S Mischler

Q Weng

Relaxation in time elapsed neuron network models in the weak connectivity regime

Keywords: Neuron networks, time elapsed dynamics, semigroup spectral analysis, weak connectivity, long time asymptotic AMS Subject Classification. 35B40, 35F15, 35F20, 92B20

 in the case without delay. Our approach uses the spectral analysis theory for semigroups in Banach spaces developed recently by the first author and collaborators.

Introduction

In nervous systems, neuronal circuits carry out tasks of information transmission and processing. Many neurons generate trains of stereotyped electrical pulses in response to incoming stimulations. Following each discharge, the neuron undergoes a period of refractoriness during which it is less responsive to inputs, before recovering its excitability [START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF]. The main carrier of information is the discharge times or some statistics of the discharge times. In this work, we consider a simple neuronal model which neglects the mechanisms underlying spike generation and focusses on describing the neuronal dynamics in terms of discharge times. More precisely, we consider a model which has been introduced and studied in [START_REF] Gerstner | Spiking neuron models[END_REF][START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF] and which describes the post-discharge recovery of neuronal membranes through an instantaneous firing rate that depends on the time elapsed since the last discharge and the inputs by neurons. We refer to these papers for biologic motivation and discussions. We also refer to [START_REF] De Masi | Hydrodynamic limit for interacting neurons[END_REF][START_REF] Fournier | On a toy model of interacting neurons[END_REF][START_REF] Robert | On the dynamics of random neuronal networks[END_REF][START_REF] Quiñinao | A microscopic spiking neuronal network for the agestructured model[END_REF] where these models (or similar ones) are obtained as a mean field limit of finite number of neuron network models.

The neuronal network is described here by the density number of neurons f = f (t, x) ≥ 0 which at time t ≥ 0 are in the state x ≥ 0. The state of a neuron is a local time (or internal clock) which corresponds to the elapsed time since the last discharge. The dynamic of the neuron network is given by the following nonlinear time elapsed (or of age structured type) evolution equation ∂ t f = -∂ x fa(x, ε m(t))f =: L εm(t) f, (.a) f (t, 0) = p(t), f (0, x) = f 0 (x).

(.b)

Here a(x, ε µ) ≥ 0 represents the firing rate of a neuron in the state x for a network activity µ ≥ 0 and a network connectivity parameter ε ≥ 0.

The function p(t) represents the total density of neurons which undergo a discharge at time t and is defined through The function m(t) represents the network activity at time t ≥ 0 resulting from earlier discharges and is defined by

m(t) := ∞ 0 p(t -y)b(dy), (.)
where the delay distribution b is a probability measure which takes into account the persistence of the electric activity in the network resulting from discharges (synaptic integration). In the sequel, we will consider the two following situations :

• The case without delay, when b = δ 0 and then m(t) = p(t).

• The case with delay, when b is a smooth function.

We observe that in both cases, the solution f of the time elapsed equation (.)-(.) satisfies

d dt ∞ 0 f (t, x)dx = f (t, 0) - ∞ 0 a(x, εm(t))f (t, x)dx = 0.
As a consequence, the total density number of neurons (also called mass in the sequel) is conserved and we can normalize that mass to be 1. In other words, we may always assume f (t, .) = f 0 = 1, ∀t ≥ 0, g := ∞ 0 g(x)dx.

A (normalized) steady state for the time elapsed evolution system of equations (.)-(.) is a couple (F ε , M ε ) of a density number of neurons F ε = F ε (x) ≥ 0 and a network activity M ε ≥ 0 such that

0 = -∂ x F ε -a(x, ε M ε )F ε = L εMε F ε , (.a) F ε (0) = M ε , F ε = 1. (.b)
It is worth emphasizing that for a steady state the associated network activity and discharge activity are two equal constants because of the normalization of the delay distribution, i.e. b = 1.

In equations (.)-(.) and (.), the connectivity parameter ε ≥ 0 corresponds to the strength of the influence of the neuronal network activity on each neuron through the functions m(t) and f (t, x) respectively. In the limit case ε = 0, equation (.)-(.) is linear which means that each neuron evolves accordingly to its own dynamic. In the other hand, when ε > 0, equation (.)-(.) is nonlinear and the dynamic of any given neuron is affected by the state (or the past states in the case of the model with delay) of all the other neurons through the global activity of the neuronal network. Finally, the weak connectivity regime, about which we are mainly concerned in the present paper, corresponds to a range of connectivity parameter ε ∈ (0, ε 0 ], ε 0 > 0 small enough, such that the nonlinearity of equations (.)-(.) and (.) is not too strong.

Our main purpose in this paper is to prove that solutions to the time elapsed evolution equation (.)-(.) converge to a stationary state under a weak connectivity assumption. Before stating that result, let us present the precise mathematical assumptions we will need on the firing rate a and on the delay distribution b.

We make the physically reasonable assumption

∂ x a ≥ 0, a ′ = ∂ µ a ≥ 0, (.) 0 < a 0 := lim x→∞ a(x, 0) ≤ lim x,µ→∞ a(x, µ) =: a 1 < ∞, (.)
as well as the smoothness assumption

a ∈ W 2,∞ (R 2 + ). (.)
In the delay case, we assume that b(dy) = b(y) dy satisfies the exponential bound and smoothness condition

∃δ > 0, ∞ 0 e δy (b(y) + |b ′ (y)|) dy < ∞. (.)
We begin by stating our main result about the stationary problem (.).

Theorem 1.1. Assume (.)-(.)-(.). For any ε ≥ 0, there exists at least one solution

(F ε , M ε ) ∈ W 1,∞ (R + ) × R + to the stationary problem (.) such that 0 ≤ F ε (x) ≤ C e -a 0 2 x , |F ′ ε (x)| ≤ C e -a 0 2 x , ∀ x ≥ 0, (.)
for a constant C ∈ (0, ∞). Moreover, there exists ε 0 > 0, small enough, such that the above solution is unique for any ε ∈ [0, ε 0 ).

For a given initial datum 0 ≤ f 0 ∈ L 1 (R + ), we say that a function f is a weak (positive and mass conserving) solution to (.)-(.) if

0 ≤ f ∈ C([0, ∞); L 1 (R)), f (t) = f 0 , ∀ t ≥ 0,
and f satisfies (.) in the distributional sense D ′ ([0, ∞) × [0, ∞)) for some functions m, p ∈ C([0, ∞))
which fulfilled the constrains (.) and (.). Under the above assumptions, existence and uniqueness of weak solutions have been established in [START_REF] Weng | General time elapsed neuron network model: well-posedness and strong connectivity regime[END_REF]Theorem 1.1]. The main concern of the present work is the following long-time asymptotic result on the solutions. Theorem 1.2. We assume that the firing rate a satisfies (.), (.) and (.). We also assume that the delay distribution b satisfies b = δ 0 or (.). There exists ε 0 > 0, small enough, and there exist some constants α < 0, C ≥ 1 and η > 0 such that for any connectivity parameter ε ∈ (0, ε 0 ) and any initial datum 0 ≤ f 0 ∈ L 1 with mass 1 and such that f 0 -F ε L 1 ≤ η/ε, the unique solution f to the evolution equation

(.)-(.) satisfies f (t, .) -F ε L 1 ≤ Ce αt , ∀ t ≥ 0.
In other words, in that weak connectivity regime, we prove that the total asynchronous firing of neurons appears exponentially fast in the large time asymptotic. Theorem 1.2 extends to firing rates a satisfying (.)-(.) some similar exponential stability results obtained in [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF] in the case without delay and for a step function firing rate a given by

a(x, µ) = 1 x>σ(µ) , σ, σ -1 ∈ W 1,∞ (R + ), σ ′ ≤ 0. (.)
It is worth mentioning that the above firing rate does not fall in the class of rates considered in the present paper because condition (.) is not met. On the other hand, we are able to tackle the case without and with delay in the same time, what it was not the case in [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF]. In the delay case, stability results were established in [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF], but not exponential stability.

Our proof follows a strategy of "perturbation of semigroup" initiated in [START_REF] Mischler | Stability, convergence to selfsimilarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF] for studying long time convergence to the equilibrium for the homogeneous inelastic Boltzmann equation and used recently in [START_REF] Mischler | On a kinetic Fitzhugh-Nagumo model of neuronal network[END_REF] for a neuron network equation based on a brownian (hypoelliptic) perturbation of the well-known FitzHugh-Nagumo dynamic. More precisely, we introduce the linearized equation for the variation functions (g,

n, q) = (f, m, p)-(F ε , M ε , M ε ) around a stationary state (F ε , M ε , M ε ), which writes ∂ t g = -∂ x g -a(x, ε M ε )g -n(t) ε(∂ µ a)(x, εM ε )F ε , (.a) g(t, 0) = q(t), g(0, x) = g 0 (x), (.b) with q(t) = ∞ 0 a(x, ε M ε )g dx + n(t) ε ∞ 0 (∂ µ a)(x, εM ε )F ε dx (.) and n(t) := ∞ 0 q(t -y)b(dy). (.)
We associate to that linear evolution equation a generator Λ ε (which acts on an appropriate space to be specified in the two cases without and with delay) and its semigroup S Λε . It turns out that we may split the operator Λ ε as

Λ ε = A ε + B ε ,
for some α-hypodissipative operator B ε , α < 0, and some bounded and B εpower regular operator A ε as defined in [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF][START_REF] Gualdani | Factorization of non-symmetric operators and exponential H-Theorem[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF]. In particular, adapted versions of the Spectral Mapping Theorem in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] and the Weyl's Theorem in [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF][START_REF] Gualdani | Factorization of non-symmetric operators and exponential H-Theorem[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] imply that the semigroup S Λε has a finite dimensional dominant part. Moreover, in the limit case when ε = 0, the term n(t) disappears from equation (.) and the resulting semigroup S Λ 0 becomes positive. That allows us to use the Krein-Rutman Theorem established in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] in order to get that the stationary state (F 0 , M 0 , M 0 ) is unique and exponentially stable. Using next a perturbative argument developed in [START_REF] Mischler | Stability, convergence to selfsimilarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF][START_REF] Tristani | Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF][START_REF] Mischler | Uniform semigroup spectral analysis of the discrete, fractional & classical fokker-planck equations[END_REF], we get that the unique stationary state (F ε , M ε , M ε ) is also exponentially stable in the weak connectivity regime. We conclude the proof of Theorem 1.2 by a somewhat classical nonlinear exponential stability argument.

The same strategy applies to the case without delay and with delay. In both cases, the boundary condition in the age structure equation is treated as a source term and, in the delay case, the delay equation (.) is replaced by a simple age equation on an auxiliary function, so that the resulting linearized equation writes as an autonomous system of two PDEs and falls in the classical framework of linear evolution equations generating a semigroup.

Our approach is thus quite different from the usual way to deal with delay equations, as introduced by I. Fredholm [START_REF] Fredholm | Sur une classe d'équations fonctionnelles[END_REF] and V. Voltera [START_REF] Volterra | Theory of functionals and of integral and integrodifferential equations[END_REF], which consists in using the specific framework of "fading memory space", which goes back at least to Coleman & Mizel [START_REF] Coleman | Norms and semi-groups in the theory of fading memory[END_REF], or the theory of "abstract algebraic-delay differential systems" developed by O. Diekmann and co-authors [START_REF] Diekmann | Delay equations[END_REF].

Our approach is also different from the previous works [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF] where the asymptotic stability analysis were performed by taking advantage of the step function structure (.) of the firing rate. That one makes possible to explicitly exhibit a suitable norm (related to the W 1 Monge-Kantorovich-Wasserstein optimal transport distance) such that some related linear age structure operator is dissipative. The present method is based on a more abstract approach but in the other hand it is somewhat more flexible because if does note require to explicitly exhibit a norm for which the underlying linear(ized) operators are dissipative. In particular, we hope that our strategy can be adapted to the large connectivity regime, to the step function firing rate (.) as well as to models including fragmentation term to describe neuronal networks with adaptation and fatigue, and thus generalize to the case with a delay term all the stability results established in [START_REF] Pakdaman | Dynamics of a structured neuron population[END_REF][START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF] in the case without delay.

Let us end the introduction by describing the plan of the paper. In Section 2, we introduce the strategy, we prove the stationary state result and we establish Theorem 1.2 in the case without delay. In Section 3, we establish Theorem 1.2 in the case with delay. As we mentioned above, the strategy of proof for the case without and with delay is rather the same. For pedagogical reason, we start presenting the method on the simplest "without delay case" in Section 2, where we prove the stationary problem result Theorem 1.1 as well as Theorem 1.2 in that case. Next, in Section 3, we only explain how the proof must be modified in order to treat the more complicated "with delay case" and thus establish Theorem 1.2 in all generality.

Acknowledgments. The research leading to this paper was (partially) funded by the French "ANR blanche" project Kibord: ANR-13-BS01-0004.

Case without delay

The present section is devoted to the proof of our main result Theorem 1.2 in the without delay case.

The stationary problem

We first deal with the stationary problem and we prove the existence of steady state as well as its uniqueness in the small connectivity regime.

Proof of Theorem 1.1. Step 1. We prove the existence of a solution. We set

A(x, m) := x 0 a(y, m)dy, ∀ x, m ≥ 0.
For any m ≥ 0, we can solve the equation (.a), by writing

F ε,m (x) := T m e -A(x,εm) , (.)
where T m ≥ 0 is chosen in order that F ε,m satisfies the mass normalized condition, namely

T -1 m = ∞ 0 e -A(x,εm) dx.
In order to conclude the existence of a solution, we just have to find a real number m = M ε such that m = F ε,m (0) = T m . Equivalently, we need to find

M ε ≥ 0 such that Φ(ε, M ε ) = 1, (.) where Φ(ε, m) = mT -1 m := m ∞ 0 e -A(x,εm) dx.
From the assumption (.) of a, there exists x 0 ∈ [0, ∞) such that a(x, µ) ≥ a 0 2 , for any x ≥ x 0 , µ ≥ 0, and therefore

a 0 2 (x -x 0 ) + ≤ A(x, µ) ≤ a 1 x, ∀ x ≥ 0, ∀ µ ≥ 0. (.)
Thanks to the Lebesgue dominated convergence theorem, we deduce that Φ(ε, •) is a continuous function. Because Φ(ε, 0) = 0, Φ(ε, ∞) = ∞ and thanks to the intermediate value theorem, we conclude to the existence of at least one real number M ε ∈ (0, ∞) such that (.) holds. Estimates (.) immediately follow from the identity (.) and the estimate (.).

Step 2. We prove the uniqueness of the solution in the weak connectivity regime. Obviously, there exists a unique

M 0 := ( ∞ 0 e -A(x,0) dx) -1 ∈ (0, ∞) such that Φ(0, M 0 ) = 1. Moreover, we compute ∂ ∂m Φ(ε, m) = ∞ 0 e -A(x,εm) (1 -mε ∂A ∂m (x, εm))dx,
which is continuous as a function of the two variables because of (.). We then easily obtain that Φ ∈ C 1 . Since moreover

∂ ∂m Φ(ε, m)| ε=0 = ∞ 0 e -A(x,0) dx > 0,
the implicit function theorem implies that there exists ε 0 > 0, small enough, such that the equation (.) has a unique solution for any ε ∈ [0, ε 0 ).

Remark 2.1. In the above proof, we do not need (.) but only the weaker smoothness assumption that A and ∂ m A are continuous.

Linearized equation and structure of the spectrum

To go one step further, we introduce the linearized equation around the stationary solution (F ε , M ε ). On the variation (g, n), the linearized equation writes

∂ t g + ∂ x g + a ε g + a ′ ε F ε n(t) = 0, g(t, 0) = n(t) = ∞ 0 (a ε g + a ′ ε F ε n(t)) dx, g(0, x) = g 0 (x), with a ε := a(x, εM ε ), a ′ ε := ε (∂ µ a)(x, εM ε ). Since there exists ε 0 > 0, small enough, such that ∀ ε ∈ (0, ε 0 ) κ := ∞ 0 a ′ ε F ε dx < 1,
we may define

M ε [g] := (1 -κ) -1 ∞ 0 a ε g dx, (.)
and the linearized equation is then equivalent to

∂ t g + ∂ x g + a ε g + a ′ ε F ε M ε [g(t, .)] = 0, (.) g(t, 0) = M ε [g(t, .)], g(0, x) = g 0 (x). (.)
To the above linear evolution equation, one can classically associate a semigroup S Lε (t) acting on X := L 1 (R + ), with generator

L ε g := -∂ x g -a ε g -a ′ ε F ε M ε [g]
and domain

D(L ε ) := {g ∈ W 1,1 (R + ); g(0) = M ε [g]}.
Here we also use another approach by considering the boundary term as a source term, and then rewriting the equation as

∂ t g = Λ ε g := -∂ x g -a ε g -a ′ ε F ε M ε [g] + δ x=0 M ε [g], (.)
acting on the space of bounded Radon measures

X := M 1 (R + ) = {g ∈ (C 0 (R)) ′ ; supp g ⊂ R + },
endowed with the weak * topology σ(M 1 , C 0 ). Here and below, C 0 (I) denotes the space of continuous functions on the closed interval I = R or I = R + which goes to 0 at infinity. We then denote by S Λε (t) the semigroup on X generated by Λ ε . It is worth emphasizing that for any g 0 ∈ X, the function g(t) = S Lε (t)g 0 ∈ C([0, ∞); X) is a weak solution to equations (.)-(.), and more precisely clearly satisfies

- ∞ 0 ϕ(0) g 0 dx + ∞ 0 ∞ 0 g { -∂ t ϕ + Λ * ε ϕ} dxdt = 0, for any ϕ ∈ C 1 c ([0, ∞); C 0 (R + )) ∩ C c ([0, ∞); C 1 0 (R + )).
Here, we have defined

Λ * ε ψ := ∂ x ψ -a ε ψ -a ε (1 -κ) -1 ∞ 0 a ′ ε (y)F ε (y)ψ(y) dy -ψ(0) ,
for any ψ ∈ C 1 0 (R + ), the space of C 1 functions which goes to 0 at infinity as well as their first derivative. As a consequence, the semigroup S Λε being defined by duality from the semigroup S Λ * ε , we have S Λε | X = S Lε .

For a generator L, we denote by Σ(L) its spectrum and by S L the associated semigroup. We refer to the classical textbooks [START_REF] Kato | Perturbation theory for linear operators[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF][START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] for an introduction to the spectral analysis of operators and the semigroup theory. Our next result deals with the structure of the spectral set Σ(Λ ε ) of Λ ε and the splitting structure of the associated semigroup S Λε .

Theorem 2.2. Assume (.)-(.)-(.) and define a * := -a 0 /2 < 0. The operator Λ ε is the generator of a weakly * continuous semigroup S Λε acting on X endowed with the weak * topology σ(M 1 , C 0 ). Moreover, there exists a finite rank projector Π Λε,a * which commutes with S Λε , an integer j ≥ 0 and some complex numbers

ξ 1 , ..., ξ j ∈ ∆ a * := {z ∈ C, ℜe z > a * }, such that on E 1 := Π Λε,a * X the restricted operator satisfies Σ(Λ ε|E 1 ) ∩ ∆ a * = {ξ 1 , ..., ξ j } (.)
(with the convention Σ(Λ ε|E 1 ) ∩ ∆ a * = ∅ when j = 0) and for any a > a * there exists a constant C a such that the remainder semigroup satisfies

S Λε (I -Π Λε,a * ) B(X ) ≤ C a e at , ∀ t ≥ 0. (.)
The proof of the result is a direct consequence of the fact that the operator Λ ε splits as Λ ε = A ε + B ε where A ε and B ε are defined on X by

A ε g := γ ε M ε [g], γ ε := δ 0 -a ′ ε F ε , (.) B ε g := -∂ x g -a ε g, (.)
for which we can adapt Weyl's Theorem of [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF][START_REF] Gualdani | Factorization of non-symmetric operators and exponential H-Theorem[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] and the Spectral Mapping Theorem of [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF]. The picture may seem not that simple, because Λ ε does not generate a strongly continuous semigroup on X and then apparently does not fit the framework developed in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF]. However, we may probably circumvent that issue in the following ways:

-We may observe that the strong continuity is little used in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF] and then the results stated therein extend to weakly * continuous semigroups. In other words, on the finite dimensional eigenspace associated to the principal part of the spectrum (.) continuity and weak * continuity are equivalent, while in the remainder part (.) we just use a decay bound which does not require the strong continuity property, see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Chapter 1] as well as [START_REF] Weng | General time elapsed neuron network model: well-posedness and strong connectivity regime[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] where such a weak * continuous framework is also discussed.

-We may apply the theory developed in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF] to the adjoint operator Λ * ε and the associated semigroup acting on C 0 (R) and then deduce the result by duality.

-We may probably reduce the space X by "sun duality", see [4, Chapter II.2.6], apply the theory developed in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF] to the resulting strongly continuous semigroup and then conclude by a density argument.

We rather follow another strategy, which has the advantage to be more self-contained and pedagogical, by adapting to our context the proofs from [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF]. It is worth emphasizing that we introduce the two spaces X ⊂ X because the splitting Λ ε = A ε + B ε has some clearer meaning in the space X while the Banach structure of X is used when we establish the adequate version of Weyl's theorem.

We recall the definition of hypodissipativity introduced in [START_REF] Gualdani | Factorization of non-symmetric operators and exponential H-Theorem[END_REF]. We say that the closed operator L on a Banach space E with dense domain

D(L) is α-hypodissipative if there exists an equivalent norm ||| • ||| on E such that ∀ f ∈ D(L), ∃ ϕ ∈ F |||•||| (f ) ℜe ϕ, (L -α) f ≤ 0,
where, for any f ∈ E, the associated dual set

F |||•||| (f ) ⊂ E ′ is defined by F |||•||| (f ) := {ϕ ∈ E ′ ; ϕ, f = |||f ||| 2 X = |||ϕ||| 2 E ′}.
We also recall that the generator L of a semigroup of bounded operators is α-hypodissipative if, and only if, there exists a constant M ≥ 1 such that the associated semigroup S L on E satisfies the growth estimate

S L (t) B(E) ≤ M e α t , ∀ t ≥ 0,
where B(E) denotes the space of linear and bounded operators on E. We will sometime abuse by saying that S L is α-hypodissipative when it satisfies the above growth estimate. We refer to [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF][START_REF] Gualdani | Factorization of non-symmetric operators and exponential H-Theorem[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] for details. We start with the properties of the two auxiliary operators.

Lemma 2.3. Assume that a satisfies conditions (.)-(.). For any ε ≥ 0, the operators A ε and B ε satisfy :

(i) A ε ∈ B(W -1,1 (R + ), X ).
(ii) S Bε is a * -hypodissipative in both X and X .

(iii) The family of operators S Bε * A ε S Bε satisfies

(S Bε * A ε S Bε )(t) B(X ,Y ) ≤ C a e a t , ∀ a > a * ,
for some some constant C a ∈ (0, ∞) and with

Y := BV (R + ) ∩ L 1 1 (R + ).
Here BV (R + ) stands for the space of functions with bounded variation and L 1 1 (R + ) stands for the weighted Lebesgue space associated to the weight function x → x .

Proof. In order to shorten notation we skip the ε dependency, we write a(x) = a(x, εM ε ), A(x) = A(x, εM ε ), S Λ = S Λε , S A = S Aε , S B = S Bε and so on.

Step 1. Proof of (i). We have

A ∈ B(W -1,1 (R + ), X ) from the fact that M ε [•] ∈ B(W -1,1 (R + ), R) because a W 1,∞ < ∞ as assumed in (.).
Step 2. Proof of (ii). We use the same notation B and S B for the generator and its associated semigroup defined in the spaces X and X . We write S B with the explicit formula S B (t)g(x) = e A(x-t)-A(x) g(xt)1 x-t≥0 =: S(t, x).

(.)

From the inequality a(z, 0) ≥ (3a 0 /4)(1 -1 0≤z≤x 1 ) for some x 1 ∈ (0, ∞) coming from (.), we deduce A(xt) -A(x) ≤ 3a 0 x 1 /4 -(3a 0 /4)t for any

x ≥ t ≥ 0, next e A(x-t)-A(x) ≤ C e 3βt , ∀ x ≥ t ≥ 0, (.) with C := e 3a 0 x 1 4
> 0, β := -a 0 /4 < 0, and finally

S B (t)g X ≤ Ce 3βt g X , ∀ t ≥ 0, ∀ g ∈ X.
We conclude by observing that 3β < a * and that the same estimate holds with X replaced by X thanks to a (weakly * ) density argument.

Step 3. Proof of (iii). For g ∈ X and with the notation of Step (ii), we have

AS B (t)g = γ ε N(t),
where γ ε is defined in (.) and

N(t) := M ε [S(t, •)] = (1 -κ) -1 ∞ 0 a(x)e A(x-t)-A(x) g(x -t)1 x-t≥0 dx.
It is worth noticing that N ∈ C b (R + ), because (x, t) → a(x)e A(x-t)-A(x) is a bounded and continuous function on the set T := {(x, t); x ≥ t ≥ 0}. Moreover, as in Step (ii), we have

|N(t)| ≤ C a 1 ∞ 0 e 3βt |g(x -t)|1 x-t≥0 dx ≤ Ca 1 e 3βt g X , (.)
for any t ≥ 0. We deduce

(S B * AS B )(t)g(x) = t 0 (S B (s)γ ε )(x)N(t -s) ds = t 0 e A(x-s)-A(x) γ ε (x -s)N(t -s)1 x-s≥0 ds (.) = e -A(x) (ν ε * Ňt )(x), (
≤ C e a * t (|γ ε |e -β ) * (|N|e -β ) C b ≤ C e a * t γ ε e -β X Ne -β C b .
Using the definition (.) and the estimates (.) and (.), we deduce

(S B * AS B )(t)g e -β L ∞ ≤ C ′ e a * t g X , ∀ t ≥ 0, (.) for some constant C ′ ∈ (0, ∞).
Next, differentiating the functions in both sides of identity (.), we get

∂ x [(S B * AS B )(t)g](x) = -a(x)e -A(x) (ν ε * Ňt )(x) -e -A(x) (ν ε * Ň′ t )(x) -e -A(x) ν ε (x -t)N(0)1 x-t≥0 + e -A(x) ν ε (x)N(t), with Ň′ t (s) = N ′ (t -s).
In order to estimate the second term, we compute

N ′ (t) = (1 -κ) -1 ∞ 0 ∂ x [a(x)e -A(x) ] e A(x-t) g(x -t)1 x-t≥0 dx = (1 -κ) -1 ∞ 0 [a ′ (x) -a(x) 2 ] e A(x-t)-A(x) g(x -t)1 x-t≥0 dx.
Using the inequality (.), we deduce

|N ′ (t)| ≤ C ′ ∞ 0 |a ′ (x)| + a(x) 2 e 3βt g(x -t)1 x-t≥0 dxdt e a * t g X .
As a consequence, using again (.), we get

e -A (ν ε * Ň′ t ) X t 0 ∞ 0 e 3βs |γ ε (x -s)| |N ′ (t -s)| dxds e a * t γ ε X N ′ X e a * t g X .
Treating in a similar way the two other terms involved in the expression of

∂ x [(S B * AS B )(t)g],
we finally obtain

∂ x [(S B * AS B )(t)g] X e a * t g X .
Together with (.) that concludes the proof of (iii).

Proof of Theorem 2.2. The main idea is to apply or adapt the versions of Weyl's Theorem [14, Theorem 3.1], [10, Theorem E.3.1] and of the Spectral Mapping Theorem [14, Theorem 2.1], [10, Theorem E.2.1] (see also the variant results in [START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF]). We start collecting the three key properties satisfied by the operators A ε and B ε involved in the splitting of Λ ε We denote by X ζ the abstract Sobolev space associated to L ε for ζ ∈ R, see e.g. [4, Section II.5], so that in particular X ζ ⊂ W ζ,1 (R + ), with equality when ζ ≤ 0. We recall that for the generator L of a semigroup S L , we define the resolvent set ρ(L) by

ρ(L) := {z ∈ C; L -z is a bijection } = C\Σ(L),
as well as the resolvent (operator) R L (z) := (Lz) -1 for any z ∈ ρ(L). We finally define the half plane ∆ a := {z ∈ C; ℜez > a} for any a ∈ R.

(A1) For any a > a * and ℓ ∈ N, there exists a positive constant C a,ℓ such that the following growth estimate holds

S Bε * (A ε S Bε ) ( * ℓ) (t) B(X) ≤ C a,ℓ e at , ∀t ≥ 0. (.)
It is obvious that (A1) is true for ℓ = 0 from Lemma 2.3-(ii) and for ℓ = 1 from (.). We then deduce (A1) for any ℓ ≥ 2 and a > a * by writing 1) and by using that

S Bε * (A ε S Bε ) ( * ℓ) = [S Bε * (A ε S Bε )] * (A ε S Bε ) ( * ℓ-
(B ε -a * ) is hypodissipative in X and A ε ∈ B(X ). (A2) There holds A ε ∈ B(X -1 , X ζ ) for any ζ ∈ (-1, 0
) and the family of operators (S Bε A ε ) ( * 2) (t) satisfies the estimate

(S Bε A ε ) ( * 2) (t) B(X -1 ,X) ≤ C ′ a * ,1 e a * t , ∀t ≥ 0, (.)
for a positive constant C ′ a,1 . The first claim is a consequence of the continuous embedding X ⊂ W ζ,1 and of Lemma 2.3-(i). The second estimate is obtained by putting together the properties (i) and (ii) in Lemma 2.3.

(A3) The family of operators (R Bε A ε ) 3 (z) satisfies the compactness estimate (R Bε (z)A ε ) 3 B(X ,Y ) ≤ C ′′ 2,a , ∀ z ∈ ∆ a , ∀ a > a * ,
for some positive constant C ′′ 2,a . Observing that z → -(R Bε (z)A ε ) 3 is nothing but the Laplace transform of the function t → (S Bε A ε ) ( * 3) (t), that clearly holds true thanks to Lemma 2.3.

We now briefly explain how the proof goes on, referring to [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] for more details and developments. In order to shorten notation we skip again the ε dependency.

Step 1. On the one hand, because of (.) and the fact that R B is nothing but the opposite of the Laplace transform of S B , we have

(R B (z)A) 2 X -1 →X ≤ C, ∀ z ∈ ∆ a . (.)
Recalling that the negative abstract Sobolev norm is defined by

g X -1 := R B (a)g X ,
and using the dissipativity property of B, we immediately have

∀ z ∈ ∆ a , ∀ g ∈ X, R B (z)g X -1 ≤ C g X -1 .
Moreover, using the resolvent identity

R B (a)R B (z) = z -1 (R B (a)BR B (z) -R B (a)), = z -1 (R B (z) -R B (a) + aR B (z)R B (a)),
we also have

R B (z)g X -1 ≤ 1 |z| { R B (z)g X + R B (a)g X + |a| R B (z)R B (a)g X } ≤ C |z| g X ,
for any g ∈ X and z ∈ ∆ a . Finally, using both estimates and an interpolation argument, we deduce

∀ z ∈ ∆ a , R B (z) X ζ →X -1 ≤ C z 1+ζ , ∀ ζ ∈ (-1, 0). (.)
We conclude with

∀ z ∈ ∆ a , (R B (z)A) 3 X →X ≤ C z 3/4 , (.) by writing (R B (z)A) 3 = (R B (z)A) 2 R B (z)A
and by using estimates (i) in Lemma 2.3 with ζ = -1/4, (.) with ζ = -1/4 and (.).

Step 2.

Here, we follow [14, Proof of Theorem 2.1] and [START_REF] Voigt | A perturbation theorem for the essential spectral radius of strongly continuous semigroups[END_REF] by taking advantage of the analysis of degenerate-meromorphic functions performed in [START_REF] Ribarič | Analytic properties of the inverse A(z) -1 of an analytic linear operator valued function A(z)[END_REF]. Iterating the factorization identity

R Λ = R B -R B AR Λ , we obtain (I -V)R Λ = U in B(X ) (.) with U = R B -R B AR B + (R B A) 2 R B , V := -(R B A) 3 .
Because of (.), (.) and of the identity

R B (AR B ) ℓ (z) = (-1) ℓ+1 ∞ 0 e -zt S B * (AS B ) ( * ℓ) (t) dt ∈ B(X)
for any z ∈ ∆ a , we see that all the terms involved in (.) act in B(X), so that we also have

(I -V)R L = U in B(X). (.)
Because of (.), we see that I -V is inversible on ∆ a \B(0, M) for M > 0 large enough and (.) tells us that R L = (I -V) -1 U is uniformly bounded in ∆ a \B(0, M), in particular Σ(L) ⊂ ∆ c a ∪ B(0, M). On the other hand, because of (A3), there holds

V(z) X→Y ≤ C,
with compact embedding Y ⊂ X. As a consequence of the fact that V is the opposite of the Laplace transform of the nice function R + → X, t → (S B A) ( * 3) (t), we also deduce that V is holomorphic in B(X) on ∆ a . We define Ψ(z) := I -V(z). Because Ψ is an holomorphic function on ∆ a , Ψ(z) is invertible for (some) z ∈ ∆ a \B(0, M) and RV(z) ⊂ Y for any z ∈ ∆ a , [21, Corollary II] asserts that the function z → Ψ(z) -1 is a degeneratemeromorphic operators valued function, that is Ψ is holomorphic on ∆ a \D, D ⊂ ∆ a is discrete, any point ξ ∈ D is an isolate pole and the coefficients of the principal part in the Laurent series are finite rank operators. Next, from [START_REF] Ribarič | Analytic properties of the inverse A(z) -1 of an analytic linear operator valued function A(z)[END_REF] and the identity R L = Ψ -1 U, we deduce that R L is also degenerate-meromorphic on ∆ a , in particular Σ(L) ∩ ∆ a is a finite set of discrete eigenvalues, which means isolated eigenvalues associated to an algebraically finite eigenspace. That is nothing but (.). We define Π := I -Π Λ,a .

Step 3. We claim that for any n ≥ 2, there holds

S L (t)Π = {S B (t) + ... + (S B A) * (n-1) * S B (t)}Π (.) +(-1) n+1 i 2π ↑a (R B A) n (z) R L (z)Π e tz dz
in B(X) where ↑ c := {c + iy, y ∈ R} is the complex line of abscissa c ∈ R. In order to prove that identity, we iterate the Duhamel formula

S Λ (t) = S B (t) + S B A * S Λ (t),
in B(X ) and for any f ∈ X and n ≥ 2. We then obtain

S L (t)Πf = {S B (t) + ... + (S B A) * (n-1) * S B (t)}Πf + g n (t), with g n (t) := (S B A) * (n) * S L (t)Πf.
Because t → g ′ n (t)e -bt = Bg n (t) e -bt + Ag n-1 (t) e -bt ∈ L 1 (R + , X ) for b > 0 large enough from (A1)-(A2) and the fact that t → S L (t)Πf e -bt ∈ L ∞ (X), the following inverse Laplace transform formula holds

g n (t) = (-1) n+1 i 2π ↑ b (R B A) n (z) R L (z)Πf e tz dz, ∀ t ≥ 0.
From the preceding steps, the function under the integral sign is holomorphic on ∆ a * and we may move the line of integration from ↑ b to ↑ a . Observing that every term in the resulting identity belongs to X, we conclude to (.). More precisely and importantly, taking n = 6, using (.) and the fact that R L (z)Π B(X) is uniformly bounded on ∆ a from Step 2, we have

g n (t) X ≤ e at 2π ↑a (R B A) 6 (z) B(X) R L (z)Π B(X) dz f X e at f X ,
for any t ≥ 0. We conclude to (.) using that the other terms involved in (.) are similarly bounded thanks to (A1) and using a weakly * density argument.

The vanishing connectivity regime

When the network connectivity parameter vanishes, ε = 0, the linearized time elapsed operator simplifies into

Λ 0 g = -∂ x g -a(x, 0)g + δ x=0 M 0 [g], (.) where M 0 [g] = ∞ 0 a(x, 0)g(x)dx.
The associated semigroup is then positive and the following version of the Krein-Rutman theorem holds.

Theorem 2.4. There exist some constants α 0 < 0 and C > 0 such that Σ(Λ 0 ) ∩ ∆ α 0 = {0} and for any g 0 ∈ X , g 0 = 0, there holds

S Λ 0 (t)g 0 X ≤ Ce α 0 t g 0 X , ∀ t ≥ 0. (.)
We denote

X + := {g ∈ M 1 (R + ); g ≥ 0},
the space of bounded and nonnegative Radon measures. We start with two elementary auxiliary results.

Lemma 2.5. S Λ 0 is positive: S Λ 0 (t)g ∈ X + for any g ∈ X + and any t ≥ 0.

Proof. We introduce a dual problem of (.) defined on the space C 0 (R) by

∂ t ϕ = Λϕ = Bϕ + Ãϕ, ϕ(0, •) = ϕ 0 , (.)
where the operators à and B are defined by

Bϕ = ∂ x ϕ -a(x, 0)ϕ, Ãϕ = a(x, 0)ϕ(0).
A solution ϕ to (.) then satisfies

ϕ(t) = S B(t)ϕ 0 + (S B * Ãϕ)(t).
Let us fix ϕ 0 ∈ C 0 (R) such that ϕ 0 ≤ 0 and let us prove that ϕ(t) ≤ 0 for any t ≥ 0. We obviously have that S B is a positive operator and it is a contraction in C 0 (R). Taking the positive part in (.), we get

ϕ + (t) ≤ S B(t)ϕ 0+ + (S B * Ãϕ + )(t) ≤ a 1 t 0 S B(t -s)ϕ + (0) ds, so that ϕ + (t) L ∞ ≤ C t 0 ϕ + (s) L ∞ ds.
From Grönwall's lemma, we deduce that ϕ + (t) = 0 for any t ≥ 0 and then ϕ ≤ 0. We conclude by observing that S Λ 0 is the dual of S Λ.

Lemma 2.6. -Λ 0 satisfies the following version of the strong maximum principle: for any given g ∈ X + and µ ∈ R, there holds

g ∈ D(Λ 0 ) \ {0} and (-Λ 0 + µ)g ≥ 0 imply g > 0.
Proof. Suppose that there holds (-Λ 0 + µ)g ≥ 0 for g satisfying the above conditions. It is only necessary to prove that g does not vanish in R + . Since g ≡ 0, there exists x * ∈ R + such that g(x * ) > 0. Rewriting the assumption as

∂ x g + a(x, 0) + µ g ≥ δ x=0 ∞ 0 a(x, 0)gdx,
we observe that

∂ x (e A(x,0)+µx g) = e A(x,0)+µx ∂ x g + a(x, 0) + µ g ≥ 0. (.) (i) For x ∈ (x * , ∞),
we have e A(x,0)+µx g ≥ e A(x * ,0)+µx * g(x * ) > 0.

(ii) For x ∈ (0, x * ), by integrating the same equation on (0, x), we obtain

e A(x,0)+µx g ≥ x 0 δ u=0 e A(u,0)+µu ∞ 0 a(y, 0)g(y)dydu + g(0) ≥ ∞ 0 a(y, 0)g(y)dy.
From the positivity assumption (.) on a and step (i), we have

∞ 0 a(y, 0)g(y)dy > a 0 2 ∞ max{x 0 ,x * } g(y)dy > 0.
Therefore, g does not vanish on (0, ∞).

Proof of Theorem 2.4. First, we know from Theorem 1.1 that there exists at least one nonnegative and non-vanishing solution F 0 to the eigenvalue problem Λ 0 F 0 = 0 and the associated dual eigenvector is ψ = 1. Next, we observe that, defining the signf operator for f ∈ D(Λ 2 0 ) by

[(signf ) * ψ](x) := 1 2|f (x)| [ f (x)ψ(x) + f (x) ψ(x)], ∀ ψ ∈ C 0 (R),
we have, for any

ψ ∈ C 0 (R) + , ℜe (signf )M 0 [f ] , ψ = ℜe M 0 [f ] , (signf ) * ψ = ℜe a 0 f dx ℜef (0) |f (0)| ψ(0) ≤ a 0 |f |dx ψ(0) = M 0 [|f |] , ψ ,
which is nothing but the complex Kato's inequality

∀ f ∈ D(Λ 2 0 ), ℜe(signf ) Λ 0 f ≤ Λ 0 |f |. (.)
We also observe that D(Λ 2 0 ) ⊂ C b (0, ∞), and, as a consequence, g ∈ D(Λ 2 0 ) and |g| > 0 implies g > 0 or g < 0. We then may use exactly the same argument as in [14, 10, Proof of Theorem 5.3] (see also [START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF]):

-Kato's inequality (.) and the strong maximum principle imply that the eigenvalue λ = 0 is simple and the associated eigenspace is Vect(F 0 ); -together with the fact that S Λ 0 is a positive semigroup, one deduces that λ = 0 is the only eigenvalue with nonnegative real part.

We conclude to the spectral gap estimate (.) for some α 0 ∈ (a * , 0) with the help of Theorem 2.2.

Weak connectivity regime -exponential stability of the linearized equation

We extend the exponential stability property which holds for a vanishing connectivity to the weak connectivity regime thanks to a perturbation argument.

Theorem 2.7. There exist some constants ε 0 > 0, α < 0 and C > 0 such that for any ε ∈ [0, ε 0 ] there hold Σ(Λ ε ) ∩ ∆ α = {0} and S Λε (t)g 0 X ≤ Ce αt g 0 X , ∀ t ≥ 0, (.)

for any g 0 ∈ X , g 0 = 0.

The proof follows the stability theory for semigroups developed in Kato's book [START_REF] Kato | Perturbation theory for linear operators[END_REF] and revisited in [START_REF] Mischler | Stability, convergence to selfsimilarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF][START_REF] Tristani | Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF][START_REF] Mischler | Uniform semigroup spectral analysis of the discrete, fractional & classical fokker-planck equations[END_REF].

Proof of Theorem 2.7. With the definitions (.), (.) and (.) of M ε , A ε and B ε , we have

(B ε -B 0 )g = (a(x, 0) -a(x, εM ε ))g and (A ε -A 0 )g = (M ε [g] -M 0 [g]) δ 0 -ε(∂ µ a)(x, εM ε ) F ε M ε [g].
Together with the smoothness assumption (.), we deduce

B ε -B 0 B(X ) + A ε -A 0 B(X ) ≤ C ε, ∀ ε ≥ 0. (.)
We then argue similarly as in the proof of [START_REF] Tristani | Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting[END_REF]Theorem 2.15] (see also [START_REF] Kato | Perturbation theory for linear operators[END_REF][START_REF] Mischler | Stability, convergence to selfsimilarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF][START_REF] Mischler | Uniform semigroup spectral analysis of the discrete, fractional & classical fokker-planck equations[END_REF]) and therefore just sketch the proof. We now define

K ε (z) := (R Bε (z)A ε ) 2 R Λ 0 (z)(Λ ε -Λ 0 ) ∈ B(X , X)
and we take any α ∈ (α 0 , 0), recalling that a * < α 0 < 0. We deduce from (.) and the estimates (i) and (ii) in Lemma 2.3 that for some η < |α|, ε 0 > 0, small enough, C > 0, and for any z

∈ ∆ α \B(0, η), ε ∈ [0, ε 0 ), we have K ε (z) B(X) ≤ Cε < Cε 0 < 1, (.)
and thus (1 -K ε (z)) -1 is well defined in B(X). Using the elementary identities

R Λε = R Bε -R Bε A ε R Bε + (R Bε A ε ) 2 R Λε =: U ε + (R Bε A ε ) 2 R Λε , (.) and R Λε = R Λ 0 + R Λ 0 (Λ ε -Λ 0 )R Λε , we get (I -K ε )R Λε = U ε + (R Bε A ε ) 2 R Λ 0 .
Because both terms are well defined in B(X) from Lemma 2.3 and the link between resolvent and semigroup, we also have

(I -K ε )R Lε = U ε + (R Bε A ε ) 2 R L 0 ,
from what we deduce

R Lε = (I -K ε ) -1 (U ε + (R Bε A ε ) 2 R L 0 ).
Since the RHS expression is clearly uniformly bounded in B(X) on the complex set ∆ α \B(0, η) for any ε ∈ [0, ε 0 ), we have

Σ(L ε ) ∩ ∆ α ⊂ B(0, η).
Using the definition of the eigenprojector Π 0 on the eigenspace associated to the spectral values of Λ 0 lying in B(0, η) by mean of Dunford integral (see [START_REF] Kato | Perturbation theory for linear operators[END_REF]Section III.6.4] or [START_REF] Gualdani | Factorization of non-symmetric operators and exponential H-Theorem[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF]) and the analogous of (.) in B(X), we have

Π 0 = i 2π |z|=η R L 0 (z) dz = i 2π |z|=η (R B 0 A 0 ) 2 R L 0 dz,
by using that the contribution of holomorphic functions vanish. In a similar way, we have

Π ε = i 2π |z|=η (I -K ε + K ε )R Lε dz = i 2π |z|=η (R Bε A ε ) 2 R L 0 dz + i 2π |z|=η K ε R Lε dz.
For g ∈ X, we next compute

(Π ε -Π 0 )g X ≤ 1 2π |z|=η ((R Bε A ε ) 2 -(R B 0 A 0 ) 2 )R Λ 0 g X dz + 1 2π |z|=η K ε R Λε g X dz ≤ C ε g X ,
where we have used the identity

(R Bε A ε ) 2 -(R B 0 A 0 ) 2 = R Bε A ε R Bε {(A ε -A 0 ) + (B 0 -B ε )R B 0 A 0 } +R Bε {(A ε -A 0 ) + (B 0 -B ε )R B 0 A 0 }R B 0 A 0
and the estimates (.) and (.). As a consequence, we deduce

Π ε -Π 0 B(X) < 1.
for any ε ∈ (0, ε 0 ), up to take a smaller real number ε 0 > 0. From the classical result [9, Section I.4.6] (or more explicitly [23, Lemma 2.18]), we deduce that there exists ξ ε ∈ ∆ α such that

Σ(Λ ε ) ∩ ∆ α = {ξ ε }, ξ ε is a simple eigenvalue,
for any ε ∈ [0, ε 0 ]. We conclude by observing that ξ ε = 0 because 1 ∈ X ′ and Λ * ε 1 = 0 (which is nothing but the mass conservation).

Weak connectivity regime -nonlinear exponential stability

Now, we focus on the nonlinear exponential stability of the solution to the evolution equation (.)-(.) in the case without delay. We start with an auxiliary result. We define the function Φ :

L 1 (R + ) × R → R by Φ[g, µ] := ∞ 0 a(x, εµ)g(x) dx -µ.
We denote by W 1 the optimal transportation Monge-Kantorovich-Wasserstein distance on the probability measures set P(R + ) associated to the distance d(x, y) = |x -y| ∧ 1, or equivalently defined by

∀ f, g ∈ P(R + ), W 1 (f, g) := sup ϕ, ϕ W 1,∞ ≤1 ∞ 0 (f -g) ϕ.
Lemma 2.8. Assume (.). There exists ε 0 > 0 and for any ε ∈ (0, ε 0 ) there exists a function ϕ ε : P(R) → R which is Lipschitz continuous for the weak topology of probability measures and such that µ = ϕ ε [g] is the unique solution to the equation

µ ∈ R + , Φ(g, µ) = 0.
Proof of Lemma 2.8.

Step 1. Existence.

For any g ∈ P(R) we have Φ(g, 0) > 0, and for any g ∈ P(R) and µ ≥ 0, we have

Φ(g, µ) ≤ a L ∞ -µ, so that Φ(g, µ) < 0 for µ > a L ∞ .
By the intermediate value theorem and the continuity property of Φ, for any fixed g ∈ P(R + ) and ε ≥ 0, there exists at least one solution µ ∈ (0, a L ∞ ] to the equation Φ(g, µ) = 0.

Step 2. Uniqueness and Lipschitz continuity. Fix f, g ∈ P(R + ) and consider µ, ν ∈ R + such that Φ(f, µ) = Φ(g, ν) = 0.

We have

ν -µ = ∞ 0 a(x, εν)(g -f ) + ∞ 0 (a(x, εν) -a(x, εµ))f, with ∞ 0 a(x, εν)(g -f ) ≤ a(•, εν) W 1,∞ W 1 (g, f ),
and

∞ 0 a(x, εν) -a(x, εµ) f ≤ a(•, εν) -a(•, εµ) L ∞ ≤ ε ∂ µ a L ∞ |µ -ν|.
We then obtain

|µ -ν| (1 -ε ∂ µ a L ∞ ) ≤ a(•, εν) W 1,∞ W 1 (g, f ), (.)
and we may fix

ε 0 > 0 such that 1 -ε 0 ∂ µ a L ∞ ∈ (0, 1), ε ∈ [0, ε 0 ].
On the one hand, for f = g, we deduce that µ = ν and that uniquely defines the mapping ϕ ε [g] := µ. On the other hand, the function is Lipschitz continuous because of (.). We also recall the following classical Grönwall's type lemma.

Lemma 2.9. Assume that u ∈ C([0, ∞); R + ) satisfies the integral inequality

u(t) ≤ C 1 e at u 0 + C 2 t 0 e a(t-s) u(s) 2 ds, ∀ t > 0,
for some constants C 1 ≥ 1, C 2 , u 0 ≥ 0 and a < 0. Under the smallness assumption a + 2C 2 u 0 < 0,

there holds u(t) ≤ 1 + C 1 u 0 C 2 |a + 2C 2 u 0 | C 1 e at u 0 , ∀ t ≥ 0.
Proof of Lemma 2.9. We fix A ∈ (C 1 u 0 , 2C 1 u 0 ), so that C 1 u(t) ≤ A at least on a small interval, that is for any t ∈ [0, τ ], τ > 0 small enough, and then the integral inequality implies on the same interval

u(t) ≤ C 1 e at u 0 + C 2 C -1 1 A t 0
e a(t-s) u(s) ds.

The classical Grönwall's lemma (for linear integral inequality) and the smallness assumption a

+ C 2 C -1 1 A ≤ 0 imply u(t) ≤ C 1 u 0 e (a+C 2 C -1 1 A)t ≤ C 1 u 0 < A
on that interval. By a continuity argument, the first above inequality holds on R + and then with A := C 1 u 0 . Next, replacing that first estimate in the integral inequality we started with, we get

u(t) ≤ C 1 e at u 0 + C 2 C 2 1 u 2 0 e at t 0 e (a+2C 2 u 0 )s ds, ∀ t > 0,
from which we immediately conclude.

We come to the proof of our main result Theorem 1.2 in the case without delay.

Proof of Theorem 1.2 in the case without delay. We split the proof into two steps.

Step 1. New formulation. We start giving a new formulation of the solutions to the evolution and stationary equations in the weak connectivity regime ε ∈ (0, ε 0 ], where ε 0 is defined in Lemma 2.8. For a given initial datum 0 ≤ f 0 ∈ L 1 (R + ) with unit mass the solution f ∈ C([0, ∞); L 1 (R + )) to the evolution equation (.) and the solution F ε to the stationary equation (.) clearly satisfy

∂ t f + ∂ x f + a(εϕ[f ])f = 0, f (t, 0) = ϕ[f (t, •)], ∂ x F + a(εM)F = 0, F (0) = M = ϕ[F ],
where here and below the ε and x dependency is often removed without risk of misleading. We introduce the variation function g := f -F which satisfies the PDE

∂ t g = -∂ x g -a(εM)g -εa ′ (εM)F M[g] -Q[g] (.) with Q[g] := a(εϕ[f ])f -a(εϕ[F ])F -a(εϕ[F ])g -εa ′ (εϕ[F ])F M[g],
where M = M ε is defined in (.). The above PDE is complemented with the boundary condition

g(t, 0) = ϕ[f (t, •)] -ϕ[F ],
and we may write again

ϕ[f ] -ϕ[F ] = ∞ 0 a(εϕ[f ])f - ∞ 0 a(εϕ[F ])F = ∞ 0 a(εM)g + εa ′ (εM)F M[g] + ∞ 0 Q[g] dx = M[g] + Q[g], Q[g] := Q[g] .
As a consequence, we have proved that the variation function g satisfies the equation

∂ t g = Λ ε g + Z[g], Z[g] := -Q[g] + δ 0 Q[g]. (.)
Step 2. The nonlinear term. On the one hand, we obviously have

Z[g] = 0, ∀ g ∈ M 1 (R + ). (.)
On the other hand, in order to get an estimate on the nonlinear term Z[g], we introduce the notation

ψ(u) = a(x, εm u )f u ,
where, for some fixed g ∈ P(R + ), g = 0, we have set

f := F + g, f u := uf + (1 -u)F, m u := ϕ[f u ].
We first notice that ψ(0) = a(εϕ[F ])F and ψ(1) = a(εϕ[f ])f . Second, we have

ψ ′ (u) = a ′ ε (m u )f u m ′ u + a ε (m u )g. (.)
In order to compute m ′ u , we differentiate with respect to u the identity

m u = ∞ 0 a ε (m u )f u dx,
and we have

m ′ u = ∞ 0 a ′ ε (m u )f u dx m ′ u + ∞ 0 a ε (m u ) gdx,
which implies

m ′ u = 1 - ∞ 0 a ′ ε (m u )f u dx -1 ∞ 0 a ε (m u ) gdx. (.)
We may thus observe that m ′ 0 = M[g], so that ψ ′ (0) = a ′ ε (M)F M ε [g] + a ε (M)g, and therefore

Q[g] = ψ(1) -ψ(0) -ψ ′ (0).
Third, from (.), we have

ψ ′′ (u) = a ′′ ε (m u )f u (m ′ u ) 2 + 2a ′ ε (m u )gm ′ u + a ′ ε (m u )f u m ′′ u ,
and from (.), we have

m ′′ (u) = 2 1 - ∞ 0 a ′ ε f u -2 ∞ 0 a ε g ∞ 0 a ′ ε g +2 1 - ∞ 0 a ′ ε f u -3 ∞ 0 a ′′ ε f ∞ 0 a ε g 2 .
In the small connectivity regime ε ∈ (0, ε 0 ], ε 0 a ′ ∞ < 1, we get the bound

ψ ′′ (u) X ≤ a ′′ ε ∞ |m ′ u | 2 + 2 a ′ ε ∞ g X |m ′ u | + a ′ ε ∞ |m ′′ u | ≤ ε 2 a ′′ ∞ a 2 ∞ (1 -ε a ′ ∞ ) 2 g 2 X + 2ε a ′ ∞ a ∞ 1 -ε a ′ ∞ g 2 X +2ε 2 a ′ 2 ∞ a ∞ (1 -ε a ′ ∞ ) 2 g 2 X + 2ε 3 a ′′ ∞ a ′ ∞ a ∞ (1 -ε a ′ ∞ ) 3 g 2 X ≤ ε K g 2 X ,
for some constant K ∈ (0, ∞). Using the Taylor expansion

Q[g] = ψ(1) -ψ(0) -ψ ′ (0) = 1 0 (1 -u)ψ ′′ (u)du,
we then obtain

Z[g] X ≤ 2 Q[g] X ≤ 1 0 (1 -u) ψ ′′ (u) X du ≤ C g 2 X .
Step 3. Decay estimate. Thanks to the Duhamel formula, the solution g to the evolution equation (.) satisfies

g(t) = S Λε (t)(f 0 -F ) + t 0 S Λε (t -s)Z[g(s)] ds.
Using Theorem 2.7 and the second step, we deduce

g(t) X ≤ C e αt g 0 X + t 0 C e α(t-s) Z[g(s)] X ds ≤ C e αt g 0 X + C ε K t 0 e α(t-s) g(s) 2 X ds,
for any t ≥ 0 and for some constant C ≥ 1, α < 0, independent of ε ∈ (0, ε 0 ]. Observing that g(t) X ∈ C([0, ∞), we conclude thanks to Lemma 2.9.

but adapting the functional framework. The main difference comes from the boundary term and it will be explained in the first subsection. We have already proved in Theorem 1.1 the existence of a unique stationary solution (F ε , M ε ) in the weak connectivity regime and we may then focus on the evolution equation.

Linearized equation and structure of the spectrum

In order to write as a time autonomous equation the linearized equation (.)-(.)-(.), we introduce the following intermediate evolution equation on a function v = v(t, y)

∂ t v + ∂ y v = 0, v(t, 0) = q(t), v(0, y) = 0, (.)
where y ≥ 0 represents the local time for the network activity. That last equation can be solved with the characteristics method v(t, y) = q(ty)1 0≤y≤t .

Therefore, equation (.) on the variation n(t) of network activity writes

n(t) = D[v(t)], D[v] := ∞ 0 v(y)b(dy),
and then equation (.) on the variation q(t) of discharging neurons writes

q(t) = O ε [g(t), v(t)], with O ε [g, v] := N ε [g] + κ ε D[v], N ε [g] := ∞ 0 a ε (M ε )g dx, κ ε := ∞ 0 a ′ ε (M ε )F ε dx.
All together, we may rewrite the linear system (.)-(.)-(.), as the autonomous system

∂ t (g, v) = L ε (g, v), (.)
where the operator L ε is defined by

L ε g v := -∂ x g -a ε g -a ′ ε F ε D[v] -∂ y v , with domain D(L ε ) := {(g, v) ∈ W 1,1 (R + ) × W 1,1 (R + , ω); g(0) = v(0) = O ε [g, v]},
where ω(x) := e -δx with δ > 0 defining in (.). The associated semigroup S Lε (t) acts on

X := X 1 × X 2 := L 1 (R + ) × L 1 (R + , ω).
Considering the boundary condition as a source term, we also introduce the semigroup S Λε (t) acting on

X := X 1 × X 2 := M 1 (R + ) × M 1 (R + , ω) with the generator Λ ε = (Λ 1 ε , Λ 2 ε ) given by Λ 1 ε (g, v) := -∂ x g -a ε g -a ′ ε F ε D[v] + δ x=0 O ε [g, v], Λ 2 ε (g, v) := -∂ y v + δ y=0 O ε [g, v].
In a similar way as in section 2.2, we have S Λε | X = S Lε . As a first step, we establish that the semigroup S Λε has a nice decomposition structure with finite dimensional principal modes and a fast decaying remainder term. The result is obtained as a consequence of the Spectral Mapping and Weyl's theory developed in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Erratum: Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Mischler | Semigroups in Banach spaces -splitting approach for spectral analysis and asymptotics estimates[END_REF] and taken over in section 2.2. For that purpose, we ontroduce the convenient splitting of the operator Λ ε on X as Λ

ε = A ε + B ε defined by B ε (g, v) = B 1 ε (g, v) B 2 ε (g, v) = -∂ x g -a ε g -∂ y v and A ε (g, v) = A 1 ε (g, v) A 2 ε (g, v) = -a ′ ε F ε D[v] + δ x=0 O ε [g, v] δ y=0 O ε [g, v] .
It is only necessary to establish the following adequate properties of the operators A ε and B ε . We skip the rest of the proof and refer to the proof of Theorem 2.2 for more details.

Lemma 3.2. Assume that a satisfies conditions (.)-(.) and that b satisfies (.). For any ε ≥ 0, the operators A ε and B ε satisfy :

(i) A ε ∈ B(W -1,1 (R + ) × W -1,1 (R + , ω), X ).
(ii) S Bε (t) is a ♯ -hypodissipative in both X and X ;

(iii) the family of operators S Bε * A ε S Bε satisfies

(S Bε * A ε S Bε )(t) B(X ,Y ) ≤ Ce αt , ∀a > a ♯ ,
for some constant C a > 0 and with

Y := Y 1 ×Y 2 , where Y 1 = BV (R + )∩ L 1 1 (R + ) and Y 2 = BV (R + , ω) ∩ L 1 1 (R + , ω). Proof. (i) It is an immediate consequence of the fact that D ∈ B(W -1,1 (R + , ω), R) and N ε ∈ B(W -1,1 (R + ), R) because of (.) and (.).
(ii) Since S B 1 ε is nothing but the semigroup S Bε defined in (.) which is a * -dissipative thanks to Lemma 2.3-(ii), we just have to prove the dissipativity of the translation semigroup S B 2 ε which is defined through the explicit formula

[S B 2 ε (t)v](y) = v(y -t)1 y-t≥0 . That follows from S B 2 ε (t)v X 2 ≤ ∞ 0 |v(y -t)|1 y-t≥0 e -δy dy ≤ e -δt v X 2 ,
for any v ∈ X 2 and any t ≥ 0.

(iii) For (g, v) ∈ X , we have

A 1 S B (t)(g, v)(x) = γ(x)D(t) + δ x=0 N(t), (.) A 2 S B (t)(g, v)(y) = δ y=0 κD(t) + δ y=0 N(t), (.) with γ(x) := κδ x=0 -a ′ (x)F ε (x), N(t) := N [S B 1 (t)g] = ∞ 0 a(x)e A(x-t)-A(x) g(x -t)1 x-t≥0 dx, D(t) := D[S B 2 (t)v] = ∞ 0 v(y -t)1 y-t≥0 b(dy),
where here and below the ε dependency is removed without risk of ambiguity. We observe that

|N(t)| ≤ Ca 1 e 3βt g X 1 , |D(t)| ≤ Ce δt v X 2 ,
for any t ≥ 0. We then compute

N ′ (t) = ∞ 0 ∂ x [a(x)e -A(x) ]e A(x-t) g(x -t)1 x-t≥0 dx = ∞ 0 [a ′ (x) -a(x) 2 ]e A(x-t)-A(x) g(x -t)1 x-t≥0 dx, D ′ (t) = ∞ 0 v(y -t)1 y-t≥0 b ′ (dy) = b ′ * v(t),
from what we get the estimates

|N ′ (t)| e a * t g X 1 , |D ′ (t)| e -δt v X 2 .
Denoting

T 1 (t)(g, v)(x) := (S B 1 * A 1 S B )(t)(g, v)(x), T 2 (t)(g, v)(y) := (S B 2 * A 2 S B )(t)(g, v)(y), we compute T 1 (t)(g, v)(x) = t 0 S B 1 (s) γ(x)D(t -s) + δ x=0 N(t -s) ds = t 0 e A(x-s)-A(x) γ(x -s)D(t -s) + δ x-s=0 N(t -s)1 x-s≥0 ds = e -A(x) (ν * Ďt )(x) + e -A(x) N(t -x)1 x≤t , T 2 (t)(g, v)(y) = t 0 S B 2 (s)δ y=0 κD(t -s) + N(t -s) ds = t 0 δ y-s=0 κD(t -s) + N(t -s) ds = κD(t -y) + N(t -y) 1 y≤t ,
where we use the notation ν := γ e A . We next differentiate the above identity, and we get ∂ x T 1 (t)(g, v)(x) = -a(x)e -A(x) (ν * Ďt )(x) + N(tx)1 x≤t -e -A(x) ν * Ď′ t (x) + N ′ (tx)1 x≤t -e -A(x) ν(xt)D(0)1 x-t≥0e -A(x) N(0)δ x=t +e -A(x) ν(x)D(t), ∂ y T 2 (t)(g, v)(y) = -(κD ′ + N ′ )(ty)1 y≤t -(κD + N)(0)δ y=t All together, we deduce

∂ x T 1 (t)(g, v)(x) X 1
e a ♯ t (g, v) X , ∂ y T 2 (t)(g, v)(y) X 2 e a ♯ t (g, v) X , and the similar estimate for (S B * AS B )(t)(g, v) X . As a consequence, the announced estimate holds for the family of operators S B * AS B .

The vanishing connectivity regime

When the network connectivity parameter vanishes, ε = 0, the linearized operator simplifies as

Λ 0 g v = -∂ x g -a(x, 0)g + δ x=0 O 0 [g, v] -∂ y v + δ y=0 O 0 [g, v] , (.)
where O 0 [g, v] = N 0 [g] = ∞ 0 a(x, 0)g(x)dx. The associated semigroup is exponentially stable as shown in the following theorem. Theorem 3.3. There exist some constants α < 0 and C > 0 such that Σ(Λ 0 ) ∩ ∆ α = {0} and for any (g 0 , v 0 ) ∈ X , g 0 = 0, there holds S Λ 0 (t)(g 0 , v 0 ) X ≤ Ce αt (g 0 , v 0 ) X , ∀ t ≥ 0.

(.)

Proof of Theorem 3.3. Since Λ 1 0 = Λ 0 , from Theorem 2.4 we have already proved that g(t) := S Λ 1 0 (t)g 0 satisfies g(t) X 1 ≤ Ce at g 0 X 1 for any t ≥ 0 and any a ∈ (a * , 0). We then focus on Λ 2 0 . The Duhamel formula associated to the equation ∂ t v = Λ 2 0 (g, v) writes v(t) = S B 2 0 (t)v 0 + t 0 S B 2 0 (ts)A 2 0 g(s), v(s) ds.

Using the already known estimate on g(t), we deduce

S Λ 2 0 v 0 (t) X 2 = v(t) X 2 ≤ S B 2 0 (t)v 0 X 2 + t 0 S B 2 0 (t -s)δ 0 N 0 [g(s)] X 2 ds ≤ e -δt v 0 X 2 + t 0
e -δ(t-s) C e as g 0 X 1 ds ≤ C e αt (g 0 , v 0 ) X for max{a, -δ} < α < 0, which yields our conclusion.

Weak connectivity regime -exponential stability of the linearized equation

In this part, we shall discuss the geometry structure of the spectrum of the linearized time elapsed equation in weak connectivity regime taking delay into account and using again a perturbation argument.

Theorem 3.4. There exists some constants ε 0 > 0, C ≥ 1 and α < 0 such that for any ε ∈ [0, ε 0 ] there holds Σ(Λ ε ) ∩ ∆ α = {0} and S Λε (t)(g 0 , v 0 ) X ≤ Ce αt (g 0 , v 0 ) X , (.)

for any (g 0 , v 0 ) ∈ X such that g 0 = 0.

We start presenting a technical result needed in the proof below.

Lemma 3.5. The operator Λ ε is continuous with respect to ε, and more precisely Λ ε -Λ 0 B(X ) ≤ O(ε).

(.)

Proof. For all (g, v) ∈ X , we have

Λ ε g v = -∂ x g -a ε g -a ′ ε F ε D ε [v] + δ x=0 O ε [g, v] -∂ y v + δ y=0 O ε [g, v]
, (.a)

Λ 0 g v = -∂
(Λ ε -Λ 0 ) g v = (a(x, 0) -a ε )g -a ′ ε F ε D ε [v] + δ x=0 (O ε [g, v] -O 0 [g, v]) δ y=0 (O ε [g, v] -O 0 [g, v]
) .

We then compute

(Λ ε -Λ 0 )(g, v) X = (a(x, 0) -a ε )g X 1 + a ′ ε F ε D ε [v] X 1 + 2|O ε [g, v] -O 0 [g, v]| ≤ 3 (a ε -a 0 )g X 1 + 2 a ′ ε F ε D ε [v] X 2 ≤ 3ε a ′ ∞ g X 1 + 2εa 1 a ′ ∞ (1 -ε a ′ ∞ ) F ε X 1 v X 2 = Cε (g, v) X ,
which is nothing but (.).

Proof of Theorem 3.4. With the help of Lemma 3.5, we may proceed similarly as in the proof of Theorem 2.7 (see also again [START_REF] Tristani | Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting[END_REF][START_REF] Mischler | Uniform semigroup spectral analysis of the discrete, fractional & classical fokker-planck equations[END_REF]) and we conclude that Σ(Λ ε ) ∩ ∆ α = {ξ ε }, with |ξ ε | ≤ O(ε) and ξ ε is algebraically simple. We observe that

Λ * ε ϕ ψ = ∂ x ϕ -a ε ϕ + a ε (ϕ(0) + ψ(0)) ∂ y ψ + κ ε b ψ(0) + κ ε b ϕ(0) -b a ′ ε F ε ϕ dx
, from which we deduce that Λ * ε (1, 0) = 0. Then 0 ∈ Σ(Λ * ε ) and ξ ε = 0. Moreover, the orthogonality condition g 0 = (g 0 , v 0 ), (1, 0) X ,X ′ = 0 implies that the exponential estimate (.) holds.

Weak connectivity regime -nonlinear exponential stability

We finally come back on the nonlinear problem and we present the proof of the second part of our main result for the case with delay.

Proof of Theorem 1.2 in case with delay. We write the system as We introduce the variation g := f -F and v = u -U. The equation on g is

∂ t f = -∂ x f
∂ t g = -∂ x g -a ε (D[u])f + a ε (M)F + δ 0 (P[f, D[u]] -P[F, D[U]]) = -∂ x g -a ε (M)f -a ′ ε F D[v] -Q[g, v] + δ 0 O[g, v] + δ 0 Q[g, v] = Λ 1 ε (g, v) + Z 1 [g, v],

p

  (t) := P[f (t); m(t)], (.) where P[g, µ] = P ε [g, µ] := ∞ 0 a(x, εµ)g(x)dx. (.)

0 e

 0 .) with ν ε := γ ε e A and the classical notation Ňt (s) = N(ts). Starting from identity (.) and denoting e -β (x) := e -βx , we have |(S B * AS B )(t)g(x)| e -β (x) ≤ e -βx t 0 C e 3βt |γ ε (xs)||N(ts)| ds = C e a * t t 2β(t-s) e -β(x-s) |γ ε (xs)| e -β(t-s) |N(ts)| ds

Theorem 3 . 1 .

 31 Assume (.)-(.)-(.) and (.). The conclusions of Theorem 2.2 holds true with a ♯ := max{a * , -δ} < 0.

  x ga(x, 0)g + δ x=0 O 0 [g, v] -∂ y v + δ y=0 O 0 [g, v].(.b) From (.a)-(.b), we deduce

  a ε (D[u])f + δ 0 P[f, D[u]] ∂ t u = -∂ y u + δ 0 P[f, D[u]], with P[f, m] = a(m)f, D[u] = bu.We recall that the steady state (F, U), U := M1 y≥0 , satisfies0 = -∂ x Fa ε (M)F + δ 0 M 0 = -∂ y U + δ 0 M, M = D[U] = P[F, D[U]].

This section is devoted to the proof of our main result, Theorem 1.2, in the case with delay by following the same strategy as in the case without delay

where

The equation on v is

where

We then write the associated Duhamel formula

Refering to

Step 2 in the proof of Theorem 1.2 gives a similar estimate Z[g, v] X ≤ C (g, v) 2 X , we then conclude the rest part of Theorem 1.2 in the case with delay.


Julien Vercueil

Emerging Economies. Genealogy, evolutions and vulnerabilities

Genealogy

. Unfortunately, these attempts did not help much clarifying this fuzzy concept. For instance, the IMF World Economic Outlook repeatedly used the notion of "emerging Asia", or "emerging Europe", to qualify some countries of these continents. But Africa and Latin America apparently did not deserve this qualification. Neither were Georgia and CIS countries included in the "emerging Europe". In one IMF publication certain Asian countries are sometimes qualified of "emerging countries" (IMF, 2010, p. 64), sometimes not (Id., p. 67).

Last but not least, the notion of emerging economy has been completed by a range of acronyms that did not help, in turn, to clarify the economic meaning of the whole. BRICS, Frontier, CIVETS,CEMENT,BRIICS,NEST,MINT,Eagles,BENIVM,CARB are some examples of those acronyms that flourished in the financial community since the tremendous success of the initial "BRIC" expression used by Jim O'Neil, of Goldman Sachs, in 2001 in order to characterize four economic powers emerging out of the G--7 circle, in the beginning of this century [START_REF] O'neil | Building Better Global Economic BRICs[END_REF].

Definition

The fact that the notion of "emerging economy" has not been clarified at an official level is understandable. In order to create a new official category of countries, the number of justifications that an international organisation like the World Bank and the IMF would have to provide has certainly been seen as problematic. Does it follow that any attempt to rationalize this category is doomed to fail? Would it be preferable to let aside this concept, because of its lack of consistency, and go on using old official categories as "middle income" economies? Our hypothesis is that the phenomenon involved in the various meanings of the concept of "emerging economies" deserves attention, especially from economists who are aware of the importance of institutions in the process of economic and social development. While being diverse, emerging economies have some characteristics in common. They have been experiencing deep transformations for several decades that are far from being identical, but that are pointing toward the same direction: all of these countries have undergone major changes in their economic institutions since the 1990's, and all of them are now much more integrated in the world economy than twenty years ago. They have become actors of the globalization, they have contributed to it in an increasingly large extent, and in return their economies have been profoundly shaped by it. Economic emergence is intrinsically a dynamic notion. It is a process of economic and institutional transformation of middle--income countries that entails an acceleration of their economic growth and an increased participation to international trade and capital flows. It can be qualified by the combination of three criteria:

--First criterion: intermediate income. Per capita GDP, measured in PPP terms, is comprised between 10% and 75% of the average income in the European Union (obviously, those two limits are to be interpreted with a margin of flexibility). --Second criterion: economic growth and catching--up process. During the past decades, these countries have enjoyed an average economic growth that surpasses European Union's growth. --Third criterion: institutional transformations and economic opening. The countries belonging to the "emerging" category can also be characterized by the recent changes in their institutional framework concerning economic activities, particularly as far as external economic relations are concerned. Therefore, their economic growth is now far more heavily influenced by external developments than before.

In the beginning of the 2010's, we used this set of criteria to propose a indicative list of countries 1 that could be considered as "emerging economies" [START_REF] Vercueil | Les pays émergents. Brésil -Russie -Inde -Chine… Mutations économiques et nouveaux défis[END_REF].

Evolution

Five years later, has the meaning -and, eventually, the list -of emerging economies changed? The 2008 financial crisis and the following slowing down of the world economy have unevenly impacted these countries, depending on their growth model and their vulnerabilities [START_REF] Vercueil | Les pays émergents. Brésil -Russie -Inde -Chine… Mutations économiques, crises et nouveaux défis[END_REF]. But all of the countries in our initial pool of "emerging economies" are still, in 2016 situated on a long--term catching up trajectory vis--à--vis more advanced (EU average or OECD average) countries (Table 1). Therefore, we claim that even in the current atmosphere of doubt about the current economic perspectives of some of them, it is simply to early to speak of the "end of emerging economies". Source: The World Bank, 2016. Author's calculations. However, if we use a narrower time scope, we must reckon that several "emerging countries" have struggled against deep economic turbulences since 2008. For the past five years Latvia, Ukraine, Serbia, Venezuela and Russia have been showing economic trajectories that are less dynamic than the European Union, disrupting, for the moment, their long--term catching up trend. For the first four of them, their level of GDP is now even lower than it was in 2008. Meanwhile, by contrast, Argentina, Brazil and South Africa, who are also experiencing various degrees of economic and social difficulties nowadays, are still far better off in 2016 (between 11 and 13 % respectively) than they were just before the international financial crisis. Vulnerabilities This is not to say that the "emerging economies" category is a stable one. First, some countries can leave this category by becoming advanced countries. This is the case, for instance, of the Slovak Republic, whose per capita GDP in PPP terms is now significantly higher than 75 % of the average EU per capita income. Conversely, it is also possible that some institutional, political or economical changes in the situation of a given country (more probably if it is situated at the lower limit) prevent it to meet simultaneously the three criteria proposed before for defining an emerging economy. Here come the vulnerabilities. Financial - overexposure to capital inflows and outflows for instance, or over--evaluation of the national currency --, economical - dependency on raw materials exports, structural deficit of the current account --, institutionalpervasive corruption, lack of proper regulation of a given sector, weaknesses in the rule of law -, social - income and wealth inequalities, socio--political conflicts in the society--, environmental -degradation of the living and working conditions for workers and their families, due to various pollutions, exhaustion of some raw materials --, and geopolitical vulnerabilities -unresolved territorial conflicts, political unrest of social groups that are fostered by a foreign country -, all these risks are present, to a certain extent, in each region where emerging economies are located. Obviously, the political leadership's ability to tackle each of these vulnerabilities without jeopardising their growth potential will be key for the future of these emerging economies.

Table 1 . Cumulative GDP growth in selected emerging countries (1999--2015)

 1

		1999 Index	2005	2010	2015
	World	100	121,31	137,97	156,93
	OECD members	100	115,66	121,59	131,42
	European Union	100	114,17	119,39	125,18
	Albania	100	141,86	182,55	200,83
	Angola	100	137,88	245,99	309,94
	Argentina	100	109,19	138,98	148,96
	Armenia	100	188,37	227,50	281,19
	Azerbaijan	100	209,17	447,08	498,82
	Bosnia and Herzegovina	100	139,18	160,10	170,88
	Brazil	100	120,40	149,85	157,35
	Bulgaria	100	139,35	162,17	174,88
	Chile	100	128,39	152,39	183,89
	China	100	172,75	294,50	429,00
	Colombia	100	124,73	155,70	194,85
	Ecuador	100	128,11	151,35	187,49
	Egypt, Arab Rep.	100	125,33	169,11	191,39
	Estonia	100	155,74	152,92	182,79
	Georgia	100	145,05	186,46	236,37
	India	100	143,74	211,96	293,69
	Indonesia	100	132,18	174,73	228,51
	Jordan	100	142,00	192,19	219,61
	Kazakhstan	100	179,82	243,16	303,90
	Latvia	100	156,29	152,79	181,34
	Lithuania	100	149,83	158,85	189,57
	Macedonia, FYR	100	115,26	140,15	157,73
	Malaysia	100	137,22	170,81	220,97
	1 See

Table 1 .

 1 Some countries are absent of this list due to a lack of data in 2011 in the Groeningen University database used for its elaboration:Algeria, Costa Rica, Dominican Republic, Guatemala, Honduras, Jamaica, Lebanon, Libya, Panama, El Salvador.

	Mexico	100	114,22	125,65	144,50
	Moldova	100	143,72	168,43	203,03
	Morocco	100	129,37	165,16	200,50
	Peru	100	126,62	176,72	223,05
	Philippines	100	130,67	166,25	221,03
	Poland	100	120,68	151,95	175,67
	Romania	100	135,46	156,41	175,90
	Russian Federation	100	148,12	176,28	186,83
	Serbia	100	145,63	166,11	169,10
	Slovak Republic	100	129,04	162,79	183,10
	South Africa	100	125,75	146,46	162,44
	Thailand	100	136,15	163,57	188,29
	Tunisia	100	126,76	158,28	169,02
	Turkey	100	133,40	156,08	193,53
	Ukraine	100	153,22	161,11	142,92
	Uruguay	100	98,97	132,20	157,03
	Venezuela, RB	100	117,62	141,09	142,59
	Vietnam	100	149,06	202,47	269,84


Income inequalities, productive structure and macroeconomic dynamics. A regional approach to the Russian case Julien Vercueil CREE, INALCO,

Working paper, comments welcome / Version May, 15 th 2016. international comparison being carried using IMF PPP exchange rates. The main results of this comparison are summarised in Table 1. Sources: Crédit Suisse Research Institute (2015a), p. 32, and(2015b), p. 120.

As far as Russia is concerned, this definition poses several problems. First, the data on which the abovementioned analysis is conducted are questionable. Second, such an analysis seems to underestimate the size of the Russian middle class when compared to other countries. In relative terms, this group is presented as comparable to its Indian counterpart (resp. 4,1 % and 3 % of the population), whereas in terms of per capita GDP, real income and consumption expenditures, the share of middle--income Russian households is far higher. Therefore, in the following analysis we use these figures for time--series comparison (and not cross--sections analysis). Our aim is to compare the impact of recent economic fluctuations on the evolutions of income groups in BRICS population. During the 2000--2007 period, all countries enjoyed an increase of the population belonging to the wealth--defined middle class (Table 2). The most spectacular growth occurred in China, which managed to increase the group by more than 100 millions people. In Russia, with a population of 143 millions people, the increase was of 11 millions people, approximately the same amount as for United States (over 300 millions people, of which more than a third was already part of the middle class) and India (with a total population of around 1 billion). The crisis period (identified by the author of the Credit Suisse Research Institute study as 2007--2008, although for number of countries like Russia, the most acute period of GDP drop was 2008--2009) provoked a sudden fall of the middle class population. In China 67 millions people were expelled from this category in only two years while in the United States and in India, the figures are of 6,8 and 5,5 millions respectively. In Russia, the fall was less acute (2,5 millions people quitted the group), a paradoxical results that should be explained by the period chosen for the comparison. Indeed, after 2008 Russia was the only country in which the drop of the middle class population continued. Russia's wealth--defined middle class lose 9,2 millions people when in China, it gained 8,5 millions, in the United States -- 17 millions, and in India --0,7 million. These contrasting evolutions confirm the "divergence hypothesis". According to this hypothesis, due to the varying growth models of BRICS countries their macroeconomic trajectories tended to diverge as soon as the financial crisis of 2007 began to affect the world economy (Vercueil (2015)).

Starting from these general comparative observations, we intend to dig into the possible explanations regarding these uneven vulnerability of income distribution in BRICS countries to the changing world macroeconomic conditions. We focus on Russia, whose growth model is qualitatively different from those of China or India. Russia's growth model can be characterizes broadly as "rent based", relying on energy exports to the rest of the world that contributes for a half to State budget earnings, while helping to stabilize the exchange rate of the rouble and providing the domestic economy with the necessary amount of foreign currency to import manufactured goods and equipments.

In order to link the growth model with external vulnerability, we use the regional dataset provided by Rosstat (2015), featuring value added sectorial composition, incomes evolution, structure and distribution. Our main hypothesis is that the regional level of analysis can help us to understand the recent changes affecting income distribution at the macro--level.

2. Inside Russia: regional development and productive structure Russia is a largely urbanized society: almost three quarters of the population live in urban area. The urbanization process is still at works, accompanied with a westernization of the population. Eastern regions are loosing people regularly, at a pace that is higher than the natural demographic trend, hence indicating an emigration trend. The largest western cities have benefitted from this trend. Since 2005 Moscow, St Petersburg and their region's population grew 20 times faster than the rest of the country. They now account for 18,3 % of the national population. More generally, the thirty--seven biggest cities (those that have more than 500000 habitants) gather about 30% of the country's population. Among them, 24 are situated in the European part of the territory. However, this urbanization and westernization trend does not totally match the evolution of the overall production distribution. For instance, the regional repartition of per capita GDP doesn't show that big cities (dark blue bars in Chart 1) produce more wealth than some relatively remote, poorly populated regions (green bars). This is because some of the latter regions are well endowed with natural resources that require capital intensive, labour saving productions technologies. Hence, the amount of per capita regional GDP can be very high since it largely reflects the level of productivity of extractive industries. More importantly, this productivity level is directly and positively tied to the world prices of raw materials that are exported by Russia.

Graph 1. Per capita regional output (2013, current roubles)

Source: Rosstat (2015) 3. A productive typology of regions A first observation can illustrate the abovementioned link at the regional level: it shows a strong correlation between the intensity in raw materials production of the region and its per capita GDP (Graph 2). This correlation implies that, given the relatively modest number of regions that benefit from a rich natural resources endowment, strong fiscal and redistributive policies are needed to correct the consecutive distortion effects on household incomes. In order to examine the relation between the productive structure of Russian regions and their characteristics in terms of income distribution, we construct a simple regional productive typology. For this purpose we use the Rosstat regional dataset presenting the structure of the value added by sectors. Starting from the 15 different sectors distinguished by Rosstat, we divide them according to their relation to world markets and competition. "Rent--providing" sectors are those that can directly benefit from the rise of raw materials prices. When world prices are high, they provide a rent for producers who enjoy the positive outcomes of an oligopolistic situation as they have an exclusive access to deposits. We include financial services (representing only 1,5 % of the total value added) into this category because of the existing links between oil and gas majors companies and some Russian banks. "Non tradable" sectors are sheltered from the world competition either because of their very nature (some services cannot be imported), either because of national regulation regarding their activities (public services). On the contrary, "tradable" sectors' products can be exported or imported. Firms belonging to these sectors are exposed to foreign competition even if, in some cases, a varying degree of protection is provided by the Government. Table 2 presents the result of our re--arranging of these sectors according to this typology. Tradable sectors Source: author's elaboration At the national level, the Russian economy exhibits the following structure: rent-providing sectors represents 15,2% of the domestic value added; non tradable sectors, 63%; tradable sectors, 28%. We use this distribution of weight and the national average per capita income as benchmarks for the regional analysis. Starting from this, we distribute Russian regions among the 3 following types: Type 1. Outward oriented, rent--providing regions: this type regroups regions in which rent--providing sectors accounts for more than 25% of the regional value added (that is, almost the double than the national average). Type 2. Non--tradables intensive regions: in these regions, non--tradable sectors account for at least 60% of the value added (equal to, or higher than the national average). Type 3. Exposed regions: in these regions, tradable sectors represent more than 30% of the national value added (more than the national average). Analysing 84 Russian regions with these criteria, we end up with a regional typology presented in Table 3. Sixteen regions fall into the "rent providing" group, 36 are considered as "non--tradables intensive" ones and 32 are of the "exposed" type. It can be noted that the first group is composed of some of the richest regions in terms of GDP. Its 2014 average per capita GDP is two times higher than the national average, and three times higher than in the two other groups.

From productive structure to income distribution: a first look into Russian regions

According to Rosstat statistics and using implied PPP exchange rates of the IMF, at the national level 16% of the population earn more than 2000 US $ per month, a result that tend to confirm the limits abovementioned of the Credit Suisse Research Institute's indicator for defining the middle class, that lead to the underestimation of the size of the Russian "middle class" (only 4% according to the wealth--based criteria, cf. supra). On the other hand, like in other emerging countries, income distribution in Russia is strongly concentrated: less than 37% of the population earn more than the national average. This leads us to ask whether the productive type of region can be associated with a specific pattern of income distribution. As we combine the Gini coefficient with per capita GDP, we obtain a picture that confirms our first findings (Graph 2): rent--providing regions are more productive than the others, but also more unequal. The Graph 4 distibutes Russian regions in four quadrants. Nine of them are situated in Q4, in which both per capita GDP and Gini coefficient are higher than national average. On the opposite side, only two exposed regions (out of 32) enjoy a relatively high level of per capita GDP. They are both situated in Q4 and have a share of rent--providing sectors in their value added that, while being inferior to the 30 % limit, is superior to the national average. Most (more than twenty) of the remaining exposed regions are situated in Q2, where Gini coefficient and per capita GDP levels are lower than the national average. As regards per capita GDP, only seven non--tradables intensive regions are equal or above the average. Among them are Moscow, St Petersburg and the oblast of Leningrad.

  Graph 4. Types of regions, per capita GDP and Gini coefficient : rent--providing regions : non--tradable intensive regions : exposed regions Source: author's elaboration based on Rosstat (2015) 5. Who earns what? Sources of income and productive types of region In the last step of this study, we use Rosstat data on households' income sources to shed more light on our typology of Russian regions. The data set distinguishes five sources of revenues: salaries, business incomes, property revenues, social transfers and other sources. This last category is composed of hidden revenues, remittances and currency sales. During the last decade, the fundamental structure of households' incomes sources did not change. Salaries remained the main source of income, accounting for more than 40% of the total. Two other important sources are social transfers and other sources. Put together, they represented almost 45% of total revenues in 2014, showing a slightly rising tendency since 2005. The remaining 14--15% are divided into business incomes (8--9%) and property revenues (little less than 6%) (Graph 5

Table 1 . Wealth--based definition of middle class: an international comparison Wealth lower bound of middle class, US $ (PPP)

 1

	Middle class,	Above middle
	% of the	class, % of the
	population	population

Table 2 . Comparative wealth--defined middle class dynamics, 2000--2015 2000--2007 2007--2008 2008--2015 Total 2000--2015

 2

	..of which middle class	..of which beyond middle class

2. Raw materials intensity of value added and per capita GDP of the Russian regions, 2014

	Graph
	Ненецкий автономный	Ханты--Мансийский автономный	Сахалинская область	Чукотский автономный округ	Республика Саха (Якутия)	Республика Коми	Красноярский край	Камчатский край	Мурманская область	Томская область	Белгородская область	Московская область	Оренбургская область	Иркутская область	Республика Башкортостан	Краснодарский край	Архангельская область	Калининградская область	Новгородская область	Нижегородская область	Республика Карелия	Республика Хакасия	Астраханская область	Амурская область	Кемеровская область	Курская область	Смоленская область	Еврейская автономная область	Тверская область	Ростовская область	Орловская область	Забайкальский край	Пензенская область	Республика Мордовия	Чувашская Республика	Брянская область	Алтайский край	Кировская область	Республика Северная Осетия -	Ивановская область	Республика Калмыкия	Карачаево--Черкесская Республика	Республика Ингушетия

Source: author's elaboration based on Rosstat database

(2015)

Table 3 . A productive typology of Russian regions

 3

	Source: author's elaboration

Graph 5. Structure of households' revenues, 2005--2014

Source: Rosstat ( 2015) At the regional level, the distribution of incomes sources varies largely. For instance, salaries represent 80% of the total in Chukotka, against less than 12% in the Daghestan Republic. Conversely, in the Tyva Republic, social transfers amount to a third of the average income of households, whereas in the Yamalo--Nenets Okrug, they represent only 12%. Other revenues show even wider variations across regions: in Daghestan and Adygues republics, they represent more than 50% of the total revenues, whereas in Yamalo--Nenets Okrug, Chukotka and Magadan the proportion is not higher than 2%. In order to link these regional variations to the structure of income inequality in Russia, we study the relation between the productive typology of regions and their main sources of incomes. We find that rent--providing regions are associated with a higher share of salaries in the total income. This can be attributed to the role played in the local economy by the main raw materials extracting companies that dominates the local economy: they usually represent a high proportion of local employment and distribute attractive wages in order to retain workers in these remote, desert and harsh--climate regions. At the same time, exposed regions tend to benefit more than the average from social transfers. Some of them -notably the poorest ones, in which the agricultural sector is prominent -rely also to other sources of revenues (remittances, currency sales and revenue from the grey economy). Finally, within the non--tradable intensive regions, two sub--groups must be distinguished: in the first one, salaries are the main source of revenues, far above the national average as for rent--providing regions. These regions (Kamtchatka, Moscow city, St Petersburg among others) are also the wealthiest of their category. They have managed to benefit from the development of non--tradable sectors (services, construction and real estate mainly) by attracting part of the revenues produced by the rent--providing regions, with whom they are closely linked. It is in these regions that a large part of the Russian middle class is living. The second sub--group is constituted of the poorest non--tradable intensive regions, in which other activitiesgrey economy, remittances, currency sales --provide the main source of income for local Conclusion: the productive structure as a challenge for redistribution policies Albeit very preliminary, this study provides several interesting empirical findings regarding the link between the productive structure of Russia and the income distribution.

First, it shows that the Russian middle class has suffered more than its counterparts of BRICS countries from the recent crises. One of the reasons of this impact is the dependence of the Russian economy on raw material prices, which reveals the role of the productive structure and the growth model of the economy in the income distribution. Second, it shows that, at the regional level, income inequalities remain important in Russia. Part of these inequalities can be explained by the uneven natural resources endowment of regions in Russia. Third, a simple productive typology of regions permits to shed light upon the forces that tend to drive income distribution in Russia. Rent--providing regions are not only wealthier than others: they are also associated with a higher share of salaries in households' income. But this source of income can prove vulnerable since world raw material prices are unstable and largely beyond the control of the Russian firms that produce them. Moreover, relatively poor non--tradable intensive and exposed regions rely more extensively on remittances and hidden revenues that are particularly vulnerable to the macroeconomic conjuncture. Fourth, social transfers can theoretically buffer these vulnerabilities. But they depend on the ability of the State to lesser the dependency of the budgetary resources from the price of raw materials. This has not been done in Russia during the past quarter of century. Therefore, as world oil prices are bound to be subject to high uncertainty in the next decade, fiscal and redistributive policy will remain one of the most pressing challenges of economic policy in Russia.