Tony Lelièvre
email: lelievre@cermics.enpc.fr

Loucas Pillaud-Vivien
email: loucas.pillaud-vivien@inria.fr

Julien Reygner
email: julien.reygner@enpc.fr

Central Limit Theorem for stationary Fleming-Viot particle systems in finite spaces

Keywords: ] and Cérou, Delyon, Guyader and Rousset [ 1991 Mathematics Subject Classification. 60F05, 60J27 Central Limit Theorem, Fleming-Viot particle system, stationary distribution

We consider the Fleming-Viot particle system associated with a continuoustime Markov chain in a finite space. Assuming irreducibility, it is known that the particle system possesses a unique stationary distribution, under which its empirical measure converges to the quasistationary distribution of the Markov chain. We complement this Law of Large Numbers with a Central Limit Theorem. Our proof essentially relies on elementary computations on the infinitesimal generator of the Fleming-Viot particle system, and involves the so-called π-return process in the expression of the asymptotic variance. Our work can be seen as an infinite-time version, in the setting of finite space Markov chains, of results by Del Moral and Miclo [ESAIM:

A probability measure π on D is called a quasistationary distribution (QSD) for (x t ) t≥0 in D if for any t ≥ 0,

P π (x t ∈ •|τ D > t) = π(•).
Here and throughout the paper, we use the notation P µ , E µ (respectively P x , E x ) to indicate that x 0 is distributed according to the probability measure µ (respectively equal to x almost surely).

As soon as the restriction of the matrix P to the set D is irreducible, there exists a unique QSD in D, which possesses a natural spectral characterisation, recalled in Proposition 2.1 below. Furthermore, [START_REF] Darroch | On quasi-stationary distributions in absorbing continuous-time finite Markov chains[END_REF] proved that the QSD is the so-called Yaglom limit

lim t→+∞ P µ (x t ∈ •|τ D > t) = π(•),
for any initial distribution µ in the set of probability measures on D. We refer to [START_REF] Collet | Quasi-stationary distributions[END_REF] for a standard reference on QSDs.

1.2. Fleming-Viot particle system. Consider n particles evolving independently in E according to the infinitesimal generator L, with the supplementary condition that as soon as one particle attempts to jump to a point y ∈ E \ D, it is instantly moved to the location of one the n -1 remaining particles, picked uniformly at random. Clearly, this dynamics induces a Markov chain (x 1 t , . . . , x n t ) t≥0 in D n , which is called the Fleming-Viot particle system 1 [START_REF] Asselah | Quasistationary distributions and Fleming-Viot processes in finite spaces[END_REF].

Assume that the particles are initially iid according to some probability measure µ on D. [START_REF] Asselah | Quasistationary distributions and Fleming-Viot processes in finite spaces[END_REF] proved that, for any t ≥ 0, the empirical distribution η n t of (x 1 t , . . . , x n t ), defined by

∀x ∈ D, η n t (x) := 1 n n i=1 1 {x i t =x} ,
converges, when n → +∞, to the probability measure P µ (x t ∈ •|τ D > t).

On the other hand, the process (x 1 t , . . . , x n t ) t≥0 is irreducible in D n , and therefore it possesses a unique stationary distribution. Let (x 1 ∞ , . . . , x n ∞ ) be a random vector in D n distributed according to this stationary distribution, and consider the random probability measure η n ∞ defined on D by

∀x ∈ D, η n ∞ (x) := 1 n n i=1 1 {x i ∞ =x} .
It was also proved by [START_REF] Asselah | Quasistationary distributions and Fleming-Viot processes in finite spaces[END_REF] that η n ∞ converges, when n → +∞, to the QSD π. We refer to [START_REF] Burdzy | A Fleming-Viot particle representation of the Dirichlet Laplacian[END_REF]; Del [START_REF] Del Moral | Particle approximations of lyapunov exponents connected to schrödinger operators and feynman-kac semigroups[END_REF]; [START_REF] Grigorescu | Hydrodynamic limit for a Fleming-Viot type system[END_REF]; [START_REF] Rousset | On the control of an interacting particle estimation of Schrödinger ground states[END_REF]; [START_REF] Ferrari | Quasi stationary distributions and Fleming-Viot processes in countable spaces[END_REF]; [START_REF] Löbus | A stationary Fleming-Viot type Brownian particle system[END_REF]; [START_REF] Villemonais | General approximation method for the distribution of Markov processes conditioned not to be killed[END_REF]; Cloez and Thai (2016b); [START_REF] Oçafrain | Non-failable approximation method for conditioned distributions[END_REF] for general 1 The terminology Fleming-Viot particle system for such processes, with a finite number of particles and space-inhomogeneous exit rate, was popularised by [START_REF] Burdzy | Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions[END_REF][START_REF] Burdzy | A Fleming-Viot particle representation of the Dirichlet Laplacian[END_REF] -the second reference specifically addressing the case of d-dimensional Brownian motions. A discussion of the differences between the evolution of the empirical measure of this n-particle model and the original Fleming-Viot superprocess Etheridge (2000, Chapter 1) can be found in the introduction of [START_REF] Grigorescu | Hydrodynamic limit for a Fleming-Viot type system[END_REF].

approximation results for either the conditional distribution P µ (x t ∈ •|τ D > t) or the QSD π by Fleming-Viot-like particle systems, in various contexts.

In particular, the convergence of η n ∞ to π provides an effective numerical procedure to approximate the QSD, by simulating the Fleming-Viot particle system over sufficiently long time intervals, and with a sufficiently large number of particles (see [START_REF] Aldous | Two applications of urn processes: the fringe analysis of search trees and the simulation of quasi-stationary distributions of Markov chains[END_REF]; [START_REF] Groisman | Simulation of quasi-stationary distributions on countable spaces. Markov Process[END_REF]; [START_REF] Benaïm | A stochastic approximation approach to quasi-stationary distributions on finite spaces[END_REF]; [START_REF] Benaim | Stochastic approximation of quasi-stationary distributions on compact spaces and applications[END_REF] for some alternative methods). To assess the quality of this approximation, it is necessary to obtain error estimates, both on the time needed for the Fleming-Viot particle system to reach its equilibrium, and on the rate of convergence of η n ∞ to π. In this article, we address the latter question by proving a Central Limit Theorem for η n ∞ , stated in Theorem 2.4.

Notation and main results

2.1. Functions and measures. We denote by C(D) the set of functions D → R and by M(D) the set of finite signed measures on D (both sets can be identified with R D , but we introduce distinct notations to make our statements more clear). The duality bracket between M(D) and C(D) is written

∀ρ ∈ M(D), ∀f ∈ C(D), ρ, f := x∈D ρ(x)f (x).
Let H and K be subspaces of C(D) or M(D), with respective dual spaces H ′ and K ′ for the bracket •, • . The adjoint of an operator R :

H → K is denoted by R * : K ′ → H ′ . If H ⊂ C(D) and K ⊂ M(D) (or alternatively H ⊂ M(D) and K ⊂ C(D)) are each other's dual, the operator R is called symmetric if it coincides with R * .
The identity of a space H is denoted by I H . We denote by P(D) the set of probability measures on D, namely the set of measures η ∈ M(D) such that η(x) ≥ 0 for all x ∈ D and η, 1 = 1, where 1 ∈ C(D) is defined by 1(x) := 1 for all x ∈ D. (2.1)

Throughout the article, we work under the following assumption, which we shall not recall in the statement of our results.

(Irr) The substochastic matrix P D is irreducible on the finite space D.

The following result is a consequence of the Perron-Frobenius Theorem (Seneta (2006, Theorem 1.5, p. 22)), and its proof is omitted. We recall that (x t ) t≥0 is the continuous-time Markov chain with infinitesimal generator L introduced in (1.1).

Proposition 2.1 (Existence and uniqueness of the QSD). The following properties hold.

(i) There exists λ ∈ [0, 1) such that 1 -λ is the spectral radius of P D . (ii) The eigenvalue 1 -λ of P D is simple, and any other eigenvalue σ satisfies Re σ < 1 -λ. (iii) There is a unique π ∈ P(D) such that P * D π = (1 -λ)π. (iv) For all x ∈ D, π(x) > 0.

(v) π is the unique QSD of (x t ) t≥0 in D. (vi) π and q satisfy the relation π, q = λ.

2.3. Fleming-Viot particle system. For all n ≥ 1, let P n (D) be the subset of probability measures η ∈ P(D) with atoms of mass proportional to 1/n; namely such that for all x ∈ D, there exists k(x) ∈ {0, . . . , n} such that η

(x) = k(x)/n. Of course, x∈D k(x) = n.
For all x, y ∈ D, let us define the measure θ x,y ∈ M(D) by ∀z ∈ D, θ x,y (z) := 1 {y=z} -1 {x=z} .

(2.2) Definition 2.2 (Empirical distribution of the Fleming-Viot particle system). Let n ≥ 2. The empirical distribution of the Fleming-Viot particle system with n particles is the continuous-time Markov chain (η n t ) t≥0 in P n (D) with infinitesimal generator

L n φ(η) := x,y∈D nη(x) p D (x, y) + q(x) nη(y) n -1 φ η + θ x,y n -φ(η) ,
for any function φ : P n (D) → R.

That the measure-valued process (η n t ) t≥0 actually describes the evolution of the empirical distribution of the particle system introduced in Subsection 1.2 is immediate. Indeed, in the configuration η ∈ P n (D), there are nη(x) particles located at x ∈ D, each of which can jump 'directly' to y ∈ D at rate p D (x, y), or try to exit D at rate q(x) and be moved to y with probability nη(y) (the number of particles located at y) divided by n -1 (the total number of remaining particles).

2.4. Statement of the main result. By Assumption (Irr), the process (η n t ) t≥0 introduced in Definition 2.2 is irreducible in the finite space P n (D), therefore it possesses a unique stationary distribution. Let η n ∞ be a random variable in P(D) distributed according to this stationary distribution. The following Law of Large Numbers was obtained by Asselah et al. (2011, Theorem 2).

Proposition 2.3 (LLN for η n ∞ ). Let • be any norm on M(D). Then lim n→+∞ E[ η n ∞ -π ] = 0.
Our main result is a Central Limit Theorem complementing Proposition 2.3. For any f ∈ C(D), we denote

Var π (f ) := x∈D π(x) (f (x) -π, f ) 2 .
For all t ≥ 0 and f ∈ C(D), we also define In the course of the proof, we shall provide an equivalent formulation of the covariance operator K, which involves the semigroup of the so-called π-return process [START_REF] Groisman | Simulation of quasi-stationary distributions on countable spaces. Markov Process[END_REF]), see Lemma 5.4. Through a spectral argument on this semigroup, we shall in particular check that the integral in the right-hand side above is finite.

∀x ∈ D, Q t f (x) := E x f (x t )1 {τ D >t} = e t(P D -I C(D) ) f (x). (2.3) Theorem 2.4 (CLT for η n ∞ ). The fluctuation field √ n(η n ∞ -π) converges in distri- bution, in M(D),
Remark 2.5 (Case λ = 0). By Proposition 2.1, λ = 0 if and only if q(x) = 0 for all x ∈ D, in which case the Markov chain (x t ) t≥0 never leaves D. In this case, π is in fact the stationary distribution of the chain in D. This case is formally covered by the results of this article, but it is actually trivial because the Fleming-Viot particle system merely consists in n independent copies of the Markov chain (x t ) t≥0 , to which the standard Central Limit Theorem immediately applies and yields the covariance operator K characterised by Kf, f = Var π (f ).

In a much more general framework, Del [START_REF] Del Moral | Particle approximations of lyapunov exponents connected to schrödinger operators and feynman-kac semigroups[END_REF], and more recently [START_REF] Cérou | A central limit theorem for Fleming-Viot particle systems with soft killing[END_REF][START_REF] Cérou | A central limit theorem for Fleming-Viot particle systems with hard killing[END_REF], established a Central Limit Theorem for η n t , t ∈ [0, +∞). In the particular case of Markov chains with a finite space, and assuming for the sake of simplicity that in the Fleming-Viot particle system, the particles are initially iid according to π, the corresponding asymptotic variance for It is immediate to check that in the t → +∞ limit, one recovers the covariance operator K from Theorem 2.4.

η n t reads K t f, f = Var π (f ) + 2λ
Owing to the remark that the Central Limit Theorem for η n t , t ∈ [0, +∞), holds for quite a large class of continuous-time Markov processes in general state spaces (see Cérou et al. (2017, Theorem 2.6)), it is natural to ask whether our approach can be generalised to such cases. Although our arguments can heuristically be formulated in an abstract enough setting, we expect their practical implementation to become much more technical, in particular because of the infinite-dimensional nature of η n t . For the sake of clarity, we therefore chose to focus on finite space Markov chains and keep the formalism elementary.

Proposition 2.3 naturally raises the question of determining a rate of convergence for η n ∞ to π, and Theorem 2.4 indicates that, at least asymptotically, √ n is the correct answer. At the nonasymptotic level, an inequality of the form

∀n ≥ 2, E η n ∞ -π 2 ≤ C n , (2.4)
for some norm • on M(D), may therefore be expected to hold. In the terms of [START_REF] Ferrari | Quasi stationary distributions and Fleming-Viot processes in countable spaces[END_REF] and [START_REF] Asselah | Quasistationary distributions and Fleming-Viot processes in finite spaces[END_REF], such a control essentially amounts to asserting that, under the stationary distribution of the Fleming-Viot particle system, 'the correlations between the particles are of order 1/n'. At finite time t, this estimate was proved by Asselah et al. (2011, Proposition 2) and is an important ingredient of the proof of Proposition 2.3. However, to the best of our knowledge, the control of the correlations under the stationary distribution is known to hold only under specific mixing conditions [START_REF] Ferrari | Quasi stationary distributions and Fleming-Viot processes in countable spaces[END_REF]; Cloez and Thai (2016b)). It also appears in [START_REF] Rousset | On the control of an interacting particle estimation of Schrödinger ground states[END_REF] for the case of diffusions with soft killing. Our proof relies on a weaker form of (2.4), further comments on this aspect are provided in Remark 4.3.

2.5. Sketch of the proof and outline of the article. In Section 3, we introduce the π-return process, which describes the trajectorial behaviour of the stationary Fleming-Viot particle system in the n → +∞ limit, and with which several of the notions that we shall manipulate are related. The proof of Theorem 2.4 follows the standard approach of: (i) proving the tightness of the fluctuation field √ n(η n ∞ -π); (ii) identifying the law of its limits. The first step is detailed in Section 4. It rests on a moment estimate derived from algebraic manipulations on the infinitesimal generator L n , and the use of an appropriate Lyapunov equation. The second step is detailed in Section 5, where the law of any limit of √ n(η n ∞ -π) is shown to be the stationary distribution of a linear diffusion process, for which uniqueness and identification follow from standard arguments.

Elementary linear algebra results, which are suited to our framework, are collected in Appendix A.

Apart from the use of Proposition 2.3 made in the proof of our tightness result, we emphasise that our arguments are entirely static, in the sense that they merely involve estimates on the law of η n ∞ , which stem from elementary manipulations of the infinitesimal generator L n . At the technical level, we thereby avoid resorting to graphical constructions of the process [START_REF] Asselah | Quasistationary distributions and Fleming-Viot processes in finite spaces[END_REF]; [START_REF] Groisman | Simulation of quasi-stationary distributions on countable spaces. Markov Process[END_REF]), coupling techniques (Cloez and Thai (2016b,a)) or martingale arguments [START_REF] Cérou | A central limit theorem for Fleming-Viot particle systems with soft killing[END_REF][START_REF] Cérou | A central limit theorem for Fleming-Viot particle systems with hard killing[END_REF]).

Throughout the article, we take the convention to call 'Theorem' and 'Lemma' the results which are proper to our arguments, while we call 'Proposition' the results which are either proved elsewhere or essentially standard.

3. The π-return process 3.1. Definition. By exchangeability, the convergence of the empirical distribution η n ∞ to π stated in Proposition 2.3 is known to be equivalently formulated as a chaoticity result [START_REF] Sznitman | Topics in propagation of chaos[END_REF]); namely, under the stationary distribution of the Fleming-Viot particle system, any finite subset of particles asymptotically behaves, when n → +∞, like independent realisations of π. Groisman and Jonckheere (2013) provided a trajectorial description of this chaoticity phenomenon (see also [START_REF] Grigorescu | Tagged particle limit for a Fleming-Viot type system[END_REF]; [START_REF] Löbus | A stationary Fleming-Viot type Brownian particle system[END_REF] for related results), based on the following process.

Definition 3.1 (π-return process). The π-return process is the continuous-time Markov chain (x π t ) t≥0 in D with jump rates ∀x, y ∈ D, p π D (x, y) := p D (x, y) + q(x)π(y).

The process (x π t ) t≥0 can be simply described as a copy of (x t ) t≥0 in which the transitions to points in E \ D are replaced with transitions to points in D independently drawn according to π. By Assumption (Irr) and Proposition 2.1, the QSD π is the unique stationary distribution of (x π t ) t≥0 . In Groisman and Jonckheere (2013, Theorem 2.10), it was proved that if the Fleming-Viot particle system is started under its stationary distribution, then when n → +∞, the marginal evolution of each particle converges in distribution, in the space of sample-paths, to the process (x π t ) t≥0 with initial distribution π.

We denote by

P π D : C(D) → C(D) the operator defined by ∀x ∈ D, ∀f ∈ C(D), P π D f (x) := y∈D p π D (x, y)f (y), so that P π D f = P D f + π, f q. (3.1) The infinitesimal generator L π D : C(D) → C(D) of the π-return process writes L π D = P π D -I C(D)
, and the associated semigroup is denoted by

∀t ≥ 0, ∀x ∈ D, ∀f ∈ C(D), P π t,D f (x) := E x [f (x π t )] = e tL π D f (x). (3.2)
Finally, the Dirichlet form of the π-return process is the quadratic form A π D on C(D) defined by

A π D (f ) := 1 2 x,y∈D π(x)p π D (x, y) [f (y) -f (x)] 2 = - x∈D π(x)f (x)L π D f (x).
3.2. Spectral estimates. In this subsection, we describe some spectral properties of the π-return process. We first introduce the spectral gap of the operator P D , which by Proposition 2.1 is positive.

Definition 3.2 (Spectral gap of P D ). The spectral gap of P D is defined by

γ := 1 -λ -max σ =1-λ Re σ > 0,
where the max is taken over the eigenvalues σ of P D .

Since π is the unique stationary distribution of the π-return process, 1 is a single eigenvalue for P π D , and

P π D 1 = 1, (P π D ) * π = π. (3.
3) As a consequence, we have the direct sum decomposition We may now state some useful properties of P π D . We recall the definition (2.3) of the semigroup of operators (Q t ) t≥0 .

C(D) = R1 ⊕ C 0 (D), C 0 (D) := {f ∈ C(D) : π, f = 0}, ( 3
Lemma 3.4 (Spectral properties of the π-return process). The operator P π D possesses the following properties. (iii) For any f ∈ C(D), x ∈ D and t ≥ 0,

P π t,D f (x) = Q t (f -π, f 1)(x) + π, f .
Proof : All three assertions of the lemma follow from the observation that, by (3.4) and (3.1), the restriction of P π D to the stable subspace C 0 (D) coincides with P D . Then any eigenvalue of P π D which is not equal to 1 is necessarily an eigenvalue of P D and thus satisfies Assertion (i) as a consequence of Definition 3.2. The latter assertion implies that, for any δ > 0, the family of operators {e t(λ+γ-δ) P π t,D , t ≥ 0} is bounded on C 0 (D) and thereby yields Assertion (ii). One finally obtains Assertion (iii) by noting that the operators Q t = e t(P D -I C(D) ) and P π t,D = e t(P π

D -I C(D) )
coincide on C 0 (D).

3.3. Two related operators. To prove Theorem 2.4, we shall show that √ n(η n ∞ -π) converges to the stationary distribution of a linear diffusion process in M 0 (D), with drift and diffusion operators expressed in terms of quantities related to the π-return process. We introduce these operators in the present subsection and will use them in Sections 4 and 5.

B 0 := (P π D ) * -(1 -λ)I M0(D) = (L π D ) * + λI M0(D) .
Remark 3.6 (Spectrum). By Lemma 3.4, any eigenvalue τ of B 0 satisfies Re τ ≤ -γ.

The diffusion operator.

Recall the definition of the Dirichlet form A π D of the π-return process in Subsection 3.1. Since, by Remark 3.3, the dual of C 0 (D) is identified with M 0 (D), by Riesz' Theorem, there exists a symmetric operator

A π D : C 0 (D) → M 0 (D) such that, for any f ∈ C 0 (D), A π D f, f = A π D (f ).
We call A π D the diffusion operator ; notice that it can be expressed as the restriction to C 0 (D) of the symmetric part of -L π D in L 2 (π). The irreducibility of the π-return process implies the following result (elementary material on symmetric and positive definite operators is gathered in Appendix A).

Lemma 3.7 (On the operator A π D ). The operator A π D is positive definite, in the sense that it satisfies A π D f, f > 0, for all f ∈ C 0 (D) \ {0}.

Tightness of the fluctuation field

In this section, we prove the following result. For all ǫ > 0, there exists r ǫ ∈ (0, +∞) such that, for all n ≥ 2,

P( √ n η n ∞ -π > r ǫ ) ≤ ǫ.
An auxiliary moment estimate is established in Subsection 4.1. The proof of Lemma 4.1 is detailed in Subsection 4.2. 4.1. Moment estimate. We first state a moment estimate for η n ∞ . We recall the Definition 3.5 of the drift operator B 0 : M 0 (D) → M 0 (D), and the definition (2.1) of q ∈ C(D).

Lemma 4.2 (Moment estimate). Let R : M 0 (D) → C 0 (D) be a symmetric operator. There exists C(R) ∈ [0, +∞) such that, for all n ≥ 2, |E [ B 0 (η n ∞ -π) + η n ∞ -π, q (η n ∞ -π), R(η n ∞ -π) ]| ≤ C(R) n .
Proof : In the proof, we shall employ the operator Q : M(D) → M(D) defined by ∀ρ ∈ M(D), ∀x ∈ D, Qρ(x) := q(x)ρ(x).

Let R : M 0 (D) → C 0 (D) be a symmetric operator. Let us define the function φ : P(D) → R by

∀η ∈ P(D), φ(η) := 1 2 η -π, R(η -π) .
For all x, y ∈ D, for all η ∈ P(D),

φ η + θ x,y n -φ(η) = 1 n θ x,y , R(η -π) + 1 2n 2 θ x,y , Rθ x,y ,
where we recall the definition (2.2) of θ x,y . Since the definition of η n ∞ implies that E[L n φ(η n ∞ )] = 0, we get the identity

0 = E   x,y∈D η n ∞ (x) p D (x, y) + q(x) nη n ∞ (y) n -1 θ x,y , R(η n ∞ -π)   + 1 2n E   x,y∈D η n ∞ (x) p D (x, y) + q(x) nη n ∞ (y) n -1 θ x,y , Rθ x,y   . (4.1)
We first make explicit the value of the first expectation in the right-hand side of (4.1). By (2.2) and (2.1), for all η ∈ P(D), for all z ∈ D,

x,y∈D

η(x) p D (x, y) + q(x) nη(y) n -1 θ x,y (z) = x∈D η(x) p D (x, z) + q(x) nη(z) n -1 - y∈D η(z) p D (z, y) + q(z) nη(y) n -1 = P * D η(z) + n n -1 η, q η(z) -η(z)(1 -q(z)) - n n -1 η(z)q(z) = P * D -I M(D) - Q n -1 η(z) + n n -1 η, q η(z) = P * D -I M(D) η(z) + η, q η(z) + 1 n -1 {-Qη(z) + η, q η(z)} , so that E   x,y∈D η n ∞ (x) p D (x, y) + q(x) nη n ∞ (y) n -1 θ x,y , R(η n ∞ -π)   = E P * D -I M(D) η n ∞ + η n ∞ , q η n ∞ , R(η n ∞ -π) + 1 n -1 {E [ -Qη n ∞ + η n ∞ , q η n ∞ , R(η n ∞ -π) ]} . (4.2)
Using Proposition 2.1, we now rewrite, for all η ∈ P(D),

(P * D -I M(D) )η + η, q η = (P * D -(1 -λ)I M(D) )(η -π) + η -π, q (η -π) + η -π, q π, so that P * D -I M(D) η + η, q η, R(η -π) = B 0 (η -π) + η -π, q (η -π), R(η -π) ,
where we have used the Definition 3.5 of B 0 as well as the fact that R(η -π) ∈ C 0 (D) to justify that η -π, q π, R(η -π) vanishes. As a consequence, the first expectation in the right-hand side of (4.2) rewrites

E P * D -I M(D) η n ∞ + η n ∞ , q η n ∞ , R(η n ∞ -π) = E [ B 0 (η n ∞ -π) + η n ∞ -π, q (η n ∞ -π), R(η n ∞ -π) ] . (4.3)
Combining (4.1), (4.2) and ( 4.3), we finally get the identity

E [ B 0 (η n ∞ -π) + η n ∞ -π, q (η n ∞ -π), R(η n ∞ -π) ] = - 1 n -1 E [ -Qη n ∞ + η n ∞ , q η n ∞ , R(η n ∞ -π) ] - 1 2n E   x,y∈D η n ∞ (x) p D (x, y) + q(x) nη n ∞ (y) n -1 θ x,y , Rθ x,y   ,
the right-hand side of which is bounded in modulus by C(R)/n for some constant C(R) ∈ [0, +∞) depending on R.

4.2.

Proof of Lemma 4.1. We are now ready to present the proof of Lemma 4.1.

Proof of Lemma 4.1: Let us fix a norm • on M(D), and ǫ > 0. The proof is divided into 3 steps.

Step 1. Recall the Definition 3.2 of the spectral gap γ > 0 of P D . By Proposition 2.3, there exists n 0 ≥ 2 depending on ǫ such that, for any n ≥ n 0 ,

P(| η n ∞ -π, q | ≥ γ/2) ≤ ǫ/2.
As a consequence, using Markov's inequality, we get that for any r ∈ (0, +∞), for any n ≥ n 0 ,

P( √ n η n ∞ -π ≥ r) ≤ 1 r 2 E n η n ∞ -π 2 1 {| η n ∞ -π,q |<γ/2} + ǫ 2 .
In the next step, we use Lemma 4.2 and Proposition A.3 to control the expectation in the right-hand side.

Step 2. Let us rewrite the result of Lemma 4.2 as

E B 0 + η n ∞ -π, q I M0(D) (η n ∞ -π), R(η n ∞ -π) ≤ C(R) n , (4.4)
for any symmetric operator R : M 0 (D) → C 0 (D). Now, for any η ∈ P(D),

B 0 + η -π, q I M0(D) = B 0 - γ 2 -η -π, q I M0(D) , (4.5)
with B 0 := B 0 + (γ/2)I M0(D) . By Remark 3.6, all eigenvalues of B 0 have a negative real part, therefore applying Proposition A.3 for any choice of a symmetric and positive definite operator A : C 0 (D) → M 0 (D), we obtain that there exists a symmetric and positive definite operator K : C 0 (D) → M 0 (D) such that B 0 K + K B * 0 + 2A = 0. By Proposition A.2, K is invertible and R := K -1 is symmetric and positive definite. We thus get, for any η ∈ P(D),

B 0 (η -π), R(η -π) = -AR(η -π), R(η -π) .
Let • C0(D) be a norm on C 0 (D), and let • M0(D) be the norm induced by • on M 0 (D). Since A is symmetric and positive definite, by Proposition A.2 there exists c A ∈ (0, +∞) such that for all

f ∈ C 0 (D), Af, f ≥ c A f 2 C0(D) . As a consequence, AR(η -π), R(η -π) ≥ c A R(η -π) 2 C0(D) ≥ c A ||| K||| 2 η -π 2 M0(D) ,
where we denote by ||| • ||| the operator norm between the spaces (C 0 (D), • C0(D) ) and (M 0 (D), • M0(D) ). We deduce that

B 0 (η -π), R(η -π) ≤ - c A ||| K||| 2 η -π 2 M0(D) . (4.6) On the other hand, if η ∈ P(D) is such that | η -π, q | < γ/2, then γ 2 -η -π, q η -π, R(η -π) ≥ 0, (4.7)
where we have used the fact that R is symmetric and positive definite. Combining (4.5), (4.6) and (4.7), we deduce that, for all η ∈ P(D),

η -π 2 1 {| η-π,q |<γ/2} ≤ - ||| K||| 2 c A B 0 + η -π, q I M0(D) (η -π), R(η -π) .
Evaluating this inequality for η = η n ∞ , taking the expectation and applying (4.4), we get

E n η n ∞ -π 2 1 {| η n ∞ -π,q |<γ/2} ≤ C ′ := ||| K||| 2 c A C( K -1 ).
Step 3. The final estimates of Steps 1 and 2 show that any choice of r ≥ 2C ′ /ǫ implies P( √ n η n ∞ -π ≥ r) ≤ ǫ for all n ≥ n 0 . On the other hand, it is known that any finite family of random variables in M(D) is tight (Billingsley (1999, Theorem 1.3, p. 8)), therefore there exists r ′ ǫ ∈ (0, +∞) such that P(

√ n η n ∞ -π ≥ r ′ ǫ ) ≤ ǫ
, for all n < n 0 . Taking r ǫ as the maximum between r ′ ǫ and 2C ′ /ǫ completes the proof.

Remark 4.3 (Variance estimates). Assume that q and γ satisfy the condition

α := max x∈D q(x) -min x∈D q(x) < γ. (4.8)
Then employing the decomposition

B 0 + η -π, q I M0(D) = B 0 -(α -η -π, q ) I M0(D) , B 0 := B 0 + αI M0(D) ,
in Step 2 of the proof of Lemma 4.1 leads to the variance estimate (2.4), which holds without any smallness condition on η n ∞ -π . Such an estimate was also obtained by Cloez and Thai (2016b, Theorem 1.3), under an assumption similar to (4.8). The latter assumption is in particular satisfied if q(x) = λ for any x ∈ D, that is to say, the rate at which the Markov chain (x t ) t≥0 exits D does not depend on its current position. An example of such a chain is studied by the same authors in Cloez and Thai (2016b, Section 3.1) and Cloez and Thai (2016a, Section 2).

Asymptotic normality of the fluctuation field

In this section, we complete the proof of Theorem 2.4 by identifying the law of any limit of √ n(η n ∞ -π). Let us sketch our argument. In Subsection 5.1, we interpret the law of √ n(η n ∞ -π) as the stationary distribution of a continuous-time Markov chain (ξ n t ) t≥0 in M 0 (D). In Subsection 5.2, we describe the n → +∞ limit of the infinitesimal generator of (ξ n t ) t≥0 . In Subsection 5.3, we show that this limit is the infinitesimal generator of a linear diffusion process in M 0 (D), the unique stationary distribution of which is the Gaussian measure introduced in Theorem 2.4. The proof of the latter theorem is completed in Subsection 5.4. 5.1. The process (ξ n t ) t≥0 . Let Φ n : P n (D) → M 0 (D) be defined by

∀η ∈ P n (D), Φ n (η) := √ n(η -π),
and let M n 0 (D) ⊂ M 0 (D) denote the range of Φ n . For all t ≥ 0, we define ξ n t := Φ n (η n t ) = √ n (η n t -π) . Since Φ n is a one-to-one map between P n (D) and M n 0 (D), (ξ n t ) t≥0 is a continuoustime Markov chain, with infinitesimal generator

M n ψ(ξ) := L n (ψ • Φ n )((Φ n ) -1 (ξ)) = x,y∈D ϑ n (x, y, ξ) ψ ξ + θ x,y √ n -ψ(ξ) ,
where

ϑ n (x, y, ξ) := n π(x) + ξ(x) √ n p D (x, y) + q(x) n n -1 π(y) + ξ(y) √ n .
Besides, the law of the random variable

ξ n ∞ := Φ n (η n ∞ ) = √ n (η n ∞ -π) (5.1)
is the unique stationary distribution of (ξ n t ) t≥0 , so that Theorem 2.4 reduces to proving the convergence in distribution, in M 0 (D), of ξ n ∞ . where we recall the definition (2.2) of θ x,y ∈ M 0 (D). Proof : Let ψ : M 0 (D) → R be a C ∞ function with compact support. There exists a compact set K ⊂ M 0 (D) such that for any ξ ∈ M 0 (D) \ K, for all n ≥ 2,

Convergence

M n ψ(ξ) = Mψ(ξ) = 0.
Thus, we restrict our attention to ξ ∈ K and define On the other hand, for any ξ ∈ K, we write

r x,y n (ξ) := ψ ξ + θ x,y √ n -ψ(ξ) - 1 √ n θ x,y , ∇ψ(ξ) - 1 2n θ x,y , ∇ 2 ψ(ξ)θ x,
M n ψ(ξ) = √ nA (1) n (ξ) + A (2) n (ξ) + 1 √ n A (3) n (ξ),
where, letting

p π,n D (x, y) := p D (x, y) + n n -1 q(x)π(y),
we write

A (1) n (ξ) := x,y∈D π(x)p π,n D (x, y) θ x,y , ∇ψ(ξ) , A (2)
n (ξ) :=

x,y∈D n n -1 π(x)q(x)ξ(y) + ξ(x)p π,n D (x, y) θ x,y , ∇ψ(ξ)

+ 1 2 x,y∈D π(x)p π,n D (x, y) θ x,y , ∇ 2 ψ(ξ)θ x,y , A (3) n (ξ) := 1 2 x,y∈D n n -1 π(x)q(x)ξ(y) + ξ(x)p π,n D (x, y) θ x,y , ∇ 2 ψ(ξ)θ x,y + n 3/2 x,y∈D π(x) + ξ(x) √ n p π,n D (x, y) + √ n n -1 π(x)q(x)ξ(y) r x,y n (ξ).
A short computation shows that, for any z ∈ D,

x,y∈D

π(x)p π,n D (x, y)θ x,y (z) = 1 n -1 (λπ(z) -q(z)π(z)), so that lim n→+∞ sup ξ∈K | √ nA (1) n (ξ)| = 0.
Likewise, it follows from (5.3) and the compactness of K that |A Using the convergence of p π,n D (x, y) to p π D (x, y), we may first observe that lim

n→+∞ x,y∈D n n -1 π(x)q(x)ξ(y) + ξ(x)p π,n D (x, y) θ x,y , ∇ψ(ξ) = x,y∈D (π(x)q(x)ξ(y) + ξ(x)p π D (x, y)) θ x,y , ∇ψ(ξ) ,
and that the limit is uniform in ξ ∈ K. For any z ∈ D,

x,y∈D

(π(x)q(x)ξ(y) + ξ(x)p π D (x, y)) θ x,y (z) = x∈D (π(x)q(x)ξ(z) + ξ(x)p π D (x, z)) - y∈D (π(z)q(z)ξ(y) + ξ(z)p π D (z, y)) = λξ(z) + (P π D ) * ξ(z) -ξ(z)
, where we have used Proposition 2.1 to write π, q = λ, the fact that ξ ∈ M 0 (D) to make the first sum over y vanish, and the fact that P π D is a stochastic matrix. By the Definition 3.5 of B 0 , we therefore conclude that

x,y∈D (π(x)q(x)ξ(y) + ξ(x)p π D (x, y)) θ x,y = B 0 ξ, so that the first term in the definition of A

(2) n (ξ) converges to B 0 ξ, ∇ψ(ξ) , uniformly in ξ ∈ K. With similar arguments, it is immediate to show that the second term converges, uniformly in ξ ∈ K, to A π D :: ∇ 2 ψ(ξ) defined by (5.2), which completes the proof.

Identification of M.

Combining the results of Lemma 3.7 and Propositions A.1 and A.2, we deduce that there exist

ζ 1 , . . . , ζ k-1 ∈ M 0 (D) and c 1 , . . . , c k-1 > 0 such that ∀f ∈ C 0 (D), A π D f, f = k-1 l=1 c l ζ l , f 2 ,
where k ≥ 1 is the cardinality of D (see Appendix A). Let (w 1 t ) t≥0 , . . . , (w k-1 t ) t≥0 be independent Brownian motions, and let us consider the linear stochastic differential equation

dξ t = B 0 ξ t dt + k-1 l=1 √ 2c l ζ l dw l t ,
which defines a diffusion process (ξ t ) t≥0 in M 0 (D).

Lemma 5.2 (Infinitesimal generator of (ξ t ) t≥0 ). The infinitesimal generator of (ξ t ) t≥0 is the operator M defined in Lemma 5.1.

Proof : Applying Ito's Formula to ψ(ξ t ) for a smooth function ψ : M 0 (D) → R, we observe that the claimed result reduces to checking the identity

∀ξ ∈ M 0 (D), k-1 l=1 c l ζ l , ∇ 2 ψ(ξ)ζ l = A π D :: ∇ 2 ψ(ξ).
For a fixed ξ ∈ M 0 (D), Proposition A.1 shows that there exist g 1 , . . . , g k-1 ∈ C 0 (D) and d

1 , . . . , d k-1 ∈ R such that ∀l = 1, . . . , k -1, ζ l , ∇ 2 ψ(ξ)ζ l = k-1 m=1 d m ζ l , g m 2 .
As a consequence,

k-1 l=1 c l ζ l , ∇ 2 ψ(ξ)ζ l = k-1 l,m=1 c l d m ζ l , g m 2 = k-1 m=1 d m A π D g m , g m .
Using the definition of the operator A π D and of the measure θ x,y , we now write, for m ∈ {1, . . . , k -1},

A π D g m , g m = A π D (g m ) = 1 2 x,y∈D π(x)p π D (x, y) [g m (y) -g m (x)] 2 = 1 2 x,y∈D π(x)p π D (x, y) θ x,y , g m 2 ,
so that, by the definition of g m and d m ,

k-1 m=1 d m A π D g m , g m = 1 2 x,y∈D π(x)p π D (x, y) k-1 m=1 d m θ x,y , g m 2 = 1 2 x,y∈D π(x)p π D (x, y) θ x,y , ∇ 2 ψ(ξ)θ x,y = A π D :: ∇ 2 ψ(ξ),
where we have used (5.2) to obtain the last identity. A.3. Lyapunov equation. In this section, we recall a standard result regarding Lyapunov equations, see for instance Abou-Kandil et al. (2012, Theorem 1.1.7, p. 6).

  Quasistationary distribution. Let E be a finite set, and P be a stochastic matrix on E with coefficients p(x, y), x, y ∈ E. Denote by (x t ) t≥0 the continuous-time Markov chain in E with infinitesimal generator∀x ∈ E, ∀f : E → R, Lf (x) := y∈E p(x, y)[f (y) -f (x)]. (1.1)For any nonempty subset D of E, define the random time τ D by τ D := inf{t ≥ 0 : x t ∈ D}.

2. 2 .

 2 Quasistationary distribution. Let us write p D (x, y) := p(x, y), x, y ∈ D, and define P D : C(D) → C(D) by ∀f ∈ C(D), ∀x ∈ D, P D f (x) := y∈D p D (x, y)f (y

  to a centered Gaussian random variable with covariance operator K : C(D) → M(D) defined by Kf, f := Var π (f ) + 2λ +∞ s=0 e 2λs Var π (Q s f ) ds, for any f ∈ C(D) such that π, f = 0.

e

  2λs Var π (Q s f ) ds, for any f ∈ C(D) such that π, f = 0 (see Del Moral and Miclo (2003, Proposition 3.7) and Cérou et al. (2016, Corollary 2.2)).

  .4) where both subspaces R1 and C 0 (D) are stable by P π D . Likewise, M(D) = Rπ ⊕ M 0 (D), M 0 (D) := {ξ ∈ M(D) : ξ, 1 = 0}, (3.5) and both subspaces Rπ and M 0 (D) are stable by (P π D ) * . Remark 3.3 (Dual spaces of M 0 (D) and C 0 (D)). Throughout the article, we identify the spaces M 0 (D) and C 0 (D) as each other's dual.

  (i) Any complex eigenvalue σ = 1 of P π D satisfies Re σ ≤ 1 -λ -γ. (ii) For any δ > 0, for any norm • on C(D), there exists C δ ∈ [0, +∞) such that, for any f ∈ C 0 (D), f (x) ≤ C δ e -t(λ+γ-δ) f .

  3.3.1. The drift operator. We recall that by (3.3) and (3.5), the subspace M 0 (D) of M(D) is stable by the adjoint (P π D ) * : M(D) → M(D) of the operator P π D : C(D) → C(D). Definition 3.5 (Drift operator). The drift operator is the operator B 0 : M 0 (D) → M 0 (D) defined by

Lemma 4. 1 (

 1 Tightness of the fluctuation field). Let • be any norm on M(D).

  of M n . For any smooth function ψ : M 0 (D) → R, the gradient ∇ψ(ξ) ∈ C 0 (D) and the Hessian matrix ∇ 2 ψ(ξ) : M 0 (D) → C 0 (D) are defined by the Taylor expansion∀ξ, ζ ∈ M 0 (D), ψ(ξ + ǫζ) = ψ(ξ) + ǫ ζ, ∇ψ(ξ) + ǫ 2 2 ζ, ∇ 2 ψ(ξ)ζ + o(ǫ 2 ).Besides, for any symmetric operator R : M 0 (D) → C 0 (D), we introduce the notation A π D :: R := 1 2 x,y∈D π(x)p π D (x, y) θ x,y , Rθ x,y , (5.2)

Lemma 5. 1 (

 1 Convergence of M n ). Let ψ : M 0 (D) → R be a C ∞function with compact support. We have lim n→+∞ sup ξ∈M0(D) M n ψ(ξ) -Mψ(ξ) = 0, where Mψ(ξ) := B 0 ξ, ∇ψ(ξ) + A π D :: ∇ 2 ψ(ξ).

  y , for x, y ∈ D. By the Taylor-Lagrange inequality,

  n (ξ) -Mψ(ξ) = 0.

Lemma 5. 3 (

 3 Stationary distribution of (ξ t ) t≥0 ). The unique stationary distribution of (ξ t ) t≥0 is the centered Gaussian measure on M 0 (D) with the covariance operator K : C 0 (D) → M 0 (D) defined byK := 2 +∞ s=0 e sB0 A πD e sB * 0 ds.Proof : Lemma 3.7 and Proposition A.2 that the diffusion process (ξ t ) t≥0 is uniformly elliptic on M 0 (D), implies that transition semigroup of (ξ t ) t≥0 has a positive density with respect to the Lebesgue measure on M 0 (D)(Abou- Kandil et al. (2012, Corollary 1.1.6, p. 6) andPavliotis (2014, Eq. (3.108), p. 82)). This 'irreducibility' condition ensures uniqueness of stationary distributions (see for instance the arguments in Pagès (2001, Section 3.1)).On the other hand,Pavliotis (2014, Proposition 3.5, p. 80) shows that a Gaussian measure on M 0 (D) is stationary for (ξ t ) t≥0 if and only if it is centered and its covariance operator K : C 0 (D) → M 0 (D) satisfies the Lyapunov equationB 0 K + KB * 0 + 2A π D =0, which by Remark 3.6 and Proposition A.3 completes the proof. Lemma 5.4 (Identification of the variance). For any f ∈ C 0 (D), Kf, f = Var π (f ) + 2λ +∞ s=0 e 2λs Var π (P π s,D f )ds = Var π (f ) + 2λ +∞ s=0 e 2λs Var π (Q s f )ds. Proof : Let us fix f ∈ C 0 (D) and write Kf, f = 2 +∞ s=0 A π D e sB * 0 f, e sB * 0 f ds. For any s ≥ 0, it follows from Definition 3.5 and (3.2) that e sB * 0 f = e s(P π D -(1-λ)I C 0 (D) ) f = e λs P π s,D f, whence Kf, f = 2 +∞ s=0 e 2λs A π D (P π s,D f )ds. On the other hand, it follows from the definition of A π D that for any s ≥ 0, π P π s,D f , (i) N is positive definite. (ii) The numbers c 1 , . . . , c k-1 provided by Proposition A.1 are positive. (iii) For any norm • C0(D) on C 0 (D), there exists c N ∈ (0, +∞) such that for all f ∈ C 0 (D), N f, f ≥ c N f 2 C0(D) . Besides, under any of these conditions, N is invertible and its inverse N -1 : M 0 (D) → C 0 (D) is a symmetric and positive definite operator. With obvious adjustments, a similar statement holds for symmetric and positive definite operators R : M 0 (D) → C 0 (D).

  Proposition A.3 (Solution to Lyapunov equations). Let B 0 : M 0 (D) → M 0 (D) be an operator of which all the eigenvalues have a negative real part, and let A : C 0 (D) → M 0 (D) be a symmetric operator. (i) The integral K := 2 +∞ s=0 e s B0 Ae s B * 0 ds is finite and defines a symmetric operator C 0 (D) → M 0 (D). (ii) K is the unique symmetric solution to the Lyapunov equation B 0 K + K B * 0 + 2A = 0. (iii) If A is positive definite, then so is K.

Acknowledgements

We thank Arnaud Guyader, Laurent Miclo and Mathias Rousset for fruitful discussions.

This work is partially supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492 and the French National Research Agency (ANR) under the program ANR-17-CE40-0030 -EFI -Entropy, flows, inequalities.

1

where in the last equality we used the fact that π, P π s,D f = π, f = 0. As a consequence, Kf, f = -+∞ s=0 e 2λs d ds Var π P π s,D f ds, and using Lemma 3.4 (ii) to integrate the right-hand side by parts leads to the first claimed expression. The second expression follows from the fact that by Lemma 3.4 (iii), the operators P π s,D and Q s coincide on C 0 (D).

5.4. Proof of Theorem 2.4. We may now complete the proof of Theorem 2.4.

Proof of Theorem 2.4: We recall from (5.1) that the fluctuation field

Therefore by Prohorov's Theorem (Billingsley (1999, Theorem 5.1, p. 59)), any subsequence possesses a further subsequence, which we shall index by n m , which converges in distribution to a random variable ξ ∞ in M 0 (D). In particular, for any

, where we have used Lemma 5.1 to get that E[(M nm -M)ψ(ξ nm ∞ )] converges to 0. Thus, we deduce from Lemma 5.2 that the law of ξ ∞ is a stationary distribution for (ξ t ) t≥0 , which by Lemma 5.3 entails its identification and yields the convergence of the whole sequence (ξ n ∞ ) n≥2 to ξ ∞ , with the asymptotic covariance given by Lemma 5.4.

Appendix A. Complements on operators and Lyapunov equations

In this appendix, we recall some elementary results of linear algebra which are useful in our framework. We denote by k ≥ 1 the cardinality of D, so that M 0 (D) and C 0 (D) are linear spaces of dimension k -1.


Floréal Cabanettes

Jean-François Dufayard

Vincent Berry

Vincent Lefort

Frédéric De Lamotte

Anne-Muriel Arigon Chifolleau

CompPhy2: a flexible and real time collaborative web platform for editing and comparing phylogenies

CompPhy2 is a real time collaborative web platform for visualization, comparison and annotation of phylogenetic trees with a responsive user friendly interface. It is the only phylogenetic web platform allowing several users to work synchronously or asynchronously on a same dataset. CompPhy2 brings new features such as side by side comparison, tree annotation, custom display, supertree inference, ...

Introduction

Online collaborative and real-time tools are widely used in various domains for projects involving remote partners. In the last twenty years, web technologies have enabled the development of such tools to jointly edit office documents. However, only few such tools exist in scientific domains, but prior to the introduction of the CompPhy website [10], no collaborative tool was available to process phylogenies. CompPhy⣙s purpose is to fulfill this gap in the phylogeny domain. Although a large number of studies and projects in evolutionary biology and systematics involve collaborations between scientists from different institutes, or between students, most existing tools for visualizing and comparing phylogenies offer single-user access or focus on high-level features for a single tree (TreeView [START_REF] Page | Visualizing phylogenetic trees using TreeView[END_REF], TreeDyn [START_REF] Chevenet | TreeDyn: towards dynamic graphics and annotations for analyses of trees[END_REF], Dendroscope [START_REF] Huson | Dendroscope: An interactive viewer for large phylogenetic trees[END_REF], Archeopteryx [START_REF] Han | phyloXML: XML for evolutionary biology and comparative genomics[END_REF], FigTree [Rambaut, 2016], iTOL [START_REF] Letunic | Interactive Tree Of Life v2: online annotation and display of phylogenetic trees made easy[END_REF], EvolView [START_REF] Zhang | EvolView, an online tool for visualizing, annotating and managing phylogenetic trees[END_REF] and PhyloIO [START_REF] Robinson | io: interactive viewing and comparison of large phylogenetic trees on the web[END_REF]).

To fulfill this gap, we have introduced CompPhy in 2014, a web platform providing a collaborative approach to store, edit, visualize, compare phylogenetic trees [START_REF] Fiorini | CompPhy: a web-based collaborative platform for comparing phylogenies[END_REF]. This platform allows multiple users to manage shared phylogenetic trees stored on a common server. The access can be done synchronously, when several partners edit and discuss data or trees at the same time, or asynchronously, partners then accessing the work of others done since their last access. CompPhy is a unique tool that combines tree comparison operations such as restricting two trees to their common taxa, automatically swapping branches to match leaves of two trees, the computation of consensus trees and supertrees. The web platform provides two ways to handle trees: either by choosing two trees for a detailed visualization side by side, or in batch mode by operations on several hundred trees. Finally, it offers an interface with the usual editing features of trees, such as leaf or sub-tree coloring, annotation management, subtree swapping and re-rooting.

CompPhy introduced several key features. Data are stored online on a secured platform and all collaborators have access to an always up-to-date dataset, avoiding data loss or desynchronisation between partners. Since CompPhy is a web platform, it does not require any installation, a web "main" -2018/9/6 -page 2 -#2 browser is enough. The real time approach facilitates collaborative works between remote partners. This makes it also appropriate for teaching.

Comment

Students can easily work in groups, lecturers can share data, perform phylogenetic analyses in real time and show phylogenetic trees to students or give them assignments on the platform. Finally, the side by side tree comparaison is useful to visually highlight tree differences or common parts and customize each tree by annotating it. It opens the door to a visual comparison of trees inferred from different hypothesis or analyses. However, the first version of CompPhy suffered from several drawbacks: a non-smooth process when several partners edited trees simultaneously, several tree edit operations had to be performed in nonintuitive ways and finally only few annotations could be put in regards of the trees. We thus developed a new version, that aims to improve the existing functionalities and to add new ones, in particular to answer the needs of the biologist users from feedback we had.

Features

CompPhy2 offers a friendly and responsive interface developed with D3.js [START_REF] Bostock | D3 Data-Driven Documents[END_REF]. A user can easily create a project to gather phylogenetic trees and invite collaborators to cooperate on CompPhy2. Once a project is created, several actions are available, including :

• importing trees (Newick, NEXUS and NHX format), all trees are available online at anytime and can be visualized via thumbnails; • importing annotations in a tabular format to visualize them next to the trees; • visualizing and manipulating imported phylogenetic trees on the main CompPhy2 page; • export trees in various formats.

The main CompPhy2 page is divided into three areas (see Figure 1). The area at the top ("Tree Collections 1/2" in Figure 1) displays a list of imported trees, that can be splitted into two collections. From the thumbnail pictures of the trees in these collections, the user easily selects trees to be sent to the central workbench (allowing side by side display of two trees) for close inspection, comparison, annotation or discussion purposes. CompPhy2 accommodates big trees by automatically collapsing some selected subtrees.

The workspaces in the central area ( "Workbenches" in Figure 1) provide a set of features for editing and comparing two trees. Within a project, each user can take control of desired trees. Thus, the other project collaborators can see in live the changes made to these trees. They can also manipulate other trees. A set of relevant tools is available between the two workspaces:

• to customize the tree display (branch color and thickness, subtree coloring, subtree masking by a collapse operation, nodes swapping); • to set up the zoom applicable to the displayed trees;

• several tree comparison features: consensus trees computation, distance between the two trees topologies, or automatic branch rotating. The methods implemented in these tools comply with the state of art [START_REF] Fiorini | CompPhy: a web-based collaborative platform for comparing phylogenies[END_REF].

In each workspace, users can edit tree annotations ("Side annotations⣞ in Figure 1) and manually edit the tree in Newick format (" Manual newick editing " in Figure 1). The box at the bottom of the page ("Miscellaneous tools" in Figure 1) contains project management tools and tools that can be simultaneously applied to a bunch of trees, including renaming trees or taxa, re-rooting trees based on a specific out-group and inferring supertrees. For the latter operation, CompPhy uses the phylogenomics services provided by the ATGC platform (http://www.atgc-montpellier.fr/). Finally, a lateral toolbar The main page has three areas: tree collections ("Tree collection 1 and 2"), workspaces ("Workbenches") and an area containing tools for processing tree sets and managing the project ("Miscellaneous tools"). facilitates collaborative aspects. It allows a user to quickly have access to notifications and queries waiting to be answered (such as control requests) or to modify certain user parameters applicable to the project. An instant messenger is also available to facilitate the real time collaboration.

Two connection modes are available: a user can create a project while remaining anonymous (its data are not saved) or create an account (its data are then preserved). In the later mode, a user can share projects with others and beneficiate from the collaborative features to coordinate multiple accesses to the project by several partners at the same time.

Conclusion

CompPhy was a first step toward: a collaborative web platform in phylogeny. CompPhy2 improves significantly the first version and marks a major breakthrough. It presents new features and great enhancements of existing ones. The main developments concern the interactivity and the fluidity of the interface as well as the collaborative aspects. For the latter, the granularity is particularly refined to allow management at each tree. This online platform is available through a simple web browser and offers solutions to modern problems by facilitating collaborative work in phylogeny domain. CompPhy2⣙s interface was entirely reimplemented taking advantages of new technologies to present a modern, reactive and user-friendly interface. In 2014, CompPhy was integrated the ATGC platform (processing more than 25000 hours of computation per month), which coordinates the bioinformatics services activities in the South of France for the French Institute of Bioinformatics (IFB). CompPhy2 is been in production on the ATGC platform since 2017. "main" -2018/9/6 -page 3 -#3 CompPhy2

Comment

  citer ce document : Cabanettes, F. (Co-premier auteur), Dufayard, J.-F., Berry, V. (Collaborateur), Lefort, V. (Collaborateur), De Lamotte Guéry, F. (Co-dernier auteur), Arigon Chifolleau, A.-M. (Auteur de correspondance) (2018). CompPhy2: a flexible and real time collaborative web platform for editing and comparing phylogenies . Bioinformatics, In Press, 1-3.

Fig. 1 .

 1 Fig. 1. CompPhy2⣙s interface. The main page has three areas: tree collections ("Tree

  citer ce document : Cabanettes, F. (Co-premier auteur), Dufayard, J.-F., Berry, V. (Collaborateur), Lefort, V. (Collaborateur), De Lamotte Guéry, F. (Co-dernier auteur), Arigon Chifolleau, A.-M. (Auteur de correspondance) (2018). CompPhy2: a flexible and real time collaborative web platform for editing and comparing phylogenies . Bioinformatics, In Press, 1-3.

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Funding

This work benefited from a grant from the French state managed by the French National Research Agency under an Investment for the Future program through the LabEx NUMEV (reference ANR-10-LABX-20).


Tanguy Pouzol

Yves Lévi

Jean-Luc Bertrand-Krajewski

A dynamic pharmaceuticals loads source generator *

Keywords: Stochastic modelling, Individual behaviour, Pharmaceuticals sales

As pharmaceuticals and metabolites are assumed to reach the environment after human consumption and excretion, it is important to model their discharge in wastewater. Due to the difficulty to collect detailed data on pharmaceuticals sales and the diversity of processes influencing the fate of pharmaceuticals, modelling pharmaceutical fate is difficult and models in the literature struggle to accurately and reliably predict pharmaceuticals loads. In such a context, a new stochastic model is proposed. Pharmaceuticals loads in wastewater are predicted by modelling the discharge from sick persons in a catchment at a minute time step. Posology, time-use, metabolism and excretions are generated using different distributions. The model is tested for 9 molecules on an urban catchment (16 000 inhabitants), using monthly sales data. 24 h average samples and 24 consecutives 1 h average samples campaigns are used to assess the model performance. Results show that the model can predict with reasonable accuracy and reliability daily and hourly loads for an urban catchment.

INTRODUCTION

Pharmaceuticals in the environment are an important field of study and a great concern in terms of environmental and public health risks. Their modelling is paramount to fully understand their fate. However monitoring pharmaceuticals with analytical chemistry is difficult and costly. A significant part of pharmaceuticals reach the environment through human consumption and excretion [START_REF] Heberer | Occurrence, fate, and removal of pharmaceutical residues in the aquatic environment: a review of recent research data[END_REF][START_REF] Qian | Occurrence, sources and fate of pharmaceuticals and personal care products in the groundwater: A review[END_REF] and pharmaceuticals fate models almost always include a source model in which loads discharged into wastewater are proportional to sales data [START_REF] Kümmerer | Biodegradability of the anti-tumour agent ifosfamide and its occurrence in hospital effluents and communal sewage[END_REF][START_REF] Heberer | Contribution of effluents from hospitals and private households to the total loads of diclofenac and carbamazepine in municipal sewage effluentsmodelling versus measurements[END_REF][START_REF] Singer | Rapid Screening for Exposure to "Non-Target" Pharmaceuticals from Wastewater Effluents by Combining HRMS-Based Suspect Screening and Exposure Modeling[END_REF]. The results of such models are difficult to interpret since, from one molecule to another and from one study to another, pharmaceuticals loads may be greatly over-or under-estimated (de Garcia et al., 2013). However, they are widely used because they are quick and easy to use and do not require detailed data, compared to other more elaborated models (e.g. [START_REF] Coutu | Integrated stochastic modelling of pharmaceuticals in sewage networks[END_REF]. In this context, we propose and assess a new pharmaceuticals fate stochastic model in wastewater based on detailed pharmaceutical sales data and monitoring campaigns.

MATERIALS AND METHODS

Experimental site and data

The model is tested on the Bellecombe wastewater treatment plant (WWTP) (France) collecting effluents from 16 000 inhabitants through 230 km of sewers which were previously modelled [START_REF] Pouzol | Stochastic modelling of dry weather flow in an extensive sewer network using individual behaviour[END_REF]. Census data indicate that 14 500 inhabitants are active people and that only 7 000 jobs exist on the catchment; then other active people work out of the catchment during the day. Monthly pharmaceuticals sales over 2.5 years were collected. Nine molecules were monitored with state-of-theart sampling strategy, method (materials and handling) and analyses (LC-MS/MS). Twenty 24 h average samples and four 24 consecutives 1 h average samples campaigns were performed between 2012 and 2016.

Model

The successive processes leading to the discharge of pharmaceuticals in sewers are stochastic by nature. The model thus aims generating discharges of sick persons over the catchment, who can be either inhabitants or outside persons who work in the catchment. Their number is determined by i) the total mass M of pharmaceuticals consumed in one day that is randomly picked according to sales data and ii) the standard individual posology. For each sick person, the model follows 4 steps: Posology. It is determined according to pharmaceutical data and websites and consists of time consumption pattern, number of intakes, mass for each intake and duration between intakes. The last three values are randomly picked from literature distributions. The time consumption pattern consists of generic descriptions, e.g. "Awake and not out from household" for presence or "33% chance for each meal" for intakes time.

Time-Use. Periods of time when the person is asleep, out of household and eating are generated according to time-use survey data. These periods are weighted according to the time consumption pattern and then all intakes are randomly picked.

Metabolism. Pharmaceuticals intakes are transformed into a mass of pharmaceutical stored in the bladder over time, according to medical literature. Fractions of pharmaceuticals are transferred through three "boxes" (gastro-intestinal system, blood and bladder) at certain rates (exponential decay). All parameters are randomly picked from medical literature distributions (e.g. www.eurekasante.vidal.fr, https://pubchem.ncbi.nlm.nih.gov/).

Excretion. When persons excrete urine in toilets, pharmaceuticals stored in the bladder are discharged. The times of toilet uses are randomly picked using the time-use generated in step 2 (see Pouzol et al., 2015 for details). For local inhabitants, it is assumed that excretion during "out of household" times are made outside the catchment. Conversely, for outside persons working in the catchment, these excretions are the only ones considered.

RESULTS AND DISCUSSION

Daily loads

Measured and modelled daily loads are compared using the ratio average modelled over average measured daily loads (R). The results of the model for one molecule are considered satisfactory whenever the ratio R is between 0.5 and 2. Also, the model is considered reliable if it has satisfactory results for every molecule.

Eight out of the nine molecules show satisfactory ratios R (average: 1.3, min: 0.5, max: 2.19). Ketoprofen is the only molecule with an unsatisfactory ratio R. However, its metabolic parameters are not well defined (wide range of excretion) and it is not sold solely as oral forms (Ketoprofen: 17 % dermal forms), as such the fraction directly discharged into the sewer is not well known and approximated by the model (from 25 to 75 %). This leads to an over-estimation of its daily loads.

The ratios between the coefficients of variation of the modelled over the measured daily loads indicate that the model underestimates the dispersion of the daily loads (averagely 0.28). This seems to be mainly the result of the temporal scale used for the pharmaceutical sales. Indeed, daily sales derived from monthly data are understandably unable to represent daily sales and so daily consumption and loads in wastewater.

Hourly loads

Usually, measured and modelled hourly dynamics are compared using the Nash-Sutcliffe efficiency (NSE) indicator. However, as only 4 hourly loads campaigns are available, average normalized measured hourly loads time series are not sufficiently representative of the dynamics. As an alternative, it is proposed to apply a modified NSE indicator, named 𝑵𝑺𝑬 𝒇𝒖𝒛𝒛𝒚 , for each measured time series accounting for the distribution of all modelled time series. 𝑵𝑺𝑬 𝒇𝒖𝒛𝒛𝒚 is calculated as follows and is interpreted like the usual NSE (satisfactory model performance if above 0.5): The model is considered reliable if it gives satisfactory results for every molecule.

𝑵𝑺𝑬 𝒇𝒖𝒛𝒛𝒚 = 𝟏 - ∑ (𝑳 ̃𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 (𝒕) -𝑳 ̃𝒇𝒖𝒛𝒛𝒚 (𝒕)) 𝟐 𝑻 𝒕=𝟏 ∑ (𝑳 ̃𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 (𝒕) -𝑳 ̃𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 ̅̅̅̅̅̅̅̅̅̅̅̅̅ ) 𝟐 𝑻 𝒕=𝟏 (
Seven out of the nine molecules show satisfactory 𝑁𝑆𝐸 𝑓𝑢𝑧𝑧𝑦 values (average: 0.5, min: 0.18, max: 0.72). Results for Ibuprofen are displayed in Figure 1. The low 𝑁𝑆𝐸 𝑓𝑢𝑧𝑧𝑦 values for Atenolol (0.18) and Carbamazepine (0.19) can be explained by the combination of i) stochastic nature of pharmaceutical consumption and ii) low consumption levels (only a few patients per day for Carbamazepine in the entire catchment). More hourly campaigns would help to determine the average dynamics of hourly loads with more confidence. Assessing the model performance would then be less sensitive to random peaks in measured hourly loads which do reflect the average dynamics.

CONCLUSIONS

The proposed model is able to predict reliably both the daily and hourly loads of pharmaceuticals at the WWTP inlet with an acceptable accuracy considering the available data and the analytical uncertainties. Daily loads are either over or underestimated depending on the molecule. The variability of the daily loads is under-estimated. Hourly loads are sensitive to random peaks due to the stochastic nature of the processes.

  1) 𝒊𝒇 𝑸 𝟏 (𝑳 ̃𝒎𝒐𝒅𝒆𝒍𝒍𝒆𝒅 ≤ 𝑳 ̃𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 (𝒕) 𝒕𝒉𝒆𝒏 𝑳 ̃𝒇𝒖𝒛𝒛𝒚 (𝒕) = 𝑸 𝟏 (𝑳 ̃𝒎𝒐𝒅𝒆𝒍𝒍𝒆𝒅 (𝒕)) 𝒊𝒇 𝑸 𝟑 (𝑳 ̃𝒎𝒐𝒅𝒆𝒍𝒍𝒆𝒅 (𝒕)) ≤ 𝑳 ̃𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 (𝒕) 𝒕𝒉𝒆𝒏 𝑳 ̃𝒇𝒖𝒛𝒛𝒚 (𝒕) = 𝑸 𝟑 (𝑳 ̃𝒎𝒐𝒅𝒆𝒍𝒍𝒆𝒅 (𝒕)) 𝒆𝒍𝒔𝒆 𝑳 ̃𝒇𝒖𝒛𝒛𝒚 (𝒕) = 𝑳 ̃𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 (𝒕) with: 𝑡: time 𝐿 ̃𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 (𝑡): normalized measured pharmaceutical hourly load at time 𝑡 𝐿 ̃𝑓𝑢𝑧𝑧𝑦 (𝑡): normalized modelled hourly load at time t for 𝑁𝑆𝐸 𝑓𝑢𝑧𝑧𝑦 𝐿 ̃𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 ̅̅̅̅̅̅̅̅̅̅̅̅̅ : average normalized measured hourly load at time 𝑡 𝑄 1 (𝑋), 𝑄 3 (𝑋): first and third quartiles of a list of values 𝑋 𝐿 ̃𝑚𝑜𝑑𝑒𝑙𝑙𝑒𝑑 (𝑡): distribution of the normalized modelled hourly loads at time 𝑡.

Figure 1 .

 1 Figure 1. Comparison of the measured and modelled hourly loads.

Future works should focus on: i) measuring more hourly loads in order to better determine the average dynamics of pharmaceuticals, ii) testing the model on other catchments and for more molecules and iii) simplifying the model structure in order to facilitate its implementation and use.