Vlad Bally
email: bally@univ-mlv.fr

Lucia Caramellino
email: caramell@mat.uniroma2.it

Tubes estimates for diffusion processes under a local Hörmander condition of order one

Keywords: Hörmander condition, Tube estimates, Diffusion processes, Caratheodory metric. 2000 MSC: 60H07, 60H30

We consider a diffusion process X t and a skeleton curve x t (φ) and we give a lower bound for P (sup t≤T d(X t , x t (φ)) ≤ R). This result is obtained under the hypothesis that the strong Hörmander condition of order one (which involves the diffusion vector fields and the first Lie brackets) holds in every point x t (φ), 0 ≤ t ≤ T. Here d is a distance which reflects the non isotropic behavior of the diffusion process which moves with speed √ t in the directions of the diffusion vector fields but with speed t in the directions of the first order Lie brackets. We prove that d is locally equivalent with the standard control metric d c and that our estimates hold for d c as well.

Introduction

We consider the diffusion process solution of dX t = d j=1 σ j (t, X t ) • dW j t + b(t, X t )dt where the coefficients σ j , b are three times differentiable and verify the strong Hörmander condition on order one (involving σ j and the first order Lie brackets [σ i , σ j ]) locally around a skeleton path dx t (φ) = d j=1 σ j (t, x t (φ))φ j t dt + b(t, x t (φ))dt. The aim of this paper is to give a lower bound for the probability that X t remains in a tube around x t (φ) for t ≤ T. This problem has already been addressed in the literature. The first result was given by Stroock and Varadhan in their celebrated paper [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]. They obtain a lower bound for P (sup t≤T X t -x t (φ) ≤ R) and use it in order to prove the support theorem for diffusion processes. Here X t -x t (φ) is the Euclidian norm. Later, one has considered other norms which reflect the degree of regularity of the trajectories of the diffusion process X t : Ben Arous and Gradinaru [START_REF] Arous | Normes Hölderiennes et support des diffusions[END_REF] and Ben Arous, Gradinaru and Ledoux [START_REF] Arous | Hölder norms and the support theorem for diffusions[END_REF] obtained similar results for the Hölder norm. And more recently Friz, Lyons and Stroock [START_REF] Friz | Lévy's area under conditioning[END_REF] use a norm related to the rough path theory. All these results hold without any non degeneracy assumption. Tubes estimates has also been considered in connection with the Onsager-Machlup functional for diffusion processes. There is an abundant literature on this subject: see e.g. [START_REF] Capitaine | Onsager-Machlup functional for some smooth norms on Wiener space[END_REF], [START_REF] Capitaine | On the Onsager-Machlup functional for elliptic diffusion processes[END_REF], [START_REF] Hara | Lagrangian for pinned diffusion processes[END_REF], [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF], [START_REF] Zeitouni | On the Onsager-Machlup functional for diffusion processes around non C 2 curves[END_REF]. In this case one considers strong ellipticity conditions and the norm which describes the tube is the Euclidian norm or some Hölder norm. Notice that these are asymptotic results whether in our paper we give estimates which are non asymptotic.

Finally, in [START_REF] Bally | Estimates for the probability that a Itô process remains near a path[END_REF] and [START_REF] Bally | Tube estimates for Itô processes and density minorations for log-normal type diffusions[END_REF] one obtains similar lower bounds for general Itô processes under an ellipticity assumption. The specific point in our paper is that we use a distance which reflects the non isotropic structure of the problem: the diffusion process X t moves with speed √ t in the direction of the diffusion vector fields σ j and with speed t = √ t × √ t in the direction of [σ i , σ j ]. Let us be more precise. For R > 0 and x ∈ R n we construct the matrix A R (t, x) with columns √ Rσ i (t, x), [ √ Rσ j , √ Rσ p ](t, x), 1 ≤ i, j, p ≤ d. If the above vectors span R n the matrix A R A * R (t, x) is invertible, so we are able to define the norm |y| 2 A R (t,x) = (A R A * R ) -1 (t, x)y, y .

Our main result is the following (see Theorem 3 for a precise statement): we assume that the non-degeneracy condition holds along the curve x t (φ), 0 ≤ t ≤ T and we prove

P (sup t≤T |X t -x t (φ)| A R (t,xt(φ)) ≤ 1) ≥ exp -C 1 R + T 0 |φ t | 2 dt .
Computations involving the above norms are generally not easy -so we give some estimates which seem to be more explicit. In Proposition 1 we prove that |y| A R (t,x) describes (roughly speaking) ellipsoids with semi-axes of length √ R in the directions of σ j (t, x) and of length R in the directions of [σ i , σ j ](t, x). Moreover we associate to the above norms the following semi-distance: d(x, y) < R if and only if |y| A R (x) < 1. With this definition we have {sup t≤T |X t -x t (φ)| A R (t,xt(φ)) ≤ 1} = {sup t≤T d(x t (φ), X t ) ≤ R}. In Proposition 28 we prove that the semi-distance d is equivalent with the standard control metric d c (see [START_REF] Hara | Lagrangian for pinned diffusion processes[END_REF] for the definition) so the estimates of the tubes hold in the control metric as well. In Proposition 6 we give local lower and upper bounds for d and d c in terms of some semi-distances which describe in a more explicit way the ellipsoid structure we mentioned above. The paper is organized as follows. In Section 2 we give the statements of the main results. In Section 3 we consider a process Z t which is a linear combination of W j t , j = 1, ..., d and of t 0 W i s dW j s , 1 ≤ i, j ≤ d. And we give a decomposition of such a process -this decomposition represents the main ingredient in our approach. Roughly speaking the idea is the following: we consider a small interval of time [0, δ] and we split it in d subintervals I i = (t i-1 , t i ] with t i = i d δ. We fix i and for t ∈ I i we take conditional expectation with respect to W j t , j = i so all these processes appear as "controls". And the only process which is at work is W i t . Then the vector (W i t i -W i t i-1 ), t i t i-1 (W j s -W j t i-1 )dW i s , j = i is Gaussian (with respect to the above mentioned conditional probability). And we may choose the trajectories (controls) (W j s -W j t i-1 ) s∈I i , j = i in such a way that the covariance matrix of the above Gaussian vector is non degenerated (this is a support property proven in Section 7). Then we are able to use estimates for non degenerated Gaussian random variables. The process Z t appears as the principal part in the development in stochastic series of order two of the diffusion process X t . In Section 4 we use the estimates for Z t in order to obtain estimates for X t and so to finish the proof of the main theorem stated in Section 2. The fact that one may choose (W j s -W j t i-1 ) s∈I i , j = i in an appropriate way is due to the support theorem for the Brownian motion. But the quantitative property that we use employs in a crucial way the estimates of the variance (with respect to the time) of the Brownian motion obtained in [START_REF] Donati-Martin | Fubini's theorem for double Wiener integrals and the variance of the Brownian path[END_REF].

Acknowledgments. We are grateful to Arturo Kohatsu-Higa and to Peter Friz for useful discussions on this topic.

Notations and main results

We consider the n dimensional diffusion process

dX t = d j=1 σ j (t, X t ) • dW j t + b(t, X t )dt (1)
where W = (W 1 , ..., W d ) is a standard Brownian motion, •dW j t denotes the Stratonovich integral and σ j , b : R + × R n → R n are three time differentiable in x ∈ R n and one time differentiable with respect to the time t ∈ R + . We also assume that the derivatives with respect to the space x ∈ R n are one time differentiable with respect to t. And for (t, x) ∈ R + ×R n we denote by n(t, x) a constant such that for every s ∈ [(t-1)∨0, t+1], y ∈ B(x, 1) and for every multi index α of length less or equal to three

|∂ α x b(s, y)| + |∂ t ∂ α x b(s, y)| + d j=1 |∂ α x σ j (s, y)| + |∂ t ∂ α x σ j (s, y)|) ≤ n(t, x). (2)
Here, α = (α 1 , ..., α k ) ∈ {1, ..., n} k is a multi index and |α| = k is the length of α and

∂ α x = ∂ xα 1 ...∂ xα k .
In the following we assume that for external reasons one produces a continuous adapted process X which solves equation (1) on the time interval [0, T ] and we give estimates for this process. More precisely, for φ ∈ L 2 ([0, T ]; R d ), we assume there exists a solution of

dx t (φ) = d j=1 σ j (t, x t (φ))φ j t dt + b(t, x t (φ))dt (3)
and we want to estimate the probability that X t remains in a tube around the deterministic curve x t = x t (φ). We need some more notations. First, we use the following notation of directional derivatives: for f, g : R

+ × R n → R n we define ∂ g f (t, x) = n i=1 g i (t, x)∂ x i f (t,
x) and we recall that the Lie bracket (with respect to the space variable x) is defined as [f, g](t, x) = ∂ g f (t, x) -∂ f g(t, x). Moreover, let M ∈ M n×m be a matrix (which generally may be not square) such that MM * is invertible (M * denotes the transposed matrix). We denote by λ * (M) (respectively λ * (M)) the smaller (respectively the larger) eigenvalue of MM * and we consider the norm on

R n |y| M = (MM * ) -1 y, y . (4)
We are concerned with the matrix A(t, x) ∈ M n×m with columns σ i (t, x), [σ j , σ p ](t, x), 1 ≤ i, j, p ≤ d, j = p. Here and all along the paper m = d 2 .

We will write A(t, x) = (σ i (x), [σ j , σ p ](t, x)) i,j,p=1,...,d,j =p .

We denote by λ(t, x) the lower eigenvalue of A(t, x) that is

λ(t, x) = inf |ξ|=1 m i=1 A i (t, x), ξ 2 , (6)
A i (t, x), i = 1, . . . , m, denoting the columns of A(t, x). Moreover for R > 0 we define

A R (t, x) = ( √ Rσ i (t, x), [ √ Rσ j , √ Rσ p ](t, x)) i,j,p=1,...,d,j =p . Consider now some x ∈ R n , t ≥ 0 such that (σ i (t, x), [σ j , σ p ](t, x)) i,j,p=1,...,d,j =p span R n . Then A R A * R (t, x
) is invertible and we may define |y| A R (t,x) . We give some lower and upper bounds for |y| A R (t,x) . We denote by S(t, x) the space spanned by σ 1 (t, x), ..., σ d (t, x) and by S ⊥ (t, x) the orthogonal of S(t, x). We also denote by Π t,x the projection on S(t, x) and by Π ⊥ t,x the projection on S ⊥ (t, x). Moreover we denote

λ t,x = inf ξ∈S(t,x),|ξ|=1 d i=1 σ i (t, x), ξ 2 , λ ⊥ t,x = inf ξ∈S ⊥ (t,x),|ξ|=1 i<j [σ i , σ j ](t, x), ξ 2 . ( 7
)
By the very definition λ t,x > 0 (which is different from λ(t, x)) and under our hypothesis λ ⊥ t,x > 0 also. Then Proposition 26 gives:

Proposition 1 If R ≤ λ t,x /(4m × n 4 (t, x)) then 1 4Rn 2 (t, x) |Π t,x y| 2 + 1 4R 2 n 2 (t, x) Π ⊥ t,x y 2 ≤ |y| 2 A R (t,x) ≤ 4 Rλ t,x |Π t,x y| 2 + 4 R 2 λ ⊥ t,x Π ⊥ t,x y 2 . ( 8
)
For µ ≥ 1 and 0 < h ≤ 1 we denote by L(µ, h) the class of non negative functions f : R + → R + which have the property

f (t) ≤ µf (s) for |t -s| ≤ h.
We will make the following hypothesis: there exists some functions n : [0, T ] → [1, ∞) and λ : [0, T ] → (0, 1] such that for some µ ≥ 1 and 0 < h ≤ 1 we have

(H 1 ) n(t, x t (φ)) ≤ n t , ∀t ∈ [0, T ], (H 2 ) λ(t, x t (φ)) ≥ λ t > 0, ∀t ∈ [0, T ], (H 3 ) n . , λ . ∈ L(µ, h). (9)
Remark 2 The hypothesis (H 2 ) implies that for each t ∈ (0, T ), the space R n is spanned by the vectors (σ i (t, x t ), [σ j , σ p ](t, x t )) i,j,p=1,...,d,j<p , so the Hörmander condition holds along the curve x t (φ).

The main result in this paper is the following.

Theorem 3 Suppose that (H 1 ), (H 2 ) and (H 3 ) hold and that X 0 = x 0 (φ). Let ρ ∈ (0, 1).

There exists a universal constant C (depending on d and ρ only) such that for every R ∈ (0, 1) one has

P (sup t≤T |X t -x t (φ)| A R (t,xt(φ)) ≤ 1) ≥ exp -Cµ 9 T h + T 0 n 6(1+dρ) t λ 1+2dρ t 1 R + |φ t | 2 dt . ( 10
)
Remark 4 Suppose that X t = W t is just the Brownian motion and that x t = 0, so that

n t = 1, λ t = 1, µ = 1 and φ t = 0. Then |X t -x t | A R (xt(φ)) = R -1/2 W t so we obtain P (sup t≤T |W t | ≤ √ R) ≥ exp(-CT /R
) which is coherent with the standard estimate (see [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF]).

Remark 5 Since ∂ t x t (φ) -b(t, x t (φ)) = σ(t, x t (φ))φ(t) we immediately obtain 1 dn(t, x t (φ)) |∂ t x t (φ) -b(t, x t (φ))| ≤ |φ(t)| ≤ 1 λ t,xt(φ) |∂ t x t (φ) -b(t, x t (φ))| with λ t,xt(φ)
given in [START_REF] Capitaine | Onsager-Machlup functional for some smooth norms on Wiener space[END_REF].

We establish now the link between the norm |z| A R (t,x) and the control (Caratheodory) distance. We will use in a crucial way the alternative characterizations given in [START_REF] Nagel | Balls and metrics defined by vector fields, I. Basic properties[END_REF] for this distance -and these results hold in the homogeneous case: the coefficients of the equations do not depend on time: σ j (t, x) = σ j (x) and b(t, x) = b(x). Consequently now on we have a matrix A R (x) instead of A R (t, x). We define the semi-distance d : R n × R n → R + by d(x, y) < √ R if and only if |y| A R (x) < 1 (see page 37 for the definition of a semi-distance). We also consider the standard control distance d c (Caratheodory distance) associated to σ 1 , ..., σ d in the following way. Let y t (φ) be the solution of the equation dy t (φ) = d j=1 σ j (y t (φ))φ j t dt (notice that here b = 0). We denote C(x, y) = {φ ∈ L 2 (0, 1) : y 0 (φ) = x, y 1 (φ) = y} and we define

d c (x, y) = inf 1 0 |φ s | 2 ds 1/2 : φ ∈ C(x, y) . (11)
In Section 8 Theorem 28 we prove that d is locally equivalent with d c . Moreover we obtain the following bounds for them. We define d(x, y) and d(x, y) as follows:

• d(x, y) < √ R if and only if 4 Rλ x |Π x (y -x)| 2 + 4 R 2 λ ⊥ x Π ⊥ x (y -x) 2 < 1; • d(x, y) < √ R if and only if 1 4Rn 2 x |Π x (y -x)| 2 + 1 4R 2 n 2 x Π ⊥ x (y -x) 2 < 1.
Then as an immediate consequence (we give a detailed proof at the end of Appendix 4) of Proposition 1 and Theorem 28 we obtain:

Proposition 6 Let x, y ∈ R n be such that |y -x| ≤ λ x λ * (A(x)) (4m)n 4 (x) . ( 12
)
Then d(x, y) ≤ d(x, y) ≤ d(x, y). ( 13
)
Moreover for every compact set K ⊂ R n there exists some constants C K , r K such that for ever x, y ∈ K which satisfy [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF] and such that d(x, y) ≤ r K one has

1 C K d(x, y) ≤ d c (x, y) ≤ C K d(x, y). (14)
As an immediate consequence of the definition of d and of the local equivalence of d c with d we obtain the following:

Proposition 7 Suppose that (H i ), i = 1, 2, 3 hold and X 0 = x 0 (φ). Let ρ ∈ (0, 1). There exists a universal constant C (depending on d and ρ only) such that for every R ∈ (0, 1) one has

P (sup t≤T d(x t (φ), X t ) ≤ R) ≥ exp(-Cµ 9 ( T h + T 0 n 6(1+dρ) t λ 1+2dρ t ( 1 R + |φ t | 2 )dt)).
Moreover there exists a constant C (depending on d and ρ but also on x t (φ) and on the coefficients σ i (x t (φ)), b(x t (φ)) and on their derivatives up to order three) such that

P (sup t≤T d c (x t (φ), X t ) ≤ R) ≥ exp(-Cµ 9 ( T h + T 0 n 6(1+dρ) t λ 1+2dα t ( 1 R + |φ t | 2 )dt)). ( 15
)
We finish this section with two simple examples.

Example 1. We consider the two dimensional diffusion process

X 1 t = x 1 + W 1 t , X 2 t = x 2 + t 0 X 1 s dW 2 s .

Straightforward computations give

|ξ| 2 A δ (x) = |T x,δ ξ| 2 with T x,δ ξ = ( 1 √ δ ξ 1 , 1 δ(δ + x 2 1 ) ξ 2 ).
In particular, if

x 1 = 0 then T 0,δ ξ = ( 1 √ δ ξ 1 , 1 δ ξ 2
) and consequently {ξ : |ξ| A δ (x) ≤ 1} is an ellipsoid. But if x 1 = 0 and δ is small, then the distance given by |ξ| A δ (x) is equivalent with the Euclidian one. If we take a path x t which keeps far from zero then we have ellipticity along the path and so we may use estimates for elliptic processes (see [START_REF] Bally | Estimates for the probability that a Itô process remains near a path[END_REF] and [START_REF] Bally | Tube estimates for Itô processes and density minorations for log-normal type diffusions[END_REF]). But if x 1 (t) = 0 for some t ∈ [0, T ] then we may no more use them. Let us compare the norm here and the norm in the elliptic case: if x 1 > 0 the diffusion matrix is not degenerated so we may consider the norm |ξ| B δ (x) with B δ (x) = δσσ * (x). We have

|ξ| 2 B δ (x) = 1 δ ξ 2 1 + 1 δx 2 1 ξ 2 2 ≥ 1 δ ξ 2 1 + 1 δ(δ + x 1 ) ξ 2 2 = |ξ| 2 A δ (x) .
So the estimates obtained using the Lie brackets are sharper even if ellipticity holds.

Let us now take x 1 = x 2 = 0, x t (φ) = (0, 0). We have n s = 1 and λ s = 1 and X t -

x t = (W 1 t , t 0 W 1 s dW 2 s
). And we obtain

P (sup t≤T 1 δ W 1 t 2 + 1 δ 2 t 0 W 1 s dW 2 s 2 ≤ 1) = P (sup t≤T (|X t -x t | 2 A δ (0) ≤ 1) ≥ e -C/δ .
Example 2. The principal invariant diffusion on the Heisenberg group. We consider the diffusion process

X 1 t = x 1 + W 1 t , X 2 t = x 2 + W 2 t , X 3 t = x 3 + 1 2 t 0 X 1 s dW 2 s - 1 2 t 0 X 2 s dW 1 s .

Direct computations give

|ξ| 2 A δ (x) = A -1 δ (x)ξ 2 = 1 δ ξ 1 -ξ 3 × x 2 2 √ δ 2 + 1 δ ξ 2 -ξ 3 × x 1 2 √ δ 2 + ξ 2 3 δ 2 .
In particular for x = 0 we obtain

P sup t≤T /δ W 1 t 2 + W 2 t 2 + A 2 t (W ) ≤ 1 = P sup t≤T 1 δ W 1 t 2 + 1 δ W 2 t 2 + 1 δ 2 A 2 t (W ) ≤ 1 ≥ e -CT δ where A t (W ) = t 0 W 1 s dW 2 s - t 0 W 2 s dW 1 s .
3 Multiple stochastic integrals

Decomposition

We consider the stochastic process

Z(t) = d i=1 a i W i t + d i,j=1 a i,j t 0 W i s • dW j s ( 16
)
with a i , a i,j ∈ R n . Our aim is to give a decomposition for this process. In order to do it we have to introduce some notation. We fix δ > 0 and we denote s k (δ) = k d δ and

∆ i k (δ, W ) = W i s k (δ) -W i s k-1 (δ) , ∆ i,j k (δ, W ) = s k (δ) s k-1 (δ) (W i s -W i s k-1 ) • dW j s .
Notice that ∆ i,j k (δ, W ) is the Stratonovich integral, but for i = j it coincides with the Ito integral. When now confusion is possible we use the short notation

s k = s k (δ), ∆ i k = ∆ i k (δ, W ), ∆ i,j k = ∆ i,j k (δ, W
). Moreover for p = 1, ..., d we define

µ p (δ, W ) = i =p ∆ p i ψ p (δ, W ) = i =j,i =p,j =p a i,j ∆ i,j p + d l=p+1 i =p d j =l a i,j ∆ j l ∆ i p + 1 2 d i =p a i,i ∆ i p 2 ε p (δ, W ) = d l>p d j =l a p,j ∆ j l + d p>l d j =l a j,p ∆ j l + j =p a p,j ∆ j p η p (δ, W ) = 1 2 a p,p ∆ p p 2 + d l>p a p,l ∆ l l ∆ p p + ∆ p p ε p . (17)
We denote η(δ, W ) = d p=1 η p (δ, W ) and ψ(δ, W ) = d p=1 ψ p (δ, W ) and [a] i,p = a i,p -a p,i .

Our aim is to prove the following decomposition.

Proposition 8

Z(δ) = d p=1 a p (∆ p p (δ, W ) + µ p (δ, W )) + d p=1 i =p [a] i,p ∆ i,p p (δ, W ) + η(δ, W ) + ψ(δ, W ) (19) Remark 9
The reason of being of this decomposition is the following. We split the time interval (0, δ) in d sub intervals of length δ/d. And we also split the Brownian motion in corresponding pieces:

(W i s -W i s p-1 ) s p-1 ≤s≤sp , i = 1, ..., d. Let us fix i. For s ∈ (s i-1 , s i ) we have the processes (W j s -W j s i-1 ) s i-1 ≤s≤s i , j = 1, ..., d.
Our idea is to settle a calculus which is based on W i and to take conditional expectation with respect to W j , j = i. So (W j s -W j s i-1 ) s i-1 ≤s≤s i , j = i will appear as parameters (or controls) which we may choose in an appropriate way. And the random variables on which the calculus is based are

∆ i i = W i s i - W i s i-1 and ∆ j,i i = s i s i-1 (W j s -W j s i-1 )dW i s , j = i.
These are the random variables that we have emphasized in the decomposition of Z(δ). Notice that, conditionally to the controls (W j s -W j s i-1 ) s i-1 ≤s≤s i , j = i, this is a centered Gaussian vector and, under appropriate hypothesis on the controls this Gaussian vector is non degenerated (we treat in the Appendix 3 the problem of the choice of the controls). But there is another term which appear and which is difficult to handle by a choice of the controls W j : this is ∆ i,j i =

s i s i-1 (W i s -W i s i-1 )dW j s . So we use the identity ∆ i,j i = ∆ j i ∆ i i -∆ j,i
i in order to eliminate this term -and this is the reason for which (a i,j -a j,i ) = [a] i,j appears.

Proof. We decompose

Z(δ) = d l=1 Z(s l ) -Z(s l-1 ) = d l=1 d i=1 a i ∆ i l + d i,j=1 a i,j s l s l-1 W i s • dW j s
and we write

s l s l-1 W i s • dW j s = W i s l-1 ∆ j l + ∆ i,j l = ( l-1 p=1 ∆ i p )∆ j l + ∆ i,j l .
Then

Z(δ) = d l=1 d i=1 a i ∆ i l + d l=1 d i,j=1 a i,j ( l-1 p=1 ∆ i p )∆ j l + d l=1 d i,j=1 a i,j ∆ i,j l =: S 1 + S 2 + S 3 .
Notice first that

S 1 = d l=1 a l ∆ l l + d l=1 i =l a i ∆ i l .
We treat now S 3 . We will use the identities

∆ i l 2 = 2∆ i,i l and ∆ i l ∆ j l = ∆ i,j l + ∆ j,i l . Then .S 3 = d l=1 d i=1 a i,i ∆ i,i l + d l=1 i =j a i,j ∆ i,j l = d l=1 d i=1 a i,i ∆ i,i l + d l=1 i =l a i,l ∆ i,l l + d l=1 j =l a l,j ∆ l,j l + d l=1 i =j,i =lj =l a i,j ∆ i,j l = 1 2 d l=1 d i=1 a i,i ∆ i l 2 + d l=1 i =l a i,l ∆ i,l l + d l=1 j =l a l,j ∆ j l ∆ l l -∆ j,l l + d l=1 i =j,i =l,j =l a i,j ∆ i,j l = 1 2 d i=1 a i,i ∆ i i 2 + 1 2 d l=1 d i =l a i,i ∆ i l 2 + d l=1 i =l (a i,l -a l,i )∆ i,l l + d l=1 j =l a l,j ∆ j l ∆ l l + d l=1 i =j,i =l, =j = a i,j ∆ i,j l .
We treat now S 2 . We want to emphasis terms which contain ∆ i i . We have

S 2 = d l>p d i,j=1 a i,j ∆ i p ∆ j l = S ′ 2 + S ′′ 2 + S ′′′ 2 + S iv 2 with d l>p = d p=1 d l=p+1 and
S ′ 2 = d l>p a p,l ∆ p p ∆ l l , S ′′ 2 = d l>p d j =l a p,j ∆ p p ∆ j l S ′′′ 2 = d l>p d i =p a i,l ∆ i p ∆ l l , S iv 2 = d l>p d i =p,j =l a i,j ∆ i p ∆ j l .
We have

S ′′ 2 = d p=1 ∆ p p d l=p+1 d j =l a p,j ∆ j l and S ′′′ 2 = d l=1 ∆ l l l-1 p=1 d i =p a i,l ∆ i p = d p=1 ∆ p p p-1 l=1 d j =l a j,p ∆ j l so that S ′′ 2 + S ′′′ 2 = d p=1 ∆ p p d l=p+1 d j =l a p,j ∆ j l + p-1 l=1 d j =l a j,p ∆ j l .
Finally

Z(δ) = d l=1 a l ∆ l l + d l=1 i =l a i ∆ i l + d l>p a p,l ∆ p p ∆ l l + d p=1 ∆ p p d l>p d j =l a p,j ∆ j l + d p>l d j =l a j,p ∆ j l + d l>p d i =p,j =l a i,j ∆ i p ∆ j l + 1 2 d i=1 a i,i ∆ i i 2 + 1 2 d l=1 d i =l a i,i ∆ i l 2 + d l=1 i =l (a i,l -a l,i )∆ i,l l + d l=1 j =l a l,j ∆ j l ∆ l l + d l=1 i =j,i =l,j =l a i,j ∆ i,j l .
We want to compute the coefficient of ∆ p p : this term appears in

d p=1 ∆ p p (a p + ε p ) with ε p = d l>p d j =l a p,j ∆ j l + d p>l d j =l a j,p ∆ j l + j =p a p,j ∆ j p .
We consider now ∆ i,p p . It appears in

d p=1 i =p (a i,p -a p,i )∆ i,p p
The other terms are

d l=1 i =l a i ∆ i l + d l>p d i =p,j =l a i,j ∆ i p ∆ j l + 1 2 d i=1 a i,i ∆ i i 2 + 1 2 d l=1 d i =l a i,i ∆ i l 2 + d l=1 i =j,i =l,j =l a i,j ∆ i,j l + d l=p+1 a p,l ∆ p p ∆ l l .
We put everything together and ( 19) is proved.

Main estimates

Throughout this section we will assume that

Span{a i , [a] j,p , i, j, p = 1, ..., d, j = p} = R n . ( 20
)
Let us introduce some notation. We consider the matrix A = (a i , [a] j,p , i, j, p = 1, ..., d, j = p) to be the matrix with columns a i and [a] j,p . For R ∈ (0, 1] we define the matrix

A R = ( √ Ra i , R[a] j,p , i, j, p = 1, ..., d, j = p) and we denote λ * (A R ), λ * (A R )

the lower and the larger eigenvalue of

A R A * R . We just write λ * (A), λ * (A) if R = 1. We associate the norms |y| 2 A R = (A R A R ) -1 y, y .
In Proposition 25 from the Appendix 4 we prove the following basic properties. For every 0

< R ≤ R ′ ≤ 1 R R ′ |y| A R ≥ |y| A R ′ ≥ R R ′ |y| A R (21) and 1 √ R λ * (A) |y| ≤ |y| A R ≤ 1 R λ * (A) |y| . ( 22
) Finally |A R y| A R ≤ |y| . ( 23
)
Lemma 10 Suppose that (20) holds. There exists an universal constant C 0 such that for every R ≥ δ > 0 and r > 0

P (sup t≤δ |Z t | A R ≥ r) ≤ exp - rR C 0 δ r ∧ λ * (A) a ( 24
)
with a = 1 ∨ max i,j |a i,j | . ( 25
)
Remark 11 One might think to use directly Bernstein's inequality in order to estimate P (sup t≤δ |Z t | A R ≥ r) but then one would not obtain the right inequality. Indeed one writes

|Z t | A R ≤ (R λ * (A)) -1 |Z t |
and then the above probability is bounded by

P (sup t≤δ |Z t | ≥ rR λ * (A)) ≤ exp(- r 2 R 2 λ * (A) δ ).
So one obtains R 2 δ instead of R δ and this is not in the right scale. The reason is that in the above argument we just use the lower eigenvalue λ * (A) in order to upper bound |Z t | A R since in the proof of our lemma we use the more subtle inequality |A R y| A R ≤ |y| .

Proof. Let t ≤ δ. We decompose Z(t) instead of Z(δ) and similarly to (19) we obtain

Z(t) = d p=1 a p (∆ p p (t, W ) + µ p (t, W )) + d p=1 i =p [a] i,p ∆ i,p p (t, W ) + η(t, W ) + ψ(t, W ),
in which η(t, W ) and ψ(t, W ) are defined as in (17) with ∆ i p and ∆ ij p replaced by ∆ i p (t, W ) and ∆ ij p (t, W ) respectively, and these last quantities are defined as follows:

for t ∈ [0, T ], ∆ i p (t, W ) = W i sp∧t -W i s p-1 ∧t and ∆ ij p (t, W ) = sp∧t s p-1 ∧t (W i s -W i s p-1 ∧t )dW j s . We denote by u(t) ∈ R m the vector with component u p (t) = t -1/2 (∆ p p (t, W ) + µ p (t, W )) = t -1/2 W p
t , p = 1, ..., d and u i,j (t) = 0, i = j and we also denote

U(t) = d p=1 i =p [a] i,p ∆ i,p p (t, W ) + η(t, W ) + ψ(t, W ).
Then we have

Z(t) = d p=1 t 1/2 a p u p (t) + d p=1 i =p t[a] i,p × 0 + U(t) = A t u(t) + U(t).
Using the norm inequalities given above

|U(t)| A R ≤ 1 R λ * (A) |U(t)| ≤ Ca R λ * (A) d i,j=1 ( ∆ i j (t, W ) 2 + d p=1 ∆ i,j p (t, W ) ) so that P sup t≤δ |U(t)| A R ≥ r 2 ≤ d i,j=1 P sup t≤δ ∆ j i (t, W ) 2 ≥ rR λ * (A) Ca + d i,j,p=1 P sup t≤δ ∆ i,j p (t, W ) ≥ rR λ * (A) Ca .
It is easy to check that

P sup t≤δ ∆ p p (t, W ) 2 ≥ rR λ * (A) Ca ≤ C ′ exp - rR λ * (A) C ′ aδ . Moreover, sup t≤δ ∆ i,j p (t, W ) ≤ 2 sup t≤δ t 0 W i s dW j s + 2 sup t≤δ ( W i t 2 + W j t 2 ).
Using (43) from the Appendix 1 we obtain

P sup t≤δ t 0 W i s dW j s ≥ rR λ * (A) Ca ≤ C exp - rR λ * (A) Caδ .
So we have proved that

P sup t≤δ |U(t)| A R ≥ r 2 ≤ C exp - rR λ * (A) Caδ .
Using (21) (recall that t ≤ δ ≤ R) and ( 23)

|A t u(t)| A R ≤ t R |A t u(t)| At ≤ t R |u(t)| ≤ C √ R sup t≤δ |W t | .
It follows that

P sup t≤δ |A t u(t)| A R ≥ r 2 ≤ P sup t≤δ |W t | ≥ r √ R C ≤ C exp - r 2 R Cδ .
We give the main result in this section.

Proposition 12 Suppose that λ * (A) > 0. Let ρ ∈ (0, 1) be fixed. There exists an universal constant C * (depending on d and on ρ only) such that for every

r ≤ λ 1/2 * (A) C * a (26)
one has

P (|Z δ | A δ ≤ r) ≥ r m C * × λ 2d 3 * (A) a d 3 × exp(- C * λ d 2 ρ * (A) a 2 ). ( 27
)
Proof.

Step 1. Scaling. Let B t = δ -1/2 W tδ . Then B is a standard Brownian motion and we denote

∆ j i (B) = B j i -B j i-1 , ∆ i,j p (B) = p p-1 (B j s -B j p )dB i s , i = j.
We also denote by ∆(B) the vector (∆ j i (B), ∆ i,j p (B), i, j, p = 1, ..., d) and we define Θ(B) = (Θ 1 (B), ..., Θ d (B)) with Θ p (B) = (∆ p p (B), ∆ j,p p (B), j = p). We consider the σ field

G := σ(W j s -W j s p-1 (δ) , s p-1 (δ) ≤ s ≤ s p (δ), p = 1, ...d, j = p). Conditionally to G the random variable Θ p (B) is Gaussian with covariance matrix Q p (B) given by Q p,j p (B) = p p-1 (B j s -B j i-1 )ds, j = p, Q i,j p (B) = p p-1 (B j s -B j i-1 )(B i s -B i i-1 )ds, j = p, i = p, Q p,p p (B) = 1.
Since the random variables Θ 1 (B), ..., Θ d (B) are independent Θ(B) is a Gaussian random variable. We denote by Q(B) the covariance matrix of Θ(B) and by λ * (B), λ * (B) the smaller and the larger eigenvalues of Q(B). Since this matrix is built with the blocks Q p (B), p = 1, ..., d we have

λ * (B) =
Z δ = d p=1 √ δa p (∆ p p (B) + µ p (∆(B))) + d p=1 i =p δ[a] ip ∆ i,p p (B) + δη(∆) + δψ(∆).
We define now the vector µ(∆(B)) = (µ p (∆(B)), µ i,j (∆(B) ∈ R m , i = j) by µ i,j (∆(B)) = 0 and then we may write the above decomposition in matrix notation

Z δ = A δ (Θ(B) + µ(∆(B))) + δη(Θ(B)) + δψ(∆(B)) (28) = y + .A δ Θ(B) + η δ (Θ(B)) with y = A δ µ(∆(B)) + δψ(∆(B)), η δ (θ) = δη(θ).
Step 2. Localization. We take

ε ≤ λ * (A) C 1 a 2 (29)
where C 1 is an universal constant to be chosen in the sequel. For each p = 1, ..., d we define the sets

Λ ρ,ε,p = det Q p (B) ≥ ε ρ , sup p-1≤t≤p j =p B j t -B j p-1 ≤ ε -ρ , q p (B) ≤ ε with q p (B) = j =p B j p -B j p-1 + j =p,i =p p p-1 (B j s -B j i-1 )dB i s .
By (61) in Appendix 3 we may find some constants c and ε * depending on d and ρ only such that

P (Λ ρ,ε,p ) ≥ cε 1 2 d(d+1) for ε ≤ ε * (30)
And using the independence we obtain

P ∩ d p=1 Λ ρ,ε,p ≥ c d × ε 1 2 d 2 (d+1) . (31)
On the set ∩ d p=1 Λ ρ,ε,p we have det Q p (B) ≥ ε ρ so that det Q(B) ≥ ε dρ . We also have λ * (B) ≤ ε -ρ and this gives λ * (B) ≥ ε d 2 ρ . And we also have det

Q(B) ≤ ε -dρ so ∩ d p=1 Λ ρ,ε,p ⊂ det Q(B) ≤ ε -dρ , λ * (B) ≥ ε d 2 ρ , d p=1 q p (B) ≤ dε (32)
Step 3. Inverse function theorem. We will use (55) with G = Z δ so we have to estimate the parameters associated to η δ and A δ . Notice first that λ * (A δ ) ≥ δ 2 λ * (A), c 3,η δ = 0 and c 2,η δ ≤ C 2 aδ. So the first inequality in (54) reads

r ≤ λ 1/2 * (A) C 2 a ≤ λ 1/2 * (A δ ) 16(c 2,η δ + c 3,η δ ) .
And this is verified by our hypothesis. Moreover

c * (η δ , r) ≤ C 3 a(|θ| + d p=1 |ε p (∆(B))|) ≤ C 4 a(r + d p=1 q p (B)) ≤ C 4 a( λ 1/2 * (A) C 2 a + dε).
If we choose C 1 in (29) sufficiently large and C 2 large also we obtain c * (η δ , r) ≤ 1 2 which is the second restriction in (54). Let p G,Z δ (z) be the density of Z δ conditionally to G. Then, using (55

), if |z -y| A δ ≤ r ≤ 1 we obtain p G,Z δ (z) ≥ (4λ * (B)) (m-n)/2 (8π) m/2 det Q(B) det A δ A * δ exp(- 1 4λ * (Q(B)) |z -y| 2 A δ ) ≥ ε d 3ρ (8π) m/2 det A δ A * δ exp(- 1 4ε d 2 ρ )
the second inequality being true on ∩ d p=1 Λ ρ,ε,p . On this set we also have

|µ(∆(B))| + |ψ(∆(B))| ≤ C 5 a d p=1 q p (B) ≤ C 6 aε so that |y| A δ ≤ |A δ µ(∆(B))| A δ + δ |ψ(∆(B))| A δ ≤ |µ(∆(B))| + 1 λ * (A) |ψ(∆(B))| ≤ C 7 a λ * (A) ε ≤ r 2 . So, if |z| A δ ≤ r 2 then |z -y| A δ ≤ r. It follows that P G (|Z δ | A δ ≤ r 2 ) = {|z| A δ ≤ r 2 } p G,Z δ (z)dz ≥ ε d 3ρ (8π) m/2 exp(- 1 4ε d 2 ρ ) {|z| A δ ≤ r 2 } 1 det A δ A * δ dz = ε d 3ρ (8π) m/2 exp(- 1 4ε d 2 ρ ) × r m 2 m
the last equality being obtained by a change of variable. Finally using (31)

P (|Z δ | A δ ≤ r 2 ) ≥ P (P G (|Z δ | A δ ≤ r), ∩ d p=1 Λ ρ,ε,p ) ≥ r m ε 2d 3 C 8 exp(- 1 4ε d 2 ρ ).
We replace now ε by the expression in the RHS of (29) and we obtain (27).

Corollary 13 Suppose that λ * (A) > 0. Let ρ ∈ (0, 1) be fixed. There exists some universal constant C (depending on d and on ρ only) such that for every r, R > 0 the following holds. Suppose that

δ ≤ rR C ln 1 r r ∧ λ * (A) a × λ dρ * (A) a 2dλ . ( 33
)
Then

P (sup t≤δ |Z t | A R ≤ r, |Z δ | A δ ≤ r) ≥ r m 2C * exp(- C * a 2dρ λ dλ * (A) ) ( 34
)
with C * the constant from (27).

Proof. We use ( 24) and ( 27) in order to obtain

P (sup t≤δ |Z t | A R ≤ r, |Z δ | A δ ≤ r) ≥ P (|Z δ | A δ ≤ r) -P (sup t≤δ |Z t | A R > r) ≥ r m C 3 exp(- C 3 a 2dρ λ dλ * (A) ) -exp(- rR C 0 δ r ∧ λ * (A) a ) ≥ r m 2C 3 exp(- C 3 a 2dρ λ dλ * (A)
)

the last inequality being a consequence of our restriction on δ.

4 Diffusion processes

Short time behavior

We consider the diffusion process X t solution of (1) and the skeleton x t = x t (φ) solution of (3) and we give for them an estimate which is analogous to (34). Using a development in stochastic Taylor series of order two we write

X t = X 0 + Z t + b(0, X 0 )t + R t
where Z t is defined in [START_REF] Zeitouni | On the Onsager-Machlup functional for diffusion processes around non C 2 curves[END_REF] with

a i = σ i (0, X 0 ), a i,j = ∂ σ i σ j (0, X 0 ) so that [a] i,j = [σ i , σ j ](0, X 0 ), and
R t = d j,i=1 t 0 s 0 (∂ σ i σ j (u, X u ) -∂ σ i σ j (0, X 0 )) • dW i u • dW j s + d i=1 t 0 s 0 ∂ b σ i (u, X u )du • dW i s + d i=1 t 0 s 0 ∂ u σ j (u, X u )du • dW i s + d i=1 t 0 s 0 ∂ σ i b(u, X u ) • dW i u ds + t 0 s 0 ∂ b b(u, X u )duds.
We denote A(t, x) = (σ i (t, x), [σ j , σ p ](t, x)) i,j,p=1,...,d,j =p and

A δ (t, x) = ( √ δσ i (t, x), [ √ δσ j , √ δσ p ](t, x)) i,j,p=1,...,d,j =p .
In particular λ * (A(t, x)) = λ(t, x).

We will need the following estimate for the skeleton x t = x t (φ) as in [START_REF] Bally | Tube estimates for Itô processes and density minorations for log-normal type diffusions[END_REF]. And for φ ∈ L 2 ([0, T ], R d ), we set

ε φ (δ) = δ 0 |φ s | 2 ds 1/2 . ( 35
)
Lemma 14 Let δ be such that

ε φ (δ) + √ δ ≤ 1, δ < 1 4n(0,x 0 ) and n(0, x 0 )(ε φ (δ) + √ δ) + √ δ ≤ λ(0, x 0 ) 8d 3 n 2 (0, x 0 ) . ( 36
)
Then for every 0 ≤ t ≤ δ and z ∈ R n ,

|z| 2 A δ (0,x 0 ) ≤ 4 |z| 2 A δ (t,xt) ≤ 16 |z| 2 A δ (0,x 0 ) . (37)
Moreover,

sup t≤δ |x t -x 0 -b(0, x 0 )t| A δ (0,x 0 ) ≤ 4ε φ (δ) + 1 n(0, x 0 ) δ. (38)
Proof. First, one has x s ∈ B(x 0 , 1) for every s ≤ δ. In fact, setting τ = inf{t > 0 :

|x t -x 0 | > 1}, for s ≤ δ ∧ τ one has x s -x 0 ≤ n(0, x 0 ) √ δ(ε φ (δ) + √ δ) ≤ 1 2
because ε φ (δ) + √ δ ≤ 1 and δ < 1 4n(0,x 0 ) . This gives s < τ . This means that δ < τ , so that |x s -x 0 | < 1 for every s ≤ δ. Moreover, by using (36),

|x s -x 0 | + |s| ≤ n(0, x 0 ) √ δ(ε φ (δ) + √ δ) + δ ≤ λ(0, x 0 )) 8d 3 n 2 (0, x 0 ) × √ δ. (39)
Now, (37) follows immediately from Proposition 27 in Appendix 4 (see page 36).

We prove now (38). For t ≤ δ, we write now

J t := x t -x 0 -b(0, x 0 )t = t 0 (∂ s x s -b(s, x s ))ds + t 0 (b(s, x s ) -b(0, x 0 ))ds.
By using inequality (65) in Lemma 25 from Appendix 4 (see page 33), we get

|J t | 2 A δ (0,x 0 ) ≤ 2t t 0 |∂ s x s -b(s, x s )| 2 A δ (0,x 0 ) ds + 2t t 0 |b(s, x s ) -b(0, x 0 )| 2 A δ (0,x 0 ) ds =: I ′ t + I ′′ t
As for I ′ t , we use (37): for s ≤ t ≤ δ we have

|∂ s x s -b(s, x s )| 2 A δ (0,x 0 ) ≤ 4|∂ s x s -b(s, x s )| 2 A δ (s,xs) .
Moreover, we can write

∂ s x s -b(s, x s ) = d j=1 σ j (s, x s )φ j (s) = A δ (s, x s )ψ(s), with ψ j (s) = 1 √ δ φ j , ψ i,j (s) = 0 so that |∂ s x s -b(s, x s )| A δ (s,xs) = |A δ (s, x s )ψ(s)| A δ (s,xs) ≤ |ψ(s)| = 1 √ δ |φ(s)| .
Then, for t ≤ δ we can write

I ′ t ≤ 8δ δ 0 |∂ s x s -b(s, x s )| 2 A δ (s,xs) ds ≤ 8 δ 0 |φ(s)| 2 ds = 8ε φ (δ) 2 .
We estimate now I ′′ t : by using (39),

I ′′ t ≤ 2δ δ 0 1 λ * (A δ (0, x 0 )) |b(s, x s ) -b(0, x 0 )| 2 ds ≤ 2 n 2 (0, x 0 ) λ(0, x 0 ) t 0 (|s| + |x s -x 0 |) 2 ds ≤ 1 n 2 (0, x 0 ) × δ 2 .
By inserting the estimates for I ′ t and I ′′ t , we get

sup t≤δ |J t | A δ (0,x 0 ) ≤ 8ε φ (δ) 2 + 1 n 2 (0, x 0 ) δ 2 1/2 ≤ 4ε φ (δ) + 1 n(0, x 0 ) δ.
The main estimate in this section is the following proposition.

Proposition 15 Let (9) hold and let ρ ∈ (0, 1) be fixed. Then there exist some universal constants C 1 , C 2 (depending on d and ρ only) such that the following holds. Let 0 < δ ≤ R ≤ 1 and r ∈ (0, 1) be such that

ε φ (δ) ≤ r ∧ λ(0, x 0 ) C 1 n 3 (0, x 0 ) , δ ≤ r 5 R C 1 × λ 1+3dρ (0, x 0 ) n 6+6dρ (0, x 0 ) ( 40
)
and suppose that

|X 0 -x 0 | A δ (0,x 0 ) ≤ r 8 . (41)
Then

P sup t≤δ |X t -x t | A R (t,xt) ≤ 2r, |X δ -x δ | A δ (δ,x δ ) ≤ r ≥ r m C 2 exp - C 2 n 2dρ (0, x 0 ) λ dρ (0, x 0 ) . (42)
Proof. For t ≤ δ, by using (37) we obtain

|X t -x t | A δ (t,xt) ≤ 4 |X t -x t | A δ (0,x 0 ) ≤ 4 6 j=1 |I j | A δ (0,x 0 )
with

I 1 = X 0 -x 0 , I 2 = Z t , I 3 = R t I 4 = x t -x 0 -b(0, x 0 )t, I 5 = (b(0, X 0 ) -b(0, x 0 ))t,
We have to estimate the above terms for t ≤ δ. First

|I 5 | A δ (0,x 0 ) ≤ n(0, x 0 ) λ(0, x 0 ) |X 0 -x 0 | ≤ n 2 (0, x 0 ) λ(0, x 0 ) |X 0 -x 0 | A δ (0,x 0 ) × √ δ ≤ r 8
and by ( 38)

|I 4 | A δ (0,x 0 ) ≤ 4ε φ (δ) + 1 n(0, x 0 ) δ ≤ r 8
And by our assumption

|I 1 | A δ (0,x 0 ) ≤ r 8 . So we have |X -x t | A δ (t,xt) ≤ r 2 + 4(|Z t | A δ (0,x 0 ) + |R t | A δ (0,x 0 ) ).
Since R ≥ δ, by (62) in Lemma 25 from Appendix 4 (see page 33) we have |y| A R (0,x 0 ) ≤ |y| A δ (0,x 0 ) so |I i | A R (0,x 0 ) ≤ r 8 for i = 1, 4, 5. And this gives

|X t -x t | A R (t,xt) ≤ r 2 + 4(|Z t | A R (0,x 0 ) + |R t | A R (0,x 0 ) ).
Using the above inequalities we easily obtain

P sup t≤δ |X t -x t | A R (t,xt) ≤ 2r, |X δ -x δ | A δ (δ,x δ ) ≤ r ≥ P sup t≤δ |Z t | A R (0,x 0 ) + sup t≤δ |R t | A R (0,x 0 ) ≤ r 4 , |Z δ | A δ (0,x 0 ) + |R δ | A δ (0,x 0 ) ≤ r 8 ≥ P sup t≤δ |Z t | A R (0,x 0 ) ≤ r 8 , |Z δ | A δ (0,x 0 ) ≤ r 16 -P sup t≤δ |R t | A δ (0,x 0 ) > r 8 .
We upper bound now the last term. First, using the norms inequalities

P sup t≤δ |R t | A δ (0,x 0 ) > r 8 ≤ P sup t≤δ |R t | > K with K = rδ √ λ(0,x 0 ) 8
. We define now τ = inf{t : |X t -X 0 | ≥ 1 2 ). Using the norms inequalties, (40) and (41) we obtain |x 0 -X 0 | ≤ 1 2 so that for t ≤ τ we have |X t -x 0 | ≤ 1. It follows that up to τ the diffusion process X coincides with a diffusion process X which has the coefficients and their derivatives up to order three bounded by n(x 0 ). We denote by R the reminder in which X is replace with X and we write

P sup t≤δ |R t | > K ≤ P sup t≤δ R t > K + P (τ ≤ δ). Since τ = τ := inf{t : X t -X 0 ≥ 1
2 ) a standard reasoning based on Bernstin's inequality gives P (τ ≤ δ) = P (τ ≤ δ) ≤ exp(-1/Cδn 2 (x 0 )). In order to estimate the last first we use (43) from the Appendix 1 (see Lemma 18 at page 24) with k = 3, p 3 2 3 and with k = 1, p 1 = 2, and K = rδ 4 λ(0, x 0 ). A straightforward computation gives

P sup t≤δ R t > rδ λ(0, x 0 ) 8 ≤ C exp - r 2 λ(0, x 0 ) Cδn 4 (0, x 0 ) + C exp - r 2/3 λ 1/3 (0, x 0 ) Cδ 1/3 n 2 (0, x 0 ) ≤ C exp - r 2/3 λ 1/3 (0, x 0 ) Cδ 1/3 n 2 (0, x 0 )
the last inequality being a consequence of (40). Using (34)

P sup t≤δ |Z t | A R (0,x 0 ) ≤ r 8 , |Z δ | A δ (0,x 0 ) ≤ r 16 ≥ r m 2C * exp - C * n 2dλ (0, x 0 ) λ dλ (0, x 0 )
with C * the universal constant in (34). Our assumption on δ gives

C exp - r 2/3 λ 1/3 (0, x 0 ) Cδ 1/3 n 2 (0, x 0 ) ≤ 1 2 × r m 2C * exp - C * n 2dρ (0, x 0 ) λ dλ (0, x 0 )
so we have proved that

P sup t≤δ |X t -x t | A R (t,xt) ≤ 2r, |X δ -x δ | A δ (δ,x δ ) ≤ r ≥ r m 4C * exp - C * n 2dρ (0, x 0 ) λ dλ (0, x 0 ) .

Chain argument

We recall that, by the hypothesis (9) we have some functions λ, n ∈ L(µ, h) such that λ(t) ≤ 1 ∧ λ(t, x t ) and n t ≥ 1 ∨ n(t, x t ) such that λ, n ∈ L(µ, h) for some h > 0 and µ ≥ 1.

We also consider some R, r, ρ ∈ (0, 1) and we define (with C 1 the constant in (40))

f h (t) = 2 h + C 1 (ln 1 r ) 3 n 6+4dρ t Rr 2 λ 1+dρ t + C 2 1 n 6 t r 2 ∧ λ t |φ t | 2 .
Notice that, if dρ ≤ 1 5 then f h ∈ L(µ 8 , h). We define

δ(t) = inf δ > 0 : t+δ t f h (s)ds ≥ 1 µ 8 Lemma 16 i) One has δ(t) ≤ h 2 ∧ Rr 2 t λ 1+dρ t C 1 (ln 1 r ) 3 n 6+4dρ t , ε φ (δ(t)) ≤ r ∧ λ 1/2 t C 1 n 3 t . ii) If |t -t ′ | ≤ δ(t) then 1 4µ 16 |y| A δ(t) (t,xt) ≤ |y| A δ(t ′ ) (t ′ ,x t ′ ) ≤ 4µ 8 |y| A δ(t) (t,xt) . Proof. i) Since t+h/2 t 2 h ds = 1 ≥ 1/µ 8 we have δ(t) ≤ 1
2 h. So we may use the properties L(µ, for t ≤ s ≤ t + δ(t). Consequently, for 0 < δ ≤ δ(t)

1 µ 8 ≥ t+δ t C 1 (ln 1 r ) 3 n 6+4dρ s Rr 2 λ 1+dρ s ds ≥ 1 µ 8 × C 1 (ln 1 r ) 3 n 6+4dρ t Rr 2 λ 1+dρ t × δ which gives δ(t) ≤ Rr 2 λ 1+dρ t C 1 (ln 1 r ) 3 n 6+4dρ t .
We also have

1 µ 8 ≥ t+δ t C 2 1 n 6 s r 2 ∧ λ s |φ s | 2 ds ≥ 1 µ 8 × C 2 1 n 6 t r 2 ∧ λ t t+δ t |φ s | 2 ds so that ε 2 φ (t) ≤ r 2 ∧ λ t C 2 1 n 6 t .
This proves i).

ii) We use here next Proposition 27 from Appendix 4 (see page 36). 73) is verified and we may use (74) to obtain

If |t -t ′ | ≤ δ(t), then |x t -x t ′ | ≤ δ 1/2 (t)(dε φ (δ(t)) + δ 1/2 (t))n t so (
1 4 |y| A δ(t) (t,xt) ≤ |y| A δ(t) (t ′ ,x t ′ ) ≤ 4 |y| A δ(t) (t,xt) .

It remains to compare |y|

A δ(t) (t,xt) with |y| A δ(t ′ ) (t,xt) . Since δ(t ′ ) ≤ 1 2 h and |t -t ′ | ≤ 1 2 h we have |t -s| ≤ h for every s ∈ (t ′ , t ′ + δ(t ′
). We use the property L(µ 8 , h) for f h and we obtain

µ 8 f h (t)δ(t) ≥ t+δ(t) t f h (s)ds = 1 µ 8 = t ′ +δ(t ′ ) t ′ f h (s)ds ≥ µ -8 f h (t)δ(t ′ ).
So (δ(t)/δ(t ′ )) 1/2 ≥ µ -8 . Suppose now that δ(t) ≤ δ(t ′ ). We use then (21) and we obtain

1 µ 16 |y| A δ(t) (t,xt) ≤ |y| A δ(t ′ ) (t,xt) ≤ 1 µ 8 |y| A δ(t) (t,xt) .
We construct now a time grid in the following way. We put t 0 = 0 and

t k = t k-1 + δ(t k-1 )
and we denote

Θ k = sup t k-1 ≤t≤t k |X t -x t | A R (t,xt) ≤ r , Γ k = |X t k -x t k | A δ(t k ) (t k ,xt k ) ≤ r 8 .
Proposition 17 i) Suppose that (9) holds and let R, r ∈ (0, 1) and ρ ∈ (0, 1 5d ). There a universal constant C (depending on d and on ρ) such that

P ∩ k i=1 Θ i ∩ Γ i ≥ P ∩ k-1 i=1 Θ i ∩ Γ i exp - Cn 2dρ t k-1 λ dρ t k-1
.

ii) Moreover there exists an universal constant C such that

P sup 0≤s≤T |X s -x s | A R (s,x(s)) ≤ r ≥ exp -Cµ 9 T 0 f h (t) n 2dρ s λ dρ t dt ≥ exp -Cµ 9 T h + 1 r 2 T 0 n 6+6dρ t λ 1+2dρ t (ln 1 r ) 3 R + |φ t | 2 dt . Proof i) Let Γ k = |X t k -x t k | A δ(t k-1 ) (t k-1 ,xt k-1 ) ≤ 1 32µ 8 r .
Using ii) from the previous lemma we obtain Γ k ⊂ Γ k so by (42)

P t k-1 (Θ k ∩ Γ k ) ≥ P t k-1 (Θ k ∩ Γ k ) ≥ exp -C n 2dρ t k-1 (x 0 ) λ dρ t k-1 (A(x 0 ))
.

The above inequality holds if

X t k-1 -x t k-1 A δ(t k-1 ) (t k-1 ,xt k-1 )
≤ r 8 and this is true on the set Γ k-1 . ii) Let N T = min{k : t k > T }. Since X 0 = x 0 we may use the recursively the inequality from i) and we obtain

P sup t≤T |X t -x t | A R (xt) ≤ r ≥ P ∩ N T i=1 Θ i ∩ Γ i ≥ exp -C N T k=1 n 2dρ t k-1 λ dρ t k-1
.

We write

T 0 f h (s) n 2dρ s λ dρ s ds ≥ N T -1 i=1 t i t i-1 f h (s) n 2dρ s λ dρ s ds ≥ 1 µ 3dρ N T -1 i=1 n 2dρ t i-1 λ dρ t i-1 t i t i-1 f h (s)ds = 1 µ 8+3dρ N T -1 i=1 n 2dρ t i-1 λ dρ t i-1
the last equality being a consequence of the definition of δ(t k ).

Appendix 1. Exponential decay for multiple stochastic integrals

In this section W = (W 1 , ..., W d ) is a standard Brownian motion and α = (α 1 , ..., α k ) ∈ {1, ..., d} k denotes a multi index. We use the notation α = (α 1 , ..., α k-1 ). We consider an adapted and bounded stochastic process a and we denote by a ∞ a constant such that sup t≤T |a(t, ω)| ≤ a ∞ almost surely. Then we define the iterated stochastic integrals

I 0 (a, W )(t) = a(t), I α k (a, W )(t) = t 0 I α k-1 (a, W )(s)dW α k s .
Lemma 18 There exist some universal constants C k , C ′ k such that for each T, K ≥ 0 and every multi-index α = (α 1 , ..., α k ) one has

P sup t≤T |I α k (a, W )(t)| ≥ K ≤ C k exp -C ′ k K T k/2 a ∞ p k (43) with p 1 = 2, p k+1 = 2p k 2 + p k .
Proof. We assume that a ∞ = 1 almost surely (if not we normalize with a ∞ ) and T = 1 (if not we use a scaling argument). We proceed by recurrence. We take some Q ≥ 0 and we write

P (sup t≤1 |I α k (a, W )(t)| ≥ K) ≤ I + J with I = P (sup t≤1 |I α k (a, W )(t)| ≥ K, sup t≤1 I α k-1 (a, W )(t) ≤ Q) J = P (sup t≤1 I α k-1 (a, W )(t) ≥ Q).
Using the recurrence hypothesis

J ≤ C k-1 exp(-C ′ k-1 Q p k-1 ). We set h(t) = t 0 I α k-1 (a, W )(s)
2 ds and we write

I α k (a, W )(t) = b(h t )
where b is a standard Brownian motion. So, we obtain

I ≤ P (sup t≤h 1 |b(t)| ≥ K, h(1) ≤ Q 2 ) ≤ P ( sup t≤Q 2 |b(t)| ≥ K) ≤ C exp(- C ′ K 2 Q 2 ). We choose Q solution of Q p k-1 = K 2 /Q 2 that is Q = K 2 2+p k-1 .
Then we obtain

P (sup t≤1 |I α k (a, W )(t)| ≥ K) ≤ C k exp(-C ′ k K 2p k-1 2+p k-1 ) with C k = C ∨ C k-1 , C ′ k = C ′ ∧ C ′ k-1 .
6 Appendix 2. Small perturbations of Gaussian random variables

The inverse function theorem

We give first a quantitative version of the inverse function theorem. We consider a three time differentiable function η : R d → R d and Φ(θ) := θ + η(θ).

We assume that η(0) = 0 and ∇η(0) ≤ 1 2 .

In particular this implies that ∇Φ(0) is invertible and

|∇Φ(0)x| 2 ≥ 1 2 |x| 2 -|∇η(0)x| 2 ≥ 1 2 |x| 2 - 1 4 |x| 2 = 1 4 |x| 2 .
We also have

|∇Φ(0)x| ≤ √ 3 |x| so 1 2 |x| ≤ |∇Φ(0)x| ≤ √ 3 |x| .
We denote

c 2 (η) = max i,j=1,d sup |x|≤1 ∂ 2 ij η(x) , c 3 (η) = max i,j,k=1,d sup |x|≤1 ∂ 3 ijk η(x)
and we take h η > 0 such that

h η ≤ 1 2 and h η ≤ 1 4d 3 (c 2 (η) + c 3 (η)) . ( 44
)
Proposition 19 Suppose that η ∈ C 3 (R d , R d ), η(0) = 0 and ∇η(0) ≤ 1 2 . Then there exists a neighborhood V (hη) ⊂ B(0, 2h η ) of zero such that Φ : V (hη) → B(0, 1 2 h η ) is a diffeomorphism. In particular, one has

Φ -1 : B 0, 1 2 h η → B(0, 2h η )
and for every y ∈ B(0, 1 2 h η ) the following estimates hold:

1 4 Φ -1 (y) ≤ |y| ≤ 4 Φ -1 (y) . ( 45
)
Proof. The existence and the differentiability property of the inverse function Φ -1 in a neighborhood of the origin is a well known result from the Inverse Function Theorem. What we aim to prove is that Φ -1 : B(0, 1 2 h η ) → B(0, 2h η ) and the estimates in (45). Since η(0) = 0 we have (1 -t) ∇ 2 η(tθ)θ, θ dt .

η(θ) = ∇η(0)θ + 1 0 (1 -t) ∇ 2 η(tθ)θ, θ dt with ∇ 2 η k = (∂ 2 ij η k ) i,
Recall that 1 2 |x| ≤ |∇Φ(0)x| . Then, for θ 1 , θ 2 ∈ B(0, 2h η ) we have

|U y (θ 1 ) -U y (θ 2 )| = (∇Φ(0)) -1 1 0 (1 -t)( ∇ 2 η(tθ 1 )θ 1 , θ 1 -∇ 2 η(tθ 2 )θ 2 , θ 2 )dt ≤ 2 1 0 (1 -t) ∇ 2 η(tθ 1 )θ 1 , θ 1 -∇ 2 η(tθ 2 )θ 2 , θ 2 dt ≤ 2d 3 h η c 2 (η) + c 3 (η) |θ 1 -θ 2 | ≤ 1 2 |θ 1 -θ 2 | , so that |U y (θ 1 ) -U y (θ 2 )| ≤ 1 2 |θ 1 -θ 2 | . ( 46
)
Notice also that for y ∈ B(0, 1 2 h η ) and θ ∈ B(0, 2h η ) the above inequality gives

|U y (θ)| ≤ |U y (θ) -U y (0)| + |U y (0)| ≤ 1 2 |θ| + 2 |y| ≤ h η + h η = 2h η . (47)
We define now θ 0 = 0, θ k+1 = U y (θ k ).

From (47) we know that θ k ∈ B(0, 2h η ), k ∈ N and consequently

|U y (θ k+1 ) -U y (θ k )| ≤ 1 2 |θ k -θ k-1 | .
So the sequence θ k , k ∈ N converges to the solution of the equation θ = U y (θ), that is Φ(y) = θ. We have thus proved that for any y ∈ B(0, 1 2 h η ) there exists a unique θ ∈ B(0, 2h η ) such that Φ(θ) = y, that is Φ -1 : B(0, 1 2 h η ) → B(0, 2h η ) is well defined. Finally, for y ∈ B(0, 1 2 h η ) let θ = Φ -1 (y). Then θ = U y (θ) so, using (47

) |θ| = |U y (θ)| ≤
|θ| = |U y (θ)| ≥ |U y (0)| -|U y (θ) -U y (0)| ≥ 1 2 |y| - 1 2 |θ| which proves that |θ| ≥ 1 3 |y| ≥ 1 4 |y|.
Let us consider a more specific variant of the local inversion theorem we will need in next Section 8. We consider a matrix B ∈ M d×d with columns B i ∈ R d , i = 1, ..., d and we suppose that B is invertible. Then we consider the equation

y = Bθ + r(θ) (48)
where r ∈ C 3 (R d , R d ). Our aim is to prove that for small y the above equation has a unique solution and to obtain some precise estimates for θ and its projection on a suitable subspace of R d in terms of y. In order to do it we have to introduce some more notations.

We fix d ′ ∈ {1, ..., d -1} and we denote

d ′′ = d -d ′ . For x = (x 1 , ..., x d ) ∈ R d we denote ← - x = (x 1 , ..., x d ′ ) and -→ x = (x d ′ +1 , ..., x d ).
We denote by ← -B ∈ M d×d ′ (respectively by -→ B ∈ M d×d ′′ ) the matrix with columns B 1 , ..., B d ′ (respectively the matrix with columns B d ′ +1 , ..., B d ). Let S = V ect{B 1 , ..., B d ′ ). Since B is invertible the columns B i , i = 1, ..., d are linearly independent and so dim S = d ′ . We denote by S ⊥ the orthogonal of S by Π the projection on S and by Π ⊥ the projection on S ⊥ . We define -→ B ⊥ to be the matrix with columns

B ⊥ i := Π ⊥ B i , i = d ′ + 1, ..., d. Since B 1 , ..., B d span R d it follows that B ⊥ i , i = d ′ + 1, ..., d span S ⊥ which has dimension d ′′ . So B ⊥ i , i = d ′ + 1, .
.., d are linearly independent. We conclude that the matrices

B * B, ← - B * ← - B ∈ M d ′ ×d ′ and -→ B ⊥, * -→ B ⊥ ∈ M d ′′ ×d ′′
are all invertible, and as usual we denote by λ * (B),

λ * ( ← - B ) and λ * ( -→ B ⊥ ) the smaller eigenvalue of B * B, ← - B * ← - B and -→ B ⊥, * -→ B ⊥ respectively.
Theorem 20 We assume that the matrix B is invertible and that r(0) = ∇r(0

) = 0. Set |B| ∞ = sup i,j=1,...,d |B ij |. Then for every y ∈ R d such that |y| < λ * (B) 1/2 4 and |y| < λ * (B) 8d 3 (c 2 (r) + c 2 (r)) , (49)
the equation ( 48) has a unique solution θ and

|θ| ≤ 4 λ 1/2 * (B) |y| , -→ θ ≤ |B| ∞ λ * ( -→ B ⊥ ) Π ⊥ y + 16c 2 (r) |B| ∞ λ * ( -→ B ⊥ )α * (B) |y| 2 . (50)
In particular if

Π ⊥ y ≤ |Πy| then |y| ≤ 2 |Πy| so |θ| ≤ 8 λ 1/2 * (B) |Πy| , -→ θ ≤ |B| ∞ λ * ( -→ B ⊥ ) Π ⊥ y + 64c 2 (r) |B| ∞ λ * ( -→ B ⊥ )λ * (B) |Πy| 2 . ( 51
)
Proof. We write the equation (48) as B -1 y = θ+B -1 r(θ) and we use Proposition 19 with

η(θ) = B -1 r(θ). Since ∂ α B -1 r(θ) = B -1 ∂ α r(θ) we have c 2 (η) + c 3 (η) ≤ λ * (B) -1/2 (c 2 (r) + c 3 (r)
). So our assumption (49) ensures that for some h η fulfilling (44), one has |B -1 y| ≤ 1 2 h η and we may use Proposition 19 in order to produce the solution θ of our equation. And moreover, by (45) one has

|θ| ≤ 4 B -1 y ≤ 4 λ 1/2 * (B) |y| .
In particular this proves the first inequality in (50). Using (49) we also have |θ| ≤ 1. Since r(0) = ∇r(0) = 0 we obtain

|r(θ)| ≤ max |α|=2 sup |θ ′ |≤1 |∂ α θ r(θ ′ )| × |θ| 2 = c 2 (r) |θ| 2 ≤ 16c 2 (r) λ * (B) |y| 2 .
We multiply our equation with (

-→ B ⊥, * -→ B ⊥ ) -1 -→ B ⊥, * and we obtain ( -→ B ⊥, * -→ B ⊥ ) -1 -→ B ⊥, * y = -→ θ + ( -→ B ⊥, * -→ B ⊥ ) -1 -→ B ⊥, * r(θ).
Notice that

-→ B ⊥, * y = -→ B ⊥, * Π ⊥ y so ( -→ B ⊥, * -→ B ⊥ ) -1 -→ B ⊥, * y ≤ λ -1 * ( -→ B ⊥ ) |B| ∞ Π ⊥ y and this gives -→ θ ≤ λ -1 * ( -→ B ⊥ ) |B| ∞ Π ⊥ y + λ -1 * ( -→ B ⊥ ) |B| ∞ |r(θ)| ≤ |B| ∞ λ * ( -→ B ⊥ ) Π ⊥ y + 16c 2 (r) |B| ∞ λ * ( -→ B ⊥ )λ * (B) |y| 2 .

Estimates of the density

For h > 0 we denote

c * (η, h) = sup |x|≤2h max i,j ∂ i η j (x) . (52)
Let Θ be a m dimensional centered Gaussian random variable with covariance matrix Q. We assume that Q is invertible and we denote by λ(Q) and λ(Q)) the lower and the upper eigenvalue of Q respectively. We also consider a matrix Γ ∈ M n×m with n ≤ m and we recall that |x| 2 Γ = ΓΓ * x, x , λ * (Γ) is the smaller eigenvalue of ΓΓ * and B Γ (y, r) = {z : |y -z| Γ < r}.

Lemma 21 Suppose that ΓΓ * is invertible. Let η ∈ C 3 b (R m , R n ) such that η(0) = 0. Set G = y + ΓΘ + η(Θ) (53)
and assume there exists r > 0 such that

r ≤ 1 2 λ * (Γ) 1/2 h η and c * (η, 4r) λ * (Γ) 1/2 ≤ 1 2m , (54)
h η being defined in (44). Then the law of G has a density p G on B Γ (y, r) and for z ∈ B Γ (y, r) one has

p G (z) ≥ λ(Q) (m-n)/2 π n/2 8 m/2 2 (m-n)/2 (det Q det ΓΓ * ) 1/2 exp - 2 λ(Q) |z -y| 2 Γ (55) p G (z) ≤ λ(Q) (m-n)/2 8 (m-n)/2 2 m/2 π n/2 (det Q det ΓΓ * ) 1/2 exp - 1 8 λ(Q) |z -y| 2 Γ ( 56
)
In particular, (55) and (56) imply that, for z ∈ B Γ (y, r),

λ(Q) 16 λ(Q) m/2 p N(y, 1 4 λ(Q)ΓΓ * ) (z) ≤ p G (z) ≤ 16 λ(Q) λ(Q) m/2 p N(y,4λ(Q)ΓΓ * ) (z)
where p N(y,BB * ) denotes the Gaussian density with mean y and covariance matrix BB * .

Proof.

Step 1. We assume first that n = m, y = 0 and Γ is the identity matrix. We denote Φ(θ) = θ + η(θ), so that Φ(Θ) = G. Let f : R m → R be a non negative measurable function with the support included in the (Euclidian) ball B(0, r), with r fulfilling (54). Using a change of variable and Proposition 19, we obtain

E(f (Φ(Θ))) = {θ∈Φ -1 (B(0,r))} f (Φ(θ)) 1 (2π) m/2 (det Q) 1/2 exp(- 1 2 Q -1 θ, θ )dθ = B(0,r) f (z) p Φ(Θ) (z) dz,
where we have set, for z ∈ B(0, r),

p Φ(Θ) (z) = 1 (2π) m/2 |det ∇Φ(Φ -1 (z))| (det Q) 1/2 exp(- 1 2 Q -1 Φ -1 (z), Φ -1 (z) ) Since r ≤ h η , if z ∈ B(0, r) one has θ = Φ -1 (z) ∈ B(0, 4r
) and for x ∈ B(0, 4r) we have

1 2 |x| 2 ≤ (1 -mc * (η, h η )) |x| 2 ≤ | ∇Φ(θ)x, x | ≤ (1 + mc * (η, h η )) |x| 2 ≤ 2 |x| 2 , because c * (η, 4r) ≤ 1 2m . Therefore, if z ∈ B(0, r) then 2 -m ≤ det ∇Φ(Φ -1 (z)) ≤ 2 m .
Moreover, using (45) we obtain

Q -1 Φ -1 (z), Φ -1 (z) ≤ 1 λ(Q) Φ -1 (z) 2 ≤ 4 λ(Q) |z| 2 and Q -1 Φ -1 (z), Φ -1 (z) ≥ 1 λ(Q) Φ -1 (z) 2 ≥ 1 4λ(Q) |z| 2
So, as z ∈ B(0, r) we get

1 (8π) m/2 √ det Q exp(- 2 λ(Q) |z| 2 ) ≤ p Φ(Θ) (z) ≤ 2 m/2 π m/2 √ det Q exp(- 1 8λ(Q) |z| 2 ) (57)
Step 2. We still assume that n = m but now y and Γ are general, with Γ invertible. We write

G = y + Γ(θ + η Γ (θ)) with η Γ (θ) = Γ -1 η(θ) and denote Φ Γ (θ) = θ + η Γ (θ). One has c 2 (η Γ ) + c 3 (η Γ ) ≤ λ * (Γ) -1/2 (c 2 (η) + c 3 (η)), so that h η Γ ≥ λ * (Γ) 1/2 h η and then (54) gives r ≤ 1 2 h η Γ . Moreover, since c * (η Γ , 4r) ≤ λ * (Γ) -1/2 c * (η,
p G (z) = 1 |det Γ| p Φ Γ (Θ) (Γ -1 (z -y)).
Since |Γ -1 (z -y)| = |z -y| Γ we use (57) and we obtain

p G (z) ≥ 1 (8π) m/2 √ det Q |det Γ| exp(- 2 λ(Q) |z -y| 2 Γ ) p G (z) ≤
Step 3. Now we allow n to be strictly smaller than m. Since ΓΓ * is invertible the lines Γ 1 , ..., Γ n ∈ R m of Γ are linearly independent. We denote S = V ect{Γ 1 , ..., Γ n } and we take Γ n+1 , ..., Γ m to be an orthonormal basis in the orthogonal of S. Then we define Γ ∈ M m×m to be the matrix with lines Γ 1 , ..., Γ n , Γ n+1 , ..., Γ m and we notice that

Γ Γ * = ΓΓ * 0 0 I m-n
where I m-n ∈ M (m-n)×(m-n) is the identity matrix. In particular det Γ = √ det ΓΓ * and for

z = (z 1 , z 2 ), z 1 ∈ R n , z 2 ∈ R m-n we have |z| 2 Γ = |z 1 | 2 Γ + |z 2 | 2 .
Moreover, for y ∈ R n we denote y = (y, 0) and we also set η(θ) = (η(θ), 0). So, we define H = y + ΓΘ + η(Θ), and we notice that h η = h η and c * ( η, 4r) = c * (η, 4r). For the density of H = (H 1 , H 2 ) we can use the estimate from the previous step. Notice that since H 2 is an orthogonal transformation of a Gaussian random variable, one easily gets that the estimates hold for (z, u) ∈ R m such that z ∈ B(0, r) and u ∈ R m-n . Now, since

H 1 = G we obtain p G (z) = R m-n p H (z, u)du ≥ R m-n 1 (8π) d/2 √ det Q det Γ exp(- 2 λ(Q) |z -y| 2 Γ + |u| 2 du = (4λ * (Q)) 1 2 (m-n) (8π) m/2 √ det Q |det ΓΓ * | exp - 2 λ(Q) |z -y| 2 Γ .
The proof of the other inequality is the same.

Appendix 3. Support Property

In this section we prove (30). Let B = (B 1 , ..., B d-1 ) be a standard Brownian motion. We consider the analogues of the covariance matrix Q i (B) considered in the previous sections: we define a symmetric square matrix of dimension d × d by We need the following two preliminary lemmas.

Q d,d = 1, Q d,j = Q j,d = 1 0 B j s ds, j = 1, ..., d -1, Q j,p = Q p,j =
Lemma 22 With g(s) = B s , s ∈ [0, 1] we have

Qξ, ξ = α 2 B (ξ) + β B (ξ).
As a consequence, one has

λ(Q) = inf |ξ|=1 (α 2 B (ξ) + β B (ξ)) and λ(Q) ≤ sup |ξ|=1 (α 2 B (ξ) + β B (ξ)) ≤ 1 + sup t≤1 |B t | 2 .
Taking ξ * = 0 and ξ d = 1 we obtain Qξ, ξ = 1 so that λ(Q) ≤ 1 ≤ λ(Q).

Proof. By direct computation

Qξ, ξ = ξ 2 d + 2ξ d 1 0 B s , ξ * ds + 1 0 B s , ξ * ds) 2 + 1 0 B s , ξ * 2 ds - 1 0 B s , ξ * ds 2 = ξ d + 1 0 B s , ξ * ds 2 + 1 0 B s , ξ * 2 ds - 1 0 B s , ξ * ds 2 .
The remaining statements follow straightforwardly.

Proposition 23 For each p ≥ 1 one has

E(|det Q| -p ) ≤ C p,d < ∞ ( 58
)
where C p,d is a constant depending on p, d only.

Proof. By Lemma 7-29, pg 92 in [START_REF] Bichteler | Malliavin Calculus for Processes with Jumps[END_REF], for every p ∈ (0, ∞) one has

1 |det Q| p ≤ 1 Γ(p) R d |ξ| d(2p-1) e -Qξ,ξ dξ. Let θ(ξ * ) := 1 0 B s , ξ * ds. Using the previous lemma R d |ξ| d(2p-1) e -Qξ,ξ dξ = R d (ξ 2 d + |ξ * | 2 ) d(2p-1)/2 e -(ξ d +θ(ξ * )) 2 -β B (ξ * ) dξ ≤ C R d-1 ((1 + θ 2 (ξ * )) d(2p-1)/2 + |ξ * | d(2p-1) )e -β B (ξ * ) dξ * ≤ C R d-1 sup t≤1 1 ∨ |B t | d(2p-1) (1 + |ξ * | d(2p-1)+1 )e -β B (ξ * ) dξ * .
We integrate and we use Schwartz inequality in order to obtain

E 1 |det Q| p ≤ C + C {|ξ * |≥1} (E((1 + |ξ * | d(2p-1)+1 ) 2 e -2β B (ξ * ) )) 1/2 dξ * .
For each fixed ξ * the process b ξ * (t) := |ξ * | -1 B t , ξ * is a standard Brownian motion and

β B (ξ * ) = |ξ * | 2 1 0 (b ξ * (t) - 1 0 b ξ * (s)ds) 2 dt =: |ξ * | 2 V ξ *
where V ξ * is the variance of b ξ * with respect to the time. Then it is proved in [START_REF] Donati-Martin | Fubini's theorem for double Wiener integrals and the variance of the Brownian path[END_REF] (see (1.f), p. 183) that

E(e -2β B (ξ * ) ) = E(e -2|ξ * | 2 V ξ * ) = 2 |ξ * | 2 sinh 2 |ξ * | 2 .
We insert this in the previous inequality and we obtain E(|det Q| -p ) < ∞.

We are now able to give the main result in this section. We define

q(B) = d-1 i=1 B i 1 + j =p 1 0 B j s dB p s ( 59
)
and for ε, ρ > 0 de denote

Λ ρ,ε (B) = {det Q(B) ≥ ε ρ , sup t≤1 |B t | ≤ ε -ρ , q(B) ≤ ε}. ( 60
)
Proposition 24 There exist some universal constants c ρ,d , ε ρ,d ∈ (0, 1) (depending on ρ and d only) such that for every ε ∈ (0, ε ρ,d ) one has

P (Λ ρ,ε (B)) ≥ c ρ,d × ε 1 2 d(d+1) . (61)
Proof. Using the previous proposition and Chebyshev's inequality we get

P (det Q < ε ρ ) ≤ ε pρ E |det Q| -p ≤ C p,d ε pρ and P (sup t≤1 |B t | > ε -ρ ) ≤ exp(- 1 Cε 2ρ ). Let q ′ (B) = d-1 i=1 |B i 1 | + j<p 1 0 B j s dB p s . Since 1 0 B j s dB p s ≤ B j 1 |B p 1 | + 1 0 B p s dB j s
we have q(B) ≤ 2q ′ (B) + q ′ (B) 2 so that {q ′ (B) ≤ 1 3 ε} ⊂ {q(B) ≤ ε}. We will now use the following fact: consider the diffusion process X = (X i , X j,p , i = 1, ..., d, 1 ≤ j < p ≤ d) solution of the equation dX i t = dB i t , dX j,p t = X j t dB p t . The strong Hörmander condition holds for this process and the support of the law of X 1 is the whole space. So the law of X 1 is absolutely continuous with respect to the Lebesgue measure and has a continuous and strictly positive density p. This result is well known (see for example [START_REF] Kusuoka | Applications of the Malliavin calculus[END_REF] or [START_REF] Bally | Positivity and lower bounds for the density of Wiener functionals[END_REF]). We denote c d := inf |x|≤1 p(x) > 0 and this is a constant which depends on d only. Then, by observing that q

′ (B) ≤ √ m |X 1 |, where m = 1 2 d(d + 1
) is the dimension of the diffusion X, we get

P (q(B) ≤ ε) ≥ P q ′ (B) ≤ ε 3 ≥ P |X 1 | ≤ ε 3 √ m ≥ ε m (3 √ m) m × cd ,
with cd > 0. So finally we obtain

P (Λ ρ,ε (B)) ≥ cd ε 1 2 d(d+1) -C p,d ε pρ -exp(- 1 Cε 2ρ ). Choosing p > 1 2ρ d(d + 1
) and ε small we obtain our inequality.

Appendix 4. Norms and distances

In this section we use the notation from Section 3 and 4. We consider the matrix A with columns a i , [a] j,p = a j,p -a p,j , i = 1, ..., d, j = p defined in Section 3.2 and we assume that the non degeneracy condition (20) holds. For notational convenience we denote A i = a i , i = 1, ..., d and A i , i = d + 1, ..., m will be an enumeration of [a] j,p , 1 ≤ j, p ≤ d, j = p. We work with the norm

|y| 2 A R = (A R A * R ) -1 y, y , y ∈ R n .
We have the following simple properties:

Lemma 25 i) For every y ∈ R n and 0 < R ≤ R ′ ≤ 1 one has R R ′ |y| A R ≥ |y| A R ′ ≥ R R ′ |y| A R and (62) 1 √ R λ * (A) |y| ≤ |y| A R ≤ 1 R λ * (A) |y| . (63)
ii) For every z ∈ R m and R > 0 one has

|A R z| A R ≤ |z| . (64) iii) For every µ ∈ L 2 ([0, T ]; R m ) and R > 0 one has t 0 µ s ds 2 A R ≤ t t 0 |µ s | 2 A R ds, t ∈ [0, T ]. ( 65
) Proof. i) It is easy to check that R ′ R A R A * R ≤ A R ′ A * R ′ ≤ R ′ R 2 A R A * R
which is equivalent with (62). This also implies (one takes

R ′ = 1 so A R ′ = A) that 1 R λ * (A R ) ≤ λ * (A) ≤ 1 R 2 λ * (A R ) and 1 R λ * (A R ) ≤ λ * (A) ≤ 1 R 2 λ * (A R ) which immediately gives (63). ii) For z ∈ R m , we write z = A * R y + w with y ∈ R n and w ∈ (ImA * R ) ⊥ = KerA R . Then A R z = A R A * R y so that |A R z| 2 A R = |A R A * R y| 2 A R = (A R A * R ) -1 A R A * R y, A R A * R y = z, A R A * R z = A * R y, A * R y = |A * R y| 2 ≤ |z| 2 and (64) holds. iii) For µ ∈ L 2 ([0, T ]; R m ) and t ∈ [0, T ] one has t 0 µ s ds 2 A R = A R -1 t 0 µ s ds, t 0 µ s ds = t 0 t 0 A R -1 µ s , µ u dsdu = 1 2 t 0 t 0 A R -1 (µ s -µ u ), µ s -µ u -A R -1 µ s , µ s -A R -1 µ u , µ u dsdu = 1 2 t 0 t 0 |µ s -µ u | 2 A R -2|µ s | 2 A R dsdu ≤ t 0 t 0 |µ u | 2 A R dsdu = t t 0 |µ u | 2 A R du.
We give now some lower and upper bounds for |y| A R . We denote S = V ect{A i , i = 1, ..., d} and Π S is the projection on S. S ⊥ is the orthogonal of S and Π S ⊥ is the projection on S ⊥ . Moreover we denote

λ S = inf ξ∈S,|ξ|=1 d i=1 A i , ξ 2 , λ S = sup ξ∈S,|ξ|=1 d i=1 A i , ξ 2 (66)
λ S ⊥ = inf ξ∈S ⊥ ,|ξ|=1 m i=d+1 A i , ξ 2 , λ S ⊥ = sup ξ∈S ⊥ ,|ξ|=1 m i=d+1 A i , ξ 2 .
By the very definition λ S > 0 and under assumption (20) we also have λ S ⊥ > 0. And λ S ≤ λ * (A), λ S ⊥ ≤ λ * (A).

Proposition 26 Suppose that (20) holds and let

R < λ S 4 m i=d+1 |Π S A i | 2 . ( 67
)
Then for every y

∈ R n 1 4R λ S |Π S y| 2 + 1 4R 2 λ S ⊥ |Π S ⊥ y| 2 ≤ |y| 2 A R ≤ 4 R λ S |Π S y| 2 + 4 R 2 λ S ⊥ |Π S ⊥ y| 2 . ( 68
) In particular, if |A| ∞ = max i=1,...,m |A i | and R ≤ λ S /4m |A| ∞ then 1 4R |A| ∞ |Π S y| 2 + 1 4R 2 |A| ∞ |Π S ⊥ y| 2 ≤ |y| 2 A R ≤ 4 R λ S |Π S y| 2 + 4 R 2 λ S ⊥ |Π S ⊥ y| 2 . ( 69
)
Proof. In a first stage we assume that A i ⊥A j for i ≤ d < j. We will drop out this restriction in the second part of the proof. Let T S and T S ⊥ be the restriction of y → A R A * R y to S and to S ⊥ respectively. Since

A R A * R y = R d i=1 A i , y A i + R 2 m i=d+1 A i , y A i our orthogonality hypothesis implies that T S y = R d i=1 A i , y A i ∈ S for y ∈ S and T S ⊥ y = R 2 m i=d+1 A i , y A i ∈ S ⊥ for y ∈ S ⊥ . Since A R A * R is invertible it follows that T S (respectively T S ⊥ ) is an invertible operator from S (respectively from S ⊥ ) into itself. For y ∈ S we have R λ S |y| 2 ≥ T S y, y = R d i=1 A i , y 2 ≥ R λ S |y| 2 and since |y| 2 A R = T -1 S y, y for y ∈ S, we obtain 1 R λ S |y| 2 ≤ |y| 2 A R ≤ 1 R λ S |y| 2 , if y ∈ S.
Similarly, we get

1 R 2 λ S ⊥ |y| 2 ≤ |y| 2 A R ≤ 1 R 2 λ S ⊥ |y| 2 , if y ∈ S ⊥ . Let y ∈ R n . Since (A R A * R ) -1 Π S y ∈ S we have (A R A * R ) -1 Π S y, Π S ⊥ y = 0 so that |y| 2 A R = |Π S y| 2 A R + |Π S ⊥ y| 2 A R . We obtain 1 R λ S |Π S y| 2 + 1 R 2 λ S ⊥ |Π S ⊥ y| 2 ≤ |y| 2 A R ≤ 1 R λ S |Π S y| 2 + 1 R 2 λ S ⊥ |Π S ⊥ y| 2 . (70)
We drop now out the orthogonality assumption. For j > d we consider the decomposition A j = Π S A j + Π S ⊥ A j and we define the matrices A R = ( √ RA 1 , ..., √ RA d , RΠ S ⊥ A d+1 , ..., RΠ S ⊥ A m ) and A R = (0, ..., 0, RΠ S A d+1 , ..., RΠ S A m ) so that A R = A R + A R . We will check that under the restriction (67) we have

4 A * R y 2 ≥ |A * R y| 2 ≥ 1 4 A * R y 2 ∀y ∈ R n . (71)
We suppose for the moment that the above inequality is true and we prove (68). Since

|A * R y| 2 = A R A * R y, y the above inequality means that 4A R A * R ≥ A R A * R ≥ 1 4 A R A * R and this gives 1 4 |y| 2 A R ≤ |y| 2 A R ≤ 4 |y| 2 A R . (72)
Since the columns of A R verify the orthogonality assumption we may use the result from the first step with A replaced with A = (A 1 , . . . , A d , A d+1 , . . . A m ), with A j = A j if j ≤ d and A j = Π S ⊥ A j for j > d. Here, we have S = V ect{A 1 , . . . , A d } = V ect{A 1 , . . . , A d } = S, so that λ S = λ S and λ S = λ S . Moreover, since S ⊥ = S ⊥ , the computations in (66) are actually done with ξ ∈ S ⊥ , and thus we obtain λ S ⊥ = λ S ⊥ and λ

S ⊥ = λ S ⊥ . So (70) gives 1 R λ S |Π S y| 2 + 1 R 2 λ S ⊥ |Π S ⊥ y| 2 ≤ |y| 2 A R ≤ 1 R λ S |Π S y| 2 + 1 R 2 λ S ⊥ |Π S ⊥ y| 2
which together with (72) imply (68).

It remains to prove (71). We have

A * R y 2 ≥ R d j=1
A j , y 2 = R From now on we consider the specific situation when a i = σ i (t, x), [a] i,j = [σ i , σ j ](t, x) and we denote by A(t, x) respectively A R (t, x) the matrices associated to these coefficients. We will need the following locally equivalent with d c . So our aim is to prove that d * is locally dominated by d. Let us be more precise: we fix x ∈ Ω and we look for two constants C x , δ x > 0 such that the following holds: if 0 < δ ≤ δ x and d(x, y) ≤ √ δ then one may construct a control φ ∈ R m such that v • (φ) ∈ P * (x, y) and |φ i | < C x √ δ, |φ i,j | < C 2 x δ. This implies d * (x, y) ≤ C x √ δ, and the statement will hold. Notice that we discuss local equivalence, that is why we may take C x , δ x depending on x. We recall that A i (x), i = 1, ..., m is an enumeration of σ i (x), [σ j , σ p ](x), i, j, p = 1, ..., d and that they span R n because x ∈ Ω. So, we choose i 1 < ... < i d ′ ≤ d < i d ′ +1 < ... < i n ≤ m such that A i k (x), k = 1, ..., d ′ span V ect{A 1 (x), ..., A d (x)} and A i k (x), k = 1, ..., n span R n . In particular all of them are linearly independent. We denote B k (x) = A i k (x) and we want to use Theorem 20 for them. Notice that V ect{A 1 (x), .. Since d and d c are locally equivalent for every compact K ⋐ Ω and for every x ∈ K there exists C x , ε x > 0 such that for y ∈ B d (x, ε x ) we have d c (x, y) ≤ C x d(x, y). Since the set K is compact we may find x 1 , ..., x N ∈ K such that K ⊂ ∪ N i=1 B d (x i , ε x i ). We denote C max = max i=1,...,N C x i . Let us prove that there exists r * > 0 such that for every x ∈ K and every y ∈ B d (x, r * ) we have d c (x, y) ≤ C max d(x, y). For x ∈ K one may find i such that x ∈ B d (x i , ε x i ) and r > 0 such that B d (x, r) ⊂ B d (x i , ε x i ). We define r x = sup{r > 0 : ∃ i ∈ {1, ..., N} such that B d (x, r) ⊂ B d (x i , ε x i )}. We claim that r * := inf x∈K r x > 0. Indeed suppose that this is not true. Then one may find a sequence y n → y 0 such that r yn → 0. Since r y 0 > 0 there exists n * such that for n ≥ n * one has B d (y n , 1 2C K r y 0 ) ⊂ B d (y 0 , r y 0 ) ⊂ B d (x i , ε t i ) for some i. Here C K is the constant in the triangle inequality iii) at page 37. And this means that r yn ≥ 1 2 r y 0 > 0 which is in contradiction with our hypothesis. So we have proved that r * > 0. Consider now y ∈ B d (x, r * ). There exists i such that B d (x, r * ) ⊂ B d (x i , ε x i ) and this means that y, x ∈ B d (x i , ε x i ) and consequently d c (x, y) ≤ C x i d(x, y) ≤ C max d(x, y).

Finally we give:

Proof of Proposition 6. We will first prove that under our hypothesis d(x, y) ≤ λ x /(4m)n

λ

  * ,p (B) and λ * (B) = d p=1 λ * p (B) where λ * ,p (B), λ * p (B) are the smaller and the larger eigenvalues of Q p (B). We come now back to our problem. Let η(∆(B)), ψ(∆(B)), ε(∆(B)), µ(∆(B)) be the quantities defined in (17) with ∆ = ∆(δ, W ) replaced by ∆(B). Then δη(∆(B)) = η(δ, W ). The same is true for ψ and ε and finally √ δµ(∆(B)) = µ(δ, W ). So using (19)

  j=1,d , k = 1, ..., d. So, given y ∈ R d and recalling that ∇Φ(0) = I + ∇η(0), the equation Φ(θ) = y reads θ = U y (θ), with U y (θ) := (∇Φ(0)) -1 y -1 0

1 2 |θ| + 2

 12 |y| which gives |θ| ≤ 4 |y| . Moreover, again by (47),

  4r), (54) gives also c * (η Γ , 4r) ≤ 1 2m . And since |Γx| Γ = |x|, one has G ∈ B Γ (y, r) iff Φ Γ (Θ) ∈ B(0, r). Then by a change of variable, for z ∈ B Γ (y, r) we have

1 0 1 0g s , ξ * 2 ds - 1 0

 111 s ds, j, p = 1, ..., d -1 and we denote by λ(Q) (respectively by λ(Q)) the lower (respectively larger) eigenvalue of Q. For a measurable function g : [0, 1] → R d-1 we denote α g (ξ) = ξ d + g s , ξ * ds, β g (ξ) = g s , ξ * ds 2 with ξ = (ξ 1 , ..., ξ d ) ∈ R d and ξ * = (ξ 1 , ..., ξ d-1 ).

AΠ|Π S A j | 2 . 2 ≥ 4 A * R y 2 . 2 .

 22422 j , Π S y 2 ≥ R λ S |Π S y| 2 S A j , y 2 = R 2 m j=d+1 Π S A j , Π S y 2 ≤ R 2 |Π S y| 2 m j=d+1 Then (67) gives A * R y Using the inequality (a + b) 2 ≥ 1 2 a 2 -b 2 we obtain |A * R y| 2 = A * R y + A * R y a + b) 2 ≤ 2a 2 + 2b 2 we get |A * R y| 2 ≤ 4 A * R y

Lemma 27 4 ( 2 = m k=1 ( 2 ≥ 1 2 m k=1 A 2 ≥ 1 2 m k=1 A

 2742k=1212k=1212k=1 Let x, y ∈ R n be such that |x -y| ≤ 1 and let s, t∈ [0, 1]. Assume that |x -y| + |t -s| ≤ λ * (A(t, x)) (8dm)n 2 (t, x)Then for every z ∈ R n and δ ≤ 1 one has1 4 |z| 2 A δ (t,x) ≤ |z| 2 A δ (s,y) ≤ 4 |z| 2 A δ (t,x) . (74)Proof. The inequality (74) is equivalent to4(A δ A * δ )(t, x) ≥ (A δ A * δ )(s, y) ≥ 1 A δ A * δ )(t, x).We use the numerical inequality (a + b) 2 ≥ 1 2 a 2 -b 2 , the hypothesis (73) and we obtain(A δ A * δ )(s, y)z, z = m k=1 A δ,k (s, y), z A δ,k (t, x), z + A δ,k (s, y) -A δ,k (t, x), z ) δ,k (t, x), z 2 -m k=1 ( A δ,k (s, y) -A δ,k (t, x), z ) δ,k (t,x), z 2 -(2dm) 2 n 4 (t, x)δ(|x -y| 2 + |t -s| 2 ) × |z| 2 .

BA

  ., A d (x)} = V ect{B 1 (x), ..., B d ′ (x)} so the projections Π and Π ⊥ considered in Theorem 20 and in Proposition 62 coincide.In particular if d(x, y) ≤ √ δ then |Π(y -x)| ≤ |A(x)| ∞ √ δ and Π ⊥ (y -x) ≤ |A(x)| ∞ δ. And this also implies that |y -x| ≤ 2 |A(x)| ∞ √ δ. As θ ∈ R n ,we look for a solution to the equationy = ξ 1 (θ), with ξ t (θ) = x + k (ξ s (θ))ds = x + i k (ξ s (θ))ds.So, we write it asy = x + B(x)θ + r(θ) with r(θ) = (ξ s (θ)) -B k (x))ds.Clearly r ∈ C 3 (R n , R n ) and r(0) = ∇r(0) = 0. Then,|y -x| ≤ 2 |A(x)| ∞ √ δ ≤ 2 |A(x)| ∞ δ xand we suppose that δ x is sufficiently small in order that |y -x| satisfies (49), that is|y -x| < λ * (B(x)) 1/2 4 and |y -x| < λ * (B(x)) 8d 3 (c 2 (r) + c 2 (r)).We use then (51) and we obtain|θ i | ≤ C x √ δ, i = 1, ..., d ′ and |θ i | ≤ C 2 x δ, i = d ′ + 1, ..., n. This proves that d * (x, y) ≤ C x √ δ. C.For x ∈ Ω, we denote B d (x, r) := {y ∈ Ω : d(x, y) < r} and this is an open set.

  4 (x). Let R be such that d(x, y) ≥ √ R so that |y -x| A R (x) ≥ 1. Then by (63)λ x λ * (A) (4m)n 4 (x) ≥ |y -x| ≥ R λ * (A) |y -x| A R ≥ R λ * (A).It follows that R ≤ λ x /(4m)n 4 (x) which proves our assertion.We suppose now that d(x, y) ≥ √ R. Since R ≤ λ x /(4m)n 4 (x) we may use[START_REF] Capitaine | On the Onsager-Machlup functional for elliptic diffusion processes[END_REF] and we obtain 4Rλ x |Π x (y -x)| 2 + 4 R 2 λ ⊥

				√	R. Then
	1 4Rn 2 (x)	|Π x (y -x)| 2 +	1 4R 2 n 2 (x)	Π ⊥ x (y -x)

x Π ⊥ x (y -x) 2 ≥ |y -x| A R (x) ≥ 1 which gives d(x, y) ≥ √ R. So d(x, y) ≥ d(x, y). Suppose that d(x, y) < 2 ≤ |y -x| 2 A R (x) < 1

and this reads d(x, y) < √ R which gives d(x, y) ≤ d(x, y).

m/2 π m/2 √ det Q |det Γ| exp(-1 8λ(Q) |z -y| 2 Γ ).

Since λ * (A δ (t, x)) ≥ δ 2 λ * (A(t, x)) our hypothesis says that (2dm) 2 n 4 (t, x)δ(|x

A δ,k (t, x), z 2 so that

Using (a + b) 2 ≤ 2a 2 + 2b 2 one proves the other inequality.

In the last part of this section we establish the link between the norm |z| A R (t,x) and the control (Caratheodory) distance. We will use in a crucial way the alternative characterizations given in [START_REF] Nagel | Balls and metrics defined by vector fields, I. Basic properties[END_REF] for this distance -and these results hold in the homogeneous case: the coefficients of the equations do not depend on time any more, so that we suppose now σ j (t, x) = σ j (x). Consequently, we handle the matrix A R (x) instead of A R (t, x).

We first introduce a semi-distance d on an open set Ω ⊂ R n which is naturally associated to the family of norms

It is straightforward to see that d is a semi-distance on Ω, in the sense that d verifies the following three properties (see [START_REF] Nagel | Balls and metrics defined by vector fields, I. Basic properties[END_REF]): i) for every r > 0, the set {y ∈ Ω :

iii) for every compact set K ⋐ Ω there exists C > 0 such that for every x, y, z

Moreover, one says that

In particular if d 1 is a distance and d 2 is equivalent with d 1 then d 2 is a semi-distance. And one says that d 1 is locally equivalent with d 2 if for every x 0 ∈ Ω there exists a neighborhood V of x 0 and a constant C such that (75) holds for every x, y ∈ V.

We introduce now the control metric. For x, y ∈ R n we denote by C(x, y) the set of controls ψ ∈ L 2 ([0, 1]; R n ) such that the corresponding skeleton du t (ψ) = d j=1 σ j (u t (ψ))ψ j t dt with u 0 (ψ) = x satisfies u 1 (ψ) = y. Notice that the drift b does not appear in the equation of u t (ψ). Then we define

: ψ ∈ C(x, y) .

.

Then for every x, y ∈ Ω such that d c (x, y) ≤ 1 4 α 2 (x) one has d(x, y) ≤ 9α 2 (x)d c (x, y). B. d is locally equivalent to d c on Ω. C. In particular for every compact set K ⋐ Ω there exists r K and C K such that for every x, y ∈ K with d(x, y) ≤ r K one has d c (x, y) ≤ C K d(x, y).

Proof A. Let δ > 0, x, y ∈ Ω and ψ ∈ C(x, y). Setting x t = u t/δ (ψ), we obtain dx t = d j=1 σ j (x t )φ j t dt with φ(t) = δ -1 ψ(tδ -1 ), which means that x t = u t (φ). Notice also that

As a consequence of the previous computations, one has

Suppose now that d c (x, y) ≤ 1 4 α 2 (x). We take δ = 1 4 α 2 (x) so that d c (x, y) ≤ 1 2 α(x) √ δ. Then one may find a control φ ∈ C δ (x, y) such that ε φ (δ) ≤ 1 2 α(x) and y = x δ (φ). Since ε φ (δ) + √ δ ≤ α(x) we may use (38) and we obtain |y -

and this gives d(x, y) ≤ 9α 2 (x)δ = 9α 2 (x) × 1 4 α 2 (x). And this guarantees that d(x, y) ≤ 9α 2 (x) × d c (x, y). B. We prove now the converse inequality. We use the results from [START_REF] Nagel | Balls and metrics defined by vector fields, I. Basic properties[END_REF] so we recall the definition of the semi-distance d * (which is denoted by ρ 2 in [START_REF] Nagel | Balls and metrics defined by vector fields, I. Basic properties[END_REF]). Given φ i , φ k,j ∈ R, i = 0, ..., d, 1 ≤ k < j ≤ d we consider the equation

Notice that φ j , φ i,j are now real constants (in contrast with the time depending controls in the standard skeleton) and we have added the vector fields [σ i , σ j ] which does not appear in skeletons. And the drift term b does not appear. We denote by P * (x, y) the family of paths v t (φ) which satisfy (76) and such that v 1 (φ) = y. We define d * by: d * (x, y) < δ if and only if one may find φ i , φ k,j ∈ R, 1 ≤ i, k, j ≤ d, j = k such that v • (φ) ∈ P * (x, y) and |φ i | < δ, |φ i,j | < δ 2 . As a consequence of Theorem 2 and Theorem 4 from [START_REF] Nagel | Balls and metrics defined by vector fields, I. Basic properties[END_REF] d * is