Tomasz Komorowski

Stefano Olla
email: olla@ceremade.dauphine.fr

Lenya Ryzhik
email: ryzhik@math.stanford.edu

Asymptotics of the solutions of the stochastic lattice wave equation

come

1 Introduction {intro} Energy transport and dispersion in dynamics of oscillators in a lattice have been investigated in many situations in order to understand macroscopic thermal conductivity properties. Typical example is the Fermi-Pasta-Ulam chain under the Hamiltonian evolution corresponding to a quartic interaction potential

H = N y=-N p 2 y 2m + 1 2 ω 2 0 q 2 y + N y=-N +1
1 2 (q yq y-1 ) 2 + γ(q yq y-1 ) 4 (1.1)

{eq:fpu}

Here q y is the displacement of the y-th particle from its equilibrium position, p y is its momentum and m is the mass. When ω 0 = 0, the particle is confined, this breaks translation invariance, and correspondingly the conservation of the total momentum, and we say that the chain is pinned. When γ = 0 the Hamiltonian dynamics is given by the discrete in space linear wave equation, and the energy evolution is purely ballistic and dispersive. If γ > 0 and ω 0 = 0, we expect that wave scattering due to the presence of the non-linearity gives a finite thermal conductivity and consequently a diffusive macroscopic evolution of the energy. If the chain is 1 unpinned, ω 0 = 0, and γ > 0, long waves scatter rarely, giving rise to a superdiffusive behavior of the energy [START_REF] Lepri | Thermal conduction in classical low-dimensional lattices[END_REF].

The mathematical analysis of the macroscopic behavior of the energy is difficult in the case of deterministic nonlinear dynamics, and recently various models considering stochastic perturbations of the dynamics have been proposed. They simulate, at least qualitatively, the effect of the scattering by the non-linearity. Such noisy perturbations should conserve energy and be local in space [START_REF] Bernardin | Fourier's law for a microscopic model of heat conduction[END_REF]. In the unpinned case it is also important that they conserve the momentum [START_REF] Basile | A momentum conserving model with anomalous thermal conductivity in low dimension[END_REF][START_REF] Basile | Thermal conductivity for a momentum conserving model[END_REF]. The perturbations considered in these papers are given by a random exchange of momentum so that the total kinetic energy is constant (consequently, the total energy is preserved as well, since the position components are untouched by the noise) and the total momentum is also conserved. This is achieved by adding, to each triple of adjacent particles, a diffusion on the corresponding circle of constant energy and momentum. Another example of a noisy perturbation having similar properties appears in a discontinuous in time model in which momenta of pairs of adjacent particles are exchanged at independent random times that are exponentially distributed.

When the interaction is linear, the thermal diffusivity of the energy in these models can be explicitly computed -it is finite for the pinned model but diverges with the size of the system in the unpinned case (corresponding to superdiffusive energy transport for the unpinned model).

The limit dynamics for the spectral measure of the energy in these stochastic models is investigated in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF], where the noise is also rescaled in such a way that there are only finitely many wave collisions in the unit macroscopic time. In a sense, this weak noise limit is similar to the regime where phonon-Boltzmann equation is valid in weakly nonlinear models (cf. [START_REF] Spohn | The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics[END_REF]). The dynamics is defined in the following way. Consider the infinite lattice Z with the Hamiltonian associated to the linear evolution (1.1) (γ = 0), with N = ∞, perturbed by a conservative noise. Formally, it is given by the solution of the stochastic differential equations: qy (t) = p y (t) dp y (t) = ∆q yω 2 0 q y dt + dη y (ǫt), (1.2) {eq:sde1}

where ∆q y = q y+1 + q y-1 -2q y is the lattice Laplacian. The noise dη y (ǫt) will be chosen so as to model random exchange of momenta between the adjacent sites so that the total kinetic energy and momentum of the system are conserved (see (2.1) for the precise form of the noise).

The small parameter ǫ > 0 slow down its effect. The total Hamiltonian can be formally written as

H(q, p) = y∈Z p 2 y 2 +
x,y∈Z α x-y q x q y , (

{eq:2}

with α 0 = 1 2 ω 2 0 + 2, α -1 = α +1 = -1, and α y = 0 otherwise. The dispersion relation ω(k) for this system is .4) {eq:dr0}

ω(k) = α(k) = ω 2 0 2 + 4 sin 2 (πk), k ∈ T. ( 1
The Fourier transform α(k) is defined as in (2.3) below. Let us introduce the complex wave function ψ y (t) := (ω * q) y (t) + ip y (t), (1.5) {eq:wf0}

where ωy is the inverse Fourier transform of ω(k). Its Fourier transform is given by ψ(t, k) := ω(k)q(k, t) + ip(t, k) (1.6) {eq:fwf0}

and satisfies the equation

d ψ(t, k) := -iω(k) ψ(t, k)dt + idη(ǫt, k), (1.7)
{eq:fwf01}

where dη(t, k) is the Fourier transform of the noise. Due to the conservation properties of the dynamics, if the initial configuration has finite total energy H(q(0), p(0)) < +∞, then all the functions introduced in (1.3) and (1.5)- (1.6) exists in the sense of distributions, and is the solution of the linear kinetic equation

∂ t Ē (t, k) = T R(k, k ′ ) Ē t, k ′ -Ē (t, k) dk ′ (1.9)
eq:phbolhom}

with the initial condition Ē (0, k) = | ψ(0, k)| 2 . The scattering kernel R(k, k ′ ) is given by (3.2) below.
The goal of the present article is to obtain a direct information on the wave function ψ(t/ǫ, k), as was done in [START_REF] Bal | Asymptotics of the solutions of the random Schrdinger equation[END_REF] for the Schrödinger equation, and not only for the average energy.

It follows from (1.7) that the unperturbed (by noise) evolution of this function is governed by the highly oscillating factor e -iω(k)t/ε (after we rescale the time). It is therefore reasonable to consider, in case of the perturbed system, the compensated wave function of the form ψ(ǫ) (t, k) := e iω(k)t/ε ψ(t/ε, k).

We show that once we compensate for fast oscillations, the wave function converges in law to the solution a Langevin equation driven by (1.9). More precisely, we prove in Theorem 3.1 below, existence of the limit (in law and pointwise in k):

lim ε→0 ψ(ǫ) (t, k) = ψ(t, k).
(1.10)

{eq:comp0}
The limit ψ(t, k) is a complex valued stochastic process satisfying the linear (time inhomogeneous) Ornstein-Uhlenbeck equation

d ψ(t, k) = - β(k) 4 ψ(t, k)dt + R(t, k)dw k (t), (1.11)
{eq:lan0}

with the initial condition ψ(0, k) = ψ(0, k). Here β(k) is given by (2.5) below,

R(t, k) = T Ē(t, k ′ )R(k, k ′ )dk ′ , (1.12)
{eq:beta0}

and {w k (t)} is a family of pairwise independent standard complex valued Brownian motions parametrized by k ∈ T. That is, they are complex valued, jointly Gaussian, centered processes satisfying .13) {eq:3}

E[w k (t)w k ′ (s)] = 0 and E[w * k ′ (t)w k (s)] = δ k,k ′ t ∧ s for all t, s ≥ 0 and k, k ′ ∈ T. Here δ k,k ′ = 0 for k = k ′ and δ k,k = 1. Equation (1.11) has the explicit solution ψ(t, k) = e -1 4 β(k)t ψ(0, k) + t 0 e -1 4 β(k)(t-s) R(s, k)dw k (s). ( 1
In particular, we have

E| ψ(t, k)| 2 = e -1 2 β(k)t | ψ(0, k)| 2 + t 0 e -1 2 β(k)(t-s) R(s, k)ds which is equivalent to (1.9), since Ē(t, k) = E| ψ(t, k)| 2 and β(k) = 2 T R(k, k ′ )dk ′ , (1.14)

{R-B}

as can be seen by a direct calculation from (2.5) and (3.2). Initial conditions such that T | ψ(0, k)| 2 dk < ∞ correspond to a local perturbation of the zero temperature equilibrium. We are also interested in the macroscopic evolution of the equilibrium states at a positive temperature T > 0, starting with a random distribution given by the Gibbs measure at temperature T . In the mode space this is a centered, complex valued, Gaussian random field with distribution valued ψ(k). Its covariance is given by .15) eq:gibbscov}

E[ ψ * (k) ψ(k ′ )] = T δ(k -k ′ ), E[ ψ(k) ψ(k ′ )] = 0. ( 1
Here δ(kk ′ ) is Dirac's delta function. The distributions are invariant under the dynamics, due to the conservation of energy. Actually, in Section 3.2 we consider more general class of space homogeneous Gaussian, random initial conditions that are not necessarily stationary in law in time. More precisely, we show (see Theorem 3.3) that if the law of the initial condition is a homogeneous, centered Gaussian field with the covariance given by

E ψ(k) * ψ(k ′ ) = E 0 (k)δ(k -k ′ ), E ψ(k) ψ(k ′ ) = 0,
then the compensated wave function converges in law, as a continuous in time process taking values in an appropriate distribution space, to the solution of the time inhomogeneous stochastic equation: .16) {eq:spdet1}

d ψ(t, k) = - β(k) 4 ψ(t, k)dt + R(t, k)dW (t, k). ( 1
Here, R(t, k) is given by (1.12) and Ē(t, k) is the solution of the deterministic equation (1.9) with the initial condition Ē(0, k) = E 0 (k), while dW (t, k) is a white noise on R × T, a complex valued Gaussian process with the covariance

E[dW (t, k)dW * (s, k ′ )] = δ(k -k ′ ) ⊗ δ(t -s)dtds
and R(t, k) is given by (1.12). The solution of (1.16) is also explicit: ψ(t) is the distribution

ψ(t) = e -βt/4 ψ + t 0 e -β(t-s)/4 R 1/2 (s)dW (s).
In particular, in the case of the initial condition distributed according to a Gibbs measure, the solution ψ(t, k) of (1.7) has the same law for all times, therefore Ē(t, k) = T for all t ≥ 0. In this case, (1.14) shows that R(t, k) = β(k)T /2. Therefore, as a consequence of (1.16), the limit of the compensated wave function is the solution of the linear infinite dimensional stochastic differential equation:

d ψ(t, k) = - β(k) 4 ψ(t, k)dt + T β(k) 2 dW (t, k).
(1.17)

{eq:spde0}

In the general case, when E 0 (k) is not constant, we have

lim t→∞ Ē(t, k) = T E 0 (k ′ )dk ′ = T,
hence, equation (1.17) describes the asymptotic stationary regime of (1.16) where the temperature is given by the average of the initial energy over all the modes k. Recall that the microscopic noise conserves the total energy and that the resulting temperature T depends only on the law of the initial condition.

Let us also comment on the difference between the square integrable and distributionvalued initial data. While the Ornstein-Uhlenbeck equations (1.11) and (1.16) are essentially identical, the limits hold in a different sense. For the square integrable data, the limit equation holds point-wise in k. If you consider the limit in the sense of distributions (that is, integrated against a test function) for such initial data, it is described simply by attenuation of the initial condition by an exponential factor e -β(k)t/4 (see part (ii) of Theorem 3.1) -that is, by (1.11) with no stochastic forcing. This result stands in sharp contrast with the case of spatially homogeneous initial data (note that then the energy has to be infinite) when the respective limit in the sense of distributions is stochastic, see (1.16), and fluctuations can not be averaged out by integration.

The results of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] also concern the Wigner transform, which is the spatially localized energy spectrum. The spatially inhomogeneous version of (1.9) is (after a simultaneous rescaling of space-time by (ǫ -1 x, ǫ -1 t))

∂ t Ē (t, x, k) + ω ′ (k)∂ x Ē (t, x, k) = T R(k, k ′ ) Ē t, x, k ′ -Ē (t, x, k) dk ′ .
(1.18)

eq:inophbol}

This gives the probability distribution at time t of the phonons in the (x, k) space. The behavior at small k of the velocity ω ′ (k) is responsible for the superdiffusive behavior of the energy for the unpinned chain (cf. [START_REF] Basile | A momentum conserving model with anomalous thermal conductivity in low dimension[END_REF][START_REF] Basile | Thermal conductivity for a momentum conserving model[END_REF][START_REF] Komorowski | Limit theorems for a additive functionals of a Markov chain[END_REF][START_REF] Komorowski | Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension[END_REF]). It would be interesting to understand the relation of an inhomogeneous version of the result of the present paper to this superdiffusive phenomenon.

The organization of the paper is as follows. In Section 2 we introduce the basic notions that shall be used throughout the article. We formulate rigorously the stochastic differential equation for the Fourier transform of the wave function, see (2.8). The equation for the compensated wave function is formulated in (2.13).

The paper is organized as follows. Section 2 contains the precise mathematical formulation of the problem and necessary definitions. We formulate the results for the convergence of compensated wave function in in Section 3, see Theorem 3.1 for square integrable initial data, and Theorem 3.3 for spatially homogeneous, Gaussian initial distributions. The proofs of these results are presented in Sections 4 and 5, respectively.

Acknowledgement. T.K. acknowledges the support of Polish Ministry of Higher Education grant NN201419139, S.O. acknowledges the support by the ERC AdG 246953 (MALADY) and by ANR-10-BLAN 0108 (SHEPI), L.R. acknowledges the support by NSF grant DMS-0908507. This work was also supported by NSSEFF fellowship by AFOSR.

2 Preliminaries {prelim}

Infinite system of interacting harmonic oscillators

The dynamics of the system of oscillators can be written formally as a system of Itô stochastic differential equations indexed by y ∈ Z dq y (t) = p y (t)dt (2.1)

{eq:bas} dp y (t) = -(α * q(t)) y dt - ǫ 2 (β * p(t)) y dt + √ ǫ k=-1,0,1 (Y y+k p y (t))dw y+k (t).
Here and {w y (t), t ≥ 0}, y ∈ Z is a family of i.i.d. one dimensional, real valued, standard Brownian motions, that are non-anticipative over the filtered probability space (Ω, F, {F t , t ≥ 0}, P). In addition,

Y x := (p x -p x+1 )∂ p x-1 + (p x+1 -p x-1 )∂ px + (p x-1 -p x )∂ p x+1 (2.2)
β y = ∆β (0) y := β (0) y+1 + β (0) y-1 -2β (0) y with β (0) y =    -4, y = 0 -1, y = ±1 0, if otherwise.
Recall that the lattice Laplacian of g : Z → C is given by ∆g y := g y+1 + g y-1 -2g y .

The Fourier transform of a square integrable sequence of complex numbers {γ y , y ∈ Z} is defined as γ We assume also (cf [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]) that a1) {α y , y ∈ Z} is real valued and there exists C > 0 such that |α y | ≤ Ce -|y|/C for all y ∈ Z, a2) α(k) is also real valued and α(k) > 0 for k = 0 and in case α(0) = 0 we have α′′ (0) > 0.

(k) = y∈Z γ y e y (k), k ∈ T. ( 2
The above conditions imply that both functions y → α y and k → α(k) are even. In addition, α ∈ C ∞ (T) and in case α(0) = 0 we have α(k) = k 2 φ(k 2 ) for some strictly positive φ ∈ C ∞ (T).

Recall that the function ω(k) := α(k) is the dispersion relation.

Evolution of the wave function

For a given m ∈ R we define the space H m (T) as the completion of C ∞ (T) under the norm

f 2 m := y∈Z (1 + y 2 ) m | fy | 2 .
We shall denote by •, • the scalar product on L 2 (T). By continuity it extends in an obvious way to H m (T) × H -m (T) for an arbitrary m ∈ R.

It is convenient to introduce the wave function that, adjusted to the macroscopic time, is given by

ψ (ǫ) (t) := ω * q t ǫ + ip t ǫ . (2.6)

{011307}

Here {ω y , y ∈ Z} is the inverse Fourier transform of ω(k) := α(k). We shall consider the Fourier transform of the wave function

ψ(ǫ) (t, k) := ω(k)q t ǫ , k + ip t ǫ , k . (2.7) {011307a}
Using (2.1) as a motivation, we obtain formally, by considering the Fourier transform of (2.1), that

d ψ(ǫ) (t) = A[ ψ(ǫ) (t)]dt + Q[ ψ(ǫ) (t)]dW (t), (2.8)
basic:sde:2}

ψ(ǫ) (0) = ψ,
where ψ ∈ L 2 (T), and mapping A :

L 2 (T) → L 2 (T) is defined by A[f ](k) := - i ǫ ω(k)f (k) - β(k) 4 [f 1 (k) -f -1 (k)], ∀ f ∈ L 2 (T). (2.9) {040607} Here f 1 (k) := f (k) and f -1 (k) := f * (-k). (2.10) {020906} In addition, Q[g] : L 2 (T) → L 2 (T) is a linear mapping that for any g ∈ L 2 (T) is given by Q[g](f )(k) := i T r(k, k ′ )[g 1 (k -k ′ ) -g -1 (k -k ′ )]f (k ′ )dk ′ , ∀ f ∈ L 2 (T), (2.11) {053009} where r(k, k ′ ) := sin(2πk) + sin[2π(k -k ′ )] + sin[2π(k ′ -2k)] = 4 sin(πk) sin[π(k -k ′ )] sin (2k -k ′ )π , k, k ′ ∈ T.
The cylindrical Wiener process on L 2 (T) appearing in (2.8) is dW (t) := y∈Z e y dw y (t).

It can be easily checked that y∈Z Q

[g](e y ) 2 L 2 ≤ C g 2 L 2 for some C > 0 and all g ∈ L 2 (T) so Q[g] is Hilbert-Schmidt, which ensures that Q[ ψ(ǫ) (t)]dW (t) := y∈Z Q[ ψ(ǫ) (t)](e y )dw y (t)
is summable in L 2 (T), both in the L 2 and a.s. sense. It is also obvious that the mapping A is Lipschitz. Using Theorem 7.4, p. 186, of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF] one concludes therefore that there exists an L 2 (T)-valued, adapted process { ψ(ǫ) (t), t ≥ 0} that is a unique solution to (2.8), understood in the mild sense. In addition, see Section 2 of [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF], the total energy is conserved:

ψ(ǫ) (t) L 2 = const, ∀ t ≥ 0 (2.12)
onservation} for a.s. realization of Brownian motions and an initial condition from L 2 (T).

Compensated wave function eudo-wigner}

Let us define the compensated wave function

ψ(ǫ) (t, k) := ψ(ǫ) (t, k) exp it ω(k) ǫ .
From (2.8) we obtain the following equation

d ψ(ǫ) (t, k) = A t ǫ , ψ(ǫ) (t) (k)dt + d M(ǫ) t (k), ψ(ǫ) (0) = ψ, (2.13) llified-eqt} where ψ ∈ L 2 (T), A[t, •] : L 2 (T) → L 2 (T) is Lipschitz A[t, f ](k) := - β(k) 4 [f (k) -exp {2iω(k)t} f * (-k)] . (2.14)

{012808}

The martingale term is

d M(ǫ) t := Q t ǫ , ψ(ǫ) (t) dW (t), (2.15)

{060410}

where for any g ∈ L 2 (T) and t ≥ 0, the operator Q

[t, g] : L 2 (T) → L 2 (T), given by Q[t, g](f )(k) := i σ=±1 σ T r(k, k ′ )g σ (k -k ′ )f (k ′ ) exp i[ω(k) -σω(k -k ′ )]t dk ′ , (2.16) {022808} is Hilbert-Schmidt. Its Hilbert-Schmidt norm satisfies Q[t, g 1 ] -Q[t, g 2 ] (HS) ≤ C g 1 -g 2 L 2 (T) , ∀ t ≥ 0, g 1 , g 2 ∈ L 2 (T),
for some constant C > 0. In addition, it can be verified by a direct computation that t → Ã[t, •] is Lipschitz, uniformly in t. The solution of (2.13) exists in the mild sense, see [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF], Theorem 7.4, p. 186, and is unique. Moreover, if the initial condition belongs to C(T) so does ψ(ǫ) (t) for any t ≥ 0 and a.s. realization of the Wiener process.

3 Convergence of the compensated process Before formulating the result we introduce some auxiliaries. First, for any

k 0 1 , k 0 2 ∈ T, σ 1 , σ 2 ∈ {-1, 1} let us denote K σ 1 ,σ 2 (k 0 1 , k 0 2 ) := [k : ω(k 0 1 ) -ω(k 0 1 -k) = σ 1 [ω(k 0 2 ) -σ 2 ω(k 0 2 -k)]] and K 0 (k 0 1 , k 0 2 ) := [k : ω(k 0 1 ) + ω(k 0 1 -k) = ω(k 0 2 ) + ω(k 0 2 -k)]. Let also K(k 0 1 , k 0 2 ) = σ 1 ,σ 2 =±1 K σ 1 ,σ 2 (k 0 1 , k 0 2 ) ∪ K 0 (k 0 1 , k 0 2 ).
We shall require that:

Condition ω) for any k 0 1 = k 0 2 the one dimensional Lebesgue measure m 1 (K(k 0 1 , k 0 2 )) = 0.
Define the scattering operator L :

L 1 (T) → L 1 (T) by Lf (k) := T R(k, k ′ )[f (k ′ ) -f (k)]dk ′ , f ∈ L 1 (T), (3.1)

{L}

where the scattering kernel is given by

R(k, k ′ ) := r 2 (k, k -k ′ ) + r 2 (k, k + k ′ ) (3.2) {kernel} = 16 sin 2 (πk) sin 2 (πk ′ ) sin 2 π(k + k ′ ) + sin 2 π(k -k ′ ) . Suppose that ψ ∈ L 2 (T). Let R(t, k) := T R(k, k ′ ) Ē(t, k ′ )dk ′ , (3.3)

{010810}

where Ē(t, k) is the solution of an equation

Ē(t, k) = | ψ(k)| 2 + t 0 L Ē(s, k)ds. (3.4)

{100510}

Assume also that {w k (t), t ≥ 0} is a family of pairwise independent standard, one dimensional, complex valued Brownian motions indexed by k ∈ T. Our first principal result can be stated as follows.

{main-thm1}

Theorem 3.1 (i) Suppose that ψ ∈ C(T) and k 1 , . . . , k n ∈ T for a given integer n ≥ 1. Then {( ψ(ǫ) (t, k 1 ), . . . , ψ(ǫ) (t, k n )), t ≥ 0} converge in law over C([0, +∞); C n ), as ǫ → 0+, to {( ψ(t, k 1 ), . . . , ψ(t, k n )), t ≥ 0}
, where { ψ(t, k), t ≥ 0} is a complex valued, non-homogeneous in time Ornstein-Uhlenbeck process that is the solution of the equation

d ψ(t, k) = - β(k) 4 ψ(t, k)dt + R 1/2 (t, k)dw k (t), ψ(0, k) = ψ(k). (3.5) {limit-eqt1} (ii) If ψ ∈ L 2 (T) then for any f ∈ L 2 (T) and t * > 0 we have lim ǫ→0+ sup t∈[0,t * ] ψ(ǫ) (t) -ψ(t), f = 0 (3.6)

{040510}

in probability. Here ψ(t) is given by

ψ(t, k) := ψ0 (k) exp - t β(k) 4 . (3.7)

{012909}

Remark. Note that L1 = 0 and

Lf, f = - 1 2 T 2 R(k, k ′ )[f (k ′ ) -f (k)] 2 dkdk ′ ≤ 0, ∀ f ∈ L 2 (T).
Since the operator L is symmetric and compact on L 2 (T), and 0 is its simple eigenvalue we conclude that R(t, k) → ( β(k)/2)T , where T = ψ0 L 2 (T) , as t → +∞, uniformly on T. As a result we obtain that lim

t→+∞ E ψ(t, k) -ψs (t, k) 2 = 0, (3.8)
{040510a}

where ψs (t, k) is a time homogeneous Ornstein-Uhlenbeck process given by

d ψs (t, k) = - β(k) 4 ψs (t, k)dt + β(k)T 2 dw k (t), ψs (0, k) = ψ(k). (3.9) limit-eqt1a}
Let us also comment briefly on condition ω). An important fact used in the proof of part (i) of Theorem 3.1, is that the energy | ψ(ǫ) (t)| 2 converges in probability, weakly to Ē(t), as ǫ → 0+. The fact that the mean E | ψ(ǫ) (t)| 2 , f converges to Ē(t), f has already been shown in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]. It is clear from (2.13) that | ψ(ǫ) (t)| 2 , f is a semimartingale, for any test function f . Condition ω) is used to prove that the martingale part of the semimartingale vanishes, see Lemma 4.6 below. This, in turn implies convergence in probability. The following simple criterion is useful for verification of condition ω), e.g. for dispersion relation ω(k) of the form (1.4). {lmK} Lemma 3.2 Suppose that the dispersion relation satisfies the following condition: the equation

ω ′′ (k) = 0 (3.10)
{sec-der} has no solution in T\{0}. Then, for any

(k 0 1 , k 0 2 ) such that k 0 1 = k 0 2 we have m 1 (K(k 0 1 , k 0 2 )) = 0. In consequence the hypothesis ω) holds. Proof. From the assumptions made we know that ω ∈ C ∞ (T \ {0}). Fix (k 0 1 , k 0 2 ) such that k 0 1 = k 0 2 .
We only prove that the corresponding section of K 1 :

K 1,1 (k 0 1 , k 0 2 ) := [k : ω(k 0 1 ) -ω(k 0 2 -k) = ω(k 0 2 ) -ω(k 0 1 -k)]
is of null Lebesgue measure. To simplify our considerations we assume that σ 1 = σ 2 = 1.

The remaining cases can be dealt with similarly. Suppose on the contrary that the Lebesgue measure of the set is positive. We can find an increasing sequence of {ℓ n , n ≥ 1} ⊂ K 1,1 (k 0 1 , k 0 2 ) so that the signs of {k 0 2ℓ n , n ≥ 1} and {k 0 1ℓ n , n ≥ 1} are definite. In consequence, we get

ω(k 0 2 -ℓ n ) -ω(k 0 2 -ℓ n+1 ) = ω(k 0 1 -ℓ n ) -ω(k 0 1 -ℓ n+1 ). Since k 0 1 = k 0 2 and {ℓ n , n ≥ 1} is converging we conclude the existence of ℓ ′ = ℓ ′′ such that ω ′ (ℓ ′ ) = ω ′ (ℓ ′′ ).
When the signs of the arguments are the same we conclude easily the contradiction with (3.10). Assume therefore that they are opposite and ℓ ′ < ℓ ′′ . Since ω ′ (k) (understood as a function on R) is 1-periodic and differentiable except at integer lattice points, the above implies that

ω ′ (1 + ℓ ′ ) = ω ′ (ℓ ′′ )
and 0 < ℓ ′′ < 1 + ℓ ′ < 1. Hence, we conclude the existence of a point k ∈ T \ {0} where (3.10) holds. This is also a contradiction. Therefore,

m 1 (K 1,1 (k 0 1 , k 0 2 )) = 0 for all k 0 1 = k 0 2 .

Statistically homogeneous initial data

{sec2.3.2}
For a given non-negative m ≥ 0, we assume that the initial data ψ is an H -m (T) valued Gaussian random element. More precisely, suppose that E 0 (•) is a non-negative function from C(T), {ξ y , y ∈ Z} are i.i.d. complex Gaussian random variables such that Eξ 0 = 0 and

E|ξ 0 | 2 = 1, and ψ(k) = y∈Z ξ y E 1/2 0 (k)e y (k). (3.11) {053110} It is supported in H -m (T), provided that m > 1/2.
Observe that the covariance form equals

C(J 1 , J 2 ) := E J 1 , ψ J 2 , ψ * = T E 0 (k)J 1 (k)J * 2 (k)dk (3.12) {021310aa}
for any J 1 , J 2 ∈ C ∞ (T). The Gibbs equilibrium states described in the introduction correspond to

E 0 (k) ≡ const. Suppose that ω(•) ∈ C ∞ b (T \ {0}
) is bounded with all derivatives on T \ {0} and such that ω ′ (0-) and ω ′ (0+) are defined. Then, for any m < 3/2 and t * > 0 there exists C > 0 such that operators A[t, •], given by (2.14), satisfy

A[t, g 1 ] -A[t, g 2 ] H -m (T) ≤ C g 1 -g 2 H -m (T) , ∀ t ∈ [0, t * ]. Likewise, the Hilbert-Schmidt norm of Q[t, g] : L 2 (T) → H -m (T) satisfies Q[t, g 1 ] -Q[t, g 2 ] (HS) ≤ C g 1 -g 2 H -m (T) , ∀ g 1 , g 2 ∈ H -m (T) for t ∈ [0, t * ].
Using again the results of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF], it is easy to conclude that equation (2.13) has a unique mild solution { ψ(ǫ) (t), t ≥ 0} whose realizations belong to C([0, +∞); H -m (T)).

Let R(t, k) be given by (3.3) with Ē(t, k) the solution of (3.4) satisfying Ē(0, k)

= E 0 (k). Since the operator f (k) → R 1/2 (t, k)f (k) is Hilbert-Schmidt, when considered from L 2 (T) to H -m (T), and f (k) → -( β(k)/4)f (k) is bounded on H -m (T), the equation d ψ * (t, k) = - β(k) 4 ψ * (t, k)dt + R 1/2 (t, k)dW (t, k), ψ * (0, k) = ψ(k) (3.13)
{limit-eqt} has a unique H -m (T)-valued mild solution, see Theorem 7.4, p. 186 of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF]. It is given by the formula ψ * (t, k) = e -β(k)t/4 ψ +

t 0 e -β(k)(t-s)/4 R 1/2 (s, k)dW (s, k).
We denote by H -m w (T) the Hilbert space equipped with the weak topology. Our main result is as follows.

{main-thm2}

Theorem 3.3 Suppose that m ∈ (1/2, 3/2) and condition ω) holds. Then, under the above assumptions, the processes { ψ(ǫ) (t), t ≥ 0} converge in law over C([0, +∞), H -m w (T)), as ǫ → 0+, to { ψ * (t), t ≥ 0}.

Remark. As in the remark made after Theorem 3.1 we can also conclude that

lim t→+∞ E ψ * (t) -ψs (t), f 2 = 0, (3.14)
{040510b}

where ψs (t) is a time homogeneous, distribution valued Ornstein-Uhlenbeck process given by

d ψs (t, k) = - β(k) 4 ψs (t, k)dt + β(k)T 2 dW (t, k), ψs (0, k) = ψ(k), (3.15)
limit-eqt1b}

where

T = E 0 L 1 (T) .
4 Proof of Theorem 3.1

{sec4}
We explain the idea of the proof in the case n = 1 (that is, the process ψ(t, k) for a fixed k), the independence of the compensated wave function for various k is handled in the same manner. Since the coefficients appearing in the stochastic differential equation describing the evolution of ψ(ǫ) (t) (see (2.13)) are of the order O(1), it is easy to conclude that for each k the laws of processes { ψ(ǫ) (t, k), t ≥ 0} are tight over C([0, +∞); C), as ǫ → 0+. In order to identify the weak limit, thus proving part i) of the theorem, we have to deal with the rapidly oscillating terms. First, we show that the rapidly oscillating part of the bounded variation term in (2.13) (with the factor exp{2iω(k)t/ǫ} in (2.14)) vanishes in the limit, because of the following result.

{cormar82} Proposition 4.1 For given t * > 0, a ≥ 0, k, k ′ ∈ T, k = 0 and function f ∈ C 1 [0, t * ] we have lim ǫ→0+ E sup t∈[0,t * ] t 0 exp -i as ǫ f (s) ψǫ (s, k)ds = 0. (4.1) 070410-mar8}
This result is a part of Corollary 4.5 below.

Next, we show that the limit of the martingale part M(ǫ) t (k) in (2.13) is a complex Gaussian martingale with the quadratic variation equal to t 0 R(s, k)ds. To this purpose we prove the following convergence of the quadratic variation:

lim ǫ→0+ sup t∈[0,t * ] M(ǫ) (k) t - t 0 R(s, k)ds = 0 (4.2) {mar0802}
in probability, for any t * > 0. This is done in Proposition 4.2. The method of proof of (4.2) is as follows. The terms of the following form appear in the quadratic variation of M(ǫ) t (k):

V (0) ǫ (t) := t 0 | ψ(ǫ) (s)| 2 , f ds, V (1) ǫ (t) := t 0 ds T ψ(ǫ) (s, k) ψ(ǫ) (s, -k)f * (k)dk, (4.3) {mar804} V (2) ǫ (t) := t 0 [ ψ(ǫ) (s)] 2 , f ds.
Here f (k) is a certain explicit function related to the scattering kernel. As ψ(ǫ) (t, k) (without the compensation) is rapidly oscillating as e -iω(k)t/ǫ , we expect that only V (0) ǫ (t) has a nontrivial limit. This term contains no oscillation and is essentially the time integral of scattered energy | ψ(ǫ) (t, k)| 2 . It has been shown in [START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] that the expectation of the energy converges to the solution of (1.9). We need to strengthen this result to convergence in probability.

The proof of part ii) uses the same ideas. Integrating against a test function results in the formula for the quadratic variation, see (4.35), containing only terms with fast oscillating factors, so the stochastic part vanishes in the limit.

We now turn to the proof of the theorem. An application of the Itô formula to (2.8) yields, see Theorem 4.17 of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF],

d| ψ(ǫ) (t, k)| 2 = [I ǫ (t, k) + II ǫ (t, k)] dt + dM (ǫ) t (k), (4.4)
{wigner-eqt}

where

I ǫ (t, k) := (A[ ψ(ǫ) (t)]) * (k) ψ(ǫ) (t, k) + ( ψ(ǫ) ) * (t, k) A[ ψ(ǫ) (t)] (k) , II ǫ (t, k) := y∈Z (Q[ ψ(ǫ) (t)](e y )) * (k) Q[ ψ(ǫ) (t)](e y ) (k) , and
M (ǫ)
t is an F t -adapted local martingale, given by M (ǫ)

t = M (1,ǫ) t + M (2,ǫ) t , with M (1,ǫ) t (k) := t 0 (Q[ ψ(ǫ) (s)]dW (s)) * (k) ψ(ǫ) (s, k) , (4.5)
{030207} M (2,ǫ) t (k) := t 0 ( ψ(ǫ) ) * (s, k) (Q[ ψ(ǫ) (s)]dW (s))(k).
We obtain from (2.9) that

I ǫ (t, k) = - β(k) 2 | ψ(ǫ) (t, k)| 2 - β(k) 4 ψ(ǫ) 2 (t, k), (4.6)
{030207c} where ψ(ǫ)

2 (t, k) := ψ(ǫ) (t, k) ψ(ǫ) (t, -k) + ( ψ(ǫ) ) * (t, k)( ψ(ǫ) ) * (t, -k), while equation (2.11) yields II ǫ (t, k) = T R(k, k ′ )| ψ(ǫ) (t, k ′ )| 2 dk ′ + 1 2 T R(k, k ′ ) ψ(ǫ) 2 (t, k ′ )dk ′ . (4.7) {030207b}
We can derive analogous equations for

d[ ψ(ǫ) (t, k)] 2 and d[ ψ(ǫ) (t, k) ψ(ǫ) (t, -k)].
The corresponding terms shall be denoted by

I (i) ǫ (t, k), II (i) ǫ (t, k) and the martingale N (ǫ) t,i (k) = N (1,ǫ) t,i (k) + N (2,ǫ) t,i (k), i = 1, 2.
We have

I (1) ǫ (t, k) = - 2iω(k) ǫ [ ψ(ǫ) (t, k)] 2 + P 1 [ ψ(ǫ) (t), ( ψ(ǫ) ) * (t)], I (2) ǫ (t, k) = - 2iω(k) ǫ ψ(ǫ) 2 (t, k) + P 2 [ ψ(ǫ) (t), ( ψ(ǫ) ) * (t)],
and

II (i) ǫ (t, k) = Q i [ ψ(ǫ) (t), ( ψ(ǫ) ) * (t)], i = 1,
V (i) ǫ (t) = 0, i = 1, 2 (4.10)

{023009}

in probability.

Proof. The proof of this proposition shall be obtained at the end of a series of lemmas. We start with the following.

E sup t∈[0,t * ] ψ(ǫ) (t) p L p (T) ≤ Ce Ct * ψ p L p (T) , ∀ t * > 0. (4.11) {110510} Proof. Let T (ǫ) t ψ(k) := exp -i ω(k)t ǫ ψ(k), ψ ∈ L p (T), t ∈ R.
We obviously have

T (ǫ) t ψ L p (T) = ψ L p (T) , ∀ t ≥ 0, p ∈ [1, +∞]. ( 4
.12)

{013108}

Using the Duhamel formula, the solution of (2.8) can be written as

ψ(ǫ) (t, k) = ψ(k) + t 0 T (ǫ) t-s B[ ψ(ǫ) (s)](k)ds + y∈Z t 0 T (ǫ) t-s Q[ ψ(ǫ) (s)
](e y )(k)dw y (s). (4.13)

{023108}

Hence, for a given ǫ ∈ (0, 1] and t 0 > 0 to be adjusted later on, we can write

E sup t∈[0,t 0 ] | ψ(ǫ) (t, k)| p ≤ C | ψ(k)| p + t p-1 0 t 0 0 E| ψ(ǫ) (s, k)| p ds + E    sup t∈[0,t 0 ] t 0 y∈Z T (ǫ) -s Q[ ψ(ǫ) (s)](e y )(k)dw y (s) p       . ( 4

.14)

{010410}

To estimate the martingale term on the right hand side we use Burkholder-Davis-Gundy inequality which allows to bound it by

4 p/2 E t 0 0 T R(k, k ′ )| ψ(ǫ) (s, k -k ′ )| 2 dk ′ ds p/2 ≤ C 1 t p/2-1 0 t 0 0 E ψ(ǫ) (s) p L p (T) ds, (4.15)
{020410} for some constant C 1 > 0. Choosing t 0 sufficiently small, so that

Ct p 0 + CC 1 t p/2 0 < 1/2, we conclude that E sup t∈[0,t 0 ] ψ(ǫ) (t) p L p (T) ≤ 2C ψ p L p (T) . (4.16)

{011411}

The argument leading to (4.16) can be used on each of the intervals [jt 0 , (j + 1)t 0 ) for any j ≥ 1 and yields

E sup t∈[jt 0 ,(j+1)t 0 ] ψ(ǫ) (t) p L p (T) ≤ CE ψ(ǫ) (jt 0 ) p L p (T) ≤ CE sup t∈[(j-1)t 0 ,jt 0 ] ψ(ǫ) (t) p L p (T) , (4.17)
{011411j}

for some constant C > 0 independent of j and ǫ ∈ (0, 1]. Hence, after j iterations of the above estimate, we conclude

E sup t∈[jt 0 ,(j+1)t 0 ] ψ(ǫ) (t) p L p (T) ≤ C j ψ p L p (T) (4
[ ψ(ǫ) (s, k) ψ(ǫ) (s, k ′ )] and d[ ψ(ǫ) (s, k)( ψ(ǫ) ) * (s, k ′ )]
. Then, we repeat the argument used above.

The following lemma shall be crucial for us. Proof. We can write

E M (1,ǫ) t , f 2 ≤ 2    j∈Z t 0 ds E T 2 r(k, k ′ )f * (k)( ψ(ǫ) ) * (s, k -k ′ )e * j (k ′ ) ψ(ǫ) (s, k)dk 2 + j∈Z t 0 dsE T 2 r(k, k ′ )f * (k)( ψ(ǫ) )(s, k ′ -k)e * j (k ′ ) ψ(ǫ) (s, k)dk 2    (4.25) {030207a}
Here, for abbreviation sake, we wrote dk = dkdk ′ . The estimate of E M

T 3 r(k, k ′ )r(k 1 , k ′ )f * (k)f (k 1 ) × E ( ψ(ǫ) ) * (s, k -k ′ ) ψ(ǫ) (s, k) ψ(ǫ) (s, k 1 -k ′ )( ψ(ǫ) ) * (s, k 1 ) +E ψ(ǫ) (s, k -k ′ ) ψ(ǫ) (s, k)( ψ(ǫ) ) * (s, k 1 -k ′ )( ψ(ǫ) ) * (s, k 1 ) dk, (4.26) {030207aa}
where dk = dkdk 1 dk ′ . Consider the term of (4.26) corresponding to the first expectation (the other can be dealt with in a similar fashion). Recall that

K 1 := [(k, k ′ , k 1 ) : ω(k) + ω(k ′ -k 1 ) = ω(k ′ ) + ω(k -k 1 )]. We claim that for k = (k, k ′ , k 1 ) ∈ K 1 we have lim ǫ→0+ t 0 Ψ (ǫ) (s, k)ds = 0, (4.27) {phi} where Ψ (ǫ) (s, k) := E ( ψ(ǫ) ) * (s, k -k ′ ) ψ(ǫ) (s, k 1 -k ′ ) ψ(ǫ) (s, k)( ψ(ǫ) ) * (s, k 1 ) .
Using (2.8) and Itô formula we conclude that

i ǫ ω(k -k ′ ) + ω(k 1 ) -ω(k 1 -k ′ ) -ω(k) t 0 Ψ (ǫ) (s, k)ds = Ψ (ǫ) (t, k) -Ψ (ǫ) (0, k) + t 0 P[ ψ(ǫ) (s), ( ψ(ǫ) ) * (s)](k)ds, (4.28)

{010110}

where P is a fourth degree polynomial formed over the wave function ψ(ǫ) (s), ( ψ(ǫ) ) * (s). Dividing both sides of (4.28) by the factor in front of the integral on the left hand side and subsequently using Corollary 4.4 with p = 4 we conclude (4.27). The lemma then follows, provided we can substantiate the following interchange of the limit with integral lim

ǫ→0+ t 0 ds T 3 r(k, k ′ )r(k 1 , k ′ )f * (k)f (k 1 )Ψ (ǫ) (s, k)dk = T 3 r(k, k ′ )r(k 1 , k ′ )f * (k)f (k 1 )dk lim ǫ→0+ t 0 Ψ (ǫ) (s, k)ds .
The latter however is a consequence of the Lebesgue dominated convergence theorem and Corollary 4.4. This ends the proof of (4.23). The proof of (4.24) is analogous.

Going back to the proof of Proposition 4.2 we conclude that (4.10) holds, thanks to (4.20) and the Lebesgue dominated convergence theorem.

On the other hand from the Itô formula for d| ψ(ǫ

) (t, k)| 2 we obtain | ψ(ǫ) (t)| 2 , f -| ψ(0)| 2 , f = t 0 L| ψ(ǫ) (s)| 2 , f ds + 1 2 t 0 L ψ(ǫ) 2 (s), f ds + M (ǫ) t , f .
Denote by {Q ǫ , ǫ ∈ (0, 1]} the family of the laws of {| ψ(ǫ) (t)| 2 , t ≥ 0} over C([0, +∞), L 2 w (T)). Here L 2 w (T) stands for the space L 2 (T) equipped with the weak topology.

Using Lemma 4.3 we conclude from the above equality that for any t * > 0 there exists a constant C > 0 such that

E | ψ(ǫ) (t)| 2 , f -| ψ(ǫ) (s)| 2 , f 4 ≤ C(t -s) 2 , ∀ ǫ ∈ (0, 1], t, s ∈ [0, t * ].
This implies tightness of the family of the laws of { | ψ(ǫ) (t)| 2 , f , t ≥ 0}, as ǫ → 0+ over C[0, +∞). From the above and estimate (4.11) we conclude weak pre-compactness of Q ǫ , ǫ ∈ (0, 1], see Theorem 3.1, p. 276 of [START_REF] Jakubowski | On the Skorochod topology[END_REF]. Thanks again to Lemma 4.6 and the already proved formula (4.10) we conclude that the limiting law is a δ-type measure supported on Ē(t) -the solution of (3.4). This, in particular, implies that lim

ǫ→0+ sup t∈[0,t * ] | ψ(ǫ) (t)| 2 -Ē(t), f = 0
in probability. Hence (4.9) follows.

With the results proved above in hand, we return to the proof of Theorem 3.1. Assume first that n = 1 and we consider the process ψ(ǫ) (t, k) evaluated at a single k. From (2.13) and Corollary 4.4 we conclude easily that for any t * > 0 there exists a constant C > 0 such that

E| ψ(ǫ) (t, k) -ψ(ǫ) (s, k)| 4 ≤ C(t -s) 2 , ∀ ǫ ∈ (0, 1], s, t ∈ [0, t * ].
This implies tightness of the laws of { ψ(ǫ) (t, k), t ≥ 0} over C[0, +∞).

In the next step we identify the limiting law

P k of { ψ(ǫ) (t, k), t ≥ 0} over C[0, +∞). Denote by Π t (f ) := f (t), f ∈ C[0, +∞) the canonical coordinate map.
Consider the quadratic variation of the complex valued martingale given by (2.15)

M(ǫ) (k), ( M(ǫ) ) * (k) t = σ 1 ,σ 2 =±1 σ 1 σ 2 T r 2 (k, k ′ )ψ (ǫ) σ 1 (t, k -k ′ )(ψ (ǫ) σ 2 ) * (t, k -k ′ )dk ′ , (4.29) {050410} and M(ǫ) (k), M(ǫ) (k) t = σ 1 ,σ 2 =±1 σ 1 σ 2 T r 2 (k, k ′ )ψ (ǫ) σ 1 (t, k -k ′ )ψ (ǫ) σ 2 (t, k -k ′ )dk ′ . ( 4
M(ǫ) (k), ( M(ǫ) ) * (k) t - t 0 R(s, k)ds = 0 and lim ǫ→0+ sup t∈[0,t * ] M(ǫ) (k), M(ǫ) (k) t = 0.
Then by virtue of Theorem 5.4 of [START_REF] Helland | Central limit theorems for martingales with discrete, or continuous time[END_REF] we conclude that { M(ǫ) t , t ≥ 0} converge in law over C[0, +∞) to a complex valued Gaussian process { Mt , t ≥ 0} given by Mt (k) :

= t 0 R 1/2 (s, k)w(ds), (4.31) {010510}
where {w(t), t ≥ 0} is a complex valued standard Brownian motion.

Assume now that k = 0 and P k is a limiting law of { ψ(ǫ) (t, k), t ≥ 0} obtained from a certain sequence ǫ n → 0+. Denote by Π t the coordinate mapping, given by Π t (g) := g(t) for g ∈ C[0, +∞).We conclude from (2.13) and (4.19) that

Π t + β(k) 4 t 0 Π s ds, t ≥ 0
is a P k -martingale whose law coincides with that of the process described by (4.31). The conclusion extends also to the case when k = 0 and ω(0) > 0. If, on the other hand, ω(0) = 0 we have β(0) = 0 and R 1/2 (s, 0) = 0 and therefore Π t ≡ Π 0 a.s. Suppose now that k 1 , . . . , k n ∈ T are pairwise distinct. Denote by Q ǫ the law of

{( ψ(ǫ) (t, k 1 ), . . . , ψ(ǫ) (t, k n )), t ≥ 0} over C([0, +∞), C n ).
Then, following the argument made in the previous part of the proof, we conclude lim

ǫ→0+ sup t∈[0,t * ] M(ǫ) (k i ), ( M(ǫ) ) * (k j ) t -δ i,j t 0 R(s, k i )ds = 0 and lim ǫ→0+ sup t∈[0,t * ] M(ǫ) (k i ), M(ǫ) (k j ) t = 0, ∀ i, j = 1, . . . , n.
Combining this observation with (4.19) we obtain from equation (2.13) that any limiting point of the family of laws of Q ǫn as ǫ n → 0+ is a measure P k 1 ,...,kn such that

M t = (M (1)
t , . . . , M

t ) := Π t + β(k) 4 t 0 Π s ds, t ≥ 0 (n)
is C n -valued martingale, whose quadratic covariation is given by

M (i) , (M (j) ) * t = δ i,j t 0 R(s, k j )ds and M (i) , (M (j) ) t = 0, ∀ i, j = 1, . . . , n.
This of course implies that P k 1 ,...,kn = P k 1 ⊗ . . . ⊗ P kn . Let f ∈ L 2 (T). We shall prove that lim

ǫ→0+ E| M(ǫ) t , f | 2 = 0. (4.32) {050410a1}
Assuming this result we show how to finish the proof of part (ii). Denote

δψ (ǫ) (t) := ψ(ǫ) (t) -ψ(t).
Using Lemma 4.3 and Theorem 3.1, p. 276 of [START_REF] Jakubowski | On the Skorochod topology[END_REF] we can conclude weak pre-compactness of P ǫ , ǫ ∈ (0, 1] -the family of the laws of {δψ (ǫ) (t), t ≥ 0} -in C([0, +∞), L 2 w (T)). With the help of Corollary 4.20 and (4.32) we conclude that the limiting measure, as ǫ → 0+, is supported on the solution of the equation

g(t), f - 1 4 t 0 βg(s), f ds = 0, ∀ f ∈ L 2 (T).
This of course shows that it is the δ-measure supported on g(t) ≡ 0. Hence, in particular we get lim

ǫ→0+ sup t∈[0,t * ] | δψ (ǫ) (t), f | = 0 (4.33) {060510}
in probability and (3.6) follows.

Coming back to the proof of (4.32) note that by the definition of the martingale M(ǫ) t , see (2.15), we only need to show that lim

ǫ→0+ E t 0 y∈Z T 2 exp is ω(k) ǫ r(k, k ′ )f * (k) ψ(ǫ) σ (s, k -k ′ )e y (k ′ )w y (ds) 2 = 0 (4.34) {080510}
for σ = ±1. We consider only the case σ = 1, the other one can be dealt in a similar manner.

The expression under the limit in (4.34) equals

t 0 T 3 exp is ω(k) -ω(k 1 ) ǫ r(k, k ′ )r(k 1 , k ′ )f * (k)f (k 1 )E ψ(ǫ) (s, k -k ′ )( ψ(ǫ) ) * (s, k 1 -k ′ ) dsdk, (4.35)
{080510a}

with dk = dkdk 1 dk ′ . Observe that exp it ω(k) -ω(k 1 ) ǫ E ψ(ǫ) (t, k -k ′ )( ψ(ǫ) ) * (t, k 1 -k ′ ) -E ψ(k -k ′ ) ψ * (k 1 -k ′ ) = t 0 d ds exp is ω(k) -ω(k 1 ) ǫ E ψ(ǫ) (s, k -k ′ )( ψ(ǫ) ) * (s, k 1 -k ′ ) ds = i ǫ ω(k) -ω(k -k ′ ) -ω(k 1 ) + ω(k 1 -k ′ ) × t 0 exp is ω(k) -ω(k 1 ) ǫ E ψ(ǫ) (s, k -k ′ )( ψ(ǫ) ) * (s, k 1 -k ′ ) ds + t 0 P ψ(ǫ) (s), ( ψ(ǫ) ) * (s) (k, k ′ )ds,
where P is a second degree polynomial in ψ(ǫ) (s), ( ψ(ǫ) ) * (s). Using Corollary 4.4 and an argument identical with the one used in the proof of Lemma 4.6 we conclude that lim

ǫ→0+ t 0 exp is ω(k) -ω(k 1 ) ǫ E ψ(ǫ) (s, k -k ′ )( ψ(ǫ) ) * (s, k 1 -k ′ ) ds = 0 for all k = k 1 . Equality in (4.34
) is a consequence of the Lebesgue dominated convergence theorem.

Spatially homogeneous initial data {sec5}

Tightness of the family of laws { ψ(ǫ) (t), t ≥ 0}, in the space of continuous functionals taking values in a space of distributions is again due to the fact that the evolution equation (2.13) contains no terms that are large in magnitude. This is done in Sections 5.1 and 5.2. However, we have no estimates of the H -m (T) norm of ψ(ǫ) (t) analogous to the ones in Lemma 4.3, that have played an important role in the limit identification argument of Section 4 for square integrable data. Therefore, instead of considering the quadratic variation of the martingale term as we did in the proof of Theorem 3.1, for the proof of Theorem 3.3 we identify the limit of all moments of ψ(ǫ) (t). Accordingly, we first write equations for time evolution of an arbitrary moment of ψ(ǫ) (t) in Section 5.3. Using standard averaging argument we show (see Proposition 5.3) the convergence of moments, as ǫ → 0+, to a solution of the limiting equation obtained simply by discarding the oscillatory terms from the moment equation. Finally in Section 5.5 we prove that the solutions of the limiting equation coincide with the respective moments of the non-homogeneous Ornstein-Uhlenbeck equation (3.13) concluding in this way the proof of Theorem 3.3.

5.1 Properties of spatially homogeneous solutions of (2.8)

{sec5.1}
Recall that the initial data ψ is random and takes values in the Hilbert space of distributions H -m (T) for some m ∈ (1/2, 3/2). In fact, in Sections 5.1-5.3 we shall not make any use of the assumption that the data is Gaussian and we use only the fact that E ψ 2 -m < +∞. Consider the random field {ψ y := ψ, e y , y ∈ Z}. Note that

y∈Z (1 + y 2 ) -m E|ψ y | 2 = E ψ 2 -m < +∞,
and Eψ 0 = 0. Moreover, {ψ y+z , y ∈ Z} and {ψ y , y ∈ Z} have identical laws for all z ∈ Z. The latter is equivalent to the fact that ψ(k) and e z (k) ψ(k) are identically distributed in H -m (T) for any z ∈ Z.

Since the covariance function of the field

S x-y := E[ψ x ψ * y ], ∀ x, y ∈ Z
is positive definite, there exists a finite measure Ê(dk) such that

S x = T e ixk Ê(dk), ∀ x ∈ Z.
We assume that Ê(dk) = E 0 (k)dk for some non-negative density E 0 ∈ C(T). In addition, we also suppose that the covariance function depends sufficiently fast in space so that When the field ψ x is a complex valued Gaussian, the quantity in the left side vanishes. Assumption (5.1) implies, in particular, that

Y = x∈Z e x E[ψ x ψ 0 ]
belongs to C(T).

We note that the translation invariance of the solution persists in time. Indeed, let ψ (ǫ)

x (t) := ψ(ǫ) (t), e x and z ∈ Z. A direct computation shows that e z ψ(ǫ) (t) is also a solution of (2.13). Since the laws of the initial conditions e z ψ and that of ψ are identical, we conclude from the uniqueness in law of solutions that the same holds for the processes {e z ψ(ǫ) (t), t ≥ 0} and { ψ(ǫ) (t), t ≥ 0}. In consequence, the laws of {ψ x (t)(ψ

(ǫ) 0 ) * (t) and Y (ǫ) t,x = E ψ (ǫ) x (t)ψ (ǫ) 0 (t)
and introduce two distributions on H -m (T) 5.2 Tightness of solutions of (2.13)

f, Ŝ (ǫ)
( Ŝ(ǫ) t L 1 (T) + Ŷ (ǫ) t L 1 (T) ) < +∞, (5.2)
{sec5.2}
Given f ∈ H m (T), we denote by Q ǫ and Q ǫ,f the laws of the processes { ψ(ǫ) (t), t ≥ 0} and { f, ψ(ǫ) (t) , t ≥ 0} over C([0, +∞), H -m w (T)) and C([0, +∞), C), respectively, and by { Qǫ , ǫ ∈ (0, 1]} the family of laws of { ψ(ǫ) (t), t ≥ 0} over C([0, +∞), H -m w (T)). According to [START_REF] Mitoma | On the sample continuity of S ′ processes[END_REF], see Remark R1, p. 997, to verify the tightness of Qǫ , it suffices to show the following two conditions:

(UC) for any σ, M, t * > 0 there exists a δ > 0 such that

P sup t∈[0,t * ] | ψ(ǫ) (t), f | ≥ M < σ, ∀ f m < δ, ǫ ∈ (0, 1],
and (FDT) for any f ∈ H m (T) the family of the laws of the processes { ψ(ǫ

) (t), f , t ∈ [0, t * ]}, ǫ ∈ (0, 1] is tight over C[0, t * ].
As in (3.12) we conclude that for any f 1 , f 2 ∈ H m (T), where m > 1/2, the covariance

E f 1 , ψ(ǫ) t f 2 , ψ(ǫ) t * = T Ŝ(ǫ) t (k)f 1 (k)f * 2 (k)dk.
(5.5)

{031310}

From (2.13) and Doob's inequality there exists a constant C > 0 such that

E sup t∈[0,t * ] | ψ(ǫ) (t), f | 2 ≤ C E| ψ, f | 2 + t * 0 E A t ǫ , ψ(ǫ) (t) , f 2 dt + E M(ǫ) t * , f 2 .
(5.6)

{041310}

Using (5.5), (5.2) and the definitions of A[t/ǫ, •], and the martingale M(ǫ) t (see (2.14) and (2.15)) we conclude that the right hand side of (5.6) can be estimated from above by C f 2 ∞ , which can be made less than σ > 0, provided we choose δ > 0 sufficiently small.

To show condition (FDT) consider Q(M) ǫ,f -the law of the stopped process

{( ψ(ǫ) (t ∧ τ (ǫ) M ), f , ψ(ǫ) (t ∧ τ (ǫ) M ), f 0 ) t ∈ [0, t * ]} over C([0, t * ]; C 2 ). Here f 0 (k) := f (-k) and τ (ǫ) M := inf[t ∈ [0, t * ] : | ψ(ǫ) (t), f | 2 + | ψ(ǫ) (t), f 0 | 2 ≥ M 2 ].
We adopt the convention that τ M := t * if the set is empty. Thanks to (UC) we conclude that lim M →+∞ τ (ǫ) M = t * , a.s. for each ǫ ∈ (0, 1]. Denote also by Qǫ,f the law of the process without the stopping condition.

From (2.13) we conclude that for a fixed M and an arbitrary non-negative function φ :

C 2 → R, of class C 1 c (R 4 ), one can choose a constant K φ , independent of spatial translations of φ, such that φ( ψ(ǫ) (t ∧ τ (ǫ) M ), f , ψ(ǫ) (t ∧ τ (ǫ) M ), f 0 ) + K φ t, t ∈ [0, t * ]
is a non-negative submartingale. This proves tightness of { Q(M) ǫ,f , ǫ ∈ (0, 1]} for a fixed M , by virtue of Theorem 1.4.3 of [START_REF] Stroock | Multidimensional diffusion processes[END_REF]. Since for any σ > 0 one can find a sufficiently large M > 0 such that B M -the ball centered at 0 and of radius

M in C([0, t * ]; C 2 ) -satisfies Q(M) ǫ,f (B c M ) + Qǫ,f (B c M ) < σ and Q(M) ǫ,f (B M ∩ A) = Qǫ,f (B M ∩ A) for all Borel measurable subsets A of C([0, t * ]; C 2 )
, we conclude tightness of { Qǫ,f , ǫ ∈ (0, 1]}, see step (vi) of the proof of Theorem 3 of [START_REF] Kesten | A limit theorem for turbulent diffusion[END_REF] for details of this argument.

Evolution of moments {sec5.3}

To describe the evolution of moments we rewrite equation (2.13) in a more compact form. Denote by

C(t, k) = [C ij (t, k)], i, j = ±1, the 2 × 2 hermitian matrix C(t, k) := C 1,1 C 1,-1 C -1,1 C -1,-1 ,
with the entries

C p,q (t, k) := pq β(k) 4 exp {ipω(k)(1 -pq)t} . Let also Q(t, k, k ′ ) = [Q pq (t, k, k ′ )], p, q = ±1, be the 2 × 2 matrix Q p,q (t, k, k ′ ) := ipqr(k, k -k ′ )e ip[ω(k)-pqω(k ′ )]t
and W (t, k) := y e y (k)w y (t). Let us recall that ψ(ǫ) -1 (t, k) = ψ(ǫ) * (t, -k). Then, equation for

Ψ (ǫ) (t, k) =   ψ(ǫ) (t, k) ψ(ǫ) -1 (t, k)   is dΨ (ǫ) (t, k) = -C t ǫ , k Ψ (ǫ) (t, k)dt + T Q t ǫ , k, k -k ′ Ψ (ǫ) (t, k -k ′ )W (dt, k ′ )dk ′ , (5.7)
{031710} Ψ (ǫ) (0, k) = Ψ(k),
with the initial data

Ψ(k) =   ψ(k) ψ-1 (k)   .
Let {S ǫ (s, t, k), s, t ∈ R} be the Hermitian matrices solving the deterministic system of ODE's

dS ǫ (s, t, k) dt = -C t ǫ , k S ǫ (s, t, k) S ǫ (s, s, k) = I 2 .
Here I 2 is the 2 × 2 identity matrix. Existence and uniqueness of solutions to (5.7) in the strong sense (thus implying the result in the mild, or weak sense as well) follows from an argument used in Chapter 6 of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF] (because the generators for the evolution family S ǫ (s, t) are bounded), see Proposition 6.4 there. Although the case considered here differs slightly because the coefficients are time dependent, this does not influence the results. Given a nonnegative integer p ≥ 1, define a tensor valued distribution on H -m/p (T p )

M (ǫ) (t) := M (ǫ) i (t) , i = (i 1 , . . . , i p ) ∈ {-1, 1} p , by M (ǫ) i (t) = E ψ(ǫ) i 1 (t) ⊗ . . . ⊗ ψ(ǫ) ip (t) .
Suppose that the components of the tensor M = [ Mi ] belong to M(T 2n ). Similarly to part 1) of Proposition 5.2 we conclude that the initial value problem The measure is supported in H := F H(F) where

d dt Mi (t) = - 1 4 2n ℓ=1 β (k ℓ ) Mi (t) + 1≤ℓ<m≤2n j=±1 R ℓ,m Mi ℓ,m (j,-j) (t), M (
H(F) := [k : k ℓ + k m = 0, ∀ (ℓ, m) ∈ F].
Suppose that the components of the tensor ρ

(k) = [ρ i (k)], i ∈ {-1, 1} 2n belong to L 1 (µ). Consider the initial value problem d dt ρ i (t, k) = - 1 4 2n ℓ=1 β (k ℓ ) ρ i (t, k) + 1≤ℓ<m≤2n j=±1 T r 2 (k ℓ , k ℓ -k ′ )1 D ℓ,m (k)ρ i ℓ,m (j,-j) (t, k ′ ℓ,m )dk ′ , ρ i (0, k) = ρ i (k), i ∈ {-1, 1} 2n , (5.12)
{031910a1} with k ′ ℓ,m := (k 1 , . . . , k ℓ-1 , k ′ , . . . , k m-1 , -k ′ , . . . , k 2n ). It is straightforward to conclude that the above system possesses a unique continuous solution ρ(t, k) = [ρ i (t, k)] whose components belong to L 1 (µ). The next proposition gives the convergence of even moments to the solution of (5.11).

{012410}

Proposition 5.3 Suppose that all the components of the tensor [ Mi (dk)] are absolutely continuous with respect to µ, i.e. Mi (dk) = ρ i (k)µ(dk), and the dispersion relation satisfies hypothesis ω). Then, the following are true: 1) Mi (t, dk) is absolutely continuous with respect to µ(dk) and

Mi (t, dk) = ρ i (t, k)µ(dk), ∀ i ∈ {-1, 1} 2n
(5.13)

{012910}

where {ρ i (t), t ≥ 0} satisfy (5.12).

2) For any T > 0 there exists a constant C > 0 such that

lim ǫ→0+ i∈{-1,1} 2n sup t∈[0,t * ] M (ǫ) i (t) -Mi (t) TV = 0.
(5.14)

{063110}

Proof. The conclusion of part 1) follows from uniqueness of solutions of (5.11) and (5.12), and the fact that the right hand side of (5.13) defines a solution of (5.11). From (5.9) and (5.11) we conclude that

M (ǫ) i (t) -Mi (t) TV ≤ 2n ℓ=1 j=±1 t 0 C i ℓ ,j s ǫ [ M (ǫ) i ℓ (j) (s) -Mi ℓ (j) (s)] TV ds + 1≤ℓ<m≤2n j 1 ,j 2 =±1 t 0 R j 1 ,j 2 i ℓ ,im s ǫ [ M (ǫ) i ℓ,m (j 1 ,j 2 ) (s) -Mi ℓ,m (j 1 ,j 2 ) (s)] TV ds + 2n ℓ=1 j=±1 t 0 T 2n E i ℓ ,j s ǫ , k ℓ ρ i ℓ (j) (s, k)dsµ(dk) + 1≤ℓ<m≤2n j 1 ,j 2 =±1 t 0 T 2n+1 Rj 1 ,j 2 i ℓ ,im s ǫ , k, k ′ ρ i ℓ,m (j 1 ,j 2 ) (s, k)dsµ(dk)dk ′ .
The matrix E(t, k) = [E p,q (t, k)], p, q = ±1 is given by

E(t, k) := C(t, k) -( β(k)/4)I 2 , (5.15)

{042910}

where I 2 is the 2 × 2 identity matrix. In addition,

Rj 1 ,j 2 i ℓ ,im s ǫ , k, k ′ := R j 1 ,j 2 i ℓ ,im s ǫ , k ℓ , k m , k ′ -δ -im i ℓ δ -j 2 j 1 r 2 (k ℓ , k ℓ -k ′ )1 D ℓ,m (k).
Denote the terms appearing on the right hand side of (5. IV (t) = 0.

(5.18)

{102910}

It implies the conclusion of part 2) of the proposition, via an application of the Gronwall's inequality.

We write IV (t) = IV 1 (t) + IV 2 (t), where the terms IV i (t), i = 1, 2 correspond to the integration over D ℓ,m and its complement. In the latter case, we have to deal with terms of the form Using (5.12) and Gronwall's inequality, we conclude that Using condition ω) we conclude therefore, by virtue of Lebesgue dominated convergence theorem, that lim ǫ→0+ I ǫ = 0. Likewise, we conclude that lim ǫ→0+ J ǫ = 0. Part 2) of the proposition follows then from another application of Gronwall's inequality after substituting for ρ ′ i (s, k) from (5.12). Summarizing, we have shown so far that lim

2 ) = -(σ

(2) 1 , σ (2)
2 ) and σ (j)

p ∈ {-1, 1}. To these terms we can apply the integration by parts argument as before, to conclude that lim ǫ→0+ sup t∈[0,t * ] IV 1 (t) = 0. Summarizing, we have shown that (5.18) holds, and the proof of part 2 of the proposition is therefore complete.

Proof of Theorem 3.3 {sec5.5}

In this section, and in this section only, we make use of the assumption that ψ is Gaussian. We show that the limiting measure for Qǫ , as ǫ → 0+, coincides with the law Q of the process given (3.13) by proving that for any N ≥ 1, 0 ≤ t 1 < . . . < t N , any non-negative integers ℓ j , m j , test functions f j , g j ∈ H m (T), j = 1, . . . , N we have [ ψ(t j ), f j ℓ j ( ψ(t j ), g j * ) m j ]   .

(5.22)

{011710}

To simplify the notation, we prove (5.22) only in the case N = 1. The general case can be handled in the same manner, using Markov property of the process { ψ(ǫ) (t), t ≥ 0}, at the expense of some additional complications in the notation. We recall (see Section 3.2) that the initial data { ψ(k), k ∈ T} is a δ-correlated Gaussian random field given by (3.11). Therefore, for the odd moments we have M (ǫ) i (0) = 0, ∀ i ∈ {-1, 1} 2n-1 , where n ≥ 1 is an integer. By uniqueness of solutions of (5.9) we conclude that in this case M (ǫ) (t) ≡ 0 for all t ≥ 0. When i ∈ {-1, 1} 2n we can use the conclusion (5.14) of Proposition 5.

{011210}

{sec3} 3 . 1

 31 Square integrable initial data {sec3.1}

{lm023108} Lemma 4 . 3

 43 For any p ∈ [2, +∞) there exists C > 0 such that sup ǫ∈(0,1]

{lm013009} Lemma 4 . 6

 46 For any f ∈ L 2 (T), t * > 0 we have lim

  except f * on the right hand side should be replaced by f . Using Parseval identity we can further transform the right hand side of (4

x∈Z |E[ψ x ψ 0 ]

 0

x

  (t), x ∈ Z} and that of {ψ

  x+z (t), x ∈ Z} are identical for any z ∈ Z. We can now define the correlation functionsS (ǫ) t,x = E ψ (ǫ)

{011310}( 2 )

 2 for any f ∈ L ∞ (T) we havelim ǫ→0+ sup t∈[0,t * ] Ŝ(ǫ) t -Ē(t), f = 0,(5.3){021310}where Ē(t) is given by (3.4) with the initial condition replaced by E 0 (k)(3) for any f such that f ω -1 ∈ L ∞ (T) we have lim Parts 1) and 2) of the lemma are contained in Lemma 12 and Theorem 10 of[START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF], respectively. Part 3) follows easily from part 1) and the arguments used in the proof of Corollary 4.5.

  0) = M . (5.11) {031910} possesses a unique solution in C([0, +∞), M(T 2n )).Any partition of the set {1, . . . , 2n} into a disjoint set of pairs is called a pairing. Defineµ(dk) = F (ℓ,m)∈F δ(k ℓ + k m )dk,where dk = dk 1 . . . dk 2n and the summation extends over all possible pairings of {1, . . . , 2n}.

1 [

 1 k ℓ =-km] r(k ℓ , k ′ )r(k m , -k ′ )ρ i (s, k) )dk ′ for some i ∈ {-1, 1} 2n , σ (j) p ∈ {-1, 1}. Here k (1) ℓ = k ℓ and k (2) ℓ = k m .Using integration by parts over the s variable we can estimate the supremum of the above expression over t ∈ [0, t * ] by the sum ofI ǫ := T 2n+1 µ(dk)dk ′ 1 [k ℓ =-km] |r(k ℓ , k ′ )r(k m , -k ′ )| sup t∈[0,t * ] |ρ i (t, k)| )dk ′ 1 [k ℓ =-km] r(k ℓ , k ′ )r(k m , -k ′ ) d ds ρ i (s, k)

  |ρ i (t, k)|dk < +∞.

T 2n+1 r 2

 2 k ′ ρ i ℓ,m (j 1 ,j 2 ) (s, k)dsµ(dk)dk ′ .The non-vanishing terms appearing in the above sum are of the formt 0 (k ℓ , k ℓk ′ )1 D ℓ,m (k)

  ǫ) (t j ), f j ℓ j ( ψ(ǫ) (t j ), g j * ) m j ]

  , i = (i 1 , . . . , i 2n ) ∈ {-1, 1} 2n ,

  Therefore we can identify | ψ(t, k)| 2 with the energy density in the mode space. In the zero noise case, | ψ(t, k)| 2 is conserved for any k ∈ T (i.e. ∂ t | ψ(t, k)| 2 = 0). The stochastic conservative perturbation mixes the energies between different modes k, and | ψ(t, k)| 2 becomes a random variable. The evolution of the average energy E(t, k) := E| ψ(t, k)| 2 was considered in[START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF]. Since the stochastic perturbation is of order ǫ, to have a visible effect of mixing of different modes we have to look at the time scale ǫ -1 t. It was shown in[START_REF] Basile | Energy transport in stochastically perturbed lattice dynamics[END_REF] that the limit

				are well defined and
	H(q(t), p(t)) =
	lim ǫ→0	E	t ǫ	, k = Ē (t, k)	(1.8)

y |ψ y (t)| 2 = T | ψ(t, k)| 2 dk {eq:bos0}

  Integrating by parts we obtain that the expression above can be bounded from above by (s, k)1 D ℓ,m (k)dsµ(dk) .The first term can be easily estimated by Cǫ, due to the fact that sup k∈T β(k)ω -1 (k) < +∞. To estimate the second term, we use equation(5.11). As a result,, we conclude that for any t * > 0 we can find a constant C(t * ) > 0 such that

		Finally we show that
							lim ǫ→0+	sup t∈[0,t * ]
							0	t	sup i∈{-1,1} 2n	M (ǫ) i (s) -Mi (s)	TV	ds	(5.16)
		t ρ ǫ 0 T 2n β(k ℓ ) exp 2iω(k ℓ ) s ǫ T 2n β(k ℓ ) 2iω(k ℓ ) exp 2iω(k ℓ ) t ǫ -1 ρ i (t, k)1 D ℓ,m (k)µ(dk)
	{0512910}	+ǫ	0	t	T 2n	β(k ℓ ) 2iω(k ℓ )	exp 2iω(k ℓ ) ρ i sup t ǫ -1 d ds t∈[0,t * ] III(t) ≤ C(t * )ǫ.	(5.17)

[START_REF] Mitoma | On the sample continuity of S ′ processes[END_REF]

by I(t), II(t), III(t) and IV (t) respectively. It is easy to see that

I(t) + II(t) ≤ C {0312910}

for some constant C > 0. To estimates the term III we need to bound terms of the form i (s, k)dsµ(dk) for some ℓ and i.

Note that also M (ǫ) i (0) = Mi := E ψi 1 ⊗ . . . ⊗ ψip (5.8)

{021910}

For a given multi-index i we define the multi-indices i ℓ (j) = (i ′ 1 , . . . , i ′ p ), i ℓ,m (j 1 , j 2 ) = (i ′′ 1 , . . . , i ′′ p ) given by: i ′ q = i q for q = ℓ and i ′ ℓ = j, and i ′′ q = i q for q = ℓ, m and i ′′ ℓ = j 1 , i ′′ m = j 2 . Denote by M(T p ) the space of all complex valued Borel measures ν on T p whose total variation norm ν TV is finite. 1) M (ǫ) (t) is the unique solution in H -m/p (T p ) of the system of equations

with i ∈ {-1, 1} p and the initial data given by (5.8). Here

2) If the initial condition is from M(T p ) then the solution also belongs to M(T p ) and for any

(5.10)

{062910}

Proof. The fact that M (ǫ) (t) is a solution of (5.9) follows by an application of Itô formula and equation (5.7). Since the operators appearing on the right hand side of the equation in question are uniformly Lipschitz, on any compact time interval, both in H -m/p (T p ) and M(T p ) the proof of uniqueness of solutions in these spaces is standard. Estimate (5.10) follows by an application of Gronwall's inequality.

Asymptotics of even moments

Let us now describe the limit moment equations. Assume that p = 2n is even, then for any

We define a bounded linear operator R ℓ,m : M(T 2n ) → M(T 2n ) by

for any bounded, measurable f : T 2n → C and ν ∈ M(T 2n ). Here

and ψ1 (t) = ψ(t) is the solution of (3.13) and ψ-1 (t, k) = ψ * (t, -k). The conclusion of Theorem 3.3 will follow provided that we show that M (2n) (t), satisfies (5.11). Note that for n = 1 we obtain that

From (3.13) and Itô formula we conclude that

and ( ψi 1 (t), . . . , ψi 2n (t)) is jointly Gaussian, we infer that the last term on the right hand side of the first equation in (5.23) can be rewritten as being equal to the last term on the right hand side of the first equation of (5.11). Thus the conclusion of Theorem 3.3 has been shown.


Benjamin Girard

ON THE EXISTENCE OF ZERO-SUM SUBSEQUENCES OF DISTINCT LENGTHS by

In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland and Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao's conjecture is solved using Alon's Combinatorial Nullstellensatz. To conclude, we show that, assuming every integer satisfies Property B, this conjecture holds in the case of finite Abelian groups of rank two.

Introduction

Let P be the set of prime numbers and let G be a finite Abelian group, written additively. By exp(G) we denote the exponent of G. If G is cyclic of order n, it will be denoted by C n . In the general case, we can decompose G as a direct product of cyclic groups C n 1 ⊕ • • • ⊕ C nr where 1 < n 1 | . . . | n r ∈ N. For each g in G, we denote by ord(g) its order in G, and by g the subgroup it generates.

By a sequence over G of length ℓ, we mean a finite sequence of ℓ elements from G, where repetitions are allowed and the order of elements is disregarded. We use multiplicative notation for sequences. Let

S = g 1 • . . . • g ℓ = g∈G g vg(S)
be a sequence over G, where, for all g ∈ G, v g (S) ∈ N is called the multiplicity of g in S. We call Supp(S) = {g ∈ G | v g (S) > 0} the support of S, and σ(S) = ℓ i=1 g i = g∈G gv g (S) the sum of S. In addition, we say that s ∈ G is a subsum of S when s = i∈I g i for some ∅ I ⊆ {1, . . . , ℓ}.

If 0 is not a subsum of S, we say that S is a zero-sumfree sequence. If σ(S) = 0, then S is said to be a zero-sum sequence. If, moreover, one has σ(T ) = 0 for all proper subsequences T | S, then S is called a minimal zero-sum sequence.

By D(G) we denote the smallest integer t ∈ N * such that every sequence S over G of length |S| ≥ t contains a non-empty zero-sum subsequence. The number D(G) is called the Mathematics Subject Classification (2010): 11R27, 11B75, 11P99, 20D60, 20K01, 05E99, 13F05.

Davenport constant of the group G. Even though its definition is purely combinatorial, the invariant D(G) found many applications in number theory (see for instance the book [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF] which presents the various aspects of non-unique factorization theory and [START_REF]Geroldinger Additive group theory and non-unique factorizations[END_REF] for a recent survey). Thus, many direct and inverse problems related to D(G) have been studied during last decades, and even if numerous results were proved (see [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF]Chapter 5] and [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups : a survey[END_REF] for a survey), its exact value is known for very special types of groups only.

A sequence S over a finite Abelian group G is said to be dispersive if it contains two nonempty zero-sum subsequences S 1 and S 2 of distinct length, and non-dispersive otherwise, that is, when all non-empty zero-sum subsequences of S have same length. One can readily notice, using the very definition of the Davenport constant, that every sequence S over G with |S| ≥ 2D(G) is dispersive, since S has to contain at least two disjoint non-empty zero-sum subsequences. So, one can ask for the smallest integer t ∈ N * such that every sequence S over G with |S| ≥ t is dispersive. The associated inverse problem is then to make explicit the structure of non-dispersive sequences over a finite Abelian group. Concerning this problem, Gao, Hamidoune and Wang recently proved [START_REF] Gao | Distinct lengths modular zero-sum subsequences : a proof of Graham's conjecture[END_REF] that every non-dispersive sequence of n elements in C n has at most two distinct values, solving a conjecture of Graham reported in a paper of Erdős and Szemerédi [4]. This result was then generalized by Grynkiewicz in [START_REF] Grynkiewicz | Note on a conjecture of Graham[END_REF].

In this article, we study a still widely open conjecture, proposed by Gao in [START_REF] Gao | Zhuang Sequences not containing long zero-sum subsequences[END_REF], on the structure of the so-called normal sequences over a finite Abelian group G. A sequence S over G with |S| ≥ D(G) is said to be normal if all its zero-sum subsequences S ′ satisfy |S ′ | ≤ |S| -D(G) + 1. Gao's conjecture is the following (see also [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups : a survey[END_REF]Conjecture 4.9]).

Conjecture 1. -Let G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | . . . | n r ∈ N,
be a finite Abelian group. Let also S be a normal sequence over G of length |S| = D(G) + i -1, where i ∈ 1, n 1 -1 . Then S is of the form S = 0 i T , where T is a zero-sumfree sequence.

This notion of a normal sequence, first introduced in [START_REF] Gao | Zhuang Sequences not containing long zero-sum subsequences[END_REF], happens to be crucial in the characterization of sequences S over G with |S| = D(G) + |G| -2 and which do not contain any zero-sum subsequence S ′ satisfying |S ′ | = |G| (see [START_REF] Gao | Zhuang Sequences not containing long zero-sum subsequences[END_REF]Theorem 1.7]). The following two theorems, due to Gao (see [START_REF] Gao | Zhuang Sequences not containing long zero-sum subsequences[END_REF]Theorems 1.5 and 1.6]), are the only results known concerning the structure of normal sequences over a finite Abelian group. Before stating these two results on Conjecture 1, we recall that an integer n ≥ 2 is said to satisfy Property B if every minimal zero-sum sequence over C 2 n with |S| = 2n -1 contains some element repeated n -1 times (see [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF]Section 5.8], and [START_REF] Gao | Grynkiewicz Inverse zero-sum problems III[END_REF][START_REF] Gao | Inverse zero-sum problems[END_REF][START_REF] Schmid | Inverse zero-sum problems II[END_REF] for recent progress). It is conjectured that every n ≥ 2 satisfies Property B.

(i) G is a finite cyclic group. (ii) G ≃ C 2 n ,

New results and plan of the paper

In this paper, we use the notion of a dispersive sequence to obtain a characterization of short normal sequences over a finite Abelian p-group, thus improving both Gao's results on this problem. The main theorem (Theorem 2.3) is proved in Section 3 yet, before giving this general result, we would like to emphasize its consequences.

Theorem 2.1. -Let G be a finite Abelian p-group, and let S be a normal sequence over G with |S| = D(G) + i -1, where i ∈ 1, p -1 . Then S is of the form S = 0 i T , where T is a zero-sumfree sequence. The main result of this paper is deduced from a theorem of Alon, Friedland and Kalai (see [START_REF] Alon | Regular subgraphs of almost regular graphs[END_REF]Theorem A.1]), originally proved so as to study the existence of regular subgraphs in almost regular graphs. Our result is the following.

Theorem 2.3. -Let G be a finite Abelian p-group, and let S be a sequence over G with

|S| = D(G) + i -1, where i ∈ 1, p . Let also A be any (i -1)-subset of 1, p -1 . Then S contains a non-empty zero-sum subsequence S ′ such that |S ′ | ≡ b (mod p), for all b ∈ A.
Consequently, Theorem 2.3 gives the existence of non-empty zero-sum subsequences whose length avoids certain remainders modulo p. In addition, this result applies to 'short' sequences, the length of which is close to D(G), thus allowing one to tackle Gao's conjecture, whereas other existing results with a similar flavor (see for instance [19, Theorem 1.2]) hold for longer sequences only. In particular, Theorem 2.3 provides the following insight into dispersive sequences. (i

) If i ≥ 2 and S contains a zero-sum subsequence S ′ with p ∤ |S ′ |, then S is dispersive. (ii) If S contains no non-empty zero-sum subsequence S ′ with p | |S ′ |, then i ≤ p-1
, and S has to contain at least i non-empty zero-sum subsequences with pairwise distinct lengths.

In Section 4, we give a proof of Theorem 2.3, in the special case of elementary p-groups, using the polynomial method. Since this proof is short and may be relevant in its own right, we will present it in full.

In Section 5, we then prove the following theorem, which extends Statement (ii) of Theorem 1.1 to every finite Abelian group of rank two. The proof of this theorem relies on a structural result obtained by Schmid [START_REF] Schmid | Inverse zero-sum problems II[END_REF], which is a characterization of long minimal zero-sum sequences over these groups, provided that a suitable divisor of their exponent satisfies Property B. Finally, in Section 6, we propose two general conjectures suggested by the results proved in this paper.

The case of finite Abelian p-groups

As stated in Section 2, we prove our Theorem 2.3 by using the following theorem of Alon, Friedland and Kalai (see [START_REF] Alon | Regular subgraphs of almost regular graphs[END_REF]Theorem A.1]). Before stating this result, we need to introduce the following notation. Let Z be the set of integers. For S ⊆ Z and m ∈ Z, we denote by card m (S) the number of distinct elements in S modulo m. Theorem 3.1. -Let p be a prime, and let

1 ≤ d 1 ≤ • • • ≤ d n be n integers. For 1 ≤ j ≤ n, let S j ⊆ Z be a set of integers containing 0. For 1 ≤ i ≤ m, let (a i,1 , . . . , a i,n ) be a vector with integer coordinates. If m ≥ n j=1 p d j -card p (S j ) + 1,
then a subset ∅ I ⊆ {1, . . . , m} and numbers s j ∈ S j (1 ≤ j ≤ n) exist such that i∈I a i,j ≡ s j (mod p d j ), for all 1 ≤ j ≤ n.

For instance, it may be observed that Theorem 3.1 provides the exact value for the Davenport constant of a finite Abelian p-group, which was originally obtained by van Emde Boas, Kruyswijk and Olson (see [START_REF] Van Emde Boas | Kruyswijk A combinatorial problem on finite abelian groups[END_REF][START_REF] Olson | A combinatorial problem on finite abelian groups, I[END_REF]).

Indeed, let G ≃ C p d 1 ⊕ • • • ⊕ C p dr , where 1 ≤ d 1 ≤ • • • ≤ d r ∈ N,
be a finite Abelian p-group, and let us set D * (G) = r i=1 p d i -1 + 1. On the one hand, an elementary construction (see [START_REF] Geroldinger | Halter-Koch Non-unique factorizations. Algebraic, combinatorial and analytic theory[END_REF]Proposition 5.1.8]) implies that D(G) ≥ D * (G). On the other hand, let (e 1 , . . . , e r ) be a basis of G, where ord(e i ) = p d i for all i ∈ 1, r , and let Using Theorem 3.1, we can now prove the main theorem of this paper. Let also A be a (i -1)-subset of 1, p -1 , and Ā = 0, p -1 \A. For all j ∈ 1, n , we set

S = g 1 • . . . • g m be a sequence over G of length m = D * (G). Setting g i = a i,1 e 1 + • • • + a i,
Proof of Theorem 2.3. -Let G ≃ C p d 1 ⊕ • • • ⊕ C p dr , where 1 ≤ d 1 ≤ • • • ≤ d r ∈ N,
S j = Ā if j = n, {0} otherwise. Since p dn -card p (S n ) = p -Ā = |A| = i -1, one obtains n j=1 p d j -card p (S j ) + 1 = D(G) + i -1 = m.
Therefore, using Theorem 3.1, there exists a subset ∅ I ⊆ {1, . . . , m} such that

       i∈I a i,j ≡ 0 (mod p d j ) for all 1 ≤ j ≤ r, i∈I 1 = |I| ≡ s (mod p) for some s ∈ Ā.
Consequently, the sequence S ′ = i∈I g i is a non-empty zero-sum subsequence of S such that |S ′ | = |I| ≡ b (mod p) for all b ∈ A, which is the desired result.

We can now prove Theorem 2.1 and Corollary 2.

| ≤ k = i -|S ′ |,
that is, T is also a normal sequence. Therefore, the induction hypothesis applied to T implies that S is of the form S = 0 k U. One has |U| = D(G) + ℓ -1, where 1 ≤ ℓ = ik ≤ i -1, and since S is a normal sequence, every non-empty zero-sum subsequence U ′ | U has to satisfy |U ′ | ≤ ℓ = ik, that is, U is also a normal sequence. Finally, the induction hypothesis applied to U implies that S is of the form S = 0 k 0 i-k V = 0 i V , where V is a zero-sumfree sequence over G, which completes the proof. which contradicts the assumption made on S. Now, we can prove the second part of the assertion, by induction on k ∈ 1, i . If k = 1, then since |S| ≥ D(G), S has to contain a zero-sum subsequence S 1 such that p ∤ |S 1 |, and we are done. Now, let k ∈ 2, i , and let us assume the assertion is true for k -1, that is, S contains at least k -1 non-empty zero-sum subsequences S 1 , . . . , S k-1 with pairwise distinct lengths. By hypothesis, one has p ∤ |S j | for all j ∈ 1, k -1 , and we can write |S j | = q j p + r j , where q j ≥ 0 and r j ∈ 1, p -1 . Now, let A be a (i -1)-subset satisfying {r 1 , . . . , r k-1 } ⊆ A ⊆ 1, p -1 . Then, by Theorem 2.3, we obtain the existence of a non-empty zero-sum subsequence

S k | S such that |S k | ≡ b (mod p) for all b ∈ A. In particular, |S k | = |S j | for all j ∈ 1, k -1 ,
and consequently, S contains at least k non-empty zero-sum subsequences S 1 , . . . , S k with pairwise distinct lengths, and the proof is complete.

The special case of elementary p-groups

In this section, we propose an alternative proof of Theorem 2.3, in the special case of elementary p-groups, which uses an algebraic tool introduced by Alon and called the Combinatorial Nullstellensatz (see [START_REF] Alon | Combinatorial Nullstellensatz[END_REF] for a survey on this method). This polynomial method uses the fact that a non-zero multivariate polynomial over a field cannot vanish on 'large' Cartesian products so as to derive a variety of results in combinatorics, additive number theory and graph theory. This method relies on the following theorem. -Let F be a field and let f be a polynomial in F[x 1 , . . . , x n ] of total degree deg(f ), admitting a monomial of the following form:

x α 1 1 x α 2 2 • • • x αn n of degree n i=1 α i = deg(f ).
Then, for any choice of n subsets S 1 , . . . , S n ⊆ F such that |S i | > α i for all i ∈ 1, n , there exists an element (s 1 , . . . ,

s n ) ∈ S 1 × • • • × S n such that one has f (s 1 , . . . , s n ) = 0.
Proof of Theorem 2.3 in the special case of elementary p-groups.

Let p be a prime, and let G ≃ C r p be an elementary p-group of rank r. Let also (e 1 , . . . , e r ) be a basis of G, and let S = g 1 • . . . • g m be a sequence over G of length m = D(G) + k -1, where k ∈ 1, p . The elements of S can be written in the following way:

g 1 = a 1,1 e 1 + . . . + a 1,r e r , . . . . . . . . . g m = a m,1 e 1 + . . . + a m,r e r .
Let A be a (k -1)-subset of 1, p -1 , and let also P ∈ F p [x 1 , . . . , x m ] be the following polynomial over the finite field F p of order p,

P (x 1 , . . . , x m ) = r h=1 p-1 j=1 m i=1 a i,h x p-1 i -j j∈A m i=1 x p-1 i -j -δ m i=1 x p-1 i -1 ,
where δ ∈ F p is chosen such that P (0, . . . , 0) = 0. In particular, since no element of A is a multiple of p, one has δ = 0. Moreover, the total degree of

r h=1 p-1 j=1 m i=1 a i,h x p-1 i -j j∈A m i=1 x p-1 i -j being (r(p -1) + (k -1)) (p -1) = (D(G) + k -2) (p -1) < m(p -1)
, we deduce that deg(P ) = m(p-1). Now, since the coefficient of m i=1 x p-1 i is -δ = 0, Theorem 4.1 implies that there exists a non-zero element x = (x 1 , . . . , x m ) ∈ F m p such that P (x 1 , . . . , x m ) = 0. Consequently, setting I = {i ∈ 1, m | x i = 0}, we obtain that S ′ = i∈I g i is a nonempty zero-sum subsequence of S satisfying |S ′ | = |I| ≡ b (mod p) for all b ∈ A, which completes the proof.

The case of finite Abelian groups of rank two

In this section, we prove a result extending Statement (ii) of Theorem 1.1 to every finite Abelian group of rank two. The proof of this theorem relies on the following result of Schmid [START_REF] Schmid | Inverse zero-sum problems II[END_REF], which gives a structural characterization of minimal zero-sum sequences of length D(G) = m + mn -1 over the group G ≃ C m ⊕ C mn , where m, n ∈ N * and m ≥ 2, under the hypothesis that m satisfies Property B. x i ≡ -1 (mod ord(e j )).

(ii) S = g sm-1

1 (n+1-s)m i=1 (-x i g 1 + g 2 ) where s ∈ 1, n , {g 1 , g 2 } is a generating set of
G with ord(g 2 ) = mn and such that s = 1 or mg 1 = mg 2 , and

x i ∈ N with (n+1-s)m i=1 x i = m -1.
In addition, if m satisfies Property B, then all minimal zero-sum sequences of maximal length over G are of this form.

Moreover, Gao and Zhuang proved a useful structural result on normal sequences (see [START_REF] Gao | Zhuang Sequences not containing long zero-sum subsequences[END_REF]Theorem 1.2]). In this section, we will use the following corollary of this theorem.

Theorem 5.2. -Let G be a finite Abelian group, and let S be a normal sequence over G. Then S is of the form S = 0 k T U, where T is a zero-sumfree sequence with |T | = D(G)-1, and U is a sequence over G such that Supp(U) ⊆ Supp(T ).

Using the above two results, we can now prove the following theorem.

S = g ℓ 1 1 ℓ 2 i=1 (-x i g 1 + g 2 ) ,
where {g 1 , g 2 } is a generating set of G with ord(g 2 ) = mn, and ℓ 1 , ℓ 2 ∈ N are such that ℓ 1 + ℓ 2 = |S|. In particular, one has mg 1 ∈ mg 2 , and ag 1 ∈ g 2 if and only if m | a. Since |S| ≥ D(G), it has to contain a non-empty zero-sum subsequence S ′ . Now, let us write S ′ = V W where V | g ℓ 1 1 and W = i∈I (-x i g 1 + g 2 ), for some I ⊆ 1, ℓ 2 . We obtain where i ∈ 1, m -1 . By Theorem 5.2, S is of the form S = 0 k T U, where T is a zero-sumfree sequence with |T | = D(G) -1, and U is a sequence over G such that Supp(U) ⊆ Supp(T ). Since m > i, S does not contain any zero-sum subsequence S ′ with |S ′ | ≥ m. So, it follows from Theorem 5.3 that U is empty, which implies k = i and completes the proof.

σ(S ′ ) = |V | g 1 - i∈I x i g 1 + |W | g 2 = 0. Since σ(S ′ ) = 0 ∈ g 2 ,

Two concluding remarks

In this section, we would like to present two conjectures suggested by the results proved in this paper. The first one may be interpreted as a more general version of Theorem 2.3, and would imply Conjecture 1 in the same way as Theorem 2.3 implies Theorem 2.1. Let G be a finite Abelian group of exponent m. By s mN (G) we denote the smallest t ∈ N * such that every sequence S over G with |S| ≥ t contains a non-empty zerosum subsequence S ′ with |S ′ | ≡ 0 (mod m). The exact value of the invariant s mN (G) is currently known for finite Abelian groups of rank r ≤ 2, and finite Abelian p-groups only (see [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups : a survey[END_REF]Theorem 6.7]). For instance, Conjecture 2 implies that for all integers n, r ≥ 1, one has s nN (C r n ) = (r + 1)(n -1) + 1. This conjecture would also help to tackle the inverse problem associated to s mN (G), by giving an account of the variety of zero-sum subsequences contained in a long sequence without any non-empty zero-sum subsequence of length congruent to 0 modulo m. Finally, any progress on this conjecture would provide a new insight into the structure of sequences over C n without any zero-sum subsequence of length n (see [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups : a survey[END_REF]Theorem 7.5 and Conjecture 7.6], as well as [START_REF] Savchev | Long n-zero-free sequences in finite cyclic groups[END_REF]).

Conjecture 2. -Let G ≃ C n 1 ⊕ • • • ⊕ C nr , with 1 < n 1 | . . . | n r ∈ N,
Let G be a finite Abelian group of exponent m, and let ℓ ≥ 1 be an integer. By η ℓm (G) we denote the smallest t ∈ N * such that every sequence S over G with |S| ≥ t contains a non-empty zero-sum subsequence S ′ with |S ′ | ≤ ℓm. It may be observed that, for every ℓ ≥ ⌈D(G)/m⌉, one has the equality η ℓm (G) = D(G). Now, our second conjecture is the following. Let G be a finite Abelian group of order n and exponent m. For instance, if ℓ = 1 in Conjecture 3, one obtains a generalization of Conjecture 6.5 in [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups : a survey[END_REF], due to Gao. If ℓ = n/m, then, since it is known that D(G) ≤ n = ℓm, we obtain η ℓm (G) = D(G). Therefore, Conjecture 3 implies that every sequence S over G such that |S| = D(G) + i -1, where i ∈ 1, n , has to contain a zero-sum subsequence S ′ with i ≤ |S ′ | ≤ n, which can be seen as a generalization of Gao's theorem (see [START_REF] Gao | A combinatorial problem on finite abelian groups[END_REF]Theorem 1]). Finally, if ℓ = ⌈D(G)/m⌉ and i = ℓm, then Conjecture 3 implies that every sequence S over G with |S| = D(G) + ℓm -1 has to contain a zero-sum subsequence S ′ with |S ′ | = ℓm, which would provide an answer to a problem of Gao (see [START_REF] Gao | On zero-sum subsequences of restricted size II[END_REF]Section 3] and [START_REF] Gao | Geroldinger Zero-sum problems in finite abelian groups : a survey[END_REF]Theorem 6.12]).

Note added in proof

Shortly after this paper was accepted for publication, Reiher [START_REF] Reiher | A proof of the theorem according to which every prime number possesses Property B[END_REF] proved that Property B holds for all primes. It follows from the results of [START_REF] Gao | Grynkiewicz Inverse zero-sum problems III[END_REF] that all integers n ≥ 2 satisfy Property B. Thus, our Theorem 2.5 holds unconditionally, which gives a positive answer to Conjecture 1 for all Abelian groups of rank two. Further progress has since been made on this conjecture [START_REF] Guan | Normal sequences over finite abelian groups[END_REF].

Theorem 1 . 1 .

 11 -Conjecture 1 holds whenever:

Corollary 2 . 4 .

 24 -Let G be a finite Abelian p-group, and let S be a sequence over G with |S| = D(G) + i -1, where i ≥ 1. The following two statements hold.

Theorem 2 . 5 .

 25 -Let G ≃ C m ⊕ C mn , where m, n ∈ N * and m ≥ 2. Let also S be a normal sequence over G with |S| = D(G) + i -1, where i ∈ 1, m -1 . If m satisfies Property B, then S is of the form S = 0 i T , where T is a zero-sumfree sequence.

  r e r for all i ∈ 1, m , and S j = {0} for every j ∈ 1, r , one readily obtains the converse inequality D(G) ≤ D * (G) by Theorem 3.1, thus D(G) = D * (G) for every finite Abelian p-group.

  be a finite Abelian p-group. Let (e 1 , . . . , e r ) be a basis of G, where ord(e i ) = p d i for all i ∈ 1, r , and let S = g 1 • . . . • g m be a sequence over G of length m = D(G) + i -1, where i ∈ 1, p . The elements of S can be written in the following way:g 1 = a 1,1 e 1 + . . . + a 1,r e r , . . . . . . . . . g m = a m,1 e 1 + . . . + a m,r e r . Now, let us set n = r + 1 and d n = 1.

4 .

 4 Proof of Theorem 2.1. -We prove this theorem by induction on i ∈ 1, p -1 . If i = 1, the desired result is straightforward. Now, let i ≥ 2, and let us assume that the assertion of Theorem 2.1 is true for every 1 ≤ k ≤ i -1. Let S be a normal sequence over G with |S| = D(G) + i -1, and let A be a (i -1)-subset satisfying {i} ⊆ A ⊆ 1, p -1 . Then, Theorem 2.3 gives the existence of a non-empty zero-sum subsequence S ′ | S such that |S ′ | ≡ i (mod p). In particular, one has |S ′ | = i, and since S is a normal sequence, we must have |S ′ | ≤ i -1. Now, let T | S be the sequence such that S = T S ′ . Then |T | = D(G)+k -1, where 1 ≤ k = i-|S ′ | ≤ i-1. Moreover, since S is a normal sequence, every non-empty zero-sum subsequence T ′ | T has to satisfy |T ′

Proof of Corollary 2 . 4 .

 24 -(i) Let S be a sequence over G with |S| = D(G) + i -1, where i ≥ 2, such that S contains a zero-sum subsequence S ′ with p ∤ |S ′ |. Then, one has |S ′ | = qp + r for some integers q ≥ 0 and r ∈ 1, p -1 . Now, let T be any subsequence of S such that |T | = D(G) + 2 -1. Specifying A = {r} in Theorem 2.3, we obtain the existence of a non-empty zero-sum subsequence T ′ | T such that |T ′ | ≡ r (mod p). In particular, one obtains |T ′ | ≡ |S ′ | (mod p), which implies |T ′ | = |S ′ |. Since T ′ is also a zero-sum subsequence of S, the desired result is proved. (ii) Let S be a sequence over G with |S| = D(G) + i -1, where i ≥ 1, such that S contains no non-empty zero-sum subsequence S ′ with p | |S ′ |. If one had i ≥ p, we would obtain, specifying A = 1, p -1 in Theorem 2.3, that every subsequence T | S with |T | = D(G) + p -1 contains a non-empty zero-sum subsequence T ′ such that p | |T ′ |,

Theorem 4 . 1 (

 41 Combinatorial Nullstellensatz).

Theorem 5 . 1 .

 51 -Let G ≃ C m ⊕ C mn , where m, n ∈ N * and m ≥ 2, be a finite Abelian group of rank two. The following sequences are minimal zero-sum sequences of maximal length. (i) S = e ord(e j )-1 j ord(e k ) i=1 (-x i e j + e k ) where (e 1 , e 2 ) is a basis of G with ord(e 2 ) = mn, {j, k} = {1, 2}, and x i ∈ N with ord(e k ) i=1

Theorem 5 . 3 .

 53 -Let G ≃ C m ⊕ C mn , where m, n ∈ N * and m ≥ 2, be a finite Abelian group of rank two. Let T be a zero-sumfree sequence over G with |T | = D(G) -1, and let U be a non-empty sequence over G such that Supp(U) ⊆ Supp(T ). If m satisfies Property B, then every non-empty zero-sum subsequence S ′ of S = T U has length |S ′ | ≥ m. Proof of Theorem 5.3. -Let G and S = T U be as in the statement of the theorem. By Theorem 5.1, and since T ′ = T (-σ(T )) is a minimal zero-sum sequence over G, S = T U can be written in the following fashion

  we have m | |V | -i∈I x i which implies that, for some a ∈ N, one has |V | -i∈I x i = amg 2 . Thus, σ(S ′ ) = (am + |W |) g 2 = 0, which gives m | ord(g 2 ) | am + |W |. Consequently, m | |W |. If |W | ≥ m, then we are done. Otherwise, one has |W | = |I| = 0, and σ(S ′ ) = |V | g 1 = 0 ∈ g 2 . Thus, m | |V | = |S ′ |, and since S ′ is a non-empty zero-sum subsequence of S, we obtain |S ′ | ≥ m, which is the desired result. Theorem 2.5 is now an easy corollary of Theorem 5.3. Proof of Theorem 2.5. -Let G ≃ C m ⊕ C mn , where m, n ∈ N * and m ≥ 2, be a finite Abelian group of rank two. Let also S be a normal sequence over G with |S| = D(G)+i-1,

  be a finite Abelian group, and let S be a sequence over G with |S| = D(G) + i -1, where i ∈ 1, n 1 . Let also A be any (i -1)-subset of 1, n 1 -1 . Then S contains a non-empty zero-sum subsequence S ′ such that |S ′ | ≡ b (mod n 1 ), for all b ∈ A.

Conjecture 3 .

 3 -Let G be a finite Abelian group of exponent m, and let S be a sequence over G with |S| = η ℓm (G) + i -1, where i ∈ 1, ℓm . Then S contains a zero-sum subsequence S ′ of length i ≤ |S ′ | ≤ ℓm.

  Firstly, Theorem 2.1 improves Theorem 1.2, by showing that the assumption i ≤ 6 is unnecessary for finite Abelian p-groups. Secondly, it improves Statement (iii) of Theorem 1.1, by settling Conjecture 1 for every elementary p-group. More generally, we can derive immediately from Theorem 2.1 the following corollary on Gao's conjecture. Corollary 2.2. -Conjecture 1 holds for all groups of the form G ≃ C p ⊕ H, where H is any finite Abelian p-group.

Acknowledgments

This work was supported by a postdoctoral grant awarded by the École polytechnique, Paris. I am also grateful to Melvyn Nathanson and to the City University of New York's Graduate Center for all their hospitality and for providing an excellent atmosphere for research.


Wolfgang Horn
email: wolfgang.horn@bam.de

Oliver Jann

Elevtheria Juritsch

Olivier Ramalho

Pawel Wargocki

Henrik Knudsen

Birgit Muller

Birgit Müller

Detecting odorous compounds emitted from building and consumer products within the European Project SysPAQ

Keywords: Olfactory detection port, VOC analysis, Thermal extraction, Building material, Consumer products

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In 2006 the European research project Innovative Sensor System for Measuring Perceived Air Quality and Brand Specific Odours (SysPAQ / see www.syspaq.eu) started with 10 partners involved. The main goal of this project is to develop an innovative system to measure indoor air quality as it is perceived by humans to be used as an indicator and a control device for the indoor air quality. Within this project many different building and consumer products were tested. Some of these products were several years old. The odorous Evaluation of these products has been done so far. Here the emission tests with an olfactory detection port are described.

Methods

The emission tests were done with a thermal extractor (Gerstel GmbH). At 30°C, an airflow of 100 ml/min and a sample volume of 1 litre the emissions of small sample pieces about 20 mg of 5 carpets, 1 linoleum, 2 newspapers, 2 white t-shirt (clean, used), 5 filter bags (clean, used) were sampled on Tenax sampling tubes. The analysis was performed with Tenax thermal desorption, gas chromatography and the eluate is splitted 1:1 into mass spectrometry detection and odour detection port (ODP). At the ODP test persons smell the eluate of the column and press a button in case of smelling anything during GC run. Usually 5 different samples on Tenax were prepared of each material and 5 persons smell the eluate. If 3 or more persons have an impression of odour at the same retention time the compound is considered to be an odorous compound.

Results

VOCs emit from all materials and products. Odorous compounds can be detected from carpets, linoleum newspaper and t-shirt. From the filter bags no odorous compound can be detected. Most odorous compounds emit from the linoleum. In Table 1 the identified compounds from all products are listed which are suspected to be odorants.

Conclusions

With the technique described here the identification of odorous compounds is often possible. Unfortunately for many odorous signals from the persons that smell no VOC can be identified at that retention time. With the detected odorous compounds an odour space will be defined to test the sensory system which will be developed in this study.

Indoor Air 2008, 17-22 August 2008, Copenhagen, Denmark -Paper ID: 230

Table 1 .

 1 List of identified odorous compounds.

	Group	Compound
	aldehydes	benzaldehyde, hexanal, heptanal, octanal, nonanal, octenal, nonenal, decenal
	ketones	heptanone, octanone, nonanone, decanone, MIBK
	alcohols	heptanol, octanol, 2-ethylhexanol, propanediol, phenol
	carbonic acids	acetic acid, propanoic acid, butanoic acid, pentanoic acid, hexanoic acid, hepatanoic acid, octanoic acid, nonanoic acid
	other	longifolene, camphor, benzothiazole, phenylcyclohexene


Olivier Ramalho
email: olivier.ramalho@cstb.fr

Stéphane Moularat

François Loubet

Joachim Goschnick

Daniel Haeringer

Birgit Muller

Birgit Müller

Discrimination of gas-phase emissions from building material samples by combining two gas-sensor arrays

Keywords: Gas sensor array, Building material, Discrimination, Feature extraction

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

As part of the European project SYSPAQ, aiming at the conception of sensor systems to measure perceived air quality, experiments have been performed to assess the discrimination power of two gas-sensor systems as regards typical building material emissions. The first is based on individual metal oxide sensors and the second more compact constituted by a segmented sputtered layer of Pt-doped tin oxide differentiated by a temperature gradient and a thickness variation of a gas permeable SiO 2 coating. The 2 systems are intended to be integrated in a larger system in the framework of the project. The objective of the preliminary experiments presented here is to check the discrimination performance of the merged system to emissions of 20 building materials.

Methods

Experiments with each system were performed separately with different sample conditioning. A pulse from the static headspace of samples mildly heated at 35 °C was introduced in the first system, whereas the second system was exposed continuously to a dynamic headspace of the samples. Each sample provided a significant change in the sensor signals based on conductivity variations. A method has been developped to select optimal signal features in order to increase discrimination power. Three extracted features were selected : t max , the time needed to reach maximum signal change, ΔR/R and dR/dt, respectively relative change in resistance and resistance change rate chosen at a time corresponding to the maximum variance between sensors. This pre-coonditioning resulted in the determination of a single pattern associated to each sample. Multivariate statistical analysis such as principal component analysis with a subsequent hierarchical classification resulted in several discriminated groups of samples.

Results

The best discrimination level (9 classes) was achieved by combining normalized ΔR/R values of the two gas-sensors array. This selection was performed by first chosing the partition with the maximum number of classes from the three best automatically selected. Finally, the best discrimination level was the partition with the maximum value in the inter-class on max(intraclass) inertia ratio.

Conclusion

A coherent statistical discrimination level was achieved. However, in order to relate it with human perception of air quality, the perceivable olfactory differences between samples need to be assessed. This will be achieved in the next step of the SYSPAQ project.

Indoor Air 2008, 17-22 August 2008, Copenhagen, Denmark -Paper ID: 595

Figure 1 .

 1 Figure 1. PCA plot of the 20 building materials.


Olivier Ramalho

Stéphane Moularat

Wolfgang Horn

Henrik Knudsen

Pawel Wargocki

Birgit Muller

Occurrence of odorous compounds in the emission of building materials measured by GC-olfactometry

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Occurrence of odorous compounds in the emission of building materials measured by GC-olfactometry

O. Ramalho* CSTB -Scientific and Technical Center for Building, Health Division (France) olivier.ramalho@cstb.fr

In order to identify the most frequent odorous compounds associated with indoor background odour, the static headspace of 16 building material samples (painted and unpainted gypsum boards, wood boards, linoleum and vinyl flooring materials, carpets and sealants) has been characterized by gas chromatography -olfactometry.

Material samples were conditioned in 5 ml-vials and mildly heated at 50 °C during 24 hours. Sampling was realized through the septum cap by solid phase microextraction on a carboxen/pdms fiber during 1 hour. The fiber was then immediately injected on a GC equipped with a polar capillary column. The double detection provided consisted of a mass spectrometer and an olfactometry sniffing port. The perceived odour intensity of the sniffing subject was continuously monitored by a potentiometer associated with a data acquisition system (Fig. 1). Two male subjects performed the experiment. The number of identified odorous compounds vary from 0 (gypsum board) to 60 (linoleum). A total of 118 odorous compounds were detected. Carboxylic acids and aldehydes were the most frequent (Fig. 2). Specific compounds like 4phenylcyclohexene or 4-cyanocyclohexene were exclusively associated with carpet samples. Unsaturated carbonyl compounds and lactones (e.g. gammahexalactone) were mostly associated with linoleum samples. The identified olfactory part of building material emissions is dominated by oxygenated compounds. A large part of odorous events remains unidentified: 67% for a vinyl flooring material and up to 89% for an alkoxysilicone sealant.The occurrence of these major odorous compounds is related to oxidative ageing of the materials in contact with atmosphere, but also to biomass presence (wood products and linoleum). Further research is needed to relate these parameters with background odour in closed spaces.

Fig. 1 :

 1 Fig.1: Typical chromatogram (relative ion count) and monitored odour intensity for a linoleum sample.

Fig. 2 :

 2 Fig. 2: Frequency of detection of the 20 major odorous compounds in the 16 materials.