Elma Nassar
email: elma_nassar@hotmail.com

Etienne Pardoux
email: etienne.pardoux@univ-amu.fr

Small jumps asymptotic of the moving optimum Poissonian SDE

Keywords: Poissonian SDE, Law of large numbers, Central limit theorem, Approximation of invariant measure, canonical equation of adaptive dynamics, moving optimum model

We consider a Poissonian SDE for the lack of fitness of a population subject to a continuous change of its environment, and an accumulation of advantageous mutations. We neglect the time of fixation of new mutations, so that the population is monomorphic at all times. We consider the asymptotic of small and frequent mutations. In that limit, we establish a law of large numbers and a central limit theorem. For small enough mutations, the original process is Harris recurrent and ergodic. We show in which sense the limits as t → ∞ of the law of large and number and central limit theorem give a good approximation of the invariant probability measure of the original process.

Introduction

The present work is motivated by the moving optimum model in theoretical biology, which aims at evaluating the possibility for mutations to rescue a given population undergoing a linear change in its environment which deteriorates its survival conditions. We refer the reader to [START_REF] Kopp | Hermisson The genetic basis of phenotypic adaptation II: The distribution of adaptive substitutions in the moving optimum model[END_REF] and [START_REF] Matuszewski | Fisher's geometric model with a moving-optimum[END_REF] for the presentation of this 1 model. The authors have set in [START_REF] Nassar | On the long time behavior of an SDE driven by a Poisson point process[END_REF] a rigorous mathematical study of the moving optimum model by introducing a stochastic differential equation driven by a Poisson point process describing the evolution of a quantitative one-dimensional phenotypic trait in accordance with the biological description of this evolutionary rescue model. They studied the large time behavior of its solution, which is Harris recurrent when the speed of the environment v is smaller than the mean effect of the beneficial mutations m per unit time, transient if v > m. In the case of equality between the two parameters, the solution of the stochastic differential equation can either be transient or Harris recurrent depending upon additional technical conditions.

One is mainly interested in the positive recurrent case. However, the limitation of the ability to draw biological conclusions from these result is due to the difficulty to compute explicitly any quantity related to the invariant probability measure. This led us to study the small jumps limit, which is obtained by multiplying the jumps' sizes by ε, dividing the rates by ε 2 , and then letting ε → 0. In this paper, we study the limit as ε → 0 in the such rescaled multidimensional version of the SDE from [START_REF] Nassar | On the long time behavior of an SDE driven by a Poisson point process[END_REF]. More precisely, we consider the SDE in R d

X ε t = X ε 0 -vt + t 0 R d 1 0 εαΓ(X ε s -, εα, ξ)N ε (ds, dα, dξ),
where N ε is a Poisson random measure on R + × R d × [0, 1], with the intensity 1 ε 2 ds × ν(dα) × dξ.

Our first main result is Theorem 1. If X ε 0 → x 0 in probability as ε → 0, then X ε t --→ ε→0 xt in probability, locally uniformly in t, where, with a given d × d matrix L,

dx t dt = -v + Lx t , x0 = x 0 .
The above limiting ODE can be interpreted as the canonical equation of adaptive dynamics in the context of a changing environment, see [START_REF] Dieckmann | The dynamical theorey of coevolution: a derivation from stochastic ecological processes[END_REF] and [START_REF] Champagnat | Polymorphic evolution sequence and evolutionary branching[END_REF]. This is discussed in greater detail in [START_REF] Kopp | Phenotypic lag and population extinction in the moving-optimum model : insights from a small-jumps submitted[END_REF]. The next step is to establish a Central Limit Theorem. Indeed, we define U ε t = ε -1/2 (X ε t -xt ), and show Theorem 2. Assume that X ε 0 = x0 . Then U ε ⇒ U , where U is an Ornstein-Uhlenbeck process:

dU t = LU t dt + Λ 1 2 (x t )dB t , U 0 = 0,
with B being a d-dimensional standard Brownian motion. In other words,

U t = t 0 e L(t-s) Λ 1 2 (x s )dB s .
Moreover it is not hard to show that as t → ∞, both xt → x∞ and the law of U t converges to a Gaussian law N (0, S2 ).

Both x∞ and S2 can be easily computed with high accuracy, given the parameters of the model. On the other hand, we show that there exists ε 0 > 0 such that, for any 0 < ε ≤ ε 0 , the process X ε t is Harris recurrent and possesses a unique invariant probability measure µ ε . Since our motivation for studying the small jumps limit is to get informations about µ ε , it is desirable to show that the pair (x ∞ , S2 ) gives a precise approximation of the invariant measure µ ε , for small ε. This is a delicate question, since it amounts in a sense to invert the two limits ε → 0 and t → ∞.

We first show that the collection of probability measures {µ ε , ε ≤ ε 0 } is tight. It is then not too difficult to deduce that µ ε ⇒ δ x∞ , as ε → 0. We want to prove more, namely that µ ε is close to ν ε , which is the law of x∞ + √ εξ, if ξ N (0, S2 ). This is done by analysing the probability measure

µ ε t (A) = 1 t t 0 1 A (X ε s )ds
for large t and small ε.

We believe that those results are original, and have an interest, not only for the specific model which we study, but could also be useful in different frameworks, where a process converges in law to a limiting process, and one wants to compare large time behaviors. In our case, numerical simulations tend to indicate that the approximation is valid even for not very small values of ε, see [START_REF] Kopp | Phenotypic lag and population extinction in the moving-optimum model : insights from a small-jumps submitted[END_REF].

The proofs of the first two theorems follow the following scheme : prove tightness, and the identify the unique possible limit point. Harris recurrence for small enough ε is proved using a criterion due to Meyn and Tweedie [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF].

The paper is organized as follows: After giving some useful notations, section 1 gives a detailed presentation of the model, from the biological literature. Furthermore we present the stochastic differential equation that describes the evolution of the vector phenotypic lag between the population and its environment, explaining the fixation mechanism of mutations.

In section 2, we prove Theorem 1 and Theorem 2.

Section 3 is dedicated to the study of the large time behavior of X ε t which will turn out to be positive Harris recurrent for ε sufficiently small, admitting thus a unique invariant probability measure. Then we proceed to prove that the sequence of invariant measures is tight and converges in law. We finally give a precise statement which describes in which sense the invariant measure is well approximated by a combination of the limit as t → ∞ of the LLN and the CLT limits. This involves a sort of interchange of limits as ε → 0 and as t → ∞, which seems to be new.

Notation

We will deal with processes X t with "càdlàg" paths, i.e. paths which are rightcontinuous and have a left limit everywhere. This left limit is denoted X t-:= lim s↑t,s<t X s , and for any t such that X t = X t-, we denote by ∆X t = X t -X t-the jump of X at time t. Note that a càdlàg path has at most countably many jumps on R + .

We remind that the quadratic variation of a scalar discontinuous bounded variation locally square integrable martingale M t is the sum of the squares of its jumps and is denoted by :

[M ] t = s≤t |∆M s | 2 .
Its predictable quadratic variation M t is the unique increasing predictable process such that [M ] t -M t , and hence also M 2 t -M t is a local martingale. In the d-dimensional case, we define the quadratic variation of a discontinuous bounded variation locally square integrable martingale M t as :

[[M ]] t = s≤t ∆M s ⊗ ∆M s .
Its predictable quadratic variation M t is the unique S d -valued predictable increasing process such that both [[M ]] t -M t and M t ⊗M t -M t are S d -valued local martingales. Here S d denotes the set of symmetric positive semi-definite

d × d matrices. Note that [[M ]] t (resp. M t ) is the matrix whose i, j element is [M i , M j ] t (resp. M i , M j t ). We shall use the notation [M ] t = 0<s≤t ∆M s 2 = Tr[[M ]] t , and M t = Tr M t ,
so that M t 2 -M t is an R-valued local martingale, and the notations in the scalar and vector case are coherent. See [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF] for more details. Suppose now that X t is a d-dimensional process which takes the form

X t = X 0 + t 0 A s ds + M t ,
where A t is d-dimensional and M t is a d-dimensional martingale with paths of bounded variation (hence discontinuous). Then, if • and (•, •) denote the Euclidean norm and scalar product in R d , we have the following

X t 2 = X 0 2 + 2 t 0 (X s , A s )ds + 2 t 0 (X s-, dM s ) + [M ] t . ( 1
)
This is a change of variable formula which is the content (in the case where X t is scalar) of Theorem 31 in [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF]. Its proof is quite elementary and does not require any stochastic calculus.

Finally in this paper we will repeatedly write integrals of vector-valued (or matrixvalued) functions of their argument, as we already did in the definition of X t above, where the vector t 0 A s ds was such that its i-th coordinates equals t 0 A i s ds if A i s stands for the i-th coordinate of A s . Below in [START_REF] Kimura | On the probability of fixation of mutant genes in a population[END_REF], (x,α)≤0 α ⊗ α ν(dα) (where {α, (x, α) ≤ 0} is a subset of R d ) stands for the d × d matrix whose (i, j) entry is (x,α)≤0 α i α j ν(dα), and (x,α)≤0 α ν(dα) stands for the d-dimensional vector whose i-th coordinate is (x,α)≤0 α i ν(dα).

The model

The model from Matuszewski et al. [START_REF] Matuszewski | Fisher's geometric model with a moving-optimum[END_REF] is set up as follows: a population of constant size N is subject to Gaussian stabilizing selection with a moving optimum that increases linearly with speed vector v ∈ R d . That is, at time t, the phenotypic lag between an individual with trait value z ∈ R d and the optimum equals x = z -vt ∈ R d , and the corresponding fitness is

W(x) = exp -x Σ -1 x , (2)
where Σ describes the shape of the fitness landscape. For the adaptive-walk approximation, the population is assumed to be monomorphic at all times (i.e., its state is completely characterized by x). Mutations arise at rate Θ/2 = N µ (where µ is the per-capita mutation rate and Θ = 2N µ is a standard population-genetic parameter), and their phenotypic effects α are drawn from a distribution p(α). We neglect the possibility of fixation for deleterious mutations. Yet even beneficial mutations have a significant probability of being lost due to the effects of genetic drift while they are rare. A mutation with effect α that arises in a population with phenotypic lag x has a probability of fixation

g(x, α) =    1 -exp(-2s(x, α)) if s(x, α) > 0, 0 otherwise (3)
where

s(x, α) = -(2x + α) Σ -1 α (4)
is the selection coefficient. Once a mutation gets fixed, it is assumed to do so instantaneously (which is of course a simplification which is not realistic), and the phenotypic lag x of the population is updated accordingly.

We make the following assumptions (see [START_REF] Matuszewski | Fisher's geometric model with a moving-optimum[END_REF]):

1. v is a horizontal vector, v = (v 1 , 0, . . . , 0) with v 1 > 0. 2. Σ is isotropic, i.e. Σ = σ 2 I R d .
It is always possible to reduce the situation to our assumptions, via a change of variables.

The evolution of the phenotypic lag of the population can be described by the following equation:

X t = x 0 -vt + [0,t]×R d ×[0,1] αΓ(X s -, α, ξ)N (ds, dα, dξ). (5)
Here, N is a Poisson point process over R + × R d × [0, 1] with intensity ds ν(dα) dξ where ν(dα) is the measure of new mutations and

Γ(x, α, ξ) = 1 {ξ≤g(x,α)} ,
where the fixation probability g(x, α) of a mutation of size α that hits the population when the lag is x, as defined by ( 3) and ( 4), can be expressed as

g(x, α) = 1 -exp 2σ -2 (2x + α, α) × 1 {(2x+α,α)≤0} .
Following the model by [START_REF] Matuszewski | Fisher's geometric model with a moving-optimum[END_REF], we consider that

ν(dα) = Θ 2 p(α)dα, ( 6
)
where p is the density of a centered multidimensional Gaussian distribution N (0, M ), M being a positive definite symmetric matrix. Under the above assumptions about the speed vector v and the fitness matrix Σ, M is generally not an isotropic matrix. Θ/2 is the rate at which new mutations are "proposed".

The points of this Poisson Point Process (T i , A i , Ξ i ) are such that the (T i , A i ) form a Poisson Point Process over R + ×R d of the mutations that hit the population with intensity dsν(dα), and the Ξ i are i.i.d. U[0, 1], globally independent of the Poisson Point Process of the (T i , A i ). T i 's are the times when mutations are proposed and A i 's are the effect sizes of those mutations. The Ξ i are auxiliary variables determining fixation: a mutation gets instantaneously fixed if Ξ i ≤ g(X T i , A i ), and is lost otherwise.

Note that the limit of the probability of fixation as x → ∞, while x x remains a constant unit vector, is 1 {(x,α)≤0} . This means that when the process is sufficiently far away from 0, the fixation mechanism tends to accept all mutations inside the half space x

x , α ≤ 0. Define the covariance matrix of fixed mutations :

V (x) = (x,α)≤0 α ⊗ α ν(dα). ( 7
)
Proposition 1. Under the definition of ν given by (6), V (x) is independent of the direction of x and

V = Θ 4 M.
Proof. The additional factor 1/2 comes from the fact that we integrate over a half space. What is not obvious a priori is that V does not depend upon x.

We assume without loss of generality that we have chosen as orthonormal basis of R d a basis made of eigenvectors of M , the covariance matrix of p. In other words,

M = P DP,
where D is a diagonal matrix and P is the matrix representing the change of basis such that P = P -1 = P * . First, we will show that 1

(2π) d 2 (det D) 1 2 (x,α)≤0 α ⊗ α e -1 2 α D -1 α dα = D 2 . ( 8
)
This is equivalent to showing that for X 1 , . . . , X d being mutually independent zero mean Gaussian random variables, and for any vector

a = (a 1 , . . . , a d ) ∈ R d , with the notation E[Z; A] = E(Z1 A ), E X 2 j ; (X, a) ≤ 0 = 1 2 E X 2 j , for any j ∈ {1, . . . , d} and E [X j X ; (X, a) ≤ 0] = 0, for any j = ∈ {1, . . . , d}.
The first of these two identities follows from the fact that if X and Y are two mutually independent zero mean Gaussian random variables, then

E[X 2 ; Y < X] = 1 2 E[X 2 ].
Indeed, if F Y denotes the distribution function of the zero mean Gaussian r.v. Y , and σ 2 is the variance of X, since

F Y (x) + F Y (-x) = 1 for all x ∈ R, E[X 2 ; Y ≤ X] = E[X 2 F Y (X)] = 1 σ √ 2π ∞ 0 x 2 [F Y (x) + F Y (-x)]e -x 2 /2σ 2 dx = 1 2 E[X 2 ].
We now establish the second formula. All we have to compute is the following quantity, where X, Y, Z are mutually independent zero mean Gaussian random variables, and a, b are arbitary real numbers,

E[XY ; Z ≤ aX + bY ] = E[XY F Z (aX + bY )] = 1 στ 2π R R xyF Z (ax + by)e -x 2 2σ 2 -y 2 2τ 2 dxdy = 1 στ 2π ∞ 0 ∞ 0 xy[F Z (ax+by)+F Z (-ax-by) -F Z (-ax+by)-F Z (ax-by)]dxdy = 0, since clearly F Z (ax + by + F Z (-ax -by) = F Z (-ax + by) + F Z (ax -by) = 1.
Now that (8) is established, the change of variables α = P α in the integral formula for V yields

V = Θ 2 P D 2 P = Θ 4 M.

Small Jumps Limit

We now introduce the rescaling α = εα and s = s ε 2 with ε > 0 of the jumps and the time, respectively. In other words, we rewrite our process (5) as

X ε t = X ε 0 -vt + t 0 R d 1 0 εαΓ(X ε s -, εα, ξ)N ε (ds, dα, dξ),
where the intensity measure of the Poisson Point Process N ε is

1 ε 2 ds × ν(dα) × dξ.
The above SDE can be rewritten as

X ε t = X ε 0 -vt + t 0 1 ε 2 m ε (X ε s )ds + M ε t , (9)
with

m ε (x) = R d εαg(x, εα)ν(dα), where g(x, εα) ≤ 2σ -2 (2x + εα, εα) 1 {(2x+εα,εα)≤0} ≤ 4σ -2 ε (x, α) 1 {(x,α)≤0} ,
and, if N ε denotes the compensated Poisson measure N ε , i.e.

N ε (ds, dα, dξ) = N ε (ds, dα, dξ) - 1 ε 2 dsν(dα)dξ, M ε t = t 0 R d 1 0 εαΓ(X ε s -, εα, ξ)N ε (ds, dα, dξ)
is a martingale. We will sometimes consider the above SDE with a random initial condition X ε 0 . We insist upon the fact that in such a case the Poisson Point Process N ε will be assumed to be independent of the initial condition X ε 0 . The goal of this section is to prove Theorem 1 below, which says that if X ε 0 → x0 , then X ε t converges to the solution xt of a linear ODE, and Theorem 2 below, which says the fluctuations U ε t := ε -1/2 (X ε t -xt ) converge to an Ornstein-Uhlenbeck process. We prepare the proofs of those two main results by first establishing two Lemmas and one Proposition. Lemma 1, Lemma 2, 1. and Proposition 2, 1. will be used in the proof of Theorem 1, Lemma 2, 2. Proposition 2, 2. in the proof of Theorem 2. The general idea of the proof of both Theorems is to prove tightness and identify a unique possible limit. In Theorem 1, we do that for the sequence X ε t , while in Theorem 2 we apply this strategy not directly to U ε t , but rather to the right-hand side of the linear SDE whose solution is U ε t .

Lemma 1. If the collection {X ε 0 , 0 < ε ≤ 1} of d-dimensional random vectors is tight, then the following two properties hold:

1. the collection {X ε , 0 < ε ≤ 1} is tight in D(R + , R d ); 2. we have M ε t --→ ε→0
0 in probability, locally uniformly in t.

Proof.

Step 1 Proof of the Lemma under a stronger condition We assume in this step that the following holds

sup 0<ε≤1 E( X ε 0 2 ) < ∞. ( 10
)
It is plain that

1 ε 2 m ε (X ε t ) ≤ 4 X ε t σ -2 R d α 2 ν(dα), (11)
and

M ε t = 1 ε 2 t 0 R d εα 2 g(X ε s , εα)ν(dα)ds ≤ 4σ -2 ε t 0 (α,X ε s )≤0 α 2 (α, X ε s ) ν(dα)ds ≤ 4σ -2 ε R d α 3 ν(dα) t 0 X ε s ds. ( 12
)
Moreover for all x ∈ R d , we have

1 ε 2 (m ε (x), x) ≤ 0.
On the other hand, from (1),

X ε t 2 = X ε 0 2 -2 t 0 (v, X ε s ) ds + 2 ε 2 t 0 (m ε (X ε s ), X ε s ) ds + 2 t 0 X s -dM ε s + [M ε ] t
Hence, for fixed T , we have for all 0 ≤ t ≤ T

E X ε t 2 ≤ E X ε 0 2 + 2v 1 t 0 E X ε s ds + E M ε t ≤ E X ε 0 2 + v 1 t + t 0 E X ε s 2 ds + 2 ε σ 2 R d α 3 ν(dα) t + t 0 E X ε s 2 ds ≤ E X ε 0 2 + v 1 + 2 ε σ 2 R d α 3 ν(dα) t + v 1 + 2 ε σ 2 R d α 3 ν(dα) t 0 E X ε s 2 ds ≤ E X ε 0 2 + v 1 + 2 ε σ 2 R d α 3 ν(dα) T e [v1+2 ε σ 2 ( R d α 3 ν(dα))]T , ( 13
) since 2E X ≤ 1 + E( X 2 )
, and where we have used Gronwall's Lemma (see e.g. Proposition 6.59 in [START_REF] Pardoux | Stochastic Differential Equations, Backward SDEs, Partial Differential Equations[END_REF]). We deduce from ( 11), ( 12) and ( 13) that the process X ε is tight in D R + , R d . Indeed, as noted in Remark 14, part 2 page 119 in [START_REF] Pardoux | Probabilistic Models of Population Evolution[END_REF], if

X ε t = X ε 0 + t 0 F ε s ds + M ε t , where M ε t = t 0 G ε s ds, tightness of the collection of real valued r.v.'s { X ε 0 + T 0 ( F ε t 2 + G ε t 2 )dt, 0 < ε ≤ 1} for each T > 0 implies tightness of X ε in D(R + ; R d ) (
this makes use of Aldous' tightness criterion [START_REF] Aldous | Stopping times and tightness[END_REF]). Note that [START_REF] Pardoux | Probabilistic Models of Population Evolution[END_REF] treats real valued processes, but the proof is identical in the vector valued case.

Finally from [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF],

M ε t --→ ε→0 0 in probability.
This convergence is locally uniformly in t since t → M ε t is increasing.

Step 2 Proof of the Lemma under the original assumption For each n ≥ 1 let ϕ n be a bounded function from R d into itself such that ϕ n (x) = x whenever x ≤ n. Let now for each ε > 0 and n ≥ 1 {X ε,n t , t ≥ 0} denote the solution of equation ( 9) starting with the initial condition X ε,n 0 := ϕ n (X ε 0 ), and

M ε,n t = t 0 R d 1 0 εαΓ(X ε,n s -, εα, ξ)N ε (ds, dα, dξ). On the event { X ε 0 ≤ n}, X ε t ≡ X ε,n t and M ε t ≡ M ε,n t .
From the original assumption, for any δ > 0, there exists n δ ≥ 1 such that P( X ε 0 > n δ ) ≤ δ/2 for each ε > 0. Let T > 0 be arbitrarily fixed. From step 1 of the proof, there exists a compact subset

K δ ⊂⊂ D([0, T ], R d ) such that P(X ε,n δ • ∈ K δ ) ≤ δ/2,
and ε δ > 0 small enough such that P(sup 0≤t≤T M ε,n δ t ≥ δ) ≤ δ/2 for all 0 < ε ≤ ε δ . It follows readily that P(X ε ∈ K δ ) ≤ δ for all 0 < ε ≤ 1 and

P(sup 0≤t≤T M ε t ≥ δ) ≤ δ for all 0 < ε ≤ ε δ . QED Lemma 2. For all x ∈ R d , we have that 1. ε -2 m ε (x) --→ ε→0 Lx, where Lx = 4σ -2 (x,α)≤0 α (x, α) ν(dα) = -4σ -2 V x = -Θσ -2 M x. ( 14
) 2. α ⊗ α g(x,εα) ε ν(dα) --→ ε→0 Λ(x), where
Λ(x) = 4σ -2 (x,α)≤0 (x, α) α ⊗ αν(dα), ( 15
)
Remark 1. Note that the above two Lemmas remain true if the measure ν is not Gaussian, but satisfies the following moment condition :

R d α 4 ν(dα) < ∞. ( 16
)
In this case the limit L(x) is given by the expression

L(x) = 4σ -2 (x,α)≤0 α (x, α) ν(dα) = -4σ -2 V (x)x,
where V depends upon the direction of x. The advantage of taking a Gaussian measure (up to a multiplicative constant) of new mutations is the resulting linear behavior of the limit Lx, due to the fact that V (x) is a constant matrix. More generally, V (x) would depend upon x x .

Proof of Lemma 2. Let for all ε > 0 and x, α ∈ R d y ε = -2σ -2 (2x + εα, α) , and y = -4σ -2 (x, α) .

Note that ε -2 m ε (x) = yε≥0
1-e -εyε ε αν(dα). We have that

1 -e -εyε ε 1 yε≥0 -y1 y≥0 = 1 -e -εyε ε 1 yε≥0 -y ε 1 yε≥0 + (y ε 1 yε≥0 -y1 y≥0 ) . ( 17
)
In addition, since for all z > 0, z -

z 2 2 ≤ 1 -e -z ≤ z, -ε y 2 ε 2 1 yε≥0 ≤ 1 -e -εyε ε 1 yε≥0 -y ε 1 yε≥0 ≤ 0, (18)
From y ε = y -2ε σ 2 α 2 , y ε ≤ y and 1 yε≥0 ≤ 1 y≥0 . This combines with Cauchy-Schwarz entails

y 2 ε 1 yε≥0 ≤ y 2 1 y≥0 ≤ 16σ -4 x 2 α 2 .
Combining the last inequality with [START_REF] Stroock | Multidimensional Diffusion Proceses[END_REF], we obtain

-8σ -4 ε x 2 α 2 ≤ 1 -e -εyε ε 1 yε≥0 -y ε 1 yε≥0 ≤ 0. ( 19
)
Again {y ε ≥ 0} ⊂ {y ≥ 0}. It follows that

y ε 1 yε≥0 -y1 y≥0 = (y ε -y)1 y≥0 -y ε 1 y≥0\yε≥0 = -2σ -2 ε α 2 1 y≥0 -y ε 1 y≥0\yε≥0 ≥ -2σ -2 ε α 2 .
We have proved that

-2σ -2 ε α 2 ≤ y ε 1 yε≥0 -y1 y≥0 ≤ 0. ( 20
)
We deduce from ( 17), ( 19) and (20) that

-ε 8σ -4 x 2 + 2σ -2 α 2 ≤ 1 -e -εyε ε 1 yε≥0 -y1 y≥0 ≤ 0.
Hence,

α 1 -e -εyε ε 1 yε≥0 -y1 y≥0 ≤ ε 8σ -4 x 2 + 2σ -2 α 3 . ( 21
)
By integrating (21) with respect to ν, we obtain

1 ε 2 m ε (x) -L(x) ≤ ε 8σ -4 x 2 + 2σ -2 R d α 3 ν(dα). (22) Hence, ε -2 m ε (x) --→ ε→0 Lx.
By a similar argument, we have that

α ⊗ α g(x, εα) ε ν(dα) -Λ(x) ≤ ε 8σ -4 x 2 + 2σ -2 R d α 4 ν(dα), ( 23
)
and we deduce the second result of the Lemma.

Under the assumption of Lemma 1, we can extract a subsequence which we still denote X ε by an abuse of notation such that X ε ⇒ X (this notation means here and below that X ε converges in law towards X as D(R + , R d )-valued random variables), and we have the following result:

Proposition 2. If X ε ⇒ X, then 1. 1 ε 2 m ε (X ε • ) ⇒ L X• in D(R + , R d ), and
2. 1 ε d dt M ε • ⇒ Λ( X• ) in D(R + , R d ),
Proof of 1. It follows from ( 22) that for all δ, C > 0, there exists ε δ,C such that if ε < ε δ,C and for all x ≤ C,

1 ε 2 m ε (x) -Lx ≤ δ,
thus, for an arbitrary T > 0 and for ε < ε δ,C ,

P sup t≤T 1 ε 2 m ε (X ε t ) -L(X ε t ) > δ ≤ P sup t≤T X ε t > C .
It follows that for all δ, C > 0, lim sup

ε→0 P sup t≤T 1 ε 2 m ε (X ε t ) -L(X ε t ) > δ ≤ sup ε P sup t≤T X ε t > C .
From the tightness of X ε , for all η > 0, we can choose C > 0 such that sup

ε P sup t≤T X ε t > C ≤ η.
Hence for all δ, η > 0 lim sup

ε→0 sup t≤T 1 ε 2 m ε (X ε t ) -L(X ε t ) > δ ≤ η. Moreover L(X ε ) ⇒ L( X) in D(R + , R d ) since L is a continuous function. Conse- quently, 1 ε 2 m ε (X ε ) = 1 ε 2 m ε (X ε ) -L(X ε ) + L(X ε ) ⇒ L( X). Proof of 2. Note that d dt M ε t = R d α ⊗ α g(X ε t , ε α)ν(dα).
By a similar argument as in the first part of the proof, using this time (23), we establish the second claim, thanks to the fact that

α 4 ν(dα) < ∞.
We can now establish Theorem 1. If X ε 0 → x 0 in probability as ε → 0, then X ε t --→ ε→0 xt in probability, locally uniformly in t, where

dx t dt = -v + Lx t , x0 = x 0 . ( 24
)
Proof.

The assumption and Lemma 1 imply tightness of the collection of processes {X ε • , 0 < ε ≤ 1}. Hence X ε converges weakly to a limit along a subsequence. From (9), Proposition 2, 1. and Lemma 1, 2., we deduce that the limit solves the ODE (24). Since the limit xt is uniquely determined and deterministic, the whole collection X ε t converges in probability as ε → 0 towards xt , locally uniformly in t.

Remark 2. In case X ε 0 ⇒ 0 , where X 0 is a d-dimensional random vector, the same result still holds, except that the limit is random and the convergence is in law.

The differential equation (24) represents a deterministic approximation for the stochastic process X ε in the limit of small jumps. We note that xt ---→ t→∞ x∞ = -

M -1 v Θσ -2 . ( 25
)
To estimate the fluctuations of the process in the small-jumps limit, we now consider the following process

U ε t = X ε t -xt √ ε . ( 26
)
Theorem 2. Assume that X ε 0 = x0 . Then U ε ⇒ U , where U is an Ornstein-Uhlenbeck process:

dU t = LU t dt + Λ 1 2 (x t )dB t , U 0 = 0, ( 27
)
with B being a d-dimensional standard Brownian motion. In other words,

U t = t 0 e L(t-s) Λ 1 2 (x s )dB s .
Proof. We have that

U ε t = t 0 ε -2 m ε (X ε s ) -Lx s √ ε ds + 1 √ ε M ε t , = L t 0 X ε s -xs √ ε ds + 1 √ ε t 0 1 ε 2 m ε (X ε s ) -LX ε s ds + 1 √ ε M ε t = L t 0 U ε s ds + 1 √ ε t 0 1 ε 2 m ε (X ε s ) -LX ε s ds + 1 √ ε M ε t .
Thus by the formula for the solution of a linear ODE and an integration by parts,

U ε t = 1 √ ε t 0 e L(t-s) 1 ε 2 m ε (X ε s ) -LX ε s ds+ 1 √ ε M ε t + 1 √ ε t 0 e L(t-s) LM ε s ds. ( 28
)
We deduce from ( 22) that

1 √ ε t 0 e L(t-s) 1 ε 2 m ε (X ε s ) -LX ε s ds --→ ε→0 0 (29)
in probability. Furthermore, 1 √ ε M ε are tight martingales by a similar argument as before since sup

{t>0, ε} R 2 α 2 g(X ε t , εα)ν(dα) ≤ R 2 α 2 ν(dα) < ∞.
Let us admit for a moment the Lemma 3. For any T > 0, as ε → 0,

sup t≤T 1 √ ε (M ε t -M ε t -) → 0 in probability.
Hence every converging subsequence of 1 √ ε M ε converges to a continuous process M as ε goes to 0, and using Proposition 2, 2., we have (recall ( 15))

1 √ ε M ε t = t 0 α ⊗ α g(X ε s , εα) ε ν(dα)ds --→ ε→0 t 0 Λ(x s )ds.
Let us now explain why these facts imply that M t is a martingale such that

M t = t 0 Λ(x s )ds.
We first note that since X ε 0 (ω) = x0 for all ε > 0 and ω, we deduce from ( 12) and ( 13) that for each t > 0 the collections indexed by ε of random matrices and vectors

1 ε M ε t , 1 ε M ε t ⊗ M ε t and 1 √ ε M ε t are uniformly integrable. On the other hand, for any n ≥ 1, 0 < s 1 < • • • < s n = s < t, Φ ∈ C b (R nd ), we have E Φ M ε s 1 √ ε , . . . , M ε sn √ ε M ε t √ ε - M ε s √ ε = 0, E Φ M ε s 1 √ ε , . . . , M ε sn √ ε 1 ε M ε t ⊗ M ε t - 1 ε M ε t - 1 ε M ε s ⊗ M ε s + 1 ε M ε s = 0.
We may take the limit as ε → 0 in these two identities, which yields identities which (thanks to the freedom of choice of n, s 1 , . . . , s n and Φ) show that both M t and M t ⊗ M t -t 0 Λ(x s )ds are continuous martingales.

We deduce thanks to a representation theorem of continuous martingales, see e.g. Theorem 4.5.2 from [START_REF] Stroock | Multidimensional Diffusion Proceses[END_REF], that there exists a d-dimensional Brownian motion B t such that

M t = t 0 Λ 1 2 (x s )dB s , t ≥ 0.
This being true for any converging subsequence of 1 √ ε M ε , the limit is unique (in law). Finally, combining this with (28) and (29), we deduce that

U ε t ⇒ t 0 e L(t-s) Λ 1 2 (x s )dB s .

It remains to turn to the

Proof of Lemma 3 It is clearly sufficient to prove the result for any coordinate of the d-dimensional process M ε t , which means that it suffices to prove the result in case d = 1, which we assume for the rest of this proof. In other words, p is the density of the Gaussian law N (0, γ 2 ), with some γ > 0. Let us define

M ε t = [0,t]×R×[0,1] εαN ε (ds, dα, dξ).
It is plain that

sup t≤T 1 √ ε (M ε t -M ε t -) ≤ sup t≤T 1 √ ε M ε t -M ε t - .
Hence all we need to show is that the above right hand side tends to 0 in probability, as ε → 0. Let a > 0 be arbitrary. We need to estimate

P sup t≤T | M ε t -M ε t -| > a √ ε .
It is plain that, with

Z ε := N ε ([0, T ] × R × [0, 1]), P sup t≤T | M ε t -M ε t -| > a √ ε ≤ P Z ε > Kε -2 + P Z ε ≤ Kε -2 , sup t≤T | M ε t -M ε t -| > a √ ε = A(K, ε, T ) + B(K, ε, T ).
Since the law of Z ε is Poisson with mean (2ε 2 ) -1 ΘT , it follows from Chebychef's inequality that

A(K, ε, T ) ≤ ΘT 2K . ( 30
)
On the other hand, if {Y k , k ≥ 1} are i.i.d. r.v.'s with the common distribution N (0, 1), with the notation

a ε = a γ √ ε , B(K, ε, T ) ≤ P sup 0≤k≤Kε -2 |Y k | ≥ a ε ≤ 2P sup 0≤k≤Kε -2 Y k ≥ a ε = 2 -2P sup 0≤k≤Kε -2 Y k ≤ a ε = 2 -2 (1 -P(Y 1 > a ε )) Kε -2 .
Note that, since exp(a ε Y 1 -a2 ε /2) has mean 1,

P(Y 1 > a ε ) = P(e aεY 1 -a 2 ε /2 > e a 2 ε /2 ) ≤ e -a 2 ε /2 . Hence B(K, ε, T ) ≤ 2 -2 1 -e -a 2 ε /2 Kε -2 For ε small enough, e -a 2 ε /2 ≤ 1/2, from which follows that 1-e -a 2 ε /2 ≥ exp(-e -a 2 ε /2 ), hence B(K, ε, T ) ≤ 2 1 -exp - K ε 2 e -a 2 2γ 2 ε . ( 31
)
For any δ > 0, we first choose K δ = ΘT δ , so that from (30), A(K δ , ε, T ) ≤ δ/2 for all ε > 0, and then thanks to (31) ε small enough such that B(K δ , ε, T ) ≤ δ/2. The result follows. Remark 3. We have exploited the fact that p is the density of a Gaussian distribution. In fact it would suffice that P(|Y 1 | > a) ≤ C a 4+δ for some C, δ > 0. We note that this is exactly what is already necessary for [START_REF] Peischl | Establishment of new mutations in changing environments[END_REF] to hold.

The Ornstein-Uhlenbeck process U t which appears in the last Theorem is a centered Gaussian vector. It is easy to compute its covariance matrix

E(U t ⊗ U t ) = t 0 e L(t-s) Λ(x s )e L(t-s) ds.
as well as the limit of the latter as t → ∞

E(U t ⊗ U t ) ---→ t→∞ S2 = ∞ 0 e Lt Λ(x ∞ )e Lt dt. ( 32
)
It follows readily that the law of U t converges, as t → ∞, to N (0, S2 ). In case xt = x∞ for all t ≥ 0, then U t solution of the linear SDE dU t = LU t dt+Λ 3 Large time behavior of X ε t for small ε > 0

In this section, we shall first prove that provided ε is small enough, X ε t is positive Harris recurrent with a unique invariant probability measure, next that the collection of invariant probability measures indexed by ε is tight, and finally that the large time behavior of xt + √ εU t is a good approximation of the large time behavior of X ε t , thus justifying a sort of interchange of the limits as ε → 0 and as t → ∞.

Large time behavior of the process X ε t

The aim of this subsection is to prove (in the next statement, • stands for the total variation norm): Theorem 3. There exists ε 0 > 0 such that for any 0 < ε ≤ ε 0 , the process X ε t is positive Harris recurrent, with a unique invariant probability measure µ ε . Moreover

X ε t is ergodic, in the sense that P(X ε t ∈ •|X ε 0 = x) -µ ε → 0, as t → ∞, for all x ∈ R d .
Thanks to Theorem 4.2 from [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF], the first part Theorem 3 follows from a positive recurrence condition (which is called (CD2) in [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF]), and the fact that a certain ball centered at 0 is a petite set. We first establish in the next Lemma and the subsequent Corollary an a priori estimate. The proof of the first part of the Theorem will then consist in two steps : conclude (CD2) from the Corollary, and verify the petite set property. The second part will follow from Theorem 5.1 in [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF]. Lemma 4. There exists three constants c, C, a > 0 (whose values depend explicitly upon the parameters of our model) such that for all 0 < ε ≤ 1 4 , t > 0,

X ε t ≤ X ε 0 + C t -c t 0 X ε s ∧ a ε ds + N ε t ,
where the local martingale N ε t , which is a martingale as soon as E[ X ε 0 ] < ∞, is given as

N ε t = [0,t]×R d ×[0,1] ( X ε s-+ εα -X ε s-)Γ(X ε s-, εα, ξ) Mε (ds, dα, dξ).
Proof. We have

X ε t = X ε 0 - t 0 v, X ε s X ε s ds + s≤t ( X ε s -+ ∆X ε s -X ε s -) = X ε 0 - t 0 v, X ε s X ε s ds + [0,t]×R d ×[0,1] X ε s-+ εα -X ε s- Γ(X ε s-, εα, u)N ε (ds, dα, du) = X ε 0 + t 0 R d X ε s + εα -X ε s ε × g(X ε s , εα) ε ν(dα)-v, X ε s X ε s ds + N ε t ( 33
)
Recall that only mutations which improve the fitness may get fixed, since from ( 3) and ( 4) g(x, α) > 0 requires s(x, α) > 0, i.e. (2x + α, α) < 0, hence, with again the notation ∆X ε s = X ε s -X ε s-, X ε s -+ ∆X ε s -X ε s -≤ 0, and moreover we shall restrict ourselves below to the jumps such that

X ε s -+ ∆X ε s -X ε s -≤ - 1 2 ∆X ε s .
For x ∈ R d , let C(x) denote the set of vectors α such that

x + α -x ≤ - 1 2 α .
We define γ, the angle between the two directions -x and α, as follows. If x and α are colinear, then γ = 0. Otherwise, we consider the plane spanned by x and α. If we restrict ourselves to those α such that Γ(x, εα, 0) > 0, then in the above plane, the angle between the directions -x and α is between -π/2 and π/2. We denote by γ that angle, whose sign is unimportant, since only cos γ will enter our computations. From the identity (again we consider only the case where (x, α) < 0)

x + α = x 2 + α 2 -2 x α cos γ, we deduce that α ∈ C(x) iff α ≤ 4 3 (2 cos γ -1) x .
We note in particular that whenever α ∈ C(x) and ε ≤ 1, then εα ∈ C(x). Now

R d X ε s + εα -X ε s ε × g(X ε s , εα) ε ν(dα) ≤ - 1 2 C(X ε s )/ε α g(X ε s , εα) ε ν(dα) ≤ -(1 -e -1 ) C(X ε s-)/ε α (2X ε s + εα, α) - σ 2 ∧ 1 2ε ν(dα)
where (y, z) -stands for the negative part the the scalar product of the two vectors y and z, and we have exploited the elementary inequality 1 -e -u ≥ (1 -e -1 )(u ∧ 1), for all u ≥ 0.

We now need to lower bound the factor of -(1 -e -1 ) in the last right hand side.

For that sake, we consider the expression

α∈C(x)/ε α (2x + εα, α) - σ 2 ∧ 1 2ε ν(dα).
For x ≤ 2, we lower bound this integral by 0. We now consider the case x > 2.

We lower bound the integral by reducing the integration to the set

A ε (x) = C(x) ε ∩ {1 ≤ α ≤ 2}.
It is not hard to see that whenever α ∈ A ε (x), -2 ≤ α, x x < -1/2. Indeed, the lower bound is clear and from εα ∈ C(x), we deduce that 2(εα,

x) + α 2 ≤ α 2 4 -x × α , whence (α, x x ) ≤ -α] 2 ≤ -1 2 , since α ≥ 1. Moreover, since x > 2, if α ∈ A ε (x), -2 x < (x, α) < - x 2 < -1, because ε α 2 ≤ 1,
provided ε ≤ 1/4. Consequently (x, α) + ε α 2 ≤ 0 and

(2x + εα, α) -≥ (x, α) ≥ x 2 , so that Aε(x) α (2x + εα, α) - σ 2 ∧ 1 2ε ν(dα) ≥ 1 2σ 2 x ∧ σ 2 ε Aε(x) α ν(dα) ≥ β 2σ 2 x ∧ σ 2 ε ,
where β = inf x >2,ε≤1 Aε(x) α ν(dα) > 0. We have proved that, with a = σ 2 and c = (1 -e -1 )β/(2σ 2 ),

X ε t ≤ X ε 0 + v 1 × t -c t 0 1 X ε s >2 X ε s ∧ a ε ds + N ε t .
The result follows with C = v 1 + 2c.

Corollary 1. There exist 0 < ε 0 ≤ 1/4, b, k > 0 and B a compact subset of R d such that for any 0 < ε ≤ ε 0 ,

X ε t ≤ X ε 0 -bt + k t 0 1 B (X ε s )ds + N ε t . ( 34
)
Proof. We choose B = {x; x ≤ 2C/c} and ε 0 = ca 2C . (34) with b = C, k = 2C now follows from Lemma 4.

We finally turn to the Proof of Theorem 3. Step 1. Verification of the positive recurrence condition The process X ε t satisfies (34) which is exactly condition (CD2) from [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF]. Indeed, (33) implies that the function V (x) := x belongs to the domain of the extended generator A ε of the Markov process X ε t and that

A ε V (x) = R d x + εα -x ε × g(x, εα) ε ν(dα) -v, x x .
The proof of Corollary 1 implies that

A ε V (x) ≤ -b + k1 B (x), again for V (x) = x .
Step 2. B is a petite set In order to deduce positive Harris recurrence from Theorem 4.2 in [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF], it remains to show that B = {x; x ≤ K}, with K = 2C/c is a closed petite set. This will follow if we show that there exist two constants t B , C B > 0 such that for all x ∈ B,

P(X ε t B ∈ dy|X ε 0 = x) ≥ C B 1 y ≤1 dy. ( 35
)
We choose 0 < η < 1 and define the event

A ε K,η =    X ε t does not jump on the time interval 0, 2K-η v , and
X ε t jumps exactly once on the time interval 2K-η v , 2K v .    It is plain that P(A ε K,η ) > 0. Let T ε denote the first jump time of X ε t . We note that, provided that X ε 0 = x ∈ B, on the event A ε K,η , X ε T ε -∈ B K,η , where B K,η = {x ∈ R d , -3K ≤ x 1 ≤ -K + η, -K ≤ x i ≤ K, 2 ≤ i ≤ d}.
. Let Λ ε denote a random vector which is such that the law of ε -1 Λ ε has the density p defined by [START_REF] Karatzas | Brownian motion and stochastic calculus 2nd[END_REF]. Denote by f ε (y) = εp(εy) the density of the law of Λ ε . We define

Σ ε = Λ ε 1 { X ε T ε -+Λε ≤1+η} . On the event Σ ε = 0, since K > 2 > 1 + η, g(X ε T -, Σ ε ) ≥ c ε,K,η , where c ε,K,η = inf x∈B K,η , x+α ≤1+η g(x, α) > 0.
We denote by ξ the random variable with the uniform distribution on the interval [0, 1], which is such that whenever

ξ ≤ g(X ε T -, Σ ε ), the "proposed" jump Σ ε happens at time T ε . Recall that 0 ≤ 2K v -T ε < η v . On the event A ε K,η ∩ {Σ ε = 0} ∩ {ξ ≤ c ε,K,η }, X2K v = X ε T ε -+ Σ ε + v 2K v -T ε , and for any h ∈ C(R d , R + ), Eh X ε 2K v ≥ P(A ε K,η ∩ {Σ ε = 0})c ε,K,η a ε,K,η y ≤1 h(y)dy, with a ε,K,η = inf x∈B K,η , x+y ≤1+η f ε (y). (35) follows, with t B = 2K v and C B = P(A ε K,η ∩ {Σ ε = 0})c ε,K,η a ε,K,η .
Step 3. Ergodicity We apply Theorem 5.1 from [START_REF] Meyn | Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes[END_REF]. All we have to show is that their condition (S) holds, namely that all compacts sets are petite for some skeleton chain. In fact in Step 2 of the proof, we have established that B is petite for a skeleton chain. It is easy to verify that the same proof works when replacing B by an arbitrary compact set.

Tightness of the invariant probability measure of X ε t

It follows from Theorem 3 that for any ε ≤ ε 0 , X ε t possesses a unique invariant probability measure µ ε . The aim of this subsection is to prove Theorem 4. For any sequence ε n ↓ 0, the sequence of invariant measures {µ εn , n ≥ 1} is tight.

The result will follow from the following statement Proposition 3. There exist two constants a, C > 0 such that for any

M > 0, if ε ≤ a M ∧ 1 4 , R d ( x ∧ M )µ ε (dx) ≤ C .
Let us first show how Theorem 4 follows from Proposition 3.

Proof of Theorem 4. We deduce from the above Proposition that whenever

ε ≤ a M ∧ 1 4 , µ ε ( x > M ) ≤ C M .
Fix δ > 0 arbitrarily small. Let us from now on fix M ≥ C /δ. Let n 0 be such that ε n 0 ≤ a M ∧ 1 4 . It follows from the above that for any n ≥ n 0 ,

µ εn ( x > M ) ≤ δ.
It is finally easy to find M ≥ M such that

µ εn ( x > M ) ≤ δ
for any 1 ≤ n ≤ n 0 , hence the result.

Now return to the

Proof of Proposition 3. It follows readily from Lemma 4 that whenever

X ε 0 = x ε 0 is deterministic, M < a/ε, E( X ε t ) ≤ x ε 0 + Ct -c t 0 E [ X ε s ∧ (a/ε)] ds ≤ x ε 0 + Ct -c t 0 E [ X ε s ∧ M ] ds.
Since the function f (x) = x ∧ M is bounded and continuous, it follows from the last part of Theorem 3 that E [ X ε t ∧ M ] → R d ( x ∧ M )µ ε (dx) as t → ∞. Hence, since the first part of the proof implies that

c t t 0 E [ X ε s ∧ M ] ds ≤ x ε 0 t + C
for all t > 0, the result follows by letting t → ∞.

Asymptotic analysis of the large time behavior of X ε t

We now want to analyze the large time behavior of X ε t , for small ε. We first show Theorem 5. As ε → 0, µ ε ⇒ δ x∞ .

Proof We consider the X ε t equation started with X ε 0 ∼ µ ε . Choose a sequence ε n which converges to 0 as n → ∞. Since the collection {µ εn } n≥1 is tight, along a subsequence still denoted the same way, µ εn ⇒ µ 0 . It follows from Theorem 1 (in fact its extension explained in Remark 2) that X εn t ⇒ X t , where X t solves the ODE dX t dt = LX t -v, X 0 ∼ µ 0 .

But for any f ∈ C b (R d ), n ≥ 1, t → Ef (X εn t ) is a constant, so this is true in the limit, which implies that µ 0 = δ x∞ . This shows that the whole collection µ ε ⇒ δ x∞ as ε → 0.

We expect from the above results that for small enough ε, the invariant measure µ ε is close to ν ε , which is the law of x∞ + √ εξ, where ξ ∼ N (0, S2 ) (recall (32)). If we interpret µ ε as the mass of X ε t for large t, this is not really correct, since for some large t, X ε t will make a large deviation from xt , see [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF]. Therefore, we prefer to interpret µ ε (A) as lim t→∞ 1 t t 0 1 A (X ε s )ds.

We now want to give a precise description of µ ε for small ε > 0. For that sake, let us introduce some notation. For any Borel set A and for all ε > 0, let A ε := x∞ + √ εA and denote by λ S2 = N (0, S2 ) the invariant Gaussian distribution of the Ornstein-Uhlenbeck process U t . Theorem 6. Consider the process X ε t , starting at time t = 0 from X ε 0 = x∞ . Let A be an arbitrary element of B d such that its boundary ∂A has zero d-dimensional Lebesgue measure. For any δ > 0 there exist t δ > 0 large enough such that for any t ≥ t δ , there exists ε t,δ > 0 such that for all ε ≤ ε t,δ , with a probability larger than 1 -δ, the fraction of the time in the interval [0, t] which X ε s spends in the set Hence there exists ε t,δ such that for any 0 < ε ≤ ε t,δ , P

A ε = x∞ + √ εA
1 t t 0 1 A (U ε s )ds -λ S2 (A) > δ ≤ δ.
Recall that

1 A (U ε s ) =    1, if X ε s ∈ x∞ +
√ εA, 0, otherwise, implying the result.

  belongs to the interval [λ S2 (A) -δ, λ S2 (A) + δ]. A (U s )ds → λ S2 (A) in probability, as t → ∞.Hence for all δ > 0, there exists t δ > 0 such that for any t ≥ t δ , A (U s )ds -λ S2 (A) ≥ δ ≤ δ/2.

	Proof. As ε → 0, for any fixed t > 0, U ε t being defined by (26), 1 t t 0 1 Aε (X ε s )ds = 1 t t 0 1 A (U ε s )ds ⇒ 1 t t 0 1 A (U s )ds, and 1 t t 0 1 P 1 0 t t 1 From the Portmanteau theorem, lim sup ε→0 P 1 t t 0 1 A (U ε

s )ds -λ S2 (A) ≥ δ ≤ δ/2.

(x ∞ )dB t is a time homogeneous Gauss-Markov process, and N (0, S2 ) is its unique invariant probability measure, see Theorem 5.6.7 in Karatzas and Shreve[START_REF] Karatzas | Brownian motion and stochastic calculus 2nd[END_REF]. Moreover (x t , U t ) is a time homogeneous Markov process and δ x∞ × N (0, S2 ) is its unique invariant probability measure.

Acknowledgements

The support of CNRS -EGP/Mission pour l'interdisciplinarité, within the project Évolution et génétique des populations of G. Didier, is gratefully acknowledged. The authors want to express their gratitude to Micheal Kopp, who introduced them to the model and suggested the study of the small jumps limit, to Nicolas Champagnat, whose clever question during the thesis defense of the first author, led them to the developments of the above section 3, and to two anonymous Referees, whose very detailed and relevant comments helped them to improve significantly the first version of this paper.


Hélène Lavanant
email: helene.lavanant@univ-rouen.fr.

Michael Groessl

Carlos Afonso

Collision cross sections of negative cluster ions of phosphoric acid in N2 determined by drift tube ion mobility and their use in travelling wave ion mobility

Keywords: Drift tube ion mobility, collision cross section, phosphoric acid, cluster ions, travelling wave ion mobility, calibration Graphical

Travelling wave ion mobility usually uses a calibration procedure to determine collision cross sections. Here, we investigate phosphoric acid cluster anions as possible calibrants to increase the pool of reference ions for singly and multiply charged anions with nitrogen as drift gas.

Phosphoric acid solutions (5 to 50 mM) in water and acetonitrile were analyzed using electrospray ionization in the negative ion mode. Collision cross sections were obtained using an IMS-TOF drift tube ion mobility mass spectrometer (Tofwerk, Thun, Switzerland) (DTIMS-MS) on which we varied the electric field. Twenty-six collision cross sections from DTIMS-MS in nitrogen gas, termed DT CCS(N2), were obtained for singly, doubly and triply charged phosphoric acid cluster anions which range from 146 to 530 Å 2 . Reduced mobilites K0 ranged from 0.767 to 1.435 cm 2 V -1 s -1 for singly charged cluster anions, 1.177 to 1.006 cm 2 V -1 s -1 and 1.250 to 1.0143 cm 2 V -1 s -1 for doubly and triply charged cluster anions, respectively. The DT CCS(N2) were found fitted to a power function 80n 0.51 where n is the aggregation number.

Ion mobility mass spectrometry measurements were also carried out on a travelling wave ion mobility mass spectrometry (TWIMS-MS) device (SYNAPT G2 HDMS, Waters Corp., Manchester, UK). The DT CCS(N2) were well correlated to the drift times measured on the TWIM instrument (R 2 above 0.9999). The correlation function was found to depend on the charge state of the ions. Accuracy tests evidenced a systematic error of 9 to 11 Å 2 (4-6%) on singly charged ions of polyalanine, dextran and hexakis(fluoroalkoxy)phosphazine compared to literature values. Although absolute values were shifted, CCS differences were preserved.

INTRODUCTION

With the development of commercial instruments, the coupling of ion mobility and mass spectrometry is spreading as supplemental method to be used for compound identification or structural characterization to mass spectrometry. [START_REF] Campuzano | Ion mobility in the pharmaceutical industry: an established biophysical technique or still niche?[END_REF][START_REF] May | Ion mobility-mass spectrometry: time-dispersive instrumentation[END_REF] Indeed, ion mobility depends largely on the charge and the collision cross section (CCS or Ω) of the ions that is related to their conformations in the gas phase.

Conventional uniform electric field drift tube ion mobility spectrometry (DTIMS) enables the direct determination of ion mobility K and CCS, usually by stepping the electric field. Other ion mobility techniques, such as travelling wave ion mobility (TWIMS) [START_REF] Pringle | An investigation of the mobility separation of some peptide and protein ions using a new hybrid quadrupole/travelling wave IMS/oa-ToF instrument[END_REF] or trapped ion mobility (TIMS) [START_REF] Ridgeway | Trapped ion mobility spectrometry: A short review[END_REF] require the tuning of a higher number of parameters and use reference ions and calibration to determine the CCS. [START_REF] Smith | Deciphering drift time measurements from travelling wave ion mobility spectrometry-mass spectrometry studies[END_REF][START_REF] Ruotolo | Ion mobility-mass spectrometry analysis of large protein complexes[END_REF][START_REF] Hernandez | Ion dynamics in a trapped ion mobility spectrometer[END_REF][START_REF] Lippens | Modular calibrant sets for the structural analysis of nucleic acids by ion mobility spectrometry mass spectrometry[END_REF] For positive ions, many reference calibrants exist, [START_REF] May | Ion Mobility Collision Cross Section Compendium[END_REF] for which CCS were previously determined experimentally with DTIMS and usually helium as the drift gas (noted DT CCSHe). The pool of reference ions is smaller for the negative ion mode [START_REF] Forsythe | Collision cross section calibrants for negative ion mode traveling wave ion mobility-mass spectrometry[END_REF][START_REF] Hofmann | Estimating Collision Cross Sections of Negatively Charged N-Glycans using Travelling Wave Ion Mobility-Mass Spectrometry[END_REF][START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF] , especially for multiply charged ions and with nitrogen as drift gas. Furthermore, there still remains, at this day, many questions and discussions on the different choice of CCS reference (ion and drift gas) and on the calibration procedure for travelling wave ion mobility. [START_REF] Bush | Ion mobility mass spectrometry of peptide ions: effects of drift gas and calibration strategies[END_REF][START_REF] Haler | Comprehensive Ion Mobility Calibration: Poly(ethylene oxide) Polymer Calibrants and General Strategies[END_REF][START_REF] Gabelica | Recommendations for reporting ion mobility Mass Spectrometry measurements[END_REF] Cluster ions are readily produced using direct infusion of salt solutions with an electrospray ion source and are routinely used for calibration of mass analyzers. [START_REF] Zhang | Doubly charged cluster ions [(NaCl)m(Na)2] 2+ : magic numbers, dissociation, and structure[END_REF][START_REF] Lermyte | A broader view on ion heating in traveling-wave devices using fragmentation of CsI clusters and extent of H migration as molecular thermometers[END_REF][START_REF] Hao | Electrospray ionization tandem mass spectrometric study of salt cluster ions: part 2--salts of polyatomic acid groups and of multivalent metals[END_REF][START_REF] Galhena | Influence of cluster size and ion activation method on the dissociation of cesium iodide clusters[END_REF][START_REF] Sobott | A tandem mass spectrometer for improved transmission and analysis of large macromolecular assemblies[END_REF] A number of ionic clusters derived from alkali halides salts, [START_REF] Fernandez-Lima | Experimental and theoretical studies of (CsI)n Cs + cluster ions produced by 355 nm laser desorption ionization[END_REF][START_REF] Ouyang | The collision cross sections of iodide salt cluster ions in air via differential mobility analysis-mass spectrometry[END_REF][START_REF] Ohshimo | Compact non-rock-salt structures in sodium fluoride cluster ions at specific sizes revealed by ion mobility mass spectrometry[END_REF] ionic liquids [START_REF] Ku | Relation between Electrical Mobility, Mass, and Size for Nanodrops 1-6.5 nm in Diameter in Air[END_REF][START_REF] Larriba | The Mobility-Volume Relationship below 3.0 nm Examined by Tandem Mobility-Mass Measurement[END_REF][START_REF] Larriba-Andaluz | Collision cross section calculations for polyatomic ions considering rotating diatomic/linear gas molecules[END_REF][START_REF] Larriba-Andaluz | Gas molecule scattering & ion mobility measurements for organic macro-ions in He versus N2 environments[END_REF] and transition metal oxides [START_REF] Ohshimo | Isomer separation of iron oxide cluster cations by ion mobility mass spectrometry[END_REF][START_REF] Ohshimo | Structures and CO-Adsorption Reactivities of Nickel Oxide Cluster Cations Studied by Ion Mobility Mass Spectrometry[END_REF][START_REF] Ohshimo | Stable compositions and geometrical structures of titanium oxide cluster cations and anions studied by ion mobility mass spectrometry[END_REF][START_REF] Wu | Compositions and Structures of Vanadium Oxide Cluster Ions VmOn (+/-) (m = 2-20) Investigated by Ion Mobility Mass Spectrometry[END_REF][START_REF] Moriyama | Geometrical Structures of Gas Phase Chromium Oxide Cluster Anions Studied by Ion Mobility Mass Spectrometry[END_REF][START_REF] Wu | Compositions and structures of niobium oxide cluster ions, NbmOn (+/-) , (m = 2-12), revealed by ion mobility mass spectrometry[END_REF] have been studied by drift tube ion mobility mainly and differential mobility analysis-time-of-flight mass spectrometry. Since a first study in 2014, [START_REF] Lavanant | Traveling Wave Ion Mobility Mass Spectrometry and Ab Initio Calculations of Phosphoric Acid Clusters[END_REF] we have started studying negative phosphoric acid cluster ions in view of investigating their potential as reference for the calibration of the TWIMS.

Here, we determined the CCS values of twenty-six negatively charged phosphoric acid cluster ions, singly, doubly and triply charged with a drift-tube ion mobility-time-of-flight mass spectrometer, at 32.6°C, and using nitrogen as the drift gas. The DT CCS(N2) values were then used as reference for calibration of a TWIMS device and the resulting calibration was tested for accuracy.

METHODS

The phosphoric acid used was an 85 % weight solution in water (d=1.680) from Acros Organics (through Fisher Scientific France, Illkirch, France). LC-MS grade acetonitrile were purchased from VWR (Fontenay-sous-bois, France). Deionized water (18 MΩ) was obtained from a Milli-Q apparatus (Millipore, Bedford, MA, USA). 50 mM and 5 mM solutions were prepared in water/acetonitrile 50/50 and (0.35 % to 0.035 % volume of the phosphoric acid solution). Solutions of Poly-DL-alanine (mol wt 1000-5000 purchased from Sigma-Aldrich, St. Louise, USA) at a concentration of 10 µM and dextran enzymatic synthesis (Mr 1500, Fluka Analytical, Sigma-Aldrich, Denmark) at a concentration of 100 µM were prepared in water/acetonitrile 50/50 v/v. The solution of tune mix (from Agilent Technologies, Santa Clara, CA) was obtained by diluting to 1:4 with 95 % acetonitrile in water.

All drift tube ion mobility -mass spectrometry measurements (DTIMS-MS) were carried out on a Tofwerk IMS-TOF instrument (Thun, Switzerland). The system comprises an electrospray ionization (ESI) source, a 10 cm desolvation tube, a 20.56 cm drift tube (both made from resistive glass) and a Tofwerk time of flight HTOF analyzer (TOF-MS) with a two-stage interface to the IMS, as described in detail elsewhere. [START_REF] Kwasnik | Performance, resolving power, and radial ion distributions of a prototype nanoelectrospray ionization resistive glass atmospheric pressure ion mobility spectrometer[END_REF] Desolvation and drift tubes were thermostated at 32.6 °C with nitrogen as the buffer gas. Drift-tube pressure was set at 1000 mbar (N2). The pressure sensor is a JUMO MIDAS C08 (JUMO GmbH & Co. KG,Fulda, Germany) whose accuracy is given as 0.5 % and the temperature sensor is a generic k-type thermocouple with an accuracy of +-0.75 %. Ion mobility separation was carried out at field strengths of ca. 300 to 460 V cm -1 (reduced electric field strength between 1 and 2 Td). Samples were introduced directly into the ESI source using a syringe pump at 1 μl min -1 . The instrument was operated from m/z 50 to 1500 in negative ion mode with a measurement time of 2 to 5 min at each field strength. The mass spectrometer was calibrated internally. Ion optics were set as to minimize dissociation between the drift tube and the TOF analyzer (Table S1). Raw IMS-TOF data was post-processed using IMS Viewer and Tofware (Tofwerk, Switzerland) with a denoise and sharpness features set to 1. The instrument uses a Hadamard-type multiplexing method. In addition to increasing sensitivity, properties of the time domain data can be exploited to increase both signal to noise ratio and effective resolving power.

For reduced ion mobility and collision cross section measurements, drift-tube ion mobility drift times were measured at five different field strengths (potential differences from 6.2 to 9.4 kV). With a uniform field, the slope of the drift time vs the reciprocal of the potential difference is the square of the tube length divided by the ion mobility K (L 2 /K). The ion mobility K can then be converted to a reduced ion mobility K0 and, using the Mason Schamp equation, [START_REF] Mason | Mobility of gaseous lons in weak electric fields[END_REF] to a collision cross section in N2, that we will note DT CCS(N2). Standard uncertainties for the measurement of instrumental parameters were given as follows: pressure ±5 mbar (0.5 %), temperature ±3 K (0.75 %), drift voltage ±2 V (0.03 %). The observed relative standard errors for the slopes of the drift time vs the reciprocal of the potential difference were less than 0.5%. The exact length of the drift tube is 20.555 cm but due to possible edge effects of the electric field, we cautiously estimated an expanded uncertainty of ±3 mm for the effective electric field length, by doubling results observed by Stow et al on effective length. [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF] Then, applying a type B evaluation of standard uncertainty and propagation of error, we found this latter expanded uncertainty had a dominant effect and resulted in a combined relative expanded uncertainty of 3 % on K0 and CCS values (level of confidence of approximately 95 %).

Travelling wave ion mobility and mass spectra were recorded using a SYNAPT G2 HDMS (Waters Corp., Manchester, UK) fitted with an electrospray ionization source. ESI mass spectra were acquired over the m/z 50-1600 or 50-4000 range, in the negative ion modes. Values for the capillary emitter were 2.7 kV and sampling cone 60 V. The source temperature was set at 90°C. Nitrogen gas at a temperature of 250°C and a flow rate of 500 L/h was used to assist desolvation in the ESI source. The solution was infused at a flow rate of 5 µL/min using the spectrometer sample pump. Gas flow rates were set at 20 mL/min in the source stacked ring ion guide, 180 mL/min in the helium cell and 70 mL/min in the ion mobility cell. Trap and transfer collision energies were set to 5 V and 0 V respectively. Travelling wave velocities were set to 350 m/s and the wave height at 15 or 20 V, to minimize hydrogen bond disruption. Direct current voltages in the trap and IMS cells were also adjusted to minimize fragmentation (Table S2). Data treatment was carried out with Mass Lynx (4.1).

Measurements were duplicated on different days to eliminate aberrant values; the results presented here are from single experiments with type B uncertainties.

All ion mobility spectra (DTIMS and TWIMS) were fitted to Gaussian curves with OriginPro 9.1 to obtain experimental drift times with maximum accuracy.

RESULTS AND DISCUSSION

Drift-tube ion mobility of phosphoric acid cluster ions

With phosphoric acid cluster ions, the process of assignment of ion mobility peaks to the correct ion was complicated by two facts. First, phosphoric cluster ions, as they are hydrogen bonded cluster ions, tend to dissociate very easily. When dissociation happens after the ion mobility device, the extracted ion mobility spectrum (EIM, also called arrival time distributions or ATD) of a given m/z corresponding to a cluster size with n phosphoric acid molecules (that we will refer to a P#n z-) also contains peak of larger clusters (n+1 , n+2…). Indeed, such larger cluster ions are precursor ions that dissociated by losing one or two neutral H3PO4 molecules after the ion mobility device but before the time of flight analyzer to give the P#n z-cluster ion. This process is classically observed for many cluster ions [START_REF] Fernandez-Lima | Experimental and theoretical studies of (CsI)n Cs + cluster ions produced by 355 nm laser desorption ionization[END_REF] and has been termed neutral evaporation by Juan Fernandez de la Mora et al. [START_REF] Hogan | Tandem ion mobility-mass spectrometry (IMS-MS) study of ion evaporation from ionic liquid-acetonitrile nanodrops[END_REF][START_REF] Hogan | Ion-pair evaporation from ionic liquid clusters[END_REF] Dissociation involving loss of charge may also occur from large multiply charged cluster ions, and in that case it may be termed ion evaporation. Second, a given m/z classically corresponds to multimers of mass m, 2m and 3m with charge states [M -H ] -, [2M -2H ] 2-and [3M -3H ] 3- respectively and each multimer appears as a different ion mobility peak. These multimers may of course also be distinguished by their isotopic pattern on the mass spectrum, which for phosphoric acid cluster ions, mainly arises from the 0.2 % relative abundance of [START_REF] Hao | Electrospray ionization tandem mass spectrometric study of salt cluster ions: part 2--salts of polyatomic acid groups and of multivalent metals[END_REF] O, as P has one unique isotope [START_REF] Wu | Compositions and Structures of Vanadium Oxide Cluster Ions VmOn (+/-) (m = 2-20) Investigated by Ion Mobility Mass Spectrometry[END_REF] P. The peaks of the isotopic distribution are thus spaced by m/z 2, 1, 0.67 for the singly, doubly and triply charged, respectively (Figure S1).

Figure 1 shows the EIM spectra of two series of six and eight m/z values corresponding to a set of phosphoric acid cluster ions from [(H3PO4)3 -H] -to [(H3PO4)33 -3H] 3-(P#3 -to P#33 [START_REF] Pringle | An investigation of the mobility separation of some peptide and protein ions using a new hybrid quadrupole/travelling wave IMS/oa-ToF instrument[END_REF]-), at a potential difference of 8.6 kV. The resolving power (calculated form the FWHM) of all ion mobility signals was around 130. By overlaying the different EIM spectra, the signals of dissociation products of larger clusters could be assigned. Indeed, correct peak assignment could be unambiguously obtained by comparing the EIM spectra of the m/z values of clusters ions of adjacent aggregation numbers (i.e. n and n+1). The peak that is uniquely present in the EIM spectrum from the m/z value of P#n z-and not P#(n+1) z-may be assigned to the intact P#n z-cluster ion. For example, the EIM spectrum of m/z 488.88 (or m/z value of P#5 -represented in magenta in Figure 1a), included a signal at 39.04 ms from the intact P#5 - cluster ion, as well as a significant peak at a higher drift time (42.67 ms). This signal at 42.67 ms was also clearly visible in the EIM spectrum of m/z 586.85 (m/z value of P#6 -in green in Figure 1a). The signal at 42.67 ms corresponding to the P#6 -cluster ion is present in the EIM spectrum of m/z 488.88 because P#6 -is a precursor ion of P#5 -, and it fragments to P#5 - (i.e. m/z 488.88) after the ion mobility device.

Figure 1. (a) Extracted ion mobility spectra of m/z 293 (thin black), 391 (thick black), 489 (magenta), 586 (green), 587(blue), and 685 (red) corresponding phosphoric acid cluster ions from [(H3PO4)3 -H] -to [(H3PO4)8 -H] -(P#3 -to P#8 -) respectively. (b)

EIM spectra of m/z 881 (green) (P#9 -and P#18 2-), 978 (blue) (P#10 -and P#20 2-), 1077 (red) (P#11 -, P#22 2-and P#33 3-), 1175 (thick black) (P#24 [START_REF] May | Ion mobility-mass spectrometry: time-dispersive instrumentation[END_REF]-and P#36 3-), 832 (P#17 2-), 930(P#19 2-), 1028 (P#21 2-), 1109 (P#34 3-), and 1142 (P#35 3-) (all in thin black). EIM spectra are overlapped and were recorded at a nominal potential difference of 8.6 kV in the ion mobility region. Asterisks (*) refer to peaks that could not be attributed. The omitted y axis is the signal intensity in arbitrary units normalized to the highest peak on display.

The amount of dissociation deduced from these observations depends on the size of the cluster ion. The EIM spectrum of m/z 292.92 contained many different peaks beside the intact P#3 -cluster ion at 31.02 ms. These peaks at higher drift times could not be attributed except for the signal corresponding to P#4 -at 34.96 ms which appeared in the EIM of m/z 390.90. Although these unattributed peaks were a problem at low m/z, it was not the case for the higher m/z as the amount of dissociation was found to decrease with increasing cluster size. In the EIM spectrum of m/z 684.83 (blue trace in Figure 1a) the amount of P#8 -, as precursor ion of P#7 -, was very small and in the EIM spectrum of m/z 782.81, one signal for P#8 -was detected at 49.56 ms and no peaks at higher drift times were observed.

For larger cluster sizes, multiply charged cluster ions appeared on the EIM spectra. The EIM spectrum of m/z 880.78 (represented in green in figure 1b) very distinctively showed two peaks corresponding to P#9 -(52.16 ms) and P#18 2-(38.53 ms). A P#17 2-cluster ion could also be detected at 37.72 ms. Here again, the peaks could be unambiguously assigned by comparing the extracted ion mobility spectra of the m/z values of clusters ions of adjacent aggregation numbers: this overlay highlighted a pattern for the drift times of cluster ions of consecutive aggregation numbers and similar charge state, increasing confidence in the peak assignments. Noteworthy, in the EIM spectrum of m/z 831.80, a P#17 2-cluster was observed, making 17 the threshold cluster number for which doubly charged cluster ions could be unambiguously detected. The EIM spectrum of m/z 1076.73 showed peaks corresponding to P#11 -, P#22 2-and P#33 3-. In fact, 33 was the threshold cluster number for which triply charged cluster ions could be detected in our experiments. Charge threshold and multiple peaks due to post-ion mobility dissociation may also be visualized in twodimensional IMS-MS color map (Figure S2).

To determine collision cross sections in nitrogen gas in the drift tube ion mobility instrument, termed DT CCS(N2), experiments were carried out at five different potential differences on the ion mobility drift-tube (Figure 2 and Figure S3-S10). After unambiguous assignment of the peaks corresponding to the intact cluster ions, the drift time of each cluster ions were measured by fitting a Gaussian function on the assigned ion mobility peaks. For lower field strengths, however, the intensities of doubly and triply charged cluster ions were too low for reliable drift time measurements. The plots of measured drift times vs the reciprocal of the potential difference thus include five points for singly charged, four points for doubly charged and three points for triply charged cluster ions. Nevertheless, the data always showed the expected linear correlation with a determination coefficient above 0.9999 (Figures S3-S10). As the slope of the linear regression of the drift times vs the reciprocal of the potential difference is the square of the drift tube length divided by the ion mobility (L 2 /K), the ion mobility K could be derived and converted to a reduced ion mobility K0 or DT CCS(N2) using the Mason Schamp equation. The results are presented in table 1.

Table 1. Reduced ion mobilities K0 and collision cross section values in N2 gas measured by drift-tube ion mobility mass spectrometry ( DT CCS(N2)) of twenty-six phosphoric acid cluster anions. The B type relative expanded uncertainties on these values was estimated at ±3 %. The type B relative expanded uncertainty on these values were cautiously estimated to be ±3 % (level of confidence of approximately 95 %). Indeed, studies by Stow et al [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF] showed "effective" length of the drift tube could differ from the geometrical length by 1.6 mm. We chose to double this number to 3 mm, as a rough evaluation of any edge effects of the electric field. The reduced ion mobility K0 range was much larger for singly charged cluster anions (0.77 to 1.4 cm 2 V -1 s -1 ) while the doubly and triply charged cluster anions cover only 30 % and 16 % of that range respectively. As expected, the DT CCS(N2) values were observed to increase with the number n of phosphoric acid molecules in the cluster. A graph of the DT CCS(N2) vs the number of phosphoric acid molecules in the cluster is represented in Figure 3. We found that several functions could be fitted to our experimental values: a second order polynomial or a power function both produced an R 2 of 0.998.

Ion type P# m/z K0 (cm 2 V -1 s -1 ) DT CCS(N2) (Å 2 ) [(H3PO4)n -H] -

Travelling wave ion mobility of phosphoric acid cluster ions

The ion mobility and mass spectra obtained with travelling wave ion mobility were found comparable to the results obtained with drift-tube ion mobility mass spectrometry but with a lower resolving power. Figure 4 shows the EIM spectra of m/z values corresponding to singly charged phosphoric acid cluster ions from P#3 -to P#8 -and to a set of doubly (P#18 2- to P#24 2-) and triply charged (P#35 [START_REF] Pringle | An investigation of the mobility separation of some peptide and protein ions using a new hybrid quadrupole/travelling wave IMS/oa-ToF instrument[END_REF]-and P#36 3-) cluster ions, at wave velocity of 350 m/s and wave height 15 V. The resolving power (calculated form the FWHM) of all ion mobility signals was around 17 for the signals of the singly charged cluster ions, 19 for the doubly charged cluster ions and 27 for the triply charged cluster ions. With such resolving power, only the singly charged cluster ions had their signals well separated for adjacent clusters with one phosphoric acid molecule difference in aggregation number. The signals from the EIM spectra of adjacent doubly and triply charged cluster ions all partially overlapped. The presence under each peak in the EIM spectra of unresolved precursor ions of higher cluster numbers that fragmented after the IMS separation cannot be completely ruled out. In this case, this could potentially result in shifts of the peak centroids. Indeed, as in the DTIMS-MS experiments, cluster dissociation in the transfer area between the ion mobility drift tube and the time-of-flight analyzer could be deduced from the comparison the EIM spectra of the m/z values of clusters ions of adjacent aggregation numbers. Indeed, the EIM spectrum of m/z 488.88 (represented in magenta in Figure 5a), included a signal at 4.52 ms from the intact P#5 -cluster ion, as well as a peak at a higher drift time (5.41 ms) corresponding to the P#6 -cluster ion. A signal at 5.41 ms was also clearly visible in the EIM spectrum of m/z 586.85 (in green in figure 4a) which came here from the intact P#6 -cluster ion. The amount of dissociation was observed to be larger than in the DTIMS-MS experiments, as the peak from the precursor ion had a relative intensity between 40 to 50 % compared to the intact cluster ions. This proportion of precursor ion signal was observed to decrease less with increasing cluster size: the precursor signal from P#12 -was still observed at 30 % relative intensity compared to the intact P#11 -cluster ion in the EIM spectrum of m/z 1077. For multiply charged cluster ions, the amount of dissociation was impossible to estimate as the signals of adjacent cluster ions overlapped. We nevertheless supposed the signals to be largely attributable to the intact cluster ions. Regardless of the amount of dissociation, we hypothesized that the intact clusters should have similar conformations to those observed in the DTIMS-MS experiments.

We did observe some differences as to the range of phosphoric acid cluster ions detected. For singly charged ions, we could observe cluster ions up to P#12 -with traces of P#13 -. Such large singly charged cluster ions which were not well observed in our DTIMS-MS experiments. Conversely, P#17 2-was very low intensity and low signal/noise ratio in the EIM spectrum of m/z 832 in TWIMS-MS. The smallest doubly charged cluster ion distinguishable was P#18 2-(Figure 4b). For triply charged ions, P#33 3-and P#34 3-were not visible and the smallest triply charged cluster ion observed was P#35 3-. The TWIM-MS experiments also showed larger cluster ions with charges up to 6-(Figure S11-12). Out of the twenty-six cluster ions measured by DTIMS-MS, twenty-three could be used in the calibration of the TWIMS-MS. Nevertheless, we hypothesized that the intact cluster ions observed in both devices should have similar conformations.

For the calibration of the TWIMS, we chose to use a linearized version of the method which uses a power function (Equation 1) to fit reference DT CCS values with drift times measured on the TWIM device for the reference compounds, as proposed by Smith et al. [START_REF] Smith | Deciphering drift time measurements from travelling wave ion mobility spectrometry-mass spectrometry studies[END_REF] In equation 3, Ω′ is defined as the CCS divided by the charge number z and multiplied by the square root of the reduced mass µ of the ion and the drift gas N2 (equation 2). A and B are found by fitting the curve of reference DT CCS derived Ω′ values vs TWIMS drift times td. Here, we have linearized this equation by using natural logarithms, in order to have a linear regression where ln(A) and B are the intercept and the slope respectively (Equation 3). We did not apply any correction to the drift times and used the TWIMS drift time values given by a fit to a Gaussian function on the assigned peaks (with OriginPro 9.1).

Ω′ =

(1)

Ω′ = √ Ω (2)
Ω = + (3)
The resulting correlations are shown on Figure 5, for the three set of values obtained from the singly, doubly and triply charged phosphoric acid cluster ions. The determination coefficient R 2 observed for the singly charged phosphoric cluster ions was remarkably high with a value higher than 0.9999. The determination coefficients R 2 were not as good for doubly and triply charged cluster ions with values of 0.998 and 0.939 respectively. A repeated experiment at 350 m/s wave velocity and 20 V wave height led to similar results (Figure S13). Note that these high determination coefficients were obtained with low wave velocities, which are the parameters for which good calibration correlations have been observed before by Zhong et al. [START_REF] Zhong | Characterizing the resolution and accuracy of a secondgeneration traveling-wave ion mobility separator for biomolecular ions[END_REF] Recently, Richardson et al, [START_REF] Richardson | Fundamentals of travelling wave ion mobility revisited: I. Smoothly moving waves[END_REF] who studied velocity relaxation effects in TWIMS, noted calibration to have the highest R 2 with low wave velocity. Finally, Mortensen et al [START_REF] Mortensen | Collisional Cross-Sections with T-Wave Ion Mobility Spectrometry without Experimental Calibration[END_REF] who recently proposed a computational method to measure collision cross sections with TWIMS without experimental calibration noted that low wave velocities and gentle instrument settings should be used to minimize uncertainties in their method. Indeed, low wave velocities and gentle instrument setting in the TWIMS-MS were key parameters for the cluster ions to survive the ion mobility separation and ensure that ion conformations would be similar to the conformations in the DTIMS-MS experiments. We do not believe the ion source settings to be as important as the only effect was to change the range and abundance of the cluster ions detected. When the cluster ions enter the ion mobility device, intact cluster ions of given aggregation numbers are expected to adopt always the same isotropic conformations and growth. In fact, monitoring the shape of peaks and amount of dissociation from the EIM spectra of phosphoric acid clusters might help to achieve the desirable experimental conditions that minimize cleavage of hydrogen bonds and should thus preserve labile biomolecule conformations throughout the ion mobility device and during the transfer to the TOF analyzer.

We observed the multiply charged cluster ions led to different correlation functions compared to singly charged cluster ions. Indeed, both the intercept lnA and slope B from the multiply charged cluster ions differed by 13 to 30 % compared to the values obtained from the singly charged cluster ions. Doubly and triply charged cluster ions both resulted in very similar correlation functions with intercepts and slopes that differed by less than 0.2 %. Such difference between the calibration curves obtained for TWIMS from multiply and singly charged ions has been observed before for positively charged ions by Gelb et al [START_REF] Gelb | A study of calibrant selection in measurement of carbohydrate and peptide ion-neutral collision cross sections by traveling wave ion mobility spectrometry[END_REF] who recommend matching the charge states of the calibrant and analyte ions. They also recommended matching molecular classes.

With these calibrations, we proceeded to test the resulting accuracy on estimated TW CCS(N2) values using a set of ten ions obtained from negative ions derived from polyalanine, dextran and the hexakis(fluoroalkoxy)phosphazine (HFAP) at m/z 602 from the Agilent tune mix, for which DT CCS(N2) are published [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF][START_REF] Forsythe | Collision cross section calibrants for negative ion mode traveling wave ion mobility-mass spectrometry[END_REF][START_REF] Hofmann | Estimating collision cross sections of negatively charged N-glycans using traveling wave ion mobility-mass spectrometry[END_REF] and are therefore often used as calibrants. These chosen test ions are all singly charged anions in the CCS range of the singly charged phosphoric acid cluster anions. To examine multiply charged anions, we also tested a set of five anions derived from polyoxometalates for which we had measured the DT CCS(N2) on the Tofwerk instrument in a previous study. [START_REF] Hupin | A calibration framework for the determination of accurate collision cross sections of polyanions using polyoxometalate standards[END_REF] The results with 350 m/s IMS wave velocity and 15 V IMS wave height are presented in Table 2 (and Table S3 for with 350 m/s IMS wave velocity and 20 V IMS wave height). The TW CCS(N2) were estimated using four different calibration correlations: the correlations obtained from the singly, doubly and triply charged phosphoric anions as three separate sets and one correlation from all phosphoric cluster anions assembled regardless of the charge state. The comparison of the values obtained from all these different calibrations showed that the TW CCS(N2) differed by up to 20 Å 2 from one calibration to the other, with the highest difference observed for the multiply charged ions, with the largest CCS. In fact, a maximum relative difference of 6 % was observed between the different calibration sets which was surprisingly low given the 20 % difference observed for the ln(A) and B values from the correlation functions.

Focusing on the values obtained using the set of singly charged cluster anions, we found the TW CCS(N2) values of singly charged anions to be systematically higher by 9 to 11 Å 2 compared to published DT CCS(N2) values, which amounts to 5 or 6 % relative error, with the notable exception of [Glc2 -H] -for which the TW CCS(N2) showed only a 4 Å 2 (2 %) difference compared to its DT CCS(N2) value. Such 9 to 11 Å 2 discrepancy between DT CCS(N2) and TW CCS(N2) using a high quality correlation must arise from a systemic error. An offset of 5 Å 2 on average has been observed before between the DT CCS(N2) and TW CCS(N2) of lipids. [START_REF] Groessl | High resolution ion mobility-mass spectrometry for separation and identification of isomeric lipids[END_REF] Temperature could, in part, explain this bias. Litterature DT CCS(N2) values for polyalanine [START_REF] Groessl | High resolution ion mobility-mass spectrometry for separation and identification of isomeric lipids[END_REF] were determined at room temperature and around 25°C for the tune mix [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF] . For dextran [START_REF] Hofmann | Estimating collision cross sections of negatively charged N-glycans using traveling wave ion mobility-mass spectrometry[END_REF] , the authors discuss the fact that ion temperature was uncertain. In fact, higher temperature decreases the value of the CCS because of the greater average speed of the gas molecules in the drift cell. [START_REF] Revercomb | Theory of plasma chromatography/gaseous electrophoresis[END_REF] Our larger values thus suggested that the ion temperature in our drift tube experiments was lower than during the experiments associated with the published DT CCS(N2) values of polyalanine, dextran and the hexakis(fluoroalkoxy)phosphazine (HFAP) at m/z 602 from the Agilent tune mix, Another explanation may reside in the approximate nature of the Mason Schamp equation which is a zero field limit. Improvement of the equation have been studied by Hauck and Siems et al. [START_REF] Siems | Correcting the fundamental ion mobility equation for field effects[END_REF][START_REF] Hauck | Determination of E/N Influence on K0 Values within the Low Field Region of Ion Mobility Spectrometry[END_REF] They show how an increase in electric field strength may lead to larger CCS values.

The errors of the singly charged anions were slightly lower when the TW CCS(N2) were estimated using the other calibrations with higher or all charge states. The lower errors in this case, were most probably the result of two or more systematic errors that happened to compensate. These lower CCS values closer to the reference DT CCS could be the result of a smaller difference in temperature and electric field with the reference measurements. Indeed, multiply charged ions might be considered "hotter" due to the larger speeds resulting of their higher influence of the electric field.

Table 2. Accuracy test on ions from dextran, polyalanine , the hexakis(fluoroalkoxy)phosphazine at m/z 602 (Agilent tune mix) and polyoxometalate derived anions. Calibrations using singly, doubly triply charged phosphoric cluster anions are compared as well as the calibration obtained from ions of charge states. DT CCS(N2) values are reported from references [START_REF] Stow | An Interlaboratory Evaluation of Drift Tube Ion Mobility-Mass Spectrometry Collision Cross Section Measurements[END_REF][START_REF] Forsythe | Collision cross section calibrants for negative ion mode traveling wave ion mobility-mass spectrometry[END_REF][START_REF] Hofmann | Estimating collision cross sections of negatively charged N-glycans using traveling wave ion mobility-mass spectrometry[END_REF] The accuracy of the measurement of the polyoxometalate derived anions were quite good with relative errors of 2% or less, which represent differences of -10 Å 2 between TW CCS(N2) and DT CCS(N2) for these ions of higher CCS values. This result was expected considering that the DT CCS(N2) measurement were carried out on the same instrument at a similar temperature (the CCS polyoxometalate derived anions were measured at 31°C). Any systematic error arising from instrumental conditions should therefore be absent. It can be noted that structurally very different compounds could be used here as reference for CCS measurements which is not the commonly recommended practice. [START_REF] Gabelica | Recommendations for reporting ion mobility Mass Spectrometry measurements[END_REF]

CONCLUSION

We herein presented the first determination of twenty-six new DT CCS(N2) for phosphoric clusters anions. The values come in three sets: singly deprotonated clusters with an aggregation numbers 3 to 11, doubly charged clusters anions with aggregation numbers 17 to 25 and triply charged phosphoric acid cluster anions with aggregation numbers 33 to 40.

The DT CCS(N2) ranged from three to 146 to 264 Å 2 , 337 to 402 Å 2 and 486 to 530 Å 2 , respectively. Reduced mobilities K0 range from 0.77 to 1.4 cm 2 V -1 s -1 for singly charged cluster anions. Doubly and triply charged cluster anions cover only 30 % and 16 % of that range (1.177 to 1.006 cm 2 V -1 s -1 and 1.250 to 1.0143 cm 2 V -1 s -1 ).

Although these measurements will need to be further reproduced on different instruments to establish these DT CCS(N2) more firmly and complete them, we found these DT CCS(N2) values to be self-consistent and that the three sets combined could be fitted to a power function with an exponent of 0.51.

The values of DT CCS(N2) of the singly charged phosphoric acid anions were found remarkably well correlated (determination coefficient R 2 above 0.9999) to the drift times measured on a TWIMS-MS device thereby confirming that the conformation of the phosphoric acid cluster ions were comparable in the two very different DTIMS and TWIMS devices and they thus could be well suited compounds for calibration. Indeed, we expect that a given cluster ion will dissociate rather than undergo significant conformational change thus guarantying that a given cluster ion will present a unique and reproducible CCS value.

Comparing the correlations obtained in TWIMS using the multiply and singly charged anions, we observed that there was a charge state effect on the calibration. Accuracy tests on singly charged ions evidenced a bias of 9 to 11 Å 2 for TW CCS(N2) values compared to reference DT CCS(N2) values of ions from polyalanine, dextran and the hexakis(fluoroalkoxy)phosphazine at m/z 602. Note that although absolute values were shifted, CCS differences were preserved.

This bias was most probably instrument-related. Indeed, the accuracy test on polyoxometalate derived anions, for which DT CCS(N2) were measured on the same instrument showed errors for the TW CCS(N2) values below 2 %, even though the compounds were structurally very different from the phosphoric acid cluster anions. We could not identify the cause of the bias for polyalanine, dextran and the hexakis(fluoroalkoxy)phosphazine at m/z 602 within this study. Temperature and field strengths could play a role; a compound class effect may also intervene although that did not seem to be the case with polyoxometalates.

Collision cross section values for singly and multiply charged anions can be obtained with a single experiment using one very simple and easily available compound. One drawback is that phosphoric acid cluster ions are very fragile and care must be taken to avoid dissociation between the drift tube and the mass analyzer, as such dissociations can lead to difficulties in the assignment of the signals in the ion mobility spectra. However, comparison of the extracted ion mobility spectra of the m/z values of clusters ions of adjacent aggregation numbers enables unambiguous ion mobility peak attribution and allows a pattern of drift times depending on the aggregation number and charge state to be gathered. Moreover, the lability of phosphoric acid cluster ions could be turned to an advantage because it enables to visualize and monitor the cleavage of hydrogen bonds and thus help achieve the desirable experimental conditions that should preserve labile biomolecule conformations.

ELECTRONIC SUPPLEMENTARY INFORMATION

Voltage parameters on the Tofwerk IMS-TOF and SYNAPT G2 instruments, isotopic distribution of phosphoric cluster ions, a two-dimensional IMS-MS color map of the ESI in the negative ion mode of a phosphoric acid solution, extracted ion mobility spectra and graphs of the drift time vs the reciprocal of the potential difference in the ion mobility drift tube, m/z vs drift time maps of the TWIMS MS analyses, graph of the calibration correlation and accuracy tests with 350 m/s IMS wave velocity and 20 V IMS wave height.

Figure 2 .

 2 Figure 2. (a) Extraction ion mobility spectra of m/z 489 (P#5 -) at five different electric field strengths corresponding to 9.4, 8.6, 7.8, 7.0 and 6.2 kV potential difference. (b) Graph of drift times vs 1/V for m/z 489 (P#5 -). (c) EIM spectra of m/z 930 (P#19 2-) at four field strength corresponding to 9.4, 8.6, 7.8 and 7.0 kV. (d) Graph of drift times vs 1/V for m/z 930 (P#19 2-). (e) EIM spectra of m/z 1142 (P#35 3-) at four field strength corresponding to 9.4, 8.6, 7.8 and 7.0 kV. No signal appeared at 7.0 kV. (d) Graph of drift times vs 1/V for m/z 930 (P#35 3-). The omitted y axis is the signal intensity in arbitrary units normalized to the peak observed at 9.4 kV potential difference.

Figure 3 .

 3 Figure 3. Graph of DT CCS(N2) (Å 2 ) of phosphoric acid cluster ions vs the number of phosphoric acid molecules in the cluster. The error bars are set at 3 % of the value.

Figure 4 .

 4 Figure 4. (a) EIM spectra of m/z 293 (thin black), 391 (thick black), 489 (magenta), 586 (green), 587(blue), and 685 (red) corresponding phosphoric acid cluster ions from [(H3PO4)3 H] -to [(H3PO4)8 -H] -(P#3 -to P#8 -) respectively. (b) EIM spectra of m/z 881 (green) (P#9 -and P#18 2-), 978 (purple) (P#10 -and P#20 2-), 1077 (red) (P#11 -, P#22 2-and P#33 3-), 1175 (thick black) (P#12 -, P#24 2-and P#36 3-), 930(P#19 2-), 1028 (P#21 2-), 1126 (P#23 2-), and 1142 (P#35 3-) (all in thin black). EIM spectra are overlapped and were recorded at a wave velocity 350 m/s and wave height 20 V for a m/z range of 50 to 4000.

Figure 5 .

 5 Figure 5. Calibration correlations obtained from singly charged (red), doubly charged (blue) and triply charged (green) phosphoric cluster anions and TWIM drift time values obtained with 350 m/s IMS wave velocity and 15 V IMS wave height.

  . Values in bold highlight the CCS measurement using calibration with reference ions of charges states matching that of the measured ions. TWIMS was carried out at 350 m/s IMS wave velocity and 15 V IMS wave height.

	Ion	DT CCS( N2) (Ų)	m/z	TW CCS(N2) (Ų) estimated with different calibrations	ΔCCS/CCS with different calibrations (%)
				1-	2-	3-	all charge	1-2-3-	all charge
	[(Ala)3 -H] -	150.8	230.2	159.5	151.3	152.2	154.3	6	0	1	2
	[(Ala)4 -H] -	165.5	301.3	175.3	168.7	168.3	170.7	6	2	2	3
	[(Ala)5 -H] -	180.6	372.4	191.4	186.6	184.9	187.5	6	3	2	4
	HFAP m/z 602	180.1	602.0	190.3	185.7	183.9	186.5	6	3	2	4
	[(Ala)6 -H] -	196.2	443.5	207.2	204.4	201.2	204.2	6	4	3	4
	[(Ala)7 -H] -	210.3	514.6	221.4	220.5	216.0	219.1	5	5	3	4
	[(Ala)8 -H] -	223.7	585.6	234.4	235.3	229.5	232.9	5	5	3	4
	[(Glc)2 -H] -	174.6	341.1	178.3	172.1	171.5	173.9	2	-1 -2 -0.4
	[(Glc)3 -H] -	202.3	503.2	213.3	211.3	207.6	210.6	5	4	3	4
	[(Glc)4 -H] -	234.0	665.2	243.6	245.9	239.1	242.6	4	5	2	4
	[PMo12O40 + TBA] 2-	348	1033.0	353.8	342.7	340.8	345.6	2	-2 -2 -0.7
	[PW12O40 + TBA] 2-	349	1559.7	356.1	345.4	343.2	348.1	2	-1 -2 -0.3
	[P2W18O62 + TBA + 2H] 3-	438	1535.8	451.7	428.4	430.9	436.9	3	-2 -2 -0.2
	[P2W18O62 + 2TBA + H] 3-	481	1617.4	492.1	472.2	471.9	478.6	2	-2 -2 -0.5
	[P2W18O62 + 3TBA] 3-	520	1696.9	530.9	514.6	511.5	518.8	2	-1 -2 0.2

ACKNOWLEDGMENTS

The authors gratefully acknowledge the Région Normandie, the Labex SynOrg (ANR-11-LABX-0029) and the European Regional Development Fund (ERDF 31708) for financial support. HL thanks Sébastien Hupin for the TWIMS-MS data of polyalanine, dextran and the polyoxometalates, and Valentina Calabrese for helpful discussions.