Pierre Gabriel
email: pierre.gabriel@uvsq.fr

ON AN IRREDUCIBILITY TYPE CONDITION FOR THE ERGODICITY OF NONCONSERVATIVE SEMIGROUPS

Keywords: 2010 Mathematics Subject Classification. Primary 47A35, Secondary 35B40, 47D06, 60J80, 92D15, 92D25 Positive semigroups, measure solutions, ergodicity, Krein-Rutman theorem, evolutionary model

We propose a simple criterion, inspired from the irreducible aperiodic Markov chains, to derive the exponential convergence of general positive semigroups. When not checkable on the whole state space, it can be combined to the use of Lyapunov functions. It differs from the usual generalization of irreducibility and is based on the accessibility of the trajectories of the underlying dynamics. It allows to obtain new existence results of principal eigenelements, and their exponential attractiveness, for a nonlocal selection-mutation population dynamics model defined in a space-time varying environment.

The main result

We are interested in the long time behavior of positive semigroups in weighted signed measures spaces. Before dealing with these semigroups, let us start by defining more precisely what we mean by weighted signed measures.

We consider positive semigroups (M t ) t≥0 acting on M(V ) on the left (µ → µM t ) and on B(V ) on the right (f → M t f ), which enjoy the classical duality relation µ(M t f ) = (µM t )(f ).

Let X be a measurable space. We work in weighted signed measures spaces on X . More precisely for a weight function V : X → (0, ∞) we denote by M + (V ) the set of positive measures on X which integrate V and we define the space of weighted signed measures as

M(V ) = M + (V ) -M + (V )
the smallest vector space with positive cone M + (V ), see [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] for a rigorous construction as a quotient space. Basically, an element µ of M(V ) is the difference µ = µ + -µ -of two positive measures µ + , µ -∈ M + (V ) which are mutually singular (Hahn-Jordan decomposition). It acts on the Banach space

B(V ) = f : X → R measurable, f B(V ) := sup x∈X |f (x)|/V (x) < +∞ through µ(f ) = X f dµ + - X f dµ -.
The space M(V ) is a Banach space for the weighted total variation norm

µ M(V ) = sup f B(V ) ≤1 µ(f ).
Notice that, with these definitions, the standard total variation norm is

• TV = • M(1)
, where 1 stands for the mapping x → 1.

We present sufficient irreducibility type conditions which, possibly combined to Lyapunov type conditions, ensure the so-called asynchronous exponential behavior of the semigroup:

µM t ∼ e λt µ(h)γ as t → ∞,
where λ ∈ R, h is a positive function, and γ is a positive measure.

We start with the global case, i.e. conditions verified on the whole state space X .

Hypothesis 1. The weight function is given by V = 1. There exist τ > 0 and C ≥ 1 such that C -1 ≤ M s 1 ≤ C for all s ∈ [0, τ ], and there exist c > 0 and a family of probability measures (σ x,y ) x,y∈X over [0, τ ] such that

(H1) ∀x, y ∈ X , δ x M τ (•) ≥ c τ 0 δ y M τ -s (•) σ x,y (ds), (H2) sup x,x ′ ∈X inf y∈X σ x,y -σ x ′ ,y TV < 2.
Let us do some comments on these two assumptions. Condition (H1) ensures the existence of a time τ such that, with positive probability and uniformly with respect to any initial positions x and y in X , the trajectories issued from x intersect at time τ the trajectories issued from y at some random times s ∈ [0, τ ]. Even though they are not comparable in general, this path crossing condition has connections with the notion of irreducibility of Markov processes, which means basically that for any x and y there is a deterministic time at which the trajectories issued from x reach y with positive probability [START_REF] Norris | Markov chains, volume 2 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF].

Consider for instance the semigroup (M t ) t≥0 of a time continuous finite Markov chain (X t ) t≥0 :

M t f (x) = E[f (X t ) | X 0 = x].
Irreducibility induces the existence of a random time T (x, y) such that X T (x,y) = y and the strong Markov property gives

M t f (x) ≥ E[f (X t )1 T (x,y)≤t | X 0 = x] = E[M t-T (x,y) f (y)1 T (x,y)≤t | X 0 = x].
Assumption (H1) then holds with σ x,y being the law of T (x, y) conditioned on being positive. For Markov processes on infinite state spaces however, irreducibility does not imply (H1) in general, due to the requirement in (H1) for the probability of intersection and the crossing maximal time τ to be uniform in x and y. Actually there are few examples of Markov processes on infinite state spaces which are irreducible since in most cases the probability of reaching a point y at a deterministic time starting from an arbitrary position x is zero. Similarly, the concept of irreducibility for positive semigroups, see [2, Definition C-III.3.1 p.306] for instance, which proves very useful when working in spaces of functions, is rarely applicable in spaces of measures, exactly for the same reasons as for Markov processes. Condition (H1) can then be seen as a useful alternative to irreducibility since it allows to cover many more interesting examples.

Consider for instance a Markov process on a compact topological state space X . If the random hitting time T (x, y) has a positive density s(x, y, •) with respect to the Lebesgue measure which is continuous on its three components, then the associated semigroup satisfies (H1) and (H2). A simple and typical example satisfying (H1) but not (H2) is given by the periodic (and irreducible) semigroup (M t ) t≥0 defined for every f ∈ B(1), t ≥ 0 and x ∈ X = [0, 1] by

M t f (x) = f (x + t -⌊x + t⌋) ,
where ⌊•⌋ stands for the integer part. For this semigroup, (H1) is satisfied only with (σ x,y ) 0≤x,y≤1 a family of Dirac masses, so that (H2) does not hold. A more sophisticated example (still irreducible) can be found in [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF], see also the comments in Section 2.

Condition (H2) is thus an aperiodicity assumption, reminiscent from the aperiodicity of Markov chains. It can also be seen as a coupling condition since it means that, uniformly in x and x ′ in X , the trajectories issued from x and x ′ intersect at time τ with positive probability (the intersection point being on a trajectory issued from some y ∈ X in the time interval [0, τ ]).

In Hypothesis 1 the conditions are satisfied on the whole state space, which can be too much to ask in many applications, typically when the state space is not compact. In that case Assumptions (H1) and (H2) can be localized through the use of Lyapunov functions, in the spirit of [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. It leads to the following set of assumptions where, for two functions f, g : Ω → R, the notation f g means that f ≤ Cg on Ω for some C > 0.

Hypothesis 2. There exists a function ψ : X → (0, ∞), a subset K ⊂ X , a time τ > 0, and constants β > α > 0 and θ > 0 such that

(A0) ψ ≤ V on X and V ψ on K; M V V and M ψ ψ on [0, τ ] × X , (A1) M τ V ≤ αV + θ1 K ψ, (A2) M τ ψ ≥ βψ,
and there exist c > 0 and a family of probability measures (σ x,y ) x,y∈K over [0, τ ] such that (H1') for all f ∈ B(V ), f ≥ 0, and all x, y ∈ K,

M τ f (x) ψ(x) ≥ c τ 0 M τ -s f (y) ψ(y) σ x,y (ds), (H2') sup x,x ′ ∈K inf y∈K τ 0 M τ -s (ψ1 K )(y) ψ(y) (σ x,y ∧ σ x ′ ,y )(ds) > 0.
We are now ready to state our main result.

Theorem 1.1. Assume that Hypothesis 1 or Hypothesis 2 is verified. Then there exists a unique triplet (γ, h, λ)

∈ M + (V ) × B + (V ) × R of eigenelements of (M t ) t≥0 with γ(h) = h B(V ) = 1,
i.e. satisfying for all t ≥ 0 γM t = e λt γ and M t h = e λt h.

Moreover, there exist C, ω > 0 such that for all t ≥ 0 and µ ∈ M(V ),

e -λt µM t -µ(h)γ M(V ) ≤ C µ M(V ) e -ωt . (1.1)
In [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF], the authors give a set Assumption A of conditions which they prove to be equivalent to the convergence (1.1). These conditions are the same as Hypothesis 2 where (H1') and (H2') are replaced by a Doeblin type coupling condition (A3) and an assumption (A4) which is basically a control on the ratio M t ψ(x)/M t ψ(y) uniform in x, y ∈ K and t ≥ 0. The main advantage of Hypothesis 2 compared to Assumption A in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] is that (H1') is much easier to check than (A4) on examples. It will be illustrated on an example coming from evolutionary biology in Section 2.

The idea for proving Theorem 1.1 is to show that (H1') implies a slightly stronger control than (A4) and, combined with (H2'), a slightly weaker version of (A3). Then we verify that the proof in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] can be adapted to these small variations of (A3)-(A4).

Note that, contrary to Assumption A in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF], Hypothesis 2 is sufficient but not necessary for ensuring the convergence (1.1), see for instance the comments on the example in Section 2.

It is similar to the situation of finite Markov chains for which irreducibility and aperiodicity are a powerful sufficient but not necessary condition for proving convergence to an equilibrium. Perron-Frobenius theory states that it becomes equivalent to the convergence only by adding the assumption that the Markov chain is undecomposable, in the sense that it has only one recurrent class.

Proof of Theorem 1.1. We will prove that Hypothesis 2 implies the existence of a constant d > 0 such that for all x, y in K and all t ≥ 0

M t ψ(x) ψ(x) ≥ d M t ψ(y) ψ(y) , (1.2)
and the existence of a constant c > 0 and a family (ν x,x ′ ) x,x ′ ∈K of probability measures such that for all x, x ′ in K and all y ∈ {x,

x ′ } δ y M τ (ψ •) M τ ψ(y) ≥ c ν x,x ′ , (1.3)
and that Hypothesis 1 implies these same inequalities with K replaced by X and ψ = 1. Such estimates, which first appeared in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF], are a powerful tool to derive the exponential ergodicity of nonconservative semigroups, as attested by the further developments in [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF][START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. When (1.2) and (1.3) are verified on X with ψ = 1, and t → M t 1 ∞ is locally bounded, we can use the proof of [3, Theorem 3.5] to get (1.1) with V = 1. Indeed (1.2) readily implies [3, Assumption (H2)] and (1.3) is similar to [3, Assumption (H1)] with the difference that the coupling measure ν is independent of x, x ′ . But it is not an issue since the proof in [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF] also works with a family (ν x,x ′ ). This was already noticed in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF], where Assumption (A1') is the analogue to (1.3).

When (1.2) and (1.3) are verified in K ⊂ X and ψ is as in Hypothesis 2, we can use the proof in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] to obtain (1.1). Similarly as above, (1.2) implies [4, Assumption (A4)] and (1.3) is the same as [4, Assumption (A4)] except that in (1.3) the coupling measure depends on x, x ′ . The proof in [START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] can be adapted to this slight variation by: replacing the quantity ν(M kτ ψ/ψ) by inf K (M kτ ψ/ψ) in the definition of the Lyapunov functions V k and using accordingly (1.2) instead of [4, Assumption (A4)]; letting ν depend on x, x ′ in the proof of [4, Lemma 3.2] and using (1.3)

instead of [4, Assumption (A3)]

It thus only remains to check that Hypothesis 2 implies (1.2) and (1.3), and that Hypothesis 1 implies the same inequalities with K = X and ψ = 1. Since the proof for Hypothesis 1 is simpler, we only give it for Hypothesis 2 (replace below K by X and ψ by 1 for Hypothesis 1).

By virtue of Assumption

(A0), the condition (1.2) is clearly verified for t ∈ [0, τ ]. When t > τ , using f = M t-τ ψ in (H1') ensures that for all x, y ∈ K M t ψ(x) ψ(x) ≥ c τ 0 M t-s ψ(y) ψ(y) σ x,y (ds). (1.4)
By Assumption (A0), there exists a constant C > 0 such that for all y ∈ X , t ≥ 0 and s ∈ [0, τ ]

M t ψ(y) M t-s ψ(y) ≤ C M kτ V (y) M kτ ψ(y) , (1.5)
where k = t-s τ . Using Assumptions (A1) and (A2) we can prove as in [4, Lemma 4.1] that for all y ∈ X M kτ V (y)

M kτ ψ(y) ≤ α β k V (y) ψ(y) + θ β -α . (1.6)
Since β > α and V ψ on K by (A0), we deduce from (1.4), (1.5) and (1.6) that Condition (1.2) is verified for a constant d > 0 independent of x, y ∈ K and t ≥ 0.

We turn now to (1.3). Due to (A0), there exists C > 0 such that M τ ψ ≤ Cψ on K. Hypothesis (H1') then yields that for every x, y ∈ K

δ x M τ (ψ •) M τ ψ(x) ≥ c C τ 0 δ y M τ -s (ψ •) ψ(y) σ x,y (ds). (1.7)
By Hypothesis (H2') there exists ǫ > 0 such that for all x, x ′ ∈ K we can find y ∈ K such that

m x,x ′ ,y := τ 0 M τ -s (ψ1 K )(y) ψ(y) (σ x,y ∧ σ x ′ ,y )(ds) ≥ ǫ. Setting ν x,x ′ = 1 m x,x ′ ,y τ 0 M τ -s (ψ1 K •)(y) ψ(y) (σ x,y ∧ σ x ′ ,y )(ds),
we deduce from (1.7) that (1.3) is satisfied with c = cǫ/C.

An application in population dynamics

We apply our method to the following nonlocal equation with drift:

∂ t u(t, x) + ∂ x u(t, x) = R u(t, y)Q(y, dx) dy + a(x)u(t, x), x ∈ R. (2.1)
The solutions describe the density of traits in a population with nonlocal mutation in a space-time varying environment, encoding for instance the influence of a climate change through a spatial shift of the coefficient, see [START_REF] Berestycki | Can a species keep pace with a shifting climate[END_REF][START_REF] Coville | On generalized principal eigenvalues of nonlocal operators witha drift[END_REF] for motivations. In the recent work [START_REF] Coville | On generalized principal eigenvalues of nonlocal operators witha drift[END_REF], Coville and Hamel address the Perron spectral problem associated to Equation (2.1) in a slightly more general case, namely with a possibly non constant transport speed q(x) and on a domain Ω which can be an open subinterval of R. In the present paper we choose to limit ourselves to the case q ≡ 1 and Ω = R in order to remain concise. However, the method can be applied to other situations: to the price of additional technicalities for q ≡ 1 since the flow of the transport part may not be explicit, or on the contrary with less calculations when Ω = (r 1 , r 2 ) is a bounded interval (-∞ < r 1 < r 2 < +∞) since no Lyapunov condition is needed in that case (i.e. Hypothesis 1 is verified).

We assume that a : R → R is a continuous function such that

ā = sup x∈R a(x) < +∞ and lim x→±∞ a(x) = -∞ (2.2) and x → Q(x, •) is a weak* continuous function R → M + (1) which satisfies ∃ ǫ, κ 0 > 0, ∀x ∈ R, Q(x, dy) ≥ κ 0 1 (x-ǫ,x+ǫ) (y)dy (2.3) Q = sup x∈R Q(x, R) < +∞. (2.4)
The assumptions on Q are typically verified by a convolution kernel Q(x, y) = J(x -y) with J a finite positive measure such that J(dz) ≥ κ 0 1 (-ǫ,ǫ) (z)dz. They are less restrictive than in [START_REF] Coville | On generalized principal eigenvalues of nonlocal operators witha drift[END_REF], since we do not require the measures Q(x, •) to have a bounded and compactly supported Lebesgue density. Our potential function a satisfies a confining assumption a(±∞) = -∞ (no longer required if working in a bounded subinterval of R), while it is supposed to be bounded in [START_REF] Coville | On generalized principal eigenvalues of nonlocal operators witha drift[END_REF]. The reason is that in [START_REF] Coville | On generalized principal eigenvalues of nonlocal operators witha drift[END_REF], only the existence of (λ, h) is addressed, whereas we prove the existence of (λ, h, γ) and the asynchronous exponential behavior of the solutions to the Cauchy problem. The well-posedness of Equation (2.1) completed with an initial condition u(0, •) = u 0 in M(1) or in the subspace L 1 (R, dx) is a standard result. Our method allows us to prove the following result stated in the L 1 framework, which is more usual in the partial differential equations community, but it is also valid for measure solutions by replacing the L 1 norm by the total variation norm.

Theorem 2.1. Under Assumptions (2.2),(2.3) and (2.4), there exist constants C, ω > 0 and a unique eigentriplet (λ, γ, h) ∈ R × L 1 × C 1 b with γ ≥ 0, h > 0 and hγ = h ∞ = 1, such that for any initial condition u 0 ∈ L 1 (R, dx) the corresponding solution u(t, x) of Equation (2.1) verifies

u(t, •)e -λt - R h(x)u 0 (x) dx γ L 1 ≤ C u 0 L 1 e -ωt .
The method of proof consists in applying our general result in Theorem 1.1 to the semigroup (M t ) t≥0 associated to Equation (2.1), which is defined through the Duhamel formula

M t f (x) = f (x + t)e x+t x a(s)ds + t 0 e x+s x a(s ′ )ds ′ R M t-s f (y)Q(x + s, dy) ds. (2.5)
It is a standard result, see for instance [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF][START_REF] Bansaye | A non-conservative Harris ergodic theorem[END_REF] for more details on a similar model, that the Duhamel formula (2.5) indeed defines a positive semigroup

(M t ) t≥0 in B(1), but also in C b (R) and in C 1 b (R). When f ∈ C 1 b (R) it additionally holds that (t, x) → M t f (x) is continuously differentiable and satisfies ∂ t M t f = LM t f = M t Lf, where Lf (x) = f ′ (x) + a(x)f (x) + R f (y)Q(x, dy).
This right action of the semigroup provides the unique solutions to the dual equation

∂ t ϕ(t, x) = ∂ x ϕ(t, x) + a(x)ϕ(t, x) + R ϕ(t, y)Q(x, dy)
and thus the left action, defined by duality through (µM t )(f ) = µ(M t f ), yields the unique measure solutions to the direct equation (2.1). The well-posedness of the direct equation in L 1 (R, dx) ensures that if µ has a Lebesgue density, then so does µM t for all t ≥ 0.

For applying Theorem 1.1, we check that (M t ) t≥0 verifies Hypothesis 2. Equation (2.1) is a nice example to illustrate the conditions (H1') and (H2'), as we will explain now. The drift term, which represents the spatial shift of the space-time varying environment, is crucial for verifying (H1'). If we delete it, the situation is more involved: the result of Theorem 2.1 is still true when typically 1/(ā -a) is not locally integrable [START_REF] Alfaro | Confining integro-differential equations from evolutionary biology: ground states and long time dynamics[END_REF][START_REF] Bürger | Perturbations of positive semigroups and applications to population genetics[END_REF][START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Li | On eigenvalue problems arising from nonlocal diffusion models[END_REF], otherwise it may not hold due to the presence of a Dirac mass in γ, see [START_REF] Bürger | Stationary distributions under mutation-selection balance: structure and properties[END_REF][START_REF] Coville | Singular measure as principal eigenfunctions of some nonlocal operators[END_REF]. Assumption (2.3) is crucial for verifying (H2'). If we consider for instance the singular kernel Q(x, •) = δ x-1 + δ x+1 , then the convergence in Theorem 2.1 does not hold and a periodic asymptotic behavior takes place, similarly as in [START_REF] Bernard | Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts[END_REF][START_REF] Gabriel | Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation[END_REF].

Before proving Theorem 2.1, with start with a useful strong positivity result about M t . Lemma 2.2. Let x 1 , x 2 ∈ R with x 1 < x 2 . Then for any y 1 , y 2 ∈ R with y 1 < y 2 and any τ > 0, there exists η > 0 such that

M τ 1 [x1,x2] ≥ η1 [y1,y2] .
Proof of Lemma 2.2. Iterating once the Duhamel formula (2.5) we get for all t > 0

M t 1 [x1,x2] (x) ≥ e t inf (x,x+t) a t 0 R 1 [x1,x2] (y + t -s)Q(x + s, dy)ds ≥ κ 0 t e t inf (x,x+t) a x+t+ǫ x+t-ǫ 1 [x1,x2] (z)dz from which we deduce that M t 1 [x1,x2] (x) ≥ const > 0 for all x ∈ [x 1 -t -ǫ/2, x 2 -t + ǫ/2].
Letting n be a positive integer and considering t = τ /n we get that

M t 1 [x1,x2] (x) ≥ c 0 > 0 for all x ∈ [x 1 -τ /n -ǫ/2, x 2 -τ /n + ǫ/2],
and by iteration

M τ 1 [x1,x2] ≥ c n 0 1 [x1-τ -nǫ/2,x2-τ +nǫ/2]
. The conclusion follows by choosing n large enough.

Proof of Theorem 2.1. We proceed in two steps. First we check that on any bounded set K, Hypotheses (H1') and (H2') are satisfied for any τ > 0 (and any function ψ bounded from above and below by a positive constant over K). Then for Assumptions (A0)-(A1)-(A2), we use V = 1 and build a suitable function ψ such that the sublevel sets of V /ψ are bounded.

Step #1. Let K be a bounded set and τ ∈ (0, ǫ/2). We will build by induction two families (σ t,n

x,y ) and

(c t,n x,y ) indexed by n ∈ N, t ∈ [0, τ ], x ∈ R and y ∈ [x + t -nǫ/2, x + t + nǫ/2] such that: σ t,n
x,y is a probability measure on [0, t] which has a positive Lebesgue density s t,n x,y when n ≥ 1, c t,n

x,y is a positive constant, (x, y, t, s) → s t,n x,y (s) and (x, y, t) → c t,n x,y are continuous functions, and for all f ≥ 0

M t f (x) ≥ c t,n x,y t 0 M t-s f (y) σ t,n
x,y (ds).

(

The integer n essentially represents a number of jumps required for reaching y by starting from x. Once these families are built, we choose n large enough so that K ⊂ [x + τ -nǫ/2, x + τ + nǫ/2] for every x ∈ K. Positivity and continuity of (x, y, t, s) → s t,n x,y (s) and (x, y, t) → c t,n x,y then guarantee, together with Lemma 2.2, that Hypotheses (H1') and (H2') are satisfied with σ x,y = σ τ,n x,y and c = inf

x,y∈K c τ,n

x,y for any function ψ bounded from above and below by a positive constant on K. Clearly if (H1') and (H2') are satisfied for any τ ∈ (0, ǫ/2), they are also satisfied for any τ > 0. Let us now give the details of the induction.

For n = 0, y = x + t ≥ x and the Duhamel formula (2.5) ensures that for any f ≥ 0

M t f (x) ≥ f (y) e y x a(s)ds . (2.7)
This transcribes that y can be reached from x by only following the drift, without doing any jump, and it gives (2.6) with σ t,0 x,y = δ t and c t,0 x,y = e y x a(s)ds .

For n = 1, we start by noting that replacing f by M t ′ -(y-x) f in (2.7) gives

M t ′ f (x) ≥ e y x a(s)ds M t ′ -(y-x) f (y)
for any t ′ ≥ y -x ≥ 0. Using this inequality with t ′ = t -s and x = z together with Assumption (2.3) in the Duhamel formula (2.5) we get for all y ∈ [x + t -ǫ/2, x + t + ǫ/2], t ∈ [0, τ ], For n → n + 1, the induction hypothesis ensures that for any 0 ≤ s ≤ t ≤ τ and z ∈

M t f (x) ≥ t 0 e x+s x a(s ′ )ds ′ R M t-s f (z)Q(x + s, dz
[y -t + s -nǫ/2, y -t + s + nǫ/2] M t-s f (z) ≥ c t-s,n z,y t s M t-s ′ f (y) s t-s,n z,y (s ′ -s)ds ′ .
Injecting this inequality in the Duhamel formula (2.5)

c t-s ′ ,n z,y s t-s ′ ,n z,y (s -s ′ )dzds ′ ds > 0 and σ t,n+1
x,y (ds) = s t,n+1

x,y (s)ds with

s t,n+1 x,y (s) = κ 0 c t,n+1 x,y s 0 e x+s ′ x a min(x+s ′ +ǫ,y-t+s ′ +nǫ/2) max(x+s ′ -ǫ,y-t+s ′ -nǫ/2) c t-s ′ ,n z,y s t-s ′ ,n z,y (s -s ′ )dzds ′ > 0.
Step #2. Let us fix τ > 0. We use drift conditions, similarly as in [4, Section 2.2]. More precisely, we will prove that the functions

V = 1 and ψ 0 (x) = ((1 -(x/x 0 ) 2 ) + ) 2
satisfy LV ≤ α 0 V + θ 0 ψ 0 and Lψ 0 ≥ β 0 ψ 0 for some x 0 ≥ ǫ and some constants α 0 = β 0 -1 and θ 0 > 0. From these drift conditions, we get that the function φ = V -θ 0 ψ 0 verifies Lφ ≤ α 0 V + θ 0 ψ 0 -θ 0 β 0 ψ 0 = α 0 φ and, since ∂ t M t φ = M t Lφ, we deduce from Grönwall's lemma that M τ φ ≤ e α0τ φ, which gives

M τ V ≤ e α0τ (V -θ 0 ψ 0 ) + θ 0 M τ ψ 0 ≤ e α0τ V + θ 0 M τ ψ 0 .
We then prove the existence of ζ > 0 such that M τ ψ 0 ≤ ζV and we set ψ = ζ -1 M τ ψ 0 , so that ψ > 0 due to Lemma 2.2, ψ ≤ V , and

M τ V ≤ e α0τ V + θ 0 ζψ. (2.8)
Besides, Lψ 0 ≥ β 0 ψ 0 yields M 2τ ψ 0 ≥ e β0τ M τ ψ 0 , which also reads Finally we check that {V ≤ Rψ} is bounded, which ends the verification of Assumptions (A0)-(A1)-(A2). Now we show that Lψ 0 ≥ β 0 ψ 0 for some β 0 . We have Next, for V = 1, we have

M τ ψ ≥ e β0τ ψ. Choosing R > θ 0 ζ e α0τ (e τ -1) and K = {V ≤ Rψ} we obtain from (2.8) that M τ V ≤ e α0τ + θ 0 ζ R V + θ 0 ζ1 K ψ and
Lψ 0 (x) = -4 x x 2 0 (1 -(x/x 0 ) 2 ) + + a(x)ψ 0 (x) + x0 -x0 (
LV (x) = a(x) + Q(x, R) ≤ a(x) + Q.
Choosing r 0 > 0 such that a(x) < -Q + β 0 -1 := -Q + α 0 when |x| > r 0 we get LV (x) ≤ α 0 V (x) + (ā + Q)1 (-r0,r0) (x).

So if we choose x 0 ≥ √ 2r 0 in the definition of ψ 0 we get 1 (-r0,r0) ≤ 4ψ 0 and thus LV ≤ α 0 V + 4(ā + Q)ψ 0 := α 0 V + θ 0 ψ 0 .

Besides, since LV ≤ ā + Q, we have by Grönwall's lemma that

M τ ψ 0 ≤ M τ V ≤ e -(ā+ Q)τ V := ζV.

Conclusion.

We have checked Hypothesis 2 so the result is a direct application of Theorem 1.1, the regularity of γ following from the convergence in total variation norm and the invariance of L 1 under the semigroup, and the regularity of h being obtained by usual arguments from Equation (2.5).

κ 0 t 0 M

 0 t-s-(y-z) f (y)dz ds ≥ t-s ′ f (y) ′ -s a ds ′ ds > 0 and σ t,1x,y (ds) = s t,1x,y (s)ds with s t,1x,y (s) = ′ -s a ds ′ > 0.

  Assumptions (A1) and (A2) are verified with θ = θ 0 ζ and α = e α0τ + θ 0 ζ R < e (α0+1)τ = e β0τ = β.

1 - 1 - 1 - 1 ( 1 -y 2 ) 2 1- 4 x 0 ( 1 - 1 1-ǫ/ x0 ( 1 - 4 x 0 ( 1 -ǫ 3 15 x 2 0 1 (

 1111122011x01121 (y/x 0 ) 2 ) 2 Q(x, dy) (x/x 0 ) 2 ) + + inf (-x0,x0) a ψ 0 (x) + κ 0 x 0 (x-ǫ,x+ǫ) (x 0 y) dy ≥ (x/x 0 ) 2 ) + + inf (-x0,x0) a ψ 0 (x) + κ 0 1 (-x0,x0) (x)x 0 -y) 2 (1 + y) 2 dy ≥ (x/x 0 ) 2 ) + + inf (-x0,x0) a ψ 0 (x) + 8κ 0 -x0,x0) (x),since for r ∈ [0, 1] we have

1 1-r ( 1 -y) 2 ( 1 + y) 2 dy = r 0 z 2 ( 2 -z) 2 dz = r 3 4 3 -r + r 2 5 ≥ 8r 3 15 . If 2κ 0 ǫ 3 ≥

 112122323 15x 0 we get Lψ 0 ≥ inf (-x0,x0) a ψ 0 .Otherwise we split, for |x| ≥ 1 -(2κ 0 ǫ 3 )/(15x 0 ) x 0 ,Lψ 0 (x) ≥ inf (-x0,x0) a ψ 0 (x),and for |x| ≤ 1 -(2κ 0 ǫ 3 )/(15x 0 ) x 0 , we have ψ 0 (x) ≥ 2κ0ǫ 3 15x0 which yields ψ 0 (x) ≤ 15x0 2κ0ǫ 3 ψ 0 (x) and thusLψ 0 (x) ≥ -30 κ 0 ǫ 3 + inf (-x0,x0)At the end, in any case and for any x 0 ≥ ǫ,Lψ 0 ≥ -30 κ 0 ǫ 3 + inf (-x0,x0)a ψ 0 := β 0 ψ 0 .

Finally

  K = {V ≤ Rψ} is bounded because ψ tends to zero at ±∞: from the Duhamel formula (2.5) we haveM τ ψ 0 (x) ≤ ψ 0 (x + t)e ′ )ds ′ e (t-s)(ā+ Q) dswhich yields the result since lim x→±∞ a(x) = -∞.

Acknowledgments

The authors are very grateful to Vincent Bansaye and Aline Marguet for useful discussions about the results of the present note.

B.C. has received the support of the Chair "Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole Polytechnique-MnHn-FX and the ANR project MESA (ANR-18-CE40-006), funded by the French Ministry of Research.


G Rival

E Dantras
email: eric.dantras@univ-tlse3.fr

T Paulmier

Electronic irradiation ageing in the vicinity of glass transition temperature for PEEK space applications

Keywords: Irradiation, PEEK, Structure-Properties Relationship, Temperature, Thermoplastic Polymer

Satellites in space environment are subject to several environmental constraints. In the case of polymer materials, electronic irradiations can induce premature ageing which can lead to a shorter lifetime for the satellite. Moreover, synergistic effects between temperature and irradiation can occur and enhance this ageing. A previous study showed that PolyEtherEther-Ketone (PEEK), a newly used polymer in space industry, presents great radiation tolerance properties. Nonetheless, it was necessary to investigate the influence of temperature factor on its ageing. This study focuses on the comparison of PEEK samples aged at two temperatures: room temperature and 165 °C. Irradiations have been performed under high-vacuum and with a mono-energetic electron beam. They led to doses representative of 15 years of exposure to geostationary electronic environment. The comparison between both types of samples showed that increase in irradiation temperature have a significant impact on ageing kinetics. In particular, analyses revealed a higher cross-linking density for 165 °C irradiated samples which has been associated with a higher recombination rate of radicals due to amorphous phase mobility increase above glass transition.

Introduction

Polymers used in satellite manufacturing are exposed, in Geostationary Earth Orbit (GEO), to several environmental stresses like high-vacuum, UV and gamma radiations, thermal cycling or irradiation by high-energy particles (e.g. protons, electrons, . . . ). Thus, it is essential to understand how these different ageing factors will impact the polymer properties. In the case of electronic irradiations, two major issues can be pointed out. The first one is due to low-energy electrons which induce electrical charging of polymer surface and lead to electrostatic discharge phenomena responsible of many spacecraft failures [START_REF] Leach | Failures and Anomalies Attributed to Spacecraft Charging[END_REF]. The second one is due to high-energy electrons which depose energy in polymer volume and induce different phenomena like physico-chemical ageing [START_REF] Roggero | Dynamic glass transition of filled polysiloxane upon electron irradiation[END_REF][START_REF] Roggero | Inorganic fillers influence on the radiation-induced ageing of a space-used silicone elastomer[END_REF] or radiation-induced conductivity [START_REF] Hanna | Radiation induced conductivity in space dielectric materials[END_REF][START_REF] Paulmier | Charging Properties of Space Used Dielectric Materials[END_REF].

Moreover, polymers have to cope with temperature variations which can induce synergistic effects with electronic irradiations. For example, an increase in the polymer temperature enhances its electrical conductivity and therefore, limits the accumulation of charges due to low-energy electrons. But also, increasing temperature can also exacerbate the radiative ageing of a polymer and significantly reduce its lifetime.

In the literature, several studies report the influence of temperature on radiative ageing [START_REF] Sun | Modification of polytetrafluoroethylene by radiation-1. Improvement in high temperature properties and radiation stability[END_REF][START_REF] Li | Preparation of the crosslinked polyethersulfone films by hightemperature electron-beam irradiation[END_REF][START_REF] Ferry | Aliphatic/Aromatic Systems under Irradiation: Influence of the Irradiation Temperature and of the Molecular Organization[END_REF]. Kudoh et al. showed on different polymeric materials (epoxy-based GFRP, PMMA and PTFE) that irradiation temperature has an influence on ageing even by remaining below the glass transition temperature [START_REF] Kudoh | Low temperature gamma-ray irradiation effects of polymer materials on mechanical property[END_REF]. Indeed, a decrease in irradiation temperature delays the evolution of mechanical properties due to the lowest mobility of macromolecules. In their study, Mélot et al. followed, by Fourier Transform InfraRed spectroscopy, the ageing of aliphatic polymers irradiated at low-temperature (i.e. at 8 K) [START_REF] Mélot | Very low temperature irradiation of aliphatic polymers: Role of radical migration on the creation of stable groups (O-127)[END_REF]. Like Kudoh et al., they observed a significant influence of temperature on radiative ageing but studied as well the effect of annealing after irradiation at 8 K. As a result, they showed that, even after the end of irradiation, an increase in temperature can activate the macromolecule mobility and promote the creation of new chemical bonds due to remaining radicals. Moreover, the evolution of ageing with temperature can be very non-monotonous as showed by Oshima et al. for PTFE [START_REF] Oshima | Radiation induced crosslinking of polytetrafluoroethylene[END_REF][START_REF] Tabata | Temperature effects on radiation induced phenomena in polymers[END_REF]. They observed a chain-scission phenomenon when irradiations are performed below melting temperature, a cross-linking phenomenon just above the melting temperature and a depolymerisation phenomenon at higher temperature. However, in case of space applications, the authors did not find radiative-ageing studies which take into account the influence of the temperature factor.

This study focuses on a newly-used polymer in space industry: PolyEtherEtherKetone (PEEK).

In particular, this polymer is used for structural applications in the form of PEEK / Carbon fibre composites such as in the articulated arm of the International Space Station [START_REF] Lanouette | Residual mechanical properties of a carbon fibers/PEEK space robotic arm after simulated orbital debris impact[END_REF] or for mechanical support applications in scientific instruments such as the search-coil magnetometer (SCM) used in the Parker Solar Probe mission [START_REF] Bale | The FIELDS Instrument Suite for Solar Probe Plus: Measuring the Coronal Plasma and Magnetic Field, Plasma Waves and Turbulence, and Radio Signatures of Solar Transients[END_REF]. In a previous study [START_REF] Rival | Influence of electronic irradiations on the chemical and structural properties of PEEK for space applications[END_REF], PEEK has been exposed, under high-vacuum and at room temperature, to ionising electronic radiation in order to investigate its ageing behaviour for space applications. This study showed that electronic irradiations induce changes of both the amorphous phase (cross-linking) and the crystalline phase (amorphisation). In the literature, only one study tried to investigate the synergistic effects of thermal-ageing and radiative-ageing for PEEK in case of nuclear applications [START_REF] Yang | Aging of poly(ether ether ketone) by heat and gamma rays -Its degradation mechanism and effects on mechanical, dielectric and thermal properties[END_REF].

However, the study being focused on the comparison of thermal-ageing effect with or without irradiation, no real interpretation is made on the influence of temperature on the radiative ageing. Through this work, we propose to compare the ageing of PEEK at two irradiation temperatures chosen below and above the glass transition temperature: room temperature and 165 °C respectively. For this purpose, a heating sample-holder was developed to maintain samples at a given temperature during the irradiation. The resulting ageing has been analysed through different experimental approaches in order to probe changes at the macromolecular scale and link them to property modifications.

Materials and methods

Material

PEEK samples used in this study are semi-crystalline films with a thickness of 100 µm. They have been supplied by Victrex and correspond to the Aptiv 1000 grade.

Experimental simulation of electronic irradiation combined with high-temperature

In order to irradiate samples at a controlled temperature, a heating sample-holder has been developed. Figure 1 shows a schematic representation of the experimental set-up made for this purpose. supply. Irradiations being performed under high-vacuum, the use of halogen lamps seemed to be the best choice for heating the sample-holder. To measure and control the temperature of the sample-holder, a PT100 sensor was coated with thermal transfer grease and inserted in the copper plate. Finally, samples were placed on curved copper plates to maximise contact with the support and are maintained at the edge by screws. Moreover, a conductive thermal transfer grease was deposited between samples and copper support to optimise heat transfer.

The procedure of irradiations at room temperature and dose calculations have been already

described in a previous study [START_REF] Rival | Influence of electronic irradiations on the chemical and structural properties of PEEK for space applications[END_REF]. In the case of irradiations at 165 °C, irradiation conditions (apart from sample temperature) and procedure are the same. They are summarised thereafter.

Irradiations were carried out in the SIRENE facility (ONERA, Toulouse, France). Samples were exposed under high-vacuum to a mono-energetic electron beam of 350 keV. Irradiations were achieved in less than 60 h thanks to a high beam current density (up to 60 nA.cm -2 ). The resulting ionising doses D (Gy) were calculated using equation 1. All doses mentioned in this study correspond to the average dose over the thickness of samples.

D (Gy = J.kg -1 ) = 1.6 10 -9 × Φ 1 ρ d E d x (1)
Where Φ represents the total electronic fluence (electron/cm 2 ), ρ the material density (g.cm -3 )

and d E d x the electronic stopping power of the material (keV.µm -1 ).

1.2 × 10 7 Gy and 3.4 × 10 7 Gy doses have been deposited in each type of sample (i.e. irradiated at room temperature and 165 °C). In these conditions, the dose profiles are homogeneous over the sample thickness (the ratio between front and back face doses is about 1.5), allowing to limit uncertainties due to a dose gradient inside samples.

Differential Scanning Calorimetry

Differential Scanning Calorimetry (DSC) analyses were performed on a DSC 7 from Perkin Elmer. Analyses were carried out under nitrogen flow and involved two heating runs and one cooling run from 50 °C to 400 °C at a rate of 10 °C.min -1 . The temperature of first-order transitions (i.e. melting temperature T m and crystallisation temperature T c ) was taken at the maximum of the peaks and the glass transition temperature T g was determined by the tangent method as well as the heat capacity jump ∆C p . Crystallinity ratios χ c were calculated from melting enthalpy using equation 2.

χ c = ∆H m ∆H ∞ × 100 (2)
Where ∆H m is the sample melt enthalpy (J.g -1 ), ∆H ∞ is the theoretical melt enthalpy of a fully crystalline sample reported for PEEK at 130 J.g -1 [START_REF] Blundell | The morphology of poly(aryl-ether-ether-ketone)[END_REF].

Acetone absorption test

Absorption tests has already proven that they can be used to study the effect of ageing on polymer materials, especially in order to probe changes in the physico-chemical structure [START_REF] Rival | Influence of electronic irradiations on the chemical and structural properties of PEEK for space applications[END_REF].

In the case of PEEK, absorption tests were performed in acetone due to the high equilibrium weight gain in regard of water: Stober and Seferis found a weight gain of ≈7 % m with acetone [START_REF] Stober | Fluid sorption characterization of PEEK matrices and composites[END_REF] while Mensitieri et al. found a weight gain of ≈0.5 % m with water [START_REF] Mensitieri | Water sorption kinetics in poly(aryl ether ether ketone)[END_REF]. Moreover, Stober and Seferis also verified that acetone penetration does not induce sample crystallisation.

The experimental procedure for weighing has already been described in our previous study [START_REF] Rival | Influence of electronic irradiations on the chemical and structural properties of PEEK for space applications[END_REF]. We present hereafter the equation for weight gain calculations.

The weight gain (WG) is calculated by equation 3.

WG = m -m 0 m 0 × 100 (3)
Where m 0 is the initial dried mass of the sample.

Moreover, diffusion phenomena in polymer only take place in amorphous phase. Thus, the weight gain is normalised by the quantity of amorphous phase in order to consider a possible crystallinity ratio evolution. This normalised weight gain (NWG) is obtained by equation 4.

NWG = m -m 0 (1 -χ c 100) × m 0 × 100 (4)
With χ c the crystallinity ratio (%) obtained by DSC analyses.

Dynamic Mechanical Analysis

Dynamic Mechanical Analysis (DMA) tests were carried out on an ARES G2 strain-controlled rheometer from TA Instruments. Sample dimensions are 35 mm × 10.5 mm. As sample thickness does not allow a good signal-to-noise ratio in shear deformation geometry, they were analysed in tensile geometry mode. Each sample have been studied over the temperature range [-130 ; 250] °C at a rate of 3 °C.min -1 . Strain and frequency were fixed at 0.02 % and 1 Hz respectively allowing the determination of storage E ′ ω (T) and loss E ′′ ω (T) moduli. For samples irradiated at room temperature, the strain used in the initial study was 0.01 %. The damping factor tan δ is then calculated by the ratio of loss modulus E ′′ ω (T) over storage modulus E ′ ω (T).

Broadband Dielectric Spectroscopy

Broadband Dielectric Spectroscopy (BDS) is a common technique for studying dielectric behaviour of polymers and has already been extensively described [START_REF] Kremer | Broadband Dielectric Spectroscopy[END_REF].

Experiments were carried out on a Novocontrol BDS 4000 spectrometer associated with an Alpha-A impedance analyser. The sample diameter was 30 mm for pristine and room temperature irradiated samples, 20 mm for 165 °C irradiated samples. Analyses were performed over a frequency range of [10 -2 ; 10 6 ] Hz and for isotherms going from -150 °C to 250 °C by 5 °C steps. The complex impedance Z * (ω,T) is used to determine the values of complex permittivity ε * (ω,T) thanks to equation 5.

ε * = ε ′ -i ε ′′ = 1 i ωC 0 Z * (5)
With C 0 = ε 0 A e in which ε 0 is the vacuum permittivity, A is the area of the sample and e its thickness.

Havriliak-Negami equation was used to analyse dielectric relaxation phenomena observed on loss permittivity [START_REF] Havriliak | A complex plane analysis of α-dispersions in some polymer systems[END_REF][START_REF] Havriliak | A complex plane representation of dielectric and mechanical relaxation processes in some polymers[END_REF]:

ε * (ω) = ε ∞ + ε s -ε ∞ (1 + (i ωτ HN ) α HN ) β HN (6)
ε ∞ and ε s represent the instantaneous and the equilibrium permittivities respectively, α HN and β HN the Havriliak-Negami parameters which show the deviation from Debye's relaxation model (α HN is related to the peak width and β HN to its asymmetry at high-frequency) and τ HN (s) the Havriliak -Negami relaxation time.

Sub-glass relaxations were fitted with an Arrhenius law (equation 7).

τ(T) = τ 0,a exp ∆H RT (7)
Where τ 0,a (s) is the arrhenian pre-exponential factor, ∆H (J.mol -1 ) is the activation enthalpy of the relaxation and R the ideal gas constant.

Above glass transition, dielectric relaxations were fitted with a Vogel-Tammann-Fulcher (VTF) equation [START_REF] Vogel | The law of the relation between the viscosity of liquids and the temperature[END_REF][START_REF] Fulcher | Analysis of recent measurements of the viscosity of glasses[END_REF][START_REF] Tammann | Die Abhängigkeit der Viscosität von der Temperatur bie unterkühlten Flüssigkeiten[END_REF]:

τ(T) = τ 0,v exp 1 α f (T -T ∞ ) (8)
With τ 0,v (s) the vogelian pre-exponential factor, α f (°C -1 ) the free volume thermal expansion coefficient and T ∞ (°C) the free volume activation temperature.

Results and discussion

Impact of irradiation temperature on PEEK physical and chemical structures

To investigate the irradiation temperature influence on the physico-chemical structure of PEEK, DSC analyses were carried out. For purpose of clarity, DSC thermograms are not presented here. Instead, all the data extracted from these thermograms are reported in Table 1 and Figure 2 summarises the evolution of the different calorimetric parameters. Due to internal stresses which increase T g during the first run, the T g value presented in Figure 2 melting temperature [START_REF] Oshima | Radiation induced crosslinking of polytetrafluoroethylene[END_REF][START_REF] Tabata | Temperature effects on radiation induced phenomena in polymers[END_REF]. Nonetheless, looking at the evolutions of glass transition, melting and crystallisation temperatures, it is evident that 165 °C irradiations accelerate ageing kinetics.

• For the glass transition temperature increase, it has been previously associated with a chemical cross-linking phenomenon which occurs in the amorphous phase [START_REF] Rival | Influence of electronic irradiations on the chemical and structural properties of PEEK for space applications[END_REF]. With increasing ageing temperature, the faster T g increase is explained by a higher cross-link creation. Indeed, irradiations being performed above glass transition temperature, the higher mobility of macromolecules induces a higher recombination rate of radicals and therefore, a greater number of cross-links. By considering a three-phase model (crystalline phase, mobile amorphous phase and rigid amorphous phase), one can calculate the mobile amorphous fraction MAF from equation 9.

MAF = ∆C p ∆C 0 p (9)
Where ∆C 0 p is the heat capacity jump of a 100 % amorphous sample reported for PEEK at 0.27 J.g -1 .K -1 [START_REF] Cheng | Glass transition and melting behavior of poly(oxy-1,4phenyleneoxy-1,4-phenylenecarbonyl-1,4-phenylene) (PEEK)[END_REF]. Finally, the rigid amorphous fraction RAF is calculated by using equation 10.

RAF = 1 - χ c 100 -MAF (10)
By using the ∆C p value of the second heating run, the calculation of RAF gives a value of 8 % for the pristine sample, 18 % for the sample irradiated at room temperature (34 MGy), and 24 % for the sample irradiated at 165 °C (34 MGy). These results show that irradiation tends to increase the rigid amorphous fraction. The rigid amorphous phase is described as a highly entangled amorphous region where the macromolecules don't relax at the glass transition temperature. Thus, the increase in RAF can be explained by cross-links which play the role of chemical entanglements and immobilise a fraction of amorphous phase. Moreover, 165 °C irradiation induces a higher increase of the RAF which further confirms the higher crosslinking density.

The influence of this higher density of cross-linking nodes can also be seen through the sample crystallisation; i.e. T c decreases faster for high-temperature irradiated samples. The higher cross-linking density further limits the crystallisability of macromolecules.

• Concerning the crystalline phase evolution, either in room temperature or in 165 °C irradiated samples, T m and χ c decreases are due to the creation of defects in crystallites which can be explained by a decrease in their size. This is consistent with the study of Yoda who observed by X-Ray Diffraction a 15 % decrease in the size of PEEK crystallites after electronic irradiation [START_REF] Yoda | The Crystallite Size and Lattice-Distortions in the Chain Direction of Irradiated Poly(aryl-Ether-Ketone)[END_REF]. However, after 165 °C irradiations, only the decrease in T m is accelerated.

It means that the increase in irradiation temperature leads to less stable crystalline entities but does not affect the crystallinity ratio. As irradiations being performed below the melting temperature, the surface of crystallites (i.e. at the amorphous / crystal interface) is probably mostly impacted due to the increase in amorphous phase mobility. Thus, the higher decrease in crystallite stability can be explain by a higher creation of defects mostly at their surface.

However, the decrease in χ c remains the same. During the second heating run, T m value at a same dose remains lower for high-temperature irradiations. The samples having already melted once, these evolutions have been associated with chemical modifications. Thus, it can be associated with the higher cross-linking density which limits the crystallisation of macromolecules and leads to smaller crystallites. were the first to observe this phenomenon [START_REF] Bagley | Two-stage Sorption and Desorption of Organic Vapors in Cellulose Acetate[END_REF]. This two-stage mechanism is described as follow: the first part corresponds to a quasi-equilibrium which is reached at the surface of the sample and the second part to the diffusion of the penetrant through the polymer sheet which is assumed to be allowed by the rearrangement of macromolecules in the amorphous phase, equilibrium (diffusion and retention processes taking place in amorphous phase), it has been shown that cross-linking has opposite effects [START_REF] Perrin | Sorption and Diffusion of Solvent Vapours in Poly(vinylalcohol) Membranes of Different Crystallinity Degrees[END_REF][START_REF] Chou | Water transport in crosslinked 2-hydroxyethyl methacrylate[END_REF]. As already discussed in our previous study [START_REF] Rival | Influence of electronic irradiations on the chemical and structural properties of PEEK for space applications[END_REF], the absorption behaviour of room temperature irradiated sample reveals a greater diffusion kinetics in regard to pristine sample. Therefore, the data suggest that amorphisation effects prevail over cross-linking phenomena after room temperature irradiations. However, DSC analyses presented before showed that cross-linking density is higher after irradiation at 165 °C while crystallinity ratio evolution remains similar for both irradiation conditions. Thus, the decrease in absorption kinetics and in equilibrium weight gain can be associated with the fact that cross-linking effects have become predominant over amorphisation effects. Two mechanical relaxations can be pointed out. At low temperature, the γ relaxation is associated with localised mobility of polar groups. Studies have concluded that this relaxation is allowed by the presence of water molecules absorbed in the polymer [START_REF] Harris | Miscible blends of poly(aryl ether ketone)s and polyetherimides[END_REF][START_REF] Coulson | Dynamic rheological behavior of poly(ether ketone ketone) from solid state to melt state[END_REF]. At higher temperature, the relaxation named α is associated with the mechanical manifestation of the glass transition. Mechanically, it corresponds to the transition from a glassy state to a rubbery state. In our previous study [START_REF] Rival | Influence of electronic irradiations on the chemical and structural properties of PEEK for space applications[END_REF], mechanical analyses showed that γ relaxation is not affected by the ageing. The cross-linking density was too low to have an effect on localised mobility. Moreover, even after irradiations at 165 °C, no evolution of this localised mobility can be observed. It indicates that the cross-linking density is not enough to modify γ mobility. NMR- 13 C analyses were performed on pristine and 165 °C irradiated samples. No change has been observed between both NMR spectra. This result confirms the hypothesis of a low-density cross-linking.

Cross-linking density influence on absorption behaviour

Evolution of mechanical behaviour

In contrast, the α relaxation is significantly modified, especially after irradiations at 165 °C.

Firstly, the faster increase in T α for high-temperature irradiated samples is the mechanical counterpart of the faster increase in T g observed by DSC analyses.

In addition, the rubber modulus noted ∆E' 220 °C (E' irradiated -E' pristine at 220 °C) and α relaxation normalised area are reported in Figure 5 as a function of dose. For both irradiation temperatures, the E rubber value increase is directly related with the crosslink creation in the amorphous phase: cross-linking nodes limit the mobility of macromolecules and therefore, increase E rubber . However, it is evident that irradiations at 165 °C induce a faster increase in E rubber , despite a higher uncertainty. The equation 11 allows us to estimate an entanglement density from the rubber modulus value [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF][START_REF] Doi | The theory of polymer dynamics[END_REF].

1 M e = 1 g ′ 0 E rubber ρRT (11)
Where 1 M e is the density of entanglements (mol.g -1 ), g ′ 0 is a factor equal to 3, E rubber is the storage rubber modulus (Pa), ρ the density of PEEK (g.m -3 ), R the ideal gas constant (J.K -1 .mol -1 ) and T the temperature (K).

Thus, the evolution of entanglement density between 34 MGy irradiated samples and pristine sample give an increase of 4.6 × 10 -3 mol.g -1 at room temperature and 7.3 × 10 -3 mol.g -1 at 165 °C respectively. Since irradiations do not modify the ease entanglement density, this increase can only be due to cross-link density. Moreover, it shows that this increase is greater after 165 °C irradiations which confirms the hypothesis of a higher ageing kinetics with hightemperature irradiations. Nonetheless, these values need to be handled carefully and only give an estimation of the evolution of the number of entanglements / cross-links.

Finally, the normalised area, which is related to the energy dissipated by α relaxation, shows an opposite evolution between both irradiation temperatures. While a significant increase is evident after room temperature irradiations, a slight decrease is observed for samples irradiated at 165 °C. For room temperature irradiated samples, the increase in α normalised area can be linked with amorphous phase quantity increase as observed by DSC (i.e. decrease in χ c ). Indeed, cross-linking does not rigidify sufficiently the system in front of the mobility improve granted by the increase in amorphous quantity. For 165 °C irradiated samples, the greater cross-linking density counterbalances the effect of amorphisation and therefore, induces a decrease in energy dissipation by stiffening the amorphous phase. These conclusions are consistent with those made by studying acetone diffusion in samples and once again, show the importance of the competition between cross-linking and amorphisation.

Study of dielectric relaxations

The dielectric loss permittivity ε ′′ of a pristine sample is presented in Figure 6 as a function of frequency and temperature.

Three dielectric phenomena can be observed. The γ and α relaxations are consistent with relaxations observed by mechanical analyses. In addition, at low frequency and high temperature, a third dielectric phenomenon can be guessed but it is hidden by the predominance of conductivity. This last phenomenon has been associated with a Maxwell-Wagner-Sillars (MWS) polarisation [START_REF] Perrier | Maxwell-Wagner-Sillars relaxations and crystallinity in PEEK[END_REF]. It corresponds to the interfacial polarisation of crystallite induced by the difference of conductivity between crystalline and amorphous phases. In order to fit this last phenomenon with equation 6, it was necessary to remove the conductivity contribution by recalculating the loss permittivity from the derivative of storage permittivity [START_REF] Wübbenhorst | Analysis of complex dielectric spectra. I. One-dimensional derivative techniques and three-dimensional modelling[END_REF].

For each of these relaxation phenomena, equation 6 was used to extract the associated relaxation times. They are plotted as a function of 1000/T for each dose and irradiation temperature in the Figure 7. As expected, the sub-glass relaxation γ follows an Arrhenius behaviour while the α relaxation can be fitted with a VTF behaviour law. Curves resulting from these fits are represented in dashed lines and fit parameters are reported in Table 2. For the α relaxation, the fragility index m has been calculated with the following equation [START_REF] Böhmer | Nonexponential relaxations in strong and fragile glass formers[END_REF]: [START_REF] Tabata | Temperature effects on radiation induced phenomena in polymers[END_REF] In this equation, the T g value for the second heating run is used in order to prevent the effect of internal stresses.

m = d log(τ) d (T g T) T=Tg
For the γ relaxation, the value of 49 kJ.mol -1 for ∆H is consistent with the literature [START_REF] Mourgues-Martin | Dielectric relaxation phenomena in PEEK[END_REF][START_REF] Audoit | Thermal, mechanical and dielectric behaviour of poly(aryl ether ketone) with low melting temperature[END_REF].

For the α relaxation, as predicted by the VTF theory, the value of T ∞ ≃ T g -50 °C confirms its attribution to the dielectric manifestation of the glass transition. The fragility index values reported in the literature lie between 150 and 280, depending on the analysis method and on the material state [START_REF] Al Lafi | Structural development in ion-irradiated poly(ether ether ketone) as studied by dielectric relaxation spectroscopy[END_REF][START_REF] Govinna | Melt-electrospinning of poly(ether ether ketone) fibers to avoid sulfonation[END_REF]. In this study, a value of 127 was calculated for pristine PEEK which is in the same order of magnitude as literature values.

Finally, the MWS relaxation times can be fitted by a VTF behaviour law. It indicates that this dielectric process is governed by the amorphous phase mobility. This relaxation is not affected at all by the sample ageing, whether after room temperature irradiations or 165 °C irradiations. Like for mechanical analysis results, it indicates that the cross-linking of the amorphous phase is not sufficiently dense to modify the localised mobility response.

α relaxation

For this relaxation, Figure 7 shows a shift of characteristic times to higher temperatures after irradiation. Like for thermal and mechanical analyses, it indicates an increase in the thermal activation of glass transition. It is due to the cross-linking of amorphous phase which limits the macromolecule mobility. This can be view as well from the point of view of relaxation times: after irradiation, the α relaxation is shifted to longer times which is consistent with a slower relaxation kinetic. Between room temperature and 165 °C irradiations, a greater shift is observed with increasing irradiation temperature which further confirms the higher density of cross-links induced by irradiation above glass transition. This α relaxation dynamic slowdown with an increasing cross-link density has already been observed for different polymer systems [START_REF] Glatz-Reichenbach | Influence of Crosslinking on the Molecular Relaxation of an Amorphous Copolymer Near Its Glass-Transition Temperature[END_REF][START_REF] Kramarenko | Influence of cross-linking on the segmental dynamics in model polymer networks[END_REF]. Moreover, Kramarenko et al. observed an increase in fragility index with crosslink density [START_REF] Kramarenko | Influence of cross-linking on the segmental dynamics in model polymer networks[END_REF]. In this work, fragility index increases as well with ionising dose. However, room temperature irradiated samples show a moderate increase in contrast to 165 °C irradiated samples. Therefore, this higher increase in m is consistent with a higher cross-linking density.

This shift to higher temperatures can also be noticed through the evolution of T ∞ which is represented, as well as the evolution of α f , in Figure 8 versus ionising dose. Indeed, an increase in T ∞ value is observed with an increasing dose which is consistent with the increase in T g observed by DSC and in T α by DMA. These data also confirm the faster ageing induced by hightemperature irradiations with a faster increase in T ∞ . For the α f parameter, opposite evolutions are observed between both irradiation temperatures.

Like for the evolutions of diffusion kinetics and α relaxation area, these opposite tendencies can be link to the competition between amorphisation and cross-linking. For room temper-ature irradiated samples, the slight increase in α f is linked to the predominant influence of amorphisation. Since this parameter is associated with the free volume expansion coefficient, less crystallites induce less stresses on the amorphous phase and therefore lead to a higher expansion coefficient. Nonetheless, this effect is probably partially balanced by the crosslinking of amorphous phase which limits its mobility and consequently its possible expansion.

In 165 °C irradiated samples, DSC analyses showed an equivalent amorphisation of the polymer while the density of cross-linking increases faster with ionising dose. Thus, it is assumed that the cross-linking influence becomes predominant in front of amorphisation and induces a α f decrease.

Finally, the study of the α relaxation loss permittivity peak width can also provide information about the environment close to relaxing entities [START_REF] Roggero | Dynamic glass transition of filled polysiloxane upon electron irradiation[END_REF]. In this way, the loss permittivity peaks have been recalculated for each type of samples using the imaginary part of equation 6, which is developed in [START_REF] Havriliak | A complex plane analysis of α-dispersions in some polymer systems[END_REF], and the Havriliak-Negami parameters obtained by the fit of α relaxation.

This method allows us to eliminate the conductivity in order to better discern any change. The data have been normalised on position and amplitude to compare the width evolution at one isotherm and for different samples. The resulting curves, calculated at 180 °C, are represented in Figure 9 as well as the value of α HN parameter which governs the peak width.

The graph shows a broadening of the α relaxation with increasing dose, either after room temperature or 165 °C irradiations. These evolutions are confirmed by the decrease in α HN observed on the histogram. In the dielectric relaxation theory, this broadening is associated with an increase in the relaxation time distribution. In the case of the α relaxation, this wider distribution can be linked to a more heterogeneous environment in the vicinity of relaxing entities, i.e. a more heterogeneous amorphous phase. This heterogeneity increase is associated with the creation of cross-linking nodes in this phase. This figure also shows a greater heterogeneity increase for 165 °C than for room temperature irradiated samples which is the consequence of a higher cross-linking density as assumed above.

MWS relaxation

Figure 7 shows no evolution of MWS relaxation times after irradiation at room temperature but a shift to higher temperature (longer relaxation times) once irradiated at 165 °C. It is known that the relaxation times of a MWS process is proportional to the storage permittivity ε ′ of the two phases (i.e. crystalline and amorphous) and inversely proportional to their electrical conductivity [20, p. 498]. The ε ′ value is dependent from the dipole density which is not modified by irradiation. Thus, the relaxation time shift can be associated with a conductivity decrease of one of them. As crystal conductivity remains stable after irradiation, this shift to longer relaxation times can indicate that irradiation at 165 °C decreases the amorphous phase conductivity. This conductivity evolution can be associated with the increase in defects (cross-links) which limits the mobility of electrical charges in the amorphous phase. For room temperature irradiated samples, this decrease is probably also present but less significant due to the lower cross-linking density. Moreover, it can be assumed that the increase in MWS relaxation times is compensated by the decrease in crystallite size, i.e. crystallites being smaller, their polarisation occurs at higher frequencies / shorter times.

Like for α relaxation, α HN parameter can give some information on the homogeneity in the vicinity of relaxing entities. Figure 10 presents the value of this parameter at 230 °C for each sample. Crystallinity ratios obtained by DSC analyses are reported as well on the same figure. For room temperature irradiated samples, the value of α HN decreases with increasing ionising dose as well as the crystallinity ratio. It can be assumed that by decreasing crystallite size (associated with the decrease in T m ), irradiations also increase their size distribution and lead to a wider distribution of relaxation times. After irradiations at 165 °C, 12 MGy irradiated sample follows the same behaviour with a α HN and χ c decrease. However, for 34 MGy, the decrease in homogeneity (i.e. in α HN ) is significantly lower compared to the decrease in crystallinity ratio.

It has been shown in previous studies that polymers of PolyArylEtherKetone family are subject to secondary crystallisation phenomenon; this phenomenon occurs at a temperature between glass transition and melting thanks to the higher mobility of amorphous phase [START_REF] Marand | On the observation of a new morphology in poly(arylene ether ether ketone). A further examination of the double endothermic behavior of poly(arylene ether ether ketone)[END_REF][START_REF] Quiroga Cortés | Morphology and dynamical mechanical properties of poly ether ketone ketone (PEKK) with meta phenyl links[END_REF]. Thus, it can be assumed that this phenomenon promotes a reorganisation at the edge of crystallites during irradiation and leads to a crystalline phase homogenisation.

Conclusion

The influence of irradiation temperature on radiative ageing of PolyEtherEtherKetone by electronic irradiations has been studied. For this purpose, two irradiation campaigns have been carried out in the SIRENE facility installed at the ONERA: the first was performed at room temperature and the second at 165 °C, just above the PEEK glass transition temperature.

For each irradiation temperature, two average ionising doses were deposited: 1.2 × 10 7 Gy and 3.4 × 10 7 Gy. Thanks to the high energy of electrons (350 keV), these doses are homogeneous over sample thickness.

DSC analyses showed a real influence of the temperature on the ageing kinetics. In the amorphous phase, a faster increase in T g with dose was observed. It was associated with a higher cross-linking density due to a faster recombination rate of radicals above glass transition. In the crystalline phase, the increase in irradiation temperature only affects the evolution of T m while the amorphisation of samples remains similar. It was associated with the fact that at 165 °C, only the mobility of amorphous phase is impacted.

These assumptions on the influence of irradiation temperature on the physico-chemical ageing were then studied and corroborated by using different probes: molecular (i.e. acetone diffusion), mechanical and dielectric probes.

• The higher cross-linking density in 165 °C irradiated samples has been confirmed by slower diffusion kinetics, a higher increase in rubber modulus and a shift of α relaxation times to higher temperatures. Moreover, the study of the α relaxation peak width showed that 165 °C irradiations induce a more heterogeneous medium in the vicinity of relaxing entities associated with the higher cross-linking density. Nonetheless, no evolution of γ relaxation times was observed and suggests that cross-linking density remains too low to modify localised macromolecules dynamics.

• The crystalline phase ageing was studied through the MWS relaxation. The comparison of crystallinity ratio with α HN values showed that room temperature irradiations increase the heterogeneity of crystalline phase proportionally to the decrease in crystallinity ratio.

Nonetheless, maintaining samples above glass transition during 165 °C irradiations looks to homogenise crystalline phase.

• Finally, it has been shown that the competition between amorphisation and cross-linking density has a significant influence on the polymer behaviour after irradiations.

Figure 1 :

 1 Figure 1: Schematic representation of the heating sample-holder used for irradiations at 165 °C

  are extracted from the second runs. This figure shows that room temperature irradiations and 165 °C irradiations induce similar evolutions of the PEEK physico-chemical structure. It means that the irradiation temperature does not affect the ageing mechanism under electronic irradiations. It is consistent with Oshima et al. studies on PTFE for which the degradation mechanisms only change above

Figure 2 :

 2 Figure 2: Calorimetric parameters extracted from DSC analyses as a function of dose. Full symbols represent room temperature irradiated samples while empty symbols represent 165 °C irradiated samples.

Figure 3

 3 Figure3shows the normalised weight gain curves for PEEK versus time square root. To emphasise the influence of irradiation temperature on the acetone absorption behaviour, only samples irradiated at 34 MGy have been studied with this method.

  i.e. by breaking interchain physical links. However, in this case, for which the second stage is faster than the first one, none of the previous studies provided an equation allowing to model this behaviour. For example, the study of Loh et al.[START_REF] Loh | Modelling anomalous moisture uptake, swelling and thermal characteristics of a rubber toughened epoxy adhesive[END_REF] suggests a model consisting in the sum of two fickian diffusion phenomena but can't fit correctly our data.Data show that the two studied irradiation temperatures have an opposite influence on the acetone diffusion. It can be explained by the opposite influence of amorphisation and crosslinking. While amorphisation accelerates diffusion kinetics and increases weight gain at

Figure 3 :

 3 Figure 3: Weight gain data as a function of time square root for pristine PEEK and for samples irradiated at room temperature and 165 °C.

Figure 4

 4 Figure 4 presents the damping factor thermograms of samples irradiated at room temperature (left graph) and 165 °C (right graph).

Figure 4 :

 4 Figure 4: Damping factor thermograms of samples irradiated at room temperature and at 165 °C.

Figure 5 :

 5 Figure 5: Rubber modulus and α relaxation normalised area as a function of dose. Full symbols represent room temperature irradiated samples while empty symbols represent 165 °C irradiated samples.

Figure 6 :

 6 Figure 6: 3D cartography of dielectric loss permittivity ε" for the pristine sample. Arrows are intended to highlight dielectric relaxation modes.

Figure 7 :

 7 Figure 7: Arrhenius diagram of relaxation times associated with dielectric phenomena observed in PEEK samples. Full symbols represent room temperature irradiated samples while empty symbols represent 165 °C irradiated samples.

Figure 8 :

 8 Figure 8: VTF parameters α f and T ∞ as a function of dose for room temperature irradiated samples (full symbols) and 165 °C irradiated samples (empty symbols).

Figure 9 :

 9 Figure 9: Normalised loss permittivity peak for α relaxation at a temperature of 180 °C (left graphs) and value of α HN parameter at the same temperature (right graph).

Figure 10 :

 10 Figure 10: Value of α HN parameter at 230 °C (bars) obtained for MWS mode and crystallinity ratio χ c (symbols) obtained from DSC analyses.

Table 1 :

 1 Data extracted from DSC analyses of samples irradiated at Room Temperature (RT) and irradiated at 165 °C. 1 st and 2 nd refers to first and second heating runs respectively.

		T g		∆C p		T m		χ c		T c
		(°C)		(J.g -1 .°C -1 )	(°C)		(%)		(°C)
		1 st	2 nd	1 st	2 nd	1 st	2 nd	1 st	2 nd
	Pristine	158	147	0.15	0.14	339	339	34.1	40.1	297
	RT Irradiation
	Dose = 12 MGy	157	150	0.11	0.12	331	333	32.6	33.7	286
	Dose = 34 MGy	159	153	0.16	0.16	318	320	25.8	23.0	269
	165 °C Irradiation
	Dose = 12 MGy	158	151	0.13	0.12	327	328	30.9	30.6	280
	Dose = 34 MGy	163	156	0.12	0.14	309	312	26.3	24.3	255

Table 2 :

 2 Fit parameters obtained for dipolar relaxations and fragility indexes m of pristine, room temperature irradiated and 165°C irradiated PEEK samples.

		γ relaxation		α relaxation
		τ 0,a	∆H	τ 0,v	α f	T ∞	m
		(s)	(kJ.mol -1 )	(s)	(°C -1 )	(°C)
	Pristine	1.1 × 10 -15	49.0	4.2 × 10 -17	4.42 × 10 -4	90	127
	RT Irradiation
	Dose = 12 MGy	1.9 × 10 -15	48.0	4.5 × 10 -17	4.50 × 10 -4	94	128
	Dose = 34 MGy	9.6 × 10 -16	49.6	6.2 × 10 -17	4.55 × 10 -4	98	133
	165 °C Irradiation
	Dose = 12 MGy	3.3 × 10 -15	46.5	6.3 × 10 -18	4.12 × 10 -4	99	165
	Dose = 34 MGy	1.4 × 10 -15	48.6	4.8 × 10 -18	3.69 × 10 -4	102	172
	γ relaxation

Acknowledgement

The authors would like to thank the CNES for technical support (funding of the SIRENE facility) and the Région Occitanie for financial support in this project.


Dr Servane Gey
email: servane.gey@parisdescartes.fr

This paper gives a very complete overview of the state of the art on minimal penalties and slope heuristics developed these past 20 years. Sylvain Arlot has to be congratulated for providing a survey which clearly relies theory and heuristics, and which proposes several ways to proceed heuristics in practice. My comments will focus on binary classification with the 0-1 loss, for which, as mentioned several times by Sylvain Arlot, there is still no clear results on slope heuristics. We propose to illustrate the behavior of two slope heuristics by an empirical simulation study, and on the benchmark well-known spam data set. We focus on classification trees model selection with the 0-1 loss, and we use the CART algorithm proposed by Breiman et al. in 1984 [3] to build the models. The results of the slope heuristics are compared with the ones obtained by the classical 10-fold cross-validation via the prediction errors of the selected classifiers, and the dimensions of the corresponding classification tree models. Some final comments are given at the end of the discussion.

CART and Slope Heurisitics

The pruning procedure used in CART is based on a penalized empirical misclassification rate criterion, with a penalty proportional to the classification trees' number of leaves. It allows to reduce drastically the collection of candidate tree classifiers by providing a collection of nested models m K . . . m 1 , associated with an increasing sequence of complexity parameters C 1 < • • • < C K , with C 1 = 0 and D m K = 1. Here the dimension D m of a classification tree model m is the number of its leaves. Gey et al. [START_REF] Gey | Risk bounds for cart classifiers under a margin condition[END_REF][START_REF] Gey | Risk bounds for embedded variable selection in classification trees[END_REF] obtained oracle-type inequalities with the 0-1 loss under the strong margin assumption (denoted by SMA in the following) proposed by Massart and Nédélec [START_REF] Massart | Risk bounds for statistical learning[END_REF] (see also the paper by Bartlett et al. [START_REF] Bartlett | Convexity, classification, and risk bounds[END_REF]), which is a particular case of the margin assumptions proposed by Mammen and Tsybakov [START_REF] Mammen | Smooth discrimination analysis[END_REF], and later generalized by Koltchinskii [START_REF] Koltchinskii | Local Rademacher complexities and oracle inequalities in risk minimization[END_REF]. These results show that the penalty term to use in classification tree model selection is of the form (C/h)(D m /n), where C is a large enough unknown constant, and h ∈ (0, 1] is the unknown margin parameter of the n observations' common distribution. Hence each complexity parameter ( C i ) 1 i K is a data-driven calibration of the unknown penalty constant (C/h). Let us mention that subadditive penalties can be used in the slope heuristics to select a final tree after pruning (see the paper by Scott [START_REF] Scott | Tree pruning with subadditive penalties[END_REF]). To be able to select automatically a model via the slope heuristics, we use heuristics proposed by Bar-Hen, Gey and Poggi [START_REF] Bar-Hen | Spatial cart classification trees[END_REF], well-adapted to CART classification trees: a first tree model mjump is selected via a modified version of Algorithm 5 with pen 0 = pen 1 (see Appendix); a second tree model mplateau is selected by taking plateau = argmax i C i+1 -C i , corresponding to the elbow selection. A typical example of the behavior of C i → D m i is given in Figure 1.

Simulation study

The simulation designs are described in Table 1. Since all the covariates have symmetrical roles, only designs with 2 covariates are considered. For each, a representation of one sample's realization of size n = 1000 is given in Figure 2, with fixed global or local margin parameter h = 0.9. The represented partitions are the ones of the true underlying conditional distribution. Checkerbord and Crux designs are easy for CART: the Bayes classifiers are trees, and the SMA is fulfilled. The two other ones are difficult for CART: in the Line design, the SMA is fulfilled, but the Bayes classifier's model is very difficult to approximate by trees; in the Square design, the Bayes classifier is a tree, but the SMA is not fulfilled. The CART models are built using the R packages rpart for 10-fold cross validation (denoted by CV), and tree for the two slope heuristics (denoted by Jump and Plateau respectively). For each design, the methods are compared with respect to the value of the global or local margin parameter h: for each value h ∈ [0, 1] on a regular grid, the average CART tree classifiers' risks and model's dimensions over 400 samples of size n = 1000 are computed. The risk of a tree is computed as its expected misclassification rate (estimated on a large test sample of size 2000), minus the Bayes error. Designs' Bayes errors can be found in Table 1.

The results are represented in Figure 3 for the easy designs, and in Figure 4 for the difficult ones. The calibration parameter α for the Jump heuristic is taken as α = 10% for the Checkerboard, Crux and Square designs, and α = 5% for the Line design. The black lines on the dimensions' graphs represent the dimension to take under the true underlying observations' distribution if it were known. When h is close to 0, the risk of every selected model is close to 0 since the underlying true labels' distribution depends slightly on the covariates' one, and is Bernoulli with parameter close to 1/2. Nevertheless, one can see that the slope heuristics select more intuitive low dimensional models, while CV selects much larger dimensional ones to better separate labels. As soon as h 0.75, the three methods give similar results on the easy designs, and recover the true models, what is encouraging for the use of slope heuristics for classification trees. Let us just mention that there are border artefacts on the regions' limits, leading CV to select slightly larger dimensional trees. These artefacts seem to be automatically compensated by the slope heuristics, at a small cost on the risk. One can also see that all risks present a maximum. The shapes' decreasing part confirms the penalty term's dependency on 1/h as soon as h is sufficiently large. The shapes' increasing part might correspond to margin values for which the penalty term is not a linear function of the dimension (see [START_REF] Massart | Risk bounds for statistical learning[END_REF][START_REF] Gey | Risk bounds for embedded variable selection in classification trees[END_REF]). When there is an approximation bias (see Line on Figure 4), the slope heuristics differ from CV by choosing smaller dimensional models rather than decreasing bias. CV and Jump heuristic's risks stabilize with increasing margin parameter, while Plateau heuristic's risk keeps on increasing, what is characteristic of overpenalization. For the Square design (see Square on Figure 4), the risks behave the same way as for the easy designs, but the models' dimension differ largely as soon as h 0.25: the risks are almost equal, but CV selects high dimensional models to cope with the region where the SMA is not fulfilled, while the slope heuristics "forget" it and recover the true 2-dimensional intuitive model.

Application on the spam data set

The spam data set consists of information from 4601 email messages, in a study to screen email for "spam" (i.e. junk email). The data are presented in details in [6, p. 301]. The response has values nonspam or spam, and there are 57 covariates relative to specific words and characters indicators in the email. This data set can be found in the kernlab R package, or on the UCI Machine Learning data base: https://archive.ics.uci.edu/ml/datasets/spambase. The three methods described in Section 2 are applied on the spam data set, with a calibration parameter α = 1% for the Jump heuristic. The function D m i → C i is represented in Figure 5, and the CART trees computed on the whole data set are represented in Figure 6. Let us mention that the probability for an email to be a spam is estimated far from 0.5 in every leaf of the corresponding probability tree, what might indicate that the SMA is fulfilled for these data. The methods are compared through Monte-Carlo average prediction errors and models' dimensions computed on 400 random drawing of a learning set to build the trees, and a test set representing 10% of the data to estimate the prediction error. The results are presented in Table 2: CV and the Jump heuristic select almost always the same 7-dimensional tree, while the Plateau heuristic always selects the 2-dimensional tree. The prediction errors indicate that the Plateau heuristic overpenalizes too much compared to the two other methods, which have comparable and much better performance on the data.

Comments

The empirical results obtained for classification trees are very encouraging for the use of slope heuristics for classification with the 0-1 loss. They show that, even if the slope heuristics do not perform as well as cross-validation, they seem to adapt better to the strong margin assumption. It has to be noticed that one drawback of the Plateau elbow heuristic seems to be overpenalization. Further investigations should be made with models more regular than classification trees, and with model selection methods allowing to visit all models' dimensions. Also, as for classification with the quadratic loss, it could be interesting to investigate theory under margin assumptions to have a better handle of the slope heuristics' performance with respect to margin parameters.

Figures

  Figures

Figure 1 :

 1 Figure 1: Typical example of the behavior of dimension (Dm i ) 1 i K with respect to complexity parameter ( Ci) 1 i K for CART classification trees.represents the tree mjump, while represents the tree mplateau .

Figure 2 :

 2 Figure 2: Examples of one sample's realization of size n = 1000 for each design, with common global or local margin parameter h = 0.9. From left to right, and up to bottom: Checkerboard, Crux, Line, Square.

Figure 3 :

 3 Figure 3: Average risk (left) and model's dimension (right) obtained by CV, Jump and Plateau heuristics over 400 simulations of the Checkerboard (up) and Crux (bottom) simulated data sets. Black line: dimension of the Bayes classifier's model.

Figure 4 :

 4 Figure 4: Average risk (left) and model's dimension (right) obtained by CV, Jump and Plateau heuristics over 400 simulations of the Line (up) and Square (bottom) simulated data sets. Black line: dimension to take if the true underlying observations' common distribution were known for the Square design.

Figure 5 :

 5 Figure 5: Dimension (Dm i ) 1 i K with respect to complexity parameter ( Ci) 1 i K for the spam data set.represents the tree mjump, while represents the tree mplateau .

Figure 6 :

 6 Figure 6: Classification trees for the spam data set. Left: tree selected via CV and Jump heuristic; Right: tree selected via Plateau heuristic.

Discussion on "Minimal penalties and the slope heuristics: a survey" by Sylvain Arlot Appendix mjump selection for CART classification trees The pruning step of the CART algorithm does not visit all possible configurations, nor all possible dimensions. Hence, if there exist negligible plateaus in the graph of C m i → D m i , then the corresponding dimensions are considered as artefacts of the pruning procedure. These negligible plateaus, and therefore the corresponding tree dimensions, are then removed, and the final tree model is selected through the classical maximal jump in this new collection of dimensions. In practice, only plateaus occurring before the largest one are considered, and a plateau is set negligible if it represents less than a proportion α of the largest one (where α is chosen with respect to the relative size of the largest plateau).

Tables

Data Covariates Response's Distribution Bayes error

On a regular 3 by 3 blue and red checkerboard;

Checkerboard

Ua and U b delimited by the line x2 = 1/2;


Servane Gey

Jean-Michel Poggi

Discussion of Parallel Construction of Decision Trees with Consistently Non-Increasing Expected Number of Tests

Introduction

We would like to congratulate the authors for their interesting paper in which they propose a new parallel construction of decision trees and show how their new method relates to other methods that have been previously proposed. We found the paper to be stimulating and found that it offers insightful ideas related to the new construction. The algorithm permits to remove false alarms occurring at the beginning of the tree, thus reduces the number of tests. Indeed, the construction of a decision tree needs in general to compensate early false alarms by adding more tests leading to deeper trees. It is part of the renewed interest for tree-based methods; we can indeed quote numerous survey papers, as for example [START_REF] Patil | Issues in optimization of decision tree learning: A survey[END_REF] in computer science or [START_REF] Loh | Fifty Years of Classification and Regression Trees[END_REF] in statistics, which contain useful extensive bibliographies. The paper is clearly motivated by applications and implementation issues. As statisticians, we will try to highlight this contribution from a statistical perspective, so our comments center on questions related to how to generate and how to select tree models.

About using decision trees for time series segmentation

It may be interesting to use decision trees for time series segmentation, in particular when the number of observations is huge. Indeed, an exhaustive search is computationally too greedy in this case, whereas algorithms like CART or C4.5 are well adapted. The main issue of the use of decision trees is false alarms, which cannot be removed if appearing at the beginning of the tree. [START_REF] Gey | Using CART to Detect Multiple Change-Points in the Mean for Large Samples[END_REF] proposed an hybrid algorithm which uses CART to preselect a family of change points, and then removes false alarms with an exhaustive search. The main drawback of this hybrid algorithm is that the final predictor's configuration is no more a tree. Hence SF-GOTA is a good alternative in this case since it removes false alarms and provides a tree at the same time.

About model selection and decision trees

L-curve

As shown in Figures 12 to 15 of the paper, ENT is a decreasing function of the parameter s in SF-GOTA having an « L » form. This is a classical good behaviour in the model selection field when one aims at selecting the optimal value of some calibration parameter occuring in the complexity penalty.

Hence, a good alternative to select the optimal value of s could be to use L-curve heuristic (as proposed in [START_REF] Hansen | Analysis of discrete ill-posed problems by means of the Lcurve[END_REF], [START_REF] Engl | Using the L-curve for determining optimal regularization parameters[END_REF] or [START_REF] Hanke | Limitations of the L-curve method in ill-posed problems[END_REF]) : find the point of the L-curve for which there is a corner, and take the corresponding value of s.

Connexion between misclassification rate and ENT minimization

The authors draw on ENT to select optimal trees in SF-GOTA. Nevertheless, even if ENT is directly relied to the tree's complexity, taking only ENT into account means forgetting the important compromise to be made between misclassification rate and complexity. It is shown in [START_REF] Gey | Risk Bounds for Embedded Variable Selection in Classification Trees[END_REF] that this compromise is necessary to obtain good decision tree predictors. The authors claim that the tree selected at the end of SF-GOTA makes this compromise, but with s ranging from 1 to at most 6, the trees provided by SF-GOTA are undeep by construction. Then it could be interesting to compare prediction performance between trees provided by SF-GOTA and trees provided by C4.5 or CART to quantify the loss of prediction performance with respect to the huge gain of computational time.

About generating tests using trees or forests

The next remark is related to what can be understood as a restriction for the application since the discussed paper focuses on binary datasets to illustrate the value of the proposed algorithm. Indeed, starting from data described by an array of the classical form individuals x variables, one can use a CART model, which includes an optimal pruning, to automatically generate splits, that is a variable and a threshold value (for numerical variables) or a subset of values (for categorical variables). If the dataset is too large, this can applied to a smaller pilot sample. Then the selected splits can be used as candidates to generate tests and to encode the data as done in the paper. More generally, Random Forests and related methods can help to rank the variables (see [START_REF] Genuer | Variable selection using random forests[END_REF]) and construct the former tests according to the most important ones or by selecting splits generated according to the previous strategy. Alternative ideas could be to consider deterministic splits on each variable as performed in dyadic trees (see [START_REF] Blanchard | Optimal dyadic decision trees[END_REF] or purely RF where random splits are selected (see [START_REF] Biau | Consistency of random forests and other averaging classifiers[END_REF]), to construct the list of tests.

A final remark about Big Data issues

Many tree-based schemes including bagging and Random Forests admit reformulations based on Hadoop programming (see [START_REF] Prajapati | Big Data Analytics with R and Hadoop[END_REF]). One of the issues discussed at the beginning of the discussed paper relates to Big Data, which is one of the motivations of such proposals. The paper mainly addresses the massive data issue but the problem of how to manage online updating of trees, which is the second characteristic of Big Data problems, is not considered. It could be of interest to explore this potential source to speed up the data processing, see for example [START_REF] Gama | Accurate decision trees for mining highspeed data streams[END_REF] for adapting decision trees to high-speed data streams.