Didier Robert
email: didier.robert@univ-nantes.fr

Laurent Thomann
email: laurent.thomann@univ-nantes.fr

On random weighted Sobolev inequalities on R d and applications

Keywords: 1991 Mathematics Subject Classification. 35R60, 35P05, 35Q55, 35J10, 33D45, 58J51 Harmonic oscillator, spectral analysis, concentration of measure, Hermite functions, quantum ergodicity, nonlinear Schrödinger equation with random initial conditions

. We focus here on properties of the harmonic oscillator and Hermite functions for simplicity. The general idea is that several well known deterministic results like Sobolev type inequalities are improved in a large extent by introducing some randomness. This is very useful in particular for N.L.S with supercritical initial data.

Introduction

We present here the results we have obtained in [START_REF] Poiret | Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator[END_REF][START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF] in collaboration with Aurélien Poiret and the results of [START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF]. Using ideas of Shiffman-Zelditch [START_REF] Shiffman | Random polynomials of high degree and Lévy concentration of measure[END_REF] and Burq-Lebeau [START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF] who developed a randomisation method based on the Laplace operator on a compact Riemannian manifold, we give a randomisation method on L 2 (R d ) associated with the Laplace operator with confining potentials. We are able to construct probability measures on L 2 (R d ), on the support of which a typical function enjoys better Sobolev estimates than expected. These measures and estimates have an interest in themselves, but we can moreover use them to

• construct orthonormal bases of L 2 (R d ) of Hermite functions which enjoy good L ∞ bounds. These bounds are shown to be optimal ; • prove quantum ergodicity results ; • show a.s. local and global well-posedness results for supercritical nonlinear Schrödinger equations with quadratic potential ; • show a.s. global well-posedness results for supercritical nonlinear Schrödinger equations (without potential).

To simplify the exposition, we will focus on estimates in the Sobolev spaces based on the harmonic oscillator in L 2 (R d )

H = -∆ + |x| 2 = d j=1 (-∂ 2 j + x 2 j ).
We get optimal stochastic weighted Sobolev estimates on R d using the Burq-Lebeau method. In [START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF], the construction of the measures relied on Gaussian random variables but it is possible to consider general random variable which satisfy concentration of measure estimates (like Bernoulli random variables). However, the optimal estimates (including lower bounds) are obtained only for the Gaussian law.

Let us emphasis here that even if there exist many similarities between the spectral properties of the Laplace operators a compact Riemannian manifolds and Schrödinger Hamiltonians with a confining potential on R d , there exists a big difference due to the complex behaviour at high energy of the spectral function on a non-compact configuration space.

In [START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF] we prove that most of the results can be extended to more general Schrödinger Hamiltonians -+ V (x) with confining potentials V .

Spectral estimates and harmonic Sobolev spaces

Let us recall the well known spectral analysis of the quantum harmonic oscillator H on R d . This is explained in many text books on quantum mechanics. For mathematical details, we refer to Helffer [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF] or to the course of Ramond [START_REF] Ramond | Analyse semiclassique, Résonance et Contrôle de l'équation de Schrödinger[END_REF].

The operator H is self-adjoint on the Hilbert space L 2 (R d ), and has the discrete spectrum {2n + d} n∈N .

• For d = 1 each eigenvalue is simple, Hh n = (2n + d)h n where h n is the n th Hermite function. • For d ≥ 2, an orthonormal basis is obtained by tensor products.

Let

α = (α 1 , • • • , α d ) ∈ N d , x = (x 1 , • • • , x d ) ∈ R d , (2.1) h α (x) = h α1 (x 1 ) • • • h α d (x 1 ), α 1 + • • • + α d = n.
It is sometimes convenient to use different notations. Any orthonormal basis of ker(H -(2n + d)) can be denoted by {ϕ n,k } 1≤k≤mn , so that Hϕ n,k = (2n + d)ϕ n,k .

In the sequel, {λ j } j∈N denotes the non-decreasing sequence of eigenvalues where each eigenvalue (2n+d) is repeated according its multiplicity, such that Hϕ j = λ j ϕ j where {ϕ j } j≥0 is an orthonormal basis of L 2 (R d ). By (2.1) the multiplicity of the eigenvalue 2n + d is of order n d-1 , therefore λ j is of order j 1/d .

The scale of harmonic Sobolev spaces is defined as follows: s ≥ 0, p ≥ 1.

(2.2)

W s,p = W s,p (R d ) = u ∈ L p (R d ), H s/2 u ∈ L p (R d ) , H s = H s (R d ) = W s,2 .
The natural norms are denoted by u W s,p and up to equivalence of norms we have (see [START_REF]Local smoothing property and strichartz inequality for Schrödinger equations with potentials superquadratic at infinity[END_REF]Lemma 2.4]) for 1 < p < +∞

u W s,p = H s/2 u L p ≡ (-∆) s/2 u L p + x s u L p .
For s ≥ 0, H s is the domain of the self-adjoint operator H s/2 and we have

H s = u ∈ L 2 (R d ), j≥0 λ s j | ϕ j , u | 2 < +∞ and u 2 H s ≈ j≥0 λ s j | ϕ j , u | 2 .
In the sequel we fix d ≥ 2. Let us consider spectral windows I h depending on a small parameter h > 0 (this notation is convenient to make use of semiclassical results here and in the more general setting considered in [START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF]).

Let us denote by

I h = [ a h
h , b h h [ and assume that a h and b h satisfy, for some a, b, D > 0 and δ ∈ [0, 1],

(2.3) lim h→0 a h = a, lim h→0 b h = b, 0 < a ≤ b and b h -a h ≥ Dh δ ,
with any D > 0 if δ < 1 and D ≥ 2 in the case δ = 1. This condition ensures that N h , the number (with multiplicities) of eigenvalues of H in I h tends to infinity when h → 0. Indeed, we can check that

N h ∼ ch -d (b h -a h ), in particular lim h→0 N h = +∞, since d ≥ 2. In the sequel, we write Λ h = {j ≥ 1, λ j ∈ I h } and E h = span{ϕ j , j ∈ Λ h }, so that N h = #Λ h = dim E h . Finally, we denote by S h = u ∈ E h : u = 1
the unit sphere of E h (which is a complex linear space of dimension N h ).

A very useful tool to get local and global estimates in the deterministic setting as well as in the probabilistic setting are L p estimates of spectral projectors and of their kernels, the so-called spectral function. On compact manifolds this is obvious in [START_REF] Sogge | Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds[END_REF] for example.

The spectral function is then defined as

π H (λ; x, y) = λj ≤λ ϕ j (x)ϕ j (y)
(recall that this definition does not depend on the choice of {ϕ j , j ∈ N}). When the energy λ is localized in I ⊆ R + we denote by Π H (I) the spectral projector of H on I. The range E H (I) of Π H (I) is spanned by {ϕ j ; λ j ∈ I} and Π H (I) has an integral kernel given by

π H (I; x, y) = [j : λj ∈I] ϕ j (x)ϕ j (y).
We will also use the notation

E H (λ) = E H ([0, λ]), N H (λ) = dim[E H (λ)].
The relationship between Sobolev type estimates and the spectral function is illustrated by the following elementary result:

(2.4) |u(x)| ≤ (π H (I; x, x)) 1/2 u L 2 (R d ) .
We have the following uniform estimate:

π H (λ; x, x) ≤ C θ λ (d+θ)/2 x -θ ,
where C θ depends only on θ > 0. Now using (2.4) we get (with the semiclassical parameter h = λ -1 )

x θ/2 h (d+θ)/4 |u(x)| ≤ C θ u L 2 (R d ) , ∀u ∈ E H (h -1 ).
For smaller spectral windows like I h we need more accurate estimates on the spectral function

e x := π H ( b h h ; x, x) -π H ( a h h ; x, x).
For any θ ≥ 0 there exists C θ > 0 such that (2.5)

x θ e x ≤ C θ N h h (d-θ)/2 .
where C θ depends only on θ ≥ 0. Using (2.4) and interpolation inequalities we get Sobolev type inequalities for

u ∈ E h , θ ≥ 0, p ≥ 2 (2.6) u L ∞,θ/2 (R d ) ≤ C N h h (d-θ)/2 1/2 u L 2 (R d ) ,
which in turn implies using interpolation

(2.7) u L p,θ(p/2-1) (R d ) ≤ C N h h (d-θ)/2 1 2 -1 p u L 2 (R d ) .
The previous inequality can be written as

u L p,θ(p/2-1) (R d ) ≤ C(b h -a h ) 1 2 -1 p h -( d+θ 2 )( 1 2 -1 p ) u L 2 (R d ) , ∀p ∈ [2, +∞], ∀θ ∈ [0, d].
We shall see that these Sobolev type inequalities in a large extent can be improved in a random sense that we shall explain now.

3. Probabilistic weighted estimates for frequency localized functions 3.1. Probabilistic setting. Let us introduce now our probabilistic setting. Consider a probability space (Ω, F, P) and let {X n , n ≥ 1} be a sequence of i.i.d random variables, centered and normalized following a law ν. We assume for simplicity in all this paper that ν is either the standard complex Gaussian N C (0, 1) or the Bernoulli law 1 2 δ -1 + 1 2 δ 1 (see [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF] for more general laws satisfying a concentration of measure property). Let γ := (γ n ) n∈N be a sequence of complex numbers and define the random vector in E h

(3.1) v γ (ω) := j∈Λ h γ j X j (ω)ϕ j .
The probability law of v γ is denoted by ν γ . We define a probability measure P γ on the sphere S h by: for all measurable and bounded function f :

S h -→ R, S h f (u)dP γ (u) = Ω f v γ (ω) v γ (ω) L 2 (R d ) dP(ω).
It is not difficult to see that in the isotropic case (γ j = 1

√ N h
for all j ∈ Λ h ) and when X j ∼ N C (0, 1), then P γ is the uniform probability on the sphere S h .

For more general sequences γ we need to assume the following squeezing conditions. There exists

K 0 > 0, K 1 > 0 such that (3.2) γ ∞ (Λ h ) ≤ K 0 √ N h γ 2 (Λ h ) , ∀h ∈]0, 1].
We also need the stronger condition

(3.3) K 1 √ N h γ 2 (Λ h ) ≤ |γ j | ≤ K 0 √ N h γ 2 (Λ h ) , ∀h ∈]0, 1], ∀j ∈ Λ h .
Notice that

E( v γ 2 L 2 (R d ) ) = j∈Λ h |γ j | 2 = |γ| 2 2 (Λ h ) .
Let us comment now our assumptions.

Remark ). (iv) The Gaussian complex law is natural because it gives the uniform probability measure on the sphere S h , but in this case the H s -norm of v γ (3.1) depends on ω. This is not the case if ν is a centred Bernoulli law. Moreover the probabilistic information we get clearly depends on ν and it may be interesting to consider also discrete probability laws. Actually, our results are proven under the assumption that the family (ν ⊗N , R N ) N ≥1 satisfies the concentration property of measures. Finally, we point out that the results of Section 6 hold true under the weaker assumption that ν is sub-gaussian: there exists σ ≥ 0 such that R e sx dν(x) ≤ e σ 2 2 s 2

, ∀s ∈ R.

For a discussion about these conditions we refer to [33, Section 2.1].

A very useful tool here is a measure concentration property satisfied by the probability P γ , proved by P. Lévy for the uniform law. For a study of this notion, we refer to the book [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]. Proposition 3.1. Suppose that Assumption (3.3) is satisfied. Then there exist constants K > 0, κ > 0 such that for every Lipschitz function

F : S h -→ R satisfying |F (u) -F (v)| ≤ F Lip u -v L 2 (R d ) , ∀u, v ∈ S h ,
we have

(3.4) P γ u ∈ S h : |F -M F | > r ≤ Ke - κN h r 2 F 2 Lip , ∀r > 0, h ∈]0, 1],
where M F is a median for F .

Recall that a median M F for F is defined by

P γ u ∈ S h : F ≥ M F ≥ 1 2 , P γ u ∈ S h : F ≤ M F ≥ 1 2 .
The factor N h in the exponential of r.h.s of (3.4) is crucial in applications to get large deviation estimates.

< C 0 < C 1 , K > 0, c 1 > 0 , h 0 > 0 such that for all r ∈ [2, K| log h|] and h ∈]0, h 0 ] such that (3.5) P γ,h u ∈ S h : u L p,d(p/2-1) -M p > Λ ≤ 2 exp -c 2 N 2/p h Λ 2 . Moreover if condition (3.3) is satisfied and if δ ∈ [0, 2/3[, then we have C 0 √ p ≤ M p ≤ C 1 √ p, ∀p ∈ [2, K log N h ].
This result shows that u L p,d(p/2-1) has a Gaussian concentration around its median. The first part of the theorem is a direct application of Proposition 3.1 to

F (u) = u L p,d(p/2-1)
. The estimate of M p is more involved and needs the following estimate for the spectral function.

Lemma 3.1 ([33], Lemma 4.9). Let δ ∈ [0, 2/3[ and -d/(p -1) < θ ≤ 1. There exist 0 < C 0 < C 1 and h 0 > 0 such that C 0 N h h d-θ 2 (1-1 p ) ≤ R d x θ(p-1) e p x dx 1/p ≤ C 1 N h h d-θ 2 (1-1 p ) , for every p ∈ [1, ∞[ and h ∈]0, h 0 ].
Let 0 < η < 1, then we are able to precise the concentration estimate (3.5) in the regime p ∼ | ln h| 1-η . Assume that for all j ∈ Λ h , γ j = N -1/2 h and that X j ∼ N C (0, 1), so that P := P γ is the uniform probability on S h . Theorem 3.2. Let δ ∈ [0, 2/3[, 0 < η < 1 and set p h = | ln h| 1-η . Then there exists constants C = C (d) and c > 0 such that for all ε > 0 there exists h 0 > 0 so that for all h ∈]0, h 0 ]

P γ,h u ∈ S h : u L p h ,d(p h /2-1) -C √ p h > ε √ p h ≤ 2e -cε 2 | ln h| .
We give the main lines of the proof in the appendix.

Theorem 3.3 ([33], Theorem 4.1). There exist h

0 ∈]0, 1], c 2 > 0 and C > 0 such that if c 1 = d(1 + d/4), we have P γ,h u ∈ S h : h -d-θ 4 u L ∞,θ/2 (R d ) > Λ ≤ Ch -c1 e -c2Λ 2 , ∀Λ > 0, ∀h ∈]0, h 0 ].
We can deduce probabilistic estimates for the derivatives as well. Recall that the Sobolev spaces W s,p (R d ) are defined in (2.2). The following result say that Theorem 3.3 is sharp for large enough spectral windows.

Theorem 3.4 ([33], Theorem 1.1). Let d ≥ 2. Assume that 0 ≤ δ < 2/3 in (2.
3) and that condition (3.3) holds. Then there exist 0 < C 0 < C 1 , c 1 > 0 and h 0 > 0 such that for all h ∈]0, h 0 ].

P γ,h u ∈ S h : C 0 | log h| 1/2 ≤ u W d/2,∞ (R d ) ≤ C 1 | log h| 1/2 ≥ 1 -h c1 .
It is clear that under condition (3.3), there exist 0 < C 2 < C 3 , so that for all u ∈ S h , and s ≥ 0

C 2 h -s/2 ≤ u H s (R d ) ≤ C 3 h -s/2 ,
since all elements of S h oscillate with frequency h -1/2 . Thus Theorem 3.4 shows a gain of d/2 derivatives in L ∞ , and this induces a gain of d derivatives compared to the usual deterministic Sobolev embeddings. This can be compared with the results of [START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF] where the authors obtain a gain of d/2 derivatives on compact manifolds: this comes from different behaviours of the spectral function, see Section 2. Notice that the bounds in Theorem 3.4 (and in the results of [START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF] as well) do not depend on the length of the interval of the frequency localisation I h (see (2.3)), but only on the size of the frequencies. This is a consequence of the randomisation, and from the bound (2.5).

In our works we give estimates for the eigenfunctions in the configuration space, while Feng, Shiffman and Zelditch [START_REF] Shiffman | Random polynomials of high degree and Lévy concentration of measure[END_REF][START_REF] Feng | Median and mean of the supremum of l 2 normalized random holomorphic fields[END_REF] give similar estimates in the Bargmann representation for holomorphic fields.

We now state a result which gives optimal L ∞ bounds in the general case 0 ≤ δ ≤ 1 Theorem 3.5. Let d ≥ 2. Assume that 0 ≤ δ ≤ 1 in (2.3) and that condition (3.3) holds. Then there exist 0 < C 0 < C 1 , c 1 > 0 and h 0 > 0 such that for all h ∈]0, h 0 ].

P γ,h u ∈ S h : C 0 | log h| 1/2 ≤ h -d/4 u L ∞ (R d ) ≤ C 1 | log h| 1/2 ≥ 1 -h c1 .
We will give the main lines of the proof of this result in the Appendix B. The key obervation is that the estimates of the L p norm of the spectral function (which are not optimal when δ is close to 1), become optimal in the regime p = p h ∼ c| ln h|.

Hermite functions estimates

In this section the previous results are applied to obtain L ∞ estimates for Hermite functions.

ϕ n L ∞ (R d ) ≤ M λ -d 4 n (1 + log λ n ) 1/2 .
In other words, all elements of this basis are decreasing in L ∞ (R d ) norm. Let us compare (4.1) with the general known bounds on Hermite functions. We have Hϕ n = λ n ϕ n , with λ n ∼ cn 1/d , therefore (4.1) can be rewritten

(4.2) ϕ n L ∞ (R d ) ≤ M n -1/4 (1 + log n) 1/2 , ∀n ≥ 0.,
For a general orthonormal basis of Hermite functions, with d ≥ 2, Koch and Tataru [START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF] (see also [START_REF] Koch | Semiclassical L p estimates[END_REF]) prove that

ϕ n L ∞ (R d ) ≤ Cλ d 4 -1 2 n ,
whichshows that (4.2) induces a gain of d -1 derivatives compared to the general case.

Notice that for the tensorial Hermite basis h α (x), using the bound for d = 1 [START_REF] Yajima | Smoothing property for Schrödinger equations with potentials superquadratic at infinity[END_REF][START_REF] Koch | L p eigenfunction bounds for the Hermite operator[END_REF] we have

h α L ∞ (R d ) ≤ Cλ -1 12 n , for α 1 + • • • + α d = λ n .
Observe also that the basis of radial Hermite functions does not satisfy (4.1) in dimension d ≥ 2, as we have recently proved in [START_REF] Imekraz | On random Hermite series[END_REF]. We will see in Theorem 4.3 that the log term in (4.1) can not be avoided.

Theorem 4.1 is a consequence of a more powerful result, obtained following an idea in [START_REF]Quantum ergodicity on the sphere[END_REF]. Here we follow the main lines of [START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF]Section 3]. In this part the upper bounds estimates of Section 3 are used in their full strength.

Firstly, we assume that for all j ∈ Λ h ,

γ j = N -1/2 h
and that X j ∼ N C (0, 1), so that P := P γ is the uniform probability on S h . We set h k = 1/k with k ∈ N * , and

a h k = 2 + dh k , b h k = 2 + (2 + d)h k .
Then (2.3) is satisfied with δ = 1 and D = 2. In particular, each interval

I h k = a h k h k , b h k h k = [2k + d, 2k + d + 2[
only contains the eigenvalue 2k + d with multiplicity N h k ∼ ck d-1 , and E h k is the corresponding eigenspace of the harmonic oscillator H. We can identify the space of the orthonormal basis of E h k with the unitary group U (N h k ) and we endow U (N h k ) with its Haar probability measure ρ k . Then the space B of the Hilbertian bases of eigenfunctions of H in L 2 (R d ) can be identified with

(4.3) B = × k∈N U (N h k ),
which can be endowed with the measure

(4.4) dρ = ⊗ k∈N dρ k . Denote by B = (ϕ k, ) k∈N, ∈ 1,N h k ∈ B a typical orthonormal basis of L 2 (R d ) so that for all k ∈ N, (ϕ k, ) ∈ 1,N h k ∈ U (N h k ) is an orthonormal basis of E h k .
Then the main result of the section which implies Theorem 4.1, is the following. Then, if M > 0 is large enough, there exist c, C > 0 so that for all r > 0

ρ B = (ϕ k, ) k∈N, ∈ 1,N h k ∈ B : ∃k, ; ϕ k, W d/2,∞ (R d ) ≥ M (log k) 1/2 + r ≤ Ce -cr 2 .
The previous result relies on the following proposition Then, if M > 0 is large enough, there exist c, C > 0 so that for all r > 0 and k ≥ 1

ρ k B k = (ψ ) ∈ 1,N h k ∈ U (N h k ) : ∃ ∈ 1, N h k ; ψ W d/2,∞ (R d ) ≥ M (log k) 1/2 +r ≤ Ck -2 e -cr 2 .
Let us show how this result implies Theorem 4.2: We set

F k,r = B k = (ψ ) ∈ 1,N h k ∈ U (N h k ) : ∀ ∈ 1, N h k ; ψ W d/2,∞ (R d ) ≤ M (log k) 1/2 +r ,
and

F r = ∩ k≥1 F k,r . Then for all r > 0 ρ(F c r ) ≤ k≥1 ρ k (F c k,r ) ≤ C k≥1 k -2 e -cr 2 = C e -cr 2 ,
and this completes the proof.

Then from the Borel-Cantelli Lemma we get

(ϕ k, ) k∈N, ∈ 1,N h k of eigenfunctions of H we have (i) For 2 ≤ p < +∞ ϕ k, L p (R d ) ≤ C p k -d 2 ( 1 2 -1 p ) , ∀ k ∈ N, ∀ ∈ 1, N h k . (ii) For p = +∞ ϕ k, L ∞ (R d ) ≤ (M + 1)k -d/4 (1 + log k) 1/2 , ∀ k ∈ N, ∀ ∈ 1, N h k .
We now state a result we shows that the log factor in the previous bounds is optimal

Theorem 4.3. Let d ≥ 2. Then for all M > 0 ρ B = (ϕ k, ) k∈N, ∈ 1,N h k ∈ B : lim inf k→+∞ inf =1,...,N k k d/4 ϕ k, L ∞ (R d ) ≤ M = 0.
This means that, from an orthonormal basis, one cannot extract a subsequence which satisfies better bounds than in Corollary 4.1.

This result is a consequence of Theorem 3.5. For the proof, we can follow the main lines of [START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF]Théorème 8] where an analogous result for the spherical harmonics is proven.

Application to Quantum ergodicity

Recall that "quantum ergodicity" for a quantum Hamiltonian H usually means that in some semiclassical regime almost all eigenfunctions of H are distributed according to an invariant measure with a support equal to the classical energy shell.

Assume that

I h =]a h , b h ] is such that lim h→0 a h = lim h→0 b h = η > 0 and lim h→0 b h -a h h = +∞.
The Liouville measure L η associated with the classical Hamiltonian H c (x, ξ) = |x| 2 + |ξ| 2 is here the uniform probability on the sphere √ η S 2d-1 . We define the class of symbols

S(1) = A ∈ C ∞ (R 2d ), ∀α, ∀β, sup (x,ξ) |∂ α x ∂ β ξ A(x, ξ)| < +∞ ,
and S(1, 1) ⊂ S(1) the class of symbols such that A ∈ C ∞ (R 2d ) and A is homogeneous of degree 0 outside a small neighbourhood of (0, 0

) in R d x × R d ξ : A(λx, λξ) = A(x, ξ) for every λ ≥ 1 and |(x, ξ)| ≥ ε.
For A ∈ S(1) let us denote by  the Weyl quantization of A (here h = 1). Notice that if u h L 2 (R d ) = 1 then A → u h , Âu h defines a semiclassical measure on √ η S 2d-1 (see e.g. [START_REF] Burq | Mesures semi-classiques et de défaut[END_REF]).

Then we have

Theorem 5.1 ([35], Theorem 1.4). Assume that we are in the isotropic case (γ j = 1

√

N h for all j ∈ Λ h ). Then there exist c, C > 0 so that for all r ≥ 1 and A ∈ S(1, 1), (5.1)

P h u ∈ S h : | u, Âu -L η (A)| > r ≤ Ce -cN h r 2 , ∀h ∈]0, 1].
This result can be related with quantum ergodicity (see [START_REF] Zelditch | Uniform distribution of eigenfunctions on compact hyperbolic surfaces[END_REF][START_REF] De Verdière | Ergodicité et fonctions propres du Laplacien[END_REF][START_REF] Helffer | Ergodicité et limite semi-classique[END_REF][START_REF] Maples | Quantum unique ergodicity for random bases of spectral projections[END_REF] and the book [START_REF] Zworski | Semiclassical analysis[END_REF]Chapter 15] for an introduction to this subject) which concerns the semi-classical behavior of ϕ j , Âϕ j when the classical flow is ergodic on the energy hyper surface H -1 c (η). Then, for "almost all" eigenfunctions ϕ j , we have ϕ j , Âϕ j j→+∞ -→ L η (A). The meaning of Theorem 5.1 is that we have u, Âu h→0 -→ L η (A) for almost all u ∈ S h such that all modes (ϕ j ) j∈Λ h are "almost uniformly distributed " in u. For related results on compact manifolds see Zelditch [START_REF]Random orthonormal bases of spaces of high dimension, Complex patterns in wave functions-drums, graphs and disorder[END_REF].

As a consequence of Theorem 5.1 we get easily that almost all bases of Hermite functions is Quantum Uniquely Ergodic (see Theorem 5.2 for a precise statement). In [START_REF]Quantum ergodicity on the sphere[END_REF] the author proved that on the standard sphere a random orthonormal basis of eigenfunctions of the Laplace operator is ergodic.

Burq-Lebeau [6, Théorème 3] obtained a similar result for the Laplacian on a compact manifold. A modification of their proof allows to consider more general random variables satisfying the Gaussian concentration assumption instead of the uniform law.

From (5.1) we directly deduce that there exists C > 0 so that for all p ≥ 2 and

h ∈]0, 1], u, Âu -L η (A) L p P h ≤ CN -1/2 h √ p.

Therefore, if one denotes by

u ω h = 1 √ N h j∈Λ h X j (ω)ϕ j , then we have u ω h , Âu ω h h→0 -→ L η (A) in L p (Ω)
-norm with a remainder estimate.

For example taking the Bernoulli law for the X j then

u ω h = 1 √ N h j∈Λ h (-1) εj (ω) ϕ j
where {ε j } j≥1 are i.i.d Bernoulli variables.

We are also able to prove that a random orthonormal basis of eigenfunctions of the Harmonic oscillator H is Quantum Uniquely Ergodic (QUE, according the terminology used in [START_REF]Random orthonormal bases of spaces of high dimension, Complex patterns in wave functions-drums, graphs and disorder[END_REF] and introduced in [START_REF] Rudnick | The behavior of eigenstates of arithmetic hyperbolic manifolds[END_REF]).

Recall the definitions (4.3) and (4.4) of B and ρ. Denote by B = (ϕ j, ) j∈N, ∈ 1,N h j ∈ B a typical orthonormal basis of L 2 (R d ) so that for all j ∈ N, (ϕ j, ) ∈ 1,N h j ∈ U (N hj ) is an orthonormal basis of E hj . Then In other words, ρ-almost all orthonormal basis of Hermite functions is QUE.

Using estimates proved in [START_REF] Burq | Injections de Sobolev probabilistes et applications[END_REF], an analogous result to Theorem 5.2 can be proved for the Laplace operator on Riemannian compact manifolds with the same method. This holds true in particular for the sphere in any dimension d ≥ 2 and more generally for Zoll manifolds (in this last setting a random orthonormal basis of quasi-modes is obtained).

For Schrödinger operators with super-quadratic potentials a similar result can be obtained (see [START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF]), considering orthonormal basis of quasi-modes (approximated eigenfunctions) satisfying the conclusion of Theorem 5.1.

Application to supercritical nonlinear

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 -→ +∞. It might also be noticed that I(j) corresponds to exactly one eigenvalue λ n of H, with multiplicity #I(j), where λ n = 2j if d is even and λ n = 2j + 1 if d is odd. Let us consider any fixed orthonormal basis (ϕ n ) n≥0 of eigenfunctions for the harmonic oscillator H. Let s ∈ R, then any u ∈ H s (R d ) can be written in a unique fashion (6.1)

u = +∞ j=1 n∈I(j) c n ϕ n ,
and we make the following condition on the coefficients (this is actually condition (3.2)) (6.2)

|c k | 2 ≤ C #I(j) n∈I(j) |c n | 2 , ∀k ∈ I(j), ∀j ≥ 1.
We then define the random variable u ω by

u ω = +∞ j=1 n∈I(j) X n (ω)c n ϕ n ,
where {X n } n≥0 is a sequence of i.i.d random variables, following either the standard complex Gaussian law N C (0, 1) or the Bernoulli law 1 2 δ -1 + 1 2 δ 1 .

Remark 6.1. Here condition (6.2) allows us to use the optimal estimates on the spectral function e x (see Lemma 3.1 and inequality (6.3) below). This will induce a gain of Sobolev regularity for the random series u ω as it is stated in Proposition 6.1, compared to the deterministic Strichartz estimates (6.4).

We then have the following probabilistic improvement of the Strichartz estimates. Proposition 6.1 ([32], Proposition 2.1). Let s ∈ R and let u ∈ H s (R d ) as in (6.1). Assume that (c n ) n∈N satisfies (6.2). Let 1 ≤ q < +∞, 2 ≤ p ≤ +∞, and set α = d(1/2 -1/p) if p < +∞ and α < d/2 if p = +∞. Then there exist c, C > 0 so that for all τ ∈ R P ω :

e -i(t+τ )H u ω L q [0,T ] W s+α,p (R d ) > K ≤ Ce - cK 2 T 2/q u 2 H s (R d ) .
The first key ingredient in the proof is the Khinchin inequality (see e.g. [9, Lemma 4.2] for a proof): There exists C > 0 such that for all real k ≥ 2 and

(a n ) ∈ 2 (N) n≥0 X n (ω) a n L k P ≤ C √ k n≥0 |a n | 2 1 2 .
The second key ingredient in the proof is 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 for d ≥ 2 (6.3) sup

x∈R n∈I(j) |ϕ n x 2 ≤ Cj d/2-1 ,
and which does not depend on the choice of the (ϕ n ) n≥1 .

Let us recall the deterministic Strichartz estimates for the harmonic oscillator, which can be established using the Mehler formula (see [START_REF]Long time dynamics for the one dimensional non linear Schrödinger equation[END_REF]Lemma 5.1] for the argument in 1D which can be extended). We say that a couple (q, p)

∈ [2, +∞] 2 is admissible if 2 q + d p = d 2 and (d, q, p) = (2, 2, +∞).
Then for all T > 0 there exists C T > 0 so that for all u 0 ∈ H s (R d ) we have

(6.4) e -itH f X s T ≤ C T f H s (R d ) , where
X s T := (q,p) admissible L q [-T, T ] ; W s,p (R d ) .
When p is close to +∞, the result of Proposition 6.1 expresses a gain P-a.s. of almost d/2 derivatives in space compared to the bound (6.4).

Before we state our well-posedness results, we need to define the measures on the space of initial conditions. We define the set A s ⊂ H s (R d ) by

A s = u = +∞ j=1 n∈I(j) c n ϕ n ∈ H s (R d ) s.t. condition (6.
2) holds for some C > 0 .

Let γ ∈ A s . We define the probability measure µ γ on H s via the map

Ω -→ H s (R d ) ω -→ γ ω = +∞ j=1 n∈I(j) c n X n (ω)ϕ n ,
in other words, µ γ is defined by: for all measurable F :

H s -→ R H s (R d ) F (v)dµ γ (v) = Ω F (γ ω )dP(ω).
In particular, we can check that µ γ satisfies

• If γ ∈ H s \H s+ε , then µ γ (H s+ε ) = 0.
• Assume that for all j ≥ 1 such that I(j) = ∅ we have c j = 0. Then for all nonempty open subset B ⊂ H s , µ γ (B) > 0. Finally, we denote by M s the set of all such measures

M s = γ∈As {µ γ }.
For more properties of these measures, we refer to the introduction of [START_REF] Poiret | Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator[END_REF].

Each element γ ∈ A s defines a probability measure µ γ . We will see in the next section that for a typical u 0 ∈ H s in the support of µ γ , the nonlinear Schrödinger equation with initial condition u 0 is well-posed.

By the Kakutani theorem (see also Remark 3.1), the space M s contains mutually singular measures, and this extends the set of initial conditions for which we are able to solve NLS. 6.2. Application to NLS. We now consider the Cauchy problem for nonlinear Schrödinger equations, where the initial condition is random. We apply the estimates developed in the previous sections, in particular Proposition 6.1, in order to show local and global well-posedness results, for problems with Sobolev supercritical regularity.

Much work has been done on dispersive PDEs with random initial conditions since the papers of Burq-Tzvetkov [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local existence theory[END_REF][START_REF]Random data Cauchy theory for supercritical wave equations II: A global existence result[END_REF]. In these articles, the authors showed that thanks to a randomisation of the initial condition one can prove well-posedness results even for data with supercritical Sobolev regularity. We also refer to [START_REF]Probabilistic well-posedness for the cubic wave equation[END_REF][START_REF] Thomann | Random data Cauchy for supercritical Schrödinger equations[END_REF][START_REF]Long time dynamics for the one dimensional non linear Schrödinger equation[END_REF][START_REF]Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions[END_REF][START_REF] Poiret | Solutions globales pour des équations de Schrödinger cubique en dimension 3[END_REF][START_REF] De Suzzoni | Large data low regularity scattering result for the wave equation on the euclidian space[END_REF][START_REF] Burq | Global infinite energy solutions for the cubic wave equation[END_REF][START_REF]Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system[END_REF][START_REF] Nahmod | Almost sure well-posedness for the periodic 3d quintic nonlinear Schrödinger equation below the energy space[END_REF] and references therein for further developments.

Let us consider nonlinear Schrödinger equation with harmonic potential (6.5)

   i ∂u ∂t + ∆u -|x| 2 u = ±|u| p-1 u, (t, x) ∈ R × R d . u(0) = u 0 ,
with d ≥ 2, p ≥ 3 an odd integer. This is a model used in the description of the Bose-Einstein condensates.

Before we state our results, let us recall some facts concerning the deterministic study of the nonlinear Schrödinger equation (6.5). We say that (6.5) is locally wellposed in H s (R d ), if for all initial condition u 0 ∈ H s (R d ), there exists a unique local in time solution u ∈ C([-T, T ]; H s (R d )), and if the flow-map is uniformly continuous. We denote by

s c = d 2 - 2 p -1 ,
the critical Sobolev index. Then one can show that NLS is well-posed in H s (R d ) when s > max(s c , 0), and ill-posed when s < s c . We refer to the introduction of [START_REF] Thomann | Random data Cauchy for supercritical Schrödinger equations[END_REF] for more details on this topic.

Local existence results.

We are now able to state our first result on the local well-posedness of (6.5). Theorem 6.1 ([32], Theorem 1.1). Let d ≥ 2, p ≥ 3 an odd integer and fix µ = µ γ ∈ M 0 . Then there exists Σ ⊂ L 2 (R d ) with µ(Σ) = 1 and so that:

(i) For all u 0 ∈ Σ there exist T > 0 and a unique local solution u to (6.5) with initial data u 0 satisfying

(6.6) u(t) -e -itH u 0 ∈ C [-T, T ]; H s (R d ) , for some d 2 -2 p-1 < s < d 2 . (ii) More precisely, for all T > 0, there exists Σ T ⊂ Σ with µ(Σ T ) ≥ 1 -C exp -cT -δ γ -2 L 2 (R 2 )
, C, c, δ > 0, and such that for all u 0 ∈ Σ T the lifespan of u is larger than T .

Denote by γ

= +∞ n=0 c n ϕ n (x), then u ω 0 := +∞ n=0 g n (ω)c n ϕ n (x) is a typical element
in the support of µ γ . Another way to state Theorem 6.1 is : for any T > 0, there exists an event Ω T ⊂ Ω so that

P(Ω T ) ≥ 1 -C exp -cT -δ γ -2 L 2 (R d ) , C, c, δ > 0,
and so that for all ω ∈ Ω T , there exists a unique solution of the form (6.6) to (6.5) with initial data u ω 0 .

The key argument in the proof is the use of Proposition 6.1 which yields a gain of d/2 derivatives compared to the deterministic theory. To prove Theorem 6. The deterministic Cauchy problem for (6.5) was studied by Oh [START_REF] Oh | Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials[END_REF] (see also Cazenave [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]Chapter 9] for more references). In [START_REF] Thomann | Random data Cauchy for supercritical Schrödinger equations[END_REF], Thomann has proven an almost sure local existence result for (6.5) in the supercritical regime (with a gain of 1/4 of derivative), for any d ≥ 1. This local existence result was improved by Burq-Thomann-Tzvetkov [START_REF]Long time dynamics for the one dimensional non linear Schrödinger equation[END_REF] when d = 1 (gain of 1/2 derivatives), by Deng [START_REF] Deng | Two dimensional nls equation with random radial data[END_REF] when d = 2, and by Poiret [START_REF]Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions[END_REF][START_REF] Poiret | Solutions globales pour des équations de Schrödinger cubique en dimension 3[END_REF] in any dimension. 6.2.2. Global existence and scattering results for NLS. As an application of the results of the previous part, we are able to construct global solutions to the non-linear Schrödinger equation without potential, which scatter when t → ±∞. Consider the following equation (6.7)

   i ∂u ∂t + ∆u = ±|u| p-1 u, (t, x) ∈ R × R d . u(0) = u 0 .
The well-posedness indexes for this equation are the same as for equation (6.5). Namely, (6.7) is well-posed in H s (R d ) when s > max(s c , 0), and ill-posed when s < s c .

Then we can prove Theorem 6.2 ([32], Theorem 1.3). Let d ≥ 2, p ≥ 3 an odd integer and fix µ = µ γ ∈ M 0 . Then there exists Σ ⊂ L 2 (R d ) with µ(Σ) > 0 and so that:

(i) For all u 0 ∈ Σ there exists a unique global solution u to (6.7) with initial data

u 0 satisfying u(t) -e it∆ u 0 ∈ C R; H s (R d ) , for some d 2 -2 p-1 < s < d 2 . (ii) For all u 0 ∈ Σ there exist states f + , f -∈ H s (R d ) so that when t -→ ±∞ u(t) -e it∆ (u 0 + f ± ) H s (R d ) -→ 0.
(iii) If we assume that the distribution of ν is symmetric, then

µ u 0 ∈ L 2 (R d ) : the assertion (ii ) holds true u 0 L 2 (R d ) ≤ η -→ 1, when η -→ 0.
We can show [START_REF]Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions[END_REF]Théorème 20], that for all s > 0, if u 0 / ∈ H σ (R d ) then µ(H σ (R d )) = 0. This shows that the randomisation does not yield a gain of derivative in the Sobolev scale; thus Theorem 6.2 gives results for initial conditions which are not covered by the deterministic theory.

There is a large literature for the deterministic local and global theory with scattering for (6.7). We refer to [START_REF] Banica | Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space[END_REF][START_REF] Carles | Rotating points for the conformal nls scattering operator[END_REF] for such results and more references.

One of the key points in the proof is to use the lens transform L defined as

u(t, x) -→ L u(t, x) = 1 √ 1 + 4t 2 d/2 u arctan(2t) 2 , x √ 1 + 4t 2 e ix 2 t 1+4t 2 ,
which permits link equation (6.7) to equation (6.5). In particular, local in time results for (6.5) on the time interval [-π/4, π/4] imply global in time results for (6.7).

In Theorem 6.2 we assumed that d ≥ 2 and p ≥ 3 was an odd integer, so we had p ≥ 1 + 4/d, or in other words we were in a L 2 -supercritical setting. Our approach also allows to get global in time results in an L 2 -subcritical context, i.e. when 1 + 2/d < p < 1 + 4/d. Theorem 6.3 ([32], Theorem 1.4). Let d = 2 and 2 < p < 3 and fix µ = µ γ ∈ M 0 . Then there exists Σ ⊂ L 2 (R 2 ) with µ(Σ) > 0 and so that for all 0 < ε < 1 (i) For all u 0 ∈ Σ there exists a unique global solution u to (6.7) with initial data u 0 satisfying u(t) -e it∆ u 0 ∈ C R; H 1-ε (R 2 ) .

(ii) For all u 0 ∈ Σ there exist states f + , f -∈ H 1-ε (R 2 ) so that when t -→ ±∞

u(t) -e it∆ (u 0 + f ± ) H 1-ε (R 2 ) -→ 0.
(iii) If we assume that the distribution of ν is symmetric, then

µ u 0 ∈ L 2 (R 2 ) : the assertion (ii ) holds true u 0 L 2 (R 2 ) ≤ η -→ 1,
when η -→ 0.

In the case p ≤ 1 + 2/d, Barab [START_REF] Barab | Nonexistence of asymptotically free solutions for nonlinear Schrödinger equation[END_REF] showed that a non trivial solution to (6.7) never scatters, therefore even with a stochastic approach one can not have scattering in this case. When d = 2, the condition p > 2 in Theorem 6.3 is therefore optimal. Usually, deterministic scattering results in L 2 -subcritical contexts are obtained in the space H 1 ∩ F(H 1 ). Here we assume u 0 ∈ L 2 , and thus we relax both the regularity and the decay assumptions (this latter point is the most striking in this context). Again we refer to [START_REF] Banica | Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space[END_REF][START_REF] Antonelli | Scattering for nonlinear Schrödinger equation under partial harmonic confinement[END_REF] for an overview of scattering theory for NLS.

   i ∂u ∂t -Hu = |u| 2 u, (t, x) ∈ R × R d , u(0) = u 0 ,
and denote by E the energy of (6.8), namely

E(u) = u 2 H 1 (R d ) + 1 2 u 4 L 4 (R d ) .
Deterministic global existence for (6.8) has been studied by Zhang [START_REF] Zhang | Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential[END_REF] and by Carles [START_REF]Nonlinear Schrödinger equation with time dependent potential[END_REF] in the case of time-dependent potentials.

When d = 3, our global existence result for (6.8) is the following Theorem 6.4 ([32], Theorem 1.5). Let d = 3, 1/6 < s < 1 and fix µ = µ γ ∈ M s . Then there exists a set Σ ⊂ H s (R 3 ) so that µ(Σ) = 1 and so that the following holds true (i) For all u 0 ∈ Σ, there exists a unique global solution to (6.8) which reads

u(t) = e -itH u 0 + w(t), w ∈ C R, H 1 (R 3 ) .
(ii) The previous line defines a global flow Φ, which leaves the set Σ invariant Φ(t)(Σ) = Σ, for all t ∈ R.

(iii) There exist C, c s > 0 so that for all t ∈ R,

E w(t) ≤ C(M + |t|) cs+ ,
where M is a positive random variable so that

µ(u 0 ∈ H s (R 3 ) : M > K) ≤ Ce - cK δ γ 2 H s (R 3 ) .
Here the critical Sobolev space is H 1/2 (R 3 ), thus the local deterministic theory combined with the conservation of the energy, immediately gives global wellposedness in H 1 (R 3 ). To prove Theorem 6.4, we use the high/low frequency decomposition method of Bourgain [4, page 84]), which relies on the almost well-posedness result of Theorem 6.1 and the global well-posedness in H 1 (R 3 ). As a result, we get a global well-posedness result in a supercritical context. This strategy has been successful in different contexts, and has been first used together with probabilistic arguments by Colliander-Oh [START_REF] Colliander | Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2 (T)[END_REF] for the cubic Schrödinger below L 2 (S 1 ) and later on by Burq-Tzvetkov [START_REF]Probabilistic well-posedness for the cubic wave equation[END_REF] for the wave equation.

With a similar approach, in dimension d = 2, we can prove the following result Theorem 6.5 ([32], Theorem 1.6). Let d = 2, 0 < s < 1 and fix µ = µ γ ∈ M s . Then there exists a set Σ ⊂ H s (R 2 ) so that µ(Σ) = 1 and so that for all u 0 ∈ Σ, there exists a unique global solution to (6.8) which reads

u(t) = e -itH u 0 + w(t), w ∈ C R, H 1 (R 2 ) .
In addition, statements (ii) and (iii) of Theorem 6.4 are also satisfied with c s = 1-s s . Here the critical Sobolev space is L 2 (R 2 ), thus Theorem 6.5 shows global wellposedness for any subcritical cubic non linear Schrödinger equations in dimension two. Proof. In the sequel, write p = p h . As in the proof of [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Theorem 4.7], we denote by A p = E h ( u L p,d(p/2-1) ) 1/p . Then from [33, estimate (4.14)] we have |A p -M p | ≤ CN -1/p √ p ≤ e -c| ln h| η √ p, hence it is enough to prove estimate (A.1) for A p . The proof then consists in tracking all the constants in [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF] and to show that they are optimal. We do not write all the details, but we give the main steps. Let us recall [33, estimate Then for ε > 0

• With an inspection of the proof of [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Theorem 2.6,(2.11)] and in [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Lemma 2.11] we can show that we can take in the previous line 1/2 -ε/8 < c 2 < c 1 < 1/2 + ε/8.

• We can construct the parametrix in [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Lemma 4.9] Assume here that δ = 1 (the general case 0 ≤ δ ≤ 1 can be treated in the same way). To begin with, we state an estimate of u L r . Theorem B.1. Let 2d/(d -1) < r ≤ ∞ and denote by M r a median of u L r . Then there exist 0 < C 0 < C 1 , K > 0, c 1 > 0 and h 0 > 0 such that for all r ∈ [2, K| log h|] and h ∈]0, h 0 ] such that This is a result similar to [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Theorem 4.7] in which we made the restriction δ < 2/3. However, the price to pay, is that the estimate (B.1) is no more optimal. The reason is that, in this case we prove the following estimate on the spectral function when p > d/(d -1) (here r = 2p)

P γ,h u ∈ S h : u L r -M r > Λ ≤ 2 exp -c 2 N 2/r h h -d 4 (1+ 2 r ) Λ 2 .
(B.2) C 0 h 1-d 2 (1-1 p ) ≤ R d e p x dx 1/p ≤ C 1 h 1-d 2 (1+ 1 p ) .
Let us prove (B.2). The upper bound is the same as in Lemma 3.1 and is proved using (2.5) (see also [START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF]Appendix A.5] for more general results). To get the lower bound, we oberve that among the family (ϕ j ) j∈Λ h there exists a radial function ϕ rad,h and which satisfies Hϕ rad,h ∼ 1 h ϕ rad,h . Therefore e x := The proof of Theorem 3.5 is analogous to [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF]Corollary 4.8]. Roughly speaking, the L ∞ norm is reached in the regime r = r h = c| ln h| and h 1. In this regime, the estimate (B.1) becomes optimal, since h c1/r h ∼ c 2 .

3. 2 . 1 p -1 2 h

 212 L p bounds, 2 ≤ p < +∞. The following result shows that for θ = d the Sobolev estimate (2.7) is improved by the factor N for u in a set of probability close to 1. Theorem 3.1 ([33], Theorem 4.7). Let p ≥ 2. Denote by M p a median of u L p,d(p/2-1) . Assume condition (3.2) and let δ ∈ [0, 1]. Then there exist 0

3. 3 . 1 2h

 31 L ∞ bounds. The next result shows that for θ = d the Sobolev estimate (2.6) is improved by the factor N -| log h| for u in a set of probability close to 1. We suppose that (3.2) and (2.3) with 0 ≤ δ ≤ 1 are satisfied.

Theorem 4 . 1 (

 41 [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF], Theorem 1.3). Let d ≥ 2. Then there exists an orthonormal basis of L 2 (R d ) of eigenfunctions of the harmonic oscillator H denoted by {ϕ n } n≥1 such that ϕ n L 2 (R d ) = 1 and so that for some M > 0 and all n ≥ 1, (4.1)

Theorem 4 . 2 (

 42 [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF], Theorem 5.1). Let d ≥ 2.

Proposition 4 . 1 (

 41 [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF], Proposition 5.2). Let d ≥ 2.

Corollary 4 . 1 (

 41 [START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF], Corolllary 5.3). For ρ-almost all orthonormal basis

Theorem 5 . 2 (

 52 [START_REF] Robert | Random weighted Sobolev inequalities and application to quantum ergodicity[END_REF], Theorem 1.6). For B ∈ B and A ∈ S(1, 1) let us denote byD j (B) = max 1≤ ≤N h j ϕ j, , Âϕ j, -L η (A) .Then we have lim j→+∞ D j (B) = 0, ρ -a.s on B.

Schrödinger equations 6 . 1 .

 61 Construction of measures on L 2 (R d ) and Strichartz inequalities. For j ≥ 1 denote by

  1 we only have to gain s c = d/2 -2/(p -1) derivatives. The solution is constructed by a fixed point argument in a Strichartz space X s T ⊂ C [-T, T ]; H s (R d ) with continuous embedding, and uniqueness holds in the class X s T .

6. 2 . 3 .

 23 Global existence results for NLS with quadratic potential. We also get global existence results for defocusing Schrödinger equation with harmonic potential. For d = 2 or d = 3, consider the equation(6.8)

2 Lemma A. 1 .

 21 Appendix A. Proof of Theorem 3.Let 0 < η < 1 and set p h = | ln h| 1-η . Then there exists a constantC = C (d) such that (A.1) (C -ε/2) √ p h ≤ M p h ≤ (C + ε/2) √ p h ,for h > 0 small enough.

2 R d x d( p 2 - 1 ) e p/2 x dx c0N 0 t p/ 2 - 1 e

 221021 (4.12)] (A.2) C 1 p(c 1 N ) -p/-t dt ≤ A p p ≤ ≤ C 2 p(c 2 N ) -p/2 R d x d( p 2 -1) e p/2x dx Γ(p/2).

2 r 1 √ rh d 4 ( 1 - 2 r

 21412 ) ≤ M r ≤ C ) , ∀r ∈ [2, K| log h| ].

  j∈Λ h |ϕ j | 2 ≥ |ϕ rad,h | 2 .Now we invoke the sharp L p bounds of radial Hermite functions, proved in[START_REF] Imekraz | On random Hermite series[END_REF] Proposition 2.4] which imply the result. Finally, to prove Theorem B.1, we proceed as in the proof of[START_REF]Random weighted Sobolev inequalities on R d and applications to Hermite functions[END_REF] Theorem 4.7], using the estimate (B.2).

  where C is a constant which only depends on the dimension.• With the Laplace method, we can show that for p ≥ 2 large enough Putting the previous estimates together with (A.2), we get(C -ε/2) √ p ≤ A p ≤ (C + ε/2) √ p,with C = (C /(2e)) 1/2 , which was the claim.We are now able to complete the proof of Theorem 3.2. By Lemma A.1, for p ≥ 2 large enoughu L p,d(p/2-1) -C √ p > ε √ p ⊂ u L p,d(p/2-1) -M p > ε 2 √ p ,and an application of Theorem 3.1 gives the result.

												in such a way that
	(	√	C -ε/8) √	N ≤	R d	x d( p 2 -1) e p/2 x dx	1/p	√ ≤ (	C + ε/8)	√	N ,
	(1 -ε/8)	p 2e	≤	0	c0N	t p/2-1 e -t dt	1/p	≤ Γ(p/2)	1/p ≤ (1 + ε/8)	p 2e	.

Appendix B. Proof of Theorem 3.5

Acknowledgements. We are grateful to the anonymous referee for very useful comments on the first version of this text.

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.

c 0000 (