Athanasios Batakis

Michel Zinsmeister

On the time schedule of Brownian Flights

We are interested in the statistics of the duration of Brownian diffusions started at distance ǫ from the boundary of a given domain and stopped when they hit back this boundary.

Introduction

The motivation of the following work has its origin in experimental physics. Some long molecules are solvable in a liquid (for instance imogolite in water or DNA in lithium) and the molecules forming the liquid show an intermittent dynamics, alternating diffusion in the bulb and adsorption on the long molecules. For the physicist's point of view, it is very important to have as precise as possible knowledge of the statistics of these brownian flights.

In [GKL + 06] a connection is established between the statistics of the long flight lengths and the geometry of the long molecules (more precisely their Minkowski dimension). This connection has been made rigorous in [START_REF] Batakis | Brownian flights[END_REF]. These two papers concern almost exclusively lengths. How does one check experimentally the results? A very powerful tool for that is relaxation methods in nuclear magnetic resonance (see [DPP + 08]): but this method only allows to compute (the statistics of the) duration of long flights. Some heuristic link between time and length was derived in [GKL + 06], [DPP + 08]. The aim of this paper is to make this heuristics rigorous.

Geometric Backgound

In the sequel, Ω will always denote a domain in R d with compact boundary. The crucial tool we need to use is the notion of Whintey cubes. We thus recall the Proposition 2.1 (cf. [START_REF] Grafakos | Classical Fourier Analysis[END_REF], p. 463) Given any non-empty open proper subset Ω of R d , there exists a family of closed dyadic cubes {Q j } j such that • j Q j = Ω and the cubes Q j 's have disjoint interiors

• √ dℓ(Q j ) ≤ dist(Q j , ∂Ω) ≤ 4 √ dℓ(Q j ) • if Q j and Q k touch then ℓ(Q j ) ≤ 4ℓ(Q k )
• for a given Whitney cube Q j there are at most 12 d Whitney cubes Q k 's that touch Q j .

In this statement, ℓ(Q) stands for the side-length of the cube Q and, for λ > 0, λQ is the cube of the same center and of sidelength λℓ(Q). For k ∈ Z, we denote by Q k , the collection of Whitney cubes Q j with ℓ(Q j ) = 2 k . We also recall the definition of the Minkowski sausage: for r > 0,

M r = {x ∈ Ω ; dist(x, ∂Ω) ≤ r} and Γ r = {x ∈ Ω ; dist(x, ∂Ω) = r}
We then define S r as the collection of Whitney cubes intersecting Γ r . Notice that S r is a finite set.

Definition 2.1 Let ε > 0. We will call Brownian flight the random process F t , t ≥ 0 consisting in picking at random with equiprobability one of the dyadic Whitney cubes of S ε and starting from the center of the cube a Brownian motion B t killed once it reaches ∂Ω. We denote by τ Ω = inf{t ; F t / ∈ Ω} the lifetime of this process.

We are interested in the asymptotics of P(τ Ω > t) as t grows, but this needs some explanation:

It is well known that, if Ω is bounded, this quantity decreases exponentially, as t → ∞, as e λt , where λ is the first eigenvalue of the Laplacian. We define

R Ω = min 1, sup x∈Ω dist(x, ∂Ω) .
Our aim is to evaluate P(τ Ω > t) in the interval ǫ 2 ≤ t ≤ R 2 Ω , and this independently of ε. In fact, the estimate we are looking for is thus an estimate with respect to ε rather than for "pure" t.

We study first a simple example, since we will need the partial result anyhow. Let [0, a] be a real segment and take x ∈ (0, a). The probability that brownian motion started at x has not exit the interval (0, a) by time t, P(τ x > t), is given by the following equivalent formulas (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF], pg. 342)

P(τ x > t) = 1 √ 2π -x/ √ t -x/ √ t exp - 1 2 y 2 dy + 2 π ∞ k=1 (-1) k (ka+x) √ t (ka-x) √ t exp - 1 2 y 2 dy and P(τ x > t) = 4 π ∞ k=1 exp - (2n + 1) 2 π 2 2a 2 t sin (2n + 1)πx a .
By symmetry we can assume x ≤ a 2 . If a √ t is not too big, say a √ t < 1/2, we have an easy estimate of P(τ x > t) using the first formula : P(τ x > t) ∼ x √ t . On the other hand, for a √ t not too small we get, by the second formula, that P(τ x > t) ≤ exp -π 2 2a 2 t . Hence, that there is a change of regime, the decay of P(τ x > t) with t going from polynomial to exponential and the quantity x √ t is relevant for small times. In a higher dimensional context, let Q be the cube centered at 0 and of side r. By the preceeding remark it follows that, T denoting the exit time from Q of the Brownian motion starting at 0, we have

P(T > t) ≤ c r √ t d (1)
where c depends only on d. Here, ε = r ≃ R Ω and the opposite inequality thus holds for

ε 2 ≤ t ≤ R 2 Ω .
If the Brownian motion is started at distance ε from the boundary then the exit time is essentially the exit time from a half space and we thus get

P(T > t) ≤ c ε √ t = c ε √ t d-1+2-d , (2)
and, as we will see, the opposite inequality is valid for t not too big.

Our goal is to extend (2) to general domains with rough boundary. In order to describe the domain of validity of our result let us recall a few definitions. Let K be a compact subset of R d . For j ≥ 0 let N j be the number of dyadic cubes of the j-th generation (i.e of size 2 -j ) that intersect K.

Definition 2.2 The Minkowski dimension of K is d M (K) = lim sup j→∞ log 2 (N j ) j
Returning to our situation, we can define similarly the Whitney dimension of ∂Ω as

d W = d W (∂Ω) = lim sup j→∞ log 2 (W j ) j , (3)
where W j is the number of elements of Q j .

Under very mild conditions (see [START_REF] Bishop | Minkowski dimension and the Poincaré exponent[END_REF], [START_REF] Jerison | Boundary behaviour of harmonic functions in nontangentially accessible domains[END_REF], [START_REF] Batakis | Brownian flights[END_REF]) these two dimensions coincide. If the boundary of Ω has some self similarity we can moreover say that there is a constant

c > 0 such that 1 c ε d M ≤ #S ε ≤ cε d M , (4)
for all ε ≤ R Ω , where d M = d M (∂Ω).

In our main theorem we will assume that our domain Ω satisfies (4).

We also suppose that the domain Ω satisfies so-called ∆-regularity condition (see also [START_REF] Jones | Hausdorff dimension of harmonic measures in the plane[END_REF], [START_REF] Ancona | On strong barriers and an inequality of Hardy for domains in R n[END_REF], [START_REF] Heinonen | Nonlinear Potential Theory of Degenerate Elliptic Equations[END_REF]): there exists L > 0 such that for all x ∈ Ω, if

d x = dist(x, ∂Ω) < R Ω then ω x B(x,2dx)∩Ω (∂Ω) ≥ L, (5)
where ω x B(x,2dx)∩Ω is the distribution law of the hitting point of Brownian motion starting at x and killed when reaching the boundary of B(x, 2d x ) ∩ Ω). This is a very mild condition (satisfied, for instance, by all domains in R 2 with non-trivial connected boundary) that appears frequently in related literature in various forms (for instance "uniform capacity condition" or Hardy inequality).

Time and length estimates for Brownian flights

We can now state the main result. Let Ω be a bounded domain in R d satisfying (4) and (5). If τ Ω denotes the life-time of a Brownian flight F t with parameter ε we have Theorem 3.1 There exists c > 0 depending only on constants in (4), (5) (and in particular not on ε) such that

1 c ε √ t d M +2-d ≤ P(τ Ω > t) (6)
and

P(τ Ω > t) ≤ c ε √ t d M +2-d log ε √ t 2d , ( 7
)
for all ε 2 < t < R 2 Ω .
This theorem has a "cousin" theorem, which was proved in [START_REF] Batakis | Brownian flights[END_REF].

Theorem 3.2 Let ε < r < R Ω . The probability that the hitting point of F is at distance greater than r from the starting point x is comparable to

#S r #S ε d M r ε d-2 (8)
Notice that we do not assume (4) for this theorem. If we do, we have

#S r #S ε d M r ε d-2 ∼ r ε d M -(d-2) (9)
Notice that the quantity on the left of (9) is the same as the one in (6) where we have replaced r by √ t which is coherent with standard behaviour of Brownian motion.

4 Proof of the theorem 3.1

For s > 0 we denote by β s the total time spent by Brownian flight F t in the Minkowski sausage {x ∈ Ω ; dist(x, ∂Ω) ≤ s} and δβ s = β sβ s/2 .

We define analogue quantities more adapted to the Whitney decomposition ; namely, δ β2 k will denote the time spent by

F t inside Mk = {Q ; Q ∈ Q k }.
If Q is a Whintney cube we define the "vicinity" of Q as

Q = Q ∪ {Q ′ ; Q ′ ∈ E}, where E = {Q ′ ∈ k Q k ; λQ ∩ Q ′ = ∅ and λQ ′ ∩ Q = ∅},
and λ = 8 √ d satisfy that for all Whitney cubes

λQ ⊃ B(x Q , 2 dist(x Q , ∂Ω)),
x Q being the center of Q.

We may now start the proof and we begin with

The upper bound

We separate the event {τ Ω > t} by the partition {τ Ω = β √ t } and {τ Ω > β √ t }, that is whether the process goes or does not go at distance √ t from the boundary. From theorem 3.2 we get

P(τ Ω > t and τ Ω > β √ t ) ≤ P(τ Ω > β √ t ) ≤ c ε √ t d M +2-d
.

In order to estimate the term P(τ Ω = β √ t > t) we begin by writing

τ Ω = log 2 √ t k=-∞ δβ 2 k ,
and thus, putting

k 0 = log 2 ǫ and k 1 = log 2 √ t, P   log 2 √ t k=-∞ δβ 2 k > t   ≤ k 1 k=k 0 +1 P δβ 2 k > t 2(k 1 -k 0 ) + P β 2 k 0 > t 2 (10)
We now invoke the following lemma whose proof is post poned to the next section.

Lemma 4.1 There exists a number k * depending only on d and constants C , 0 < p < 1 depending only on d and L such that for all t > 0, k ∈ Z and N ∈ N we have

P(δβ 2 k > t) ≤ Cp N + P ∃Q ∈ Q k ∪ ... ∪ Q k-k * ; ∃ 0 < s 1 < s 2 < t with F [s 1 ,s 2 ] ⊂ Q and s 2 -s 1 > t/N
Using this lemma, we get

P δβ 2 j > t 2(k 1 -k 0 ) ≤ cp N + P ∃Q ∈ Q j ∪ ... ∪ Q j-k * ∃s 1 < s 2 < τ Ω ; F [s 1 ,s 2 ] ⊂ Q , s 2 -s 1 > t 2(k 1 -k 0 )N
By (1), (9) and the strong Markov property of Brownian motion we then have :

P δβ 2 j > t 2(k 1 -k 0 ) ≤ cp N + P ∃Q ∈ Q j ∪ ... ∪ Q j-k * ∃s 1 τ Ω ; F s 1 ∈ Q × P ∃s 2 > s 1 , F [s 1 ,s 2 ] ⊂ Q and s 2 -s 1 > t 2(k 1 -k 0 )N ≤ cp N + c   2 j t 2(k 1 -k 0 )N   d ε 2 j d M +2-d
Suming up the first term in (10) we get

k 1 k=k 0 +1 P δβ 2 k > t 2(k 1 -k 0 ) ≤ c(k 1 -k 0 )p N + c k 1 k=k 0 +1   2 k t 2(k 1 -k 0 )N   d ε 2 k d M +2-d ≤ c(k 1 -k 0 )p N + c(2(k 1 -k 0 )N) d ε d M +d-2 1 √ t d k 1 k=k 0 +1 2 2d-d M +2 ≤ c(k 1 -k 0 )p N + c(2(k 1 -k 0 )N) d ε d M +d-2 1 √ t d 2 k 1 (2d-d M +2) ≤ cp N log 2 √ t ε + c log 2 √ t ε N d ε √ t d M +2-d Take N ≃ (d M + 2 -d) log p ε √ t to obtain k 1 k=k 0 +1 P δβ 2 k > t 2(k 1 -k 0 ) ≤ c log ε √ t 2d ε √ t d M +2-d (11)
To bound the second term P β 2 k 0 > t 2 of the sum (10) we need a lemma of the same nature as lemma 4.1. The proof of this lemma is also post poned to the next section. Lemma 4.2 Let R k 0 be the collection of all dyadic cubes of sidelength 2 k 0 intersecting ∂Ω. There exist constants C , 0 < p < 1 depending only on d and L such that for all t > 0,

k 0 ∈ Z and N ∈ N P(δβ 2 k 0 > t) ≤ Cp N + P ∃Q ∈ R k 0 ; ∃ 0 < s 1 < s 2 < t with F [s 1 ,s 2 ] ⊂ NQ and s 2 -s 1 > t/N
We therefore deduce

P(δβ 2 k 0 > t 2 ) ≤ cp N + 2 k 0 N √ t d .
We minimize on N ≃ log p (ε) to get

P(δβ 2 k 0 > t 2 ) ≤ c log(ε) ε √ t d .
Combining this last inequality and (11) we get the upper bound, since

d M + d -2 ≤ d.

The Lower Bound

Following the same reasoning for k 1 = [log 2 √ t] + 1 we get

P(τ Ω > t) ≥ P(∃s 1 > 0 s.t. F s 1 ∈ Q∈Q k 1 Q and ∃s 2 > s 1 + t s.t. F [s 1 ,s 2 ] ∈ Q∈Q k 1 2Q)
Using strong Markov property the this probability can be written as the product of P(∃s 1 > 0 s.t.

F s 1 ∈ Q k 1 ) with P(∃s 2 > s 1 + t s.t. F [s 1 ,s 2 ] ∈ Q∈Q k 1 2Q).
The second term of the product is greater than the probability that Brownian motion exits a cube of size 2 k 1 +1 ≃ √ t at time greater that t which is bounded below by a positive constant depending only on d. The first one is simply the probability that Brownian flight gets to

Q k 1 which is equivalent to ǫ √ t d M +2-d
and the proof is complete.

Proofs of lemmas

Let un first deal with lemma 4.1. The proof of 4.2 is quite similar and will hence be abridged.

Proof of lemma 4.1

We need the following Lemma 5.1 Under the ∆-regularity hypothesis the probability that BM touches more than N Whitney cubes of a given size decreases as Cp N , with 0 < p < 1, C a positive constant.

The proof of the lemma relies on an annuli reasoning.

Proof Le (B t ) t>0 be Brownian motion started at any point x ∈ Ω and choose k ∈ Z. Choose any Q ∈ Q k and let λQ be the cube of the same center but λ times the side-length ℓ(Q) of Q. By the definition of Whitney cubes, there is a λ = 8 √ d depending only on d such that

λ 2 ℓ(Q) ≤ dist(Q, ∂Ω) ≤ 2λℓ(Q).
Suppose that there exists t 0 > 0 such that B t 0 ∈ Q. By the ∆-regularity condition (5), the probability that there exists t 1 > t 0 with B [t 0 ,t 1 ] ⊂ Ω and B t 1 / ∈ λQ is bounded above by p < 1 depending only on L, λ:

P ∃t 1 > t 0 ; B [t 0 ,t 1 ] ⊂ Ω and B t 1 / ∈ λQ ∃t 0 > 0 ; B t 0 ∈ Q < p (12)
On the other hand, the number of Whitney cubes of Q k lying inside λQ is bounded by a constant c 1 = c 1 (d). The probability that there exists a Whitney cube Q 1 ∈ Q k outside λQ that is visited by Brownian motion is hence bounded above by p < 1.

We study probability that there exist Whitney cubes Q

1 , ..., Q m ∈ Q k such that Q 1 ∩λQ = Q 2 ∩ λQ 1 = ... = Q m ∩ λQ m-1 = ∅ all
visited by Brownian motion. It is sufficient to prove that this probability decays exponentially with m.

By the strong Markov property the probability that there exists t m > t m-1 > ... > t 0 such that

B t 0 ∈ Q , B t 1 ∈ Q 1 ... , B tm ∈ Q m is given by P (∃ t m > t m-1 > ... > t 0 ; and Q, ...Q m as above such that B tm ∈ Q m , ..., B t 0 ∈ Q) = P ∃ t m > t m-1 ; B tm ∈ Q m |∃ t m-1 > ... > t 0 B t m-1 ∈ Q m-1 , ..., B t 0 ∈ Q • P ∃ t m-1 > ... > t 0 B t m-1 ∈ Q m-1 , ..., B t 0 ∈ Q = P ∃ t m > t m-1 ; B tm ∈ Q m with Q m ∩ λQ m-1 = ∅|∃ t m-1 ; B t m-1 ∈ Q m-1 • P ∃ t m-1 > ... > t 0 B t m-1 ∈ Q m-1 , ..., B t 0 ∈ Q Now, by (12), P ∃ t m > t m-1 ; B tm ∈ Q m with Q m ∩ λQ m-1 = ∅|∃ t m-1 ; B t m-1 ∈ Q m-1 < p.

By induction we get that

P (∃ t m > t m-1 > ... > t 0 ; and Q, ...Q m as above such that B tm ∈ Q m , ..., B t 0 ∈ Q) < p m
and hence the lemma. • Recall that for a given dyadic Whitney cube Q we have defined the vicinity Q of Q as the union of all Whitney cubes Q ′ verifying

Q ′ ∩ λQ = ∅ and Q ∩ λQ ′ = ∅.
We can easily check that there are less than (100 √ d) d such cubes Q ′ of size at most ℓ(Q)/12 (the constants are not optimal). We say that the k-level layers are visited at least n times if there exist t 0 < s 1 < t 1 < ... < s n < t n satisfying

B t j ∈ Q∈S 2 k Q and B s j / ∈ Q∈S 2 k Q,
for all j = 1, ..., n. For any k ∈ Z note ν k = sup{n ∈ N ; the k-level layers are visited at least n times} Lemma 5.2 There exists 0 < p < 1 and a positive constant C such that, given k ∈ Z, for all n

∈ N P(ν k > n) ≤ p n P (∃t 0 > 0 and Q ∈ S 2 k ; B t 0 ∈ Q) .
Proof The arguments as similar as in lemma 5.1. We only need to prove that P(ν k > 1) < p and apply strong Markov property. We have

P(ν k > 1) ≤ P   ∃0 < t 0 < s 1 < t 1 , Q ∈ S 2 k ; B t 0 ∈ Q , B s 1 / ∈ Q∈S 2 k Q , B t 1 ∈ Q∈S 2 k Q   = P   ∃t 1 > s 1 > t 0 ; B s 1 / ∈ Q∈S 2 k Q , B t 1 ∈ Q∈S 2 k Q ∃t 0 > 0 ; B t 0 ∈ Q ∈ S 2 k   × P (∃t 0 > 0 and Q ∈ S 2 k ; B t 0 ∈ Q)
To abbreviate formulas we note P c (.) = P (.|∃t 0 > 0 ; B t 0 ∈ Q ∈ S 2 k ). With this notation,

P c   ∃t 1 > s 1 > t 0 ; B s 1 / ∈ Q∈S 2 k Q , B t 1 ∈ Q∈S 2 k Q)   = P c   ∃t 1 > s 1 ; B t 1 ∈ Q∈S 2 k Q A   P c (A) + P c   ∃t 1 > s 1 ; B t 1 ∈ Q∈S 2 k Q B   P c (B)
where

A =    ∃s 1 > t 0 ; B s 1 / ∈ Q∈S 2 k λQ    and B =    ∃s 1 > t 0 ; B s 1 ∈ Q ′ , λQ ′ ∩ Q∈S 2 k Q = ∅ , B s 1 ∈ Q∈S 2 k λQ    form a partition of the event ∃s 1 > t 0 ; B s 1 / ∈ Q∈S 2 k Q .
By (12), P c (A) ≤ p. Similarly, by the strong Markov property of Brownian motion and by (12),

P c   ∃t 1 > s 1 ; B t 1 ∈ Q∈S 2 k Q B   = P   ∃t 1 > s 1 ; B t 1 ∈ Q∈S 2 k Q B s 1 ∈ Q ′   ≤ p.
We deduce that

P c   ∃t 1 > s 1 > t 0 ; B s 1 / ∈ Q∈S 2 k Q , B t 1 ∈ Q∈S 2 k Q)   ≤ P c (A) + p(1 -P c (A))
The function t → t + p(1t) being increasing on [0, p] we get

P c   ∃t 1 > s 1 > t 0 ; B s 1 / ∈ Q∈S 2 k Q , B t 1 ∈ Q∈S 2 k Q)   ≤ 2p -p 2 < 1
and the lemma is proven. • Remark that by definition of dyadic Whitney cubes there exist k * depending only on the dimension of the space (k * = [log 2 (8 √ d)] + 3 will do) such that for all k ∈ Z,

{x ∈ Ω ; 2 k-1 ≤ dist(x, ∂Ω) ≤ 2 k } ⊂ k j=k-k * Q∈Q j Q. ( 13
)
Proof of lemma 4.1 We clearly have

P(δβ 2 k > t) ≤ P(δβ 2 k > t , ν k + ... + ν k-k * > N) + P(δβ 2 k > t , ν k + ... + ν k-k * ≤ N) (14)
Given t > 0, k ∈ Z, by lemma 5.2 we have, for all N,

P(δβ 2 k > t , ν k + ... + ν k-k * > N) < k * p 1 k * N P (∃t 0 > 0 and Q ∈ S 2 k-k * ; B t 0 ∈ Q) < cp N
Let us estimate the second term of the sum (14). By (13) and the definition of ν k we get

P(δβ 2 k > t , ν k + ... + ν k-k * ≤ N) ≤ P(∃l ≤ N , Q 1 , ..., Q l ∈ k j=k-k * Q j ; Q s ∩ Qs-1 = ∅ , ∀s = 2, ..., l and
∃t 1 < s 1 ≤ t 2 < s 2 < ... ≤ t l < s l ; B [t i ,s i ] ⊂ Qs ∀s = 1, ..., l and l i=1 s i -t i > t).
Proof The proof of this lemma is a straightforward application of the ∆-regularity condition. It suffices to show that there exists 0 < p < 1 such that

Further Comments

We should point out that hypothesis (4) in theorem 3.1 can be dropped; in this case, the same reasoning as in subsection 4.2 gives the lower bound

#S √ t #S ε d M √ t ε d-2 .
The best upper bound is less evident; nevertheless a slight improvement of the above proofs gives

P(τ Ω > t) ≤ c log √ t ǫ cd sup N 2 -N d + N k=0 #S √ t/2 k #S ε d M 2 -k √ t ε d-2
.

  P((∃t1 > s 1 > t 0 ; B t 0 ∈ M r , B s 1 / ∈ M 4r , B t 1 ∈ M r ) < p,and then apply the Markov property. Remark that, the probabilityP((∃t 1 > s 1 > t 0 ; B t 0 ∈ M r , B s 1 / ∈ M 4r )is smaller than the probability that brownian motion started at t 0 exits a ball of radius 2r > 2 dist(B t 0 , ∂Ω) without hitting ∂Ω. By the ∆-regularity this last probability is bounded by a constant p < 1. • Proof of lemma 4.2 As before we get :P(δβ 2 k 0 > t) = P(δβ 2 k 0 > t , ξ 2 k 0 > N) + P(δβ 2 k 0 > t , ξ 2 k 0 ≤ N)(15)By lemma 5.4 we get P(δβ 2 k 0 > t , ξ 2 k 0 > N) ≤ p N , where 0 < p < 1, for all N ∈ N. Let us now deal with the second term of the sum.P(δβ 2 k 0 > t , ξ 2 k 0 ≤ N) ≤ P(∃s 1 < s 2 < τ Ω ; s 2s 1 > t/N , B [s 1 ,s 2 ] ⊂ M 4r )Using lemma 5.3 we get that, for R > 4r, this probability is bounded bycp R r + P ∃s 1 < s 2 < τ Ω ; s 2s 1 > t/N , B [s 1 ,s 2 ] ⊂ M 4r ∩ B(B s 1 , R) ,and the statement of the lemma 4.2 follows on taking N = R

r

•

Since l ≤ N we get that P(δβ 2 k > t , ν k + ... + ν k-k * ≤ N) is bounded above by

The ideas are the same but we will work with cubes touching the boundary instead of Whitney cubes.

Lemma 5.3 Under the ∆-regularity hypothesis, the probability that BM started at distance ε < r from the boundary gets at distance greater than R from the starting point without leaving the Minkowski sausage M r = {x ∈ Ω ; dist(x, ∂Ω) ≤ r} is bounded above by cp R/r where c > 0 and 0 < p < 1 are constants (depending only on L of the ∆-regularity hypothesis and on d).

The proof, similar to the one of lemma 5.1, is therefore abridged.

Proof Let x ∈ M r and consider the annuli centered at x of inner radii 4ℓr and outer radii 4(ℓ + 1)r where ℓ = 0, ..., R 4r . Brownian motion started at x and moving at distance R from x before exiting Ω must go through all these annuli. The probability of going through such an annulus while staying at distance at most r from the boundary is bounded by a p 0 ∈ (0, 1) by the ∆-regularity hypothesis. To see this take any point y in the middle of the annulus (i.e. at distance 6ℓ+6 r from x) and consider the ball of center y and radius 2r. If d y < r, the probability to exit the ball without touching ∂Ω is uniformly bounded away from 1 by the same hypothesis. This probability being greater than the probability of going through the annulus we have the statement. By the independence of the "crossing annuli" events we get that the probability that Brownian motion goes through all the annulli is smaller that

We say that the Minkowki sausage M r is visited by the Brownian motion at least k times if there exist t 0 < s 1 < t 1 < ... < s n < t n < τ Ω satisfying B t i ∈ M r for all i = 0, ..., n and

In a similar way with ν k we define ξ k as the ξ r = sup{n ∈ N ; M r has been visited at least k times}.

Lemma 5.4 There exists 0 < p < 1 depening only on the ∆-regularity's L such that, given r > 0, for all n ∈ N P(ξ r > n) ≤ p n .


François Baudin
email: francois.baudin@upmc.fr

Jean-Robert Disnar

Philippe Martinez

Bernard Dennielou

Distribution of the organic matter in the channel-levees systems of the Congo mud-rich deep sea fan (West Africa). Implication for deep offshore petroleum source rocks and global carbon cycle

Keywords: South-East Atlantic, Turbidites, Carbon isotopes, Nitrogen isotopes, Biomarkers, Source rocks

published or not. The documents may come     Distribution of the organic matter in the channel-levees systems of the Congo mud-rich deep-sea fan (West Africa). Implication for deep offshore petroleum source rocks and global carbon cycle

Introduction

With deep-water exploration for petroleum purposes becoming more intensive and successful, it is important to have a better knowledge of the depositional processes of source rocks, reservoirs and seals in the deep sea. Up to now, deep-water reservoirs have received most of the attention from the scientific community whereas the source rocks have been less studied. According to conventional views, deep water settings are unsuitable for source rock deposition because organic material is intensely degraded during settling through the water column. However several transport processes may lead to deposition of organic-rich sediments in the deep sea, which can now be regarded as areas favourable for hydrocarbon source rocks [START_REF] Huc | Organic sedimentation in deep offshore settings: the Quaternary sediments approach[END_REF][START_REF] Stow | Depositional processes of black shales in deep water[END_REF][START_REF] Saller | Leaves in turbidite sands: the main source of oil and gas in the deep-water Kutei Basin, Indonesia[END_REF]. Sediment mass transport processes such as turbidity currents provide one efficient way for accumulating both continental and marine organic-rich fine-grained sediments in the deep offshore.

Clay rich fine-grained turbidites are particularly well developed as levees in the deepsea turbidite systems and are therefore also closely associated to sandy reservoirs that are turbiditic channels. These shaly facies usually act as seals in the petroleum system, but their potential as source rock is poorly known. Furthermore, these facies are widely distributed from the upper slope to the distal part of the deep sea turbidite systems, in the abyssal plains, and represent huge volumes of sediment in which organic matter can be preserved. The contribution of such facies to the global carbon budget is still poorly constrained.

We attempt here to quantify the organic matter preserved in the channel-levees systems from the Congo mud-rich deep-sea turbidite system as well as to determine its origin (continental versus marine) in order to discuss the importance of such facies in terms of potential petroleum source rocks and their contribution for carbon storage.

General setting and studied sites

General setting

The Zaire/Congo River drains the second largest catchment area of the world (3.7 10 6 km², Fig. 1) and about 38% (42,800 m 3 /s or 1350 km 3 /year) of the yearly run-off from Africa occurs through this river [START_REF] Kinga-Mouzeo | Transport particulaire actuel du fleuve Congo et de quelques affluents : enregistrement quaternaire dans l'éventail détritique profond (sédimentologie, minéralogie et géochimie[END_REF][START_REF] Probst | Hydrology and geochemistry of the Congo River system[END_REF]. Although the Congo River shows a small variability in its flow (22,000 to 76,500 m 3 /s), floods are recorded in May and December.

The Congo River delivers yearly ~55 10 6 t of suspended sediment [START_REF] Wetzel | The transfer of river load to deep-sea fans: a quantitative approach[END_REF]. This is 15 to 17 times lower than the Amazon River, which has a drainage basin only 2 times higher and a fluid discharge of 130,000 m 3 /s. This difference is attributed to the flat morphology of the catchment area of the Congo River and to the presence of several lakes through its course that trap an important part of the suspended material (N'Kounkou and [START_REF] Probst | Hydrology and geochemistry of the Congo River system[END_REF].

Despite its relatively low sedimentary load, the Congo River contributes to 3.9% (13 10 6 t C/year) of the global annual supply of terrigenous organic carbon into the ocean [START_REF] Martins | Biogeochemistry of major African rivers: carbon and minerals transport, in Biogeochemistry of Major World Rivers[END_REF]. The dissolved organic carbon and particulate organic carbon near the mouth are 8.5 and 1.0 mg/L, respectively [START_REF] Meybeck | River discharges to the oceans: an assessment of suspended solids, major ions, and nutrients, environment information and assessment report[END_REF][START_REF] Coynel | Spatial and seasonal dynamics of total suspended sediment and organic carbon species in the Congo River[END_REF].

Consequently, the particulate organic carbon/suspended material ratio delivered by the Congo River is high (1/20) compared to other major rivers with higher sedimentary loads, e.g. 1/50 for the Amazon River.

The freshwater outflow of the Congo River into the Atlantic Ocean is detectable by reduced salinity in a 5-10 m thick surface layer as far as 800 km offshore (van Bennekom and [START_REF] Van Bennekom | Hydrography and silica budget of the Angola Basin[END_REF]. This freshwater outflow, rich in phosphate and nitrate, induces upwelling and two nearshore coastal cells are seasonally observed just south and north of the Congo mouth at 5 and 7° S (Fig. 1). They are considered to be produced by the piling up of eastwardflowing equatorial undercurrents as they arrive at the continental margin off Congo and Gabon [START_REF] Voituriez | Primary production in the tropical Atlantic Ocean mapped from oxygen values of Equalant 1 and 2 (1963)[END_REF][START_REF] Servain | Evidence of remote forcing in the equatorial Atlantic Ocean[END_REF]. As a result of oceanic or riverinduced upwellings and the supply of nutrients by the Congo River, the primary productivity is high in the surface waters in that part of the Atlantic Ocean. [START_REF] Berger | Global maps of ocean productivity[END_REF] gives values of 90-125 g C/m²/year for the area off the Congo mouth and slightly higher values, between 125 and 180 g C/m²/year, for the oceanic upwelling area off Angola.

The strong influence of the river sediment load on the adjacent continental margin is evident from the existence of the Congo mud-rich deep sea turbidite system. This edifice is one of the world's largest active deep sea turbidite systems, with an area estimated of 330,000 km 2 [START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF]. Owing to the narrow (~50 km wide) continental shelf in front of the Congo mouth and the presence of a canyon that begins 30 km within the estuary of the river (Fig. 1 and Fig. 2), a large part of the sediment load, including organic matter, is transported directly into the deep-sea.

The modern active canyon/channel system extends 760 km westward off the Congo-Angola margin (Fig. 2), but more than 80 inactive paleo-channels have been identified [START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF][START_REF] Savoye | Geological overview of the Angola-Congo Margin, the Congo deep-sea fan and its submarine valleys[END_REF]. The canyon/channel deeply incises the continental shelf and slope, and is very meandering [START_REF] Droz | Morphology and recent evolution of the Zaire turbidite system (Gulf of Guinea)[END_REF][START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF]Fig. 2). The sinuosity of the modern channel decreases downslope from high to moderate [START_REF] Babonneau | Morphology and architecture of the present canyon and channel system of the Zaire deep-sea fan[END_REF]. A transition zone starts at about 3300 m-depth and links the canyon area to the channel-levees system [START_REF] Babonneau | Morphology and architecture of the present canyon and channel system of the Zaire deep-sea fan[END_REF]. The channel-levees system shows an east-west trend, but abrupt changes of the direction are observed locally [START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF]. The slope of the channel floor decreases from 1.5 to 0.2% down channel. The channel is over-incised, with an incision depth ranging from 200 m at 3000 m-depth to 100 m at 4300 m-depth [START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF].

Evidence for the occurrence of turbidity currents in the canyon/channel includes submarine cable breaks between 500 and 2300 m-depth, which were attributed to turbidity currents initiated by Congo floods [START_REF] Heezen | Congo submarine canyon[END_REF]. During long-term monitoring along the channel using a series of moorings with currentmeters and sediment traps, several turbiditic events were recorded between 2001 and 2004 [START_REF] Khripounoff | Direct observation of intense turbidity current activity in the Zaire submarine valley at 4000 m water depth[END_REF][START_REF] Vangriesheim | Turbidity events observed in situ along the Congo submarine channel[END_REF]. The frequency of turbiditic events is estimated at about 60 per century [START_REF] Heezen | Congo submarine canyon[END_REF], but it is unclear where those turbidity currents start and what causes them.

Turbidity currents are supposed to remain within the canyon/channel all the way down, but in the meander section the upper part of the thick turbidity currents overflows the channel flanks and built levees. Levees are evidenced by seismic surveys all along the present-day active channel. Levees are made up of overflow sedimentation of the turbidity currents on both sides of the channel with a relatively symmetrical morphology; the Coriolis force being negligible in this area given the proximity to the Equator. The lateral extent of levees decreases from 50 to 10 km down slope [START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF]. The presence of sediment waves has also been reported on both right-hand and left-hand levees [START_REF] Droz | Morphology and recent evolution of the Zaire turbidite system (Gulf of Guinea)[END_REF][START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF][START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. The frequency of overbank depositional events at the lower channellevees system is estimated at 2 per century [START_REF] Savoye | Geological overview of the Angola-Congo Margin, the Congo deep-sea fan and its submarine valleys[END_REF].

To the best of our knowledge, information about the nature of organic matter from the channel-levees systems is lacking. The main objective of the present paper is to provide insight into the nature and distribution of the organic matter at different scales in the modern channel-levees system.

Studied areas

Two areas were selected along the modern active channel in the lower part of the channel-levees system (Fig. 2). Both are composed of three cores that allow study of a transect from the channel to the external part of the levees.

Transect KZAI [4][5][6] This transect is located at 712 km from the apex of the canyon, following the main active channel, and is between 4000 and 4150 m-depth. The cores are from an area where the channel is relatively straight, with a width around 1.6 km and a depth nearly 150 m [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF]Fig. 2). Lateral extension of the south levee, studied here, is around 7 km. Levees are symmetrical and show sediment waves with NW-SE orientation, i.e. approximately at 45° of the channel direction (Fig. 2; [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF]. One core (KZAI 06) was collected in the channel at 4150 m-depth, one core (KZAI-05) was collected on the proximal part of the levee at 4012 m-depth and one core (KZAI-04) has been collected on the distal part of the levee at 4047 m-depth (Fig. 3). A seismic profile (Fig. 3) shows that core KZAI 05 is close to the crest of the south levee and that core KZAI 04 is 5 km apart where seismic facies are more continuous and bedded. It should be noted that 10 km upstream of the studied area the Congo channel forms a well-developed meander (Fig. 2) that may be a subordinate source of sediments in core KZAI 05, as suggested by the orientation of the field of sediment waves connected to this meander. [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF] indicate that sediments from cores KZAI 04 (16.9 m long) and KZAI 05 (13.3 m long) are made of clayey and silty clayey sequences showing millimetric silt beds layered with millimetric to decimetric clay beds (Fig. 3). These small sequences are fining-up and correspond to Td and Te terms of the Bouma sequence or the fine-grained turbidites defined by [START_REF] Piper | Turbidite muds and silts on deep sea fans and abyssal plains[END_REF]. Turbiditic clay cannot easily be distinguished from hemipelagic clay. Most of the components of the sediment (quartz, micas, plant fragments, benthic foraminifers) have an allochthonous origin, whereas some diatoms and sponge spicules constitute a minor autochthonous component. Both cores show the same type of sequences, although core KZAI 04 (more distant from the channel compared to core KZAI 05) shows a lower proportion of fine to medium sand fraction. This may reflect the lower sediment accumulation rate, which is about 0.79 m/ka for core KZAI 05 to 0.44 m/ka for KZAI 04 [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF].

Sediments from core KZAI 06 (3.3 m long) are made of medium (250-500 µm) to fine (250-125 µm) homogeneous sands, containing some mud clasts and large plant debris. The 20 cm at the top of this core includes silty clays with some sandy lenses [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF].

Transect KZAI 13-15

This transect is 855 km from the apex of the canyon at about 4350 to 4450 m-depth (Fig. 2). The studied levees are the left-hand flank of a meander (1 km of curvature) and show marked sediment waves. The relief is 90 m between the channel and the crest of the levee.

Crests of proximal sediment wave mimic the curvature of the meander, and then become straighter down-levee (Fig. 2).

Core KZAI 15 (1.9 m long) was collected in the channel and consists of very fine (100 µm) to very coarse (1-2 mm) sand beds with abundant scattered mud clasts (Fig. 3). Accumulation rates are very low, 0.25 m/ka according to [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF], confirming that the channel is a zone of sediment bypassing and/or erosion rather than deposition.

Cores KZAI 13 (17.9 m long) and KZAI 14 (17.3 m long) were collected respectively on the downstream and upstream flank of a sediment wave (Fig. 2). Core KZAI 13 is 0.2 km downstream from core KZAI 14, the latter being located at 1 km from the axis of the channel.

Both are made of clayey and silty clayey sequences showing millimetric silt beds layered with millimetric to decimetric clay beds (Fig. 3). Accumulation rate is difficult to estimate because some inversions of the 14 C ages are observed in both cores [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. Nevertheless, the average accumulation rate seems fluctuate from 6.5 m/ka in core KZAI 13 and 4.4 m/ka in core KZAI 14. Such values suggest that the mean accumulation rate of levees sharply increases from the area of transect KZAI 04-06 to this area, although the lithology of sediments does not change a lot (Dennielou and Jouanneau, in press;[START_REF] Savoye | Geological overview of the Angola-Congo Margin, the Congo deep-sea fan and its submarine valleys[END_REF].

Material and methods

Samples

Three hundred and sixteen (316) samples were collected from these six cores with spacing between samples varying from 2 to 20 cm (Tab. 1). Only the first 10 m of cores KZAI 04, 05, 13 and 14 were sampled (Fig. 3).

In the laboratory, the samples were washed with deionizated water to eliminate salt that could affect pyrolysis analyses. Indeed, salt generates an artefact at the end of the S 2 peak of the Rock-Eval pyrolysis (see below), which is erroneously interpretated as a fraction of refractory organic matter coming from soils [START_REF] Holtvoeth | Soil organic matter as an important contributor to late Quaternary sediments of the tropical West African continental margin[END_REF]. Then, samples were dried in an oven at 50 °C for 24 hours before being pulverized in an agate mortar.

Methods

Grain-size of the bulk sediment was measured using a Coulter LS200 (93 channels from 0.4 µm to 2 mm).

Carbonate content was measured using a Bernard calcimeter, with an analytical precision of ± 0.2% CaCO 3 .

Pyrolytic analyses were carried out using both an Oil Show Analyser and a Rock-Eval 6 Turbo device (Vinci Technologies), operating in free cycle mode which is devoted to Recent sediments [START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications[END_REF][START_REF] Lafargue | Rock-Eval 6 Applications in Hydrocarbon Exploration, Production and Soils Contamination Studies[END_REF][START_REF] Béhar | Rock-Eval 6 Technology: Performances and Developments[END_REF]. This technique provides five fundamental parameters: S 1 (representing free and adsorbed hydrocarbons released during 3 min at 180 °C, in mg/g); S 2 (pyrolytic hydrocarbons generated from 180 to 650 °C, in mg/g); S 3 (CO 2 released during the pyrolysis phase of the analysis, in mg CO 2 /g); S 4 (CO 2 released during the oxidation phase of the analysis, in mg CO 2 /g); and

Tmax (the temperature of maximum pyrolytic hydrocarbon yield, in °C). Hydrocarbons are detected by a flame ionization detector, and CO 2 by a thermal conductivity detector or infrared detector.

These fundamental parameters are used to obtain derived parameters that are: TOC (Total Organic Carbon, in wt %); Hydrogen Index (HI = S 2 /TOC × 100, in mg HC/g TOC); Oxygen Index (OI = S 3 /TOC × 100, in mg CO 2 /g TOC); and Petroleum Potential (PP = S 1 +S 2 , in mg HC/g or kg HC/ton of rock). The precision for the parameters is ± 0.1% for TOC, ± 1 °C for Tmax, ± 10 mg HC/g TOC for HI, ± 5 mg CO 2 /g TOC for OI, and ± 1 kg/t for PP. isotopic measurements. Total Organic Carbon (TOC) and Total Nitrogen (TN) were measured by high temperature combustion on a Carlo Erba NC 2500. The average standard deviation of each measurement, determined by replicate of the same sample is ± 5% Stable carbon and nitrogen isotope composition of the sedimentary organic matter was determined by on-line combustion in a Carlo Erba NC 2500 interfaced with a Isoprime isotope ratio mass-spectrometer. 13 C/ 12 C and 15 N/ 14 N are expressed by the conventional δ notation in ‰ relative to PDB and air, respectively. Analytical precision is ± 0.15 ‰ for δ 13 Corg and ± 0.2 ‰ for δ 15 N.

Thermochemolysis was carried out on 9 samples from cores KZAI 04, 05, 13 and 15 following the procedure described by [START_REF] Disnar | Assessment of peat quality by molecular and bulk geochemical analysis; application to the Holocene record of the Chautagne marsh (Haute Savoie, France)[END_REF]. Briefly, about 25 mg of dried and crushed kerogens were introduced in SVL® screw-cap glass tubes with 50 µL of standard solution (heptylbenzoic acid 21 µg/100 µL MeOH) and 100 µL tetramethyl ammonium hydroxide solution (TMAH; 25 % in MeOH). The tubes were placed open in an oven at 75 °C for 3 to 5 hours to evaporate the excess methanol, then cooled and closed under vacuum. Then they were placed vertically in a sand-bath and heated at 220 °C during 20 min. After cooling in ambient air, 1.5 mL diethylether was introduced to extract the pyrolysis products. After evaporation of the ether, the extracts were diluted in 50 or 100 µL CH 2 Cl 2 and analysed by GC-MS with a Thermo-Finnigan TRACE-Polaris GCQ gas chromatograph-mass spectrometer. The gas chromatograph was fitted with an Rtx™-5Sil MS capillary column (30 m x 0.32 mm i.d., 0.25 µm film thickness) with 5 m of guard column. The GC operating conditions were as follows: temperature hold at 40 °C for 1 min, then increase from 40 to 120 °C at 30 °C/min, 120 to 300 °C at 3 °C/min with final isothermal hold at 300 °C for 30 min.

The samples were injected splitless, with the injector temperature set at 280 °C. Helium was the carrier gas. The mass spectrometer was operated in the electron ionization (EI) mode at 70 eV ionization energy and scanned from 50 to 650 Daltons. Compounds were tentatively identified by comparison with library (NIST) mass spectra and relative retention times.

Forty-five samples were selected and prepared for palynofacies analysis, which is the microscopical study of all the particulate organic matter present in sediment. Neither oxidation nor ultrasonic probe was carried out during processing as all the particles were of interest in this study and may selectively be destroyed by such procedures. The residue was directly mounted on glass microscope slides.

The classification of palynofacies has always been rather subjective [START_REF] Tyson | Sedimentary organic matter: organic facies and palynofacies[END_REF], and for this study, the following particles were identified:

-Phytoclasts: comprise all opaque and translucent land-plant debris. Different subgroups can be observed within this fraction and they were distinguished during the counting.

-Sporomorphs: comprise the land-derived pollen grains and spores.

-Fungal debris: comprise all the filamentous segmented particles and sclerotes.

-Marine algae: comprise the dinoflagellate cysts and other organic walled algae.

-Amorphous organic matter (AOM): comprise all particulate organic components that appear structureless at the scale of light microscopy.

Results

Organic matter distribution and characteristics in the clayey-silty facies of the levees

Grain-size determinations indicate that most of the sediment is made of clayey and silty fractions (up to 80 to 90%) with a median grain-size around 10 µm for cores KZAI 04 and 05 and 15 µm for cores KZAI 13 and 14, which are located at 1 km from the axis of the channel (Fig. 4). The sandy fraction usually represents less than 15% in cores KZAI 04 and 05 and less than 20 % in cores KZAI 13 and 14, except for various short-term spikes (30% in cores KZAI 04 and 05, 40 % in core KZAI 14 and up to 50% in core KZAI 13), which correspond to coarser turbiditic events.

Calcium carbonate contents remain generally low (<6 %, Tab. 2) with an average around 2.5 %. Thus, these sediments can be regarded as virtually devoid of carbonate.

Total Organic Carbon (TOC) contents are extremely homogeneous in the four studied cores with a value around 3% (Tab. 2). Some samples, however, are richer in organic carbon with a TOC content reaching 5.6% in core KZAI 04, whereas others show values as low as 0.1 % in core KZAI 13 (Fig. 4). These organic-lean samples correspond to the sandy turbiditic facies. Except for these rare sandy facies, no relationship appears between TOC and the granulometric characteristics of the sediment.

Hydrogen Index (HI) values fluctuate between 82 and 236 mg HC/g TOC (Tab. 2) with a mean value around 145 mg HC/g TOC in the four studied cores (Fig. 4). The HI-values are extremely constant in cores KZAI 04 and 05 and show a wider range and more abrupt fluctuations in cores KZAI 13 and 14 (Fig. 4). Nevertheless, these variations are not very important if we consider that HI-values may fluctuate from 0 to 1000 [START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications[END_REF].

C/N ratios are also very homogeneous with a mean C/N ratio around 13 (range between 10.7 and 15.7). Similarly, isotopic ratios do not vary greatly with a mean δ 13 Corg around -26 ‰ (range between -23.8 and -27.1 ‰) and a mean δ 15 N around 5.5 ‰ (range between 5 and 6.3 ‰; Tab. 2 and Fig. 4).

Palynofacies analyses indicate that amorphous organic matter (AOM) represents almost the half of total particles (between 33 and 62% with an average around 46.5%). The AOM particles can be classified into two main subgroups. The first subgroup is composed of fluffy AOM (Fig. 5A) which is probably derived from phytoplankton. The second subgroup is represented by AOM particles with remains of phytoclasts inside (Fig. 5B). Variations in AOM aspect probably highlight the origin of the organic matter (marine versus continental) but also the role of the depositional setting, i.e. the intensity of oxidation.

The structured particles consist exclusively of higher plant debris at different states of preservation. Different types can be recognized from yellow coloured cuticles or woody fibres with well preserved cellular structures (Fig. 5C) to dark-brown and gelified ligno-cellulosic debris (Fig. 5D). Ligno-cellulosic particles represent between 20 and 53% (34% on average) of total particles. Black particles with different shapes, rounded or bladed, are a subordinate group of phytoclasts and represent char or highly oxidized particles from soil (Fig. 5E).

Cuticles are present at a low level (0.5 to 6%), whereas sporomorphs are rare (< 2% on average; Fig. 5F). Fungal remains are common and may represent few percents in some samples. Marine palynomorphs are scarce in the palynofacies.To summarize, the palynofacies analysis indicates that half of the organic matter is clearly continental in origin, and part of the amorphous organic matter is also derived from higher plants.

At the scale of the bulk sediment of the levees the organic matter content appears very homogeneous, both in terms of quantity (TOC) and quality (HI, C/N and isotopic ratios, palynofacies).

Organic matter distribution and characteristics in the turbiditic beds

Using detailed lithological descriptions, X-radiographs, and grain size analyses, six types of turbiditic beds have been distinguished within the levees [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF][START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. We sampled several of these beds with a 2 cm-spacing in cores KZAI 04, 13 and 14 in order to investigate the variations of organic matter content and quality within different types of small fining-up sequences.

Type 1-beds consist of highly bioturbated dark silty clays grading upward into grey clays, with thickness ranging from several centimeters to several decimeters (Fig. 6A andB).

Dark silty clays are probably true turbidite deposits whereas grey clays are probably hemipelagic deposits, but discriminating between them is difficult [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. In most cases, the grain size is near constant whatever the facies. The organic matter content of both the dark and grey clays is very homogeneous (3% in average), but dark clays may contain a little bit more organic matter than grey hemipelagic clays (Fig. 6B). Such beds resulted from fallout of suspended flocks and characterize dilute and low-velocity spillovers from the uppermost part of channelized turbidity currents [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF].

Type 2-beds show a basal unit with the alternation of infra-millimetric to millimetric silty laminae and millimetric to centimetric clayey laminae, and an upper unit with dark silty clays grading upward into grey clays (Fig. 6C andD). The basal contact is always sharp and beds present thickness ranging from 5 to 30 cm. The contact between the two units is gradational, the whole sequence is normally graded, and grain-size analyses display several normally graded sub-intervals. Bioturbation is always absent from the basal unit but commonly observed in the upper unit. The absence of bioturbation in the basal unit suggests rapid deposition by a single channelized turbidity current. Such type 2-beds result from the spillover of the upper muddy part and deeper parts of channelized flows, where silts are carried in suspension, and correspond to divisions Td and Te of the Bouma sequence [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF][START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. The organic matter content of such type 2-beds are usually high (3% TOC) except in the first coarser centimeters of the basal unit where lower TOC content are noted (1.5 to 2%; Fig. 6D).

Type 3-beds consist of three units: from base to top, dark silty clays 1-3 cm thick (unit 1), alternating silty and clayey laminae (unit 2), and grey clays (unit 3; Fig. 6E). All contacts between the three units are gradational. In unit 2, silty laminae first coarsen and thicken upward, then fine and thin upward. The whole bed exhibits a basal inverse grading and an upper normal grading. The basal contact is sharp or gradational. Bed-type 3 is uncommon and its thickness varies between 10 and 20 cm [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. Organic carbon content of such beds shows a wider range in accordance with the median grain size. As coarser the sediment as lower the organic matter content, with values as low as 0.2% in fine sands (Fig. 6E).

Type 4-beds consist of three units from base to top: a centimeter-scale silty bed passing upward to alternating silty and clayey laminae, and finally grey to beige clays (Fig. 6F). The basal silty bed is structureless, and rarely laminated or cross-laminated. The whole bed is normally graded, and the basal contact is sharp. This type 4-bed has thickness varying between 10 and 30 cm and result from processes similar to those described for deposition of type 2-beds [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. Organic carbon content is almost constant (3 % TOC) within this type 4-beds except at the basal part of the small sequence when grain size increases (Fig. 6F).

Types 5 and 6-beds are rare, in contrast to the classical fine-grained deposits commonly described on levees as they correspond to structureless Ta and/or laminated Tb of the Bouma sequence with maximum grain-size up to 400 µm [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. They have not studied so far for their organic matter content.

Whatever the small changes in organic matter content within the small-scale fining-up sequences described above, they are not associated with lower HI-values. On the contrary, both HI and OI-values are rather constant (Fig. 6), which indicates a similar type of organic matter and probably a comparable preservational state of the organic particles in these different facies.

Organic matter in the sandy facies of the channel

The sandy facies from the channel have been analyzed in core KZAI 15 and only 4 samples were investigated. Even if we cannot generalize these observations, it appears that the sands contain two times less organic carbon than the clayey facies (Tab. 2). The HI-values of the sandy facies from the channel are comparable to that of levee facies (Tab. 2) and geochemical data indicate the same molecular proportion of continental organic material and the same state of degradation compounds from lignin (see TMAH results in section 5.1 and Tab. 4). This suggests that size sorting, which explain the difference in TOC between channel and levee, did not influence the quality of particulate organic sediment. Any generalization of these observations for the entire Congo deep-sea turbiditic system is limited because of the small number of the samples in our study.

Discussion

Sources of organic matter and estimation of the continental organic carbon contribution

The identification and quantification of sources, whether continental or marine, of organic carbon in the sediment are always difficult because several parameters influence the bulk and molecular characteristics of the preserved organic matter. In order to estimate the amount of continental organic matter relative to total organic matter in the Congo deep sea fan sediments we used binary mixing models with different sets of proxies (HI, C/N ratio, δ 13 Corg and δ 15 N), defined as follow:

Fter = [(Xsample-Xmar)/(Xter-Xmar)] x 100
Where Fter is the terrestrial organic carbon fraction and Xsample depends on the proxy analysed. Xmar and Xter are the marine and terrestrial end-member values, respectively, of these proxies.

Information on the origin of organic matter can be determined by pyrolytic measurements if the thermal evolution of organic matter is low [START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications[END_REF][START_REF] Peters | Guidelines for evaluating petroleum source rock using programmed pyrolysis[END_REF]. Measured Tmax values are always low (< 420°C), indicating that organic matter did not experience strong thermal maturation and therefore contains very little charcoals or recycled material from older mature rocks.

Although the type of organic matter is usually defined by the mean of elemental analysis, the Hydrogen Index (HI) parameter approximates the H/C atomic ratio, which determines the organic matter type [START_REF] Tissot | Petroleum formation and occurrence[END_REF]. According to the low range of HI values (82 to 236 mg HC/g TOC, Tab. 2), the organic matter of the studied samples could be attributed mainly to Types III to IV (Fig. 7). Type IV, however, appears as a subordinate type for some samples (Fig. 7). Type III is usually related to continental higher plants debris, whereas Type IV corresponds to residual deeply altered organic matter; the origin of which is difficult to determine.

If we assume that HI value for fresh marine organic matter is around 400 mg HC/g TOC [START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications[END_REF] and around 100 for the terrigeneous end-member (mixture of Types III-IV; [START_REF] Espitalié | La pyrolyse Rock-Eval et ses applications[END_REF], the mean Fter value calculated in the studied cores would indicate that 85% of the organic matter is derived from the detrital source (Tab. 3). Taking into consideration the highest HI values measured on every core (Tab. 2), the continental contribution drops to 67% on average (Tab. 3), which still indicates a major contribution of continental organic matter to the Congo deep sea fan sediments.

The carbon isotopic composition remains also remarkably constant with a range of δ 13 Corg between -23.8 and -27.1 ‰ (Fig. 4) and an mean-value around -26 ‰ (Tab. 2). Such values illustrate the dominant influence of organic matter inherited from plants with C3 photosynthetic pathway and are in the same range of the particulate organic matter measured in the Congo River (δ 13 Corg = -26.7 ± 0.4 ‰, [START_REF] Mariotti | Carbon isotope composition and geochemistry of particulate organic matter in the Congo River (Central Africa): application to the study of Quaternary sediments off the mouth of the river[END_REF][START_REF] Holmes | Stable nitrogen isotopes in Angola Basin surface sediments[END_REF]. If we assume that δ 13 Corg of marine organic matter is -21 ‰ for that part of the equatorial Ocean [START_REF] Tyson | Sedimentary organic matter: organic facies and palynofacies[END_REF], and if the detritial end-member presents a mean δ 13 Corg = -26.7 ‰ as reported by [START_REF] Mariotti | Carbon isotope composition and geochemistry of particulate organic matter in the Congo River (Central Africa): application to the study of Quaternary sediments off the mouth of the river[END_REF], the Fter values calculated in the studied cores would indicate that 93% of the organic matter derives from the detrital source (Tab. 3). This assumption is probably not fully valid, because terrigeneous organic matter fraction is more resistant than the marine fraction. Consequently this terrigeneous input, although important, is probably overestimated.

The relationship between total N (TN) and TOC reveals an intercept that is close to the origin of the cross-plot (Fig. 8), consistent with the idea that most of the nitrogen is associated with organic matter. If we assume an organic origin for nitrogen, the C/N ratios for the studied area range between 10.7 and 15.5 (Tab. 2; Fig. 4). Such values are also indicative of a mixture of marine phytoplankton (C/N~6.7 for Redfield ratio) and terrigeneous sources, which contain lower nitrogen than marine organic matter. The almost constant average C/N ratio (~13) in the different studied cores would require that the relative proportion of these two sources remain constant.

Molar C/N ratios of samples and end members are often used in linear mixing equations to estimate the fraction of continentally derived organic carbon in sedimentary environments. Several authors have shown that this calculation actually yields the fraction of terrestrially derived organic nitrogen (Perdue and Koprivnjak, 2007 and references therein).

Because continental organic matter is relatively depleted in nitrogen, the fraction of terrestrially derived organic carbon has been seriously and systematically underestimated by this misinterpretation of C/N mixing lines. Only the mixing equation based on N/C yields the true fraction of continentally derived organic carbon.

We choose N/C ratios of 0.0653 and 0.142 as end-members for terrestrial and marine sources, Xter and Xmar respectively. The Xter value chosen here is derived from the C/N ratio (15.3) reported by [START_REF] Mariotti | Carbon isotope composition and geochemistry of particulate organic matter in the Congo River (Central Africa): application to the study of Quaternary sediments off the mouth of the river[END_REF] for suspended particulate organic matter in the load of the Congo River near its estuary, whereas the marine end-member is close to the reverse of the C/N for Redfield ratio. Taking these values, the organic matter preserved in sediments from the levees is clearly dominated by continental organic matter which represents almost 85 % of the preserved organic carbon (Tab. 3).

As δ 13 Corg and N/C ratio, the nitrogen isotopic composition of sedimentary organic matter (δ 15 N) has the potential to provide information on the source of organic matter to the ocean [START_REF] Calvert | Geochemical and isotopic evidence for postglacial paleoceanographic changes in Saanich Inlet[END_REF]. δ 15 N records the isotopic composition of the substrate (nitrate, ammonium or dinitrogen) and the fractionation between the two isotopes 14 N and 15 N that occurs during photosynthetic pathways so that δ 15 N will depend on the relative utilisation of the nitrogen source [START_REF] Altabet | Sedimentary nitrogen isotopic ratio as a recorder for surface ocean nitrate utilization[END_REF]. Since nitrate and atmospheric dinitrogen have different isotopic signature (respectively ~5‰ and 0‰) and are the main sources of nitrogen respectively for marine and terrestrial plants, the nitrogen isotopic signature of marine plants and algae is heavier (>4-5‰) than for terrestrial plants (<1‰). Source identification can be complicated in oceanic sedimentary systems when bacterial denitrification occurs in oxygen-depleted waters or when dinitrogen fixation occurs in oceanic waters. However, these processes are negligible in our area off Congo (see [START_REF] Holmes | Stable nitrogen isotopes in Angola Basin surface sediments[END_REF]. Finally, the same conclusion may be inferred from the nitrogen isotopic ratios (δ 15 N around 5.5‰) which indicate a mixture of marine plants and C3 plants (Fig. 9). The continental fraction deduced from δ 15 N data represents around 44% of the total organic matter (Tab. 3). This inference for a dominance of land-derived organic matter in levee facies from the Congo fan, based on low δ 13 Corg and HI-values, is in conflict with the nitrogen isotope values which are too heavy (<15) for such a dominance. The nitrogen isotopic composition of sedimentary organic matter in the deep-sea fan of Congo must be considered with caution. In a previous study in the same area, [START_REF] Müller | Late Quaternary pCO2 variations in the Angola Current: evidence from organic carbon δ 13 C and alkenone temperatures[END_REF] and [START_REF] Holmes | Stable nitrogen isotopes in Angola Basin surface sediments[END_REF] reported already δ 13 Corg typical of a mixing between continental and marine plants whereas δ 15 N values were close to a marine end-member. In addition, higher than expected δ 15 N values were reported in the Congo estuary [START_REF] Holmes | Stable nitrogen isotopes in Angola Basin surface sediments[END_REF]. This apparent contradiction in source characterization between carbon isotopic composition, N/C ratio and Rock-Eval, and nitrogen isotopic ratios is evidence that the marine organic matter contributes significantly to the organic content of these sediments. As algal marine hydrogen-rich organic matter (Type II) is oxidized, its hydrogen content decreases while its oxygen content increases, and it may take on Rock-Eval characteristics of Type III organic matter. The fact that samples having high TOC content show HI-value reaching 256 mg HC/g TOC supports this hypothesis and suggests that the algal organic matter was deeply oxidized by microbial reworking. Finally, optical investigations of palynofacies reveal high proportion of fluffy amorphous organic matter (Fig. 5A) which is usually derived from phytoplanktonic sources [START_REF] Tyson | Sedimentary organic matter: organic facies and palynofacies[END_REF]. The quantification using HIvalues underestimates the real contribution of the marine source to the enrichment in organic matter.

The thermochemolysis of 9 samples provides another way to estimate the continental versus marine contribution to the organic matter. All chromatograms are similar to that shown on Fig. 10; the main quantitative data are given in Table 4. Compound distributions are dominated by fatty acid methyl esters (FAMEs) and lignin degradation products: vanillic and syringic acids and aldehydes, coumaric and ferulic acids, plus non lignitic pHO-benzoic acid and aldehyde. FAME distributions are dominated by even carbon numbered FA from C 22 to C 34 , with a mode at n-C 24 (Fig. 10). These compounds are also accompanied by lower but still notable proportions of odd-numbered n-alkanes from C 27 to C 31 , with a predominance of n-C 29 (Fig. 10). All these compounds are classically assumed to be typical components of higher plants, the C20 + FAMEs and the high molecular weight n-alkanes being common constituents of aerial plants cuticular waxes. Relatively high proportions of syringic moieties (i.e. acid plus aldehyde) over their vanillic counterparts (S/V = 0.3-0.61; Tab. 4) reveal that lignin originates from Angiosperms, with a contribution from Gymnosperms being not to be excluded [START_REF] Hedges | The characterization of plant tissues by their lignin oxidation products[END_REF]. Rather extensive lignin alteration is revealed by the predominance of vanillic and syringic acids over the corresponding aldehydes [(Vac+Sac)/(Vald+Sald) = 1.74 to 2.56; Tab. 4), by the high contribution of acid moieties to the total lignin (Vac+Sac)/Ltot = 0.55 to 0.65; table 4) and also by very low proportions of coumaric and ferulic acids (data not shown), these two compounds originally linked to the lignin backbone by labile ester bonds being rather easily degraded during early diagenesis [START_REF] Bourdon | Organic matter sources and early diagenetic degradation in a tropical peaty marsh (Tritrivakely, Madagascar). Implications for environmental reconstruction during the Sub-Atlantic[END_REF]. The importance of the continental organic matter contribution to the sediments is also assessed by rather high lignin and C20 + even-carbon numbered FAMEs concentrations, amounting to 32-48 mg.g -1 TOC and 5.7 -26.4 mg g -1 TOC, respectively (table 4). For comparison, recent analysis of Holocene peat by the same methods [START_REF] Disnar | Assessment of peat quality by molecular and bulk geochemical analysis; application to the Holocene record of the Chautagne marsh (Haute Savoie, France)[END_REF] yielded lignin and C20 + FAME concentrations of 4-20 mg.g -1 TOC and 2-7 mg g -1 TOC, respectively. The higher yields obtained with Congo sediments is very probably due the loss of the more labile constituents (e.g. polysaccharides), during diagenesis.

The presence of FAMEs from C 16 and C 18 is noted in all chromatograms. Such compounds are usually produced by both algae and higher plants but they are rapidly destroyed in aerial condition and soils [START_REF] Marseille | n-Alkanes and free fatty acids in humus and A1 horizons of soils under beech, spruce and grass in the Massif Central (Mont-Lozère) France[END_REF]. Then, in sediments they can be assumed to derive from autochthonous (marine) production. Hopanoids components are ubiquitous in all analyzed samples, although in very low proportion. This suggests a contribution of bacterial biomass, which may have developed in soil as well as within marine sediments.

To summarize, the recent mud-rich sediments from the Congo deep-sea turbidite system contain high amounts of organic matter, the origin of which is a mixture of terrestrial higher-plant debris and deeply oxidized phytoplanktonic material. Although we have evidence that both sources contribute to the organic matter sedimentation in the Congo deepsea fan, their relative importance cannot be precisely quantified. In any case, the continental fraction of the organic matter appears very important (at least 70-80%) especially for such depths and distance from the coast. A similar proportion (60 %) of continental organic matter was recently reported in the sediments from the GeoB6518-1 core recovered by 962 m of water-depth close to the canyon/upper channel-levees transition (Weijers et al., 2009). This similarity is surprising in that the sampling sites are at very different depths and processes that allow the accumulation of sediments on the edge of the canyon and along the middle part of channel-levees system are quite different [START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF]. This suggests a homogeneous distribution of the particulate organic matter delivered by the Congo River in the different parts of the deep-sea fan.

Controls on the distribution and accumulation of organic matter in channel-levees

Turbidite emplacement is discontinuous and usually produces layers of sediment in which organic matter quantity, type and preservational state are heterogeneous [START_REF] Cowie | Elemental and major biochemical changes across an oxidation front in a relict turbidite: An oxygen effect[END_REF][START_REF] Meyers | Effects of turbidity flows on organic matter accumulation, sulfate reduction, and methane generation in deep-sea sediments on the Iberia Abyssal Plain[END_REF][START_REF] Watanabe | Characterization of organic matter in the Miocene turbidites and hemipelagic mudstones in the Niigata oil field, central Japan[END_REF][START_REF] Lindblom | Organic geochemistry of lipids in marine sediments in the Canary Basin: Implications for origin and accumulation of organic matter[END_REF][START_REF] Saller | Leaves in turbidite sands: the main source of oil and gas in the deep-water Kutei Basin, Indonesia[END_REF][START_REF] Caja | Significance of organic matter in Eocene turbidite sediments (SE Pyrenees, Spain)[END_REF]. By contrast, in the levees from the Congo mud-rich deep-sea fan, the organic matter appears very homogeneous regardless of the scale.

At the scale of a turbiditic small sequence, the organic matter shows more or less a constant quantity (3 %) and quality according to the pyrolysis parameters HI and OI. This is mainly due to the fine-grain composition of the turbidite off Congo which are mainly made by clay and fine silt fractions (10 µm as median grain-size). As organic matter particles are less dense than minerals, they are mainly associated with fine-grain component of the sediment.

Organic matter particles have larger size in sandy facies but they are less numerous; then the organic content is usually lower at the base of sandy turbidites. Because the sandy fraction is a minor component of the studied levees, the organic matter content is near constant from the base to the top of a single turbiditic bed. Although, turbiditic clays cannot easily be distinguished from hemipelagic clays in such fine mud-rich system, the latter have been analysed at least in several levels and they show changes neither in quantity nor in type of organic matter. This implies that the hemipelagic sedimentary flux has more or less a similar characteristic for organic matter than the turbiditic input or, at least, that the preserved organic matter from hemipelagic sedimentation finally achieves the same characteristics.

Indeed, [START_REF] Treignier | Terrestrial and marine n-alcohol inputs and degradation processes relating to a sudden turbidity current in the Zaire canyon[END_REF] studied the organic matter content and composition of sediment trapped 30 m above the channel before and just after a turbiditic event, as well as the surficial sediment sampled nine months after the event at the same water depth (~ 4000 m) than our studied area. This sediment contained 4.2 % TOC and showed a predominance of long-chain n-alcohols typical for higher plant waxes, in the free lipid extracts. Despite the predominance of these compounds in cuticular waxes, and especially over long chain fatty acids and n-alkanes, they were absent among the thermochemolysis products of the nine samples we analyzed (Fig. 10).

This absence is undoubtedly due to the microbial degradation of these labile compounds during early diagenesis. [START_REF] Treignier | Terrestrial and marine n-alcohol inputs and degradation processes relating to a sudden turbidity current in the Zaire canyon[END_REF] estimated a degradation constant value comprised between 0.6 and 1.2 y -1 for the C22 + n-alcohols that is rather considerable. The fate of these compounds fully illustrates the considerable changes that can affect molecular signatures during sedimentation and diagenesis and the importance of the choice of markers used for assessing the origin and relative importance of original organic matter inputs to sediments.

The quantity of organic matter is near constant in the transect 700 km from the apex of the canyon (KZAI 04-05) compared to the transect located 150 km downstream (KZAI 13-14), although the sediment accumulation rate increases by one order of magnitude between these two areas (0.4 to 0.8 m/ka against 4.4 to 6.5 m/ka; [START_REF] Savoye | Geological overview of the Angola-Congo Margin, the Congo deep-sea fan and its submarine valleys[END_REF]. However, these sedimentation rates are both important for deep-sea environments. Such values indicate that, despite long distance transportation and/or deposition at rather great water depth, sediment emplacement occurred rapidly which is a suitable condition for organic matter preservation.

Although the difference in the channel depth (140 m at KZAI 04-05 site and 90 m at KZAI 14-15 site) generates ten times more spillovers downstream [START_REF] Savoye | Geological overview of the Angola-Congo Margin, the Congo deep-sea fan and its submarine valleys[END_REF], the near constant quantity of organic matter suggests that the composition of the spillovers of the channelized flows, where organic matter, clay minerals and silts are carried in suspension, have a more or less similar composition in these different type of particles and that in the channelized flow, between 90 m and 140 m height (50 m thickness), the turbulence does not generate a significant segregation of organic particles. As the lithology of sediments does not change a lot between the two areas, this assumption seems valid.

More surprising is the fact that changes in sediment accumulation rates do not imply changes in the quality of the preserved organic matter. Usually, low sedimentation rates determine a longer residence time of the organic particles in the oxygenated zone near the water-sediment interface and consequently the organic matter is more easily remineralised. In contrast, high sedimentation rate led to a better preservation of organic particles. Here the organic matter quality parameters (HI, OI, C/N, δ 13 Corg and δ 15 N) appear very constant along both the upstream and downstream transect.

This may be due to the fact that continental organic particles, which are the main component of the organic matter in Congo deep sea fan sediments, are more resistant than marine organic matter. Marine organic matter may represent 40% of the preserved organic matter but is also deeply and rapidly altered after deposition as shown by [START_REF] Treignier | Terrestrial and marine n-alcohol inputs and degradation processes relating to a sudden turbidity current in the Zaire canyon[END_REF] on the n-alcohol fraction. Thus, changes in marine contribution during settling have little impact on the final composition of the organic matter of the sediment. It seems that after its deposition, the organic mixture of continental and marine particles is homogenised by oxidative alteration and exhibits constant quality parameters.

A minor change is noted, however, in the HI and OI values with the distance from the channel. Samples from cores KZAI 05 and 14 which are located closer to the channel display higher HI-and lower OI-values compared to samples from cores KZAI 04 and 13 which are 500 to 5000 m separated from the counterpart. The changes are minor (Δ = -40 mg HC/g TOC for HI and +40 mg CO 2 /g TOC for OI) but significant if we consider the distribution of analysed samples on a HI-OI diagram (Fig. 11). Moreover, the dispersion of samples is more important in the IH-IO diagram in cores located far from the channel. This is maybe the indication that changes in accumulation rates have already a little impact on the quality of organic matter.

Implication for deep-sea petroleum source rocks analogs

TOC is the primary parameter of source rock appraisal, with a threshold of 1 wt% at the immature stage for potential source rocks. With an average TOC value that largely exceeds this threshold, the fine-grained turbiditic sediments in the Congo deep-sea fan could be regarded as good future gas-prone source rocks. Mean petroleum potential range between 4 and 4.7 kg HC per t rock and are mainly gas-prone as the organic matter is primarily of continental origin.

The shaly levees are the main sedimentary facies that built the present-day Congo deep sea fan. The volume of potential source rock is here very important, especially if we consider their proximity to the sandy reservoirs located in the channel filling. Such organicrich facies may also a source for biogenic gas, which is not yet fully characterized. Methanerich vents are numerous in the Congo deep sea fan and the degassing of studied cores has been noted on board the research vessel just after their recovery.

Fossil analogs of the present-day deep sea fan are distributed along the Atlantic passive margins throughout the geological times, especially in the Congo-Angola basin.

Ancient deep-sea fans are now buried and such facies may be regarded as contributors to the regional petroleum systems. A recent study of the Oligocene succession off Angola demonstrates the petroleum potential of deep-sea claystones deposited as levee facies (Disnar et al., in press).

Implication for the global carbon cycle

The transfer of carbon from land to sea has been recognized as an important pathway in the global carbon cycle [START_REF] Milliman | Flux and fate of fluvial sediment and water in coastal seas[END_REF][START_REF] Burdige | Burial of terrestrial organic matter in marine sediments; a re-assessment[END_REF]. Rivers play a major role and transport most of the ~500 Tg per year of organic carbon carried from land to the global ocean [START_REF] Spitzy | Dissolved and particulate organic matter in rivers[END_REF]. One critical aspect in the ocean carbon budget is the particulate export flux from the coastal zone to the open ocean [START_REF] Biscaye | Fluxes of particulate matter on the slope of the southern Middle Atlantic Bight: SEEP-II[END_REF][START_REF] Goni | Sources and contribution of terrigenous organic carbon to surface sediments in the Gulf of Mexico[END_REF]Hedges et al., 1997;[START_REF] Andersson | Shallow-water oceans: a source or sink of atmospheric CO 2 ? Front[END_REF][START_REF] Dagg | Transformation of dissolved and particulate materials on continental shelves influenced by large rivers: plume processes[END_REF]. Several programs have addressed this question on different types of oceanic margin and have shown the diversity of the modes of transfer (i) in nature with continental versus marine carbon, (ii) in space with canyons playing a major role in channelling the particulate flux in some places, (iii) in time with storms, instabilities, current surge playing a significant role in transporting particles and carbon from the shelf to the slope and open ocean.

In the case of the Congo deep-sea fan, the main part of the sedimentary load, including organic matter, is transported directly from the river mouth to the deep-sea because the canyon starts within the estuary. Our results indicate that high concentration of organic matter (3% on average) is preserved in the shaly facies developed all along the lower channel/levee systems. These facies are the main deposits through time, as channel sandy facies represent only 20% on seismic profiles [START_REF] Droz | Morphology and recent evolution of the Zaire turbidite system (Gulf of Guinea)[END_REF][START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF]. Thus, huge amount of organic matter is preserved in the deep-sea fan system of the Congo. [START_REF] Anka | Reconnaissance study of the ancient Zaire (Congo) deep-sea fan (ZaiAngo Project)[END_REF] estimate the volume of ancient fan related to the Congo River from Oligocene to Present to a minimum of 0.7 10 6 km 3 . Taking an average of 3% of Corg and 1500 kg/m 3 for the density of dry sediment, we may calculate that 3 10 13 tons of carbon are stored in that part of the Atlantic margin.

If we assume that the present-day delivery of particulate organic matter by the Congo River (1 mg C/L and a mean flow around 50 000 m 3 /s) was constant through time and that this particulate organic flux was totally preserved in the sediment of the margin, we calculate that 5.4 10 13 t of carbon should be stored since 34 Ma. Despite our roughly assumptions, both estimations are in the same range and the magnitude is probably correct.

The Congo deep-sea turbiditic system is one of the largest in the world still affected by turbidite sedimentation during the interglacial high sea-level [START_REF] Savoye | Structure et évolution récente de l'éventail turbiditique du Zaire : premiers résultats scientifiques des missions d'exploration Zaiango1 & 2 (marge Congo-Angola)[END_REF]. Such a phenomenon is unique along the present-day African margin as most of the canyons are not connected with the river. During the glacial periods, when sea-level was low, other huge rivers, including the Amazon River, were directly connected to their submarine canyons and carbon transfer from land to deep sea was much more higher than today. Indeed, the presentday situation of the Congo River cannot be generalized to low seal-level period and further studies are needed to evaluate the efficiency of such systems for the storage of continental organic matter into the deep ocean at that time.

Conclusion

The claystones and siltstones deposited in the lower channel/levees system of the Congo deep sea fan are virtually devoid of carbonate and contain a high proportion of organic matter, with total organic carbon content around 3 wt%. This organic richness appears very homogeneous at different scales, although smaller quantities are noted in more sand-rich facies corresponding to the basal part of a turbiditic event.

The identification and quantification organic carbon sources are difficult because several parameters influence the bulk and molecular characteristics of the preserved organic matter. Hydrogen Index values, carbon isotopic ratios and molecular data, all indicate a strong influence from the detrital source, with proportion as high as 90%. C/N ratios and nitrogen isotopic ratios, on the contrary, suggest that marine contribution to the organic matter sedimentation may represent up to 40 %. Although evidence exist that both sources contribute to the organic matter sedimentation in the Congo deep-sea fan, their relative importance cannot be precisely quantified. In any case, the continental fraction of the organic matter appears very important (at least 60%) for such depth and distance from the coast.

The quantity and quality of organic matter is near constant in cores located at 700 km from the apex of the canyon compared to cores located 150 km downstream, although the sediment accumulation rate increases by one order of magnitude between these two areas. This suggests that the spillovers of the channelized flows, where organic matter, clay minerals and silts are carried in suspension, have a more or less similar composition in these different types of particles. Change in accumulation rates has already a little impact on the quality of organic matter, as higher plant and soil organic matter particles, which seem to be dominant, are more resistant than marine organic matter.

With a TOC content of 3% and petroleum potential around 4.5 kg HC per t rock, the fine-grained turbiditic sediments in the Congo deep-sea fan could be regarded as good future gas-prone source rocks. Analogs of the present-day deep sea fan are distributed along the Atlantic passive margins throughout the geological times, and should be considered as potential source rocks for the deep offshore realm.

Finally, the Congo deep-sea fan is a major conveyor of organic carbon to the deep Atlantic Ocean in the present-day high sea-level situation. This system where the canyon is connected to the estuary can be generalized to other deep sea fans during low seal-level periods. Further studies are needed to evaluate the efficiency of such systems for the storage of continental organic matter into the deep ocean. [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF][START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF]. Bottom: core logs showing the sediment grain-size changes and the sampled interval along each core, redrawn after [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF], [START_REF] Gervais | Recent processes of levee formation on the Zaire deep-sea fan[END_REF] and Migeon (2000). A and B: type1-bed, C and D: type 2-bed, E: type 3-bed, F: type 4-bed following the nomenclature defined by [START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]. Tab. 2. Bulk inorganic and organic characteristics of the studied samples in each studied core.

Captions of figures and tables

Minimum, mean and maximum values are given for CaCO 3 (in %), TOC (in %), Hydrogen Index (HI, in mg HC/g TOC), C/N, δ 13 Corg (in ‰) and δ 15 N (in ‰).

(in ‰) and δ 15 N (in ‰). Sac, Sald & S = syringic acid, aldehyde and total, respectively; Vac, Vald & V = vanillic acid, aldehyde and total, respectively; Ltot = total lignin = S + V + (coumaric and ferulic acids); *Compound concentrations in mg g -1 TOC. Note that sample from core KZAI 15 corresponds to a sandy facies from the channel.

Fig. 1 .

 1 Fig. 1. General map of central Africa and the Gulf of Guinea in eastern equatorial Atlantic Ocean, showing the course of the Congo River, its main distributaries and its drainage area (watershed), the surface and subsurface currents (open arrows: cold currents; black arrows: warm currents), and highly productive areas (the latters modified from Schneider et al., 1994).

Fig. 2 .

 2 Fig. 2. Top: bathymetric map (contour interval: 100 m) showing the general morphology of the Congo canyon and the modern meandering channel along the Congo fan. Limits of the canyon, upper fan, channel/levee systems and lobes are from Babonneau et al. (2002). Studied areas are located in the lower channel/levee systems. Bottom: Detailed bathymetric map (contour interval: 10 m) and location of studied cores within the two studied areas, modified after Gervais et al. (2001),[START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF].

Fig. 3 .

 3 Fig.3. Top: line drawings of 3.5 kHz profiles showing the internal structures of the studied levees and location of studied cores along these schematic cross-sections[START_REF] Migeon | Processes of sedimentwave contruction along the present-day Zaire deep-sea meandering channel: role of meanders and flow stripping[END_REF]

Fig. 4 .

 4 Fig. 4. Vertical distribution of grain-size parameters (granulometric composition, median grain-size), Total Organic Carbon (TOC, in weight %), Hydrogen Index (in mg HC/g TOC), C/N and δ 13 Corg (in ‰) of the four cores studied in the levee facies with a 20 cm-spacing. Core location is shown on figs. 3 and 4.

Fig. 5 .

 5 Fig. 5. Microphotographs of palynofacies of the fine grained sediments of the levees from the Congo deep sea fan near 4000 m-depth. a: Amorphous organic matter (AOM) having a fluffy aspect. b: AOM containing small ligno-cellulosic debris. c: Structured phytoclast, apparently derived from root cortex tissues. d: Partly-gelified ligno-cellulosic debris on which vegetal fibre are still discernible. e: Opaque particle with corroded outlines on the right and small piece of structurless phytoclast on the left. f: Spore.

Fig. 6 .

 6 Fig.6. Vertical distribution of median grain-size (in µm), Total Organic Carbon (TOC, in weight %), Hydrogen and Oxygen Indexes (in mg HC/g TOC and mg CO 2 /g TOC, respectively) in several elemental turbiditic beds from cores KZAI 04, 14 and 15 with a 2 cmspacing. Core location is shown on figs 3 and 4.

Fig. 7 .

 7 Fig. 7. Kerogen type in the silty-clayey sediments of the lower channel/levee systems from the Congo fan as defined by the cross-plot of TOC and pyrolysis S 2 parameters. Nearly all samples are located in the domain of Type III organic matter which usually derives from higher plant debris.

Fig. 8 .

 8 Fig. 8. Relationship between total organic carbon and total nitrogen for three core sediments from the lower channel-levees system of the Congo fan. The linear regression line based on all data points (R²=0.81) is indicated by the dashed line.

Fig. 9 .

 9 Fig. 9. Kerogen type in the silty-clayey sediments of the lower channel/levee systems from the Congo fan as defined by the cross-plot of δ 13 Corg (in ‰) and δ 15 N (in ‰). Here the organic matter derives from a mixture of C3 land plants and marine algae.

Fig. 10 .

 10 Fig. 10. Example of partial reconstituted Total Ion Current (TIC) chromatogram of the products of thermochemolysis with TMAH of sample from the silty-clayey sediments of the lower channel/levee systems from the Congo fan (sample KZAI 05 -540-541 cm). The dominance of lignin-derived compounds (L), long-chain and odd-numbered n-alkanes, and long-chain fatty acid methyl esters (FAMEs) indicate the predominance of terrestrial organic matter. Nevertheless, nC16:0 and nC18:0 FAMEs may be interpreted as derived from algal biomass whereas the presence of hopanoid testifies from a bacterial origin of some part of the organic matter. L = lignin derivatives and others phenolic compounds, all as Me ester and ethers (in increasing elution order) : pHO-benzaldehyde, pHO-benzoic acid, vanillin, vanillic acid (dominant), syringaldehyde, coumaric acid, syringic acid, ferulic acid.

Fig. 11 .

 11 Fig. 11. van Krevelen diagram showing the homogeneity of the organic matter in the siltyclayey sediments of the lower channel/levee systems from the Congo fan. The samples which are more distant from the axis of the channel show lower HI-values and higher OI-values indicating a little bit stronger oxidation effect.

  Figure 1

Figure 6 Figure 7 Figure 8

 678 Figure 6

Figure 9 Figure 10

 910 Figure 9

Figure 11

 11 Figure 11

Table 1 .

 1 Location of the studied cores (Lat., Long., water depth), penetration into the sediments, length of studied interval and number of studied samples.

		Core	Lat. (°S)	Long. (°E)	Water depth (mbsl)	Situation	Penetration (mbsf)	Studied interval (m)	Number of samples	Mean spacing (m)
	Upstream	KZAI 04	5° 48.04	8° 08.96	4047	Levee	16.87	10.0	83	0.12
		KZAI 05	5° 44.50	8° 08.29	4012	Crest of levee	13.34	9.8	49	0.20
		KZAI 06	5° 44.10	8° 08.23	4150	Channel axis	3.24	0.2	1	-
	Downstream	KZAI 13	5° 47.30	7° 13.89	4447	Levee	17.92	9.6	79	0.12
		KZAI 14	5° 47.09	7° 13.98	4343	Crest of levee	17.3	10.0	100	0.10
		KZAI 15	5° 46.62	7° 14.19	4433	Channel axis	1.93	0.6	4	0.15

  Tab. 3. Estimates of continental organic matter fraction (in %) of the total organic matter in marine sediments from KZAI cores based on the binary mixing models of different proxies. . Tab. 4 . Main data on lignin-derived products and on FAMEs released by thermochemolysis Sac, Sald & S = syringic acid, aldehyde and total, respectively Vac, Vald & V = vanillic acid, aldehyde and total, respectively Ltot = total lignin = S + V + (coumaric and ferulic acids) *Compound concentrations in mg.g -1 TOC. Note that sample from core KZAI 15 corresponds to a sandy facies from the channel.Main data on lignin-derived products and on FAMEs released by thermochemolysis.

	Cor e	Dept h (cm)	TO C (%)	Cor e S/ V	Situati on (Vac + Sac)/L Numbe r of sample s tot	CaCO 3 (%) (Vac + Sac)/(Vald + S ald)	TOC (%) Ltot (nC16 + nC18)FAM HI (mg HC/g TOC) C20 + evenFAM Numbe r of sample * Es* Es*	C/N			δ 13 C (%o)		δ 15 N (%o)
	KZA I 04	297-298	2.60	0.3 3	0.60	analys ed	1.74			36.2 5.4				5.7	s
	KZA I 05	2-3	2.40	0.4 5	0.62			Mi n 2.02	Mea n	Ma x	Mi n 40.1 8.2 Mea n	Ma x	Mi n	Mea n	Ma x 16.5	analys ed	Mi n	Mea n	Ma x	Min	Mea n	Max	Mi n	Mea n	Ma x
	Upstream 540-541 3.28 KZAI 04 89 KZA I 05 Core Proxy KZ AI 04 0.5 9 KZ AI 05 Mean HI Max HI N/C δ 13 C (%o) δ 15 N (%o) Levee 83 1.5 2.7 6.0 1.4 3.0 0.56 1.83 45.0 7.6 67 84 94 45 Crest of levee 49 2.0 2.6 3.3 2.4 3.1 KZA I 05 960-961 3.45 0.3 3 0.62 2.05 33.9 7.2	5.6 3.7	93 115 148 134	200 17 26.4 189 16 12.5	10. 7 12. 6	12.9 13.1 6	15. 5 13. 7	-27.1 2 -26.9 6	-26.3 7 26.61 -26.3 -23.8 5	5.1 2 5.2 3	5.44 5.61	6.0 9 1 6.3
	KZAI 05 84 KZA I 13 540-541 3.20	0.6 1	70 0.59		86	98 2.37		42	46.3 7.4				18.7
	Downstre am KZAI 13 88 KZA I 13 926-927 2.42	KZ AI 13 0.5 7	Levee 76 0.55	79	86	0.8 2.6 87 1.75	3.3 44	0.1 2.97 48.2 3.8	4.0 4	82	137	171 16 13.7	11. 8	13.0 9	15. 7	-26.1 7	-25.9 4	-25.4 6	4.9 8	5.47	5.9 8
	KZAI 14 79 KZAI 15 85 KZA I 13 928-929 2.42	KZ AI 14 0.5 6	Crest of levee 55 73 0.55	100	1.3 3 1.81	2.5	3.2 5	0.8 5 48.4 5.4 3.08 3.9	128 163	236 15.8
	Average 85 KZA I 15 4-5 1.59	KZ AI 15 0.3 6	67 Channe l 0.61	4	85	93 0.7 1.88 1.98	44 3.1 8	0.0 1.48 34.4 7.5	3 3.6	118 145	182 13.0
	Cor e	Dept h (cm)	TO C (%)	S/ V	(Vac + Sac)/L tot	(Vac + Sac)/(Vald + S ald)	Ltot *	(nC16 + nC18)FAM Es*	C20 + evenFAM Es*
	KZA I 04	1-2	1.54	0.3 0	0.65			2.56			32.2 8.9				19.5

Acknowledgements

The present study was funded by the GDR Marges Program. We acknowledge TOTAL Company for the permission to use core material. We acknowledge Marielle Hatton, Didier Kéravis and Florence Savignac for their analytical help and Alexandre Lethiers for drawings.

We are also grateful to Phil Meyers and an anonymous reviewer for constructive comments.