Maxence Phalempin

AVERAGING THEOREMS FOR SLOW FAST SYSTEMS IN Z-EXTENSIONS (DISCRETE TIME)

We study the averaging method for flows perturbed by a dynamical system preserving an infinite measure. Motivated by the case of perturbation by the collision dynamic on the finite horizon Z-periodic Lorentz gas and in view of future development, we establish our results in a general context of perturbation by Z-extension over chaotic probability preserving dynamical systems. As a by product, we prove limit theorems for non-stationary Birkhoff sums for such infinite measure preserving dynamical systems.

introduction

We call perturbed differential equation or equivalently slow fast system the following Cauchy problem

dx t (x, ω) dt = F (x t (x, ω), T t (ω)), x 0 (x, ω) = x, ∀(t, x, ω) ∈ [0, +∞) × R d × M , (1)
where F : R d × M → R d is a function, Lipschitz in its first coordinate (in R d ) and where T : M → M is a measurable map on a measurable space M and preserving a measure µ (i.e. (M, T, µ) is a measure preserving dynamical system). In (??), the term T t/ represents the fast motion when → 0 + , whereas x t is considered as a slow motion. The behavior of (x t ) t≥0 has been studied in many cases when the map T preserves a probability measure µ and the resulting dynamical system has rapid mixing properties, we can notably cite the works done for exponentially mixing flows and transformations as Anosov flows or Billiard maps or ergodic toral automorphisms by Anosov [?], Arnold [?], Khasminskii [?], Kifer [?] and Pène [?], Dolgopyat[?] Chevyrev, Friz, Korepanov and Melbourne in [?], among others. In these settings they proved that while the dynamic accelerates ( → 0 + ), the perturbed solution (x t ) t≥0 converges to some averaged motion (w t ) t≥0 solution of an ordinary averaged differential equations

dw t (x) dt = F (w t (x)), w 0 (x) = x, ∀(t, x) ∈ [0, +∞) × R d , (2)
where F (x) := M F (x, •)dµ. Actually they go further proving limit theorems for the error term (e t ) t≥0 between the slow motion (x t ) t≥0 and the averaged solution (w t ) t≥0 . In the present article, our motivation is to study slow fast systems with fast dynamics T t/ preserving an infinite measure µ. To our knowledge, this article contains the first averaging results for perturbation by infinite measure preserving dynamical systems. In this context, the previous definition of the averaged function F either doesn't make sense in general (indeed it makes sense only if F (x, •) is µ-integrable, but then F (x, •) -F (x) is not µ-integrable). Thus, the classical technique consisting in decomposing the F in the sum of a drift F and a centered part F -F in order to study the oscillations of the latter to obtain the speed of convergence and a limit theorem for the error term (e t ) t≥0 needs to be adapted. At least two settings naturally arise with two radically different behaviours. A first one is to consider some case where F is integrable which is investigated in [?, Chapter 8] (where a limit theorem is established, without averaging). A second one, that we investigate here, is to consider a differential equation (??) obtained by a perturbation of a differential equation (??), by considering a map F given by a sum of a null-integral function F and of the drift F appearing in (??) (note the F is independent of the dynamics). In this context, we consider the solution (x t ) t of (??) and the solution (w t ) t of (??) and the error e t defined by e t (x, ω) := x t (x, ω) -w t (x), ∀(t, x, ω) ∈ [0, +∞)R d × M .

for all T > 0, Theorem 1.1 (See Theorem ?? for a more general statement). Let > 0, and (M, T, µ) be the collision dynamics associated to the Z-periodic Lorentz gas with finite horizon (see Section ??) and suppose F (x, ω) = F (x) + F (x, ω) with F : R d → R d and F : R d × M → R d such that M F (x, •)dµ = 0 for all x ∈ R d and F (•, ω) is uniformly C 2 b (uniformly bounded, and with uniformly bounded first and second order derivatives) and F (x, •) is Hölder continuous (with respect to the euclidean metric, or dynamically Hölder) and sharply vanishing at infinity. [START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF] Then, for any x ∈ R d , the error term (e t (x, •)) t∈[0,T ] satisfies a limit theorem for the strong convergence in law:

( -3/4 e t (x, •)) t∈[0,T ] Lµ, . ∞ → →0 (y t (x)) t∈[0,T ] ,
where L µ denotes the strong convergence in law 2 with respect to the measure µ and where (y t (x)) t≥0 is the solution of the following stochastic differential equation dy t (x) = a(w s (x))dB L s (0) + DF (w s (x))y s ds, with (B t ) t≥0 a multi-dimensional standard Brownian motion i.e a stochastic process of which the coordinates B (i) are independent uni-dimensional Brownian motions (B.m) of variance 1. The process (L s ) s≥0 is a local time associated to a Brownian motion (B t ) t≥0 independent of the previous process whereas a : R d → R d 2 is a map such that for any x ∈ R d , a(x) is a non negative symmetric matrix that can be interpreted as a local asymptotic variance in x (see (??)). This result relies on the following key statement which is interesting in itself since it provides the stochastic convergence for perturbed ergodic sums and in particular implies the central limit theorem on Z-periodic Lorentz gas as stated in [?] (for a more general class of observables than the one considered in [?] which were locally constant) by considering the case when F (x, ω) = F (ω) does not depends on its first coordinate. Theorem 1.2 (see Theorem ?? page ??). Under the same assumptions on the map F as in the above Theorem ??, for any t ∈ [0, +∞), we denote, for any (x, ω) ∈ R d × M , by (v t (x, ω)) t≥0 the perturbed Birkhoff sum defined as follows v t (x, ω) := 1/4 t/ 0 F (w s (x), T s ω) ds .

Then the process (v t (x, •)) t≥0 satisfies the following functional central limit theorem for any T ≥ 0 and x ∈ R d ,

(v t (x, •)) t∈[0,T ] Lµ, . ∞ → →0 t 0 a(w s (x))dB L s (0) t∈[0,T ] ,
with (B t ) t≥0 and (L t ) t≥0 the same processes as in the previous theorem. [START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF] The precise assumptions on F are stated in Theorem ?? where • lip can be replaced by the Hölder norm. [START_REF] Arnold | Chapitres supplémentaires de la théorie des équations différentielles ordinaires[END_REF] This means the convergence in distribution with respect to every probability measure absolutely continuous with respect to µ.

The present article is organized as follows. The first section is dedicated to the detailed example of the collision dynamics associated to the Lorentz gas. Section ?? presents the general settings and the results under abstract hypotheses. Section ?? contains the proof of the limit theorem for perturbed Birkhoff sums (proof of Theorem ?? which generalizes Theorem ??) and section ?? the averaging result for transformations (proof of Theorem ?? which generalizes Theorem ??).

2.

Collision dynamics associated to Z-periodic Lorentz gas with finite horizon.

We present here the Billard transformation associated to the collision dynamics of the Zperiodic Lorentz gas (M, T, µ) (this dynamical system is also called the discrete time Z-periodic Lorentz gas). Consider a domain R 0 corresponding to the infinite flat cylinder R × T (where T := R/Z) doted of open convex obstacles belonging to a periodic, locally finite, family {O m + (l, 0), l ∈ Z, m ∈ I} with C 3 boundary, positive curvature and with pairwise disjoint closures, placed along the cylinder. So

R 0 := (T × R)\ m,l (O m + (l, 0)).
We assume furthermore that the horizon is finite horizon, i.e that R 0 contains no line. The dynamical system (M, T, µ) describes the dynamics at collision times of a point particle moving in R 0 , going straight inside R 0 and colliding against the obstacles according to the Snell-Descartes reflection law (the reflected angle is equal to the incident angle). The phase space M is the set of post-collisional unit vectors, i.e. the set of couples of (position,direction of its speed) based on ∂R 0 and pointing into R 0 . Formally, M is the set of (q, v) ∈ ∂R 0 × S 1 (where S 1 is the unit circle) such that v, n(q) ≥ 0 where n(q) denotes the unit normal vector to ∂R 0 in q directing into R 0 .

O 1 + 0 O 1 -1 O 1 + 1 O 2 + 1 O 2 + 1 O 2 + 2 O 2 + 2 O 2 + 0 O 2 + 0 O 2 -1 O 2 -1 x T (x)
Figure 1. Illustration of the map T corresponding to the dynamics of the discrete time Z-periodic Lorentz gas in finite horizon, with two periodic shape of obstacles.

The transformation T maps a couple (position/direction of the momentum) to the new couple (position/direction of the momentum) after one collision against an obstacle. This transformation T preserves an infinite measure µ absolutely continuous with respect to the Lebesgue measure with an explicit Z-periodic density (as recalled in [?, Section 2.1]) The fact that the horizon is finite ensures that the free flight between two collisions is uniformly bounded. A crucial fact is that the dynamical system (M, T, µ) can be represented as a Z-extension with centered bounded step function φ : M → Z over the probabilistic dynamical system (M , T , µ) called the Sinai Billiard. The Sinai billiard (M , T , µ) is the quotient modulo Z of the previous system (M, T, µ). Thus T is the billiard transformation with obstacles Ōi on the two-dimensional flat torus T 2 := R 2 × Z 2 (with Ōi the image of O i by the canonical quotient map). The ergodicity and mixing of this Sinai Billiard (M , T , µ) has been proved by Sinai in [?], exponential mixing has been proved by Young via the construction of towers in [?]. The stochastic properties of the Lorentz gas (M, T, µ) have been studied in previous decades using the chaotic properties of (M , T , µ). This system was proved to be recurrent in [?], ergodic (conservatively ergodic) in [?, ?] and to satisfy a central limit theorem for some class of observables in [?]. Given some β ∈ (0; 1), we endow the dynamical system (M , T , µ) with the dynamical metric d = d β given by ∀x, y ∈ M , d(x, y) := β s 0 (x,y) ,

where s 0 : M × M → N is the separation time such that for x, y ∈ M , s 0 (x, y) ≥ n iff x and y are in the same continuity component of T i for any i ∈ {-n, . . . , n}. For any η ∈ (0, 1), there exists β ∈ (0, 1) such that the class of Lipschitz functions with respect to d β comprises the class of η-Hölder functions for the usual euclidean metric on M (see [?, Exercise 5.49 page 124]). We also introduce, for k, n ∈ Z, the partitions ξ n k as of M such that for any A ∈ ξ n k two elements x, y ∈ A if their iterates T i x and T i y share the same obstacle (∂O m i × S for some m i ∈ I) for any i ∈ {k, . . . , n}.

In particular if k = n and A ∈ ξ n k , x, y ∈ A ⇒ s 0 (x, y) ≥ n.
Thus these sets are related to the the separation time s 0 , which dominates the separation time in the "coding" of the Young Towers on which our main theorem are stated, Indeed maps f : M → R d that are constant on elements of ξ n -n can be lifted to maps onto a Young Towers structure (see [?] or [?, Section 4]). Other stochastic properties have already been stated for the Sinai billiard system (M , T , µ) such as Central limit theorem for smooth observables (including the step function Φ) by Bunimovich, Sinai and Chernov (see [?], [?]) and the already mentioned works of Kifer, Khasminiskii and Pène on perturbed differential equations (see [?], [?], [?]) among others.

General results for Z-extensions

In this section, we introduce a general context of Z-extension and establish general results in this context. The fact that the Z-periodic Lorentz gas in finite horizon introduced in Section ?? satisfies our general assumptions has been proved in [?, Section 4] and will be recalled in remark ?? below.

General model of Z-extension.

In this paper we consider a infinite measure preserving dynamical system (M, T, µ) described as a Z-extension of an ergodic probabilistic dynamical system (M , T , µ) with a centered bounded step function φ : M → Z. The dynamics on such a system is described by the following skew product for any (ω, m) ∈ M × Z by :

T (ω, m) = (T (ω), m + φ(ω)).
When iterated, the dynamic brings the following identity

T n (ω, m) = (T n (ω), m + S n φ(ω)),
where S n φ := n-1 k=0 φ • T k . We endow (M, T ) with the T -invariant measure µ defined on the

Z-extension M × Z is a product measure µ := µ ⊗ m where m is the counting measure on Z.
The assumption that φ is centered (i.e M φdµ = 0) combined with the ergodicity of (M , T , µ) ensures that the Birkhoff sum S n φ is recurrent as a random walk on Z. This implies that (M, T, µ) is a recurrent dynamical system (see [?]) then additional properties such as mixing (which will be implied by our abstract settings ??) will ensure the ergodic conservativity of such dynamical systems.

The main result here is given under some abstract hypotheses : The quotient probability invariant dynamical system (M , T , µ) satisfies the following hypotheses ?? in the sense that it can be modeled to an exponentially mixing Young tower (described in hypothesis ??). Hypothesis 3.1. Let ( M , T , μ) be a probability preserving dynamical system and an application φ : M → Z. There exists β ∈]0, 1[ and ε > 0 and a complex Banach space B = B β, such that (0) the constant function 1 M is in B and B is continuously included in L 1 μ.

(1) there is a partition (∆ i,k ) i,k∈N of M and a splitting time s : M × M → N such that for any i, k ∈ N, and any x, y ∈ ∆ i,k , s(x, y) ≥ 1. In addition for any x, y ∈ M such that s(x, y) ≥ 1, s( T x, T y) = s(x, y) -1. (2) the Banach space (B, . B ) is described as follows: a function

f : M → R is in B if f B = f ∞ + f lip < ∞ with f lip = sup l,k∈N e -l f lip,∆ l,k et f ∞ = sup l,k∈N e -l f ∞,∆ l,k .
where

f lip,∆ l,k := sup x,y∈∆ l,k |f (x)-f (y)| β s(x,y) and f ∞,∆ l,k := f ∆ l,k ∞ .
We will denote l(x) = l if there is k ∈ N such that x ∈ ∆ l,k . (3) the transfer operator P of ( M , T , μ) satisfies for f ∈ B

P f (x) = w∈ T -1 ({x}) e -h(w) f (w)
with h ∈ B a function Lipschitz with respect to β s(•,•) , i.e.

h lip = sup i,k∈N sup x,y∈∆ i,k |h(x) -h(y)| β s(x,y)
is a finite quantity. Furthermore, for any i, k ∈ N and x, y ∈ ∆ i,k there is a bijection

X 0 : T -1 ({x}) → T -1 ({y})
preserving the partition of M , i.e. s(z, X 0 (z)) = 1 + s(x, y). (4) For any x, y ∈ M such that s(x, y) > 1, φ(x) = φ(y). Denote S n := n-1 k=0 φ • T k . There is β 0 > 0 such that or any b ∈]0, β 0 [, there is α ∈]0, 1[, C > 0 such that for any f ∈ B, any positive integer n and any u ∈ [-b, b], the operator P u := P (e iu φ•) satisfies

P n u (f ) := P (e iuSn f ) = λ n u Π u (f ) n + N n u (f ), (4)
where

• λ . ∈ C 4 ([-b, b], C) satisfies λ u = 1 -Σ u 2 2 + O(u 3 ) with Σ := k≥0 E μ( φ • T k φ) • u → Π u ∈ C 2 ([-b, b], L(B)) satisfies Π 0 (•) = E μ(•)1, in addition, for any u ∈ [-b, b],
Π u is a projection, thus

Π n u = Π u . • N u ∈ L(B) satisfies N n u ≤ Cα n . In addition, for any u ∈ [-π, π]\[-b, b], P n u ≤ Cα n .
We now state the main hypothesis ?? which guarantees that (M , T , µ) has a Young tower structure with Lipschitz observables that can be approximated onto ( M , T , μ). Hypothesis 3.2. The set M is a metric space with metric d. We define as follow the Lipschitz norm of an application g : M → R, g lip := sup x,y∈M |g(x)-g(y)| d(x,y) . We suppose that (M , T , µ) and ( M , T , μ) are common factors of a dynamical system (M 1 , T 1 , µ 1 ) with respective projections π 1 and π 2 . We suppose that there is a map φ : M → Z such that ( M , T , μ) and φ satisfy Hypothesis ?? and φ • π 1 = φ • π 2 , which implies by induction that

S n • π 1 = S n • π 2 ,
for any n ∈ N where S n := n-1 k=0 φ • T k and where the notation S n comes from hypothesis ??). In addition there is 0 < β 0 < 1 and C > 0 such that for any n ∈ N, and for any Lipschitz bounded function g : M → R there is, for any n ∈ N, a function g n μ-centered (in the sense that E μ(g n (x, .a)) = 0) satisfying :

g • T n • π 1 -g n • π 2 ∞ ≤ Cβ n 0 g lip (5)
g n ∞ ≤ g ∞ and such that for any x, y ∈ M , s(x, y) > 2n ⇒ g n (x) = g n (y).

Remark 3.3 (Application to the periodic Lorentz gas). We consider here the Z-periodic Lorentz gas (M, T, µ) introduced in section ??. The fact that Hypothesis ?? applies with ( M , T , μ) the symbolic factor of Young Tower constructed by Young [?] for the Billiard transformation has been explained in [?, section 4]). More precisely, there is β ∈]0, 1[ such that for any β ∈] β, 1[, there exists ¯ such that Hypothesis ?? holds true for this β and for any ∈]0, ¯ [. Let β ∈] β, 1[. We consider the dynamical metric d = d β on M given by (??). Let g : M → R be a Lipschitz function with respect to d. For any integer n ≥ 0, set

h n := E µ (g|ξ n -n ).
Then the map h n (x, T n (•)) is constant over the atoms of ξ 2n 0 and there is a map

g n (x, •) : M → R d such that h n (x, •) • T n • π 1 = g n (x, •) • π 2 .
Furthermore this map satisfies

g • T n • π 1 -g n • π 2 ∞ ≤ β n g lip , since sup A∈ξ n -n diam(A) ≤ β n , by definition of d = d β . In addition g n ∞ = h n ∞ ≤ g ∞ and (??) is satisfied.
3.2. Perturbed ergodic Birkhoff sum, result for transformation. The next statement provides a limit theorem for perturbed Birkhoff sums k≥0 F (s, T k •) for some perturbed observable F : R d × M → R d on the Z-extension (M, T, µ) : Theorem 3.4 (Convergence of perturbed Birkhoff sums for the map). Fix 0 > 0 and a dynamical system (M, T, µ) described as a Z-extension over a system (M , T , µ) satisfying Hypothesis ?? above with bounded step function φ :

M → Z. Let F : R d × M × Z → R d satisfying • sup x∈R d a∈Z F (x, ., a) lip < ∞ • for any x ∈ R d , F (x, •) has zero µ-integral over M . • sup x∈R d a∈Z |1 + a| 2(1+ 0 ) F (x, ., a) ∞ < ∞ • these functions F (•, ω) are uniformly lipschitz (uniformly in ω ∈ M ).
For x ∈ R d , let s → w s (x) be a Lipschitz function. Set for any t ∈ [0, T ] and ω ∈ M ,

v t (x, ω) := 1/4 t/ 0 F (w s (x), T s ω)ds. ( 7
)
Then for T ≥ 0 and x ∈ R d ,

(v t (x, •)) t∈[0,T ] Lµ, . ∞ → →0 t 0 a(w s (x))dB L s (0) t∈[0,T ]
,

where L µ means 3 the strong convergence in distribution with respect to µ, (B t ) t≥0 the standard Brownian motion and (L s ) s≥0 is the local time of a Brownian motion (B t ) t≥0 with variance 3 recall that this means the convergence in distribution with respect to every probability measure absolutely continuous with respect to µ.

Σ (with Σ defined as in Hypothesis ??) independent of (B t ) t≥0 . a(•) is the positive symmetric operator given 4 by the following Green Kubo formula :

a i,j (x) := 1 2 l∈Z M F i (x, ω)F j (x, T |l| (ω)) + F j (x, ω)F i (x, T |l| (ω)) dµ(ω). (8)
Observe that when (M, T, µ) is invertible (as for the periodic Lorentz gas) then (??) becomes simply

a i,j (x) := l∈Z M F i (x, ω)F j (x, T l (ω))dµ(ω). ( 9
)
Remark 3.5.

• The process (B L t (0) ) t≥0 is a martingale for its natural filtration, this is a consequence of the independence between (B t ) t≥0 and (L s ) s≥0 . Indeed for any t ≥ 0,

B L t (0) is integrable with E(|B L t (0) |) = E(E(|B L t (0) | |(L s (0)) s≥0 )) = E (L t (0)) 1/2 E(|B 1 |), and E B L t (0) -B L s (0) |(B L s (0) ) s≤u = 0: E B L t (0) -B L s (0) |(B L s (0) ) s≤u = E E B L t (0) -B L s (0) |(L s (0)) s≥0 , (B L s (0) ) s≤u |(B L s (0) ) s≤u The latter term E B L t (0) -B L s (0) |(L s (0)) s≥0 , (B L s (0) ) s≤u is 0 by independence of (L s ) s≥0
and (B s ) s≥0 : to see this it is enough to prove that for any measurable map f :

C 0 (R, R)× C 0 (R, R d ), E f ((L s (0)) s≥0 , (B u ) u≤s ) (B L t (0)-L s (0) ) s≥0 = 0. This is a consequence of the following : E f (L s (0)) s≥0 , (B s ) s≥0 (B L t (0) -B L s (0) ) s≥0 = E f (L s (0)) s≥0 , (B u ) u≤s (B L t (0) -B L s (0) ) s≥0 |(L s (0)) s≥0 = E E f (L s (0)) s≥0 , (B u ) u≤s (B L t (0) -B L s (0) ) s≥0 |(L s (0)) s≥0 = E (f ((h u ) u≥0 , (B hu ) u≤s ) (B ht -B hs )) dP (L s (0)) s≥0 ((h s ) s≥0 ) = 0,
by independence of (B hu ) u≤s with respect to the random variable B ht -B hs .

• When d = 1, the law of the process

t 0 a(w s (x))dB L s (0) t≥0
can be identified with the law of the process B t 0 a(ws(x))dL s (0) t≥0

.

To check this, it is enough to verify that both processes have same law conditionally to (L s (0)) s≥0 . The law of each process conditioned to (L s (0)) s≥0 = (h s ) s≥0 for some continuous map (h s ) s≥0 satisfies Dubins-Schwarz theorem (see [?]) with the same quadratic variation ( t 0 a(w s (x))dB hs = a(w s (x))dh s ) thus they have same law.

4 a(ws(x)) makes sense because as a variance matrix, a(ws(x)) is symmetric non negative definite and its square root is the unique symmetric non negative matrix whose square power identifies with reduction through orthogonal matrix of the diagonal reduced matrix of a(ws(x))

3.3. Slow-fast systems (perturbation by the map). We consider the following slow fast system for the transformation : Let F : R d × M → R d and F : R d → R d , both Lipschitz for their coordinates on R d . We say that (x t ) t∈R + solves the perturbed differential equation if for any (x, ω) ∈ R d × M , the map (x t (x, ω)) t∈R + is a solution of the following equation :

dx t dt (x, ω) = F (x t (x, ω), T t/ ω) + F (x t (x, ω)) x 0 (x, ω) = x (10)
The next theorem states a limit theorem over the convergence of the solutions (x t ) t∈R + of a perturbed differential equation (??) to the averaged map (w t ) t∈R + defined as follows : for any

x ∈ R d , (w t (x)) t≥0 is a solution of dwt dt (x) = F (w t (x)) w 0 (x) = x. (11)
More precisely it provides a limit theorem for the error term (e t ) t≥0 defined for any

(x, ω) ∈ R d × M by e t (x, ω) := x t (x, ω) -w t (x).
Theorem 3.6. Let > 0, and (M, T, µ) a Z-extension over a probability preserving dynamical system (M , T , µ) satisfying Hypothesis ??, with bounded step function φ : M → Z. Let F : R d ×M ×Z → R d satisfying the hypotheses of theorem ??, such that for any (ω, a) ∈ M , F (•, ω, a) is C 2 and its derivative D 1 F is itself uniformly bounded with uniformly bounded derivative and satisfies :

sup

x∈R d a∈Z (1 + |a|) 2(1+ 0 ) D 1 F (x, ., a) ∞ < ∞ and sup x∈R d a∈Z D 1 F (x, ., a) lip < ∞.
Let F : R d → R d be a C 2 bounded map with bounded derivative. Then for any x ∈ R d , the error term (e t (x, •)) t∈[0,T ] satisfies a limit theorem for the strong convergence in law :

( -3/4 e t (x, •)) t∈[0,T ] Lµ, . ∞ → →0 (y t (x)) t∈[0,T ] ,
where L µ denotes the strong convergence in law for the measure µ and where (y t (x)) t≥0 is the random process given by

(y t (x)) t∈[0,T ] := t 0 a(w s (x))dB L s (0) (12)
+ t 0 DF (w s (x))B s 0 a(wu(x))dL u (0) exp t s DF (w u (x))du ds t∈[0,T ] , (13)
where (B t ) t≥0 is a standard Brownian motion (with variance 1) and (L s ) s≥0 is the local time associated to a Brownian motion (B t ) t≥0 with variance Σ (where

Σ = k∈Z E μ(φ • T |k| φ) is
determined in Hypothesis ??), independent of (B t ) t≥0 , and where a :

x ∈ R d → a(x) is the map defined as a(x) := 1 2 l∈Z I µ F i (x, T |l| •)F j (x, •) + F j (x, T |l| •)F i (x, •) i,j=1,...,d (14)
where I µ denotes the integral with respect to µ, i.e.

∀g ∈ L 1 µ , I µ (g) := M gdµ.
The map t → t 0 a(w s (x))dL s (0) derived from this theorem corresponds to a perturbed variance closed to the one obtained in averaging method for probability invariant dynamical systems. Indeed in the case where (M, T, µ) is the probability preserving Sinai billiard this variance is given by t 0 a(w s (x))ds (see[?], [?]). The local time (L s (0)) s≥0 is also expected since it is linked to the recurrence of S n and already appeared naturally in the central limit theorem in [?] which was proved for maps on M M × Z depending only on the second coordinate, f (w, l) := β(l).

4. Perturbed ergodic sums : proof of the limit theorem.

In order to ease its reading, we prove the statement of theorem ?? in dimension d = 1. The proof in higher dimension follows the same path and we will only mention the alternative quantities on which the techniques apply when necessary. We first prove the following intermediate lemma (stated in dimension d ≥ 1).

Convergence in law.

Proposition 4.1. Let S > 0, on a Z-extension (M, T, µ) and a map F : R d × M → R d both satisfying the same hypotheses as their counterpart in theorem ??, we define the variable ṽ t for t ∈ [0, S], x ∈ R d and ω ∈ M as

ṽ t (x, ω) := 1/4 t/ k=1 F (w k (x), T k ω).
When t is fixed, this family of random variables satisfies the following convergence in law with respect to the probability measure µ on M M × {0} ,

ṽ t (x, •) L µ → →0 N t 0 a(w s (x))dL s (0) 1/2
, with a(x) given by (??) and where N is a standard (variance 1) centered normal law and ((L s (u)) s≥0 ) u∈R is a continuous version of the local time of a Brownian motion (B t ) t≥0 of variance Σ (with Σ as defined in Hypotheses ??) independent of N .

In order to prove Proposition ?? we introduce some classical results about the regularity of the Transfer operator P of the factor ( M , T , μ) (stated in Lemma ??) and mixing properties of the infinite measure preserving dynamical system ( M , T , μ) (related to the mixing local limit theorem, see Proposition ??) based on the spectral gap satisfied by the perturbed operator P u (see Hypothesis ??). Lemma 4.2. Let A n := {g ∈ B, ∀x, y ∈ M , g(x) = g(y) whenever s(x, y) ≥ n}. There is C > 0 such that, for any bounded map g ∈ A n ,

P 2n (g•) L(B,B) ≤ C g ∞ .
Proof. Let f ∈ B, we first determine P 2n (gf ) ∞ : For some ∆ i,k we denote by l(•) ∈ B the map defined by l(w) = i whenever there is j ∈ N such that w ∈ ∆ i,j . The positivity of P implies that for x ∈ ∆ i,k ,

|P 2n (gf )(x)| ≤ |P 2n ( g ∞ |f |)(x)| ≤ g ∞ P 2n |f | ∞ e l(x) .

Since P 2n is uniformly bounded for any n ∈ N

There is a constant C > 0 such that

P 2n (gf ) ∞ ≤ C g ∞ f B . (15)
We now give an estimate of P 2n (gf ) lip . Let i, k ∈ N, for any x, y ∈ ∆ i,k , we denote by X a bijection between T -2n ({x}) and T -2n ({y}) such that for any z ∈ T -2n ({x}), s(z, X(z)) = 2n + s(x, y). Once this set, we have for any f ∈ B :

P 2n (gf )(x) -P 2n (gf )(y) = w∈ T -2n ({x}) e -2n-1 i=0 h( T i w) (gf )(w) -e -2n-1 i=0 h( T i X(w)) (gf )(X(w)) ≤ w∈ T -2n ({x}) (G w + H w ) , (16)
with

G w := e 2n-1 i=0 h( T i w) -e 2n-1 i=0 h( T i X(w)) g ∞ f ∞ e l(w) ≤ g ∞ f ∞ e -2n-1 i=0 h( T i w) 2n-1 i=0
h lip β 2n-i+s(x,y) e l(w) β s(x,y) , and

H w := e -2n-1 i=0 h( T i X(w)) g ∞ |f (w) -f (X(w))| ≤ e -2n-1 i=0 h( T i X(w)) g ∞ f lip e l(w) β s(x,y)+2n .
Thus, it follows from (??) and from the previous estimates of G w and H w that

P 2n (gf )(x) -P 2n (gf )(y) ≤ P 2n e l(•) ∞,∆ i,k β s(x,y) g ∞ β 1 -β h lip f ∞ + f lip .
Since e l(•) B ≤ 1 and (P n ) n∈N is a sequence of uniformly bounded operators, There is C 2 > 0 such that

P 2n (gf ) lip ≤ C 2 g ∞ f B .
This concludes the lemma. Proposition 4.3. Let 0 > 0. Let ( M , T , μ) be a probabilistic dynamical system with transfer operator P and B satisfying ??. For l ∈ Z and n ∈ N,

P n (1 {Sn=l} •) = 1 (2πΣn) 1/2 Π 0 (•) + G n,l (•) where G n,l (•) B = O (1+|l|) 1+ 0 n 1+ 0 2
, uniformly in the choice of l.

Proof. Notice that on M ,

1 {Sn=l} = π -π e iu(Sn-l) du.
Fix b such that β 0 > b > 0 where β 0 is fixed from Hypothesis ??, using the identity P n u (•) = P n u (e iuSn •) and a change of variable u = v n 1/2 ,

P n (1 {Sn=l} •) = 1 2π π -π P n (e iu(Sn-l) •)du = 1 2π b -b e -iul λ n u Π u (•)du + O(α n ) = 1 2πn 1/2 bn 1/2 -bn 1/2 e -iul/n 1/2 λ n u/n 1/2 Π 0 (•) + un -1/2 Π 0 (•) du + O(n -3/2 ), ( 17
)
where the approximations O(•) are uniforms in l for the operator norm . L(B,B) . We then use the following identity (??) from Feller's book [?]; for any A, B ∈ R and r ∈ N,

e A - r m=0 B m m! ≤ e A -e B + e B - B m m! ≤ e A∨B |A -B| + B r+1 (r + 1)! . ( 18
)
We approximate λ t as some exponential form,

λ n n -1/2 t = e -Σ 2 t 2 +nψ( t n 1/2 )
where ψ(t [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF] (0) and an error term ψ continuous with ψ(0) = 0. Then applying equation (??) with A := nψ(un -1/2 ), B := u 2 ψ 2 (un -1/2 ) and r = 2 :

) := t 2 ψ 2 (t) + t 4 ψ(t) has a polynomial part ψ 2 (t) := t 3! ψ (3) (0) + t 2 4! ψ
λ n un -1/2 -e -Σ 2 u 2 r m=0 u 2 ψ 2 (un -1/2 ) m m! = O e -Σu 2 4 u 4 ψ(un -1/2 ) n + u 9 n 3/2 .
And thus,

λ n un -1/2 = e -Σ 2 u 2 1 + u 3 n 1/2 ψ (3) (0) 3! + O e -Σu 2 4 Q(u)1/n .
Where Q is polynomial. We inject this equation into formula (??) and get the following expansion for the norm . L(B,B) ,

P n (1 {Sn=l} •) = 1 2πn 1/2 βn 1/2 -βn 1/2 e -iul/n 1/2 λ n u/n 1/2 Π 0 (•) + un -1/2 Π 0 (•) du + O(n -3/2 ) = 1 2πn 1/2 βn 1/2 -βn 1/2 1 - iul n 1/2 + O ul n 1+ 0 e -Σ 2 u 2 2 Π 0 (•) + u n 1/2 Π 0 (•) + u 3 n 1/2 ψ (3) (0) 3! Π 0 (•) du + O(n -3/2 ) = 1 2πn 1/2 βn 1/2 -βn 1/2 1 - iul n 1/2 Π 0 (•) + u n 1/2 Π 0 (•) + u 3 n 1/2 ψ (3) (0) 3! Π 0 (•) e -Σ 2 u 2 2 du + O (1 + |l|) 1+ 0 n 1+ 0 2 . Indeed, R |u| m e -(Σu) 2 2 du < ∞ for all m ∈ N and |e -ix -1 -ix| ≤ x 2 2 ≤ x 1+ 0 2 si x ≤ 1 |e -ix -1 -ix| ≤ (1 + |x|) ≤ |x| 1+ 0 si x > 1,
with O(•) uniform in l ∈ Z. The terms in 1 n 1/2 with symmetric factors make the integral null. Thus we conclude the lemma,

P n (1 {Sn=l} •) = 1 (2πΣn) 1/2 Π 0 (•) + G n,l (•) with G n,l (•) L(B,B) = O (1 + |l|) 1+ 0 n 1+ 0 2 .
Proof of Proposition ??. To prove Proposition ?? we use a criterion of convergence of moments.

Inside the proof of ?? we will state in Lemma ?? some approximation of key quantities into product that can be lifted to ( M , T , μ). Then the convenient mixing properties on ( M , T , μ) will allow us to decorrelate the different products involved and to treat them as i.i.d variable.

In order to ease further the reading, we also introduce the following notations all along the proof : Given X a subset of some vector space, we represent with thick writing a a vector of at most countable coordinate( a ∈ X N or a ∈ X N ), we denote by a v u the subvector of coordinates (a u , . . . , a v ) ∈ X v-u and by a q the sum of the coordinates up to q ∈ N, a q = q k=0 a k . We also state as convention that a -1 = 0. As stated before, Proposition ?? will be proved using a convergence of moment technique and applying Carleman's criterion from [?, Chapter VII, 3.14].Thus we investigate here an expression of the moment 5 of N -th order of ṽ t (x, •) for some fixed x ∈ R.

E µ ṽ t (x, •) N = E µ      1/4 t k=0 p∈Z F (w k (x), T k (w), p)1 {S k =p}   N    = N/4 0≤k 1 ,...,k N ≤ t/ p∈Z N E µ N i=1 F (w k i (x), T k i (•), p i )1 {S k i =p i } = N/4 N q=1 N∈(N * ) q ,N q =N C N A ,q,N (F ),
In the product we gather the indexes k i which are identical and we set C N the number of vectors k having N i coordinates of value i for i ∈ [|1, q|] and N ∈ {N ∈ N q , q i=1 N i = N }:

C N := |{φ ∈ [|1, q|] [|1,N |] , |φ -1 ({i})| = N i }|.
As for the quantity A ,q,N (F ), it is defined as

A ,q,N (F ) := 1≤n 1 <•••<nq≤ t/ a∈Z q E µ q i=1 F (w n i (x), T n i (•), a i ) N i 1 {Sn i =a i }
= l∈(N * ) q ,l q ≤ t/ a∈Z q Ξ ,q,l i ,N (F, w -1 l 1 (x), . . . , w -1 l q (x), a)

5 when the dimension of the space is d > 1 we look at the moment of u, ṽ t (x, •) for any u ∈ R d instead, we then replace the quantity ϕ(x, •) appearing in the moment by u, ϕ(x, •) through the rest of the proof with for F ∈ B, z ∈ R q and a ∈ Z q , Ξ ,q,l,i,N (F, z, a)

:= E µ     q j=1 F (z i , T l i (•), a i ) N i 1 {S l i =a i -a i-1 }   • T l i-1   .
In what follows, we introduce the quantity

k := (log( S )) 2
this quantity will allow us to approximate with arbitrary precision when → 0 the terms A ,q,N (F ) in the following Lemma ?? in order to place the quantities on ( M , T , μ) and use the decorrelation properties.

Lemma 4.4. For any x ∈ R d , a ∈ Z, let (g k (x, ., a)) the map defined by g n in equation (??) as an approximation of g := f (x, ., a) and satisfying Hypothesis ??. Then

sup z∈(R d ) q a∈Z q Ξ ,q,l,i,N (F, z, a) -Ξ ,q,l,i,N (g k , z, a) = →0 O   qβ (ln(1/ )) 2 0 sup x∈R a∈Z ( f (x, ., a) lip + f (x, ., a) ∞ ) N ( 19
)
with Ξ ,q,l,i,N (g k , z, a)

:= E μ q i=1 g k (z i , T l i (•), a i ) N i 1 {S l i =a i -a i-1 } • T k • T l i-1 . Thus, when k = (log( S )) 2 , lim →0 A ,q,N (f ) -Â ,q,N (g k ) = 0, (20)
where

 ,q,N (g k ) := l∈(N * ) q ,l q ≤ t/ a∈Z q E μ q i=1 g k (x i , T l i (•), a i ) N i 1 {S l i =a i -a i-1 } • T k • T l i-1 .
Proof. Thanks to the invariance of µ by T , we can compose by T k in the expectancy of A ,q,N (F )

and lift the quantity inside on (M 1 , T 1 , µ 1 ) using the projections π 1 and π 2 . Thus applying the formula (??), we get for any 1 ≤ i 0 ≤ q,

a∈Z q E µ 1 i 0 -1 i=1 F (x i , π 1 • T l i +k 1 (•), a i ) N i 1 {S l i =a i -a i-1 } • π 2 • T k 1 • T l i-1 1 F (x i 0 , T l i 0 +k 1 (•), p i 0 ) N i 0 -g k (x i 0 , π 2 • T l i 0 1 (•), a i 0 ) N i 0 )1 {S l i 0 =a i 0 -a i 0 -1 } • π 2 • T k 1 • T l i 0 -1 1 q i=i 0 +1 g k (x i , π 2 • T l i 1 (•), a i ) N i 1 {S l i =a i -a i-1 } • π 2 • T k 1 • T l i-1 ∞ ≤ β k 0 a∈Z F (x i 0 , ., a) lip l∈(N * ) q ,l q ≤ t/ a∈Z q i =i 0 f (x i , ., a i ) N i ∞ = O β k 0 N i=1 a∈Z ( f (x i , ., a) lip + f (x i , ., a) ∞ ) .
Then, by chain rule, we get the upper bound (??) we looked for and we deduce (??).

Thanks to Lemma ??, it remains to estimate the following quantity  ,q,N (g k ) =

1≤l 1 ,...,lq≤n ,l q ≤n b a 0 ,m, ,l,N (g k )

where the quantity b a 0 ,m, ,l,N (f ) is defined for any f ∈ B and l ∈ N q by b a 0 ,m, ,l,N (f ) := a 1 ,...,aq∈Z

E μ f q i=1 g k (w -1 (l i +m) (x), T l i (•), a i ) N i 1 {S l i =a i -a i-1 } • T k • T l i-1 .
We use some classical trick : since μ is T -invariant, then E μ(P (•)) = E μ(•), and we can apply P l q +k in the expectancy above and iterate the identity

P m (G • T m f ) = GP m (f ) to obtain the following, b a 0 ,m, ,l,N (f ) := E μ P k (g k (w -1 (l q +m) (x), ., a q ) Nq H k ,lq,aq-a q-1 ) • . . . •(g k (w -1 (l 1 +m) (x), ., a 1 ) N 1 H k ,l 1 ,a 1 -a 0 )(f ) , (21)
with

H k ,m,a (•) := P m (1 {Sm=a} • T k •)
for any m ≥ 2k . This latter term may be written as

H k ,m,a (•) = l∈Z 1 {S k =l} P m-k (1 {S m-k =a-l} P k (•)) = l∈Z l ∈Z 1 {S k =l} P m-2k 1 {S m-2k =a-l-l } P k (1 {S k =l } P k (•)) .
As in [?], we split H k ,m,a (•) as follows in order to apply the mixing property ?? : For any m ≥ 2k , we denote by Q m (•) the operator defined as

Q m,a (•) := P m (1 {Sm=a} •).
This operator can be split according to Proposition ?? into

Q m,a (•) := Q (0) m,a (•) + Q (1) m,a (•) where Q (0) m,a (•) = 1 (2πΣm) 1/2 Π 0 (•) and Q (1) m,a (•) = G m,a (•).
We also introduce the following operator

H (1) k ,m,a (•) := H k ,m,a if m ≤ 2k l∈Z 1 {S k =l} l ∈Z G m-2k ,a-l (P k (1 {S k =l } (P k •)) otherwise and when 2k ≤ m, H (0) k ,m,a (•) := l∈Z 1 {S k =l} Q (0) m-k ,a-l (P k (•)) = Q (0) m-k ,0 .
The last notations we introduce are the two following sets E(l) := {(e i ) 1≤i≤q , if e i = 0 then l i ≥ 2k } and G(e) := {l, i l i ≤ t/ , and e ∈ E(l)}, and we can now decompose further  ,q,N (g k ) as follows,  ,q,N (g k ) = 1≤l 1 ,...,lq≤n ,l q ≤n b a 0 ,m, ,l,N (g k ) = e∈{0,1} q l∈G(e) b e a 0 ,m, ,l,N (g k ),

where for each e = (e 1 , . . . , e q ) and f ∈ B, b e a 0 ,m, ,l,N (f ) := a 1 ,...,aq∈Z E μ P 2k (g k (w (l q +m) (x), ., a q ) Nq H (eq) k ,lq,aq-a q-1 ) • . . .

•(g k (w (l 1 +m) (x), ., a 1 ) N 1 H (e 1 ) k ,l 1 ,a 1 -a 0 )(f ) . (22)
In what follows, we look for the couples (N, e) that will provide some weight in the computation of the N th moment and get rid of the others. The term

Q (0) m,a (•) = 1 (2πΣn) 1/2 E μ(•)1 appearing in H (0)
k ,m,a (•) will enable us to apply decorrelation. Notice from the expression in (??) that for 0 ≤ i ≤ q : b (e,0,e') a 0 ,m, ,(l,l i ,l ),N (G) = b e a 0 ,m, ,l,N (G)b

(0,e ) 0,m+l i-1 , ,(l i ,l ),(N i ,N ) (1) = 1 (2πΣl i ) 1/2 b e a 0 ,m, ,l,N (G) a i ∈Z b e a i ,m+l i , ,l ,N g k (w (m+l i ) (x), ., a i ) N i . ( 23
)
Let e ∈ {0, 1} q , we denote by α 1 < • • • < α K the indexes such that e α i = 0, we fix the following conventions α K+1 = q + 1 and e q+1 = 0. Now we fix l ∈ G(e) and we decompose 1 n N/4 b e 0,0, ,l,N (1) in

N/4 b e a 0 ,0, ,l,N (1) = 1 (2πΣ) K/2 N 1 +•••+N α 1 -1 4 b (1,...,1) a 0 ,0, ,l α 1 -1 1 ,N α 1 -1 1 (1) × K i=1 1 l 1/2 α i N α i+1 -1 α i 4 a∈Z b (1,...,1) a,l α i , ,l α i+1 -1 α i +1 ,N α i+1 -1 α i +1 g k (w l α i (x) , ., a) Nα i . (24)
In what follows, we prove that the mass a∈Z b

(1,...,1)

a,l α i , ,l α i+1 -1 α i +1 ,N α i+1 -1 α i +1 (•) is negligible. Let K 1 , . . . , K r
be the indexes in [|α i + 1, . . . , α i+1 -1|] such that l K i ≥ 2k (with the convention K 0 := α i + 1 and K r+1 := α i+1 ) , and

B i,j,m (a j i-1 ) : = g k (w ( j k=i l k +m) (x), ., a j ) N j H k ,l j ,a j -a j-1 • . . . • g k (w (l i +m) (x), ., a i ) N i H k ,l i ,a i -a i-1 (•) . For h ∈ B, b (1,...,1) a,m+l α i , ,l α i+1 -1 α i +1 ,N α i+1 -1 α i +1 (h) = a α i+1 -1 α i +1 ∈Z α i+1 -1-α i -1 E μ P 2k B Kr+1,K r+1 -1,l Kr (a K r+1 -1 Kr )• l∈Z 1 {S k =l} g k (w (l Kr +m) (x), ., a Kr ) N Kr l ∈Z Q (1) l Kr -2k ,a Kr -a Kr -1 -l-l • P k (1 {S k =l } P k (•))) • . . . • B K 1 +1,K 2 -1,l K 1 (a K 2 -1 K 1 ) l∈Z 1 {S k =l} g k (w (l K 1 +m) (x), ., a K 1 ) N K 1 l ∈Z Q (1) l K 1 -k ,a K 1 -a K 1 -1 -l-l P k (1 {S k =l } P k (•)) • B K 0 ,K 1 -1,m+l α i (a, a K 1 -1 K 0 )(h)
Applying the finite horizon hypothesis in the sense that there is D > 0 such that φ ∞ ≤ D, and we deduced that 1 {S k =l} = 0 for any |l| > Dk , we can rewrite the above quantity as follows

a α i+1 -1 α i +1 ∈Z α i+1 -1-α i -1 l K 1 ...,l K r+1 ∈Z E μ Q k ,l K r+1 • P k • B Kr+1,K r+1 -1,l Kr (a K r+1 -1 Kr )•   |l|≤Dk 1 {S k =l } g k (w (l Kr +m) (x), ., a Kr ) N Kr Q (1) l Kr -2k ,a Kr -a Kr -1 -l-l Kr     • . . . • Q k ,l K 2 • P k • B K 1 +1,K 2 -1,l K 1 (a K 2 -1 K 1 )•   |l|≤Dk 1 {S k =l } g k (w (l K 1 +m) (x), ., a K 1 ) N K 1 Q (1) l K 1 -2k ,a K 1 -a K 1 -1 -l-l K 1     •Q k ,l K 1 • P k • B K 0 ,K 1 -1,m+l α i (a, a K 1 -1 K 0 )(h) , (25)
As in (??), we can use the identity P r (G

• T r f ) = GP r (f ) to express Q k ,l K i+1 • P k • B K i +1,K i+1 -1,l K i (a K i+1 -1 K i )(1 {S k =l} g k (w (l K i +m) (x), ., a K i ) N K i •) as Q k ,l K i+1 • P k •B K i +1,K i+1 -1,l K i (a K i+1 -1 K i )(1 {S k =l} g k (w (l K i +m) (x), ., a K i ) N K i •) := (26)
P 2k + K i+1 u=K i +1 lu h ,m (a K i+1 -1 K i ). , (27)
with h ,m (a

K i+1 -1 K i
) defined as follows h ,m (a

K i+1 -1 K i ) := 1 {S k =l K i+1 } • T k +l K i+1 K i +1 K i+1 -1 j=K i g k (w -1 (l j +m) (x), T l j -l j-1 (•), a j ) N j 1 {S l j =a j -a j-1 } • T k N j • T l j-1 K i +1 1 {S k =l} g k (w -1 (l K i +m) (x), ., a K i ) N K i .
The map h ,m (a

K i+1 K i ) is an element of B such that for any x, y ∈ M , s(x, y) ≥ 2k + K i+1 u=K i +1 l u ⇒ h ,m (a K i+1 K i )(x) = h ,m (a K i+1 K i )(y)
and for any , m and a K i ∈ Z, h ,m (a

K i+1 K i ) ∞ ≤ K i+1 j=K i g k (w (l j +m) (x), ., a j ) N j ∞ . (28)
We apply Lemma ?? on equation (??), and using Proposition ??, along with equation (??) we deduce the following estimate of the terms in formula (??):

l∈Z l K i+1 ∈Z Q k ,l K i+1 -1 • P k • B K i +1,K i+1 -1,l K i (a K i+1 -1 K i )• l∈Z 1 {S k =l} g k (w (l K i +m) (x), ., a K i ) N K i Q (1) l K i -2k ,a K i -a K i -1 -l-l K i L(B,B) (29)
≤ |l|≤Dk |l K i+1 |≤Dk K i+1 j=K i g k (w (l j +m) (x), ., a j ) N j ∞ O |a K i -a K i -1 -l -l K i | 1+ 0 |l K i -2k | 1+ 0 2 .
It follows that

|(??)| ≤ (2Dk ) 2 K i+1 j=K i g k (w (l j +m) (x), ., a j ) N j ∞ O (||a K i | + 1|||a K i -1 | + 1||1 + k | 2 ) 1+ 0 |l K i -2k | 1+ 0 2 .
We now estimate equation (??) using the above majoration as well and the convergence (by assumption) of the sum a∈Z (1 + |a|) 2(1+ 0 ) g k ∞ :

a∈Z l∈G α i +1,α i+1 -1 (e) b (1,...,1) a,m+l α i , ,l α i+1 -1 α i +1 ,N α i+1 -1 α i +1 ( l∈Z Q k ,l (g k (w l α i (x) , ., a) Nα i )) B ≤ k q (2D(1 + k )) r(2+1+ 0 ) sup x∈R a α i+1 -1 α i +1 ∈Z α i+1 -2-α i α i+1 -1 u=α i +1 (1 + |a u |) 2(1+ 0 ) g k (x, ., a u ) Nu ∞ k ≤l K 1 ...,l K r+1 O( Kr i=K 1 (l i -2k ) -(1+ 0 2 ) ) = O(k q+r(2+1+ 0 ) ). ( 30
)
Where G i,j (e) := {(l j i ), l ∈ G(e)}. Thus for any 1

≤ i ≤ K, N α i+1 -1 α i 4 l∈G α i +1,α i+1 -1 (e) a∈Z b (1,...,1) a,m+l α i , ,l α i+1 -1 α i +1 ,N α i+1 -1 α i +1 ( l∈Z Q k ,l (g k (w l α i (x) , ., a) Nα i )) B = O(k q+r(2+1+ 0 ) N α i+1 -1 α i 4 - 1
2 ).

The latter converges to 0 with rate in O( 1/8 ) whenever N

α i+1 -1 α i ≥ 3 and is in O(k q+r(2+1+ 0 ) ) when N α i+1 -1 α i = 2.
finally, in the special case where N α i+1 -1 α i = 1 (equivalently α i+1 = 1 + α i and N α i = 1), we use the fact that g k has by assumption zero mean. b (0,0,e) a,m, ,(lα i ,l ,l),(1,N 2 ,N) (G) = E μ g k (w (lα i +m) (x), ., a q )E μ(. . . ) = 0. Thus, summing formula (??) over all possible elements l,

1 q/4 l q 1 ,l q ≤ t/
|b e a 0 ,0, ,l,N (1)|

L 1 μ → →0 0, (31)
as soon as there is an index 1 ≤ i ≤ K such that N α i+1 -1 α i ≥ 3. The couples (e, N) such that q/4 l 1 ,...,lq,l q ≤ t/ b e a 0 ,0, ,l,N (1) is not negligible are thus those satisfying for any 1 ≤ i ≤ K (with K = N 2 ), whether

2 + α i = α i+1 and N α i = N α i+1 = 1, (32)
or

1 + α i = α i+1 and N α i = 2. ( 33
)
We call a couple satisfying one of these two relations an admissible couple. For such a couple (e, N), we denote by J 2 the set of indexes i satisfying relation (??) and J 1 those satisfying relation (??). For l ∈ G(e), b e a 0 ,0, ,l,N (1) can be decomposed by applying (??) while noticing that the terms b (0,e) 0,l i-1 , ,(l i ,l i+1 ), [START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF][START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF] are independent of the first index a 0 (since

Q (0) m,a (•) does not depend of a), b e 0,0, ,l,N (1) =   i∈J 2 b (0,1)
0,l i-1 , ,(l i ,l i+1 ), [START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF][START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF] (1)

    i∈J 1 b (0) a 0 ,l i-1 , ,(l i ),(2) (1)   ,
with the convention l 0 = 0. We recall that g k has zero mean (by Hypothesis ??), thus

Q (0)
l i+1 ,a (g k ) = 0 and we can apply the following relation :

b (0,1) 0,l i-1 , ,(l i ,l i+1 ),(1,1) (1) = 1 (2πΣl i ) 1/2 a 1 ∈Z b 0,l i , ,(l i+1 ),(1) (g k (w l i (x), ., a 1 )) = l i+1 ≤ t/ 1 (2πΣl i ) 1/2 a 1 ,a 2 ∈Z E μ g k (w l i (x), ., a 1 )1 {S l i+1 =a 1 -a 2 } • T k g k (w l i+1 (x), T l i+1 (•), a 2 ) .
To gather the last terms b e 0,0, ,l,N (1) and make a direct computation of the moments, we pull back the expression onto the dynamical system (M, T, µ) using the following lemma : Lemma 4.5. For any

x 1 , x 2 ∈ R d , I µ F (x 1 , T l (•))F (x 2 , •) = O(l -(1+ 0 /4) )
uniformly in x 1 and x 2 . In particular,

σ 2 (f )(x 1 , x 2 ) := l∈Z I µ F (x 1 , T |l| (•))F (x 2 , •) converges and lim →0 0≤l≤ t/ a 1 ,a 2 ∈Z E μ g k (w (l i ) (x 1 ), T l (•), a 1 )1 {S l =a 1 -a 2 } • T k g k (w (l i +l) (x 2 ), ., a 2 ) = lim →0 l≥0 I µ (F (w (l i +l) (x 1 ), T l (•))F (w (l i ) (x 2 ), •)). ( 34
)
Proof. To prove the first point, we can notice that the proof of Lemma ?? actually states that for any l ∈ N * sup

x 1 ,x 2 ∈R I µ F (x 1 , T l (•))F (x 2 , •) - a 1 ,a 2 ∈Z E μ g k (x 1 , T l (•), a 1 )1 {S l =a 1 -a 2 } • T k g k (x 2 , ., a 2 ) = O(l 2 β k 0 f 2 ∞
). Instead of approximating f by g k in the above formula, we approximate it by g k l with k l := log(l) 2 , then the difference above is summable and finally we only need to check the following convergence to ensure the summability of sup

x 1 ,x 2 ∈R I µ F (x 1 , T l (•))F (x 2 , •) : lim →0 l≤ t/ sup x 1 ,x 2 ∈R a 1 ,a 2 ∈Z E μ g k l (x 1 , T l (•), a 1 )1 {S l =a 1 -a 2 } • T k l g k l (x 2 , ., a 2 ) < ∞. ( 35
)
In addition the rate of convergence of the terms in the above sum will provide the rate of convergence of the terms I µ F (x 1 , T l (•))F (x 2 , •) . We thus apply lemmas ?? and ?? page ?? to estimate the terms in the sum above: for any

x 1 , x 2 ∈ R, a 1 ,a 2 ∈Z E μ g k l (x 1 , T l (•), a 1 )1 {S l =a 1 -a 2 } • T k l g k l (x 2 , ., a 2 ) = a 1 ,a 2 ∈Z E μ u∈Z v∈Z Q k l ,u g k l (x 1 , ., a 1 )Q l-2k l ,a 2 -a 1 -u-v Q k l ,v P k l g k l (x 2 , ., a 2 ) (36) = a 1 ,a 2 ∈Z E μ u∈Z v∈Z P k l Q k l ,u (g k l (x 1 , ., a 1 )G l-2k l ,a 2 -a 1 -u-v Q k l ,v P k l g k l (x 2 , ., a 2 ) (37) ≤ a 1 ,a 2 ∈Z u∈Z |v|≤Dk l P k l Q k l ,u (g k l (x 1 , ., a 1 )•) L(B,B) G l-2k l ,a 2 -a 1 -u-v L(B,B) Q k l ,v P k l (g k l (x 2 , ., a 2 )) B ≤ (Dk l ) 2 sup x 1 ,x 2 ∈R a 1 ,a 2 ∈Z F (x 1 , ., a 2 ) ∞ F (x 2 , ., a 2 ) B O ((|a 2 | + 1)(|a 1 | + 1)(2k l )) 1+ 0 (l -2k l ) 1+ 0 2 = O (k l ) 3+ 0 (l -2k l ) 1+ 0 2 .
Where G n,l is the operator defined in Proposition ??. and we went from line (??) to (??) by noticing that

a 2 ∈Z Q (0) l-k l ,a 2 -a 1 -u P k l g k l (x 2 , ., a 2 ) = a 2 ∈Z Q (0) l-k l ,0 P k l g k l (x 2 , ., a 2 ) = 0,
and the majoration at the last line is done using the Proposition ?? on spectral gap for perturbed operator and the assumption on the decay rate of F from the Proposition ?? we want to prove :

sup x∈R a∈Z (1 + |a|) 2(1+ 0 ) F (x, ., a) ∞ < ∞.
Thus the convergence in formula (??) holds and we can conclude the first point of our lemma: sup

x 1 ,x 2 ∈R I µ F (x 1 , T l (•))F (x 2 , •) = O(l -(1+ 0 /4) ) and l∈Z sup x 1 ,x 2 ∈R I µ F (x 1 , T |l| (•))F (x 2 , •) < ∞.
The above convergence along with the convergence stated in Lemma ?? then provides the conslusion of the second point of the lemma: the convergence in formula (??).

From Lemma ?? we deduce that for (e, N) satisfying (??) or (??), lim

→0 N/4 l∈G(e)
b e a 0 ,0, ,l,N (1) = lim

→0 N/4 l∈G(e)   i∈J 2 b (0,1)
0,l i-1 , ,(l i ,l i+1 ), [START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF][START_REF] Anosov | Averaging in systems of ordinary differential equations with rapidly oscillating solutions[END_REF] (1)

    i∈J 1 b (0)
a 0 ,l i-1 , ,(l i ), [START_REF] Arnold | Chapitres supplémentaires de la théorie des équations différentielles ordinaires[END_REF] (1)

  = lim →0 N/4 (2πΣ) N/4 l∈G(e) K i=1 1 l 1/2 α i   i∈J 2 I µ F (w l i (x), •)F (w l i+1 (x), T l i+1 (•)     i∈J 1 I µ (F (w l i (x), •) 2 )   = lim →0 N/4 (2πΣ) N/4 1≤l 1 ,...,l K ,l K ≤ t/ K i=1 1 l 1/2 i (38)   i∈J 2 ∞ l=1 I µ F (w l i (x), •)F (w l i (x), T l (•)     i∈J 1 I µ (F (w l i (x), •) 2 )   , (39)
where (??) is obtained by noticing that Lemma ?? allows us to neglect some perturbations in the first coordinate of f : Indeed, fix the parameter α > 0 such that α

→ →0 ∞ and α = →0 o 1 ;
first one can split G(e) = A ∪ i∈J 2 A i+1 between a central part A := {l ∈ G(e), j,j-1 / ∈J 2 l j ≤ t/ -Kα } and marginal ones composed by A i := {l ∈ G(e), t/ -α ≤ j =i l j ≤ t/ }. Thanks to Lemma ??, the terms in the sum in (??) may be estimated from above by the term

K i=1 1 l 1/2 α i   i∈J 2 l -(1+ 0 4 ) i+1     i∈J 1 I µ (F (w l i (x), •) 2 )   . ( 40
)
The convergence of Riemann sums then implies that the sum over each marginal term

A i for i -1 ∈ J 2 N/4 l∈A i K j=1 1 l 1/2 α j j∈J 2 l -(1+ 0
4 ) j+1 tends to 0. We are left with the sum over the central part A which provides the line (??) through the following reasoning : indeed, in one hand

(1/ ) 1/4 ≤l≤ t/ I µ F (w l i (x), •)F (w l i +l (x), T l (•)) converges to 0 (uniformly in l i ) and on the other hand, since s → F (w s (x), •) is uniformly Lipschitz, there is C F > 0 such that l≤(1/ ) 1/4 I µ F (w l i (x), •)F (w l i +l (x), T l (•) -I µ F (w l i (x), •)F (w l i (x), T l (•) ≤ l≤(1/ ) 1/4 I µ |F (w l i (x), •)| C F l = O( 1/2 ),
uniformly in l. Thus, we can substitute (??)

1≤l i+1 ≤ t/ -l i I µ F (w l i (x), •)F (w l i+1 (x), T l i+1 (•)) by 1≤l i+1 ≤ t/ -l i I µ F (w l i (x), •)F (w l i+1 (x), T l i+1 (•))
and since l ∈ A, t/ -l i ≥ α → ∞ and thus it can be replaced by its limit

∞ l=1 I µ F (w l i (x), •)F (w l i (x), T l (•)) .
We introduce the following map G corresponding to a continuous version of the terms in line (??) with respect to u ∈ R K + instead of l ∈ N K . Lemma 4.6. The map G described a follows is uniformly continuous on

R K + G(u) :=   i∈J 2 ∞ l=1 I µ F (w i k=1 u k (x), .)F (w i k=1 u k (x), T l (.)     i∈J 1 I µ (F (w i k=1 u k (x), .) 2 )   .
Proof. Since the map w is Lipschitz, it is enough to prove that x ∈ R d → ∞ l=1 I µ F (x, .)F (x, T l (.)) is uniformly continuous. This is a consequence of the uniform convergence of I µ F (x, .)F (x, T l (.) in O(l -(1+ 0 /4) ) in Lemma ?? and the Lipschitz regularity of F (., .) :

For any compact K, any N ∈ N * and any x, y ∈ K,

∞ l=1 I µ F (x, .)F (x, T l (.)) - ∞ l=1 I µ F (y, .)F (y, T l (.)) ≤ C 1 |l|≥N |l| -1+ 0 4 + CN x -y .
The latter inequation depends only on x -y thus we deduce the uniform continuity Notice that for any u, v ≥ ,

| 1 u -1 v | = u-v uv ≤ 1 2
|u -v|, thus the following Riemann sums converge :

lim →0 K/2 l∈(N * ) K ,l K ≤ t/ K i=1 1 l 1/2 i   i∈J 2 ∞ l=1 I µ F (w l i (x), •)F (w l i (x), T l (•)     i∈J 1 I µ (F (w l i (x), •) 2 )   = lim →∞ K l,l K ≤ t/ l 1 -1<u 1 ≤l 1 • • • l K -1<u K ≤l K K i=1 1 ( u i ) 1/2 G( u 1 , . . . , u K )du = lim →∞ 0<u 1 ,...,u K ≤ t/ , i u i / ≤ t/ K i=1 1 ( u i / ) 1/2 G( u 1 / , . . . , u K / )du (41) = 0<u 1 ,...,u K , i u i ≤t K i=1 1 u 1/2 i G(u 1 , . . . , u n )du 1 , . . . , du K . (42)
The last line is obtained through dominated convergence of the term (??) which is upper bounded by C K i=1

1 {0<u i ≤t+1} u 1/2 i
, where C > 0 comes from the boundedness of G(•) as a continuous function on a compact. We denote by C(N, q) the set of admissible couple (N, e) of length q such that N q = N . When N is odd, we notice that

C(N, q) = ∅ thus lim →0 E µ v t (x, •) N = 0.
We consider now the case where the integer N is even,

lim →0 E µ v t (x, •) N = lim →0 N/4 N q=1 N i ≥1, q i=1 N i =N C N A ,q,N (F ) = lim →0 N/4 N q=1 (e,N)∈C(N,q) C N l 1 ,...,lq,l q ≤ t/ b e a 0 ,0, ,l,N (1), (43)
and so

lim →0 E µ v t (x, •) N = N q=1 (N,e)∈C(N,q) C N 1 (2πΣ) N/4 0≤u 1 ,...,u K ≤1, i u i ≤1 K i=1 1 u 1/2 i   i∈J 2 (N) ∞ l=1 I µ F (w i k=1 u k (x), •)F (w i k=1 u k (x), T l (•)     i∈J 1 (N) I µ (F (w i k=1 u k (x), •) 2 )   du.
In that case q, N ∈ C(q), C N et J 1 (N) Notice that given N , the set J 2 (N) determines entirely the quantities q, (N, e) ∈ C(N, q) and J 1 (N) :

N/2 = |J 1 | + |J 2 |, q = |J 1 | + 2|J 2 | and ψ : (N, e) ∈ C(N, q) → J 2 (N) is injective.
Notice also that by recurrence on the cardinality of

J 2 (N), C N = 2 |J 2 | N ! 2 N/2 . lim →0 E µ ṽ t (x, •) N = J 2 ∈P([|1;N/2|]) (2πΣ) -N/4 N ! 2 N/2 (N/2)! 2 |J 2 | (N/2)! 0≤u 1 ,...,u N/2 , i u i ≤t   N/2 i=1 1 u 1/2 i     i∈J 2 (N) ∞ l=1 I µ F (w i k=1 u k (x), •)F (w i k=1 u k (x), T l (•)     i∈J 1 (N) I µ (F (w i k=1 u k (x), •) 2 )   du = (2πΣ) -N/4 N ! 2 N/2 (N/2)! J 2 (N)∈P([|1;N/2|]) (N/2)! 0≤u 1 ,...,u N/2 , i u i ≤t   N/2 i=1 1 u 1/2 i     i∈J 2 (N) 2 ∞ l=1 I µ F (w i k=1 u k (x), •)F (w i k=1 u k (x), T l (•)     i∈J 1 (N) I µ (F (w i k=1 u k (x), •) 2 )   du = N ! 2 N/2 (N/2)! (2πΣ) -N/4 (N/2)! 0≤u 1 ,...,u N/2 , i u i ≤t   N/2 i=1 1 u 1/2 i   l∈Z I µ F (w i k=1 u k (x), •)F (w i k=1 u k (x), T |l| (•) du. = N ! 2 N/2 (N/2)! (2πΣ) -N/4 (N/2)! u∈R N 2 + , i u i ≤t   N/2 i=1 1 u 1/2 i   l∈Z I µ F (w i k=1 u k (x), •)F (w i k=1 u k (x), T |l| (•) du
The term in N ! 2 N/2 (N/2)! match with the moment of order N/2 of a standard normal law (which is the N 2 th moment of B 1 for (B t ) t≥0 Brownian motion of law of variance 1). The second factor can be identified using [?, proposition B.1.17] with the moment of order N of the variable t 0 a(w s (x))dL s (0)

1/2
with a(x) given by (??) and (L s ) s≥0 a local time process (L s (0)) t≥0 of a Brownian motion (B t ) t≥0 of variance Σ. The limit law is thus the law of a Gaussian vector conditionally to the local time process (L s (0)) t≥0 (of a Brownian motion (B t ) t≥0 of variance Σ ) [START_REF] Dolgopyat | Limit theorems for partially hyperbolic systems[END_REF] . We now check that this limit law corresponds to the law of t 0 a(w s (x))dB L s (0) [START_REF] Dolgopyat | Limit theorems for partially hyperbolic systems[END_REF] with variance t 0 u, a(ws(x))u dL s (0)

1/2
when treating the dimension d > 1 from the conclusion of the theorem ?? we want to prove 7 . To do so we only need to prove that t 0 a(w s (x))dB L s (0) is a Gaussian vector conditionally to (L s (0)) t≥0 and compute its variance. We fix (L s (0)) s≥0 := (h(s)) s≥0 , since B is independent of that process, its law is still driven by the law of the process (B h(s) ) s≥0 which is then a martingale with quadratic variation B h(s) = h(s). Thus Dubins-Schwarz relation [?] ensures that the law of the martingale t 0 a(w s (x))dB L s (0) t≥0 matches with the law of the process β t 0 a(ws)df (s) t≥0

with β a standard Brownian motion. In particular, the law of the random variable t 0 a(w s (x))dB L s (0)

corresponds to the Gaussian law of β t 0 a(ws)df (s) whose variance is t 0 a(w s )df (s)

1/2
. Thus we deduce the convergence of moments, for any N ∈ N :

lim →0 E µ v t (x, •) N = E   N t 0 a(w s (x))dL s (0) 1/2 N   .
We deduce from the convergence above of moments of v t (x, •) to those of N t 0 a(w s (x))dL s (0)

1/2
the corresponding convergence in law using Carleman's criterion (see [?]) whose assumptions are checked in the following Lemma ??. Proof. We deduce the moments of order 2N from the independence between the process (L s ) s≥0 and the gaussian law N :

E   N t 0 a(w s (x))dL s (0) 1/2 N   = E N 2N E t 0 a(w s (x))dL s (0) N ≤ (2N )! 2 N N ! a(•) ∞,w [0,T ] (x) E(L N T ). ( 44
)
Stirling formula states that (2N

)! = O (2N ) 2N +1 e 2N
and thus ((2N )!) N is in order O(N ) and thus its inverse is the general term of a divergent series.

5.

Proof of theorem ?? : finite dimensional distributions.

We now focus on proving the following lemma consisting on the convergence of the finite dimensional distribution (f.d.d) in the conclusion of theorem ??. Lemma 5.1. For any x ∈ R, the process (v t (x, •)) t∈[0,T ] converges in law for the f.d.d (with respect to measure µ) to the process t 0 a(w s (x))dB L s (0) t∈[0,S] ,

7 when d > 1, we check in the same way that conditionally to the process (L s (0)) s≥0 , the marginals of u, t 0 a(ws(x))dB L s (0) are Gaussian and of variance t 0 u, a(ws(x))u dL s (0).

Proof. Let x ∈ R, It is enough to prove the convergence of (ṽ t (x, •)) t∈[0,S] . Indeed since F is uniformly Lipschitz and bounded in its first coordinate and (w s (x)) s is a Lipschitz map,

sup t∈[0,T ] v t (x, •) -ṽ t (x, •) L 1 µ = 1/4 sup t∈[0,T ] t/ k=1 k s=k-1 f (w k (x), T k •) -f (w s (x), T s •)ds + t/ t/ f (w s (x), T s ω)ds L 1 µ ≤ 1/4 (T C f + f ∞ ). Thus (v t (x, •)) t∈[0,S] converges in law iff (ṽ t (x, •)) t∈[0,S] converges. Let 0 = t 0 < t 1 < • • • < t p ∈ [0, S]
, by linear combinations, it is enough to prove that 1/4 ṽ t i (x, •) -1/4 ṽ t i-1 (x, •) p i=1 converges to

t i t i-1 a(w s (x))dB (i) L s (0) i=1,...,p L = t i 0 a(w s (x))dB L s (0) - t i-1 0 a(w s (x))dB L s (0) i=1,...,p
where the random processes B (i) are mutually independent standard motions independent from (B t ) t∈[0,T ] .

Carleman criterion applies to these random variable (see Lemma ?? page ??) thus we only need to check the convergence of the moments of 1/4 ṽ t i (x, •) -

ṽ t i (x, •) -ṽ t i-1 (x, •) M i = E   p i=1 t i t i-1 a(w s (x))dB (i) L s (0) M i   .
Proof. The term E µ p i=1 ṽ t i (x, •) -ṽ t i-1 (x, •) M i can be expanded as follows,

E µ p i=1 ṽ t i (x, •) -ṽ t i-1 (x, •) M i = M 1 q 1 =1 • • • Mp qp=1 1≤N 1 ,...,Nq p ,M i = q i j=q i-1 +1 N j p i=1
C (N q j-1 +1 ,...,Nq j ) A ,q,N ;

(45) with q i := i j=1 q j , as in Proposition ?? and

A ,q,N (F ) = 1≤n 1 <•••<nq 1 ≤ t 1 / • • • t p-1 / <n q p-1 +1 <•••<nq p ≤ tp/ a∈Z q p E µ   q p i=1 F (w n i (x), T n i (•), a i ) N i 1 {Sn i =a i }   = l∈E q (t 1 ,...,tp)
b 0,0, ,l,N (1), [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF] When d > 1 the convergence of moment it is enough to prove the limit in Lemma ?? identity for lim →0 Eµ p i=1 θi, ṽ t i (x, •) -

ṽ t i-1 (x, •) M i = E p i=1 θi, t i t i-1
a(ws(x))dB

(i) L s (0) M i for any θi ∈ R d
where [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF] for any map f satisfying the same assumptions as the map F in Theorem ??, b a 0 ,m, ,l,N (f ) :=

a∈Z q p E µ   f   q p i=1 f (w -1 (l i +m) (x), T l i (•), a i ) N i 1 { Sl i =a i -a i-1 } • T k   • T l i-1   ,
and E q (t 1 , . . . , t p ) := {l, l i ≥ 1 : t j-1 / < l i ≤ t j / , ∀i, 1 + q j-1 ≤ i ≤ q j , ∀j, 1 ≤ j ≤ p}.

We denote N := p i=1 M i . According to equation (??), the admissible couples (N, e) are among those satisfying relations (??) or (??) at page ??. So for some fixed e, the set C(N, p i=1 q i ) (introduced page ??) of admissible couples satisfies the following inclusion, C(N,

p i=1 q i ) = C(M 1 , q 1 ) × • • • × C(M p , q p ) ∪ p-1
j=2 A j with A j := {(N, e) ∈ C(N, p i=1 q i ), N q j = N 1+q j = 1, e q j = 0, e q j +1 = 1}. We now prove that for any couple (N, e) ∈ A j , N/4 A ,q,N (F ) is negligible (i.e A j makes no contribution). Fix j ∈ {2, . . . , p -1} and (N, e) ∈ A j , then according to equation (??) where we introduced for k ∈ {1, . . . , N 2 } notation α k to represent the indexes such that e α k = 0, N/4 l∈E q (t 1 ,...,tp)

|b e 0,0, ,l,N (1)| ≤ N/4 l∈E q (t 1 ,...,tp)

  N/2 k=1 l -1/2 α k   (46)   i∈J 2 (N) I µ F (w l i (x), •)F (w l i+1 (x), T l i+1 (•)   (47)   i∈J 1 (N) I µ F (w l i (x), •) 2   (48)
From Lemma ?? page ?? we deduce that the terms (??), (??) and (??) are bounded. Notice that by definition of A j , q j ∈ J 2 (N) and l q j ≤ t j / < l 1+q j . Thus, having e 1+q j = 1 means that either l 1+q j ≥ k and then the marginal sum l 1+q j ≥k I µ F (w l i (x), •)F (w l i+1 (x), T l i+1 (•) tends to 0 when tends to 0 and so does N/4 A ,q,N (F ), or t j / -l q j ≤ k in which case the upper estimate of the terms given in formula (??) applies to the terms in the marginal sums over the elements l ∈ E q (t 1 , . . . , t p ) such that t j / -l q j ≤ k in (??). Then these marginal sums converge as a Riemann approximations to 0. Thus the term N/4 A ,q,N (F ) is negligible and the contribution of the couple (N, e) ∈ A j is negligible; the set of admissible couple C(N, p i=1 q i ) can be identified with C(M 1 , q 1 ) × • • • × C(M p , q p ). In particular, if there is i ≤ p such that M i is odd then (??) tends to 0 which is consistent with the fact that

E   p i=1 t i t i-1 a(w s (x))dB (i) L s (0) M i   = 0.
We recall that (B (i) t ) t≥0 i∈N are mutually independent and independent of (L t ) t≥0 , with symmetric law.We suppose now that M 1 , . . . , M p are all even and choose (N, e) ∈ C(M 1 , q 1 ) × • • • × C(M p , q p ). Equation (??) page ?? implies, lim

→0 N/4 A ,q,N (F ) = lim →0 N/4   p j=1 1 (2πΣ) M j /4   l∈ Ẽq (t 1 ,...,tp)   N/2 i=1 1 l 1/2 i     i∈J 2 (N) ∞ l=1 I µ F (w l i (x), •)F (w l i (x), T l (•)     i∈J 1 (N) I µ (F (w l i (x), •) 2 )   =   p j=1 1 (2πΣ) M j /4   F M 1 ,...,Mp   N/2 i=1 1 u 1/2 i     i∈J 2 (N) ∞ l=1 I µ F (w t j-1 + i k=1 u k (x), •)F (w t j-1 + i k=1 u k (x), T l (•)     i∈J 1 (N) I µ (F (w t j-1 + i k=1 u k (x), •) 2 )   du,
Where Ẽq (t 1 , . . . , t p ) :

= {1 ≤ l 1 , . . . , l M 1 +•••+Mp 2 : t j-1 / < l k ≤ t j / , ∀k, M 1 +•••+M j-1 2 + 1 ≤ k ≤ M 1 +•••+M j 2 } and F M 1 ,...,Mp := {1 ≤ u 1 , . . . , u M 1 +•••+Mp 2 : t j-1 / < u k ≤ t j / , ∀k, M 1 +•••+M j-1 2 + 1 ≤ k ≤ M 1 +•••+M j 2 }.
We introduce the notation C (M, q) for the non negligible admissible couples,

C (M, q) := C(M 1 , q 1 ) × • • • × C(M p , q p ), then relation (??) sums up to lim →0 E µ p i=1 ṽ t i (x, •) -ṽ t i-1 (x, •) M i = M 1 q 1 =1 • • • Mp qp=1 (N,e)∈C (M,q) p i=1 C (N q i-1 +1 ,...,Nq i ) F M 1 ,...,Mp 1 (2πΣ) M i /2   N/2 i=1 1 u 1/2 i     i∈J 2 (N) ∞ l=1 I µ F (w t j-1 + i k=1 u k (x), •)F (w t j-1 + i k=1 u k (x), T l (•)     i∈J 1 (N) I µ (F (w t j-1 + i k=1 u k (x), •) 2 )   du.
The same reasoning from formulas (??) page ??, gives still with the even quantity N :

= p i=1 M i , lim →0 E µ p i=1 ṽ t i (x, •) -ṽ t i-1 (x, •) M i = M 1 q 1 =1 • • • Mp qp=1 p i=1 M i ! 2 M i /2 (M i /2)!(2πΣ) M i /4 M i 2 ! F M 1 ,...,Mp N/2 j=1 a(w j i=1 u i (x)) N/2 j=1 u 1/2 j du (49) = p i=1 E(B M i 1 )E   p i=1 t i t i-1 a(w s (x))dL s (0) M i /2   = E   p i=1 t i t i-1 a(w s (x))dB (i) L s (0) M i   (50)
Equation (??) corresponds to a product of moment of independent variable. We refer to [?, Appendix B.1.2] for the explicit computation of the moments of an integral driven by a Local time. This leads to the identification made at line 10 (??).

6. Proof of theorem ??: tightness.

In order to complete the proof of theorem ?? we will prove the tightness of (v t (x, •)) t∈[0,T ] thanks to the following Lemma ??. Lemma 6.1. Let p ∈ N * , and h 1 , . . . , h p : R + × M → R maps such that for any i = 1, . . . , p,

• M h i (s, •)dµ = 0 for all s ≥ 0, • sup s≥0 a∈Z (1 + |a|) 2(1+ 0 ) h i (s, ., a) ∞ < ∞, • h i (s, ., a) is Lipschitz on M and sup s≥0 a∈Z (1 + |a|) 2(1+ 0 ) h i (s, ., a) lip < ∞. Then a ≤u 1 ≤•••≤up≤ b E µ p i=1 h i (u i , T u i ) du 1 . . . du p = O b -a 1 2 p 2 . (51)
Proof. Let k := ln b-a 2 and h i,k (x, ., a) : M → R an uniform approximation from formula (??) page ?? of a map h i (x, ., a) satisfying the Hypothesis ??. According to Lemma ??, we will prove the asymptotic behaviour (??) by estimating the following term

1 2 p 2 a ≤u 1 ≤•••≤up≤ b E µ p i=1 h i,k (u i , T u i ) du 1 . . . du p .
This term may be bounded as follows,

1 2 p 2 a ≤u 1 ≤•••≤up≤ b E µ p i=1 h i,k (u i , T u i ) du ≤ 1 2 p 2 a ≤n 1 ≤•••≤np≤ b sup u E µ p i=1 h i,k (u i , T n i ) ,
We re-use the notation b e a 0 ,m,n,l,N (•) of section ?? where we substitute g k (w (l p +m) (x), ., a p ) Np by h i,k (x, ., a q ), i.e for h ∈ B : b e a 0 ,m,n,l (h) := sup

u∈Z p a∈Z p E μ P 2k (h p,k (u p , ., a p )H (ep) k ,lq,aq-a p-1 ) • • • • • (h 1,k (u 1 , ., a 1 )H (e 1 )
k ,l 1 ,a 1 -a 0 )(h) .

Relations (??) and (??) still apply, thus the only admissible vectors e are those satisfying relation (??). Thus an element e satisfies e 2i = 1 and e 2i-1 = 0 for any 1 ≤ i ≤ p (otherwise b e a 0 ,m,n,l,N (h) := sup u 1 ,...,up∈Z a 1 ,...,ap∈Z E μ P 2k h p,k (u p , ., a p ) E µ (•) = 0). Thus when introducing the notation l(u) for the vector with coordinates l

i (u) = u i -u i-1 , lim →0 1 2 2p 2 a ≤u 1 ≤•••≤u 2p ≤ b E µ 2p i=1 h i,k (u i , T u i ) du ≤ lim →0 1 2 p a ≤u 1 ≤•••≤u 2p ≤ b p i=1 b e 0,0, ,l(u) (1)du ≤ lim →0 1 2 p a ≤u 1 ≤•••≤u 2p ≤ b p i=1 1 2πΣ( u 2i-1 -u 2(i-1) ) 1/2 sup t 2i ,t 2i-1 ∈R a∈Z Ξ ,2,(0, u 2i -u 2i-1 ),(1,1) (h i,k , (t 2i-1 , t 2i ), (0, a)) du ≤ lim →0 1 (2πΣ) 1/2 l∈Z sup t 2i ,t 2i-1 ∈R Ξ ,2,(0,l),(1,1) (h i,k , (t 2i-1 , t 2i ), (0, a)) (52) 1 2 p a ≤s 1 ≤ b 0≤s 3 ≤•••≤s 2p-1 ≤ b-a p i=1 1 s 1/2 2i-1 du, (53)
where Ξ ,2,(0,l),(1,1) (h i,k , (t 2i-1 , t 2i ), (0, a)) is a notation corresponding to the following quantity Ξ ,2,(0,l),(1,1)

(h i,k , (t 2i-1 , t 2i ), (0, a)) = E μ q i=1 g k (z i , T l i (•), a i ) N i 1 {S l i =a i -a i-1 } • T k • T l i-1 .
As done in equation (??), the sum l∈Z sup t 2i ,t 2i-1 ∈R E µ h i,k (t 2i , T |l| •)h i,k (t 2i-1 , •) converges, thus according to Lemma ??, the sum appearing in (??) converges, the asymptotic behavior of formula (??) is thus given by the asymptotic behavior from formula (??). To conclude, we compute the integral in formula (??) using the following estimates: in one hand Thus we obtain formula (??) proving the lemma.

Proof. We first prove relation (??). We recall that F , (x t (x, •)) t∈[0,T ] and (w t (x)) t∈[0,T ] satisfy respectively relations (??) and (??). For any ω ∈ M , e t (x, ω) = x t (x, ω) -w t (x, •) We deduce from equation (??) and the inequality of finite increased, According to Lemma ??, v t (x, •) L p µ is uniformly bounded in , thus we deduce (??). To prove the tightness, it is enough to prove that for any p ∈ N * there is C > 0 such that sup 0≤s≤t≤T -3/4 (e t (x, •) -e s (x, •)) L p µ ≤ C(t -s) ≤ K(t -r) 2α .

This relation corresponds to the tightness condition of theorem 15.6 from [?, theorem 13.5] when choosing β = 2 and α = β 2 > 1 2 thus proving the tightness of (e t ) t≥0 in Lemma ??. To prove relation (??), we fix 0 ≤ s ≤ t ≤ T and obtain the following relation by applying the mean value theorem within formula (??) : du.

Then we conclude from Lemma ?? and the estimate (??) from theorem ?? that for any p ∈ N * , -3/4 e t (x, •) -e s (x,

•) L 2p µ ≤ v t (x, •) -v s (x, •) L 2p µ + (t -s) D 1 F ∞ + D 1 F ∞ sup u∈[0,T ]
e u (x, ω)

3/4
≤ C(t -s) We will now compare through variational analysis the error term (e t ) t≥0 with the solution (y t (x, ω)) t≥0 of the following differential equation : y t (x, ω) = v t (x, ω) + t 0 DF (w s (x)).y s (x, ω)ds.

(60)

An explicit expression for (y t (x, ω)) t≥0 is given by (y t (x, ω)) t∈[0,T ] = F (v t (x, ω)) t∈[0,T ] ,

where F is a bounded linear operator on C 0 ([0, T ], R d ) defined as

We then deduce from Lemma ?? that follows the convergence of (e t ) t≥0 for the law L µ to the same process (y t ) t≥0 . Thus, according to Lemma ??. (67)

The condition given by equation (??) is then a consequence of the following estimate of u t (x, ω) using the variational formulation of (x t ) t≥0 and the estimate (??) above :

|u t (x, ω)| =
x t (x 0 , T (ω)) -x t-(x (x 0 , ω), T (ω)) This ends the proof of theorem ??.

Aknowledgement. This article complete some work initiated during my PhD thesis under the supervision of Françoise Pène whom I want to thank for leading me into this study providing precious advises as well as a strong editorial support.

Lemma 4 . 7 .

 47 For any t ∈ [0, S], the sequence(M N ) N ∈N of moments of t 0 a(w s (x))dB L s (0) defined by M N := E t 0 a(w s (x))dB L s (0) N satisfies Carleman's criterion, i.e N ≥0 M -1/(2N ) 2N diverges (see [?]).

1 2N=

 1 O(N ) whereas the last term (??) is in O(N ) according to the technical computation in [?, Corollaire 1.5.5]. thus M 1 2N

2 -

 2 and on the other hand, we use inequality b2 -a 2 ≤ (b -a) 2 for b ≥ a ≥ 0, i.e √ b

= 3 / 0 F(ω) ds + t 0 F

 300 [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF] v t (x, ω) + t x s (x, ω), T s (ω) -F w s (x), T s (x s (x, ω)) -F (w s (x))ds.(58)

- 3 / 0 D 1 F

 301 [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF] e t (x, w)≤ |v t (x, w)| + t ∞ + D 1 F ∞ -3/4 e s (x, w) ds.Then we apply Grönwall inequality and take the L p µ norm,-3/4 e t (x, ω) ≤ |v t (x, ω)|e T ( D 1 F ∞+ D 1 F ∞) . Thus for p ∈ N * , sup t∈[0,T ] -3/4 e t (x, •) L p µ ≤ sup t∈[0,T ] v t (x, •) L p µ e T ( D 1 F ∞+ D 1 F ∞)

  relation (??) holds, we can then apply Cauchy-Schwarz formula with p = 2β to obtain the following relationE µ |e s -e r | 2β |e t -e s | 2β ≤ K(s -r) 4β 8 (t -s)

- 3 / 4 D 1 F

 341 |e t (x, •) -e s (x, •)| ≤ |v t (x, •) -v s (x, •)| + t s ∞ + D 1 F ∞ e u (x, ω) 3/4

F

  ((v t ) t≥0 ) := v t + t 0 DF (w s (x))v s exp t s DF (w u (x))du ds t∈[0,S] . The convergence in law L µ on the space (C 0 ([0, T ], R d ), . ∞ ) of (v t (x, •)) t∈[0,T ] to the process t 0 a(w s (x))dB L s (0) t≥0defined in theorem ?? implies that(v t (x, •), y t (x, •)) t∈[0,T ] s (x))dB L s (0) , y t (x, •)) t∈[0,T ] .

Lemma 7 . 2 . 4 t 0 D 1 F 4 t 0 F≤ - 3 / 4 T 0 D 2 1 F≤ 3

 724014034013 Under the assumptions of theorem ??, for any p ≥ 1,sup t∈[0,T ] -3/4 e t (x, •) -y t (x, •) L p µ = →0 O( 3/4 ).Proof. We introducea t (x, ω) := -3/4 e t (x, ω) -v t (x, ω) --3/(w s (x), T s (ω)).e s (x, ω)ds.According to equation (??) and some Taylor expansion,|a t (x, ω)| = -3/x s (x,ω), T s (ω)-F w s (x), T s (ω) -D 1 F (w s (x), T s (ω)).e s (x, ω)ds ∞ (e s (x, ω)) 2 ds

≤ 3 / 4 D 2 1 F 0 D 1 F 0 D 1 FK 0 [

 34101010 sup t∈[0,T ] a t (x, ω) L p µ ∞ T sup t∈[0,T ] e t (x, •) the process (b t (x, •)) t≥0 as follows, b t (x, ω) := y t (x, ω) -v t (x, ω) -t (w s (x), T s (ω)).y s (x, ω)ds = t (w s (x), u,s D 1 F (w s (x), T s (ω)).F (w u (x), T u (ω))duds,From the perturbed differential equation ??, (x t ) t≥0 satisfies the following variational inequality|x s (x 0 , T (ω)) -x s (x (x 0 , ω), T (ω))| ≤ |x 0 -x (x 0 , ω)| + s F ]|x u (x 0 , T (ω)) -x u (x (x 0 , ω), T (ω))|du,and thus applying Grönwall inequality on |x s (x 0 , T (ω)) -x s (x (x 0 , ω), T (ω))|, we deduce the following estimate sup s∈[0,S] |x s (x 0 , T (ω)) -x s (x (x 0 , ω), T (ω))| = O(|x 0 -x (x 0 , ω)|) = O( ).

F

  (x s (x, T (ω), T s (T ω))ds+ -1/2 t-0 [F ]|x s (x 0 , T (ω)) -x s (x (x 0 , ω), T (ω))|ds = O( 1/4 ).Thus theorem [?,Theorem 1] applies and the convergence in law in (??)holds for any measure P absolutely continuous w.r to µ :( -3/4 e t (x, ω)) t∈[0,T ] L P , . ∞ → →0 (y t ) t∈[0,T ] .

  1/4 ṽ t i-1 (x, •) For any 8 M 1 , . . . , M p ≥ 0,

			p
			through the
	following Lemma ??.	i=1
	Lemma 5.2. lim E µ →0	p i=1

when d > 1, we replace F in the formula (??) by Fi = θj, F for all i and j such that 1 + q j-1 ≤ i ≤ q j .

Proposition 6.2. For any S > 0, the family of processes (v t ) t∈[0,T ] >0 is tight in D([0, T ])

with metric . ∞ . Furthermore [START_REF] Korepanov | Superdiffusive limits for deterministic fast-slow dynamical systems[END_REF] , there is some constant K > 0 such that for any p ∈ N * ,

Proof. According to Billingsley Theorem [?, theorem 13.5], it is enough to show that β ≥ 0, α > 1 2 and C > 0, such that for any 0 ≤ r ≤ s ≤ t ≤ T ,

Then Cauchy-Schwarz inequality gives,

Then we apply Lemma ??

Then we introduce this inequality in relation (??) and we conclude the tightness by checking the assumption from [?, theorem 13.5] with parameters β = 2, α = β 2 > 1 2 by choosing p = 2β :

≤ K(t -r) 2α .

7.

Averaging : Proof of theorem ??.

The proof of the main theorem ?? relies on the statements of theorem ?? and Lemma ??. To control the error term e t , we will give a variational expression of its that we will control using classical Grönwall lemma and the behavior of the perturbed sum v t that we recall here

Proof of theorem ??. Through this proof, we keep the notations from theorem ?? and we first provide a loose control on (e t ) t≥0 with the following Lemma ??. Lemma 7.1. For any S > 0 and p ≥ 1, the error term e t (x,

and the family

11 When d ≥ 2 it is enough to prove tightness of ( θ, v t ) t≥0 for any θ ∈ R d . The proof follows the same path replacing (v t ) t≥0 by ( θ, v t ) t≥0 .

where for any u ≤ s, the quantity

Thus Lemma ?? applies and we can conclude that for any

Thus, [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF] ).

(63)

Now we apply Grönwall lemma on the following variational equation

and we obtain the following estimate

Then estimates (??) and (??) provide the conclusion (??) of the lemma, [START_REF] Bunimovich | Statistical properties of Lorentz gas with periodic configuration of scatterers[END_REF] ).

From Lemma ?? above and the convergence in of (y t ) t≥0 in (??), we deduce the conclusion of theorem ?? in the particular case where the probability law P in the theorem is precisely the law µ on M M × {0} :

To prove the full theorem and get the conclusion for any probability measure P absolutely continuous with respect to µ, it is enough to prove that the assumptions from Zweimuller theorem [?, Theorem 1] hold. In our case, those assumptions can be summarized as the following condition :

• Condition : for any x ∈ R d and ω ∈ M , We introduce the notation u t (x, ω) := e t (x 0 , T (ω)) -e t (x 0 , ω)

3/4 = x t (x 0 , T (ω)) -x t (x 0 , ω)


Y Aeschlimann

⋆ A Calissano

⋆ A Hansen

T Papadopoulo

⋆ S Deslauriers-Gauthier

GRAPH MATCHING OF STRUCTURAL AND FUNCTIONAL BRAIN NETWORKS

Keywords: Brain network, Brain connectivity, Structure-Function, Graph matching

By subdividing the cortex into regions according to a given atlas, we assume that all the regions match across different subjects. However, because of inter-subject variability, it may not be possible to produce a single brain atlas for all subjects. Previous works have proposed to use structural connectomes constructed from diffusion magnetic resonance imaging tractography to align brain regions across a group of subjects, leading to an improved similarity of connectomes. In this work, we propose to simultaneously exploit structural and functional information in the alignment process. This allows us to explore multiple perspective of brain networks. Our results show the permutations induced by one type of connectivity (structural, functional) are not always supported by the other connectivity network, but when considering combined alignment, it is possible to find permutations of regions which are supported by both connectome modalities, leading to an increased similarity of functional and structural connectivity across subjects.

INTRODUCTION

Due to the large number of neurons in the brain [START_REF] Herculano-Houzel | The human brain in numbers: a linearly scaled-up primate brain[END_REF], detecting all of their individual activation patterns responsible for human behavior is technically impossible. A reasonable approach consists in subdividing the gray matter surface into homogeneous neuronal areas [START_REF] Cabezas | A review of atlas-based segmentation for magnetic resonance brain images[END_REF]. The regions and their relationships can be represented as a graph, with the regions being the nodes and the connectivity between them being the edges [START_REF] Sporns | The human connectome: a structural description of the human brain[END_REF][START_REF] Yeh | Connectomes from streamlines tractography: Assigning streamlines to brain parcellations is not trivial but highly consequential[END_REF][START_REF] Cao | Topological organization of the human brain functional connectome across the lifespan[END_REF]. The connectivity, corresponding to the edges of the graph, is usually represented as a square matrix called a connectome [START_REF] Sporns | The human connectome: a structural description of the human brain[END_REF], of dimensions N × N , with N being the number of brain regions. In our case, we consider symmetric connectome matrices. It is generally assumed that the regions of connectomes correspond across subjects, meaning regions with the same label are expected to have the same structural and functional roles in the brain network. However, spatial organization of both cortex and white matter is shaped by environmental and genetic factors [START_REF] Thompson | Genetic influences on brain structure[END_REF]. This leads to spatial neuronal inter-subject variability and consequently in structural and functional variability. As a result, it is challenging to produce a single brain atlas for all subjects [START_REF] Salehi | There is no single functional atlas even for a single individual: Functional parcel definitions change with task[END_REF]. The inter-subject variability can be observed during neurosurgery, where patients undergo highly invasive awake-surgery with electrical cortical stimulation to precisely localize given functional areas [START_REF] Tate | Probabilistic map of critical functional regions of the human cerebral cortex: Broca's area revisited[END_REF]. The problem of producing a single brain atlas is exacerbated for atlases with very small regions, as the spatial variability across subjects may be larger than the region size [START_REF] Vigneau | Meta-analyzing left hemisphere language areas: phonology, semantics, and sentence processing[END_REF]. In particular, it is problematic when we want to carry out group-level analysis or modeling of brain networks, because a particular label of the chosen atlas may not represent the same neuronal structure across subjects [START_REF] Suárez | Linking structure and function in macroscale brain networks[END_REF], therefore limiting the specificity of the results. A previous approach used connectivity fingerprint to find a matching of regions [START_REF] Mars | Comparing brains by matching connectivity profiles[END_REF]. Another one [START_REF] Calissano | Graph alignment exploiting the spatial organisation improves the similarity of brain networks[END_REF] proposes to increase the similarity between graphs of structural connectivity across subjects, computed from diffusion magnetic resonance imaging (dMRI) tractography, by allowing permutations of regions (the nodes of the graph). Two subject's brain atlas can be aligned by selecting the connectome of a subject as the reference to align the other subject's connectome. A matching of the regions across subjects is found by maximizing the graph similarity, a process known as graph alignment [START_REF] Conte | Thirty years of graph matching in pattern recognition[END_REF]. Graph alignment was shown to improve the inter-subject similarity, as it captures the inter-subject variability of connectivity in insect brains [START_REF] Bordes | Huge steps on a tiny brain: Unraveling the fruit fly connectome and approaches for comparative connectomics[END_REF]. It was observed that brain function connectivity often differs spatially across subjects [START_REF] Mueller | Individual variability in functional connectivity architecture of the human brain[END_REF], but the previous approach has only been tested using structural connectomes (SCs) and never on functional connectomes (FCs) derived from resting-state functional MRI (rs-fMRI). In this work, we present the results of graph matching using both structural and functional networks. More specifically, we investigate whether it is possible to find permutations that are consistent in structural and functional graphs, resulting in an improvement of similarity for both SCs and FCs across subjects.

MATERIALS AND METHODS

Network alignment, FAQ algorithm

As mentioned previously, the regions of an atlas may not be mapped coherently from one subject to another. To find the mismatched regions, we can compare the connectivity of regions of the atlas to the rest of the cortex, i.e. one row of the symmetric structural connectome, across subjects. We will then permute the labels of the regions if the permutation increases the similarity of the region connectivity with respect to the reference graph. More precisely, for a given subject, a region labeled l a is permuted with a region labeled l b if the permuted connectome is more similar to the connectome of reference. Finding the optimal permutation, i.e. finding the permutation matrix that maximizes the similarity between two graphs is formulated mathematically as: minimize

P ∈P ∥P CP T -C ref ∥ 2 F , (1)
where P is a permutation matrix belonging to the permutation group P = {P ∈ {0, 1} n×n : P T 1 = P 1 = 1} and induces a permutation of rows and columns of the connectome C to align it with the reference connectome C ref . The similariry of the connectomes is based on the Frobenius norm ∥.∥ F . Unfortunately, solving this problem is NP-hard. The Fast Approximate Quadratic Assignment Problem (FAQ) algorithm [START_REF] Vogelstein | Fast approximate quadratic programming for graph matching[END_REF] tackles this difficulty by solving a relaxed version of the above optimization problem: P is allowed to be in the set of bistochastic matrices instead of just the set of permutation matrices to enable optimization by gradient descent.

The obtained bistochastic matrix is then projected onto the set of permutation matrices [START_REF] Crouse | On implementing 2d rectangular assignment algorithms[END_REF], yielding an approximate solution to Eq. ( 1). The gradient descent is initialized as the bistochastic projection of the adjacency matrix of the spatially adjacent regions of the atlas, as it was shown to be the best choice for initializing the algorithm [START_REF] Calissano | Graph alignment exploiting the spatial organisation improves the similarity of brain networks[END_REF] by favoring local permutations.

To combine the information of two brain networks coming from different modalities, we modified the FAQ algorithm such that it solves the optimization problem minimize

P ∈P α ∥P C A P T -C ref A ∥ 2 F + (1 -α) ∥P C B P T -C ref B ∥ 2 F ( 2
)
with C A and C B being the symmetric connectomes of modality A and B, and

C ref A and C ref B the corresponding ref- erence connectomes.
The same permutation matrix P is applied to both connectomes. The weight term α is used to favor one modality over the other. Indeed, if one modality is more noisy than the other, we may disfavor it with respect to the less noisy one. Specific cases are α = 1, which corresponds to an alignment with modality A only and α = 0 for an alignment considering only modality B. Both are equivalent to the original FAQ algorithm of Eq. (1) for C A or C B respectively. We kept the same initialisation depicted above as a starting point.

In the work of [START_REF] Calissano | Graph alignment exploiting the spatial organisation improves the similarity of brain networks[END_REF], the FAQ algorithm is applied on structural connectomes, and shows significant alignment of structural networks. In our work, C A will correspond to SC and C B to FC. The connectomes of references C ref A and C ref B are the Fréchet means of the SCs and FCs of the cohort. Using Frobenius distance, the Fréchet mean corresponds to the connectome that minimizes the sum of the variance to all the other connectomes [START_REF] Fréchet | Les éléments aléatoires de nature quelconque dans un espace distancié[END_REF].

Note that the network alignment is carried out on the left and right hemisphere of the brain independently. Otherwise the algorithm would be very likely to permute regions across hemispheres because of relative symmetry of the brain, which is undesirable.

MRI acquisition and preprocessing, HCP data

In this work, we consider 1050 subjects of the HCP diffusion MRI and resting-state MRI young-adults dataset. For each subject, 288 dMRI volumes have been acquired: 18 volumes at b-value 0 s/mm 2 , 90 volumes at b-values 1000 s/mm 2 , 90 volumes at b-values 2000 s/mm 2 , and 90 volumes at b-values 3000 s/mm 2 . Each volume is composed of 145 × 174 × 145 voxels with a resolution of 1.25 × 1.25 × 1.25 mm 3 . Diffusion data has been preprocessed according to the Minimal Preprocessing HCP pipeline [START_REF] Glasser | The minimal preprocessing pipelines for the human connectome project[END_REF], including EPI distortion and eddy-current correction. Functional data are originated from FIX-Denoised (Extended) rs-fMRI dataset of HCP [START_REF] Wu-Minn | 1200 subjects data release reference manual[END_REF]. The rs-fMRI data is rescaled in MNI space.

Data processing

fMRI

To filter out irrelevant parts of the BOLD signal, we used a Butterworth bandpass filter of order 8 and with cut-off frequencies of 0.06 and 0.125 Hz [START_REF] Romascano | Multicontrast connectometry: A new tool to assess cerebellum alterations in early relapsing-remitting multiple sclerosis[END_REF]. The processed BOLD signals are averaged over the regions of a multimodal atlas, and their respective temporal mean of every region averaged signal is removed. The correlation matrix is computed on processed BOLD signals using Pearson's coefficient. Regarding the atlas, we chose a multimodal atlas [START_REF] Glasser | A multimodal parcellation of human cerebral cortex[END_REF] with the FreeSurfer surface of subjects registered in the MNI space. The subcortical region of the hippocampus has been removed, leading to 179 regions per hemisphere. Because FCs are correlation matrices, there are by definition symmetric.

dMRI

We used the TractoFlow [START_REF] Theaud | Tractoflow: A robust, efficient and reproducible diffusion mri pipeline leveraging nextflow singularity[END_REF] pipeline to reconstruct tractograms from preprocessed dMRI. Spherical harmonics of order 10 maximum are fitted to the preprocessed dMRI data. Then, 120 seeds per voxels at the intersection of the gray matter and white matter are positioned as starting points of streamlines. The streamlines are reconstructed using a step size for Particle Filter Tracking (PFT) of 0.3125 mm. Only valid streamlines with a length ranging from 20 to 300 mm are kept. The other parameters of TractoFlow script are set to their default values. The structural connectome is constructed by counting the number of streamlines included in the tractogram that connects the two regions of the given matrix entry [START_REF] Yeh | Connectomes from streamlines tractography: Assigning streamlines to brain parcellations is not trivial but highly consequential[END_REF]. A local search is performed to find the closest cortical region to the endpoints of the streamline, with a upper bound distance of 5.0 mm. The same atlas as for the FC is used, but with the FreeSurfer surface of subjects in the original space. The connectomes are symmetrized by adding their transpose and dividing by 2. Then they are normalized such that they become dimensionless and have the same Frobenius norm as the FC of the same subject.

RESULTS AND DISCUSSION

Our implementation of the FAQ algorithm, sometimes, does not find a permutation matrix that increases the connectome similarity with respect to the reference connectomes. It then returns the identity matrix, meaning no permutations. Moreover, the permutation matrix may increase the similarity of one connectome modality, but not the similarity of the other. A relevant criterion to evaluate the performance of our algorithm is, therefore, the percentage of subjects in the dataset where the connectome similarity strictly increased with respect to the respective references for both structural and functional modality.

Hyperparameter alpha

To empirically find the α value, we track the percentage of subjects whose connectomes are more similar to the Frechet mean connectomes after the permutation, for the SCs and FCs (See Fig. 1). The percentage of subjects that have an increased similar-ity after graph matching depends on the value of the hyperparameter α (See Fig. 1). Recall that the particular cases where α = 0 and α = 1 correspond to aligning the FCs and SCs independently. We observe that matching networks using only one modality (α = 0 or 1) does not improve the connectome similarity of the other modality. However, there is a trade-off for α = 0.65, where we improve the structural and functional similarity in more than 73% of the subjects. This result shows that it is possible to find a permutation matrix that strictly increases the similarity for both structural and functional brain networks for a majority of subjects.

Comparison of alignment on SC, on FC, and on combined SC-FC

We take a particular look at the specific cases where we align only on SC (α = 1) or only on FC (α = 0). Indeed, those two specific cases give very different results on the set of permutations, which explains why alignment with SC does not align properly the FCs and conversely. On average, over the 1050 subjects, 18.5% of labels have been permuted with the alignment on SC only (α = 1) and 61.3% of labels have been permuted with the alignment on FC only (α = 0), but only 15.6% of the labels are permuted for a combined alignment with α = 0.65. Moreover, 57.7% of the permutations (projection of one label onto another) from the alignment with α = 0.65 are either given by the alignment of SC (α = 1) or by the alignment of FC (α = 0). More precisely, 42.7% of the permutations for the combined alignment are shared with the alignment on SC and only 16.3% are shared with the alignment on FC. The numbers of permutations given by the 3 approaches (3 values of α) averaged per subject are illustrated in Fig. 2d. The permutations can be illustrated as edges on the brain cortex networks, linking two regions of the atlas when they are permuted. Those edges are weighted by their frequencies of permutations in the dataset of 1050 subjects (See Fig. 2a-c).

The alignments on SC only (Fig. 2a) or FC only (Fig. 2b) induce many permutations throughout the cortical surface. Most of them are discarded in the combined alignment (Fig. 2c). We observe that the combined alignment shares more permutations with the SC alignment than with the FC alignment (Fig. 2b,d). The edges are mostly located in the frontal and in the occipital lobes.

The alignment on FC gives more permutations than the alignment on SC because FCs represent statistical correlation of BOLD signals which contains a non-negligible level noise, leading to quite similar connectivity profiles across the regions, that can be permuted without a great change in the FC.

One important feature of the permutations to notice is that they are almost all short-range permutations, which we explicitly favored with the initialization based on the adjacency matrix. This is consistent with the fact that it is very unlikely to have distant matching regions across subjects. Moreover, a lower number of permutations is provided by the FAQ algorithm when considering SC and FC together, than when considering only one modality. This suggests that irrelevant permutations are removed by the addition of one modality. Those irrelevant permutations proposed in the one-modality alignment might be the results of remaining noise, artifacts or lack of information.

Interestingly, Fig. 2a shows permutations which agreed with the variability in the left-hemisphere of language areas in the meta-analysis of Vigneau et al. [START_REF] Vigneau | Meta-analyzing left hemisphere language areas: phonology, semantics, and sentence processing[END_REF] The permutations localized in the visual cortex are unexpected because of the strong assumptions about their spatial organization. Indeed, those suggested permutations supported by both structural and functional connectivity information might occur due to the similar connectivity of the visual regions to the rest of the cortex and may only align remaining noise. This needs further investigation. A confidence index might be used on the permutations, such as the connectivity variability of the permuted regions of the same subject. If two permuted regions have similar connectivity profile on the same subjects (as for the regions in the visual cortex) we are less confident in the permutation as for permuted regions that present weak connectivity similarity.

Often, two regions would not be entirely mislabeled, but their boundaries might be rotated, shrunk or shifted in certain directions and thus do not really delineate a homogeneous neuronal region, and overlap more of another region than the one it is supposed to delineate. In that case, the permutation of labels is a way to partially improve the similarity across subjects of regions with same labels.

CONCLUSION

In this work, we propose a method which improved the alignment of atlas regions, incorporating functional connectivity information. We modified the FAQ algorithm in order to find a single permutation matrix that aligns the brain atlas using both structural and functional connectivity. Using only structural connectivity entails permutations often not consistent in the functional data, whereas using only functional connectivity seems noisy and suggests a lot of irrelevant permutations. Our novel approach of aligning the cortex regions across subjects finds permutations of regions that remain consistent both in structural and functional networks. Our results suggest that, before any group processing of brain networks, this alignment step should be performed to avoid consideration of mismatched regions across subjects.

COMPLIANCE WITH ETHICAL STANDARDS

This research study was conducted retrospectively using human subject data made available in open access by the Human Connectome Project Dataset [START_REF] Wu-Minn | 1200 subjects data release reference manual[END_REF]. Ethical approval was not required as confirmed by the license attached with the open access data.

Fig. 1 .

 1 Fig.1. The percentage of subjects where the connectome similarity increased after graph matching, depending on the hyperparameter α. The percentage of SC (blue curve) increases, whereas the percentage of FC (yellow curve) decreases with α because the alignment on SCs is favored. The red curve represents the percentage of subjects who have both their SC and FC more similar to the respective references with the alignment. The highest percentage of subjects is reached for α = 0.65, and is equal to 73.4%.

Fig. 2 .

 2 Fig. 2. Distribution of permutations for alignment on SC (a) (α = 1), FC (b) (α = 0) and combined alignment (c) (α = 0.65) in the left hemisphere. The gravity centers of the atlas regions are represented by green dots. The width of the edges represents the frequency of permutations of the two nodes that are incident with the given edge. (d) presents a Venn diagram, showing the total amount and overlap of permutations that are common in the 3 cases (averaged over left and right hemispheres).

ACKNOWLEDGMENTS

The authors are grateful to the OPAL infrastructure from Université Côte d'Azur for providing resources and support.