Céline Lacaux
email: celine.lacaux@iecn.u-nancy.fr

Renaud Marty
email: renaud.marty@iecn.u-nancy.fr

From invariance principles to a class of multifractional fields related to fractional sheets

Keywords: Multifractional anisotropic random fields, invariance principle, local self-similarity, sample path properties MSC classification (2010): 60F17, 60G22, 60G17, 60G52, 60G60

In this paper, we study some invariance principles where the limits are Gaussian random fields sharing many properties with multifractional Brownian sheets. In particular, they satisfy the same type of self-similarity and Hölder regularity properties. We also extend the invariance principles mentioned above in a stable setting.

Introduction

Fractional Brownian motion has been extensively studied because it is a relevant model for many problems where fractal properties occur, which is due in particular to its selfsimilarity, the stationarity of its increments and its regularity properties. It is also universal in the class of fractional processes because it satisfies the following invariance principle. Consider X = {X n , n ∈ N} a centered, stationary and Gaussian sequence and the process

S N H =    1 N H [N t] n=1 X n , t ∈ [0, +∞)   
where H ∈ (0, 1) and [y] is the integer part of the real number y. Then, S N H converges in distribution to a fractional Brownian motion of Hurst index H (see Theorem 7.2.11 of [START_REF] Samorodnitsky | Stable non-Gaussian random processes, Stochastic models with infinite variance[END_REF]) as soon as one of the three following properties is fulfilled:

• If H ∈ (1/2, 1), there exists σ H > 0 such that E[X 0 X n ] ∼ σ H n 2H-2 as n → ∞ • If H ∈ (0, 1/2), there exists σ H < 0 such that E[X 0 X n ] ∼ σ H n 2H-2 as n → ∞ and ∞ n=-∞ E[X 0 X n ] = 0. • If H = 1/2, ∞ n=1 |E[X 0 X n ]| < ∞ and ∞ n=-∞ E[X 0 X n ] > 0.
The most famous generalization of the fractional Brownian motion is the fractional Brownian field B H = (B H (u)) u∈R d (see [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF][START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum[END_REF]). This isotropic Gaussian random field B H , whose increments are stationary, is self-similar of order H, which means that

∀ε ∈ [0, +∞), B H (εu), u ∈ R d dist. = ε H B H (u), u ∈ R d (1)
where dist.

= stands for the equality of the finite-dimensional distributions. However, a drawback of the model B H is the strong homogeneity of its properties. For instance, the selfsimilarity property (1) is a global property, which is too restrictive for some applications, and the pointwise Hölder exponent of B H at any point t equals to its Hurst parameter H. Thus, many generalizations of fractional Brownian fields have been introduced to model phenomena whose Hölder regularity may vary along the trajectories. The most famous of them is the class of multifractional Brownian fields (see [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF][START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF]). Each of them is defined from a Hurst function h instead of a Hurst index, and has locally but not globally (in general) the same properties as a fractional Brownian motion.

Another drawback of fractional Brownian fields is their isotropy property, which is not suitable for applications in medecin [START_REF] Brunet-Imbault | A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform[END_REF] or hydrology [START_REF] Benson | Aquifer operatorscaling and the effect on solute mixing and dispersion[END_REF]. Thus, some fractional anisotropic generalizations of the fractional Brownian motion have also been introduced. In particular, a fractional Brownian sheet (see [START_REF] Kamont | On the fractional anisotropic Wiener field[END_REF][START_REF] Ayache | Drap brownien fractionnaire[END_REF]), that we still denote by B H , of order H = (H 1 , . . . , H d ) ∈ (0, 1) d exhibits different scaling properties in the d orthogonal directions, which is summarized in the following self-similarity property:

∀ε ∈ [0, +∞) d , B H (ε 1 u 1 , . . . , ε d u d ), u ∈ R d dist. = d k=1 ε H k k B H (u), u ∈ R d . (2)
This property is more general than (1) but is still a global property. Moreover, the rectangular increments of B H are stationnary and its pointwise Hölder regularity may vary with the direction but not along a trajectory (except on the axes). Then, to allow more flexibility, a local version of the property (2) has been introduced in [START_REF] Léger | Analyse stochastique de signaux multi-fractaux et estimations de paramètres[END_REF], following the way [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF] has introduced a local version of [START_REF] Ayache | Drap brownien fractionnaire[END_REF]. As multifractional Brownian fields have been defined, replacing H by a function t → h(t) = (h 1 (t 1 ), . . . , h d (t d )), multifractional Brownian sheets have been defined and studied in [START_REF] Léger | Analyse stochastique de signaux multi-fractaux et estimations de paramètres[END_REF][START_REF] Herbin | From N parameter fractional Brownian motions to N parameter multifractional Brownian motions[END_REF]. Their pointwise Hölder exponent can vary with the point t and with the direction. Observe that multifractional Brownian fields are not in general isotropic but their Hölder regularity do not depend on the directions. As mentioned previously for fractional processes, a question of interest is the existence of universal multifractional processes or, in other words, invariance principles for multifractional processes. This question in the case of long-range dependence is addressed in [START_REF] Cohen | Invariance principle, multifractional Gaussian processes and long-range dependence[END_REF] whose we describe the main result. We consider a centered Gaussian field X = {X n (H), H ∈ (1/2, 1), n ∈ N}, a continuous function h : R → [a, b] ⊂ (1/2, 1) and for every N ∈ N,

S N h =    [N t] n=1 X n (h(n/N )) N h(n/N ) , t ∈ [0, +∞)    . (3)
As N goes to ∞, under an assumption of long-range dependence, [START_REF] Cohen | Invariance principle, multifractional Gaussian processes and long-range dependence[END_REF] proves that the finitedimensional distributions of S N h converge to those of a multifractional Gaussian process S h . More precisely, S h is locally asymptotically self-similar in the sense of [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF] and its pointwise Hölder exponent may vary along the trajectories. This theorem is a generalization of the classical invariance principle ( [START_REF] Samorodnitsky | Stable non-Gaussian random processes, Stochastic models with infinite variance[END_REF]) in a framework of long-range dependence.

In this paper we generalize to different directions the invariance principles we mention above. Our main results deal a multidimensional version of the sequence (3). More precisely, we consider a sequence of the form

S N h =    [N t 1 ] n 1 =1
. . .

[N t d ] n d =1 X n h N n N r N n , t ∈ [0, +∞) d    where X = X n (H), n ∈ N d , H ∈ (0, 1) d is a
Gaussian or an α-stable (0 < α < 2) random field defined by a harmonizable integral representation. For suitable families h N n n,N and r N n n,N we prove the convergence in distribution of S N h to a multifractional field S h as N goes to ∞. Moreover, in the Gaussian case, we study the local self-similarity and the pointwise Hölder regularity properties of the limit S h . These properties are the same as those satisfied by multifractional Brownian sheets. We also introduce pointwise multi-Hölder exponents which are related to the regularity of the rectangular increments.

The paper is organized as follows. Section 2 is devoted to establish the invariance principle in the Gaussian case. We get limit fields whose we study local self-similarity and regularity properties in Section 3. We extend the invariance principle in a stable setting in Section 4. Some technical lemmas are postponed to the Appendix.

Invariance principle

Let {W H (x), x ∈ R d , H ∈ (0, 1) d } be a real centered Gaussian field such that for every

H ∈ (0, 1) d , K ∈ (0, 1) d , ζ ∈ R d and η ∈ R d , E[W H (ζ)W K (η)] = C(H, K) d j=1 inf(ζ j , η j ), (4)
with C : (0, 1) d × (0, 1) d → R a symmetric and smooth function. Let us emphasize that

{W H (x), x ∈ R d , H ∈ (0, 1) d } exists if and only if C is a covariance function. We also remark that, if we fix H ∈ (0, 1) d , then {W H (x), x ∈ R d } is a Brownian sheet.
Consider the set

F = f : R d → C, f ∈ L 2 (R d ), ∀ξ ∈ R d , f (ξ) = f (-ξ)
and for each H ∈ (0, 1) d , let W H (dx) be the Fourier transform of the real Gaussian random measure associated with W H (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes, Stochastic models with infinite variance[END_REF] for example). Then, for each H, for any f, g ∈ F,

E R d f (x) W H (dx) R d g(x) W K (dx) = C(H, K) R d f (x)g(x)dx. ( 5
)
Let a : (0, 1) d × R d → C be a measurable and bounded function satisfying

∀H ∈ (0, 1) d , a(H, •) ∈ F. (6)
Then, the random field

X = {X n (H), n ∈ N d , H ∈ (0, 1) d } such that X n (H) = R d a(H, x) exp(i n, x ) d j=1 exp(ix j ) -1 |x j | 1/2+H j W H (dx)
is well-defined and is a centered real-valued Gaussian field. Using [START_REF] Benson | Aquifer operatorscaling and the effect on solute mixing and dispersion[END_REF], the covariance between X n (H) and X m (K) is

C(H, K) R d a(H, x)a(K, x)e i n-m,x d j=1 |exp(ix j ) -1| 2 |x j | 1+H j +K j dx := r H,K (n -m). ( 7
)
Our aim is to consider a field S N h defined as a weighted sum of X which converges to a random field sharing many properties with a multifractional Brownian sheet. Before we introduce the weighted sum S N h , let us introduce several notations we use throughout the paper.

Notation

• For any t = (t 1 , . . . , t d ) ∈ [0, +∞) d and N ∈ N * , let

D N t = k = (k 1 , . . . , k d ) ∈ N d / 1 ≤ k j ≤ [N t j ], ∀1 ≤ j ≤ d .
• For any integer 1 ≤ j ≤ d, let us consider a function h j : [0, ∞) → (0, 1) and set

∀t = (t 1 , . . . , t d ) ∈ [0, +∞) d , h(t) = (h 1 (t 1 ), . . . , h d (t d )). (8)
• In the following, for k

= (k 1 , . . . , k d ), h N k := (h N k,1 , . . . , h N k,d ) where for 1 ≤ j ≤ d, h N k,j = h j (k j /N ).
• In order to simplify some notation, for α = (α 1 , . . . , α d ) ∈ [0, +∞) d , we write |α| = d j=1 α j .

In this section, we study the behavior as N tends to +∞ of the random field

S N h = S N h (t), t ∈ [0, +∞) d defined by S N h (t) := k∈D N t X k (h N k ) N |h N k | = [N t 1 ] k 1 =1 [N t 2 ] k 2 =1 • • • [N t d ] k d =1 X k (h N k ) N d j=1 h N k,j . (9)
We assume that the function a satisfies the following technical condition.

Condition (A).

There exist two measurable functions a 0 : (0, 1) d → C and a 1 : (0, 1) d × R d → C satisfying the following properties.

• For every (H, x) ∈ (0, 1)

d × R d , a(H, x) = a 0 (H) + a 1 (H, x) .
• The functions a 0 and a 1 are of class C d with respect to H.

• For every κ = (κ 1 , • • • , κ d ) ∈ {0,
(H,x)∈K×R d {|∂ κ H a 1 (H, x)|} < ∞ with ∂ κ H = ∂ κ 1 H 1 . . . ∂ κ d H d .
These properties imply that for every κ ∈ {0, 1} d and every compact set K of (0, 1) d , sup

(H,x)∈K×R d {|∂ κ H a(H, x)|} < ∞.
They also imply that a 0 is a real-valued function and that a 1 (H, •) ∈ F for any H ∈ (0, 1) d . Now we state the main result of this paper.

Theorem 1. Assume that for any 1 ≤ j ≤ d, h j is a C 1 -function and that Condition (A) is fulfilled. Then, the finite-dimensional distributions of S N h converge to those of the realvalued centered Gaussian field S h whose covariance is given, for all s and t ∈ [0, +∞) d , by [START_REF] Feyel | On fractional Brownian processes[END_REF] with D u = d j=1 [0, u j ] and

E[S h (t)S h (s)] = R d dy Dt×Ds C 0 (h(θ), h(σ))e i y,θ-σ d j=1 |y j | 1-h j (θ j )-h j (σ j ) dθdσ,
C 0 (H, K) = C(H, K)a 0 (H)a 0 (K). ( 11
)
In the special case where C ≡ 1,

S h f.d.d. =    R d W (dy) Dt a 0 (h(θ))e i y,θ d j=1 |y j | 1/2-h j (θ j ) dθ, t ∈ [0, +∞) d    (12)
with W the Fourier transform of the real Gaussian measure associated with a Brownian sheet.

Observe that S h (t) = 0 if there exists j such that t j = 0. This is also true for multifractional sheets.

Before we prove this theorem, let us relate it with some previous works [START_REF] Cohen | Invariance principle, multifractional Gaussian processes and long-range dependence[END_REF][START_REF] Samorodnitsky | Stable non-Gaussian random processes, Stochastic models with infinite variance[END_REF][START_REF] Sly | Integrated fractional white noise as an alternative to multifractional Brownian motion[END_REF].

Remark 1. Assume h ≡ H ∈ (0, 1) d . Then according to Theorem 1, the random field

t → N -d j=1 H j [N t 1 ] k 1 =1 [N t 2 ] k 2 =1 • • • [N t d ] k d =1 X k (H)
converges to a fractional Brownian sheet B H of order H. If d = 1, this is the classical principle invariance described in [START_REF] Samorodnitsky | Stable non-Gaussian random processes, Stochastic models with infinite variance[END_REF]. If h is not a constant function, the limit is not a multifractional Brownian sheet. For instance, if d = 2 and C ≡ 1,

S h (t) = a 0 (h 1 (t 1 ))a 0 (h 2 (t 2 ))B h (t) -R(t)
where B h is the multifractional Brownian sheet defined by

B h (t) = R 2 W (dx) e it 1 x 1 -1 ix 1 |x 1 | h 1 (t 1 )-1/2 e it 2 x 2 -1 ix 2 |x 2 | h 2 (t 2 )-1/2 and R(t) = R 2 W (dx)f 1 (t, x)g 2 (t, x) + R 2 W (dx)f 2 (t, x)g 1 (t, x) - R 2 W (dx)g 1 (t, x)g 2 (t, x) with f k (t, x) = a 0 (h k (t k )) e it k x k -1 ix k |x k | h k (t k )-1/2 and g j (x, t) = t j 0 h j (θ j ) a 0 (h j (θ j )) -a 0 (h j (θ j )) ln |x j | e iθ j x j -1 ix j |x j | h j (θ j )-1/2 dθ j .
Remark 2. Assume that d = 1 and that Condition (A) is fulfilled. Let r H,K be defined by [START_REF] Biermé | Multi-operator scaling random fields[END_REF]. Then one can prove that for any compact set K of (1/2, 1), sup

H,K∈K j 2-H-K r H,K (j) -C(H, K)a 0 (H)a 0 (K) R exp(ix)|x| 1-H-K dx → j→∞ 0.
Hence, the Gaussian process {X H (t), t ∈ R, H ∈ (1/2, 1)} satisfies the assumptions of [START_REF] Cohen | Invariance principle, multifractional Gaussian processes and long-range dependence[END_REF], in which h takes its values in a compact subset of (1/2, 1). Theorem 1 is then a generalization of Theorem 1 of [START_REF] Cohen | Invariance principle, multifractional Gaussian processes and long-range dependence[END_REF], with some restrictions on the form of X and the regularity of h, to the case where h takes its values in (0, 1).

Remark 3. Let d = 1. Assume that a ≡ 1 and W H = W does not depend on H. Then under the assumptions of Theorem 1, the finite-dimensional distributions of S N h (t), t ∈ [0, +∞) converge to those of the process {Y h (t), t ∈ [0, +∞)} defined by

Y h (t) = R W (dy) t 0 exp(iyθ) |y| h(θ)-1/2 dθ
with W the Fourier transform of a real Gaussian measure. This process, called integrated fractional white noise, has first been introduced in [START_REF] Sly | Integrated fractional white noise as an alternative to multifractional Brownian motion[END_REF] as an alternative to multifractional Brownian motion because its magnitude is much less sensitive to the variations of the Hölder exponent. Here we give a new motivation to the introduction of the process Y h : it is the limit of an invariance principle, so it can be chosen as a universal multifractional Gaussian model. Note that this has already been noticed in [START_REF] Cohen | Invariance principle, multifractional Gaussian processes and long-range dependence[END_REF] in the case of a function h with values in a compact set of (1/2, 1).

Let us now conclude this section by proving Theorem 1.

Proof. Let t, s ∈ [0, T ] d . Then, E[S N h (t)S N h (s)] = k∈D N t l∈D N s 1 N |h N k | N |h N l | E X l (h N l )X k (h N k ) = R d dx k∈D N t l∈D N s b l N , k N , x e i l-k,x N |h N k | N |h N l | d j=1 |exp(ix j ) -1| 2 |x j | 1+h N l,j +h N k,j
where b(θ, σ, x) = C(h(θ), h(σ))a(h(θ), x)a(h(σ), x). Then, a change of variables leads to

E[S N h (t)S N h (s)] = R d Φ N (s, t, x)dx where Φ N (s, t, x) := k∈D N t l∈D N s b l N , k N , x N d j=1 e ix j /N -1 2 e i(l j -k j )x j /N |x j | 1+h N l,j +h N k,j
.

Let us define the function q by

q(ξ) = 1 -e iξ ξ , ξ ∈ R\{0}.
We can write

Φ N (s, t, x) := q x 1 N N [N s 1 ] l 1 =1 e il 1 x 1 /N |x 1 | h 1 (l 1 /N )-1/2 q x 1 N N [N t 1 ] k 1 =1 e -ik 1 x 1 /N |x 1 | h 1 (k 1 /N )-1/2 • • • q x d N N [N s d ] l d =1 e il d x d /N |x d | h d (l d /N )-1/2 q x d N N [N t d ] k d =1 e -ik d x d /N |x d | h d (k d /N )-1/2 b l N , k N , x N .
Because of the regularity of h, a and C, we can apply 2d times Lemma 3 (stated and proved in the Appendix) with α = 2. This leads to

Φ N (s, t, x) ≤ max κ∈{0,1} 2d sup (θ,σ)∈[0,T ] 2d ∂ κ b θ, σ, x N d j=1 g(x j ) 2
where

∂ κ b = ∂ κ 1 θ 1 • • • ∂ κ d θ d ∂ κ d+1 σ 1 • • • ∂ κ 2d σ d b and g(ξ) = c T,h (1 + |ln |ξ||) 1 |ξ| H + -1/2 I |ξ|<1 + 1 |ξ| H -+1/2 I |ξ|≥1 (13)
with

H + = max 1≤j≤d max [0,T ] h j , H -= min 1≤j≤d min [0,T ]
h j and c T,h a finite positive constant. which only depends on T and the infinite norm of each

h j on [0, T ]. Since h is C 1 , C is smooth and since Condition (A) is fulfilled, sup y∈R d sup (θ,σ)∈[0,T ] 2d |∂ κ b (θ, σ, y)| < +∞.
The sequence of functions {x → Φ N (s, t, x)} is then uniformly bounded by an L 1 (R d )-function.

In order to use the bounded convergence theorem, it remains to prove the convergence of Φ N (s, t, x) for almost every x. We write

Φ N (s, t, x) = 3 n=0 Φ N n (s, t, x)
where, for n ∈ {0, 1, 2, 3},

Φ N n (s, t, x) = 1 N 2d k∈D N t l∈D N s b n l N , k N , x N d j=1 |q x j N | 2 e i(l j -k j )x j /N |x j | h N l,j +h N k,j -1 with b 0 (θ, σ, y) = C(h(θ), h(σ))a 0 (h(θ))a 0 (h(σ)) := C 0 (h(θ), h(σ)), b 1 (θ, σ, y) = C(h(θ), h(σ))a 1 (h(θ), y)a 0 (h(σ)), b 2 (θ, σ, y) = C(h(θ), h(σ))a 0 (h(θ))a 1 (h(σ), y),
and b 3 (θ, σ, y) = C(h(θ), h(σ))a 1 (h(θ), y)a 1 (h(σ), y).
For each n ∈ {1, 2, 3}, using 2d times Lemma 3 as previously, we obtain that

Φ N n (s, t, x) ≤ max κ∈{0,1} 2d sup (θ,σ)∈[0,T ] 2d ∂ κ b n θ, σ, x N g(x j ) 2
with g defined by [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum[END_REF]. Thanks to Condition (A), for almost every x,

lim N →∞ 3 n=1 Φ N n (s, t, x) = 0.
Moreover, by the Riemann sum theorem, for almost every x,

lim N →∞ Φ N 0 (s, t, x) = Dt dθ Ds dσ C 0 (h(θ), h(σ)) d j=1 e i(θ j -σ j )x j |x j | 1-h j (θ j )-h j (σ j ) .
Hence, by the bounded convergence theorem we get lim

N →+∞ E[S N h (t)S N h (s)] = R d dy Dt×Ds C 0 (h(θ), h(σ))e i y,θ-σ d j=1 |y j | 1-h j (θ j )-h j (σ j ) dθdσ.
Since S N h is a real-valued centered Gaussian random field, this proves the convergence of the finite-dimensional distributions of S N h to those of a real-valued centered Gaussian random field S h whose covariance is given by [START_REF] Feyel | On fractional Brownian processes[END_REF]. Moreover, if C ≡ 1, the integral representation [START_REF] Kamont | On the fractional anisotropic Wiener field[END_REF] of S h is a direct consequence of the isometry property of W on the space of square integrable functions.

Sample path properties of the limit field

This section is devoted to the smoothness of the sample paths and the self-similarity properties of the limit field S h . To this aim, let us first introduce and study some directional increments of S h .

Covariance structure of directional increments

As usual, (e 1 , . . . , e d ) denotes the canonical basis of R d . Let us consider a random field

Y = Y (t), t ∈ R d and δ ∈ R. Then for every k = 1, • • • , d and δ ∈ R, the directional increment field ∆ (k) δ Y (t), t ∈ R d is defined by ∆ (k) δ Y (t) = Y (t + δe k ) -Y (t).
Then, the rectangular increment of Y at point t in direction (k 1 , . . . , k p ) of size δ = (δ 1 , . . . , δ p ) is defined by

∆ (k 1 ,...,kp) Y (t, δ) = ∆ (kp) δp ∆ (k p-1 ) δ p-1 • • • ∆ (k 1 ) δ 1 Y (t).
Observe that the increment ∆ (1,...,d) Y (t, s-t) is the classical rectangular increment between t and s of the random field Y . For the sake of simplicity, we set

∆Y (t, s) := ∆ (1,...,d) Y (t, s -t) = (-1) d 1 κ 1 =0 . . . 1 κ d =0 (-1) d l=1 κ l Y (t + κ (s -t))
where for a and b

∈ R k , a b = (a 1 b 1 , a 2 b 2 • • • , a k b k ).
All the properties of S h we study are consequences of the Proposition 1 stated below, which focuses on the behavior of the covariance

E ∆ (k 1 ,...,kp) S h (t, ν u)∆ (k 1 ,...,kp) S h (t, ν u) (14)
as ν tends to 0 R p + for any (k 1 , . . . , k p ). In particular, since S h is a real-valued Gaussian random field, these behaviors are related to some local asymptotic self-similarity properties and to some Hölder smoothness properties, see Sections 3.2 to 3.5.

Proposition 1. Let p ∈ {1, • • • , d} and k = (k 1 , . . . , k p ) ∈ {1, • • • , d} p with k l = k j if l = j. Then, for every T ∈ R + , lim ν→0 R p + sup u,v∈[0,T ] p t∈[0,T ] d E[∆ (k 1 ,••• ,kp) S h (t, ν u)∆ (k 1 ,••• ,kp) S h (t, ν v)] ν 2h k 1 (t k 1 ) 1 • • • ν 2h kp (t kp ) p -E k h,1 (t, u, v)E k h,2 (t) = 0
where

E k h,1 (t, u, v) = R p dy p j=1
(e iy j u j -1)(e -iy j v j -1)

|y j | 2h k j (t k j )+1 (15)
and

E k h,2 (t) = R d-p dy D k t ×D k t dξdη j∈J(k)
e iy j (ξ j -η j )

|y j | h j (ξ j )+h j (η j )-1 C 0 h ξ k,t , h η k,t (16)
with

J(k) = {1, . . . , d}\{k 1 , . . . , k p }, D k t = j∈J(k) [0, t j ] and ζ k,t j = ζ j if j ∈ J(k) t j else.
Here, by convention, E

(1,...,d) h,2 (t) = C 0 (h(t), h(t)) = a 0 (h(t)) 2 C(h(t), h(t)). Remark 4. E k h,1 (t, •, •)
is the covariance function of the fractional Brownian sheet B (hk 1 (tk 1 ),...,hk p (tk p )) defined by

B (hk 1 (tk 1 ),...,hk p (tk p )) (u) = R p p j=1 e iy j u j -1 iy j |y j | h k j (t k j )-1/2 W (dy), u ∈ R p + . (17)
Moreover, if we denote

J(k) = {k p+1 , . . . k d } with k l < k l+1 , then E k h,2 (t) = Var S(t k p+1 , . . . , t k d ) ∈ R +
where S satisfies the same kind of invariance principle as S h (that is, which can be introduced by Theorem 1 replacing d by d -p and with suitable functions C, a and h).

Let us now prove Proposition 1.

Proof. Let u, v ∈ [0, T ] p , t ∈ [0, T ] d and ν ∈ (0, 1] p .
Because of the symmetry properties of (10), we can assume without loss of generality that k = (1, . . . , p). Then, we study

I ν (t, u, v) = E[∆ (1,••• ,p) S h (t, ν u)∆ (1,••• ,p) S h (t, ν v)] ν 2h 1 (t 1 ) 1 • • • ν 2hp(tp) p
.

By definition of ∆ (1,••• ,p) S h and by [START_REF] Feyel | On fractional Brownian processes[END_REF], we have:

I ν (t, u, v) = p j=1 ν -2h j (t j ) j R d dx Dt,ν u×Dt,ν v dθdσ d j=1 e i(θ j -σ j )x j |x j | h j (θ j )+h j (σ j )-1 C 0 (h(θ), h(σ))
where for all δ ∈ R p + , D t,δ = p j=1 [t j , t j + δ j ] × d j=p+1 [0, t j ]. Some substitutions lead to

I ν (t, u, v) = R d dy Dt,u×Dt,v dξdη Φ 1,t,ν (y, ξ, η)Φ 2 (y, ξ, η)C 0 (h(ξ(t p , ν)), h(η(t p , ν)))
where for all w ∈ R d + , w p = (w 1 , . . . , w p ), Φ 1,t,ν (y, ξ, η) = p j=1 e i(ξ j -η j )y j ν h j (ν j ξ j +t j )+h j (ν j η j +t j )-2h j (t j ) j |y j | h j (ν j ξ j +t j )+h j (ν j η j +t j )-1

,

Φ 2 (y, ξ, η) = d j=p+1 e i(ξ j -η j )y j |y j | h j (ξ j )+h j (η j )-1
and

∀ϑ ∈ R p + , (ζ(ϑ, ν)) j = ν j ζ j + ϑ j if 1 ≤ j ≤ p ζ j if p + 1 ≤ j ≤ d.
Note that Φ 1,t,ν (y, ξ, η) (respectively Φ 2 (y, ξ, η)) does not depend on (y j , ξ j , η j , t j ) p+1≤j≤d (respectively (y j , ξ j , η j , t j ) 1≤j≤p ).

Let us fix y ∈ R d with y j = 0 for every j ∈ {1, . . . , d}. First let us consider C y ∈ C(R p × R p , C) the function defined by

C y (ϑ, ς) = D (1,...,p) t ×D (1,...,p) t dξ p+1 . . . dξ d dη p+1 . . . dη d Φ 2 (y, ξ, η)C 0 (h(ξ(ϑ, 0)), h(η(ς, 0))) with D (1,...,p) t = d j=p+1 [0, t j ]. Let j ∈ {1, • • • , p}. We define the operator I ρ j,1 from C([0, 2T ] j ×[0, 2T ] j-1 , C) to C([0, 2T ] j-1 × [0, 2T ] j-1 , C) by I ρ j,1 ϕ(ϑ, ς) =            u j 0 dθ ρ h j (ρθ+t j )-h j (t j ) e iy j θ
|y j | h j (ρθ+t j )-1/2 ϕ(ϑ, t j + ρθ, ς)

if ρ ∈ (0, 1] u j 0 dθ e iy j θ
|y j | h j (t j )-1/2 ϕ(ϑ, t j , ς) = e iy j u j -1

iy j |y j | h j (t j )-1/2 ϕ(ϑ, t j , ς) if ρ = 0 for any ϑ ∈ [0, 2T ] j-1 , ς ∈ [0, 2T ] j-1 and ϕ ∈ C([0, 2T ] j × [0, 2T ] j-1 , C).
For the sake of simplicity, we do not emphasize the dependence of the operator I ρ j,1 on (y, u).

We also define the operator

I ρ j,2 from C([0, 2T ] j × [0, 2T ] j , C) to C([0, 2T ] j × [0, 2T ] j-1 , C) by I ρ j,2 ϕ(ϑ, ς) =            v j 0 dσ ρ h j (ρσ+t j )-h j (t j ) e -iy j σ |y j | h j (ρσ+t j )-1/2 ϕ(ϑ, ς, t j + ρσ) if ρ ∈ (0, 1] v j 0 dσ e -iy j σ |y j | h j (t j )-1/2 ϕ(ϑ, ς, t j ) = 1 -e -iy j v j iy j |y j | h j (t j )-1/2 ϕ(ϑ, ς, t j ) if ρ = 0 for any ϑ ∈ [0, 2T ] j , ς ∈ [0, 2T ] j-1 and ϕ ∈ C([0, 2T ] j × [0, 2T ] j , C).
As for I ρ j,1 we do not emphasize the dependence of the operator I ρ j,2 on (y, v).

We also consider the operator I ρ j = I ρ j,1 I ρ j,2 . Then the Fubini Theorem leads to

I ν (t, u, v) = R d dy I ν 1 1 • • • I νp p C y .
By the Fubini Theorem, we also have

R d dy I 0 1 • • • I 0 p C y = E k h,1 (t, u, v)E k h,2 (t)
with E k h,1 (t, u, v) defined by ( 15) and E k h,2 (t) by [START_REF] Léger | Analyse stochastique de signaux multi-fractaux et estimations de paramètres[END_REF]. Therefore

I ν (t, u, v) -E k h,1 (t, u, v)E k h,2 (t) = R d dy I ν 1 1 • • • I νp p -I 0 1 • • • I 0 p C y ≤ p j=1 R d dy I ν 1 1 • • • I y,u,ν j-1 j-1 I ν j j,1 -I 0 j,1 I 0 j,2 I 0 j+1 • • • I 0 p C x + R d dy I ν 1 1 • • • I ν j-1 j-1 I ν j j,1 I ν j j,2 -I 0 j,2 I 0 j+1 • • • I 0 p C x
Thanks to Lemma 5 we have the following properties for I ρ j,1 and I ρ j,2 . Let y ∈ R d + with y j = 0 and ρ ∈ (0, 1].

(i) Let ϕ ∈ C([0, 2T ] j × [0, 2T ] j-1 , C). Then, for every ϑ, ς ∈ [0, 2T ] j-1 . (I ρ j,1 -I 0 j,1 )ϕ(ϑ, ς) ≤ c 1 ρ| ln(ρ)|g(y j ) max max w∈[0,2T ] |ϕ(ϑ, w, ς)|, max w∈[0,2T ] |∂ w ϕ(ϑ, w, ς)|
where c 1 = c 1 (T, h) only depends on T and h and g(r) = 1

|r| 1/2+H - 1 |r|≥1 + 1 |r| H + -1/2 1 |r|<1 (1 + |ln |r||) (18)
with

H + = max 1≤k≤d max [0,2T ] h k and H -= min 1≤k≤d min [0,2T ] h k . Moreover, I ρ j,1 ϕ(ϑ, ς ) ≤ c 1 g(y j ) max max w∈[0,2T ] |ϕ(ϑ, w, ς)|, max w∈[0,2T ] |∂ w ϕ(ϑ, w, ς)|
Let us remark that this last inequality also holds for ρ = 0 choosing c 1 ≥ 1.

(ii

) Let ϕ ∈ C([0, 2T ] j × [0, 2T ] j , C). Then for every ϑ ∈ [0, 2T ] j and ς ∈ [0, 2T ] j-1 , (I ρ j,2 -I 0 j,2 )ϕ(ϑ, ς) ≤ c 1 ρ| ln(ρ)|g(y j ) max max w∈[0,2T ] |ϕ(ϑ, ς, w)|, max w∈[0,2T ]
|∂ w ϕ(ϑ, ς, w)| and

I ρ j,2 ϕ(ϑ, ς) ≤ c 1 g(y j ) max max w∈[0,2T ] |ϕ(ϑ, ς, w)|, max w∈[0,2T ] |∂ w ϕ(ϑ, ς, w)| .
Let us remark that this last inequality also holds for ρ = 0.

Since

∂ w I ρ j,k ϕ = I ρ j,k ∂ w ϕ (k ∈ {1, 2})
, by applying (i) and (ii), we obtain:

I ν (t, u, v) -E k h,1 (t, u, v)E k h,2 (t) ≤ 2c p 1 p j=1 ν j | ln ν j | R d dy p j=1 g(y j ) 2 max ε∈{0,1} 2p sup ϑ,ς∈[0,2T ] p ∂ ε 1 ϑ 1 • • • ∂ εp ϑp ∂ ε p+1 ς 1 • • • ∂ ε 2p ςp C x
Moreover, applying Lemma 4, we obtain:

max ε∈{0,1} 2p sup ϑ,ς∈[0,2T ] p ∂ ε 1 ϑ 1 • • • ∂ εp ϑp ∂ ε p+1 ς 1 • • • ∂ ε 2p ςp C x ≤ c 2 d j=p+1 g(x k ) 2 max ε∈{0,1} 2d sup ϑ,ς∈[0,2T ] d ∂ ε 1 ϑ 1 • • • ∂ ε d ϑ d ∂ ε d+1 ς 1 • • • ∂ ε 2d ς d C0
where C0 (ϑ, ς) = C 0 (h(ϑ), h(ς)) and where the finite positive constant c 2 only depends on (T, h). Therefore,

I ν (t, u, v) -E k h,1 (t, u, v)E k h,2 (t) ≤ c 3 p j=1 ν j | ln ν j | with c 3 = 2c p 1 c 2 g 2d L 2 (R) max ε∈{0,1} 2d sup ϑ,ς∈[0,2T ] d ∂ ε 1 ϑ 1 • • • ∂ ε d ϑ d ∂ ε d+1 ς 1 • • • ∂ ε 2d ς d C0 .
One can easily check that c 3 < ∞, which concludes the proof.

Local asymptotic self-similarity properties

In general, a multifractional Brownian sheet satisfies neither a global self-similarity property (1) or (2) nor a local asymptotically self-similarity property in the sense of [START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF]. However, restricted on a line directed by e k , it is locally self-similar. Moreover, it also satisfies a local version, introduced in [START_REF] Léger | Analyse stochastique de signaux multi-fractaux et estimations de paramètres[END_REF], of the self-similarity property (2). This local property, which involves its rectangular increments, takes account of its local anisotropic behavior. The next proposition proves, in particular, that all these self-similarity properties also hold for the limit field S h . Since S h is a centered real-valued Gaussian random field, it is a direct consequence of Proposition 1.

Proposition 2. Let p ∈ {1, . . . , d} and k = (k 1 , . . . , k p ) ∈ {1, . . . , d} p with k l = k j if l = j. Then, for any t ∈ [0, +∞) d ,

lim ε→0 + R p      p j=1 ε -h k j (t k j ) j   ∆ (k 1 ,...,kp) S h (t, ε u), u ∈ [0, +∞) p    dist. = {T h,k,t (u), u ∈ [0, +∞) p } where T h,k,t (u) = E k h,2 (t) 1/2 B (hk 1 (t k 1 ),...,h kp (t kp )) (u)
with E k h,2 (t) defined by ( 16) and B (hk 1 (t k 1 ),...,h kp (t kp )) the fractional Brownian sheet defined by [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF].

Observe that the limit field T h,k,t is non-degenerate if and only if E k h,2 (t) = 0. In the following remark, we assume that T h,k,t is a non-degenerate random field. Remark 5. Choosing p = 1 and k ∈ {1, • • • , d}, Proposition 2 can be written as

lim ε→0 + ε -h k (t k ) (S h (t + εue k ) -S h (t)), u ∈ [0, +∞) dist.
= {T h,k,t (u), u ∈ [0, +∞)}, [START_REF] Meerschaert | Fernique-type inequalities and moduli of continuity of anisotropic Gaussian random fields[END_REF] where the tangent process T h,k,t is a fractional Brownian motion of order h k (t k ). In other words, S h is locally asymptotically self-similar with respect to k th -component at point t with tangent field a fractional Brownian motion.

Modulus of continuity and Hölder exponents

Since S h is a centered real-valued Gaussian random field, its sample path properties are related to the behavior of its variogram. Hence, we start by giving an upper bound this variogram, that is we give a Kolmogorov-type estimate for the increments of S h which takes account of the anisotropic behavior of S h . Lemma 1. Let T ∈ [0, +∞). There exists two finite positive constants c > 0 and η > 0 such that, for every t and s in [0, T ] d with t -s ≤ η,

E[(S h (t) -S h (s)) 2 ] ≤ c   d j=1 |t j -s j | h j (min (t j ,s j ))   2 . ( 20
)
Proof. Let s, t ∈ [0, T ] d . Observe that we can assume without loss of generality that t j = s j for all 1 ≤ j ≤ d. Otherwise S h (t) = S h (s) and ( 20) holds for any c ∈ (0, +∞).

Let us now consider the sequence u (j) 0≤j≤d of [0, T ] d defined by u(0) = t and u(j + 1) = u(j) + (s j+1 -t j+1 )e j+1 .

Observe that u(d) = s and for any 0 ≤ j ≤ d -1, u(j + 1) -u(j) = (s j+1 -t j+1 )e j+1 .

Then using the Minkowski inequality, we have

E (S h (t) -S h (s)) 2 ≤   d j=1 E {S h (u(j)) -S h (u(j -1))} 2 1/2   2 .
Let 1 ≤ j ≤ d. By Proposition 1 (applied with p = 1 and k = j), there exists r j such that for all ν ∈ (0, r j ),

sup v∈[0,T ] d ν -2h j (v j ) E (S h (v) -S h (v + νe j )) 2 -E j h,1 (v, e j , e j )E j h,2 (v) ≤ 1.
with E j h,1 and E j h,2 defined by ( 15) and ( 16). Since for any 1 ≤ j ≤ d, v → E j h,1 (v, e j , e j ) and v → E j h,2 (v) are continuous on the compact set [0, T ] d , there exists a finite positive constant c such that for all 1 ≤ j ≤ d, for all ν ∈ (0, min 1≤k≤d r k ) and for all v ∈ [0, T ] d ,

E (S h (v) -S h (v + νe j )) 2 ≤ cν 2h j (v j ) .
Applying this inequality with v = u(j -1) and ν = s j -t j (respectively v = u(j) and ν = t j -s j ) if s j > t j (respectively if s j < t j ), one obtains that there exists η ∈ (0, 1) such that E (S h (u(j)) -S h (u(j -1))) 2 ≤ c|t j -s j | 2h j (min(t j ,s j ))

for s, t ∈ [0, T ] d with t -s ≤ η. This leads to the conclusion.

Observe that, by Lemma 1, for any compact set K of R d , there exists a finite positive constant c = c(K) such that

∀s, t ∈ K, E (S h (t) -S h (s)) 2 ≤ cρ K (s, t) 2 (21)
with ρ K the metric defined on R d by

ρ K (s, t) := d j=1 |s j -t j | min K h j , ∀s, t ∈ R d . ( 22
)
Henceforth S h admits a continuous version on R d , which we still denote by S h .

To conclude this section, we are interested in the modulus of continuity of the continuous Gaussian random field S h . Many authors have studied moduli of continuity in a Gaussian setting, e.g. see [START_REF] Kôno | On the modulus of continuity of sample functions of Gaussian processes[END_REF][START_REF] Orey | Sample functions of the n-parameter Wiener process[END_REF][START_REF] Benassi | Gaussian processes and Pseudodifferential Elliptic operators[END_REF][START_REF] Ayache | Asymptotic properties and hausdorff dimensions of fractional Brownian sheets[END_REF][START_REF] Xiao | Sample path properties of anisotropic gaussian random fields[END_REF][START_REF] Meerschaert | Fernique-type inequalities and moduli of continuity of anisotropic Gaussian random fields[END_REF]. In particular, [START_REF] Ayache | Asymptotic properties and hausdorff dimensions of fractional Brownian sheets[END_REF] establishes a sharp modulus of continuity for fractional Brownian sheets using a wavelet expansion. Here, we cannot follow the approach of [START_REF] Ayache | Asymptotic properties and hausdorff dimensions of fractional Brownian sheets[END_REF] since it is based on an integral representation. Nevertheless, the upper bound we give is sufficient to obtain the Hölder regularity properties of S h . Proposition 3. For any non-empty compact set K of R d and any η > 0,

lim δ→0 sup s,t∈K 0< t-s ≤δ |S h (t) -S h (s)| ρ K (s, t)|log ρ K (s, t)| 1/2+η = 0 almost surely
where the metric ρ K is defined by [START_REF] Samorodnitsky | Stable non-Gaussian random processes, Stochastic models with infinite variance[END_REF].

Proof. Let K be a non-empty compact set of R d and E be the diagonal matrix given by

E = diag 1 min K h 1 , . . . , 1 min K h d .
Since the eigenvalues of E are positive, according to Chapter 6 of [START_REF] Meerschaert | Limit distributions for sums of independent random vectors[END_REF], there exists an unit sphere S E (for a suitable norm) such that we can write any ξ ∈ R d \{0} uniquely as

ξ = τ E (t) E E (t)
with τ E (t) > 0 and

E (t) ∈ S E . By convention τ E (0) = 0. Then, for any t, s ∈ R d such that t = s, ρ K (t, s) = ρ K (0, t -s) = τ E (t -s)ρ K (0, E (t -s))
by definition of E and ρ K . Therefore by continuity and positiveness of u → ρ K (0, u) on the compact set S E , there exists two finite positive constants c 1 and c 2 such that

∀s, t ∈ R d , c 1 τ E (t -s) ≤ ρ K (t, s) ≤ c 2 τ E (t -s). (23)
The upper bound in [START_REF] Sly | Integrated fractional white noise as an alternative to multifractional Brownian motion[END_REF] and Lemma 1 prove that the centered Gaussian random field S h satisfies the assumption of Proposition 5.3 of [START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF] (with β = 0). Both this proposition and (23) lead to the conclusion.

The following corollary is a direct consequence of the previous proposition.

Corollary 1. Let K = d j=1 [a j , b j ] be a non-empty compact set of R d . Let H j = min [a j ,b j ] h j for 1 ≤ j ≤ d and let H = min 1≤j≤d H j .
1. Then for any ε ∈ (0, H), S h is Hölderian on K of order H -ε.

2. Let t ∈ K and K j = {t + re j ∈ K / r ∈ R}. Then, for any ε ∈ (0, H j ), S h is Hölderian on K j of order H j -ε.

Multi-Hölder continuity property

The notion of double Hölder continuity has been introduced in [START_REF] Feyel | On fractional Brownian processes[END_REF]. This property, which is defined in the framework of functions of two variables, is related to rectangular increments and can be readily extended to functions of d variables as follows.

Definition 1. Let U ⊂ R d . A function ϕ : U → R is said to be multi-Hölder continuous on U of indexes (α 1 , • • • , α d ) ∈ (0, ∞) d if there exists a finite constant c > 0 such that ∀s, t ∈ U, |∆ϕ(t, s)| ≤ c d j=1 |s j -t j | α j
where ∆ϕ(t, s) is the rectangular increment of ϕ between s and t (see Section 3.1).

D. Feyel and A. De La

Pradelle have also introduced in [START_REF] Feyel | On fractional Brownian processes[END_REF] a framework to study double Hölder continuity property for a two-variable random field Y . More precisely, they prove a Kolmogorov-type lemma (Lemma 17 page 283 of [START_REF] Feyel | On fractional Brownian processes[END_REF]) by means of Liouville spaces. By following the same lines as their proof, one obtains the lemma below.

Lemma 2. Let Y = (Y (t), t ∈ [0, ∞) d ) be a continuous real-valued random field and let K ⊂ [0, ∞) d be a non-empty compact set. Assume that there exist c ∈ (0, ∞), q ∈ (0, ∞), η ∈ (0, 1] and a vector of indexes

(β 1 , • • • , β d ) ∈ (0, ∞) d such that, for every (s, t) ∈ K × K satisfying t -s ≤ η, E[(∆Y (s, t)) q ] ≤ c d j=1 |t j -s j | 1+β j . (24)
Then, for every

(α 1 , • • • , α d ) ∈ d j=1 [0, β j /q), almost surely, Y is multi-Hölder on K of indexes (α 1 , • • • , α d ).
We can now establish a multi-Hölder continuity property for S h .

Proposition 4. Let K = d j=1 [a j , b j ] be a non-empty compact set of [0, ∞) d . Then, almost surely, the random field S h is multi-Hölder on K of indexes (α 1 , • • • , α d ) for every (α 1 , • • • , α d ) ∈ d j=1 [0, min [a j ,b j ] h j ). Proof. Let s, t ∈ K. We can check that |∆S h (t, s)| = |∆S h (min(s, t), max(s, t))|
where min(s, t) = (min(t 1 , s 1 ), . . . , min(t d , s d )) and max(s, t) = (max(t 1 , s 1 ), . . . , max(t d , s d )). Hence, without loss of generality, we assume that t j ≥ s j for each j = 1, • • • , d. Because of Lemma 1, there exists η ∈ (0, 1] so that for every ν ∈ (0, ∞) d with ν ≤ η and every

v ∈ K E[(∆S h (v, v + ν)) 2 ] ≤ (1 + E (1,...,d) h,1 (v, 1, 1)E (1,...,d) h,2 (v)) d j=1 ν 2h j (v j ) j ≤ sup u∈K (1 + E (1,...,d) h,1 (u, 1, 1)E (1,...,d) h,2 (u)) d j=1 ν 2 min [a j ,b j ] h j j where 1 = (1, 1, • • • , 1) ∈ R d . Observe that C = sup u∈K (1 + E (1,...,d) h,1 (u, 1, 1)E (1,...,d) h,2 (u)) < ∞.
Then, choosing v = t and ν = t -s and since S h is Gaussian, Y = S h satisfies (24) on K with β j = min [a j ,b j ] h j -1/q. The conclusion follows from Lemma 2.

Hölder exponents

Let us first focus on the pointwise Hölder exponent

H S h ,k (t) := sup H > 0 : lim r→0 S h (t + re k ) -S h (t) |r| H
of S h at point t on the direction e k and on the global pointwise Hölder exponent

H S h (t) := sup H > 0 : lim s→0 S h (t + s) -S h (t) s H
of S h at point t. We briefly explain how to derive them from the previous sections. On one hand, Corollary 1 gives a lower bound of these exponents, which is improved by applying Proposition 4 when t has at a least one null coordinate. On the other hand, the upper bound is derived from Proposition 2 in which the limit field has to be non-degenerate.

Proposition 5. Let 1 ≤ k ≤ d, t = (t 1 , • • • , t d ) ∈ R d + and N t = {1 ≤ l ≤ d; t l = 0}. 1. (a) If N t \{k} = ∅, H S h ,k (t) = +∞ almost surely. (b) If N t \{k} = ∅ and E k h,2 (t) = 0, then H S h ,k (t) = h k (t k ) almost surely. 2. (a) Assume that h k (t k ) = min 1≤j≤d h j (t j ). If E k h,2 (t) = 0, then almost surely H S h (t) = h k (t k ) = min 1≤j≤d h j (t j ). (b) Assume that N t = {k 1 , . . . , k p } with 1 ≤ p ≤ d. If E (k 1 ,...,kp) h,2 (t)
= 0, then almost surely

H S h (t) = p j=1 h k j (0) = l∈Nt h l (t l ).
Let us now comment the assumption of the previous proposition.

Remark 6. The condition

E k h,2 (t) = 0. ( 25
)
is equivalent to assuming that the tangent field T h,k,t is non-degenerate. Then, in view of [START_REF] Léger | Analyse stochastique de signaux multi-fractaux et estimations de paramètres[END_REF], k has to be chosen (as done in the previous proposition) such that t j = 0 for j / ∈ {k 1 , . . . , k p }. Moreover, (25) also implies that there exist θ, σ ∈ D t , such that

a 0 (h(θ))a 0 (h(σ))C(h(θ), h(σ)) = 0. ( 26
)
It would be interesting to exhibit necessary and sufficient conditions on t and the functions C, a and h such that condition (25) holds. However, we have chosen not to focus on this question. We can easily check that E k h,2 (t) = 0 in many cases, as for instance when (26) holds for any θ, σ ∈ D t or when for t small enough if (26) holds for θ = σ = 0.

Before we prove the previous proposition, observe that the pointwise Hölder exponents may vary with the position and that the directional pointwise Hölder exponent may also depend on the direction as in the framework of multifractional Brownian sheets (see [START_REF] Léger | Analyse stochastique de signaux multi-fractaux et estimations de paramètres[END_REF][START_REF] Herbin | From N parameter fractional Brownian motions to N parameter multifractional Brownian motions[END_REF]). Let us also mention that our proof holds for multifractional or fractional Brownian sheets. The statements of Theorem 6.11 of [START_REF] Léger | Analyse stochastique de signaux multi-fractaux et estimations de paramètres[END_REF] and Proposition 13 of [START_REF] Herbin | From N parameter fractional Brownian motions to N parameter multifractional Brownian motions[END_REF] have to be slightly rectified in the case where N t = ∅.

Proof.

1. If N t \{k} = ∅, then S h (t + re k ) = S h (t) = 0 for any r and H S h ,k (t) = +∞. Assume now that N t \{k} = ∅ and E k h,2 (t) = 0. By proposition 2

S h (t + εe k ) -S h (t) ε h k (t k )
dist.

----→

ε→0 + T h,k,t (1) = 0 almost surely since T h,k,t (1) = E k,1 h,2 (t)B h k (t k ) (1) with B h k (t k )
the fractional Brownian motion defined by [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF]. This implies that H S h ,k (t) ≤ h k (t k ) almost surely (see for example the proof of Proposition 2.3 of [START_REF] Benassi | Identification and properties of real harmonizable fractional Lévy motions[END_REF]). We conclude the proof of assertion 1. by applying Corollary 1.

By definition, H S h (t)

≤ min 1≤j≤d H S h ,j (t). Then if h k (t k ) = min 1≤j≤d h j (t j ) and E k h,2 (t) = 0, assertion 2.(a) is a direct consequence of assertion 1. and of Corollary 1. Let us now assume that N t = {k 1 , . . . , k p } with 1 ≤ p ≤ d.

• Let us prove, using the multi-Hölder continuity property, that H S h (t) ≥ p j=1 h k j (0). To this aim, observe that for any s, S h (t + s) = ∆S h (0, t + s) since S h vanishes on the axes. Then let T ∈ (0, 1). Applying Proposition 4, one can find a finite positive random variable c(t p+1 , . . . , t d ) such that

∀s ∈ [0, T ] d , |S h (t + s)| = |∆S h (0, t + s)| ≤ c p j=1 |s j | min [0,T ] h k j -ε ,
which proves that H S h (t) ≥ p j=1 min [0,T ] h k j -2pε for ε small enough. Therefore, letting ε and T tends to 0, by continuity of each h j , we obtain that

H S h (t) ≥ p j=1 h k j (0)
• Since S h vanishes on the axes and t k j = 0 for 1 ≤ j ≤ p, we have:

∀δ ∈ R p + , ∆ (k 1 ,...,kp) S h (t, δ) = S h   t + p j=0 δ j e k j   Then, since E (k 1 ,...,kp) h,2
(t) = 0, Proposition 2 implies that

S h (t + εu) ε p j=1 h k j (t k j ) = ∆ (k 1 ,...,kp) S h t, ε 1 R p ε p j=1 h k j (t k j )
dist.

----→ ε→0 + T h,(k 1 ,...,kp),t 1 R p = 0 almost surely.

where we have set u = p j=0 e k j . Therefore, H S h (t) ≤ p j=1 h k j t k j almost surely, which concludes the proof.

One can also consider the local Hölder exponent at point t, which is related to the Hölder property fulfilled around t. In our framework and under the assumptions of Proposition 5, one can prove that it is equal to min 1≤j≤d h j (t). Observe that the pointwise and the local Hölder exponent at point t are different if t = 0. This is also true for multifractional or fractional sheets.

Observe that pointwise and local exponents of S h do not directly involve all the values of h j (t j ) with t j ∈ [0, +∞) and 1 ≤ j ≤ d. However, the knowledge of all these values is required to obtain pointwise multi-Hölder exponents which are introduced below. This proves that all the range of h is involved in the regularity of the sample paths. ∆ϕ(t,s) d j=1 |t j -s j | α j does not converge when s → t. In our framework and also in the framework of multifractional sheets, this exponent is given by the function h. Proposition 6. Assume a 0 (h(t))C(h(t), h(t)) = 0. Then the pointwise multi-Hölder exponent of S h at point t is almost surely equal to h(t).

Proof. Since ∆S h (t, t + ε) = ∆ (1,...,d) S h (t, ε), by Proposition 2,

lim ε→0 + R d ∆S h (t, t + ε) d j=1 ε h j (t j ) j dist. = |a 0 (h(t))| C(h(t), h(t))B h(t) (1, . . . , 1)
where B h(t) is the fractional Brownian sheet defined by [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF]. Then, since B h(t) (1, . . . , 1) = 0 almost surely, for any α ∈ R d + \ d j=1 (0, h j (t j )], there exists a sequence (ε (n) ) n∈N which converges to 0 + R d and such that

∆S h (t, t + ε (n) ) d j=1 ε (n) j α j - → ∞
almost surely. Therefore property (ii) of the definition 2 holds with H j = h j (t j ), 1 ≤ j ≤ d.

Then, one concludes applying Proposition 4.

Extension to stable fields

Let 0 < α ≤ 2 and W α (dξ) be a complex isotropic α-stable random measure on R d with Lebesgue control measure (see [START_REF] Samorodnitsky | Stable non-Gaussian random processes, Stochastic models with infinite variance[END_REF] p.281). If α = 2, W α (dξ) is a complex isotropic Gaussian random measure. We recall that, for any function g :

R d → C in L α (dx), W α (g) = R d g(x)W α (dx)
is well-defined and is an isotropic α-stable random variable. Moreover, if for any function g : R d → C in L α (dx), we have:

∀z ∈ C, E exp i z R d g(ξ)W α (dξ) = exp -c α α |z| α R d |g(x)| α dx (27) with c α = 1 2π π 0 |cos (x)| α dx 1/α .
Let a : (0, 1) d × R d → C be a measurable and bounded function. Then, the random field

X α = {X n,α (H), n ∈ N d , H ∈ (0, 1) d } such that X n,α (H) =   R d a(H, x) exp(i n, x ) d j=1 exp(ix j ) -1 |x j | 1/α+H j W α (dx)  
is well-defined and is a real-valued symmetric α-stable random field. Observe that if a satisfies (6) and if α = 2, then the field X α has the same distribution as the field defined in Section 3 with C ≡ 1/2.

For any integer 1 ≤ j ≤ d, let us consider a function h j : [0, ∞) → (0, 1). Then, let us define h : [0, ∞) d → (0, 1) d by [START_REF] Brunet-Imbault | A new anisotropy index on trabecular bone radiographic images using the fast Fourier transform[END_REF]. We now consider

∀t ∈ [0, +∞) d , S N h,α (t) := k∈D N t X k,α (h N k ) N |h N k |
where D N t and h N k are defined in Section 2. The next theorem is the analogous of Theorem 1 for S N h,α .

Theorem 2. Assume that the function a satisfies Condition (A). Then, the finite dimensional distributions of S N h,α converge to those of S h,α defined by

∀t ∈ [0, +∞) d , S h,α (t) =   R d W α (dx) Dt a 0 (h(θ))e i x,θ d j=1 |x j | 1-1/α-h j (θ j ) dθ   , where D t = d j=1 [0, t j ].
Observe that if h ≡ H, the limit field S h,α is a real harmonizable fractional stable field of order H. and λ l ∈ R. In order to simplify the notation, we set D N,l := D N t (l) . Then, using the linearity of the integral with respect to W α and (27), we obtain that

E exp i m l=1 λ l S N h,α (t (l) ) = exp (-c α α I α (N )) (28)
with

I α (N ) := R d m l=1 λ l k∈D N,l a(h N k , x) exp (i k, x ) d j=1 exp (ix j ) -1 N h j (k j /N ) |x j | 1/α+h j (k j /N ) α dx.
The change of variables y = N x leads to

I α (N ) = R d Φ N α (y) α dy with Φ N α (y) := m l=1 λ l k∈D N,l a(h N k , y N )e i k,y/N d j=1 e iy j /N -1 |y j | 1/α+h j (k j /N ) .
As in the proof of Theorem 1, we apply The sequence of functions {x → Φ N α (s, t, x)} α } N is then uniformly bounded by an

L 1 (R d )-function.
Then, in order to obtain the convergence I α (N ), it remains to prove the convergence of each Φ N α,l (y) for almost every y. To this aim, we write:

Φ N α,l (y) = Φ N α,l,0 (y) + Φ N α,l,1 (y)
where

Φ N α,l,0 (y) = k∈D N,l a 0 h k N d j=1
e ik j y j /N e iy j /N -1

|y j | 1/α+h j (k j /N ) and Φ α,l,1 (y) = k∈D N,l a 1 h k N , y N d j=1
e ik j y j /N e iy j /N -1 |y j | 1/α+h j (k j /N ) .

As in the proof of Theorem 1, Lemma 3 and Condition (A) lead to lim and concludes the proof.

Let us mention that one can prove an analogous of Theorem 2 for the stable limit field S h,α studying the characteristic function of each quantity of the form m l=1 λ l ∆ (k 1 ,...,kp) S h,α (t, ε u l ).

However, the arguments used in the Gaussian framework cannot be applied to study the regularity of the limit field S h,α . Since this field is α-stable, such study can certainly be done using a series representation (see [START_REF] Kôno | Hölder continuity of sample paths of some self-similar stable processes[END_REF][START_REF] Biermé | Hölder regularity for operator scaling stable random fields[END_REF][START_REF] Biermé | Multi-operator scaling random fields[END_REF] for example) but this is out of the scope of this paper. e i(k-1)x/N -1

Technical lemmas

ix|x| -1+1/α φ k N 1 |x| h(k/N ) - 1 |x| h((k-1)/N ) ≤ max T 2 , 2T h ∞ φ ∞ g α (x).( 32
)
Applying the mean value theorem to the continuously differentiable function φ, we also obtain

[N t] k=1 e i(k-1)x/N -1 ix|x| h((k-1)/N )-1+1/α φ k N -φ k -1 N max T 2 , 2T φ ∞ g α (x). (33)
Equations ( 30), (31), ( 32) and (33) lead to the conclusion. Proof. This is a direct consequence of Lemma 3 in which we make N go to infinity. |y| h(νθ+t)-1/2 φ(νθ + t)dθ ≤ c T,h g(y)

with g = g 2 defined by (29) and c T,h is a finite positive constant which only depends on T and on max [0,2T ] |h |.

Proof.

  Let m, M ∈ N\{0} and T > 0. For any l = 1 . . . m, let t (l) = t

e

  d times Lemma 3 to each function Φ N α,l (1 ≤ l ≤ m) defined by Φ N α,l (y) :=k∈D N,l ik j y j /N e iy j /N -1|y j | 1/α+h j (k j /N ) with b(θ, x) = a(h(θ), x). ) = c T,h (1 + |ln |ξ||) 1 |ξ| H + -1+1/α I |ξ|<1 + 1 |ξ| H -+1/α I |ξ|≥1with H + = max 1≤j≤d max [0,T ] h j , H -= min 1≤j≤d min [0,T ] h j and c T,h a finite positive constant which only depends on T and of each infinite norm of h j on [0, T ]. Moreover, using the smoothness of h and Condition (A), we see that sup θ∈[0,T ] d

iy j e iy j θ j |y j | 1 /e iy j θ j |y j | 1 /

 11 α+h j (θ j ) dθ.Hence, by the bounded convergence theorem we get the convergence of I α (N ), which leads (combined with (28)α+h j (θ j )-1 dθ

Lemma 3 .e-N 2

 32 Let φ : [0, T ] → C and h : [0, T ] → (0, 1) be two continuously differentiable functions and α ∈ (0, 2]. Let H + = max [0,T ] h and H -= min [0,T ] h. Then, for every x ∈ R\{0}, for every N ∈ N\{0} and every t ∈ [0, T ],1 -e -ix/N ix[N t] k=1 φ k N e ikx/N |x| 1-1/α-h(k/N ) ≤ c T,h max φ ∞ , φ ∞ g α (x)whereg α (x) = (1 + |ln |x||) 1 |x| H + -1+1/α I |x|<1 + 1 |x| H -+1/α I |x|≥1(29)and c T,h is a finite positive constant which only depends on T and on max [0,T ] |h |.Proof. Let x ∈ R\{0}, N ∈ N\{0} and t ∈ [0, T ]. Let us now setg N (t, x) = 1 -e -ix/N ix ikx/N -e i(k-1)x/N ix|x| h(k/N )-1+1/α .Then,g N (t, x) = φ [N t] N e i[N t]x/N -1 ix|x| h([N t]/N )-1+1/α remark that φ [N t] N e i[N t]x/N -1 ix|x| h([N t]/N )-1+1/α ≤ max (2, T ) φ ∞ g α (x)(31) with g α defined by (29). Moreover, since h is continuously differentiable, by the mean value theorem, |x| H + -1+1/α + I |x|≥1 2[N t]| ln |x|| N |x| H -+1/α

Lemma 4 .

 4 Let φ : [0, T ] → C and h : [0, T ] → (0, 1) be two continuously differentiable functions. Let α ∈ (0, 2]. Then, for every x ∈ R\{0} and every t ∈ [0, T ],t 0 φ(θ)e iθx |x| 1-1/α-h(θ) dθ ≤ c T,h max φ ∞ , φ ∞ g α (x)with g α defined by (29) and c T,h a finite positive constant which only depends on T and on max [0,T ] |h |.

Lemma 5 . 0 e 0 e

 500 Let φ : [0, 2T ] → C and h : [0, 2T ] → (0, 1) be two continuously differentiable functions. LetH + = max [0,2T ] h and H -= min [0,2T ] h. Then, for all u ∈ [0, T ], y ∈ R\{0}, t ∈ [0, T ] and ν ∈ (0, 1] u iyθ |ν| h(νθ+t)-h(t) |y| h(νθ+t)-1/2 φ(νθ + t)dθ -φ(t) e iyu -1 iy|y| h(t)-1/2 ≤ c T,h ν|ln(ν)| max φ ∞ , φ ∞ g(y)(34)and u iyθ |ν| h(νθ+t)-h(t)

  Definition 2. Let us consider ϕ a real-valued function defined on a neighborhood U of t. The pointwise multi-Hölder exponent of ϕ at point t ∈ R d + is the only multi-indexes (H 1 , . . . , H d ) such that (i) for any α ∈ d j=1 (0, H j ) , lim s→t

	∆ϕ(t,s) j=1 |t j -s j | α j = 0 d
	(ii) and for any α ∈ R d + \ d j=1 (0, H j ],

Proof. Let u ∈ [0, T ], y ∈ R\{0}, t ∈ [0, T ] and ν ∈ (0, 1]. Let us consider

An integration by parts leads to

Then,

Let us remark that for every θ ∈ [0, u],

with g 2 defined by (29). Therefore,

we have:

ρ|ln ρ| + 1 max φ ∞ , φ ∞ g 2 (y), which concludes the proof.


David Flavigné
email: flavigne@isir.upmc.fr

Michel Taïx
email: taix@laas.fr

Interactive Locomotion Animation using Path Planning

This paper presents a method aimed at enhancing interaction between a path planning algorithm and a user to guide a robot motion task in a virtual environment. Existing works use a two-step decomposition which limits the interaction between the user and the ongoing process. We propose a modification of a classic motion planning method, the Rapidly-exploring Random Tree to build a Interactive-RRT. This method is based on exchanging forces between the algorithm and the user, and on data gathering (labels) from the virtual scene. The Interactive-RRT is combined with a locomotion controller and then applied to a virtual character for generating a locomotion movement along a path suggested by the user.

Introduction

This work concerns the realization of a new approach to solve motion problems in numeric models (digital mockup) based on interaction between a user and a motion planning algorithm. The problem of moving an object inside the digital mockup can be solved generally by two approaches : interaction device or automatic path planning.

The classical approach uses an interaction device (3D mouse or a haptic arm) in order to interact with CAD software. Haptics is a recent enhancement that provides an additional perceptual modality in virtual environments [START_REF] Burdea | Haptics issues in virtual environments[END_REF]. A force feedback device allows the user to test collisionfree paths in assembly tasks during the process of industrial design or benchmark maintenance. The contribution of haptics can improve the use of Virtual Reality [START_REF] Ye | A comparative study of assembly planning in traditional and virtual environments[END_REF][START_REF] Chabal | Emm-3d : a virtual environment for evaluating maintenability from cad models[END_REF]. The use of a 3D mouse for interaction, is less expensive than a haptic device and can give very good results in many design processes [START_REF] Chen | A 3d mouse for interacting with virtual objects[END_REF][START_REF] Su | Virtual assembly platform based on pc[END_REF]. The main applications concern the operations of dis-assembly for maintenance checks as well as during final phase assembly in car and aeronautical industries Application in animation by controlling a virtual char-acter or humanoid using a joystick is a simple an intuitive approach when the user wishes to control the walking direction [START_REF] Chestnutt | An intelligent joystick for biped control[END_REF][START_REF] Chestnutt | Interactive control of humanoid navigation[END_REF]. But in complex scenes the user may get lost and need help in finding the solution easily.

In robotics, a lot of work in motion planning has been done to compute free paths in digital models for mechanical systems. With the recent results in random planning algorithm [START_REF] Lavalle | Planning Algorithms[END_REF] it is possible to solve automatically problems for systems with many degrees of freedom. The algorithm computes a collision-free roadmap in a configuration space (CS) where the object is reduced to a point. This point represents the robot's model in the environment and the dimension of CS is equal to the number of degrees of freedom of the mechanical system. The algorithm searches a free path for a point in the roadmap. Now, companies uses automatic path planning to solve PLM applications [START_REF] Ferré | Progresses in assembly path planning[END_REF]. There are mainly two families of methods for building a roadmap, the Probabilistic Roadmap (PRM) [START_REF] Kavraki | Probabilistic roadmaps for path planning in highdimensional configuration spaces[END_REF] and the Rapidly-exploring Random Tree (RRT) [START_REF] Lavalle | Rapidly-exploring random trees: Progress and prospects[END_REF]. The roadmap in the PRM's case is a graph and in RRT's case a tree. The RRT approach is more interesting to our study because it is faster in the single query case, when the roadmap computation must be limited in time.

These methods are very effective but does not allow interaction with a user who wants to guide movement. Moreover, it suffers from the difficulty in finding narrow passages (see for example [START_REF] Boor | The Gaussian sampling strategy for probabilistic roadmap planners[END_REF][START_REF] Amato | OBPRM: An obstacle-based PRM for 3D workspaces[END_REF][START_REF] Sun | Narrow passage sampling for probabilistic roadmap planning[END_REF][START_REF] Liangjun | A hybrid approach for complete motion planning[END_REF]). When the algorithm finds the narrow passage entrance in the CS, it will quickly progress to find a solution. Random algorithms progress easily in narrow passage but have no global vision of the solution. On the other hand, a user will have much more difficulties to progress in narrow passages.

For a free or low-constrained environment, a user will easily find a solution (for example, we quickly steer the key towards the keyhole). Thus, the user has a good global vision of the problem, but the algorithm will find it difficult to explore and find narrow passage.

It is noticeable that the algorithm and a user with an interaction device, can both work in complementary fashion. The complementarity between user and algorithm is the essential idea of this work.

The idea of mixing a planning algorithm with user interaction is also interesting in the character animation context. The automatic part can take care of tedious tasks (creating the walking animation for example), while the user takes care of higher level constraints such as determining the path and the upper body movements.

The main contribution of the method is to allow simultaneously cooperation between the loop of roadmap search and the loop of interaction for the user to guide a virtual character.

The paper is organized as follows. We begin by describing previous work in interactive motion planning and using motion planning technique for animation. The general interaction loop between user and motion planning algorithm which is at the basis of our work and the problem statement are summarized in section 3. The following section describes the Interactive locomotion planner that is decomposed into two parts : semi-interactive planner and interactive planner Several experiments are illustrated in the various sections.

Previous work

The idea of taking into account user input has been studied by various authors specialized in motion planning. In [START_REF] Bayazit | Enhancing randomized motion planners: Exploring with haptic hints[END_REF] the authors use randomized techniques to transform a colliding user-input path into a collision-free path.

In [START_REF] Rosell | Motion planning for haptic guidance[END_REF], authors use harmonic functions, computed over a channel solution generated by a cell decomposition, to generate guiding forces that aid the user in the virtual environment. The idea of [START_REF] Ladeveze | Haptic assembly and disassembly task assistance using interactive path planning[END_REF] is to find a free volume between start and goal using octree decomposition and A * . The path is planned in this volume using an RRT approach.

In all these studies, it is necessary to note that the interaction between the automatic search by motion planning algorithm and the user is simplified by a decomposition in two stages. In our work we do not separate the two stages.

Among other applications of path planning, virtual character animation was also explored. In [START_REF] Bottesi | A motion planning based video game[END_REF] authors illustrate in a video game the advantages of using path planning for virtual animation. Some works introduced a multi-level roadmap that represent movements on different ground elevations [START_REF] Salomon | Interactive navigation in complex environments using path planning[END_REF] or for different types of locomotion [START_REF] Shiller | Planning motion patterns of human figures using a multi-layered grid and the dynamics filter[END_REF]. But path planning based methods are difficult to customize for a flexible use in character animation. On the other hand, motion capture is widely used to generate character animations [START_REF] Kovar | Motion graphs[END_REF], but can be tedious.

In [START_REF] Pettré | Planning human walk in virtual environments[END_REF], the author proposed a method to generate automatically an animation of a walking virtual character. The method is divided in a planning step and an animation step. The planning step is done using the visibility-PRM algorithm. The standard linear local method is replaced by a method that uses bezier curves to generate a more natural looking path. The resulting path is resampled to create intermediate configurations that are equally separated from each other in time. This is done to simulate a speed profile along the path.

The animation step is done by interpolating a limited set of motion capture samples. The samples are recorded with constant linear speed v and angular speed ω. Each sample is identified by its (v,ω) values. Then, for each configuration on the planned path, its (v,ω) values are computed and projected into (v,ω) space containing the motion capture samples. The configuration of the virtual character is interpolated from the nearest library samples in the two dimensional (v,ω) space.

In [START_REF] Truong | Reactive synthesizing of human locomotion combining nonholonomic and holonomic behaviors[END_REF], the author stated that the human locomotion has some holonomic parts. To illustrate that fact he proposed a variant of the locomotion controller including a lateral speed v l corresponding to the holonomic behaviour of human walk. A new motion capture library was recorded to include a lateral speed in the samples.

While these works greatly ease virtual character animation, they are based on probabilistic algorithms that are not controllable by the user. Usually, a user may want to choose the path among obstacles himself according to constraints that can not easily be taken into account by automatic algorithms. The method proposed in section 4 is based on this observation and on the work in [START_REF] Pettré | Planification de mouvements de marche pour acteurs digitaux[END_REF] and [START_REF] Truong | Un modele de locomotion humaine unifiant comportements holonomes et nonholonomes[END_REF].

Interactive Motion Planning System

Approach overview

The proposed approach aims at improving the operator guidance with the help of an automatic path planning algorithm with an interactive device, I-Device. We consider that the user wants to guide the trajectory of the object or virtual character through I-Devices such as space mouse or haptic arm. If I-Device is an haptic device the method uses force through the haptic feedback, in the case of a 3D-mouse, the motion of the mouse is transformed in a pseudo-force. We use the same name to represent the real force with haptic device.

The approach has to take into account two constraints:

• The user's direction of movement has to lead the planner development.

• The planner has to return some useful information to the user (haptic and/or visual information) concerning obstacles proximity and if this area has already been explored.

The general plan is constituted by two main loops which will work in parallel (see figure 1). In one loop, the path planner runs to explore the free configuration space, build a roadmap with RRT path planning method and search an automatic solution . The sampling and extension of the initial RRT method is modified to take into account the user interaction. In the other loop, the user moves the object in the virtual environment in six dimensions (position and orientation) with the I-Device as he wishes. The user motion creates a pseudo-force, F u , and the algorithm returns the pseudo-force, F a , that must be seen as disturbance. • The user does not move the I-Device: the IRRT informs the user of the preferential direction to follow via visual and pseudo-force information.

• Both user and planner are not working at the same time: there is a synchronization between both loops of the main plan, which are working on different frequencies. The difference of frequency between both loop is not a problem because each loop is nonblocking. They're only sharing few data, i.e. pseudoforces.

Interactive RRT Algorithm

Our algorithm is based on Rapidly-exploring Random Tree (RRT) approach [START_REF] Lavalle | Rapidly-exploring random trees: Progress and prospects[END_REF]. Starting at a given initial configuration, RRTs incrementally search the configuration space for a path connecting the initial and the goal configuration. At each iteration a new configuration is sampled (RANDOM CONFIG) and the extension from the nearest node in the tree (NEAREST NODE) toward this sample is attempted. If the extension succeeds (CONNECT ) a new node in the roadmap is created at a distance ε of the nearest node. This extension process will be called classical extension in this article.

In our case it is necessary to take into account the interaction with the user. The main modifications of the basic RRT are to be made on the following stages:

• By definition, the state of the node is binary. If the node is free and the connection path is also collision free, the node/edge is created in the roadmap, otherwise, any information is memorized. It is interesting to memorise information for the user when the CONNECT function return false.

• The sampling area in RANDOM CONFIG is a crucial point to extend efficiently the roadmap. User's interaction must be taken into account in the definition of the sampling area.

• The NEAREST NODE function has not only to take into account the Euclidian distance but also the most interesting nodes with regard to the task of the user.

• At each iteration, the new I-Device position must be taken into account to compute these differents steps

To take into account these various points we propose a new algorithm called Interactive Rapidly-exploring Random Tree, IRRT, which is presented in detail in [START_REF] Flavigné | Interactive motion planning for assembly tasks[END_REF].

The algorithm is shown in Algorithm 1 and the principal steps are:

• Compute an attractive pseudo-force F a from the current tree using the roadmap data (nodes) to influence the user movement via an I-Device and haptic and/or visual hints.

• Compute an interaction pseudo-force F u from the user's input (position and/or movement) to take into account the user's intention.

• Compute sampling from a pseudo-force F r : choose an efficient way to combine user's intentions and algorithm automatic search (from F a and F u ).

• Choose nearest neighbor in the tree.

• Extend the roadmap and label the nodes.

Algorithm 1 Interactive-RRT T (q init ) for i = 0 to N do q rand ← SAMPLED CONFIG(F a ,F u ) q near ← NEAREST NODE(q rand , T ) L near ← NEAREST NEIGHBORS(q user , T ) if CONNECT(T, q rand , q near , q new ) then Add Vertex(T , q new ) Add Edge(T , q near , q new ) Update Labels(T , q near , q new ) COM ← Compute CenterOfMass(L near ) F a ← Compute FAlgo(COM, q user ); F u ← Compute FUser(q user ); end if end for

Attractive pseudo-force computation

We first retrieve the user's position in the virtual scene and then take some roadmap nodes (the p nearest nodes) near this position (NEAREST NEIGHBORS). After getting their respective labels, we compute the center of mass (COM). The attractive pseudo-force (F a ) is given by the vector that goes from the user's position (q user ) to the center of mass (Compute FAlgo).

Interaction pseudo-force computation

Then we have to compute another pseudo-force from the user input intended to guide the algorithm development via a specific sampling method. We first compute the user's movement (i.e.: difference between the current position and the previous recorded position) and then convert it into a pseudo-force vector (F u ) in the configuration space (Compute FUser).

Sampling method

Allowing the user to control the sampling is allowing him to lead the search. With the two previous computed pseudo-forces, we can compute a new one by :

F r = α.F u + (1 -α).F a .
The parameter α allows to tune the part of user's intention compared with the algorithm search. The sampling is realized by shooting along the F r direction. The aim is to have a sampling area deformed along the pseudo-force direction (SAMPLED CONFIG).

Nearest neighbor

The nearest neighbor search is done by comparing the Euclidean distance between the sampled configuration and each node of the roadmap (NEAREST NEIGHBORS). Then we can compute COM using the previously computed labels as weights for each node (Compute CenterO f Mass). We take a fixed number p of neighbors (the nearest ones) for this.

Extension

First we compute a classical extension (CONNECT ). Then, we analyze the result and classify it in one of these three categories : Collision (extension has failed), Reached (extension succeeds and reaches the last shot sample q rand ) and Obstacle (extension reaches the node q new but did not reach the shot sample q rand ).

Node Labelling

We compute collision labels for the new node and the extended node. The only information we can use comes from the collision tests. In the interactive algorithm, we used them to label roadmap nodes (U pdate Labels).

There are two types of information we used in the algorithm: the distance from the node to the goal configuration and the number of collisions that occur from trying to extend the node (towards a shot configuration, when this node was chosen as the nearest neighbor).

Results

We use a haptic device (haptionvirtuose) to move the object in CAD model. During all the tests the α parameter is equal to 0.5 for balancing user and algorithm part. The number of nearest neighbors p is fixed to ten.

The algorithm was implemented in C++ in the software platform HPP developed at LAAS based on KineoWorks 1The experiments were performed on a PC Dual Core, 2.1 Ghz and 2 GB ram.

This example is a real use-case from the automotive industry. The problem is to find a collision free path dismounting a part of a car (a silencer) to check the manufacturing process or the maintainability of the assembly.

We show only a part of the real environment for visualization problem and the environment is limited to represent the reality. This example is very constrained and it is very difficult to disassembly the silencer for a user who does not have good practice on haptic arm. To help the user, we did not forbid the penetration of the virtual object moved by the user into obstacles, because without a real 3D display the user can be stuck in a narrow passage in a complex environment without seeing why he cannot move. It is not a problem because the algorithm develops a roadmap in the free-collision space, the soluton is always without collision. The input of the algorithm is a CAD model with initial and final configuration for the silencer (figure 2-a). The goal of the user is to take out the silencer by the top.

The example is solved in a few minutes (solution in figure 2-c) and much faster than basic RRT. If the user does not move the haptic device the computation time is the RRT time computation.

The computation time to find the solution is strongly dependent on the user dexterity with the I-Device (analysis is done in [START_REF] Flavigné | Planification de mouvement interactive: coopration humain-machine pour la recherche de trajectoire et l'animation[END_REF]).

The roadmap build by the I-RRT with the user's motion is shown in figure 2-d.

This industrial case is a direct application of the motion planning system that allows a user to cooperate with the algorithm to find a feasible path for an object during a disassembly operation. But the system can also be applied to other kind of problem such as character animation, as we shall see in the next section.

Animating a virtual character to imitate human behaviour is a common problem in computer animation. Common solutions are key frames and motion capture. The first one allows users to specify key configurations of the virtual character and then, intermediate positions are interpolated. The drawback is that the user has to specify each key frame manually. The second one allows to copy a human movement to reproduce it on the character, but each movement that we want the character to perform must be recorded.

The main idea of the interactive locomotion planner is to combine the interactive motion planning system presented in section 3 with the work in [START_REF] Truong | Reactive synthesizing of human locomotion combining nonholonomic and holonomic behaviors[END_REF] on a nonholonomic locomotion controller, described in section 2. This controller use a limited number of motion capture samples to generate the animation of a virtual character along any given path.

We decided to separate the planner into two parts to propose a solution to two different problems of character animation : animation with a predefined trajectory, and real-time animation. In the next sections we will describe these two planners and show some examples of their applications.

The semi-interactive planner 4.1.1 Principle

The aim of the semi-interactive version of the planner is to perform a specific planning query with a known goal configuration. The main steps of the algorithm are shown by algorithm 2.

Algorithm 2 Semi-Interactive Locomotion Planner

P bezopt ← IRRT(q init ,q goal ) T root ← PATH TO TRAJ(P bezopt ) T anim ← ANIMATE(T root )

It can be divided into three parts:

• A planning step using the interactive motion planning system. The user is guiding a path search between the initial configuration (q init ) and the final configuration (q goal ). The local method used to link two configurations is modified to make a more human-like path. The path search is done with the bounding box of the virtual character. The bounding box has three degrees-of-freedom (X, Y , θ ).

• A transformation step from the path obtained in the previous step (P bezopt ) to a trajectory T root containing time data (which was not the case with P bezopt ).

• An animation step: for each configuration of T root , the whole-body configuration of the virtual character is computed using the motion capture-based algorithm presented in [START_REF] Truong | Reactive synthesizing of human locomotion combining nonholonomic and holonomic behaviors[END_REF]. In section 3, we described a system to compute a path interactively for a free-flyer type object. In the present case, the object is not a free-flyer and has non-holonomic constraints. With a simple local method using bezier curves, the path would result as on Fig. 3-a. We implemented a local method similar to the one used in [START_REF] Truong | Reactive synthesizing of human locomotion combining nonholonomic and holonomic behaviors[END_REF], but the method to choose the nearest neighbour was modified to promote human-like walk, unlike the interactive motion planner. This smoothing step can be divided into three parts :

• The weight of rotations in the distance computation was increased.

• A coefficient representing the relative positions of the two configurations to link is applied on the distance computation. This coefficient favorizes forward movements more than backward movements.

• The last step uses common path optimization methods (shortcut optimization, etc.) to smooth the path resulting from the two previous steps as they result in a winding path (Fig. 3-b).

Results

In this example (shown on Fig. 4), the user is guiding the trajectory search through a cluttered environment. On Fig. 4-a), we can see the user's path in green. It is approximative (the red frames are showing colliding parts of this path) because the user is not really experienced with the interactive device. The uniform component of the sampling method allows us to avoid the collisions on the user's path. The roadmap build by the interactive algorithm is shown in black. On Fig. 4-b), the computed trajectory is displayed. Another run (which implies another user path in this case) of the semi-interactive planner in the same environment is shown on Fig. 5-a). In this case, the algorithm chooses another path for avoiding the table in the central room. The Fig. 5-b) shows the animation of the virtual character along the trajectory computed on Fig. 5-a). The main goal of the interactive version of the locomotion planner is to animate a walking virtual character while the user is guiding the path. In this version, the goal configuration can be omitted as the position of the user is replacing it. The idea is to animate the character along the path traveled by the user during one iteration of the algorithm main loop. This idea is described in the algorithm 3:

Steps

These five steps are computed in parallel with another part of the algorithm which does not appear in the implementation : the user side. During one iteration of the algorithm, the user moves the interactive device to specify the path that the virtual character should follow. 1. The first step is the planning step (IRRT(q pv )). This step plans the path of the character bounding box between the character start position q pv and the current user's position (specified with the interactive device). For this step, the interactive motion planner described in section 3 was used with some modifications aimed at reduce the computation time. First, the local method was simplified to a simple linear interpolation method. Secondly, the degrees-of-freedom are reduced to 2 : the orientation of the bounding box is not taken into account anymore. The resulting path P linear is collision free.

2. The second step is the transformation step. The goal of this step is to get a human-like path P opt from the planned path P linear , using bezier curves (BEZIER OPTIMIZER(P linear )). The different steps of this transformation are detailed on Fig. 6. q q q q q init goal 1 2 3 q m 1 q m 2 q m 3 q m 4 q init q 1 q 2 q 3 q goal (a) The path resulting from the planning step.

(b) Intermediate configurations are added. Orientations of q init and q goal are not changed.

q init q goal q m 1 q m 2 q m 3 q m 4 (c) Bezier local paths are created between the intermediate configurations. 4. The fourth step is the animation step (ANIMATE(T root )).

Using the trajectory computed in the previous step, the actual configurations of the character along the path are generated by interpolating motion captures samples. These samples are chosen in the library depending on the speed along the trajectory.

The fifth step is the playback step (PLAY(T anim )).

The trajectory previously computed is applied to the virtual character. This step is aimed at giving a visual feedback to the user during the computation.

The five main steps of the algorithm are repeated until the user stops the algorithm.

Results

This section show some examples to describe and illustrate the interactive version of the locomotion planner.

On Fig. 7, different steps of the interactive locomotion planner are shown. Fig. 8 shows that the collision avoidance properties of the semi-interactive planner are also working with the interactive planner. The user seems to be unexperienced with the device and draws a colliding path. The algorithm manages to avoid the colliding parts and reach the user's position. One of the problems that occurs with the interactive method is that the collision avoidance is only local and can create blocking situations. For example, when the user is in collision (to ease the guidance by the user, collisions are not forbidden) or is crossing walls, the algorithm may not be able to reach his current position in the virtual scene, and will use the nearest non-colliding position to generate the animation. The user's path may not be followed exactly.

Conclusions

In this work, we described a simple interactive method designed for interaction between user and a path planning algorithm. This method is based on pseudo-forces exchange between the algorithm and the user, and on data gathering (labels) from the virtual scene. We saw an illustration on a real industrial case. The method was applied on virtual character animation to compute a locomotion movement. Two cases were covered. The first one, when the goal of the planning query is known, is solved with a semi-interactive planner which can avoid collisions and help an unexperienced user to find a path. The second one uses a fully interactive method to track the user position while animating the character at the same time, locally avoiding collisions.

We should integrate a real haptic device in order to use real forces and contact perception from the virtual scene instead of using only movements speeds. We have to define a normalization forces, in order to let the user decide all along the process. Visual hints have to be inserted from the labels to have a visual feedback. Concerning the interactive locomotion planner, the algorithm performances depend on the length of the path to animate. In the interactive version, since the path is modified at each iteration, the animation has to be regenerated and the computation time grows with the length of the path. It can be interesting to modify the algorithm so that the computation time is constant with the length of the path by reusing previously computed animations.

Figure 1 .

 1 Figure 1. General loop

Figure 2 .

 2 The automotive industrial case.

  (a) In green, the path imposed for the virtual character by the user using the interactive device. In black, the roadmap build during the planning.(b) The path resulting from transformation using bezier curves.

Figure 4 .

 4 Figure 4. Guiding character movement

Algorithm 3

 3 Interactive Locomotion Planner loop P linear ← IRRT(q pv ) P opt ← BEZIER OPTIMIZER(P linear ) T root ← PATH TO TRAJ(P opt ) T anim ← ANIMATE(T root ) q pv ← PLAY(T anim ) end loop The algorithm can be divided into five steps: • A planning step • A transformation step • A trajectory step • An animation step • A playback step These steps will be detailed in the next section.

  (a) In black, the path obtained with the semiinteractive planner. Axes are indicating the character orientation along the path. (b) Character animation along the path.

Figure 5 .

 5 Figure 5. Semi-interactive Locomotion Planner case

Figure 6 .

 6 Transformation step 3. The third step is the generation of a trajectory with information. The path resulting from the previous step does not have any speed or time information.For animating the character, speed information is computed acoording to speed and acceleration limitations provided by the samples of the motion capture library (PATH TO TRAJ(P opt )).

  Fig. (a), (b) and (c) shows the three first iterations of the algorithm. No optimization is visible because the path is to small. The algorithm is tracking the user position. On Fig. (d) and (e), the path is slightly optimized to be more natural looking.

Figure 7 .

 7 Figure 7. Different steps of the interactive planner, in chronological order

Figure 8 .

 8 Figure 8. Interactive Locomotion Planner case

KineoWorks is the path planning dedicated Software Development Kit developed by KineoCAM.


Michel Cristofol

Lionel Roques
email: lionel.roques@inra.fr

Partial Differential Equations

This Note deals with a uniqueness and stability result for a nonlinear reaction-diffusion equation with heterogeneous coefficients, which arises as a model of population dynamics in heterogeneous environments. We obtain a Lipschitz stability inequality which implies that two non-constant coefficients of the equation, which can be respectively interpreted as intrinsic growth rate and intraspecific competition coefficients, are uniquely determined by the knowledge of the solution on the whole domain at two times t0 and t1 and on a subdomain during a time interval which contains t0 and t1. This inequality can be used to reconstruct the coefficients of the equation using only partial measurements of its solution.

Version française abrégée

L'équation de réaction-diffusion [START_REF] Cristofol | Biological invasions: Deriving the regions at risk from partial measurements[END_REF], dans laquelle u(t, x) correspond à une densité de particules à un temps t et une position x, fut introduite par Fisher et Kolmogorov et al. [START_REF] Fisher | The wave of advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] pour modéliser un problème de génétique des populations. Depuis, ce modèle est utilisé dans des domaines allant de l'écologie à la cytologie. Nous nous intéressons ici à sa version hétérogène. Alors que le coefficient de diffusion est supposé constant, nous faisons en effet l'hypothèse que les coefficients de croissance intrinsèque, µ(x), et de compétition intraspécifique, γ(x), peuvent dépendre de la variable d'espace. Ces deux coefficients jouent un rôle essentiel en dynamique des populations. Nous les supposons non connus, notre objectif étant de les retrouver à partir de mesures partielles de u(t, x), via une inégalité de stabilité. A notre connaissance, dans la littérature existante, de tels résultats de stabilité portent uniquement sur un coefficient. En se plaçant dans un domaine borné Ω, et en utilisant des résultats de régularité parabolique (voir [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]), le principe du maximum ainsi que le lemme de Hopf, nous démontrons tout d'abord trois lemmes préliminaires (1.1, 2.1 et 2.2) donnant des estimations a priori des solutions de [START_REF] Cristofol | Biological invasions: Deriving the regions at risk from partial measurements[END_REF], indépendantes des coefficients de l'équation. Nous établissons ensuite trois inégalités de Carleman (avec des poids spéciaux) associées aux systèmes (4), ( 5) et [START_REF] Fisher | The wave of advance of advantageous genes[END_REF]. Le principal résultat [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] en découle. Il implique que si deux coefficients μ et γ sont tels que la solution ũ de (1), où µ et γ sont respectivement remplacés par μ et γ, est proche de u sur Ω aux temps t 0 et t 1 et dans un sous-domaine ω durant un intervalle de temps ]t 0 -δ, t 1 + δ[, alors μ est proche de µ et γ est proche de γ.

Introduction and main results

The idea of modelling population dynamics with reaction-diffusion models has begun to develop at the beginning of the 20 th century, with random walk theories of organisms. Then independently, Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky, Piskunov [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] used a reaction-diffusion equation as a model for population genetics. The corresponding equation is u t = D∇ 2 u + u(µ -γu), where u = u(t, x) is the population density at time t and space position x, D is the diffusion coefficient, and µ and γ respectively correspond to the constant intrinsic growth rate and intraspecific competition coefficients. In the 80's, this model has been extended to heterogeneous environments by Skellam [START_REF] Skellam | Random dispersal in theoretical populations[END_REF]:

u t = D∇ 2 u + u(µ(x) -γ(x)u), for t > 0, (1)
in a bounded and smooth domain Ω ⊂ R N . Recently, this model revealed that the heterogeneous character of the environment played an essential role on species persistence and spreading, in the sense that for different spatial configurations of the environment, a population can survive or become extinct and spread at different speeds, depending on the habitat spatial structure ( [START_REF] El Smaily | Homogenization and influence of fragmentation in a biological invasion model[END_REF], [START_REF] Roques | Modelling the impact of an invasive insect via reaction-diffusion[END_REF], [START_REF] Roques | On population resilience to external perturbations[END_REF], [START_REF] Roques | Mathematical analysis of the optimal habitat configurations for species persistence[END_REF], [START_REF] Roques | A population facing climate change: joint influences of Allee effects and environmental boundary geometry[END_REF], [START_REF] Roques | Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments[END_REF], [START_REF] Shigesada | Traveling periodic-waves in heterogeneous environments[END_REF]). In a previous work [START_REF] Cristofol | Biological invasions: Deriving the regions at risk from partial measurements[END_REF] assuming that γ was constant, we stated a stability inequality which enabled us to successfully recover µ(x), using the following measurements: (i) µ(x) is known and equal to a constant near the boundary ∂Ω; (ii) the density u(0, x) is known in Ω at t = 0; (iii) the density u(t, x) is known and equal to 0 for (t, x) ∈ [0, T ] × ∂Ω for some T > 0; (iv) the density u(t, x) is known for (t, x) ∈ (t 0 , t 1 ) × ω, for some times 0 < t 0 < t 1 < T and a subset ω ⊂⊂ Ω; (v) the density u( t0+t1 2 , x) is known for all x ∈ Ω. Here, our aim is to obtain a stability inequality which enables to simultaneously recover both coefficients µ(x) and γ(x), and to prove their uniqueness, provided they belong to a particular subset of C ∞ (Ω), given the following information: (i') µ(x) and γ(x) are known near the boundary ∂Ω; (ii') the density u i (x) = u(0, x) is known in Ω at t = 0; (iii') u is known and satisfies Neumann boundary conditions in [0, ∞) × ∂Ω; (iv') the density u(t, x) is known in a finite time interval and in a subset ω ⊂⊂ Ω; (v') the densities u(t 0 , x) and u(t 1 , x) are known at two fixed times t 0 , t 1 and for all x ∈ Ω.

Few works are related to the reconstruction of several coefficients of reaction-diffusion equations. Furthermore, those works only deal with the reconstruction of source terms and initial conditions (see e.g. [START_REF] Choulli | Some stability estimates in determining sources and coefficients[END_REF], [START_REF] Yamamoto | Simultaneous reconstruction of the initial temperature and heat radiative coefficient[END_REF]), or provide uniqueness results without stability [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF].

The main tools used to establish these new results are Carleman estimate with special weights and parabolic estimates together with parabolic maximum principle and Hopf's lemma.

Let us make our hypotheses more precise. We define two subsets of C ∞ (Ω) by M 1 := {μ s.t. μ(x) = µ * (x) if d(x, ∂Ω) < ε, for all x ∈ Ω}, and Γ 1 := {γ s.t. γ(x) = γ * (x) if d(x, ∂Ω) < ε, for all x ∈ Ω}, for given functions µ * ∈ C ∞ (Ω) and γ * ∈ C ∞ (Ω), and a small parameter ε > 0; d(x, ∂Ω) corresponds to the distance from x to ∂Ω. Let M 2 and Γ 2 be two other subsets of C ∞ (Ω), defined by: M 2 := {μ s.t. µ -≤ μ ≤ µ + on Ω}, for two given functions µ + , µ -in M 1 with 0 < µ -≤ µ + , and Γ 2 := {γ s.t. γ -≤ γ ≤ γ + on Ω}, for two given functions γ -and γ + in Γ 1 with 0 < γ -≤ γ + . Lastly, let K ⊂ C ∞ (Ω) be defined by K := {ρ s.t. ρ C 5 (Ω) ≤ m}, for some m > 0.

We then define the state spaces M and Γ by M := M 1 ∩ M 2 ∩ K, and Γ := Γ 1 ∩ Γ 2 ∩ K. Note that, if ε is chosen small enough and m is chosen large enough, these sets are not empty. Let us fix two couples (µ, γ) and (μ, γ) in M × Γ and let u, ũ be, respectively, the solutions of

         ∂ t u = D∆u + u(µ -γu) in (0, ∞) × Ω, ∂ ν u = 0 on [0, ∞) × ∂Ω, u(0, •) = u i in Ω,
and

         ∂ t ũ = D∆ũ + ũ(μ -γ ũ) in (0, ∞) × Ω, ũ = u on [0, ∞) × ∂Ω, ũ(0, •) = u i in Ω.
(

) 2
for some constant D > 0 and some function u i in C ∞ (Ω) which verifies:

u i > 0 in Ω, ∂ ν u i = 0 on ∂Ω and 6 sup Ω u i < inf Ω µ -/ sup Ω γ + . (3)
The functions u, ũ belong to

C 2 1 ([0, ∞) × Ω) ∩ C ∞ ([σ, ∞) × Ω)
, for any σ > 02 . Before stating our main theorem, let us state a preliminary lemma: Lemma 1.1 It exists an interval T in (0, ∞) such that, for any couple (t 0 , t 1 ) with 0 < t 0 ≤ inf T < sup T ≤ t 1 , and for all (μ, γ)

∈ M × Γ, 6 sup x∈Ω ũ(t 0 , x) < inf x∈Ω ũ(t 1 , x).
The interval T can be computed explicitly. The proof of this lemma uses hypothesis (3) on u i .

Our main result is: Theorem 1.2 For any ω ⊂⊂ Ω and any time interval (t 0 , t 1 ) containing T it exists δ ∈ (0, t 0 ) and a constant C such that for all µ, μ ∈ M , γ, γ ∈ Γ,

µ -µ 2 L 2 (Ω) + γ -γ 2 L 2 (Ω) ≤ C G(u, ũ), with G(u, ũ) = u -u 2 H 2 ((t0-δ,t1+δ),L 2 (ω)) + (u -u)(t 0 , .) 2 H 2 (Ω) + (u -u)(t 1 , .) 2 H 2 (Ω)
. A straightforward corollary is a uniqueness result for the couple (μ, γ), given u(t 0 , x), u(t 1 , x) for x ∈ Ω and u(t, x) for t ∈ (t 0 -δ, t 1 + δ) and x ∈ ω. Another practical consequence of Theorem 1.2 is to enable a numerical reconstruction of the unknown coefficients µ and γ, given the partial measurements (i)', (ii)', (iii)', (iv)', (v)' (see [START_REF] Cristofol | Biological invasions: Deriving the regions at risk from partial measurements[END_REF]).

In the sequel, we give a very schematic proof of this result. More details will be given in the forthcoming paper [START_REF] Cristofol | Simultaneous reconstruction of two coefficients in a nonlinear parabolic equation[END_REF].

Let us recall a classical Carleman estimate (see [START_REF] Fernández | Global Carleman inequalities for parabolic systems and application to controllability[END_REF]): Theorem 2.3 Let ρ ∈ R, 0 < τ 0 < τ 1 and 0 < δ < τ 0 . Then it exists a constant K > 0, a function K < ζ(x) < 2K in C 2 (Ω), λ 0 ≥ 0, s 0 > 0 and a positive constant C 0 such that, for any λ ≥ λ 0 , any s ≥ s 0 , and any function q ∈ C 2 (Q i ) with q ≡ 0 on [τ i -δ, τ i + δ] × ∂Ω, the following estimate holds:

I i (ρ, q) ≤ C 0 s Qω i e -2sηi λ 4 (sϕ i ) ρ+3 |q| 2 dtdx + Qi e -2sηi (sϕ i ) ρ |∂ t q -∆q| 2 dtdx , (7)
where

I i (ρ, q) = s Qi e -2sηi (sϕ i ) ρ-1 (|∂ t q| 2 + |∆q| 2 ) dtdx +s 2 λ 2 Qi e -2sηi (sϕ i ) ρ+1 |∇q| 2 dtdx + sλ 4 Qi e -2sηi (sϕ i ) ρ+3 |q| 2 dtdx.
Using [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] applied to the solution y of ( 4), together with Lemmata 2.1 and 2.2 we get that, for any 0 < τ 0 < τ 1 , 0 < δ < τ 0 , λ 0 ≤ λ, and for s large enough, it exist C(s, λ) > 0 and C > 0 such that, independently of the choice of (μ, γ) ∈ M × Γ,

I i (0, y) ≤ C(s, λ) Qω i ϕ 3 i |y| 2 e -2sηi dtdx + Cs Qi |α -β ũ| 2 e -2sηi dtdx. (8)
Similarly, using [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] applied to the solution z of ( 5) and Lemma 2.1 of [START_REF] Klibanov | Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate[END_REF], we get that it exist C(s, λ) > 0 and C > 0 such that:

I i (0, z) ≤ C(s, λ) Qω i ϕ 3 i |z| 2 e -2sηi dtdx + Cs Qi |β| 2 e -2sηi dtdx + C(s, λ) (y(τ i , .) 2 H 1 (Ω) . (9)
Then, from (6) and using once more Lemma 2.1 of [START_REF] Klibanov | Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate[END_REF] we can write:

I i (0, w) ≤ C(s, λ) Qω i e -2sηi ϕ 3 i (|w| 2 + |z| 2 )dtdx + C 1 λ 2 Qi |β| 2 e -2sηi dtdx + C(s, λ) (y(τ i , .) 2 H 1 (Ω) . (10)
Multiplying ( 4) by e -sη0 , evaluating at time t = τ 0 and taking the L 2 norm in space we can get an upper bound for αe -sη0(τ0,.) 2 L 2 (Ω) . Then, using Carleman estimates (8), ( 9) and [START_REF] Roques | Modelling the impact of an invasive insect via reaction-diffusion[END_REF] we also obtain an upper bound for (α -β ũ(τ 1 , .))e -sη1(τ1,.) 2 L 2 (Ω) . Combining these two upper bounds, and using the fact that η 0 (τ 0 , .) = η 1 (τ 1 , .), we obtain, for sufficiently large s, the existence of C > 0, such that: αe -sη0(τ0,.) 2 L 2 (Ω) + β ũ(τ 1 , .)e -sη0(τ0,.) 2 L 2 (Ω) ≤ C

We then use Lemma 1.1 to find two times t 0 and t 1 such that 6 sup x∈Ω ũ(t 0 , x) < inf x∈Ω ũ(t 1 , x), and we fix δ ∈ (0, t 0 ). We finally obtain the existence of a constant C > 0 such that, for any couple (μ, γ) ∈ M × Γ,

α 2 L 2 (Ω) + β 2 L 2 (Ω) ≤ C t1+δ t0-δ ω ϕ 3 max e -
) where η min (., .) = min (t0-δ,t1+δ)×ω (η 0 , η 1 )(., .) and ϕ max (., .) = max (t0-δ,t1+δ)×ω (ϕ 0 , ϕ 1 )(., .).

Qω 1 ϕ 3 1 e

 1 -2sη1 |z| 2 dtdx + C Qω 0 ϕ 3 0 e -2sη0 (|w| 2 + |z| 2 )dtdx +C y(τ 0 , .) 2 H 2 (Ω) + C y(τ 1 , .) 2 H 2 (Ω) + 6 sup Ω |ũ(τ 0 , .)| 2 βe -sη0(τ0,.) 2 L 2 (Ω) .

  2sηmin (|z| 2 + |w| 2 )dtdx + C y(t 0 , .) 2 H 2 (Ω) + C y(t 1 , .) 2 H 2 (Ω) ≤ C u -ũ 2 H 2 ((t0-δ,t1+δ),L 2 (ω)) + (u -ũ)(t 0 , .) 2 H 2 (Ω) + (u -ũ)(t 1 , .) 2

	H 2 (Ω)

The spaces C i j ([σ, ∞) × Ω) are spaces of functions on [σ, ∞) × Ω whose derivatives up to order i in x and order j in t are continuous. The regularity of ũ follows from the hypothesis on u i and from the definition of M 1 and Γ 1 which lead to proper compatibility conditions.

Proof of Theorem 1.2.

We begin with a priori estimates on ũ, independent of the choice of the couple (μ, γ) ∈ M × Γ. Lemma 2.1 For any T > 0 it exists r > 0 such that, for all (μ, γ) ∈ M × Γ, ũ ≥ r and ∂ t ũ ≥ r on [0, T ] × Ω. The next lemma states C α α/2 boundary estimates on ũ and its time derivatives. Lemma 2.2 For any T > 0 and σ ∈ (0, T ) it exists a constant C > 0, independent of the choice of (μ, γ) ∈ M × Γ, such that:

The main tools used to prove Lemmata 1.1 and 2.1 are comparison principles and Hopf's lemma.

Let u (resp. u) be the solution of (2) associated to (µ, γ) (resp. (μ, γ)). We set

where α = µ -μ and β = γ -γ. Using Lemma 2.1, we can set y = U u and the previous system becomes

We set z = ∂ t y.

z(0, x) = (D∆y + 2D u ∇ u • ∇y + Ay(0, .) + α -β u(0, .) on Ω.

(5)

Using Lemma 2.1, we set z = z ∂t u and w = ∂ t z. We obtain w(t, x) = 0 on (0, ∞) × ∂Ω and:

where C and E are functionals depending on ũ and its time and space derivatives until order two. Given any couple 0 < τ 0 < τ 1 and 0 < δ < τ 0 , we set

and some constant K > 0, we may also define:

, η i (x, t) = e 2λK -e λζ(x) (t -(τ i -δ))(τ i + δ -t) for i = 0, 1. Note that η 0 (τ 0 , .) = η 1 (τ 1 , .) on Ω.