Functional inequalities on path space of sub-Riemannian manifolds and applications

For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a natural gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequalities on path space analogous to what has been established in Riemannian geometry by Aaron Naber, such as gradient inequalities, log-Sobolev and Poincar\'e inequalities.


Introduction
Stochastic analysis on the path space over a complete Riemannian manifold has been well developed ever since B. K. Driver [12] proved the quasi-invariance theorem for the Brownian motion on compact Riemannian manifolds in 1992. A key point of the study is to first establish an integration by parts formula for the associated gradient operator induced by the quasi-invariant flows, then prove functional inequalities for the corresponding Dirichlet form (see e.g. [14,24] and references within). For more analysis on Riemannian path spaces we refer to [13,25,28] and references within. Recently, there has been an extensive study by A. Naber [27] on the equivalence of bounded Ricci curvature and certain inequalities on path space. R. Haslhofer and A. Naber [21] extended these results to characterize solutions of the Ricci flow, see also [22].
In the present article, we develop this formalism in the framework of hypoelliptic operators and diffusions in sub-Riemannian geometry. Let (M, H, g) be a sub-Riemannian manifold, meaning that H is a subbundle of T M with a metric tensor g. Let ∇ be an affine connection on T M compatible with (H, g) in the sense that it preserves H and its metric g under parallel transport. We define an operator (1.1) L = tr H ∇ 2 ×,× , as the trace of the Hessian ∇ 2 over H with respect to the inner product g. We assume that the subbundle H is bracket-generating, meaning that its sections and their iterated brackets span the entire tangent bundle. This makes L into a hypoelliptic operator on functions by Hörmander's theorem [23]. Let B x t be a standard Brownian motion in the inner product space H x . Then the solution of the SDE, x is a diffusion on M with 1 2 L as infinitesimal generator, with // t : T x M → T X x t M denoting ∇-parallel transport along X x t . We note that for the case when H = T M and ∇ is the Levi-Civita connection, the operator L is the Laplacian and X x t is the Brownian motion in M .
The analysis of path space of sub-Riemannian manifolds has been earlier considered in [2,3] for the case where the sub-Riemannian structure (H, g) is the restriction to the transverse bundle of a foliation that is Riemannian, totally geodesic and of Yang-Mills type. In this present paper, we will generalize the approach in [3] to arbitrary sub-Riemannian manifolds with a metric preserving complement, which include sub-Riemannian manifolds coming from Riemannian foliations, but does not require anything of the metric along the foliation or even any extension of the sub-Riemannian metric.
The derivative formula and integration by parts formula established in this paper correspond to a generalization of the gradient in [3] to a more general class of sub-Riemannian manifolds. This gradient is defined in terms of a connection ∇ compatible with the sub-Riemannian structure which is canonical in the sense that any choice of complement V to the subbundle H determines it uniquely. To motivate the reasonability of the definition, we first review the smooth path space and the development map with respect to an arbitrary connection. The underlying idea is that if we have a variation of curves {γ s } that are all tangent to H, then the corresponding variational vector field Y = ∂ s γ s | s=0 will not be in H in general, yet it can not be arbitrary in the sense that is determined by pr H Y for any choice of projection pr H : T M → H. We will construct the sub-Riemannian gradient on path space to reflect this property.
Our formula for the damped gradient appears more similar to the definition in Riemannian geometry, however, it uses the adjoint connection∇ of ∇ which will not be compatible with the sub-Riemannian structure. We will show that there are nice formulas relating the two operators with the gradient operator defined by the adjoint connection, which help us to establish the derivative and integration by parts formulae for both the gradient and the damped gradient.
Having set up this formalism, we extend the approach of Naber to the sub-Riemannian case in our main result in Theorem 4.1. We establish functional inequalities on the path space of the stochastic flow x → X x t including gradient inequalities, log-Sobolev inequalities and Poincaré inequalities. These inequalities are shown to be equivalent to bounds on the horizontal Ricci operator Ric H : H → H which is defined taking the trace of the curvature tensor only over H. We want to emphasize that it is quite surprising that we can establish almost identical relations between bounded Ricci curvature and functional inequalities in the sub-Riemannian path space as in the Riemannian case. By contrast, the relationship between lower Ricci curvature bounds and functional inequalities for the heat semigroup is much more complicated in the sub-Riemannian case compared to the Riemannian one, see e.g. [4,1,5,18,19] for details.
The structure of the paper is as follows. In Section 2 we first consider the smooth path space and development with respect to an arbitrary connection. We review the basic definitions of sub-Riemannian manifolds and connections compatible with such structures. Unlike in the Riemannian case, we do not have torsion-free compatible connections on such spaces, however, we give analogues of the Levi-Civita connection by defining a canonical connection with minimal torsion relative to a chosen complement V to the horizontal bundle H. We finally use these connections to define corresponding vector fields on smooth path space.
We generalize the definition of these vector fields in Section 3 in order to define a gradient and a damped gradient for functions on path space. We relate these concepts and look at their properties in Theorems 3.1, 3.3 and 3.4. In particular, we establish integration by parts formulas for both the gradient and the damped gradient, generalizing the Riemannian case and the case treated in [3]. Finally, in Section 4, we show that several functional inequalities related to functions on path space are equivalent to the analogue of bounded Ricci curvature. We state our main result in Theorem 4.1. From this result, we also obtain a spectral gap estimate in Corollary 4.5 for the Ornstein-Uhlenbeck operator corresponding to the gradient.
In Section 5, we look closer at how such results can be interpreted geometrically. Intuitively, we show that if one uses the canonical connection ∇ corresponding to a metric preserving complement V , then the sub-Riemannian path space has geometry "similar to M/V ". This latter concept is well defined in the case when V is an integrable submanifold corresponding to a regular foliation Φ in which M/Φ has an induced Riemannian structure, but our formalism is valid for non-integrable choices of complements as well.
For the main results of this paper, we need to choose a complement which is metric preserving. To explain the reasons behind this assumptions for this choice and for later references, we include some formulas related to a general choice of connection and complement in Appendix A, which show the additional complications that appear if the complement is not metric preserving.
and let R denote its curvature We define its adjoint∇ as the connection Observe that the torsion of∇ is −T and hence ∇ is the adjoint of∇. We remark that if (s, t) → ω s t is a two-parameter function with values in M , then where D s andD t denote covariant derivatives of respectively ∇ in the direction of s and∇ in the direction of t.
Conversely, we say that ω is the development of u. We write Dev(u) = ω and Dev −1 (ω) = u. We note that Dev −1 is defined for any element in W ∞ x (M ), however, for a general u ∈ W ∞ 0 (T x M ), t → Dev(u) t might be only defined for short time. If ω t = Dev(u) t is defined for all time for any u ∈ W ∞ 0 (T x M ), x ∈ M , then ∇ is called complete. For the rest of this subsection, we assume that ∇ is complete. For the general case, see Remark 2.2. The next lemma describes the derivative of Dev.
then y t andŷ t are the unique solutions of We remark that in the above statement, we used the notation . We will use this notation for tensors in general throughout the paper.
Proof. Let D andD be the covariant derivative of respectively ∇ and∇. Write e 1,s (t), . . . , e n,s (t) for an orthonormal ∇-parallel basis along t → ω s t with e j,0 (t) = e j (t) and e j,s (0) = e j (0), and use the same basis to define u t + sk t = n j=1 (u j (t) + sk j (t))e j (0). Then By definition, we have D s e j,s (0) = 0. Furthermore, we have It follows that at s = 0, and hence, Remark 2.2 (Non-complete connections). Let ω ∈ W ∞ x (M ) be any given curve with u = Dev −1 (ω). Then for arbitrary k ∈ W ∞ 0 (T x M ) and any T > 0, there is some ε > 0 such that t → Dev(u + sk) t has a solution on [0, T ] for |s| < ε. Hence, we have that t → Y t can still be defined as a derivative of a two-parameter family as in (2.4) for any t ≥ 0.

2.2.
Sub-Riemannian manifolds. We consider a sub-Riemannnian manifold as a triple (M, H, g) where M is a connected manifold, H ⊆ T M is a subbundle of the tangent bundle and g = · , · g is a metric tensor on H. The sub-Riemannian structure (H, g) induces a map ♯ : We can then define a (degenerate) sub-Riemannian cometric g * by g * (α, β) = α, β g * = ♯α, ♯β g .
We remark that in what follows, we use g, the map ♯ as well as the cometric g * to state our results. For v ∈ H and α ∈ T * M , we also use the notation |v| g = v, v 1/2 g and |α| g * = v, v 1/2 g * and ask the reader to keep in mind that |α| g * may vanish for non-zero covectors.
As usual, we assume that H is bracket-generating, meaning that sections of H and their iterated brackets span the entire tangent bundle. A curve ω t is called horizontal if it is absolutely continuous and satisfiesω t ∈ H ωt for almost every t. The bracket-generating condition implies that any pair of points can be connected by a horizontal curve. We hence have a well defined distance on M given by The topology induced by the metric d g coincides with the manifold topology. We say that (M, H, g) is complete if (M, d g ) is a complete metric space. For more details on sub-Riemannian geometry, see e.g. [26].

2.3.
Connections compatible with the sub-Riemannian structure and development. Let ∇ be an affine connection on T M for a sub-Riemannian manifold (M, H, g). We are interested in the following types on connections.
We say that ∇ is horizontal compatible with (H, g) if H is horizontally parallel with respect to ∇ and (2.6) holds for Z ∈ Γ(H).
Unlike what holds in Riemannian geometry, there exists no affine connection that is both compatible with the sub-Riemannian structure and also torsion free when H is bracket-generating and a proper subbundle of T M , see e.g. [20].
Let t → ω t be any smooth horizontal curve with ω 0 = x. If H is horizontally parallel relative to ∇, then the corresponding anti-development u = Dev −1 (ω) is a smooth curve H x , and the converse is also true for any curve u ∈ W ∞ x (H x ) if only for short time in general. We say that ∇ is horizontally complete if Dev(u) is defined for all time for every u ∈ W ∞ 0 (H x ). We note the following relation. Proposition 2.4. Let (M, H, g) be a complete sub-Riemannian manifold and let ∇ be a connection that is horizontally compatible with the sub-Riemannian structure (H, g). Then ∇ is horizontally complete. is a linear isometry by our assumptions, we have that ω t , t ∈ [0, T ] has to be contained in the ball B g (x, ϕ(T ) + ε), ε > 0, centered at x with radius ϕ(T ) + ε defined relative to the sub-Riemannian distance d g defined in (2.5). Since we are assuming that (M, H, g) is complete, all such balls have compact closures, see e.g. [6]. Hence, for any T > 0, we can solve the development equation in B g (x, ϕ(T ) + ε). It follows that Dev(u) is well defined.
We finally note that the map Dev restricted to W ∞ 0 (H x ) only depends on parallel transport along horizontal curves. For this reason, we will consider the concept of partial connections. Investigating this concept also allows us to find a unique choice of horizontally compatible connection relative to a choice of complement.

Partial connections on sub-Riemannian manifolds. A partial connection
In other words, covariant derivatives are only defined in the direction of H. A partial connection will give us a well defined parallel transport along H-horizontal curves. For more on partial connections, see [10].
Let (M, H, g) be a sub-Riemannian manifold. A partial connection on H in the direction of H is compatible with (H, g) if , from definition of torsion and using that (2.7) implies, Hence, it follows that κ is determined by Hence κ is uniquely determined by t H . Furthermore, to prove (c), if we take then t H = 0 and this is the unique such choice.
Given a sub-Riemannian manifold (M, H, g), let V be a choice of complement. Let pr H and pr V be the corresponding projections. We write ∇ g,V for the unique compatible partial connection with t(H, H) ⊆ V . We will also write ∇ = ∇ g,V for an affine connection on the following form and where ∇ |V |V can be an arbitrary partial connection on V in the direction of V . We allow this ambiguity in covariant derivatives, since, in what follows, the values of covariant derivatives of sections of V in the directions of V will not affect our formulas. We note the following.
Proposition 2.6. The connection on the form ∇ = ∇ g,V with torsion T satisfies the following properties: We also note that ∇ g,V is always horizontally compatible with (H, g), and hence is horizontally complete if (H, g) is complete. In order to get compatible connections, we consider the following property introduced in [18]. Definition 2.7. For any Z ∈ Γ(T M ), let L Z denote the corresponding Lie derivative. Let V be a complement to a sub-Riemannian manifold (M, H, g) with corresponding projection pr H and pr V . The complement V is then called metric preserving if for any Z ∈ Γ(V ) and X ∈ Γ(H), we have Remark 2.8. Let (M, H, g) be a sub-Riemannian manifold. We say that a Riemannian metricḡ tames g ifḡ| H×H = g. Assume that we have chosen a taming metricḡ for which V = H ⊥ is the orthogonal complement of H. Assume further that V is integrable with corresponding foliation Φ. Then the assumption of V being metric-preserving is equivalent to assuming that the metricḡ is bundle-like, or, in a different terminology, assuming that Φ is a Riemannian foliation. We refer to [18] for details. We emphasize that none of these properties depend onḡ| V ×V .
2.5. The smooth horizontal path space seen from a metric preserving complement. Let (M, H, g) be a complete sub-Riemannian manifold with a chosen complement V . We assume that V is metric preserving such that ∇ = ∇ g,V is a connection compatible with the sub-Riemannian structure. For discussion of the general case, see Appendix A.
Let ∇ have torsion T and curvature R and define Dev = Dev ∇ relative to this connection. Then the following result holds. .
The statement is a special case of Lemma A.2, Appendix. Based on this result, we make the following definition.
where // t denotes parallel transport along ω with respect to ∇.
We note the following immediate consequence of Proposition 2.9.
Corollary 2.11. For any horizontal curve ω with u = Dev −1 (ω), we have We note that d ds Dev(u + sk)| s=0 = D h | ω where k and h are related by (2.9). In the case when ∇ is a flat connection, i.e. if R ≡ 0, then k = h. We will generalize such vector fields to functions h with values in the Cameron-Martin space in the next section.

Diffusions and gradients on path space
Throughout this section, we assume that M is compact for a simpler presentation. We hence have that all tensors are bounded and that all local martingales are indeed martingales. The same results hold in the non-compact case under some additional assumptions, see Section 3.7 for details.
3.1. Sub-Riemannian diffusions and notation. Let M be a compact manifold and let W x = W x (M ) be the space of continuous maps ω : [0, ∞) → M with ω 0 = x. Let (H, g) be a sub-Riemannian structure on M and let ∇ be a horizontally compatible connection. Recall that the Hessian of ∇ is defined as We write L = tr H ∇ 2 ×,× for the connection sub-Laplacian of ∇ and let x → X x t ∈ W x be the stochastic flow with generator 1 2 L and X x 0 = x defined on the filtered probability space (W x , F . , P x ).
Hence, X x t can be considered as the development of the Brownian motion in H x .
For any T > 0, we define W T x as the curves in W x restricted to [0, T ]. We write the induced structure of a filtered probability space as (W T x , F T . , P T x ) and the corresponding stochastic process as X More generally, we define as a Hilbert space with inner product h, k L 2 = E h, k H . As usual, we write 3.2. Gradient on path space. Let V be an arbitrary complement to H and define . be the corresponding stochastic flow with generator equal to the trace of the Hessian of ∇.
x (M ) from Definition 2.10. As parallel transport is well defined along a path in W x almost surely, we can consider D h as a P-almost surely defined vector field on W x . We want to make this definition more precise and valid for functions h in the Cameron-Martin space.
We first introduce the following notation. Let T be the torsion of ∇ and define We remark that by the defining properties of ∇ in Proposition 2.6, we have that We define D h acting on a cylindrical function F : ω → f (ω t1 , ω t2 , . . . , ω tn ) by This is consistent with our definition in the smooth case in Section 2.5. If we define We define the gradient DF ∈ H T W,x by the relation DF, h H = D h F . We like to show that the operator D : F → DF can be closed on path space in the case when V is a metric preserving complement. For this case, we need the following integration by parts formula. Recall that R is the curvature of ∇. Introduce the corresponding Ricci operator Ric : T M → T M by where Q t is the solution to the following equation: In particular, for F (X [0,T ] ) = f (X t ), our result reduces to the following form, see the end of Section 3.5 for more details.
We note that the result in (a) has already appeared in [20]. We show this formula by proving the corresponding derivative formula and integration by parts formula in Theorem 3.4 for the damped gradient.
3.3. The damped gradient on path space. We define the damped gradientDF similarly to the formula in Riemannian geometry, but using parallel transport of the adjoint connection. We use the connection ∇ = ∇ g,V and define Ric as in (3.1). Define // s,t : T Xs M → T Xt M as parallel transport along X t with respect to∇, the adjoint of ∇, and write // t = // 0,t . We first introduceQ s,t : We note that ifQ t =Q 0,t , thenQ s,t = // sQtQ −1 s // −1 s and for s ≤ r ≤ t, Q s,t = // −1 s,rQ r,t // s,rQs,r .
and furthermore, for any k ∈ H T x , The next result clarifies the relationship between DF andDF .
Next, with respect to the damped gradient, we establish the gradient formula and integration by parts formula on path space as follows.

In particular, for any
We will prove Theorem 3.3 and Theorem 3.4 in the next subsections. Now we show how Theorem 3.1 follows from these results.
Proof of Theorem 3.1. We can prove (a) directly by using Theorem 3.4 (a) and Theorem 3.3 (b). For (b), where the last equation follows from dk t = dh t + 1 2 Ric //t h t dt. 3.4. Proof of Theorem 3.3. Let (M, H, g) be a sub-Riemannian manifold with a metric preserving complement V . In all the steps below, we will consider the connection ∇ = ∇ g,V .
Define the tensor Ric relative to ∇ as in (3.1). To prove Theorem 3.3, we first observe the Ric vanish outside the horizontal bundle H. Proof. We note that since H is parallel with respect to ∇, we have R( · , · )v ∈ H x for any v ∈ H x , x ∈ M . It follows that Ric(T M ) ⊆ H. From the proof of Lemma A.2, we also have that R(v 1 , z)v 1 , v 2 g = 0 whenever V is metric preserving, giving us Ric(V ) = 0.
Next, we relate the damped gradient and the gradient by the following conversion formula.
We then have that Proof. Define U t = // −1 t // t . We note first that dU t = T //t (•dB t , U t · ) and U 0 = id, giving us that U t = id +A t . Since A 2 t = 0, we have that U −1 t = id −A t . We will use this to find a formula forQ t by We use these identities to compute and hence, Using this lemma, we prove Theorem 3.3 as follows.

Proof of Theorem 3.3. We consider
Hence, we have that T 0 D t F (ω),k g ds = T 0 D t F,ḣ g ds by Lemma 3.6, which then gives (a).
The relationship between D t F andD t F can be observed from which then implies (b).
3.5. Proof of Theorem 3.4. We first prove Theorem 3.4 (a) for the case n = 1.
3) and the fact that V is metric preserving, it follows thatÑ s is a local martingale dÑ s =Q * s // −1 s∇//sdBs dP t−s f | Xs , and from our compactness assumption, it is a true martingale.
By Lemma 3.7, the desired assertion holds for n = 1. Assume that it holds for n ≥ 1. We will prove that the assertion also holds for n + 1. First let By our induction hypothesis, for all x ∈ M . Using the strong Markov property of X, we look at the process X starting from X x t1 at time t 1 . From (3.7) and from the result at n = 1, ) .
By the strong Markov properties, From this equality, we obtain For part (b), we first observe that E[F | F t ] is a martingale according to the definition. By martingale representation, we have Integrating from 0 to T gives For part (c), we first take the formula about F inside the term E [F k, B x H ]. By (3.9), giving the formula in (c). By letting F (X [0,T ] ) = f (X t ) we have the equality in (3.3).

Proof of Corollary 3.2.
For (a), we first observe from Lemma 3.7 that The formula in (b) follows from Theorem 3.1 (b) by taking F (X [0,T ] ) = f (X t ) directly.
3.6. Quasi-invariance. We want to link D F, k H to the directional derivative induced by some quasi-invariant flow. We use techniques from [29,Chapter 4.2]. For k ∈ H T x and s ∈ (−ε, ε), let X s = X x,s solve the SDE where // s is the parallel transport along X s with respect to ∇. This flow is quasiinvariant, i.e., the distribution of X s [0,T ] is absolutely continuous with respect to that of If P denotes the Wiener measure on the path space of H x , d = rank H x , then by the Girsanov theorem, Proof. Let B s t = B t − sk t , which is the d-dimensional Brownian motion under P s . By the weak uniqueness of (3.10), we conclude that the distribution of X under P s is consistent with that of X s under P. In particular,

3.7.
Comments for the non-compact case. In order to keep the exposition simpler, we assumed throughout this section that we are working over a compact manifold. This has the advantage that we can be assured that processes such as X x t , A t and Q t have infinite lifetime, all tensors are bounded and all local martingales that appear in our proofs are indeed true martingales. If one can find alternative ways to show that the same properties hold, our results hold without the compactness assumption. One way to ensure this on a non-compact manifold, is to verify that the following assumptions hold. (A) Assume that X x t has infinite lifetime, i.e. assume that P t 1 = 1.
(B) We must be able to pick a taming Riemannian metricḡ such that T and Ric are bounded and such that sup 0≤t≤T |// t |ḡ ⊗ḡ * is finite for every T . We note that since // t = // t (id +A t ), such assumptions are sufficient to bound parallel transport with respect to∇ as well. (C) Our cylindrical functions will now be defined as F C ∞ 0 consisting of functions is of compact support. In order to differentiate expectations of such functions and in order that they remain bounded, we need to assume that sup t∈[0,T ] |dP t f |ḡ * is bounded for every finite T > 0 and every compactly supported function f ∈ C ∞ 0 (M ). For sufficient conditions for these assumptions to be satisfied on non-compact manifolds, more specifically on complete sub-Riemannian manifolds coming from totally geodesic Riemannian foliations, see [20].
For alternate approaches of dealing with path space analysis on non-compact spaces in the Riemannian case, we also refer to [8]. For the rest of the paper, we will assume that properties (A) to (C) are satisfied or some other conditions are satisfied in order to ensure that the results of this section hold.

Bounded curvature and functional inequalities on path space
4.1. Inequalities equivalent to bounded curvature. Inspired by Naber's work [27], we have the following characterization formulae for the boundedness of Ric H . Let V be a metric preserving complement with ∇ = ∇ g,V the corresponding connection. Recall that Ric H := Ric | H where Ric is defined as in (3.1). We state the main result of the paper with proof given in the next section.  (i) for any F ∈ F C ∞ 0 , (iv) (Poincaré inequality) for any F ∈ F C ∞ 0 and t > 0 in [0, T ], II. The following functional inequalities on the manifold M are equivalent to the curvature bound (4.1): Remark 4.2. Theorem 4.1 describes equivalent statements for a symmetric bound of Ric H . For non-symmetric bounds, i.e. if K 1 ≤ Ric H ≤ K 2 for some constants K 1 ≤ K 2 , we can also give corresponding equivalent functional inequalities as in [9] by modifying the gradient operator on the path space to: The functional inequalities will then be written in terms ofDF . We refer to [9, Proposition 2.2] for a more detailed discussion.

4.2.
Proof of Theorem 4.1. We will show equivalence of the properties in Theorem 4.1 by proving the relations We divide the proof into two parts.
Proof of Theorem 4.1, Part I. In this part, we prove the implications "(4.1) ⇒ (i) ⇒ (ii) ⇒ (v)", "(4.1) ⇒ (iii)" and "(4.1) ⇒ (iv)". "(4.1)⇒ (i)". By Theorem 3.1 (a), we have Then using the condition (4.1), we can prove (i) by controlling Q t and Ric H : "(4.1)⇒(iii)(iv)". We now prove (iii) and (iv) by using the estimate (4.2) above and the Itô formula, where M t is a local martingale such that Integrating from 0 to t and taking expectation of both sides, we prove the inequality (iii). Similarly, we can prove (iv) by taking into consideration of the process E x [F |F t ] 2 and using the following Itô formula: whereM t is a local martingale such that Integrating from 0 to t and taking expectation of both sides, we prove the inequality (iv).
To give the second part of the proof, we first need to include the following lemmas. For the Riemannian manifold case, the corresponding results can be found for instance in [29,9].
In particular, ×, ♯df )). Proof. The result follows from Lemma A.1, Appendix, for the case of a metric preserving complement and from the property of the torsion for our choice of connection.
Then the following limits hold: and hence Z s is a supermartingale. Therefore, If we take δ = 1/4, then (a) By Itô's formula and the estimate for σ r , for small t and f ∈ C ∞ 0 (M ) such that ∇df (x) = 0, we get From this equality, the result is easily derived using Taylor expansion: where again we have used the Weitzenböck identity and that (δ H T) * df | x = 0 since df (V x ) = 0. Next, we observe from the Weitzenböck identity that Ldf |V = dLf |V . Hence, the process N s in T * x M given by , that df vanishes at V x and that ∇df = 0, we have As a consequence, (c) Since ∇df (x) = 0, we have |∇df | g * ⊗g * (x) = 0 and tr H ∇ × df (T(×, ♯df ))(x) = 0.
By the Weitzenböck formula in Proposition 4.3, Thus, by the Taylor expansions at the point x (we drop x below for simplicity): and we obtain On the other hand, Proof of Theorem 4.1, Part II. We finish the remaining implications "(iii)⇒(iv)⇒(ii)" and "(v) ⇒ (4.1)". "(iv)⇒(ii)". By Itô's formula, we have For any function F (ω) = f (ω t1 , . . . , ω tn ) with t 1 > 0, the function r → E x |D r F | 2 g and s → E x |D s E x [F |F s ]| 2 are continuous at time 0. Hence, we can divide both sides of (4.9) by t and take the limit as t goes to 0 to obtain (ii).
We note that then (δ H T) * (df ), df (x) = 0. Observe also that the inequalities of (v) are equivalent to: Letting t tend to 0 and using Lemma 4.4, we obtain Then (E , F C ∞ 0 ) is a positive bilinear form on L 2 (W T x ; P x,T ). It is standard that the integration by parts formula in Theorem 3.1 implies closability of the form (see e.g. the argument in [29,Lemma 4.3.1.]). We shall use (E , Dom(E )) to denote the closure of (E , F C ∞ 0 ). Let (L , Dom(E )) be the analogue of the Ornstein-Uhlenbeck operator as the generator of the Dirichlet form E . Let gap(L ) denote the spectral gap of the Ornstein-Uhlenbeck operator. The following is then a consequence of Theorem 4.1.
(c) the spectral gap has the following estimate: Proof. The inequalities in (a) and (b) are derived by using Theorem 4.1 (iii) and (iv) with t = T : The estimate in (c) is derived according to the definition of the spectral gap.

5.
On the geometry of path space and complements 5.1. Integrable complements. Let (M, H, g) be a sub-Riemannian manifold and let V be a metric preserving complement that is also Frobenius integrable, i.e.
[V, V ] ⊆ V . Let Φ be the corresponding foliation of V . Then M/Φ locally has the structure of a Riemannian manifold. More precisely, any x ∈ M has a neighborhood U such that π U : U → U/Φ| U is a submersion of differentiable manifolds. Since V is metric preserving, there exists a Riemannian metricǧ Consider the special case when Φ is a regular foliation, i.e. whenM = M/Φ is a differentiable manifold. Write the corresponding projection as π : M →M . Letǧ be the corresponding complete Riemannian metric onM . For the sake of simplicity, we assume that (M, H, g) is complete, which implies that (M ,ǧ) is complete, as this is a distance decreasing map. Write∇ for the Levi-Civita connection ofǧ. Let x ∈ M be a given point withx = π(x). Let W ∞ x,H and W ∞ x be the space of smooth curves with domain [0, ∞) starting at x andx, respectively, where the curves starting at x are required to be horizontal. Then since H is an Ehresmann connection on π, curves starting atx have unique horizontal lifts to x. Hence, the map W ∞ x,H → W ∞ x , ω → π(ω), is a bijection. Next, let hY denote the horizontal lift of a vector field Y onM , that is hY is the unique section of H satisfying π * hY = Y . We can then describe the connection ∇ = ∇ g,V as for Y, Y 2 ∈ Γ(TM ), Z ∈ Γ(V ). Hence, if Devx : TxM →M is the development map of∇, then we have the following commutative diagram: with every map in the diagram being a bijection. Going from smooth curves to continuous curves, the concept of horizontal curves will no longer be well defined. However, ifBx t is the standard Brownian motion in TxM andXx t denotes the Brownian motion inM , then we can still make sense of the following diagram We finally note that by (5.1) we have Ric H = π * Ř ic| H whereŘic denotes the Ricci operator onM . In summary, if we consider the path space W x (M ) with the probability distribution given by the sub-Riemannian Brownian motion, then, viewed from the connection ∇ = ∇ g,V , the path space has a geometry similar to that of the path space of M/ exp(V ). See [18] for more details.
We can consider elements q ∈ M as linear isometries q : (2) is mapped to x (1) and x (2) , respectively. We can then define a subbundle H ⊆ T M by H = q t : π (1) * qt = π (2) * qt , q t Y t is a parallel vector field for every parallel Y t along π(q t ) .

5.3.
A non-integrable complement. We will include an example from [7]. Consider the Lie algebra so (4). If e 1 , . . . , e 4 is the standard basis of R 4 , we write e ij ∈ so(4) for the matrices satisfying e ij e k = δ ik e j − δ jk e i . Consider the inner product on so(4) given by Y 1 , Consider an orthogonal decomposition so(4) = h ⊕ v where v = span{e 12 , e 23 }. On SO(4), define subbundles T SO(4) = H ⊕ V where H and V are respective left translations of h and v. We define a sub-Riemannian metric g by left translation of the restriction of inner product of so(4) to h.
The subbundle V is not integrable, but it is metric preserving from the biinvariance of the inner product on so (4). Furthermore, if we define ∇ = ∇ g,V , then See [7, Example 3.1] for detailed calculations.

5.4.
How to understand the curvature bounds. Let (M, H, g) be a given sub-Riemannian manifold. As the above calculations show, our curvature bounds will in general depend on the choice of complement V which determines the connection ∇ = ∇ g,V . This dependence can be understood in the following way. Firstly, our connection sub-Laplacian L = tr H ∇ 2 ×,× depends on the choice of complement, and hence the same is true for the underlying diffusion X x t . See e.g. [15,18] for more details relating connections and sub-Laplacians. Furthermore, even for complements that define the same sub-Laplacian L, the derivatives d ds Dev(B x . − sk . )| s=0 on cylindrical functions will differ. In this sense, the curvature Ric H can be seen as a curvature of the development map.
Appendix A. General geometric formulas In most of our previous result, we restricted ourselves to sub-Riemannian manifolds (M, H, g) equipped with a choice of metric preserving complement V . In this appendix, we include formulas without this assumption to point out additional complications that exist in general and for the benefit of future research. Assume that H is parallel with respect to H, and hence ∇g is well defined. For any vector field Z ∈ Γ(T M ), define q Z : H → H by the formula We note that Z → q Z is a tensorial map, so we can consider q ∈ Γ(T * M ⊗ End H) as a tensor. Proof. For a given point x and any elements v ∈ H x and w ∈ T x M , choose arbitrary vector fields Y ∈ Γ(H), Z ∈ Γ(T M ) such that Y (x) = v, Z(x) = w, ∇Y (x) = 0 and ∇Z(x) = 0. We remark that this is possible since we assumed that H was parallel with respect to ∇. Then at x ∈ M , T(Y, Z))).
Next, let us insert an orthonormal basis Y 1 , . . . , Y k of H. We can choose this orthonormal basis such that ∇ Z Y i (x) = q Z Y i (x) for some given point x. Evaluated at x ∈ M , we have Summing over this basis and using the symmetry of q Z gives us (A.2). The result in (A.3) then follows from the identitŷ A.2. The smooth horizontal path space seen from an arbitrary complement. Let (M, H, g) be a complete sub-Riemannian manifold and let V be an arbitrary choice of complement. Let ∇ = ∇ g,V be the corresponding connection horizontally compatible with (H, g) and with torsion T and curvature R. Define the development map Dev relative to this connection. For any Z ∈ Γ(T M ), define q Z as in (A.1) and note that q Z = q pr V Z since the connection i horizontally compatible. We note the following result.
Lemma A.2. Let t → ω t be a smooth horizontal curve with u = Dev −1 (ω) ∈ Consider ω s t = Dev(u + sk) t and define Y t = d ds ω s t | s=0 . Then Y t = // t y t = // tŷt with where h t = pr H y t is the solution of  Note that for arbitrary z ∈ V x and v 1 , v 2 , v 3 ∈ H x , we have that and hence from the first Bianchi identity and subtracting the second line from the sum of the first and the third, we obtain In conclusion Combining this with the formula (A.5), we prove (A.4).