The H\"ormander condition for delayed stochastic differential equations

In this paper, we are interested in path-dependent stochastic differential equations (SDEs) which are controlled by Brownian motion and its delays. Within this non-Markovian context, we prove a H\"ormander-type criterion for the regularity of solutions. Indeed, our criterion is expressed as a spanning condition with brackets. A novelty in the case of delays is that noise can"flow from the past"and give additional smoothness thanks to semi-brackets. Our proof follows the general lines of Malliavin's proof, in the Markovian case, which led to the development of Malliavin calculus. In order to handle the non-Markovian aspects of this problem and to treat anticipative integrals in a path-wise fashion, we heavily invoke rough path integration.

This is a first paper on the general question of smoothness for marginals of solutions to (non-Markovian) SDEs. Here, we fix a time maturity T > 0 and (h i ) 0≤i≤N −1 ∈ R N + is an increasing sequence of delays satisfying: 0 =h 0 < h 1 < h 2 < · · · < h N −1 < T . (1.0.1) We also fix m, d > 0 two integers and consider the random variable X T ∈ R d where (X t ) 0≤t≤T is the solution to a delayed SDE: The process (W k ) 1≤k≤m is an m-dimensional Brownian motion. The vector fields are of the form and depend smoothly on delayed values of the path X. By convention, the additional index k = 0 will refer to time and W 0 t = t. Notice that the Stratonovich stochastic integration •dW k · is ill-defined as the integrand has no reason to be a semi-martingale. At this point, we leave to later the discussion regarding which theory of stochastic integration is invoked. Also N = 1 recovers the usual Markovian setting.
The Markovian setting i.e when V k = V k (t, X t ) has a quite beautiful answer in the form of Hörmander's spanning condition [Hör67]. Of course, the language of Hörmander was functional analysis and PDEs. The translation from probability to PDEs is readily obtained when one remembers that densities are fundamental solutions to the forward Fokker-Planck PDE. Malliavin's proof pushed further by giving a probabilistic approach. We recommend [Hai11] for a pedagogical review.
We choose to work under the Stratonovich convention and such a choice is not inoccuous. Indeed, it is well-known that the Stratonovich reformulation in terms of vector fields is the right language for "geometric" arguments (see for e.g [Hsu02]). The Hörmander condition itself is very geometric by nature, since it morally says that heat dissipates along the vector fields V k and their brackets, due to the erratic movement of Brownian motion. Another reason is our use of "geometric rough paths" (see [FH14,Chapter 2.2] for a definition and [FH14, Chapter 3] for the discussion) thanks to which the Itô formula looks similar to the usual chain rule.
In the literature, there two ways of understanding the word "non-Markovian" regarding the topic of Hörmander's hypoellipticity. On the one hand, certain authors as [CF10,HP13,CHLT15] mean that the SDE's driving noise is a fractional Brownian motion. On the other hand, another legitimate direction of investigation is to consider a source of non-Markovianity which is the path-dependence of our SDE. To the best of the authors' knowledge, regularity under a non-Markovian nature originating from path-dependence has been treated by [Bel04], in the form of delays and in [Str83] with a dependence via a kernel. Both Bell and Stroock rely heavily on Malliavin calculus.
Ultimately, we are interested in SDEs with the most general path-dependence. However, in the present paper, we choose to focus on the setting of SDEs with delays. As such, we mainly revisit Section 4 of [Bel04], and [BM91], [BM95] with a rough path approach and aiming for a Hörmander-type spanning condition.
Setting: Let α be a real satisfying 1 3 < α < 1 2 and let C α be the Banach space of α-Hölder continuous functions and for a given a, C α a is the subset of C α that contains paths are equal to a at time 0. For a vector field F : and a path (t, x) ∈ [0, T ] × C α we define the partial derivatives of the functional F as the elements given by: Note that for all i = 0, . . . N − 1 these partial derivatives measure the sensitivity of the vector field F with respect to the i-th delay. The action of ∂ i F on a vector a vector v ∈ R d will be denoted The delay case is also the setting of [NNT + 08] where the authors prove well-posedness for SDEs driven by rough paths and in particular for fractional Brownian motion with Hurst parameter H > 1 3 . Also, given the right integration framework, which will be given in the preliminaries, we shall see that there is global existence and uniqueness of solutions under the following analytic assumptions.
Assumption 1.1 (The analytic assumptions). The family of functions are smooth with bounded derivatives at all order and satisfy for all k = 1 . . . m, In our main Theorem 1.5, we give a criterion in the form of spanning conditions, which is a geometric assumption. We now define the analog of Hörmander's condition for delayed diffusions.
Definition 1.2. 1) We introduce first the Lie brackets of the vector fields with respect to the end-point of X: where ∂ 0 stands for the derivative defined at (1.0.4).
2) Given the SDE (1.0.2), we define sets of vector fields which span the Lie algebra generated by the V k : 3) We also define extensions of these sets by the contribution of V 0 and semi-brackets: , ∂ t F and ∂ i F are well defined for all F ∈ V j and k ≥ 0 since both F (t, x) and V k (t, x) can be expressed as smooth functions of (t, x t , . . . x t−h N−1 ).
ii) Note that for j > 0, the set V j is smaller than its Markovian counter-part that also contains the brackets with V 0 . The bracket with V 0 is introduced at V j and we are able to infer regularity results for diffusions such as the Langevin equation with delay (treated in Section 5).
iii) The fundamental difference between V j and V j is the fact that the elements of the first are functions of (t, x t , . . . x t−h N−1 ) but not the elements of the latter.
Notice that in the non-Markovian case the functionals V k are necessarily depending on t in a peculiar manner as time plays a special role. Thus, unlike [Hai11] for example, we made the choice of not treating the time variable as an additional dimension and adjust the brackets with respect to V 0 by adding the time derivative. However, the estimate on the drift part does not allow us to give separate contributions for each of the different terms in the sum We can only rely on the contribution of the sum to produce smoothness.
Assumption 1.4 (The geometric assumption -Hörmander's hypoellipticity condition). We assume either of the following hypotheses: (1) Weak hypoellipticity: ∃j 0 such that (2) Strong hypoellipticity : ∃j 0 such that (3) The bounded case : The process (X t ) t∈[0,T ] is uniformly bounded by a deterministic constant and there exists j 0 ≥ 0 such that for all x ∈ C α X 0 and for all |η| = 1 the pointwise Hormander condition hold We are now ready to state the main result of the paper.
Theorem 1.5. Let X T be the marginal of the solution to the SDE (1.0.2). If the analytic assumption 1.1 is satisfied, as well as either of the geometric assumptions in 1.4, then X T has a smooth density with respect to the Lebesgue measure.
We give the proof of this theorem only at Subsection 4.2 after handling all the prerequisites.
Structure of the paper. In the Preliminaries of Section 2, we start by making precise rough path integration against Brownian motion and its delays. This will show that Eq.
(1.0.2) is well posed in Stratonovich form with unique solutions.
Section 3 defines and collects results on the Malliavin derivative in our non-Markovian context.
Section 4 proves our main result by taking Malliavin's original approach. We define the classical Malliavin Gram matrix, quickly review how its control yields smoothness and relate it to tangent flows.
Finally we conclude with examples in Section 5. These are interesting in their own right. We give a treatment of smoothness for the Langevin equation with delay, as well as cases where smoothness comes from "semi-brackets". Acknowledgements. R.C. and I.E. would like thank Y. Bruned, L. Coutin and J. Teichmann for fruitful conversations. Research of I.E. is partly supported by the Swiss National Foundation Grant SNF 200021 153555.

Preliminaries
From now on, m will refer to the number of Brownian motions we will be working with and d is the dimension of the process X we will study. {e j } j=1,··· ,d is the canonical basis of R d and {f k } k=1,··· ,m the canonical basis of R m .
Throughout the paper V will stand for a finite dimensional vector space. For any V , we denote by C([0, T ], V ) the space of continuous V -valued paths. C α ([0, T ], V ) will denote the subspace of α-Hölder continuous functions. We will drop the dependence in V if it isobvious from context. Also, given a path X : [0, T ] → V and (s, t) ∈ [0, T ] 2 , we write the increment between s and t as X s,t = X t − X s . Let us start by giving a meaning to the Equation (1.0.2) and a solid foundation to its treatment.
2.1. Stochastic integration and rough paths.
2.1.1. Enhancing Brownian motion to a rough path. In this section we give statements for any given (h i ) 0≤i≤N −1 ∈ R N + increasing sequence of delays. The results will be valid upon changing h to another sequence of delays if necessary. We set V (h) := R m×N and consider the V (h)-valued process: where each component is understood as a vector in R m . When h is understood from context, we drop the dependence in h and write W t instead of W t (h). Also, there is no loss of generality in assuming that W is two-sided: (W t ; t ≥ 0) and (W −t ; t ≥ 0) are independent Brownian motions. Taking W to be two-sided will avoid problems of boundary effects and delays can be arbitrarily large. A relevant quantity will be the first delayed date before maturity The goal of this subsection is to establish that W is a bona-fide rough path against which we can integrate. We now give a lemma concerning the quadratic covariation of the process W.
Lemma 2.1. Consider two indices i, j, and two reals h, h ′ . For a partition P of [s, t] with mesh size going to zero, we have the limit in L 2 and in probability: Proof. The case h = h ′ is obvious. By symmetry and time shifting, we can assume h > 0 and h ′ = 0. For shorter notations, set Also: Indeed, in the double sum, the right-most interval among [u, v], We fix α, θ satisfying 1/3 < α < 1/2 < θ < 2α. Recall that an α-Holder path X is lifted to a rough by adjoining another path X which is 2α-Holder. We recall the following definition from [FH14] of the topology we use: Definition 2.2. We say that the pair (X, X) is an α-Holder rough path on a Banach We first need to define the first order iterated integrals of W in order to form the lift W also known as Lévy stochastic areas. With V = R m×N , it is an (V ⊗ V )-valued path and it is given for s < t: The matter at hand is to give a precise meaning of the above integrals in such a way that the "first order calculus" condition (2.2.2) holds. We have two possibilities.
The first possibility is to define the iterated integrals as limits in probability of Riemann sums. More precisely, if P is a partition of [s, t] with mesh size |P| → 0 and (X, Y ) is a pair of paths: The above limit is well-defined for (X, as soon as h i 2 ≤ h i 1 . Indeed, the left-centered Riemann sum converges by standard adapted Itô integration and we can use Lemma 2.1 to pass to the Stratonovich case. In the other case h i 2 > h i 1 , notice that we have the first order calculus rule at the discrete level: and as such the second term converges to a limit as soon as the first does. Therefore, (2.2.3) is well-defined as limits in probability of Riemann sums and gives a geometric rough path, as the first order calculus rule is built-in at the discrete level already.
The second possibility is to invoke an anticipative integration theory such as Skororhod's. In their paper [NP88] Section 4, Nualart and Pardoux form anticipative Riemann sums which are centered "A la Stratonovich", prove that they converge and relate them to Skorohod's integral. In any case limits in probability of Riemann sums and anticipative Stratonovich integrals "A la Nualart-Pardoux" coincide. See [FH14,Exercice 5.17] as well as [OP89].
2.1.2. Integration with respect to W. Thanks to this paragraph, for systems controlled by delays, we will give a proper meaning to the integration in Eq. (1.0.2).
We denote the space of controlled rough paths by We recall the following Gubinelli integration theorem from [FH14, Theorem 4.10].
2.1.3. Roughness of W. For the convenience of the reader, we recall the concept of roughness for rough paths as in [FH14, Definition 6.7]. Our goal in this subsection is to prove roughness for W. This will be crucial in order to use the so-called Norris lemma, a quantitative version of the Doob-Meyer decomposition.
The largest such L is called the modulus of θ-Holder roughness of X.
Lemma 2.6. We can choose a version of the Brownian motion such that W is θ-Holder rough at scale T 2 on [0, T ]. Proof. The proof of roughness is exactly the same as the proof of [FH14, Proposition (6.11)]. The only ingredient that is missing is the small ball estimate for W, which we now prove.
Set ∆(h) := min 0≤i≤N −2 |h i − h i+1 | with the convention that it is infinity when N = 1. We shall prove that there are constants c, C > 0 such that for all ε > 0, δ > 0 and ϕ = (ϕ i,k ) ∈ V * : where |ϕ| is the Euclidian norm. This estimate is sufficient to replace [FH14, Eq. (6.11) p. 91] so that all the arguments carry verbatim. To prove Eq. (2.6.1), we start by using the translation invariance and symmetry of Brownian motion increments: Now notice that W being a two sided Brownian motion, the family of processes Brownian motions as we have increments over disjoint intervals when changing i and independent Brownian motions when changing k. As such by packaging them into a single Brownian motion B and then invoking the standard small balls estimates, there exists constants c, C such that: Remark 2.7. The use of the two sided Brownian motion instead of the Brownian motion allows us to cancel the boundary effects in Lemma 2.6. However there is small price to pay here. In order to make the formulas work in the sequel, for all k = 0, . . . , m, we extend V k to negative times: Note that this extension is continuous on (−∞, 0) and on (0, ∞). We will need to take care of the discontinuity at time 0.
which solves both the SDE (2.8.2), formulated in term of Itô integrals and the RDE: For completeness, we show that it is possible to reformulate the SDE in an Itô form as the vector fields are adapted. To do so, one should define the Itô lift W Itô from W by taking into account quadratic variations: s,t . The covariation of Brownian motion against its own delay is zero which is known as absence of autocorrelation. This was already formalized in Lemma 2.1. It is classical to see that rough integration against adapted processes and with the Itô lift W Itô coincides with the usual adapted stochastic integration. As such Eq. (1.0.2) is readily reformulated as an Itô integral: where V 0 (r, X) = V 0 (r, X) + 1 2 m k=1 ∂ 0 V k (r, X) · V k (r, X) and, Notice that X is a semi-martingale although the integrands in the SDE are not necessarily semi-martingales. The reader more familiar with the Itô framework rather than rough paths can establish well-posedness of Eq. (2.8.2). Indeed, the vector fields V k are Lipschitz continuous in the variable x for every fixed t, uniformly. We have existence and uniqueness of strong solutions to the Equation (1.0.2) via a standard implementation of the Picard iteration scheme, only in the function space C [0, T ], R d (see the more general theorem 4.6 in [LS01]). Via standard arguments in this framework, solutions are global with E sup 0≤s≤T |X s | p 1 p < ∞. We now want some sort of Itô formula for processes of the form It is easy to see that this process has no reason to be a semi-martingale and as such, we can only give a rough integral formulation of the Itô lemma. Also after formal computations one notices that the Gubinelli derivative of this process are not controlled by W and they cannot be integrated with respect to W. However, they are controlled by Thus we define the family of double delays h := {h j 1 + h j 2 : j 1 , j 2 = 0 . . . N − 1} and choose a family of index J ⊂ {0, . . . , N − 1} 2 with minimal cardinality such that h = {h j 1 + h j 2 : j = (j 1 , j 2 ) ∈ J}. By the construction at Subsection 2.1, we obtain a rough path W t (h) := {W t−(h j 1 +h j 2 ) } j∈J along with its first order iterated integral W(h).
The following holds: Proposition 2.9 (Itô formula). Let F : R + × V (h) → R d be a smooth function of time and (X t , X t−h 1 , . . . , X t−N −1 ). The path t → F (t, X) is controlled by W(h) as the following control equation holds, for all 0 ≤ s ≤ t ≤ T with s / ∈ {h j : j = 0 . . . N − 1}, we have: ] < ∞ for all p > 0. Moreover, the Gubinelli derivatives of t → F (t, X) are controlled by W(h) as for all i = 0, . . . N − 1, k = 1 . . . , m and 0 ≤ s ≤ t ≤ T such that s / ∈ h we have: Finally, we have the following rough integrals against W(h), W(h) : (2.9.3) Proof. Both control equations hold by virtue of a Taylor

The Malliavin derivative
In the context of performing probabilistic constructions and estimating densities, one needs to be able to differentiate with respect to Brownian trajectories. This contribution of Malliavin brought functional analysis to probability.  (3.1.1) when it exists. For ease of notation, if f takes as input functions in C d , i.e from [0, T ] to R d , then for all h ∈ C d , we also define the Fréchet derivative matrix Df (x)(h) ∈ M d (R) as the matrix whose j-th column is where (h) j ∈ C is the path of the j-th column of h. The functional f is said to be Fréchet differentiable if the Fréchet derivative exists and is a bounded linear operator. The operator norm is the supremum norm.
In the particular case of Brownian motion, we obtain the Malliavin derivative. In order to define the Malliavin derivative we introduce the Cameron-Martin space Let F be an R d -valued smooth functional that depends on the Brownian motion W . The Malliavin derivative of F applied to h ∈ H is defined as the Fréchet differential of F : The iterated Malliavin derivatives D j are defined in the same fashion from higher order Fréchet differentials. Then D j acting on random variables is extended to the domain D j,p in L p (Ω), p ≥ 1, with respect to the norm: Moreover, we write: D j,∞ := ∩ p≥1 D j,p . For further details we refer to [Nua06, Section 1.2].
A standard notation is to represent the Mallavin derivative as an element in the Cameron-Martin space DF = D 1 t F, . . . , D m t F 0≤t≤T and write: Morally, at time t and for j = 1, · · · , m, the operator D j t is given by: We denote by D t F ∈ M d,m (R) the matrix whose j-th column is D j t F . The following proposition sums up the properties of the Malliavin derivative in our context. Proposition 3.2 (Kusuoka-Stroock [KS84]). For all t ≤ r, the random variable X r belongs to the space D 1,∞ . Moreover, for all j = 1 . . . m, the Malliavin derivative D j r X t of the random variable X t satisfies: One also has D j r X t = 0 if t < r, as X r is adapted. In matrix notation one has where V (r, X) is the matrix whose columns are V j (r, X) for j = 1 . . . m.
Pointers to the proof. This is essentially [KS84, Lemma (2.9)]. Note that the latter reference uses Ito's formulation. Thus we start with the equation (2.8.2) and see that it satisfies the assumptions in [KS84, Lemma (2.9)]. Due to the analytical assumptions 1.1 the functions V k admits Fréchet derivatives at all order and for all h ∈ H the Malliavin derivatives (DX s (h)) s∈[0,T ] solves the SDE Additionally, as proven by Kusuoka and Stroock, the mapping D j X t is a Hilbert-Schmidt operator on H hence the existence of {D j Note that the equality which implies by identification (3.2.1).

3.2.
Factorization of the Malliavin derivative. The main result of this section concerns a factorization of the Malliavin derivative.
Proposition 3.3. Define the family of processes (J r,t ; 0 ≤ r ≤ t ≤ T ) as the solution to the SDE: Then, for 0 ≤ r ≤ t ≤ T , the tangent process and the Malliavin derivative satisfy where on the LHS, the product denotes a matrix product.
Proof . Inspecting equations (3.2.1) and (3.3.1), we recognize the same stochastic differential equation with a different initial condition. The starting condition V j (r, X) in equation Remark 3.4. In the Markovian setting, let X t,x · be the solution to (1.0.2) such that X t,x t = x. By uniqueness of the solution, there exists flow maps Φ t,s : Remark 3.7. The splitting property (3.6.2) is one of the main limitations of this paper. This property gives the invertibility of {J s,T } s∈[T h ,T ] and its regularity in s. This property does not hold for s ≤ T h since there would be an extra noise coming from the delays. Additionally, when V k is a general path-dependent functional there is no obvious way to obtain the invertibility of J s,T and its regularity in s.

Malliavin's argument: smoothing by Gaussian noise
The gist of Malliavin's argument is that the random variable X T is a complicated function of the Brownian motion W . Provided that such a map is smooth enough, and because Gaussian noise is smooth, one expects X T to have a smooth density. The quantity that encodes this dependence is the Malliavin matrix M 0,T ∈ M d (R) which is defined as: It is morally a Gram matrix or a covariance matrix of the sensitivities of X T to the Brownian motion W . The norm of its inverse will control the smoothness of the map W → X T . Let η ∈ R d such that |η| R d = 1. Thanks to Proposition 3.3 we have: Remark 4.1. In the Markovian case, it is very convenient to introduce the reduced Malliavin matrix C 0,T such that M 0,T = J 0,T C 0,T J * 0,T . In that case, tangent processes have the multiplicative property J s,T = J 0,T J −1 0,s , and one obtains: which is an adapted process. This classical trick allows to use Itô calculus to study the matrix C 0,T and relate its evolution to iterated Lie brackets, thus to the Hörmander's condition. See the general guidelines of Theorem 4.5 in [Hai11].
However, in our setting, such an approach is not possible because the infinitesimal flow property (Eq. (3.3.1)) takes a more complicated form. It is a priori not obvious to find a reduced Malliavin matrix which is the integral of an adapted process. This is the reason why, we perform an analysis only on the segment [T h , T ]. 4.1. The evolution of Z F and its derivatives. In this subsection, we fix a functional of time and X t , X t−h 1 , . . . , X t−h N−1 denoted by F : R + × (R d ) N → R d and compute the expansion as a rough integral of {η * J t,T F (t, X)} t∈[T h ,T ] on the path W. For notational simplicity we define Z F (t) := η * J t,T F (t, X) .
The underlying assumption is that F is smooth and all of its derivatives at any order are bounded.
Recall that the rough path W(h), W(h) is the lift of W taken with the family of double delays h and defined at subsection 2.2. We also mentioned at Remark 2.7 that the functionals V k have discontinuities at time h i . In order to avoid problems due to this lack regularity and to be able to use the Norris' lemma we define Note that on the interval the analysis above concerning the Malliavin derivative holds. We also have the following lemma where all the integrands are free of discontinuities on for T h < s ≤ t ≤ T and s / ∈ h satisfies Proof. Apply the Leibniz rule on the product Z F (s) = ηJ s,T F (s), and then use the rough integral expansions for F (Proposition 2.9) and J ·,T (Proposition 3.6).
Note that we can apply Lemma 2.6 for the rough path W(h) and obtain that this path is θ-Holder rough. We can apply the Norris' Lemma in [HP13,Theorem 3.1] in the following form.
where the constant C depends only on T, {h i } and m and . The choice of T h is mainly motivated by the fact that B C α ,I might become infinite if the interval I contains an element of h.
For notational simplicity we define the key quantity for all n ∈ N which satisfies E[R p n ] < ∞ for all p > 0 and n ∈ N. Recall the definition of j 0 in our Hörmander assumption 1.4. It is the rank such that V j 0 has the uniform spanning condition.
Lemma 4.4. Fix j 0 ∈ N. There exist deterministic constants p 0 , q 0 , C > 0 depending on j 0 , T and {h i }, such that for all F ∈ V j 0 , we have Proof. We reason by induction over the index j 0 .
For initial step j 0 = 0, we start by the fact that there exists a constant C h,T such that for all j = 1 . . . m We finish the proof of Eq. (4.4.2) with the obvious inequalities Now for the induction step, we assume that the result holds true for j 0 . Consider F ∈ V j 0 +1 . Due to the definition of the brackets at (1.2.1), there exists G ∈ V j 0 such that G is a function of the form G : R + × (R d ) N → R d and Z F is a Gubinelli derivative of Z G or Z F is the absolutely continuous part in the decomposition of Z G . We apply the Norris Lemma 4.3 to Equation (4.2.1) for Z G on [T h , T ] to have the existence of C 1 , p 1 and q 1 such that This implies by induction hypothesis that there are p 0 and q 0 such that The fact that this inequality is in particular true for F ∈ V j 0 +1 is what we need to iterate.

4.2.
Proof of Theorem 1.5. We first prove the theorem under the assumptions (i) or (ii). It is classical that E |M −1 0,T | p < ∞ for all p ≥ 2 is a sufficient condition for the existence of smooth densities for X T (see for example [NP88, Theorem 2.1.4]). As shown in [Hai11,Lemma 4.7], this latter statement is itself implied by the existence for all p ∈ N of a constant C p such that: We now use the inequality (4.4.1) at time T and obtain Due to the Hörmander condition in assumption 1.4 the left hand side is a positive deterministic constant that we denote δ > 0. We obtain η, M 0,T η ≥ δ 1/r 0 (CR p 0 j 0 ) 1/r 0 . Using the integrability of R j 0 we easily obtain (4.4.3).
Remark 4.5 (Special case of bounded diffusion). Note that the classical Hörmander theorem requires a pointwise spanning condition. This is due to the fact that in the Markovian case the derivative of the flow J t,T is invertible for all t ∈ [0, T ] and the spanning condition is only required at the initial point of the diffusion. We do not have any hope of obtaining this invertibility. Thus we are only able to reason at time T and check a spanning condition at the random variable (T, X T ) via our uniform condition 1.4. Note that if we know a priori that the diffusion is bounded we can still have a more pointwise statement of the Hörmander condition. In order to state this result we give a weaker version of Assumption 1.4.
We now prove the theorem under assumption (iii). Denote C the constant bounding X t . j 0 is finite and the functions F ∈ V j 0 are continuous on a finite dimensional space. Thus there exists η * with |η * | = 1 and x * ∈ C α with such that We now use the Hormander condition 1.4.1 at the point x * to obtain that the existence of F * ∈ V j 0 such that |η * F * (T, x * )| = δ > 0. Similarly to the beginning of this section we obtain that η, M 0,T η ≥ δ 1/r 0 (CR p 0 j 0 ) 1/r 0 . and finish the proof.

Examples
5.1. Uniformly elliptic diffusions. At this subsection we assume that there exists ε > 0 such that for the order of symmetric matrices we have V V * ≥ εid.
Note that under this assumption the uniform spanning condition holds for j 0 = 0 and we obtain the smoothness of X T . However this results is not new. Indeed in [KS84] and also in [BCC16], it is shown in a general path-dependent framework that this condition implies the smoothness of X T .

Langevin Equation with
Delay. Consider the diffusion in R 2 dp t = V 0 (p t , q t )dt + V 1 (p t , q t , p t−h , q t−h ) • dW t dq t = p t dt .
with V 1 uniformly elliptic. By checking the spanning condition, one realizes that (V j ; j ≥ 0) is stationary from the index j = 0 and for all j ≥ 0: We compute V 0 Hence the uniform spanning condition is satistied. Notice that the semi-brackets play absolutely no role in this case, only the standard case is extended "as is".
5.3. Noise flowing from the past and semi-brackets. We now consider the following diffusion  Again, we check the spanning condition. We have: We compute the semi-brackets ∂ 1 V 2 (t)V 1 (t − h), ∂ 1 V 1 (t)V 2 (t − h) hence finding a subset of V 0 . We have: Again, the uniform spanning condition is satisfied and X s has smooth densities for s > h.
As the previous computation shows, semi-brackets are crucial in this case.