Fabrice Baudoin
email: fbaudoin@math.purdue.edu

AND Cheng Ouyang

Samy Tindel
email: tindel@iecn.u-nancy.fr

UPPER BOUNDS FOR THE DENSITY OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

In this paper we study upper bounds for the density of solution of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/3. We show that under some geometric conditions, in the regular case H > 1/2, the density of the solution satisfy the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case H > 1/3 and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper sub-Gaussian bound.

(0, 1). Recall that it means that B is a centered Gaussian process indexed by R + , whose coordinates are independent and satisfy [START_REF] Baudoin | Operators associated with a stochastic differential equation driven by fractional Brownian motions[END_REF] E B j t -B j s 2 = |t -s| 2H , for s, t ∈ R + .

In particular, by considering the family {B H ; H ∈ (0, 1)}, one obtains some Gaussian processes with any prescribed Hölder regularity, while fulfilling some intuitive scaling properties. This converts fBm into the most natural generalization of Brownian motion to this day. We are concerned here with the following class of equations driven by B:

(2)

X x t = x + t 0 V 0 (X x s )ds + d i=1 t 0 V i (X x s )dB i s ,
where x is a generic initial condition and {V i ; 0 ≤ i ≤ d} is a collection of smooth vector fields of R d . Owing to the fact that fBm is a natural generalization of Brownian motion, this kind of model is often used by practitioners in different contexts, among which we would like to highlight recent sophisticated models in Biophysics [START_REF] Kou | Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule[END_REF][START_REF] Szymanski | Elucidating the origin of anomalous diffusion in crowded fluids[END_REF][START_REF] Tejedor | Quantitative Analysis of Single Particle Trajectories: Mean Maximal Excursion Method[END_REF].

As far as mathematical results are concerned, equation ( 2) is now a fairly well understood object: existence and uniqueness results are obtained for H > 1 2 thanks to Young integral type tools [START_REF] Zähle | Integration with respect to fractal functions and stochastic calculus I[END_REF][START_REF] Nualart | Differential equations driven by fractional Brownian motion[END_REF], while rough paths methods [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF][START_REF] Lyons | System control and rough paths[END_REF] are required for 1 4 < H < 1 2 . Numerical schemes can be implemented for this kind of systems [START_REF] Deya | A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion[END_REF][START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF], and a notion of ergodicity is also available [START_REF] Hairer | Ergodicity of stochastic differential equations driven by fractional Brownian motion[END_REF][START_REF] Hairer | Ergodicity theory of SDEs with extrinsic memory[END_REF]. Finally, the law of X x t has been analyzed by means of semi-group type methods [START_REF] Baudoin | Operators associated with a stochastic differential equation driven by fractional Brownian motions[END_REF][START_REF] Neuenkirch | Trees and asymptotic developments for fractional diffusion processes[END_REF] and its density has also been investigated in [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF][START_REF] Cass | Non-degeneracy of Wiener functionals arising from rough differential equations[END_REF][START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF][START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF].

In spite of these advances, concentrations results and Gaussian bounds for the solution to [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF] are scarce: we are only aware of the large deviation results [START_REF] Millet | Large deviations for rough paths of the fractional Brownian motion[END_REF] in this line of investigation. The current article is thus an attempt to make a step in this direction, by analyzing a special but nontrivial situation.

Indeed, we consider here equation [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF] driven by a fBm with Hurst parameter H ∈ ( 1 3 , 1), and we suppose that our vector fields V 0 , . . . , V d fulfill either of the following nondegeneracy and antisymmetric hypothesis:

Hypothesis 1.1. The vector fields V 0 , . . . , V d are C ∞ -bounded, and V 1 , . . . , V d satisfy (i) For every x ∈ R d , the vectors V 1 (x), • • • , V d (x) form a basis of R d . (ii) There exist smooth and bounded functions ω k ij such that:

(3) [V i , V j ] = d k=1 ω k ij V k , and ω k ij = -ω j ik .
The second assumption (ii) is of geometric nature and actually means that the Levi-Civita connection associated with the Riemannian structure given by the vector fields

V i 's is ∇ X Y = 1 2 [X, Y ].
In a Lie group structure, this is equivalent to the fact that the Lie algebra is of compact type, or in other words that the adjoint representation is unitary. Such geometric assumption already appeared in the work [START_REF] Baudoin | Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions[END_REF] where it was used to prove a small-time asymptotics of the density.

Hypothesis 1.2. The Hypothesis 1.1 is satisfied and moreover, the vector fields V 1 , ..., V d form moreover a uniform elliptic system. That is

|v T V V T v| ≥ λ|v| 2 , for all v ∈ R d .
Here V = (V i j ) i,j=1,...,d and λ is a positive constant. When H > 1 2 , under Hypothesis 1.1 our main result can be loosely summarized as follows (see Theorem 3.14 for a precise statement): Theorem 1.3. Fix H > 1 2 . Let X x be the solution to equation [START_REF] Baudoin | A version of Hörmander's theorem for the fractional Brownian motion[END_REF], and suppose Assumption 1.1 is satisfied. Then for any t ∈ R * + , the random variable X x t admits a smooth density p X (t, •). Furthermore, there exist 3 positive constants c

(1)

t , c (2)
t , c (3)
t,x such that p X (t, y) ≤ c

(1)

t exp -c (3) t |y| -c (2) t,x 2 ,
for any y ∈ R d .

We don't claim any optimality in the quantities c

(1) t , c

(2) t and c

(3) t,x above (whose exact definitions are postponed to Section 3.3). Nevertheless, this is (to the best of our knowledge) the first Gaussian type bound available for solutions of differential equations driven by fBm.

Let us say a few words about the strategy we have followed in order to prove Theorem 1.3. It is mostly based on stochastic analysis tools, and particularly on a general integration by parts formula giving an exact expression for the density p X (t, •) in terms of Malliavin derivatives in the non-degenerate case we are dealing with. In this context, it is crucial to bound the first Malliavin derivative of X x t (called DX x t in the sequel) efficiently. This is where our asymmetry hypothesis on the vector fields V 1 , . . . , V n enter into the picture, and we shall see (at Theorem 3.2) how asymmetry properties yield an easy deterministic bound on DX x t . This result enables to get concentration results for the law of X x t , and is the key to our density bounds as well. As another interesting consequences of the deterministic bound on DX x t , we also obtain Log-Sobolev inequality and Poincaré inequality for the law of X t .

Once the picture for the smooth case (when H > 1 2 ) becomes clear, we are able to extend some of our results described above to the irregular case when 1 3 < H < 1 2 . In particular, we are able to prove

Theorem 1.4. Fix H ∈ ( 1 3 , 1
2 ). Assume Hypothesis 1.2. Let X x be the solution to equation ( 2) and γ Xt the Malliavin matrix of X x t , t > 0. We have | det γ Xt | -1 ∈ L ∞ (P). The random variable X x t admits a smooth density p X (t, •) and for any δ < H there exist 2 positive constants c

(1) t , c (2) t such that (4) p X (t, y) ≤ c (1)
t exp -c

(2)

t |y| δ , for all y ∈ R d .
The existence of a density for solutions to stochastic differential equations of the form (2) under Hörmander's condition has been obtained by Cass and Friz [6] for any 1 4 < H < 1 2 .

While finishing the current article, an important step towards the study of regular densities in the rough case 1 4 < H < 1 2 has been accomplished in [START_REF] Cass | Integrability estimates for Gaussian rough differential equations[END_REF], where integrability estimates for the Jacobian of equation ( 2) are established. Nevertheless, as of today, besides the result of P. Driscoll [START_REF] Driscoll | Smoothness of density for the area process of fractional Brownian motion[END_REF], when H < 1 2 , to the best of our knowledge, Hypothesis 1.2 is a first wide class of examples where we have an affirmative answer for the smoothness of the density. Our bound on the inverse of the Malliavin matrix together with polynomial bounds on the Hölder norm of the Malliavin derivative allows then to obtain the sub-Gaussian upper bound [START_REF] Besalú | Estimates for the solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter[END_REF].

Notations: Throughout this paper, unless otherwise specified we use | • | for Euclidean norms and • L p for the L p norm with respect to the underlying probability measure P.

Consider a finite-dimensional vector space V . The space of V -valued Hölder continuous functions defined on [0, 1], with Hölder continuity exponent γ ∈ (0, 1), will be denoted by C γ (V ), or just C γ when this does not yield any ambiguity. For a function g ∈ C γ (V ) and 0 ≤ s < t ≤ 1, we shall consider the semi-norms [START_REF] Capitaine | Martingale representation and logarithmic Sobolev inequality[END_REF] g s,t,γ = sup

s≤u<v≤t |g v -g u | V |v -u| γ ,
The semi-norm g 0,1,γ will simply be denoted by g γ .

Stochastic calculus for fractional Brownian motion

For some fixed H ∈ ( 1 3 , 1), we consider (Ω, F , P) the canonical probability space associated with the fractional Brownian motion (in short fBm) with Hurst parameter H. That is, Ω = C 0 ([0, 1]) is the Banach space of continuous functions vanishing at 0 equipped with the supremum norm, F is the Borel sigma-algebra and P is the unique probability measure on Ω such that the canonical process B = {B t = (B 1 t , . . . , B d t ), t ∈ [0, 1]} is a fractional Brownian motion with Hurst parameter H. In this context, let us recall that B is a d-dimensional centered Gaussian process, whose covariance structure is induced by equation ( 1). This can be equivalently stated as

R (t, s) := E B j s B j t = 1 2 s 2H + t 2H -|t -s| 2H , for s, t ∈ [0, 1] and j = 1, . . . , d.
In particular it can be shown, by a standard application of Kolmogorov's criterion, that B admits a continuous version whose paths are γ-Hölder continuous for any γ < H. This section is devoted to give the basic elements of stochastic calculus with respect to B which allow to understand the remainder of the paper.

2.1. Malliavin calculus tools. Gaussian techniques are obviously essential in the analysis of fBm, and we proceed here to introduce some of them (see [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] for further details): let E be the space of R d -valued step functions on [0, 1], and H the closure of E for the scalar product:

(1 [0,t 1 ] , • • • , 1 [0,t d ] ), (1 [0,s 1 ] , • • • , 1 [0,s d ] ) H = d i=1 R(t i , s i ).
We denote by K * H the isometry between H and L 2 ([0, 1]). When H > 1 2 it can be shown that L 1/H ([0, 1], R d ) ⊂ H, and when

1 3 < H < 1 2 one has C γ ⊂ H ⊂ L 2 ([0, 1])
for all γ > 1 2 -H. Some isometry arguments allow to define the Wiener integral B(h) = 1 0 h s , dB s for any element h ∈ H, with the additional property E[B(h 1 )B(h 2 )] = h 1 , h 2 H for any h 1 , h 2 ∈ H. A F -measurable real valued random variable F is then said to be cylindrical if it can be written, for a given n ≥ 1, as

F = f B(h 1 ), . . . , B(h n ) = f 1 0 h 1 s , dB s , . . . , 1 0 h n s , dB s ,
where h i ∈ H and f : R n → R is a C ∞ bounded function with bounded derivatives. The set of cylindrical random variables is denoted S.

The Malliavin derivative is defined as follows: for F ∈ S, the derivative of F is the R d valued stochastic process (D t F ) 0≤t≤1 given by

D t F = n i=1 h i (t) ∂f ∂x i B(h 1 ), . . . , B(h n ) .
More generally, we can introduce iterated derivatives. If F ∈ S, we set

D k t 1 ,...,t k F = D t 1 . . . D t k F.
For any p ≥ 1, it can be checked that the operator D k is closable from S into L p (Ω; H ⊗k ).

We denote by D k,p (H) the closure of the class of cylindrical random variables with respect to the norm

F k,p = E (F p ) + k j=1 E D j F p H ⊗j 1 p
, and

D ∞ (H) = p≥1 k≥1 D k,p (H).
2.2. Differential equations driven by fBm. Recall that we consider the following kind of equation:

(6) X x t = x + t 0 V 0 (X x s )ds + d i=1 t 0 V i (X x s )dB i s ,
where the vector fields V 0 , . . . , V n are C ∞ -bounded. When equation ( 6) is driven by a fBm with Hurst parameter H > 1 2 it can be solved, thanks to a fixed point argument, with the stochastic integral interpreted in the (pathwise) Young sense (see e.g. [START_REF] Gubinelli | Controlling rough paths[END_REF]). Let us recall that Young's integral can be defined in the following way: Proposition 2.1. Let f ∈ C γ , g ∈ C κ with γ + κ > 1, and 0 ≤ s ≤ t ≤ 1. Then the integral t s g ξ df ξ is well-defined as limit of Riemann sums along partitions of [s, t]. Moreover, the following estimation is fulfilled:

(7) t s g ξ df ξ ≤ C f γ g κ |t -s| γ ,
where the constant C only depends on γ and κ. A sharper estimate is also available:

(8) t s g ξ df ξ ≤ |g s | f γ |t -s| γ + c γ,κ f γ g κ |t -s| γ+κ .
With this definition in mind, we can solve our differential system of interest, and the following moments bounds are proven in [START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF]

|X x t | ≤ |x| + c V,T B 1/β 0,T,β .
(2) If we only assume that vector fields V have linear growth, with ∇V, ∇ 2 V , bounded, the following estimate holds true:

(9) sup t∈[0,T ] |X x t | ≤ (1 + |x|) exp c V,T B 1/β 0,T,β .
Remark 2.3. The framework of fractional integrals is used in [START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF] in order to define integrals with respect to B. It is however easily seen to be equivalent to the Young setting we have chosen to work with.

When the Hurst parameter 1 3 < H < 1 2 , equation ( 6) can be solved, again by fixed point argument, with the stochastic integral interpreted in the (pathwise) rough path theory (see e.g. [START_REF] Gubinelli | Controlling rough paths[END_REF] and [START_REF] Lyons | System control and rough paths[END_REF]). In this case, we obtain Proposition 2.4. (Besalú-Nualart [START_REF] Besalú | Estimates for the solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter[END_REF]) Consider equation [START_REF] Cass | Densities for rough differential equations under Hörmander condition[END_REF] driven by a fBm B with Hurst parameter 1 3 < H < 1 2 . Denote by X x its unique β-Hölder continuous solution, for any β < H. If the vector fields V are C ∞ -bounded, then for any λ > 0 and δ < H

E exp λ sup 0≤t≤T |X t | δ < ∞.
Once equation ( 6) is solved, the vector X x t is a typical example of random variable which can be differentiated in the Malliavin sense. In fact, fix H ∈ ( 1 3 , 1), one gets the following results (see [START_REF] Cass | Densities for rough differential equations under Hörmander condition[END_REF] and [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF] for further details): Proposition 2.5. Let X x be the solution to equation [START_REF] Cass | Densities for rough differential equations under Hörmander condition[END_REF] and suppose V i 's are C ∞ -bounded vector fields on R d . Then for every i = 1, . . . , d, t > 0, and

x ∈ R d , we have X x,i t ∈ D ∞ (H) and D j s X x t = J 0→t J -1 0→s V j (X s ), j = 1, . . . , d, 0 ≤ s ≤ t, where D j s X x,i t is the j-th component of D s X x,i t , and J 0→t = ∂X x t ∂x .
Finally the following approximation result, which can be found for instance in [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF], will also be used in the sequel: Proposition 2.6. For m ≥ 1 and T > 0, let B m = {B m t ; t ∈ [0, T ]} be the sequence of linear interpolations of B along the dyadic subdivision of [0, T ] of mesh m; that is if

t m i = i2 -m T for i = 0, ..., 2 m ; then for t ∈ (t m i , t m i+1 ], B m t = B t m i + t -t i m t m i+1 -t m i (B t m i+1 -B t m i ).
Consider X m the solution to equation ( 6) restricted to [0, T ], where B has been replaced by B m . Set also

D j s X m t = J 0→t J -1 0→s V j (X m s ), for j = 1, . . . , d and 0 ≤ s ≤ t.
Then almost surely, for any γ < H and t ∈ [0, T ] the following holds true:

(10) lim m→∞ X x -X m γ + D j X x t -D j X m t γ = 0.

Estimates for solutions of SDEs driven by fBm: the smooth case

Throughout this section, we fix H ∈ ( 1 2 , 1). Recall that X x designates the solution to [START_REF] Cass | Densities for rough differential equations under Hörmander condition[END_REF]. This section is devoted to get some further bounds for X x t and its Malliavin derivatives, under Assumption 1.1.

Notice that among our set of hypothesis, the antisymmetric property (3) for the vector fields V 1 , . . . , V n is the most specific one. It will be mainly used through the following lemma:

Lemma 3.1. Let A 1 , A 2 be n × n matrices, whose exponential are defined by e A j = ∞ n=0 A n j /n! for j = 1, 2. If we assume that A 2 is skew symmetric, then e A 1 +A 2 ≤ e A 1 ,
where A stands for the Euclidean norm of a matrix A.

Proof. Let us first prove the following (presumably classical) identity:

(11) e t(A 1 +A 2 ) = e tA 2 - t 0 e (t-s)(A 1 +A 2 ) Ae sA 2 ds. Indeed, consider the function s → ϕ(s) defined on [0, 1] by ϕ(s) = e (t-s)(A 1 +A 2 ) e sA 2 .
Then it is easily seen that ϕ is differentiable and

ϕ ′ (s) = e (t-s)(A 1 +A 2 ) A 1 e sA 2 .
By writing

e tA 2 -e t(A 1 +A 2 ) = ϕ(t) -ϕ(0) = t 0 ϕ ′ (s) ds, relation (11)
is now easily obtained. Let us see now the implications of ( 11): according to the fact that A 2 is skew-symmetric, we have e sA 2 ≤ 1 for any s ≥ 0. Therefore,

e t(A 1 +A 2 ) ≤ 1 + A 1 t 0 e (t-s)(A 1 +A 2 ) ds.
By denoting f (t) = e t(A 1 +A 2 ) we thus get

f (t) ≤ 1 + A 1 t 0 f (s)ds.
This implies f (t) ≤ e A 1 t by a standard application of Gronwall's lemma and finishes our proof.

We are now ready to prove the main result of this section, which is an almost sure deterministic bound for the Malliavin derivative of X x t : Theorem 3.2. Under Assumption 1.1, the Malliavin derivative of the solution X x T to equation ( 6) can be bounded as follows for any T ∈ [0, 1] (in the almost sure sense):

(12) DX x T ∞ ≤ M exp (CT ) , with M = sup x∈R n sup λ ≤1 d i=1 λ i V i (x) 2 ,
and where the constant C linearly depends on V 0 . In particular, one also has

DX x T H ≤ M exp (CT ).
Proof. Let us focus on the proof of [START_REF] Driscoll | Smoothness of density for the area process of fractional Brownian motion[END_REF]. Indeed, since f H is dominated by the supremum norm when H > 1 2 , this will be sufficient in order to prove the second claim of our theorem. We now split our proof in two steps.

Step1: Matricial expression for the derivative. Let us first restate Proposition 2.5 in the following form:

DX x T is solution to D j s X x T = J 0→T (Φ * s V j )(x), 0 ≤ s ≤ T
, where Φ * s V j denotes the pullback action of the diffeomorphism Φ s = X • s : R d → R d on the vector field V j . Now, a simple application of the change of variable formula for Young type integrals yields

d(Φ * s V j )(x) = (Φ * s [V 0 , V j ])(x)ds + d i=1 (Φ * s [V i , V j ])(x)dB i s .
Moreover, recall that the Lie brackets [V i , V j ] can be decomposed, according to Assumption 1.1, into

[V 0 , V j ] = d k=1 ω k 0j V k , and [V i , V j ] = d k=1 ω k ij V k , with ω k ij = -ω j ik for i, j, k ≥ 1. Hence, d(Φ * s V j )(x) = d k=1 ω k 0j (X x s )(Φ * s V k )(x)ds + d k=1 d i=1 ω k ij (X x s )(Φ * s V k )(x)dB i s .
By denoting M s the d × d matrix with columns

M j s = D j s X x T = J 0→T (Φ * s V j )(x),
we therefore obtain the equation

dM s = M s ω 0 (X x s )ds + d i=1 ω i (X x s )dB i s , M T = V (X x T ), (13)
where V (X x T ) is the matrix with columns V j (X x T ), 1 ≤ j ≤ d and where ω i (X x s ) is the skew symmetric matrix with entries ω k ij (X x s ). Step2: Approximation procedure. In order to show that the process M is uniformly bounded, consider the dyadic approximation introduced at Proposition 2.6. By applying [START_REF] Davie | Differential equations driven by rough paths: an approach via discrete approximation[END_REF] to the couple (X, M), it is sufficient to prove our uniform bounds on D j X m T , uniformly in m. Let us thus consider M m the solution of ( 13) where B is replaced by B m , that is

dM m s = M m s ω 0 (X m s )ds + d k=1 ω k (X m s )dB m,k s , M m T = V (X m T ).
In the sequel set also

∆B k t m n-1 t m n := B k t m n -B k t m n-1 t m n -t m n-1 , for 1 ≤ n ≤ 2 m and 1 ≤ k ≤ d. Then, for s ∈ [t m n-1 , t m n ), we have dM m s = M m s ω 0 (X m s )ds + d k=1 ω k (X m s ) ∆B k t m n-1 t m n ds , Therefore, for s ∈ [t m n-1 , t m n ), we obtain M m s = M m t m n e -t m n s ω 0 (X m u )du+ d k=1 t m n s ω k (X m u )∆B k t m n-1 t m n du
Proceeding inductively, we end up with the following identity, valid for t ∈ [t m n-1 , t m n ) and n = 0, ..., 2 m :

(14) M m t = V (X m T )e - t m 2 m t m 2 m -1 ω 0 (X m s )ds+ d k=1 t m 2 m t m 2 m -1 ω i (X m s )∆B k t m 2 m -1 t m 2 m ds × • • • × e - t m n t ω 0 (X m s )ds+ d k=1 t m n t ω k (X m s )∆B k t m n-1 t m n ds .
Owing to the skew-symmetry of ω k for k ≥ 1, we can now apply Lemma 3.1 to expression [START_REF] Gubinelli | Controlling rough paths[END_REF] in order to get

M m t ≤ exp T 2 m ω 0 (X m t m 2 m -1 ) • • • exp (t m n -t) ω 0 (X m t m n-1 ) V (X m T ) ≤ M exp (CT ) . ( 15
)
This is our claimed uniform bound on M m t , from which the end of our proof is easily deduced.

Once the bound (12) on DX x T ∞ is obtained, one can also retrieve some information on the Hölder norms of DX x T improving the general estimate [START_REF] Coutin | Stochastic analysis, rough path analysis and fractional Brownian motions[END_REF]. This is the content of the following proposition: Proposition 3.3. Consider 1 2 < γ < H and set c T 0 = M exp(CT ), which is the constant appearing in relation [START_REF] Driscoll | Smoothness of density for the area process of fractional Brownian motion[END_REF]. Under Assumption 1.1, the Malliavin derivative of the solution X x T to equation ( 6) can be bounded as follows for any T ∈ [0, 1]:

(16) DX x T γ ≤ c T,V,d 1 + |x| + B 1/γ γ B (1-γ)/γ γ ,
for a strictly positive constant c T,V,d .

Proof. We have shown at Theorem 3.2 that DX x T is governed by equation [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF], and that M ∞ ≤ c T 0 . We will now separate our proof into a local and a global estimate, and notice that the constants appearing in the computations below might change from line to line.

Step1: Local estimate. Consider 0 ≤ s < t ≤ T and set ε = ts. Let u < v be two generic elements of [s, t]. Applying relation [START_REF] Cass | Integrability estimates for Gaussian rough differential equations[END_REF] to the expression of M v -M u given by equation ( 13), we obtain

|M v -M u | ≤ c T 0 c V |v -u| + d i=1 |M u ||ω i (X x u )| B γ |v -u| γ + Mω i (X x ) s,t,γ B γ |v -u| 2γ .
We thus obtain, for a constant c V depending on the vector fields V ,

M s,t,γ ≤ c T 0 c V |t -s| 1-γ + d c T 0 c V B γ + c γ c T 0 c V X x γ + c V M s,t,γ B γ |t -s| γ ≤ c T 0 c V T 1-γ + d B γ + d c T 0 c V X x γ B γ |t -s| γ + d c V B γ |t -s| γ M s,t,γ . (17)
Take now t -s = ε such that d c V B γ ε γ = 1/2, namely ε = [2d c V B γ ] -1/γ . Recall also that X x γ ≤ c(1 + B 1/γ
γ ) according to [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF]. It is then easily seen that relation (17) yields

M s,t,γ ≤ c T 0 c T,V,d 1 + |x| + B 1/γ γ := a T,V,d,B ,
for a strictly positive constant c T,V,d .

Step2: Global estimate. We consider now s, t ∈ [0, T ] such that iε ≤ s < (i + 1)ε ≤ jε ≤ t < (j + 1)ε, where ε has been defined at Step 1. Set also

t i = s, t k = kε for i + 1 ≤ k ≤ j, and t j+1 = t. Then |M t -M s | = j k=i M t k+1 -M t k ≤ a T,V,d,B j k=i (t k+1 -t k ) γ ≤ a T,V,d,B (j -i + 1) 1-γ (t -s) γ , ( 18
)
where we have used the fact that r → r λ is a concave function. Note that the indices i, j above satisfy (ji + 1) ≤ 2T /ε. Plugging this into the last series of inequalities, we end up with our claim (16).

3.1. Log-Sobolev inequality. In this section, we present some interesting functional inequalities which are usually studied in a Markov setting; namely, the logarithmic-Sobolev inequality and Poincaré inequality. As we will see, these inequalities become available in our non-Markov case when we have uniform boundedness for the Malliavin derivative of X T (see Theorem 3.2).

We start with the following version of logarithmic Sobolev inequality for the law of X T .

Theorem 3.4. Let C and M be in Theorem 3.2. We have for all f ∈ C 1 and T ∈ [0, 1],

Ef (X T ) 2 ln f (X T ) 2 -Ef (X T ) 2 ln Ef (X T ) 2 ≤ 2M 2 e 2CT T 2H E|∇f (X T )| 2 .
provided the right hand side in the above is finite.

Proof. The proof is standard by applying Clark-Ocone formula (see e.g. [START_REF] Capitaine | Martingale representation and logarithmic Sobolev inequality[END_REF]). First recall the representation of fractional Brownian motion

B t = t 0 K H (t, s)dW s .
Here W is a d-dimensional Wiener process. Denote by D W the Malliavin derivative with respect to the Wiener process W . We have

K * H D = D W , (19)
where K * H is the isometry from H, the reproducing kernel space of B, to L 2 . By Clark-Ocone formula we have

f (X T ) -Ef (X T ) = T 0 E D W s f (X T )|F s dW s = T 0 E K * H (Df (X T )) s |F s dW s .
Hence, if we denote

M s = E(f (X T )|F s ), 0 ≤ s ≤ T , we have dM s = E K * H (Df (X T )) s |F s dW s .
For simplicity we may assume that f ≥ ε for some ε > 0, which can be removed afterwards by letting ε tend to 0. Applying Itô's formula to M s ln M s , we get

Ef (X T ) ln(f (X T )) -Ef (X T ) ln (Ef (X T )) = E(M T ln M T ) -E(M 0 ln M 0 ) = 1 2 E T 0 1 M s E K * H (Df (X T )) s |F s 2 ds. (20)
Replace now f by f 2 in the above. By the Cauchy-Schwarz inequality,

E K * H (Df (X T ) 2 ) s |F s 2 = 4 E f (X T ) ∇f (X T ), K * H (DX T ) s |F s 2 ≤ 4E f (X T ) 2 |F s E ∇f (X T ), K * H (DX T ) s 2 |F s .
Substituting the above to ( 20), together with Theorem 3.2, we obtain the desired result.

As a corollary of the logarithmic Sobolev inequality obtained above, we have the following Poincaré inequality (see e.g. [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]Theorem 8.6.8]).

Theorem 3.5. Let C and M be in Theorem 3.2. We have for all

f ∈ C 1 , Ef (X T ) 2 -Ef (X T ) 2 ≤ M 2 e 2CT T 2H E|∇f (X T )| 2 ,
provided the right hand side in the above is finite.

Remark 3.6. In the above, assume further that the vector fields V 1 , ..., V d form an uniform elliptic system, we obtain the following natural expression of logarithmic-Sobolev inequality and Poincaré inequality when working on a Riemannian manifold

Ef (X T ) 2 ln f (X T ) 2 -Ef (X T ) 2 ln Ef (X T ) 2 ≤ 2aM 2 e 2CT T 2H d i=1 E|V i f | 2 ,
and

Ef (X T ) 2 -Ef (X T ) 2 ≤ aM 2 e 2CT T 2H d i=1 E|V i f | 2 . 3.2.
E(e θF ) ≤ e E(F )θ+ 1 2 C 2 θ 2
and therefore for every x ≥ 0,

P (F -E(F ) ≥ x) ≤ e -x 2 2C 2 . (21)
As a corollary of this lemma, we deduce Proposition 3.8. Assume that the Assumption 1.1 is satisfied, then there exist C and M such that for every T ≥ 0 and λ ≥ 0,

P sup 0≤t≤T |X x t | -E sup 0≤t≤T |X x t | ≥ λ ≤ exp - λ 2 2M 2 e 2CT T 2H . Proof. Let F = sup 0≤t≤T |X x t |.
By Theorem 3.2, it is not hard to see that (see e.g. [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF])

DF H ≤ Me CT T H ,
where M and C are the same as in Theorem 3.2. Now an easy application of Lemma 3.7 completes our proof.

Remark 3.9. Notice that this relation can only be obtained for H > 1 2 . Indeed, denote by C 0 the space of continuous functions endowed with the supremum norm. Then the norm involved in [START_REF] Üstünel | Analysis on Wiener Space and Applications[END_REF]Theorem 9

.1.1] is DX t L ∞ (H) . This norm is dominated by DX t L ∞ (C 0 ) only if H > 1 2 .
3.3. Gaussian upper bound. One natural way to estimate the density of X t is to apply the results in [26, Chapter 2]. More precisely, we first have the following integration by parts formula for non-degenerate random vectors.

Proposition 3.10. Let F = (F 1 , ..., F d ) be a non-degenerate random vector and γ F the Malliavin matrix of F . Let G ∈ D ∞ and ϕ be a function in the space C ∞ p (R d ). Then for any multi-index α ∈ {1, 2, ..., d} k , k ≥ 1, there exists an element

H α ∈ D ∞ such that E[∂ α ϕ(F )G] = E[ϕ(F )H α ].
Moreover, the elements H α are recursively given by

H (i) = d j=1 δ G(γ -1 ) ij DF j H α = H α k (H (α 1 ,...,α k-1 ) ),
and for 1 ≤ p < q < ∞ we have

H α L p ≤ C p,q γ -1 F DF k k,2 k-1 r G k,q
, where 1 p = 1 q + 1 r . Remark 3.11. By the estimates for H α above, one can conclude that there exist constants β, γ > 1 and integers m, n such that

H α L p ≤ C p,q det γ -1 F m L β DF n k,γ G k,q . (22)
Remark 3.12. In what follows, we use H α (F, G) to emphasize its dependence on F and G.

As a consequence of the above proposition, one has Proposition 3.13. Let F = (F 1 , ..., F d ) be a non-degenerate random vector. Then the density p F (x) of F belongs to the Schwartz space, and

p F (x) = E[I {F >x} H (1,2,...,d) (F, 1)].
Now we state and prove a global Gaussian upper bound for the density function of X x t . Theorem 3.14. Denote by p X (t, y) the density function of X x t . There exist positive constants c

(1)

t , c (2)
t,x , c 3 , and c 4 such that for all t ∈ [0, ∞),

p X (t, y) ≤ c (1) t exp    - |y| -c (2) t,x 2 c 3 e c 4 t t 2H    . ( 23
)
Here c

(1) t is of the form

c (1) t = O 1 t α as t ↓ 0,
for some positive number α; and c

(2)

t,x converges to a constant as t ↓ 0.

Proof. Fix β ∈ ( 1 2 , 1). By [START_REF] Lyons | System control and rough paths[END_REF] and Proposition 3.13, we have p X (t, y) ≤ C(P{X t > y})

1 q det γ -1 Xt m L α DX t n k,γ , (24)
for some constants α, γ > 1 and integers m, n. Without lose of generality, we may assume y i ≥ 0 for 1 ≤ i ≤ d. Since otherwise, for example y j < 0, we can consider the alternative expression for the density p X (t, y) = E Π i =j I {y i <X i t } I {y j >X j t } H (1,2,...,d) (X t , 1) , and deduce similar estimate. In the following, we provide estimate for

DX t n k,γ , det γ -1 Xt m L α
and P{X t > y} respectively. By Proposition 2.2 there exist constants C > 0, depending on V, x and k, such that

sup 0≤t≤T |X i t | ≤ |x i | + CT B 1/β 0,T,β , (25)
sup 0≤t≤T γ -1 Xt ≤ C T 2Hd 1 + e CT B 1/β 0,T,β , (26)
sup 0≤t,r i ≤T |D j k r k ...D j 1 r 1 X i t | ≤ Ce CT B 1/β 0,T,β . (27)
On the other hand we have, for some constant M β (cf. [START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF]):

P { B 0,T,β > r} ≤ M β e - r 2 2T 2(H-β) . (28)
Hence

det γ -1 Xt m L α < ∞ and DX t n k,γ < ∞.
Moreover, by [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF], [START_REF] Nualart | Differential equations driven by fractional Brownian motion[END_REF] and the tail estimate [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF], there exists constant C > 0 independent of t such that

det γ -1 Xt m L α ≤ C t 2Hmd 1 + t m α F (t, 1/β -1, 1/β, C) m α
, and

DX t n k,γ ≤ C 1 + t nH(k+1))+ n γ F (t, 1/β -1, 1/β, C) n γ .
In the above

F (t, a, b, M) = ∞ 0 r a e - r 2
P {X t > y} ≤ P ξ t > |y| √ d ≤ exp    - |y| √ d -Eξ t 2 2Me Ct t 2H    ≤ exp    - |y| -c (2) t,x 2 2 √ dMe Ct t 2H    . (30)
Now ( 24), ( 29) and [START_REF] Tejedor | Quantitative Analysis of Single Particle Trajectories: Mean Maximal Excursion Method[END_REF] give us the desired upper bound for the density p X (t, y).

Remark 3.15. There is also an upper bound for the constant c

(1) t in the above theorem as t ↑ ∞. Indeed, by some elementary computation one can show that c (1) t ≤ t α e t β as t ↑ ∞, for some α, β > 0.

Extension to the irregular case

From now on, our purpose is to extend the previous to the case of a fBm with Hurst index 1 3 < H < 1 2 . This requires the introduction of some rough paths tools, which is the aim of the current section. We shall use in fact the language of algebraic integration theory, which is a variant of the rough paths theory introduced in [START_REF] Gubinelli | Controlling rough paths[END_REF] (we also refer to [START_REF] Gubinelli | Rough evolution equations[END_REF] for a detailed introduction of the topic).

4.1.

Increments. The extended pathwise integration we will deal with is based on the notion of increments, together with an elementary operator δ acting on them. The algebraic structure they generate is described in [START_REF] Gubinelli | Controlling rough paths[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF], but here we present directly the definitions of interest for us, for sake of conciseness. First of all, for an arbitrary real number T > 0, a vector space V and an integer k ≥ 1 we denote by C k (V ) the set of functions g : [0, T ] k → V such that g t 1 •••t k = 0 whenever t i = t i+1 for some i ≤ k -1. Such a function will be called a (k -1)-increment, and we set C * (V ) = ∪ k≥1 C k (V ). We can now define the announced elementary operator δ on C k (V ):

(31) δ : C k (V ) → C k+1 (V ), (δg) t 1 •••t k+1 = k+1 i=1 (-1) k-i g t 1 ••• ti •••t k+1 ,
where ti means that this particular argument is omitted. A fundamental property of δ, which is easily verified, is that δδ = 0, where δδ is considered as an operator from

C k (V ) to C k+2 (V ). We denote ZC k (V ) = C k (V ) ∩ Kerδ and BC k (V ) = C k (V ) ∩ Imδ.
Some simple examples of actions of δ, which will be the ones we will really use throughout the paper, are obtained by letting g ∈ C 1 and h ∈ C 2 . Then, for any s, u, t ∈ [0, T ], we have [START_REF] Zähle | Integration with respect to fractal functions and stochastic calculus I[END_REF] (δg) st = g tg s , and (δh

) sut = h st -h su -h ut .
Furthermore, it is easily checked that ZC k (V ) = BC k (V ) for any k ≥ 1. In particular, the following basic property holds:

Lemma 4.1. Let k ≥ 1 and h ∈ ZC k+1 (V ). Then there exists a (non unique)

f ∈ C k (V ) such that h = δf .
Proof. This elementary proof is included in [START_REF] Gubinelli | Controlling rough paths[END_REF], and will be omitted here. However, let us mention that f t 1 ...t k = (-1) k+1 h 0t 1 ...t k is a possible choice.

Observe that Lemma 4.1 implies that all the elements h ∈ C 2 (V ) such that δh = 0 can be written as h = δf for some (non unique) f ∈ C 1 (V ). Thus we get a heuristic interpretation of δ| C 2 (V ) : it measures how much a given 1-increment is far from being an exact increment of a function, i.e., a finite difference.

Notice that our future discussions will mainly rely on k-increments with k ≤ 2, for which we will make some analytical assumptions. Namely, we measure the size of these increments by Hölder norms defined in the following way:

for f ∈ C 2 (V ) let (33) f µ = sup s,t∈[0,T ] |f st | |t -s| µ , and C µ 2 (V ) = {f ∈ C 2 (V ); f µ < ∞} .
Obviously, the usual Hölder spaces C µ 1 (V ) will be determined in the following way: for a continuous function g ∈ C 1 (V ), we simply set (34) g µ = δg µ , and we will say that g ∈ C µ 1 (V ) iff g µ is finite. Notice that • µ is only a semi-norm on C 1 (V ), but we will generally work on spaces of the type

(35) C µ 1,a (V ) = {g : [0, T ] → V ; g 0 = a, g µ < ∞}
, for a given a ∈ V , on which g µ defines a distance in the usual way. For h ∈ C 3 (V ) set in the same way

h γ,ρ = sup s,u,t∈[0,T ] |h sut | |u -s| γ |t -u| ρ (36) h µ = inf i h i ρ i ,µ-ρ i ; h = i h i , 0 < ρ i < µ ,
where the last infimum is taken over all sequences {h i ∈ C 3 (V )} such that h = i h i and for all choices of the numbers ρ i ∈ (0, µ). Then • µ is easily seen to be a norm on C 3 (V ), and we set

C µ 3 (V ) := {h ∈ C 3 (V ); h µ < ∞} . Eventually, let C 1+ 3 (V ) = ∪ µ>1 C µ 3 (V )
, and notice that the same kind of norms can be considered on the spaces ZC 3 (V ), leading to the definition of some spaces ZC µ 3 (V ) and ZC 1+ 3 (V ). With these notations in mind the following proposition is a basic result, which belongs to the core of our approach to pathwise integration. Its proof may be found in a simple form in [START_REF] Gubinelli | Rough evolution equations[END_REF].

Proposition 4.2 (The Λ-map).

There exists a unique linear map Λ :

ZC 1+ 3 (V ) → C 1+ 2 (V ) such that δΛ = Id ZC 1+ 3 (V )
and Λδ = Id C 1+ 2 (V ) . In other words, for any h ∈ C 1+ 3 (V ) such that δh = 0 there exists a unique g = Λ(h) ∈ C 1+ 2 (V ) such that δg = h. Furthermore, for any µ > 1, the map Λ is continuous from ZC µ 3 (V ) to C µ 2 (V ) and we have

(37) Λh µ ≤ 1 2 µ -2 h µ , h ∈ ZC µ 3 (V ).
Let us mention at this point a first link between the structures we have introduced so far and the problem of integration of irregular functions.

Corollary 4.3. For any 1-increment g ∈ C 2 (V ) such that δg ∈ C 1+ 3 , set δf = (Id -Λδ)g. Then (δf ) st = lim |Πst|→0 n-1 i=0 g t i t i+1 ,
where the limit is over any partition Π st = {t 0 = s, . . . , t n = t} of [s, t], whose mesh tends to zero. Thus, the 1-increment δf is the indefinite integral of the 1-increment g. (gh) t 1 ,...,t m+n+1 = g t 1 ,...,tn h tn,...,t m+n-1 , t 1 , . . . , t m+n-1 ∈ [0, T ].

In this context, we have the following useful properties. The iterated integrals of smooth functions on [0, T ] are obviously particular cases of elements of C, which will be of interest for us. Let us recall some basic rules for these objects: consider f ∈ C ∞ 1 and g ∈ C ∞ 1 , where C ∞ 1 denotes the set of smooth functions on [0, T ]. Then the integral f dg, which will be denoted indistinctly by f dg or J (f dg), can be considered as an element of

C ∞ 2 . Namely, let S 2,T denote the simplex {(s, t) ∈ [0, T ] 2 : 0 ≤ s < t ≤ T }, for (s, t) ∈ S 2,T we set J st (f dg) = f dg st = t s f u dg u .
The multiple integrals can also be defined in the following way: given a smooth element h ∈ C ∞ 2 and (s, t) ∈ S 2,T , we set

J st (h dg) ≡ hdg st = t s h su dg u .
In particular, for

f 1 ∈ C ∞ 1 , f 2 ∈ C ∞ 1 and f 3 ∈ C ∞ 1 the double integral J st (f 3 df 2 df 1 ) is defined as J st (f 3 df 2 df 1 ) = f 3 df 2 df 1 st = t s J su f 3 df 2 df 1 u .
Now suppose that the nth order iterated integral of

f n+1 df n • • • df 2 , which is denoted by J (f n+1 df n • • • df 2 ), has been defined for f j ∈ C ∞ 1 . Then, if f 1 ∈ C ∞ 1 , we set (39) J st (f n+1 df n • • • df 2 df 1 ) = t s J su f n+1 df n • • • df 2 df 1 u ,
which recursively defines the iterated integrals of smooth functions. Observe that an nth order integral J (df n • • • df 2 df 1 ) can be defined along the same lines, starting with

J (df ) = δf, J st (df 2 df 1 ) = t s J su (df 2 ) df 1 u = t s δf 2 su df 1 u ,
and so on.

The following relations between multiple integrals and the operator δ will also be useful. The reader is sent to [START_REF] Gubinelli | Rough evolution equations[END_REF] for its elementary proof.

Proposition 4.5. Let f ∈ C ∞ 1 and g ∈ C ∞ 1 . Then it holds that δg = J (dg), δ (J (f dg)) = 0, δ (J (df dg)) = (δf )(δg) = J (df )J (dg),
and

δ J (df n • • • df 1 ) = n-1 i=1 J df n • • • df i+1 J df i • • • df 1 .

4.3.

Weakly controlled processes. The rough path theory allows to define and solve differential equations driven by a generic Hölder continuous path B provided enough iterated integrals of this function can be defined. We shall briefly recall how this is done, in the simplest nontrivial case of a Hölder continuity exponent 1 3 < γ < 1 2 . Observe that we keep here the notation B for the underlying path as in the fBm case for notational sake, while the theory can be applied to much more general situations.

The basic assumption one has to add in order to define our objects when γ > 1 3 is the existence of an (abstract) double iterated integral of B with respect to itself, which can be defined as follows:

Hypothesis 4.6. The path B is R d -valued γ-Hölder with γ > 1 3 and admits a Lévy area, that is a process

B 2 ∈ C 2γ 2 (R d,d ) satisfying δB 2 = B 1 ⊗ B 1 , i. e. B 2,ij sut = [B 1,i ] su [B 1,j
] ut , for s, u, t ∈ S 3,T and i, j ∈ {1, . . . , d}. We also assume that B 2 can be obtained in the following way: consider the sequence of linear dyadic approximation B m of B defined like in Proposition 2.6. For 0 ≤ s < t ≤ 1 and i 1 , i 2 ∈ {1, . . . , d}, set B 2,m,i 1 i 2 st = s<u 1 <u 2 <t dB m,i 1 u 1 dB m,i 2 u 2 , which is defined as a Riemann-Stieljes integral. Then we suppose that B 2,m converges to B 2 in the norm of C 2γ 2 . It should be noticed at this point that fBm satisfies the above assumption, as shown in [START_REF] Coutin | Stochastic analysis, rough path analysis and fractional Brownian motions[END_REF][START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF][START_REF] Neuenkirch | Discretizing the Lévy area[END_REF]:

Let us now turn to the integration of weakly controlled paths, which is summarized in the following theorem. Theorem 4.10. For a given 1 3 < γ ≤ 1 2 , let B be a process satisfying Hypothesis 4.6.

Furthermore, let m ∈ Q γ,b (R d ) with m 0 = b ∈ R d and decomposition (43) δm i = µ ii 1 B 1,i 1 + r i , where µ ∈ C γ 1 (R d,d ), r ∈ C 2γ 2 (R d ). Define z by z 0 = a ∈ R and (44) δz = m i B 1,i + µ ii 1 B 2,i 1 i -Λ r i B 1,i + δµ ii 1 B 2,i 1 i .
Finally, set

J st (m dB) = t s m u , dB u R d δz st .
Then:

(1) z is well-defined as an element of Q γ,a (R), and coincides with the Riemann-Stieltjes integral of m with respect to B whenever these two functions are smooth.

(2) The semi-norm of z in Q γ,a (R) can be estimated as

(45) N [z; Q γ,a (R)] ≤ c B 1 + N [m; Q γ,b (R d )] ,
for a positive constant c B which can be bounded as follows:

c B ≤ c N [B 1 ; C γ 2 (R d )] + N [B 2 ; C 2γ 2 (R d 2 )] ,
for a universal constant c.

(3) It holds

(46) J st (m dx) = lim |Πst|→0 n-1 q=0 m i tq B 1 tq,t q+1 (i) + µ ii 1 tq B 2,i 1 i tq,t q+1 ,
for any 0 ≤ s < t ≤ T , where the limit is taken over all partitions Π st = {s = t 0 , . . . , t n = t} of [s, t], as the mesh of the partition goes to zero.

4.4.

Rough differential equations. Recall that we are concerned with equations of the form (2). In our algebraic setting, we will rephrase this as follows: we will say that X x is a solution to (2) if X x 0 = x, X x ∈ Q γ,x (R d ) and for any 0 ≤ s ≤ t ≤ 1 we have (47)

δX x st = t s V 0 (X x u )du + J st V i (X x ) dB i ,
where the integral J (V i (X x ) dB i ) has to be understood in the sense of Proposition 4.10.

The following existence and uniqueness result is then classical in rough paths theory.

Theorem 4.11. Let B be a process satisfying Hypothesis 4.6 and V 0 , . . . , V n a collection of vector fields which fulfill Hypothesis 1.1. Then (i) Equation (47) admits a unique solution X x in Q γ,x (R d ).

(ii) Consider the linear approximation B m of B introduced in Hypothesis 4.6, and set Xm for the solution of the (ordinary) differential equation (47) driven by the piecewise smooth function B m . Then Xm converges to X x in C γ 1 norm.

(iii) Consider the sequence of dyadic partitions of Proposition 2.6. Define a process X n on the points t n k by X n 0 = x and (48)

δX n t k t k+1 = V 0 (X n t k ) 2 n + V i (X n t k )B 1,i t k t k+1 + V i 1 V i 2 (X n t k )B 2,i 1 i 2 t k t k+1 .
Complete the definition of X n on [0, 1] by linear interpolation. Then as n → ∞, the process X n converges to X x in C γ 1 norm. Proof. We refer to [START_REF] Lyons | System control and rough paths[END_REF] for the proof of the existence and uniqueness part, as well as to [START_REF] Gubinelli | Controlling rough paths[END_REF] for the same result in the algebraic integration setting. Part (ii) of our proposition stems from the continuity of the Itô map, which is also stated and proved in both [START_REF] Lyons | System control and rough paths[END_REF][START_REF] Gubinelli | Controlling rough paths[END_REF]. The approximation statement (iii) has first been stated by Davie [START_REF] Davie | Differential equations driven by rough paths: an approach via discrete approximation[END_REF] and then been generalized in [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF].

In the sequel we will simply try to relate the decomposition of the solution to Equation (47) as a controlled process and the numerical scheme given by (48), a relation which turns out to be useful in the sequel. For this, we shall denote by r any increment in C 2γ 2 and by r ♯ any increment in C 1 + 2 in the computations below, independently of their values. Observe then that, according to the right hand side of (47), the decomposition of X x as a controlled process is given by

δX x,j = ζ jj 1 B 1,j 1 + r, with ζ jj 1 = V j j 1 (X x
). Hence, owing to Proposition 4.9, one has

(49) δV i (X x ) = ζij 1 B 1,j 1 + r i , with ζij 1 = ∂ j V i (X x )ζ jj 1 = V j 1 V i (X x ).
Now, if one desires an expansion of δX x up to increments of regularity 1 + , consider again the right hand side of (47), and compute it by a direct application of Theorem 4.10. This yields

δX x st = V 0 (X x s )(t -s) + V i (X x s )B 1,i st + ζij 1 B 2,j 1 i st + r ♯ = V 0 (X x s )(t -s) + V i (X x s )B 1,i st + V j 1 V i (X x s )B 2,j 1 i st + r ♯ .
Thanks to identity (46), it is now easily seen that ( 48) is a natural candidate for our numerical scheme.

We show now how to get efficient bounds on the solution to equation (47) out of its numerical scheme. This step is understood as a warmup for the same kind of estimates concerning the Malliavin derivative of the solution.

Proposition 4.12. Under the assumptions of Theorem 4.11, the solution to equation (47) satisfies:

(50) X x γ ≤ c V 1 + B 1 1/γ γ + B 2 1/2γ 2γ ,
where c V is a constant which only depends on the vector fields V 0 , . . . , V n .

Remark 4.13. This proposition is shown in [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF] by identifying the signature of B with the signature of a certain finite variation process, plus some easy estimates for ordinary differential equations. We have included here a direct elementary proof of (50) because we haven't been able to find them in the literature under this form, and mostly because the generalization of our estimates to linear cases will be obvious from the considerations below.

Proof of Proposition 4.12. Theorem 4.11 part (iii) asserts the convergence of the approximation X n towards X x as n → ∞. It is thus sufficient to prove relation (50) for X n , uniformly in n. One can also be easily reduced to prove

sup 0≤i<j≤2 n |δX n t n i t n j | |t n j -t n i | γ ≤ c V 1 + B 1 1/γ γ + B 2 1/2γ 2γ ,
which is what we shall proceed to do. We now divide our proof in several steps.

Step 1: Expression for δX n . Set (51)

q n st = V 0 (X n s ) (t -s) + V i (X n s )B 1,i st + V i 1 V i 2 (X n s )B 2,i 1 i 2 st
, so that δX n t n i t n j = j-1 l=1 q n t n l t n l+1 . For i < j, we also construct a dyadic partition {τ k l ; 0 ≤ k ≤ K, 0 ≤ l ≤ 2 k } of the set {t n i , . . . , t n j } inductively in the following way: set τ 0 0 = t i and τ 0 1 = t j . Now, if the τ k l 's are known, we set

τ k+1 2l = τ k l . Furthermore, if τ k+1 2l = t m and τ k+1 2l+2 = t m ′ , then take τ k+1 2l+1 = t m * , with m * = ⌊(m + m ′ )/2⌋
. This procedure is then non trivial as long as ji ≥ 2 k , which means that we stop at K = ⌈log 2 (ji)⌉. Here is then a simple example of construction: consider i = 1, j = 4. Then we have:

τ 0 0 = 1, τ 0 1 = 4; τ 1 0 = 1, τ 1 1 = 2, τ 1 2 = 4; τ 2 0 = 1, τ 2 1 = 1, τ 2 2 = 2, τ 2 3 = 3, τ 2 4 = 4. With these notations in hand, it is easily checked that the relation δX n t n i t n j = j-1 l=1 q n t n l t n l+1
can also be written as δX n

t n i t n j = 2 K -1 k=0 q n τ K k τ K k+1 . Furthermore, q n τ k 2l τ k 2l+1 + q n τ k 2l+1 τ k 2l+2 = q n τ k-1 l τ k-1 l+1 -(δq n ) τ k 2l τ k 2l+1 τ k
2l+2 , and summing this equality for k = K and l = 0, . . . , 2 K -1 we get

2 K -1 l=0 q n τ K l τ K l+1 = 2 K -1 l=0 q n τ K-1 l τ K-1 l+1 - 2 K -1 l=0 (δq n ) τ K-1 2l τ K-1 2l+1 τ K-1 2l+2 .
Iterating, we obtain (52)

δX n t n i t n j = q n t n i t n j - K k=1 2 k -1 l=0 δq n τ k 2l τ k 2l+1 τ k 2l+2 .
Step 2: Expression for δq n . Denote by I the identity function on R, so that δI st = ts.

Start then from expression (52) and use Proposition 4.4 in order to get, for any s, u, t in the dyadic partition,

δq n sut = δV 0 (X n ) su (t -u) + δV i (X n ) su B 1,i ut + δ [V i 1 V i 2 (X n )] su B 2,i 1 i 2 ut -V i 1 V i 2 (X n s )δB 2,i 1 i 2 sut , or otherwise stated thanks to convention (38), (53) δq n = δV 0 (X n ) δI + δV i (X n )B 1,i + δ [V i 1 V i 2 (X n )] B 2,i 1 i 2 -V i 1 V i 2 (X n )δB 2,i 1 i 2 .
Observe now that one can prove, as for (49), that

δV i (X n ) = V i 1 V i (X n )B 1,i 1 + r i,n , with |r i,n st | ≤ c V |δX n st | 2 .
Thus, according to the fact that δB 2,i

1 i 2 = B 1,i 1 B 1,i 2 , one can recast (53) into (54) δq n = δV 0 (X n ) δI + r i,n B 1,i + δ [V i 1 V i 2 (X n )] B 2,i 1 i 2 := ρ 1,n + ρ 2,n + ρ 3,n .
Notice that for j = 1, 2, 3, the increment ρ j,n lies into C 3γ 3 . Furthermore, it is readily checked that (55)

|ρ 1,n sut | ≤ c V |δX n su | |t -u| 1+γ , |ρ 2,n sut | ≤ c V |δX n su | 2 B 1 γ |t -u| γ , and |ρ 3,n sut | ≤ c V |δX n su | B 2 2γ |t -u| 2γ .
Step 3: An induction procedure. Let us consider an integer ℓ ≥ 1 and the quantity:

N n ℓ = sup 0≤i<j≤ℓ |δX n t n i t n j | |t n j -t n i | γ .
We localize now our study to an interval of the form [a, a + η] with an arbitrary positive number a, and η small enough. We will prove that if η is of order

(1 + B 1 1/2γ γ + B 2 1/2γ 2γ ) -1 , then N ℓ ≤ c V (1 + B 1 γ + B 2 1/2 2γ
) by induction. The case ℓ = 1 being trivial, let us assume that the hypothesis is true up to a given ℓ ≥ 1. Take now 1 ≤ i ≤ ℓ and j = ℓ + 1. According to (52), write

δX n t n i t n j -q n t n i t n j = - K k=1 2 k -1 l=0 δq n τ k 2l τ k 2l+1 τ k 2l+2 .
In the right hand side of this decomposition, all the points τ k 2l , τ k 2l+1 are of the form t n p with p ≤ ℓ. Thus (54), our bounds (55) on ρ j,n and the induction hypothesis entail

δX n t n i t n j -q n t n i t n j ≤ c V N ℓ + N 2 ℓ B 1 γ + N ℓ B 2 2γ |t n j -t n i | 3γ . Furthermore, it is obvious from (51) that |q n t n i t n j | |t n j -t n i | γ ≤ c V 1 + B 1 γ + B 2 2γ |t n j -t n i | γ .
Hence, putting together the last two inequalities, taking into account that we work on an interval of size η and that we have chosen j = ℓ + 1, we end up with the following induction relation: N ℓ+1 ≤ F η (N ℓ ), where the function F η : R + → R + is defined by

F η (ξ) = ξ ∨ c V 1 + B 1 γ + B 2 2γ η γ + 1 + B 2 2γ ξ + B 1 γ ξ 2 η 2γ . By separating the cases η γ ξ ≤ 1 and η γ ξ > 1, one can also prove that F η (ξ) ≤ ξ ∨ ϕ η (ξ), with ϕ η (ξ) = c V B 2 2γ η 3γ + B 1 γ η 2γ ξ 2 + 1 + B 1 γ + B 2 2γ η γ := a ξ 2 + c.
In order to obtain a bound of the form N ℓ ≤ M which remains valid for all ℓ, it is now sufficient to have the interval [0, M] left invariant by ϕ η .

We let the reader check the following elementary fact: whenever 4ac is of order 1, the interval [0, M] is left invariant by the application ξ → a ξ 2 +c, with M of order c. Applying this to ϕ η , we obtain that (56)

η ≍ 1 + B 1 γ + B 2 1/2 2γ -1/γ =⇒ N ℓ B 1 γ + B 2 1/2 2γ .
Step 4: Conclusion. We have thus obtained that on any interval of length η given by (56), we have X n γ B 1 γ + B 2 1/2 2γ . Our claim (50) is now easily deduced by dividing an arbitrary interval [s, t] into subintervals of length η as in [START_REF] Hsu | Stochastic Analysis on Manifolds[END_REF]. 4.5. Estimates for the Malliavin derivative. We are now interested in extending Proposition 3.3 beyond the Young setting. Recall thus that we are concerned with equation [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF], which can be written in our algebraic integration setting as (57)

δM st = t s M u ω 0 (X x u ) du + J st M ω i (X x ) dB i , 0 ≤ s < t ≤ T, with final condition M T = V (X x T ). Then we have the following equivalent of Theorem 4.11: Proposition 4.14. Theorem 4.11 holds true for equation (57) under Hypothesis 1.1 and 4.6. The discretization scheme for M can be written as: (58)

δM n t k t k+1 = M n t k ω 0 (X n t k ) 2 n δI t k t k+1 + ω i (X n t k )B 1,i t k t k+1 + ω i 1 ω i 2 (X n t k ) + V i 1 ω i 2 (X n t k ) B 2,i 2 i 1 t k t k+1 .
Proof. As in the case of X x , we only justify expression (58). Note that, according to equation (57), we have δM st = ζ l s B 1,l st + r, with ζ l s = M s ω l (X x s ). Hence, invoking Proposition 4.4 and relation (49), we get

δ[M ω i 2 ] = M [ω i 1 ω i 2 (X x ) + V i 1 ω i 2 (X x )] B 1,i 1 + r.
Expanding now the right hand side of equation (57) with the help of Theorem 4.10, one easily gets δM = M ω 0 (X x ) δI + ω i (X x )B 1,i + (ω i 1 ω i 2 (X x ) + V i 1 ω i 2 (X x )) B 2,i 2 i 1 + r ♯ , which gives the desired justification of our scheme (58).

We are now ready to state and prove our bounds for the process M: Proposition 4.15. Let M be the unique solution to (57) under Hypothesis 1.1 and 4.6. Then (i) The bound [START_REF] Driscoll | Smoothness of density for the area process of fractional Brownian motion[END_REF] on M ∞ still holds true in our irregular context. (ii) M γ satisfies inequality (50).

Proof. By the continuity property of Itô's map for M, the approximation procedure of M by M m described in the proof of Theorem 3.2 is still valid. The desired bound for M ∞ is thus obtained just like in [START_REF] Gubinelli | Rough evolution equations[END_REF]. Once a bound on M ∞ is available, M γ can be bounded by considering the Davie type scheme (58), along the same lines as for X x . Details are left to the reader for sake of conciseness. 4.6. Density upper bound. We finish this section by extending the density estimate and functional inequalities obtained in the smooth case (when H > 1 2 ) to the irregular case (when 1 3 < H < 1 2 ). We first show that in the rough case, we can obtain the smoothness of density of X t under Hypothesis 1.2. Indeed, for this purpose, we only need to show the following integrability of Malliavin matrix. Theorem 4.16. Fix H ∈ ( 1 3 , 1 2 ). Assume Hypothesis 1.2. Let γ X be the Malliavin matrix of X x 1 , we have | det γ X | -1 ∈ L ∞ (P). Therefore X x 1 admits a smooth density. Proof. It suffices to show that there exists C > 0 such that for all v ∈ R d ,

v T γ X v ≥ C|v| 2 .
Recall that γ ij X = D . X i , D . X j H and that the d × d matrix M is such that its j-th column M j is given by In the above, we used that H ⊂ L 2 ([0, 1]) for the second step, and M m is that described in Theorem 3.2.

M j t = D j t X 1 . We have v T γ X v = d j=1 v T M j
On the other hand one can show, according to the description of M m in Theorem 3.2

v T M m s (M m s ) T v ≥ C|v| 2 uniformly for some constant C > 0, if we assume Hypothesis 1.2. The proof is therefore completed.

Another consequence of the above boundedness of the Malliavin Matrix γ X is the following.

Theorem 4.17. Fix H ∈ (

4. 2 .

 2 Computations in C * . Let us specialize now to the case V = R, and just write C γ k for C γ k (R). Then (C * , δ) can be endowed with the following product: for g ∈ C n and h ∈ C m let gh be the element of C n+m-1 defined by (38)

Proposition 4 . 4 . 1 ) 2 ) 3 )

 44123 The following differentiation rules hold true: (Let g ∈ C 1 and h ∈ C 1 . Then gh ∈ C 1 and δ(gh) = δg h + g δh. (Let g ∈ C 1 and h ∈ C 2 . Then gh ∈ C 2 and δ(gh) = -δg h + g δh. (Let g ∈ C 2 and h ∈ C 1 . Then gh ∈ C 2 and δ(gh) = δg h + g δh.

M j s ) 2 ds = C 1 0v T M s M T s vds ≥ C min m 1 0v

 11 T M m s (M m s ) T vds.

  :

	Proposition 2.2. (Hu-Nualart) Consider equation (6) driven by a fBm B with Hurst
	parameter H > 1 2 . Let us call X x its unique β-Hölder continuous solution, for any β < H. Then
	(1) When vector fields V are C ∞ -bounded, we have
	sup
	t∈[0,T ]

  Concentration inequality. It is classical that the boundedness of the Malliavin derivative in H implies the Gaussian concentration inequality, more precisely (see[START_REF] Üstünel | Analysis on Wiener Space and Applications[END_REF] Theorem 9.1.1] or[START_REF] Ledoux | The Concentration of Measure Phenomenon[END_REF]):

	Lemma

3.7. Let F ∈ D 1,2 (H) such that almost surely DF H ≤ C, where C is a non random constant. Then, for every θ ≥ 0,

  Me Ct , since the Malliavin derivative |DX i t | ≤ Me Ct for each i = 1, 2, ...d. Hence ξ t has the same concentration property as specified in Proposition 3.8. By (25) and the concentration property for ζ t , we conclude

	First note that
			|Dξ t | ≤
		(1) t = det γ -1 Xt	m L α DX t	n k,γ .
	Elementary computation shows, for some constant α > 0
	(29)	c	(1) t ∼ O	1 t α	as t ↓ 0.
	Finally we estimate P{X t > y}. Define
		ξ t = max 1≤i≤d	|X 1 t |, |X 2 t |, ..., |X d t | .

2t 2(H-β) e M tr b dr, for any a, M > 0 and 2 > b > 0. Now set c

  1 3 , 1 2 ). Assume Hypothesis 1.2. Let p X (t, y) denote the density function of X x t . There exist constants c Proof. The proof is similar to that of Theorem 3.14. We apply Theorem 4.15 and Theorem 4.16 to obtain bounds for DX t n k,γ and det γ -1 Xt m L α . The tail estimate for P{X t > y} is derived by Proposition 2.4. The rest of the proof is then clear.

		(1) t	and c	(2) t	such that
	p X (t, y) ≤ c	(1) t exp -c	(2) t |y| δ	y ∈ R d ,
	for any δ < H.

Proposition 4.7. Let B be a d-dimensional fBm with H > 1 3 as defined in Section 2. For the linear dyadic approximation B m of B defined at Proposition 2.6, the increment B 2,m almost surely converges to an element B 2 satisfying Hypothesis 4.6. The convergence holds in any C 2γ 2 norm for γ < H.

The first difference between the Young case and the situation of a Hölder continuity exponent 1 3 < γ ≤ 1 2 is that a restriction has to be imposed on the class of allowed integrands with respect to B. This class is called the class of weakly controlled processes, and is defined as follows:

). We say that z is a weakly controlled path based on B and starting from a if z 0 = a, which is a given initial condition in R n , and δz ∈ C γ 2 (R n ) can be decomposed into (40)

), and r is a regular part such that r ∈ C 2γ 2 (R n ). The space of weakly controlled paths starting from a will be denoted by Q γ,a (R n ), and a process z ∈ Q γ,a (R n ) can be considered in fact as a couple

Two basic steps in order to define and solve differential equations with respect to B are then:

(1) Study the decomposition of f (z) as weakly controlled process, when f is a smooth function and z a weakly controlled process. (2) Define rigorously the integral z u dB u = J (zdB) for a weakly controlled path z and compute its decomposition (40). We shall now detail a little this program.

Let us see then how to decompose f (z) as a controlled process when f is a smooth enough function, a step for which we first introduce a convention which will hold true until the end of the paper: for any smooth function f : R n → R, k ≥ 1, (i 1 , . . . , i k ) ∈ {1, . . . , d} k and ξ ∈ R n , we set (41)

With this notation in hand, our decomposition result is the following: Proposition 4.9. Let f : R n → R be a C 2 b function such that f (a) = â, z a controlled process as in Definition 4.8 and set ẑ = f (z). Then ẑ ∈ Q γ,â (R), and it can be decomposed into δẑ = ζi 1 B 1,i 1 + r, with

Furthermore,

Remark 4.18. We are not able to obtain log-Sobolev inequality as in Theorem 3.4 in the rough case, since when H < 1 2 the Hilbert norm H is not controlled by L ∞ ([0, 1]). On the other hand, by reproducing the proof in Theorem 3.4 together with the following interpolation inequality

and estimates in Proposition 4.15, we are able to prove the following version of a log-Sobolev type inequality: for T ∈ [0, 1] we have

Ef (X T ) 2 ln f (X T ) 2 -Ef (X T ) 2 ln Ef (X T ) 2 ≤ C p,T E|∇f (X T )| 2p 1/p , For all p > 1. Here C p,T is a universal constant independent of f .


Md Z Hoque

M Lapert

E Hertz

F Billard

D Sugny

B Lavorel

O Faucher

Observation of laser-induced field-free permanent planar alignment of molecules

Keywords: numbers: 37.10.Vz, 42.50.Hz, 42.50.Md

Permanent planar alignment of gas-phase linear molecules is achieved by a pair of delayed perpendicularly polarized short laser pulses. The experiment is performed in a supersonic jet ensuring a relatively high number density of molecules with moderately low rotational temperature. The effect is optically probed on a femtosecond time scale by the use of a third short pulse, enabling a time-resolved birefringence detection performed successively in two perpendicular planes of the laboratory frame. The technique allows for an unambiguous estimation of the molecular planar delocalization produced within the polarization plane of the pulse pair after the turn-off of the field. The measurements are supported by numerical simulations which lead to the quantification of the observed effect and provide more physical insights into the phenomenon.

I. INTRODUCTION

The space handling of molecules is a fascinating topic that has received during these last years a growing attention. In particular, molecular alignment induced by a laser field was shown to be practically useful in various fields extending to high-harmonic generation and attophysics [1,2], molecular tomography [3,4], molecularframe photoelectron angular distribution [5][6][7], ultrafast all-optical switching of electrical junctions [8], laser filamentation [9], ultrafast optical imaging [10,11], generation of terahertz radiation [12,13], and so forth. Although the mechanisms at play in molecular alignment are now well understood, as documented in both experimental [14,15] and theoretical [16] works, valuable strategies that aim at optimizing angular focusing [17,18] or at preparing angular distributions designed for specific applications [19] remain a current challenge.

Recently, the ultrafast orientation of the angular momentum of a symmetric top molecule has been achieved using a sequence of two short laser pulses, properly delayed and polarized at 45 • with respect to each other [20]. The field arrangement enables the production of an asymmetric distribution of the helicity quantum number M , defined as the projection of the total angular momentum J along the quantization axis, supporting the rotation of the molecules in one sense rather than in the other [21,22]. At the same time, a strategy to produce a field-free permanent molecular planar alignment was proposed [23]. The two previous approaches mainly differ by the relative intensity, the delay, and more importantly, the mutual polarization between the two pulses. In the latter, the two pulses form an angle of 90 • in order to optimize the confinement of the molecular axis in the polarization plane, which is performed to the detriment of the orientation of the angular momentum achieved in the former.

Permanent planar alignment is relevant for various applications including stereodynamics effects of elementary processes. For instance, it can play an important role on the adsorption of molecules on a surface [24,25]. The scattering of molecules through electrostrictive forces [START_REF] Boyd | Nonlinear Optics[END_REF] is also influenced by the rotation of the molecules [START_REF] Gershnabel | [END_REF]. Very recently, it has been shown that the deflection angles can be controlled by tilting the plane of rotation of the molecule prepared in a state of planar alignment [28].

II. METHODS

Hereby, we provide an experimental realization of permanent planar alignment by demonstrating the effect for a linear molecule. We consider two short laser pulses copropagating along the z axis and linearly polarized along two different coordinate axes of the laboratory frame, e.g. x and y, respectively. Both pulses, of same wavelength, are temporally delayed by T r /2, where T r is the rotational period of the molecule. Each pulse individually produces a non-adiabatic alignment of the linear molecule along its respective polarization axis [14], featured by field-free revivals occurring at fractional times of the rotational period superimposed on a permanent alignment component. However, the collective action of the two pulses results on a transient alignment of the molecular axis that alternates between the two x and y directions during each revival. More importantly, it also leads to a quasi steady confinement of the molecular axis in the (xy) plane [23].

This permanent planar alignment can be understood in a first approach in terms of selection rules. The alignment proceeds from the non-resonant impulsive Raman excitation of the molecular polarizability resulting in a rotational wave packet produced in the ground vibronic state of the molecule. Considering first a single linearly polarized pulse, the electric-dipole-allowed Raman transitions satisfy the selection rules ∆J = 0, ±2 and ∆M = 0. Starting from the initial condition |J 0 , M 0 , the |J ≫ J 0 , M 0 states are populated through stepwise excitation during the strong pulse interaction. The angular distribution of these unequally populated helicity states correspond to a confinement of the molecular axis along the quantization axis, i.e. the field direction, together with a delocalization of the total angular momentum in the equatorial plane. Considering now the sequence of cross-polarized pulses introduced above, the total field, in addition, leads to ∆M = ±2 transitions, enabling the population of the |J ≫ J 0 , |M | ≫ |M 0 | levels. These spatially anisotropic states, in particular those for which M ≃ J and M ≃ -J, represent respectively the molecules with an angular momentum J pointing along the two opposite directions of the quantization axis z. For the pulse delay T r /2, the superposition of the freeevolving states populated during the pulse interactions leads to a quasi-symmetric distribution in M , with an angular momentum aligned along the z axis, and to a confinement of the molecular axis in the plane of polarization spanned by the two field vectors [23].

III. RESULTS AND DISCUSSION

A. Experiment

In order to track the free evolution of the system prepared by the bi-pulse, we optically probe the molecules with a third delayed pulse [29]. The experiment is performed on the CO 2 molecule. For a better efficiency of the process, the molecules are rotationally cooled down to a temperature of the order of 100 K in a supersonic expansion of a continuous jet. The number density estimated at the interaction region is close to 10 19 cm -3 . For this study, the relevant observable is cos 2 θ z , with • indicating the quantum expectation value and θ z the angle of the molecular axis with respect to the propagation z axis. A planar alignment in the polarization (xy) plane is achieved when cos 2 θ z < 1/3, whereas cos 2 θ z > 1/3 reflects an alignment along the z axis, 1/3 being the degree of alignment at thermal equilibrium. In principle, a cross-Kerr defocusing technique allows to reveal the value of cos 2 θ z from the spatial modification of the refractive index measured by a probe field polarized along the z axis [30,31]. However, it has been shown that experiments performed with this technique at large peak intensity suffer from an additional signal produced by the free electrons due to the ionization of the gas sample [32]. For this reason, we have employed a time-resolved polarization technique immune to plasma effects, and particularly sensitive for probing molecular alignment of linear [33][34][35][36] and asymmetric top molecules [37]. It consists in measuring the transient birefringence experienced by a probe pulse linearly polarized at 45 • with respect to a pair of space-fixed axes i, j. In heterodyne detection [38], the birefringence signal denoted as S ij is proportional to n i -n j ≃ ̺∆α 2nε0 cos 2 θ i -cos 2 θ j , where n l=x,y,z is the optical Kerr index produced along the l axis, ̺ the number density, n = 1/2(n i +n j ), ε 0 the vacuum permittivity, and ∆α = α -α ⊥ the polarizability anisotropy with α and α ⊥ being respectively the polarizability components parallel and perpendicular to the internuclear axis [37]. Using the geometric relation l cos 2 θ l = 1 between the director cosines, any expectation value cos 2 θ i can be deduced from a combination of birefringence signals recorded between two distinct pairs of space-fixed axes. It is easy to show for instance that the value of interest cos 2 θ z can be obtained from a combination of S xz and S yz .

Aligning and probing beams are produced by a chirped pulse amplified Ti:sapphire femtosecond laser. The system provides 100 fs-duration pulses at a repetition rate of 100 Hz, with a wavelength centered at 800 nm. At first, a series of measurements is conducted by setting the three pulses as depicted in Fig. 1. The aligning pulse P 1 (resp., P 2 ) of intensity I 1 (resp., I 2 ) propagates along the z axis with a field linearly polarized along the x (resp. y) direction, while the probe pulse, polarized at 45 • with respect to the x axis, propagates along the y axis. All pulses are focused and intersect at a right angle on the molecular jet axis. The delay between P 1 and P 2 is set to half of the rotational period T r = 42.74 ps of the CO 2 molecule. The data are recorded with a heterodyne detection [38].

Figure 2(a) shows the birefringence signal S xz ∝ n xn z measured by tuning the probe delay τ over one rota-tional period for an estimated peak intensity I 2 ≃ 140 TW/cm 2 and an intensity ratio I 2 /I 1 = 2.2. The data recorded for 0 < τ < T r /2 correspond to molecules exposed to a single pulse P 1 . The latter being linearly polarized along the x direction, it is straightforward to show that S xz is proportional to cos 2 θ x -1/3 [33]. The positive and negative transients feature hence molecules aligned along the field ( cos 2 θ x > 1/3, n x > n z ) and delocalized in the (yz) plane perpendicular to the field ( cos 2 θ x < 1/3, n x < n z ), respectively. The elevated baseline observed between the revivals reflects the permanent alignment induced along the field, whereas the negative signal produced at zero delay stems from the inversion of the electronic Kerr effect occurring at large intensity [39][40][41]. The second part of the trace, recorded for τ T r /2, is related to the combined action of the cross-polarized pulses. The permanent alignment is enhanced by the second pulse P 2 as resulting from the delocalization of the molecules in the (xy) plane.

In order to assess the alignment achieved along the y axis, both horizontal polarizations depicted in Fig. 1 are rotated by 90 • , so as from now on P 1 (resp., P 2 ) points along the y axis (resp., x axis). This is equivalent to having a probe beam propagating along the x axis, providing hence a signal given by S yz ∝ n y -n z . The experimental result is presented in Fig. 2(b) for the same delay range and intensity ratio as in the previous measurement. For τ < T r /2, we see that the signal is not affected by the first pulse P 1 which is perpendicular to the probe wavefront. Conversely, for τ T r /2, due to the anisotropy induced by P 2 on the wavefront of the probe, the signal exhibits the same features as in Fig. 2(a) but for τ < T r /2.

The value cos 2 θ z -1/3 reconstructed from S xz and S yz is shown in Fig. 2(c). More precisely, we plot the value -S xz -S yz ∝ cos 2 θ z -1/3 dV versus the delay, with [•]dV denoting the integration over the interaction volume V (omitted in the previous discussion for simplicity's sake). First, we should notice that, for symmetry reasons, we have cos 2 θ z = cos 2 θ y between the two aligning pulses, since P 1 vibrates along the x direction. This relation together with l cos 2 θ l = 1 leads to cos 2 θ z -1/3 = -cos 2 θ x -1/3 /2 during this time lag. The effect produced along the z and x axes are therefore out of phase for τ < T r /2 , as attested by the comparison of Figs. 2(c) and (a), respectively. Secondly, the application of the second pulse P 2 results in a strong field-free delocalization of the molecules in the polarization plane of the bi-pulse for τ T r /2, as predicted in Ref. [23]. In fact, we can see that the remaining transient components present a modulation which does not exceed the amplitude of the permanent one. This provides clear evidence that, even at moderately low temperature, the bi-pulse strategy allows to produce a significant planar alignment.

B. Theory

In order to support the experimental findings, we numerically investigate the system under the present experimental conditions. For that purpose, we consider a linear molecule modelizing CO 2 in its ground vibronic state which is subject to a non-resonant interaction with a laser field of the form

E(t) = E x (t) cos(ωt) e x + E y (t -T r /2) cos(ωt + φ(t)) e y , (1)
where ω is the angular frequency and φ(t) the phase difference between the two polarization directions. We denote respectively by E x (t) and E y (t -T r /2) the temporal envelopes of the electric field along the x and y axes, T r /2 being the temporal delay between the two pulses. Within the rigid rotor approximation, the Hamiltonian of the system obtained after averaging over the rapid oscillations of the field can be written as

H(t) = BJ 2 - 1 4 [E 2 x (t)(α ⊥ + ∆α cos 2 θ x ) +E 2 y (t -T r /2)(α ⊥ + ∆α cos 2 θ y )].
In the expression of the Hamiltonian, the first term represents the field-free rotational Hamiltonian, with the rotational constant B and the angular momentum operator J 2 . For the numerical applications, we use for the CO 2 molecule the following parameters B = 0.3902 cm -1 , α = 27.2984 and α ⊥ = 12.7908 in atomic units. We consider for E x and E y gaussian envelopes written as

E 0 exp[-t 2 /σ 2 ]
, where E 0 is the maximum amplitude and √ 2 ln 2σ is the pulse duration [full width at half maximum (FWHM)], which is assumed to be equal to 100 fs to simulate the experiments.

For a non-zero temperature, the system is described by a density operator ρ(t) whose dynamics is governed by the von Neumann equation

i ∂ρ(t) ∂t = [H(t), ρ(t)] (2)
with the Boltzmann distribution at initial condition as

1 Z ∞ j=0 j m=-j
|j, m e -Bj(j+1)/(kBT ) j, m|,

where Z = ∞ j=0 j m=-j e -Bj(j+1)/(kBT ) is the partition function, with k B the Boltzmann constant, and |j, m 's the eigenvectors of J 2 . In this formalism, the expectation value of any observable A is given by A (t) = Tr[ρ(t)A].

We present the numerical results in Fig. 3 at T = 100 K and for the peak intensities I y ∝ E 2 0y = 120 TW/cm 2 and I x = I y /2.2. Figure 3 displays the time evolution of the expectation values of the three director cosines during the interaction with the bi-pulse and over a fieldfree rotational period. In spite of a moderately low temperature, a substantial permanent planar alignment is achieved with a value of cos 2 θ z -1/3 = -0.13. Note also the symmetry of the dynamics of the two other expectation values cos 2 θ x and cos 2 θ y which is a signature of the planar alignment. This computation confirms the efficiency of the bi-pulse process which is not the optimal solution, but sufficiently simple to be experimentally implemented. A complete numerical study of the different strategies constructed from optimal control theory [42,43] and evolutionary algorithm [17,18] can be found in [23]. In particular, it can be shown that the ratio I y /I x = 2.2 is a good compromise for the bipulse strategy considering the laser intensity distribution and the temperature achieved in the experiment. The angular distribution computed at t = T r /8 after switch off of the second pulse (corresponding to τ = 26.71 ps in Fig. 2) is represented in Fig. 4. We recall that, when no field is applied, this distribution is spherical which means that any space direction is equivalent. When the molecule is aligned in a given direction, the probability distribution is peaked along the corresponding axis. In the case of planar alignment, the probability is delocalized in a plane as can be seen in Fig. 4. Note that the x and the y axes are not equivalent since the alignment along the y direction is better than along the x one, giving rise to a rectangularlike shape. This dissymmetry between the two axes can also be observed in Fig. 3 where the permanent values of cos 2 θ x and cos 2 θ y are clearly different. The qualitative discussion of Sec. II about the effect of the bipulse strategy on the population of the states |J, M is quantitatively illustrated in Fig. 5. As could be expected, the two pulses, when successively applied, populate levels with higher and higher J and M quantum numbers. With respect to the initial Boltzmann distribution, the bipulse process leads however to a depletion of the population in the center of the diagram where J ≫ |M |, and produces an increase of this population in the boundaries of the diagram where J ≃ |M |. This observation is in agreement with the fact that states with J ≃ |M | maximize the planar alignment as shown in [23].

The numerical simulations described above are used to fit the experimental data presented in Fig. 2. Two free parameters have to be adjusted in order to compare the theoretical and the experimental results. The first one is a scaling factor and the second is the laser peak intensity. In order to account for the volume effects, the simulated signals are averaged over the spatial distributions of the laser beams. Doing so, a reasonable match between theory and experiment has been obtained in Fig. 2.

IV. CONCLUSION

We have presented an experimental demonstration of permanent planar alignment in a linear molecule. We have used a simple control strategy composed of two delayed short pulses linearly polarized in two orthogonal directions. This study calls for further investigations about the extensions of the concept of planar alignment. Having confined the molecular axis in a plane, the next step could be the control of the planar dynamics of the molecule by using a third laser pulse so as to control the sense of rotation of the molecular axis [22] or to transform the molecule in a quantum cogwheel [44].

FIG. 1 :

 1 FIG. 1: (Color online) Schematic diagram depicting the arrangement of the probe and the two pulses P1 and P2 used to generate a planar alignment in the (xy) plane. M: Mirrors, L: Lens, P: Polarizer, A: Analyzer, BS: Beam stop, PM: Photomultiplier. The relative polarizations of the probe (P) and signal-field (A) are shown in the inset.

FIG. 2 :FIG. 3 :

 23 FIG. 2: (a) (Color online) Transient birefringence signal Sxz ∝ nx -nz of CO2 measured between the space-fixed x and z axes. (b) Syz ∝ ny -nz. (c) -Sxz -Syz proportional to the space averaged value cos 2 θz -1/3. The half of the rotational period of CO2 is Tr/2 = 21.37 ps (vertical dashed lines). Experimental and calculated data are represented respectively by blue dots and red lines.

FIG. 4 :FIG. 5 :

 45 FIG. 4: (Color online) Angular distribution of the CO2 molecule computed at t = Tr/8 after the interaction with the bi-pulse of intensities Iy = 120 TW/cm 2 and Ix = Iy/2.2. The temperature is T = 100 K.

Acknowledgments

We thank R. Saint-Loup, V. Tissot, B. Sinardet, S. Pernot, J.-M. Müller, and A. Javelle for valuable technical support. This work was supported by the Conseil Régional de Bourgogne, the ANR COMOC, and the FASTQUAST ITN Program of the 7th FP.