Dominique Lépingle
email: dominique.lepingle@univ-orleans.fr

A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES

Keywords: Singular stochastic differential equation, electrostatic repulsion, Bessel process, product form stationary distribution, skew-symmetry condition, obliquely reflected Brownian motion AMS classification: 60H10

We consider a Brownian motion forced to stay in the quadrant by an electrostatic oblique repulsion from the sides. We tackle the question of hitting the corner or an edge, and find product-form stationary measures under a certain condition, which is reminiscent of the skew-symmetry condition for a reflected Brownian motion.

Introduction

In the present paper we study existence and properties of a new process with values in the nonnegative quadrant S = R + ×R + where R + := [0, ∞). It may be seen as a two-dimensional extension of a usual Bessel process. It is a two-dimensional Brownian motion forced to stay in the quadrant by electrostatic repulsive forces, in the same way as in the one-dimensional case where a Brownian motion which is prevented from becoming negative by an electrostatic drift becomes a Bessel process. Note here and now that the corner 0 = (0, 0) will play a crucial role and in some cases it will be necessary to restrict the state space to the punctured nonnegative quadrant S 0 = S \ {0}.

Let (Ω, F, (F t ) t≥0 , P) be a complete probability space endowed with a filtration (F t ) t≥0 satisfying the usual conditions. Let (B t , C t ) be an adapted driftless Brownian motion in the plane starting from 0, with covariance matrix 1 ρ ρ 1 and ρ ∈ [-1, +1]. Definition 1. Let α, β, γ, δ be four real constants with α > 0, δ > 0. We say that an (F t )adapted continuous process (X, Y ) with values in S is an Oblique Two-dimensional Bessel Process (O2BP) if for any t ≥ 0 (1)

X t = X 0 + B t + α t 0 ds Xs + β t 0 ds Ys ≥ 0 Y t = Y 0 + C t + γ t 0 ds Xs + δ t 0 ds Ys ≥ 0
where X 0 and Y 0 are nonnegative F 0 -measurable random variables, and

t 0 1 {Xs=0} ds = 0 t 0 1 {Ys=0} ds = 0 t 0 1 {Xs>0} ds Xs < ∞ t 0 1 {Ys>0} ds Ys < ∞ .
This stochastic differential system is very singular at the edges of the quadrant and the question of existence and uniqueness of a solution is not simple. The particular case when

β = γ = 0 (2) U t = X 0 + B t + α t 0 ds Us ≥ 0 V t = Y 0 + C t + δ t 0 ds Vs ≥ 0 .
is already known: the processes U and V are Bessel processes. Actually, U is a Bessel process of dimension 2α + 1, and the point 0 is instantaneously reflecting for U if α < 1 2 and polar if α ≥ 1 2 . If ρ = 0, U 2 + V 2 is the square of a Bessel process of dimension 2α + 2δ + 2 [START_REF] Shiga | Bessel diffusions as a one-parameter family of diffusion processes[END_REF], and so the corner 0 is polar for (U, V ) in this case. Comparison between X and U , Y and V will play a key role in the construction of the solution (X, Y ) and the study of its behavior close to the edges of the quadrant. The process (U, V ) is an example of Brownian motion perturbed by a drift deriving from a convex potential. More generally, stochastic differential systems including such a singular drift have been studied in [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF][START_REF] Pettersson | Yosida approximations for multivalued stochastic differential equations[END_REF][START_REF] Storm | Stochastic differential equations with convex constraint[END_REF][START_REF] Rȃşcanu | Deterministic and stochastic differential equations in Hilbert spaces involving multivalued maximal monotone operator[END_REF], where strong existence and uniqueness were obtained. They are examples of so-called multivalued stochastic differential equations, also called stochastic variational inequalities in convex analysis.

We will use these results to study the solutions to [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF] in the oblique case where β or γ do not vanish. We obtain strong existence and uniqueness for a large set of parameters and initial conditions, but not for all possible values. In the proofs we naturally fall into the crucial question of hitting the corner, that is the non-smooth part of the boundary. Using McKean's argument on the asymptotic behavior of continuous local martingales obtained by time change from the real driftless Brownian motion, we are able to state several sufficient conditions to prevent the processes from hitting the corner. Our methods are not powerful enough to allow for necessary conditions. However we shall not restrict to processes avoiding the corner and, depending on the parameters, we will get existence and uniqueness (in a strong sense) sometimes in the whole quadrant, sometimes in the punctured quadrant, that is, the quadrant without a corner.

We will also obtain some partial results about the attainibility of the edges of the quadrant. It is interesting to see whether the boundary behavior of one component may be modified by the interaction with the other component.

In the one-dimensional case, the so-called scale functions transform the solution of a stochastic differential equation into local martingales and one may infer some information on the boundary behavior of the process. This technique was very successful in the study of Bessel processes. Here we still obtain functions of an O2BP which are local martingales or supermartingales for some values of the parameters and we derive some information on its asymptotic behavior.

The laws of Bessel processe with different parameters [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] are mutually abolutely continuous when their paths do not reach the origin. Here we obtain two partial results about the absolute continuity of the laws of O2BPs for some values of the parameters.

Finally we follow the way explored in [START_REF] O'connell | Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition[END_REF], where a drifted Brownian motion is endowed with another drift term that is continuous and depends obliquely, via a regular potential function, on the position of the process relative to an orthant. Under an additional condition, which is called a skew-symmetry condition, an invariant density was given in an explicit product form. We obtain the same result for an O2BP (where the Brownian motion is now drifted) under an anologous condition. This time, the invariant density is the product of two gamma densities, which is consistent with the one-dimensional case where the invariant measure of a drifted Bessel process is a gamma distribution.

1.1. Comparison with obliquely reflected Brownian motion. Studying O2BPs makes appear a strong connection with the properties of a semimartingale reflecting Brownian motion in the quadrant. We briefly recall the definition. Let (B t , C t ) t≥0 be the Brownian motion in Definition 1. Definition 2. An semimartingale reflecting Brownian motion (SRBM) in the quadrant is a continuous adapted process (X t , Y t ) which is a solution to the system

(3) X t = X 0 + B t + L 1 t + r 1 L 2 t ≥ 0 Y t = Y 0 + C t + r 2 L 1 t + L 2 t ≥ 0
where L 1 and L 2 are two continuous adapted nondecreasing processes with L 1 0 = L 2 0 = 0 such that for any t ≥ 0

t 0 1 {Xs>0} dL 1 s = t 0 1 {Ys>0} dL 2 s = 0.
Here X 0 and Y 0 are nonnegative F 0 -measurable random variables, r 1 and r 2 are real numbers.

There was an extensive literary output on that topic in the eighties, with a more general domain : a wedge, an orthant or a convex polyhedron. We mention the works of Harrison, Reiman, Varadhan, Williams, Dai [START_REF] Harrison | The diffusion approximation for tandem queues in heavy traffic[END_REF][START_REF] Harrison | Reflected Brownian motion in an orthant[END_REF][START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF][START_REF] Williams | Reflected Brownian motion with skew symmetric data in a polyhedral domain[END_REF][START_REF] Dai | Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedra[END_REF] to cite a few of them. For a more complete bibliography we refer to [START_REF] Williams | Semimartingale reflecting Brownian motions in the orthant. A survey[END_REF].

We note a first analogy between their results and ours: we prove a necessary condition of existence of an O2BP is the existence of a convex combination of the directions of interaction (4)

r x = α γ and r y = β δ
that points into the quadrant from the corner. In the reflection setting it was proved in [START_REF] Williams | Reflected Brownian motion in a wedge: semimartingale property[END_REF] that a necessary an sufficient condition for the existence of a SRBM is the existence of a convex combination of the directions of reflection 1 r 2 and r 1 1

with the same property.

Another analogy is met in the delicate and important question of attainibility of the corner. The authors in [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF] have found an explicit harmonic function that provides a full answer to the question of hitting the corner for a reflected Brownian motion in a wedge of angle ξ ∈ (0, 2π) with the identity matrix as covariance matrix. We do not have any such convenient function. However, our second condition [START_REF] Dupuis | Lyapunov functions for semimartingale reflecting Brownian motions[END_REF] 2ρ ≤ β δ + γ α in Corollary 10 is reminiscent of the necessary and sufficient condition in [START_REF] Sarantsev | Triple and simultaneous collisions of competing Brownian particles[END_REF] which writes [START_REF] Fernholz | Stochastic portfolio theory[END_REF] 2ρ ≤ r 1 + r 2 in the setting of Definition 2. Restricting to ξ ∈ (0, π), we may consider a linear transformation in the plane that changes the wedge in [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF] into the nonnegative quadrant and the initial Brownian motion into a Brownian motion with covariance coefficient ρ = -cos ξ. Then the non-attainibility condition in [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF] is transformed into the above condition [START_REF] Fernholz | Stochastic portfolio theory[END_REF].

For theoretical as well as practical reasons, a great deal of interest was taken in the question of recurrence of the Brownian motion with a constant drift vector and oblique reflection, and in the computation of the invariant measure [START_REF] Williams | Recurrence classification and invariant measure for reflected Brownian motion in a wedge[END_REF][START_REF] Williams | On reflecting Brownian motion -a weak convergence approach[END_REF][START_REF] Harrison | Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution[END_REF][START_REF] Sarantsev | Reflected Brownian motion in a convex polygonal cone: tail estimates for the stationary distribution[END_REF]. Under the assumption that the directions of reflection satisfy a skew-symmetry condition, it was proved that the invariant measure has exponential product form density [START_REF] Harrison | Multidimensional reflected Brownian motion having exponential stationary distributions[END_REF][START_REF] Harrison | Brownian models of open queueing networks with homogeneous customer populations[END_REF][START_REF] Williams | Reflected Brownian motion with skew symmetric data in a polyhedral domain[END_REF][START_REF] Williams | Semimartingale reflecting Brownian motions in the orthant. A survey[END_REF]. This result has a practical interest because of the interpretation of the SRBM as approximation of the queue length processes for networks of queues in heavy traffic [START_REF] Harrison | The diffusion approximation for tandem queues in heavy traffic[END_REF][START_REF] Harrison | Multidimensional reflected Brownian motion having exponential stationary distributions[END_REF]. There is also a financial reason for studying SRBMs. Motivated by the so-called Atlas model of equity market presented in [START_REF] Fernholz | Stochastic portfolio theory[END_REF], some authors [START_REF] Ichiba | On collisions of Brownian particles[END_REF][START_REF] Ichiba | Hybrid Atlas models[END_REF][START_REF] Ichiba | Strong solutions to stochastic equations with rank-biased coefficients[END_REF][START_REF] Ichiba | Diffusions with rank-based characteristics and values in the nonnegative quadrant[END_REF][START_REF] Ichiba | Convergence rates for rank-based models with applications to portfolio theory[END_REF][START_REF] Sarantsev | Triple and simultaneous collisions of competing Brownian particles[END_REF] have recently studied competing Brownian particles on the line with rank dependent local characteristics. In fact the gaps between adjacent particles are proportional to the components of a SRBM in an orthant. There is an invariant probability density with an explicit exponential product form when the volatility coefficients are constant [START_REF] Pal | One-dimensional Brownian particle systems with rank-dependent drifts[END_REF]. More generally, this is still true if their squares depend on rank linearly since one may infer from the statements in Section 2 of [START_REF] Sarantsev | Triple and simultaneous collisions of competing Brownian particles[END_REF] that the skew-symmetry condition is still satisfied. Following the way in [START_REF] Harrison | Brownian models of open queueing networks with homogeneous customer populations[END_REF], [START_REF] Dupuis | Lyapunov functions for semimartingale reflecting Brownian motions[END_REF] and [START_REF] Harrison | Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution[END_REF], a more general question is the recurrence or transience of an O2BP. Answering this question does not seem to be an easy task. The method in [START_REF] Hobson | Recurrence and transience of reflecting Brownian motion in the quadrant[END_REF], which provides a full answer for the obliquely reflected Brownian motion in the quadrant, appears to break down here. Mimicking the computation in [START_REF] O'connell | Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition[END_REF], we just calculate an invariant measure in product form under a skew-symmetry condition which is the equality condition in the inequality [START_REF] Dupuis | Lyapunov functions for semimartingale reflecting Brownian motions[END_REF], whereas the skew-symmetry condition in the reflecting case is the equality condition in [START_REF] Fernholz | Stochastic portfolio theory[END_REF]. Now the terms of the product are gamma distributions with explicit parameters.

Another topic of interest in [START_REF] O'connell | Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition[END_REF] lies in its Remark 4.12: when a scale parameter goes to zero, the exponentially reflected Brownian motion should converge to the obliquely reflected Brownian motion. It is typically a penalty method. This kind of approximation by a sequence of diffusions with regular drifts living on the whole Euclidean space has been used for instance in [START_REF] Lions | Stochastic differential equations with reflecting boundary conditions[END_REF][START_REF] Sarantsev | Penalty method for obliquely reflected diffusions[END_REF] for oblique reflection in domains with smooth boundary and in [START_REF] Menaldi | Stochastic variational inequality for reflected diffusion[END_REF][START_REF] Slomiński | Weak and strong approximations of reflected diffusions via penalization methods[END_REF] for normal reflection in convex domains. With regard to our framework, it can be shown that a sequence of real Bessel processes with dimension decreasing to 1 pathwise decreases to a reflected real Brownian motion. It could be interesting to consider a sequence of oblique two-dimensional Bessel processes with interaction matrices

n -1 1 r 1 r 2 1
and to let n go to infinity. It should converge to the SRBM in Definition 2. This time, this would be an interior approximation, as it was done in [START_REF] Williams | On reflecting Brownian motion -a weak convergence approach[END_REF] and [START_REF] Pardoux | Symmetric reflected diffusions[END_REF] in the particular setting of stationary Markov processes associated with Dirichlet forms. We leave aside this point for further investigation.

1.2. Organization of the paper. The rest of the paper is organized as follows. In Section 2 we recall some trajectorial properties of usual Bessel processes. In Section 3 we state and prove three main lemmas of repeated use in the sequel. We also recall an existence and uniqueness result for a special case of multivalued stochastic differential equation that will be useful in our construction of an OB2P. Section 4 is devoted to the proof of sufficient conditions to avoid the corner of the quadrant. The main theorems of existence and uniqueness of an O2BP are given in Section 5. In Section 6 we discuss the question of hitting the edges of the quadrant. In Section 7 we consider two particular cases where there exist simple functions of O2BPs that are local martingales and obtain some information on the asymptotic behavior of the paths. We leave the trajectorial point of view in Section 8 to tackle questions of absolute continuity of the law of an O2BP with respect to the product of laws of real Bessel processes. The final Section 9 introduces a skew-symmetry condition that allows us to obtain existence of a stationary probability in form of the product of two gamma distributions.

Some properties of Bessel processes

A Bessel process of dimension d > 1, starting at r ≥ 0, is the unique solution to the stochastic differential equation ( 7)

R t = r + W t + d -1 2 t 0 1 R s ds.
where W is a standard driftless real-valued Brownian motion starting at 0. We know that:

• d ≥ 2 :
P(R t > 0, ∀t > 0) = 1, P(R t → ∞, t → ∞) = 1, R 2-d is a local martingale. • Dimension d = 2: P(R t > 0, ∀t > 0) = 1, P(sup t>a R t = ∞, inf t>a R t = 0) = 1 for any a > 0, ln R is a local martingale.
• Dimension 1 < d < 2:

P (R t > 0, ∀t > a) = 0 for any a > 0. Corollary 3. When d ≥ 2, for any a > 0, ∞ a ds R 2 s = ∞ a.s. Proof. From Itô's formula ln R t = ln R a + t a dWs Rs + d-2 2 t a ds R 2 s .
From ( [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] We shall also use an absolute continuity result ( [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Exercise XI.1.22 or [START_REF] Jeanblanc | Mathematical Methods for Financial Markets[END_REF], Proposition 6.1.5.1). On the canonical space

Ω = C([0, ∞), [0, ∞)), we denote by R the canonical map R t (ω) = ω(t), by R t = σ(R s , s ≤ t)
P d r | Rt = R t r d-2 2 exp - (d -2) 2 8 t 0 ds R 2 s . P 2 r | Rt .

Four useful tools

The following simple lemma is a comparison lemma. It will play an ubiquitous role in our proofs.

Lemma 4. For T > 0, α > 0, let x 1 and x 2 be nonnegative continuous solutions on [0, T ] to the equations

x 1 (t) = v 1 (t) + α t 0 ds x 1 (s) x 2 (t) = v 2 (t) + α t 0 ds x 2 (s) where v 1 , v 2 are continuous functions such that 0 ≤ v 1 (0) ≤ v 2 (0), and v 2 -v 1 is nondecreas- ing. Then x 1 (t) ≤ x 2 (t) on [0, T ]. Proof. Assume there exists t ∈ (0, T ] such that x 2 (t) < x 1 (t). Set τ := max{s ≤ t : x 1 (s) ≤ x 2 (s)} . Then, x 2 (t) -x 1 (t) = x 2 (τ ) -x 1 (τ ) + (v 2 (t) -v 1 (t)) -(v 2 (τ ) -v 1 (τ )) + α t τ ( 1 x 2 (s) -1 x 1 (s) )ds ≥ 0 , a contradiction.
The following elementary lemma will also be repeatedly used.

Lemma 5. Let Q(x, y) = ax 2 + bxy + cy 2 be a second degree homogeneous polynomial. Then Q is nonnegative on the whole S if and only if a ≥ 0, c ≥ 0 and b ≥ -2 √ ac.

Proof. Taking x = 0 (resp. y = 0) we see that a (resp. c) must be nonnegative. Then we rewrite

Q(x, y) = ( √ ax - √ cy) 2 + (b + 2 √ ac)xy. So Q is nonnegative for all x ≥ 0 and y ≥ 0 iff b ≥ -2 √ ac.
Another main tool will be the following convergence result, whose argument goes back to McKean ( [START_REF] Mckean | Stochastic Integrals[END_REF], p.31 and p.47). The statement and the proof below are borrowed from ( [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Theorem V.1.7 and Proposition V.1.8). The only change lies in the introduction of a stopping time τ up to which the local martingale M is now defined. Lemma 6. Let M be a continuous local martingale defined on [0, τ ) where τ is a stopping time. Let ( M t ) 0≤t<τ be its quadratic variation and let M τ := lim t→τ M t ≤ ∞.

(1) On { M τ < ∞}, lim t→τ M t exists a.s. in R.

(

) On { M τ = ∞}, lim sup t→τ M t = -lim inf t→τ M t = +∞ a.s. 2
Proof.

(1)For any p ≥ 1 let

σ p = inf{t ≥ 0 : M t ≥ p}.
In order every term in the following to be well defined we introduce a nondecreasing sequence of stopping times (τ n ) n≥1 with limit τ such that each stopped process (M τn∧t ) t≥0 is a uniformly integrable martingale for any n ≥ 1. The stopped process (M t∧τn∧σp ) t≥0 is a

L 2 -bounded martingale and as m, n → ∞ E[(M τn∧σp -M τm∧σp ) 2 ] = E[| M τn∧σp -M τm∧σp |] → 0. We set M (p) := lim n→∞ M τn∧σp = lim t→τ M t∧σp .
On { M τ < ∞}, the stopping times σ p are a.s. infinite from some p on and we can set M τ := lim p→∞ M (p) . Thus on this set

M t → M τ as t → τ.
(2) Let for any t ≥ 0

T t = inf{0 ≤ s ≤ τ : M s > t}.
There exist an enlargement ( Ω, ( Ft ), P) of (Ω, (F Tt ), P) and a Brownian motion β on Ω independent of M such that the process

B t = M Tt if t < M τ M τ + βt-M τ if t ≥ M τ is a standard linear Brownian motion. As lim sup t→∞ B t = -lim inf t→∞ B t = +∞ a.s. we obtain on { M τ = ∞} lim sup t→∞ M Tt = -lim inf t→∞ M Tt = +∞ and therefore lim sup t→τ M t = -lim inf t→τ M t = +∞ .
We will also need the following consequence of the results in [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF] on multivalued stochastic differential systems, completed with the method used in [START_REF] Cépa | Diffusing particles with electrostatic repulsion[END_REF] and developed in [START_REF] Lépingle | Boundary behavior of a constrained Brownian motion between reflecting-repelling walls[END_REF] to check the lack of additional boundary process.

Proposition 7. Let α > 0, δ ≥ 0, σ = (σ i j ; i, j = 1, 2) a 2 × 2-matrix, (B, C
) a Brownian motion in the plane, b 1 and b 2 two Lipschitz functions on R 2 , Z 1 0 and Z 2 0 two F 0 -measurable nonnegative random variables. There exists a unique strong solution (Z 1 , Z 2 ) to the system

(9) Z 1 t = Z 1 0 + σ 1 1 B t + σ 1 2 C t + α t 0 ds Z 1 s + t 0 b 1 (Z 1 s , Z 2 s )ds Z 2 t = Z 2 0 + σ 2 1 B t + σ 2 2 C t + δ t 0 ds Z 2 s + t 0 b 2 (Z 1 s , Z 2 s )ds with the conditions Z 1 t ≥ 0 if δ = 0 and Z 1 t ≥ 0, Z 2 t ≥ 0 if δ > 0.
It is worth noticing that the solutions to (1) enjoy the Brownian scaling property. It means that if (X, Y ) is a solution to (1) starting from (X 0 , Y 0 ) with driving Brownian motion (B t , C t ), then for any c > 0 the process (

X t := c -1 X c 2 t , Y t := c -1 Y c 2 t ; t ≥ 0) is a solution to (1) starting from (c -1 X 0 , c -1 Y 0 ) with driving Brownian motion (c -1 B c 2 t , c -1 C c 2 t ).

Avoiding the corner

We shall see in Section 5 that existence and uniqueness of the solution to (1) are easily obtained as soon as the solution process keeps away from the corner. Thus the question of attaining the corner in finite time is of great interest. For some class of reflection matrices, a necessary and sufficient condition is given in [START_REF] Sarantsev | Triple and simultaneous collisions of competing Brownian particles[END_REF]. Unfortunately, we are not able to provide such a complete answer and we have to be content with a collection of sufficient conditions ensuring nonattainability. We will just see at the end of Section 5 a degenerated case where the corner is reached in finite time with full probability.

Our sufficient conditions are stated in the following theorem. In some cases (conditions C 2a and C 2b below), the comparison with Bessel processes suffices to conclude. In other cases (conditions C 1 and C 3 below), we are first looking for a C 2 -function f on the punctured quadrant S 0 with limit -∞ at the corner. Then we use Lemma 6 to show that f (X t , Y t ) cannot converge to -∞ in finite time.

Theorem 8. Let (X, Y ) be a solution to [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF]. We set

τ 0 := inf{t > 0 : (X t , Y t ) = 0}
with the usual convention inf ∅ = ∞. Then P(τ 0 < ∞) = 0 if one of the following conditions is satisfied:

(1)

C 1 : β ≥ 0, γ ≥ 0 and -1 ≤ ρ ≤ α + δ. (2) C 2a : α ≥ 1 2 and β ≥ 0 (3) C 2b : δ ≥ 1
2 and γ ≥ 0 (4) C 3 : There exist λ > 0 and µ > 0 such that

• λα + µγ ≥ 0 • λβ + µδ ≥ 0 • ( λ(λα + µγ) + µ(λβ + µδ)) 2 ≥ 1 2 (λ 2 + µ 2 + 2ρλµ). Proof. Condition C 1 . For > 0 let σ = 1 {(X 0 ,Y 0 )=0} inf{t > 0 : X t + Y t ≥ } τ 0, = inf{t > σ : (X t , Y t ) = 0} . As ↓ 0, σ ↓ 0 and τ 0, ↓ τ 0 . We set R t = X 2 t + Y 2 t . From Itô's formula we get for t ∈ [σ , τ 0, ) ln R t = ln R σ + 2 t σ XsdBs+YsdCs Rs + 2 t σ (α+δ)ds Rs + 2 t σ (β Xs Ys + γ Ys Xs ) ds Rs -4ρ t σ XsYs R 2 s ds ≥ ln R σ + M t + 2 t σ P 1 (Xs,Ys) R 2 s ds
where M is a continuous local martingale on [σ , τ 0, ) and P 1 (x, y) is the second degree homogeneous polynomial

P 1 (x, y) = (α + δ)(x 2 + y 2 ) -2ρxy.
Using Lemma 5 we check that this polynomial is nonnegative on

S if ρ ≤ α + δ. Therefore 0 ≤ τ 0, σ P 1 (X s , Y s ) R 2 s ds ≤ ∞
From Lemma 6 we know that depending on whether M τ 0, is finite or not, the local martingale M t either converges in R as t → τ 0, or oscillates between +∞ and -∞. It cannot converge to -∞. Thus R τ 0, > 0 on {τ 0, < ∞}, which contradicts the definition of the moment τ 0, . This proves that τ 0, = ∞ for every > 0, and therefore, τ 0 = ∞ a.s. where M is a continuous local martingale on [σ , τ 0, ) and P 2 (x, y) is the second degree homogeneous polynomial

P 2 (x, y) = λ(λβ + µδ)x 2 + µ(λα + µγ)y 2 + [λ(λα + µγ) + µ(λβ + µδ) -1 2 (λ 2 + µ 2 + 2ρλµ)]xy . Using again Lemma 5, we see that P 2 is nonnegative on S if 1 2 (λ 2 + µ 2 + 2ρλµ) -[λ(λα + µγ) + µ(λβ + µδ)] ≤ 2 λ(λβ + µδ)µ(λα + µγ).
This is exactly the condition C 3 . Therefore

0 ≤ t σ P 2 (X s , Y s ) X s Y s S 2 s ds < ∞ and so 0 ≤ τ 0, σ P 2 (X s , Y s ) X s Y s S 2 s ds ≤ ∞ .
Similarly, using Lemma 6 again, we see that the continuous local martingale M either converges to a finite limit or oscillates between +∞ and -∞ when t → τ 0, . It cannot converge to -∞ and thus S τ 0, > 0 on {τ 0, < ∞}, proving P(τ 0, < ∞) = 0. Letting finally → 0 we obtain P(τ 0 < ∞) = 0.

Condition C 3 is not explicit. We give two concrete examples when this condition holds true.

Corollary 9. Assume ρ = 0 , α = δ and |β| = |γ|. Then the condition C 3 is satisfied if

(10) • β 2 ≤ α -1 4 when β = -γ • -β ≤ α -1 4 when β = γ < 0 .
Proof. In both cases we take λ = µ > 0. When β = -γ ≥ 0, the condition C 3 writes

• α -β ≥ 0 • 1 ≤ ( √ α + β + √ α -β) 2 = 2α + 2 α 2 -β 2 . If β 2 ≤ α -1 4 , then α 2 -β 2 ≥ α 2 -α + 1 4 ≥ 0 and (1 -2α) 2 = 1 -4α + 4α 2 ≤ 4(α 2 -β 2 ).
When β = γ < 0, the condition C 3 writes

• α + β ≥ 0 • 1 ≤ (2 √ α + β) 2 = 4(α + β) and this is -β ≤ α -1 4 . Corollary 10. Assume • max{α, δ} ≥ 1 2 • 2ρ ≤ β δ + γ α .
Then P(τ 0 < ∞) = 0 .

Proof. We may assume α ≥ 1 2 . If β ≥ 0, then condition C 2a holds true and the conclusion follows. If β < 0, we will use condition C 3 . We take λ = δ and µ = -β. Then λβ + µδ = 0,

λα + µγ = δα -βγ ≥ δα -βα(2ρ -β δ ) = α δ (δ 2 + β 2 -2ρβδ) ≥ α δ (δ -ρβ) 2 ≥ 0 and ( λ(λα + µγ) + µ(λβ + µδ)) 2 -1 2 (λ 2 + µ 2 + 2ρλµ) = λ(λα + µγ) -1 2 (λ 2 + µ 2 + 2ρλµ) ≥ α(δ 2 + β 2 -2ρβδ) -1 2 (δ 2 + β 2 -2ρβδ) ≥ (α -1 2 )(δ -ρβ) 2 ≥ 0 .

Existence and uniqueness

We now proceed to the question of existence and uniqueness of a global solution to [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF]. We consider separately the three cases: first β ≥ 0 and γ ≥ 0, second βγ < 0, third β ≤ 0 and γ ≤ 0. In the first and second cases, we construct the solution by switching from one edge to the other and patching the paths together. Thus it is essential to avoid the corner, as it was supposed in [START_REF] Sarantsev | Penalty method for obliquely reflected diffusions[END_REF] in order to weakly approximate an obliquely reflected Brownian motion. The third case uses a different proof and does not requires avoiding the corner. All three proofs heavily use the comparison method of Lemma 4. (1) There is a unique solution to (1) in S 0 .

(2) There is a solution to (1) in S starting from 0.

(3) If αδ ≥ βγ, there is a unique solution to (1) in S.

Proof. 1. Let a > 0, > 0 and define for (x,

z) ∈ R + × R ψ (x, z) := 1 max(γx + z, α ) .
This is a Lipschitz function. From Proposition 7 we know that the system (11)

X t = X 0 + B t + α t 0 ds X s + αβ t 0 ψ (X s , Z s )ds ≥ 0 Z t = -γX 0 + α(Y 0 + 1 {Y 0 =0} a) -γB t + αC t + α(αδ -βγ) t 0 ψ (X s , Z s )ds has a unique solution. Let τ Y := inf{t > 0 : γX t + Z t < α } .
If 0 < η < < a we deduce from the uniqueness that (X , Z ) and (X η , Z η ) are identical on [0, τ Y ]. Patching together we can set

X t := lim →0 X t Y t := lim →0 1 α (γX t + Z t ) on {(ω, t) ∈ Ω × [0, ∞) : Y 0 (ω) > 0 and 0 ≤ t < τ 0 Y (ω)}, where τ 0 Y := lim →0 τ Y .
On this set, (X, Y ) is the unique solution to (1). As we already noted, we have

X t ≥ U t and Y t ≥ V t . Therefore, on {Y 0 > 0} ∩ {τ 0 Y < ∞},
T 1 = τ 0 Y , T 2 = inf{t > T 1 : X t = 0}. Iterating, we get a solution on {Y 0 > 0} × [0, lim n→∞ T n ) where T 2p := inf{t > T 2p-1 : X t = 0} T 2p+1 := inf{t > T 2p : Y t = 0} .
On {Y 0 > 0} ∩ {lim n→∞ T n < ∞} we set X limn→∞ Tn := lim p→∞ X T 2p = 0 and Y limn→∞ Tn := lim p→∞ Y T 2p+1 = 0. The polarity of 0 entails this is not possible in finite time and thus lim n→∞ T n = ∞. So we have obtained a unique global solution on {Y 0 > 0}. In the same way we obtain a unique global solution on {X 0 > 0} and as P((X 0 , Y 0 ) = 0) = 0 the proof is complete.

2. Assume now X 0 = Y 0 = 0. Let (y n ) n≥1 be a sequence of real numbers (strictly) decreasing to 0. From the above paragraph it follows there exists for any n ≥ 1 a unique solution (X n , Y n ) with values in S 0 to the system

X n t = B t + α t 0 ds X n s + β t 0 ds Y n s Y n t = y n + C t + γ t 0 ds X n s + δ t 0 ds Y n s . Let τ := inf{t > 0 : X n+1 t < X n t } . Using Lemma 4 we obtain Y n+1 t ≤ Y n t on [0, τ ]. We note that (X n τ , Y n τ ) ∈ S 0 on {τ < ∞}. On {Y n+1 τ = Y n τ } ∩ {τ < ∞}, since X n+1 τ = X n
X t := lim n→∞ ↑ X n t Y t := lim n→∞ ↓ Y n t .
As Y n t ≥ V t where (U, V ) is the solution to (2) with X 0 = Y 0 = 0, we have

X t = B t + α lim n→∞ t 0 ds X n s + β lim n→∞ t 0 ds Y n s = B t + α t 0 ds Xs + β t 0 ds Ys < ∞
and also

Y t = lim n→∞ y n + C t + γ lim n→∞ t 0 ds X n s + δ lim n→∞ t 0 ds Y n s = C t + γ t 0 ds Xs + δ t 0 ds Ys < ∞ .
3. Assume finally αδ -βγ ≥ 0. As the conclusion holds true if β = γ = 0, we may also assume β > 0. Let (X, Y ) be the solution to [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF] with X 0 = Y 0 = 0 obtained in the previous paragraph and let (X , Y ) be another solution. Considering (X n , Y n ) again and replacing (X n+1 , Y n+1 ) with (X , Y ), the previous proof works and we finally obtain X t ≥ X t and Y t ≤ Y t . Then, [START_REF] Ichiba | On collisions of Brownian particles[END_REF] 0

≤ δ(X t -X t ) -β(Y t -Y t ) = t 0 (αδ -βγ)( 1 X s - 1 X s )ds ≤ 0.
Thus X t = X t and Y t = Y t , proving uniqueness. Replacing (δ, β) with (γ, α) in equation ( 13) we obtain the same conclusion if γ > 0.

Remark 12. The statement in Theorem 11 is not complete since the problem of uniqueness when starting at the corner and αδ < βγ is not solved. When considering the solution (X, Y ) in the above proof of existence, we have noted that X ≥ X and Y ≤ Y for any other solution (X , Y ). Thus uniqueness in law would be sufficient to obtain path uniqueness. A possible way to prove weak uniqueness could be the method in [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF]. This would be far from our trajectorial methods and we don't go further in that direction.

5.2.

Case βγ < 0.

Theorem 13. Assume βγ < 0 and one of the conditions C 2a or C 3 is satisfied. Then, there exists a unique solution to (1) in S 0 .

Proof. Assume first β > 0, γ < 0. The proof is similar to the proof of 1 in Theorem 11. The only change is that now

Y t ≤ V t . Therefore, on {Y 0 > 0} ∩ {τ 0 Y < ∞}, δ τ 0 Y 0 ds Y s ≤ V τ 0 Y -Y 0 -C τ 0 Y -γ τ 0 Y 0 ds U s < ∞
and we can define X τ 0 Y and Y τ 0 Y as previously done. The application of Theorem 8 to the process on the time interval [0, τ 0 Y ] shows that X τ 0 Y > 0 and we can iterate the construction as in Theorem 11. The proof if β < 0, γ > 0 is analogous. 5.3. Case β ≤ 0 and γ ≤ 0. In this case we can give a full answer to the question of existence and uniqueness. When |ρ| < 1, our condition of existence is analogous to the condition found in [START_REF] Williams | Reflected Brownian motion in a wedge: semimartingale property[END_REF] for the reflected Brownian in a wedge being a semimartingale, i.e. there is a convex combination of the directions of reflection that points into the wedge from the corner. It amounts to saying that the interaction matrix α β γ δ is completely-S in the terminology of [START_REF] Taylor | Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant[END_REF][START_REF] Dupuis | Lyapunov functions for semimartingale reflecting Brownian motions[END_REF][START_REF] Williams | Semimartingale reflecting Brownian motions in the orthant. A survey[END_REF][START_REF] Dai | Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedra[END_REF][START_REF] Sarantsev | Triple and simultaneous collisions of competing Brownian particles[END_REF][START_REF] Sarantsev | Penalty method for obliquely reflected diffusions[END_REF][START_REF] Sarantsev | Reflected Brownian motion in a convex polygonal cone: tail estimates for the stationary distribution[END_REF]. Theorem 14. Assume β ≤ 0 and γ ≤ 0.

(1) If αδ > βγ, there exists a unique solution to (1) in S.

(2) If αδ ≤ βγ and 1 + ρ + |α + γ| + |β + δ| > 0, there is no solution.

(3) If 1 + ρ = α + γ = β + δ = 0 and (X 0 , Y 0 ) = 0 there exists a unique solution.

(4) If 1 + ρ = α + γ = β + δ = 0 and (X 0 , Y 0 ) = 0 there is no solution.

Proof. 1. Assume first αδ > βγ. a) Existence. Let (h n , n ≥ 1) be a (strictly) increasing sequence of bounded positive nonincreasing Lipschitz functions converging to 1/x on (0, ∞) and to +∞ on (-∞, 0]. For instance we can take

h n (x) = (1 -1 n ) 1 x on [ 1 n , ∞) = n -1 on (-∞, 1 n ] . We consider for each n ≥ 1 the system (14) X n t = X 0 + B t + α t 0 ds X n s + β t 0 h n (Y n s )ds Y n t = Y 0 + C t + γ t 0 h n (X n s )ds + δ t 0 ds Y n s .
From Proposition 7 it follows there exists a unique solution to this system. We set Then we may let n go to ∞ in [START_REF] Ichiba | Strong solutions to stochastic equations with rank-biased coefficients[END_REF] proving that (X, Y ) is a solution to (1). b) Uniqueness. Let (X , Y ) be another solution to [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF]. Replacing (X n+1 , Y n+1 ) with (X , Y ) we follow the above proof to obtain for t ∈ [0, ∞) and n ≥ 1

τ := inf{s > 0 : X n+1 s > X n s } . We have h n+1 (X n+1 t ) ≥ h n (X n t ) on [0, τ ]. A
X t ≤ X n t and Y t ≤ Y n t Letting n → ∞ we conclude X t ≤ X t and Y t ≤ Y t .
With the same λ > 0 and µ > 0 as above,

0 ≤ λ(X t -X t ) + µ(Y t -Y t ) = t 0 [(λα + µγ)( 1 X s - 1 X s ) + (µδ + λβ)( 1 Y s - 1 Y s )]ds ≤ 0 and therefore X t = X t , Y t = Y t .
2. If αδ ≤ βγ there exist λ > 0 and µ > 0 such that λα + µγ ≤ 0 and µδ + λβ ≤ 0. For that, just take

(16) α -γ ≤ µ λ ≤ -β δ .
Let us consider the nonnegative quadratic (λ

2 + µ 2 + 2ρλµ). It is positive if ρ ≥ 0. It is larger than (1 -ρ 2 )λ 2 > 0 if -1 < ρ < 1.
It may be positive if ρ = -1 and |α + γ| + |β + δ| > 0 because from ( 16) we may take λ = µ. Thus, if (X, Y ) is a solution to (1), for any t ≥ 0,

0 ≤ λX t + µY t ≤ λX 0 + µY 0 + λB t + µC t .
This is not possible since the paths of the Brownian martingale (λ 2 +µ 2 +2ρλµ) -1/2 (λB t +µC t ) are not bounded below. So there is no global solution.

3. Assume now 1 + ρ = |α + γ| + |β + δ| = 0 and X 0 + Y 0 > 0. The system becomes ( 17)

X t = X 0 + B t + α t 0 ds Xs -δ t 0 ds Ys ≥ 0 Y t = Y 0 -B t -α t 0 ds Xs + δ t 0 ds Ys ≥ 0.
This entails for any t ≥ 0

X t + Y t = X 0 + Y 0
and the first equation in [START_REF] Ichiba | Hybrid Atlas models[END_REF] reduces to

(18) 0 ≤ X t = X 0 + B t + α t 0 ds X s -δ t 0 ds X 0 + Y 0 -X s ≤ X 0 + Y 0 ..
Clearly pathwise uniqueness holds for equation [START_REF] Jeanblanc | Mathematical Methods for Financial Markets[END_REF] since if there are two solutions X and X , for any t ≥ 0

(X t -X t ) 2 = 2α t 0 (X s -X s )( 1 Xs -1 X s )ds -2δ t 0 (X s -X s )( 1
X 0 +Y 0 -Xs - 1 X 0 +Y 0 -X s )ds ≤ 0.
Consider now for any c > 0 and 0 ≤ c 0 ≤ c the equation [START_REF] Lépingle | Boundary behavior of a constrained Brownian motion between reflecting-repelling walls[END_REF] 0

≤ Z t = c 0 + B t + α t 0 ds Z s -δ t 0 ds c -Z s ≤ c.
The solution Z is a Brownian motion perturbed by a drift deriving from the concave potential

α ln z + δ ln(c -z) for 0 < z < c.
From [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF] we know that equation [START_REF] Lépingle | Boundary behavior of a constrained Brownian motion between reflecting-repelling walls[END_REF] has a unique solution living on the interval [0, c]. Thus there is a weak and then a unique strong solution to [START_REF] Jeanblanc | Mathematical Methods for Financial Markets[END_REF]. Setting

Y t = X 0 + Y 0 -X t
we obtain a unique strong solution to [START_REF] Ichiba | Hybrid Atlas models[END_REF]. 4. We must have for any t ≥ 0

X t + Y t = X 0 + Y 0 = 0
and thus X t = Y t = 0. But this is not possible for a solution to [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF].

We end this section by considering a degenerate case where the solution hits the corner with probability one. Proposition 15. Assume ρ = 1, αδ > βγ, max{α, δ} < 1 2 and max{β, γ} ≤ 0. Then,

P(τ 0 < ∞) = 1.
Proof. We set τ 0 X := inf{t > 0 :

X t = 0} τ 0 Y := inf{t > 0 : Y t = 0}
. The dimension of each Bessel process U and V in ( 2) is less than 2, and X ≤ U , Y ≤ V .

Then P(τ 0 X < ∞ = 1) and P(τ 0

Y < ∞ = 1). Assume first α + β ≤ γ + δ. Using Lemma 4 we obtain X t ≤ Y t ≤ V t on the time interval [τ 0 X , ∞). Therefore P(τ 0 < ∞) = 1. Same result when α + β ≥ γ + δ.
In the following pictures, we display the directions of interaction r x and r y defined in (4) in three illustrative instances. We now consider the question of hitting an edge of the quadrant. Remember the definitions (20)

• E T I £ £ £ £ £ £ £ £ # r x r y x y β > 0, γ > 0, αδ > βγ • E T x y I f f f f w r x r y β < 0, γ > 0, αβ + γδ = 0 • E T x y r r
τ 0 X := inf{t > 0 : X t = 0} τ 0 Y := inf{t > 0 : Y t = 0} .
We already know that P(τ 0

X < ∞) = 0 if α ≥ 1 2 and β ≥ 0.
Conversely, a comparison with the Bessel process U shows that P(τ 0

X < ∞) = 1 if α < 1 2
and β ≤ 0. If we know that the corner is not hit and α ≥ 1 2 , we can get rid of the nonnegativity assumption on β. Since we are only interested in one coordinate, we are looking for a function that is C 2 on (0, ∞) and goes to -∞ when approaching 0. A natural candidate is the logarithmic function.

Proposition 16. Assume P(τ 0 < ∞) = 0. If α ≥ 1 2 , then P(τ 0 X < ∞) = 0. Proof. For η > 0 let θ η X = 1 {X 0 =0} inf{t > 0 : X t ≥ η} τ 0,η X = inf{t > θ η X : X t = 0} .
As η ↓ 0, θ η X ↓ 0 and τ 0,η X ↓ τ 0 X . For t ∈ [θ η X , τ 0,η X ), from Itô's formula [START_REF] Mckean | Stochastic Integrals[END_REF] ln

X t = ln X θ η X + t θ η X dBs Xs + (α -1 2 ) t θ η X ds X 2 s + β t θ η X ds
XsYs .

Since P(τ 0 < ∞) = 0, on the set {τ 0,η X < ∞} we have Y τ 0,η X > 0 because X τ 0,η X = 0. On this set, from the definition of an O2BP,

τ 0,η X θ η X ds X s < ∞ , τ 0,η X θ η X ds Y s < ∞.
As Y s > 0 on some interval [χ, τ 0,η X ] with positive measure and X s > 0 on [θ η X , χ], we see that

|β| τ 0,η X θ η X ds X s Y s < ∞ .
As t → τ 0,η X , the local martingale in the r.h.s. of ( 21) cannot converge to -∞. This entails that P(τ 0,η X < ∞) = 0 and therefore P(τ 0

X < ∞) = 0.
We may also be interested in hitting either edge. This time we are looking for a function that is C 2 in the interior of S and goes to -∞ when approaching either edge.

Proposition 17. Assume α ≥ 1 2 and δ ≥ 1 2 . Then P(τ 0 X < ∞) = P(τ 0 Y < ∞) = 0 if one of the following conditions is satisfied: (1) β ≥ 0 (2) γ ≥ 0 (3) 0 < βγ ≤ (α -1 2 )(δ - 1
2 ). Proof. The proofs in the cases (1) and ( 2) ) are direct consequences of Theorem 8 (under conditions C 2a or C 2b ) and Proposition 16. Assume now the conditions in (3) hold true. For

> 0 let σ = 1 {X 0 Y 0 =0} inf{t > 0 : X t Y t ≥ } τ = inf{t > σ : X t Y t = 0} . For λ > 0 and µ > 0 we set R t = λ ln X t + µ ln Y t . From Itô's formula we get for t ∈ [σ , τ ) R t = R σ + t σ ( λ Xs dB s + µ Ys dC s ) + t σ [ λ(α-1 2 ) X 2 s + µ(δ-1 2 ) Y 2 s + (λβ+µγ) XsYs ]ds = R σ + N t + t σ P 3 (Xs,Ys) X 2 s Y 2 s ds
where N is a continuous local martingale and P 3 (x, y) is the second degree homogeneous polynomial

P 3 (x, y) = µ(δ - 1 2 )x 2 + λ(α - 1 2
)y 2 + (λβ + µγ)xy.

If the conditions (3) are satisfied, we may take

λ = 2(α -1 2 )(δ -1 2 ) -βγ > 0 µ = β 2 > 0.
Then, using again conditions (3), we check that

P 3 (x, y) = β 2 δ - 1 2 x 2 + 2 α - 1 2 δ - 1 2 -βγ α - 1 2 y 2 +2 α - 1 2 δ - 1 2 βxy
is nonnegative on S. The proof terminates as previously in Theorem 8 .

Associated local martingales

We easily check that if max {β, γ, αδ -βγ} ≥ 0 and if ρ > -1, there exist some λ ≥ 0 and µ ≥ 0 with λ + µ > 0 such that λX t + µY t is not less than λ(X 0 + B t ) + µ(Y 0 + C t ) which is proportional to a real driftless Brownian motion. Therefore lim sup t→∞ (λX t + µY t ) = +∞.

An usual way in the study the asymptotic behavior of the solutions to stochastic differential equations is introducing associated martingales. This is carried out through scale functions ( [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] Section VII.3). There is no equivalent functions on the plane. However, in some particular cases we can find simple functions of O2BPs that are supermartingales or local martingales.

Proposition 18. Assume the following set of conditions:

(22) α > 1/2 δ > 1/2 β 2δ-1 + γ 2α-1 ≥ ρ ρ > -1. Then M t := X 1-2α t Y 1-2δ
t is a positive supermartingale on (0, ∞) which tends to 0 as t → ∞. It is a local martingale if the third inequality in ( 22) is an equality.

Proof. For > 0 let

σ = 1 {X 0 Y 0 =0} inf{t > 0 : X t Y t ≥ } τ = inf{t > σ : X t Y t = 0} .
As ↓ 0, σ ↓ 0. Up to τ , we get from Itô's formula on {σ < ∞}

M t = M σ + t σ (1-2α)YsdBs+(1-2δ)XsdCs X 2α s Y 2δ s +[(1 -2α)β + (1 -2δ)γ + ρ(2α -1)(2δ -1)] t σ ds X 2α s Y 2δ s .
The function f (x, y) = x 1-2α y 1-2δ is C 2 in the interior of S and goes to +∞ when approaching the edges of the quadrant, whereas the finite variation part in the semimartingale decomposition of M t is nonincreasing. We apply again Lemma 6 and obtain that P(τ < ∞) = 0.

Then, letting → 0, we see that M is a positive supermartingale on (0, ∞). As such it tends to a limit H ≥ 0 when t → ∞ ( [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Corollary II.2.11 ). If β ≥ 0 and γ ≥ 0, then X ≥ U and Y ≥ V where U and V are the Bessel processes in [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF]. We have seen in Section 2 that U t → ∞ and V t → ∞ as t → ∞; so M t → 0 as t → ∞. If now min{β, γ} < 0, say γ < 0, then we have Y ≤ V . From Corollary 3 we deduce that for any > 0

(23) ∞ σ ds Y 2 s ≥ ∞ σ ds V 2 s = ∞.
We consider the quadratic variation M given by

M t = t σ X -4α s Y -4δ s [(1 -2α) 2 Y 2 s + (1 -2δ) 2 X 2 s + 2ρ(1 -2α)(1 -2δ)X s Y s ]ds = t σ M 2 s [ (1-2α) 2 X 2 s + (1-2δ) 2 Y 2 s + 2ρ (1-2α)(1-2δ) XsYs ]ds. If ρ ≥ 0, M t ≥ t σ M 2 s (1 -2δ) 2 Y 2 s ds and if ρ 2 < 1, M t ≥ (1 -ρ 2 ) t σ M 2 s (1 -2δ) 2 Y 2 s ds.
In both cases, using [START_REF] O'connell | Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition[END_REF] we should have M ∞ = ∞ on the set {H > 0} ∩ {σ < ∞}, and then lim sup t M t = -lim inf t M t = ∞, a contradiction with M t → H. Thus H = 0 a.s.

During the proof we have seen that P(τ 0 X < ∞) = P(τ 0 Y < ∞) = 0. In fact this is not a new result since here the conditions in Proposition 17 are in force. There is another case of interest.

Proposition 19. Assume α = δ = 1/2, ρβγ < |βγ|, and there exists a solution to [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF] satisfying P(τ 0 < ∞) = 0. Then

M t := γ ln X t -β ln Y t is a continuous local martingale on (0, ∞) and lim sup t→∞ M t = -lim inf t→∞ M t = ∞ .
Proof. We deduce from Proposition 17 that X t > 0 and Y t > 0 for any t > 0. Let again for > 0

σ = 1 {X 0 Y 0 =0} inf{t > 0 : X t Y t ≥ }. Now Itô's formula gives M t = M σ + t σ γ dB s X s -β dC s
Y s and the quadratic variation is

M t = t σ γ 2 X 2 s + β 2 Y 2 s -2ρ βγ X s Y s ds.
Then, if ρβγ ≤ 0,

M t ≥ t σ γ 2 X 2 s + β 2 Y 2 s ds,
and if ρ 2 < 1,

M t ≥ (1 -ρ 2 ) max t σ γ 2 X 2 s ds, t σ β 2 Y 2 s ds .
If β > 0, then X ≥ U where U is the Bessel process of dimension two in (2) and lim sup t→∞

(ln X t ) ≥ lim sup t→∞ (ln U t ) = ∞ If β < 0, then X ≤ U and lim inf t→∞ (ln X t ) ≤ lim inf t→∞ (ln U t ) = -∞ But ln X t = ln X σ + t σ dB s X s + β t σ ds X s Y s . and t σ ds X s Y s ≤ 1 2 t σ 1 X 2 s + 1 Y 2 s ds.
Thus ln X t would converge a.s. as t → ∞ if

∞ σ ds X 2 s < ∞ and ∞ σ ds Y 2 s < ∞.
Therefore M ∞ = ∞ on {σ < ∞} and the assertion is proved.

Absolute continuity properties

In this section we suppose ρ = 0. In some cases we easily obtain an absolute continuity property between the laws of O2BPs with various parameters. When there exists a unique solution to (1), we denote by P α,β,γ,δ

x,y the law on C(R + , S) = C(R + , R + ) × C(R + , R + ) of the solution starting at (x, y) ∈ S. We denote by (U, V ) the canonical map (U t (u, v), V t (u, v)) = (u(t), v(t)) and by U t = σ((U s , V s ), s ≤ t) the canonical filtration. Recall that we denote by P d r the law of one-dimensional Bessel process of dimension d starting at r ≥ 0. Proposition 20. Assume ρ = 0, δ ≥ 1 2 and y > 0. Then

P α,β,0,δ

x,y

| Ut = exp β t 0 dU s V s -αβ t 0 ds U s V s - β 2 2 t 0 ds V 2 s . P 2α+1 x ⊗ P 2δ+1 y | Ut .
Proof. Under P 2α+1

x ⊗ P 2δ+1 y the process

B t := U t -x -α t 0 ds U s
is a one-dimensional Brownian motion. The assumptions on δ and on y imply that V t > 0 for any t ≥ 0 and thus

t 0 ds v 2 (s) < ∞ P 2δ+1 y -a.s.
Therefore, for P 2δ+1 y -almost every v, We see that

Z t := exp β t 0 dB s V s - β 2 2 t 0 ds V 2
s is a P 2α+1

x ⊗ P 2δ+1 y -positive martingale with expectation 1. Setting for any T > 0

Q T := Z T . P 2α+1 x ⊗ P 2δ+1 y | U T we check that under Q T U t -x -α t 0 ds U s -β t 0 ds V s , 0 ≤ t ≤ T is a real Brownian motion independent of the Brownian motion {V t -y -δ t 0 ds Vs , 0 ≤ t ≤ T }. Thus, Q T = P α,β,0,δ x,y | U T .
A second set of conditions is obtained using Novikov's criterion. Proposition 21. Assume ρ = 0 and the following set of conditions is satisfied:

(25) |β| ≤ δ -1/2 |γ| ≤ α -1/2 x > 0 y > 0.
The process Proof. Under P 2α+1

Z t := exp β t 0 dU s V s -γ t 0 dV s U s -(αβ + γδ) t 0 ds U s V s - β 2 2 t 0 ds V 2 s - γ 2 2 t 0 ds U 2
x ⊗ P 2δ+1 y the processes

B t := U t -x -α t 0 ds U s and C t := V t -y -δ t 0 ds
V s are independent Brownian motions. Then Z t may be written

Z t = exp β t 0 dB s V s + γ t 0 dC s U s - β 2 2 t 0 ds V 2 s - γ 2 2 t 0 ds U 2 s , which shows that Z t is a positive local martingale. Using (8) we compute exp { γ 2 2 t 0 ds U 2 s } dP 2α+1 x ⊗ dP 2δ+1 y = exp { γ 2 2 t 0 ds R 2 s } dP 2α+1 x = ( Rt x ) α-1/2 exp {-(α-1/2) 2 -γ 2 2 t 0 ds R 2 s } dP 2 x < ∞
since a Bessel process of dimension two has finite moments of any order. Finally,

exp { β 2 2 t 0 ds V 2 s } exp { γ 2 2 t 0 ds U 2 s } dP 2α+1 x ⊗ dP 2δ+1 y = ( exp { β 2 2 t 0 ds R 2 s } dP 2δ+1 x )( exp { γ 2 2 t 0 ds R 2 s } dP 2α+1
= Γ(a + b) Γ(a)Γ(b) x a-1 (1 -x) b-1 .
It is well-known that the function g(x) = x d-1 is an invariant measure density on [0, ∞) for the Bessel process of dimension d. To get a stationary probability we have to introduce a negative drift that entails positive recurrence. Then, the process

X t = X 0 + B t + d -1 2 t 0 ds X s -θt,
with θ > 0, has γ(x; d, 2θ) as stationary density. As for the Brownian motion reflected at 0 with constant drift -θ, it has the stationary exponential density 2θ exp{-2θx}. In the bidimensional case, the drifted obliquely reflected Brownian motion has a stationary density in the form of product of two exponential densities if and only if it satisfies at once [START_REF] Harrison | Multidimensional reflected Brownian motion having exponential stationary distributions[END_REF][START_REF] Williams | Semimartingale reflecting Brownian motions in the orthant. A survey[END_REF]:

• an invertibility condition on the reflection matrix;

• a positivity condition on the exponential coefficients;

• a skew-symmetry condition.

Therefore it is natural to ask wether one can find similar conditions for drifted O2BPs ensuring existence of a stationary distribution in the form of product of two gamma distributions.

The answer is positive and given in the next theorem. In fact, this may be considered as a consequence of the study in [START_REF] O'connell | Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition[END_REF] which introduced a generalised reflected Brownian motion associated with a potential U regular on the whole real line. The difference is that here the logarithmic potential is defined only on the positive axis. The proof below is an adaptation of the proof in [START_REF] O'connell | Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition[END_REF] to this special case.

We introduce an additional constant drift (-θ, -η) in order to make the solution a recurrent process in the nonnegative quadrant. We consider the system [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] X t = X 0 + B t + α ds Ys -ηt with the conditions X t ≥ 0, Y t ≥ 0. Changing probability through a Girsanov transformation, we easily check that Theorem 8 in Section 4 is still valid for this drifted system. Moreover, the proofs in Section 5 do not bother whether the Brownian motions B and C are drifted or not drifted. Therefore existence and uniqueness results in Section 5 hold true for the solution to [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]. Under these conditions, the unique solution is given by

• a = 1 + 2α • b = 1 + 2δ • c = 2α δθ-βη αδ-βγ
• d = 2δ αη-γθ αδ-βγ . Proof. We first remark that under the above conditions αδ -βγ cannot be < 0, because in that case there is no solution if β and γ are < 0 and if β and γ are > 0 the skew-symmetry condition is not satisfied since 2ρ ≤ 2 < β/δ + γ/α. So we could have written αδ -βγ > 0 as first condition. Let now q(x, y) = γ(x; a, c)γ(y; b, d) for x ≥ 0, y ≥ 0 .

The infinitesimal generator of the diffusion [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] Assume (X 0 , Y 0 ) has density q and characteristic function E[e i(λX 0 +µY 0 ) ] = φ(λ; a, c) φ(µ; b, d) .

We set f (x, y) = e i(λx+µy) . We want to prove that for any t ≥ 0, λ, and µ, It was proved in [START_REF] Williams | Reflected Brownian motion with skew symmetric data in a polyhedral domain[END_REF] (see also [START_REF] Sarantsev | Triple and simultaneous collisions of competing Brownian particles[END_REF]) that under a skew-symmetry condition the obliquely reflected Brownian motion does not reach the non-smooth part of the boundary. In the same way the above skew-symmetry equality is reminiscent of the second condition in Corollary 10, which is an inequality. We therefore get a partial but handy statement. Proof. This is a straightforward consequence of Corollary 10, Theorems 11, 13, 14 and 22.

Remark 24. When ρ = 0, the skew-symmetry condition αβ + γδ = 0 means that the directions of interaction r x and r y are orthogonal. Then the second condition in Theorem 22 means that (θ, η) points into the interior of the quadrant designed by r x and r y . This condition ensures recurrence of the process, while the first condition is now a consequence of the skew-symmetry condition.

Remark 25. With the same proof, we may check that under the skew-symmetry condition, when θ = η = 0, the function q(x, y) = x 2α y 2δ is a non-integrable invariant density that does not depend on the other parameters.

A simple change of variables (beta-gamma algebra) provides the following result.

  the canonical filtration and by P d r the law of the Bessel process of dimension d ≥ 2 starting at r > 0. Then,[START_REF] Harrison | Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution[END_REF]

5. 1 .

 1 Case β ≥ 0 and γ ≥ 0. Theorem 11. Assume β ≥ 0, γ ≥ 0 and one of the conditions C 1 . C 2a , C 2b , C 3 is satisfied.

  We have Y τ 0 Y = 0. From Theorem 8 we know that X τ 0 Y > 0. In exactly the same way we can construct a solution on {Y 0 > 0} in the interval [T 1 , T 2 ], where

6 .

 6 Avoiding the edges

|

  Ut = Z t . P 2α+1 x ⊗ P 2δ+1 y | Ut .

9 ..

 9 s criterion ([START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Proposition VIII.1.15) proves that Z t has expectation 1 with respect to P 2α+1 x ⊗ P 2δ+1 y . We easily see that {B t -β t 0 ds Vs , 0 ≤ t ≤ T } and {C t -γ t 0 ds Us , 0 ≤ t ≤ T } are independent Brownian motions under the probability with density Z T and this proves that the new probability is P α,β,γ,δ x,y | U T . Product form stationary distribution For a > 0 and c > 0, let Γ(a, c) be the probability measure on [0, ∞) with density γ(x; a, c) For a > 0, b > 0, let B(a, b) be the probability measure on [0, 1] with density β(x; a, b) :

  Y t = Y 0 + C t + γ

Theorem 22 .

 22 Assume there exists a unique solution to[START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] in S 0 or S. This process has an invariant distribution in the form Γ(a, c) ⊗ Γ(b, d) with a > 1 and b > 1 if and only if at once• αδ -βγ = 0(invertibility of the interaction matrix) • αη -γθ > 0 and δθ -βη > 0 (positivity of exponents) • 2ρ = β δ + γ α (skew-symmetry)

E 0 ∞ 0 Lf 2 (λ 2 + µ 2 +0 1 .

 002221 [f (X t , Y t )] = E[f (X 0 , Y 0 )] (= φ(λ; a, c)φ(µ; b, d) ) .It is enough to prove that ∞ (x, y)q(x, y) dxdy = 0 for any λ and µ. LetR(λ, µ) := [-1 Lf (x, y)q(x, y) dxdy = ∞ 0 ∞ 0 R(λ, µ)e i(λx+µy) γ(x; a, c)γ(y; b, d) dxdy = S(λ, µ) φ(λ; a, c) φ(µ; b, d)where S(λ, µ) is a second degree polynomial. In the last computation we used formula[START_REF] Rȃşcanu | Deterministic and stochastic differential equations in Hilbert spaces involving multivalued maximal monotone operator[END_REF] several times. Setting to zero the coefficients of λ, µ, λ 2 , µ 2 and λµ in polynomial S we obtain the set of conditions: 0 = -θ + αc aThe solution is given by the specified values for a, b, c, d and the skew-symmetry condition.

Corollary 23 . 2 •

 232 Assume • max{α, δ} ≥ 1 αδ -βγ > 0 • αη -γθ > 0 and δθ -βη > 0 • 2ρ = β δ + γ α .Then the unique solution to[START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] has an invariant distribution given byΓ 1 + 2α, 2α δθ -βη αδ -βγ ⊗ Γ 1 + 2δ, 2δαη -γθ αδ -βγ .

Corollary 26 .

 26 With the conditions and notations of Theorem 22, the two-dimensional process(29) W t := cXt cXt+dYt Z t := cX t + dY t has B(a, b) ⊗ Γ(a + b, 1) for invariant distribution on [0, 1] × [0, ∞).

  , Proposition IV.1.26) we know that on the set { Rs converges as t → ∞. But we have seen that with unit probability ln R t does not converge.

	martingale	a t	dWs	∞ a	ds R 2 s	< ∞}, the continuous local

  If αδ > βγ, there is a convex combination of the directions of repulsion pointing into the positive quadrant, i.e. there exist λ > 0 and µ > 0 such that λα + µγ > 0 and µδ + λβ > 0. + µY 0 + λB t + µC t + (λα + µγ)

	first application of Lemma 4 shows that Y n+1 t τ ) on {τ < ∞}, we deduce from the continuity of ≤ ) > h n (Y n solutions that there exists σ > 0 such that h n+1 (Y n+1 Y n t on [0, τ ]. Since h n+1 (Y n+1 τ t ) ≥ h n (Y n t ) on [τ, τ + σ]. A second application of Lemma 4 shows that X n+1
	Y n+1 t	≤ Y n t . Then we can set for any t ∈ [0, ∞)		n+1 t	≤ X n t and
		X t := lim n→∞	X n t	and Y t := lim n→∞	Y n t .
	For n ≥ 1 and t ≥ 0,
	(15)	λU t + µV t ≥ λX n t + µY n t ≥ λX 0 t 0	ds X n s	+ (µδ + λβ)	t 0	ds Y n s	.
	Letting n → ∞ in (15) we obtain
		0	t	ds X s	< ∞ and	0	t	ds Y s	< ∞ .

t ≤ X n t on [τ, τ + σ], a contradiction to the definition of τ . Thus P(τ = ∞) = 1 proving that on the whole [0, ∞) we have X

  is given by

	L =	1 2	(	∂ 2 ∂x 2 +	∂ 2 ∂y 2 + 2ρ	∂ 2 ∂x∂y	) + (	α x	+	β y	-θ)	∂ ∂x	+ (	γ x	+	δ y	-η)	∂ ∂y	.

Acknowledgments. The author is deeply indebted to the anonymous referee for careful reading, encouragements and sound advice. They have given rise to fundamental improvements in the content of this paper.


Erwann Delay
email: erwann.delay@univ-avignon.fr

53c21

CONFORMALLY COVARIANT PARAMETRIZATIONS FOR INITIAL DATA IN SOME MODIFIED EINSTEIN GRAVITY THEORIES

Keywords: Conformal Riemannian geometry, non-linear elliptic systems, f (R) gravity, Lovelock gravity, Einstein-Gauss-Bonnet gravity, higher order gravity, constraint equations, Cauchy problem 53A45, 53A30, 58J05, 35J61

We propose further conformal parametrizations for initial data in some modified Einstein gravity theories. Some of them give rise to conformally covariant systems.

Introduction

Motivated by the problem of dark matter, early-time inflation or late-time acceleration of the universe, a multitude of modified gravity theories are currently studied: f (R), Lovelock, Einstein-Gauss-Bonnet, cubic, quadratic,... As in general relativity, the space and time correlation in theses theories implies that the initial data for the evolution problem cannot be chosen freely.

In general, in vacuum for instance, the initial data for such modified Einstein gravity theories are given by a manifold M of dimension n, equipped with a Riemannian metric g (spatial geometry at a time 0) and a symmetric 2-tensor field K (infinitesimal deformation of the spatial geometry at time 0), satisfying some constraint equations of the form (for details, see eg. [START_REF]The Einstein Equations and the Large Scale Behavior of Gravitational Fields (50 Years of the Cauchy Problem in General Relativity[END_REF], [START_REF] Bartnik | The constraint equations, The Einstein equations and the large scale behavior of gravitational fields[END_REF], [START_REF] Lefloch | Mathematical Validity of the f (R) Theory of Modified Gravity[END_REF], [START_REF] Torii | N + 1 formalism in Einstein-Gauss-Bonnet gravity[END_REF], and the appendix 5.2 )

ρ( g, K) = 0 , J( g, K) = 0 . (C)
In this system, the first equation, the Hamiltonian constraint, is scalar and the second one, the momentum constraint, is vectorial. Such a system is known as the Cauchy problem for the related gravity theory. This system is highly under-determined because it contains (n + 1) equations for n(n + 1) unknowns. As a consequence, it is natural to Date: November 21, 2016.

1 look for (n + 1) unknowns, fixing the remaining ones.

The goal of this note is to point out some different possibilities to parametrize (and then different ways to construct) solutions of (C). Some of them give rise to an interesting mathematical property, namely the conformal covariance.

Acknowledgements: I am grateful to P.T. Chruściel and Ph. Delanoë for comments.

The usual conformal method

On a smooth manifold M of dimension n, given a Riemannian metric g, we denote by ∇ its Levi-Civita connection. If h is a symmetric covariant 2-tensor field, we define its divergence as the 1-form given by

(div g h) i = -∇ k h ki .
The classical method (York's method A) starts from a given metric g together with a trace-free and divergence-free symmetric 2-tensor field σ (a TT-tensor) and a real function 1 τ . It consists in looking for solutions of (C) of the form

g = φ N -2 g , K = τ n g + φ -2 (σ + Lg W ) (P)
where N = 2n n-2 , and the unknowns are a function φ > 0 and a one form W and where

( Lg W ) ij = ∇ i W j + ∇ j W i - 2 n ∇ k W k g ij .
We infer from (C) and (P) a coupled system of the form (see eg. [START_REF] Bartnik | The constraint equations, The Einstein equations and the large scale behavior of gravitational fields[END_REF], [START_REF] Torii | N + 1 formalism in Einstein-Gauss-Bonnet gravity[END_REF])

(S) ρ( g, K) =: L g,τ,σ (φ, W ) =0 J( g, K) =: V g,τ,σ (φ, W ) =0 (L) (V)
where the scalar equation (L) is a generalisation of the Lichnerowicz one (see [START_REF] Lichnerowicz | L'intégration des équations de la gravitation relativiste et le problème des n corps[END_REF]) and the equation (V) is usually called the vector equation (see [START_REF] York | Conformally invariant orthogonal decomposition of symmetric tensors on riemannian manifolds and the initial value problem of general relativity[END_REF]).

In order to produce further interesting parametrization we recall some basic fact. Firstly, the operators div g and Lg are conformally covariant (see appendix 5.1 for a precise definition). Indeed, for any positive function ψ,

if g = ψ N -2 g then div g h = ψ -N div g (ψ 2 h) , L g W = ψ N -2 Lg (ψ 2-N W ). (C)
Secondly, we recall the York decomposition [START_REF] York | Conformally invariant orthogonal decomposition of symmetric tensors on riemannian manifolds and the initial value problem of general relativity[END_REF] valid for instance if M is compact and g has no conformal Killing vector fields (i.e. ker Lg is trivial): any covariant symmetric trace free 2-tensor field h splits in a unique way as

h = σ + Lg W, (Y)
where σ is a TT-tensor and W is a 1-form.

A conformally covariant parametrization

Getting back to (C), we use now the York decomposition relative to g, namely

K - τ n g = σ + L g W .
From (C), if we are looking for a solution of (C) in a conformal class g = φ N -2 g, it is natural to introduce σ = φ -2 σ, where σ is a TT-tensor. Still using (C), and expressing L g W in terms of g = φ 2-N g, we are also prompted to set W := φ N -2 W . Sticking to the same fixed g, σ, τ , we can thus parametrize the solutions of the constraint (C) by

   g = φ N -2 g , K = τ n g + φ -2 (σ + φ N Lg W ) .
With this parametrization in (C) we obtain a new system of the form (see eg. [START_REF] Delay | Conformally covariant parametrizations for relativistic initial data[END_REF])

(S ′ g,τ,σ )    ρ( g, K) =: L ′ g,τ,σ (φ, W ) = 0 , J( g, K) =: V ′ g,τ,σ (φ, W ) = 0 .
If we follow the same method but, instead of g, starts from a conformal metric g = ψ N -2 g and from the related symmetric 2-tensor σ = ψ -2 σ (which is div g free by (C)), sticking to the same given real function τ , we see from (C) that the couple ( φ = ψ -1 φ, W = ψ N -2 W ) solves the resulting system (S ′ g,τ, σ ) iff (φ, W ) solves (S ′ g,τ,σ ). In other words,the system (S ′ g,τ,σ ) is conformally covariant. This simple observation shows that York's method B (see [1, Section 4.1] or the original paper [9, Page 461]), also called the physical TT method, stays valid for every modified gravity theory, regardless of the explicit form of the equations (C).

Further parametrizations

In this section, we propose more general but still natural parametrizations. Let g = ψ N -2 g be another conformal metric. The York decomposition (Y) relative to g of the tensor

φ 2 ψ -2 ( K - τ n g), leads to the parametrization    g = φ N -2 g , K = τ n g + φ -2 (σ + ψ N Lg W ).
Putting this parametrization in (C), we obtain a new interesting system. Of course, we could let ψ depend on φ and possibly on some other parameters in the system obtained. It can be seen (compare [START_REF] Bartnik | The constraint equations, The Einstein equations and the large scale behavior of gravitational fields[END_REF] for the GR initial data) that the conformal method A consists in choosing ψ = 1, the conformal method B arises when ψ = φ, and for ψ a fixed positive function, we obtain the conformal thin sandwich method. But many other choices can be made, and some of them also yield conformally covariants systems (see [START_REF] Delay | Conformally covariant parametrizations for relativistic initial data[END_REF]).

Remarks 4.1.

• It is probable that, depending on the gravity theory studied, an adapted choice of ψ will be judicious.

• For an arbitrary function f , we could use in the same way the York decomposition related to g of the tensor f ( Kτ n g). This will produce a system where f and ψ can be choosen freely (possibly depending on some other parameters and/or variables).

• In [START_REF] Torii | N + 1 formalism in Einstein-Gauss-Bonnet gravity[END_REF], a discussion is made about a choice of the different powers of the conformal factor that can be used to define the conformal parametrization. With our parametrizations, only the A ij there is changed. But we can see that in that case the natural choice to make, with the notations of [START_REF] Torii | N + 1 formalism in Einstein-Gauss-Bonnet gravity[END_REF] is τ = 0, m = 2/(N -2) and l = -2 there.

E = E 1 × ... × E k , F = F 1 × ... × F l , G = G 1 × ... × G m ,
and a differential operator acting on the sections :

P g : Γ(E) -→ Γ(F ),
with coefficients determined by g = (g 1 , ..., g m ) ∈ G. We will say that P g is conformally covariant if there exist a = (a 1 , .., a k ) ∈ R k , b = (b 1 , .., b l ) ∈ R l and c = (c 1 , .., c m ) ∈ R m such that for each smooth section e of E, and every smooth function ψ on M, we have

ψ b ⊙ P ψ c ⊙g (ψ a ⊙ e) = P g (e),
where ψ a ⊙ e = (ψ a 1 e 1 , ..., ψ a k e k ). A differential system will be said conformally covariant if it can be written in the form P g (e) = f , for a conformally covariant operator P g . 5.2. The equations. We specify here the equations related to the Hamiltonian constraint ρ(g, K) = 0 and the momentum constraint J(g, K) = 0, in some modified gravity theories.

The constraints equations for the general Lovelock gravity can be found in [START_REF] Bartnik | Relativity and the Einstein Equations[END_REF] pages 692-693 (a reedition of [START_REF] Choquet-Bruhat | Gravitation with Gauss Bonnet terms[END_REF] pages 56-57). We do not reproduce them here but choose to only details the Einstein-Gauss-Bonnet particular case.

Einstein-Gauss-Bonnet constraint: The following expressions can be found in [START_REF] Torii | N + 1 formalism in Einstein-Gauss-Bonnet gravity[END_REF] for instance:

ρ(g, K) = M + α GB (M 2 -4M ij M ij + M ijkl M ijkl ), -1 2 J i (g, K) = N i + 2α GB (MN i -2M j i N j + 2M kl N ikl -M i jkl N klj ),
where

M ijkl = R ijkl (g) + K ik K jl -K il K jk , M ij = R ij (g) + Tr g KK ij -K il K l j , M = R(g) + (Tr g K) 2 -K ij K ij , N ijk = ∇ i K jk -∇ j K ik N i = ∇ j K j i -∇ i
Tr g K, and α GB is a coupling constant, equal to zero in the Einstein theory. f(R) gravity constraint: The following equations can be found in [START_REF] Lefloch | Mathematical Validity of the f (R) Theory of Modified Gravity[END_REF] :

ρ(g, K) = R(g) -K ij K ij + (Tr g K) 2 -f ′ (R) -1 [2∆ g f ′ (R) + 2 Tr g Kf ′′ (R) Ṙ -f (R) -Rf ′ (R)], 1 2
J j (g, K) = -∇ i K ij +∇ j Tr g K-f ′ (R) -1 ∇ j (f ′′ (R) Ṙ)-K i j ∇ i (ln(f ′ (R))), where ∆ = ∇ i ∇ i , and R, Ṙ are respectively the initial data2 of the scalar curvature and the time derivative of the scalar curvature of the space time. More precisely Ṙ will be equal to L n R, where n will be the futur unit normal to the initial data of the space-time when evolved.

5 . Appendix 5 . 1 .

 551 Conformal covariance. Let us consider three products of tensor bundles over M,

to be chosen


Dissimilarity measure for collections of objects and values

Petko Valtchev, Jérôme Euzenat

To cite this version:


Isabelle Crampé
email: isabelle.crampe@inrialpes.fr

Jérôme Euzenat
email: jerome.euzenat@inrialpes.fr

come

Object knowledge base revision

Isabelle Crampé and Jérôme Euzenat * Abstract. A revision framework for object-based knowledge representation languages is presented. It is defined by adapting logical revision to objects and characterised both semantically and syntactically. The syntactic analysis of revision shows that it can be easily interpreted in terms of object structures (e.g. moving classes or enlarging domains). This is the source of the implementation and it enables users to be involved in the revision process.

There has been comprehensive work done on revision in logical formalisms. However, due to several factors (complexity of revision and decision procedures, needs not well-stated [START_REF] Friedman | Belief revision: a critique[END_REF] and mismatch between logical formalisms and implemented knowledge representation systems), little have been implemented. Moreover, very few studies (with the notable exception of [START_REF] Nebel | Reasoning and revision in hybrid representation systems[END_REF]) have been devoted to revision in object formalisms.

The present work attempts to transfer the results of logical revision to the field of object-based knowledge representation systems. Moreover, its goal is to come up with an implemented system in the context of error correction during knowledge base editing.

Two noticeable points are thus taken into account: • the revision must take advantage of the structure of object formalisms (the developed operator is grounded on a language which enables a localised syntactic characterisation of inconsistency and a natural interpretation of minimal revisions in terms of objects);

• the user can be interactively asked about the possible revisions.

The paper begins by formalising a basic object-based language ( §1) in order to formally characterise the kind of revision which is being performed. Then we introduce a syntactic manipulation of the knowledge bases required in order to transpose usual revision notions such as contraction and minimality ( §2). In section 3, revision is defined and minimality criterion is introduced. They are found to closely follow the object-based structure. In the last section, we finally present the implementation of revision and its embedding in an actual knowledge representation system.

Proof of propositions, extensions and extensive examples can be found in [START_REF] Crampé | Révision interactive dans une base de connaissance à objets[END_REF].

OBJECT REPRESENTATION LANGUAGE

The definition of a language corresponding to object-based notions has two purposes: formally defining what objects mean and allowing an easier transposition from logical revision t o object models. Although the language presented here might seem simple, it covers in fact a large part of object representation systems.

Language

As expected, the representation language contains objects and classes. They have attributes which are typed in the classes and valued in the objects. The values of attributes can be either objects or basic values (integers in the examples). Moreover, there is a notion of sub-class relation between classes and of attachment of an object to a class. The grammar of the language is presented below.

Definition 1 (Grammar).

Semantics

The semantics of the language is given by reference to a domain of interpretation. The objects are interpreted as the elements of this domain while classes are subsets of it. The specialisation relation is interpreted as subset and attachment as set membership. This semantics is not that of terminological logics [START_REF] Nebel | Reasoning and revision in hybrid representation systems[END_REF] because the language presented here is purely descriptive (there is no definition of the classes as necessary and sufficient Isabelle Crampé and Jérôme Euzenat conditions) and not all formulas that can be assigned a truth statement are reducible to subsumption. Moreover, the language has been designed to correspond to those many people work with every day. Another semantics that comes to mind is that of frame logics [START_REF] Kifer | Logical foundations of object-oriented and frame-based languages[END_REF]. It is not used here because our semantics of specialisation is more precise (it is strictly identified with setinclusion and not with an arbitrary order relation).

Definition 2 (Interpretation).

Let B be a knowledge base (with O, C, A as set of object, class and attribute names respectively and % as the set of possible values) and D a domain. An interpretation is a pair <D,I>, i n which I is called interpretation function:

I: O & D injective C & 2 D A & (D&% 'D) I(a) is total
Injectivity of I on O ensures the unique name assumption for objects.

Definition 3 (Assertion satisfaction).

An interpretation <D,I> satisfies an assertion ( (written

|= <D,I> () if |= <D,I> c!c' iff I c I c ( ) ( ) # ) |= <D,I> c.a#d iff I a c I d ( | ) ( ) # |= <D,I> o c " iff I o I c ( ) ( ) " |= <D,I> o a v . = iff I a I o I v ( )( ( )) ( ) = |= <D,I> c.a#c' iff I(a|c)#I(c') |= <D,I> o.a=o' iff I(a)(I(o))=I(o') I(a|c)
is the domain of the interpretation of slot a when the codomain is restricted to c.

As usual, a knowledge base model

interpretation which satisfies all assertions, an assertion is a consequence of a knowledge base if it is satisfied in every models, and a knowledge base is consistent if it has at least one model (and inconsistent if it has none).

Deduction

Modelling deduction is useful for revision since inconsistency is sometimes raised by inferring in the knowledge base and since the set of deduction rules can help to revise through an abductive process.

A deduction procedure is provided as a set of inference rules specific to object representation systems. Some rules (!reflexivity) are technical rules (enabling to obtain completeness), and others are well-known (!-transitivity, inheritance). "-inference is an original rule (suggested in [START_REF] Ducournau | Les incertitudes de la classification incertaine[END_REF]) providing a kind of type-inference not provided by many systems but required by the above semantics. * c, c', c", c i "C, o"O, a"A, v"V andd, d', d i "T:

Definition 4 (Inference rules).

(1) (!-reflexivity) c c ! (3) ("-closure) c c o c o c ! ) " " ) (4) (d-intersection) c a d c a d c a d d . . . # # ) # + ) (5) (d-inheritance) c a d c c c a d . . # ) ! ) # (5') (c-inheritance) c a c c c c a c . . # )) ) ! ) # )) (6) (d-closure) c a d c a d . . # # ) d d < ) : (6') (c-closure) c a c c c c a c . ' ' " . " # ! # (7) (o-valuation) { . } . c a d o c o a v i i i # " = { } v d i i = + (8) ("-inference) c a c o c o a o o c . ' . ' ' ' # " = "
The closure (Cn(B)) of a knowledge base B by the rules i s defined by the set of assertions obtainable by applying the rules and deducibility by membership of an assertion to the closure.

Definition 5 (Deduction).

An assertion ( is deducible from a knowledge base B (written

B |-(), iff ("Cn(B).

Completeness

As expected the deduction is sound and complete with regard t o the semantics.

Proposition 1 (Soundness).

Let B be a knowledge base and ( an assertion,

if B |-( then B |= (.

Proposition 2 (Partial completeness).

Let B be a consistent knowledge base and ( an assertion,

if B |= ( then B |-(.
It is noteworthy that the completeness property is only partial because it requires the consistency of the knowledge base. As usual, when a knowledge base is inconsistent (i.e. has no model) every assertion is true in all models (since there i s none). In the present language, total completeness could be achieved (with the help of results below) by adding inference rules enabling to infer anything from evidence of inconsistency.

However, retaining partial consistency allows to keep the inconsistency local (at its source) instead of spreading it to the whole base. This enables the revision process to start from a relevant position.

NORMAL FORM AND SYNTACTIC TREATMENT

While a theoretical presentation of an object representation language has been provided, actual implementations are very different. A normalisation process is usually applied to the knowledge base enabling to compact it and to retrieve knowledge efficiently.

A normal form for knowledge bases is presented here because it allows to define syntactic treatments on the knowledge bases used in revision. A constructive characterisation of the normal form is given before syntactically characterising inconsistency.

Normal form

The definition of the normal form is first provided axiomatically through three properties. Basically, it is a form equivalent to the initial knowledge base (1 and 2) in which the deletions are effective (3, i.e. if an assertion is deleted from the base, it is not deducible from it anymore). This property is invaluable when revising because it discards some non minimal revisions.

Inconsistency

Among the advantages of the normal form is the opportunity t o syntactically isolate the source of inconsistency. This, combined with the partial completeness property above, allows for the immediate localisation of inconsistency and the triggering of abductive repairing mechanisms.

Proposition 4 (Syntactic inconsistency).

A knowledge base B is inconsistent iff one of the following propositions is true:

(1) (o.a=x)"B, (o.a=x')"B with x,x'.

Proposition 5 (Complexity).

The complexity of obtaining the normal form and detecting inconsistency is polynomial.

REVISION

The definition of revision on the object representation language is provided below ( §3.1). Minimality is defined and can be characterised both semantically and syntactically ( §3.2). Moreover, this minimality has a natural interpretation in the context of object languages ( §3.3). This allows to apply i t straightforwardly to actual knowledge bases ( §4).

Revised knowledge bases

Here are classically introduced the notions of contracted base (a consistent subset of a knowledge base inconsistent with a particular assertion) and revised base (a contracted base to which the problematic assertion has been added).

Definition 7 (Contracted knowledge base).

Let B be a consistent base and ( an assertion such that B'{(} i s inconsistent. B' is a contracted base of (B,() iff Cn(B').Cn(B), (that is B' |, B and B |= B') and B''{(} is consistent.

Definition 8 (Revised knowledge base).

Let B be a consistent base and ( an assertion such that B'{(} i s inconsistent. B' is a revised base of (B,() iff B'=B"'{(} with B" a contracted base of (B,()

Minimality

Among the various revised knowledge bases, a partial order can be provided enabling to select the closest revisions to the initial knowledge base. This order is based on consequence (or entrenchment).

Definition 9 (Order between contracted k n o w l e d g e b a s e s ) .

Let B' and B" be contracted knowledge bases of (B,(), B'/B" iff any of the following equivalent propositions is satisfied:

(1) B' |= B"

(2) Cn(B")#Cn(B')

Thus the notion of minimality is strictly based on the conservation of the maximum of deductions (or the limitation of new models).

Definition 10 (Minimal contraction).

B' is a minimal contraction of (B,() iff B' is a contracted knowledge base of (B,() and there is no B",B' such that B" is a contracted knowledge base of (B,() and B"/B'.

In order to manipulate contracted knowledge bases, we need to write them as finite assertion sets, namely using normal form. The normal form of a contracted knowledge base of (B,()

B

Proposition 7 (Syntactic characterisation).

Let B' and B" contracted knowledge bases of (B,(), then B' is smaller than B" (B'/B") iff ) # )) 0 0

B B and )

))

+ B B |- .
The two characterisations are equivalent.

Object interpretation

One strength of the resulting revision is that it can be interpreted in terms of objects instead of logics. In this particular case, object terms can be substituted for logic ones. In fact, the revision of an object-based knowledge base is a set of the following modifications:

• moving an object upward in the class hierarchy;

• deleting a value of an object slot; Isabelle Crampé and Jérôme Euzenat

• enlarging a class slot restriction (by enlarging the domain or using a more general class); • moving a class upward in the class hierarchy.

Minimality itself has a behaviour that translates into objects.

Proposition

(Object interpretation o f m i n i m a l i t y ) .

B' is a minimal contracted knowledge base of (B,() iff:

(1) B' is a contracted knowledge base of (B,().

WORKING OUT REVISION

The final goal of this work is to build a practical revision system for an actual object-based system. The results provided so far (because they are always syntactically characterised on the normal form) allow to implement this revision.

Below, the context in which revision is to be applied i s provided ( §4.1). This context is important in the feasibility of the system. Some not self-explained implementation details are then provided ( §4.2).

Framework

The revision operation is used in the context of a shared knowledge base. It aims at helping users to sort out the conflicts which can result from a modification they attempt. Revision i s thus triggered when the users interact with the system.

Since there can exist many different minimal revised bases, the task of choosing the appropriate one is left to the user (while there are some general strategies such as not modifying terminological knowledge or -conversely -not modifying assertional knowledge). In order to be synthetic, the choices are presented to the user in a levelled way (following the abduction processed by the algorithm).

Thus, the revision protocol is the following: 1) the user attempts a modification; 2) the system detects an inconsistency, points it out to the user and proposes some assertions to withdraw (including the modification); 3) the user selects the assertion to withdraw; this might be sufficient or require more iterations in order to select from the various ways to withdraw the assertion. Detecting an inconsistency related to a new assertion is very efficient (polynomial) due to the results of §2.2. However, the finding of all the revisions remains intractable in this system. Taking advantage of the user in order to choose between the various alternatives to explore enables to obtain any revised base in polynomial time (because only the abduction branch which leads to it is explored and testing consistency i s efficient). However, if the users decide to explore several branches the revision process can be very long.

Implementation

The described system has been implemented on top of the Troeps object-based knowledge representation system [START_REF]Tropes 1.0 reference manual[END_REF]. It traps all the errors raised by Troeps when the system is operated and, if the error is revisable (e.g. it is not a spelling error), the revision system deals with the problem.

When inconsistency is detected, the assertions involved are precisely known. For each kind of inconsistency, we have listed the possible global strategies (e.g. move a class, move an instance under the most specific class possible). When a particular inconsistency arises, the adapted strategy is applied to the concerned assertions. The choices made by the user are ordered such that the first modifications made cannot be influenced by the following ones, since for example assertions on class hierarchy are modified before ones on instances. So that, the minimality of the global solution is generally ensured by the minimality of every steps. There are a few exceptions when a choice has an impact on previous identified modifications for which an a posteriori verification is needed, especially to verify if no assertion has been unnecessarily removed.

Interaction with the user is handled through HTTP (as is the rest of the system). At any moment the user can decide t o abandon the current track and to come back to a previous choice. Isabelle Crampé and Jérôme Euzenat This is implemented by stacking the alternatives (of course this can take polynomial space).

DISCUSSION

The apparent simplicity of the language proposed is a point deserving discussion. In fact, the language presented here covers a large part of the needs in object knowledge bases whose purpose is not to solve any possible problem but rather t o organise knowledge and answer to some simple queries (e.g. [START_REF] Euzenat | A knowledge base for D. melanogaster gene interactions involved in pattern formation[END_REF]). In the examples, values are integers and domains are intervals, but this is a restriction which is not required by the work presented. The implementation takes advantage of an independent type system [START_REF] Capponi | Type extensibility of a knowledge representation system with powersets[END_REF] that provides the required operations (emptiness test, intersection, generalisation). However, the expressiveness of the language could be enhanced in other ways (e.g. by adding constructors such as list or set). A more dramatic and required improvement would be to consider cyclic references among objects and classes.

Another important point is complexity. As a matter of fact, the complexity of the revision process exists but it i s transferred to the users. The rationale behind this is that users are those who know the revision to be done and that it cannot be achieved automatically: the knowledge base does not contain the knowledge required for taking the decision [START_REF] Friedman | Belief revision: a critique[END_REF]. In the tests we carried out, the compromise seems satisfactory. Although, the users could be overloaded with the number of solutions they have to consider. We have tried to overcome this problem through the design of the user interface and are currently working on an argumentative framework allowing several users to collaborate in the management of the revision.

RELATED WORKS

The revision obtained satisfies the basic AGM postulates (the first six ones) of [START_REF] Alchourrón | On the logic of theory change: partial meet contraction and revision functions[END_REF] and a maximal consistent set of Hansson constraints [START_REF] Hansson | A test battery for rational database updating[END_REF]. It can be characterised as a maxi-choice operator [START_REF] Alchourrón | On the logic of theory change: partial meet contraction and revision functions[END_REF].

The main related piece of work concerns revision i n description logics [START_REF] Nebel | Reasoning and revision in hybrid representation systems[END_REF][START_REF] Nebel | Syntax-based approaches to belief revision[END_REF]. However, the complexity of the decision problem in description logics did not allow the design of operational systems. The presented system is more alike feature algebras, however, we are not aware of any attempts t o revise in this context.

Recently, Norman Foo [START_REF] Foo | Ontology revision[END_REF] proposed methods to revise conceptual graph bases, others have considered using scripts for repairing errors in problem solving methods [START_REF] Gil | A script-based approach to modifying knowledge bases[END_REF] and numerous other works have studied reorganisation of object databases (with the help of invariant properties and rules for restoring them). These studies give an idea on the methods for revising a base but do not explicitly consider the semantics of the knowledge representation formalism.

Theodoratos [START_REF] Theodoratos | Updating object-oriented schema structures viewed as logical theories[END_REF] studied the same problem as ours. His language is more powerful than the one used here in the expression of the links between classes, but slots are not considered. From a logical modelling of the class definition language, he defined two classical revision procedures, showed the co-NP-completeness of the general problem and several polynomial restrictions interesting for the language presented here.

CONCLUSION

We have presented a framework for interactive knowledge revision in object-based knowledge representation systems. It applies the results of automatic revision in logic to interactive revision in object knowledge representation systems. The work takes advantage of the structure of objects in order to constrain the space (the inconsistency does not spread, the source of inconsistencies can be syntactically tracked and the revisions are naturally ordered) and to interact with the users in an object style.

The complete treatment presented here has also been successfully applied to incoherent classes (classes whose interpretation is the empty set in all models) [START_REF] Crampé | Révision interactive dans une base de connaissance à objets[END_REF]. This has not been presented due to restricted space.

There are three lines of research that can be developed from this point: • Extending the language capabilities (this covers cyclic references, type constructors -set, list, etc. -and perspectives and bridges -Troeps features [START_REF]Tropes 1.0 reference manual[END_REF]). • Introducing the notion of preference in the implementation (obvious choices are protecting the « conceptual scheme » and protecting the data). • Exporting that model to typed object-oriented programming languages (for providing the possible repairs to the compiling errors in such languages).

Figure 1 :

 1 Figure1: B={c 1 !c 0 ; c 2 !c 1 ; c 3 !c 2 ; c 4 !c 2 ; i"c 1 ; i'"c 3 ; i""c 4 ; c 0 .a#[-$ +$] ;c 1 .a#[0 100] ; c 2 .a#[10 25] ; c 4 .a#[5 15] ; i.a=24 ; i'.a=18}.

  Isabelle Crampé and Jérôme Euzenat Definition 6 (Normal form). The normal form of a base B, written NF(B), is such that: (1) B |-NF(B), (2) NF(B) |-B, (3) for each ( of NF(B), NF(B)-{(} |/ -(. Proposition 3 states that the normal form exists, is unique and can be obtained by a simple procedure. It roughly consists of applying the deduction rules backward, starting with the knowledge base closure. Of course, the implementation does not start with the deductive closure but rather mixes inference and computation of the normal form. Proposition 3 (Characterisation of normal form). Let B be a base, the normal form of B is computed by: NF(B) = [Cn(B)-{(c!c)"Cn(B)}] -{(c!c") ; (c!c'), (c'!c")"Cn(B)} -{(o"c) ; (o"c'), (c'!c)"Cn(B)} -{(c.a#d) ; (c.a#d')"Cn(B) et d'<:d} -{(c.a#c') ; (c.a#c"), (c"!c')"Cn(B)} -{(o.a=v) ; (c i .a#d i )"Cn(B) et { } v d i i = + } -{(c.a#d) ; {c i .a#d i , c!c i }"Cn(B), + = i i d d} -{(c.a#c') ; (c".a#c'), (c!c")"Cn(B)} -{(o"c) ; (o'"c' ), (c'.a#c), (o'.a=o)"Cn(B)}

( 2 )

 2 B |-o"c, B |c a d . # , (o.a=v)"B with ¬v:d ( there exists I, such that for each i"I B |-o"c i , (c i .a 1 .a 2 .…a n #-d i )"B and + " i I i d =Ø with c.a 1 .a 2 .…a n #-d standing for "there exist c 1 ,…c n , c' 1 ,…c' n such that for each 1!j!n (c j !c' j )"B and (c' j .a j #c' j+1 )"B (with c=c 1 and (c' n .a n #d) "B)".

BBProposition 6 (

 6 ' can always be written as B'=NF(B')= (NF(B)-# NF(B) and ) + B # Cn(B) This corresponds to the normal form of B from which a set ) 0 is deleted and another set ) + B , deducible from B, is added. The minimality can be characterised semantically and syntactically. Semantic characterisation). B' is a minimal contracted knowledge base of (B,() iff *3 such that B |= 3 and B''{(} |, 3 then B''{(}'{3} i s inconsistent.

( 2 )B

 2 * B" a contracted knowledge base of (B,(), if )) , then B'=B". (3) *(c!c')" ) + B , *k,c', if B' |-k!c' and (c!k)"Cn(B), then (c!k) cannot be added in B''{(}. (4) *(o"c)" ) + B , *c',c, if B' |-c'!c and (o"c')"Cn(B), then (o"c') cannot be added in B''{(}. (5) * (c.a#d)" ) + B , *(c',d'),(c,d), if d'!:d, B' |-c!c' and (c'.a#d')"Cn(B), then (c'.a#d') cannot be added in B''{(}. (6) * (c.a#k)" ) + B , *(c',k'),(c,d), k,k'"C, if B' |-k'!k, B' |-c!c' and (c'.a#k')"Cn(B), then (c'.a#k') cannot be added in B''{(}.

( 7 )Figure 2 .

 72 Figure 2. Examples of the different cases of the proposition 8. Proposition 8 basically states that the notion of minimality can be interpreted on specialisation hierarchies as it is shown i n figure 2. Its interpretation is that: if an assertion (3: specialisation relation, 4,7: class membership, 5,6: slot restriction, 8: slot valuation) is added ( ) + B ) in a minimal knowledge base, some other assertion (3: specialisation relation, 4: class membership, 5,6: slot restriction, 7: slot valuation, class membership or slot restriction, 8: class membership or slot restriction) which was necessary to its inference in the initial knowledge base (B) has to be discarded i n order to avoid inconsistency ( ) 0 B ).

ACKNOWLEDGEMENTS

We thank Roland Ducournau for helpful comments on a previous version of the article.