Delphine Salort
email: delphine.salort@sorbonne-universite.fr

Didier Smets
email: didier.smets@sorbonne-universite.fr

Convergence towards equilibrium for a model with partial diffusion

come

Convergence towards equilibrium for a model with partial diffusion

Introduction

The study of partial differential equations (PDE) associated to neurosciences has seen a growing interest in recent years. These models often involve a mixture of effects leading to singular terms, heterogeneities, nonlocal terms, jumps, partial diffusion mechanisms and long term memory effects, which overall make their qualitative study somewhat challenging (see for instance [START_REF] Perthame | On a voltage-conductance kinetic system for integrate and fire neural networks. Kinetic and related models[END_REF][START_REF] Fonte | Long time behavior of an age-and leaky memory-structured neuronal population equation[END_REF][START_REF] Brunel | Fast global oscillations in networks of integrate-and-fire neurons with long firing rates[END_REF][START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Dumont | The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states[END_REF][START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF][START_REF] Crevat | Rigorous derivation of the nonlocal reactiondiffusion Fitzhugh-Nagumo system[END_REF][START_REF] Fournier | On a toy model of interacting neurons[END_REF][START_REF] Höpfner | Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin-Huxley model[END_REF]). In this paper, we propose to study the following toy model, which is directly derived from a model in [START_REF] Hertäg | Analytical approximations of the firing rate of an adaptive exponential integrate-and-fire neuron in the presence of synaptic noise[END_REF]. It reads

∂ t u(t, x, y) -∂ x (xu) -∂ y (yu) -∂ xx u = δ x=1 N (t, y), (1.1)
N (t, y) = ∂ x u(t, 0, y -1), t ≥ 0, x ≥ 0, y ∈ R,

u(t, 0, y) = 0, u(t, x, y) = 0 if y ≤ 0, u(0, x, y) ≥ 0, Modulo a change of sign in the variable x, the solution u(t, x, y) can be interpreted as the probability density of a neuron to be at the membrane potential x with an adaptation current y at time t. The point x = 0 models the activation potential, and x = 1 the reset value. The function N (t, y) models the flux of neurons which discharge at time t with an adaptation current y -1. If we integrate the equation in the y variable, we find that w(t, x) := +∞ y=0 u(t, x, y)dy is a solution to

∂ t w(t, x) -∂ x (xw) -∂ xx w = δ x=1 N (t),
N (t) = ∂ x w(t, 0), w(t, 0) = 0, w(0, x) ≥ 0, +∞ x=0 w(0, x)dx = 1.

In other words w is solution to the classical LIF Fokker Planck equation, which was widely studied in the linear and non linear case in several articles (see for instance [START_REF] Delarue | Global solvability of a networked integrate-and-fire model of mckean-vlasov type[END_REF][START_REF] Carrillo | Classical solutions for a nonlinear fokker-planck equation arising in computational neuroscience[END_REF][START_REF] Carrillo | Qualitative properties of solutions for the noisy integrate & fire model in computational neuroscience[END_REF][START_REF] Cormier | Long time behavior of a mean-field model of interacting neurons[END_REF][START_REF] Cormier | Hopf bifurcation in a Mean-Field model of spiking neurons[END_REF][START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF][START_REF] Ikeda | Theoretical study of the emergence of periodic solutions for the inhibitory nnlif neuron model with synaptic delay[END_REF][START_REF] María | Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF][START_REF] Roux | Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: blow-up and global existence[END_REF]).

In particular, it was proven in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF] that, in the linear case, the solution converges exponentially fast to its stationary state, in an exponential weighted norm in L 2 . The proof consists of the use of an entropy inequality combined with Poincaré type estimates. The adaptation of this strategy fails in the context of Equation (1.1), due to the degenerate diffusion. The so-called hypocoercivity method [START_REF] Bouin | Hypocoercivity without confinement[END_REF]28,[START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF][START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF] are other possible methods which do not apply immediately here. For this reason, we propose here an entirely different approach based on an adaptation of the so-called Doeblin-Harris method. The latter has proved very performant in a number of contexts recently, as a growing number of papers have focused on the use of this method for many equations from neuroscience or physics [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF][START_REF] José | Spectral gap for the growth-fragmentation equation via Harris's theorem[END_REF][START_REF] José | Hypocoercivity of linear kinetic equations via harris's theorem[END_REF][START_REF] Dumont | The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states[END_REF][START_REF] Fonte | Long time behavior of an age-and leaky memory-structured neuronal population equation[END_REF][START_REF] Torres | A multiple time renewal equation for neural assemblies with elapsed time model[END_REF][START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF]. Following these ideas, we eventually prove exponential convergence to the stationary state for Equation (1.1). More precisely, the next theorem holds where, here and in the sequel, we denote by S(t)u 0 (x, y) := u(t, x, y) the solution operator1 :

Theorem 1.1 There exist two constants C > 0, µ > 0, such that for all initial data u 1 , u 2 satisfying (1.3), the following estimate holds

(1 + |y| + |x| 2 )|S(t)u 1 -S(t)u 2 |(x, y)dxdy ≤ Ce -µt (1 + |y| + |x| 2 )|u 1 -u 2 |(x, y)dxdy. (1.4)
As a consequence, there exists a unique stationary state u * of (1.1), (1.2), (1.3), with

(1 + |y| + |x| 2 )u * (x, y)dxdy < +∞,
and for all initial data u 0 satisfying (1.3),

(1 + |y| + |x| 2 )|u(t, x, y) -u * (x, y)|dxdy ≤ Ce -µt (1 + |y| + |x| 2 )|u(0, x, y) -u * (x, y)|dxdy,
for all t ≥ 0.

The Doeblin method (see e.g. [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF]) relies on the following simple and efficient result. Assume that Ω ⊂ R d , d ≥ 1, and that S(t) : L 1 (Ω) → L 1 (Ω) is a linear semi-group for t ≥ 0 which preserves mass and positivity: for all f ∈ L 1 (Ω)

Ω S(t)f (x)dx = Ω f (x)dx, and f ≥ 0 ⇒ S(t)f ≥ 0.
Assume also that for some time t 0 > 0, one is able to prove a uniform non trivial lower bound on the solution, relative to its initial mass. More precisely, assume that there exists t 0 > 0, α ∈]0, 1[, and a non negative ν ∈ L 1 (Ω) with total mass 1 such that for all non negative f ∈ L 1 (Ω)

S(t 0 )f ≥ αν Ω f.
(1.5)

Then S(t) possesses a unique non negative stationary state f * ∈ L 1 (Ω) with unit mass, and moreover there exists C > 0 and µ > 0 such that for all f ∈ L 1 (Ω), and for all t ≥ 0

S(t)f -f * L 1 (Ω) ≤ Ce -µt f -f * L 1 (Ω) .
The key observation leading to this result is the following. Take f ∈ L 1 (Ω) such that Ω f = 0 and decompose f = f + -f -, where f ± are respectively the positive and the negative part of f , in particular

|f | = 2 f + = 2 f -.
We write then

S(t 0 )f = S(t 0 )f + - α 2 Ω |f |(x)dx ν -S(t 0 )f -- α 2 Ω |f |(x)dx ν =: g 1 -g 2 ,
where ν and α are given in (1.5). Since the functions g 1 and g 2 are non negative by (1.5), it follows that

|S(t 0 )f |(x)dx ≤ g 1 (x)dx + g 2 (x)dx = (1 -α) |f |(x)dx,
where we have used the fact that S(•) is mass preserving. The existence of a unique stationary state and exponential convergence towards it can then be proved following standard semi-group theory arguments, mainly through iteration, the details of which can be found e.g. in [START_REF] José | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF].

In order to prove Theorem 1.1, we will have to adapt the previous strategy because the equivalent of Estimate (1.5) clearly cannot hold for the semi-group associated to Equation (1.1), in particular due to the unboundedness of our domain. To overcome this difficulty, we shall first prove some uniform lower bounds in the spirit of (1.5), provided some control of the initial data in suitable weighted norms holds (see Theorem 3.1). For that purpose, and because the diffusion term only operates in the variable x, we need to understand the delicate mechanism that allows mass to spread along y as well. The next difficulty for the proof of Theorem 1.1, assuming Theorem 3.1 holds, is that we cannot iterate easily the contraction argument as one would do in a classical Doeblin method. Instead, in the spirit of Doeblin-Harris type method (see [START_REF] José | Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups[END_REF]), we need to understand the global dynamics of the solution with respect to the added weighted norm, combined with the information of the uniform non trivial lower-bound on the solution obtained in Theorem 3.1.

The organisation of the paper is as follows. In the next section we obtain uniform lower-bounds on the solutions on compact subsets K ⊂ ]0, +∞[×[0, +∞], at the cost of integrating in the variables x and y and assuming that the second moment in the variable x and the first moment on the variable y are a priori bounded by a given constant C > 0. In the third section, still assuming these uniform bounds on the moments, we get rid of integration and obtain pointwise lower-bounds estimates. This part is complicated due to the degenerate diffusion in Equation (1.1). Finally in the last section, we finish the proof of Theorem 1.1, by combining the pointwise estimates with controls on the moments of the solution.

Uniform local mass bounds

Our main goal in this section is to prove that the solution can neither concentrate too much near the boundary in the x variable, nor have too much mass flowing to infinity. We rely on weighted norms, and define the quantities

M (t) := +∞ x=0 +∞ y=0 ((x -1) 2 + y)S(t)u 0 (x, y)dxdy. (2.1)
We shall prove Proposition 2.1 Let C > 0 be given. Then, there exists σ > 0, and for each t ≥ σ there exist ε > 0, A > 0, and R > 0 such that, for all initial data u 0 satisfying (1.3) with M (0) ≤ C, the following estimate holds

K ε,R S(t)u 0 (x, y)dxdy ≥ A, where K ε,R = [ε, R] × [0, R 2 ].
Proof of Proposition 2.1. We proceed in two steps. First, we establish a control on the growth M (t), which allows us to ensure that the mass of the solution does not too much concentrate near infinity. Next, we prove on control on the possible mass concentration near the boundary at x = 0. More precisely, we shall prove

Lemma 2.2 Let u 0 satisfying (1.3) with M (0) < +∞. Then for t ≥ 0, M (t) ≤ M (0)e -t + 3, and
Lemma 2.3 Let C > 0.
Then, there exists σ > 0, and for each t ≥ σ there exists B > 0 such that for all u 0 satisfying

(1.3) with M (0) ≤ C, X 1 (t) := +∞ x=0 +∞ y=0 xS(t)u 0 (x, y)dxdy ≥ B.
We postpone the proofs of these lemmas and proceed with the proof of Proposition 2.1. Let C > 0 be fixed as in the statement, and let t ≥ σ be given by Lemma 2.3 for that same value of C. Lemma 2.2 implies that M (t) ≤ C + 3. Hence, comparing the weights and assuming R ≥ 2, we deduce that

4 C + 3 R ≥ +∞ x=R +∞ y=0 xu(t, x, y)dxdy and C + 3 R 2 ≥ +∞ x=0 +∞ y=R 2 u(t, x, y)dxdy. (2.2)
Thanks to Lemma 2.3, we obtain that, for all R > 0 and ε > 0,

B ≤ +∞ x=0 +∞ y=0 x S(t)u 0 (x, y)dxdy = ε x=0 +∞ y=0 x S(t)u 0 (x, y)dxdy + R x=ε R 2 y=0
x S(t)u 0 (x, y)dxdy

+ R x=ε +∞ y=R 2
x S(t)u 0 (x, y)dxdy

+ +∞ x=R +∞ y=0
x S(t)u 0 (x, y)dxdy.

From the latter and (2.2), we deduce that when

R ≥ 2, R R x=ε R 2 y=0 S(t)u 0 (x, y)dxdy ≥ B -ε -5 C + 3 R •
Taking ε sufficiently small (e.g. ε = B/3), and then and R ≥ 2 sufficiently large (e.g. R = max(2, 15(C + 3)/B)) , the conclusion follows.

We proceed now with the proofs of Lemma 2.2 and Lemma 2.3.

Proof of Lemma 2.2. The function M satisfies the identity

M (t) = -M (t) + +∞ x=0 +∞ y=0 (x -1)(-x + 3) S(t)u 0 (x, y)dxdy + 2 +∞ x=0 +∞ y=0 S(t)u 0 (x, y)dxdy.
Using that (x -1)(-x + 3) ≤ 0 for x ≥ 3, and that (x -1)(-x + 3) ≤ 1 for 0 ≤ x ≤ 3, we obtain the inequality

M (t) ≤ -M (t) + 3 +∞ x=0 +∞ y=0 S(t)u 0 (x, y)dxdy = -M (t) + 3,
and therefore the announced

M (t) ≤ M (0)e -t + 3 which gives Lemma 2.2.
Proof of Lemma 2.3. We have

X 1 (t) = -X 1 (t) + N 0 (t),
where N 0 (t) = +∞ y=0 N (t, y)dy, and therefore

X 1 (t) = X 1 (0)e -t + e -t t 0 e s N 0 (s)ds. (2.3) If X 1 (0) ≥ 1 2 , we simply derive that X 1 (t) ≥ 1 2 e -t . (2.4)
Assume now that X 1 (0) ≤ 1 2 , and set, for µ > 0, Φ µ (t) := e -µx u(t, x, y)dxdy.

Multiplying Equation (1.1) by e -µx , and integrating, we obtain that

Φ µ (t) = µ 2 (t)Φ µ (t) + µ xe -µx u(t, x, y)dxdy + (e -µ -1)N 0 (t) and therefore Φ µ (t) ≥ µ 2 (t)Φ µ (t) + (e -µ -1)N 0 (t).
Mass conservation implies that for all t ≥ 0, Φ µ (t) ≤ 1, and so

t s=0 e -µ 2 s N 0 (s)ds ≥ 1 (1 -e -µ ) Φ µ (0) -e -µ 2 t .
On the other hand, as M (0) ≤ C, we deduce that there exists C 1 > 0 such that

C 1 x=0 C 1 y=0 u(0, x, y)dxdy ≥ 1 2
and so Φ µ (0) ≥ 1 2 e -C 1 µ , which gives

t s=0 N 0 (s)ds ≥ t s=0 e -µ 2 s N 0 (s)ds ≥ 1 (1 -e -µ ) 1 2 e -C 1 µ -e -µ 2 t .
Taking µ = 1, we obtain that there exists a constant C 1 > 0 and σ > 0 such that for all t ≥ σ,

t s=0 N 0 (s)ds ≥ C 1 .
Coming back to Estimate (2.3), we obtain that, for t ≥ σ,

X 1 (t) ≥ e -t C 1 ,
which concludes the proof of Lemma 2.3.

Uniform pointwise lower bounds

The aim of this section is to prove the following theorem.

Theorem 3.1 Let C > 0. Then, there exists t * > 0, and for each t 0 ≥ t * there exist α ∈]0, 1[ and an integrable function

ν : R + × R + → R + with ν(x, y)dxdy = 1,
such that for all initial data satisfying (1.3), with M (0) ≤ C, where M is defined by (2.1), the following estimate holds u(t 0 , x, y) ≥ αν(x, y).

The main difficulty of the proof lies in the fact that we have a diffusion only in the variable x. It is therefore necessary to understand how the source term, coupled with the drift term in the variable y, allows to gain a sufficiently diffusive dynamics in y. The sketch of the proof goes as follows: First, by a suitable change of variable, we observe that Equation (1.1) can be written as infinitely many heat equations, parametrized by y, with a source term that couples all equations together. For the 1D linear heat equation, with a Dirichlet boundary term, it can be explicitly verified that the derivative in x of the solution at 0 can, for any time, be bounded from below using a weighted average of the initial data at time 0 (this is the object of Lemma 3.2 below). Using Duhamel's formula for our new set of equations, which involves a time integral of the source term, we obtain a control from below on the solution via the coupling term, which itself can be controlled using Lemma 3.2. One key point is that the integral in time obtained via Duhamel's formula also provides in the original variables some implicit integration in the y variable, because the derivative of the velocity of the drift term with respect to y is non trivial. We begin with the explicit boundary derivative control for the heat equation, and the proceed with the actual proof of Theorem 3.1 Lemma 3.2 Let v : R + × R + → R + be the solution of the equation

∂ t v -∂ 2 x v = 0, v(t, 0) = 0, v(0, x) ≥ 0.
Let S(t) := ∂ x v(t, 0). Then,

S(t) = 1 t √ 4πt +∞ 0 xe -x 2 4t v(0, x)dx.
Proof of Lemma 3.2. We have

v(t, x) = 1 √ 4πt +∞ 0 (e -(x-y) 2 4t -e -(x+y) 2 4t
)v(0, y)dy.

Hence,

∂ x v(t, x) = 1 t √ 4πt +∞ 0 ((y -x)e -(x-y) 2 4t + (x + y)e -(x+y) 2 4t
)v(0, y)dy, which yields the result, taking x = 0.

Proof of Theorem 3.1.

First observe that by direct computation, if u is a solution to (1.1), then

n(t, x, y) := 1 (2t + 1) u(ln( √ 2t + 1), x √ 2t + 1 , y √ 2t + 1 ) (3.1) satisfies ∂ t n -∂ 2 x n = δ x= √ 2t+1 G(t, y), G(t, y) = ∂ x n(t, 0, y - √ 2t + 1). (3.2)
By Duhamel formula, it follows that n(t, x, y) = e -t∆ n(0, x, y) )G(s, y)ds.

+ t 0 e -(t-s)∆ δ x= √ 2s+1 G(s,
From the latter, we deduce that for any compact set K ⊂]0, +∞[, and for any t ≥ 0, there exists a positive constant C(t, K) such that

n(t, x, y)I x∈K ≥ C(t, K) t 0 G(s, y)ds. (3.3)
We now estimate the term involving G. To this end, observe first that by the comparison principle G(t, y) ≥ S(t, y), where S(t, y) is given by S(t, y) := ∂ x v(t, 0, y), where

∂ t v -∂ 2 x v = 0, v(t, 0, y) = 0, v(0, x, y) = n(0, x, y) ≥ 0.
Hence, applying Lemma 3.2 in Estimate (3.3), we obtain that for all t > δ > 0

n(t, x, y)I x∈K ≥ C(t, K, δ) t s=δ +∞ x=0 xe -x 2 4s n(δ, x, y - √ 2s + 1)dxds,
for some positive C(t, K, δ). In turn this implies that for any compact interval K ⊂]0, +∞[, we have

n(t, x, y)I x∈K ≥ C(t, K, K , δ) y- √ 2δ+1 z=y- √ 2t+1 x∈K n(δ, x, z)dxdz,
for some positive C(t, K, K , δ). Using Relation (3.1) between u and n, and rescaling the variable z, we infer that for all compact interval K and K of ]0, +∞[,

u(ln( √ 2t + 1), x √ 2t + 1 , y √ 2t + 1 )I x∈K ≥C(t, K, δ, K ) y √ 2δ+1 -1 z= y- √ 2t+1 √ 2δ+1
x∈K u(ln( √ 2δ + 1), x, z)dxdz.

(3.4)

Let A > 0, ε > 0, R > 0 be given by Proposition 2.1, where in the statement of Proposition 2.1, we take the value of C identical to its value in the statement of Theorem 3.1 and with the choice t = σ. We next choose δ in such a way that ln( √ 2δ + 1) = σ, and let K = [ε, R]. The interval

K := y ≥ 0 s.t. y ≤ √ 2t + 1 and y ≥ (R 2 + 1)e σ
has positive length provided t > t * := 1 2 ((R 2 + 1) 2 e 2σ -1), and by construction we have

[0, R 2 ] ⊂ [ y - √ 2t + 1 √ 2δ + 1 , y √ 2δ + 1 -1],
∀y ∈ K , so that by (3.4) and Proposition 2.1, for all t > t * ,

u(ln( √ 2t + 1), x √ 2t + 1 , y √ 2t + 1 )I x∈K I y∈K ≥ C(t, σ, K) K ε,R u(σ, x, z)dxdz ≥ C(t, σ, K)A.
This implies that for all t > t * , there exists a compact set of positive measure K ⊂ R + × R + , such that u(ln( √ 2t + 1), x, y)I (x,y)∈K ≥ C(t, σ, K)A.

As for all t > 0, ln( √ 2t + 1) < t, we deduce that for all t 0 ≥ t * , there exists a compact set of positive measure K ⊂ R + × R + , such that u(t 0 , x, y)I (x,y)∈K ≥ C(t 0 , σ, K)A, which concludes the proof of Theorem 3.1 by taking ν = We first recall for completeness the classical argument showing that Estimate (1.4) implies that there exists a unique stationary state u * of (1.1), (1.2), (1.3), such that

+∞ x=0 +∞ y=0 (1 + |x| 2 + |y|)u * (x, y)dxdy < +∞.
Let L 1 m denote the Banach space of integrable functions f : R + × R + → R, endowed with the norm

• L 1 m , where f L 1 m = +∞ x=0 +∞ y=0 (1 + |x| 2 + |y|)|f |(x, y)dxdy.
For t > 0, let Ψ t : L 1 m → L 1 m defined by Ψ t (f ) = S(t)f . The application Ψ t is well defined because of Lemma 2.2, and if t is sufficiently large, it follows from Estimate (1.4) that there exists κ < 1 such that for all (f, g)

∈ L 1 m × L 1 m , Ψ t (f ) -Ψ t (g) L 1 m ≤ κ f -g L 1 m
, and Picard fixed point theorem yields a unique u * ∈ L 1 m such that

S(t)u * = u * .
On the other hand, we have for all t ≥ 0,

S(t)(S(t )u * ) = S(t )(S(t)u * ) = S(t )u * .
Hence, for all t ≥ 0, S(t )u * is a fixed point of Ψ t which implies that for all t ≥ 0, S(t )u * = u * .

It remains to prove Estimate (1.4). For that purpose, we consider u 1 and u 2 two initial data satisfying (1.3), and set

m 0 (t) := +∞ x=0 +∞ y=0 |S(t)(u 1 -u 2 )|(x, y)dxdy M (t) := +∞ x=0 +∞ y=0 ((x -1) 2 + y)|S(t)(u 1 -u 2 )|(x, y)dxdy.
We note that by conservation of mass and positivity, the quantity m 0 is a non increasing function.

The next lemma, based on Theorem 3.1, provides a first contraction type result, assuming a uniform control on M (t) with respect to m 0 (t).

Lemma 4.1 Let γ > 0. There exists t * > 0, and for any t 0 ≥ t * there exists α ∈]0, 1[, such that for any t ≥ 0, if M (t) ≤ γm 0 (t) then m 0 (t + t 0 ) ≤ αm 0 (t).

Proof of Lemma 4.1. We decompose S(t)(u 1 -u 2 ) =: µ + -µ -, where µ ± denote respectively the positive and the negative parts of S(t)(u 1 -u 2 ). We have

+∞ x=0 +∞ y=0 µ + (x, y)dxdy = +∞ x=0 +∞ y=0 µ -(x, y)dxdy = 1 2 m 0 (t).
We also have, for all (x, y)

∈ R + × R + , |S(t)(u 1 -u 2 )(x, y)| ≥ µ + (x, y) and |S(t)(u 1 -u 2 )(x, y)| ≥ µ -(x, y),
and therefore from the assumption M (t) ≤ γm 0 (t) we deduce that

+∞ x=0 +∞ y=0 ((x -1) 2 + y)µ ± (x, y)dxdy ≤ 2γ +∞ x=0 +∞ y=0 µ ± (x, y)dxdy.
Applying Theorem 3.1 with initial data µ ± , it follows that there exists a time t * > 0 and for any t 0 ≥ t * there exists α ∈]0, 1[ and a function ν as in Theorem 3.1, such that

S(t 0 )(µ ± ) ≥ α m 0 (t)ν 2 •
We write

S(t 0 + t)(u 1 -u 2 ) = S(t 0 )(µ + ) - m 0 (t) 2 ν + m 0 (t) 2 ν -S(t 0 )(µ -),
and taking the absolute value and integrating yields

+∞ x=0 +∞ y=0 |S(t + t 0 )(u 1 -u 2 )|(x, y)dxdy ≤ α +∞ x=0 +∞ y=0 |S(t)(u 1 -u 2 )|(x, y)dxdy
which ends the proof of Lemma 4.1.

Because it requires an a priori bound on M (t), it is not clear a priori that it can be iteratively used in order to derive and exponential decay. As a matter of fact, even assuming that there exists t≥ 0 such that M m 0 (t) ≤ γ, we do not know, for example, if for k ≥ 1 we still have M m 0 (t + kt 0 ) ≤ γ. In order to overcome this difficulty, we shall split the time space into two regions determined by the ratio M m 0 with respect to a parameter γ: in one region we will directly make use of that ratio bound, while in the other will make use of Lemma 4.1.

We being with the case where M is large with respect to m 0 .

Lemma 4.2 Let γ > 0. Assume that M ≥ γm 0 on some interval I = [T 1 , T 2 ]. Then M (t) ≤ e (-1+ 3 γ )t M (T 1 ), for all t ∈ I. Proof of Lemma 4.2. Set p(t) := |S(t)(u 1 -u 2 )|. Then, ∂ t p -∂ x (xp) -∂ y (yp) -∂ xx p ≤ δ x=1 |N 1 (t, y) -N 2 (t, y)|
where N i (t, y) = ∂ x S(t)u i (0, y -1). Multiplying the above equation by (x -1) 2 + y and integrating, we derive that

M (t) ≤ -M (t) + 3m 0 (t) ≤ -M (t) + 3 γ M (t), (4.1)
and the conclusion follows from Gronwall's lemma.

We now proceed with the region where M ≤ γm 0 . To deal with that situation, we define a suitable area where one can iterate over Lemma 4.1, in order to control m 0 , and where, at the same time, a sufficient control on M can be established. More precisely, we have Lemma 4.3 Let γ > 0, let t * be given by Lemma 4.1 for this choice of γ, and let t 0 > t * .

Assume that for some T ≥ 0, M (T ) ≤ γm 0 (T ), and set

N = N (T, γ, t 0 ) := sup k ≥ 0 s.t. M (T + jt 0 ) ≤ γm 0 (T + jt 0 ) ∀ 0 ≤ j ≤ k ∈ N ∪ {+∞}.
There exists 0 < β ≤ 1 2 , depending only on γ and t 0 , such that

m 0 (t) ≤ 2e -β(t-T ) m 0 (T ), (4.2)
and

M (t) ≤ e -(t-T ) M (T ) + 12e -β(t-T ) m 0 (T ), (4.3)
for all t ∈ I N , where I N := [T, T + (N + 1)t 0 ) if N is finite, and I N := [T, +∞) otherwise.

Proof of Lemma 4.3. Inequality (4.2) follows from Lemma 4.1 and the fact that m 0 is a decreasing function. Indeed, for 0 ≤ k ≤ N we can apply Lemma 4.1 at the point T + kt 0 and therefore obtain that m 0 (T + (k + 1)t 0 ) ≤ αm 0 (T + kt 0 ).

Using that m 0 is a decreasing function, it follows that

m 0 (t) ≤ α k m 0 (T ), ∀t ∈ [T + kt 0 , T + (k + 1)t 0 ], ∀0 ≤ k ≤ N,
and therefore (4.2) holds provided e -βt 0 ≥ max( 1 2 , α). We choose β := max( 1 2 , -1 t 0 log(max( 1 2 , α)). Estimate (4.3) is then a direct consequence of the fist inequality in (4.1). Indeed, we have

M (t) ≤ e -(t-T ) M (T ) + 3e -t t T
e s m 0 (s)ds, and using (4.2) as well as β ≤ 1 2 , we bound

3e -t t T e s m 0 (s)ds ≤ 6e -β(t-T ) m 0 (T ) t T e -(1-β)(t-s) ds ≤ 12e -β(t-T ) m 0 (T ).
This concludes the proof of Lemma 4.3.

Proof of Theorem 1.1 completed. Let γ = 24 and let t * be given by Lemma 4.3 for this choice of γ. We then set t 0 = max(t * , 2 log(2)), and denote by β the value provided by Lemma 4.3 for such values of γ and t 0 . We split the argument in three steps.

Step 1: Initialization. We distinguish two cases.

Case 1: M (0) ≤ γm 0 (0). We may then apply Lemma 4.3 with T = 0, and we distinguish again two subcases depending on N := N (T, γ, t 0 ). Case 1a: N = +∞. In that case, summation of (4.2) and (4.3) leads to

M + m 0 (t) ≤ 14e -βt M + m 0 (0), ∀ t ≥ 0, (4.4)
which immediately implies (1.4). Case 1b: N < +∞. In that case, we set T 0 = (N + 1)t 0 and similarly summation of (4.2) and (4.

3) yields M + m 0 (t) ≤ 14e -βt M + m 0 (0), ∀t ∈ [0, T 0 ]. (4.5)
By definition of N , we also have M (T 0 ) > γm 0 (T 0 ). Case 2: M (0) > γm 0 (0). In that case, we simply set T 0 = 0. Obviously, (4.5) also holds.

Step 2: Induction. Assume that for some k ∈ N we have constructed T k ≥ 0 such that

M (T k ) > γm 0 (T k ), (4.6)
M (T k ) ≤ 14e -βT k M + m 0 (0), (4.7)
(M + m 0 )(t) ≤ 28e -βt M + m 0 (0), ∀ 0 ≤ t ≤ T k . (4.8) Let T k = sup{t ≥ T k s.t. M (s) > γm 0 (s) ∀ s ∈ [T k , t]}.
We distinguish two cases. Case 1: T k = +∞. Then by Lemma 4.2 and in view of the inequality -1

+ 3 γ = -7 8 ≤ -1 2 ≤ -β we obtain that for all t ≥ T k (M + m 0 )(t) ≤ (1 + 1 γ )M (t) ≤ (1 + 1 γ )e -β(t-T k ) M (T k ) ≤ 28e -βt M + m 0 (0),
where we have used (4.7) for the last inequality. It follows that the equivalent of (4.8) holds for all t ≥ 0, and there is nothing left to prove in that case. Case 2: T k < +∞. By continuity it holds M (T k ) = γm 0 (T k ). By Lemma 4.2, and using once more that -1

+ 3 γ = -7 8 ≤ -1 2 ≤ -β, we have M (t) ≤ e -β(t-T k ) M (T k ) ∀ t ∈ [T k , T k ], (4.9)
and therefore by definition of

T k M + m 0 (t) ≤ 1 + 1 γ M (t) ≤ 1 + 1 γ e -β(t-T k ) M (T k ) ∀ t ∈ [T k , T k ]. (4.10)
We distinguish once more two subcases depending on N := N (T k , γ, t 0 ). Case 2a: N = +∞. Summation of (4.2) and (4.3), together with (4.9) (evaluated at t = T k ) yields

M + m 0 (t) ≤ 14e -β(t-T k ) M (T k ) ≤ 14e -β(t-T k ) M (T k ), ∀ t ≥ T k . (4.11)
Using (4.7) to control M (T k ) in both (4.11) and in (4.10), and combining the control so obtained for all t ≥ T k with the existing control (4.8) for t ≤ T k finishes the proof in that case. Case 2b: N < +∞. This is the only case that will potentially repeat itself, it is therefore important that it yields an exponential decay without a pre-multiplying factor. We set Since by construction t 0 ≥ 2 log(2) and β ≤ 1 2 , we have e -(t-T k ) ≤ e -1 2 (t-T k ) e -β(t-T k ) ≤ 1 2 e -β(t-T k ) ∀ t ≥ T k = T k + t 0 .

T k = T k + t 0 and T k+1 = T k + (N + 1)t 0 . Inequality (4.3) combined with m 0 (T k ) = M (T k )/γ yield M (
It follows therefore from (4.12) and the choice γ = 24 that

M (t) ≤ e -β(t-T k ) M (T k ) ∀ t ∈ [T k , T k+1 ]. (4.13)
Combining (4.13) and (4.9) (evaluated at t = T k ) we deduce that

M (t) ≤ e -β(t-T k ) M (T k ) ∀ t ∈ [T k , T k+1 ] \ (T k , T k ), (4.14)
and therefore by the inductive assumption (4.7) we obtain

M (t) ≤ 14e -βt M + m 0 (0) ∀ t ∈ [T k , T k+1 ] \ (T k , T k ). ( 4

.15)

The latter evaluated at t = T k+1 reproduces the equivalent of (4.7) at t = T k+1 . Note also that by definition of N we have M (T k+1 ) > γm 0 (T k+1 ), i.e. After summation with (4.16) or (depending on the value of t) (4.15), and using that ( 3 2 + 2 γ )14 ≤ 28, we finally derive (4.8) for all t ≤ T k+1 .

Step 3: Conclusion. If the algorithm presented in Step 2 exits early (i.e. if either case 1 or case 2b holds at some point), then we already showed that there was nothing left to prove. If it doesn't then the sequence (T k ) k≥0 is necessarily unbounded, since T k+1 -T k ≥ t 0 for all k ≥ 0. Inequality (4.8) therefore eventually holds for all t ≥ 0, and the proof of Theorem 1.1 is hereby completed.

Aknowledgments: Delphine Salort is supported by the ANR project ChaMaNe (ANR-19-CE40-0024). Didier Smets is supported by the ANR project ODA (ANR-18-CE40-0020-01).

  I K |K | , where |K | denotes the Lebesgue measure of K . 4 Proof of Theorem 1.1

  t) ≤ e -(t-T k ) M (T k ) + 12 γ e -β(t-T k ) M (T k ), ∀ T k ≤ t ≤ T k+1 . (4.12)

( 4 . 6 )

 46 also holds at time T k+1 . It remains to prove the inequality in (4.8) holds for all t ≤ T k+1 . In view of(4.15), this amounts to bound M (t) for t ∈ (T k , T k ) and m 0 (t) for t ∈ [T k , T k+1 ]. For the first one, we simply refer to (4.12) and combine it with (4.15) (evaluated at t = T k ) to obtainM (t) ≤ 3 2 14e -βt M + m 0 (0) ∀ t ∈ [T k , T k ]. (4.16) Regarding m 0 (t), first by definition of T k and (4.15)m 0 (t) ≤ 1 γ M (t) ≤ 1 γ 14e -βt M + m 0 (0) ∀ t ∈ [T k , T k ].Second, by (4.2) we havem 0 (t) ≤ 2e -β(t-T k ) m 0 (T k ) = 2 γ e -β(t-T k ) M (T k ) ≤ 2 γ 14e -βt M + m 0 (0) ∀ t ∈ [T k , T k+1 ].

Although we shall not study the existence part for the Cauchy problem here, our estimates below could actually be used to do so.


Arafet Bouhalleb
email: abouhalleb@edcparis.edu

Ali Smida
email: alismida@aol.com

Exploring the relationship between scenario planning and strategic flexibility and complexity

Keywords: scenario planning, strategic flexibility, strategic complexity

An organisation's ability to deal with uncertainty and to adapt to changes is a key strategic capability in dynamic environments. Thus, firms need a flexible planning in order to adapt and survive. Scenario planning is one of the tools that stimulates strategic thinking and offers strategic choices by creating multiple futures. However, little evidence is available about its effects on organisational competencies. This paper explores the direct contributions of scenario planning to strategic flexibility and complexity. A structural equation modelling (SEM) is used to evaluate the causal links between concepts. Data analysis derived from a sample of 108 European manufacturing firms.

Introduction

Organisational competencies such as strategic flexibility, innovation and strategic complexity result from the firm's ability to negotiate and capitalise on uncertainty and dynamism in its environment (Neill and Rose, 2006;[START_REF] Sinkula | Market information processing and organizational learning[END_REF]. Under such conditions, firms need a flexible strategic planning in order to adapt and survive [START_REF] Chakravarthy | A new strategy framework for coping with turbulence[END_REF]Jarzabkowski and Balogun, 2009;[START_REF] Phelps | Does scenario planning affect performance? Two exploratory studies[END_REF][START_REF] Titus | Aligning strategic processes in pursuit of firm growth[END_REF][START_REF] Wiltbank | What to do next? The case for nonpredictive strategy[END_REF]. Scenario planning is a tool for considering the future through the understanding of determinant factors and logical paths that orient strategic decision making. A large body of literature on scenario planning has focused on the widespread of this tool in companies and government [START_REF] Derbyshire | Preparing for the future: development of an 'antifragile' methodology that complements scenario planning by omitting causation[END_REF]Bowmen, 2015). The most cited literature presents scenario planning as an innovative activity that deals with uncertainties and thus improves strategic decisions.

While there is a wealth of literature in scenario planning (cf. [START_REF] Amer | A review of scenario planning[END_REF]Rawland and Sapaniol, 2017;Bowmen, 2015), research on the theory development and its benefits in practice have not yet become developed [START_REF] Chermack | Studying scenario planning: theory, research suggestions, and hypotheses[END_REF]Derbyshire, 2016, Bouhalleb and[START_REF] Bouhalleb | Scenario planning: an investigation of the construct and its measurement[END_REF]. In fact, a large body of research focuses on the description and improvement of different methods (see, for example, [START_REF] Amer | A review of scenario planning[END_REF][START_REF] Bradfield | The origins and evolution of scenario techniques in long range business planning[END_REF]. However, some progress has been made in addressing this issue, for example, [START_REF] Meissner | Cognitive benefits of scenario planning: its impact on biases and decision quality[END_REF], who examined its impact on biases and decision quality. Similarly, some researches have focused on scenario planning role on performance [START_REF] Maarten | Perceptions of the relationship between scenario planning and firm performance: a qualitative study[END_REF][START_REF] Phelps | Does scenario planning affect performance? Two exploratory studies[END_REF]. Yet, despite this, little is known about its effects on organisational competencies. In this paper, we address these questions and analyse the scenario planning's effect on strategic flexibility and complexity. As such, this work is designed to contribute to areas of scenario planning and strategic orientation in three ways. Firstly, in the relation to scenario planning, this study partially fulfils the recommendations of researchers who have suggested additional studies to better understand scenario planning outcomes (Bowmen, 2015;[START_REF] Chermack | Improving decision-making with scenario planning[END_REF], De Smedt et al., 2013, Visser and Chermack, 2009;[START_REF] Meissner | Cognitive benefits of scenario planning: its impact on biases and decision quality[END_REF]. Secondly, in the relation to strategic orientation, it seeks to achieve an in-depth understanding of scenario planning's contribution to firm competitiveness by studying how this strategic tool shapes strategic flexibility and complexity. Thirdly, the strategic planning literature has received serious criticism based on reliance on empirical studies that investigated direct and bivariate relationships [START_REF] Rudd | Strategic planning and performance: extending the debate[END_REF]. Our research addresses this criticism by investigating the mediating effect of strategic flexibility on scenario planning and strategic complexity.

This paper has the following structure. Section 2 presents the conceptual model and hypotheses. Section 3 outlines the study's methodological approach based on a sample of 108 European manufacturing firms. Section 4 offers results. Finally, section 5 and 6 present discussion and principal conclusions.

Conceptual model and hypotheses

About scenario planning

Scenario planning literature increasingly emphasises on the strategic role of this tool. As a process, it contributes at various levels of the organisation by generating suitable inputs for planning and decisions and facilitating interaction between stakeholders [START_REF] Becker | Scenarios: a tool of growing importance to policy analysts in government and industry[END_REF]De Smedt et al., 2013). The composition of this process varies across authors and approaches (see, for example, [START_REF] Bradfield | The origins and evolution of scenario techniques in long range business planning[END_REF], but generally speaking they deal with three phases: a preparatory phase, a development phase and a use phase. The first stage is related to preparation, where the purpose is to identify driving forces. It deals with the macro-external environment using tools such as PEST or SWOT. The objective is to highlight crucial uncertainties that may impact the decision making process. The second stage deals with the generation and development of scenarios. In this vein, the purpose of scenario planning is to develop several features in order to assess strategic options and capabilities (Shoemaker, 1995). The last step concerns scenario use where the aim is to enrich strategic initiatives that lead to concrete action. As advocated by Chermack and van der Merwe (2003), scenarios are used to test decisions against a wide range of futures. Such framework promotes strategic thinking within firms. In fact, strategic choices are the result of interactions and dialogue that occur between stakeholders. O'briens and Meadows (2013) emphasise on the role of scenario planning on strategic thinking. They conclude that scenario role is oriented toward strategy development. Furthermore, the scenario planning process deals with environmental uncertainty and complexity, it combines the generation of stories and images about the environment and its potential evolution and the plausible scenario leading to strategic choices and actions. As such, it is seen as a tool for considering the future [START_REF] Derbyshire | Potential surprise theory as a theoretical foundation for scenario planning[END_REF] and supporting strategic decision making [START_REF] Bootz | Strategic foresight and organizational learning: A survey and critical analysis[END_REF]. In this vein, [START_REF] Wilson | From scenario to strategic action[END_REF] suggested four types of link between scenario planning and strategy: the first is sensitivity/risk assessment in which the main role of these activities -particularly scenario work -is to evaluate a specific strategic decision. This approach is used to assess a specific strategic process by comparing the key environmental conditions such as market growth rate, technological developments and the future environment evolution with the decision. It is then possible to assimilate scenario's conditions with the desired future and make decisions. The second approach deals with strategy evaluation with scenarios as a tool to evaluate the viability of an existing strategy. The aim is to check the strategy's effectiveness in a range of business conditions, especially opportunities and threats. The third approach consists of selecting one of the scenarios as a strategic point in order to develop strategy; however, other scenarios are used to test the strategy's resilience. The fourth one focuses on strategy development, and aims at the development of a resilient strategy which is able to deal with variations and changes.

Figure 1 Theoretical model

Strategic complexity

Scenario planning

Strategic flexibility

Product orientation

Customer orientation

Competitor orientation Macroenvironment orientation

Scenario planning and flexibility

Strategic flexibility refers to the extent to which organisations identify changes in their environment, commit resources to new courses of actions in response to these changes and be reactive when necessary to stop or invest their commitment of current resources (Simizu and Hilt, 2004). Flexible firms have diversity in strategic responses and rapid shifts from one strategic choice to another [START_REF] Sanchez | Strategic flexibility in product competition[END_REF][START_REF] Stalk | Competing on capabilities: the new rules of corporate strategy[END_REF]. As argued by Santos-Vijandi et al. (2012), the increased rates of change in product and process technology and also in competitor's strategic actions make difficulties to build sustainable competitive advantages. Thus, to survive in such conditions, firms need flexibility in their strategic actions and planning [START_REF] Cingoz | Strategic flexibility, environmental dynamism, and innovation performance: an empirical study[END_REF]Eisenhart, 1989;[START_REF] Krupp | Strategic flexibility and exchange rate uncertainty[END_REF][START_REF] Nadkarni | Strategic schemas, strategic flexibility, and firm performance: the moderating role of industry clockspeed[END_REF][START_REF] Sanchez | Strategic flexibility in product competition[END_REF][START_REF] Zhang | Is support for strategic flexibility, environmental dynamism, and firm performance: an empirical investigation[END_REF][START_REF] Zhou | Technological capability, strategic flexibility, and product innovation[END_REF]. As such, firms' competitiveness depends on their ability to manage information and knowledge in order to innovate and to adapt to market through new products and services [START_REF] Ferraris | Open innovation in multinational companies' subsidiaries: the role of internal and external knowledge[END_REF][START_REF] Martinez | An empirical study about knowledge transfer, entrepreneurial orientation and performance in family firms[END_REF].

The existing literature proposes two elements of strategic flexibility: resource deployment and competitive actions [START_REF] D'aveni | Hypercompetition: Managing the Dynamics of Strategic of Strategic Maneuvering[END_REF][START_REF] Eisenhardt | Dynamic capabilities: what are they?[END_REF]. Firstly, resource deployment concerns the firm's ability to allocate and control resources in order to adapt to changes and move quickly into new niches [START_REF] Eisenhardt | Dynamic capabilities: what are they?[END_REF]. Secondly, competitive actions are barriers through which firms protect their advantages. These actions determine the ability to respond quickly to competitor threats [START_REF] Young | Austrian' and industrial organization perspectives on firm-level competitive activity and performance[END_REF].

By examining the strategic foresight literature, scenario planning is a suitable tool for promoting strategic flexibility. In fact, the use of scenario planning is intended to understand the business environment and to cope with uncertainty [START_REF] Tapinos | Perceived environmental uncertainty in scenario planning[END_REF]. Similarly, O' Briens and Meadows (2013) suggest that this strategic tool improves the portfolio of possible strategic initiatives that facilities adaptation to environmental changes and action. It permits also to create new possibilities for actions and to develop a more comprehensive strategy [START_REF] Wack | Scenarios: shooting the rapids[END_REF][START_REF] Schoemaker | Scenario planning: a tool for strategic thinking[END_REF][START_REF] Porter | Competitive Advantage: Creating and Sustaining Superior Performance[END_REF]. Strategic flexibility is promoted through strategic conversation. Moreover, scenario planning is largely based on dialogue and strategic conversation. As mentioned by [START_REF] Chermack | Technological forecasting and social change[END_REF], the effectiveness of scenario planning is related to the ability of actors to dialogue and discuss alternative futures of their company. As a conversation tool, it permits to change current assumptions and manager's mental models, to improve decision making and to enhance human and organisational learning.

Scenario planning involves different steps that enable firms to accept uncertainty and incorporate it into the forecasting process [START_REF] Tapinos | Perceived environmental uncertainty in scenario planning[END_REF]. Therefore, it improves information processing activities that allow a more effective adjustment to changing environment and market conditions [START_REF] Amer | A review of scenario planning[END_REF][START_REF] Hiltunen | Scenarios: process and outcome[END_REF]Shoemaker, 1991). Similarly, scenario planning reduces two major biases in strategic decision making: discounting and cognitive inertia. The discounting bias occurs when the decision making process is the result of a narrow focus on specific events and the ignorance of important environmental variables; it arises from the gaps between environmental factors and the strategic schemas adapted to interpret the environment. According to [START_REF] Weick | Sensemaking in Organizations[END_REF], "Organisations with access to more varied image will engage in sense making that is more adaptive than organisations with more limited vocabularies". Concerning cognitive inertia, scenario planning develops strategic flexibility by preventing firms from getting locked into cognitive inertia during strategic diagnosis and the consideration of alternative choices.

In light of these arguments, our first hypothesis is as follows:

H1: Scenario planning relates positively with the development of strategic flexibility within firms.

Scenario planning as a tool for developing strategic complexity

Strategic complexity is the "range and concentration of concern and activities within a firm's strategic repertoire" (Miller et al., 1996, p.863). It refers to the number of elements in a strategy and the interactions among these elements. The complexity of a strategy is a source of competitive advantage and constitutes a barrier to its imitation. It is mainly related to the strategist's ability to find an effective combination of choices, reinforce interactions among decisions and create conflicting constraints [START_REF] Rivkin | Imitation of complex strategies[END_REF]. It is also associated with the firm's effort in knowledge acquisition [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Houghton | The influence of social capital on strategic choice: an examination of the effects of External and internal network relationships on strategic complexity[END_REF], the firm's distinctive competency (Hofer and Schendel, 1978) and the firm's capacity to exploit its internal strengths through responding to different opportunities, to neutralise external threats and to avoid internal weaknesses [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Barney | Gaining and Sustaining Competitive Advantage[END_REF]Reed and Defillibi, 1990). According to Miller (1993), organisations that simplify their strategies over long periods are likely to lose their competitive advantage. Thus, in order to cope with multidimensional challenges, firms must "complicate their selves" [START_REF] Weick | The Social Psychology of Organizing[END_REF]. He further states that "the importance of complication is difficult to overemphasize" (Weick, 1979, p.261).

Past researches have focused on two aspects of barriers to imitation. The first approach is grounded in the resource based-view and considers social complexity, causal ambiguity, tacit knowledge, economies of scale and scope, adjustment costs as factors that deter imitation [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Dierickx | Asset stock accumulation and sustainability of competitive advantage[END_REF]Lippmann and Rumelt, 1982). Second, the economic approach is mainly based on game theoretic models and explains the inimitability of strategies by the firm's ability to undertake costly commitment that alters their own future incentives and fend off copy costs [START_REF] Rivkin | Imitation of complex strategies[END_REF]. Studies have shown that a complex strategic orientation improves decision making, while strategic orientations dominated by a single dimension are incomplete [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Barney | Gaining and Sustaining Competitive Advantage[END_REF]Day and Nedundagi, 1994;[START_REF] Houghton | The influence of social capital on strategic choice: an examination of the effects of External and internal network relationships on strategic complexity[END_REF]. Strategically, complex organisations consider a variety of factors and trends in their decision making process, while decisions based on a partial understanding of the environment and predetermined trends are unable to survive in a turbulent environment. Thus, decision makers have to invest in promoting organisational learning, to adapt a more decentralised hierarchy, less formalised organisational structure, an innovative culture and open communication between actors (Neill and Rose, 2006).

Scenario planning is recommended in such conditions. Plausibly based scenarios are designed to deal with the complexity of a variety of environmental factors. Scenario planning methods constitute a suitable tool to understanding complexity by encouraging participants to analyse logical paths and develop strategic options leading to action (Wiklinson et al., 2013). As such, scenarios constitute a response to a more complex environment. Indeed, scenario builders include global aspects related to both micro environment and macro environment such as economic growth, inflation, interest rates and social indicators. As stated by Shoemaker (Shoemaker, 1991), good scenarios should stretch people's beliefs, degrees of confidence and problem perceptions. Furthermore, it analyses competitor behaviour, industry structure and firms positioning [START_REF] Phelps | Does scenario planning affect performance? Two exploratory studies[END_REF]. It requires interaction, debate, dialog and challenging different assumptions. As such, it is a method to figure out the actions, beliefs in a new cultural context without missing the present one.

In light of these arguments, our second hypothesis is as follows:

H2: Scenario planning has a positive impact on developing strategic complexity.

Mediating effects of strategic flexibility

Firms operate in a highly competitive environment where rapid changes in technologies, product innovation and customers' needs have influenced strategies and decision-making process [START_REF] Oliver | Managing future uncertainty: re-evaluating the role of scenario planning[END_REF][START_REF] Santos-Vijande | How organizational learning affects a firm's flexibility, competitive strategy, and performance[END_REF]. This dynamic environment increases the need of flexible planning that enables managers to develop competitive strategy. Scenario planning is considered as management tool that permits to move away from a single vision of the future and in doing so develop strategic flexibility. Strategic management literature argues that strategic flexibility enables firms to maintain competitiveness [START_REF] Eisenhardt | Dynamic capabilities: what are they?[END_REF]. Indeed, it contributes to competitive advantage at different levels. At a functional level, strategic flexibility is a tool that improves operations and manufacturing strategies, including time-to-market and operational excellence [START_REF] Stalk | Competing on capabilities: the new rules of corporate strategy[END_REF][START_REF] Byrd | Information technology: core competencies, and sustained competitive advantage[END_REF][START_REF] Zhang | Information systems, strategic flexibility and firm performance: an empirical investigation[END_REF]. At the business level, strategic flexibility promotes market orientation by introducing new products and technologies and carrying out frequent strategic and organisational changes [START_REF] Eisenhardt | Dynamic capabilities: what are they?[END_REF]. At the corporate level, strategic flexibility is a means to involve constant improvement in the firm's organisational process, organisational learning and capabilities and skills [START_REF] Zhang | Information systems, strategic flexibility and firm performance: an empirical investigation[END_REF]. From this perspective, this strategic tool enables firms to permanently renew their skills, to integrate, build and reconfigure internal and external competencies in order to deal with complex environments [START_REF] Santos-Vijande | How organizational learning affects a firm's flexibility, competitive strategy, and performance[END_REF]. [START_REF] Grewal | Building organizational capabilities for managing economic crisis: The role of market orientation and strategic flexibility[END_REF] suggest that this dynamic capability enhance firms' ability to modify their operating routines and review their resources to achieve adequacy to a rapidly changing environment. Similarly, it is a tool with great potential to develop competitive strategies by enabling firms to have flexible resource pools and diverse portfolios of strategic options (Dreyer and Gronbaug, 2004). According to Nadkarni and Narayaman (2007), the ability of a firm to respond quickly and to change competitive posture promotes "action complexity".

As a conclusion, strategic flexibility is a tool to establish and protect a competitive advantage as well as eroding competitor's advantages [START_REF] Ferrier | Navigating the competitive land scape: the drivers and consequences of competitive aggressiveness[END_REF][START_REF] Ferrier | The role of competitive action in market share erosion and industry dethronement: a study of industry leaders and challenger[END_REF][START_REF] Young | Austrian' and industrial organization perspectives on firm-level competitive activity and performance[END_REF].

In light of these arguments, our third hypothesis is as follows:

H3: Strategic flexibility positively mediates the scenario planning-strategic complexity relationship.

3 Methodology and research design

Sample and data collection

The data used in this research come from a cross-sectional study that attempts to analyse the effect of scenario planning on strategic flexibility and complexity. Our sample consists of European firms operating in Tunisia. The sample of this research was identified using a list of firms obtained from different sources such as Enterprise Europe Network, CEPEX and CCI Tunisia. The criterion of selection is the international dimension of the firm. As such, 514 European firms in different industries are selected.

After collecting information about their activities, their location and their managers, each firm were contacted in order to solicit their participation in this study. The aim was to explain our research purposes and to ask them questions related to the decision making process and if they have introduced scenario planning or scenarios activities in their decision-making process. Following this step, 213 firms are dropped from our sample because they affirm that they are not concerned with strategy formulation. Finally, we have received 108 valid responses. Responds held the position of general manager (15%), chief financial officer (25%), exports sales director (30%), site director (27%) and production manager (3%). In terms of size, 11.1% had an employee base of between 1 and 50, 58.3% between 50 and 500 and the remaining 30, 6 % over 500 employees.

Measures scales

The questionnaire was developed after an extensive review of literature on scenario planning, strategic flexibility and strategic complexity. All constructs are measured using reflective indicators. Multi-item scales were used for each three constructs. The items are measured via a seven-point scale. The Appendix A presents these scales in full details.

The scenario planning scale includes six items derived from previous research by [START_REF] Malaska | Scenarios in Europe -who uses them and why?[END_REF]. Respondents were asked to rate to what extend their organisations are engaged in the environment diagnosis, the definition of the basic assumptions about the environment, the scenario development and strategy formulation on a seven-point scale (where 1 = strongly disagree and 7 = strongly agree). The strategic flexibility scale used in this study is developed and validated by [START_REF] Theoharakis | Organizational resources enabling service responsiveness: evidence from Greece[END_REF], [START_REF] Tsai | The impact of marketing knowledge among managers on marketing capabilities and business performance[END_REF], Vorhies and Morgan (2003) and [START_REF] Santos-Vijande | How organizational learning affects a firm's flexibility, competitive strategy, and performance[END_REF]. This construct was measured by asking respondents to rate the firm's level expertise to respond to environmental change. This dimension included six items measured via seven point scale (where 1 = strongly disagree and 7 = strongly agree).

Regarding strategic complexity, it is referring to the organisation's emphasis on its customers, competitors, products and macro-environment [START_REF] Boulding | Understanding managers' strategic decision-making process[END_REF][START_REF] Houghton | The influence of social capital on strategic choice: an examination of the effects of External and internal network relationships on strategic complexity[END_REF]Neill and Rose, 2006). The customer orientation concentrates on interests of target buyers. The competitor orientation focuses on current and potential competitors. The product orientation deals with product quality, value offering and innovation, and the macro-environmental orientation focuses on analysing and capitalising on issues and trends beyond the organisation's immediate environment. This construct was measured using an existing scale developed by [START_REF] Boulding | Understanding managers' strategic decision-making process[END_REF]. In fact, the customer and competitor orientation are developed by [START_REF] Narver | The effect of a market orientation on business profitability[END_REF]. The product and macro-environmental orientation are developed and validated by Neill and Rose (2006). Respondents were asked to rate their firm's level of orientation toward product, customers, competitors and macro-environment on a seven-point scale (where 1 = strongly disagree and 7 = strongly agree).

Common method variances

Our sample is potentially impacted by common method bias because we used a single key informant approach [START_REF] Podsakoff | Common method biases in behavioral research: a critical review of the literature and recommended remedies[END_REF]. To deal with this bias, we used several survey design methods. Thus, we have guaranteed anonymity and full confidentially of information collected. In addition, we adapted the structure of our survey by changing the order of independent, mediator and dependent variables. Furthermore, we tried to keep questions simple and avoid vague concepts in order to minimise item ambiguity [START_REF] Tourangeau | The Psychology of Survey Response[END_REF].

Reliability and validity

This study uses the following measures: scenario planning, strategic flexibility and strategic complexity. Following [START_REF] Churchill | A paradigm for developing better measures or marketing constructs[END_REF] and Gerbing and Anderson (1988) recommendations, the assessment of measures employs three steps: first, an exploratory factor analysis; second, a confirmatory factor analysis; and third, an examination of the psychometric properties.

The measurement model analyses were performed using the robust maximum likelihood (ML) estimation in order to avoid issues related to normality. Item reliability is insured if Cronbach's alpha coefficient exceeds the suggested value of 0.7 (Cronbach, 1971;Nunnally, 1978). Regarding composite reliability, it is verified through the Average Variance Extracted (AVE) where values greater than 0.5 are considered significant (Fornel and Larcker, 1981). All constructs meet the requirement of construct reliability, since their Composite Reliability (CR) and Cronbach's alpha coefficient are greater than 0.7 (see Table 1). In a next step, convergent validity was assessed through an examination of the average variance extracted (AVE) where value is greater than 0.5 are considered significant to demonstrate convergent validity [START_REF] Chinn | An empirical test of taxonomy of responses to anomalous data in science[END_REF]. All measures were found significant (see Table 1). Furthermore, discriminant validity was assured by using Fornell and Larcker approach (1981), where AVE must exceed the sum of the measures squared correlations. In fact, for every pair of latent variables, the square root of the AVE exceeded the correlations between the latent variables, demonstrating the discriminant validity of the model (see Table 2). In summary, by interpreting statistical tests of measure reliability, validity and fit, all measures are considered significant for the theory development purposes. Finally, we used bootstrap procedure [START_REF] Efron | An Introduction to the Bootstrap[END_REF]. This procedure is used where the sample size is insufficient for straightforward statistical inference. It offers a way to take into consideration the distortions caused by the initial model that may not be fully representative of the population (n = 108). The idea behind Bootstrap is to insure the stability of the results by resampling the sample data and performing inference on (resample → sample). Bootstrapping is used to estimate proprieties for an estimator such as variances, correlations, etc. It enables to control and check the stability of results. Based on the results obtained after resampling (2000 samples), we find that the risk of instability of the model is very low because all regressions weights and variances are significant with p<0.005.

Model testing

Results

The model and hypotheses were tested using structural equation modelling in Amos 20, using bath analysis [START_REF] Hoyle | The structural equation modeling approach: basic concepts and fundamental issues[END_REF]. The analysis includes maximum likelihood estimation techniques to test the hypotheses. Different indices were used to evaluate the fit of the measurement of the model such as Incremental Fit Index (IFI), Comparative Fit Index (CFI), and the Root Mean Square Error of Approximation (RMSEA). The results suggest that the theoretical model is a good representation of the data collected. In fact, measures of absolute fit are acceptable, (χ 2 = 322.196, df = 268, and p = .001) and (CFI = 0.950, IFI = 0.952).

To provide greater confidence in the model, Table 3 demonstrates the findings of testing the theoretical model (M0) against an alternative model (M1) that treats strategic flexibility as an intermediate variable between scenario planning and strategic complexity, and omits the direct effect of scenario planning on strategic complexity. Anderson and Gerbing (1988) suggest this method; they recommend the use of chisquare difference test (CDT) to test the following null hypothesis: M0-M1 = 0. A nonsignificant CDT would lead to acceptance of the more parsimonious M0. Table 3 reports a change in chi-square between the proposed model ( M1) and the theoretical model (M0). Compared to the alternative model (M1), theoretical model (M0) presents a significantly best fit. So M0 is preferred as a better alternative. Table 4 lists the standardised path coefficients and critical ratio (C. R). Thus, a value of 1.96 or higher indicates significance at the customary of 5% level. In our confirmatory factor analysis, the critical ratio tests indicate that the first loading is significant with a value of 3.474 (Scenario planning → Strategic Flexibility). Whereas, the second loading related to the positive effect of scenario planning on strategic complexity is not significant with a value of -1.356. Finally, the third loading (Strategic Flexibility → Strategic Complexity) is significant with a C.R value higher than 1.96. The results support hypotheses H1 and H3 relating to the positive effect of scenario planning on strategic flexibility and the impact of strategic flexibility on strategic complexity, while H2 is not supported. Scenario planning is associated negatively with the strategic complexity with a correlation (-0.182) and a critical ratio of -1.356 (CR< 1.96) (Table 4).

Mediating effect testing

Additional testing was conducted in order to test the potential mediator effect of strategic flexibility on the scenario planning-strategic complexity relationship using Baron and Kenny approach. [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations[END_REF] recommended structural equation modelling (SEM) in order to introduce measurement error and validate the absence of statistical bias. This approach is based on three steps: (1) the impact of independent on mediator;

(2) the impact of independent on dependent; (3) and the impact of both the independent and mediator on dependent. In order to identify mediation, (a) the independent variable must influence the mediator; (b) the independent variable must influence the dependent variable, and (c) the mediator must influence the dependent variable. Furthermore, the effect of the independent variable on the dependent variable must be less in the third test than in the second. The results of mediation test are summarised in Table 5. According to our results; the mediating effect predicted in hypothesis 3 is not supported.

Discussion

Scenario planning is a tool for considering the future and developing strategic options. Yet, despite this, there has been a little work examining its outcomes and benefits [START_REF] Bouhalleb | Scenario planning: an investigation of the construct and its measurement[END_REF]Bowmen, 2015;[START_REF] Jarzabkowski | Strategy tools-in-use: a framework for understanding 'technologies of rationality' in practice[END_REF][START_REF] Meissner | Cognitive benefits of scenario planning: its impact on biases and decision quality[END_REF]. In this paper, we have tried to fill this gap by presenting scenario planning benefits in practice. Thus, the primary purpose of this research is to provide evidence on scenario planning on the development of organisational competencies such as strategic flexibility and complexity. In fact, the predicted positive relationship between scenario planning and strategic flexibility is supported. Our results demonstrate that this strategic tool enhances the firm's ability to respond to environmental contingencies. In particular, the use of multiple futures strengths the firm's ability to quickly act to new business opportunities or threats. Notably, our finding offer insights about scenario planning benefits in firms. As a practice scenario planning contributes to the enrichment of strategic choices and enables decision makers to be more flexible in their actions. As advocated by [START_REF] Rowe | Enhancing horizon scanning by utilizing predeveloped scenarios: analysis of current practice and specification of a process improvement to aid the identification of important 'weak signals[END_REF], scenario planning is a strategic foresight tool that is designed to deal with uncertainty by exploring and anticipating changes. As such, strategic flexibility is enhanced through the formulation of strategies that takes into consideration the consequences of current actions, the anticipation of important shifts and events and the development of scenarios.

Our finding extends the work of Hit et al. (1998) and Santos-Vijande et al. ( 2012) who indicated that the success of a company is mainly related to its ability to learn quickly and to be flexible in order to anticipate changes and face unpredictable conditions. In fact, empirical evidence on the antecedents of firm's strategic flexibility is scarce [START_REF] Schoemaker | When and how to use scenario planning: a heuristic approach with illustration[END_REF]. Only [START_REF] Santos-Vijande | How organizational learning affects a firm's flexibility, competitive strategy, and performance[END_REF] who consider the mediating effect of firm's strategic flexibility on the organisational learning, competitive strategy and performance relationship, and also [START_REF] Rudd | Strategic planning and performance: extending the debate[END_REF] who consider strategic flexibility as mediator in the strategic planning-performance relationship. These authors focus on the importance of managerial ability to consider unfamiliar decisions and uncertainty. They confirmed that strategic flexibility requires a strong managerial ability, a decentralised structure and innovative culture, characteristics in which scenario planning has a considerable role [START_REF] Dibrell | Linking the formal strategic planning process, planning flexibility, and innovativeness to firm performance[END_REF][START_REF] Tapinos | Perceived environmental uncertainty in scenario planning[END_REF].

This study also shows that strategic flexibility enhances the firm's ability to deter imitation through its strategic complexity. This tool enables them to concentrate on the development of new products, processes, organisational arrangements, customer's needs and competitor behaviour. Indeed, dynamic forces in hypercompetitive environment call organisations to be proactive, innovative and flexible. Addressing these competitive forces is related to the firm's ability to introduce internal and external adjustments, to precipitate intentional changes and respond to unanticipated events. Our results are consistent with [START_REF] Houghton | The influence of social capital on strategic choice: an examination of the effects of External and internal network relationships on strategic complexity[END_REF] work that suggest that richer management capabilities would be reflected in more competitive strategy, as measured by strategic complexity. Our finding contributes also to the managerial cognition literature which suggested a positive relation between strategic complexity and strategic flexibility through the consideration of multiple dimensions of environment, diverse strategic alternatives and speedy diagnosis [START_REF] Nadkarni | Strategic schemas, strategic flexibility, and firm performance: the moderating role of industry clockspeed[END_REF]. Such finding reinforces this relationship by considering strategic flexibility as an antecedent of strategic complexity.

Finally, the predicted positive relationship between scenario planning and strategic complexity is not supported. The finding contradicts the rationale of the primary hypothesis which suggested a positive relationship. Results demonstrate empirically that scenario planning worsened the strategic complexity. Our finding is consistent with Wilkonson et al. (2013) work that confirms that scenario planning methods (for example, the intuitive logics school) are an on-ramp to complexity thinking.

Besides, empirical implications may emerge from the current research. A challenge for managers is to combine the benefits of scenario planning and flexibility to deal with uncertainty. This alignment should strength the firm's ability to quickly respond to environment threats and opportunities. As a strategic tool, scenario planning allow practitioners to move from a single forecasts of the future to a more competitive strategies based on environmental evolution. This research also proposes scenario planning as an important determinant of strategic flexibility. Thus, to promote this strategic tool [START_REF] Tapinos | Perceived environmental uncertainty in scenario planning[END_REF] within firms, managers have to engage in changes related to organisational structure such as decentralisation and communication between different departments.

In addition, an implication of our finding is that managers in international contexts should develop strategic competencies in order to survive. In fact, multinational firms operate in changing and unpredictable market. Their strategic decision-making largely depends on local context. Thus, they should not rely on their home country practices [START_REF] Bos-Nehles | HRM implementation in multinational companies: the dynamics of multifaceted scenarios[END_REF]. Finally, the specificities of the Tunisian context (emerging countries) require organisational skills that facilitate the adaptability to best meet stakeholders' exigencies. It is safe to say, however, that the results of this study are a contribution to managers as they define how to make decisions in a hostile environment. They encourage them to opt for flexible planning and to question projective and deterministic techniques.

Conclusion and direction for further research

Scenario planning is one of the tools used by firms and government to consider the future and to enrich their strategic decisions. In this paper we have focused on the scenario planning benefits in practice. We began our analyses with a review of the extant literature describing scenario planning relation with other organisational competencies such as learning, innovation and performance. We then presented an empirical data, using structural equation modelling approach, in order to test the relationship between scenario planning, strategic flexibility and complexity. The overall conclusion driving from this research is that scenario planning enhances the firm's ability to adapt to environmental changes. This work fills the gap by providing empirical evidence on how firms may use scenario planning in order to develop flexibility and strategic complexity orientation. It is also a response to calls for examining strategic complexity antecedents (Neill and Rose, 2006). Drawing from these findings, we have proposed a framework that could serve as a guide for future works on scenario planning outcomes and benefits.

The present research findings should be treated in light of several limitations. In fact, the impact of scenario planning on organisational competencies in European firms is analysed. The results are however, hypothetical and explorative and require further studies to generalise them. One limitation is that the study is based on collecting answers from one person per organisation. We require them to transform their perceptions on seven-point scale. Further researches have to seek confirmatory and negative answers from different respondents in the same organisation in order to achieve significant conclusions. -Drafting of multiple scenarios.

-Evaluating draft scenarios and selecting one for strategy formulation.

-Formulation and appraisal of strategy -Cross-checking strategy against other scenarios -Final selection of strategies for operations

Measurement of strategic flexibility

This construct was measured by asking responds to rate the firm's level expertise to provide a quick response in the face of environment changes on a seven-point scale (where 1 = strongly disagree and 7 = strongly agree) [START_REF] Theoharakis | Organizational resources enabling service responsiveness: evidence from Greece[END_REF], [START_REF] Tsai | The impact of marketing knowledge among managers on marketing capabilities and business performance[END_REF]

Figure 2

 2 Figure 2 Structural model

  and criteria for the business activities and spelling out the basic assumptions about the environment.

Table 1

 1 Measures used: summary statistics

	Variable name	Mean CA	CR AVE	1	2	3	4	5	6
	1. Scenario planning	3.76 .814 .85 .500 1.00
	2. Strategic flexibility 4.61 .83	.75 .501 .428* 1.00
	3. Product orientation 5.11 .826 .75 .500 .002* .35* 1.00
	4. Customer orientation	4.95 .806 .74 .500 .028* .21* .522* 1.00
	5. Competitor orientation	5.15 .838 .80 .505 .046* .09* .397* .537* 1.00
	6. Macro environment orientation	5.16 .842 .83 .500 .049* .32* .488* .571* .482* 1.00
	Notes: *Significant result 0.01 level.
	CR = Composite reliability, CA = Cronbach's alpha, AVE = Average variance
	extracted.

Table 3

 3 Sequential chi-square tests

	Model	Chi-square	Degrees of freedom	Chi-square difference	Degree of freedom difference	Probability
	Proposed Model (M0)	322.196	268	+1.895	1	0.000
	Alternative Model (1)	324.091	269

Table 4

 4 Structural model: standardised path coefficients and critical ratio

	Hypotheses	Path specified	Expected sign Estimates S.E	CR	P
	H1 supported	-scenario planning -strategic flexibility	+	0.35	0.101 3.951	***
	H2 unsupported	-scenario planning -strategic complexity	-	-0.182	0.126 -1.356 0.175
	H3 supported	-strategic flexibility -strategic complexity	+	0.587	0.208 2.727 0.005

Table 5

 5 Results of testing for mediating effects

		Condition 1	Condition 2		Condition 3
		Independent	Independent	Independent		Mediator
	Variables	on mediator	on dependent	on dependent	on dependent
		Path co-efficient	t-values	Path co-efficient	t-values	Path co-efficient	t-values	Path co-efficient	t-values
	Strategic flexibility	0.53	3.92*	-0.201	-1.75	-0.16	-1.35	0.35	2.727*