Didier Rafik Imekraz

Laurent Thomann

ON RANDOM HERMITE SERIES by

Keywords: 2000 Mathematics Subject Classification. -60G50 ; . -Harmonic oscillator, Hermite functions, random series, harmonic analysis

We study integrability and continuity properties of random series of Hermite functions. We get optimal results which are analogues to classical results concerning Fourier series, like the Paley-Zygmund or the Salem-Zygmund theorems. We also consider the case of series of radial Hermite functions, which are not so well-behaved. In this context, we prove some L p bounds of radial Hermite functions, which are optimal when p is large.

Introduction

In this paper we prove some optimal integrability and regularity results on the convergence of random Hermite expansions, i.e. on series of eigenfunctions of the harmonic oscillator with random coefficients.

Before we enter in the details, let us recall an old result on the 1-D torus T = R/2πZ. Let u(x) = n∈Z c n e inx and define the Sobolev space H s (T) by the norm u 2 H s (T) = n∈Z (1 + |n|) 2s |c n | 2 . By the usual Sobolev embeddings, if u ∈ H 1/2-1/p (T) with p ≥ 2 then u ∈ L p (T), but in general u / ∈ C(T). Paley and Zygmund (1930) have improved this result allowing random coefficients. Moreover if for some α > 1,

n∈Z ln α (1 + |n|)|c n | 2 < +∞ then a.s u ω ∈ C(T).
Many other results concerning random trigonometric series were obtained by Paley and Zygmund, as it is detailed in the book of J-P. Kahane [START_REF] Kahane | Some random series of functions[END_REF]. The study has been extended to random Fourier series on Lie groups (see Marcus-Pisier [START_REF] Marcus | Pisier -Random Fourier series with applications to harmonic analysis[END_REF]) and to Riemannian compact manifolds for orthonormal basis of eigenfunctions of the Laplace-Beltrami operator (see Tzvetkov [START_REF] Tzvetkov | Riemannian analogue of a Paley-Zygmund theorem[END_REF] and references therein).

On the torus T d = R d /(2πZ) d , there is a natural choice of the basis for the expansion, namely the (e in•x ) n∈Z . In our context (or more generally, if one study expansions on eigenfunctions of the Laplacian on a compact manifold) it is not clear which basis to choose, and the convergence properties of the random series u ω (x) = c n X n (ω)ϕ n (x) might depend on the choice of the basis (ϕ n ) n≥0 . For instance, an analogous result to Theorem 1.1 has been obtained by Tzvetkov [START_REF] Tzvetkov | Riemannian analogue of a Paley-Zygmund theorem[END_REF]Theorem 5] in compact manifolds with a condition depending on the L ∞ bound of the ϕ n .

Here we show that by adding a squeezing condition (see condition (1.5) below), we can use the intrinsic estimates of the spectral function, and obtain a convergence condition on the (c n ) which does not depend on the choice of the basis of Hermite functions. The idea to take profit of the bounds of the spectral function and of the Weyl law comes from [START_REF] Shiffman | Random polynomials of high degree and Lévy concentration of measure[END_REF][START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF] and has been fruitful in different contexts (see [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF][START_REF] Poiret | Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator[END_REF][START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF]), where results have been obtained for a large class of probability laws.

Here we extend this approach by working in a space Z s ϕ (instead of using condition (1.5)) and which also enables to exploit the estimates of the spectral function and which is compatible with the Lévy contraction principle of random series. We refer to the next paragraph for more details.

Let us now briefly describe our main contribution in this paper:

We first study integrability properties of the random series u ω . We then detail the case of series of radial Hermite functions, for which the situation is different than in the general case.

In a second time, we prove regularity results of the random series. We prove a Salem-Zygmund theorem which describes the behaviour of partial sums. We are then able to obtain an analogous result to Theorem 1.1 in our context, and we show that the ln factor is optimal. Finally, we state in Theorem 2.7 some more precise regularity results. Notice that due to dispersive effects of the harmonic oscillator on R d , the randomisation yields better estimates than on the torus.

In Proposition 2.4 we state some L p bounds of radial Hermite functions and which are optimal at least for p ≥ 2 large enough. Even if the proof is elementary, using the well-known asymptotic estimates of Laguerre functions, we did not find the result in the literature. Therefore, we have written down the details, since the estimates we obtain are better than the bounds of general Hermite functions.

Finally, we point out that the previous results have analogues for random series of eigenfunctions of the Laplacian on a Riemannian compact manifold or for the Laplacian on R d with a confining potential. These results can be obtained with the same strategy by using the corresponding bounds of the spectral function.

1.1. Functional analysis. -1.1.1. Some elements on the harmonic oscillator. -We consider the multidimensional harmonic oscillator H := -∆ + |x| 2 on L 2 (R d ) with d ≥ 1. The spectrum of H is d + 2N and we consider the sequence of eigenvalues (λ n ) n≥0 by counting multiplicities:

d = λ 0 < λ 1 ≤ λ 2 ≤ λ 3 ≤ . . .
Fix any orthonormal basis (ϕ n ) n≥0 of normalized eigenfunctions for the harmonic oscillator H such that Hϕ n = λ n ϕ n . For j ≥ 1 denote by

I(j) = n ∈ N, 2j ≤ λ n < 2(j + 1) .
Observe that for all j ≥ d/2, I(j) = ∅ and that #I(j) ∼ C d j d-1 when j -→ +∞, and therefore λ n ∼ c d n 1/d . Though (ϕ n ) n≥0 is arbitrary, the vector space spanned by {ϕ n , n ∈ I(j)} is independent of the choice of the Hilbert basis. Now, we recall what are the natural Sobolev spaces for H:

∀s ≥ 0 ∀p ∈ [1, +∞) ∪ {+∞} W s,p (R d ) := {u ∈ S ′ (R d ), H s/2 u ∈ L p (R d )}.
Therefore, we define

(1.1) u W s,p (R d ) := H s 2 u L p (R d ) .
It turns out (see [START_REF] Yajima | Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity[END_REF]Lemma 2.4]) that a functional characterisation of W s,p (R d ) for 1 ≤ p < +∞ and s ≥ 0 is given by

u ∈ W s,p (R d ) ⇔ (I -∆) s 2 u L p (R d ) + x s u L p (R d ) < +∞.
In the Hilbertian framework, we have

H s (R d ) := W s,2 (R d ) = {u ∈ H s (R d ), x s u ∈ L 2 (R d )}
where

H s (R d ) = Dom((I -∆) s/2
) is the classical Sobolev space. Thus, up to an equivalence of norm, one can define

(1.2) u H s (R d ) = H s/2 u L 2 (R d ) = u H s (R d ) + x s u L 2 (R d ) .
Consequently, one can check that

H s (R d ) is an algebra if s > d 2 and is included in L ∞ (R d ).
We will need the L ∞ estimate of the spectral function given by Thangavelu/Karadzhov (see [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Lemma 3.5]) which reads

(1.3) Π j 2 L 2 (R d )→L ∞ (R d ) = sup x∈R d n∈I(j) |ϕ n x 2 ≤ Cj γ(d) ,
with γ(1) = -1/6 and γ(d) = d/2 -1 for d ≥ 2 and where Π j is the spectral projector of H on the eigenspace associated to the unique eigenvalue which belongs to I(j). It is classical that the function defined in (1.3) does not depend on the choice of the (ϕ n ) n≥0 . For d = 1, (1.3) comes from the simplicity of the spectrum of H and the classical estimate of the normalized Hermite functions:

ϕ j L ∞ (R) j -1 12 .
In the sequel we will also need the notation

β(d) = d -1 -γ(d) as follows (1.4) γ(d) β(d) d = 1 -1 6 1 6 d ≥ 2 d 2 -1 d 2 1.1.2. The space Z s ϕ (R d ).
-Given a Hilbertian basis of Hermite functions (ϕ n ) n≥0 and s ∈ R, any u ∈ H s (R d ) can be written in a unique fashion

u = n≥0 c n ϕ n , n≥0 λ s n |c n | 2 < +∞.
We define the space Z s ϕ (R d ) by the norm

u 2 Z s ϕ = j≥1 j s+d-1 max n∈I(j) |c n | 2 ,
and we stress that this space depends on the choice of the basis (ϕ n ). It is clear that we have the strict embeddings

H s+d-1 (R d ) ⊂ Z s ϕ (R d ) ⊂ H s (R d ).
In the works [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF][START_REF] Poiret | Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator[END_REF][START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF], the following assumption on the coefficients of u ∈ H s (R d ) was made

(1.5) |c k | 2 ≤ C #I(j) n∈I(j) |c n | 2 , ∀k ∈ I(j), ∀j ≥ 1.

Let us explain why the condition

u ∈ Z s ϕ (R d ) is more natural. Firstly, observe that if the coefficients of u ∈ H s (R d ) satisfy (1.5) then u ∈ Z s ϕ (R d ). Secondly, consider two functions u, v ∈ H s (R d ) u = +∞ n=0 c n ϕ n , v = +∞ n=0 γ n c n ϕ n ,
where (γ n ) is a real bounded sequence. The contraction principle for the random series (see Theorem 5.5) states roughly that if one can prove an almost sure convergence for the random series coming from u (see below (1.9)), then the same is true for v. But it is easy to see that condition (1.5) is not stable by multiplication by bounded sequences whereas u ∈ Z s ϕ (R d ) is the most general condition which is implied by (1.5) and stable by multiplication by bounded sequences.

Sometimes, we also need the stronger condition

(1.6) C 1 #I(j) n∈I(j) |c n | 2 ≤ |c k | 2 ≤ C 2 #I(j) n∈I(j) |c : n | 2 , ∀k ∈ I(j), ∀j ≥ 1.
1.2. Probabilistic setting. -Consider a probability space (Ω, F, P) and let (X n ) n≥0 be independent and identically distributed random variables which are not constant almost surely. All random variables are real valued. In all the paper (except in the annex 5), we will make two different assumptions depending on whether we study integrability or regularity results:

(1.7) E[X 1 ] = 0 and ∀k ≥ 1 E[|X 1 | k ] < +∞. (1.8) ∃σ > 0, ∀r > 0, E[e rX 1 ] ≤ e 1 2 σ 2 r 2 .
One checks that (1.8) implies (1.7). The usual laws we have in mind fulfill (1.8): the real Gaussian law N R (0, 1) or the Rademacher law (in that case, we will write X n = ε n ). More generally, any centered and bounded r.v. satisfies (1.8).

We explain now the way we introduce randomness in Sobolev spaces. Let (c n ) n≥0 be such that

n≥0 λ s n |c n | 2 < +∞.
Then we can define a random variable u ω by (1.9)

u ω = +∞ n=0 X n (ω)c n ϕ n .
It is clear that we have

E u ω 2 H s (R d ) = E +∞ n=0 λ s n |c n | 2 |X n | 2 ≤ E[X 2 1 ] +∞ n=0 λ s n |c n | 2 < +∞,
In other words ω → u ω belongs to L 2 (Ω, H s (R d )) and almost surely u ω belongs to H s (R d ).

2. Main results of the paper 2.1. Integrability results for random Hermite series. -We state here convergence results in the L p (R d ) scale with p ∈ [2, ∞). The following result (used in a slightly weaker form in [START_REF] Grivaux | Almost sure convergence of some random series[END_REF]) will play a key role. It is a combination of results of Hoffman-Jorgensen, Maurey-Pisier [START_REF] Maurey | Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach[END_REF] and the fact that L p (R d ) has finite cotype.

Proposition 2.1. -Let p ∈ [2, +∞) and (f n ) n≥0 be a sequence of L p (R d ).
Assume that the sequence (X n ) n≥0 fulfills (1.7), the following statements are equivalent:

(i) the series ε n f n converges almost surely in L p (R d ), (ii) the series X n f n converges almost surely in L p (R d ), (iii) the function n≥0 |f n | 2 belongs to L p 2 (R d ).
This proposition is a synthesis of known results on the convergence of random series in Banach spaces. For the reader's convenience, we have gathered the elements of the proof in Section 5.

Here is our first result involving random Hermite series. Recall the definition (1.4) of γ and β. Theorem 2.2. -Let d ≥ 1 and 2 ≤ p < +∞. We assume that the r.v. (X n ) n≥0 fulfill (1.7) and that

u = n≥0 c n ϕ n belongs to Z -2β(d) 1 2 -1 p ϕ (R d ), i.e. the sequence (c n ) n≥0 is such that (2.1) +∞ j=1 j γ(d)+ 2β(d) p max n∈I(j) |c n | 2 < +∞. Then u ω = n≥0 X n c n ϕ n converges almost surely in L p (R d ).
We will see in the proof that the exponent γ(d)

+ 2β(d) p which appears in (2.1) is such that n∈I(j) |ϕ n (x)| 2 L p/2 (R d ) ≤ Cj γ(d)+ 2β(d) p ,
We refer to [START_REF] Poiret | Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator[END_REF]Proposition 2.1] where a result similar to Theorem 2.2 was given using the condition (1.5).

By considering radial functions as in Ayache-Tzvetkov [START_REF] Ayache | L p properties for Gaussian random series[END_REF] and in Grivaux [START_REF] Grivaux | Almost sure convergence of some random series[END_REF], we introduce now a natural example for which the gain of integrability may not hold in all the spaces L p (R d ), in this case condition (1.5) does not hold true and we may have

u ∈ Z s (R d ) and u ∈ H s+d-1 (R d )\H s+d-1+ε (R d ).
Let d ≥ 2 and L 2 rad (R d ) be the subspace of L 2 (R d ) invariant by the action of the rotation group SO(d). One can prove that there exists a Hilbertian basis (ψ n ) n≥0 of L 2 rad (R d ) of eigenfunctions of H. Indeed, we have Hψ n = (4n + d)ψ n , each eigenspace has dimension 1 and ψ n may be expressed with Laguerre polynomials (see Section 3 for more details). where

α ⋆ (c) := inf α > 0 : N n=0 n d 2 -1 |c n | 2 = O(N α ) .
Let us give some examples:

• If d = 1, then by Theorem 2.2, the series u ω rad (defined in the obvious way) converges a.s. in

L p (R d ) for all p < ∞. • If (c n ) is such that n≥0 n d 2 -1 |c n | 2 < +∞, then d α⋆(c) = +∞. Therefore u ω rad converges a.s. in L p (R d ) for all p < ∞. • Assume that c n ∼ n -κ with κ ≥ 1/2, then α ⋆ (c) = max d 2 -2κ, 0 and then d α ⋆ (c) =    min 2d d -4κ , +∞ , if κ < d 4 , +∞, if κ ≥ d 4 .
An analogous result to Theorem 2.3, but with a different numerology, was first obtained in [START_REF] Ayache | L p properties for Gaussian random series[END_REF][START_REF] Grivaux | Almost sure convergence of some random series[END_REF] for the family of the radial eigenfunctions of the Laplacian on the unit disc in R d where the analogue value of d α⋆(c) is called the critical convergence exponent of c. We will follow the main lines of [START_REF] Grivaux | Almost sure convergence of some random series[END_REF], the difference in the proof involves the study of L p (R d ) bounds of the radial Hermite functions.

c p n d 2 ( 1 2 -1 p )-1 2 ≤ ψ n L p (R d ) ≤ C p n d 2 ( 1 2 -1 p )- 1 2 .
(ii) Assume that p = 2d d-1 . Then

ψ n L p (R d ) ≤ C p n -1 4 ln 1 p (n).
(iii) Assume that 2 ≤ p < 2d d-1 . Then

ψ n L p (R d ) ≤ C p n -d 2 ( 1 2 -1 p ) .
The proof uses asymptotic estimates of Laguerre functions proved by Erdelyi (such a method has been used in [4, Lemma 3.1] for d = 2 and is indicated in [START_REF] Thangavelu | Lectures on Hermite and Laguerre expansions[END_REF]Chapter 1]).

We do not know if the estimates stated in (ii) and (iii) are optimal or not. To get the lower bound in (i) we show that there exist ε, c > 0 such that for all n ≥ 1 and all |x| ≤

ε √ n , |ψ n (x)| ≥ cn d 4 -1 2
, and the result follows by integrating this estimate.

In the figures below, we represent the estimates of Proposition 2.4. The dashed lines represent the bounds of Koch-Tataru [10, Corollary 3.2] obtained for general Hermite functions as defined in Section 1.1. We see that in the range 2 < p < d-2 2d the radial functions enjoy better bounds than in the general case, but not in the regime d-2 2d < p ≤ +∞.

1/p θ 0 • 3/10 • 1/4 • 1/2 • -1/4 • -1/10 • • • L p estimates of radial Hermite functions: the case d = 2 1/p θ 1/p 3 • 1/p 2 • 1/p 1 • 1/2 • d/4 -1/2 • -1/4 • -1/(2d + 6) • • • L p estimates of radial Hermite functions: the case d ≥ 3
In the second figure we have set

2 < p 1 := 2d d -1 ≤ p 2 := 2(d + 3) d + 1 ≤ p 3 := 2d d -2 .
2.2. Continuity results for random Hermite series. -We are concerned with the random behavior of the partial sums of (1.9) in the space L ∞ (R d ). Let us define for any λ ≥ d

(2.2) u ω λ (x) = λn≤λ c n X n (ω)ϕ n (x).
There is not an equivalent of Proposition 2.1 for the space L ∞ (R d ) (the reason is that L ∞ (R d ) is not a Banach space with finite cotype, see Annex 5). Hence, we will use other methods to get probabilistic results, like the following one which is in the spirit of the Salem-Zygmund inequality (see [9, Theorem, page 55]).

Theorem 2.5. -Assume that (X n ) n≥0 is an i.i.d. family of r.v. which satisfies the subnormality condition (1.8) with a real number σ > 0. For any positive integer N > 0, there is C := C(N, d, σ) > 0 such that for any λ ≫ 1 one has for any sequence

(c n ) n≥0 (2.3) P   λn≤λ c n X n ϕ n 2 L ∞ (R d ) ≤ C ln(λ) j≤[λ/2] j γ(d) max n∈I(j) |c n | 2   ≥ 1 - 1 λ N ,
where γ(d) is defined in (1.4). Furthermore, if d ≥ 2 holds and if the (X n ) n≥0 are independant Gaussians N R (0, 1), then one can find a sequence (c n ) n≥0 such that we cannot replace the function λ → ln(λ) with a slower function of order o(ln(λ)).

In particular the previous result shows that there exists C > 0 such that almost surely we have lim sup

λ→+∞ λn≤λ c n X n ϕ n L ∞ (R d ) ln(λ) ≤ C j≥0 j γ(d) max n∈I(j) |c n | 2 1/2 .
Furthermore there exists a sequence {c n } and c > 0 such that j≥0 j γ(d) max n∈I(j) |c n | 2 = 1 and

lim inf λ→+∞ λn≤λ c n X n ϕ n L ∞ (R d ) ln(λ) ≥ c. It is straightforward that if the coefficients (c n ) n≥0 satisfy (1.5), then (2.3) implies (2.4) P   λn≤λ c n X n ϕ n 2 L ∞ (R d ) ≤ C ln(λ) λn≤λ λ -β(d) n |c n | 2   ≥ 1 - 1 λ N , with β(1) = 1/6 and β(d) = d/2 for d ≥ 2.
The Salem-Zygmund inequality in the classical case of random trigonometric polynomials is similar to (2.4) but holds for β(d) = 0. Thus, (2.4) shows that randomness for Hermite series has a much more smoothing effect than for Fourier series. Indeed, this is a consequence of a better behavior of the spectral function

(1.3) of H in the space L ∞ (R d ).
Let us add that the proof of the classical Salem-Zygmund inequality [9, Theorem 1, page 55] uses in an essential way that the torus T is compact. In our setting, the non-compactness of R d is counterbalanced by the localization of (2.2) in any subset or R d which contains strictly the ball B(0, √ λ) (here we will choose the closed ball B(0, λ) which is much bigger than B(0, √ λ)).

Our next result gives a sufficient condition to get almost surely continuity as in Theorem 1.1.

Theorem 2.6. -Let γ(d) be defined by (1.4), and let (c n ) n∈N be such that

(2.5) ∃ α > 1, +∞ j=1 j γ(d) (ln j) α max n∈I(j) |c n | 2 < +∞.
Assume that (X n ) n≥0 is an i.i.d. family of symmetric r.v. such that (1.8) holds. Denote by

u ω λ = λn≤λ c n X n (ω)ϕ n . Then u ω λ -→ u ω in L ∞ (R d ) almost surely when λ -→ +∞. In particular for almost all ω ∈ Ω, u ω is a bounded continuous function on R d .
In the particular case where (c n ) n≥0 are such that (1.5) holds, then the assumption (2.5) becomes

∃ α > 1, +∞ n=0 λ -β(d) n (ln λ n ) α |c n | 2 < +∞, with β(1) = 1/6 and β(d) = d/2 for d ≥ 2. This shows that for d ≥ 2, u is in a slightly smaller space, denoted by H -d/2+ (R d ) (with a log correction), than H -d/2 (R d ).
In other words, under condition (1.5) almost all series u ω in the very irregular distribution space

H -d/2+ (R d ) is actually a continuous function on R d .
It is interesting to notice that if we forget the logarithmic term in in the assumption (2.5), we find exactly the assumption (2.1) of Theorem 2.2 as p tends to infinity although methods of proofs are different.

The symmetry assumption of the r.v. is only needed for the convergence of the partial sums, but the continuity result holds without this assumption.

We shall give two different proofs of Theorem 2.6: one is an application of the Salem-Zygmund inequality (Theorem 2.5), and the other relies on an entropy criterion (see Section 6).

From the Salem-Zygmund inequality we can also get a sufficient condition so that u ω (x) satisfies a global Hölder continuity condition. Recall the definition of the modulus of continuity of u : R d → C,

m u (h) = sup |x-y|≤h x,y∈R d |u(x) -u(y)|, h > 0. Theorem 2.7. -Let (c n ) n≥0 such that there exists C > 0 such that (2.6) 2 j+1 k=2 j max n∈I(k) |c n | 2 ≤ C2 (-γ(d)-µ)j j 2ν , ∀j ≥ 0,
with C > 0, (ν ∈ R and 0 < µ ≤ 1) or (ν < -1 and µ = 0). Assume that (X n ) n≥0 is an i.i.d. family of r.v. such that (1.8) holds. Then we have, almost surely in ω,

m u ω (h) = O(h µ | ln h| θ )
where

• θ = 1 2 + ν if 0 < µ < 1 • θ = 1 + ν if µ = 0 • if µ = 1 then    θ = 1 + ν, if ν ≥ -1 2 θ = 1 2 if -1 ≤ ν ≤ -1 2 u ω is a.e differentiable if ν < -1.
In particular, if (c n ) n≥0 is a sequence which satisfies (1.5) and such that there exists C > 0 such that

n : 2 j ≤λn<2 j+1 |c n | 2 ≤ C2 (β(d)-µ)j j 2ν , ∀j ≥ 0, then (2.6) is satisfied. Remark 2.8.
-With a slight modification of the proof of Theorem 2.7 we can get the following extension of Theorem 2.6.

If +∞ j=1 j γ(d)+µ (ln j) α max n∈I(j) |c n | 2 < +∞, then almost surely in ω, m u ω (h) = O(h µ | ln h| θ )
where The rest of the paper is organised as follows. In Section 3 we prove the integrability results on the Hermite series. Section 4 is devoted to the proof of the regularity results (Theorems 2.5, 2.6 and 2.7). In Section 5 we review some results we need about the convergence of random series in Banach spaces. Finally, in Section 6 we give an alternative proof of Theorem 2.6.

• θ = 1 2 -α + ε for all ε > 0 if 0 < µ < 1 • θ = 1 -α + ε for all ε > 0 if µ = 0.

Proof of the integrability results

Proof of

n≥0 |c n | 2 |ϕ n | 2 ∈ L p 2 (R d ).
By interpolating the L p 2 norm and using that I(j) ∼ C j j d-1 , we get n∈I(j)

|ϕ n | 2 L p 2 (R d ) ≤ n∈I(j) |ϕ n | 2 2 p L 1 (R d ) n∈I(j) |ϕ n | 2 1-2 p L ∞ (R d ) j 2 p (d-1)+ 1-2 p γ(d) = j γ(d)+2β(d)/p , as a consequence n≥0 |c n | 2 |ϕ n | 2 L p 2 (R d ) ≤ j≥1 max n∈I(j) |c n | 2 n∈I(j) |ϕ n | 2 L p 2 (R d ) j≥1 j γ(d)+2β(d)/p max n∈I(j) |c n | 2 < +∞.
We get (3.1) and hence conclude. of type α and degree n ≥ 0 is defined by

(3.2) e -r r α L (α) n (r) = 1 n! d n
dr n e -r r n+α , x ∈ R. We need the following identities (see [19, lines (5.1.1),(5.1.3),(5.1.7) and (5.1.14)]):

(3.3) +∞ 0 L (α) n (r)L (α) m (r)e -r r α dr = Γ(n + α + 1) Γ(n + 1) δ nm , (3.4)
L (α) n (0) = Γ(n + α + 1) Γ(n + 1)Γ(α + 1) ≈ n α , (3.5) ∀n ≥ 1, d dr L (α) n (r) = -L (α+1) n-1 (r), (3.6) r d 2 L (α) n dr 2 + (α + 1 -r) dL (α) n
dr + nL (α) n = 0. We will need the following lemma Lemma 3.1. -For any α > -1 there are c, ε > 0 such that

∀n ≥ 1, ∀r ∈ 0, ε 2 n , |L (α) n (r)| ≥ cn α .
Proof. -As in [19, p. 176], we introduce the function

r → nL (α+1) n-1 (r) 2 + r d dr L (α+1) n-1 (r) 2 whose derivative is 2 r -3 2 -α d dr L (α+1) n-1
2 thanks to (3.6). Thus, one has

∀r ∈ 0, α + 3 2 , |L (α+1) n-1 (r)| ≤ |L (α+1)
n-1 (0)| n α+1 . By using (3.5), we have

∀r ∈ 0, α + 3 2 , |L (α) n (r) -L (α) n (0)| rn α+1 .
We can conclude by using a new time (3.4).

Because of the orthogonality condition (3.3), it is usual to introduce the Laguerre functions normalized in L 2 (0, +∞)

(3.7) L α n (r) := √ n! Γ(n + α + 1) L (α) n (r)e -r/2 r α/2 , √ n! Γ(n + α + 1) ≈ n -α/2 .
Those functions satisfy the following uniform estimates (see [START_REF] Erdélyi | Asymptotic forms for Laguerre polynomials[END_REF][START_REF] Askey | Mean convergence of expansions in Laguerre and Hermite series[END_REF][START_REF] Muckenhoupt | Mean convergence of Hermite and Laguerre series. II[END_REF]).

Proposition 3.2. -For any α > -1, there are C = C(α) and γ = γ(α) > 0 such that, by denoting ν = 4n + 2α + 2, one has Let us estimate the L p (R d ) norm of ψ n for p ≥ 2 by using Proposition 3.2 with ν ∼ n and α = d 2 -1. The case p = ∞ is the easiest, and we get directly that |L

|L (α) n (r)| ≤              C(rν) α/2 if 0 ≤ r ≤ 1 ν C(rν) -1/4 if 1 ν ≤ r ≤ ν 2 Cν -1/4 (ν 1/3 + |ν -r|) -1/4 if ν 2 ≤ r ≤ 3ν 2 Ce -γr if 3ν 2 ≤ r.
(α) n (r)|r -α 2 ≤ Cν α 2 , in other terms ψ n L ∞ (R d ) n d 4 -1 2 .
To get the lower bound, it is sufficient to combine Lemma 3.1 with (3.8) and the equivalent in (3.7).

We now consider p ∈ [2, +∞). Then we have (3.9)

ψ n p L p (R d ) = c(d) p Vol(S d-1 ) +∞ 0 |L (d/2-1) n (r 2 )| p r -p( d 2 -1) r d-1 dr = c(d) p-2 +∞ 0 |L (d/2-1) n (r)| p r -( p 2 -1)( d 2 -1) dr.
We begin by the following integrals:

(3.10)

1 ν 0 |L (d/2-1) n (r)| p r -( p 2 -1)( d 2 -1) dr n p 2 ( d 2 -1) 1 ν 0 r d 2 -1 dr n p 2 ( d 2 -1)-d 2 = n pd 2 1 2 -1 p -p 2 +∞ 3ν 2 |L (d/2-1) n (r)| p r -( p 2 -1)( d 2 -1) dr +∞ 3ν 2 e -γr r -( p 2 -1)( d 2 -1) dr = O(n -∞ ).
To study the integrals over the others intervals given by Proposition 3.2, we have to consider several subcases.

• If p > 2d d-1 holds, one has obviously 1 2 -1 p > 1 2d and the comparison of different exponents of n will rely on:

(3.11) -d 2 1 2 - 1 p < d 2 1 2 - 1 p - 1 2 .
Notice that one also has (3.12)

p 4 + p 2 -1 d 2 -1 > d 2(d -1) + d d -1 -1 d -2 2 = 1,
which implies that the following integral is of interest near r = 0:

(3.13) ν 2 1 ν |L (d/2-1) n (r)| p r -( p 2 -1)( d 2 -1) dr n -p 4 ν 2 1 ν r -p 4 -( p 2 -1)( d 2 -1) dr n -p 4 + p 4 +( p 2 -1)( d 2 -1)-1 ≈ n pd 2 1 2 -1 p -p 2 .
The integral over [ ν 2 , 3ν 2 ] is bounded by .

3ν 2 ν 2 |L (d/2-1) n (r)| p r -( p 2 -1)( d 2 -1) dr n -p 4 3ν 2 ν 2 dr ( 3 √ ν + |ν -r|) p 4 r ( p 2 -1)( d 2 -1) n -p 4 -( p 2 -1)( d 2 -1)
We have to use now the following fact if p > 4 holds (which is necessary for d = 2):

d 2 1 2 - 1 p - 1 2 - 1 6 + d 2p - d 4 - 2 3p = d 2 - 2 3 -d - 2 3 1 p > d 4 - 1 2 ≥ 0.
That brings us to

(3.14) 3ν 2 ν 2 |L (d/2-1) n (r)| p r -( p 2 -1)( d 2 -1) dr          n p 6 + d 2 -pd 4 -2 3 ≤ n p d 2 1 2 -1 p -1 2 if p > 4 n -d 2 ln(n) n d 2 = n -pd 2 1 2 -1 p if p = 4 n p 6 + d 2 -pd 4 -2 3 + 2 3 (1-p 4 ) = n -pd 2 1 2 -1 p if p < 4.
Thanks to (3.11), the comparison of exponents in (3.10), (3.13) and (3.14) gives

ψ n L p (R d ) n d 2 1 2 -1 p -1 2 . • If p < 2d d-1 holds, one has 1 2 -1 p < 1 2d
and the contrary of (3.11) and (3.12) hold:

(3.15) d 2 1 2 - 1 p - 1 2 < -d 2 1 2 - 1 p p 4 + p 2 -1 d 2 -1 < 1.
Hence, the integral over 1 ν , ν 2 is of interest for r ≫ 1:

(3.16)

ν 2 1 ν |L (d/2-1) n (r)| p r -( p 2 -1)( d 2 -1) dr n -p 4 ν 2 1 ν r -p 4 -( p 2 -1)( d 2 -1) dr n -p 4 +1-p 4 -( p 2 -1)( d 2 -1) n -pd 2 1 2 -1 p .
We deal with the integral over ν 2 , 3ν 2 by the same way with the help of (3.14) and by noticing that p < 2d d-1 ≤ 4 holds. Hence we get (3.17)

3ν 2 ν 2 |L (d/2-1) n (r)| p r (1-p 2 )( d 2 -1) dr n -pd 2 1 p -1 2 .
Once again, we compare the exponents in (3.10), (3.16) and (3.17) with the help of (3.15) and we get

ψ n L p (R d ) n -d 2 1 2 -1 p . • If p = 2d
d-1 holds, we follow the previous analysis and we see that ψ n p L p n -p 4 ln(n). We have finished the proof of Proposition 2.4. We consider p > d α⋆(c) and we write

Proof of

R d n≥0 |c n | 2 |ψ n (x)| 2 p 2 dx ≥ sup N ≥1 |x|≤ ε √ N N n=1 |c n | 2 |ψ n (x)| 2 p 2 dx ≥ C sup N ≥0 ε d N d 2 N n=1 |c n | 2 n d 2 -1 p 2 ≥ +∞.
We consider p < d α⋆(c) and we write

n≥0 |c n | 2 |ψ n | 2 L p/2 (R d ) ≤ n≥0 |c n | 2 ψ 2 n L p/2 (R d ) = n≥0 |c n | 2 ψ n 2 L p (R d ) .
If p belongs to (2, 2d d-1 ], then ψ n L p (R d ) is less than n -ε for some ε > 0 (see Proposition 2.4). By using that

n≥0 c n ψ n belongs to ε>0 H -ε (R d ), it is clear that the series |c n | 2 ψ n 2 L p (R d ) converges.
If p is greater than 2d d-1 , we use first an Abel summation and then two times the inequality α ⋆ (c) < d p to bound the sum of the series

|c n | 2 ψ n 2 L p (R d ) by C|c 0 | 2 + n≥1 |c n | 2 n d 2 -1 n -d p ≤ ≤ C|c 0 | 2 + lim N →+∞ N -d p N n=1 |c n | 2 n d 2 -1 + N ≥1 N n=1 |c n | 2 n d 2 -1 |N -d p -(N + 1) -d p | C|c 0 | 2 + 0 + N ≥1 1 N 1+ d p N n=1 |c n | 2 n d 2 -1 < +∞.
Remark. If we define for any sequence

(c n ) n≥0 ∀p > 2d d -1 c d,p := |c 0 | + sup N ≥1 1 N d 2p N n=1 |c n | 2 n d 2 -1 1 2
, then the previous proof shows that one has

∀ν > 0 C c d,p ≤ n≥0 |c n | 2 ψ 2 n L p ≤ C(ν) c d,p+ν .
It is not clear if one can find a more precise norm on the sequence (c n ) n≥0 which is equivalent to

n≥0 |c n | 2 ψ 2 n L p
. Indeed, this is essentially equivalent to decide whether or not the almost sure convergence in L p (R d ) holds if p is the critical convergence exponent d α⋆(c) .

∀u ∈ E H (λ), |u(x)| ≤ Cλ d 4 exp -c |x| 2 2λ u L 2 (R d ) ,
The first tool we need to prove Theorem 2.5 is a Bernstein inequality for the harmonic oscillator. In the Hilbertian framework, it is easy to check that one has

∀λ ≥ 1 ∀u ∈ E H (λ) ∂ x u L 2 (R d ) ≤ C √ λ u L 2 (R d ) .
We need a version of the previous inequality by replacing the space L 2 (R d ) with L ∞ (R d ).

(4.2) ∀λ ≥ 1 ∀u ∈ E H (λ) ∇u L ∞ (R d ) ≤ Cλ s(d) u L ∞ (R d ) .
Proof. -For any real number s > d 2 , the Sobolev embedding H s (R d ) ⊂ L ∞ (R d ) allows us to write for each ℓ ∈ {1, 2, . . . , d}:

∂ x ℓ u L ∞ (R d ) ≤ C ∂ x ℓ u H s (R d ) ≤ C u H s+1 (R d ) ≤ C   λ j ≤λ λ s+1 j R d ϕ j (x)u(x)dx 2   1 2 ≤ Cλ s+1 λ d u L ∞ (R d ) sup λ j ≤λ ϕ j L 1 (R d ) .
In view to get a bound of ϕ j L 1 (R d ) we just use the Cauchy-Schwarz inequality:

R d |ϕ j (x)|dx = R d x -d+1 2 x d+1 2 |ϕ j (x)|dx ≤ C x d+1 2 ϕ j (x) L 2 (R d ) ≤ C ϕ j H d+1 2 (R d ) ≤ Cλ d+1 4 j .
Thus (4.2) is proved.

It is not clear for us if the exponent s(d) can be chosen to be independent of d or if we can find the optimal value of s(d).

Corollary 4.2. -If λ is enough large, there is a constant c > 0 which is independent of λ such that for any u ∈ E H (λ) there is y ∈ B(0, λ) for which we have Proof.

(i) u L ∞ (B(0,λ)) = u L ∞ (R d ) , (ii) ∀x ∈ B(y, cλ -s(d) ) ∩ B(0, λ), |u(x)| ≥ 1 2 u L ∞ (B(0,λ)) , (
-By the same argument we used in the proof of Lemma 4.1, we claim that there is a constant ν > 0, independent of λ, such that

(4.3) ∀u ∈ E H (λ) u L 2 (R d ) ≤ Cλ ν u L ∞ (R d ) .
By combining (4.1) and (4.3), we understand that if λ is enough large and if |x| > λ holds then we have

|u(x)| ≤ Cλ d 4 +ν exp - cλ 2 u L ∞ (R d ) ≤ 1 2 u L ∞ (R d ) .
That proves Point (i). Let us check Point (ii). By a compactness argument, there is y ∈ B(0, λ) which maximizes u on the whole space R d . For any x ∈ B(0, λ), Lemma 4.1 gives us

|u(x) -u(y)| ≤ C|x -y|λ s(d) u L ∞ (R d ) . If |x -y| < C 2 λ -s(d) holds then |u(x)| ≥ 1 2 u L ∞ (R d ) . Point (iii) is a consequence of a geometric fact. Indeed, it is quite clear that we have lim inf R→0 inf z∈B(0,1) Vol B(z, R) ∩ B(0, 1) Vol B(z, R) = 1 2 .
Consequently, if λ is enough large then Point (iii) holds.

We can prove Theorem 2.5 by following [9, Theorem 1, page 55]. Our preliminaries allow us to deal with the non-compactness of R d . We define the random maximum

M ω λ := u ω λ (x) L ∞ x (R d ) = u ω λ (x) L ∞ x (B(0,λ)) .
We apply Lemma 4.2 to the random function u ω λ ∈ E H (λ). If x belongs to the random set A ω λ := B(y ω , cλ -s(d) ) ∩ B(0, λ) then we have

u ω λ (x) ≥ 1 2 M ω λ or -u ω λ (x) ≥ 1 2 M ω λ .
Thus, it comes for any r > 0

E exp 1 2 rM ω λ ≤ E 1 Vol(A ω λ ) A ω λ exp (ru ω λ (x)) + exp (-ru ω λ (x)) dx .
From Point (iii) of Lemma 4.2, we get

E exp 1 2 rM ω λ ≤ Cλ ds(d) E A ω λ exp (ru ω λ (x)) + exp (-ru ω λ (x)) dx ≤ Cλ ds(d) B(0,λ) E exp (ru ω λ (x)) + exp (-ru ω λ (x)) dx.
By coming back to the definition (1.9) of u ω λ , we can use the independence of the random variables X n :

E exp ru ω λ (x) = λn≤λ E exp rc n X n (ω)ϕ n (x) .
Now we use (1.8) and ( 1.3) to get

E exp ru ω λ (x) ≤ exp Cr 2 λn≤λ |c n | 2 |ϕ n (x)| 2 ≤ exp Cr 2 j≤λ max n∈I(j) |c n | 2 n∈I(j) |ϕ n (x)| 2 ≤ exp Cr 2 ρ λ , (4.4)
where we have set

ρ λ := j≤λ j γ(d) max n∈I(j) |c n | 2 .
Obviously, a similar argument gives the same bound for E expru ω λ (x) , and we have obtained

E exp 1 2 rM ω λ ≤ Cλ ds(d)+d exp C r 2 2 ρ λ ,
which is totally equivalent to

∀L ≥ 1 ∀r > 0 E exp r 2 M ω λ -Crρ 2 λ - 2 r ln(Cλ ds(d)+d L) ≤ 1 L .
From Markov's inequality, it comes

P M ω λ -Crρ 2 λ - 2 r ln(Cλ ds(d)+d L) ≥ 0 ≤ 1 L .
Now we just have to optimize in r by choosing r 2 = 1 ρ 2 λ ln(Cλ ds(d)+d L). For another constant C > 0, we have

P M ω λ ≥ Cρ λ ln(Cλ ds(d)+d L) ≤ 1 L .
The conclusion comes with the choice L = λ N . Finally, we have to see that the term ln(λ) is optimal in (2.3) if d ≥ 2 holds, and when the (X n ) n≥0 are independent Gaussians N R (0, 1).

Let us suppose the contrary and consider a function ϑ(λ) = o(ln(λ)) such that Theorem 2.5 holds true by replacing ln(λ) with ϑ(λ).

To see that implies a contradiction, let us recall a result proved in [16, Theorem 1.1] (1) (with the sequence d j = λ -d 4 j c j and assuming (1.6)), there are real numbers C 0 > 0 and c > 0 such that for any j ≫ 1 one has

P C 0 ln(j) n∈I(j) λ -d 4 n c n X n ϕ n 2 L 2 (R 2 ) ≤ n∈I(j) λ -d 4 n c n X n ϕ n 2 W d 2 ,∞ (R d ) ≥ 1 - 1 (j + 2) c .
(1) [16, Theorem 1.1] is stated for complex Gaussians, but the result also holds for real r.v. (see [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Assumption 1]).

From the definition (1.1) and Theorem 2.5 with the function ϑ and any chosen positive integer N , we have with probability greater than 1 -1 (j+2) N -1

j N n∈I(j) λ -d 4 n c n X n ϕ n W d 2 ,∞ (R d ) = n∈I(j) c n X n ϕ n L ∞ (R d ) ≤ λn<2j c n X n ϕ n L ∞ (R d ) + λn<2j+2 c n X n ϕ n L ∞ (R d ) ≤ C ϑ(2j) + ϑ(2j + 2) λn<2j+2 λ -d 2 n |c n | 2 1 2 .
We have now to make use of the condition (1.6):

n∈I(j) λ -d 2 n |c n | 2 × 1 #I(j) n∈I(j) |X n | 2 ≤ C n∈I(j) |λ -d 4 n c n X n | 2 = C n∈I(j) λ -d 4 n c n X n ϕ n 2 L 2 (R 2 )
.

By combining these arguments, we have with probability greater than 1

-1 (j+2) c -1 (j+2) N -1 j N 1 #I(j) n∈I(j) |X n | 2 ≤ C ϑ(2j) + ϑ(2j + 2) ln(j) λn<2j+2 λ -d 2 n |c n | 2 n∈I(j) λ -d 2 n |c n | 2 -1
.

One can obviously choose the sequence (c n ) n≥0 such that (1.6) and the two following properties hold:

u ∈ H -d 2 (R d ), j≥1 n∈I(j) λ -d 2 n |c n | 2 < +∞, # j ≥ 1, n∈I(j) λ -d 2 n |c n | 2 ≥ ϑ(2j) + ϑ(2j + 2) ln(j) 1 2 = +∞.
Hence, we get for probability greater than 1

-1 (j+2) c -1 (j+2) N -1 j N (4.5) 1 #I(j) n∈I(j) |X n | 2 ≤ ε(j)
where lim inf j→+∞ ε(j) = 0. Since lim j→+∞ #I(j) = +∞ holds, the Law of Large Numbers ensures that the left side of (4.5) converges almost surely to E |X 1 | 2 > 0. Since the almost sure convergence implies the convergence in probability, we understand that (4.5) cannot hold. That proves that Theorem 2.5 is optimal for the function ln(λ). 4.2. Proof of Theorem 2.6. -We give here an argument which uses the Salem-Zygmund theorem. In Section 6 we will present an alternative proof relies on an entropy argument. 4.2.1. Proof of Theorem 2.6 using the Salem-Zygmund Theorem 2.5. -For any positive integer K, we introduce J(K)

:= n ∈ N, λ n ∈ [2 2 K , 2 2 K+1 -1] and u ω K := n∈J(K) c n X n (ω)ϕ n .
By using Theorem 2.5, we have

P u ω K L ∞ (R d ) ≥ C2 K/2 2 2 K+1 -1 -1 j=2 2 K -1 j γ(d) max k∈I(j) |c n | 2 1 2 ≤ 1 2 2 K+1 -1 .
The Borel-Cantelli lemma ensures that almost surely there is C ω > 0 such that

u ω K L ∞ (R d ) ≤ C ω 2 K/2 2 2 K+1 -1 -1 j=2 2 K -1 j γ(d) max k∈I(j) |c n | 2 1 2 ≤ C ω 1 2 K(α-1)/2 2 2 K+1 -1 -1 j=2 2 K -1 (ln j) α j γ(d) max k∈I(j) |c n | 2 1 2
. Now by (2.5) and by the Cauchy-Schwarz inequality since α > 1 holds we get

K≥1 u ω K L ∞ (R d ) < +∞ a.s.
As a consequence, we have shown that a sub-sequence of the partial sum converges uniformly, a.s. This implies that u ω is a continuous and bounded function, a.s. Now if we moreover assume that the (X n ) are symmetric, we can apply [11, Theorem II.5, p.120] which yields that

u ω λ = λn≤λ c n X n (ω)ϕ n , also converges in L ∞ (R d ), a.s. for λ → +∞. 4.3. Proof of Theorem 2.7.
-The proof will follow the proof of J.-P. Kahane [9, Theorem 2, p. 66], with the necessary modifications in our context. Let κ ≥ 1 and let us introduce the notations:

ν j = κ2 j-1 , N j = 2 ν j , u ω 0 (x) = λn<N 1 c n X n (ω)ϕ n (x), for j ≥ 1, u ω j (x) = N j ≤λn<N j+1 c n X n (ω)ϕ n (x).
Using the triangle inequality and the fundamental calculus theorem we have

m u ω (h) ≤ h ∇ x u ω 0 L ∞ (R d ) + 2 1≤j<+∞ u ω j L ∞ (R d ) .
From Theorem 2.5 we have for j ≥ 1, (

P   u ω j L ∞ (R d ) ≥ C(ln N j+1 ) 1/2 N j /2≤ℓ<N j+1 /2 ℓ γ(d) max n∈I(ℓ) |c n | 2 1/2   ≤ 1 N 2 j+1 . 4.6)
The j = 0 term satisfies the following Lemma 4.3. -There exists C > 0 large enough such that

(4.7) P   ∇ x u ω 0 L ∞ (R d ) ≥ C(ln N 1 ) 1/2 ℓ<N 1 /2 ℓ 1+γ(d) max n∈I(ℓ) |c | 2 1/2   ≤ 1 N 2 1
.

The lemma will be proved later.

Remark 4.4.

-More generally, we can get a similar bound for a(x, D)u ω 0 , when a(x, ξ) is a polynomial in (x, ξ) ∈ R 2d . We leave the details to the reader.

Using this lemma we can prove Theorem 2.7. Let us denote Ω j (κ) the event in (4.6), Ω 0 (κ) the event in (4.7) and Ω ∞ (κ) = j≥0 Ω j (κ).

Using the definition of N j we have

P Ω ∞ (κ) ≤ 2 1-2κ .
Hence using the Borel-Cantelli lemma we get that

P lim sup κ→+∞ Ω ∞ (κ) = 0.
On the other side denote by

E 0 = (ln N 1 ) 1/2 ℓ<N 1 /2 ℓ 1+γ(d) max n∈I(ℓ) |c n | 2 1/2 E j = (ln N j+1 ) 1/2 N j /2≤ℓ<N j+1 /2 ℓ γ(d) max n∈I(ℓ) |c n | 2 1/2 .
Using assumption (2.6) we have

E 0 ≤ (ln N 1 ) 1/2 κ-1 ℓ=1 2 ℓ k=2 ℓ-1 k 1+γ(d) max n∈I(k) |c n | 2 1/2 ≤ Cκ 1/2 κ-1 ℓ=1 2 ℓ(1+γ(d)) 2 ℓ k=2 ℓ-1 max n∈I(k) |c n | 2 1/2 ≤ Cκ 1/2 κ-1 ℓ=1 2 ℓ(1-µ) ℓ 2ν 1/2
, and for all j ≥ 1

E j ≤ C(κ2 j ) 1/2 ν j ≤ℓ<ν j+1 2 ℓ k=2 ℓ-1 k γ(d) max n∈I(k) |c n | 2 1/2 ≤ C(κ2 j ) 1/2 ν j ≤ℓ<ν j+1 2 ℓγ(d) 2 ℓ k=2 ℓ-1 max n∈I(k) |c n | 2 1/2 ≤ C(κ2 j ) 1/2 ν j ≤ℓ<ν j+1 2 -µℓ ℓ 2ν 1/2 .
• Assume that 0 < µ < 1. We easily compute the following estimates

E 0 ≤ Cκ 1 2 +ν 2 (1-µ)κ/2 , E j ≤ C(κ2 j ) 1 2 +ν 2 -µκ2 j-2 and j≥1 E j ≤ C j≥1 (κ2 j ) 1 2 +ν 2 -µκ2 j-2 ≤ Cκ 1 2 +ν 2 -κµ/2 .
Now taking h = h κ = 2 -κ we have proved that for every ω / ∈ lim sup κ→+∞ Ω ∞ (κ) and for every κ large enough

m u ω (h κ ) ≤ Ch µ κ | ln(h κ )| 1 2 +ν .
Using that m u ω (h) is non increasing in h we have proved Theorem 2.7 for 0 < µ < 1.

• Assume that µ = 0 and ν < -1. Then in this case we get

E 0 ≤ Cκ 1 2 +ν 2 κ/2 , E j ≤ Cκ ν+1 2 (1+ν)j , j≥1 E j ≤ Cκ ν+1 ,
and the end of the proof is similar.

• The other cases are proved in the same way (see [START_REF] Kahane | Some random series of functions[END_REF]) excepted the last one (µ = 1, ν < -1) where the result is obtain by applying Theorem 2.6 to the partial derivatives ∂ x j u ω , 1 ≤ j ≤ d.

Now we prove the Lemma 4.3.

Proof of Lemma 4.3. -It is more convenient here to index the Hermite basis by N d . So we have

u ω 0 (x) = 2|α|+d≤N 1 c α X α (ω)ϕ α (x)
where we have denoted

|α| = α 1 + • • • + α d . We have Hϕ α = λ α ϕ α , with λ α = 2|α| + d.
It is easier to consider first the tensor basis:

ϕ α (x) = h α (x) = h α 1 (x 1 ) • • • h α d (x d ).
Recall that in 1D the Hermite functions satisfy for all t ∈ R

d dt h k (t) = 2 -1/2 √ kh k-1 (t) - √ k + 1h k+1 (t) . So we get √ 2∂ x 1 u ω 0 (x) = 2|α|+d≤N 1 √ α 1 c α X α (ω)h α-e 1 (x) - 2|α|+d≤N 1 √ α 1 + 1c α X α (ω)h α+e 1 (x)
where {e j } 1≤j≤d is the canonical basis of R d . Applying the Theorem 2.5 to each term of the sum we have proved the Lemma 4.3 for the tensor basis h α . For a general orthonormal basis (ϕ α ) α∈N d of Hermite functions, we write

ϕ α (x) = |α|=|β| t α,β h β (x)
where {t α,β } is a unitary matrix. So we have

√ 2∂ x 1 u ω 0 (x) = = 2|α|+d≤N 1 √ α 1 c α X α (ω) |β|=|α| t α,β h β-e 1 (x) - 2|α|+d≤N 1 √ α 1 + 1c α X α (ω) |β|=|α| t α,β h β+e 1 (x).
Now we estimate separately the two sums by revisiting the proof of Theorem 2.5.

It is enough to consider the first one denoted v ω λ (x) where λ = N 1 . We have to estimate:

E exp(rv ω λ ) ≤ exp Cr 2 λα≤λ α 1 |c α | 2 |β|=|α| |t α,β h β-e 1 | 2 .
For λ α ∈ I(j) we have λα∈I(j)

α 1 |c α | 2 |β|=|α| t α,β h β-e 1 (x) 2 ≤ C(1 + j) max λα∈I(j) |c α | 2 |β|=|α| λα∈I(j) t α,β h β-e 1 (x) 2 .
Using that the matrix {t α,β } is unitary we have

|β|=|α| λα∈I(j) t α,β h β-e 1 (x) 2 ≤ λα∈I(j-1) |h α (x)| 2 ≤ Cj γ(d)
and

E exp(rv ω λ ) ≤ exp Cr 2 λα≤λ λ γ(d)+1 α |c α | 2 .
This inequality has the same form as (4.4), hence we can conclude here as in the proof of Theorem 2.5.

Annex: about random series in Banach spaces

We present here some elements on the theory of random series in Banach spaces. We refer the reader to the books [START_REF] Hoffmann-Jørgensen | Probability in Banach space[END_REF], [START_REF] Diestel | Tonge -Absolutely summing operators[END_REF] and [START_REF] Li | Queffélec -Introduction à l'étude des espaces de Banach[END_REF] for more elements on this subject.

Let B be a Banach space on the field of real or complex numbers. Let (ε n ) n≥0 be a sequence of Rademacher i.i.d. random variables and let us define For instance, if B is a Hilbert space, the previous theorem can be used to see that Σ(B) is nothing else than ℓ 2 (B) (see also [START_REF] Kahane | Some random series of functions[END_REF]Chapter 3]).

A natural question is to study what happens for the almost sure convergence of X n b n if (X n ) n≥0 is i.i.d. with another reference law. A part of this question is solved by the following result proved by Hoffman-Jorgensen. We emphasize the fact that no integrability assumption is made on the law of X n . We do not know any published reference of Theorem 5.2 and we give below a proof we learned from Hervé Queffélec. The converse question is not easy and needs assumptions on the geometry of the Banach space B. It is worthwhile now to recall Kahane-Khintchine's inequalities. For any real numbers q, p ≥ 1 and any finite sequence (b n ) n≥0 in B there is a constant K(p, q) which depends only on p and q such that (5.2)

E n≥0 ε n b n q 1/q ≤ K(p, q)E n≥0 ε n b n p 1/p .
For the specific case B = R, those inequalities are called Khinthine's inequalities and we have

E n≥0 ε n b n 2 1/2 =   n≥0 |b n | 2   1/2
.

We can now define the notion of cotype of a Banach space.

Definition 5.3. -A Banach space B has cotype p ≥ 2 if there are real numbers q ≥ 1 and C q > 0 such that for any finite sequence (b n ) n≥0 in B one has

(5.3) n≥0 b n p 1/p ≤ C q E n≥0 ε n b n q 1/q .
Thanks to (5.2), notice that if (5.3) holds then it holds for any q ≥ 1. For instance, one can prove that for any p ≥ 1 the Banach space B := L p (R d ) has cotype max(2, p). To see this, we can make use of Kahane-Khintchine's inequalities for q = p:

E N n=1 ε n f n p L p (R d ) = R d E N n=1 ε n (ω)f n (t) p dt ∼ C p R d N n=1 |f n (t)| 2 p 2 dt.
In the case p ≤ 2, by denoting • 2/p the obvious norm of R N , we can write

R d N n=1 |f n (t)| 2 p 2 dt = R d |f 1 (t)| p , . . . , |f N (t)| p 2/p dt ≥ R d (|f 1 (t)| p , . . . , |f N (t)| p )dt 2/p ≥ N n=1 f n 2 L p (R d ) p 2 .
In the case p ≥ 2, we write

R d N n=1 |f n (t)| 2 p 2 dt ≥ R d N n=1 |f n (t)| p dt = N n=1 f n p L p (R d ) .
As used in [START_REF] Grivaux | Almost sure convergence of some random series[END_REF] for Gaussian random variables, we have the following astonishing result of Maurey and Pisier: To see (i) ⇒ (iii), we begin by assuming that the random variables X n are symmetric. The proof of [13, Corollaire 1.3, a) ⇒ b), page 69] shows that there is a positive constant C which involves a moment E[|X 1 | q ] (for some q > 0) such that for any sequence (b n ) n≥0 we have

∀k, ℓ ≥ 1 E ℓ n=k X n b n 2 ≤ CE ℓ n=k ε n b n 2 .
Since the series ε n b n converges almost surely, it converges in L 2 (Ω, B) (see Theorem 5.1), so does X n b n . Now assume that X n are merely centered. Clearly, Z n (ω, ω ′ ) = X n (ω)-X n (ω ′ ) is symmetric on the probability space Ω × Ω ′ . Therefore, the previous analysis shows that Z n (ω, ω ′ )b n converges in L 2 (Ω × Ω ′ , B) and also in L 1 (Ω × Ω ′ , B). Now we use that random variables X n are centered:

∀ℓ ≥ k E ω ℓ n=k X n (ω)b n ≤ E ω,ω ′ ℓ n=k X n (ω)b n -X n (ω ′ )b n .

That means that

X n b n converges in L 1 (Ω, B), so converges in probability and almost surely in B (see [START_REF] Li | Queffélec -Introduction à l'étude des espaces de Banach[END_REF]Théorème II.3]).

5.1. Proof of Proposition 2.1. -Equivalence of (i) and (ii) comes from Theorem 5.2, Theorem 5.4 and the fact that L p (R d ) has finite cotype. In view to check the link with (iii), it is necessary and sufficient to study convergence in L p (Ω, L p (R d )) (see Theorem 5.1). Cauchy criterion leads to handle terms of the following form:

Ω R d ℓ n=k ε n (ω)f n (x) p dP(ω)dx = R d E ω ℓ n=k ε n (ω)f n (x) p dx.
By Khintchine's inequalities (5.2), there exists C p ≥ 1 so that

1 C p R d ℓ n=k |f n (x)| 2 p/2 dx ≤ Ω R d ℓ n=k ε n (ω)f n (x) p dP(ω)dx ≤ C p R d ℓ n=k |f n (x)| 2 p/2 dx,
and we conclude easily.

We can now prove Theorem 5.2.

Proof of Theorem 5.2. -Step 1. It is well known that we can realize any sequence of independent real random variables on the probability space [0, 1] endowed with the Lebesgue measure [START_REF]Williams -Probability with martingales[END_REF] (p. 34 and p.43). For any n ≥ 0, we consider a sequence ( Z n,ℓ ) ℓ≥1 of i.i.d. random variables on [0, 1] and such that X n = Z n,0 for any n ≥ 0. The following random variables

Z n,ℓ : [0, 1] N → R (ω 0 , ω 1 , . . . ) → Z n,ℓ (ω n ) are i.i.d
≥ 1 E exp(iξZ n,2ℓ-1 -iξZ n,2ℓ ) = E[exp(iξX 1 )] 2 ,
we see that Z n,2ℓ-1 -Z n,2ℓ is not constant almost surely. By using Lemma 5.7 with the sequence Y ℓ = Z n,2ℓ-1 -Z n,2ℓ , we see that there is an integer N ≥ 1 which depends only on the law of

X 1 such that 1 2 ≤ P [|Z n,1 -Z n,2 + • • • + Z n,2N -1 -Z n,2N | ≥ 1]
and is independent of n.

By setting S

n := Z n,1 -Z n,2 + • • • + Z n,2N -1 -Z n,2N
, we have the three properties:

(i) the series n≥0 S n b n converges almost surely in B in ω ∈ [0, 1] N , (ii) (S n ) n≥0 is a sequence of real, non-constant, symmetric and i.i.d. random variables, (iii) for any n ≥ 0 one has

P[|S n | ≥ 1] ≥ 1 2 . By construction, S n (ω) = S n (ω n ) with S n := Z n,1 -Z n,2 + • • • + Z n,2N -1 -Z n,2N .
Step 2. On the probability space [0, 1] N × [0, 1], one checks that the sequence (S n (ω)ε n (ω ′ )) n≥0 is i.i.d. and has the same common law than S 1 . From (i) and (ii), the series S n (ω)ε n (ω ′ )b n converges almost surely in (ω, ω ′ ) ∈ [0, 1] N × [0, 1]. Fubini's theorem ensures that almost surely in ω ∈ [0, 1] N the sequence (S n (ω)b n ) n≥0 belongs to Σ(B) (see definition (5.1)). Since S n (ω) = S n (ω n ), we also have P(| S n | ≥ 1) = P(|S n | ≥ 1) ≥ 1 2 . Thus, we can consider a Borel subset A n ⊂ [0, 1] such that

P(A n ) = 1 2 and A n ⊂ ω n ∈ [0, 1], | S n (ω n )| ≥ 1] .
Let us define ρ n (ω) := 1 An (ω n ) ≤ |S n (ω)| for each ω ∈ [0, 1] N . It is obvious that (ρ n ) n≥0 is a sequence of i.i.d. random variables with the 1 2 -Bernoulli law. From the contraction principle (Theorem 5.5), we know that almost surely in ω the sequence (ρ n (ω)b n ) n≥0 belongs to Σ(B).

Step 3. Let us identify Z/2Z with {0, 1} and introduce the compact group G := (Z/2Z) N which becomes now our reference probability space. It is clear that the maps g ∈ G → g n ∈ {0, 1} seen as random variables are independent and identically distributed with a 1 2 -Bernoulli law. Let us define G 0 ⊂ G the subset of elements (g n ) n≥0 such that (g n b n ) n≥0 belongs to Σ(B). Since Σ(B) is a vector space, G 0 is a subgroup of G. We directly get from the previous analysis in Step 2 that G 0 has a full Haar measure in G. Furthermore, G 0 contains obviously the everywhere dense subgroup of G of elements (g n ) n≥0 which satisfy g n = 0 for n ≫ 1. We use Lemma 5.8 to conclude that (1, 1, . . . ) belongs to G 0 , in other words (b n ) n≥0 belongs to Σ(B).

6. Annex: An alternative proof of Theorem 2.6 inspired by [START_REF] Tzvetkov | Riemannian analogue of a Paley-Zygmund theorem[END_REF] We give here a different proof of Theorem 2.6 we learnt from [START_REF] Tzvetkov | Riemannian analogue of a Paley-Zygmund theorem[END_REF], which we decided to detail for pedagogical reasons. For α > 1, we define the function Φ α : (0, +∞) -→ (0, +∞) Φ α (t) = (-ln t) α/2 if 0 < t < 1/a, Φ α (1/a) if t ≥ 1/a, where a > 1 is chosen in such a way that the function t → tΦ α (t) is increasing on (0, +∞). Observe also that t → Φ α (t) is non-increasing on (0, +∞). Then we have a result similar to [START_REF] Tzvetkov | Riemannian analogue of a Paley-Zygmund theorem[END_REF]Theorem 5]. We split the previous sum in two parts. Then, by Lemma 6.1 Now we use that Φ α is non-increasing and for aj 1/2 > |y -x| -1 we get Φ α (yx) ≤ Φ α (a -1 j -1/2 ) ≤ C(ln j) α/2 . As a consequence, from (6.3) and the assumption (2.5) on the c n , we deduce that

Theorem 1 . 1 (

 11 Paley-Zygmund). -Let u ω (x) = n∈Z ε n (ω)c n e inx where (ε n ) n∈Z is a sequence of independent Rademacher random variables. If u ∈ L 2 (T) then for all 2 ≤ p < +∞, a.s u ω ∈ L p (T).

Theorem 2 . 3 .

 23 -Let d ≥ 2, assume that (X n ) n≥0 verifies (1.7) and that u rad := n≥0 c n ψ n belongs to ε>0 H -ε (R d ). The random series u ω rad (x) = +∞ n=0 X n (ω)c n ψ n (x). converges almost surely in L p (R d ) for any p ∈]2, d α⋆(c) [ and diverges almost surely for any p > d α⋆(c)

Proposition 2 . 4 .

 24 -Let d ≥ 2. Consider the family (ψ n ) n≥0 of the L 2 -normalized radial Hermite functions which satisfies Hψ n = (4n + d)ψ n . Then (i) Assume that 2d d-1 < p ≤ +∞. Then

2. 3 .

 3 Notations and plan of the paper. -Notations. -In this paper c, C > 0 denote constants the value of which may change from line to line. These constants will always be universal, or uniformly bounded with respect to the other parameters. We write a b if a ≤ Cb and a ≈ b if ca ≤ b ≤ Ca, for some c, C > 0.

Now, denote by ψ n the nth L 2 ((|x| 2 2 √

 222 R d )-normalized radial Hermite function for d ≥ 2. One can prove that Hψ n = (4n + d)ψ n holds and that ψ n is proportional to L (d/2-1) n (|x| 2 )e -|x| 2 /2 (see for instance [20, Corollary 3.4.1]). By using the orthogonality of Laguerre functions L )e -|x| 2 /2 with c(d) := √ Vol(S d-1 )(see below (3.9) for p = 2).

p 4

 4

  Theorem 2.3. -We will use Proposition 2.1 and Proposition 2.4.

4. Proof of the regularity results 4 . 1 .

 41 Proof of Theorem 2.5. -Let us begin by introducing the following notation:∀λ > 0 E H (λ) := Span{ϕ j , λ j ≤ λ },and let us recall the following bound on the spectral function of H (see [16, Lemmas 3.1, 3.2 and 3.5]: there are constants C, c > 0 such that for any λ ≥ 1 and x ∈ R d one has (4.1)

Lemma 4 . 1 .

 41 -For any dimension d ≥ 1, there are s(d) ≥ 0 and C = C(d) > 0 such that the following inequalities hold

3

 3 iii) by denoting Vol the volume function, we have Vol B(y, cλ -s(d) ) ∩ B(0, λ) ≥ 1 Vol B(y, cλ -s(d) ) .

  (5.1) Σ(B) := (b n ) n≥0 , ε n b n converges a.s. It is clear that Σ(B) is a vector subspace of B N . The following theorem is well-known in the theory of Banach random series (see for instance [11, Chapitre 3, IV.2]): Theorem 5.1. -Let B a Banach space and consider a sequence (b n ) n≥0 in B. The following facts are equivalent (i) the sequence (b n ) n≥0 belongs to Σ(B), (ii) the random series ε n (ω)b n converges in probability, (iii) the random series ε n (ω)b n converges in law, (iv) there is some p ≥ 1 such that the random series ε n (ω)b n converges in L p (Ω, B), (v) for any p ≥ 1, the random series ε n (ω)b n converges in L p (Ω, B).

Theorem 5 . 2 (

 52 Hoffman-Jorgensen). -Let (X n ) n≥0 be a sequence of real, non-constant and i.i.d. random variables and (b n ) n≥0 be a sequence which takes values in a general Banach space B, we assume that the series X n (ω)b n converges almost surely in B. Then the series ε n (ω)b n converges almost surely in B, in other words (b n ) n≥0 belongs to Σ(B).

Theorem 5 . 4 (

 54 Maurey-Pisier). -The following assertions are equivalent (i) the Banach space B has finite cotype (that means that there is p ≥ 2 such that B has cotype p), (ii) for any sequence (b n ) n≥0 of B, the almost sure convergence of ε n b n implies the almost sure convergence of G n b n , where (G n ) n≥0 is a sequence of i.i.d. N R (0, 1) Gaussian random variables, (iii) for any sequence (b n ) n≥0 of B the almost sure convergence of ε n b n implies the almost sure convergence of X n b n where (X n ) n≥0 is any sequence of real, centered and i.i.d random variables with finite moments of any order. Proof. -The equivalence (i) ⇔ (ii) is done in [13, Corollaire 1.3]. Obviously, (iii) ⇒ (ii) is true by choosing X n = G n . Let us explain arguments which are not explicitly written in [13, Corollaire 1.3].

Lemma 6 . 1 . 1 0 2 ≤ |y -x| 2 1 0≤ Cj|y -x| 2 1 0ϕ 1 0≤ 1 0H 1 / 2 ϕ 2 dt = λ n 1 0ϕn z 2 ≤

 6112111112212 -Let (ϕ n ) n≥0 be any Hilbertian basis of eigenfunctions for the harmonic oscillator H.Let γ(1) = -1/6 and γ(d) = d/2 -1 for d ≥ 2.Then for all j ≥ 1 and x, y ∈ R d we haven∈I(j) |ϕ n (y)ϕ n (x)| 2 1/2 ≤ C|y -x|j γ(d)/2+1/2 .Proof.-By the Taylor formula and Cauchy-Schwarz we get, for n ∈ I(j)|ϕ n (y)ϕ n (x)| 2 ≤ |y -x| 2 ∇ϕ n x + (yx)t dt ∇ϕ n x + (yx)t 2 dt n x + (yx)t 2 dt, (6.1)where in the last line we used∇ϕ n x + (yx)t 2 dt n x + (yx)t n x + (yx)t 2 dt.Now we sum up the inequalities (6.1) and get with (1.3) n∈I(j) |ϕ n (y)ϕ n (x)| 2 ≤ Cj|y -x| 2 sup z∈R n∈I(j) |ϕ Cj γ(d)+1 |y -x| 2 , which was the claim We follow the main lines of the proof of N. Tzvetkov [21, Theorem 5]. We define the pseudodistance δ by δ(x, y) = n≥0 |c n | 2 |ϕ n (y)ϕ n (x)| 2 1/2 .

Lemma 6 . 2 .

 62 -Assume that the coefficients (c n ) satisfy (2.5), thenδ(x, y) ≤ C Φ α (|y -x|) .Proof. -We clearly have(δ(x, y)) 2 ≤ C +∞ j=1 max k∈I(j) |c k | 2 n∈I(j)|ϕ n (y)ϕ n (x)| 2 .

I 1 2 . ( 6 . 2 ) 2 ≤

 12622 (x, y) := j: aj 1/2 ≤|y-x| -1 max k∈I(j)|c k | 2 n∈I(j) |ϕ n (y)ϕ n (x)| 2 ≤ C j: aj 1/2 ≤|y-x| -1 j γ(d)+1 max k∈I(j) |c k | 2 |y -x| 2 = C Φ 2 α (yx) j: aj 1/2 ≤|y-x| -1 j γ(d)+1 max k∈I(j) |c k | 2 |y -x|Φ α (yx)Now we use that the function t -→ tΦ α (t) is increasing, thus for aj 1/2 ≤ |y -x| -1 we have |y -x|Φ α (yx) j -1 (ln j) α , therefore from (6.2) and the assumption (2.5) on the c n , we getI 1 (x, y) ≤ CΦ -2 α (yx) +∞ j=1 j γ(d) (ln j) α max k∈I(j) |c k | 2 ≤ CΦ -2 α (yx).Next, by (1.3)I 2 (x, y) := j: aj 1/2 >|y-x| -1 max k∈I(j) |c k | 2 n∈I(j) |ϕ n (y)ϕ n (x)| 2 ≤ C j: aj1/2 >|y-x| -1 j γ(d) max k∈I(j) |c k | 2 . (6.3)

I 2

 2 (x, y) ≤ CΦ -2 α (yx) +∞ j=1 j γ(d) (ln j) α max k∈I(j) |c k | 2 ≤ CΦ -2 α (yx),which completes the proof.

  3.2. Proof of Proposition 2.4. -Let us first recall some results concerning the Laguerre polynomials, see[START_REF] Thangavelu | Lectures on Hermite and Laguerre expansions[END_REF] Chapter 1] or[START_REF] Szegö | Orthogonal polynomials[END_REF]. For α > -1, the Laguerre polynomial L

	(α) n

  with the same law than the random variables X n . The assumption of Theorem 5.2 ensures that the series n≥0 Z n,ℓ (ω)b n = n≥0 Z n,ℓ (ω n )b n converges in B almost surely in ω ∈ [0, 1] N . By combining Lemma 5.6 and the equations ∀ℓ

Acknowledgments. The authors would like to thank warmly Hervé Queffélec for very interesting discussions about random Banach series, and in particular for the proof of Theorem 5.2.

D. R. was partly supported by the grant "NOSEVOL" ANR-2011-BS01019 01. L.T. was partly supported by the grant "HANDDY" ANR-10-JCJC 0109 and by the grant "ANA É" ANR-13-BS01-0010-03.

5.2. Proof of Theorem 5.2. -We need the contraction principle (see for instance [START_REF] Li | Queffélec -Introduction à l'étude des espaces de Banach[END_REF]Théorème III.1] or [START_REF] Kahane | Some random series of functions[END_REF]Chapter 2.6 in the Rademacher framework]) and a few lemmas.

Theorem 5.5 (contraction principle). -Let (X n ) n≥0 be a sequence of symmetric independent random variables which takes values in a Banach space B. If X n converges almost surely in B then, for any bounded real sequence (λ n ) n≥0 , the series λ n X n converges almost surely in B.

Let us recall a classical lemma in the probability theory.

Lemma 5.6. -Let X be a real random variable, the following statements are equivalent:

Proof. -The implications (iii) ⇒ (ii) and (ii) ⇒ (i) are obvious. Suppose now (i) and let

The same is true by replacing ξ 0 with ξ 1 and α 0 with α 1 . Because X is not constant almost surely, there are at least two numbers x = y which both belong to

-For any sequence of real, non-constant and i.i.d. random variables (Y ℓ ) ℓ≥1 we have lim

Proof. -Let µ be the law of Y 1 and ϕ ∈ L 1 (R) be a function such that ϕ(x) ≥ 1 holds for any x ∈ (-1, +1). It comes

Point (iii) of Lemma 5.6 ensures that | µ(ξ)| < 1 holds for almost all ξ in the sense of Lebesgue. We conclude by the dominated convergence theorem if N tends to infinity.

Lemma 5.8. -Let G be a locally compact Abelian group, consider a subgroup G 0 ⊂ G which has a positive Haar measure and is everywhere dense. Then G 0 is the whole group G.

Proof. -It is sufficient to prove that G 0 is closed. Steinhaus theorem states that G 0 -G 0 ⊂ G 0 contains an open neighbourhood of the origin. By using translations of G 0 , it turns out that G 0 is an open subgroup of G. A classical argument from the theory of topological groups asserts that G 0 is also closed: we just write G = ⊔ i∈I (G 0 + g i ) where (g i ) i∈I is a family of elements of G and g i = 0 for one i ∈ I, it appears that the complementary subset of G 0 is open.

Proof of Theorem 2.6. -It is enough to prove that on every compact set K ⊂ R d , a.e. in ω, u ω is continuous on K. Hence we can follow the proof given in [21, Theorem 5] using an entropy argument (Dudley-Fernique criterion), together with the result of Lemma 6.2.

Remark 6.3. -Let's compare the two different proofs. This proof relies on both a decomposition in space and in frequencies, while in the other proof one only needs a decomposition in frequencies.

Observe also that in the first proof one moreover gets that for almost all ω ∈ Ω, u ω is bounded.