The Lie Bracket of Adapted Vector Fields on Wiener Spaces

Transcription

1 Appl Math Optim 39: (1999) 1999 Springer-Verlag New York Inc. The Lie Bracket of Adapted Vector Fields on Wiener Spaces B. K. Driver Department of Mathematics, 112, University of California, San Diego, La Jolla, CA , USA Communicated by M. Röckner Abstract. Let W (M) be the based (at o M) path space of a compact Riemannian manifold M equipped with Wiener measure ν. This paper is devoted to considering vector fields on W (M) of the form Xs h(σ ) = P s(σ )h s (σ ) where P s (σ ) denotes stochastic parallel translation up to time s along a Wiener path σ W (M) and {h s } s [,1] is an adapted T o M-valued process on W (M). It is shown that there is a large class of processes h (called adapted vector fields) for which we may view X h as first-order differential operators acting on functions on W (M). Moreover, if h and k are two such processes, then the commutator of X h with X k is again a vector field on W (M) of the same form. Key Words. Wiener measure, Itô development map, Lie bracket, Integration by parts. AMS Classification. Primary 6H7, 6D5, Secondary 58D Introduction Let (M d,,,, o) be given, where M is a compact connected manifold (without boundary) of dimension d,, is a Riemannian metric on M, is a, -compatible covariant derivative, and o is a fixed base point in M. Let T = T and R = R denote the torsion and curvature of, respectively. This research was partially supported by NSF Grant No. DMS

2 18 B. K. Driver Standing Assumption. The covariant derivative ( ) is assumed to be Torsion Skew Symmetric or TSS for short. That is to say T (X, Y ), Y for all vector fields X and Y on M. (With the TSS condition, the Laplacian on functions ( f = tr( grad f ) associated to is the same as the usual Levi Civita Laplacian.) Let ν denote Wiener measure on the path space W (M) {σ C ([, 1], M) σ() = o}. To be more explicit, let s (σ ) σ(s) for σ W (M) and let F s be the σ -field on W (M) generated by { s : s s}. Then ν is the unique probability measure on (W (M), {F s }) such that { s } s [,1] is a diffusion process with 1 as the generator. 2 Let P s denote stochastic parallel translation along up to time s relative to the covariant derivative. Given h in the Cameron Martin space H, { 1 } H h :[, 1] T o M h is absolutely continuous and h (s) 2 ds <, let X h denote the Cameron Martin vector field on W (M) given by Xs h P s h(s). It was shown in [15] that X h may indeed be considered as a vector field on W (M) in the sense that X h generates a quasi-invariant flow, at least when h is C 1. This theorem was extended by Hsu [34], [35] to include all h H. See also [44] and [24] for other approaches. It was also shown in Theorem 9.1, p. 363, of [15] (where X h was written as h ), that X h may be viewed as a densely defined closed operator on L 2 (W (M), ν). This last result relies on an integration by parts formula which in the special case of X h acting on functions of the form f (σ ) = F(σ (s)) is due to Bismut [9]. There have been numerous proofs and extensions of integration by parts formulas on W (M), see, for example, [4], [7], [22], [28], [29], [31], [4] [44], and the references therein for some of the more recent articles. See also Proposition 4.1 below. The purpose of this paper is to consider the commutator [X h, X k ] of two vector fields X h and X k. It has been known for some time that, in general, the commutator between two Cameron Martin vector fields on W (M) is no longer a Cameron Martin vector field. This is explicitly pointed out in Section 6.5 of [14] and in the case that M is a homogeneous manifold by Aida in [4]. Since so much of differential geometry relies on the use of the commutator of vector fields, it is highly desirable to have a class of vector fields which is stable under the Lie bracket operation. In this paper we study the adapted vector fields on W (M) introduced in [18], see also [14]. (Cruzeiro and Malliavin call the adapted vector fields by the suggestive name of tangent processes.) Intuitively, X is an adapted vector field (or tangent process) on W (M) iff X = (d/dt) ϕ t, where ϕ t : W (M) W (M) is a one-parameter family of quasi-invariant adapted maps on W (M) such that ϕ = id, see Definitions 3.2 and 4.1 for the precise definition. We call {ϕ t } t R as above an approximate flow for X. The main result of this paper is Theorem 7.4 and Corollary 8.4 both of which state: Theorem 1.1 (Informal Version). The Lie bracket of sufficiently regular adapted vector fields on W (M) is an adapted vector field.

3 Wiener Space Lie Brackets 181 We may give an informal proof of this theorem as follows. Let X and Y be two adapted vector fields on W (M) and let ϕ t and ψ t be approximate flows for X and Y, respectively. Then (formally) [X, Y ] = (d/dt) +η t where η t is the approximate flow on W (M) defined by η t ψ t ϕ t t ϕ t. (1.1) Thus [X, Y ] is also an adapted vector field. The remainder of this paper is devoted to a precise formulation and proof of the above stability result. Along the way we develop explicit formulas for the bracket [X, Y ]. Our approach here is at the infinitesimal level, viewing vector fields as first-order differential operators. Making the argument given in the above paragraph rigorous would involve a more delicate global analysis. See [1] for the beginnings of such an analysis. In [1] the differential of the flow e tx is computed for certain adapted vector fields X. As stated above, the reason for wanting vector fields to be stable under the Lie bracket operation is related to the desire to develop calculus and geometry on W (M). Recall that the Lie bracket typically enters into the coordinate-free definition of differential and geometric objects. At this time, the coordinate-free approach seems to be essential when working on path and loop spaces. For some of the recent developments on the calculus and geometry of path and loop spaces, see [1] [6], [8], [11] [18], [2] [22], [24] [36], [38], and [44]. 2. Background and Notation 2.1. Geometric Notation Let O(M) denote the orthogonal frame bundle of M. We choose T o M (o M is the fixed base point) as the model fiber of TM so that the fiber of O(M) above m M is O m (M) {u : T o M T m M u is an isometry}. The structure group of this bundle is the group O(d) of isometries of T o M. Let so(d) be the Lie algebra of O(d) consisting of skew-symmetric linear transformations on T o M. Given smooth paths u in O(M) and σ in M such that u(s) O σ(s) (M), let u(s)/ds : T o M T σ(s) M denote the linear operator defined by ( u(s)/ds)a = (u(s)a)/ds for all a T o M. Notice that V (s) u(s)a is a vector field along σ so that V (s)/ds = (u(s)a)/ds makes sense. Definition 2.1 (Connection 1-Form). Let ω be the so(d)-valued connection 1-form on O(M) given by ω u () u() 1 u(s)/ds s=, where s u(s) is any smooth curve in O(M). Notice that a path u is parallel or horizontal in O(M) iff ω u. Definition 2.2 (Horizontal Vector Fields). For a T o M and u O(M) let B a (u) T u O(M) be defined by ω B a (u) = (i.e., B a (u) is horizontal) and π B a (u) = ua, where π: O(M) M is the canonical projection map. We often write B a (u) instead of B a (u).

4 182 B. K. Driver Let T = T and R = R denote the torsion and curvature of, respectively, and let {e i } d i=1 be an orthonormal basis for T om. Notation 2.3. define For a, b T o M and an isometry u: T o M T m M (i.e., u O m (M)), u a, b u 1 R ua, ub u End(T o M), Ric u a d u a, e i e i, i=1 u a, b u 1 T ua, ub T o M, and u a d B ei (u)( a, e i ). i=1 So, Ric, and are the equi-invariant forms of the curvature tensor, the Ricci tensor, and the torsion tensor, respectively. Similarly, u is the equi-invariant form of a contraction of T Path Spaces and Development Maps In this subsection we introduce a number of path spaces and connecting maps between these path spaces. The reader is referred to [15] and [19] for a more leisurely discussion of this material. Definition 2.4 (Path Spaces). Let W W (T o M) {ω C ([, 1], T M) ω() = }, W (so(d)) {A C ([, 1], so(d)) A() = }, W (M) {σ C ([, 1], M) σ() = o}, and W (O(M)) {u C ([, 1], O(M)) u() = id}. Let µ denote the Wiener measure on W and let ν denote the Wiener measure on W (M). A map b: W (M) W may be viewed as a continuous process on W (M) with

5 Wiener Space Lie Brackets 183 values in T o M. This is done by writing b s for the function from W (M) to T o M given by b s (σ ) = b(σ )(s). Conversely, given a continuous process b s on W (M) with values in T o M, we may define a map b: W (M) W by b(σ )(s) b s (σ ). Similar identifications may be made for maps between any of the two path spaces defined above. In what follows we use both points of view interchangeably. Finally recall from the Introduction that : W (M) W (M) is the process defined by s (σ ) = σ(s) for all σ W (M). Definition 2.5 (Connecting Maps). path spaces: There are the following maps connecting the above 1. (Canonical Brownian Motion) Let β: W W be the canonical Brownian motion on W given by β s (ω) = ω(s) for all ω W and s [, 1]. 2. (Itô Map) I : W W (O(M)) is defined by the Stratonovich differential equation: δi s = B δβ s (I s ) with I = id O o (M), where the stochastic integrals are computed relative to the Wiener measure µ on W. Notice that I is horizontal, in the sense that ω δi =. 3. (Projection) π: W (O(M)) W (M) is an abuse of notation by which we mean π(u) π u. 4. (Horizontal Lift/Parallel Translation) P: W (M) W (O(M)) is the process defined by π(p) =, P = id T M, and ω δ P =, where the stochastic integrals are computed relative to the Wiener measure ν on W (M). 5. (Development Map) Let ϕ: W W (M) denote the composite map ϕ = π I. 6. (Inverse Brownian Motion) Let b: W (O(M)) W be defined as a version of θ δ P (relative to ν), where θ ξu u 1 π ξ for ξ T u O(M). The next theorem is well known. It recalls how the maps above are all related. The proof may be found in many places, see, for example, Theorem 3.3, p. 297, in [15] and also [19] and [23]. Theorem 2.6. The following identities hold: 1. π P = id W (M) ν-a.s., 2. ϕ b = id W (M) ν-a.s., 3. b ϕ = id W µ-a.s., 4. I b = P ν-a.s., and 5. I = P ϕµ-a.s. Moreover ϕ µ = Law(ϕ) = ν and b ν = Law(b) = µ A Collection of Norms In this section (, F, P) is a probability space, 1 p, and (V, ) is a finitedimensional normed vector space.

6 184 B. K. Driver Definition 2.7. Given a jointly measurable function f : [, 1] V we define: 1. The root mean square norm in L p : ( 1 1/2 f R p (V ) f (, s) ds) 2 2. The supremum norm in L p : f S p (V ) f L p, L p where f (ω) is the essential supremum of s f (ω, s) relative to Lebesgue measure on [,1]. (Notice that f (ω) = lim n f (ω, ) L n (ds) so that f (ω) is measurable.) Let R p (V ) denote those f : [, 1] V such that f R p (V ) <. Let S p (V ) denote those f : [, 1] V such that s f (s,ω)is continuous for almost every ω and f S p (V ) <. In what follows, we write R p for R p (V ) and S p for S p (V ) since the appropriate vector space V may be determined by looking at the range of the function f. We write f s (ω), f (ω, s), f (ω)(s), and f (s)(ω) interchangeably. Now suppose that (, {F s }, {X s }, F, P) is a filtered probability space equipped with a T o M-valued Brownian motion X. A function f : [, 1] V is a Brownian semimartingale if f may be represented in the form f (s, ) = s Q s dx(s ) + s. r s ds, (2.1) where (Q s, r s ) is a predictable process with values in Hom(T o M, V ) V. (Hom(T o M, V ) denotes the set of linear transformations from T o M to V.) We call the processes Q s and r s the kernels of f and write Qs f for Q s and rs f for r s if we are considering more than one Brownian semimartingale at a time. The Brownian semimartingale f is said to have continuous kernels if s (Q s (ω), r s (ω)) is continuous for almost every ω. Definition 2.8. Let H p (X) denote the set of Brownian semimartingales f such that f H p (X) Q R p + r R p <, and let B p denote the set of Brownian semimartingales f such that f has continuous kernels (Q s, r s ) and f B p (X) Q S p + r S p <. Remark 2.9. The B p -norms are the same as those used in [15]. However, the H p -norm differs slightly from the H p -norm used in [15]. In [15] the H p -norm was the weaker

7 Wiener Space Lie Brackets 185 norm given by ( 1 1/2 Q s ds) r s ds L p. To avoid notational clutter, if (, X, P) = (W,β,µ)and f : W [, 1] V, then we let f H p and f B p denote f H p (β) and f B p (β), respectively, where β is the canonical Brownian motion on W given by β s (ω) = ω(s). Similarly, if (, X, P) = (W (M), b,ν) and f : W (M) [, 1] V, then we let f H p and f B p denote f H p (b) and f B p (b), respectively, where b is the Brownian motion on W (M) defined in Definition 2.5. The next lemma is proved by unwinding the definitions and applying Theorem 2.6. Lemma 2.1. Suppose that f s : W V is a process and f s : W (M) V is the process defined by f s = f s b, then f R p = f R p, f S p = f S p, f H p = f H p, and f B p = f B p. Note, for all p [1, ], that f R p f S p and f H p f B p. Also it follows from Burkholder s inequality that for each p [1, ) there is a constant c p < such that f S p c p f H p. Notation Given p [1, ), let L p+ q>p Lq and L p q<p Lq. We say that lim t f (t) = f in L p+ (resp. L p )iflim t f (t) = f in L q for some q > p (resp. for all q < p). Analogous definitions for R p±, S p±, H p±, and B p± are also used. 3. Vector Fields on W For motivational purposes, recall Theorem 2.1 on p. 48 of [18]. Theorem 3.1 (Structure Theorem). Let : W W be an adapted map (i.e., s s is an adapted process) such that µ is equivalent to µ. Also assume there is an adapted map 1 : W W such that 1 and 1 are both equal to the identity map µ-a.s. Then there exist (O(d) T o M)-valued predictable processes (O,γ)on W such that (ω) = O(ω) dω + γ(ω)ds, (3.1) and 1 γ s 2 ds < µ a.s. With this in mind, if t : W W is an adapted flow on W, then s s t (ω)(s) = O t (ω)(s ) dω(s ) + γ t (ω)(s ) ds, (3.2)

8 186 B. K. Driver where, for each t R, (O t,γ t ) are (O(d) T o M)-valued predictable processes on W. Differentiating this equation at t = gives the form of the vector fields which can generate adapted flows on W. This motivates the following definition. Definition 3.2 (Adapted Vector Fields on W ). An adapted vector field h is a T o M- valued Brownian semimartingale on W with predictable kernels Qs h so(d) and r s h T o M such that 1 r s h 2 ds < a.s. Let V denote the collection of adapted vector fields on W and V p = V H p. Such processes were called adapted tangent vector fields on W in Definition 2.2, p. 41, of [18] and tangent processes in [12] [14]. Given an adapted vector field h as above, following Fang and Malliavin [31], let E (th)(s) s e tqh s dβ(s ) + t s r h s ds. (3.3) Notice that formally, (d/dt) E (th) = h, hence if f L p (W, dµ) is a function, it is reasonable to try to define the directional derivative h f of f by h by h f (d/dt) f E (th). A minimal requirement for this to make sense is that the law of E (th) must be equivalent to µ, since otherwise the composition f E (th) is not well defined. Proposition 3.3. Suppose that h V p (an adapted vector field on W in H p ) and rs h(ω) is bounded by a nonrandom constant k. Then E (th) has its law equivalent to µ and if f L p+ (µ), then f (E (th)) L p (µ) for all t R. Moreover, (d/dt) E (th) = h in H p. (3.4) Proof. Girsanov s theorem shows that, for all f L 1 (W, µ), where E µ ( f (E (th))e F(t) ) = E µ f, (3.5) F(t) t 1 r h e tqh dβ (t 2 /2) 1 r h 2 ds. (3.6) From this it follows that E (th) has its law equivalent to Brownian motion, see Lemma 8.2, p. 347, of [15] for details. Let q > 1 and q be the conjugate exponent to q. Then, by Hölder s inequality, E µ f (E (th)) p = E µ ( f (E (th)) p e F(t)/q e F(t)/q ) (E µ { f (E (th)) pq e F(t) }) 1/q e F(t)/q q = f p pq e F(t)/q q, where in the last line we have used (3.5). Since f L p+, q may be chosen sufficiently close to 1 such that f pq p <. By Remark 8.1 of [15], e ±F(t) L and hence e F(t)/q q <. Thus f (E (th)) L p for all t R.

9 Wiener Space Lie Brackets 187 By the fundamental theorem of calculus 1 e tqh I = tq h e utqh du. (3.7) Therefore 1 (e tqh I )/t Q h R p = Qh (e utqh I ) du R p ( 1 1/2 = Qs h R s(t) ds) 2 where R s (t) = 1 (e utqh s I ) du. (3.8) Since R s (t) ast and R s (t) is uniformly bounded (e utqh s is orthogonal since Q h s is in so(d)), the dominated convergence theorem shows that (e tqh I )/t Q h R p as t, that is, (d/dt) e tqh = Q h in R p. (3.9) This proves the proposition since L p (E (th) E (h))/t h H p = [(1/t)(e tqh I ) Q h ] R p. For h V 2 set z(h) 1 rs h dβ(s). (3.1) With this notation we have the following integration by parts formula. The idea of the proof already appears in [9], see also [4] and [31]. Theorem 3.4 (Integration by Parts). Suppose that h V 8 with rs h (ω) bounded by a nonrandom constant k, f, g L 4+ (dµ), and the directional derivatives h f (d/dt) f E (th) and h g (d/dt) g E (th) exist in L 4 (dµ). Then where E µ ( h f g) = E µ ( f h g), h g [ hg + z(h)g]., Proof. The idea of the proof is simply to compute the derivative of the identity E µ ( f (E (th))g(e (th))e F(t) ) = E µ ( fg), with respect to t at t =, where F(t) is given in (3.6).

10 188 B. K. Driver Set j = fg, then by standard arguments involving Hölder s inequality, j L 2+ and the derivative h j (d/dt) j (E (th)) exists in L 2 (µ) and is given by h j = h f g + f h g. Using Lemma 3.5 below and standard Hölder s inequality arguments it follows that = (d/dt) E µ ( j (E (th))e F(t) ) = E µ ( h j + j ( z(h))) = E µ ( h f g + f h g fgz(h)). Lemma 3.5. Suppose that p 1, h V 4p, and assume that r h is bounded by a nonrandom constant k, then with F(t) as (3.6), (d/dt) e F(t) = z(h) = 1 r h s dβ(s) in L p. Proof. To simplify notation, let f p denote the L p (µ) norm of f. By Burkholder s inequality, 1 F(t)/t r h dβ = r h (I e tqh ) dβ (t/2) r h 2 ds 4p 4p ( 1 1/2 k 2 t /2 + c 4p k e tqh s 1 ds) 2, 4p where A 2 tr A A if A is a matrix. By the fundamental theorem of calculus we have 1 2 e tqh 1 2 = tqh e utqh du t 2 Q h 2, where we have used the fact that e utqh is orthogonal in this last inequality. Combining the two above displayed inequalities shows that 1 F(t)/t + r h dβ t (k 2 /2 + c 4p k h H4p ) 4p as t. (3.11) By the fundamental theorem of calculus and Hölder s inequality the quantity 1 ε p (t) (ef(t) e F() )/t + r h dβ is bounded by 1 1 ε p (t) = (F(t)/t) e uf(t) du + r h dβ p 1 F(t)/t + 1 r h dβ + (F(t)/t) [ e uf(t) du 1] p p p

11 Wiener Space Lie Brackets F(t)/t + 1 r h dβ + F(t)/t 2p e uf(t) du 1 p 2p 1 F(t)/t + 1 r h dβ + F(t)/t 2p e uf(t) 1 2p du. p Because of (3.11) it follows that lim sup ε p (t) K lim sup t t K lim sup t 1 1 e uf(t) du 1 2p where K is a bound on F(t)/t 2p for t near. Now u e uf(t) 1 2p = F(t)e vf(t) dv F(t) 4p 2p e uf(t) 1 2p du, (3.12) u e vf(t) 4p dv and by (3.11) F(t) 4p C t for some constant C. These observations coupled with Remark 8.1, p. 348, of [15], which shows that e vf(t) 4p is bounded (by say K 1 ) for v 1 and t near zero, shows that e uf(t) 1 2p K 1 C t. Hence the limit in (3.12) is zero Pull-Back Vector Fields In the next section certain vector fields on W (M) are studied. These vector fields may be pulled back to W by the map ϕ: W W (M) in Definition 2.5. In this section we study these pulled-back vector fields. As above δβ is used to denote the Stratonovich differential of β and dβ the Itô differential. Definition 3.6 (Pull-Back Fields). For an adapted vector field h V, let Y h denote the adapted vector field on W given by Y h s where s C h s δβ(s ) + h s, (3.13) C h s A s h + Is h s, (3.14) and A s h s Is h s,δβ(s ). (3.15) In what follows, we abbreviate (3.13) (3.15) by Y h C h δβ + h, C h A h + I h,,

12 19 B. K. Driver and A h I h,δβ, respectively. For future reference, the Itô form of Y h is Y h (C h + Q h ) dβ [ ( + r h + 1 Ric 2 I h + I h + )] I Q h e i, e i ds, (3.16) i where {e i } i=1 d is an orthonormal basis for T om and Ric,,and are defined in Notation 2.3. The proof of (3.16) is straightforward, for details see the proof of Proposition 6.1, p. 323 in [15]. Equation (3.16) is equivalent to Y h Q Y h dβ + r Y h ds, (3.17) where Q Y h = C h + Q h = A h + I h, + Q h (3.18) and ( r Y h = r h + 1 Ric 2 I h + I h + i ) I Q h e i, e i. (3.19) Lemma 3.7. For all p [2, ) there is a constant K depending only on p and the geometry of M such that 1. A h S p K h H p for all h H p, 2. Y h H p K h H p for all h H p, and 3. Y h B p K h B p for all h B p. Proof. In the following argument K denotes a generic finite constant which only depends on p and the geometry of M. Express A h in Itô form as A h = I h, dβ + 1 { 2 I Q h e i, e i +(B i ) I h, e i } ds, (3.2) i where (B i ) u (d/dt) e tb i (u), B i B e i, and {e i } i=1 d is an orthonormal basis for T o M. Using the compactness of M and Burkholder s inequality, we show that A h S p = K { h S p + Q h R p} K h H p. (3.21) Similarly, r Y h R p r h R p + K ( h S p + Q h R p) K h H p, (3.22)

13 Wiener Space Lie Brackets 191 r Y h S p r h S p + K ( h S p + Q h S p) r h S p + K (c p h H p + Q h S p) K h B p, (3.23) and, by (3.14) and (3.18), Q Y h R p = A h + I h, + Q h R p A h S p + h S p + Q h R p K h H p (3.24) and Q Y h S p = A h + I h, + Q h S p A h S p + h S p + Q h S p K h H p + h B p K h B p. (3.25) Item 2 of the lemma follows from (3.22) and (3.24) while item 3 follows from (3.23) and (3.25). 4. Geometric Vector Fields Definition 4.1 (Geometric Vector Fields). Given an adapted vector field (h) on W, let X h denote the adapted vector field on W (M) given by X h (σ ) P(σ )h(b(σ )), i.e., X h (σ ) is the vector field along σ such that X h s (σ ) = P s(σ )h s (b(σ )) for all s [, 1]. We wish to have the vector fields X h act as first-order differential operators on a large class of functions on W (M). Our starting point will be to differentiate functions along an approximate flow to X h for nice h CV V S B. The next lemma guarantees that CV is sufficiently large. Lemma 4.2. For each p [2, ), CV V S B is dense in V p. Proof. First suppose that h V B, i.e., (Q h, r h ) is an (so(d) T o M)-valued continuous and bounded adapted process. For each integer n choose ϕ n C (T o M, T o M) such that ϕ n (x) = x if x n, ϕ n (x) = if x 2n, ϕ n(x) 2n, and sup x,n { ϕ n (x) + ϕ n (x) } <. To construct such functions, let C c (R) such that (x) 1 for all x and (x) Then define { 1 if x 1, if x 2. ϕ n (x) ( x 2 /n 2 )x.

14 192 B. K. Driver It is now a simple matter to verify that the functions ϕ n satisfy all of the requirements. Set h n ϕ n (h), then, by Itô s lemma, dh n = ϕ n (h){qh dβ + r h ds}+ 1 ϕ 2 n (h) Qh e i, Q h e i ds. So h n is bounded with bounded kernels: Q hn = ϕ n (h)qh and r hn = ϕ n (h)r h + 1 ϕ 2 n (h) Qh e i, Q h e i, i i.e., h n CV. It easily follows from the construction of the ϕ n s that (Q hn, r hn ) converges boundedly to (Q h, r h ) as n. Therefore, by the dominated convergence theorem, h n h in H p as n. Because of the above paragraph, it suffices to show that bcp (the bounded continuous adapted processes) are dense in P R p (the predictable processes in R p ). Using the fact that continuous adapted processes generate the predictable σ -algebra (see Remark 2.3, p. 16, of [37]) one may mimic the proof of Theorem 2 on p. 126 of [45] to show that bcp is dense in bp R p the bounded predictable processes endowed with the R p -norm. This proves the theorem, since it is easily shown that bp R p is dense in P R p using a standard truncation argument and the dominated convergence theorem Differentials of I and P Notation 4.3. Suppose that I W (O(M)) and A W (so(d)), let I A W (TO(M)) be defined by (I A)(s) (d/dt) I (s)e ta(s). (4.1) Also if I W (O(M)) and h W, let B h (I ) W (TO(M)) be defined by (B h (I ))(s) B h(s) (I (s)). (4.2) i Theorem 4.4. Suppose that h CV V S B. For t R let u(t) I E (ty h ): W W (O(M)). (4.3) Then the process u(t) is B p -differentiable for all 1 p < and u() = I A h +B h (I ). (4.4) We summarize this formula by writing I Y h = I A h +B h (I ). (The B p norms for manifold-valued processes are defined with the aid of an embedding of the manifold, see Definition 4.1, Proposition 4.1 (p. 31), and Notation 5.1 (p. 319) of [15] for more details.) The following lemma is needed in the proof of Theorem 4.4.

15 Wiener Space Lie Brackets 193 Lemma 4.5. Suppose that h B V, then (d/dt)e tqh = Q h e tqh in S (4.5) and (d/dt)e (th) = Q h e tqh dβ + r h ds in B. (4.6) Moreover, E (th) and d E (th)/dt are B p -Lipschitz in t for all p [1, ). Proof. Using (3.7) and Hölder s inequality, for any q < we have [(1/t)(e tqh I ) Q h ] S q = Q h R (t) S q Q h S r R (t) S p, (4.7) where 1/q = 1/r + 1/p and R s (t) is given in (3.8). Since e tqh is an orthogonal matrix, it easily follows from (3.7) and (3.8) that 1 R (t) S p utq h du 1 2 t Qh S p as t. (4.8) S p Therefore (d/dt) e tqh = Q h in S. For general t R, (d/dt) t e tqh = (d/dt) e (t +t)q h = (d/dt) e tqh e t Q h. (4.9) Since e t Q h is an orthogonal matrix-valued process, it follows from the case t = above that the derivative in (4.9) exists in S p and is given by Q h e t Q h. This proves (4.5) and because (d/dt)(tr h ) = r h in S p for all p [, ) holds trivially we have also proved (4.6). Again since e tqh is an orthogonal matrix, it follows from (4.6) that (d/dt)e (th) B p = h B p < for all 1 p <. Hence t E (th) is B p -Lipschitz. By similar calculations to above, we show that d 2 E (th)/dt 2 = (Q h ) 2 e tqh dβ in B and hence by Hölder s inequality, d 2 E (th)/dt 2 B p = (Q h ) 2 S p Q h 2 S 2p h 2 B 2p <. This shows that t (d/dt)e (th) is also B p -Lipschitz. Proof of Theorem 4.4. Because h CV, it follows by Lemma 3.7 and (3.19) that Y h B and r Y h is bounded. By Proposition 3.3, w(t) E (ty h ) B hasalaw equivalent to µ so that u(t) in (4.3) is well defined. By Lemma 4.5 ẇ(t) dw(t)/dt = Y h in B and w(t) and ẇ(t) are B p -Lipschitz. Using these observations, the proof of (4.4) may now be given using exactly the same argument as the proof of Theorem 5.2, p. 321, of [15]. It is only necessary for the reader to replace w(t) used in [15] by w(t) E (ty h ) and then to evaluate all t s appearing in the proof at t =. Corollary 4.6 (Approximate Flow for X h ). Assume that h CV. For each t R, let E(tX h ) ϕ E (ty h ) b. (4.1)

16 194 B. K. Driver Then 1. (d/dt) P E(tX h ) = A h b P + B h b (P) in B (b), and 2. (d/dt) E(tX h ) = X h in B (b). Proof. By Theorem 2.6, P E(tX h ) = P ϕ E (ty h ) b = I E (ty h ) b = u(t) b. This equation shows that the first assertion is a direct consequence of Theorem 4.4 and Lemma 2.1. The second assertion follows from the first assertion, since π P = id W (M), π (A P) =, and π B a (u) = ua so that (d/dt) E(tX h ) = (d/dt) π P E(tX h ) = P(h b) = X h First-Order Differential Operators We now wish to have X h act as a first-order differential operator on functions on W (M). We begin with the action of X h on smooth cylinder functions (see Definition 4.9 below) based on W (O(M)). We first need some more notation. Notation 4.7. Suppose that V is a finite-dimensional vector space, F: O(M) n V is a smooth function, and Q { < s 1 < s 2 < < s n 1} is a partition of [, 1]. 1. For each path u:[, 1] O(M) set F Q (u) F(u(s 1 ), u(s 2 ),...,u(s n )). 2. Suppose also that A:[, 1] so(d) and h:[, 1] T o M. Let F Q (u) A + h (d/dt) F Q (ue ta ) + (d/dt) F Q (e tb h (u)), where (ue ta )(s) u(s)e ta(s), and (e tb h (u))(s) e tb h(s) (u(s)). (We view A + k as a path from [, 1] so(d) T o M.) 3. Similarly, if C:[, 1] so(d) and k:[, 1] T o M, set F Q (u) A + h, C + k (d/dt) F Q (ueta ) C + k + (d/dt) F Q (etb h (u)) C + k. Remark 4.8. The notion of B p -differentiability used in Theorem 4.4 is very strong. In particular, with F as in the above notation and u(t) as in Theorem 4.4, (d/dt) F Q (u(t)) exists in L p (dµ) and (d/dt) F Q (u(t)) = F Q (I ) A h +h. (4.11) Definition 4.9 (Cylinder Functions). A function f : W (M) R is a smooth cylinder function if there is an integer n, a C -function F: O(M) n R, and a partition Q = { < s 1 < s 2 < < s n 1} of [, 1] such that f = F Q (P s1,...,p sn ) = F Q P a.s. (4.12) We denote the collection of smooth cylinder functions by FC.

17 Wiener Space Lie Brackets 195 Suppose that f : M n R is a smooth function, then f (σ ) f (σ s1,...,σ sn ) (4.13) is in FC. Indeed, f = F Q (P s1,...,p sn ) a.s. where F f π, i.e., F(u 1,...,u n ) = f (π(u 1 ),...,π(u n )). We call a cylinder function f as in (4.13) a restricted cylinder function and denote the collection of restricted cylinder functions by RFC. The integration by parts formula in the next proposition is a slight generalization of Theorem in [31]. For the special case that h is in the Cameron Martin space see Theorem 9.1 in [15]. Also see [22], [24], [34], [4], [41] and [44]. Proposition 4.1. Given f FC as in (4.12) and h V 2, then X h f F Q (P) (A h ) b + h b (4.14) is well defined. Moreover, if g FC, then where E ν (X h f g) = E ν ( f (X h ) g), (4.15) (X h ) g X h g + (z(y h ) b)g (4.16) and z(y h ) is defined in (3.1). We view X h and (X h ) as unbounded operators on L 2 (W (M), ν) each with FC as its domain. Proof. First suppose that h CV. Because of Corollary 4.6, (d/dt) f E(tX h ) exists and is given by the right-hand side of (4.14). Therefore, for such h, X h f is well defined. Moreover, since f E(tX h ) = F Q P ϕ E (ty h ) b = F Q I E (ty h ) b, we see by Theorem 4.4 that X h f = ( Y h (F Q I )) b = ((d/dt) F Q I E (ty h )) b, (4.17) where the derivative exists in L. Let G C c (O(M)n ) and g = G Q P FC. Now apply Theorem 3.4 with h replaced by Y h, f by F Q I, and g by G Q I to find E µ [( Y h (F Q I )) (G Q I )] = E µ ((F Q I ) { Y h (G Q I ) + z(y h ) (G Q I )}), which owing to (4.17) and the fact that Law(b) = µ shows that (4.15) holds. For general h V 2, choose h n CV such that h n h in H 2 as n. Then it is easy to check that L 2 lim X hn f = F n Q (P) (A h ) b + h b, (4.18)

18 196 B. K. Driver which shows that X h f defined in (4.14) is well defined. By (3.19) z(y h ) = 1 r Y h dβ is linear in h and by (3.22) E µ z(y h ) 2 = E µ 1 r Y h s 2 ds = r Y h 2 R 2 K h 2 H 2. Hence z(y h n ) z(y h ) in L 2 as n when h n h in H 2 as n. From this fact and (4.18), it is easy to conclude that (4.15) holds for general h V 2. We now define a closed extension X h of X h using the weak derivative formulation in the next definition. Definition Given h V 2 let X h ((X h ) ). Explicitly, f L 2 (ν) is in the domain D( X h ) of X h and X h f = k L 2 (ν) iff E ν (k g) = E ν ( f (X h ) g) for all g FC. It is reasonable to conjecture that X h is the closure of X h on RFC. We do not pursue this here. However, the following elementary lemma gives a sufficient condition for a function f on W (M) to be in the domain of X h. The proof is similar to that of Proposition 4.1. Lemma Suppose that f L 4+ (ν), and, for h CV,(d/dt) f E(tX h ) exists in L 4 (ν) and there is a constant K < such that (d/dt) f E(tX h ) 2 K h H 4, h CV. Then f h V D( X h ), X h f = L 2 (d/dt) 4 f E(tX h ) for all h CV, and for h V 4 we have X h f = lim n X hn f, where h n CV is any sequence such that h n hinh 4 as n. Proof. Let h CV, g FC, and put f f ϕ and g g ϕ. Since ϕ µ µ ϕ 1 = ν, the map f L p (ν) f L p (µ) is an isometric isomorphism with inverse f f = f b. Since f E (ty h ) = f E(tX h ) ϕ, it follows that Y h f (d/dt) f E (ty h ) = S ϕ, where S (d/dt) f E(tX h ). Hence, by Theorem 3.4, E µ ((S ϕ) g) = E µ ( f (Y h ) g) = E µ ( f ( Y h g + z(y h ) g)). (4.19) From the proof of Proposition 4.1, (Y h ) g = ((X h ) g) ϕ. Hence (4.19) shows that E ν [Sg] = E ν ( f ((X h ) g)). Since g FC was arbitrary, it follows that f D( X h ) and S = X h f. For general h V 4, choose h n CV such that h n h in H 4 as n. Using the above paragraph, we have for any g FC that E ν [ X hn f g] = E ν [ f (X hn ) g]. (4.2)

19 Wiener Space Lie Brackets 197 With S L 2 lim n X hn f, it follows by letting n in (4.2) that E ν [S g] = E ν [ f (X h ) g]. (4.21) Therefore, f D( X h ) and X h f = L 2 lim n X hn f. 5. Product of Two Vector Fields on Restricted Cylinder Functions Definition 5.1. An adapted vector field k on W is said to be p-smooth if (i) k H p+, (ii) for all h V B with r h bounded the derivative (d/dt) k E (th) =: h k exists in H p, and (iii) there is a constant C < such that h k H p/2 C h H p holds for all h V B. If k is p-smooth we may and do extend the definition of h k to all h V p by continuity. In what follows, to simplify notation we write Y h k for Y h k. Theorem 5.2. Suppose that k is 4-smooth, h V 4, f RFC is given as in (4.13), and F(u 1,...,u n ) = f (π(u 1 ),...,π(u n )). Then X k f D( X h ) and (using Notation 4.7) X h X k f = F Q (P) (Y h k)) b +F Q (P) ( A h +h) b, k b. (5.1) Proof. Lemma 3.7 implies that Y h V 4. By (4.14) and the assumption f RFC, X k f = F Q (P) k b. For the moment assume that h CV and consider (X k f ) E(tX h ) = F Q (P E(tXh )) k b E(tX h ) = F Q (P E(tXh )) k E (ty h ) b. Because F is a smooth function on the compact manifold O(M) n, the assumption that k is 4-smooth, and Corollary 4.6 one may show by standard arguments that (d/dt) (X k f ) E(tX h )= F Q (P) (Y h k) b +F Q (P) ( A h +h) b, k b, (5.2) where the derivative exists in L 4. It follows from this equation and the use of Hölder s and Burkholder s inequalities that there are constants K and K depending on the bounds on F Q and F Q such that (d/dt) (X k f ) E(tX h ) 2 K ( Y h k S 2 + k S 4{ h S 4 + A h S 4}) K ( Y h k H2 + k H 4 h H 4), where in the second inequality we have used Lemma 3.7 to bound A h S 4. Hence, using the assumption that k is 4-smooth, it follows that there is a constant C = C(k, f ) such that (d/dt) (X k f ) E(tX h ) 2 C h H 4 for all h CV. The theorem now follows by an application of Lemma 4.12.

20 198 B. K. Driver 6. Lie Bracket on Restricted Cylinder Functions Our goal in this section is to compute the commutator of two geometric vector fields, see Theorem 6.2 below. The following lemma is in preparation for this result. Lemma 6.1. Let A, C W (so(d)), u W (O(M)), and h, k W W (T o M)). Then in the notation of Notation 4.7, where F Q (u) A + h, C + k F Q (u) C + k, A + h = F Q (u) ([A, C] u h, k ) + (Ah Ck u h, k ), (6.1) ([A, C] u h, k )(s) [A(s), C(s)] u(s) h(s), k(s) is in W (so(d)) and is in W. (Ah Ck u h, k )(s) A(s)h(s) C(s)k(s) u(s) h(s), k(s) Proof. Forα so(d), let ˆα be the vertical vector field on O(M) defined by ˆα(u) = (d/dt) ue tα. For A W (so(d)) and h W (T o M) let Â(s i ) and B h(si ) denote the vector fields on O(M) n given by and (Â(s i )F)(u 1, u 2,...,u n ) (d/dt) F(u 1,...,u i 1, u i e ta(s i ), u i+1,...,u n ) (B h(si )F)(u 1, u 2,...,u n ) (d/dt) F(u 1,...,u i 1, e tb h(s i ) (u i ), u i+1,...,u n ), respectively. Also for u W (O(M)), let u Q (u(s 1 ), u(s 2 ),...,u(s n )). Then with this notation: F Q (u) A + h, C + k F Q (u) C + k, A + h n = ([Â(s i ) + B h(si ), Ĉ(s j ) + B k(sj )]F)(u Q ) = i, j=1 n ([Â(s i ) + B h(si ), Ĉ(s i ) + B k(si )]F)(u Q ). i=1 Using the commutator formulas in Lemma A.2 in the Appendix gives n [Â(s i ) + B h(si ), Ĉ(s i ) + B k(si )] i=1

21 Wiener Space Lie Brackets 199 = = n {[Â(s i ), Ĉ(s i )] + [B h(si ), Ĉ(s i )] + [Â(s i ), B k(si )] + [B h(si ), B k(si )]} i=1 n {[A, Cˆ](s i ) + B )Ĉ(s (Ah Ck)(si i) ˆ h(s i ), k(s i ) B h(si ),k(s i ) }. i=1 Combining the two above displayed equations proves (6.1). We are now ready to compute the commutator of two geometric vector fields. Theorem 6.2. Let h and k be 4-smooth adapted vector fields on W, let f RFC be as in (4.13), and let F(u 1,...,u n ) = f (π(u 1 ),...,π(u n )). Then [ X h, X k ] f = F (P) {Y h k Y k h + c h, k } b, (6.2) where c h, k is the process on W given by c h, k A h k + A k h I h, k. (6.3) Proof. By Theorem 5.2, [ X h, X k ] f = F (P) (Y h k Y k h) b +F (P) ( A h +h) b, k b F (P) ( A k +k) b, h b. Because of Lemma 6.1, this equation shows [ X h, X k ] f = F (P) (Y h k Y k h A h k + A k h I h, k I h, k ) b = F (P) (Y h k Y k h A h k + A k h I h, k ) b, where in the second equality we have used the assumption that f RFC, so that F (P) ( I h, k ) b =. 7. The Lie Bracket Preserves Adapted Vector Fields Lemma 7.1. Suppose that k is a 2p-smooth adapted vector field on W, then, for all h V B with r h bounded, h Q k := (d/dt) Q k (E (th)) and h r k := (d/dt) r k (E (th)) exist in R (2p) and there exists a constant C = C(k) (depending only on k and p) such that 1 ( h Q k ) s ds C h H 2p, (7.1) L p (µ) and 1 ( h r k ) s ds C h H 2p. (7.2) L p (µ)

22 2 B. K. Driver Because of the estimates in (7.1) and (7.2), we may extend by continuity the definitions of h Q k and h r k to all h V 2p. Then with this notation the kernels of h k (for all h V 2p ) are given by and Q hk = h Q k + Q k Q h (7.3) r hk = h r k + Q k r h. (7.4) Proof. Let h V B with r h bounded. To simplify notation, let dβ = dβ(s), w(t) E (th) = e tqh dβ + t r h ds, and dw(t) = d s w s (t) so that dw(t) e tqh dβ + r h ds. (Please note well that dw(t) is the differential of w(t) in the suppressed s variable not the t variable.) The assumption that k(w(t)) = Q k (w(t)) dw(t) + r k (w(t)) ds = Q k (w(t))e tqh dβ + {tq k (w(t))r h + r k (w(t))} ds is H 2p differentiable is equivalent to and Q(t) := Q k (w(t))e tqh r(t) := {tq k (w(t))r h + r k (w(t))} being R 2p differentiable. Using Lemma 4.5 and Hölder s inequality on the identity we find [Q k (w(t)) Q k ]/t = Q(t)(e tqh I )/t + [Q(t) Q()]/t h Q k = (d/dt) Q k (w(t)) = Q k Q h + Q() = Q k Q h + Q hk Similarly, (r k (w(t)) r k )/t = (r(t) tq k (w(t))r h r())/t = (r(t) r())/t Q k (w(t))r h ṙ() Q k r h in R (2p) as t, in R (2p). (7.5)

23 Wiener Space Lie Brackets 21 where in taking the limit as t we have used the continuity of t Q k (w(t)) in R (2p) at t = implied by (7.5) and the boundedness of r h. The above displayed equation is equivalent to h r k = (d/dt) r k (w(t)) = r hk Q k r h in R (2p) as t. (7.6) So for h V B with r h bounded we have proved the differentiability assertions of the lemma and identities in (7.3) and (7.4). To finish the proof it suffices to prove the estimates in (7.1) and (7.2). Using the definition of k being 2p-smooth and Hölder s inequality, it follows from (7.3) that L ( h Q k ) s ds (Q hk ) s ds + Q k s Qh s ds L p L p p wherein we have used Q hk R p + Q k R 2p Q h R 2p ( C(k) + Q k R 2p) h H 2p = C(k) h H 2p, Q hk R p h k H p C(k) h H 2p. Here C(k) is a finite constant guaranteed to exist because k is 2 p-smooth, see Definition 5.1. The estimate in (7.2) is proved similarly. Lemma 7.2. For h, k H 4, A h k H 2, Q A h k = I h, k + A h Q k (7.7) and ( r A h k = R h k + A h r k + i ) I h, e i Q k e i, (7.8) where R h 1 { 2 I Q h e i, e i +(B i ) I h, e i }. (7.9) i Proof. ByItô s lemma and (3.2), d(a h k) = I h, dβ k + R h kds + A h (Q k dβ + r k ds) + i I h, e i Q k e i ds = I h, dβ k + A h Q k dβ ( + R h k + A h r k + i ) I h, e i Q k e i ds which proves (7.7) and (7.8).

24 22 B. K. Driver To prove A h k H 2 notice: Q A h k R 2 I h, k R 2 + A h Q k R 2 C{ h S 4 k S 4 + A h S 4 Q k R 4} C h H 4 k H 4 <, where we have used Lemma 3.7 in the last inequality. One similarly shows that r A h k R 2 <. Lemma 7.3. For h, k H 4 such that either Q h or Q k in S 4, then I h, k H 2 and Q I h,k = I Q h, k + I h, Q k + I, h, k (7.1) and r I h,k = { I Q h e i, Q k e i +(B i ) I Q h e i, k +(B i ) I h, Q k e i } i + I r h, k + I h, r k + 1 (B 2 2 i ) I h, k, (7.11) where {e i } i=1 d is an orthonormal basis for T M. Proof. ByItô s lemma we have d( I h, k ) = I δh, k + I h,δk +(B δβ ) I h, k =: α + γ + κ. We now work on the three terms separately: α = I δh, k = I dh, k { I dh, dk +(B dβ ) I dh, k } d = I dh, k + 1 { 2 I Q h e i, Q k e i +(B i ) I Q h e i, k } ds so that i=1 α + β = I dh, k + I h,δk d {2 I Q h e i, Q k e i +(B i ) I Q h e i, k +(B i ) I h, Q k e i } ds i=1 Similarly, κ = (B dβ ) I h, k {(B dβ ) I dh, k +(B dβ ) I h, dk +(Bdβ 2 ) I h, k } = (B dβ ) I h, k d {(B i ) I Q h e i, k +(B i ) I h, Q k e i +(Bi 2 ) I h, k } ds i=1 i

25 Wiener Space Lie Brackets 23 and hence d( I h, k ) = ( I Q h dβ,k + I h, Q k dβ +(B dβ ) I h, k ) + ( I r h, k + I h, r k ) ds + 1 (B 2 2 i ) I h, k ds + { I Q h e i, Q k e i +(B i ) I Q h e i, k i + (B i ) I h, Q k e i } ds. This formula implies (7.1) and (7.11). The assertion that I h, k H 2 is easily verified using (7.1) and (7.11). The term I Q h e i, Q k e i in (7.11) is the one which requires the assumption that either Q h or Q k is in S 4. Theorem 7.4. Suppose that h and k are 4-smooth adapted vector fields on W and that either Q h or Q k is in S 4. Then {Y h k Y k h + c h, k } V 2 and [ X h, X k ] = X {Y h k Y k h+c h,k } on RFC, (7.12) where c h, k A h k + A k h I h, k as in (6.3). Proof. Once we know that J {Y h k Y k h + c h, k } V 2, (7.12) follows from (4.14) and (6.2). Using Lemma 3.7 and the assumption that h and k are 4-smooth, it follows that Y h k and Y k h are in H 2. By Lemmas 7.2 and 7.3 and the definition of c h, k, we see that c h, k H 2. Therefore J = Y h k Y k h + c h, k H 2 and it suffices to show that J V, i.e., Q J is an so(d)-valued process. From (7.3) and (3.18), Q Y hk = Y h Q k + Q k Q Y h = Y h Q k + Q k (A h + I h, + Q h ) and hence Q (Y h k Y kh) = (Y h Q k Y k Q h ) + [Q k, Q h ] + Q k A h Q h A k +Q k I h, Q h I k,. (7.13) By Lemmas 7.2 and 7.3, Q c h,k = ( I h, k I k, h) + (A h Q k A k Q h ) + I Q h, k + I h, Q k + I, h, k. (7.14) We write h I for I h, and note that h I is an so(d)-valued process because of the standing TSS assumption on the covariant derivative. Combining (7.13) and (7.14) gives Q J = (Y h Q k Y k Q h ) + [Q k, Q h ] + [Q k, A h ] [Q h, A k ] + [Q k, h I ] [Qh, k I ] ( I h, k I k, h + I, h, k ). i

26 24 B. K. Driver The first two lines in the above formula for Q J are clearly in so(d), so to finish the proof it suffices to show that Q I h, k + I k, h I, h, k = I, h k I, k h I, h, k is an so(d)-valued process, i.e., Qb, b = for all b T o M. However, this follows directly from Theorem A.4 in the Appendix with a = h and c = k. 8. Lie Bracket on General Cylinder Functions By taking limits one expects (7.12) to hold on a much larger class of functions than RFC. In this section, we outline how to show that (7.12) is valid on FC. For the sake of brevity only the algebraic aspects of the proofs are given; leaving the analytic details and even the precise statements of the theorems to the reader. The next theorem is the analogue of Theorem 5.2 which is stated without proof. Theorem 8.1. Suppose h, k V 4 are sufficiently smooth, and f FC is given as in (4.12). Then X k f D( X h ) and X h X k f = F (P) Y h ( A k +k) b + F (P) ( A h +h) b,( A k +k) b, where Y h ( A k +k) denotes the directional derivative of ( A k +k) by Y h. Combining this theorem with lemma 6.1 gives the following commutator formula on FC. Theorem 8.2. Let f FC be represented as in (4.12), and let h, k V 4 be sufficiently smooth, then [ X h, X k ] f = F (P) (Y h ( A k +k) Y k ( A h +h) + [A h, A k ] A h k + A k h I h, k I h, k ) b = F (P) (Y h k Y k h A h k + A k h I h, k ) b + F (P) ( Y h (A k ) + Y k (A h ) + [A h, A k ] I h, k ) b. The next lemma is the key identity which enables us to recognize the far right member in the above equation as X {Y h k Y k h+c h,k } f, see Corollary 8.4 below. Lemma 8.3. Again assuming that h, k V 4 are sufficiently smooth, Y h (A k ) + Y k (A h ) + [A h, A k ] I h, k = A Y h k Y k h A h k + A k h I h, k, (8.1)

27 Wiener Space Lie Brackets 25 where c h, k A h k + A k h I h, k as in (6.3). The desired commutator formula is now a direct consequence of Theorem 8.2, Lemma 8.3, and (4.14). Corollary 8.4. Let h, k V 4 be sufficiently smooth, then [ X h, X k ] = X {Y h k Y k h+c h,k } on FC, where as before c h, k A h k + A k h I h, k. Proof of Lemma 8.3. (Sketch! Again only the algebraic aspects of the proof are given here.) We start with the identity: ( ) Y h (A k ) = Y h I k,δβ = {(Y h I ) k,δβ + I Y h k,δβ + I k,δy h } = I A h +h, k,δβ +A Y h k + I k, A h δβ + I h,δβ +δh, where, for A so(d) and a, b, c T o M, and u A + a, b, c Â(u) b, c +B a (u) b, c Â(u) b, c (d/dt) ue ta b, c =(d/dt) Ad e ta u e ta b, e ta c = [A, u b, c ] + u Ab, c + u b, Ac. Therefore, I A h, k,δβ = [A h,δ(a k )] A A h k I k, A h δβ which combined with the first displayed equation in the proof gives Y h (A k = [A h,δ(a k )] + A Y h k A h k + I h, k,δβ + I k, I h,δβ +δh.

28 26 B. K. Driver Hence using δ( I h, k ) = I δh, k + I h,δk + I δβ, h, k we find Y h (A k ) Y k (A h ) = {[A h,δ(a k )] + [δ(a h ), A k ]} + A Y h k Y k h A h k + A k h + { I h, k,δβ + I k, I h,δβ + I k, h,δβ I h, I k,δβ } { I k,δh + I δk, h } = [A h, A k ] + A Y h k Y k h A h k + A k h + { I h, k,δβ + I k,δβ,h } + { I k, I h,δβ + I h, I δβ, k } I h, k + I δβ, h, k = [A h, A k ] + A Y h k Y k h A h k + A k h I h, k + {( I h, k,δβ + I k, I h,δβ ) + (cyclic)} I h, k + I δβ, I h, k, where in the last equality the term (cyclic) indicates there are two more terms obtained from the preceding term by performing cyclic permutation in h, k, and δβ. Combining this equation with the geometric identity in Lemma A.3 shows that Y h (A k ) Y k (A h ) = [A h, A k ] + A Y h k Y k h A h k + A k h I h, k + I δβ, I h, k, which is the same as (8.1) upon noting that I δβ, I h, k = A I h, k. Acknowledgments It is a pleasure to thank Ana-Bela Cruzeiro and Paul Malliavin for many useful discussions. I am grateful to the Mittag-Leffler Institute and Bernt Oksendal for the opportunity to participate in the Winter session of the Probability year where this work was stimulated.

29 Wiener Space Lie Brackets 27 A.ppendix. Geometric Identities Remark A.1. Let θ denote the T o M-valued 1-form on O(M) defined by θ ξ u = u 1 π ξ u, where ξ u T u O(M). Set dω + ω ω (A.1) and dθ + ω θ, (A.2) then u a, b = B a (u), B b (u) and u a, b = B a (u), B b (u). For A so(d), let  be the vertical vector field on O(M) defined by Â(u) = (d/dt) ue ta T u O(M) for all u O(M). The tangent vector Â(u) is also denoted by u A when convenient. We also write a, b and a, b for the functions u u a, b and u u a, b. Lemma A.2 (Commutator Formulas). Let a, b T o M and A, C so(d), then 1. [Â, B a ] = B Aa, 2. [Â, Ĉ] = [A, Cˆ], and 3. [B a, B b ](u) = ˆ u a, b B u a, b (u), where ˆ u a, b (d/dt) ue t u a,b T u O(M). We abbreviate this last equation as [B a, B b ] = ˆ a, b B a,b. (A.3) Proof. The proof may be found in [39]. The short proof is given here for the readers convenience. The proof of the first two assertions relies on the fact that e t  = R e ta, where R g (u) ug for u O(M) and g O(d). It is easy to verify that R e ta B a R e ta = B e ta a, and hence [Â, B a ] = (d/dt) B e ta a = B Aa. Similarly, [Â, Ĉ] = (d/dt) R e ta Ĉ R e ta = (d/dt) (d/ds) R e ta R e sc R e ta = (d/dt) (d/ds) R e ta e sc e ta = (d/dt) (d/ds) R e sad e ta C = (d/dt) (Ad e tacˆ) = ([A, C]ˆ). For the proof of the third item recall that all tangent vectors η u T u O(M)) can be written as u ω η u +B θ η u. Using the structure equation = dω + ω ω it is easy to conclude that ω η u = u a, b. Similarly, using the structure equation = dθ + ω θ, we show that θ η u = u a, b.

30 28 B. K. Driver Lemma A.3. For all a, b, c T o M and u O(M), ( u a, b, c + u u a, b, c ) + (cyclic permutations in a, b, c) =, (A.4) where u a, b, c (B a ) u b, c =(d/dt) e tba (u) b, c. Proof. The Bianchi identity states that (d ) H =. That is, = d B a, B b, B c =B a B b, B c [B a, B b ], B c +(cyclic). Because of (A.3) and the fact that annihilates vertical vectors, the above equation may be written as = B a B b, B c + B a, B b, B c, which, in view of Remark A.1, is equivalent to (A.4). Theorem A.4. Let a, b, c T o M, then b, a c b b, c a b B b a, c b =, (A.5) where B b B b and a b a, b. Proof. The identity (see [39]) d + ω = θ applied to B a, B b, and B c gives ={B a b, c [B a, B b ], B c a, b c}+cyclic{a, b, c} ={B a b, c + a, b, c a, b c}+cyclic{a, b, c}, (A.6) where (A.3) and the fact that annihilates vertical vectors was used in the second equality. Take the inner product of (A.6) with b and use the TSS assumption on to conclude that ={B a b, c + a, b, c a, b c} b +{B b c, a + b, c, a b, c a} b +{B c a, b + c, a, b c, a b} b = b, c a, b a, b c b + B b c, a b b, a b, c b, c a b + = a, b b c B b c, b a c, b b a = b, a c b B b a, c b b, c a b.

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