M Besalú

A Kohatsu-Higa

Samy Tindel
email: samy.tindel@univ-lorraine.fr

GAUSSIAN TYPE LOWER BOUNDS FOR THE DENSITY OF SOLUTIONS OF SDES DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

Keywords: Mathematics Subject Classification. Primary 60G22; Secondary 34K50, 60H07 Fractional Brownian motion, stochastic equations, density function estimates

In this paper we obtain Gaussian type lower bounds for the density of solutions to stochastic differential equations (sde's) driven by a fractional Brownian motion with Hurst parameter H. In the one dimensional case with additive noise, our study encompasses all parameters H ∈ (0, 1), while the other cases are restricted to the case H > 1/2. We rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools.

 [START_REF] Baudoin | Hitting probabilities for equations driven by a fractional Brownian motion[END_REF]. Upper and lower bounds on J 1,i 5.5. Upper bounds for J 2,i . 5.6. Upper bounds for J 3,i . 5.7. Lower bound 6. Appendix: Some properties of stochastic derivatives References

Introduction

Let B = (B 1 , . . . , B d ) be a d dimensional fractional Brownian motion (fBm in the sequel) defined on a complete probability space (Ω, F , P), with Hurst parameter H ∈ (0, 1). Recall that this means that B is a centered Gaussian process indexed by R + , whose coordinate processes are independent and their covariance structure is defined by 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. [START_REF] Alòs | Stochastic integration with respect to the fractional Brownian motion[END_REF] This implies that the variance of an increment is given by

E B j t -B j s 2 = |t -s| 2H , for s, t ∈ R + . (2)
In particular, this process is γ-Hölder continuous a.s. for any γ < H and is an H-self similar process. This converts fBm into a very natural generalization of Brownian motion and explains the fact that it is used in applications [START_REF] Kou | Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule[END_REF][START_REF] Odde | Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth[END_REF][START_REF] Willinger | Stock market prices and long-range dependence[END_REF].

We are concerned here with the following class of stochastic differential equations (sde's) in R m driven by B on the time interval [0, 1]:

X t = a + t 0 V 0 (X s )ds + d i=1 t 0 V i (X s )dB i s , (3)
where a ∈ R m is a generic initial condition and {V i ; 0 ≤ i ≤ d} is a collection of smooth and bounded vector fields of R m . Though equation ( 3) can be solved thanks to rough paths methods in the general case H > 1/4, d ≥ 1, we shall consider in the sequel three situations which can be handled without recurring to this kind of technique:

(1) The one-dimensional case with additive noise, which can be treated via simple ODE techniques.

(2) The one-dimensional situation, namely m = d = 1, where the equation can be solved thanks to a Doss-Sussman type methodology as mentioned in [START_REF] Nourdin | On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion[END_REF]. [START_REF] Baudoin | Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions[END_REF] The case of a Hurst exponent H > 1/2, for which Young integration methods are available (see e.g [START_REF] Gubinelli | Controlling rough paths[END_REF][START_REF] Nualart | Differential equations driven by fractional Brownian motion Collect[END_REF][START_REF] Zähle | Integration with respect to fractal functions and stochastic calculus[END_REF]). Hence, we always understand the solution to equation (3) according to the settings mentioned above. We shall see however that rough path type arguments shall be involved in some of our proofs.

The process defined as the solution of ( 3) is obviously worth studying, and a natural step in this direction is to analyze the density of the random variable X t for a fixed t > 0. To this respect, the following results are available in our cases of interest:

(1) For m = d = 1, the existence of density for L(X t ) has been examined in [START_REF] Nourdin | On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion[END_REF].

(2) Whenever H > 1/2 and in a multidimensional setting, the existence of density is established in [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF], while smoothness under elliptic assumptions is handled in [START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF]. Let us also mention that for a multidimensional equation [START_REF] Baudoin | Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions[END_REF] and H ∈ (1/4, 1/2), rough paths techniques also enable the study of densities of the solution. We refer to [START_REF] Cass | Densities for rough differential equations under Hörmander condition[END_REF][START_REF] Cass | Non-degeneracy of Wiener functionals arising from rough differential equations[END_REF] for existence and [START_REF] Cass | Hörmander's theorem for Gaussian rough differential equations[END_REF] for smoothness results for L(X t ). However, the only Gaussian type estimate for the density we are aware of is the one contained in [START_REF] Baudoin | Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions[END_REF], which relies heavily on a skew-symmetric assumption for the vector fields V 1 , . . . , V d .

The current article is thus dedicated to give Gaussian type lower bounds for the density of X t . More specifically, we work under the following assumptions on the coefficients of equation (3):

Hypothesis 1.1. The coefficients V 0 , . . . , V d of equation (3) satisfy the following conditions:

(1) If m = d = 1, then V 0 , V 1 ∈ C 3 b and we also assume λ ≤ |V 1 | ≤ Λ. (2) In the multidimensional case, the vector fields V 0 , . . . , V d belong to the space C ∞ b of smooth functions bounded together with all their derivatives. Furthermore, if V (x) denotes the matrix (V 1 (x), . . . , V d (x)) ∈ R m×d for all x ∈ R m , then we assume the following uniform elliptic condition:

λ Id m ≤ V (x)V * (x) ≤ Λ Id m , for all x ∈ R m , (4)
where the inequalities are understood in the matrix sense and where λ, Λ are two given strictly positive constants which are independent of x.

With these hypotheses in hand, our main goal is to prove the following result:

Theorem 1.2. Consider equation (3), under the following three specific situations: I. m = d = 1, H ∈ (0, 1), V 0 ∈ C 1 b and the noise is additive (i.e., V 1 is a non vanishing real constant). II. m = d = 1, H ∈ (1/2, 1) and Hypothesis 1.1(1) is satisfied for V 0 , V 1 . III. Arbitrary m, d ∈ N, H ∈ (1/2, 1) and V 0 . . . V d satisfying Hypothesis 1.1 [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF]. Then, the solution X t of equation (3) possesses a density p t (x) such that for every x ∈ R m and t ∈ (0, 1] we have:

p t (x) ≥ c 1 t mH exp - c 2 |x -a| 2 t 2H , (5)
for some constants c 1 , c 2 only depending on d, m and V 0 , . . . , V d .

As mentioned above, this is (to the best of our knowledge) the first Gaussian lower bounds obtained for equations driven by fBm in a general setting. It should also be mentioned that the lower bound [START_REF] Cass | Hörmander's theorem for Gaussian rough differential equations[END_REF] can be complemented by a similar upper bound contained in [START_REF] Baudoin | Hitting probabilities for equations driven by a fractional Brownian motion[END_REF].

Let us say a few words about the methodology we rely on in order to obtain our lower bound [START_REF] Cass | Hörmander's theorem for Gaussian rough differential equations[END_REF]. Generally speaking it is based on Malliavin calculus tools, but the three results mentioned in Theorem 1.2 are proved in different ways: [START_REF] Alòs | Stochastic integration with respect to the fractional Brownian motion[END_REF] In the one dimensional additive case, we invoke a recent formula for densities introduced in [START_REF] Nourdin | Density estimates and concentration inequalities with Malliavin calculus[END_REF] which yields an easy way to estimate p t in the case of additive stochastic equations. We thus include this study for didactical purposes, and also because we obtain (slightly non optimal) Gaussian upper and lower bounds with elegant methods.

(2) The one dimensional case with multiplicative noise is based on the Doss-Sussmann's transform and Girsanov type arguments. It is rather easy to implement and yields results when the criterion of [START_REF] Nourdin | Density estimates and concentration inequalities with Malliavin calculus[END_REF] can not be applied.

(3) As far as the general case is concerned, it will be basically handled thanks to the decomposition of random variables strategy introduced in [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF][START_REF] Kohatsu-Higa | Lower bounds for densities of uniformly elliptic random variables on Wiener space[END_REF]. However, let us point out two important differences between the fBm and the diffusion case:

(i) In the case of the sde (3) without drift coefficient V 0 , the first step of the method implemented (for a fixed t ∈ (0, 1]) in [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF][START_REF] Kohatsu-Higa | Lower bounds for densities of uniformly elliptic random variables on Wiener space[END_REF] amounts to introduce a partition {t j ; 0 ≤ j ≤ n} such that t 0 = 0 and t n = t, with n large enough, and then split X t into small contributions of the form

X t j+1 -X t j = d i=1 V i (X t j ) B i t j+1 -B i t j + d i=1 t 0 V i (X s ) -V i (X t j ) dB i s . (6)
Then a main conditionally Gaussian contribution V i (X t j )[B i t j+1 -B i t j ] is identified in the right hand side of equation ( 6), while the other terms are a small remainder in the Malliavin calculus sense in comparison with the first. Roughly speaking, the Gaussian lower bound ( 5) is then obtained by adding those main contributions and proving that the remainder does not significantly modify the estimate. However, let us highlight the fact that this general scheme does not fit to the fractional Brownian motion setting.

Indeed, due to the fBm dependence structure, the main contributions to the variance of X t in the current situation come from the cross terms

E[(B i t j+1 -B i t j )(B i t k+1 -B i t k )] for j = k.
We have thus decided to express equation ( 3) as an anticipative Stratonovich type equation with respect to the Wiener process induced by B. This is known to be an inefficient way to solve the original equation, but turns out to be very useful in order to analyze the law of X t . We shall detail this strategy at Section 5.1.

(ii) In the case of an equation driven by usual Brownian motion, the Malliavin-Sobolev norms involved in the computations give deterministic contributions after conditioning, due to the independence of increments of the Wiener process. This is not true anymore in the fBm case, and we thus need to add a proper localization to the arguments of [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF][START_REF] Kohatsu-Higa | Lower bounds for densities of uniformly elliptic random variables on Wiener space[END_REF].

Our article is structured as follows: Section 2 is devoted to recall some useful facts on fractional Brownian motion and stochastic differential equations. We handle the one dimensional case with additive noise at Section 3 and the one dimensional case with multiplicative noise in Section 4 with different methodologies. Finally, the bulk of our article focuses on the general multidimensional case contained in Section 5. Some auxiliary results used in Section 5 dealing with stochastic derivatives are given in an Appendix. Notations: Throughout this paper, unless otherwise specified we use | • | for Euclidean norms and • L p for the L p (Ω) norm with respect to the underlying probability measure P. For a random variable X, L(X) denotes its law and for a σ-field F , X ∈ F denotes the fact that X is F -measurable.

Consider a finite-dimensional vector space V and a subset U ⊂ R d . The space of V -valued Hölder continuous functions defined on U, with k-derivatives which are γ-Hölder continuous with γ ∈ (0, 1), will be denoted by

C k+γ (U; V ), or just C γ when U = [0, 1]. For a function g ∈ C γ (V ) and 0 ≤ s < t ≤ 1, we shall consider the semi-norms 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 . Similarly, for an open set U, C 1 b (U; V ) denotes the space of bounded continuously differentiable functions with bounded first derivative. For x, y ∈ R m , we set 1 {y≥x} := m k=1 1 {y k ≥x k } . Vectors x ∈ R m denote column vectors, their j-th component is denoted by x j and the transpose of x is denoted by x * . The identity matrix of order m × m is denoted by Id m .

Finally, let us mention that generic constants will be denoted by c, c H , c V , etc. independently of their actual value which may change from one line to the next. This rule will also apply for the constants M and M ′ which will appear as localization parameters, with the following additional convention: each time a localization constant appears, it increases its value by the addition of a fixed universal constant from the previous value. For a detailed explanation, see [START_REF] Nourdin | On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion[END_REF].

Stochastic calculus for fractional Brownian motion

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. For some fixed H ∈ (0, 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 (2).

2.1. Malliavin calculus tools. Gaussian techniques are obviously essential in the analysis of fBm driven differential equations like (3), and we proceed here to introduce some of them (see Chapter 5 in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] for further details).

2.1.1. Wiener space associated to fBm. Let E be the space of R d -valued step functions on [0, 1], and H the closure of E under the distance defined by through 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 ).
The space H is isometric to the reproducing kernel Hilbert space associated to R.

Furthermore, if (e 1 , . . . , e d ) designates the canonical basis of R d , one constructs an isometry

K * ≡ K * H,1 : H → L 2 ([0, 1]; R d ) such that K * (1 [0,t] e i ) = 1 [0,t] K H (t,
•) e i , where the kernel K = K H is given by

K(t, s) = c H s 1 2 -H t s (u -s) H-3 2 u H-1 2 du, H > 1 2 , (7)
K(t, s) = c H,1 s t 1/2-H (t -s) H-1/2 + c H,2 s 1/2-H t s (u -s) H-1 2 u H-3 2 du, H < 1 2 ,
for 0 ≤ s ≤ t and some explicit universal constants c H , c H,1 , c H,2 . With a slight abuse of notation we will denote the associated integral operator by Kf (x) =

x 0 f (s)K(x, s)ds. Note that we have that R(s, t) = s∧t 0 K(t, r)K(s, r) dr. Moreover, let us observe that K * ≡ K * 1 can be represented in the following form: for H ∈ (1/2, 1) we have

[K * ϕ] t = 1 t ϕ r ∂ r K(r, t) dr while for H ∈ (0, 1/2) it holds that [K * ϕ] t = K(1, t) ϕ t + 1 t (ϕ r -ϕ t ) ∂ r K(r, t) dr.
When H ∈ (1/2, 1) it can be shown that L 1/H ([0, 1], R d ) ⊂ H, and when H ∈ (0, 1/2) one has C γ ⊂ H ⊂ L 2 ([0, 1]) for all γ > 1 2 -H. We shall also use the following representations of the inner product in H:

(i) For H ∈ (1/2, 1) and φ, ψ ∈ H we have

K * φ, K * ψ L 2 ([0,1]) = φ, ψ H = c H 1 0 1 0 |s -t| 2H-2 φ s , ψ t R d dsdt . (8)
(ii) For H ∈ (0, 1/2), consider any family of partitions π = (t j ) of [0, 1], and set

Q jk = d i=1 E[∆ i j (B)∆ i k (B)] with ∆ i j (B) = B i t j -B i t j-1 . Then, for φ, ψ ∈ H we have φ, ψ H = lim |π|→0 j,k φ t j-1 , ψ t k-1 R d Q jk . (9)
Let us also recall that there exists a d-dimensional Wiener process W defined on (Ω, F , P) such that B can be expressed as

B t = t 0 K(t, r) dW r , t ∈ [0, T ]. (10)
This formula will be referred to as Volterra's representation of fBm. Formula (10) has various important implications. For example, it is readily checked that F t ≡ σ{B s ; 0 ≤ s ≤ t} = σ{W s ; 0 ≤ s ≤ t}. This filtration will appear in the sequel.

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 by S.

The Malliavin derivative with respect to B 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 D D k,p 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
, for k ≥ 0 and p ≥ 1. In particular, F 0,p ≡ F p = (E [F p ]) 1/p . The dual operator of D is denoted by δ δ δ, which corresponds to the Skorohod integral with respect to the fBm B on the interval [0, 1]. The set of smooth integrands is defined as D D D ∞ = ∩ k,p≥1 D D D k,p (H), the set of smooth processes is L L L k,p (H) and the Malliavin covariance matrix of F is denoted by Γ F .

As mentioned in the introduction, our lower bound (5) will be obtained by considering equation ( 3) as an equation driven by the underlying Wiener process W defined in [START_REF] Garsia | Continuity properties of Gaussian processes with multidimensional time parameter[END_REF], meaning that we shall also use stochastic analysis estimates with respect to W . We refer to Chapter 1 in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] for this classical setting, and just mention here a few notations: we denote by D the differentiation operator with respect to W and by δ the corresponding dual operator (Skorohod integral). The respective norms in the Sobolev spaces D k,p (L 2 ([0, 1])) are denoted by • k,p and the space of smooth integrands by L k,p . The following simple relation between D and D is then shown in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF], Proposition 5.2.1 : Proposition 2.1. Let D 1,2 is the Malliavin-Sobolev space corresponding to the Wiener process W . Then D D D 1,2 = (K * ) -1 D 1,2 and for any F ∈ D 1,2 we have DF = K * DF whenever both members of the relation are well defined.

In fact the above proposition says that the derivatives D D D and D are somewhat interchangeable. Indeed, using formula (5.14) in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] which gives an explicit formula for (K * ) -1 one obtains such a property. In particular, we will use that for F ∈ F t with F ∈ D k,p and for u = (u 1 , . . . , u k ) ∈ [0, 1] k we have

|D k u 1 ,...,u k F | ≤ ess sup u i ≤r i ; i=1,...,k |D k r 1 ,...,r k F |K(t, u 1 )...K(t, u k ). (11)
For the proof of [START_REF] Gubinelli | Controlling rough paths[END_REF] and other needed properties see Appendix 6. Some of our computations in Section 5 will rely on some conditional Malliavin calculus arguments, for which some definitions need to be recalled. First, for a given t ∈ [0, 1] and F ∈ L 2 (Ω), we shorten notations and write

E t [F ] := E[F | F t ],
and also set P t for the respective conditional probability and Cov t (G) for the conditional covariance matrix of a Gaussian vector G. We shall only use conditional Malliavin calculus with respect to the underlying Wiener process W , for which we recall the following definitions: For a random variable F and t ∈ [0, 1], let F k,p,t and Γ F,t be the quantities defined (for k ≥ 0, p > 0) by:

F k,p,t = E t [F p ] + k j=1 E t D j F p (L 2 t ) ⊗j 1 p
, and

Γ F,t = DF i , DF j L 2 t 1≤i, j≤d , (12)
where we have set

L 2 t ≡ L 2 ([t, 1]).
With this notation in hand, we give a conditional version of the integration by parts formula with respect to the Wiener process W borrowed from [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]Proposition 2.1.4]. Proposition 2.2. Fix n ≥ 1. Let F, Z s , G ∈ (D ∞ ) d be three random vectors where Z s ∈ F smeasurable and (det Γ F +Zs ) -1 has finite moments of all orders. Let g ∈ C ∞ p (R d ). Then, for any multi-index α = (α 1 , . . . , α n ) ∈ {1, . . . , d} n , there exists a r.v.

H s α (F, G) ∈ ∩ p≥1 ∩ m≥0 D m,p such that E [(∂ α g)(F + Z s ) G|F s ] = E [g(F + Z s ) H s α (F, G)|F s ] , (13)
where H s α (F, G) is recursively defined by

H s (i) (F, G) = d j=1 δ s G Γ -1 F,s ij DF j , H s α (F, G) = H s (αn) (F, H s (α 1 , ..., α n-1 ) (F, G)).
Here δ s denotes the Skorohod integral with respect to the Wiener process W on the interval [s, 1]. Furthermore, the following norm estimates with

1 p = 1 q 1 + 1 q 2 + 1 q 3 hold true: H s α (F, G) p,s ≤ c det(Γ F,s ) -1 n 2 n-1 q 1 ,s F 2(dn+1)
n+2,2 n q 2 ,s G n,q 3 ,s . We will also resort to a localized version of the above bounds. Namely, we introduce a family of functions Φ M,ǫ : R + → R + indexed by M, ǫ > 0, which are regularizations of 1 {x≤M } . Specifically, we define a function

φ ǫ = ǫ -1 φ : R → R with φ(x) := c φ exp - 1 1 -x 2 1 {|x|<1} ,
where c φ is a normalization constant chosen in order to have R φ(x) dx = 1. Then, we define

Φ M,ǫ (y) := 1 - y -∞ φ ǫ (x -M ) dx. ( 14
)
It is then readily checked that Φ M,ǫ (z) = 0

for |z| > M + ǫ, Φ M (z) = 1 on [0, M -ǫ] and Φ M,ǫ ∈ C ∞ b .
We will use the above localization function in two situations: one for M >> 1, ǫ = 1 and in that case we simplify the notation using Φ M ≡ Φ M,1 . In a second case M will not be a large quantity and therefore we will have to choose ǫ accordingly.

Consider now Z ∈ D ∞ . Under the same conditions as for Proposition 2.2 we get a conditional integration by parts formula of the form (13) localized by Z, with the following modification on the estimation of the norms of H s α :

H s α (F, G Φ M (Z)) p,s ≤ c det(Γ F,s ) -1 Φ M ′ (Z) k 3 p 3 ,s F Φ M ′ (Z) k 4 k 2 ,p 2 ,s G Φ M ′ (Z) k 1 ,p 1 ,s , (15)
for some appropriate positive integers k 1 , p 1 , k 2 , p 2 , k 3 , p 3 , k 4 , and where we recall our convention on increasing constants M ′ > M. Notice that ( 15) is valid for localizations of the form Φ M,ǫ (Z) as well.

Differential equations driven by fBm.

Recall that X is the solution of (3), and that our working assumptions are summarized in Hypothesis 1.1. We have distinguished 3 situations:

(1) The one dimensional additive case, for which equation ( 3) can be reduced to an ordinary differential equation by considering the process Z = X -B.

(2) The one dimensional multiplicative case, handled thanks to the Doss-Sussman transform (see e.g [START_REF] Nourdin | On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion[END_REF]). (3) The multidimensional case with H ∈ (1/2, 1), solved in a pathwise way by interpreting stochastic integrals as generalized Riemann-Stieljes type integrals.

In this section we give a brief account on the known results in the last situation.

In the case H ∈ (1/2, 1), (3) is solved thanks to a fixed point argument, after interpreting the stochastic integral 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 2.3. 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 a Riemann-Stieltjes integral. Moreover, the following estimation is fulfilled:

t s g ξ df ξ ≤ C f γ g κ |t -s| γ ,
where the constant C only depends on γ and κ.

With this definition in mind and under Hypothesis 1.1, we can solve (3). Specifically, it is proven in [START_REF] Nualart | Differential equations driven by fractional Brownian motion Collect[END_REF] that equation (3) driven by B admits a unique γ-Hölder continuous solution X, for any 1 2 < γ < H. Moreover, the following moments bounds are shown in [START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF]: Proposition 2.4. Let H ∈ (1/2, 1) and assume that V 0 , . . . , V d satisfy Hypothesis 1.1. Then for t ∈ [0, 1] and 1 2 < γ < H, we have

X 0,t,∞ ≤ |a| + c V B 1/γ 0,t,γ . (16)
Moreover X t ∈ D D D ∞ and for all n ≥ 1, i 1 , . . . , i n ∈ {1, . . . , d} n and 0 ≤ s < t ≤ 1 the following bound holds true:

sup s≤u, r 1 ...rn≤t |D i 1 ...in r 1 ...rn X u | ≤ C V,n exp c V,n B 1/γ s,t,γ . (17)
We remark here that due to Proposition 5.3 (iii), the good definition of the supremum in ( 17) can be justified. Furthermore, a bound for γ-Hölder norms with 1 2 < γ < H is provided by [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF]Eq.(10.15)] for X together with its Malliavin derivatives: Proposition 2.5. Under the same assumptions as for Proposition 2.4 we have

X s,t,γ ≤ c 1,V B s,t,γ ∨ B 1 γ s,t,γ , D i 1 ...in r 1 ...rn X u s,t,γ ≤ c 2,V,n exp c 3,V,n B 1 γ
s,t,γ .

One dimensional additive case

This section is devoted to prove our main Theorem 1.2 in the particular case m = d = 1 with additive noise. In this context, one can take advantage of the results obtained by Nourdin and Viens in [START_REF] Nourdin | Density estimates and concentration inequalities with Malliavin calculus[END_REF] in order to derive Gaussian type upper and lower bounds for p t . Let us then first recall what those results are.

3.1. General bounds on densities of one-dimensional random variables. Recall that we denote the Malliavin-Sobolev spaces with respect to the fBm B by D D D k,p , and consider a real-valued centered random variable F ∈ D D D 1,2 . We define a function g on R by:

g(z) := E DF, -DL -1 F H |F = z ,
where the operator L is the Ornstein-Uhlenbeck operator associated to the fBm B (see [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] for further details), which can be defined using the chaos expansion by the formula L = -∞ n=0 nJ n . Based on the function g, the following simple criterion for Gaussian type bounds has been obtained in [START_REF] Nourdin | Density estimates and concentration inequalities with Malliavin calculus[END_REF]:

Proposition 3.1. Let F ∈ D D D 1,2 with E[F ] = 0. If there exist c 1 , c 2 > 0 such that c 1 ≤ g(F ) ≤ c 2 , P -a.s,
then the law of F has a density ρ satisfying, for almost all z ∈ R,

E[|F |] 2c 2 exp - z 2 2c 1 ≤ ρ(z) ≤ E[|F |] 2c 1 exp - z 2 2c 2 .
Interestingly enough, [START_REF] Nourdin | Density estimates and concentration inequalities with Malliavin calculus[END_REF]Proposition 3.7] also gives an alternative formula for g(F ) which is suitable for computational purposes. Indeed, if we write DF = Φ F (B), where Φ F : R H → H is a measurable mapping, then the following relation holds true:

g(F ) = ∞ 0 e -θ E Φ F (B), Φ F (e -θ B + 1 -e -2θ B ′ ) H F dθ, (18)
where B ′ stands for an independent copy of B, and is such that B and B ′ are defined on the product probability space (Ω × Ω ′ , F ⊗ F ′ , P × P ′ ). Here we abuse the notation by letting E be the mathematical expectation with respect to P × P ′ , while E ′ is the mathematical expectation with respect to P ′ only. One can thus recast relation [START_REF] Nualart | Regularization of differential equations by fractional noise[END_REF] as

g(F ) = ∞ 0 E E ′ DF, DF θ H F dθ, (19)
where, for any random variable X defined in (Ω, F , P), X θ denotes the following shifted random variable in Ω × Ω ′ :

X θ (ω, ω ′ ) = X e -θ ω + 1 -e -2θ ω ′ , ω ∈ Ω, ω ′ ∈ Ω ′ .

3.2.

Main result in the additive one-dimensional case. Before stating our result let us point out that we assume through this subsection V 1 ≡ σ. That is, X is the solution of

X t = x + t 0 V 0 (X s )ds + σ B t , t ∈ [0, 1] (20)
where σ > 0 is a strictly positive constant, V 0 satisfies V ′ 0 ∞ ≤ M for some constant M > 0 and B is a fBm with H ∈ (0, 1). Under this setting, we are able to get the following bounds: Theorem 3.2. Assume that V 0 satisfies that V ′ 0 ∞ ≤ M, for some constant M > 0, σ > 0 and H ∈ (0, 1). Then, for all t ∈ (0, 1], X t possesses a density p t and there exist some constants c 1 and c 2 depending only on M and H such that for all z ∈ R

E[|X t -m|] c 1 σ 2 t 2H exp - (z -m) 2 c 2 σ 2 t 2H ≤ p t (z) ≤ E[|X t -m|] c 2 σ 2 t 2H exp - (z -m) 2 c 1 σ 2 t 2H , (21)
where m := E[X t ].

Remark 3.3. The advantage of the Nourdin-Viens method of estimating densities is that upper and lower bounds are obtained with similar proofs. The drawback is the restriction to one dimensional additive situations. Also notice that the exponents in equation ( 21) are optimal if one can prove that E[|X t -m|] ≍ σ t H . This easy step is left to the reader for the sake of conciseness.

Strategy of the proof. Obviously, we shall mainly rely on Proposition 3.1. We thus define

F = X t -E[X t ]
, where X t is the solution of [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF]. We get a centered random variable, and we shall prove that there exists two constants 0 < K 1 < K 2 such that

K 1 σ 2 t 2H ≤ g(F ) ≤ K 2 σ 2 t 2H . ( 22
)
Notice first that in the present case, it is easily seen that for any t > 0 we have X t ∈ D 1,2 (this is a particular case of [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF]). Furthermore, the Malliavin derivative of X t satisfies the following equation for r ≤ t:

D r X t = t r V ′ 0 (X s )D r X s ds + σ.
This equation can be solved explicitly, and we obtain

D r X t = σe t r V ′ 0 (Xs)ds . (23)
We shall now bound g(F ) according to this explicit expression, and we separate the cases H ∈ (1/2, 1) and H ∈ (0, 1/2). Notice that the Brownian case, i.e. H = 1/2, is well known and it is thus omitted here for the sake of conciseness.

Case H > 1

2 . Recall that we wish to prove [START_REF] Ll | Pathwise definition of second order SDEs[END_REF], for which we can use the explicit expression of D r X t obtained in [START_REF] Willinger | Stock market prices and long-range dependence[END_REF]. Furthermore, owing to expression [START_REF] Friz | Victoir: Multidimensional dimensional processes seen as rough paths[END_REF] for the inner product in H we can write g(F ) as

g(F ) = c H ∞ 0 E E ′ t 0 t 0 D u X t D v X θ t |u -v| 2H-2 dudv F dθ (24) = c H σ 2 ∞ 0 e -θ E E ′ t 0 t 0 e t u V ′ 0 (Xs)ds e t v V ′ 0 (X θ s )ds |u -v| 2H-2 dudv F dθ.
The lower and upper bounds follow from applying

e -M ≤ e t u V ′ 0 (X θ s )ds ≤ e M , for any 0 ≤ u ≤ t ≤ 1. 3.4. Case 0 < H < 1 2 .
As in the case H > 1 2 , we prove [START_REF] Ll | Pathwise definition of second order SDEs[END_REF]. We thus go back to equation ( 19) and we observe that we can reduce the problem to the existence of two constants 0

< c 1 < c 2 such that c 1 t 2H ≤ DX t , DX θ t H ≤ c 2 t 2H . ( 25
)
The proof of these inequalities will rely on the following quadratic programming lemma, which is a slight variation of [5, Lemma 6.2]:

Lemma 3.4. Let Q ∈ R n ⊗
R n be a strictly positive symmetric matrix such that n j=1 Q ij ≥ 0 for all i = 1, . . . , n. For two positive constants a and b, consider the sets

A = [a, ∞) n and B = [b, ∞) n . Then inf {x * Qx; x ∈ A, x ∈ B} = ab n i,j=1 Q ij . Proof. Set a = a 1 ∈ R n and b = b 1 ∈ R n . The Lagrangian of our quadratic programming problem is a function L : R n × R n × R n + × R n + → R defined as L(x, x, λ 1 , λ 2 ) = x * Qx -λ * 1 (x -b) -λ * 2 (x -a) . It is readily checked that ∇ x L(x, x, λ 1 , λ 2 ) = Qx -λ 1 and ∇ xL(x, x, λ 1 , λ 2 ) = Qx -λ 2 , which vanishes for x = Q -1 λ 2 and x = Q -1 λ 1 . Therefore, inf {L(x, x, λ 1 , λ 2 ); x, x ∈ R n } = L Q -1 λ 2 , Q -1 λ 1 , λ 1 , λ 2 = -λ * 1 Q -1 λ 2 + λ * 1 b + λ * 2 a =: G(λ 1 , λ 2 ). We have thus obtained a dual problem of the form max G(λ 1 , λ 2 ); λ 1 , λ 2 ∈ R n + . (26)
Let us now solve Problem (26). We first maximize G without positivity constraints on λ 1 and λ 2 : we get

∇ λ 1 G(λ 1 , λ 2 ) = -Q -1 λ 2 + b and ∇ λ 2 G(λ 1 , λ 2 ) = -λ * 1 Q -1 + a, which vanishes for λ • 1 = Qa and λ • 2 = Qb . Observe now that our assumption n j=1 Q ij ≥ 0 for all i = 1, . . . , n implies λ • 1 , λ • 2 ≥ 0, so that λ • 1 and λ • 2 are feasible for the dual problem. Hence max G(λ 1 , λ 2 ); λ 1 , λ 2 ∈ R n + = G(λ • 1 , λ • 2 ) = ab n i,j=1 Q ij ,
which finishes the proof.

Importantly enough, Lemma 3.4 can be applied in order to get a lower bound on H norms:

Proposition 3.5. Let B be a 1-dimensional fBm on [0, τ ], let H ≡ H τ be the associated reproducing kernel Hilbert space and f, f ∈ H such that f u ≥ b and fu ≥ a for any u ∈ [0, τ ]. Then f, f H ≥ a b τ 2H .
Proof. Recall that, owing to relation (9), we have f, f H = lim |π|→0 I π (f, f ), where π stands for a generic partition

{0 = t 0 < • • • < t n = τ } and I π (f, f ) = n i,j=1 f t i-1 Q ij ft j-1 , with Q ij = E[∆ i (B)∆ j (B)],
where we recall that ∆ i (B) = B t i -B t i-1 . We assume for the moment that Q satisfies the hypothesis of Lemma 3.4, and we get

I π (f, f ) ≥ ab n i,j=1 Q ij = ab n i,j=1 E [∆ i (B)∆ j (B)] = ab E B 2 τ = ab τ 2H ,
which is our claim.

Let us now prove that Q satisfies the hypothesis of Lemma 3.4. First, the strict positivity of Q stems from the local non determinism of B (see e.g [START_REF] Xiao | Properties of local-nondeterminism of Gaussian and stable random fields and their applications[END_REF]). Indeed, for u ∈ R n we have

u * Qu = Var n-1 j=0 u j ∆ j (B) ≥ c n n j=1 u 2 j |t j -t j-1 | 2H ,
where the lower bound is the definition of local nondeterminism. Thus u * Qu > 0 as long as u = 0.

Let us now check that for a fixed i we have n j=1 Q ij ≥ 0. To this aim, write

n j=1 Q ij = E [∆ i (B)B τ ] = t i+1 t i ∂ u R(τ, u) du.
Going back to expression (1), it is now easily seen that for u < τ we have

∂ u R(τ, u) = H u 2H-1 + (τ -u) 2H-1 > 0,
which completes the proof.

We can now go back to the proof of relation [START_REF] Zähle | Integration with respect to fractal functions and stochastic calculus[END_REF], which is divided again in two steps:

Step 1: Lower bound. Thanks to relation [START_REF] Willinger | Stock market prices and long-range dependence[END_REF] and since V ′ 0 ∞ ≤ M, we have that σe -tM ≤ D r X t .

Thus we just have to apply Proposition 3.5 to the Malliavin derivative in order to obtain

DX t , DX θ t H ≥ σ 2 t 2H e -2M , (27)
which is our desired lower bound.

Step 2: Upper bound. In order to obtain an upper bound for g(F ) we will use the representation of H through fractional derivatives. Indeed, apply first Cauchy-Schwarz inequality in order to get

DX t , DX θ t H ≤ DX t H DX θ t H . (28)
We then invoke Lemma 6.1 to bound DX θ t H . This boils down to estimate

a = sup r∈[0,t] |D r X θ t |, and b = sup r,v∈[0,t] D r X θ t -D v X θ t (v -r) γ , with 1/2 -H < γ < 1/2.
Now starting from expression [START_REF] Willinger | Stock market prices and long-range dependence[END_REF] and owing to the fact that V ′ 0 is uniformly bounded by M, we trivially get a ≤ σ e M . As far as b is concerned, we write

D r X θ t -D v X θ t ≤ σ e t v V ′ 0 (X θ s )ds 1 -e v u V ′ 0 (X θ s )ds ≤ σ M e 2M (v -r).
We thus end up with the inequalities a ≤ σ e M , and b ≤ σ M e 2M t 1-γ .

We now apply Lemma 6.1 with constants a and b and we obtain

DX t H ≤ c H σ e M t H + σ M e 2M t 1+H ≤ c H σ e M t 2H ,
and hence

DX t , DX θ t H ≤ c H σ 2 M 2 e 4M t 2H
. Finally, putting together the last bound and (27), we get [START_REF] Ll | Pathwise definition of second order SDEs[END_REF] in the case H ∈ (0, 1/2), which finishes the proof of Theorem 3.2.

One dimensional non-vanishing diffusion coefficient case

We turn now to the case m = d = 1, H ∈ ( 1 2 , 1) for a non constant elliptic coefficient σ. Observe that this special case is treated in a separate section because: (i) The Gaussian bound is obtained with weaker conditions on the coefficients than in the multidimensional case. (ii) The proof is shorter due to specific one-dimensional techniques based on Doss-Sussman's transform and Girsanov's theorem. This is detailed below.

4.1. Doss-Sussmann transformation. The idea of the method is to first consider a one dimensional equation of Stratonovich type without drift and then apply Girsanov's theorem for fBm in order to obtain a characterization of the density.

In order to carry out this plan, we start by using an independent copy of (Ω, F , P) called (Ω ′ , F ′ , P ′ ) supporting a fBm denoted by B ′ . On (Ω ′ , F ′ , P ′ ), let Y be the unique solution to:

Y t = a + t 0 V 1 (Y s ) • dB ′ s , (29)
where [START_REF] Alòs | Stochastic integration with respect to the fractional Brownian motion[END_REF]. We also call W ′ the underlying Wiener process appearing in the Volterra type representation [START_REF] Garsia | Continuity properties of Gaussian processes with multidimensional time parameter[END_REF] for B ′ . We now recall here some details from Doss and Sussmann's classical computations adapted to our fBm context. Indeed, as in [START_REF] Nourdin | On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion[END_REF], let us recall that the solution of equation ( 29) can be expressed as

V 1 ∈ C 1 (R; R), V 1 = 0 and H ∈ ( 1 2 ,
Y t = F (B ′ t , a), t > 0, where F : R 2 → R is the flow associated to V 1 : ∂F ∂x (x, y) = V 1 (F (x, y)), F (0, y) = y. ( 30
) We remark that if V 1 is bounded then F satisfies |F (x, y)| ≤ c(1 + |x| + |y|).
Next we relate the solution X of equation ( 3) to the process Y defined by (29). This step is partially borrowed from [START_REF] Nualart | Regularization of differential equations by fractional noise[END_REF], and we refer to that paper for further details. Indeed, thanks to a Girsanov type transform, the following characterization of the law of the solution to (3) is shown for m = d = 1: For any bounded measurable function U : R → R, one has

E P [U(X t )] = E P ′ [U (F (B ′ t , a)) ξ] , (31)
where ξ ≡ ξ t = dP dP ′ is the random variable defined by

ξ = exp t 0 M s dW ′ s - 1 2 M 2 s ds , (32)
where we have set

M = K -1 H ( • 0 V 0 V -1 1 (Y u )du).
Notice that in definition (32), the operators K, K -1 are respectively defined (with a slight abuse of notation), for H ≥ 1 2 and an appropriate function h, by:

[K H h](s) = I 1 0 + (s H-1 2 (I H-1 2 0 + (s 1 2 -H h)))(s), and [K -1 H h](s) = s H-1 2 (D H-1 2 0 + (s 1 2 -H h ′ ))(s).
We also recall that in the last equation, I α 0 + and D α 0 + denote the fractional integral and fractional derivative, whose expressions are:

I α 0 + f (x) = 1 Γ(α) x a (x -y) α-1 f (y)dy, and
D α 0 + f (x) = 1 Γ(1 -α) f (x) x α + α x a f (x) -f (y) (x -y) α+1 dy .
Notice that in order for (31) to be satisfied it is required that

• 0 V 0 V -1 1 (Y u )du ∈ I H+ 1 2 0+ (L 2 [0, T ]
). This condition is satisfied due to the γ-Hölderianity of Y for any γ < H.

Actually one should prove that Novikov's type conditions are satisfied for ξ in order to apply Girsanov's transform and get relation (31). This is achieved in the following lemma: Lemma 4.1. Let ξ be the random variable defined by (32). Then

M s ≤ c V β s , with β s := s 1 2 -H + B ′ H-1 2 . ( 33
)
Furthermore E P ′ [ξ] = 1, which justifies the Girsanov identity (31). That is, under

P, B = B ′ + • 0 V 0 V -1 1 (Y u )du is a H-fBm.
Proof. According to the expression of K -1 H we have

M s = s H-1 2 Γ(H -1 2 ) s 1 2 -H V 0 V -1 1 (Y s ) s H-1 2 + H - 1 2 s 0 s 1 2 -H V 0 V -1 1 (Y s ) -u 1 2 -H V 0 V -1 1 (Y u ) (s -u) H+ 1 2 du .
We now invoke the uniform ellipticity of V 1 and the regularity of V 0 and V 1 , which yields

M s ≤ c V s 1 2 -H + s 0 |F (B ′ s , a) -F (B ′ u , a)| (s -u) H+ 1 2 du ≤ c V β(s).
Now let us have a closer look at the process β: it is readily checked that B ′ γ admits quadratic exponential moments for any γ < H (see Theorem 3 in [START_REF] Nualart | Regularization of differential equations by fractional noise[END_REF]). Hence there exists λ > 0 such that the expected value E[exp(λ t 0 β 2 (s)ds)] is a finite quantity. Owing to a version of Novikov's condition stated in [9, Theorem 1.1] we deduce that E[ξ] = 1. This concludes the proof.

Main result.

As in the additive case of Section 3, we are able to get both upper and lower Gaussian bounds in a one dimensional context:

Theorem 4.2. Assume that V 0 , V 1 ∈ C 1 b , λ ≤ |V 1 | ≤ Λ and H ∈ (1/2, 1
). Then, there exist constants C 1 and C 2 such that for all t ∈ (0, 1], the solution X t to equation (29) possesses a density p t satisfying for all x ∈ R:

1 C 1 √ 2πt 2H exp -C 1 (x -a) 2 2t 2H ≤ p t (x) ≤ 1 C 2 √ 2πt 2H exp -C 2 (x -a) 2 2t 2H . ( 34
)
Proof.

Step 1: Upper bound. We start from an equivalent of (31) for densities, which is justified by [START_REF] Nourdin | On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion[END_REF] and a duality argument:

p t (x) = E P ′ [δ x (F (B ′ t , a)) ξ] , ( 35
)
where ξ is the random variable defined in (32). We now integrate by parts in order to get

p t (x) = E P ′ 1 {F (B ′ t , a)≤x} H (F (B ′ t , a), ξ) , with H (F (B ′ t , a), ξ) = δ ξ DF (B ′ t , a) DF (B ′ t , a) 2 L 2 ([0,t]) , (36)
where D, δ respectively stand (with a slight abuse of notation) for the Malliavin derivative and divergence operator for the Brownian motion W ′ under P ′ . Let us further simplify the expression for the random variable H(F (B ′ t , a), ξ): setting K t (u) ≡ K(t, u)1 [0,t] (u), it is readily checked that we have

D u F (B ′ t , a) = ∂ x F (B ′ t , a) K t (u) and DF (B ′ t , a) 2 L 2 ([0,t]) = |∂ x F (B ′ t , a)| 2 t 2H
. Plugging this information into (36), and defining Z := ξ (∂ x F (B ′ t , a)) -1 , we end up with

H (F (B ′ t , a), ξ) = δ (Z K t ) t 2H = K 1 -K 2 ,
where

K 1 = Z B ′ t t 2H , and K 2 = DZ, K t L 2 ([0,t]) t 2H .
We have thus obtained

p t (x) = E P ′ 1 {F (B ′ t , a)≥x} K 1 -E P ′ 1 {F (B ′ t , a)≥x} K 2 =: p 1 t (x) -p 2 t (x), (37)
and we shall upper bound these two terms separately. The term p 1 t (x) can be bounded as follows: for q 1 , q 2 , q 3 > 1 large enough and a parameter 1 < q 4 = 1 + ε with an arbitrary small ε > 0 we have

p 1 t (x) ≤ E 1/q 1 P ′ [|B ′ t | q 1 ] t 2H P ′ 1/q 2 (F (B ′ t , a) ≥ x) E 1/q 3 P ′ [|∂ x F (B ′ t , a)| q 3 ] E 1/q 4 P ′ [ξ q 4 ] (38)
We now bound the right hand side of this inequality:

(i) We obviously have

E 1/q 1 P ′ [|B ′ t | q 1 ] t 2H ≤ ct -H , since B ′ is a P ′ -fBm.
(ii) Let us prove that there exists two positive constants c 1 and c 2 such that

c 1 (x -a) ≤ F -1 (x, a) ≤ c 2 (x -a). (39)
Indeed, for a fixed a ∈ R, F is a continuously differentiable function and ∂ x F (•, a) = 0 for any a ∈ R. Then by the inverse function theorem F -1 (•, a) exists and it is a continuously differentiable function verifying

∂ x F -1 (z, a) = 1 V 1 (F (x, a)) = 1 V 1 (z)
, where F (x, a) = z.

Since F -1 is continuous we can now resort to the mean value theorem to get that for any x 1 , x 2 ∈ R we have 39) is now easily proven once we recall that λ ≤ |V 1 (u)| ≤ Λ.

F -1 (x 2 , a) -F -1 (x 1 , a) = x 2 -x 1 V 1 (θ) , for a certain θ ∈ [F -1 (x 1 , a), F -1 (x 2 , a)]. Moreover, it is easily seen that F -1 (x, x) = 0 for all x ∈ R, so that |F -1 (x, a)| = 1 |V 1 (θ)| |x -a|. Our claim (
Using (39) and the Gaussian density for B ′ t , it is now readily checked that

P ′ 1/q 2 (F (B ′ t , a) ≥ x) ≤ c 1 exp - c 2 (x -a) 2 t 2H .
(iii) Equation (30) reveals that ∂ x F is bounded by a constant, so that

E 1/q 3 P ′ [|∂ x F (B ′ t , a)| q 3 ] ≤ c. (iv) Set S = t 0 M s dW ′ s and D = t 0 M 2 s ds, where M ≡ K -1 H ( • 0 V 0 V -1 1 (Y u )du)
as above, and where we recall that q 4 = 1 + ε with an arbitrarily small ε > 0. It is readily checked that ξ q 4 = exp q 4 S -

q 4 2 D = exp q 4 S - q 2 4 2 D exp q ε 2 D ,
where q ε = q 2 4 -q 4 = ε(1 + ε). Now observe that the term exp(q 4 S -

q 2 4 2 D
) is a Girsanov change of measure which corresponds to a shift on B ′ of the form

B = B ′ -q 4 • 0 V 0 V -1 1 (Y u )du = B -(q 4 -1) • 0 V 0 V -1 1 (Y u )du.
Calling P′ the probability under which B is a fractional Brownian motion, we get

E P ′ [ξ q 4 ] = E P′ exp q ε 2 D . ( 40
)
Now plug the estimate (33) into (40). This yields

D ≤ c V 1 + B 2 H-1 2 ≤ c V 1 + B + q 4 • 0 V 0 V -1 1 (Y u )du 2 H-1 2 ≤ c V 1 + B 2 H-1 2 .
Going back to relation (40) and taking into account the fact that q ε can be chosen arbitrarily small, we get

E P ′ [ξ q 4 ] < ∞.
Gathering all the above estimates into (38), we have thus obtained that

p 1 t (x) ≤ c 1 t H exp - c 2 (x -a) 2 t 2H .
The same kind of bound can be deduced for p 2 t (x) in (37), which gives our global upper bound in (34).

Step 2: Lower bound. Our strategy to obtain the lower bound in (34) is based on the following decomposition:

p t (x) = E P ′ [δ x (F (B ′ t , a))(ξ t -ξ c 1 t )] + E P ′ [δ x (F (B ′ t , a))ξ c 1 t ] =: ρ 1 t + ρ 2 t , (41)
where c 1 is a constant to be determined later. Observe that the main term will be ρ 2 t , which means that we consider a two point partition of the interval [0, t] and we perform a one-step decomposition of X t (or Y t ) on [0, c 1 t] and [c 1 t, t], as opposed to the general time interval partition in Section 5.

First, we start studying the main term ρ 2 t : Note that due to (10), we have by Girsanov's Theorem

ρ 2 t = E P ′ E P ′ δ x (F (B ′ t , a)) F c 1 t ξ c 1 t = E P ′   exp - (F -1 (x, a) - c 1 t 0 K(t, s)dW ′ s ) 2 2 t c 1 t K(t, s) 2 ds ∂ x F -1 (x, a) 2π t c 1 t K(t, s) 2 ds ξ c 1 t   = E P [L c 1 ,t ],
where we have set

L c 1 ,t := exp - (F -1 (x, a) - c 1 t 0 K(t, s)dW s + c 1 t 0 V 0 V -1 1 (X s )ds) 2 2 t c 1 t K(t, s) 2 ds ∂ x F -1 (x, a) 2π t c 1 t K(t, s) 2 ds .
In order to determine a lower bound for the above expression, we use the following information:

(i) We have ∂ x F -1 (x, a) ≥ Λ -1 . (ii) We apply the inequality (x + a) 2 ≥ 1 2 x 2 -2a 2 to a 2 := ( c 1 t 0 V 0 V -1 1 (X s )ds) 2 ≤ c V t 2 .
(iii) Gaussian convolution identities can be invoked in order to compose the quadratic exponential term defining L c 1 ,t with the expected value with respect to the Gaussian random variable

c 1 t 0 K(t, s)dW ′
s . These ingredients easily entail that

ρ 2 t ≥ c ε √ 2πσ 2 exp - F -1 (x, a) 2 2σ 2 , for σ2 = 2 t c 1 t K(t, s) 2 ds + c 1 t
0 K(t, s) 2 ds, and we observe that σ 2 ≤ σ2 ≤ 2σ 2 . Now we estimate the first term ρ 1 t in (41) and prove that it is upper bounded by a quantity which is smaller than half of the lower bound we have just obtained. For this term we need to use again the integration by parts estimates carried out in (35). In order not to repeat arguments we just mention the main steps: we start by writing

ρ 1 t = E P ′ [δ x (F (B ′ t , a))(ξ t -ξ c 1 t )] = E P ′ 1 {F (B ′ t , a)≤x} H (F (B ′ t , a), ξ t -ξ c 1 t ) ,
and we decompose this expression into p 1 -p 2 like in (37), except for the fact that this time Z is replaced by Z t := ((ξ t -ξ c 1 t ) ∂ x F (B ′ t , a)) -1 . We wish to take advantage of the fact that ξ t -ξ c 1 t is a small quantity whenever c 1 is close to 1. For this, define the process

M c 1 t,• as M c 1 t,s = K -1 H ( • c 1 t V 0 V -1 1 (Y u )du), consider θ ∈ [0, 1] and define ξ t (θ) := ξ c 1 t exp θ t c 1 t M c 1 t,s dW ′ s - θ 2 2 t c 1 t M 2 c 1 t,s ds .
Then by the mean value theorem, we have

ξ t -ξ c 1 t = 1 0 dθ ξ t (θ) t c 1 t M s dW ′ s -θ t c 1 t M 2 s ds .
Applying Fubini's theorem, one sees that the same estimates as in (38), appear again with the exception that (i) The last term in the decomposition becomes E

1/q 4 P ′ [(ξ t (θ)) q 4
] which is handled in the same fashion as before. (ii) There is another term appearing in the decomposition, namely

E 1/q 5 P ′ t c 1 t M s dW ′ s -θ t c 1 t M 2 s ds q 5 .
Using (33) and usual estimate methods for stochastic integrals, one obtains that the latter term is upper bounded by c(1 -c 2-2H 1 )t 2-2H . Therefore taking c 1 sufficiently close to 1 one obtains that this upper bound is smaller than 1/2 of the lower bound previously obtained. The proof is now complete.

General lower bound

We now wish to obtain Gaussian type lower bounds for the multi-dimensional case of equation [START_REF] Baudoin | Upper bounds for the density of solutions of stochastic differential equations driven by fractional Brownian motions[END_REF]. However, the computations in this section will be performed on the following simplified version for notational sake (adaptation of our calculations to the drift case are straightforward):

X t = a + d i=1 t 0 V i (X s ) • dB i s , (42)
where a ∈ R m is a generic initial condition, V i : R m → R m i = 1, . . . , d is a collection of smooth and bounded vectors fields and B 1 , . . . , B d are d independent fBm's with H ∈ (1/2, 1). Recall that our goal is then to prove relation [START_REF] Cass | Hörmander's theorem for Gaussian rough differential equations[END_REF] in this context. To this aim, we shall assume that Hypothesis 1.1 (especially relation ( 4)) is satisfied for the remainder of the article.

Preliminary considerations.

Let us recall briefly the strategy used in [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF][START_REF] Kohatsu-Higa | Lower bounds for densities of uniformly elliptic random variables on Wiener space[END_REF] in order to obtain Gaussian lower bounds for solutions of stochastic differential equations. The argument starts with some additional notation: Recall that the natural filtration of B, which is also the natural filtration of the underlying Wiener process W defined by [START_REF] Garsia | Continuity properties of Gaussian processes with multidimensional time parameter[END_REF], is denoted by F t . As we have introduced in section 2.1, we write E t for the conditional expectation with respect to F t . Under our working Hypothesis 1.1, let us also mention that the following result is available (see [START_REF] Baudoin | Hitting probabilities for equations driven by a fractional Brownian motion[END_REF][START_REF] Hu | Differential equations driven by Hölder continuous functions of order greater than 1/2[END_REF] for further details):

Proposition 5.1. Under Hypothesis 1.1, let X be the unique solution to (42). Then for any t ∈ (0, 1], the random variable X t is non degenerate in the sense of Definition 2.1.1 in [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF].

In particular, the density of X t admits the representation p t (x) = E[δ x (X t )], where δ x stands for the Dirac measure at point x.

With this preliminary result in hand, the quantity E[δ x (X t )] will be analyzed by means of the succesive evaluation of conditional densities of an approximation sequence {F j ; 0 ≤ j ≤ n} such that X t = F n . We thus consider p t (x) = E[δ x (F n )]. The discretization procedure is based on a corresponding partition of the time interval as π : 0 = t 0 < • • • < t n = t, and the sequence of random variables F j which satisfy the relation

F j ∈ F t j .
Let us give some hints about the general strategy for the discretization: it is designed to take advantage of conditional Malliavin calculus, which allows to capture the convolution property of Gaussian distributions. We shall thus assume for the moment a structure of the form

F j = F j-1 + I j + R j , (43)
where we recall that F j-1 ∈ F t j-1 . In formula (43), the term I j will stand for a Gaussian random variable (conditionally to F t j-1 ) and R j refers to a small remainder term, whose contribution to the density of F j can be neglected with respect to the one induced by I j just like in the argument in (41). The local Gaussian bound (5) will be obtained from the density of the sum n j=1 I j . The argument will finish by an application of the Chapman-Kolmogorov formula.

As suggested by equation ( 6), and setting ∆ i j+1 (B) := B t i j+1 -B t i j , a natural candidate consists in taking F j = X t j , which yields

I j = d i=1 V i (X t j ) ∆ i j+1 (B), and R j = d i=1 t 0 V i (X s ) -V i (X t j ) dB i s . (44)
However, this simple and natural guess is not suitable for the fBm case. Indeed, the analysis of the variances of I j induced from the decomposition (44) reveals that a significant amount is generated by the covariances between the increments ∆ i j (B). Now, if we write

t 2H = E B i t 2 = E   n j=1 ∆ i j (B) 2   = n j,k=1 E ∆ i j (B)∆ i k (B) , (45)
we realize that the diagonal terms in the right hand side expression only accounts for a term of the form j |t j -t j-1 | 2H , which vanishes as the mesh of the partition goes to 0 when H ∈ (1/2, 1). This means that our decomposition (44) will not be able to capture the correct amount of variance contained in X t , and has to be modified. There are at least two natural generalizations of the Euler type scheme method described above:

(1) Take into account the off-diagonal terms in (45), and perform a block type analysis.

(2) Express the equation as an equation driven by the Wiener process W defined by relation [START_REF] Garsia | Continuity properties of Gaussian processes with multidimensional time parameter[END_REF] and take advantage of the independence of the increments of W . In the current paper we have chosen to follow the second approach above, and thus we first recall how to define equation (42) as a Stratonovich equation with respect to W .

Fractional equations as Stratonovich type equations.

In order to handle equation (42) as an equation with respect to W , let us first introduce the following functional space: Definition 5.2. Let |H| be the space of measurable functions φ : [0, T ] → R n such that

φ 2 |H| := α H 1 0 1 0 |φ r ||φ u ||r -u| 2H-2 dr du < +∞.
Note that |H| endowed with the norm • |H| is a Banach space of functions, which is also a subspace of H.

In the sequel we also consider random elements with values in |H|. In particular, the norm of φ in D D D 1,2 (|H|) is given by

φ D D D 1,2 (|H|) = E φ 2 |H| + E Dφ 2 |H|⊗|H| .
For these elements, the following result from [1, Proposition 3] allows to define Stratonovich type integrals (see [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] for a complete definition), which turn out to coincide with Young integrals in our cases of interest. In the sequel, we will use

1 0 u t • dB t ≡ d i=1 1 0 u i t • dB i t to denote Stratonovich integrals. Proposition 5.3. Let u = {u t , t ∈ [0, 1]} ∈ D D D 1,2 (|H|), such that 1 0 1 0 |D s u t ||t -s| 2H-2 dsdt < ∞. (46)
Then (i) The Stratonovich integral 1 0 u t • dB t in the sense of [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] exists and we also have

1 0 u t • dB t = δ δ δ(u) + α H 1 0 1 0 D s u t |t -s| 2H-2 dsdt. (47)
(ii) Whenever u ∈ C γ a.s. with γ > 1/2 and H ∈ (1/2, 1), the Stratonovich integral

1 0 u t • dB t coincides with the Young integral 1 0 u t dB t .
The next Proposition will allow us to interpret the stochastic integral appearing in (42) as a Stratonovich type integral. Proposition 5.4. Let X = {X t , t ∈ [0, 1]} be the solution to (42), and assume Hypothesis 1.1. Then the process X ∈ D D D 1,2 (|H|) and satisfies the equation

X t = a + d k=1 t 0 V k (X u ) • dB k u ,
where the Stratonovich integral can be decomposed as a Skorohod integral plus a trace term as in (47).

Proof. According to Proposition 5.3, we just have to prove that X ∈ D D D 1,2 (|H|) and satisfies relation (46). We first focus on proving the relation

E X 2 |H| + E DX 2 |H|⊗|H| < ∞.
In order to see the first part of this inequality, invoke relation [START_REF] Nourdin | Density estimates and concentration inequalities with Malliavin calculus[END_REF] and write

E X 2 |H| = α H 1 0 1 0 E [|X r ||X s |] |r -s| 2H-2 drds ≤ c E X 2 ∞ 1 0 1 0 |r -s| 2H-2 drds < c 1 .
Along the same lines and owing to [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF], it is also readily checked that E[ DX 2 |H|⊗|H| ] < ∞ and that relation (46) holds true, which ends the proof. Note that due to Proposition 5.3 (ii) and Proposition 2.5, we obtain the other assertions.

Finally, the following corollary is the key to the the effective decomposition we shall use in order to get our Gaussian lower bound on p t : Corollary 5.5. Under the same assumptions as for Proposition 5.4, the process K * t (V k (X)) ∈ dom(δ) and satisfies the equation

X t = a + d k=1 t 0 [K * t (V k (X))] s • dW k s = a + d k=1 t 0 t s ∂ u K(u, s)V k (X u ) du • dW k s , (48)
where the anticipative Stratonovich integral with respect to W can be decomposed as a Skorohod integral plus a trace term.

Proof. With Proposition 5.4 in hand, relation ( 48) is easily deduced from [17, Proposition 5.2.2] and Proposition 2.1.

5.

3. Discretization procedure. We now proceed to the decomposition of F n := X t as announced in (43), starting from the expression of F j for j = 0, . . . , n. Indeed, according to expression (48), a natural approximation sequence for X t based on a partition 0 = t 0 < . . . < t n = t of [0, t] is the following:

F i = F i-1 + I i + R i , (49)
where, introducing the additional notations η i (u) := inf(u, t i ), and g k i,s :=

t s ∂ u K(u, s)V k (X η i (u) ) du (50)
we set (note that

g k i-1,s ∈ F t i-1 ) F i-1 := d k=1 t i-1 0 g k i-1,s • dW k s , I i := d k=1 t i t i-1 g k i-1,s • dW k s = d k=1 V k (X t i-1 ) t i t i-1 K(t, s) dW k s , (51)
where the last integral above is simply a Wiener integral with respect to W . We also introduce a family of random variables R i defined by:

R i := d k=1 t i t i-1 Q k s • dW k s , (52)
where Q is the process defined by

Q k s := t s ∂ u K(u, s)[V k (X η i (u) ) -V k (X t i-1 )]du. (53)
Observe that if V is elliptic and bounded, it is clear from expression (51) that i Cov t i-1 (I i ) ≍ t 2H Id m up to a constant, independently of the particular values of the t i 's. We shall see however how to choose those values at Condition 5.8. Finally we introduce some random variables Φ M (N i γ,p (B)) for i = 1, . . . , n which allow us to control the supremum norm of the solution of the equation (42) and of their stochastic derivatives. This argument needs to be added in the methodology of [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF][START_REF] Kohatsu-Higa | Lower bounds for densities of uniformly elliptic random variables on Wiener space[END_REF] and therefore we have to tailor the arguments therein to our situation. The localization random variables are based on the family of functionals N i γ,p (B) defined by

N i γ,p (B) = t i t i-1 t i t i-1 |B v -B u | 2p |v -u| 2γp+2 dudv,
which can be compared to Hölder type norms and have the advantage that they can be differentiated with respect to B. In fact, we can see the aim of introducing this functional in the following proposition which is direct consequence of Garsia-Rodemich-Rumsey's Lemma (see e.g [START_REF] Garsia | Continuity properties of Gaussian processes with multidimensional time parameter[END_REF]).

Proposition 5.6. Let H > 1 2 and p such that 0 < γ < H -1 2p , then we have

B t i-1 ,t i ,γ ≤ c γ,p [N i γ,p (B)] 1/2p
. The next step is to study the conditional densities of the approximation sequence F i . To this aim, one has to control various terms for which the localization technique of Malliavin Calculus turns out to be useful. Specifically, recall that we have introduced families of functions Φ M , Φ M,ε given by expression [START_REF] Kou | Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule[END_REF]. In the sequel we localize our expectations using functionals of the type Φ M (N i γ,p (B)) and

Φ c i ,ǫ ( d j=1 t i t i-1 |D j r R i | 2 dr)
for some constants c i , ǫ of the form:

c i := λ 4 t i t i-1 K(t, s) 2 ds > 0, and
ǫ i := c i 2 > 0. (54)
Furthermore, in order to ease notations, notice that we will simply write:

Φ M ≡ Φ M (N i γ,p (B)), and
Φ c i ,ǫ i ≡ Φ c i ,ǫ i d j=1 t i t i-1 |D j r R i | 2 dr . (55)
With this additional notation in hand, we can proceed to the first step of our approximation scheme: since F i is F t i-1 conditionally non-degenerate and the localizations Φ M and Φ c i ,ǫ i ∈ D ∞ , we can write

E t i-1 [δ x (F i )] = E t i-1 [δ x (F i ) Φ M Φ c i ,ǫ i ] + E t i-1 [δ x (F i ) (1 -Φ M Φ c i ,ǫ i )],
and due to the non-negativity of the second term, we have

E t i-1 [δ x (F i )] ≥ E t i-1 [δ x (F i ) Φ M Φ c i ,ǫ i ].
Recalling that F i = F i-1 + I i + R i , we then obtain the following decomposition:

E t i-1 [δ x (F i )Φ M Φ c i ,ǫ i ] = J 1,i + J 2,i + J 3,i , (56)
where

J 1,i = E t i-1 [δ x (F i-1 + I i )], J 2,i = E t i-1 [δ x (F i-1 + I i )(Φ M Φ c i ,ǫ i -1)], (57) and
J 3,i = m j=1 E t i-1 Φ M Φ c i ,ǫ i 1 0 ∂ x j δ x (F i-1 + I i + ρR i ) R j i dρ . ( 58
)
Our aim is now to prove that in this decomposition J 1,i should yield the main contribution, while J 2,i is small because of the quantity (Φ M Φ c i ,ǫ i -1) whenever M and n are large enough and J 3,i is small due to the presence of the difference between X t i -X t i-1 in R i . We shall implement this strategy in the next subsections.

5.4. Upper and lower bounds on J 1,i . The main information which will be used about J 1,i is the following:

Proposition 5.7. Let J 1,i be defined by (57). Then under Hypothesis 1.1 we have

J 1,i = E t i-1 [δ x (F i-1 + I i )] = exp -1 2 (x -F i-1 ) * Σ -1 i-1 (x -F i-1 ) (2π) m/2 |Σ i-1 | 1/2 , ( 59
)
where Σ i-1 is a deterministic (conditionally to F t i-1 ) matrix such that

λ t i t i-1 K 2 (t, u) du Id m ≤ Σ i-1 ≤ Λ t i t i-1 K 2 (t, u) du Id m ,
and where the two strictly positive constants λ, Λ satisfy (4).

Proof. The fact that I i-1 is conditionally Gaussian is clear from expression (51), and this immediately yields our claim (59). Furthermore,

Σ i-1 := Cov t i-1 (I i ) = E t i-1 [I i I * i ] = E t i-1 d k=1 V k (X t i-1 ) t i t i-1 K(t, u) dW k u d l=1 V * l (X t i-1 ) t i t i-1 K(t, u) dW l u = d k=1 V k (X t i-1 )V * k (X t i-1 ) t i t i-1 K 2 (t, u) du,
which finishes the proof of our second claim thanks to Hypothesis 1.1.

The previous proposition induces a natural choice for the partition (t i ) in terms of the kernel K: Condition 5.8. We choose the partition 0 = t 0 < . . . < t n = t of [0, t] such that we have

t i t i-1 K 2 (t, u) du = t 2H
n =: σ 2 n for all i = 1, . . . , n. With this choice in hand, let us note the following properties for further use: Lemma 5.9. Let t 0 , . . . , t n be the partition of [0, t] defined by Condition 5.8. Then (i) The partition is constructed in a unique way.

(ii) We have 0 ≤ t i -t i-1 ≤ c H n -1/(2H) for all i = 1, . . . , n.

(iii) The parameters c i defined at (54) are all equal to λt 2H 4n .

Here, we have used that

1 {F i-1 +I i >x} ≤ 1.
In order to bound the right hand side of (62) we start by computing the Malliavin derivatives of I i . Recall that due to (51), we have for j = 1, . . . , d, α > 1 and r, r 1 , . . . , r α > t i-1 that

D j r I i = V j (X t i-1 )K(t, r) 1 [t i-1 ,t i ] (r)
, and D α r 1 ...rα I i = 0.

As far as Γ I i ,t i-1 is concerned, it is a conditionally deterministic quantity such that for i, j = 1, . . . , d, we can write

Γ I i ,t i-1 = d j=1 D j I i , D j I * i L 2 ([t i-1 ,t i ]) = d j=1 V j (X t i-1 )V * j (X t i-1
)

t i t i-1 K 2 (t, s)ds = σ 2 n V (X t i-1 )V * (X t i-1 ).
Using the ellipticity condition of Hypothesis 1.1(2) for V , we thus obtain that

0 ≤ Γ -1 I i ,t i-1 ≤ 1 λσ 2 n Id m . Therefore I i k 4 k 2 ,p 2 ,t i-1 ≤ C (σ 2 n Λ) k 4 2 and det(Γ I i ,t i-1 ) -1 k 3 p 3 ,t i-1 ≤ 1 λσ 2 n mk 3
.

Substituting these inequalities in (62), our proof is now finished.

From the above Proposition 5.11, we see that in order to get a convenient bound for J 2,i we need to study the random variable 1 -Φ M Φ c i ,ǫ i k 1 ,p 1 ,t i-1 . A suitable information for us will be the following bound: Proposition 5.12.

Let L i n (k 1 , p 1 ) = 1 -Φ M Φ c i ,ǫ i k 1 ,p 1 ,t i-1
be the random variable defined at Proposition 5.11. Then for any p (recall that

Φ M ≡ Φ M (N i γ,p (B))) large enough so that 2p(1 -γ H ) -k 1 -2 H > 0 with H -1 2 < γ < H -1 2p
, the following holds true: For any η > 0 there exists C M,η,p > 0 such that

E[L i n (k 1 , p 1 )] ≤ C M,η,p n -η . ( 63
)
Proof. We sketch the proof for p 1 = 2. The general case follows similarly. We start noting that it is enough to find a proper bound for 1-Φ M k 1 ,p 1 ,t i-1 and 1-Φ c i ,ǫ i k 1 ,p 1 ,t i-1 separately. We start with the first one. Using Chebyshev inequality we have, for any k 2 ≥ 1 and γ ∈ (0, H-1 2p )

E |1 -Φ M | 2 ≤ P(N i γ,p (B) > M -1) ≤ E |N i γ,p (B)| k 2 (M -1) k 2 . ( 64
) Evaluate now E[|N i γ,p (B)| k 2 ].
A simple application of Jensen's inequality yields:

E |N i γ,p (B))| k 2 = E t i t i-1 t i t i-1 |B v -B u | 2p |v -u| 2pγ+2 dudv k 2 ≤ c |t i -t i-1 | 2(k 2 -1)
t i t i-1 t i t i-1 E |B v -B u | 2pk 2 |v -u| (2pγ+2)k 2 dudv ≤ c k 2 ,p,γ,H |t i -t i-1 | 2k 2 p(H-γ) . ( 65
)
Recalling from that Lemma 5.9(ii) that |t i -t i-1 | ≤ c H n -1/(2H) and taking k 2 large enough, we obtain that

E[|N i γ,p (B))| k 2 ] ≤ c M,η n -η
for an arbitrary large η. This is consistent with relation (63).

In order to achieve the first claim, we also have to get a bound for the derivatives of Φ M . In fact, we shall only detail the calculations for the first derivative, since estimates for the higher order derivatives are similar. Now we resort to the explicit expression of the Malliavin derivative of Φ M given in [START_REF] Ll | Pathwise definition of second order SDEs[END_REF]Remark 3.2], which can be computed as:

|D j s Φ M | ≤ |Φ ′ M (N i γ,p (B))| µ s , where s ∈ [t i-1 , t i ]
and the process µ is defined by

µ s = 4p s t i-1 t i s |B ξ -B η | 2p-1 |ξ -η| 2γp+2 dξ dη 1 [t i-1 ,t i ] (s).
We thus get

DΦ M 2 L 2 ([t i-1 ,t i ]) = DΦ M 2 H([t i-1 ,t i ]) ≤ c H |Φ ′ M (N i γ,p (B))| 2 t i t i-1 t i t i-1 µ u |u -v| 2H-2 µ v dudv.
Furthermore, taking into account the fact that p is large enough, we obtain that for any exponent κ > 0 the following hold true:

E[|µ s | κ ] ≤ (2p) κ (t i -t i-1 ) 2(κ-1) s t i-1 t i s E |B ξ -B η | (2p-1)κ |ξ -η| (2γp+2)κ dξdη ≤ c p,κ,γ,H |t i -t i-1 | 2pκ(H-γ)-κH .
Applying this inequality with κ = 4, recalling from ( 14) that

Φ ′ M ∞ = φ ∞ ≤ c φ e -1
, and plugging (64), this yields

E DΦ M 2 L 2 [t i-1 ,t i ] ≤ c H Φ ′ M 2 ∞ t i t i-1 t i t i-1 |u -v| 2H-2 E[µ u µ v 1 {N i γ,p (B)>M -1} ]dudv ≤ c H,p t i t i-1 t i t i-1 |u -v| 2H-2 E 1 4 [µ 4 u ] E 1 4 [µ 4 v ] E |N i γ,p (B)| k 3 (M -1) k 3 dudv,
for an arbitrary large k 3 . Proceeding as in (65), for an arbitrary large k 4 we end up with:

E DΦ M 2 L 2 [t i-1 ,t i ] ≤ c H,p,M |t i -t i-1 | 2p(k 4 +2)(H-γ) ,
which corresponds again to our assertion (63

) since |t i -t i-1 | ≤ c H n -1/(2H) .
Finally, let us mention that along the same lines, we get a bound of the following form for the ℓ-th derivative:

E D j 1 ,...,j ℓ Φ M 2 L 2 ([t i-1 ,t i ] ℓ ) ≤ c H,p max j=1,...,ℓ Φ (j) M ∞ |t i -t i-1 | 2p(2+k 5 )(H-γ) ≤ c H,p,M |t i -t i-1 | 2p(2+k 5 )(H-γ) ,
for an arbitrary large k 5 . This yields our claim (63), invoking again the upper bound |t i -

t i-1 | ≤ c H n -1/(2H) .
The calculation for 1 -Φ c i ,ǫ i k 1 ,p 1 ,t i-1 is similar and we skip details for sake of conciseness. It is based on the fact that for any k 6 > 0, Chebyshev's inequality and Lemma 6.4 (postponed to the Appendix) imply that:

P d j=1 t i t i-1 |D j r R i | 2 dr > λ 8 t i t i-1 K(t, s) 2 ds ≤ λ 8 t i t i-1 K(t, s) 2 ds -k 6 E   d j=1 t i t i-1 |D j r R i | 2 dr k 6   ≤ c λσ 2 n -k 6 (t i -t i-1 ) (2γ+1)k 6 ≤ cn (1-2γ+1 2H )k 6 .
Here we have used the result in Lemma 5.9 (ii) once more and the fact that γ > H -1 2 . 5.6. Upper bounds for J 3,i . We now turn to the main technical issue in our computations, namely the bound on J 3,i . Our aim is thus to prove the following proposition: Proposition 5.13. Let J 3,i be the quantity defined by (58). Then there exist c > 0 and k > 0 such that for any H -1 2 < γ < H.

|J 3,i | ≤ c M,V,m (t i -t i-1 ) γ (σ 2 n ) m/2 ≤ c M,V,m n γ/2H (σ 2 n ) m/2 . ( 66
)
Proof. We start from expression (58) and normalize I i + ρR i in the following way: we just set

I i + ρR i = σ n U i , where U i = σ -1 n (I i + ρR i ). We thus have J 3,i = m j=1 E t i-1 Φ M Φ c i ,ǫ i 1 0 ∂ x j δ x (F i-1 + σ n U i ) R j i dρ .
Along the same lines as in (62), the integration by parts formula (15) now yields

J 3,i = σ -(m+1) n m j=1 1 0 E t i-1 1 {I i +ρR i >x-F i-1 } H t i-1 (j,1,...,m) (U i , R j i Φ M Φ c i ,ǫ i ) dρ.
Hence the following bound holds true (see [17, p. 102]):

|J 3,i | ≤ c 1,q σ -(m+1) n A 1 1 0 A 2 (ρ) A 3 (ρ) dρ,
where the quantities A 1 , A 2 (ρ), A 3 (ρ) are respectively defined by

A 1 = max j=1,..,m R j i Φ M ′ k 1 ,p 1 ,t i-1 , A 2 (ρ) = det(Γ -1 U i ,t i-1 )Φ M ′ Φ c i ,ǫ i k 3 p 3 ,t i-1 and A 3 (ρ) = U i Φ M ′ k 4 k 2 ,p 2 ,t i-1
, and where we also recall that R j i is defined by (52). Then the first inequality in (66) follows from Lemmas 6.4, 6.5 and 6.6 which have been postponed to the Appendix, and by choosing γ such that H -1 2 < γ. In order to go from the first inequality in (66) to the second one, we simply apply Lemma 5.9. 5.7. Lower bound. Let us first summarize the considerations of the previous section: starting from decomposition (56) and applying Corollary 5.10, Propositions 5.11, 5.12 and 5.13 and the forthcoming relation (76), we have obtained the following facts: The inequality

E t i-1 [δ x (F i )] ≥ J 1,i + J 2,i + J 3,i
holds true, and thus

E t i-1 [δ x (F i )] ≥ 1 (2π) m/2 (Λσ 2 n ) m/2 exp - |x -F i-1 | 2 2λ σ 2 n -c λ,Λ σ 2 n -k 2 L i n (k 1 , p 1 ) - c M,V,m n γ/2H (σ 2 n ) m/2 , (67) with the additional information E[L i n (k 1 , p 1 )] ≤ C M,η n -η
for an arbitrarily large exponent η. We are now ready to prove the main theorem of this article:

Proof of Theorem 1.2. With equation (67) in hand, we shall follow the strategy designed in [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF][START_REF] Kohatsu-Higa | Lower bounds for densities of uniformly elliptic random variables on Wiener space[END_REF]: Fix x-a throughout the proof and define the balls B i = B(y i , c 1 σ n ) for i = 1, . . . , n where y i = a + i n (x -a). We also define below an additional sequence {x i ; i = 1 . . . , n}, such that x i ∈ B i and x n = x. The constant c 1 will be fixed later on. We shall now proceed in a backward recursive way on the index i. For instance in order to go from n to n -1, we resort to (67) in order to write:

a y 0 y 1 • • • y i-1 B i x i y i c 1 σ n y i+1 • • • y n-1 x y n x-a n
E [δ x (F n )] = E[E t n-1 [δ x (F n )]] ≥ c V,m σ m n E exp - |x -F n-1 | 2 2λσ 2 n -c M,V,m n -κ ,
for a certain strictly positive constant κ. Hence

E [δ x (F n )] ≥ c V,m σ m n R E exp - |x -F n-1 | 2 2λσ 2 n -c M,V,m n -κ δ x n-1 (F n-1 ) dx n-1 ≥ c V,m σ m n B n-1 E exp - |x -F n-1 | 2 2λσ 2 n -c M,V,m n -κ δ x n-1 (F n-1 ) dx n-1 .
We now observe the following: if we wish the term δ x n-1 (F n-1 ) to give a non null contribution, the relations

x n-1 ∈ B(y n-1 , c 1 σ n ), x -y n-1 = x -a n , σ n = t H n 1/2 , |F n-1 -x n-1 | ≤ c 1 σ n must be satisfied. Moreover, from these conditions, it is easily seen that |x -F n-1 | ≤ 4c 1 σ n whenever n ≥ |x-a| 2 c 1 t 2H . We thus define a constant c 2 ≥ 1 4c 1 such that n = c 2 |x -a| 2 t 2H . ( 68
)
Then if we take c 1 such that exp -

8c 2 1 λ ≥ 1 2 and n such that c M,V,m n -κ ≤ 1/4, we obtain E [δ x (F n )] ≥ c V,m 4σ m n B n-1 E δ x n-1 (F n-1 ) dx n-1 .
These arguments can now be iterated backward from i = n-1 to 1, and the reader can easily check that the only additional required condition is the compatibility relation y i+1 -y i ≤ c 1 σ n (this will be verified below). Denoting by α m the volume of a unit ball in R m (namely α m = π m+2 /Γ( m 2 + 1)), we end up with

E [δ x (F n )] ≥ c V,m 4 σ m n n |B(0, c 1 σ n )| n-1 = c V,m 4 n n 1/2 t H nm c 1 t H n 1/2 m(n-1) α n-1 m (69) = c V,m 4 n (c m 1 α m ) n-1 n 1/2 t H m = 1 α m (c 1 t H ) m exp n ln c V,m c m 1 α m 4 + m 2 ln(n) .
Once here, we are reduced to tune our parameters according to the following constraints: (i) Recalling (68), we have that if c 1 is taken small enough so that ρ ≡ -ln(c V,m c m 1 α m /4) > 0 and (as alluded to above) such that exp(-8c 2 1 /λ) ≥ 1 2 and n ln(ρ) + m ln(n) ≥ 0 for all n ∈ N, we get

exp n ln c V,m c m 1 α m 4 = exp - ρc 2 x -a 2 t 2H .
We remark here that the values of c 1 , c 2 and c M,V,m are fixed independently of n. It is now easily seen that our bound (69) is of the form (5).

(ii) We now choose the constant c 2 in (68) so that the compatibility relation y i+1 -y i ≤ c 1 σ n is satisfied. Towards this aim, recall that

|y i+1 -y i | = |x -a| n = |x -a| n 1/2
|y i+1 -y i | = |x -a| n 1/2 c -1/2 2 t H |x -a| = c -1/2 2 σ n .
It is thus sufficient to take c

-1/2 2 ≤ c 1 ∧ (2c 1/ 2
1 ), which also satisfies that n ≥ |x-a| 2 4c 1 t 2H . This completes our proof.

Appendix: Some properties of stochastic derivatives

We start this technical section with a general bound on the space H related to fractional Brownian motion. Lemma 6.1. Let H ∈ (0, 1/2), t ∈ (0, 1] and consider the space H defined on [0, t] as in Section 2.1. Let f be an element of C γ ([0, t]) for 1/2 -H < γ < 1/2, with f ∞ ≤ a and f 0,t,γ ≤ b. Then f H ≤ c H a t H + b t γ+H .

Proof. For a function g defined on [0, t], recall that its fractional derivative is given by

D 1/2-H t - g u = g u (t -u) 1/2-H + t u g u -g v (v -u) 3/2-H dv. ( 70
)
Consider now f ∈ C γ ([0, t]) satisfying the conditions above, and set g u = u -(1/2-H) f u . According to [17, formula 5.31], we have

f 2 H ≤ c H t 0 s 1-2H D 1/2-H t - g s 2 ds. (71)
We now proceed to estimate the right hand side of relation (71). Indeed, plugging definition (70) into (71), it is readily checked that

f 2 H ≤ c H t 0 A 2 s ds + t 0 B 2 s ds , with A s = f s (t -s) 1/2-H , B s = t s f s -ψ v f v (v -s) 3/2-H dv,
where we have set ψ v = (s/v) 1/2-H . It is then easily seen that t 0 A 2 s ds ≤ c H a 2 t 2H . In order to bound B, notice that the function ψ is well defined on [s, t] and satisfies ψ s = 1,

ψ v ≤ 1 and |ψ ′ v | ≤ v -1 . |f s -ψ v f v | ≤ |f s -f v | ψ v + |f s ||1 -ψ v | ≤ b (v -s) γ + a|1 -ψ v | γ ≤ b + a s γ (v -s) γ .
Dividing this inequality by (v -s) Let us now state a bound on Malliavin derivatives.

Proof of relation [START_REF] Gubinelli | Controlling rough paths[END_REF]. We focus on the first derivative case, the other ones being handled in a similar fashion. We will thus prove that We now turn to the bounds on the process Q featuring in the definition of our remainders R i (see decomposition (49) of X t ): Lemma 6.2. Let X be the solution to (42), let η i be the function defined by (50) and Q the process given by (53). If r 1 , s ∈ (t i-1 , t i ) then the following bounds hold true:

Q k s ≤ c V K(t, s)|t i -t i-1 | γ Z i 0 (72) D l r 1 Q k s ≤ c V K(t, s)K(t, r 1 )Z i 1 , (73)
for F 1 -measurable random variables Z i 0 , Z i 1 defined by Z i 0 = B t i-1 ,t,γ ∨ B γ t i-1 ,t,γ and

Z i 1 = sup |D l r 1 (X v -X t i-1 )|, t i-1 ≤ r 1 ≤ v ≤ t i , (74)
admitting moments of all orders. In general, we can extend these results to Malliavin derivatives of arbitrary order ℓ ≥ 1 in the following way: for r 1 , s ∈ (t i-1 , t i ) and r 2 , . . . , r ℓ < t i we have

|D j 1 ...j ℓ r 1 ...r ℓ Q k s | ≤ c V K(t, s) Z i ℓ n j=1 K(t, r j ), (75)
for Z i ℓ ≡ sup |D j 1 ...j ℓ r 1 ...r ℓ (X v -X t i-1 )|, t i-1 ≤ r i ≤ v ≤ t i , i = 1, . . . , n , which is a F 1 -measurable random variable with moments of all orders.

Proof. The bound (72) is an easy consequence of (53), Proposition 2.5 and the fact that ∂ u K(u, s) ≥ 0. Moreover, observe that whenever r 1 > t i-1 we have D r 1 V k (X t i-1 ) = 0. Hence, using Proposition 2.1 we get:

|D l r 1 Q k s | = t s∨r 1 ∂ u K(u, s) D l r 1 V k (X η i (u) )du = t s∨r 1 ∂ u K(u, s)[K * t D l • V k (X η i (u) )] r 1 du = t s∨r 1 ∂ u K(u, s) t r 1 D l r 2 V k (X η i (u) )∂ r 2 K(r 2 , r 1 )dr 2 du .
It is thus readily checked that

|D l r 1 Q k s | ≤ c V Z i 1 t s∨r 1
∂ u K(u, s)K(t, r 1 )du ≤ c V Z i 1 K(t, s)K(t, r 1 ).

The general result (75) is now obtained by means of an induction argument and resorting to the same techniques as in the case of the first order derivative (namely ℓ = 1).

Remark 6.3. Note that due to the definition (74) of Z i l and Proposition 2.5 which controls the derivatives of X using the Hölder norms of B, the random variables Z verify:

|Z i j | ≤ C V exp C V B 1 γ t i-1 ,t i ,γ ,
for any γ ∈ ( 1 2 , H). Hence, applying Proposition 5.6 we obtain

|Z i j | ≤ C V exp C V,γ N i γ,p (B) 1 2γp
, for any p such that 0 < γ < H -1 2p . This relation yields in particular that Z i j ∈ ∩ q≥1 L q (Ω). Furthermore, once we localize by the random variables Φ M or Φ M ′ , we end up with:

max 0≤l≤k Z i l Φ M ′ ≤ c M,V,m , with c M,V,m = c V,m exp(c V,m (M ′ ) 1 2γp ). (76)
In the next proposition, we give norm estimates for the remainder terms R i needed in the upper bound for J 3,i . Lemma 6.4. In the setting of Proposition 5.4 and Corollary 5.5, with the definition (52) and (55), the following estimate is valid:

R j i Φ M ′ k 1 ,p 1 ,t i-1 ≤ c V,M (t i -t i-1 ) γ σ n . (77)
Proof. This result obviously involves the control of many derivative terms. For the sake of conciseness, we only sketch the bound for DR i . Now recall that

R i = d k=1 t i t i-1 Q k s • dW k s .
We now apply a small variant of [START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF]Proposition 1.3.8] to Stratonovich integrals, which states that for r ∈ [t i-1 , t i ] we have

D j r R i = Q j r + d k=1 t i t i-1 D j r Q k s • dW k s . (78)
Let us now evaluate the L 2 [t i-1 , t i ] norm of D j r R i . The main contribution for this norm comes from the term Q in the right hand side of (78), for which we obtain, according to (72),

t i t i-1 Q j r 2 dr ≤ c V |t i -t i-1 | 2γ Z i 0 2 t i t i-1 K 2 (t, r) dr = c V Z i 0 2 |t i -t i-1 | 2γ σ 2
n , and thus

E 1/p 1 t i-1 Q p 1 L 2 ([t i-1 ,t i ]) Φ M ′ ≤ c V |t i -t i-1 | γ σ n E p 1 t i-1 Z i 0 p 1 Φ M ′ ≤ c V,M |t i -t i-1 | γ σ n ,
which is consistent with our claim (77). Let us give another example of term which has to be analyzed in order to bound the norm of D j r R i : the term A defined as

A := E 1/p 1 t i-1 t i t i-1 dr t i t i-1 ds D j r Q k s 2 p 1 2 Φ M ′ .

Figure 1 .

 1 Figure 1. Space partition for the lower bound, with sequence y 1 , . . . , y n and x i .

|D u F

 u | ≤ ess sup u≤r |D r F |K(t, u).Indeed, according to Proposition 2.1, we have that forF ∈ F t |D u F | = |[K * t DF ] u | = | t u D r F ∂ r K(r, u)dr| ≤ ess sup u≤r≤t |D r F |K(t, u),which is exactly our claim.

1 , h 2 ∈ H. A F -measurable real valued random variable F is then said to be cylin- drical if it can be written, for a given n ≥ 1, as

	2.1.2. Malliavin calculus for B. Isometry arguments allow to define the Wiener integral B(h) = 1 0 h s , dB s for any element h ∈ H, such that it satisfies E[B(h 1 ) B(h 2 )] = h 1 , h 2 H for any h

  3/2-H , recalling that γ ≤ 1/2 and integrating over [s, t], we get |B s | ≤ c H b + a s γ (t -s) γ-(1/2-H) , which entails that H a 2 t 2H + b 2 t 2(γ+H) .

	t
	0 s ds ≤ c Gathering our bounds on B 2 t 0 A 2 s ds and

t 0 B 2 s ds, our proof is now finished.

grants of the Japanese government.

Proof. Our first claim stems from the fact that t 0 K 2 (t, u) du = t 2H and v → τ v K 2 (t, u) du is a strictly decreasing function for all 0 ≤ v ≤ τ ≤ t.

In order to prove our item (ii) recall expression [START_REF] Cass | Non-degeneracy of Wiener functionals arising from rough differential equations[END_REF], from which we easily deduce the bound

Consider now a fixed point τ ∈ (0, t] and 0

n . Thanks to the bound (60) we have v τ ≥ w τ where w τ ≡ w is defined by

In addition, since

for w < τ < t, which means that w τ ≥ x τ where x τ is defined by the equation

n . The latter equation can be solved explicitly as x τ = τ -c H t n 1/(2H) , and summarizing our last considerations we end up with the relation ) , which easily yields our assertion (ii). The proof of (iii) is straightforward. Now we state the following Corollary to Proposition 5.7, whose immediate proof is left to the reader: Corollary 5.10. Let J 1,i be defined by (57). Then under Hypothesis 1.1 and Condition 5.8 we have for σ

Summarizing the considerations of this section, we have obtained that the main contribution to E t i-1 [δ x (F i )], J 1,i , is of the order given by (61). Most of our work is now devoted to prove that the contributions of J 2,i and J 3,i are smaller than a fraction of (61) if M, n are conveniently chosen. 5.5. Upper bounds for J 2,i . We start the control of J 2,i by stating a bound in terms of the localization we have chosen: Proposition 5.11. Let J 2,i be the quantity defined by (57). Then there exists positive constants c λ,Λ , k 1 , k 2 and p 1 independent of n such that:

n , and where we recall that the norms • k,p,t have been introduced at equation (12) and the random variables Φ M , Φ c i ,ǫ i at equation (55).

Proof. Our strategy hinges on the conditional integration by parts formula we have introduced in Proposition 2.2 , which gives for some constants k i , p i , i = 1, ..., 4,

Along the same lines as above, using (72), we find

which is a remainder term with respect to (77). Notice that many other higher order terms have to be evaluated in order to complete the proof. We omit these cumbersome but routine developments for sake of conciseness.

We now turn to the bound on A 2 (ρ):

. Then this quantity is uniformly bounded in n, ρ and ω ∈ Ω.

Proof. First remark that using Proposition 4 in [START_REF] Bally | Lower bounds for the density of locally elliptic Itô processes[END_REF], we have that

Moreover, we have localized

, from which the result follows.

The estimates for A 3 (ρ) are obtained in a similar fashion. In fact, we have: Lemma 6.6. The same conclusion as in Lemma 6.5 holds true for the quantity A 3 (ρ) = U i Φ M ′ k 4 k 2 ,p 2 ,t i-1 . Proof. With respect to Lemma 6.4, we only need to consider additionally the bound for

The above follows from Hölder's inequality. Therefore the result follows from straightforward calculations for I i as in the proof of Proposition 5.11.