Research Article Malliavin Differentiability of Solutions of SPDEs with Lévy White Noise

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1 Hidawi Ieraioal Sochasic Aalysis Volume 17, Aricle ID , 9 pages hps://doi.org/1.1155/17/ esearch Aricle Malliavi Differeiabiliy of Soluios of SPDEs wih Lévy Whie Noise aluca M. Bala ad Cheikh B. Ndogo Deparme of Mahemaics ad Saisics, Uiversiy of Oawa, 585 Kig Edward Aveue, Oawa, ON, Caada K1N 6N5 Correspodece should be addressed o aluca M. Bala; rbala@uoawa.ca eceived Jauary 17; Acceped 1 February 17; Published 1 March 17 Academic Edior: Bohda Maslowski Copyrigh 17 aluca M. Bala ad Cheikh B. Ndogo. his is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. We cosider a sochasic parial differeial equaio (SPDE) drive by a Lévy whie oise, wih Lipschiz muliplicaive erm σ.we prove ha, uder some codiios, his equaio has a uique radom field soluio. hese codiios are verified by he sochasic hea ad wave equaios. We iroduce he basic elemes of Malliavi calculus wih respec o he compesaed Poisso radom measure associaed wih he Lévy whie oise. If σ is affie, we prove ha he soluio is Malliavi differeiable ad is Malliavi derivaive saisfies a sochasic iegral equaio. 1. Iroducio I his aricle, we cosider he sochasic parial differeial equaio (SPDE): Lu (, x) =σ(u (, x)) L (, x), [, ], x, (1) wih some deermiisic iiial codiios, where L is a secod-order differeial operaor o [, ], L deoes he formal derivaive of he Lévy whie oise L (defied below), ad he fucio σ: is Lipschiz coiuous. Aprocessu = {u(, x); [, ], x } is called a(mild)soluio of (1) if u is predicable ad saisfies he followig iegral equaio: u (, x) =w(, x) () + G( s,x y)σ(u(s,y))l(ds,dy), where w is he soluio of he deermiisic equaio Lu = wih he same iiial codiios as (1) ad G is he Gree fucio of he operaor L. he sudy of SPDEs wih Gaussia oise is a welldeveloped area of sochasic aalysis, ad he behaviour of radom field soluios of such equaios is well udersood. We refer he reader o [1] for he origial lecure oes which led o he developme of his area ad o [, 3] for some rece advaces. I paricular, he probabiliy laws of hese soluios ca be aalyzed usig echiques from Malliavi calculus, as described i [4, 5]. O he oher had, here is a large lieraure dedicaed o he sudy of sochasic differeial equaios (SDE) wih Lévy oise, he moograph [6] coaiig a comprehesive accou o his opic. Oe ca develop also a Malliavi calculus for Lévy processes wih fiie variace, usig a aalogue of he Wieer chaos represeaio wih respec o uderlyig Poisso radom measure of he Lévy process. his mehod was developed i [7] wih he same purpose of aalyzigheprobabiliylawofhesoluioofasdedrive by a fiie variace Lévy oise. More recely, Malliavi calculus for Lévy processes wih fiie variace has bee used i fiacial mahemaics, he moograph [8] beig a very readable iroducio o his opic. herearewoapproachesospdesihelieraure.oe is he radom field approach which origiaes i Joh Walsh s lecure oes [1]. Whe usig his approach, he soluio is viewed as a real-valued process which is idexed by ime ad space. he oher approach is he ifiie-dimesioal approach, due o Da Prao ad Zabczyk [9], accordig o

2 Ieraioal Sochasic Aalysis which he soluio is a process idexed by ime oly, which akesvaluesiaifiie-dimesioalhilberspace.iiso always possible o compare he soluios obaied usig he wo approaches (see [1] for several resuls i his direcio). SPDEs wih Lévy oise were sudied i he moograph [11], usig he ifiie-dimesioal approach. I he prese aricle, we use he radom field approach for examiig a SPDE drive by he fiie variace Lévy oise iroduced i [1], wih he goal of sudyig he Malliavi differeiabiliy of he soluio. As meioed above, his sudy ca be useful foraalyzigheprobabiliylawofhesoluio.wepospoe his problem for fuure work. We begi by recallig from [1] he cosrucio of he Lévy whie oise L drivig (1). We cosider a Poisso radom measure (PM) N o he space U=[,] of iesiy μ(d, dx, dz) = ddx](dz) defied o a complee probabiliy space (Ω, F,P),where] is a Lévy measure o = \{}; ha is, ] saisfies (1 z ) ] (dz) <. (3) Iaddiio,weassumeha] saisfies he followig codiio: V fl z ] (dz) <. (4) We deoe by N he compesaed PM defied by N(A) = N(A) μ(a) for ay A U wih μ(a) <,where U is he class of Borel ses i U.WedeoebyF he σ-field geeraed by N([, s] B Γ) for all s [, ], B B b (), ad Γ B b ( ). We deoe by B b () he class of bouded Borel ses i ad by B b ( ) he class of Borel ses i which are bouded away from. A Lévy whie oise wih iesiy measure ] is a collecio L={L (B); [, ], B B b ()} of zero-mea squareiegrable radom variables defied by L (B) = z N (ds, dx, dz). (5) B hese variables have he followig properies: (i) L (B) = a.s. for all B B b (). (ii) L (B 1 ),...,L (B k ) are idepede for ay >ad for ay disjoi ses B 1,...,B k B b (). (iii) For ay <s ad for ay B B b (), L (B) L s (B) is idepede of F s ad has characerisic fucio E(e iu(l (B) L s (B)) ) = exp {( s) B (e iuz 1 iuz)] (dz)}, u. (6) We deoe by F L he σ-field geeraed by L(s) for all s [,]. For ay h L ([, ] ), we defie he sochasic iegral of h wih respec o L: L (h) = h (, x) L (d, dx) = h (, x) z N (d, dx, dz). Usig he same mehod as i Iô s classical heory, his iegral ca be exeded o radom iegrads, ha is, o he class of predicable processes X = {X(, x); [, ], x }, suchhae X(, x) dx d <. heiegralhas he followig isomery propery: E X (, x) L (d, dx) = VE X (, x) dx d. ecall ha a process X = {X(, x);, x d } is predicable if i is measurable wih respec o he predicable σ-field o Ω +, hais,heσ-field geeraed by elemeary processes of he form X (ω,, x) =Y(ω) 1 (a,b] () 1 A (x), ω Ω, [, ], x, where <a<b, Y is a bouded ad F L a-measurable radom variable, ad A B b (). his aricle is orgaized as follows. I Secio, we iroduce he basic elemes of Malliavi calculus wih respec o he compesaed Poisso radom measure N. I Secio 3, we prove ha, uder a cerai hypohesis, (1) has a uique soluio. his hypohesis is verified i he case of he wave ad hea equaios. I Secio 4, we examie he Malliavi differeiabiliy of he soluio, i he case whe he fucio σ isaffie.fially,iheappedix,weiclude a versio of Growall s lemma which is eeded i he sequel.. Malliavi Calculus o he Poisso Space I his secio, we iroduce he basic igredies of Malliavi calculus wih respec o N, followig very closely he approach preseed i Chapers 1 1 of [8]. he differece compared o [8] is ha our parameer space U has variables (, x, z) isead of (, z). Forhesakeofbreviy,wedoo iclude he proofs of he resuls preseed i his secio. hese proofs ca be foud i Chaper 6 of he docoral hesis [13] of he secod auhor. We se H =L (U, U,μ)ad H =L (U, U,μ ).We deoe by H he se of all symmeric fucios f H.We deoe by H C, H C, H C he aalogous spaces of C-valued fucios. (7) (8) (9)

3 Ieraioal Sochasic Aalysis 3 Le S = {(u 1,...,u ) U ; u i = ( i,x i,z i ) wih 1 < < }. For ay measurable fucio f:s wih f L (S ) fl S f(u 1,...,u ) dμ (u 1,...,u ) <, (1) we defie he -fold ieraed iegral of f wih respec o N by J (f) =... N(du ),...( f(u 1,...,u ) N(du 1 )) (11) where u i =( i,x i,z i ).he,e[j (f)j m (g)] = for all =m ad E J (f) = f L (S ). For ay f H,wedefiedhemuliple iegral of f wih respec o N by I (f) =!J (f). Ifollowsha E[I (f)i m (g)] = for all =mad E I (f) =! f H f H. (1) If f H C wih f=g+ih, we defie I (f) = I (g) + ii (h). Le L C (Ω) be he se of C-valued square-iegrable radom variables defied o (Ω, F,P).Byheorem7of [14], ay F L -measurable radom variable F L C (Ω) admis he chaos expasio F= I (f ) i L C (Ω), (13) where f H for all 1ad f = E(F). he chaos expasio plays a crucial role i developig he Malliavi calculus wih respec o N. Iparicular,he Skorohod iegrals wih respec o N ad L are defied as follows. Defiiio 1. (a) Le X = {X(u); u U} be a squareiegrable process such ha X(u) is F L -measurable for ay u U.Foreachu U,leX(u) = I (f (, u)) be he chaos expasio of X(u), wihf (, u) H. Oe deoes by f (u 1,...,u,u)he symmerizaio of f wih respec o all +1variables. Oe says ha X is Skorohod iegrable wih respec o N (ad oe wries X Dom(δ))if E I +1 ( f ) = (+1)! f <. H (+1) (14) I his case, oe defies he Skorohod iegral of X wih respec o N by δ (X) = X (, x, z) N (δ, δx, δz) fl I +1 ( f ). (15) (b) Le Y = {Y(, x); [, ], x } be a squareiegrable process such ha Y(, x) is F L -measurable for ay [,]ad x. OesayshaY is Skorohod iegrable wih respec o L (ad oe wries Y Dom(δ L ))ifheprocess {Y(, x)z; (, x, z) U} is Skorohod iegrable wih respec o N. I his case, oe defies he Skorohod iegral of Y wih respec o L by δ L (Y) = Y (, x) L (δ, δx) fl Y (, x) z N (δ, δx, δz). (16) he followig resul shows ha he Skorohod iegral ca be viewed as a exesio of he Iôiegral. heorem. (a) If X = {X(u); (u) U} is a predicable process such ha E X U <,hex is Skorohod iegrable wih respec o N ad X (, x, z) N (δ, δx, δz) = X (, x, z) N (d, dx, dz). (17) (b) If Y = {Y(, x); [, ], x } is a predicable process such ha E Y(, x) dx d <, hey is Skorohod iegrable wih respec o L ad Y (, x) L (δ, δx) = Y (, x) L (d, dx). (18) We ow iroduce he defiiio of he Malliavi derivaive. Defiiio 3. Le F L (Ω) be a F L -measurable radom variable wih he chaos expasio F= I (f ) wih f H.OesayshaF is Malliavi differeiable wih respec o N if! f H <. (19) 1 I his case, oe defies he Malliavi derivaive of F wih respec o N by D u F= I 1 (f (, u)), u U. () 1 Oe deoes by D 1, he space of Malliavi differeiable radom variables wih respec o N. Noe ha E DF H = 1! f H <. heorem 4 (closabiliy of Malliavi derivaive). Le (F ) 1 D 1, ad F L (Ω) such ha F Fi L (Ω) ad (DF ) 1 coverges i L (Ω; H). he,f D 1, ad DF DF i L (Ω; H).

4 4 Ieraioal Sochasic Aalysis ypical examples of Malliavi differeiable radom variables are expoeials of sochasic iegrals: for ay h L ([, ] ), Moreover, he se D 1, E D,x,z (e L(h) ) =e L(h) (e h(,x)z 1). (1) of liear combiaios of radom variables of he form e L(h) wih h L ([, ] ) is dese i D 1,. he followig resul shows ha he Malliavi derivaive is a differece operaor wih respec o N, o a differeial operaor. heorem 5 (chai rule). For ay F D 1, ad ay coiuous fucio g: such ha g(f) L (Ω) ad g(f + DF) g(f) L (Ω; H), g(f) D 1, ad Dg (F) =g(f+df) g(f) i L (Ω; H). () Similar o he Gaussia case, we have he followig resuls. heorem 6 (dualiy formula). If F D 1, ad X Dom(δ), he E[F X (, x, z) N (δ, δx, δz)] =E[ X (, x, z) D,x,z F] (dz) dx d]. (3) heorem 7 (fudameal heorem of calculus). Le X = {X(s, y, ζ); s [, ], y, ζ } be a process which saisfies he followig codiios: (i) X(s, y, ζ) D 1, for ay (s, y, ζ) U. (ii) E X(s,y,ζ) ](dz)dy ds <. (iii) {D,x,z X(s, y, ζ); (s, y, ζ) U} Dom(δ) for ay (, x, z) U. (iv) {δ(d,x,z X); (, x, z) U} L (Ω; H). he, X Dom(δ), δ(x) D 1, ad D[δ(X)] = X + δ(dx);hais, D,x,z ( X(s,y,ζ) N(δs,δy,δζ)) =X(, x, z) + D,x,z X(s,y,ζ) N(δs,δy,δζ) i L (Ω; H). (4) As a immediae cosequece of he previous heorem, we obai he followig resul. heorem 8. Le Y = {Y(s, y); s [, ], y } be a process which saisfies he followig codiios: (i) Y(s, y) D 1, for all s [,]ad y. (ii) E Y(s, y) dy ds <. (iii) {D,x,z Y(s, y); s [, ], y } Dom(δ L ) for ay (, x, z) U. (iv) E D,x,zY(s, y)l(δs, δy) ](dz)dx d <. he, Y Dom(δ L ), δ L (Y) D 1, ad he followig relaio holds i L (Ω; H): D,x,z (δ L (Y)) =Y(, x) z+ D,x,z Y(s,y)L(δs,δy). 3. Exisece of Soluio (5) I his secio, we show ha (1) has a uique soluio. We recall ha w is he soluio of he homogeeous equaio Lu =wih he same iiial codiios as (1), ad G is he Gree fucio of he operaor L o +. We assume ha, for ay [,], G(, ) L 1 () ad we deoe by FG(, ) is Fourier rasform: FG (, )(ξ) = e iξx G (, x) dx. (6) We suppose ha he followig hypoheses hold. Hypohesis H1. w is coiuous ad uiformly bouded o [, ]. Hypohesis H. (a) G (, x)dx d < ;(b)hefucio FG(, )(ξ) is coiuous o [, ], forayξ d ;(c) here exis ε>ad a oegaive fucio k ( ) such ha FG (+h, )(ξ) FG (, )(ξ) k (ξ) (7) for ay [,]ad h [,ε],ad k (ξ) dξ d <. Sice σ is a Lipschiz coiuous fucio, here exiss a cosa C σ >such ha, for ay x, y, I paricular, for ay x, σ (x) σ(y) C σ x y. (8) σ (x) D σ (1+ x ), (9) where D σ = max{c σ, σ() }. he followig heorem is a exesio of heorem 1.1.(a) of [15] o a arbirary operaor L. heorem 9. Equaio (1) has a uique soluio u = {u(, x); [, ], x } which is L (Ω)-coiuous ad saisfies sup E u (, x) <. (3) (,x) [,]

5 Ieraioal Sochasic Aalysis 5 Proof. Exisece. Weusehesameargumeasiheproofof heorem 13 of [16]. We deoe by (u ) hesequeceof Picard ieraios defied by u (, x) = w(, x) ad u +1 (, x) =w(, x) + G( s,x y)σ(u (s,y))l(ds,dy),. (31) By iducio o, i ca be proved ha he followig properies hold: (P) (i) u (, x) is well defied for ay (, x) [, ]. (ii) K fl sup (,x) [,] E u (, x) <. (iii) (, x) u (, x) is L (Ω)-coiuous o [, ]. (iv) u (, x) is F -measurable for ay [,]ad x. Hypoheses (H1) ad (H) are eeded for he proof of propery (iii). From properies (iii) ad (iv), i follows ha u has a predicable modificaio, deoed also by u.his modificaio is used i defiiio (31) of u +1 (, x).usighe isomery propery (8) of he sochasic iegral ad (8), we have E u +1 (, x) u (, x) = VE G ( s,x y) σ(u (s, y)) σ (u 1 (s, y)) dy ds VC σ G ( s,x y) E u (s, y) u 1 (s, y) dy ds VC σ H (s) ( G ( s,x y)dy)ds, (3) where H () = sup x E u (, x) u 1 (, x).foray [, ], we deoe J () = G (, x) dx. (33) akig he supremum over x i he previous iequaliy, we obai ha H +1 () VC σ H (s) J ( s) ds, (34) for ay [,]ad. By applyig Lemma A.1 (he Appedix) wih C =ad p=,weiferha sup H () 1/ <. (35) [,] his shows ha he sequece (u ) coverges i L (Ω) o a radom variable u(, x),uiformlyi[, ] ;hais, sup E u (, x) u(, x). (36) (,x) [,] o see ha u is a soluio of (1), we ake he limi i L (Ω) as i (31). I paricular, his argume shows ha K fl sup sup 1 (,x) [,] E u (, x) <. (37) Uiqueess. LeH() = sup x E u(, x) u (, x),whereu ad u arewosoluiosof(1).asimilarargumeasabove shows ha H () VC σ H (s) J ( s) ds, (38) for ay [,].ByGrowall slemma,h() = for all [, ]. Example 1 (wave equaio). If Lu = u/ u/ x for [,]ad x,heg(, x) = (1/)1 { x }.Hypohesis (H) holds sice FG (, )(ξ) = si ( ξ ) ξ. (39) Example 11 (hea equaio). If Lu = u/ (1/)( u/ x ) for [,]ad x,heg(, x) = (π) 1/ exp( x / ). Hypohesis (H) holds sice FG (, )(ξ) = exp ( ξ ). (4) 4. Malliavi Differeiabiliy of he Soluio I his secio, we show ha he soluio of (1) is Malliavi differeiable ad is Malliavi derivaive saisfies a cerai iegral equaio. For his, we assume ha he fucio σ is affie. Our firs resul shows ha he sequece of Picard ieraios is Malliavi differeiable wih respec o N ad he correspodig sequece of Malliavi derivaives is uiformly bouded i L (Ω; H). Lemma 1. Assume ha σ is a arbirary Lipschiz fucio. Le (u ) be he sequece of Picard ieraios defied by (31). he, u (, x) D 1, for ay (,x) [,] ad,ad Proof. A fl sup sup (,x) [,] E Du (, x) H <. (41) Sep 1. We prove ha he followig propery holds for ay : u (, x) D 1, for ay (, x) [, ], A fl sup E Du (, x) H <. (,x) [,] (Q)

6 6 Ieraioal Sochasic Aalysis Forhis,weuseaiducioargumeo.Propery(Q) is clear for =. We assume ha i holds for ad we prove ha i holds for +1. By he defiiio of u +1 ad he fac ha he Iôiegral coicides wih he Skorohod iegral if he iegrad is predicable, i follows ha, for ay (, x) [, ], u +1 (, x) =w(, x) + G( s,x y)σ(u (s,y))l(δs,δy). (4) We fix (,x) [,]. We apply he fudameal heorem of calculus for he Skorohod iegral wih respec o L (heorem 8) o he process: Y(s,y)=G( s,x y)σ(u (s, y)) 1 [,] (s), s [, ], y. (43) We eed o check ha Y saisfies he hypoheses of his heorem. o check ha Y saisfies (i), we apply he chai rule (heorem 5) o F=u (s, y) ad g=σ.noeha,foray (s,y) [,], E σ(u (s, y)) D σ (1 + E u (s, y) ) D σ (1 + K ) <, E σ(u (s, y) + D r,ξ,z u (s, y)) σ (u (s, y)) ] (dz) dξ dr C σ E ] (dz) dξ dr C σ A <, D r,ξ,zu (s, y) (44) (45) by he iducio hypohesis. We coclude ha Y(s, y) D 1, ad D r,ξ,z Y(s,y)=G( s,x y) [σ(u (s, y) + D r,ξ,z u (s, y)) σ (u (s, y))] 1 [,] (s). We oe ha Y saisfies hypohesis (ii) sice, by (44), E Y(s,y) dy ds D σ (1 + K ) G ( s,x y)dyds <. (46) (47) o check ha Y saisfies hypohesis (iii), i.e., he process {D r,ξ,z Y(s, y); s [, ], y } is Skorohod iegrable wih respec o L for ay (r,ξ,z) U,isufficesoshow ha his process is Iô iegrable wih respec o L.Noeha D r,ξ,z u (s, y) = if r>sad i is F s -measurable if r s. Hece, he process {D r,ξ,z Y(s, y); s [, ], y } is predicable. By (46) ad (8), E D r,ξ,zy(s,y) dy ds C σ E ad, hece, (48) G ( s,x y) D r,ξ,zu (s, y) dy ds, (E D r,ξ,zy(s,y) dy ds) ] (dz) dξ dr C σ E Du (s, y) H dy ds G ( s,x y) C σ A G ( s,x y)dyds<. (49) his proves ha E D r,ξ,zy(s, y) dy ds < for almos all (r,ξ,z) [,]. By heorem (b), {D r,ξ,z Y(s, y); s [,], y } is Skorohod iegrable wih respec o L ad D r,ξ,z Y(s,y)L(δs,δy) = D r,ξ,z Y(s,y)L(ds,dy). (5) Fially, Y saisfies hypohesis (iv) sice, by (5) ad he isomery properies (8) ad (49), we have E ] (dz) dξ dr =E ] (dz) dξ dr = V D r,ξ,z Y(s,y)L(δs,δy) ] (dz) dξ dr <. D r,ξ,z Y(s,y)L(ds,dy) (E D r,ξ,zy(s,y) dy ds) (51) By heorem 8, we ifer ha Y Dom(δ L ), δ L (Y) D 1,, ad D r,ξ,z (δ L (Y)) =Y(r, ξ) z + D r,ξ,z Y (s, y) L (δs, δy). (5)

7 Ieraioal Sochasic Aalysis 7 Sice u +1 (, x) = w(, x)+δ L (Y),hismeashau +1 (, x) D 1,. Usig (5) ad (46), we ca rewrie relaio (5) as follows: D r,ξ,z u +1 (, x) =G( r,x ξ) σ(u (r, ξ))z + G( s,x y) [σ(u (s, y) + D r,ξ,z u (s, y)) σ (u (s, y))] L(ds,dy). I remais o prove ha A +1 = (53) sup E Du +1 (, x) H <. (54) (,x) [,] Usig (53), he isomery propery (8), relaio (44), ad he fac ha σ is Lipschiz, we see ha E D r,ξ,zu +1 (, x) z G ( r,x ξ) where V () = sup x E Du (, x) H ad J() is give by (33). his shows ha V +1 () 4VD σ ] (1+K) +VC σ V (s) J ( s) ds, (59) where K is give by (37). By Lemma 15 of [16], sup 1 sup [,] V () <. We are ow ready o sae he mai resul of he prese aricle. heorem 13. Assume ha σ is a affie fucio; ha is, σ(x) = ax + b for some a, b. Ifu is he soluio of (1), he, for ay [,]ad x, u (, x) D 1, (6) E σ(u (r, ξ)) +VE G ( s,x y) σ(u (s, y) + D r,ξ,z u (s, y)) σ(u (s, y)) dy ds 4z D σ (1 +K )G ( r,x ξ) +VC σ E G ( s,x y) (55) ad he followig relaio holds i L (Ω; H) (ad hece, almos surely, for μ-almos all (r, ξ, z) U): D r,ξ,z u (, x) =G( r,x ξ) σ (u (r, ξ)) z + G( s,x y) [σ(u(s,y)+d r,ξ,z u (s, y)) σ (u (s, y))] L(ds,dy). (61) D r,ξ,zu (s, y) dy ds. We perform iegraio wih respec o drdξ](dz) o [, ]. We deoe We obai ] = G (s, y) dy ds. (56) E Du +1 (, x) H 4VD σ (1 + K ) ] +VC σ G ( s,x y) E Du (s, y) H dy ds 4VD σ (1 + K ) ] (57) Proof. Sep 1.Foray [,]ad,le M () = supe Du (, x) Du 1 (, x) H. (6) x Noe ha, by Lemma 1, u (, x) D 1, for ay (, x) [, ] ad. Fix (r,ξ,z) U. We wrie relaio (53) for D r,ξ,z u +1 (, x) ad D r,ξ,z u (, x). We ake he differece bewee hese wo equaios. We obai D r,ξ,z u +1 (, x) D r,ξ,z u (, x) =G( r,x ξ) +VC σ A ]. elaio (54) follows akig he supremum over (, x) [, ]. [σ(u (r, ξ)) σ(u 1 (r, ξ))] z + G( s,x y){[σ(u (s, y) + D r,ξ,z u (s, y)) σ (u (s, y))] (63) Sep. We prove ha sup 1 A <.By(57),wehave [σ(u 1 (s, y) + D r,ξ,z u 1 (s, y)) σ (u 1 (s, y))]} E Du +1 (, x) H 4VD σ (1 + K ) ] +VC σ V (s) J ( s) ds, (58) L(ds,dy). A his poi, we use he assumpio ha σ is he affie fucioσ(x) = ax+b. (A explaaio of why his argume

8 8 Ieraioal Sochasic Aalysis does o work i he geeral case is give i emark 14.) I hiscase,relaio(63)hashefollowigsimplifiedexpressio: D r,ξ,z u +1 (, x) D r,ξ,z u (, x) =ag( r,x ξ) [u (r, ξ) u 1 (r, ξ)]z +a G( s,x y) [D r,ξ,z u (s, y) D r,ξ,z u 1 (s, y)] L (ds, dy). (64) emark 14. Uforuaely, we were o able o exed heorem 13 o a arbirary Lipschiz fucio σ. oseewhere he difficuly comes from, recall ha we eed o prove ha {Du (, x)} 1 coverges i L (Ω; H), ad he differece D r,ξ,z u +1 (, x) D r,ξ,z u (, x) isgiveby(63).foraarbirary Lipschiz fucio σ,byrelaio(8),wehave σ(u (s, y) + D r,ξ,z u (s, y)) σ(u 1 (s, y) + D r,ξ,z u 1 (s, y)) Usig Iô s isomery ad he iequaliy (a + b) (a +b ), we obai E D r,ξ,zu +1 (, x) D r,ξ,z u (, x) a z G ( C σ (u (s, y) + D r,ξ,z u (s, y)) (u 1 (s, y) + D r,ξ,z u 1 (s, y)) (71) r,x ξ) b +a VE G ( s,x y) D r,ξ,zu (s, y) D r,ξ,z u 1 (s, y) dy ds, (65) where b = sup (s,y) [,] E u (s, y) u 1 (s, y).noeha boh sides of he previous iequaliy are zero if r>.akig he iegral wih respec o drdξ](dz) o [, ],we obai E Du +1 (, x) Du (, x) H a V] b +a VE G ( s,x y) (66) E Du (s, y) Du 1 (s, y) H dy ds, where ] is give by (56). ecallig he defiiio of M (), we ifer ha M +1 () C +a V M (s) J ( s) ds, (67) where C =a V] b ad he fucio J is give by (33). By relaio (35), we kow ha 1 b <,whichmeasha 1 C 1/ <. By Lemma A.1 (he Appedix), we coclude ha supm () 1/ <. (68) 1 Hece, he sequece {Du (, x)} 1 coverges i L (Ω; H) o avariableu(, x),uiformlyi(, x) [, ] ;hais, sup E Du (, x) U(, x) H. (69) (,x) [,] Sep. We fix (, x) [, ]. By(36),u (, x) u(, x) i L (Ω).BySep1, {Du (, x)} 1 coverges i L (Ω; H).We apply heorem 4 o he variables F =u (, x) ad F=u(,x). We ifer ha u(, x) D 1, ad Du (,x) Du(,x) i L (Ω; H). Combiig his wih (69), we obai sup E Du (, x) Du(, x) H. (7) (,x) [,] elaio (61) follows by akig he limi i L (Ω; H) as i (53). C σ { u (s, y) u 1 (s, y) + D r,ξ,zu (s, y) D r,ξ,z u 1 (s, y) }. Usig (63), he isomery propery (8), he iequaliy (a + b) (a +b ), ad he previous iequaliy, we have E D r,ξ,zu +1 (, x) D r,ξ,z u (, x) z C σ G ( r,x ξ) E u (r, ξ) u 1 (r, ξ) +4VC σ G ( s,x y) E u (s, y) u 1 (s, y) dy ds +4VC σ G ( s,x y) E D r,ξ,zu (s, y) D r,ξ,z u 1 (s, y) dy ds. (7) he problem is ha he secod erm o he righ-had side of he iequaliy above does o deped o (r, ξ, z) ad hece is iegral wih respec o drdξ](dz) o [, ] is equal o. Appedix A Varia of Growall s Lemma he followig resul is a varia of Lemma 15 of [16], which is used i he proof of heorem 13. Lemma A.1. Le (f ) be a sequece of oegaive fucios defied o [, ] such ha M = sup [,] f () < ad, for ay [,]ad, f +1 () C + f (s) g ( s) ds, (A.1) where g is a oegaive fucio o [, ] wih g()d < ad (C ) is a sequece of oegaive cosas. he, here exiss a sequece (a ) of oegaive cosas which

9 Ieraioal Sochasic Aalysis 9 saisfy a 1/p < for ay p>1,suchha,foray [, ] ad, 1 f () C + I paricular, if 1 C 1/p 1 [,] j=1 C j a j +C a M. < for some p>1,he sup f () 1/p <. (A.) (A.3) Proof. Le G() = g()d, (X i) i 1 be a sequece of i.i.d. radom variables o [, ] wih desiy fucio g()/g(), ad S = i=1 X i. Followig exacly he same argume as i he proof of Lemma 15 of [16], we have f () C +C 1 G () P(S 1 )+ +C 1 G () 1 P(S 1 ) +C G () E[1 {S }f ( S )]. (A.4) elaio (A.) follows wih a = G() P(S )for 1. he fac ha 1 a 1/p < for all p 1was show i he proof of Lemma 17 of [16]. o prove he las saeme, we le a = 1 ad M 1 = max(m, 1). he, f () M 1 j= C ja j ad, hece, sup f () 1/p M 1/p 1 j= C1/p j a 1/p j.wecocludeha supf k () 1/p M 1/p k= Coflics of Ieres 1 C 1/p j a 1/p k j j= k=j M 1/p 1 C 1/p j a 1/p k j k fl C<. (A.5) he auhors declare ha here are o coflics of ieres regardig he publicaio of his paper. [3] D. Khoshevisa, Aalysis of Sochasic Parial Differeial Equaios, vol. 119 of CBMS egioal Coferece Series i Mahemaics, AMS, Providece, I, USA, 14. [4] D. Nualar,Malliavi Calculus, Spriger, d ediio, 6. [5] M. Saz-Solé, Malliavi Calculus wih Applicaios o Sochasic Parial Differeial Equaios, EPFLPress,CCPress,Boca ao, Fla, USA, 5. [6] D. Applebaum, Lévy Processes ad Sochasic Calculus, vol.116 of Cambridge Sudies i Advaced Mahemaics, Cambridge Uiversiy Press, Cambridge, UK, d ediio, 9. [7] K. Bicheler, J.-B. Gravereaux, ad J. Jacod, Malliavi Calculus for Processes wih Jumps, vol. of Sochasics Moographs, Gordo ad Breach Sciece, New York, NY, USA, [8]G.DiNuo,B.Oksedal,adF.Proske,Malliavi Calculus for Lévy Processes wih Applicaios o Fiace, Uiversiex, Spriger, Berli, Germay, 9. [9] G. Da Prao ad J. Zabczyk, Sochasic Equaios i Ifiie Dimesios, Cambridge Uiversiy Press, Cambridge, UK, 199. [1]. C. Dalag ad L. s. Quer-Sardayos, Sochasic iegrals for spde s: a compariso, Exposiioes Mahemaicae, vol.9, o. 1, pp , 11. [11] S. Pesza ad J. Zabczyk, Sochasic Parial Differeial Equaios wih Lévy Noise, vol.113ofecyclopedia of Mahemaics ad is Applicaios, Cambridge Uiversiy Press, Cambridge, UK, 7. [1]. M. Bala, Iegraio wih respec o Lévy colored oise, wih applicaios o SPDEs, Sochasics,vol.87,o.3,pp , 15. [13]C.B.Ndogo,Equaios aux dérivées parielles sochasiques avec brui blac de Lévy [Ph.D. hesis], UiversiyofOawa, 16 (Frech). [14]. M. Bala ad C. B. Ndogo, Io formula for iegral processes relaed o space-ime Lévy oise, Applied Mahemaics, vol. 6, o. 1, pp , 15. [15].M.BalaadC.B.Ndogo, Iermiecyforhewave equaio wihlévy whie oise, Saisics & Probabiliy Leers, vol. 19, pp. 14 3, 16. [16]. C. Dalag, Exedig marigale measure sochasic iegral wih applicaios o spaially homogeeous s.p.d.e. s, Elecroic Probabiliy,vol.4,o.6,9pages,1999,Erraum i Elecroic Probabiliy,vol.6,5pages,1. Ackowledgmes his research suppored by a gra from he Naural Scieces ad Egieerig esearch Coucil of Caada. efereces [1] J. B. Walsh, A iroducio o sochasic parial differeial equaios, i École d Éé deprobabiliés de Sai Flour XIV 1984, vol. 118ofLecure Noes i Mah., pp , Spriger, Heidelberg, Germay, [].Dalag,D.Khoshevisa,C.Mueller,D.Nualar,adY. Xiao, A miicourse o sochasic parial differeial equaios, vol. 196 of Lecure Noes i Mahemaics, Spriger, Heidelberg, Berli, 9.

10 Advaces i Operaios esearch Volume 14 Advaces i Decisio Scieces Volume 14 Applied Mahemaics Algebra Volume 14 Probabiliy ad Saisics Volume 14 he Scieific World Joural Volume 14 Ieraioal Differeial Equaios Volume 14 Volume 14 Submi your mauscrips a hps:// Ieraioal Advaces i Combiaorics Mahemaical Physics Volume 14 Complex Aalysis Volume 14 Ieraioal Mahemaics ad Mahemaical Scieces Mahemaical Problems i Egieerig Mahemaics Volume 14 Volume 14 Volume 14 Volume 14 Discree Mahemaics Volume 14 Discree Dyamics i Naure ad Sociey Fucio Spaces Absrac ad Applied Aalysis Volume 14 Volume 14 Volume 14 Ieraioal Sochasic Aalysis Opimizaio Volume 14 Volume 14