Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada-Watanabe principle

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1 Coucio of Malliavi diffeeiable og oluio of SD ude a iegabiliy codiio o he dif wihou he Yamada-Waaabe icile David R. Baño davidu@mah.uio.o Side Duedahl ided@mah.uio.o Thilo Meye-Badi Meye-Badi@mah.lmu.de ad Fak Poke oke@mah.uio.o Abac: I hi ae we aim a emloyig a comace cieio of Da Pao, Malliavi, Nuala 3] fo uae iegable Bowia fucioal o couc uiue og oluio of SD ude a iegabiliy codiio o he dif coefficie. The obaied oluio u ou o be Malliavi diffeeiable ad ae ued o deive a Bimu-lwohy-Li fomula fo oluio of he Kolmogoov euaio. MSC 21 ubjec claificaio: 6H1, 6H7, 6H4, 6J6.. Keywod ad hae: Sog oluio of SD, Malliavi calculu, Kolmogoov euaio, Bimu-lwohy-Li fomula, igula dif coefficie.. 1. Ioducio The objec of udy of hi ae i he ochaic diffeeial euaio SD X = x b, X x d B, T, x R d, 1 whee B i a d-dimeioal Bowia moio o ome comlee obabiliy ace Ω, F, µ wih eec o a µ-comleed Bowia filaio {F } T ad whee b :, T ] R d R d i a Boel-meauable fucio. I hi aicle we ae ieeed i he aalyi of og oluio X of he SD 1, ha i a {F } T -adaed oluio ocee o Ω, F, µ whe he dif coefficie i iegula, e.g. o-lichizia o dicoiuou. A widely ued coucio mehod fo og oluio i hi cae i he lieaue i baed o he o-called Yamada-Waaabe icile. Uig hi icile, a oce couced weak oluio, ha i a oluio which i o eceaily a fucioal of he divig oie, combied wih ahwie uiuee give a uiue og oluio. So Weak oluio Pahwie uiuee Uiue og oluio. 2 Hee, ahwie uiuee mea he followig: If X 1 ad X 2 ae {F 1 } T - ad eecively {F 2 } T -adaed weak oluio o a obabiliy ace, he hee oluio mu 1

2 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio2 coicide a.. See 31]. I he mileoe ae fom ], A.K. Zvoki ued he Yamada- Waaabe icile i he oe-dimeioal cae i coecio wih PD echiue o couc a uiue og oluio o 1, whe b i meely bouded ad meauable. Subeuely, he lae eul wa geealied by A.Y. Veeeikov 3] o he mulidimeioal cae. Imoa ohe ad moe ece eul i hi diecio ae e.g. 15], 9] ad 14]. See alo he ikig wok 2] i he Hilbe ace eig, whee he auho ue oluio of ifiie-dimeioal Kolmogoov euaio o obai uiue og oluio of ochaic evoluio euaio wih bouded ad meauable dif fo a.e. iiial value. I hi aicle we wa o emloy a coucio icile fo og oluio develoed i 22]. Thi mehod which elie o a comace cieio fom Malliavi Calculu fo uae iegable fucioal of he Bowia moio 3] i i diameical ooiio o he Yamada- Waaabe icile 2 i he ee ha Sog exiece Uiuee i law Sog uiuee, ha i he exiece of a og oluio o 1 ad uiuee i law of oluio imly he exiece of a uiue og oluio. A cucial coeuece of hi aoach i he addiioal iigh ha he couced oluio ae egula i he ee of Malliavi diffeeiabiliy. We meio ha hi mehod ha bee ecely alied i a eie of ohe ae. See e.g. 2], whee he auho obai Malliavi diffeeiable oluio whe he dif coefficie i R d i bouded ad meauable. Ohe alicaio eai o he ochaic ao euaio wih igula coefficie 23], 24] o ochaic evoluio euaio i Hilbe ace wih bouded Hölde-coiuou dif 8]. See alo 1] i he cae of ucaed α-able ocee a divig oie ad 1] i he cae of facioal Bowia moio fo Hu aamee H < 1/2, which i a o-makovia divig oie. Uig he above meioed ew aoach, oe of he objecive of hi ae i o couc Malliavi diffeeiable uiue og oluio o 1 ude he iegabiliy codiio fo 2, > 2 uch ha b L, T ], L R d, R d 3 d 2 < 1. The idea fo he oof e o a mixue of echiue i 2] ad 6]. Moe eciely, we aoximae i he fi e he dif coefficie b by mooh fucio b wih comac o ad aly he Iô-Taaka-Zvoki ick by afomig he oluio X,x of 1 aociaed wih he coefficie b o ocee whee he ocee Y,x Y,x := X,x U, X,x, aify a euaio wih moe egula coefficie ha 1 give by dy,x = λu, X,x fo oluio U o he backwad PD U d I d U, X,x db 1 2 U b U = λu b, U T, x =. 4 I he ecod e we ue he comace cieio fo L 2 Ω i 3] alied o he euece Y,x, 1 i coecio wih Schaude-ye of eimae of oluio of 4 ad echiue fom whie oie aalyi o how ha Y,x Y x i L 2 Ω fo all ad ha X x = ϕ, Y x,

3 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio3 whee ϕ, i he ivee of he fucio x x U, x fo all ad U a oluio of 4, i a Malliavi diffeeiable uiue og oluio of 1. Ou ae i ogaied a follow: I Secio 2 we ee ou mai eul o he coucio of og oluio Theoem 2.1 ad Theoem A a alicaio of he eul obaied i Secio 2 we eablih i Secio 3 a Bimu-lwohy-Li fomula fo he eeeaio of fi ode deivaive of oluio of Kolmogoov euaio. 2. Mai eul I hi ecio, we wa o fuhe develo he idea ioduced i 6] ad 22] o deive Malliavi diffeeiable og oluio of ochaic diffeeial euaio wih iegula coefficie. Moe eciely, we aim a aalyzig he SD of he fom dx = b, X d db, 1, X = x R d, 5 whee he dif coefficie b :, T ] R d R d i a Boel meauable fucio aifyig ome iegabiliy codiio ad B i a d-dimeioal Bowia moio wih eec o he ochaic bai Ω, F, µ, {F } T 6 fo he µ augmeed filaio {F } T geeaed by B. A he ed of hi ecio we hall alo aly ou echiue o euaio wih moe geeal diffuio coefficie Theoem Coide he ace L := L, T ], L R d, R d fo, R aifyig he followig codiio > 2, > 2 ad d 2 < 1 7 ad deoe by he uclidea om i R d. The Baach ace L i edowed wih he om / 1/ f L = f, x dx d < 8 R d fo f L. The mai goal of he ae i o how ha SD of he ye 5 wih dif coefficie b aifyig he iegabiliy codiio give i 8 admi og oluio ha ae uiue ad i addiio, Malliavi diffeeiable. So, ou mai eul i he followig heoem: Theoem 2.1. Suoe ha he dif coefficie b :, T ] R d R d i 5 belog o L. The hee exi a uiue global og oluio X o euaio 5 uch ha X i Malliavi diffeeiable fo all T. A imoa e of he oof of Theoem 2.1 i diecly baed o he udy of he egulaiy of oluio o he followig aociaed PD o euaio SD 5. U, x b, x U, x 1 U, x λu, x b =,, T ], UT, x =, 9 2 whee U :, T ] R d R d, λ > ad b L. The followig eul i due o 5] ad ablihe he well-oede of he above PD oblem i a ceai ace.

4 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio4 ad Fi, ecall he defiiio of he followig fucioal ace H α, = L, T ], W α, R d, The om i H α, ca be ake o be H α, = H α, H 1,. u H α, u H α, u L. H β, = W β,, T ], L R d Theoem 2.2. Le, be uch ha 2, > 2 ad d 2 < 1 ad λ >. Coide wo veco field b, Φ L. The hee exi a uiue oluio of he backwad aabolic yem belogig o he ace u 1 u b u λu Φ =,, T ], ut, x = 1 2 H 2, := L, T ], W 2, R d W 1,, T ], L R d, i.e. hee exi a coa C > deedig oly o d,,, T, λ ad b L uch ha u H 2, C Φ L. 11 The followig eul i a a of 15, Lemma 1.2] ha give u ome oeie o he egulaiy of u H 2, ha we will eed fo he oof of Theoem 2.1. Lemma 2.3. Le, 1, uch ha d 2 < 1 ad u H 2,, he u i Hölde coiuou i, x, T ] R d, amely fo ay ε, 1 aifyig ε d 2 < 1 hee exi a coa C > deedig oly o, ad ε uch ha fo all,, T ] ad x, y R d, x y u, x u, x C ε/2 u 1 1/ ε/2 H 2, u, x whee deoe ay om i R d d u 1/ε/2, 12 u, x u, y x y ε CT 1/ u H 2, T u L, 13 Ou mehod o couc og oluio i acually moivaed by he followig obevaio i 17] ad 21] ee alo 22]. Pooiio 2.4. Suoe ha he dif coefficie b :, T ] R d R d i 5 i bouded ad Lichiz coiuou. The he uiue og oluio X = X 1,..., X d of 5 ha he exlici eeeaio ϕ, Xω i = µ ϕ, ] ω i T b 14 fo all ϕ :, T ] R R uch ha ϕ, B i L 2 Ω fo all T, i = 1,..., d,. The adom eleme T b i give by T bω, ω := ex d 1 2 L W j ω b j, ω d ω j W j ω b j, ω 2 d. 15

5 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio5 Hee Ω, F, µ, i a coy of he uadule Ω, F, µ, B i 6. Fuhe µ deoe a Pei iegal of adom eleme Φ : Ω S wih eec o he meaue µ. The Wick oduc i he Wick exoeial of 15 ee 6 i ake wih eec o µ ad W j i he whie oie of B j i he Hida ace S ee 57. The ochaic iegal φ, ωd ω j i 15 ae defied fo edicable iegad φ wih value i he couclea ace S. See e.g. 12] fo defiiio. The ohe iegal ye i 15 i o be udeood i he ee of Pei. Remak 2.5. Le = 1 < 2 <... < m = T be a euece of aiio of he ieval, T ] wih max m 1 i=1 i1 i. The he ochaic iegal of he whie oie W j ca be aoximaed a follow: W j ωd j ω = lim i=1 m j ω j i1 ωw j i ω i i L 2 λ µ; S. Fo moe ifomaio abou ochaic iegaio o couclea ace he eade i efeed o 12]. I he euel we hall ue he oaio Y i,b fo he execaio o he igh had ide of 14 fo ϕ, x = x, ha i fo i = 1,..., d. We e Y i,b Y b = ] := µ i T b Y 1,b,..., Y d,b. 16 The fom of Fomula 14 i Pooiio 2.4 acually give ie o he cojecue ha he execaio o he igh had ide of Y b i 16 may alo defie oluio of 5 fo dif coefficie b lyig i L. Ou mehod o couc og oluio o SD 5 which ae Malliavi diffeeiable i eeially baed o hee e. Fi, we coide a euece of comacly oed mooh fucio b :, T ] R d R d, uch ha b := b ad b L < aoximaig b L a.e. wih eec o he Lebegue meaue ad he we ove ha he euece of og oluio X = Y b, 1, i elaively comac i L 2 Ω; R d Coollay 2.9 fo evey, T ]. The mai ool o veify comace i he boud i Lemma 2.6 i coecio wih a comace cieio i em of Malliavi deivaive obaied i 3] ee Aedix B. Thi e i oe of he mai coibuio of hi ae. Secodly, give a meely meauable dif coefficie b i he ace L, we how ha Y b,, T ] i a geealized oce i he Hida diibuio ace ad we ivoke he S- afom 58 o ove ha fo a give euece of a.e. aoximaig, mooh coefficie b wih comac o uch ha b L, a ubeuece of he coeodig og oluio X j = Y b j fulfil Y b j Y b i L 2 Ω; R d fo T Lemma Fially, uig a ceai afomaio oey fo Y b Lemma 2.14 we diecly how ha Y b i a Maaliavi diffeeiable oluio o 5. We u ow o he fi e of ou ocedue. The ucceful comleio of he fi e elie o he followig eeial lemma: Lemma 2.6. Le b :, T ] R d R d, 1 be a euece of fucio i C R d ace of ifiiely ofe diffeeiable fucio wih comac o aoximaig b L a.e. uch ha b := b ad b L <. Deoe by X,x he og oluio of SD 5 wih dif

6 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio6 coefficie b fo each. The fo evey, T ], hee exi a < δ < 1 ad a fucio C : R, deedig oly o,, d, δ ad T uch ha wih D X,x Hee deoe ay om i R d d. Moeove, fo all 2. D X,x 2] C b L, δ 17 C b L <. 1 1,T ] D X,x ] < 18 Poof. Thoughou he oof we will deoe by C : R, ay fucio deedig o he aamee. We will alo ue he ymbol o deoe le o eual u o a oiive eal coa ideede of. We will ove he above eimae by coideig he oluio of he aociaed PD eeed i 9 wih b, i lace of b which we deoe by U, ad he uig he eul ioduced a he begiig of hi ecio o he egulaiy of i oluio. Fi, le u ioduce a ew oce ha will be ueful fo hi uoe. Coide fo each ad, T ] he fucio γ, : R d R d defied a γ, x = x U, x. I u ou, ee 5, Lemma 3.5], ha he fucio γ,,, T ], defie a family of C 1 -diffeomohim o R d. Fuhemoe, coide he auxiliay oce := γ, X,x,, T ], 1. Oe check,x uig Iô fomula ad 9 ha X aifie he followig SD,x d X = λu, γ, 1,x X d I d U, γ, 1,x X db, X,x = x U, x 19 which i euivale o SD 5 if we elace b by b, 1. Uig he chai ule fo Malliavi deivaive ee e.g. 25] we ee ha fo, D X,x X,x = γ, X,x D X,x.,x Becaue of Lemma B.4 i uffice o ove he eimae 17 ad 18 fo he oce X. Sice b ae ow mooh we have ha 19 admi a uiue og oluio which ake he fom X,x = x U, x λ The he Malliavi deivaive of X,x D X,x U, γ, 1,x X d = I d U, γ 1 λ I d U, γ, 1,x X db. fo, which exi ee e.g. 25], i, U, γ 1, 2 U, γ 1, X,x,x X γ, 1,x X γ, 1 X,x D X,x d,x X D X,x db.

7 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio7 Deoe fo imliciy, Z, := D X,x. The fo < we ca wie Z, Z, = U, γ 1 λ λ, U, γ 1, U, γ 1, 2 U, γ 1, 2 U, γ 1, = Z, Z, λ U, γ 1, 2 U, γ 1,,x X U, γ 1,x, X,x X γ, 1,x X Z,d,x X γ, 1,x X Z, Z, d,x X γ, 1,x X Z,dB,x X γ, 1,x X Z, Z, db,x X γ, 1,x X Z, Z, d,x X γ, 1,x X Z, Z, db. By di of Lemma B.3 we kow ha U i bouded uifomly i ad Lemma B.2 how ha 2 U belog, a lea, o L uifomly i. Thi imlie ha he ochaic iegal i he exeio fo Z, Z, i a ue maigale, which we hee deoe my M. A a eul, ice he iiial codiio Z, Z, i F -meauable fo each, fo a give α 2, by Iô fomula we have Z, Z, α Z, Z, α Z, Z, α 2 T 2 U, γ, 1 Z, Z, α d M 2 U, γ 1,x X γ, 1,,x X γ, 1 ],x X Z, Z, d,x X Z, Z, 2 whee hee T ad fo he ace ad fo he aoiio of maice. We oceed he uig he fac ha he ace of he maix aeaig i 2 ca be bouded by a coa C,d ideede of, ime Z, Z, 2 2 U, γ, 1,x X γ, 1,x X 2. Alogehe, Z, Z, α Z, Z, α Coide hu he oce Z, Z, α d M Z, Z, α 2 U, γ 1,,x X γ, 1,x X 2 d 21 The oce V V := 2 U, γ 1, The Lemma B.2 i coecio wih Theoem 2.2 we have ha The Iô fomula yield,x X γ, 1,x X 2 d. 22 i a coiuou o-deceaig ad {F },T ] -adaed oce uch ha V =. V ] <.

8 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio8 e V Z, Z, α Z, Z, α The akig execaio e V Z, Z, α] Z, Z, α] e V Z, Z, α d e V dm. 23 e V Z, Z, α] d. 24 The Gowall ieualiy give e V Z, Z, α] Z, Z, α]. 25 A hi oi, i i eay o ee, followig imila e, ha fo he oce Z, oe ha e V Z, α] Z, α], whee Z, = I d U, γ 1,T ] e V,,x X. So Z, α] 1,T ] becaue of Lemma B.3 ii fo a ufficiely lage λ R. The, he Cauchy-Schwaz ieualiy ad Lemma B.5 give,t ] Z, α],t ] e 2V We coiue o ove he eimae 17. Recall ha Z, Z, = U, γ 1 λ, U, γ 1, 2 U, γ 1, U, γ, 1,x X α] < 26 Z, 2α] 1/2 ] 1/2 e 2V T <.,x X U, γ 1,x, X,x X γ, 1,x X Z,d 27,x X γ, 1,x X Z,dB. The akig om ad uig Bukholde-Davi-Gudy ieualiy we ge Z, Z, α] U, γ 1,x, X U, γ 1,x, X α] α ] 28 λ α U, γ, 1,x X γ, 1,x X Z, d ] α/2 2 U, γ, 1,x X γ, 1,x X Z, 2 d. =: i ii iii The aim ow i o fid Hölde boud i he ee of 17 fo he exeio aeaig i 28. Fo i we may wie i = U, γ 1, U, γ 1, U, γ 1,,x X U, γ 1,x, X α],x X U, γ 1,,x X α] X,x U, γ 1, X,x α].

9 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio9 The by Lemma 2.3 hee exi a ε, 1/α ad a coa C,,d,α > ideede of uch ha ] U, γ 1,x, X U, γ 1,x, X α α C,,d,α ε/2 U 1 1/ ε/2 U 1/ε/2 ad U, γ 1,,x X U, γ 1 C,,d,α T α/, γ 1, X,x α ],x X γ 1, H 2, L,x X αε] α U H T U 2, L. The above boud i coecio wih ieualiy 11 i Theoem 2.2 give i C,,d,α,T b L αε/2 γ 1,x, X γ 1,x, X αε] fo ome coiuou fucio C,,d,α,T ad hece C,,d,α,T b L <. Moeove, uig Giaov heoem, we obai ha ] γ 1,x, X γ 1,x, X = X,x X,x ] ] T b, x B d b u, x B u db u B B ] 1/2 1/2 ] 1/2 b, x B 2 d 1/2, whee we ued, Cauchy-Schwaz ieualiy ad boh ha 2 T b u, x B u db u < ad ] 1/2 b, x B 2 d <, ee 15, Lemma 3.2] o Lemma B.1. By Jee ieualiy fo cocave fucio ad he eviou eimae we have γ 1,x, X γ 1,x, X αε] αε γ 1,x, X γ 1,x, X ] αε/2. Alogehe, i C,,d,α,T b L δ fo a δ, 1. Fo he ecod em, ii, we ue Hölde ieualiy, Lemma B.3 ii fo a ufficiely lage λ R, Lemma B.4 ad he eimae 18 o obai

10 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 1 ii λ α α 1 C,,d,α,T δ,] U, γ 1,,x X γ, 1,x X 2α] d 1/2 fo a δ, 1. Fially, fo he hid em, fo α 2, we ue Hölde ieualiy o obai iii α 2 2 ] 2 U, γ, 1 X α γ, 1 X α Z, α d. The chooe α = 21 δ wih δ, 1/4 ad ue Lemma B.4 o ge ] iii δ 2 U, γ, 1 X 21δ Z, 21δ d. Z, 2α] 1/2 d The Fubii heoem, Hölde ieualiy oce moe wih eec o µdω, wih exoe 1δ, δ, 1/4 ad Cauchy-Schwaz yield ] 2 U, γ, 1 X 21δ Z, 21δ d = 2 U, γ, 1 X 21δ Z, 21δ] d 2 U, γ, 1 X ] 1/1δ 21δ1δ,] Z, 21δ 1δ δ ] δ 1δ Z 2 U, γ 1, X 21δ1δ ] 1/1δ d, 21δ 1δ δ ] δ 1δ d 2 U, γ, 1 X ] 21δ1δ 1/1δ d whee he la e follow fom 18. Fo he la faco, ice < 1/1 δ < 1, uig he ivee Jee ieualiy ad he fac ha 1 < 1 δ1 δ < 2 fo uiable δ, δ, 1/4 i coecio wih Lemma B.2 we have 2 U, γ 1, X 21δ1δ T 1 1/1δ fo evey, w... a coa M. A a ummay, i follow fom 25 ha ] 1/1δ d 2 U, γ 1, X 21δ1δ d ] 1/1δ M < e V Z, Z, 21δ] C,,d,α,T b L δ. The by Hölde ieualiy wih exoe 1δ, δ, 1 ogehe wih Lemma B.5 we obai Z, Z, 2] = e 1 1δ V e 1 1δ V Z, Z, 2] ] e 1 δ V δ 1δ e V Z, Z, 21δ] 1 1δ wih C,,d,α,T b L δ/1δ C,,d,α,T b L <.

11 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 11 Remak 2.7. The boud give i 18 i i fac uifom i x R d. Ideed, by Lemma B.3 iem ii we have ha he boud give i 26 i alo uifom i x R d. Moeove, ice U L fo all, he by Lemma B.3 iem iii i coecio wih Lemma B.1 we have ha fo ay k R e kv T ] <. Hece, fo ay α 1 x R d x R d,t ] D X,x α ] <. Remak 2.8. Oe alo check ha he ame hold fo he aial deivaive, ha i fo ay α 1 ] x X,x α < x R d,t ] by uig he fac ha x X,x olve he ame SD a D X,x, aig a =. A a eecuio of Lemma 2.6 we have he followig eul which i ceal i he oof of he exiece of og oluio of 5. Coollay 2.9. Le {b } be a euece of comacly oed mooh fucio aoximaig b i L. Deoe, a befoe, X x, he oluio o euaio 5 wih dif coefficie b. The fo each, T ] he euece of adom vaiable X,x, i elaively comac i L 2 Ω. Poof. Thi i a diec coeuece of he comace cieio ha ca be foud i Aedix C, Lemma C.1 ad C.2, which i due o 3], ogehe wih Lemma 2.6. Oe ca check ha he double iegal i Lemma C.2 i fiie. Namely fo ay < δ < 1 ad 2β 1 δ < 1. Z, Z, 2] 12β d d belog o he Hida dii- The followig lemma give a cieio ude which he oce Y b buio ace. Lemma 2.1. Suoe ha µ ex 36 b, B 2 d 1 2β1 δ d d < ] <, 29 whee he dif b :, 1] R d R d i meauable i aicula, 29 i valid fo b L becaue of Lemma B.1. The he coodiae of he oce Y b, defied i 16, ha i ] Y i,b ae eleme of he Hida diibuio ace. = µ i T b, 3 Poof. See 22] fo a imila oof. Lemma Le ε, 1 ad defie ε := 1 ε ad ε := 1ε ε. Le b :, T ] R d R d be a euece of Boel meauable fucio wih b = b uch ha ] ex 16 ε 8 ε 1 b, B 2 d < 31

12 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 12 hold. The SY i,b Y i,b φ co J ] 1 ε ex 28 ε 1 φ 2 d fo all φ S C, 1] d, i = 1,..., d, whee S deoe he S-afom ee Secio A.1 i Aedix A ad whee he faco J i defied by J = d 2 b, b, 2 d ε 2 b, 2 b, 2 d Hee S C, 1] i he comlexificaio of he Schwaz ace S, 1] o, 1], ee Secio A.1 i Aedix A. I aicula, if b aoximae b i he followig ee a, i follow ha a fo all 1, i = 1,..., d. Y b ε. 32 J ] 33 Y b i S Poof. Fo i = 1,..., d we obai by Pooiio 2.4 ad 59 ha { d SY i,b Y i,b i φ µ ex Re b, φ d 1 ] } b, φ 2 d 2 { d ex 1 2 b, b, φ b, b, 2 b, b, 2 d d d } ] 1. Sice ex{z} 1 z ex{ z } i follow fom Hölde ieualiy wih exoe ε = 1 ε ad ε = 1ε ε, fo a aoiae ε >, ha SY i,b Y i,b φ µ Q ε ] 1 ε µ 1 2 b, i ex { d Re b, φ d] } ] 1 ε ε 2 ex { ε Q }, φ d whee Q = d b, φ b, b, b, d 1 2 d. b, 2 b, 2 d

13 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 13 The uig he Cauchy-Schwaz ieualiy o he la iegal ad he fac ha x e x ad 1 e x fo x we may wie ε/2 T d T ε µ Q ε ] C ex φ 2 d µ b, b, d 1 T ε b, 2 b, 2 d 2 ε T 2 ] b, b, 2 d ε/2 T d ε T 2 = C ex φ 2 d µ 2 b, b, 2 d T ε ] b, 2 b, 2 d, whee i he la ieualiy we ued he Bukholde-Davi-Gudy ieualiy fo he ochaic iegal. The µ Q ε ] 1 ε ε/2 1 T C ex φ 2 d µ J ] 1 ε ε, whee J = d 2 b, b, 2 d ε 2 b, 2 b, 2 d ε. Fuhe we ge ha µ 1 2 i ex µ 1 2 { d b, i ex b, Re b, φ d φ 2 d] } ε ex { ε Q } { d Re b, ] } ] 1 2ε 2ε φ 2 d ] 1 ε φ d µ ex {2 ε Q } ] 1 2ε. The fo z C oe ha ex{ z } 1 2 ex{2re z} ex{ 2Re z} ex{2im z} ex{ 2Im z}. Thu µ ex {2 ε Q } ] 1 2ε ] 1 ] 1 µ 1 2ε 2ε ex {4 ε Re Q } µ ex { 4 ε Re Q } 2 2ε µ ex {4 ε Im Q } ] 1 ] 1 2ε 2ε µ ex { 4 ε Im Q }.

14 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 14 By he Cauchy-Schwaz ieualiy ad he emaigale oey of Doléa-Dade exoeial we ge ] µ µ ex {4 ε Re Q } { d ex 32ε 2 4 ε b, b, 2 b, 8 ε Re φ b, { } T L ex 2 ε φ 2 d, b, 2 d b, 2 d ] }] 1 2 d whee he la e follow fom he fac ha f, g 1 2 f 2 g 2, f, g L 2, T ] ad whee { d L = µ ex 4 ε 8 ε 1 b, b, 2 d 4 ε b, 2 b, ] }] d. Similaly, oe alo obai ] { } T µ ex { 4 ε Re Q } L ex 2 ε φ 2 d. I he ame way, oe alo obai he ame boud fo µ ex{4 ε Im Q }] ad µ ex{ 4 ε Im Q }]. Fially, fo he emaiig faco we ee ha µ 1 2 i ex { d b, Re b, φ d ] } ] 1 2ε 2ε φ 2 d { ] 1 i µ 4ε 4ε µ ex 4 ε 1 2 b, ] 1 i µ 4ε 4ε µ ex d ] }] 1 4ε φ 2 d { d Re b, 4 ε 8 ε 1 Re b, φ d φ 2 d }] 1 4ε Now, ice Re z 2 Re z 2, z C we have ha Re b φ 2 b Re φ 2 he uig Mikowki ieualiy, i.e. f g 2 1 f g fo ay 1 ad Cauchy-Schwaz ieualiy w... µ oe fially obai.

15 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 15 µ 1 2 i ex { d b, C µ ex { Alogehe, we obai SY i,b Y i,b Re b, φ d ] } ] 1 2ε 2ε φ 2 d 16 ε 8 ε 1 b, 2 d φ co J ] 1 }] 1 8ε 1ε ex {2 ex { 8 1 ε ε 28 ε 1 φ 2 d } 1 φ 2 d. }. Lemma Le b :, T ] R d R d be a euece of mooh fucio wih comac o wih b := b which aoximae he coefficie b :, T ] R d R d i L. The fo ay T hee exi a ubeuece of he coeodig og oluio X j, = Y b j, j = 1, 2..., uch ha Y b j Y b fo j i L 2 Ω. I aicula hi imlie Y b L 2 Ω, T. Poof. By Coollay 2.9 we kow ha hee exi a ubeuece Y b j, j 1, covegig i L 2 Ω. Fuhe, we eed o how ha J j ] a j wih J j a i 32. To hi ed, obeve ha fo a fucio f L oe ha ] T f, d = 2π d/2 f, ze z 2 /2 dzd. R d The by uig Hölde ieualiy wih eec o z ad he o we ee ha fo ay, 1, ] aifyig d 2 < 2, we have ] T f, d C f L, whee C i a coa deedig o T, d,,. The fom codiio 7, ice, > 2 we ca fid a δ, 1 mall eough o ha, > 21 δ. Fo hee, defie := 21δ 1 ad := 21δ > 1 ad aly he above eimae o f 21δ o obai ] T f, 21δ d C f L. 34 Now ice b b L fo evey j = 1,..., d ad < 1ε 2 < 1 we have b, b, 2 d 1ε 2 b, b, 2 d ] 1ε 2

16 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 16 which goe o zeo by he above eimae 34 by ju akig he cae whee δ =. Fially, fo he he ecod em i J j ] we have b, T ε b, T ε b, T ε b, 2 b, b, 2 d 1ε ] 1ε b, b, b, 21ε] 1/2 b, 1/2 b, d] 21ε The ice b b L fo evey we have b, 1ε d b, ] 21ε] 1/2 d 1/2 b, d] 21ε. b, 1/2 b, d] 21ε < fo a ufficiely mall ε, 1 by Lemma B.2 ad b, 1/2 b, d] 21ε a by eimae 34 fo a ufficiely mall ε >. Y b j Thu, by Lemma 2.11, Y b j Y b i L 2 Ω. Y b Remak I follow fom he above oof ha Y b ad x. a j i S. Bu he, by uiuee of he limi, alo Y b a i L 2 Ω; R d fo all I fac, Lemma 2.12 eable u ow o ae he followig afomaio oey fo Y b. Lemma Aume ha b :, T ] R d R d i i L. The ϕ i, Y b = µ ϕ i, ] T b a.e. fo all T, i = 1,..., d ad ϕ = ϕ 1,..., ϕ d uch ha ϕb L 2 Ω; R d. Poof. See 29, Lemma 16] o 21]. Uig he above auxiliay eul we ca fially give he oof of Theoem 2.1. Poof of Theoem 2.1. We wa o ue he afomaio oey 35 of Lemma 2.14 o how ha Y b i a uiue og oluio of he SD 5. To hoe oaio we e ϕ, ωdb := d ϕ, ωdb ad x =. Alo, le b, = 1, 2,..., be a euece of fucio a euied i Lemma We comme o ha Y b ha a coiuou modificaio. The lae ca be ee a follow: Sice each Y b i a og oluio of he SD 5 wih eec o he dif b we obai fom Giaov heoem ad ou aumio ha ] 4 4 ] T µ Y i,b Yu i,b = µ i i u b, d co u 2 35

17 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 17 { fo all u, T, 1, i = 1,..., d. The above coa come fom he fac ha b, d i bouded i L } 1 2 Ω; R d wih eec o he meaue µ, ee Lemma 3.2. i 15] o Lemma B.1. By Remak 2.13 we kow ha Y b Y b i L 2 Ω; R d ad hece we have almo ue covegece fo a fuhe ubeuece, T. So we ge ha by Faou lemma ] 4 µ Y i,b Yu i,b co u 2 36 fo all u, T, i = 1,..., d. The Kolmogoov lemma guaaee a coiuou modificaio of Y b. Sice i a weak oluio of 5 fo he dif b, x φ wih eec o he meaue dµ T = b, φ d dµ we ge ha SY i,b φ = µ i ] b, φ d ] = i µ = µ b i, φ i = µ b i, ] d Thu he afomaio oey 35 alied o b yield SY i,b φ = S ] bu, u φu d u d S b i u, Y i,b u The i follow fom he ijeciviy of he S-afom ha Y b = duφ SB i φ. b, Y b d B. B i φ. See Secio A i he Aedix. The Malliavi diffeeiabiliy of Y b come fom he fac ha Y i,b Y i,b i L 2 Ω ad Y i,b D 1,2 M < 1 fo all i = 1,..., d ad 1. See e.g. 25]. O he ohe had, uig uiuee i law, which i a coeuece of Lemma B.2 ad Pooiio 3.1, Ch. 5 i 13] we may aly, ude ou codiio, Giaov heoem o ay ohe oluio. The he oof of Pooiio 2.4 ee e.g. 28, Pooiio 1] how ha ay ohe oluio eceaily coicide wih Y b. We coclude hi ecio wih a geealiaio of Theoem 2.1 o a cla of o-degeeae d dimeioal Iô-diffuio. Theoem Aume he ime-homogeeou R d valued SD dx = bx d σx db, X = x R d, T, 37

18 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 18 whee he coefficie b : R d R d ad σ : R d R d R d ae Boel meauable. Suoe ha hee exi a bijecio Λ : R d R d, which i wice coiuouly diffeeiable. Le Λ x : R d L R d, R d ad Λ xx : R d L R d R d, R d be he coeodig deivaive of Λ ad aume ha Λ x yσy = id R d fo y a.e. a well a Λ 1 i Lichiz coiuou. Reuie ha he fucio b : R d R d give by b x := Λ x Λ 1 x bλ 1 x ] 1 2 Λ xx Λ 1 x d σλ 1 x e i ], i=1 ] d σλ 1 x e i ] aifie he codiio of Theoem 2.1, whee e i, i = 1,..., d, i a bai of R d. The hee exi a Malliavi diffeeiable oluio X o 37. Poof. The oof ca be diecly obaied fom Iô Lemma. See 22]. i=1 3. Alicaio 3.1. The Bimu-lwohy-Li fomula A a alicaio we wa o ue Theoem 2.1 o deive a Bimu-lwohy-Li fomula fo oluio v o he Kolmogoov euaio d v, x = b j, x x j v, x 1 2 d 2 x 2 i=1 i v, x 38 wih iiial codiio v, x = Φx, whee b :, T ] R d R d belog o L. I i kow ha, ee 15] o 7], ha whe Φ i coiuou ad bouded hee exi a oluio o 38 give by v, x = ΦX x ], 39 whee v i a oluio o he Kolmogoov uaio 38 which i uiue amog all bouded oluio i he ace H 2,, a ioduced i Theoem 2.2, wih, > 2 aifyig 7. Moeove, x v L, T ] R d. I he euel, we aim a fidig a eeeaio fo xv wihou uig deivaive of Φ. See 2] i he cae of b L, T ] R d. Theoem 3.1 Bimu-lwohy-Li fomula. Aume Φ C b R d ad le U be a oe, bouded ube of R d. The he deivaive of he oluio o 38 ca be eeeed a x v, x = ΦXx a x Xx db ] 4 fo almo all x U ad all, T ], whee a = a i ay bouded meauable fucio uch ha a d = 1 ad whee deoe he aoiio of maice. Poof. The oof i imila o Theoem 2 i 22] i he cae of b L, T ] R d. Fo he coveiece of he eade we give he full oof. Aume ha Φ Cb 2Rd he geeal cae of Φ C b R d ca be oved by aoximaio of Φ i elaio 42 ad le b ad X,x be a i he eviou ecio. If we elace b by b i 38 we have he uiue oluio give by v, x = ΦX,x ].

19 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 19 By uig Remak 2.13 we ee ha v, x v, x fo each ad x. By 25, Page 19] we have ha D X,x x X,x = x X,x, whee he above oduc i he uual maix oduc. So i follow ha x X,x = ad X,x x X,x d. 41 Iechagig iegaio ad diffeeiaio i coecio wih he chai ule we fid ha x v, x = Φ X,x x X,x ] = = ΦX,x ad ΦX,x a x X,x x X,x d] db ], whee we alied he chai ule ad he dualiy fomula fo he Malliavi deivaive o he la eualiy. Chooe ϕ C U. I wha follow, we will ove ha R d ϕxv, xdx = ϕxφx x x R d I fac, domiaed covegece combied wih Remak 2.13 give R d ϕxv, xdx = lim ϕxφx,x x R d = lim ϕxφx,x R d lim ϕxφx x R d = lim i lim ii. a x Xx db ] dx. 42 a ΦX x a x X,x a db ] dx x X,x db ] dx x X,x db ] dx A fo he fi em we ge i R d ϕx x Φ X,x X x L 2 Ω;R d a 1/2 k 1,,T ] x Xk,x 2 R d d] dx, which goe o zeo a ed o ifiiy by Lebeue domiaed covegece heoem, Remak 2.13 ad Remak 2.8. Fo he ecod em, ii ice X x i Malliavi diffeeiable ad Φ Cb 2Rd i follow fom he Clak-Ocoe fomula ha ee e.g. 25] ΦX x = ΦX x ] D ΦX x F ]db.

20 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 2 So ii = ϕxφx x R d = ϕx R d = a ΦX x ] a x X,x db ] dx 43 D ΦX x F ]db a x X,x db ] dx 44 R d ϕxd ΦX x x X,x ]dxd. 45 Oe check by mea of Lemma 2.6 ha ϕ D ΦX = ϕ Φ X D X belog o L 2 R d Ω; R d o ha fo each, he fucio g = ϕxd ΦX x R x X,x ]dx d covege o ϕxd R d ΦX x x Xx ]dx by he weak covegece of x X,x i L 2, T ] U Ω fo a ubeuece i viue of Remak 2.8. Fuhe, g ϕx D ΦX x L 2 Ω;R d R x X,x L 2 Ω;R dx d d D u ΦX y L2 Ω;R d y R d, u, k N x Xk,y u L2 Ω;R d ϕx dx R d o ha Lebegue domiaed covegece heoem give lim ii = a ϕxd ΦX x R x Xx ]dxd. d By eveig euaio 43, 44 ad 45 wih x Xx i lace of x X,x we obai he eul. Aedix A: Famewok I hi aedix we collec ome fac fom Gauia whie oie aalyi ad Malliavi calculu, which we hall ue i Secio 2 o couc og oluio of SD. See 11, 27, 16] fo moe ifomaio o whie oie heoy. A fo Malliavi calculu he eade may coul 25, 18, 19, 26]. A.1. Baic Fac of Gauia Whie Noie Theoy A cucial e i ou oof fo he coucio of og oluio ee Secio 3 elie o a geealied ochaic oce i he Hida diibuio ace which i how o be a SD oluio. Le u fi ecall he defiiio of hi ace which i due o T. Hida ee 11]. Fom ow o we fix a ime hoizo < T <. Le A be a oiive elf-adjoi oeao o L 2, T ] wih SecA > 1. Reuie ha A i of Hilbe-Schmid ye fo ome > ad le {e j } j be a comlee ohoomal bai of L 2, T ] i DomA ad le λ j >, j be he eigevalue of A uch ha 1 < λ λ Suoe ha each bai eleme e j i a coiuou fucio o, T ]. Fuhe le O λ, λ Γ, be a oe coveig of, T ] uch ha fo αλ. j λ αλ j e j < O λ

21 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 21 I he euel le S, T ] be he adad couably Hilbeia ace couced fom L 2, T ], A. See 27]. The S, T ] i a uclea ubace of L 2, T ]. The oological dual of S, T ] i deoed by S, T ]. The he Boche-Milo heoem eail he exiece of a uiue obabiliy meaue π o BS, T ] Boel σ algeba of S, T ] uch ha e i ω,φ πdω = e 1 2 φ 2 L 2,T ] S,T ] fo all φ S, T ], whee ω, φ ad fo he acio of ω S, T ] o φ S, T ]. Defie fo i = 1,..., d. The he oduc meaue Ω i = S, T ], F i = BS, T ], µ i = π, µ = d µ i 46 i=1 o he meauable ace d Ω, F := Ω i, i=1 i called d-dimeioal whie oie obabiliy meaue. Coide he Doléa-Dade exoeial ẽφ, ω = ex d i=1 F i ω, φ 1 2 φ 2 L 2,T ];R d, 47 fo ω = ω 1,..., ω d S, T ] d ad φ = φ 1,..., φ d S, T ] d, whee ω, φ := d i=1 ω i, φ i. Now le S, T ] d be he h comleed ymmeic eo oduc of S, T ] d wih ielf. Oe check ha ẽφ, ω i holomohic i φ aoud zeo. Hece, hee exi geealied Hemie olyomial H ω S, T ] d uch ha ẽφ, ω = 1 H ω, φ 48! fo φ i a ceai eighbouhood of zeo i S, T ] d. Oe ove ha { H ω, φ : φ S, T ] d, N } 49 i a oal e of L 2 Ω. Fuhe, i ca be how ha he geealied Hemie olyomial aify he ohogoaliy elaio H ω, φ H m ω, ψ m µdω = δ,m! φ, ψ 5 L 2,T ] ;R d S fo all, m N, φ S, T ] d, ψ m S, T ] d m whee δ,m = { 1 if = m ele. Deoe by L 2, T ] ; R d he ace of uae iegable ymmeic fucio fx 1,..., x wih value i R d. The i follow fom elaio 5 ha he maig φ H ω, φ

22 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 22 fom S, T ] d o L 2 Ω have uiue coiuou exeio I : L 2, T ] ; R d L 2 Ω fo all N. Thee exeio I φ ca be ideified a -fold ieaed Iô iegal of φ L 2, T ] ; R d wih eec o a d dimeioal Wiee oce o he whie oie ace B = B 1,..., B d 51 Ω, F, µ. 52 We meio ha uae iegable fucioal of B admi a Wiee-Iô chao eeeaio which ca be egaded a a ifiie-dimeioal Taylo exaio, ha i L 2 Ω = I L 2, T ] ; R d. 53 The defiiio of he Hida ochaic e fucio ad diibuio ace i baed o he Wiee- Iô chao decomoiio 53: Se A d := A,..., A. 54 Uig a ecod uaiaio agume, he Hida ochaic e fucio ace S i defied a he ace of all f = H, φ L 2 Ω uch ha f 2, :=! A d φ 2 < 55 L 2,T ] ;R d fo all. I fac, he ace S i a uclea Féche algeba wih eec o mulilicaio of fucio ad i oology i iduced by he emiom,,. Fuhe oe how ha ẽφ, ω S 56 fo all φ S, T ] d. O he ohe had, he oological dual of S, deoed by S, i called Hida ochaic diibuio ace. Uig hee defiiio we oai he Gel fad ile S L 2 Ω S. I u ou ha he whie oie of he coodiae of he d dimeioal Wiee oce B, ha i he ime deivaive W i := d d Bi, i = 1,..., d, 57 belog o S. We alo ecall he defiiio of he S-afom. See 28]. The S afom of a Φ S, deoed by SΦ, i defied by he dual aiig SΦφ = Φ, ẽφ, ω 58 fo φ S C, T ] d. Hee S C, T ] he comlexificaio of S, T ]. The S afom i a moomohim fom S o C. I aicula, if SΦ = SΨ fo Φ, Ψ S he Φ = Ψ.

23 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 23 A a examle oe fid ha SW i φ = φ i, i = 1,..., d 59 fo φ = φ 1,..., φ d S C, T ] d. Fially, we ecall he coce of he Wick o Wick-Gama oduc. The Wick oduc defie a eo algeba mulilicaio o he Fock ace ad i ioduced a follow: The Wick oduc of wo diibuio Φ, Ψ S, deoed by Φ Ψ, i he uiue eleme i S uch ha SΦ Ψφ = SΦφSΨφ 6 fo all φ S C, T ] d. A a examle, we ge H ω, φ H m ω, ψ m = H m ω, φ ψ m 61 fo φ S, T ] d ad ψ m S, T ] d m. The lae i coecio wih 48 imlie ha ẽφ, ω = ex ω, φ 62 fo φ S, T ] d. Hee he Wick exoeial ex X of a X S i defied a ex X = 1! X, 63 whee X = X... X, ovided ha he um o he igh had ide covege i S. A.2. Baic eleme of Malliavi Calculu I hi ecio we a i eview ome baic defiiio fom Malliavi calculu. Fo coveiece we coide he cae d = 1. Le F L 2 Ω. The we kow fom 53 ha F = H, φ 64 fo uiue φ L 2, T ]. Suoe ha! φ 2 1 L 2,T ] <. 65 The he Malliavi deivaive D of F i he diecio of B ca be defied a D F = H 1, φ, We deoe by D 1,2 he ace of all F L 2 Ω uch ha 65 hold. The Malliavi deivaive D i a liea oeao fom D 1,2 o L 2, T ] Ω. We meio ha D 1,2 i a Hilbe ace wih he om 1,2 give by F 2 1,2 := F 2 L 2 Ω,µ D F 2 L 2,T ] Ω,λ µ. 67 We ge he followig chai of coiuou icluio: whee D 1,2 i he dual of D 1,2. S D 1,2 L 2 Ω D 1,2 S, 68

24 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 24 Aedix B: Techical eul We give a li if echical eul eeded fo he oof of Secio 2 ad 3. Lemma B.1. Le {f } be a bouded euece of fucio i L. The, fo evey k R { }] x R d ex k I aicula, hee exi a weak oluio o SD 5. Poof. See 15, Lemma 3.2] f, x B 2 d <. Lemma B.2. Le {f } a euece of eleme i L, ha covege o ome f L,. The hee exi ε > 1 uch ha ] f, φ 2ε d <. 69 Hee φ : x X x, deoe he ochaic flow aociaed o he oluio of he SD 5 wih dif coefficie b Cb Rd. Poof. See 6, Lemma 15]. We alo eed he followig cucial lemma, which ca be foud i 5], Lemma 3.4. Lemma B.3. Le U be he oluio of he PD 1 wih Φ = b = b Cb R. Le X x, be he oluio of he SD 5 wih dif coefficie b Cb Rd. The he followig hold ue i Fo each > hee exi a fucio f wih lim f = uch ha ad x B,T ] x B,T ] ii Thee exi a λ R fo which,t ] x R d U, x U, x f U, x U, x f U, x iii U, x L, <. iv A a coeuece of he boudede of U ad U we have 1 2. γ x a ] C 1 x a.,t ] The followig lemma give a boud fo he deivaive of he ivee of he family of diffeomohim γ. See 5], Lemma 3.5 fo i oof. Lemma B.4. Le γ, : R d R d be he C 1 -diffeomohim defied a γ, x := x U, x fo x R d aociaed o X x, he oluio of SD 5 wih dif coefficie b C b Rd. The,T ] The ex eul wa how i 4], Coollay 13. Lemma B.5. Le V γ 1, CRd 2. be he oce defied i 22. The fo evey α R ] e αv T C. Obeve ha he ame eimae hold fo ay, T ] ice V i a iceaig oce.

25 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 25 Aedix C The followig eul which i due o 3, Theoem 1] give a comace cieio fo ube of L 2 Ω; R d uig Malliavi calculu. Theoem C.1. Le {Ω, A, P ; H} be a Gauia obabiliy ace, ha i Ω, A, P i a obabiliy ace ad H a eaable cloed ubace of Gauia adom vaiable of L 2 Ω, which geeae he σ-field A. Deoe by D he deivaive oeao acig o elemeay mooh adom vaiable i he ee ha Dfh 1,..., h = i fh 1,..., h h i, h i H, f Cb R. i=1 Fuhe le D 1,2 be he cloue of he family of elemeay mooh adom vaiable wih eec o he om F 1,2 := F L 2 Ω DF L 2 Ω;H. Aume ha C i a elf-adjoi comac oeao o H wih dee image. The fo ay c > he e G = {G D 1,2 : G L 2Ω C 1 D G } c L2Ω;H i elaively comac i L 2 Ω. A ueful boud i coecio wih Theoem C.1, baed o facioal Sobolev ace i he followig ee 3]: Lemma C.2. Le v, be he Haa bai of L 2, T ]. Fo ay < α < 1/2 defie he oeao A α o L 2, T ] by A α v = 2 kα v, if = 2 k j fo k, j 2 k ad A α T = T. The fo all β wih α < β < 1/2, hee exi a coa c 1 uch ha T A α f c 1 f L 2,T ] f f 2 1/2 d d 12β. A diec coeuece of Theoem C.1 ad Lemma C.2 i ow he followig comace cieio which i eeial fo he oof of Coollay 2.9. Coollay C.3. Le a euece of F T -meauable adom vaiable X D 1,2, = 1, 2..., be uch ha hee exi coa α > ad C > wih fo T ad X 2 ] C, D X D X 2] C α T D X 2] C. The he euece X, = 1, 2..., i elaively comac i L 2 Ω.

26 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 26 Refeece 1] D. R. Baño, T. Nile, ad F. Poke. Pahwie uiuee ad highe ode féche diffeeiabiliy of ochaic flow of facioal bowia moio dive de wih igula dif. wok i oge. 2] G. Da Pao, F. Fladoli,. Piola, ad M. Röcke. Sog uiuee fo ochaic evoluio euaio i hilbe ace wih bouded ad meauable dif. Aal of Pobabiliy, 415: , ] G. Da Pao, P. Malliavi, ad D. Nuala. Comac familie of wiee fucioal. C. R. Acad. Sci. Pai, 315Séie I: , ] G. Di Nuo, B. Økedal, ad F. Poke. Malliavi Calculu fo Lévy Pocee wih Alicaio o Fiace. Sige, 28. 5]. Fedizzi ad F. Fladoli. Pahwie uiuee ad coiuou deedece fo de wih oegula dif. Sochaic, 833: , ]. Fedizzi ad F. Fladoli. Hölde flow ad diffeeiabiliy fo de wih oegula dif. Sochaic Aalyi ad Alicaio, 314:78 736, ]. Fedizzi ad F. Fladoli. Noie eve igulaiie i liea ao euaio. Joual Of Fucioal Aalyi, 2646: , ] F. Fladoli. Radom Peubaio of PD ad Fluid Dyamic Model. École d Éé de Pobabiliié de Sai-Flou XL -21, Lecue Noe i Mahemaic, Sige, ] F. Fladoli, T. Nile, ad F. Poke. Malliavi diffeeiabiliy ad og oluio fo a cla of de i hilbe ace. ei, ] I. Gyögy ad N. V. Kylov. xiece of og oluio fo iô ochaic euaio via aoximaio. Pobab. Theoy Rela. Field, 15: , ] S. Haadem ad F. Poke. O he coucio ad malliavi diffeeiabiliy of oluio of levy oie dive de wih igula coefficie. Joual of Fucioal Aalyi, ] T. Hida, H.-H. Kuo, J. Pohoff, ad L. Sei. Whie Noie: A Ifiie Dimeioal Calculu. Kluwe Academic, ] G. Kalliau ad J. Xiog. Sochaic Diffeeial uaio i Ifiie Dimeioal Sace. IMS Lecue Noe. Moogah Seie, ] I. Kaaza ad S.. Sheve. Bowia Moio ad Sochaic Calculu. 2d d. Sige, ] N. V. Kylov. Some oeie of ace fo ochaic ad deemiiic aabolic weighed obolev ace. Joual of Fucioal Aalyi, 1831:1 41, ] N.V. Kylov ad M. Röcke. Sog oluio of ochaic euaio wih igula ime deede dif. Pob. Theoy Rel. Field, 1312: , ] H.-H. Kuo. Whie Noie Diibuio Theoy. Soch. Seie, Boca Rao, FL: CRC Pe, ] A. Lacoelli ad F. Poke. O exlici og oluio of iô-de ad he doke dela fucio of a diffuio. Aal. Qua. Pob. elaed Toic, 73, ] P. Malliavi. Sochaic calculu of vaiaio ad hyoelliic oeao. Poc. Ie. Sym. o Soch. Diff. uaio, Kyoo 1976, Wiley, age , ] P. Malliavi. Sochaic Aalyi, volume. Gudlehe de Mahemaiche Wiechafe. Sige-Velag, Beli, ] O. Meoukeu-Pame, T. Meye-Badi, T. Nile, F. Poke, ad T. Zhag. A vaiaioal aoach o he coucio ad malliavi diffeeiabiliy of og oluio of de. Mah. A., 3572: , ] T. Meye-Badi ad F. Poke. O he exiece ad exlici eeeabiliy of og oluio of lévy oie dive de. Commuicaio i Mahemaical Sciece, 41, ] T. Meye-Badi ad F. Poke. xlici eeeaio of og oluio of de dive by ifiie dimeioal lévy ocee. Joual of Theoeical Pobabiliy, 231:31 314, ] S-. A. Mohammed, T. Nile, ad F. Poke. Sobolev diffeeiable ochaic flow of de wih meauable dif ad alicaio. o aea i Aal of Pobabiliy ISSN , 215.

27 D. Baño, S. Duedahl, F. Poke ad T. Meye-Badi/Coucio of Malliavi diffeeiable og oluio 27 25] T. Nile. Oe-dimeioal de wih dicoiuou, ubouded dif ad coiuouly diffeeiable oluio o he ochaic ao euaio. ei,. 26] D. Nuala. The Malliavi Calculu ad Relaed Toic. 2d d. Sige, ] N. Obaa. Whie Noie Calculu ad Fock Sace. LNM 1577 Sige, ] J. Pohoff ad L. Sei. A chaaceizaio of hida diibuio. Joual of Fucioal Aalyi, 11: , ] F. Poke. Sochaic diffeeial euaio- ome ew idea. Sochaic, 79:563 6, 27. 3] A.Y. Veeeikov. O he og oluio of ochaic diffeeial euaio. Theoy Pobab. Al., 24: , ] T. Yamada ad S. Waaabe. O he uiuee of oluio of ochaic diffeeial euaio. J. Mah. Kyoo Uiv., II: , ] A.K. Zvoki. A afomaio of he ae ace of a diffuio oce ha emove he dif. Mah.USSR Sboik, 22: , 1974.