Clément Foucart
email: foucart@math.univ-paris13.fr

Pei-Sen Li
email: peisenli@ruc.edu.cn

Xiaowen Zhou
email: xiaowen.zhou@concordia.ca

On the entrance at infinity of Feller processes with no negative jumps

Keywords: Coming down from infinity, entrance boundary, Feller property, weak convergence

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space [0, ∞]. Moreover, when started from infinity, the extended Markov process on [0, ∞] leaves infinity instantaneously and stays finite, almost-surely. Arguments are adapted from a proof given by O. Kallenberg [10] for diffusions. We also show that the process started from x converges weakly towards that started from infinity in the Skorokhod space, when x goes to infinity.

Introduction

In the last decade, a regain of attention in the literature has been paid to the study of one-dimensional Markov processes at their boundaries. Such studies are of interest for instance when the process is representing the size of a random population. There is a large body of literature on this topic and we refer the reader for instance to the recent articles of Bansaye et al. [START_REF] Bansaye | Diffusions from Infinity[END_REF], [START_REF] Bansaye | Speed of coming down from infinity for birth-and-death processes[END_REF], Döring and Kyprianou [START_REF] Döring | Entrance and exit at infinity for stable jump diffusions[END_REF], Foucart [START_REF] Foucart | Continuous-state branching processes with competition: duality and reflection at infinity Electron[END_REF], Le and Pardoux [START_REF] Le | Height and the total mass of the forest of genealogical trees of a large population with general competition[END_REF] and [START_REF] Le | Extinction time and the total mass of the continuous-state branching processes with competition[END_REF], Li [START_REF] Li | A continuous-state polynomial branching process[END_REF] and Li et al. [START_REF] Li | A general continuous-state nonlinear branching process[END_REF]. See also the monography of Pardoux [START_REF] Pardoux | Probabilistic models of population evolution, Scaling limits, genealogies and interactions[END_REF]Chapters 6 and 8].

In the latter cited works, the authors share the same interest to understand how the population behaves when the initial size takes arbitrarily large values. Several behaviors are possible. When the dynamics prevents the process to explode (i.e. to hit ∞ in finite time) and yet allows the process to "start from infinity"; the boundary is called an entrance. Such a phenomenon is ubiquitous in population models with self-regulation properties or in models of statistical physics.

Several definitions of entrance boundary are given in the literature. The process is often said to come down from infinity, when the first entrance times of compact sets are, in some sense, uniformly bounded with respect to the initial state of the process.

A rigorous definition in this vein can be found in Revuz and Yor [START_REF] Revuz | Yor Continuous martingales and Brownian motion[END_REF]Chapter 3] and Kallenberg [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Chapter 23] and is stated as follows.

Definition 1.1. Let (X t , t ≥ 0) be a positive Markov process. The boundary ∞ is said to be an instantaneous entrance boundary for the process (X t , t ≥ 0) if the process does not explode and ∀t > 0, lim where P x (•) := P(•|X 0 = x) and T b := inf{t ≥ 0; X t < b} for all b ∈ R + , with the convention inf ∅ := ∞.

Although the meaning of condition (1.1) is intuitively clear, it does not guarantee a priori, that the process X can be started at ∞. Kallenberg [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF] has designed elegant arguments for diffusions ensuring that indeed, if the assumption (1.1) holds, then the diffusion process (X t , t ≥ 0) has a regular version started from infinity. The main purpose of this note is to observe that, more generally if the process under study has no negative jumps and satisfies a Feller property on E := [0, ∞), then the assumption (1.1) ensures that its semigroup can be extended into a Feller semigroup on Ē = [0, ∞]. As a direct consequence, any Feller process (X t , t ≥ 0) on E, with no negative jumps, satisfying (1.1) admits a càdlàg extension started from ∞.

The following equivalent conditions for (1.1) to hold, have already appeared in the literature, see e.g. [17, Proposition 2.12] and [START_REF] Li | A continuous-state polynomial branching process[END_REF]Theorem 1.11] Consider a Markov process (X t , t ≥ 0) with state space E. Denote by (P t ) t≥0 its semigroup. Assume that

(i) for any f ∈ C b (E), P t f ∈ C b (E), (ii) for any f ∈ C b (E) and x ∈ E, P t f (x) -→ t→0 f (x).
We shall say in the sequel that a process is Feller on the space Ē, if its semigroup satisfies (i) and (ii) for any function f ∈ C( Ē) and any x ∈ Ē. Remark 2.1. Several definitions of "Feller processes" coexist in the literature. We are following the definition given in Kallenberg's book [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Chapter 19,p.369]. In order to deal with an entrance boundary at ∞, the usual class C 0 of continuous functions vanishing at ∞ cannot be used. Recall however that when the state space is compact, as the case for Ē, C 0 can be replaced by C( Ē). Moreover, according to [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Theorem 19.6], when the state-space is compact, the strong continuity at 0 of the semigroup is equivalent to the pointwise one.

Theorem 2.2. Consider a non-explosive Markov process (X t , t ≥ 0) on E, with no negative jumps, satisfying (i), (ii) and the entrance boundary condition (1.1). Then (X t , t ≥ 0) can be extended into a Feller process over Ē such that under P ∞ , it starts from ∞, leaves it instantaneously and stays finite, almost-surely:

P ∞ (X 0 = ∞) = 1 and P ∞ (∀t > 0, X t < ∞) = 1.
Remark 2.3. Theorem 2.2 holds true also for a process (X t , t ≥ 0) valued in N with the skip-free property (i.e. negative jumps of size at most -1). It is also worth mentioning that the state-space E = [0, ∞) does not play any specific role here and can be any real interval ( , r) by replacing ∞ by the right-end boundary r everywhere in Definition 1.1, in Theorem 2.2, and in the forthcoming Proposition 2.4, Theorem 2.5 and Corollary 2.7. Theorem 2.2 applies for instance to diffusions, see e.g. [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Chapter 23] where an explicit condition equivalent to (1.1) is given in terms of the scale function and the speed measure of the diffusion process. It is also invoked in Foucart et al. [9, Section 2.1] for defining certain time-changed spectrally positive Lévy processes, started from infinity. An explicit condition equivalent to (1.1) is found for this class of processes in [START_REF] Foucart | Time-changed spectrally positive Lévy processes started from infinity[END_REF]Theorem 3.1]. Other examples will be discussed in the sequel, see Corollary 2.9.

The next proposition states a regularity property of the first entrance times T b under the probability measures (P x , x ≥ 0). Denote by E ∞ the expectation under P ∞ . Proposition 2.4. Suppose that the assumptions of Theorem 2.2 hold. Let h be a continuous function on [0, ∞) that is either bounded or nonnegative and increasing. Then (a) for any θ > 0, there exists b θ > 0, such that for all b ≥ b θ , E ∞ (e θT b ) < ∞;

(b) for any b > 0, if T b < ∞ P ∞ -a.s. and E ∞ (h(T b )) < ∞, then E x (h(T b )) -→ x→∞ E ∞ (h(T b )).
(2.2)

In particular, we see from Proposition 2.4-(a) that, under P ∞ , one can find a large enough b such that T b has moments of all orders. The convergence (2.2) holds for instance with the function h(x) = x n for any n ∈ N. Such a convergence of moments is crucial when one wants to study the process started from infinity, from the sample paths started from large but finite levels. We refer to [START_REF] Bansaye | Diffusions from Infinity[END_REF], [START_REF] Bansaye | Speed of coming down from infinity for birth-and-death processes[END_REF] and [START_REF] Foucart | Time-changed spectrally positive Lévy processes started from infinity[END_REF] where it is used for studying the speed of coming down from infinity of different Feller processes with no negative jumps.

The last theorem states a general result for Feller processes valued in a compact state space and clarifies in which sense the laws (P x ) x≥0 converge as x goes to ∞ towards P ∞ . Theorem 2.5. Assume that X is a Feller process on [0, ∞]. Let X (x) be the Markov process started from x ∈ [0, ∞] with càdlàg sample paths. Then the family of processes (X (x) ) x∈[0,∞) converges weakly, in the Skorokhod topology, as x → ∞ towards X (∞) . Remark 2.6. Theorem 2.5 does not require the assumption of absence of negative jumps.

A direct consequence of Theorem 2.2 and Theorem 2.5 is the following convergence in law of the process started from x towards that started from ∞, when ∞ is an entrance boundary.

Corollary 2.7. Suppose that the assumptions of Theorem 2.2 hold. Denote by (X (x) ) x∈[0,∞) the family of processes started from x. Then, (X (x) ) x∈[0,∞) converges in law, in the Skorokhod topology, as x → ∞ towards X (∞) . Such a weak convergence was previously shown for different Markov processes, satisfying the conditions (i) and (ii), with ∞ as entrance boundary. See for instance Donnelly [START_REF] Donnelly | Weak convergence to a Markov chain with an entrance boundary: ancestral processes in population genetics[END_REF], Lambert [13, Corollary 3.10], Bansaye [1, Proposition 4.4] and Bansaye et al. [START_REF] Bansaye | Diffusions from Infinity[END_REF], [3, Lemma 2.1]. The arguments for establishing the weak convergence in these latter works were requiring the stochastic monotonicity of the process in its initial values.

We now show how our main results apply to the so-called continuous-state nonlinear branching processes. Those processes were defined and studied in [START_REF] Li | A general continuous-state nonlinear branching process[END_REF]. Let {B t } t≥0 be an (F t )-Brownian motion. Write ν for a σ-finite nonnegative measure on (0, ∞) such that ∞ 0 (z ∧ z 2 ) ν(dz) < ∞. Let {N (ds, dz, du) : s, z, u > 0} be an independent (F t )-Poisson random measure on (0, ∞) 3 with intensities ds ν(dz) du, and { Ñ (ds, dz, du) : s, z, u > 0} be the corresponding compensated measure, namely Ñ (ds, dz, du) := N (ds, dz, du)ds ν(dz) du. Consider the following stochastic differential equation (SDE):

X (x) t = x + t 0 γ 0 (X (x) s ) ds + t 0 γ 1 (X (x) s ) dB s + t 0 ∞ 0 γ 2 (X (x) s-) 0 z Ñ (ds, dz, du). (2.3)
Any solution to (2.3), whenever it exists, is called a general continuous-state nonlinear branching process. In the two next results, we shall make the following general assumption:

Condition H: there is a pathwise unique nonnegative càdlàg solution (X

(x) t , t ≥ 0) to (2.
3) which is non-explosive and satisfies for any x ≤ y, Any càdlàg process solution to (2.3) has no negative jumps and satisfies the condition (ii). In the following proposition, we provide a sufficient condition on the drift γ 0 entailing the Feller property (i).

X (x) t ≤ X (y) t for all t ≥ 0 a.s. ( 2
Proposition 2.8. Assume that Condition H holds. If there exists θ > 0 such that for any 0 ≤ x ≤ y, γ 0 (y) -γ 0 (x) ≤ θ(y -x), (2.5)

then the semigroup of (X t , t ≥ 0) satisfies the Feller property (i).

Combining Condition H, the one-sided Lipschitz condition (2.5) and the entrance boundary condition (1.1), one can apply Theorem 2.2 and Corollary 2.7 to define processes solution to (2.3) started from ∞. We sum up those results in the following corollary.

Corollary 2.9. Assume that Condition H holds. If (1.1) and (2.5) are satisfied, then the flow of càdlàg strong solutions to (2.3), (X (x) ) x∈[0,∞) , converges weakly, in the Skorokhod topology, as x → ∞, towards a non-explosive Feller process X (∞) started from infinity.

Remark 2.10. Condition H encompasses the comparison property (2.4), so that under the assumptions of Corollary 2.9, for any t > 0, X

(x) t increases almost-surely towards X (∞) t < ∞, as x goes to ∞.
Sufficient conditions for continuous-state nonlinear branching processes to satisfy (1.1) are given in [START_REF] Li | A general continuous-state nonlinear branching process[END_REF]Theorem 2.13] and [START_REF] Le | Extinction time and the total mass of the continuous-state branching processes with competition[END_REF]Theorem 2.2]. The problem of defining the process started from infinity was left unanswered in these works. Corollary 2.9 enables us to address this question and to give a sense, in terms of sample paths, to the phenomenon of coming down from infinity.

To exemplify Corollary 2.9, consider the case for which γ i (x) = x for all x ≥ 0, i = 1, 2 and γ 0 is continuous on R + . In this setting, the process solution to (2.3) is also called continuous-state branching process with competition. The nonlinear drift γ 0 is interpreted as modelling interactions between individuals, see Le and Pardoux [START_REF] Le | Extinction time and the total mass of the continuous-state branching processes with competition[END_REF]. The following corollary combines some results established in [START_REF] Le | Extinction time and the total mass of the continuous-state branching processes with competition[END_REF] and Corollary 2.9.

Corollary 2.11. Assume that γ i (x) = x for all x ≥ 0, i = 1, 2 and that γ 0 is continuous on R + , with γ 0 (0) = 0. If γ 0 satisfies (2.5) and ∞ b dx |γ 0 (x)| < ∞ for some b > 0, then Condition H and (1.1) are satisfied, and the process (X (∞) t , t ≥ 0), started from ∞, is well-defined, comes down from infinity instantaneously and then stays finite almost-surely. Remark 2.12.

1. An explicit example is the logistic continuous-state branching process for which for all x ≥ 0, γ i (x) = x, i = 1, 2 and γ 0 (x) = -c 2 x 2 with c > 0. It satisfies (1.1), starts from infinity, "comes down" instantaneously, and then stays finite almost-surely. We refer to [START_REF] Lambert | The branching process with logistic growth[END_REF], see also [START_REF] Foucart | Continuous-state branching processes with competition: duality and reflection at infinity Electron[END_REF]Lemma 6.4] where (1.1) is established. 2. The non-explosion of the process in Corollary 2.11 will be guaranteed by the assumption

∞ 0 (z ∧ z 2 )ν(dz) < ∞.
When the latter integral is infinite, different behaviors at ∞ may occur, see for instance [START_REF] Foucart | Continuous-state branching processes with competition: duality and reflection at infinity Electron[END_REF] for the case γ 0 (x) = -c 2 x 2 for all x ≥ 0. 3. Examples of continuous-state nonlinear branching processes for which γ i (x) = x, i = 1, 2 and Corollary 2.9 applies can be found in [17, Page 2536]. For instance, if ν(dz) = c α 1 {z>0} z -1-α dz, with α ∈ (1, 2), c α > 0, γ 0 = γ 1 = 0 and γ 2 (x) = x r 2 with r 2 > α, then the process solution to (2.3) can be started from infinity and comes down from infinity. We refer to [START_REF] Li | A continuous-state polynomial branching process[END_REF] for a different method allowing one to define this process started from infinity.

P x (T b > t) ≤ P x b (T b > t) < 1. Therefore, if b is large enough then, for all x ≥ b, P x (T b > t) ≤ sup x≥b P x (T b > t) < 1. (3.6)
In particular, we see that P x (T b < ∞) > 0 for all x ≥ b and that there is t > 0 and a large enough b such that

α b := sup x≥b P x (T b > t) < 1.
We now work with those t and b. Let x > b, and n ∈ N, by using the Markov property at time t

P x (T b > nt) = E x 1 {T b >t} 1 {T b •θt>(n-1)t} = E x 1 {T b >t} E Xt 1 {T b >(n-1)t} ≤ P x (T b > t) sup x≥b P x (T b > (n -1)t).
Hence for all n ≥ 1, sup

Proof of Theorem 2.2

Recall the statement of Theorem 2.2 and its assumptions. The arguments to prove Theorem 2.2 are adapted from those of Kallenberg in [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Theorem 23.13]. Let y > x > b.

Since the process (X t , t ≥ 0) has no negative jumps, T b > T x almost surely under P y . Moreover for any t ≥ 0, {T b < t} ⊂ {T x < t, T b -T x < t} and

P y (T b < t) ≤ P y (T x ≤ t, T b -T x < t) = P y (T x < t)P x (T b ≤ t) ≤ P x (T b < t).
This implies that (P x (T b < t), x ≥ 0) admits a limit as x goes to ∞ and that (E (3.7)

For any function f ∈ C b (E), one denotes by ||f || the supremum norm of f . We now show that (P t f (x), x ≥ 0) admits a limit as x goes to ∞ for any f ∈ C b (E) with E = [0, ∞). Fix t ≥ 0, for any x > b,

P t f (x) = E x [1 {T b >t} f (X t )] + E x [1 {T b ≤t} f (X t )] = E x [1 {T b >t} f (X t )] + E x [1 {T b ≤t} P t-T b f (X T b )] = E x [1 {T b >t} f (X t )] + E x [1 {T b ≤t} P t-T b f (b)],
where we have used the strong Markov property at T b ∧ t in the second equality and in the third, the absence of negative jumps which implies X T b = b a.s. Set g(a, s) := 1 s≤t P t-s f (a) + 1 s>t f (a).

For any x and y larger than b,

|P t f (y) -P t f (x)| = |E x [1 {T b >t} f (X t )] -E y [1 {T b >t} f (X t )] + E x [1 {T b ≤t} P t-T b f (X T b )] -E y [1 {T b ≤t} P t-T b f (X T b )]| ≤ ||f ||(P x (T b > t) + P y (T b > t)) + E x g(b, T b ) -f (b)1 {T b >t} -E y g(b, T b ) -f (b)1 {T b >t} ≤ 2||f ||(P x (T b > t) + P y (T b > t)) + |E x [g(b, T b )] -E y [g(b, T b )]| .
From (1.1), we see that for x, y → ∞ and then b → ∞, The family of laws of T b under P x for x ≥ 0, (P x •T -1 b , x ≥ 0), is therefore tight and admits a unique limit since (P x (T b < t), x ≥ 0) converges. Thus, (P x • T -1 b , x ≥ 0) converges weakly and since the map s → g(b, s) is bounded and continuous, therefore,

2||f ||(P x (T b > t) + P y (T b > t)) -→ 0.
|E x [g(b, T b )] -E y [g(b, T b )]| -→ x,y→∞ 0. Since |P t f (y) -P t f (x)| -→
x,y→∞ 0, then for any sequence (x n , n ≥ 1) such that x n -→ n→∞ ∞, (P t f (x n ), n ≥ 1) is a Cauchy sequence and admits a limit in R. This limit does not depend on the sequence (x n , n ≥ 1) and we set P t f (∞) := lim x→∞ P t f (x). By the assumption of non-explosiveness, for all x ∈ E, the transition kernels P t (x, •) := P(X t ∈ •|X 0 = x) are probability measures over E = [0, ∞). Since the convergence P t f (∞) := lim x→∞ P t f (x), holds for any f ∈ C b (E), the probability measures P t (x, •) over E converge weakly towards P t (∞, •) as x goes to ∞. Taking the constant function f = 1, we see that P t (∞, •) is a probability measure over E.

We proceed to check that (P t ) forms a Feller semigroup on Ē = [0, ∞]. Recall C( Ē). Let f ∈ C( Ē). By the assumption (ii), P t f is continuous on E. By the definition, P t f (∞) = lim x→∞ P t f (x) < ∞ and P t f is continuous at ∞. Therefore, P t maps C( Ē) into itself. By the assumption (ii), for any x ∈ [0, ∞), one has P t f (x) -→ t→0 f (x), and it only remains to show that

P t f (∞) -→ t→0 f (∞). Let ε > 0 and choose b large enough such that sup x≥b |f (x) -f (∞)| ≤ ε. Note that there is t small enough such that P 2b (T b ≤ t) ≤ ε. Then, |P t f (∞) -f (∞)| ≤ lim x→∞ E x [|f (X t ) -f (∞)|1 {T b ≤t} ] + E x [|f (X t ) -f (∞)|1 {T b >t} ] ≤ 2||f ||P 2b (T b ≤ t) + ε ≤ (2||f || + 1)ε.
The pointwise continuity of the semigroup at t = 0 is therefore satisfied on [0, ∞] and finally the semigroup on Ē satisfies (i) and (ii) and is Feller. Theorem 19.15 in [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF] provides the existence of a strong Markov process (X t , t ≥ 0) with càdlàg paths started from ∞. We denote its law by P ∞ . Note that for any t > 0, P t (∞, E) = P ∞ (X t < ∞) = 1. It remains to show that P ∞ (∀t > 0, X t < ∞) = 1. By the assumption for any x ∈ E, (X t , t ≥ 0) does not explode under P x . Let s > 0. By the Markov property at time s under P ∞ ,

P ∞ (∃t > s; X t = ∞) = E ∞ (P Xs (∃t > 0; Xt = ∞))
where X is a copy of X. Since P ∞ (X s < ∞) = 1, we have that for any s > 0, P ∞ -a.s. P Xs (∃t > 0; Xt = ∞) = 0. Hence P ∞ (∃t > s; X t = ∞) = 0, which allows one to conclude since s is arbitrary.

Proof of Proposition 2.4

Assume that (1.1) holds and recall E ∞ the corresponding expectation under P ∞ . We show first the exponential moment property stated in (a). Recall that under P ∞ , T b decreases as b goes to ∞. Denote by T ∞ = inf{t > 0; X t < ∞} its limit. By Theorem 2.2, T ∞ = 0 P ∞ -a.s. Let θ > 0. Fix t > 0. By Lebesgue's theorem

lim b→∞ P ∞ (T b > t) = 0. (3.8)
Hence, there exists b θ > 0, such that for all b ≥ b θ , e θt P ∞ (T b > t) < 1. Moreover for any n ≥ 1, by the Markov property, we get

P ∞ (T b > nt) ≤ P ∞ (T b > t) n and for any b ≥ b θ E ∞ (e θT b ) = E ∞ θ T b 0 e θs ds + 1 = θt ∞ 0 e θts P ∞ (T b > ts)ds + 1 = θt ∞ n=0 n+1 n e θts P ∞ (T b > ts)ds + 1 ≤ θte θt ∞ n=0 e θt P ∞ (T b > t) n + 1 < ∞.
We establish now the convergence in (2.2). Denote by θ the shift operator, see e.g. [10, p.146]. For any x > b > 0,

T b • θ Tx = T b -T x -→ T b as x → ∞, P ∞ -a.s. (3.9)
By the strong Markov property,

E ∞ [h(T b )] = E ∞ [h(T b ) -h(T b • θ Tx )] + E x [(h(T b ))].
Recall (3.9) and that h is either bounded or nonnegative and increasing. By applying Lebesgue's theorem, we get

E ∞ [h(T b )] = lim x→∞ E x [h(T b )].

Proof of Theorem 2.5

Define the metric ρ on Ē = [0, ∞] by ρ(x, y) = |e -x -e -y | for any x, y ∈ Ē and let D be the space of càdlàg functions f : E → Ē. We endow D with the Skorokhod topology, for which we refer, for instance, to Ethier and Kurtz's book Let (P t ) t≥0 be a Feller semigroup on C( Ē). Let A : D → C( Ē) be the generator of (P t ) t≥0 with D its domain. Then by [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Theorem 19.4], we see that D is dense in C( Ē) with respect to the supremum norm || • ||. For each x ∈ Ē, denote by X (x) = (X (x) t ) t≥0 the corresponding Feller process with initial value x > 0 and càdlàg paths. From Dynkin's formula, see [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Theorem 19.21], we have for any bounded stopping time τ , and any

f ∈ D E[f (X (x) τ )] = f (x) + E τ 0
Af (X (x) s )ds .

Lemma 3.1. The sequence of processes (X (x) ) x>0 is relatively compact in distribution, i.e. every subsequence has a subsequence that converges in distribution in D.

Proof. Since D is dense in C( Ē), for any ε > 0 and f ∈ D, there exists g ∈ D such that ||f 2 -g|| < ε. Applying the Markov property and Dynkin's formula we have for any bounded stopping time τ and constant h > 0,

E[(f (X (x) τ +h ) -f (X (x) τ )) 2 ] = E[f 2 (X (x) τ +h ) -f 2 (X (x) τ )] -2E f (X (x) τ )E[f (X (x) τ +h ) -f (X (x) τ )|F τ ] ≤ 2ε + E[g(X (x) τ +h ) -g(X (x) τ )] + 2||f || E E[f (X (x) τ +h ) -f (X (x) τ )|F τ ] = 2ε + E τ +h τ Ag(X (x) s )ds + 2||f || E τ +h τ Af (X (x) s )ds ≤ 2ε + ||Ag||h + 2||f || • ||Af ||h.
Therefore, by Jensen's inequality and the arbitrariness of ε,

lim δ→0 lim sup x→∞ sup τ ≤t sup h∈[0,δ] E|f (X (x) τ +h ) -f (X (x) τ )| = 0 t > 0,
where the second supremum is over all stopping time τ ≤ t for the process X (x) . Then by Aldous' criterion of tightness, see e.g. [START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF]Theorem 16.11 Lemma 3.2. The finite-dimensional distributions of X (x) converges weakly to that of X (∞) as x → ∞.

Proof. We are going to prove that for any sequence

{t 1 < t 2 < • • • < t n } ⊂ E and {f 1 , f 2 , . . . , f n } ⊂ C( Ē), lim x→∞ E[f 1 (X (x) t 1 )f 2 (X (x) t 2 ) . . . f n (X (x) tn )] = E[f 1 (X (∞) t 1 )f 2 (X (∞) t 2 ) . . . f n (X (∞) tn )]. ( 3
E[F n (X (x) t 1 , X (x) t 2 , . . . , X (x) tn )] = E[F n (X (∞) t 1 , X (∞) t 2 , . . . , X (∞) tn )].
Obviously, (3.10) holds for n = 1. Assume that (3.10) holds for n = m. Then by the Markov property

E[f 1 (X (x) t 1 )f 2 (X (x) t 2 ) . . . f m+1 (X (x) t m+1 )] = E[f 1 (X (x) t 1 )f 2 (X (x) t 2 ) . . . f m (X (x) tm )E[f m+1 (X (x) t m+1 )|F tm ]] = E[f 1 (X (x) t 1 )f 2 (X (x) t 2 ) . . . f m (X (x) tm )P t m+1 -tm f t m+1 (X x tm )].
By the Feller property, the map x → f m (x)P t m+1 -tm f t m+1 (x) is continuous and bounded.

Then we can finish the proof of (3.10) by induction.

Proof of Proposition 2.8

Recall (X (x) t , t ≥ 0) the solution to (2.3) and the statement of Proposition 2.8. Let bL be the set of bounded Lipschitz continuous functions on E and L(f ) denotes the Lipschitz constant of f ∈ bL . For f ∈ C b (E) and M > 0, let

f M (x) = f (x), 0 ≤ x ≤ M, f (M ), x > M.
It follows from the Stone-Weierstrass theorem that there exists a sequence of bounded Lipschitz functions (f

(n) M ) n≥1 such that ||f (n) M -f M || → 0 as n → ∞. (3.11)
Since by the assumption, X

≤ X (y) t a.s. for any t ≥ 0 and 0 ≤ x < y, for any 0 ≤ x < y, by (2.5)

E(|X (y) t -X (x) t |) = E(X (y) t -X (x) t ) = y -x + t 0 E(γ 0 (X (y) s ) -γ 0 (X (x) s ))ds ≤ y -x + θ t 0 E(X (y) s -X (x) s )ds. By Gronwall's Lemma, E(|X (y) t -X (x)
t |) ≤ e θt |x -y| for all t. Thus, for any f ∈ bL ,

|P t f (x) -P t f (y)| ≤ L(f )e θt |x -y|, (3.12)
which entails that P t f ∈ bL . Define

g M (x) =      0, 0 ≤ x ≤ M/2, 2x M -1, M/2 ≤ x ≤ M, 1,
x > M.

Fix t and x and denote by P t (x, •) the law of X

t . For any ε > 0, we can choose M, δ > 0, such that for each y ∈ ((x -δ) ∨ 0, x + δ)

|P t f (y) -P t f M (y)| ≤ 2||f ||P t (y, [M, ∞)) ≤ 2||f || P t g M (x) + |P t g M (y) -P t g M (x)| ≤ 2||f || P t (x, [M/2, ∞)) + 2e θt M |x -y| < ε/4 (3.13)
where, for any Borel set A, P t (y, A) := P t 1 A (y) and for the third inequality we used (3.12). From (3.11), we see that there exists n ∈ N such that

||f M -f (n) M || ≤ ε/8. (3.14) Since P t f (n) M ∈ bL , there is δ < δ such that for y ∈ ((x -δ ) ∨ 0, y + δ ), P t f (n) M (x) -P t f (n) M (y) ≤ ε/4 (3.15)
Combining (3.13), (3.14) and (3.15), we see that for y ∈ ((x -δ ) ∨ 0, y + δ ),

|P t f (x) -P t f (y)| ≤ |P t f M (x) -P t f M (y)| + |P t f (x) -P t f M (x)| + |P t f (y) -P t f M (y)| ≤ P t f (n) M (x) -P t f (n) M (y) + 2||f M -f (n) M || +|P t f (x) -P t f M (x)| + |P t f (y) -P t f M (y)| ≤ ε/4 + ε/4 + ε/4 + ε/4 = ε.
Thanks to the arbitrariness of ε, we can complete the proof.

3.6 Proof of Corollaries 2.9 an 2.11 3.6.1 Proof of Corollary 2.9

Recall Corollary 2.9. Assume that (1.1) is satisfied. Under Condition H, the solution of (2.3) has no negative jump and is right-continuous, so that condition (ii) is satisfied. By Proposition 2.8, under the assumption (2.5), the process (X t , t ≥ 0) satisfies (i). Assumptions of Theorem 2.2 are therefore fulfilled and the process admits a Feller extension started from infinity. The weak convergence is a consequence of Corollary 2.7.

Proof of Corollary 2.11

Consider now the continuous-state branching processes with competition. These processes are solution to the SDE (2.3) with γ i (x) = x for all x ≥ 0 and i = 1, 2, with a continuous function γ 0 satisfying γ 0 (0) = 0.

Assume that γ 0 satisfies the one-sided Lipschitz condition (2.5). We establish that (2.5) combined with ∞ 0 (z ∧ z 2 )ν(dz) < ∞, entails Condition H. Dawson and Li [4, Theorem 2.1] ensures that Equation (2.3) admits a unique nonnegative càdlàg solution (X (x) t , t ≥ 0), started from x ∈ [0, ∞). Moreover, the assumption γ 0 (0) = 0, combined with (2.5) entails that γ 0 (x) ≤ θx for all x ≥ 0. Assumptions of [4, Theorem 2.2] are thus satisfied and the comparison property (2.4) holds true. A second application of [START_REF] Dawson | Stochastic equations, flows and measure-valued processes[END_REF]Theorem 2.2] ensures that for all x ≥ 0 and all t ≥ 0, X (x) t ≤ Z (x) t where (Z (x) t , t ≥ 0) is a supercritical continuous-state branching process with Lévy measure ν, started from x. By the assumption ∞ 0 (z ∧ z 2 )ν(dz) < ∞, the process (Z (x) t , t ≥ 0) has finite mean and thus is non-explosive. By comparison, the process (X x) t , t ≥ 0) is thus also non-explosive for any x ∈ R + , and finally Condition H holds true.

It remains to establish (1.1). Note that (2.5) is hypothesis (H1) in [START_REF] Le | Extinction time and the total mass of the continuous-state branching processes with competition[END_REF]. Assume that γ 0 satisfies γ 0 (0) = 0 and (2.5). By [15, Theorem 2.2] and its proof, see [15,

  . The proof is deferred to Section 3.1. Denote by E x the corresponding expectation under P x . Lemma 1.2. Consider a positive strong Markov process (X t , t ≥ 0) with no negative jumps. For any b ∈ E, recall T b its first passage time below b. The following statements are equivalent: (a) Condition (1.1) holds. (b) For all large enough b, sup x≥b E x (T b ) < ∞. (c) lim b→∞ lim x→∞ E x (T b ) = 0. 2 Main results Recall E := [0, ∞) and Ē := [0, ∞]. Denote by C b (E) the space of continuous bounded functions on E and by C( Ē) the space of continuous functions on Ē, namely those in C b (E) with finite limits at ∞.

2

 2 Recall the statement of Lemma 1.2. We show that (a) =⇒ (b). The condition (1.1) entails that for any fixed t, there is b large enough from which, lim sup x→∞ P x (T b > t) < 1. This implies that there exists x b ≥ b such that sup x≥x b P x (T b > t) < 1. Since there are no negative jumps, for any x ∈ [b, x b ],

  x≥b P x (T b > nt) ≤ α b sup x≥b P x (T b > (n-1)t) and sup x≥b P x (T b > nt) ≤ α n b . This entails sup x≥b E x (T b ) ≤ ∞ n=0 α n b < ∞. We now show that (b) =⇒ (c). Note that the limits in (c) are monotone increasing in x and decreasing in b. According to (b), for large enough b, E x (T b ) < ∞ for any x ≥ b so that P x (T b < ∞) = 1. Let x > x > b. By the absence of negative jumps and the strong Markov propertyE x (T b ) = E x (T x ) + E x (T b ).Letting x go towards ∞ and then x go towards ∞, we getlim x→∞ E x (T b ) = lim x →∞ lim x→∞ E x (T x ) + lim x →∞ E x (T b ).By (b), lim x→∞ E x (T b ) < ∞ and thus lim x →∞ lim x→∞ E x (T x ) = 0. The last implication (c) =⇒ (d) is a simple consequence of the Markov inequality.

  x (T b ), x ≥ b) is non-decreasing in x. The inequality (3.6) ensures that lim x→∞ P x (T b < t) > 0. Recall also that Lemma 1.2 states that (1.1) holds if and only if sup x≥b E x (T b ) -→ b→∞ 0.

Moreover, ( 3 . 7 )

 37 entails sup x E x (T b ) < ∞ which provides for any fixed b lim s→∞ sup x≥0 P x (T b > s) = 0.

[ 7 ,

 7 Chapter 3, p.116]. The proof of Theorem 2.5 follows by combining Lemma 3.1 (tightness) and Lemma 3.2 (convergence of finite dimensional laws), given below, and by applying [7, Corollary 9.3].

  page 10], for any b > 0, if ∞ b dx |γ 0 (x)| < ∞, then sup x≥b E x (T b ) < ∞. Thus, by Lemma 1.2, if ∞ b dx |γ 0 (x)| < ∞for some b > 0, then (1.1) is satisfied. Hence Theorem 2.2 and Corollary 2.9 apply.

  and Lemma 16.12, p.314], for any f ∈ D, (f • X (x) ) x>0 is relatively compact in distribution. Finally, since D is dense in C( Ē), applying [7, Theorem 9.1 p.142] we see that (X (x) ) x>0 is relatively compact in distribution. Note that Ē is compact, thus the compact containment condition, needed for [7, Theorem 9.1 p.142] to apply, holds.

Acknowledgements: C.F's research is partially supported by LABEX MME-DII (ANR11-LBX-0023-01). P.L. and X.Z.'s research is supported by Natural Sciences and Engineering Research Council of Canada (RGPIN-2016-06704). P.L.'s research is supported by Natural Sciences and Engineering Research Council of Canada (RGPIN-2012-07750) and National Natural Science Foundation of China (No. 11901570 and No. 11771046). X.Z.'s research is supported by National Natural Science Foundation of China (No. 11731012). The authors are grateful to Donald Dawson for his support and encouragement. C.F would like to thank Leif Döring for mentioning the proof of Kallenberg during the 3th Workshop on Branching Processes and Related Topics held in 2017 in Beijing.


Aline Bertin
email: aline.bertin@inrae.fr

Arielle Beraud
email: beraud.arielle@gmail.com

Léa Lansade
email: lea.lansade@inrae.fr

Baptiste Mulot
email: baptiste.mulot@zoobeauval.com

Cécile Arnould
email: cecile.arnould@inrae.fr

Bill covering and nape feather ruffling as indicators of calm states in the Sulphurcrested cockatoo (Cacatua galerita)

Keywords: Animal sentience, Calmness, Facial expression, Positive emotions, Emotional expression

Parrots are highly social birds that are recognized for their primate-like cognitive abilities but their way to express emotions remain overlooked. Herein we explored potential facial indicators of emotions in cockatoos. We predicted that facial feather ruffling is an indicator of a cockatoo's emotional state and hypothesized that specific facial feather positions would be present more during positive valence and low arousal situations. We observed feather position on the crest, cheek and nape during the daily routine of a group of five captive, non-breeding, Sulphur-crested cockatoos. The data show that cheek and nape feather ruffling occurred significantly more during activities associated with low arousal levels and positive valence such as maintenance behaviours, positive and quiet social contact and resting. Our data suggest that ruffling feathers over the bill (i.e. cheek feather ruffling) and nape ruffling may provide visual indicators of calm/relaxed states in cockatoos. Subtle movement of facial feathers may be an effective close-ranged visual signal to communicate birds' affective states or their intention to engage in specific activities. This work provides a novel approach to assessing the positive welfare of captive cockatoos and to understanding emotional communication in non-mammalian species.

Introduction

Positive emotions have been recognized as adaptive as negative ones and as primary component of animal well-being [START_REF] Mendl | An integrative and functional framework for the study of animal emotion and mood[END_REF][START_REF] Panksepp | Affective neuroscience: The foundations of human and animal emotions[END_REF][START_REF] Boissy | Assessment of positive emotions in animals to improve their welfare[END_REF]). Despite their importance, positive emotions have been less studied than negative ones because they are more subtle and difficult to assess. Emotions are subjective experiences composed of cognitive processes, neurophysiological and behavioural responses [START_REF] Boissy | Assessment of positive emotions in animals to improve their welfare[END_REF]). In mammals, movement of facial muscles provides species-specific repertoires of facial expressions, which are one of the most studied and reliable behavioural tools to access the emotional world of animals [START_REF] Waller | Facial expression in nonhuman animals[END_REF]. As birds lack a complex facial musculature [START_REF] Diogo | From fish to modern humans-comparative anatomy, homologies and evolution of the head and neck musculature[END_REF], they have been discarded from this field of research.

However, birds do have the capacity to move their facial feathers due to the contractile properties of the feather-bearing integument [START_REF] Homberger | The role of mechanical forces on the patterning of the avian feather bearing skin: A biomechanical analysis of the integumentary musculature in birds[END_REF]. According to anecdotal observations, facial/head feather movement including the crest may communicate emotions or moods, like play mood in cockatoos [START_REF] Kaplan | Bird minds: cognition and behaviour of Australian native birds[END_REF]. In the few other crested-birds studied so far, crest displaying was mainly observed in contexts of high arousal levels like intra or interspecific aggression, courtship or defence from predators [START_REF] Kumar | Communication value of displays and postures in Red-vented Bulbul Pycnonotus cafer (Aves: Pycnonotidae)[END_REF][START_REF] Graves | Function of crest displays in royal flycatchers (Onychorhynchus)[END_REF][START_REF] Goodwin | Further observations on the behaviour of the jay Garrulus glandarius[END_REF][START_REF] Ruiz-Rodríguez | Multi-functional crest display in hoopoes Upupa epops[END_REF].

Emotions are both characterized by their valence (positive or negative) and arousal level (high or low) [START_REF] Mendl | An integrative and functional framework for the study of animal emotion and mood[END_REF]. Two types of positive emotions or states are commonly distinguished. Those characterized by high arousal levels such as joy, excitement, consummatory or appetite motivational states and, those characterized by low arousal levels such as calm, relaxed, safeness, social bonding or post-consummatory behaviours [START_REF] Mendl | An integrative and functional framework for the study of animal emotion and mood[END_REF][START_REF] Richardson | Joy and Calm: How an Evolutionary Functional Model of Affect Regulation Informs Positive Emotions in Nature[END_REF][START_REF] Carver | Affect and the functional bases of behavior: On the dimensional structure of affective experience[END_REF]. In a previous experiment on a group of captive blue-and-yellow macaws (Ara aranauna), we followed the birds' routine activities and their associated facial display. We showed that crown and nape ruffling were associated with activities with positive valence and low arousal levels, like quiet positive social interactions, maintenance or resting [START_REF] Bertin | Facial display and blushing: Means of visual communication in blue-and-yellow macaws (Ara Ararauna)?[END_REF]. As these behavioural categories are commonly associated with post-consummatory behaviours and positive welfare in mammals or birds [START_REF] Mattiello | How Can We Assess Positive Welfare in Ruminants?[END_REF][START_REF] Riters | Why do birds flock? A role for opioids in the reinforcement of gregarious social interactions[END_REF][START_REF] Mendl | An integrative and functional framework for the study of animal emotion and mood[END_REF][START_REF] Richardson | Joy and Calm: How an Evolutionary Functional Model of Affect Regulation Informs Positive Emotions in Nature[END_REF][START_REF] Luescher | Manual of parrot behavior[END_REF], we aimed to expand this research on emotional communication in Psittaciformes by following the routine activities of a group of cockatoos. Identifying indicators of positive welfare or calmness may help improve the well-being of the millions of parrots and cockatoos (two families of the order Psittaciformes) kept in captivity as pets, or for conservation programs.

Herein we provide a first exploration of potential facial indicators of emotions of Sulphur-crested cockatoos -highly social Australian native birds living in stable family groups and forming large feeding or resting flocks [START_REF] Styche | Distribution and behavioural ecology of the sulphur-crested cockatoo (Cacatua galerita L.) in New Zealand[END_REF]. We observed a group of non-breeding captive cockatoos in their aviary and recorded their crest, cheek and nape feather ruffling during their routine daily activities. According to our previous findings on macaws, we hypothesized that there would be a higher probability to observe feather ruffling (i.e. erection of feathers), except for the crest, in activities associated with low arousal levels and positive valence. The cognitive abilities of Psittaciformes showed striking convergence with mammals [START_REF] Emery | Bird Brain: An exploration of avian intelligence[END_REF]) and the present work provides new insights into facial emotional communication systems across vertebrates.

Material and methods

Birds and housing conditions

We observed five hand-reared Sulphur-crested cockatoos (Cacatua galerita) (2 males, 1 female and 2 undetermined sex, between 3 and 4 years old), not exposed to the public but part of a free-flying show (i.e. unrestrained outdoor flight), at the Zooparc de Beauval (Saint Aignan, 41110, France). These birds were housed in an aviary with an indoor area (250 cm x 520 cm x 260 cm) freely connected to an outdoor area (250 cm x 850 cm x 260 cm) and mixed with one citron-crested cockatoo (Cacatua sulphurea citrinocristata). The aviary contained several tree branches, perches and cords. Enrichment was provided daily (cardboard and journal paper). Birds were fed daily with fresh fruits and vegetables, germinated seeds (wheat, corn, sunflower, rice, and oat), millet seeds, oyster shells, and a commercial mix for exotic birds.

Behaviour and feather postures

To determine during which activity feathers were ruffled we used a focal sampling method with a handheld video camera recorder (Sony HDRP PJ410) capturing 24 images per second. Each day, we followed one focal bird's behaviour for twenty minutes, and successively followed another bird, in a random order, until all the five birds had been observed. These observations were repeated during three consecutive weeks until 5 hours of recording per bird had been obtained. As birds were trained for free flight daily in the morning, we observed the birds in the afternoon between 2 and 5 PM, outside of any disturbance. The experimenter was familiar to them and moved around the aviary very quietly only when necessary.

We used a scan sampling method to analyse the videos. Every 5 s, the experimenter recorded the bird's feather position and behavioural activity: locomotion (walking, flying, climbing on the grid); alimentation (ground-foraging and eating or drinking); chewing (actively cutting and/or chewing pieces of paper, branches, cardboards, cords or the grid); maintenance (preening, scratching, stretching); social contact (allopreening, perched in body contact without or with interactions i.e. gently touching the conspecific with the beak or the feet; resting (the bird is perched, the body is immobile with little or no movement of the head). Agonistic interactions such as threat displays were extremely rarely observed.

We determined three areas where feathers can move independently from one another: the crest, the cheek and the nape (Fig. 1). For the crest and the nape, when ruffled, individual feathers can be distinguished. For the cheek, the feathers were over the bill when ruffled and we considered there to be two possible positions: "lower" when part of the lower mandible was covered and "upper" when any part of the upper mandible was covered in addition to the entire lower mandible (Fig. 1 and supplementary data). When some parameters were unobservable, the scan was not included. We obtained a mean total number of 3247.6 ± 114.6 scans per individual (minimum: 2915; maximum: 3566). The same experimenter conducted all observations. We assessed observer reliability for the occurrence of feather ruffling and activities by rescoring twice 20 min of video per bird. Intra-observer accordance was 96 % (percentage of scans where the scans were scored the same).

Statistical analyses

For each behavioural category, we calculated the mean proportion of ruffled feather scans per bird for the crest, the nape and the two categories of cheek feather positions. As the data were not normally distributed, we used the function aovp of the lmPerm package in R 3.4.2, to run permutation tests on the proportion of scans containing ruffled feathers during each activity.

Activity type was considered a fixed factor while the individual was considered as a random factor. Two activity types were defined according to their presumed arousal levels (high or low). Behavioural categories without locomotion including: "maintenance", "social contact" and "resting" were grouped and considered low arousal and positive valence activity type.

These three relatively motionless behavioural categories are commonly considered quiet activities with low arousal levels and indicators of positive welfare across vertebrates [START_REF] Mattiello | How Can We Assess Positive Welfare in Ruminants?[END_REF][START_REF] Riters | Why do birds flock? A role for opioids in the reinforcement of gregarious social interactions[END_REF][START_REF] Mendl | An integrative and functional framework for the study of animal emotion and mood[END_REF][START_REF] Richardson | Joy and Calm: How an Evolutionary Functional Model of Affect Regulation Informs Positive Emotions in Nature[END_REF][START_REF] Luescher | Manual of parrot behavior[END_REF]). Behavioural categories with active behaviours and locomotion including "locomotion", "alimentation" and "chewing" were grouped and considered high arousal activity type. These behavioural categories might be associated with consummatory behaviours and/or appetite motivational states with neutral or positive emotional valence [START_REF] Mendl | An integrative and functional framework for the study of animal emotion and mood[END_REF]) but the precise valence remains understudied so far in birds. As agonistic interactions were rarely observed this category was not included in the analysis. With our limited sample size, post-hoc analyses lead to a drastic loss of statistical power and a high risk of type II error. However, a descriptive view of the data on behavioural categories is represented in the form of median and interquartile distribution ranges. To evaluate correlations between the proportions of ruffled feather scans observed on the different areas we used Spearman correlations (N=5).

Ethical note

The Zooparc de Beauval (41110, Saint Aignan) kindly provided access to their birds.

Only video-recorded observations were conducted. These birds were trained since weaning to perform unrestrained outdoor flights with their animal caretakers and the observer spent several months in presence of the birds beforehand. No signs of avoidance behaviours to the presence of the observer or the animal caretakers were observed.

Results

Crest displaying was rarely observed and did not show any significant variation according to the bird's activity type (Median [1 st quartile-3 rd quartile], lower arousal level vs. higher arousal level, 0 [0-0] vs. 0 [0-0]; df = 1; Mean Square (MS) < 0.01; P = 0.38) (Fig. 2A).

The proportion of scans where feathers ruffling was observed was significantly higher during activities with low arousal level (maintenance, social contact, resting) than during activities with higher arousal level (locomotion, alimentation, chewing) for the two positions of cheek feathers (lower mandible: 0.31 [0.05-0.80] vs. 0.05 [0.01-0.06]; df = 1; MS = 0.29; P = 0.001; upper mandible: 0.48 [0.28-0.94] vs. 0.03 [0.01-0.06]; df = 1; MS = 1.45; P < 0.001) (Fig. 2B) and for nape feathers (0.84 [0.66-0.91] vs. 0.05 [0.02-0.09]; df = 1; MS = 3.13; P < 0.001) (Fig. 2C).

We found no significant effect of the random factor individual on the fixed effect activity type for the crest (df = 4; MS = 0.06; P = 1), the cheek (lower; df = 4; MS = 0.02; P = 1; upper; df = 4; MS = 0.12; P = 1) or the nape (df = 4; MS = 0.06; P = 1).

We found a positive correlation between nape feather ruffling and both lower (rho = 0.53, P < 0.01) and upper mandible cheek position (rho = 0.89, P < 0.01) but not between the two cheek feather positions lower and upper (rho = 0.29, P = 0.11). Two few crest displaying events were recorded to perform correlations.

Discussion

This study is the first to show variation in facial feather displays according to cockatoos' activity and potential emotional state. Cheek and nape feather ruffling occurred significantly more frequently during activities with low arousal levels and positive valence; suggesting that calm or relaxed states may be indicated by bill covering and nape feather ruffling.

Previously we found that activities with low arousal levels were associated with facial and nape feather ruffling in blue-and-yellow macaws [START_REF] Bertin | Facial display and blushing: Means of visual communication in blue-and-yellow macaws (Ara Ararauna)?[END_REF]. In the current work with Sulphur-crested cockatoos we found, as hypothesized, a higher probability of ruffling feathers over the bill and nape ruffling during low arousal positive valence activities such as maintenance behaviours, social contact and resting. In the same direction, finches adopt a more spheroid body posture by ruffling their feathers when clumping or resting, which may be an effective signal inducing appeasement, clumping or allopreening in birds [START_REF] Morris | The Feather Postures of Birds and the Problem of the Origin of Social Signals[END_REF][START_REF] Moynihan | Hostile, Sexual, and Other Social Behaviour Patterns of the Spice Finch (Lonchura Punctulata) in Captivity[END_REF]. In our study, the activity "resting" was particularly associated with nape ruffling and ruffling feathers over the upper mandible. Although speculative, this position may provide a cryptic shape during resting by covering the black bill and decreasing its contrast with the white plumage. The primary functions of a bird's feathers are flight and thermoregulation of the body [START_REF] Morris | The Feather Postures of Birds and the Problem of the Origin of Social Signals[END_REF]). In addition to these functions, more subtle and localized feather movements may provide social information about a bird's arousal level or intention to engage in specific activities.

In the context of presumed higher arousal level activities such as locomotion, foraging, actively interacting with enrichments, eating and drinking, nape and cheek feather ruffling occurred significantly less. These behavioural categories might be associated with consummatory behaviours and/or appetite motivational states with neutral or positive emotional valence but this remains understudied so far in birds so caution must be taken. In cockatoos, crest raising is probably more characteristic of states of high arousal levels as it was reported in contexts of alertness, agonistic interactions or play readiness in cockatoos [START_REF] Kaplan | Bird minds: cognition and behaviour of Australian native birds[END_REF]. Accordingly, we did observe crest raising when birds were excited after a free flight or during the rare agonistic interactions we witnessed (pers. obs).

As we observed a stable group of healthy birds kept in a particularly enriched environment, we cannot rule out that in other situations head feather ruffling, in combination with specific body postures could also signal sickness or negative emotions in parrots. Our sample size was low and the inter-individual variability was important therefore, additional investigations with larger, diverse samples will be required to determine with precision the function of bird facial displays. At term, it would be of interest to observe more extreme emotional valences and different sensory channels (acoustic, visual) to understand better parrots' emotional language. For example, contact calls or long distance calls may carry information about parrots' intention to engage in specific activities and/or their emotional states (New Zealand kea parrot, Nestor notabilis; [START_REF] Schwing | Vocal repertoire of the New Zealand kea parrot Nestor notabilis[END_REF].

So far, avian visual communication is almost exclusively studied in ultimate, evolutionary contexts such as sexual selection and focused on conspicuous signals like bright colours or ornaments (e.g., [START_REF] Wachtmeister | Display in monogamous pairs: a review of empirical data and evolutionary explanations[END_REF][START_REF] Gomes | Multimodal signalling in estrildid finches: song, dance and colour are associated with different ecological and lifehistory traits[END_REF]). Our study is complementary and calls attention to the overlooked function of more subtle visual displays (non-conspicuous facial feather movements). In social groups of birds emotion expression is probably adaptive, contributing to social cohesion by regulating approach and avoidance behaviours. Head feather displays may have similar function to facial expression in mammals, which convey close-range public information regarding individuals' intention to engage in specific activities or emotions [START_REF] Waller | Facial expression in nonhuman animals[END_REF].

Conclusion

As the Psittaciformes are highly social with primate-like cognitive capacities [START_REF] Olkowicz | Birds have primate-like numbers of neurons in the forebrain[END_REF][START_REF] Gutiérrez-Ibáñez | Parrots have evolved a primate-like telencephalic-midbrain-cerebellar circuit[END_REF], they are very popular as companion animals. However, captive parrots and cockatoos are particularly sensitive to feather plucking or stereotypic behaviours, which are signs of negative welfare [START_REF] Van Zeeland | Feather damaging behaviour in parrots: A review with consideration of comparative aspects[END_REF]. Wild parrots normally spend a large amount of their diurnal time engaged in comfort, affiliative or resting behaviours (Rowley 1987;[START_REF] Bergman | Comfort behavior and sleep[END_REF]. These behavioural categories are commonly considered as reflecting calm and relaxed states and low level of threat in birds and vertebrates [START_REF] Mattiello | How Can We Assess Positive Welfare in Ruminants?[END_REF][START_REF] Riters | Why do birds flock? A role for opioids in the reinforcement of gregarious social interactions[END_REF][START_REF] Mendl | An integrative and functional framework for the study of animal emotion and mood[END_REF][START_REF] Richardson | Joy and Calm: How an Evolutionary Functional Model of Affect Regulation Informs Positive Emotions in Nature[END_REF][START_REF] Luescher | Manual of parrot behavior[END_REF]. Therefore, assessing these positive low arousal states in captive birds could help to evaluate their wellbeing and prevent damaging behaviours. Low arousal positive affect states have been less studied due to practical limitations in assessment. Our data show that subtle facial feather movements may provide reliable tools to assess calm or relaxed states. As for mammals, species-specific repertoires of facial expressions could provide useful tools to better assess well-being of captive birds and, more broadly, to better understand emotional communication across vertebrate species.

Figure captions Fig 1 .

 captions1 Figure captions

Fig 2 :

 2 Fig 2: Median and interquartile distribution ranges of the proportions of scans where

Acknowledgements

We are grateful to the Association Beauval Nature pour la Recherche et la Conservation and the Zooparc de Beauval who provided access to the birds and financial support. This funding source had no role in the study or the preparation of the article. We thank all the caretakers of the Zooparc de Beauval and particularly Amandine Diot for the care provided to the birds. We are grateful to Dr. Scott Love for improving the use of the English language in our manuscript