¸ois Bolley
email: bolley@ceremade.dauphine.fr.

QUANTITATIVE CONCENTRATION INEQUALITIES ON SAMPLE PATH SPACE FOR MEAN FIELD INTERACTION

Keywords: Mean field limits, Particle approximation, Transportation inequalities

We consider the approximation of a mean field stochastic process by a large interacting particle system. We derive non-asymptotic large deviation bounds measuring the concentration of the empirical measure of the paths of the particles around the law of the process. The method is based on a coupling argument, strong integrability estimates on the paths in Hölder norm, and a general concentration result for the empirical measure of identically distributed independent paths.

This paper is devoted to the study of the particle approximation of a mean field stochastic process. In the models to be considered, the evolution is governed by a random diffusive term, an exterior force field and a mean field interaction depending on the law of the process itself.

Quantitative estimates on the approximation have been obtained at the level of the time marginals. Let indeed µ t be the law of the considered process at time t, and (X i t ) 1≤i≤N be the position of the N particles in the phase space R d ; let also μN t denote their empirical measure. First, [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] adapted concentration of measure ideas to obtain non-asymptotic bounds on the deviation of observables

1 N N i=1 ϕ(X i t ) = R d ϕ dμ N t around R d
ϕ dµ t , where ϕ is a given Lipschitz function on R d . Then, transposing Sanov's large deviation argument to their setting, [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] got non-asymptotic bounds on the deviation of μN t around µ t at the very level of the measures, namely for a distance which induces a topology stronger that the narrow topology.

In this work we go one step further again by considering the law µ [0,T ] of the paths of the process on a given time interval [0, T ]. A natural object to consider in the particle approximation is now the empirical measure μN [0,T ] of the N trajectories (X i t ) 0≤t≤T . We shall give a precise meaning and estimates on the convergence of μN [0,T ] to µ [0,T ] ; we shall see that they imply previously mentionned results by projection at time t, but above all they give concentration estimates at the level of the paths.

In the first section we state our main results and give an insight of the proofs, which will be given in more detail in the following sections.

Statement of the results

We are interested in the particle approximation of the R d -valued process (X t ) t≥0 evolving according to the mean field stochastic differential equation

dX t = σ dB t -b(X t ) dt -c * µ t (X t ) dt. ( 1
)
Here σ is a d × d real matrix, (B t ) t≥0 a standard Brownian motion on R d , b and c are R d to R d maps, * stands for the convolution and µ t is the law on R d of the random variable X t .

Two instances of such processes are particularly interesting. First of all, when R d is the phase space of positions x ∈ R d ′ and velocities v ∈ R d ′ with d = 2d ′ , one is interested in the process (X t ) t≥0 = ((x t , v t )) t≥0 solution to the diffusive Newton's equations

dx t = v t dt dv t = √ 2 db t -λ v t dt -∇ x U * x ρ t (x t ) dt. (2)
Here (b t ) t≥0 is a Brownian motion in the velocity space R d ′ , U = U (x) is an interaction potential in the position space and ρ t is the law of x t on R d ′ ; moreover ∇ x and * x respectively stand for the gradient and convolution with respect to the position variable x ∈ R d ′ . By Itô's formula the distribution µ t of X t is solution to the Vlasov-Fokker-Planck equation

∂µ t ∂t + v • ∇ x µ t -∇ x U * x ρ t • ∇ v µ t = ∆ v µ t + λ∇ v • (vµ t ), t > 0, x, v ∈ R d ′ .
Here a • b denotes the scalar product of two vectors a and b in R d ′ , whereas ∇ v , ∇ v • and ∆ v respectively stand for the gradient, divergence and Laplace operators with respect to the velocity variable v ∈ R d ′ . This equation is used in the modelling of diffusive stellar matter. We are also concerned with the process (X t ) t≥0 = ((x t , v t )) t≥0 solution to

dx t = v t dt dv t = √ 2 db t -∇ v V (v t ) dt -∇ v W * ν t (v t ) dt (3)
where V and W are respectively exterior and interaction potentials in the velocity space and ν t is the law of v t on R d ′ . By Itô's formula, the distribution µ t of X t is solution to

∂µ t ∂t + v • ∇ x µ t = ∆ v µ t + ∇ v • (µ t (∇ v V + ∇ v W * v µ t )), t > 0, x, v ∈ R d ′ .
This equation is used in the modelling of granular media.

For position homogeneous distributions, we are brought to study the solution (v t ) t≥0 to

dv t = √ 2 db t -∇ v V (v t ) dt -∇ v W * v µ t (v t ) dt (4)
in R d with d = d ′ ; here µ t is the law of v t and is solution to the McKean-Vlasov equation

∂µ t ∂t = ∆ v µ t + ∇ v • (µ t (∇ v V + ∇ v W * v µ t )), t > 0, v ∈ R d ′ . (5)
The particle approximation of such a process consists in introducing N processes (X i t ) t≥0 , with 1 ≤ i ≤ N , which evolve no more according to the distribution µ t of the physical system, but according to its discrete counterpart, namely the empirical measure

μN t = 1 N N i=1
δ X i t of the particle system (X 1 t , . . . , X N t ). In other words we let the processes (X i t ) t≥0 solve

dX i t = σ dB i t -b(X i t )dt - 1 N N j=1 c(X i t -X j t )dt, 1 ≤ i ≤ N. (6)
Here the (B i t ) t≥0 's are N independent standard Brownian motions on R d . Under regularity and growth assumptions on σ, b and c, and if the particles are initially distributed in a chaotic way, for instance as independent and identically distributed variables, then μN t indeed converges as N tends to infinity to the distribution µ t of X t . The convergence of μN t is strongly linked with the phenomenon of propagation of chaos for the N interacting particles (X i t ) t≥0 , as we shall see below more in detail, and both issues have been studied in [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF], [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non uniformly case[END_REF], [START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF] and [START_REF] Sznitman | Topics in propagation of chaos[END_REF] for instance. Then quantitative estimates on this convergence have been obtained in [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] at the level of observables, and at the very level of the law in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF].

In this work we go one step further and give precise estimates on the approximation of the law µ [0,T ] of the path X = (X t ) 0≤t≤T on a time interval [0, T ] by the empirical measure

μN [0,T ] = 1 N N i=1 δ X i of the N trajectories X i = (X i t ) 0≤t≤T .
The convergence of μN [0,T ] has been proved in works mentionned above, and here we extend to this new setting the techniques developed in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF].

We shall assume that b and c are Lipschitz on R d . Then global existence and uniqueness, pathwise and in law, of the solutions to (1) and ( 6) are proven in [START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF] for instance for square integrable initial data; moreover the paths are continuous (in time).

To state our main theorem on the particle approximation, we first give some notation. If (S, d) is a separable and complete metric space, and p is a real number ≥ 1, the Wasserstein distance of order p between two Borel probability measures µ and ν on S is

W p (µ, ν) := inf X,Y E d(X, Y ) p 1/p
where X and Y are S-valued random variables with respective law µ and ν. W p induces a metric on the set of Borel probability measures on S with moment S d(x 0 , x) p dµ(x) finite for some (and thus any) x 0 in S; convergence in this metric is equivalent to narrow convergence plus some tightness condition on the moments (see for instance [START_REF] Villani | Topics in optimal transportation[END_REF]).

In this work (S, d) will be the space C := C([0, T ], R d ) of R d -valued continuous functions on [0, T ], equipped with the uniform norm

f ∞ := sup 0≤t≤T |f (t)|;
for this space W p will be denoted W p,[0,T ] .

Theorem 1. Let µ 0 be a probability measure on R d such that R d e a 0 |x| 2 dµ 0 (x) be finite for some a 0 > 0, and let b and c be Lipschitz functions on R d . Given T ≥ 0, let µ [0,T ] be the law of the solution to (1) on [0, T ] for some initial value distributed according to µ 0 . Let also (X i 0 ) 1≤i≤N be N independent random variables with common law µ 0 and μN [0,T ] be the empirical measure of the solutions to (6) on [0, T ], with respective initial value X i 0 . Then, for any α ∈ (0, 1/2), there exist positive constants K and N 0 such that

P W 1,[0,T ] (µ [0,T ] , μN [0,T ] ) > ε ≤ e -K N ε 2 for all ε > 0 and N ≥ N 0 ε -2 exp (N 0 ε -1/α ).
The constants K and N 0 depend on T, b, c, α and a finite square exponential moment of µ 0 .

Kantorovich-Rubinstein dual formulation of the W 1 distance on a general space (S, d) reads

W 1 (µ, ν) = sup [ϕ] 1 ≤1 S ϕ dµ - S ϕ dν (7)
where [ϕ] 1 := sup

x =y |ϕ(x) -ϕ(y)| d(x, y)
• Then the bound in Theorem 1 can be written as

P sup [ϕ] 1 ≤1 1 N N i=1 ϕ(X i ) - C ϕ dµ [0,T ] > ε ≤ e -K N ε 2
where the supremum runs over all 1-Lipschitz functions ϕ on C. By projection at time t, it implies the concentration inequalities

P sup [ϕ] 1 ≤1 1 N N i=1 ϕ(X i t ) - R d ϕ dµ t > ε ≤ e -λN ε 2 (8)
given in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] for the time marginals, provided N ≥ N 0 ε -(d+2) for some N 0 ; here the supremum runs over all 1-Lipschitz functions ϕ on R d . But above all it gives error bounds in the approximation by 1 N N i=1 ϕ(X i ) of the expectation of quantities ϕ(X) which depend on the whole path X. In return we impose a stronger condition on the required size of the sample. An example is the distance d(X, A) = inf{|X t -y|; t ∈ [0, T ], y ∈ A} of the trajectory to a given set A in R d , which measures how close X t has been to A; under the assumptions of Theorem 1, for any α ∈ (0, 1/2) there exist constants K and N 0 such that

P E[d(X, A)] - 1 N N i=1 d(X i , A) > ε ≤ e -KN ε 2
for any Borel set A in R d , ε > 0 and N ≥ N 0 ε -2 exp(N 0 ε -1/α ).

A more involved example of such error bounds will be discussed in detail in Section 5.

As pointed out in [START_REF] Sznitman | Topics in propagation of chaos[END_REF], the convergence of the empirical measure μN [0,T ] towards the distribution µ [0,T ] is strongly linked with the phenomenon of propagation of chaos, namely, that the interacting particles X i tend to behave like independent variables with law µ [0,T ] , as N goes to infinity. For instance, letting μN,2

[0,T ] := 1 N (N -1) i =j δ (X i ,X j )
be the empirical measure of pairs of paths, the asymptotic independence of two paths (among N ) can be estimated as Theorem 2. With the same notation and assumptions as in Theorem 1, for all T ≥ 0 and α ∈ (0, 1/2) there exist positive constants K and N 0 such that

P W 1,[0,T ] (µ [0,T ] ⊗ µ [0,T ] , μN,2 [0,T ] ) > ε ≤ e -K N ε 2 for all ε > 0 and N ≥ N 0 ε -2 exp(N 0 ε -1/α ).
Here the constants K and N 0 depend on T, b, c, α and a finite square exponential moment of µ 0 , and W 1,[0,T ] stands for the Wasserstein distance of order 1 on the product space C × C.

In turn this leads to error bounds in the approximation of functions of the paths of two independent solutions to [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF]. Let for instance (X t ) t≥0 and ( Xt ) t≥0 be the solutions to (1) respectively driven by two independent Brownian motions B and B and with independent initial data. Then, in average, the minimal distance d(X,

X) = inf{|X t -Xt |; t ∈ [0, T ]} between X t and Xt on [0, T ] is approximated by 1 N (N -1) i =j d(X i , X j
) with an error controlled by

P E[d(X, X)] - 1 N (N -1) i =j d(X i , X j ) > ε ≤ e -KN ε 2 for ε > 0 and N ≥ N 0 ε -2 exp(N 0 ε -1/α ).
The proof of Theorem 1 is based on this phenomenon of propagation of chaos: as N goes to infinity, the interacting particles (X i t ) t≥0 tend to behave like the N independent and identically distributed processes (Y i t ) t≥0 solution to

dY i t = σ dB i t -b(Y i t ) dt -c * µ t (Y i t ) dt Y i 0 = X i 0 1 ≤ i ≤ N . (9)
Here µ t is the law of X t , but is also the law of any Y i t and, for each i, (B i t ) t≥0 is the Brownian motion driving the evolution of (X i t ) t≥0 . Then the paths (Y i t ) t≥0 are close to the paths (X i t ) t≥0 and Proposition 3 ensures the existence of a constant C (depending only on b, c and

T ) such that W 1,[0,T ] (µ [0,T ] , μN [0,T ] ) ≤ C W 1,[0,T ] (µ [0,T ] , νN [0,T ] )
holds almost surely, where νN

P W 1,[0,T ] (µ [0,T ] , μN [0,T ] ) > ε ≤ P W 1,[0,T ] (µ [0,T ] , νN [0,T ] ) > ε C ≤ e -K N ε 2
under a condition on ε and N . The proof will be given in greater detail in Section 4 and above all in [START_REF] Bolley | Quantitative concentration inequalities on sample path space for mean field interaction[END_REF]. It is an adaptation of the argument given in [5, Section 2.6] of estimates ( 8) for time marginals. The current proof turns out to be simpler since it consists in fewer steps; in return each step of the present infinite dimensional case, where µ [0,T ] is a measure on the functional space C, exhibits new difficulties with respect to the finite dimensional case of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] where µ t is a measure on R d .

The proof of Theorem 2 consists in writing this coupling argument for pairs of paths and

comparing µ [0,T ] ⊗ µ [0,T ] and νN,2 [0,T ] := 1 N (N -1) i =j δ (Y i ,Y j ) by means of 1 N 2 i,j δ (Y i ,Y j ) .
Let us finally note that it would be desirable to relax the assumptions made on the drift terms b and c: first of all to include the interesting case of the cubic potential W (z) = |z| 3 /3 on R in the case of equation ( 3), which models the interaction among one-dimensional granular media (see [START_REF] Benedetto | A non-Maxwellian steady distribution for one-dimensional granular media[END_REF] for instance); then to treat the fundamental cases of the Coulomb and gravitational potentials in the diffusive Newton's equations (2) (see [START_REF] Dolbeault | Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (large time behavior and steady states)[END_REF] for instance).

It would also be interesting to consider the whole trajectories (X t ) t≥0 on [0, +∞), and derive concentration bounds on functionals such as hitting times for instance.

Before turning to the proofs we briefly give the plan of the paper. In the coming section we reduce our concentration issue on interacting particles to the same issue for independent variables by a coupling argument. In Section 3 we prove a general concentration result for C-valued independent variables and in Section 4 we check that it applies to our framework. An example of error bounds implied by Theorem 1 is finally discussed in Section 5.

Coupling

Let us recall that we want to measure the distance between the law µ [0,T ] of the solution X = (X t ) 0≤t≤T on [0, T ] to (1) and the empirical measure μN [0,T ] of the N solutions X i = (X i t ) 0≤t≤T on [0, T ] to [START_REF] Bolley | Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities[END_REF].

In the following proposition we reduce this issue to measuring the distance between µ [0,T ] and the empirical measure

νN [0,T ] := 1 N N i=1 δ Y i (10) of the N independent solutions Y i = (Y i t ) 0≤t≤T to (9) on [0, T ] :
Proposition 3. Let us assume that there exist real numbers β, γ and Γ such that

(b(x) -b(y)) • (x -y) ≥ β |x -y| 2 , (c(x) -c(y)) • (x -y) ≥ γ |x -y| 2 , |c(x) -c(y)| ≤ Γ|x -y|
for all x and y in R d . Then there exists a constant C depending only on β, γ, Γ and T such that

W 1,[0,T ] (µ [0,T ] , μN [0,T ] ) ≤ C W 1,[0,T ] (µ [0,T ] , νN [0,T ]
) almost surely in the above notation.

Proof. We first follow the lines of the proof of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] or [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] in the case when b = ∇V and c = ∇W , but in the end we want an estimate on the paths. Since for each i both processes X i and Y i are driven by the same Brownian motion B i , the process

X i -Y i satisfies d(X i t -Y i t ) = -b(X i t ) -b(Y i t ) dt -c * μN t (X i t ) -c * µ t (Y i t ) dt. In particular 1 2 d dt |X i t -Y i t | 2 = -b(X i t ) -b(Y i t ) • (X i t -Y i t ) -c * μN t (X i t ) -c * µ t (Y i t ) • (X i t -Y i t ). ( 11
)
We decompose the last term according to

c * μN t (X i t ) -c * µ t (Y i t ) = (c * μN t -c * µ t )(X i t ) + c * µ t (X i t ) -c * µ t (Y i t ) . Then the map c(X i t -• ) is Γ-Lipschitz, so the Kantorovich-Rubinstein formulation (7) of the Wasserstein distance of order 1 ensures that c * (μ N t -µ t )(X i t ) = R d c(X i t -y) d(μ N t -µ t )(y) ≤ Γ w 1 (μ N t , µ t )
where w p is the Wasserstein distance of order p ≥ 1 between measures on R d equipped with the Euclidean distance | • |. Then [START_REF] Dolbeault | Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (large time behavior and steady states)[END_REF] and the assumptions on b and c imply 1 2

d dt |X i t -Y i t | 2 ≤ Γ w 1 (μ N t , µ t ) |X i t -Y i t | -(β + γ) |X i t -Y i t | 2 . ( 12
)
In particular, by integration,

|X i s -Y i s | ≤ Γ s 0 w 1 (μ N u , µ u ) du -(β + γ) s 0 |X i u -Y i u | du (13)
since initially

X i 0 = Y i 0 . But w 1 (μ N u , µ u ) ≤ W 1,[0,u] (μ N [0,u] , µ [0,u] ) since μN
u and µ u are the respective image measures of μN [0,u] and µ [0,u] by the 1-Lipschitz projection

π u from C([0, u], R d ) into R d defined by π u (f ) = f (u). Moreover W 1,[0,u] (μ N [0,u] , µ [0,u] ) ≤ 1 N N i=1 sup 0≤s≤u |X i s -Y i s | + W 1,[0,u] (ν N [0,u] , µ [0,u] ).
by triangular inequality. Hence, by averaging (13) over i, and by Gronwall's lemma,

1 N N i=1 sup 0≤s≤t |X i s -Y i s | ≤ Γ t 0 e (Γ+m)(t-u) W 1,[0,u] (ν N [0,u] , µ [0,u] ) du (14)
where m := max 0, -(β + γ) . On the other hand, for 0

≤ u ≤ t, W 1,[0,u] (ν N [0,u] , µ [0,u] ) ≤ W 1,[0,t] (ν N [0,t] , µ [0,t] ) since νN [0,u] and µ [0,u] are the respective image measures of νN [0,t] and µ [0,t] by the 1-Lipschitz map defined from C([0, t], R d ) into C([0, u], R d ) as the restriction to [0, u]. Hence W 1,[0,t] (μ N [0,t] , νN [0,t] ) ≤ 1 N N i=1 sup 0≤s≤t |X i s -Y i s | ≤ Γ(Γ + m) -1 (e (Γ+m)T -1) W 1,[0,t] (ν N [0,t] , µ [0,t] )
by [START_REF] Kulkarni | A general classification rule for probability measures[END_REF]. This concludes the argument by triangular inequality.

Remark 4. In the case of the granular media equation ( 4), and under convexity assumptions on V and W , such as β > 0, β + 2 γ > 0, it has been proven in [START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF], [START_REF] Cattiaux | Probabilistic approach for granular media equations in the non uniformly case[END_REF], [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] that the time marginal µ t converges, as t goes to infinity, to the stationary solution to [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF]; one can also prove that in expectation observables of the particle system are bounded in time.

Hence, under this kind of assumptions, one can hope for uniform in time constants in this coupling argument. This was obtained in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] for the time marginals of the granular media equation; here, for the whole processes, if β +γ > Γ, then we can let C be (β +γ) (β +γ -Γ) -1 in Proposition 3, independently of T . Indeed, if β + γ > 0, then [START_REF] Fernique | Régularité des trajectoires des fonctions aléatoires gaussiennes[END_REF] leads to

|X i s -Y i s | ≤ Γ s 0 e -(β+γ)(s-u) w 1 (μ N u , µ u ) du by integration. Consequently W 1,[0,t] (μ N [0,t] , νN [0,t] ) ≤ Γ sup 0≤s≤t s 0 e -(β+γ)(s-u) du sup 0≤u≤t w 1 (μ N u , µ u ) ≤ Γ β + γ W 1,[0,t] (μ N [0,t] , µ [0,t] ).
If moreover β + γ > Γ, then by triangular inequality

W 1,[0,t] (μ N [0,t] , µ [0,t] ) ≤ β + γ β + γ -Γ W 1,[0,t] (ν N [0,t] , µ [0,t] ).

A preliminary result on independent variables

Our main theorem on the particle approximation is based on a general concentration result for the empirical measure of C-valued independent and identically distributed random variables. In this section we state this result in a more general formulation and for this purpose we first introduce some notation.

If µ and ν are two measures on C, the relative entropy of ν with respect to µ is defined by

H(ν|µ) = C dν dµ ln dν dµ dµ
if ν is absolutely continuous with respect to µ, and H(ν|µ) = +∞ otherwise. This notion is linked with the Wasserstein distances by the family of transportation inequalities: given p ≥ 1 and λ > 0, a probability measure µ on C satisfies the inequality T p (λ) if

W p (µ, ν) ≤ 2 λ H(ν|µ)
holds true for any measure ν. Moreover, given a Borel probability measure µ on C and N independent random variables (X i ) 1≤i≤N with law µ, we let μN denote their empirical measure, defined as

μN := 1 N N i=1 δ X i .
Given a real number α ∈ (0, 1], we let C α := C α ([0, T ], R d ) be the space of functions in C which moreover are Hölder of order α, equipped with the Hölder norm

f α := sup ( f ∞ , [f ] α ) where [f ] α := sup 0≤t,s≤T |f (t) -f (s)| |t -s| α • C α is
a Borel set of the space C equipped with the topology induced by the uniform norm, and for Borel measures on C, concentrated on C α , we have in the above notation: Proposition 5. Let µ be a Borel probability measure on C satisfying a T p (λ) inequality for some λ > 0 and p ∈ [1, 2], and such that C e a x 2 α dµ(x) be finite for some a > 0 and α ∈ (0, 1].

Then, for any α ′ < α and λ ′ < λ, there exists a constant N 0 such that

P W p,[0,T ] (µ, μN ) > ε ≤ e -βp λ ′ 2 N ε 2 (15)
for any ε > 0 and N ≥ N 0 ε -2 exp (N 0 ε -1/α ′ ), where

β p = 1 if 1 ≤ p < 2 (1 + λ/a) -2 if p = 2.
Here the constant N 0 depends on µ only through λ, a, α and C e a x 2 α dµ(x).

This proposition will be applied with p = 1 to the distribution µ [0,T ] of the process X. For p = 1, a T 1 (λ) inequality is equivalent to the existence of a > 0 such that C e a x 2 ∞ dµ(x) be finite (see [START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF]). Numerical relations between such a and λ are given in [START_REF] Bolley | Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities[END_REF][START_REF] Gozlan | Principe conditionnel de Gibbs pour des contraintes fines approchées et inégalités de transport[END_REF]. In particular this condition if fulfilled if C e a x 2 α dµ(x) is finite, as will be the case for µ [0,T ] .

For p = 1 again, a result by S. Bobkov and F. Götze [START_REF] Bobkov | Exponential integrability and transportation cost related to logarithmic Sobolev inequalities[END_REF], based on [START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF], ensures that a T 1 (λ) inequality for µ is equivalent to the following precise version of the central limit theorem:

sup [ϕ] 1 ≤1 P 1 N N i=1 ϕ(X i ) - C ϕ dµ > ε ≤ e -λ 2 N ε 2 , N ≥ 1 .
By comparison, the bound given in Proposition 5 implies 7), but a modification of the proof would also lead to

P sup [ϕ] 1 ≤1 1 N N i=1 ϕ(X i ) - C ϕ dµ > ε ≤ e -λ ′ 2 N ε 2 , λ ′ < λ , N large enough by (
P sup [ϕ] 1 ≤1 1 N N i=1 ϕ(X i ) - C ϕ dµ > ε ≤ C(ε) e -λ ′ 2 N ε 2 , λ ′ < λ , N ≥ 1
for some computable large constant C(ε). Thus we control a much stronger quantity, up to some loss on the constant in the right-hand side or some condition on the sample size. This result is reasonable in view of Sanov's theorem (stated in [START_REF] Dembo | Large deviations techniques and applications[END_REF] for instance). Indeed, by this theorem, one can hope for an upper bound like

P [W p,[0,T ] (µ, μN ) > ε] ≤ exp (-N inf{H(ν|µ); ν ∈ A})
for large N , where A := {ν; W p,[0,T ] (ν, µ) > ε}. But µ satisfies T p (λ), so that

inf{H(ν|µ); ν ∈ A} ≥ λ 2 ε 2 ,
and one indeed obtains an upper bound like [START_REF] Lorentz | Approximation of functions[END_REF], but only asymptotically, whereas Proposition 5 gives a sufficient size of the sample for the deviation bound to hold. Sanov's theorem does not actually give such an upper bound here: indeed, the space C being unbounded, the closure A of A (for the narrow topology) contains µ itself; in particular inf{H(ν|µ); ν ∈ A} = 0 and Sanov's theorem only gives the trivial bound

P[μ N ∈ A] ≤ exp(-N inf{H(ν|µ); ν ∈ A}) = 1.
Finally this result can be seen as an extension of the following similar concentration result given in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF]Theorem 2.1] in the case of measures on R d : if m satisfies T p (λ), then

P [w p (m, mN ) > ε] ≤ e -γp λ ′ 2 N ε 2 , ε > 0 , N ≥ N 0 max(ε -(d ′ +2) , 1). ( 16
)
Let indeed m be such a measure on R d . Then the law µ of a constant process on [0, T ] initially distributed according to m satisfies the assumptions of Proposition 5 (one can take any a < λ/2), and the bound [START_REF] Malrieu | Logarithmic Sobolev inequalities for some nonlinear PDE's[END_REF] follows by [START_REF] Lorentz | Approximation of functions[END_REF] with the constant γ p obtained in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF]. Note however that the required size of the sample is here larger for small ε.

The proof goes in four steps: we first reduce the problem to a ball B α R of C α , compact in C, which has almost full µ measure by the integrability assumption on µ; then we cover B α R and then P(B α R ) by small balls on which we develop Sanov's argument as in [START_REF] Kulkarni | A general classification rule for probability measures[END_REF] and [10, Exercises 4.4.5 and 6.2.19]; we estimate the cardinality of these covering balls; then we conclude the proof by optimizing the introduced parameters. The argument has the same plan as in the finite dimensional case of [5, Section 3.1], so we shall only sketch it, stressing only on the bounds specific to the present infinite dimensional case. We refer to [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] for further details.

1. Truncation. Given R > 0, to be chosen later on, we let B α R denote the ball {f ∈ C α ; f α ≤ R} of center 0 and radius R in C α . This set B α R is a compact subset of C for the topology induced by the uniform norm

• ∞ . Letting 1 B α R be the indicator function of B α R , we truncate µ into a probability measure µ R on the ball B α R , defined as µ R := 1 B α R µ/µ[B α R ].
Note that µ[B α R ] is positive for R larger than some R 0 depending only on

E := C e a x 2 α dµ(x)
and a. In this step we reduce the concentration issue for C to the same issue for the compact ball B α R , by bounding the quantity P W p,[0,T ] (µ, μN ) > ε in terms of µ R and an associated empirical measure μN

R := 1 N N k=1 δ X k R
where the X k R are independent variables with law µ R .

Bounding by above the • ∞ norms by • α norms when necessary, we obtain the bound

P W p,[0,T ] (µ, μN ) > ε ≤ P W p,[0,T ] (µ R , μN R ) > η ε -2 E 1 p R e -a p R 2 + exp -N θ(1 -η) p ε p -E e (a 1 -a) R 2 ; ( 17
)
here p is any real number in [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF][START_REF] Benedetto | A non-Maxwellian steady distribution for one-dimensional granular media[END_REF], η in (0, 1), ε, θ > 0, a 1 < a and R is constrained to be larger than R 2 max(1, θ 1 2-p ) for some constant R 2 depending only on E, a, a 1 and p. In the case when p = 2, we obtain

P W 2,[0,T ] (µ, μN ) > ε ≤ P W 2,[0,T ] (µ R , μN R ) > η ε -2 E 1 2 R e -a 2 R 2 + exp -N a 1 2 (1 -η) 2 ε 2 -2E 2 e (a 1 -a) R 2 . ( 18
)
2. Sanov's argument on small balls. In view of (17) for p < 2 or (18) for p = 2, we now aim at bounding P [μ N R ∈ A] where

A := ν ∈ P(B α R ); W p,[0,T ] (µ R , ν) ≥ η ε -2 E 1 p R e -a p R 2 .
For this we let δ > 0 and cover A with N p (A, δ) balls B i with radius δ/2 in W p,[0,T ] distance.

Then we develop Sanov's argument on each of these compact and convex balls to obtain

P μN R ∈ A ≤ P μN R ∈ Np(A,δ) i=1 B i ≤ Np(A,δ) i=1 P μN R ∈ B i ≤ Np(A,δ) i=1 exp -N inf ν∈B i H(ν|µ R ) . (19)
Remark 8. The order of magnitude of this covering number in an infinite dimensional setting constitutes a main change by comparison with the finite dimensional setting of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF], and will influence the final condition on the size N of the sample.

Conclusion of the argument.

We first focus on the case when p ∈ [START_REF] Benachour | Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos[END_REF][START_REF] Benedetto | A non-Maxwellian steady distribution for one-dimensional granular media[END_REF]. By estimates ( 17), ( 20) and (21), and given λ 2 < λ and a 1 < a, we obtain the existence of positive constants K 1 , K 2 , K 3 and R 3 depending on E, a, a 1 , α, λ and λ 2 such that

P W p,[0,T ] (µ, μN ) > ε ≤ exp -N K 3 θ ε p -K 4 e (a 1 -a) R 2 + exp K 2 (R ε -1 ) d 3 K 2 (R ε -1 ) 1/α ln max(1, K 2 R ε -1 ) -N λ 2 2 ε 2 -K 1 R 2 e -αR 2 (22)
for all ε, θ > 0 and R ≥ R 3 max(1, θ 1 2-p ), and for a constant K 4 = K 4 (θ, a 1 ). Then let λ 3 < λ 2 . One can prove that the second term in the right-hand side in ( 22) is bounded by exp -

λ 3 2 N ε 2 provided R 2 ≥ A max(1, ε 2 , ln(ε -2 )) , N ε 2 ≥ B 3 C(Rε -1 ) 1/α (23)
for positive constants A, B and C depending also on λ 3 . Moreover, for θ = ε 2-p λ 3 2 K 3 , also the first term in the right-hand side in ( 22) is bounded by exp -

λ 3 2 N ε 2 as soon as R 2 ≥ R 4 max(1, ln(ε -2 )), for a constant R 4 depending on λ 3 . Letting R = ε 1 C ln 3 ln N ε 2 B α if ε ∈ (0, 1) and R = √ A ε otherwise, and α ′ < α, both
conditions in (23) hold true as soon as N ≥ N 0 ε -2 exp(N 0 ε -1/α ′ ) for a constant N 0 depending on E, a, λ, λ 3 , α and α ′ . Finally, given λ ′ < λ 3 < λ, this condition ensures that

P W p,[0,T ] (µ, μN ) > ε ≤ 2 exp - λ 3 2 N ε 2 ≤ exp - λ ′ 2 N ε 2 ,
for a possibly larger N 0 . This concludes the argument in the case when p ∈ [1, 2).

In the case when p = 2, given 0 < η < 1, λ 3 < λ 2 and a 2 < a 1 , the same condition on N and ε (for some N 0 ) is sufficient for the bound

P W 2,[0,T ] (µ, μN ) > ε ≤ exp - λ 3 2 η 2 N ε 2 + exp - a 2 2 (1 -η) 2 N ε 2
to hold (by ( 18)). One optimizes this bound by letting

a 2 = a λ 3 λ (∈ (0, a)) and η = √ a 2 √ a 2 + √ λ 3 • Given λ ′ < λ 3 < λ, this ensures the existence of N 0 such that P W 2,[0,T ] (µ, μN ) > ε ≤ 2 exp - λ 3 2 a ( √ a + √ λ) 2 N ε 2 ≤ exp - λ ′ 2 1 (1 + λ/a) 2 N ε 2 .
for any ε > 0 and N ≥ N 0 ε -2 exp(N 0 ε -1/α ′ ). This concludes the proof of Proposition 5 in this second and last case.

Integrability in Hölder norm

In Section 2 we have reduced the issue of measuring the distance between µ [0,T ] and μN

[0,T ]
to measuring the distance between µ [0,T ] and the empirical measure νN [0,T ] of N independent random variables drawn according to µ [0,T ] .

We now solve the latter issue by proving that the measure µ [0,T ] fulfills the hypotheses of Proposition 5 with p = 1, namely, that there exist α ∈ (0, 1] and a > 0 such that E exp(a X 2 α ) := C e a x 2 α dµ [0,T ] (x) < +∞.

Proposition 9. Let µ 0 be a probability measure on R d with a finite square exponential moment and let X 0 be with law µ 0 . Given T ≥ 0, let X be the solution on [0, T ] to (1) starting at X 0 , where b and c are Lipschitz on R d . Then, for any α ∈ (0, 1/2), there exists a > 0, depending on µ 0 only through a finite square exponential moment, such that E exp(a X 2 α ) be finite.

Assuming this result for the moment we can now conclude the proof of Theorem 1. Let indeed α be given in (0, 1/2) and α 0 ∈ (α, 1/2). Then, by Propositions 5 and 9, applied with α = α 0 and α ′ = α, there exist positive constants K and Ñ0 , depending on α 0 , α, T and a square exponential moment of µ 0 , such that

P [W 1,[0,T ] (µ [0,T ] , νN [0,T ] ) > ε] ≤ e -KN ε2
for any ε > 0 and N ≥ Ñ0 ε-2 exp( Ñ0 ε-1/α ), where νN [0,T ] is defined by ( 9) and [START_REF] Dembo | Large deviations techniques and applications[END_REF]. Then, by Proposition 3, there exist some constants C, depending only on T , and then K and N 0 , depending on α 0 , α, T and a finite square exponential moment of µ 0 , such that

P [W 1,[0,T ] (µ [0,T ] , μN [0,T ] ) > ε] ≤ P [W 1,[0,T ] (µ [0,T ] , ν[0,T ] ) > ε/C] ≤ e -KN ε 2
for any ε > 0 and N ≥ N 0 ε -2 exp(N 0 ε -1/α ). This concludes the argument. 2

Proof of Proposition 9. We separately prove the existence of positive constants a 1 and a 2 such that E exp(a 1 X 2 ∞ ) and E exp(a 2 [X] 2 α ) be finite. We only sketch the proof for the uniform norm and give it in detail for the Hölder seminorm, refering to [START_REF] Bolley | Quantitative concentration inequalities on sample path space for mean field interaction[END_REF] for further details.

1. For the expectation in uniform norm: 1.1. we first prove that E|X t | 2 is bounded on [0, T ] by applying Itô's formula to (|X t | 2 ) t≥0 ; 1.2. then we prove the existence of K > 0 such that E exp(K|X t | 2 ) be bounded on [0, T ]. In fact we prove the existence of a positive and decreasing function k on [0, T ] such that E exp(k(t) |X t | 2 ) be finite and bounded on [0, T ]; for that purpose we apply Itô's formula to the process (exp k(t)|X t | 2 ) t≥0 . Then we let K = k(T ) > 0;

1.3. we finally use Doob's inequalities to obtain the existence of a 1 > 0, depending on µ 0 only through a square-exponential moment, such that E exp(a 1 X 2 ∞ ) be finite. 2. We now turn to the expectation in Hölder seminorm. We first write the solution as

X t = X 0 + σ B t - t 0 b(X s ) + c * µ s (X s ) ds, so that [X] α ≤ [σ B] α + . 0 b(X s ) + c * µ s (X s ) ds
E exp(a 2 [X] 2 α ) ≤ E exp(4 a 2 σ 2 [B] 2 α ) 1/2 E exp 4 a 2 . 0 (b(X s ) + c * µ s (X s )) ds 2 α 1/2 . But E exp(4 a 2 σ 2 [B] 2 α
) is finite for a 2 small enough (see [START_REF] Fernique | Régularité des trajectoires des fonctions aléatoires gaussiennes[END_REF]Theorem 1.3.2] for instance). On the other hand, by step 1.1 and assumption on b and c, there is a constant A such that

(X s )+c * µ s (X s ) ds α ≤ sup 0≤s<t≤T 1 |t -s| α t s A+(B+Γ)|X u | du ≤ T 1-α A+(B+Γ) X ∞
almost surely, and

E exp 4 a 2 . 0 b(X s )+c * µ s (X s ) ds 2 α ≤ exp(8 a 2 T 2-2α A 2 ) E exp 8 a 2 T 2-2α (B +Γ) 2 X 2 ∞
which by step 1 is finite as soon as 8 a 2 T 2-2α (B + Γ) 2 ≤ a 1 .

On the whole, E exp(a 2 [X] 2 α ) is indeed finite for a 2 small enough, depending on µ 0 only through a finite square exponential moment, which concludes the argument.

An example of application

In this section we give an instance of error bound in the approximation by 1 N N i=1 ϕ(X i ) of the expectation of a quantity ϕ(X) depending on the whole path of the considered process.

Let 0 < t 1 < • • • < t n ≤ T and for 1 ≤ j ≤ n let B j be the (for instance closed) ball B(x j , r j ) of center x j and radius r j > 0 in the Euclidean space R d . Then we can approximate the probability P X t j ∈ B j ; 1 ≤ j ≤ n for X t to be in B j at t = t j for each j as follows:

In the notation and assumptions of Theorem 1, assume moreover that all partial derivatives ∂ m b and ∂ m c of b and c are continuous and bounded on R d , for any multi-index m ∈ N d with 1 ≤ |m| ≤ s where s is the smallest integer number larger than d/2, and that the diffusion matrix σ is (for instance) a nonzero multiple of the identity, as in the example (4). Then, for any α ∈ (0, 1/2), there exist positive constants K and N 0 such that

P   1 N N i=1 n j=1 χ j (X i t j ) -P X t j ∈ B j ; 1 ≤ j ≤ n > ε   ≤ e -K N ε 4
for all ε ∈ (0, 1) and N ≥ N 0 ε -4 exp (N 0 ε -2/α ).

Here χ j is the indicator function of the ball B j defined by χ j (x) = 1 if x ∈ B j and 0 otherwise, and let us note that only the ε ∈ (0, 1) have to be considered.

Indeed 1 N N i=1 n j=1 χ j (X i t j ) -P X t j ∈ B j ; 1 ≤ j ≤ n = C n j=1 χ j (f (t j )) dμ N (f ) - C n j=1 χ j (f (t j )) dµ(f ) ≤ C n j=1 χ j (f (t j )) - n j=1 ϕ j (f (t j )) dμ N (f ) + C n j=1 ϕ j (f (t j )) d(μ N -µ)(f ) + C n j=1 ϕ j (f (t j )) - n j=1 χ j (f (t j )) dµ(f ). ( 24
)
Here ϕ j is the R d to R map defined by ϕ j (x) = 1 -d(x, B j ) δ + for some 0 < δ ≤ min 1≤j≤n r j to be chosen later, where d(x, A) is the distance of a point x to a set A and u + = max(u, 0) for all real u. For simplicity we write μN , µ and W 1 instead of μN [0,T ] , µ [0,T ] and W 1,[0,T ] respectively. The second term in (24), which is the main term, will be bounded by the Kantorovich-Rubinstein formulation [START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF]. Indeed, if f and g are two functions in C, then

n j=1 ϕ j (f (t j )) - n j=1 ϕ j (g(t j )) ≤ n j=1 |ϕ j (f (t j )) -ϕ j (g(t j ))|
since the maps ϕ j take values in [0, 1] and because of the following elementary bound: Then the maps ϕ j are δ -1 -Lipschitz, so

n j=1 |ϕ j (f (t j )) -ϕ j (g(t j ))| ≤ nδ -1 f -g ∞ .
Consequently the map f → n j=1 ϕ j (f (t j )) is nδ -1 -Lipschitz on C, and by [START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF] the second term in (24) is bounded by nδ -1 W 1 (μ N , µ).

Then the first term in (24) is bounded by Moreover, if for any j we let µ t j be the marginal at time t j of the distribution µ, then

C γ j (f (t j ))dµ(f ) = R d γ j (x) dµ t j (x) ≤ µ t j [C j ]
where C j = B(x j , r j + δ) \ B(x j , r jδ), since γ j is zero outside of C j and bounded by 1 on C j .

In the same way the third term in (24) is bounded by

n j=1 µ t j [D j ]
where D j = B(x j , r j +δ)\B(x j , r j ), since ϕ j -χ j is zero outside of D j and bounded by 1 on D j . Now, under our assumptions, we can adapt the techniques in [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF]Theorem B.1] to prove that for any t > 0 the time marginal µ t belongs to the Sobolev space H s (R d ), with s > d/2, whence to L ∞ (R d ). Moreover there exists a constant K 1 , depending only on t 1 , t n , d, b, c, σ and a square exponential moment of µ 0 , such that sup

t 1 ≤t≤tn µ t L ∞ ≤ K 1 .
In particular the first and third terms in (24) are together bounded by In the case when ε ≤ 2K 2 Rr we choose δ = ε 2K 2 R ∈ (0, r], so that, by Theorem 1, for any α ∈ (0, 1/2) there exist some constants K and N 0 , independent of ε and N , such that As a conclusion, there exist two new constants K and N 0 , depending on µ 0 only through a square exponential moment, such that

P W 1 (μ N , µ) > δ 2n (ε -K 2 Rδ) = P W 1 (μ N , µ) > ε 2 8K 2 Rn ≤ exp -KN ε 2 8K 2 Rn
P 1 N N i=1 n j=1
χ j (X i t j ) -P X t j ∈ B j ; 1 ≤ j ≤ n > ε ≤ e -KN ε 4 for all ε ∈ (0, 1) and N ≥ N 0 ε -4 exp(N 0 ε -2/α ). This concludes the argument.

δ

  Y i and Y i = (Y i t ) 0≤t≤T ; hence controlling the distance between µ [0,T ] and μN [0,T ] reduces to the same issue with µ [0,T ] and νN [0,T ] . But, by definition, the N paths Y i for 1 ≤ i ≤ N are independent and distributed according to µ [0,T ] . Then Propositions 5 and 9 ensure good concentration estimates for the empirical measure νN [0,T ] around the common law µ [0,T ] . In the end we obtain the bound

  ) ds is an antiderivative of ϕ. Hence, by Cauchy-Schwarz' inequality,

  |b(x) + c * µ s (x)| ≤ A + (B + Γ)|x| for all x ∈ R d and s ∈ [0, T ]; here B and Γ are the Lipschitz seminorms of b and c. In particular . 0 b

Lemma 10 .

 10 If a j and b j are real numbers in [0, 1] for 1 ≤ j ≤ n, then jb j |.

+.

  f (t j ))ϕ j (f (t j ))| dμ N (f ) by Lemma 10 again. Now, for all 1 ≤ j ≤ n and x ∈ R d ,|χ j (x)ϕ j (x)| = ϕ j (x)χ j (x) ≤ γ j (x) where γ j (x) = 1 -d(x, B j ) + d(x, R d \ B j ) δ The map x → d(x, B j ) + d(x, R d \ B j ) is1-Lipschitz, so γ j is δ -1 -Lipschitz, and by[START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF] again the first term in (24) is bounded byn j=1 C γ j (f (t j ))dμ N (f ) ≤ n j=1 δ -1 W 1 (μ N , µ) + C γ j (f (t j ))dµ(f ) .

nδ - 1

 1 W 1 (μ N , µ) + K 1 n j=1 Leb[C j ] + Leb[D j ]where Leb[A] stands for the Lebesgue measure of a Borel set A in R d .MoreoverLeb[C j ] = ω d (r j + δ) d -(r jδ) d ≤ 2 d ω d r d-1 j δ and Leb[D j ] = ω d (r j + δ) dr d j ≤ (2 d -1) ω d r d-1 jδ for any 1 ≤ j ≤ n and 0 < δ ≤ r j , where ω d is the Lebesgue measure of the Euclidean unit ball in R d . Hence it follows from (24) that for a constant K 2 , independent of N and the r j ,X i t j ) -P X t j ∈ B j ; 1 ≤ j ≤ n ≤ 2nδ -1 W 1 (μ N , µ) + K 2 i t j ) -P X t j ∈ B j ; 1 ≤ j ≤ n > ε ≤ P W 1 (μ N , µ) > δ 2n (ε -K 2 Rδ)for any 0 < δ ≤ r, in the notation r = min 1≤j≤n r j and R =

2 for

 2 when ε > 2K 2 Rr we choose δ = r, so thatP W 1 (μ N , µ) > δ 2n (ε -K 2 Rδ) ≤ P W 1 (μ N , µ) > εr 4n ≤ exp -KN εr 4n

Acknowledgments. The author thanks Professor M. Ledoux for his kind interest in this work, in particular pointing out Reference [12], and Professors A. Guillin and C. Villani for stimulating discussions during the preparation of [5].

Then, from the T p (λ) inequality for µ, one establishes an approximate T p (λ) inequality for µ R : namely, for any λ 1 < λ there exists K 1 such that

for any measure ν on B α R . With this inequality in hand, given 1 ≤ p < 2 and λ 2 < λ 1 < λ, one deduces from [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF] the existence of positive constants δ 1 , η 1 and K 1 such that

(20) where we have chosen δ := δ 1 ε and η := η 1 .

In the case when p = 2, we do not choose η at this stage, and simply obtain

where δ := δ 1 ε.

3. Estimate on the covering number N p (A, δ). Given r > 0, the covering number N (S, r) of a compact metric space (S, d) is the smallest integer n such that S can be covered by n balls centered in S and of radius r in d metric. On N (B α R , r) we have the following bound: Lemma 6. Given the space R d with d ≥ 1, some positive numbers T , R, r and α with r < R and α ≤ 1, the covering number N (B α R , r) of B α R , equipped with the uniform norm, satisfies

The proof and a lower bound of the same order are given in detail in [START_REF] Bolley | Quantitative concentration inequalities on sample path space for mean field interaction[END_REF]; in our case these bounds make more precise those given in [START_REF] Gozlan | Principe conditionnel de Gibbs pour des contraintes fines approchées et inégalités de transport[END_REF], [START_REF] Lorentz | Approximation of functions[END_REF] or [START_REF] Van Der Vaart | Weak convergence and empirical processes[END_REF].

Moreover, given a compact metric space (S, d), p ≥ 1 and δ > 0, the following result of [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] (see also [START_REF] Dembo | Large deviations techniques and applications[END_REF], [START_REF] Kulkarni | A general classification rule for probability measures[END_REF]) gives an upper bound on the covering number N p (P(S), δ) of the compact metric space (P(S), W p ) of probability measures on S: Lemma 7. Let (S, d) be a compact metric space with finite diameter D, and p and δ be positive numbers with p ≥ 1. Then the covering numbers of S and P(S) satisfy

Since B α R equipped with the metric defined by the uniform norm is a compact metric space with finite diameter 2R, it follows that the space P(B α R ) of probability measures on B α R can be covered by N p (P(B α R ), δ) balls of radius δ in Wasserstein distance W p , with

Then A is a subset of P(B α R ), so, for a constant K 2 depending neither on ε nor on R, N p (A, δ 1 ε) ≤ exp K 2 (R ε -1 ) d 3 K 2 (R ε -1 ) 1/α ln max(1, K 2 R ε -1 ) .

(21)


L Verleye
email: leenverleye@yahoo.com

I Vergote
email: ignace.vergote@uz.kuleuven.ac.be

M.D Ate G J Van Der Zee
email: a.g.j.van.der.zee@og.umcg.nl

Patterns of care in surgery for ovarian cancer in Europe

Keywords: ovarian cancer, surgery, quality of health care, patterns of care

Quality of surgery is one of the most important determinants of the outcome in ovarian cancer patients. Surgery by a gynaecological oncologist in a specialised, highvolume environment and removal of all visible tumours are associated with a higher likelihood of favourable outcome for patients with advanced-stage ovarian cancer. Population-based studies in Europe however show that a substantial number of patients do not receive optimal surgical care. Less than half of the patients suffering from advanced-stage ovarian cancer are operated by a gynaecological oncologist. Also the proportion of patients operated in a high-volume or specialised hospital is lower than 50%. In a substantial number of patients, minimum standard procedures are not performed and optimal tumor debulking is not achieved. To improve the quality of care, efforts are needed to develop and implement robust evidence-based European guidelines, provide surgical training for gynaecological oncologist and establish comprehensive cancer networks with sufficient resources.

Introduction

In Europe, approximately 66.700 new ovarian cancer cases are diagnosed yearly, with the highest incidence in the Nordic countries and the UK [1;2]. The majority of ovarian cancer patients present with advanced-stage disease as more specific symptoms only occur when the disease has spread throughout the peritoneal cavity. State-of-the-art treatment for advanced ovarian cancer (FIGO stage III-IV) consists of cytoreductive surgery and platinum-based chemotherapy [START_REF] Du Bois | consensus statements on the management of ovarian cancer: final document of the 3rd International Gynecologic Cancer Intergroup Ovarian Cancer Consensus Conference (GCIG OCCC 2004)[END_REF]. The therapeutic value of debulking surgery has been suggested already early in the previous century [START_REF] Meigs | Tumors of the Female Pelvic Organs[END_REF]. In 1975, Griffiths confirmed the role of cytoreduction showing survival benefit when all tumour lesions larger than 1.5cm were removed. Ever since, a plethora of evidence demonstrated a better outcome for patients with residual disease after surgery smaller than 2cm, 1cm or no macroscopic disease [5;6]. A more recent meta-analysis of the Arbeitsgemeinschaft Gynäkologische Onkologie (AGO) Study Group included the data of 3 prospective randomized trials with 3126 patients. These data have clearly shown that there is still a significant benefit for residuals up to 1 cm compared with larger remaining tumor masses, but the largest benefit could be reached by complete resection and therefore complete resection is now regarded as the aim of surgery in ovarian cancer [START_REF] Harter | Prognostic factors for complete debulking in first-and second-line ovarian cancer[END_REF]. Although extent of residual disease at the end of surgery has been recognised as one of the most important prognostic factors for advanced ovarian cancer already a long time ago [8;9], an ongoing matter of debate has been whether the extent of residual disease at the end of surgery either reflects the extent and quality of surgery or an intrinsic, biological characteristic of the tumours [10;11]. In the past these uncertainties lead to different approaches to ovarian cancer surgery with the use of advanced techniques such as diaphragmatic surgery and partial pancreatectomy [12;13] on one side and pleas for a more modest approach on the other [START_REF] Crawford | Does aggressive surgery only benefit patients with less advanced ovarian cancer? Results from an international comparison within the SCOTROC-1 Trial[END_REF]. However, more recent studies clearly indicate that surgical outcome is more related with surgeon and hospital related factors than with spreading pattern of the tumour [START_REF] Du Bois | Variations in institutional infrastructure, physician specialization and experience, and outcome in ovarian cancer: a systematic review[END_REF]. Data on the benefit of surgery performed by gynaecological oncologists, specialised hospitals and surgical volume have led to initiatives such as centralisation of care and subspecialty training in some European countries [START_REF]Accounts commission for Scotland. Fighting the silent killer. Optimising ovarian cancer management in Scotland[END_REF][START_REF] Vergote | How to organize gynecologic oncology in the future: results of an international questionnaire?[END_REF][START_REF] Marx | Organisation and quality of primary surgical intervention for ovarian cancer in Denmark[END_REF][START_REF] Vernooij | Specialized care and survival of ovarian cancer patients in The Netherlands: nationwide cohort study[END_REF][START_REF]Guidance on Commissioning Cancer Services. Improving Outcomes in Gynaecological Cancers[END_REF].

As a result, current surgical practice and its organization differ substantially in Europe. National and regional cancer registries all over Europe collect data on cancer incidence and survival and are used to compare mortality and survival rates between countries and over time in Europe. However, these European studies have insufficient data available to assess treatment related factors [1;21]. In this paper, we aim at giving an overview of patterns of care in surgery for ovarian cancer based on population based studies from different European countries with special attention for the advanced stages (FIGO stage III-IV).

Specialisation of surgeons

Population based data on the number of advanced ovarian cancer patients operated by a gynaecological oncologist are available in the literature for Sweden [START_REF] Skirnisdottir | Prognostic factors for surgical outcome and survival in 447 women treated for advanced (FIGO-stages III-IV) epithelial ovarian carcinoma[END_REF], Finland [START_REF] Kumpulainen | Surgical treatment of ovarian cancer in different hospital categories--a prospective nation-wide study in Finland[END_REF], Switzerland [START_REF] Petignat | Surgical management of epithelial ovarian cancer at community hospitals: A population-based study[END_REF], United Kingdom [START_REF] Junor | Specialist gynaecologists and survival outcome in ovarian cancer: a Scottish national study of 1866 patients[END_REF][START_REF] Bailey | Stage III and IV ovarian cancer in the South West of England: five-year outcome analysis for cases treated in 1998[END_REF][START_REF] Shylasree | Survival in ovarian cancer in Wales: Prior to introduction of all Wales guidelines[END_REF], Denmark [START_REF] Soegaard | The results of treatment of epithelial ovarian cancer after centralisation of primary surgery. Results from North Jutland, Denmark[END_REF], the Netherlands [START_REF] Engelen | Surgery by consultant gynecologic oncologists improves survival in patients with ovarian carcinoma[END_REF] and Norway [START_REF] Paulsen | Improved short-term survival for advanced ovarian, tubal, and peritoneal cancer patients operated at teaching hospitals[END_REF]. Data are summarised in Table 1. The proportion of patients operated on by a gynaecological oncologist varies between 0% in Valais, Switzerland and 95% in North Jutland, Denmark.

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Hospital volume

Population based data on the number of patients with advanced-stage ovarian cancer treated in high-volume hospitals have been published for Austria [31;32], Finland [START_REF] Kumpulainen | The effect of hospital operative volume, residual tumor and first-line chemotherapy on survival of ovarian cancer -a prospective nation-wide study in Finland[END_REF] and Germany [START_REF] Du Bois | Pattern of care and impact of participation in clinical studies on the outcome in ovarian cancer[END_REF]. Data are summarized in Table 2. The proportion of patients operated in high-volume hospitals varies between 25% in Finland and 50% in Tyrol, Austria.

Type of hospital

Population based data on the number of patients with advanced-stage ovarian cancer treated in specialized hospitals have been published for the Netherlands [START_REF] Vernooij | Specialized care and survival of ovarian cancer patients in The Netherlands: nationwide cohort study[END_REF], Germany [35;36], United Kingdom [27;37;38], Norway [START_REF] Paulsen | Improved short-term survival for advanced ovarian, tubal, and peritoneal cancer patients operated at teaching hospitals[END_REF]. Data are summarized in Table 3. The proportion of patients operated in specialized hospitals varies between 18% in the Netherlands and 64% in Hesse, Germany.

Surgical outcome

Population-based data on the number of optimally debulked (advanced) ovarian cancer patients are available for Austria [START_REF] Marth | Influence of department volume on survival for ovarian cancer: results from a prospective quality assurance program of the Austrian Association for Gynecologic Oncology[END_REF], Sweden [22;39], Switzerland [START_REF] Petignat | Surgical management of epithelial ovarian cancer at community hospitals: A population-based study[END_REF], United Kingdom [START_REF] Junor | Specialist gynaecologists and survival outcome in ovarian cancer: a Scottish national study of 1866 patients[END_REF], Germany [34;35], Denmark [18;28], Finland [START_REF] Kumpulainen | Surgical treatment of ovarian cancer in different hospital categories--a prospective nation-wide study in Finland[END_REF], Norway [START_REF] Paulsen | Improved short-term survival for advanced ovarian, tubal, and peritoneal cancer patients operated at teaching hospitals[END_REF], Belgium [START_REF] Vergote | VWOG-Oncologieregistratie 1998-2002 deel 5: Ovariumcarcinoom[END_REF] and the Netherlands [START_REF] Engelen | Surgery by consultant gynecologic oncologists improves survival in patients with ovarian carcinoma[END_REF]. Data are summarized in Table 4.

Procedures

Ovarian tumours tend to spread throughout the peritoneal cavity and pelvic and paraaortic lymph nodes with possible metastases on every serosal surface within the abdomen. Hysterectomy, bilateral salpingo-oophorectomy and infracolic omentectomy are considered the basis of cytoreductive surgery and can be supplemented with many other procedures necessary to remove all visible tumours. As such, it is difficult to define a standard operation for advanced ovarian cancer. However, some population studies report on the proportion of patients in whom minimal procedures were performed Data from the UK [START_REF] Wolfe | Management and survival of ovarian cancer patients in south east England[END_REF], Switzerland [START_REF] Petignat | Surgical management of epithelial ovarian cancer at community hospitals: A population-based study[END_REF], Germany [START_REF] Munstedt | Shortcomings and deficits in surgical treatment of gynecological cancers: a German problem only?[END_REF], Finland [START_REF] Kumpulainen | Surgical treatment of ovarian cancer in different hospital categories--a prospective nation-wide study in Finland[END_REF], Belgium [START_REF] Vergote | VWOG-Oncologieregistratie 1998-2002 deel 5: Ovariumcarcinoom[END_REF], the Netherlands [START_REF] Engelen | Surgery by consultant gynecologic oncologists improves survival in patients with ovarian carcinoma[END_REF], Denmark [START_REF] Marx | Organisation and quality of primary surgical intervention for ovarian cancer in Denmark[END_REF] and Greece [START_REF] Pectasides | Epithelial ovarian cancer in Greece: a retrospective study of 1,791 patients by the Hellenic Cooperative Oncology Group (HeCOG)[END_REF] are summarized in Table 5.

Discussion

The extent of residual disease after surgery is the most important prognostic factor for survival of patients with advanced-stage ovarian cancer. As such, the quality of surgery appears to have an important impact on survival. A meta-analysis by Bristow et al. has shown a 5.5% survival benefit for every 10% increase of optimally debulked tumours [START_REF] Bristow | Survival effect of maximal cytoreductive surgery for advanced ovarian carcinoma during the platinum era: a meta-analysis[END_REF]. Several factors related to the quality of surgery have been identified on a structural, outcome and process level. In summary, one could postulate that surgery leading to the highest chance of survival is performed by an experienced gynaecological oncologist in a high-volume, specialised hospital. At the start of surgery, a full assessment of peritoneal cavity and pelvic and para-aortic lymph nodes should be performed and if possible no macroscopic tumour should be left in situ at the end of In contrast with the expert opinions, population-based studies as summarised in this manuscript show that optimal surgical care is achieved for only a minority of patients in Europe. A limitation of our review is, however, that our findings may not be fully representative because the scarcity of published data and limited access to local journals in local languages. Furthermore, the majority of studies date from several years ago and may no longer reflect the current situation.

Nevertheless, available data from many different European countries show a discrepancy between the ideal surgical care and daily practice in the European population and also a significant variation between regions is observed. Several factors may possibly explain this gap between the ideal and day-to-day practice. First, many gynaecologists are reluctant to refer patients suspected for ovarian because of fear that the risks and morbidity of 'aggressive' debulking of advanced-stage ovarian cancer overweigh the potential benefits as no randomised controlled trial is available. Second, the majority of gynaecologists received no formal training in the surgical techniques frequently used in debulking ovarian cancer. Third, some patients need emergency surgery without time for referral or are considered inoperable due to severe co-morbidity or poor performance status. Finally, the local health care system and available resources may be insufficient to deliver high quality care to all patients.

To improve quality of care, physicians first of all have to be aware of and agree with best practice guidelines [START_REF] Cabana | Why don't physicians follow clinical practice guidelines? A framework for improvement[END_REF]. Indeed, in the case of debulking surgery for ovarian cancer, this may be an important barrier [50;51]. The lack of prospective, randomized trials is often quoted as a reason to refrain from 'aggressive' surgery to achieve complete cytoreduction [10;50]. This may be inappropriate as the evidence delivered by well designed, large observational studies and meta-analyses should not be considered as inferior [9;15;52;53]. Further discussion with statisticians regarding the correct interpretation and value of available evidence may be more useful.

To achieve complete cytoreduction in the majority of patients, advanced surgical techniques are needed. Bowel resections, diaphragmatic surgery, pelvic and paraaortic lymphadenectomy, supracolic omentectomy, splenectomy and peritonectomy should ideally be part of the skills of the surgeon, hence specialized training is needed. The European Society of Gynaecological Oncology (ESGO) has developed a training program and certification of training centres. Postgraduate training in and recognition of gynaecological oncology as a subspeciality is now available in a growing number of European countries [17;54]. Not only is the training of the operating gynaecologist important to ensure favourable outcome, the expertise of all involved care providers plays a role in optimizing the quality of care [START_REF] Verleye | Surgical program building in advanced ovarian cancer: European perspective[END_REF]. The European Commission has engaged itself to fight the cancer burden in Europe with the concentration of complex interventions in comprehensive cancer centres as an important part of their strategy [56;57]. Being a relatively rare disease, ovarian cancer is can certainly benefit from centralized care in dedicated centres. It is important however that a centralisation policy is accompanied with solid diagnostic and referral pathways and ensured accessibility of care. Therefore,

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6 governments should provide sufficient resources and infrastructure [START_REF] Albreht | Making progress against cancer in Europe in 2008[END_REF][START_REF] Crawford | Centralisation of services for the management of ovarian cancer: arguments against[END_REF][START_REF] Petignat | Why do ovarian cancer patients not consult a gynecologic oncologist?[END_REF]. Some regional or national initiatives for centralization have already been taken and appear to lead to the desired improvement in surgical care for ovarian cancer patients [16;60-62]

Conclusion

Quality of surgery is one of the most important determinants of the outcome of ovarian cancer patients. Surgery by a gynaecological oncologist in a specialized environment and removal of all visible tumour are associated with a high likelihood of favourable outcome for patients with advanced ovarian cancer. Population-based studies in Europe however show that a substantial number of women currently do not receive optimal surgical care. To improve the quality of care, efforts are needed to develop robust evidence-based guidelines, provide excellent surgical training for gynaecological oncologist and establish comprehensive cancer networks with sufficient resources. 3. Details of population-based studies of (advanced) ovarian cancer: surgery in specialized hospitals

  8;15;43]. These findings are reflected in published guidelines and quality indicators agreed on by experts from all over the world[3;44-48].

Table 1 .

 1 Details of population-based studies of (advanced) ovarian cancer: Surgery by a Gynaecological Oncologist For voluntary registration, the percentage of patients included is indicated between (). b. Surgery for ovarian cancer is centralized in one university hospital in the region, effectively covering 95.5% of the cases during the reported period

	Author	Year Country	Data source a	Time period	Definition gynecological	Number of patients operated by
					oncologist	Gynecological Oncologist
	Junor	1999 UK (Scotland)	Regional cancer	1987 and	Defined by committee of	351/1866 (18.8%)
			registry	1992-1994	gynecologists
	Petignat	2000 Switzerland	Regional cancer	1989-1995	Not specified	0/102 (0%)
		(Valais)	registry
	Soegaard	2005 Denmark (North	Central hospital	1999-2002	Not specified	95.5%
		Jutland)	database b
	Kumpulainen 2006 Finland	Voluntary national	1999	Not specifed	124/307 (40%)
			survey (88%)
	Bailey	2006 UK (South West	Regional database	1998	Not specified	70%
		England)
	Engelen	2006 The Netherlands	Regional database	1994-1997	Not specified	184/512 (36%)
		(Northern
		region)
	Paulsen	2006 Norway	National cancer	2002	> 1 year of training at	75/198 (37.8%)
			registry		Norwegian Radium
					Hospital
	Shylasree	2006 Wales	Regional audit	1999	Not specified	32/250 (13%)
			project
	Skirnisdottir 2007 Sweden	Regional cancer	1975-1993	Not specified	137/447 (31%)
		(Örebro)	registry
	a.

Table 2 .

 2 Details of population-based studies of (advanced) ovarian cancer: Surgery in high-volume hospitals

	Author	Year Country	Data source a	Time period Definition large hospital	Number of patients operated in
						high-volume hospitals
	Kumpulainen 2002 Finland	National	1983-1994	Upper quartile	990/3851 (25.7%)
			cancer registry
	Du Bois	2005 Germany	Voluntary	2001	≥ 16 OvCa pts yearly	171/352 (48.6%)
			national QA
			program (48%)
	Oberaigner	2006 Austria	Regional	1988-2000	≥ 24 OvCa pts yearly	455/911 (50%)
		(Tyrol)	cancer registry
	Marth	2009 Austria	Voluntary	1999-2004	≥ 24 OvCa pts yearly	492/1948 (25.5%)
			national QA
			program (40%)

a. For voluntary registration, the percentage of patients included is indicated between ().

Acknowledgements

The authors would like to thank Fonds Cancer (FoCa) from Belgium for supporting the research fellowship of Dr. Leen Verleye.

Conflict of interest

The authors declare that there are no conflicts of interest.

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