 ) -Φ n (f 1 ))) • ∇(g 1 -g 2 ) m(u) dxdu ≤ -ν (∇ x , ∂ u )(g 1 -g 2 ) 2 2,m + cn (f 1 -f 2 ) 2,m ∇ x (g 1 -g 2 ) 2,m g 2 2,m ≤ cn 2 4ν 2 (f 1 -f 2 ) 2 2,m g 2 2 2,m ≤ cn 2 M 4ν 2 (f 1 -f 2 ) 2 2,m
And so

g 1 -g 2 L ∞ (0,T ;L 2 m ) ≤ cT e 2(ν+β)T f 1 -f 2 L ∞ (0,T ;L 2 m )
. Hence there is a unique fixed point of the map S on [0, T 1 ] for T 1 small enough. The bounds used for defining T 1 being independent of T 1 , a unique solution of the problem can be determined globally in time by iteration. The fact that this unique solution satisfy the a-priori estimates of the next section is clear since these estimates only depends on the bound satisfied by Φ and not on its precise form.

Proof of Theorem

thm:existence thm:existence

1.1 The sequence (f n ) is compact in L 2 loc,u ((0, T ) × Ω). Indeed, it is bounded in L ∞ (0, T ; L 2 u (Ω U )) ∩ L 2 (0, T ; H 1 u (Ω U ))
for any bounded subset U > 0 (Recall of Ω U = T 2 × R + ). It follows from the interpolation theory that (f n ) is bounded in L 10 3 u ((0, T ) × T 2 × U ). Together with the boundedness of (∇ x Φ n (f n )) in L 2 loc,u ((0, T ) × Ω), this implies that ( ∂fn ∂t ) is bounded in W -1, 5 4 loc,u ((0, T ) × Ω U ). By the Aubin lemma Lions-Magenes [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF], it holds that (f n ) is compact in L 2 u ((0, T )×Ω U ), so converges up to a subsequence to some function f in L 2 u . It remains to pass to the limit when n → +∞ in the weak formulation satisfied by f n . A weak form of ( eq:gyroFP2D eq:gyroFP2D 1.1)-( eq:indata eq:indata 1.4) is that for every smooth test function α with compact support in [0, T [×Ω,

f i (x, u)α(0, x, u) u dxdu + t 0 f n ∂α ∂t + ∇ ⊥ x (J 0 u Φ n (f n )) • ∇ x α u dxduds = t 0 uν∇ x f n • ∇ x α + ∂ u f n ∂ u α + βu 2 f n ∂ u α dxduds . (4.
2) eq:weakform

The passage to the limit in ( eq:weakform eq:weakform 4.2) when n → +∞ can be performed if

lim n→∞ t 0 uf n ∇ ⊥ x (J 0 u (Φ n (f n )) • ∇ x αdxduds = t 0 uf ∇ ⊥ x (J 0 u (Φ(f )) • ∇ x αdxduds. This holds since (f n ) (resp. (∇ x (J 0 u (Φ n )(f n ))) strongly (resp. weakly) converges to f (resp. ∇ x (J 0 u (Φ(f ))) in L 2
loc,u ((0, T ) × Ω). And since the f n satisfy all the a priori bound, the limit f also satisfies them.

5 Short time uniqueness and stability of the solution.

uniqueness

In this section we prove the shot time uniqueness and stability theorem

d dt δf 2 2,m ≤ -ν (∇ x , ∂ u )δf 2 2,m + (4ν + 2β) δf 2 2,u + δf ∇ ⊥ x (δΦ) • ∇ x f 2 m(u) dxdu ≤ -ν (∇ x , ∂ u )δf 2 2,m + 4ν δf 2 2,m + δf ∇ x (δΦ) 2,um ∇ x f 2 2, m u . To estimate δf ∇(δΦ) 2,u m , apply the inequality ab ≤ e a -b + b ln b , a, b > 0, to (a, b) = ∇xδΦ 6 ∇ 2 x δΦ 2 2 , δf δf 2 2
for every nonnegative u and apply the Trüdinger inequality (See

Moser [9]) T 2 e ∇x δΦ 6 ∇ 2 x ψ 2 2 dz ≤ 2 . ( 5
δf ∇ x (δΦ) 2 2 = C ∇ 2 δΦ 2 2 δf 2 2 δf δf 2 2 |∇ x δΦ| 6 ∇ 2 δΦ 2 2 dx ≤ C ∇ 2 δΦ 2 2 δf 2 2 1 + (δf ) 2 δf 2 ln (δf ) 2 δf 2 2 dx . ≤ Cc 2 T u ∇ x δf 2 2,m δf 2 2 1 + ln δf 4 δf 4 2
Integrating in u with the weight um, it holds using again Jensen inequality that

δf ∇(δΦ) 2 2,um ≤ 2Cc 2 T ∇ x δf 2 2,m δf 2 2,m δf 2 2 δf 2 2,m 1 + ln δf 2 4 δf 2 2 m(u) du ≤ Cc 2 T ∇ x δf 2 2,m δf 2 2,m 1 + ln δf 2 L 2 m (L 4 ) δf 2 2,m . Consequently, 1 2
d dt δf 2 2,m ≤ Cc T ∇ x δf 2,m δf 2,m 1 + ln δf 2 L 2 m (L 4 ) δf 2 2,m ∇f 2 2, m u -ν (∇ x , ∂ u )δf 2 2,m + (4ν + 2β) δf 2 2,m ,
and finally using the inequality and from that inequality we get the short time uniqueness and stability. Remark that the previous calculation do not use ∇ x f 1 and this is why we do not need an assumption on this quantity in the stability result.

f (t) L 2 m (L 4 ) ≤ e (β-2ν)t f i L 2 m (L 4 ) from lemma lem:disp lem:disp 3.6 1 2 d dt δf 2 2,m ≤ Cc 2 T 4ν δf 2 2,m ln 2e δf 2 L 2 m (L 4 ) δf 2,m ∇f 2 2 2, m u + (4ν + 2β) δf 2 2,m , ≤ Cc 2 T 4ν δf 2 2,m ln r 2 e 2(β+2ν)t+1 δf 2 2,m ∇f 2 2 2, m u + (4ν + 2β) δf 2 2,m , where r = e 1 2 f 1,i L 2 m (L 4 ) + f 2,i L 2 m (L 4 ) . Defining s(t) = 1 r 2 δf 2 2,m e -2(
It remains to prove that H is bounded on [0, τ * ]. In fact,

τ * 0 ∇f 2 2 2,(1+u 2 ) 7 2 dt ≤ 2 5 2 τ * 0 ∇f 2 2 2,(1+u 7 ) dt, since (1+u 2 ) 7 2 ≤ 2 5 2 (1+u 7 ). Moreover, τ * 0 ∇f 2 2
2,u 0 dt may be bounded using Lemma lem:grad_xf lem:grad_xf 3.7 and the inequality ( eq:trick eq:trick 3.15) (we emphasize that the form used here is slighty different because we are in here in the 1D setting for the variable u) A Appendix: Some controls on the Bessel function of zero-th order.

App:A

The first Bessel function J 0 is much used in this paper. Indeed, in Fourier space, J 0 1 that appears in the definition of the gyroaverage of the electric field, is the multiplication by J 0 . Some properties of the function J 0 are given in Watson [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF]. In this appendix, some bounds on J 0 and its derivative are proven. The bound by ( √ 2k) -1 2 is obtained as follows. J 0 is solution of the ordinary differential equation k 2 (J 0 ) ′′ + k(J 0 ) ′ + k 2 J 0 = 0 , J 0 (0) = 1 , (J 0 ) ′ (0) = 0 .

(A.4)

The new unknown u = √ kJ 0 is solution to

u ′′ + 1 + 1 4k 2 u = 0 .
There are no exact initial conditions for u. However,

u(k) = k→0 + √ k[1 + O(k 2 )] , u ′ (k) = k→0 + 1 2 √ k [1 + O(k 2 )].
The second equation admits the k-dependent energy,

H(k) = H(k, u, u ′ ) = u ′2 2 + u 2 2 1 + 1 4k 2 ,
that satisfies

H(k) -H(k 0 ) = - k k0 u 2 (l) 4l 3 dl.
It follows from the behaviour of u near 0 that

H(k) = k→0 + 1 4k + O(k) .
Moreover the series expansion of J 0 near k = 0,

J 0 (k) = ∞ j=0 (-1) j k 2j 2 2j (j!) 2
and its alternate character if k ≤ 2 imply that u 2 (k) ≥ k -k 3 2 (valid for k ≤ √ 2). Using, the inequality and the behavior of H near 0, we get if 0 < k 0 ≤ k ≤ √ 2

H(k) ≤ 1 4k 0 + O(k 0 ) - k k0 1 4l 2 - 1 8 dl, H(k) ≤ 1 4k + k 8 , (A.5)
since the first line is satisfied for any k 0 > 0. Therefore,

u 2 (k) ≤ k k 2 + 2 4k 2 + 1 , k ≤ √ 2 .
A simple calculation shows that the function appearing in the right hand side is increasing in k, so that

u 2 (k) ≤ 1 √ 2 , k ∈ [0, √ 2] .
For k ≥ √ 2, simply remark that H is decreasing and that from ( eq:Hk eq:Hk A.5)

u 2 (k) ≤ 2H(k) ≤ 2H( √ 2) ≤ 1 √ 2
In any case we get u 2 (k) ≤ 2 -1 2 which gives the desired inequality.

Second inequality : It is a consequence of the first, for k ≥ 1. For k ≤ 1, it may be obtain from a comparison of the entire development of J 0 and (1 + k) -1/4 around the origin. We get

J 0 (k) ≤ 1 - k 2 4 + k 4 64 ≤ 1 - k 2 4 + 5k 4 32 - 15k 6 128 ≤ (1 + k 2 ) -1/4
Third inequality : Taking the derivative of J 0 in the definition ( eq:J0 eq:J0 (-1) j s j , where (h i ) 1≤i≤j are the points where sin(kθ) vanishes and 1,

h 0 = 0 < h 1 = π k < h 2 = 2π k < . . . < h j-1 = (j -1)π k < h j = 1.
The previous sum has alterned sign, the larger terms occuring for large i. Its terms are with increasing absolute values, except for the last one which is incomplete and may be smaller than the next to last term. However,

-s 1 ≤ s 0 -s 1 ≤ j i=0
(-1) j s j ≤ s 0 -s 1 + s 2 ≤ s 0 , so that

|(J 0 ) ′ (k)| ≤ max(s 0 , s 1 ) ≤ 2 π 1 1-π/k αdα √ 1 -α 2 ≤ 1 π 2π k - π 2 k 2 ≤ 2 πk , k ≥ π.
This ends the proof of the third inequality.

The proof of iv) is similar to the proof of ii), since

2 √ T |x| . ( 1 . 6 )

 216 eq:dfH We also used the notation b ⊥ = (-b 2 , b 1 ), for any vector b = (b 1 , b 2 ) of R 2 .

lem:phireg lem:phireg 3 . 2 .

 32 Proof of Lemmalem:rhoreg lem:rhoreg 3.3. Denote by ρ(k) the k-th Fourier term of ρ with respect to the space variable, i.e. ρ(k) = 2π J 0 (|k|w) f (k, w)w dw. By lem:boundJ lem:boundJ A.1,

GheHauNou09 [ 4 ]

 4 for more details), the operator H T = I -T L T is the multiplication by ĤT (k) = 2 T +∞ 0 J 0 (ku) 2 e -u 2 /T u du. lem:boundL Lemma 3.4. The Fourier multipliers ĤT (k) satisfy,

1. 2 )

 2 is replaced by another definition which still satisfies the bound given in Lemma lem:rhoreg lem:rhoreg 3.3 and lem:phireg lem:phireg 3.2. Precise bounds by below for τ * are given at the end of the proof (only in the case β = 0). Proof of Lemma lem:grad_xf lem:grad_xf 3.7 : We take the x-gradient of equation ( eq:VFP4D eq:VFP4D 3.1), written in 2D in u (with u = | u|), and obtain

3 2 +ǫ a 3 2

 33 with C(ǫ) = CcT ǫ √ c s . Now, with the temporary notations a = ∇ x f 2, m and b = ∇ x, u ∇ x f 2, m, what we need is to eliminate all the b in the right hand side, with the help of the b of the left hand side. Precisely, in the right hand side, we have the two terms b 1+ǫ a 2-ǫ , and b -ǫ .

1 . 2 .

 12 Denote by f 1 (resp. f 2 ) a solution to eq:gyroFP2D eq:gyroFP2D 1.1 for the field Φ 1 (resp. Φ 2 ), by δf = f 1 -f 2 and by δΦ = J 0 u (Φ 1 -Φ 2 ). Multiplying the equation satisfied by (1 + u 2 )δf by δf and integrating w.r.t. (x, u) with the weight u leads to 1 2

. 1 )

 1 eq:exp-int Therefore, using ∇ 2x δΦ 2 ≤ CcT √ u ∇δf 2,m (Lemmas lem:rhoreg lem:rhoreg

  f (t) 2,u ∇ x f (t)7 dt is bounded by Lemma lem:u-moment lem:u-moment 3.5, with n = 3.

2π 2π 0 e

 0 lem:boundJLemma A.1. J 0 satisfies the following estimates for all k ∈ R i)|J 0 (k)| ≤ min 10 ) ′ (k)| ≤ (1 + k 2 ) First Inequality : The bound |J 0 (k)| ≤ 1 is clear from the definition of J 0 , J 0 (k) = 1 ik cos θ dθ = 1 π π 0 cos(k cos θ) dθ . (A.3) eq:J0bis

1. 5

 5 ),(J 0 ) ′ (k) = i 2π 2π 0 cos θe ik cos θ dθ = -1 π π 0 cos θ sin(k cos θ) dθ , from which it is clear that |(J 0 ) ′ (k)| ≤ 1 for all k.Next we transform the previous integral in (J 0 ) ′ (k) = -

2 π < 2

 22 -1 4 .

2000 Mathematics Subject Classification. 41A60, 76P05, 82A70, 78A35.

Key words. Plasmas, gyrokinetic model, electroneutrality equation, Cauchy problem.