A class of integration by parts formulae in stochastic analysis I

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1 A class of inegraion by pars formulae in sochasic analysis I K. D. Elworhy and Xue-Mei Li Mahemaics Insiue Universiy of Warwick Covenry CV4 7AL,U.K. 1 Inroducion Consider a Sraonovich sochasic differenial equaion dx = X(x ) db + A(x )d (1) wih C coefficiens on a compac Riemannian manifold M, wih associaed differenial generaor A = 1 2 M + Z and soluion flow {ξ : of random smooh diffeomorphisms of M. Le ξ : M M be he induced map on he angen bundle of M obained by differeniaing ξ wih respec o he iniial poin. Using an observaion by A. halmaier we will exend he basic formula of [EL94] o obain ) EdF ( ξ (h )) = EF (ξ (x)) ξ s (ḣs, X (ξ s (x)) db s where F FCb (C x(m)), he space of smooh cylindrical funcions on he space C x (M) of coninuous pahs γ : [, ] M wih γ() = x, df is is derivaive, and h is a suiable adaped process wih sample pahs in he Cameron-Marin space L 2,1 ([, ]; x M). Se F x = σ{ξ s (x) : s. aking condiional expecaion wih respec o F x, formula (2) yields inegraion by pars formulae on C x (M) of he form where V h is he vecor field on C x (M) (2) EdF (γ)( V h ) = EF (γ)δv h (γ) (3) V h (γ) = E { ξ (h ) ξ (x) = γ 1

2 and δv h : C x (M) R is given by { δv h (γ) = E < ξ s (ḣs), X(ξ s (x))db s > ξ (x) = γ. When h is adaped o F x resuls from [ELJL95] exending [EY93] give explici expressions for V h and δ V h in erms of he Ricci curvaure of he LeJan-Waanabe connecion associaed o (1). Equaion (3) hen reduces o a Driver s inegraion by pars formula, heorem 3.3 below, bu no hypohesis of orsion skew symmery of he connecion is required: he inegraion by pars formulae follow for he adjoin of any meric connecion. In paricular for any such connecion here is a Hilber angen space of good direcions obained by parallel ranslaion of he Cameron-Marin space of pahs in x M. (In fac i is he Ricci flow or Dohrn-Guerra parallel ranslaion (see Nelson [Nel84]), leading o he damped gradien ([FM93]) which occurs more naurally). However, in Remark 2.4, we show ha in his case V h is in he class for which inegraion by pars formulae are known, so ha he resuls of 2.3, 3.3, 3.5 are no claimed o be new in subsance. Alhough his filering ou of he exraneous noise gives inrinsic resuls comparable o hose of Driver [Dri92], his viewpoin hrows away a lo of he srucure we have. Moreover inegraion by pars formulae such as (2) should have some connecion wih quasi-invariance properies of flows associaed o he vecor fields. Flows for he V h on C x (M) do no appear o be easy o analyse in general. However in 3 we show ha in he conex of Diff M valued processes here are very naural flows associaed and (2) has a raher naural geomeric inerpreaion. his leads o anoher elemenary proof of (2) and in heorem 4.1 we use his mehod o obain inegraion by pars formulae for he free pah space. here are a leas 3 proofs of (2). he firs given here is via Iô s formula and elemenary maringale calculus (i requires F o be cylindrical), he second given here is based on he Girsanov-Maruyama heorem (and works for more general F ), and a hird mehod would be o deduce i from he sandard inegraion by pars formula on Wiener space applied o he funcional F ξ. Indeed his work was simulaed by D. Bell and D. Nualar poining ou ha his hird approach could be used o deduce he basic formula of [EL94]. he poin made (and carried ou) in [Elw92] and [EL94] ha he firs approach can be applied direcly o Ricci flows insead of derivaive flows o give inrinsic formulae wihou sochasic flows, also needs o be emphasized: see also [SZ]. 2

3 here are also now many proofs of Driver s resuls for C x (M) and for he free pah space and heir exensions. See [Hsu95], [ES95], [LN] (wih a very concise proof), [AM], [Aid], and [CM]. Acknowledgmen: his research was suppored by SERC gran GR/H67263 and simulaed and helped by our conacs wih A. halmaier. 2 he inegraion by pars formula from finie dimensional manifolds o pah spaces In his secion we deduce by inducion an inegraion by pars formula on he pah space from a formula on he base manifold M. he key is o obain formula (1) for M. Le h : Ω [, ] x M be an adaped process wih h(ω) : [, ] x M in L 2,1 for almos all ω. Lemma 2.1 If h : Ω [, ] x M is adaped, L 2,1 for a.s. ω and ( 1/2 ds) ḣs 2 L 1+ɛ for some ɛ >. hen for <, { E < ξ s(ḣs), X(ξ s (x))db s > ξ (x) { (4) = E < ξ s ( ), X(ξ s (x))db s > h h ξ (x). If furhermore h is non-random hen for, { E < ξ s(ḣs), X(ξ s (x))db s > ξ (x) { = E < ξ s( ), X(ξ s (x))db s > ( h h ) ξ (x). Proof. Firs by he Burkholder-Davis-Gundy inequaliy, for some consan c 1, E ( < ξ s (ḣs), X(ξ s (x))db s > c 1E c 1 (E sup x ξ s 1+ɛ ɛ s ) [ ɛ ( 1+ɛ E ḣs 2 ds ) 1+ɛ 2 ) 1 ξ s (ḣs) 2 2 ds ] 1 1+ɛ his is finie since sup s x ξ s L q for all 1 q <, e.g. see [Li94]. Moreover, since he adaped processes in L (Ω, F, P; C 1 ([, ]; x M)) are 3. (5)

4 dense in he subspace of adaped processes in L 1+ɛ (Ω, F, P; L 2,1 ([, ]; x M)), his esimae allows us o assume ha h belongs o he former space. Se M = < xξ s ( ), X(ξ s (x))db s >. hen {M is a x M valued local maringale. If = < 1 <... < l = is a pariion of [, ], j = j+1 j, and j M = M j+1 M j, hen l 1 j M(ḣ j ) ḣ s dm s = < ξ s (ḣs), X(ξ s (x))db s > (6) and he convergence is in L 1. On he oher hand if v x M and P is he probabilisic semigroup associaed o he S.D.E. and f a bounded measurable funcion hen d(p f)(v ) = 1 Ef(ξ (x)) ξ s (v ), X(ξ s (x))db s. (7) See [EL94]. However by an observaion of halmaier: he same proof shows ha for any r, h [, ] wih h > and r + h d(p f)(v ) = 1 h Ef(ξ (x)) { 1 { 1 r+h h r r+h r ξ s (v ), X(ξ s (x))db s c.f. [SZ]. From hese wo formulae we obain: E < ξ s(v ), X(ξ s (x))db s > ξ (x) = E < ξ s (v ), X(ξ s (x))db s > ξ (x). (8) For any r, le {ξ r s(x) : r s, x M be he soluion flow o (1) saring from x a ime r. he flow ξ r can be aken o be adaped o a filraion {F r s : r s independen of F r, and hen we have ξ r sξ r = ξ s, almos surely, r s. From his, ime homogeneiy, and (8), E { l 1 = E = E j M(ḣ j ) ξ (x) { l 1 { l 1 j 1 j+1 ( )) ξ j s ξ j ), X (ḣj ( ( ξ j s ξj (x) )) ξ db j s j (ξ j (x)) j 1 j ( ) ξ j s ξ j (ḣ j ), X ( ( ξ j s ξj (x) )) ξ db j s (ξ j (x)) 4

5 { l 1 1 = E j { E < ξ s (ḣ j ), X(ξ s (x))db s > ξ (x) < ξ s ( ), X(ξ s (x))db s > h h ξ (x). Comparing wih (6) his gives he firs required ideniy. When h is non-random he second follows immediaely from (8). Remark: As in [SZ] a furher modificaion is possible replacing (8) by: { 1 E 1 = Ψ(r)dr E < ξ s (v ), X(ξ s (x))db s > ξ (x) { Ψ(s) < ξ s (v ), X(ξ s (x))db s > ξ (x) for Ψ : [, ] R inegrable wih Ψ(r)dr. he argumen leads o, for non-random h, { E < ξ s(ḣs), X(ξ s (x))db s > ξ (x) { ( ) = E Ψ(s) < ξ (9) h s( ), X(ξ s (x))db s > R h ξ Ψ(r)dr (x). Corollary 2.2 Under he condiions of he lemma, for any C 1 funcion f : M R, Ef (ξ (x)) < ξ s (ḣs), X(ξ s (x))db s >= Edf ( ξ (h h )). (1) Proof. Firs by he composiion propery of soluion flows, { E { = E < ξ s ( ), X(ξ s (x))db s > h h ξ (x) < ξ s( ), X(ξ s (ξ (x)))db s > ξ (h h ) ξ (ξ (x)). 5

6 As in he proof of he lemma, (4) yields Ef(ξ (x)) < ξ s (ḣs), X(ξ s (x))db s > = Ef(ξ (ξ (x)) ξ s ( ), X(ξs(ξ ξ (h h ) (x))db s = E {dp (f) ( ξ (h h )) by [EL94], since F is independen of F. Now le increase o and he required resul follows. Nex consider a cylindrical funcion F on C x (M), he space of coninuous pahs wih base poin x. Wrie F (γ ) = f(γ 1,..., γ k ), for ( 1,..., k ) [, ] k, γ C x (M) and f a smooh funcion on M k. Suppose h = and consider he angen vecor field V h (ξ (x)) along {ξ (x) : on C x (M) given by V h (ξ ) = x ξ (h ). hen df (V h (ξ )) = k ( ) d j f ξ V h (ξ ) j. (11) Here ξ = (ξ 1,..., ξ k ) and d j f is he parial derivaive of f in he jh direcion. Le δv h (ξ ) = < x ξ s (ḣs), X(ξ s (x))db s >. heorem 2.3 Le h : [, ] Ω x M be an adaped sochasic process ( ) 1+ɛ wih almos surely all h(ω) L 2,1 and E ḣs 2 2 ds < for some ɛ >. hen EdF (V h (ξ )) = EF (ξ (x))δv h (ξ ). (12) Proof. We prove by inducion on k. When k = 1, his is jus (1), he formula for funcions. Le Ω = C ([, ]; R n ) be he canonical probabiliy space. We se Ω 1 = C ([, 1 ]; R n ) and Ω 2 = C ([ 1, ]; R n ). here is hen he sandard decomposiion of filered spaces {Ω, F, F,, P = {Ω 1, F, F, 1, P {Ω 2, F, F 1, 1, P 6

7 in he sense ha F = F Ω 2 if 1, and F = F 1 F 1 if 1. As before le ξ 1 (y ), 1, y M be he soluion flow o (1) saring a ime 1, i.e. ξ 1 1 (y ) = y. We will consider i as a funcion of ω 2 Ω 2, adaped o F 1, while {ξ : 1 will be considered on Ω 1, and {ξ : 1 on Ω 1 Ω 2 = Ω. he composiion propery for flows gives ξ 1 (ξ 1 (x, ω 1 ), ω 2 ) = ξ (x, (ω 1, ω 2 )), each 1, a.s. Assume he required resul holds for cylindrical funcions depending on k 1 imes, some k {2, ake y M and define f y 1 : M k 1 R and F y 1 : Ω 2 R by: f y 1 (x 1,..., x k 1 ) = f(y, x 1,..., x k 1 ) and F y 1 (ω 2 ) = f(y, ξ 1 2 (y, ω 2 ),..., ξ 1 k (y, ω 2 )). ( ) 1+ɛ ake h 1 : Ω 2 L 2,1 ([ 1, ]; y M), adaped o F 1, and wih E ḣ1 1 s 2 2 ds finie. By ime homogeneiy our inducive hypohesis gives k j=2 Ω 2 d j f ( y, ξ 1 2 (y, ω 2 ),..., ξ 1 k (y, ω 2 ) ) ( ) ξ 1 j (h 1 j (ω 2 ), ω 2 ) dp (ω 2 ) = Ω 2 f ( y, ξ 1 2 (y, ω 2 ),..., ξ 1 k (y, ω 2 ) ) 1 ξ 1 r (ḣ1 r(ω 2 ), ω 2 ), X(ξ 1 r (y, ω 2 ))db r (ω 2 ) dp (ω 2 ). (13) Now for ω 1 Ω 1 (ouside of a cerain measure zero se) we can ake y = ξ 1 (x, ω 1 ) and h 1 (ω 2 ) = ξ 1 (h (ω 1, ω 2 ) h 1 (ω 1 ), ω 1 ). hen, for almos all ω 1 Ω 1, we have h 1 adaped o F 1. Subsiue his in (13). Using he composiion propery, and hen inegraing over Ω 1 yields k j=2 Edj f(ξ ) ( ξ j (h j h 1 ) ) = Ef(ξ (x)) (14) 1 ξ r (ḣr), X(ξ r (x))db r. On he oher hand we can define g : M R 1 by g(x) = f ( x, ξ 1 2 (x, ω 2 ),..., ξ 1 k (x, ω 2 ) ) Ω 2 and apply formula (1) o g o obain: 7

8 dg( ξ 1 (h 1 ))dp (ω 1 ) = g(ξ 1 (x)) Ω 1 Ω 1 Bu noe ha and herefore Ω 1 dg( ξ 1 (h 1 ))dp (ω 1 ) = 1 ξ r (ḣr)), X(ξ r (x ))db r dp (ω 1 ). k Ed k f ξ ( ξ j (h 1 ))dp (ω 1 ), k 1 Ed j f ξ ( ξ j (h 1 )) = Ef(ξ ) ξ r (ḣr), X(ξ r (x))db r, Adding (14) we arrive a (12): (15) k Ed j f ξ ( ξ j (h j )) = Ef(ξ (x)) ξ r (ḣr), X(ξ r (x))db r. B. Le be a meric connecion for he manifold M wih orsion, and is adjoin connecion defined by V 1 V 2 = V1 V 2 (V 1, V 2 ). Here V 1, V 2 are vecor fields. Le R be he curvaure ensor of and define Ric # : M M by Ric # (v) = race R(v, ). If {x s is a diffusion on M wih generaor 1race grad + L 2 Z denoe by // s he parallel ranspor along {x s, and { B s : s he maringale par of he ani-developmen of {x s : s using // s, a Brownian moion on x M. Le v s = W s Z (v ) be he soluion o D s v s = 1 Ric 2 # (v s ) + Z(v s ) saring from v x M. Here D denoes he covarian differeniaion along he pahs of {x using he adjoin connecion. We will show ha (12) implies Driver s inegraion by pars formula. However we do no need o assume (or equivalenly ) is orsion skew symmeric. Corollary 2.4 Le F be a cylindrical funcion on C x (M). Suppose h : [, ] Ω x M is adaped o he filraion of {x s : s < and such 8

9 ha h(ω) is in L 2,1 for almos all ω and h L ( 1+ɛ Ω, F, P; L 2,1 ([, ]; x M) ) for some ɛ >. hen EdF ( W Z (h )) = EF (ξ (x )) < W Z s (ḣs), // s d B s >. (16) When is meric for some Riemannian meric on M, i suffices o have h L 1 ( Ω, F, P; L 2,1 ([, ]) ). Proof. By a resul of [ELJL95] we can choose X such ha equals he Le Jan-Waanabe connecion induced from he sochasic differenial equaion dx = X(x ) db + Z(x )d and he soluion flow {ξ (x) has generaor 1race grad + L 2 Z (c.f. Corollary 3.4 of [ELJL95]). Moreover he condiioned process of he derivaive flow ξ (v ) wih respec o he naural filraion of {ξ (x ) is given by { W Z (v ): E{ ξ (v ) F x = W Z (v ), by heorem 3.2 of [ELJL95] exending [EY93]. he resul follows since B equals // 1 s X(ξ s(x ))db s. If is meric for some Riemannian meric hen sup s W s Z is in L (Ω, F, P) and so he Burkholder-Davis-Gundy inequaliy used as in he proof of Lemma 2.1 allows us o ake ɛ =. Remarks 2.5. (i). Le S : M M M be a ensor fields of ype (1,2), and le refer o he Levi-Civia connecion of M. hen, by [KN69] p.146, a connecion can be defined by V1 (V 2 ) = V1 (V 2 ) + S(V 1, V 2 ) for vecor fields V 1, V 2. and all linear connecions on M can be obained his way. I is easy o see ha is meric if and only if < S(W, U), V >= < U, S(W, V ) > for all vecor fields U, V, W, i.e. if and only if S(W, ) is skew symmeric. On he oher hand he adjoin connecion is given by V 1 (V 2 ) = V1 (V 2 ) + S(V 2, V 1 ) so ha i is orsion skew symmeric if also S(, W ) is skew symmeric. In erms of he Levi-Civia connecion our vecor fields V h for which he inegraion by pars formula hold herefore saisfy an equaion of he form 9

10 D V h = S( V h, dx ) + Λ ( V h )d + W h (ḣ)d + A( v h )d where Λ is linear (also depending on S). In paricular hey are angen processes in he sense proposed by Driver, for which inegraion by pars formulae are known: see [Dri95b], [CM], [AM], and [Aid], [Dri95a]. (ii) For cylinder funcions depending on one ime only such inegraion by pars formulae go back o Bismu [Bis84]. 3 Geomeric inepreaion and a shorer proof A. he processes x ξ (h ) canno sricly speaking be considered as angen vecors or vecor fields on C x (M). In some sense hey form angen vecors a ξ (x, ) o he space of processes (or semi-maringales) [, ] Ω M since x ξ (h (ω), ω) ξ(x,ω)m for (, ω) [, ] Ω or equivalenly as angen vecors o he space of random variables Ω C x (M) a ω ξ (x, ω). However c.f. [Dri92] here is sill no naural associaed flow. In fac he mos naural inerpreaion akes ino accoun he variable x and replaces C x (M) by P id DiffM he space of pahs on he diffeomorphism group of M, as we now describe. Le DiffM be he space of C diffeomorphisms of M. We can consider i wih a raher formal differenial srucure or if he reader prefers i can be replaced by a suiable Sobolev space of diffeomorphisms, o give a Hilber manifold (as in [Elw82] following [EM7]). In any case he angen space α (DiffM) will be idenified wih all vecor fields on M over α i.e. smooh v : M M such ha v(x) α(x) M for all x M. If P DiffM refers o coninuous pahs φ : [, ] DiffM wih φ() = id M hen φ P DiffM will be idenified wih coninuous v : [, ] DiffM vanishing a =, such ha v() φ() DiffM, or equivalenly v : [, ] M M wih v()(x) φ()(x) M. B. Given our S.D.E. (1) now ake h L 2,1 ([, ]; R n ). here is X h, he ime dependen vecor field X( )(h ) on M. From his we obain a field U h on P DiffM by U h (φ) (x) = x φ (X(x)h ). (17) 1

11 his is jus he lef invarian vecor field on P DiffM corresponding o X h e P DiffM for e() = id M,. For each le H τ : M M, τ R be he soluion flow o he vecor field X( )(h ) so { τ Hτ (x) = X(H τ (x))h H (18) (x) = x. Lemma 3.1 he vecor field U h on P DiffM has soluion flow Φ τ : P DiffM P DiffM, τ R given by Φ τ (φ) (x) = φ (H τ (x)). Proof. By lef invariance we can suppose φ = e. We hen need only o observe ha τ Hτ (x) = H τ (X(x)h ) for each : a sandard propery of ordinary, ime-independen dynamical sysems which is seen by differeniaing he ideniy H τ+σ = H τ H σ (x) wih respec o σ a σ =. C. In he case where h is random, wih h : Ω L 2,1 ([, ]; R d ) adaped, we can use he same noaion o obain a variaion of our sochasic flow {ξ : on M generaed by he vecor field V h, and given explicily by = Φ τ (ξ ), i.e. In paricular ξ τ ξ τ (x) = ξ (H τ (x)). (19) τ ξτ (x) τ= = ξ (X(x)h ). (2) Using he srucure of C x (M) as a C Banach manifold le BC 1 (C x (M)) be he space of C 1 maps F : C x (M) R such ha here is a consan df wih df (v) df sup v (21) for all angen vecors v : [, ] M o C x (M). Se V X(h) (x) = ξ (X(x)(h )), which gives rise o a vecor field along {ξ (x) on C x (M). 11

12 Proposiion 3.2 Suppose h : [, ] Ω x M is adaped, belongs o L 2,1 ( ) 1+ɛ a.s. and such ha E ḣs 2 2 ds < for some ɛ >. hen for each x M he processes ξ τ (x), τ R have muually equivalenly laws P x τ, τ R on C x (M) wih dp x τ dp { = exp ( ) < X(ξs τ (x)) ξ s s Hτ s (x), db s > 1 ( ) ξ s 2 s Hτ s (x) 2 ds. Moreover, for any F BC 1 (C x (M)), EdF (V X(h) ) = EF (ξ ) X(ξ s (x))db s, V X(ḣ) s (x)). Proof. For he equivalen par noe ha {ξ τ : saisfies he equaion: ( ) dξ τ (x) = X (ξ τ (x)) db + A(ξ τ (x))d + ξ Hτ (x) d. A sraighforward argumen shows ha herefore if we se M τ = ( ) 2 X(ξτ s (x)) ξ s s Hτ s (x) <, a.s. ( ) X(ξs τ (x)) ξ s s Hτ s (x), db s, hen by he Girsanov-Maruyama heorem, P x τ is equivalen o P x and dp x τ dp = e M τ 1 2 <M>τ. (22) Consequenly, EF (ξ τ (x)) = EF (ξ (x)) dpx τ dp Now suppose h and ḣs 2 ds are bounded on [, ] Ω. Differeniaing wih respec o τ a τ = and using (18) gives EdF ( ξ (X(x)h )) = EF (ξ (x)) τ. ( dp x τ dp ), τ= 12

13 since df is bounded and sup s ξ s 1 p L p. he ( second ) saemen follows from differeniaion of (22), using he fac ha dp x τ = 1 and Hτ (x) τ= = : τ dp ( dp x τ dp τ= ) τ= = = = = ( dp x τ dp ) τ= [( ) τ M τ 1 ( ) τ= 2 τ M τ 2 [ ( )] ξ s s Hτ s (x) X(ξs τ (x)db s, D τ τ= X(ξ s (x))db s, ξ s ( D s X(Hτ s (x))h s ) τ= X(ξ s (x))db s, ξ s (X(x)ḣs). For general h ake a sequence of bounded h n which converges o h in L 1+ɛ 2 (Ω, L 2,1 ([, ])) o finish he proof. See he proof of heorem 4.1. he following is an analogue of Corollary 2.4: here is any meric connecion and W Z is as in Corollary 2.4, heorem 3.3 Le F BC 1 (C x (M)) and h(ω) L 2,1 ([, ]; R n ) a.s.. Suppose h is adaped o he filraion of {F x ( ) 1+ɛ and such ha E ḣs 2 2 ds < for some ɛ >. hen EdF ( W Z (h )) = EF (ξ (x)) τ= < W Z s (ḣs), // s d B s >. (23) If is meric for some Riemannian meric, we can ake ɛ =. 4 Inegraion by pars for he free pah space I is easy o modify he proof of Proposiion 3.2 o he case where h() and so obain an inegraion by pars formula for he free pah space P M = x M P x M wih uniform opology and measure given by he Riemannian measure of M ogeher wih he laws of {ξ (x) : x M. In fac i is sraighforward o generalize o he case of an x-dependen h. For his le C 1 ( M) be he space of C 1 vecor fields on M wih is usual opology: heorem 4.1 Le h : [, ] Ω C 1 ( M) be a cadlag adaped process such ha he x M valued process h (x) has sample pahs in L 2,1 ([, ]; x M) 13 ]

14 for each x M wih h ( ) + ḣs( ) 2 ds in L 1+ɛ (Ω M; R) for some ɛ >. Le F be in BC 1 (P M; R). hen E M df ( xξ (h (ω)(x))) dx = E M F (ξ (x)) { divh (x) + ξ s (ḣs(x)), X(ξ s (x))db s dx. (24) Proof. Proceed as for Proposiion 3.2 bu wih X(x)h replaced by h (x). In paricular he definiion (6) of H τ becomes τ Hτ (x) = h (H τ (x)) H (x) = x. while ξ τ is defined by (19). However now ξ τ (x) = ξ (H τ (x)): he saring poin is ranspored by he flow of h (x). We firs assume h and ḣs 2 ds are bounded on Ω M. hen he Girsanov-Maruyama heorem gives us equivalence beween he measures Pτ x and P Hτ (x) wih EF (ξ τ (x)) dx = EF (ξ (H τ dp x τ (x))) dx. M M dp Hτ (x) On differeniaing his here is he exra erm ( df ξ ( ) M τ Hτ (x) ) τ= = df ( x ξ (h (x))) dx M = d x (F ξ ) (h (x)) dx M where d x (F ξ ) refers o he derivaive in M of F ξ : M Ω R. Now apply he classical Sokes heorem on M o ge: E = E M M df ( x ξ (h (ω)(x)))dx { F (ξ (x)) divh (x) + < x ξ s (ḣs(x)), X(ξ s (x))db s > dx. 14

15 For general h le τ R be he firs exi ime of h C 1 + h s(x) 2 ds from [, R). Se h R (x) = h τr (x)χ { h C 1 <R. We have: E M df ( x ξ (h R (ω)(x)))dx = Eχ { h C 1 <R F (ξ (x)) M { τr divh (x) + < x ξ s (ḣs(x)), X(ξ s (x))db s > Now le R. he lef hand side converges o E df ( ξ (h (ω)(x)))dx M since df ( ξ (h R (ω)( ))) c sup ξ (ω) sup h (, ω) and sup x E ( sup ξ M sup h (x, ω) dx ) < from sup h (x) h (ω) + ḣs(ω) ds dx. h (ω) + ḣs(ω) 2 ds L 1+ɛ (Ω M) Using Burkholder-Davies-Gundy inequaliy o jusify he inegraion on he righ hand side we see ha i converges o he righ hand side of (24). Jus as before he inrinsic formulae can be deduced using [ELJL95]: heorem 4.2 Le F be in BC 1 (P M; R) and h be as in heorem 4.1 bu wih h (x) adaped o he filraion of {F x, and divh L 1 (Ω M, R). hen for any meric connecion on M, E ( ) df M W Z (h (ω)(x)) dx = E { F (ξ (x)) divh M (x) + W Z s (ḣs(x)), // s d B (25) s dx. If furhermore is meric wih respec o a Riemannian meric, we can ake ɛ =. Proof. he proof is jus as ha of heorem

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