Navier Stokes equations and forward backward SDEs on the group of diffeomorphisms of a torus

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1 Sochaic Pocee and hei Applicaion 9 (9) Navie Soke equaion and fowad backwad SDE on he goup of diffeomophim of a ou Ana Bela Cuzeio a,b,, Evelina Shamaova b,c a Dep. de Maemáica, IST-UTL, Pougal b Gupo de Fíica Maemáica da Univeidade de Liboa, Pougal c Ceno de Maemáica da Univeidade do Poo, Pougal Received 3 Ocobe 8; eceived in evied fom July 9; acceped 7 Sepembe 9 Available online 5 Sepembe 9 Abac We eablih a connecion beween he ong oluion o he paially peiodic Navie Soke equaion and a oluion o a yem of fowad backwad ochaic diffeenial equaion (FBSDE) on he goup of volume-peeving diffeomophim of a fla ou. We conuc epeenaion of he ong oluion o he Navie Soke equaion in em of diffuion pocee. c 9 Elevie B.V. All igh eeved. Keywod: Navie Soke equaion; Fowad backwad SDE; Diffeomophim goup. Inoducion The claical Navie Soke equaion ead a follow: u(, x) = (u, )u(, x) ν u(, x) p(, x), div u =, u(, x) = u (x), () Coeponding adde: Gupo de Fíica Maemáica, Complexo Inediciplina da Univeidade de Liboa, Av. Pof. Gama Pino,, PT Liboa, Pougal. Tel.: ; fax: addee: abcuz@mah.i.ul.p (A.B. Cuzeio), evelina@fc.up.p (E. Shamaova) /$ - ee fon mae c 9 Elevie B.V. All igh eeved. doi:.6/j.pa.9.9.

2 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) whee u (x) i a divegence-fee mooh veco field. We fix a ime ineval [, T ], and ewie Eq. () wih epec o he funcion ũ(, x) = u(t, x). Poblem () i equivalen o he following: ũ(, x) = (ũ, )ũ(, x) ν ũ(, x) p(, x), div ũ =, ũ(t, x) = u (x), whee p(, x) = p(t, x). In wha follow, yem () will be efeed o a he backwad Navie Soke equaion. To hi yem we aociae a ceain yem of fowad backwad ochaic diffeenial equaion on he goup of volume-peeving diffeomophim of a fla ou. Fo impliciy, we wok in wo dimenion. Howeve, he genealizaion of mo of he eul o he cae of n dimenion i aighfowad. The neceay conucion and non-aighfowad genealizaion elaed o he n-dimenional cae ae conideed in he Appendix. Auming he exience of a oluion of () wih he final daa in he Sobolev pace H α fo ufficienly lage α, we conuc a oluion of he aociaed yem of FBSDE. Conveely, if we aume ha a oluion of he yem of FBSDE exi, hen he oluion of he Navie Soke equaion can be obained fom he oluion of he FBSDE. In fac, he conuced FBSDE on he goup of volume-peeving diffeomophim can be egaded a an alenaive objec o he Navie Soke equaion fo udying he popeie of he lae. The connecion beween fowad backwad SDE and quai-linea PDE in finie dimenion ha been udied by many auho, fo example in [9,8,]. Ou conucion ue he appoach oiginaing in he wok of Anold [3] which ae ha he moion of a pefec fluid can be decibed in em of geodeic on he goup of volume-peeving diffeomophim of a compac manifold. The neceay diffeenialgeomeic ucue wee developed in lae wok by Ebin and Maden []. We noe hee ha [3,] deal only wih diffeenial geomey on he goup of map wihou involving pobabiliy. The aociaed yem of FBSDE i olved uing he exience of a oluion o (), and by applying eul fom he wok of Gliklikh [ 5]. The lae wok ue, in un, he appoach o ochaic diffeenial equaion on Banach manifold developed by Dalecky and Belopolkaya [5], and aed by McKean [9]. Conveely, a oluion of () i obained uing he exience of a oluion o he aociaed FBSDE a well a ome idea and conucion fom [9]. Howeve, unlike [9], we wok in an infinie-dimenional eing. Repeenaion of he Navie Soke velociy field a a dif of a diffuion poce wee iniiaed in [4,]. A diffeen yem of ochaic equaion (bu no a yem of wo SDE) aociaed o he Navie Soke yem wa inoduced and udied in [4]. Thi yem alo include an SDE on he goup of volume-peeving diffeomophim, bu i no a yem of fowad backwad SDE. Alo, we menion hee he wok [,] dicuing pobabiliic epeenaion of oluion o he Navie Soke equaion, and he wok [6] eablihing a ochaic vaiaional pinciple fo he Navie Soke equaion. Diffeen pobabiliic epeenaion of he oluion o he Navie Soke equaion wee udied fo example in [7,7]. We noe ha he li of lieaue on pobabiliic appoache o he Navie Soke equaion a ()

3 436 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) well a connecion beween finie-dimenional FBSDE and PDE cied in hi pape i by no mean complee. The mehod of applying infinie-dimenional fowad backwad SDE in connecion o he Navie Soke equaion i employed, o he auho knowledge, fo he fi ime.. Geomey of he diffeomophim goup of he wo-dimenional ou Le T = S S be he wo-dimenional ou, and le H α (T ), α >, be he pace of H α -Sobolev map T T. By G α we denoe he ube of H α (T ) whoe elemen ae C - diffeomophim. Le G α V be he ubgoup of Gα coniing of diffeomophim peeving he volume meaue on T. Lemma. Le g be an H α -map and a local diffeomophim of a finie-dimenional compac manifold M, F be an H α -ecion of he angen bundle T M. Then, F g i an H α -map. Poof. See [4] (p. 39) o [] (p. 8). Le R g denoe he igh anlaion on G α, i.e. R g (η) = η g. Lemma. The map R g i C -mooh fo evey g G α. Fuhemoe, fo evey η G α, he angen map T R g eiced o he angen pace T η G α i defined by he fomula: T R g : T η G α T η g G α, X X g. Poof. The poof eaily follow fom he α-lemma (ee [,4,5]). Lemma 3. The goup G α and G α V ae infinie-dimenional Hilbe manifold. The goup G α V i a ubgoup and a mooh ubmanifold of G α. Lemma 4. The angen pace T e G α i fomed by all H α -veco field on T. The angen pace T e G α V i fomed by all divegence-fee H α -veco field on T. The poof of Lemma 3 and 4 can be found fo example in [,4,5]. Lemma 5. Le X T e G α be an H α -veco field on T. Then he veco field ˆX on G α defined by ˆX(g) = X g i igh-invaian. Fuhemoe, ˆX i C k -mooh if and only if X H αk. Poof. The fi aemen follow fom Lemma. The poof of he econd aemen can be found in []. The veco field ˆX on G α defined in Lemma 5 will be efeed o below a he igh-invaian veco field geneaed by X T e G α. Le g G α, X, Y T e G α. Conide he weak (, ) and he ong (, ) α Riemannian meic on G α (ee [5]): ( ˆX(g), Ŷ (g)) = (X g(θ), Y g(θ))dθ, (3) T ( ˆX(g), Ŷ (g)) α = (X g(θ), Y g(θ))dθ T ((d δ) α X g(θ), (d δ) α Y g(θ))dθ (4) T

4 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) whee d i he diffeenial, δ i he codiffeenial, ˆX and Ŷ ae he igh-invaian veco field on G α geneaed by he H α -veco field X and Y. Meic (3) give ie o he L -opology on he angen pace of G α, and meic (4) give ie o he H α -opology on he angen pace of G α (ee [5]). If g GV α, hen cala poduc (3) and (4) do no depend on g. Moeove, fo he ong meic on GV α, we have he following fomula: ( ˆX(g), Ŷ (g)) α = (X g(θ), ( ) α Y g(θ))dθ T whee = (dδ δd) i he Laplace de Rham opeao (ee [3]). Le u inoduce he noaion: Z = {(k, k ) Z : k > o k =, k > }; k = (k, k ) Z, k = (k, k ), k = θ = (θ, θ ) T, ( =, θ and he veco Ā k (θ) = k Ā = ( ) k co(k θ) α k ( ), B = ( ). ), ( k, ) = k θ k k, k θ = k θ k θ, k, θ θ, B k (θ) = ( ) k in(k θ), k α k Le {A k (g), B k (g)} k Z {} be he igh-invaian veco field on G α geneaed by {Ā k, B k } k Z {}, i.e. A k (g) = Ā k g, B k (g) = B k g, g G α, A = Ā, B = B. By ω-lemma (ee [4]), A k and B k ae C -mooh veco field on G α. Lemma 6. The veco A k (g), B k (g), k Z {}, g G V α, fom an ohogonal bai of he angen pace T g GV α wih epec o boh he weak and he ong inne poduc in T ggv α. In paicula, he veco Ā k, B k, k Z {}, fom an ohogonal bai of he angen pace T e GV α. Moeove, he weak and he ong nom of he bai veco ae bounded by he ame conan. Poof. I uffice o pove he lemma fo he ong nom. Le u compue α Ā k. Noe ha he veco k k and k k fom an ohonomal bai of R. Le u obeve ha by he ideniy ( k, ) co(k θ) =, δ Ā k =. Hence dδ Ā k = which implie Ā k = δd Ā k. We obain: Ā k = k co(k θ) k α k, d Ā k = k in(k θ) k α k k k, Ā k = δd Ā k = k co(k θ) k α k = k Ā k,

5 438 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) α Ā k = k α co(k θ) k k = k α Ā k. Thi and he volume-peeving popey of g GV α imply ha (B m (g), A k (g)) α = ( B m, Ā k ) α = ( k α )( B m, Ā k ) L =, A k (g) α = Ā k α = ( k α) ( ) Ā k L = π k α whee α i he nom coeponding o he cala poduc (, ) α. Thu, π A k (g) α 4π. Clealy, fo he B k (g) α we obain he ame. I ha been hown, fo example, in [] and [5] ha he weak Riemannian meic ha he Levi-Civia connecion, geodeic, he exponenial map, and he pay. Le and denoe he covaian deivaive of he Levi-Civia connecion of he weak Riemannian meic (3) on G α and GV α, epecively. In [] (ee alo [5,4]), i ha been hown ha = P whee P : T G α T G α V i defined in he following way: on each angen pace T gg α, P = P g whee P g = T R g P e T R g, T R g and T R g ae angen map, and P e : T e G α T e G α V i he pojeco defined by he Hodge decompoiion. Lemma 7. Le Û be he igh-invaian veco field on G α geneaed by an H α -veco field U on T, and le ˆV be he igh-invaian veco field on G α geneaed by an H α -veco field V on T. Then ˆV Û i he igh-invaian veco field on Gα geneaed by he H α -veco field V U on T. Lemma 8. Le Û be he igh-invaian veco field on GV α geneaed by a divegence-fee H α -veco field U on T, and le ˆV be he igh-invaian veco field on G α geneaed by a divegence-fee H α -veco field V on T. Then ˆV Û i he igh-invaian veco field on G V α geneaed by he divegence-fee H α -veco field P e V U on T. The poof of Lemma 7 and 8 follow fom he igh-invaiance of covaian deivaive on G α and GV α (ee [5]). Remak. The bai {Ā k, B k } k Z {} of T egv α can be exended o a bai of he enie angen pace T e G α. Indeed, le u inoduce he veco: A k (θ) = ( ) k co(k θ), B k α k k (θ) = ( ) k in(k θ), k Z k α k. The yem Ā k, B k, k Z {}, A k, B k, k Z, fom an ohogonal bai of T eg α. Fuhe le A k and B k denoe he igh-invaian veco field on G α geneaed by A k and B k. 3. The FBSDE on he goup of diffeomophim of he wo-dimenional ou Le h : T R be a divegence-fee H α -veco field on T, and le ĥ be he ighinvaian veco field on G α geneaed by h. Fuhe le he funcion V (, ) be uch ha hee exi a funcion p : [, T ] H α (T, R) aifying V (, ) = p(, ) fo all

6 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) [, T ]. Fo each [, T ], ˆV (, ) denoe he igh-invaian veco field on G α geneaed by V (, ) H α (T, R ). Le E be a Euclidean pace panned on an ohonomal, elaive o he cala poduc in E, yem of veco {ek A, eb k, e A, eb } k Z, k N. Conide he map σ (g) = A k (g) ek A B k(g) ek B, g Gα, k Z {}, k N i.e. σ (g) i a linea opeao E T g G α fo each g G α. Le (Ω, F, P) be a pobabiliy pace, and W, [, T ], be an E-valued Bownian moion: W = (βk A ()e k A β k B ()eb k ) k Z {}, k N whee {βk A, β k B} k Z {}, k N i a equence of independen Bownian moion. We conide he following yem of fowad and backwad SDE: dz,e dy,e = Y,e d ɛσ (Z,e )dw, )d X,e dw, (5) = ĥ(z,e T ). = ˆV (, Z,e Z,e = e; Y,e T The fowad SDE of (5) i an SDE on GV α whee G V α i conideed a a Hilbe manifold. Sochaic diffeenial and ochaic diffeenial equaion on Hilbe manifold ae undeood in he ene of Dalecky and Belopolkaya appoach (ee [5]). Moe peciely, we ue he eul fom [4] which inepe he lae appoach fo he paicula cae of SDE on Hilbe manifold. The ochaic inegal in he fowad SDE can be explicily wien a follow: σ (Z,e )dw = k Z {}, k N Le u conide he backwad SDE: Y,e = ĥ(z,e T ) ˆV (, Z,e )d A k (Z,e )dβk A () B k(z,e )dβk B (). (6) X,e dw. (7) Noe ha he pocee ˆV (, Z,e ) = V (, ) Z,e and ĥ(z,e T ) = h Z,e T ae H α -map by Lemma. Theefoe, i make ene o undeand SDE (7) a an SDE in he Hilbe pace H α (T, R ). Le F = σ (W, [, ]). We would like o find an F -adaped iple of ochaic pocee (Z,e, Y,e, X,e ) olving FBSDE (5) in he following ene: a each ime, he poce (Z,e, Y,e ) ake value in an H α -ecion of he angen bundle T GV α. Namely, fo each [, T ] and ω Ω, Z,e GV α, Y,e T Z,e G α V. Theefoe, he fowad SDE i well-poed on boh G α and GV α, and can be wien in he Dalecky Belopolkaya fom: dz,e = exp Z,e exp Z,e dz,e = {Y,e d ɛσ (Z,e )dw } o {Y,e d ɛσ (Z,e )dw } whee exp and exp ae he exponenial map of he Levi-Civia connecion of he weak Riemannian meic (3) on G α and ep. GV α. Below, we will how ha uing eihe of hee epeenaion lead o he ame oluion of FBSDE (5).

7 44 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) Finally, he poce X,e X,e = X k A k Z {}, k N whee he pocee X k A ake value in he pace of linea opeao L(E, H α (T, R )), i.e. e A k X k B e B k (8) and X k B ake value in H α (T, R ). Remak. The eul obained below alo wok in he iuaion when he Bownian moion W i infinie dimenional (a in [8]). Namely, when W = k Z {} a kβ A k e A k b kβ B k eb k whee a k, b k, k Z {}, ae eal numbe aifying k Z {} a k b k <. Howeve, hi equie an addiional analyi on he olvabiliy of he fowad SDE baed on he appoach of Dalecky and Belopolkaya [5] ince he eul of Gliklikh [,4,5] applied below ae obained fo he cae of a finie-dimenional Bownian moion. 4. Conucing a oluion of he FBSDE 4.. The fowad SDE Le u conide he backwad Navie Soke equaion in R : y(, θ) = h(θ) divy(, θ) = [ p(, θ) ( y(, θ), ) y(, θ) ν y(, θ) ] d, (9) whee [, T ], θ T, and ae he Laplacian and he gadien. Aumpion. Le u aume ha on he ineval [, T ] hee exi a oluion ( y(, ), p(, ) ) o (9) uch ha he funcion p : [, T ] H α (T, R) and y : [, T ] H α (T, R ) ae coninuou. Clealy, y(, ) T e GV α ;k A. Le {Y epec o he bai {Ā k, B k } k Z {}, i.e. y(, θ) = Y ;k A k Z {} Ā k (θ) Y ;k B, Y ;k B } k Z {} B k (θ). be he coodinae of y(, ) wih Le Ŷ ( ) denoe he igh-invaian veco field on G α geneaed by he oluion y(, ), i.e. Ŷ (g) = y(, ) g. On each angen pace T g G α, he veco Ŷ (g) can be epeened by a eie conveging in he H α -opology: Ŷ (g) = Y ;k A k Z {} A k (g) Y ;k B B k (g). In hi paagaph we will udy he SDE: d Z,e = Ŷ (Z,e )d ɛσ (Z,e )dw. () Lae, in Theoem 6, we will how ha he oluion Z,e, X,e ae he fi wo pocee in he iple (Z,e, Y,e o () and he poce Y,e ) ha olve FBSDE (5). () = Ŷ (Z,e )

8 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) Theoem. Thee exi a unique ong oluion Z,e condiion Z,e = e., [, T ], o () on GV α, wih he iniial Poof. Below, we veify he aumpion of Theoem 3.5 of [5]. The lae heoem will imply he exience and uniquene of he ong oluion o (). Noe ha, if um (6) epeening he ochaic inegal σ (Z,e )dw conain only he em A (β A() β A()) and B (β B() β B ()), i.e., infomally peaking, if he Bownian moion un only along he conan veco A and B, hen he aemen of he heoem follow fom Theoem 8.3 of [5]. If um (6) conain alo em wih A k and B k, k Z, o, infomally, when he Bownian moion un alo along non-conan veco A k and B k, k Z, hen he aumpion of Theoem 3.5 of [5] equie he boundedne of A k and B k wih epec o he ong nom. The lae fac hold by Lemma 6. Hence, all he aumpion of Theoem 3.5 of [5] ae aified. Indeed, he poof of Theoem 8.3 of [5] how ha he Levi-Civia connecion of he weak Riemannian meic (3) on GV α i compaible (ee Definiion 3.7 of [5]) wih he ong Riemannian meic (4). The funcion σ (g) = k Z {}, k N A k(g) ek A B k(g) ek B i C -mooh ince A k and B k ae C -mooh. Moeove, by Lemma 6, σ (g) i bounded on GV α. Nex, ince y : [, T ] H α (T, R ) i coninuou, hen i i alo bounded wih epec o (a lea) he H α -nom. Hence, he geneaed igh-invaian veco field Ŷ (g) i bounded in wih epec o he ong meic (4), and i i a lea C -mooh in g. The boundedne of Ŷ in g follow fom he volume-peeving popey of g. Theoem. Thee exi a unique ong oluion Z,e, [, T ], o () on G α, wih he iniial condiion Z,e = e. Thi oluion coincide wih he oluion o SDE () on GV α. Poof. Conide he idenical imbedding ı : GV α Gα. By eul of [5] (Popoiion.3, p. 46; ee alo [5], p. 64), he ochaic poce ı(z,e ) = Z,e, [, T ], i a oluion o SDE () on G α, i.e. wih epec o he exponenial map exp. Thi eaily follow fom he fac ha T ı : T GV α T Gα, whee T i he angen map, i he idenical imbedding, and ha ı ( exp(x) ) = exp(t ı X). The oluion Z,e o () on G α i unique. Thi follow fom he uniquene heoem fo SDE () conideed on he manifold G α equipped wih he weak Riemannian meic. Indeed, σ (g) and Ŷ (g) ae bounded wih epec o he weak meic (3) ince he funcion Ā k, B k, k Z {}, ae bounded on T, and y(, ) i bounded on [, T ] T. Moeove σ (g) i C -mooh and Ŷ i a lea C -mooh on G α. One can alo conide () a an SDE wih value in he Hilbe pace H α (T, R ). Theoem 3. Thee exi a unique ong oluion Z,e o he H α (T, R )-valued SDE () on [, T ], wih he iniial condiion Z,e = e whee e i he ideniy of GV α. Thi oluion coincide wih he oluion o SDE () on GV α o Gα. Poof. By Theoem, SDE () on GV α,e ha a unique ong oluion Z on [, T ]. Le u pove ha he oluion Z,e o () olve hi SDE conideed a an SDE in H α (T, R ). Conide he idenical imbedding ı V : GV α H α (T, R ), g g. Applying Iô fomula o ı V, and aking ino accoun ha A k (g) ı V (g) = Āk θ g = A k (g) and ha A k (g)a k (g) ı V (g) = A k (g)a k (g) =, we obain ha he oluion Z,e o () on GV α olve he H α (T, R )-valued SDE (). Noe ha by he uniquene heoem fo SDE in Hilbe pace, SDE () can have

9 44 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) only one oluion in L (T, R ). Thi pove he uniquene of i oluion in H α (T, R ) a well. Thu he oluion o () on G α, GV α, and in H α (T, R ) coincide. Le u find he epeenaion of SDE () in nomal coodinae on G α and GV α. Fi, we pove he following lemma. Lemma 9. The following equaliy hold: σ (Z,e ) dw = σ (Z,e )dw, i.e. inead of he Iô ochaic inegal in () we can wie he Saonovich ochaic inegal σ (Z,e ) dw. Poof. We have: σ (Z,e ) dw = σ (Z,e )dw da k (Z,e )dβk A () db k(z,e )dβk B (). k Z {}, k N Hence, we have o pove ha da k (Z,e )dβk A() = and db k(z,e )dβk B () =. Fo impliciy of noaion we ue he noaion A ν fo boh of he veco field A k and B k and he noaion Ā ν fo Ā k and B k, k Z {}. Alo, we ue he noaion β ν() fo he Bownian moion {βk A(), β k B()} k Z {}, k N. We obain: d(ā ν Z,e ) = γ Thi implie d(ā ν Z,e A γ (Z,e ) dβ ν = A ν (Z,e ) ( Ā ν Z,e ) dβγ () Y,e ) ( Ā ν Z,e ) d = (Āν Z,e ) d. which hold by he ideniy ( k, ) co(k θ) = ( k, ) in(k θ) = o by diffeeniaing of conan veco field. Le Z ;k A = {Z, Z ;k B,e } k Z {} be he veco of local coodinae of he oluion Z o () on GV α, i.e. he veco of nomal coodinae povided by he exponenial map exp : T egv α GV α. Le U e be he canonical cha of he map exp. Theoem 4 (SDE () in Local Coodinae). Le τ = inf{ [, T ] : Z,e U e }. () On he ineval [, τ], SDE () ha he following epeenaion in local coodinae: Z,k A τ = Z,k B τ = τ τ Y ;k A d δ k ɛ(βk A ( τ) β k A ()), Y ;k B d δ k ɛ(βk B ( τ) β k B ()) (3) whee δ k = if k N, and δ k = if k > N. Poof. Le ḡ = {g k A, g k B } k Z {} be local coodinae in he neighbohood U e povided by he map exp. Le f C (G α V ), and le f : T e GV α R be uch ha f = f exp. Since exp i a

10 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) C -map (ee []), hen f C (U ), whee U = and g k B f (ḡ) = B k (g) f (g). By Iô fomula, we obain: f (Z τ,e ) f (e) = f ( Z τ, ) f () τ f = d g k A ( Z ;k A )Y = τ τ d k Z {} d k Z {} k Z {} τ ɛ τ f τ g k B ( Z ;k B )Y ( Y ;k A k Z {} δ k A k (Z,e ( Ak (Z,e Uing epeenaion () and (6) we obain: f (Z,e τ ) f (e) = Thi how ha he poce { exp Z,k A τ k Z {} τ ) f (Z,e ) f (Z,e Ŷ (Z,e ) f (Z,e )d Ā k Z,k B τ B k } exp U e. Noe ha ɛ k Z {} δ k ɛ ) Y ;k B k Z {} δ k g k A f g k A ( Z f (ḡ) = A k (g) f (g) )Y ;k A dβ A k () f g k B ( Z ;k B )Y dβk B () B k (Z,e ) f (Z,e ) ) ) dβk A () B k(z,e ) f (Z,e ) dβk B ()). τ ɛσ (Z,e ) f (Z,e ) dw. olve SDE () on he ineval [, τ]. Le ˇ Z = {Ž ;k A, Ž ;k B, Ž ;ka, Ž ;kb, Ž ;A, Ž ;B } k Z be he veco of local coodinae of he oluion Z,e o () on G α, i.e. he veco of nomal coodinae povided by he exponenial map exp : T e G α G α. Fuhe le Ǔ e be he canonical cha of he map exp. Theoem 5. Le ˇτ = inf{ [, T ] : Z,e Ǔ e }. Then, a.. ˇτ = τ, whee he opping ime τ i defined by (), and on [, τ], Ž ;k A Ž ;k B = Z ;k B, k Z ;ka {}, Ž = Ž ;kb =, k Z, a.. = Z ;k A, Poof. Le u inoduce addiional local coodinae g ka, g kb, k Z, and pefom he ame compuaion a in he poof of Theoem 4. We have o ake ino accoun ha Y ka = Y kb =, k Z, and ha he componen of he Bownian moion ae non-zeo only along divegencefee and conan veco field. We obain ha he coodinae poce ˇ Z veifie SDE (3) and he equaion Ž ;ka = Ž ;kb =, k Z.

11 444 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) The backwad SDE and he oluion of he FBSDE We have he following eul: Theoem 6. Le Ŷ be he igh-invaian veco field geneaed by he oluion y(, ) o he backwad Navie Soke equaion (9). Fuhe le Z,e be he oluion o SDE () on GV α. Then hee exi an ɛ > uch ha he iple of ochaic pocee Z,e, Y,e = Ŷ (Z,e ), X,e olve FBSDE (5) on he ineval [, T ]. Remak 3. The expeion σ (Z,e σ (Z,e )Ŷ (Z,e ) = = ɛσ (Z,e )Ŷ (Z,e ) )Ŷ (Z,e ) mean he following: A k (Z,e k Z {}, k N )Ŷ (Z,e ) e A k B k(z,e )Ŷ (Z,e ) ek B whee Ŷ ( ) i egaded a a funcion G α V H α (T, R ), and A k (g)ŷ (g) mean diffeeniaion of Ŷ : G α V H α (T, R ) along he veco field A k a he poin g G α V. Le γ ξ be he geodeic in G α V uch ha γ = e and γ = Ā k. We obain: Thu, A k (g)ŷ (g)(θ) = d dξ Ŷ(γ ξ g)(θ) ξ= = R g d dξ y(, γ ξ θ) ξ= X,e = ɛ = R g Āk y(, θ) = Ak Ŷ (g)(θ). (4) k Z {}, k N [ Āk y(, ) e A k B k y(, ) e B k ] Z,e and he ochaic inegal in (7) can be epeened a = ɛ X,e dw k Z {}, k N In paicula, if N =, X,e dw = ɛ Āk y(, ) Z,e dβk A () ( T y(, ) Z,e dβ A θ () A eul imila o Lemma wa obained in [6]., B k y(, ) Z,e dβk B (). θ y(, ) Z,e dβ B () ). Lemma (The Laplacian of a Righ-invaian Veco Field). Le ˆV be he igh-invaian veco ( field on G α geneaed by) an H α -veco field V on T. Fuhe le ɛ > be uch ha ɛ k Z, k N = ν. Then fo all g G α, k α ɛ k Z {}, k N ( Ak Ak Bk Bk ) ˆV (g) = ν V g. Hee α i an inege which i no neceay equal o α. (5) (6)

12 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) Poof. By he igh-invaiance of he veco field Ak Ak ˆV and Bk Bk ˆV (Lemma 7), i uffice o how (6) fo g = e. We obeve ha ( k, ) co(k θ) = ( k, ) in(k θ) =. Then, fo k Z, θ T, Ak Ak ˆV (e)(θ) = k α co(k θ)( k, ) [ co(k θ)( k, )V (θ) ] = k α co(k θ) ( k, ) V (θ). Similaly, Bk Bk ˆV (e)(θ) = k α in(k θ) ( k, ) V (θ). Hence, fo each k Z, ( Ak Ak Bk Bk ) ˆV (e)(θ) = Noe ha fo each k Z, eihe k o k i in Z, and ( k, ) (k, ) = k. k α ( k, ) V (θ). (7) Summaion ove k Z, k N, in (7), and coupling he em numbeed by k and k (o k) give: ( Ak Ak Bk Bk ) ˆV (e)(θ) = V (θ). k α k Z, k N k Z, k N Noe ha ( A A B B ) ˆV (e)(θ) = V (θ). Finally, we obain: ( ( Ak Ak Bk Bk ) ˆV (e)(θ) = k Z {}, k N The lemma i poved. k Z, k N ) k α V (θ). Coollay. Le he funcion ϕ : T R be C -mooh. Fuhe le A k (g)[ϕ g] and B k (g)[ϕ g], k Z, mean he diffeeniaion of he funcion G α L (T, R ), g ϕ g along A k and ep. B k. Then fo all g G α, ɛ k Z {}, k N ( Ak (g)a k (g) B k (g)b k (g) ) [ϕ g] = ν ϕ g. Poof. The compuaion ha we made in (4) bu applied o ϕ g implie ha [ ] A k (g)[ϕ g] = k α co(k θ)( k, )ϕ(θ) g. Similaly, we compue B k (g)[ϕ g]. Now we ju have o epea he poof of Lemma o come o (8). Lemma. Le Φ, [, T ], [, T ), be an H α (T, R )-valued ochaic poce whoe ajecoie ae inegable, and le φ T be an H α (T, R )-valued andom elemen o ha (8)

13 446 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) boh Φ and φ T poe finie expecaion. Then hee exi an F -adaped H α (T, R ) L ( E, H α (T, R ) ) -valued pai of ochaic pocee (Y, X ) olving he BSDE Y = φ T Φ d X dw (9) on [, T ]. The Y -pa of he oluion ha he epeenaion [ ] Y = E φ T Φ d F, () and heefoe i unique. The X -pa of he oluion i unique wih epec o he nom X = X L(E,H α (T,R )) d. The poof of he lemma ue ome idea fom []. Poof. Repeenaion () follow fom (9). Le u exend he poce Y o he enie ineval [, T ] by eing Y = Y fo [, ], and noe ha he exended poce Y i a oluion of he SDE Y = φ T I [,T ] Φ d X dw on [, T ]. Le X L ( E, H α (T, R ) ), [, T ], be uch ha E [ ] φ T I [,T ] Φ d Y F = X dw. () The poce X exi by he maingale epeenaion heoem. Indeed, on he igh-hand ide of () we have a Hilbe pace valued maingale. By Theoem 6.6 of [6], each componen of he H α (T, R )-valued maingale on he ighhand ide of () can be epeened a a um of eal-valued ochaic inegal wih epec o he Bownian moion {β A k (), β B k ()} k Z {}, k N. Hence, hee exi F -adaped ochaic pocee {X k A, X k B } k Z {}, k N uch ha [ ] E φ T I [,T ] Φ d Y F = k Z {}, k N Le he poce X be defined by (8) via he pocee X k A iomey how ha E X k A dβk A () X k B dβk B (). and X k B, k Z {}, k N. Iô X dw = X L(E,H α (T,R )) <. Noe ha fo all [, ], X dw. Thi how ha X = fo almo all ω Ω and almo all [, ], and heefoe can be choen equal o zeo on [, ]. Thu, () ake he fom: E [ ] φ T Φ d Y F = X dw. () I i eay o veify ha he pai (Y, X ) defined by () and () olve BSDE (9). To pove he uniquene, noe ha any F -adaped oluion o (9) ake he fom (), (). Moeove, if he

14 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) pocee X and X aify (), hen T X X L(E,H α (T,R )) d = (X X ) dw =. H α (T,R ) Poof of Theoem 6. Le u conide BSDE (7) a an L (T, R )-valued SDE, and Ŷ a a funcion GV α L (T, R ). Since fo each [, T ], y(, ) H α (T, R ) and α > by aumpion, hen Ŷ : GV α L (T, R ) i a lea C -mooh. Eq. () how ha he funcion y(, ) : [, T ] L (T, R ) i coninuou ince p, y, and (y, y) ae coninuou funcion [, T ] L (T, R ) by Aumpion. Taking ino accoun ha he diffeomophim of GV α ae volume-peeving, we conclude ha fo each fixed g G V α, Ŷ(g) : [, T ] L (T, R ) i a coninuou funcion. Hence, Ŷ : [, T ] GV α L (T, R ) i C -mooh in [, T ] and C -mooh in g GV α. Iô fomula i heefoe applicable o Ŷ (Z,e ). Below we ue he fac ha Z,e i a oluion o fowad SDE () and he ideniy Ŷ ) = y(, ) Z,e.,e (Z Fo he lae deivaive we ubiue he igh-hand ide of he fi equaion of (). The noaion ˆX(g)[Ŷ (g)] (omeime wihou quae backe) mean diffeeniaion of he funcion Ŷ : GV α L (T, R ) along he igh-invaian veco field ˆX on GV α a he poin g G V α. The ame agumen a in Remak 3 implie ha ˆX(g)[Ŷ (g)] = ˆX Ŷ(g). Taking ino accoun hi agumen, we obain: Noe ha Ŷ (Z,e ) ĥ(z,e T ) = Ŷ (Z,e d ɛ Alo, le u obeve ha ɛ k Z {}, k N Ŷ (Z,e ) d [ Ak (Z,e )A k (Z,e d Ŷ (Z,e )[Ŷ (Z,e )] )Ŷ (Z,e ) B k (Z,e )B k (Z,e )Ŷ (Z,e ) ] ɛ σ (Z,e )Ŷ (Z,e ) dw. (3) )[Ŷ (Z,e )] = [(y(, ), )y(, )] Z,e. k Z {}, k N = ɛ [ Ak (Z,e k Z {}, k N = ν[ y(, )] Z,e )A k (Z,e )Ŷ (Z,e ) B k (Z,e [ Ak Ak Ŷ (Z,e ) Bk Bk Ŷ (Z,e ) ] )B k (Z,e )Ŷ (Z,e ) ] whee he lae equaliy hold by Lemma, and ɛ > i choen o ha ) k Z, k N k α = ν. Noe ha he em Ak Ak Ŷ (Z,e ( ɛ ) and Bk Bk Ŷ (Z,e ) ae elemen of T G α, and heefoe ae well defined in L (T, R ). Coninuing (3), we obain: Ŷ,e (Z ) ĥ(z,e T )

15 448 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) = = d [ ˆV (, Z,e ) [(y(, ), )y(, )] Z,e [(y(, ), )y(, )] Z,e d ɛ σ (Z,e )Ŷ (Z,e ) dw ˆV (, Z,e ) d ] ν[ y(, )] Z,e ν[ y(, )] Z,e d ɛ σ (Z,e )Ŷ (Z,e ) dw. (4) Thu he pai of ochaic pocee (Ŷ (Z,e ), ɛ σ (Z,e )Ŷ (Z,e )) i a oluion o BSDE (7) in L (T, R ). I i F -adaped ince Z,e i F -adaped. By Lemma, we know ha hee exi a unique F -adaped oluion (Y,e, X,e ) o (7) in H α (T, R ). Clealy, (Y,e, X,e ) i alo a unique F -adaped oluion o (7) in L (T, R ). Hence, Y,e = Ŷ (Z,e ) and X,e ɛ σ (Z,e )(Ŷ (Z,e )) d =, and heefoe he pai of ochaic pocee L(E,H α (T,R )) (Ŷ (Z,e ), ɛ σ (Z,e )Ŷ (Z,e ) ) i a unique F -adaped oluion o BSDE (7) in H α (T, R ). The heoem i poved. 5. Some ideniie involving he Navie Soke oluion The backwad SDE allow u o obain he epeenaion below fo he Navie Soke oluion. Alo, i eaily implie he well-known enegy ideniy fo he Navie Soke equaion. 5.. Repeenaion of he Navie Soke oluion Theoem 7. Le [, T ], and le Z,e be he oluion o SDE () on [, T ] wih he iniial condiion Z,e = e. Then he following epeenaion hold fo he oluion y(, ) o (9). ],e y(, ) = E [ĥ(z T ) p(, ) Z,e d. Poof. Noe ha Ŷ (Z,e ) = y(, ), and E[ X,e dw ] =. Taking he expecaion fom he boh pa of (7) a ime = we obain he above epeenaion. 5.. A imple deivaion of he enegy ideniy Iô fomula applied o he quaed L (T, R )-nom of Y,e Y,e L = ĥ(z,e T ) L (Y,e (Y,e, X,e dw ) L, ˆV (Z,e )) L d give: X,e L d. (5) Uing epeenaion (5) fo he poce X,e we obain: X,e L = ɛ Āk y(, ) L B k y(, ) L k Z {}, k N

16 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) = ɛ k α ( k, y(, )) L y(, ) L k Z, k N = ɛ [ ( ( k, k α y(, )) L (k, y(, )) ) ] L y(, ) L = ɛ ( k Z, k N k Z, k N ) k α y(, ) L = ν y(, ) L. Taking he expecaion in (5) and uing he volume-peeving popey of Z,e y(, ) L ν y(, ) L d = h L., we obain: 6. Conucing he oluion o he Navie Soke equaion fom a oluion o he FBSDE Le u pove now a eul which i, in ome ene, a convee of Theoem 6. In hi ecion we conide (5) a a yem of fowad and backwad SDE in he Hilbe pace H α (T, R ), whee α 3. A befoe, le ˆV (, Z,e ) denoe p(, ) Z,e, and le F denoe he filaion σ {W, [, ]}. Theoem 8. Aume, fo an H α -mooh funcion p(, ), [, T ], and fo any (, T ), he exience of an F -adaped oluion (Z,e, Y,e, X,e ) o (5) on [, T ] uch ha he pocee Z,e and Y,e have a.. coninuou ajecoie and uch ha Z,e ake value in GV α. Then hee exi T > uch ha fo all T < T hee exi a deeminiic funcion y(, ) T e GV α on [, T ], uch ha a.. on [, T ] he elaion Y,e = y(, ) Z,e hold. Moeove, he pai of funcion (y, p) olve he backwad Navie Soke equaion (9) on [, T ]. Lemma 8 ae he ep in he poof of Theoem 8. Lemma. Fo all [, T ) and fo any F -meauable GV α -valued andom vaiable ξ, he iple of ochaic pocee (Z,ξ, Y,ξ, X,ξ ) = (Z,e ξ, Y,e ξ, X,e ξ) (6) i F -adaped and olve he FBSDE Z,ξ = ξ Y,ξ d σ (Z,ξ ) dw Y,ξ = h(z,ξ T ) ˆV (, Z,ξ )d on he ineval [, T ] in he pace H α (T, R ). X,ξ dw (7) Poof. Le u apply he opeao R ξ of he igh anlaion o he boh ide of FBSDE (5). We only have o pove ha we ae allowed o wie R ξ unde he ign of boh ochaic inegal in (5). Le u pove ha i i ue fo an F -meauable epwie funcion ξ = i= g i I Ai, whee g i G α V and he e A i ae F -meauable. Indeed, le and S be uch ha < S T, and le Φ be an F -adaped ochaically inegable poce. We obain:

17 45 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) S Φ dw g i I Ai = i= = S I Ai Φ g i dw = i= S Φ g i I Ai dw. i= i= S I Ai Φ g i dw Nex, we find a equence of F -meauable epwie funcion conveging o ξ in he pace of coninuou funcion C(T, R ). Thi i poible due o he epaabiliy of C(T, R ). Indeed, le u conide a counable numbe of dijoin Boel e Oi n coveing C(T, R ), and uch ha hei diamee in he nom of C(T, R ) i malle han n. Le An i = ξ (Oi n) and gn i Oi n GV α. Define ξ n = i= gi ni Ai n. Then i hold ha fo all ω Ω, ξ ξ n C(T,R ) < n. Le I (Φ) and I (Φ ξ) denoe S Φ dw and ep. S Φ ξ dw. We have o pove ha a.. I (Φ) ξ = I (Φ ξ). Fo hi i uffice o pove ha lim E I (Φ) ξ n I (Φ) ξ n L (T,R =, (8) ) lim E I (Φ ξ n) I (Φ ξ) n L (T,R =. (9) ) Due o he volume-peeving popey of ξ and ξ n, I (Φ) ξ n L (T,R ) = I (Φ) ξ L (T,R ) = I (Φ). Hence, by Lebegue heoem, in (8) we can pa o he limi unde he L (T,R ) expecaion ign. Relaion (8) hold hen by he coninuiy of I (Φ) in θ T. To pove (9) we obeve ha by Iô iomey, he limi in (9) equal o lim n E S Φ ξ n Φ ξ d. The ame agumen ha we ued o pove (8) implie ha we can pa o he L (T,R ) limi unde he expecaion and he inegal ign. Relaion (9) follow fom he coninuiy of Φ in θ T. Hence, (Z,e ξ, Y,e ξ, X,e Lemma 3 7 ue ome idea and conucion fom [9]. ξ) i a oluion o (7). Thi oluion i clealy F -adaped. Lemma 3. The map [, T ] T R, (, θ) Y,e (θ) i deeminiic. Poof. Le u exend he oluion (Z,e = Y,e, X,e Y,e Z,e = e Y,e, Y,e, X,e ) o he ineval [, ] by eing Z,e = fo all [, ]. The exended poce olve he poblem: = h(z,e T ) I [,T ] ()Y,e d I [,T ] () ˆV (, Z,e ) d I [,T ] ()σ (Z,e ) dw X,e dw. = e, The andom veco Y,e i F -meauable, and hence i deeminiic by Blumenhal zeo one law. Since Y,e = Y,e, he eul follow. Lemma 4. Thee exi a conan T > uch ha fo T < T he funcion [, T ] H (T, R ), Y,e i coninuou. Poof. Le (Z,e, Y,e, X,e ) and (Z,e, Y,e, X,e ) be oluion o (7) which a a he ideniy e a ime and ep., and le <. Thee oluion can be egaded a oluion of (3) if (3)

18 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) we exend hem o he enie ineval [, T ] a i wa decibed in Lemma 3. The applicaion of Iô fomula o Y,e and he backwad SDE of (7) imply ha he expecaion E Y,e L (T,R ) i bounded. The fowad SDE of (3), Gonwall lemma, and uual ochaic L (T,R ) inegal eimae imply ha hee exi a conan K > uch ha [ E Z,e Z,e ] L (T,R ) < K I [,T ] E Y,e Y,e L (T,R ) d ( ). Le u apply Iô fomula o Y,e Y,e when uing he backwad SDE of (3). L (T,R ) Again, Gonwall lemma, uual ochaic inegal eimae and he above eimae fo E Z,e Z,e L (T,R ) imply ha hee exi a conan K > uch ha [ ] E Y,e Y,e L (T,R ) < K E Y,e Y,e L (T,R ) d ( ). We ake T malle han K. Then hee exi a conan K > uch ha up E Y,e Y,e L (T [,T ],R ) < K ( ). (3) Evaluaing he igh-hand ide a he poin =, and aking ino accoun ha Y,e = Y,e we obain ha Y,e Y,e L (T,R ) < K ( ). (3) Diffeeniaing (3) wih epec o θ we obain he following yem of fowad and backwad SDE: Z,e = I I [,T ] () Y,e d I [,T ] () σ (Z,e ) Z,e dw Y,g = h(z,e,e T ) ZT I [,T ] () ˆV (, Z,g ) Z,e d X,e dw. Again, andad eimae imply he boundedne of E Z,e,e and E Y L (T,R ) L (T,R ). The ame agumen ha we ued o obain (3) a well a he eimae fo he up [,T ] E Z,e Z,e, which eaily follow fom (3), and he fowad SDE imply ha hee exi a L (T,R ) conan L > uch ha fo all and fom he ineval [, T ], Y,e Y,e L (T,R ) < L. (33) Diffeeniaing (3) he econd ime and uing he ame agumen once again we obain ha hee exi a conan M > uch ha fo all and belonging o [, T ], Y,e Y,e L (T,R ) < M. (34) Now (3) (34) imply he coninuiy of he map Y,e wih epec o he H (T, R )- opology. Eveywhee below we aume ha T < T whee T i he conan defined in Lemma 4.

19 45 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) Lemma 5. Fo evey [, T ) and fo evey F -meauable andom vaiable ξ, he oluion (Z,ξ, Y,ξ, X,ξ ) o (7) i unique on [, T ]. Poof. Le u aume ha hee exi anohe oluion ( Z,ξ, Ỹ,ξ The ame agumen a in he poof of Lemma 4 implie he uniquene of oluion o (7). Specifically, he agumen ha we applied o he pai of oluion (Z,e (Z,e, Y,e, X,e ) ha o be applied o (Z,ξ, Y,ξ, X,ξ ) and ( Z,ξ, Ỹ,ξ, X,ξ aken ino accoun ha =. Lemma 6. Le he funcion y : [, T ] T R be defined by he fomula:, X,ξ ) o (7) on [, T ]., Y,e, X,e ) and ), and i ha o be y(, θ) = Y,e (θ). Then, fo evey [, T ], y(, ) i H α -mooh, and a.. Y,e u = y(u, ) Z,e u. (35) (36) Poof. Noe ha (6) implie ha if ξ i F -meauable hen Y,ξ = y(, ) ξ. Fuhe, fo each fixed u [, T ], (Z,e [u, T ]: Z,e Y,e = Zu,e Y,e d u u = h(z,e T ), Y,e ˆV (, Z,e )d σ (Z,e )dw (37), X,e ) i a oluion of he following poblem on X,e dw. By uniquene of oluion, i hold ha Y,e = Y a.. on [u, T ]. Nex, by (37), we obain,e u,zu ha Yu = y(u, ) Zu,e. Thi implie ha hee exi a e Ω u (which depend on u) of full P-meaue uch ha (36) hold eveywhee on Ω u. Clealy, one can find a e Ω Q, P(Ω Q ) =, uch ha (36) hold on Ω Q fo all aional u [, T ]. Bu he ajecoie of Z,e and Y,e ae a.. coninuou. Fuhemoe, Lemma 4 implie he coninuiy of y(, ) in wih epec o (a lea) he L (T, R )-opology. Theefoe, (36) hold a.. wih epec o he L (T, R )- opology. Since boh ide of (36) ae coninuou in θ T i alo hold a.. fo all θ T. u,z,e u Lemma 7. The funcion y defined by fomula (35) i C -mooh in [, T ]. Poof. Le δ >. We obain: y( δ, ) y(, ) = Y δ,e δ Y,e = Y δ,e δ Y,e δ Y,e δ Y,e. Le Ŷ be he igh-invaian veco field on G α geneaed by y(, ). Lemma 6 implie ha a.. Y,e δ = Ŷ δ (Z,e δ ). Thu we obain ha a.. y( δ, ) y(, ) = ( Ŷ δ (e) Ŷ δ (Z,e δ )) (Y,e δ Y,e ).

20 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) We ue he backwad SDE fo he econd diffeence and apply Iô fomula o he fi diffeence when conideing Ŷ δ a a C -mooh funcion GV α L (T, R ). We obain: Ŷ δ (Z,e δ ) Ŷ δ (e) = δ d δ [A k (Z,e k Z {} d Ŷ,e )A k (Z,e The ame agumen a in Theoem 6 implie: Ŷ δ (Z,e δ ) Ŷ δ (e) = δ Fuhe we have: Y,e δ d ν y( δ, ) Z,e δ Y,e δ = Finally we obain ha d p(, ) Z,e ( ) y( δ, ) y(, ) = δ ν y( δ, ) p(, )] Z,e (Z,e )[Ŷ δ (Z,e )] ) B k (Z,e d y(, ) y( δ, ) Z,e δ δ ɛ σ (Z,e )B k (Z,e )] Ŷ δ (Z,e ). ɛ σ (Z,e ) Ŷ δ (Z,e ) dw. δ X,e dw. [ δ δ E d [ (y(, ), ) y( δ, ) ) Ŷ δ (Z,e ) dw ]. (38) Noe ha Z,e, p(, ), and (y(, ), ) y( δ, ) Z,e ae coninuou in a.. wih epec o he L (T, R )-opology. By Lemma 4, y(, ) and y(, ) ae coninuou in wih epec o he L (T, R )-opology. Fomula (38) and he fac ha Z,e = e imply ha in he L (T, R )-opology y(, ) = [ y(, ) y(, ) ν y(, ) p(, )]. (39) Since he igh-hand ide of (39) i an H α -map, o i he lef-hand ide. Thi implie ha y(, ) i coninuou in θ T. Relaion (39) i obained o fa fo he igh deivaive of y(, θ) wih epec o. Noe ha he igh-hand ide of (39) i coninuou in which implie ha he igh deivaive y(, θ) i coninuou in on [, T ). Hence, i i unifomly coninuou on evey compac ubineval of [, T ). Thi implie he exience of he lef deivaive of y(, θ) in, and heefoe, he exience of he coninuou deivaive y(, θ) eveywhee on [, T ]. Lemma 8. Fo evey [, T ], he funcion y(, ) : T R i divegence-fee. Moeove, he pai (y, p) veifie he backwad Navie Soke equaion. Poof. Fix a >, and conide he T e G α V -valued cuve γ ζ = E[exp Z,e ζ ], ζ, in a neighbohood of he oigin of T e G α V. The fowad SDE of (7) can be epeened a an SDE on G α : { dz,e = exp{ŷ (Z,e )d σ (Z,e )dw }, Z,e = e,

21 454 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) whee Ŷ i he igh-invaian veco field on G α geneaed by y(, ). Thi implie ha ζ γ ζ = y(, ), ζ = and heefoe y(, ) T e G α,e V. Nex, he backwad SDE of (7) implie ha YT = h(z,e T ). Thi and elaion (36) imply ha y(t, ) = h. Since we aleady obained (39) in Lemma 7 he poof of he lemma i now complee. 7. The backwad SDE a an SDE on a angen bundle Le (Z,e, Y,e, X,e ) be a oluion o FBSDE (5). We will how ha he backwad SDE can be epeened a an SDE on he angen bundle T GV α a well a an SDE on T Gα. We will conuc a backwad SDE in he Dalecky Belopolkaya fom (ee [5]) and how ha he poce i i unique oluion. Y,e 7.. The epeenaion of he backwad SDE on T G α V Le y(, ), [, T ], be he oluion o he backwad Navie Soke equaion (9). Le Ŷ be he igh-invaian veco field on GV α geneaed by y(, ). The connecion map on he manifold GV α geneae he connecion map on he manifold T G V α a i wa hown in [5], p. 58 (ee alo []). A befoe, we conide he Levi-Civia connecion of he weak Riemannian meic (3) on GV α. Le exp denoe he exponenial map of he geneaed connecion on T G V α. Moe peciely, exp i given a follow: ( ) ( ) α γα () exp ( x a ) = β η β () ( ) γα () whee η β () i he geodeic cuve on T GV α wih he iniial daa γ α () = α, η β () = β, γ α () = x, η β () = a. Le he veco field Ak H and Bk H be he hoizonal lif of A k and B k ono T T GV α. Fuhe le Ŷ l be he veical lif of Ŷ ono T T GV α. Le u conide he backwad SDE on T GV α: dy,e Y,e T = exp Y,e ɛ = ĥ(z,e T ) { Ŷ l,e (Y k Z {}, k N )d S(Y,e )d [ A H k (Y,e ) ek A BH k,e (Y ) ek A ] } dw, (4) whee S i he geodeic pay of he Levi-Civia connecion of he weak Riemannian meic (3) on GV α,e (ee [4] o [5]), and Z, [, T ], i he oluion o () on GV α wih he iniial condiion Z,e = e. Theoem 9. Thee exi a oluion o (4) on [, T ]. Moeove, if y(, ) H α (T, R ), hen hi oluion i unique and coincide wih he Y,e -pa of he unique F -adaped oluion (Y,e, X,e ) o (7). Poof. Fom he poof of Theoem 6 we know ha he pai of ochaic pocee (Ŷ (Z,e ), ɛ σ (Z,e )Ŷ (Z,e )) i he unique F -adaped oluion o (7) in H α (T, R ). Le u

22 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) pove ha Ŷ (Z,e ) i a ong oluion o (4). Fi we decibe a yem of local coodinae (g k A, X k A, g k B, X k B ) k Z {} in a neighbohood U eg T e GV α of he poin ˆX(g) T GV α whee U e GV α i he canonical cha. The veco ḡ = (gk A, g k B ) k Z {} i he veco of nomal coodinae in he neighbohood U e g, g GV α. The veco X = (X k A, X k B ) k Z {} epeen he coodinae of he decompoiion of he veco ˆX(g) T GV α in he bai {A k, B k } k Z {} : ˆX(g) = k Z {}(X k A A k (g) X k B B k (g)). Le f be a mooh funcion on T G α V, and le f ( X, ḡ) = f ( ˆX(g)), whee ˆX(g) T GV α,e. Le τ be he exi ime of he poce Z fom he neighbohood U e Z,e Le ( Z, Ȳ ) = (Z k A Ŷ (Z,e. We will compue he diffeence f (Y,e, Z k B, Y k A, Y k B ) k Z {} ) on [, τ]. Uing SDE (4), we obain: f (Y,e ) f (Yτ,e ) = τ [ k Z {} Y k A f (Ȳ, Z ) Z k A ɛ k Z {}, k N (Y k A Y k B f (Ȳ, Z ) τ Z k B [ f (Ȳ, Z ) Z k A ) f (Yτ,e ) uing Iô fomula. be he veco of local coodinae of he poce ) f (Ȳ, Z ) Y k A ɛ ( δ k (Z k A (Y k B ) (Z k B ek A f (Ȳ, Z ) Z k B ek A ) f (Ȳ, Z ) Y k B ) ) ] f (Ȳ, Z ) d ] dw (4) whee δ k = if k N, and δ k = ohewie. Since f i a mooh funcion on T GV α, all i eicion o he angen pace of GV α ae mooh. Hence, one can alk abou deivaive of f eiced o a angen pace along he veco of hi angen pace. Namely, he following elaion hold: f (Ȳ, Z ) Y k A = f (Ŷ (Z,e ))A k (Z,e ). Noe ha he diffeeniaion of f wih epec o Z k A and Z k B can be egaded a he diffeeniaion of he compoie funcion f Ŷ along he veco A k and B k. Namely, f (Ȳ, Z ) = A k (Z,e )[( f Ŷ )(Z,e )]. Thi implie: f (Y,e ) f (ĥ(z,e T )) = ɛ ɛ k Z {}, k N k Z {}, k N ( Ak (Z,e d [A k (Z,e [ )A k (Z,e ( f Ŷ )(Z,e ) B k (Z,e )( f Ŷ )(Z,e ) ek A ) Ŷ (Z,e )( f Ŷ )(Z,e ) )B k (Z,e ) ) ( f Ŷ )(Z,e ) Z k A B k (Z,e )( f Ŷ )(Z,e ) ek B ]dw. (4) We exended he inegaion o he enie ineval [, T ] ince he local coodinae no longe appea unde he inegal ign. Thi i alo poible ince (4) hold alo wih epec o he local coodinae in he neighbohood U e Zτ,e and a new exi ime τ. The ame agumen can be

23 456 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) epeaed wih epec o he local coodinae in he neighbohood U e Zτ,e, ec. Le u conide now f Ŷ a a ime-dependen funcion of g GV α. Applying Iô fomula o ( f Ŷ )(Z,e ) on he ineval [, T ] and uing SDE () on GV α, we obain exacly he above ideniy. Thi pove ha Y,e = Ŷ (Z,e ) i a ong oluion o (4) on T GV α. By eul of [4], Ŷ l i C -mooh. Moeove S, Ak H and BH k, k Z, ae C -mooh. Again, by eul of [4], he oluion of BSDE (4) on T GV α i unique. 7.. The epeenaion of he backwad SDE on T G α Applying Popoiion.3 (p. 46) of [5] (ee alo [5], p. 64) o he manifold T GV α and T Gα and he idenical imbedding ı V : T GV α T Gα, we obain ha he poce ı V (Ŷ (Z,e ) ) = Ŷ (Z,e ) olve he following backwad SDE on T G α : { dy,e = exp Y,e Ŷ l (Y,e )d S(Y,e )d [ ɛ A H k (Y,e ) ek A B H k (Y,e ) ek A ] } dw, Y,e T = ĥ(z,e T ) k Z {}, k N whee S i he geodeic pay of he Levi-Civia connecion of he weak Riemannian meic on G α, Ŷ l denoe he veical lif of Ŷ ono T T G α, A H k and B H k denoe he hoizonal lif of A k and B k ono T T G α, he poce Z,e, [, T ], i he oluion o () on G α wih he iniial condiion Z,e = e. The exponenial map exp on T T G α i defined imilaly o he map exp on T T GV α. Namely, he Levi-Civia connecion of he weak Riemannian meic on Gα geneae a connecion on T G α. The lae give ie o he exponenial map exp on T T G α a i wa decibed in Secion 7.. We acually have obained he following heoem. Theoem. Backwad SDE (43) ha a unique ong oluion. Moeove, hi oluion coincide wih he unique ong oluion o BSDE (4) on T GV α,e, and wih he Y -pa of he unique F -adaped oluion (Y,e, X,e ) o (7). Poof. We have aleady hown ha he poce Ŷ (Z,e ) olve BSDE (43). The uniquene of oluion can be poved in exacly he ame way a he uniquene of oluion o (4) on T GV α (ee he poof of Theoem 9). Acknowledgemen We would like o hank he efeee fo meaningful queion. Thi wok wa uppoed by he Pouguee Foundaion fo Science and Technology (FCT) unde he pojec PTDC/MAT/69635/6 and SFRH/BPD/4874/8. Appendix A.. Geomey of he goup of volume-peeving diffeomophim of he n-dimenional ou Le T n = S S }{{} n (43) denoe he n-dimenional ou. Le u decibe he bai of he angen pace T e G α V of he goup G α V of volume-peeving diffeomophim of Tn. We inoduce

24 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) he following noaion: Z n = {(k, k,..., k n ) Z n : k > o k = = k i =, k i >, i =,..., n}; k = (k,..., k n ) Z n, k = n n ki, k θ = k i θ i, θ = (θ,..., θ n ) T n, = i= ( θ, θ,..., ). θ n Fo evey k Z n, ( k,..., k n ) denoe an ohogonal yem of veco of lengh k which i alo ohogonal o k. Inoduce he veco field on T n : Ā i k = k α co(k θ) k i, B i k = k α in(k θ) k i, i =,..., n, k Z n, and he conan veco field Ā i, i =,..., n, whoe ih coodinae i and he ohe coodinae ae. Le A i k, Bi k, i =,..., n, k Z n, denoe he igh-invaian veco field on GV α geneaed by Āi k, B k i, i =,..., n, k Z n, epecively, and le Ai = Āi, i =,..., n, and fo conan veco field on GV α. The following lemma i an analog of Lemma 6. Lemma 9. The veco A i k (g), Bi k (g), k Z n, i =,..., n, g G V α, Ai, i =,..., n, fom an ohogonal bai of he angen pace T g GV α wih epec o boh he weak and he ong inne poduc in T g GV α. In paicula, he veco Āi k, B k i, k Z n, i =,..., n, Āi, i =,..., n, fom an ohogonal bai of he angen pace T e GV α. Moeove, he weak and he ong nom of he bai veco ae bounded by he ame conan. The ohe lemma of Secion hold in he n-dimenional cae, wih epec o he yem A i k, Bk i, k Z n, i =,..., n, Ai, i =,..., n, wihou change. The index α of he Sobolev pace H α ha o be choen bigge han n. A.. The Laplacian of a igh-invaian veco field on G α (T n ) One of he mo impoan ep in he poof of Theoem 6 and 8 i Lemma, i.e. he compuaion of he Laplacian of a igh-invaian veco field on G α wih epec o he ubyem {A k, B k } k Z {}, k N whee N can be fixed abiay. Below we pove an n-dimenional analog of hi lemma. Lemma. Le ˆV be he igh-invaian veco field on G α (T n ) geneaed by an H α -veco field V on T n. Fuhe le ɛ > be uch ha ɛ ( n ) n k α = ν. Then fo all g G α, ɛ [ k Z n, k N i= k Z n, k N n ( A i k i= ) n ] A i k B i k B i k A i A i ˆV (g) = ν V g. i=

25 458 A.B. Cuzeio, E. Shamaova / Sochaic Pocee and hei Applicaion 9 (9) Poof. A i wa menioned in he poof of Lemma 7, i uffice o conide he cae g = e. We obeve ha fo all i =,..., n, ( k i, ) co(k θ) = in(k θ)( k i, k) =. Similaly, ( k i, ) in(k θ) =. Then, fo k Z n, θ Tn, n i= A i k A i k ˆV (e)(θ) = = = n k α co(k θ)( k i, ) [ co(k θ)( k i, )V (θ) ] i= n co(k θ) ( k i, ) V (θ) k α i= k α co(k θ) ( k (k, ) )V (θ). The lae equaliy hold by he ideniy n i= ( k i, ) (k, ) = k ha follow, in un, fom he fac ha he yem { k i k, k } k, i =,..., n, fom an ohonomal bai of R n. Similaly, n i= B i k B i k ˆV (e)(θ) = Hence, fo each k Z n, i= k α in(k θ) ( k (k, ) )V (θ). n ( A i k A i k B i k B i ) ˆV (e)(θ) = k k α ( k (k, ) )V (θ). (44) Fuhe we have: k α (k, ) = k Z n, k N = k Z n, k N k α k Z n, k N n ki i i= (k, ) k α k Z n, k N k α k i k j i j whee i = θ i, and due o he faco we pefom he ummaion ove all k Z n. Clealy, he econd um i zeo. To how hi, we have o pecify he way of ummaion. Le u collec in a goup he em k i k j i j aibued o hoe k Z n whoe coodinae excep he ih and he jh coincide, while he ih and he jh coodinae aify he following ule: hey ae obained fom k i and k j aibued o one of he veco of he goup by mean of an abiay aignmen of a ign. Thi opeaion pecifie fou veco. The ohe fou veco ae obained fom he fi fou veco of he goup by mean of he pemuaion of he ih and he jh coodinae. In oal, we ge eigh veco in he goup. Clealy, he ummand k i k j i j aibued o hee veco cancel each ohe. Le u compue he fi um. k Z n, k N k α n ki i = i= n [ i= k Z n, k N i j ] k α k i i.

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