Lahcen Boulanba
email: l.boulanba@ucam.ac.ma

Mohamed Mellouk
email: mohamed.mellouk@parisdescartes.fr

ON A HIGH-DIMENSIONAL NONLINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATION

Keywords: 2000 Mathematics Subject Classification. 60H15, 47G30 Stochastic partial differential equations, non-Lipschitz coefficients, martingale problem, Gaussian noises, Skorohod representation theorem

In this paper we investigate a nonlinear stochastic partial differential equation (spde in short) perturbed by a space-correlated Gaussian noise in arbitrary dimension d ≥ 1, with a non-Lipschitz coefficient noisy term. The equation studied coincides in one dimension with the stochastic Burgers equation. Existence of a weak solution is established through an approximation procedure.

Introduction

Let d ≥ 1 be an integer and D : = [0, 1] d . In this paper, we study the following nonlinear stochastic partial differential equation on

R + × D ∂u ∂t (t, x) = ∂ ∂x i ∂ ∂x i u(t, x) + 1 2 u 2 (t, x)
+ σ(u(t, x)) Ḟ (t, x), t > 0, x ∈ D, i = 1, ..., d

with Dirichlet boundary conditions u(t, x) = 0, t > 0, x ∈ ∂D,

and the initial condition

u(0, x) = u 0 (x), x ∈ D, (1.3)
where u 0 is a continuous function on D with values in the interval [0, 1] and Ḟ is a noise on R + × D that is white in time and colored in space. The noisy coefficient term σ is a real-valued function which will be described precisely later. We will refer to the problem (1.1) -(1.3) by Eq(d, u 0 , σ).

It is worth noting that the equation Eq(d, u 0 , 0) gives the classic Burgers equation, which arises in physics and have extensively been studied in the literature, see e.g. [START_REF] Ladyzhenskaya | Linear and quasilinear equations of Parabolic Type[END_REF] and [START_REF] Matsumura | Asymptotic stability of travelling waves for scalar viscous conservation laws with non-convex nonlinearity[END_REF].

A class of equations of type Eq(d, u 0 , σ), with noise depending only on time, was studied by Gyöngy and Rovira [START_REF] Gyöngy | On L p -solutions of semilinear stochastic partial differential equations[END_REF]. The authors took the space variable in a bounded convex domain of R d , and the class investigated contains a version of the one-dimensional Burgers equation [START_REF] Bertini | The Stochastic Burgers Equation[END_REF][START_REF] Burgers | The nonlinear diffusion equation. Asymptotic solutions and statistical problems[END_REF][START_REF] Prato | Stochastic Burgers equation[END_REF][START_REF] Prato | Stochastic Burgers equation with correlated noise[END_REF][START_REF] Gyöngy | Existence and uniqueness results for semilinear stochastic partial differential equations[END_REF][START_REF] Gyöngy | On the stochastic Burgers equation in the real line[END_REF][START_REF] Hopf | The partial differential equation ut + uux = µuxx[END_REF] as a special case. They proved existence, uniqueness and a comparison theorem under assumptions on coefficients which include global Lipschitz condition on the diffusion coefficient σ.

However, the present work is concerned with a non-Lipschitz diffusion coefficient. Since the F is a space-time noise and σ is non-Lipschitz the results of [START_REF] Gyöngy | On L p -solutions of semilinear stochastic partial differential equations[END_REF] cannot be applied here.

The equation Eq(1, u 0 , σ) was studied by Kolkovska [START_REF] Kolkovska | On a Stochastic Burgers Equation with Dirichlet Boundary Conditions[END_REF]. The author proved the existence of a weak solution by using an argument of tightness and solving a martingal problem. The case considered corresponds to the stepping stone model which describes migrating populations consisting of two types when the total population size does not change over time. For more details on the genealogy of a variation of this model, one can see Chapter 7 in [START_REF] Sturm | On spatially structured population processes and relations to stochastic partial differential equations[END_REF]. Our study extends this of [START_REF] Kolkovska | On a Stochastic Burgers Equation with Dirichlet Boundary Conditions[END_REF] to a multidimensional case, and with a more general class of diffusion coefficients.

It is also worth mentioning that a one dimensional version of (1.1), perturbed with a space-time white noise, was investigated by Adler and Bonnet [START_REF] Adler | The Burgers superprocess[END_REF][START_REF] Bonnet | The Burgers Superprocess[END_REF] on the whole real line and with the non-Lipschitz noisy coefficient σ(r) = √ r. The authors established the existence of a weak solution and its Hölder continuity. Moreover, they discussed (but did not prove) the uniqueness of the solution.

In the present paper, we deal with a d-dimensional spdes. As it is well known, we can no longer use the space-time white noise for the perturbation; indeed, in dimension d ≥ 2, the fundamental solution associated with the operator ∂ ∂t -∆ is not square integrable when d ≥ 2, while the one associated with the operator ∂ ∂t 2 -∆ becomes less smooth as the dimension increases. Hence, the martingale measure approach, introduced by Walsh [START_REF] Walsh | An Introduction to stochastic partial differential equations[END_REF], to investigate multidimensional SPDEs driven by a space-time white noise gives solutions only in the space of random distributions (see e.g. [START_REF] Albeverio | Trivial solutions for a non-linear two spacedimensional wave equation perturbed by a space-time white noise[END_REF][START_REF] Dawson | Spatially homogeneous random evolutions[END_REF][START_REF] Peszat | Nonlinear stochastic wave and heat equations[END_REF]).

In order to circumvent this difficulty, Dalang and Frangos [START_REF] Dalang | The stochastic wave equation in two spatial dimensions[END_REF] suggested to replace the space time white noise by some Gaussian noise which is white in time and which has some space correlation in order to obtain solutions in the space of real-valued stochastic processes. See [START_REF] Dalang | Extending martingale measure stochasti c integrals with applications to spatially homogeneous SPDEs[END_REF] and [START_REF] Sanz-Solé | Malliavin Calculus, with Applications to Stochastic Partial Differential Equations[END_REF] and the references therein for more literature in the subject.

The aim of this paper is to study the existence of solution of a non-linear equation Eq(d, u 0 , σ) in arbitrary dimension d ≥ 1, with a non-Lipschitz coefficient noisy term. The idea to prove our main result is to adopt an approach going back to Funaki [START_REF] Funaki | Random motion of strings and related stochastic evolution equations[END_REF].

The paper is organized as follows. In the second section we give a formulation of the problem. Section 3 is devoted to define a spatial discretization scheme of (1.1) and obtain a system of stochastic differential equations. The existence and uniqueness of a unique strong solution for this system. Section 4 is devoted to the tightness of these approximating solutions. The existence of a weak solution of Eq(d, u 0 , σ) is established in Section 5 by solving a martingale problem. We conclude with an Appendix containing some technical results needed along the paper.

Note that all real positive constants are denoted by c regardless of their values and some of the standing parameters are not mentioned.

Framework

Let (Ω, F , P ) be a complete probability space. Let F = {F (ϕ), ϕ ∈ D(R d+1 } be a mean-zero L 2 (Ω, F , P)-valued Gaussian process with the covariance functional

E(F (ϕ)F (ψ)) = +∞ 0 ds D dx D dyϕ(s, x)f (x, y)ψ(s, y), (2.1)
where f : R d × R d -→ R denotes the correlation kernel of the noise F .

Several authors discussed the weaker assumptions in such a way that (2.1) defines a covariance functional, see e.g. the references [START_REF] Dalang | The stochastic wave equation in two spatial dimensions[END_REF][START_REF] Dalang | Extending martingale measure stochasti c integrals with applications to spatially homogeneous SPDEs[END_REF][START_REF] Sturm | On spatially structured population processes and relations to stochastic partial differential equations[END_REF].

In this paper, we will take f symmetric and belonging to C b (D × D), the space of continuous bounded functions. Following the same approach in [START_REF] Dalang | Extending martingale measure stochasti c integrals with applications to spatially homogeneous SPDEs[END_REF], the Gaussian process F can be extended to a worthy martingale measure M = {M (t, A) := F ([0, t] × A) : t ≥ 0, A ∈ B b (D)} which shall acts as integrator, in the sense of Walsh [START_REF] Walsh | An Introduction to stochastic partial differential equations[END_REF], where B b (D) denotes the bounded Borel subsets of D. Let G t be the completion of the σ-field generated by the random variables σ(u(s, x))ϕ(x)F (ds, dx).

{M (s, A), 0 ≤ s ≤ t, A ∈ B b (D)}. The properties of F ensure that the process M = {M (t, A), t ≥ 0, A ∈ B b (D)},
In order to formulate our main result, we assume the following hypothesis :

(H) The function σ is Hölder continuous of order 1 2 ≤ α < 1 i.e. there exists a positive constant c such that for all x, y ∈ [0, 1]

|σ(x) -σ(y)| ≤ c|x -y| α .
Moreover, we assume that σ is not identidically zero and satisfies σ(0) = σ(1) = 0. Theorem 2.1. Assume that u 0 is a continuous function on D with values in the interval [0, 1], and that σ satisfies (H). Then there exists a weak solution to the equation Eq(d, u 0 , σ).

Remark 2.1. The hypothesis (H) implies the existence of a positive constant c such that for all x ∈ [0, 1]

σ(x) ≤ c min(x α , (1 -x) α ).
Examples 1.: The function σ 1 : x -→ x(1x), which corresponds to the stepping stone model, satisfies (H) with α = 1 2 . 2.: Let γ ∈] 1 2 , 1]. The function σ 2 defined on the interval [0, 1] by σ 2 (0) = 0 and σ 2 (x) = -x γ log(x) for x ∈]0, 1] satisfies (H). It is a concave function with values in [0, 1]. Indeed, σ 2 is twice continuously differentiable on ]0, 1[ and σ ′′ 2 (x) = x γ-2 (1 -2γ + γ(1γ) log(x)), for all x ∈]0, 1[. Moreover, taking into account the continuity of σ 2 on [0, 1] we deduce its concavity on

[0, 1]. It follows that |σ 2 (x) -σ 2 (y)| ≤ sup{|σ 2 (x -y) -σ 2 (0)|, |σ 2 (1) - σ 2 (1 -x + y)|}, for all x, y ∈ [0, 1] such that x ≥ y. Finally, we point out that |σ 2 (x -y) -σ 2 (0)| = |σ 2 (x -y)| ≤ c|x -y| γ-ǫ , where ǫ ∈]0, γ -1 2 [, and |σ 2 (1) -σ 2 (1 -x + y)| = |σ 2 (1 -x + y)| ≤ c|x -y| η for any η ∈]0, 1[. The result follows.

The approximating processes

Let n ≥ 2 be an integer. For

Ik := (k 1 , ..., k d ) ∈ {1, ..., n -1} d , set x n Ik = k1 n , k2 n , ..., k d n
and define the interval

II n Ik = Π d j=1 [ k j n , k j + 1 n ).
Consider the discretized version of Eq(d, u 0 , σ):

       du n (t, x n Ik ) = ∆ n u n (t, x n Ik )dt + 1 2 d i=1 ∇ n i (u n (t, x n Ik ) 2 )dt + n d σ(u n (t, x n Ik ))F (dt, II n Ik ), u n (0, x n Ik ) = u 0 (x n Ik ), (3.1)
where, the operator in the first term is given by

∆ n u n (t, x n Ik ) = n 2 d i=1 u n (t, x n Ik + e i n ) + u n (t, x n Ik - e i n ) -2u n (t, x n Ik ) ,
and in second term by

∇ n i (u n (t, x n Ik ) 2 ) = n u n (t, x n Ik + e i n ) 2 -u n (t, x n Ik ) 2 ,
with {e i , 1 ≤ i ≤ d} denotes the canonical basis of R d , and

F (t, II n Ik ) = t 0 II n I k F (ds, dx).
The noises {F (t, II n Ik ), t ≥ 0}, derived from colored noise F on R d with covariance given by (2.1), are one dimensional correlated Brownian motions. Moreover, the boundedness of f implies that

F (•, II n Ik ) t ≤ ctn -2d
, where c is a constant depending on f . Indeed

F (•, II n Ik ) t = t 0 R d R d 1 II n I k (x)f (x, y)1 II n I k (y)dsdxdy = t II n I k II n I k f (x, y)dxdy ≤ c t n -2d ,
where c = max{f (x, y); (x, y) ∈ D × D}. Now, we proceed as in [START_REF] Millet | On implicit and explicit schemes for parabolic SPDEs in any dimension[END_REF] and we consider the bijection between the grid

D d n := {x n Ik = k1 n , k2 n , ..., k d n , (k 1 , ..., k d ) ∈ {1, ..., n-1} d }, which assigns to each k1 n , k2 n , ... k d n the integer k 1 + (k 2 -1)(n -1) + ... + (k d -1)(n -1) d-1 .
Set N := (n -1) d , and for each i ∈ {1, .., N }, let i n be the unique element

k1 n , k2 n , ..., k d n such that i = k 1 + (k 2 -1)(n -1) + ... + (k d -1)(n -1) d-1 and 1 ≤ k j ≤ n -1 for each j ∈ {1, .., d}.
Finally, let u n i (t) denotes u n (t, i n ). Hence, the system (3.1) can be written, for i = 1, ..., d,

       du n i (t) = N j=1 [a i,j u n j (t) + 1 2 b i,j u n j (t) 2 (t)]dt + n d σ(u n i (t))F (dt, II n i n ) u n i (0) = u 0 (i n ). (3.2) Let A (d) n and B (d) n be the N × N matrices such that A (d) n : = (a i,j ) 1≤i,j≤N and
B (d) n : = (b i,j ) 1≤i,j≤N .
For d = 1, it is well known that we have

a i,j =    n 2 if |i -j| = 1, -2n 2 if j = i, 0 otherwise, (3.3)
and,

b i,j =    n if j = i + 1, -n if j = i, 0 otherwise. (3.4) For d ≥ 2, it is known from [24] that A (d)
n can be obtained by induction using a relation of recurrence and the fact that A

(1) n is known. For the coefficients b i,j , let us first introduce some notations. For an integer ℓ ≥ 1, let I ℓ be the ℓ × ℓ identity matrix and set M := (n -1) d-1 . One can easily show that

B (d) n =              B (d-1) n nI M 0 • • • • • • 0 0 B (d-1) n nI M 0 • • • . . . . . . 0 . . . . . . . . . . . . . . . • • • . . . . . . . . . 0 . . . • • • • • • . . . . . . nI M 0 • • • • • • • • • 0 B (d-1) n              -nI N . Note that B (1)
n is the (n -1) × (n -1)-matrix defined by (3.4). The main result of this section is the following Proposition 3.1. For any integer n ≥ 1 and for any initial random condition

u n (0) = (ξ n 1 , ..., ξ n N ) ∈ [0, 1] N , the system        du n i (t) = N j=1 [a i,j u n j (t) + 1 2 b i,j u n j (t) 2 ]dt + n d σ(u n i (t))F (dt, II n i n ), u n i (0) = ξ n i , (3.5)
admits a unique strong solution u n (.

) ∈ C([0, +∞[, [0, 1] N ).
Proof .

We consider first the following system

       du n i (t) = N j=1 a i,j u n j (t) + 1 2 b i,j g(u n j (t)) dt + h(u n i (t))F (dt, I n i ), u n i (0) = ξ n i , (3.6)
where h : R -→ R is defined by h(x) = n d σ(x)1 {0≤x≤1} , and g : R -→ R is defined by g(x) = x 2 1 {-1≤x≤1} . In order to show that the continuous coefficients of the finite dimensional system (3.6) satisfy the growth linear condition, we write it under the following form

   dU n (t) = A (d) n U n (t)dt + 1 2 B (d) n Σ(U n (t))dt + K(U n (t))dF t U n (0) = u n (0), (3.7) with U n (t) := (u n i (t), 1 ≤ i ≤ N ) , Σ(U n (t)) := (g(u n i (t)), 1 ≤ i ≤ N ) , K(U n (t)) := diag(h(u n i (t)), 1 ≤ i ≤ N ), and
F t := F (t, II n i n ), 1 ≤ i ≤ N .
Since the first term in the right hand side of (3.7) is linear, then it satisfies the linear growth condition. For the second one, using an elementary property of matrix norms and the equivalence of norms on R N , we can write

B (d) n Σ(U n (t)) ≤ B (d) n L(U n (t)) ≤ c N i=1 |u n i (t)| ≤ c U n (t) ,
where the constant c depends on n and d. Concerning the noise coefficient, owing to Remark 2.1 we obtain a.s. for t ≥ 0,

N i=1 h(u n i (t)) 2 ≤ cn 2d N i=1 u n i (t) 2α 1 {0≤u n i (t)≤1} ≤ cn 2d N i=1 (1 + (u n i (t)) 2 ) ≤ cn 2d N 1 + U n (t) 2 .
Thus, the existence of a weak solution with continuous sample paths for (3.6) is given by Theorem 6.1 in the Appendix. Now, we will show that 0 ≤ u n i (t) ≤ 1 for all i = 1, ..., n and t ≥ 0. Taking ρ(x) = x β with 1 ≤ β ≤ 2α, and using the hypothesis (H) and the continuity of the trajectories of u n i , we get fot 0 ≤ t ≤ T t 0

1 {u n i (s)>0} ρ(u n i (s)) d u n i s = n 2d t 0 1 {u n i (s)>0} h 2 (u n i (s)) ρ(u n i (s)) ds ≤ n 2d T < +∞.
Then, by Lemma 6.1 we obtain that the local time, L 0 t (u n i ), of the semi-martingale u n i satisfies L 0 t (x n i ) = 0 for all t > 0, a.s, and i = 1, • • • , N .

Moreover, set x -: = max(0, -x) and by applying the Tanaka formula and summing over all indices i = 1, .., N , we get for t ≥ 0,

N i=1 u n i (t) -= - t 0 N i=1 1 {u n i (s)≤0} N j=1 a i,j u n j (s)ds - 1 2 t 0 N i=1 1 {u n i (s)≤0} N j=1
b i,j g(u n j (s))ds

:= I 1 + I 2 .
Since -x ≤ x -and taking into account the fact that a i,j ≥ 0 for all j = i and

N i=1,i =j a i,j ≤ 2dn 2 I 1 = - t 0 N j=1 u n j (s) N i=1 a i,j 1 {u n i (s)≤0} ds = - t 0 N j=1 u n j (s) -2n 2 1 {u n j (s)≤0} + N i=1,i =j a i,j 1 {u n i (s)≤0} = - t 0 N j=1 u n j (s) -2n 2 1 {u n j (s)≤0} + u n j (s) N i=1,i =j a i,j 1 {u n i (s)≤0} = t 0 N j=1 u n j (s) -(-2n 2 )1 {u n j (s)≤0} -u n j (s) N i=1,i =j a i,j 1 {u n i (s)≤0} ≤ t 0 N j=1 u n j (s) - N i=1 a i,j 1 {u n i (s)≤0} ds ≤ 2dn 2 t 0 N i=1 u n i (s) -ds. (3.8)
For I 2 , notice first that by using the definition of g,

I 2 = 1 2 t 0 N i=1 1 {-1≤u n i (s)≤0}   ndu n i (s) 2 - N j=1,j =i b i,j u n j (s) 2   ds.
Taking into account the fact that b i,j ≥ 0 for all j = i, we infer that

I 2 ≤ 1 2 nd t 0 N i=1 1 {-1≤u n i (s)≤0} u n i (s) 2 ds.
Moreover, for 0 ≤ s ≤ t such that -1 ≤ u n i (s) ≤ 0, we have

u n i (s) 2 = (-u n i (s)) 2 ≤ -u n i (s) = u n i (s) -.
Hence,

I 2 ≤ 1 2 nd t 0 N i=1 u n i (s) -ds. (3.9)
From the estimates (3.8) and (3.9), we deduce

N i=1 u n i (t) -≤ c t 0 N i=1 u n i (s) -ds.
Then, by Gronwall's Lemma, we obtain that N i=1 u n i (t) -= 0. Consequently, the solution is non-negative for each t ≥ 0. Concerning the fact that u n i (t) ≤ 1, it can be easily checked by using the same arguments as above for (1u n i (t)).

Therefore, the system (3.5) admits a weak solution with trajectories in C([0, +∞[, [0, 1] N ).

To show the pathwise uniqueness for (3.5), we assume that u 1,n (t) : = (u 1,n i (t), 1 ≤ i ≤ N ) and u 2,n (t) : = (u 2,n i (t), 1 ≤ i ≤ N ) are two weak solutions to (3.5), with the same noise F and the same initial data.

Set y

n i := u 1,n i -u 2,n i , then for t > 0, y n i (t) = N j=1 a i,j t 0 y n j (s)ds + 1 2 N j=1 b i,j t 0 u 1,n j (s) 2 -u 2,n j (s) 2 ds +n d t 0 σ(u 1,n i (s)) -σ(u 2,n i (s)) F (ds, II n i n ).
Consequently, by the boundedness of the correlation kernel f ,

y n i (•) t ≤ c t 0 σ(u 1,n i (s)) -σ(u 2,n i (s)) 2 ds.
Owing to Remark 2.1 and Lemma 6.1, we obtain that a.s. L 0 t (y n i ) = 0, for all i ∈ {1, ..., N }. Applying the Tanaka's formula for the continuous semimartingale y n i , it follows that

|y n i (t)| = N j=1 a i,j t 0 sgn(y n j (s))y n j (s)ds + 1 2 N j=1 b i,j t 0 sgn(y n j (s)) u 1,n j (s) 2 -u 2,n j (s) 2 ds +n d t 0 sgn(y n i (s)) σ(u 1,n i (s)) -σ(u 2,n i (s)) F (ds, II n i n ).
Noting that the sign function is defined by

sgn(x) = +1, x ≥ 0; -1, x < 0.
By summing over all i ∈ {1, ..., N } and taking the expectation, we get

E N i=1 |y n i (t)| = E   N i,j=1
a i,j t 0 sgn(y n j (s))y n j (s)ds

  + 1 2 E   N i,j=1 b i,j t 0 sgn(y n j (s)) u 1,n j (s) 2 -u 2,n j (s) 2 ds   := J 1 + J 2 .
(3.10)

For J 1 , the boundedness of the coefficients a i,j yields

J 1 = E t 0 N j=1 sgn(y n j (s))y n j (s) N i=1 a i,j ds ≤ c E t 0 N j=1 |y n j (s)|ds, (3.11)
where c is a constant depending on n. Concerning J 2 , using the boundedness of the coefficients b i,j and the fact that the solutions are with values in the interval [0, 1], we can write

J 2 = 1 2 E t 0 N j=1 sgn(y n j (s))y n j (s)(u 1,n i (s) + u 2,n i (s)) N i=1 b i,j ds ≤ cE t 0 N j=1
|y n j (s)|ds.

(3.12)

Combining the estimates (3.10)-(3.12) and applying Gronwall's Lemma, we obtain for all t ≥ 0

E N i=1 |y n i (t)| = 0.
Finally, the well-known theorem of Yamada and Watanabe [START_REF] Yamada | On the uniqueness of solutions of stochastic differential equations[END_REF] (see also Corollary 3.23 in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]) implies that (3.5) has a unique strong solution.

The tightness of the approximating processes

As in [START_REF] Sturm | On convergence of population processes in random environments to the stochastic heat equation with colored noise[END_REF], let Y n be a simple random walk whose generator is the discrete Laplacian ∆ n on the lattice D d n . To get the system (3.2) in its mild form we define the fundamental solution p n d associated with ∆ n .

For i n , j n ∈ D d n , set

p n d (t, i n , j n ) = n d P(Y n t = j n | Y n 0 = i n ).
The system (3.2) can be now written in its variation of constant form

u n i (t) = 1 n d N j=1 p n d (t, i n , j n )u 0 (j n ) + 1 2n d t 0 N j=1 p n d (t -s, i n , j n )b i,j u n j (s) 2 ds + t 0 N j=1 p n d (t -s, i n , j n )σ(u n j (s))F (ds, II n i n ).
In the remainder of this paper, we denote u n i (t) by u n (t, i n ), where the relationship between i and i n is described in the previous section. Moreover, in order to define u n (t, •) for all x ∈ D, we set κ n (x) = i n for x ∈ I n i , and pn d (t, x, y) = p n d (t, κ n (x), κ n (y)) for x, y ∈ D. Using a polygonal interpolation like as in [START_REF] Sturm | On convergence of population processes in random environments to the stochastic heat equation with colored noise[END_REF], we can write for all

x ∈ D u n (t, x) = D p n d (t, x, κ n (y))u 0 (κ n (y))dy + 1 2 t 0 D p n d (t -s, x, κ n (y))u n (s, y) 2 dyds + t 0 D p n d (t -s, x, κ n (y))σ(u n (s, y))F (ds, dy) ≡ u n 1 (t, x) + u n 2 (t, x) + u n 3 (t, x). (4.1)
The main result of this section is as follows. To handle the stochastic integral, we will use the factorization method introduced by Da Prato el al. in [START_REF] Prato | Regularity of solutions of linear stochastic equations in Hilbert spaces[END_REF]. Indeed, it is well known that one can write for β ∈ (0, 1),

t 0 D pn d (t -s, x, y)σ(u n (s, y))F (ds, dy) = sin(βπ) π t 0 D (t -s) β-1 pn d (t -s, x, y)Y (s, y)dyds, (4.3)
where

Y (s, y) : = s 0 D (s -r) -β pn d (s -r, y, z)σ(u n (r, z))F (dr, dz).
In the sequel, we will take 0 < β < 1 2 . Hence, by Burkholder's inequality, the boundedness of σ and f and the estimate i) of Lemma 6.2, we obtain for any T > 0 and p > 1, that there exists a constant c(T, p) > 0 such that

sup s∈[0,T ] sup y∈D E|Y (s, y)| p ≤ c(p, T ). (4.4)
Let us now prove an estimate on the increments in both the space and time variables. Firstly, we are concerned with the third term in (4.1). Clearly, for all x, y ∈ D and

0 ≤ s ≤ t ≤ T E|u n 3 (t, x) -u n 3 (s, y)| p ≤ c (E|u n 3 (t, x) -u n 3 (t, y)| p + E|u n 3 (t, y) -u n 3 (s, y)| p ) .
By Burkholder's inequality and the fact that σ and the kernel noise f are bounded, we can write

E|u n 3 (t, x) -u n 3 (t, y)| p = E t 0 D [p n d (t -s, x, z) -pn d (t -s, y, z)] σ(u n (s, z))F (dz, ds) p ≤ c t 0 D |p n d (t -s, x, z) -pn d (t -s, y, z)| dz 2 ds p/2 ≤ c x -y ( 1 q -1 2 ) p 2 , (4.5)
where 1 < q < 2. Note that in the last inequality, we have used the estimate ii) of Lemma 6.2. On the other hand, we have

E|u n 3 (t, y) -u n 3 (s, y)| p = E t 0 D (t -r) β-1 pn d (t -r, y, z) -(s -r) β-1 pn d (s -r, y, z)1 [0,s] (r) Y (r, z)dzdr p ≤ c E s 0 D (s -r) β-1 (p n d (t -r, y, z) -pn d (s -r, y, z))Y (r, z)dzdr p + E s 0 D ((t -r) β-1 -(s -r) β-1 )p n d (t -r, y, z)Y (r, z)dzdr p + E t s D (t -r) β-1 pn d (t -r, y, z)Y (r, z)dzdr p ≡ A 1 + A 2 + A 3 .
By Hölder's inequality, (4.4) and i) of the Lemma 6.2, we get

A 3 ≤ c|t -s| (β-λd)p , (4.6)
where 0 < β < 1 2 and 0 < λ < β d . Concerning A 2 , the same arguments as above yield

A 2 ≤ c s 0 [(s -r) β-1 -(t -r) β-1 ](t -r) -λd dr p ≤ c|t -s| -2λdp s 0 [(s -r) β-1 -(t -r) β-1 ]dr p ≤ c|t -s| p(β-λd) . (4.7)
For A 1 , we use firstly Burkholder's inequality

A 1 = E s 0 D (p n d (t -r, y, z) -pn d (s -r, y, z))σ(u n (r, z))F (dr, dz) p ≤ cE s 0 D D (p n d (t -r, y, z) -pn d (s -r, y, z))σ(u n (r, z))f (z, x) ×(p n d (t -r, y, x) -pn d (s -r, y, x))σ(u n (r, x))drdzdx p/2
, and since σ and f are bounded, and by iv) of the Lemma 6.2

A 1 ≤ c s 0 D D (p n d (t -r, y, z) -pn d (s -r, y, z))|z -x| -α (4.8) × (p n d (t -r, y, x) -pn d (s -r, y, x))) p/2 ≤ c s 0 pn d (t -r, y, •) -pn d (s -r, y, •) 2 α dr p/2 ≤ c|t -s| (1-α 2 ) p 2 . (4.9)
Therefore, taking into account (4.5)-(4.8), there exists a positive constant c and 0 < δ 1 < 1 2 and 0 < δ 2 < 1 4 such that

E|u n 3 (t, y) -u n 3 (s, y)| 2p ≤ c |t -s| δ1p + x -y δ2p . (4.10)
From the Proposition 3.2 in [START_REF] Millet | On implicit and explicit schemes for parabolic SPDEs in any dimension[END_REF], we deduce that (4.10) holds for u n 1 . Concerning u n 2 , using Hölder's inequality, (4.2) and the estimates on pn d given by the Lemma 6.2, we deduce that (4.10) holds for u n 2 . Finally, we deduce the tightness of the sequence {u n , n ≥ 1} by the Totoki-kolmogorov criterion, see Theorem 4.10 and Problem 4.11 in [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF].

The weak solution

Since the sequence {u n , n ≥ 1}, is tight in C ([0, T ], H), it is relatively compact in this space. Thus, there exists a subsequence, that we still denote by u n , which converges weakly in C ([0, T ], H) to a process u. By Skorohod's Representation Theorem, there exists a probability space Ω, and on it a sequence ũn , as well as a noise F and a process ũ such that (u n , u, F ) D ≡ (ũ n , ũ, F ) and ũn converges almost surely to ũ in C ([0, T ], H). We will show, by solving the corresponding martingale problem, that ũ is a weak solution of the equation Eq(d, u 0 , σ). Proposition 5.1. For any ϕ ∈ C 2 (D), such that ϕ ≡ 0 on ∂D, we have

M ϕ (t) : = D ũ(t, x)ϕ(x)dx - D u 0 (x)ϕ(x)dx - t 0 D ũ(s, x)∆ϕ(x)dxds + 1 2 d i=1 t 0 D ũ2 (s, x) ∂ϕ ∂x i (x)dxds (5.1)
is a martingale with the quadratic variation

M ϕ t = t 0 D D
σ(ũ(s, x))ϕ(x)f (x, y)σ(ũ(s, y))ϕ(y)dxdyds.

(5.2)

Proof. We consider the scheme (3.1) and by multiplying both its sides by 1 n d ϕ(x n Ik ) and summing over all the elements of the grid D d n , we obtain

M n ϕ (t) : = 1 n d x n I k u n (t, x n Ik )ϕ(x n Ik ) - 1 n d x n I k u(0, x n Ik )ϕ(x n Ik ) - t 0 1 n d x n I k ∆ n u n (s, x n Ik )ϕ(x n Ik )ds - 1 2 t 0 1 n d x n I k d i=1 ∇ n i (u n (s, x n Ik )) 2 ϕ(x n Ik )ds = 1 n d x n I k ũn (t, x n Ik )ϕ(x n Ik ) - 1 n d x n I k u(0, x n Ik )ϕ(x n Ik ) - t 0 1 n d x n I k ũn (s, x n Ik )∆ n ϕ(x n Ik )ds + 1 2 t 0 1 n d x n I k d i=1 (ũ n (s, x n Ik )) 2 ∇ n i ϕ(x n Ik )ds = t 0 x n I k ϕ(x n Ik )σ(ũ n (s, x n Ik ))F (ds, II n Ik ).
Note that {M n ϕ (t), t ≥ 0} is a martingale as it is a finite sum of martingales. Moreover, by (2.1) and the boundedness of σ, we can write

E(M n ϕ (t)) 2 = x n I k ,x n I k ′ t 0 ds II n I k dx II n I k ′ dyϕ(x)σ(ũ(s, x))ϕ(y)σ(ũ(s, y))f (x, y) ≤ t x n I k ,x n I k ′ II n I k dx II n I k ′ dyϕ(x)f (x, y)ϕ(y) ≤ t D dx D dyϕ(x)f (x, y)ϕ(y).
Hence, there exists a constant c depending only on ϕ, k and T such that

sup 0≤t≤T sup n E(M n ϕ (t)) 2 ≤ c < ∞.
Consequently, {M n ϕ , n ≥ 1} converges a.s., as n -→ ∞, to the martingale M ϕ given by (5.1). Also, we have

M ϕ t = lim n-→∞ M n ϕ t ,
where

M n ϕ t = x n I k • 0 II n I k ϕ(x)σ(ũ n (s, x))F (ds, dx) t = t 0 x n I k ,x n I k ′ II n I k II n I k ′ ϕ(κ n (x))σ(ũ n (s, κ n (x)))f (x, y)ϕ(κ n (y))σ(ũ n (s, κ n (y)))dsdxdy,
which converges a.s. to (5.2) as n -→ ∞.

Proof of the Theorem 2.1 Following Walsh [START_REF] Walsh | An Introduction to stochastic partial differential equations[END_REF], there exists a martingale measure M with quadratic variation ν(dxdyds) = σ(ũ(s, x))f (x, y)σ(ũ(s, y))dxdyds, which corresponds to the quadratic variation M ϕ t . Furthermore, we consider a measure martingale N independent of M and characterized by (2.1), and we set

F (t, ϕ) = t 0 D 1 σ(ũ(s, x)) 1 {ũ(s,x) ∈{0,1}} ϕ(x)M (dxds) + t 0 D 1 {ũ(s,x)∈{0,1}} ϕ(x)N (dxds).
Therefore, F (t, ϕ) satisfies

F (•, ϕ) t = t 0 D D ϕ(x)f (x, y)ϕ(y)dxdyds.
Hence, because of the form of its quadratic variation, it has the same distribution as N , and by Proposition 2.5.7 in [START_REF] Sturm | On spatially structured population processes and relations to stochastic partial differential equations[END_REF], we can write

M ϕ (t) = t 0 D σ(ũ(s, x))ϕ(x)F (dx, ds),
therefore, ũ is a weak solution of the equation Eq(d, u 0 , σ).

Appendix

Firstly, we recall the Lemma 1.0 in [START_REF] Gall | Applications du temps local aux équations différentielles stochastiques unidimensionnelles[END_REF], Lemma 6.1. Let Z ≡ {Z(t), t ≥ 0} be a real-valued semi-martingale. Suppose that there exists a function ρ :

[0, +∞) -→ [0, +∞) such that ε 0 dx ρ(x) = +∞,
for all ε > 0, and

t 0 1 {Zs>0} ρ(Z s ) d Z s < +∞,
for all t > 0 a.s. Then, the local time of Z at zero, L 0 t (Z) is identically zero for all t > 0 a.s.

The following lemma gives some useful estimates satisfied by pn d . Lemma 6.2. i) There exists a constant c > 0 such that for any t > 0 and λ > 0,

sup n≥1 sup x∈D pn 1 (t, x, •) 1 ≤ ct -λ exp(-ct)
ii) There exist constants c > 0, 1 < q < 2 such that for any t > 0 and x, y ∈ D,

sup n≥1 t 0 D |p n d (t -s, x, z) -pn d (t -s, y, z)| dz 2 ds ≤ c x -y 1 q -1 2 ;
this estimate holds with q = 1 when d = 1.

iii) For any T > 0, there exists a constant c > 0 such that for h > 0 Proof. The estimates i), iii) and iv) were proved in [START_REF] Millet | On implicit and explicit schemes for parabolic SPDEs in any dimension[END_REF], that are respectively, (A.16) of the Lemma A.4, (A.36) and (A.38) of the Lemma A.6. Concerning ii), using the fact that pn d (t, x, z) = |p n 1 (ts, x i , z i ) -pn 1 (ts, y i , z i )| 2 dz i ds.

sup n≥1 sup x∈D sup t∈[0,T ] t 0 pn d (t -s, x, •) -pn d (t + h -s, x, •) 1 ds ≤ ch
On the other hand, using notations of Gyöngy [START_REF] Gyöngy | Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations Driven by Space-Time White noise I[END_REF] and Hölder's inequality, we get 1/q (j 2 |x i -yi| 2 ∧ 1)

≤ c

n-1 j=1 1 j 2/q (j 2 |x i -yi| 2 ∧ 1)

≤ c|x i -y i | 1 q -1 2 .
Note that we choose λ and q such that 1 < q < 2 and 0 < 2λ(d -1)ξ < 1, where

1 q + 1 ξ = 1.
We have also used the following result in Section 3, for the proof see Theorem 3.10 of Chapter 5 in [START_REF] Ethier | Markov Process : characterization and convergence[END_REF]. ≤ c(1 + x ), for all t ≥ 0 and x ∈ R d , then for all initial condition probability measure µ on R d , there exists a weak solution of the SDE (6.1).

Proposition 4 . 1 .

 41 For each T > 0, the sequence {u n , n ∈ N} is tight in the space C ([0, T ], H), where H : = C (D, [0, 1]). Proof. Note firstly that u n (t, x) are in [0, 1]. Consequently, there exists a constant c(p, T ) > 0 such that for any p ≥ 1, sup n E sup 0≤t≤T sup x∈D |u n (t, x)| p ≤ c(p, T ).(4.2)

1 2 iv)

 12 For any T > 0, there exists a constant c > 0 such that for h > 0sup n≥1 sup x∈D sup t∈[0,T ] t 0 pn d (ts, x, •) -pn d (ts + h, x, •) 2 α ds ≤ ch 1-α where 0 < α < 2 ∧ d,and • α is defined on an appropriate class of functions ϕ by ϕ 2 (α) = D D |ϕ(x)||x -y| -α |ϕ(y)|dxdy

d i=1 pn 1 1

 11 (ts, x i , z i ), where x = (x i , 1 ≤ i ≤ d) and z = (z i , 1 ≤ i ≤ d), the inequality d i=1 a i -d i=1 b i ≤ d i=1 |a ib i | d j=i+1 |a j | i-1 j=1 |b j |for all real numbers a 1 , ..., a d , b 1 ..., b d , the estimate i) and the Cauchy-Schwarz inequality, we can write t 0 D |p n d (ts, x, z) -pn d (ts, y, z)| dz (ts, x j , z j ) -

d j=1 pn 1

 1 (ts, y j , z j ) dz s, x i , z i ) -pn 1 (ts, y i , z i )|dz i s, x i , z i ) -pn 1 (ts, y i , z i )|dz i

n 1

 1 (ts, x i , z i ) -pn 1 (ts, y i , z i )| 2 dz i ds ≤ s) -2λ(d-1) exp(-8j 2 (ts))ds|ϕ n j (x i )ϕ n j (y i )| 2 8q(ts)j 2 )

Theorem 6 . 1 .

 61 Consider the SDEX t = X 0 + t 0 b(s, X s )ds + t 0 σ(s, X s )dM s ,(6.1)where t ≥ 0, b : R+ × R d -→ R d and σ : R + × R d -→ R d ⊗ R d , are Borel continuous functions, M is a d-dimensional martingale in M 2 c and X is a d-dimensional process. Assume further that the R d -valued random variable X 0 is F 0 -measurable such that PX -1 0 = µ.If there exists a constant c such that b(t, x) ≤ c(1 + x )

  is a martingale with respect to the filtration {G t : t ≥ 0}.

													Then one
	can give a rigorous meaning to solution of the formal equation (1.1). A stochastic
	process u : Ω × R + × D → R, which is jointly measurable and G t -adapted, is said to be a weak solution to the equation Eq(d, u 0 , σ) if there exists a noise F of the form
	(2.1) such that for each ϕ ∈ C 2 (D) such that ϕ = 0 on ∂D, and a.s. for almost all t ≥ 0 and x ∈ D, t
	u(t, x)ϕ(x)dx =			u 0 (x)ϕ(x)dx +		u(s, x)∆ϕ(x)dxds
	D	D					0	D
		-	1 2	0	t	D	u 2 (s, x)	d i=1	∂ϕ ∂x i	(x)dxds +	0	t	D


Marie-Laure Ancelin

Isabelle Carrière

Pascale Barberger-Gateau

Sophie Auriacombe

Olivier Rouaud

Spiros Fourlanos

Claudine Berr

Anne-Marie Dupuy

Karen Ritchie

Isabelle Carri

Lipid lowering agents, cognitive decline, and dementia: the three

Keywords: Fibrate, Statin, Cognitive aging, Alzheimer s disease ', Elderly, Apolipoprotein E, Cholesteryl Exchange Transfer Protein, Prospective cohort

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

INTRODUCTION

Although there is some evidence from experimental studies that lowering cholesterol may slow the expression of Alzheimer s disease ' (AD), systematic reviews from epidemiological and clinical studies provide, however, little evidence that low cholesterol may be associated with a decrease risk of cognitive decline or AD in the elderly (see for reviews 1, 2 ). Similarly, studies of the potential reduce risk of Alzheimer s disease risk but not cognitive decline, other studies have been unable to confer this association. Inconsistencies ' could notably result from heterogeneity in study design, analytic methods, sample size, cognitive evaluation (generally limited to global cognition), and LLA characteristics ( use of statin non-statin LLA, and lipophilicity influencing capacity to cross the blood brain e.g. vs.

barrier), with very few studies specifically examining fibrates. A common problem has been failure to take into account the numerous potential vascular confounders for cognitive decline. Of particular importance is the clinical context in which lipid treatment occurs; statins being generally prescribed in the first instance for the management of low density lipoprotein-cholesterol (LDL-cholesterol), whereas fibrate prescription generally following statin treatment failure, intolerance, or in cases of atherogenic dyslipidemia 4 . This point needs to

[ ] be taken into account in clinical analyses as persons receiving fibrates are therefore likely to constitute a separate clinical sub-group from those receiving statins.

Genetic vulnerability has not been systematically considered despite a possible interactive effect between lipid levels and apolipoprotein E (ApoE). Surprisingly, cholesteryl ester transfer protein has not been examined although it is not only involved in the aetiology of atherogenic dyslipidemia phenotype but has also been associated with slower memory decline and lower incident dementia 5

[ ]
. Cholesteryl Exchange Transfer Protein promotes the exchange of triglycerides for cholesteryl ester in lipoprotein, and statin and fibrate may inhibit its activity.

In addition, cholesterol is a precursor for male and female steroids. However, gender differences have not been examined although they have been reported in relation to cognition and progression to dementia 6 , to the association between lipid levels and LLA effects 7 ,

[ ] [ ]
risk factors and progression of cardiovascular disease, and therapeutic recommendations for cholesterol management 4, 8,

[ ]

This large prospective study aims to examine the relationship between LLA use and cognitive decline and dementia onset in community-dwelling elderly taking into account LLA type, gender, genetic vulnerability, and multiple competing factors associated with of cognitive decline.

METHODS

Study participants

Subjects were recruited as part of a multi-site cohort study of community-dwelling persons aged 65 years and over, randomly selected from the electoral rolls of three French cities between 1999 and 2001 10 . The 3C study protocol was approved by the Ethical Committee

[ ]
of the University-Hospital of Bic tre (France). Written informed consent was obtained from each participant. Participants were ê administered standardized questionnaires and underwent clinical examinations at baseline and at 2, 4, and 7-year follow-up (respectively wave 1, 2 and 3). Of the 9080 dementia-free participants included at baseline, 1181 subjects having only one cognitive evaluation at baseline, 1062 subjects having missing data for at least one adjustment variable and seven subjects treated with both fibrate and statin were excluded leaving 6830 subjects in the analysis. Among the subjects excluded 693 deceased. The mean (SD) age of the analyzed sample was 73.6 (5.3) for men and 73.8 (5.2) for women. Subjects not included in the analysis were significantly older, with lower education levels, and worse physical and mental health, lower baseline cognitive scores, and were less frequently statin users (p<0.005 for each comparison).

Cognitive measures and dementia

The Isaacs Set Test provided a measure of verbal fluency or semantic access (number of items produced within 30 seconds), the administered at baseline, and waves 1, 2, and 3 of the follow-up, except the TMT which was not given in wave 1.

A three-step procedure was used to diagnose cases of dementia 10 . First, screening was based on a thorough neuropsychological Disorders Association criteria. The onset of dementia was the date of the follow-up interview when dementia was diagnosed.

Socio-demographic and clinical variables

The standardized interview included questions on socio-demographic and lifestyle characteristics with evaluation of hypertension and diabetes as well as an inventory of all drugs (including LLA and anticholinergic drugs as defined previously 12 ) used over the preceding polymorphisms) was performed.

Statistical analyses

The Chi2 test for categorical variables and the Student t test for continuous variables were used to identify treatment group differences.

Cognitive decline was evaluated over the 7-year follow-up period. The score distribution being not normal for the use of criteria based on standard deviation, cognitive decline was defined as being in the first quintile of the distribution of the differences, in the worst i.e. quintile. More specifically, a difference score was first calculated between cognitive scores obtained at any follow-up and the baseline score. The number of difference scores for a given subject thus depended on the number of follow-ups he/she had data for, from only one difference (the subject was only followed up once in the 7-year period) up to 3 differences (the subject had data on each follow-up). The quintile of the individual maximum of the difference scores was then calculated. The group of decliners was then defined as those participants belonging to the worst quintile of the worst difference scores and the time of decline was the first visit when the subject fell below the cut-off. Decliners were thus defined as persons having decreased from baseline by at least 3 points on the MMSE or on the Benton test or at least 10 points on the Isaacs test and an increase of at least 22 seconds on the TMTA or 57 seconds on the TMTB. Cox proportional hazards models with delayed entry taking age as the basic time scale and birth as the time origin 14 were used to determine [ ] whether baseline LLA use was associated with risks of cognitive decline and dementia incidence. This model enabled to take into account the full information until time of censoring, either by death or loss to follow-up, or the time of observed cognitive decline (or dementia), thus minimizing selection bias due to cohort attrition. After gender stratification, Hazard Ratios (HRs) were adjusted for centre, educational level, and baseline cognitive performance in addition to age, which was already taken into account in the model (minimally adjusted model 0). Multivariate analyses further included selected covariates associated with cognitive decline (at p<0.15), namely marital status, body mass index (BMI), mobility, as well as alcohol intake for men and anticholinergic use, depression, chronic respiratory disorders, and hormone treatment for women (model 1). Final multi-adjusted models were further adjusted for ischemic pathologies, diabetes, hypertension, LDL-cholesterol and triglyceride levels (categorized into quartiles), APOE 4, and Cholesteryl Exchange Transfer ε Protein for both gender (model 2). For incident dementia, the analysis was undertaken on 7056 subjects without missing data for baseline adjustment variables but with possibly missing repeated cognitive testing. Analyses were carried out using SAS software (version 9.2) with a significant level of p <0.05. In the Tables, we reported data without correcting for multiple testing as we consider Bonferroni adjustment to be overly conservative given that the cognitive tests independently explore different aspects of cognitive functioning involving different cerebral areas (see the Discussion section).

RESULTS

Subject characteristics

Within this elderly community-dwelling sample, 936 of the 6830 subjects (13.7 ) were taking fibrates at baseline and 1119 ( fluvastatin (3.8 ). Women were more frequent fibrate users (14.9 ) than men (11.8 , p 0.001). For both men and women, LLA use was

% % % =
higher in subjects overweight, with ischemic pathologies, diabetes, high blood pressure, low LDL-cholesterol levels, and carrying ApoE4 allele (p<0.005) (Table 1). Men with higher triglyceride levels but women with lower triglyceride levels were more frequent LLA users (p<0.001), whereas women using LLA had less frequently chronic respiratory disorders (p 0.01). There was no significant association = between LLA use and other non-vascular factors more related to general health such as depressive symptoms, mobility, and use of drugs with anticholinergic properties.

LLA use and cognitive decline

Cox models adjusted for age, center, education level and baseline cognitive performance indicated that women taking fibrates but not statins at baseline showed greater decline over 7 years on the Benton visual recall test (HR 1.24, 95 CI 1.05 1.46, p 0.04) (Table 2).

% = - = =
The same association was observed when considering the 636 women who were taking fibrate continuously throughout the 7-year follow-up (defined as a baseline intake and at least 2 consecutive fills within the 3 follow-up examinations) compared with never users of LLA (HR 1.26, 95 CI 1.02 1.56, p 0.03, in fully adjusted model 2). No significant effect was observed in men regardless of the

= % = - =
cognitive domain and the LLA type. We did not observe significant associations as a function of statin lipophilicity in either sex (data not shown).

Dementia incidence

Within the 7056 subjects included in the analysis, 483 incident cases of all dementias were diagnosed during the 7-year follow-up, of whom 332 had Alzheimer s disease (AD). Adjusted Cox models failed to find a significant association between fibrate or statin use at ' baseline and the incidence of dementia or AD in women or men (Table 3).

DISCUSSION

Our results indicate a significant 1.3-fold increased risk of decline in visual memory in elderly women treated with fibrates at baseline or continuously during the 7-year follow-up. The same results were obtained in the minimally and the fully adjusted model, which was able to take into account multiple co-determinants of cognitive decline, including pathologies associated with LLA treatment and genetic vulnerability related to both lipid levels and cognitive decline and dementia ( ApoE and Cholesteryl Exchange Transfer Protein). It e.g. should be noted that baseline global cognitive functioning was not significantly different between LLA users and non users and the same results were obtained in minimally and fully adjusted models for cognitive scores and incident dementia which suggests that an indication bias in which LLA prescribed for patients at higher risk of dementia could have confounded the associations would be highly unlikely. The association between fibrate use and decline in visual memory remained significant even after applying the conservative Bonferroni correction for multiple comparisons (2 genders x 2 LLA types, p <0.0125). No significant association between statin or fibrate use and cognitive decline was observed in men. There was no significant association between fibrate use at baseline and risk of developing dementia or AD over 7-years in men or women.

Very few studies have evaluated the specific effect of fibrates on cognitive function; being generally assimilated with other LLA (e.g. bile-acid binding resins or nicotinic acid and derivatives) as non-statin cholesterol-lowering drugs which collectively have been reported

" "

as having no significant association with dementia incidence 15 18 although one study reports an association in younger elderly subjects [ -] only 19 . In the present study, we found that only women with both relatively high LDL-cholesterol levels (> 3.1 mmol/l) and taking [ ] fibrates were at higher risk for cognitive decline in visual memory but not those with lower LDL-cholesterol levels. This notably precludes a neurotoxic effect of fibrate Fibrates and statins differ by their mechanism of action. Statins inhibit hydroxymethylglutaryl-CoA per se.

reductase the rate-limiting step in cholesterol biosynthesis producing a lowering of LDL-cholesterol levels. Fibrates are agonists for the nuclear transcription factor peroxisome proliferator-activated receptor-alpha with effects on ApoA-I, A-II, and C-III, lipoprotein lipase, and fatty acid oxidation, notably leading to a number of anti-atherogenic effects including triglyceride lowering and HDL-cholesterol increase. In various animal models, neuronal and vascular protective effects of fibrate have been reported which have recently been related to their capacity to modulate oxidative stress and inflammation 20, 21 . The negative effect found in this study on cognitive decline in memory and fibrate use remained significant after controlling for triglycerides, vascular factors, and genetic vulnerability. We cannot exclude however, the possibility that asymptomatic subclinical atherosclerosis could be involved to confound this association. The specific effect observed in elderly women treated with fibrate remains to be explained but may involve pharmacodynamic or metabolism properties as well as differences in patterns of vascular risk factors and lipid lowering capacity which can be modulated by hormonal status 4, 7 .

[ ] Current hormonal treatment is known to be associated with decreased LDL-cholesterol levels in postmenopausal women 7 . Hormonal LDL-cholesterol and not treated with hormonal treatment may be at increased risk of decline in visual memory remains to be further examined.

For statins, we only observed non significant associations with cognitive function in the minimally adjusted models but not in the fully adjusted model, suggesting that this was more likely related to the underlying burden of vascular illness rather than the statin itself (indication bias) 24, 25 . The absence of effect of statins on cognitive decline is in agreement with a number of clinical and A limitation of our study could have been the exclusion of participants, those lost to follow-up being more likely to have cognitive decline or dementia, low education level, to be older, and thus with worse physical and mental health. This may limit the generalizability of our results, and associations may have thus been underestimated. We did not consider treatment compliance, which may have caused classification bias. We did not have information on the reasons for prescribing fibrate or on precise initiation time and duration of medication use, which could not definitively address the question of the modifying role of prolonged use and of midlife optimal therapeutic window. We cannot exclude the possibility that there are other unknown factors including subclinical disease in addition to that detectable through the analysis of lipids, glycemia, and hypertension, which may confound the associations.

The strengths of this study relate to its prospective, community-based design, large size, and extensive information obtained on clinical status. LLA use was ascertained, thus minimizing exposure misclassification and evaluated at baseline and throughout the 7-year follow-up. We could examine separately fibrates and statins as well as statin lipophilicity related to blood brain barrier permeability.

Finally, we have taken into account a wide range of competing causes of cognitive dysfunction in the elderly, notably lifestyle, genetic, and health covariates especially those related to vascular disorders, thus limiting any potential confounding including prescription bias. We excluded cases of prevalent dementia from the analyses and were able to distinguish AD from other forms of dementia using a validated clinical diagnosis of dementia.

In conclusion, our study, adjusting for multiple confounders, found no evidence that LLA given in late life reduced the risk over 7 years of cognitive decline and incident dementia. It may on the other hand suggest that elderly women with treatment-resistant high LDL-cholesterol may be at increased risk of decline in visual memory, for which women were reported to be less resistant to insult and ageing than in verbal skills 35 Further studies with a more comprehensive battery of cognitive tests are required to determine whether this

[ ]
risk is indeed specific to visual memory or may be observed in other areas of information processing. Whether these women constitute a subgroup which has given rise to cognitive compromise or even a pre-dementia syndrome perhaps decades before dementia diagnosis, remains to be evaluated.

Acknowledgements:

The 3C Study is conducted under a partnership agreement between Inserm, the Victor Segalen Bordeaux II University and Sanofi-Synth

Footnotes:

Joint first authors : No conflict of interest.

Declaration of interest

=

The non significant associations concerning the other cognitive tasks were not reported (p > 0.40). Model 0: adjusted for age (time scale), centre, education, and baseline cognitive performance. Model 1: adjusted for age (time scale), centre, education, base-line cognitive performance, marital status, BMI, mobility (for men and women) as well as alcohol intake for men and depression, anticholinergic use, chronic respiratory disorder, and hormonal treatment for women. Model 2: adjusted for all the covariates in model 1, plus ischemic pathologies, diabetes, hypertension, LDL-cholesterol and triglyceride levels, ApoE4, and Cholesteryl Exchange Transfer Protein. a The median Interquartile range follow-up was 6.7 3.8 7.2 years. This analysis was undertaken on 7056 subjects without missing data for baseline adjustment variables but with possibly missing

  lipid lowering agents (LLA) remain inconclusive 3 . While some studies indicate that statins may moderately [ ]

  psychologists (see above). Data on activities of daily living, severity of cognitive disorders, and, where possible, magnetic resonance images or computed tomography scans were collected. In addition, in Montpellier and Bordeaux all participants were examined by a neurologist. In Dijon only persons suspected of having a cognitive deficit on the basis of their neuropsychological performance underwent further examination by a specialist. Finally, in the 3 centers, all suspected dementia cases were analyzed by a common independent committee of neurologists according to DSM-IV 11 criteria. This committee reviewed all potential cases of [ ] dementia to obtain a consensus on its diagnosis and etiology based on all available information. Alzheimer s disease was diagnosed ' according to National Institute of Neurological and Communicative Disorders and Stroke and the Alzheimer s Disease and Related'

  The association was strengthened in a model further adjusted for other confounders including lipid levels, ischemic pathologies associated with LLA prescription, and genetic risk factors (HR 1.29, 95 CI 1.09 1.54, p 0.004, model 2). No significant interactions were found= % = -=for decline in visual recall in women between LLA use and triglyceride (p 0.89), ApoE (p 0.26) or Cholesteryl Exchange Transfer Protein = = (p 0.29). A significant interaction was found between fibrate use and LDL-cholesterol levels (p 0.03). Women with moderate or high = = LDL-cholesterol levels (> 3.1 mmol/l, corresponding to the three upper quartiles) and treated with fibrate (n 397) were at higher risk for = decline in visual memory (HR 1.39, 95 CI 1.08 1.79, p 0.01 in model 2) compared to those with low LDL-cholesterol levels (< 3.1 ) not treated with fibrate (n 469) whereas risk was not significantly modified for those having low LDL-cholesterol and treated = with fibrate (HR 0.94, 95 CI 0.67 1.30, p 0.69, n 218) or those with moderate or high LDL-cholesterol not treated by fibrate (HR , 95 CI 0.77 1.15, p 0.54, n 2378).

  thus reflect a more severe form of hypercholesterolemia persisting despite fibrate treatment, considering that statin is generally the first-line drug to achieve LDL-cholesterol treatment goals, whereas fibrate prescription would only generally follow statin treatment failure or intolerance or elevated triglyceride levels and atherogenic dyslipidemia 4 . The association between decline in visual [ ]

  found to be specifically associated with better visual memory in the large Women s Health Initiative Memory Study ' randomized controlled trial 22 as well as in a subgroup of women of the 3C study 23 . Whether women with treatment-resistant high [ ] [ ]

curit

  Fondation pour la Recherche M dicale funded the preparation and first phase of the study. The 3C-Study is also supported by the é Caisse Nationale Maladie des Travailleurs Salari s, Direction G n rale de la Sant , MGEN, Institut de la Long vit , Agence Fran aise de S Sanitaire des Produits de Sant , the Regional Governments of Aquitaine, Bourgogne and Languedoc-Roussillon and, the Fondation de é é France, the Ministry of Research-Inserm Programme Cohorts and collection of biological material . The Lille G nop le received an " " é ô unconditional grant from Eisai. Part of this project is financed by two grants from the Agence Nationale de la Recherche (projects 07 LVIE 004 and 06-PNRA-005). We thank the G nop le of Lille, the Laboratories of Biochemistry of the University Hospitals of Dijon and é ô Montpellier, the Town Council of Dijon and the Conseil G n ral of C te d Or. Neither funding source provided scientific input to the study.

  Abbreviations: ApoE, Apolipoprotein E; BMI, body mass index; MMSE, Mini Mental State Examination; TMT, Trail Making Test. a Decline corresponded to lowest quintile of performances, a decrease from baseline of at least 3 points on the Benton test (and MMSE) or at least 10 points on the Isaacs score and an increase from i.e. baseline of at least 57 seconds on the TMTB (22 for TMTA). b Except for TMT, where n 3532 for women and 2258 for men.

b

  repeated cognitive testing; 290 women and 193 men were diagnosed with incident dementia, of whom 206 women and 126 men had incident Alzheimer s disease. The 110 women and 87 men with other ' types of dementia (of whom 21 men and 22 women with vascular dementia) were excluded from the analysis of Alzheimer s disease.' Adjusted for age (time scale), centre, and education (model 0).

  16.4 )

	%	%	%	%	%

% % statins. No subjects were taking nicotinic acid and derivatives and only 11 subjects were taking bile acid sequestrants. Fibrates consisted of fenofibrate (72.0 of fibrates), ciprofibrate (15.7 ), bezafibrate (9.6 ) and gemfibrozil (2.7 ). The lipophilic statins were simvastatin % % % %

(36.7 of statins), cerivastatin (15.0 ) and lovastatin (0.1 ) and hydrophilic statin were pravastatin (26.9 ), atovastatin (17.5 ) and

  epidemiological studies. Two large randomized controlled trials conducted in mid-to late-life adults at high risk for vascular disease did not show differential effect on cognitive function of the lipophilic simvastatin and the hydrophilic pravastatin which were taken by 64 of

	[ -]
	statin use, but it was not adjusted for baseline cognitive performance, lipid levels, and/or vascular pathologies 29, 30 . Regarding incident [ ]
	AD, several prospective cohort studies did not find significant associations even when considering lipophilicity 17, 18, 31 . A cohort study [ ]
	reported a significant negative association after 9.2 years of follow-up independent of statin lipophilicity but it did not consider lipid
	components as confounders 16 . A negative association was recently reported in early old but not the very old 32 . No large randomized [ ] [ ]
	controlled trials with statins have so far reported an AD preventive effect 26, 28 , nor an effect on AD evolution 33, 34 . [ ] [ ]

[ ] % our sample 26 28 . Two prospective cohort studies reported a significant negative association between global cognition and dementia and

Table 1

 1 Characteristics of the study population as a function of lipid lowering agent use at baseline Apolipoprotein E; BMI, body mass index; LLA, Lipid lowering agent; MMSE, Mini Mental State Examination. a The Chi2 test and the Student t test were used for categorical and continuous variables.b Mobility: assistance required to perform at least one of the three Rosow Breslau items (relating to confinement to home and neighborhood). The presence of depressive symptoms was assessed using the Center for Epidemiological Studies-Depression Scale 13 with a cut-off of 16. Chronic respiratory disorders including corticosteroid drug intake and self-reported bronchitis, wheezing, tachypnea, and asthma attacks (over the last 12months). e History of stroke, myocardial infarction, angina pectoris, or arteritis and cardio-vascular surgery f Diabetes defined as glucose 7 mmol/l or treated.

			Men				Women
		Lipid lowering agent			Lipid lowering agent
		No (n 1928) =	Yes (n 781) =	p	a	No (n 2847) =	Yes (n 1274) =	p	a
	Characteristic	%	%			%	%
	Age, y
	65 69 -	25.4	27.3	0.006	24.9	24.6	<0.0001
	70 74 -	33.7	37.4			31.1	36.7
	75 80 -	25.0	25.5			28.2	29.1
	80+	15.9	9.8			15.8	9.6
	Education, y
	5	21.2	22.4	0.30	24.5	26.5	0.13
	9	29.9	32.5			39.9	41.4
	12	19.9	19.1			21.2	20.0
	12 +	29.0	26.0			14.4	12.2
	Marital status
	Married	81.4	85.0	0.07	44.1	49.3	0.005
	Single or divorced	8.5	6.4			20.4	17.4
	Widowed	10.1	8.6			35.6	33.4
	BMI (kg/m ) 2
	Normal (<25)	40.4	32.3	0.0004	55.5	50.0	0.002
	Overweight 25 30 [ -[	47.8	54.0			31.3	36.7
	Obese ( 30) ≥	11.8	13.7			13.2	13.3
	Mobilityb	3.3	2.3	0.40	6.6	5.2	0.20
	Alcohol (g/day)
	0	8.6	6.0	0.07	27.9	24.7	0.05
	1 36 -	72.8	74.4			70.8	73.4
	> 36	18.6	19.6			1.3	1.9
	Depressive symptomsc	13.8	13.1	0.61	28.0	28.1	0.96
	Anticholinergic use	4.2	4.6	0.59	9.6	9.9	0.79
	Chronic respiratory disordersd	8.3	6.5	0.13	7.2	5.2	0.01
	Ischemic pathologiese	17.2	33.9	<0.0001	10.9	15.2	0.0001
	Diabetesf	11.0	16.1	0.0002	5.5	9.6	<0.0001
	High Blood Pressureg	58.0	66.1	0.0001	51.3	58.3	<0.0001
	At least 1 ApoE 4 ε	18.1	27.3	<0.0001	17.2	24.4	<0.0001
	Cholesteryl Exchange Transfer Protein1 (rs1800775)
	AA	30.1	30.9	0.56	30.7	31.0	0.49
	CA	40.6	38.4			41.5	39.7
	CC	29.3	30.7			27.8	29.3

c ≥ g High blood pressure defined as 160/95 mm Hg or treated. ≥ h Mean (SD) except for triglycerides (geometric mean).

Table 2

 2 Base-line lipid lowering agent use and cognitive decline over the 7-year follow-up perioda Abbreviations: ApoE, Apolipoprotein E; BMI, body mass index; MMSE, Mini Mental State Examination; TMT, Trail Making Test

					Model 0				Model 1				Model 2
			HR 95 CI [ % ]		Global p-value	p	HR 95 CI [ % ]		Global p-value	p	HR 95 CI [ % ]		Global p-value	p
	n 4121b) WOMEN ( =
	ΔBenton -3 ≤	Fibrate	1.24 1.05 1.46 [ -	]	0.04	0.01 1.23 1.04 1.45 [ -	]	0.05	0.01 1.29 1.09 1.54 [ -	]	0.01	0.004
		Statin	1.05 0.89 1.24 [ -	]		0.57 1.06 0.90 1.26 [ -	]		0.50 1.06 0.89 1.26 [ -	]	0.54
	ΔTMT B 57 ≥	Fibrate	1.16 0.95 1.41 [ -	]	0.12		1.15 0.94 1.40 [ -	]	0.15		1.03 0.83 1.28 [ -	]	0.59
		Statin	1.19 0.98 1.46 [ -	]			1.18 0.97 1.45 [ -	]			1.11 0.91 1.37 [ -	]
	n 2709b) MEN ( =
	ΔTMT A 22 ≥	Fibrate	1.20 0.91 1.59 [ -	]	0.11		1.20 0.91 1.59 [ -	]	0.10		1.14 0.84 1.55 [ -	]	0.46
		Statin	0.81 0.61 1.08 [ -	]			0.81 0.60 1.07 [ -	]			0.89 0.67 1.20 [ -	]

Table 3

 3 Baseline lipid lowering agent use and 7-year incidence of dementiaa (Cox model with delayed entry)

	HRb 95 CI [ % ]	Global p-value


Marine Agogué

Osman Akin

Armand Kazakçi

Pascal Le Hatchuel

Benoit Masson

Nicolas Weil

Mathieu Poirel

Cassotti

Akin Kazakçi

Benoit Weil

Mathieu Cassotti

THE IMPACT OF EXAMPLES ON CREATIVE DESIGN:EXPLAINING FIXATION AND STIMULATION EFFECTS

Keywords: Concept-Knowledge Design Theory, fixation effects, example in creativity

come

INTRODUCTION

Today there are large expectations regarding innovation and creativity, and some sectors seem to be less innovative than others. Trendsin cognitive psychology have clarified some obstacles that most people are likely to face in creative situations. More specifically, numerous reports have indicated that recently activated knowledge canconstrain the ability to generate creative ideas [START_REF] Abraham | Creative cognition: The diverse operations and the prospect of applying a cognitive neuroscience perspective[END_REF]. For example, creative problem solving can be inefficient when the solution requires subjects to generate an atypical object function and when the object"s typical function has been primed [START_REF] Defeyter | Acquiring an understanding of design: Evidence from children"s insight problem solving[END_REF]. In the psychology literature this effect is labeled as thefunctional fixedness or fixation. In the same vein, converging evidence reports a fixation bias when subjects are asked to generate new ideas after being exposed to solution examples. Nevertheless, in the specific field of neuroimaging,recentstudies have made significant contributions to our understanding of the neuro-cognitive influence of ideas produced by other people by demonstrating anopposing trend [START_REF] Fink | Enhancing creativity by means of cognitive stimulation: Evidence from an fMRI study[END_REF]. Indeed, in contrast with the fixation phenomena, this work highlightsthat creative cognition can be improved effectively by means of the ideas of others. Although the discrepancies in the results of these studies underscore the need for further and focused research inthis field, little is known about the potential impact of the nature of examples in the realmcreativity. This lack of knowledge is regrettable if one considers that examining the precise influence of different types of examples on idea generation may provide experimental arguments enabling to reconcile studies showing either provocative or constraining effects of examples. In the field of design science, the development of C-K theory modeling creative reasoning [START_REF] Hatchuel | A New Approach to lnnovative Design : An Introduction to C-K Theory[END_REF], [START_REF] Hatchuel | C-K design theory: an advanced formulation[END_REF]offers a new and rich interpretation of both the fixation and the stimulation effects.Our hypothesis is therefore that there are two types of examples that C-K theory helps to characterize: [START_REF] Abraham | Creative cognition: The diverse operations and the prospect of applying a cognitive neuroscience perspective[END_REF] restrictive examples that do not change the definition or the attributes of the object, and ( 2) expansive examples that modify the identity of the initial task by adding unexpected attributes. We build therefore a ICED 11 2 framework that allows us to characterize the nature of examples that provoke either fixation or stimulation effects. The purpose of this paper is to examine the impact on creativity in relation to the nature ofthe examples. We do this via a creative task where the aim is to design a way to drop a hen's egg from a height of 10m so that it does not break. We choose this familiar task because it requires minimal engineering expertise and allows many possible solutions.We then use C-K theory to chartresponses that do not include only fixation effects and to characterize the nature of the type of responses that can be possible solutions. In this way, we will test how restrictive examples andexpansive examples impact the originality of the solutionsto the design task.

LITERATURE REVIEW

Fixation and stimulation effect

Over the last six decades, numerous reports in the psychological literature have indicated that the ability to generate creative ideas could be limited by recently activated knowledge. One of the most well known examples of this failure in the domain of creative problem solving is the phenomenon of functional fixedness. Here, people fail to see new ways of using objects that could lead to an original solution to a problem, because they remain bound topreviously activated prescription of theobject. For example, in the "candle problem" [START_REF] Duncker | On problem-solving[END_REF], [START_REF] Adamson | Functional fixedness as related to problem solving: A repetition of three experiments[END_REF], subjects are provided with a candle, a book of matches and a box of tacks, and asked to fix the candle to a vertical screen in such a way that it will burn without dripping wax on the table below. The solution tothis problem requiressubjectsto use the tack box in an unfamiliar way:as a platform. Adults failed to spontaneously find this solution when the box was presented full of tacks. In other words, priming the box"s typical function blocked the subjects" ability to use the box in an atypical manner. Traditional interpretations of functional fixedness suggest that accumulated knowledge about thetypical properties of the object isautomatically activated and blocks alternative uses of the object.

Another famous example of the fixation effect was reported by Smith, Ward and Schumacher [START_REF] Smith | Constraining effects of examples in a creative generation task[END_REF], who observed and described the constraining effects of the examples used in generating creative ideas. In the classical task, participants were required to imagine and draw animals that lived on another planet that were very different to those on Earth. Prior to the drawing of the animal, participants areexposed to examples that have fundamental elements in common (such as eyes, antennas,…). Two independent assessorsnoted then whether the subjects" drawings contained any of these elements. Results showed that participants tended to incorporate these elements in their owndrawing, despite an explicit warning to avoid replicatingfeatures given inthe examples. (see also [START_REF] Jansson | Design fixation[END_REF]). Interestingly, exposure to examples did not systematically lead to fixation. Indeed, in the domain of brainstorming, exposure to ideas from others can stimulate associations that lead to the generation of additional ideas. In one experiment, Dugosh and Paulus [START_REF] Dugosh | Cognitive and social comparison processes in brainstorming[END_REF] reported that participants generated more unique solutions when a list of eight ideas was given prior to the brainstorming session. The neural basis of this stimulation effect has been recently investigated in a functional Magnetic ResonanceImaging(fMRI) study. Using a creative idea generation task where participants wererequested to generate alternative uses of conventional everyday object such as an umbrella, Fink et al. [START_REF] Fink | Enhancing creativity by means of cognitive stimulation: Evidence from an fMRI study[END_REF]reported that creative performance increases after the exposure toexamples of solutions. At a neurophysiological level, cognitive stimulation with ideas from others is associated with activation increase in a neural network involved in memory retrievaland attentional processes. Altogether these results indicate an interaction between an external idea and participants" memory and therefore suggest that exposure to examples of a solution could modulate the bottom-up attention and retrieval of novel associations, allowing participants to produce more original ideas. Nevertheless, these studies provideno cluestoaid usin understanding why examples lead to fixation effectsin some situationswhereasin otherstheystimulate creativity.

Characterizing restrictive and expansive examples

According to us, theoretical work in design science provides insights on describing different features of solutions to a design task. C-K design theory or concept-knowledge theory models the creative process as the interrelated expansion of two spaces [START_REF] Hatchuel | A New Approach to lnnovative Design : An Introduction to C-K Theory[END_REF], [START_REF] Hatchuel | C-K design theory: an advanced formulation[END_REF]. One space (the Concept space) is tree structured and describes the progressive and stepwise generation of "temptative alternatives", which are usually indecidable propositions before a"conjunction" can beinterpreted as "a solution"; the other space (the Knowledge space) is formed by the network of memorized and activated knowledge that is used for the generative process of the C-Space. One of the recognizedbreakthroughs provided by C-K theory is that it revealsthat to obtain a creative solution both processes have to reinforce each other: thinking of an alternative changes the activated sets of knowledge and also vice versa. C-K theory sets the framework of a design process based on refining and expanding the initial concept by adding attributes stemmingfrom the knowledge space. The initial concept-set is thus partitioned step-by-step in several increasingly more refined sub-concepts. There are however two kinds of partitions [START_REF] Hatchuel | A New Approach to lnnovative Design : An Introduction to C-K Theory[END_REF]. The restrictive partition is a partition where we restrict the space of possibilities without changing the definition or the attributes of the object or process to design. However, sticking to restrictive partitions does not allow the redefinition of objects; an expansive partition is a partition that modifies the identity of the initial design object by adding unexpected attributes to that of the initial concept. It is precisely because of those expansions that the true innovation, including surprises, is possible.

In this context, C-K theory explainsthe fixation effect as arising from both a limitation of C and K expansions. Interestingly, the theory predicts that people tend to generate ideas that are most accessible in memory. This activation of common knowledge could lead to fixation effects. Consequently, we hypothesized that external cues reinforcing the activation of these common knowledge should increase the fixation effect. On the contrary, example of solutions that activate knowledge that are less spontaneouslyaccessible should reduce the fixation effect and therefore stimulate creativity.

GENERAL PROCEDURE

Our investigation is divided in two phases: (1) identifying restrictive and expansive examples using C-K reasoning, (2) examining the potential impacts on originality of both types of examples based on the results of the first phase. Therefore, we first used C-K theory to explore possible paths of innovation, capturing the possible generation of new knowledge and objects on the matter of dropping an egg without breaking it. In a first study we aimed to determine both fixation path based on spontaneous knowledge and expansive path activating knowledge that is less accessible. In a pre-test session, a group of participants was asked to give solutions to the egg task. We then used this natural distribution of solutions in order to control that expansive solutions require an expansion of knowledge (ie the activation of a set of knowledge that is not spontaneously activated when designing a solution to the egg task). Then, using results of this first study, we identified restrictive and expansive examples. We then examined their specific impact on originality in a second study, where three groups of participants were asked to solve the egg task while given either of those examples. A fourth group was then given both restrictive and expansive examples.

STUDY 1: METHOD TO IDENTIFY RESTRICTIVE AND EXPANSIVE EXAMPLES

The use of C-K theory

To use C-K theory in our case, we first gathered the knowledge and expertise and the solutions usually proposed on the topic (such as shells, mattresses, parachutes). Our hypothesis was that this knowledge basis is the one that is classically activated while solving this task (see table 1). We then expanded in the C-space by making explicit the hidden partition that had been chosen to achieve the design of devices that change either the reception or the fall: those solutions design the drop of an egg from a 10-meter height without breaking it, using a device that is inert (eg not living). The expansions (a) without using a device and (b) using a living device pushed us to expand in the K-space, to look for new knowledge (on the egg properties, on living devices, etc), that then allowed us to think of new expansions in the C-space (using the natural properties of the egg or modifying them).

Phase

In the K-space In the C-space

Phase 1 Gathering of spontaneous knowledge

Model of dropping an egg from a 10meters height. Knowledge on gravity, fall, reception, forces. Benchmark of existing solutions: shells, mattress, parachutes. Some knowledge about the fragility of the egg Choice between a focus on the reception (damping the shock or protecting the egg) or a focus on the fall (slowing it)

Phase 2 Expansion in Cspace

Partition on device / with no device, and inert device / living device.

Phase 3 Expansion in Kspace

Knowledge on the properties of the egg, properties of the drop, of the environment Knowledge on living devices that break falls, or that reception well.

Phase 4 Expansion in Cspace

Using the natural properties of the egg, modifying them

And going … …

The use of C-K theory allowed us to map the possible solutions axis, as it is shown on the Figure 1. Thus, our hypothesis is that the use of C-K theory helped us to think of possible paths of solution that do not come spontaneously to mind (training an eagle to catch the egg, using the natural robustness of the longitudinal axis of an egg, freezing the egg, etc). We therefore examined in this first study whether the paths of solutions generated by the C-K reasoning are more than the fixation effects paths.

Participants

Twenty-eight undergraduate students in psychology from Paris Descartes University were recruited for this study. Subjects were between 17 and 22years old (mean age: 18,8years). Participants didn"t have previous experience with design project.

Procedure

Each participant was given 10 minutes to generate as many original solutions to the following problem: "You are a designer and you are asked to propose as many original solutions as possible to the following problem: Ensure that a hen's egg dropped from a height of 10m does not break."

Results

As illustrated in Figure 1, analysis of responses given by the participants revealed a strong fixation effect. Indeed, 81 % of the responses consist in using an inert device in order to damp the shock (33 %), protect the egg (26 %), or slow the fall (22 %)( 2 significant at p <.0005). One good example of solutions in this fixation path based on common knowledge is to slow the fall with a parachute. On the contrary, solutions belonging to expansive paths were less often proposed by the participants ( 2 significant at p <.0005). Interestingly, less than 10 % of the responses consist in using a living device (4%) or in modifying the natural properties of the egg (3%). Examples of solutions belonging to these expansive paths are to train an eagle to catch the egg during the fall or to freeze the egg before dropping it.

ICED 11 5 The key aim is now to determine whether examples belonging to the fixation path (restrictive examples) and examples belonging to expansive paths (expansive example) have an opposite effect on participants" ability to generate creative ideas.

STUDY 2 : IMPACTS OF EXAMPLES ON ORIGINALITY

In the present study, we explored the impact of restrictive and expansive examples on creativity in the egg task. Based on C-K theory, we hypothesized that the introduction of example during the egg task would provoke participants to propose more original solutions when the provided example was expansive (i.e. belonging to the expansive paths evidenced in the first study). In contrast, when the example provided was restrictive (i.e. belonging to the fixation paths evidenced in the first study),we hypothesized that the originality of the given solutions would be reduced. We have then conducted two analyses, one on the mean number of solutions and one on an objective originality score of those solutions.

Participants

One hundred and thirty two undergraduate students from Paris Descartes University were recruited for this study. Subjects were between 17 and 28years old (mean age: 19,1years). Participants were tested at the same time in a large room. All participants were naïve regarding the experimental aims and didn"t have previous experience with design project.

Procedure

Each participant was randomly assigned to one of the four experimental conditions and was given 10 minutes to generate as many original solutions to one of the following problem.

 Control group (n=28): Without example "You are a designer and you are asked to propose as many original solutions as possible to the following problem: Ensure that a hen's egg dropped from a height of 10m does not break."

 Group A (n=25): Restrictive example "You are a designer and you are asked to propose as many original solutions as possible to the following problem: Ensure that a hen's egg dropped from a height of 10m does not break. The most often proposed solution is to slow the fall with a parachute"  Group B (n= 27): Expansive example 1 "You are a designer and you are asked to propose as many original solutions as possible to the following problem: Ensure that a hen's egg dropped from a height of 10m does not break. One possible solution is to train an eagle to catch the egg during the fall."

 Group C (n= 26): Expansive example 2 "You are a designer and you are asked to propose as many original solutions as possible to the following problem: Ensure that a hen's egg dropped from a height of 10m does not break. One possible solution is to freeze the egg before dropping it."

 Group D (n= 26): Restrictive example and Expansive example 1 "You are a designer and you are asked to propose as many original solutions as possible to the following problem: Ensure that a hen's egg dropped from a height of 10m does not break. Your company specializes in parachute slowing the fall of the eggs. A competitor recently proposed as a solution the training of an eagle so that the eagle catches the egg in the air."

Results

In order to examine whether the number of solutions proposed varied according to the experimental conditions, we conducted one factor analysis of variance (ANOVA) with the experimental groups (A, B, C, and D) as a between-subjects factor. This analysis revealedthat main effect of experimental groups is close to reach significance (F(4,129) = 2.09 p = .08), indicating that our experimental conditions affect the number of solutions given by the participants. Interestingly, the two groups exposed to restrictive examples proposedfewer solutions than the others groups (see Figure 2). It should be noted that we measured the simple count of solutions generated and not a count of solution types.

Figure 2 :Mean number of solutions according to the experimental groups

As usual in divergent thinking literature, we computed an objective measurement of originality of solutions by taking the frequency of responses given across all the subjects in each group. In this score, originality of a solution is defined as the statistical infrequency of that particular solution. For example, if less than 5 % respondents report a response, that response is considered to be of high originality. On the contrary if more than 95 % respondents report a response, that response is considered to be of low originality. Thus, we obtained a mean originality score between 0 and 1 for each participant where 0 represents the lower level of originality and 1 the higher one (see Figure 3). These scores were subjected to one factor ANOVA with the experimental groups (A, B, C, and D) as a between-subjects factor. This analysis showed a main effect of experimental groups (F(4,129) = 11.219, p < .0005), indicating that the presence of examples have a significant impact on the originality of solutions. Post-hoc comparisons revealed that solutions proposed by group A are less original than those given by the control group (p < .005), or group B (p <.005) and the group C (p <.005). This suggested that the exposure to restrictive example reduces the originality of the solutions. On the contrary, our data showed that the two groups exposed to expansive examples proposed more original solutions than the control group without example (p = .07). Interestingly, participants who received both a restrictive and an expansive examples are more original than those in group A (p <.0005) with restrictive example solely. Note that there was no significant difference between group D and the two other groups with expansive examples (all p"s >.05). Beside our data also revealed that globally the first solution given by the participants is less original than the last one (F(1,129) = 28.59, p < .0005) even if there is no significant interaction between experimental groups and the first/last factor (F < 1).

DISCUSSION AND CONCLUSION

In the two studies, we explored the impact of restrictive (i.e. belonging to the fixation path) and expansive (i.e. belonging to the expansive paths) examples on a creative task that is to design a way to drop a hen's egg from a height of 10 m so that it does not break. Four major findings emerged from this investigation: (i) participants" ability to generate a creative idea is blocked by solutions that come spontaneously to mind, leading to fixation effect (ii) the introduction of expansive example during the egg taskled participants to propose more original solutions (iii) in contrast, when the example provided was restrictive, both the number and the originality of solutions were reduced; (iv) interestingly when both a restrictive and an expansive examples were presented in the design brief, participants gave fewer solutions without affecting the originality. As predicted by C-K theory, our first study revealed strong fixation effects indicating that participants gave solutions based on common knowledge spontaneously activated. This result is in keeping with studies in the domain of reasoning that demonstrated the involvement of an intuitive/automatic system in problem solving [START_REF] Kahneman | Frames and brains: Elicitation and control of response tendencies[END_REF]. In other words, when people have to solve a problem in a creative way they are constrained by automatic activation of common knowledge leading them to give usual solutions.

Interestingly enough, when the activation of this common knowledge is reinforced by the presentation of possible solutions belonging to the fixation path, participants proposed fewer solutions that were also less original. This finding is in line with several previous works and confirms that recently activated knowledge can strongly constrain the ability to generate creative solution [START_REF] Abraham | Creative cognition: The diverse operations and the prospect of applying a cognitive neuroscience perspective[END_REF], [START_REF] Smith | Constraining effects of examples in a creative generation task[END_REF].

One major result of the present investigation is that expansive example produces the opposite effect on originality. Indeed, providing example belonging to an expansive path in the C-space increases creative performance. Although this result confirms that a stimulation effect of example is possible [START_REF] Dugosh | Cognitive and social comparison processes in brainstorming[END_REF], [START_REF] Fink | Enhancing creativity by means of cognitive stimulation: Evidence from an fMRI study[END_REF], see also [START_REF] Ward | The role of specificity and abstraction in creative idea generation[END_REF], [START_REF] Howard | The use of creative stimuli at early stages of industrial product innovation[END_REF] it can appear contradictory with regardsto the vast number of studies mentioned previously reporting that examples block creativity. However thanks to C-K theory we can suggest explanation of this apparent discrepancy. The exposure to an expansive example unveils a possible expansion in the C-space to the participants, which is not the case for a restrictive example. Thus, expansive example provides with a set of alternatives in the C-space: using a living device, modifying the properties of the egg. This expansion could be however limited to the C-space in the sense that participants did not have access to ways to expand their knowledge (for instance via the internet). In addition our results suggest that participants did not transfer the fixation from the common path to the expansive examples given in the present experiment.

A limitation of our study is thatwe computed an objective measurement of originality of solutions without taking into account the appropriateness of the solutions given by the participants. Yet, a creative idea is original, but an original idea is not always creative. Therefore, it would be interesting in future works to use an external rating procedure similar to the Consensual Assessment Technique (CAT) proposed by Amabile. Although the marked interest for the study of the impact of external cues on creativity, there are many loose ends in the field and little is known about the neurocognitive processes allowing (1) to control routine strategies and (2) to generate new creative ones. Using both new modeling of creative reasoning and recent advance in neuroimaging techniques it would be possible in future work to directly observe the neural activity associated with (i) thefixation effect (ii) the neuro-cognitive impact of both restrictive and expansive examples.

Figure 1 :

 1 Figure 1 : C-K diagram for Study 1

Figure 3 :

 3 Figure 3 :Originality score according to the experimental groups

Table 1 :

 1 Sequences to build the C-K diagram