ON SOME FRACTIONAL PARABOLIC EQUATIONS DRIVEN BY FRACTIONAL GAUSSIAN NOISE

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1 IJRRAS 6 3) Februry ON SOME FRACIONAL ARABOLIC EQUAIONS RIVEN BY FRACIONAL GAUSSIAN NOISE Mhmoud M. El-Bori & hiri El-Sid El-Ndi Fculy of Sciece Alexdri Uiversiy Alexdri Egyp Emil: yhoo.com homil.com ABSRAC Some frcio prbolic pril differeil euios drive by frcio Gussi oise re cosidered. Iiil-vlue problems for hese euios re sudied. Some properies of he soluios re give uder suible codiios d wih urs prmeer less h hlf. eywords: Frciol prbolic sochsic pril differeil euios frciol clculus frcio Browi moio. AMS Subjec Clssificios: 6G INROUCION I his oe sochsic pril differeil euios of he form: re cosidered dv db f Lu ) d.) < < > x R u v Lu.) L u x) u L u b x) u m m j < < x R is he -dimesiol Euclide spce... ) is -dimesiol muli idex... B is frciol Browi moio wih urs prmeer [ ] B ) E[ B ] for ll R ) d E[ B B ] s s s R E [X ] deoes he expecio of rdom vrible X ). If he B coicides wih clssicl Browi moio B. For B is o semi mrigle so oe co use he geerl heory of sochsic clculus for semi mrigle o B see[][][3]). * eoe by he lier operor defied o he se of ll sep fucios o subse of he se of ll sure iegrble fucio L [ ] such h: * r ) [ r) ] d r s r ) j F ) s

2 IJRRAS 6 3) Februry El-Bori & ElNdi O Some Frciol rbolic Euios deoes he gmm fucio d F b c z)is he Guss hyper geomeric fucio. he process iegrl represeio: B.3) B { B : [ is he Browi moio defied by ]} [] ) is he idicor fucio). Le f : R R such h E[ f B )] < he f B * B B[ ) [] )].4) ) E[ f B ))] ) d B ) E x f x B x B ).5) see []. I is supposed h: ) All he coefficies b sisfy uiform ölder codiio o R )All he coefficies 3)he operor his mes h b re bouded o R m for ll y R y...) x) ) is uiformly prbolic o m m y x) y R. c y m c > y... y y y... y 4) he fucio f is coiuous o R [ ] R. I is ssumed h u ) u ) u x) u x).5) u u re give sufficiely smooh bouded fucios o R. Wihou loss of geerliy we c ssume h u x) u ) x d c is posiive cos B hs I secios 3 he soluio of he sochsic Cuchy problem.).5) is sudied. he frciol Browi moio hs my differe impoe pplicios wih mzig rge. his mzig rge mes frciol Browi moio very ieresig objec o sudy see [4-7]).. FORMAL SOLUIONS he soluio of euio.) is formlly give by: u v B f Lu ) d.) R G is he fudmel soluio of he prbolic euio: u x) he fucio G sisfies he followig ieuliy: ) G ) v d d.) m G exp[ c c u. ].3) 37

3 IJRRAS 6 3) Februry El-Bori & ElNdi O Some Frciol rbolic Euios m m m x m m m c m m d c re posiive coss [8-]. he defiiio of he fucio ) c be foud i [8]. 3. FRACIONAL INEGRAL RERESENAION Le I be he frciol iegrl operor defied by I f ) I L [ b he imge of [ ] eoe by ]) b ssocied wih erel is isomorphism from d i c be expressed i erms of frciol iegrls by he iverse operor is give by L by he operor f >. I. he operor L[ ] oo I L[ ]) g) I s I s g g) g s f ds. s for ll g I L [ ]). If g is bsoluely coiuous i c be proved h is he frciol derivive defied by g s I s g g dg ds g 3.) d g g ) d o L ) see [3][6]. A we soluio of euio.) is defied by couple of dped processes B v) for every fixed x o filered probbiliy spce F { F : [ ]}) such h ) B is F - frciol Browi moio b) v d B sisfy.). Suppose h euio.5) hs we soluio. he usig he defiiios of he operors represeio.) oe c wrie euio.) i he form v 3.) B B d he 38

4 IJRRAS 6 3) Februry El-Bori & ElNdi O Some Frciol rbolic Euios g.) g ) f s Lu ) ds. heorem 3.. Le < d v be we soluio of euio.5). If f is Borel fucio o R [ ] R d sisfies he lier growh codiio f u) C u )3.3) C is posiive cos he g.) I L [ ]) for ll u R x R [ ] proof. From.).).3) d 3.3) i c be deduced h > V B C is cos d Sup v hus from 3.4) we ge C C V V x. he ls ieuliy leds o C ) C V B C e B ) d C e V ds C3 B ds C43.5) C > re coss. From 3.3) d 3.5) we ge > C3 C 4C5 d C 6 re posiive coss. I is esy o see h g ) d C4 C5 B ds C63.6) I g ds ) ) d he reuired resul follows from 3.5) d 3.6). g d.. ).3.4) 3.7) I is cler h g.) L [ ].s. if d oly if g.) I L [ ]).s. Le ) exp[ ds]. If f is bouded he ) defies for every x R rdom vrible such h he mesure give by d ) d is probbiliy mesure euivle o. If E deoes he expecio wih respec o he [ E ).3.8) From 3.) 3.7) heorem 3.) d Girsov heorem we see h v is urs prmeer uder he probbiliy see [7]). Lemm 3.. If f is bouded he E C is posiive cos. roof. We c deduce from he resuls i [ 7 ] h [ )] exp[ C ) ) ] F - frciol Browi moio wih 39

5 IJRRAS 6 3) Februry El-Bori & ElNdi O Some Frciol rbolic Euios for ll Now R E [ E O he oher hd usig 3.) we ge E exp d )] E exp ds exp d s I s f s Lu ) s s s ) d M ) M is posiive cos f M). hus E exp[ ds exp[ M ]] M is posiive cos. Usig he fc h we ge he reuired resul. We c deduce from 3.) h he operor is dped. Le b be posiive Borel fucio defied o exiss > >. I his cse we sy h b belogs o L E [ )] E direcly geerlized o obi he followig esimios [ )] preserves he dpbiliy propery. I oher words he process [ ] R such h he followig iegrl. r [ b v) dv) ] R b d he by usig lemm 3.) he resuls of Nulr d Ouie i [7] c be E b v d C b E exp[ b v d Q b C is posiive cos d Q is rel lyic fucio []. heorem 3.. If f is coiuous o R [ ] R d sisfies he Lipschiz codiio; f u) f v) C u v for ll Moreover x R [ ] u vr C is posiive cos he here is we soluio v of euio.5). E [ v ] <. roof. We shll use he mehod of successive pproximios. r )] 4

6 IJRRAS 6 3) Februry El-Bori & ElNdi O Some Frciol rbolic Euios Se v B f L u )) d hus u i follows h he seuece } R ) G )) v v ). d d C v v ) B ) d. )! v uiformly coverges wih respec o x o sochsic process v. I is esy o see { h E[ v ] ) his complee he proof of he heorem see [-5]). E[ ) { v v } ]. 4. REFERENCES [] C. Beder: A Ioˆ formul for geerlized fuciols of frciol Browi moio wih rbirry urs prmeer. Soch. roc. d heir Appl. 4 3) 8-6. [] F. Bigii B. Osedl A. Sulem d N. Wller: A iroducio o whie oise heory d Mllivi clculus for frciol Browi moio. he proceedigs of he royl sociey 46 4) [3] L.ecreuse fod d A.S. V suel: Sochsic lysis of he frciol Browi moio. oeil lysis 999) [4] Y. u d B. Osedl: Fuciol whie oise clculus d pplicio o fice. If. im. Al. Quum prob.. Rel. op. 63) - 3. [5] R. Ellio d J. V er oc: A geerl frciol whie oise heory l pplicios o fice. Mh. Fice 3 3) [6] S.G. Smo A.A. ilbs l O.I. Mrichece: Frciol iegrls d derivives. Gordo d Brech Sciece 993). [7]. Nulr d Y. Ouie: Regulrizio of differeil euios by frciol oise Soch. roc. Appl. ) 3-6. [8] Mhmoud M. El-Bori Some probbiliy desiies d fudmel soluio of frciol evoluio euios ChosSolio d Frcls4) [9] hiri El-Sid El- Ndi: O some sochsic prbolic differeil euios i ilber spce. Jourl of Appl. Mh. d Soch. Alysis Vol.5) [] Mhmoud M. E-Bori hiri El-Sid El- Ndi O.L. Mosf d mdy M. Ahmed Volerr euios wih frciol sochsic iegrls. Mhemicl problems i egieerig 4: 54) [] hiri El-Sid El-Ndi O some sochsic differeil euios d frciol Browi moio I. J. of ure d Applied Mh. Vol4 No.3 5) [] hiri El-Sid El-Ndi Asympoic mehods d differece frciol differeil euios. I. J. Comp. Mh. Sci. Vol. No. 6) 3-4. [3] W. Wyss: he frciol diffusio euio Jourl Mh. hys ) o [4] hiri El-Sid El-Ndi O some sochsic models of ccer Cdi J. o Biomedicl Egieerig d echology Vol. No. ) [5] A. Ju. Vereemiove: O srog soluios d explici formuls for soluios of sochsic iegrl euios. Mh USSR Sb )