Michel Bonnefont
email: michel.bonnefont@math.u-bordeaux.fr

Nicolas Juillet
email: nicolas.juillet@math.unistra.fr

COUPLINGS IN L p DISTANCE OF TWO BROWNIAN MOTIONS AND THEIR L ÉVY AREA

Keywords: Mathematics Subject Classification. 60H10, 60J60, 60J65, 53C17, 22E25 Heisenberg group, Co-adapted coupling, Wasserstein distance, Hypoelliptic diffusion

We study co-adapted couplings of (canonical hypoelliptic) diffusions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brownian motions and are the joint laws of a planar Brownian motion with its Lévy area. We show that contrary to the situation observed on Riemannian manifolds of non-negative Ricci curvature, for any co-adapted coupling, two Heisenberg Brownian motions starting at two given points can not stay at bounded distance for all time t ≥ 0. Actually, we prove the stronger result that they can not stay bounded in L p for p ≥ 2.

We also prove two positive results. We first study the coupling by reflection and show that it stays bounded in L p for 0 ≤ p < 1. Secondly, we construct an explicit static (and in particular non co-adapted) coupling between the laws of two Brownian motions, which provides L 1 -Wasserstein control uniformly in time.

Finally, we explain how the results generalise to the Heisenberg groups of higher dimension.

1. Introduction 1.1. L ∞ control. The motivation for this paper is a question, concerning heat diffusion on the Heisenberg group, that is implicitly raised by Kuwada in [START_REF] Kuwada | Duality on gradient estimates and Wasserstein controls[END_REF]Remark 4.4], and that we reproduce at page 2 after Theorem 1.1. Before we reach this question let us start with some background and a few definitions. All remaining material will be introduced later in the paper. In the literature, L ∞ -Wasserstein control for a diffusion has been used to deduce L 1 -gradient estimates of its associated semigroup (see for instance [START_REF] Wang | On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups[END_REF] and the references therein). Kuwada extends this result to L p -Wasserstein control and L q -gradient estimates for all p, q ≥ 1 with 1 p + 1 q = 1 and, using Kantorovich duality, proves that, conversely, L q -gradient estimates allow one to obtain L p -Wasserstein control for the diffusion.

We recall that, on a metric space (M, d), for p ∈ (0, ∞], the L p -Wasserstein distance between two probability measures µ and ν is given by

W p (µ, ν) = inf π∈Π(µ,ν)
d(x, y) p dπ(x, y)

1/p = inf X∼µ, Y ∼ν d(X, Y ) p . ( 1
)
Here Π(µ, ν) is the set of probability measures on M × M with marginals µ and ν.

For p = ∞ the first expression is replaced by the essential supremum of d. Note that W p is a distance only for p ≥ 1. For 0 < p < 1, it is only a quasidistance, in the sense that the triangle inequality only holds up to a multiplicative constant.

Using Hölder inequality, it is clear that W p (µ, ν) ≤ W q (µ, ν) if 0 < p ≤ q.

On the Heisenberg group H, the following L 1 -gradient bound was established by H.Q. Li [START_REF] Li | Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg[END_REF] (see also [START_REF] Bakry | On gradient bounds for the heat kernel on the Heisenberg group[END_REF]) generalising [START_REF] Driver | Hypoelliptic heat kernel inequalities on the Heisenberg group[END_REF] ∀f ∈ C ∞ c (H), ∀t ≥ 0, ∀a ∈ H,

|∇ h P t f (a)| ≤ CP t (|∇ h f |)(a)
, where C > 1 is constant, P t denotes the heat semigroup associated to half the sub-Laplacian and ∇ h the horizontal gradient (see Section 2.1 for the definitions). Consequently, Kuwada's result implies that the heat diffusion of the Heisenberg group possesses a L ∞ -Wasserstein control for its diffusion: Theorem 1.1 (H.Q. Li, Kuwada). There exists C > 0 such that for every t ≥ 0 and a, a

′ ∈ H W ∞ (µ a t , µ a ′ t ) ≤ Cd H (a, a ′ ), (2)
where µ a t = Law(B a t ) and µ a ′ t = Law(B a ′ t ) and (B a s ) s≥0 , (B a ′ s ) s≥0 are two Heisenberg Brownian motions, starting respectively in a, a ′ . Moreover,

W p (µ a t , µ a ′ t ) ≤ Cd H (a, a ′ ) ( 3
)
holds for every p < ∞.

In other words, for each a, a ′ ∈ H and each t ≥ 0, there exists a coupling (B a s , B a ′ s ) s≥0 of the two Heisenberg Brownian motions such that d H (B a t , B a ′ t ) ≤ Cd H (a, a ′ ) almost surely.

Please note: Firstly the time t ≥ 0 is fixed; secondly B a t and B a ′ t are conveniently defined on the same probability space (and the remaining random variables (B a s ) s =t and (B a ′ s ) s =t of the Heisenberg Brownian motions are defined without paying attention to their correlation). Kuwada's problem is precisely on inverting the quantifiers ∀ and ∃, namely, he asks whether it is possible to define a coupling of the two Heisenberg Brownian motions (B a t ) t≥0 and (B a ′ t ) t≥0 such that (4) holds for all t ≥ 0.

In this paper we answer negatively and show that (4) can not hold for all t ≥ 0 for co-adapted couplings (see Definition 2.1), probably the most usual couplings in the literature for our type of problem, (see, e.g. [START_REF] Ben Arous | Coupling constructions for hypoelliptic diffusions: two examples[END_REF][START_REF] Kendall | Coupling all the Lévy stochastic areas of multidimensional Brownian motion[END_REF][START_REF] Kendall | Brownian couplings, convexity, and shy-ness[END_REF][START_REF] Kendall | Coupling time distribution asymptotics for some couplings of the Lévy stochastic area[END_REF][START_REF] Cranston | Gradient estimates on manifolds using coupling[END_REF][START_REF] Von Renesse | Intrinsic coupling on Riemannian manifolds and polyhedra[END_REF][START_REF] Kuwada | A counterexample for the optimality of Kendall-Cranston coupling[END_REF][START_REF] Pascu | Shy and fixed-distance couplings of Brownian motions on manifolds[END_REF]). Informally, a coupling of two processes (B t ) and (B ′ t ) is said co-adapted if the interaction in the coupling only depends on the common past of the process (B t , B ′ t ). See Definition 2.1 fo the rigorous definition.

Our results hold for the Heisenberg groups of higher dimension, as explained in Section 6, but we only prove them thoroughly in the first Heisenberg group where all the significative ideas are present and the notation is lighter. Theorem 1.2. Let (B t ) t and (B ′ t ) t be any two co-adapted Heisenberg Brownian motions starting respectively in a = (x, y, z) and a ′ = (x ′ , y ′ , z ′ ) with (x ′x) 2 + (y ′y) 2 > 0. Then, for every C > 0,

P (∀t ≥ 0, d H (B t , B ′ t ) ≤ C) = 1.
Moreover, using the homogeneous dilations (defined in Section 2.1), we also obtain the following slightly improved version.

Theorem 1.3. For every T > 0 and every C > 0 there exists two points a, a ′ ∈ H with a = a ′ such that for every co-adapted coupling (B a t , B a ′ t ) 0≤t≤T , there exists

t ≤ T such that essup (Ω,P) d H (B a t , B a ′ t ) > Cd H (a, a ′ ). Remark 1.4. Another way to state Theorem 1.3 is as follows: Let T > 0, then sup a =a ′ ∈H inf A (a,a ′ ) T sup 0≤t≤T essup (Ω,P) d H (B a t , B a ′ t ) d H (a, a ′ ) = +∞ where A (a,a ′ ) T
denotes the set of all co-adapted couplings (B a t , B a ′ t ) 0≤t≤T of two Heisenberg Brownian motions starting respectively in a and a ′ .

1.2.

Comparison with the Riemannian case. These results show a significative difference with the Riemannian case. Indeed, on a Riemannian manifold M, it is well known (see e.g. [START_REF] Wang | On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups[END_REF] and [START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy, and Ricci curvature[END_REF]) that if the Ricci curvature is bounded from below by k ∈ R, for every a, a ′ ∈ M, there exists a Markovian coupling of two Brownian motions such that almost surely

d(B a t , B a ′ t ) ≤ e -(k/2)t d(a, a ′ ) for every t ≥ 0. (5)
Here we call Brownian motions the diffusion processes starting at a and a ′ respectively, having generator half the Laplace-Beltrami operator. We make clear that Markovian coupling is a type of co-adapted coupling. Note moreover that the motivation for proving [START_REF] Borodin | Handbook of Brownian motion-facts and formulae. Probability and its Applications[END_REF] is exactly to provide estimates on the heat semi-group (see, e.g. [START_REF] Cranston | Gradient estimates on manifolds using coupling[END_REF][START_REF] Cranston | A probabilistic approach to gradient estimates[END_REF]), so that the historical L p -Wasserstein controls have been established for co-adaptive processes whereas L p -Wasserstein controls at fixed time may first appear unusual from a stochastic perspective. We note further that • the Heisenberg group can be thought as the first sub-elliptic model space of curvature 0 (e.g. [START_REF] Montgomery | A Tour of Subriemannian Geometries, Their Geodesics and Applications[END_REF]) but, its behaviour with respect to couplings of coadapted Brownian motions is therefore completely different from the case of Riemannian manifolds with curvature bounded from below by k = 0.

• the Heisenberg group is also classically presented as the limit space for a sequence of Riemannian metrics on the Lie group, the optimal lower bound on the Ricci curvature of which tends to -∞. On this topic see [START_REF] Juillet | Geometric inequalities and generalized Ricci bounds in the Heisenberg group[END_REF][START_REF] Baudoin | Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries[END_REF]. This fact is coherent with the interpretation of Theorem 1.2 as a special case of [START_REF] Borodin | Handbook of Brownian motion-facts and formulae. Probability and its Applications[END_REF] where the best bound for the L ∞ control is C = e -kt with k = -∞: There is no possible control for t > 0.

1.3. L p control for 0 < p < ∞. To go further, given two diffusion processes (B t ) t≥0 and (B ′ t ) t≥0 on a metric space (M, d), we shall consider the function

t ∈ [0, ∞) → E [d(B t , B ′ t ) p ] 1 p ∈ [0, ∞].
and try to bound it from above uniformly in time for some well-chosen co-adapted coupling. If we denote by µ t and ν t the law of the processes (B t ) t≥0 and (B ′ t ) t≥0 , we clearly have for each t ≥ 0:

E [d(B t , B ′ t ) p ] 1 p ≥ W p (µ t , ν t ).
On the Heisenberg group, we will prove the result stronger than Theorem 1.3 that any co-adapted coupling (B t ) t and (B ′ t ) t of Brownian motions do not stay bounded in L 2 : Theorem 1.5. Let (B t ) t≥0 and (B ′ t ) t≥0 be any two co-adapted Heisenberg Brownian motion starting respectively in a = (x, y, z) and a

′ = (x ′ , y ′ , z ′ ) such that (x ′ -x) 2 + (y ′ -y) 2 > 0. Then, lim sup t→+∞ E d H (B t , B ′ t ) 2 → +∞.
Moreover, as before, using the homogeneous dilations, we also obtain:

Theorem 1.6. Let p ≥ 2. For every T > 0 and every C > 0 there exists two points a, a ′ ∈ H with a = a ′ such that for every co-adapted coupling (B a t , B a ′ t ) 0≤t≤T , there exists t ≤ T such that

E d p H (B a t , B a ′ t ) > Cd H (a, a ′ ) p .
Remark 1.7. Equivalently Theorem 1.6 can be stated as follows

: Let p ≥ 2. Let T > 0, then sup a =a ′ ∈H inf A (a,a ′ ) T sup 0≤t≤T E[d p H (B a t , B a ′ t )] 1 p d H (a, a ′ ) = +∞
where A (a,a ′ ) T denotes the set of all co-adapted coupling (B a t , B a ′ t ) 0≤t≤T of two Heisenberg Brownian motions starting respectively in a and a ′ . 1.4. Two positive results. To complete the picture, we provide two positive results. We first show that the coupling by reflection on the Heisenberg group stays bounded in L p for 0 < p < 1. We recall that for 0 < p < 1, the quantity

E [d p (B t , B ′ t )]
1 p is not a distance, but only a quasidistance, in the sense that the triangle inequality only holds up to a multiplicative constant. Theorem 1.8. Let (B t ) t≥0 and (B ′ t ) t≥0 be a coupling by reflection of two Heisenberg Brownian motions starting in (x, y, z) and (x ′ , y ′ , z ′ ). Then, for every p ∈ (0, 1), sup

t≥0 E [d H (B t , B ′ t ) p ] < +∞. ( 6
)
Moreover, for the coupling by reflection, for every p ∈ (0, 1), we also have:

sup a =a ′ ∈H sup t≥0 E[d H (B a t , B a ′ t ) p ] d H (a, a ′ ) p < +∞ (7)
Unfortunately, the above result is false for the reflection coupling for p ≥ 1 (as a close look at Proposition 4.1 shows).

In the general context of co-adapted coupling, we were not able to obtain any results for p ∈ [1, 2): we ignore whether there exist co-adapted couplings satisfying (6) or [START_REF] Cranston | A probabilistic approach to gradient estimates[END_REF], or not, for p ∈ [START_REF] Bakry | On gradient bounds for the heat kernel on the Heisenberg group[END_REF][START_REF] Banerjee | Coupling in the Heisenberg group and its applications to gradient estimates[END_REF]. One difficulty in this study is to obtain estimates for the expectation of nonnegative (nonconvex) functionals of martingales as typically x → |x| 1/2 , see Remark 3.5.

The next remark recalls the situation of L p -Wasserstein control in the case of Riemannian manifolds. Remark 1.9. On a Riemannian manifold M, for p ≥ 1, the L p version of the L ∞ control (2), namely W p (µ a t , µ a ′ t ) ≤ e -kt/2 d(a, a ′ ) for all a ∈ M, a ′ ∈ M and t ≥ 0; is satisfied if and only if the Ricci curvature is bounded from below by k (see [START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy, and Ricci curvature[END_REF] and [START_REF] Kuwada | Duality on gradient estimates and Wasserstein controls[END_REF]Remark 2.3]). Therefore all the above L p -Wasserstein controls, for p ∈ [1, ∞] are equivalent and only depend on the Ricci curvature lower bound. Moreover in this situation, as said before, for every a ∈ M, a ′ ∈ M one has the existence of some appropriate Markovian coupling such that almost surely d(B a t , B a ′ t ) ≤ e -kt/2 d(a, a ′ ) for all t ≥ 0. We now turn to the second positive result. We propose an explicit coupling of the laws µ a t = Law(B a t ) and µ a ′ t = Law(B a ′ t ). This coupling is not at all dynamical and is made at a given fixed time t. This coupling has thus no interpretation in terms of co-adapted coupling. It provides a new proof of the case p = 1 in Theorem 1.1. This is the weakest result in the spectrum of L p -Wasserstein controls ; the strongest result is for p = ∞. However, we stress that, apparently, our static coupling provides the first direct proof (that is, not obtained by duality) of the case p = 1.

Theorem 1.10. There exists C > 0 such that for every t ≥ 0 and a = (x, y, z), a

′ = (x ′ , y ′ , z ′ ) ∈ H, there is a random vector (X, Y, Z, X ′ , Y ′ , Z ′ ) of marginals µ a t = Law(B a t ) = Law(X, Y, Z) and µ a ′ t = Law(B a ′ t ) = Law(X ′ , Y ′ , Z ′ ) such that W 1 (µ a t , µ a ′ t ) ≤ E(d H ((X, Y, Z), (X ′ , Y ′ , Z ′ ))) ≤ Cd H (a, a ′ )
and

X ′ = X + (x ′ -x) almost surely, Y ′ = Y + (y ′ -y) almost surely.
The idea of the coupling is the following. The goal is to construct a random vector ((X, Y, Z); (X ′ , Y ′ , Z ′ )) of marginals µ a t and µ a ′ t . First we perform a coupling of the horizontal parts (X, Y ) and (X ′ , Y ′ ) of the two Brownian laws by a simple translation. It appears that the conditional laws of the last coordinates L(Z|(X, Y )) and L(Z ′ |(X ′ , Y ′ )) differ also only by a translation; but which depends on the value of (X, Y ). Eventually, we use a coupling of the last coordinates which is well adapted to optimal transport for the cost (z, z ′ ) → |zz ′ | on the real line, and is better than the simple translation. Observe that our proof requires at the end an analytic estimate on the heat kernel, see (41). We stress that this estimate is not directly related to the L ∞ -gradient estimate which is through Kuwada duality equivalent to a L 1 -Wasserstein control. Moreover Theorem 1.10 provides more than only this control but a rather explicit and constructive coupling.

We mention the interesting recent work by S. Banerjee, M. Gordina and P. Mariano [START_REF] Banerjee | Coupling in the Heisenberg group and its applications to gradient estimates[END_REF] where the authors also use non co-adapted couplings to study the decay in total variation for the laws of Heisenberg Brownian motions and obtain gradient estimates for harmonic functions. This work and our work seems to deliver a common message namely that co-adapted couplings are not the unique relevant couplings, what concerns obtaining gradient estimates.

The paper is organised as follows. In Section 2.1, we recall the notion of coadapted coupling and describe quickly the geometry of the Heisenberg group, its associated Brownian motions and their coupling. We also discuss some classical couplings. The proofs of the main theorems on the non-existence of co-adapted Heisenberg Brownian motions which stay at bounded distance are given Section 3. The reflection coupling on H is studied in Section 4. The construction of the static coupling between the Brownian laws is given in Section 5. The results are then generalised to the Heisenberg groups of higher dimension in the final section.

2.1. The Heisenberg group. The Heisenberg group can be identified with R 3 equipped with the law:

(x, y, z) • (x ′ , y ′ , z ′ ) = x + x ′ , y + y ′ , z + z ′ + 1 2 (xy ′ -yx ′ ) .
The left invariant vector fields are given by

         X(f )(x, y, z) = d dt |t=0 f ((x, y, z) • (t, 0, 0)) = ∂ x -y 2 ∂z f (x, y, z) Y (f )(x, y, z) = d dt |t=0 f ((x, y, z) • (0, t, 0)) = ∂ y + x 2 ∂z f (x, y, z) Z(f )(x, y, z) = d dt |t=0 f ((x, y, z) • (0, 0, t)) = ∂ z f (x, y, z). Note that [X, Y ] = Z and that Z commutes with X and Y . We are interested in half the sub-Laplacian L = 1 2 (X 2 + Y 2 )
. This is a diffusion operator that satisfies the Hörmander bracket condition and thus the associated heat semigroup P t = e tL admits a C ∞ positive kernel p t .

From a probabilistic point of view, L is the generator of the following stochastic process starting in (x, y, z):

B (x,y,z) t := (x, y, z) • B 1 t , B 2 t , 1 2 t 0 B 1 s dB 2 s - t 0 B 2 s dB 1 s where (B 1 t ) t≥0 and (B 2 t ) t≥0 are two standard independent 1-dimensional Brownian motions. The quantity t 0 B 1 s dB 2 s -t 0 B 2 s dB 1 s that we denote by A t is one of the first stochastic integrals ever considered. It is the Lévy area of the 2-dimensional Brownian motion (B t ) t≥0 := (B 1 t , B 2 t ) t≥0 .
It is easily seen that (B t ) t≥0 is a continuous process with independent and stationary increments. We simply call it the Heisenberg Brownian motion.

The sub-Laplacian L is strongly related to the following subRiemmanian distance (also called Carnot-Carathéodory) on H:

d H (a, a ′ ) = inf γ 1 0 | γ(t)| h dt
where γ ranges over the horizontal curves connecting γ(0) = a and γ(1) = a ′ . We remind the reader of the fact that a curve is said horizontal if it is absolutely continuous and γ(t

) ∈ Vect(X(γ(t)), Y (γ(t))) almost surely holds. The horizontal norm | • | h is a Euclidean norm on Vect(X, Y ) obtained by asserting that (X, Y ) is an orthonormal basis of Vect(X(a), Y (a)) at each point a ∈ H. Finally the horizontal gradient ∇ h f is (Xf )X + (Y f )Y .
The Heisenberg group admits homogeneous dilations adapted both to the distance and the group structure. They are given by

dil λ (x, y, z) = (λx, λy, λ 2 z) for λ > 0. They satisfy d H (dil λ (a), dil λ (a ′ )) = λd H (a, a ′ ) and, in law: dil 1 √ t B 1 t , B 2 t , 1 2 t 0 B 1 s dB 2 s - t 0 B 2 s dB 1 s Law = B 1 1 , B 2 1 , 1 2 1 0 B 1 s dB 2 s - 1 0 B 2 s dB 1 s .
The distance is clearly left-invariant so that trans p : q ∈ H → p.q is an isometry for every p ∈ H. In particular

d H (a, a) = d H (e, a -1 a ′ )
with e = (0, 0, 0). Another isometry is the rotation rot θ : (x + iy, z) ∈ C × R ≡ H → (e iθ (x + iy), z), for every θ ∈ R. Since the explicit expression of d H is not so easy, it is often simpler to work with a homogenous quasinorm (still in the sense that the triangle inequality only holds up to a multiplicative constant). We will use

H : a = (x, y, z) ∈ H → x 2 + y 2 + |z| ∈ R,
and the attached homogeneous quasidistance

d H (a, a ′ ) = H(a -1 a ′ ). It satisfies c -1 d H (a, a ′ ) ≤ d H (a, a ′ ) ≤ cd H (a, a ′ ) (8)
for some constant c > 1. We finally mention d H ((0, 0, 0), (x, y, 0)

) = √ x 2 + y 2 and d H ((x, y, z), (x, y, z + h)) = 2 π|h|.

Co-adapted couplings.

We first recall the notion of co-adapted coupling of two processes. Indeed, in this study, we only want to consider couplings built solely knowing the past of the two processes. The definition below is taken from [16, Definition 1.1.].

Definition 2.1. Given two continuous-time Markov processes (X

(1) t ) t≥0 , (X (2) t ) t≥0 , we say that ( X(1) t , X(2) t ) t≥0 is a co-adapted coupling of (X (1)
t ) t≥0 and (X

t ) t≥0 if X(1) and X(2) are defined on the same filtered probability space (Ω, (F t ) t≥0 , P), satisfy Law(X

(i) t ) t≥0 = Law( X(i) t ) t≥0 for i = 1, 2, and
P (i) t f : z → E f ( X(i) t+s )| F s , X(i) s = z equals P (i) t f : z → E f (X (i) t+s )| X (i) s = z , Law(X (i) s )-almost surely for i = 1, 2, for each bounded measurable function f , each z, each s, t ≥ 0.
If we moreover assume that the full process ( X(1) t , X(2) t ) t≥0 is Markovian, we say that the co-adapted coupling is Markovian.

The next lemma describes more explicitly co-adapted couplings in the case of Brownian motion in R 2 (see [START_REF] Kendall | Brownian couplings, convexity, and shy-ness[END_REF]Lemma 6]). Lemma 2.2. Let (B t ) t and (B ′ t ) t be two co-adapted Brownian motions on R 2 × R 2 defined on some filtered probability space (Ω, (F t ) t≥0 , P). Then, enriching the filtration if necessary, there exists a Brownian motion ( Bt ) t≥0 defined on the same filtration (F t ) t≥0 and independent of (B t ) t≥0 such that

dB ′ (t) = J(t)dB t + Ĵ(t)d Bt (9)
where (J t ) t≥0 = ((J i,j t ) 1≤i,j≤2 ) t≥0 and Ĵ are matrices satisfying JJ T + Ĵ ĴT = I 2 [START_REF] Hardy | Inequalities. Cambridge Mathematical Library[END_REF] and J(t), Ĵ(t) ∈ F t .

In the following • may denote the operator norm of a matrix attached to the Euclidean norm, or the Euclidean norm of a vector.

Lemma 2.3. Let J be a 2 × 2 real matrix J = ( a b c d ). Then 0 ≤ J T J ≤ I 2 ⇐⇒ J ≤ 1 ⇐⇒ 0 ≤ JJ T ≤ I 2 ,
where ≤ is the ordering of symmetric matrices. In particular

• a 2 + b 2 , a 2 + c 2 , c 2 + d 2 and b 2 + d 2 are smaller or equal to 1, • all the four entries of J are in [-1, 1]. Proof. Let S 1 = {(cos(θ), sin(θ)) ∈ R 2 : θ ∈ R} be the Euclidean sphere of R 2 and Q : x ∈ S 1 → (x, J T Jx) = Jx 2 . Therefore, Q is bounded by 1 if and only if Jx ≤ 1, for all x ∈ S 1 . The bound 0 ≤ J T J is trivially satisfied. The proof is completed by J = sup x,y∈S 1 (Jx, y) = J T .
Remark 2.4. A necessary and sufficient condition can be found considering λ, the greatest eigenvalue of J T J. It writes

2λ = (a 2 + b 2 + c 2 + d 2 ) + (a 2 + b 2 + c 2 + d 2 ) 2 -4(bc -ad) 2 ≤ 2 × 1.
Paradoxically, it not easy to deduce |a|, |b|, |c|, |d| ≤ 1 from this condition.

dA t = 1 2 B 1 t dB 2 t -B 2 t dB 1 t and dA ′ t = 1 2 B ′1 t dB ′2 t -B ′2 t dB ′1 t ).
For the following, we denote by J and Ĵ the matrices appearing in Lemma 2.2. A computation gives:

B ′-1 t B t = B 1 t -B ′1 t , B 2 t -B ′2 t , A t -A ′ t + 1 2 B 1 t B ′2 t -B 2 t B ′1
t and thus:

d(B ′-1 t B t ) =     dB 1 t -dB ′1 t dB 2 t -dB ′2 t (B 1 t -B ′1 t ) dB ′2 t +dB 2 t 2 -(B 2 t -B ′2 t ) dB ′1 t +dB 1 t 2 + 1 2 (d B 1 t , B ′2 t -d B 2 t , B ′1 t )     ,
where we used:

d(X t Y t ) = X t dY t + Y t dX t + d X t , Y t .
We denote by R t the horizontal distance between the two Brownian motions B t and 2 and by Z t the third coordinate, the relative Lévy area. Hence

B ′ t in R 2 , that is R 2 t = (B 1 t -B ′1 t ) 2 +(B 2 t -B ′2 t )
Z t = (B ′-1 t B t ) 3 . The homogeneous distance d H (B t , B ′ t ) is thus given by R 2 t + |Z t |.
In the following, when R t > 0, we choose to work in the direct orthonormal (random moving) frame (v 1 , v 2 ) defined by taking v 1 (t) the normalised vector of R 2 directed by B t -B ′ t . Let Q t be the matrix whose columns are respectively v 1 (t) and v 2 (t). In this new basis, for (α, β) ∈ R 2 and (• | •) the usual scalar product on R 2 , we have:

(αv 1 + βv 2 | dB ′ t ) = Q t α β | J t dB t + Ĵt d Bt = α β | (Q T t J t Q t )Q T t dB t + (Q T t Ĵt Q t )Q T t d Bt = α β | K t dW t + Kt d Ŵt for K t = Q T t J t Q t and Kt = Q T t Ĵt Q t ,
and where W and Ŵ are the two standard independent 2-dimensional Brownian motions defined by

dW t = Q T t dB t , d Ŵt = Q T t d Bt .
This can be summed up as follows:

Q T t dB ′ t = (Q T t J t Q t ) Kt Q T t dB t + (Q T t Ĵt Q t ) Kt Q T t d Bt .
The next easy lemma describes the relation between the matrices J and K.

Lemma 2.5. With the above notation, when R t > 0,

• Equation (10) is satisfied for (K, K) if and only if it is satisfied for (J, Ĵ ).

• tr K = tr J.

• K 1,2 -K 2,1 = J 1,2 -J 2,1 .
Proof. The first two relations follow from the fact that Q is an orthogonal matrix.

For the last relation, one can note that J 1,2 -J 2,1 = tr(JM) = tr(KQ T MQ) with M the matrix ( 0 -1 1 0 ). Now a computation gives Q T MQ = (det Q)M and the last relation follows from the fact Q is actually a rotation matrix.

The stochastic processes R 2 t and Z t are semimartingales defined for all time t ≥ 0. In the next statement, we provide stochastic differential equations for their evolution.

Lemma 2.6. With the above notation, when R t = 0, the processes R 2 t and Z t solve the stochastic differential equation:

       d(R 2 t ) = 2R t 2(1 -K 1,1 ) dC t + 2(1 -K 1,1 ) + 2(1 -K 2,2 ) dt dZ t = R t 2 2(1 + K 2,2 ) d Ct - 1 2 (K 1,2 -K 2,1 )dt
where (C t ) t≥0 and ( Ct ) t≥0 are some 1-dimensional Brownian motions whose covariation satisfies:

2(1 -K 1,1 )dC t , 2(1 + K 2,2 )d Ct = -(K 1,2 -K 2,1 )dt. ( 11
)
Remark 2.7. Actually the stochastic process (R 2 t , Z t ) t≥0 is perfectly defined for all t ≥ 0 (even when R t = 0). The technical problem in Lemma 2.6 is that the matrix Q t and thus the matrix K t are only defined for R t = 0. However, the matrix J t is defined for every value of R t and we have:

     d(R 2 t ) = σ R (B t , B ′ t , J t ) dC t + 2(1 -J 1,1 ) + 2(1 -J 2,2 ) dt dZ t = σ Z (B t , B ′ t , J t ) d Ct + 1 2 (J 1,2 -J 2,1 )dt
where σ R and σ Z are defined by:

σ R (B t , B ′ t , J t ) = 0 if B t = B ′ t 2R t 2(1 -(Q T t J t Q t ) 1,1 ) if B t = B ′ t and σ Z (B t , B ′ t , J t ) = 0 if B t = B ′ t Rt 2 2(1 + (Q T t J t Q t ) 2,2 ) if B t = B ′ t .
Note finally that the fact that σ R and σ Z vanish for R t = 0 is rather clear from their expressions in Lemma 2.6.

Proof of Lemma 2.6. The computations are done in [START_REF] Kendall | Coupling all the Lévy stochastic areas of multidimensional Brownian motion[END_REF] but we repeat them for the sake of completeness.

First by Itô formula and with the previous notation:

dR 2 t = d (B 1 t -B ′1 t ) 2 + (B 2 t -B ′2 t ) 2 = 2R t (v 1 | (dB t -dB ′ t )) + d (B 1 t -B ′1 t ), (B 1 t -B ′1 t ) + d (B 2 t -B ′2 t ), (B 2 t -B ′2 t )
. We turn to the martingale part and write

(v 1 | (dB t -dB ′ t )) = (1 -K 1,1 )dW 1 t -K 1,2 dW 2 t -K1,1 d Ŵ 1 t -K1,2 d Ŵ 2 t = 2(1 -K 1,1 )dC t
for some 1-dimensional Brownian motion (C t ) t where we used Lemma 2.5 for

(K 1,1 ) 2 + (K 1,2 ) 2 + ( K1,1 ) 2 + ( K1,2 ) 2 = 1.
The quadratic variation writes

d (B 1 t -B ′1 t ), (B 1 t -B ′1 t ) = (J 1,1 -1) 2 + (J 1,2 ) 2 + ( Ĵ1,1 ) 2 + ( Ĵ1,2 ) 2 = 2 -2J 1,1 and similarly d (B 2 t -B ′2 t ), (B 2 t -B ′2 t ) = (J 2,1 ) 2 + (J 2,2 -1) 2 + ( Ĵ2,1 ) 2 + ( Ĵ2,2 ) 2 = 2 -2J 2,2 , thus d (B 1 t -B ′1 t ), (B 1 t -B ′1 t ) + d (B 2 t -B ′2 t ), (B 2 t -B ′2 t ) = 2tr(I -J) = 2tr(I -K).
We turn now to Z t . Using the basis (v 1 , v 2 ), we can rewrite

dZ t = R t 2 (v 2 | (dB ′ t + dB t )) + 1 2 (d B 1 t , B ′2 t -d B 2 t , B ′1 t ).
As before, we get:

(v 2 | (dB ′ t + dB t )) = K 2,1 dW 1 t + (K 2,2 + 1)dW 2 t + K2,1 d Ŵ 1 t + K2,2 d Ŵ 2 t = 2(1 + K 2,2 )d Ct
for some 1-dimensional Brownian motion ( Ct ) t . Moreover:

d B 1 t , B ′2 t -d B 2 t , B ′1 t = (J 2,1 -J 1,2 )dt = (K 2,1 -K 1,2 )dt.
The equation on the covariation [START_REF] Hsu | Maximal coupling of Euclidean Brownian motions[END_REF] follows since by [START_REF] Hardy | Inequalities. Cambridge Mathematical Library[END_REF] and Lemma 2.5,

K 1,1 K 2,1 + K 1,2 K 2,2 + K1,1 K2,1 + K1,2 K2,2 = 0.
Remark 2.8. Since we will use them in the following we also write stochastic differential equations satisfied by R t , R 4 t and Z 2 t , obtained for R t = 0 using Itô's formula in Lemma 2.6:

                   dR t = 2(1 -K 1,1 ) dC t + 1 -K 2,2 R t dt, dR 4 t = 4R 3 t 2(1 -K 1,1 ) dC t + (4R 2 t (1 -K 2,2 ))dt + 12R 2 t (1 -K 1,1 )dt, d(Z 2 t ) = Z t R t 2(1 + K 2,2 ) d Ct + -Z t (K 1,2 -K 2,1 ) + R 2 t 2 (1 + K 2,2 ) dt.
As in Remark 2.7, an expression for dZ 2 t is possible in the canonical basis with the matrix J in place of K. According to Lemma 2.5, K 1,2 -K 2,1 is replaced by J 1,2 -J 2,1 and the factor R t make the undefined terms vanish when R t = 0. The same holds for the semimartingale (R 4 t ) t≥0 . On the contrary R t is not a semimartingale, but only locally when R t > 0.

2.4. Description of some couplings. In this section, we describe some interesting couplings.

The synchronous coupling (K

1,1 = 1, K 2,2 = 1, K 1,2 = K 2,1 = 0.)
The coupling is called synchronous because the planar trajectories (B t ) t≥0 and (B ′ t ) t≥0 are parallel. Here R t ≡ R 0 and Z t = Z 0 + W R 0 t with W a Brownian motion.

The reflection coupling (K

1,1 = -1 K 2,2 = 1, K 1,2 = K 2,1 = 0.
) For the reflection coupling the planar trajectories of (B t ) t≥0 and (B ′ t ) t≥0 evolve symmetrically with respect to the bisector of line segment [B 0 , B ′ 0 ]. We stop the coupling when R t hits 0 and continue synchronously with J 1,1 = J 2,2 = 1. Denote by τ this hitting time. One thus has

R t = R 0 + 2C t∧τ and Z t = Z 0 + t∧τ 0 (R 0 + 2C s∧τ )d Cs
where (C s ) s and ( Cs ) s are two independent Brownian motions (starting in 0) and with τ = inf{s ≥ 0, 2C s = -R 0 }. This coupling is studied in Section 4. On Euclidean and Riemannian manifolds, the efficiency of reflection coupling has been studied in [START_REF] Hsu | Maximal coupling of Euclidean Brownian motions[END_REF][START_REF] Kuwada | On uniqueness of maximal coupling for diffusion processes with a reflection[END_REF].

Kendall's coupling: (K

1,1 = ±1, K 1,2 = K 2,1 = 0 and K 2,2 = 1)
. In [START_REF] Kendall | Coupling all the Lévy stochastic areas of multidimensional Brownian motion[END_REF], Kendall describes a coupling which alternates between synchronous coupling and reflection coupling. In order to avoid the use of local times the strategy of Kendall is defined with hysteresis. The regime swaps when the process (R t , |Z t |) hits a certain parabola [START_REF] Kendall | Coupling all the Lévy stochastic areas of multidimensional Brownian motion[END_REF]Theorem 4]), depending for the synchronous or the reflection coupling. Thus the process is not Markovian, but it is co-adapted. The author proves that this coupling is successful: this means T := inf{s ≥ 0, B s = B ′ s } is almost surely finite, or, equivalently, the process (R t , Z t ) hits almost surely (0, 0) in finite time.

{8Z 2 t = κ 2 R 4 t } or {8Z 2 t = (κ -ε) 2 R 4 t } (see

The perverse coupling: K

1,1 = 1, K 2,2 = -1, K 1,2 = K 2,1 = 0. We assume R 0 > 0. It satisfies, dR(t) = 2 R t dt and dZ t = 0.
Thus the distance R t and Z t are deterministic and given by: R t = R 2 0 + 4t and Z t = Z 0 . The name perverse coupling is given by Kendall in [START_REF] Kendall | Brownian couplings, convexity, and shy-ness[END_REF] as a generic name for a repulsive coupling. Here, the planar components of (B t ) t and (B ′ t ) t are coupled in a perverse way. This particular method to produce a perverse coupling appears in [23, Section 5] in a Riemannian setting.

Non-existence of co-adapted Heisenberg Brownian motions at bounded distance.

In this section, we provide the proofs that co-adapted Brownian motions do not stay at bounded distance on the Heisenberg group. Of course Theorems 1.2 and 1.3 are immediate corollaries of Theorems 1.5 and 1.6. However since Theorem 1.2 already shows a clear difference with the Riemannian case and is easier to prove, we shall present a direct proof in Section 3.1. The proof of Theorem 1.5 is given in Section 3.2. This section also contains the core of the proof of Theorem 1.6. Its proof is finished in Section 3.3.

Proof of Theorem 1.2.

Proof of Theorem 1.2. Assume that (B t ) t and (B ′ t ) t are two co-adapted Heisenberg Brownian motions starting in a = (x, y, z) and a ′ = (x ′ , y ′ , z ′ ) with R 0 = (x ′x) 2 + (y ′y)

E[R 2 t ] = R 2 0 + E 2 T 0 (1 -J 1,1 (s)) + (1 -J 2,2 (s))ds and R t ≤ C gives E[R 2 t ] ≤ C 2 and E 2 t 0 (1 -J 1,1 (s)) + (1 -J 2,2 (s))ds ≤ C 2 . ( 12
)
Recall from Lemma 2.5 that K 1,1 + K 2,2 = J 1,1 + J 2,2 and from Lemma 2.3 that the matrix entries are ≤ 1. Therefore

max i∈{1,2} E t 0 (1 -K i,i (s)) 1 {Rs>0} ds ≤ C 2 /2 (13)
and R t ≤ C, again, gives

E t 0 (1 -K 2,2 (s)) R 2 s 2 ds ≤ C 4 /4. ( 14
)
Until now we have used E(R 2 t ) ≤ C 2 and R t ≤ C. We turn to exploit E(Z 2 t ) ≤ C 4 . Lemma 2.5 and Remark 2.8 give

E[Z 2 t ] = Z 2 0 + E -2 t 0 Z s (J 1,2 (s) -J 2,1 (s))ds + t 0 R 2 s 2 (1 + K 2,2 (s))ds (15)
Adding ( 14) and ( 15), and using E(Z 2 t ) ≤ C 4 , we obtain

E -2 t 0 Z t (J 1,2 (s) -J 2,1 (s))ds + t 0 R 2 s ds ≤ (1 + 1/4)C 4 . ( 16
)
Next, we aim to compare E 2 t 0 Z s (J 1,2 (s) -J 2,1 (s))ds with E t 0 R 2 s ds that both appear in [START_REF] Kendall | Coupling time distribution asymptotics for some couplings of the Lévy stochastic area[END_REF]. On the one hand, since Z t stays bounded E[Z 2 t ] is also bounded. Hence by Cauchy-Schwarz inequality (for the product measure on Ω × [0, t]):

E t 0 Z s (J 1,2 (s) -J 2,1 (s))ds ≤ t 0 E[Z 2 t ]ds 1/2 t 0 E(J 1,2 -J 2,1 ) 2 ds 1/2 ≤ t 0 E[Z 2 t ]ds 1/2 2 t 0 E((J 1,2 ) 2 + (J 2,1 ) 2 )ds 1/2 ≤ √ C 4 t 2 t 0 E(J 1,2 ) 2 + E(J 2,1 ) 2 ds 1/2 ≤ √ C 4 t • √ 2C 2 = C 3 √ 2t. ( 17
)
The last estimate follows from Lemma 2.3 (the rows and columns of J have L 2 -norm smaller than 1), 1 -

J 2 i,i ≤ (1 -J i,i )(1 + J i,i
) and ( 12):

t 0 E(J 1,2 ) 2 + E(J 2,1 ) 2 ds ≤ t 0 E[(1 -(J 1,1 ) 2 ) + (1 -(J 2,2 ) 2 )]ds ≤ t 0 2E[(1 -J 1,1 ) + (1 -J 2,2 )]ds ≤ C 2 .
On the other hand, since (R

2 t ) t≥0 is a submartingale, E[R 2 s ] ≥ R 2 0 and t 0 E[R 2 s ]ds ≥ R 2 0 t. ( 18
)
Since R 0 > 0, ( 17) and ( 18) provide a contradiction in [START_REF] Kendall | Coupling time distribution asymptotics for some couplings of the Lévy stochastic area[END_REF]

E[R 4 t ] = R 4 0 + E t 0 12R 2 s (1 -K 1,1 (s)) + 4R 2 s (1 -K 2,2 (s))ds .
This quantity is uniformly bounded by C 4 for every t ≥ 0 so that [START_REF] Kendall | Coupling all the Lévy stochastic areas of multidimensional Brownian motion[END_REF] holds. (the bound in [START_REF] Kendall | Coupling all the Lévy stochastic areas of multidimensional Brownian motion[END_REF] can even be divided by two: C 4 /8 in place of C 4 /4).

Proof of Theorem 1.5.

To go beyond Theorem 1.2, we conduct a precise study of the expected total variation (or length in L 1 ) of the martingale part and of the drift part of (Z t ) t≥0 , the relative Lévy area. The core of the proofs of Theorems 1.6 and 1.5 is the same. As before, the proofs will be by contradiction and the principle is the following. We derive an upper bound for the drift part of Z t similar to [START_REF] Kuwada | On uniqueness of maximal coupling for diffusion processes with a reflection[END_REF]

′ = (x ′ , y ′ , z ′ ) such that (x ′ -x) 2 + (y ′ -y) 2 > 0. Assume that C := sup 0≤t≤T f max(E(R 2 t ), E(|Z t |)) < +∞. ( 19
)
Let us decompose the semimartingale (Z t ) t = M t -A t into its martingale M t and its bounded variation part -A t . Denote by V t s (A) the total variation of A on [s, t]. Then, there exists h > 0 depending only on C (independent of T f ) such that for all 0 ≤ t ≤ T fh,

E(|M t+h -M t |) ≥ 10C 2 (20)
and for all 0

≤ t ≤ T f EV t 0 (A) ≤ C 2 t 2 . ( 21
)
Before we turn to the proof of Lemma 3.2, we present two lemmas. The first lemma is crucial to obtain the lower bound for absolute value of the martingale part of Z t . The proof of Lemma 3.3 is postponed at the end of the section. The second lemma relies on the probabilistic variant of the rearrangement inequality [10, §10.12]. It is directly obtained from the Hoeffding-Fréchet bound on bivariate cumulative distribution functions through the Hoeffding formula for the covariance (see [25, §2.2] and [START_REF] Mcneil | Quantitative risk management[END_REF]Lemma 5.24]). Lemma 3.3. Let (N t ) t≥0 be a continuous martingale with N 0 = 0 and p be in ]0, 1[. Then there exists a p > 0 such that for every positive real numbers β and h, the estimate

P ( N h ≥ β) ≥ p implies E[|N h |] ≥ a p β.
Lemma 3.4. Let X be a non-negative random variable admitting a density. Then for every measurable set A, one has:

E [X1 A ] ≥ E [X1 A * ]
where A * is defined by

A * = {w, X(ω) ≤ r} where r ∈ [0, ∞] is chosen such that P(A * ) = P(A).
We now turn to the proof of Lemma 3.2.

Proof of Lemma 3.2. First recall

E R 2 t = R 2 0 + E t 0 2[(1 -J 1,1 ) + (1 -J 2,2 )]ds ≥ 0,
which gives

E T f 0 [(1 -J 1,1 ) + (1 -J 2,2 )]ds ≤ C 2 2 < +∞. ( 22
)
We study now the bounded variation part of (Z t ) t = M t -A t . From Lemma 2.6, we recall:

A t = 1 2 t 0 (J 1,2 -J 2,1 )ds. ( 23
)
Applying Cauchy-Schwarz inequality and following the same track as for ( 17) we obtain

E t 0 |J 1,2 -J 2,1 |ds ≤ √ t E t 0 (J 1,2 -J 2,1 ) 2 ds ≤ √ t √ 2C 2 . ( 24
)
Remark now that the quantity on the left hand side is two times the expected total variation of A t on [0, t].

We turn now to the study of the martingale part of (Z t ) t . First we show that under condition [START_REF] Kuwada | A counterexample for the optimality of Kendall-Cranston coupling[END_REF], with positive probability, some trajectories of R t stay far from 0 (see [START_REF] Villani | Topics in optimal transportation[END_REF] below for the precise statement). We set

T := inf{0 ≤ t ≤ T f , R t = R 0 2 } the hitting time of R 0
2 (before T f ). We recall the convention inf ∅ = +∞ and we introduce the random time

S = T 0 2(1 -K 1,1 )ds on {T < +∞} +∞ on {T = +∞}.
Because of the non-negative drift in the stochastic differential equation of R t , using the Dambins-Dubins-Schwarz theorem (see e.g. [START_REF] Revuz | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]), the random variable S is greater in stochastic order than the hitting time τ BM of R 0 2 for a Brownian motion starting in R 0 . This hitting time is almost surely finite but non integrable. Now, if we set p := P (T = +∞), Lemma 3.4 gives

E T 0 2(1 -K 1,1 )ds 1 {T <+∞} ≥ inf A,P(A)=1-p E τ BM 1 Ap = α 1-p 0 u f τ BM (u)du
with f τ BM the density of τ BM and α 1-p its quantile of order 1p. But conversely, since by [START_REF] Montgomery | A Tour of Subriemannian Geometries, Their Geodesics and Applications[END_REF] (and Lemma 2.5), the left hand side is bounded from above; thus with C the constant appearing in [START_REF] Kuwada | A counterexample for the optimality of Kendall-Cranston coupling[END_REF], one gets

α 1-p 0 u f τ BM (u)du ≤ C 2 2 .
Of course, since α 1-p → +∞ as p → 0; under the condition [START_REF] Kuwada | A counterexample for the optimality of Kendall-Cranston coupling[END_REF], P(T = +∞) has to be uniformly bounded from 0. As a consequence, there exists q > 0 (q depending only on C) such that for all T f > 0 and all co-adapted coupling on [0, T f ] satisfying ( 19)

P R t ≥ R 0 2 for all 0 ≤ t ≤ T f ≥ q > 0. (25)
To obtain the desired result [START_REF] Li | Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg[END_REF] on the martingale part of (Z t ) t , we study its bracket. We will prove that for h ≥ 5C 2 q ,

P M t+h -M t ≥ R 2 0 h 8 ≥ 9q 10 . ( 26
)
We hence obtain [START_REF] Li | Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg[END_REF] taking h large enough in [START_REF] Von Renesse | Intrinsic coupling on Riemannian manifolds and polyhedra[END_REF] and applying Lemma 3.3 to (M t+h -M t ) h≥0 . Proof of (26): considering only the event {T = +∞} for the martingale part of Z t described in Lemma 2.6, one has

M t+h -M t ≥ 1 {T =+∞} R 0 2 2 t+h t 1 + K 2,2 2 ds. ( 27
)
Now, since by [START_REF] Montgomery | A Tour of Subriemannian Geometries, Their Geodesics and Applications[END_REF] it holds

E t+h t 1 {T =+∞} 1 -K 2,2 2 ds ≤ C 2 /4, taking the complementary set of { t+h t 1+K 2,2 2
ds ≥ h 2 } in {T = +∞} and using Markov inequality, one obtains:

P 1 {T =+∞} t+h t 1 + K 2,2 2 ds ≥ h 2 =q -P 1 {T =+∞} t+h t 1 -K 2,2 2 ds > h 2 ≥ q - C 2 4 • 2 h . ( 28
)
Hence, in [START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy, and Ricci curvature[END_REF] we consider the probability that the right-hand side is greater than (R 0 /2) 2 • (h/2), which, with [START_REF] Wang | On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups[END_REF], gives the wanted estimate [START_REF] Von Renesse | Intrinsic coupling on Riemannian manifolds and polyhedra[END_REF] for every h ≥ 5C 2 q . We can now turn to the proof of Theorem 1.5.

Proof of Theorem 1.5. We work here with T f = +∞. Since, we have assumed by contradiction that E[Z t ] = E(|M t -A t |) ≤ C 2 for every t ≥ 0, the estimate 20 of Lemma 3.2 and the triangle inequality imply

E(|A t+h -A t |) ≥ 8C 2 for every t ≥ 0.
The control of the expected total variation of (A t ) expressed in [START_REF] Mcneil | Quantitative risk management[END_REF] and the lower estimate just proved give

8C 2 n ≤ n k=1 E(|A kh -A (k-1)h |) ≤ 1 2 E nh 0 |J 1,2 -J 2,1 |ds ≤ C 2 hn 2 , (29)
which, as n tends to ∞, provides a contradiction.

Finally, we present the proof of Lemma 3.3 on the lower bound for the absolute value of a martingale.

Proof of Lemma 3.3. Let φ be the quadratic variation of N φ(t) = N t , and consider the hitting time τ = inf{t ≥ 0 : φ(t) ≥ β}. Set ψ(t) = φ(t) ∧ β. The Dambins theorem shows that there exists a standard Brownian motion (W t ) t≥0 such that for every t ≥ 0,

E [|N t |] ≥ E [|N t∧τ |] = E |W ψ(t) | . ( 30
)
Let now A be the event {ω ∈ Ω : φ(h) ≥ β} and recall the assumption P(A) ≥ p. One has

E |W ψ(h) | ≥ E |W ψ(h) | • 1 A = E [|W β | • 1 A ] ≥ E |W β | √ β • 1 A β ≥ a p β (31)
where we use Lemma 3.4 for the last inequality and the constant a p is defined as

a p = E |G| 1 {|G|≤Φ -1 ( 1+p 2 )} = Φ -1 ( 1+p 2 ) Φ -1 ( 1-p 2 ) |x| e -x 2 /2 √ 2π dx;
with G a standard normal random variable and Φ its cumulative distribution function.

Remark 3.5. The major constraint for generalising Theorem 1.5 and its proof to a L p -Wasserstein control for p < 2 is that

E(|Z t |) is replaced by E(|Z t | p/2
). Here x → |x| p/2 is not convex when p < 2 and Jensen's inequality does not apply.

3.3. Proofs of Theorems 1.3 and1.6. In this section we finish the proof of Theorems 1.3 and1.6. As said before, 1.3 is a clear consequence of Theorem 1.6. Contrary to Theorem 1.2, we do not present a direct proof. Now, the key point for deducing Theorem 1.6 from Lemma 3.2 is the use of the homogeneous dilations on the Heisenberg group. For a fixed time T > 0 and p ∈ (0, ∞], we introduce C T,p to be the constant:

C T,p := sup a =a ′ ∈H inf A (a,a ′ ) T sup 0≤t≤T E[d p H (B a t , B a ′ t )] 1 p d H (a, a ′ ) ∈ [0, +∞];
where, as before in Remarks 1.4 and 1.7, A (a,a ′ ) T denotes the set of co-adpated couplings of (B a t ) 0≤t≤T and (B a ′ t ) 0≤t≤T , starting respectively in a and a ′ . When p = +∞, the numerator is essup (Ω,P) d H (B a t , B a ′ t ). As noticed in these remarks we aim at proving C T,p = +∞ for p ≥ 2. The next lemma is crucial. Proof. Let S, T > 0 be two different times. The point is that if (B a t , B a ′ t ) 0≤t≤T is a co-adapted coupling of two Heisenberg Brownian motions on [0, T ] starting respectively in a and a ′ ; setting for b ∈ {a, a ′ }: We can now complete the proof of Theorem 1.6.

Proof of Theorem 1.6. Suppose by contradiction that C T 0 ,2 < +∞ for some T 0 > 0. Lemma 3.6 implies that C T,2 is finite and constant for all T > 0. Let a = (0, 0, 0) and a ′ = (x ′ , 0, 0) with x ′ = 0 be fixed. Thus, there exists a constant C (one can take C = C 2 T 0 ,2 d H (a, a ′ ) 2 ) such that for each final time T f > 0, there is a coadapted coupling satisfying

E[d 2 H (B a t , B a ′ t )] ≤ C for t ∈ [0, T f ]. Lemma 3.
2 applies and similarly as in the proof of Theorem 1.5, the chain of inequalities (29) holds for 0 ≤ nh ≤ T f . Now, since the value of h does not depend on T f , letting T f and n tend to infinity gives the contradiction. Thus C T 0 ,2 = +∞.

Coupling by reflection

In this section, we study precisely the coupling by reflection. We recall that (B t ) t≥0 and (B ′ t ) t≥0 are two Heisenberg Brownian motions coupled by reflection if and only if their horizontal parts (B t ) t≥0 and (B ′ t ) t≥0 are two Brownian motions on R 2 coupled by reflection. This means that the coupling matrices are given by K 1,1 = -1, K 2,2 = 1, K 1,2 = K 2,1 = 0 for t < τ and by the matrix J = Id 2 for t ≥ τ where τ = inf{s ≥ 0 : R s = 0} is the hitting time of 0 for (R t ) t≥0 . We recall

R t = R 0 + 2C t∧τ and Z t = Z 0 + t∧τ 0 (R 0 + 2C s∧τ )d Cs
where (C s ) s and ( Cs ) s are two independent Brownian motions (starting in 0) and with τ = inf{s ≥ 0 :

C s = -R 0 /2}.
For simplicity, in the following we only consider the case R 0 > 0 and Z 0 = 0.

Proposition 4.1. With the above notation, assume R 0 > 0 and Z 0 = 0, Let p > 0, then there exists some constants

C p , C ′ p , C ′′ p > 0 such that E[R t ] = R 0 and      E[|Z t | p ] ∼ t→∞ C p R 0 t p-1 2 if p > 1 2 E[|Z t | p ] ∼ t→∞ C ′ p R 0 ln t if p = 1 2 E[|Z t | p ] → t→+∞ C ′′ p R 2p 0 if 0 < p <
E(R t ) α ] ≤ [E(R t )] α .
Proof. We assume R 0 = 1. Let t > 0 be fixed. By the Dambins-Dubins-Schwarz theorem, Z is a changed time Brownian motion:

Z t = W T (t) with T (t) = t 0 R 2
s ds with W a Brownian motion independent of (R t ) t≥0 . Set τ = inf{s ≥ 0 : R s = 0}. As (R s /2) s≥0 is a Brownian motion starting in R 0 /2 and stopped in 0, it is known that τ is almost surely finite and that its density f τ is given by

f τ (u) = R 0 /2 √ 2πu 3/2 e -R 2 0 4u , u ≥ 0. ( 32
)
Using τ , we compute

E [|Z t | p ] = E(|W T (t) | p ) (33) = +∞ 0 E(|W T (t) | p |τ = u)f τ (u)du = t 0 E(|W T (t) | p |τ = u)f τ (u)du h 1 (t) + +∞ t E(|W T (t) | p |τ = u)f τ (u)du h 2 (t)
.

In the last line, we split the integral between the trajectories of R that have hit 0 before t and those which will hit 0 after t. Let us estimate h 1 (t), the first integral in the decomposition (33). Hence we set u ≤ t. Since W and R are independent, with c p = E(|W 1 | p ), one has:

E(|W T (t) | p |τ = u) = c p E(|T (t)| p/2 |τ = u) (34) = c p E u 0 R 2 s ds p/2 |τ = u = c p 2 p u p E 1 0 R2 λ dλ p/2 |τ = u
where we have introduced the normalised process ( Rλ ) λ∈[0,1] (defined almost surely, since the hitting time τ is almost surely finite) in such a way it hits 0 at time 1:

Rλ = 1 2 √ τ R τ λ , λ ∈ [0, 1].
Note that R0 = R 0 2 √ τ . It is then well-known that, conditioned on τ = u, the entire process ( Rλ ) λ∈[0,1] converges in law when u → ∞ to a normal positive Brownian excursion (X s ) s∈[0,1] . Moreover, as proven in Lemma 4.3, when u → ∞,

E 1 0 R2 λ dλ p/2 |τ = u → E 1 0 X 2 s ds p/2 . ( 35
)
Finally with (34) denoting the limit in (35) by E p , the first integral in (33) satisfies the following equivalence:

h 1 (t) = t 0 E(|W T (t) | p |τ = u)f τ (u)du ∼ t→∞ 2 p c p E p t 0 u p f τ (u)du
From the density estimate of f τ in (32) we have u p f τ (u) ∼ +∞ R 0 2 √ 2π u p-3/2 . Therefore:

• If p > 1/2, the function h 1 (t) is equivalent to 2 p-3/2 R 0 cpEp (p-1/2) √ π u 3/2 t p-1/2 at +∞, • if p = 1/2, it is equivalent to R 0 cpEp 2 √ π u 3/2 ln t, • if 0 < p < 1/2,
it converges to a positive constant. We now turn to h 2 . As before,

h 2 (t) = +∞ t E(|W T (t) | p |τ = u)f τ (u)du = c p +∞ t E t 0 R 2 s ds p/2 |τ = u f τ (u)du = 2 p c p +∞ t u p E   t u 0 R2 λ dλ p/2 |τ = u   f τ (u)du = 2 p c p t p+1 +∞ 1 v p E   1 v 0 R2 λ dλ p/2 |τ = tv   f τ (tv)dv = 2 p-3/2 c p R 0 t p-1/2 √ π +∞ 1 v p-3/2 E   1 v 0 R2 λ dλ p/2 |τ = tv   e -
v p-3/2 E   1 v 0 R2 λ dλ p/2 |τ = tv   e -R 2 0 4tv dv → +∞ 1 v p-3/2 E   1 v 0 X 2 s ds p/2   dv.
As a consequence, denoting by I p the last integral,

h 2 (t) ∼ t→∞ 2 p-3/2 c p R 0 √ π t p-1/2 I p .
This with the treatment of h 1 above gives the complete result in case R 0

= 1. Next, if R 0 > 0 one infers E R 0 [|Z t | p ] = R 2p 0 E {R 0 =1} [|Z t/R 2 0 | p ]
from the classical dilations of Subsection 2.1, so that the general case follows.

The two next lemmas complete the proof of Proposition 4.1. Lemma 4.3. Let ( Rt ) t∈[0,1] be a Brownian motion starting in r 0 > 0 conditioned to hit 0 for the first time at time 1. Let α ∈ [0, 1] and p be positive. As r 0 → 0,

E α 0 R2 λ dλ p/2 → E α 0 X 2 s ds p/2 (36)
where (X s ) s∈[0,1] is a Brownian excursion.

Proof. The process ( Rt ) t∈[0,1] converges in law to the Brownian excursion (X s ) s∈[0,1] .

To obtain the convergence of the moments of α 0 R2 λ dλ, we use a uniform integrability property. Let 0≤ α ≤ 1. We bound

P α 0 R2 λ dλ ≥ y| τ = u ≤ P sup 0≤s≤1 W s ≥ √ y| T 0 = 1
where (W s ) s≥0 is a Brownian motion starting in r 0 and T 0 its hitting time of 0. Next, by [5, Formula 2.1.4 (1) p.198], still for W starting in r 0 , we have for every t > 0

P sup 0≤s≤T 0 W s < y| T 0 = t = +∞ k=-∞ (r 0 + 2ky) √ 2πt 3/2 exp - (r 0 + 2ky) 2 2t √ 2πt 3/2 r 0 exp r 2 0 2t ,
In particular, reorganising the terms,

P sup 0≤s≤1 W s < y| T 0 = 1 = 1 -2 +∞ k=1 4k 2 y 2 sinh(2kyr 0 ) 2kyr 0 -cosh(2kyr 0 ) exp -2k 2 y 2 ,
and, since for u ≥ 0, sinh u u ≤ e u , uniformly on 0 < r 0 ≤ 1 and y ≥ 1,

P sup 0≤s≤1 W s ≥ y| T 0 = 1 ≤ 8 +∞ k=1 k 2 y 2 exp(2ky) exp -2k 2 y 2 ≤ a exp(-by 2 )
for some a, b > 0.

Thus, for all 0 < r 0 ≤ 1, the random variables

α 0 R2 λ dλ r 0 ≤1
admit some uniformly bounded exponential moment. As a consequence, the corresponding α 0

R2

λ dλ p 2 are uniformly integrable and the desired convergence follows.

Lemma 4.4. With the above notation, there exists a coupling of ( Rt ) t∈[0,1] and of a 3-Bessel process (V t ) t∈[0,1] on the same probability space such that both start in r 0 ≥ 0 and such almost surely: Rt ≤ V t , for all 0 ≤ t ≤ 1.

In particular, for R 0 fixed and p > 0 there exists a constant D p > 0 such that for all v ≥ 1 and r 0 ≤ R 0 √ v it holds

v p E   1 v 0 R2 λ dλ p/2   ≤ D p
where the process Rλ starts in r 0 .

Proof. ) ≤ C ′ d H ((0, 0, 0), (x ′ , 0, 0)) = C ′ |x ′ | for some C ′ ≥ 1, we have finally

W 1 (µ 0 1 , µ (x ′ ,0,z ′ ) 1
) ≤ Cd H ((0, 0, 0), (x ′ , 0, 0)) + d H ((0, 0, 0), (0, 0, z ′ ))

= C ′ |x ′ | + 2 π|z ′ | ≤ max(C ′ , √ 2π) (x ′ ) 2 + |z ′ |
≤ Cd H ((0, 0, 0), (x ′ , 0, z ′ )), with C = c • max(C ′ , √ 2π) with c defined as in [START_REF] Driver | Hypoelliptic heat kernel inequalities on the Heisenberg group[END_REF]. Finally the proof amounts to the case t = 1, a = (0, 0, 0) and a ′ = (x ′ , 0, 0), as we announced.

Main body of the proof: We set We want to estimate W 1 (µ, ν) from above and start with

µ = µ 0 1 = Law(X, Y, Z) (37
W 1 (µ, ν) ≤ W 1 (µ, μ) + W 1 (μ, ν)
The coupling in (38) and (39) yields W 1 (μ, ν) ≤ |x ′ | = d H ((0, 0, 0), (x ′ , 0, 0)). The coupling suggested in (37) and (39) yields where the two last inequalities are in fact equalities. The order |x ′ | is such that |x ′ |/d H ((x ′ , 0, 0), (0, 0, 0)) = |x ′ | -1/2 → ∞ as x ′ goes to zero.

W 1 (µ, μ) ≤ E [E [d H ((X, Y, Z), (X, Y, Z + x ′ Y ))|(X, Y )]] (40)
Remark 5.1. Note that the coupling in (40) is exactly the synchronous coupling of subsection 2.4.1 evaluated at time t = 1. The estimate from above can easily be checked with Lemma 2.6.

We modify the computation above just a little based on the knowledge that the translation is not the optimal transport plan on the real line when considering costs that are increasing concave functions of the distance.

The following lemma will be in order: Proof. The left-hand side is the 1-Wasserstein distance W R 1 (η, (trans R t ) # η) on R for the distance (x, y) → |y -x|. This is also the Kantorovich norm η -(trans R t ) # η) 1 , in the sense of [START_REF] Kantorovič | On a space of completely additive functions[END_REF]. Recall that the Kantorovich norm of a signed Radon measure σ = σ +σ -of mass zero is

σ 1 = W R 1 (σ + , σ -) = inf π∈Π(σ + ,σ -)
|y -x|dπ(x, y)

where the mass of σ + can be different from 1. Let σ = η -(trans R t ) # η) and for the computation of σ 1 assume without loss of generality that t ≥ 0. We have σ = t 0 (trans R u ) # γ du where γ is the Radon measure of density f ′ . Therefore

σ 1 = t 0 (trans R u ) # γ du ≤ t 0 (trans R u ) # γ 1 du = t 0 γ 1 du = t γ 1 ≤ t × |f ′ (x)| |x| dx .
Note that for the last inequality, we use the triangle inequality by transporting γ + , of density f ′ + , to the atomic measure ( f ′ + )δ 0 and then from this measure to γ -, of density f ′ -.

We can now conclude by coupling Z and Z ′ , conditionally on (X, Y ). From (40): Here f X,Y,Z is the density of µ = Law(X, Y, Z) with respect to the Lebesgue measure and f X,Y is the density of (X, Y ). Note that we have f Z|x,y (z) = f X,Y,Z (x, y, z)/f X,Y (x, y) so that the last inequality is an equality.

Since f X,Y,Z is the density of the heat kernel. It is well known that there exists a constant C > 0 such that for all (x, y, z) ∈ H,

∂ z f X,Y,Z (x, y, z) f X,Y,Z (x, y, z) ≤ C. ( 41)

Lemma 3 . 6 .

 36 Let p ∈ (0, ∞]. The value of the constant C T,p is independent of T > 0.

,

  for 0 ≤ s ≤ S; then, ( Bã s , B ã′ s ) 0≤s≤S is a co-adapted coupling of two Heisenberg Brownian motions on [0, S] starting respectively in ã = dil √ S T (a) and ã′ = dil √ S T (a ′ ), Moreover, d H ( Bã s , a, a ′ ). This easily gives C S,p ≤ C T,p and by symmetry of S and T : C S,p = C T,p .

  above, s = uλ and Rλ = R τ λ / √ τ and where we set the change of variable u = tv in the next to last line. Now, Lemma 4.3 and the dominated convergence, which is completely justified by Lemma 4.4, give as t → +∞, +∞ 1

1 =

 1 )ν = µ (x ′ ,0,0) Law [(x ′ , 0, 0).(X, Y, Z)] = Law(X + x ′ , Y, Z + (1/2)x ′ Y ) (38) μ = Law [(x ′ , 0, 0).(X, Y, Z).(-x ′ , 0, 0)] = Law(X, Y, Z + x ′ Y ) (39)

≤

  E [E [d H ((0, 0, 0), (0, 0, x ′ Y ))|(X, Y )]] ≤ E 2 π|Y x ′ | = |x ′ | × [2 √ π E( |Y |)]

Lemma 5 . 2 .

 52 If η is a probability measure on R with rapidly decreasing and smooth density f , theninf π∈Π(η,(trans R t ) # η) |y -x| dπ ≤ |t| × |f ′ (x)| |x| dx

E

  [d H ((X, Y, Z), (X, Y, Z + x ′ Y ))|(X, Y )] ≤ 2 √ π|x ′ Y | × 2 |f ′ Z|X,Y (z)| |z| dzwhere f Z|x,y is the density of Law(Z| X = x, Y = y). Finally,E [d H ((X, Y, Z), (X, Y, Z + x ′ Y ))] ≤ |x ′ | f X,Y (x, y)2 √ π|y| × |f ′ Z|x,y (z)| |z| dz dx dy ≤ |x ′ | 2 √ π|y| |z| |∂ z f X,Y,Z (x, y, z)| f X,Y,Z (x, y, z)f X,Y,Z (x, y, z) dx dy dz.

  We now describe co-adapted Heisenberg Brownian motions. As seen before, a Brownian motion B is entirely determined by its two first coordinates (B t ) ′2 t ) t are two co-adapted Brownian motions on R 2 and if moreover their third coordinates satisfy

2.3. Co-adapted couplings on H. t = (B 1 t , B 2 t ) t ; the third one being (A t ) t the Lévy area swept by this 2-dimensional process (B t ) t . Thus two Heisenberg Brownian motions (B t ) t = (B 1 t , B 2 t , A t ) t and (B ′ t ) t = (B ′1 t , B ′2 t , A ′ t ) t on H are co-adapted if and only if B = (B 1 t , B 2 t ) t and B ′ = (B ′1 t , B

  as t goes to infinity.

	E[R 4 t ] ≤ C 4 for every t ≥ 0 is a sufficient assumption for (14) and, hence, for the proof.
	Indeed, by Lemma 2.8 one has
	Remark 3.1. The proof of Theorem 1.2 can be improved to show that any co-
	adapted Heisenberg Brownian motions can not stay bounded in L 4 . In this proof,
	the only place where we fully use the fact that R t is uniformly bounded almost surely is (14). At the other places we merely need that E[R 2 t ] and E[Z 2 t ] are bounded. But

  from the proof of Theorem 1.2 and we provide a lower bound for the martingale part of Z t . These key bounds are presented in Lemma 3.2 below. Theorem 1.5 is then deduced easily. To reach Theorem 1.6, another argument is employed: the use of homogeneous dilations on the Heisenberg group.

Lemma 3.2. Let T f ∈ (0, ∞] be a final time. Let (B t ) 0≤t≤T f and (B ′ t ) 0≤t≤T f be any two co-adapted Heisenberg Brownian motion starting respectively in a = (x, y, z) and a

  H (B t , B ′ t ) α ] < +∞ is satisfied by the coupling by reflection. This is obtained by recalling that d H (B t , B ′ t ) is equivalent to the homogeneous distance R 2 t + |Z t | using Proposition 4.1 for p = α/2, (a + b) p ≤ a p + b p , and Jensen's inequality [

		1 2 .
	Remark 4.2. In particular for 0 < α < 1, the upper bound
	sup	E [d
	t≥0

  The process Rt can be thought as a Bessel bridge (see e.g.[24, Chapter XIII]) and thus satisfies

	If we can prove W 1 (µ 0 1 , µ	(x ′ ,0,0) 1
		d Rt = dW t +	1 Rt	-	Rt 1 -t	dt

Acknowledgment. We thank the anonymous referees of this paper for their remarks and comments which improve the presentation of the paper.

for some Brownian motion (W t ) t∈ [0,[START_REF] Bakry | On gradient bounds for the heat kernel on the Heisenberg group[END_REF] . The coupling is obtained by considering the same Brownian motion in the stochastic differential equation defining (V t ) t∈ [0,[START_REF] Bakry | On gradient bounds for the heat kernel on the Heisenberg group[END_REF] :

The other conclusion follows since the 3-Bessel process shares the same scaling property as the Brownian motion: (V λt / √ λ) t has the same law as the 3-Bessel process starting in r 0 / √ λ.

The static coupling, a transport problem

In this section, we turn to the proof of Theorem 1.10. Recall that it gives a direct proof of the following L 1 -Wasserstein control: There exists C > 0 such that for every t ≥ 0, and every a, a ′ ∈ H,

where

Proof of Theorem 1.10 . Reduction of the problem: Let us first see how, using the symmetries of H presented in Subsection 2.1, the proof can be reduced to t = 1, a = (0, 0, 0) and a ′ = (x ′ , 0, 0). First, the isometries of H induced isometries for W 1 . In particular (trans p ) # and (rot θ ) # are isometries for W 1 , for every p ∈ H and θ ∈ R, that stabilise the family {µ a t } a∈H .

Hence, we can assume a = (0, 0, 0). Using rot θ we can moreover assume a ′ = (x ′ , 0, z ′ ). As moreover dil λ , defined in Subsection 2.1 satisfies

we can assume t = 1. Finally

) ≤ d H ((0, 0, 0), (0, 0, z ′ )) = 2 π|z ′ | comes from the fact that trans (0,0,z ′ ) is not only the left-translation but also the right-translation of vector (0, 0,

The proof of this fact is analytic and is based on the explicit representation of the heat kernel as an oscillatory integral. See [1, equation (14)] and the reference to [START_REF] Li | Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg[END_REF] there. We recall that it is different to the L ∞ -gradient estimate. The proof of Theorem 1.10 is then finished since the quantity |y| |z| is clearly integrable with respect to the heat kernel f X,Y,Z .

6. Generalisation to the Heisenberg groups of higher dimension.

In this section, we prove that Theorems 1.3, 1.2, 1.6, 1.5, 1.8 and 1.10 also hold in the case of the Heisenberg groups of higher dimension. For n ≥ 1 the Heisenberg group H n can be identified with R 2n+1 equipped with the law:

The corresponding Brownian motion starting from 0 R 2n+1 is given by:

where When R t > 0 we introduce the following basis: let e 1 be 1

, set e 2 = (ia 1 , . . . , ia n ) and complete (e 1 , e 2 ) into a direct orthonormal basis of R 2n . This basis is well adapted for studying couplings in H n . Indeed, with L and L being the coupling matrices in this new basis in place of J, Ĵ in the canonical basis, a computation gives:

where (C t ) t≥0 and ( Ct ) t≥0 are some 1-dimensional (possibly correlated) standard Brownian motions. Now, since trL = trJ, and since each

and one can directly adapt the proof of Theorems 1.2 and 1.5 to this setting.

Similarly as before, one can deduce that Theorems 1.3 and 1.6 are also satisfied for higher dimensional Heisenberg groups.

Generalisation of Theorem 1.8: The coupling by reflection can also be done on H n . It corresponds to the matrix L defined by

In this case, a computation easily gives that C t and Ct are independent. Moreover since L is symmetric, J is also symmetric and n i=1 (J 2i-1,2i -J 2i,2i-1 ) = 0. As a consequence, R 2 t and Z t satisfy the same stochastic differential system as in the case of H 1 . Thus Proposition 4.1 holds in H n with constants C p , C ′ p and C ′′ p independent of the dimension and Theorem 1.8 also holds for H n . Generalisation of Theorem 1.10: We turn to the static coupling. As before, using the symmetries and the dilatation of the higher dimensional Heisenberg groups, it suffices to study the case t = 1 and a = 0, a ′ = ((x ′ , 0), (0, 0), . . . , (0, 0), 0).

1 . An analogue coupling as the one in Section 5 can be defined. The horizontal part is translated by ((x, ′ 0), . . . , (0, 0)) and conditionally on ((X 1 , Y 1 ), . . . (X d , Y d )), we perform a coupling between the law of Z and the law of Z + 1 2 x ′ Y 2 as described in Lemma 5.2. Recall that it is adapted to the non-convex transport cost (z, z ′ ) → |zz ′ |.

Since the heat kernel estimate corresponding to (41) also holds in higher dimension (see [START_REF] Eldredge | Precise estimates for the subelliptic heat kernel on H-type groups[END_REF]), the proof finishes analogously to the one in H 1 . Therefore Theorem 1.10 is satisfied for higher dimensional Heisenberg groups too.


Gaganpreet Singh
email: gaganpreet.singh@isae-supaero.fr

Raphaëlle N Roy
email: raphaelle.roy@isae-supaero.fr

Caroline P Carvalho
email: caroline.chanel@isae-supaero.fr

Towards Multi-UAV and Human Interaction Driving System Exploiting Human Mental State Estimation

Keywords: Unmanned Aerial Vehicles (UAV), Multi-UAVs and Human Interaction, Manned-Unmanned Teaming (MUM-T), Mental Workload, Engagement, Physiological Computing

This paper addresses the growing human-multi-UAV interaction issue. Current active approaches towards a reliable multi-UAV system are reviewed. This brings us to the conclusion that the multiple Unmanned Aerial Vehicles (UAVs) control paradigm is segmented into two main scopes: i) autonomous control and coordination within the group of UAVs, and ii) a human centered approach with helping agents and overt behavior monitoring. Therefore, to move further with the future of human-multi-UAV interaction problem, a new perspective is put forth. In the following sections, a brief understanding of the system is provided, followed by the current state of multi-UAV research and how taking the human pilot's physiology into account could improve the interaction. This idea is developed first by detailing what physiological computing is, including mental states of interest and their associated physiological markers. Second, the article concludes with the proposed approach for Human-multi-UAV interaction control and future plans.

INTRODUCTION

Recent automation progress in terms of control, navigation, and decision making brings the speculation of autonomous decision-making multi-UAV systems' deployment closer to reality [START_REF] Schulte | Humansystem interaction analysis for military pilot activity and mental workload determination[END_REF]. However, keeping the human in the (decisional) loop is still a compulsory point [START_REF] Valavanis | Future of unmanned aviation[END_REF][START_REF] Schurr | Improving adjustable autonomy strategies for timecritical domains[END_REF]. In particular, unmanned aircraft's engineering for optimal control strategies are still evolving and the idea of having better control of an unmanned aircraft is shifting to the desire of controlling several UAVs at once.

The actual ratio factor between UAV operators (O) and UAV units (N) is O≥N. For example, in the US army, a UAV is managed by several operators: one is in charge of following the flight parameters, other is in charge of payload, and the last one is responsible for the mission supervision. In the next future, this ratio would probably be inverted (O<N) (Gangl et al., 2013a).

Indeed, UAVs are getting more and more automated, taking decisions by themselves, which lightens the need for such a number of operators. The idea is that UAVs could explore safety automation to ensure a completely autonomous navigation and even a completely autonomous mission planning. How-ever, the human operator is still vital, and unfortunately, considered as a providential agent [START_REF] Schurr | Improving adjustable autonomy strategies for timecritical domains[END_REF][START_REF] Casper | Human-robot interactions during the robot-assisted urban search and rescue response at the world trade center[END_REF], who gets over the autonomous or automatic system when some hazardous event occurs. Yet, it is known that, in UAV operations, Human Factors represent the most important part of accidents [START_REF] Williams | A summary of unmanned aircraft accident/incident data: Human factors implications[END_REF][START_REF] Haddal | Homeland security: Unmanned aerial vehicles and border surveillance[END_REF]. Nevertheless, in some cases, it is mandatory to handle a mission from close proximity, by keeping human agents in the loop, which are in charge of taking the difficult decisions.

However, to leverage the advantage of being smart and adaptable like humans, and consistent and precise like machines, the idea is to bring them in close contact beforehand, and so, to design a system that provides authority and integrity to both (non unerring) actors -human and machine -while helping a single human to work collaboratively with multiple UAVs. In particular, considering that neither of them is leading, but both of them are helping each other in accomplishing the mission goal(s). This interesting positioning is also known as mixed-initiative for Human-Machine Interaction (HMI) [START_REF] Jiang | Mixed-initiative humanrobot interaction: definition, taxonomy, and survey[END_REF], which should consider that each of the agents could seize (e.g. relinquish) the initiative from the other(s). From a human point of view, such a system that takes over us is not always acceptable (or even desirable), but should at least be welcome when human capabilities (cognitive or physical) are not forthcoming.

Therefore, this paper proposes a multi-UAV interaction driving concept. Where multiple UAVs are interacting with a single human agent from a plane's cockpit while the latter is being part of the mission, and its physiological measurements are exploited to enhance coordination between man and machine. In other words, the human and UAVs can be seen as a Manned-Unmanned Teaming (MUM-T). In missions scenarios, where a MUM-T is suitable, hot events can occur and the human agent, which is in charge of harder decisions, can experience degraded mental states. In this context, the main idea is to explore the use of mental state estimation in real time [START_REF] Singh | Physiologically attentive user interface for robot teleoperation: real time emotional state estimation and interface modification using physiology, facial expressions and eye movements[END_REF] to drive the Human and Multi-UAVs Interaction.

The paper is organized as follows: section 2 presents multi-UAV applications that take (or not) into account the human operator inputs, directly in the supervisory control-loop, or to well-drive human and multi-UAV interaction. In section 3 promising physiological parameters that could be useful to estimate the cognitive state of human operator are reviewed. The proposed approach is presented in section 4, as well as, a brief description of the experiment scenario currently being designed. At the end, conclusions and future works in section 5 closes this paper.

MULTI-UAVS IN ACTION

With the emergence of unmanned aircrafts and the evolution of automation, researchers are pushing the limits to attain the capability of controlling multiple UAVs. But there is still a debate between having a fully autonomous group of UAVs performing different sub-tasks to achieve one common goal or having one human controlling several UAVs all together towards the success of a mission. Moreover, both of these ideologies settle in one common urge of having a fully capable multi-UAV system.

Several multi-UAV applications have already been designed and demonstrated. The COMETS project [START_REF] Ollero | Architecture and perception issues in the comets multi-uav project. multiple eyes in the skies[END_REF] is one of them with several research organizations involved in design and implementation of a new control architecture for multiple heterogeneous UAVs working cooperatively in forest fire missions. Maza and collaborators [START_REF] Maza | Experimental results in multiuav coordination for disaster management and civil security applications[END_REF] designed a multi-UAV distributed decisional architecture to autonomously cooperate and coordinate with UAVs while accomplishing high level tasks by dividing and assigning low level tasks to each UAV with respect to the capabilities of each one of them.

Perez and collaborators [START_REF] Perez | A ground control station for a multi-uav surveillance system[END_REF] developed a ground control station for dynamically assigning tasks to several UAVs, and Schere and collaborators [START_REF] Scherer | An autonomous multi-uav system for search and rescue[END_REF] created a distributed control system to coordinate multiple UAVs and override autonomy when required. Brisset and collaborators [START_REF] Brisset | Multi-uav control with the paparazzi system[END_REF] have conducted two multi-UAV experiments using Paparazzi (a free autopilot) [START_REF] Brisset | Paparadziy: do-ityourself uav[END_REF][START_REF] Brisset | The paparazzi solution[END_REF]. In these experiments, a formation flight using first 3 UAVs, and secondly 2 UAVs at different locations in Germany and France were controlled by the same Ground Control Station in Germany by two operators. Franchi and collaborators [START_REF] Franchi | Shared control: Balancing autonomy and human assistance with a group of quadrotor uavs[END_REF] also studied the involvement of human in control loop of multi-UAVs with self arranged autonomy. Whereas, Muller and collaborators [START_REF] Mueller | A human-system interface with contingency planning for collaborative operations of unmanned aerial vehicles[END_REF] targeted 2 problems of multi-UAVs operation: first an effective Human System Interface for better understanding, control, and monitoring of overall activity; and second to authorize UAVs to plan, verify, and act when the connection to the human operator is lost.

Such systems are trying to diminish the need of a human operator by making systems capable enough to take decisions and accomplish the mission without any human intervention, or just with little supervisory control, or takeover when required.

On the other hand, recent works are heading towards an integrated system with human involvement in critical situations [START_REF] Donath | Cognitive assistant system concept for multi-uav guidance using human operator behaviour models[END_REF]Gangl et al., 2013a;[START_REF] Gangl | Single-seat cockpit-based management of multiple ucavs using on-board cognitive agents for coordination in mannedunmanned fighter missions[END_REF][START_REF] Schulte | Humansystem interaction analysis for military pilot activity and mental workload determination[END_REF]. The main idea behind these researches is the involvement of a human operator not just in the supervisory, or control loop, but in the mission itself. The human operator is not remotely controlling the UAVs to perform tasks, but one is also involved in the mission plan and performs the required supervision of accompanying UAVs from the cockpit of a plane. The advantages of this kind of system over those that keep the human operator only for supervision from ground control stations are tremendous. For instance, with this setup there is less physical limitation between the UAVs and the control station (Gangl et al., 2013a), there are long range mission possibilities. Moreover, there is a better availability of the human operator for critical decision making considering better situation awareness, along with real time task (re)planning.

Donath and collaborators [START_REF] Donath | Cognitive assistant system concept for multi-uav guidance using human operator behaviour models[END_REF]) worked on assistant systems to help human pilot in managing multiple UAVs from a manned aircraft, and also evaluated the workload experienced by the pilot. They used human behavior models to represent the workload experienced and to provide pos-sible solutions to balance it. Their work was further evaluated by Gangl and collaborators [START_REF] Gangl | Single-seat cockpit-based management of multiple ucavs using on-board cognitive agents for coordination in mannedunmanned fighter missions[END_REF] with Unmanned Combat Aerial Vehicles (UCAVs). In this particular study, Artificial Cognitive Units (ACUs) were used to control each UCAV separately along with the presence of a human pilot in a manned aircraft's cockpit working hand-in-hand with the UCAVs.

A very critical issue of mental workload experienced by human pilots in a Manned-Unmanned Teaming (MUM-T) scenarios has been raised in these works [START_REF] Donath | Cognitive assistant system concept for multi-uav guidance using human operator behaviour models[END_REF]Gangl et al., 2013a;[START_REF] Gangl | Single-seat cockpit-based management of multiple ucavs using on-board cognitive agents for coordination in mannedunmanned fighter missions[END_REF][START_REF] Schulte | Humansystem interaction analysis for military pilot activity and mental workload determination[END_REF]. However, a promising way, in our point of view, is to use physiological features over subjective or behavioral human measurements or models in order to better estimate mental workload, this could provide a unique advantage. Indeed, estimating humans' mental state using subjective and behavioral measures can only tell what might have occurred but cannot measure and reveal what actually went on. Hence, physiological measures can help extracting pilots' mental states in real time and can provide better estimates of such states than overt measurements which are relatively sparse [START_REF] Mehta | Neuroergonomics: a review of applications to physical and cognitive work[END_REF].

In this sense, this work's position aims to explore the benefits of using physiological computing to estimate pilots' mental state in order to drive humanmulti-UAVs interaction.

PHYSIOLOGICAL COMPUTING

Physiological computing provides a revolutionized way towards Human-Machine Interaction (HMI) by directly monitoring, analyzing, and responding to covert physiological features of a user in real time [START_REF] Fairclough | Fundamentals of physiological computing[END_REF]. A system that explores physiological computing, works through reading and transforming psychophysiological signals as inputs to a control signal without going through any direct communication channel with the human operator [START_REF] Byrne | Psychophysiol-ogy and adaptive automation[END_REF]. It brings in an efficient HMI, or rather opens up a communication channel which was left unused before [START_REF] Hettinger | Neuroadaptive technologies: applying neuroergonomics to the design of advanced interfaces[END_REF]. The use of such physiological features or markers is a great means to provide sixth sense for research and other applications that are willing to peak into the core of human activity and want to get insight of how a person is actually experiencing the world in their cognitive realm without putting the operator in any direct or indirect conversation. Previous research supports the use of physio-logical features to better understand human mental state and enhance a task's outcome [START_REF] Roy | Operator engagement during prolonged simulated uav operation[END_REF][START_REF] Senoussi | Pre-stimulus antero-posterior eeg connectivity predicts performance in a uav monitoring task[END_REF]Drougard et al., 2017a;[START_REF] Drougard | Mixed-initiative mission planning considering human operator state estimation based on physiological sensors[END_REF]. For aeronautical applications, several mental states are particularly relevant to try and estimate such as: Mental Workload, Engagement, Fatigue, and Drowsiness. The work presented here focuses on Mental Workload and Engagement since these states are great contributors to human performance modulation in risky settings.

Mental Workload and Engagement

Workload was defined in several different ways, but it could be understood as ones' information processing capacity or amount of resources required to meet system demand [START_REF] Eggemeier | Workload assessment in multi-task environments[END_REF] or the difference in the capacity of an information processing system to satisfy a task's performance and the available capacity at any given time [START_REF] Gopher | Workload-an examination of the concept[END_REF]. A closely related concept is therefore the Engagement, or attentional/cognitive resource engagement. The Engagement level of an operator varies depending on several factors such as time-on-task (and therefore vigilance/alertness), task demands, and motivation [START_REF] Berka | Eeg correlates of task engagement and mental workload in vigilance, learning, and memory tasks[END_REF][START_REF] Chaouachi | Mental workload, engagement and emotions: an exploratory study for intelligent tutoring systems[END_REF][START_REF] Mcmahan | Evaluating player task engagement and arousal using electroencephalography[END_REF]. In our understanding, trying and estimating mental workload is akin to estimate engagement if one thinks in terms of mental resources. Hence, these two concepts will be considered as one in the remaining parts of this paper.

Associated Physiological Markers

Mental workload and mental resource engagement have been widely studied [START_REF] Mehta | Neuroergonomics: a review of applications to physical and cognitive work[END_REF], which allowed to reveal several physiological parameters that can enable to effectively estimate human engagement state. Both cerebral and peripheral physiological measures can be used to infer the engagement state. Hence, in order to perform engagement estimation, electroencephalographical (EEG) features in the temporal and frequency domain can be used [START_REF] Pope | Biocybernetic system evaluates indices of operator engagement in automated task[END_REF].

In the temporal domain, an example is the use of Event Related Potentials (ERP) which are the timelocked cerebral responses to specific events [START_REF] Fu | Event-related potentials (erps) in neuroergonomics[END_REF]. The amplitude of these voltage variations can be extracted at various time points and is known to fluctuate with engagement. For example, after 300 ms post-event (e.g. after an alarm) there is a lower positive deflection at posterior electrode sites if the operator has not engaged enough resources to correctly process this event.

Next, in the frequency domain one can use modulations in the Power Spectral Density (PSD) of different EEG frequency bands (e.g. θ: 4-8 Hz, α: 8-12 Hz and β: 13-30 Hz) (Roy et al., 2016a;Roy et al., 2016c;Heard et al., 2018a). For instance, a widely accepted and evaluated Engagement Index (EI) developed by Pope and collaborators [START_REF] Pope | Biocybernetic system evaluates indices of operator engagement in automated task[END_REF] can be used to modulate task allocation in a closed-loop system and is computed using band power as follows: EI = β/(α + θ) [START_REF] Chaouachi | Mental workload, engagement and emotions: an exploratory study for intelligent tutoring systems[END_REF][START_REF] Berka | Eeg correlates of task engagement and mental workload in vigilance, learning, and memory tasks[END_REF].

As for peripheral measures that can be useful to estimate operator's mental state, markers can be extracted from the electrocardiogram (ECG). Well known ones are the Heart Rate (HR; time domain metric) and the Heart Rate Variability (HRV; can be computed both in the time and the frequency domains). These metrics are sensitive to workload and engagement but not specific to it, indeed they are also modulated by physical activity (Heard et al., 2018a). Another way of recording peripheral activity is to use an eye-tracker device, which records ocular activity. Thanks to this device, one can for instance extract Blink Frequency (BF; i.e. number of blinks per minute), Fixation Duration (FD; i.e. amount of time the eyes fixated a particular area) and Blink Latency (BL; i.e. amount of time between two blinks) which variate with engagement (Heard et al., 2018a;[START_REF] Heard | A diagnostic human workload assessment algorithm for human-robot teams[END_REF].

THE PROPOSED APPROACH

Machine intelligence still does not have far reaching capabilities to match human intelligence and abilities, in particular to work in unpredictable and continuously changing environments. Since the human brain does have far reaching capabilities, a better integration of its capabilities with machines could bring unmatched results.

As before highlighted, research has already taken place to achieve high levels of autonomy in UAVs and to make their control system (in terms of flying) capable enough to handle flight parameters without much human intervention. Therefore, this work is not directed towards controlling the dynamics and autonomy of the UAVs. It is neither directed towards achieving a super smart interface to enable human operators to supervise several UAVs through that.

Rather, this work focuses on unveiling and estimating human pilot's mental states involved in the mission to enhance Human-UAVs interaction. Secondly, it focuses on developing decisional systems that understand the situation and choose appropriate actions. Here, understanding the situation means how the overall system's coordinator estimates the state of both human and UAV units, being able to predicts its influence on mission achievement. The appropriate actions could be: managing the amount of work the human pilot could (or should) handle; managing the tasks that could done by the UAV; when and how information should be transferred to the human pilot; and how tasks should be shared between human and UAVs. These concepts should bring to a coordinative approach towards mission success.

Researchers have been trying to improve mission success, focusing in three main problems associated with Unmanned Aerial Vehicles (UAVs): higher accident rates [START_REF] Haddal | Homeland security: Unmanned aerial vehicles and border surveillance[END_REF], higher human to machine ratio [START_REF] Gangl | Single-seat cockpit-based management of multiple ucavs using on-board cognitive agents for coordination in mannedunmanned fighter missions[END_REF], and state awareness of human counterpart [START_REF] Schulte | Humansystem interaction analysis for military pilot activity and mental workload determination[END_REF]. The approach, here proposed, will try to tackle these issues by increasing state awareness for both human and machine, eventually decreasing human errors and increasing system's performance. The aim is to invert the higher human-to-machine ratio into a higher machine-to-human ratio.

The overall system will contain a human pilot and several UAVs working as a team (MUM-T) on a common mission, see Figure 1. There will be a main system i.e. Mission and Interaction Coordinator (MIC) that has knowledge of overall mission plan and goals, see Figure 2. A search and rescue mission [START_REF] Souza | Towards human-robot interaction: A framing effect experiment[END_REF][START_REF] Gateau | Considering human's non-deterministic behavior and his availability state when designing a collaborative human-robots system[END_REF] will be the core of the scenario. In such a mission several actions would be considered: UAV requests to perform identification and confirmation of possible targets; which agent should visit dangerous or accidental zones; communicate the targets position; present or not to present the information to the human pilot; etc. The challenge behind it is to choose when to launch a request to the pilot. The system's coordinator should decide based on the availability of the human pilot i.e. inverse of workload [START_REF] Gateau | Considering human's non-deterministic behavior and his availability state when designing a collaborative human-robots system[END_REF], ethical commitment [START_REF] Souza | Towards human-robot interaction: A framing effect experiment[END_REF], or based on degraded mental states like attentional tunneling (de [START_REF] De Souza | Momdp-based target search mission taking into account the human operator's cognitive state[END_REF]. On the basis of these, the overall mission coordinator has to estimate human pilot's mental state, to assign tasks to UAVs, and to change the level of autonomy to help maintain a human pilot's engagement within a suitable window i.e. neither too high nor too low [START_REF] Ewing | Evaluation of an adaptive game that uses eeg measures validated during the design process as inputs to a biocybernetic loop[END_REF], while maintaining an acceptable system's performance to achieve mission's goal(s).

Research Milestones

This research will be carried out in three phases: i) data collection, ii) implementation, and iii) closedloop validation.

In the first phase, the hard-coded experiment should allow behavioral and physiological data collection from the human pilot, equipped with an EEG, an ECG, and an ET. Therefore, for this phase, a controlled environment where a human pilot will handle a simulated flight along with interactions with the UAVs is considered. The experiment is designed to take place considering four experimental conditions: low workload (L), high workload (H), low to high workload transition (LH), and high to low workload transition (HL) (see Fig. 3 for an example). The four conditions will be split in a pseudo-random manner: L-LH-H-HL or H-HL-L-LH, and are expected to bring up engagement-workload variations. The human pilot's workload level will be manipulated by means of:

• choosing from time to time way-points to meet the requirement of staying in a given distance range that allows to maintain communicating with UAVs. These way-points will transform into Air Traffic Control (ATC) instructions for flying the plane and will be given to pilot in the form of audio messages. [START_REF] Risser | Interference effects on the recall of words heard and read: Considerations for atc communication[END_REF] and [START_REF] Gateau | In silico versus over the clouds: On-the-fly mental state estimation of aircraft pilots, using a functional near infrared spectroscopy based passive-bci[END_REF] showed that, depending on their length and complexity, recalling ATCs instruction (e.g. speed, altitude, heading) can create a high cognitive load. • answering to pop-ups containing UAVs' requests that concern the identification and recognition of detected targets as in [START_REF] Gateau | Considering human's non-deterministic behavior and his availability state when designing a collaborative human-robots system[END_REF]. Particularly in a search and rescue mission, where lives are at stake [START_REF] Souza | Towards human-robot interaction: A framing effect experiment[END_REF]. Errors of identification or recognition may be avoided, which implies an important human involvement; • and performing checklists related, for instance, to malfunctioning of flight equipment or UAVs embedded systems. Note that, interruptions during checklists can be considered as an issue [START_REF] Loukopoulos | Cockpit interruptions and distractions: A line observation study[END_REF], and can potentially increase the workload while decreasing performance [START_REF] Loft | Prospective memory and task interference in a continuous monitoring dynamic display task[END_REF]); Currently we have designed the application shown in Figure 4, that lets the human pilot interact with UAVs during the experiment (e.g. like sending requests for areas to search; answer UAV queries for validation of recognized objects). Following this first experiment (data collection) under the conditions listed, it is expected to design a smart tool (implementation phase) that could estimate mental states of human pilots in real time based on the physiological markers reviewed in Sec. 3. This real time estimation would serve as an input to an overall system's state estimator (i.e. mental state of human and UAVs' states).

Once the overall system's state estimator is designed, a decisional framework, called Mission and Interaction Coordinator (MIC) would reason in a long-term way. In other words, it will predict future states of the whole system (i.e. agents and mis- sion's states) and will have to choose an appropriate action, bringing coordination between all involved agents (i.e. human and UAVs) while ensuring mission success. This will constitute the closed-loop phase of this research. The evaluation of the closed-loop framework will be handled with an experiment similar to the one described above.

Conclusion and Future Work

This paper presents our current research towards a multi-UAV and human interaction driving system that would exploit human's mental state estimation. The main idea is to integrate the latest advances of physiological computing into a high-level mission coordinator. State-of-the-art approaches were presented as well as promising physiological markers. A mission scenario was also proposed, in which a human pilot should coordinate his actions along with UAVs' requests (MUM-T). In this scenario, four conditions would be evaluated in order to study the variations of engagement of human pilot. In particular, such conditions would constitute labels to the collected data therefore allowing the design of the subsequent smart estimation system.

The next step of this work is to define the experimental protocol of the proposed mission scenario in details, to implement a rigorous experimental setup, hence ensuring the validity of the expected results. Such results will be used in the forthcoming stages of this work, in which an intelligent artificial system will have to reason in a long-term manner. In other words, it should predict future states of agents (i.e. mental state of the human pilot, and UAVs' states) and requirements of overall mission (i.e. needs and possible future actions of the human pilot or of UAVs) in order to take optimal actions to balance the load between human pilot and UAVs while maximizing system's performance.

Figure 1 :

 1 Figure 1: Project layout

Figure 2 :

 2 Figure 2: Project architecture

Figure 3 :

 3 Figure 3: Experiment structure with conditions designed to elicit variations of pilots' engagement.

Figure 4 :

 4 Figure 4: Developed application that enables the human pilot to interact with UAVs (e.g. to choose regions to search, to answer UAVs requests, or to define way-points to meet).