An optimal transport approach of hypocoercivity for the 1d kinetic Fokker-Plank equation
AAN OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY FOR THE 1DKINETIC FOKKER-PLANK EQUATION
SAMIR SALEM
Abstract.
A quadratic optimal transport metric on the set of probability measure over R is introduced.The quadratic cost is given by the euclidean norm on R associated to some well chosen symmetricpositive matrix, which makes the metric equivalent to the usual Wasserstein-2 metric. The dissipationof the distance to the equilibrium along the kinetic Fokker-Planck flow, is bounded by below in terms ofthe distance itself. It enables to obtain some new type of trend to equilibrium estimate in Wasserstein-2like metric, in the case of non-convex confinement potential. Introduction
The kinetic Fokker-Planck equation (in one dimension of space) ∂ t f t + v∂ x f t − U (cid:48) ( x ) ∂ v f t = ∂ v ( vf t + ∂ v f t ) , (1.1)is a kinetic model which describes the time evolution of the law of some particle in the phase space,submitted to some Brownian force, velocity friction and evolving in some confinement potential U ∈ C ( R ) dX t = V t dtdV t = − U (cid:48) ( X t ) dt − V t dt + √ dW t (1.2)where ( W t ) t ≥ is a Brownian motion.The well posedness theory for (1.1) is classical, since all the coefficients are smooth.As the diffusion operator is degenerated and acts only on velocity, the theories of regularity and trend toequilibrium are more delicate. The latter is known as hypocoercivity since Villani’s eponymous memoir[11].In order to roughly sketch the core idea of this theory, let us sat that for some a, b, c > f ∞ ( x, v ) = Z − e − U ( x ) − | v | , Z = √ π (cid:90) R e − U ( y ) dy, (the confusion between probability measures and densities may be throughout the paper), is defined as (cid:107) · (cid:107) H = (cid:107) · (cid:107) L ( f ∞ ) + a (cid:107) ∂ x · (cid:107) L ( f ∞ ) + b (cid:104) ∂ x · , ∂ v ·(cid:105) L ( f ∞ ) + c (cid:107) ∂ x · (cid:107) L ( f ∞ ) . It is equivalent to the usual norm on H ( f ∞ ), provided that the quadratic form associated to the triplet( a, b, c ) is positive definite.But compared to the usual norm, it includes the cross term b (cid:104) ∂ x · , ∂ v ·(cid:105) which, when differentiated along(1.1), enables to conjugate the effects of the diffusion ∂ v in velocity and the free transport v∂ x in thephase space. The result (see [11, Theorem 27]) is that equation (1.1) is a contraction for this specificnorm, i.e. 12 ddt (cid:107) f t /f ∞ − (cid:107) H ≤ − κ (cid:107) f t /f ∞ − (cid:107) H . Date : February 23, 2021.
Key words and phrases.
Hypocoercivity, optimal transport, kinetic Fokker-Planck equation, functional inequalities. a r X i v : . [ m a t h . A P ] F e b SALEM
It is obtained under the assumption that the confinement potential U (up to an additive constant) issuch that the probability measure e − U satisfies a Poincar´e inequalityUnder the stronger assumption that e − U satisfies a logartihmic Sobolev inequality (see [10, Section 9.2]),a similar result is also derived, in terms of relative entropy w.r.t. the equilibrium measure. A typicalsufficient condition for log-Sobolev inequality to be satisfied, is for instance that U can be decomposedas the sum of some strictly convex and a bounded functions (see [10, Theorem 9.9 ]). More precisely, thedecay estimate H ( f t | f ∞ ) := (cid:90) R f t ln (cid:18) f t f ∞ (cid:19) ≤ C e − λ ( t − t ) t H ( f | f ∞ ) , ∀ t ∈ (0 , , (1.3)is obtained (see [11, Remark 41 (7.11)]. Since e − U satisfies a λ -log-Sobolev inequality, f ∞ satisfies ( λ ∧ f ∈ P ( R ) W ( f, f ∞ ) ≤ H ( f | f ∞ ) λ ∧ . In particular, it is clear that W ( f t , f ∞ ) = O ( e − κ t ), and the time asymptotic behavior of the distance ofthe solution to (1.1) and the equilibrium measure is well understood.Under the more restrictive condition that U is strictly convex, Bolley et al. [3], extended in the coreidea of hypocoercivity to the framework of coupling metric, by introducing a twisted Wasserstein-2 metric,under which equation (1.1) is a contraction.More precisely for r, s > µ, ν ∈ P ( R ), a distance d Q is defined as d Q ( µ, ν ) = inf γ ∈ Π( µ,ν ) (cid:90) R × R ( r | x − x | + 2( x − x )( v − v ) + s | v − v | ) γ ( dz , dz ) , where Π( µ, ν ) is the choice of probability measures on R × R admitting respectively µ and ν as marginals.Compared to the usual Wasserstein-2 metric, there is a cross term between position and velocity, similarlyas what is done in the L theory.This enables the authors to show that there is κ > f t ) t ≥ , ( g t ) t ≥ are the solutions to (1.1)starting from f , g respectively, there holds for any t ≥ d Q ( f t , g t ) ≤ e − κt d Q ( f , g ) . In particular, since for the set of r, s the quadratic form associated to ( r, , s ) is positive definite, d Q ismetrically equivalent to the usual Wasserstein-2 metric, it provides a new way to estimate of the expo-nential convergence of W ( f t , f ∞ ). But the estimate is somehow smoother than (1.3) for short time.This result is obtained by so-called synchronous coupling, which means that the quantity of interest isthe expectation of the squared distance between two solutions to (1.2) for two distinct initial conditionsof law f and g , and the same Brownian motion.The fact that the Brownian motion is the same for both trajectories, roughly means that the techniqueis actually indifferent to the presence of diffusion. And that is why the strict convexity assumption on U is needed. In the case with no diffusion, the physical example of a bullet evolving in a gap with friction isinstructive to realize that strict convexity of the confinement is needed to ensure uniqueness of stationarystate. And since no advantage is taken form the diffusion in velocity, the analogy with the standard L theory is limited.In this paper, we aim at presenting through a simple 1d toy example, how to extend this idea of twistedWasserstein metric to obtain a W hypocoercivity theory closer to the standard L theory (and also tothe relative entropy theory by some aspects). For that purpose, we consider some locally non convexpotential U satisfying : U ( x ) = α | x | ψ ( x ) (1.4) N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 3
In that case, the fact that W ( f t , f ∞ ) converges to 0 exponentially fast is well known (as noted above).But the non convexity of the confinement prevents from obtaining contraction estimate in equivalentmetric, by using standard coupling techniques. We then extend the idea of [1, 2], which consists in esti-mating the dissipation of the optimal transport distance itself, and not some quantity which bounds itby above. The argument strongly relies on the optimal transportation map between the solution and theequilibrium measure, and the fact that, in this quadratic framework, this map derives from some convexpotential `a la Brenier.Obtaining some exponential stability estimates in optimal transport or coupling distances, for kineticFokker-Planck in non convex landscape, is a problem that has dragged a lot of attention from the kineticcommunity lately. We can mention the recent [8], which relies on some entropy method, as roughlydescribed above. In [5], the authors study the equation set on the torus in space, with no confinement,and obtain some exponential stability estimate in usual W distance, up to some multiplicative constant,with a probabilistic approach.Finally, let us mention that for the weaker W metric, recent works [6],[7] based on sophisticated mar-kovian coupling, have proved their efficiency to solve the problem of non convex landscape.2. Preliminaries and main results
Optimal transport metric on R × R . We begin with some optimal transport considerations.The usual Wasserstein-2 metric on P ( R × R ) is defined as W ( µ, ν ) = inf T =( T , T ) , T ν = µ (cid:90) R × R (cid:12)(cid:12) T ( x, v ) − x (cid:12)(cid:12) + |T ( x, v ) − v | ν ( dx, dv ) , for µ and ν admitting smooth enough densities (see for instance [10, Theorem 2.12]). There exists aunique map T which achieves the infimum, and it is given as the gradient of some (up to an additiveconstant) convex potential.As noted in the above introduction, in order to mimic the L hypocoercivity theory, we need some crossterm between position and velocity. In that purpose we introduce some positive definite matrix A = (cid:18) a bb c (cid:19) with A − = det( A ) − (cid:18) c − b − b a (cid:19) . We denote the eigenvalues of this matrix ν = c + a − (cid:112) ( c − a ) + 4 b , ν = c + a + (cid:112) ( c − a ) + 4 b , and define the equivalent norm on R | ( x, v ) | A := ( x, v ) · A ( x, v ) = a | x | + 2 bxv + c | v | . This twisted euclidean norm is the cost function associated to the optimal transport metric given in the
Definition 2.1.
For A a symmetric positive definite matrix,we define W A for any µ, ν ∈ P ( R ) as W A ( µ, ν ) = inf γ ∈ Π( µ,ν ) (cid:90) R | ( x , v ) − ( x , v ) | A γ ( dx , dv , dx , dv ) , where Π( µ, ν ) = (cid:110) γ ∈ P ( R × R ) , γ ( K × R ) = µ ( K ) , γ ( R × K ) = ν ( K ) , ∀ K ⊂ R (cid:111) . Moreover, W A is a distance on P ( R ) . See for instance [10, Theorem 1.3] for the well posedness and [10, Theorem 7.3] for the metric aspect.The main result of this section, is a result of polarization of the optimal transport map given in the
SALEM
Theorem 2.1 (Brenier’s theorem for twisted Wasserstein metric) . For any µ, ν ∈ P ( R ) having C ,strictly positive densities : ( i ) there is (unique up to additive constant) ϕ convex such that A − ∇ ϕ ν = µ and W A ( µ, ν ) = (cid:90) R (cid:12)(cid:12) ( x, v ) − A − ∇ ϕ ( x, v ) (cid:12)(cid:12) A ν ( dx, dv ) , ( ii ) if ϕ ∗ denotes the convex conjugate of ϕ , then ∇ ϕ ∗ ( A · ) µ = ν and W A ( µ, ν ) = (cid:90) R | ( x, v ) − ∇ ϕ ∗ ( A ( x, v )) | A µ ( dx, dv ) , ( iii ) ϕ is C ( R ) , (in particular ∂ x ϕ, ∂ v ϕ > ) and ∀ ( x, v ) ∈ R there holds ∇ ϕ ∗ ( ∇ ϕ ( x, v )) = ( x, v ) . The proof of this result can be found in the below Appendix. But it mostly consists in a simpleapplication of [9, Theorem 1.17] or [10, Theorem 2.44], and regularity theory for the Monge-Amp`ereequation. One of the main feature of this metric is given in the
Proposition 2.1.
The distance W A is metrically equivalent to W .Proof. Let µ, ν ∈ P ( R ). Let T be the optimal transport map from ν toward µ w.r.t. W (resp. T (cid:48) theoptimal transport map from ν toward µ w.r.t. W A ). Since for any z ∈ R we have ν | z | ≤ | z | A ≤ ν | z | W A ( µ, ν ) = (cid:90) R |T (cid:48) ( x, v ) − ( x, v ) | A ν ( dx, dv ) ≥ ν (cid:90) R |T (cid:48) ( x, v ) − ( x, v ) | ν ( dx, dv ) ≥ ν (cid:90) R |T ( x, v ) − ( x, v ) | ν ( dx, dv )= ν W ( µ, ν ) , and the converse comes with a similar argument. (cid:3) Main result.
We are now in position to state the main result of the paper.We begin with some smallness assumption on the perturbation, which is consistent with
W J type ofinequality (see for instance [1, Definition 3.1]).More precisely, we assume that the perturbation ψ is compactly supported on [ − R, R ] for some
R > (cid:107) ψ (cid:48) (cid:107) L ∞ < α < (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ < − , γ := (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ − α, c ∗ > b ∗ ,b ∗ := ( γ + (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ ) 21 + (cid:112) − γ + (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ ) ,c ∗ = 120 e − α ( R +2) e −(cid:107) ψ (cid:107) L ∞ (cid:18) ∧ ( α − (cid:107) ψ (cid:48) (cid:107) L ∞ )4 (cid:19) . (2.1)This condition is quite technical, and rather complicated to check. But at this cost, it enables a simplechoice of the coefficients of the matrix A, in a manner of speaking. Nevertheless, this restriction is non void(see Lemma A.3). The part of this restriction which must be emphasized, is that γ = (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ − α > U (cid:48)(cid:48) is non positive, and classical coupling method fails for quadratic optimal transport metrics.The confinement potential being set, we define the vector field B as B ( x, v ) = (cid:18) v − U (cid:48) ( x ) − v (cid:19) = (cid:18) v − αx − ψ (cid:48) ( x ) − v (cid:19) . For f, g ∈ P ( R ), and ϕ the unique (up to an additive constant) convex potential such that A − ∇ ϕ transports g onto f , optimally w.r.t. W A (as given by Theorem 2.1 above) ,we introduce now the key N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 5 functional J A ( f | g ) := − (cid:90) R (cid:104) B ( A − ∇ ϕ ( x, v )) − B ( x, v ) , A − ∇ ϕ ( x, v ) − ( x, v ) (cid:105) A g ( x, v ) dxdv + (cid:90) R ( ∂ v ϕ ) − (cid:32)(cid:0) ∂ v ϕ − c (cid:1) + (cid:0) b (cid:0) ∂ v ϕ − c (cid:1) − c (cid:0) ∂ x,v ϕ − b (cid:1)(cid:1) det( ∇ ϕ ) (cid:33) g ( x, v ) dxdv. (2.2)The main feature of this functional is given in the Proposition 2.2.
Let ( f t ) t ≥ , ( g t ) t ≥ be two solutions to (1.1) . Then for any symmetric positive definitematrix and any t > there holds ddt W A ( f t , g t ) ≤ −J A ( f t | g t ) . This result is no more than a careful application in the kinetic case of classical results (see for instance[10, Theorem 23.9]). Note that it would be possible to be even more careful and obtain an equalityinstead of an upper bound, but this is not required for our purpose.Next we obtain a key functional inequality in the
Proposition 2.3.
Assume that U is of the form (1.4) , for some α > and some compactly supportedon [ − R, R ] function ψ , satisfying (2.1) . There is a symmetric positive definite matrix A ∈ M ( R ) and κ > , such that for any probability f ∈ P ( R ) with smooth enough density, there holds κW A ( f, f ∞ ) ≤ J A ( f | f ∞ ) . This result is greatly inspired by [1, Proposition 3.4], where a similar ”entropy”-”entropy dissipation”inequality is derived, between the Wasserstein 2 distance to the equilibrium and its dissipation alongsome (non degenerate) Fokker-Planck equation.Let us briefly sketch the proof of this result, which structure might remind acquainted readers the ”cas-cade” structure of the proof of the L result.Frictions in velocity always provide some contraction effects in velocity variable. Due to the local lack Figure 1.
Decomposition of the phase space used in the proof of Proposition 2.3.
SALEM of convexity of U , we can not always benefit of contraction effects in the position variable. Of course, at v ∈ R fixed, if x or the position component of the transported of ( x, v ) lies far enough of the support ofthe perturbation ψ , contraction effects are obvious.Problems may arise if these two positions lie in the support of the perturbation. But since in this case, westill enjoy the contraction in velocity, we may solve the issue if the velocity component of transportationvector dominates the position component.On the remaining case, we crucially rely on the non negative quantity in the definition of J A , involvingsecond order derivatives of ϕ . Thanks to a surprising matricial inequality (see Lemma A.2)), a simplegeometric interpretation of this quantity is given, and with help of the contraction already found in pre-vious regions, some contraction effects in the position variable are derived on the set of non convexityof U . We emphasize that, the quantity involving second order derivative in J A is obtained because onthe one hand we rely on the optimal transport map for quadratic cost function, thus admitting somegradient structure by Brenier’s Theorem, and on the other hand by taking into account the effects ofthe diffusion in velocity. And optimal transport theory and hypocoercivity theory are mixed in some sense.A simple application of these two propositions, together with Gronwall’s inequality yields to the Theorem 2.2.
Assume that U is of the form (1.4) , for some α > and some compactly supported on [ − R, R ] function ψ , satisfying (2.1) . For some c ∈ (2 b ∗ , c ∗ ) , let A = c (cid:18) α +
12 1212 (cid:19) There is κ > depending only on U such that for any f ∈ P ( R ) it holds for any t > W A ( f t , f ∞ ) ≤ e − κt W A ( f , f ∞ ) , where ( f t ) t ≥ is the solution to (1.1) starting from f . Discussion.
We make here some comments about the main result of the paper.A first simple consequence of Theorem 2.2 and Proposition 2.1, is that there is a constant C α > W ( f t , f ∞ ) ≤ C α e − κt W ( f , f ∞ ) . (2.3)In [5], the authors prove that on a torus in position with no confinement, such an inequality can not holdwith C α = 1 and is obtained for some C α >
1. It could be of some interests to investigate whether thetechniques developed here extend to the case of a torus, to obtain a (slightly) stronger contraction in anequivalent Wasserstein metric.It is important to note that Theorem 2.2 only provides stability around equilibrium. It is a consequenceof the fact that our analysis (more precisely Proposition 2.3) strongly relies on the explicit knowledge ofthe density of the stationary measure. As noted in the introduction, a consequence of [3] is that, underthe assumption of strict convexity of U , there is a symmetric positive definite matrix A and κ > f , g W A ( f t , g t ) ≤ e − κt W A ( f , g ) . For a non degenerated diffusion, it is well known that the strict convexity of the potential is a necessaryand sufficient condition for a contraction inequality in usual Wasserstein-2 metric (see [1, Remark 3.6]).Therefore it could be interesting to wonder if the converse to the result of Bolley et al is true, that iswhether there can exist a matrix A such that the kinetic Fokker-Planck equation is a contractive in W A distance, only if the potential U is strictly convex.Under assumption (2.1), it is clear in view of [10, Theorem 9.9], that f ∞ satisfies a (1 ∧ αe − (cid:107) ψ (cid:107) L ∞ )-log-Sobolev inequality (as it is some bounded perturbation of some strictly convex function). Hence for N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 7 any f ∈ P ( R ) there holds W ( f , f ∞ ) ≤ ∧ αe − (cid:107) ψ (cid:107) L ∞ H ( f | f ∞ ) . On the other hand, we can use some regularization result (see for instance [8, Proposition 15]), to obtainsome constant C U > U such that for any t ∈ (0 ,
1) and t > t , H ( f t | f ∞ ) ≤ C U t − W ( f t − t , f ∞ ) . Using then together (2.3) and the above remark yields H ( f t | f ∞ ) ≤ C U C α t − e − κ ( t − t ) W ( f , f ∞ ) ≤ C U e − κ ( t − t ) t H ( f | f ∞ ) , which is (1.3) (up to the value of the constants on which we will not comment here). Therefore Theorem2.2 is in some sense ”stronger” than [11, Theorem 39] for the specific case on which we are focusing. Therelation between these results may yield to think that there is here some common structure to understand,as indicated in [11, Chapter 6].Finally, it can safely by conjectured that under assumption (2.1), ( e − U , U (cid:48) ) satisfies a WJ inequalityin the sense of [1, Definition 3.1]. The result can be obtained either by checking that assumption (2.1)implies the sufficient condition of [1, Proposition 3.8], either by straightforward computations. It couldbe interesting to obtain Theorem 2.2 under the assumption that ( e − U , U (cid:48) ) satisfies a WJ inequality, inthe vein of the entropy method of [11, Theorem 37] which is based on the assumption that e − U satisfiesa log-Sobolev inequality, or the L method based on Poincar´e inequality. Another possible direction ofinvestigation, would be to obtain a more abstract and general result, based on commutator theory, in thespirit of [11, Theorem 28].These questions are delayed to some future works, as is the possible extension to larger dimension (whichseems mostly technical), or the extension to non-linearity and particles systems.3. Proof of Theorem 2.2
Proof of Proposition 2.2.
Let f ∈ P ( R ) and ( f t ) t ≥ be the unique solution to (1.1) for theinitial condition f . In view of [11, Theorem A.15, Theorem A.19] it admits for any t > f t ) t ≥ solves thetransport equation ∂ t f t + ∇ x,v · ( ξ t f t ) = 0 , (3.1)with advection field ξ t ( x, v ) := (cid:18) v − v − U (cid:48) ( x ) − ∂ v ln f t ( x, v ) (cid:19) = B ( x, v ) − (cid:18) ∂ v ln f t ( x, v ) . (cid:19) (cid:5) Step one :We first check that for any σ ∈ P ( R ) there holds12 ddt W A ( f t , σ ) ≤ (cid:90) R A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · ξ t ( A − ∇ ϕ t ( x, v )) σ ( x, v ) dxdv. = (cid:90) R A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · B ( A − ∇ ϕ t ( x, v )) σ ( x, v ) dxdv − (cid:90) R A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · (cid:18) − ∂ v ln f t ( A − ∇ ϕ t ( A ( x, v )) (cid:19) σ ( x, v ) dxdv = I − I . SALEM
Indeed, let t > h > ϕ t the unique (up to an additive constant) convex potential such that A − ∇ ϕ t transport σ onto f t optimally w.r.t. W A , so that W A ( f t , σ ) = (cid:90) R | ( x, v ) − A − ∇ ϕ t ( x, v ) | A σ ( dx, dv ) . We denote Ξ t + ht ( z ) the solution at time t + h to the ODE dds Ξ t + st = ξ s (Ξ t + st ) , Ξ tt ( z ) = z, so that Ξ t + ht ◦ A − ∇ ϕ t transports, not optimally, σ onto f t + h and W A ( f t + h , σ ) − W A ( f t , σ )2 h ≤ (cid:90) R | ( x, v ) − Ξ t + ht ◦ A − ∇ ϕ t ( x, v ) | A − | ( x, v ) − A − ∇ ϕ t ( x, v ) | A h σ ( dx, dv ) . We conclude this step by observing that by definition of Ξ t + ht there holdslim h → | ( x, v ) − Ξ t + ht ◦ A − ∇ ϕ t ( x, v ) | A − | ( x, v ) − A − ∇ ϕ t ( x, v ) | A h = A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · ξ t ( A − ∇ ϕ t ( x, v )) . (cid:5) Step two :We choose σ = f ∞ . Since for any ( x, v ) there holds( x, v ) = A − ∇ ϕ t ( ∇ ϕ ∗ t ( A ( x, v )) , (3.2)by point ( iii ) of Theorem 2.1 and since A − ∇ ϕ t transports f ∞ to f t , we have I = (cid:90) R A (( x, v ) − ∇ ϕ ∗ t ( A ( x, v ))) · (cid:18) − ∂ v ln f t ( x, v ) (cid:19) f t ( x, v ) dxdv == (cid:90) R (( x, v ) − ∇ ϕ ∗ t ( A ( x, v ))) · (cid:18) − b∂ v ln f t ( x, v ) − c∂ v ln f t ( x, v ) (cid:19) f t ( x, v ) dxdv = (cid:90) R − b ( x − ∂ x ϕ ∗ t ( A ( x, v ))) ∂ v f t ( x, v ) dxdv + (cid:90) R − c ( v − ∂ v ϕ ∗ t ( A ( x, v ))) ∂ v f t ( x, v ) dxdv. Then integrating each term by parts, using (3.2) and the fact that ∇ ϕ ∗ t ( A ( x, v )) transports f t onto f ∞ yields I = (cid:90) R (cid:0) − b ∂ x ϕ ∗ t ( A ( x, v )) − cb∂ v,x ϕ ∗ t ( A ( x, v )) (cid:1) f t ( x, v ) dxdv + (cid:90) R c (1 − b∂ x,v ϕ ∗ t ( A ( x, v )) − c∂ v ϕ ∗ t ( A ( x, v ))) f t ( x, v ) dxdv = (cid:90) R (cid:0) − b ∂ x ϕ ∗ t ( AA − ∇ ϕ t ( ∇ ϕ ∗ t ( A ( x, v ))) − cb∂ v,x ϕ ∗ t ( AA − ∇ ϕ t ( ∇ ϕ ∗ t ( A ( x, v ))) (cid:1) f t ( x, v ) dxdv + (cid:90) R c (1 − b∂ x,v ϕ ∗ t ( AA − ∇ ϕ t ( ∇ ϕ ∗ t ( A ( x, v ))) − c∂ v ϕ ∗ t ( AA − ∇ ϕ t ( ∇ ϕ ∗ t ( A ( x, v )))) f t ( x, v ) dxdv = (cid:90) R (cid:0) − b ∂ x ϕ ∗ t ( ∇ ϕ t ( x, v )) − cb∂ v,x ϕ ∗ t ( ∇ ϕ t ( x, v )) (cid:1) f ∞ ( x, v ) dxdv + (cid:90) R c (1 − b∂ x,v ϕ ∗ t ( ∇ ϕ t ( x, v )) − c∂ v ϕ ∗ t ( ∇ ϕ t ( x, v ))) f ∞ ( x, v ) dxdv. By point ( iii ) of Theorem 2.1, ϕ t is C and ∇ ϕ ∗ t ( ∇ ϕ t ( x, v )) = ( x, v ) we obtain by differentiation ∇ ϕ ∗ t ( ∇ ϕ t ( x, v )) = (cid:0) ∇ ϕ t ( x, v ) (cid:1) − = (cid:18) ∂ x ϕ t ∂ x,v ϕ t ∂ v,x ϕ t ∂ v ϕ t (cid:19) − = 1det( ∇ ϕ t ( x, v )) (cid:18) ∂ v ϕ t − ∂ x,v ϕ t − ∂ v,x ϕ t ∂ x ϕ t (cid:19) . N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 9
Therefore, identifying term by term yields I = − (cid:90) R ∇ ϕ t ( x, v )) (cid:0) b ∂ v ϕ t ( x, v ) − cb∂ x,v ϕ t ( x, v ) + c ∂ x ϕ t ( x, v ) − c (cid:1) f ∞ ( x, v ) dxdv. (cid:5) Step three :Next observe that since f ∞ is a stationary solution to (3.1), we easily find that for any t >
00 = (cid:90) R | A − ∇ ϕ t ( x, v ) − ( x, v ) | A ∇ x,v · ( ξ ∞ f ∞ ) dxdv = (cid:90) R A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · ξ ∞ ( x, v ) f ∞ ( x, v ) dxdv. = (cid:90) R A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · B ( x, v ) f ∞ ( x, v ) dxdv − (cid:90) R A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · (cid:18) − ∂ v ln f ∞ ( x, v ) (cid:19) f ∞ ( x, v ) dxdv = J − J . By integration by parts, we obtain J = − (cid:90) R ( ∂ v ϕ ( x, v ) − ( bx + cv )) ∂ v f ∞ ( x, v ) dxdv = − (cid:90) R ( c − ∂ v ϕ ( x, v )) f ∞ ( x, v ) dxdv. (cid:5) Step three :Gathering all the above results we have12 ddt W A ( f t , f ∞ ) ≤ (cid:90) R A ( A − ∇ ϕ t ( x, v ) − ( x, v )) · (cid:0) B ( A − ∇ ϕ t ( x, v ) − B ( x, v ) (cid:1) ) f ∞ ( x, v ) dxdv + (cid:90) R ∇ ϕ t ( x, v )) (cid:0) b ∂ v ϕ t ( x, v ) − cb∂ x,v ϕ t ( x, v ) + c ∂ x ϕ t ( x, v ) − c (cid:1) f ∞ ( x, v ) dxdv + (cid:90) R ( ∂ v ϕ ( x, v ) − c ) f ∞ ( x, v ) dxdv. Now, note that c ∂ x ϕ t det( ∇ ϕ t ) − c ∂ v ϕ t = c ∂ v ϕ t − ( ∂ x,v ϕ t ) ∂ x ϕ t − ∂ v ϕ t = c ∂ x,v ϕ t ) ∂ x ϕ t ∂ v ϕ t (cid:16) ∂ v ϕ t − ( ∂ x,v ϕ t ) ∂ x ϕ t (cid:17) = c ( ∂ x,v ϕ t ) ∂ v ϕ t det( ∇ ϕ t )) , and the result is proved since1det( ∇ ϕ t ) (cid:0) b ∂ v ϕ t − cb∂ x,v ϕ t + c ∂ x ϕ t − c (cid:1) + ( ∂ v ϕ t ( x, v ) − c )= ∂ v ϕ t ( x, v ) − c + c ∂ v ϕ t + c ∂ x ϕ t det( ∇ ϕ t ) − c ∂ v ϕ t + b ∂ v ϕ t − cb∂ x,v ϕ t det( ∇ ϕ t ) = (cid:0) ∂ v ϕ t − c (cid:1) ( ∂ v ϕ t ) − + c ∂ x,v ϕ t ) ∂ v ϕ t + b ∂ v ϕ t − cb∂ x,v ϕ t det( ∇ ϕ t )= (cid:0) ∂ v ϕ t − c (cid:1) ( ∂ v ϕ t ) − + 1det( ∇ ϕ t ) (cid:32) b ( ∂ v ϕ t ) / − c ∂ v,x ϕ t ∂ v ϕ / t (cid:33) = ( ∂ v ϕ ( x, v )) − (cid:32)(cid:0) ∂ v ϕ − c (cid:1) + (cid:0) b∂ v ϕ − c∂ x,v ϕ (cid:1) det( ∇ ϕ ) (cid:33) . Proof of Proposition 2.3.
We choose A of the form A = (cid:18) b + cα bb c (cid:19) , for c = 2 b ∈ (2 b ∗ , c ∗ ). For v ∈ R we define X v := (cid:110) x ∈ R , | x | ≤ R + 1 , det( A ) − | c∂ x ϕ ( x, v ) − b∂ v ϕ ( x, v ) | ≤ R + 1 (cid:111) Y v := (cid:110) x ∈ X v , | det( A ) − ( c∂ x ϕ ( x, v ) − b∂ v ϕ ( x, v )) − x | ≥ | det( A ) − (( b + cα ) ∂ v ϕ ( x, v ) − b∂ x ϕ ( x, v )) − v | (cid:111) . We decompose − (cid:90) R (cid:104) B ( A − ∇ ϕ ( x, v )) − B ( x, v ) , A − ∇ ϕ ( x, v ) − ( x, v ) (cid:105) A e − U ( x ) dx = − (cid:90) R \X v ∪X v \Y v ∪Y v (cid:104) B ( A − ∇ ϕ ( x, v )) − B ( x, v ) , A − ∇ ϕ ( x, v ) − ( x, v ) (cid:105) A e − U ( x ) dx := I + I + I (cid:5) Estimate of I and I :Using ( i ) and ( ii ) of Lemma A.1 with z = ∇ ϕ ( x, v ) and z = A (cid:18) xv (cid:19) we have that for any x ∈ R \ Y v − (cid:90) R \Y v (cid:104) B ( A − ∇ ϕ ( x, v )) − B ( x, v ) , A − ∇ ϕ ( x, v ) − ( x, v ) (cid:105) A e − U ( x ) dx ≥ c min( κ , κ ) (cid:90) R \Y v | A − ∇ ϕ ( x, v ) − ( x, v ) | e − U ( x ) dx (cid:5) Estimate of I :For each v ∈ R we define x + v = sup Y v , and x (cid:48) ∈ [ x + v , R + 2] (which is not necessarily unique) such that (cid:12)(cid:12) A − ∇ ϕ ( x (cid:48) , v ) − ( x (cid:48) , v ) (cid:12)(cid:12) = inf y ∈ [ x + v ,R +2 ] (cid:12)(cid:12) A − ∇ ϕ ( y, v ) − ( y, v ) (cid:12)(cid:12) . For any x ∈ Y v , there holds by Taylor’s expansion ∂ v ϕ ( x, v ) − ( bx + cv ) = ∂ v ϕ ( x (cid:48) , v ) − ( bx (cid:48) + cv ) + (cid:90) ( ∂ x,v ϕ − b )( x s , v )( x − x (cid:48) ) ds, where x s = sx + (1 − s ) x (cid:48) . Therefore | ∂ v ϕ ( x, v ) − ( bx + cv ) | ≤ | ∂ v ϕ ( x (cid:48) , v ) − ( bx (cid:48) + cv ) | + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ∂ x,v ϕ − b )( x s , v )( x − x (cid:48) ) ds (cid:12)(cid:12)(cid:12)(cid:12) := K + K , • Estimate of K :Since for any z ∈ R we have | Az | ≤ ρ ( A ) | z | A ≤ ρ ( A ) | z | , N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 11 we obtain by definition of x (cid:48) , x + v | ∂ v ϕ ( x (cid:48) , v ) − ( bx (cid:48) + cv ) | ≤ |∇ ϕ ( x (cid:48) , v ) − A ( x (cid:48) , v ) | = (cid:12)(cid:12) A (cid:0) A − ∇ ϕ ( x (cid:48) , v ) − ( x (cid:48) , v ) (cid:1)(cid:12)(cid:12) ≤ ρ ( A ) (cid:12)(cid:12) A − ∇ ϕ ( x (cid:48) , v ) − ( x (cid:48) , v ) (cid:12)(cid:12) ≤ ρ ( A ) (cid:90) R +2 R +1 (cid:12)(cid:12) A − ∇ ϕ ( x, v ) − ( x, v ) (cid:12)(cid:12) dx ≤ ρ ( A ) e sup y ∈ [ R +1 ,R +2] U ( y ) (cid:90) R +2 R +1 (cid:12)(cid:12) A − ∇ ϕ ( x, v ) − ( x, v ) (cid:12)(cid:12) e − U ( x ) dx ≤ ρ ( A ) e α ( R +2) e (cid:107) ψ (cid:107) L ∞ (cid:90) R \Y v (cid:12)(cid:12) A − ∇ ϕ ( x, v ) − ( x, v ) (cid:12)(cid:12) e − U ( x ) dx • Estimate of K :By Cauchy-Schwarz’s inequality we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ( ∂ x,v ϕ − b )( x s , v )( x − x (cid:48) ) ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ x,v ϕ − b∂ x ϕ / ( x s , v ) e − U ( xs )2 e U ( xs )2 ( ∂ x ϕ ( x s , v )) / ( x − x (cid:48) ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:32)(cid:90) ( ∂ x,v ϕ − b ) ∂ x ϕ ( x s , v ) e − U ( x s ) ds (cid:33) (cid:18)(cid:90) ∂ x ϕ ( x s , v ) e U ( x s ) ( x − x (cid:48) ) ds (cid:19) . Since x, x (cid:48) ∈ [ − ( R + 2) ; ( R + 2)] we have (cid:90) ∂ x ϕ ( x s , v ) e U ( x s ) ( x − x (cid:48) ) ds ≤ e α ( R +2) e (cid:107) ψ (cid:107) L ∞ ( ∂ x ϕ ( x, v ) − ∂ x ϕ ( x (cid:48) , v ))( x − x (cid:48) ) . Next observe that | ∂ x ϕ ( x, v ) − ∂ x ϕ ( x (cid:48) , v ) | = | ( ∂ x ϕ ( x, v ) − ( bx + cv )) − ( ∂ x ϕ ( x (cid:48) , v ) − ( bx (cid:48) + cv )) + b ( x (cid:48) − x ) |≤ |∇ ϕ ( x, v ) − A ( x, v ) | + |∇ ϕ ( x (cid:48) , v ) − A ( x (cid:48) , v ) | + b ( | x | + | x (cid:48) | ) ≤ ρ ( A ) (cid:0)(cid:12)(cid:12) A − ∇ ϕ ( x, v ) − ( x, v ) (cid:12)(cid:12) + (cid:12)(cid:12) A − ∇ ϕ ( x (cid:48) , v ) − ( x (cid:48) , v ) (cid:12)(cid:12)(cid:1) + b ( | x | + | x (cid:48) | ) ≤ ρ ( A ) (cid:0)(cid:12)(cid:12) A − ∇ ϕ ( x, v ) − ( x, v ) (cid:12)(cid:12) + (cid:12)(cid:12) A − ∇ ϕ ( x ∗ v , v ) − ( x ∗ v , v ) (cid:12)(cid:12)(cid:1) + b ( R + 2) . But since x ∈ Y v we have (cid:12)(cid:12) A − ∇ ϕ ( x, v ) − ( x, v ) (cid:12)(cid:12) = (cid:113) (det( A ) − ( c∂ x ϕ − b∂ v ϕ ) − x ) + (det( A ) − (( b + cα ) ∂ v ϕ − b∂ x ϕ ) − v ) ≤ | c∂ x ϕ − b∂ v ϕ − x | √ ≤ ( R + 2) √ , and by definition of x ∗ v we similarly obtain (cid:12)(cid:12) A − ∇ ϕ ( x ∗ v , v ) − ( x ∗ v , v ) (cid:12)(cid:12) ≤ ( R + 2) √ . Gathering all these estimates yields (cid:90) ∂ x ϕ ( x s , v ) e U ( x s ) ( x − x (cid:48) ) ds ≤ ce α ( R +2) e (cid:107) ψ (cid:107) L ∞ ( R + 2) (cid:18) ρ ( A ) c √ bc (cid:19) Therefore for any x ∈ Y v there holds | ∂ v ϕ ( x, v ) − ( bx + cv ) | ≤ c (cid:18) ρ ( A ) c (cid:19) e α ( R +2) e (cid:107) ψ (cid:107) L ∞ (cid:90) R Y v (cid:12)(cid:12) A − ∇ ϕ ( y, v ) − ( y, v ) (cid:12)(cid:12) e − U ( y ) dy + 2 ce α ( R +2) e (cid:107) ψ (cid:107) L ∞ ( R + 2) (cid:18) ρ ( A ) c √ bc (cid:19) (cid:32)(cid:90) R ( ∂ x,v ϕ − b ) ∂ x ϕ ( y, v ) e − U ( y ) dy (cid:33) =: c C (cid:90) R \Y v (cid:12)(cid:12) A − ∇ ϕ ( y, v ) − ( y, v ) (cid:12)(cid:12) e − U ( y ) dy + cC (cid:90) R ( ∂ x,v ϕ − b ) ∂ x ϕ ( y, v ) e − U ( y ) dy Hence integrating over Y v and using Lemma A.2, we obtain (cid:90) Y v | ∂ v ϕ ( x, v ) − ( bx + cv ) | e − U ( x ) dx ≤ c C (cid:90) R \Y v (cid:12)(cid:12) A − ∇ ϕ ( y, v ) − ( y, v ) (cid:12)(cid:12) e − U ( y ) dy + cC (cid:90) R ( ∂ v ϕ ) − (cid:32)(cid:0) ∂ v ϕ − c (cid:1) + (cid:0) b∂ v ϕ − c∂ x,v ϕ (cid:1) det( ∇ ϕ ) (cid:33) ( x, v ) e − U ( x ) dx. (cid:5) Conclusion :By definition of J A we have Z J A ( f | f ∞ ) = (cid:90) R (cid:32)(cid:90) R \Y v ∪Y v −(cid:104) B ( A − ∇ ϕ ( x, v )) − B ( x, v ) , A − ∇ ϕ ( x, v ) − ( x, v ) (cid:105) A e − U ( x ) dx (cid:33) e − | v | dv + (cid:90) R (cid:32)(cid:90) R ( ∂ v ϕ ) − (cid:32)(cid:0) ∂ v ϕ − c (cid:1) + (cid:0) b (cid:0) ∂ v ϕ − c (cid:1) − c (cid:0) ∂ x,v ϕ − b (cid:1)(cid:1) det( ∇ ϕ ) (cid:33) e − U ( x ) dx (cid:33) e − | v | dv. Using the above steps yields Z J A ( f | f ∞ ) ≥ c ( κ ∧ κ )2 (cid:90) R (cid:32)(cid:90) R \Y v | A − ∇ ϕ ( x, v ) − ( x, v ) | e − U ( x ) dx (cid:33) e − | v | dv + c ( κ ∧ κ )2 (cid:90) R (cid:32)(cid:90) R \Y v | A − ∇ ϕ ( x, v ) − ( x, v ) | e − U ( x ) dx (cid:33) e − | v | dv + (cid:90) R (cid:32)(cid:90) R ( ∂ v ϕ ) − (cid:32)(cid:0) ∂ v ϕ − c (cid:1) + (cid:0) b (cid:0) ∂ v ϕ − c (cid:1) − c (cid:0) ∂ x,v ϕ − b (cid:1)(cid:1) det( ∇ ϕ ) (cid:33) e − U ( x ) dx (cid:33) e − | v | dv + (cid:90) R (cid:18)(cid:90) Y v −(cid:104) B ( A − ∇ ϕ ( x, v )) − B ( x, v ) , A − ∇ ϕ ( x, v ) − ( x, v ) (cid:105) A e − U ( x ) dx (cid:19) e − | v | dv. Since c < c ∗ there holds cC ≤ ( κ ∧ κ )2 , cC ≤ , therefore, by the above steps, (cid:90) R (cid:18)(cid:90) Y v | ∂ v ϕ ( x, v ) − ( bx + cv ) | e − U ( x ) dx (cid:19) e − | v | dv ≤ c ( κ ∧ κ )2 (cid:90) R (cid:32)(cid:90) R \Y v | A − ∇ ϕ ( x, v ) − ( x, v ) | e − U ( x ) dx (cid:33) e − | v | dv + (cid:90) R (cid:32)(cid:90) R ( ∂ v ϕ ) − (cid:32)(cid:0) ∂ v ϕ − c (cid:1) + (cid:0) b (cid:0) ∂ v ϕ − c (cid:1) − c (cid:0) ∂ x,v ϕ − b (cid:1)(cid:1) det( ∇ ϕ ) (cid:33) e − U ( x ) dx (cid:33) e − | v | dv, and Z J A ( f | f ∞ ) ≥ c ( κ ∧ κ )2 (cid:90) R (cid:32)(cid:90) R \Y v | A − ∇ ϕ ( x, v ) − ( x, v ) | e − U ( x ) dx (cid:33) e − | v | dv + (cid:90) R (cid:18)(cid:90) Y v | ∂ v ϕ ( x, v ) − ( bx + cv ) | e − U ( x ) dx (cid:19) e − | v | dv + (cid:90) R (cid:18)(cid:90) Y v −(cid:104) B ( A − ∇ ϕ ( x, v )) − B ( x, v ) , A − ∇ ϕ ( x, v ) − ( x, v ) (cid:105) A e − U ( x ) dx (cid:19) e − | v | dv N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 13
Finally, since b > b ∗ , we have by point ( iii ) of Lemma A.1 Z J A ( f | f ∞ ) ≥ c ( κ ∧ κ )2 (cid:90) R (cid:32)(cid:90) R \Y v | A − ∇ ϕ ( x, v ) − ( x, v ) | e − U ( x ) dx (cid:33) e − | v | dv + cκ (cid:90) R (cid:18)(cid:90) Y v | A − ∇ ϕ ( x, v ) − ( x, v ) | e − U ( x ) dx (cid:19) e − | v | dv, and J A ( f | f ∞ ) ≥ c min (cid:18) ( κ ∧ κ )2 , κ (cid:19) (cid:90) R | A − ∇ ϕ ( x, v ) − ( x, v ) | f ∞ ( x, v ) dxdv ≥ ρ ( A ) − c min (cid:18) ( κ ∧ κ )2 , κ (cid:19) (cid:90) R | A − ∇ ϕ ( x, v ) − ( x, v ) | A f ∞ ( x, v ) dxdv, and the result is proved with κ = α + + (cid:113)(cid:0) α − (cid:1) + 12 − min (cid:18) ( κ ∧ κ )2 , κ (cid:19) . Acknowledgements
The author was supported by the Fondation Math´ematique Jacques Hadamard, and warmly thanksPatrick Cattiaux and Arnaud Guillin for many advices, comments and discussions which have made thiswork possible.
Appendix A. Toolbox
Proof of Theorem 2.1 :( i ) Since the cost function c ( z ) = | z | A , thanks to which W A is defined, is strictly convex (since A ispositive definite) and superlinear (i.e. lim | z |→ + ∞ | z | A | z | = + ∞ ), we may invoke [10, Theorem 2.44] (seealso [9, Theorem 1.17]), and for any µ, ν ∈ P ( R ) with smooth enough densities, we obtain that hereexists a unique map T : R (cid:55)→ R such that T ν = µ and W A ( µ, ν ) = (cid:90) R | ( x, v ) − T ( x, v ) | A ν ( x, v ) dxdv. Moreover T is given as T ( z ) = z − ∇ c ∗ ( ∇ Φ( z )) , where Φ is some c -concave function (see [10, Definition 2.33 ]. But since for any z ∈ R , ∇ c ∗ ( z ) = A − z we have T ( z ) = A − ( Az − ∇ Φ( z )) = A − ∇ (cid:18) | z | A − Φ( z ) (cid:19) . We set ϕ ( z ) = | z | A − Φ( z ) , and observe that it is a convex functional since Φ is | · | A -concave (see [9, Proposition 1.21]).( iii ) follows from the regularity theory for Monge-Amp`ere equation (see for instance [4, Theorem 1])and the fact that µ and ν have C , non vanishing densities.( ii ) follows from point ( i ) and the fact that, since ∀ ( x, v ) ∈ R we have ∇ ϕ ∗ ( ∇ ϕ ( x, v )) = ( x, v ), then( A − ∇ ϕ ) − = ∇ ϕ ∗ ( A · ) transports µ onto ν , and point ( i ). (cid:3) Lemma A.1.
Assume that U is of the form (1.4) , for some α > and some compactly supported on [ − R, R ] function ψ , satisfying (2.1) .Let c = 2 b > , and A = (cid:18) b + cα bb c (cid:19) . Assume that b > ( (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b ) b + γ, i.e. b > b ∗ . Then there are κ , κ , κ > such that for any z = ( x , v ) , z = ( x , v ) ∈ R , there holds, with z (cid:48) i = ( x (cid:48) i , v (cid:48) i ) = A − z i for i = 1 , i ) in the case | x (cid:48) | ≥ R + 1 or | x (cid:48) | ≥ R + 1 −(cid:104) B ( z (cid:48) ) − B ( z (cid:48) ) , z (cid:48) − z (cid:48) (cid:105) A ≥ cκ | z (cid:48) − z (cid:48) | with κ = min (cid:18) − (cid:107) ψ (cid:48) (cid:107) L ∞ ,
12 ( α − (cid:107) ψ (cid:48) (cid:107) L ∞ ) (cid:19) , ( ii ) in the case | x (cid:48) | ≤ R + 1 and | x (cid:48) | ≤ R + 1 and | x − x | < | v − v |−(cid:104) B ( z (cid:48) ) − B ( z (cid:48) ) , z (cid:48) − z (cid:48) (cid:105) A ≥ cκ | z (cid:48) − z (cid:48) | with κ = 12 (cid:18) − (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ − γ (cid:19) , ( iii ) in the case | x | ≤ R + 1 and | x | ≤ R + 1 and | x − x | ≥ | v − v |−(cid:104) B ( z (cid:48) ) − B ( z (cid:48) ) , z (cid:48) − z (cid:48) (cid:105) A + | v − v | ≥ κ | z (cid:48) − z (cid:48) | . Proof.
Recall that U (cid:48) ( x ) = αx + ψ (cid:48) ( x ) and observe that (cid:104) B ( z (cid:48) ) − B ( z (cid:48) ) , z (cid:48) − z (cid:48) (cid:105) A =( b + cα )( v (cid:48) − v (cid:48) )( x (cid:48) − x (cid:48) ) + b ( v (cid:48) − v (cid:48) )( v (cid:48) − v (cid:48) ) − b (( v (cid:48) − v (cid:48) ) + ( U (cid:48) ( x (cid:48) ) − U (cid:48) ( x (cid:48) ))) ( x (cid:48) − x (cid:48) ) − c (( v (cid:48) − v (cid:48) ) + ( U (cid:48) ( x (cid:48) ) − U (cid:48) ( x (cid:48) ))) ( v (cid:48) − v (cid:48) )= − ( c − b ) | v (cid:48) − v (cid:48) | − bα | x (cid:48) − x (cid:48) | − b ( ψ (cid:48) ( x (cid:48) ) − ψ (cid:48) ( x (cid:48) ))( x (cid:48) − x (cid:48) ) − c ( ψ (cid:48) ( x (cid:48) ) − ψ (cid:48) ( x (cid:48) ))( v (cid:48) − v (cid:48) ) (cid:5) Proof of ( i )W.l.o.g. assume that | x (cid:48) | ≥ R + 1, and | x (cid:48) | ≤ R (otherwise the result is obvious since ψ (cid:48) is compactlysupported on [ − R, R ])Since | ψ (cid:48) ( x (cid:48) ) − ψ (cid:48) ( x (cid:48) ) | ≤ | ψ (cid:48) ( x (cid:48) ) | ≤ (cid:107) ψ (cid:48) (cid:107) L ∞ ≤ (cid:107) ψ (cid:48) (cid:107) L ∞ | x (cid:48) − x (cid:48) | , by Young’s inequality −(cid:104) B ( z (cid:48) ) − B ( z (cid:48) ) , z (cid:48) − z (cid:48) (cid:105) A ≥ ( c − b ) | v (cid:48) − v (cid:48) | + b ( α − (cid:107) ψ (cid:48) (cid:107) L ∞ ) | x (cid:48) − x (cid:48) | − c (cid:107) ψ (cid:48) (cid:107) L ∞ | x (cid:48) − x (cid:48) || v (cid:48) − v (cid:48) | = c (cid:18) | v (cid:48) − v (cid:48) | + 12 ( α − (cid:107) ψ (cid:48) (cid:107) L ∞ ) | x (cid:48) − x (cid:48) | − (cid:107) ψ (cid:48) (cid:107) L ∞ | x (cid:48) − x (cid:48) || v (cid:48) − v (cid:48) | (cid:19) ≥ c (cid:18)(cid:18) − (cid:107) ψ (cid:48) (cid:107) L ∞ (cid:19) | v (cid:48) − v (cid:48) | + 12 ( α − (cid:107) ψ (cid:48) (cid:107) L ∞ ) | x (cid:48) − x (cid:48) | (cid:19) , and the result is proved with κ = min (cid:18) − (cid:107) ψ (cid:48) (cid:107) L ∞ ,
12 ( α − (cid:107) ψ (cid:48) (cid:107) L ∞ ) (cid:19) , due to condition (2.1) N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 15 (cid:5)
Proof of ( ii )Since | x (cid:48) − x (cid:48) | ≤ | v (cid:48) − v (cid:48) |−(cid:104) B ( z (cid:48) ) − B ( z (cid:48) ) , z (cid:48) − z (cid:48) (cid:105) A ≥ ( c − b ) | v (cid:48) − v (cid:48) | − γb | x (cid:48) − x (cid:48) | − c (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ | x (cid:48) − x (cid:48) || v (cid:48) − v (cid:48) |≥ c (cid:18) − (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ − γ (cid:19) | v (cid:48) − v (cid:48) | , and the result is proved with κ = 12 (cid:18) − (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ − γ (cid:19) . (cid:5) Proof of ( iii )We denote r = x − x , s = v − v and r (cid:48) = x (cid:48) − x (cid:48) , s (cid:48) = v (cid:48) − v (cid:48) . Since A (cid:18) r (cid:48) s (cid:48) (cid:19) = (cid:18) rs (cid:19) = (cid:18) ar (cid:48) + bs (cid:48) br (cid:48) + cs (cid:48) (cid:19) , where a = b + cα . We have −(cid:104) B ( z (cid:48) ) − B ( z (cid:48) ) , z (cid:48) − z (cid:48) (cid:105) A + | v − v | ≥ ( c − b ) | s (cid:48) | − γb | r (cid:48) | − c (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ | r (cid:48) || s (cid:48) | + b | r (cid:48) | + 2 bcr (cid:48) s (cid:48) + c | s (cid:48) | ≥ ( c − b + c ) | s (cid:48) | + b ( b − γ ) | r (cid:48) | − (2 bc + c (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ ) | r (cid:48) || s (cid:48) | = c (cid:18)(cid:18)
12 + c (cid:19) | s (cid:48) | + 12 ( b − γ ) | r (cid:48) | − ( (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b ) | r (cid:48) || s (cid:48) | (cid:19) . By Young’s inequality − ( (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b ) | r (cid:48) || s (cid:48) | = ( (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b ) (cid:115) (cid:0) + c (cid:1) (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b | s (cid:48) | (cid:115) (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b (cid:0) + c (cid:1) | r (cid:48) |≥ − (cid:18)
12 + c (cid:19) | s (cid:48) | − ( (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b ) (cid:0) + c (cid:1) | r (cid:48) | , and provided that 12 ( b − γ ) − ( (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b ) b > , the result is proved with κ = 12 (cid:18)
12 ( b − γ ) − ( (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ + b ) b (cid:19) (cid:3) Lemma A.2.
Let M = (cid:18) m , m , m , m , (cid:19) ∈ M ( R ) be a symmetric positive definite matrix. For any b, c ∈ R , there holds (cid:18) ( m , − c ) + ( b ( m , − c ) − c ( m , − b )) det( M ) (cid:19) ( m , ) − ≥ ( m , − b ) ( m , ) − , with equality if and only if m , = c + ( m , − b ) m , m , . Proof.
First observe that since M is symmetric positive definite it holds det( M ) > m , > m , m , .Then the claimed inequality is equivalent to( m , − c ) det( M ) + ( b ( m , − c ) − c ( m , − b )) − ( m , − b ) m , m , det( M ) ≥ . Therefore the claimed inequality will follow from the sign study on (cid:16) m , m , , + ∞ (cid:17) the function g ( x ) = ( x − c ) ( xm , − m , ) + ( bx − cm , ) − ( m , − b ) xm , ( xm , − m , )= ( x − xc + c )( xm , − m , ) + ( b x − bcxm , + c m , ) − ( m , − b ) ( x − x m , m , )= m , x + (cid:0) − cm , − m , + b − ( m , − b ) (cid:1) x + (cid:32) c m , + 2 cm , − bcm , + m , m , ( m , − b ) (cid:33) x − c m , + c m , = m , x − cm , + ( m , − b ) m , ) x + (cid:32) c m , + 2 cm , ( m , − b ) + m , m , ( m , − b ) (cid:33) x = x (cid:32) m , x − cm , + ( m , − b ) m , ) x + (cid:32) c m , + 2 cm , ( m , − b ) + m , m , ( m , − b ) (cid:33)(cid:33) = xp ( x ) . The discriminant ∆ of the second order polynomial p is given by∆ = 4 (cid:32) ( cm , + ( m , − b ) m , ) − m , (cid:32) c m , + 2 cm , ( m , − b ) − m , m , ( m , − b ) (cid:33)(cid:33) = 4 (cid:16) ( cm , + ( m , − b ) m , ) − (cid:0) c m , + 2 cm , m , ( m , − b ) + ( m , − b ) m , (cid:1)(cid:17) . = 0Therefore g ( x ) = m , x (cid:18) x − cm , + ( m , − b ) m , m , (cid:19) , and g is non negative on (cid:16) m , m , , + ∞ (cid:17) which concludes the proof. (cid:3) Lemma A.3.
There exist
R > , ψ ∈ C ( R ) compactly supported on [ − R, R ] and α > satisfyingcondition (2.1) .Proof. We set R = 1 and Ψ( x ) = ( x − ( x + 1) x ∈ [ − , , which is C . We numerically check that (cid:107) Ψ (cid:107) L ∞ = 1 , (cid:107) Ψ (cid:48) (cid:107) L ∞ (cid:39) . < , (cid:107) Ψ (cid:48)(cid:48) (cid:107) L ∞ = 8 . Let A ∈ (2 (cid:107) Ψ (cid:48) (cid:107) L ∞ , (cid:107) Ψ (cid:48)(cid:48) (cid:107) L ∞ ) and set Γ = (cid:107) Ψ (cid:48)(cid:48) (cid:107) L ∞ − A . Let φ > α = φA, ψ ( x ) = φ Ψ( x ) , γ = φ Γ . We impose that γ < (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ , i.e. φ > (cid:107) Ψ (cid:48)(cid:48) (cid:107) L ∞ − A (cid:107) Ψ (cid:48)(cid:48) (cid:107) L ∞ . In view of (2.1), we also need to impose that φ < (cid:107) Ψ (cid:48)(cid:48) (cid:107) − L ∞ − .Therefore c ∗ in (2.1) is (brutally) bounded by below by c ∗ > e − e − φ (cid:18) ∧ φ ( A − (cid:107) Ψ (cid:48) (cid:107) L ∞ )4 (cid:19) > e − e − φ φ, N OPTIMAL TRANSPORT APPROACH OF HYPOCOERCIVITY 17 since we can always choose A close enough to (cid:107) Ψ (cid:48)(cid:48) (cid:107) so that A − (cid:107) Ψ (cid:48) (cid:107) L ∞ > b ∗ is bounded by above by b ∗ < ( γ + (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ ) 43 < (cid:107) ψ (cid:48)(cid:48) (cid:107) L ∞ = 83 φ (cid:107) Ψ (cid:48)(cid:48) (cid:107) L ∞ . We can now choose A close enough to (cid:107) Ψ (cid:48)(cid:48) (cid:107) , so that φ maybe small enough so that c ∗ > b ∗ (cid:3) References [1] F. Bolley, I. Gentil, A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, Journalof Functional Analysis, Vol 236(8), (2012), 2430-2457[2] F. Bolley, I. Gentil, A. Guillin, Uniform convergence to equilibrium for granular media, Archive for Rational MechanicalAnalysis, Vol 208(2), (2013), 429-445.[3] F. Bolley, A. Guillin, F. Malrieu, Trend to equlibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, Mathematical Modeling and Numerical Analysis, Vol 44(5), (2010), 867-884[4] D. Cordero-Erausquin, A. Figalli Regularity of monotone maps between unbounded domains. Discrete & Contin. Dyn.Syst. 39 (2019), 7101–7112.[5] H. Dietert, J. Evans, and T. Holding, Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation onthe torus, Kinetic & Related Models, vol. 11, no. 6, pp. 1427–1441, 2018[6] J. Evans, Hypocoercivity in Wasserstein-1 for the kinetic Fokker-Planck equation via Malliavin Calculus, preprinthttps://arxiv.org/abs/1810.01324 .[7] A. Eberle, A. Guillin, R. Zimmer, Couplings and quantitative contraction rates for Langevin dynamics, Ann. Probab.Volume 47, Number 4 (2019), 1982-2010.[8] A. Guillin, P. Monmarch´e, Uniform long-time and propagation of chaos estimates for mean field kinetic particles innon-convex landscapes , preprint https://arxiv.org/pdf/2003.00735.pdf[9] F. Santambrogio, Optimal Transportfor Applied Mathematicians[10] C. Villani, Optimal Transport, Old and New, Grundlehren Math. Wiss., vol. 338, Springer, Berlin, 2009.[11] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202, 950 (2009).(Samir Salem)
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