L Amour

J Faupin

T Raoux

Inverse spectral results for Schrödinger operators on the unit interval with partial informations given on the potentials

We pursue the analysis of the Schrödinger operator on the unit interval in inverse spectral theory initiated in [AR]. Whereas the potentials in [AR]

we consider here potentials in W k,1 spaces having their difference in W k,p where 1 ≤ p ≤ +∞, k ∈ {0, 1, 2}. It is proved that two potentials in W k,1 ([0, 1]) being equal on [a, 1] are also equal on [0, 1] if their difference belongs to W k,p ([0, a]) and if the number of their common eigenvalues is sufficiently high. Naturally, this number decreases as the parameter a decreases and as the parameters k and p are increasing.

Introduction and statement of the results

In this paper we consider the Schrödinger operator

A q,h,H = - d 2 dx 2 + q (1)
defined on [0, 1] associated with the following boundary conditions, u ′ (0) + hu(0) = 0, u ′ (1) + Hu(1) = 0.

(2)

In ( 2) and throughout the paper we use the abbreviated notation ′ for the derivative with respect to x and h, H are real numbers. In (1) the potential q is a real-valued function belonging to L 1 ([0, 1]). For each (q, h, H) ∈ L 1 ([0, 1]) × R 2 it is known that the operator A q,h,H is self-adjoint in L 2 ([0, 1]) and we denote by σ(A q,h,H ) the spectrum of this operator. Moreover, σ(A q,h,H ) is an increasing sequence of eigenvalues (λ j (q, h, H)) j∈N∪{0} , each eigenvalue being of multiplicity one. The asymptotic expansion of the eigenvalues is as follows ( [LG]), λ j (q, h, H) = j 2 π 2 + 2(Hh) + 1 0 q(x)dx + o(1) as j → +∞.

(3)

For any sequence α = (α j ) j∈N∪{0} , with α j ∈ C, and for any t ≥ 0, let n α (t) denotes

n α (t) = ♯{j ∈ N ∪ {0} | |α j | ≤ t}. (4)
The main result of the paper is the following.

Theorem 1.1. Let k ∈ {0, 1, 2}. Fix q 1 , q 2 ∈ W k,1 ([0, 1]) and h 1 , h 2 , H ∈ R. Consider an infinite set S S ⊆ σ(A q1,h1,H ) ∩ σ(A q2,h2,H ).

(5)

Fix a ∈ (0, 1 2 ] and p ∈ [1, +∞]. Suppose that q 1 = q 2 on [a, 1] and q 1q 2 ∈ W k,p ([0, a]). Assume that

n S (t) ≥ 2a n σ(A) (t) - k 2 + 1 2p - 1 2
a, t ∈ σ(A), t large enough, (H) where the operator A denotes either A q1,h1,H or A q2,h2,H . Then h 1 = h 2 and q 1 = q 2 .

In the case p = +∞ the term 1 p in the hypothesis (H) is omitted.

Remark 1.2. Theorem 1.1 remains true when replacing the assumption (H) by the hypothesis:

there exists a real number C such that

2a n σ(A) (t) + C ≥ n S (t) ≥ 2a n σ(A) (t) - k 2 + 1 2p
-2a, t ∈ S, t large enough, (H ′ ).

2

For a = 1 2 the lower bounds in the assumptions (H) and (H ′ ) are the same. Nevertheless, when a < 1 2 and if the known spectrum is in some sense regularly spaced then (H ′ ) becomes useful. For example, in [AR] we show that in the L 1 case the even (respectively odd) spectrum determines the potential on [0, 1 4 ] using (H ′ ) (with k = 0 and p = 1) whereas it is not possible with (H).

One recovers Theorem 1.1 in [AR] from Theorem 1.1 above with the assumption (H ′ ) setting k = 0 and 1 ≤ p < +∞. We remark that the case p = +∞ is excluded in [AR] whereas it is allowed here, the reason being that the details of the two proofs are different.

Let us mention some already known results related to Theorem 1.1. In 1978, [HL] proved that if

σ(A q1,h,H ) = σ(A q2,h,H ) and if q 1 = q 2 on [ 1 2 , 1] then q 1 = q 2 on [0, 1] (for L 1 potentials). This is Theorem 1.1 setting (a, k, p) = ( 1 2 , 0, 1) in the particular case h 1 = h 2 = h.
In 1980 it is derived in [H] when q 1 and q 2 are continuous near x = 1 2 that q 1 = q 2 on [0, 1] and h 2 = h 1 under the assumptions σ(A q1,h1,H ) = σ(A q2,h2,H ) excepted for at most one eigenvalue and q 1 = q 2 on [ 1 2 , 1] . In 2000, these two results were largely extended in [GS].

Theorem 1.1 with (k, p) = (0, 1) (which is also Theorem 1.1 in [AR] setting p = 1) is related to Theorem 1.3 in [GS]. See [AR] for comparisons between these two results. In another result it is proved in [GS] that q 1 = q 2 on [0, 1] and h 1 = h 2 if q 1 and q 2 are C 2k near x = a, if q 1 = q 2 on [a, 1] and assuming (H) and (H ′ ) it suffices to consider t ∈ σ(A) and S respectively, which can be useful (see the proof of corollary 1.2 in [AR]). In particular (a = 1 2 ), the potential already known on one half of the interval together with its spectrum except possibly k + 1 eigenvalues determine uniquely the potential on the other half of the interval when the potential is C 2k near the middle of the interval.

that n S (t) ≥ 2an σ(A) (t) -(k + 1) + 1 2 -a, t ∈ R large enough. Note that instead of t ∈ R, in
We now emphasize that Theorem 1.1 admits the following corollary.

Corollary 1.3. Fix H, h 1 , h 2 ∈ R. Suppose that q 1 and q 2 belongs to L 1 ([0, 1]) and are equal on 1 2 , 1 . If σ(A q1,h1,H ) = σ(A q2,h2,H ) excepted for at most one eigenvalue and if the difference q 1q 2 belongs to

L ∞ ( 0, 1 2 ) then h 1 = h 2 and q 1 = q 2 .
It is an immediate consequence of Theorem 1.1 in the particular case a = 1 2 , k = 0, p = +∞.

The above corollary is already known ( [H]) when q 1 and q 2 are continuous near x = 1 2 (see above) whereas the condition here is

q 1 -q 2 ∈ L ∞ ([0, 1 2 ]). Similarly, Theorem 1.1 implies that if σ(A q1,h1,H ) = σ(A q2,h2,H ) excepted for k + 1 eigenvalues, if q 1
and q 2 belong to W 2k,1 ([0, 1]) and are equal on 1 2 , 1 , if the difference q 1q 2 is in W 2k,∞ ( 0, 1 2 ), then h 1 = h 2 and q 1 = q 2 . This holds here for k = 0, 1 and we believe that it should be valid for all k ∈ N∪{0}. This type of results (k + 1 eigenvalues missing from the known part of the spectrum) appears in [GS] for potentials being in C 2k near x = 1 2 (see above). More generally (for any a ∈ (0, 1 2 ] and not only for a = 1 2 ) one also notes that Theorem 1.1 with p = +∞ and the result in [GS] have exactly the same assumption on n S (t). This common assumption is n S (t) ≥ 2an σ(A) (t)k -1 2a (t ∈ σ(A) in Theorem 1.1 and t ∈ R in [GS]). They differ from the hypotheses on the potential: q 1 , q 2 ∈ W 2k,1 , q 1q 2 ∈ W 2k,∞ in Theorem 1.1 and q 1 , q 2 ∈ C 2k near x = a in [GS]. Therefore Theorem 1.1 in the particular case p = +∞ and the result in [GS] are close but different.

At this point we note that our proof for Theorem 1.1 is different from the proofs of the results in [GS].

As it is explained in [H], Theorem 1.1 with a = 1 2 and k = 0 is closely related to inverse problem for the Earth which consists in the determination of the density, the incompressibility and the rigidity in the lower mantle, the upper mantle and the crust. However it is supposed in [H] that the density, the incompressibility and the rigidity are twice differentiable. Then one may think that Theorem 1.1 with a = 1 2 and k = 2 could be used for further analysis of this problem and in particular to remove some eigenvalues of the torsional spectrum, as it is mentioned in [H].

The starting point of the proof of Theorem 1.1 is related to the proof found in [L] that two spectra determine the potential, that is to say σ

(A q1,h1,H ) = σ(A q2,h2,H ′ ) and σ(A q1,h ′ 1 ,H ) = σ(A q2,h ′ 2 ,H ′ ) imply q 1 = q 2 , h 1 = h 2 , h ′ 1 = h ′ 2 and H = H ′ .
Let us recall the main steps of the proof of this result. First an entire function f is introduced having the property to vanish on the known part of the common spectra.

Next it is derived that f is identically vanishing from the maximum modulus principle. Then it is proved that f ≡ 0 imply that q 1 = q 2 (see also step 5 in [AR] for a quick proof of this point). Here we start similarly and introduce the entire function f (see ( 7)). Then the goal of the paper is to derive that if f vanishes on S then f is identically vanishing. This is established thanks to some precise estimates on the growth of f proved in Proposition 3.1. These estimates are related to the Paley-Wiener Theorem. In other words, the entire function f of a given growth vanishes if it has a sufficiently large number of zeros. This point is effectuated using Jensen formula.

Note that the case of Dirichlet boundary conditions where formally h and/or H equal ∞ (see [GS]) would demand a separated treatment. Nevertheless, one expect that this method could be applied in these situations. This article is organized as follows. In Section 2 we recall elementary results to be used in Section 3.

Section 3 is concerned with the proof of a global estimate of the entire function f . In Section 4 we derive Theorem 1.1.

Preliminaries

The asymptotic expansion (3) of the eigenvalues for q ∈ L 1 ([0, 1]) gives the spectral invariant 1 0 q(x)dx + 2(Hh). In particular, since S is an infinite set,

1 0 q 1 (x)dx - 1 0 q 2 (x)dx = 2(h 1 -h 2 ). (6) Let q ∈ L 1 ([0, 1]) and H, h ∈ R. For z ∈ C, let ψ(•, z, q, h) defined on [0, 1] as the solution to -d 2 ψ dx 2 + qψ = z 2 ψ, ψ(0) = 1, ψ ′ (0) = -h. It is known that ψ(x, •, q, h) is an entire function ([LG]). For all z ∈ C set f (z) = a 0 ψ(x, z, q 1 , h 1 )ψ(x, z, q 2 , h 2 ) - 1 2 (q 1 (x) -q 2 (x))dx. ( 7
)
In order to globally estimate the entire function f on C let us define y 1 (x, z, q) and y 2 (x, z, q) as the solutions to

- d 2 y 1 dx 2 (x, z, q) + q(x)y 1 (x, z, q) = z 2 y 1 (x, z, q), x ∈ [0, 1] y 1 (0, z, q) = 1, y ′ 1 (0, z, q) = 0 and - d 2 y 2 dx 2 (x, z, q) + q(x)y 2 (x, z, q) = z 2 y 2 (x, z, q), x ∈ [0, 1] y 2 (0, z, q) = 0, y ′ 2 (0, z, q) = 1 for q ∈ L 1 ([0, 1]), z ∈ C. It is known that y l (•, •, q) are analytic on [0, 1] × C for l = 1, 2.
In particular, one observes that

ψ(x, z, q, h) = y 1 (x, z, q) -hy 2 (x, z, q) (8) for all (x, z) ∈ [0, 1] × C. Moreover it is clear that z 2 is an eigenvalue of A q,h,H if and only if ψ ′ (1, z, q, h) + Hψ(1, z, q, h) = 0. Furthermore, if z 2 is an eigenvalue of A q,h,H then ψ(•, z, q, h) is up to a normalization coefficient the corresponding eigenfunction.
It is known (see [PT] for similar computations with potentials in L 2 ([0, 1])) that there exist c

(l) j (x, z, q)'s such that y l (x, z, q) = +∞ j=0 c (l) j (x, z, q)
for l = 1, 2 and

y l (x, z, q) = k j=0 c (l) j (x, z, q) + O e |ℑz|x |z| k+l (9) for l = 1, 2, k ∈ N and all (x, z, q) ∈ [0, 1] × C × L 1 ([0, 1]). Let us mention that c (1) 0 (x, z, q) = c (1) 0 (x, z) = cos zx, c (2) 0 (x, z, q) = c (2) 0 (x, z) = sin zx z and c (l) j (x, z, q) = 0≤t1≤•••≤tj+1=x c (l) 0 (t 1 , z) j m=1 c (2) 0 (t m+1 -t m , z)q(t m ) dt 1 . . . dt j .
The coefficients c

(l) j (x, z, q) above satisfy

|c (1) 0 (x, z)| ≤ e |ℑz|x , |c (2) 0 (x, z)| ≤ e |ℑz|x and for j ≥ 1 |c (l) j (x, z, q)| ≤ ||q|| j L 1 ([0,1]) e |ℑz|x j!|z| j+l-1 .
Note that the factor 1 |z| j+l-1 is not explicitly mentioned in [PT] but it will be useful for our purpose.

We shall modify (9) for k = 1, 2 in order to prove Theorem 1.1. Integrating by parts the c (l) j (x, z, q) for l = 1, 2 one deduce (see also section 1 in [PT] ) the expansions of y 1 and y 2

y 1 (x, z, q) = k j=0 C j (x, z, q) + O e |ℑz|x |z| k+1 (10)
and

y 2 (x, z, q) = k-1 j=0 D j (x, z, q) + O e |ℑz|x |z| k+1 (11)
uniformly in x ∈ R and z ∈ C under the assumption q ∈ W k,1 ([0, 1]). The coefficients C j (x, z, q) and D j (x, z, q) are defined below.

Let us first define the following functions Q (resp. R) by

Q(x) = x 0 q(t) dt, (resp. R(x) = q(x) -q(0) - 1 2 Q 2 (x)), for x ∈ [0, 1] provided that the potential q ∈ L 1 (resp. q ∈ W 1,1 ).
The coefficients C j (x, z, q) and D j (x, z, q) for j ≤ k when q ∈ W k,1 (k = 0, 1, 2) are defined by

C 0 (x, z, q) = C 0 (x, z) = cos zx C 1 (x, z, q) = sin zx 2z Q(x) C 2 (x, z, q) = cos zx 4z 2 R(x) (12)
and

D 0 (x, z, q) = D 0 (x, z) = sin zx z D 1 (x, z, q) = - cos zx 2z 2 Q(x) (13) for x ∈ R and z ∈ C. Using the inequalities | cos zx| ≤ e |ℑz|x , | sin zx| ≤ e |ℑz|x
for x ∈ R and z ∈ C we verify that

C j (x, z, q) = O e |ℑz|x |z| j (14)
for j = 0, 1, 2 and

D j (x, z, q) = O e |ℑz|x |z| j+1 (15)
for j = 0, 1 and uniformly in x ∈ R and z ∈ C.

The computations (integrations by parts) in order to get useful expressions for the coefficients C 3 (x, z, q)'s and D 2 (x, z, q)'s when q ∈ W 3,1 would be rather complicated. Furthermore the expression for the corresponding k 3 would be even more complicated (see next section for the definitions of k 0 , k 1 , k 2 ).

Then we restrict ourselves to the three cases k = 0, 1, 2.

Estimation of the entire function f

This section is devoted to the proof of the next proposition which states a global estimate related to the parameters k, p, a on the function f . Let us recall that throughout this section 1 p + 1 p ′ = 1 including the cases p or p ′ being infinite.

Proposition 3.1. Fix k ∈ {0, 1, 2} and let p ∈ [1, +∞]. Fix q 1 , q 2 ∈ W k,1 ([0, 1]) such that q 1 -q 2 ∈ W k,p ([0, a]) and h 1 , h 2 , H ∈ R.
In the case k = 2 we assume that σ(A q1,h1,H ) ∩ σ(A q2,h2,H ) is infinite. Also suppose that q 1 = q 2 on [a, 1]. There is a real positive number C depending only on p and ||q 1 -

q 2 || W k,p ([0,a]
) such that for any ε > 0 there exists a real positive number δ ε depending only on ε, p, a and

||q 1 -q 2 || W k,p ([0,a]) verifying lim ε→0 δ ε = 0 and |f (z)| ≤ C e 2|ℑz|a |ℑz| k+ 1 p ′ (e -ε|ℑz| + δ ε )
for all z ∈ C and where

1 p + 1 p ′ = 1.
Let us mention here that the above inequality is also valid when ε = 0 and that the role of ε > 0 will appear in next section.

The fact that supp (q 1q 2 ) ⊂ [0, a] appears in the term e 2|ℑz|a above. The regularity of the potentials q 1 , q 2 ∈ W k,1 gives the power k in denominator and the regularity of the difference q 1q 2 ∈ W k,p provides an additional power 1 p ′ . This estimate is in some sense related to the Paley-Wiener theorem. Nevertheless the factor e 2|ℑz|a will not be sufficient for our purpose and an improved estimate is needed.

Therefore the factor e -ε|ℑz| is introduced.

Set

k 0 (x, z) = C 0 (x, z)C 0 (x, z) - 1 2 k 1 (x, z, q 1 , h 1 , q 2 , h 2 ) = C 0 (x, z)C 1 (x, z, q 2 ) + C 1 (x, z, q 1 )C 0 (x, z) -h 1 D 0 (x, z)C 0 (x, z) -h 2 C 0 (x, z)D 0 (x, z) k 2 (x, z, q 1 , h 1 , q 2 , h 2 ) = C 0 (x, z)C 2 (x, z, q 2 ) + C 2 (x, z, q 1 )C 0 (x, z) + C 1 (x, z, q 1 )C 1 (x, z, q 2 ) -h 2 C 0 (x, z)D 1 (x, z, q 2 ) -h 2 D 1 (x, z, q 1 )C 0 (x, z) -h 2 C 1 (x, z, q 1 )D 0 (x, z) -h 1 D 0 (x, z)C 1 (x, z, q 2 ) + h 1 h 2 D 0 (x, z)D 0 (x, z), (16)
for all (x, z) ∈ [0, 1] × C.

Then we define

K l (z) = a 0 k l (x, z)(q 1 (x) -q 2 (x))dx for all z ∈ C.
From (8) together with (10) ( 11) and ( 14) (15) (16), one can easily check that

ψ(x, z, q 1 , h 1 )ψ(x, z, q 2 , h 2 ) - 1 2 = k l=0 k l (x, z, q 1 , h 1 , q 2 , h 2 ) + O e 2|ℑz|x |z| k+1 . ( 17
)
We shall use the result below appearing in the proof of Theorem A.III.1.3 in [L]. This result follows directly from Hölder inequality (including the case p = ∞). Indeed, |w(z)| ≤

Lemma 3.2. Let a ∈ (0, 1]. Suppose that the function u defined on [0, 1]×C satisfy |u(x, z)| = O e 2|ℑz|x and let v ∈ L p ([0, 1]) with 1 ≤ p ≤ +∞. Set w(z) = a 0 u(x, z)v(x)
C ′ I e 2|ℑz|x |v(x)|dx+C ′ J e 2|ℑz|x |v(x)|dx ≤ C ′ (||v|| L p ([0,a]) e 2|ℑz|(a-ε) |ℑz| 1 p ′ +||v|| L p (J) e 2|ℑz|a |ℑz| 1 p ′
) where I = [0, aε] and J = [aε, a] with 0 < ε < a. Here C ′ is a real number which may vary from line to line. Thus, if v is not identically vanishing then δ ε equals ||v|| L p (J) up to a numerical multiplicative factor.

We easily deduce the following proposition.

Proposition 3.3. Fix a ∈ (0, 1 2 ], k = 0, 1, 2 and suppose that q 1 , q 2 ∈ W k,1 ([0, 1]). Then there is C depending only on ||q 1q 2 || L 1 such that for any ε > 0 there is δ ε depending only on ε and ||q 1q 2 || L 1 satisfying lim ε→0 δ ε = 0 and

f (z) - k l=0 K l (z) ≤ C e 2|ℑz|a |ℑz| k+1 (e -ε|ℑz| + δ ε ) (18)
for all z ∈ C.

Proof of Proposition 3.3:

Following the definition of f together with equality (17) we have

f (z) = a 0 k l=0 k l (x, z, q 1 , h 1 , q 2 , h 2 ) + O e 2|ℑz|x |z| k+1
{q 1 (x)q 2 (x)} dx for all z ∈ C. Applying Lemma 3.2 with v = q 1q 2 we finish the proof (Note that Lemma 3.2 is only used to get a factor e -ε|ℑz| ).

Proof of Proposition 3.1. The case k = 0:

Following Proposition 3.3 it remains to check that |K 0 (z)| ≤ C e 2|ℑz|a |ℑz| 1 p ′ (e -ε|ℑz| +δ ε ) when q 1 , q 2 ∈ L 1 ([0, 1])
and q 1q 2 ∈ L p ([0, a]). This clearly follows from k 0 (x, z) = cos 2zx 2 , the definition of K 0 together with Lemma 3.2 applied with u(x, z) = cos 2zx and v = q 1q 2 .

Proof of Proposition 3.1. The case k = 1:

From Proposition 3.3 it suffices to verify that |K l (z)| ≤ C e 2|ℑz|a |ℑz| 1+ 1 p ′ (e -ε|ℑz| +δ ε ) when q 1 , q 2 ∈ W 1,1 ([0, 1])
and q 1q 2 ∈ W 1,p ([0, a]) for l = 0, 1. Since q 1q 2 ∈ W 1,1 ([0, 1]) we may integrate by parts K 0 (z) to obtain

K 0 (z) = sin 2zx 4z (q 1 (x) -q 2 (x))| a 0 - a 0 sin 2zx 4z (q ′ 1 (x) -q ′ 2 (x))dx.
Since q 1 (a) = q 2 (a) then the first term in the r.h.s. of the above equality vanishes and one concludes using Lemma 3.

2 with v = q ′ 1 -q ′ 2 ∈ L p ([0, a]) that |K 0 (z)| ≤ C e 2|ℑz|a |ℑz| 1+ 1 p ′ (e -ε|ℑz| + δ ε ). Set Q j (x) = x 0 q j (t)dt, j = 1, 2. Similarly K 1 (z) = a 0 sin 2zx 4z (Q 1 (x) + Q 2 (x) -2(h 1 + h 2 ))(q 1 (x) -q 2 (x))dx also verify |K 1 (z)| ≤ C e 2|ℑz|a |ℑz| 1+ 1 p ′ (e -ε|ℑz| +δ ε ) using again Lemma 3.2 with v = (Q 1 +Q 2 -2(h 1 +h 2 ))(q 1 - q 2 ) ∈ L p being the product of function in L ∞ with a function in L p .
It remains to consider the case k = 2 which is much longer.

We shall first prove the following proposition.

Proposition 3.4. Under the assumptions of Proposition 3.1 (with k = 2) there exists a real number L(q 1 , h 1 , q 2 , h 2 ) and there is a real positive number C depending only on p and ||q 1q 2 || W 2,p ([0,a]) such that for any ε > 0 there exists a real positive number δ ε depending only on ε, p, a and ||q 1 -

q 2 || W 2,p ([0,a])
verifying lim ε→0 δ ε = 0 and

f (z) - L(q 1 , h 1 , q 2 , h 2 ) z 2 ≤ C e 2|ℑz|a |ℑz| 2+ 1 p ′ (e -ε|ℑz| + δ ε ).
Proposition 3.4 follows from Propositions 3.3, 3.5, 3.7 and 3.9 below.

In order to derive proposition 3.4 let us estimate K l for l = 0, 1, 2 in the next three propositions.

Proposition 3.5. Let p ∈ [1, +∞]. Fix q 1 , q 2 ∈ W 2,1 ([0, 1]) with q 1 -q 2 ∈ W 2,p ([0, a]) and h 1 , h 2 , H ∈ R. Suppose that q 1 = q 2 on [a, 1]. Set L 0 (q 1 , q 2 ) = - 1 8 (q ′ 1 (0) -q ′ 2 (0)).
There is a real positive number C depending only on p and ||q ′′ 1q ′′ 2 || L p ([0,a]) such that for any ε > 0 there is a real positive number δ ε depending only on ε, p, a and ||q ′′ 1q ′′ 2 || L p ([0,a]) satisfying lim ε→0 δ ε = 0 and

K 0 (z) - L 0 (q 1 , q 2 ) z 2 ≤ C e 2|ℑz|a |ℑz| 2+ 1 p ′ (e -ε|ℑz| + δ ε ).
Note that since q j ∈ W 2,1 ([0, 1]) then q ′ j has a continuous representant for all j = 1, 2. Therefore L 0 (q 1 , q 2 ) is well-defined.

Proof of Proposition 3.5: Since q j ∈ W 2,1 ([0, 1]) we may integrate by parts twice K 0 (z) for each z ∈ C. Using q 1 (a) = q 2 (a) and q ′ 1 (a) = q ′ 2 (a) we remark using two integrations by parts that

K 0 (z) = - q ′ 1 (0) -q ′ 2 (0) 8z 2 - a 0 cos 2zx 8z 2 (q ′′ 1 (x) -q ′′ 2 (x))dx.
Using Lemma 3.2 and q 1q 2 ∈ W 2,p ([0, a]) for each ε > 0 there is δ ε (δ ε → 0 as ε → 0) such that the second term in the r.h.s. of the above equality is now bounded by e 2|ℑz|a |ℑz| 2+ 1 p ′ (e -ε|ℑz| + δ ε ). .

In the proof of Proposition 3.7 we shall need the following lemma.

Lemma 3.6. Suppose q 1 , q 2 ∈ W 2,1 ([0, 1]) with their difference q 1q 2 ∈ W 2,p ([0, 1]) and assume that

q 1 (a) = q 2 (a) (a ∈ [0, 1]). Then (Q 1 + Q 2 )(q 1 -q 2 ) ∈ W 1,p ([0, 1]).
Here Q j (x) =

x 0 q j (t)dt, j = 1, 2.

Proof of Lemma 3.6:

Since (Q 1 + Q 2 ) and (q 1 -q 2 ) belongs to W 1,1 ([0, 1]) then (Q 1 + Q 2 )(q 1 -q 2 ) ∈ W 1,1 ([0, 1]) and ((Q 1 + Q 2 )(q 1 -q 2 )) ′ = (q 1 + q 2 )(q 1 -q 2 ) + (Q 1 + Q 2 )(q 1 -q 2 ) ′ . Using Q 1 + Q 2 ∈ L ∞ and (q 1 -q 2 ) ′ ∈ W 1,1 ⊂ L p we deduce that the second term above is in L p ([0, 1]).
Let us check that the first term also belongs to L p ([0, 1]).

We have for all x ∈ [0, 1],

q 2 1 (x) -q 2 2 (x) = 2 x a q 1 (t)q ′ 1 (t) -q 2 (t)q ′ 2 (t)dt (19) = x a (q 1 (t) -q 2 (t))(q ′ 1 (t) + q ′ 2 (t)) + (q 1 (t) + q 2 (t))(q ′ 1 (t) -q ′ 2 (t))dt, (20)
since q 2 1 (a) = q 2 2 (a).

Since q j ∈ W 2,1 then q j ∈ W 1,∞ for j = 1, 2 and we have

|q 2 1 (x) -q 2 2 (x)| ≤ C x a |q 1 (t) -q 2 (t)| + |q ′ 1 (t) -q ′ 2 (t)|dt for all x ∈ [0, 1].
Here C is a positive real number which may vary from line to line. Then Hölder inequality shows

|q 2 1 (x) -q 2 2 (x)| ≤ C(||q 1 -q 2 || L p ([0,1]) + ||q ′ 1 -q ′ 2 || L p ([0,1]) ), for all x ∈ [0, 1].
Thus, a convexity inequality shows that

|q 2 1 (x) -q 2 2 (x)| p ≤ C(||q 1 -q 2 || p L p ([0,1]) + ||q ′ 1 -q ′ 2 || p L p ([0,1]) )
for all x ∈ [0, 1] and integrate this inequality over [0, 1] to obtain Lemma 3.6.

Proposition 3.7. Let p ∈ [1, +∞]. Fix q 1 , q 2 ∈ W 2,1 ([0, 1]) with q 1 -q 2 ∈ W 2,p ([0, a]) and h 1 , h 2 , H ∈ R. Set L 1 (q 1 , q 2 , h 1 , h 2 ) = -1 4 (h 1 + h 2 )(q 1 (0) -q 2 (0)).
There is a real positive number C depending only on p and ||q 1q 2 || W 1,p ([0,a]) such that for any ε > 0 there exists a real positive number δ ε depending only on ε, p, a and ||q 1q 2 || W 1,p ([0,a]) satisfying lim ε→0 δ ε = 0 and

K 1 (z) - L 1 (q 1 , q 2 , h 1 , h 2 ) z 2 ≤ C e 2|ℑz|a |ℑz| 2+ 1 p ′ (e -ε|ℑz| + δ ε ).
Proof of Proposition 3.7:

Following (12)( 13) and ( 16)

k 1 (x, z) = sin 2zx 2z A(x)
where

A(x) = Q 1 (x) + Q 2 (x) 2 -(h 1 + h 2 ),
for all x ∈ [0, 1] and all z ∈ C. From Lemma 3.6 we have A(q 1q 2 ) ∈ W 1,p . Since A(q 1q 2 ) ∈ W 1,1 we may integrate by parts K 1 (z) for each z ∈ C to obtain,

K 1 (z) = - 1 4z 2 (h 1 + h 2 )(q 1 (0) -q 2 (0)) + a 0 cos 2zx 4z 2 (A(x)(q 1 (x) -q 2 (x))) ′ dx, for all (x, z) ∈ [0, 1] × C.
Since A(q 1q 2 ) ∈ W 1,p ([0, a]) we derive from Lemma 3.2 that for each ε > 0 there is δ ε going to zero as ε → 0 such that the second term above is now bounded by e 2|ℑz|a

|ℑz|

2+ 1 p ′ (e -ε|ℑz| + δ ε ). . Set B 1 = R 1 + R 2 + 2h 1 Q 1 + 2h 2 Q 2 B 2 = Q 1 Q 2 -2h 2 Q 1 -2h 1 Q 2 + 4h 1 h 2 B = B 1 + B 2 (21)
on [0, 1], where R j (x) = q j (x)q j (0) -1 2 Q 2 j (x), for x ∈ [0, 1] and j = 1, 2.

Lemma 3.8. Suppose that q 1 , q 2 ∈ W 2,1 ([0, 1]) with q 1q 2 ∈ W 2,p ([0, 1]). Then we have B j (q 1q 2 ) ∈ L p ([0, 1]) for j = 1, 2.

Proof of Lemma 3.8:

Since Q 1 , Q 2 ∈ L ∞ ([0, 1]) it suffice to check that (R 1 + R 2 )(q 1 -q 2 ) ∈ L p ([0, 1]). From (R 1 (x) + R 2 (x))(q 1 (x) -q 2 (x)) = q 2 1 (x) -q 2 2 (x) -(q 1 (0) + q 2 (0))(q 1 (x) -q 2 (x)) - 1 2 (Q 2 1 (x) + Q 2 2 (x))(q 1 (x) -q 2 (x))
and similarly as in the proof of Lemma 3.6 we derive Lemma 3.8.

Proposition 3.9.

Let p ∈ [1, +∞]. Fix q 1 , q 2 ∈ W 2,1 ([0, 1]) with q 1 -q 2 ∈ W 2,p ([0, a]) and h 1 , h 2 , H ∈ R. Set L 2 (q 1 , q 2 , h 1 , h 2 ) = 1 8 a 0 B(x)(q 1 (x) -q 2 (x))dx.
There is a real positive number C depending only on p and ||q 1q 2 || L p ([0,a]) such that for any ε > 0 there exists a real positive number δ ε depending only on ε, p, a and ||q 1q 2 || L p ([0,a]) satisfying lim ε→0 δ ε = 0 and

K 2 (z) - L 2 (q 1 , q 2 , h 1 , h 2 ) z 2 ≤ C e 2|ℑz|a |ℑz| 2+ 1 p ′ (e -ε|ℑz| + δ ε ).
Proof of Proposition 3.9:

From ( 16) and ( 21)

k 2 (x, z) = cos 2 zx 4z 2 B 1 (x) + sin 2 zx 4z 2 B 2 (x) = B 1 (x) + B 2 (x) 8z 2 + cos 2zx 8z 2 B 1 (x) - sin 2zx 8z 2 B 2 (x).
From Lemma 3.6 and Lemma 3.8

a 0 cos 2zx 8z 2 B 1 (x)(q 1 (x) -q 2 (x))dx ≤ C e 2|ℑz|a |ℑz| 2+ 1 p ′ (e -ε|ℑz| + δ ε ) and a 0 sin 2zx 8z 2 B 2 (x)(q 1 (x) -q 2 (x))dx ≤ C e 2|ℑz|a |ℑz| 2+ 1 p ′ (e -ε|ℑz| + δ ε )
for all z ∈ C and where C and δ ε are defined in the statement of the proposition.

This and the definition of L 2 prove Proposition 3.9.

In particular Proposition 3.5, 3.7 and 3.9 imply Proposition 3.4 with L(q 1 , h 1 , q 2 , h 2 ) = L 0 (q 1 , q 2 ) + L 1 (q 1 , h 1 , q 2 , h 2 ) + L 2 (q 1 , h 1 , q 2 , h 2 ).

In order to get Proposition 3.1 in the case k = 2 from Proposition 3.4 it suffices to verify the next result which follows directly from Riemann-Lebesgue lemma.

Proposition 3.10.

If σ(A q1,h1,H ) ∩ σ(A q2,h2,H ) is infinite then L(q 1 , h 1 , q 2 , h 2 ) = 0.
Proof of Proposition 3.10:

We first remark that the following estimates holds (for z ∈ R)

f (z) = L(q 1 , h 1 , q 2 , h 2 ) z 2 + o 1 z 2 as z → +∞. (22)
Indeed, in order to get ( 22) one may use Riemann-Lebesgue lemma instead of using Lemma 3.2 in the proofs of Proposition 3.5, 3.7 and 3.9.

Let (t j ) j∈N be an infinite increasing sequence in σ(A q1,h1,H ) ∩ σ(A q2,h2,H ). It is known that

∀ j ∈ N, f (t j ) = 0. (23)
Indeed let z 2 = t j and ψ(x, z, q 1 , h 1 ) and ψ(x, z, q 2 , h 2 ) be the corresponding eigenfunctions, multiply d 2 dx 2 + q 1 (x)z 2 ψ(x, z, q 1 , h 1 ) = 0 by ψ(x, z, q 2 , h 2 ), multiply -d 2 dx 2 + q 2 (x)z 2 ψ(x, z, q 2 , h 2 ) = 0 by ψ(x, z, q 1 , h 1 ) and integrate their difference on [0, 1] to obtain that f (z) equals 2(h 1 -h 2 )+ 1 0 q 2 (x)q 1 (x)dx. This term is zero from (6). Consequently, (6) shows (23). Therefore ( 22) and ( 23 As a complementary result we have

Proposition 3.11. Let q ∈ W 2,1 ([0, 1]) and h ∈ R. Define L ♯ (q, h) = -q ′ (0) -4hq(0) + 1 0 q 2 (t)dt + 4h 2 1 0 q(t)dt -2h 1 0 q(t)dt 2 + 1 3 1 0 q(t)dt 3 . Then 8L(q 1 , h 1 , q 2 , h 2 ) = L ♯ (q 1 , h 1 ) -L ♯ (q 2 , h 2 ).
In particular L ♯ (q, h) is a spectral invariant.

This means that if q 1 , q 2 ∈ W 2,1 ([0, 1]), H, h 1 , h 2 ∈ R and if σ(A q1,h1,H ) = σ(A q2,h2,H ) then L ♯ (q 1 , h 1 ) = L ♯ (q 2 , h 2 )
. This spectral invariant is very probably related to the spectral invariant c 3 (q, h, H) appearing in the asymptotic expansion of the eigenvalues λ j (q, h, H) when q ∈ W 2,1 ([0, 1])

λ j (q, h, H) = jπ + c 1 (q, h, H) jπ + c 3 (q, h, H) j 3 π 3 + o 1 j 3 as j → +∞ in the sense that c 3 (q 1 , h 1 , H) = c 3 (q 2 , h 2 , H) ⇒ L ♯ (q 1 , h 1 ) = L ♯ (q 2 , h 2 ) in the same way that c 1 (q, h, H) = H -h + 1 2 1 0 q(x)dx is a spectral invariant and that c 1 (q 1 , h 1 H) = c 1 (q 2 , h 2 , H) ⇒ 1 0 q 1 (x)dx + 2h 1 = 1 0 q 2 (x)dx + 2h 2 .
The spectral invariant L ♯ (q, h) is probably well-known but we are not able to indicate an exact reference.

In particular, we have not seen the coefficient c 3 written down explicitly for the boundary conditions (2)

(for the existence of c 3 and more generally for the existence of c 2j+1 when the potential is sufficiently regular, see Section 5.6.1 in [LS], Problems in Section 5 of [M], or [LG]. See also [KP] and the references therein for the spectral invariants in the periodic case related to the KdV hierarchy). The proof of Proposition 3.11 is obtained by direct computations and we omit it. Furthermore, if the coefficient c 3 was known, Proposition 3.11 would provide an alternative proof (instead of the use of Riemann-Lebesgue lemma) of the equality L(q 1 , h 1 , q 2 , h 2 ) = 0 in Proposition 3.10.

4 Proof of Theorem 1.1

The purpose of this section is to deduce Theorem 1.1 from Proposition 3.1 and the hypothesis (H) or the hypothesis (H ′ ).

Define the s j , j ∈ N as the strictly increasing sequence being in S. For any c ∈ R the map q → q + c acts as λ j (q, h, H) → λ j (q, h, H) + c for each j ∈ N. Therefore we suppose without loss of generality that all the s j are strictly positive numbers. Therefore we may define the sets S 1 2 = {± √ s j , j ∈ N}, S 1 2 ,+ = { √ s j , j ∈ N}.

We also set for any sequence of numbers α,

N α (R) = R 0 n α (t) t dt,
for any R > 0 and where n α (t) is defined in (4).

for j ∈ N. From (3) we see that there exist a positive real number C satisfying

jπ - C j ≤ λ j ≤ jπ + C j , j ∈ N. (25)
Therefore, j ln λ j = j(ln j + ln π) + O(1), ln λ j = ln j + O(1),

as j → +∞.

Moreover, using Stirling asymptotic expansion ln j! = j + 1 2 ln jj + O(1) as j → +∞ we obtain,

as j → +∞.

Combining ( 24) with ( 26) and ( 27) we obtain the following estimate

N S 1 2 ( λ j ) ≥ 4aj + 1 p -1 -k ln j + O(1), (28)

  dx. There is a real positive number C depending only on p and ||v|| L p ([0,a]) such that for any ε > 0 there is a real positive number δ ε depending only on ε, p, a and ||v|| L p ([0,a]) ε|ℑz| + δ ε ).

  Proof of Proposition 3.1. The case k = 2: It now follows from Proposition 3.4 and Proposition 3.10.

Proposition 4. 1 .

 1 The hypothesis (H ′ ) implies that the sequenceN ln λ k+1 -ln λ k ) + 1 p -1 -2ak (ln λ k+1 -ln λ k ) + O(1) ln λ k+1k ln λ k ) -4a 2ak ln λ j + O(1)= 4aj ln λ j -4a 2ak ln λ j + O(1),

  ln k + ln 1 + C k 2 ) + O(1) ≤ j ln π + ln(j -1)! + O(1) = j ln π + j -1 2 ln jj + O(1),

is bounded from below.

Proof of Proposition 4.1: This is a straightforward modification of the arguments found in [AR], where k = 0. It suffice to replace (H 1 ), (H 2 ) and (H L ) in [AR] (where 1 p + 1 p ′ = 1). We also note that p = +∞ is allowed.

In the following, A is the operator chosen in Theorem 1.1. We also denote by (λ j ), j ∈ N the increasing sequence of its eigenvalues and σ(A)

1 2 ,+ denote the sequence ( λ j ), j ∈ N.

Proposition 4.2.

The assumption (H) implies that the sequence

In the proof of the Proposition 4.1 the assumption (H ′ ) is mainly used in order to derive an asymptotic expansion of the sequence ( √ s j ) (see (H 2 ) in [AR]). In particular, it is at this point that we need an estimate from above for the s j 's provided by the left inequality in (H ′ ). Then N S 1 2 is evaluated on the √ s j 's. In the proof of the Proposition 4.2 we proceed slightly differently even if the computations are similar. We first use (H) to minorize N S 1 2 (R) for all R > 0. We evaluate N S 1 2 on the λ j 's. The point being that asymptotic expansion of the λ j 's is available without any supplementary assumption. Consequently, we do not need any estimate from above for the s j 's and there is no estimation from above in (H) for n S (t).

Proof of Proposition 4.2:

We have for any R > 0,

where the last inequality follows from (H).

as j → +∞.

Turning back to the λ j 's, (28) reads as

as j → +∞. This concludes the proof.

Proposition 4.3. Suppose that the function f is not identically vanishing on C. Then

Proof of Proposition 4.3:

This is a consequence of Proposition 3.1 with Jensen's Theorem. This argument is borrowed to [L].

Let n f (t) be the number of zeros of the entire function f in the closed ball centered at the origin with radius t > 0. In the proof of Proposition 3.10 we recall that if z 2 belongs to S then f (z) = 0. Thus n S 1 2 (t) ≤ n f (t) for all t > 0. Moreover, using the estimates in Proposition 3.1 in Jensen's Theorem we obtain

If R is large enough and ε is small enough (> 0) the third term in the r.h.s. of ( 29) is smaller than any negative number.

Proof of Theorem 1.1:

If f is not identically vanishing then Proposition 4.1 or Proposition 4.2 together with Proposition 4.3 lead to a contradiction. Thus f (z) = 0 for all z ∈ C. This implies using the short argument in [START_REF] Amour | Inverse spectral results for Schrödinger operators on the unit interval with potentials in L p spaces[END_REF]step 5] that q 1 = q 2 and h 1 = h 2 .


Quang-Khoi Nguyen

Shih-Kang Lin

Chi-Shou Wu

Yeh-Wei Yu

Benoit Glorieux

Tung-Hsun Yang

Benoit A Glorieux

Benoît Glorieux

Ching-Cherng Sun
email: ccsun@dop.ncu.edu.tw

A novel solution for solving the blue light hazard problem in phosphor-converted white light emitting diodes

Keywords: pcW-LEDs, blue light hazard, CNCII, optical fuse, blue light leakage

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A novel solution for solving the blue light hazard problem in phosphor-converted white light emitting diodes

Introduction

Phosphor-converted white LEDs (pcW-LEDs), based on a blue die with yellow phosphor [START_REF] Shih | Study of temperature distributions in pc-WLEDs with different phosphor packages[END_REF], well-known with advantages including high efficiency, a wide color range, fast response, long life, and potential environmental benefits. One of the problems associated with phosphor-converted white LEDs (pcW-LEDs) for long-time stability is the thermal effect of degradation on the external quantum efficiency of LED dies and the phosphor, where the thermal effect plays a crucial role. The thermal effect causes more heat flow on the phosphor and, consequently, the thermal quenching of the phosphor particles induces a marked drop in external quantum efficiency. When the yellow light emitted by the phosphor decreases, the balance between the blue light and yellow light at a fixed correlated color temperature (CCT) in a pcW-LED is lost. Accordingly, the thermal effect can ultimately induce blue light leakage, which can cause the pcW-LED to emit white light at an unexpectedly high CCT in the defined operation lifetime; moreover, the thermal effect could be worsened if the affected pcW-LED is still deemed workable and used by a user. In this paper, we propose a novel solution for preventing the blue light leakage hazard induced by the thermal effect at the final stage, when the blue-yellow balance is lost. A so-called optical fuse is implemented by introducing a special thermal chromatic chemical to disable the pcW-LED when blue light leakage can occur.

Principle of solution

Principle of solution based on the big difference of temperature distribution in the packaging volume of pcW-LEDs that corresponds to different operation stage as shown in Fig. 1. A thermal chromatic chemical (CNCII) is selected to implants in the packaging volume of pcW-LEDs so that as it is darkened by overheat in pcW-LEDs that corresponds to occurring of the blue light leakage. Once it's color is turned to darkening, a significant amount of the Blue light is absorbed and thus, the blue light leakage hazard successfully is prevented.

Results and discussion

In considering the temperature effect, the CNCII is placed at an appropriate location in the packaging volume. Fig. 2 shows the packaging volume of pcW-LEDs in which the CNCII is placed above the conformal phosphor layer. The performance of preventing blue light hazard is evaluated through the related optical characteristics that were studied through an aging process with over drive. As shown in Fig. 2, the relative intensity and CCT were monitored, as indicated in the top views of the LED with or without the CNCII hemisphere at different stages in the aging process. In the second half of the steady-state period, the CCT of the pcW-LED without CNCII increased slowly, but the CCT with CNCII remained stable and even decreased because of the initial action of the thermal chromatic effect on CNCII. When the breakdown process started and went through L70, the CNCII darkened, and more blue light was absorbed so that the maximum CCT stopped at 6452K. By contrast, in the sample without CNCII, the CCT dramatically increased to as high as 14000K after passing the period by L70. The introduction of the CNCII was proven to block blue leakage and to darken the pcW-LED effectively in the period passing through L70. The trade-off is the degradation of the brightness of the pcW-LED under normal conditions.

Conclusion

In summary, we propose a novel solution for preventing the blue light leakage hazard induced by the thermal effect at the final stage, when the blue-yellow balance is lost. A so-called optical fuse is implemented by introducing a special thermal chromatic chemical to disable the pcW-LED when blue light leakage can occur. The experimental results indicated the passive optical fuse successfully prevents the blue light leakage hazard. The proposed novel approach with the passive optical fuse provides people with a safer lighting environment through the use of pcW-LEDs.

Acknowledgment

This research was funded by the Ministry of Science and Technology of the Republic of China, grant numbers MOST 109-2221-E-008 -087 -MY2 and 109-2622-E-008 -014 -CC2.

Fig. 1 (

 1 Fig. 1 (a) Geometry of a conformally coated pcW-LED shown in half. The blue, yellow, and grey regions indicate the LED chip, phosphor, and silicone lens, respectively. Case (b) can be considered a normal operation condition. Cases (c) and (d) represent the pcW-LED at the early and late stages of failure.

Fig. 2 .

 2 Fig. 2. (a) Schematic of the packaging structure embedded with CNCII above the phosphor layer. (b) An example of an aging process involves showing the emitting flux and CCT, of a pcW-LED with or without introducing CNCII in the LED package. The red (black) line corresponds to with (without) the CNCII. In the aging experiment, the over-drive current caused less emission flux in the steady state than during normal operation. (c) Corresponding spectra with or without the CNCII at selected four stages, as indicated by the triangles in (b). (d) Top views of the pcW-LEDs with CNCII above the phosphor layer.


Julien Berestycki
email: julien.berestycki@stats.ox.ac.uk

Eric Brunet
email: eric.brunet@lps.ens.fr

Aser Cortines
email: aser.cortinespeixoto@math.uzh.ch

Bastien Mallein
email: mallein@math.univ-paris13.fr

Éric Brunet

A simple backward

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Spatial branching processes, and in particular, the behaviour of their extremal particles, have been at the centre of a wide research activity over the past few years, both in the physics [BD09, [START_REF] Brunet | Phenomenological theory giving the full statistics of the position of fluctuating pulled fronts[END_REF][START_REF] Derrida | Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion[END_REF] and in the mathematical literature [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF][START_REF] Aïdékon | Branching Brownian motion seen from its tip[END_REF][START_REF] Arguin | The extremal process of branching Brownian motion[END_REF][START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF]. These models have a rich and complex structure that is of intrinsic interest, but they are also representatives of an intriguing "universality" class, the so-called log-correlated fields which includes the two-dimensional Gaussian free field [START_REF] Bramson | Convergence in law of the maximum of the twodimensional discrete Gaussian free field[END_REF][START_REF] Biskup | Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field[END_REF], Gaussian multiplicative chaos [START_REF] Rhodes | Gaussian multiplicative chaos and applications: a review[END_REF], random matrices [START_REF] Arguin | Maximum of the characteristic polynomial of random unitary matrices[END_REF] and others.

Perhaps the simplest model in this class is the branching Brownian motion, in which particles move in R as Brownian motions, branch into two particles at rate one and behave independently of each others. For the system started with a single particle at the origin, let N t be the set of particles alive at time t and for u ∈ N t let X t (u) ∈ R be its position. For s ≤ t we will also write X s (u) for the position of the unique ancestor of u at time s so that (X s (u), s ≤ t) is the path followed by the particle u. Then, it was proved in [START_REF] Aïdékon | Branching Brownian motion seen from its tip[END_REF][START_REF] Arguin | The extremal process of branching Brownian motion[END_REF] that the point measure

E t := u∈Nt δ Xt(u)- √ 2t+ 3 2 √ 2 log t
(1.1) converges in law, as t → ∞ toward a random intensity decorated Poisson point process (DPPP for short) E ∞ .

In general, the law of a DPPP E is characterized by a pair (ν, D) where ν is a random sigma-finite measure on R and D is the law of a random point process on R. The point measure E can be constructed, conditionally on ν, by first taking a realisation of a Poisson point process on R with intensity ν, whose atoms are listed as (x i , i ∈ I), and an independent family of i.i.d. point processes (D i , i ∈ I) with law D. Then, each atom x i is replaced by the point process D i , shifted by x i (this action is called the decoration of x i with a point process of law D). In other words, writing (d j i , j ∈ J i ) the atoms of the point process D i , we have

E = i∈I j∈Ji δ xi+d j i .
(1.2)

We refer to [START_REF] Subag | Freezing and decorated Poisson point processes[END_REF] for an in-depth study of random intensity decorated Poisson point processes, and their occurrences as limit of extremal point measures.

With this notation, E ∞ := lim t→∞ E t is the following DPPP

E ∞ = DPPP(κZ ∞ e - √ 2x dx, D 1 ) (1.3)
where κ is an implicit constant, Z ∞ is the a.s. positive limit of the so-called derivative martingale

Z t := u∈Nt √ 2t -X t (u) e √ 2Xt(u)-2t , (1.4)
and where the decoration law D 1 is the law of a point measure supported on (-∞, 0], with an atom at 0 defined by the following weak limit

D 1 (•) := lim t→∞ P u∈Nt δ {Xt(u)-Mt} ∈ • M t ≥ √ 2t , (1.5)
where M t := max u∈Nt X t (u). Moreover, it is well-known that max E t converges in distribution toward max E ∞ , where max E is the position of the largest atom in a point process E (see Lalley and Selke [START_REF] Lalley | A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion[END_REF]).

The decoration law D 1 belongs to the family (D , ∈ [1, ∞]), defined, for < ∞ by the weak limits

D (•) := lim t→∞ P u∈Nt δ {Xt(u)-Mt} ∈ • M t ≥ √ 2 t .
(1.6)

We denote by D ∞ the law of the Dirac mass at 0. The family D (•) was introduced by Bovier and Hartung [START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF] as the decorations appearing in the extremal processes of variable speed branching Brownian motions.

A detailed statement of the result of Bovier and Hartung is given in Section 4. This convergence in law can be seen as an extension of the Yaglom limit [CR88, Theorem 3], that is the convergence of the number of particles above level √ 2 t conditionally on M t ≥ √ 2 t. The decoration law D (•) can also appear in the context of multitype branching Brownian motions [START_REF] Belloum | Anomalous spreading of reducible multitype branching Brownian motion[END_REF].

Note that the law D is constructed by conditioning the branching Brownian motion on a large deviation event for its maximum. For ∈ (1, ∞) we define C( ) := lim t→∞ t 1/2 e ( 2 -1)t P(M t ≥ √ 2 t).

(1.7)

The asymptotic behaviour of P(M t > √ 2 t) was first studied in the seminal paper [START_REF] Chauvin | KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees[END_REF] (where the existence of the limit C( ) is implicit) and the function C( ) plays a key role in [START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF] where it is proven that C(1) = 0 and that lim →∞ C( ) = (4π) -1/2 . We plot in Figure 1 More recently, the same function C(•) is the focus of [START_REF] Derrida | Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion[END_REF] where, in particular, the asymptotic behaviour of C( ) as → ∞ and → 1 are conjectured. Using non-rigorous analytic computations, they conjectured that C( ) ∼ →1+ α , where, using our notation

α = 2 lim z→∞ e √ 2z √ 2z lim t→∞ P(M t ≥ √ 2t -3 2 √ 2 log t + z). (1.8)
The behavior of C( ) as → 1 is deeply linked to moderate and large deviations of the maximal displacement of the branching Brownian motion, see [START_REF] Derrida | Large Deviations for the Rightmost Position in a Branching Brownian Motion[END_REF][START_REF] Gantert | Large deviations for the maximum of a branching random walk[END_REF][START_REF] Buraczewski | Large deviation estimates for branching random walks[END_REF] for further recent developments on this topic. The goal of this article is to study both the function → C( ) and the family (D , ∈ (1, ∞]). We provide a new construction of these quantities, that do not rely on the conditioning on a vanishing event but uses a spine decomposition. Recall that a sequence of random point measures (P t ) t≥0 on R converges to P in law for the topology of vague convergence if and only if, for every compactly supported continuous function ϕ, the real valued random variables P t , ϕ := ϕ(x)P t (dx)

(1.9) converge in law to P, ϕ as t → ∞. We prove in this article that C is continuous on [1, ∞], and that → D is continuous on (1, ∞] for the topology of vague convergence. This can be used to extend the main theorem of [START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF]. Let (B t , t ≥ 0) be a standard Brownian motion, (σ k , k ∈ N) be the ranked atoms of a Poisson point process with intensity 2 dx on R + and (X

(k) t (u), u ∈ N (k)
t , t ≥ 0) for k ∈ N be i.i.d. branching Brownian motions. We shall assume that B, (σ k , k ≥ 1) and (X (k) , k ≥ 1) are independent of one another. Given ∈ (1, ∞) and t ≥ 0, we define the point process

D = δ 0 + k∈N u∈N (k) σ k δ Bσ k - √ 2 σ k +X (k)
σ k (u) .

(1.10)

In words, D is the point process constructed using a Brownian motion with drift -√ 2 , that spawns branching Brownian motions at rate 2. A branching Brownian motion spawned at time σ k then starts evolving backward in time until it hits time 0, the particles alive at that time are added to the point process.

Then (i) C( ) = 1 √ 4π P D ((0, ∞)) = 0 for all > 1. The function C is continuous on [1, ∞]. It also satisfies C(1) = 0, C( ) > 0 for > 1 and C(∞) = 1/ √ 4π. (ii) P D ∈ • = P D ∈ • | D ((0, ∞)) = 0 . The family of point processes (D , ∈ (1, ∞]
) is continuous in the space of Radon point measures equipped with the topology of vague convergence.

The rest of the article is organized as follows. In Section 2 we introduce the spinal decomposition of the branching Brownian motion, and its application to the extremal process of the branching Brownian motion, seen from the rightmost particle. We then prove Theorem 1.1 in Section 3. Then, as an application of Theorem 1.1, in Section 4 we show how the results of Bovier and Hartung [START_REF] Bovier | The extremal process of two-speed branching Brownian motion[END_REF][START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF] about variable speed branching Brownian motion also describe the extremal point process of a branching Ornstein-Uhlenbeck. We conclude this article with some open questions.

Spinal decomposition at the maximum

We apply the so-called spinal decomposition of the branching Brownian motion to obtain the joint law of the maximum and the extremal process of the branching Brownian motion. The spinal decomposition is an alternative description of the process constructed via the probability tilting by the additive martingale W √ 2 , which is defined for all t ≥ 0 by W

√ 2 t := u∈Nt e √ 2 Xt(u)-( 2 +1)t .
(2.1)

This idea was already at the heart of the approach taken in [START_REF] Chauvin | KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees[END_REF] to study branching Brownian motions conditioned to have large displacements (i.e. the event M t > ct with c > √ 2). In particular, they give in [CR88, Theorem 5] a description of the law of the branching Brownian motion biased by W t up to the first branching time. Spine decomposition techniques were expended by Lyons, Peamantle and Peres in [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF] to study Galton-Watson processes, then generalized to branching random walks by Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF] and to general branching processes in [START_REF] Biggins | Measure change in multitype branching[END_REF]. Let (F t ) be the natural filtration of the branching Brownian motion, defined by

F t = σ N s , (X s (u), u ∈ N s ), s ≤ t .
For ∈ R and t ≥ 0, we introduce the size-biased law as

P Ft = W √ 2 t • P Ft , (2.2)
and call X under P the size biased process. The law P is identified in [CR88, Theorem 4] as the limit law conditioning on the maximal displacement begin large at large times, precisely for all ≥ 1,

∀s ≥ 0, ∀A ∈ F s , lim t→∞ P(A|M t ≥ √ 2 t) = P (A).
The spinal decomposition links the size biased process with the so-called branching Brownian motion with spine. It describes the evolution of a branching particle system with a distinguished particle ξ t , which behaves differently from the others. The system starts with the spine particle at position 0. This particle moves according to a Brownian motion with drift √ 2 and produces children at rate 2. Each of its children starts an independent (standard) branching Brownian motion from its birth place. We shall use the same notation N t for the set of particles alive at time t in this process (it is not a Yule process anymore), and write ξ t ∈ N t for the label of the spine particle. The law of this branching Brownian motion with spine is denoted by P . The spinal decomposition can be stated as follows.

Theorem A (Spinal decomposition [START_REF] Berestycki | Topics on Branching Brownian motion[END_REF][START_REF] Chauvin | KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees[END_REF]). For all ∈ R, with the above notation we have P Ft = P Ft for all t ≥ 0. Moreover, for all u ∈ N t ,

P (ξ t = u | F t ) = e √ 2 Xt(u)-t( 2 +1) W √ 2 t .
In words: the law of the marked tree ((X s (u), u ∈ N s ), s ≤ t) has same law under probability P and P. Moreover, conditionally on this marked tree, one can choose to distinguish at random an individual u ∈ N t with probability proportional to e √ 2 Xt(u) to construct the law of the branching Brownian motion with spine. Using this result, we can describe the joint law of the extremal process and the maximal displacement of the branching Brownian motion.

Lemma 2.1. Let ≥ 1 and t ≥ 0, we denote by

E * t = u∈Nt δ Xt(u)-Mt
the extremal process of the branching Brownian motion seen from the rightmost individual, and we introduce the point process

D t = δ 0 + k∈N:σ k ≤t u∈N (k) σ k δ Bσ k - √ 2 σ k +X (k) σ k (u) , (2.3)
where B is a Brownian motion, (σ k , k ≥ 1) are the jump times of a Poisson process with intensity 2 and (X (k) (u) s , u ∈ N s , s ≥ 0) k∈N are i.i.d. branching Brownian motions. For all non-negative measurable function f, F , we have

E F (E * t )f (M t - √ 2 t) = e (1-2 )t E e √ 2 Bt f (-B t )F D t 1 D t ((0,∞))=0 .
Proof. For t ≥ 0, denote by u tip t ∈ N t the label of the largest particle alive at time t (which is a.s. unique). We observe that we can write

E F (E * t )f (M t - √ 2 t) = E u∈Nt F E * t (u) 1 {u=u tip t } f (M t - √ 2 t) ,
where E * t (u) := v∈Nt δ Xt(v)-Xt(u) is the extremal point measure seen from particle u ∈ N t . Thanks to the spinal decomposition and using (2.2), the above reads

E F (E * t )f (M t - √ 2 t) = E 1 W √ 2 t u∈Nt F E * t (u) 1 {u=u tip t } f (M t - √ 2 t) = E e - √ 2 Xt(ξt)+( 2 +1)t F (E * t (ξ t ))1 {ξt=u tip t } f (X t (ξ t ) - √ 2 t) .
Next, we use the definition of the branching Brownian motion with spine to rewrite the above expression. For s ∈ [0, t] let B s = Z t-s -Z t where Z s = X t (ξ s ) -√ 2 s and for all k ∈ N, σ k is the kth instant at which the spine gives birth to a new particle when running time backward from t (i.e. t -σ 1 is the last time before t at which the spine branches).Then, under E , B is a standard Brownian motion and (σ k ) are the atoms of a Poisson point process on R + with intensity measure 2dx. For each branching event σ k , the spine gives birth to a standard branching Brownian motion that we call

X (k) ≡ (X (k) s (u), u ∈ N (k) s ; s ∈ R + ). With these notation we then get E * t (ξ t ) = k∈N:σ k ≤t u∈N (k) σ k δ Bσ k - √ 2 σ k +X (k) σ k (u) . (2.4)
All that is left to do is thus to note that under E , the pair of variables (E * t (ξ t ), X t (ξ t )) jointly have the same law as ( D t , (-

B t + √ 2 t)) from (2.3). Thus substituting 1 {ξt=u tip t } by 1 D t ((0,∞))=0 and f (X t (ξ t ) - √ 2 t) by f (-B t ) we conclude that E F (E * t )f (M t - √ 2 t) = e (1-2 )t E e √ 2 Bt f (-B t )F ( D t )1 D t ((0,∞))=0 .
We are now going to show that for all > 1, the point measure D given in (1.10) is well-defined as the increasing limit of D t as t → ∞. Recall that, with the notation of Lemma 2.1, we have

D = δ 0 + k∈N u∈N (k) σ k δ Bσ k - √ 2 σ k +X (k) σ k (u) . (2.5)
Our first step is to prove that D is a well-defined sigma-finite point measure, meaning that for every

a < b ∈ (-∞, ∞], we have D [a, b] < ∞ a.s..
Lemma 2.2. For all > 1, D is a well-defined point measure. Moreover, we have

lim t→∞ D t = D a.s.
for the topology of the vague convergence.

Remark 2.3. Note that the above result would not hold for = 1, as in that case one can prove that lim t→∞ D 1 t (0, 1) = ∞ a.s.

Proof. Let > 1, the point measure D can be rewritten as

D = δ 0 + k∈N u∈N (k) σ k δ (Bσ k - √ 2( -1)σ k )+ X (k) σ k (u)- √ 2σ k . (2.6) We observe that B σ k - √ 2( -1)σ k , k ≥ 0 is a random walk with negative drift - √ 2( -1)/2. Moreover, for all k ∈ N the position of the largest atom in the point measure u∈N (k) σ k δ X (k) σ k (u)- √ 2σ k is, for large values of k, typically around position -3 2 √ 2 log σ k ≈ -3 2 √
2 log k. Thus, heuristically, if > 1, the random walk drifts to -∞ at positive speed such that only a finite number of branching Brownian motions put particles in any given compact set. On the other hand, when = 1, the random walk B σ k has drift zero and we show that it implies that an infinite number of particles are to be found in any finite neighbourhood of 0.

To make the above argument rigorous, we write M t for the maximal displacement at time t in a branching Brownian motion. Setting

m t = √ 2t -3 2 √ 2 log t, It is well-known [ABBS13, ABK13
] that (M t -m t , t ≥ 0) is tight and has uniform exponential tails. More precisely, it is proved in [START_REF] Fang | Tightness for maxima of generalized branching random walks[END_REF] (in a much more general settings) there exists C > 0 and λ > 0 such that

P (|M t -m t | ≥ x) ≤ Ce -λx for all t, x > 0.
(2.7)

Given k ∈ N, we denote by k) at time σ k . Using the bounds from (2.7), we observe immediately, using the Borel-Cantelli Lemma and the fact that

M (k) = max u∈N (k) σ k X (k) σ k (u) the maximal displacement of X (
σ k ∼ k→∞ k/2 that, with probability one, lim sup k→∞ M (k) - √ 2σ k log k ≤ 3 2 √ 2 + λ -1 . (2.8)
In view of (2.8) and the law of large numbers, we deduce that

lim k→∞ 1 k B σ k + M (k) - √ 2 σ k = - √ 2( -1) 2 < 0 a.s.
(2.9)

In particular, it implies that given A > 0 one can find a random T ∈ R + such that

∀σ k ≥ T, B σ k + M (k) - √ 2 σ k ≤ -A, in which case D ((x, ∞)) = D t ((x, ∞)
) for all t > T and x > -A. This proves that D is locally finite a.s. and that D t D as t → ∞, as claimed.

Next we show the weak continuity of the family ( D , > 1).

Lemma 2.4. The family of point processes ( D , > 1) is a.s. continuous in > 1. Moreover, for all > 1, P( D ((0, ∞) = 0) > 0 and

lim t→∞ P D 1 t (0, ∞) = 0 = 0 and lim →1 P D (0, ∞) = 0 = 0.
Proof. To prove the a.s. continuity of ( D , > 1), it is enough to show that for all continuous function ϕ with compact support, the function → ϕ, D is continuous a.s. This is a direct consequence of the fact that there are only finitely many atoms in any compact interval, and that the position of these atoms in D are decreasing and continuous with , by (1.10). Hence, for any 0 > 1, there is only a finite number of atoms to follow as increases to compute ∈ [ 0 , ∞) → ϕ, D . Hence this function is continuous, which completes the proof of the first statement. For the second statement, it suffices to observe that for T > 0 there is positive probability that σ 1 > T and that D (0, ∞) -D T (0, ∞) = 0. We now focus on the case = 1. By law of iterated logarithms for the random walk, we have that

lim sup k→∞ k -1/2 B σ k = ∞ a.s.,
which together with (2.8) yields lim sup

k→∞ B σ k + M (k) - √ 2σ k k 1/2 = ∞ a.s.
This shows that the event {B σ k + M (k) -√ 2σ k ≥ a infinitely often} has probability 1 for every a > 0. In particular it implies that D 1 t ((a, ∞)) ↑ ∞ a.s. as t → ∞. To conclude the proof, we observe that for all ε > 0, there exists t > 0 such that P(

D 1 t ((0, ∞)) = 0) < ε. At the same time it follows from (2.3) that D t is continuous in ∈ R, hence for all > 1 small enough, we have P D ((0, ∞)) = 0 ≤ P D t ((0, ∞)) = 0 ≤ 2ε,
which shows that lim →1 P D ((0, ∞)) = 0 = 0, completing the proof.

Probabilistic representation of the extremal point process conditioned on a large maximum

In this section, we prove the weak continuity in of the cluster point process D as well as the continuity of the function → C( ) and their spine representation. To prove this, we show that the cluster law D and the function C can be computed as continuous functionals of D defined in (1.10). This connection is based an application of Lemma 2.1 to the study of the extremal process of the branching Brownian motion conditioned on having a maximum larger than √ 2 . Those results in combination complete the proof of Theorem 1.1. We begin with the following computation of the extremal process of the branching Brownian motion conditioned on satisfying {M t ≥ √ 2 t}.

Lemma 3.1. Let > 1 and ϕ : R → R + be a continuous function whose support is bounded from the left. Then

lim t→∞ t 1/2 e ( 2 -1)t E e -E * t ,ϕ 1 {Mt≥ √ 2 t} = 1 √ 4π E e -D ,ϕ 1 D ((0,∞))=0 .
Proof. Fix ϕ as in the lemma and > 1. Using Lemma 2.1, we may write for t > 0

e ( 2 -1)t E e -E * t ,ϕ 1 {Mt≥ √ 2 t} =E e √ 2 Bt 1 {Bt≤0} e -D t ,ϕ 1 D t ((0,∞))=0 .
We compute the right hand side by first conditioning on B t = x. Introducing the point measure D ,x t as D t conditioned on {B t = x}, one gets

e ( 2 -1)t E e -E * t ,ϕ 1 {Mt≥ √ 2 t} = 0 -∞ dx √ 2πt e √ 2 x-x 2 2t E e -D ,x t ,ϕ 1 D ,x t ((0,∞))=0
.

We are going to show that, for any fixed x < 0,

lim t→∞ E e -D ,x t ,ϕ 1 D ,x t ((0,∞))=0 = E e -D ,ϕ 1 D ((0,∞))=0 (3.1)
then, the lemma follows by a simple application of the dominated convergence Theorem. We shall couple the processes D ,x t and D in such a way, that for any fixed x ∈ R,

lim t→∞ D ,x t , ϕ = D , ϕ a.s. (3.2)
Then, this gives (3.1) (and the lemma) by dominated convergence. Fix x ∈ R, recall that B is the Brownian underlying the construction of D and introduce for 0 ≤ s ≤ t

β (t) s := B s + s t (x -B t ).
It is well-known that (β

(t) s ; s ∈ [0, t]) is a Brownian bridge from β (t) 0 = 0 to β (t) t =
x. Almost surely, there exists a random constant C such that |B s | ≤ 1 + Cs 0.51 for all s ≥ 0.

Then, with the same constant C, one checks that we have the following uniform bound:

|β (t)
s | ≤ 2 + |x| + Cs 0.51 + sCt -0.49 for all t ≥ 0 and all s ∈ [0, t].

(3.3)

Recall that ϕ has bounded support on the left. Therefore, there exists a ∈ R such that ϕ(x) = 0 for all x < a. Let us fix 0 < ε < √ 2( -1). For all t large enough so that Ct -0.49 < ε, observe that

β (t) s - √ 2 s ≤ -( √ 2 -ε)s + 2 + |x| + Cs 0.51 for all s ∈ [0, t].
As in the proof of Lemma 2.2, since √ 2 -ε > √ 2, we conclude that there exists T < ∞ a.s. such that uniformly in t, all the points in D ,x t on the right of a come from branching events on the spine that occurred at times σ k ≤ T .

Therefore, in computing D ,x t , ϕ , one only needs to consider finitely many points: those that branched from the spine at a time smaller than T . These points converge, as t → ∞ to the corresponding points in D (because β (t) s → B s as t → ∞) and, as ϕ is continuous, (3.2) holds and the lemma is proved.

Using that last result, we now prove Theorem 1.1.

Proof of Theorem 1.1. We recall from (1.7) that for all > 1, we have

C( ) = lim t→∞ t 1/2 e ( 2 -1)t P(M t ≥ √ 2 t).
Therefore applying Lemma 3.1 with ϕ ≡ 0, we can rewrite C( ) as

C( ) = 1 √ 4π P D ((0, ∞)) = 0 . (3.4)
We deduce from Lemma 2.4 that C is a continuous function on [1, ∞) such that C(1) = 0. Additionally, it can be seen from the proof of [BH15, Lemma 3.3] that lim →∞ C( ) = 1 √ 4π =: C(∞), which completes the proof of the first part of Theorem 1.1.

We now turn to the proof of the second part. We recall that by the definition (1.6), given D a point process of law D , for all continuous function ϕ with compact support, we have

E e -D ,ϕ = lim t→∞ E e -E * t ,ϕ M t ≥ √ 2 t .
At the same time, by Lemma 3.1 we get

lim t→∞ E e -E * t ,ϕ M t ≥ √ 2 t = lim t→∞ E e -E * t ,ϕ 1 {Mt≥ √ 2 t} P(M t ≥ √ 2 t) = E e -D ,ϕ 1 D ((0,∞))=0 P( D ((0, ∞)) = 0) .
This shows that for all continuous compactly supported function

ϕ : R → R + E e -D ,ϕ = E e -D ,ϕ D ((0, ∞)) = 0 , proving that P(D ∈ •) = P D ∈ • | D ((0, ∞)) = 0 .
The weak continuity of D for ∈ (1, ∞) follows readily from Lemma 2.2 and the continuity of C. This concludes the proof.

An alternative proof for the first part of Theorem 1.1

We sketch here an alternative proof for the representation of C( ) in terms of the point processes D defined in (1.10). This proof is based on PDE analysis rather than tight probabilistic estimates, and can thus be of independent interest. Let M t be the maximum at time t in a branching Brownian motion, and set u(x, t) = P(M t > x) its tail distribution. We recall that u is solution to the Fisher-KPP equation

∂ t u = 1 2 ∂ 2 x u + u -u 2 with step initial condition u(x, 0) = 1 {x<0} .
We can thus compute C( ) from its definition (1.7) using the Feynman-Kac representation to evaluate P(M t > √ 2 t) = u( √ 2 t, t). Recall from Feynman-Kac that, given a function K(x, t), the solution to ∂ t h = 1 2 ∂ 2 x h + Kh can be written as

h(x, t) = E x h(B t , 0) exp t 0 ds K(B s , t -s) .
We apply this not to u(x, t), but to ∂ x u(x, t), the derivative of the solution to the Fisher-KPP equation, which is solution to

∂ t [∂ x u] = 1 2 ∂ 2 x [∂ x u] + (1 -2u)[∂ x u] with initial condition ∂ x u(x, 0) = -δ(x). This gives ∂ x u(x, t) = -e t E x δ(B t )e -2 t 0 ds u(Bs,t-s) = - 1 √ 2πt e t-x 2 2t E x→0 e -2 t 0 ds u(Bs,t-s)
where in the last expression B is a Brownian bridge from x to 0. We write B s = x(1 -s t ) -B t-s , so that B is a Brownian bridge from 0 to 0, we make the change of variable s = t -s and we drop the tildas:

∂ x u(x, t) = - 1 √ 2πt e t-x 2 2t E 0→0 e -2 t 0 ds u(x s t -Bs,s) .
Then, by setting x = √ 2 t + z and integrating over z > 0, one gets

u( √ 2 t, t) = e (1-2 )t √ 2πt ∞ 0 dz e - √ 2 z-z 2 2t E 0→0 e -2 t 0 ds u(z s t + √ 2 s-Bs,s) .
For > 1, the quantity u(z s t + √ 2 s -B s , s) goes exponentially fast to 0 as s → ∞, (unless B has wild fluctuations, but these events have a vanishingly small probability). Then, using the fact that B s (the value at time s of a Brownian bridge over a time t) looks, as t → ∞ for fixed s, more and more like a Brownian motion at time s, it is not very difficult (and akin to what was done in the proof of Lemma 3.1) to show that

lim t→∞ E 0→0 e -2 t 0 ds u(z s t + √ 2 s-Bs,s) = E 0 e -2 ∞ 0 ds u( √ 2 s-Bs,s) for > 1,
where B on the right hand side is a Brownian motion. In fact, the convergence also holds for = 1, as one can check that the quantities on either side are then equal to zero. Then, by dominated convergence,

∞ 0 dz e - √ 2 z-z 2 2t E 0→0 e -2 t 0 ds u(z s t + √ 2 s-Bs,s) ---→ t→∞ 1 √ 2 E 0 e -2 ∞ 0 ds u( √ 2 s-Bs,s) for ≥ 1 and C( ) = 1 √ 4π E 0 e -2 ∞ 0 ds u( √ 2 s-Bs,s) .
Observe that in the point process (1.10) the probability that there are no particles on the right of 0 is then

P D ((0, ∞)) = 0 = E k [1 -u( √ 2 σ k -B σ k , σ k )] = E e -2 ∞ 0 ds u( √ 2 s-Bs,s)
and therefore C( ) = 1 √ 4π P D ((0, ∞)) = 0 , as claimed.

Application to branching Ornstein-Uhlenbeck processes

As an application of Theorem 1.1, we study the asymptotic behavior, as t → ∞, of a branching Ornstein-Uhlenbeck process with a pulling parameter that decay to 0 as t → ∞. The main motivation to study this process is the article of Cortines and Mallein [START_REF] Cortines | The genealogy of an exactly solvable Ornstein-Uhlenbeck type branching process with selection[END_REF], in which it is conjectured that such a process, when undergoing selection, should exhibit unusual behaviour. In particular, the genealogy of these processes could be given by Beta coalescents, a family that interpolates between the Kingman and Bolthausen-Sznitman coalescents. Let us begin by introducing the branching Ornstein-Uhlenbeck process. An Ornstein-Uhlenbeck process X with spring constant µ is the solution of the stochastic differential equation dX µ s = -µX µ s ds + dB s , (

where B is a Brownian motion. It is well-known that Ornstein-Uhlenbeck processes may be represented, if µ > 0, as a space-time scaled Brownian motion: given W a standard Brownian motion, the process defined by ∀s ≥ 0, X µ s = X 0 e -µs + e -µs √ 2µ W e 2µs -1 , (4.2) is an Ornstein-Uhlenbeck with spring constant µ and initial condition X 0 . Equation (4.2) shows that, if µ > 0, the law of X s , conditionally on

{X 0 = x}, is N (xe -µs , 1-e -2µs 2µ
). In particular, X s is then strongly recurrent and its invariant measure is N (0, 1 2µ ). In a branching Ornstein-Uhlenbeck, since the genealogical structure of the process is independent of the motion of the particles, we continue to denote by N t the set of particles alive in a branching Ornstein-Uhlenbeck process with spring constant µ and we write (X µ s (u), u ∈ N s ) for the positions of such particles. It will be convenient to work with a normalized version X µ s (u) of X µ s (u) that has variance t so that things happen on the same scale as for the branching Brownian motion. This can be easily obtained by setting

X µ s (u) = 2µs 1 -e -2µs X µ s (u). (4.3)
With this notation, we define the extremal point process:

E µ t = u∈Nt δ X µ t (u)- √ 2t+ 1 2 √ 2 log t . (4.4)
Note that here the logarithmic correction is

1 2 √ 2 instead of 3 2 √
2 as in the branching Brownian motion case (µ = 0, see (1.1)). The aim of this section is to study the asymptotic behaviour of E µ t as µ → 0 and t → ∞ simultaneously.

Throughout this section, we will choose the spring constant µ as depending on the time-horizon t at which we observe the positions of particles, in the sense that µ = µ t is kept fixed for the evolution of the branching process at all times s ∈ [0, t]. For reasons that will become clear later on, one should choose µ t such that µ t t → γ ∈ (0, ∞] as t → ∞, which trivially covers the standard case where µ is fixed for all t's.

The particular case µ t = γ/t for some γ ∈ (0, ∞) is a direct application of the results of Bovier and Hartung [START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF]. Hence we start by recalling their result.

Extremal processes of variable speed branching Brownian motions and of branching Ornstein-Uhlenbeck processes

For each σ b ∈ [0, 1) and σ e > 1 let E σ b ,σe ∞ be a decorated Poisson point process defined as

E σ b ,σe ∞ := DPPP( √ 2C(σ e )W √ 2σ b ∞ e - √ 2x dx, σ e D σe ), (4.5)
with the parameters of the process being described as follows: Let (X t (u), u ∈ N t ) be a branching Brownian motion and M t its maximal displacement at time t. Then, -W β ∞ is the limit of the additive martingale, previously defined in (2.1). As (W β t , t ≥ 0) is a non-negative martingale, it converges a.s. to a limit W β ∞ . Moreover, it is well known that a.s.

W β ∞ > 0 if, and only if, β ∈ (- √ 2, √ 2).
-The function C is the one defined in (1.7).

-The family of laws (D , ≥ 1) is the family of point processes introduced in (1.6), and cD is the image measure of D by the application D → dj ∈D δ cdj , scaling the positions of the atoms by a factor c.

Let us now introduce the variable speed branching Brownian motion. Let A : [0, 1] → [0, 1] be a twice differentiable increasing function with A(0) = 0 and A(1) = 1. Then, the variable speed branching Brownian motion with variance profile A and time horizon t is defined in the same way as a branching Brownian motion, except that particles move as Brownian motions with time-dependent variance σ 2 t (s) = A (s/t) where s ∈ [0, t] is the time of the process. In particular, the position of a particle at time s is a Gaussian random variable with variance tA(s/t).

The main result in [START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF] is the following:

Theorem B (Bovier and Hartung [BH15] Theorem 1.2). Assume that the twice differentiable increasing function

A : [0, 1] → [0, 1] satisfies 1. A(0) = 0, A(1) = 1 and A(x) < x for all x ∈ (0, 1) ; 2. σ 2 b := A (0) < 1 and σ 2 e := A (1) > 1. Let (Y s (u); u ∈ N s ; s ∈ [0, t

]) denote the variable speed branching Brownian motion with variance profile A and E

A t = u∈Nt δ Yt(u)- √ 2t+ 1 2 √ 2 log t
be its extremal point measure at time t. Then (i) the extremal process E A t converges in law for the topology of the vague convergence to E σ b ,σe ∞ .

(ii) the maximal displacement of the process converges in law, and for all x ∈ R,

lim t→∞ P max E A t ≤ x = P (max E σ b ,σe ∞ ≤ x) .
This Theorem is the basis for obtaining a similar result for branching Ornstein-Uhlenbeck processes. More precisely, we will see that the case µ t = γ/t is a direct consequence and that more generally the case tµ t → γ as t → ∞ cane be deduced through comparison arguments. For each γ > 0, we define two constants, c γ and d γ by

c γ := 2γ e 2γ -1
and

d γ := 2γ 1 -e -2γ . (4.6) Now for γ > 0 let E γ ∞ := E cγ ,dγ ∞ . (4.7) In the γ = ∞ case, we set c ∞ = 0 and d ∞ = ∞, thus W √ 2c∞ ∞ = W 0
∞ is an exponential random variable with mean 1, the limit of the martingale associated to the Yule process (#N t , t ≥ 0). As is shown in [START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF] (see also Theorem 1.1), C(d ∞ ) = C(∞) = 1 √ 4π and a point measure drawn from D d∞ = D ∞ is a.s. δ 0 . We prove the following result in the rest of the section. Theorem 4.1. Assume that lim t→∞ tµ t = γ ∈ (0, ∞], then, with the above notations, we have that

lim t→∞ (E µt t , max E µt t ) = (E γ ∞ , max E γ ∞ ) jointly in law,
where the convergence of the point process is in the sense of the topology of vague convergence.

Remark 4.2. We prove in the forthcoming Lemma 4.4 that the convergence in law of a random point measure (for the topology of vague convergence) jointly with that of its maximum is equivalent to the convergence in law of P t , ϕ to P, ϕ for all continuous functions ϕ with support bounded from the left. This notion of convergence forms a thinner topology on the space of point measures.

Remark 4.3. In the simplest case where µ t = µ is a constant, the theorem with (4.3) and (4.4) implies the following behaviour for the non-normalised positions X µ t (u): the position of the rightmost particle is almost surely given by max u∈Nt

X µ t (u) = t µ - log t 4 √ µt + O(t -1/2 ),
and the next particles are at distance of order t -1/2 from the rightmost.

We shall call the case tµ t → ∞ the uncorrelated case, because the extremal particles have the same distribution as the extremal particles of an i.i.d. sample of Gaussian random variables. Indeed, in this regime, the dilation factor 2µ t t/(1 -e -2µtt ) diverges as t → ∞, which prevents the existence of local correlations (decorations) in the limiting picture.

The µ t = γ/t case

We start with the proof in the case µ t = γ/t, since it is a direct application of Theorem B.

Proof of Theorem 4.1 in the µ t = γ/t case. Recall from (4.2) that, an Ornstein-Uhlenbeck X µt s at time s with spring-constant µ t = γ/t started from 0 can be written as

X µt s = e -γs/t 2γ/t W e 2γs/t -1 . For any u ∈ N t , we define Y s (u), s ∈ [0, t] by Y s (u) = 2γ e 2γ -1 e γs/t X γ/t s (u). (4.8)
Clearly, Y s (u) has variance t e 2γs/t -1 e 2γ -1 . It is easily checked that the whole process (Y s (u), s ≤ t) u∈Nt is then a variable speed branching Brownian motion, with variance profile A(x) := e 2γx -1 e 2γ -1 , which is a function satisfying the assumptions of Theorem B with

σ 2 b = A (0) = 2γ e 2γ -1 = c 2 γ , σ 2 e = A (1) = 2γ 1 -e -2γ = d 2 γ .
Therefore, the extremal point process u∈Nt δ Yt(u)-

√ 2t+ 1 2 √ 2 log t converges in distribution as t → ∞ to a DPPP( √ 2C(σ e )W √ 2σ b ∞ e - √ 2x dx, σ e D σe ),
and the maximal atom converges as well. Since Y t (u) = X γ/t t (u) by (4.3), and using the forthcoming Lemma 4.4, we conclude in the joint convergence (E

γ/t t , max E γ/t t ) toward (E γ ∞ , max E γ ∞ ) in law, completing the proof of Theorem 4.1 when µ t = γ/t.

Comparison of extremal processes of branching Ornstein-Uhlenbeck processes with different spring constants

We use here Slepian-type computations to compare the extremal measures of Gaussian processes with different correlation structures. We begin with a general result on the joint convergence of point measures and their largest atom.

Lemma 4.4. Let (P t , P ∞ ) be point processes on R such that P ∞ ((0, ∞)) < ∞ a.s. The four following statements are equivalent: as t → ∞, (i) (P t , max P t ) → d (P ∞ , max P ∞ ) jointly;

(ii) P t → d P ∞ and max P t → d max P ∞ ;

(iii) E e -Pt,ϕ → E e -P∞,ϕ for all continuous function ϕ with support bounded from the left.

(iv) E e -Pt,ϕ → E e -P∞,ϕ for all C ∞ non-decreasing function ϕ with support bounded from the left and such that for some a ∈ R, ϕ(x) is constant for x > a.

The proof of this lemma being rather classical and straightforward, we postpone it to the appendix. A consequence of the above lemma is that to prove Theorem 4.1, it is enough to prove the convergence in distribution of random variables of the form E µt , ϕ , where ϕ is a generic non-decreasing bounded function with support bounded from the left.

We now recall that Kahane's theorem is a more general version of Slepian's lemma that allows to compare Gaussian processes with different variances. We refer to [Bov16, Chapter 3.1] for a self-contained proof of Kahane's Theorem.

Theorem C (Kahane's Theorem [START_REF] Kahane | Une inégalité du type de Slepian et Gordon sur les processus gaussiens[END_REF]). Let (X j , j ≤ n), (Y j , j ≤ n) be two centred Gaussian vectors. Let F be a twice differentiable function on R n with bounded second derivatives, that satisfies

∂ 2 F ∂x i ∂x j (x) ≥ 0 if E(X i X j ) > E(Y i Y j ) and ∂ 2 F ∂x i ∂x j (x) ≤ 0 if E(X i X j ) < E(Y i Y j ).
Then we have E(F (X)) ≥ E(F (Y )).

From Kahane's Theorem C, we obtain Lemma 4.5 below, which is useful when comparing the Laplace transform of the extremal point measures of branching Ornstein-Uhlenbeck processes with different spring constants.

Lemma 4.5. Let ϕ : R → R be a continuous non-negative non-decreasing function. Then, for all µ ≤ ν ≤ ∞ and t > 0, we have

E exp -ϕ, E µ t ≥ E exp -ϕ, E ν t ,
where E µ t and E ν t are the normalized, centred extremal point measures of branching Ornstein-Uhlenbeck processes as defined in (4.4) when ν < ∞, and E ∞ t is the point measure defined as

E ∞ t = u∈Nt δ X ∞ t (u)- √ 2t+ 1 2 √ 2 log t ,
where ( X ∞ t (u), u ∈ N t ) is a family of i.i.d. centred Gaussian random variables with variance t.

Remark 4.6. Note that as the spring constant µ increases toward ∞, the vector of normalized leaves ( X µ t (u), u ∈ N t ) converges in law toward i.i.d. Gaussian random variables with variance t. This can be checked by computing the covariance function of this vector, conditionally on N t . Therefore, we have lim µ→∞ E µ t = E ∞ t in law, for the topology of weak convergence, justifying the notation.

Proof. Remember that, since the branching events are independent of the spatial displacements, one can construct a branching Ornstein-Uhlenbeck with spring constant µ by first drawing its genealogical Yule tree (N s , s ≥ 0) then, conditionally on (N s , s ≥ 0) the spatial positions (X µ s (u), u ∈ N s , s ≥ 0). Thus, given two spring constants µ, ν, we can construct the two branching Ornstein-Uhlenbeck processes X µ and X ν using the same (N s , s ≥ 0). In the rest of the proof we work conditionally on (N s , s ≥ 0) to study the extremal processes.

For u, v ∈ N t , we denote by τ u,v the time of the most recent common ancestor of u and v. The covariance matrix of the Gaussian vectors X µ is given by

Var(X µ t (u)) = 1 -e -2µt 2µ and Cov(X µ t (u), X µ t (v)) = e -2µ(t-τu,v) 1 -e -2µτu,v 2µ .
Recall that we normalize positions to have variance t, setting as in (4.3)

X µ t (u) = X µ t (u) 2µt 1 -e -2µt .
As a result, we have that

Cov( X µ t (u), X µ t (v)) = t e 2µτu,v -1 e 2µt -1 . (4.9)
Observe that when µ ≤ ν (including the case ν = ∞), we have that

Cov( X µ t (u), X µ t (v)) ≥ Cov( X ν t (u), X ν t (v)),
for all u, v ∈ N t . Indeed, it is easy to verify that for all 0 < s < t fixed the function µ → e 2µs -1 e 2µt -1 is non-increasing in µ ∈ R.

We start by showing the result for ϕ : R → R, a smooth non-negative non-decreasing function, such that ϕ has compact support. Then the function

F : x ∈ R Nt → exp - u∈Nt ϕ(x u ) ,
is twice differentiable and constant outside of a compact, hence its second derivatives are bounded. It satisfies

∂ 2 F ∂x i ∂x j (x) = ϕ (x i )ϕ (x j ) exp - u∈Nt ϕ(x u ) ≥ 0, for all i = j ∈ N t ,
by monotonicity of ϕ. Thus, we can apply Kahane's Theorem C, and we have that for all µ ≤ ν ≤ ∞,

E F ( X µ t ) N t ≥ E F ( X ν t ) N t .
Therefore, averaging over the genealogical tree (N t , t ≥ 0), we obtain that

µ ∈ (-∞, ∞] → E exp -E µ t , ϕ is non-increasing.
To conclude, note that any continuous non-decreasing non-negative function ϕ can be approached from below by a sequence (ϕ n , n ≥ 1) of smooth non-decreasing functions with derivatives having compact support. Moreover, lim

n→∞ E exp -E µ t , ϕ n = E exp -E µ t , ϕ
by monotone convergence. Hence, we conclude that µ → E exp -E µ t , ϕ is non-increasing.

Proof of Theorem 4.1

We complete the proof of Theorem 4.1 in this section. We start with the observation that the family of limiting point measures (E γ ∞ , γ ∈ (0, ∞]), defined in (4.7), is continuous in distribution. Proposition 4.7. The family (E γ ∞ , max E γ ∞ ); γ ∈ (0, ∞] is continuous in law. Otherwise said, as per Lemma 4.4, for all continuous ϕ : R → R + non-decreasing with bounded support from the left, the function

γ ∈ (0, ∞] → E e -E γ ∞ ,ϕ is continuous.
Proof. Let ϕ be a continuous non-decreasing function, with support bounded from the left. For any γ > 0, by Campbell's formula, we have

E e -E γ ∞ ,ϕ = E exp - R E 1 -e -D dγ ,ϕ(dγ • +z) √ 2C(d γ )W √ 2cγ ∞ e - √ 2z dz . (4.10) We observe that C(d γ ), W √ 2cγ ∞
as well as E 1 -e -D dγ ,ϕ(dγ • +z) are non-negative for all γ > 0 and hence the exponential term on the right-hand side of (4.10) is bounded by 1. Therefore, by dominated convergence, it is enough to prove that each of the above functions is continuous.

It is obvious from the definition that both functions γ → c γ and γ → d γ are continuous in γ with c γ ∈ (0, 1) and d γ > 1 for all γ > 0. At the same time, Theorem 1.1 says that both γ → C( ) and

→ E 1 -e -D ,ϕ( •+z)
are continuous in > 1, by dominated convergence. Finally, Biggins [START_REF] Biggins | Uniform convergence of martingales in the branching random walk[END_REF] proved that the convergence of the additive martingale W √ 2 is uniform on compact subsets of (-1, 1), i.e. for all ε ∈ (0, 1), we have

lim t→∞ sup ∈[ε-1,1-ε] W √ 2 t -W √ 2 ∞ = 0 a.s.
As a result, we deduce that W

√ 2 ∞
is continuous in , completing the proof.

We now show that the point process E ∞ t defined in Lemma 4.5 converges in law, as t → ∞, to the Poisson point process E ∞ ∞ defined in (4.7), jointly with its maximum. Recall that

E ∞ t = u∈Nt δ X ∞ t (u)- √ 2t+ 1 2 √ 2 log t and E ∞ ∞ ∼ P P P 1 √ 2π W 0 ∞ e - √ 2x dx ,
where ( X ∞ t (u), u ∈ N t ) are i.i.d. centred Gaussian random variables with variance t. Lemma 4.8. We have lim t→∞

(E ∞ t , max E ∞ t ) = (E ∞ ∞ , max E ∞ ∞ ) in law.
Proof. Note this result can be straightforwardly deduced from standard extreme values theory for Gaussian processes. We include a direct self-contained proof which furthermore demonstrates how our toolbox can be used. Recall from Lemma 4.4 that to prove the joint convergence of E ∞ t and its maximum, it is enough to prove the convergence of E exp -E ∞ t , f for all non-decreasing continuous functions f with support bounded from the left.

Observe, by Campbell's formula for Poisson point processes, that

E exp -E ∞ ∞ , f |W 0 ∞ = exp -W 0 ∞ 1 -e -f (y) √ 2e - √ 2y √ 4π dy .
Therefore, as W 0 ∞ is distributed as a standard exponential random variable, we have

E exp -E ∞ ∞ , f = 1 + 1 √ 2π 1 -e -f (y) e - √ 2y dy -1 . (4.11)
On the other hand, conditioning with respect to #N t the number of leaves at time t, and writing X t for a Gaussian random variable with variance t and

m t = √ 2t -1 2 √ 2 log t, we have E exp -E ∞ t , f = E E e -f (Xt-mt) #Nt .
As #N t is a geometric random variable with parameter e -t , we have

E exp -E ∞ t , f = e -t E e -f (Xt-mt) 1 -(1 -e -t )E e -f (Xt-mt)
= E e -f (Xt-mt) e t E 1 -e -f (Xt-mt) + E e -f (Xt-mt) (4.12) Therefore, to complete the proof, it is enough to prove that lim

t→∞ e t E 1 -e -f (Xt-mt) = 1 √ 2π (1 -e -f (y) )e - √
2y dy, (4.13) which implies that (4.12) converges to (4.11) as t → ∞.

We now turn to the proof of (4.13). By change of variables, we have

E 1 -e -f (Xt-mt) = (1 -e -f (x-mt) ) e -x 2 /2t √ 2πt dx = (1 -e -f (y) ) √ 2πt e - √ 2y e -t+ 1 2 log t e -y 2 2t +y √ 2 log t 4t - (log t) 2 16t
dy.

Hence, as the support of y → 1 -e -f (y) is bounded from the left, we can apply the dominated convergence theorem in the above equation yielding, as t → ∞,

E 1 -e -f (Xt-mt) ∼ e -t √ 2π
(1 -e -f (y) )e - √ 2y dy concluding the proof.

Finally, we use the Kahane estimate to control the branching Ornstein-Uhlenbeck process with pulling strength µ t by branching Ornstein-Uhlenbeck processes with pulling strength (γ ± ε)/t.

Proof of Theorem 4.1. We denote by (X µt t (u), u ∈ N t ) the positions at time t of a branching Ornstein-Uhlenbeck process with spring constant µ t (recall that the spring constant µ t remains constant throughout the process, up to time t) and assume that lim

t→∞ tµ t = γ ∈ (0, ∞].
We first consider the case γ < ∞. Let 0 < γ < γ < γ. For t large enough γ/t < µ t < γ/t. Thus, by Lemma 4.5,

E exp -ϕ, E γ/t t ≤ E exp -ϕ, E µt t ≤ E exp -ϕ, E γ/t t , for all ϕ continuous non-decreasing functions R → R + .
As a result, taking t → ∞, and supposing furthermore that ϕ has bounded support on the left, combining Lemma 4.4 and Theorem B, we obtain that

lim inf t→∞ E exp -ϕ, E µt t ≥ E exp -ϕ, E γ ∞ lim sup t→∞ E exp -ϕ, E µt t ≤ E exp -ϕ, E γ ∞ .
Now, letting γ ↑ γ and γ ↓ γ, using Proposition 4.7 we obtain

lim t→∞ E exp -ϕ, E µt t = E exp -ϕ, E γ .
We conclude by Lemma 4.4 that (E µt t , max E µt t ) converge toward (E γ , max E γ ). We now consider the case γ = ∞. If lim t→∞ tµ t = ∞, then for all γ > 0, one has µ t ≥ γ/t for all t large enough. One the other hand, (X µt t (u), u ∈ N t ) is straightforwardly "more correlated" than i.i.d. Gaussian random variables (formally corresponding to the case γ = ∞). Hence, using again Lemma 4.5, then Lemma 4.4 and Theorem B for the lower bound, and Lemma 4.8 for the upper bound, we obtain

lim inf t→∞ E exp -ϕ, E µt t ≥ E exp -ϕ, E γ ∞ , lim sup t→∞ E exp -ϕ, E µt t ≤ E exp -ϕ, E ∞ ∞
for all smooth increasing function ϕ : R → [0, 1] such that ϕ has compact support. Letting γ → ∞ concludes the proof of Theorem 4.1.

Open questions and future work

The cases tµ t → γ ∈ (0, ∞) interpolate between the uncorrelated case and the branching Brownian motion regime (µ t = 0). Notice, though, that the multiplicative factor of the logarithmic correction remains equal to 1 2 √ 2 (as in the uncorrelated case) and not 3 2 √ 2 (as in the branching Brownian motion). We believe that there is a second transition when tµ → 0 where one gradually goes from the 1 2 √ 2 log t correction to 3 2 √ 2 log t while the decoration measure always is D 1 , which is the decoration of the branching Brownian motion.

More precisely, it is predicted in [START_REF] Derrida | Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion[END_REF] that C( ) ∼ κ( -1) as → 1, with the same constant κ as in (1.3). Note that κ ≈ 1.18 is also the constant such that lim

t P(M t ≥ √ 2t -3 2 √ 2 log t + y) ∼ κ √ 2 ye - √ 2y , as y → ∞.
This constant is proved to exist for all branching random walks in [Aïd13, Proposition 4.1]. Note that in [START_REF] Derrida | Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion[END_REF] the function Φ defined by

u(ct, t) ∼ e -t(c 2 /4-1) √ 4πt Φ(c) as t → ∞
where u is the solution of the Fisher-KPP equation ∂ t u = ∂ 2 x u + u(1 -u) started from the Heavyside initial condition is the analogue of C. The exact correspondence between the functions Φ and C is

C( ) = √ 4π Φ(2 ).
Our factor κ is thus given by the constant denoted 2α in [START_REF] Derrida | Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion[END_REF] (see Equation (73) there).

On the other hand, we also know from [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF], that for the additive martingale

W β lim β→ √ 2- W β ∞ √ 2 -β = √ 2Z ∞ ,
with Z ∞ the limit of the derivative martingale. Since d γ 1 + γ/2 and c γ 1 -γ/2 when γ → 0, we see that

C(d γ )W √ 2cγ ∞ κγ 2 2 Z ∞ as γ → 0. Since γ 2 e - √ 2x = e - √ 2(x- √ 2 log γ)
, the extremal point process E γ ∞ is roughly E ∞ , the centred extremal point process of the standard branching Brownian motion see (1.3), shifted to the left by √ 2| log γ| + O(1) (as γ → 0). This might suggest that the above-mentioned intermediate logarithmic corrections between 1 2 √ 2 and 3 2 √ 2 should appear for µ t = t -α with α ∈ (1, 3/2) , and the extremal point measure would be the same as for the branching Brownian motion as soon as µ t = o(t -3/2 ). This would complement the recent work [START_REF] Bovier | From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion[END_REF] on a similar phenomenon for branching Brownian motion with piecewise constant variance.

It may be worth noting that our model is notably different from the one studied by Kiestler and Schmidt [START_REF] Kistler | From Derrida's random energy model to branching random walks: from 1 to 3[END_REF] which yields a different interpolation between the uncorrelated case and the branching Brownian motion. In that later model, the extremal model is a Poisson point process without decoration, but the logarithmic correction of the median of the maximal displacement interpolates between -1 2 √ 2 and -3 2 √ 2 . On the contrary, in our case, the decoration of the extremal processes interpolate continuously between the absence of decoration of the uncorrelated case and the decoration of the branching Brownian motion. However, the logarithmic correction does not interpolate continuously on the scale of parameters we are considering.

The case µ < 0 is also interesting and is not covered in the present work. Notice that in the case µ > 0 we rely heavily on the results from Bovier and Hartung [START_REF] Bovier | Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime[END_REF]. However we think that the µ < 0 case corresponds to that of decreasing variances for the variable speed branching Brownian motion for which results concerning the position of the maximum are known (see e.g. Maillard and Zeitouni [START_REF] Maillard | Slowdown in branching Brownian motion with inhomogeneous variance[END_REF]), but not concerning the full extremal point process.

A Proof of Lemma 4.4

Proof. Obviously, (i) implies (ii) and (iii) implies (iv). It remains to proove that (ii) implies (iii) and (iv) implies (i).

We start by proving that (ii) implies (iii). First consider the case of a non-negative continuous function ϕ with support bounded from the left, and introduce for A ∈ R

ϕ A : x →      ϕ(x) if x < A (A + 1 -x)ϕ(A) if x ∈ [A, A + 1] 0 if x > A + 1.
The function ϕ A is continuous compactly supported, hence by (ii) we have lim t→∞ E e -Pt,ϕ A = E e -P∞,ϕ A .

By triangular inequality, E e -Pt,ϕ -E e -P∞,ϕ ≤ E e -Pt,ϕ -E e -Pt,ϕ A + E e -Pt,ϕ A -E e -P∞,ϕ A + E e -P∞,ϕ A -E e -P∞,ϕ .

Moreover, as ϕ is non-negative, we have for all t ≥ 0 and also for t = ∞: E e -Pt,ϕ -E e -Pt,ϕ A ≤ P (max P t ≥ A) .

Hence, by convergence of max P t , we have lim sup t→∞ E e -Pt,ϕ -E e -P∞,ϕ ≤ 2P (max P ∞ ≥ A) .

As the right hand side goes to zero as A → ∞, we have proved (iii) for non-negative functions. Now consider an arbitrary continuous function ϕ with support bounded on the left, and write ϕ = ϕ + -ϕ - where ϕ + (x) = max ϕ(x), 0 and ϕ -(x) = max -ϕ(x), 0 .

Then, for any α, β ≥ 0, the function αϕ + + βϕ -is continuous non-negative with support bounded on the left and, therefore, lim t→∞ E e -α Pt,ϕ+ -β Pt,ϕ-= E e -α P∞,ϕ+ -β P∞,ϕ-.

We conclude that ( P t , ϕ + , P t , ϕ -) jointly converge in law toward ( P ∞ , ϕ + , P ∞ , ϕ -). Therefore, P t , ϕ converges as well toward P ∞ , ϕ , which implies that (iii) holds.

We now prove that (iv) implies (i). Let f be a C ∞ non-decreasing function such that f (x) = 0 for x < 0 and f (x) = 1 for x > 1. For any y ∈ R and ε > 0, we set f ε,y (x) = f ε -1 (x -y) .

Noting that f ε,y (x) ≤ 1 {x>y} ≤ f ε,y-ε (x), we have for all (y 1 , . . . , y n ) ∈ R n , (λ 1 , . . . , λ n ) ∈ R n + and ε > 0: We conclude that (P t ((y i , ∞)), i ≤ n) jointly converge in law to (P ∞ ((y i , ∞)), i ≤ n) as t → ∞, except at discontinuity points y i where P ∞ ({y i }) > 0 with positive probability. Hence, P t converges in law to P ∞ for the topology of vague convergence. In Hence (P t , max P t ) converges to (P ∞ , max P ∞ ) in law jointly.

E e -

Figure 1 :

 1 Figure 1: Monte-Carlo approximation for the function C together with its right derivative at = 1.

Figure 2 :

 2 Figure 2: Construction of the point process D .

i

  λi Pt,fε,y i -ε≤ E e - i λiPt((yi,∞)) ≤ E e - i λi Pt,fε,y i .As t → ∞, the two bounds converge by (iv) applied to the functionsi λ i f ε,yi and i λ i f ε,yi-ε E e - i λi P∞,fε,y i -ε yi,∞)) ≤ E e - i λi P∞,fε,y i . Note that f ε,y (x) → 1 {x>y} and f ε,y-ε (x) → 1 {x≥y} as ε → 0. Hence one gets E e - i λiP∞([yi,∞)) ≤ lim inf t→∞ E e - i λiPt((yi,∞)) ≤ lim sup t→∞ E e - i λiPt((yi,∞)) ≤ E e - i λiP∞((yi,∞)) .(A.1)

  (A.1), add one extra pair (λ, y) to the λ i , y i , and send λ to infinity. Noticing that for A ≥ 0 thatE(A1 {max Pt≤y} ) ≤ E Ae -λPt((y,∞)) ≤ E(A1 {max Pt≤y} ) + e -λ E(A), one gets E e - i λiP∞([yi,∞)) 1 {max P∞≤y} ) ≤ lim inf t→∞ E e - i λiPt((yi,∞)) 1 {max Pt≤y} ) ≤ lim sup t→∞ E e - i λiPt((yi,∞)) 1 {max Pt≤y} ) ≤E e - i λiP∞((yi,∞)) 1 {max P∞≤y} ) .

Acknowledgements: A.C.'s work is supported by the Swiss National Science Foundation 200021 163170. B.M. and É.B. are partially funded by ANR-16-CE93-0003 (ANR MALIN). B.M. is also partially funded by a PEPS JCJC 2019 grant from CNRS. J.B. is partially supported by ANR grants ANR-14-CE25-0014 (ANR GRAAL) and ANR-14-CE25-0013 (ANR NONLOCAL).


Quang-Khoi Nguyen

Shih-Kang Lin

Chi-Shou Wu

Tung-Hsun Yang

Hsu-Yi Huang

Benoit Glorieux

Xuan-Hao Lee

Yeh-Wei Yu

Ching-Cherng Sun
email: ccsun@dop.ncu.edu.tw

Chi-Sou Wu

Benoît Glorieux

Sensing phosphor temperature in pc-WLEDs using noncontact detection

Keywords: pc-WLEDs, phosphor temperature, noncontact and instant detection

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Phosphor-converted white light-emitting diodes (pc-WLEDs) are a key light source because they have much advantage such as low cost , enegy efficiency, high color performance and high reliability [1][2][3][4][5]. A pc-WLED is usually made with a yellow (or green with red) phosphor layer that ecited by a blue die to gennerate white light with different Corelated Color Temperature and Color rendering index. An inavoidable phenomenon in operating pc-WLEDs is the high temperature in the package volume of a pc-WLED that can cause thermal quenching of phosphor, resulting in color drift, which induces blue-light leakage, efficiency droop, and lifetime shortening [6]. It was reported that the hottest spot in the package volume was located at the top of the blue chip or in the phosphor layer (a package with remote phosphor). Thus, managing the thermal effect is critical in extending the lifetime of a pc-WLED. Since the temperature on the surface of samples is primarily measured, rather than the temperature inside the samples, an effective method for measuring the phosphor temperature in pc-WLEDs is a requirement. So far, the thermal dynamics of the phosphor layer/volume in pc-WLEDs still lacks useful support. In this paper, we proposes a novel noninvasive method for building a database for a specific phosphor with a specific LED bonding that has been proven to detect phosphor temperature precisely for different pumping blue lights and different phosphor concentrations or CCTs.

In first phase of the method, the analysis of thermaldependent spectra was conducted. Variations in the emission spectra of the phosphors indicated the phosphor temperature, but a useful signal must be extracted from the noises and overlapping phenomena. For that purpose, the emission spectrum of YAG: Ce 3+ , which was a yellow phosphor with the maximum peak at approximately 540 nm was analyzed. From the band structure, the down conversion comprised two separate Gaussian contributions in the energy range. The two subbands corresponded to 580 nm (called long band) and 520 nm (called short wavelength band) as shown Fig. Nextly, the experimental data for the fitting process are prepared. In the experiment, we attached a blue die to a metal core printed circuit board (MCPCB) and covered it with a silicone layer embedded with YAG:Ce 3+ . An IR camera was used to monitor the surface temperature of the phosphor layer. No heat sink was used to dissipate the heat of the MCPCB; hence, the temperature would increase when the injection current increased.

The fitting results from the experimental data are illustrated in Figs 2 and3, where the associations between all four factors are well defined, including the fitting peak wavelengths and the full-width at half-maximums (FWHMs) of the short and long wavelength bands, and the phosphor temperature. This fitting results suggests that the phosphor temperature can be precisely measured by checking variations of the emission spectrum Fig. 4 shows the experimental measurements, obtained using the proposed detecting model, of the phosphor temperatures with and without lens encapsulation. The conformal coating packaging without lens encapsulation was used to build the database. Then the pc-WLEDs were encapsulated with a silicone lens. The database was used to indicate the temperature of the phosphor layer. The lens encapsulation blocked possible air convection of the phosphor layers; hence, the temperature was higher than the packaging without lens encapsulation. Based on the proposed modeling and the database, we found that the temperature of the sample with lens encapsulation increased by more than 160% compared with the packaging without lens encapsulation under different injection currents. Note that the phosphor temperature as a function of the injection current (Fig. 4) can be adopted only for the same thermal dissipation design. In conclusion, based on model of spectrum fitting, it can remotely measure the phosphor temperature of a pc-WLED. We successfully demonstrated a correspondence between the yellow spectrum and phosphor temperature from a white light spectrum with large cross-talk. This study provides a highly effective method for monitoring phosphor temperatures, which can be adopted by the soldstate lighting industry to develop more light sources that are reliable and efficient and have high color stability.

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Fig. 1

 1 Fig.1 (a) Normalized emission spectrum of phosphor consisting of the transition between two subbands. (b) Two subbands corresponding to two specific down conversions.

Fig. 2

 2 Fig.2 Fitting results of normalized spectrum for the short wavelength band. Correspondence between the (b) peak wavelength and (c) bandwidth of the phosphor.

Fig. 3

 3 Fig.3 Fitting results of normalized spectrum for the long wavelength band. Correspondence between the (b) peak wavelength and (c) bandwidth of the phosphor temperature.

Fig. 4 .

 4 Fig.4. Detection of phosphor temperature for the package with and without lens encapsulation under different injection currents. Black (red) dash line indicates the pc-WLED with (without) lens encapsulation.