ON RELATIONS BETWEEN INFINITESIMAL GENERATORS AND MEAN DERIVATIVES OF STOCHASTIC PROCESSES ON MANIFOLDS *

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1 UDС 59.6.; ON RELATIONS BETWEEN INFINITESIMAL GENERATORS AND MEAN DERIVATIVES OF STOCHASTIC PROCESSES ON MANIFOLDS Yu. E. Gllh Voronezh Se Unversy Поступила в редакцию г. Аннотация. Описываются соотношения между инфинитезимальными генераторами слева и справа с одной стороны и производными в среднем слева и справа с другой стороны для случайных процессов на многообразиях. Эти соотношения формулируются в терминах порожденного аффинной связностью отображения из расслоения векторов второго порядка в обычное (т.е. первого порядка) касательное расслоение. Также показано, что так называемая квадратичная производная в среднем может быть получена из генератора с помощью другого морфизма соответствующих расслоений. Ключевые слова: Случайные процессы; инфинитезимальные генераторы; производные в среднем; связности на многообразиях; касательные векторы второго порядка Absrc. We descrbe he relon beween he forwrd (bcwrd) nfnesl generor on he one hnd nd forwrd (bcwrd, respecvely) en dervve on he oher hnd for sochsc process on nfold. The relon s foruled n ers of ppng fro he second order ngen bundle o he frs order one, genered by gven connecon. I s lso shown h he so-clled qudrc en dervve cn be obned fro he generor by noher orphs beween he correspondng bundles. Key words: Sochsc processes; nfnesl generors; en dervves; connecons on nfolds; second order ngen vecors. There re wo ypes of dfferenl operors ssoced o sochsc process: he nfnesl generors (forwrd nd bcwrd) nd en dervves (forwrd, bcwrd nd qudrc). Recll h he generors re deerned nvrnly s he so-clled second order ngen vecors, qudrc en dervves re nvrn s well nd e vlues n (, 0) -ensors whle he forwrd nd bcwrd en dervves on nfold re well defned for gven connecon nd hen e vlues n frs order vecors. In hs pper we show h gven connecon, one cn consruc forwrd (bcwrd) en dervve fro forwrd (bcwrd, respecvely) generor by pplcon of nurl ppng fro second order ngen bundle o he frs order one, genered by he connecon. The qudrc en dervve cn be obned fro he generor by noher specl operor beween he correspondng bundles h s ndependen of he choce of connecon. Gllh Yu. E., 008 The reserch s suppored n pr by RFBR Grns nd Le M be sooh nfold of denson n. Le U be chr on M. Denoe by q,, q n he locl coordnes n U nd by º,, n he correspondng bss vecors n ngen spces o U (we do no dsngush n noons he ngen vecors nd he correspondng frs order dfferenl operors). Consder dfferenl operor n U of order no greer hn whou consn er of he for B (, )= b j j b, () where he coeffcens b j for syerc r ( b j ). Defnon. A second order ngen vecor o nfold M pon Œ M s dfferenl ope r or of he order no greer h whou consn er s n () h hs syerc r of coeffcens second order dervves. The lner spce of second order ngen vecors Œ M s clled he second order ngen spce nd s denoed by M. The second order ngen bundle s denoed by ( M ). ВЕСТНИК ВГУ, СЕРИЯ: ФИЗИКА. МАТЕМАТИКА, 008, 97

2 Yu. E. Gllh Noce h every Œ M he frs order ngen spce T Ms subspce n M he frs order vecors hve zero r ( b j ). On he oher hnd, f h r s no zero, he colun ( b ) s no frs order ngen vecor snce hs noher rule for rnsforons: under coordne chnges he pure second order er rnsfors no he one wh ddonl frs order er. Neverheless nyhow he feld of rces ( b j ) s syerc (, 0) -ensor feld. A every Œ M here s cnoncl soorphss beween he spce TM TM (where TM s he frs order,.e., ordnry ngen spce, denoes he syerc ensor produc) nd he quoen spce MTM /, nd so beween TM TM nd M/ TM. Denoe by Q : M Æ TM TM he feld of lner projecors Q : M Æ T M T M deerned by he bove fcorzon. Noe h he secons of TM TM re syerc (, 0) - ensor felds nd h by consrucon QB (, )= Ê j ˆ () j = Q b =( ). j b b Specfy n rbrry connecon H on M wh Chrsoffel sybols of second nd G j. I deernes he orphs H : M Æ TM (.e., he sooh feld of lner operors H : M Æ T M ) of he for j HB(, )= b G jb (3) (see [ 3]). One cn esly verfy h he rghhnd sde of (3) rnsfors le frs order ngen vecor under he chnges of coordnes. Le R N be cern N -densonl lner spce. Denoe by L( R N, R n ) he spce of lner operors sendng R N o R n where n s he denson of M s bove. Defnon. (cf. [4 7]) The Iô bundle I( M) over M s he bundle such h over chr U on M s presened s drec produc U ( R n ( R N, R n n N n L ))(.e., R L( R, R ) s he sndrd fber ofi( M)) nd under he chnge of coordnes j b fro U o noher chr U b rnsfors ccordng o he rule (,( X, A)) Ê Ê ˆˆ (4) jb, jb X rjb ( A, A), jb A. where j b nd j b re he frs nd he second dervves of j b, respecvely. The (generlly speng non-uonoous) secons of I( M) re clled Iô equons. The bcwrd Iô bundle I ( M) over M s he bundle such h over chr U on M s presened s drec produc U ( R n L ( R N, R n )) n N n (.e., R L( R, R ) s he sndrd fber of I( M)) nd under he chnge of coordnes j b fro U o noher chr U b rnsfors ccordng o he rule (,( X, A)) Ê Ê ˆˆ (5) jb, jb X - rjb ( A, A), jb A. where j b nd j b re he frs nd he second dervves of j b, respecvely. The (generlly speng non-uonoous) secons of I ( M) re clled bcwrd Iô equons. In he chr U n Iô equon s presened s couple ( â (, ), A (, )) where A (, ) s lner operor fro R N o he ngen spce TM. We eep he noon of Iô equon s he couple everywhere n spe of he fc h s well-posed only n chr. Noe h â(, ) s no ngen vecor o M. Neverheless wh respec o he gven rvlzon of I( M) over U we cn represen â(, ) s colun of coordnes ( â j ). Denoe by ( A )(, ) he r of A (, ) wh respec o he se rvlzon. Inroduce Wener process w() n R N gven on cern probbly spce. Then we cn consder n Iô sochsc dfferenl equon d()= ˆ (, ()) d A (, ()) dw() (6) n U. Tng no ccoun he I ô forul, one cn esly see h soluon ( ) of (6) s presened n U b n he for djb()= jb d ˆ rjb ( A, A) d (7) j Adw(). b Thus fro (4) nd (7) obvously follows h soluon ( ) s well-posed on he enre nfold (see dels n [4 7]). We shll cll soluon of ( â (, ), A (, )) on M. Te relvely copc chr U. Denoe by qu he rndo e of he frs enrnce o he boundry of U by sochsc process () wh () Œ U nd by q U s ls e e fro he boundry. 98 ВЕСТНИК ВГУ, СЕРИЯ: ФИЗИКА. МАТЕМАТИКА, 008,

3 On relons beween nfnesl generors nd en dervves of sochsc processes on nfolds We defne he nfnesl generor of () s he feld of dfferenl operors G(, ) h cs on sooh enough rel vlued funcon f : M Æ R by he forul Gf (, ) = E f ((( ) f U ) (()) = l Ê D Ÿq - ˆ () = 0 D Æ D nd he bcwrd nfnesl generor G(, ) s G (, ) f = Ê E f (()) - f ((( - D ) q U ) ˆ = l () =. DÆ 0 D By roune clculons one cn esly see h n U he generor L of soluon ( ) of Iô equon ( â (, ), A (, )) es he for j L (, )= ˆ j (8) where he r ( j )(, ) s he r produc ( A )( A ) l l nd ( A ) l s he rnsposed r o (A l ),.e., he r of conjuge operor. I s lso esy o see h operor L fro (8) s second order ngen vecor feld on M. A couple ( (, ), A (, )) where (, ) s (frs order) vecor feld nd A (, ) s feld of lner operors fro R N o T M, s clled n Iô vecor feld. Noce h he rnsforon rule under chnges of coordnes on M for n Iô vecor feld concdes wh h for ordnry ngen vecors. Of course Iô equons nd Iô vecor felds cnno concde snce hey hve dfferen rnsforon rules under chnges of coordnes. They y concde for gven rvlzons over cern chr bu hs concdence wll becoe fled n oher rvlzons (over he oher chrs). Specfy connecon H on M. Defnon 3. We sy h n Iô vecor feld ( (, ), A (, )) nd n Iô equon ( ˆ(, ), A ˆ(, )) cnonclly correspond o ech oher wh respec o connecon H f every Œ M nd for ll fro he don, (, )= ˆ (, ) nd A (, )= A ˆ (, ) wh respec o rvlzon of norl chr of H. Fro defnon of A (, ) nd  (, ) follows h f A (, )= A ˆ (, ) n cern chr, he equly rens rue n every chr. Theore 4. An Iô vecor feld ( (, ), A (, )) nd n Iô equon ( â (, ),  (, )) cnonclly correspond o ech oher wh respec o connecon H f nd only f A (, )= A ˆ (, ) nd n every chr for every =,, n he equly j â (, )= (, ) - Gj holds where ( j )=( )( ) A A j nd G j re Chrsoffel sybols of second nd for H n he chr. Theore s reforulon of le 9.8 fro [5] (see lso le. n [7]). j The objec wh coordnes G j s rg ( A, A) where G (, ) s he so clled locl connecor of H n he chr (we shll no descrbe hs noon here, see dels, e.g., n [5 7]). Thus for gven connecon equon (6) cn be presened n equvlen for d()= (, ()) d - - rg() A A d A dw (, ) (, ( )) ( ) (9) where (, A ) s he Iô vecor feld cnonclly correspondng o ( âa, ) wh respec o H. Equon (9) s nown s Iô equon n Bcsendle s for. I s presenon n locl chr of nvrn equons nown s Iô equon n Belopolsy Dles for (see [4 7]). Theore. Le he Iô vecor feld ( (, ), A (, )) cnonclly correspond o n Iô equon ( â (, ), A (, )) wh respec o connecon H nd le L be he generor of soluons of ( â (, ), A (, )). Then (, )=H L where H s gven by (3). Proof. Indeed, by (0) n chr U we hve j L (, )= ˆ j. Then by (3) j HL = ˆ G j. j Bu by Theore â (, )= (, ) - Gj. Ths coplees he proof. Le ( ) be sochsc process wh vlues n M gven on cern probbly spce ( W, F, P ). Denoe by E he condonl epecon wh respec o he s -sublgebr N genered by preges of Borel ses n M relve o he ppng ( ): WÆ M (he presen of process ( )). In [8 3] he noon of forwrd en dervve, bcwrd en dervve nd qudrc en dervve were nroduced s follows. Te relvely copc norl chr U of connecon H specfed pon Œ M. Denoe by q U he rndo e of he frs enrnce of ВЕСТНИК ВГУ, СЕРИЯ: ФИЗИКА. МАТЕМАТИКА, 008, 99

4 Yu. E. Gllh () o he boundry of U wh ( ) Œ U. Inroduce he noon D()= (( D) ŸqU )- (). For Œ U we cn clcule he regresson 0 ÈÊ D() ˆ Y (, ) U = l E ( ). 0 Í = D Ø Î D Consruc he vecor feldy 0 (,), ssgnng o ech Œ M he correspondng vecor Y 0 (, ) U n U. Thus by consrucon Y 0 s esurble secon of ngen bundle TM,.e., vecor feld. Defnon 4. The rndo vecor D H 0 ()= Y (, ()) s clled forwrd en dervve of process ( ) on M e nsn wh respec o H. In nlogy wh bove consrucon we gve he defnon of bcwrd en dervve. For relvely copc norl chr U of H connng pon Œ M denoe by q U he rndo e of he ls e of () fro he boundry of U wh ( ) Œ U. Inroduce he noon D ()= () - (( - D ) q U ). Then for Œ U clcule he regresson 0 ÈÊ D () ˆ Y (, ) U = l E ( ). 0 Í = D Ø Î D Consruc he vecor feld Y 0 (,), ssgnng o ech Œ M he correspondng vecor Y 0 (, ) U n U. Thus by consrucon Y 0 s esurble secon of ngen bundle TM,.e., vecor feld. Defnon 5. The rndo vecor D H 0 ()= Y (, ()) s clled becwrd en dervve of process ( ) on M e nsn wh respec o H. Defnon 6. The l Ê D() ƒ D() ˆ D()= l E DØ 0 (0) D where ƒ s he ensor produc n odel spce connng chr, s clled he qudrc en dervve of ( ) on M e nsn. Noe h he qudrc en dervve s wellposed ndependenly of ny connecon. I should be lso poned ou h f we defne n objec by (0) where D( ) s replced by D (), for soluon of ( âa, ) we shll obn D () s well. The ne seen follows fro he resuls of [ 3]. Theore 3. Le he Iô vecor feld ( (, ), A (, )) cnonclly correspond o n Iô equon ( â (, ), A (, )) wh respec o connecon H. Then for soluon ( ) of ( â (, ), A (, )) we hve: D H ( ) = (, ( )), D ()= (, ()), where (, ) s he syerc (, 0) -ensor feld (, )= AA (, ) (, ), nd A (, ) s he conjuge operor o A (, ). Thus fro Theores nd 3 nd fro forul () follows h for bove ( ) wh generor L we hve H D ( ) = ( HL)(, ( )) () nd D ()=( Q L)(, ()) (6) where H s nroduced n (3) nd Q n (). If H s specfed, we shll no ndce n he noon of en dervves. Rer. Le f : M Æ M be sooh ppng of nfolds. Noe h, snce he vlue of en dervve depends on he now s -lgebr of he process, he ngen ppng Tf sends en dervves of process h( ) o en dervves of he process ()= f( h()) only n he followng for: h Tf ( Dh( ))= D ( ( )) or Tf ( D h( ))= D ( ) bu generlly speng Tf ( Dh( )) π D( ). Anlogous fc n rue for bcwrd en dervves: h Tf ( D h( ))= D ( ( )) Tf ( D h( ))= D ( ) bu generlly speng Tf ( D h( )) π D ( ). Noce h f we pply he se connecon boh for rnson fro ( âa, ) o ( A, ) nd for deernng he en dervve, we obn for soluon ( ) h D()= (, ()). Moreover, f we chnge he connecon, he Iô vecor fled (, A ) cnonclly correspondng o ( âa,, ) nd he forwrd en dervve D( ) wll be chnged bu he equly D()= (, ()) for hose new vlues wll ren rue. Now le us urn o relons beween bcwrd en dervves nd bcwrd nfnesl generors. For he se of splcy, f ( ) s soluon of ( âa, ) we rene he vecor Y 0 (, ) s (, ), hus D () = (, ()). Inroduce Dw () by forul w ()-w ( - D) Dw () = l E where E s he DÆ 0 D condonl epecon wh respec o he «presen» s -lgebr of ( )(see bove). 00 ВЕСТНИК ВГУ, СЕРИЯ: ФИЗИКА. МАТЕМАТИКА, 008,

5 On relons beween nfnesl generors nd en dervves of sochsc processes on nfolds Defnon 7 The process w ()= Ú D w() s ds w ()- wt ( ) s clled he bcwrd Wener process wh respec o ( ). Specfy connecon H on M. Le ( A, ) be n I ô vecor feld on M. Denoe by A he covrn dervve of he feld A (, ) wh respec o connecon H ; A s feld of blner n operors A (, )(,): T M R ÆT M. Consder he feld A (, )( A,):R R ÆT M n n nd he reled vecor feld r A( A)(, ) = r A(, )( A(, )( ), ). Deerne on M he followng equon n he locl coordnes of chr U s follows: d()= (, ()) d r A( A)(, ()) d - -A (, ()) D wd () - () - G () ( AAd, ) A (, ( )) dw ( ). Drec verfcon shows h () rnsfors correcly (covrnly) under chnges of coordnes. Ths ens h equon () s well defned on enre M. Theore 4. Le ( ), (0) = 0, be srong soluon o (). Then D ()= (, ()) for Œ(0, l]. Theore 4 s proved s heore.3 n [7]. Specfy cern e oen. Fro he bove forule follows h he process h( ) such h ()= h() nd ssfyng for s < he relon h() - h()= s (, h( )) d s Ú G h ( )( AAd, ) A(, h( )) d Ú w(), s Ú s T (3) where (, ) = (, ) - r A( A, ) A (, ) D w( ) nd he ls sund n he rgh-hnd sde s he bcwrd sochsc negrl, hs he se bcwrd en dervve s ( ). Relon (3) s clled Iô equon n bcwrd dfferenls. Thus for sll enough s < such h( s ) pproes ( s ). Inroduce he noon â (, ) = (, ()) G () ( AA, ). Tng no ccoun he nerrelons beween he rnsforon rule for locl connecor nd he second dervve of chnge of coordnes j b (see, e.g., [7]) we obn h under he chnge of coordnes j b beween he chrs U nd U b he rple (,( ˆ, A)) rnsfors by he rule (,( ˆ, A )) Ê ˆ jb, jb ˆ - rjb ( A, A ), jb ( A ). Thus ( â, A ) s bcwrd Iô equon ccordng o Defnon (forul (5)). Defnon 8. The Iô equon ( âa, ) nd he bcwrd I ô equon ( â, A ) nroduced bove, re clled coupled o ech oher. Denoe by L he bcwrd generor of ( â, A ) coupled wh ( âa, ) h descrbes he process ( ). In locl coordnes obvously epressed n he for j L = -ˆ ( AA ). j Theore 5. H D ()= - H( L ). (4) Proof. By consrucon D H ()= (, ()) nd ˆ (, ())= G() ( A, A) = j = ˆ G AA j( ). On he oher hnd, we obn h H( L ) =-ˆ j Gj ( AA ) = - - G ( AA ) j j G j ( AA ) = -. j Forul (4) s «syerc» o (). REFERENCES. Eery M. Sochsc clculus on nfolds / M. Eery. Berln e l.: Sprnger, Meyer P.A. A dfferenl geoerc forls for he Io clculus / P. A. Meyer // Lecure Noes n Mhecs, 98. Vol. 85. P Schwrz L. Serngles nd her sochsc clculus on nfolds / L. Schwrz. Monrel: Monrel Unversy Press, Belopolsy Y.I. Sochsc processes nd dfferenl geoery / Y. I. Belopolsy, Yu. L. Dlecy. Dordrech: Kluwer Acdec Publ shers, Gllh Yu.E. Ordnry nd Sochsc Dfferenl Geoery s Tool for Mhecl Physcs / Yu. E. Gllh. Dordrech: Kluwer, 996. ВЕСТНИК ВГУ, СЕРИЯ: ФИЗИКА. МАТЕМАТИКА, 008, 0

6 Yu. E. Gllh 6. Gllh Yu.E. Globl Anlyss n Mhecl Physcs. Geoerc nd Sochsc Mehods / Yu. E. Gllh. N.Y.: Sprnger-Verlg, Gllh Yu.E. Sochsc nd globl nlyss n probles of hecl physcs / Yu. E. Gllh. Moscow: KoKng, 005. (n Russn) 8. Nelson E. Dervon of he Schrödnger equon fro Newonn echncs / E. Nelson // Phys. Revews, 966. Vol. 50, 4. P Nelson E. Dyncl heory of Brownn oon / E. Nelson. Prnceon: Prnceon Unversy Press, Nelson E. Qunu flucuons / E. Nelson. Prnceon: Prnceon Unversy Press, 985. Azrn S.V. Dfferenl equons nd nclusons wh forwrd en dervves n R n / S. V. Azrn, Yu. E. Gllh // Proceedngs of Voronezh Se Unversy, P (n Russn). Azrn S.V. Dfferenl nclusons wh en dervves / S. V. Azrn, Yu. E. Gllh // Dync Syses nd Applcons Vol. 6,. P Azrn S.V. On dfferenl equons nd nclusons wh en dervves on copc nfold / S. V. Azrn, Yu. E. Gllh // Dscussones Mhece. DICO Vol. 7.. P Гликлих Юрий Евгеньевич доктор физико-математических наук, профессор кафедры алгебры и топологических методов анализа Воронежского государственного университета, тел , e-l: yeg@h.vsu.ru. Gllh Yur Evgenevch Docor of Scences n Physcs nd Mhecs, Professor of he Depren of lgebr nd opologcl ehods of nlyss, Voronezh Se Unversy, Phone: 08-64, e-l: yeg@h.vsu.ru. 0 ВЕСТНИК ВГУ, СЕРИЯ: ФИЗИКА. МАТЕМАТИКА, 008,