Marie Doumic

Marc Hoffmann
email: hoff-mann@ceremade.dauphine.fr

Nathalie Krell
email: nathalie.krell@univ-rennes1.fr

Lydia Robert
email: lydia.robert@upmc.fr

Statistical estimation of a growth-fragmentation model observed on a genealogical tree

Keywords: Growth-fragmentation, cell division equation, Nonparametric estimation, Markov chain on a tree. Mathematical Subject Classification: 35A05, 35B40, 45C05, 45K05, 82D60, 92D25, 62G05, 62G20

Introduction

1.1 Size-structured models and their inference Growth-fragmentation models and structured population equations describe the temporal evolution of a population characterised by state variables such as age, size, growth, maturity, protein content and so on -see [START_REF]The dynamics of physiologically structured populations[END_REF][START_REF] Perthame | Transport equations arising in biology[END_REF] and the references therein. This field continues to grow in interest as its applications appear to be wider and wider, ranging from the internet TCP/IP window size protocol [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF] to neuronal activity [START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF], protein polymerization [START_REF] Engler | Analysis of a model for the dynamics of prions ii[END_REF], cell division cycle [START_REF] Banks | Estimation of cell proliferation dynamics using cfse data[END_REF], phase transitions [START_REF] Niethammer | Non-self-similar behavior in the LSW theory of Ostwald ripening[END_REF], parasite proliferation [START_REF] Bansaye | Proliferating parasites in dividing cells: Kimmels branching model revisited[END_REF] etc.

In order to quantitatively fit experimental observations and thus validate the relevance of the models, developing new and well-adapted statistical methods appears to be one of the major challenge for the coming years. A paradigmatic example, which can serve both as a toy model and a proof of concept for the methodology we develop here, is given by the growthfragmentation or size-structured cell division equation [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF]. When applied to the evolution of a bacterial population it reads    ∂ t n(t, x) + τ ∂ x x n(t, x) + B(x)n(t, x) = 4B(2x)n(t, 2x) n(0, x) = n (0) (x), x ≥ 0, [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF] and it quantifies the concentration n(t, x) of individuals (cells) having size x (the state variable) at time t. A common stochastic mechanism for every single cell is attached to Equation (1):

1. The size x = x(t) of a cell at time t evolves exponentially according to the deterministic evolution dx(t) = τ x(t)dt, where τ > 0 is the growth rate of each cell, that quantifies their ability to ingest a common nutrient.

2. Each cell splits into two offsprings according to a division rate B(x) that depends on its current size x.

3. At division, a cell of size x gives birth to two offsprings of size x/2 each, what is called binary fission.

Model (1) is thus entirely determined by the parameters τ, B(x), x ∈ [0, ∞) . Typically, the growth rate τ is assumed to be known or guessed [START_REF] Doumic | On the calibration of a sizestructured population model from experimental data[END_REF], and thus inference about (1) mainly concerns the estimation of the division rate B(x) that has to be taken from a nonparametric perspective.

By use of the general relative entropy principle, Michel, Mischler and Perthame showed that the approximation n(t, x)e -λ 0 t ≈ N (x) is valid [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF], with λ 0 > 0, and where (λ 0 , N ) is the dominant eigenpair related to the corresponding eigenvalue problem, see [START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF][START_REF] Perthame | Transport equations arising in biology[END_REF][START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF][START_REF] Laurencot | Exponential decay for the growthfragmentation/cell-division equation[END_REF][START_REF] Cáceres | Rate of convergence to the remarkable state for fragmentation and growth-fragmentation equations[END_REF][START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF]. The "stationary" density N (x) of typical cells after some time has elapsed enables to recover (B(x), x ∈ D for a compact D ⊂ (0, ∞) by means of the regularisation of an inverse problem of ill-posedness degree 1. From a deterministic perspective, this is carried out in [START_REF] Perthame | On the inverse problem for a sizestructured population model[END_REF][START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF][START_REF] Doumic | Estimating the division rate for the growthfragmentation equation[END_REF]. From a statistical inference perspective, if an n-sample of the distribution N (x) is observed and if B(x) has smoothness s > 0 in a Sobolev sense, it is proved in [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] that B(x) can be recovered in squared-error loss over compact sets with a rate of convergence n -s/(2s+3) . Both deterministic and stochastic methodology of [START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF] and [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] are motivated by experimental designs and data such as in [START_REF] Kubitschek | Growth during the bacterial cell cycle: Analysis of cell size distribution[END_REF][START_REF] Doumic | On the calibration of a sizestructured population model from experimental data[END_REF]. However, they do not take into account the following two important aspects:

• Bacterial growth exhibits variations in the individual growth rate τ as demonstrated for instance in [START_REF] Sturm | The cost of virulence: Retarded growth of salmonella typhimurium cells expressing type iii secretion system 1[END_REF]. One would like to incorporate variability in the growth rate within the system at the level of a single cell. This requires to modify Model [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF].

• Recent evolution of experimental technology enables to track the whole genealogy of cell populations (along prescribed lines of descendants for instance), affording the observation of other state variables such as size at division, lifetime of a single individual and so on [START_REF] Wang | Robust growth of Escherichia coli[END_REF]. Making the best possible use of such measures is of great potential impact, and needs a complementary approach.

The availability of observation schemes at the level of cell individuals suggests an enhancement of the statistical inference of B(x), x ∈ D , enabling to improve on the rates of convergence obtained by indirect measurements such as in [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF][START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF]. This is the purpose of the present paper. We focus on bacterial growth, for which we apply our method on experimental observations. This serves as a proof of concept for the relevance of our modelling and statistical methodology, which could adapt to other application fields and growth-fragmentation types.

Results of the paper

Statistical setting

Let

U = ∞ k=0 {0, 1} k
denote the binary genealogical tree (with {0, 1} 0 := {∅}). We identify each node u ∈ U with a cell that has a size at birth ξ u and a lifetime ζ u . In the paper, we consider the problem of estimating B(x), x ∈ [0, ∞) over compact sets of (0, ∞). Our inference procedure is based on the observation of (ξ u , ζ u ), u ∈ U n .

(

) 2
where U n ⊂ U denotes a connected subset of size n containing the root u = ∅. Asymptotics are taken as n → ∞. Two important observation schemes are considered: the sparse tree case, when we follow the system along a given branch with n individuals, and the full tree case, where we follow the evolution of the whole binary tree up to the N n -th generation, with N n ≈ log 2 n. In this setting, we are able to generalise Model [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF] and allow the growth rate τ to vary with each cell u ∈ U. We assume that a given cell u has a random growth rate τ u = v ∈ E ⊂ (0, ∞) (later constrained to live on a compact set). Moreover, this value v is inherited from the growth rate v of its parent according to a distribution ρ(v , dv). Since a cell splits into two offsprings of the same size, letting u -denote the parent of u, we have the fundamental relationship

2 ξ u = ξ u -exp τ u -ζ u - (3)
that enables to recover the growth rate τ u of each individual in U n since U n is connected by assumption, possibly leaving out the last generation of observed individuals, but this has asymptotically no effect on a large sample size approach.

Variability in growth rate

In the case where the growth rate can vary for each cell, the density n(t, x) of cells of size x at time t does not follow Eq. ( 1) anymore and an extended framework needs to be considered. To that end, we structure the system with an additional variable τ u = v, which represents the growth rate and depends on each individual cell u. We construct in Section 2 a branching Markov chain (ξ u , τ u ), u ∈ U that incorporates variability for the growth rate in the mechanism described in Section 1.1. Equivalently to the genealogical tree, the system may be described in continuous time by a piecewise deterministic Markov process

X(t), V (t) = X 1 (t), V 1 (t) , X 2 (t), V 2 (t) , . . . ,
which models the process of sizes and growth rates of the living particles in the system at time t, with value in ∞ k=0 S k , where S = [0, ∞) × E is the state space of size times growth rate. Stochastic systems of this kind that correspond to branching Markov chains are fairly well known, both from a theoretical angle and in applications; a selected list of contributions is [START_REF] Bansaye | Limit theorems for Markov processes indexed by supercritical Galton Watson tree[END_REF][START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF][START_REF] Meyn | Markov chains and stochastic stability[END_REF] and the references therein.

By fragmentation techniques inspired by Bertoin [START_REF] Bertoin | Random fragmentation and Coagulation Processes[END_REF], see also Haas [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF], we relate the process (X, V ) to a growth-fragmentation equation as follows. Define

n(t, •), ϕ = E ∞ i=1 ϕ X i (t), V i (t)
as the expectation of the empirical measure of the process (X, V ) over smooth test functions defined on S. We prove in Theorem 1 that, under appropriate regularity conditions, the measure n(t, •) that we identify with the temporal evolution of the density n(t, x, v) of cells having size x and growth rate v at time t is governed (in a weak sense 1 ) by

           ∂ t n(t, x, v) + v ∂ x x n(t, x, v) + B(x)n(t, x, v) = 4B(2x) E ρ(v , v)n(t, 2x, dv ), n(0, x, v) = n (0) (x, v), x ≥ 0. (4)
If we assume a constant growth rate τ > 0, we then take ρ(v , dv) = δ τ (dv) (where δ denotes the Dirac mass) and we retrieve the standard growth-1 For every t ≥ 0, we actually have a Radon measure n(t, dx, dv

) on S = [0, ∞) × E: If ϕ(x, v
) is a function defined on S, we define n(t, •), ϕ = S ϕ(x, v)n(t, dx, dv) whenever the integral is meaningful. Thus (4) has the following sense: for every sufficiently smooth test function ϕ with compact support in E, we have

S ∂tn(t, dx, dv)ϕ(x, v) -vxn(t, dx, dv)∂xϕ(x, v) + B(x)n(t, dx, dv)ϕ(x, v) = 4 S B(2x) E ρ(v , dv)n(t, 2dx, dv )ϕ(x, v) .
fragmentation equation [START_REF] Baccelli | A mean-field model for multiple tcp connections through a buffer implementing red[END_REF]. The proof of Theorem 1 is obtained via a socalled many-to-one formula, established in Proposition 3 in Section 5.1. Indeed, thanks to the branching property of the system, it is possible to relate the behaviour of additive functionals like the mean empirical measure to the behaviour of a so-called tagged cell (like a tagged fragment in fragmentation process), that consists in following the behaviour of a single line of descendants along a branch where each node is picked at random, according to a uniform distribution. This approach, inspired by fragmentation techniques, is quite specific to our model and enables to obtain a relatively direct proof of Theorem 4.

Nonparametric estimation of the growth rate

In Section 3 we take over the problem of estimating (B(x), x ∈ D) for some compact D ⊂ (0, ∞). We assume we have data of the form (2), and that the mean evolution of the system is governed by [START_REF] Bansaye | Proliferating parasites in dividing cells: Kimmels branching model revisited[END_REF]. The growth rate kernel ρ is unknown and treated as a nuisance parameter. A fundamental object is the transition kernel

P B (x, dx ) = P (ξ u , τ u ) ∈ dx (ξ u -, τ u -) = x
of the size and growth rate distribution (ξ u , τ u ) at the birth of a descendant u ∈ U, given the size of birth and growth rate of its parent (ξ u -, τ u -). We define in Section 3.3 a class of division rates and growth rate kernels such that if (B, ρ) belongs to this class, then the transition P B is geometrically ergodic and has a unique invariant measure ν B (dx) = ν B (x, dv)dx. From the invariant measure equation

ν B P B = ν B
we obtain in Proposition 2 the explicit representation

B(x) = x 2 ν B (x/2) E ν B 1 τ u -1 {ξ u -≤x, ξu≥x/2} . ( 5
)
where ν B (x) = E ν B (x, dv) denotes the first marginal of the invariant distribution ν B . A strategy for constructing and estimator B consists in replacing the right-hand size of (5) by its empirical counterpart, the numerator being estimated via a kernel estimator of the first maginal of ν B (dx). Under local Hölder smoothness assumption on B of order s > 0, we prove in Theorem 2 that for a suitable choice of bandwidth in the estimation of the invariant density, our estimator achieves the rate n -s/(2s+1) in squared-error loss over appropriate compact sets D ⊂ (0, ∞), up to an inessential logarithmic term when the full tree observation scheme is considered. We see in particular that we improve on the rate obtained in [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF]. Our result quantifies the improvement obtained when estimating B(x) from data (ξ u , ζ u ), u ∈ U n , as opposed to overall measurements of the system after some time has elapsed as in [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF]. We provide a quantitative argument based on the analysis of a PDE that explains the reduction of ill-posedness achieved by our method over [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] in Section 4.2.

In order to obtain the upper bound of Theorem 2, a major technical difficulty is that we need to establish uniform rates of convergence of the empirical counterparts to their limits in the numerator and denominator of (5) when the data are spread along a binary tree. This can be done via covariance inequalities that exploit the fact that the transition P B is geometrically ergodic (Proposition 4) using standard Markov techniques, see [START_REF] Meyn | Markov chains and stochastic stability[END_REF][START_REF] Baxendale | Renewal theory and computable convergence rates for geometrically ergodic Markov chains[END_REF]. The associated chain is however not reversible, and this yields an extraneous difficulty: the decay of the correlations between ϕ(ξ u , τ u ) and ϕ(ξ v , τ v ) for two nodes u, v ∈ U n are expressed in terms of the sup-norm of ϕ, whenever |ϕ(x)| ≤ V(x) is dominated by a certain Lyapunov function V for the transition P B . However, the typical functions ϕ we use are kernels that depend on n and that are not uniformly bounded in sup-norm as n → ∞. This partly explains the relative length of the technical Sections 5.5 and 5.6.

Organisation of the paper

In Section 2, we construct the model (ξ u , τ u ), u ∈ U of sizes and growth rates of the cells as a Markov chain along the genealogical tree. The discrete model can be embedded into a continuous time piecewise deterministic Markov process (X, V ) of sizes and growth rates of the cells present at any time within the system. In Theorem 1 we explicit the relation between the mean empirical measure of (X, V ) and the growth-fragmentation type equation 4. In Section 3, we explicitly construct an estimator B n of B by means of the representation given by (5) in Section 3.2. Two observation schemes are considered and discussed in Section 3.1, whether we consider data along a single branch (the sparse tree case) or along the whole genealogy (the full tree case). The specific assumptions and the class of admissible division rates B and growth rate kernels ρ are discussed in Section 3.3, and an upper bound for B n in squared-error loss is given in our main Theorem 2. Section 4 shows and discusses the numerical implementation of our method on simulated data. In particular, ignoring the variability in the reconstruction dramatically deterioriates the accuracy of estimation of B. We also explain from a deterministic point perspective the rate improvement of our method compared with [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] by means of a PDE analysis argument in Section 4.2. The parameters are inspired from real data experiments on Escherichia coli cell cultures. Section 5 is devoted to the proofs.

A Markov model on a tree 2.1 The genealogical construction

Recall that U := ∞ n=0 {0, 1} n (with {0, 1} 0 := {∅}) denotes the infinite binary genealogical tree. Each node u ∈ U is identified with a cell of the population and has a mark

(ξ u , b u , ζ u , τ u ),
where ξ u is the size at birth, τ u the growth rate, b u the birthtime and ζ u the lifetime of u. The evolution

ξ u t , t ∈ [b u , b u + ζ u )
of the size of u during its lifetime is governed by

ξ u t = ξ u exp τ u (t -b u ) for t ∈ [b u , b u + ζ u ). (6)
Each cell splits into two offsprings of the same size according to a division rate B(x) for x ∈ (0, ∞). Equivalently

P ζ u ∈ [t, t + dt] ζ u ≥ t, ξ u = x, τ u = v = B x exp(vt) dt. (7)
At division, a cell splits into two offsprings of the same size. If u -denotes the parent of u, we thus have

2 ξ u = ξ u -exp τ u -ζ u - (8)
Finally, the growth rate τ u of u is inherited from its parent τ u -according to a Markov kernel

ρ(v, dv ) = P(τ u ∈ dv | τ u -= v), (9)
where v > 0 and ρ(v, dv ) is a probability measure on (0, ∞) for each v > 0. Eq. ( 6), ( 7), ( 8) and ( 9) completely determine the dynamics of the model (ξ u , τ u ), u ∈ U , as a Markov chain on a tree, given an additional initial condition (ξ ∅ , τ ∅ ) on the root. The chain is embedded into a piecewise deterministic continuous Markov process thanks to (6) by setting

(ξ u t , τ u t ) = ξ u exp τ u (t -b u ) , τ u for t ∈ [b u , b u + ζ u )
and (0, 0) otherwise. Define

X(t), V (t) = X 1 (t), V 1 (t) , X 2 (t), V 2 (t) , . . .
as the process of sizes and growth rates of the living particles in the system at time t. We have an identity between point measures

∞ i=1 1 {X i (t)>0} δ (X i (t),V i (t)) = u∈U 1 {bu≤t<bu+ζu} δ (ξ u t ,τ u t ) (10)
where δ denotes the Dirac mass. In the sequel, the following basic assumption is in force.

Assumption 1 (Basic assumption on B and ρ). The division rate x B(x) is continuous. We have

B(0) = 0 and ∞ x -1 B(x)dx = ∞. The Markov kernel ρ(v, dv ) is defined on a compact set E ⊂ (0, ∞).
Proposition 1. Work under Assumption 1. The law of (X(t), V (t)), t ≥ 0 or (ξ u , τ u ), u ∈ U or (ξ u t , ζ u t ), t ≥ 0, u ∈ U is well-defined on an appropriate probability space with almost-surely no accumulation of jumps.

If µ is a probability measure on the state space S = [0, ∞) × E, we shall denote indifferently by P µ the law of any of the three processes above where the root (ξ ∅ , τ ∅ ) has distribution µ. The construction is classical (see for instance [START_REF] Bertoin | Random fragmentation and Coagulation Processes[END_REF] and the references therein) and is outlined in Appendix 6.1.

Behaviour of the mean empirical measure

Denote by C 1 0 (S) the set of real-valued test functions with compact support in the interior of S.

Theorem 1 (Behaviour of the mean empirical measure). Work under Assumption 1. Let µ be a probability distribution on S. Define the distribution n(t, dx, dv) by

n(t, •), ϕ = E µ ∞ i=1 ϕ X i (t), V i (t) for every ϕ ∈ C 1 0 (S).
Then n(t, •) solves (in a weak sense)

           ∂ t n(t, x, v) + v ∂ x xn(t, x, v) + B(x)n(t, x, v) = 4B(2x) E ρ(v , v)n(t, 2x, dv ), n(0, x, v) = n (0) (x, v), x ≥ 0.
with initial condition n (0) (dx, dv) = µ(dx, dv).

Theorem 1 somehow legitimates our methodology: by enabling each cell to have its own growth rate and by building-up new statistical estimators in this context, we still have a translation in terms of the approach in [START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF]. In particular, we will be able to compare our estimation results with [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF]. Our proof is based on fragmentation techniques, inspired by Bertoin [START_REF] Bertoin | Random fragmentation and Coagulation Processes[END_REF] and Haas [START_REF] Haas | Loss of mass in deterministic and random fragmentations[END_REF]. Alternative approaches to the same kind of questions include the probabilistic studies of Chauvin et al. [START_REF] Chauvin | Growing conditioned trees[END_REF], Bansaye et al. [START_REF] Bansaye | Limit theorems for Markov processes indexed by supercritical Galton Watson tree[END_REF] or Harris and Roberts [START_REF] Harris | The many-to-few lemma and multiple spines[END_REF] and references therein.

Statistical estimation of the division rate 3.1 Two observation schemes

Let U n ⊂ U denote a subset of size n of connected nodes: if u belongs to U n , so does its parent u -. We look for a nonparametric estimator of the division rate

y B n (y) = B n y, (ξ u , τ u ), u ∈ U n ) for y ∈ (0, ∞)
Statistical inference is based on the observation scheme

(ξ u , τ u ), u ∈ U n
and asymptotic study is undertaken as the population size of the sample n → ∞. We are interested in two specific observation schemes.

The full tree case. We observe every pair (ξ u , τ u ) over the first N n generations of the tree: andN n is chosen such that that 2 Nn has order n.

U n = {u ∈ U, |u| ≤ N n } with the notation |u| = n if u = (u 0 , u 1 , . . . , u n ) ∈ U,
The sparse tree case. We follow the first n offsprings of a single cell, along a fixed line of descendants. This means that for some u ∈ U with |u| = n, we observe every size ξ u and growth rate τ u of each node (u 0 ), (u 0 , u 1 ), (u 0 , u 1 , u 2 ) and so on up to a final node u = (u 0 , u 1 , . . . , u n ).

Remark 1. For every n ≥ 1, we tacitly assume that there exists a (random) time T n < ∞ almost surely, such that for t ≥ T n , the observation scheme U n is well-defined. This is a consequence of the behaviour of B near infinity that we impose later on in (17) below.

Estimation of the division rate

Identification of the division rate

We denote by x = (x, v) an element of the state space S = [0, ∞) × E.

Introduce the transition kernel

P B (x, dx ) = P (ξ u , τ u ) ∈ dx (ξ u -, τ u -) = x
of the size and growth rate distribution (ξ u , τ u ) at the birth of a descendant u ∈ U, given the size at birth and growth rate of its parent (ξ u -, τ u -). From [START_REF] Bertoin | Random fragmentation and Coagulation Processes[END_REF], we infer that

P(ζ u -∈ dt ξ u -= x, τ u -= v) is equal to B x exp(vt) exp - t 0 B x exp(vs) ds dt.
Using formula [START_REF] Cáceres | Rate of convergence to the remarkable state for fragmentation and growth-fragmentation equations[END_REF], by a simple change of variables

P ξ u ∈ dx ξ u -= x, τ u -= v = B(2x ) vx 1 {x ≥x/2} exp - x x/2 B(2s)
vs ds dx .

Incorporating [START_REF] Chauvin | Growing conditioned trees[END_REF], we obtain an explicit formula for

P B (x, dx ) = P B (x, v), x , dv )dx , with P B (x, v), x , dv ) = B(2x ) vx 1 {x ≥x/2} exp - x x/2 B(2s)
vs ds ρ(v, dv ). ( 11)

Assume further that P B admits an invariant probability measure ν B (dx), i.e. a solution to

ν B P B = ν B , (12)
ν B (y) = B(2y) y E ν B 1 τ u - 1 {ξ u -≤2y, ξu≥y} (13)
where E ν B [•] denotes expectation when the initial condition (ξ ∅ , τ ∅ ) has distribution ν B and where we have set ν B (y) = E ν B (y, dv ) in (13) for the marginal density of the invariant probability measure ν B with respect to y.

We exhibit below a class of division rates B and growth rate kernels ρ that guarantees the existence of such an invariant probability measure.

Construction of a nonparametric estimator

Inverting [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] and applying an appropriate change of variables, we obtain

B(y) = y 2 ν B (y/2) E ν B 1 τ u -1 {ξ u -≤y, ξu≥y/2} , (14)
provided the denominator is positive. This formula has no easy interpretation: it is obtained by some clever manipulation of the equation

ν B = P B ν B .
A tentative interpretation in the simplest case with no variability (so that τ u = τ for some fixed τ > 0 and for every u ∈ U is proposed in Section 4.2. Representation ( 14) also suggests an estimation procedure, replacing the marginal density ν B (y/2) and the expectation in the denominator by their empirical counterparts. To that end, pick a kernel function

K : [0, ∞) → R, [0,∞) K(y)dy = 1, and set K h (y) = h -1 K h -1 y for y ∈ [0, ∞) and h > 0.
Our estimator is defined as

B n (y) = y 2 n -1 u∈Un K h (ξ u -y/2) n -1 u∈Un 1 τ u -1 {ξ u -≤ y, ξ u ≥ y/2} , (15)
where > 0 is a threshold that ensures that the estimator is well defined in all cases and x y = max{x, y}. Thus ( B n (y), y ∈ D) is specified by the choice of the kernel K, the bandwidth h > 0 and the threshold > 0.

Assumption 2. The function K is bounded with compact support, and for some integer

n 0 ≥ 1, we have [0,∞) x k K(x)dx = 1 {k=0} for 0 ≤ k ≤ n 0 .

Error estimates

We assess the quality of B n in squared-loss error over compact intervals D. We need to specify local smoothness properties of B over D, together with general properties that ensure that the empirical measurements in [START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF] converge with an appropriate speed of convergence. This amounts to impose an appropriate behaviour of B near the origin and infinity.

Model constraints

For λ > 0 such that 2 λ > sup E/ inf E > 0 and a vector of positive constants c = (r, m, , L), introduce the class F λ (c) of continuous functions

B : [0, ∞) → [0, ∞) such that r/2 0 x -1 B(2x)dx ≤ L, r r/2 x -1 B(2x)dx ≥ , (16)
and

B(x) ≥ m x λ for x ≥ r. ( 17
)
Remark 2. Similar conditions on the behaviour of B can also be found in [START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF], in a deterministic setting.

Remark 3. Assumption 1 is satisfied as soon as B ∈ F λ (c) (and ρ is defined on a compact E ⊂ (0, ∞) of course).

Let ρ min , ρ max be two positive finite measures on E such that ρ max -ρ min is a positive measure and ρ min (E) > 0. We define M(ρ min , ρ max ) as the class of Markov transitions ρ(v, dv ) on E such that

ρ min (A) ≤ ρ(v, A) ≤ ρ max (A), A ⊂ E, v ∈ E. ( 18
)
Remark 4. Control [START_REF] Fort | Convergence of adaptive and interacting Markov chain Monte Carlo algorithms[END_REF] ensures the geometric ergodicity of the process of variability in the growth rate.

Let us be given in the sequel a vector of positive constants c = (r, m, , L) and 0 < e min ≤ e max such that E ⊂ [e min , e max ]. We introduce the Lyapunov function

V(x, v) = V(x) = exp m e min λ x λ for (x, v) ∈ S. (19)
The function V controls the rate of the geometric ergodicity of the chain with transition P B and will appear in the proof of Proposition 4 below. Define

δ = δ(c) := 1 1 -2 -λ exp -(1 -2 -λ ) m emaxλ r λ ρ max (E).
Assumption 3 (The sparse tree case). Let λ > 0. We have δ(c) < 1.

In the case of the full tree observation scheme, we will need more stringent (and technical) conditions on c. Let γ B,V denote the spectral radius of the operator P B -1 ⊗ ν B acting on the Banach space of functions g :

S → R such that sup{|g(x)|/V(x), x ∈ S} < ∞,
where V is defined in [START_REF] Gobet | Nonparametric estimation of scalar diffusion from low frequency data[END_REF] above.

Assumption 4 (The full tree case). We have δ(c) < 1 2 and moreover sup

B∈F λ (c) γ B,V < 1 2 . ( 20
)
Remark 5. It is possible to obtain bounds on c so that (20) holds, by using explicit (yet intricate) bounds on γ B,V following Fort et al. or [START_REF] Fort | Convergence of adaptive and interacting Markov chain Monte Carlo algorithms[END_REF], Douc et al. [START_REF] Douc | Quantitative bounds on convergence of time-inhomogeneous Markov chains[END_REF], see also Baxendale [START_REF] Baxendale | Renewal theory and computable convergence rates for geometrically ergodic Markov chains[END_REF].

Rate of convergence

We are ready to state our main result. For s > 0, with s = s + {s}, 0 < {s} ≤ 1 and s an integer, introduce the Hölder space H s (D) of functions f : D → R possessing a derivative of order s that satisfies

|f s (y) -f s (x)| ≤ c(f )|x -y| {s} . ( 21
)
The minimal constant c(f ) such that (21) holds defines a semi-norm |f | H s (D) . We equip the space H s (D) with the norm

f H s (D) = f L ∞ (D) + |f | H s (D)
and the Hölder balls

H s (D, M ) = {B, B H s (D) ≤ M }, M > 0.
Theorem 2. Work under Assumption 3 in the sparse tree case and Assumption 4 in the full tree case. Specify B n with a kernel K satisfying Assumption 2 for some n 0 > 0 and

h n = c 0 n -1/(2s+1) , n → 0.
For every M > 0, there exist c 0 = c 0 (c, M ) and d(c) ≥ 0 such that for every 0 < s < n 0 and every compact interval

D ⊂ (d(c), ∞) such that inf D ≥ r/2, we have sup ρ,B E µ B n -B 2 L 2 (D) 1/2 -1 n (log n) 1/2 n -s/(2s+1) ,
where the supremum is taken over

ρ ∈ M(ρ min , ρ max ) and B ∈ F λ (c) ∩ H s (D, M ),
and E µ [•] denotes expectation with respect to any initial distribution µ(dx)

for (ξ ∅ , τ ∅ ) on S such that S V(x) 2 µ(dx) < ∞.
Several remarks are in order: 1) Since n is arbitrary, we obtain the classical rate n -s/(2s+1) (up to a log term) which is optimal in a minimax sense for density estimation. It is presumably optimal in our context, using for instance classical techniques for nonparametric estimation lower bounds on functions of transition densities of Markov chains, see for instance [START_REF] Gobet | Nonparametric estimation of scalar diffusion from low frequency data[END_REF].

2) The extra logarithmic term is due to technical reasons: we need it in order to control the decay of correlations of the observations over the full tree structure. 3) The knowledge of the smoothness s that is needed for the construction of B n is not realistic in practice. An adaptive estimator could be obtained by using a data-driven bandwidth in the estimation of the invariant density ν B (y/2) in [START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF]. The Goldenschluger-Lepski bandwidth selection method [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF], see also [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] would presumably yield adaptation, but checking the assumptions still requires a proof in our setting. We implement data-driven bandwidth in the numerical Section 4 below.

Numerical implementation 4.1 Protocol and results

Generating simulated data

Given a division rate B(x), a growth rate kernel ρ, an initial distribution µ(dx) for the node (ξ ∅ , τ ∅ ) (as in Theorem 2) and a dataset size n = 2 Nn , we simulate the full tree and the sparse tree schemes recursively:

1. Given (ξ u -, τ u -), we select at random its lifetime ζ u -(by a rejection sampling algorithm) with probability density

t B ξ u -exp(τ u -t) exp - t 0 B ξ u -exp(τ u -s) ds .
following the computations of Section 3.2.

2. We derive the size at birth ξ u for the two offsprings (with u = (u -, 0) and (u -, 1) with obvious notation) by Formula (8).

3. We simulate at random the growth rates τ u (by the rejection sampling algorithm) according to the distribution ρ(τ u -, dv).

4.

For the sparse tree case, we select only one offspring (either (u -, 0) of (u -, 1)), whereas we keep both for the full tree case.

In order to stay in line with previous simulations of [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] we pick B(x) = x 2 . We fix µ(dx) as the uniform distribution over

[1/3, 3] × E, with E = [0.2, 3].
As for the growth rate kernel, we implement

ρ(v, dv ) = g(v -v)dv
where g is a uniform distribution over [1 -α, 1 + α] for some α > 0, and dilated by a scaling factor so that (v -v) 2 ρ(v, dv ) 1/2 = 1/2. We also condition the values of τ u to stay in E (by rejection sampling).

Implementing B n

We implement B n using Formula [START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF]. We pick a standard Gaussian kernel K(x) = (2π) -1/2 exp(-x 2 /2), for which n 0 = 1 in Assumption (2); henceforth we expect a rate of convergence of order n -1/3 at best. We evaluate B n on a regular grid x 1 = ∆x, • • • x m , = m∆x with ∆x = n -1/2 and x m = 5. Thus x m is large enough so that ν B (x/2) becomes negligible for x ≥ x m and ∆x is smaller than n -1/3 to avoid numerical discrepancies. For tractability purposes, we wish to avoid the use of any relationship between the nodes u ∈ U n . Indeed, whereas it is quite easy to label u -and u in the sparse tree case, it is a bit more difficult to track the parent of each individual in the full tree case if we do not want to double the memory. As a consequence, we simply reformulate (15) into

B n (y) = y 2 n -1 u∈Un K h (ξ u -y/2) n -1 u∈Un 1 τu 1 {ξ u ≤ y ≤ ξ u e τuζu } . (22)
We take h n = n -1/3 for the bandwidth according to Theorem 2 to serve as a proof of concept. Data-driven choices could of course be made, such as the Goldenschluger and Lepski's method [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality[END_REF][START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF], and improve the already fairly good results shown in Figure 2. Finally, we also test whether taking into account variability in the growth rate improves significantly or not the estimate of B, replacing τ u by its mean value n -1 u∈Un τ u everywhere in Formula ( 22), thus ignoring growth variability in that case.

Numerical results

We display our numerical results as specified above in Figures 1, 2 and3. Figure 1 displays the reconstruction of B on the full tree scheme for a simulated sample of size n = 2 17 . At a visual level, we see that the estimation deteriorates dramatically when the variability is ignored in the region where ν B is small, while our estimator [START_REF] Harris | The many-to-few lemma and multiple spines[END_REF] still shows good performances. In Figure 2, we plot on a log-log scale the empirical mean error of our estimation procedure for both full tree and sparse tree schemes. The numerical results agree with the theory. The empirical error is computed as

e i = B -B ∆x,m B ∆x,m, , i = 1, . . . , M, (23)
where • ∆x,m, denotes the discrete norm over the numerical sampling described above, conditioned on the fact that the denominator in [START_REF] Harris | The many-to-few lemma and multiple spines[END_REF] is larger than / log(n). We end up with a mean-empirical error e = M -1 M i=1 e i . The number of Monte-Carlo samples is chosen as M = 100. In Figure 3, we explore further the degradation of the estimation process on the region where ν B is small, plotting 95% confidence intervals of the empirical distribution of the estimates, based on M = 100 Monte-Carlo samples. Finally, Table 4.1 displays the relative error for the reconstruction of B according to [START_REF] Kaern | Stochasticity in gene expression: from theories to phenotypes[END_REF]. The standard deviation is computed as (M -1 M i=1 e i -e 2 ) 1/2 . We also carried out control experiments for other choices of variability kernel ρ(v, dv ) for the growth rate. These include ρ(v, dv ) = g(v )dv , so that the variability of an individual is not inherited from its parent, a Gaussian density for g with the same prescription for the mean and the variance as in the uniform case, conditioned to live on [e min , e max ]. We also tested the log 2 (n) Relative error e for B and its standard deviation, with respect to n (on a log scale). The error is computed using (23) with = 1/ log(n). absence of variability, with ρ(v, dv ) = δ τ (dv ), with τ = 1. None of these control experiments show any significant difference from the case displayed in Figures 1, 2 and3.

Analysis on E. coli data

Finally, we analyse a dataset obtained through microscopic time-lapse imaging of single bacterial cells growing in rich medium, by Wang, Robert et al. [START_REF] Wang | Robust growth of Escherichia coli[END_REF] and by Stewart et al. [START_REF] Stewart | Aging and death in an organism that reproduces by morphologically symmetric division[END_REF]. Thanks to a microfluidic set-up, the experimental conditions are well controlled and stable, so that the cells are in a steady state of growth (so-called balanced growth). The observation scheme corresponds to the sparse tree case for the data from Wang, Robert et al.: at each generation, only one offspring is followed. On the contrary, data corresponds to the full tree case for the data by Stewart et al., where the cells grow in a culture. The growth and division of the cells is followed by microscopy, and image analysis allows to determine the time evolution of the size of each cell, from birth to division. We picked up the quantities of interest for our implementation: for each cell, its size at birth, growth rate and lifetime. We consider that cells divide equally into two daughter cells, neglecting the small differences of size at birth between daughter cells. Each cell grows exponentially fast, but growth rates exhibit variability.

Our data is formed by the concatenation of several lineages, each of them composed with a line of offsprings coming from a first single cell picked at random in a culture. Some of the first and last generations were not considered in order to avoid any experimental disturbance linked either to non stationary conditions or to aging of the cells.

We proceed as in the above protocol. Figure 4 shows the reconstructed B and ν B for a sample of n = 2335 cells for the sparse tree data, n = 748 for the full tree data. Though much more precise and reliable, thanks both to the experimental device and the reconstruction method, our results are qualitatively in accordance with previous indirect reconstructions carried out in [START_REF] Doumic | On the calibration of a sizestructured population model from experimental data[END_REF] on old datasets published in [START_REF] Kubitschek | Growth during the bacterial cell cycle: Analysis of cell size distribution[END_REF] back in 1969.

The reconstruction of the division rate is prominent here since it appears to be the last component needed for a full calibration of the model. Thus, our method provides biologists with a complete understanding of the size dependence of the biological system. Phenotypic variability between genetically identical cells has recently received growing attention with the recognition that it can be genetically controlled and subject to selection pressures [START_REF] Kaern | Stochasticity in gene expression: from theories to phenotypes[END_REF]. Our mathematical framework allows the incorporation of this variability at the level of individual growth rates. It should allow the study of the impact of variability on the population fitness and should be of particular importance to describe the growth of populations of cells exhibiting high variability of growth rates. Several examples of high variability have been described, both in genetically engineered or natural bacterial populations [START_REF] Sturm | The cost of virulence: Retarded growth of salmonella typhimurium cells expressing type iii secretion system 1[END_REF][START_REF] Tan | Emergent bistability by a growth-modulating positive feedback circuit[END_REF].

Link with the deterministic viewpoint

Considering the reconstruction formula [START_REF] Doumic | Numerical solution of an inverse problem in size-structured population dynamics[END_REF], let us give here some insight from a deterministic analysis perspective. For the sake of clarity, let us focus on the simpler case when there is no variability, so that for all u ∈ U n we have τ u = τ > 0 a fixed constant. Formula (15) comes from [START_REF] Doumic | On the calibration of a sizestructured population model from experimental data[END_REF], which in ) and ν B (solid line and dash-dotted line resp.) on experimental data of E. coli dividing cells, for resp. a sparse tree and a full tree experiment. n = 2335 for the sparse tree, n = 748 for the full tree. The experimental conditions being different (temperature and nutrient), these results are not supposed to be identical, yet a generic pattern appears for B, that could serve as a basis for future biological studies.

the case τ u = τ simplifies further into

B(y) = τ y 2 ν B (y/2) E ν B 1 {ξ u -≤y, ξu≥y/2} = τ y 2 ν B (y/2) y y/2 ν B (z)dz . ( 24
)
We also notice that, in this particular case, we do not need to measure the lifetime of each cell in order to implement [START_REF] Kubitschek | Growth during the bacterial cell cycle: Analysis of cell size distribution[END_REF]. Define N (y) = 1 2 ν B (y/2) B(y) , or equivalently ν B (x) = 2B(2x)N (2x). Differentiating [START_REF] Kubitschek | Growth during the bacterial cell cycle: Analysis of cell size distribution[END_REF], we obtain

∂ x (τ xN ) = 2B(2x)N (2x) -B(x)N (x)
which corresponds to the stationary state linked to the equation

   ∂ t n(t, x) + τ ∂ x x n(t, x) = 2B(2x)n(t, 2x) -B(x)n(t, x), n(0, x) = n (0) (x), x ≥ 0. ( 25
)
Eq. ( 25) exactly corresponds to the behaviour of the tagged cell of Section 5.1 below, in a (weak) sense:

n(t, dx) = P(χ(t) ∈ dx)
where χ(t) denotes the size at time t along a branch picked at random, see Section 5.1. Existence and uniqueness of an invariant measure ν B has an analogy to the existence of a steady state solution for the PDE [START_REF] Laurencot | Exponential decay for the growthfragmentation/cell-division equation[END_REF], and the convergence of the empirical measure to the invariant rejoins the stability of the steady state [START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF]. The equality ν B (x) = 2B(2x)N (2x) may be interpreted as follows: N (x) is the steady solution of Eq. ( 25), and represents the probability density of a cell population dividing at a rate B and growing at a rate xτ , but when only one offspring remains alive at each division so that the total quantity of cells remains constant. The fraction of dividing cells is represented by the term B(x)N (x) in the equation, with distribution given by 1 2 ν B (x/2), whereas the fraction of newborn cells is 2B(2x)N (2x). Eq. ( 24) can be written in terms of BN as

B(y) = τ yBN (y) 2y y B(z)N (z)dz . (26)
This also highlights why we obtain a rate of convergence of order n -s/(2s+1) rather than the rate n -s/(2s+3) obtained with indirect measurements as in [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF]. In that latter case, we observe a n-sample with distribution N . As shown in [START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF], one differentiation is necessary to estimate B therefore we have a degree of ill-posedness of order 1. In the setting of the present paper, we rather observe a sample with distribution BN , and B can be recovered directly from [START_REF]The dynamics of physiologically structured populations[END_REF] and we have here a degree of ill-posedness of order 0.

Proofs

We first prove Theorem 1 in Sections 5.1 and 5.2. The strategy consists in obtaining a so-called many-to-one formula (Proposition 3) that enables to relate additive functionals of the whole Markov tree to a special onedimensional process that consists of following at random a branch on the tree. It suffices to check in Section 5.2 that this randomly tagged branch integrated against appropriate test functions satisfies the desired transportfragmentation equation. Section 5.3 studies at length the Markov transition P B . We first prove the key representation formula for B obtained in Proposition 2. We then quantify the geometric ergodicity of the model by a standard Lyapunov technique (Proposition 4). In Section 5.5 we subsequently apply the geometric ergodicity of the transition P B by establishing covariance inequalities on a tree in Propositions 5 and 6; these are the crucial tools to later control the convergence rate of the estimator. We need in particular to study the covariance of delta-like functions with supremum norm increasing to infinity with our asymptotic, and this explains the relative technical length of our estimates. This enables to further control in Section 5.6 a rate of convergence for the empirical measure in Propositions 7 and 8. The fact that we work on a tree with an non-reversible Markov transition and delta-like test functions is an extra technical difficulty. Finally, we can prove Theorem 2 in Section 5.7 for the rate of convergence of our estimator with a classical trade-off technique between a bias and a variance term, thanks to the tools developed in the preceding sections and in particular in Section 5.4 where some useful estimates for the invariant measure are established.

The notation means inequality up to a constant that does not depend on n. We set a n ∼ b n when a n b n and b n a n simultaneously. A mapping

f : E → R or g : [0, ∞) → R is implicitly identified as a function on S via f (x, v) = f (x) and g(x, v) = g(v).

A many-to-one formula via a tagged cell

For u ∈ U, we set m i u for the i-th parent along the genealogy of u. Define

τ u t = |u| i=1 τ m i u ζ m i u + τ u t (t -b u ) for t ∈ [b u , b u + ζ u )
and 0 otherwise for the cumulated growth rate along its ancestors up to time t. In the same spirit as tagged fragments in fragmentation processes (see the book by Bertoin [START_REF] Bertoin | Random fragmentation and Coagulation Processes[END_REF] for instance) we pick a branch at random along the genealogical tree at random: for every k ≥ 1, if ϑ k denotes the node of the tagged cell at the k-th generation, we have

P(ϑ k = u) = 2 -k for every u ∈ U such that |u| = k,
and 0 otherwise. For t ≥ 0, the relationship

b ϑ C t ≤ t < b ϑ C t + ζ ϑ C t
uniquely defines a counting process (C t , t ≥ 0) with C 0 = 0. The process C t enables in turn to define a tagged process of size, growth rate and cumulated growth rate via

χ(t), V(t), V(t) = ξ ϑ C t t , τ ϑ C t t , τ ϑ C t t for t ∈ [b ϑ C t , b ϑ C t + ζ ϑ C t )
and 0 otherwise. We have the representation

χ(t) = xe V(t) 2 Ct ( 27
)
and since V(t) ∈ [e min , e max ], we note that

e min t ≤ V(t) ≤ e max t. (28)
The behaviour of χ(t), V(t), V(t) can be related to certain functionals of the whole particle system via a so-called many-to-one formula. This is the key tool to obtain Theorem 1.

Proposition 3 (A many-to-one formula). Work under Assumption 1. For x ∈ (0, ∞), let P x be defined as in Lemma 1. For every t ≥ 0, we have

E x φ χ(t), V(t), V(t) = E x u∈U ξ u t e -τ u t x φ ξ u t , τ u t , τ u t for every φ : S × [0, ∞) → [0, ∞). Proof of Proposition 3. For v ∈ U, set I v = [b v , b v + ζ v )
. By representation [START_REF] Meyn | Markov chains and stochastic stability[END_REF], we have

E x φ χ(t), V(t), V(t) = E x φ xe V(t) 2 C t , V(t), V(t) = E x v∈U φ xe τ v t 2 |v| , τ v t , τ v t 1 {t∈Iv,v=ϑ C t } .
Introduce the discrete filtration H n generated by (ξ u , ζ u , τ u ) for every u such that |u| ≤ n. Conditioning with respect to H |v| and noting that on {t ∈ I v }, we have

P ϑ Ct = v | H |v| = 1 2 |v| = ξ v e -τ v bv x ,
we derive

E x v∈U φ xe τ v t 2 |v| , τ v t , τ v t 1 {t∈Iv,v=ϑ C t } = E x v∈U ξ v e -τ v bv x φ xe τ v t 2 |v| , τ v t , τ v t 1 {t∈Iv} = E x u∈U ξ u t e -τ u t x φ ξ u t , τ u t , τ u t .

Proof of Theorem 1

We fix x ∈ (0, ∞) and prove the result for an initial measure µ x as in Proposition 3. Let ϕ ∈ C 1 0 (S) be nonnegative. By [START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF] we have

n(t, •), ϕ = E x ∞ i=1 ϕ X i (t), Z i (t) = E x u∈U ϕ(ξ u t , τ u t )
and applying Proposition 3, we derive

n(t, •), ϕ = x E x ϕ χ(t), V(t) e V(t) χ(t) . ( 29
)
For h > 0, introduce the difference operator

∆ h f (t) = h -1 f (t + h) -f (t) .
We plan to study the convergence of ∆ h n(t, •), ϕ as h → 0 using representation [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF] in restriction to the events {C t+h -C t = i}, for i = 0, 1 and {C t+h -C t ≥ 2}. Denote by F t the filtration generated by the tagged cell χ(s), V(s), s ≤ t . The following standard estimate proved in Appendix 6.2 will be later useful.

Lemma 1. Assume that B is continuous. Let x ∈ (0, ∞) and let µ x be a probability measure on S such that µ x ({x} × E) = 1. Abbreviate P µx by P x . For small h > 0, we have

P x (C t+h -C t = 1 | F t ) = B χ(t) h + h ε(h),
with the property |ε(h)| ≤ (h) → 0 as h → 0, for some deterministic (h), and

P x (C t+h -C t ≥ 2) h 2 .
Since ϕ ∈ C 1 0 (S), there exists c(ϕ) > 0 such that ϕ(y, v) = 0 if y ≥ c(ϕ). By (28), we infer

ϕ χ(t), V(t) e V(t) χ(t) ≤ sup y,v ϕ(y, v) exp(e max t) c(ϕ) (30)
By Lemma 1 and (30), we derive

E x ∆ h ϕ χ(t), V(t) e V(t) χ(t) 1 {C t+h -Ct≥2} h. (31)
On the event {C t+h -C t = 0}, the process

V(s) is constant for s ∈ [t, t + h)
and so is e V(s) χ(s) thanks to [START_REF] Meyn | Markov chains and stochastic stability[END_REF]. It follows that

∆ h ϕ χ(t), V(t) e V(t) χ(t) = ∆ h ϕ χ(t), V(s) s=t e V(t) χ(t)
on {C t+h -C t = 0} and also

∆ h ϕ χ(t), V(s) s=t e V(t) χ(t) ≤ sup y,v |∂ y ϕ(y, v)|xe max exp(2e max t) c(ϕ) on {C t+h -C t = 0} likewise. Since P x (C t+h -C t = 0) → 1 as h → 0, by dominated convergence x E x ∆ h ϕ χ(t), V(t) e V(t) χ(t) 1 {C t+h -Ct=0} → x E x ∂ 1 ϕ χ(t), V(t) V(t)e V(t) as h → 0. (32)
By Proposition 3 again, this last quantity is equal to n(t, dx, dv), xv ∂ x ϕ . On {C t+h -C t = 1}, we successively have

χ(t + h) = 1 2 χ(t) + ε 1 (h), ϕ χ(t + h), V(t + h) = ϕ χ(t)/2, V(t + h) + ε 2 (h) and exp V(t + h) = exp V(t) + ε 3 (h)
with the property |ε i (h)| ≤ 1 (h) → 0 as h → 0, where 1 (h) is deterministic, thanks to ( 27) and [START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF]. Moreover,

V(t + h) = τ ϑ C t +1 on {C t+h -C t = 1}.
It follows that

E x ϕ χ(t + h), V(t + h) e V(t+h) χ(t + h) 1 {C t+h -Ct=1} = E x ϕ χ(t)/2, τ ϑ C t +1 2e V(t) χ(t) 1 {C t+h -Ct=1} + 2 (h) = E x ϕ χ(t)/2, τ ϑ C t +1 2e V(t) χ(t) 1 {C t+h -Ct≥1} + 3 (h)
where 2 (h), 3 (h) → 0 as h → 0, and where we used the second part of Lemma 1 in order to obtain the last equality. Conditioning with respect to F t τ ϑ C t +1 and using that {C t+h -C t ≥ 1} and τ ϑ C t +1 are independent, applying the first part of Lemma 1, this last term is equal to

E x ϕ χ(t)/2, τ ϑ C t +1 2e V(t) χ(t) B χ(t) h + 4 (h) = E x E ϕ χ(t)/2, v ρ V(t), dv 2e V(t) χ(t) B χ(t) h + 4 (h)
where 4 (h) → 0 as h → 0. Finally, using Lemma 1 again, we derive

E x ∆ h ϕ χ(t), V(t) e V(t) χ(t) 1 {C t+h -Ct=1} → E x E 2ϕ χ(t)/2, v ρ V(t), dv -ϕ χ(t), V(t) e V(t) χ(t) B χ(t) (33)
as h → 0. By Proposition 3, this last quantity is equal to

n(t, dx, dv), E 2ϕ(x/2, v )ρ(v, dv ) -ϕ(x, v) B(x)
which, in turn, is equal to

n(t, 2dx, dv), E 4ϕ(x, v )ρ(v, dv )B(2x) -n(t, dx, dv), ϕ(x, v)B(x)
by a simple change of variables. Putting together the estimates (31), ( 32) and ( 33), we conclude

∂ t n(t, dx, dv), ϕ -n(t, dx, dv), xv∂ x ϕ + n(t, dx, dv)B(x), ϕ = n(t, 2dx, dv), E 4ϕ(x, v )ρ(v, dv )B(2x) ,
which is the dual formulation of (4). The proof is complete.

Geometric ergodicity of the discrete model

We keep up with the notations of Sections 2 and 3. We first prove Proposition 2.

Proof of Proposition 2

The fact that ν B (dx) = ν B (x, dv)dx readily follows from the representation P B (x, dx ) = P B (x, v), x , dv )dx together with the invariant measure equation [START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF]. It follows that for every y ∈ (0, ∞),

ν B (y, dv ) = S ν B (x, dv)dx P B (x, v), y, dv = B(2y) y E 2y 0 ν B (x, dv) exp - y x/2 B(2s) vs ds ρ(v, dv ) v dx.
By Assumption 1, we have

∞ x/2 B(2s) s ds = ∞ hence exp - y x/2 B(2s) vs ds = ∞ y B(2s) vs exp - s x/2 B( 2s
)
vs ds ds,

therefore ν B (y, dv ) is equal to B (2y) y E 2y 0 ν B (x, dv)dx ∞ y B(2s) vs exp - s x/2 B(2s )
vs ds ds

ρ(v, dv ) v = B(2y) y S [0,∞) 1 {x ≤ 2y, s ≥ y} v -1 ν B (x, dv)dx P B (x, v), s, dv ds.
Integrating with respect to dv , we obtain the result.

Geometric ergodicity

We extend P B as an operator acting on functions f : S → [0, ∞) via

P B f (x) = S f (y)P B (x, dy) If k ≥ 1 is an integer, define P k B = P k-1 B • P B .
Proposition 4. Let c satisfy Assumption 3. Then, for every B ∈ F λ (c) and ρ ∈ M(ρ min ), there exists a unique invariant probability measure of the form ν B (dx) = ν B (x, dv)dx on S. Moreover, there exist 0 < γ < 1, a function

V : S → [1, ∞) and a constant R such that sup B∈F λ (c),ρ∈M(ρ min ) sup |g|≤V P k B g(x) - S g(z)ν B (dz) ≤ RV(x)γ k (34)
for every x ∈ S, k ≥ 0, and where the supremum is taken over all functions g : S → R satisfying |g(x)| ≤ V(x) for all x ∈ S. Moreover, under Assumption 4, we can take γ < 1 2 . Finally, the function V is ν B -integrable for every B ∈ F λ (c) and (34) is well defined.

We will show in the proof that the function V defined in [START_REF] Gobet | Nonparametric estimation of scalar diffusion from low frequency data[END_REF] satisfies the properties announced in Proposition 4.

Proof of Proposition 4

We follow the classical line of establishing successively a condition of minorisation, strong aperiodicity and drift for the transition operator P B (see for instance [START_REF] Meyn | Markov chains and stochastic stability[END_REF][START_REF] Baxendale | Renewal theory and computable convergence rates for geometrically ergodic Markov chains[END_REF][START_REF] Fort | Convergence of adaptive and interacting Markov chain Monte Carlo algorithms[END_REF]. We keep in with the notation of Baxendale [START_REF] Baxendale | Renewal theory and computable convergence rates for geometrically ergodic Markov chains[END_REF]). Recall that 0 < e min ≤ e max is such that E ⊂ [e min , e max ].

Minorisation condition. Let B ∈ F λ (c). Define ϕ B (y) = B(2y) e max y exp - y 0 B (2s)
e min s ds .

Set C = (0, r) × E, where r is specified by c. For any measurable X × A ⊂ S and (x, v) ∈ C, we have

P B (x, v), X × A = A ρ(v, dv ) X ∩[x/2,∞] B (2y) vy exp - y x/2 B(2s)
vs ds dy

≥ ρ min (A) X ∩[r/2,∞] ϕ B (y)dy. Define Γ B (dy, dv) = c -1 B 1 [r/2,∞) (y)ϕ B (y)dy ρ min (dv)
ρ min (E) , where

c B = ρ min (E) ∞ r/2 ϕ B (y)dy ≥
e min ρ min (E) e max exp -L e min ) =: β > 0 by [START_REF] Doumic | Estimating the division rate for the growthfragmentation equation[END_REF] since B ∈ F λ (c). We have thus exhibited a small set C, a probability measure Γ B and a constant β > 0 so that the minorisation condition

P B (x, v), X × A ≥ β Γ B (X × A) (36)
holds for every (x, v) ∈ C and X × A ⊂ S, uniformly in B ∈ F λ (c).

Strong aperiodicity condition. We have where we applied [START_REF] Doumic | Estimating the division rate for the growthfragmentation equation[END_REF] for the last inequality.

β Γ B (C) = β c -1 B E ρ min (dv)
Drift condition. Let B ∈ F λ (c). Let V : S → [1, ∞) be continuously differentiable and such that for every v ∈ E,

lim y→∞ V(y, v) exp -2 λ m vλ y λ = 0. (38)
For x ≥ r, by [START_REF] Engler | Analysis of a model for the dynamics of prions ii[END_REF] and integrating by part with the boundary condition [START_REF] Wang | Robust growth of Escherichia coli[END_REF], we have, for every v ∈ E,

P B V(x, v) = E ρ(v, dv ) ∞ x/2 V(y, v ) B(2y) vy exp - y x/2 B(2s)
vs ds dy

≤ E ρ(v, dv ) ∞ x/2 ∂ y V(y, v ) exp -m2 λ v y x/2 s λ-1 ds dy ≤ exp m vλ x λ E ρ(v, dv ) ∞ m2 λ vλ (x/2) λ V y vλ m2 λ
1/λ , v e -y dy.

Pick V(x, v) = V(x) = exp m e min λ x λ defined in [START_REF] Gobet | Nonparametric estimation of scalar diffusion from low frequency data[END_REF] and note that ( 38) is satisfied for an apprioriate choice of e min and e max since 2 λ > sup E/ inf E. With this choice, we further infer

P B V(x, v) ≤ V(x, v) E ρ(v, dv ) ∞ m2 λ vλ (x/2) λ exp -(1 -2 -λ )y dy ≤ V(x, v) 1 1 -2 -λ exp -(1 -2 -λ ) m vλ r λ ρ max (E) since x ≥ r. Recall that δ(c) = 1 1 -2 -λ exp -(1 -2 -λ ) m emaxλ r λ ρ max (E).
We obtain, for x ≥ r and v ∈ E

P B V(x, v) ≤ δ(c)V(x, v) (39)
and we have δ(c) < 1 by Assumption 3. We next need to control P B V outside x ∈ [r, ∞), that is on the small set C. For every (x, v) ∈ C, we have

P B V(x, v) ≤ E ρ(v, dv ) r/2
x/2

V(y, v ) B(2y) vy dy

+ ∞ r/2 V(y, v ) B(2y) vy exp - y r/2 B(2s)
vs ds dy

≤ ρ max (E) e -1 min sup y∈[0,r] V(y)L + δ(c)V(r/2) =: Q < ∞ (40)
where we used ( 16) and the fact that B ∈ F λ (c). Combining together (39) and (40), we conclude

P B V(x) ≤ δ(c)V(x)1 {x / ∈C} + Q1 {x∈C} . (41)
Completion of proof of Proposition 4. The minorisation condition [START_REF] Sturm | The cost of virulence: Retarded growth of salmonella typhimurium cells expressing type iii secretion system 1[END_REF] together with the strong aperiodicity condition [START_REF] Tan | Emergent bistability by a growth-modulating positive feedback circuit[END_REF] and the drift condition (41) imply inequality (34) by Theorem 1.1 in Baxendale [START_REF] Baxendale | Renewal theory and computable convergence rates for geometrically ergodic Markov chains[END_REF], with R and γ that explicitly depend on δ(c), β, β, V and Q. By construction, this bound is uniform in B ∈ F λ (c) and ρ ∈ M(ρ min ). More specifically, we have

γ < min{max{δ(c), γ V,B }, 1}
therefore under Assumption 3 we have γ < 1 and under Assumption 4, we obtain the improvement γ < 1 2 .

Further estimates on the invariant probability

Lemma 2. For any c such that Assumption 3 is satisfied and any compact interval D ⊂ (0, ∞), we have

sup B∈F λ (c) ∩ H s (D,M ) sup x∈2 -1 D ν B (x) < ∞, with ν B (x) = E ν B (x, dv).
Proof. Since B ∈ F λ (c), ν B is well-defined and satisfies

ν B (x, dv) = B(2x) x E 2x 0 ν B (y, dv )dy exp - x y/2 B(2s) v s ds ρ(v , dv) v .
Hence ν B (x, dv) ≤ B(2x)(e min x) -1 ρ max (dv) and we also have

ν B (x) ≤ B(2x)(e min x) -1 ρ max (E). Since B ∈ H s (D, M ) implies sup x∈2 -1 D B(2x) = B L ∞ (D)
≤ M , the conclusion follows.

Lemma 3. For any c such that Assumption 3 is satisfied, there exists a constant d(c) ≥ 0 such that for any compact interval D ⊂ (d(c), ∞), we have

inf B∈F λ (c) inf x∈D ϕ B (x) -1 ν B (x) > 0,
where ϕ B (x) is defined in [START_REF] Stewart | Aging and death in an organism that reproduces by morphologically symmetric division[END_REF].

Proof. Let g : [0, ∞) → [0, ∞) satisfy g(x) ≤ V(x) = exp( m e min λ x λ ) for every x ∈ [0, ∞). By Proposition 4, we have sup B∈F ν λ (c) [0,∞) g(x)ν B (x)dx < ∞, (42)
as a consequence of [START_REF] Perthame | On the inverse problem for a sizestructured population model[END_REF] with n = 1 together with the property that sup B∈F ν λ (c) P B V(x) < ∞ for every x ∈ S, as follows from (41) in the proof of Proposition 4. Next, for every x ∈ (0, ∞), we have

Z(ξ u -, τ u -, ξ u ) = G(ξ u -, τ u -)H(ξ u ) -E ν B [G(ξ u -, τ u -)H(ξ u )].
For any u, w ∈ U with |u|, |w| ≥ 1, we have

E µ Z(ξ u -, τ u -, ξ u )Z(ξ w -, τ w -, ξ w ) γ D(u,w) (45)
uniformly in B ∈ F λ (c), with γ and ν B defined in (34) of Proposition 4.

Proof. In view of (45), with no loss of generality, we may (and will) assume that for every (

x, v) ∈ S |G(x, v)| ≤ V(x) and |H(x)| ≤ V(x) (46)
Applying repeatedly the Markov property along the branch that joins the nodes a -(u, w) := a(u -, w -) and w, we have

E µ G(ξ u -, τ u -)H(ξ u )| ξ a -(u,w) , τ a -(u,w) = P |u -|-|a -(u,w)| B (G P B H) ξ a -(u,w) , τ a -(u,w) = P |u|-|a(u,w)| B (G P B H) ξ a -(u,w) , τ a -(u,w)
with an analogous formula for G(ξ w -, τ w -)H(ξ w ). Conditioning with respect to ξ a -(u,w) , τ a -(u,w) , it follows that

E µ Z(ξ u -, τ u -, ξ u )Z(ξ w -, τ w -, ξ w ) = E µ P |u|-|a(u,w)| B (G P B H)(ξ a -(u,w) , τ a -(u,w) ) -E ν B [G P B H(ξ ∅ , τ ∅ )] P |w|-|a(u,w)| B (G P B H)(ξ a -(u,w) , τ a -(u,w) ) -E ν B [G P B H(ξ ∅ , τ ∅ )] .
Applying Proposition 4 thanks to Assumption 3 and (46), we further infer w) .

E µ Z(ξ u -, τ u -, ξ u )Z(ξ w -, τ w -, ξ w ) ≤ R 2 sup x H(x) 2 E µ V(ξ a -(u,w) ) 2 γ D(u,w) S P |a -(u,w)| B V 2 (x)µ(dx) γ D(u,
We leave to the reader the straightfoward task to check that the choice of V in [START_REF] Gobet | Nonparametric estimation of scalar diffusion from low frequency data[END_REF] implies that V 2 satisfies (41). It follows that Proposition 4 applies, replacing V by V 2 in (34). In particular, sup

B∈F λ (c) P |a -(u,w)| B V 2 (x) 1 + V(x) 2 . (47)
Since V 2 is µ-integrable by assumption, inequality (45) follows.

Proposition 6. Work under Assumption 3. Let µ be a probability on S such that S V(x) 2 µ(dx) < ∞. Let x 0 be in the interior of 1 2 D. Let H : R → R be bounded with compact support. Set

H ξu-x 0 h = H ξu-x 0 h -E ν B H ξ ∅ -x 0 h . For any u, w ∈ U with |u|, |w| ≥ 1, we have E µ H ξu-x 0 h H ξw-x 0 h γ D(u,w) hγ D(u,a(u,w))∨ D(w,a(u,w)) (48) uniformly in B ∈ F λ (c) ∩ H s (D, M ) for sufficiently small h > 0.
Proof. The first part of the estimate in the right-hand side of (48) is obtained by letting G = 1 in (45). We turn to the second part. Repeating the same argument as for (45) and conditioning with respect to ξ a(u,w) , we obtain

E µ H ξu-x 0 h H ξw-x 0 h = E µ P |u|-|a(u,w)| B H ξ a(u,w) -x 0 h -E ν B H ξ ∅ -x 0 h P |w|-|a(u,w)| B H ξ a(u,w) -x 0 h -E ν B H ξ ∅ -x 0 h . (49)
Assume with no loss of generality that |u| ≤ |w| (otherwise, the same subsequent arguments apply exchanging the roles of u and w). On the one hand, applying [START_REF] Perthame | On the inverse problem for a sizestructured population model[END_REF] of Proposition 4, we have

P |w|-|a(u,w)| B H ξ a(u,w) -x 0 h -E ν B H ξ ∅ -x 0 h ≤ RV ξ a(u,w) γ |w|-|a(u,w)| .
(50) On the other hand, identifying H as a function defined on S, for every (x, v) ∈ S, we have

P B H x-x 0 h = ∞ x/2 H h -1 (y -x 0 ) B(2y) vy exp - y x/2 B(2s) vs ds dy ≤ [0,∞) H h -1 (y -x 0 ) B (
E ν B H ξ a(u,w) -x 0 h ≤ [0,∞) H x-x 0 h ν B (x)dx h (53)
using Lemma 2. We have E ν B H ξu-x 0 h h likewise. Putting together (52) and (53) we derive

P |u|-|a(u,w)| B H ξ a(u,w) -x 0 h -E ν B H ξu-x 0 h h and this estimate is uniform in B ∈ F λ (c) ∩ H s (D, M ).
In view of ( 49) and (50), we obtain

E µ H ξu-x 0 h H ξw-x 0 h hγ |w|-|a(u,w)| E µ V ξ a(u,w) .
We conclude in the same way as in Proposition 5.

Rate of convergence for the empirical measure

For every y ∈ (0, ∞) and u ∈ U with |u| ≥ 1, define

D(y) = E ν B 1 τ u -1 {ξ u -≤2y, ξu≥y} , (54)
D n (y) = n -1 u∈Un 1 τ u -1 {ξ u -≤ 2y, ξ u ≥ y} , (55)
and D n (y) = D n (y) .

Proposition 7. Work under Assumption 3 in the sparse tree case and Assumption 4 in the full tree case. Let µ be a probability on S such that

S V(x) 2 µ(dx) < ∞. If 1 ≥ = n → 0 as n → ∞, we have sup y∈D E µ D n (y) n -D(y) 2 n -1 (56) uniformy in B ∈ F λ (c) ∩ H s (2 -1 D, M ) and ρ ∈ M(ρ min , ρ max ).
We first need the following estimate For k = 0, we have

N 0 = 1 + 2γ + 4γ 2 + . . . + 2 Nn γ Nn = 1-(2γ) Nn+1 1-2γ =: φ γ (N n ).
Under Assumption 4, by Proposition 4, we have γ < 1 2 therefore φ γ (N n ) is bounded as n → ∞. For k = 1, if we start with the node u = (∅, 0), then the contribution of its descendants in (58) is given by φ γ (N n -1), to which we must add γ for its ancestor corresponding to the node u = ∅ and also γφ γ (N n ) for the contribution of the second lineage of the node u = ∅. Finally, we must repeat the argument for the node u = (∅, 1). We obtain

N 1 = 2 φ γ (N n -1) + γ + γ 2 φ γ (N n -1) .
More generally, proceeding in the same manner, we derive

N k = 2 k φ γ (N n -k) + γ + γ 2 φ γ (N n -k) + . . . + γ i + γ i+1 φ γ N n -k + (i -1) + . . . + γ k + γ k+1 φ γ (N n -1) (59)
for k = 1, . . . , N n , and this last quantity is of order 2 k . It follows that Putting together the sparse and full tree case, we obtain the proposition.

Let ν n (dy) = n -1 u∈Un δ ξu (dy) denote the empirical measure of the observation (ξ u , u ∈ U n ). The full tree case. The computations are a bit more involved. Let us evaluate |u|=k w∈Un γ D(u,w) h n γ D(u,a(u,w))∨ D(w,a(u,w)) .

We may repeat the argument displayed in (59) in order to evaluate the contribution of the term involving γ D(u,a(u,w)) . However, in the estimate N k , each term γ i + γ i+1 φ γ N n -k + (i -1) in formula (59) may be replaced by h n γ i + γφ γ N n -k + (i -1) up to constants. This corresponds to the correction given by h n γ D(u,a(u,w))∨ D(w,a(u,w)) . As a consequence, we obtain |u|=k w∈Un γ D(u,w) h n γ D(u,a(u,w))∨ D(w,a(u,w))

2 k k i=1 h n γ i + γφ γ (N n -k + i -1) γ i 1 + γφ γ (N n -k + i -1) 2 k k i=1 h n ∧ γ i .
where D(y) and D n (y) are defined in (54) and (55) respectively. It follows that The term IV. We get rid of the term y 2 D(y) 2 by Lemma 4 and the fact that D is bounded. By Assumption 2 and classical kernel approximation, we have for every 0 < s ≤ n 0 The term V. We have

B n -B 2 L 2 (D) = 2
IV K hn ν B -ν B 2 L 2 (2 -1 D) |ν B | 2 H s (2 -1 D) h 2s n . (62
E µ [V ] ≤ -2 n |D| sup y∈2 -1 D
y 2 E µ K hn ν n (y) -K hn ν B (y) 2 .

By (60) of Proposition 8 we derive Applying (56) of Proposition 7, we conclude

E µ [V ]
E µ V I -2 n n -1 (66)
uniformly in B ∈ F λ (c).

Completion of proof of Theorem 2. We put together the three estimates (63), (64) and (66). We obtain

E µ B n -B 2 L 2 (D) h 2s n + -2 n | log h n |(nh n ) -1 + -2 n n -1
uniformly in B ∈ F λ (c) ∩ H s (D, M ). The choice h n ∼ n -1/(2s+1) and the fact that -2 n grows logarithmically in n yields the rate n -s/(2s+1) up to log terms and the inessential supplementary multiplicative error factor -1 n . The proof is complete. the lemma thanks to the representation [START_REF] Meyn | Markov chains and stochastic stability[END_REF] and the uniform continuity of B over compact sets. For the second part, introduce the (F t )-stopping time B(y) P(Υ t < t + h).

Υ t =
In the same way, P(Υ t < t + h) h and the conclusion follows.

Proposition 2 .

 2 where µP B (dy) = S µ(dx)P B (x, dy) denotes the left action of positive measures µ(dx) on S for the transition P B . Work under Assumption 1. If P B admits an invariant probability measure ν B of the form ν B (dx) = ν B (x, dv)dx then we have

Figure 1 :

 1 Figure 1: Reconstruction for n = 2 17 . When the variability in the growth rate is ignored, the estimate reveals unsatisfactory. The parameter values are the reference ones.

  3 log(n) average error, sparse tree

Figure 2 :

 2 Figure2: Error v.s. n for the full tree and the sparse tree case on a log-log scale. The error actually proves better than the upper rate of convergence announced in Theorem 2, and may be taken smaller than log(n). Estimates are comparable for both schemes. The parameter values are the reference ones.

Figure 3 :

 3 Figure 3: Reconstruction for n = 210 , error band for 95%, full tree case, over M = 100 simulations, with = 1/n in order to emphasise that the larger x, the smaller ν B and the larger the error estimate.

Figure 4 :

 4 Figure 4: Estimation of B (dotted line and dashed line resp.) and ν B (solid line and dash-dotted line resp.) on experimental data of E. coli dividing cells, for resp. a sparse tree and a full tree experiment. n = 2335 for the sparse tree, n = 748 for the full tree. The experimental conditions being different (temperature and nutrient), these results are not supposed to be identical, yet a generic pattern appears for B, that could serve as a basis for future biological studies.

e

  min y dy ≥ β(1 -exp(e min ) =: β > 0[START_REF] Tan | Emergent bistability by a growth-modulating positive feedback circuit[END_REF]

∞e

  2x ν B (y)dy ≤ exp(-m e min λ (2x) λ ) [0,∞)V(y)ν B (y)dy and this bound is uniform in B ∈ F λ (c) by (42). Therefore, for every x ∈ (0, ∞), we have supB∈F ν λ (c) ∞ 2x ν B (y)dy ≤ c(c) exp(-m e min λ (2x) λ ) (43)for some c(c) > 0. Letd(c) > e min λ2 -λ m log c(c) 1/λ 1 {c(c)≥1} .(44)By definition of ν B , for every x ∈ (0, ∞), we now haveν B (x, dv) = Bmin s ds 1 -c(c) exp(-m e min λ (2x) λ ) ρ min (dv)where we used (43) for the last inequality. By (44), for x ≥ d(c) we have 1 -c(c) exp(-m e min λ (2x) λ ) > 0 and the conclusion readily follows by integration.

5. 5 Proposition 5 .

 55 Covariance inequalitiesIf u, w ∈ U, we define a(u, w) as the node of the most recent common ancestor between u and w. Introduce the distance D(u, w) = |u| + |w| -2|a(u, w)|. Work under Assumption 3. Let µ be a probability distribution on S such that S V(x) 2 µ(dx) < ∞. Let G : S → R and H : [0, ∞) → R be two bounded functions. Define

  |H(x)|dx h. (51)Indeed, since x 0 is in the interior of1 2 D we have {x 0 + h supp(H)} ⊂ 1 2 D for small enough h hence sup y∈{x 0 +h supp(H)} B(2y) ≤ M . Now, since P B is a positive operator and P B 1 = 1, we derive P |u|-|a(u,w)| B H ξ a(u,w) -x 0 h h (52) as soon as |u| -|a(u, w)| ≥ 1, uniformly in B ∈ F λ (c) ∩ H s (D, M ). If |u| = |a(u, w)|, since E ν B (dx, dv) = ν B (x)dx, we obtain in the same way

Lemma 4 .e

 4 Work under Assumption 3. Let d(c) be defined as in Lemma 3. For every compact interval D ⊂ (d(c), ∞) such that inf D ≤ r/2, we have inf B∈F λ (c)∩H s (2 -1 D,M ) inf y∈D D(y) > 0.Proof of Lemma 4. By[START_REF] Doumic | Nonparametric estimation of the division rate of a size-structured population[END_REF] and the definition of ϕ B in[START_REF] Stewart | Aging and death in an organism that reproduces by morphologically symmetric division[END_REF], we readily haveD(y) = 1 e max ϕ B (y) -1 ν B (y) exp -y 0 B(2s)e min s ds .Since B ∈ F λ (c) ∩ H s (2 -1 D, M ), by applying (16) and (17) successively, weobtain min (L + M log sup D r/2 ) < ∞where we used that inf D ≤ r/2. min s ds ≥ exp -e -1 min (L + M log sup D r/2 ) > 0 and Lemma 4 follows by applying Lemma 3.

2 k 2

 2 Nn n and the conclusion follows likewise.

Proposition 8 . 2 | 2 = 2 = (nh n ) - 2 u 2 n -2 h - 1 n

 8222221 Work under Assumption 3 in the sparse tree case and Assumption 4 in the full tree case. Let µ be a probability on S such that S V(x) 2 µ(dx) < ∞. We havesup y∈D E µ K hn ν n (y) -K hn ν B (y) log h n |(nh n ) -1 (60) uniformly in B ∈ F λ (c).Proof. We have, with the notation of Proposition 6E µ K hn ν n (y) -K hn ν B (y) (nh n ) -2 E µ u∈ Un K ξu-y hn -E ν B K ξu-y hn ,w ∈Un E µ K ξu-y hn K ξw-y hn (nh n ) -2u,w∈ Un γ D(u,w) h n γ D(u,a(u,w))∨ D(w,a(u,w)) (61) by applying (48) of Proposition 6. It remains to estimate (61).The sparse tree case. We have a(u, w) = u if |u| ≤ |w| and a(u, w) = w otherwise. It follows thatE µ K hn ν n (y) -K hn ν B (y) u,w∈ Un γ D(u,w) ,and since u,w∈ Un γ D(u,w) = 1≤|u|,|w|≤n γ ||u|-|w|| is of order n as soon as γ < 1, we obtain the result.

1 2 DKK 2 D

 122 B n (2y) -B(2y)2 dy IV + V + V I, hn ν B (y) -ν B (y) hn ν n (y) -K hn ν B (y) 2 D n (y) -2 y 2 dy V I = 1 D n (y) -D(y) 2 K hn ν B (y) 2 D n (y) D(y)-2 y 2 dy.

) Lemma 5 .

 5 Let D ⊂ (0, ∞) be a compact interval. Let B ∈ F λ (c) for some c satisfying Assumption 3. We haveν B H s (2 -1 D) ≤ ψ e min , e max , D, B H s (D)for some continuous function ψ.Proof of Lemma 5. Define Λ B (x, y) = E ν B (y, dv ) v exp -. If B ∈ H s (D), then x Λ B (x, y) ∈ H s (2 -1 D) for every y ∈ [0, ∞),and we have Λ B (•, y) H s (2 -1 D) ≤ ψ 1 (y, B H s (D) , e min , e max ) for some continous function ψ 1 . The result is then a consequence of the representation ν B (x) = B(2x) x 2x 0 Λ B (x, y)dy. Going back to (62) we infer from Lemma 5 that |ν B | H s (2 -1 D) is bounded above by a constant that depends on e min , e max , D and B H s (D) only. It follows that IV h 2s n (63) uniformly in B ∈ H s (D, M ).

- 2 n- 1 D

 21 | log h n |(nh n ) -1 (64) uniformly in B ∈ F λ (c).The term VI. First, by Lemma 4, the estimate infB∈F λ (c) inf y∈2 -1 D D n (y) D(y) n holds. Next, sup y∈2 |K hn ν B (y)| = sup y∈2 -1 D [0,∞) K hn (z -y)ν B (z)dz ≤ sup y∈2 -1 D hn ν B (y) K L 1 ([0,∞))(65)where 2 -1 D hn = {y + z, y ∈ 2 -1 D, z ∈ supp(K hn )} ⊂ D, for some compact interval D since K has compact support by Assumption 2. By Lemma 2, we infer that (65) holds uniformly in B ∈ F λ (c). We deriveE µ V I -2 n sup y∈2 -1 D E µ D n (y) n -D(y) 2 .

Table 1 :

 1

		5	6	7	8	9	10
	e	0.2927 0.1904 0.1460 0.1024 0.0835 0.0614
	std. dev. 0.1775 0.0893 0.0627 0.0417 0.0364 0.0241

  inf{s > t, C s -C t ≥ 1} and note that {C t+h -C t ≥ 1} = {Υ t ≤ t + h} ∈ F Υt . Writing {C t+h -C t ≥ 2} = {Υ t < t + h, Υ Υt ≤ t + h}and conditioning with respect to F Υt , we first haveP(C t+h -C t ≥ 2)

	= E	t	t+h-Υt	B xe	V(b ϑ C t 2 C t )+sV(s)	e -s 0 B xe	)+s V(s ) 2 C t V(b ϑ C t	ds ds 1 {Υt<t+h}
	≤ h		sup
		y≤x exp(2emaxt)

Acknowledgements

The research of M. Doumic is partly supported by the Agence Nationale de la Recherche, Grant No. ANR-09-BLAN-0218 TOPPAZ. The research of M. Hoffmann is partly supported by the Agence Nationale de la Recherche, (Blanc SIMI 1 2011 project CALIBRATION). We are grateful to V. Bansaye, A. Dalalyan, A. Marguet, E. Moulines, A. Olivier, B. Perthame, V. Rivoirard and V.C. Tran for helpful discussion and comments.

Proof of Proposition 7. Since D n (y) is bounded, we have

Next, take n sufficiently large, so that 0

a choice which is possible thanks to Lemma 4. Since

{D n (y) < n } ⊂ {D n (y) -D(y) < -q}, integrating (57), we have that

and H(x) = 1 {x≥y} and note that G and H are bounded on S (and also uniformly in y ∈ D). It follows that

We then apply (45) of Proposition 5 to infer, with the same notation that

uniformly in y ∈ D and B ∈ F λ (c). We further separate the sparse and full tree cases.

The sparse tree case. We have u,w∈Un γ D(u,w) = 1≤|u|,|w|≤n γ ||u|-|w|| and by Proposition 4, this last quantity is of order n.

The full tree case. We have n ∼ 2 Nn , where N n is the number of generations used to expand U n . We evaluate

Define

and the conclusion follows in the full case.

Putting together the sparse and full tree cases, we obtain the proposition.

Proof of Theorem 2

From

, we plan to use the following decomposition

6 Appendix

Construction of the discrete model

Fix an initial condition x = (x, v) ∈ S. On a rich enough probability space, we consider a Markov chain on the binary tree (τ u , u ∈ U) with transition ρ(v, dv ) and initial condition v: if u = (u 1 , . . . , u k ) ∈ U, we write ui = (u 1 , . . . , u k , i), i = 0, 1 for the two offsprings of u; we set τ ∅ = v and τ u0 ∼ ρ(τ u , dv ) and τ u1 ∼ ρ(τ u , dv ) so that conditional on τ u , the two random variables τ u0 and τ u1 are independent. We also pick a sequence of independent standard exponential random variables e u , u ∈ U , independent of (τ u , u ∈ U). The model (ξ u , τ u ), u ∈ U is then constructed recursively. We set

where F x,v (t) = t 0 B x exp(vs) ds. For u ∈ U and i = 0, 1, we put

To each node u ∈ U, we then associate the mark (ξ i , b u , ζ u , τ u ) of the size, date of birth, lifetime and growth rate respectively of the individual labeled by u. One easily checks that Assumption 1 guarantees that the model is well defined. In particular, since B is locally bounded, we see that there is no accumulation of jumps almost-surely.

Proof of Lemma 1

Note first that

, it follows by [START_REF] Bertoin | Random fragmentation and Coagulation Processes[END_REF] that ) 2 -Ct within the integral.

Noting that V(b ϑ C t ) + V(t)(t -b ϑ C t ) = V(t) we obtain the first part of