Mathematische Annalen

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1 Mah. Ann. 39, (997) Mahemaiche Annalen c Springer-Verlag 997 Inegraion by par in loop pace Elon P. Hu Deparmen of Mahemaic, Norhweern Univeriy, Evanon, IL 628, USA ( elon@@mah.nwu.edu) Received: 28 April 996 / Revied verion: 6 Sepember 996 Mahemaic Subjec Claificaion (99): 6D58, 28D5. Inroducion We aume hroughou hi paper ha M i a n-dimenional compac Riemannian manifold and O(M ) i orhonormal frame bundle. We ue H o denoe he R d - valued Cameron-Marin pace over he inerval [, ] wih zero iniial value and H he ubpace of H wih zero value a. We fix a poin o M and a frame u o O(M ) over o. WeueW o (M) o denoe he e of M -valued pah (of ime lengh ) aring from o and L o (M ) he e of loop a o, i.e., he e of pah γ in W o (M ) uch ha γ() = o. The Levi-Civia connecion deermine a Laplace-Belrami operaor on M.Weueνo denoe he Wiener meaure on W o (M ) generaed by /2. The meaure ν o defined inuiively by ν o ( ) =ν( ω() = o) i a meaure on he loop pace L o (M ), which we call he Wiener meaure on L o (M ). For a mooh or a ypical Brownian γ W o (M )orl o (M), le U (γ) behe horizonal lif of γ uch ha U (γ) o = u o. Fix an h H (or H ), he vecor field D h on W o (M ) (or L o (M )) i defined by () D h (γ) = U (γ) h. There i a complee heory of inegraion by par for D h on W o (M ), developed by Driver[] and upplemened by Hu[5]. See alo Enchev and Sroock[3] for The reearch wa uppored in par by he NSF gran DMS

2 332 E. P. Hu anoher approach. In he cae of he loop pace L o (M ), Driver[2] proved an inegraion by par formula for vecor field D h wih lipchizian h and he complee reul for all Cameron-Marin vecor field wa proved in Enchev and Sroock[4]. The purpoe of hi paper i o give an alernaive approach o inegraion by par in loop pace. Armed wih an upper eimae on he 2 log p(, x, y) due o Sheu[8] (ee (7) below), we prove an inegraion by par formula in he loop pace L o (M ) hrough he correponding formula for he pah pace W o (M ). Such an approach avoid he quai-invariance of he Wiener meaure in he loop pace hu providing a more direc roue o he reul. 2. Inegraion by par in pah pace Le µ be he uual Wiener meaure on he pah W o (R n ). Le {U } be he oluion of he ochaic differenial equaion on O(M ) (2) du = H U dω, U = u o. Here H = {H i, i =,...,d} are he canonical horizonal vecor field on O(M ) and {ω } i he coordinae proce on W o (R n ). Le γ = π(u ) be he projecion of U in W o (M ). The Iô map J : W o (R n ) W o (M ) i defined by J ω = γ. I i well known ha he law of γ i ν, he Wiener meaure on W o (M ), i.e., J carrie he Wiener meaure µ on W o (R d ) o he Wiener meaure ν on W o (M ). The invere J : W o (M ) W o (R d ) i he ochaic developmen map. A funcion F : W o (M ) R i called cylindrical if here i a poiive ineger l, a e of l poin < < l and a mooh funcion f : M M R uch ha (3) F(γ) =f ( γ,,γ l ). The e of cylindrical funcion on W o (M ) i denoed by C. We will ue L 2 (ν) o denoe he Hilber pace of meaurable funcion F on W o (M ) uch ha F 2 L 2 (ν) = F(γ) 2 Eν(dγ) <. W o(m ) The inner produc on L 2 (ν) i denoed by (, ) L 2 (ν) or imply (, ). Le F C be given by (3). From he definion of he vecor field D h in () i i naural o define (4) D h F(γ) = l (p) F(γ), U (γ) p h p, p= where (p) F denoe he gradien of f wih repec o he ph variable. Le h H, define

3 Inegraion by par in loop pace 333 l h (γ) = ḣ + 2 Ric U h, dω, where ω = J γ, U = U (γ) i he horizonal lif of γ o O(M ), and Ric u : R n R n i he Ricci ranform a u O(M ). Theorem 2.. (Inegraion by par in pah pace) Le F, G be wo cylindrical funcion on W o (M ). Then (5) where (D h F, G) = ( F, D h G ), D h = D h + l h. The aumpion ha h H implie ha here exi a conan c > uch c lh 2 ha E ν e <. By a andard funcional analyi argumen, he inegraion by par formula implie ha D h i cloable and he adjoin Dh i denely defined (he cloabiliy of D h require only l h L 2 (ν)). There are pleny of funcion in Dom (Dh ). More preciely, we have he following reul. Le L2+ (ν) = p>2 Lp (ν). Theorem 2.2. Le h H. Then D h : C L 2 (ν) i cloable in L 2 (ν) and ha a denely defined adjoin Dh. Furheremore, Dom (D h ) L 2+ (ν) Dom ( Dh ) and for all G Dom (D h ) L 2+ (ν) we have D h G = D h G + l h G. 3. Some preliminary reul In hi ecion we collec ome reul which will be ued in he proof of inegraion by par formula on he loop pace in he nex ecion. We denoe by p(, x, y) he hea kernel of he half Laplacian /2 onm. Propoiion 3.. There exi a conan depending only on M uch ha for all (, x, y) (, ) M M, { d(x,y) log p(, x, y) C + } (6) (7) { d(x,y) 2 2 log p(, x, y) C 2 + Proof. A far a we know, hee reul are due o Sheu[8]. See alo Hu[7] and Sroock and Trubeky[9] for furher dicuion. }.

4 334 E. P. Hu Lemma 3.2. For each poiive ineger N here i a conan C N depending only on N and M uch ha { E νo d(γ, o) N C N min N /2, ( ) N /2}. Proof. Thi inequaliy i inuiively clear and can be proved baed on he eimae (6). See Driver[2] or Hu[6] for deail. Lemma 3.3. (Hardy inequaliy) Le h H, hen 2 h d 4 ḣ 2 d. Proof. We have for any (, ), 2 h [ ] d = h 2 d h ḣ = 2 d + h 2 2 h 2 d +2 ḣ 2 d + h 2. In he la ep we have ued inequaliy 2ab 2 a2 +2b 2. Therefore 2 h d 4 ḣ 2 d + 2 h 2. The deired inequaliy follow by leing in he above inequaliy becaue h 2 = 2 ḣ d ḣ 2 d. 4. Inegraion by par on loop pace Recall ha in pah pace W o (M ) he adjoin of D h i given by (8) D h = D h + l h, where l h : W o (M ) R n i defined by l h (γ) = ḣ + 2 Ric U h, dω. Here U i he horizonal lif of γ and ω = J γ i he ochaic developmen of γ. On he loop pace L o (M ), we define D h F for a cylindrical funcion by

5 Inegraion by par in loop pace 335 he ame formula (4) a in he pah pace. The nex propoiion how ha l h i well defined under he meaure ν o. Thi ep i neceary becaue ν and ν o are muually ingular. Le {l h, } be he ν-maringale l h, = ḣ τ + 2 Ric U τ h τ, dω τ. Le {B, } be he andard filraion of σ-field on W o (M ). Then he meaure ν o and ν are muually aboluely coninuou on B for all <. Hence he proce {l h,, < } i well defined under he meaure ν o. The nex lemma concern he limi of l h, a under he meaure ν o. Propoiion 4.. The limi l h, l h L 2 (ν o ). l h exi in L (ν o ) a. Furhermore Proof. Under he meaure ν, he ochaic developmen ω = J γ i a Brownian moion. Under he meaure ν o, i i a local emimaringale before ime and i maringale par {b } i a Brownian moion. The meaure ν o i characerized by he fac ha Le ω = b + Uτ log p( τ,γ τ,o)dτ. Q = U log p(,γ,o), F =h 2 for impliciy. We have for < Ric Uτ h τ dτ l h, = = Ḟ τ, db τ + Ḟ τ, db τ Ḟ τ, Q τ dτ = I, I 2, +I 3, Q,F. F τ,dq τ + Q, F Q,F Now Ric u h τ i uniformly bounded, and ḣ L 2 [, ]. Thee fac imply ha he limi I, I exi in L 2 (ν o ) and I = Ḟ, db. For I 3, we have F C{ h +( )}and uing (6) and Lemma 3.2 we have

6 336 E. P. Hu E νo F, Q C { h +( )}E νo log p(,γ,o) { } Eνo d(γ,o) C { h +( )} + C 2 ḣ τ 2 dτ+( ). Thi how ha I 3, inl (ν o ). For I 2, we ue Iô formula on he R n -valued funcion Q = U log p(,γ,) = H log P(, U ) of (, U ) (, ) O(M ), where P(, u) =p(,πu,o). Uing he ochaic differenial equaion (2) for U we have for he ih componen (9) dh i log P(, U ) = H i H log P(, U ), db + 2 Ric U e i, H log P(, U ) d +H i { H log P(, U )+ 2 H log P(, U ) 2 } d, where H = 2 H +, and H = n j = H j 2 i Bochner horizonal Laplacian. Noe ha in he above compuaion we need o ue he econd rucural equaion [H i, H j ]=Ω(H i,h j ) o exchange H i and H j (Ω i he canonical verical vecor field correponding o Ω o(n)). The la erm in (9) vanihe becaue p(, x, y) aifie he hea equaion. Hence we have I 2, = + 2 F τ, Uτ 2 log p( τ,γ τ,o),db τ Ric Uτ F τ, Uτ log p( τ,γ τ,o)dτ. To how ha he limi I 2, I 2 exi in L 2 (ν o ) i i enough o how ha () and () E νo F 2 2 log p(,γ,o) 2 d < E νo F 2 log p(,γ,o) 2 d <.

7 Inegraion by par in loop pace 337 From he definiion of F here exi a conan C uch ha (2) F C{ h +( )}. Uing he eimae (7) and Lemma 3.2 we ee ha here exi a conan C uch ha E νo 2 log p(,γ,o) 2 C ( ) 2. I follow from Lemma 3.3 ha E νo F 2 2 log p(,γ,o) 2 d { } 2 h +( ) C d ( ) { } 8C ḣ 2 d +. Thi prove (). From (6) and Lemma 3.2 here i a conan C uch ha E νo log p(,γ,o) 2 C. Uing hi inequaliy and (2) we have E νo F 2 log p(,γ,o) 2 d { h +( )} 2 C d { } 8C ḣ 2 d +. Thi prove (). I follow ha he limi I 2, I 2 exi in L 2 (ν o ) and I 2 = + 2 F, U 2 log p(,γ,o),db Ric U F, U log p(,γ,o)d. To ummarize, we have l h, = I, I 2, + I 3, Q,F ; I, I,I 2, I 2, boh in L 2 (ν o ), and I 3, inl (ν o ). I follow ha he ochaic inegral l h = ḣ + 2 Ric U h, dω (ω = J γ) exi a he L (ν o )-limi of l h, a and l h L 2 (ν o ).

8 338 E. P. Hu We now prove he main heorem. Theorem 4.2. (Inegraion by par formula in loop pace) Le F, G be wo cylindrical funcion on L o (M ). Then (D h F, G) L2 (ν o) = ( F, D h G ) L 2 (ν o), where and l h L 2 (ν o ) i defined by D h = D h + l h l h (γ) = ḣ + 2 Ric U h, dω. Here ω = J γ i he ochaic developmen of γ and U i he horizonal lif of γ. Proof. Suppoe ha F and G depend on he pah up o ime <. Then we have for all (, ), (D h F, G) L2 (ν o) = C o (D h F, Gp(,γ,o)) L2 (ν), where C o = p(, o, o). By he inegraion by par formula (5) for he pah pace, we have (D h F, G) L 2 (ν o) = C o ( F, D h {Gp(,γ,o)} ) L 2 (ν) = C o (F,D h Gp(,γ,o)) L 2 (ν) C o (F,GD h p(,γ,o)) L2 (ν) +C o (F,l h Gp(,γ,o)) L2 (ν) = (F,D h G) L 2 (ν o) (F,GD h log p(,γ,o)) L2 (ν o) + ( F,l h, G ) L 2 (ν o). Since F and G are uniformly bounded, by Propoiion 4. we have ( F, lh, G ) L 2 (ν o) (F, l hg) L2 (ν o). I i herefore enough o how (F, GD h log p(,γ,o)) L2 (ν o). Thi i implied by (3) We have E νo D h log p(,γ,o). D h log p(,γ,o)= h,u log p(,γ,o). By (6) and Lemma 3.2 we have

9 Inegraion by par in loop pace 339 E νo D h log p(,γ,o) = h E νo p(,γ,o) { } Eνo d(γ,o) C h + C h C = ḣ τ dτ C ḣ τ 2 dτ. Thi how (3) and he heorem i proved. A a conequence of he above inegraion by par formula and he fac ha l h L 2 (ν o ), we have he following reul parallel o Theorem 2.2. Le B(ν o )be he pace of ν o -eenially bounded meaurable funcion on L o (M ). Theorem 4.3. Le h H. Then D h : C L 2 (ν o ) i cloable in L 2 (ν o ) and ha a denely defined adjoin D h. Furhermore Dom(D h ) B(ν o ) Dom(D h ) and for all G Dom(D h ) B(ν o ) we have (4) D h G = D h G + l h G. Acknowledgemen. I wan o hank Profeor M. Cranon for hi generou help hroughou he work. Reference. Driver, B.: A Cameron-Marin ype of quaiinvarance for he Brownian moion on a compac manifold. J. Func. Anal., (992), Driver, B.: A Cameron-Marin ype of quaiinvarance heorem for pinned Brownian moion on a compac manifold. TAMS, 342, No. (994), Enchev, O., Sroock, D. W.: Toward a Riemannian geomery on he pah pace over a Riemannian manifold. J. Func. Anal., 34 (995). 4. Enchev, O., Sroock, D. W.: Inegraion by par for pinned Brownian moion. To appear in Advance in Mahemaic. 5. Hu, E. P.: Quaiinvariance of he Wiener meaure and inegraion by par in he pah pace over a compac Riemannian manifold. J. Func. Anal., 34 (995), Hu, E. P.: Sochaic Gau-Bonne-Chern formula. J. Theore. Probabiliy (995). 7. Hu, E. P.: Eimae of he derivaive of he hea kernel, preprin (996). 8. Sheu, S.-Y.: Some eimae of he raniion deniy funcion of a nondegenerae diffuion Markov proce. Annal of Probabiliy, 9, No. 2 (99), Sroock, D. W., Trubeky, J.: Upper bound for he derivaive of he logarihm of he hea kernel, preprin (996)