INTEGRATION by PARTS and QUASI-INVARIANCE for the HORIZONTAL WIENER MEASURE on a FOLIATED COMPACT MANIFOLD Contents 1. Introduct

INTEGRATION BY PARTS AND QUASI-INVARIANCE FOR THE HORIZONTAL WIENER MEASURE ON A FOLIATED COMPACT MANIFOLD

FABRICE BAUDOIN††, QI FENG††, AND MARIA GORDINA†

Abstract. We prove a version of Driver’s integration by parts formula [16] on the horizontal path space of a totally geodesic Riemannian foliation and prove that the horizontal Wiener measure is quasi-invariant with respect to the ﬂows generated by suitable tangent processes.

Contents 1. Introduction 2 1.1. Main results and organization of the paper 3 n 1.2. Cameron-Martin theorem on R and its lift to the Heisenberg group 5 2. Notation and preliminaries 7 2.1. Bott connection 9 2.2. Generalized Levi-Civita connections and adjoint connections 10 2.3. Horizontal Brownian motion and stochastic parallel transport 11 2.4. Weitzenb¨ock formulas 12 3. Integration by parts formula on the horizontal path space 15 3.1. Malliavin and directional derivatives 17 3.2. Gradient formula 20 3.3. Integration by parts formula for the damped Malliavin derivative 20 3.4. Integration by parts formula for the directional derivatives 24 3.5. Examples 27 4. Quasi-invariance of the horizontal Wiener measure 28 4.1. Normal frames 28 4.2. Construction of the horizontal Brownian motion from the orthonormal frame bundle 29

1991 Mathematics Subject Classiﬁcation. Primary 58G32; Secondary 58J65, 53C12, 53C17. Key words and phrases. Wiener measure; CameronMartin theorem; Quasi-invariance; sub-Riemannian geometry, Riemannian foliation. † Research was supported in part by NSF Grant DMS-1660031. † Research was supported in part by NSF Grant DMS-1660031. † Research was supported in part by NSF Grant DMS-1007496 and the Simons Fellowship. 1 2 BAUDOIN, FENG, AND GORDINA

4.3. Quasi-invariance of the horizontal Wiener measure 31 References 35

1. Introduction In this paper we study quasi-invariance properties and related integration by parts formulas for the horizontal Wiener measure on a foliated Rie- mannian manifold equipped with a sub-Riemannian structure. These are most closely related to the well-known results by B. Driver [16] who established such properties for the Wiener measure on a path space over a compact Riemannian manifold. Quasi-invariance in such settings can be viewed as a curved version of the classical Cameron-Martin theorem for the Euclidean space. While the techniques developed for path spaces over Riemannian manifolds are not easily adapted to the sub-Riemannian case we consider, we take advantage of the recent advances in this field. The geometric and stochastic analysis of sub-Riemannian structures on foliated manifolds has attracted a lot of attention in the past few years (see for instance [2, 6, 19, 27–30, 43]). In particular, we make use of the tools such as Weitzenböck formulas for the sub-Laplacian to extend results by J.-M. Bismut, B. Driver et al to foliated Riemannian manifolds. More precisely, the first progress in developing geometric techniques in the sub-Riemannian setting has been made in [4], where a version of Bochner’s formula for the sub-Laplacian was established and generalized curvature-dimension conditions were studied. This Bochner- Weitzenböck formula was then used in [2] to develop a sub-Riemannian stochastic calculus. One of the difficulties in this case is that, a priori, there is no canonical connection on such manifolds such as the Levi-Civita connection in the Riemannian case. However, [2] introduces a one-parameter family of metric connections associated with Bochner’s formula proved in [4] and shows that the derivative of the sub-Riemannian heat semigroup can be expressed in terms of a damped stochastic parallel transport. It should be noted that these connections do not preserve the geometry of the foliation in general. In particular, the corresponding parallel transport does not necessarily transforms a horizontal vector into a horizontal vector, that is, these connections in general are not horizontal. As a consequence, estab- lishing an integration by parts formulas for directional derivatives on the path space of the horizontal Brownian motion, similarly to Driver’s integration by parts formula [16] for the Riemannian Brownian motion, is not straightforward. As a result, the integration by parts formula we prove in the current paper can not be simply deduced from the derivative formula for the corresponding semigroup by applying the standard techniques of covariant stochastic analysis on manifolds as presented for instance in [21, Section 4], in particular [21, Theorems 4.1.1, 4.1.2]. A different approach to proving INTEGRATION BY PARTS 3 quasi-invariance in an infinite-dimensional sub-Riemannian setting has been used in [5]. The history of analysis on path and loop spaces have been developed over several decades, and we will not be able to refer to all the relevant publica- tions, but we mention some which are closer to the subject of this paper. In particular, J.-M. Bismut’s book [8] contains an integration by parts formula on a path space over a compact Riemannian manifold. His methods were based on the Malliavin calculus and Bismut’s motivation was to deal with a hypoelliptic setting as described in [8, Section 5]. A breakthrough has been achieved by B. Driver [16], who established quasi-invariance properties of the Wiener measure over a compact Riemannian manifold, and as a consequence an integration by parts formula. This work has been simplified and extended by E. Hsu [31], and also approached by O. B. Enchev and D. W. Stroock [23]. A review of these techniques can be found in [33]. In [34, 36] the noncompact case has been studied. A different approach to analysis on Riemannian path space can be found in [14], where tangent processes, Markovian connections, structure equations and other elements of what the authors call the renormalized differential geometry on the path space have been introduced.

1.1. Main results and organization of the paper. We now explain in more detail the main results and the organization of the paper. Let (M, g) be a smooth connected and compact Riemannian manifold equipped with a totally geodesic and bundle-like foliation as described in the beginning of Section 2. On such manifolds, one can define a horizontal Laplacian L according to [1, Sections 2.2 and 2.3]), and then the horizontal Brownian 1 motion is defined as the diffusion on M with generator 2 L. The distribution of the horizontal Brownian motion is called the horizontal Wiener measure and its support the horizontal path space. Our main goal is to define a suitable class of tangent processes generating flows such that the horizontal Wiener measure is quasi-invariant under these flows, and to study related integration by parts formulas. As we will see these tangent processes have bounded variation in the horizontal directions but in addition they have a martingale part in the vertical directions. This can be seen even in the simplest case of M being the Heisenberg group as we explain in Section 1.2. In Section 2, we mainly survey known results and introduce the notation and conventions used throughout the paper. Most of this material is based on [6] for the geometric part and [2] for the stochastic part. The most relevant result that will be used later is the Weitzenböck formula given in ∞ Theorem 2.3. It asserts that for every f ∈ C0 (M), x ∈ M and every ε > 0

(1.1) dLf(x) = εdf(x), where ε is a one-parameter family of sub-Laplacians on one-forms indexed by the parameter ε > 0. These sub-Laplacians on one-forms are constructed from a family of metric connections ∇ε whose adjoint connections ∇ˆ ε in the 4 BAUDOIN, FENG, AND GORDINA sense of B. Driver [16] are also metric. Even though Section 2 introduces mostly preliminaries, we present a number of new results there such as Lemma 2.5. In Section 3, we prove integration by parts formulas for the horizontal Wiener measure with the main result being Theorem 3.12. Suppose F is a cylinder function on the horizontal path space and v is a tangent process in TxM as defined in Definition 3.8, then we have Z 1 0 1 −1 (1.2) Ex (DvF ) = Ex F vH(t) + //0,t RicH//0,t, dBt , 0 2 H where x is the starting point of the horizontal Brownian motion, DvF is the directional derivative of F in the direction of v, and //0,t is the stochastic parallel transport for the Bott connection, and RicH is the horizontal Ricci curvature of the Bott connection. The Bott connection is defined in Section 2 and corresponds to the adjoint connection ∇ˆ ε as ε → ∞. In the integration by parts formula (1.2), the tangent process v is a TxM–valued process such that its horizontal part v is absolutely continuous and satisfies H R 1 kv0 (t)k2 dt < +∞ and its vertical part is given by E 0 H TxM Z t −1 (1.3) vV (t) = − //0,sT (//0,s ◦ dBs,//0,svH(s)), 0 where T is the torsion tensor of the Bott connection. Observe that (1.2) looks similar to the integration by parts formulas by J.-M. Bismut and B. Driver. This is not too surprising if one thinks about the special case where the foliation comes from a Riemannian submersion with totally geodesic fibers. We consider this case in Section 3.5.1, and we prove that then Equation (3.12) is actually a horizontal lift of Driver’s formula from the base space of the fibers to M. However, in general foliations do not come from submersions (see for instance [24] for necessary and sufficient conditions) and one therefore needs to develop an intrinsic horizontal stochastic calculus on M to prove (1.2). Developing such a calculus is one of the main accomplishments of the current paper. The proof of Theorem 3.12 proceeds in several steps. As in [2], the Weitzenböck formula (1.1) yields a stochastic representation for the derivative of the semigroup of the horizontal Brownian motion in terms of a damped stochastic parallel transport associated to the connection ∇ε (see Lemma 3.13). By using techniques of [3], Lemma 3.13 implies an integration by parts formula for the damped Malliavin derivative as stated in Theorem 3.11. The final step is to prove Theorem 3.12 from Theorem 3.11. The main difficulty is that the connection ∇ε is in general not horizontal. How- ever, it turns out that the adjoint connection ∇ˆ ε is not only metric but also horizontal. As a consequence, one can use the orthogonal invariance of the horizontal Brownian motion (Lemma 3.18) to filter out the redundant noise which is given by the torsion tensor of ∇ε . INTEGRATION BY PARTS 5

It is remarkable that the integration by parts formula for the directional derivatives (3.12) is actually independent of the choice of a particular connection and therefore is independent of ε in the one-parameter family of connections used to define the damped Malliavin derivative. While integration by parts formulas for the damped Malliavin derivative may be used to prove gradient bounds for the heat semigroup (as in [2]) and log-Sobolev inequalities on the path space (as in [3]), we prove that the integration by parts formula (1.2) comes from a quasi-invariance property of the horizontal Wiener measure as we show in Section 4. Quasi-invariance of the horizontal Wiener measure is proven in Theorem 4.12, namely, we show that if v is a tangent processes on TxM whose horizontal part is deterministic, then v generates a flow in the horizontal path space such that the horizontal Wiener measure is quasi-invariant under this flow. To prove this, it is convenient to work in a bundle over M which is constructed from horizontal orthonormal frames. This allows us to construct an n Itômap between the path space of R (where n is the rank of the horizontal bundle) and the horizontal path space of M (see Corollary 4.6). Following the methods by B. Driver in [16] and E.P. Hsu in [31], one can compute the pullback of the directional derivative Dv by this Itômap when v is a tangent process and finally apply Girsanov’s theorem in the form of [16, Lemma 8.2].

Remark 1.1. In the current paper, we restrict ourselves to the case of compact manifolds mainly for the sake of conciseness. It is reasonable to conjecture that as in [36], our results may be extended to complete manifolds. Moreover, our ﬁrst example in Section 1.2 is the path space of the Heisenberg group which is a connected nilpotent (non-compact) Lie group.

n 1.2. Cameron-Martin theorem on R and its lift to the Heisenberg group. To conclude the introduction it is instructive to show in the simple case of the Heisenberg group, where the vertical part (1.3) comes from. We d start by reminding the classical setting of the Wiener space. Let W(R ) be d the Wiener space of continuous functions w : [0,T ] → R vanishing at 0, d and ν be the Wiener measure on W(R ). In what follows we take d = 2n. The associated Cameron–Martin space H is deﬁned

Z T d 0 2 H := H R = h ∈ W : h (s) d ds < ∞ , 0 R

R T 0 2 wherein |h (s)| d ds := ∞ if h is not absolutely continuous. The Cameron- 0 R d Martin theorem says that the Wiener measure ν on W R is quasi-invariant with respect to the translations in the directions of the Cameron-Martin subspace H (see for instance [41, Theorem 2.2, p. 339], [9, Corollary 2.4.3], or the original work in [11, 12, 38]). More precisely, ﬁrst we deﬁne the map for every h ∈ H by 6 BAUDOIN, FENG, AND GORDINA

d d ph : W R −→ W R , ω 7−→ w + h, h h −1 and denote by ν the pushforward of ν under ph (that is, ν (A) = ν(ph (A)). By the Cameron-Martin theorem νh is equivalent to ν and the Radon- Nikodym density is explicitly given by dνh Z T 1 Z T = exp hh0(s), dω(s)i − |h0(s)|2 ds , Rd dν 0 2 0 R T 0 where 0 hh (s), dω(s)i is the Itˆostochastic integral. Now we would like to consider a similar setting on the path space over the Heisenberg group. Recall that the Heisenberg group is the set 2n+1 n n H = {(x, y, z), x ∈ R , y ∈ R , z ∈ R} endowed with the group law

(x1, y1, z1) ? (x2, y2, z2) := (x1 + x2, y1 + y2, z1 + z2 + hx1, y2iRn − hx2, y1iRn ). The vector fields ∂ ∂ Xi = − yi , 1 6 i 6 n, ∂xi ∂z ∂ ∂ Yi = + xi , 1 6 i 6 n, ∂yi ∂z ∂ Z = ∂z 2n+1 form a basis for the space of left-invariant vector fields on H . We 2n+1 choose a left-invariant Riemannian metric on H in such a way that {X1, ..., Xn,Y1, ..., Yn,Z} are orthonormal with respect to this metric. Note that these vector fields satisfy the following commutation relations

[Xi,Yj] = 2δijZ, [Xi,Z] = [Yi,Z] = 0, i = 1, ..., n. Then the map

2n+1 2n π : H −→ R (x, y, z) 7−→ (x, y) is a Riemannian submersion with totally geodesic ﬁbers as explained in 2n Section 3.5.1. Let W R be the Wiener space of continuous functions 2n [0,T ] → R vanishing at 0. We denote by (Bt, βt)06t6T the coordinate maps 2n 2n on W(R ) and by ν the Wiener measure on W(R ), so that (Bt, βt)06t6T is a 2n-dimensional Brownian motion under ν. By using the submersion π, the

Brownian motion (Bt, βt)06t6T can be horizontally lifted to the horizontal 2n+1 Pn R t i i Brownian motion on H which is given explicitly by Xt = (Bt, βt, i=1 0 Btdβt− i i βtdBt). The distribution of (Xt)06t6T is denoted by µH and referred to as 2n+1 the horizontal Wiener measure on H . The support of µH is called the INTEGRATION BY PARTS 7

2n+1 2n+1 horizontal path space of H and denoted by WH(H ). The measure space isomorphism 2n 2n+1 Π:(W(R ), ν) −→ (WH(H ), µH) (Bt, βt)06t61 7−→ (Xt)06t6T is the horizontal lift map induced by the submersion π. 2n Now we use the map Π to lift the Cameron-Martin theorem from W(R ) 2n+1 2n to WH(H ). Indeed, for any h ∈ H R the translation ph induces a map peh so that the following diagram is commutative.

2n+1 peh 2n+1 WH(H ) WH(H )

Π Π

2n ph 2n W R W R 2n For h ∈ H R the map peh can be computed explicitly and is given for 2n+1 w ∈ WH(H ) by peh(w)t n X i i+n i+n i =wt + h(t), h (t)wt − h (t)wt+ i=1 Z t i i i+n i+n i+n i (h (s) + 2ws)dh (s) − (h (s) + 2ws )dh (s) 0 Observe that the non-trivial vertical part in peh(w)t is consistent with formula h (1.3). Let µH be the pushforward of µH under peh, then it is immediate that h µH is equivalent to µH and the density is explicitly given by 2n ! dµh X Z T 1 Z T H = exp h0 (s)dwi − |h0(s)|2 ds . dµ i s 2 R2n H i=1 0 0

2. Notation and preliminaries Let M be a smooth connected and compact manifold of the dimension n+ m. We assume that M is equipped with a Riemannian foliation structure, F, with a bundle-like metric g and totally geodesic m-dimensional leaves. We refer to [1,39,40,44] for details about the geometry of Riemannian foliations. The subbundle V formed by the vectors tangent to the leaves is referred to as the set of vertical directions. The subbundle H which is normal to V is referred to as the set of horizontal directions. We assume that H is bracket generating, that is, the Lie algebra of vector ﬁelds generated by global C∞ sections of H has the full rank at each point in M. Example 2.1 (Riemannian submersions). Let (M, g) and (B, j) be smooth and connected Riemannian manifolds. A smooth surjective map π :(M, g) → (B, j) is called a Riemannian submersion if its derivative maps Txπ : TxM → 8 BAUDOIN, FENG, AND GORDINA

∗ Tπ(x)B are orthogonal projections, i.e. for every x ∈ M, the map Txπ(Txπ) : Tp(x)B → Tp(x)B is the identity map. The foliation given by the fibers of a Riemannian submersion is then bundle-like (see [1, Section 2.3] ). Example 2.2 (K-contact manifolds). Another important example of Rie- mannian foliation is obtained in the context of contact manifolds. Let (M, θ) be a 2n+1-dimensional smooth contact manifold, where θ is a contact form. On M there is a unique smooth vector field Z, called the Reeb vector field, satisfying θ(Z) = 1, LZ (θ) = 0, where LZ denotes the Lie derivative with respect to Z. On M there is a foliation, the Reeb foliation, whose leaves are the orbits of the vector field Z. As it is well-known (see [42]), that it is always possible to find a Riemannian metric g and a (1, 1)-tensor field J on M so that for every vector fields X,Y g(X,Z) = θ(X),J 2(X) = −X + θ(X)Z, g(X,JY ) = (dθ)(X,Y ). The triple (M, θ, g) is called a contact Riemannian manifold. We see then that the Reeb foliation is totally geodesic with bundle-like metric if and only if the Reeb vector field Z is a Killing field, that is,

LZ g = 0, as is stated in [10, Proposition 6.4.8]. In this case, (M, θ, g) is called a K- contact Riemannian manifold. Observe that the horizontal distribution H is then the kernel of θ and that H is bracket generating because θ is a contact form. We refer to [7, 42] for further details on this class of examples. We will use the following standard notation. ∗ Notation 2.1. By T M we denote the tangent bundle and by T M the ∗ cotangent bundle, and by TxM (Tx M) the tangent (cotangent) space at x ∈ M. The inner product on T M induced by the metric g will be denoted by g (·, ·), and its restrictions to H and V will be denoted by gH (·, ·) and gV (·, ·) respectively. As always, for any x ∈ M we denote by g (·, ·)x (or h·, ·ix), gH (·, ·)x (or h·, ·iHx ), gV (·, ·)x (or h·, ·iVx ) the inner product on the ﬁbers TxM, Hx and Vx correspondingly. The space of smooth functions on ∞ M will be denoted by C (M). The space of smooth and compactly supported ∞ functions will be denoted C0 (M). The space of smooth sections of a vector ∞ bundle E over M will be denoted Γ (E). The space of smooth and compactly ∞ supported sections will be denoted Γ0 (E). We say that a one-form is horizontal (resp. vertical) if it vanishes on the vertical bundle V (resp. on the horizontal bundle H). Then the splitting of the tangent space TxM = Hx ⊕ Vx induces a splitting of the cotangent space ∗ ∗ ∗ Tx M = Hx ⊕ Vx. INTEGRATION BY PARTS 9

The sub-bundle H∗ of the cotangent bundle will be referred to as the co- horizontal bundle. Similarly, V∗ will be referred to as the covertical bundle.

2.1. Bott connection. On the Riemannian manifold (M, g) there is the Levi-Civita connection that we denote by ∇R, but this connection is not adapted to the study of foliations because the horizontal and the vertical bundle may not be parallel with respect to ∇R. We will rather make use of the Bott connection on M which is deﬁned as follows.

 R ∞ πH(∇X Y ),X,Y ∈ Γ (H),  ∞ ∞ πH([X,Y ]),X ∈ Γ (V),Y ∈ Γ (H), ∇X Y = π ([X,Y ]),X ∈ Γ∞(H),Y ∈ Γ∞(V),  V  R ∞ πV (∇X Y ),X,Y ∈ Γ (V), where πH (resp. πV ) is the projection on H (resp. V). One can check that the Bott connection is metric-compatible, that is, ∇g = 0, though unlike the Levi-Civita connection it is not torsion-free. Let T be the torsion of the Bott connection ∇. Observe that for X,Y ∈ Γ∞(H)

T (X,Y ) = ∇X Y − ∇Y X − [X,Y ]

= πH(∇X Y − ∇Y X) − [X,Y ]

= πH([X,Y ]) − [X,Y ]

= −πV ([X,Y ]). Similarly one can check that the Bott connection satisﬁes the following properties that we record here for later use ∞ ∞ ∇X Y ∈ Γ (H) for any X,Y ∈ Γ (H), ∞ ∞ ∇X Y ∈ Γ (V) for any X,Y ∈ Γ (V), (2.1) T (X,Y ) ∈ Γ∞(V) for any X,Y ∈ Γ∞(H), T (U, V ) ∈ Γ∞(H) for any U, V ∈ Γ∞(V), T (X,U) = 0 for any X ∈ Γ∞(H),U ∈ Γ∞(V),

Example 2.3 (Example 2.2 revisited). Let (M, θ, g) be a K-contact Rie- mannian manifold. The Bott connection coincides with Tanno’s connection that was introduced in [42] and which is the unique connection that satisﬁes the following. (1) ∇θ = 0; (2) ∇Z = 0; (3) ∇g = 0; (4) T (X,Y ) = dθ(X,Y )Z for any X,Y ∈ Γ∞(H); (5) T (Z,X) = 0 for any vector ﬁeld X ∈ Γ∞(H). 10 BAUDOIN, FENG, AND GORDINA

We deﬁne the horizontal Ricci curvature RicH of the Bott connection as the ﬁberwise symmetric linear map on one-forms such that for all smooth functions f, g on M

hRicH(df), dgi = Ric (∇Hf, ∇Hg) = RicH(∇f, ∇g), where Ric is the Ricci curvature of the Bott connection ∇ and RicH its horizontal Ricci curvature (horizontal trace of the full curvature tensor R of the Bott connection).

2.2. Generalized Levi-Civita connections and adjoint connections. When studying Weitzenböck type identities for the horizontal Laplacian (see Section 2.4), one can not make use of the Bott connection because the adjoint connection of the Bott connection is not a metric connection. We refer to [6, 20, 28, 29] and especially the books [21, 22] for a discussion on Weitzenböck-type identities and adjoint connections. We shall rather make use of a family of connections first introduced in [2] and only keep the Bott connection as a reference connection. This family of connections is constructed from the canonical variation of the metric that we recall now. The metric g can be split as

(2.2) g = gH ⊕ gV . Using the splitting of the Riemannian metric g in (2.2) we can introduce the following one-parameter family of Riemannian metrics

1 g = g ⊕ g , ε > 0. ε H ε V One can check that for every ε > 0, ∇gε = 0. The metric gε then induces a metric on the cotangent bundle which we still denote by gε. By using similar notation and conventions as before we have 2 2 2 ∗ kηkε = kηkH + εkηkV , for every η ∈ Tx M. For each Z ∈ Γ∞(V) there is a unique skew-symmetric endomorphism JZ : Hx → Hx, x ∈ M such that for all horizontal vector ﬁelds X,Y ∈ Hx

(2.3) gH(JZ (X),Y )x = gV (Z,T (X,Y ))x, where T is the torsion tensor of ∇. We then extend JZ to be 0 on Vx. Also, to ensure (2.3) holds also for Z ∈ Γ∞(H), taking into account (2.1) we set JZ ≡ 0. We now introduce the following family of connections (see [2]) 1 ∇ε Y = ∇ Y − T (X,Y ) + J X,X,Y ∈ Γ∞( ). X X ε Y M ε ε It is easy to check that ∇ gε = 0. The torsion of ∇ is equal to 1 1 T ε(X,Y ) = −T (X,Y ) + J X − J Y,X,Y ∈ Γ∞( ). ε Y ε X M INTEGRATION BY PARTS 11

The adjoint connection of ∇ε (in the sense of B. Driver [16], see [21, Section 1.3] for a discussion about adjoint connections) is then given by

ε ε ε 1 ∇b Y := ∇ Y − T (X,Y ) = ∇X Y + JX Y, X X ε thus ∇b ε is also a metric connection. Moreover, it preserves the horizontal and vertical bundles. For later use, we record that the torsion of ∇b ε is

ε ε 1 1 Tb (X,Y ) = −T (X,Y ) = T (X,Y ) − JY X + JX Y. ε ε The Riemannian curvature tensor of ∇b ε also can be computed easily in terms of the Riemannian curvature tensor R of the Bott connection and it is given by the following lemma.

∞ Lemma 2.2. For X,Y,Z ∈ Γ (M),

ε 1 1 Rb (X,Y )Z =R(X,Y )Z + JT (X,Y )Z + (JX JY − JY JX )Z+ ε ε2 1 1 (∇ J) Z − (∇ J) Z, ε X Y ε Y X where R is the curvature tensor of the Bott connection. Proof.

ε ε ε ε ε ε Rb (X,Y )Z = ∇b X ∇b Y Z − ∇b Y ∇b X Z − ∇b [X,Y ]Z 1 1 1 1 1 =(∇ ∇ + (∇ J) + J ∇ + J ∇ + J + J J )Z X Y ε X Y ε X Y ε Y X ε ∇X Y ε2 X Y 1 1 1 1 1 − (∇ ∇ + (∇ J) + J ∇ + J + J ∇ + J J )Z Y X ε Y X ε Y X ε ∇Y X ε X Y ε2 Y X 1 − ∇ Z − J Z [X,Y ] ε [X,Y ] 1 =R(X,Y )Z + (J J − J J )Z+ ε2 X Y Y X 1 1 1 (∇ J) Z − (∇ J) Z + J Z. ε X Y ε Y X ε T (X,Y )

2.3. Horizontal Brownian motion and stochastic parallel transport. We define the horizontal gradient ∇Hf of a function f as the projection of the Riemannian gradient of f on the horizontal bundle H. Similarly, we define the vertical gradient ∇V f of a function f as the projection of the Riemannian gradient of f on the vertical bundle V. Consider the pre-Dirichlet form Z ∞ EH(f, h) = gH (∇Hf, ∇Hh) d Vol, f, h ∈ C0 (M), M 12 BAUDOIN, FENG, AND GORDINA where d Vol is the Riemannian volume measure. Then there exists a unique ∞ diffusion operator L on M such that for all f, h ∈ C0 (M) Z Z EH(f, h) = − fLh d Vol = − hLf d Vol . M M The operator L is called the horizontal Laplacian of the foliation. If X1, ··· ,Xn is a local orthonormal frame of horizontal vector fields, then we can write L in this frame n X 2 L = Xi − ∇Xi Xi. i=1 Observe that the subbundle H satisfies Hörmander’s(bracket generating) condition, therefore by Hörmander’stheorem the operator L is locally subelliptic (for comments on this terminology introduced by Fefferman-Phong we refer to [26], see for instance the survey paper [37] or [18, p. 944]). By [1, Proposition 5.1] the completeness of the Riemannian metric g im- ∞ plies that L is essentially self-adjoint on C0 (M) and thus that EH is closable. t L Then we can define the semigroup Pt = e 2 by using the spectral theorem. The diffusion process {X } corresponding to the semigroup {P } t t>0 t t>0 will be called the horizontal Brownian motion on the Riemannian foliation (F, g). Since M is assumed to be compact, 1 ∈ dom(EH) and thus Pt1 = 1. This implies that {X } is a non-explosive diffusion. At the same time the t t>0 horizontal Brownian motion {X } is a semimartingale on which can t t>0 M be constructed from a globally defined stochastic differential equation on a bundle over M (see [19, Theorem 3.8] or Corollary 4.6). 2.4. Weitzenböck formulas. A key ingredient in studying the horizontal Brownian motion {X } is the Weitzenböck formula that has been proven t t>0 in [2,6]. We recall here this formula. If Z1,...,Zm is a local vertical frame, then the (1, 1) tensor

m 2 X J := JZ` JZ` `=1 does not depend on the choice of the frame and may be deﬁned globally.

Example 2.4 (Example 2.2 revisited). If M is a K-contact manifold equipped with the Reeb foliation, then, by taking Z to be the Reeb vector field, one 2 2 gets J = JZ = −IdH. The horizontal divergence of the torsion T is the (1, 1) tensor which in a local horizontal frame X1,...,Xn is defined by n X δHT (X) := − (∇Xj T )(Xj,X). j=1 Example 2.5 (Example 2.2 revisited). If M is a K-contact manifold equipped with the Reeb foliation, then δHT can be expressed as the horizontal trace INTEGRATION BY PARTS 13 of the Tanno tensor (see [7,42]). Therefore if (M, θ) is a CR manifold, then δHT = 0 By using the duality between the tangent and cotangent bundles with 2 respect to the metric g, we can identify the (1, 1) tensors J and δHT with ∗ linear maps on the cotangent bundle T M. ∗ Namely, let ] : T M → T M be the standard musical (raising an index) isomorphism which is defined as the unique vector ω] such that for any x ∈ M

] g ω ,X = ω (X) for all X ∈ TxM, x while in local coordinates the isomorphism ] can be written as follows

n+m n+m n+m n+m X i ] X j X X ij ω = ωidx 7−→ ω = ω ∂j = g ωi∂j. i=1 j=1 j=1 i=1 The inverse of this isomorphism is the (lowering an index) isomorphism ∗ [ : T M → T M deﬁned by [ X = g (X, ·)x ,X ∈ TxM and in local coordinates

n+m n+m n+m n+m X i [ X i X X j i X = X ∂i 7−→ X = Xidx = gijX dx . i=1 i=1 i=1 j=1 If η is a one-form, we define the horizontal gradient in a local adapted frame of η as the (0, 2) tensor n X ∇Hη = ∇Xi η ⊗ θi, i=1 where θi, i = 1, . . . , n is the dual of Xi. Finally, for ε > 0, we consider the following operator which is defined on one-forms by (2.4) n X ε 2 ε 1 2 1 ε := (∇Xi − TX ) − (∇∇X Xi − T∇ X ) − J + δHT − RicH, i i Xi i ε ε i=1 where Tε is the (1, 1) tensor defined by

1 Tε Y = −T (X,Y ) + J X,X,Y ∈ Γ∞( ). X ε Y M Similarly as before, we will use the notation n X Tε Tε Hη := Xi η ⊗ θi. i=1 14 BAUDOIN, FENG, AND GORDINA

The expression (2.4) does not depend on the choice of the local horizontal frame and thus ε may be globally deﬁned. Formally, we have 1 1 (2.5) = −(∇ − Tε )∗(∇ − Tε ) − J2 + δ T − Ric , ε H H H H ε ε H H 2 where the adjoint is understood with respect to the L (M, gε, µ) inner prod- R uct on sections, i.e. h·, ·iεdµ (see [1, Lemma 5.3] for more detail). The M main result of [6] is the following.

∞ Theorem 2.3 (Lemma 3.3, Theorem 3.1 in [6]). Let f ∈ C0 (M), x ∈ M and ε > 0, then

(2.6) dLf(x) = εdf(x).

Remark 2.4. From [6, Lemma 3.4], we see that for ε1, ε2 > 0, the operator

ε1 − ε2 vanishes on exact one-forms. It is therefore no surprise that the left hand side of (2.6) does not depend of ε.

To conclude this section we remark, and this is not a coincidence (see Sec- tion 4), that the potential term in the Weitzenb¨ock identity can be identiﬁed with the horizontal Ricci curvature of the adjoint connection ∇b ε.

Lemma 2.5. The horizontal Ricci curvature of the adjoint connection ∇b ε is given by ε 1 1 Ricˆ = Ric − δ∗ T + J2, H H ε H ε ∗ where δHT denotes the adjoint of δHT with respect to the metric g. ∞ Proof. Let X,Y ∈ Γ (T M) and X1, ··· ,Xn be a local horizontal orthonormal frame. By the deﬁnition of the horizontal Ricci curvature and Lemma 2.2 we have ˆ ε RicH(X,Y ) n X ε = gH(Rˆ (Xi,X)Y,Xi) i=1 n n X X 1 = g (R(X ,X)Y,X ) + g J Y,X H i i H ε T (Xi,X) i i=1 i=1 n X 1 1 + g (∇ J) Y − (∇ J) Y,X . H ε Xi X ε X Xi i i=1 For the ﬁrst term, we have

n X gH(R(Xi,X)Y,Xi) = RicH(X,Y ). i=1 INTEGRATION BY PARTS 15

For the second term, we easily see that n n X X gH JT (Xi,X)Y,Xi = − gV (T (X,Xi),T (Y,Xi)) i=1 i=1 2 = gH(J X,Y ).

For the third term, we ﬁrst observe that gH((∇X J)Xi Y,Xi) = 0. Then, we have n n X X gH ((∇Xi J)X Y,Xi) = − gH ((∇Xi J)X Xi,Y ) i=1 i=1 n X = − gV ((∇Xi T )(Xi,Y ),X) i=1 = gV (δHT (Y ),X) .

3. Integration by parts formula on the horizontal path space Let x ∈ and {X } be a horizontal Brownian motion on started M t t>0 M at x, defined on a probability space (Ω, µ). As usual we will denote µx = µ (· | X0 = x). We fix ε > 0 throughout the section. Our goal in this section is to prove integration by parts formulas on the path space of the horizontal Brownian motion. Some of the integration by parts formulas for the damped Malliavin derivative have been already announced in a less general and slightly different setting in [3]. The integration by part formulas for the intrinsic Malliavin derivative are new. We point out a significant difference of our techniques from what have been used in [1–3]. Namely, we shall mostly make use of the adjoint connection ∇b ε instead of the Bott connection. Three important properties of the connection ∇b ε are as follows. ε metric connection: ∇b gε = 0; ∞ ∞ ε horizontal connection: if X ∈ Γ (H) and Y ∈ Γ (M) then ∇b Y X ∈ Γ∞(H); skew-symmetry: the torsion tensor Tbε of ∇b ε satisfies Driver’s total skew-symmetry condition ( [16, p. 272])as follows. For X,Y,Z ∈ ∞ Γ (M) ε ε hTb (X,Y ),Ziε = −hTb (X,Z),Y iε. The latter can be seen from the formula

ε 1 1 Tb (X,Y ) = T (X,Y ) − JY X + JX Y ε ε and the deﬁnition of J. We need to use a stochastic parallel transport on forms. Let ∇e be a general connection on , and {X } be a semimartingale on . We M t t>0 M denote by //e0,t : TX0 M → TXt M the stochastic parallel transport of vector 16 BAUDOIN, FENG, AND GORDINA

ﬁelds along the paths of {X } (see [35, p. 50]). Then by duality we can t t>0 deﬁne the stochastic parallel transport on one-forms as follows. We have // : T ∗ → T ∗ such that for α ∈ T ∗ e0,t Xt M X0 M Xt M

(3.1) h//e0,tα, vi = hα,//e0,tvi, v ∈ TX0 M.

In particular, the stochastic parallel transport for the adjoint connection ∇ε = ∇ + 1 J along the paths of the horizontal Brownian motion {X } b ε t t>0 ε ε will be denoted by Θb t . Since the adjoint connection ∇b is horizontal, the ε map Θb t : TxM → TXt M is an isometry that preserves the horizontal bundle, ε that is, if u ∈ Hx, then Θb t u ∈ HXt . We see then that the anti-development of {X } defined as t t>0 Z t ε −1 Bt := (Θb s) ◦ dXs, 0 is a Brownian motion in the horizontal space Hx. The usual completion of the natural filtration of {B } will be denoted by {F } , and this is the t t>0 t t>0 filtration we will be using in what follows.

Remark 3.1. Observe that on one-forms the process Θε : T ∗ → T ∗ b t Xt M x M is a solution to the following covariant Stratonovich stochastic diﬀerential equation

ε ε ε d[Θb t α(Xt)] = Θb t ∇b ◦dXt α(Xt), where α is any smooth one-form. Since ∇ε = ∇ + 1 J = ∇ , b ◦dXt ◦dXt ε ◦dXt ◦dXt we deduce that Θb ε is actually independent of ε and is therefore also the stochastic parallel transport for the Bott connection. As a consequence, the Brownian motion (Bt)t>0 and its ﬁltration are also independent of the particular choice of ε.

We deﬁne a damped parallel transport τ ε : T ∗ → T ∗ by the formula t Xt M x M

ε ε ε (3.2) τt = Mt Θt , where the process Θε : T ∗ → T ∗ is the stochastic parallel transport of t Xt M x M one-forms with respect to the connection ∇ε = ∇ − Tε along the paths of {X } . The multiplicative functional Mε : T ∗ → T ∗ , t 0, is deﬁned t t>0 t x M x M > as the solution to the following ordinary diﬀerential equation

dMε 1 1 1 (3.3) t = − MεΘε J2 − δ T + Ric (Θε)−1, dt 2 t t ε ε H H t ε M0 = Id. INTEGRATION BY PARTS 17

Observe that the process τ ε : T ∗ → T ∗ is a solution of the following t Xt M x M covariant Stratonovich stochastic differential equation ε d[τt α(Xt)](3.4) 1 1 1 = τ ε ∇ − Tε − J2 − δ T + Ric dt α(X ), t ◦dXt ◦dXt 2 ε ε H H t τ0 = Id, where α is any smooth one-form. ε Also observe that Mt is invertible and that its inverse is the solution of the following ordinary differential equation d(Mε)−1 1 1 1 (3.5) t = Θε J2 − δ T + Ric (Θε)−1(Mε)−1. dt 2 t ε ε H H t t ε In particular, it implies that τt is invertible. 3.1. Malliavin and directional derivatives. The horizontal Wiener measure on Cx([0,T ], M) is defined as the distribution of the horizontal Brownian motion. The support of this measure will be called the horizontal path space and will be denoted by WH(M). The coordinate process on Cx([0,T ], M) will be denoted by (wt)06t6T . k Definition 3.2. A function F : WH(M) → R is called a C -cylinder function if there exists a partition

π := {0 = t0 < t1 < t2 < ··· < tn 6 T } k n of the interval [0,T ] and f ∈ C (M ) such that

(3.6) F (w) = f (wt1 , ..., wtn ) for all w ∈ WH(M). The function F is called a smooth cylinder function on WH(M) of M, if there ∞ n exists a partition π and f ∈ C (M ) such that (3.6) holds. k k We denote by FC (WH(M)) the space of C -cylinder functions, and by ∞ ∞ FC (WH(M)) the space of C -cylinder functions. Remark 3.3. The representation (3.6) of a cylinder function is not unique. ∞ However, let F ∈ FC (WH(M)) and n > 0 be the minimal n such that there exists a partition

π := {0 = t0 < t1 < t2 < ··· < tn 6 T } k n of the interval [0,T ] and f ∈ C (M ) such that

(3.7) F (w) = f (wt1 , ..., wtn ) for all w ∈ WH(M). In that case, if ˜ ˜ ˜ ˜ π˜ = {0 = t0 < t1 < t2 < ··· < tn 6 T } k n is another partition of the interval [0,T ] and f˜ ∈ C (M ) is such that F (w) = f˜w , ..., w for all w ∈ W ( ), t˜1 tñ H M 18 BAUDOIN, FENG, AND GORDINA then π =π ˜ and f = f˜. Indeed, since f (w , ..., w ) = f˜w , ..., w t1 tn t˜1 tñ we first deduce that t1 = t˜1. Otherwise d1f = 0 or d1f˜ = 0, where d1 denotes the differential with respect to the first component. This contradicts the fact that n is minimal. Similarly, t2 = t˜2 and more generally tk = t˜k. The representation (3.3) will be referred to as the minimal representation of F . We now turn to the definition of directional derivative on the horizontal path space. ∞ Definition 3.4. Let F = f (wt1 , ..., wtn ) ∈ FC (WH(M)). For an F- adapted and TxM-valued semimartingale (v(t))06t6T such that v(0) = 0, we define the directional derivative n X D ε E DvF = dif(Xt1 , ··· ,Xtn ), Θb ti v(ti) i=1 ∞ Definition 3.5. For F = f (wt1 , ..., wtn ) ∈ FC (WH(M)) we define the damped Malliavin derivative by n ε X ε −1 ε Det F := 1[0,ti](t)(τt ) τti dif(Xt1 , ··· ,Xtn ), 0 6 t 6 T. i=1 Observe that from this definition DεF ∈ T ∗ . et Xt M Remark 3.6. Note that the directional derivative D is independent of ε, but the damped Malliavin derivative depends on ε. In addition, both the directional derivatives and damped Malliavin derivatives are independent of the representation of F . Indeed, let F = f (wt1 , ..., wtn ) be the minimal representation of F . If f˜w , ..., w is another representation of F , then t˜1 tÑ ˜ for every 1 6 j 6 N, we have either that there exists i such that ti = tj in which case dif = djf˜, or for all i, ti 6= t˜j in which case djf˜ = 0. Before we can formulate the main result, we need to define an analog of the Cameron-Martin subspace.

Deﬁnition 3.7. An Ft-adapted absolutely continuous Hx-valued process R T 0 2 (γ(t))06t6T such that γ(0) = 0 and Ex 0 kγ (t)kHdt < ∞ will be called a horizontal Cameron-Martin process. The space of horizontal Cameron- Martin processes will be denoted by CMH(M, Ω).

Deﬁnition 3.8. An Ft-adapted TxM-valued continuous semimartingale (v(t))06t6T R T 2 such that v(0) = 0 and Ex 0 kv(t)k dt < ∞ will be called a tangent process if the process Z t ε −1 ε ε v(t) + (Θb s) T (Θb s ◦ dBs, Θb sv(s)) 0 INTEGRATION BY PARTS 19 is a horizontal Cameron-Martin process. The space of tangent processes will be denoted by TxWH(M, Ω). ε Remark 3.9. By Remark 3.1 the stochastic parallel transport Θb s is independent of ε, therefore the notion of a tangent process is itself independent of ε as well.

Remark 3.10. As the torsion T is a vertical tensor, then an Ft-adapted and T -valued continuous semimartingale {v(t)} such that xM 06t6T Z T 2 Ex kv(t)k dt < ∞, v(0) = 0 0 is in TWH(M) if and only if (1) The horizontal part vH ∈ CMH(M, Ω); (2) The vertical part vV is given by Z t ε −1 ε ε vV (t) = − (Θb s) T (Θb s ◦ dBs, Θb svH(s)). 0 The main results of this section are the following two theorems. Theorem 3.11 (Integration by parts for the damped Malliavin de- ∞ rivative). Suppose F ∈ FC (WH(M)) and γ ∈ CMH(M, Ω), then

Z T Z T ε ε 0 0 (3.8) Ex hDesF, Θb sγ (s)ids = Ex F hγ (s), dBsiH . 0 0 Theorem 3.12 (Integration by parts for the directional derivatives). ∞ Suppose F ∈ FC (WH(M)) and v ∈ TxWH(M, Ω), then

Z T 0 1 ε −1 ε Ex (DvF ) = Ex F vH(t) + (Θb t ) RicHΘb t vH(t), dBt . 0 2 H Even though these two integration by parts formulas seem similar, they are quite diﬀerent in nature. The damped derivative is used to derive gradient bounds and functional inequalities on the path space (see [2, 3]). The directional derivative, however, is more related to quasi-invariance properties (see Section 4), and the expression

Z T 0 1 ε −1 ε vH(t) + (Θb t ) RicHΘb t vH(t), dBt 0 2 H can be viewed as a horizontal divergence on the path space. The remainder of the section is devoted to the proof of Theorem 3.11 and Theorem 3.12. We shall adapt the methods from Markovian stochastic calculus developed by Fang-Malliavin [25] and E. Hsu [32] in the Riemannian case to our framework. 20 BAUDOIN, FENG, AND GORDINA

3.2. Gradient formula. In this preliminary section we recall the gradient formula for Pt. In the case δHT = 0, the operator ε is essentially self- adjoint on the space of one-forms, and the gradient representation was first proved in [2]. Lemma 3.13 (Theorem 4.6 and Corollary 4.7 in [2], Theorem 2.7 in [30]). ∞ Let T > 0. For f ∈ C0 (M), the process ε (3.9) Ns = τs (dPT −sf)(Xs), 0 6 s 6 T, is a martingale, where dPT −sf denotes here the exterior derivative of the function PT −sf. As a consequence, for every t > 0, ε (3.10) dPtf(x) = Ex(τt df(Xt)). Proof. From Itô’sformula and the definition of τ ε, we have

dNs = 1 1 1 τ ε ∇ − Tε − J2 − δ T + Ric ds (dP f)(X ) s ◦dXs ◦dXs 2 ε ε H H T −s s d + τ ε (dP f)(X )ds. s ds T −s s We now see that d 1 1 1 (dP f) = − dP Lf = − dLP f = − dP f, ds T −s 2 T −s 2 T −s 2ε T −s where we used Theorem 2.3. Observe that the bounded variation part of 1 1 1 τ ε ∇ − Tε − J2 − δ T + Ric ds (dP f)(X ) s ◦dXs ◦dXs 2 ε ε H H T −s s

1 ε ε d is given by 2 τs εdPT −sf(Xs)ds which cancels out with the expression τs ds (dPT −sf)(Xs)ds. The martingale property follows from the bound as in Lemma 4.3 in in [2] or [28, 29, Theorem 2.7].

3.3. Integration by parts formula for the damped Malliavin derivative. We prove Theorem 3.11 in this section. Some of the key arguments may be found in [2, 3], however since our framework is more general here (for example, we do not assume the Yang-Mills condition δHT = 0 ) and we now use the adjoint connection instead of the Bott connection, for the sake of self-containment, we give a complete proof.

∞ Lemma 3.14. For f ∈ C0 (M), and γ ∈ CMH(M, Ω) Z T 0 Ex f(XT ) hγ (s), dBsiH = 0 Z T ε ε,∗ −1 ε 0 Ex τT df(XT ), (τs ) Θb sγ (s)ds . 0 INTEGRATION BY PARTS 21

Proof. Consider again the martingale process Ns deﬁned by (3.9). We have ∞ then for f ∈ C0 (M) Z t Z t 0 ε 0 ε Ex f(Xt) hγ (s), dBsiH = Ex f(Xt) hΘb sγ (s), Θb sdBsiH 0 0 Z t ε 0 ε = Ex (f(Xt) − Ex (f(Xt))) hΘb sγ (s), Θb sdBsiH 0 Z t Z t ε ε 0 ε = Ex hdPt−sf(Xs), Θb sdBsi hΘb sγ (s), Θb sdBsiH 0 0 Z t ε 0 = Ex hdPt−sf(Xs), Θb sγ (s)ids 0 Z t ε ε,∗ −1 ε 0 = Ex hτs dPt−sf(Xs), (τs ) Θb sγ (s)ids 0 Z t ε,∗ −1 ε 0 = Ex hNs, (τs ) Θb sγ (s)ids 0 Z t ε,∗ −1 ε 0 = Ex Nt, (τs ) Θb sγ (s)ds , 0 where we integrated by parts in the last equality. ∞ Remark 3.15. A similar proof as above actually yields that for f ∈ C0 (M), γ ∈ CMH(M, Ω) and t 6 T , Z T 0 Ex f(XT ) hγ (s), dBsiH | Ft = t Z T ε ε,∗ −1 ε 0 Ex τT df(XT ), (τs ) Θb sγ (s)ds | Ft . t Lemma 3.14 shows that integration by parts formula (3.8) holds for cylinder functions of the type F = f(Xt). We now turn to the proof of Theorem 3.11 by using induction on n in a representation of a cylinder function F . To run the induction argument we need the following fact. ∞ Proposition 3.16. Let F = f(Xt1 , ··· ,Xtn ) ∈ FC (WH(M)). We have n ! X ε dEx(F ) = Ex τti dif(Xt1 , ··· ,Xtn ) . i=1 Proof. We will proceed by induction. Consider a cylinder function F = f(Xt1 , ··· ,Xtn ). For n = 1 the statement follows from Lemma 3.13, which implies that ε dEx(f(Xt1 )) = dPt1 f(x) = Ex(τt1 df(Xt1 )).

Now we assume that the claim holds for any cylinder function F = f(Xt1 , ··· ,Xtk ) for any k 6 n − 1. By the Markov property we have

Ex(F ) = Ex(E(F | Ft1 )) = Ex(g(Xt1 )), 22 BAUDOIN, FENG, AND GORDINA

where g(y) = Ey(f(y, Xt2−t1 , ··· ,Xtn−t1 )). Therefore

ε dEx(F ) = E(τt1 dg(Xt1 )). By using the induction hypothesis, we obtain

dg(y) = Ey(d1f(y, Xt2−t1 , ··· ,Xtn−t1 ))+ n ! X ε Ey τti−t1 dif(y, Xt2−t1 , ··· ,Xtn−t1 ) i=2 n ! X ε = Ey τti−t1 dif(y, Xt2−t1 , ··· ,Xtn−t1 ) . i=1 By the multiplicative property of τ ε and the Markov property of X we have

τ ε d f(y, X , ··· ,X ) = EXt1 ti−t1 i t2−t1 tn−t1 ε −1 ε (τt1 ) E τti dif(Xt1 , ··· ,Xtn ) | Ft1 . Therefore we conclude

n ! X ε dEx(F ) = Ex τti dif(Xt1 , ··· ,Xtn ) . i=1

Remark 3.17. As expected, the expression

n ! X ε Ex τti dif(Xt1 , ··· ,Xtn ) i=1 is independent of the choice of the representation of the cylinder function F (see Remark 3.6).

Proof of Theorem 3.11. We use induction on n in a representation of the cylinder function F . More precisely, we would like to show that for any ∞ F = f(Xt1 , ··· ,Xtn ) ∈ FC (WH(M)) and t 6 t1 we have

Z tn 0 Ex F hγ (s), dBsiH Ft =(3.11) t n ! X Z ti hd f(X , ··· ,X ), τ ε,∗ (τ ε,∗)−1γ0(s)dsi | F . Ex i t1 tn ti s t i=1 t

The case n = 1 is Lemma 3.14 and Remark 3.15. Assume that (3.11) holds for any cylinder function F represented by a partition of size n − 1 for INTEGRATION BY PARTS 23

Ex(F | Ft1 ) = g(Xt1 ), where g(y) = Ey(f(y, Xt2−t1 , ··· ,Xtn−t1 )). Thus by Lemma 3.14 and Re- mark 3.15

Z t1 0 Ex Ex(F | Ft1 ) hγ (s), dBsiH | Ft = t Z t1 0 Ex g(Xt1 ) hγ (s), dBsiH | Ft = t Z t1 ε ∗ ε,∗ −1 ε 0 Ex dg(Xt1 ), (τt1 ) (τs ) Θb sγ (s)ds | Ft t Now according to Proposition 3.16

n ! X ε dg(y) = Ey τti−t1 dif(y, Xt2−t1 , ··· ,Xtn−t1 ) . i=1 Using the fact that

τ ε d f(y, X , ··· ,X ) = EXt1 ti−t1 i t2−t1 tn−t1 ε −1 ε (τt1 ) Ex τti dif(Xt1 , ··· ,Xtn ) | Ft1 , we conclude

Z t1 0 Ex Ex(F | Ft1 ) hγ (s), dBsiH | Ft t n ! X Z t1 = hd f(X , ··· ,X ), τ ε,∗ (τ ε,∗)−1Θεγ0(s)dsi | F . Ex i t1 tn ti s b s t i=1 t 24 BAUDOIN, FENG, AND GORDINA

Using the induction hypothesis that (3.11) holds for n − 1 we see that

Z tn 0 Ex F hγ (s), dBsiH Ft1 = t1 n ! X Z ti hd f(X , ··· ,X ), τ ε,∗ (τ ε,∗)−1γ0(s)dsi | F . Ex i t1 tn ti s t1 i=1 t1 3.4. Integration by parts formula for the directional derivatives. In this section we prove Theorem 3.12. One of the main ingredients B. Driver used in [16] in the Riemannian case was the orthogonal invariance of the Brownian motion to ﬁlter out redundant noise. As a complement to Lemma 3.14, we ﬁrst prove the following result.

Lemma 3.18. Let (Os)s>0 be a continuous and F-adapted process taking values in the space of skew-symmetric endomorphisms of Hx such that R T 2 2 ∗ ∞ E 0 kOsk ds < +∞, where kOsk = Tr(Os Os) . For f ∈ C0 (M), we have Z T ε ε,∗ −1 ε 1 ε Ex τT df(XT ), (τs ) Θb s OsdBs − TOs ds = 0, 0 2 where T ε is the tensor given in a horizontal frame e , ··· , e by Os 1 n n ε X ε −1 ε ε ε −1 TOs = (Θb s) T (ei, Θb sOs(Θb s) ei). i=1 Proof. Recall that we considered the following martingale in (3.9) ε Ns = τs (dPT −sf)(Xs), 0 6 s 6 T. We have then Z T ε ε,∗ −1 ε Ex τT df(XT ), (τs ) Θb sOsdBs = 0 Z T ε,∗ −1 ε Ex NT , (τs ) Θb sOsdBs . 0 From the proof of Lemma 3.13, we have

dNs 1 1 1 = τ ε ∇ − Tε − J2 − δ T + Ric ds (dP f)(X ) s ◦dXs ◦dXs 2 ε ε H H T −s s d + τ ε (dP f)(X )ds s ds T −s s ε ε ε ε = τs ∇ ε − T ε (dPT −sf)(Xs) = τs ∇ ε dPT −sf(Xs). Θb sdBs Θb sdBs Θb sdBs where, as before, ∇ε denotes the connection ∇−Tε. Let us denote by Hessε the Hessian for the connection ∇ε. One has therefore INTEGRATION BY PARTS 25

Z T ε,∗ −1 ε Ex NT , (τs ) Θb sOsdBs = 0 Z T ε ε ε Ex Hess PT −sf(Θb sdBs, Θb sOsdBs)(Xs) 0

∞ Due to the skew symmetry of O and the fact that for h ∈ C0 (M), X,Y ∈ ∞ Γ (M), Hessεh(X,Y ) − Hessεh(Y,X) = T ε(X,Y )h, we deduce

Z T ε,∗ −1 ε Ex NT , (τs ) Θb sOsdBs = 0 Z T 1 D ε ε E Ex dPT −sf, Θb sTOs ds = 2 0 Z T 1 D ε,∗ −1 ε ε E Ex Ns, (τs ) Θb sTOs ds . 2 0 integrating by parts the right hand side yields the conclusion.

We are now in position to prove the integration by parts formula for cylinder functions of the type F = f(Xt).

∞ Lemma 3.19. Let v ∈ TxWH(M, Ω). For f ∈ C0 (M),

D ε E Ex df(XT ), Θb T v(T ) = Z T 0 1 ε −1 ε Ex f(XT ) vH(t) + (Θb t ) RicHΘb t vH(t), dBt . 0 2 H

Proof. Let v ∈ TxWH(M, Ω). We deﬁne Z t ε −1 ε ε h(t) = v(t) + (Θb s) T (Θb s ◦ dBs, Θb sv(s)). 0

By deﬁnition of TxWH(M, Ω), we have h = vH ∈ CMH(M, Ω). From the formula

ε 1 1 Tb (X,Y ) = T (X,Y ) − JY X + JX Y, ε ε we have

ε ε ε ε 1 ε Tb (◦dXt, Θb t v(t)) = T (Θb t ◦ dBt, Θb t v(t)) − J ε (Θb t ◦ dBt). ε Θb t v(t) 26 BAUDOIN, FENG, AND GORDINA

We get therefore ε −1 ε 1 1 2 1 ∗ ε dv(t) + (Θb ) Tb (◦dXt, ·) + J − δ T + RicH dt Θb v(t) t 2 ε ε H t 1 ε −1 ε 1 ε −1 1 2 1 ∗ ε =dh(t) − (Θb t ) J ε (Θb t ◦ dBt) + (Θb t ) J − δHT + RicH Θb t h(t)dt ε Θb t v(t) 2 ε ε

1 ε −1 ε 1 ε −1 ε =dh(t) − (Θb t ) J ε Θb t dBt + (Θb t ) (RicH) Θb t h(t)dt. ε Θb t v(t) 2 It is then a consequence of Itˆo’sformula that Z t ε −1 ε,∗ ε,∗ −1 ε v(t) = (Θb t ) τt (τs ) Θb s ◦ dMt, 0 where

1 ε −1 ε 1 ε −1 ε dMt = dh(t) − (Θb t ) J ε Θb t dBt + (Θb t ) (RicH) Θb t h(t)dt. ε Θb t v(t) 2 Converting the Stratonovich integral into Itˆo’sintegral ﬁnally yields Z t ε −1 ε,∗ ε,∗ −1 ε 1 ε −1 ε v(t) = (Θb t ) τt (τs ) Θb s dh(t) + (Θb t ) (RicH) Θb t h(t)dt 0 2 1 +O dB − T ε dt , t t 2 Ot with 1 ε −1 ε Ot = − (Θb t ) J ε Θb t . ε Θb t v(t) Since Ot is a skew-symmetric horizontal endomorphism, one can conclude from Lemmas 3.14 and 3.18 that D ε E Ex df(Xt), Θb t v(t) Z T 0 1 ε −1 ε = Ex f(Xt) vH(t) + (Θb t ) RicHΘb t vH(t), dBt 0 2 H because h(t) = vH(t). Now Theorem 3.12 can be proven using induction on n in the presentation of a cylinder function F . The case n = 1 is Lemma 3.19, and showing the induction step is similar to how Theorem 3.11 has been proven, so for the sake of conciseness of the paper, we omit the details. As a direct corollary of Theorem 3.12, we obtain the following. ∞ Corollary 3.20. Let F,G ∈ FC (WH(M)) and v ∈ TxWH(M). We have ∗ Ex(F DvG) = Ex(GDvF ), where Z T ∗ 0 1 ε −1 ε Dv = −Dv + vH(t) + (Θb t ) RicHΘb t vH(t), dBt . 0 2 H INTEGRATION BY PARTS 27

Proof. By Theorem 3.12, we have Z T 0 1 ε −1 ε Ex(Dv(FG)) = Ex FG vH(t) + (Θb t ) RicHΘb t vH(t), dBt . 0 2 H

Since Dv(FG) = F Dv(G) + GDv(F ), the conclusion follows immediately.

3.5. Examples.

3.5.1. Riemannian submersions. In this section, we check that the integration by parts formula we obtained for the directional derivatives is consistent with and generalizes the formulas known in the Riemannian case. Let us assume here that the foliation on M comes from a totally geodesic submersion π :(M, g) → (B, j) (see example 2.1). Since the submersion has totally geodesic ﬁbers, π is harmonic and the projected process:

B Xt = π(Xt) is a Brownian motion on B started at π(x). Observe that from the deﬁnition of submersion, the derivative map Txπ is an isometry from Hx to TxB. ε The stochastic parallel transport Θb t projects down to the stochastic parallel B transport for the Levi-Civita connection along the paths of (Xt )06t6T . More precisely, ε −1 //0,t = TXt π ◦ Θb t ◦ (Txπ) , where // : T → T B is the stochastic parallel transport for the 0,t π(x)B Xt B B Levi-Civita connection along the paths of (Xt )06t6T . Consider now a Cameron-Martin process (h(t))06t6T in Tπ(x)B and a cylinder function F = B B f(Xt1 , ··· ,Xtn ) on B. The function F = f(π(Xt1 ), ··· , π(Xtn )) is then in ∞ FC (WH(M)). Using Theorem 3.12, one gets Z T 0 1 ε −1 ε Ex (DvF ) = Ex F vH(t) + (Θb t ) RicHΘb t vH(t), dBt , 0 2 H −1 where vH is the horizontal lift of h, that is, vH = (Txπ) h. By deﬁnition, we have n X D E D F = d f(XB , ··· ,XB ), (T π) ◦ Θε v(t ) v i t1 tn Xti b ti i i=1 n X D E = d f(XB , ··· ,XB ),// h(t ) i t1 tn 0,ti i i=1

It is easy to check that RicH is the horizontal lift of the Ricci curvature RicB of B. Therefore, the integration by parts formula for the directional 28 BAUDOIN, FENG, AND GORDINA derivative DvF can be rewritten as follows. n ! X D E d f(XB , ··· ,XB ),// h(t ) Ex i t1 tn 0,ti i i=1 ! Z T 1 = F h0(t) + //−1 RicB // h(t), dBB , Ex 2 0,t 0,t t 0 Tπ(x)B

B B where B is the Brownian motion on Tπ(x)B given by B = Txπ(B). This is exactly Driver’s integration by parts formula [16] for the Riemannian Brownian motion XB.

3.5.2. K-contact manifolds. In this section, we assume that the Riemannian foliation on M is the Reeb foliation of a K-contact structure. The Reeb vector ﬁeld on M will be denoted by R and the complex structure by J. The torsion of the Bott connection is then

T (X,Y ) = hJX,Y iHR. Therefore with the previous notation, one has

JZ X = hZ,RiJX. and the vertical part of a tangent process is given by Z t ε −1 ε ε vV (t) = − (Θb s) T (Θb s ◦ dBs, Θb svH(s)) 0 Z t ε −1 ε ε = ((Θb s) R)hJΘb svH(s), Θb s ◦ dBsiH 0

4. Quasi-invariance of the horizontal Wiener measure To prove quasi-invariance we adapt to our framework a method due to B. Driver [16] and later simplified by E. Hsu [31]. The crucial step is to use a construction similar to the one by Eells-Elworthy-Malliavin relying on the orthonormal frame bundle. In the section we fix 0 < ε 6 +∞. For ε = +∞, it will be understood ε 1 that the adjoint connection ∇b = ∇ + ε J is equal to the Bott connection. Interestingly, we will show that the flows we construct are independent of ε.

4.1. Normal frames. For computations in local coordinates, it will be convenient to work in the normal frame setting introduced in [6].

Lemma 4.1. Let x0 ∈ M. Around x0, there exist a local orthonormal horizontal frame {X1, ··· ,Xn} and a local orthonormal vertical frame {Z1, ··· ,Zm} such that the following structure relations hold n m X k X k [Xi,Xj] = ωijXk + γijZk k=1 k=1 INTEGRATION BY PARTS 29

m X j [Xi,Zk] = βikZj, j=1 k k j where ωij, γij, βik are smooth functions such that: j k βik = −βij.

Moreover, at x0 we have k k ωij = 0, βij = 0. We record the fact that in this frame the Christofel symbols of the Bott connection ∇ are given by  j ∇ X = 1 Pn ωk + ω + ωi X  Xi j 2 k=1 ij ki kj k

∇Zj Xi = 0  Pm k ∇Xi Zj = k=1 βijZk ε 1 Thus, the Christofel symbols of the adjoint connection ∇b = ∇ + ε J are given by  ε 1 Pn k j i ∇b Xj = ω + ω + ω Xk  Xi 2 k=1 ij ki kj ε 1 1 Pn j ∇b Xi = JZ Xi = − γ Xk  Zj ε j ε k=1 ik ∇ε Z = Pm βk Z b Xi j k=1 ij k 4.2. Construction of the horizontal Brownian motion from the orthonormal frame bundle. An orthonormal map at x ∈ M is an isometry n+m u :(R , h·, ·i) → (TxM, g). The orthonormal map bundle will be denoted ε by O(M). The adjoint connection ∇b allows to lift vector ﬁelds on M to vector ﬁelds on O(M). Let e1, ··· , en, f1, ··· , fm be the canonical basis of n+m R .

Notation 4.2. We denote by Ai the vector field on O(M) such that Ai(x, u) is the lift of u(ei), and we denote by Vi the vector field on O(M) such that Vi(x, u) is the lift of u(fi). To take into account the foliation structure on M, we shall be interested in a special sub-bundle of O(M), the horizontal frame bundle. An isometry n+m u :(R , h·, ·i) −→ (TxM, g) will be called horizontal if n • u(R × {0}) ⊂ Hx; m • u({0} × R ) ⊂ Vx. The horizontal frame bundle is then defined as the set of (x, u) ∈ O(M) such that u is horizontal. It will be denoted OH(M). For notational convenience, n n n+m m m when needed, we identify R with R ×{0} ⊂ R and R with {0}×R ⊂ n+m R . We can write the vector fields Ais locally in terms of the normal frames described in Section 4.1. For x0 ∈ M we let {X1 ··· ,Xn,Z1, ··· ,Zm} be a normal frame around x0. 30 BAUDOIN, FENG, AND GORDINA

n+m If u : R → TxM is a horizontal isometry, we can find an orthogonal j Pn j Pm j ¯ matrix ei such that u(ei) = j=1 ei Xj and u(fi) = j=1 fi Zj. Let Xj be the vector field on OH(M) defined by f(etXj (x), u) − f(x, u) X¯jf(x, u) = lim . t→0 t

Lemma 4.3. Let x0 ∈ M and (x, u) ∈ OH(M) n X j ¯ Ai(x, u) = ei Xj j=1 n n m X j l ε ∂ X X j l ε ∂ − e e h∇b Xl,Xki − e f h∇b Zl,Zki . i r Xj ∂ek i r Xj ∂f k j,k,l,r=1 r j=1 k,l,r=1 r

In particular, at x0 we have n X j ¯ Ai(x0, u) = ei Xj. j=1

n+m Proof. Let u : R → TxM be a horizontal isometry and x(t) be a smooth 0 ∗ curve in M such that x(0) = x and x (0) = u(ei). We denote by x (t) = 0 0 (x(t), u(t)) the lift to O(M) of x(t) and by x1(t), ··· , xn(t) the components 0 of x (t) in the horizontal frame X1, ··· ,Xn. Since the adjoint connection preserves the horizontal and vertical bundles, the curve x∗(t) takes its values in OH(M). By deﬁnition of Ai, one has n n m X 0 X 0 ∂ X 0 ∂ Ai = x (0)X¯j + u (0) + v (0) , j kl ∂el kl ∂f l j=1 k,l=1 k k,l=1 k where ukl(t) = hu(t)(ek),Xli and vkl(t) = hu(t)(fk),Zli. Since u(t)(ek) and u(t)(fk) are parallel along x(t), one has ε ε ∇b x0(t)u(t)(ek) = 0, ∇b x0(t)u(t)(fk) = 0. At t = 0, this yields the expected result.

Remark 4.4. Since the vector ﬁelds Xj are horizontal, the covariant derivatives ∇ε X and ∇ε Z are independent of ε. As a consequence, the vector b Xj l b Xj l ﬁelds Ai are themselves independent of a particular choice of ε. In particular, Lemma 4.3 implies the following statement.

Proposition 4.5. Let π : O(M) → M be the bundle projection map. For a smooth f : M → R, and (x, u) ∈ OH(M), n ! X 2 Ai (f ◦ π)(x, u) = Lf ◦ π(x, u). i=1 INTEGRATION BY PARTS 31

Proof. It is enough to prove this identity at x0. Using the fact that at x0 we have h∇ε X ,X i = h∇ε Z ,Z i = 0, we see that b Xj l k b Xj l k n n X 2 X ¯ 2 Ai = Xj . i=1 j=1

The conclusion follows. As a straightforward corollary, we obtain the corresponding statement about Brownian motion.

Corollary 4.6. Let (Bt)t>0 be an n-dimensional Brownian motion and let (Ut)t>0 be a solution to the stochastic diﬀerential equation n X i dUt = Ai(Ut) ◦ dBt,U0 ∈ OH(M), i=1 then Xt = π(Ut) is a horizontal Brownian motion on M, that is, a Markov 1 process with generator 2 L.

4.3. Quasi-invariance of the horizontal Wiener measure. Let (Bt)t≥0 be an n-dimensional Brownian motion and let (Ut)t≥0 be the solution to the stochastic diﬀerential equation n X i dUt = Ai(Ut) ◦ dBt,U0 ∈ OH(M). i=1

By Corollary 4.6, Xt = π(Ut) is a horizontal Brownian motion on M which is started at x = π(U0). The map φH : B 7−→ X will be referred to as the horizontal development map. Our goal in this section is to prove quasi-invariance of the law of X with respect to a suitable family of flows. Our argument follows relatively closely the one by B. Driver [16] and then E. Hsu [31] (see also [13, 14, 23]). The only difference is that, in [16, 31], the authors consider the Itômap associated to the stochastic differential equation n m X X i dUt = Ai(Ut) ◦ dBt + Vi(Ut) ◦ dβt,U0 ∈ O(M), i=1 i=1 n+m where (Bt, βi) is a R -valued Brownian motion. All of the formulas and results in [16,31] can essentially be used by changing the driving path (Bt, βt) into the driving path (Bt, 0). The only difference is that we need to consider adapted vector fields on the horizontal path space whose pullbacks by the horizontal development map are horizontal processes. We now introduce some notation and formulate the precise framework in which we establish quasi-invariance. We will mainly follow the presentation in [15, 17]. 32 BAUDOIN, FENG, AND GORDINA

n+m Notation 4.7. We work in the probability space (C0(R ), B, µ), where n+m n+m C0(R ) is the space of continuous functions ω : [0,T ] → R such that n+m ω(0) = 0, B is the Borel σ-field on C0(R ), and µ is the Wiener measure. n+m The coordinate process (ωt)06t6T is therefore a Brownian motion in R . The usual completion of the natural filtration generated by (ωt)06t6T will be denoted by Bt. We decompose H V ωt = ωt + ωt , H n n V m m where ωt ∈ R × {0}' R and ωt ∈ {0} × R ' R . The process H n+m (ωt )06t6T will be called the horizontal Brownian motion in R and the law of the horizontal Brownian motion will be called the horizontal Wiener n+m measure on R and will be denoted by µH. Definition 4.8. We define the horizontal Cameron-Martin space denoted n+m n by CMH(R ) as the space of absolutely continuous R -valued (deterministic) functions (γ(t))06t6T such that γ(0) = 0 and Z T 0 2 |γ (t)|Rn dt < ∞. 0 n+m Pn If h ∈ R is a vector, we denote by Ah = i=1 hiAi and V h = Pm i=1 hi+nVi, where Ai and Vi are defined in Notation 4.2. Ah and V h are therefore vector fields on O(M) whose values at some u ∈ O(M) will be denoted respectively by Auh and Vuh. We consider the solution (Ut)06t6T to the Stratonovitch stochastic differential equation H dUt = AUt ◦ dωt ,U0 ∈ OH(M), and the horizontal Brownian motion on M given by Xt = π(Ut). The hor- H izontal Itômap IH is defined as the map IH : ω → U and the horizon- n tal development map φH is defined as the map φH : C0(R ) → WH(M), ωH → X. We refer to Definition 2.5 in [15] and the associated comments for a discussion of the Itôand development map in the classical Riemannian setting. In view of Definition 3.8, we define a set of tangent processes which is suitable in the frame bundle setting. As we will see, intuitively, tangent processes are the adapted vector fields on WH(M) (in the sense of [15, Def- inition 4.1]) whose pullbacks by the horizontal development map φH are horizontal processes. n+m Definition 4.9. A B-adapted R -valued continuous semimartingale (v(t))0 t T 6 6 such that v(0) = 0 and R T |v(t)|2 dt < ∞ will be called a tangent E 0 Rn+m process if the process Z t H v(t) + TUs (A ◦ dωs , Av(s)) 0 is a horizontal Cameron-Martin path. The space of tangent processes will be denoted by TWH(M). INTEGRATION BY PARTS 33

Remark 4.10. In Deﬁnition 4.9 we denote by T the torsion of the Bott connection. Observe that since T is a vertical tensor, a B-adapted and n+m R -valued continuous semimartingale (v(t))06t6T such that v(0) = 0 and R T 2 |v(t)| n+m dt < ∞ is in TWH( ) if and only if E 0 R M

n+m (1) The horizontal part vH is in CMH(R ); (2) The vertical part vV is given by

Z t H vV (t) = − TUs (A ◦ dωs , AvH(s)). 0

If v ∈ TWH(M) is a tangent process, we denote

Z t ε H pv(t) = v(t) + TbUs (A ◦ dωs , Av(s) + V v(s))+ 0 Z t Z s ε H H ΩbUτ (A ◦ dωτ , Av(τ) + V v(τ)) ◦ dωs , 0 0

ε ε where Ωb is the curvature form of ∇b . We observe that pv(t) can be seen as the pullback by the the horizontal development map φH of the adapted vector ﬁeld v (see Section 3.1 in [15]). Since ∇ε is a horizontal metric connection, the stochastic integranl R s Ωε (A◦ b 0 bUτ H dωτ , Av(τ) + V v(τ)) takes values in the space of skew-symmetric endomor- n phisms of R . Also, we have

Z t ε H TbUs (A ◦ dωs , Av(s) + V v(s)) = 0 Z t Z t H 1 H TUs (A ◦ dωs , Av(s)) − JV v(s)(A ◦ dωs )Us . 0 ε 0

n As a consequence, pv(t) is actually a horizontal process, that is, it is R - 1 2 1 ∗ valued. Moreover, by Lemma 2.5 the tensor ε J − ε δHT + RicH is equal to the horizontal Ricci curvature of the adjoint connection. As a consequence, we can rewrite pv(t) by using Itˆo’sintegral, and we obtain pv(t) = 1 Z t 1 1 v (t) + J2 − δ∗ T + Ric (Av(s) + V v(s))ds H 2 ε ε H H 0 Us Z t Z t Z s 1 H ε H H − JV v(s)(A ◦ dωs )Us + ΩbUτ (A ◦ dωτ , Av(τ) + V v(τ)) dωs . ε 0 0 0 34 BAUDOIN, FENG, AND GORDINA

We can further simplify this expression as follows. H JV v(s)(A ◦ dωs )Us = n X i JV v(s)(Ai)Us ◦ dωs = i=1 n n X 1 X J (A ) dωi + A J (A ) ds V v(s) i Us s 2 i V v(s) i Us i=1 i=1 n 1 X − J (A ) ds = 2 T (Ai,AvH(s)) i Us i=1 1 1 J (AdωH) − (δ∗ T ) (V v(s)) + J2(Av (s))ds. V v(s) s Us 2 H Us 2 H As a result, we see that

pv(t) = Z t Z t 1 1 H vH(t) + (RicH)Us (AvH(s))ds − JV v(s)(Adωs )Us 2 0 ε 0 Z t Z s ε H H + ΩbUτ (A ◦ dωτ , Av(τ) + V v(τ)) dωs . 0 0 More concisely, one can thus write Z t Z t H (4.1) pv(t) = qv(s)dωs + rv(s)ds, 0 0 n where qv is an so(n)-valued adapted process and rv is an R -valued adapted R t 2 process such that 0 |rv(s)| n ds < +∞ a.e. R n The process pv is therefore an adapted vector field on W (R ) in the sense of [15, Definition 3.2]. In particular, one deduces the following analogue of [17, Theorem 7.28] (see [15] for the details). Theorem 4.11 (Differential of the horizontal development map). Let v ∈ 0 TWH(M) be a tangent process such that |vH|Rn 6 K, where K is a non- random constant. Let Z s Z s t tqv(u) H bs = e dωu + t rv(u)du, 0 0 where qv and rv are defined as in (4.1). Then, there exists a version of t φH(b )s which is continuous in (s, t) differentiable in t and such that d | φ (bt) = Θˆ εv, dt t=0 H where, as in Sections 2 and 3, Θˆ ε denotes the parallel transport along the paths of the horizontal Brownian motion X for the connection ∇ˆ ε. INTEGRATION BY PARTS 35

The following theorem is an analogue of in [17, Theorem 7.29] and can be proved similarly to [16, 31] (the main ingredients are Girsanov theorem in the form of [16, Lemma 8.2] and the previous theorem). We recall that, H under µ, φH(ω ) is a horizontal Brownian motion on M started at x with distribution denoted µX (the horizontal Wiener measure on M). Theorem 4.12 (Quasi-invariance of the horizontal Wiener measure). Let n+m h ∈ CMH(R ). Consider the µ-a.e. defined process given by Z t H H vt(ω) = h(t) − TIH(ω )s (A ◦ dωs , Ah(s)). 0 v Consider the µ-a.e. defined process ηt (ω), given by parallel transport of H ˆ ε v −1 v(ω) along φH(ω ) for the connection ∇ . Then, η ◦ φH generates a flow on the horizontal path space WH(M) and this flow leaves the measure µX quasi-invariant.

v −1 Remark 4.13. Observe that η ◦ φH is indeed µX -a.e. well deﬁned since µ-a.e. ηv(ω) = ηv(ωH).

In the case where the foliation on M comes from a totally geodesic submersion (M, g) → (B, j) (see example 2.1 and section 3.5.1) then, as expected, theorem 4.12 projects down to [17, Theorem 7.29]. In the Heisenberg group, the situation is described in section 1.2 .In the general case, with the notations of theorem 4.12, and denoting Π the horizontal lift, one has a commutative diagram

tηv◦φ−1 e H WH(M) WH(M)

Π Π etζ W (B) W (B)

v −1 −1 tη ◦φH v tζ where e is the flow generated by η ◦φH . The induced flow e on the path space W(B) lets then the Wiener measure on W (B) quasi-invariant. Since the torsion T is vertical and the connection ∇ˆ ε projects down to the tζ Levi-Civita connection on B, the generator ζ of the flow e is the µB-a.e. defined process given by parallel transport (for the Levi-Civita connection on B) of the Cameron-Martin path h along the Brownian motion on B .

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† Department of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A.

† Department of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A. E-mail address: [email protected]