Tommi Sottinen

Ciprian A Tudor

On the equivalence of multiparameter Gaussian processes

Keywords: Mathematics Subject Classification (2000): Primary 60G15; Secondary 60G30, 60H05 Brownian sheet, Fractional Brownian sheet, equivalence of Gaussian processes, Hitsuda representation, Shepp representation, canonical representation of Gaussian processes, Girsanov theorem, stochastic differential equations

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.

Introduction

The question of the equivalence in law of Gaussian processes has been widely studied in the sixties-seventies (see [START_REF] Hitsuda | Representation of Gaussian processes equivalent to Wiener process[END_REF], [START_REF] Hida | Gaussian Processes[END_REF], [START_REF] Kallianpur | Non-anticipative representation of equivalent Gaussian processes[END_REF], [START_REF] Shepp | Radon-Nikodym derivatives of Gaussian measures[END_REF]). Recently, the problem has been reopened by several authors, due to the intensive study of the fractional Brownian and of stochastic calculus with respect to this process. Precisely, the Wiener integral representation of the fractional Brownian motion with respect to the Brownian motion and the explicit form of its kernel, allows us to find concrete expressions of Gaussian processes that are equivalent in law to the fractional Brownian motion. We refer to [START_REF] Baudoin | Equivalence of Volterra processes[END_REF], [START_REF] Sottinen | On Gaussian processes equivalent in law to fractional Brownian motion[END_REF] and [START_REF] Cheridito | Regularizing fractional Brownian motion with a view towards stock price modelling[END_REF] for recent works on the Gaussian equivalence. We investigate in this paper the equivalence in law of multiparameter Gaussian processes (Note that lately the stochastic integration and stochastic equations with fractional Brownian sheet have been considered in e.g. [START_REF] Erraoui | Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet[END_REF] and [START_REF] Tudor | Itô formula and local time for the fractional Brownian sheet[END_REF]). More precisely, we will characterize all processes that are equivalent in law with a Brownian sheet and to a fractional Brownian sheet. We study both Hitsuda-type (or nonanticipative) representation and Shepp-type representation. At least former seems to be closely related to the stochastic calculus and Itô's formula (see [START_REF] Hida | Gaussian Processes[END_REF], Section 6.4.). Since the stochastic calculus in the multiparameter case is more complicated than in the one-parameter case, we ask ourselves if the representation theorems using stochastic integrals can yet be obtained. The answer is positive and it is due to the fact that actually the main tool used is the Gohberg-Krein factorization theorem (see [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF]). Our study surveys different types of representations, canonical or non-canonical, for the equivalent Gaussian sheets. In the "fractional" case, we show a Hitsuda characterization theorem for Gaussian sheets equivalent to the fractional Brownian sheet only in the case when all parameters are bigger than 1/2. The reason of the lack of a such characterization in the other cases consists in the structure of the linear Gaussian space generated by the fractional Brownian sheet, see [START_REF] Pipiras | Integration questions related to fractional Brownian motion[END_REF] and [START_REF] Pipiras | Are classes of deterministic integrands for fractional Brownian motion on an interval complete?[END_REF] for the one-parameter case.

We organized our paper as follows. Section 2 contains the notation and preliminaries on Hilbert-Schmidt operators. In Section 3 we give a general (and rather abstract) result on the Gaussian equivalence. In Section 4 we characterize the Gaussian processes that are equivalent in law to the Brownian sheet and in Section 5 we consider the same topics for the fractional Brownian sheet. Section 6 is an application of our results to a special kind of stochastic equation with (fractional) Brownian sheet as noise term.

Preliminaries and notation

Let t = (t 1 , t 2 , . . . , t d ) and s = (s 1 , s 2 , . . . , s d ) be vectors in R d . By s ≤ t we mean the partial ordering s i ≤ t i for all i = 1, 2, . . . , d. By [0, t] we denote [0,

t 1 ] × [0, t 2 ] × • • • × [0, t d ]. So, e.g. [0,t] f (s) ds = t 1 0 t 2 0 • • • t d 0 f (s 1 , s 2 , . . . , s d ) ds d • • • ds 2 ds 1 . Similarly, [s, t] = [s 1 , t 1 ] × [s 2 , t 2 ] × • • • × [s d , t d ]. A kernel b : [0, 1] 2d → R is of Volterra type if b(t, s) = 0 implies s ≤ t. Here t, s ∈ R d . The set of square integrable Volterra kernels is denoted by L 2 V ([0, 1] 2d ). A kernel b : [0, 1] 2d → R is symmetric if b(t, s) = b(s, t). The set of symmetric square integrable kernels is denoted by L 2 S ([0, 1] 2d ). The tensor product f ⊗ g : [0, 1] d+d ′ → R of functions f : [0, 1] d → R and g : [0, 1] d ′ → R is simply (f ⊗ g)(t, s) = f (t)g(s)
. The tensor product K ⊗ H of operators K and H is defined as (K⊗H)f (t, s) = (Kg s )(t), where g s = t → Hf (t, •)(s). Multiple tensor products are defined through iteration.

Let k be a kernel in L 2 ([0, 1] 2d ). The corresponding capital roman letter (K in this case) denotes the Hilbert-Schmidt operator on L 2 ([0, 1] d ) induced by k as

Kf (t) = [0,1] d f (s)k(t, s) ds.
A Hilbert-Schmidt operator on L 2 ([0, 1] d ) always admits a kernel.

The trace of the operator K with kernel k is simply

Tr(K) = [0,1] d k(t, t) dt,
if it exists, and in this case we say that K is of trace class.

If K and H are two Hilbert-Schmidt operators on L 2 ([0, 1] d ) with kernels k and h then in is easy the see that the composition HK is again a Hilbert-Schmidt operator. Indeed, the kernel associated to HK is the convolution

(h ⋆ k)(t, s) = [0,1] d h(t, u)k(u, s) du.
The adjoint K * of a Hilbert-Schmidt operator K is again a Hilbert-Schmidt operator, and its kernel k * is simply k * (t, s) = k(s, t).

Let I 0 denote the identity operator (this notation is in accordance to the fractional integral operators to be defined later in Section 5) and let σ(K) = {λ 1 , λ 2 , . . .} denote the (countable) set of the eigenvalues of K, or its spectrum. If k ∈ L 2 ([0, 1] 2d ) and 1 ∈ σ(K) then there exists a unique kernel k 1 ∈ L 2 ([0, 1] 2d ) called the resolvent of k , such that the corresponding operators satisfy

I 0 -K 1 = (I 0 -K) -1 , or, equivalently, KK 1 = K + K 1 = K 1 K.
Note that we have (K 1 ) 1 = K. The operator K 1 is also called the resolvent of K.

Remark 2.1 In the literature sometimes the operator -K 1 (and correspondingly -k 1 ) is called the resolvent. Some authors even define the resolvent to be I 0 -K 1 (which is no longer Hilbert-Schmidt).

The resolvent kernel k 1 may be represented as the Neumann series

k 1 (t, s) = - ∞ n=1 k ⋆n (t, s), that is K 1 = - ∞ n=1 K n .
Also, if k has the Mercer expansion along its eigenvalues and eigenfunctions

k(t, s) = ∞ j=1
λ j e j (t)e j (s) then the resolvent kernel k 1 may be represented as

k 1 (t, s) = - ∞ j=1 λ j 1 -λ j e j (t)e j (s). The Carleman-Fredholm determinant of K is Det 2 (K) = ∞ j=1 (1 -λ j )e λ j .
It exists for any Hilbert-Schmidt operator since ∞ j=1 |λ j | 2 < ∞. A Hilbert-Schmidt operator K is of Volterra type if its kernel k is a Volterra kernel. So, in this case we have

Kf (t) = [0,t] f (s)k(t, s) ds.
The resolvent K 1 of a Volterra operator K always exists since in this case the spectral radius sup{|λ| : λ ∈ σ(K)} is zero. The resolvent is also of Volterra type.

General results on equivalence

Let X = (X t ) t∈[0,1] d be a centered multiparameter Gaussian process on the probability space (Ω, F, P) with covariance

E [X t X s ] = R P (s, t) for every s, t ∈ [0, 1] d .
We assume that X is mean square continuous and denote by H(X) the reproducing kernel Hilbert space (RKHS, for short) of the process X. That is, for every f ∈ H(X) we have

f, R P (t, •) H(X) = f (t)
for every t ∈ [0, 1] d . The Hilbert space H(X) is the closure of the subspace spanned by R P (t, •), t ∈ [0, 1] d , with respect to the inner product •, • H(X) . Note that the continuity of R P , i.e. the mean square continuity of X, implies that H(X) is separable.

For every t ∈ [0, 1] d , let us denote by L(X, t) the closed linear subspace of L 2 (Ω) generated by the random variables X u , u ≤ t, and let us consider the isometric isomorphism Φ from L(X, 1) onto the RKHS H(X) such that, for every t ∈ [0, 1] d

ΦX t = R P (t, •).
We will use the following notation

F (t) = {f ∈ H(X) such that f (u) = 0 if 0 ≤ u ≤ t} ,
and let M (t) be the orthogonal complement of F (t) in H(X). Clearly,

M (t) = {f ∈ H(X) such that f (u) = 0 if u i > t i for some i = 1, . . . , d} .
We have the following lemma.

Lemma 3.1 For every t ∈ [0, 1] d it holds that Φ (L(X, t)) = M (t). (3.1)
Proof: It holds that f ∈ F (t) if and only if Φ -1 f belongs to the complement of L(X, t) in L(X, 1).

Indeed, for all u ∈ [0, t] we have

0 = f (u) = f, R P (t, •) H(X) ⇐⇒ E Φ -1 f • X t = 0.
Let P be another probability measure on (Ω, F) such that (X, P) is a centered Gaussian sheet with covariance R P. Using the same type of arguments as in [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF] or [START_REF] Kallianpur | Non-anticipative representation of equivalent Gaussian processes[END_REF], we can prove the following canonical nonanticipative representation. Theorem 3.2 Every mean square continuous Gaussian sheet (X, P) which is equivalent in law to a given Gaussian sheet (X, P) admits a nonanticipative representation with respect to (X, P) . That is, there exists a centered Gaussian sheet (Y, P) with covariance R P such that Y t ∈ L(X, t) for every t ∈ [0, 1] d .

In the next section we show that in the case when the sheet X above is equivalent in law to a Brownian sheet, Theorem 3.2 takes a less abstract form.

4 Gaussian processes equivalent in law to a Brownian sheet

Hitsuda representation

Let W = (W t ) t∈[0,1] d be a Brownian sheet on the probability space (Ω, F, P). That means, W is a centered multiparameter Gaussian process with the covariance

E [W t W s ] = (s ∧ t) = d i=1 (s i ∧ t i ).
To prove the stochastic integral representation of processes, we will follow the lines of [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF], Section 9.

Lemma 4.1 Let b be a kernel in L 2 ([0, 1] 2d ) satisfying [0,t] d b(u, x)f (u) du = 0 implies x ≤ u (4.1) for every f ∈ L 2 ([0, 1] d ). Then b is a Volterra kernel, that is, b ∈ L 2 V ([0, 1] 2d ).
Proof: Equation (4.1) implies that

[0,1] 2d b(u, x)f (u)1 [0,t] (u) d i=1 1 -1 [0,t i ] (x i ) g(x) du dx = 0 for any f, g ∈ L 2 ([0, 1] d ). Let us choose f (u) = 1 (a,b] (u) and g(x) = 1 (c,d] (x)
where for every i = 1 . . . d, we have

(either a i ≥ d i or b i ≤ c i ) .
So we have, since

a i ≥ d i or b i ≤ c i , [0,1] 2d b(u, x) d i=1 1 -1 [0,u i ] (x i ) 1 (a,b] (u)1 (c,d] (x) du dx = 0. Since the family {1 (a,b] (x)1 (c,d] (u), b ≤ c or a ≥ d} spans the space L 2 ([0, 1] 2d ), we obtain b(u, x) = 0 if there exists a coordinate i with x i > u i .
The representation theorem for the Gaussian processes equivalent in law to the Brownian sheet is given below.

Wt = W t - [0,t] [0,s] b(s, u) dW u ds (4.2) for some b ∈ L 2 V ([0, 1] 2d .
Proof: By Theorem 3.2, we have that for every t

∈ [0, 1] d , Wt ∈ L(W, t) ⊂ L(W, 1).
It follows from Kallianpur [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF], pp. 225-233 that Φ Wt can be written as

Φ Wt = (I 0 -B)ΦW t = (I 0 -B)1 [0,t] (•)
where I 0 (identity) and B are operators on L 2 ([0, 1] d ). Moreover, B is a Volterra operator. Therefore, we can write

(I 0 -B)1 [0,t] (•) = 1 [0,t] (•) - [0,1] d b(s, •)1 [0,t] (s) ds = 1 [0,t] (•) - [0,t] b(s, •)1 [0,t] (s)
P (t)f (s) = f (s)1 [0,t] (s) or BP (t)f (u) = [0,t] b(s, u)f (s) ds
implies that b is a Volterra kernel.) The conclusion follows by Lemma 4.1.

The non-centered case is now obvious. Indeed, we know that admissible drifts are of the form

A(t) = [0,t] a(s) ds.
for some a ∈ L 2 ([0, 1] d ). Also, by using the resolvents we may invert the relation (4.2). So, we have the following result:

Theorem 4.3 A Gaussian sheet W is equivalent in law to the Brownian sheet W if and only if there exists a Volterra kernel b ∈ L 2 V ([0, 1] 2d ) and a function a ∈ L 2 ([0, 1] d ) such that W can be represented as Wt = W t - [0,t] [0,s] b(s, u) dW u ds - [0,t] a(s) ds. (4.4)
The Volterra kernel b and the function a are unique and the representation (4.4) is canonical in the sense that the sheets W and W generate the same filtration. Moreover, the Brownian sheet W in (4.4) is constructed from the fractional one W by

W t = Wt - [0,t] [0,s] b 1 (s, u) d Wu ds - [0,t] a(s) - [0,s] b 1 (s, u)a(u) du ds (4.5)
where

b 1 ∈ L 2 V ([0, 1] 2d ) is the resolvent of b.
The Hitsuda representation (4.4) is connected to the Girsanov representation of the density between P W and P W in the following way: Theorem 4.4 The Radon-Nikodym derivative between the laws of W and W is

dP W dP W F t = exp [0,t] [0,s] b(s, u) dW u + a(s) dW s - 1 2 [0,t] [0,s] b(s, u) dW u + a(s) 2 ds    . (4.6)
In terms of W the Radon-Nikodym derivative takes the form

dP W dP W F t = exp [0,t] [0,s] b 1 (s, u) d Wu + a(s)- [0,s] b 1 (s, u)a(u) du d Ws - 1 2 [0,t] [0,s] b 1 (s, u) d Wu + a(s)- [0,s] b 1 (s, u)a(u) du 2 ds    . (4.7)
Proof:

The formula (4.6) is an immediate consequence of the multiparameter Girsanov formula (see [START_REF] Dozzi | Stochastic processes with a multidimensional parameter[END_REF] or [START_REF] Erraoui | Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet[END_REF]) and the isometry

E [0,t] Y s dW s 2 = [0,t] EY 2 s ds.
The formula (4.7) follows from (4.6) by symmetry.

Shepp representation

The Shepp representation of Gaussian measures equivalent in law with the Wiener measure can be extended to the multiparameter case.

Cov( Wt , Ws ) = s ∧ t - [0,t] [0,s] k(u, v) du dv, (4.8)
and

E[ Wt ] = [0,t] m(u) du. (4.9)
The kernel k is unique and symmetric and is given by

k(s, t) = - ∂ 2d ∂s∂t Cov( Ws , Wt )
for almost every s and t. The function m is also unique and it is given by

m(t) = ∂ d ∂t E[ Wt ]
for almost all t.

The Radon-Nikodym derivative

dP W dP W is given by dP W dP W = 1 Det 2 (K)Tr(K 1 K) × exp - 1 2 [0,1] d [0,1] d k 1 (t, s) d Ws d Wt + [0,1] d m(t) d Wt + 1 2 [0,1] d m(t) 2 dt .
(4.10)

Proof: The proof given in [START_REF] Shepp | Radon-Nikodym derivatives of Gaussian measures[END_REF] can be applied exactly.

For the proof of next Corollary in the one-parameter case we refer to [START_REF] Cheridito | Regularizing fractional Brownian motion with a view towards stock price modelling[END_REF]. The multiparameter case may be proved similarly.

∈ L 2 ([0, 1] d ) and k ∈ L 2 S ([0, 1] 2d ) such that E[Z t ] = [0,t] m(s) ds, and
Cov(Z t , Z s ) = [0,t] [0,s] k(u, v) du dv. Example 4.7 Let H ∈ (0, 1) and K ∈ (0, 1]. Set R H,K (s, t) = 1 2 K (t 2H + s 2H ) K -|t -s| 2HK . (4.11)
It has been proved in [START_REF] Houdre | An example of infinite dimensional quasi-helix[END_REF] that the function (4.11) is positive definite. So we may define a centered sheet B = (B H,K t

) t∈[0,1] d with the covariance R H,K (t, s) = d i=1 R H i ,K i (t i , s i ).
According to Corollary 1, we have that W + B H,K is equivalent in law to a Brownian sheet if and only if 2) we obtain that this is true if and only if H i K i > 3/4 for every i. In the case K i = 1 (B H,1 is a standard fractional Brownian sheet, cf. next section) we have that W + B H,1 is equivalent in law to a Brownian sheet if and only if min i=1,...d H i > 3/4.

∂ 2d ∂s∂t R H,K (s, t) = d i=1 2H i K i 2 K i 2H i (K i -1)(t 2H i i + s 2H i i ) K i -2 t 2H i -1 i s 2H i -1 i +(2H i K i -1)(t i -s i ) 2H i K i -2 belongs to L 2 ([0, 1] 2d ). Majorizing (t 2H i + s 2H i ) K-2 by (t i s i ) H(K-
We end this section by comparing the Hitsuda and Shepp representations. The situation is similar to the one-parameter case studied by Cheridito [START_REF] Cheridito | Regularizing fractional Brownian motion with a view towards stock price modelling[END_REF]. Indeed, by comparing (4.7) and (4.10), we obtain the following.

E exp - 1 2 [0,1] d [0,1] d k 1 (t, s) d Ws Wt = exp 1 2 [0,1] d [0,t] b 1 (t, s) ds dt ,
and for the kernels b and k we have the relation

k = b + b * -b ⋆ b * . (4.12)
By symmetry, we have for the resolvent kernels b 1 and k 1 that

k 1 = b 1 + b * 1 -b 1 ⋆ b * 1 .
Remark 4.9 In general, the equation (4.12) does not admit a unique solution in b. Indeed, given a kernel b ∈ L 2 ([0, 1]) 2d the sheet

Wt = W t - [0,t] [0,1] d b(t, s) dW s dt
is equivalent in law to a Brownian sheet and it k in the Shepp representation in given by (4.12). If b is a Volterra kernel, as it is in the Hitsuda representation, then (4.12) is the Gohberg-Krein factorization of k, cf. [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF] where it is written in the symmetric form

I 0 -K = (I 0 -B)(I 0 -B * ).
5 Gaussian processes equivalent in law to fractional Brownian sheet

Wiener integrals and Hitsuda representation

The fractional Brownian sheet

Z = (Z H t ) t∈[0,1] d with multiple Hurst index H = (H 1 , H 2 , . . . , H d ) ∈ (0, 1) d on (Ω, F, P) is a centred Gaussian field with covariance function R H (t, s) = d i=1 R H i (t i , s i ) = d i=1 1 2 t 2H i i + s 2H i i -|t i -s i | 2H i .
We want to derive a representation of the fractional Brownian sheet with respect to the ordinary one. To this end we recall some preliminaries of fractional calculus. For details we refer to [START_REF] Samko | Fractional integrals and derivatives[END_REF].

Let f be a function over the interval [0, 1] and α > 0. Then

I α ± f (t) = 1 Γ(α) 1 0 f (s) (t -s) 1-α ± ds
are the Riemann-Liouville fractional integrals of order α. For α ∈ (0, 1),

D α ± f (t) = ±1 Γ(1 -α) d dt 1 0 f (s) (t -s) α ± ds.
are the Riemann-Liouville fractional derivatives of order α; I 0 ± and D 0 ± are identity operators.

If one ignores the troubles concerning divergent integrals and formally changes the order of differentiation and integration one obtains

I -α ± = D α ± .
We shall take the above as the definition for fractional integral of negative order.

Remark The operators I α ± are not Hilbert-Schmidt, that is, they do not admit kernels; Hence the use of calligraphic letters.

By [START_REF] Samko | Fractional integrals and derivatives[END_REF], Theorem 2.5, the composition formula

I α ± I β ± f = I α+β ± f (5.1)
is valid in any of the following cases.

(i) β ≥ 0, α + β ≥ 0 and f ∈ L 1 ([0, 1]), (ii) β ≤ 0, α ≥ 0 and f ∈ I -β ± L 1 ([0, 1]), (iii) α ≤ 0, α + β ≤ 0 and f ∈ I -α-β ± L 1 ([0, 1]).
Let us now briefly consider Wiener integrals with respect to fractional Brownian sheet. Details for the one-parameter case may be found in Pipiras and Taqqu [START_REF] Pipiras | Integration questions related to fractional Brownian motion[END_REF][START_REF] Pipiras | Are classes of deterministic integrands for fractional Brownian motion on an interval complete?[END_REF].

Let us introduce operators

K β f (t) = c β t 1 2 -β I β-1 2 - s β-1 2 f (s) (t), K -1 β f (t) = 1 c β t 1 2 -β I 1 2 -β - s β-1 2 f (s) (t),
where

c β = 2β(β -1 2 )Γ(β -1 2 ) 2 B(β -1 2 , 2 -2β)
. Now, for i = 1, 2, . . . , d we have

R H i (t i , s i ) = K H i 1 [0,t i ] , K H i 1 [0,s i ] L 2 ([0,1]) .
So, if we set

K H = K H 1 ⊗ K H 2 ⊗ • • • ⊗ K H d
we see by using the Fubini theorem that

R H (t, s) = d i=1 R H i (t i , s i ) = d i=1 K H i 1 [0,t i ] , K H i 1 [0,s i ] L 2 ([0,1]) = [0,1] d d i=1 K H i 1 [0,t i ] (u i ) • d i=1 K H 1 1 [0,s i ] (u i ) du = K H 1 [0,t] , K H 1 [0,s] L 2 ([0,1] d ) .
Consequently, we have the representation

Z t = [0,t] K H 1 [0,t] (s) dW s . (5.2)
Also, the operator K H is injective and has simple functions in its image. Thus we have a reverse representation

W t = [0,t] K -1 H 1 [0,t] (s) dZ s . (5.3)
Actually, the inverse operator K -1 H is given as

K -1 H = K -1 H 1 ⊗ K -1 H 2 ⊗ • • • ⊗ K -1 H d .
Also, note that by Proposition 3.4. of [START_REF]On the multiparameter fractional Brownian motion[END_REF] the representations (5.2) and (5.3) may be understood pathwise.

We drop the Hurst indices in the notations now.

Since the integral with respect to the Brownian sheet is defined for any f ∈ L 2 ([0, 1] d ) we may define:

Definition 5.1 Set Λ = f : Kf ∈ L 2 ([0, 1] d ) .
The Wiener integral of f ∈ Λ with respect to fractional Brownian sheet is

[0,1] d f (t) dZ t = [0,1] d Kf (t) dW t .
The integral in Definition 5.1 can be considered as a limit of elementary functions. Indeed, by Theorems 4.1 and 4.2 of [START_REF] Pipiras | Are classes of deterministic integrands for fractional Brownian motion on an interval complete?[END_REF] the set Λ is a Pre-Hilbert space with

f, g Λ = Kf, Kg L 2 ([0,1] d )
and the set of step functions is dense in Λ.

Obviously, if f is a step function then KK -1 f = f = K -1 Kf. Also, from the composition formula (5.1) and tensorization it follows that for any

f ∈ L 2 ([0, 1] d ) we have K -1 Kf = f if min i≤d H i ≥ 1 2 and KK -1 f = f if max i≤d H i ≤ 1 2
. However, we have the following.

Lemma 5.2 Let max i≤d H i > 1 2 . Then there exist functions f ∈ L 2 ([0, 1] d ) such that the equation Kg = f (5.4)
has no solution in g.

The reason for the lack of solutions in (5.4) is that for H i > 1/2 the operator K is a weighted fractional integral operator on coordinate i. So, one can partially differentiate f fractionally (K -1 is a weighted fractional differential operator on coordinate i). However, there are "arbitrarily non-smooth" functions in L 2 ([0, 1] d ).

An example of such a function is

f (t) = d i=1 t 1 2 -H i i ψ i (t i ) (5.5)
where ψ i is the real part of the Weierstrass function

ψ * i (t) = ∞ n=1
b -p i n e ib n t , b > 1 and p i ∈ (0, H i -1/2). For the rigorous proof of Lemma in the one-parameter case 5.2 see [START_REF] Pipiras | Are classes of deterministic integrands for fractional Brownian motion on an interval complete?[END_REF], Lemma

The reproducing kernel Hilbert space H(Z) of the fractional Brownian sheet is easy to identify.

Proposition 5.3 A function f ∈ H(Z) if and only if it can be represented as

f (t) = [0,t] K1 [0,t] (s) f (s) ds (5.6) for some f ∈ L 2 ([0, 1] d ). The scalar product •, • H(Z) on H(Z) is given by f, g H = f , g L 2 ([0,1] d )
.

(5.7)

Moreover, as a vector space

H(Z) = d i=1 I H i + 1 2 + L 2 ([0, 1] d ) . (5.8)
Proof: The forms (5.6) and (5.7) follow immediately from the relation (5.2). The equality (5.8) follows from the isomorphism in [START_REF] Samko | Fractional integrals and derivatives[END_REF], p. 187.

Recall that Λ is isometric to H(Z). It is desirable that it is also isometric to the linear space L(Z, t), i.e. one wants, as in the case of the Brownian sheet, to identify any F ∈ L(Z, t) with a single function f ∈ Λ so that

F = [0,1] d f (t) dZ t .
Obviously, this is possible if and only if Λ is complete. Otherwise Λ is isometric to a proper subspace of L(Z, t). Indeed, if H i ≤ 1/2, for all i = 1, 2, . . . , d then

Λ = K -1 f : f ∈ L 2 ([0, 1] d ) .
In particular, Λ is complete and hence isometric to L(Z, t). On the other hand, if H i > 1/2 for some i then Lemma 5.2 implies that Λ is not complete and hence isometric to a proper subspace of L(Z, t).

Let us summarize the discussion above as a lemma: Lemma 5.4 For H i ≤ 1 2 for all i = 1, 2, . . . , d then the equality

[0,1] d g(t) dW t = [0,1] d K -1 g (t) dZ t (5.9)
holds for any g ∈ L 2 ([0, 1] d ). If H i exceeds 1 2 for some i = 1, 2, . . . , d then the equality (5.9) holds only for g ∈ K(Λ) and the inclusion K(Λ) ⊂ L 2 ([0, 1] d ) is strict.

As the fractional Brownian sheet Z is mean square continuous we know now by the generalisation of the Kallianpur-Oodaira representation that any Gaussian sheet Z that is equivalent in law to it can be represented canonically in terms of Z in the sense that Zt ∈ L(Z, t).

In the case of ordinary Brownian sheet W the elements in the linear space L(W, t) were Wiener integrals of the form

[0,t] f (s) dW s for some f ∈ L 2 ([0, 1] d ).
In the case of fractional Brownian sheet this is not generally true, as Lemma 5.2 suggests. Therefore, in general we do not have the Hitsuda representation for the fractional Brownian sheet as we have for the ordinary Brownian sheet. Indeed, the situation is similar to the one-parameter case studied in [START_REF] Sottinen | On Gaussian processes equivalent in law to fractional Brownian motion[END_REF].

where W is a Brownian sheet constructed from Z by (5.3), a ∈ L 2 ([0, 1] d ) and b ∈ L 2 V ([0, 1] 2d ). Moreover, the fractional Brownian sheet Z is constructed from Z by

Z t = Zt - [0,t] K1 [0,t] (s) [0,s] b 1 (s, u) d Wu ds + [0,t] K1 [0,t] (s) a(s) - [0,s] b 1 (s, u)a(u) du ds where W is constructed from Z by Wt = [0,t] K -1 1 [0,t] (s) d Zs and b 1 ∈ L 2 V ([0, 1] d ) is the resolvent of b.
By using Proposition 5.3 we can rephrase (5.12) in Proposition 5.7.

Proposition 5.8 A Gaussian sheet Z is equivalent in law to a fractional Brownian sheet Z with index H if and only if it can be represented as

Zt = Z t - [0,t] f (t, s) dW s -A(t) (5.13)
where W is a Brownian sheet constructed from Z by (5.3),

A = d i=1 I H i + 1 2 + a for some a ∈ L 2 ([0, 1] d ) and for all s ∈ [0, 1] d f (•, s) = d i=1 I H i + 1 2 + b(•, s) for some b ∈ L 2 V ([0, 1] 2d ).
The proofs of Theorem 5.5 and Propositions 5.7 and 5.8 are easy search-andreplace modifications of the corresponding one-parameter case studied in [START_REF] Sottinen | On Gaussian processes equivalent in law to fractional Brownian motion[END_REF]. It would be too boring to reproduce them here.

The obvious difference between the representations (4.4) and (5.12) (or (5.13)) is that in the latter one needs to construct a Brownian sheet from Z in order to represent Z. Nevertheless, (5.12) and (5.13) are Kallianpur-Oodaira representations of Z with respect to Z, since obviously Zt ∈ L(Z, t). In the case max i≤d H i ≤ 1/2 (and in this case only) we can use Lemma 5.4 to obtain a representation without the auxiliary Brownian sheet. This is the essence of Theorem 5.5.

Theorem 5.11 If max i≤d H ≤ 1/2 then any sheet Z that is equivalent to the fractional Brownian sheet is given by (5.10) and we have If max i≤d H i > 1/2 then there are sheets Z that are equivalent to the fractional Brownian sheet Z that do not admit the representation (5.15) even if Z admits the representation (5.10).

dP Z dP Z F t = exp [0,t] [0,t] K -1 2 b (s, u) dZ u dZ s + 0,t] K -1 a (s) dZ s - 1 2 [0,t] [0,t] K -1 b(s, •) (u) dZ u + K -1 a (s)
The first claim of Theorem 5.11 is obvious. To see the latter claim take g = g 1 ⊗ g 2 ∈ L 2 ([0, 1] 2d ) such that g 2 is smooth but g 1 is not. Then apply Lemma 5.2.

Shepp representation

The Shepp representation of a Gaussian sheet Z that is equivalent in law to the fractional Brownian sheet Z is easily obtained from the Shepp representation for the Brownian sheet and the connection (5.2). Theorem 5.12 A Gaussian sheet Z is equivalent in law to a fractional Brownian sheet Z if and only if there exists a kernel k ∈

L 2 S ([0, 1] 2d ) such that 1 ∈ σ(K) and a function m ∈ L 2 ([0, 1] d ) such that Cov( Zt , Zs ) = Cov(Z t , Z s ) - d i=1 I H i + 1 2 + ⊗ d i=1 I H i + 1 2 + k (t, s), and
E[ Zt ] = d i=1 I H i + 1 2 + m (t).
The kernel k and the function m are unique. Indeed, they may be obtained, almost everywhere, from the covariance and mean functions by applying the inverses of the tensorized operators above. ). The form of covariance follows now from the result in [START_REF] Samko | Fractional integrals and derivatives[END_REF], p. 187.

The claims concerning the Radon-Nikodym derivative are obvious given the results of Girsanov representation.

Application to stochastic equations

We apply the Hitsuda representation to discuss the existence and uniqueness of the solution of a stochastic equation of functional type where the noise term is a (non)fractional Brownian sheet. A rather similar equation (but of a non-functional type) has been studied in [START_REF] Erraoui | Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet[END_REF].

In this section we consider the canonical space Ω = C([0, 1] d ) of continuous sheets (the Brownian and the fractional Brownian sheets admit sample-continuous versions).

Let α = (α(t, x) : t ∈ [0, 1] d , x ∈ Ω) be a causal or nonanticipative functional. Let ξ be a Gaussian sheet satisfying dξ t (ω) = α t, ξ(ω) dt + dW t (ω), P W -a.s., (6.1) where W is a Brownian sheet. Assume also that the induced Gaussian measure P ξ satisfies P ξ ω ∈ Ω : In any case, if a Gaussian weak solution of (6.4) exists, then a unique Gaussian strong solution exists.

Proposition 4 . 2

 42 A centered Gaussian sheet W = ( Wt ) t∈[0,1] d is equivalent in law to a Brownian sheet W if and only if it admits the representation

Theorem 4 . 5 A

 45 Gaussian sheet W is equivalent in law with the Brownian sheet W if and only if there exists a kernel k ∈ L 2 S ([0, 1] 2d ) such that 1 / ∈ σ(K) and a function m ∈ L 2 ([0, 1] d ) and

Corollary 4 . 6

 46 Let W a Brownian sheet and Z be an independent Gaussian sheet. Then the the sheet W + Z is equivalent to a Brownian sheet if and only if there exists m

Proposition 4 . 8

 48 Let b and a be as in Theorem 4.4 and k and m as in Theorem 4.5. Then a = m,

2 ds ,( 5 . 15 )

 2515 where b = Vf and A = [0,•] K1 [0,•] (s)a(s) ds.

(5. 16 )

 16 Proof: By Theorem 5.5 we already know the form of the mean function. Let us then consider the covariance. From (5.12) we see thatCov( Zt , Zs ) = Cov(Z t , Z s ) -[0,t] [0,s] K1 [0,t] (u)K1 [0,s] (v)k(u, v) du dv,where k is connected to b by (4.12): k = b + b * -b ⋆ b * , and b ∈ L 2 ([0, 1] 2d

  The Radon-Nikodym derivative dP Z / dP Z is just (4.10), where W is constructed from Z by Wt =If max i≤d H i ≤ 1/2, and in this case only, we may write (4.10) in terms of Z :

			dP Z dP Z	=	1 Det 2 (K)Tr(K 1 K)	×
	exp -	1 2 [0,1] d [0,1] d	K -1 2 k 1 (t, s) d Zs d Zt +	[0,1] d	K -1 m(t) d Zt +	1 2 [0,1] d	m(t) 2 dt .
					[0,t]

K -1 1 [0,t] (s) d Zs .

  If H i ≤ 1/2 for all i = 1 . . . d, then we can write with f (t, •) = K -1 b(t, •). Otherwise, if max i=1...d H i > 1/2,then there exist functionals that cannot be represented as Wiener integrals with respect to ω.

	where we denoted by
	ω * (t) =	[0,t]	K -1 1 [0,t] (s) dω(s).
	α(t, ω) =	f (t, s) dω(s)
			[0,t]
		α(t, ω) 2 dt < ∞	= 1.	(6.2)
		[0,1] d

From September

the authors's address will be SAMOS-MATISSE, Université de Paris 1, 90 rue de Tolbiac, 75634 Paris Cedex 13, France.

Acknowledgments

Sottinen was financed by the European Commission research training network DYN-STOCH.

Let us introduce operators acting on Volterra kernels:

Note that V -1 is defined for kernels b ∈ L 2

V ([0, 1] 2d ) with the property that b(t,

V ([0, 1] 2d ) then Vb exists. In particular, b = VV -1 b.

Theorem 5.5 Let Z be a fractional Brownian sheet with index H ∈ (0, 1) d .

(i) A Gaussian sheet Z given by the equation

is equivalent in law to Z if and only if

Moreover, the fractional Brownian sheet Z is constructed from Z by

V -1 (Vf ) 1 (t, s) dA(s) (5.11) where (Vf ) 1 is the resolvent of Vf.

(ii) If max i≤d H i ≤ 1/2 then a Gaussian sheet is equivalent in law to a fractional Brownian sheet if and only if it is given by (5.10). If max i≤d H i > 1/2 then there are Gaussian sheets equivalent in law to the fractional Brownian sheet that do not admit the representation (5.10).

Remark 5.6 The terms in (5.10) are of different form than in (4.5), i.e. they are not differentiable. The reason for this lies in the structure of the reproducing kernel Hilbert space of fractional Brownian sheet, cf. Proposition 5.3.

We have the following Kallianpur-Oodaira representations.

Proposition 5.7 A Gaussian sheet Z is equivalent in law to a fractional Brownian sheet Z if and only if it can be represented as

K1 [0,t] (s)a(s) ds (5.12) Remark 5.9 Note that since each one of the variables W t , Wt , Z t , and Zt can be constructed from any of the corresponding sheets by using only the information upto multi-time t the corresponding filtrations coincide:

Double Wiener integrals and Girsanov representation

Suppose that a Gaussian sheet Z is equivalent in law to a fractional Brownian sheet Z. If Z is given by (5.12) then the density is of course given by (4.6):

where the Brownian sheet W is constructed from from the fractional one by (5.3). If Z is given by (5.10) then we know that the inner integral in (4.6) can be represented in terms of Z. However, to consider the outer integral we have to define what we mean by a stochastic or double Wiener integral with respect to fractional Brownian sheet.

In one-parameter case in [START_REF] Pérez-Abreu | A transfer principle for multiple stochastic fractional integrals[END_REF] the authors defined where K 2 = K ⊗ K and the right hand side of (5.14) is a double Wiener integral in the sense of Itô. In the multiparameter case the analogous definition is obvious.

Definition 5.10 Set K 2 = K ⊗ K and let

The double Wiener integral of f ∈ Λ 2 with respect to fractional Brownian sheet is

The situation is now similar to the single Wiener integral case: If max i≤d H i ≤ 1/2 then Λ 2 is complete and otherwise it is not. Indeed, in this case

where

As in the one-parameter case the condition (6.2) implies that P ξ ≪ P W . To see this on just rewrites the proof Theorem 7.3.1 of [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF] in multidimensional time and use the multidimensional Girsanov theorem in [START_REF] Dozzi | Stochastic processes with a multidimensional parameter[END_REF], p. 89. Now, since the measures are Gaussian, we have that P ξ ∼ P W . Hence, by using the Hitsuda representation we have ξ ′ , a weak solution of (6.1), given by

where k 1 is a suitable kernel depending on α. As in the one-parameter case, cf [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF], p. 234ff, we can now show that if (6.1) admits a strong solution, ξ ′′ say, then ξ ′ = ξ ′′ P W (and hence P ξ ) almost surely. So, we have a complete analogue of Theorem 9.4.2 of [START_REF] Kallianpur | Stochastic Filtering Theory[END_REF].

Theorem 6.1 Let α be a nonanticipative functional satisfying (6.2). Then the stochastic equation ( 6.1) has a Gaussian weak solution if and only if α is of the form

where b ∈ L 2 V ([0, 1] 2d ). Moreover, if a Gaussian weak solution exists, then so does a Gaussian strong solution, and the latter is unique.

Using Theorem 6.1 and multiparameter version of the technique used in [START_REF] Sottinen | On Gaussian processes equivalent in law to fractional Brownian motion[END_REF] we may prove the following. Theorem 6.2 Let Z be a fractional Brownian sheet, and let α be a nonanticipating functional satisfying Leb × P Z -a.e., (6.5)


M El

A non-unital algebra has UUNP iff its unitization has UUNP

Keywords: unitization, unique uniform norm property, regular norm. Mathematics Subject Classification 2010. 46H05

Let A be a non-unital Banach algebra, S. J. Bhatt and H. V. Dedania showed that A has the unique uniform norm property (UUNP) if and only if its unitization has UUNP. Here we prove

Let A be a non-unital algebra and let A e = {a + λe : a ∈ A, λ ∈ C} be the unitization of A with the identity denoted by e. For an algebra norm . on A, define a + λe op = sup{ (a + λe)b : b ∈ A, b ≤ 1} and a + λe 1 = a + |λ| for all a + λe ∈ A e . . op is an algebra seminorm on A e , and . 1 is an algebra norm on A e . An algebra norm . on A is called regular if . op = . on A. A uniform norm . on A is an algebra norm satisfying the square property a 2 = a 2 for all a ∈ A; and in this case, . is regular and . op is a uniform norm on A e . An algebra has the unique uniform norm property (UUNP) if it admits exactly one uniform norm. If either (i) or (iv) is satisfied, then . = |||.|||. By noting that (ii) and (iii) are similar by interchanging the roles of . and |||.|||, it is enough to assume (ii). Let (c(A), . ∼ ) be the completion of (A, . ), we distinguish two cases:

(1) c(A) has not an identity:

. ∼ is regular since it is uniform.

By [1, Corollary 2], . ∼ op ≤ . ∼ 1 ≤ 3 . ∼ op on c(A) e (unitization of c(A)). Let a + λe ∈ A e ⊂ c(A) e , a + λe ∼ 1 = a ∼ + |λ| = a +|λ| = a+λe 1 and a+λe ∼ op = sup{ (a+λe)b ∼ : b ∈ c(A), b ∼ ≤ 1} = sup{ (a + λe)b : b ∈ A, b ≤ 1} = a + λe op . Therefore . op ≤ . 1 ≤ 3 .

  Theorem . A non-unital algebra A has UUNP if and only if its unitization A e has UUNP. Proof: Let . and |||.||| be two uniform norms on A e , then . = |||.||| on A since A has UUNP, and so . op = |||.||| op on A e . By [3, Corollary 2.2(1)] and since two equivalent uniform norms are identical, it follows that ( . = . op or . ∼ = . 1 ) and (|||.||| = |||.||| op = . op or |||.||| ∼ = |||.||| 1 = . 1 ); equivalently, at least one of the following holds: (i) . = . op and |||.||| = |||.||| op = . op ; (ii) . = . op and |||.||| ∼ = |||.||| 1 = . 1 ; (iii) . ∼ = . 1 and |||.||| = |||.||| op = . op ; (iv) . ∼ = . 1 and |||.||| ∼ = |||.||| 1 = . 1 .

op . 1 By

 1 (ii), . and |||.||| are equivalent uniform norms, and so . = |||.|||. (2) c(A) has an identity e : Let (c(A e ), |||.||| ∼ ) be the completion of (A e , |||.|||). Since . = |||.||| on A, c(A) can be identified to the closure of A in (c(A e ), |||.||| ∼ ) so that . ∼ = |||.||| ∼ on c(A). Let a + λe ∈ A e ⊂ c(A), a + λe = a + λe op by (ii) = sup{ (a + λe)b : b ∈ A, b ≤ 1} = sup{ (a + λe)b ∼ : b ∈ c(A), b ∼ ≤ 1} = a + λe ∼ since c(A) is unital = |||a + λe||| ∼ = |||a + λe|||. Thus . = |||.|||. Conversely, let . and |||.||| be two uniform norms on A, then . op and |||.||| op are uniform norms on A e , hence . op = |||.||| op since A e has UUNP. Therefore . = . op = |||.||| op = |||.||| on A since . and |||.||| are regular.


Olivier Bonnefon
email: olivier.bonnefon@avignon.inra.fr

Jérôme Coville
email: jerome.coville@avignon.inra.fr

Guillaume Legendre
email: guillaume.legendre@ceremade.dauphine.fr

CONCENTRATION PHENOMENON IN SOME NON-LOCAL EQUATION

Keywords: 1991 Mathematics Subject Classification. Primary:35R09, 45K05; Secondary: 35B40, 35B44, 92D15 Non-local equation, Demo-genetics, Concentration phenomenon, Asymptotic behaviour

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation ∂tu(t, x) = ˆΩ m(x, y) (u(t, y) -u(t, x)) dy+ a(x) -ˆΩ k(x, y)u(t, y) dy u(t, x), supplemented by the initial condition u(0, •) = u 0 in Ω. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values Ω is a bounded subset of R N , the kernels k and m respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait, we construct a positive stationary solution which belongs to the space of Radon measures on Ω. Moreover, in the case where this measure is regular and bounded, we prove its uniqueness and show that, for any non-negative initial datum in L 1 (Ω)∩L ∞ (Ω), the solution of the Cauchy problem converges to this limit measure in L 2 (Ω). We also exhibit an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. The numerical simulations seem to reveal a dependence of the limit measure with respect to the initial datum.

Introduction

In this paper, we are interested in the evolution of a clonal population structured with respect to a phenotypic trait and essentially subjected to three processes: mutation, growth, and competition. As an example, one can think of a virus population structured by its virulence, as this trait can be easily quantified from experimental data. For such type of population, a mathematical model commonly used (see for instance [START_REF] Bürger | Mutation load and mutation-selection-balance in quantitative genetic traits[END_REF][START_REF] Bürger | The mathematical theory of selection, recombination, and mutation, Wiley series in mathematical and computational biology[END_REF][START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF][START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF][START_REF] Calsina | Stationary solutions of a selection mutation model: the pure mutation case[END_REF][START_REF]Asymptotic stability of equilibria of selection-mutation equations[END_REF][START_REF] Perthame | From differential equations to structured population dynamics[END_REF][START_REF] Champagnat | Individual-based probabilistic models of adaptive evolution and various scaling approximations[END_REF][START_REF] Calsina | Asymptotics of steady states of a selection-mutation equation for small mutation rate[END_REF][START_REF] Raoul | Long time evolution of populations under selection and vanishing mutations[END_REF][START_REF]Local stability of evolutionary attractors for continuous structured populations[END_REF]) is based on the following partial integro-differential equations:

∀(t, x) ∈ R * + × Ω, ∂ t u(t, x) = M[u](t,
x) + a(x) -ˆΩ k(x, y)u(t, y) dy u(t, x), [START_REF] Asher | Implicit-explicit methods for time-dependent partial differential equations[END_REF] ∀x ∈ Ω, u(0, x) = u 0 (x), [START_REF] Asher | Implicit-explicit Runge-Kutta methods for timedependent partial differential equations[END_REF] in which the non-negative function u is the density of individuals of the considered population characterized by the trait x, the functions k and a are respectively a kernel accounting for interaction between individuals through competition and a growth rate, the set Ω is a bounded domain of R N , and M denotes a linear diffusion operator modelling the mutation process. Depending on the context, several kinds of diffusion operators have been considered in the literature, see [START_REF] Bürger | Mutation load and mutation-selection-balance in quantitative genetic traits[END_REF][START_REF] Bürger | The mathematical theory of selection, recombination, and mutation, Wiley series in mathematical and computational biology[END_REF][START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF][START_REF] Calsina | Stationary solutions of a selection mutation model: the pure mutation case[END_REF][START_REF] Barles | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Champagnat | Individual-based probabilistic models of adaptive evolution and various scaling approximations[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF][START_REF] Raoul | Long time evolution of populations under selection and vanishing mutations[END_REF][START_REF] Mirrahimi | Dynamics of sexual populations structured by a space variable and a phenotypical trait[END_REF][START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF] among others. In the present work, our attention is focused on models for which the operator M is integral and of the form

(3) ∀v ∈ L 1 (Ω) ∩ L ∞ (Ω), M[v](x) := ˆΩ m(x, y) (v(y) -v(x)) dy,
the function m being a positive kernel satisfying some integrability conditions. Lately, this type of integro-differential equation has attracted a lot of interest and much effort has been made to analyse the solutions of [START_REF] Asher | Implicit-explicit methods for time-dependent partial differential equations[END_REF]. In particular, existence of global solutions in C 1 (R + , L 1 (Ω) ∩ L ∞ (Ω)) for any non-negative initial datum in L ∞ (Ω) and fairly general assumptions on the functions k, m, a and the domain Ω was established in [START_REF] Calsina | Stationary solutions of a selection mutation model: the pure mutation case[END_REF][START_REF]Asymptotic stability of equilibria of selection-mutation equations[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Calsina | Asymptotics of steady states of a selection-mutation equation for small mutation rate[END_REF], and the local stability of bounded continuous stationary solutions in unidimensional domains Ω ⊂ R was investigated in [START_REF] Raoul | Long time evolution of populations under selection and vanishing mutations[END_REF][START_REF] Calsina | Asymptotics of steady states of a selection-mutation equation for small mutation rate[END_REF][START_REF]Local stability of evolutionary attractors for continuous structured populations[END_REF]. However, the analysis of stationary solutions of (1) in higher dimension remains to be done, whereas the long time behaviour of positive solutions of problem (1)-( 2) is still not fully understood.

When mutations are neglected (that is, when m ≡ 0), equation ( 1) is reduced to (4) ∀(t, x) ∈ R * + × Ω, ∂ t u(t, x) = a(x) -ˆΩ k(x, y)u(t, y) dy u(t, x), and, for any generic positive initial datum u 0 , the solution to (4)-( 2) is known to converge weakly to a positive Radon measure µ [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF]. This measure is, in some sense, a stationary solution of (4) representing an evolutionarily stable strategy for the system. For example, when the kernel k is positive and does not depend on the trait, the support of µ lies in the set A := {x ∈ Ω | ∀y ∈ Ω, a(y) ≤ a(x)}, and one may check in such a situation that the sum of Dirac masses

µ = xi∈A a(x i ) k(x i ) δ xi ,
is indeed a stationary solution of the equation. Moreover, when the measure µ is unique, any positive solution u(t, •) of ( 4)-( 2) converges weakly to it as t tends to infinity (see [START_REF] Jabin | On selection dynamics for competitive interactions[END_REF] for a proof).

Since the mutation process can be seen as a diffusion operator on the trait space, it is expected that the long time behaviour of a positive solution to problem (1)-( 2) is simple and that concentration phenomena do not occur. This is verified when the mutation operator M is a classical elliptic operator [START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF][START_REF]Convergence to the equilibria on some mutation selection model[END_REF]. For an integral operator, as in the present case, the existence of bounded equilibria when the domain Ω is unidimensional seems to give credit to such a conjecture. However, we prove that it is false in higher dimension, by exhibiting a class of situations for which a positive singular measure µ, solution of (1), can be constructed and by investigating numerically the long time behaviour of positive solutions of the corresponding Cauchy problem.

1.1. Main results. Let us state precisely the assumptions on the domain Ω, the kernels k and m and the function a under which the results are obtained. First, we suppose that the domain Ω is an open bounded connected set of R N with a Lipschitz boundary, that the function a is such that [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF] a is continuous on Ω and positive, and that the function m is a non-negative symmetric Carathéodory kernel function, that is,

     ∀(x, y) ∈ Ω × Ω, m(x) ≥ 0, m(x, y) = m(y, x), ∀x ∈ Ω, m(x, •) is measurable, for a. e. y in Ω, m(•, y) is continuous. (6)
Finally, we assume that the kernel k is independent of the trait, that is, for any (x, y) in Ω × Ω, k(x, y) = k(y), and that it satisfies the following condition: [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF] there exist positive constants c 0 and

C 0 such that c 0 1 Ω ≤ k ≤ C 0 1 Ω ,
where 1 Ω denotes the characteristic function of the set Ω.

We start by considering stationary solutions of (1), that is, satisfying the equation

(8) ∀x ∈ Ω, M[u](x) + a(x) -ˆΩ k(y)u(y) dy u(x) = 0.
Under the above assumptions, we prove that there exists a positive Radon measure solving (8) in a weak sense.

Theorem 1.1. Assume that the functions a, m and k respectively satisfy conditions (5) to [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF]. Then, there exists a positive Radon measure µ such that ∀ϕ ∈ C c (Ω), ( 9)

ˆΩ (M[ϕ](x) + a(x)ϕ(x)) dµ(x) = ˆΩ ϕ(x) dµ(x) ˆΩ k(x) dµ(x) .
Let λ p be the principal eigenvalue of the operator M + a defined by

λ p (M + a) := sup {λ ∈ R | ∃ϕ ∈ C(Ω), ϕ > 0 and M[ϕ] + (a + λ)ϕ ≤ 0 in Ω}.
We have the following characterisation for the measure µ.

• If λ p is associated with an eigenfunction ϕ p which belongs to L 1 (Ω), then µ is an absolutely continuous measure, that is, there exists ū in L 1 (Ω) such that dµ(x) = ū(x)dx, and the unique strong solution to equation [START_REF]Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF]. Moreover, the function ū is in L ∞ (Ω) when the principal eigenfunction ϕ p belongs to L ∞ (Ω). • Otherwise, µ is a singular measure.

As a consequence of this theorem, the existence of a singular measure solution to [START_REF] Bürger | The mathematical theory of selection, recombination, and mutation, Wiley series in mathematical and computational biology[END_REF] is strongly related to the non-existence of an integrable principal eigenfunction for the non-local operator M + a, a result which was proved recently [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Shen | Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats[END_REF][START_REF]Singular measure as principal eigenfunction of some nonlocal operators[END_REF].

Next, the global stability of the measure µ and the long time behaviour of positive solutions of the problem (1)-( 2) are analysed. When the measure is regular, we have the following convergence result.

Theorem 1.2. Assume that the functions a, m and k respectively satisfy conditions (5) to [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF] and that there exists an absolutely continuous positive Radon measure µ, such that dµ(x) = ū(x) dx, solution to [START_REF]Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF]. Assume further that the function ū belongs to L ∞ (Ω). Then, for any non-negative initial datum u 0 in L 1 (Ω)∩L ∞ (Ω), the positive solution u of problem (1)-( 2) satisfies

lim t→∞ u(t, •) -ū L 2 (Ω) = 0.
Note that this last global stability result implies the uniqueness of the measure solution of (8). On the contrary, when no regular positive Radon measure exists, the convergence of a positive solution of (1) to a stationary one is delicate to analyse. To shed light on the possible dynamics prevailing in such a situation, we have conducted a few numerical experiments.

1.2. Numerical simulations. In order to gain some insight on the long time behaviour of solutions of problem (1)-( 2), we compute approximate solutions, for different choices of growth function a and initial datum u 0 , using a numerical method. Limiting ourselves to preliminary computations, the retained configuration is such that the two-dimensional domain Ω = B 1/4 (0) the open ball of radius 0.25 centred at the origin, and the competition and mutation kernels are two constant functions, such that k ≡ 1 and m ≡ ρ, with ρ a positive constant, respectively. The problem to be solved thus reduces to the equation [START_REF] Bürger | Mutation load and mutation-selection-balance in quantitative genetic traits[END_REF] ∂ t u(t, x) = ρ ˆΩ(u(t, y) -u(t, x)) dy + a(x) -ˆΩ u(t, y) dy u(t, x), for any (t, x) in R * + × Ω, completed by the initial condition [START_REF] Calsina | Stationary solutions of a selection mutation model: the pure mutation case[END_REF] ∀x ∈ Ω, u(0, x) = u 0 (x).

1.2.1.

A simple growth rate. We first consider a configuration in which the growth rate achieves its maximum at a single point, a case for which the uniqueness of the stationary solution can be shown.

Proposition 1. For any positive value ρ, there exist a unique positive measure µ which is a stationary solution of (10), and a critical value ρ * such that this measure is singular for ρ < ρ * or regular otherwise. In addition, for any non-negative initial datum u 0 in L 1 (Ω) ∩ L ∞ (Ω), the solution of problem (10)- [START_REF] Calsina | Stationary solutions of a selection mutation model: the pure mutation case[END_REF] converges weakly to µ.

This proposition is a direct consequence of Theorem 1.1 and of the uniform L 1 estimates obtained in Section 3. To illustrate its conclusions, we take a(x) = 1 -

x 2 , where • 2 denotes the Euclidean norm in R 2 , and solve numerically the problem. The obtained results, presented in Figures 1 and2, provide a clear picture of the dynamics of the solution with respect to the constant value of the mutation rate. 10)-( 11) at different times, starting from a uniform initial density with value 1. The mutation and competition rates are constant and set to 2 and 1 respectively, while the growth rate function achieves its maximum at the origin. We observe a convergence of the approximate solution towards a regular stationary state, the stationarity being attained numerically around t = 590. 10)-( 11) at different times, starting from a uniform initial density with value 1. The mutation and competition rates are constant and set respectively to 0.01 and 1, while the growth rate function achieves its maximum at the origin. We observe a very slow convergence towards a stationary state (as the approximate solution continues to take increasing values in a single element at t = 3000) characteristic of a singular concentration phenomenon.

1.2.2.

A complex growth rate. We next explore a situation for which the growth rate reaches its maximum at multiple points. In such a setting, we expect the stationary measure to be non-unique. In order to numerically investigate this conjecture, we consider a growth rate function of the form

a(x) = 1-4 (x 1 -0.1) 2 + x 2 2 4 (x 1 + 0.1) 2 + x 2 2 4 x 2 1 + (x 2 -0.1) 2 4 x 2 1 + (x 2 + 0.1) 2 .
With this choice, for a sufficiently small constant value ρ of the mutation rate, we can show that there are at least four different positive Radon measures which are stationary solutions of the problem. The impact of the non-uniqueness can be seen in the simulations presented in Figures 3 and4. Indeed, as soon as the regime allows several singular stationary measures to exist, we observe that the outcome of the simulation may drastically differ depending on the initial datum. In contrast, when the mutation rate is such that the stationary measure is regular, the stationary state is a global attractor (see Figure 5). 10)-( 11) at different times. The mutation and competition rates are constant and set respectively to 0.01 and 1, the growth rate function achieves its maximum at four points and the initial datum u 0 is such that it vanishes on three of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing a single Dirac mass.

1.3. Outline. The paper is organised as follows. We start by recalling important facts about the spectral properties of the class of non-local operators considered in Section 2. Uniform estimates are derived by means of non-linear relative entropy formulas in Section 3, leading to proofs of Theorems 1.1 and 1.2 in Section 4. Finally, the numerical method used for the simulations is briefly described in an appendix.

Spectral properties of non-local operators

Consider the eigenproblem

(12) M[ϕ] + a ϕ + λ ϕ = 0 in Ω,
in which M is the integral operator defined by (3), whose kernel satisfies condition [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF]. When the function a is not constant, neither the operator M + a + λ nor its inverse are compact, and the Krein-Rutman theory fails in providing existence of the principal eigenvalue of M + a. However, a variational formula, introduced in [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF] to 10)-( 11) at different times. The mutation and competition rates are constant and set respectively to 0.01 and 1, the growth rate function achieves its maximum at four points and the initial datum u 0 is such that it vanishes on two of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing two Dirac masses.

(a) t = 0 (b) t = 5 (c) t = 20 (d) t = 50 (e) t = 100 (f) t = 250 (g) t = 500 (h) t = 1000
characterise the principal eigenvalue of elliptic operators, can be transposed to the operator M + a. Namely, the quantity

(13) λ p (M + a) := sup {λ ∈ R | ∃ϕ ∈ C(Ω), ϕ > 0 and M[ϕ] + a ϕ + λ ϕ ≤ 0 in Ω}
is well defined and called the generalised principal eigenvalue of M + a. It is also well-known (see [START_REF] Coville | On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators[END_REF][START_REF] Shen | Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats[END_REF][START_REF] Coville | Pulsating fronts for nonlocal dispersion and KPP nonlinearity[END_REF][START_REF] Berestycki | Properties of the principal eigenvalue of some nonlocal operators[END_REF][START_REF] Shen | On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications[END_REF]) that λ p (M + a) is not always an eigenvalue of M + a in a reasonable Banach space, which means there is not always a positive continuous eigenfunction associated with it. Nevertheless, as shown in [START_REF]Singular measure as principal eigenfunction of some nonlocal operators[END_REF], there always exists an associated positive Radon measure.

Theorem 2.1 ([17]).

Let Ω ⊂ R N be a bounded domain, a be a continuous function over Ω, M be the integral operator defined by (3), whose kernel m satisfies condition [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF], and define

M (x) := ˆΩ m(x, y) dy, σ := sup x∈Ω (a(x)-M (x)), A := x ∈ Ω | a(x) -M (x) = σ .
Then, there exists a positive Radon measure µ p such that, for any function ϕ in C c (Ω), we have

ˆΩ ϕ(x) ˆΩ m(x, y) dµ p (y) dx + ˆΩ ϕ(x)(a(x) -M (x) + λ p ) dµ p (x) = 0.
In addition, we have the following dichotomy: to (H)) of these points. In both cases, rapid convergence of the approximate solution towards an identical regular stationary state is observed, the numerical stationarity being attained around t = 85.

(a) t = 0 (b) t = 5 (c) t = 20 (d) t = 100 (e) t = 0 (f) t = 5 (g) t = 20 (h) t = 100
• either there exists ϕ p in L 1 (Ω), ϕ p > 0, such that dµ p (x) = ϕ p (x) dx,

• or there exists g p in C(Ω), g p > 0, and ν a positive singular measure with respect to the Lebesgue measure, whose support lies in the set A, such that

dµ p (x) = g p (x) σ -(a(x) -M (x)) dx + dν(x).
The measure µ p can be characterised more precisely and there exists a simple criterion guaranteeing its regularity.

Proposition 2 ([15, 19, 4, 18]). Under the assumptions of the preceding theorem, one has dµ p (x) = ϕ p (x) dx with ϕ p in C(Ω), ϕ p > 0, if and only if λ p (M + a) < -σ.

We conclude with an alternate characterisation of λ p (M + a). In the spirit of the works in [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF]Reaction-diffusion equations for population dynamics with forced speed. I. The case of the whole space[END_REF], let us define the quantity

λ p (M + a) := inf{λ ∈ R | ∃ϕ ∈ C(Ω) ∩ L ∞ (Ω), ϕ ≥ 0, M[ϕ] + (a + λ)ϕ ≥ 0 in Ω}.
As in the case of elliptic operators, we have the following result.

Theorem 2.2 ( [START_REF] Coville | Pulsating fronts for nonlocal dispersion and KPP nonlinearity[END_REF][START_REF] Berestycki | Properties of the principal eigenvalue of some nonlocal operators[END_REF][START_REF]Nonlocal refuge model with a partial control[END_REF]). Let Ω ⊂ R N be a bounded domain, a be a continuous function over Ω, M be the integral operator defined by (3), whose kernel m satisfies condition [START_REF] Berestycki | The principal eigenvalue and maximum principle for second-order elliptic operators in general domains[END_REF]. Then, we have λ p (M + a) = λ p (M + a).

A priori estimates

In this section, given a non-negative initial datum u 0 in L 1 (Ω)∩L ∞ (Ω), we establish some uniform in time a priori estimates on the solution of problem (1)-(2). To do so, we start by proving a non-linear relative entropy identity satisfied by any solution of (1).

Proposition 3 (general identity). Let Ω ⊂ R N be a bounded domain and assume that the functions a, m and k respectively satisfy conditions (5) to [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF]. Let H be a smooth (of class C 1 at least) function. Finally, let the function ū in L 1 (Ω) ∩ L ∞ (Ω) be a positive stationary solution of (1), and the function u in C 1 ((0, +∞), L ∞ (Ω)) be a solution of (1). Then we have

(14) ∀t ∈ R * + , d dt H H,ū [u](t) = -D H,ū [u](t) + Γ(t) ˆΩ ū(x)H u ū (t, x) u(t, x) dx,
where H H,ū [u], D H,ū (u) and Γ are respectively defined by

H H,ū [u](t) := ˆΩ H u(t, x) ū(x) (ū(x)) 2 dx, Γ(t) := ˆΩ k(y)(ū(y) -u(t, y)) dy, D H,ū [u](t) := ˆΩ2 H u(t, x) ū(x) -H u(t, y) ū(y) + H u(t, x) ū(x) u(t, x) ū(x) - u(t, y) ū(y) dν,
where ν is the measure on Ω × Ω defined by ν := m(x, y)ū(x)ū(y) dx dy.

Proof:

Since the kernel k satisfies condition [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF] and defining Γ as above, we have from equation (1) that, for all t in R * + and almost every x in Ω,

∂ t u(t, x) = M[u](t, x) + a(x) -ˆΩ k(y)ū(y) dy u(t, x) + Γ(t) u(t, x).
The function ū being a positive stationary solution of (1), we also have that, for almost every x in x in Ω,

a(x) -ˆΩ k(y)ū(y) dy = - 1 ū(x) M[ū](x),
and we can rewrite the above equation as follows:

∀t ∈ R * + , for a. e. x ∈ Ω, ∂ t u(t, x) = M[u](t, x) - u(t, x) ū(x) M[ū](x) + Γ(t) u(t, x).
Multiplying this identity by the function ū(x)H u(t,x)

ū(x)
and integrating over Ω, we find that, for all

t in R * + , ˆΩ ū(x)H u(t, x) ū(x) ∂ t u(t, x) = ˆΩ H u(t, x) ū(x) (ū(x)M[u](t, x) -u(t, x)M[ū](x)) + Γ(t) ˆΩ ū(x)H u(t, x) ū(x) u(t, x).
Using definition (3) and rearranging the terms, we get, for all

t in R * + , ˆΩ ū(x)H u(t, x) ū(x) ∂ t u(t, x)dx = ˆΩ2 H u(t, x) ū(x) u(t, y) ū(y) - u(t, x) ū(x) dν + Γ(t) ˆΩ ū(x)H u(x) ū(t, x) u(t, x)dx,
where ν stands for the positive measure ν := m(x, y)ū(x)ū(y)dxdy.

Due to the symmetry property of the kernel m, we straightforwardly see that

∀t ∈ R * + , ˆΩ2 H u(t, x) ū(x) -H u(t, y) ū(y) dν = 0,
and, by combining the above equalities and using the remaining definitions, we finally reach

∀t ∈ R * + , d dt H H,ū [u](t) = -D H,ū [u] + Γ(t) ˆΩ ū(x)H u(t, x) ū(x) u(t, x) dx.
Remark 1. When k ≡ 0, equation ( 1) is linear and relative entropy formulas are well-known in this case [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF].

Remark 2. When the function H is non-decreasing and ū is only assumed to be a stationary super-solution of (1), we clearly see from the above proof that

∀t ∈ R * + , d dt H H,ū [u](t) ≤ -D H,ū [u](t) + Γ(t) ˆΩ ū(x)H u ū (t, x) u(t, x) dx.
Similarly, if ū is a positive stationary sub-solution of (1), we have

∀t ∈ R * + , d dt H H,ū [u](t) ≥ -D H,ū [u](t) + Γ(t) ˆΩ ū(x)H u ū (t, x) u(t, x) dx.
Equipped with this general relative entropy identity, we are in a position to derive useful differential inequalities. Proposition 4. Let Ω ⊂ R N be a bounded domain and assume that the functions a, k and m respectively satisfy conditions (5) to [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF]. Let the functions ū and u be defined as in Proposition 3, q in [1, +∞) and H q be the smooth convex function H q : s → s q .

Then the functional F[u] := log

H Hq , ū[u] (HH 1 , ū[u]) q satisfies (15) ∀t ∈ R * + , d dt F[u](t) = - 1 H Hq,ū [u](t) D Hq,ū [u](t) ≤ 0.
Moreover, we have for all t ∈ R * + ,

(16) D Hq,ū [u](t) = q 2 ˆΩ2 u(t, x) ū(x) - u(t, y) ū(y) u(t, x) ū(x) q-1 - u(t, y) ū(y) q-1 dν,
where ν is the positive measure in Ω × Ω defined by ν := m(x, y)ū(x)ū(y)dxdy.

Remark 3. When q = 2, we observe that H H2,ū [u](t) = u(t, •) 2 L 2 (Ω) , yielding a Lyapunov functional which involves the L 2 (Ω)-norm of u(t, •) instead of a weighted L q (Ω)-norm. Indeed, we have, for all t in R * + ,

d dt log u(t, •) 2 L 2 (Ω) (H H1,ū [u](t)) 2 = - 1 u(t, •) 2 L 2 (Ω) ˆΩ2 u(t, x) ū(x) - u(t, y) ū(y) 2 dν.

Proof of Proposition 4:

Owing to Proposition 3, we have, for

H = H q , ∀t ∈ R * + , d dt H Hq,ū [u](t) = -D Hq,ū [u](t) + q Γ(t) ˆΩ ū(x) u(t, x) ū(x) q-1 u(t, x) dx,
which, by definition of H Hq,ū [u], can be written

∀t ∈ R * + , d dt H Hq,ū [u](t) = -D Hq,ū [u](t) + q Γ(t)H Hq,ū [u](t).
A quick computation showing that D H1,ū (u) = 0, we therefore obtain, setting q = 1,

∀t ∈ R * + , d dt H H1,ū [u](t) = Γ(t)H H1,ū [u](t). Since ∀q ∈ [1, +∞), ∀t ∈ R + , H Hq,ū [u](t) > 0, we have, for all t in R * + , d dt (log(H H1,ū [u](t))) = Γ(t), ( 17
) ∀q ∈ (1, +∞), d dt log(H Hq,ū [u](t)) = - 1 H Hq,ū [u](t)
D Hq,ū [u](t) + q Γ(t), [START_REF]Nonlocal refuge model with a partial control[END_REF] and, combining these two relations, we end up with

∀t ∈ R * + , d dt log H Hq,ū [u](t) (H H1,ū [u](t)) q = - 1 H Hq,ū [u](t) D Hq,ū [u](t).
Equality [START_REF]Convergence to the equilibria on some mutation selection model[END_REF] then follows from direct computations, by using symmetry and an obvious change of variables.

We may now obtain uniform in time a priori bounds on the L 1 (Ω)-norm of a solution to problem (1)-( 2). Lemma 3.1. Let Ω ⊂ R N be a bounded domain and assume that the functions a, m and k respectively satisfy conditions (5) to [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF]. Let the function u in C 1 (R * + , L 1 (Ω) ∩ L ∞ (Ω)) be a non-negative solution of the Cauchy problem (1)-( 2) with non-negative initial datum u 0 in L 1 (Ω) ∩ L ∞ (Ω), such that u 0 L 1 (Ω) = 0. Then, there exist two positive constants c 1 and C 1 , depending on u 0 , such that

∀t ∈ R + , c 1 ≤ u(t, •) L 1 (Ω) ≤ C 1 .

Proof:

We first observe that large (respectively small) constants are stationary supersolutions (respectively sub-solutions) of equation [START_REF] Asher | Implicit-explicit methods for time-dependent partial differential equations[END_REF]. Indeed, given a constant C

such that C ≥ sup x∈Ω a(x) ´Ω k(z) dz , one has ∀x ∈ Ω, M[C] + C a(x) -C ˆΩ k(z) dz = C a(x) -sup x∈Ω a(x) ≤ 0,
and similarly, for c ≤ inf

x∈Ω a(x) ´Ω k(z) dz , ∀x ∈ Ω, M[c] + c a(x) -c ˆΩ k(z) dz = c a(x) -inf x∈Ω a(x) ≥ 0.
Therefore, from Proposition 3 and Remark 2, choosing the stationary solution ū as a large, respectively small, constant and considering the convex function H = H 1 , we obtain, for all t in R * + and C ≥ sup x∈Ω a(x)

´Ω k(z) dz , d dt u(t, •) L 1 (Ω) ≤ C ˆΩ k(z) dz -ˆΩ k(z)u(t, z) dz u(t, •) L 1 (Ω) , respectively, for all t in R * + and c ≤ inf x∈Ω a(x) ´Ω k(z) dz , d dt u(t, •) L 1 (Ω) ≥ c ˆΩ k(z) dz -ˆΩ k(z)u(t, z) dz u(t, •) L 1 (Ω) .
Using the fact that the kernel k satisfies condition [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF], we finally get, for all

t in R * + , C ≥ sup x∈Ω a(x) ´Ω k(z) dz , and c ≤ inf x∈Ω a(x) ´Ω k(z) dz d dt u(t, •) L 1 (Ω) ≤ C ˆΩ k(z) dz -c 0 u(t, •) L 1 (Ω) u(t, •) L 1 (Ω) , d dt u(t, •) L 1 (Ω) ≥ c ˆΩ k(z) dz -C 0 u(t, •) L 1 (Ω) u(t, •) L 1 (Ω) .
Let us c 1 and C 1 be the following positive constants,

c 1 := min u 0 L 1 (Ω) ; ´Ω k(z) dz C 0 , C 1 := max u 0 L 1 (Ω) ; ´Ω k(z) dz c 0 .
From the logistic character of these differential inequalities and since u 0 L 1 (Ω) > 0, we deduce that, for all t in R + ,

c 1 ≤ u(t, •) L 1 (Ω) ≤ C 1 .
Remark 4. One may observe that the above proof relies on the fact that the function k is bounded above and below by positive constants. As a consequence, such an uniform L 1 (Ω)-norm estimate holds more generally if the kernel k depends on the trait x.

Proofs

We are now in a position to prove Theorems 1.1 and 1.2.

4.1. Construction of a stationary state. To construct a solution to stationary equation ( 8) in the space of Radon measures (and thus prove Theorem 1.1), we look for a measure µ satisfying [START_REF] Bürger | The mathematical theory of selection, recombination, and mutation, Wiley series in mathematical and computational biology[END_REF]. Owing to Theorem 2.1, consider a positive measure µ p associated with λ p (M + a) and normalised in order to have ´Ω dµ p (x) = 1. We are reduced to proving the following assertion.

Claim 4.1. There exists a unique positive real number θ such that θ µ p satisfies (9).

Proof:

Set θ = -λ p (M + a) ´Ω k(x) dµ p (x)
. We have, for all ϕ in C c (Ω),

ˆΩ (M[ϕ](x) + a(x) ϕ(x)) θ dµ p = -λ p (M + a) ˆΩ ϕ(x)θ dµ p = - λ p (M + a) ´Ω k(x) dµ p ˆΩ k(x) dµ p ˆΩ ϕ(x)θ dµ p = ˆΩ k(x)θ dµ p ˆΩ ϕ(x)θ dµ p ,
so that θ µ p is a solution to [START_REF] Bürger | The mathematical theory of selection, recombination, and mutation, Wiley series in mathematical and computational biology[END_REF]. To conclude, it remains to show that -λ p (M+a) > 0.

We first notice that the couple λ = -inf x∈Ω a(x) and ϕ ≡ 1 satisfies

∀x ∈ Ω, M[ϕ](x) + (a(x) + λ) ϕ(x) = a(x) -inf y∈Ω a(y) ≥ 0.
It then follows directly from the alternate characterisation of the principal eigenvalue, given by Theorem 2.2, that

λ p (M + a) ≤ -inf x∈Ω a(x) < 0.
Remark 5. From the above computation, it is clearly seen that the uniqueness of the stationary state follows from the uniqueness of the measure associated with the principal eigenvalue λ p (M + a).

4.2.

Long time behaviour. This final subsection is devoted to the proof of Theorem 1.2. We now assume that the positive measure µ constructed above is absolutely continuous and bounded, that is, there exists a function ū in L 1 (Ω) ∩ L ∞ (Ω) such that dµ(x) = ū(x) dx. This measure being linked to the principal eigenvalue λ p (M + a) through Claim 4.1, we have in addition that ū = θ ϕ p , with ϕ p a function in

L 1 (Ω) ∩ L ∞ (Ω).
From the regularity of ϕ p , we infer that the function ū is a strong solution of equation ( 8). Thus, knowing that a positive continuous stationary solution exists, we may derive further a priori estimates on the solution u of problem (1)-(2).

Lemma 4.2. Let Ω ⊂ R N be a bounded domain and assume that the functions a, m and k respectively satisfy conditions (5) to [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF]. Let u in C 1 ((0, +∞), L 1 (Ω) ∩ L ∞ (Ω)) be a non-negative solution of the Cauchy problem (1)-( 2) with non-negative initial datum u 0 in L 1 (Ω) ∩ L ∞ (Ω), u 0 ≥ 0. Then, there exist two positive constants c 2 and C 2 , depending on u 0 , such that

∀t ∈ R * + , c 2 ≤ u(t, •) L 2 (Ω) ≤ C 2 .
Proof:

The uniform lower bound follows directly from Hölder's inequality combined with the estimates in Lemma 3.1. Indeed, since the set Ω is bounded, we have

∀t ∈ R * + , c 1 ≤ u(t, •) L 1 (Ω) ≤ |Ω| u(t, •) L 2 (Ω)
, where |Ω| denotes the Lebesgue measure of the set Ω.

On the other hand, the uniform upper bound is a straightforward application of Proposition 4. More precisely, the function ū in L 1 (Ω) ∩ L ∞ (Ω) being a positive stationary solution of (1), the quantity

F(t) = log u(t,•) 2 L 2 (Ω) (H H 1 , ū[u](t)) 2
is a decreasing function of t. Therefore, we have

∀t ∈ R + , u(t, •) L 2 (Ω) ≤ u 0 L 2 (Ω) H H1,ū [u 0 ] H H1,ū [u](t) ≤ C(u 0 ) ū L ∞ (Ω) u(t, •) L 1 (Ω) ,
and, using Lemma 3.1, we finally reach

∀t ∈ R + , u(t, •) L 2 (Ω) ≤ C(u 0 )C 1 ū L ∞ (Ω) .
To show that the solution u converges to a stationary state, we introduce the decomposition [START_REF] Coville | Pulsating fronts for nonlocal dispersion and KPP nonlinearity[END_REF] ∀t ∈ R * + , u(t, •) = λ(t) ū + h(t, •), in which the function h is such that ∀t ∈ R * + , ˆΩ ϕ p (x)h(t, x) dx = 0.

Proof:

Let us denote by •, • L 2 the standard inner product over L 2 (Ω). It stems from the decomposition of the solution u that

∀t ∈ R + , u(t, •), ū L 2 = θ u(t, •), ϕ p L 2 = λ(t)θ 2 ϕ p , ϕ p L 2 .
Since the function ϕ p is positive and bounded over Ω, we have from Lemma 3.1

∀t ∈ R + , c 1 (u 0 ) inf x∈Ω ϕ p (x) ≤ ˆΩ u(t, x)ϕ p (x) dx ≤ C 1 (u 0 ) ϕ p L ∞ (Ω) , which gives (20) ∀t ∈ R + , c 1 (u 0 ) inf x∈Ω ϕ p (x) θ ≤ λ(t) ≤ C 1 (u 0 ) ϕ p L ∞ (Ω) θ .
We next observe that an obvious upper bound for h(t, •) L 2 (Ω) can be derived from Lemma 4.2 since, by construction, one has [START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF] ∀t

∈ R + , h(t, •) 2 L 2 (Ω) ≤ u(t, •) 2 L 2 (Ω) ≤ C 2 .
Substituting to the solution u its decomposition into equation (1), we moreover have The function H H2,ū [h] being non-negative, we either have that it is positive at all times or there exists t 0 in R + such that H H2,ū [h](t 0 ) = 0. In the latter case, this means that u(t 0 , x) = λ(t 0 )θϕ p (x) for almost every x in Ω. Setting w(t, •) := η(t)θϕ p , the function η satisfying the initial value problem

∀(t, x) ∈ R * + × Ω, λ (t) ū(x) + ∂h ∂t (t, x) = λ(t)M[ū](x) + M[h](t, x) + a(x) -ˆΩ k(y)u(t,
d dt η(t) = -λ p η(t)(1 -η(t)), η(t 0 ) = λ(t 0 ),
so that it tends to 1 as t tends to infinity, we can check that w is a solution of (1) for all times larger than or equal to t 0 . Since w(t 0 , •) = u(t 0 , •), we have, by uniqueness of the solution of the Cauchy problem [START_REF] Asher | Implicit-explicit methods for time-dependent partial differential equations[END_REF], that u(t, • = w(t, •) for all t ≥ t 0 , which yields that λ(t) = η(t) and h(t, •) = 0 for all t ≥ t 0 . In the former case, we resort to the following result.

∈ R + , H H1,ū [u](t) = u(t, •), ū L 2 (Ω) = λ(t) ū, ū L 2 (Ω) ,
and we deduce from Proposition 4 that

∀t ∈ R * + , λ (t) = -λ p (1 -λ(t)) -ˆΩ k(y)h(t, y) dy λ(t).
The Claim implying that h(t, •) 2 L 2 (Ω) = H H2,ū [h](t) vanishes as t tends to infinity, we then have that

∀t ∈ R * + , λ (t) = -λ p (1 -λ(t))λ(t) + λ(t) o(1), with o(1) ≤ C h(t, •) L 2 (Ω)
, and an elementary analysis of this ordinary differential equation finally shows that λ(t) tends to 1 as t tends to infinity.

Let us now conclude by giving the postponed proof of the above Claim. Proof of Claim 4.4:

The function H H2,ū [h] being positive, we see, using [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF] and following the proof of Proposition 4, that the function

F(t) = log H H 2 , ū [h](t) (HH 1 , ū[u](t)) 2 verifies (23) ∀t ∈ R * + , d dt F(t) = - D H2,ū [h](t) H H2,ū [h](t) ≤ 0
and is thus both smooth and non-increasing. The initial value function of this semi-discrete problem is the projection of the function u 0 onto the chosen finite element space.

While the mass matrices are sparse, that is most of their entries are zero, the matrix K(u) associated with the non-linear term is a priori dense, which prevents the use of fully implicit time-integration schemes to solve the previous differential system when the integer N is large. A common strategy is thus to consider a combination of implicit and explicit (linear multistep or Runge-Kutta) schemes (see [START_REF] Asher | Implicit-explicit methods for time-dependent partial differential equations[END_REF][START_REF] Asher | Implicit-explicit Runge-Kutta methods for timedependent partial differential equations[END_REF] for instance), the most simple case of being that of the forward and backward Euler methods, yielding the system of algebraic equations

M u (n+1) h -u (n) h ∆t = (M a -M m ) u (n+1) h -K(u (n) h )u (n) h + b(u (n) h ),
where ∆t is the length of the fixed time step (which has to be chosen small enough due to stability restrictions) and u

(n) h

is the approximation of u h at time n ∆t, n ∈ 1, . . . , T ∆t , to be solved at each iteration. This method was implemented using FreeFem++ [START_REF] Hecht | New development in FreeFem++[END_REF] to obtain the numerical simulations presented in Subsection 1.2. The same mesh was used in all the simulations and composed of approximately 30000 triangles. The length of the time step was chosen constant and equal to 0.1. Note that the piecewise constant numerical solutions were interpolated continuously for the visual representations in Figures 1 to 5.

Figure 1 .

 1 Figure 1. Numerical approximation of the solution of problem (10)-(11) at different times, starting from a uniform initial density with value 1. The mutation and competition rates are constant and set to 2 and 1 respectively, while the growth rate function achieves its maximum at the origin. We observe a convergence of the approximate solution towards a regular stationary state, the stationarity being attained numerically around t = 590.

(a) t = 5 Figure 2 .

 52 Figure 2. Numerical approximation of the solution of problem (10)-[START_REF] Calsina | Stationary solutions of a selection mutation model: the pure mutation case[END_REF] at different times, starting from a uniform initial density with value 1. The mutation and competition rates are constant and set respectively to 0.01 and 1, while the growth rate function achieves its maximum at the origin. We observe a very slow convergence towards a stationary state (as the approximate solution continues to take increasing values in a single element at t = 3000) characteristic of a singular concentration phenomenon.

Figure 3 .

 3 Figure 3. Numerical approximation of the solution of problem (10)-(11) at different times. The mutation and competition rates are constant and set respectively to 0.01 and 1, the growth rate function achieves its maximum at four points and the initial datum u 0 is such that it vanishes on three of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing a single Dirac mass.

Figure 4 .

 4 Figure 4. Numerical approximation of the solution of problem (10)-(11) at different times. The mutation and competition rates are constant and set respectively to 0.01 and 1, the growth rate function achieves its maximum at four points and the initial datum u 0 is such that it vanishes on two of these four points. We observe a slow convergence of the numerical solution towards the approximation of a singular stationary measure containing two Dirac masses.

Figure 5 .

 5 Figure 5. Numerical approximation of the solution of problem (10)-(11) at different times for two configurations, which differ only in their initial datum. The mutation and competition rates are constant and set respectively to 2 and 1, the growth rate function achieves its maximum at four points and the initial datum u 0 is such that it vanishes on three (subfigures (A) to (D)) or two (subfigures (E) to (H)) of these points. In both cases, rapid convergence of the approximate solution towards an identical regular stationary state is observed, the numerical stationarity being attained around t = 85.

Claim 4 . 3 .

 43 One has lim t→+∞ λ(t) = 1 and lim t→+∞ h(t, •) L 2 (Ω) = 0.

D

  H2,ū [h(t n , •)] = 0, which enforces that h = τ ū for some real constant τ . Recalling that that the function h(t n , •) is orthogonal to ū for any integer n, we then find thatτ ū 2 L 2 (Ω) = ˆΩ h(x)ū(x) dx = lim n→+∞ ˆΩ h(t n , x)ū(x) dx = 0,implying that τ = 0. Using (25) and (26) with the fact that h ≡ 0 thus leads to the following contradictionκ ≤ lim n→+∞ ˆΩ(h(t n , x)) 2 dx ≤ C -1 lim n→+∞ ˆΩ(h(t n , x)) 2 g(x) dx = 0. the matrix K(u) is such that ∀(i, j) ∈ {1, . . . , N } 2 , (K(u)) ij = N l=1 u l ˆΩ ˆΩ k(x,y)w l (y) dy w i (x)w j (x) dx, and the last term is the vector defined by ∀i ∈ {1, . . . , N }, (b(u)) i = N l=1 u l ˆΩ ˆΩ m(x, y)w l (y) dy w i (x) dx.

  To prove the Claim, it is thus sufficient, due to this monotonicity property, to exhibit a sequence (t n ) n∈N tending to infinity such that the corresponding sequence (H H2,ū [h](t n )) n∈N has a vanishing limit, which amounts to proving that inf t∈R+ H H2,ū [h](t) = 0. Now, let us assume that inf t∈R+ H H2,ū [h](t) = κ > 0. Then, using (21) and (20) respectively, we infer there exist three positive constants α, β and γ such that∀t ∈ R * + , 0 < κ ≤ H H2,ū [h](t) ≤ α and 0 < β ≤ H H1,ū [u](t) ≤ γ.As a consequence from the above, there exists a constant c 0 such that

	(24)	lim t→+∞	F(t) = c 0 and lim t→+∞	d dt	F(t) = 0.
	Let (t n ) n∈N be a sequence that tends to infinity. On the one hand, it follows from
	(23) and (24) that
	(25)		lim
	We have
	(26)

n→+∞ D H2,ū [h(t n , •)] = 0.

On the other hand, the sequence (h(t n , •)) n∈N being bounded in L 2 (Ω), it has a weakly convergent subsequence in L 2 (Ω), which we still denote (h(t n , •)) n∈N , with limit h.

D H2,ū [h(t n , •)] = 2 ˆΩ(h(t n , x)) 2 g(x) dx -ˆΩ ˆΩ m(x,

y)h(t n , x)h(t n , y) dx dy , where the function g(x) := ˆΩ m(x, y) ū(y) ū(x) dy is well defined over Ω and bounded from below by a positive constant C, the functions ū and m being both positive and bounded. By Fatou's lemma and the weak convergence of (h(t n , •)) n∈N in L 2 (Ω), we thus get ˆΩ( h(x)) 2 g(x) dx ≤ lim inf n→+∞ ˆΩ(h(t n , x)) 2 g(x) dx and lim n→+∞ ˆΩ ˆΩ m(x, y)h(t n , x)h(t n , y) dx dy = ˆΩ ˆΩ m(x, y) h(x) h(y) dx dy.

The research leading to these results has received funding from the french ANR program under the "ANR JCJC" project MODEVOL: ANR-13-JS01-0009 held by Gael Raoul and the "ANR DEFI" project NONLOCAL: ANR-14-CE25-0013 held by François Hamel. J. Coville wants to thank G. Raoul for interesting discussions on this topic.

Hence, one has inf t∈R+ H H2,ū [h](t) = 0, which implies that lim inf t→+∞ H H2,ū [h](t) = 0, the function being non-negative.

The proof is complete.

Appendix A. The numerical method

To investigate numerically the asymptotic behaviour of solutions to problem (1)-( 2), we assume that we are in the setting of Lemma 4.2, so that solutions belong to C 1 ((0, T ], L 2 (Ω)) for all positive times T . We use a spatial discretisation based on the finite element method and solve the resulting ordinary differential equation with an implicit-explicit (IMEX) time-discretisation scheme. In this way, the linear terms in the equation are treated implicitly in time, while the non-linear reaction term is dealt with explicitly in time.

The first step in deriving the method is to write the problem in variational form. To this end, we multiply equation ( 1) by a test function w in L 2 (Ω) and integrate over the domain Ω to obtain [START_REF] Méléard | Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity[END_REF]

Next, given a mesh T h of the domain Ω, that is a collection of geometrically simple elements (we used triangles in the simulations) partitioning * the set Ω, the positive real parameter h being related to the size of the elements in T h , an approximation space V h , which is a finite dimensional subspace of L 2 (Ω) composed of functions which are constant on each element K of the partition T h , is constructed. Setting N = dim V h and considering a basis {w i } i=1,...,N of V h , the restriction of the above variational problem to the set V h gives rise to a system of N ordinary differential equations, of the form

where the components of the vector u h are the coefficients at time t of the semidiscrete approximate solution u h (t) in the basis of V h , i.e. u h (t, x) = n i=1 u i (t) w i (x), the matrices M , M a and M m are three, in some cases weighted † , so called mass matrices, respectively defined by

and (M m ) ij = ˆΩ ˆΩ m(x, y) dy w i (x)w j (x) dx, * In practice, the mesh may produce only an approximation of the domain, which can possibly entice several technical difficulties when analysing the convergence of the method. † In practice, quadrature rules are used to compute the integrals appearing in the definitions of the mass matrices. As a consequence, these integrals may be only approximately known, depending on both the form of the kernel functions k and m and the quadrature rule which is used.