Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf

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Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf

Benjamin Jourdain, Florent Malrieu

To cite this version:

Benjamin Jourdain, Florent Malrieu. Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2008, 18 (5), pp.1706-1736. �10.1214/07-AAP513�. �hal-00127988v2�

2008, Vol. 18, No. 5, 1706–1736 DOI:10.1214/07-AAP513

c

Institute of Mathematical Statistics, 2008

PROPAGATION OF CHAOS AND POINCAR´E INEQUALITIES FOR A SYSTEM OF PARTICLES INTERACTING

THROUGH THEIR CDF

By Benjamin Jourdain and Florent Malrieu Ecole des Ponts and Universit´e Rennes 1´

In this paper, in the particular case of a concave flux function, we are interested in the long time behavior of the nonlinear process as- sociated in [Methodol. Comput. Appl. Probab.2(2000) 69–91] to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by replacing the cumulative distribu- tion function in the drift coefficient of this nonlinear process by the empirical cumulative distribution function. We first obtain a trajec- torial propagation of chaos estimate which strengthens the weak con- vergence result obtained in [8] without any convexity assumption on the flux function. Then Poincar´e inequalities are used to get explicit estimates concerning the long time behavior of both the nonlinear process and the particle system.

Introduction. In this paper, we are interested in the viscous scalar con- servation law withC¹ flux function−A

∂_tF_t(x) =σ²

2 ∂_xxF_t(x) +∂_x(A(F_t(x)), F₀(x) =H∗m(x), (1)

wheremis a probability measure on the real line andH(x) = 1_{x≥0} denotes the Heaviside function. As a consequence,H∗mis the cumulative distribu- tion function of the probability measure m. Since A appears in this equa- tion through its derivative, we suppose without restriction that A(0) = 0.

According to [8], one may associate the following nonlinear process with the conservation law:







X_t=X₀+σB_t− Z t

0 A^′(H∗P_s(X_s))ds,

∀t≥0,the law of X_t is P_t, (2)

Received January 2007; revised December 2007.

AMS 2000 subject classifications.65C35, 60K35, 60E15, 35K15, 46N30.

Key words and phrases.Viscous scalar conservation law, nonlinear process, particle sys- tem, propagation of chaos, Poincar´e inequality, long time behavior.

This is an electronic reprint of the original article published by the Institute of Mathematical StatisticsinThe Annals of Applied Probability, 2008, Vol. 18, No. 5, 1706–1736. This reprint differs from the original in pagination and typographic detail.

1

where (B_t)_t≥0is a real Brownian motion independent from the initial random variable X0 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 lawP_tofX_t. More precisely, according to [8], this nonlinear stochastic differential equation admits a unique weak solution. Moreover, H∗P_t(x) is the unique bounded weak solution of (1). Fort >0, by the Girsanov theorem, Ptadmits a densityptwith respect to the Lebesgue measure on the real line.

We want to address the long time behavior of the nonlinear process solv- ing (2) by studying convergence of the density p_t (see [2] and [3] for a simi- lar study in a different setting). Since the cumulative distribution function x→H∗P_s(x) which appears in the drift coefficient is nondecreasing, con- vexity ofA 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 [8], we show that trajectorial uniqueness holds for (2) under convexity of A.

Then we introduce a simulable system of nparticles 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.

WhenAis 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 [8]. Unfortunately, be- cause 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 ^σ₂²∂_xxF_∞(x) +∂_x(A(F_∞(x)) = 0 obtained by removing the time derivative in (1). For this result, no convexity hypothesis is made on A. Instead, one assumesA(u)<0 foru∈(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 numbern 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 [14], 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 timet to this measure. In the present paper, the key step to derive quantitative convergence to equilibrium consists in obtaining a Poincar´e inequality for the stationary density of the particle system uniform inn. This density has exponential-like tails and therefore does not satisfy a logarithmic Sobolev inequality. So the derivation of the Poincar´e inequality cannot rely on the curvature criterion, used, for instance, in [5,6,12] or [13]

for the granular media equation. Instead we make a direct estimation of the Poincar´e 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 Poincar´e inequality but no logarithmic Sobolev inequality holds uniformly in the numbern of particles.

Assumption. Throughout the paper, we assume thatAis aC¹function 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 (2).

Theorem 1.1. The nonlinear stochastic differential equation (2) admits a unique weak solution ((Xt, Pt))t≥0. For t >0, Pt admits a density pt with respect to the Lebesgue measure on R. The function (t, x)7→H∗P_t(x) is the unique bounded weak solution of the viscous scalar conservation law (1).

Moreover,

∀t≥0 X_t−X₀ is integrable andE(X_t−X₀) =−A(1)t.

(3)

Last, if the function A is convex on [0,1], (2) admits a unique strong solu- tion.

Proof. The first and third statements are consequences of Proposition 1.2 and Theorem 2.1 of [8] [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 state- ment, it is enough to check that whenAis convex, then trajectorial unique- ness holds for the standard stochastic differential equation

dXt=σ dBt−A^′(H∗Qt(Xt))dt

where (Q_t)_t≥0 is the flow of time-marginals of a probability measureQ on C([0,+∞),R). Since for eacht≥0 the functionx7→A^′(H∗Qt(x)) is nonde- creasing, if (X_t)_t≥0and (Y_t)_t≥0 both solve this standard SDE, then|X_t−Y_t| is bounded by

|X₀−Y₀|+ Z t

0 sign(X_s−Y_s)(A^′(H∗Q_s(Y_s))−A^′(H∗Q_s(X_s)))ds, and then by|X₀−Y₀|which concludes the proof of trajectorial uniqueness.

Existence of the densityp_t for t >0 follows from the boundedness of the drift coefficient and the Girsanov theorem. To prove (3), one first remarks

that by boundedness of the drift coefficient, for each t≥0, the random variableX_t−X₀ is integrable and

E(Xt−X0) =− Z _t

0

E(A^′(H∗Ps(Xs)))ds

=− Z _t

0

Z

R

A^′ Z _x

−∞Ps(dy)

Ps(dx)ds.

Fors >0, since by the Girsanov theorem P_s does not weight points, Z

R

A^′ Z _x

−∞Ps(dy)

Ps(dx) = [A(H∗Ps(x))]⁺_−∞^∞=A(1).

Corollary 1.2. Assume thatA is C² on [0,1]. Then the function H∗ P_t(x) is C^1,2 on (0,+∞)×R and solves (1) in the classical sense on this domain.

Proof. By the Girsanov theorem, fort₀>0, the lawP_t0 ofX_t0 admits a density with respect to the Lebesgue measure onR. Hence (t, x)7→H∗Pt(x) is a continuous function on (0,+∞)×R with values in [0,1]. According to [11], Theorem 8.1, page 495, Remark 8.1, page 495 and Theorem 2.5, page 18, there exists a functionuwith values in [0,1], continuous on [0,+∞)×R and C^1,2 on (0,+∞)×Rsuch that







∀x∈R, u(0, x) =H∗P_t0(x),

∀(t, x)∈(0,+∞)×R, ∂_tu(t, x) = σ²

2 ∂_xxu(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≥t₀, H∗P_t(x) =u(t−t₀, x).

The conclusion follows sincet₀ is arbitrary.

1.2. Study of the particle system. For n∈N∗, let (a_n(i))_1≤i≤n be a se- quence of real numbers. In this section, we are interested in then-dimensional stochastic differential equation

dX_t^i,n=σ dBⁱ_t−a_n

n

X

j=1

1{X_t^j,n≤X_t^i,n}

!

dt, X₀^i,n=X₀ⁱ, 1≤i≤n, (4)

where (Bⁱ)_i≥1 are independent standard Brownian motions independent from the sequence (X₀ⁱ)i≥1 of initial random variables.

In the next section devoted to the approximation of the nonlinear stochas- tic differential equation (2), we will choosea_n(i) equal to the finite difference approximationn(A(i/n)−A((i−1)/n)) ofA^′(_nⁱ). For this particular choice, the nondecreasing assumption made in the following proposition is implied by convexity ofA.

Proposition 1.3. Assume that the sequence (a_n(i))_1≤i≤n is nonde- creasing. Then the stochastic differential equation (4) has a unique strong so- lution. Let(Y_t^1,n, . . . , Y_t^n,n)denote another solution starting from(Y₀¹, . . . , Y₀ⁿ) and driven by the same Brownian motion (B¹, . . . , Bⁿ). Then

a.s.,∀t≥0

n

X

i=1

(X_t^i,n−Y_t^i,n)²≤

n

X

i=1

(X₀ⁱ−Y₀ⁱ)². (5)

In addition, if the initial conditions (X₀¹, . . . , X₀ⁿ) and (Y₀¹, . . . , Y₀ⁿ) are s.t.

a.s., ∀i∈ {1, . . . , n}, X₀ⁱ< Y₀ⁱ (resp. X₀ⁱ ≤Y₀ⁱ), then

a.s., ∀t≥0,∀i∈ {1, . . . , n} X_t^i,n< Y_t^i,n (resp. X_t^i,n≤Y_t^i,n).

(6)

Existence of a weak solution to (4) is a consequence of the Girsanov the- orem. 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.

Lemma 1.4. Let (a(i))1≤i≤n and (b(i))1≤i≤n denote two nondecreasing sequences of real numbers. Then for any permutation τ ∈ Sn,

n

X

i=1

a(i)b(τ(i))≤

n

X

i=1

a(i)b(i).

(7)

Proof. For n= 2, the result is an easy consequence of the inequality (a(2)−a(1))(b(2)−b(1))≥0.

For n >2, we define τ₁ asτ if τ(1) = 1 and as τ composed with the trans- position between 1 andτ⁻¹(1) otherwise. This way, τ₁(1) = 1. In addition, using the result forn= 2, we get ^Pn_i=1a(i)b(τ(i))≤^Pni=1a(i)b(τ₁(i)).

For 2≤j≤n−1, we define inductively τ_j as τ_j−1 if τ_j−1(j) =j and as τj−1composed with the transposition betweenjandτ_j⁻₋¹₁(j) otherwise. This way, for 1≤i≤j,τj(i) =i. Again by the result forn= 2, one has

n

X

i=1

a(i)b(τ(i))≤

n

X

i=1

a(i)b(τ₁(i))≤

n

X

i=1

a(i)b(τ₂(i))≤ · · · ≤

n

X

i=1

a(i)b(τ_n−1(i)).

We conclude by remarking thatτ_n−1 is the identity.

We are now ready to complete the proof of Proposition1.3.

Proof of Proposition1.3. Let (X^1,n, . . . , X^n,n) and (Y^1,n, . . . , Y^n,n) denote two solutions. The difference

n

X

i=1

(X_t^i,n−Y_t^i,n)²−

n

X

i=1

(X₀ⁱ−Y₀ⁱ)²

is equal to 2

Z _t

0 n

X

i=1

(X_s^i,n−Y_s^i,n) a_n

n

X

j=1

1{Ys^j,n≤Ys^i,n}

!

−a_n

n

X

j=1

1{Xs^j,n≤Xs^i,n}

!!

ds.

(8)

By the Girsanov theorem, for anys >0 the distributions of (X_s^1,n, . . . , X_s^n,n) and (Y_s^1,n, . . . , Y_s^n,n) admit densities w.r.t. the Lebesgue measure onRⁿand thereforedP⊗ds a.e. the positionsX_s^1,n, . . . , X_s^n,n (resp. Y_s^1,n, . . . , Y_s^n,n) are distinct and there is a unique permutation τ_s^X ∈ Sn (resp. τ_s^Y ∈ Sn) such thatXs^τsX(1),n< Xs^τsX(2),n<· · ·< Xs^τsX(n),n (resp. Ys^τsY(1),n< Ys^τsY(2),n<· · ·<

Ys^τsY(n),n). Therefore dP⊗ds a.e.,

n

X

i=1

(X_s^i,n−Y_s^i,n) a_n

n

X

j=1

1{Ys^j,n≤Ys^i,n}

!

−a_n

n

X

j=1

1{Xs^j,n≤Xs^i,n}

!!

is equal to

n

X

i=1

a_n(i)((X_s^τsY(i),n−Y_s^τsY(i),n)−(X_s^τsX(i),n−Y_s^τsX(i),n)).

The sequence (a_n(i))_1≤i≤nis nondecreasing. Applying Lemma1.4withb(i) = Xs^τsX(i),n andτ = (τ_s^X)⁻¹◦τ_s^Y then withb(i) =Ys^τsY(i),n andτ= (τ_s^Y)⁻¹◦τ_s^X, one obtains that the integrand in (8) is nonpositive dP⊗ds a.e. Hence (5) holds.

Let us now suppose that a.s. ∀i∈ {1, . . . , n}, X₀ⁱ < Y₀ⁱ and define ν = inf{t >0 :∃i∈ {1, . . . , n}, X_t^i,n≥Y_t^i,n}with the convention inf∅= +∞. From now on, we restrict ourselves to the event {ν <+∞}. Let i∈ {1, . . . , n} be such that Y_ν^i,n = X_ν^i,n. There is an increasing sequence (s_k)_k≥1 of positive times with limit ν such that ∀k ≥ 1, a_n(^Pn_j=11_{Xj,n

sk≤X_sk^i,n}) <

a_n(^Pn_j=11_{Yj,n

sk ≤Y_sk^i,n}). 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 j6=i such that ∀k≥1, X_s^i,n_k < X_s^j,n_k and Y_s^j,n_k ≤Y_s^i,n_k . Since s_k < ν, X_s^i,n_k < X_s^j,n_k < Y_s^j,n_k ≤Y_s^i,n_k . By continuity of the paths, one obtainsX_ν^i,n=X_ν^j,n=Y_ν^j,n=Y_ν^i,n. Now since the probability of the event

∃i₁, i₂, i₃ dist. in {1, . . . , n},∃t >0 X₀ⁱ¹+σB_tⁱ¹=X₀ⁱ²+σB_tⁱ²=X₀ⁱ³+σBⁱ_t³ is equal to 0, the Girsanov theorem implies that a.s.∀l∈ {1, . . . , n} \ {i, j}, X_ν^l,n 6=X_ν^i,n=X_ν^j,n. In the same way, Y_ν^l,n6=Y_ν^i,n=Y_ν^j,n. By continuity of the paths and definition of ν one deduces that for k large enough, and for everyt∈[s_k, ν],

n

X

l=1 l6=i,j

1{Y_t^l,n≤Y_t^i,n}≤

n

X

l=1 l6=i,j

1{X_t^l,n≤X_t^i,n};

n

X

l=1 l6=i,j

1{Y_t^l,n≤Y_t^j,n}≤

n

X

l=1 l6=i,j

1{X_t^l,n≤X_t^j,n}.

Since a.s.dt a.e.,Y_t^i,n6=Y_t^j,nand (a_n(i))_1≤i≤nis nondecreasing, one obtains that a.s.dt a.e. on [s_k, ν],

a_n

n

X

l=1

1_{Yl,n t ≤Y_t^i,n}

! +a_n

n

X

l=1

1_{Yl,n t ≤Y_t^j,n}

!

≤a_n

n

X

l=1

1{X_t^l,n≤X_t^j,n}

! +a_n

n

X

l=1

1{X_t^l,n≤X_t^i,n}

! .

By integration with respect toton [s_k, ν], this implies that a.s.Y_ν^i,n−X_ν^i,n+ Y_ν^j,n−X_ν^j,n≥Y_s^i,n_k −X_s^i,n_k +Y_s^j,n_k −X_s^j,n_k >0. Therefore P(ν <+∞) = 0.

When a.s. for i∈ {1, . . . , n}, X₀ⁱ ≤Y₀ⁱ, one obtains that for ε >0 the solution (Y_t^1,n,ε, . . . , Y_t^n,n,ε) to (4) starting from (Y₀¹+ε, . . . , Y₀ⁿ+ε) is such that

a.s., ∀t≥0 ∀i∈ {1, . . . , n} X_t^i,n< Y_t^i,n,ε. Since by (5), Y_t^i,n,ε≤Y_t^i,n+√

nε, one easily concludes by letting ε→0.

1.3. Trajectorial propagation of chaos. From now on, we set

∀n∈N^∗,∀i∈ {1, . . . , n} a_n(i) =n

A i

n

−A i−1

n (9)

and assume that the initial positions (X₀ⁱ)i≥1of the particles are independent and identically distributed according tom. We prefer to define a_n(i) with the above finite difference approximation of the choice A^′(i/n) made in [8]

because the sum^Pn_i=1a_n(i) which plays a role in the long time behavior of the particle system is then simply equal to nA(1). One could also obtain trajectorial propagation of chaos estimates similar to Theorem1.5below for the choice an(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_tⁱ=X₀ⁱ +σB_tⁱ− Z t

0

A^′(H∗P_s(X_sⁱ))ds,

∀t≥0,the law ofX_tⁱ is Pt, (10)

has a unique solution and for allt≥0, the law P_tof X_tⁱ 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]→Ris convex and A^′ is Lipschitz continuous with constant K, then

∀n≥1,∀1≤i≤n,∀t≥0 E

sup

s∈[0,t]

(X_s^i,n−X_sⁱ)²

≤K²t² 6n .

Proof. Let us write ^Pn_i=1(X_t^i,n−X_tⁱ)² as 2

Z _t

0 n

X

i=1

(X_s^i,n−X_sⁱ) a_n

n

X

j=1

1{Xs^j≤X_sⁱ}

!

−a_n

n

X

j=1

1{Xs^j,n≤Xs^i,n}

!!

ds

+ 2 Z _t

0 n

X

i=1

(X_s^i,n−X_sⁱ)C(s, X_s¹, . . . , X_sⁿ)ds whereC(s, X_s¹, . . . , X_sⁿ) is equal to

A^′(H∗Ps(X_sⁱ))−n A 1 n

n

X

j=1

1_{X^j

s≤Xⁱ_s}

!

−A 1 n

n

X

j=1

1_{X^j

s≤X_sⁱ}− 1 n

!!

. Like in the proof of trajectorial uniqueness for (4), because of the convex- ity of A, the first term of the r.h.s. is nonpositive. Moreover, by Lipschitz continuity ofA^′,

A^′(H∗P_s(X_sⁱ))−n A 1 n

n

X

j=1

1{X^js≤X_sⁱ}

!

−A 1 n

n

X

j=1

1{Xs^j≤X_sⁱ}− 1 n

!!!2

= Z 1

0

A^′(H∗P_s(X_sⁱ))−A^′ 1 n

n

X

j=1

1_{Xj

s≤Xⁱ_s}+θ−1 n

! dθ

!2

≤K² n²

Z 1 0

X

j6=i

H∗P_s(X_sⁱ)−1

{Xs^j≤X_sⁱ}

!

+ (H∗P_s(X_sⁱ)−θ)

!2

dθ.

Fors >0, as the variablesX_sⁱ are i.i.d. with common law P_s which does not weight points and H∗P_s(X_sⁱ) is uniformly distributed on [0,1],

Z 1 0

E

X

j6=i

(H∗P_s(X_sⁱ)−1_{X^j

s≤Xⁱ_s}) + (H∗P_s(X_sⁱ)−θ)

!2! dθ

=^X

j6=i

E((H∗P_s(X_sⁱ)−1

{Xs^j≤X_sⁱ})²) + Z ₁

0

E((H∗P_s(X_sⁱ)−θ)²)dθ

= (n−1)E((H∗P_s(X_sⁱ))(1−H∗P_s(X_sⁱ))) + 1/6

=n/6.

Using the Cauchy–Schwarz inequality, one obtains E sup

s∈[0,t]

n

X

i=1

(X_s^i,n−X_sⁱ)²

!

≤2 Z t

0

v u u tK²

6nE _n

X

i=1

(Xs^i,n−X_sⁱ)

!2! ds

≤2K

√6 Z _t

0

v u u

tE sup

u∈[0,s]

n

X

i=1

(X_u^i,n−X_uⁱ)²

! ds.

By comparison with the ordinary differential equationα^′(t) = 2K^qα(t)₆ , one concludes that

∀t≥0 E sup

s∈[0,t]

n

X

i=1

(X_s^i,n−X_sⁱ)²

!

≤K²t² 6 .

Exchangeability of the couples ((X^i,n, Xⁱ))_{i∈{1,...,n}} completes the proof.

Remark 1.6. One could think that assuming thatA is uniformly con- vex:

∃α >0,∀0≤x≤y≤1 A^′(y)−A^′(x)≥α(y−x) (11)

would lead to a better estimation. Indeed, then for everyi∈ {1, . . . , n−1}, a_n(i+ 1)−a_n(i) =n

Z (i+1)/n i/n

A^′(x)−A^′

x−1 n

dx≥α n. But since even in this situation, the nonpositive term

n

X

i=1

(X_s^i,n−X_sⁱ) a_n

n

X

j=1

1_{Xj s≤X_sⁱ}

!

−a_n

n

X

j=1

1_{Xj,n s ≤X_s^i,n}

!!

vanishes as soon as the order between the coordinates of (X_s^1,n, . . . , X_s^n,n) is the same as the order between the coordinates of (X_s¹, . . . , X_sⁿ), 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 variableU₁ uniform on[0,1]independent from(Bⁱ)_i≥1, X₀¹= inf{x:H∗m(x)≥U1} and (Y_t¹)t≥0 denotes the solution of the nonlinear stochastic differential equation







Y_t¹=Y₀¹+σB_t¹− Z _t

0

A^′(H∗P˜_s(Y_s¹))ds,

∀t≥0,the law ofY_t¹ is P˜_t, (12)

withY₀¹= inf{x:H∗m(x)˜ ≥U₁}, then

P(∀t≥0, X_t¹≤Y_t¹) = 1.

Moreover ∀t≥0, ∀x∈R, H∗P˜t(x)≤H ∗Pt(x). Last, the function t7→

E|Y_t¹−X_t¹| is constant.

Remark 1.8. At least when m and ˜m do not weight points, one has a.s. A^′(H∗P₀(X₀¹)) =A^′(H∗P˜₀(Y₀¹)) since H∗m(X₀¹) =H∗m(Y˜ ₀¹) =U₁. Therefore a.s. d(Y¹ −X¹)₀ = 0 and one may wonder whether a.s. Y_t¹ −

X_t¹ does not depend on t. If this property holds, necessarily, a.s. dt a.e.

A^′(H∗P_t(X_t¹)) =A^′(H∗P˜_t(Y_t¹)). If A^′ is increasing, a.s. for all t >0, H∗ p_t(X_t¹) =H∗p˜_t(Y_t¹) with p_t and ˜p_t denoting the respective densities of P_t and ˜P_t. IfA isC², the Brownian contribution ind(H∗p_t(X_t¹)−H∗p˜_t(Y_t¹)) given by Itˆo’s formula vanishes, that is, p_t(X_t¹) = ˜p_t(Y_t¹) and ∀u∈]0,1[, p_t((H∗p_t)⁻¹(u)) = ˜p_t((H∗p˜_t)⁻¹(u)) or equivalently ((H∗p_t)⁻¹)^′(u) = ((H∗

˜

p_t)⁻¹)^′(u). HenceY_t¹=X_t¹+cfor a deterministic constantcwhich does not depend ont according to (3). Letting t→0, one obtains Y₀¹=X₀¹+c. This necessary condition turns out to be sufficient as (X_t¹+c)_t≥0 obviously solves the nonlinear stochastic differential equation (2) starting fromX₀¹+c.

Proof of Corollary1.7. For (U_i)_i≥2 a sequence of independent uni- form random variables independent from (U1,(Bⁱ)i≥1), we set

∀i≥2 X₀ⁱ= inf{x:H∗m(x)≥U_i} and Y₀ⁱ= inf{x:H∗m(x)˜ ≥U_i}. SinceH∗m˜ ≤H∗m, a.s.∀i≥1,Y₀ⁱ≥X₀ⁱ. From Proposition1.3, one deduces that the solutions (X_t^1,n, . . . , X_t^n,n) and (Y_t^1,n, . . . , Y_t^n,n) to (4) respectively starting from (X₀¹, . . . , X₀ⁿ) and (Y₀¹, . . . , Y₀ⁿ) are such that

a.s., ∀n≥1,∀i∈ {1, . . . , n},∀t≥0 Y_t^i,n≥X_t^i,n.

Since, by Theorem1.5, for fixedt≥0, one may extract from (X_t^1,n, Y_t^1,n)_n≥1 a subsequence almost surely converging to (X_t¹, Y_t¹), one easily deduces that P(∀t≥0, X_t¹≤Y_t¹) = 1. Hence

∀t≥0,∀x∈R H∗P˜_t(x) =P(Y_t¹≤x)≤P(X_t¹≤x) =H∗P_t(x).

Since|Y_t¹−X_t¹| − |Y₀¹−X₀¹|=Y_t¹−Y₀¹−(X_t¹−X₀¹), (3) ensures thatE|Y_t¹− X_t¹| ∈[0,+∞] does not depend ont.

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 ^Pn_i=1an(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 densityp_tofX_tby first studying the convergence of the associated cumulative distribution function Ft 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 numbern of particles, we make the stronger assumption of

uniform convexity onA. The key step in the proof is to obtain a Poincar´e in- equality uniform inn for the stationary density of the particle system. This density has exponential-like tails and therefore does not satisfy a logarith- mic Sobolev inequality. So the derivation of the Poincar´e inequality cannot rely on the curvature criterion, used, for instance, by Malrieu [12,13] 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 Poincar´e inequality relies on the Hardy inequality stated in Lemma 2.18 which is a one-dimensional Poincar´e-like inequality. To our knowledge, our study provides the first example of a par- ticle system, for which a Poincar´e inequality but no logarithmic Sobolev inequality holds uniformly in the numbern 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 time- derivative in the nonlinear Fokker–Planck equation

∂_tp_t=σ²

2 ∂_xxp_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 Poincar´e inequality.

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

σ²

2 ∂_xxµ+∂_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 C¹ density f which satisfies

∀x∈R f(x) =− 2

σ²A(H∗f(x)) and (14)

f^′(x) =− 2

σ²A^′(H∗f(x))f(x).

If A^′(0)<0 and A^′(1)>0, then

f(x)∼











−2A^′(0) σ²

Z _x

−∞

f(y)dy, when x→ −∞, 2A^′(1)

σ²

Z +∞ x

f(y)dy, when x→+∞,

(15)

Z _x

0

dy f(y) ∼











−σ²

2A^′(0)f(x), when x→ −∞, σ²

2A^′(1)f(x), when x→+∞,

and all the solutions satisfy a Poincar´e inequality and have a finite expecta- tion. Last, if the function A is C² on [0,1], thenf is C² and satisfies

f^′′(x) =− 2

σ²A^′′(H∗f(x))f²(x) +f^′2(x) f(x) . (16)

Proof. Let µ be a probability measure on R solving the stationary Fokker–Planck equation. The equality ^σ₂²∂_xxµ=−∂_x(A^′(H∗µ(x))µ) ensures that µ does not weight points. Hence the stationary equation is equivalent to ∂_xx(^σ₂²µ+A(H∗µ(x))) = 0. One deduces that µpossesses a C¹ density f such that

∀x∈R f(x) =− 2

σ²A(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)6= 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 measureµsolves the stationary Fokker–

Planck equation if and only if its cumulative distribution functionH∗µ(x) is aC² solution to the differential equation

ϕ^′(x) =− 2

σ²A(ϕ(x)), x∈R. (18)

By the Cauchy–Lipschitz theorem, for each v∈[0,1] this equation admits a unique solution ϕ_v defined onR with values in [0,1] such thatϕ_v(0) =v.

Moreover, asA(0) =A(1) = 0, ϕ₀≡0 andϕ₁≡1 and

∀v∈(0,1),∀x∈R 0< ϕ_v(x)<1.

(19)

For v∈(0,1), since ϕv is nondecreasing andϕv(x) =v−_σ^22R0^xA(ϕv(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 fromRto (0,1) with inverse denoted by ϕ⁻_v¹. The uniqueness result for (18) implies that ∀v∈(0,1),∀x∈R, ϕ_v(x) =ϕ_1/2(x+ ϕ⁻_1/2¹(v)). Therefore the solutions to the stationary Fokker–Planck equation

are the probability measures obtained by spatial translation of the proba- bility measure with densityf(x) =ϕ^′_1/2(x) which satisfies (14) according to (18).

Let us now suppose thatA^′(0)<0 and A^′(1)>0. Whenx→+∞, f(x) =− 2

σ²A

1− Z _+∞

x f(y)dy

∼2A^′(1) σ²

Z _+∞

x f(y)dy.

By (14), ^f_f(x)^′(x) = (logf(x))^′=−_σ²²A^′(ϕ_1/2(x)) converges to −^2A_σ^′2(1) as x→ +∞. This implies that ^log(f_x^(x)) converges to−^2A_σ^′2⁽¹⁾ and thatxf(x)1_{x≥0} is integrable. Moreover, since ^R₀^+∞_f(y)^dy = +∞, ^R₀^x _f(y)^dy ∼ _2A^σ′2₍₁₎^R₀^x−_f^f2^′(y)(y)dy∼

σ²

2A^′(1)f(x), asx→+∞. In the same way, one obtains the equivalents given in (15) whenx→ −∞and checks the integrability of the functionxf(x)1_{x≤0}. From (15), one has

x→−∞lim Z _x

−∞

f(y)dy Z ₀

x

dy

f(y)= σ⁴ 4(A^′(0))² and

x→lim+∞

Z _+∞

x

f(y)dy Z _x

0

dy

f(y) = σ⁴ 4(A^′(1))².

By Theorem 6.2.2, page 99 of [1], one concludes that the measure with densityf satisfies a Poincar´e inequality.

By (14), the function f is C² as soon as the function A is C² on [0,1].

Moreover,f^′′(x) =−_σ²²A^′′(H∗f(x))f²(x)−_σ²²A^′(H∗f(x))f^′(x) which com- bined with (14) implies (16).

Remark 2.2. WhenAis aC¹convex function on [0,1] such thatA(0) = A(1) = 0 andA^′(u)<0 for some u∈(0,1), then the necessary and sufficient condition in Lemma2.1is obviously satisfied. Since (14) implies

(logf(x))^′′=

f^′(x) f(x)

_′

=

−2/σ²A^′(H∗f(x))f(x) f(x)

_′

=− 2

σ²A^′′(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. Log- concavity is a property stronger than the existence of a Poincar´e inequality (see [7]).

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

• ifA(x) =¹₂x(x−1), one gets log(₁₋^ϕ1/2_ϕ ^(x)

1/2(x)) =x/σ², that is, ϕ_1/2(x) = e^x/σ2

1 +e^x/σ2 and ϕ^′_1/2(x) = 1

4σ²cosh²(x/2σ²);

• ifA(x) =x³−x=x(x−1)(x+ 1), ϕ√1/2(x) = 1

p1 +e^−4x/σ2 and ϕ^′√

1/2(x) = 2e^−4x/σ2 σ²(1 +e^−4x/σ2)^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 marginalm. When^RR|x|m(dx)<+∞, by (3), for allt≥0,E(X_t¹) =E(X₀¹).

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 byF_t=H∗P_t its cumulative distribution function.

Theorem 2.4. Let A be C² on [0,1] satisfying (H). Assume that m admits a density p0 such that ^RR|x|p0(x)dx <+∞ and ^RR

(p0(x)−p∞(x))² p∞(x) dx is small enough where p_∞ denotes the stationary distribution with same expectation asp₀. Last, we suppose thatAandp₀ are such thatpis a smooth solution of (13). Then ^RR

(pt(x)−p∞(x))²

p∞(x) dx converges to 0 exponentially fast as t→+∞.

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

Example 2.5. WhenA(x) =¹₂(x²−x), one easily checks that the func- tionφ(t, x) =−F_t(x+₂^t) solves Burgers’ equation

∂_tφ=σ²

2 ∂_xxφ−1

2∂_xφ², φ(0, x) =−F₀(x).

By the Cole–Hopf transformation, ψ(t, x) = exp(−_σ^12R_−∞^x φ(t, y)dy) solves the heat equation

∂_tψ=σ²

2 ∂_xxψ, ψ(0, x) = exp 1

σ² Z x

−∞F₀(y)dy

. Since F_t(x) =σ^2∂x_ψ^ψ(t, x−₂^t), one deduces that

Ft(x) = R

Re^{−(x−t/2−y)2/2σ2t}F₀(y)ψ(0, y)dy/(σ√ 2πt) R

Re^{−(x−t/2−y)2/2σ2t}ψ(0, y)dy/(σ√

2πt) . (20)