Benjamin Jourdain

Florent Malrieu

PROPAGATION OF CHAOS AND POINCAR É INEQUALITIES FOR A SYSTEM OF PARTICLES INTERACTING THROUGH THEIR CDF

Keywords: AMS 2000 subject classifications. 65C35, 60K35, 60E15, 35K15, 46N30 Viscous scalar conservation law, nonlinear process, particle system, propagation of chaos, Poincaré inequality, long time behavior

 without any convexity assumption on the flux function. Then Poincaré inequalities are used to get explicit estimates concerning the long time behavior of both the nonlinear process and the particle system.

 

A ′ (H * P s (X s )) ds, ∀t ≥ 0, the law of X t is P t ,

(2)

Introduction. In this paper, we are interested in the viscous scalar conservation law with C 1 flux function -A

∂ t F t (x) = σ 2 2 ∂ xx F t (x) + ∂ x (A(F t (x)), F 0 (x) = H * m(x), (1)
where m is a probability measure on the real line and H(x) = 1 {x≥0} denotes the Heaviside function. As a consequence, H * m is the cumulative distribution function of the probability measure m. Since A appears in this equation through its derivative, we suppose without restriction that A(0) = 0. According to [START_REF] Jourdain | Diffusion processes associated with nonlinear evolution equations for signed measures[END_REF], one may associate the following nonlinear process with the conservation law:

where (B t ) t≥0 is a real Brownian motion independent from the initial random variable X 0 with law m and σ a positive constant. The process X is said to be nonlinear in the sense that the drift term of the SDE depends on the entire law P t of X t . More precisely, according to [START_REF] Jourdain | Diffusion processes associated with nonlinear evolution equations for signed measures[END_REF], this nonlinear stochastic differential equation admits a unique weak solution. Moreover, H * P t (x) is the unique bounded weak solution of [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]. For t > 0, by the Girsanov theorem, P t admits a density p t with respect to the Lebesgue measure on the real line.

We want to address the long time behavior of the nonlinear process solving (2) by studying convergence of the density p t (see [START_REF] Benachour | Nonlinear selfstabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF] and [START_REF] Benachour | Nonlinear self-stabilizing processes. II. Convergence to invariant probability[END_REF] for a similar study in a different setting). Since the cumulative distribution function x → H * P s (x) which appears in the drift coefficient is nondecreasing, convexity of A is a natural assumption in order to ensure ergodicity. Then the flux function -A in the conservation law (1) is concave.

In the first section of the paper, after recalling results obtained in [START_REF] Jourdain | Diffusion processes associated with nonlinear evolution equations for signed measures[END_REF], we show that trajectorial uniqueness holds for (2) under convexity of A. Then we introduce a simulable system of n particles obtained by replacing in the drift coefficient the cumulative distribution function by its empirical version and the derivative A ′ by a suitable finite difference approximation. When A is convex, existence and trajectorial uniqueness hold for this system. Moreover, we prove a trajectorial estimation of propagation of chaos which strengthens the weak convergence result obtained in [START_REF] Jourdain | Diffusion processes associated with nonlinear evolution equations for signed measures[END_REF]. Unfortunately, because the empirical cumulative distribution function is a step function and therefore not an increasing one, this estimation is not uniform in time.

The second and main section deals with the long time behavior of both the nonlinear process and the particle system. We address the convergence of the density p t of X t by first studying the convergence of the associated solution H * p t of (1) to the solution F ∞ with the same expectation of the stationary equation σ 2 2 ∂ xx F ∞ (x) + ∂ x (A(F ∞ (x)) = 0 obtained by removing the time derivative in [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]. For this result, no convexity hypothesis is made on A. Instead, one assumes A(u) < 0 for u ∈ (0, 1), A ′ (0) < 0, A(1) = 0 and A ′ (1) > 0. In contrast, to prove exponential convergence of the density of the particle system uniform in the number n of particles, we suppose that the function A is uniformly convex. This hypothesis ensures the existence of an invariant distribution for the particle system. In [START_REF] Pal | One-dimensional Brownian particle systems with rank dependent drifts[END_REF], a necessary and sufficient condition on the drift sequence is established for existence of the invariant measure and convergence in total variation norm for the law of the particle system at time t to this measure. In the present paper, the key step to derive quantitative convergence to equilibrium consists in obtaining a Poincaré inequality for the stationary density of the particle system uniform in n. This density has exponential-like tails and therefore does not satisfy a logarithmic Sobolev inequality. So the derivation of the Poincaré inequality cannot rely on the curvature criterion, used, for instance, in [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates[END_REF][START_REF] Cattiaux | Probabilistic approach for granular media equations in the non-uniformly convex case[END_REF][START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] or [START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF] for the granular media equation. Instead we make a direct estimation of the Poincaré constant using the specific analytic form of the invariant density. To our knowledge, our study provides the first example of a particle system, for which a Poincaré inequality but no logarithmic Sobolev inequality holds uniformly in the number n of particles.

Assumption. Throughout the paper, we assume that A is a C 1 function on [0, 1] s.t. A(0) = 0.

1. Propagation of chaos.

1.1. The nonlinear process. Let us first state existence and uniqueness for the nonlinear stochastic differential equation [START_REF] Benachour | Nonlinear selfstabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF].

Theorem 1.1. The nonlinear stochastic differential equation ( 2) admits a unique weak solution ((X t , P t )) t≥0 . For t > 0, P t admits a density p t with respect to the Lebesgue measure on R. The function (t, x) → H * P t (x) is the unique bounded weak solution of the viscous scalar conservation law [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]. Moreover,

∀t ≥ 0 X t -X 0 is integrable and E(X t -X 0 ) = -A(1)t. (3)
Last, if the function A is convex on [0, 1], (2) admits a unique strong solution.

Proof. The first and third statements are consequences of Proposition 1.2 and Theorem 2.1 of [START_REF] Jourdain | Diffusion processes associated with nonlinear evolution equations for signed measures[END_REF] [uniqueness follows from uniqueness for (1) and existence is obtained by a propagation of chaos result].

According to the Yamada-Watanabe theorem, to deduce the last statement, it is enough to check that when A is convex, then trajectorial uniqueness holds for the standard stochastic differential equation

dX t = σ dB t -A ′ (H * Q t (X t )) dt where (Q t ) t≥0 is the flow of time-marginals of a probability measure Q on C([0, +∞), R). Since for each t ≥ 0 the function x → A ′ (H * Q t (x)) is nonde- creasing, if (X t ) t≥0 and (Y t ) t≥0 both solve this standard SDE, then |X t -Y t | is bounded by |X 0 -Y 0 | + t 0 sign(X s -Y s )(A ′ (H * Q s (Y s )) -A ′ (H * Q s (X s ))) ds,
and then by |X 0 -Y 0 | which concludes the proof of trajectorial uniqueness.

Existence of the density p t for t > 0 follows from the boundedness of the drift coefficient and the Girsanov theorem. To prove [START_REF] Benachour | Nonlinear self-stabilizing processes. II. Convergence to invariant probability[END_REF], one first remarks that by boundedness of the drift coefficient, for each t ≥ 0, the random variable X t -X 0 is integrable and

E(X t -X 0 ) = - t 0 E(A ′ (H * P s (X s ))) ds = - t 0 R A ′ x -∞ P s (dy) P s (dx) ds.
For s > 0, since by the Girsanov theorem P s does not weight points,

R A ′ x -∞ P s (dy) P s (dx) = [A(H * P s (x))] +∞ -∞ = A(1). Corollary 1.2. Assume that A is C 2 on [0, 1]. Then the function H * P t (x) is C 1,2
on (0, +∞) × R and solves [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] in the classical sense on this domain.

Proof. By the Girsanov theorem, for t 0 > 0, the law P t 0 of X t 0 admits a density with respect to the Lebesgue measure on R. Hence (t, x) → H * P t (x) is a continuous function on (0, +∞) × R with values in [0, 1]. According to [START_REF] Ladyzenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF], Theorem 8.1, page 495, Remark 8.1, page 495 and Theorem 2.5, page 18, there exists a function u with values in [0, 1], continuous on [0, +∞) × R and C 1,2 on (0, +∞) × R such that

   ∀x ∈ R, u(0, x) = H * P t 0 (x), ∀(t, x) ∈ (0, +∞) × R, ∂ t u(t, x) = σ 2 2 ∂ xx u(t, x) + ∂ x (A(u(t, x))).
By the uniqueness result for bounded weak solutions of this viscous scalar conservation law recalled in Theorem 1.1, ∀t ≥ t 0 , H * P t (x) = u(t -t 0 , x).

The conclusion follows since t 0 is arbitrary.

1.2. Study of the particle system. For n ∈ N * , let (a n (i)) 1≤i≤n be a sequence of real numbers. In this section, we are interested in the n-dimensional stochastic differential equation

dX i,n t = σ dB i t -a n n j=1 1 {X j,n t ≤X i,n t } dt, X i,n 0 = X i 0 , 1 ≤ i ≤ n, (4)
where (B i ) i≥1 are independent standard Brownian motions independent from the sequence (X i 0 ) i≥1 of initial random variables. In the next section devoted to the approximation of the nonlinear stochastic differential equation (2), we will choose a n (i) equal to the finite difference approximation n(A(i/n) -A((i -1)/n)) of A ′ ( i n ). For this particular choice, the nondecreasing assumption made in the following proposition is implied by convexity of A.

Proposition 1.3. Assume that the sequence (a n (i)) 1≤i≤n is nondecreasing. Then the stochastic differential equation ( 4) has a unique strong solution. Let (Y

(X i,n t -Y i,n t ) 2 ≤ n i=1 (X i 0 -Y i 0 ) 2 . ( 5
)
In addition, if the initial conditions (X 1 0 , . . . , X n 0 ) and

(Y 1 0 , . . . , Y n 0 ) are s.t. a.s., ∀i ∈ {1, . . . , n}, X i 0 < Y i 0 (resp. X i 0 ≤ Y i 0 ), then a.s., ∀t ≥ 0, ∀i ∈ {1, . . . , n} X i,n t < Y i,n t (resp. X i,n t ≤ Y i,n t ). ( 6
)
Existence of a weak solution to ( 4) is a consequence of the Girsanov theorem. Therefore, according to the Yamada-Watanabe theorem, it is enough to prove (5) which implies trajectorial uniqueness to obtain existence of a unique strong solution. To do so, we will need the following lemma. For n > 2, we define τ 1 as τ if τ (1) = 1 and as τ composed with the transposition between 1 and τ -1 (1) otherwise. This way, τ 1 (1) = 1. In addition, using the result for n = 2, we get n i=1 a(i)b(τ (i)) ≤ n i=1 a(i)b(τ 1 (i)). For 2 ≤ j ≤ n -1, we define inductively τ j as τ j-1 if τ j-1 (j) = j and as τ j-1 composed with the transposition between j and τ -1 j-1 (j) otherwise. This way, for 1 ≤ i ≤ j, τ j (i) = i. Again by the result for n = 2, one has

n i=1 a(i)b(τ (i)) ≤ n i=1 a(i)b(τ 1 (i)) ≤ n i=1 a(i)b(τ 2 (i)) ≤ • • • ≤ n i=1 a(i)b(τ n-1 (i)).
We conclude by remarking that τ n-1 is the identity.

We are now ready to complete the proof of Proposition 1.3.

Proof of Proposition 1.3. Let (X 1,n , . . . , X n,n ) and (Y 1,n , . . . , Y n,n ) denote two solutions. The difference

n i=1 (X i,n t -Y i,n t ) 2 - n i=1 (X i 0 -Y i 0 ) 2
and τ = (τ Y s ) -1 • τ X s , one obtains that the integrand in (8) is nonpositive dP ⊗ ds a.e. Hence [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates[END_REF] holds.

Let us now suppose that a.s. ∀i ∈ {1, . . . , n}, X i 0 < Y i 0 and define ν = inf{t > 0 : ∃i ∈ {1, . . . , n}, X i,n t ≥ Y i,n t } with the convention inf ∅ = +∞. From now on, we restrict ourselves to the event {ν < +∞}. Let i ∈ {1, . . . , n} be such that Y i,n ν = X i,n ν . There is an increasing sequence (s k ) k≥1 of positive times with limit ν such that ∀k ≥ 1, a n ( n j=1 1 {X j,n

s k ≤X i,n s k } ) < a n ( n j=1 1 {Y j,n s k ≤Y i,n
s k } ). Since (a n (i)) 1≤i≤n is nondecreasing, by extracting a subsequence still denoted by (s k ) k for simplicity, one deduces the existence of j ∈ {1, . . . , n} with j = i such that ∀k ≥ 1,

X i,n s k < X j,n s k and Y j,n s k ≤ Y i,n s k . Since s k < ν, X i,n s k < X j,n s k < Y j,n s k ≤ Y i,n s k . By continuity of the paths, one obtains X i,n ν = X j,n ν = Y j,n ν = Y i,n ν . Now since the probability of the event ∃i 1 , i 2 , i 3 dist. in {1, . . . , n}, ∃t > 0 X i 1 0 +σB i 1 t = X i 2 0 +σB i 2 t = X i 3 0 +σB i 3 t
is equal to 0, the Girsanov theorem implies that a.s. ∀l ∈ {1, . . . , n} \ {i, j}, X l,n ν = X i,n ν = X j,n ν . In the same way, Y l,n ν = Y i,n ν = Y j,n ν . By continuity of the paths and definition of ν one deduces that for k large enough, and for every

t ∈ [s k , ν], n l=1 l =i,j 1 {Y l,n t ≤Y i,n t } ≤ n l=1 l =i,j 1 {X l,n t ≤X i,n t } ; n l=1 l =i,j 1 {Y l,n t ≤Y j,n t } ≤ n l=1 l =i,j 1 {X l,n t ≤X j,n t } .
Since a.s. dt a.e., Y i,n t = Y j,n t and (a n (i)) 1≤i≤n is nondecreasing, one obtains that a.s. dt a.e. on [s k , ν],

a n n l=1 1 {Y l,n t ≤Y i,n t } + a n n l=1 1 {Y l,n t ≤Y j,n t } ≤ a n n l=1 1 {X l,n t ≤X j,n t } + a n n l=1 1 {X l,n t ≤X i,n t } .
By integration with respect to t on [s k , ν], this implies that a.s.

Y i,n ν -X i,n ν + Y j,n ν -X j,n ν ≥ Y i,n s k -X i,n s k + Y j,n s k -X j,n
s k > 0. Therefore P(ν < +∞) = 0. When a.s. for i ∈ {1, . . . , n}, X i 0 ≤ Y i 0 , one obtains that for ε > 0 the solution (Y

(i) = n A i n -A i -1 n (9)
and assume that the initial positions (X i 0 ) i≥1 of the particles are independent and identically distributed according to m. We prefer to define a n (i) with the above finite difference approximation of the choice A ′ (i/n) made in [START_REF] Jourdain | Diffusion processes associated with nonlinear evolution equations for signed measures[END_REF] because the sum n i=1 a n (i) which plays a role in the long time behavior of the particle system is then simply equal to nA [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]. One could also obtain trajectorial propagation of chaos estimates similar to Theorem 1.5 below for the choice a n (i) = A ′ (i/n).

In the present section, we also suppose that A is a convex function on [0, 1]. By Theorem 1.1, for each i ≥ 1, the nonlinear stochastic differential equation

   X i t = X i 0 + σB i t - t 0 A ′ (H * P s (X i s )) ds, ∀t ≥ 0, the law of X i t is P t , (10)
has a unique solution and for all t ≥ 0, the law P t of X i t does not depend on i. Under a Lipschitz regularity assumption on A ′ , we obtain the following trajectorial propagation of chaos estimation.

Theorem 1.5. If A : [0, 1] → R is convex and A ′ is Lipschitz continuous with constant K, then ∀n ≥ 1, ∀1 ≤ i ≤ n, ∀t ≥ 0 E sup s∈[0,t] (X i,n s -X i s ) 2 ≤ K 2 t 2 6n . Proof. Let us write n i=1 (X i,n t -X i t ) 2 as 2 t 0 n i=1 (X i,n s -X i s ) a n n j=1 1 {X j s ≤X i s } -a n n j=1 1 {X j,n s ≤X i,n s } ds + 2 t 0 n i=1 (X i,n s -X i s )C(s, X 1 s , . . . , X n s ) ds where C(s, X 1 s , . . . , X n s ) is equal to A ′ (H * P s (X i s )) -n A 1 n n j=1 1 {X j s ≤X i s } -A 1 n n j=1 1 {X j s ≤X i s } - 1 n .
Like in the proof of trajectorial uniqueness for (4), because of the convexity of A, the first term of the r.h.s. is nonpositive. Moreover, by Lipschitz continuity of A ′ ,

A ′ (H * P s (X i s )) -n A 1 n n j=1 1 {X j s ≤X i s } -A 1 n n j=1 1 {X j s ≤X i s } - 1 n 2 = 1 0 A ′ (H * P s (X i s )) -A ′ 1 n n j=1 1 {X j s ≤X i s } + θ -1 n dθ 2 ≤ K 2 n 2 1 0 j =i H * P s (X i s ) -1 {X j s ≤X i s } + (H * P s (X i s ) -θ) 2 dθ.
For s > 0, as the variables X i s are i.i.d. with common law P s which does not weight points and H * P s (X i s ) is uniformly distributed on [0, 1],

1 0 E j =i (H * P s (X i s ) -1 {X j s ≤X i s } ) + (H * P s (X i s ) -θ) 2 dθ = j =i E((H * P s (X i s ) -1 {X j s ≤X i s } ) 2 ) + 1 0 E((H * P s (X i s ) -θ) 2 ) dθ = (n -1)E((H * P s (X i s ))(1 -H * P s (X i s ))) + 1/6 = n/6.
Using the Cauchy-Schwarz inequality, one obtains

E sup s∈[0,t] n i=1 (X i,n s -X i s ) 2 ≤ 2 t 0 K 2 6n E n i=1 (X i,n s -X i s ) 2 ds ≤ 2K √ 6 t 0 E sup u∈[0,s] n i=1 (X i,n u -X i u ) 2 ds.
By comparison with the ordinary differential equation α

′ (t) = 2K α(t) 6 , one concludes that ∀t ≥ 0 E sup s∈[0,t] n i=1 (X i,n s -X i s ) 2 ≤ K 2 t 2 6 .
Exchangeability of the couples ((X i,n , X i )) i∈{1,...,n} completes the proof.

Remark 1.6. One could think that assuming that A is uniformly convex:

∃α > 0, ∀0 ≤ x ≤ y ≤ 1 A ′ (y) -A ′ (x) ≥ α(y -x) (11)
would lead to a better estimation. Indeed, then for every i ∈ {1, . . . , n -1},

a n (i + 1) -a n (i) = n (i+1)/n i/n A ′ (x) -A ′ x - 1 n dx ≥ α n .
But since even in this situation, the nonpositive term

n i=1 (X i,n s -X i s ) a n n j=1 1 {X j s ≤X i s } -a n n j=1 1 {X j,n s ≤X i,n s }
vanishes as soon as the order between the coordinates of (X 1,n s , . . . , X n,n s ) is the same as the order between the coordinates of (X 1 s , . . . , X n s ), we were not able so far to improve the estimation.

Corollary 1.7. Under the hypotheses of Theorem 1.5, let m be a probability measure on

R such that ∀x ∈ R, H * m(x) ≤ H * m(x). If for some random variable U 1 uniform on [0, 1] independent from (B i ) i≥1 , X 1 0 = inf{x : H * m(x) ≥ U 1 } and (Y 1 t ) t≥0 denotes the solution of the nonlinear stochastic differential equation    Y 1 t = Y 1 0 + σB 1 t - t 0 A ′ (H * Ps (Y 1 s )) ds, ∀t ≥ 0, the law of Y 1 t is Pt , (12)
with

Y 1 0 = inf{x : H * m(x) ≥ U 1 }, then P(∀t ≥ 0, X 1 t ≤ Y 1 t ) = 1. Moreover ∀t ≥ 0, ∀x ∈ R, H * Pt (x) ≤ H * P t (x). Last, the function t → E|Y 1 t -X 1 t | is constant.
Remark 1.8. At least when m and m do not weight points, one has a.s.

A ′ (H * P 0 (X 1 0 )) = A ′ (H * P0 (Y 1 0 )) since H * m(X 1 0 ) = H * m(Y 1 0 ) = U 1 . Therefore a.s. d(Y 1 -X 1 ) 0 = 0
t (X 1 t )) = A ′ (H * Pt (Y 1 t )). If A ′ is increasing, a.s. for all t > 0, H * p t (X 1 t ) = H * pt (Y 1 t
) with p t and pt denoting the respective densities of P t and Pt .

If A is C 2 , the Brownian contribution in d(H * p t (X 1 t ) -H * pt (Y 1 t )) given by Itô's formula vanishes, that is, p t (X 1 t ) = pt (Y 1 t ) and ∀u ∈ ]0, 1[, p t ((H * p t ) -1 (u)) = pt ((H * pt ) -1 (u)) or equivalently ((H * p t ) -1 ) ′ (u) = ((H * pt ) -1 ) ′ (u). Hence Y 1 t = X 1
t + c for a deterministic constant c which does not depend on t according to (3). Letting t → 0, one obtains Y 1 0 = X 1 0 + c. This necessary condition turns out to be sufficient as (X 1 t + c) t≥0 obviously solves the nonlinear stochastic differential equation ( 2) starting from X 1 0 + c.

Proof of Corollary 1.7. For (U i ) i≥2 a sequence of independent uniform random variables independent from (U 1 ,

(B i ) i≥1 ), we set ∀i ≥ 2 X i 0 = inf{x : H * m(x) ≥ U i } and Y i 0 = inf{x : H * m(x) ≥ U i }. Since H * m ≤ H * m, a.s. ∀i ≥ 1, Y i 0 ≥ X i 0 .
From Proposition 1.3, one deduces that the solutions (X 1,n t , . . . , X n,n t ) and (Y 1,n t , . . . , Y n,n t ) to (4) respectively starting from (X 1 0 , . . . , X n 0 ) and (Y 1 0 , . . . , Y n 0 ) are such that a.s., ∀n ≥ 1, ∀i ∈ {1, . . . , n}, ∀t ≥ 0 Y i,n t ≥ X i,n t . Since, by Theorem 1.5, for fixed t ≥ 0, one may extract from (X 1,n t , Y 1,n t ) n≥1 a subsequence almost surely converging to (X 1 t , Y 1 t ), one easily deduces that P(∀t ≥ 0,

X 1 t ≤ Y 1 t ) = 1. Hence ∀t ≥ 0, ∀x ∈ R H * Pt (x) = P(Y 1 t ≤ x) ≤ P(X 1 t ≤ x) = H * P t (x). Since |Y 1 t -X 1 t | -|Y 1 0 -X 1 0 | = Y 1 t -Y 1 0 -(X 1 t -X 1 0 ), (3) ensures that E|Y 1 t - X 1 t | ∈ [0, +∞] does not depend on t.
2. Long time behavior. In this section we are interested in the long time behavior of both the nonlinear process and the particle system. According to (3) and the equality n i=1 a n (i) = nA(1) which follows from (9), we have to suppose A(1) = 0 in order to obtain convergence of the densities as t tends to infinity. We address the convergence of the density p t of X t by first studying the convergence of the associated cumulative distribution function F t under the following hypothesis denoted by (H) in the sequel:

A(0) = A(1) = 0, A ′ (0) < 0, (H)
A ′ (1) > 0 and ∀u ∈ (0, 1) A(u) < 0.

These assumptions determine the spatial behavior at infinity of the drift coefficient in (2).

To prove exponential convergence of the density of the particle system uniform in the number n of particles, we make the stronger assumption of uniform convexity on A. The key step in the proof is to obtain a Poincaré inequality uniform in n for the stationary density of the particle system. This density has exponential-like tails and therefore does not satisfy a logarithmic Sobolev inequality. So the derivation of the Poincaré inequality cannot rely on the curvature criterion, used, for instance, by Malrieu [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF][START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF] when dealing with the granular media equation. Instead, we take advantage of the following nice feature: up to reordering of the coordinates, the stationary density is the density of the image by a linear transformation of a vector of independent exponential variables. And it turns out that the control of the constant in the n-dimensional Poincaré inequality relies on the Hardy inequality stated in Lemma 2.18 which is a one-dimensional Poincaré-like inequality. To our knowledge, our study provides the first example of a particle system, for which a Poincaré inequality but no logarithmic Sobolev inequality holds uniformly in the number n of particles.

2.1. The nonlinear process. In this section, we are first going to obtain necessary and sufficient conditions on the function A ensuring existence for the stationary Fokker-Planck equation obtained by removing the timederivative in the nonlinear Fokker-Planck equation

∂ t p t = σ 2 2 ∂ xx p t + ∂ x (A ′ (H * p t )p t ) (13)
satisfied by the density of the solution of (2). Under a slightly stronger condition, the solutions satisfy a Poincaré inequality.

Lemma 2.1. A necessary and sufficient condition for the existence of a probability measure µ solving the stationary Fokker-Planck equation

σ 2 2 ∂ xx µ + ∂ x (A ′ (H * µ(x))µ) = 0
in the distribution sense is A(1) = 0 and A(u) < 0 for all u ∈ (0, 1). Under that condition, all the solutions are the translations of a probability measure with a C 1 density f which satisfies

∀x ∈ R f (x) = - 2 σ 2 A(H * f (x)) and (14) f ′ (x) = - 2 σ 2 A ′ (H * f (x))f (x). If A ′ (0) < 0 and A ′ (1) > 0, then f (x) ∼        - 2A ′ (0) σ 2 x -∞ f (y) dy, when x → -∞, 2A ′ (1) σ 2 +∞ x f (y) dy, when x → +∞, ( 15
) x 0 dy f (y) ∼          -σ 2 2A ′ (0)f (x) , when x → -∞, σ 2 2A ′ (1)f (x) , when x → +∞,
and all the solutions satisfy a Poincaré inequality and have a finite expectation. Last, if the function

A is C 2 on [0, 1], then f is C 2 and satisfies f ′′ (x) = - 2 σ 2 A ′′ (H * f (x))f 2 (x) + f ′2 (x) f (x) . ( 16
)
Proof. Let µ be a probability measure on R solving the stationary Fokker-Planck equation. The equality

σ 2 2 ∂ xx µ = -∂ x (A ′ (H * µ(x))µ) ensures that µ does not weight points. Hence the stationary equation is equivalent to ∂ xx ( σ 2 2 µ + A(H * µ(x))) = 0. One deduces that µ possesses a C 1 density f such that ∀x ∈ R f (x) = - 2 σ 2 A(H * f (x)) + αx + β, (17)
for some constants α and β. Since A(0) = 0, letting x → -∞ then x → +∞ in the last equality, one obtains α = β = A(1) = 0. For u ∈ (0, 1), since u = H * f (x) for some x ∈ R and H * f is not constant and equal to u, the Cauchy-Lipschitz theorem and (17) imply that A(u) = 0. Since f is nonnegative, A(u) < 0. Hence A(1) = 0 and A(u) < 0 for all u ∈ (0, 1) is a necessary condition.

Under that condition, a probability measure µ solves the stationary Fokker-Planck equation if and only if its cumulative distribution function H * µ(x) is a C 2 solution to the differential equation

ϕ ′ (x) = - 2 σ 2 A(ϕ(x)), x ∈ R. ( 18
)
By the Cauchy-Lipschitz theorem, for each v ∈ [0, 1] this equation admits a unique solution ϕ v defined on R with values in [0, 1] such that ϕ v (0) = v. Moreover, as A(0) = A(1) = 0, ϕ 0 ≡ 0 and ϕ 1 ≡ 1 and

∀v ∈ (0, 1), ∀x ∈ R 0 < ϕ v (x) < 1. (19) For v ∈ (0, 1), since ϕ v is nondecreasing and ϕ v (x) = v -2 σ 2
x 0 A(ϕ v (y)) dy, necessarily lim y→+∞ ϕ v (y) = 1. In the same way, lim y→-∞ ϕ v (y) = 0 and ϕ v is an increasing function from R to (0, 1) with inverse denoted by ϕ -1 v . The uniqueness result for (18) implies that ∀v ∈ (0, 1), ∀x ∈ R,

ϕ v (x) = ϕ 1/2 (x + ϕ -1 1/2 (v)
). Therefore the solutions to the stationary Fokker-Planck equation are the probability measures obtained by spatial translation of the probability measure with density f (x) = ϕ ′ 1/2 (x) which satisfies [START_REF] Pal | One-dimensional Brownian particle systems with rank dependent drifts[END_REF] according to (18).

Let us now suppose that A ′ (0) < 0 and A ′ (1) > 0. When x → +∞,

f (x) = - 2 σ 2 A 1 - +∞ x f (y) dy ∼ 2A ′ (1) σ 2 +∞ x f (y) dy. By (14), f ′ (x) f (x) = (log f (x)) ′ = -2 σ 2 A ′ (ϕ 1/2 (x)) converges to -2A ′ (1)
σ 2 as x →
+∞. This implies that log(f (x))

x converges to -2A ′ (1)

σ 2 and that xf (x)1 {x≥0} is integrable. Moreover, since +∞ 0 dy f (y) = +∞, x 0 dy f (y) ∼ σ 2 2A ′ (1) x 0 -f ′ (y) f 2 (y) dy ∼ σ 2
2A ′ (1)f (x) , as x → +∞. In the same way, one obtains the equivalents given in (15) when x → -∞ and checks the integrability of the function xf (x)1 {x≤0} . From (15), one has

lim x→-∞ x -∞ f (y) dy 0 x dy f (y) = σ 4 4(A ′ (0)) 2 and lim x→+∞ +∞ x f (y) dy x 0 dy f (y) = σ 4 4(A ′ (1)) 2 .
By Theorem 6.2.2, page 99 of [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF], one concludes that the measure with density f satisfies a Poincaré inequality.

By [START_REF] Pal | One-dimensional Brownian particle systems with rank dependent drifts[END_REF], the function f is C 2 as soon as the function

A is C 2 on [0, 1]. Moreover, f ′′ (x) = -2 σ 2 A ′′ (H * f (x))f 2 (x) -2 σ 2 A ′ (H * f (x))f ′ (x)
which combined with ( 14) implies (16).

Remark 2.2. When A is a C 1 convex function on [0, 1] such that A(0) = A(1) = 0 and A ′ (u) < 0 for some u ∈ (0, 1), then the necessary and sufficient condition in Lemma 2.1 is obviously satisfied. Since ( 14) implies

(log f (x)) ′′ = f ′ (x) f (x) ′ = -2/σ 2 A ′ (H * f (x))f (x) f (x) ′ = - 2 σ 2 A ′′ (H * f (x))f (x)
≤ 0, the probability measures solving the stationary Fokker-Planck equation admit log-concave densities with respect to the Lebesgue measure. Logconcavity is a property stronger than the existence of a Poincaré inequality (see [START_REF] Fougères | Spectral gap for log-concave probability measures on the real line[END_REF]).

Example 2.3. Using (18) and (19), the following two choices for A lead to exact computations and different tails for the stationary densities:

• if A(x) = 1 2 x(x -1)
, one gets log(

ϕ 1/2 (x) 1-ϕ 1/2 (x) ) = x/σ 2 , that is, ϕ 1/2 (x) = e x/σ 2 1 + e x/σ 2 and ϕ ′ 1/2 (x) = 1 4σ 2 cosh 2 (x/2σ 2 ) ; • if A(x) = x 3 -x = x(x -1)(x + 1), ϕ √ 1/2 (x) = 1 1 + e -4x/σ 2 and ϕ ′ √ 1/2 (x) = 2e -4x/σ 2 σ 2 (1 + e -4x/σ 2 ) 3/2 .
When A(1) = 0 and A(u) < 0 for all u ∈ (0, 1), a natural question is how to link the translation parameter of the candidate long time limit of the marginal P t solving the stationary Fokker-Planck equation to the initial marginal m. When R |x|m(dx) < +∞, by [START_REF] Benachour | Nonlinear self-stabilizing processes. II. Convergence to invariant probability[END_REF], for all t ≥ 0, E(X 1 t ) = E(X 1 0 ). Therefore the translation parameter is chosen in order to ensure that the invariant measure has the same mean as the initial measure m.

Let us denote by p t the density of P t and by F t = H * P t its cumulative distribution function.

Theorem 2.4. Let A be C 2 on [0, 1] satisfying (H). Assume that m admits a density p 0 such that R |x|p 0 (x) dx < +∞ and R

(p 0 (x)-p∞(x)) 2 p∞(x)
dx is small enough where p ∞ denotes the stationary distribution with same expectation as p 0 . Last, we suppose that A and p 0 are such that p is a smooth solution of [START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF]. Then R (pt(x)-p∞(x)) 2 p∞(x)

dx converges to 0 exponentially fast as t → +∞.

By a smooth solution of (13), we mean that p possesses enough regularity and integrability so that the formal computations made in the proof below are justified.

Example 2.5. When A(x) = 1 2 (x 2 -x), one easily checks that the function φ(t, x) = -F t (x + t

2 ) solves Burgers' equation

∂ t φ = σ 2 2 ∂ xx φ - 1 2 ∂ x φ 2 , φ(0, x) = -F 0 (x).
By the Cole-Hopf transformation, ψ(t, x) = exp(-

1 σ 2
x -∞ φ(t, y) dy) solves the heat equation

∂ t ψ = σ 2 2 ∂ xx ψ, ψ(0, x) = exp 1 σ 2 x -∞ F 0 (y) dy . Since F t (x) = σ 2 ∂xψ ψ (t, x -t 2 )
, one deduces that

F t (x) = R e -(x-t/2-y) 2 /2σ 2 t F 0 (y)ψ(0, y) dy/(σ √ 2πt) R e -(x-t/2-y) 2 /2σ 2 t ψ(0, y) dy/(σ √ 2πt) . ( 20
)
If x denotes the expectation associated with the cumulative distribution function F 0 , one has

x -∞ F 0 (z) dz = +∞ x (1 -F 0 (z)) dz. Since x -∞ F 0 (z) dz = x -∞ F 0 (z) dz - x
x (1 -F 0 (z)) dz + (x -x), one deduces that the function ψ(0, x) = e -(x-x)/σ 2 ψ(0, x) [resp. ψ(0, x)] is bounded on R + (resp. R -) and converges to 1 as x tends to +∞ (resp. -∞).

Let us deduce the limit of F t (x) as t → +∞. Writing the integral for y ∈ R as the sum of the integrals for y ∈ R -and for y ∈ R + , and making the change of variables

z = y-x+t/2 σ √ t (resp. z = y-x-t/2 σ √ t ) in the first (resp. second) integral, one obtains R e -(y-x+t/2) 2 /(2σ 2 t) F 0 (y)ψ(0, y) dy σ √ 2πt = R e -z 2 /2 1 {z≤ √ t/(2σ)-x/(σ √ t)} × F 0 σ √ tz + x - t 2 ψ 0, σ √ tz + x - t 2 dz √ 2π + e (x-x)/σ 2 R e -z 2 /2 1 {z≥- √ t/(2σ)-x/(σ √ t)} × F 0 σ √ tz + x + t 2 ψ 0, σ √ tz + x + t 2 dz √ 2π .
By the Lebesgue theorem, the first term of the right-hand side converges to 0 whereas the second term converges to e (x-x)/σ 2 . Replacing F 0 by 1 in the above computation, one obtains that the denominator in (20) converges to 1 + e (x-x)/σ 2 . Therefore

∀x ∈ R lim t→+∞ F t (x) = e (x-x)/σ 2 1 + e (x-x)/σ 2 .
Notice that in the same way, one may also obtain the limit of the density

p t (x) = R ((y + t/2 -x)/(σ 2 t))e -(x-t/2-y) 2 /(2σ 2 t) F 0 (y)ψ(0, y) dy/(σ √ 2πt) R e -(x-t 2 -y) 2 /(2σ 2 t) ψ(0, y) dy/(σ √ 2πt) - 1 σ 2 R e -(x-t/2-y) 2 /(2σ 2 t) F 0 (y)ψ(0, y) dy/(σ √ 2πt) R e -(x-t/2-y) 2 /(2σ 2 t) ψ(0, y) dy(σ √ 2πt) 2 .
One easily checks

∀x ∈ R lim t→+∞ p t (x) = 1 σ 2 e (x-x)/σ 2 1 + e (x-x)/σ 2 - e 2(x-x)/σ 2 (1 + e (x-x)/σ 2 ) 2 = 1 4σ 2 cosh 2 ((x -x)/2σ 2 )
.

In order to prove Theorem 2.4, we are first going to check exponential convergence of F t to the cumulative distribution function

F ∞ of p ∞ . Let G t = F t -F ∞ . Since for a random variable X with cumulative distribution function F , E(X) = +∞ 0 (1 -F (x)) dx -0 -∞ F (x)
dx, the equality of the expectations associated to F t and F ∞ writes R G t (x) dx = 0. This very convenient expression of the link between p t and p ∞ is one main reason for first considering the convergence of G t to 0. In order to prove this convergence, we need the following result.

Lemma 2.6. Under the assumptions of Theorem 2.4, one has

R G 2 t (x) p ∞ (x) dx ≤ c R G t (x) p ∞ (x) ′ 2 p ∞ (x) dx (21)
where c denotes the constant in the Poincaré inequality satisfied by p ∞ . Moreover

R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx (22) = R G t (x) p ∞ (x) ′ 2 p ∞ (x) dx + 2 σ 2 R G t (x) 2 A ′′ (F ∞ )(x) dx and R G t (x) 2 p ∞ (x) dx ≤ c R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx. ( 23
)
Remark 2.7. When A is convex, (23) is a consequence of ( 22) and (21).

Proof of Lemma 2.6. As R G t (x) dx = 0, (21) is the Poincaré inequality satisfied by p ∞ written for the function

G t /p ∞ . Since ( Gt(x) p∞(x) ) ′ = G ′ t (x) p∞(x) -Gt(x)p ′ ∞ (x) p∞(x) 2 , one has R G t (x) p ∞ (x) ′ 2 p ∞ (x) dx = R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx - R G 2 t (x) ′ p ′ ∞ (x) p 2 ∞ (x) dx + R G 2 t (x)p ′ ∞ (x) 2 p 3 ∞ (x) dx = R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx + R G 2 t (x)p ′′ ∞ (x) p 2 ∞ (x) dx - R G 2 t (x)p ′ ∞ (x) 2 p 3 ∞ (x) dx.

Since p ∞ solves (16), one easily deduces (22).

Writing G 2 t (y) as

2 1 {y≤0} y -∞ G t (p t -p ∞ )(x) dx -1 {y>0} +∞ y G t (p t -p ∞ )(x) dx , one obtains R G 2 t p ∞ (x) dx = -2 R G t (p t -p ∞ )(x) x 0 1 p ∞ (y)
dy dx. (24) By (15), and since 1 p∞ is bounded from below and above on each compact subset of the real line,

∃C > 0, ∀x ∈ R x 0 1 p ∞ (y) dy ≤ C p ∞ (x)
.

Using the Cauchy-Schwarz inequality in (24), and inserting the latter bound, one obtains

R G 2 t p ∞ (x) dx ≤ 2C R G 2 t p ∞ (x) dx 1/2 R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx 1/2
.

One easily deduces (23).

According to (23), the exponential convergence of R

(pt(x)-p∞(x)) 2 p∞(x)
dx to zero is a stronger result than the exponential convergence stated in the next lemma.

Lemma 2.8. Under the assumptions of Theorem 2.4, there is a positive constant

C such that if R G 2 0 p∞ (x) dx is small enough, then ∀t ≥ 0 R G 2 t p ∞ (x) dx ≤ e -Ct C R G 2 0 p ∞ (x) dx.
Proof. According to [START_REF] Pal | One-dimensional Brownian particle systems with rank dependent drifts[END_REF], one has

σ 2 2 F ′′ ∞ + (A(F ∞ )) ′ = 0 which also writes p ′ ∞ p∞ = -2 σ 2 A ′ (F ∞ ).
Combining these equations with (1), then using Young's inequality, one easily obtains for ε > 0,

1 2 d dt R G 2 t p ∞ (x) dx = - σ 2 2 R G t (x) p ∞ (x) ′ 2 p ∞ (x) dx (25) - R (A(F t ) -A(F ∞ ) -A ′ (F ∞ )G t )(x) G t (x) p ∞ (x) ′ dx ≤ ε - σ 2 2 R G t p ∞ (x) ′ 2 p ∞ (x) dx + A ′′ 2 ∞ 16ε R G 4 t (x) p ∞ (x) dx. Since G t 2 ∞ ≤ R |p t (x) -p ∞ (x)| p ∞ (x) p ∞ (x) dx 2 (26) ≤ R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx, |G t | is bounded by 1 and p ∞ A ′′ (F ∞ ) = -2 σ 2 A × A ′′ (F ∞ ) is bounded, one deduces from (22) that G t 2 ∞ ≤ 4 σ 4 AA ′′ ∞ R G 2 t p ∞ (x) dx + 1 ∧ R G t p ∞ (x) ′ 2 p ∞ (x) dx .
Inserting this bound in (25) and using Young's inequality, one deduces that for η > 0,

1 2 d dt R G 2 t p ∞ (x) dx ≤ ε - σ 2 2 R G t p ∞ (x) ′ 2 p ∞ (x) dx + AA ′′ ∞ A ′′ 2 ∞ 4εσ 4 R G 2 t p ∞ (x) dx 2 + η 1 ∧ R G t p ∞ (x) ′ 2 p ∞ (x) dx 2 + A ′′ 4 ∞ 1024ε 2 η R G 2 t p ∞ (x) dx 2 ≤ ε + η - σ 2 2 R G t p ∞ (x) ′ 2 p ∞ (x) dx + AA ′′ ∞ A ′′ 2 ∞ 4εσ 4 + A ′′ 4 ∞ 1024ε 2 η R G 2 t p ∞ (x) dx 2 .
One easily concludes with (21) and Lemma 2.10 below.

Remark 2.9. (i) After reading this proof, one may wonder whether one could replace the upper bound in (25) by

ε - σ 2 2 R G t p ∞ (x) ′ 2 p ∞ (x) dx + A ′′ 2 ∞ 16ε R G 2 t p ∞ (x) dx using G t ∞ ≤ 1. If the constant c in the Poincaré inequality (21) was smaller than σ 4 A ′′ 2 ∞ , one could deduce exponential convergence of R G 2 t p∞ (x) dx to 0 even for large values of R G 2 0 p∞ (x) dx. In case A(x) = 1 2 (x 2 -x) (see Example 2.5), one has A ′′ ∞ = 1 and c ≥ R x 2 p ∞ (x) dx - R xp ∞ (x) dx 2 = +∞ 0 x 2 2σ 2 cosh 2 (x/(2σ 2 )) dx > 4σ 4 +∞ 0 y 2 e -2y dy = σ 4 = σ 4 A ′′ 2 ∞ ,
and this approach does not work.

(ii) Convexity of A implies nonnegativity of the term A(F t ) -A(F ∞ ) -A ′ (F ∞ )G t which appears in the right-hand side of the first displayed equality in the proof. One may wonder if one could exploit this property to obtain exponential convergence of p t to p ∞ even if p 0 is not close to p ∞ . We have not been able to do so.

Proof of Theorem 2.4. By [START_REF] Pal | One-dimensional Brownian particle systems with rank dependent drifts[END_REF],

p ′ ∞ = -2 σ 2 A ′ (F ∞ )p ∞ and p ∞ ∞ ≤ 2 A ∞ σ 2 .
The Fokker-Planck equation ( 13) for p t ensures that 1 2

d dt R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx = - σ 2 2 R p t p ∞ (x) ′ 2 p ∞ (x) dx - R (A ′ (F t ) -A ′ (F ∞ ))(x)(p t -p ∞ )(x) p t p ∞ (x) ′ dx - R (A ′ (F t ) -A ′ (F ∞ ))(x)p ∞ (x) p t p ∞ (x) ′ dx.
Then, using Young's inequality and (26), one easily checks that for ε, η > 0,

1 2 d dt R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx ≤ η + ε - σ 2 2 R p t p ∞ (x) ′ 2 p ∞ (x) dx + 1 4ε R (A ′ (F t )(x) -A ′ (F ∞ )(x)) 2 (p t (x) -p ∞ (x)) 2 p ∞ (x) dx + 1 4η R (A ′ (F t )(x) -A ′ (F ∞ )(x)) 2 p ∞ (x) dx ≤ η + ε - σ 2 2 R p t p ∞ (x) ′ 2 p ∞ (x) dx + A ′′ 2 ∞ 4ε R (p t (x) -p ∞ (x)) 2 p ∞ (x) dx 2 20 B. JOURDAIN AND F. MALRIEU + A ′′ 2 ∞ 4η × 4 A 2 ∞ σ 4 R G 2 t p ∞ (x) dx.
By (23) and Lemma 2.8, for R

(p 0 (x)-p∞(x)) 2 p∞(x)
dx small enough, the last term of the r.h.s. is smaller than

ce -Ct C R (p 0 (x)-p∞(x)) 2 p∞(x) dx. Since R ( pt p∞ (x)) ′ 2 p ∞ (x) dx is greater than 1 c R (pt(x)-p∞(x)) 2 p∞(x)
dx, one easily concludes by Lemma 2.10 below.

Lemma 2.10. Assume that u :

R + → R + satisfies ∀t ≥ 0 du dt (t) ≤ βu(t)(u(t) -α) + γe -δt
for some constants α, β, δ > 0 and γ ≥ 0.

If γ = 0 and u(0) < α, then ∀t ≥ 0 u(t) ≤ αu(0)e -αβt α + u(0)(e -αβt -1)
.

If u(0) < α 2 and γ < βα 2 4 , then u(t) converges to 0 exponentially fast as t → +∞.

Proof. When γ = 0, as long as u(t) ∈ (0, α), one has

du dt (t) 1 u(t) + 1 α -u(t)
≤ -αβ and after integration one obtains the desired estimation. Since the upper bound is not greater than u(0) and u(t) = 0 ⇒ ∀s ≥ t, u(s) = 0 one easily concludes. Now when γ ∈ (0, βα 2 4 ), one has βa(α -a) = γ for some a ∈ (0, α 2 ) and

d dt u(t) ∧ α 2 -a + = 1 {a<u(t)<α/2} du dt (t) ≤ 0. Hence when u(0) < α 2 , ∀t ≥ 0, u(t) ≤ u(0) ∨ a and du dt (t) ≤ -β(α -u(0) ∨ a)u(t) + γe -δt .
For v(t) = e β(α-u(0)∨a)t u(t) one deduces

dv dt (t) ≤ γe (β(α-u(0)∨a)-δ)t
and one concludes by integration of this inequality that u(t) is bounded by

C(1 + t)e -[(β(α-u(0)∨a))∧δ]t .
2.2. The particle system (4). Let us suppose that A(1) = 0 and that the first-order moment associated with the initial probability measure m is defined and equal to x. As in the case of the granular media equation considered by Malrieu [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF][START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF], the direction (1, 1, . . . , 1) is quite singular for the particle system. Indeed,

d(X 1,n t + • • • + X n,n t ) = σ n i=1 dB i t ,
which prevents the law of (X 1,n t , . . . , X n,n t ) from converging as t → +∞. Following [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF][START_REF] Malrieu | Convergence to equilibrium for granular media equations and their Euler schemes[END_REF], one introduces the hyperplane M n = {y = (y 1 , . . . , y n ) ∈ R n : y 1 + • • • + y n = nx} orthogonal to this singular direction and denotes by P the orthogonal projection on M n and by P the orthogonal projection on {y = (y 1 , . . . , y n ) ∈ R n :

y 1 + • • • + y n = 0}. Since n i=1 a n (i) = n(A(1) - A(0)) = 0, the orthogonal projection (Y i,n t = x + X i,n t -1
n n j=1 X j,n t ) 1≤i≤n of the original particle system on M n is a diffusion on this hyperplane solving

dY i,n t = σ n -1 n dB i t - σ n j =i dB j t -a n n j=1 1 {Y j,n t ≤Y i,n t } dt. (27)
Propagation of chaos for the projected system is a consequence of the following estimate.

Proposition 2.11. Assume that A is convex, such that A ′ is Lipschitz continuous with constant K and A(1) = 0 and that the initial measure m has a finite second order moment. Then, ∀i ∈ {1, . . . , n}, ∀t ≥ 0,

E[(X i t -Y i,n t ) 2 ] ≤ 1 n K 2 t 2 6 + E[(X 0 -x) 2 ] + σ 2 t + 2 t 0 R A(F s (x)) dx ds ,
where X i is solution of [START_REF] Jourdain | Probabilistic approximation via spatial derivation of some nonlinear parabolic evolution equations[END_REF].

Proof. Denoting X n 1 (t) = (X 1 t , . . . , X n t ), X n,n 1 (t) = (X 1,n t , . . . , X n,n t ) and Y n,n 1 (t) = (Y 1,n t , . . . , Y n,n t
), one has

|X n 1 (t) -Y n,n 1 (t)| 2 = |X n 1 (t) -P X n,n 1 (t)| 2 (28) = |X n 1 (t) -P X n 1 (t)| 2 + | P X n 1 (t) -P X n,n 1 (t)| 2 ≤ 1 n n i=1 (X i t -x) 2 + n i=1 (X i t -X i,n t ) 2 . (29) Since (X t -x) 2 ≤ 3((X 0 -x) 2 + σ 2 B 2 t + A ′ 2 ∞ t 2 ), the variable X t is square integrable. As ∀x > 0 |(x -x)A(F t (x))| ≤ A ′ ∞ (1 -F t (x))(x + |x|) ≤ A ′ ∞ E(X 2 t ) x + |x|(1 -F t (x)) , one has lim x→+∞ (x -x)A(F t (x)) = 0. Similarly (x -x)A(F t (x)) also van- ishes as x → -∞ and R (x -x)A ′ (F t (x))p t (x) dx = -R A(F t (x)) dx. Com- puting (X t -x)
2 by Itô's formula and taking expectations, one deduces that

E((X t -x) 2 ) = E((X 0 -x) 2 ) + σ 2 t + 2 t 0 R A(F s (x)) dx ds.
Moreover, by ( 3), E(X t -x) = -A(1)t = 0. One concludes by taking expectations in (29) then using Theorem 1.5 and exchangeability of the particles.

Let us now study the long time behavior of the projected particle system.

Theorem 2.12. Assume that the function A is uniformly convex on [0, 1] with constant α [see [START_REF] Ladyzenskaya | Linear and Quasilinear Equations of Parabolic Type[END_REF]] and such that A(1) = 0. Then, the probability measure with density

p n ∞ (y) = 1 Z n e -2/σ 2 n i=1 an(i)y (i)
with respect to the Lebesgue measure dy on M n is invariant for the projected dynamics (27). Here y (1) ≤ y (2) ≤ • • • ≤ y (n) denotes the increasing reordering of the coordinates of y = (y 1 , . . . , y n ) and Z n = Mn e -2 σ 2 n i=1 an(i)y (i) dy. Moreover, if (Y 1,n 0 , . . . , Y n,n 0 ) admits a symmetric density p n 0 (y) with respect to the Lebesgue measure on M n , then for all t ≥ 0, (Y 1,n t , . . . , Y n,n t ) admits a symmetric density p n t (y) which is such that

∀t ≥ 0 Mn p n t p n ∞ (x) -1 2 p n ∞ (x) dx (30) ≤ e -λnt Mn p n 0 p n ∞ (x) -1 2 p n ∞ (x) dx
where the sequence (λ n ) n is bounded from below by α 2 12 3 σ 2 .

In order to deduce long time properties of the nonlinear process from long time properties of the projected system, it is not restrictive to assume that p n 0 is symmetric (see Remark 2.15 to get some intuition about this hypothesis). But the lack of uniformity in time of the estimation given in Proposition 2.11 is a real problem.

dY t = σ(dB 2 t -dB 1 t ) -sgn(Y t )(a 2 (2) -a 2 (1)) dt
and the density of Y t converges exponentially to a 2 (2)-a 2 (1) 2σ 2

e (-(a 2 (2)-a 2 (1))/σ 2 )|y| when the density of Y 0 is close enough to this limit. As (Y 1,2 t , Y 2,2 t ) = x + 1 2 (-Z t , Z t ), one easily deduces exponential convergence of the density of (Y 1,2 t , Y 2,2 t ) on the straight line M 2 to a 2 (2)-a 2 (1)

√ 2σ 2
e -(a 2 (2)/σ 2 )2y (2) e (a 2 (1)/σ 2 )(-2y (1) ) .

The proof of Theorem 2.12 relies on the following Poincaré inequality.

Proposition 2.14. Under the assumptions of Theorem 2.12, the density

pn ∞ (y) = n!1 {y 1 ≤y 2 ≤•••≤yn} Z n e -(2/σ 2 ) n i=1 an(i)y i on M n is such that for f : R n → R regular enough, Mn f (y) - Mn f (y)p n ∞ (y) dy 2 pn ∞ (y) dy (31) ≤ σ 2 λ n Mn |P ∇f (y)| 2 pn ∞ (y) dy
where the sequence (λ n ) n is bounded from below by α 2 12 3 σ 2 .

Proof of Theorem 2.12. Let us first check the following Green formula: for f : R n → R and u : R n → R n regular enough, Mn f ∇ • (P u)(y) dy = -Mn P ∇f • (P u)(y) dy. (32) Let 1 ∈ R n denote the vector with all coordinates equal to 1. For ϕ : R → R and v : R n → R n , one has

R ϕ( √ nz) Mn ∇ • (P v) y + z1 √ n dy dz = R n ϕ(x 1 + • • • + x n -nx)∇ • (P v)(x) dx = - R n ϕ ′ (x 1 + • • • + x n -nx)1 • (P v)(x) dx = 0.
The function ϕ being arbitrary, one deduces that Mn ∇ • (P v)(y) dy = 0. Since ∇

• P (f u) = ∇f • (P u) + f ∇ • (P u) = P ∇f • (P u) + f ∇ • (P u), (32)
follows for the choice v = f u. By weak uniqueness for (27), when (Y 1,n 0 , . . . , Y n,n 0 ) has a symmetric density p n 0 with respect to the Lebesgue measure on M n , the particles Y i,n , i ∈ {1, . . . , n} are exchangeable and for each t ≥ 0, (Y

(i) = n(A(1) -A(0)) = 0, setting b(y) = τ ∈Sn 1 {y τ (1) ≤y τ (2) ≤•••≤y τ (n) }      ca n (τ -1 (1)) a n (τ -1 (2))
. . . ). According to [START_REF] Jourdain | Probabilistic characteristics method for a one-dimensional inviscid scalar conservation law[END_REF], the reordered system is a diffusion process normally reflected at the boundary of the closed convex set {y ∈ M n :

a n (τ -1 (n))      ,
y 1 ≤ y 2 ≤ • • • ≤ y n }. More precisely,                    dY (i),n t = σ dβ i t -a n (i) dt + (γ i t -γ i+1 t ) d|K| t , t 0 (γ i s -γ i+1 s ) d|K| s , 1 ≤ i ≤ n t≥0 is a continuous process with finite variation equal to |K| t , γ 1 ≡ γ n+1 ≡ 0, d|K| t a.e. ∀2 ≤ i ≤ n, γ i t ≥ 0 and γ i t (Y (i),n t -Y (i-1),n t ) = 0, (34)
where (β 1 , . . . , β n ) is a Brownian motion such that

β i ,β j t t = 1 {i=j} -1/n. If the initial condition (Y (1),n 0 ≤ • • • ≤ Y (n),n 0
) admits a density pn 0 with respect to the Lebesgue measure on M n , then the law of (Y In order to prove Proposition 2.14, we take advantage of the specific form of the density pn ∞ . Remarking that pn ∞ is the density of the image of a vector of independent exponential random variables by a linear transformation, one first obtains the following result.

Lemma 2.16. The Poincaré inequality (31) holds with the constant λ n greater than α 2 4σ 2 multiplied by the smallest eigenvalue λn of the (n -1) ×

(n -1) matrix Q n defined by ∀1 ≤ i, j ≤ n -1, Q n ij = b n (i)L n ij b n (j) where b n (i) = i(n -i) n and L n =            2 -1 0 . . . . . . . . . 0 -1 2 -1 0 . . . . . . 0 0 -1 2 -1 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 -1 2 -1 0 0 . . . . . . 0 -1 2 -1 0 . . . . . . . . . 0 -1 2           
.

The last statement in Proposition 2.14 then follows from the next lemma which is obtained by interpreting Q n as a finite element rigidity matrix associated with the operator -x(1 -x)∂ xx (x(1 -x).) acting on functions on (0, 1). The Hardy inequality stated in Lemma 2.18 ensures that it is enough to bound the smallest eigenvalue of the corresponding mass matrix from below. The resort to this one-dimensional Poincaré-like inequality in order to estimate the constant in the n-dimensional Poincaré inequality (31) is striking.

Lemma 2.17. The sequence ( λn ) n is bounded from below by 1/(16 × 27).

Proof of Lemma 2.16. Let f be such that Mn f (y)p n ∞ (y) dy = 0. Since the left-hand side in the Poincaré inequality (31) only depends on the restriction of f to M n , one may assume that ∀x ∈ R n , f (x) = f ( P x), which ensures that for (x 1 , . .

. , x n ) ∈ R n such that x 1 + • • • + x n = 0, f (x + x 1 , . . . , x+x n ) = f (x 1 , . . . , x n ) and P ∇f (x+x 1 , . . . , x+x n ) = ∇f (x 1 , . . . , x n ). Therefore the Poincaré inequality (31) is equivalent to I(f ) ≤ σ 2 λn I(|∇f |) where
I(g) = R n-1 (g 2 pn ∞ )(-(x 2 + • • • + x n ), x n 2 ) dx n 2 with x n 2 = (x 2 , . . . , x n ).
To integrate the coordinates over independent domains, we make the change of variables z n 2 = M x n 2 where

M =          2 1 1 . . . . . . 1 -1 1 0 . . . . . . 0 0 -1 1 0 . . . 0 . . . . . . . . . . . . . . . . . . 0 . . . 0 -1 1 0 0 . . . . . . 0 -1 1          . One easily checks that for 2 ≤ i ≤ n, z 2 + • • • + z i = x 2 + • • • + x n + x i and deduce that (n -1)z 2 + (n -2)z 3 + • • • + 2z n-1 + z n = n(x 2 + • • • + x n ).
Therefore

M -1 = 1 n          1 2 -n 3 -n 4 -n . . . -1 1 2 3 -n 4 -n . . . -1 1 2 3 4 -n . . . -1 . . . . . . . . . . . . . . . . . . 1 2 3 . . . n -2 -1 1 2 3 . . . . . . n -1         
and denoting

N =   1 -n n 2 -n n . . . - 2 n - 1 n M -1   , one has I(f ) = n! Z n (R + ) n-1 f 2 (N z n 2 )e (-2/σ 2 ) n i=2 βn(i)z i dz n 2 |M |
where

β n (i) = 1 n [(i -1)(a n (i) + • • • + a n (n)) -(n + 1 -i)(a n (1) + • • • + a n (i -1))] = -nA((i -1)/n) > 0.
Here |M | denotes the determinant of the matrix M ; it is equal to n by an easy computation. The one-dimensional exponential density with parameter c satisfies the Poincaré inequality with optimal constant 4/c 2 . Tensorizing this inequality (see Chapters 3 and 6 in [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] for further details), one obtains

I(f ) ≤ n! Z n (R + ) n-1 n j=2 σ 4 β 2 n (j) n k=1 N kj-1 ∂ k f (N z n 2 ) 2 e (-2/σ 2 ) n i=2 βn(i)z i dz n 2 |M | = R n-1 n k,l=1 n j=2 σ 4 β 2 n (j) N kj-1 N lj-1 ∂ k f ∂ l f pn ∞ (-(x 2 + • • • + x n ), x n 2 ) dx n 2 .
Since A is uniformly convex with constant α and A(0) = A(1) = 0,

β n (i) = -nA((i -1)/n) ≥ - nα 2 × i -1 n i -1 n -1 = α 2 b n (i -1).
Therefore

I(f ) ≤ 4σ 4 α 2 R n-1 n k,l=1 n-1 j=1 N kj N lj b 2 n (j) ∂ k f ∂ l f pn ∞ (-(x 2 + • • • + x n ), x n 2 ) dx n 2 ≤ 4σ 2 α 2 λn I(|∇f |)
where λn denotes the inverse of the largest eigenvalue of the symmetric positive semidefinite matrix N N * defined by Nij = N ij bn(j) . To prove Proposition 2.14 with a possibly modified lower bound, it is enough to check that the largest eigenvalue is bounded from above uniformly in n. Unfortunately, the trace of the matrix can be bounded from below by a positive constant multiplied by log(n). Therefore one has to be more precise.

Let w be an eigenvector associated with the largest eigenvalue: N N * w = . By symmetry, 1 λn is also the largest eigenvalue of N * N . We are going to check that the latter matrix is invertible with inverse equal to Q n in order to conclude the proof. Because of the definition of N , it is enough to check that N * N is invertible with inverse equal to L n .

By construction of the matrix N , for the equation N z x n 2 = y.

One easily checks that the (n -1) × (n -1) matrix in the left-hand side is equal to Using the change of variables y = 1 -x, one easily checks that ∀i ∈ {1, . . . , n -1} R n ii -R n ii-1 -R n ii+1 = r n i + r n n-i , where by convention R n 10 = R n n-1n = 0. We are going to prove that ∀n ≥ 3 ∀i ∈ {2, . . . , n -3} r n i ≥ 1 27 , and that r n 1 and r n n-2 are nonnegative. For y ∈ R n-1 , one deduces that

         1
y * R n y = n-1 i=1 R n ii y 2 i + 2 n-2 i=1 R n ii+1 y i y i+1 = n-1 i=1 (R n ii -R n ii-1 -R n ii+1 )y 2 i + n-2 i=1 R n ii+1 (y i + y i+1 ) 2 ≥ |y| 2 27
and the conclusion follows. Let us first suppose that i ≤ ⌊ n 2 ⌋ -1, which ensures that the function f (x) = x 2 (1-x) 2 is increasing on [i/n, (i+ 1)/n]. Let g(x) = u i (u i -u i+1 )(x). One easily checks that

≥    0, if i = 1, i 2 (n -i) 2 12n 4 , if i ≥ 2.
Since there is some x i ∈ [i/n, (i + 1)/n] such that the function g(x) is nonnegative on [i/n, x i ] then nonpositive on [x i , (i + 1)/n], and f is positive and increasing, one deduces that for all x ∈ [i/n, (i + 1)/n], x

≥    0, if i = 1, i 2 (n -i) 2 12(i + 1) 2 (n -i -1) 2 ≥ 1 27 , if i ≥ 2.
Now approximating v ∈ H 1 0 (0, 1) by a sequence of C ∞ functions with compact support converging in the H 1 norm and almost everywhere, one deduces with the Fatou lemma that the inequality still holds for v ∈ H 1 0 . For u satisfying the hypotheses in the lemma, v(x) = x(1 -x)u(x) belongs to H 1 (0, 1). According to Theorem VIII.2, page 122 of [START_REF] Brezis | Analyse Fonctionnelle[END_REF], v admits a representative continuous on [0, 1] still denoted by v. Moreover, since u(x) = v(x)

x(1-x) belongs to L 2 (0, 1), necessarily, v(0) = v(1) = 0. By Theorem VIII.11, page 133 of [START_REF] Brezis | Analyse Fonctionnelle[END_REF], v belongs to H 1 0 (0, 1) and the conclusion follows from (35).

Lemma 1 . 4 .

 14 Let (a(i)) 1≤i≤n and (b(i)) 1≤i≤n denote two nondecreasing sequences of real numbers. Then for any permutation τ ∈ S n , For n = 2, the result is an easy consequence of the inequality (a(2) -a(1))(b(2) -b(1)) ≥ 0.

Remark 2 . 13 . 2 t-

 2132 In case n = 2, the process Y t = Y 2,

2 2

 2 one has P b = b and the infinitesimal generator associated with (27) is Lψ = σ ∇ • (P ∇ψ) -P b • ∇ψ. Computing dψ(Y

  by increasing reordering of the symmetric law of the solution (Y

  Of course N * w is nonzero and multiplying the previous equality by N * , one obtains that N * w is an eigenvector of N * N associated with the B. JOURDAIN AND F. MALRIEU eigenvalue 1 λn

  For i ∈ {1, . . . , n -2}, let r n i = (i+1)/n i/n u i (u i -u i+1 )(x) dx and

  1)(n -i -1) 6i(n -i)

  dy ≥ 0. This ensures that ∀x ∈ [i/n, (i + 1)/n]

  n 2 = x where x ∈ R n to have a solution z n 2 , it is necessary and sufficient thatx 1 = -(x 2 + • • • + x n ) and then z n 2 = M x n 2 . Now for fixed y ∈ R n-1 , let us find x n 2 ∈ R n-1 such that N * x = y where x = -(x 2 + • • • + x n , xn2 ). This equation writes

			N 11	N 11	. . .	N 11		
	    (M -1 ) * -	   	N 12 . . .	N 12 . . .	. . . . . .	N 12 . . .	   	   
			N 1n-1 N 1n-1 . . . N 1n-1

Acknowledgment. We warmly thank Tony Lelièvre (CERMICS) for fruitful discussions concerning the analysis of the long time behavior of the nonlinear process.

.

Combining x n

2 = Ry with the solution of the previous problem, one obtains that the unique solution of the equation N * N z n 2 = y is z n 2 = M Ry. One concludes by checking that the matrix M R is equal to L n .

Proof of Lemma 2.17. For i ∈ {1, . . . , n -1}, the functions

By the Hardy inequality stated in Lemma 2.18 below, the smallest eigenvalue of the matrix Q n is greater than the smallest eigenvalue of the (n -1) × (n -1) tridiagonal matrix R n ij = 1 0 u i (x)u j (x) dx divided by 16.

B. JOURDAIN AND F. MALRIEU

Let us now suppose that i ≥ ⌊ n+1 2 ⌋ so that the function f is decreasing on [i/n, (i + 1)/n]. We deduce that

and the left-hand side is greater than 1/12 for i ≤ n -3 and nonnegative for i = n -2.

We still have to deal with the case n odd and i = (n -1)/2. Then, f is not monotonic on

which completes the proof.

Lemma 2.18. For all u ∈ L 2 (0, 1) such that the distribution derivative (x(1 -x)u(x)) ′ belongs to L 2 (0, 1),

Proof. For v a C ∞ function with compact support on (0, 1), by the integration by parts formula,

.

Dealing with the integral on (1/2, 1) in a symmetric way, one deduces