Eduardo Abi Jaber
email: eduardo.abijaber@axa-im.com

Bruno Bouchard
email: bouchard@ceremade.dauphine.fr.

Camille Illand
email: camille.illand@axa-im.com

Eduardo Abi Jaber
email: abijaber@ceremade.dauphine.fr

Stochastic invariance of closed sets with non-Lipschitz coefficients

Keywords: Stochastic differential equation, stochastic invariance, affine diffusions, polynomial diffusions. MSC Classification: 93E03, 60H10

come

Introduction

Let b : R d → R d and σ : R d → M d be continuous functions, where M d denotes the space of d × d matrices. We assume that b and σ satisfy the following linear growth conditions: there exists L > 0 such that b(x) + σσ ⊤ (x)

1 2 ≤ L(1 + x ), ∀ x ∈ R d , ( H 1 )
and we consider a weak solution of the stochastic differential equation

dX t = b(X t )dt + σ(X t )dW t , X 0 = x, ( 1.1)
i.e. a d-dimensional Brownian motion W and an adapted process X such that the above equation holds.

The aim of this paper is to provide a characterization of the stochastic invariance of a closed set D ⊂ R d , i.e. find necessary and sufficient conditions on the instantaneous drift b and the instantaneous covariance matrix σσ ⊤ under which there exists a weak solution of (1.1) that remains in D for all t ≥ 0, almost surely, given that x ∈ D. (See Definition 2.2 below for a precise formulation.)

The first stochastic invariance results can be found in Stroock and Varadhan [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF], Friedman [START_REF] Friedman | Stochastic differential equations and applications[END_REF] and Doss [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF] . Since then, many extensions were considered in the literature. For an arbitrary closed set, the stochastic invariance was characterized through the second order normal cone in Bardi and Goatin [START_REF] Bardi | Invariant sets for controlled degenerate diffusions: a viscosity solutions approach[END_REF] and Bardi and Jensen [START_REF] Bardi | A geometric characterization of viable sets for controlled degenerate diffusions[END_REF]. Aubin and Doss [START_REF] Aubin | Characterization of stochastic viability of any nonsmooth set involving its generalized contingent curvature[END_REF] used the notion of curvature, while Da Prato and Frankowska [START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF] provided a characterization in terms of the Stratonovich drift. For a closed convex set, the distance function was used in Da Prato and Frankowska [START_REF] Da Prato | Stochastic viability of convex sets[END_REF], and the invariance was characterized for affine jump-diffusions in Tappe [START_REF] Tappe | Stochastic invariance of closed, convex sets with respect to jumpdiffusions[END_REF]. Although these approaches differ, they have at least one thing in common: the tradeoff one has to make between the assumptions on the topology/smoothness of the domain and the regularity of the coefficients b and σ. This makes all of these existing results difficult to apply in practice.

Let us start by highlighting this difficulty through the two main contributions to the literature:

(i) In Bardi and Jensen [START_REF] Bardi | A geometric characterization of viable sets for controlled degenerate diffusions[END_REF], the stochastic invariance is characterized by using Nagumo-type geometric conditions on the second order normal cone. Their main result states that the closed set D is stochastically invariant if and only if

u ⊤ b(x) + 1 2
Tr(vC(x)) ≤ 0, ∀ x ∈ D and (u, v) ∈ N 2 D (x), in which C := σσ ⊤ on D and N 2 D (x) is the second order normal cone at the point x:

N 2 D (x) := (u, v) ∈ R d × S d : u, y -x + 1 2 y -x, v(y -x) ≤ o( y -x 2 ), ∀ y ∈ D .
(1.2) Here, S d stands for the cone of symmetric d × d matrices. In practice, we face two restrictions. Prior to deriving the conditions on b and σ, we have to determine the second order normal cone at all points of a given set. When the boundary is smooth, the computation of the second order normal cone is an easy task, see e.g. [5, Example 1]. However, it is much more challenging in general, by lack of efficient techniques. This renders the result of [START_REF] Bardi | A geometric characterization of viable sets for controlled degenerate diffusions[END_REF] difficult to use in practice. This also corresponds to the positive maximum principle of Ethier and Kurtz [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF].

(ii) Building on Doss [START_REF] Doss | Liens entre équations différentielles stochastiques et ordinaires[END_REF], Da Prato and Frankowska [START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF] give necessary and sufficient conditions for the stochastic invariance in terms of the Stratonovich drift and the first order normal cone:

σ(x) ⊤ u = 0 and u, b(x) - 1 2 d j=1 Dσ j (x)σ j (x) ≤ 0, ∀ x ∈ D and u ∈ N 1 D (x), (1.3)
where σ j (x) denotes the j-th column of the matrix σ(x), Dσ j is the Jacobian of σ j , and the first order normal cone N 1 D (x) at x (sometimes simply called normal cone) is defined as

N 1 D (x) := u ∈ R d : u, y -x ≤ o( y -x ), ∀ y ∈ D . (1.4)
In practice, the first order normal cone is much simpler to compute than the second order cone used in [START_REF] Bardi | A geometric characterization of viable sets for controlled degenerate diffusions[END_REF], see [START_REF] Aubin | Set-valued analysis[END_REF] and [START_REF] Rockafellar | Variational analysis[END_REF]. However, the price to pay is to impose a strong regularity condition on the diffusion matrix σ, which is assumed to be bounded and differentiable on R d , with a bounded Lipschitz derivative. Again, this constitutes a sticking point for applications, it cannot be applied to simple models (think about square-root processes for instance, see below).

The aim of the present paper is to extend the characterization (1.3), given in terms of the easyto-compute first order normal cone, under weaker regularity conditions on the diffusion matrix σ. We make the following seemingly trivial observation: C := σσ ⊤ might be differentiable at a point x while σ is not. It is the case for the square-root process mentioned above, at the boundary point x = 0. Moreover, the terms Dσ j (x)σ j (x) can be rewritten in terms of the Jacobian of C whenever both quantities are well defined, see Proposition 2.4 for a precise formulation. This suggests to reformulate (1.3) with the Jacobian matrices of the columns of C instead of σ.

We prove that this is actually possible. Our main result, Theorem 2.3 below, states that the stochastic invariance is equivalent to the following conditions:

C(x)u = 0 and u, b(x) - 1 2 d j=1 DC j (x)(CC + ) j (x) ≤ 0, ∀ x ∈ D and u ∈ N 1 D (x). (1.5)
Here, (CC + ) j (x) is the j-th column of (CC + )(x) with C(x) + defined as the Moore-Penrose pseudoinverse of C(x), see Definition A.1 in the Appendix. We only assume that

C can be extended to a C 1,1 loc (R d , S d ) function that coincides with σσ ⊤ on D, ( H 2 )
in which C 1,1 loc means C 1 with a locally Lipschitz derivative. Note that we do not impose the extension of C to be positive semi-definite outside D, so that σ might only match with its square-root on D. Also, it should be clear that the extension needs only to be local around D.

The term CC + in (1.5) plays the role of the projection on the image of C, see Proposition A.3 in the Appendix and the discussion in Remark 2.5 below. This projection term cannot be removed. To see this, let us consider the square-root process with C(x) = x and

D = R + , so that N 1 D (0) = R -. Then, C(0)(-1) = 0 and -1, b(0) - 1 2 DC(0) ≤ 0
leads to b(0) ≥ 1/2 while the correct condition for invariance is b(0) ≥ 0, which is recovered from (1.5) by using the fact that (CC + )(0) = 0.

This extension of the characterization of Da Prato and Frankowska [START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF] provides for the first time a unified criteria for the case where the volatility matrix may not be C 1 on the whole domain, which is of importance in practical situations. In fact, many models used in practice, in mathematical finance for instance, do not have C 1 volatility maps but satisfy our conditions. This is in particular the case of affine diffusions (see [START_REF] Duffie | Affine processes and applications in finance[END_REF][START_REF] Filipović | Affine diffusion processes: theory and applications[END_REF]), or of polynomial diffusions that are characterized by a quadratic covariance matrix (see [START_REF] Cuchiero | Polynomial processes and their applications to mathematical finance[END_REF][START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF]), etc. When applied to such processes, stochastic invariance results have been so far tweaked in order to fit in the previous set up, or have been proved under limiting conditions, on a case by case basis. For instance, in their construction of affine processes on the cone of symmetric semi-definite matrices, Cuchiero et al. [START_REF] Cuchiero | Affine processes on positive semidefinite matrices[END_REF] start by regularizing the martingale problem before applying the stochastic invariance characterization of [START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF] and then pass to the limit. In Spreij and Veerman [START_REF] Spreij | Affine diffusions with non-canonical state space[END_REF], some stochastic invariance results are also derived for affine diffusions but only on convex sets with smooth boundary. More recently, the mathematical foundation for polynomial diffusions is given in Filipović and Larsson [START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF]. Necessary conditions for the stochastic invariance are derived for basic closed semialgebraic sets. However, these conditions are not sharp, their sufficient conditions differ from their necessary conditions. All the above cases can now be treated by using our characterization. See Section 5 for a generic example.

Our proof of the necessary condition is in the spirit of Buckdahn et al. [START_REF] Buckdahn | Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems[END_REF]. They use a second order stochastic Taylor expansion together with small time behavior results for double stochastic integrals. However, in our case, the stochastic Taylor expansion cannot be applied directly since σ is not differentiable and σ(X) fails to be a semi-martingale whenever an eigenvalue vanishes (see [START_REF] Mijatović | On the loss of the semimartingale property at the hitting time of a level[END_REF]Example 1.2]). We therefore need to develop new ideas. We first observe that, if σ is diagonal, then vanishing eigenvalues can be eliminated by taking the conditional expectation with respect to the path of the Brownian motion acting on the non-vanishing ones. This corresponds to the projection term CC + in (1.5). If σ is not diagonal, we can essentially reduce to the former case by considering its spectral decomposition and a suitable change of Brownian motion (based on the corresponding basis change), see Lemma 3.2 below. However, it requires a smooth spectral decomposition which is not guaranteed when repeated eigenvalues are present.

To avoid this, we need an additional transformation of the state space, see Proposition 3.5.

Conversely, we show that the infinitesimal generator of our diffusion satisfies the positive maximum principle whenever (1.5) holds, see Section 4 below. Applying [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF]Theorem 4.5.4] shows that this condition is indeed sufficient. (Note that the approach based on the comparison principle for viscosity solutions used in [START_REF] Bardi | A geometric characterization of viable sets for controlled degenerate diffusions[END_REF][START_REF] Buckdahn | Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems[END_REF] cannot be applied to our case since σ is not Lipschitz.)

The rest of the paper is organized as follows. Our main result is stated in Section 2. The proofs are collected in Sections 3 and 4. In Section 5, we exemplify our characterization by deriving explicit stochastic invariance conditions for various typical examples of applications. Finally, Section 6 provides a complementary tractable sufficient condition ensuring the stochastic invariance of the interior of a domain. For the convenience of the reader, we collect some standard results of matrix calculus and differentiation in the Appendix.

From now on, all identities involving random variables have to be considered in the a.s. sense, the probability space and the probability measure being given by the context. Elements of R d are viewed as column vectors. The vector e i ∈ R d is the i-th element of the canonical basis, and we use the standard notation I d to denote the d × d identity matrix. We denote by M d the collection of d×d matrices. We say that A ∈ S d (resp. S d + ) if it is a symmetric (resp. and positive semi-definite) element of M d . Given x = (x 1 , . . . , x d ) ∈ R d , diag [x] denotes the diagonal matrix whose i-th diagonal component is x i . If A is a symmetric positive semi-definite matrix, then A 1 2 stands for its symmetric square-root.

Main result

In this section, we state our main result, Theorem 2.3, that extends Theorem 4.1 in Da Prato and Frankowska [START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF] to weaker regularity assumptions.

Since we are dealing with general coefficients b and σ, i.e. not necessarily Lipschitz coefficients, solutions to the stochastic differential equation (1.1) should be considered in the weak sense rather than in the strong sense. Existence is guaranteed by our condition (H 1 ), together with our standing assumption of continuity of b and σ: there exist a filtered probability space (Ω, F, F =(F t ) t≥0 , P) satisfying the usual conditions, a d-dimensional F-Brownian motion W and a F-adapted process X with continuous sample paths such that (1.1) holds P-a.s. See e.g. [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF]Theorems IV.2.3 and IV.2.4]. For later use, note that (H 1 ) implies that, for any positive integer p, there exists

K p,x > 0 such that E [ X t -X s p ] ≤ K p,x |t -s| p 2 (2.1)
for all 0 ≤ s, t ≤ 1. Hence, Kolmogorov's continuity criterion ensures that the sample paths of X are (locally) η-Hölder continuous for any η ∈ (0, 1 2 ) (up to considering a suitable modification). Remark 2.1. The collection Q of possible distributions of X is entirely determined by the infinitesimal generator L defined on the space of smooth functions φ by Lφ := Dφ b+ 1 2 Tr[σσ ⊤ D 2 φ]. Therefore, Q is the same if σ is replaced by σ such that σσ ⊤ = σσ ⊤ , see e.g. [START_REF] Stroock | Multidimensional diffusion processes[END_REF]Remark 5.1.7]. Hence, we can reduce to the case where σ is the symmetric square-root of C on D, which we will assume from now on.

Before stating our main result, let us make precise the definition of stochastic invariance.

Definition 2.2 (Stochastic invariance).

A closed subset D ⊂ R d is said to be stochastically invariant with respect to the diffusion (1.1) if, for all x ∈ D, there exists a weak solution (X, W ) to (1.1) starting at X 0 = x such that X t ∈ D for all t ≥ 0, almost surely.

Our characterization of stochastic invariance reads as follows (see Propositions 3.5 and 4.2 below for the proof). From now on we use the same notation C for C defined as σσ ⊤ on D and for its extension defined in Assumption (H 2 ).

Theorem 2.3 (Invariance characterization).

Let D be closed. Assume that b, σ and C are continuous and satisfy assumptions (H 1 )-(H 2 ). Then, the set D is stochastically invariant with respect to the diffusion (1.1) if and only if

       C(x)u = 0 (2.2a) u, b(x) - 1 2 d j=1 DC j (x)(CC + ) j (x) ≤ 0 (2.2b)
for every x ∈ D and for all u ∈ N 1 D (x). Clearly, the regularity conditions of Theorem 2.3 are much weaker than those of Theorem 4.1 in Da Prato and Frankowska [START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF]. Let us immediately exemplify this by considering the case of the square-root process already mentioned in the introduction. Let D = R + , C(x) = η 2 x with η > 0, and consider the diffusion

dX t = b(X t )dt + η √ X t dW t . Since C(x)C(x) + = 1 {x>0} and N 1 R + (x) = 1 {x=0} R -, Theorem 2.3 implies that R + is stochastically invariant if and only if b(0) ≥ 0, while σ : x ∈ R + → η √ x is not differentiable at 0.
On the other hand, one can easily recover [START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF]Theorem 4.1] under their smoothness assumptions. This is the object of the next proposition (recall that, by Remark 2.1, the study can be reduced to the case C = σ 2 on D).

Proposition 2.4. Fix σ ∈ C 1,1 b (R d , S d ) (i.

e. σ is differentiable with a bounded and a globally Lipschitz derivative). Then

C := σ 2 ∈ C 1,1 loc (R d , S d + ) and u, d j=1 Dσ j (x)σ j (x) = u, d j=1 DC j (x)(CC + ) j (x) , for all x ∈ D and u ∈ Ker σ(x).
Proof. Fix x ∈ D and u ∈ Ker σ(x). By using Definition A.7 and Proposition A.8 in the Appendix, we first compute that

DC(x) = D(σ(x) 2 ) = (σ(x) ⊗ I d )Dσ(x) + (I d ⊗ σ(x))Dσ(x),
which clearly shows that C is C 1,1 loc . It then follows from Proposition A.5 and the fact that u ∈ Ker σ(x) that 2 (use the spectral decomposition of σ as in Proposition A.2). Using Proposition A.5 again, the above implies that

(I d ⊗ u ⊤ )DC(x)C(x)C(x) + = (σ(x) ⊗ u ⊤ )Dσ(x)C(x)C(x) + . Observe now that C(x)C(x) + σ(x) = σ(x) since C(x) = σ(x)
Tr (I d ⊗ u ⊤ )DC(x)C(x)C(x) + = Tr σ(x)(I d ⊗ u ⊤ )Dσ(x)C(x)C(x) + = Tr (I d ⊗ u ⊤ )Dσ(x)σ(x) .
Then, by Proposition A.5 and A.8, u,

d j=1 Dσ j (x)σ j (x) = d j=1 u ⊤ D(σ(x)e j )σ(x)e j = d j=1 u ⊤ (e ⊤ j ⊗ I d )Dσ(x)σ(x)e j = d j=1 e ⊤ j (I d ⊗ u ⊤ )Dσ(x)σ(x)e j = Tr (I d ⊗ u ⊤ )Dσ(x)σ(x) = Tr (I d ⊗ u ⊤ )DC(x)C(x)C(x) + = u, d j=1 DC j (x)(CC + ) j (x) , (2.3)
in which the last identity follows by reproducing exactly the same computations in the reverse order with C in place of σ.

The following provides another formulation of (2.2b) that highlights the notion of projection on the image of C.

Remark 2.5 (Interpretation of the projection formulation). Fix x ∈ ∂D and assume that the spectral decomposition of C at x takes the form

C(x) = Q(x)diag [λ 1 (x), . . . , λ r (x), 0, . . . , 0] Q(x) ⊤ , where Q(x)Q(x) ⊤ = I d and λ j (x) > 0 for all 1 ≤ j ≤ r.
Hence, the r-first columns of Q(x), denoted by (q 1 , . . . , q r ) = (q 1 (x), . . . , q r (x)), span the image of C(x) and the projection matrix on the image of C(x) is given by C(x)C(x) + = r j=1 q j q ⊤ j , see Propositions A.3 and A.2 in the Appendix and recall that q j is a column vector. Thus, by (2.3) in the proof of Proposition 2.4 and Proposition A.5 in the Appendix, u,

d j=1 DC j (x)(CC + ) j (x) = Tr (I d ⊗ u ⊤ )DC(x)C(x)C(x) + = r j=1 Tr (I d ⊗ u ⊤ )DC(x)q j q ⊤ j = r j=1 Tr q ⊤ j (I d ⊗ u ⊤ )DC(x)q j = r j=1 u ⊤ (q ⊤ j ⊗ I d )DC(x)q j so that, by Proposition A.8, u, d j=1 DC j (x)(CC + ) j (x) = u, r j=1 D(Cq j )(x)q j = u, r j=1 D q j (Cq j )(x)
in which D q j is the directional derivative with respect to q j :

D q j (Cq j )(x) := lim t→0 C(x + tq j )q j -C(x)q j t . Therefore (2.2b) reads u, b(x)-1 2 r j=1 D q j (Cq j )(x) ≤ 0.
Otherwise stated, C is first projected onto the basis of the image of C(x) before being derived only in the directions of (q 1 , ..., q r ). This is clearly consistent with (2.2a) that states that there cannot be any transverse diffusion of C(x) to the boundary. Therefore, the drift b(x) should only compensate the tangential diffusion given by the projection onto the image of C(x) in order to keep the diffusion in the domain.

Let us conclude this section with an additional comment for the jump-diffusion case.

Remark 2.6 (Adding jumps).

Note that jumps could be included in the dynamics of X. Based on the current work, we provide in [START_REF] Abi Jaber | Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients[END_REF] an extension of the first order characterization of Theorem 2.3 to the jump-diffusion case. We also derive an equivalent formulation in the semimartingale framework.

Necessary conditions

In this section, we prove that the conditions of Theorem 2.3 are necessary for D to be invariant. Our general strategy is similar to [START_REF] Buckdahn | Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems[END_REF].

We fix x ∈ D and we consider a smooth function

φ : R d → R such that max D φ = φ(x). Since D is stochastically invariant, let X be a D-valued solution starting from X 0 = x. In particular, φ(X t ) ≤ φ(x), for all t ≥ 0. Then, if σ is sufficiently smooth, by applying Itô's Lemma twice, we obtain t 0 Lφ(X s )ds + t 0 Dφσ(x) + s 0 L(Dφσ)(X r )dr + s 0 D(Dφσ)σ(X r )dW r ⊤ dW s ≤ 0.
Recall Remark 2.1 for the definition of the infinitesimal generator L. Given (now standard) estimates on the small time behavior of single and double stochastic integrals, see e.g. [START_REF] Buckdahn | Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems[END_REF][START_REF] Cheridito | Small time path behavior of double stochastic integrals and applications to stochastic control[END_REF], this readily implies

Dφ(x)σ(x) = 0 and Dφ(x), b(x) - 1 2 d j=1
Dσ j (x)σ j (x) ≤ 0, under appropriate regularity conditions. It remains to choose a suitable test function φ, i.e. such that Dφ(x) = u ⊤ , to deduce that (2.2a)-(2.2b) must hold when σ is differentiable, recall Proposition 2.4.

In our setting, one can however not differentiate σ j in general. To surround this problem the above can be rewritten in term of the covariance matrix C. The projection term in (2.2a)-(2.2b) will appear through a conditioning argument.

In order to separate the difficulties, we shall first consider the case where C admits a locally smooth spectral decomposition. The general case will be handled in Section 3.2 below.

The case of distinct eigenvalues

As mentioned above, we shall first make profit of having distinct eigenvalues before considering the general case. The main idea consists in using the spectral decomposition of C in the form QΛQ ⊤ in which Q is an orthogonal matrix and Λ is diagonal positive semi-definite. Then, the dynamics of X can be written as

dX t = b(X t )dt + Q(X t )Λ(X t ) 1 2 dB t in which B = • 0 Q(X s ) ⊤ dW s is a Brownian motion.
If Q and Λ are smooth enough, then we can apply the same ideas as the one exposed at the beginning of this section. An additional localization and conditioning argument will allow us to reduce to the case where Λ has only (strictly) positive entries. Note that eigenvalues and the eigenvectors can always be chosen measurable. However, multiple eigenvalues and their corresponding eigenvectors can fail to have the same regularity as C. To ensure a sufficient regularity, we therefore assume in the following Lemma that non-zero eigenvalues are distinct. The general case will be treated later, thanks to a change of variable argument, see Section 3.2 below.

Lemma 3.1. Assume that C ∈ C 1,1 loc (R d , S d ). Let x ∈ D be such that the spectral decomposition of C(x) is given by C(x) = Q(x)diag [λ 1 (x), . . . , λ r (x), 0, . . . , 0] Q(x) ⊤ (3.1)
with λ 1 (x) > λ 2 (x) > • • • > λ r (x) > 0 and Q(x)Q(x) ⊤ = I d , r ≤ d.
Then there exist an open (bounded) neighborhood N (x) of x and two measurable

M d -valued functions on R d , y → Q(y) := [q 1 (y) • • • q d (y)] and y → Λ(y) := diag [λ 1 (y), . . . , λ d (y)] such that (i) C(y) = Q(y)Λ(y)Q(y) ⊤ and Q(y)Q(y) ⊤ = I d , for all y ∈ R d , (ii) λ 1 (y) > λ 2 (y) > ... > λ r (y) > max{λ i (x), r + 1 ≤ i ≤ d} ∨ 0, for all y ∈ N (x), (iii) σ : y → Q(y) Λ(y) 1 2 is C 1,1 (N (x), M d ), in which Q := [q 1 • • • q r 0 • • • 0] and Λ = diag[λ 1 , ..., λ r , 0, ..., 0].
Moreover, we have:

u, d j=1 Dσ j (x)σ j (x) = u, d j=1 DC j (x)(CC + ) j (x) , for all u ∈ Ker(C(x)). (3.2)
Proof. Note that the fact that (q i ) i≤d can be chosen measurable is guaranteed when (C, Λ) is measurable by the fact that each eigenvector solves a quadratic minimization problem, see e.g. [

u ⊤ Q(x) = u ⊤ σ(x) = 0.
Since C := σσ ⊤ is differentiable at x, the product rule of Proposition A.8 combined with Proposition A.5 yields

(I d ⊗ u ⊤ )D C(x) = (I d ⊗ u ⊤ ) (σ(x) ⊗ I d )Dσ(x) + (I d ⊗ σ(x))Dσ(x) ⊤ = (σ(x) ⊗ u ⊤ )Dσ(x) = σ(x)(I d ⊗ u ⊤ )Dσ(x). Observing that C = σσ ⊤ = C Q Q⊤ and that Q(x) Q(x) ⊤ = C(x)C(x) + ,
we get by similar computations:

(I d ⊗ u ⊤ )D C(x) = (I d ⊗ u ⊤ ) (C(x)C(x) + ⊗ I d )DC(x) + (I d ⊗ C(x))D Q Q⊤ (x) = C(x)C(x) + (I d ⊗ u ⊤ )DC(x).
Combining the above leads to

Tr (I d ⊗ u ⊤ )Dσ(x)σ(x) = Tr (I d ⊗ u ⊤ )DC(x)C(x)C(x) + ,
which proves (3.2) by similar computations as in the proof of (2.3).

We can now adapt the arguments of [START_REF] Buckdahn | Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems[END_REF]. In the following we use the notion of proximal normals. A vector u ∈ R d is said to be a proximal normal to D at a point x if u = d D (x + u), where d D is the distance function to D. We denote by N 1,prox D (x) the cone spanned by all proximal normals. Note however that (2.2a)-(2.2b) holds at x for all proximal normals u ∈ N 1,prox D (x) if and only if it holds for all u ∈ N 1 D (x). Indeed,

N 1,prox D (x) ⊂ N 1 D (x) ⊂ co lim sup D∋y→x N 1,prox D (y) , ( 3.3)
where lim sup stands for the Painlevé-Kuratowski upper limit (see e.g. [START_REF] Aubin | Set-valued analysis[END_REF][START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF]) and co is the closed convex hull (see also [ Proof. It follows from the discussion before our lemma that it suffices to prove our claim for u ∈ N 1,prox D (x). Let (X, W ) denote a weak solution starting at

X 0 = x such that X t ∈ D for all t ≥ 0. If x / ∈ ∂D, then N 1,prox D (x) = {0}
and there is nothing to prove. We therefore assume from now on that x ∈ ∂D. We fix u ∈ N 1,prox D (x).

Step 1. We first claim that there exists a function Step 2. Since D is invariant under the diffusion X, φ(X t ) ≤ φ(x), for all t ≥ 0. From now on, we use the notations of Lemma 3.1. By the above and Itô's lemma:

φ ∈ C ∞ b (R d , R) with compact support in N (x) such that max
0 ≥ t 0 Lφ(X s )ds + t 0 Dφ(X s )σ(X s )dW s = t 0 Lφ(X s )ds + t 0 (DφQΛ 1 2 Q ⊤ )(X s )dW s in which L is the infinitesimal generator of X. Let us define the Brownian motion B = • 0 Q(X s ) ⊤ dW s , recall that Q is orthogonal, together with B = Λ(x)Λ(x) + B = (B 1 , .., B r , 0, ..., 0) ⊤ and B⊥ = (I d -Λ(x)Λ(x) + )B = (0, ..., 0, B r+1 , ..., B d ), recall Proposition A.2. Since Q Λ 1 2 = QΛ 1
2 , the above inequality can be written in the form

0 ≥ t 0 Lφ(X s )ds + t 0 Dφ(X s )σ(X s )d Bs + t 0 (DφQΛ 1 2 )(X s )d B⊥ s .
Let (F B s ) s≥0 be the completed filtration generated by B. By [28, Corollaries 2 and 3 of Theorem 5.13], [START_REF] Kurtz | Lectures on Stochastic Analysis[END_REF]Lemma 14.2], and the fact that the martingale B⊥ is independent of B, we obtain

0 ≥ t 0 E F B s [Lφ(X s )]ds + t 0 E F B s [Dφ(X s )σ(X s )]d Bs = t 0 E F B s [Lφ(X s )]ds + t 0 E F B s [Dφ(X s )σ(X s )]dB s ,
where the last equality holds because the (dr) columns of σ are 0. We now apply Lemma 3.3 below to (Dφσ)(X) and use [28, Corollaries 2 and 3 of Theorem 5.13] and [START_REF] Kurtz | Lectures on Stochastic Analysis[END_REF]Lemma 14.2] again to find a bounded adapted process η such that

0 ≥ t 0 θ s ds + t 0 α + s 0 β r dr + s 0 γ r dB r ⊤ dB s (3.4)
where

θ := E F B • [Lφ(X • )] , α ⊤ := (Dφσ)(x) = u ⊤ Q(x)Λ(x) 1 2 β := E F B • [D(Dφσ)(X • )b(X • ) + η • ] , γ := E F B • [D(Dφσ)σ(X • )] ,
recall from Step 1 that Dφ(x) = u ⊤ .

Step 3. We now check that we can apply Lemma 3.4 below. First note that all the above processes are bounded. This follows from Lemma 3.1, (H 1 ) and the fact that φ has compact support. In addition, given T > 0, the independence of the increments of B implies that

θ s = E F B T
[Lφ(X s )] for all s ≤ T . It follows that θ is a.s. continuous at 0.

Similarly, γ = E F B T [D(Dφσ)σ(X • )] on [0, T ]. Moreover, since Dφσ is C 1,1 , F := D(Dφσ)σ is
Lipschitz and Jensen's inequality combined with (2.1) implies that we can find L ′ > 0 such that

E γ s -γ r 4 ≤ E F (X s ) -F (X r ) 4 ≤ L ′ |s -r| 2 , for all 0 ≤ s, r ≤ 1.
By Kolmogorov's continuity criterion, up to considering a suitable modification, γ has ǫ-Hölder sample paths for all 0 < ǫ < 1 4 , in particular t 0 γ sγ 0 2 ds = O(t 1+ǫ ) for 0 < ǫ < 1 2 .

Step 4. In view of Step 3, we can apply Lemma 3.4 to (3.4) to deduce that α = 0 and θ 0 -

1 2 Tr(γ 0 ) ≤ 0. Multiplying the first equation by Λ(x) 1 2 Q ⊤ (x) implies that 0 = α ⊤ Λ(x) 1 2 Q ⊤ (x) = u ⊤ Q(x)Λ(x) 1 2 Λ(x) 1 2 Q ⊤ (x) = u ⊤ C(x)
+ (I d ⊗ u ⊤ )Dσσ (x) = u ⊤ b(x) - 1 2 Tr (I d ⊗ u ⊤ )Dσσ (x),
which is equivalent to (2.2b) by (3.2) and similar computations as in the proof of (2.3).

The rest of this section is dedicated to the proof of the two technical lemmas that were used above. Our first result is a slight extension of Itô's lemma to only C 1,1 function. It is based on a simple application of Komlós lemma (note that the assumption that f has a compact support in the following is just for convenience, it can obviously be removed by a localization argument, in which case the process η is only locally bounded).

Lemma 3.3. Assume that b and σ are continuous and that there exists a solution (X, W ) to

(1.1). Let f ∈ C 1,1 (R d , R) have compact support.
Then, there exists an adapted bounded process η such that

f (X t ) = f (x) + t 0 (Df (X s )b(X s ) + η s ) ds + t 0 Df (X s )σ(X s )dW s for all t ≥ 0.
Proof. Since f ∈ C 1,1 has a compact support, we can find a sequence (f n ) n in C ∞ with compact support (uniformly) and a constant K > 0 such that

(i) D 2 f n ≤ K, (ii) f n -f + Df n -Df ≤ K
n , for all n ≥ 1. This is obtained by considering a simple mollification of f . By applying Itô's Lemma to f n (X), we get

f n (X t ) = f n (x) + t 0 Df n (X s )b(X s )ds + t 0 η n s ds + t 0 Df n (X s )σ(X s )dW s in which η n := 1 2 Tr[D 2 f n σσ ⊤ ](X). Since σσ ⊤ is continuous, (i) above implies that (η n ) n is uniformly bounded in L ∞ (dt × dP). By [17, Theorem 1.3], there exists ( η n ) ∈ Conv(η k , k ≥ n) such that η n → η dt ⊗ dP almost surely. Let N n ≥ 0 and (λ n k ) n≤k≤Nn ⊂ [0, 1] be such that η n = Nn k=n λ n k η k and Nn k=n λ n k = 1. Set f n := Nn k=n λ n k f k . Then, f n (X t ) = f n (x) + t 0 D f n (X s )b(X s )ds + t 0 η n s ds + t 0 D f n (X s )σ(X s )dW s . (3.5)
By dominated convergence, t 0 η n s ds converges a.s. to t 0 η s ds. Moreover, (ii) implies that

f n (X t ) -f (X t ) ≤ Nn k=n λ n k f k (X t ) -f (X t ) ≤ Nn k=n λ n k K k ≤ K n , so that f n (X t ) converges a.s. to f (X t ). Similarly, t 0 D f n (X s )σ(X s )dW s → t 0 Df (X s )σ(X s )dW s and t 0 D f n (X s )b(X s )ds → t 0 Df (X s )b(X s )ds
in L 2 (Ω, F, P) as n → ∞, and therefore a.s. after possibly considering a subsequence. It thus remains to send n → ∞ in (3.5) to obtain the required result.

The following adapts [9, Lemma 2.1] to our setting, see also [START_REF] Bruder | Super-replication of European options with a derivative asset under constrained finite variation strategies[END_REF][START_REF] Cheridito | Small time path behavior of double stochastic integrals and applications to stochastic control[END_REF][START_REF] Cheridito | The multi-dimensional superreplication problem under gamma constraints[END_REF].

Lemma 3.4. Let (W t ) t≥0 denote a standard d-dimensional Brownian motion on a filtered probability space (Ω, F, (F t ) t≥0 , P). Let α ∈ R d and (β t ) t≥0 , (γ t ) t≥0 and (θ t ) t≥0 be adapted processes taking values respectively in R d , M d and R and satisfying

(1) β is bounded,

(2) t 0 γ s 2 ds < ∞, for all t ≥ 0,

(3) there exists a random variable η > 0 such that a.s.

t 0 γ s -γ 0 2 ds = O(t 1+η ) for t → 0, (3.6)
(4) θ is a.s. continuous at 0.

Suppose that for all t ≥ 0

t 0 θ s ds + t 0 α + s 0 β r dr + s 0 γ r dW r ⊤ dW s ≤ 0. (3.7)
Then,

(a) α = 0, (b) -γ 0 ∈ S d + , (c) θ 0 -1 2 Tr(γ 0 ) ≤ 0. Proof. Since (W i t ) 2 = 2 t 0 W i s dW i s + t, (3.7) reduces to (θ 0 - 1 2 Tr(γ 0 ))t + d i=1 α i W i t + d i=1 γ ii 0 2 (W i t ) 2 + 1≤i =j≤d γ ij 0 t 0 W i s dW j s + R t ≤ 0,
where

R t = t 0 (θ s -θ 0 )ds + t 0 s 0 β r dr ⊤ dW s + t 0 s 0 (γ r -γ 0 )dW r ⊤ dW s =: R 1 t + R 2 t + R 3 t .
In

The general case

We can now turn to the general case. Proof. If x lies in the interior of D, then N 1 D (x) = {0} and there is nothing to prove. We therefore assume from now on that x ∈ ∂D. Let Λ and Q be defined through the spectral decomposition of C, as in (3.1) but with only λ 1 (x) ≥ • • • ≥ λ d (x). We shall perform a change of variable to reduce to the conditions of Lemma 3.2. To do this, we fix 0 < ǫ < 1 and define

A ǫ = Q(x)diag (1 -ǫ), (1 -ǫ) 2 , . . . , (1 -ǫ) d Q(x) ⊤ .
Since D is invariant with respect to the diffusion X, D ǫ := A ǫ D is invariant with respect to the diffusion X ǫ := A ǫ X. Note that

dX ǫ = b ǫ (X ǫ )dt + C ε (X ǫ ) 1 2 dW in which b ǫ := A ǫ b((A ǫ ) -1 •) and C ǫ := A ǫ C((A ǫ ) -1 •)(A ǫ ) ⊤
have the same regularity and growth as b and C. Moreover, the positive eigenvalues of C ǫ are all distinct at

x ǫ := A ǫ x, as C ǫ (x ε ) = Q(x)diag (1 -ǫ)λ 1 (x), . . . , (1 -ǫ) d λ d (x) Q(x) ⊤ .
We can therefore apply Lemma 3.2 to (X ǫ , D ǫ ):

       C ǫ (x ǫ )u ǫ = 0 (3.8a) u ǫ , b ǫ (x ǫ ) - 1 2 d j=1 DC j ǫ (x ǫ )(C ǫ C + ǫ ) j (x ǫ ) ≤ 0 (3.8b)
for all u ǫ ∈ N 1 A ǫ D (x ǫ ). We now easily verify that

N 1 A ǫ D (x ǫ ) = (A ǫ ) -1 N 1 D (x)
, recall the definition in (1.4). Finally, by sending ǫ → 0 in (3.8a) and (3.8b), we get by continuity:

       C(x)u = 0 u, b(x) - 1 2 d j=1 DC j (x)(CC + ) j (x) ≤ 0,
for all u ∈ N 1 D (x), which ends the proof.

Sufficient conditions

In this section, we prove that the necessary conditions of Proposition 3.5 are also sufficient. We start by showing in Proposition 4.1 that (2.2a) and (2.2b) imply that the generator L of X satisfies the positive maximum principle: Lφ(x) ≤ 0 for any x ∈ D and any function Then, the generator L satisfies the positive maximum principle. Proof. We fix x ∈ D. For 1 ≤ j ≤ d, let us consider the following deterministic control system:

φ ∈ C 2 (R d , R) such that max D φ = φ(x) ≥ 0, see e.
y ′ (t) = C(y(t))σ(x) + e j y(0) = x, ( 4.1)
where σ(x) + is the pseudoinverse of σ(x). Since C is locally Lipschitz and verifies condition (2.2a), [15, Proposition 2.5] combined with (3.3) implies that D is invariant with respect to the deterministic control system (4.1). Then, by definition of the second order normal cone in (1.2),

u, y( √ h) -x + 1 2 v(y( √ h) -x), y( √ h) -x ≤ o(||y( √ h) -x|| 2 )
for any (u, v) ∈ N 2 D (x). On the other hand, since C is C 1,1 loc , a Taylor expansion around 0 yields

y( √ h) = x + √ hC(x)σ(x) + e j + h 2 (e ⊤ j σ(x) + ⊗ I d )DC(x)C(x)σ(x) + e j + o(h),
recall Proposition A.8 and note that (σ

+ ) ⊤ = σ + since σ is symmetric. Now observe that u ∈ N 1 D (x) whenever (u, v) ∈ N 2 D (x).
In particular, u ⊤ C(x) = 0 under (2.2a). Combining the above, and recalling Proposition A.5 then leads to

h 2 e ⊤ j (σ(x) + ⊗ u ⊤ )DC(x)C(x)σ(x) + e j + h 2 e ⊤ j σ(x) + C(x)vC(x)σ(x) + e j ≤ o(h).
Note that σ + σ + = C + and that Cσ + σ + C = CC + C = C, see e.g. Definition A.1 and Proposition A.2, and recall that (σ(x) + ⊗ u ⊤ ) = σ(x) + (I d ⊗ u ⊤ ) by Proposition A.5. Then, dividing the above by h/2 and sending h → 0 before summing over 1 ≤ j ≤ d yields

Tr (I d ⊗ u ⊤ )DC(x)C(x)C(x) + + Tr (vC(x)) ≤ 0.
In view of (2.2b) and (2.3), this shows that

b(x), u + 1 2 Tr(vC(x)) ≤ u, b(x) - 1 2 d j=1 DC j (x)(CC + ) j (x) ≤ 0 for all (u, v) ∈ N 2 D (x).
To conclude, it remains to observe that (Dφ(x), Proof. We already know from Proposition 4.1 that L satisfies the positive maximum principle. Then, [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF]Theorem 4.5.4] yields the existence of a solution to the martingale problem associated to L with sample paths in the space of càdlàg functions with values in D ∆ := D ∪ {∆}, the onepoint compactification of D. The discussion preceding [10, Proposition 3.2] and [20, Proposition 5.3.5], recall our linear growth conditions (H 1 ), then shows that the solution has a modification with continuous sample paths in D. Finally, [START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF]Theorem 5.3.3] implies the existence of a weak solution (X, W ) such that X t ∈ D for all t ≥ 0 almost surely.

D 2 φ(x)) ∈ N 2 D (x) whenever φ ∈ C 2 (R d , R) is such that max D φ = φ(x) ≥ 0. Hence, Lφ(x) ≤ 0.

A generic application

We show in this section how Theorem 2.3 can be applied in various examples of application. We restrict to a two-dimensional setting for ease of computations and notations.

We first provide a generic tractable characterization for the stochastic invariance of all state spaces D ⊂ R 2 of the following form:

D = {(x, x) ∈ R 2 , x ∈ D 1 and φ(x, x) ∈ D 2 }, (5.1)
where D 1 ⊂ R and D 2 ⊂ R are closed subsets and φ is a continuously differentiable function.

Then, D can be characterized through Φ : (x, x) → (x, φ(x, x)) by

D = Φ -1 (D 1 × D 2 ),
and [36, Exercise 6.7 and Proposition 6.41] provides the following description of the normal cone whenever Φ is differentiable at x and its Jacobian DΦ(x) has full rank (H x ) holds at any point x ∈ D.

Proposition 5.1. Fix x = (x, x) ∈ D such that (H x ) holds. Then,

N 1 D (x) = ū + ∂ 1 φ(x) u ∂ 2 φ(x) u , ū ∈ N 1 D 1 (x) and u ∈ N 1 D 2 (φ(x, x)) ,
in which ∂ i φ is the derivative with respect to the i-th component.

= ( b, b) ⊤ , C = (C ij ) ij and ∂ u = u 2 ∂ 1 -u 1 ∂ 2 .
(5.2) Proposition 5.2. Let D be as in (5.1) and x = (x, x) ∈ ∂D be such that

(H x ) holds. Fix u = (u 1 , u 2 ) ⊤ ∈ N 1 D (x)
             C(x) = C 11 (x)   1 -u 1 u 2 -u 1 u 2 u 2 1 u 2 2   , (5.3a) u, b(x) - 1 {C 11 (x) =0} 2(u 2 1 + u 2 2 ) u 1 u 2 ∂ u (C 11 -C 22 )(x) + (u 2 2 -u 2 1 )∂ u C 12 (x) ≤ 0. (5.3b) (b) Or, u = 0, u 1 = ū and          C(x)1 {ū =0} = C 22 (x) 0 0 0 1 1 {ū =0} , (5.4a) ū b(x) - 1 {C 22 (x) =0} 2 ∂ 2 C 12 (x) ≤ 0. (5.4b) Proof. Case (a), u = 0: Since DΦ(x) has full rank, ∂ 2 φ(x) = 0 and therefore u 2 = 0. Since C(x) ∈ S 2 , (2.2a) is clearly equivalent to (5.3a). If C 11 (x) = 0, (5.3a) implies that u = (ū + ∂ 1 φ(x) u, ∂ 2 φ(x) u) ⊤ spans the kernel of C(x).
Therefore, by Proposition A.3,

C(x)C(x) + = I 2 - 1 u 2 uu ⊤ = 1 u 2 1 + u 2 2 u 2 2 -u 1 u 2 -u 1 u 2 u 2 1 . Straightforward computations yield u, 2 j=1 DC j (x)(CC + ) j (x) = 1 u 2 1 + u 2 2 u 1 u 2 ∂ u (C 11 -C 22 )(x) + (u 2 2 -u 2 1 )∂ u C 12 (x) ,
recall the notations introduced in (5.2). This shows the equivalence between (2.2b) and (5.3b) when C 11 (x) = 0. If C 11 (x) = 0, then (5.3a) implies that C(x)C(x) + = 0 and (2.2b) reads u, b(x) ≤ 0.

Case (b), u = 0: If ū = 0, then u = 0 and there is nothing to prove. Otherwise,

u 1 = ū = 0. Since C(x) ∈ S 2 , (2.2a) is clearly equivalent to C 11 (x) = 0 and C 21 (x) = C 12 (x) = 0, that is (5.4a). If C 22 (x) = 0, then (5.4a) provides C(x)C(x) + = 0 0 0 1 ,
and straightforward computations yield u,

2 j=1 DC j (x)(CC + ) j (x) = ū∂ 2 C 12 (x),
which shows the equivalence between (2.2b) and (5.4b) when C 22 (x) = 0. If C 22 (x) = 0, then C(x)C(x) + = 0 and (2.2b) reads ūb (x) ≤ 0.

Note that ū = 0 when D 1 = R, which will be the case from now on. In the sequel, we impose more structure on the coefficients, as it is usually done in the construction of invariant diffusions. This permits to deduce an explicit form of (b, C) on the whole domain from the boundary conditions (5.3a)-(5.3b). As already stated, Theorem 2.3 can be directly applied to a large class of diffusions, e.g. affine diffusions [START_REF] Duffie | Affine processes and applications in finance[END_REF][START_REF] Filipović | Affine diffusion processes: theory and applications[END_REF][START_REF] Spreij | Affine diffusions with non-canonical state space[END_REF] and polynomial diffusions [START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF][START_REF] Larsson | Polynomial diffusions on compact quadric sets[END_REF], not only for closed subsets of R d , but even when D ⊂ S d (as in [START_REF] Cuchiero | Affine processes on positive semidefinite matrices[END_REF]) since S d can be identified with R

d(d+1) 2
by using the half-vectorization operator. We start by defining these two main structures.

Definition 5.3 (Affine and polynomial diffusions). X is a polynomial diffusion on D if:

(i) There exist bi , b i ∈ R, 0 ≤ i ≤ 2, and A i ∈ S 2 , 1 ≤ i ≤ 5, such that b : x → b(x) := ( b(x), b(x)) ∈ R 2 and C : x → C(x) ∈ S 2
have the following form:

     b(x) = b0 + b1 x + b2 x, b(x) = b 0 + b 1 x + b 2 x, C(x) = A 0 + A 1 x + A 2 x + A 3 x2 + A 4 x x + A 5 x 2 ,
(5.5)

for all x = (x, x) ∈ D.

(ii) C(x) ∈ S d + , for all x ∈ D. When A i = 0 for all 3 ≤ i ≤ 5, we say that X is an affine diffusion. If we now impose the structural condition (5.5), then straightforward computations lead to the characterization in [START_REF] Duffie | Affine processes and applications in finance[END_REF] for affine diffusions. The case of polynomial diffusions can be treated similarly.

Example 5.5 (Parabolic convex state space). Let us consider the following parabolic state space:

D = {(x, x) ∈ R 2 , x ≥ x2 }.
Then, with the previous notations, D 1 = R, D 2 = R + and φ(x, x) = x -x2 . Therefore, the first order normal cone given by Proposition 5.1 reads

N 1 D (x) = -2x 1 R -, for all x = (x, x2 ) ∈ ∂D.
Conditions (5.3a)-(5.3b) are therefore equivalent to

           C(x) = C 11 (x) 1 2x 2x 4x 2 , (5.6a) u, b(x) - 1 {C 11 (x) =0} 2(1 + 4x 2 ) -2x∂ u (C 11 -C 22 )(x) + (1 -4x 2 )∂ u C 12 (x) ≥ 0, (5.6b)
for all x ∈ R, x = (x, x2 ) and u = (-2x, 1) ⊤ .

If we now impose an additional affine structure on the diffusion X = ( X, X), as in Duffie et al. [START_REF] Duffie | Affine processes and applications in finance[END_REF]Section 12.2], we recover the characterization given in Gourieroux and Sufana [START_REF] Gourieroux | A Classification of Two-Factor Affine Diffusion Term Structure Models[END_REF]Proposition 2]. Indeed, Proposition 5.2 says that D is invariant if and only if there exists α ≥ 0 such that (a)

C(x) = α 1 2x 2x 4 x , for all x = (x, x) ∈ D, (5.7)
(b) b2 = 0 and

     b 2 > 2 b1 and ( b 1 -2 b0 ) 2 ≤ 4( b 2 -2 b1 )( b 0 -α) or b 2 = 2 b1 , b 1 = 2 b0 and b 0 ≥ α.
(5.8)

Let us detail the computations: (a) The covariance matrix C(x) ∈ S 2 + is of the form (5.6a) on the boundary. Since C is affine in (x, x2 ), then necessarily C 11 (x) is constant (or else C 22 (x) would have at least a polynomial dependence of order 3 in x). Therefore, there exists α such that C(x) has the form (5.7) at x = (x, x2 ), in which α ≥ 0 to ensure that C(0) ∈ S 2 + . Finally, C needs to have the same form (5.7) on the whole state space D, since it is affine. (b) We now derive the form of the drift vector b(x) = ( b(x), b(x)) ∈ R 2 by using (5.6b). From (5.7), elementary computations show that condition (5.6b) is equivalent to

-2 b2 x3 + ( b 2 -2 b1 )x 2 + ( b 1 -2 b0 )x + b 0 -α ≥ 0, for all x ∈ R,
which is equivalent to (5.8), when α > 0. If α = 0, the same conclusion holds. Conversely, (5.7) clearly implies (2.2a) and (ii) of Definition 5.3 since det(C(x)) = 4α( x-x2 ) ≥ 0 and x ≥ 0 for all (x, x) ∈ D. Moreover, (5.8) leads to (5.3b) by the same computations as above.

Example 5.6 (Parabolic concave state space). We now consider the epigraph of the concave function

x → -x 2 , D = {(x, x) ∈ R 2 , x ≥ -x 2 }. It follows that D 1 = R, D 2 = R + , φ(x, x) = x + x2 and N 1 D (x) = 2x 1 R -, for all x = (x, -x 2 ) ∈ ∂D,
from Proposition 5.1. Hence, conditions (5.3a)- (5.3b) are now equivalent to

           C(x) = C 11 (x) 1 -2x -2x 4x 2 , (5.9a) u, b(x) - 1 {C 11 (x) =0} 2(4x 2 + 1) 2x∂ u (C 11 -C 22 )(x) + (1 -4x 2 )∂ u C 12 (x) ≥ 0, (5.9b)
for all x ∈ R, x = (x, -x 2 ) and u = (2x, 1) ⊤ ∈ -N 1 D (x). Let us first note that the above shows that we can not construct an affine diffusion living in D, that is not degenerate, unless it lives on the boundary only. Indeed, if C is affine then C 11 =: α has to be constant, because of (5.9a), and C is of the form (5.9a) with (-x) in place of x2 .

Since C(x) ∈ S 2 + , we must have α ≥ 0 and det C(x) = -4α 2 ( x + x2 ) ≥ 0. Thus, α = 0 unless we restrict to points (x, x) on the boundary. If we do so, it is not difficult to derive a necessary and sufficient condition on the coefficients from the identity X = -X2 .

We now impose a polynomial structure on the diffusion X = ( X, X), such that X is affine on its own, i.e. b and C 11 are of affine form and only depend on x. This extends [START_REF] Larsson | Polynomial diffusions on compact quadric sets[END_REF]Example 5.2] and entirely characterizes the stochastic invariance of D with respect to this structure of diffusion. By Proposition 5.1, D is invariant if and only if there exist α, β ≥ 0, such that (a)

C(x) = α -2αx -2αx (4α + β)x 2 + β x , for all x = (x, x) ∈ D, (5.10)
(b) b2 = 0 and

     b 2 < 2 b1 and ( b 1 + 2 b0 ) 2 ≤ 4(-b 2 + 2 b1 )( b 0 + α) or b 2 = 2 b1 , b 1 = -2 b0 and b 0 ≥ -α . ( 5

.11)

Let us do the computations explicitly: (a) The covariance matrix C(x) ∈ S 2 + is of the form (5.9a) on the boundary. Therefore, C 11 (x) ≥ 0, for all x ∈ ∂D. Since C 11 is affine and only depends on x ∈ R, then necessarily C 11 is a non-negative constant on the whole space D. Therefore, there exists α ≥ 0 such that C 11 (.) = α on D. Moreover, (5.5) reads on the boundary

C(x) = A 0 + A 1 x + (A 3 -A 2 )x 2 -A 4 x3 + A 5 x4 , for all x ∈ R.
Therefore, comparing with (5.9a) leads to A 4 = A 5 = 0 and the existence of

β, β ′ such that C is of the form α -2αx -2αx 4αx 2 + 0 β ′ β ′ β ( x + x2 )
on the whole space D. We now use the fact that C(D) ⊂ S 2 + . In particular, taking x = 0 shows that we must have αβ x-(β ′ ) 2 x 2 ≥ 0 for all x ≥ 0, so that β ′ = 0. Similarly, 4αx 2 +β( x+x 2 ) ≥ 0 must hold for all x ∈ D, which is equivalent to β ≥ 0. (b) We now derive the form of the drift vector b(x) = ( b(x), b(x)) ∈ R 2 by using (5.9b). Since X is affine on its own, b2 = 0. From (5.10), elementary computations show that condition (5.9b) is equivalent to

(-b 2 + 2 b1 )x 2 + ( b 1 + 2 b0 )x + b 0 + α ≥ 0, for all x ∈ R,
which is equivalent to (5.11), when α > 0. If α = 0, the same conclusion holds. Conversely, (5.10)- (5.11) show that X is a polynomial diffusion such that X is affine on its own since det(C(x)) = αβ( x + x2 ) ≥ 0 and 4αx 2 + β( x + x2 ) ≥ 4αx 2 ≥ 0 for all (x, x) ∈ D. (5.10) clearly implies (2.2a). Moreover, (5.11) leads to (5.3b) by the same computations as above.

We conclude with a final remark on the interplay between the local geometry of the boundary, the coefficients b and C and the structure of the diffusion. with respect to the convexity of the domain. When the domain is convex, as in Example 5.5, the drift is necessarily inward pointing since b 0 ≥ α, with α ≥ 0 from (5.8). However, when the domain is concave, as in Example 5.6, the drift could even be outward pointing. This follows from the fact that b 0 ≥ -α, with α ≥ 0 in (5.11).

{x ≥ x2 } (i) Affine diffusion b(0) {x ≥ -x 2 } (ii) Polynomial diffusion b(0)

Additional remark on the boundary non-attainment

In this last section, we provide a sufficient condition for the stochastic invariance of the interior of D, when D has a smooth boundary. We conclude by using [37, Proposition 3.5] (after a change of the sign, since D is assumed to be a connected subset of {x, Φ(x) > 0} in [START_REF] Spreij | Affine diffusions with non-canonical state space[END_REF]Proposition 3.5]).

C ∈ C 1 (R d , S d + ). Then D is stochastically invariant if there exists v ∈ R d such that        DΦ(x)C(x) = Φ(x)v ⊤ (6.1a) DΦ(x), b(x) - 1
Example 6.2. (i) Square root process: Let us consider again the process defined by dX t = b(X t )dt+η √ X t dW t , for some η > 0, on D = R + . Then, Φ : x → -x and (6.1a)-(6.1b) are equivalent to v = η 2 and b(0) ≥ η 2 2 . These are the well known conditions for the boundary non-attainment of the square-root process.

(ii) Affine diffusions: More generally, let D ⊂ R d satisfy the assumptions of Proposition 6.1 and take C(x) = A 0 + d j=1 A j x j for some A j ∈ S d , 1 ≤ j ≤ d. Then differentiating C shows that condition (6.1b) is equivalent to DΦ(x), b(x) -1 2 d j=1 (A j ) j ≤ 0 yielding [START_REF] Spreij | Affine diffusions with non-canonical state space[END_REF]Proposition 3.7].

(iii) Jacobi diffusion: Set D = (0, 1] and consider a polynomial diffusion X on D, i.e. b is affine and C is a polynomial of degree two. Theorem 2.3 applied on [0, 1] immediately yields that de dynamics of X must be of the form dX t = κ(θ -X t )dt + η X t (1 -X t )dW t where κ, η ≥ 0 and 0 ≤ θ ≤ 1. Now a localized version of Proposition 6.1 shows that D = (0, 1] is stochastically invariant under the additional condition that κθ ≥ η 2 2 . Proposition 6.1 is important in practice since it gives, in many cases, the existence and the uniqueness of a global strong solution to (1.1) as discussed in the following remark.

A Matrix tools

For the reader's convenience, we collect in this Appendix some definitions and properties of matrix tools intensively used in the proofs throughout the article. For a complete review and proofs we refer to [START_REF] Magnus | The elimination matrix: some lemmas and applications[END_REF][START_REF] Magnus | Matrix differential calculus with applications in statistics and econometrics[END_REF][START_REF] Neudecker | Symmetry, 0-1 matrices and jacobians[END_REF].

We start by recalling the definition of the Moore-Penrose pseudoinverse which generalizes the concept of invertibility of square matrices, to non-singular and non-square matrices. In the following, we denote by M m,n the collection of m × n matrices. We now collect some useful identities on the Kronecker product.

A ⊗ B = B(A ⊗ I n 2 ) if m 1 = 1.
The following definitions extend the concept of Jacobian matrix and show how to nicely stack the partial derivatives of a matrix-valued function F : M n,q → M m,p by using the vectorization operator (see [START_REF] Magnus | Matrix differential calculus with applications in statistics and econometrics[END_REF]Chapter 9]).

Definition A.6 (Vectorization operator).

Let A ∈ M m,n . The vectorization operator vec transforms the matrix into a vector in R mn by stacking all the columns of the matrix A one underneath the other.

Definition A.7 (Jacobian matrix). Let F be a differentiable map from M n,q to M m,p . The Jacobian matrix DF (X) of F at X is defined as the following mp × nq matrix:

DF (X) = ∂ vec(F (X)) ∂ vec(X) ⊤ .
Proposition A.8 (Product rule). Let G be a differentiable map from M n,q to M m,p and H be a differentiable map from M n,q to M p,l . Then, D(GH) = (H ⊤ ⊗ I m )DG + (I l ⊗ G)DH.

Dφ

  = φ(x) = 0 and Dφ(x) = u ⊤ . Indeed, it follows from[START_REF] Rockafellar | Variational analysis[END_REF] Chapter 6.E] that one can find κ > 0 such that u, yx ≤ κ 2 yx 2 for all y ∈ D. Then, one can set ψ := u, •x -κ 2 • -x 2 and define φ := ψρ in which ρ is a C ∞ b function with values in [0, 1], compact support included in N (x), and satisfying ρ = 1 in a neighborhood of x.

2

 2 , or equivalently C(x)u = 0 since C(x) is symmetric. The second identity combined with Dφ(x) = u ⊤ and Proposition A.8 shows that 0 ≥Lφ(x) -1 Tr σ⊤ D 2 φσ

Proposition 3 . 5 (

 35 Necessary conditions of Theorem 2.3). Let the conditions of Theorem 2.3 hold and assume that D is stochastically invariant with respect to the diffusion (1.1). Then conditions (2.2a) and (2.2b) hold for all x ∈ D and u ∈ N 1 D (x).

Proposition 4 . 1 .

 41 g.[20, p165]. Then, classical arguments, mainly[START_REF] Ethier | Markov Processes: Characterization and Convergence[END_REF] Theorem 4.5.4], yield the existence of a solution to the corresponding martingale problem that stays in D, see Proposition 4.2 below.The following proposition is inspired by[START_REF] Da Prato | Invariance of stochastic control systems with deterministic arguments[END_REF] Remark 5.6]. Under the assumptions of Theorem 2.3, assume that (2.2a)-(2.2b) hold for all x ∈ D and u ∈ N 1 D (x).

Proposition 4 . 2 (

 42 Sufficient conditions of Theorem 2.3). Under the assumptions of Theorem 2.3, assume that conditions (2.2a) and (2.2b) hold for all x ∈ D and u ∈ N 1 D (x). Then, D is stochastically invariant with respect to the diffusion (1.1).

When x lies in

  the interior of D, N 1 D (x) = {0} and (2.2a)-(2.2b) are trivially verified. Hence, it suffices to control b and C on the boundary of the domain in order to ensure the stochastic invariance of D as stated by the following proposition, in which we use the notations b

as in Proposition 5 . 1 .

 51 Under the assumptions of Theorem 2.3, (2.2a)-(2.2b) are equivalent to the following: (a) Either u = 0 and

  Then, it is clear that b and C are C ∞ and satisfy the linear growth conditions (H 1 ).In what follows, we highlight the interplay between the geometry/curvature of the boundary and the coefficients b and C. The three explicit examples below characterize the invariance for flat, convex and concave boundaries.

Example 5 . 4 ( 2 ∂ 1 C

 5421 Canonical state space). Fix D 1 = R, D 2 = R + and φ(x, x) = x. Then D = R × R + and N 1 D (x) = {0} × 1 {x=0} R -.Hence, (5.3a)-(5.3b) are equivalent to C(x, 0) = C 11 (x, 012 (x, 0) ≥ 0, for all x ∈ R.

Remark 5 . 7 .

 57 (i) Curvaceous boundary and covariance matrix: the curvature of the boundary plays a crucial role in determining the covariance structure. In Example 5.4, the canonical state space, which shows no curvature, imposes strict constraints on the covariance matrix. Whereas, for curved domains, as in Examples 5.5-5.6, the first order normal cone is a more complicated object and induces a richer covariance structure on the boundary. (ii) Convexity and drift direction: Figure 1 visualizes the direction of the drift b(0) = ( b0 , b 0 )

Figure 1 :

 1 Figure 1: Interplay between the convexity of the domain and the direction of the drift: (i) Inward pointing drift for convex domains (Example 5.5). (ii) Possible outward pointing drift for concave domains (Example 5.6).

2 dj=1

 2 DC j (x)e j ≤ 0 (6.1b) for all x ∈ D.Proof. Fix x ∈ D. By differentiating (6.1a) with the help of Propositions A.8 and A.5, we obtainvDΦ(x) = (C(x) ⊗ I 1 )D 2 Φ(x) + (I d ⊗ DΦ(x))DC(x) = C(x)D 2 Φ(x) + (I d ⊗ DΦ(x))DC(x),which, combined with (6.1b), leads to DΦ(x), b(x) ≤ 1 2 Tr (I d ⊗ DΦ(x) ⊤ )DC(x) ) -1 DΦ(x)C(x)DΦ(x) ⊤ .

Remark 6 . 3 . 1 2

 631 Let D be as in Proposition 6.1. Assume that C ∈ C 2 ( D, S d + ) and that b is locally Lipschitz (which is clearly the case for affine and polynomial diffusions). By [23, Remark 1 page 131], σ = C is locally Lipschitz on D. Therefore, when the boundary is never attained, (1.1) starting from any element x ∈ D admits a global strong solution and pathwise-uniqueness holds.

Definition A. 1 ( 1 2

 11 Moore-Penrose pseudoinverse). Fix A ∈ M m,n . The Moore-Penrose pseudoinverse of A is the unique n × m matrix A + satisfying:AA + A = A, A + AA + = A + , AA + and A + A are Hermitian. Proposition A.2. If A ∈ M d has the spectral decomposition QΛQ ⊤ for some orthogonal matrix Q ∈ M d and a diagonal matrix Λ = diag [(λ i ) i≤d ] ∈ M d . Then, A + = QΛ + Q ⊤ in which Λ + = diag (λ -1 i 1 {λ i =0}) i≤d , andAA + = Qdiag (1 {λ i =0} ) i≤d Q ⊤ . If moreover A is positive semi-definite and B = A , then B + = Q(Λ + ) 1 2 Q ⊤ . Proposition A.3. If A ∈ M m,n, then AA + is the orthogonal projection on the image of A.

Definition A. 4 ( 5 .

 45 Kronecker product). LetA = (a ij ) i≤m 1 ,j≤n 1 ∈ M m 1 ,n 1 and B ∈ M m 2 ,n 2 . The Kronecker product (A ⊗ B) is defined as the m 1 m 2 × n 1 n 2 matrix A ⊗ B =    a 11 B • • • a 1n 1 B . . . . . . a m 1 1 B • • • a m 1 n 1 B Let A and B be as in Definition A.4, C ∈ M n 1 ,n 3 and D ∈ M n 2 ,n 4 . Then, (A ⊗ B)(C ⊗ D) = (AC ⊗ BD), A ⊗ B = A(I n 1 ⊗ B) if m 2 = 1,

  view of [9, Lemma 2.1], it suffices to show that R t /t → 0 almost surely. To see this, first note that R 1 t = o(t) a.s. since θ is continuous at 0. Moreover, [11, Proposition 3.9] implies that R 2 The continuity assumption (3.6) implies that M i s = O(s 1+η ) almost surely. By the Dambis-Dubins-Schwarz theorem, (M i s ) s≥0 is therefore a timechanged Brownian motion, see e.g. [35, Theorem V.1.6]. By the law of iterated logarithm for Brownian motion (M i

	• 0	d j=1 M ij r dW j

t = o(t) a.s., as β is bounded. It remains to prove that R 3 t = o(t) a.s. To see this, define

M ij = γ ijγ ij 0 and M i = r for all 1 ≤ i, j ≤ d. s ) 2 = O(s 1+ η 2 ) almost surely. Hence, R 3 t = O(t 2+ η

2 ) almost surely. By applying the Dambis-Dubin-Schwarz theorem and the law of iterated logarithm again, we obtain that R 3 t = o(t) a.s.


R Mailler
email: romain.mailler@siaap.fr

J Gasperi
email: gasperi@u-pec.fr

Y Coquet

C Derome

A Buleté

E Vulliet

A Bressy

G Varrault

G Chebbo

V Rocher

Siaap

experimental study of different activated carbons and factors influencing the adsorption of micropollutants in wastewater

come

Removal of emerging micropollutants from wastewater by activated carbon adsorption:

INTRODUCTION

The presence of a large range of emerging micropollutants, particularly pharmaceuticals and hormones, personal care products or pesticides, has been highlighted in wastewater treatment plant (WWTP) discharges [START_REF] Miège | Fate of pharmaceuticals and personal care products in wastewater treatment plants --Conception of a database and first results[END_REF][START_REF] Verlicchi | Occurrence of pharmaceutical compounds in urban wastewater: Removal, mass load and environmental risk after a secondary treatment----A review[END_REF][START_REF] Loos | EU-wide monitoring survey on emerging polar organic contaminants in wastewater treatment plant effluents[END_REF][START_REF] Mailler | Study of a large scale powdered activated carbon pilot: Removals of a wide range of emerging and priority micropollutants from wastewater treatment plant effluents[END_REF]. Even if several hydrophobic, volatile or biodegradable compounds are strongly removed by conventional wastewater treatments [START_REF] Ruel | Occurrence and fate of relevant substances in wastewater treatment plants regarding Water Framework Directive and future legislations[END_REF]Mailler et al. 2014b), most of micropollutants are not removed. Thus, various strategies of contamination reduction are assessed by the scientific community and water treatment engineers. Among them, the implementation of tertiary treatments dedicated to micropollutants elimination represents a relevant solution. In particular, adsorption on activated carbon appears to be efficient, very flexible and relatively cheap [START_REF] Abegglen | Micropolluants dans les eaux résiduaires urbaines[END_REF], together with not producing oxidation byproducts.

In this context, the Parisian public sanitation service (SIAAP) and the Water Environment and Urban Systems laboratory (LEESU) study, in collaboration with SAUR teams, the CarboPlus® process. This process is based on a fluidized bed of activated carbon (powder -PAC -or micro-grain -µGAC) to remove micropollutants by adsorption. A large-scale pilot based on this technology was set up at the Seine Centre WWTP (240 000 m 3 /days) to characterize the efficiency of activated carbon to remove a wide range of pollutants from WWTP discharges. The in-situ results are presented in [START_REF] Mailler | Study of a large scale powdered activated carbon pilot: Removals of a wide range of emerging and priority micropollutants from wastewater treatment plant effluents[END_REF] for PAC.

In parallel to the in situ study, complementary laboratory scale experiments were conducted to i) better understand the relationships between activated carbon properties and the adsorption in wastewater and ii) to improve the understanding of the micropollutant adsorption mechanisms in wastewater application, in particular with PAC.

Activated carbon is characterized by different structural and morphological properties that can affect adsorption [START_REF] Baudu | Prediction of Adsorption from Aqueous Phase of Organic Molecules as a Function of Some Physicochemical Characteristics of Activated Carbons[END_REF][START_REF] Li | Effects of activated carbon surface chemistry and pore structure on the adsorption of organic contaminants from aqueous solution[END_REF][START_REF] Yu | Adsorption characteristics of selected pharmaceuticals and an endocrine disrupting compound----Naproxen, carbamazepine and nonylphenol----on activated carbon[END_REF][START_REF] Delgado | The removal of endocrine disrupting compounds, pharmaceutically activated compounds and cyanobacterial toxins during drinking water preparation using activated carbon----A review[END_REF]. Thus, the first axis consists in studying the relationships between activated carbon properties and their efficiency to remove micropollutants. In particular, a focus is performed on the specific BET surface and the bulk density.

The second axis consists in studying the adsorption mechanism in wastewater representing a complex matrix. Indeed, the literature highlighted that the matrix where adsorption is performed plays a crucial role in the fate of molecules [START_REF] Ebie | Pore distribution effect of activated carbon in adsorbing organic micropollutants from natural water[END_REF][START_REF] Ternes | Removal of Pharmaceuticals during Drinking Water Treatment[END_REF][START_REF] De Ridder | Modeling equilibrium adsorption of organic micropollutants onto activated carbon[END_REF][START_REF] De Ridder | Influence of natural organic matter on equilibrium adsorption of neutral and charged pharmaceuticals onto activated carbon[END_REF], particularly the organic matter which competes with pollutants for adsorption through direct sites competition or pore blocking. In addition, most of the studies were conducted for drinking water production or in surface waters, and the adsorption mechanisms were poorly assessed in wastewater. Thus, the influences of both the activated carbon dose and the contact time on the performances were assessed, as well as the influence of the organic matter and chemicals that can be injected within the process for operational purposes.

This article summarizes the results from the laboratory scale experiments conducted within this project. First, the relationships between micropollutants removals and activated carbon properties are assessed. Then, the influence of the dose of carbon, the contact time, the organic matter quantity and quality, the presence of a residual concentration of methanol and the injection of FeCl3 were studied.

MATERIALS AND METHODS

Activated carbon characterization

11 adsorbents, including micro and mesoporous PACs and µGAC, have been selected based on their technical datasheets provided by the producers. Among them, 3 are commercialized by DaCarb® (PB 170®, PB 170-400® and PC 1000® -France), 3 by Chemviron® (WP 235®, Carbsorb 28® and Cyclecarb 305® -Belgium), 2 by Norit® (W 35® and SA Super® -Netherlands) and 3 by Jacobi® (LP 39®, MP 25® and Hydro XP 17® -Sweden). These activated carbons are recognized for their high organic pollutants affinity, but their use in wastewater was poorly studied in the literature.

Several structural properties such as the specific BET surface (m 2 /g), the porous volume (mL/g), the pore size distribution and the bulk density (g/cm 3 ) have been measured on the 11 activated carbons. In addition, the particle size distribution and the micropollutant elimination have been determined for 4 of them: PB 170®, WP 235®, W 35® and PC 1000®.

These structural properties have been analyzed at the University of Technology of Compiègne (UTC, France). The specific BET surface and the pore size distribution were measured with an ASAP 2010 Micromeritics analyzer equipped with a degasing station and a gas isotherm adsorption analyses station (nitrogen), according to the conventional methods used to determine these parameters. Fresh activated carbon samples (100 mg) were degased at least 12 h before to be analyzed. Results correspond to the specific surface in m 2 /g obtained with the BET (Brunauer, Emett and Teller) method.

The pore size distribution is determined with the BJH (Barrett, Joyner and Halenda) method, using the desorption curve of the same gas on the same analyzer. This method allows also calculating the microporous and mesoporous pore volumes.

The bulk density was measured by weighting, with a high precision balance (0.01 mg), 50 to 100 mL of activated carbon, measured with a 100 mL graduated cylinder. The activated carbon is introduced by small doses (10 mg) and is compacted every 10 mL to minimize the vacuum between particles.

The particles size distribution of the 4 PACs was measured with a Mastersizer 2000 Malvern laser particle sizer. Every analysis corresponds to 15 000 light diffraction measures. 3 scans were performed at least per sample.

Pollutants and analytical procedures

For every sample, several parameters were measured in the dissolved phase: UV absorbance at 254 nm (UV-254), dissolved organic carbon (DOC) and concentrations of 16 pharmaceuticals and 2 pesticides (list and limit of quantification in supporting material -Table S1). Organic micropollutants were analyzed by the Institute of Analytical Sciences (ISA -Villeurbanne, France), while UV-254 and DOC were measured by the SIAAP laboratory (French accreditation -Colombes, France). Analytical protocols are validated and are given in [START_REF] Mailler | Study of a large scale powdered activated carbon pilot: Removals of a wide range of emerging and priority micropollutants from wastewater treatment plant effluents[END_REF].

3D fluorescence spectrometry analyses were also performed by the LEESU. They allow generating a tridimensional matrix and to plot it on a 3D spectrum containing all excitation and emission spectra of the dissolved organic matter (DOM) present in a sample. This method gives the distribution of the different components of the DOM as they don't fluoresce in the same zones of the spectrum. The detailed description of 3D fluorescence spectrometry is given in [START_REF] Hudson | Fluorescence analysis of dissolved organic matter in natural, waste and polluted waters----a review[END_REF]. Analyses have been performed with a Jasco FP-8300 spectrofluorometer equipped with a 1 cm quartz cell.

Laboratory experiments protocols

The laboratory experiments have been performed in the Seine Centre WWTP (Colombes, France) between April 2013 and May 2015. Wastewater was punctually sampled from the WWTP discharges with 10 L glass bottles, properly rinsed and grilled, and used the same day for experiments. Then, samples were filtered on glass fibers filters (GF/F 0.7 µm) after experiments, before to be sent to laboratories for analyses.

Efficiency comparison of 4 PACs

The efficiency of 4 PACs (PB 170®, WP 235®, W 35® and PC 1000® -Table 1) to remove micropollutants was assessed, by contacting individually 10 mg/L of each PAC with 1 L of Seine Centre WWTP (SEC) discharges during 45 min under strong mixing. After filtering, samples were conditioned and sent to ISA laboratory to analyze pharmaceuticals and pesticides.

PAC dose and adsorption kinetic

Regarding its performances, PB 170® was selected to characterize the relationship between PAC dose, contact time and micropollutant removal. First, 10 mg/L of this PAC were contacted with 1 L of SEC discharges under strong mixing during different contact times (5 -10 -30 -45 -60 min), which are close from the pilot contact time range. Then, 3 PAC doses (5 -10 -20 mg/L) were tested with 1 L of SEC discharges and during 45 min.

DOM and adsorption

The DOM has been identified to play a crucial role in the adsorption process because it induces a competition with the organic micropollutants for adsorption [START_REF] Matsui | Effect of natural organic matter on powdered activated carbon adsorption of trace contaminants: characteristics and mechanism of competitive adsorption[END_REF][START_REF] De Ridder | Influence of natural organic matter on equilibrium adsorption of neutral and charged pharmaceuticals onto activated carbon[END_REF][START_REF] Delgado | The removal of endocrine disrupting compounds, pharmaceutically activated compounds and cyanobacterial toxins during drinking water preparation using activated carbon----A review[END_REF]). These phenomena have been principally observed with surface water and natural organic matter, but some recent studies also studied it in wastewater [START_REF] Kienle | Treatment of micropollutants in municipal wastewater: Ozone or powdered activated carbon?[END_REF][START_REF] Altmann | Direct comparison of ozonation and adsorption onto powdered activated carbon for micropollutant removal in advanced wastewater treatment[END_REF]. In this context, it is necessary to determine the role of both the quantity and the quality of the DOM in the adsorption process. To this end, 7 types of wastewater featuring by different levels of organic matter (Table 1) have been contacted under strong mixing with PAC (10 mg/L of PB 170® during-45 min). Among the tested waters, 4 were WWTP discharges, from Seine Centre (SEC), Seine Aval (SAV), Seine Morée (SEM) and Seine Amont (SAM) WWTPs. The 3 other tested waters were outlet waters from the physico-chemical lamellar settling unit (SEC settled water) and the carbon biofiltration effluent (carbon removal) of SEC, as well as SEC settled water diluted with distilled water to reach a DOC level comparable to the WWTP discharges. The layouts of the different studied WWTPs are given in supporting material -Table S1, as well as the sampling points (red circles). The organic matter characteristics of these waters are different and given in Table 1. Among the 7 wastewaters, the micropollutants adsorption has been assessed with SEC settled water, SEC carbon biofiltration effluent, SEC and SAV discharges.

A 3D fluorescence spectrometry characterization has also been performed on these 7 waters, before and after contact with PAC. The spectra interpretation follows the fluorophores defined in [START_REF] Parlanti | Combined 3D-spectrofluorometry, high performance liquid chromatography and capillary electrophoresis for the characterization of dissolved organic matter in natural waters[END_REF], as indicated for SEC settled water and discharges in supporting material -Figure S2. α, α' and β correspond to humic-like substances, γ and δ respectively represents tyrosine-like and tryptophan-like proteins. Then, I1, I2, I3 and I4 indexes, respectively corresponding to the ratios α'/α, β/α, γ/α and δ/α, were calculated to evaluate the distribution. Finally, the humification index (HIX) and the biological activity index (BIX) were also evaluated [START_REF] Zsolnay | Differentiating with fluorescence spectroscopy the sources of dissolved organic matter in soils subjected to drying[END_REF][START_REF] Parlanti | Combined 3D-spectrofluorometry, high performance liquid chromatography and capillary electrophoresis for the characterization of dissolved organic matter in natural waters[END_REF]. The different fluorophores, HIX and BIX are given in Table 1 for the 7 tested wastewaters.

3.4.

Influence of the residual methanol concentration in the WWTP discharges A residual methanol concentration is present in both SEC and SAV discharges because of the post-denitrification stage.

Indeed, denitrifying microorganisms needs a carbon source and when the denitrification is performed after the carbon removal and the nitrification, as in both SEC and SAV, the carbon content is too low [START_REF] Rocher | Municipal wastewater treatment by biofiltration: comparisons of various treatment layouts. Part 1: assessment of carbon and nitrogen removal[END_REF]. Methanol is then added during this stage as carbon source, but its consumption by the microorganisms is not complete. Operationally speaking, it is interesting to study the impact of this residual concentration on the micropollutant adsorption by activated carbon. Thus, the performances obtained with the SEC discharges have been compared to performances obtained with the same water spiked with methanol (6 and 12 mgC/L). The adsorbability of methanol has also been studied and preliminary tests demonstrated that no adsorption of methanol occurs when contacting 12 mgC/L of methanol in ultrapure water with 10 mgPAC/L.

Influence of the ferric chloride

The CarboPlus® process operating in PAC configuration requires the injection of FeCl3 to stabilize the PAC bed and prevent any leakage. This substance is known to destabilize particles by neutralization of the surface charges, which could impact the adsorption of micropollutants. To evaluate the impact of FeCl3, 1 L of SEC discharges have been contacted

with PAC alone (10 mg/L), FeCl3 alone (2.5 gFeCl3/m 3 ) and both PAC and FeCl3 under strong mixing during 45 min.

RESULTS AND DISCUSSION

Influence of PAC characteristics on its efficiency

Structural and morphological properties of the 4 tested PACs are given in the Table 2. The removals of the micropollutants with a PAC dose of 10 mg/L during 45 min are also provided. Among the 4 PACs, the PB 170® and WP 235® have a similar mesoporous structure with close pore size distribution and specific BET surface (900-1000 m 2 /g), while W 35® and especially PC 1000® have different pore surface distributions and lower specific BET surfaces (supporting material -Figure S3; Table 2). W 35® is strongly mesoporous with 45% of the BET surface resulting from mesopores, while PC 1000® is microporous with 59% of the surface resulting from micropores. 3 of the tested PACs have similar bulk densities (0.30-0.40 g/cm 3 ) while PC 1000® is heavier (0.54 g/cm 3 ). Regarding the particle size distribution, PB 170® and WP 235® are comparable while the two other PACs have a larger particle size distribution and a higher median diameter. This is particularly notable for the PC 1000® which is characterized by median and d90 values twice higher than those of the PB 170® and WP 235®.

Even if only 1 punctual experiment was performed, a clear trend can be observed with most of the compounds. For individual compounds, the PB 170® allows the highest removals for 15 of the 18 micropollutants and the PC 1000® is always the worst efficient. The WP 235® and W 35® seem to have similar performances, slightly weaker than PB 170®.

Overall, the sum of the 16 pharmaceuticals is removed by 61%, 53%, 51% and 32% respectively by the PB 170®, WP 235®, W 35® and PC 1000®.

The micropollutant removal is well correlated with the specific BET surface (Figure 1): the higher the specific BET surface, the higher the micropollutant removal. In addition, the specific BET surface is also correlated to the bulk density of the PAC, a low bulk density corresponding to a high BET surface. This relationship has been verified with 7 other PACs differing by their nature, as the bulk density is very easy to measure and could therefore be a proxy of the BET surface. Considering the 11 PAC, this link is still observed (Figure 1). Thus, the bulk density could be used as an indicator to select activated carbons and estimate their BET surface and efficiency for micropollutants removal from wastewater.

The link between the efficiency of activated carbons and their specific BET surface (600-1500 m 2 /g) has already been observed in the literature [START_REF] Çeçen | Activated carbon for water and wastewater treatment integration of adsorption and biological treatment[END_REF], but the distribution of this surface between the different categories of pores (micro < 2 nm, meso 2-50 nm and macropores > 50 nm) has also to be considered. This is particularly important in presence of organic matter because it competes with micropollutants for adsorption, directly or by pore blocking (Newcombe et al. 2002;[START_REF] Yu | Adsorption characteristics of selected pharmaceuticals and an endocrine disrupting compound----Naproxen, carbamazepine and nonylphenol----on activated carbon[END_REF]). Thus, a mesoporous structure with a large pore size distribution should limit the competition with the DOM [START_REF] Ebie | Pore distribution effect of activated carbon in adsorbing organic micropollutants from natural water[END_REF]. According to the results (Figure 2), a higher dose of PAC logically results in higher micropollutant removals, i.e.

carbamazepine (32-54-80%; removals at 5-10-20 mgPAC/L), diclofenac (22-36-60%), propranolol (56-81-95%) or sulfamethoxazole (51-61-80%). However, the relationship is not linear as doubling the PAC mass doesn't result in a doubling of the removals. Actually, the relationship is logarithmic resulting in the reaching of a plateau when increasing the dose, as displayed in supporting material -Figure S4. Thus, the PAC dose in wastewater has a great influence on the removal, as already observed in the literature [START_REF] Snyder | Role of membranes and activated carbon in the removal of endocrine disruptors and pharmaceuticals[END_REF][START_REF] Boehler | Removal of micropollutants in municipal wastewater treatment plants by powder-activated carbon[END_REF][START_REF] Kienle | Treatment of micropollutants in municipal wastewater: Ozone or powdered activated carbon?[END_REF] or at large scale with the CarboPlus® pilot [START_REF] Mailler | Study of a large scale powdered activated carbon pilot: Removals of a wide range of emerging and priority micropollutants from wastewater treatment plant effluents[END_REF].Considering the individual results of the micropollutants, this positive influence of the dose is statistically significant (test of Shapiro-Wilk, Fischer and Student -p-value < 0.05) between 5 and 10 mgPAC/L (p-value = 0.016) and between 10 and 20 mgPAC/L (p-value = 0.0005).

Furthermore, the removals notably increase with the contact time (Figure 2), i.e. ciprofloxacin (29-40-62-66-71%; removals at 5-10-30-45-60 min contact times) or diclofenac (19-23-32-36-39%). However, the PAC is very fine (< 50 µm), resulting in a relatively fast adsorption kinetic. As a consequence, although the equilibrium is not reached after 60 min, differences of removal between 30, 45 and 60 min contact time for a same PAC dose are limited (< 10%) indicating the proximity of the equilibrium. Thus, the short contact times (30-60 min) employed in PAC tertiary treatments are suitable for an efficient adsorption of micropollutants. For several compounds (8/15), such as atrazine (26-46%; removals after 5 and 60 min), norfloxacin (38-71%), ofloxacin (39-64%) or sulfamethoxazole (40-55%), the removal achieved after 5 min represents more than half of the removal after 60 min, confirming the very fast kinetic. In contrary, erythromycin (7-39%) or roxithromycin (3-38%) seem to have slower kinetics, probably because of their high molecular weight. These compounds will then be more sensitive to changes of contact time in adsorption processes.

Influence of the matrix

Dissolved organic carbon

The Figure 3 displays the removals of micropollutants observed with different types of wastewater, sorted from the highest to the lowest DOC value. undergone an intense biological carbon treatment. For instance, the carbamazepine is removed at 18% in SEC settled water, against 38 to 42% in the other matrixes. Similarly, the trimethoprim removal increases from 5% in settled water to 39-50% in the other waters.

These results indicate that DOC value has a negative impact on micropollutant adsorption, as the removals observed in the SEC settled water, featuring the highest DOC concentration (27 mgC/L), are the lowest compare to the other waters.

The negative influence of DOC on micropollutant adsorption from wastewater was also observed in the literature [START_REF] De Ridder | Influence of natural organic matter on equilibrium adsorption of neutral and charged pharmaceuticals onto activated carbon[END_REF][START_REF] Kienle | Treatment of micropollutants in municipal wastewater: Ozone or powdered activated carbon?[END_REF][START_REF] Altmann | Direct comparison of ozonation and adsorption onto powdered activated carbon for micropollutant removal in advanced wastewater treatment[END_REF]. In contrary, DOC value seems to have a limited impact on propranolol behavior, a compound which is easily adsorbed by activated carbon as positively charged [START_REF] Mailler | Study of a large scale powdered activated carbon pilot: Removals of a wide range of emerging and priority micropollutants from wastewater treatment plant effluents[END_REF].

According to what is previously stated, micropollutant removals should be lower in SAV discharges compared to those in the SEC carbon biofiltration effluent and discharges, regarding the differences of DOC (11 vs. 6.6-6.8 mgC/L). This is observed for bezafibrate, lorazepam and metronidazole. However, removals are similar for the other compounds, in spite of slightly higher influent concentrations in the SAV discharges. Besides, considering the individual removals of compounds from Figure 3, the removals from SEC and SAV discharges are significantly similar (p-value = 0.175). The organic matter of SAV discharges should then be less competitive than organic matter from other discharges, what counterbalances the higher DOC.

This result indicates that the sole DOC level isn't sufficient to explain the variations of removals from water to water, the organic matter composition and nature have also to be considered. Considering that, the DOM nature and adsorption was assessed before and after contact with PAC for these 4 waters, as well as 3 additional wastewater matrixes.

Composition and adsorption of the organic matter

Figure 4 displays the composition (indexes from 3D fluorescence analysis) of the organic matter from 7 wastewater matrixes, as well as the organic matter removal by PAC adsorption (10 mgPAC/L -45 min).

Figure 4 -DOM adsorption and distribution using the defined indexes for the different studied wastewaters

First of all, the 4 WWTP discharges (SEC, SAM, SEM and SAV) and the SEC carbon biofiltration effluent have a similar distribution of DOM with a large fraction of humic-like organic matter (peaks α, α' and β; indexes I1 and I2), and a smaller fraction of organic matter resulting from bacterial activity (γ and δ; I3 and I4), as shown on Figure 4 and in Table 1. In contrary, the DOM distribution is different in the SEC settled water with very high I3 and I4 indexes, characterizing protein molecules, compared to I1 and I2 indexes, characterizing the humic-like substances. This indicates that carbon biological treatments significantly remove DOC and fluorophores δ and γ (see SEC settled water and carbon biofiltration effluent in Table 1), resulting in a significant modification of the DOM distribution (Figure 4). Together with this difference of DOM composition, the SEC settled water has higher DOC (27 mgC/L) and UV-254 (0.859 cm -1 ) values, highlighting a higher quantity of DOM.

In addition, the HIX increases along the different steps of wastewater treatment, as shown by the low HIX in the SEC settled water (1.0) and the higher HIX after the biological treatments (> 3).This suggests that small protein molecules (δ and γ) are transformed into humic-like substances (α, α' and β) during biological treatments, explaining the HIX increase.

The HIX from SAM and SEC discharges, as well as SEC carbon biofiltration effluent, are similar (3.2-3.6), while those from SAV and SEM discharges are higher (5.0-6.4), resulting from higher hydraulic retention times.

Regarding the removal of DOM by activated carbon, DOC and UV-254 removals are notably higher in the SEC settled water compared to the other ones. This is consistent as DOC and UV-254 are much higher in this water. This suggests that in this water, less sites will be available for micropollutants as more DOM is sorbed, what is consistent with micropollutants results (Figure 3). For the other wastewaters, the DOM removal is rather comparable (10-25% for DOC and 5-10% for UV-254).

In addition, fluorophores removals are rather similar in all wastewaters except in the SEC settled water. In the WWTP discharges, all the fluorophores are removed between 10 and 30%. In contrary, in SEC settled water, fluorophores δ and γ are predominant (I3 and I4) and are better removed (35-50%) than the other fluorophores. They are also better removed than in the other water. In consequence, these fluorophores will have the highest impact on the competition with micropollutants for adsorption. This is consistent with the study of [START_REF] De Ridder | Influence of natural organic matter on equilibrium adsorption of neutral and charged pharmaceuticals onto activated carbon[END_REF] which identified the small size DOM as the most competing ones for adsorption. Moreover, the fluorophores removal pattern is the same in SEC diluted settled water, confirming that the distribution of DOM is as important in the adsorption process as the DOC value.

In this context, and considering that the carbon biological treatment significantly reduces the DOC and the quantity of fluorophores δ and γ (the most competing DOM), the lower removals of micropollutants by PAC in the SEC settled water compared to the other waters are then consistent. Indeed, the competition is stronger in this water resulting from i) a higher quantity of DOM (DOC and UV-254) and ii) a different DOM distribution, with more small protein-like substances and less humic-like substances. In addition, I3 and I4 are lower in the SAV discharges than in SEC carbon biofiltration effluent and discharges, together with a higher HIX. This means that this water has less very competitive DOM what counterbalances the higher DOC, resulting in similar micropollutants removals.

It can be concluded that, besides a low DOC, low I3 and I4 indexes, together with a high HIX, are positive for the minimization of DOM competition during the micropollutants adsorption.

Presence and concentration of methanol

The micropollutant and organic matter removals obtained in presence of a residual concentration of methanol are presented in Figure 5. The comparison of removals obtained in the carbon biofiltration effluent and in the SEC discharges allows assessing the impact of the methanol presence in the wastewater on the PAC adsorption (Figure 5). Its presence doesn't decrease the performances as the removals observed in the SEC discharges are similar or higher than those in the carbon biofiltration effluent. The better removals in the SEC discharges could be explained i) by the methanol itself, and ii) by the carbon and nitrogen treatment achieved in the second and third biofiltration stages. but further experiments are needed to bring an answer.

Moreover, no clear trend is observed between the increase of the residual concentration of methanol and the micropollutant removals, except for sulfamethoxazole and naproxen for which the removal decrease and increase respectively. However, the results indicate that the increase of the methanol concentration doesn't lead to a significant degradation of the PAC performances for most compounds.

Thus, results have shown that the residual concentration of methanol doesn't affect negatively the removal of micropollutants by activated carbon (Figure 5). In contrary, its presence in biological treatment effluents could favor a biological activity in the tertiary treatments by activated carbon.

Influence of the injection of FeCl3

The ferric chloride influence on the fate of micropollutants in presence of PAC is illustrated on figure 7.

Figure 6 -Influence of the presence of FeCl3 on the pharmaceuticals removals by PAC adsorption

Three types of behaviors are observed for micropollutants. For most compounds, such as carbamazepine, beta blockers or diclofenac, and for DOM (DOC and UV-254), a slightly higher removal (+ 5-15%) is obtained by coupling PAC and ferric chloride, probably by coagulation of the colloidal fraction. These differences are not significant taking uncertainties into account, but 10 compounds in 13, as well as DOC and UV-254, have a greater removal in presence of FeCl3. In contrary, no clear influence is observed for lorazepam and bezafibrate. Finally, sulfamethoxazole seems to be the only one with a lower removal (-30%) in presence of ferric chloride.

(Margot et al. 2013) also observed a positive effect of the coagulant on micropollutant removals. However, the mechanism at the origin of this improvement is not identified. Supplementary tests are required to determine if coagulation, complexation or direct decrease of the competition with organic matter are involved.

CONCLUSIONS

The Parisian public sanitation service (SIAAP) and the Water Environment and Urban Systems laboratory (LEESU) study

the CarboPlus® process at large scale, in collaboration with the SAUR teams. In parallel to the study of the large scale pilot, complementary laboratory scale experiments were carried out to improve the understanding of the micropollutants adsorption on activated carbon, in particular PAC.

These experiments allowed displaying the links between structural properties of activated carbons and their efficiency to remove micropollutants. In particular, it has been shown that the PAC performances are closely related to the specific BET surface, which can easily be estimated through the bulk density measure.

The strong influence of the PAC dose has also been confirmed, as well as the suitability of the adsorption kinetic (less than one hour) for WWTP application (operational requirements). In an operational point of view, the injection of FeCl3 doesn't disrupt the adsorption process. A slight improvement is even observed for most compounds (+ 10-15%), except for sulfamethoxazole, probably through coagulation of the colloidal fraction of the DOM. In contrary, the residual concentration of methanol present in the WWTP discharges due to the post denitrification process seems to have no impact on the fate of most pharmaceuticals.

Then, this study highlighted the differences of micropollutants adsorption in the various wastewater matrixes. The DOC concentration is important considering the competition of adsorption but it isn't always sufficient to explain the performances reductions from water to water. It has been shown that the distribution of the different molecular components of the DOM should be considered in addition to the quantity of DOM. Indeed, small protein-like fluorophores have been identified as the most problematic for adsorption competition in wastewater, while a high HIX, highlighting the humification of the DOM, is favorable to micropollutants adsorption. The reduction of protein-like substances in wastewater is then a way to maximize the adsorption of micropollutants by activated carbon.

ACKNOWLEDGEMENT

This study has been performed within the framework of the OPUR research program. The authors would like to thank Benjamin Cassier and Mélanie Gobin, as well as the SIAAP (Céline Briand), the SAUR (Séverine Bareilles), the ISA (Antoine Vauchez, Mikaël Tournier, Florent Lafay and Loic Spinner), the LEESU and UTC teams for their technical support and their active participation to the sampling campaigns and analyses.

  Removal of emerging micropollutants from wastewater by activated carbon adsorption: Experimental study of different activated carbons and factors influencing the adsorption of micropollutants in wastewater R. Mailler, J. Gasperi, Y. Coquet, C. Derome, A. Buleté, E. Vulliet, Adèle Bressy, G. Varrault, G. Chebbo, V. Rocher To cite this version: R. Mailler, J. Gasperi, Y. Coquet, C. Derome, A. Buleté, et al.. Removal of emerging micropollutants from wastewater by activated carbon adsorption: Experimental study of different activated carbons and factors influencing the adsorption of micropollutants in wastewater. Journal of Environmental Chemical Engineering, 2016, 4 (1), 10.1016/j.jece.2016.01.018. hal-01266245

  WWTP; SAM = Seine Amont WWTP; SAV = Seine Aval; SEM = Seine Morée. BF = biofiltration; CAS = conventional activated sludge; MBR = membrane bioreactor: C = carbon removal; N = nitrogen removal (nitrification + denitrification). DOC = dissolved organic carbon, UV-254 = UV absorbance at 254 nm, Fluo = fluorophore, HIX = humification index, BIX = biological activity index.

Figure 1 -

 1 Figure1-Correlations between the specific BET surface and the micropollutant removals obtained with a PAC, and link with its bulk density

Figure 2 -

 2 Figure 2 -Micropollutants adsorption kinetic in the Seine Centre discharges and influence of the dose on removals

Figure 3 -

 3 Figure 3 -Pharmaceuticals removals by PAC adsorption from different types of wastewater

Figure 5 -

 5 Figure 5 -Impact of a residual concentration of methanol in the WWTP discharges on the adsorption of pharmaceuticals by PAC

Table 1 -The 7 studied wastewaters and their main DOM characteristics

 1

		SEC settled	SEC settled	SEC carbon	SEC	SAM	SAV	SEM
		water	water -	biofiltration	discharges	discharges	discharges	discharges
			diluted	effluent
	Biological	-	-	BF (C)	BF (C+N) CAS (C+N)	CAS (C)	MBR
	treatment						BF (N)	C+N
	TOC	32	7.8	-	7.3	7.2	-	5.5
	(mgC/L)
	DOC	27	7.7	6.6	6.8	6.6	11	5.2
	(mgC/L)
	UV-254	0.859	0.203	0.120	0.150	0.173	0.237	0.140
	(cm -1 )
	Fluo α (UA)	1 879	455	812	1 522	822	2 097	701
	Fluo α' (UA)	2 019	442	741	1 262	712	2 041	730
	Fluo β (UA)	1 926	440	742	1 371	761	2 018	657
	Fluo δ (UA)	4 949	1 080	665	1 348	678	1 685	622
	Fluo γ (UA)	4 194	907	354	1 077	448	590	231
	HIX	1.0	1.2	3.4	3.2	3.6	5.0	6.4
	BIX	0.85	0.90	0.97	0.96	0.99	0.95	0.96
	SEC = Seine Centre

Table 2 -Characterization and performances of the 4 PACs studied

 2 Distribution of surface. Micropore = < 2 nm, mesopores = < 50 nm and macropores = > 50 nm. 2 Sum of 16 pharmaceuticals = atenolol, bezafibrate, carbamaepine, ciprofloxacin, diclofenac, erythromycin, ketoprofen, lorazepam, metronidazole, naproxen, norfloxacin, ofloxacin, propranolol, roxithromycin, sulfamethoxazole and trimethoprim.

		PB 170®	WP 235®	W 35®	PC 1000®
	Producer	DaCarb	Chemviron	Norit	DaCarb
	Raw material	Wood	Coal	Peat	Coconut
	Bulk density (g/cm 3 )	0.30	0.38	0.33	0.54
	Specific BET surface (m 2 /g)	957 ± 28	909 ± 30	768 ± 19	458 ± 14
	Pore size
	distribution (micro, meso,	54% -35% -11%	53% -31% -16% 45% -45% -10% 59% -29% -12%
	macroporous) 1
	Pore volume
	(micro + meso)	0.5066	0.4841	0.4876	0.2435
	(mL/g)
	Particle size
	distribution	3.4 -16.2 -58.9	3.4 -18.0 -58.0	3.2 -19.4 -86.2	3.7 -31.7 -129.4
	(µm)