José A Cañizo
email: canizo@ugr.es

Pierre Gabriel
email: pierre.gabriel@uvsq.fr

Havva Yoldaş
email: yoldas@math.univ-lyon1.fr

Spectral gap for the growth-fragmentation equation via Harris's Theorem

We study the long-time behaviour of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalise those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the nonconservativeness of the equation is overcome by performing an h-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.

Introduction and main result

The growth-fragmentation equation is a linear, partial integro-differential equation which is commonly used in structured population dynamics for modelling various phenomena including the time evolution of cell populations in biology such as in [START_REF] Adimy | Asymptotic behavior of a singular transport equation modelling cell division[END_REF][START_REF] Bell | Cell growth and division. III. Conditions for balanced exponential growth in a mathematical model[END_REF][START_REF] Bell | Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF][START_REF] Brikci | Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle[END_REF][START_REF] Diekmann | On the stability of the cell size distribution[END_REF][START_REF] Hall | A functional-differential equation arising in modelling of cell growth[END_REF][START_REF] Metz | The Dynamics of Physiologically Structured Populations[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF][START_REF] Rudnicki | Piecewise Deterministic Processes in Biological Models[END_REF], single species populations [START_REF] Sinko | A model for populations reproducing by fission[END_REF], or carbon content in a forest [START_REF] Broda | Convergence rates for semistochastic processes[END_REF]; some aggregation and growth phenomena in physics or biophysics as in [START_REF] Banasiak | Analytic Methods for Coagulation-Fragmentation Models[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Engler | Analysis of a model for the dynamics of prions[END_REF][START_REF] Gabriel | Global stability for the prion equation with general incidence[END_REF][START_REF] Leis | Existence of global classical and weak solutions to a prion equation with polymer joining[END_REF][START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF]; neuroscience in [START_REF] Cañizo | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF] and even TCP/IP communication protocols such as in [START_REF] Baccelli | A mean-field model for multiple TCP connections through a buffer implementing red[END_REF][START_REF] Bardet | Total variation estimates for the TCP process[END_REF][START_REF] Chafaï | On the long time behavior of the TCP window size process[END_REF]. The general form of the growthfragmentation equation is given by: ∂ ∂t n(t, x) + ∂ ∂x (g(x)n(t, x)) + B(x)n(t, x) = +∞ x κ(y, x)n(t, y) dy, t, x > 0, n(t, 0) = 0, t ≥ 0, n(0, x) = n 0 (x), x > 0,

where n(t, x) represents the population density of individuals structured by a variable x > 0 at a time t ≥ 0. The structuring variable x could be age, size, length, weight, DNA content, biochemical composition etc. depending on the modelling context. Here we refer to it as 'size' for simplicity. Equation ( 1) is coupled with an initial condition n 0 (x) at time t = 0 and a Dirichlet boundary condition which models the fact that no individuals are newly created at size 0. The function g is the growth rate and B is the total division/fragmentation rate of individuals of size x ≥ 0. The fragmentation kernel κ(y, x) is the rate at which individuals of size x are obtained as the result of a fragmentation event of an individual of size y. When fixing x, κ(x, •) is a nonnegative measure on (0, x]. The total fragmentation rate B is always obtained as

B(x) = y 0 y x κ(x, y) dy, x > 0. (2)
Important particular cases are κ(x, y) = B(x) 2 x δ {y= x 2 } , which corresponds to the mitosis process, suitable for modelling of biological cells, where individuals can only break into two equal fragments; and κ(x, y) = B(x) 2 x , which is the case with uniform fragment distribution, where fragmentation gives fragments of any size less than the original one with equal probability. This case is used for example in modelling the dynamics of polymer chains, as in [START_REF] Engler | Analysis of a model for the dynamics of prions[END_REF]. Two opposing dynamics, growth and fragmentation, are balanced through Equation [START_REF] Adimy | Asymptotic behavior of a singular transport equation modelling cell division[END_REF]. The growth term tends to increase the average size of the population and the fragmentation term increases the total number of individuals but breaks the population into smaller sizes. If the growth rate g(x) vanishes, then only fragmentation takes place and the equation is known as the pure fragmentation equation. Similarly when B and κ are both 0, Equation (1) is the pure growth equation.

We are concerned here with the mathematical theory of this equation, and more precisely with its long-time behaviour as t → +∞. Under suitable conditions on the coefficients κ and g, the typical behaviour is that the total population tends to grow exponentially at a rate e λt , for some λ > 0, and the normalised population distribution tends to approach a universal profile for large times, independently of the initial condition. This has been investigated in a large amount of previous works, of which we give a short summary. The first mathematical study of this type of equation was done in [START_REF] Diekmann | On the stability of the cell size distribution[END_REF] for the mitosis case, in a work inspired by some biophysical papers [START_REF] Bell | Cell growth and division. III. Conditions for balanced exponential growth in a mathematical model[END_REF][START_REF] Bell | Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF][START_REF] Sinko | A model for populations reproducing by fission[END_REF]. In [START_REF] Diekmann | On the stability of the cell size distribution[END_REF], the authors considered the mitosis kernel with the size variable in a bounded interval and proved exponential growth at a rate λ, and exponentially fast approach to the universal profile. In [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF], the authors considered the size variable in (0, +∞) and introduced the general relative entropy method for several linear PDEs including the growth-fragmentation equation. They proved relaxation to equilibrium in L p spaces without an explicit rate. Following [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF] and [START_REF] Laurençot | Exponential decay for the growth-fragmentation/celldivision equations[END_REF], providing an explicit rate of convergence to a universal profile under reasonable assumptions became a topic of research for many other works. New functional inequalities were proved in [START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF] in order to obtain explicit rates of convergence, see also [START_REF] Gabriel | Exponential relaxation to self-similarity for the superquadratic fragmentation equation[END_REF]. Some authors used a semigroup approach [START_REF] Arino | Some spectral properties for the asymptotic behavior of semigroups connected to population dynamics[END_REF][START_REF] Banasiak | Perturbations of Positive Semigroups with Applications[END_REF][START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF][START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF][START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF][START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF] or a probabilistic approach [START_REF] Bardet | Total variation estimates for the TCP process[END_REF][START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF][START_REF] Bertoin | On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors[END_REF][START_REF] Bertoin | Probabilistic aspects of critical growth-fragmentation equations[END_REF][START_REF] Bertoin | A probabilistic approach to spectral analysis of growthfragmentation equations[END_REF][START_REF] Bertoin | The strong Malthusian behavior of growth-fragmentation processes[END_REF][START_REF] Bouguet | A probabilistic look at conservative growth-fragmentation equations[END_REF][START_REF] Broda | Convergence rates for semistochastic processes[END_REF][START_REF] Cavalli | On a family of critical growth-fragmentation semigroups and refracted Lévy processes[END_REF][START_REF] Cavalli | A probabilistic view on the long-time behaviour of growthfragmentation semigroups with bounded fragmentation rates[END_REF][START_REF] Chafaï | On the long time behavior of the TCP window size process[END_REF][START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF], and some authors provided explicit solutions as in [START_REF] Zaidi | Solutions to an advanced functional partial differential equation of the pantograph type[END_REF]. In this paper we are able to give more general results regarding the speed of convergence to equilibrium: we obtain constructive results which cover a wide range of bounded and unbounded fragmentation rates, and which apply both in mitosis and uniform fragmentation situations.

When the equal mitosis kernel is considered, there is a special case with a linear growth rate where the solutions exhibit oscillatory behaviour in long time. This property was first proved mathematically in [START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF] when the equation is posed in a compact set. Recently, this result was extended to (0, +∞) by the general relative entropy argument in suitable weighted L 2 or measure spaces in [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF][START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF] and by means of Mellin transform in L 1 space by [START_REF] Van Brunt | On a cell division equation with a linear growth rate[END_REF].

An important tool when studying the asymptotic behaviour of (1) is the Perron eigenvalue problem: finding a positive eigenfunction for the operator which defines the equation, associated to a simple, real eigenvalue which is also equal to the spectral radius; see [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF][START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF] for general existence results. In [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF], the authors gave some estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first order behaviour close to 0 and +∞. Then they proved a spectral gap result by means of entropy-entropy dissipation inequalities, with tools similar to those of [START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF]. They assumed that the growth and the fragmentation coefficients behave asymptotically like power laws.

In this paper we use a probabilistic approach, namely Harris's theorem, for showing the spectral gap property. We give a novel approach based on estimating solutions to the PDE, and obtain results which can be applied to general growth and fragmentation rates including mitosis and uniform fragmentation cases. Detailed hypotheses and results are given later in this introduction. The method is also completely constructive and gives explicit estimates. However, in some cases these estimates depend on estimates on the first dual eigenfunction, which may be not easy to obtain, but constitute a separate question. After stating our results we also give a brief comparison to other spectral gap results in the literature.

Applications of this type of argument into biological and kinetic models which can be described as Markov processes is becoming a subject of many works recently, and has been extended to models which are not Markov processes but share similar properties. The predecessor of Harris's theorem, namely Doeblin's argument is used in [START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF] for proving exponential relaxation of solutions to the equilibrium for the conservative renewal equation. In [START_REF] Cañizo | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF] and [START_REF] Dumont | The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states[END_REF], the authors study population models which describe the dynamics of interacting neurons, structured by elapsed-time in [START_REF] Cañizo | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF] or by voltage in [START_REF] Dumont | The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states[END_REF], and existence of a spectral gap property in the 'no-connectivity' setting is proved by Doeblin's Theorem. Moreover, there are some recent works for the extension of this method into the non-conservative setting. In [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF], the authors consider several types of linear PDEs including a growth-diffusion model with time-space varying environment and some renewal equations with time-fluctuating (e.g. periodic) coefficients. They provide quantitative estimates in total variation distance for the associated non-conservative and non-homogeneous semigroups by means of generalized Doeblin's conditions. The full Harris's theorem is used in [START_REF] Bouguet | A probabilistic look at conservative growth-fragmentation equations[END_REF][START_REF] Broda | Convergence rates for semistochastic processes[END_REF] for deriving exponential convergence to the equilibrium in the conservative form of the growth-fragmentation equation. In the present work, we are interested in the long time behaviour of the more challenging non-conservative case, namely when no quantity is preserved along time. Our method is in the spirit of [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF], where a non-conservative version of Harris's Theorem is proposed and applied to the growth-fragmentation equation with constant growth rate g and increasing total division rate B, see also [START_REF] Cloez | On an irreducibility type condition for the ergodicity of nonconservative semigroups[END_REF] for an application to a mutation-selection model which is similar to growth-fragmentation. The difference here is that we first build a solution to the dual Perron eigenproblem by using Krein-Rutman's theorem and a maximum principle. Then we take advantage of the dual eigenfunction to perform a so-called (Doob) h-transform [START_REF] Doob | Conditional Brownian motion and the boundary limits of harmonic functions[END_REF], similarly as in [START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF][START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF], in order to apply Harris's theorem. It allows us to consider very general growth and fragmentation rates. The drawback is that the spectral gap is given explicitly in terms of the dual eigenfunction, for which quantitative estimates are in general hard to obtain. However, for certain specific coefficients that are worth of interest, the dual eigenfunction is known explicitly. It is the case of the so-called self-similar fragmentation equation, widely studied in the literature, for which we provide new quantitative estimates on the spectral gap.

Let us now precise the functional analytic setting of our work and what we mean by solutions to Equation [START_REF] Adimy | Asymptotic behavior of a singular transport equation modelling cell division[END_REF]. We are interested in measure solutions to this equation, which is a relevant notion in population dynamics, see e.g. [START_REF] Cañizo | Measure solutions for some models in population dynamics[END_REF][START_REF] Gabriel | Measure solutions to the conservative renewal equation[END_REF]. We say that a family (n(t, •)) t≥0 of positive measures on (0, +∞) is a solution to Equation [START_REF] Adimy | Asymptotic behavior of a singular transport equation modelling cell division[END_REF]

if for all f ∈ C 1 c ([0, +∞)) the function t → n(t, •), f is continuously differentiable and for all t ≥ 0 d dt n(t, •), f = n(t, •), L * [f ] , (3)
where

L * [f ](x) := g(x) ∂ ∂x f (x) + x 0 κ(x, y)f (y) dy -B(x)f (x)
is the dual operator of the growth-fragmentation operator

L[n](x) := - ∂ ∂x (g(x)n(x)) -B(x)n(x) + +∞ x κ(y, x)n(y) dy,
which appears in Equation [START_REF] Adimy | Asymptotic behavior of a singular transport equation modelling cell division[END_REF]. We refer to [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF][START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF] for (the method of) proof that Equation ( 1) is well-posed in the set of positive (or signed Radon) measures µ such that the weighted total variation norm

µ V = +∞ 0 V (x)|µ|(dx) (4)
is finite, when V (x) = x k + x K with k ≤ 0 and K > 1.

The Perron eigenvalue problem consists of finding suitable eigenelements (λ, N, φ) with λ > 0 and N, φ : (0, +∞) → [0, +∞), N, φ ≡ 0, satisfying the following:

L[N ] = λN, (gN )(0) = 0, (5)
L * [φ] = λφ. (6)
If such a triple exists then λ is actually the dominant eigenvalue of Equation (1), and the solution is expected to converge to a universal profile whose shape is given by the eigenfunction N (x). The convergence rate is given by the gap between the dominant eigenvalue λ > 0 and the rest of the spectrum. If we scale the equation by defining m(t, x) := n(t, x)e -λt we obtain:

∂ ∂t m(t, x) + ∂ ∂x (g(x)m(t, x)) + (B(x) + λ)m(t, x) = +∞ x κ(y, x)m(t, y) dy, t, x ≥ 0, m(t, 0) = 0, t > 0, m(0, x) = n 0 (x), x > 0. (7)
We remark that N (x) is the stationary solution of Equation [START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF] and φ(x) provides a conservation law for [START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF] since

d dt +∞ 0 φ(x)m(t, x) dx = 0.
Since the existence and uniqueness of the eigenelements provide useful information about the long time behaviour of the growth-fragmentation equation [START_REF] Adimy | Asymptotic behavior of a singular transport equation modelling cell division[END_REF], it has been a popular topic of research. We refer to [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF] for a general recent result. From now on we consider Equation ( 7) instead of Equation (1) since it is more convenient to study the long-time behaviour of the former and we can easily recover the nature of the latter.

We now list all the assumptions we need throughout the paper.

As we will explain in Section 3, Harris's method relies on a local Doeblin's minorisation condition. The computations for checking this condition strongly depend on the fragmentation kernel. In [START_REF] Cañizo | Asymptotic behaviour of neuron population models structured by elapsed-time[END_REF] a global Doeblin condition is proved (for a similar equation) for kernels κ which satisfy, for some , η, x * > 0, the condition that κ(x, y) ≥ for all x ∈ [0, η] and y ≥ x * . Here we rather consider kernels that are of self-similar form, which is commonly assumed in the literature about spectral gaps for the growth-fragmentation equation [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF][START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF][START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF][START_REF] Cavalli | A probabilistic view on the long-time behaviour of growthfragmentation semigroups with bounded fragmentation rates[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF] and includes the classical kernels appearing in applications (in particular equal or unequal mitosis and uniform fragment distribution, see below).

Hypothesis 1.1. We assume that κ(x, y), the fragmentation kernel, is of the self-similar form such that κ(x, y) = 1

x p y x B(x), for y > x > 0,
where p, the "fragment distribution", is a nonnegative measure on (0, 1] such that zp(z) is a probability measure; that is,

(0,1]
zp(z) dz = 1.

Remark 1.1. It is useful to define p k , for k ∈ R, as the k-th moment of p:

p k := 1 0 z k p(z) dz.
With this notation, Hypothesis 1.1 ensures that p 1 = 1, so the relation (2) is guaranteed.

Our next hypothesis states that we consider only the two extreme cases of the fragment distribution, namely the very singular equal mitosis case and the very smooth uniform fragment distribution. One can find conditions for our methods to work in intermediate cases, but we have preferred to give simple proofs that show both singular and smooth cases can be treated: Hypothesis 1.2. We assume that the fragment distribution p is either the one corresponding to the equal mitosis:

p( dz) = 2δ 1 2 ( dz) (8)
or the uniform fragment distribution:

p( dz) = 2 dz. ( 9
)
Remark 1.2. We restrict to these two particular fragmentation kernels because they naturally appear in the modelling of natural phenomena. They are also good representatives of two opposite mathematical situations: a very regular, strictly positive case and a singular case which is positive only at z = 1/2. However, the results which we prove to be valid for the uniform kernel can be readily extended to self-similar kernels with p satisfying p(z) ≥ c > 0 for all z in some interval (z 1 , z 2 ) ⊆ (0, 1)

and either

p 0 < +∞ if 1 0 1 g(x) dx < +∞, or ∃k < 0 with p k < +∞ if 1 0 1 g(x) dx = +∞.
In the particular case of the linear growth rate, g(x) = x, it is enough to assume that ∃k < 1 with p k < +∞.

Notice that under condition [START_REF] Bardet | Total variation estimates for the TCP process[END_REF], similarly as for ( 8) and ( 9), the function k → p k is strictly decreasing on the interval where it takes finite values. (The only case in which p k is not strictly decreasing is that of p(z) concentrated at z = 1, which actually means no fragmentation at all is happening.)

In the case of constant growth rate, a more general condition than [START_REF] Bardet | Total variation estimates for the TCP process[END_REF] is assumed in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] that covers the unequal mitosis kernels p( dz) = δ α ( dz) + δ 1-α ( dz) with 0 < α < 1. In our proofs we can also consider this generalisation with straightforward modifications when the growth rate g satisfies forthcoming Hypothesis 1.4.

Regarding non self-similar kernels, there are results of exponential convergence to the stationary distribution in the literature, but only for bounded fragmentation rates; see [START_REF] Laurençot | Exponential decay for the growth-fragmentation/celldivision equations[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF] for PDE-based arguments and [START_REF] Bertoin | On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors[END_REF][START_REF] Bertoin | A probabilistic approach to spectral analysis of growthfragmentation equations[END_REF][START_REF] Cavalli | A probabilistic view on the long-time behaviour of growthfragmentation semigroups with bounded fragmentation rates[END_REF] for a probabilistic point of view. We also point out that an optimal condition on the fragment distribution is given in [START_REF] Bertoin | On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors[END_REF] for a spectral gap to exist (for bounded fragmentation rates).

Next we have a general assumption on the growth rate g and the total fragmentation rate B: Hypothesis 1.3. We assume that g : (0, +∞) → (0, +∞) is a locally Lipschitz function such that g(x) = O(x) as x → +∞ and g(x) = O(x -ξ ) as x → 0 for some ξ ≥ 0. The total fragmentation rate B : [0, +∞) → [0, +∞) is a continuous function and the following holds

1 0 B(x) g(x) dx < +∞, xB(x) g(x) -→ x→0 0, xB(x) g(x) -→ x→+∞ +∞. ( 11
)
This assumption is very mild, and is always present in the previous works to ensure the existence of an equilibrium and a dual eigenfunction. If B behaves like a power of exponent b and g behaves like a power of exponent a, conditions [START_REF] Bell | Cell growth and division. III. Conditions for balanced exponential growth in a mathematical model[END_REF] are equivalent to the more familiar b-a+1 > 0. The condition g = O(x) for large x ensures that the characteristics corresponding to the growth part are defined for all times (i.e., clusters do not grow to infinite size in finite time). A stronger assumption which is implicit in Hypothesis 1.3 is that B is bounded above on intervals of the form [0, R] (since it is continuous there), so we do not allow fragmentation rates B which blow up at 0. This is used in the proof of Lemma 5.1.

A consequence of Hypothesis 1.3, later we will need the following:

There exists t B > 0 such that B is bounded below by a positive quantity on any interval of the form

[t B , θ] with θ > t B . ( 12
)
One sees this from the last limit in [START_REF] Bell | Cell growth and division. III. Conditions for balanced exponential growth in a mathematical model[END_REF], which implies that for large enough t B we have

B(x) ≥ g(x)
x .

This easily implies [START_REF] Bell | Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF], since g(x)/x is continuous and strictly positive, so bounded below by some positive quantity on any compact interval.

Our last assumption gives a stronger requirement on the growth rate g when the mitosis kernel is considered. In this case, some additional requirement is necessary, since when the linear growth rate with equal mitosis is considered, it is known that there is no spectral gap [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF][START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF][START_REF] Van Brunt | On a cell division equation with a linear growth rate[END_REF]. We point out that the sharp assumption of "there exists a point x > 0 with g(2x) = 2g(x)" is enough to show convergence to the profile N , without a rate and only in particular cases, as proved in [START_REF] Rudnicki | Piecewise Deterministic Processes in Biological Models[END_REF]Section 6.3.3]. Our assumption is stronger than this, but also leads to a stronger result: Hypothesis 1.4. When p is the equal mitosis kernel (8), we assume that the growth rate g satisfies ωg(x) < g(ωx)

for all x > 0 and ω ∈ (0, 1),

H(z) := z 0 1 g(x)
dx < +∞ for all z > 0,

and also H -1 (the inverse of H) does not grow too fast, in the sense that for all r > 0 we have

lim z→+∞ H -1 (z + r) H -1 (z) = 1. (13)
If we consider just powers, examples of growth and fragmentation rates which satisfy all of the above are

B(x) = x b , g(x) = x a
with:

• any b ≥ 0, -∞ < a ≤ 1 in the uniform fragment distribution case, excluding the case (b, a) = (0, 1).

• any b ≥ 0, -∞ < a < 1 in the mitosis case.

Under Hypothesis 1.1, the rescaled growth-fragmentation equation ( 7) takes the form: Our main result is given by the following theorem:

∂ ∂t m(t, x) + ∂ ∂x (g(x)m(t, x)) + c(x)m(t, x) = A(t, x), t, x ≥ 0, m(t, 0) = 0, t > 0, m(0, x) = n 0 (x), x > 0. (14)
Theorem 1.3. Assume that Hypotheses 1.1, 1.2, 1.3, and 1.4 are satisifed. Then there exists a solution (λ, N, φ) to the Perron eigenvalue problem (5)-( 6) with the normalization N = φN = 1, λ > 0, and there exist C, ρ > 0 such that the solution n = n(t, x) ≡ n t (x) to Equation (1) with initial data given by a nonnegative finite measure n 0 with n 0 V < +∞ satisfies

e -λt n t - φn 0 N V ≤ Ce -ρt n 0 - φn 0 N V for all t ≥ 0, (15)
where the weight V of the total variation norm • V defined in (4) is given by

V (x) = 1 + x K , 1 + ξ < K if 1 0 1 g(x)
dx < +∞,

V (x) = x k + x K , -1 < k < 0, 1 + ξ < K if 1 0 1 g(x) dx = +∞.
In the specific case of g(x) = x, the weight V (x) can be taken to be

V (x) = x k + x K , -1 < k < 1 < K.
It is worth noticing that we obtain a spectral gap in spaces with essentially optimal weights. Indeed it was proved in [START_REF] Bernard | Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate[END_REF] that there is no spectral gap in weighted L 1 space with the dual eigenfunction φ when B is bounded (see the estimates in Theorem 2.1 below).

To the best of our knowledge, even the existence of the Perron eigenelements in such generality is new (allowing a total fragmentation rate with any growth at infinity, and with no required connectivity condition on its support), and hence so is the existence of a spectral gap. However, since our approach for the existence of the principal eigenfunction N is a byproduct of the contraction result provided by Harris's theorem, this precludes the case of self-similar fragmentation with equal mitosis and growth rate g(x) = x, for which convergence to a universal profile does not hold, as we already mentioned. In that case the existence of a Perron eigenfunction has to be tackled with other spectral methods, as in [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF][START_REF] Hall | Functional differential equations determining steady size distributions for populations of cells growing exponentially[END_REF][START_REF] Heijmans | An eigenvalue problem related to cell growth[END_REF][START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF].

Note also that our result is valid for the measure solutions of Equation ( 1), thus improving the result in [START_REF] Dȩbiec | Relative entropy method for measure solutions of the growth-fragmentation equation[END_REF] where the general relative entropy method is extended to measure solutions, providing convergence to Malthusian behaviour but without a rate and under restrictive assumptions on the coefficients.

Regarding the assumptions on the coefficients, the only existing spectral gap results that consider general growth rates are the ones in [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF] and [START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF]. In theses papers, the fragmentation rate is assumed to behave like a power law, which we relax here by only requiring Hypothesis 1.3 on B. The other results in the literature focus on constant or linear growth rates and, except in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF], they also consider division rates that grow like power laws.

Finally, when explicit estimates are available for φ, our method allows us to derive quantitative estimates on the spectral gap. It is the case for instance when g(x) = x since then φ(x) = x. An important particular case is to consider additionally that B(x) = x b for some b > 0. This corresponds to the so-called self-similar fragmentation equation, which appears as a rescaling of the pure fragmentation equation, see e.g. [START_REF] Doumic | Time asymptotics for a critical case in fragmentation and growth-fragmentation equations[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF]. To illustrate the quantification of the spectral gap, we prove that for the homogeneous fragmentation kernel and the choice V (x) = 1 + x 2 , the inequality (15) holds true for

ρ = -log 1 - α 2(1 + 2α) 2 log 2 (16)
where

α = 2 log 2 R b+3 e -2(4R) b /b with R = 80 15 2 1 b + b 2 .
This seems to simplify the computable bound in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF]Proposition 6.7]. It can also be compared to [START_REF] Gabriel | Exponential relaxation to self-similarity for the superquadratic fragmentation equation[END_REF] where the spectral gap in L 2 (x dx) is proved to be at least 1 2 , but only for b ≥ 2. Similarly as in [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF], our method also allows for deriving explicit estimates for more general fragmentation kernels since it does not change the function φ.

Historically, the first explicit spectral gap was obtained for constant growth and division rates and the equal mitosis kernel in [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF], and then in [START_REF] Bardet | Total variation estimates for the TCP process[END_REF][START_REF] Chafaï | On the long time behavior of the TCP window size process[END_REF]. The conditions were relaxed [START_REF] Laurençot | Exponential decay for the growth-fragmentation/celldivision equations[END_REF] and in particular general fragmentation kernels were considered. Our method also allows to get explicit spectral gap in the case of constant growth rates, when the division rate is affine and the fragmentation kernel is self-similar. Indeed if g(x) = 1 and B(x) = ax+b, then we easily check that φ(x

) = αx + 1 with α = (p 0 -1)b 2 1 + 4a (p 0 -1)b 2 -1
, where we recall that p 0 is the mass of the self-similar kernel p, and the Perron eigenvalue is given by λ

= (p 0 -1)b 2 1 + 4a (p 0 -1)b 2 + 1 .
It is a particular case of the one treated in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] where B is only assumed to be non-increasing, but it extends the historical case of constant division rate. This paper is organized as follows: We devote Section 2 to showing existence of the dual eigenfuction and some bounds on it. In Section 3, we recall some introductory concepts from the theory of Markov processes and state Harris's Theorem 3.3 based on the previous literature. Eventually for the proof of Theorem 1.3 which is given by applying Harris's theorem, we need to have Hypotheses 3.2 and 3.3 satisfied for Equation [START_REF] Bernard | Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate[END_REF]. In Sections 4 and 5, we prove that Hypotheses 3.2 and 3.3 are verified for Equation [START_REF] Bernard | Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate[END_REF], respectively. Finally in Section 6 we give the proof of Theorem 1.3 and the computations leading to [START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF].

Existence of the dual eigenfunction

In this section, we prove the following theorem which implies existence and boundedness of the dual Perron eigenfunction φ, a solution to the dual eigenproblem (6): Theorem 2.1 (Existence and bounds on the eigenfunction φ). We assume that Hypotheses 1.1 and 1.3 hold true and assume also that p 0 < +∞. Then there exist a continuous function φ which is a solution to Equation (6) and C > 0 such that for any k > 1;

0 < φ(x) ≤ C(1 + x k )
for all x > 0.

Additionally we have φ(0) > 0 when

1 0 1 g < +∞ and φ(0) = 0 when 1 0 1 g = +∞.
Notice that our only assumption on p is that p 0 < +∞ (see Remark 1.2). We prove this theorem at the end of the section.

Following the idea introduced in [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF] and also used in [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF][START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF], we begin with defining a truncated version of the dual Perron eigenproblem [START_REF] Banasiak | Analytic Methods for Coagulation-Fragmentation Models[END_REF] in an interval [0, R] for some R > 0:

-g(x) ∂ ∂x φ R (x) + (B(x) + λ R )φ R (x) = B(x) x R 0 p y x φ R (y) dy, φ R (x) > 0 for x ∈ (0, R), φ R (R) = 0. (17)
Now we give some lemmas which will be used in the proof of Theorem 2.1. The existence of a solution to Equation ( 17) is a consequence of the Krein-Rutman theorem (see the appendices in [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF] and [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF]). Moreover in [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF], the authors proved that there exists R 0 > 0 large enough such that for all R > R 0 we have λ R > 0. We thus have the following result: Lemma 2.2. For any R > 0, the truncated dual Perron eigenproblem (17) admits a solution (λ R , φ R ) with φ R a Lipschitz function. Moreover there exists R 0 > 0 such that λ R > 0 for all R > R 0 .

Before proving uniform estimates on (λ R , φ R ), we first recall a maximum principle. We begin by defining an operator L * R , acting on once-differentiable functions ϕ ∈ C 1 ([0, R]):

L * R ϕ(x) := -g(x)ϕ (x) + (λ R + B(x)) ϕ(x) - B(x) x x 0 p y x ϕ(y) dy.
We have the following maximum principle, see [38, Appendix C] or [4, Section 3.2]:

Lemma 2.3. Suppose that ϕ(x) ≥ 0 for x ∈ [0, A] for some A ∈ (0, R) with ϕ(R) ≥ 0 and L * R ϕ(x) > 0 on [A, R]. Then ϕ(x) ≥ 0 on [0, R].
This maximum principle allows us to get a uniform upper bound on φ R , for a suitable normalization.

Lemma 2.4. Consider that Hypotheses 1.1 and 1.3 are satisfied, and that p 0 < +∞. For any k > 1, there exists

A > 0 such that if φ R is normalized such that sup x∈[0,A] φ R (x) = 1, (18)
then for all R > max{A, R 0 } and for all x ∈ (0, R] we have

0 < φ R (x) ≤ 1 + x k .
Additionally, φ R (0) > 0 when

1 0 1 g < +∞ and φ R (0) = 0 when 1 0 1 g = +∞.
Proof. For the bound from above we want to use the maximum principle in Lemma 2.3. Therefore we want to prove that L * R ϕ(x) > 0 for x ∈ (A, R) with A ∈ (0, R) as in Lemma 2.3. We take ϕ(x) = 1 + x k for some k > 1. Then for R ≥ R 0 we have

L * R ϕ(x) = λ R (1 + x k ) -kg(x)x k-1 + B(x) (1 + x k ) - 1 x x 0 (1 + y k )p y x dy = λ R (1 + x k ) -kg(x)x k-1 + B(x)(1 + x k -p 0 -x k p k ) > x k-1 -kg(x) -B(x)x 1-k + (1 -p k )B(x)x := (x)
since p 0 = 2 and 0 < p k < 1 = p 1 for k > 1. Moreover assuming [START_REF] Bell | Cell growth and division. III. Conditions for balanced exponential growth in a mathematical model[END_REF] gives that behaviour of will be dominated by the positive term (1 -p k )B(x)x k > 0. Therefore, we can find A(k) > 0 such that for all A(k) < x < R, we have L * R ϕ(x) > 0. We fix such a A > 0 and normalize φ R by [START_REF] Bertoin | Probabilistic aspects of critical growth-fragmentation equations[END_REF]. Then by the maximum principle in Lemma 2.3 we obtain that φ R (x) ≤ 1 + x k . The positivity or nullity of φ R (0) is a direct consequence of [4, Theorem 1.10].

Lemma 2.5. Under Hypotheses 1.1 and 1.3 with p 0 < +∞, there exists a constant C > 0 such that λ R ≤ C for all R > R 0 .

Proof. Since φ R is continuous and by [START_REF] Bertoin | Probabilistic aspects of critical growth-fragmentation equations[END_REF], there exists

x R ∈ [0, A] such that φ R (x R ) = 1. Notice that necessarily x R > 0 when 1 0 1 g = +∞, since φ R (0) = 0 is the case. Moreover, the equation L * R φ R = 0 ensures that for all x > 0 we have φ R (x) exp - x x R λ R + B(s) g(s) ds = - B(x) xg(x) exp - x x R λ R + B(s) g(s) ds x 0 p y x φ R (y) dy. By integrating this from x R to x ≥ x R ; φ R (x) exp - x x R λ R + B(s) g(s) ds -1 = - x x R B(y) yg(y) exp - y x R λ R + B(s) g(s) ds y 0 p z y φ R (z) dz dy.
By using the upper bound on φ R we obtain, for R > R 0 ,

φ R (x) exp - x x R λ R + B(s) g(s) ds ≥ 1 - x x R B(y) yg(y) exp - y x R λ R + B(s) g(s) ds y 0 p z y (1 + z k ) dz dy ≥ 1 - x x R B(y) g(y) exp - y x R λ R + B(s) g(s) ds p 0 + p k y k dy. Since φ R (R) = 0 we deduce that for all R > R 0 , R x R B(y) g(y) exp - y x R λ R + B(s) g(s) ds p 0 + p k y k dy ≥ 1, (19)
and this enforces λ R to be bounded from above. Indeed, otherwise, there would exist a sequence

(R n ) n≥0 and x ∞ ∈ [0, A] such that R n → +∞, λ Rn → +∞, x Rn → x ∞ .
But in that case, since Proof. By the equation L * R φ R (x) = 0 and bounds on φ R (x) and λ R we obtain

|φ R (x)| = λ R φ R (x) g(x) + B(x) g(x) φ R (x) - 1 x x 0 p y x φ R (y) dy ≤ λ R g(x) (1 + x k ) + B(x) g(x) 1 + x k - 1 x (1 + x k ) x 0 p y x dy ≤ λ R g(x) (1 + x k ) + B(x) g(x) 1 + x k - 1 x (1 + x k )xp 0 ≤ λ R g(x) (1 + x k ) + B(x) g(x) |1 -p 0 |,
which gives a bound on φ R (x) for all R > R 0 , taking into account that λ R is uniformly bounded for all R > R 0 thanks to Lemma 2.5.

Proof of Theorem 2.1. Lemmas 2.2, 2.4, 2.5 and 2.6 give the proof. Since there exists a solution to the truncated dual Perron eigenproblem [START_REF] Bertoin | On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors[END_REF] for any R > 0 by Lemma 2.2, it only remains to prove that the terms are bounded in order to pass to the limit as R → +∞. We provide the bounds on φ R , λ R and φ R by Lemmas 2.4, 2.5, 2.6 respectively. These bounds ensure that we can extract a subsequence of (λ R ) which converges to λ > 0 and a subsequence of (φ R ) which converges locally uniformly to a limit φ which satisfies 0 < φ(x) ≤ 1 + x k . Clearly (λ, φ) is the solution to the dual Perron eigenproblem [START_REF] Banasiak | Analytic Methods for Coagulation-Fragmentation Models[END_REF], and φ ≡ 0 since sup x∈[0,A] φ(x) = 1. Similarly, the proof of the positivity or nullity of φ(0) is a direct consequence of [4, Theorem 1.10].

Harris's Theorem

In this section, we state Harris's theorem based on [START_REF] Hairer | Convergence of Markov processes[END_REF] and [START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF]. The original idea comes from the study of discrete-time Markov processes and dates back to Doeblin and [START_REF] Harris | The existence of stationary measures for certain Markov processes[END_REF] where conditions of existence and uniqueness of having an equilibrium (or an invariant measure) for a Markov process are investigated. It is a probabilistic method which relies on both a minorisation property and a drift condition (also called Foster-Lyapunov condition), which we describe below. We use Harris's theorem applied to continuous-time Markov processes in order to show that solutions to rescaled growth-fragmentation equation ( 14), under suitable assumptions, converge towards a universal profile at an exponential rate.

We assume that Ω is a Polish space and (Ω, Σ) is a measurable space together with its Borel σ-algebra Σ, so that Ω endowed with any probability measure is a Lebesgue space. Moreover we denote the space of finite measures on Ω by M(Ω) and the space of probability measures on Ω by P(Ω).

A discrete-time Markov process x is defined through a transition probability function. A linear, measurable function S : Ω × Σ → P(Ω) is a transition probability function if S(x, •) is a probability measure for every x and x → S(•, A) is a measurable function for every A ∈ Σ. By using the transition probability function we can define the associated Markov operator S acting on the space of signed measures on Ω and its adjoint S * acting on the space of bounded measurable functions ϕ : Ω → [0, +∞) in the following way:

(Sµ)(A) = Ω S(x, A)µ( dx), (S * ϕ)(x) = Ω ϕ(y)S(x, dy).
On the other hand, a continuous-time Markov process is no longer described by a single transition function, but by a family of transition probability functions S t defined for each time t > 0, with the property that the associated operators S t satisfy

• the semigroup property: S s+t = S s S t ,

• and S 0 is the identity, or equivalently, S 0 (x, •) = δ x for all x ∈ Ω.

We notice that S t is linear, mass preserving and positivity preserving. An invariant measure of a continuous-time Markov process (S t ) t≥0 is a probability measure µ on Ω such that S t µ = µ for every t ≥ 0, and it is the main concept we need to investigate when studying the asymptotic behaviour of a Markov process.

Let us state Doeblin's and Harris's theorems along with some hypotheses. We always assume (S t ) t≥0 is a continuous-time Markov semigroup. For their proofs we refer to [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] or [START_REF] Hairer | Convergence of Markov processes[END_REF][START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF].

Hypothesis 3.1 (Doeblin's condition).

There exists a time t 0 > 0, a probability distribution ν and a constant α ∈ (0, 1) such that for any initial condition x 0 in the domain we have:

S t 0 δ x 0 ≥ αν.
Using this we prove the following theorem: Theorem 3.1 (Doeblin's Theorem). If we have a Markov semigroup (S t ) t≥0 satisfying Doeblin's condition (Hypothesis 3.1) then for any two finite measures µ 1 and µ 2 and any integer n ≥ 0 we have that S n t 0 (µ 1 -µ 2 ) TV ≤ (1 -α) n µ 1 -µ 2 TV . As a consequence, the semigroup has a unique invariant probability measure µ * , and for all probability measures µ:

S t (µ -µ * ) TV ≤ Ce -ρt µ -µ * TV , t ≥ 0,
where

C := 1 1 -α > 0, ρ := -log(1 -α) t 0 > 0.
Harris's theorem is an extension of Doeblin's theorem to situations in which one cannot prove a uniform minorisation condition as in Hypothesis 3.1. This is often the case when the state space is unbounded. Instead, we use Doeblin's condition only in a given region, and then show that the stochastic process will return to that region often enough. This is established by finding a so-called Lyapunov, or Foster-Lyapunov function. Both conditions then imply the existence of a spectral gap in a weighted total variation norm. Precisely, we need the following two hypotheses to be satisfied: Hypothesis 3.2 (Foster-Lyapunov condition). There exist γ ∈ (0, 1), K ≥ 0, some time t 0 > 0 and a measurable function

V : [0, +∞) → [1, +∞) such that (S * t 0 V )(x) ≤ γV (x) + K, (20)
for all x.

Remark 3.2. When our continuous continuous-time Markov process is obtained by solving a particular PDE we often denote

(S t m 0 )(x) ≡ m(t, x),
where m is the solution to the PDE with initial condition m 0 . Then the previous condition is equivalent to

Ω m(t 0 , x)V (x) dx ≤ γ Ω m 0 (x)V (x) dx + K,
to be satisfied for all m 0 ∈ P(Ω). One can verify this by proving the inequality

d dt Ω m(t, x)V (x) dx ≤ -λ +∞ 0 m(t, x)V (x) dx + D
for some positive constants D and λ, which then implies [START_REF] Bertoin | The strong Malthusian behavior of growth-fragmentation processes[END_REF] with γ = e -λt 0 and K = D/λ.

The next hypothesis is a minorisation condition like Hypothesis 3.1, but only on a sufficiently large region: Hypothesis 3.3 (Small set condition). There exist a probability measure ν, a constant α ∈ (0, 1) and some time t 0 > 0 such that S t 0 δ x 0 ≥ αν, for all x 0 ∈ C, where C = {x : V (x) ≤ R} for some R > 2K/(1 -γ) where K, γ are as in Hypothesis 3.2.

Finally we state Harris's theorem under these hypotheses:

Theorem 3.3 (Harris's Theorem). If we have a Markov semigroup (S t ) t≥0 satisfying Hypotheses 3.2 and 3.3 then there exist β > 0 and ᾱ ∈ (0, 1) such that

S t 0 µ 1 -S t 0 µ 2 V,β ≤ ᾱ µ 1 -µ 2 V,β .
for all nonnegative measure µ 1 = µ 2 , where the norm • V,β is defined by

µ 1 -µ 2 V,β := (1 + βV (x))|µ 1 -µ 2 | dx.
Moreover, the semigroup has a unique invariant probability measure µ * and there exist C > 0 and ρ > 0 (depending only on t 0 , α, γ, K, R and β) such that

S t (µ -µ * ) V,β ≤ Ce -ρt µ -µ * V,β for all t ≥ 0.
Explicitly if we set γ 0 ∈ [γ + 2K/R, 1) for any α 0 ∈ (0, α) we can chose β = α 0 /K and ᾱ = max {1 -α + α 0 , (2 + Rβγ 0 )/(2 + Rβ)}. Then we have C = 1/ᾱ and ρ = -(log ᾱ)/t 0 .

Proofs of Theorems 3.1 and 3.3 can be found for example in [START_REF] Hairer | Convergence of Markov processes[END_REF][START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF][START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF][START_REF] Stroock | An Introduction to Markov Processes[END_REF].

Foster-Lyapunov condition

In this section we prove that Hypothesis 3.2 is verified for the semigroup generated by rescaled growth-fragmentation equation ( 14), when we consider the evolution of f (t, x) := φ(x)m(t, x).

We divide the proof of Hypothesis 3.2 into three cases which require slightly different calculations.

Linear growth rate

First we treat the linear growth case g(x) = x with a constant fragmentation kernel. (As remarked before, we do not consider the mitosis kernel when g(x) = x since there is no spectral gap in that case). In this case the Perron eigenvalue and the corresponding dual eigenfunction are known (λ = 1 and φ(x) = x), and the rescaled growth-fragmentation equation is given by

∂ ∂t m(t, x) + ∂ ∂x (xm(t, x)) = 2 +∞ x B(y) y m(t, y) dy -(B(x) + 1)m(t, x), (21)
coupled with the usual initial and boundary conditions.

+∞ 0 V (x)f (t, x) dx ≤ e -C 1 t +∞ 0 V (x)f 0 (x) dx + C +∞ 0 f 0 (x) dx ( 22
)
for all t ≥ 0, where f (t, x)

:= xm(t, x), f 0 (x) = xm 0 (x), f 0 V < +∞ and V (x) = x k-1 + x K-1 .
Proof. Let ϕ : R → [0, 1] be a non-increasing C 1 function such that ϕ(x) = 1 for x ≤ 0 and ϕ(x) = 0 for x ≥ 1. For > 0 we define ϕ (x) = ϕ(x -). Starting from (3) we have

d dt +∞ 0 x k + x K ϕ (x)m(t, x) dx = +∞ 0 kx k-1 + Kx K-1 ϕ (x) + x k + x K ϕ (x) xm(t, x) dx + 2 +∞ 0 B(x) x m(t, x) x 0 y k + y K ϕ (y) dy dx - +∞ 0 1 + B(x) x k + x K ϕ (x)m(t, x) dx.
Since ϕ is non-increasing we get

d dt +∞ 0 x k + x K ϕ (x)m(t, x) dx ≤ +∞ 0 kx k-1 + Kx K-1 ϕ (x)xm(t, x) dx + 2 +∞ 0 B(x) x k-1 k + 1 + x K-1 K + 1 ϕ (x)xm(t, x) dx - +∞ 0 1 + B(x) x k-1 + x K-1 ϕ (x)xm(t, x) dx ≤ - 1 2 (1 -k) +∞ 0 (x k-1 + x K-1 )ϕ (x)xm(t, x) dx + +∞ 0 c 1 B(x)x K-1 + c 2 x K-1 + c 3 B(x)x k-1 + c 4 x k-1 ϕ (x)xm(t, x) dx where -1 < c 1 := 1 -K 1 + K < 0, c 2 := K - k + 1 2 > 0, c 3 := 1 -k 1 + k > 0, c 4 := k -1 2 < 0.
We define

Φ(x) := c 1 B(x)x K-1 + c 2 x K-1 + c 3 B(x)x k-1 + c 4 x k-1 . ( 23
)
Due to Hypothesis 1.3, the total fragmentation rate B : [0, +∞) → [0, +∞) satisfies B(x) → 0 as x → 0 and B(x) → +∞ as x → +∞. Hence in the latter expression the behaviour as x → +∞ is dominated by the first term; thus Φ(x) will approach -∞. Similarly when x → 0, the last term will dominate the behaviour of Φ, which is negative as well. Since B is continuous we can always bound sup x≥0 Φ(x) ≤ C 2 with some positive quantity C 2 > 0.

Therefore by denoting f (t, x) = xm(t, x) and f 0 (x) = xm 0 (x) we obtain, since ϕ ≤ 1 and

f (t, x) dx = f 0 (x) dx, d dt +∞ 0 (x k-1 + x K-1 )ϕ (x)f (t, x) dx ≤ -C 1 +∞ 0 (x k-1 + x K-1 )ϕ (x)f (t, x) dx + C 2 +∞ 0 f 0 (x) dx,
where

C 1 = (1 -k)/2 > 0. Then Grönwall's lemma implies +∞ 0 V (x)ϕ (x)f (t, x) dx ≤ e -C 1 t +∞ 0 V (x)ϕ (x)f 0 (x) dx + C +∞ 0 f 0 (x) dx with C = C 2 /C 1 .
Due to the monotone convergence theorem we deduce [START_REF] Brikci | Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle[END_REF] by letting go to +∞.

Sublinear growth rate close to 0

In this section we assume that 1 0 1 g < +∞, which we sometimes refer to as the case of sublinear growth rate at x = 0. Lemma 4.2. We consider Equation ( 14) under Hypotheses 1.1, 1.3, and 1 0 1 g < +∞. We take K > 1 + ξ. Then the following holds true for C 1 = λ (the first eigenvalue), some C 2 > 0, and any nonnegative measure solution m = m(t, x):

d dt +∞ 0 x K m(t, x) dx ≤ -C 1 +∞ 0 x K m(t, x) dx + C 2 +∞ 0 φ(x)m(t, x) dx, ( 24
)
for all t ≥ 0.

Proof. For the sake of conciseness and clarity, we skip the truncation procedure here. But the same method as for Lemma 4.1 can be used to make the calculations rigorous by using the truncation function ϕ . We have

d dt +∞ 0 x K m(t, x) dx = - +∞ 0 x K ∂ ∂x (g(x)m(t, x)) dx - +∞ 0 x K (B(x) + λ)m(t, x) dx + +∞ 0 x K +∞ x B(y) y p x y m(t, y) dy dx = -λ +∞ 0 x K m(t, x) dx + +∞ 0 (p K -1)x K B(x) + Kx K-1 g(x) m(t, x) dx.
We define

Φ(x) := (p K -1)x K B(x) + Kx K-1 g(x)
and notice that sup x≥0 Φ(x) ≤ C 2 φ(x) for some C 2 > 0 due to Hypothesis 1.3 concerning the behaviour of xB(x)/g(x) as x → +∞ and x → 0, and the fact that φ(0) > 0 since

1 0 1 g < +∞ which is a result of Theorem 2.1.
We now give a translation of this lemma in terms of f = φm, since this is needed in order to apply Harris's theorem to the evolution of f : Corollary 4.3. We consider Equation ( 14) under Hypotheses 1.1, 1.3, and

1 0 1 g < +∞. For V (x) = 1 + x K φ(x) where K > 1 + ξ and f (t, x) := φ(x)m(t, x) with f 0 (x) = φ(x)m 0 (x), f 0 V < +∞, there exist C 1 , C > 0 such that for all t ≥ 0 +∞ 0 V (x)f (t, x) dx ≤ e -C 1 t +∞ 0 V (x)f 0 (x) dx + C +∞ 0 f 0 (x) dx. (25)
Proof. By adding φ(x) of both sides of ( 24) we obtain

d dt +∞ 0 x K m(t, x) dx = d dt +∞ 0 (x K + φ(x))m(t, x) dx ≤ -C 1 +∞ 0 (x K + φ(x))m(t, x) dx + (C 1 + C 2 ) +∞ 0 φ(x)m(t, x) dx.
Therefore, we have for f (t, x) = φ(x)m(t, x);

d dt +∞ 0 1 + x K φ(x) f (t, x) dx ≤ -C 1 +∞ 0 1 + x K φ(x) f (t, x) dx + (C 1 + C 2 ) +∞ 0 f 0 (x) dx, since f (t, x)dx = f 0 (x)dx. Grönwall's lemma then implies (25) with C = 1 + C 2 /C 1 .

Superlinear growth rate close to 0

Now we assume that 1 0 1 g = +∞, which implies linear or superlinear behaviour for the growth rate x close 0. This, of course, includes the case g(x) = x from Section 4.1, but the general result we obtain now is slightly more restrictive. In the case of exact linear growth, Lemma 4.1 is slightly more precise. Lemma 4.4. We consider Equation (14) under Hypotheses 1.1, 1.3, and 1 0 1 g = +∞. We take k < 0 and K > 1 + ξ. Then the following holds true for any nonnegative measure solution m = m(t, x):

d dt +∞ 0 (x k + x K )m(t, x) dx ≤ -C 1 +∞ 0 (x k + x K )m(t, x) dx + C 2 +∞ 0 φ(x)m(t, x) dx,
for all t ≥ 0, where C 1 = λ > 0 and C 2 > 0 is some constant independent of the solution m.

Proof. Here again we skip the truncation procedure and refer to the proof of Lemma 4.1 for the method which allows making the calculations rigorous. We have

d dt +∞ 0 (x k + x K )m(t, x) dx = - +∞ 0 (x k + x K ) ∂ ∂x (g(x)m(t, x)) dx - +∞ 0 (x k + x K )(B(x) + λ)m(t, x) dx + +∞ 0 B (y) y m(t, y) 1 0 (y
k z k + y K z K )p (z) y dz dy = -λ +∞ 0 (x k + x K )m(t, x) dx + +∞ 0 (p k -1)x k B(x) + (p K -1)x K B(x) + kx k-1 g(x) + Kx K-1 g(x) m(t, x) dx
Similarly to previous proofs, we define

Φ(x) := (p k -1)x k B(x) + (p K -1)x K B(x) + kx k-1 g(x) + Kx K-1 g(x)
and notice that sup x>0 Φ(x) ≤ C 2 φ(x) for some C 2 > 0 due to Hypothesis 1.3 concerning the behaviour of xB(x)/g(x) as x → +∞ and x → 0, and the fact that p K -1 < 0 and k < 0.

Corollary 4.5. We consider Equation (14) under Hypotheses 1.1, 1.3 and

1 0 1 g = +∞. For V (x) = x k +x K φ(x) with k < 0, K > 1 + ξ, and f (t, x) := φ(x)m(t, x) with f 0 (x) = φ(x)m 0 (x), f 0 V < +∞, there exist C 1 , C > 0 such that for all t ≥ 0: +∞ 0 V (x)f (t, x) dx ≤ e -C 1 t +∞ 0 V (x)f 0 (x) dx + C +∞ 0 f 0 (x) dx. ( 26
)
Proof. The inequality in Lemma 4.4 yields, for f (t, x) := φ(x)m(t, x),

d dt +∞ 0 x k + x K φ(x) f (t, x) dx ≤ -C 1 +∞ 0 x k + x K φ(x) f (t, x) dx + (C 1 + C 2 ) +∞ 0 f 0 (x) dx, since f (t, x)dx = f 0 (x)dx.
Then Grönwall's lemma implies [START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF] with

C = 1 + C 2 /C 1 .

Minorisation condition

In this section, we show that Hypothesis 3.3 is verified for the semigroup generated by rescaled growth-fragmentation equation [START_REF] Bernard | Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate[END_REF]. We give the proof in two parts where the uniform fragment distribution and the equal mitosis are considered separately.

We start by recalling some known results on the solution of the transport part of Equation ( 14). Consider the equation

∂ ∂t m(t, x) + ∂ ∂x (g(x)m(t, x)) = -c(x)m(t, x), t, x > 0, m(t, 0) = 0, t > 0, m(0, x) = n 0 (x), x > 0, (27)
which is the same as Equation ( 14) without the positive part of the fragmentation operator. We remark that Hypothesis 1.3 ensures that the characteristic ordinary differential equation

d dt X t (x 0 ) = g(X t (x 0 )), X 0 (x 0 ) = x 0 , (28)
has a unique solution, defined for t ∈ [0, +∞), for any initial condition x 0 > 0. In fact, it is defined in some interval (t * (x 0 ), +∞), for some t * (x 0 ) < 0. The solution can be explicitly given in terms of H -1 , where

H(x) := x 1 1 g(y) dy, x ≥ 0.
We notice that H is strictly increasing with H 0 := H(0) = lim +∞ (since g grows sublinearly as x → +∞), so that it is invertible as a map from (0, +∞) to (H 0 , +∞). (We allow H 0 = -∞ if 1/g is not integrable close to x = 0.) It can easily be checked that

X t (x 0 ) = H -1 (t + H(x 0 ))
for x 0 > 0 and t > H 0 -H(x 0 ), so that that the maximal time interval where the solution of ( 28) is defined is precisely as (H 0 -H(x 0 ), +∞). Since it will be convenient later, we define

X t (0) := lim x 0 →0 X t (x 0 ) = 0 if H 0 = -∞, H -1 (t + H 0 ) if H 0 ∈ (-∞, 0).
This reflects the fact that the characteristics take a very long time to escape from 0 when 1/g is not integrable close to 0; while they escape in finite time if 1/g is integrable close to 0. For each t ≥ 0, we have thus defined the flow map X t : (0, +∞) → (X t (0), +∞), which is strictly increasing. For negative times, we may consider X -t : (X t (0), +∞) → (0, +∞) (where t > 0). Of course, X -t = (X t ) -1 .

If n 0 is a nonnegative measure, it is well known that the unique measure solution to Equation ( 27) is given by

m(t, x) = X t #n 0 (x) exp - t 0 c(X -τ (x)) dτ , t ≥ 0, x > X t (0), m(t, x) = 0, t ≥ 0, x ≤ X t (0), (29)
where we abuse notation by evaluating the measures m(t, •) and X t #n 0 at a point x > 0. For a Borel measurable map X : (0, +∞) → (0, +∞), the expression X#n 0 denotes the transport, or push forward, of the measure n 0 by the map X, defined by duality through ∞ 0 ϕ(x)X#n 0 (x) dx := ∞ 0 ϕ(X(y))n 0 (y) dy for all continuous, compactly supported ϕ : (0, +∞) → R. We use the notation T t for this flow map: T t n 0 (x) := X t #n 0 (x), for all t ≥ 0,

so T t is the semigroup associated to transport equation [START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF].

If additionally n 0 is a function and X has a left inverse X -1 : (a, b) → (0, +∞), one has

X#n 0 (x) = n 0 (X -1 (x)) d dx (X -1 )(x)) if x ∈ (a, b), 0 otherwise.
Using this for the solution to [START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF], if n 0 is a function we may write m in the equivalent form

m(t, x) = n 0 (X -t (x)) d dx X -t (x) exp - t 0 c(X -τ (x)) dτ (31)
when t ≥ 0 and x > X t (0), and m(t, x) = 0 otherwise. Using that

Y t (x) := d dx X t (x) satisfies d dt Y t (x) = g (X t (x))Y t (x), we note for later that d dx X -t (x) = exp - t 0 g (X -τ (x)) dτ , t ≥ 0, x > X t (0). (32)

Uniform fragment distribution

Let us consider the case of uniform fragment distribution p(z) = 2, corresponding to the fragmentation kernel of the form κ(x, y) = 2 x B(x)1 {0≤x≤y} . The growth-fragmentation equation in this case is widely studied and depending on some assumptions made on growth and total division rates, existence (in some cases exact values) of eigenelements are known. The rescaled growth-fragmentation equation in this case becomes

∂ ∂t m + ∂ ∂x (g(x)m) = 2 +∞ x B(y) y m(t, y) dy -(B(x) + λ)m, t, x ≥ 0, m(t, 0) = 0, t > 0, m(0, x) = n 0 (x), x > 0, (33)
where m = m(t, x) whenever variables are not explicitly written. If we consider a linear growth g(x) = g 0 x and a power like total division B(x) = b 0 x γ with γ > 0, and g 0 , b 0 > 0, the Perron eigenvalue and the corresponding dual eigenfunction are given by λ = g 0 and φ(x) = x yN (y) .

In this case, eigenelements can be computed explicitly (see for example [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF]):

λ = g 0 , N (x) = b 0 γg 0 1/γ γ Γ 1 γ exp - 1 γ b 0 g 0 x γ , φ(x) = b 0 γg 0 1/γ Γ 1 γ Γ 2 γ x.
Moreover, in [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF], the authors give the asymptotics of the profile N and accurate bounds on the dual eigenfunction φ in a more general form of the growth-fragmentation equation where growth and total division rates behave like a power law for large and small x.

Lemma 5.1 (Lower bound for the uniform fragment distribution). Assume Hypotheses 1.1 and 1.3 hold true with a constant distribution of fragments p(z) = 2 for z ∈ (0, 1]. Let (S t ) t≥0 be the linear semigroup associated to Equation [START_REF] Cloez | On an irreducibility type condition for the ergodicity of nonconservative semigroups[END_REF]. For all 0 < η < θ given, there exists t 0 > 0 such that for all t > t 0 and x 0 ∈ (η, θ] it holds that

S t δ x 0 (x) ≥ C(η, θ, t) for all x ∈ I t ,
where I t is an open interval which depends on η, the time t, and for some quantity C = C(η, θ, t) depending only on η, θ and t. If in addition we assume that

1 0 1 g(x)
dx < +∞, then the above result also holds when taking η = 0.

Proof. Recall that (T t ) t≥0 the semigroup associated to the transport equation

∂ ∂t m(t, x) + ∂ ∂x (g(x)m(t, x)) + c(x)m(t, x) = 0,
where c(x) = B(x) + λ. By Duhamel's formula we have

S t n 0 (x) = m(t, x) = T t n 0 (x) + t 0 T t-τ (A(τ, .))(x) dτ,
where A(t, x) := 2

+∞ x B (y)
y m(t, y) dy. Fix 0 ≤ η < θ, and take any x 0 ∈ (η, θ]. If n 0 = δ x 0 , a simple bound gives

S t δ x 0 ≥ T t δ x 0 = X t #δ x 0 exp - t 0 c(X t-τ (x 0 )) dτ ,
where we have used the expression of T t given in [START_REF] Cavalli | On a family of critical growth-fragmentation semigroups and refracted Lévy processes[END_REF] and the fact that the support of X t #δ x 0 is the single point {X t (x 0 )}. By Hypothesis 1.3 (in particular since B is continuous on [0, X t (θ)]), for some C 1 = C 1 (θ, t) which is increasing in t, we have

c(x) = B(x) + λ ≤ C 1 for all x ∈ (0, X t (θ)].
We deduce that

S t δ x 0 ≥ X t #δ x 0 e -C 1 t = δ Xt(x 0 ) e -C 1 t . (34)
Using this we obtain

A(t, x) ≥ 2e -C 1 t B(X t (x 0 )) X t (x 0 )
for all t > 0 and x < X t (x 0 ).

We use that there is some x B > 0 for which B is bounded below by a positive quantity on any interval of the form [x B , R]. There is some t B > 0 such that for t > t B we have X t (x 0 ) > x B for all x 0 > η (for this to hold, notice we may take η = 0 in the case that 1 0 1/g < +∞, but we need η > 0 otherwise). Hence, for some C 2 = C 2 (η, θ, t) which is decreasing in t, we obtain

A(t, x) ≥ C 2 e -C 1 t
for all t > t B and x < X t (x 0 ).

Take now t > t B , which will stay fixed until the end of the proof. The previous bound shows that A(τ, x) ≥ C 2 (η, θ, τ )e -C 1 (θ,τ )τ ≥ C 2 (η, θ, t)e -C 1 (θ,t)τ =: C2 e -C1 τ for all t > t B , t B < τ < t and all x < X τ (x 0 ). As a consequence, using ( 31) and ( 32),

T t-τ A(τ, x) ≥ C2 e -C1 τ exp - t-τ 0 c(X -s (x)) ds exp - t-τ 0 g (X -s (x)) ds
for all t B < τ < t and X t-τ (0) < x < X t (x 0 ). Since X -s (x) ≤ X t (x 0 ) in this range, we can bound this by

T t-τ A(τ, x) ≥ C2 e -2 C1 t exp - t-τ 0 g (X -s (x)) ds ,
again for all t B < τ < t and X t-τ (0) < x < X t (x 0 ). In order to find a lower bound for the last exponential we restrict to a smaller x interval. Since the bound holds for all x with X t-τ (0) < x < X t (x 0 ), it holds in particular for all x with

X t-t B (η) < x < X t (η). ( 35
)
Again this is a point where we need to take η > 0 in the case 1 0 1/g = +∞, since otherwise this gives an empty range of x. In the case 1 0 1/g < +∞, η = 0 is allowed. In this range, the quantity X -s (x) inside the exponential satisfies

X τ -t B (η) ≤ X -s (x) ≤ X t (η)
Choose δ > 0 such that t B + δ < t. Then for all x satisfying (35) and all τ ∈ (t B + δ, t) we have

X δ (η) ≤ X -s (x) ≤ X t (η).
Using that g (X) ≤ C 3 for all X ∈ [X δ (η), X t (η)] we have

T t-τ A(τ, x) ≥ C2 e -C1 τ e -C 3 (t-τ ) ≥ C2 e -C 4 t
for all x satisfying (35) and all τ ∈ (t B + δ, t). A final integration gives, for x in the same interval,

t 0 T t-τ (A(τ, •))(x) dτ ≥ C2 e -C 4 t t t B +δ dτ = C2 e -C 4 t (t -t B -δ).
Taking t 0 := t B gives the result.

Equal mitosis

We now consider the fragment distribution p(z) = 2δ 1 2 (z) which describes the process of equal mitosis, in which cells of size x split into two equal daughter cells of size x/2. In Equation ( 14), we have then A(t, x) := 4B(2x)m(t, 2x) and the rescaled growth-fragmentation equation takes the form

∂ ∂t m(t, x) + ∂ ∂x (g(x)m(t, x)) = 4B(2x)m(t, 2x) -(B(x) + λ)m(t, x), t, x ≥ 0, m(t, 0) = 0, t > 0, m(0, x) = n 0 (x), x > 0. (36)
The case where g and B are constant was the subject of numerous works in the past, most notably [START_REF] Bardet | Total variation estimates for the TCP process[END_REF][START_REF] Chafaï | On the long time behavior of the TCP window size process[END_REF][START_REF] Hall | A functional-differential equation arising in modelling of cell growth[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF][START_REF] Van Brunt | On a cell division equation with a linear growth rate[END_REF]. For g(x) = 1 and B(x) = 1, eigenelements are given by

λ = 1, N (x) = +∞ n=0 (-1) n α n e -2 n+1 x , φ(x) ≡ 1.
with α n = 2 2 n -1 α n-1 and α 0 > 0 a suitable normalization constant, and the solution m(t, x) converges exponentially fast to the universal profile N (x), which vanishes as x → 0 and x → +∞. However, when a linear growth rate g(x) = x is considered Equation (36) exhibits oscillatory behaviour in the long time. This is because instead of a dominant real eigenvalue, there are nonzero imaginary eigenvalues, so that there exists a set of dominant eigenvalues. This type of periodic long time behaviour was first observed in [START_REF] Diekmann | On the stability of the cell size distribution[END_REF] and then it was proved in [START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF] by using the theory of positive semigroups combined with spectral analysis to obtain the convergence to a semigroup of rotations. Since the method relies on some compactness arguments, the authors considered the equation in a compact subset of (0, +∞). Recently in [START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF], the authors proved the oscillatory behaviour in the framework of measure solutions for general division rates on (0, +∞). The proof relies on a general relative entropy argument combined with the use of Harris's theorem on discrete sub-problems. It provides an explicit rate of convergence in weighted total variation norm. Here we consider a sublinear growth rate and a more general division rate than those so far considered in the literature. We exclude of course the case g(x) = x, for which we know the lower bound (and the exponential convergence) does not hold.

We first need a technical lemma which gives an expression for the time integration of a measure moving in time:

Lemma 5.2. Let t > 0 and F : [0, t] → R an injective, differentiable function. Then t 0 δ F (τ ) (x) dτ = F -1 (x)1 {F (0)≤x≤F (t)} .
Proof. Integrating against a smooth test function ϕ(x) we obtain

+∞ 0 ϕ(x) t 0 δ F (τ ) (x) dτ dx = t 0 +∞ 0 ϕ(x)δ F (τ ) (x) dx dτ = t 0 ϕ(F (τ )) dτ = F (t) F (0)
ϕ(y) F -1 (y) dy. by using the change of variable y = F (τ ).

The following result will ensure a certain sublinearity of the characteristic flow X t which we will need later: Lemma 5.3. Assume that the growth rate g : (0, +∞) → (0, +∞) is locally Lipschitz and satisfies ωg(x) < g(ωx) for all x > 0 and ω ∈ (0, 1).

Then for any t > 0 the characteristic flow X t satisfies ωX t (x) < X t (ωx), for all x > 0 and ω ∈ (0, 1).

Proof. Call h 1 (t) := ωX t (x) and h 2 (t) := X t (ωx). The second one satisfies the ODE

h 2 (t) = g(h 2 (t)),
while the first one satisfies

h 1 (t) = ωg(X t (x)) < g(ωX t (x)) = g(h 1 (t)).
Since they have the same initial condition, this differential inequality implies h 1 (t) < h 2 (t) for all t > 0.

Again by Lemma 5.3 we see that this interval is nonempty. Since we need a bound which is independent of x 0 , we consider the intersection of all these intervals as x 0 moves in the interval (0, θ). That intersection is

I t := 1 2 X t (θ) , X t-t B 1 2 X t B (0) .
Condition [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF] shows that this interval is nonempty for t large enough, since

X t (θ) X t-t B 1 2 X t B (0) = H -1 (t + θ) H -1 t -t B + 1 2 X t B (0) → 1 as t → +∞.
This gives the result.

Proof of the main result

We conclude by giving the proof of Theorem 1.3. It is a direct application of Harris's Theorem 3.3. Hypotheses 3.2 and 3.3 need to be verified. We already verified Hypothesis 3.2 (Lyapunov condition) in Section 4 (see the corollary given in each case); in fact, we have proved that given any t 0 > 0 we can satisfy Hypothesis 3.2 for any t ≥ t 0 , with constants γ, K which are independent of t (since we can always take γ := e -C 1 t 0 , K := C). Regarding Hypothesis 3.3, the lower bounds we obtained in Section 5 are for m(t, x) which is a solution to Equation ( 14). However we need to satisfy the minorisation condition for f (t, x) = φ(x)m(t, x) since the equation on f conserves mass; thus the associated semigroup is Markovian, and we may apply

We define (F t ) t≥0 as the semigroup associated to Equation [START_REF] Doumic | Time asymptotics for a critical case in fragmentation and growth-fragmentation equations[END_REF], or alternatively by the relationship F t (φn 0 ) := φS t n 0 , for any nonnegative measure n 0 such that φn 0 is a finite measure on (0, +∞). Lemma 6.1 (Minorisation condition for f (t, x)). We assume Hypotheses 1.1, 1.2, 1.3 and 1.4 hold true. Let (F t ) t≥0 be the semigroup associated to Equation [START_REF] Doumic | Time asymptotics for a critical case in fragmentation and growth-fragmentation equations[END_REF]. For any 0 > η > θ there exists t 0 = t 0 (η, θ) > 0 such that for all t > t 0 and x 0 ∈ [η, θ] it holds that F t δ x 0 (x) ≥ C(η, θ, t) for all x ∈ I t , where I t is an open interval which depends on time t, and for some quantity C = C(η, θ, t) depending only on η, θ and t. If in addition we assume that

1 0 1 g(x)
dx < +∞, then the above result also holds when taking η = 0.

Proof. Let (S t ) t≥0 and (F t ) t≥0 be the semigroups associated to Equations ( 14) and (37) respectively. Under the conditions of Lemma 5.1 we have a lower bound for S t δ x 0 (x) ≥ C(η, θ, t).

It immediately translates to a lower bound on F t in all cases:

1. If 1 0 1 g(x) dx = +∞, we know from [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF] that φ(x) is bounded in each interval of the form (0, θ] (since it is continuous and tends to a positive constant at x = 0). On the other hand, under the conditions of Lemma 5.4 we know again that φ(x) is bounded above and below by positive constants in each interval of the form (0, θ].

If

Therefore we obtain for x 0 ∈ [η, θ]:

F t δ x 0 (x) = φ(x) φ(x 0 ) S t δ x 0 (x) ≥ Ĉ1 (η, θ)
Ĉ2 (η, θ) C(η, θ, t) := C(η, θ, t),

allowing η = 0 if 1 0 1/g < +∞.
Proof of Theorem 1.3. As remarked above, the semigroup (F t ) t≥0 satisfies the Lyapunov condition in Hypothesis 3.2 in all cases, for t ≥ 1, with a weight V and constants γ, K which are independent of t. In order to satisfy Hypothesis 3.3 it is enough then to find any time t ≥ 1 for which we have a uniform lower bound whenever the initial condition is a delta function supported on a region of the form

C := {x > 0 | V (x) ≤ R}
for some R > 2K/(1 -γ). Lemma 6.1 gives this in all cases. Notice that in the cases in which the lower bound is only available for x 0 ∈ [η, θ] with η > 0, the function V we give in Section 4 is unbounded at x = 0, and thus the region C is contained in an interval of that form.

Explicit calculations for the self-similar fragmentation case. We recall that the socalled self-similar fragmentation equation corresponds to a linear growth rate g(x) = x, a monomial total fragmentation rate B(x) = x b , b > 0, and a self-similar kernel (here we take the homogeneous self-similar kernel p(z) ≡ 2). In that case, all the constants appearing in Harris's theorem can be quantified. This is due to the explicit expression φ(x) = x of the dual eigenfunction when g(x) = x. For the computations we choose for instance the parameters k = 0 and K = 2, which correspond to the Lyapunov function V (x) = (x k + x K )/φ(x) = 1/x + x. We start with Hypothesis 3.2. Using that B(x) = x b we can make the proof of Lemma 4.1 more quantitative. Indeed the function Φ defined in [START_REF] Broda | Convergence rates for semistochastic processes[END_REF] reads in the present case

Φ(x) = - 1 3 x b+1 + 3 2 x + x b-1 - 1 2 x -1 .
Treating separately the cases x ≤ .

This proves that we can choose ρ as in [START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF].

2 (

 2 y) dy.According to Hypothesis 1.2, we only allow p(z) = 2 or p(z) = 2δ 1

Lemma 4 . 1 .

 41 We consider Equation[START_REF] Bouguet | A probabilistic look at conservative growth-fragmentation equations[END_REF] under Hypotheses 1.1, 1.3 with a growth rate g(x) = x and the constant fragment distribution p(z) = 2 for z ∈ (0, 1]. Then the following holds true for any K > 1 > k > -1, for some C 1 , C > 0, and any nonnegative measure solution m = m(t, x):

  Harris's theorem to it. The equation satisfied by f is∂ ∂t f (t, x) + φ(x) ∂ ∂x g(x) φ(x) f (t, x) + (B(x) + λ)f (t, x) x) = n 0 (x), x > 0.

  ) dx = +∞, then since φ(x) is continuous there exist constants Ĉ1 (η, θ), Ĉ2 (η, θ) > 0 such that Ĉ1 ≤ φ(y) ≤ Ĉ2 for all y ∈ [η, θ].

1 , x ≥ 1 , 2 for 2 and K = 10 15 2 1 b + b 2 .

 112222 and b ≤ 2, b ≥ 2, we can check that all x > 0. So Hypothesis 3.2 is verified for any t 0 > 0 with the constants γ = e -t 0We now turn to Hypothesis 3.3. We chooseR = 4K 1 -γ and we notice that since V (x) = 1/x + x C = {x : V (x) ≤ R} ⊂ [1/R, R].For φ(x) = x and p(z) ≡ 2, Equation (37) reads∂ ∂t f (t, x) + ∂ ∂x (xf (t, x)) + B(x)f (t, x) = 2 +∞ x B(y)f (t, y) x y dyand we can prove directly on this equation, proceeding similarly as in Lemma 5.1, that for any t 0 > 0 and allx 0 ∈ [1/R, R] F t 0 δ x 0 ≥ αν with ν( dy) = 2e -2t 0 R 1 [0,Re t 0 ] (y)y dy and α = R b+3 t 0 exp -2R γ e bt 0 b .We are now in position to apply Harris's theorem. Choosing in Theorem 3.3

  Lemma 2.6. Under Hypotheses 1.1 and 1.3 with p 0 < +∞, |φ R (x)| is uniformly bounded on compact intervals for all R > R 0 .

	1 [x Rn ,Rn] (y)	B(y) g(y)	exp -	y x Rn	λ Rn + B(s) g(s)	ds	p 0 + p k y k
	≤ p 0 + p k y Rn B(y) g(y) exp -y A B(s) g(s) ds x Rn B(y) g(y) exp -y x Rn λ Rn + B(s) g(s) ds p 0 + p

k

and the latter function is integrable on [0, +∞) (carry out an integration by parts and use

[START_REF] Bell | Cell growth and division. III. Conditions for balanced exponential growth in a mathematical model[END_REF]

), the dominated convergence theorem ensures that k y k dy → 0, which contradicts

[START_REF] Bertoin | A probabilistic approach to spectral analysis of growthfragmentation equations[END_REF]

.

Acknowledgements

JAC and HY were supported by the project MTM2017-85067-P, funded by the Spanish government and the European Regional Development Fund. HY was also supported by the Basque Government through the BERC 2018-2021 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2017-0718, by the "la Caixa" Foundation and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639638). JAC and HY gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics (Bonn), through the Junior Trimester Program on Kinetic Theory. PG was supported by the ANR project NOLO, ANR-20-CE40-0015, funded by the French Ministry of Research.

Our main lower bound for the mitosis case is the following: Lemma 5.4 (Lower bound for equal mitosis). Assume Hypotheses 1.1, 1.3, 1.4 hold true with the mitosis kernel p(z) = 2δ 1 2 (z). Let (S t ) t≥0 be the semigroup associated to Equation [START_REF] Doob | Conditional Brownian motion and the boundary limits of harmonic functions[END_REF]. For any θ > 0 there exists t 0 = t 0 (θ) > 0 such that for all t > t 0 and x 0 ∈ (0, θ] it holds that

where I t is an open interval which depends on time t, and for some quantity C = C(t, θ) depending only on t and θ.

Proof. Fix θ > 0 and take any x 0 ∈ (0, θ]. We follow the same strategy as in the proof of Lemma 5.1. Here the only different part is A(t, x). We consider the semigroup (T t ) t≥0 defined as in ( 30) and (S t ) t≥0 defined as the semigroup associated to [START_REF] Doob | Conditional Brownian motion and the boundary limits of harmonic functions[END_REF] with A(t, x) = 4B(2x)m(t, 2x). Using (34) we have

, increasing in t. we obtain

for all t > 0. We know that there exists some x B > 0 for which B is bounded below by a positive quantity in each interval of the form [x B , R]. Take t B > 0 such that for t > t B we have X t (x 0 ) > x B for all x 0 > 0. Hence, for some

Fix now any t > t B . For t B < τ < t we have

Xτ (x 0 ) (x). Hence using (29) we have

for all τ ∈ (t B , t). Define F (τ ) := X t-τ 1 2 X τ (x 0 ) , and notice that it is a strictly decreasing function, since Lemma 5.3 ensures that for τ


Fabien Kéfélian

Olivier Lopez

Haifeng Jiang

Christian Chardonnet

Anne Amy- Klein

Giorgio Santarelli

High-resolution optical frequency dissemination on a telecommunication network with data traffic

We transferred the frequency of an ultra-stable laser over a 108 km urban fiber link comprising 22 km of optical communications network fiber simultaneously carrying Internet data traffic. The metrological signal and the digital data signal are transferred on two different frequency channels in a dense wavelength division multiplexing scheme. The metrological signal is inserted into and extracted from the communications network by using bidirectional off-the-shelf optical add-drop multiplexers. The link-induced phase noise is measured and cancelled with round-trip technique using an all-fiber-based interferometer. The compensated link shows an Allan deviation of a few 10 -16 at one second and below 10 -19 at 10,000 seconds. This opens the way to a wide dissemination of ultra stable optical clock signals between distant laboratories via the Internet network.

OCIS codes: 060. 2360, 120.3930 The transfer of ultra-stable frequencies between distant laboratories is an important issue for a large number of high-sensitivity experiments in metrology and fundamental physics, for example tests of the fundamental constants stability [START_REF] Karshenboim | Fundamental physical constants: looking from different angles[END_REF] or high-resolution geodetic measurements. With the recent development of single ion traps and cold atoms lattices, optical clocks are expected to reach fractional frequency instability level of 10 -17 or better at one day [START_REF] Ludlow | Sr lattice clock at 1x10 -16 fractional uncertainty by remote optical evaluation with a Ca clock[END_REF][START_REF] Rosenband | Frequency ratio of Al + and Hg + single-ion optical clocks; metrology at the 17 th decimal place[END_REF], which implies the reduction of transfer induced instability well below this level.

The transmission of frequency standards over optical fiber has been investigated for a few years [START_REF] Daussy | Long-Distance Frequency Dissemination with a Resolution of 10 -17[END_REF][START_REF] Narbonneau | High resolution frequency standard dissemination via optical fiber metropolitan network[END_REF][START_REF] Foreman | Remote transfer of ultrastable frequency references via fiber networks[END_REF][START_REF] Lopez | 86-km optical link with a resolution of 2×10 -18 for RF frequency transfer[END_REF][START_REF] Grosche | Transmission of an Optical Carrier Frequency over a Telecommunication Fiber Link[END_REF][START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF][START_REF] Williams | High-stability transfer of an optical frequency over long fiber-optic links[END_REF][START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF][START_REF] Musha | Coherent optical frequency transfer over 50-km physical distance using a 120-km-long installed fiber network[END_REF][START_REF] Grosche | Measurement noise floor for a long-distance optical carrier transmission via fiber[END_REF][START_REF] Grosche | Transmission of wavelength references via a commercial transparent optical network 2/12[END_REF]. Early optical fiber links use amplitude modulation of an optical carrier to transmit RF and microwave frequencies [START_REF] Daussy | Long-Distance Frequency Dissemination with a Resolution of 10 -17[END_REF][START_REF] Narbonneau | High resolution frequency standard dissemination via optical fiber metropolitan network[END_REF][START_REF] Foreman | Remote transfer of ultrastable frequency references via fiber networks[END_REF][START_REF] Lopez | 86-km optical link with a resolution of 2×10 -18 for RF frequency transfer[END_REF]. However to take full advantage of the optical fiber media the direct optical frequency transfer is the appropriate choice. Indeed, thanks to heterodyne detection the effect of link attenuation halves in dB, and, due to the higher carrier frequency the link-induced phase noise can be detected with much higher resolution. Since 2007, optical frequency transfer over a fiber link of more than 100 km has been reported by several groups [START_REF] Grosche | Transmission of an Optical Carrier Frequency over a Telecommunication Fiber Link[END_REF][START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF][START_REF] Williams | High-stability transfer of an optical frequency over long fiber-optic links[END_REF][START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF][START_REF] Musha | Coherent optical frequency transfer over 50-km physical distance using a 120-km-long installed fiber network[END_REF][START_REF] Grosche | Measurement noise floor for a long-distance optical carrier transmission via fiber[END_REF] and has demonstrated the feasibility of a full optical link with instability in the 10 - 17 to 10 -19 range. These experiments used dedicated dark fibers (i.e. not currently used for telecommunications) in urban environment. Since last year a long link of about 1000 km has connected several German research laboratories using dark fibers from the German national telecommunication network for research and education (DFN) and the stabilization of this link is currently under development [START_REF] Grosche | Measurement noise floor for a long-distance optical carrier transmission via fiber[END_REF]. These are the first milestones towards continental scale fiber links, but this approach requires dedicated fibers between every laboratory to be connected.

2/12/2009 3

In this paper, we report a new approach for ultrastable frequency transfer which consists in using the already existing optical telecommunication network and transmitting the optical frequency reference together with the internet data traffic, as already realized for very low resolution frequency dissemination [START_REF] Grosche | Transmission of wavelength references via a commercial transparent optical network 2/12[END_REF]. We demonstrate here the transmission of an optical frequency reference over 108 km by extending 86 km of dedicated fiber with 22 km of fiber from the French National Telecommunication network for Technology, Education and Research (RENATER) carrying simultaneously digital data. First, we present the scheme of the hybrid link and the interconnection method between the dark fiber span and the RENATER network. The frequency noise and the stability performance are then reported and discussed.

The metrological reference signal (RS) transferred through the link is provided by a sub-Hz-linewidth cavity-stabilized laser emitting at 1542,14 nm (ITU channel #44) [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF]. Fig. 1 shows the overall scheme of the 108 km optical link, which starts and ends at Observatoire de Paris to characterize the link stability, and comprises four spans. The first span is a dedicated dark fiber from the urban network which connects two laboratories, LPL (at University Paris 13, in the north surroundings of Paris) and LNE-SYRTE (at Observatoire de Paris), and extends over 43 km [START_REF] Daussy | Long-Distance Frequency Dissemination with a Resolution of 10 -17[END_REF]. The second and third spans are two 11 km-long fibers connecting the information Interconnection between the 43 km dark fibers and the optical telecommunication network is not straightforward. Concerning RS transmission, the round-trip propagation of the optical signal is required on the same fiber for noise compensation. For the Internet signal transmission, it is necessary to minimize insertion losses and avoid cross-talk. An elegant solution is to use bidirectional off-the-shelf optical add-drop multiplexer (OADM). This threeport component can insert or extract a specified wavelength from the other wavelengths, with isolation better than 25 dB for an adjacent channel (100 GHz) and better than 40 dB for other channels. The losses are about 1.2 dB for the add/drop channel and below 1 dB for the other channels. Four OADMs are used along the link to insert and extract the RS on the upstream and downstream 11 km fibers. The total attenuation along the link is about 38 dB. Each 43 km fiber introduces 10 dB losses, and the 22 km round-trip between Université Paris 13 and Aubervilliers has a significant loss of 15 dB probably due to the large number of connectors and the OADMs.

The optical input power is about 2 mW at Observatoire de Paris and the power injected into the 11 km fiber at Université Paris 13 is 35 µW.

The link-induced phase noise compensation system (Fig. 2) is based on our previous system described in [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF]. The only change concerns the use of a bidirectional amplifier at the end of the link. At the remote end signal is separated into two parts, one is mixed with the laser signal to evaluate the transfer performance and the other part is sent back to the input end. At the input end, the link-induced phase noise is obtained from the beat-note between round-trip signal and the input signal. After filtering and amplification, the link phase noise is actively cancelled through a phase-lock loop using acoustooptic modulator as an actuator.

As a first result, the Internet traffic was unaffected during the whole period of the test (about 3 weeks). The bit error rate was continuously monitored and no error was detected. This was a crucial point since the University Paris 13 is the access point of a metropolitan area network of about 100 km serving hospitals, high schools and universities.

The link stability performance is evaluated from the end-to-end beat-note signal. The optical phase noise power spectral density of the 108 km link is shown in Fig. 3, without and with compensation. The link phase noise rolls down very sharply after a few hundred Hz. The phase noise reduction reaches around 65 dB at 1 Hz, which is coherent with the fact that, at low frequency, the rejection is limited by the link delay and scales as   2 trip f t where trip t 0.54ms  is the 108 km trip time [START_REF] Newbury | Coherent transfer of an optical carrier over 251 km[END_REF]. Note that the laser phase noise is sufficiently low to prevent any limitation to the performance of the stabilization [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF]. Fig. 4 shows the fractional frequency Allan deviation of the 108 km link for four days of continuous operation (red circles). It was measured with a -type dead-time free frequency counter from the end-to-end beat-note signal.

Signal was filtered with a ~10 Hz bandwidth tracking filter. The free running fiber frequency noise is measured simultaneously from the frequency instability of the compensation signal (open circles). The Allan deviation is 410 -16 at 1 s averaging time and scales down with a 1/ slope from 1 s to 5000 s. After 10000 s of averaging time, it reaches a floor around 710 -20 .

Without filtering, the Allan deviation is 5 times higher between 1 s and 5000 s.

The comparison between the 86 km dedicated link and the present 108 km hybrid link allows insight into the phase noise contribution of the two 11 km fibers from the Internet network. According to Fig. 3, the lineic free running phase noise power spectral density of the 22 km RENATER fibers at 1 Hz (14 rad 2 /Hz/km) is 10 times higher than that of the 86 km fiber. By the same time the delay-limited rejection should decrease proportionally to the square of the lengths ratio (2 dB). This is not clearly noticeable from Fig. 3 because the link noise rejection is limited by the delay only below 10 Hz or less [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF]. With compensation, the integrated phase noise from 1 Hz to 1 kHz is increased from 0.4 to 1.5 rad (rms) for the 86 km and 108 km links respectively. Concerning the Allan deviation, the additional noise of the two 11 km fibers increases the 108 km link instability by a factor slightly below 2 between 1 and 40 s compared to the 86 km link (see Fig. 4). For longer time, both stabilities are limited by the noise floor of the compensation system. Note that the 86 km long-term stability was improved compared to previous results [START_REF] Jiang | Long-distance frequency transfer over an urban fiber link using optical phase stabilization[END_REF] due to a better stability of the measurement interferometer. However, comparison with other link noise measurements [START_REF] Williams | High-stability transfer of an optical frequency over long fiber-optic links[END_REF][START_REF] Musha | Coherent optical frequency transfer over 50-km physical distance using a 120-km-long installed fiber network[END_REF][START_REF] Grosche | Measurement noise floor for a long-distance optical carrier transmission via fiber[END_REF] shows that the lineic phase noise of a dark fiber can span over three orders of magnitude (0.07.rad 2 /Hz/km [START_REF] Grosche | Measurement noise floor for a long-distance optical carrier transmission via fiber[END_REF] to 180 rad 2 /Hz/km [START_REF] Musha | Coherent optical frequency transfer over 50-km physical distance using a 120-km-long installed fiber network[END_REF] at 1 Hz), with a noise roll off law between f -2 and f -3/2 in 1Hz-100 Hz region. Moreover, it varies slightly in time. Thus any consideration concerning the noise scaling with the link length is very difficult.

This 108 km link represents a first step towards long-distance frequency transfer using fiber network infrastructure carrying simultaneously internet traffic. For longer distances, an obvious solution consists in splitting the link into shorter compensated segments and developing intermediate stations. Each one should achieve three functions, to send back part of the received signal to the previous station, to amplify and filter the received signal ("repeater" function), and to compensate the phase noise induced by the next segment. The repeater could be a laser phaselocked to the incoming signal. However, use of a standard telecommunication network brings several difficulties. As already pointed out, Internet traffic is unidirectional, whereas our transfer system requires bidirectional propagation inside each fiber segment. Second, standard telecommunication networks use optical to electrical conversion of data signal in all nodes and regenerators, whereas RS dissemination has to be purely optical from end to end. Therefore most of the network equipment would have to be equipped by a specific bypass for the RS channel.

Finally, there is highly restricted access to the fiber network infrastructure, and the remote intermediate stations have to work independently without human intervention.

We have demonstrated an ultra-stable optical link which uses partly an optical telecommunication network simultaneously carrying Internet data. Frequency transfer was demonstrated with instability of 410 -16 at 1 s and integrates down to 710 -20 after 10000 s. Such an optical link allows the transfer of present and near future optical clocks without any stability degradation. Further steps will be to take advantage of the RENATER 600 km fibers from Aubervilliers to the German border near Strasbourg, where cross-border fibers connect the two French and German scientific networks. The total attenuation of the various segments is 167 dB.

Hence several intermediate stations should be implemented in order to regenerate the signal and increase the control bandwidth. The demonstration of such a transnational 600 km dual link is very challenging for the future development of continental-scale wide-range frequency transfer. over 534 km", in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2004), paper CTuH4.

Figure captions

  services and technology center of Université Paris 13 to a node of the French National Telecommunications network for Technology, Education and Research (RENATER) located in Aubervilliers. The digital stream between Université Paris 13 and Aubervilliers node is encoded on an optical carrier on the ITU channel #34 (1550,12 nm) within these two upstream and downstream fibers using 10 GBit/s data stream technology. At Aubervilliers the RS arriving from one fiber is sent into the other 11 km fiber toward Université Paris 13 and then into a second 43 km dark fiber linking Université Paris 13 to Observatoire de Paris.

Figure 1 :Figure 2 :

 12 Figure 1: Schematic of the optical frequency transfer over dedicated and non-dedicated fibers

Figure 3 :

 3 Figure 3: Phase noise power spectral density of the (a) free running 108 km, (b) compensated

Figure 4 :

 4 Figure 4: Fractional frequency instability of the 86 km (open squares) and 108 km (open circles)

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1

The authors are deeply grateful to D. Vandromme and F. Simon from GIP RENATER, and J. F.

Florence from Université Paris 13, for their support in using the Internet network between University Paris 13 and Aubervilliers. We acknowledge funding support from the Agence Nationale de la Recherche (ANR BLAN06-3_144016).

2/12/2009


Spin dependent charge transfer in

  MoSe 2 /hBN/Ni hybrid structures H. Tornatzky, C. Robert, Pierre Renucci, B. Han, Thomas Blon, Benjamin Lassagne, G. Ballon, Yuan Lu, K. Watanabe, T. Taniguchi, et al. To cite this version: H. Tornatzky, C. Robert, Pierre Renucci, B. Han, Thomas Blon, et al.. Spin dependent charge transfer in MoSe 2 /hBN/Ni hybrid structures. Applied Physics Letters, 2021, 119 (26), pp.263103. 10.1063/5.0075554. hal-03609118