Quentin Cormier

Etienne Tanré

Romain Veltz

Long time behavior of a mean-field model of interacting neurons

Keywords: McKean-Vlasov SDE, Long time behavior, Mean-field interaction, Volterra integral equation, Piecewise deterministic Markov process Mathematics Subject Classification Primary: 60B10. Secondary 60G55, 60K35, 45D05, 35Q92

We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant probability measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic external quantity (we call it the external current). For constant current, we obtain the convergence to the invariant probability measure. Using a perturbation method, we extend this result to more general external currents. Finally, we prove the result for the non-linear McKean-Vlasov equation.

Introduction

We study a model of network of neurons. For each N ∈ N, we consider a Piecewise-Deterministic Markov Process (PDMP)

X N t = (X 1,N t , • • • , X N,N t ) ∈ R N + . For i ∈ {1, • • • , N }, X i,N
t models the membrane potential of a neuron (say neuron i) in the network. It emits spikes at random times. The spiking rate of neuron i at time t is f (X i,N t ): it only depends on the potential of neuron i. When the neuron i emits a spike, say at time τ , its potential is reset (X i,N τ + = 0) and the potential of the other neurons increases by an amount J N , where the connection strength J ≥ 0 is fixed:

∀j = i, X j,N τ+ = X j,N τ-+ J N .
Between two spikes, the potentials evolve according to the one dimensional equation

dX i,N t dt = b(X i,N t ).
The functions b and f are assumed to be smooth. This process is indeed a PDMP, in particular Markov (see [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF]). Equivalently, the model can be described using a system of SDEs driven by Poisson measures. Let (N i (du, dz)) i=1,••• ,N be a family of N independent Poisson measures on R + × R + with intensity measure dudz. Let (X i,N 0 ) i=1,••• ,N be a family of N random variables on 1 R + , i.i.d. of law ν and independent of the Poisson measures. Then (X i,N ) is a càdlàg process solution of coupled SDEs:

∀i, X i,N t =X i,N 0 + t 0 b(X i,N u )du + J N j =i t 0 R+ ½ {z≤f (X j,N u-)} N j (du, dz) - t 0 R+ X i,N u-½ {z≤f (X i,N u-)} N i (du, dz).
. [START_REF] Bardet | Total variation estimates for the TCP process[END_REF] When the number of neurons N goes to infinity, it has been proved (see [START_REF] De Masi | Hydrodynamic limit for interacting neurons[END_REF][START_REF] Fournier | On a toy model of interacting neurons[END_REF]) for specific linear functions b and under few assumptions on f that X 1,N t -i.e. the first coordinate of the solution to (1) -converges in law to the solution of the McKean-Vlasov SDE:

X t = X 0 + t 0 b(X u )du + J t 0 E f (X u )du - t 0 R+ X u-½ {z≤f (Xu-)} N(du, dz), (2)
where, L(X 0 ) := L(X 1,N 0 ) = ν and N is a Poisson measure on R + × R + with intensity measure dudz. The measure N and X 0 are independent.

Equation ( 2) is a mean-field equation and is the current object of interest. Note that the drift part of (2) involves the law of the solution in the term E f (X u ): the equation is non-linear in the sense of McKean-Vlasov. Here, we study existence and uniqueness of the solution of (2) and its long time behavior.

Let ν(t, •) be the law of X t at time t ≥ 0. It is a weak solution of the following Fokker-Planck PDE:

       ∂ ∂t ν(t, x) = - ∂ ∂x [(b(x) + Jr t )ν(t, x)] -f (x)ν(t, x), x > 0 ν(0, •) = ν, ν(t, 0) = r t b(0) + Jr t , r t = ∞ 0 f (x)ν(t, x)dx.
(

This model with a noisy threshold is known in the physic literature as the "Escape Noise" model (see [START_REF] Gerstner | Neuronal dynamics: From single neurons to networks and models of cognition[END_REF]Chap. 9] for references and biological considerations). From a mathematical point of view, it has been first studied in [START_REF] De Masi | Hydrodynamic limit for interacting neurons[END_REF] and has been the object of further developments in [START_REF] Fournier | On a toy model of interacting neurons[END_REF]. The function f : R + → R + can be considered of the type f (x) = ( x ϑ ) ξ for large ξ > 0 and some soft threshold ϑ > 0. In this situation, if the potential of the neuron is equal to x then the neuron has a small probability to spike between t and t + dt if x < ϑ and a large probability if x > ϑ. Such a choice of f mimics the standard Integrate-And-Fire model with a fixed (deterministic) threshold around ϑ.

Results on the existence of solution to [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF], in a slightly different context (in particular, with b(x) ∼ ∞ -κx for κ ≥ 0), have been obtained in [START_REF] De Masi | Hydrodynamic limit for interacting neurons[END_REF]: the authors explored the case where the initial condition ν is compactly supported. This property is preserved at any time t > 0. So, the behavior of the solution with a rate function f locally Lipschitz continuous is similar to the case with a function f globally Lipschitz continuous. When the initial condition is not compactly supported, the situation is more delicate. In [START_REF] Fournier | On a toy model of interacting neurons[END_REF], the authors proved existence and path-wise uniqueness of the solution to (2) (in a slightly different setting than ours) using an ad-hoc distance.

Note that the global existence results obtained for this model differ from those obtained for the "standard" Integrate-and-Fire model with a fixed deterministic threshold. This situation, studied for instance in [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF][START_REF] Carrillo | Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience[END_REF][START_REF] Delarue | Global solvability of a networked integrate-and-fire model of McKean-Vlasov type[END_REF][START_REF] Delarue | Particle systems with a singular mean-field self-excitation. Application to neuronal networks[END_REF], corresponds (informally) to the choice f (x) = +∞½ {x≥ϑ} , ϑ > 0 being the fixed threshold. In these papers, a diffusion part is included in the modeling. In [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF], the authors proved that a blow-up phenomenon appears when the law of the initial condition is concentrated close to the threshold ϑ: the jump rate of the solution diverges to infinity in finite time. Here, the situation is completely different: the jump rate is uniformly bounded in time (see Theorem 5). In [START_REF] Cáceres | Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states[END_REF], the authors have obtained results on the stability of the solution for the diffusive model with a deterministic threshold (see also [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] for a variant).

Very little is known about the long time behavior of the solutions to [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. One can study it by considering the long time behavior of the finite particles system [START_REF] Bardet | Total variation estimates for the TCP process[END_REF] and then apply the propagation of chaos to extend the results to the McKean-Vlasov equation [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. This strategy has been developed in [START_REF] Veretennikov | On ergodic measures for McKean-Vlasov stochastic equations[END_REF][START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF] for diffusive problems. The long time behavior of the particles system (1) has been studied in [START_REF] Duarte | A model for neural activity in the absence of external stimuli. Markov Process[END_REF][START_REF] Hodara | Non-parametric estimation of the spiking rate in systems of interacting neurons[END_REF] (again in a slightly different setting but the methods could be adapted to our case): the authors proved that the particles system is Harris-ergodic and consequently converges weakly to its unique invariant probability measure. However, transferring the long time behavior of the particles system to the McKean-Vlasov equation is possible if the propagation of chaos holds uniformly in time. In [START_REF] De Masi | Hydrodynamic limit for interacting neurons[END_REF][START_REF] Fournier | On a toy model of interacting neurons[END_REF], the propagation of chaos is only proved on compact time interval [0, T ] and their estimates diverge as T goes to infinity. Because Equation (2) may have multiple invariant probability measures, there is no hope in general to prove such uniform propagation of chaos.

Coupling methods are also used to study the long time behavior of SDEs. In [START_REF] Bardet | Total variation estimates for the TCP process[END_REF], the authors have studied the TCP (a linear PDMP) which is close to [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. The size of the jumps is -x/2 in the TCP and -x in our setting, x being the position of the process just before the jump. The main difference is the non-linearity: we failed to adapt their methods when the interactions are non-zero (J > 0).

Butkovsky studied in [START_REF] Butkovsky | On ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations[END_REF] the long time behavior of some McKean-Vlasov diffusion SDE of the form:

∀t ≥ 0, X t = X 0 + t 0 [b 1 (X u ) + ǫb 2 (X u , µ u )] du + W t , µ u = L(X u ), (4)
where (W t ) t≥0 is a Brownian motion. Here the drift terms b 1 and b 2 are assumed to be globally Lipschitz and b 2 is assumed uniformly bounded with respect to its two parameters. The author proved that if the parameter ǫ is small enough, (4) has a unique invariant probability measure which is globally stable. The case ǫ > 0 (and small) is treated as a perturbation of the case ǫ = 0 using a Girsanov transform. It could be interesting to see how this method could be adapted to SDE driven by Poisson measures, but we did not pursue this path. Another approach consists in studying the non-linear Fokker-Planck equation [START_REF] Butkovsky | On ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations[END_REF]. Such non-linear transport equations with boundary conditions have been studied in the context of population dynamics (see for instance [START_REF] Gurtin | Non-linear age-dependent population dynamics[END_REF][START_REF] Prüss | Stability analysis for equilibria in age-specific population dynamics[END_REF][START_REF] Webb | Theory of nonlinear age-dependent population dynamics, volume 89 of Monographs and Textbooks in Pure and Applied Mathematics[END_REF][START_REF] Perthame | Transport equations in biology[END_REF]). In [START_REF] Gurtin | Non-linear age-dependent population dynamics[END_REF], the authors have characterized the stationary solutions of the PDE and found a criterion of local stability for stationary solutions. They derived a Volterra integral equation and used it to obtain the stability criteria. More recently, [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] Mischler | Relaxation in Time Elapsed Neuron Network Models in the Weak Connectivity Regime[END_REF] have re-explored these models for neuroscience applications (see [START_REF] Chevallier | Microscopic approach of a time elapsed neural model[END_REF][START_REF] Chevallier | Mean-field limit of generalized Hawkes processes[END_REF] for a rigorous derivation of some of these PDEs using Hawkes processes). PDE (3) differs from theirs in the sense that we have a non-linear transport term (theirs is constant and equal to one) and our boundary condition is more complex. The long time behavior of the PDE (3) has been successfully studied in [START_REF] Fournier | On a toy model of interacting neurons[END_REF] and in [START_REF] Drogoul | Exponential stability of the stationary distribution of a mean field of spiking neural network[END_REF] in the case where b ≡ 0. In this situation, one can simplify the PDE (3) with a simpler boundary condition

ν(t, 0) = 1 J .
The authors proved that if the density of the initial condition satisfies this boundary condition and regularity assumptions, then ν(t, .) converges to the density of the invariant probability measure as t goes to infinity. The convergence holds in L 1 or in stronger norms (see [START_REF] Drogoul | Exponential stability of the stationary distribution of a mean field of spiking neural network[END_REF]). For b = 0, the boundary condition is more delicate and their methods cannot be easily applied.

Actually the long time behavior of the solution to (2) may be remarkably intricate. Depending on the choice of f , b and J, equation (2) may have multiple invariant probability measures. Even if the invariant probability measure is unique, it is not necessarily a stable one and oscillations may appear (see Examples page 7). In [START_REF] Drogoul | Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics[END_REF], the authors have numerically illustrated this phenomenon in a setting close to ours.

Our main result describes the long time behavior of the solution to (2) in the weakly connected regime (Theorem 9). If the connection strength J is small enough, we prove that (2) has a unique invariant probability measure which is globally stable. We give the explicit expression of this non-trivial invariant distribution and starting from any initial condition X 0 , we prove the convergence in law of X t to it, exponentially fast, as t goes to infinity. We argue that this result is very general: it does not depend on the explicit shape of the functions f or b. For stronger connection strengths J, such a result cannot hold true in general as equation ( 2) may have multiple invariant probability measures.

Note that we prove convergence in law, which is weaker than convergence in L 1 . On the other hand, we require very few on the initial condition, in particular, we do not assume the existence of a density for the initial condition in Theorem 9. We also provide a new proof for the existence and uniqueness of the solution to (2), based on a Picard iteration scheme (see Theorem 5). As in [START_REF] Fournier | On a toy model of interacting neurons[END_REF], we do not require the initial condition to be compactly supported. One of the main difficulty to study (2) (or its PDE version (3)) is that there is no simple autonomous equation for the jump rate t → E f (X t ). To overcome this difficulty, we introduce a "linearized" version of (2) for which we can derive a closed equation of the jump rate.

Fix a s ≥ 0 and let (a u ) u≥s be a continuous deterministic non-negative function, called the external current. It replaces the interaction JEf (X u ) in [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. We consider the linear nonhomogeneous SDE:

∀t ≥ s, Y s,ν,(a.) t = Y s,ν,(a.) s + t s b(Y s,ν,(a.) u )du+ t s a u du- t s R+ Y s,ν,(a.) u- ½ {z≤f (Y s,ν,(a.) u-
)} N(du, dz), [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] where L(Y s,ν,(a.) s ) = ν. Under quite general assumptions on b and f , this SDE has a path-wise unique solution (see Lemma 14). We denote the jump rate of this SDE by:

∀t ≥ s, r ν (a.) (t, s) := E f (Y s,ν,(a.) t ). (6)
Moreover, taking s = 0 and Y 0,ν,(a.) 0

= X 0 , it holds that (Y 0,ν,(a.) t ) t≥0 is a solution to (2) if it satisfies the closure condition ∀t ≥ 0, a t = Jr ν (a.) (t, 0). (7)
Conversely, any solution to (2) is a solution to [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] with a t = J E f (X t ). We prove that the function r ν (a.) satisfies a Volterra integral equation

∀t ≥ s, r ν (a.) (t, s) = K ν (a.) (t, s) + t s K (a.) (t, u)r ν (a.) (u, s)du, ( 8
)
where the kernels K ν (a.) and K (a.) are explicit in terms of ν, a, b and f (see [START_REF] Fournier | On a toy model of interacting neurons[END_REF] and ( 19)). Our main tool is this Volterra equation: we use it with a Picard iteration scheme to "recover" the non-linear equation [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. The McKean-Vlasov equation (2), its "linearized" non-homogeneous version [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF], the Fokker-Planck PDE (3) and the Volterra equation [START_REF] Chevallier | Microscopic approach of a time elapsed neural model[END_REF] are different ways to investigate this mean-field problem, each of these interpretations having their own strength and weakness. Here, we use mainly the Volterra equation ( 8) and the non-homogeneous SDE [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF]. To prove that equation ( 2) has a path-wise unique solution, we rely on the Volterra equation [START_REF] Chevallier | Microscopic approach of a time elapsed neural model[END_REF] and show that the following mapping:

(a t ) t≥0 → Jr ν (a.) (., 0) := [t → J E f (Y 0,ν,(a.) t )], (9)
is contracting on C([0, T ], R + ) for all T > 0. It then follows that the fixed point of this mapping satisfies the closure condition [START_REF] Chevallier | Mean-field limit of generalized Hawkes processes[END_REF] and can be used to define a solution to [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. Conversely any solution to (2) defines a fixed point of this mapping and one proves strong uniqueness for [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. Finally, we prove our main result concerning the long time behavior of the solution to (2). Let us detail the structure of the proof. First, we give in Proposition 30 the long time behavior of the solution to the linear equation ( 5) with a constant current (a t ≡ a). Any solution converges in law to a unique invariant probability measure ν ∞ a (Proposition 26). In that case, the Volterra equation ( 8) is of convolution type and it is possible to study finely its solution using Laplace transform techniques. Second, we prove, for small J, the uniqueness of a constant current a * such that ∀t ≥ 0,

a * = JEf (Y 0,ν ∞ a * ,a * t ).
Third, we extend the previous convergence result to non-constant currents (a t ) satisfying

|a t -a * | ≤ Ce -λt . ( 10
)
Using a perturbation method, we prove that

Y 0,ν,(a.) t L -→ t→∞ ν ∞ a * .
Fourth, in Theorem 9, we give the long time behavior of the solution to the non-linear equation (2) for small J. Here, we use a fixed point argument.

The layout of the paper is as follows. Our main results are given in Section 2. In Section 3, we gather technical results. In Section 4, we study the non-homogeneous linear equation [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] and derive the Volterra equation satisfied by the jump rate. In Section 5, we characterize the invariant probability measures of [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. In Section 7 we study the long time behavior of the solution to [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] with a constant current a. In Section 8, we introduce the perturbation method. Finally Section 9 is devoted to the proof of our main result (Theorem 9).

Notations and results

Let us introduce some notations and definitions. For s ≥ 0 and a probability measure ν on R + , let Y s,ν,(a.) s be a ν-distributed random variable, independent of a Poisson measure N(du, dz) on R + × R + of intensity measure dudz. We consider the canonical filtration (F s t ) t≥s associated to the Poisson measure N and to the initial condition Y s,ν,(a.) s , that is the completion of

σ{Y s,ν,(a.) s , N([s, r] × A) : s ≤ r ≤ t, A ∈ B(R + )}. Definition 1. Let s ≥ 0 and consider (a t ) : [s, ∞) → R + a measurable locally integrable function (∀t ≥ s, t s a u du < ∞). • A process (Y s,ν,(a.) t
) t≥s is said to be a solution of the non-homogeneous linear equation [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] with a current (a t ) t≥s if the law of Y )du < ∞ and (5) holds a.s.

• An (F 0 t ) t≥0 -adapted càdlàg process (X t ) t≥0 is said to solve the non-linear SDE (2) if t → E f (X t ) is measurable locally integrable and if (X t ) t≥0 is a solution of (5) with s = 0, Y 0,ν,(a.) 0 = X 0 and ∀t ≥ 0, a t = J E f (X t ).
Let t ≥ s ≥ 0. We denote by Y s,ν,(a.) t a solution to the linear non-homogeneous SDE [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] driven by (a t ) t≥s ∈ C([s, ∞), R + ) starting with law ν at time s. We denote its associated jump rate by: r ν (a.) (t, s)

:= E f (Y s,ν,(a.) t
). For any measurable function g, we write ν(g

) := ∞ 0 g(x)ν(dx) = E g(Y s,ν,(a.) s
) whenever this integral makes sense. Between its random jumps, the SDE ( 5) is reduced to a non-homogeneous ODE. Let us introduce its flow ϕ (a.) t,s (x), which by definition is the solution of:

∀t ≥ s, d dt ϕ (a.) t,s (x) = b(ϕ (a.) t,s (x)) + a t (11) ϕ (a.) s,s (x) = x.
If a t ≡ a, we denote ϕ a t (x) = ϕ (a.) t,0 (x).

Assumptions 2. We assume that b : R + → R is a locally Lipschitz function with b(0) > 0 and that b is bounded from above:

∃C b ≥ 0 : ∀x ≥ 0, b(x) ≤ C b . ( 12
)
We assume moreover that there is a positive constant C ϕ such that for all (a t ) t≥0 ,

(d t ) t≥0 ∈ C(R + , R + ) we have ∀x ≥ 0, ∀s ≤ t, |ϕ (a.) t,s (x) -ϕ (d.) t,s (x)| ≤ C ϕ t s |a u -d u |du. (13)
The assumption b(0) > 0 implies that for all x, t, s ∈ R + , we have ϕ

(a.) t,s (x) ∈ R + .
Assumptions 3. We assume that f : R + → R + is a C 1 convex increasing function with f (0) = 0 and satisfies:

3.1. there exists a constant C f such that ∀x, y ≥ 0, f (x + y) ≤ C f (1 + f (x) + f (y)) and f ′ (x + y) ≤ C f (1 + f ′ (x) + f ′ (y)).

for all

θ ≥ 0, sup x≥0 {θf ′ (x) -f (x)} < ∞. Define ψ(θ) := sup x≥0 {θf ′ (x) -1 2 f 2 (x)} < ∞.
We also assume that lim θ→+∞ ψ(θ) θ 2 = 0.

Finally we assume that there is a constant

C b,f > 0 such that ∀x ≥ 0, |b(x)| ≤ C b,f (1 + f (x)).
Note that these assumptions ensure that f (x) > 0 for all x > 0.

Assumptions 4. We assume that the law of the initial condition is a probability measure ν satisfying ν(f 2 ) < ∞.

Let us give our main results.

Theorem 5. Under Assumptions 2, 3 and 4, the non-linear SDE (2) has a path-wise unique solution (X t ) t≥0 in the sense of Definition 1. Furthermore, there is a finite constant r > 0 (only depending on b, f and J) such that:

sup t≥0 E f (X t ) ≤ max(r, E f (X 0 )), lim sup t→∞ E f (X t ) ≤ r.
The upper-bound r can be chosen to be an increasing function of J. Notation 6. Denote for all a ≥ 0 the probability measure

ν ∞ a (dx) := γ(a) b(x) + a exp - x 0 f (y) b(y) + a dy ½ {x∈[0,σa]} dx, ( 14
)
where γ(a) is the normalization

γ(a) := σa 0 1 b(x) + a exp - x 0 f (y) b(y) + a dy dx -1 . ( 15
)
The upper bound σ a of the support of ν ∞ a is given by σ a := lim t→∞ ϕ a t (0) ∈ R * + ∪ {+∞}.

Remark 7.

1. For all a ≥ 0, γ(a) = ν ∞ a (f ).

2. We prove in Proposition 26 that for any a ≥ 0, ν ∞ a is the unique invariant probability measure of (5) with a t ≡ a.

Proposition 8. The probability measure ν ∞

a is an invariant measure of (2) iff a γ(a)

= J. [START_REF] Duarte | A model for neural activity in the absence of external stimuli. Markov Process[END_REF] Moreover, define J m := sup{J 0 ≥ 0 : ∀J ∈ [0, J 0 ] equation ( 16) has a unique solution}, then J m > 0. Consequently, for all 0 ≤ J < J m the non-linear process (2) has a unique invariant probability measure.

We now state our main result: the convergence to the unique invariant probability measure for weak enough interactions. Theorem 9. Under Assumptions 2, 3, 4, there exists strictly positive constants J * and λ (both only depending on b and f ) satisfying

0 < J * < J m , 0 < λ < f (σ 0 ),
(J m and σ 0 are defined in Proposition 8) and such that for any 0 ≤ J ≤ J * , there is a constant

D > 0: ∀t ≥ 0, | E f (X t ) -γ(a * )| ≤ De -λt .
Here, (X t ) t≥0 is the solution of the non-linear SDE (2) starting with law ν and a * is the unique solution of [START_REF] Duarte | A model for neural activity in the absence of external stimuli. Markov Process[END_REF]. The constant D only depends on b, f , E f (X 0 ), J and λ. Moreover, it holds that X t converges in law to ν ∞ a * at an exponential speed. If φ : R + → R is a bounded Lipschitz-continuous function, it holds that

∃D ′ > 0, ∀t ≥ 0, | E φ(X t ) -ν ∞ a * (φ)| ≤ D ′ e -λt ,
where the constant D ′ only depends on b, f, J, ν, λ and φ through its infinite norm and its Lipschitz constant.

Note that in Theorem 9, the unique invariant probability measure is globally stable: for weak enough interactions, starting from any initial condition, the system converges to its steady state.

Examples

Given the following constants p ≥ 1, µ > 0 and κ ≥ 0, define, for all x ≥ 0:

f (x) := x p , b(x) = µ -κx.
Then (b, f ) satisfies the Assumptions 2 and 3. In that case, the flow is given by

ϕ (a.) t,s (x) = xe -κ(t-s) + µ κ [1 -e -κ(t-s) ] + t s e -κ(t-u) a u du.
We have ∀x, y ∈ R + , f (x + y) ≤ 2 p-1 (f (x) + f (y)). A similar estimate holds for f ′ . Moreover

ψ(θ) = 1 2 θ 2p p+1 (p -1)
p-1 p+1 (1 + p), so Assumption 3.2. holds. Consequently, Theorem 9 applies. When κ > 0, the invariant probability measures are compactly supported and not necessarily unique. Consider for instance b

(x) = µ -x, f (x) = x 2 .
If µ is small enough, a numerical study shows that there exists 0 < a 1 < a 2 < ∞ such that the function a → a γ(a) is increasing on [0, a 1 ], decreasing on [a 1 , a 2 ] and finally increasing on [a 2 , ∞). Thus, if J ∈ (a 1 , a 2 ), the non-linear equation (2) admits exactly 3 non-trivial invariant probability measures. A numerical study shows that only two of the three are locally stable (bi-stability).

Another interesting example is the following. Assume b(x) = 2 -2x and f (x) = x 10 . Then, a numerical study shows that the function a → a γ(a) is increasing on R + and consequently for all J ≥ 0, (2) admits a unique invariant probability measure. But if J ∈ [0.7, 1.05] a further numerical analysis shows that the law of the solution of (2) asymptotically oscillates, betraying that the invariant probability measure is not locally stable. Those examples emphasis on the fact that the condition J small enough is required for Theorem 9 to hold. Remark 10. Assumption 2 is crucial to obtain our result on the long time behavior (Theorem 9). It restricts us to κ ≥ 0. If b(x) = µ -κx with κ < 0 then Assumption 2 does not hold.

Technical notations and technical lemmas

The following standard results on the ODE [START_REF] De Masi | Hydrodynamic limit for interacting neurons[END_REF] will be useful all along: Lemma 11. Assume b satisfies Assumption 2. Then:

1. For all x ≥ 0 and s ≥ 0, the ODE [START_REF] De Masi | Hydrodynamic limit for interacting neurons[END_REF] has a unique solution t → ϕ (a.) t,s (x) defined on [s, ∞). This is the flow associated to the drift b and to the external current (a t ) t≥0 .

2. Given (a t ) and (d t ) in C(R + , R + ), the flow satisfies the following comparison principle: t,s (0) is continuous on {(t, s) : 0 ≤ s ≤ t < ∞}.

[∀t ≥ 0, a t ≥ d t ] =⇒ [∀x ≥ y ≥ 0, ∀t ≥ s ≥ 0, ϕ (a.) t,s (x) ≥ ϕ (d.) t,s (y)].

The flow grows at most linearly with respect to the initial condition:

∀a ≥ 0, ∀x ≥ 0, ∀t ≥ 0, ϕ a t (x) ≤ x + C a b t, where C a b := C b + a.

5.

For any constant current a ≥ 0, the flow converges to a limit as t goes to infinity (possibly equal to +∞):

∀a ≥ 0, ∀x ≥ 0, lim t→+∞ ϕ a t (x) := σ x a ∈ R * + ∪ {+∞}. ( 17
)
It holds that inf a,x≥0 σ x a > 0. Moreover if we define:

σ a := inf{x ≥ 0 : b(x) + a = 0} ∈ R * + ∪ {+∞},
we have: σ 0 a = σ a .

Remark 12.

1. Assumption 3.2. ensures that f does not grow too fast in the sense that for all ǫ > 0, there is a constant

C ǫ > 0, such that: ∀x ≥ 0 f (x) ≤ C ǫ e ǫx .
2. Using that f is increasing and continuous, we have, for all a ≥ 0:

lim t→∞ f (ϕ a t ) = f (σ a ) ≥ f (σ 0 ) > 0.
We show that the jump rate r ν (a.) of the non-homogeneous SDE [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF], satisfies the Volterra equation [START_REF] Chevallier | Microscopic approach of a time elapsed neural model[END_REF] where the kernels K ν (a.) and K (a.) are defined by

∀t ≥ s ≥ 0, K ν (a.) (t, s) := ∞ 0 f (ϕ (a.) t,s (x)) exp - t s f (ϕ (a.) u,s (x))du ν(dx), (18)
K (a.) (t, s) := K δ0 (a.) (t, s). ( 19
)
Given two "kernels" α and β, it is convenient to follow the notation of [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF] and define:

∀t ≥ s, (α * β)(t, s) := t s α(t, u)β(u, s)du. ( 20
)
The Volterra equation ( 8) becomes

r ν (a.) = K ν (a.) + K (a.) * r ν (a.) . ( 21
)
Similarly to ( 18) and ( 19), we define the kernels

∀t ≥ s, H ν (a.) (t, s) := ∞ 0 exp - t s f (ϕ (a.) u,s (x))du ν(dx), H (a.) := H δ0 (a.) , ∀x ≥ 0, H x (a.) := H δx (a.) . ( 22
)
From the definition, one can check directly the following relation

1 * K ν (a.) = 1 -H ν (a.) . ( 23
)
To shorten notations, we shall also write r (a.) (t, s) := r δ0 (a.) (t, s). When the input current (a t ) t≥0 is constant and equal to a, equation ( 5) is homogeneous and we write

∀t ≥ 0, Y ν,a t := Y 0,ν,a t , r ν a (t) := r ν (a.) (t, 0), K ν a (t) := K ν (a.) (t, 0), H ν a (t) := H ν (a.) (t, 0), ϕ a t (x) := ϕ (a.)
t,0 (x). Note that in this homogeneous situation, the operation * corresponds to the classical convolution operation. In particular this operation is commutative in the homogeneous setting and equation ( 21) is a convolution Volterra equation.

Remark 13.

For any (a.) ∈ C(R + , R + ) and any probability measure ν, we have

∀t ≥ s ≥ 0 : H ν (a.) (t, s) ≤ H 0 (t -s).
4 Study of the non-linear SDE (2) and of its linearized version (5)

On the non-homogeneous linear SDE (5)

Fix s ≥ 0 and let (a t ) : [s, ∞) → R + be a continuous function. We consider the non-homogeneous linear SDE [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF]. We always assume that ν, the law of the initial condition Y s,ν,(a.) s

, satisfies Assumptions 4.

Lemma 14. Grant Assumptions 2, 3 and 4. Then the SDE (5) has a path-wise unique solution on [s, ∞) in the sense of Definition 1.

Proof. We give a direct proof by considering the jumps of Y s,ν,(a.) t and by solving the equation between the jumps.

• Step 1: we grant Assumptions 2, 4 and assume that f : R + → R + is measurable and bounded. There exists a constant 0 < K < ∞ such that:

sup x≥0 f (x) ≤ K.
In this case, the solution of ( 5) can be constructed in the following way. Define by induction:

τ 0 := inf{t ≥ s : t s R+ ½ {z≤f (ϕ (a.) u,s (Y s,ν,(a.) s ))} N(du, dz) > 0}, ∀n ≥ 0, τ n+1 := inf{t ≥ τ n : t τn R+ ½ {z≤f (ϕ (a.) u,τn (0))} N(du, dz) > 0}.
Using that f ≤ K, it follows that a.s. lim n→∞ τ n = +∞. We define:

Y s,ν,(a.) t = ϕ (a.) t,s (Y s,ν,(a.) s )½ t∈[s,τ0) + n≥1 ϕ (a.) t,τn (0)½ t∈[τn,τn+1) ,
and we can directly verify that t → Y s,ν,(a.) t

is almost surely a solution of (5).

Uniqueness of equation ( 5) follows immediately from Lemma 11 (point 1): two solutions have to be equal almost surely before the first jump, from which we deduce that the two solutions have to jump at the same time. By induction on the number of jumps, the two trajectories are almost surely equal.

• Step 2: We now come back to the general case where f is not assumed to be bounded and we adapt the strategy of [START_REF] Fournier | On a toy model of interacting neurons[END_REF], proof of Proposition 2. We grant Assumptions 2, 3 and 4.

We use Step 1 with f K (x) := f (min(x, K)) for some K > 0. Let us denote Y s,ν,(a.),K t the solution of ( 5) where f has been replaced by f K . The boundedness of f K implies the pathwise uniqueness of Y s,ν,(a.),K t . We introduce ) t≥s be the solution of [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF]. The func-

ζ K := inf{t ≥ 0 : |Y s,ν,(a.),K t | ≥ K}, it holds that Y s,ν,(a.),K t = Y s,ν,(a.),K+1 t for all t ∈ [0, ζ K ] and all K ∈ N. Moreover, ζ K < ζ K+1 . We define ζ := sup K ζ K and
tions t → E f (Y s,ν,(a.) t ), t → E f ′ (Y s,ν,(a.) t ), t → E f ′ (Y s,ν,(a.) t )|b(Y s,ν,(a.) t )| and t → E f 2 (Y s,ν,(a.) t ) are locally bounded on [s, ∞). Moreover, t → E f (Y s,ν,(a.) t ) =: r ν (a.) (t, s) is continuous on [s, ∞).
Proof. Consider the interval [s, T ] for some

T > 0. Let A := sup t∈[s,T ] a t . It is clear that ∀t ∈ [s, T ], a.s. Y s,ν,(a.) t ≤ Y s,ν,(a.) s + t s [b(Y s,ν,(a.) u ) + a u ]du ≤ Y s,ν,(a.) s + C T , with C T := (C b + A)(T -s).
We used here that b is bounded from above (Assumption 2). Using that f 2 is non-decreasing and Assumption 3.1., we have:

a.s. f 2 (Y s,ν,(a.) t ) ≤ f 2 (Y s,ν,(a.) s + C T ) ≤ C 2 f (1 + f (C T ) + f (Y s,ν,(a.) s )) 2 .
Using Assumption 4, we deduce that

t → E f 2 (Y s,ν,(a.) t ) is bounded on [s, T ]. By the Cauchy-Schwarz inequality, this implies that t → E f (Y s,ν,(a.) t
) is also bounded on [s, T ]. Finally, using the Assumption 3.2. (with θ = 1), there is a constant C such that for all

x ≥ 0 f ′ (x) ≤ C + f (x). Assumption 3.3. thus yields ∀x ≥ 0 f ′ (x)|b(x)| ≤ C b,f (1 + f (x))(C + f (x)),
and so this proves that

t → E f ′ (Y s,ν,(a.) t )|b(Y s,ν,(a.) t
)| is also bounded on [s, T ]. We now apply the Itô formula (see for instance Theorem 32 of [START_REF] Protter | Stochastic integration and differential equations[END_REF]Chap.

II]) to Y s,ν,(a.) t . It gives for any ǫ > 0 f (Y s,ν,(a.) t+ǫ ) = f (Y s,ν,(a.) t )+ t+ǫ t f ′ (Y s,ν,(a.) u )[b(Y s,ν,(a.) u ) + a u ]du- t+ǫ t ∞ 0 f (Y s,ν,(a.) u- )½ {z≤f (Y s,ν,(a.) u- )} N(du, dz).
Taking the expectation, it follows that

E f (Y s,ν,(a.) t+ǫ ) -E f (Y s,ν,(a.) t ) = t+ǫ t E f ′ (Y s,ν,(a.) u )[b(Y s,ν,(a.) u ) + a u ]du - t+ǫ t E f 2 (Y s,ν,(a.) u )du, from which we deduce that t → E f (Y s,ν,(a.) t
) is locally Lipschitz and consequently continuous.

The Volterra equation

Along this section, we grant Assumptions 2, 3 and 4. Let s ≥ 0 and (a t ) t≥s ∈ C([s, ∞), R + ) be fixed. We consider (Y s,ν,(a.) t

) t≥s the path-wise unique solution of equation ( 5) driven by the current (a t ) t≥s . Following [START_REF] Fournier | On a toy model of interacting neurons[END_REF], we define:

τ s,t := sup{u ∈ [s, t] : Y s,ν,(a.) u = Y s,ν,(a.) u- },
the time of the last jump before t, with the convention that τ s,t = s if there is no jump during [s, t]. It follows directly from (5) that:

∀t ≥ s, a.s. Y s,ν,(a.) t = ϕ (a.) t,s (Y s,ν,(a.) s )½ {τs,t=s} + ϕ (a.) t,τs,t ½ {τs,t>s} .
We also define:

∀t ≥ s, J t := t s ∞ 0 ½ {z≤f (Y s,ν,(a.) u- )} N(du, dz),
the number of jumps between s and t.

Lemma 16. For all t ≥ u ≥ s ≥ 0, we have

P(J t = J u |F u ) = H Y s,ν,(a.) u (a.) (t, u) a.s.
where H x (a.) is given by [START_REF] Hodara | Non-parametric estimation of the spiking rate in systems of interacting neurons[END_REF].

Proof. We have

{J t = J u } = { t u ∞ 0 ½ {z≤f (Y s,ν,(a.) θ- )} N(dθ, dz) = 0}. Moreover, F u and σ{N([u, θ]× A) : θ ∈ [u, t], A ∈ B(R + )} are independent. It follows from the Markov property satisfied by (Y s,ν,(a.) t ) t≥s that: a.s. P(J t = J u |F u ) = Φ(Y s,ν,(a.) u )
where: Φ(x) := P(

t u ∞ 0 ½ {z≤f (ϕ (a.) θ,u (x))} N(dθ, dz) = 0) = H x (a.) (t, u).
Lemma 17 (See also [START_REF] Fournier | On a toy model of interacting neurons[END_REF], Proposition 25). For all t > s, the law of τ s,t is given by:

L(τ s,t )(du) = H ν (a.) (t, s)δ s (du) + r ν (a.) (u, s)H (a.) (t, u)½ {s<u<t} du.
Proof. First, from Lemma 16, it follows that:

P(τ s,t = s) = P(J t = J s ) = E(H Y s,ν,(a.) s (a.) (t, s)) = H ν (a.) (t, s).
Let now u ∈ (s, t] and h > 0 such that: s < u -h < u ≤ t. We have:

P(τ s,t ∈ (u-h, u]) = P(J u > J u-h , J t = J u ) = E(½ {Ju>J u-h } P(J t = J u |F u )) = E(½ {Ju>J u-h } H Y s,ν,(a.) u (a.) (t, u)).
Let A := sup u∈[s,t] a u . On the event {J u > J u-h }, the process jumps at least once during (u -h, u] and so, by Lemma 11 (point 2), we have

Y s,ν,(a.) u ∈ [0, ϕ (a.) u,u-h ] ⊂ [0, ϕ A h ]. It follows that |P(τ s,t ∈ (u -h, u]) -E(½ {Ju>J u-h } H (a.) (t, u))| ≤ sup x∈[0,ϕ A h ] |H x (a.) (t, u) -H (a.) (t, u)|P(J u > J u-h ).
From the following Lemma 18, we have:

lim h↓0 1 h P(J u > J u-h ) = r ν (a.) (u, s).
Using Lemma 11 (point 4), x → H x (a.) (t, u) is continuous at x = 0. From the continuity of h → ϕ A h at h = 0, it yields:

lim h↓0 1 h |P(τ s,t ∈ (u -h, u]) -E(½ {Ju>J u-h } H (a.) (t, u))| = 0.
Combining the two results, we obtain the stated formula:

lim h↓0 1 h P(τ s,t ∈ (u -h, u]) = r ν (a.) (u, s)H (a.) (t, u).
This proves the result.

Lemma 18 (See also [START_REF] Fournier | On a toy model of interacting neurons[END_REF], Lemma 23). For all u ∈ (s, t] we have:

lim h↓0 1 h P(J u > J u-h ) = r ν (a.) (u, s).
Proof. Again let A := sup u∈[s,t] a u < ∞. We have:

|hr ν (a.) (u, s) -P(J u > J u-h )| ≤ hr ν (a.) (u, s) -hr ν (a.) (u -h, s) + hr ν (a.) (u -h, s) -E u u-h f (ϕ (a.) θ,u-h (Y s,ν,(a.) u-h ))dθ + E u u-h f (ϕ (a.) θ,u-h (Y s,ν,(a.) u-h ))dθ -1 -E exp - u u-h f (ϕ (a.) θ,u-h (Y s,ν,(a.) u-h ))dθ =: ∆ 1 h + ∆ 2 h + ∆ 3 h .
From the continuity of u → r ν (a.) (u, s) (Lemma 15) it follows that lim h↓0

∆ 1 h h = 0. Moreover, ∆ 2 h = u u-h E f (Y s,ν,(a.) u-h )dθ -E u u-h f (ϕ (a.) θ,u-h (Y s,ν,(a.) u-h
))dθ .

Assumption 2 gives

∀y ≥ 0, ∀θ ∈ [u -h, u], 0 ≤ ϕ (a.) θ,u-h (y) -y ≤ ϕ A h (y) -y ≤ C A b h.
We deduce that

∆ 2 h ≤ h u u-h E g h (Y s,ν,(a.) u-h )C A b dθ, with g h (x) := sup y∈[0,C A b h] f ′ (x + y) = f ′ (x + C A b h). Using Assumption 3.1., we have f ′ (x + C A b h) ≤ C f (1+f ′ (C A b h)+f ′ (x)). It follows that E g h (Y s,ν,(a.) u-h ) ≤ C f (1+f (C A b h)+E f ′ (Y s,ν,(a.) u-h )). The function t → E f ′ (Y s,ν,(a.) t
) being locally bounded, we deduce that lim sup h↓0

∆ 2 h h = 0. Fi- nally, using that ∀x ≥ 0, |x -(1 -e -x )| ≤ x 2 we have ∆ 3 h ≤ E u u-h f (ϕ (a.) θ,u-h (Y s,ν,(a.) u-h ))dθ 2 .
Using the Cauchy-Schwarz inequality, we obtain Proposition 19 (See also [START_REF] Fournier | On a toy model of interacting neurons[END_REF], Theorem 12). Grant Assumptions 2, 3 and 4 . Let s ≥ 0 and ν,(a.) t be the solution of equation [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF], starting from L(Y s,ν,(a.) s

∆ 3 h ≤ h E u u-h f 2 (ϕ (a.) θ,u-h (Y s,ν,(a.) u-h ))dθ ≤ h 2 E f 2 (Y s,ν,(a.) u-h + C A b h). Using ∀x ≥ 0, ∀y ∈ [0, C A b t], f 2 (x + y) ≤ C 2 f (1 + f (C A b t) + f (x))
(a t ) t≥s ∈ C([s, ∞), R + ). Let Y s,
) = ν. Let φ : R + → R + be a continuous non-negative function. It holds that E φ(Y s,ν,(a.) t ) = t s φ(ϕ (a.) t,u (0))H (a.) (t, u)r ν (a.) (u, s)du + ∞ 0 φ(ϕ (a.) t,s (x))H x (a.) (t, s)ν(dx).
In particular, r ν (a.) (t, s) = E f (Y s,ν,(a.) t

) solves the Volterra equation ( 21)

r ν (a.) = K ν (a.) + K (a.) * r ν (a.) .
Proof. We have, for all t ≥ s

E φ(Y s,ν,(a.) t ) = E φ(Y s,ν,(a.) t )½ {τs,t=s} + E φ(Y s,ν,(a.) t )½ {τs,t>s} = E φ(ϕ (a.) t,s (Y s,ν,(a.) s ))½ {τs,t=s} + E φ(ϕ (a.) t,τs,t (0))½ {τs,t>s} := α t + β t .
Using Lemma 16, it follows that

α t = E[φ(ϕ (a.) t,s (Y s,ν,(a.) s )) P(J t = J s |F s )] = E[φ(ϕ (a.) t,s (Y s,ν,(a.) s ))H Y s,ν,(a.) s (a.) (t, s)] = ∞ 0 φ(ϕ (a.) t,s (x))H x (a.) (t, s)ν(dx).
Moreover, using Lemma 17, we have

β t = t s φ(ϕ (a.) t,u (0))r ν (a.) (u, s)H (a.) (t, u)du.
Taking φ = f we obtain the Volterra equation [START_REF] Gurtin | Non-linear age-dependent population dynamics[END_REF].

Note that using Lemma 17,

t s L(τ s,t )(du) = 1 gives: H ν (a.) + H (a.) * r ν (a.) = 1.
This last formula is interesting by itself but does not characterize the jump rate r ν (a.) . We prefer to work with (21) because, as shown in the next lemma, this Volterra equation admits a unique solution.

Lemma 20. Let s ≥ 0 be fixed, (a t ) t≥s ∈ C([s, ∞), R + ). Then equation (21) has a unique continuous solution t → r ν (a.) (t, s) on [s, ∞).
Proof. Fix T > s. It is sufficient to prove the existence and uniqueness result on [s, T ]. We consider the Banach space (C([s, T ], R), || • || ∞,T ) and define on this space the following operator:

Γ : r → K ν (a.) + K (a.) * r. Let A := sup t∈[s,T ] a t , we have: M T s = sup s≤u≤t≤T K (a.) (t, u) < ∞. This follows from: ∀s ≤ u ≤ t ≤ T, K (a.) (t, u) ≤ f (ϕ (a.) t,u (0)) ≤ f (C A b (T -s)) < ∞.
It is clear (using Assumptions 3.1. and 4) that the operator Γ :

C([s, T ], R) → C([s, T ], R)
is well defined. Given n ∈ N, the iteration Γ n is an affine operator with linear part Γ n 0 : r → (K (a.) ) * (n) * r. We prove that Γ n is contracting for n large enough, which is equivalent to proving that Γ n 0 is contracting for n large enough. By induction, it is easily shown that

∀r ∈ C([s, T ], R), ∀n ∈ N ||Γ n 0 (r)|| ∞,t := sup u∈[s,t] |(Γ n 0 (r))(u, s)| ≤ ||r|| ∞,T (M T s (t -s)) n n! . Consequently ∀r ∈ C([s, T ], R), ∀n ∈ N, ||Γ n 0 (r)|| ∞,T ≤ (M T s (T -s)) n n!
||r|| ∞,T and Γ n 0 is contracting for n large enough. We deduce that the operator Γ n is also contracting and has a unique fixed point in C([s, T ], R). It is also a fixed point of Γ. This proves that (21) has a unique solution in C([s, T ], R).

We shall need the following well-known result on Volterra equation: Lemma 21. Consider k, w : R 2 + → R two continuous kernels. The Volterra equation x = w + k * x has a unique solution given by x = w + r * w, where r : R 2 + → R is the "resolvent" of k, i.e. the unique solution of r = k + k * r.

Proof. It is clear from the proof of the preceding lemma that both Volterra equations have a unique solution. Moreover, we have:

w + k * (w + r * w) = w + k * w + (r -k) * w = w + r * w.
By uniqueness, we deduce that x = w + r * w.

) = ν. Then the functions t → E f ′ (Y s,ν,(a.) t ), t → E f ′ (Y s,ν,(a.) t )b(Y s,ν,(a.) t ) and t → E f 2 (Y s,ν,(a.) t ) are continuous on [s, ∞).
Proof. The proof relies on Proposition 19. Consider the interval [s, T ] for some fixed T > s ≥ 0 and let

A := sup t∈[s,T ] a t . Let φ ∈ {f ′ , f ′ b, f 2 }. By Lemma 11 (point 4), the function (t, u) → φ(ϕ (a.) t,u (0))H (a.) (t, u)r ν (a.) (u, s) is uniformly continuous on {(t, u) : s ≤ u ≤ t ≤ T }. Consequently t → t s φ(ϕ (a.) t,u (0))H (a.) (t, u)r ν (a.) (u, s)du is continuous on [s, T ]. The continuity of t → ∞ 0 φ(ϕ (a.) t,s (x))H x (a.) (t, s)ν(dx) follows from the Dominated Convergence Theorem. For instance, for φ ≡ f ′ , one has ∀t ∈ [s, T ], ∀x ≥ 0, f ′ (ϕ (a.) t,s (x)) ≤ f ′ (ϕ A t-s (x)) ≤ f ′ (x+C A b (t-s)) ≤ C f (f ′ (x) + 1 + f ′ (C A b (T -s))),
from which the result follows easily using Assumption 4 and Assumption 3.2.. The same method can be applied for φ(x) := f ′ (x)b(x) (using Assumption 3.3.) and for φ(x) := f 2 (x).

Proposition 23. Grant Assumptions 2, 3 and 4. Let s, J ≥ 0 be fixed. Given any κ ≥ 0, there is a constant ā ≥ κ only depending on b, f , J and κ such that:

∀(a t ) t≥s ∈ C([s, ∞), R + ), sup t≥s a t ≤ ā and Jν(f ) ≤ ā =⇒ sup t≥s Jr ν (a.) (t, s) ≤ ā.
Moreover, ā can be chosen to be an increasing function of J and κ.

Proof. Assume sup t≥s a t ≤ ā for some ā > 0 that we specify later. Applying the Itô formula and taking expectations yields

∀t ≥ s, E f (Y s,ν,(a.) t ) = E f (Y s,ν,(a.) s )+ t s E f ′ (Y s,ν,(a.) u )[b(Y s,ν,(a.) u ) + a u ]du- t s E f 2 (Y s,ν,(a.) u )du. Lemma 22 implies that t → E f (Y s,ν,(a.) t ) is C 1 and ∀t ≥ s, d dt E f (Y s,ν,(a.) t ) = E f ′ (Y s,ν,(a.) t )(b(Y s,ν,(a.) t ) + a t ) -E f 2 (Y s,ν,(a.) t
).

Using [START_REF] Delarue | Global solvability of a networked integrate-and-fire model of McKean-Vlasov type[END_REF], the Cauchy-Schwarz inequality gives

d dt E f (Y s,ν,(a.) t ) ≤ [ā + C b ] E f ′ (Y s,ν,(a.) t ) - 1 2 E f 2 (Y s,ν,(a.) t ) - 1 2 E 2 f (Y s,ν,(a.) t ) ≤ 1 2 [2ψ(ā + C b ) -E 2 f (Y s,ν,(a.) t )],
where in the last line, we used Assumption 3.2.. Setting M (ā) := 2ψ(ā + C b ) and using the sign of the right hand side, we conclude that

ν(f ) ≤ M (ā) =⇒ [∀t ≥ s E f (Y s,ν,(a.) t ) ≤ M (ā)].
To complete the proof, we need to check that for any κ ≥ 0, any J ≥ 0, there is a constant ā ≥ κ such that JM (ā) ≤ ā. This follows easily from Assumption 3.2., which gives

lim θ→∞ J 2ψ(θ) θ = 0.
It is clear that ā(J) can be chosen to be a non-decreasing function of J and κ. We deduce that:

sup t≥s a t ≤ ā and Jν(f ) ≤ ā =⇒        d dt E f (Y s,ν,(a.) t ) ≤ 1 2 ā2 J 2 -E 2 f (Y s,ν,(a.) t ) E f (Y s,ν,(a.) s ) ≤ ā J .        =⇒ sup t≥s Jr ν (a.) (t, s) ≤ ā.
We have proved that t → E f (Y s,ν,(a.) t

) is C 1 and bounded on R + . The same methods can be applied to the non-linear equation (2).

Lemma 24. Grant Assumptions 2, 3 and 4. Consider (X t ) t≥0 a solution of the non-linear equation [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF] in the sense of Definition 1. Then t → E f (X t ) ∈ C 1 (R + , R) and there is a finite constant r > 0 (only depending on b, f and J) such that:

sup t≥0 E f (X t ) ≤ max(r, E f (X 0 )), lim sup t→∞ E f (X t ) ≤ r.
Moreover, r can be chosen to be an increasing function of J.

Proof. By applying the same argument as in the proof of Lemma 15 it is clear that the functions

t → E f (X t ), t → E f ′ (X t ), t → E f 2 (X t ) and t → E |b(X t )|f ′ (X t )
are locally bounded. Applying the Itô formula and taking expectations yields

E f (X t ) = E f (X 0 ) + t 0 E f ′ (X u )b(X u )du + J t 0 E f ′ (X u ) E f (X u )du - t 0 E f 2 (X u )du. ( 24
)
We deduce that t → E f (X t ) is continuous. Define for all t ≥ 0, a t := E f (X t ). From Lemma 14, it is clear that:

a.s. ∀t ≥ 0, X t = Y 0,ν,(a.) t ,
where (Y 0,ν,(a.) t

) t≥0 is the solution of (5) driven by (a t ) t≥0 . In particular, Lemma 22 applies and the functions

t → E f ′ (X t ), t → E f 2 (X t ) and t → E f ′ (X t )b(X t ) are continuous. From equation (24), we deduce that t → E f (X t ) ∈ C 1 (R + , R + ) and d dt E f (X t ) = E f ′ (X t )b(X t ) + J E f ′ (X t ) E f (X t ) -E f 2 (X t ).
We have:

1. E f ′ (X t )b(X t ) - 1 4 E f 2 (X t ) ≤ 1 2 [2C b E f ′ (X t ) - 1 2 E f 2 (X t )] ≤ 1 2
ψ(2C b ), using Assumptions 12 and 3.2..

J

E f ′ (X t ) E f (X t ) - 1 4 E f 2 (X t ) ≤ J E f ′ (X t ) E f (X t ) - 1 4 E 2 f (X t ) ≤ E f (X t )[J E f ′ (X t ) - 1 8 E f (X t ) - 1 8 E f (X t )] ≤ 2β 2 , where β := sup x≥0 Jf ′ (x) -1 8 f (x) < ∞ (by Assumption 3.2.
). We used sup y≥0 y(β -1 8 y) ≤ 2β 2 to obtain the last inequality. Note that β is a non-decreasing function of J.

Combining the points 1 and 2 gives

d dt E f (X t ) ≤ 1 2 [(ψ(2C b ) + 4β 2 ) -E f 2 (X t )]. (25)
We define: r := ψ(2C b ) + 4β 2 and deduce that sup

t≥0 E f (X t ) ≤ max(r, E f (X 0 )), lim sup t→∞ E f (X t ) ≤ r.

Existence and uniqueness of the solution of the non-linear SDE: proof of Theorem 5

We now prove that equation ( 2) has a unique strong solution (X t ) t≥0 . Let J > 0 (the case J = 0 has already been treated in Lemma 14 by choosing (a t ) t≥0 ≡ 0). Let ν, the initial condition, satisfying Assumption 4, be fixed. We grant Assumptions 2 and 3. Let T > 0 be a fixed horizon time. Thanks to Proposition 23 with κ := max(J E f (X 0 ), J r), we build the following application:

Φ : C T ā → C T ā (a t ) t → Jr ν (a.) (•, 0), (26)
where

C T ā := {(a t ) t ∈ C([0, T ], R + ) : sup t∈[0,T ] a t ≤ ā}. The function r ν (a.) (t, 0) := E f (Y 0,ν,(a.) t
) is defined by equation (5) (using s = 0). The constant ā is given by Proposition 23: in particular ā does not depend on T . We equip

C T ā with the sup norm ||(a t ) t || ∞,T := sup t∈[0,T ] |a t |. The metric space (C T ā , || • || ∞,T
) is complete. We now prove that the application Φ defined by ( 26) is contracting. Let (a t ) t , (d t ) t ∈ C T ā ; we denote by r ν (a.) (t, s) and r ν (d.) (t, s) their corresponding jump rate, where t belongs to [s, T ]. Both r ν (a.) and r ν (d.) satisfy the Volterra equation [START_REF] Gurtin | Non-linear age-dependent population dynamics[END_REF]. It follows that the difference ∆ := r ν (a.) -r ν (d.) satisfies:

∆ = K ν (a.) -K ν (d.) + K (a.) * (r ν (a.) -r ν (d.) ) + (K (a.) -K (d.) ) * r ν (d.) = W + K (a.) * ∆ with W := K ν (a.) -K ν (d.) + (K (a.) -K (d.) ) * r ν (d.)
Consequently, ∆ solves the following non-homogeneous Volterra equation with kernel K (a.)

∆ = W + K (a.) * ∆. ( 27
)
Using Lemma 21, this equation can be solved explicitly in terms of r (a.) , the "resolvent" of K (a.)

∆ = W + r (a.) * W. ( 28
)
Lemma 25. There exists a constant Θ T only depending on T , f , b and ā such that, for all a, d ∈ C T ā :

∀ 0 ≤ s ≤ t ≤ T, ∀x ∈ R + , |K δx (a.) -K δx (d.) |(t, s) ≤ Θ T (1+f ′ (x)+f (x)+f ′ (x)f (x)) t s |a u -d u |du.
Proof. Fix (a t ) and (d t ) in C T ā . We have

|K δx (a.) -K δx (d.) |(t, s) = f (ϕ (a.) t,s (x)) exp - t s f (ϕ (a.) u,s (x))du -f (ϕ (d.) t,s (x)) exp - t s f (ϕ (d.) u,s (x))du ≤ f (ϕ (a.) t,s (x)) -f (ϕ (d.) t,s (x)) exp - t s f (ϕ (a.) u,s (x))du + f (ϕ (d.) t,s (x)) exp - t s f (ϕ (a.) u,s (x))du -exp - t s f (ϕ (d.) u,s (x))du =: M + N.
Assumptions 2 and 3.1. together with Lemma 11 [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF] give

M ≤ |f (ϕ (a.) t,s (x)) -f (ϕ (d.) t,s (x))| ≤ f ′ (x + C ā b T )|ϕ (a.) t,s (x) -ϕ (d.) t,s (x)| ≤ C f (1 + f ′ (x) + f ′ (C ā b T ))C ϕ t s |a u -d u |du.
Furthermore, using that ∀A, B ≥ 0 : |e -A -e -B | ≤ |A -B|, we have

N ≤ C f [1 + f (x) + f (C ā b T )] t s |f (ϕ (a.) u,s (x)) -f (ϕ (d.) u,s (x))|du ≤ C f [1 + f (x) + f (C ā b T )]f ′ (x + C ā b T )C ϕ t s r s |a u -d u |dudr ≤ T C ϕ C 2 f [1 + f (x) + f (C ā b T )][1 + f ′ (x) + f ′ (C ā b T )] t s |a u -d u |du.
Combining the two estimates, we get the result.

Proof of Theorem 5. We now write Θ T for any constant that depends only on T , on the initial condition ν, on b, f , J and on ā and that can change from line to line. By Assumptions 3 and 4, it follows that:

∀(a t ), (d t ) ∈ C T ā , ∀t ∈ [0, T ] : |K ν (a.) -K ν (d.) |(t, 0) ≤ Θ T t 0 |a u -d u |du.
Moreover, since sup t∈[0,T ] r (d.) (t, 0) ≤ ā J by Proposition 23, we have

|(K (a.) -K (d.) ) * r (d.) |(t, 0) = | t 0 (K (a.) -K (d.) )(t, u)r (d.) (u)du| ≤ ā J Θ T (1+f ′ (0))T t 0 |a u -d u |du.
Consequently, there is a constant Θ T such that

∀(a t ), (d t ) ∈ C T ā , ∀t ∈ [0, T ] : |W |(t, 0) ≤ Θ T t 0 |a u -d u |du.
Using the formula (28), we deduce that

|∆(t, 0)| ≤|W |(t, 0) + t 0 r (a.) (t, u)|W |(u, 0)du ≤Θ T (1 + T ā J ) t 0 |a u -d u |du.
We have proved that there is a constant Θ T such that:

∀(a t ), (d t ) ∈ C T ā , ∀t ∈ [0, T ], ||Jr ν (a.) (•, 0) -Jr ν (d.) (•, 0)|| ∞,t ≤ Θ T t 0 ||a -d|| ∞,u du.
This estimate is sufficient to prove Theorem 5 by a classical Picard/Gronwall argument. We deduce that Φ has a unique fixed point (a * t ) t . It is then easy to check that (Y 0,ν,(a * .) t

) t∈[0,T ] ,
driven by the current (a * ) and with initial condition Y 0,ν,(a * .) 0 = X 0 , defines a solution of (2) up to time T . This proves existence of a strong solution to (2). Now, if (X t ) t≥0 is a strong solution of (2) in the sense of Definition 1, let ∀t ≥ 0, a t := J E f (X t ). We have sup t≥0 a t ≤ max(J r, J E f (X 0 )) ≤ ā and consequently (a t ) t∈[0,T ] ∈ C T ā . Moreover, it is clear that (X t ) t≥0 solves (5) with a t := J E f (X t ) and Y 0,ν,(a.) 0 := X 0 . We deduce that (a t ) is the unique fixed point of Φ: ∀t ∈ [0, T ] : a t = a * t . Consequently, by Lemma 14, we have: a.s. ∀t ∈ [0, T ] X t = Y 0,ν,(a.) t . This proves path-wise uniqueness and ends the proof of Theorem 5.

The invariant probability measures: proof of Proposition 8

We now study the invariant probability measures of the non-linear process (2). We follow the strategy of [START_REF] Fournier | On a toy model of interacting neurons[END_REF]: we first study the linear process driven by a constant current a and show that it has a unique invariant probability measure. We then use this result to study the invariant probability measures of the non-linear equation [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. Let a ≥ 0 and (Y ν,a t ) t the solution of the following SDE:

Y ν,a t = Y ν,a 0 + t 0 b(Y ν,a u )du + at - t 0 R+ Y ν,a u-½ {z≤f (Y ν,a u-)} N(du, dz) (29)
Equation ( 29) is equation ( 5) with ∀t ≥ 0, a t = a and s = 0.

Proposition 26. Grant Assumptions 2 and 3. Then the SDE (29) has a unique invariant probability measure ν ∞ a given by equation ( 14):

ν ∞ a (dx) := γ(a) b(x) + a exp - x 0 f (y) b(y) + a dy ½ {x∈[0,σa]} dx,
where γ(a) is the normalizing factor given by [START_REF] Drogoul | Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics[END_REF]. Moreover we have ν ∞ a (f ) = γ(a). A proof of this result can be found in [START_REF] Fournier | On a toy model of interacting neurons[END_REF]Prop. 21] with b(x) := -κx and with slightly different assumptions on f . We give here a proof based on different arguments. Note that the general method introduced by [START_REF] Costa | Stationary distributions for piecewise-deterministic Markov processes[END_REF] to find the stationary measures of a PDMP can be applied here; we use a method introduced in this paper to prove the uniqueness part.

Proof. Let us first check that the probability measure ν ∞ a is indeed an invariant measure of ( 29). Claim 1 The probability measure ν ∞ a satisfies Assumption 4. First b(0) > 0 yields to ∀a ≥ 0, σ a ≥ σ 0 > 0. The function t → ϕ a t (0) is a bijection from R + to [0, σ a ). Consequently, the changes of variable x = ϕ a t (0) and y = ϕ a u (0) give

σa 0 f 2 (x) b(x) + a exp - x 0 f (y) b(y) + a dy dx = ∞ 0 f 2 (ϕ a t (0)) exp - t 0 f (ϕ a u (0))du dt.
This last integral is finite by Remark 12. Claim 2 We have:

K ν ∞ a a (t) = γ(a)H a (t).
We recall that H a (t) = H δ0 a (t, 0). We have, for all t ≥ 0:

K ν ∞ a a (t) = σa 0 f (ϕ a t (x)) exp - t 0 f (ϕ a u (x))du γ(a) b(x) + a exp - x 0 f (y) b(y) + a dy dx. ( 30
)
The change of variable y = ϕ a u (0) yields:

K ν ∞ a a (t) = σa 0 f (ϕ a t (x)) exp - t 0 f (ϕ a u (x))du γ(a) b(x) + a exp - t(x) 0 f (ϕ a u (0))du dx,
where t(x) is the unique t ≥ 0 such that ϕ a t (0) = x. We now make the change of variable x = ϕ a s (0) we obtain (using the semi-group property satisfied by ϕ a t (0)):

K ν ∞ a a (t) = γ(a) ∞ 0 f (ϕ a t (ϕ a s (0))) exp - t 0 f (ϕ a u (ϕ a s (0)))du exp - s 0 f (ϕ a u (0))du ds = γ(a) ∞ t f (ϕ a θ (0)) exp - θ 0 f (ϕ a u (0))du dθ = γ(a) H a (t) -lim θ→∞ exp - θ 0 f (ϕ a u (0))du .
Using Remark 2, we have: lim θ→∞ exp -θ 0 f (ϕ a u (0))du = 0 and the claim is proved. We now consider (Y ν ∞ a ,a t ) t≥0 the solution of equation [START_REF] Veretennikov | On ergodic measures for McKean-Vlasov stochastic equations[END_REF]

starting from L(Y ν ∞ a ,a 0 ) = ν ∞ a . Proposition 19 applies, so r ν ∞ a a (t) = E f (Y ν ∞ a ,a t
) is the unique solution of the Volterra equation

r ν ∞ a a = K ν ∞ a a + K a * r ν ∞ a a .
Using Claim 2 and the relation [START_REF] Mischler | Relaxation in Time Elapsed Neuron Network Models in the Weak Connectivity Regime[END_REF], we verify that the constant function γ(a) is a solution of

K ν ∞ a a + K a * γ(a) = γ(a)H a + γ(a)(1 -H a ) = γ(a).
By uniqueness (Lemma 20), we deduce that ∀t ≥ 0, r ν ∞ a a (t) = γ(a). Finally, let φ : R + → R + be a measurable function. Using Proposition 19, we have:

E φ(Y ν ∞ a ,a t ) = γ(a) t 0 φ(ϕ a t-u (0))H a (t -u)du + ∞ 0 φ(ϕ a t (x))H x a (t)ν ∞ a (dx) = γ(a) t 0 φ(ϕ a u (0))H a (u)du + σa 0 φ(ϕ a t (x)) exp - t 0 f (ϕ a u (x))du γ(a) b(x) + a exp - x 0 f (y) b(y) + a dy dx.
The change of variables y = ϕ a u (0) and x = ϕ a θ (0) yields

E φ(Y ν ∞ a ,a t ) = γ(a) t 0 φ(ϕ a u (0))H a (u)du + γ(a) ∞ 0 φ(ϕ a t (ϕ a θ (0))) exp - t 0 f (ϕ a u (ϕ a θ (0)))du exp - θ 0 f (ϕ a u (0))du dθ = γ(a) t 0 φ(ϕ a u (0))H a (u)du + γ(a) ∞ t φ(ϕ a u (0)) exp - u 0 f (ϕ a θ (0))dθ du = γ(a) ∞ 0 φ(ϕ a u (0))H a (u)du = ν ∞ a (φ).
This proves that ∀t ≥ 0, L(Y

ν ∞ a ,a t ) = ν ∞
a and consequently ν ∞ a is an invariant probability measure of [START_REF] Veretennikov | On ergodic measures for McKean-Vlasov stochastic equations[END_REF]. Moreover, we have

ν ∞ a (f ) = γ(a) σa 0 f (x) b(x) + a exp - x 0 f (y) b(y) + a dy dx = γ(a).
It remains to prove that the invariant probability measure is unique. Following [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF] and [START_REF] Costa | Stationary distributions for piecewise-deterministic Markov processes[END_REF], we define B ac (R + ) the set of bounded function h : R + → R such that for all x ∈ R + , the function t → h(ϕ a t (x)) is absolutely continuous on R + . For h ∈ B ac (R + ), we define Hh(x) :=

d dt h(ϕ a t (x)) t=0 . Claim 3 Let h ∈ B ac (R + ), then for all x ≥ 0 we have d dt E h(Y δx,a t ) t=0 = Lh(x) with Lh(x) := Hh(x) + (h(0) -h(x))f (x). Let τ x 1 = inf{t ≥ 0 : Y δx,a t = Y δx,a t-} and τ x 2 = inf{t ≥ τ x 1 : Y δx,a t
= Y δx,a t-} be the times of the first and second jumps of (Y δx,a t ). We have

E h(Y δx,a t ) = E h(Y δx,a t )½ {t<τ x 1 } + E h(Y δx,a t )½ {τ x 1 ≤t<τ x 2 } + E h(Y δx,a t )½ {t≥τ x 2 } =: α t + β t + θ t .
By Lemma 16, we have α t = h(ϕ a t (x))P(t < τ x 1 ) = h(ϕ a t (x))H x a (t). It follows that d dt α t t=0 = Hh(x) -h(x)f (x). Moreover using that the density of τ x 1 is s → K x a (s) it holds that β t = t 0 h(ϕ a t-s (0))K x a (s)H 0 a (t -s)ds. We deduce that d dt β t t=0 = h(0)f (x). Then, using that h is bounded, we have

θ t ≤ ||h|| ∞ t 0 t 0 K x a (u)K 0 a (s -u)duds ∈ O(t 2
). This proves Claim 3. Let g be a bounded measurable function. We follow the method of [START_REF] Costa | Stationary distributions for piecewise-deterministic Markov processes[END_REF] (proof of Theorem 3(a)) and define

∀x ≥ 0, λ g (x) := ∞ 0 g(ϕ a t (x)) exp - t 0 f (ϕ a r (x))dr dt.

Claim 4

The function λ g belongs to B ac (R + ) and satisfies Hλ g (x) = f (x)λ g (x) -g(x).

Using the semi-group property of ϕ a t (x) we have

λ g (ϕ a t (x)) = exp t 0 f (ϕ a u (x))du λ g (x) - t 0 g(ϕ a u (x)) exp - u 0 f (ϕ a θ (x))dθ du .
This proves that λ g is in

B ac (R + ) with d dt λ g (ϕ a t (x)) = f (ϕ a t (x))λ g (ϕ a t (x)) -g(ϕ a t (x)
) and gives the stated formula.

Consider now ν an invariant probability measure with ν(f ) < ∞. The Markov property at time t = 0 together with Claim 3 shows that

d dt E λ g (Y ν,a t ) t=0 = d dt ∞ 0 E λ g (Y δx,a t )ν(dx) t=0 = ν(Lλ g ).
The exchange of the derivative at time t = 0 and the integral on R + is legitimate thanks to the Dominated Convergence Theorem. Claim 4 and the fact that ν is an invariant measure then show that

0 = d dt E λ g (Y ν,a t ) t=0 = λ g (0)ν(f ) -ν(g).
The same computations can be done with g ≡ 1, giving λ 1 (0)ν(f ) = 1. It follows that

ν(g) = λ g (0) λ 1 (0) = ∞ 0 g(x)ν ∞ a (dx).
We deduce that necessarily ν = ν ∞ a .

The next lemma characterizes the invariant probability measures of (2).

Lemma 27. The invariant probability measures of the non-linear equation

(2) are {ν ∞ a | a = Jγ(a), a ∈ R + }.
Proof. Let ν be an invariant probability measure of (2) and L(X 0 ) = ν. We have

∀t ≥ 0, E f (X t ) = ν(f ) =: p.
Let a := Jp. The process (X t ) t≥0 solves [START_REF] Veretennikov | On ergodic measures for McKean-Vlasov stochastic equations[END_REF] and ν is an invariant probability measure of equation [START_REF] Veretennikov | On ergodic measures for McKean-Vlasov stochastic equations[END_REF]. It implies that ν = ν ∞ a . Moreover p = γ(a) and so necessarily a γ(a) = J.

Conversely, let a ≥ 0 such that a γ(a) = J. Let (Y ν ∞ a ,a t
) be the solution of [START_REF] Veretennikov | On ergodic measures for McKean-Vlasov stochastic equations[END_REF] with

L(Y ν ∞ a ,a 0 ) = ν ∞ a . We have seen that E f (Y ν ∞ a ,a t ) = γ(a), it follows that a = J E f (Y ν ∞ a ,a t ). Consequently (Y ν ∞ a ,a t
) t≥0 solves (2) and ν ∞ a is one of its invariant probability measure.

The problem of finding the invariant probability measures of the mean-field equation ( 2) has been reduced to finding the solutions of the scalar equation ( 16). When J is small enough, we can prove that it has a unique solution, which concludes the proof of Proposition 8. Lemma 28. Equation ( 16) has at least one solution a * > 0. Moreover, there is a constant J 0 > 0 such that for all J ∈ [0, J 0 ] ( 16) has a unique solution.

Proof. Recall [START_REF] Drogoul | Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics[END_REF]. By the changes of variable y = ϕ a u (0) and x = ϕ a t (0), it holds that

γ(a) -1 = ∞ 0 exp - t 0 f (ϕ a u (0))du dt. ( 31
)
In particular, the function a → γ(a) is non-decreasing. Furthermore, using that b(x) ≤ C b , we have

a γ(a) ≥ a ∞ 0 exp - t 0 f ((a + C b )u)du dt ≥ a a + C b ∞ 0 exp - 1 a + C b θ 0 f (z)dz dθ.
We deduce that lim a→+∞ aγ(a) -1 = +∞. Let U (a) := aγ(a) -1 . One has U (0) = 0, lim a→+∞ U (a) = +∞ and U is continuous on R + . It follows that the equation U (a) = J has at least one solution a * . Moreover, one can show that the function U has a derivative at a = 0 and U ′ (0) = 1/γ(0) > 0. Consequently, there is a 0 > 0 such that U is strictly increasing on [0, a 0 ]. Using lim a→+∞ U (a) = +∞, we can find a 1 such that: ∀a ≥ a 1 , U (a) ≥ 1. Finally let J 0 := min a∈[a0,a1] U (a) > 0. Let J < J 0 , it is clear that the equation U (a) = J has exactly one solution a * ∈ [0, a 0 ].

The convergence of the jump rate implies the convergence in law of the time marginals

The goal of this section is to prove that controlling the behavior of the jump rate t → E f (X t ) can be sufficient to deduce the asymptotic law of (X t ), solution of (2).

Proposition 29. Grant Assumptions 2, 3, 4. Let (X t ) t≥0 be the solution of the non-linear equation [START_REF] Bolley | Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation[END_REF]. Assume that there exist constants λ, C > 0 and a * ≥ 0 (that may depend on b, f , ν, and J) such that:

∀t ≥ 0, | E f (X t ) -γ(a * )| ≤ Ce -λt ,
and that a * satisfies equation ( 16): a * γ(a * ) = J. Then

X t L -→ t→∞ ν ∞ a * .
Moreover, if φ : R + → R is any bounded Lipschitz-continuous function, it holds that

∀0 < λ ′ < min(λ, f (σ 0 )), ∃D > 0 s.t. ∀t ≥ 0, | E φ(X t ) -ν ∞ a * (φ)| ≤ De -λ ′ t
, where the constant D only depends on b, f, J, C, ν, λ ′ and φ through its infinite norm and its Lipschitz constant.

Proof. Let (X t ) t≥0 be the solution of (2) and φ : R + → R a bounded Lipschitz-continuous function, with Lipschitz constant l φ . Consider λ ′ ∈ (0, min(λ, f (σ 0 ))). We denote by D any constant only depending on b, f, J, C, ν, λ ′ , ||φ|| ∞ and l φ which shall change from line to line. Define for all t ≥ 0, a t := J E f (X t ). It holds that (X t ) t≥0 is a solution of ( 5) with driving current (a . ). Denote r ν (a.) (t, 0) = E f (X t ). By Proposition 19, we have

E φ(X t ) = t 0 φ(ϕ (a.) t,u (0))H (a.) (t, u)r ν (a.) (u, 0)du + ∞ 0 φ(ϕ (a.) t,0 (x))H x (a.) (t, 0)ν(dx)
Using Remarks 13 and 12 (2), together with the fact that λ ′ < f (σ 0 ), we deduce that

∀t ≥ 0, ∞ 0 φ(ϕ (a.) t,0 (x))H x (a.) (t, 0)ν(dx) ≤ De -λ ′ t
for some constant D. Moreover, one has, using the change of variable x = ϕ a * v (0)

ν ∞ a * (φ) = σ a * 0 φ(x)ν ∞ a * (dx) = ∞ 0 φ(ϕ a * v (0))γ(a * )H a * (v)dv = t 0 φ(ϕ a * t,u (0))H a * (t, u)γ(a * )du + ∞ t φ(ϕ a * v (0))γ(a * )H a * (v)dv.
The last equality is obtained with the change of variable v = t-u. The second term is controlled by

∞ t φ(ϕ a * u (0))γ(a * )H a * (u)du ≤ ||φ|| ∞ γ(a * ) ∞ t f (ϕ a * u (0)) inf v≥t f (ϕ a * v (0)) exp - u 0 f (ϕ a * θ (0))dθ du = ||φ|| ∞ γ(a * ) inf v≥t f (ϕ a * v (0)) exp - t 0 f (ϕ a * θ (0))dθ ≤ De -λ ′ t ,
for some constant D. We used again Remark 12. It remains to show that

∆ := t 0 φ(ϕ (a.) t,u (0))H (a.) (t, u)r ν (a.) (u, 0)du - t 0 φ(ϕ a * t,u (0))H a * (t, u)γ(a * )du
goes to zero exponentially fast. One has

∆ ≤ t 0 φ(ϕ (a.) t,u (0)) -φ(ϕ a * t,u (0)) H (a.) (t, u)r ν (a.) (u, 0)du + t 0 H (a.) (t, u) -H a * (t, u) φ(ϕ a * t,u (0))r ν (a.) (u, 0)du + t 0 H a * (t, u)φ(ϕ a * t,u (0)) r ν (a.) (u, 0) -γ(a * ) du =: α t + β t + θ t .
Using that for all t ≥ 0, |r ν (a.) (t, 0) -γ(a * )| ≤ Ce -λ ′ t (λ ′ < λ) and Remark 13, we obtain:

θ t ≤ C||φ|| ∞ t 0 H 0 (t, u)e -λ ′ u du = C||φ|| ∞ e -λ ′ t t 0 H 0 (t -u)e λ ′ (t-u) du ≤ C||φ|| ∞ ∞ 0 H 0 (u)e λ ′ u du e -λ ′ t =: De -λ ′ t .
The fact that u → H 0 (u)e λ ′ u belongs to L 1 (R + ) follows from λ ′ < f (σ 0 ). By Theorem 5, one can find a constant p (with γ(a * ) ≤ p) such that:

∀t ≥ 0, E f (X t ) = r ν (a.) (t, 0) ≤ p.
Moreover, Assumption 2 and Remark 13 give

α t ≤ pl φ t 0 |ϕ (a.) t,u (0) -ϕ a * t,u (0)|H 0 (t, u)du ≤ pl φ C ϕ t 0 t u |a θ -a * |dθH 0 (t, u)du.
Using that

t u |a θ -a * |dθ ≤ JC t u e -λ ′ θ dθ ≤ JCe -λ ′ u λ ′
, one has

α t ≤ pl φ C ϕ JC λ ′ e -λ ′ t t 0 e λ ′ (t-u) H 0 (t -u)du ≤ pl φ C ϕ JC λ ′ ∞ 0 H 0 (u)e λ ′ u du e -λ ′ t =: De -λ ′ t .
Finally, using the inequality |e -A -e -B | ≤ e -min(A,B) |A -B| together with Remark 13, we obtain

β t ≤ ||φ|| ∞ p t 0 H 0 (t -u) t u f (ϕ (a.) θ,u (0)) -f (ϕ a * θ,u (0)) dθdu.

22

Setting ā := J p, we have moreover, using that f ′ is non-decreasing and Lemma 2

t u f (ϕ (a.) θ,u (0)) -f (ϕ a * θ,u (0)) dθ ≤ f ′ (ϕ ā t,u ) t u ϕ (a.) θ,u (0) -ϕ a * θ,u (0) dθ. Assumption 2 yields t u f (ϕ (a.) θ,u (0)) -f (ϕ a * θ,u (0)) dθ ≤ C ϕ f ′ (ϕ ā t,u (0)) t u θ u |a s -a * |dsdθ ≤ C ϕ JCf ′ (ϕ ā t,u (0)) t u θ u e -λ ′ s dsdθ ≤ C ϕ JC λ ′ f ′ (ϕ ā t,u (0))(t -u)e λ ′ (t-u) e -λ ′ t .
We used the fact that

t u θ u e -λ ′ s dsdθ = t u e -λ ′ u -e -λ ′ θ λ ′ dθ ≤ (t -u)e -λ ′ u λ ′ . Note that Lemma 3 implies that f ′ (ϕ ā t,u (0)) ≤ f ′ (C ā b (t -u)
) and using Remark 12(1) we have

∀ǫ > 0, ∃A ǫ : ∀x ≥ 0, f ′ (x) ≤ A ǫ e ǫx .
Choosing ǫ := (f (σ 0 ) -λ ′ )/2, we obtain

t u f (ϕ (a.) θ,u (0)) -f (ϕ a * θ,u (0)) dθ ≤ A ǫ C ϕ JC λ ′ (t -u)e (λ ′ +ǫ)(t-u) e -λ ′ t ,
and we deduce that

β t ≤ A ǫ JC ϕ C||φ|| ∞ ā λ ′ +∞ 0 H 0 (u)ue (λ ′ +ǫ)u du e -λ ′ t =: De -λ ′ t .
Combining the three estimates, we have proved the result.

Long time behavior with constant drift

The goal of this section is to study the rate of convergence to the invariant probability measure when J = 0 (no interaction). We use Laplace transform techniques to characterize the convergence. We state here the main result of the section.

Proposition 30. Grant Assumptions 2, 3 and 4. Let (Y ν,a t

) t≥0 be the solution of [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF], driven by a constant current (a t ) ≡ a, a ≥ 0; starting at time s = 0 with law ν. One can find a constant λ * a ∈ (0, f (σ a )] (only depending on b, f and a) such that for any 0 < λ < λ * a it holds

∀t ≥ 0, | E f (Y ν,a t ) -γ(a)| ≤ De -λt ∞ 0 [1 + f (x)]|ν -ν ∞ a |(dx), ( 32
)
where D is a constant only depending on f, b, a and λ. Moreover, one has

Y ν,a t L -→ t→∞ ν ∞ a .
Remark 31. In the above result, λ * a is explicitly known in terms of f, b and a (see its expression (35)) and is optimal (see Remark 40). Note also that (32) states explicitly the dependence on the initial distribution ν through its distance to the invariant measure ν ∞ a .

Study of the Volterra equation

In the case where (a t ) is constant and equal to a, the Volterra equation ( 21) is a linear homogeneous convolution Volterra equation. If moreover the initial condition ν is δ 0 , the kernel r a (t

) := E f (Y δ0,a t ) satisfies r a = K a + K a * r a , (33)
For such equations, it is very natural to use Laplace transform techniques as convolutions become scalar products with this transformation. Furthermore, the "kernel" K a and the "forcing term" K ν a are non-negative. Volterra equation with positive kernels have been studied in the context of Renewal theory. The main reference on this question is a paper of Feller [START_REF] Feller | On the integral equation of renewal theory[END_REF]. We refer to [START_REF] Feller | On the integral equation of renewal theory[END_REF]Th. 4] for this method. However, in our case the rate of convergence is exponential. In order to achieve the optimal rate of convergence, we use general methods from the Volterra integral equation theory, and especially the so called "Whole-line Palay-Wiener" Theorem.

Along this section, we grant Assumptions 2, 3 and 4.

Definition 32 (Laplace transform). Let g : R + → R be a measurable function. The Laplace transform of g is the following function

g(z) := ∞ 0 e -zt g(t)dt,
defined for all z ∈ C for which the integral exists.

Note that the Laplace transforms of H a and K a are well defined for all z ∈ C with R(z) > -f (σ a ). This follows from the fact that ∀λ < f (σ a ), sup t≥0 H a (t)e λt < ∞. The same holds for K a . Integrating by parts the Laplace transform of K a shows that The long time behavior of r a is related to the location of the poles of r a . Equation (33) gives

∀z ∈ C, R(z) > -f (σ a ) =⇒ K a (z) = 1 -z H a (z). (34)
∀R(z) > 0 r a (z) = K a (z) 1 -K a (z) .
This suggests to study the location of the zeros of 1 -K a (z) = z H a (z).

On the zeros of H

a Lemma 34. ∀z ∈ C, R(z) ≥ 0 =⇒ H a (z) = 0.
Proof. Remark first that H a being a real-valued function, H a (z) = 0 iff H a (z) = 0, so it is sufficient to locate the zeros of H a in the region I(z) ≥ 0. Next, it follows from for the nonnegativity of K a that

| K a (z)| ≤ ∞ 0 |e -tz |K a (t)dt < ∞ 0 K a (t)dt = 1 if R(z) > 0. It yields R(z) > 0 =⇒ H a (z) = 0. Moreover, following [17] proof of Theorem 4, (b), if z = iy, y > 0 then iy H a (iy) = 1 -K a (iy) = ∞ 0 (1 -cos(yt))K a (t)dt + i ∞ 0 sin(yt)K a (t)dt.
Consequently, K a (iy) = 1 for some y > 0 would imply that for Lebesgue almost every t ≥ 0, (1 -cos(yt))K a (t) = 0, that is, a.e. K a (t) = 0. It obviously contradicts the assumption f (x) > 0 for x > 0. It follows that ∀y > 0, H a (iy) = 0. Finally for z = 0, we have

H a (0) = ∞ 0 H a (t)dt = 0.
Lemma 35. The zeros of H a are isolated.

Proof. This directly follows from the fact that H a is an holomorphic function on R(z) > -f (σ a ) and thus its zeros are isolated.

Lemma 36. For all z ∈ C, it holds that

| K a (z)| ≤ φ a (R(z)) |I(z)| , where for all x ∈ R, φ a (x) := ||K ′ a,x || 1 and K a,x (t) := e -xt K a (t), K ′ a,x (t) := d dt K a,x (t).
Consequently, the zeros of H a are within a "cone":

∀z ∈ C, R(z) > -f (σ a ), z = x + iy, H a (z) = 0 =⇒ |y| ≤ φ a (x). Proof. Let z = x + iy, y > 0, x > -f (σ a ). We have K a (z) = ∞ 0 e -zt K a (t)dt = ∞ 0 e -iyt K a,x (t)dt = ∞ 0 e -iyt iy K ′ a,x (t)dt.
The last equality follows by an integration by part. It yields

| K a (z)| ≤ ||K ′ a,x || 1 |y| .
We deduce that for |y| > ||K ′ a,x || 1 , we have K a (z) = 1 and also H a (z) = 0.

Consequently, from Lemmas 34, 35 and 36, we can define the abscissa of the "first" zero of

H a : λ * a := -sup{R(z)| R(z) > -f (σ a ), H a (z) = 0}, (35)
with the convention that λ * a = f (σ a ) if the set of zeros is empty. We have proved that

0 < λ * a ≤ f (σ a ) ≤ ∞.
The parameter λ * a is key here as it gives the speed of convergence to the invariant probability measure. It only depends on a, b and f .

Convergence with optimal rate

Our goal in this section is to prove the following proposition Proposition 37. Eq. (33) has a unique solution r a of the form:

r a = γ(a) + ξ a with ∀λ ∈ [0, λ * a ), ξ a ∈ L λ .
The constant λ * a > 0 is defined by (35).

This result can be deduced from general theorems of the Volterra equations theory. For instance, one can apply [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF]Th. 2.4,Chap. 7]. However, this last result is written for general measure kernels in weighted spaces and its proof is somehow difficult to follow. In our setting, the proof given by [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF] simplifies a lot and we give it here for completeness. We use the following so-called "Whole Line Palay-Wiener" Theorem which is one of the most important ingredients of the convolution Volterra integral equations theory.

Theorem 38 (Whole-line Palay-Wiener). Let k ∈ L 1 (R, R). There exists a function x ∈ L 1 (R, R) satisfying the equation

∀t ≥ 0, x(t) = k(t) + R k(t -u)x(u)du if and only if ∀y ∈ R, k(iy) := R e -iyt k(t)dt = 1.
Note that here k(iy) is actually the Fourier transform of k evaluated at y ∈ R.

Proof. See [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF]Th. 4.3,Chap. 2]. We prove later, in details, an extension of this theorem (see Proposition 51).

Proof of Proposition 37. Let σ -and σ + be any real numbers such that:

-λ * a < σ -< 0 < σ + < ∞.
We first extend r a , K a and H a to the whole line by defining:

∀t ∈ R, r a (t) := r a (t)½ {t≥0} , K a (t) := K a (t)½ {t≥0} and H a (t) := H a (t)½ {t≥0} . We have from (33) ∀t ∈ R, r a (t) = K a (t) + R K a (t -u)r a (u)du. ( 36
)
For any △∈ R, we also define r a,△ (t) := e -△t r a (t), K a,△ (t) := e -△t K a (t). Note that K a,σ-∈ L 1 (R) and that ∀y ∈ R, K a,σ-(iy) = K a (σ -+ iy) = 1 (by definition of λ * a ). We can apply Theorem 38: there exists ξ a,σ-∈ L 1 (R) such that

∀t ∈ R, ξ a,σ-(t) = K a,σ-(t) + R K a,σ-(t -u)ξ a,σ-(u)du. (37)
We define ξ a (t) := e σ-t ξ a,σ-(t). We have R |ξ a (u)|e -σ-u du < ∞ and (37) reads

∀t ∈ R, ξ a (t) = K a (t) + R K a (t -u)ξ a (u)du.
Remark 39. The function ξ a is not null on R -(see formula (39) just below).

We have, using equalities (36) and ( 37)

ξ a,σ-∈ L 1 (R), ξ a,σ-(iy) = K a 1 -K a (iy + σ -), r a,σ+ ∈ L 1 (R), r a,σ+ (iy) = K a 1 -K a (iy + σ + ).
We can now use the Fourier inverse formula for L 1 (R) functions to get

ξ a,σ-(t) = 1 2π R e iyt K a 1 -K a (iy + σ -)dy and r a,σ+ (t) = 1 2π R e iyt K a 1 -K a (iy + σ + )dy,
or after the changes of variable z = iy + σ -and z = iy + σ + :

ξ a (t) = lim T →∞ 1 2πi σ-+iT σ--iT e zt K a (z) 1 -K a (z) dz and r a (t) = lim T →∞ 1 2πi σ++iT σ+-iT e zt K a (z) 1 -K a (z) dz.
Let Γ T be the closed curve in the complex plane composed of four straight lines that join the points σ --iT , σ -+ iT , σ + + iT , and σ + -iT in the anti-clockwise direction. It follows from the residue theorem that

ΓT e zt K a (z) 1 -K a (z) dz = ΓT e zt K a (z) z H a (z) dz = 2πi K a (0) H a (0) = 2πiγ(a). ( 38
)
The last equality follows from

H a (0) = ∞ 0 H a (t)dt = ∞ 0 exp - t 0 f (ϕ a u )du dt (31) = 1 γ(a)
.

By Lemma 36, for all z in the strip R(z) ∈ [σ -, σ + ], z = 0, we have

| K a (z)| ≤ φ a (σ -) |I(z)| .
We deduce that lim

T →±∞ σ++iT σ-+iT e zt K a (z) 1 -K a (z) dz = 0.
Therefore we can take the limit T → ∞ in (38) and obtain

∀t ∈ R, r a (t) = γ(a) + ξ a (t). ( 39
)
The proposition is proven by choosing σ -= -λ.

Remark 40. The speed of convergence obtained in this result is optimal if λ * a < f (σ a ) (i.e. H a has at least one complex zero with R(z) > -f (σ a )) in the sense that ∀λ > λ * a , r a -γ(a) / ∈ L λ .

To see this, assume that λ * a < f (σ a ) and choose σ -such that -f (σ a ) < σ -< -λ * a . The previous proof can be mimicked except that the residues of equation (38) now involves terms of the order e -λ * a t -corresponding to the roots of H a with real part equal to -λ * a .

Long time behavior starting from initial condition ν: proof of Proposition 30

We now come back to the general case where the initial condition can be any probability measure satisfying Assumption 4, and we give the proof of Proposition 30.

Proof of Proposition 30. Note that, we only consider here the convolutions on [0, t] denoted by * (and no more the convolution on R). Let r ν a (t) = E f (Y ν,a t ) with L(Y 0 ) = ν. The function r ν a is the unique solution of the Volterra equation

r ν a = K ν a + K a * r ν a .
If we choose ν to be the invariant probability measure ν ∞ a , we get γ(a) = K ν ∞ a a + K a * γ(a) and

r ν a -γ(a) = K ν a -K ν ∞ a a + K a * (r ν a -γ(a)).
We can solve this equation in terms of r a , the "resolvent" of K a (using Lemma 21) and obtain

r ν a -γ(a) = K ν a -K ν ∞ a a + r a * (K ν a -K ν ∞ a a ) = K ν a -K ν ∞ a a + ξ a * (K ν a -K ν ∞ a a ) + γ(a) * (K ν a -K ν ∞ a a ),
where r a = ξ a + γ(a), see (39), is the solution of the Volterra equation r a = K a + K a * r a . Using [START_REF] Mischler | Relaxation in Time Elapsed Neuron Network Models in the Weak Connectivity Regime[END_REF], we have γ(a) * K ν a = γ(a)(1 -H ν a ) and thus

r ν a -γ(a) = K ν a -K ν ∞ a a + γ(a)(H ν ∞ a a -H ν a ) + ξ a * (K ν a -K ν ∞ a a ).
We now write Θ any constant only depending on λ, f, b and a and which may change from line to line. It is clear that for any 0

< λ < f (σ a ) |H ν ∞ a a -H ν a |(t) ≤ ∞ 0 H x a (t)|ν -ν ∞ a |(dx) ≤ ∞ 0 H a (t)|ν -ν ∞ a |(dx) ≤ Θe -λt ∞ 0 |ν -ν ∞ a |(dx).
Similarly, for any 0 < λ < f (σ a ),

|K ν a -K ν ∞ a a |(t) ≤ ∞ 0 f (ϕ a t (x))H x a (t)|ν -ν ∞ a |(dx) ≤ ∞ 0 f (x + C a b t)H a (t)|ν -ν ∞ a |(dx) ≤ C f ∞ 0 [1 + f (x) + f (C a b t)]H a (t)|ν -ν ∞ a |(dx) ≤ Θe -λt ∞ 0 (1 + f (x))|ν -ν ∞ a |(dx).
We used here Assumption 3.1.. Let now 0 < λ < λ * a . Using ξ a ∈ L λ , it holds that

|ξ a * (K ν a -K ν ∞ a a )|(t) ≤ t 0 |ξ a (t -u)||K ν a -K ν ∞ a a |(u)du ≤ Θe -λt ∞ 0 (1 + f (x))|ν -ν ∞ a |(dx).
Combining the three estimates, one deduces that

|r ν a (t) -γ(a)| ≤ Θe -λt ∞ 0 (1 + f (x))|ν -ν ∞ a |(dx).
It remains to prove that lim t→∞ L(Y ν,a t ) = ν ∞ a . The process (Y ν,a t ) t≥0 is the solution of (2) with b(x) = b(x) + a and J = 0. Obviously, 0 solves [START_REF] Duarte | A model for neural activity in the absence of external stimuli. Markov Process[END_REF]. Applying Proposition 29 ends the proof.

Long time behavior with a general drift

In this section, we generalize the results obtained in Section 7 to non constant currents. We consider the process (5) driven by a current (a t ) assuming to converge exponentially fast to a. We seek to prove that the jump rate of this process is converging to γ(a) and estimate the speed of convergence. This "perturbation" analysis will be useful to study the long time behavior of the solution of the non-linear McKean-Vlasov equation ( 2) with small interactions. We consider a non-negative continuous function (a t ) t≥0 such that Assumptions 41.

1. sup t≥0 a t ≤ ā for some constant ā > 0.

2. There exist a ≥ 0, C ≥ 0 and λ ∈ (0, min(λ * a , f (σ 0 ))), where σ 0 and λ * a are defined by (17) and (35), such that ∀t ≥ 0, |a t -a| ≤ Ce -λt .

(40)

Note that the values of C and λ are important in this analysis. Any mention of C and λ in this section refer to these two constants.

Let r ν (a.) (t, s) = E f (Y s,ν,(a.) t

), where Y s,ν,(a.) t

is the solution of (5) driven by the current (a t ) and starting at time s with law ν. The goal of this section is to prove that if C is small enough, then there exists an explicit constant D such that

∀t ≥ s ≥ 0, |r ν (a.) (t, s) -γ(a)| ≤ De -λ(t-s) ,
where γ(a) is given by [START_REF] Drogoul | Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics[END_REF]. Note that the exponential decay rate λ is preserved. We make efforts to keep track of the constant D and to relate it to C. As in Section 7 it is useful to split the study in two parts: the case where the initial condition is a Dirac mass at 0 and the general case. We thus consider the unique solution r (a.) of the following Volterra equation:

r (a.) = K (a.) + K (a.) * r (a.) . ( 41
)
It is also useful to introduce a Banach space adapted to this non-homogeneous setting. This result allows us to consider elements of L λ as elements of V λ . Note that the algebra L λ is commutative whereas V λ is not.

An adapted Banach algebra

The perturbation method

Define K(a.) := K (a.) -K a and H(a.) := H (a.) -H a .

Lemma 45. Grant Assumptions 2, 3, 4 and 41. Then, there exists a continuous non-negative and integrable function η such that for all t ≥ s ≥ 0, one has

| K(a.) (t, s)| ≤ Ce -λt η(t -s), | H(a.) (t, s)| ≤ Ce -λt η(t -s).
The function η only depends on b, ā, f and λ (in particular it does not depend on C). Furthermore, we can choose η such that ||η|| 1 is a non-decreasing function of ā.

Proof. Here, to simplify the notation, we write ϕ (a.) t,s for ϕ (a.) t,s (0). We have

K(a.) (t, s) = f (ϕ (a.) t,s ) exp - t s f (ϕ (a.) u,s )du -f (ϕ a t,s ) exp - t s f (ϕ a u,s )du | K(a.) (t, s)| ≤ |f (ϕ (a.) t,s ) -f (ϕ a t,s )| exp - t s f (ϕ (a.) u,s )du + f (ϕ a t,s ) exp - t s f (ϕ (a.) u,s )du -exp - t s f (ϕ a u,s )du =: M 1 + M 2 .
Assumptions 2, 3. 1. and (40) give

|f (ϕ (a.) t,s ) -f (ϕ a t,s )| ≤ f ′ (ϕ ā t,s )|ϕ (a.) t,s -ϕ a t,s | ≤ f ′ (C ā b (t -s))C ϕ t s |a u -a|du ≤ f ′ (C ā b (t -s))C ϕ C t s e -λu du ≤ Ce -λt f ′ (C ā b (t -s))C ϕ e λ(t-s) λ .
Moreover choosing λ ′ ∈ (λ, f (σ 0 )) and using the fact that f (ϕ 0 u ) → f (σ 0 ) as u → ∞, one obtains exp -

t s f (ϕ (a.) u,s )du ≤ exp - t s f (ϕ 0 u,s )du = exp - t-s 0 f (ϕ 0 u )du ≤ D(b, f, λ ′ )e -λ ′ (t-s) , for some finite constant D(b, f, λ ′ ). Let α(u) := D(b,f,λ ′ ) λ e -(λ ′ -λ)u f ′ (C ā b u)C ϕ , we have M 1 ≤ Ce -λt α(t -s),
and α ∈ L 1 (R + ). Moreover, for A, B ≥ 0, we have |e -A -e -B | ≤ e -min(A,B) |A -B|. So,

M 2 ≤ f (ϕ ā t,s ) exp - t-s 0 f (ϕ 0 u )du t s f (ϕ (a.) u,s ) -f (ϕ a u,s )du ≤ f (C ā b (t -s))D(b, f, λ ′ )e -λ ′ (t-s) f ′ (C ā b (t -s)) t s |ϕ (a.) u,s -ϕ a u,s |du. One has t s |ϕ (a.) u,s -ϕ a u,s |du ≤ C ϕ t s u s |a θ -a|dθdu ≤ CC ϕ t s u s e -λθ dθdu ≤ Ce -λt • C ϕ λ (t-s)e λ(t-s) . Consequently M 2 ≤ Ce -λt β(t -s) with β(u) := D(b, f, λ ′ )e -(λ ′ -λ)u f (C ā b u)f ′ (C ā b u) C ϕ λ ue λu .
It holds that β ∈ L 1 (R + ) and setting η := α + β completes the proof for K(a.) . The same computations give a similar result for H(a.) .

These estimates are sharp enough to give the following result:

Lemma 46. Grant Assumptions 2, 3, 4 and 41. Let η be the function given by Lemma 45.

Denote by 1 the kernel ½ t≥s . Then

1. K(a.) ∈ V λ and || K(a.) || λ,1 ≤ C||η|| 1 . 2. K(a.) * 1 ∈ V λ and || K(a.) * 1|| λ,1 ≤ C||η|| 1 .
The exact same estimates holds for H(a.) and H(a.) * 1.

Proof or equivalently ∆ K * r = K (a.) * r -r a * K (a.) .

We substitute this equality in (46) and finally obtain r = K (a.) + K (a.) * r.

By uniqueness (Lemma 20 with ν = δ 0 ), it follows that r = r (a.) . The end of the proof follows easily.

Remark 48. Let us explain how the formula (44) was derived. The algebra V λ does not have any neutral element (in fact the neutral element would be a Dirac distribution) but assume for the sake of this heuristic that I is a neutral element of the algebra (i.e. k * I = I * k = k ∀k ∈ V λ ). Equation (41) can be rewritten as We now expand this equation -the neutral element I disappears and obtain:

r (a.) -( K(a.) + r a * K(a.) ) * r (a.) = r a + K(a.) + r a * K(a.) .
Using the definition of ∆ K we obtain r (a.) = r a + ∆ K * r (a.) + ∆ K . Solving this equation in terms of ∆ r the resolvent of ∆ K we have r (a.) = r a + ∆ K + ∆ r * (r a + ∆ K ). It gives the desired formula.

We now come back to an arbitrary initial condition ν and prove the main result of this section.

Proposition 49. Grant Assumptions 2, 3 and 4. Let (Y s,ν,(a.) t

) t≥s be the solution to the nonhomogeneous equation [START_REF] Cáceres | Global-in-time classical solutions and qualitative properties for the NNLIF neuron model with synaptic delay[END_REF] driven by current (a t ) t≥0 and with distribution ν at time s. Let r ν (a.) (t, s) = E f (Y s,ν,(a.) t

). Assume (a t ) satisfies Assumption 41 and that the constant C satisfies the inequality (42) for some α ∈ (0, 1). Then it holds that

∀t ≥ s ≥ 0, |r ν (a.) (t, s) -γ(a)| ≤ De -λ(t-s) , with D := 1 + αγ(a) + ||ξ a || λ,1 1 -α ||K ν (a.) || λ,∞ + γ(a)||H ν (a.) || λ,∞ .
Proof. The kernel r ν (a.) solves the Volterra equation r ν (a.) = K ν (a.) + K (a.) * r ν (a.) . By Lemma 21, its solution is r ν (a.) = K ν (a.) + r (a.) * K ν (a.) . Using Proposition 47, we know that r (a.) = γ(a) + ξ (a.) , with ξ (a.) ∈ V λ . Furthermore using that γ(a) * K ν (a.) = γ(a)[1 -H ν (a.) ], we deduce that:

r ν (a.) = γ(a) + K ν (a.) + ξ (a.) * K ν (a.) -γ(a)H ν (a.) .
Using that λ < f (σ 0 ) (Assumption 41) we find

||H ν (a.) || λ,∞ = sup t,s H ν (a.) (t, s)e λ(t-s) < ∞, ||K ν (a.) || λ,∞ = sup t,s K ν (a.) (t, s)e λ(t-s) < ∞. We obtain ∀t ≥ s, |r ν (a.) (t, s) -γ(a)|e λ(t-s) ≤ ||K ν (a.) || λ,∞ + γ(a)||H ν (a.) || λ,∞ + e λ(t-s) t s |ξ (a.) |(t, u)K ν (a.) (u, s)du ≤ ||K ν (a.) || λ,∞ + γ(a)||H ν (a.) || λ,∞ + ||K ν (a.) || λ,∞ t s |ξ (a.) |(t, u)e λ(t-u) du ≤ ||K ν (a.) || λ,∞ + γ(a)||H ν (a.) || λ,∞ + ||K ν (a.) || λ,∞ ||ξ (a.) || λ,1 .
Using the estimate of ||ξ (a.) || λ,1 given by Proposition 47, we deduce the result.

9 Long time behavior for small interactions: proof of Theorem 9

Some uniform estimates

We now turn to the proof of Theorem 9. It is convenient to first extend the results obtained in Section 7: we need uniform estimates in the input current a. In this section, we grant Assumptions 2, 3 and 4.

Lemma 50. Let ā > 0. It holds that inf a∈[0,ā] λ * a > 0.
Proof. We define the function g related to the first zero of H a by

∀a ∈ [0, ā], g(a) := -sup{R(z)| H a (z) = 0, R(z) > -f (σ 0 )}.
By convention, g(a) = f (σ 0 ) if H a is not null on ℜ(z) > -f (σ 0 ). By definition of λ * a and by the results of Section 7 we know that g(a) ∈ (0, λ * a ]. So, to prove the lemma, it suffices to show the following result Claim g is lower semi-continuous, that is

∀a 0 ∈ [0, ā], lim inf a→a0 g(a) ≥ g(a 0 ).
Proof of the claim. Choose a 0 ∈ [0, ā]. We have g(a 0 ) > 0. Fix λ ∈ (0, g(a 0 )). Thanks to Lemma 36, one can find R > 0, such that for all a ∈ [0, ā], for all z with R(z) ∈ [-λ, 0] and

I(z) / ∈ [-R, R], we have H a (z) = 0. Denote U = {z ∈ C, R(z) ∈ [-λ, 0], |I(z)| ≤ R}. By definition of g(a 0 ), we have H a0 = 0 on U and the continuity of z → H a0 (z) yields inf z∈U | H a0 (z)| > 0.
Moreover, (a, z) → Ĥa (z) is continuous on [0, ā] × U , so one can find δ > 0 such that for all |a -a 0 | ≤ δ, z ∈ U , we have | H a (z)| = 0. and so g(a) ≥ λ. We have proved that ∀λ ∈ (0, g(a 0 )), lim inf a→a0 g(a) ≥ λ. It ends the proof.

Proposition 51 (Whole-line Palay-Wiener, an extension). Let ā > 0 and for all a ∈ [0, ā], let

k a ∈ L 1 (R, R). Assume that 1. ∃η ∈ L 1 (R, R + ) s.t. ∀a ∈ [0, ā], ∀0 < ǫ < 1, ∀t ∈ R, |k a (t) -k a (t -ǫ)| ≤ ǫη(t). 2. ∃θ ∈ L 1 (R, R + ) s.t. ∀a ∈ [0, ā], ∀t ∈ R : |k a (t)| ≤ θ(t). 3. ∀a ∈ [0, ā], ∀y ∈ R let k a (iy) = R e -iyt k a (t)dt. We assume that inf a∈[0,ā],y∈R |1 -k a (iy)| > 0.
Then for all a ∈ [0, ā], there exists a function

x a ∈ L 1 (R, R) satisfying the equation x a = k a + k a * x a and sup a∈[0,ā] ||x a || L 1 < ∞.
Proof. We follow the proof of Theorem

[0, ā], k ∞ a (t) := k a -ζ p * k a .
Claim 1 There is an integer p > 0 such that ∀a ∈ [0, ā], ∀|y| ≥ p, we have

||k ∞ a || L 1 ≤ 1/2 and k ∞ a (iy) = k a (iy). Proof of the claim. It is clear that with this choice of ζ, ∀|y| ≥ p, k ∞ a (iy) = k a (iy). Moreover, using R ζ p (s)ds = 1, we have ||k ∞ a || L 1 = R R k a (t)ζ p (s) -k a (t -s)ζ p (s)ds dt ≤ R ζ(u) R |k a (t) -k a (t - u p )|dtdu.
We used the Tonelli-Fubini Theorem (everything is non-negative). Let R > 0 such that

R\[-R,R] ζ(u)du ≤ 1 8||θ|| L 1 . It follows that ||k ∞ a || L 1 ≤ 1/4 + R -R ζ(u) R |k a (t) -k a (t - u p )|dtdu ≤ 1/4 + R -R R | u p |η(t)dt du ≤ 1/4 + R 2 p ||η|| L 1 .
The claim is proved by choosing an integer p ≥ 4R 2 ||η|| L 1 . Along the same idea, we define β(t) := 4ζ(2t) -ζ(t) = 1 πt 2 (cos t -cos 2t). Note that ∀|y| ≤ 1, we have β(iy) = 1. Then for all δ > 0, we set β δ (t) = δβ(δt) and ∀y 0 ∈ R, ∀t ≥ 0, k y0,δ a (t) = R (β δ (t -s) -β δ (t))e iy0(t-s) k a (s)ds. Consequently, for all y with |y -y 0 | ≤ δ we have

x y0 a (iy) = k a (iy)
1 -k a (iy) .

Furthermore, still following [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF], one can find an integer m > 0 such that: ∀a ∈ [0, ā], ∀j ∈ Z, |j| ≤ mp, there exists a function x We define ψ j (t) = 1 m e -ijt/m ζ(t/m). We have ||ψ j || L 1 = 1. Its Fourier transform is given by ψ j (iy) = 0 if |y -j/m| > 1/m 1 -m|y -j/m| otherwise.

We set

x a = |j|≤mp ψ j * (x j/m a -x ∞ a ) + x ∞ a .
It is clear that x a ∈ L 1 (R) and that sup a∈[0,ā]

||x a || L 1 ≤ mp 2 ǫ ||θ|| L 1 + 2||θ|| L 1 + 2||θ|| L 1 < ∞.
With this choice of ψ j , ∀y ∈ R, x a (iy) = ka(iy)

1-ka(iy)

and by uniqueness of the Fourier transform, we conclude that x a is the solution of x a = k a + k a * x a .

As a consequence of the previous result, we have Corollary 52. Let ā > 0, define λ * = inf a∈[0,ā] λ * a (λ * > 0 by Lemma 50). Let 0 < λ < λ * and consider r a the solution of the Volterra equation r a = K a + K a * r a . By Proposition 37, it holds that r a = γ(a) + ξ a for some ξ a ∈ L λ . Then we have sup a∈[0,ā] ||ξ a || λ,1 < ∞.

Proof. Recall (see proof of Proposition 37) that ξ a (t) = e -λt ξ a,-λ (t) and so ||ξ a || λ,1 = ||ξ a,-λ || L 1 . We now prove that Proposition 51 applies to ξ a,-λ . Indeed, it solves ξ a,-λ = K a,-λ + K a,-λ * ξ a,-λ , with K a,-λ (t) := e λt K a (t)½ {t≥0} . It remains to show that K a,-λ fulfills the assumptions of Proposition 51. 1. We use sup a∈[0,ā] K a (t) ≤ f (ϕ ā t (0))H 0 (t) and sup a∈[0,ā] |ϕ a t (0) -ϕ a t-ǫ (0)| ≤ ǫC ā b . 2. For all t ≥ 0 and a ∈ [0, ā], we have K a,-λ (t) ≤ θ(t) := e λt f (C ā t )H 0 (t)½ R+ (t) ∈ L 1 (R).

3. We have K a,-λ (iy) = K a (-λ + iy). We conclude by Lemmas 50 and 36.

Proof of Theorem 9

We are now ready to give the proof of the main theorem.

• Step 1 Recall that equation [START_REF] Pakdaman | Relaxation and self-sustained oscillations in the time elapsed neuron network model[END_REF] gives

d dt E f (X t ) ≤ 1 2 [r(J) 2 -E 2 f (X t )],
where (X t ) t≥0 is the solution of the non-linear equation ( 2) and the function J → r(J) is non-decreasing. Using Proposition 23 with κ := J r(J) + 1, there is a non-decreasing function J → ā(J) such that:

∀J, s ≥ 0, ∀(a t ) t≥s ∈ C([s, ∞), R + ), [sup t≥s a t ≤ ā(J) and Jν(f ) ≤ ā(J)] =⇒ sup t≥s Jr ν (a.) (t, s) ≤ ā(J).

Moreover, it holds that ∀J ≥ 0, J r(J) < ā(J).

• where J m > 0 is defined in Proposition 8. Lemma 50 gives λ * > 0. We now fix λ such that 0 < λ < λ * .

• Step 3 -Using Corollary 52, we know that the solution of the Volterra equation r a = K a + K a * r a is r a = γ(a) + ξ a with ξ a ∈ L λ and that:

ξ ∞ (J) := sup a∈[0,ā(J)] ||ξ a || λ,1 < ∞.
It is clear that J → ξ ∞ (J) is non-decreasing (as J → ā(J) is).

-One can find a function k ∞ : R + × R + → R + , non-decreasing with respect to its two parameters, such that for all (a t ) ∈ C(R + , R + ) we have:

sup t≥0 a t ≤ ā =⇒ ||K ν (a.) || λ,∞ ≤ k ∞ (ν(f ), ā) < ∞.
Moreover, one can find a constant h ∞ (only depending on λ, b and f ) such that for all (a t ) ∈ C(R + , R + ), we have

||H ν (a.) || λ,∞ ≤ h ∞ .
These two points follow from λ < f (σ 0 ), Assumption 2, Remarks 12(2) and 13.

-The function η ā of Lemma 45 satisfies ||η ā|| 1 < ∞, ā → ||η ā|| 1 is non-decreasing, and consequently the function J → ||η ā(J) || 1 is non-decreasing.

-Finally the normalization γ is a non-decreasing function of a (see (31)) and it follows that ∀a ∈ [0, ā(J)], γ(a) ≤ γ(ā(J)).

• Step 4 Let ν be a probability measure such that ν(f ) ≤ r(J m ) + 1. Remind that for all J ∈ (0, J m ) the equation aγ -1 (a) = J has a unique solution a * (J) ∈ [0, ā(J m )]. We now apply Proposition 49 with α = 1/2. Define: C(J) := 1 2||η ā(J) || 1 (1 + ξ ∞ (J) + γ(ā(J))) D(J) := 2(1 + γ(ā(J)) + ξ ∞ (J))k ∞ (r(J m ) + 1, ā(J)) + γ(ā(J))h ∞ .

From Step 3, it is clear that the functions J → 1 C(J) and J → D(J) are non-decreasing. Consequently, we can find a constant J * ∈ (0, J m ) such that ∀J ∈ [0, J * ], JD(J) C(J) ≤ 1.

Proposition 49 tells us that for every 0 ≤ J ≤ J * , given any (a t ) t≥0 ∈ C(R + , R + ) with sup t≥0 a t ≤ ā(J) and such that ∀t ≥ 0, |a t -a * (J)| ≤ C(J)e -λt , it holds ∀t ≥ 0, |Jr ν (a.) (t, 0) -a * (J)| ≤ C(J)e -λt .

• Step 5 Let now J ∈ (0, J * ] be fixed (the case J = 0 is already treated by Proposition 30). We assume the initial condition ν of (2) satisfies Jν(f ) ≤ ā(J) and that ν(f ) ≤ r(J m ) + 1 (we shall come back to the general case in Step 6). We define recursively a n ∈ C(R + , R + ) by ∀t ≥ 0, a 0 (t) := a * (J) and ∀n ≥ 0, a n+1 (t) := Jr ν (a n .) (t, 0). By Step 4 and by induction, it holds that: ∀n ≥ 0, ∀t ≥ 0, |a n (t) -a * (J)| ≤ C(J)e -λt .

We deduce that:

∀t ≥ 0, | E f (X t ) -γ(a * (J))| ≤ | E f (X t ) -r ν (a n .) (t, 0)| + 1 J |a n+1 (t) -a * (J)| ≤ 1 J |J E f (X t ) -a n+1 (t)| + C(J) J e -λt .
The Picard iteration studied in Part 4.4 shows that ∀t ≥ 0, lim n→∞ |J E f (X t ) -a n (t)| = 0.

We have proved that ∀t ≥ 0, | E f (X t ) -γ(a * (J))| ≤ C(J) J e -λt .

• Step 6 We now prove that there exists s ≥ 0 such that E f (X s ) ≤ min( ā(J) J , r(J m ) + 1). By Step 1, we have lim sup E f (X t ) ≤ r(J). Since r(J) < a(J)/J and since r(J) ≤ r (J m ), the conclusion follows. Consequently, Step 5 can be applied to the process (X t ) t≥s starting with ν = L(X s ). This proves the convergence of the jump rate.

The convergence of the law of X t to the invariant measure then follows from Proposition 29. This ends the proof of Theorem 9.

Remark 53. There is some freedom in the above construction of the constants λ and J * . We can choose any λ in [0, λ * ) and the value of J * depends both on λ and on a parameter α ∈ (0, 1), here chosen to be equal to 1/2 (see Step 4). We may optimize this construction to get either J * or λ as large as possible.

4 .

 4 The function (t, s) → ϕ (a.)

∆ 3 h

 3 h = 0. Combining the three results ends the proof.

  It is also useful to introduce the following Banach space Definition 33. For any λ ∈ R, let L λ = {f ∈ B(R + , R) : ||f || λ,1 < ∞} the space of Borelmeasurable functions from R + to R, equipped with the norm ||f || λ,1 = R+ |f (s)|e λs ds.

t≥0 t 0

 0 Ce -λt ||η|| 1 e λ(t-s) ds = C||η|| 1 .

(

  I -K (a.) ) * (I + r (a.) ) = I. (47) In particular (taking (a t ) ≡ a), we have (I -K a ) * (I +r a ) = (I +r a ) * (I -K a ) = I. Furthermore, I -K (a.) = (I -K a ) * (I -(I + r a ) * K(a.) ), with K(a.) = K (a.) -K a ∈ V λ . Equation (47) becomes (I-K a ) * (I-(I+r a ) * K(a.) ) * (I+r (a.) ) = I. We multiply by I +r a on the left of each side, and we get (I -(I +r a ) * K(a.) ) * (I +r (a.) ) = I +r a .

Claim 2 || L 1 ≤ ǫ 2 . 1 1 -.

 21211 Given ǫ > 0, one can find a constant δ > 0 such that:∀y 0 ∈ R, ∀a ∈ [0, ā], ∀|y -y 0 | ≤ δ, k a (iy) = k a (iy 0 ) + k y0,δ a (iy) and ||k y0,δ a Proof of the claim. By definition of k y0,δ a it holds that ∀y ∈ R, k y0,δ a (iy) = β δ (i(y -y 0 ))( k a (iy) -k a (iy 0 )). Moreover, β δ (iy) = 1 if |y| ≤ δ and consequently the first point of the claim is satisfied. Furthermore, R |k y0,δ a (t)|dt ≤ R |k a (s)| R |β(t -δs) -β(t)|dtds ≤ R θ(s) R |β(t -δs) -β(t)|dtds.The right hand side does not depend on y 0 nor a and goes to zero as δ goes to zero. This proves the second point of the claim.It follows from Claim 1 that ∀a ∈ [0, ā], the equationx ∞ a = k a + k ∞ a * x ∞ a has a unique solution x ∞ a ∈ L 1 (R) with ||x ∞ a || L 1 ≤ 2||θ|| L 1 . Moreover, we have ∀a ∈ [0, ā], ∀|y| ≥ p, x ∞ a (iy) = k a (iy) 1 -k a (iy) .Similarly, we define ǫ := inf a∈[0,ā],y∈R |1 -k a (iy)| > 0 and apply the second claim. Giveny 0 ∈ R and a ∈ [0, ā], let A y0 a = ka(iy0) . We have 1 -k a (iy) = 1 -k a (iy 0 ) -k y0,Using ||A y0 a k y0,δ a || L 1 ≤ 1/2,we can define the solution of x y0 a = A y0 a k a + A y0 a k y0,δ a * x y0 a and we have ||x y0 a || L 1 ≤ 2 ǫ ||θ|| L 1 .

|| L 1

 1 ≤ 2 ǫ ||θ|| L 1 such that ∀|y -j/m| ≤ 1/m, x j/m a (iy) = k a (iy)1 -k a (iy) .

Step 2

 2 We define λ* := inf a∈[0,ā(Jm)] λ * a ,

  deduce the existence and uniqueness of a solution t → Y

	s,ν,(a.) t	≤ ϕ (a.) t,s (Y

s,ν,(a.) t of (5) on [0, ζ[ such that lim sup t→ζ Y s,ν,(a.) t = ∞ on the event {ζ < ∞}. But any solution of (5) satisfies for all t ≥ s, Y s,ν,(a.) s ) < ∞ a.s. and so it holds that ζ = +∞ a.s. Lemma 15. Grant Assumptions 2, 3 and 4. Let (Y s,ν,(a.) t

4.3 The jump rate is uniformly bounded Lemma 22.

  Grant Assumptions 2, 3 and 4 . Let s ≥ 0 and (a t ) t≥s ∈ C([s, ∞), R + ). Let Y

	s,ν,(a.) t	be the solution of equation (5), starting from L(Y s s,ν,(a.)

  Definition 42. A function K : (R + ) 2 → R is said to be a Volterra Kernel with weight λ ∈ R if: K is Borel measurable, ∀s > t : K(t, s) = 0 a.e. and ||K|| λ,1 < ∞ withWe define V λ the set of Volterra kernels with weight λ. We also define for K ∈ V λ : The space (V λ , || • || λ,1 ) is a Banach algebra. Furthermore, for all a, b ∈ V λ , ||a * b|| λ,1 ≤ ||a|| λ,1 ||b|| λ,1 .

	t≥0 ||K|| λ,1 := ess sup	R+	|K(t, s)|e λ(t-s) ds.
	||K|| λ,∞ = ess sup t,s≥0	|K(t, s)e λ(t-s) | ∈ R + ∪ {+∞}.
	Proposition 43. Proposition 43 is proved in [20], Theorem 2.4 and Proposition 2.7 (i) of Chapter 9.

Lemma 44 (Connection with the time homogeneous setting). Let g ∈ L λ . We define

∀t, s ∈ R + , g(t, s) := g(t -s)½ t≥s .

Then g ∈ V λ and ||g|| λ,1 = ||g|| λ,1 .

  4.3 in [20, Chap. 2] and emphasis on the differences. Let ζ(t) := 1 πt 2 (1 -cos(t)) be the Fejer kernel; its Fourier transform is ζ(iy) = (1 -|y|)½ {|y|≤1} . For any p ≥ 1, set ζ p (t) := pζ(pt) and ∀a ∈

Acknowledgements

The authors warmly thank one anonymous referee for his/her deep reading of a previous version of the paper. His/Her comments have strongly helped us to improve the presentation. This project/research has received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreement No. 785907 (Human Brain Project SGA2).

Proposition 47. Grant Assumptions 2, 3, 4. Assume (a t ) t≥0 satisfies Assumption 41 and that the constant C is small enough:

Define ∆ K := K(a.) + ξ a * K(a.) -γ(a) H(a.) and let ∆ r be the solution of the Volterra equation

Consequently, we have r (a.) = γ(a) + ξ (a.) with

Proof. By Lemma 46, we have

with ∆ K * 1 = ( K(a.) * 1) + ξ a * ( K(a.) * 1) + γ(a)( H(a.) * 1). It follows from Lemma 46 that

It remains to check that r (a.) given by ( 44) is indeed the solution of (41). Let r := r a +∆ r +∆ r * r a . One has

Using Proposition 37 and (23), we have ∆ K = K(a.) + r a * K(a.) . Equation (46) gives r -( K(a.) + r a * K(a.) ) * r = r a + K(a.) + r a * K(a.) .

We multiply this equation by K a on the left and obtain, using that K a * r a = r a * K a = r a -K a :

K a * r -r a * K(a.) * r = r a -K a + r a * K(a.) .

The relation K(a.) = K (a.) -K a yields K a * r -r a * K(a.) * r = r a * K (a.) ,