Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures

We use Wasserstein metrics adapted to study the action of the flow of the BBM equation on probability measures. We prove the continuity of this flow and the stability of invariant measures for finite times.


Introduction
The aim of this paper is to extend the result of [8] regarding the stability of Gaussian measures under the flow of the Benjamin-Bona-Mahony equation to more general measures.
We consider the Benjamin-Bona-Mahony (BBM) equation on the torus T: It follows from the work of Bona-Chen-Saut on Boussinesq equations [1,2] that this equation is locally well-posed in L 2 and from the work of Bona-Tzvetkov [3] that it is globally well-posed in H s , s ≥ 0. We are interested in the action of this flow on measures. We consider measures on H s , s > 0. The flow ψ(t) of the BBM equation is well defined and continuous (hence measurable on the , where Marg(µ, ν) is the set of measures on H s ×H s whose marginals are µ and ν, s ′ ≤ s corresponds to the regularity of the space where the measures can be compared and p their integrability. In other words, given a large enough probability space (Ω, A, P) this distance can be seen as where M(µ, ν) is the set of couples of random variables (X, Y) : Ω → H s × H s such that the law of X is µ and the one of Y is ν. This distance corresponds to the weak convergence of the measures combined to the convergence of the moments of order q ≤ p: In [7], the use of these distances is motivated. We prove the following theorems. Theorem 1. Let s ∈]1/4, 1[. Let µ, ν such that for all q ≥ 1, the moments of order q of µ and ν satisfies: with C µ , C ν independent from q. Let p ≥ 1 and p 1 and p 2 such that 1/p = 1/p 1 + 1/p 2 . Then, for all t, and all σ ∈] max( 1 2 , s), min(1, 2s)[, we have d 0,p (µ t , ν t ) ≤ C(µ, ν, t, p 1 , σ)d s,p 2 (µ, ν) where T = 1 + |t| and In other words, ρ → ρ t is locally Lipschitz continuous for the distances d 0,p and d s,p 2 in the set of measures satisfying certain constraints on their moments.
Let p 0 > 1 and let µ be a measure on H s such that Let p 1 , p 2 ≤ p 0 and p such that In particular, if there exists C such that for all q ≥ 1, then for all t, there exists C(t, µ) such that In other words ρ is stable in the set of measures µ such that there exists δ > 0 satisfying Remark 1.1. In [8], an invariant measure ρ 0 on H s with s < 1 2 is built. This measure is a Gaussian random variable on H s whose covariance operator is The proof of these results consists in proving a deterministic global control, Proposition 2.4, on the L 2 norm of the difference between two solutions of BBM, and then integrate the obtained inequality on the probability space where u 0,1 is ρ or ν typical, and u 0,2 is µ typical.
To prove the stability theorem, we use the invariance in the proof, which makes the result better in terms of hypothesis on µ than the continuity one.
In Section 3, we define the space of measures on which we prove Theorems 1, 2 and give alternative definitions or point of views of these spaces using large deviation estimates. Then, we prove Theorems 1, 2.

Deterministic estimates
Through all this paper, we use the fact that the BBM equation is locally well-posed in L 2 according to the following proposition, that comes from [1,2]. Besides, we also use the fact that BBM is globally well-posed in H s , as was proved in [3] .  Proof. Set N such that N ≥ (CT u 0 H s ) 1/s . We have that

On a solution of BBM
We can apply the local well-posedness proposition (Proposition 2.1) for the initial Writing u the solution of BBM with initial datum u 0 , we call w = u − v. This function satisfies with initial datum w 0 = Π N u 0 . As w 0 is in H 1 , it has been proved in [3] that w ∈ H 1 for the times [−T, T ]. We compute estimates on w H σ with σ ∈]1/2, 1]. We start by differentiating w 2 H 1 with respect to time : Using that w∂ x w = 0 and w∂ x w 2 = 0, we keep only the term w∂ x (vw). Because ∂ x is skew symmetric, we have Using the Sobolev embedding H σ ⊂ L ∞ , (σ > 1/2) , the fact that σ is less than 1, and that v L 2 T −1 , we get Using Gronwall lemma, we get We use that |t| ≤ T and w 0 H σ ≤ N σ−s u 0 H s to conclude :

On the difference of two solutions
In this subsection, we estimate the difference between two solutions of BBM with the difference between the initial datum.
with constants C and c independent from u 0 and T .
Hence, we can apply Proposition 2.1 to v 0,i , i = 1, 2. There exist two unique solutions of BBM on the times [−T, T ], v 1 and v 2 with respective initial data v 0,1 and v 0,2 , and besides, v 1 and v 2 satisfy We can write this equation and by summing this equalities and by keeping only the non null terms, we get .
Using Sobolev embedding H σ ∈ L ∞ , we get By integrating over time, we get and by using Gronwall lemma We estimate each term. Thanks to Proposition 2.1, we have Thanks to Proposition 2.2, we have Finally, we use that The initial datum w 0 = Π N (u 0,1 − u 0,2 ) satisfies Therefore, we have the inequality We estimate N. By definition, N is less than and since u 1 − u 2 = v + w and the L 2 norm of v is less than N −s u 0,1 − u 0,2 H s which is less than the above bound, we have proved the proposition.

Definitions and large deviation estimates
In this subsection, we define the spaces of probability measures where we prove the continuity and stability, along with distances on these spaces, and we prove the equivalence between large deviation estimates and estimates on the moments of these measures.

Continuity
We have an equivalence between belonging to Σ and satisfying estimates on the moments of order p.

Proposition 3.3. A measure ρ ∈ M(H s ) belongs to Σ if and only if there exists C(ρ) such that for
Proof. This is a well-known property hence we only sketch the proof. For more details, we refer to Proposition 4.4 of [6] . Assume that ρ ∈ Σ. Let X = u H s . We have, thanks to Markov's inequality where E * is the average with regard to the measure * . Hence, we get By using the change of variable λ = √ 1/2δy, we get The integral ∞ 0 py p−1 e −y 2 /2 dy does not depend on ρ and by induction we have that it is less than C p p/2 , hence Conversely, assume that Then, the probability ρ(X ≥ λ) can be bounded by By choosing p such that which ensures that e δX 2 is integrable for all δ < c.
Stability For the stability, the hypothesis on the measures is weaker, we only assume that it has a p-moment in H s .

Notation 3.5.
We call Σ p the measures on H s with a p-moment (p ≥ 1), that is : To compare measures, we use the Wasserstein metrics.
Definition 3.6. Let s ′ ≤ s and p ′ ≤ p, let µ, ν two measures in Σ p . The Wasserstein metrics d s ′ ,p is defined as where Marg(µ, ν) is the set of probability measures on H s × H s whose marginals are µ and ν, that is, for all A measurable in H s , γ(A × H s ) = µ(A) and γ(H s × A) = ν(A).
We will compare the measures transported by the BBM flow in d 0,p ′ using the d s,p distance for the initial data.

Another large deviation estimate
Proposition 3.7. Let X be a random variable on a probability space with measure ρ and let α > 0. The fact that there exists δ > 0 such that e δ(ln X) 1/α+1 1 X≥1 is ρ-integrable is equivalent to the fact that there exists E 0 and C > 1 such that for all p For λ ≥ 1, thanks to Markov's inequality, we have We minimize We get that f is minimal when with the change of variable x = ln λ, we have which ensures that it is finite as long as δ is strictly less than β(α) (ln C) 1/α . Conversely, we assume that Then, the probability ρ(X ≥ λ) is less than 1 if λ ≤ 1 and is less than otherwise. Hence, for p ≥ 1, we get that and by writing λ p−1 = e (p−1) ln λ , We have that Hence, we have, by dividing the integration between [1, λ 0 ] and [λ 0 , ∞[ The quantity I.2 is less than We have that δ(ln λ) 1/α+1 /2 ≥ 2 ln λ if and only if Therefore, we get As λ 1 does not depend on p, we get that where C depends on X and α. For I.1, we use that and by summing I.1 and I. 2, Finally, as which concludes the proof.

Continuity of the flow
In this subsection, we prove the continuity of the action of the flow of BBM.
Definition 3.8. Let µ ∈ M(H s ). For all t ∈ R we call µ t the image measure of µ under the flow of BBM ψ(t), that is, for all measurable set A,

Remark 3.1. This definition is possible because ψ(t) is continuous on H s and hence measurable on its topological σ-algebra.
Proposition 3.9. Let s ∈]1/4, 1[, p ≥ 1 and p 1 , p 2 such that 1 p 1 + 1 p 2 = 1 p and t ∈ R. Let T = 1 + |t|. Let σ ∈]1/2, 1[ such that 1 < σ/s < 2. For all µ, ν ∈ Σ, we have Proof. Let γ ∈ Marg(µ, ν), that is, γ is a measure on H s × H s whose marginals are µ and ν. Set γ t the image measure of γ under the map (ψ(t), ψ(t)). For all A measurable in H s , we have Since the marginals of γ are µ and ν, we get For the same reasons, In other words, γ t ∈ Marg(µ t , ν t ). Therefore, we get We do the change of variable (u 1 , u 2 ) = (ψ(t)u 0,1 , ψ(t)u 0,2 ) = (ψ(t), ψ(t))(u 0,1 , u 0,2 ), we get, thanks to the definition of γ t , We set u i (t) = ψ(t)u 0,i . We input the estimate of Proposition 2.4 with T = 1 + |t| (T has to be bigger than 1), We use that 1/p = 1/p 1 + 1/p 2 to write the Hölder inequality, γ . We use the estimates on u 1 in Proposition 2.2 As I.1 does not depend on u 0,2 and as γ has µ as a left marginal, we get which is well defined for all time and all p 1 since σ has been chosen in ]1/2, 2s[ and thus σ/s < 2, which is possible since s > 1/4, and since µ belongs to Σ, that is µ has large Gaussian deviation estimates in H s . Similarly and is well-defined for the same reasons. Note that the bound on I does not depend on γ ∈ Marg(µ, ν). We now have the estimate and we conclude by taking the infimum over γ ∈ Marg(µ, ν).

Stability of invariant measures
Definition 3.10. The measure that is known as ρ in the rest of the paper is an invariant measure on H s under the flow of BBM, that is, for all measurable set A in H s and all time t, we have or equivalently, for all measurable bounded function F : H s → R, we have Besides, we assume that ρ admits Gaussian large deviation estimates, that is ρ ∈ Σ.

Remark 3.2.
At least one measure of this kind exists, as was proved in [8] . For this measure, s < 1/2.
Proof. The proof begins in the same way as the one of Proposition 3.9. We start from γ with 1 p = 1 p 1 + 1 p 2 , and σ > 1/2 and γ has for marginals ρ and µ. Since ρ is invariant under the flow of BBM, we have ρ t = ρ, hence d p,0 (ρ, µ t ) = d p,0 (ρ t , µ t ) .
For I.2, we use that x → e T x is convex, hence from Jensen's inequality, We then use Minkowski's inequality As u 1 refers to ρ and ρ is invariant under the flow of BBM, we have  As d 0,p (ρ, µ t ) ≤ CI u 0,1 − u 0,2 L p 2 γ ,H s , we conclude by taking the infimum over γ. If µ is such that for all q d s,q (µ, ρ) ≤ C q ε with C bigger than 1, and ε a small parameter, we get that d 0,p (µ t , ρ) ≤ (C(t, σ)) p 1 d s,p 2 (µ, ρ) .