C Édric Bernardin

Patr Ícia

Gonc ¸alves

Milton Jara
email: mjara@impa.br

Marielle Simon
email: marielle.simon@inria.fr

INTERPOLATION PROCESS BETWEEN STANDARD DIFFUSION AND FRACTIONAL DIFFUSION

.

INTRODUCTION

Since the seminal work of Fermi-Pasta-Ulam (FPU) [START_REF] Fermi | Studies of nonlinear problems. I. Los Alamos[END_REF], heat conduction in chains of oscillators has attracted a lot of attention. In one-dimensional chains, superdiffusion of energy has been observed numerically in unpinned FPU chains, which corresponds to anomalous thermal conductivity. This anomalous thermal conductivity is generally attributed to a small scattering rate for low modes, which is due to momentum conservation. When the system has a pinning potential, destroying the conservation of momentum, normal diffusion of energy is expected. In [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF][START_REF] Basile | Thermal conductivity for a momentum conserving model[END_REF], it was proposed to perturb the Hamiltonian dynamics with stochastic interactions that conserve energy and momentum, like random exchanges of velocity between nearest neighbours. These models have the advantage to be studied rigorously keeping at the same time the features of deterministic models. For linear interactions, in dimension d 3, energy follows normal diffusion, while in dimensions d = 1, 2 energy is superdiffusive [START_REF] Basile | Thermal conductivity for a momentum conserving model[END_REF]. If a pinning potential is added to the dynamics, normal diffusivity can be proved regardless of the dimension.

In [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF] it was proved that in dimension d = 1, energy fluctuations follow the fractional heat equation ∂ t u = -c(-∆) 3/4 u, with c > 0. As mentioned above, in the presence of a pinning potential, energy fluctuations follow the usual heat equation ∂ t u = D ∆u, where D > 0 is the diffusion coefficient. Our goal is to provide a crossover between these two universality classes, aiming for a better understanding of the origin of the superdiffusivity of the energy in one-dimensional chains. In particular, we aim to clarify the role of the conservation of momentum.

The stochastic chains considered in [START_REF] Basile | Thermal conductivity for a momentum conserving model[END_REF] have three conserved quantities: the energy, the momentum and the stretch of the chain. Since we are interested in the role of the conservation of momentum, for simplicity we will consider a Hamiltonian lattice field model introduced by Bernardin and Stoltz [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF], which has only two conserved quantities, see Section 2.1, but which displays similar superdiffusion features. We call these conserved quantities energy and volume. In [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF] the authors add to the deterministic dynamics an energy and volume conservative Poissonian noise, which is discrete in nature. Here we consider instead a conservative Brownian noise, for a reason that will be explained ahead. In [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF] a similar result to [START_REF] Jara | Superdiffusion of energy in a chain of harmonic oscillators with noise[END_REF] has been obtained by different techniques for these models.

Let n ∈ N be a scaling parameter, which represents the inverse mesh of the stochastic chain. We add to the dynamics a second stochastic interaction that conserves only the energy and we scale down the strength of this second interaction by a n , with a > 0. We prove that energy fluctuations follow an evolution equation of the form

∂ t u = L a u,
where L a has the Fourier representation

L a (k) = - 4π 2 k 2 √ a + 2iπk , k ∈ R.
In particular, we note that L a → -c (-∆) 3/4 -∇(-∆) 1/4 as a → 0 and √ aL a → ∆ as a → ∞, providing in this way a crossover between anomalous and normal diffusion of energy in the model. Note as well that the interpolation between the fractional and normal Laplacians can be understood as an ultraviolet cut-off at modes of order O(a): low modes behave diffusively, while high modes behave superdiffusively.

In [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF] another version of the model of Bernardin and Stoltz of [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF] was considered. An almost complete phase diagram was obtained, although the interpolating part of the diagram described here was missing there. The interested reader may verify that the methods presented in this article allow to complete the phase diagram in [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF] as well as to prove the results stated there to the model considered here.

Energy fluctuations. Let us describe in a more precise way the main result proved in [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF] for the model considered here. Let {ω n x (t)} x∈Z ∈ R Z be the infinite dimensional diffusion process defined in Section 2.1. The (formal) conserved quantities of the model are the energy ∑ x∈Z [ω n x (t)] 2 and the volume ∑ x∈Z ω n x (t). Let {µ β ; β > 0} be the family of Gibbs homogeneous product measures which are invariant by the dynamics. Under µ β , the random variables {ω x } x∈Z are independent centered Gaussian variables with variance β -1 . The probability measure on the space of trajectories which is induced by the initial law µ β and the Markov process {ω n x (t)} x∈Z is denoted by P β and its corresponding expectation by E β . Define the energy correlation function as

S n (t, x) = β 2 2 E β [ω n x (t)] 2 -β -1 [ω n 0 (0)] 2 -β -1 .
We prove here the following scaling limit for S n (t, x): for any test functions ϕ, ψ : R → R in the usual Schwartz space S (R),

lim n→∞ 1 n ∑ x,y∈Z S n (tn 3/2 , y -x)ϕ x n ψ y n = R 2 P t (v -s)ϕ(s)ψ(v) dsdv, (1.1)
where P t (•) has the Fourier representation

P t (k) = e -t L a (k) , k ∈ R.
In other words, S n (tn 3/2 , nx) converges, in a weak sense, to the fundamental solution of the evolution equation ∂ t u = L a u. The case a = 0 is the case considered in [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF] for the model with a Poissonian noise. That result is a simple consequence of a stronger scaling limit, which is the main result of this article. To state it properly let us define the energy fluctuation field as

E n t (ϕ) = 1 √ n ∑ x∈Z ω n x (t) 2 -β -1 ϕ x n (1.2)
for test functions ϕ : R → R in S (R). We will prove that this field converges in law to the Gaussian process which is the stationary solution of the equation

∂ t E t = L a E t + 2β -2 (-S a ) ∇W t , (1.3)
where W t is a space-time white noise, L a is the adjoint of L a in L 2 (R) and S a is its symmetric part given by S a = 1 2 (L a + L a ). This convergence implies the limit lim

n→∞ E β E n t (ϕ)E n 0 (ψ) = E β E t (ϕ)E 0 (ψ) ,
which is exactly the limit stated in (1. [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF].

We point out that with respect to [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF] and [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF], the model considered in this article has a Brownian noise instead of a Poissonian noise. At the level of the correlation function S n (t, x), the choice of a Poissonian or a Brownian noise does not make a sensitive difference. In particular, the method of proof in this article allows to prove (1.1) also for Poissonian noises chosen in a proper way. However, at the level of the Gaussian fluctuations, key tightness estimates do not hold for Poissonian noises due to rare events that may introduce huge discontinuities on the observables we are interested in. We believe that at the level of finite-dimensional distributions the process (1.3) still describes the scaling limit of energy fluctuations in the model with the Poissonian noise considered in [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF]. However, it is not clear whether the obstructions in order to prove tightness are technical or intrinsic to those kind of noises.

A sketch of the proof. Our proof of the convergence of the energy fluctuation field (1.2) follows the usual scheme of convergence in law of stochastic processes: we show tightness of the processes E n t in a suitable topology, then we prove that any limit point of the sequence {E n t } n∈N satisfies a weak formulation of the equation (1.3) and then we rely on a uniqueness result for the solutions of (1. [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF].

One technical difficulty comes from what is known in the literature by the replacement lemma: it is not very difficult to write down a martingale decomposition for E n t that should heuristically converge to the martingale problem associated to E t . But the drift term of this martingale decomposition involves the energy current ω n x (t)ω n x+1 (t). This current is not a function of the energy and therefore we say that the martingale problem for E n t is not closed. To overcome that, we need to replace the current ω n x (t)ω n x+1 (t) by a function of the energy. This is accomplished by studying the relation between the energy fluctuations and the fluctuations of the correlation field given by 1

n 3/4 ∑ x,y∈Z ω n x (t)ω n y (t) -δ x,y β -1 f x+y 2n , |y-x| √ n , (1.4)
on some regular two-dimensional test function f . Above, δ x,y is the usual indicator function that equals 1 if x = y and 0 otherwise. Note that, at least heuristically, the energy current is given by the correlation field evaluated at the diagonal y = x + 1. The introduction of this field is one of the main conceptual innovations in [START_REF] Bernardin | 3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise[END_REF]. This field can be interpreted as the tensor product of the volume fluctuation field with itself. It turns out that volume fluctuations have two characteristic time scales. First, the speed of sound associated to the volume is equal to 2, and therefore, volume fluctuations evolve in the hyperbolic time scale tn following a linear transport equation. If the volume fluctuation field is modified by a Galilean transformation that drives out the transport dynamics, then it evolves in a diffusive time scale tn 2 , following an equation of the form (1.3) with the operator L replaced by the usual Laplacian operator ∆. In the definition of the correlation field (1.4), we introduced two different spatial scales. This non-homogeneous spatial scaling allows to observe both natural time scales at once. In fact, the correlation field (1.4) has a scaling limit in the hyperbolic time scale tn given by the stationary solution of

dZ t = (-∂ x + ∂ 2 yy -a) Z t dt + dM t
, where M t is an infinite-dimensional martingale (see also Section 2.3 for more details). We point out that although we do not prove neither this result 1 nor anything related to it, this limiting equation was used as a guideline for the computations below. Since the energy fluctuations evolve in the superdiffusive time scale tn 3/2 , the correlation field acts as a fast variable for the evolution of the energy.

The structure of the paper is described as follows. Below we introduce the model with notations, and we state the main result of this work, namely Theorem 2.5. Section 3 is devoted to the decomposition of the energy field into a martingale problem, using both the energy field and the correlation field. In Sections 4 and 5, we prove, respectively, tightness of the processes and characterization of their limit points, for establishing the convergence. Appendix A collects some results on the Lévy operator L , while in Appendices B and C we gather all technical details used along the proof.

PRELIMINARIES

2.1. The model. In this section we define the BS model (as introduced in [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF]) with continuous noises. For that purpose we need to introduce two real parameters: λ > 0 and γ n > 0, the latter depending on a scale parameter n ∈ N. Let us consider a system of diffusions evolving on the state space Ω := R Z , in the time scale n 3/2 , and generated by the operator n 3/2 L n , where L n is decomposed as the sum L n = A + λ S 1 + γ n S 2 , where

A = ∑ x∈Z ω x+1 -ω x-1 ∂ ∂ ω x S 1 = ∑ x∈Z (X x • X x ), S 2 = ∑ x∈Z (Y x • Y x ),
and the family of operators {X x , Y x } x∈Z is given by

X x = (ω x+1 -ω x ) ∂ ∂ ω x-1 + (ω x -ω x-1 ) ∂ ∂ ω x+1 + (ω x-1 -ω x+1 ) ∂ ∂ ω x , Y x = ω x+1 ∂ ∂ ω x -ω x ∂ ∂ ω x+1 .
The generator A is the generator corresponding to the infinite system of coupled ODE's dω x (t) = (ω x+1 (t)ω x-1 (t)) dt, x ∈ Z. A simple change of variables [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF] shows that it is equivalent to the dynamics generated by an infinite system of coupled harmonic oscillators. With this change of variables, ω x represents either the momentum of a particle or the interdistance between two nearest neighbor particles. The diffusion operator X x is nothing but the generator of a Brownian motion on the circle

{(ω x-1 , ω x , ω x+1 ) ∈ R 3 ; ω 2 x-1 + ω 2 x + ω 2 x+1 = 1, ω x-1 + ω x + ω x+1 = 0} while Y x is the generator of a Brownian motion on the circle {(ω x , ω x+1 ) ∈ R 2 ; ω 2 x + ω 2 x+1 = 1}.
We call energy the formal quantity ∑ x [ω x ] 2 and volume the formal quantity ∑ x ω x . The Liouville operator A as well as the noise S 1 conserves both energy and volume, while the operator S 2 conserves only energy. We assume that the strength of the second noise scales as

γ n = a n (2.1)
for some a > 0. We emphasize that one could easily treat the general case γ n = a n b , b 0, as in [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF], using the same methods as in this paper, but we chose here to focus on the most interesting case b = 1 where the interpolation happens.

The Markov process generated by the accelerated operator n 3/2 L n is denoted by ω n (t) = {ω n x (t)} x∈Z . This diffusion has a family {µ β ; β > 0} of invariant measures given by the Gibbs homogeneous product measures

µ β (dω) = ∏ x∈Z β 2π exp -β ω 2 x 2 dω x .
Here β represents the inverse temperature, and we denote by ϕ β the average of ϕ : Ω → R with respect to µ β .

The law of the process {ω n x (t) ; t 0} x∈Z starting from the invariant measure µ β is denoted by P β , and the expectation with respect to P β is denoted by E β . Note that under µ β , the averaged energy per site equals ω 2

x β = β -1 , and the averaged volume per site equals ω x β = 0.

Fluctuation fields.

From now on, the Markov process {ω n x (t) ; t 0} x∈Z is considered starting from µ β . The energy fluctuation field is defined as the distribution-valued process E n t given by

E n t (ϕ) = 1 √ n ∑ x∈Z ω n x (t) 2 -β -1 ϕ x n (2.2)
for any ϕ : R → R in the usual Schwartz space S (R) of test functions. For fixed t and ϕ, the random variables E n t (ϕ) satisfy a central limit theorem: they converge to a centered normal random variable of variance 2β -2 ϕ 2 2 , where • 2 denotes the usual norm of the Hilbert space L 2 (R).

Our main goal is to obtain a convergence result for the S (R)-valued process {E n t ; t 0}. It turns out that the analysis of the correlation field

1 n 3/4 ∑ x,y∈Z ω n x (t)ω n y (t) -δ x,y β -1 f x+y 2n , |y-x| √ n (2.3)
will play a fundamental role on the derivation of the scaling limit of E n t . Recall that δ x,y is the indicator function that equals 1 if x = y and 0 otherwise, and f : R × R + → R is a smooth function. The non-isotropic scaling is crucial in order to see the scaling limit of E n t .

2.3. Generalized Ornstein-Uhlenbeck equation associated to a Lévy process. First of all, let us introduce some notations: for any complex number z ∈ C, we denote by √ z its principal square root, which has positive real part: if z = re iθ with r 0 and θ ∈ (-π, π], then its principal square root is √ z = √ re iθ /2 . Let also ψ : R → C be the Fourier transform of a function ψ ∈ L 1 (R), which is defined by

ψ(k) := R e -2iπuk ψ(u) du, k ∈ R. (2.4)
For any ϕ ∈ S (R), we define L ϕ via the action of the operator L on Schwartz spaces: precisely, the operator L acts on the Fourier transform of ϕ as:

L ϕ(k) = 1 2 √ 3λ (2iπk) 2 √ a + iπk ϕ(k), k ∈ R. (2.5)
This operator has nice properties, stated in the next proposition:

Proposition 2.1. The operator L is the generator of a Lévy process. It leaves the space S (R) invariant, and its Lévy-Khintchine representation is given by

(L ϕ)(u) = R ϕ(u -y) -ϕ(u) + yϕ (u) Π a (dy), (2.6)
where Π a is the measure on R defined by

Π a (dy) = - 4a 5/2 √ 6λ π e -2ay 3 16(ay) 5/2 + 1 2(ay) 3/2 + 1 (ay) 1/2 1 (0,+∞) (y).
(2.7)

Proof. For the sake of readability, we postpone this proof to Appendix A.1.

Let us give here an alternative definition of L ϕ, which will turn out to be more tractable in the forthcoming computations. We claim that L ϕ can equivalently be defined as follows: for any u ∈ R,

(L ϕ)(u) = -2∂ u f (u, 0), (2.8)
where f : R × R + → R is the function such that its Fourier transform with respect to its first variable:

F k (v) := R e -2iπuk f (u, v)du, k ∈ R, v 0,
is given by

F k (v) = - 1 4 √ 3λ (2iπk) ϕ(k) √ a + iπk exp - a + iπk 3λ v , v 0.
(2.9)

The function f defined in this way satisfies the integrability conditions

R×R + f 2 (u, v) dudv < ∞ and R×R + ∂ v f 2 (u, v) dudv < ∞. (2.10)
Moreover the function f is solution of the Laplace equation

6λ ∂ 2 vv f -∂ u f -2a f (u, v) = 0, for u ∈ R, v > 0, 12λ ∂ v f (u, 0) = ϕ (u), for u ∈ R. (2.11)
This last claim is proved in Appendix A.2.

Let L be the adjoint of L in L 2 (R) and S := 1 2 (L + L ) be its symmetric part. Let us fix a time horizon T > 0. We are going to explain the meaning of a stationary solution of the infinite dimensional Ornstein-Uhlenbeck equation driven by L , written as follows:

∂ t E t = L E t + 2β -2 (-S ) W t , (2.12)
where {W t ; t ∈ [0, T ]} is a S (R)-valued space-time white noise.

Definition 2.2. We say that an S (R)-valued process {E t ; t ∈ [0, T ]} is β -stationary if, for any t ∈ [0, T ], the S (R)-valued random variable E t is a white noise (in space) of variance 2β -2 , namely: for any ϕ ∈ S (R), the real-valued random variable E t (ϕ) has a normal distribution of mean zero and variance 2β -2 ϕ 2 2 .

Definition 2.3. We say that the S (R)-valued process {E t ; t ∈ [0, T ]} is a stationary solution of (2.12) if:

(1)

{E t ; t ∈ [0, T ]} is β -stationary;
(2) for any time differentiable function ϕ : [0, T ] × R → R, such that for each t ∈ [0, T ] both ϕ t and ∂ t ϕ t belong to S (R), the process

E t (ϕ t ) -E 0 (ϕ 0 ) - t 0 E s (∂ s + L )ϕ s ds
is a continuous martingale of quadratic variation

2β -2 t 0 R ϕ s (u)(-S ϕ s )(u) duds.
Thanks to the fact that L is the generator of a Lévy process, the same argument used in [7, Appendix B] can be worked out here to prove the uniqueness of such solutions:

Proposition 2.4 ([7]
). Two stationary solutions of (2.12) have the same distribution.

Let us denote by C ([0, T ], S (R)) the space of continuous functions from [0, T ] to S (R). Roughly speaking, the main result of this work states that the energy fluctuations described by E n t (defined in (2.2)) satisfy an approximate martingale problem, which, in the limit n → ∞ becomes the martingale characterization of the limiting process described in Definition 2.3. It can be precisely formulated as follows:

Theorem 2.5. The sequence of processes {E n t ; t ∈ [0, T ]} n∈N converges in law, as n → ∞, with respect to the weak topology of C ([0, T ], S (R)), to the stationary solution of the infinite-dimensional Ornstein-Uhlenbeck process given by (2.12).

The proof of Theorem 2.5 follows from two steps:

(1) We prove in Section 4 that the sequence {E n t ; t ∈ [0, T ]} n∈N is tight. (2) We characterize all its limit points in Section 5 by means of a martingale problem.

First, we need to do an investigation of the fluctuation field E n t , and of the discrete martingale problem that it satisfies.

MARTINGALE DECOMPOSITIONS

In this section we fix ϕ ∈ S (R). Let f : R × R + → R be as in Section 2.3. Let us introduce the time dependent bidimensional field, defined as

C n t ( f ) := C ( f )(ω n (t)) with C ( f )(ω) := 1 n ∑ x,y∈Z ω x ω y -δ x,y β -1 f n x,y ,
where, for any x, y ∈ Z,

f n x,y := f x+y 2n , |y-x| √ n . (3.1)
Note that, for any sufficiently regular square-integrable function f , since under µ β the variables {ω x } x∈N are independent and centered Gaussian, we have, by an application of the Cauchy-Schwarz inequality, that

E β C n t ( f ) 2 C(β ) n 2 ∑
x,y∈Z

( f n x,y ) 2 ---→ n→∞ 0. (3.2)
3.1. Martingale decomposition for the energy. We first need to define the two discrete operators ∇ n and ∆ n , acting on ϕ ∈ S (R) as follows: for any x ∈ Z let

∇ n ϕ( x n ) := n ϕ( x+1 n ) -ϕ( x n ) , ∆ n ϕ( x n ) := n 2 ϕ( x+1 n ) + ϕ( x-1 n ) -2ϕ( x n )
. From Dynkin's formula, see for example [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF], for any ϕ ∈ S (R), the process

M E t,n (ϕ) := E n t (ϕ) -E n 0 (ϕ) - t 0 n 3/2 L n (E n s (ϕ)) ds (3.3)
is a martingale. A straightforward computation shows that

n 3/2 L n (E n s (ϕ)) = -2 ∑ x∈Z ω n x (s)ω n x+1 (s)∇ n ϕ x n + R n s (ϕ) (3.4)
where

R n s (ϕ) = (2γ n + 4λ ) 1 n ∑ x∈Z [ω n x (s)] 2 -β -1 ∆ n ϕ x n + (2λ ) 1 n ∑ x∈Z [ω n x (s)] 2 -β -1 n 2 ϕ( x+2 n ) + ϕ( x-2 n ) -2ϕ( x n ) + (2λ ) 1 n ∑ x∈Z ω n x (s)ω n x+2 (s)∆ n ϕ x+1 n -(4λ ) 1 n ∑ x∈Z ω n x (s)ω n x+1 (s) ∆ n ϕ x n + ∆ n ϕ x+1 n .
The second term in the right hand side of (3.4), when integrated in time between 0 and t -namely t 0 R n s (ϕ)ds -is negligible in L 2 (P β ) as a consequence of the Cauchy-Schwarz inequality (recall that ω x ω x+2 ω y ω y+2 β = 0 for x = y). Analogously, the first term in the right hand side (3. [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF], integrated in time, can be replaced thanks to Cauchy-Schwarz inequality, up to a vanishing error in L 2 (P β ), by

-2 t 0 ∑ x∈Z ω n x (s)ω n x+1 (s)ϕ x n ds = - t 0 ∑ x∈Z ω n x (s)ω n x+1 (s)24λ ∂ v f x n , 0 ds
the last equality being a consequence of (2.11). Therefore, we have

E n t (ϕ) -E n 0 (ϕ) = - t 0 ∑ x∈Z ω n x (s)ω n x+1 (s)24λ ∂ v f x n , 0 ds + M E t,n (ϕ) + t 0 ε n (s)ds, (3.5)
where M E t,n (ϕ) is a martingale, whose quadratic variation will be computed in Section 3.4. Moreover, ε n (t) satisfies two estimates: first, for any t > 0 fixed,

lim n→∞ E β t 0 ε n (s)ds 2 = 0 (3.6)
and second,

lim n→∞ sup t∈[0,T ] E β ε n (t) 2 < +∞. (3.7)
3.2. Martingale decomposition for the correlation field. Now let us turn to the bidimensional field C n t ( f ). From Dynkin's formula, for any f : R 2 → R, the process

M C t,n ( f ) := C n t ( f ) -C n 0 ( f ) - t 0 n 3/2 L n (C n s ( f )) ds (3.8)
is a martingale. The computations of Appendix B allow us to write

n 3/2 L n (C ( f )) = -2 √ n ∑ x∈Z ω 2 x -β -1 ∂ u f x n , 0 + O 1 √ n + O 1 √ n (3.9) + ∑ x∈Z ω x ω x+1 24λ ∂ v f x n , 0 + O 1 n (3.10) + 4 √ n ∑ x∈Z ω x ω x+1 a f x n , 0 -ω x+1 ω x-1 λ ∂ 2 vv f x n , 0 + O 1 √ n , (3.11)
where O(ε n ) denotes a sequence of functions in Z bounded by cε n for some finite constant c that does not depend on n. Note that (3.10) contains the same term that we made appear above in (3. [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF].

Observe that w.r.t. the computations of Appendix B an extra term has been introduced (precisely in the first display (3.9)): this term is

2β -1 √ n ∑ x∈Z ∂ u f x n , 0 = -β -1 √ n ∑ x∈Z (L ϕ) x n
where the last equality follows from (2.8). We claim that this new quantity is at most of order n -1/2 . To justify this, recall that by Proposition 2.1 the function h = L ϕ is in the Schwartz space and that its integral equals R h(u)du = ĥ(0) = 0. Moreover we have

1 n ∑ x∈Z h x n - R h(u)du = ∑ x∈Z x+1 n x n h x n -h(u) du = ∑ x∈Z x+1 n x n h (u) x+1 n -u du 1 n ∑ x∈Z x+1 n x n |h (u)|du = 1 n R |h (u)|du = O( 1 n ).
Therefore ∑ x∈Z h x n = O(1) and the claim is proved. Let us go one step further, and replace the local function ω x-1 ω x+1 that appears in (3.11) with the local function ω x ω x+1 . This is the purpose of Lemma 3.1 below: from that result we can rewrite the time integral as

t 0 n 3/2 L n (C n s ( f )) ds = -2 √ n t 0 ∑ x∈Z [ω n x (s)] 2 -β -1 ∂ u f x n , 0 ds + 24λ t 0 ∑ x∈Z ω n x (s)ω n x+1 (s) ∂ v f x n , 0 ds + 4 √ n t 0 ∑ x∈Z ω n x (s)ω n x+1 (s) a f -λ ∂ 2 vv f x n , 0 ds + t 0 ε n (s)ds,
where ε n (t) satisfies the same estimates as ε n (t), namely (3.6) and (3.7).

Lemma 3.1. Let {ψ n (x)} x∈Z be a real-valued sequence such that

1 n ∑ x∈Z |ψ n (x)| 2 < +∞. (3.12)
Then,

lim n→∞ E β t 0 1 √ n ∑ x∈Z ψ n (x)(ω n x -ω n x-1 )(s)ω n x+1 (s) ds 2 = 0. (3.13)
Proof. To prove the lemma we use a general inequality for the variance of additive functionals of Markov processes: we have

E β t 0 1 √ n ∑ x∈Z ψ n (x)(ω n x -ω n x-1 )(s)ω n x+1 (s) ds 2 C(β ) t n 3/2 Ψ 2 [tn 3/2 ] -1 ,-1 (3.14)
where

Ψ(ω) := 1 √ n ∑ x∈Z ψ n (x)(ω x -ω x-1 )ω x+1 ,
and, for any z > 0,

Ψ 2 z,-1 := Ψ, z -λ S 1 -γ n S 2 -1 Ψ β = sup g 2 Ψ g β -z g 2 β -g (-λ S 1 -γ n S 2 )g β , (3.15)
where the supremum is restricted over functions g in the domain of S . In order to prove (3. [START_REF] Whitt | Proofs of the martingale FCLT[END_REF], we first apply Lemma 3.9 of [START_REF] Sethuraman | Central Limit Theorems for Additive Functionals of the Simple Exclusion Process[END_REF], with the operator -t -1 Id + n 3/2 L and we get:

E β t 0 Ψ(ω n (s)) ds 2 C(β )t Ψ, (t -1 -n 3/2 L ) -1 Ψ β = C(β )t n 3/2 Ψ, ([tn 3/2 ] -1 -L ) -1 Ψ β C(β )t n 3/2 Ψ, ([tn 3/2 ] -1 -S ) -1 Ψ β = C(β )t n 3/2 Ψ 2 [tn 3/2 ] -1 ,-1 .
We can forget about the positive operator (z-λ S 1 ), and bound the norm (3.15) as follows:

Ψ 2 z,-1 Ψ, -γ n S 2 -1 Ψ β . One can easily check that S 2 1 4 ω x-1 ω x+1 -1 6 ω x ω x+1 = (ω x -ω x-1
)ω x+1 , which implies that (-γ n S 2 ) -1 Ψ is explicit and given by 1 6 ω x ω x+1 -1 4 ω x-1 ω x+1 , so that, finally,

-γ n S 2 -1 Ψ(ω) = 1 γ n √ n ∑ x∈Z ψ n (x)
Ψ 2 z,-1 C(β ) γ n n ∑ x∈Z |ψ n (x)| 2 .
Recall γ n = a n , and then after replacing the previous bound in (3. [START_REF] Whitt | Proofs of the martingale FCLT[END_REF] we get

E β t 0 1 √ n ∑ x∈Z ψ n (x)(ω n x -ω n x-1 )(s)ω n x+1 (s) ds 2 C(β ) t n 3/2 n a 1 n ∑ x∈Z |ψ n (x)| 2 = O 1 √ n ,
which vanishes as n → ∞.

Sum of the two decompositions.

Combining the two decompositions (3.3) and (3. [START_REF] Jara | Quadratic Fluctuations of the Simple Exclusion Process[END_REF] we get

E n t (ϕ) -E n 0 (ϕ) = - t 0 2 √ n ∑ x∈Z [ω n x ] 2 (s) -β -1 ∂ u f x n , 0 ds (3.16) + t 0 4 √ n ∑ x∈Z ω n x (s)ω n x+1 (s) a f -λ ∂ 2 vv f x n , 0 ds (3.17) + M C t,n ( f ) -C n t ( f ) -C n 0 ( f ) + M E t,n (ϕ) + t 0 ε n (s)ds, (3.18)
where

ε n (t) = ε n (t) + ε (t). Note that t 0 2 √ n ∑ x∈Z [ω n x ] 2 (s) -β -1 ∂ u f x n , 0 ds = 2 t 0 E n s ∂ u f (•, 0) ds.
Since the terms in (3.17) and (3.18) will be proved to vanish, as n → ∞, this will permit to close the martingale equation in terms of the energy field. From (3.2), the term

C n t ( f ) - C n 0 ( f ) vanishes in L 2 (P β ).
Finally, the term (3.17), which is in the same form as (3.4) (but of smaller order, since it is divided by √ n), is treated by repeating the same procedure: let g : R × R + → R be solution of the equation

6λ ∂ 2 vv g -∂ u g -2ag (u, v) = 0, for u ∈ R, v > 0, 24λ ∂ v g(u, 0) = 4 a f -λ ∂ 2 vv f u, 0), for u ∈ R, (3.19)
where f is given in Section 2.3. The function g is defined by its Fourier transform w.r.t. the first variable as it has been done to define f . Then, using the same computations as before, but with ϕ (u) replaced by 2(a fλ ∂ 2 vv f )(u, 0), we get that

t 0 4 √ n ∑ x∈Z ω n x (s)ω n x+1 (s) a f -λ ∂ 2 vv f x n , 0 ds = t 0 24λ √ n ∑ x∈Z ω n x (s)ω n x+1 (s) ∂ v g x n , 0 ds = 1 √ n C n t (g) -C n 0 (g) -M C t,n (g) (3.20)
+ t 0 2 n ∑ x∈Z [ω n x (s)] 2 -β -1 ∂ u g x n , 0 ds (3.21) - t 0 4 n ∑ x∈Z ω n x (s)ω n x+1 (s) ag -λ ∂ 2 vv g x n , 0 ds + t 0 ε n (s)ds. (3.22)
Note that in (3.21) we introduced the extra term

2β -1 n ∑ x∈Z ∂ u g x n , 0
as we did above for f . The same argument works here: one can prove that this additional quantity is of order at most O( 1 n ) since R ∂ u g(u, 0)du = 0. From the Cauchy-Schwarz inequality, both terms (3.21) and (3.22) vanish in L 2 (P β ), as n → ∞, and give a contribution ε n (t) which also satisfies the same conditions as (3.6) and (3.7) (note that this is the same argument used in Section 3.1). Besides, from (3.2), C n t (g) -C n 0 (g) also vanishes in L 2 (P β ), as n → ∞. Summarizing, the approximate discrete martingale equation can be written as

E n t (ϕ) -E n 0 (ϕ) = -2 t 0 E n s ∂ u f (•, 0) ds + M E t,n (ϕ) + M C t,n ( f ) -1 √ n M C t,n (g) + t 0 ε n (s)ds, (3.23)
where ε n (t) satisfies (3.6) and (3.7). In the following paragraph, by computing quadratic variations we prove that the only martingale term that will give a non-zero contribution to the limit is the one coming from the correlation field, namely M C t,n ( f ).

Convergence of quadratic variations. We start by showing that the quadratic variations of the martingales M E

•,n (ϕ), M C •,n ( f ) and M C •,n (g) converge in mean, as n → ∞. Lemma 3.2. For any ϕ ∈ S (R) and t > 0,

lim n→∞ E β M E •,n (ϕ) t = 0.
Proof. We have

M E •,n (ϕ) t = n 3/2 n t 0 L n (F 2 )(ω n (s)) -2F(L n F)(ω n (s)) ds (3.24) = √ n t 0 ∑ z∈Z 2λ X z (F) 2 + 2γ n Y z (F) 2 (ω n (s)) ds
where F(ω) := ∑ x∈Z ω 2 x ϕ n x and ϕ n x := ϕ( x n ). Note that (3.24) can also be written as

√ n t 0 λ Q 1 (F, F) + γ n Q 2 (F, F) (ω n (s)),
where the bilinear operators Q i (i = 1, 2) are given by

Q i ( f , g) = S i ( f g) -f S i g -gS i f .
In some contexts, the bilinear form Q i is called the carré du champ. A long but simple computation (using Appendix B. [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF] gives that

M E •,n (ϕ) t = √ n t 0 4λ ∑ x∈Z ω n x (s)ω n x+1 (s)(ϕ n x+1 -ϕ n x ) + ω n x (s)ω n x-1 (s)(ϕ n x -ϕ n x-1 ) + ω n x-1 (s)ω n x+1 (s)(ϕ n x-1 -ϕ n x+1 ) 2 + 4γ n ∑ x∈Z ω n x (s)ω n x+1 (s)(ϕ n x+1 -ϕ n x ) 2 ds. (3.25)
Therefore, taking the expectation, since ϕ ∈ S (R) we get

E β M E •,n (ϕ) t tC(β )(λ + γ n ) 1 n 3/2 ∑ z∈Z ∇ n ϕ z n 2 = O 1 √ n ,
which proves the lemma.

Lemma 3.3. Let f : R × R + → R be as in Section 2.3. Then, for t > 0 lim n→∞ E β M C •,n ( f ) t = 2tβ -2 R×R + (8a f 2 + 24λ (∂ v f ) 2 )(u, v) dudv.
Moreover the term on the right hand side of last expression equals to

2tβ -2 R ϕ(u)(-S ϕ)(u) du.
Proof. As before, we have

M C •,n ( f ) t = n 3/2 n 2 t 0 ∑ z∈Z 2λ X z (F) 2 + 2γ n Y z (F) 2 (ω n (s)) ds, (3.26)
where F(ω) := ∑ x,y∈Z ω x ω y f n x,y with f n x,y defined in (3.1). Since the computations are a bit longer, we decompose them as follows: first, note that

(X z )(F) = 2ω z+1 ω z-1 (-f n z+1,z+1 + f n z-1,z-1 -f n z-1,z + f n z,z+1 ) (3.27)
+ 2ω z ω z-1 ( f n z,z -f n z-1,z-1 -f n z,z+1 + f n z-1,z+1 ) (3.28)
+ 2ω z ω z+1 (-f n z,z + f n z+1,z+1 -f n z-1,z+1 + f n z-1,z ) (3.29) + 2 ω 2 z ( f n z,z+1 -f n z-1,z ) + ω 2 z+1 ( f n z-1,z+1 -f n z,z+1 ) + ω 2 z-1 ( f n z-1,z -f n z-1,z+1 ) (3.30) + 2 ∑ y / ∈{z-1,z,z+1} ω y ω z ( f n z+1,y -f n z-1,y ) + ω z+1 ( f n z-1,y -f n z,y ) (3.31) + ω z-1 ( f n z,y -f n z+1,y
n 3/2 n 2 t 0 ∑ z∈Z 2λ X z (F) 2 (ω n (s)) ds (3.33)
to which (I) contributes as

λtC(β ) 1 n 3/2 ∑ z∈Z ∂ v f z n , 0 2 + 1 n 5/2 ∑ z∈Z ∂ u f z n , 0 2 + O 1 n 3/2 ,
therefore it vanishes as n → ∞. The second term (II) is the only contributor to the limit. By using a Taylor expansion (see also (B.5) below), one has

f n z+1,y -f n z-1,y = -2 √ n ∂ v f z+y 2n , |y-z| √ n + O 1 n , (3.34) f n z-1,y -f n z,y = 1 √ n ∂ v f z+y 2n , |y-z| √ n + O 1 n . (3.35)
Therefore, in the estimate of (3.33) the second term (II) will contribute as

2λt ω 2 0 ω 2 1 β ∑ z∈Z ∑ y / ∈{z-1,z,z+1} 24
n 3/2 ∂ v f z+y 2n , |y-z| √ n 2 + O 1 n , which converges, as n → ∞, to 48λtβ -2 R×R + ∂ v f 2 (u, v) dudv. (3.36)
Let us now take care of the second stochastic noise that appears with Y z . We have:

(Y z )(F) =2ω z ω z+1 ( f n z,z -f n z+1,z+1 ) -2(ω 2 z -ω 2 z+1 ) f n z,z+1 -2 ∑ y / ∈{z,z+1} ω y (ω z f n z+1,y -ω z+1 f n z,y ).
Recall that γ n = a n . One can check that

E β n 3/2 n 2 t 0 ∑ z∈Z 2γ n Y z (F) 2 (ω n (s)) ds (3.37)
can be rewritten by the translation invariance of µ β as 16at ω 2 0 ω 2 1 β 1

n 3/2 ∑ z∈Z ∑ y / ∈{z-1,z,z+1} f n y,z 2 + O 1 n ,
and it converges, as n → ∞, to

16atβ -2 R×R + f 2 (u, v) dudv. (3.38)
As a consequence of (3.36) and (3.38) , we have

E β M C •,n ( f ) t ---→ n→∞ 2tβ -2 R×R + (8a f 2 + 24λ (∂ v f ) 2 )(u, v) dudv.
An explicit resolution of (2.11) via Fourier transforms given in Appendix A.2 easily gives

R×R + (8a f 2 + 24λ (∂ v f ) 2 )(u, v) dudv = R ϕ(u)(-S ϕ)(u) du
which is enough to conclude.

Remark 3.4. We note that by, similar computations to the ones of the previous lemma, we can prove that

E β M C •,n (g) t ---→ n→∞ 2tβ -2 R×R + (ag 2 + 3λ (∂ v g) 2 )(u, v) dudv,
where g has been defined before as the solution to (3.19).

Lemma 3.5 (L 2 (P β ) convergence of quadratic variations). For ϕ ∈ S (R) and f : R × R + → R as in Section 2.3, we have

lim n→∞ E β M E •,n (ϕ) t -E β M E •,n (ϕ) t 2 = 0, (3.39) lim n→∞ E β M C •,n ( f ) t -E β M C •,n ( f ) t 2 = 0, (3.40)
Proof. The proof of this lemma is postponed to Appendix C.

Conclusion.

From Lemma 3.2 and the remark above, we know that, for each fixed t > 0, the martingales M E t,n (ϕ) and 1

√ n M C t,n (g) vanish, as n → ∞, in L 2 (P β ).
Therefore the non vanishing terms remaining in the right hand side of the decomposition (3.23) are

-2 t 0 E n s ∂ u f (•, 0) ds + M C t,n ( f ). (3.41)

TIGHTNESS

The tightness of the sequence {E n t ; t ∈ [0, T ]} n∈N in the space C ([0, T ], S (R)) is proved by standard arguments.

First, Mitoma's criterion [START_REF] Mitoma | [END_REF] reduces the proof of tightness of distribution-valued processes to the proof of tightness for real-valued processes. Indeed, it is enough to show tightness of the sequence {E n t (ϕ) ; t ∈ [0, T ]} for any ϕ ∈ S (R). According to (3.23), we are reduced to prove that the processes

{E n 0 (ϕ)} n∈N , t 0 E n s (∂ u f (•, 0)) ds ; t ∈ [0, T ]
n∈N are tight, where f : R × R + → R is solution to (2.11). We will also prove that the martingales

{M E t,n (ϕ) ; t ∈ [0, T ]} n∈N , {M C t,n ( f ) ; t ∈ [0, T ]} n∈N , 1 √ n M C t,n (
C ([0, T ], R) if lim n→∞ sup t∈[0,T ] E X 2 n (t) < +∞.
One can easily check from the Cauchy-Schwarz inequality that

E β E n s (∂ u f (•, 0)) 2 C(β ) n ∑ x∈Z
∂ u f ( x n , 0) 2 ---→ n→∞ C(β )t 2 R (∂ u f (u, 0)) 2 du,
and recall that ε n (t) satisfies (3.7). Therefore, the criterion of Proposition 4.1 holds for both processes, and tightness follows.

{M E t,n (ϕ) ; t ∈ [0, T ]}, {M C t,n ( f ) ; t ∈ [0, T ]}, 1 √ n M C t,n (g)
; t ∈ [0, T ] are continuous in time. In order to prove that the sequences of martingales written in (4. [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF] are convergent as n → ∞, we use the following criterion, adapted from [14, Theorem 2.1] to the case of continuous processes: Proposition 4.2. A sequence {M n t ; t ∈ [0, T ]} n∈N of square-integrable martingales converges in distribution with respect to the uniform topology of C ([0, T ]; R), as n → ∞, to a Brownian motion of variance σ 2 if for any t ∈ [0, T ], the quadratic variation M n t converges in distribution, as n → ∞, towards σ 2 t.

From Section 3.4 we conclude that the martingales

{M E t,n (ϕ) ; t ∈ [0, T ]} n∈N , 1 √ n M C t,n (g)
; t ∈ [0, T ] n∈N vanish in distribution, as n → ∞, and from Proposition 4.2 we conclude that the martingales

{M C t,n ( f ) ; t ∈ [0, T ]} n∈N converge in distribution as n → ∞ to a Brownian motion of variance 2tβ -2 R ϕ(u)(-S ϕ)(u) du.
From this, we conclude that all the martingales are tight.

CHARACTERIZATION OF LIMIT POINTS

From the previous section, we know that the sequence {E n t ; t ∈ [0, T ]} n∈N is tight. Let {E t ; t ∈ [0, T ]} be one limit point in C ([0, T ], S (R)). For simplicity, we still index the convergent subsequence by n.

We already know that {E n 0 (ϕ)} n∈N converges in distribution, as n → ∞, towards a centered Gaussian random variable of variance 2β -2 ϕ 2 L 2 (R) . For the integral term it is easy to see that the convergence in law

t 0 E n s (L ϕ) ds ---→ n→∞ t 0 E s (L ϕ) ds
holds. The convergence for the martingale term has already been proved in Section 4.3. Putting all these elements together, we conclude that, for any ϕ ∈ S (R), we have

E t (ϕ) = E 0 (ϕ) + t 0 E s (L ϕ) ds + M t (ϕ),
where M t (ϕ) is a Brownian motion of quadratic variation

2tβ -2 R ϕ(u)(-S ϕ)(u) du.
By Proposition 2.4, the distribution of {E t ; t ∈ [0, T ]} is uniquely determined. We conclude that the sequence {E n t ; t ∈ [0, T ]} n∈N has a unique limit point, and since it is tight, it converges to this limit point. This ends the proof of Theorem 2.5.

APPENDIX A. FOURIER TRANSFORMS AND L ÉVY-KHINTCHINE DECOMPOSITION

A.1. Lévy-Khintchine decomposition. Let us first prove that L lets S (R) invariant. Since the Fourier transform is a bijection from S (R) into itself, it is sufficient to prove that if ϕ ∈ S (R) then L ϕ ∈ S (R). Since a > 0, the function

θ : k ∈ R → (2iπk) 2 √ a + iπk ∈ C
is a smooth function and we have that for any p 0, there exist constants

C p , α p > 0 such that ∀ k ∈ R, |θ (p) (k)| C p (1 + |k|) α p . (A.1)
Therefore, we have that L ϕ ∈ S (R).

Let X be a random variable distributed according to the Gamma distribution Γ( 1 2 , 1). More precisely, its density f X with respect to the Lebesgue measure is given by

f X (x) := 1 (0,+∞) (x) e -x √ πx , x ∈ R,
and its characteristic function is

Φ X (t) = E[e itX ] = 1 √ 1 -it = lim ε→0 +∞ ε e -x √ πx e itx dx, t ∈ R. (A.2)
Lemma A.1. For any t ∈ R,

H(t) := t 2 √ 1 -it = +∞ 0 (e itx -1 -itx)Π(dx),
where Π(dx) := f X (x) dx.

Proof. Note that, for ε > 0, an integration by parts gives

+∞ ε f X (x)e itx dx = 1 -e itε it f X (ε) - 1 it +∞ ε (e itx -1) f X (x) dx --→ ε→0 - 1 it +∞ 0 (e itx -1) f X (x) dx,
the last convergence holds since f X (ε)

1 √ πε as ε → 0. Therefore, we have the following identity

-itΦ X (t) = +∞ 0 (e itx -1) f X (x) dx. (A.3)
A second integration by parts can now be done in the same way, and one can check that

+∞ ε (e itx -1) f X (x) dx = 1 -e itε it + ε f X (ε) - 1 it +∞ ε (e itx -1 -itx) f X (x) dx. Since f X (ε) = -e -ε √ πε (1 + 1 2ε
), by taking the limit as ε → 0 in the previous identity, using (A.3) and recalling (A.2), Lemma A.1 follows.

The function we are interested in is the one that appears in (2. [START_REF] Bernardin | Anomalous diffusion for a class of systems with two conserved quantities[END_REF], namely:

Ψ a (t) := 1 2 √ 3λ (2iπt) 2 √ a + iπt = - 2a 3/2 √ 3λ H - πt a , a > 0,
where H is given in Lemma A.1. From that lemma we get

Ψ a (t) = - 2a 3/2 √ 3λ +∞ 0 e -i πtx a -1 + iπtx a f X (x) dx = - 4a 5/2 √ 3λ +∞ 0
(e -2iπty -1 + 2iπty) f X (2ay) dy.

A simple computation gives

f X (x) = e -x √ πx 1 + 1 x + 3 4x 2 .
Therefore

Ψ a (t) = +∞ 0 (e -2iπty -1 + 2iπty) Π a (dy),
where Π a has been defined in (2.7). Proposition 2.1 easily follows.

A.2. Aternative definition: Fourier transformation and resolution. Recall that f : R × R + → R is such that its Fourier transform with respect to the first variable is given by (2.9). For any fixed k ∈ R, the function

F k (•) is solution to 6λ F k (v) -(2a + 2iπk)F k (v) = 0, v 0, 12λ F k (0) = 2iπk ϕ(k), (A.4)
If we assume (2.8), one can easily check that

L ϕ(k) = -4iπkF k (0) = 1 2 √ 3λ (2iπk) 2 √ a + iπk ϕ(k), k ∈ R,
and therefore it coincides with (2.5). Moreover, by inverting in Fourier space the system (A. [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF], one can easily recover the partial differential equation satisfied by f and given in (2.11). Finally, the integrability conditions (2.10) follow from the Parseval identity:

R×R + 8a f 2 + 24λ (∂ v f ) 2 (u, v) dudv = R×R + 8a|F k (v)| 2 + 24λ |F k (v)| 2 dkdv = R |2iπk| 2 2 √ 6λ a + |a + iπk| |a + iπk| | ϕ(k)| 2 dk = R ϕ(-k)(-S ϕ)(k) dk = R ϕ(u)(-S ϕ)(u)
A ( f g) = f A g + gA f .
The operators S 1 , S 2 are second-order differential operators. Therefore, the relation above does not hold. Recall that the bilinear operators Q i (i = 1, 2) are given by

Q i ( f , g) = S i ( f g) -f S i g -gS i f .
In our situation, these carrés des champs have simple expressions:

Q 1 ( f , g) = 2 ∑ x∈Z (X x f )(X x g), Q 2 ( f , g) = 2 ∑ x∈Z (Y x f )(Y x g).
We will only evaluate the carré du champ on pairs of functions of the form (ω x , ω y ). In the case of Q 1 , we have four cases. First, Q 1 (ω x , ω y ) = 0 if |y -x| 3. We have that

Q 1 (ω x-1 , ω x+1 ) = 2(X x ω x-1 )(X x ω x+1 ) = 2(ω x+1 -ω x )(ω x -ω x-1 ). Using the identity 2(a -b)(b -c) = (a -c) 2 -(a -b) 2 -(b -c) 2 we can rewrite Q 1 (ω x-1 , ω x+1 ) = (ω x+1 -ω x-1 ) 2 -(ω x+1 -ω x ) 2 -(ω x -ω x-1 ) 2 .
In a similar way,

Q 1 (ω x , ω x+1 ) = 2(ω x+1 -ω x ) 2 -(ω x+2 -ω x+1 ) 2 -(ω x+2 -ω x ) 2 -(ω x+1 -ω x-1 ) 2 -(ω x -ω x-1 ) 2 , Q 1 (ω x , ω x ) = 2(ω x+2 -ω x+1 ) 2 + 2(ω x+1 -ω x-1 ) 2 + 2(ω x-1 -ω x-2 ) 2 .
In the case of Q 2 we have three different cases:

Q 2 (ω x , ω y ) = 0, |y -x| 2, Q 2 (ω x , ω x+1 ) = -2ω x ω x+1 , Q 2 (ω x , ω x ) = 2ω 2 x-1 + 2ω 2 x+1 .
B.2. The generator applied to quadratic functions. As mentioned before, the correlation field plays a fundamental role in the derivation of energy fluctuations. In order to see this, we need to make a very detailed study of the action of the generator L n over functions of the form ∑

x,y∈Z ω x ω y q x,y , where q : Z 2 → R will be chosen within a few lines and is supposed to be symmetric: q x,y = q y,x . We have

L n (ω x ω y ) = ω x L n ω y + ω y L n ω x + λ Q 1 (ω x , ω y ) + γ n Q 2 (ω x , ω y ). (B.1)
Let us introduce some notation that will be useful later on. For u : Z → R we define ∇u, ∆u : Z → R as

∇u x = 1 2 u x+1 -u x-1 , ∆u x = 1 6 u x-2 + 2u x-1 -6u x + 2u x+1 + u x+2 .
One can check that

L n ω x = 2 ∇ω x + 6λ ∆ω x -2γ n ω x .
For q : Z 2 → R define Aq : Z 2 → R as Aq x,y = q x+1,yq x-1,y + q x,y+1q x,y-1 .

In other words, Performing an integration by parts and using (B. [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF] we have that

L n ∑ x,y∈Z ω x ω y q x,y = ∑ x,y∈Z ω x ω y -A + λ S -4γ n q x,y + ∑ x,y∈Z λ Q 1 (ω x , ω y ) + γ n Q 2 (ω x , ω y ) q x,y .
The second sum on the right hand side of the last identity is what we call the stochastic interaction term, since it only appears due to the stochastic nature of the dynamics. Although the first sum also depends on the stochastic noise, it can be constructed from deterministic dynamics as well.

The computations of Section B.1 show that ∑

x,y∈Z

Q 1 (ω x , ω y )q x,y = ∑ x∈Z 2(ω x+1 -ω x-1 ) 2 q x,x + q x-1,x+1 -q x-1,x -q x,x+1 + ∑ x∈Z 2(ω x+1 -ω x ) 2 q x-1,x-1 + 2q x,x+1 + q x+2,x+2 -∑ x∈Z 2(ω x+1 -ω x ) 2 q x-1,x + q x-1,x+1 + q x,x+2 + q x+1,x+2 , = ∑ x∈Z 2ω 2 x q x-2,x-2 + 2q x-1,x-1 + 2q x+1,x+1 + q x+2,x+2 -∑ x∈Z 4ω 2 x q x-1,x+1 + q x+1,x+2 + q x-2,x-1 -∑ x∈Z 4ω x ω x+1 q x-1,x-1 + 2q x,x+1 + q x+2,x+2 -q x-1,x -q x-1,x+1 -q x,x+2 -q x+1,x+2 -∑ x∈Z 4ω x+1 ω x-1 q x,x + q x-1,x+1 -q x-1,x -q x,x+1 , (B.2)
and we also have

∑ x,y∈Z Q 2 (ω x , ω y )q x,y = ∑ x∈Z 2(ω 2 x-1 + ω 2 x+1 )q x,x -4ω x ω x+1 q x,x+1 . (B .3)
Let us go on and consider now the particular choice q x,y := f n x,y given in (3.1) where f : R × R + → R is a smooth function with enough decay at infinity. The computations are pretty involved; we consider in this section only the linear part ∑ x,y∈Z ω x ω y -A + λ S -4γ n f n x,y .

To simplify the notation we define f (u, v) := f (u, -v) for v < 0. We call this definition symmetrization. Extending f in this way, the resulting function may be no longer differentiable at v = 0 (but it is smooth in u and has left and right derivatives in ν at 0). Moreover, with this extension, and recalling the definition (3.1) of f n x,y , we have:

f n x,y = f n y,x , for any x, y ∈ Z. (B.4)
We start by computing A f n x,y and S f n x,y . Consider (x, y) situated on the upper half-plane delimited by the diagonal {x = y}, namely: y x. Then, for any i ∈ Z and for any j 0, we have

f x+y 2n + i 2n , y-x √ n + j √ n -f x+y 2n , y-x √ n (B.5) = j √ n ∂ v f x+y 2n , y-x √ n + 1 n i 2 ∂ u + j 2 2 ∂ 2 vv f x+y 2n , y-x √ n + 1 n 3/2 i j 2 ∂ 2 uv + j 3 6 ∂ 3 vvv f x+y 2n , y-x √ n + O i, j 1 n 2 ,
where O i, j ( 1 n 2 ) represents a sequence of functions in Z 2 bounded by c(i, j) n 2 for some finite constant c(i, j) and for any n ∈ N. In the following, we denote O( 1 n 2 ) when the sequence of functions is bounded by c n 2 and c does not depend on any index. From now on we denote

∂ f n x,y = ∂ f x+y 2n , y-x √ n if y > x, ∂ f x n , 0 + if y = x,
where ∂ can be any differentiate operator involving the variable v. For x = y we have from (B.5) that A f n x,y = 2 n ∂ u f n x,y + O 1 n 2 . For x = y, the expression is different due to the symmetrization of f . We have that

A f n x,x = 2 n ∂ u f n x,x + O 1 n 3/2 .
S f n x,x+1 = 4 √ n ∂ v + 12 n ∂ 2 vv f n x,x + O 1 n 3/2 . For y = x we have S f n x,x = 16 √ n ∂ v + 12 n ∂ 2 vv f n x,x + O 1 n 3/2 .
Putting together all the expressions computed above, and recalling (2. [START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF], we see that .

We now deal with Q 1 (see (B.2)). First, we consider the term with ω 2

x , and we write the Taylor expansion at ( x n , 0) as

1 2 f n x-2,x-2 + f n x-1,x-1 + f n x+1,x+1 + 1 2 f n x+2,x+2 -f n x-1,x+1 -f n x+1,x+2 -f n x-2,x-1 = -4 √ n ∂ v -3 n ∂ 2 vv f n x,x + O 1 n 3/2
. Then, we have the term with ω x ω x+1 , and we write the Taylor expansions at ( x n , 0): In this section we prove Lemma 3.5. We start by showing the L 2 (P β ) convergence for M E •,n (ϕ) t , namely (3.39). Recall the explicit formula for the quadratic variation given in (3.25). By using the inequality (x + y) 2 2x 2 + 2y 2 several times, we split the four terms appearing in (3.25) and we control each one separately by using exactly the same approach. We only give the proof of the control for one of them. We start by computing the variance of √ n For the remaining term, by developing the square of the sum and using the fact that the variables {ω x } x∈Z have mean zero and are i.i.d. under µ β we bound it from above by

f n x-1,x-1 + 2 f n x,x+1 + f n x+2,x+2 -f n x-1,x -f n x-1,x+1 -f n x,x+2 -f n x+1,x+2 = -4 √ n ∂ v -

4. 3 .

 3 Convergence of martingales. By definition, and more precisely (3.3) and (3.8), for any n ∈ N and ϕ ∈ S (R), f as in Section 2.3 and g solution of (3.19), the martingales

  Aq x,y = 2 ∇q x ↑ ,y + 2 ∇q x,y ↑ , where the arrows indicate on which variable the ∇ operator acts. Define as well Sq : Z 2 → R as Sq x,y = 6 ∆q x ↑ ,y + 6 ∆q x,y ↑ .

x ω y 2 n+ O 1 n 3 / 2 .B. 3 . 1 n 3

 2132313 ω x ω y -A + λ S -4γ n f n x,y = ∑ x,y∈Z ω -∂ u + 6λ ∂ 2 vv -2a f n x,y + O 1The carré du champ revisited. In this section we perform the same computations for both carrés des champs. It is quite easy to see from (B.3) that∑x,y∈Zγ n Q 2 (ω x , ω y ) f n x,y = ∑ x∈Z ω x ω x+1 -4a n f n x,x + O

-ϕ n x ) 2 .x ω 2 x+1 -ω 2 0 ω 2 1 βω x ω x+1 ω y ω y+1 (ϕ n x+1 -ϕ n x )(ϕ n y+1 -ϕ n y ) 2 µ

 2212 ω n x (s)ω n x+1 (s)ω n y (s)ω n y+1 (s) (ϕ n x+1ϕ n x )(ϕ n y+1ϕ n y )ds. (C.1)Note that under the equilibrium probability measure µ β the expectation of [ω n x ω n x+1 ω n y ω n y+1 ](s) is non-zero only for diagonal terms y = x, so that the expectation of (C.[START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF] is equal to 4λt √ Define χ x,x+1 := ω 2 which are centered random variables. By stationarity and the Cauchy-Schwarz inequality, the variance of (C.1) is bounded by β (dω) (C.3) for some constant C > 0. First we look at the diagonal terms. Developing the square of the sum, since the variables χ x,x+1 and χ y,y+1 are correlated only if |y -x| 1, by the Cauchy-Schwarz inequality, the term (C.2) can be bounded from above by t 2 C(β )n ∑ x∈Z (ϕ n x+1ϕ n x ) 4 = O(n -2 ).

t 2 C

 2 n → ∞.

  g) ; t ∈ [0, T ] n∈N (4.[START_REF] Basile | Momentum conserving model with anomalous thermal conductivity in low dimensional systems[END_REF] are convergent and, in particular, they are tight, and finally that the process Tightness for {E n 0 (ϕ)} n∈N . As mentioned at the beginning of Section 2.2, {E n 0 (ϕ)} n∈N converges in distribution, as n → ∞, towards a centered normal random variable of variance 2β -2 ϕ 2 L 2 (R) , and in particular the sequence is tight. For these two integral terms we use the following tightness criterion:Proposition 4.1 ([7, Proposition 3.4]).A sequence of processes of the form t 0 X n (s) ds ; t ∈ [0, T ] n∈N is tight with respect to the uniform topology in

	t
	0	ε n (s)ds ; t ∈ [0, T ]	n∈N
	is tight.
	4.1. 4.2. Tightness for t 0 E n s (∂

u f (•, 0)) ds ; t ∈ [0, T ] n∈N and for t 0 ε n (s)ds ; t ∈ [0, T ] n∈N .

  du. The carré du champ. Let f , g : Ω → R be local smooth functions. Since the operator A is a first-order operator, we have the Leibniz rule

	APPENDIX B. ASIDE COMPUTATIONS
	B.1.

  Note that the term of order O( 1 n ) is the same in both expressions, the difference appears only at order O( 1 n 3/2

  ). Now let us compute S f x,y . The lack of regularity of f at v = 0 affects the computations if |x -y| 1. In particular, we can ensure that all the differences of the formf n x+k,y+f n x,yappear in such a way that x+k y+ and x y. With this precaution, we avoid to cross the axis {x = y} where derivatives can have jumps due to the irregularity of f . For |y -x| 2 For y = x + 1 we write the Taylor expansion centered at ( x n , 0) as follows

	we have
	S f n x,y = 12 n ∂ 2 vv f n x,y + O 1 n 2 .

  Finally, the term with ω x+1 ω x-1 gives the Taylor expansion centered at ( x n , 0) as:

	Therefore,
	∑ x,y∈Z	λ Q 1 (ω x , ω y ) f n x,y = ∑ x∈Z	ω 2 x	-16λ √ n ∂ v -12λ n ∂ 2 vv f n x,x + O 1 n 3/2
				+ ∑ x∈Z	ω x ω x+1	16λ √ n ∂ v + 16λ n ∂ 2 vv f n x,x + O 1 n 3/2
				+ ∑ x∈Z	ω x+1 ω x-1 -4λ n ∂ 2 vv f n x,x + O 1 n 3/2	.
	Putting every computation together, we obtain
	L n ∑	ω x ω y f n x,y = 2 n ∑	ω x ω y -∂ u + 6λ ∂ 2 vv -2a f n x,y + O 1 n
	x,y∈Z	x,y∈Z
			+ 4 √ n ∑	ω x ω x+1 6λ ∂ v -1 √ n (a -4λ ∂ 2 vv ) f n x,x + O 1 n
				x∈Z
			+ 4 n ∑	ω 2 x	-3λ ∂ 2 vv + a f n x,x + O 1 √ n
			x∈Z
			-4λ n ∑	ω x+1 ω x-1 ∂ 2 vv f n x,x + O 1 √ n	.
			x∈Z
	and after simplifications
	L n ∑	ω x ω y f n x,y = 2 n ∑	n
		x,y∈Z
			-2 n ∑	ω 2 x ∂ u f n x,x + O 1 √ n
					x∈Z
			+ 24λ √ n ∑	ω x ω x+1 ∂ v f n x,x + O 1 n
					x∈Z
			+ 4 n ∑	ω x ω x+1 (4λ ∂ 2 vv -a) f n x,x
					x∈Z
			-4 n ∑	ω x+1 ω x-1 λ ∂ 2 vv f n x,x + O 1 √ n	.
					x∈Z
	APPENDIX C. L 2 CONVERGENCE OF QUADRATIC VARIATIONS
					4 n ∂ 2 vv f n x,x + O 1 n 3/2 .
		f n x,x + f n x-1,x+1 -f n x-1,x -f n x,x+1 = 1 n ∂ 2 vv f n x,x + O 1 n 3/2 .

x =y

ω x ω y -∂ u + 6λ ∂ 2 vv -2a f n x,y + O 1

This result can be guessed by using the computations in Appendix B.2 but its rigorous proof is not trivial and would require a paper by itself. The interested reader is invited to consult[START_REF] Jara | Quadratic Fluctuations of the Simple Exclusion Process[END_REF] for a similar result in the context of the symmetric simple exclusion process.

ACKNOWLEDGEMENTS

This work benefited from the support of the project EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR) and of the PHC Pessoa Project 37854WM. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovative programme (grant agreement No 715734). C.B. thanks the French National Research Agency (ANR) for its support through the grant ANR-15-CE40-0020-01 (LSD). P.G. thanks FCT/Portugal for support through the project UID/MAT/04459/2013. M.J. thanks CNPq for its support through the grant 401628/2012-4 and FAPERJ for its support through the grant JCNE E17/2012. M.J. was partially supported by NWO Gravitation Grant 024.002.003-NETWORKS and by MathAmSud grant LSBS-2014.

M.S. thanks CAPES (Brazil) and IMPA (Instituto de Matematica Pura e Aplicada, Rio de Janeiro) for the post-doctoral fellowship, and also the Labex CEMPI (ANR-11-LABX-0007-01) for its partial support.

We let the reader work out the same argument in order to finish the proof of (3.39). Now we turn to M C •,n ( f ) t and we prove (3.40). Recall the explicit expression (3.26) in the proof of Lemma 3.3. We use again the inequality (x + y) 2 2x 2 + 2y 2 several times and we control each term separately by using exactly the same approach. We present the proof for the contribution of the term with X z but we note that for the term with Y z the estimates are analogous. Recall (3.27)-(3.32). We note that the most demanding terms are those coming from (3.31) and (3.32). To make the exposition as simple as possible, we look only at one of these terms, which is of the form

and can be written as

We sum and subtract the mean of ω n z (s) 2 to write last term as

Now we estimate the variance of each term separately. First, we note that the mean of (C. [START_REF] Bernardin | From normal diffusion to superdiffusion of energy in the evanescent flip noise limit[END_REF] is zero so that its variance is given by

To bound from above this last expression, we expand the squares and use the independence of the centered random variables {ω x } x∈Z . Therefore last expectation is bounded from above by the sum of two terms, according to z = z and z = z. The first one is

which, by (3.34), can be bounded from above by

Last expectation is bounded from above by

and vanishes as n → ∞. Now we compute the variance of (C.5) which, by developing the square in the sum, can be written as

First note that its mean is given by 8λt

and therefore, its variance can be bounded from above by

Now, the first expectation in the previous display can be bounded from above by


Reza Abiri

Griffin Heise

Fernando Schwartz

Xiaopeng Zhao

EEG-Based Control of a Unidimensional Computer Cursor Using Imagined Body Kinematics

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

Introduction: Brain computer interface (BCI) are systems that can extract information from brainwaves to control a computer or an external device. A computer cursor controlled by brain activity, one of the interesting neuro-prosthetic devices in BCI, was first offered to the patients with spinal cord injury using invasive measurements of brain activities. More recently, techniques based on noninvasive measurements such as Electroencephalography (EEG) have become popular. Here, we investigate the activation and control of a computer cursor in one dimension by engaging brain signals collected using EEG sensors and through a novel paradigm called "imagined body kinematics." In contrast to the earlier techniques using sensorimotor rhythms, the method here intends to directly extract information of imagined body kinematics from EEG and thus can significantly reduce training time.

Materials and Methods:

The experimental procedure included three phases. During the experimental procedure, EEG signals were acquired by using an Emotiv EPOC device with 14 channels and through BCI2000 software (with high pass filter at 0.1Hz and low pass filter at 30Hz). Phase 1 was the training phase. The subject was asked to sit comfortably in a fixed chair with hands resting on the laps. The subject's face was kept at an arm's length from the monitor. The subject was instructed to track the movement (up-down/right-left) of a computer cursor, whose movement was controlled by a practitioner in a random manner. The training phase consisted of 5 trials, each of which lasted 60 seconds. Phase 2 was the calibration phase, during which a decoder model was constructed to model the velocity of the cursor as a function of the EEG waves of the subject. For more accurate reconstruction and prediction of the imagined kinematics at each point, 5 previous points (time lag) of EEG data were also included in the decoding procedure. Then, the developed decoder was fed into BCI2000 software to test the performance of the subject in phase 3 (test phase). In the test phase, the subject was asked to move the cursor using their imagination to a target that randomly appeared on the monitor.

Results and Discussion:

Five healthy subjects with no prior experience in BCI studies were recruited for this work. In the control phase, six continuous trials were illustrated to the subject and they were asked to reach the presented target in a 15 second period by imaging the movement of the cursor. The trial was marked as success if the cursor hits the target within 15 seconds; otherwise failure. Table 1 shows the results of cursor movement using imagined kinematics. 83.3% (+/-11.7%) 100% (+/-0%) Results are based on 5 subjects. Four subjects each conducted 6 trials of vertical movement and 6 trials of horizontal movement. One subject conducted 6 trials of vertical movement and did not conduct horizontal movements.

Conclusions: A noninvasive BCI system was developed using scalp EEG to decode imagined body kinematics. It was shown this paradigm required just a short amount of time in training in contrast to previous algorithms based on sensorimotor rhythms. The subjects were able to gain a high success rate in controlling and hitting targets which is noteworthy compared to previous works. The current studies included one-dimensional movements in both vertical and horizontal direction. In future work, we will extend the research to conduct 2-and 3-dimensional movements. The developed algorithms have the potentials to be used in convenient, efficient assistive devices of BCI technology.

Table 1 .

 1 Results of cursor movement using imagined body kinematics

		Vertical Direction	Horizontal Direction
	Number of Trials	30	24
	Success Rate (standard deviation)

Acknowledgements: This work was in part supported by a NeuroNET seed grant.


Reza A Abiri

Justin A Kilmarx

Mohammad A Raji

Xiaopeng

Xiaopeng A Zhao

Planar

Planar Control of a Quadcopter Using a Zero-Training Brain Machine Interface Platform

come   L'archive ouverte pluridisciplinaire

Introduction: Brain Machine interface (BMI) enables promising applications in neuroprosthesis and neurorehabilitation by controlling robotic devices based on the subject's intentions. In contrast to the earlier techniques using sensorimotor rhythms, the method here intends to directly extract information of imagined body kinematics and thus can significantly reduce training time. We developed a zero-training BMI platform that controls a quadcopter using noninvasively acquired brain signals. Scalp electroencephalogram (EEG) signals of a user's imaginary movements are collected in real-time and translated by a computer to control a quadcopter along a designated path in a two-dimensional space. The results showed a promising performance using the zerotraining paradigm that can be utilized in controlling neuroprosthetic limbs and neurorehabilitation devices by patients with BCI illiteracy. Materials and Methods: EEG signals were acquired using an Emotiv EPOC device with 14 channels and through BCI2000 software (with high pass filter at 0.1Hz and low pass filter at 30Hz). The quadcopter (Parrot AR. Drone 2.0) flies at a constant height with a fixed forward velocity. The lateral movement of the quadcopter is controlled by a subject's imagined body kinematics. The subject was instructed to use a previously developed BCI cursor control model to control the quadcopter, where the left and right movements of the cursor on a computer screen are mapped to the counterclockwise and clockwise rotations, respectively. After initial calibration and testing, one recruited subject was instructed to fly the quadcopter around obstacles (Figure 1) by using the BCI cursor task application. The quadcopter movement is controlled by Node.js software platform, and the real-time communication between BCI2000 and Node.js is established by UDP protocol through JavaScript programming language. Results: A regression model based on imagined body kinematics was developed using the cursor control task in BCI2000. Parameters of the model was obtained from fitting the training experiments of one subject. The model was used by a different subject to control the lateral movement of a quadcopter. For the subject who attempted to the control the quadcopter, there was no BCI training conducted. A minimum parameter calibration was done for the test subject to adjust the magnitude of the quadcopter rotating speeds. An experiment was designed for the subject to fly the quadcopter so that it takes an S-shaped trajectory to go beyond two obstacles. Figure 1 shows a few selected trajectories of the quadcopter. After a few trials, the subject was able to successfully control the quadcopter to fly from one end of the room to the other end and simultaneously go around the two obstacles in an S-shaped curve. The mean time of the selected trajectories was 11 seconds. Conclusions: We developed and tested a novel BMI platform, which is noninvasive and requires zero-training with a minimum calibration time. For initial testing of the platform, one subject was recruited to fly a quadcopter around obstacles. The experiments here serve as a concept proof for the feasibility of zero-training BCI and BMI platforms. In future work, we would like to further develop our zero-training model to provide more stable results in BMI systems to be used more efficiently in two-and three-dimensional robotic assistive devices. Acknowledgements: This work was in part supported by a NeuroNET seed grant.

Figure 1 .

 1 Figure 1. Example quadcopter trajectories. Three trajectories from the drone control experiments are shown. Along each trajectory, the drone position is read from a video captured during the experiment at x=0 (starting point), x=1.35 (position of first obstacle), x=2.51 (the middle point between the two obstacles), x=3.68 (position of the second obstacle), as well as the end point of the drone along the straight line at y=0.


Reza Abiri

Xiaopeng Zhao

Yang A Jiang

Real

A Real Time EEG-Based Neurofeedback platform for Attention Training

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

Introduction: A Brain Computer Interface (BCI) system as a closed-loop system can be used as a neurofeedback platform in neuro-rehabilitation programs. Here, we have developed a new attention-based neurofeedback system using noninvasive brainwaves signals. Scalp Electroencephalography (EEG) signals of a subject were collected in real-time during some minutes of sustained attention tasks in which the subject was looking at a sequence of visual stimuli pictures composited of two categories (Scene & Face). The attentional states and levels of the subject for the targeted picture will be determined by analysis of EEG data. Then, the EEG attention level classifier was fed back to our BCI platform while the same visual stimuli neurofeedback is given to the subject. It was confirmed that the transparency of a targeted picture can be controlled based on subject's attention level. The developed platform can have potential application in attention enhancement of patients with cognitive disorders. Materials and Methods: During the experimental procedure, EEG signals were acquired wirelessly by using an Emotiv EPOC device with 14 channels and through Simulink software. The experimental tests consist of eight blocks. The first four blocks are stable blocks and the next four blocks are real-time neurofeedback blocks. Each block starts with a one second cue texture displaying the targeted picture and then, continues with 50 trials each one is one second of showing a composite stimuli picture with mixture of 50% scene (Indoor or outdoor) and 50% face (male or female). The subject was instructed to have sustained attention for the task relevant stimuli (e.g. indoor) and ignore the other stimuli (e.g. outdoor or face). The collected EEG data from stable blocks was processed in MATLAB and the features of attentional states' levels were extracted and classified. In the next four neurofeedback blocks, the output of our trained classifier was employed in real-time to determine which category was being attended by the subject and the mixture of the composite stimulus was updated for the next trial. Results: A Simulink program has been developed to implement the neurofeedback platform. A schematic interface is shown in Figure 1. A support vector machine model was developed to classify the attention focused target using the following EEG features: relevant energies for delta, theta, alpha, beta, and gamma bands, as well as the spectral entropy. The classification was based on the 4 stable blocks. Using three blocks as training data and one block as test, an overall classification accuracy of 70% was achieved with sensitivity equal to 40% and specificity equal to 73%.

Figure 1. Schematic Simulink interface for the neurofeedback platform. EEG data is acquired using EpocEEG and fed as input to the system. EEG signal during the last second is analyzed to determine the attention level of the subject to the previous image, which determines the clarity level of the next target image. This forms a positive feedback that rewards the subject to focus his or her attention on the task.

Conclusions:

A new noninvasive BCI system was developed using scalp EEG to decode the sustained attention level of a user and provide an efficient neurofeedback to the subject based on his/her level of attention. The subject reported that he is able to control the transparency of the targeted picture by paying attention to that during the experiment. Whenever, the subject loses his/her attention and the classifier confirms more attention to the task irrelevant stimulus, the mixture of picture will change to show clear picture of task irrelevant picture. In this way, the subject will be alerted to come back to his/her attending task. The current developed platform will have a potential application in enhancing the attention of patients with cognitive disorders. In the future work, we will extend the research to improve the classifier and test the platform with real patients. Acknowledgements: This work was in part supported by a NeuroNET seed grant.


E N Osegi

V I E Anireh

Towards Decentralized Computer Programming Shops and its place in Entrepreneurship Development

Keywords: CPS, CCC, decentralization, entrepreneurship development, IOT

Computer programming represents one interesting skill in the sciences and art of modern computing that is not well represented in a business setting. Ranging from the creation of simple programs for personal use to more complex programs for the creation and installation of mobile apps or desktop apps for a PC or Desktop, computer programming shops (CPS) have served the purpose of providing an environment for such beneficial activities to take place. Despite these benefits, computer programming shops are very limiting and highly centralized. Thus, there is an urgent need for the decentralization of computer programming business enterprise. In this paper, we present an approach that can foster the business of computer programming in a decentralized setting. In particular, we show how using technologies such as the Internet of Things (IOT) and Cloud Computing Connections (CCC) can help further entrepreneurship development in this area without the usual boundary limitations.

Introduction

An entrepreneur is one who envisages a business idea, then designs, develops, launches and runs the business idea into a useful entity for profit or not for profit. The entity may be a product such as a computer or mobile phone or a service such as selling or operating and maintaining a computer. Entrepreneurs create organizations and exhibit certain behavioral tendencies in the process [START_REF] Gartner | Who is an Entrepreneur?" Is the Wrong Question[END_REF].

Over the years, many businesses have sprung up in the field of computing offering one service or the other and impacting on the lives of a lot of people both youthful and elderly alike. More recently, computer programming a service that results in the generation of computer programs (software applications) as its main product is gradually gaining grounds as a viable trade. In big industries computer programmers (those skilled in the art of computer programming) help automate factories and work processes by writing programs that control big and small machines, and automate the processing of large volumes of documents. While smaller industries operate mainly on smaller tasks such as data processing or mobile app installations. However, most if not all of these businesses are centralized. One problem with this is the fact that it is usually very difficult to service very distant customers leading to delay of service or product development. In addition there is a limiting value in quality of service options; this means that computer programming options are either unavailable or not easily accessible. An instance of this problem may be seen when a prospective customer seeks a programming solution to a problem but has access to only one computer programming shop at a time. This gives rise to the problem we refer to here as the Blind Programming Problem (BPP). There is also the usual bureaucratic bottlenecks associated with centralized systems that 2 constitute a delay [START_REF] Caruana | Effects of some environmental challenges and centralization on the entrepreneurial orientation and performance of public sector entities[END_REF][START_REF] Kearney | A conceptual model of public sector corporate entrepreneurship[END_REF][START_REF] Stacey | The FAB LAB network: A global platform for digital invention, education and entrepreneurship[END_REF]. Thus, it becomes necessary to move from the conventional to the unconventional business.

Decentralization is one step to overcome the obvious limitations of centralized computer programming businesses [START_REF] Gonçalves | Promoting ecommerce software platforms adoption as a means to overcome domestic crises: the cases of Portugal and Spain approached from a focus-group perspective[END_REF][START_REF] Gonçalves | Promoting ecommerce software platforms adoption as a means to overcome domestic crises: the cases of Portugal and Spain approached from a focus-group perspective[END_REF]. Within a decentralized setting, startups can take advantage of tools such as Cloud Computing Connections (CCC) and Internet-of-Things (IOT) to leverage their computer programming businesses beyond their current boundaries. Existing IT businesses can also take advantage of CCC and IOT to offer more computer programming services and products to the teeming computer market. In addition, prospective customers can access multiple computer programming shops (CPS) in a very short time frame, test sample products of the computer programming shops leading to service and product delivery satisfaction. However, there may still be need for modularity to better optimize the decentralization process [START_REF] Yetis-Larsson | Networked Entrepreneurs: How Entrepreneurs Leverage Open Source Software Communities[END_REF].

Thus, it is the objective of this paper to demonstrate how this very important functionality may be incorporated in a typical business chain by an entrepreneur.

In a subsequent part of this paper, we describe our proposed approach for building a decentralized computer programming shop. We also provide illustrative use case models on how the proposed approach may be applied. In a final part, we give our conclusions and recommendations for future work.

An approach to a Decentralized Computer Programming Shop

In this part, we present a conceptual model that can be used to advance the state of computer programming businesses, particularly for young startups. In this model, a decentralized shop offering a Decentralized Computer Programming as a Service (DECPaaS) is described. The service orientation stems from the fact that support is provided to expert developers and even non-experts (through an end-user programming model) for writing computer codes that operate internal or external systems or that assist in the design of software interfaces. An instance of this can be seen from the use of a cloud computing concept that support online coding and tests as shown in Figure 1. Some examples of such sites that can serve as reference point include www.w3schools.com which is dedicated to web programming, https://octave-online.net dedicated to technical computing on the web etc. Here, the emphasis is on training prospective clients to build real-time and interactive web solutions. Of course, prospective entrepreneurs (clients) can seize the opportunity to build their own DECPaaS shops that is not limited to a central location.

instance can be realized through a collaborative network of things within and out of the internet. This gives rise to the concept of Re-programmable Internet of Things (RIOTs) i.e. the marriage of embedded hardware with computing. Here, each CPS may be viewed as a computer offering a programming service to its prospective clients who in their own right is also a computer which may or may not offer a programming service.

Some immediate benefits can be seen from the use of such a technique. One key advantage of these approaches is that the computer programming shop is not restricted to a particular location, but accessible to all in virtually any location. Another key advantage is in the fact that there is no need to install any software thus leading to memory savings and zero cost of installation.

Fig1. Decentralized Computer Programming as a Service (DECPaaS)

Conclusion

Our idea of a decentralized computer programming shop is based on the premises that such a shop need not be limited to its environs i.e. computer programming business should not necessarily be done at one particular shop. Our approach introduces yet another business idea: One-business-many-shops. From the discussions so far it is hoped that this will herald a new generation of entrepreneurs within the country and far off. Future work should be geared towards a further decentralization of CPS by collaborating with potential DECPaaS cloud computing providers and automating DECPaaS links within a larger body of networked users.

Corresponding author: Department of Information Technology, National Open University of Nigeria (NOUN); Tel:

234-8025838364; E-mail: nd.osegi@sure-gp.com 2 Department of Computer Science, Rivers State University, Nigeria


E N Osegi

B A Wokoma

S A Bruce-Allison

Leveraging Entrepreneurship through the design of Artificial Intelligence Projects

Keywords: AIP, business, entrepreneurs, technology

Artificial Intelligence projects (AIP), is currently attracting popular attention as a viable business area for young and mature entrepreneurs. Most industries, particularly in research and development, now use AIPs for the discovery and synthesis of countless of novel products and services of incomprehensible commercial and functional value. However, the immediate benefits of using AIPs in a commercial setting are still yet to be fully realized in many African countries. In this paper, we describe how potential Artificial Intelligence (AI) entrepreneurs may use AIPs to advance their business operations by building new lines of technologically viable automated businesses. We also present an illustrative example of how this may be accomplished in a specific domain.

Introduction

Artificial intelligence (AI) may refer to a branch of study that involves the design, development and operation of human, non-human, biological or non-biological and environmental agents of behaviors which are associated with learning, cognition and motor-coordination in diverse scenarios and many different applications. For instance, an AI system that can be designed to identify the correct (or expectation) face in a multitude of faces may be said to reason or behave as a human.

Some typical areas or classes of AI include, the Neural Networks which emulates operational functions in human or biological brains, Evolutionary Computing which tries to mimic the evolution or reproductive functions in bio-organisms, Swarm Intelligence that emulates the behavior of swarm or group of bio-organisms such as bees or birds, and Ambient Intelligence which emulate the ambient characteristics in environment e.g. lighting [START_REF] Sumathi | Computational intelligence paradigms: theory & applications using MATLAB[END_REF][START_REF] Zelka | From Devices to" Ambient Intelligence[END_REF].

AI cuts across many disciplines and fields of human endeavors ranging from Biology and Neuroscience, to Physics, Engineering and Technology including industry relevant domains such as financial institutions, oil and gas organizations, Information and Communication Technology (ICT) research centers and agro-allied industries. AI projects (AIPs) on the other hand, stand out as the major driver of AI leading to the derivation of a growing list of AI capable products that solve diverse real world tasks.

In a subsequent section, we describe some existing types of AIPs that potential entrepreneurs can take advantage of. We also describe in a later section, how an instance of an AIP may be put into practice in a specified domain. Finally, we give our conclusions and recommendations for future work.

Existing types of AIPs for entrepreneurial development

In the course of our research, some existing AIPS have been identified as candidates for improved entrepreneurial activity, particularly as it pertains science, engineering and technological empowerment. Some of these include:

 Image Recognition Projects  Language Encoding and Translation Projects  Educational Datamining Projects  Real-time Embedded Database/Datamining Projects Some of the products or services generated from the Image Recognition Projects (IRP) include real-time invigilation of class examinations, vehicle license plate recognition module, and facial recognition modules [START_REF]Designing AI based image recognition research projects[END_REF]. These projects used an advanced form of image pre-processing and several intuitive approaches too complex to describe in detail here. However, the projects are achievable and can be developed using locally available ICT tools.

Products or services developed from Language Encoding and Translation Projects (LETP) include handwritten digit recognition system using mode-pooling autoencoder networks [START_REF] Osegi | Learning Representations from Deep Networks Using Mode Synthesizers[END_REF], local-dialect to English language translation service [START_REF] Osegi | A Generative Model for Multi-Dialect Representation[END_REF] and an adaptive mixed-integer programming system for intelligent encoding currently being investigated in some neural models [START_REF]Deviant Learning Algorithm[END_REF][START_REF] En Osegi | HTM-MAT: Minimalist Cortical Learning Algorithm[END_REF]. In particular, the neural models have been applied in some prediction researches in the oil and gas sector and are also currently being applied in other sectors [START_REF] Osegi | Monitoring Premium Motor Spirit Demand Nigeria: A Novel Artificial Intelligence Approach[END_REF][START_REF] Osegi | p-DLA: A Predictive System Model for Onshore Oil and Gas Pipeline Dataset Classification and Monitoring-Part 1[END_REF].

Products developed for educational administrators include the GPSFARPS which is an evolutionary tool/application based on the multi-gene genetic programming technology for predicting failure rate at school [START_REF] Orove | A Multi-Gene Genetic Programming Application for Predicting Students Failure at School[END_REF].

Products developed for Real-time Embedded Database Projects (REDP) include an SMS-SQL smart query system initially developed in [START_REF] Osegi | GOEmbed: A Smart SMS-SQL Database Management System for Low-Cost Microcontrollers[END_REF] as a research idea and now further developed into a prototype product in [START_REF] Goptimaembed | Smart SMS-SQL Datamining System SURE-GP LTD[END_REF].

A typical application area for an AIP

In this section we describe an interesting sample application area that can be attempted by anyone motivated by the business of AI and its consequent derivatives. This application area is an adaptive Water Heater/Cooler system that can serve as a public water dispenser strategic areas. The system for doing just that is exemplified in Figure 1. In this example, the target i.e. the expected function or control message is the cooling/heating control valve state or value while the inputs are the initial control valve settings provided by a human expert with the accompanying weather influences (temperature) as measured by dedicated sensors. As can be seen the system comprises of an Environment which further contains the observed weather influences (temperature in this case) and also the possible adjustments (cold or hot water settings) that may be made first by a human expert, herein, called the teacher. This environment represents the input that is fed into a Sensor system. The Sensor system reads in the Environment and transforms to equivalent electrical signals that is encoded and fed to a Cortical Learning Neural Network (CLNN). The CLNN acts as an AI expert forming memory patterns of the environmental input influences earlier mentioned at different times of observation in a temporal manner and creating predictions in the process. Over time, the CLNN learns the process of water cooling or/and heating, and without the use of the human expert, is able to predict precisely what the likely future settings for cooling or heating will be for further intelligent decision making and control.

Fig. 1

 1 Fig.1Illustrative diagram of a sample AIP using Cortical Learning Neural Networks (CLNN); the mode of operation is at first supervised with a teacher (human expert) and after a certain number of trials it becomes unsupervised.

Corresponding author: Department of Information Technology, National Open University of Nigeria (NOUN); Tel:

234-8025838364; E-mail: nd.osegi@sure-gp.com 2 Department of Electrical/Electronic Engineering, Rivers State University, Nigeria 3 Department of Electrical/Electronic Engineering, Rivers State University, Nigeria

Conclusions and Recommendations for Future Work

We have presented groundwork for Artificial Intelligence Projects (AIPs) from the pragmatic perspective. We have also introduced possible areas for AIPS and provided an illustrative example of how a typical real world problem may be solved using an AIP. However, it must be emphasized here that the number of possible applications for an AIP is endless. Thus, it is recommended that more real world problems be tried and tested by potential AI entrepreneurs in collaboration with proven AI experts in the field of interest and AIP products derived to render effective automated goods and services to prospective customers.