Dario Cordero-Erausquin

Nathael Gozlan
email: nathael.gozlan@univ-mlv.fr

TRANSPORT PROOFS OF WEIGHTED POINCAR É INEQUALITIES FOR LOG-CONCAVE DISTRIBUTIONS

Keywords: . 1991 Mathematics Subject Classification. 60E15, 32F32 and 26D10 Transport inequalities, weighted Poincaré inequalities

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Introduction

In all the paper, if X = (X 1 , . . . , X n ) is a random vector defined on some probability space (Ω, A, P) with values in R n and h : R n → R is an Borel (bounded or nonnegative) function, we use the following notation for the conditional expectations:

E i [h(X)] := E[h(X)|X 1 , . . . , X i-1 ],
with the convention that E 0 [h(X)] = E[h(X)]. To any random vector X, we associate the random vector X defined as follows:

X i = X i -E i-1 [X i ],
∀i ∈ {1, . . . , n}.

This recentering procedure will play an important role in all the paper (see also [START_REF] Sergey | Bounds on the deficit in the logarithmic sobolev inequality[END_REF] for another application). We aim at proving Poincaré and transport inequalities for X, when X is log-concave. Recall that a random vector X with values in R n is log-concave if for all non-empty compact sets A, B ⊂ R n , it holds P(X ∈ (1 -t)A + tB) ≥ P(X ∈ A) 1-t

P(X ∈ B) t , ∀t ∈ [0, 1].
According to a celebrated result of Borell [START_REF] Borell | Convex measures on locally convex spaces[END_REF][START_REF]Convex set functions in d-space[END_REF], a random vector X is log-concave if and only if there is an affine map ℓ : R k → R n , k ≤ n and a random vector Y taking values in R k such that X = ℓ(Y ) and Y has a density of the form e -V with respect to the Lebesgue measure on R k , where V : R k → R ∪ {+∞} is a convex function. In what follows, by an "n-dimensional log-concave random vector", we will understand a vector X satisfying the conditions above with k = n (and ℓ = Id).

The main result of this note is that the class of all random vectors X with X ndimensional and log-concave satisfies a general weighted Poincaré inequality. Theorem 1.1. There exists a numerical constant a > 0 such that for any n-dimensional log-concave random vector X, it holds

(1.2) Var(f (X)) ≤ a n i=1 E E[X 2 i |X 1 , . . . , X i-1 ]∂ i f (X) 2
for all locally-Lipschitz f : R n → R belonging to L 2 (X), where Var(Y

) := E[Y 2 ] -E[Y ] 2
denotes the variance of a real valued random variable Y. In particular, if X is such that E i-1 [X i ] = 0 for all i ∈ {1, . . . , n}, then X = X and it holds

(1.3) Var(f (X)) ≤ a n i=1 E E i-1 [X 2 i ]∂ i f (X) 2
Remark 1.4. If the operation X → X was preserving log-concavity then, of course, (1.2) would follow from (1.3) applied to X. It is not difficult to find examples of log-concave random vectors X such that X is not log-concave anymore. A random vector X such that X = X can be interpreted as a sequence of martingale increments (see Section 3 for more details).

Theorem 1.1 is reminiscent of recent results of Klartag [START_REF]Poincaré inequalities and moment maps[END_REF] and of Barthe and Cordero-Erausquin [START_REF] Barthe | Invariances in variance estimates[END_REF] which were based on L 2 methods. The objective of this note is to give alternative proofs of variants of some of the results from [START_REF]Poincaré inequalities and moment maps[END_REF][START_REF] Barthe | Invariances in variance estimates[END_REF] using mass transport arguments.

Recall that a random vector X is unconditional when X = (X 1 , . . . , X n ) has the same law as (ε 1 X 1 , . . . , ε n X n ) for any choice of ε i = ±1. Since unconditional random vectors satisfy E i-1 [X i ] = 0 for all 1 ≤ i ≤ n, Theorem 1.1 can be seen as an extension of the following result by Klartag [START_REF]Poincaré inequalities and moment maps[END_REF]: for any log-concave and unconditional random vector X, it holds

(1.5) Var(f (X)) ≤ c n i=1 E (X 2 i + E[X 2 i ])∂ i f (X) 2 ,
for all f : R n → R smooth enough, where c > 0 is some absolute constant. Moreover, when f is itself unconditional (i.e f (ε 1 x 1 , . . . , ε n x n ) = f (x 1 , . . . , x n ) for all ε i = ±1), then the terms E[X 2 i ] can be removed from the right hand side of (1.5). Note that in [START_REF]Poincaré inequalities and moment maps[END_REF], Klartag also obtains weighted Poincaré inequalities for a larger class of unconditional distributions with a density of the form e -φ with φ : R n → R whose restriction to R n + is p convex (i.e x → φ(x 1/p 1 , . . . , x 1/p n ) is convex). Inequalities of the form (1.5) were also investigated in details in the recent paper [START_REF] Barthe | Invariances in variance estimates[END_REF]. There, the authors establish general weighted Poincaré inequalities for classes of probability measures invariant by a subgroup of isometries, not only the coordinate reflections.

Note that (1.3) applies to random vectors having less symmetries than unconditional random vectors. For instance, if the X i are independent mean zero and variance one log-concave random variables then E i-1 [X i ] = 0 for all i, whereas X does not have any particular symmetry. In this case, the conclusion (1.3) of Theorem 1.1 is consistent with the Poincaré inequality obtained using the (elementary) tensorisation property of the Poincaré inequality.

Theorem 1.1 also easily implies some variance estimates for log-concave random vectors.

Corollary 1.6.

There exists a universal constant b > 0 such that if X is an n-dimensional log-concave random vector, then, denoting by | • | the standard Euclidean norm on R n , it holds

(1.7) Var |X| 2 ≤ b n i=1 E[X 4 i ] ≤ 16b n i=1 E[X 4 i ].
In particular, when E[X 2 i ] = 1 for all i ∈ {1, . . . , n}, we have

(1.8) Var |X| 2 ≤ cn and if in addition X satisfies E i-1 [X i ] = 0 for all i, then (1.9) Var |X| 2 ≤ cn,
for some other universal constant c.

The inequality (1.9) on the variance immediately yields to the following concentration in a thin shell estimate

P |X| - √ n ≥ t √ n ≤ b e -c n 1/4 √ t , ∀t > 0.
This type of concentration inequalities plays a central role in the proof of the central limit theorem for log-concave random vectors [START_REF] Anttila | The central limit problem for convex bodies[END_REF][START_REF] Klartag | A central limit theorem for convex sets[END_REF][START_REF]High-dimensional distributions with convexity properties[END_REF][START_REF]On concentration of distributions of random weighted sums[END_REF].

Corollary 1.6 is also motivated by the so called variance conjecture. Recall that a random vector X is said isotropic if E[X] = 0 and E[X i X j ] = δ i,j for all i, j ∈ {1, . . . , n}. The variance conjecture asserts that any log-concave and isotropic random vector X satisfies (1.9) for some universal positive constant b. This conjecture was shown to be true in restriction to the class of unconditional log-concave random vectors by Klartag [START_REF]A Berry-Esseen type inequality for convex bodies with an unconditional basis[END_REF][START_REF]Poincaré inequalities and moment maps[END_REF]. We refer to [START_REF] Barthe | Invariances in variance estimates[END_REF] and [START_REF] Alonso | The variance conjecture on some polytopes, Asymptotic geometric analysis[END_REF] for other subclasses of log-concave distributions satisfying the variance conjecture. The best (dimensional) estimate in date is due to Guédon and Milman [START_REF] Guédon | Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures[END_REF] who proved that Var(|X|) ≤ bn 2/3 for any isotropic log-concave random vector X. The variance conjecture is a weak form of a celebrated conjecture by Kannan, Lovasz and Simonovits [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF] stating that any log-concave and isotropic random vector X satisfies a Poincaré inequality

Var(f (X)) ≤ aE |∇f | 2 (X) , ∀f smooth enough,
for some universal constant a > 0. According to a remarkable recent result of Eldan [START_REF] Eldan | Thin shell implies spectral gap up to polylog via a stochastic localization scheme[END_REF], the variance conjecture implies the KLS conjecture up to some log(n) factor.

Corollary 1.6 thus shows that the variance conjecture is satisfied on the class of isotropic log-concave random vectors such that X = X (see also [START_REF] Barthe | Invariances in variance estimates[END_REF]Theorem 4] and Remark 6.1 below for a related result). It is not difficult to see that this class is strictly larger than the class of unconditional isotropic and log-concave random vectors (some informations on log-concave random vectors such that X = X can be found in Proposition 2.3 and Remark 2.4 below). For general log-concave random vectors X, let us mention that it is always at least possible to bound Var(|X| 2 ) in terms of Var(|X| 2 ) and of Var(|X ′ | 2 ), where the "reduced" random vector X ′ is defined by

X ′ i = E i-1 [X i ], ∀i ∈ {1, . . . , n}.
The basic observation behind the following elementary result is that X = X + X ′ is an orthogonal decomposition of X in the space H := L 2 (Ω, A, P; R n ) of square integrable ndimensional random vectors. More precisely, for any X ∈ H, the vector X is the orthogonal projection of X onto the linear subspace

H 0 (X) = {Y ∈ H; E[Y i |X 1 , . . . , X i-1 ] = 0, ∀i ∈ {1, .
. . , n}} (the space of random sequences that are matingale increments with respect to the filtration σ(X 1 , . . . , X i ), 0 ≤ i ≤ n -1.) We will prove the following useful identity

(1.10) Var(|X| 2 ) = Var(|X| 2 ) + Var(|X ′ | 2 ) + 2Cov(|X| 2 , |X ′ | 2 ) + 4E[(X • X ′ ) 2 ] + 4E[|X| 2 X • X ′ ] + 4E[|X ′ | 2 X • X ′ ],
from which one deduces the following result:

Corollary 1.11. If X is an isotropic and log-concave random vector in R n , and X ′ is defined as above, then

Var |X| 2 ≤ a n + Var |X ′ | 2 and Var |X ′ | 2 ≤ a n + Var |X| 2 ,
for some universal constant a.

It follows that the variance conjecture is (technically) equivalent to the existence of a universal constant b > 0 such that for any isotropic log-concave random vector X,

Var |X ′ | 2 ≤ bn.
It would be of some interest to see if for some specific classes of vectors X, the variance term Var(|X ′ | 2 ) can be estimated by some power of n.

The proof of Theorem 1.1 is based on mass transport. More precisely, we will establish a transport-entropy inequality (Theorem 4.6) which is of independent interest, of the form

T µ (µ, ν) ≤ D(ν µ), ∀ν,
where µ and ν are the laws of random vectors X and Y , with X, Y distributed according to µ and ν. The optimal transport cost T µ will be of the form

T µ (ν 0 , ν 1 ) = inf π∈C(ν 0 ,ν 1 ) c µ (x, y) π(dxdy),
for a particular cost function c µ (precise definitions will be given later). Then, Theorem 1.1 will follow from this transport-entropy inequality by a standard linearization procedure. The argument towards our transport inequality will use an above tangent lemma introduced by Cordero-Erausquin [START_REF] Cordero-Erausquin | Some applications of mass transport to Gaussian-type inequalities[END_REF] which is a handy tool to prove classical functional inequalities (Log-Sobolev, Talagrand) for uniformly log-concave random vectors and to recover the celebrated HWI inequality of Otto and Villani [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF].

Let us mention here a byproduct of this approach in terms of transport inequalities involving the classical W 2 distance (definitions are recalled below). Theorem 1.12. There exists a universal constant c such that for any n dimensional log-concave random vector X taking values in the hypercube

[-R, R] n , R > 0, it holds W 2 2 (µ, ν) ≤ cR 2 D(ν µ
), for all probability measures ν on R n , where µ and ν denote respectively the laws of X and Y , Y being distributed according to ν. Theorem 1.12 can be considered as a variant of results by Eldan and Klartag [START_REF] Eldan | Dimensionality and the stability of the brunn-minkowski inequality[END_REF]Theorem 6.1] and by Klartag [24,Theorem 4.2]. Let us mention that the present paper uses techniques of proof very similar to those involved in [START_REF] Eldan | Dimensionality and the stability of the brunn-minkowski inequality[END_REF][START_REF]Concentration of measures supported on the cube[END_REF]. To be more precise, Theorem 6.1 of [START_REF] Eldan | Dimensionality and the stability of the brunn-minkowski inequality[END_REF] gives a similar inequality when µ and ν are both unconditional and log-concave. In their statement, the relative entropy is replaced by [-R,R] n H(f, g) -1, where H(f, g) = sup x∈R n f (x + y)g(x -y), denoting by f, g the densities of µ and ν with respect to Lebesgue. This quantity is relevant in their study of the stability of the Brunn-Minkowski inequality. In Theorem 4.2 of [START_REF]Concentration of measures supported on the cube[END_REF], Klartag obtains the inequality

W 2 2 (µ, ν) ≤ cL 2 D(ν µ), ∀ν
for all log-concave probability measures µ supported on the hypercube Q = [-1, 1] n and such that in addition the density f of µ with respect to Lebesgue satisfies for some L ≥ 1

f ((1 -t)x + ty) ≤ L[(1 -t)f (x) + tf (y)], ∀t ∈ [0, 1],
for all x, y ∈ Q with x -y proportional to one of the standard basis vectors e i . This condition is for instance realized with L = e M/8 if f = e -V for some smooth convex function

V : Q → R such that sup i≤n sup x∈Q ∂ 2 i V (x) ≤ M for some M ≥ 0.
The paper is organized as follows. In Section 2, we gather various observations on the relations between X and X for log-concave random vectors. In Section 3, we give some background on the mass transportation tools that are used to establish our general transport-inequality, which is stated and proved in Section 4, together with Theorem 1.12. Then, in Section 5 we linearize this transport-entropy inequality and establish Theorem 1.1. In the final Section 6, we explain how to derive the Corollaries 1.6 and 1.11 on the variance.

2. Some observations about log-concave random vectors such that X = X First, we begin with a straightforward proposition identifying the class of random vectors such that X = X as increments of martingales.

Proposition 2.1. A random vector X = (X 1 , . . . , X n ) is such that X = X if and only if M = (M 0 , M 1 , . . . , M n ), with M 0 = 0 and M k = k i=1 X i is a martingale with respect to the increasing sequence of sub-sigma fields F k = σ(M 0 , . . . , M k ), k ∈ {0, . . . , n}.
The proof if left to the reader.

If M := (M 0 , M 1 , . . . , M n ) is a martingale, we denote by ∆ i = M i -M i-1 , i ∈ {1, . . . , n} the increments of M . The quadratic variation process of M is then defined by [M ] k = k i=0 ∆ 2
i , for all k ∈ {0, 1, . . . , n}. With these definitions, Corollary 1.7 can be restated as follows.

Proposition 2.2. There exists a universal constant c > 0 such that for all martingale

M = (M 0 , M 1 , . . . , M n ) such that M 0 = 0 and (M 1 , . . . , M n ) has a log-concave density, it holds Var([M ] k ) ≤ c k i=1 E[∆ 4 i ], ∀k ≤ n.
Proof. Since the class of log-concave random vectors is stable under affine transformations, it follows that (M 1 , . . . , M n ) has a log-concave density if and only if (X 1 , . . . , X n ) with X i = ∆ i has a log-concave density. The result then follows immediately from Corollary 1.7.

We now collect in the following proposition some elementary informations on log-concave random vectors X such that X = X.

Proposition 2.3.

(1) If X, Y are two independent log-concave random vectors (defined on the same probability space) such that X = X and Y = Y , then

X + Y = X + Y. (2) If X is a log-concave random vector with values in R n then X = X if and only if E[X] = 0 and for all k ∈ {1, . . . , n -1}, E[X|X 1 , . . . , X k ] = (X 1 , . . . , X k , 0, . . . , 0). In particular, if C ⊂ R n is a bounded convex body and X is uniformly distributed over C, then X = X if and only if the barycenter of C is at 0 and for all x = (x 1 , . . . , x n ) ∈ R n Bar(C ∩ {(x 1 , . . . , x k )} × R n-i ) = (x 1 , . . . , x k , 0 . . . , 0), ∀k ∈ {1, . . . , n -1},
(whenever this section is not empty). In particular, C is symmetric with respect to the hyperplane {x n = 0}.

(3) If C ⊂ R 2 is
a bounded convex body with barycenter at 0 and X is uniformly distributed over C, then X is uniformly distributed over the convex body C obtained from C by applying Steiner symmetrization with respect to the axis D = R × {0}.

In particular X = X if and only if C is symmetric with respect to D.

Proof.

(1) It is well known that X+Y is still log-concave. Let us show that X + Y = X+Y . Let i ∈ {2, . . . , n} and take f : R i-1 → R a bounded measurable test function, then it holds

E [X i f (X 1 + Y 1 , . . . , X i-1 + Y i-1 )] = E X [X i E Y [f (X 1 + Y 1 , . . . , X i-1 + Y i-1 )]] = 0. Similarly, E [Y i f (X 1 + Y 1 , . . . , X i-1 + Y i-1 )] = 0. Therefore, E [(X i + Y i )f (X 1 + Y 1 , . . . , X i-1 + Y i-1
)] = 0, and since this holds for all test function f , one concludes that

E i-1 [(X + Y ) i ] = 0 for all i and so X + Y = X + Y . (2)
The second point follows easily from the fact that for k

≤ i -1, E[X i |X 1 , . . . , X k ] = E [E[X i |X 1 , . . . , X i-1 ]|X 1 , . . . , X k ] .
(3) Observe that C can be written as

C = {(x 1 , x 2 ) ∈ R 2 ; x 1 ∈ [α, β], a(x 1 ) ≤ x 2 ≤ b(x 1 )}, for some α < β,
C = (x 1 , x 2 ) ∈ R 2 ; x 1 ∈ [α, β], 1 2 (a(x 1 ) -b(x 1 )) ≤ x 2 ≤ 1 2 (b(x 1 ) -a(x 1 )) . Since the function a-b is convex, the set C is convex. Moreover X = X 1 , X 2 -1 2 (a(X 1 ) + b(X 1 )
) and so for all bounded measurable test function f :

R 2 → R E f (X) = 1 Vol(C) β α b(x 1 ) a(x 1 ) f (x 1 , x 2 - 1 2 (a(x 1 ) + b(x 1 ))) dx 2 dx 1 = 1 Vol(C) β α 1 2 (b(x 1 )-a(x 1 )) 1 2 (a(x 1 )-b(x 1 )) f (x 1 , y 2 ) dy 2 dx 1 = 1 Vol(C) C f (y 1 , y 2 ) dy 1 dy 2 .
This shows that X is uniformly distributed on C.

Remark 2.4. As we already mentioned, the class of log-concave random vectors such that X = X already contains unconditional log-concave random vectors and log-concave random vectors with centered independent components. Using the properties above, it is possible to give other examples of log-concave random vectors such that X = X in arbitrary large dimension. Namely, observe that if X is a log-concave random vector taking values in R k and such that X = X, then it is easy to check that for all i ∈ {1, . . . , k + 1}, the random vector X i defined by

X i = (X 1 , . . . , X i-1 , 0, X i , . . . , X k ) ∈ R k+1
is still log-concave and verifies X i = X i . Thanks to point (1) of Proposition 2.3, one thus sees that if X 1 , . . . , X k+1 are independent log-concave random vectors with values in R k and such that X i = X i , then the random vector

Y = X 1 1 + X 2 2 + • • • + X k+1
k+1 is still logconcave and satisfies Y = Y. Initializing this construction with k = 2 with the help of point (3) of Proposition 2.3 and iterating the process gives rise to a large class of non-trivial examples of log-concave random vectors such that X = X.

Some background on mass transport

The key lemma used in [START_REF] Cordero-Erausquin | Some applications of mass transport to Gaussian-type inequalities[END_REF] is the so called above tangent lemma recalled below. In what follows, the relative entropy (also called the Kullbak-Leibler distance) of ν with respect to µ is defined by

(3.1) D(ν µ) = log dν dµ dν,
if ν is absolutely continuous with respect to µ (otherwise, we set D(ν µ) = ∞).

Lemma 3.2 ([13]

). If µ is a probability measure on R n absolutely continuous with respect to the Lebesgue measure with a density of the form µ(dx) = e -V (x) dx where V : R n → R is a function of class C 2 such that Hess V ≥ ρ, ρ ∈ R, then for all compactly supported probability measures ν 0 , ν 1 absolutely continuous with respect to µ, it holds

(3.3) D(ν 1 µ) ≥ D(ν 0 µ) + ∇ dν 0 dµ (x), T x -x µ(dx) + ρ 2 |T x -x| 2 ν 0 (dx) + Tr (DT x -I n ) -log |DT x | ν 0 (dx),
where T : R n → R n pushes forward ν 0 onto ν 1 and defines a "suitable" change of variables.

First let us recall the classical applications of (3.3). In [START_REF] Cordero-Erausquin | Some applications of mass transport to Gaussian-type inequalities[END_REF], the inequality (3.3) was applied with the Brenier map T (see [START_REF] Villani | Optimal transport, Grundlehren der Mathematischen Wissenschaften[END_REF]), that is to say the ν 0 almost surely unique map T achieving the infimum in the definition of the square Kantorovich distance W 2 :

|T x -x| 2 ν 0 (dx) = inf π∈C(ν 0 ,ν 1 ) |y -x| 2 π(dxdy) := W 2 2 (ν 0 , ν 1 ),
where C(ν 0 , ν 1 ) denotes the set of all couplings of ν 0 , ν 1 , (i.e probability measures π on R n × R n having ν 0 and ν 1 as marginals). A fundamental property of the Brenier map T is that it is the gradient of a convex function: there exists φ : R n → R convex such that T (x) = ∇φ(x) for ν 0 almost every x ∈ R n . As a consequence of the inequality log(λ) ≤ λ -1, λ > 0 and of the fact that DT x = Hess x φ has a non-negative spectrum, the last term in (3.3) is always non-negative (assuming for simplicity that T is smooth). So (3.3) becomes

(3.4) D(ν 1 µ) ≥ D(ν 0 µ) + ∇ dν 0 dµ (x), T x -x µ(dx) + ρ 2 W 2 2 (ν 0 , ν 1 ).
Inequality (3.4), which expresses in some sense that the graph of the map D( • µ) lies above its tangent, is also related to the notion of displacement-convexity of the relative entropy along W 2 geodesics (see [START_REF] Mccann | A convexity principle for interacting gases[END_REF][START_REF] Villani | Optimal transport, Grundlehren der Mathematischen Wissenschaften[END_REF]). When ρ > 0, interesting consequences can be drawn from the inequality above. For instance, choosing ν 0 = µ yields to the following transport-entropy inequality

W 2 2 (ν 1 , µ) ≤ 2 ρ D(ν 1 µ), ∀ν 1 .
This type of inequalities goes back to the works by Marton [START_REF] Marton | A simple proof of the blowing-up lemma[END_REF] and Talagrand [START_REF] Talagrand | Transportation cost for Gaussian and other product measures[END_REF] (see [START_REF] Ledoux | The concentration of measure phenomenon[END_REF][START_REF] Villani | Optimal transport, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Gozlan | Transport inequalities. A survey, Markov Process[END_REF] for an introduction to the subject). On the other hand, choosing ν 1 = µ it is not difficult to derive from (3.4) the logarithmic-Soblev inequality (see [START_REF] Cordero-Erausquin | Some applications of mass transport to Gaussian-type inequalities[END_REF][START_REF] Barthe | Transportation techniques and Gaussian inequalities, Optimal transportation, geometry and functional inequalities[END_REF][START_REF] Gozlan | Transport inequalities. A survey, Markov Process[END_REF] for details)

D(ν 0 || µ) ≤ 2 ρ |∇h 0 | 2 h 0 dµ, ∀ν 0 = h 0 µ.
We refer to [START_REF] Barthe | Mass transport and variants of the logarithmic Sobolev inequality[END_REF][START_REF] Sergey | Bounds on the deficit in the logarithmic sobolev inequality[END_REF] for other applications and variants of (3.3) and (3.4).

In this paper, we will use (3.3) with ρ = 0 and ν 0 = µ:

D(ν 1 µ) ≥ Tr (DT x -I n ) -log |DT x | µ(dx).
But as a main difference, we will rather use as T the Knothe map [START_REF] Knothe | Contributions to the theory of convex bodies[END_REF] between µ and ν 1 .

Let us recall the definition of the Knothe transport between two probability measures. If µ, ν are two Borel probability on R and µ has no atom, then there exists a unique nondecreasing and left continuous map T : R → [-∞, ∞] transporting µ on ν in the sense that f (T ) dµ = f dν for all say bounded continuous function f . This map T is given by

T (x) = F -1 ν • F µ (x), ∀x ∈ R. where, for x ∈ R and t ∈ [0, 1], F µ (x) = µ(-∞, x]
and

F -1 µ (t) = inf{x ∈ R; F µ (x) ≥ t} ∈ [-∞, ∞].
The map T takes finite values µ almost surely. Let us mention that the map T achieves the minimum value in a large class of optimal transportation problems (see for instance [START_REF] Svetlozar | Mass transportation problems[END_REF]). This fact will not be explicitly used in the sequel.

The Knothe transport map is a multidimensional extension of this one dimensional transport. To define it properly, we need to introduce the following notation. If µ is a probability measure on R n and X = (X 1 , . . . , X n ) is a random vector of law µ, we will denote by µ i the law of (X 1 , . . . , X i ). For i ≥ 2 and x 1 , . . . , x i-1 ∈ R, we denote by

µ i ( • |x 1 , . . . , x i-1 ) the conditional law of X i knowing X 1 = x 1 , X 2 = x 2 , . . . , X i-1 = x i-1 .
The conditional probability measure µ i ( • |x 1 , . . . , x i-1 ) is well defined for µ i-1 almost all (x 1 , . . . , x i-1 ) ∈ R i-1 . When µ has a positive density h with respect to the Lebesgue measure on R n , the conditional probability measures µ i ( • |x 1 , . . . , x i-1 ) have an explicit density with respect to Lebesgue measure on R it holds

f (u i ) µ i (du i |x 1 , . . . , x i-1 ) = f (u i )h(x 1 , . . . , x i-1 , u i , u i+1 , . . . , u n ) du i • • • d un h(x 1 , . . . , x i-1 , u i , u i+1 , . . . , u n ) du i • • • d un ,
for all bounded continuous f : R → R.

The Knothe map T = (T 1 , . . . , T n ) transporting a probability measure µ on R n with a positive density on another probability ν, is defined recursively as follows :

-T 1 is the optimal transport map sending µ 1 on ν 1 ; -for a given

x ∈ R n , T i (x 1 , x 2 , . . . , x i-1 , • ) is the one dimensional monotone map sending µ i ( • |x 1 , . . . , x i-1 ) on ν i ( • |T 1 (x), . . . , T i-1 (x)).
Note that in particular, T is triangular in the sense that T i (x) depends only on x 1 , . . . , x i .

The following lemma is a formally contained in Lemma 3.2; for completeness, we recall its short proof below. Lemma 3.5. Let µ be probability measure on R n with µ(dx) = e -V (x) dx with V : R n → R a convex function of class C 1 ; for all probability measure ν on R n compactly supported with a smooth density, it holds

D(ν µ) ≥ Tr(DT (x) -I) -log(|DT (x)|) µ(dx) = n i=1 ∂ i T i (x) -1 -log ∂ i T i (x) µ(dx)
where T is the Knothe map transporting µ on ν.

Proof. Write g = dν dx and h = dν dµ . First assume that T is C 1 ; according to the change of variable formula, it holds e -V (x) = h(T x)e -V (T x) |DT (x)|, so taking the log and integrating with respect to µ, we obtain

-V (x) µ(dx) = log(h(T x)) µ(dx) -V (T x) µ(dx) + log(|DT (x)|) µ(dx). So D(ν µ) = V (T x) -V (x) µ(dx) -log |DT (x)| µ(dx).
By assumption,

V (y) ≥ V (x) + ∇V (x) • (y -x), ∀x, y ∈ R n . So, D(ν µ) ≥ ∇V (x) • (T x -x) µ(dx) -log |DT (x)| µ(dx).
Note that, integrating by parts (and using that ν is compactly supported),

∇V (x) • (T x -x) µ(dx) = n i=1 (∂ i T (x) -1)e -V (x) dx = Tr(DT (x) -I) µ(dx) Thus, (3.6) D(ν µ) ≥ Tr(DT (x) -I) -log |DT (x)| µ(dx).
Actually the map T is not necessarily of class C 1 so the change of variable formula above needs to be justified. One can consult Section 3 of [START_REF]Large deviations and isoperimetry over convex probability measures with heavy tails[END_REF] and invoke for instance [8, Lemma 3.1].

A general transport inequality for log-concave probability measures

Before introducing our transport cost, we need to briefly discuss on the Cheeger constant (or equivalently, the Poincaré constant) of one-dimensional log-concave densities, a case where optimal bounds are known. If γ is a log-concave probability measure on R, denote by λ γ its Cheeger's constant, namely the largest constant for which (4.1)

λ γ |f -m(f )| dγ ≤ |f ′ | dγ
holds for all f : R → R locally-Lipschitz, where m(f ) denotes a median of f . It was proven by Bobkov [START_REF] Sergey | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF] that when γ is log-concave probability measure on R, one has

(4.2) 1 3Var(X) ≤ λ 2 γ ≤ 2 Var(X) , with X ∼ γ. Note that if X is a constant random variable, Var(X) = 0 and λ = ∞.
In what follows, µ is a log-concave probability measure on R n with full support and X = (X 1 , . . . , X n ) a random vector distributed according to µ.

According to Bobkov's estimate (4.2), for all x ∈ R n , the one dimensional (log-concave) probability µ i ( • |x 1 , . . . , x i-1 ) verifies Cheeger's inequality (4.1) with a constant (optimal up to universal factor)

(4.3) λ 2 i (x) = λ 2 i (x 1 , . . . , x i-1 ) := 1 3Var(X i |X 1 = x 1 , . . . , X i-1 = x i-1 ) ∈ (0, +∞]
where

Var(X i |X 1 = x 1 , . . . , X i-1 = x i-1 ) = u 2 µ i (du|x 1 , . . . , x i-1 ) - u µ i (du|x 1 , . . . , x i-1 ) 2 ∈ [0, +∞).
In Theorem 4.6 below, we prove that any log-concave probability measure on R n verifies some transport-entropy inequality with a cost function c µ determined by the functions λ i introduced above. In order to state the result, we need to introduce some additional notation. Recall that if Z is a random vector, we denote by Z the random vector defined by

Z i = Z i -E[Z i |Z 1 , . . . , Z i-1 ].
Note in particular that X = R(X), where the recentering map R : R n → R n is defined by R(x) = (R 1 (x), . . . , R n (x)), where (4.4)

R i (x) = x i -m i (x), with m i (x) = m i (x 1 , x 2 , . . . , x i-1 ) = u µ i (du|x 1 , . . . , x i-1 )
It is not difficult to check that the map R is invertible. We will denote by S = R -1 its inverse. The cost function c µ : R n × R n → [0, ∞] is defined as follows,

c µ (x, y) = 1 16 n i=1 N λ i (S(x))(x i -y i ) , ∀x, y ∈ R n ,
where N (t) = |t| -log(1 + |t|) (with the conventions 0 × ∞ = 0 and a × ∞ = sign of a × ∞ for a = 0). The associated optimal transport cost denoted by T µ is defined by

T µ (ν 1 , ν 2 ) = inf π∈C(ν 1 ,ν 2 ) c µ (x 1 , x 2 ) π(dx 1 dx 2 ),
where

C(ν 1 , ν 2 ) is the set of all probability measures π on R n × R n such that π(dx 1 × R n ) = ν 1 (dx 1 ) and π(R n × dx 2 ) = ν 2 (dx 2 )
Let us mention that the transport inequality below also holds with the cost function cµ :

R n × R n → [0, ∞] defined as follows (4.5) cµ (x, y) = 1 16 N   n i=1 λ i (S(x)) 2 (x i -y i ) 2   , ∀x, y ∈ R n .
Indeed the function x → N ( √ x) is subadditive, since it is concave on R + and vanishes at 0, so we have for all x, y ∈ R n , c µ (x, y) ≥ cµ (x, y).

Theorem 4.6. Let X be an n-dimensional log-concave random vector and let µ be its law; for all probability measure ν on R n with finite first moment, it holds

(4.7) T µ (μ, ν) ≤ D(ν µ),
where μ is the law of X and ν is the law of Y with Y distributed according to ν.

Note that the transport cost depends on µ, and not µ. Indeed, it is given by the values of λ i , which depend on X ∼ µ through formula (4.3).

Proof. According to a result by Bobkov and Houdré [START_REF] Sergey | Isoperimetric constants for product probability measures[END_REF], if γ is probability measure on R verifying Cheeger's inequality (4.1) with constant λ, then for all convex even function L : R → R + such that L(0) = 0, L(x) > 0 for all x > 0 and p

L := sup xL ′ (x) L(x) < +∞, it holds L(f -m(f )) dγ ≤ L(p L f ′ /λ) dγ,
where m(f ) denotes the median of f. It will be convenient to replace the median of f by its mean denoted by γ(f ). First observe that Jensen inequality yields

L(γ(f ) -m(f )) ≤ L(p L f ′ /λ) dγ.
On the other hand, the convexity of L implies that

L(f -γ(f )) dγ ≤ 1 2 L(2(f -m(f ))) dγ + 1 2 L(2(m(f ) -γ(f ))).
Finally, it is not difficult to check that the function L 1/p L is subbadditive (see for instance [START_REF] Nathael Gozlan | Characterization of Talagrand's transportentropy inequalities in metric spaces[END_REF]Lemma 4.7]). It follows that L(2a) ≤ 2 p L L(a), a ≥ 0. Therefore,

L(f -γ(f )) dγ ≤ 2 p L L(p L f ′ /λ) dγ.
As it is easy to see, for the function N , it holds p N ≤ 2. So we have the inequality

(4.8) 1 16 N (λ(f -γ(f ))) dγ ≤ L(f ′ ) dγ.
First let us assume that µ(dx) = e -V (x) dx where V : R n → R is a convex function of class C 1 and ν is compactly supported with a smooth density. If X is a random vector of law µ, then using Lemma 3.5, the inequality t -log(1 + t) ≥ N (t), t > -1 and the inequality (4.8), it holds

D(ν µ) ≥ n i=1 E [N (∂ i T i (X) -1)] = n i=1 E [ E [N (∂ i T i (X) -1) | X 1 , . . . , X i-1 ] ] = n i=1 E [ E [N (∂ i (T i -x i )(X)) | X 1 , . . . , X i-1 ] ] ≥ 1 16 n i=1 E E N (λ i (X) (T i (X) -E[T i (X)|X 1 , . . . , X i-1 ] -X i + E[X i |X 1 , . . . , X i-1 ])) X 1 , . . . , X i-1 = 1 16 n i=1 E N (λ i (X) (T i (X) -E[T i (X)|X 1 , . . . , X i-1 ] -X i + E[X i |X 1 , . . . , X i-1 ])) .
Note that, since T 1 (X), . . . , T i-1 (X) are functions of X 1 , . . . , X i-1 , it holds

E[T i (X)|X 1 , . . . , X i-1 ] = E[T i (X)|T 1 (X), . . . , T i-1 (X)]
almost surely. It follows, that the vector Y defined by

Y i = T i (X)-E[T i (X)|X 1 , . . . , X i-1 ]
has law ν. Using the definition of our cost, we see that

D(ν µ) ≥ 1 16 E n i=1 N λ i (S(X)) Y i -X i = E c µ (X, Y ) .
Therefore, by definition of T µ , we have

D(ν µ) ≥ T µ (μ, ν).
Using classical approximation arguments, one extends the inequality above to all probability measures ν with finite finite first moment. This completes the proof of Theorem 4.6 in the case where µ(dx) = e -V (x) dx with a convex function V of class C 1 on R n . The conclusion is then extended, using classical approximation arguments, to any µ(dx) = e -V (x) dx where V : R n → R ∪ {+∞} is a lower semi-continuous convex function whose domain has a non empty interior. A way to do it is to consider the family of convex functions V s , s > 0 defined by

V s (x) = inf y∈R n V (y) + 1 s |x -y| 2 , x ∈ R n , s > 0.
It is well known that for all s > 0, V s : R n → R is a C 1 smooth convex function on R n converging monotonically to V as s → 0 (see for instance [START_REF] Hiriart-Urruty | Convex analysis and minimization algorithms. II, Grundlehren der Mathematischen Wissenschaften[END_REF]Theorem 4.1.4]). Details are left to the reader.

Proof of Theorem 1.12. Assume that µ is the law of an n-dimensional log-concave random vector X taking values in the hypercube Q = [-R, R] n . This assumption on the support of µ has for consequence that for all x ∈ Q,

Var(X i |X 1 = x 1 , . . . , X i-1 = x i ) ≤ 2R 2 .
Therefore, λ i (x) ≥ 1/( √ 6R) for all i ∈ {1, . . . , n} and x ∈ Q. It is not difficult to check that there is an absolute constant c > 0 such that N (u) ≥ cu 2 for all |u| ≤ 2/ √ 6. So, if ν is a given probability measure on Q, then by Theorem 4.6 it holds

D(ν µ) ≥ c R 2 E |X -Y | 2
, which completes the proof.

Weighted Poincaré inequalities for log-concave probability measures

In this last section, we use a classical linearization technique to prove that the transport cost inequality obtained in Theorem 4.6 implies the weighted Poincaré inequality of Theorem 1.1. Such linearization depends only on the behavior of the cost for small distances. It will be more convenient, notationnaly speaking, but equivalent, to use the cost cµ defined by (4.5) in the definition of T µ and in Theorem 4.6, rather than c µ .

Let us introduce the following supremum convolution operator

P t f (x) = sup y∈R n f (y) - 1 t cµ (x, y) , ∀x ∈ R n , ∀t > 0,
which is well defined for instance for bounded continuous function f on R n . It can be shown that he function u(t, x) = P t f (x) satisfies in some weak sense the following Hamilton-Jacobi equation

∂ t u(t, x) = 8 n i=1 1 λ 2 i (S(x)) (∂ x i u) 2 (t, x).
Actually, in what follows, we will only need the following elementary inequality:

Lemma 5.1. For all differentiable bounded Lipschitz function f : R n → R, lim sup t→0 1 t P t f -f dν ≤ 8 n i=1 1 λ 2 i (S(x)) (∂ x i f ) 2 (x) ν(dx),
for all probability measure ν on R n such that λ -2 i (S) dν is finite for every i ∈ {1, . . . , n}. Let us admit the lemma for a moment and prove Theorem 1.1.

Proof of Theorem 1.1. Let g : R n → R be a bounded function such that g dµ = 0 and define for all t ≥ 0 the measure ν t (dx) = (1 + tg) µ(dx). If t is small enough, then ν t is a probability measure. Let π be a coupling of μ and ν t , and a > 0 be a parameter whose value will be fixed later on ; for all bounded differentiable Lipschitz function f :

R n → R, it holds 1 t f dν t -f dµ = 1 t f (y) -f (x) π(dxdy) = 1 t f (y) -P at f (x) π(dxdy) + 1 t P at f (x) -f (x) µ(dx) ≤ 1 at 2 cµ (x, y) π(dxdy) + 1 t P at f (x) -f (x) µ(dx),
where the last line comes from the inequality f (y) -P s f (x) ≤ 1 s cµ (x, y) for all s > 0. So optimizing over π ∈ C(μ, ν t ), we get, for all t small enough,

1 t f dν t -f dµ ≤ 1 at 2 T µ (μ, ν t ) + 1 t P at f (x) -f (x) µ(dx) ≤ 1 at 2 D(ν t µ) + 1 t P at f (x) -f (x) µ(dx),
where the last inequality comes from Theorem 4.6. A straightforward calculation shows that t -2 D(ν t µ) → 1 2 g 2 dµ when t goes to 0. Therefore, using Lemma 5.1, we get lim sup

t→0 1 t f dν t -f dµ ≤ 1 2a g 2 dµ + 8a n i=1 1 λ 2 i (S(x)) (∂ x i f ) 2 (x) μ(dx)
and optimizing over a > 0 lim sup

t→0 1 t f dν t -f dµ ≤ 4 g 2 dµ 1/2 n i=1 1 λ 2 i (S(x)) (∂ x i f ) 2 (x) μ(dx) 1/2
Now let us evaluate the left hand side. Consider the map R t defined by

R t (x) = x 1 -u 1 ν t 1 (du), x 2 -u 2 ν t 2 (du 2 |x 1 ), . . . , x n -u n ν t n (du n |x 1 , . . . , x n-1 ) .
For t = 0, R 0 = R is the map introduced in (4.4). Then ν t is the image of ν t by the map R t and μ the image of µ by the map R. Therefore,

1 t f dν t -f dµ = 1 t f (R t )(1 + tg) dµ -f (R) dµ → -∇f (R) • U dµ + f (R)g dµ, (5.2)
when t goes to 0, where U := lim t→0 1 t (R t -R). Let us calculate U . Writing the definition, it is not difficult to see that,

u i ν t i (du i |x 1 , . . . , x i-1 ) = a i + tb i c i + td i , with a i = u i e -V (x 1 ,...,x i-1 ,u i ,...,un) du i • • • du n , b i = u i g(x 1 , . . . , x i-1 , u i , . . . , u n )e -V (x 1 ,...,x i-1 ,u i ,...,un) du i • • • du n , c i = e -V (x 1 ,...,x i-1 ,u i ,...,un) du i • • • du n , d i = g(x 1 , . . . , x i-1 , u i , . . . , u n )e -V (x 1 ,...,x i-1 ,u i ,...,un) du i • • • du n .
Therefore,

U i (x) = lim t→0 1 t u i dν t i (du i |x 1 , . . . , x i-1 ) -u i dµ i (du i |x 1 , . . . , x i-1 ) = lim t→0 1 t a i + tb i c i + td i - a i c i = b i c i - a i c i d i c i = E[X i g(X)|X 1 = x 1 , . . . , X i-1 = x i-1 ] -E[X i |X 1 = x 1 , . . . , X i-1 = x i-1 ] • E[g(X)|X 1 = x 1 , . . . , X i-1 = x i-1 ].
It is easy to check that 1

t a i +tb i c i +td i -a i c i ≤ 2M 1-tM |a i |
c i for t sufficiently small, where M = sup |g|. This can be used to justify the limit in (5.2). Details are left to the reader.

According to what precedes,

U (X) = E i-1 [X i g(X)] -E i-1 [X i ]E i-1 [g(X)] = E i-1 [X i g(X)].
So putting everything together, we get

E[f (X)g(X)] ≤ 4E[g 2 (X)] 1/2 E n i=1 1 λ 2 i (S(X)) ∂ i f (X) 2 1/2 + n i=1 E E i-1 [X i g(X)]∂ i f (X) = 4 √ 3E[g 2 (X)] 1/2 E n i=1 E X 2 i |X 1 , . . . , X i-1 ∂ i f (X) 2 1/2 + n i=1 E E i-1 X i g(X) ∂ i f (X) ,
where the second line comes from the definition of the λ i 's and the identity

Var(X i |X 1 = S 1 (x), . . . , X i-1 = S i-1 (x)) = E[X 2 i |X 1 = S 1 (x), . . . , X i-1 = S i-1 (x)] = E X 2 i |X 1 = x1 , . . . , X i-1 = x i-1 ,
for all x = (x 1 , . . . , x n ) ∈ R n . Finally let us bound the last term. Using Cauchy-Schwarz, it holds

n i=1 E E i-1 [X i g(X)]∂ i f (X) = n i=1 E X i g(X)E i-1 [∂ i f (X)] ≤ E[g 2 (X)] 1/2 E   n i=1 X i E i-1 [∂ i f (X)] 2   1/2 . Now observe that if i ≤ j -1, then, since X i E i-1 [∂ i f (X)]E j-1 [∂ i f (X)] is a function of X 1 , . . . , X j-1 , it holds E X i E i-1 [∂ i f (X)] • X j E j-1 [∂ i f (X) = E X i E i-1 [∂ i f (X)]E j-1 [∂ i f (X)] • E j-1 [X j ] = 0, since E j-1 [X j ] = 0. Therefore, n i=1 E E i-1 [X i g(X)]∂ i f (X) ≤ E[g 2 (X)] 1/2 E n i=1 X 2 i E i-1 [∂ i f (X)] 2 1/2 ≤ E[g 2 (X)] 1/2 E n i=1 X 2 i E i-1 [∂ i f (X) 2 ] 1/2 = E[g 2 (X)] 1/2 E n i=1 E i-1 X 2 i ∂ i f (X) 2 1/2 . Using again that E i-1 X 2 i = E X 2 i |X 1 , . . . , X i-1 , we get E[f (X)g(X)] ≤ 4 √ 3 + 1 E[g 2 (X)] 1/2 E n i=1 E X 2 i |X 1 , . . . , X i-1 ∂ i f (X) 2 1/2
.

Taking g = f • R with f such that f dμ = 0, we obtain

E[f (X) 2 ] ≤ 4 √ 3 + 1 2 E n i=1 E X 2 i |X 1 , . . . , X i-1 ∂ i f (X) 2 .
The inequality is then extended by density to all locally Lipschitz functions f : R n → R such that f 2 dµ < ∞.

It remains to prove Lemma 5.1.

Proof of Lemma 5.1. Let f : R n → R be a differentiable bounded Lipschitz function and denote by M = 1 + sup |f |. For all x ∈ R n , we denote by • x the quantity defined by

u x = n i=1 λ 2 i (S(x))u 2 i , ∀u ∈ R n .
When x is such that λ i (x) < ∞ for all i, then • x is a norm on R n . With this notation cµ (x, y) = 1 16 N ( x -y x ). and

P t f (x) = sup y∈R n f (y) - 1 16t N ( y -x x ) .
First, note that, for all x ∈ R n , the supremum in the definition of P t f (x) is attained on the ball {y ∈ R n ; y -x x ≤ N -1 (48M t)}. Namely, if y is outside the ball, it holds

f (y) -f (x) - 1 16t N ( y -x x ) ≤ -M < 0.
Since P t f (x) ≥ f (x), we conclude that the supremum is reached inside the ball. Now let us bound from above the derivative of P t f with resp ect to the t variable. Let x ∈ R n be such that λ i (x) < ∞ for all i. Using the preceding remark and the inequality uv

≤ u 2 2 + v 2 2 , we see that 1 t (P t f (x) -f (x)) = sup y-x x≤N -1 (48M t) f (y) -f (x) t - 1 16t 2 N ( y -x x ) ≤ 4 sup y-x x ≤N -1 (48M t) (f (y) -f (x)) 2 N ( y -x x ) (5.3) = 4 sup u x ≤N -1 (48M t)    (∇f (x) • u) 2 + o( u 2 x ) u 2 x 2 + o( u 2 x )    ≤ 8 sup u x ≤N -1 (48M t)    n i=1 1 λ 2 i (S(x)) (∂ i f )(x) 2 + o(1) 1 + o(1)    → 8 n i=1 1 λ 2 i (S(x)) (∂ i f )(x) 2 ,
when t goes to 0, where the last inequality follows from Cauchy-Schwarz. So we conclude that if x is such that λ i (x) < ∞ for all i, then lim sup

t→0 1 t (P t f (x) -f (x)) ≤ 8 n i=1 1 λ 2 i (S(x)) (∂ i f )(x) 2 .
If x is such that λ i (x) = 0 for some x, then P t f (x) = f (x) and so the inequality above is still true. Moreover, denoting by λ * (x) = min{λ i (S(x))} > 0, it follows from (5.3) and from the inequality

u x ≥ λ * (x)|u|, u ∈ R n , that for all t ≤ 1/(48M ) (5.4) 1 t (P t f (x) -f (x)) ≤ 4 sup λ * (x)|y-x|≤N -1 (1) (f (y) -f (x)) 2 N (λ * (x)|y -x|) ≤ a L 2 λ * (x) 2 ,
where L the Lipschitz constant of f and a = 4 sup 0<v≤N -1 (1)

v 2 N (v) is some universal con- stant.
Now, let ν be a probability measure on R n such that 1 λ 2 i (S(x)) ν(dx) < +∞ for all i. Then 1/λ * is also square integrable with respect to ν. Therefore, thanks to (5.4), one can apply Fatou's lemma in its lim sup form:

lim sup t→0 1 t (P t f -f ) dν ≤ lim sup t→0 1 t (P t f -f ) dν ≤ 8 n i=1 1 λ 2 i (S(x)) (∂ i f )(x) 2 dν.

Variance estimates

Here we prove Corollary 1.6, identity (1.10) and Corollary 1.11.

Proof of Corollary 1.6. According to Theorem 1.1 and standard properties of conditional expectations, it holds

Var(|X| 2 ) ≤ 4a n i=1 E E[X 2 i |X 1 , . . . , X i-1 ]X 2 i = 4a n i=1 E E[X 2 i |X 1 , . . . , X i-1 ] 2 ≤ 4a n i=1 E E[X 4 i |X 1 , . . . , X i-1 ] = 4a n i=1 E X i 4 .
Observe that E[X

i ] ≤ 8E[X 4 i ] + 8E E i-1 [X i ] 4 ≤ 16E[X 4 i ]. 4
We conclude using Borell's reverse Hölder inequality [START_REF] Borell | Convex measures on locally convex spaces[END_REF]: there exists some universal constant a ′ such that for any log

-concave random variable Y , it holds E[Y 4 ] ≤ a ′ E[Y 2 ] 2 . So, Var(|X| 2 ) ≤ 64aa ′ n.
Remark 6.1. Our main result Theorem 1.1 is closely related to a result by . Namely, it follows from [START_REF] Barthe | Invariances in variance estimates[END_REF]Theorem 4] that if X is a random vector following a law µ(dx) = e -V (x) dx on R n with Hess V ≥ ρId for some ρ ≥ 0, then for all smooth function f : R n → R such that

(6.2) E[∂ i f (X)|X 1 . . . , X i-1 , X i+1 , . . . , X n ] = 0, ∀i ∈ {1, . . . , n} it holds Var µ (f (X)) ≤ n i=1 E 1 ρ + 1/C i (X) ∂ i f 2 (X) ,
where, for all

x = (x 1 , . . . , x n ) ∈ R n , C i (x) denotes the Poincaré constant of the condi- tional distribution of X i knowing X 1 = x 1 , . . . , X i-1 = x i-1 , X i+1 = x i+1 , . . . , X n = x n .
Note that the conclusion of [START_REF] Barthe | Invariances in variance estimates[END_REF]Theorem 4] is more general than what we state above. In the general formulation, to any decomposition of the identity Id = m i=1 c i P E i where c i > 0 and P E i is the orthogonal projection on a subspace E i corresponds a weighted Poincaré inequality involving the Poincaré constants of the conditional distributions of X knowing P F i (X), with F i = E ⊥ i . It is well known (see for instance Theorem 4.2 below) that Poincaré constants of one dimensional log-concave probability measures can be estimated by the variance. In particular, it holds

C i (x) ≤ 3Var(X i |X 1 = x 1 , . . . , X i-1 = x i-1 , X i+1 = x i+1 , . . . , X n = x n ),
∀i ∈ {1, . . . , n}.

Therefore, taking ρ = 0, it holds (6.3) Var µ (f (X)) ≤ 3 n i=1 E Var(X i |X 1 . . . , X i-1 , X i+1 , . . . , X n ]∂ i f 2 (X) , for all smooth f enjoying 6.2. The difference between this result and Theorem 1.1 (besides the fact that we condition only with respect to the first variables) is that our result is true for all f but for X instead of X.

Let us see how to recover the conclusion of Corollary 1.6 from (6.3). Let us assume that X is such that E[X i |X 1 . . . , X i-1 , X i+1 , . . . , X n ] = 0, ∀i ∈ {1, . . . , n}.

This condition (which is actually a bit stronger than the condition X = X) exactly amounts to require that the function f (x) = |x| 2 satisfies (6.2). So applying (6.3) to this function and reasoning as in the proof of Corollary 1.6 we thus get from (6.3) that Var(|X| 2 ) ≤ an for some universal constant a.

Proof of Corollary 1.11. Let us start with identity (1.10). For all i ∈ {1, . . . , n}, it holds

E X i E i-1 [X i ] = E (X i -E i-1 [X i ]) E i-1 [X i ] = E E i-1 [X i -E i-1 [X i ]] E i-1 [X i ] = 0.
As a result X and X ′ are orthogonal in L 2 (Ω, A, P; R n ). Therefore, it holds

E[|X| 2 ] = E[|X| 2 ] + E[|X ′ | 2 ] and E[|X| 2 ] 2 = E[|X| 2 ] 2 + 2E[|X| 2 ]E[|X ′ | 2 ] + E[|X ′ | 2 ] 2 .
Since

E[|X| 4 ] = E[|X| 4 ] + 2E[|X| 2 ]E[|X ′ | 2 ] + E[|X ′ | 4 ] + 4E[(X • X ′ ) 2 ] + 4E[|X| 2 X • X ′ ] + 4E[|X ′ | 2 X • X ′ ],
we get that

Var(|X| 2 ) = Var(|X| 2 )+Var(|X ′ | 2 )+2Cov(|X| 2 , |X ′ | 2 )+4E[(X•X ′ ) 2 ]+4E[|X| 2 X•X ′ ]+4E[|X ′ | 2 X•X ′ ]
Using Cauchy-Schwarz, and the orthogonality of X and X ′ we get

Cov |X| 2 , |X ′ | 2 ≤ Var |X| 2 Var (|X ′ | 2 ) E |X| 2 X • X ′ = E |X| 2 -E[|X| 2 X • X ′ ≤ Var(|X| 2 ) E[(X • X ′ ) 2 ] E |X ′ | 2 X • X ′ = E |X ′ | 2 -E[|X ′ | 2 X • X ′ ≤ Var(|X ′ | 2 ) E[(X • X ′ ) 2 ].
Moreover, note that if i < j the random variable X i E i-1 [X i ]E j-1 [X j ] is measurable with respect to the σ field generated by X 1 , . . . , X j-1 . Therefore Therefore, expanding the square, we see that the number √ V ′ := Var(|X ′ | 2 ) is less than or equal the positive root of the equation

E X i E i-1 [X i ]X j E j-1 [X j ] = E X i E i-1 [X i ]E j-1 [X j ]E j-1 [X j ] = 0 So, it holds E[(X • X ′ ) 2 ] = n i=1 E X 2 i E i-1 [X i ] 2 + 2 i<j E X i E i-1 [X i ]X j E j-1 [X j ] = n i=1 E X 2 i E i-1 [X i ] 2 = n i=1 E E i-1 [X 2 i ]E i-1 [X i ] 2 -E i-1 [X i ] 4 ≤ n i=1 E[X 4 i ] ≤ a ′ n
x 2 -2x V + 2 √ a ′ n + V -4 √ a ′ n V -V = 0,
with V = Var(|X| 2 ) and V = Var(|X| 2 ). An easy calculation thus gives

√ V ′ ≤ V + 2 √ a ′ n + √ 4a ′ n + V ,
which together with Corollary 1.6 easily gives the desired inequality.

  and where a : [α, β] → R is a concave function and b : [α, β] → R is a convex function. Recall that the Steiner symmetrization of C with respect to D is the set C defined by

  ,where a ′ is some universal constant such that E[Y 4 ] ≤ a ′ E[Y 2 ]2 for all log-concave random variable Y. We conclude from the inequalities above thatVar(|X| 2 ) ≤ Var(|X| 2 ) + Var(|X ′ | 2 ) Var(|X| 2 ) + Var(|X ′ | 2 )

	=	Var(|X| 2 ) + Var(|X ′ | 2 ) + 2 √	a ′ n	2	≤	Var(|X ′ | 2 ) + (2 √	a ′ + b) √	n

2 + 4a ′ n + 4 √ a ′ n 2 ≤ 2Var(|X ′ | 2 ) + 2(2 √ a ′ + b) 2 n

where in the last inequalities b is the universal constant given by Corollary 1.6. Similarly,

Var(|X| 2 ) ≥ Var(|X ′ | 2 ) -Var(|X| 2 ) 2 -4 √ a ′ n Var(|X ′ | 2 ) + Var(|X| 2 ) .

This work was partially supported by the grants ANR 2011


Clément Zinoune

Philippe Bonnifait

Javier Ibañez-Guzmán

Integrity Monitoring of Navigation Systems using Repetitive Journeys

Currently, Advanced Driving Assistance Systems (ADAS) increasingly rely on information stored in vehicle on board digital maps. The vehicle position is projected onto the map to establish oncoming road context. However, errors might exist in the road geometry stored in the maps. The integrity of this map-matched estimate must be monitored in real-time to avoid errors that can lead to hazardous situations. This paper presents a monitoring system and fault detection, isolation and adaptation formalism which benefits of multiple vehicle journeys (e.g. commuting). We demonstrate that it is possible to assert correct navigation information within the first journey to a new area and to isolate areas where the road geometry is erroneous after the second journey. The approach takes into account errors that might occur on the estimation of the global vehicle position. The proposed formalism was experimentally validated using a passenger vehicle driven in different map and GNSS conditions.

I. INTRODUCTION

Digital maps are becoming an integral part of Advanced Driving Assistance Systems (ADAS) and Autonomous Driving as they provide contextual information to facilitate decision-making functions. However, experience has found that different types of errors exist in such maps. It can result in discomfort and hazardous situations which makes their use unsafe for critical applications.

A formalism applicable to the monitoring of errors in the description of road geometry embedded in digital maps is proposed. Errors on information provided by the vehicle navigation system may be due to several reasons: errors on the estimations of the localisation system, errors in the map itself or errors when projecting the location estimation onto the digital map (map-matching) [START_REF] Quddus | Current mapmatching algorithms for transport applications: State-of-the art and future research directions[END_REF]. For the purpose of this study, the monitoring of these errors is crucial for the Navigation System Integrity. The projected vehicle position on the navigation map is used as a source of information for ADAS and automated driving.

In this paper, it is assumed that the loss of integrity originates from road geometry error. Thus, our approach is concerned on identifying whether or not a road geometry error exists taking into account the likely existence of localisation errors. The first idea for reducing the ambiguity on the source of error is based on the use of repetitive paths of the subject vehicle. A second idea is to get an independent estimate of the vehicle position by combining an additional GNSS (Global Navigation Satellites System) receiver with vehicle proprioceptive data (speed and yaw rate) [START_REF] Zinoune | A sequential test for autonomous localisation of map errors for driving assistance systems[END_REF]. The latter is used to provide information with regard to the road to compensate lateral deviation. The principles are shown in Fig. 1.

The problem is addressed as a Fault Detection, Isolation and Adaptation (FDIA) task. Since both available estimates may be affected by faults, redundancy is provided by the use of position estimates of previous vehicle journey on the same road. We show how the knowledge of effects of faults on estimates makes isolation possible.

The contribution focuses on the mathematical framework necessary to reduce the ambiguity coming from the lack of redundancy and thus isolate faults. The paper is organised as follows. An overview of the theoretical tools used for FDIA together with the problem formulation is presented in Section II. It includes notations, and the underlying assumptions. The manner in which faulty estimates are inferred based on their pairwise comparison is presented in Section III. The conditions on these estimates that make fault isolation possible are included. Section IV presents and demonstrates the properties of the proposed formalism that make relevant its application to Intelligent Vehicles. Implementation of the formalism is detailed through two examples and its interest for monitoring integrity of navigation system for ADAS is given in Section V. This includes the results from experiments that apply the formalism to navigation functions using data acquired in real conditions. Section VI eventually concludes the paper.

II. PROBLEM FORMULATION

A. Background

Low level position integrity is performed in [START_REF] Jaulin | Guaranteed nonlinear estimation using constraint propagation on sets[END_REF] with interval based methods. This kind of approach requires the 1, all but FDIA function are black boxes. Access to internal variables is not permitted. For the same reason, approaches centred on direct access to the digital map data like [START_REF] Betaille | Creating enhanced maps for lanelevel vehicle navigation[END_REF] are not possible.

In [START_REF] Agamennoni | Robust and accurate road map inference[END_REF], the authors propose a method to infer the digital map based on a large set of GNSS traces provided by probe vehicles. A server uses these data to update the digital map which is transmitted to the vehicles. However, in the proposed approach, it is assumed that there is no support of the infrastructure.

A high level Fault Detection and Isolation (FDI) approach is chosen, as presented in [START_REF] Reiter | A theory of diagnosis from first principles[END_REF] and [START_REF] Kleer | Diagnosing multiple faults[END_REF]. Both of these works employ a different lexicon but the introduced theoretical frameworks are similar. The knowledge about the system is represented by a basic model. Fault is detected when there is a conflict between the observations and the system. Isolation is performed if there is a unique explanation of the conflict.

In the proposed approach, integrity of the map matched estimate is spatially evaluated. Each location of the road network is considered as an operating point of the system to monitor. For a given location of the vehicle, the presence of fault is investigated using all the estimates recorded at this location during the previous vehicle journeys.

B. Problem statement

Let K ∈ N denote the total number of times the vehicle went on a given road. As shown in Fig. 2, the vehicle curvilinear abscissa on a given road along the carriageway with respect to its origin is written s ∈ R + . The true vehicle position at abscissa s of a given road and at the k th journey is written P s k . This can be encoded as a vector that contains the vehicle's geographic coordinates (longitude, latitude and ellipsoidal height).

Using the same notation convention, G s k and N s k are estimates of the vehicle position P s k by the vehicle position and the map matched estimate respectively. Every time the vehicle is at abscissa s of a given road and for the k th time, these two estimates are recorded. Fig. 2 shows some estimates recorded during three journeys in a difficult GNSS area.

Faults may affect the navigation system as well as the vehicle state estimator and cause their outputs to be significantly different from the ground truth (if a multipath affects a GNSS receiver for example). In this case, the estimates are said faulty. The notation ≁ stands for two quantities which are significantly different. Let us define the faults as:

f N s k def = 1 if N s k ≁ P s k and f G s k def = 1 if G s k ≁ P s k
In the following, the development of the method is done using classical equalities but the reader should bear in mind that equalities are true up to a threshold that is discussed in Section V:

f N s k def = 1 if N s k = P s k 0 otherwise (1)
f G s k def = 1 if G s k = P s k 0 otherwise (2)
Four realistic assumptions can be made on the errors that affect the estimates from one journey to the other:

• The perturbations cause random errors on vehicle pose estimate. Two faulty G estimates are then different from each other.

• Navigation faults are due to error on digital map which causes systematic errors on navigation estimates. Two faulty N are then equal.

• The method is spatially sampled with respect to the curvilinear abscissa s such that the true vehicle position at s is assumed to be discretised. FDIA at abscissa s+1 is performed independently of the FDIA at abscissa s.

• When travelling several times on a road, the vehicle follows the same path with small deviation. If the vehicle is equipped with a smart front camera, the lateral deviation can be automatically compensated. According to the latter two assumptions, one can state that:

P s i = P s i+1 , ∀i ∈ {1, . . . , K -1} (3)

III. FAULT DETECTION AND ISOLATION METHOD

This section introduces the concepts of sets of faulty states and residuals that are fundamental for the proposed formalism. After having demonstrated how these concepts are linked, the fault detection and isolation method is presented.

A. Set of faulty states

Let e be the set of faulty states of all estimates available at a given abscissa s. It is composed of all f G s k and f N s k for the considered iterations K:

e def = f G s i , f N s j , ∀i, j ∈ {1, . . . , K} (4)
The cardinality of e is 2K. Each term of e is a boolean value so there are 2 2K possible sets written e n :

e n ∈ B 2K , ∀n = 1, . . . , 2 2K (5)
Let us take an example with K = 2. There are

2 2•2 = 16 different sets. The cardinality of each one is 2 • 2 = 4. For instance, e 5 = 0 0 1 0 means f G s 2 = 0 and f N s 2 = 0 and f G s 1 = 1 and f N s 1 = 0 .

B. Residuals processing

At a given abscissa s, every available estimate at the current iteration is compared to all the others and the result is stored in a residual vector R. R is therefore made of C (2K, 2) boolean elements. C (2K, 2) = K (2K -1) stands for the number of 2-combinations from a given set of 2K elements.

The components of R are defined as:

r G s i G s j def = 1 if G s i = G s j 0 otherwise ∀i, j ∈ {1, . . . , K} , i > j (6) r G s i N s j def = 1 if G s i = N s j 0 otherwise ∀i, j ∈ {1, . . . , K} (7)
r N s i N s j def = 1 if N s i = N s j 0 otherwise ∀i, j ∈ {1, . . . , K} , i > j (8)
For example, if at the second iteration, the estimates are such as

G s 1 = N s 1 = G s 2 = N s 2 then the residual vector contains 2 • (2 • 2 -1) = 6 elements: R = r N s 2 G s 2 r G s 2 G s 1 r N s 1 G s 2 r N s 2 G s 1 r N s 1 N s 2 r N s 1 G s 1 (9) Here R = 0 1 0 1 0 1 .

C. Relationships between faults and residuals

Let us show that the residual terms are actually the result of boolean operations between the faulty states of the estimates.

Proposition 1: Let ∨ and ⊕ denote boolean or and exclusive or operators respectively, 10) is demonstrated first. Let i and j be such as i, j ∈ {1, . . . , K} and i > j. First of all, one can consider the case in which no fault affects the estimates. According to (2):

r G s i G s j = f G s i ∨ f G s j , ∀i, j ∈ {1, . . . , K} , i > j (10) r G s i N s j = f G s i ∨ f N s j , ∀i, j ∈ {1, . . . , K} (11)
r N s i N s j = f N s i ⊕ f N s j , ∀i, j ∈ {1, . . . , K} , i > j (12) Proof: Equation (
f G s i = 0 ⇐⇒ G s i = P s (13) f G s j = 0 ⇐⇒ G s j = P s (14)
Then:

G s i = G s j (15)
Secondly, the same reasoning is used if a fault affects one of the estimates:

f G s i = 1 and f G s j = 0 f G s i = 1 and f G s j = 0 ⇐⇒ G s i = G s j (16)
Finally, if both estimates are faulty:

f G s i = 1 ⇐⇒ G s i = P s (17) and f G s j = 1 ⇐⇒ G s j = P s (18)
Due to the randomness of G errors, vehicle pose faults cannot compensate each other. Then:

G s i = G s j ( 19
)
One can conclude that if there is at least one fault on

G s i or G s j , then r G s i G s
j is equal to one. This proves (10). Secondly, the same reasoning scheme is used to demonstrate (11).

A similar deduction is finally applied to justify (12). Equation ( 1) allows stating that:

f N s i = 0 and f N s j = 0 ⇐⇒ N s i = N s j (20)
Similarly, if a fault affects a navigation estimates:

f N s i = 1 and f N s j = 0 ⇐⇒ N s i = N s j (21)
Contrarily to the previous cases and due to the assumption made in Section II, two faulty N are equal, then:

f N s i = 1 and f N s j = 1 ⇐⇒ N s i = N s j (22)
Finally, r N s i N s j is equal to one if there is only one fault among f N s i and f N s j . This proves (12). Equations (10), (11) and (12) of Proposition 1 establish a link between available estimates (i.e. G and N ) and the faults which affected them (i.e. f G and f M ). The first two equations tell that if there is at least one fault on the considered estimates then the residual will be affected. In (12), the residual equals one if there is a unique fault among both estimates.

D. Fault detection and isolation

The fault detection and isolation strategy relies on listing all the possible sets of faulty states for a given K and calculating residual vectors with (10), (11) and (12). This forms the truth table for K. On the other hand, available estimates are used to compute the observed residual vector based on (6), [START_REF] Kleer | Diagnosing multiple faults[END_REF] and [START_REF] Monteil | Demonstration of the rules non isolability of sets of faulty states[END_REF]. This vector, found in the truth table, allows determining the corresponding set of faulty states. Faults affecting each estimate can be finally deduced from this set.

By definition, the truth table is exhaustive; the observed residual vector does make part of it. However, some sets of faults induce the same residual vector. In this case, isolation is not possible. These are called adverse sets. At least one new system iteration is then required to perform isolation.

f G 2 f N 2 f G 1 f N 1 r N 2 G 2 r G 2 G 1 r N 1 G 2 r N 2 G 1 r N 1 N 2 r N 1 G 1 e 1 0 0 0 0 0 0 0 0 0 0 e 2 1 0 0 0 1 1 1 0 0 0 e 3 0 1 0 0 1 0 0 1 1 0 e 4 1 1 0 0 1 1 1 1 1 0 e 5 0 0 1 0 0 1 0 1 0 1 e 6 1 0 1 0 1 1 1 1 0 1 e 7 0 1 1 0 1 1 0 1 1 1 e 8 1 1 1 0 1 1 1 1 1 1 e 9 0 0 0 1 0 0 1 0 0 1 e 10 1 0 0 1 1 1 1 0 0 1 e 11 0 1 0 1 1 0 1 1 0 1 e 12 1 1 0 1 1 1 1 1 0 1 e 13 0 0 1 1 0 1 1 1 1 1 e 14 1 0 1 1 1 1 1 1 1 1 e 15 0 1 1 1 1 1 1 1 0 1 e 16 1 1 1 1 1 1 1 1 0 1
Being adverse depends on the number of faults affecting the estimates as stated in the following proposition.

Proposition 2: A set of faulty states is adverse if and only if it complies with one of the following rules:

1) f Ni = 1 , ∀i ∈ {1, . . . , K} and ∃!j ∈ {1, . . . , K} such as

f Gj = 0 2) f Gi = 1 , ∀i ∈ {1, . . . , K}
In other words it is not possible to isolate faults if:

1) Every estimate N is faulty and there is a unique true G. 2) Every G is faulty.

Proof: Proposition 2 is demonstrated in [START_REF] Monteil | Demonstration of the rules non isolability of sets of faulty states[END_REF]. For example, let the observed residual vector calculated in Section III-B be R = 0 1 0 1 0 1 . This residual is found only once in the truth table for two iterations (in bold in Table I) and is caused by the set of faulty states e 5 = 0 0 1 0

meaning f G s 2 = 0 and f N s 2 = 0 and f G s 1 = 1 and f N s 1 = 0 .

IV. NOTEWORTHY METHOD PROPERTIES

Properties are deduced from the proposed FDI formalism. They are detailed as follows.

A. Guaranteed fault detection

The formalism always detects the presence of faulty estimates. In other words, as soon as there is a faulty estimate, the formalism detects it (but may not be able to isolate the faulty estimates). This is demonstrated by showing that the formalism is always able to isolate the only set that contains no fault. The Proposition 2 indeed shows that this set (for which f Ni = f Gj = 0 , ∀i, j ∈ {1, . . . , K}) is not adverse.

B. Isolation convergence

The ratio between the number of adverse sets of faulty states and the total number of sets goes to zero when the number of iterations increases. This means that increasing K improves fault isolation capabilities.

To justify this, let A(K) stand for the number of adverse sets of faulty states for K iterations. A (K) is the sum of the number of sets induced by the two rules of Proposition 2:

A(K) = C (K, 1) + K j=0 C (K, j) (23)
The binomial formula applied for coefficients 1 and 1 gives:

A(K) = K + 2 K (24)
The ratio q(K) between A(K) and the total number of possible sets is:

q(K) = K + 2 K 2 2K (25)
The limit of q(K) as K goes to infinity is 0.

C. Conservation of isolability

Once fault isolation is performed, fault isolation will be performed at any new iteration.

Let I k be the set of isolable e at K = k. Reciprocally, let I c k be the complement of I k , i.e. the set of adverse sets of faulty states. To prove the property, let us demonstrate its contrapositive:

e ∈ I c k+1 =⇒ e ∈ I c k (26)
Let e be an adverse at iteration k + 1 such as e ∈ I c k+1 . Then e complies with one of the rules stated in proposition 2. On the one hand, if e is adverse due to rule 1, this may be caused by two reasons:

=⇒ f Gi = 1 , ∀i ∈ {1, . . . , k} (29)
=⇒ e ∈ I c k (30)
because of the rule 2. This shows that, if e is adverse due to rule 1, then (26) is satisfied.

On the other hand, let us consider the case for which e is adverse due to rule 2.

f Gi = 1 , ∀i ∈ {1, . . . , k + 1} (31) =⇒ f Gi = 1 , ∀i ∈ {1, . . . , k} (32)
=⇒ e ∈ I c k (33)
This finally fulfils the demonstration of ( 26) and proves the conservation of isolability property.

D. Adaptation

If fault detection and isolation are performed, then adaptation is possible. Adaptation is understood here as a mechanism that consists in determining a fault-free estimate once detection and isolation have been performed.

One must notice that the only set in which every estimate is faulty is adverse (the faults are not isolable because of the second rule of Proposition 2). The consequence is that every isolable fault configuration contains at least one true estimate. As isolation is performed, the true estimate is perfectly identified within the set. Then this true estimate can replace the faulty one.

E. Conservation of Adaptation

If fault isolation is achieved at the K th iteration, whatever happens at the (K + 1)

th iteration in terms of faults, the proposed formalism allows performing adaptation. Indeed, the conservation of isolability property states that, if fault isolation is performed at iteration K, isolation will also be performed at K + 1. Moreover, the adaptation property shows that adaptation is always possible as soon as faults are isolated. Then, if faults are isolated at iteration K, adaptation will be possible at K + 1. Fig. 3: A faulty map area. Circular grey marks are for estimate for which the method has detected but not isolated a fault. Green squares are for true estimates and red triangles are the faulty estimates.

i i i i i i i i i i i i i i i i
i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i

V. APPLICATION TO NAVIGATION INTEGRITY

MONITORING

The formalism presented in Section III is applied to the integrity monitoring of the navigation vehicle position estimate. The experimental framework is presented before detailing each step of the FDIA through two simple examples. A more complete set of results finally demonstrates the consistency of the proposed formalism for intelligent vehicles.

All data employed in the following were acquired from a standard GPS receiver on board a passenger vehicle. The map matching function uses an editable digital map. The stored road geometries were locally changed to represent errors found in such digital maps. All algorithms were coded in the C++ language.

The vehicle travels several times to the same new destination using the same roads. The proposed method is run at 5 metres intervals which provides a good spatial sampling of the road geometry. For implementation purposes, estimates separated by a distance lower than 1.5 metres are considered as equal. Map road geometric errors less than this threshold are not significant for most ADAS applications.

For all the experiments, two estimates are used for the analysis: the estimation of the vehicle position G and the projection of the vehicle position onto the road network of the digital map provided by the navigation function N (as defined in Fig. 1). During the experiments, the subject vehicle travels several times over the same roads, thus several estimates of the observed variables are available.

A. Illustrative examples

In the first example, the map contains an error and we show how the method performs fault detection, isolation and adaptation. Each step of the proposed method is detailed and the properties introduced in Section IV are illustrated. In the second example, the map geometry is perfect but a large building may cause errors on vehicle state estimation. This shows how the method rejects faulty estimates.

f G 1 f N 1 r G 1 N 1 = f G 1 ∨ f N 1 e 1 0 0 0 e 2 1 0 1 e 3 0 1 1 e 4 1 1 1
1) Map road geometric error: In this example (Fig. 3), the true road goes straight whilst there is an error in the representation of the road in the map. The first vehicle journey is shown in Fig. 3a. Let us detail the proposed formalism at abscissa 25m of the first journey.

The first time the vehicle is at abscissa s = 25, position estimates are provided by the vehicle state (G 25 1 ) and by the navigation (N 25 1 ) functions (see Fig. 3a). One can compute the observed residual [START_REF] Kleer | Diagnosing multiple faults[END_REF]:

G 25 1 = N 25 1 =⇒ r G 25 1 N 25 1 = 1
This residual is found three times in the truth table for one journey FDI (Table II): the sets of faulty states e 2 , e 3 and e 4 give r G1N1 = 1. The proposed method then detects a faulty estimate among G 25 1 and N 25 1 but is not able to isolate it. The integrity monitoring system cannot state on the faultiness of N 25 1 , it sends unknown to ADAS which is represented by circular grey marks on Fig. 3a.

The second time the vehicle crosses abscissa s = 25 of the same road (Fig. 3b), a new pair of position estimates becomes available: G 25 2 and N 25 2 . The residual vector dimension increases to 6. The elements are calculated using ( 6), [START_REF] Kleer | Diagnosing multiple faults[END_REF] and [START_REF] Monteil | Demonstration of the rules non isolability of sets of faulty states[END_REF]: Fig. 4: A correct map area close to a large building. Red triangles (resp. green squares) are for estimates which the method has isolated as faulty (resp. true).

N 25 2 = G 25 2 =⇒ r N 25 2 G 25 2 = 1 G 25 2 = G 25 1 =⇒ r G 25 2 G 25 1 = 0 N 25 1 = G 25 2 =⇒ r N 25 1 G 25 2 = 1 N 25 2 = G 25 1 =⇒ r N 25 2 G 25 1 = 1 N 25 1 = N 25 2 =⇒ r N 25 1 N 25 2 = 0 G 25 1 = N 25 1 =⇒ r G 25 1 N 25 1 = 1 Then R = 1 0 1 1 0 1 .
client systems. Due to the property detailed in Section IV-E, the integrity monitoring system will be able to provide an error-free pose estimate for all future vehicle journeys on this road.

2) Isolation of GNSS faults: In the second example, shown in Fig. 4, there is no error in the map but the road is close to a large building that may cause GNSS faults. No fault affects vehicle position estimate the first time the vehicle goes in this area (Fig. 4a). The method states with certainty that there is no fault for the seven evaluation points of this example. According to the properties detailed in Sections IV-C to IV-E, one knows that fault isolation and adaptation will be performed at any future journey on this road whatever the faults encountered. Fig. 4b illustrates this with the second journey which is perturbed by the buildings. The method directly isolates the faulty vehicle position estimates and keeps confidence in current and past navigation estimates.

Two opposite cases have been shown by these examples: one erroneous map area in good GNSS conditions and one correct map area in poor GNSS conditions. The method works properly even if faults affect both estimates. In such a situation, fault detection is directly made but isolation and adaptation require more journeys.

B. Experiments

A larger dataset is employed to validate the proposed formalism. A digital map that contains geometrical errors feeds the navigation function. The rural area of this digital map was designed to represent the road network as it was before large road works a few years ago. In the urban area, the map errors were generated to show that, even in poor GNSS conditions, fault detection and isolation of navigation position estimates is possible. As the method generates a large amount of data, Fig. 5 and6 are synthetic views of the results.

In the rural environment, no fault affects the GNSS receiver. Faulty areas are detected at the first journey. As shown by Fig. 5, the second journey allows the method incriminating the navigation in these areas.

In the urban environment, due to the fault affecting some GNSS estimates, three journeys are required to fully perform isolation on all the roads (Fig. 6). Fig. 5: Results of FDI in rural environment after two journeys (blue and purple lines). In green, navigation estimates that have been identified as correct (during the first journey), in red faulty ones (during the second one).

As stated in Section V, the method is spatially triggered with a distance of 5 metres in these experiments. The GNSS receivers (the one of the navigation and the one of the monitoring system shown in Fig. 1) used in these experiments are timely synchronised and their maximum frequency is 10 Hz. The navigation fixes are rejected unless they correspond to a sampled abscissa with a 1 m tolerance. Due to the velocity of the vehicle in real traffic conditions, especially in the rural environment, some sampled abscissa can't be evaluated. We define the availability rate of method as the ratio between the total number of sampling points of the vehicle path and the actual number of times the method is run. In these experiments, the availability rate is 80% on the urban scenario and 75% on the rural scenario. Availability would be improved either by using a navigation GNSS receiver triggered on vehicle odometer or by interpolating points at required abscissa.

During the first two journeys, the method states unknown every time the situation is theoretically not isolable (according to the rules stated in Proposition 2). We have noticed for these experiments that three journeys are sufficient to isolate every faulty estimate. In the rural scenario, there are 3% of wrong isolations (i.e. some map errors samples have been declared correct and some correct vehicle estimates have been isolated as faulty).

VI. CONCLUSION This paper described a fault detection, isolation and adaptation formalism based on the use of multiple iteration of a system. Residuals were defined and the FDI strategy was detailed. A set of properties deduced from this formalism showed its interest for the intelligent vehicles application. Integrity of the navigation position estimate was monitored taking benefit of the repetitive vehicle journeys. This application allowed detailing first every step of the method through two simple examples. The method performance was then evaluated based on real data in rural and urban environments and it shows a good relevance for the monitoring of the navigation system since it detects misleading information at the first journey and performs isolation from the second journey on the same road.

Current work aims at triggering sensors on spatial vehicle position to improve the availability rate. Moreover, a combination of the method presented here and the spatial approach of [START_REF] Zinoune | A sequential test for autonomous localisation of map errors for driving assistance systems[END_REF] is being developed.

Fig. 1 :

 1 Fig. 1: Structure of fault detection isolation and adaptation in standard passenger vehicle.

Fig. 2 :

 2 Fig. 2: Vehicle pose estimates for three vehicle journeys on the same road close to large buildings

rule 1 .

 1 1.a. f Ni = 1 , ∀i ∈ {1, . . . , k + 1} and ∃!j ∈ {1, . . . , k + 1} such as f Gj = 0 and j = k + 1, or 1.b. f Ni = 1 , ∀i ∈ {1, . . . , k + 1} and ∃!j ∈ {1, . . . , k + 1} such as f Gj = 0 and j = k + 1.If e complies to rule 1.a, then:=⇒ f Ni = 1 , ∀i ∈ {1, . . . , k} ,and ∃!j ∈ {1, . . . , k} |f Gj = 0 If e complies to rule 1.b, then:

  First journey (blue line)

  Second journey (purple line)

Fig. 6 :

 6 Fig. 6: Results of FDI in a urban environment after three journeys. The red links correspond to map errors that have been fully isolated during the third journey.

TABLE I :

 I Truth table of fault detection and isolation for K = 2

	Sets of faults	Residuals

TABLE II :

 II Truth table for K = 1

	Sets of faulty states	Residuals

  Table I is the truth table for two journeys. According to the first journey observation, one knows that f G 25 1 and f N 25

																	i	i
					ii	ii	ii	ii	ii	ii	ii		ii i i	ii i i	ii i i	ii i i	ii i	ii i	ii i i
					5	10	15	20	25	30	35	S	5	10	15	20	25	30	35	S
							Building						Building
						(a) First journey (blue line)		(b) Second journey (purple line)
					1
	are not both null. The first four rows of Table I could be
	ignored. The observed residual is finally found only once
	in this table (caused by the set of faulty states e 11 ), one
	can then conclude that f G 25 2	= 0, f N 25 2	= 1, f G 25 1	= 0 and
	f N 25 1	= 1.
	The integrity monitoring system returns the instruction
	don't use the navigation position estimate (N 25 2 ). Estimates
	found faulty (resp. true) by the method are represented
	by red triangles (resp. green squares) on Fig. 3b. Since
	error-free estimates have been identified (G 25 2 and G 25 1 ),
	adaptation is possible by providing either G 25 2 or G 25 1 to

Heudiasyc UMR CNRS 7253, Université de Technologie de Compiègne, France. philippe.bonnifait@hds.utc.fr, czinoune@hds.utc.fr

Renault S.A.S, France. javier.ibanez-guzman@renault.com