Translations o f

MATHEMATICAL MONOGRAPHS

Volume 22 4

Stochastic Analysis

Ichiro Shigekaw a

Translated b y Ichiro Shigekaw a

American Mathematica l Societ y Providence, Rhod e Islan d

10.1090/mmono/224

Editoria l B o a r d

Shoshichi Kobayash i (Chair )

Masamichi Takesak i

KAKURITSU KAISEK I

(STOCHASTIC ANALYSIS )

by Ichir o Shigekaw a

Copyright ©199 8 b y Ichir o Shigekaw a Originally publishe d i n Japanes e

by Iwanam i Shoten , Publishers , Tokyo , 199 8

Translated fro n th e Japanes e b y Ichir o Shigekaw a

2000 Mathematics Subject Classification. Primar y 60H07 ; Secondary 60H30 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages /mmono-224

Library o f Congres s Cataloging-in-Publicatio n Dat a

Shigekawa, Ichiro , 1953 -[Kakuritsu kaiseki . English ] Stochastic analysi s / Ichir o Shigekaw a ; translated b y Ichiro Shigekaw a

p. cm . - (Translation s o f mathematica l monographs , ISS N 0065-928 2 ; v. 224)

Includes bibliographica l reference s an d index . ISBN 0-8218-2626- 3 (pbk . : acid-fre e paper ) 1. Stochasti c analysis . I . Title . II . Series . III . Series : Iwanam i serie s i n

modern mathematics .

QA274.2 .S481 3 200 4 519.2/2-dc22 200404772 2

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Contents

Preface t o th e Englis h Editio n v

Preface t o th e Japanes e Editio n vii

Outline o f the Theor y an d th e Objective s i x

Chapter 1 . Wiene r Spac e 1 1.1. Th e Wiene r proces s 1 1.2. Multipl e Wiene r integra l 1 3

Chapter 2 . Ornstein-Uhlenbec k Proces s 2 1 2.1. Ornstein-Uhlenbec k semigrou p 2 1 2.2. Hypercontractivit y an d logarithmi c Sobole v inequalit y 3 1

Chapter 3 . Th e Littlewood-Paley-Stei n Inequalit y 4 7 3.1. Fundamenta l inequalitie s 4 7 3.2. Th e Littlewood-Paley-Stei n inequalit y 5 6

Chapter 4 . Sobole v Space s o n a n Abstrac t Wiene r Spac e 7 9 4.1. Equivalenc e o f norm s 7 9 4.2. Sobole v space s W r*(K) 8 6

Chapter 5 . Absolut e Continuit y o f Distribution s and Smoothnes s o f Density Function s 10 3

5.1. Absolut e continuit y o f distributions an d smoothnes s 10 3 5.2. Smoothnes s o f distribution s

defined b y Wiene r functional s 11 1

Chapter 6 . Applicatio n t o Stochasti c Differentia l Equation s 12 1 6.1. Stochasti c differentia l equation s 12 1 6.2. Degenerat e stochasti c differentia l equation s 15 1 6.3. Fundamenta l estimat e 16 5

Perspectives o n Curren t Researc h 17 5

iv C O N T E N T S

Bibliography 17 9

Index 18 1

Preface t o th e Englis h Editio n

This boo k i s a translatio n o f m y boo k originall y publishe d i n Japanese b y Iwanam i Shoten , Publishers . Th e ai m o f thi s boo k i s t o provide th e reade r wit h a concise introductio n t o stochasti c analysis , in particular , th e Malliavi n calculus . I hope tha t th e materia l o f thi s book wil l reach mor e reader s b y thi s translation .

I woul d lik e t o expres s m y dee p appreciatio n t o th e America n Mathematical Societ y fo r publishin g thi s translation , an d t o thei r staff fo r excellen t support .

February 200 4 Ichiro Shigekaw a

This page intentionally left blank

Preface t o th e Japanes e Editio n

This boo k i s a n introductio n t o stochasti c analysis . Wha t sto -chastic analysi s mean s i s rather wide : her e we roughly regar d i t a s a n analysis base d o n the Wiene r space . I t take s i n various technique s o f probability theor y an d i s necessarily relate d t o othe r part s o f proba -bility. I n this book we concentrate o n infinite dimensiona l analysis , i n particular, th e Malliavi n calculus . It s main stag e i s really the Wiene r space; t o b e mor e precise , ou r objec t i s Wiener functionals . W e hav e to analyz e functional s o n a n infinit e dimensiona l space , an d w e ar e forced t o develo p a calculu s a s i n a finite dimensiona l space . A s fo r integral, w e have a n abstrac t measur e theory , whic h work s efficientl y even o n a n infinit e dimensiona l space . W e als o hav e a theor y o f dif -ferentiation o n a n infinit e dimensiona l space , bu t i t doe s no t matc h with th e integration .

In th e 1970's , Pau l Malliavi n mad e a breakthroug h i n thi s area . He presente d a ne w calculu s t o realiz e a probabilisti c approac h t o a question of hypoellipticity o f Hormander type . I t was exactly a theor y of differentiation fo r the Wiener space . Th e theory turned ou t t o hav e applications no t onl y t o partia l differentia l equation s bu t als o man y other fields. Du e t o hi s contribution , th e theor y i s usually calle d th e Malliavin calculus . Th e them e o f thi s boo k i s a stochasti c analysi s which contain s th e Malliavi n calculu s a s a mai n part . I have trie d t o make th e descriptio n elementary , whic h a t time s ma y mak e thi s boo k rather redundant .

From th e pioneerin g wor k o f P . Malliavi n thi s boo k seem s t o remain a t a fundamental level . Th e research frontier i s still far beyon d this book , bu t I hop e tha t i t wil l prepar e th e reade r t o procee d t o recent furthe r topics . I am convinced tha t th e reader wil l have enough tools a t han d t o d o s o afte r readin g thi s book .

I wis h t o dedicat e th e boo k t o Professo r Shinz o Watanabe , wh o brought m e t o thi s area . I als o wis h t o expres s sincer e thank s t o Professor Yoichir o Takahashi , wh o recommende d tha t I writ e thi s

vii

V l l l PREFACE T O TH E JAPANES E EDITIO N

book. Specia l thank s ar e du e Professo r Masanor i Hino , who rea d th e entire manuscrip t carefull y an d mad e numerou s helpfu l suggestions . Finally, I would lik e to expres s my dee p gratitud e t o al l th e editoria l staff o f Iwanami Shoten , Publishers , fo r thei r efforts .

August 199 8 Ichiro Shigekaw a

Outline o f th e Theor y an d th e Objective s

The ai m o f thi s boo k i s t o giv e a n introductio n t o stochasti c calculus, i n particular , th e Malliavi n calculus . Th e mai n ai m o f th e Malliavin calculu s i s t o analyz e functional s define d o n th e Wiene r space i n whic h a Wiener proces s i s realized .

In probabilit y theor y i t i s calle d a Wiener process , bu t i t i s usu -ally calle d a Brownian motion . I t originate d i n a discovery o f Rober t Brown in 1882, who found a n extremely irregular movemen t o f minute particles comin g ou t o f a pollen . Sinc e then , thi s phenomeno n ha s been muc h studie d b y physicists , an d Norber t Wiene r establishe d i t as a rigorou s mathematica l object . On e reaso n fo r th e importanc e of the Wiene r proces s i s that i t describe s variou s mathematica l mod -els. Amon g them , th e mos t importan t i s stochastic differentia l equa -tions, develope d b y Kiyosi Ito. I n fact , man y concret e models , e.g. , i n physics, genetics, economics, etc. , are described b y stochastic differen -tial equations . Fo r the theory o f stochastic differentia l equations , see , for instance , Ikeda and Watanabe [7] , Karatzas an d Shrev e [8] , Revuz and Yo r [22] , an d others . I n thi s boo k w e ar e dealin g wit h Wiene r functionals suc h a s those define d b y stochasti c differentia l equations .

Wiener functional s ar e realize d o n the Wiene r space , whic h i s a n infinite dimensiona l space . Wha t w e need i s a calculus o n a n infinit e dimensional space , an d consistin g o f tw o things : differentiatio n an d integration. Integratio n i s base d o n well-develope d measur e theory . It work s eve n o n infinit e dimensiona l spaces . Bu t differentiatio n i s not a s eas y a s integration . O f cours e w e had a theory o f Freche t dif -ferentiation i n Banac h spaces . Man y othe r attempt s wer e als o made , but the y ar e no t organicall y connecte d t o th e integratio n theory . Fo r instance, differentiatio n i s the invers e operatio n o f integratio n i n th e one-dimensional case . Eve n i n th e multi-dimensiona l case , Stokes ' formula i s a n exquisit e combinatio n o f integratio n an d differentia -tion. O n th e Wiene r space , suc h a harmonious theor y appeare d onl y recently.

ix

x OUTLIN E O F TH E THEOR Y AN D TH E OBJECTIVE S

In 1976 , a n internationa l symposiu m o n stochasti c differentia l equations wa s hel d a t th e Researc h Institut e o f Mathematica l Sci -ences o f Kyot o University . A t tha t symposium , P . Malliavi n men -tioned, rathe r informally , a ne w theor y o f calculu s o n th e Wiene r space. Th e ful l detail s appeare d i n th e proceeding s o f th e sympo -sium (se e [14]) . Th e preprin t ha d bee n circulate d a littl e earlier , and th e autho r cam e t o kno w it . I t wa s reall y luck y fo r th e autho r to lear n th e theor y a t suc h a n earl y stage . I n Japan , N . Iked a an d S. Watanabe notice d th e importanc e o f this work a t onc e and starte d the stud y o f thi s theory , whic h i s no w calle d th e Malliavi n calculus . Malliavin himsel f visite d Japa n man y time s an d gav e stimulatio n t o Japanese probabilists . Wit h suc h a n opportunity , th e stud y o f th e Malliavin calculu s i n Japa n i s still active . Thi s i s another exampl e of the importanc e o f internationa l communication .

The Malliavin calculus is based on the Ornstein-Uhlenbeck opera -tor, which is a second order differentia l operator . I t i s a counterpart o f the Laplac e operato r i n Euclidean space . Malliavi n als o captured th e gradient operato r throug h th e squar e fiel d operato r (calle d operateu r de carre du champ in French literature ) associate d wit h th e Ornstein -Uhlenbeck operator . Th e gradient operato r i s connected t o the notio n of differentiation an d nowaday s i s formulated i n the framework o f H-differentiation o n th e Wiene r space . Her e H i s the Cameron-Marti n space, whic h w e wil l explai n later . Th e differentiatio n i s considere d only i n a subspac e if , an d derivative s ar e extende d b y mean s o f th e completion i n LP spaces. S o t o speak , derivative s ar e formulate d i n the sens e o f distributions . Throug h thi s procedur e w e could develo p a flexibl e theory . Th e ide a i s simple, bu t wha t i t brough t i s big .

In th e Malliavi n calculus , ther e appea r tw o fundamenta l opera -tors: on e i s the Ornstein-Uhlenbec k L an d th e othe r i s Jf-differentia -tion D. I n accordanc e wit h them , w e can define tw o kinds of Sobole v spaces. The y ar e closel y relate d t o eac h other , an d i n fac t w e ca n prove equivalence in the L p (p > 1 ) setting. Thi s is due to P. A. Meyer and brought a neat basi s in the Malliavin calculus. W e discuss Meyer' s result i n Chapter s 3 and 4 . A t present , tw o kinds o f proof ar e known ; we give a proo f alon g th e origina l ide a o f Meye r i n th e framework o f Littlewood-Paley theory . Th e proo f i s probabilistic, an d i t i s an goo d example tha t show s the power of martingale theor y an d Ito' s formula .

To develo p a calculu s o n a n infinit e dimensiona l space , i t i s a natural an d effectiv e wa y t o exten d result s fro m finite dimensiona l space. A s a matte r o f fact , ther e ar e som e analogie s betwee n them .

OUTLINE O F TH E THEOR Y AN D TH E OBJECTIVE S

We lis t som e o f the m below ; rigorou s definition s wil l b e give n later . Our object i s the Wiener space , and a typical finite dimensional mode l is the Euclidea n space .

Euclidean spac e Lebesgue measur e dx

Brownian motio n Laplace operato r A gradient operato r V

Hr>*> = (l-A)- r/2LP(dx) Sobolev inequalit y

Wiener spac e Wiener measur e \i

Ornstein-Uhlenbeck proces s Ornstein-Uhlenbeck operato r L

^-differentiation D Wr* = ( l - L ) - r / 2 L ^ ( / i )

logarithmic Sobole v inequalit y

We mak e som e remark s o n Sobole v spaces . I n finite dimensiona l space, better differentiability improve s the degree of integrability. Bu t in th e Wiene r space , bette r differentiabilit y improve s th e integrabil -ity onl y b y the logarithmi c order . W e cannot expec t mor e tha n this , but i t i s stil l powerful . Application s o f thi s fac t ar e no t give n i n thi s book, bu t it s importance is stressed in recent literature not only on the Wiener spac e bu t als o o n th e genera l finite dimensiona l spaces . His -torically E. Nelson first proved the hypercontractivity o f the Ornstein -Uhlenbeck semigroup , an d the n L . Gros s foun d th e equivalenc e be -tween hypercontractivit y an d th e logarithmi c Sobole v inequality . In -cidentally, a primitive notio n o f indifferentiatio n i s due t o Gross .

In this framework, w e can build a theory of differential calculu s on the Wiene r space , e.g. , th e chai n rul e o f the composit e function , inte -gration b y parts formula . O n the othe r hand , th e powerfu l metho d o f the Fourier transfor m i s available in Euclidean space . Ther e i s no cor-respondence o n the Wiene r space . Thi s remain s a s a futur e problem . Along th e way , ther e ar e som e aspect s o f the Fourie r transform . On e is tha t i t give s a spectra l decomposition . Fro m thi s viewpoint , th e spectrum o f th e Ornstein-Uhlenbec k operato r i s completel y known , and i n fact , th e eigenspaces ar e exactly the spaces of multiple Wiene r integrals. W e discuss thi s topi c i n §1.2 .

If we consider the Hilbert transform i n connection with the Fourie i transform, it s LP theory correspond s t o Meyer' s equivalence . I n fact , Meyer prove d hi s equivalenc e b y usin g martingal e theory , whic h ha s its origin in Fourier analysis . Th e spiri t o f analysis, including classica l Fourier analysis , flows here a s well .

The centra l par t o f thi s boo k i s Chapter 6 , where we discuss th e issue o f hypoellipticit y o f Hormander type . Malliavi n originall y buil t his theor y t o giv e a probabilisti c proo f t o thi s problem . Hi s wor k was followe d u p b y S . Kusuok a an d D . Stroock . The y investigate d

xii OUTLIN E O F TH E THEOR Y AN D TH E OBJECTIVE S

Malliavin's work and sharpened i t to its present form . W e discuss thi s problem followin g th e Kusuoka-Strooc k method . The y pile d u p enor -mous estimates toward s th e non-degenerac y o f Malliavin's covarianc e matrix. Th e reade r wil l find here th e rea l tast e o f analysis .

Some Frequently-use d Notation . • N , Z , Q , M , C denot e th e se t o f natura l numbers , integers ,

rational numbers , rea l numbers , comple x numbers , respec -tively. A suffix + refer s to non-negative numbers . Fo r instance, Z+ = {0 ,1 ,2 , . . . } .

• V means "fo r all " an d 3 means "ther e exist. " • C n(M.d) stand s fo r th e spac e of al l functions o n Rd o f class C n.

A suffi x b refers t o bounde d functions , a suffi x 0 to function s with compact suppor t an d a suffix + t o non-negative functions . In general , function s ar e rea l valued . T o specif y th e spac e o f values, we write, e.g. , C n(R - • R k). L p stand s fo r th e spac e of all p-th integrabl e functions , an d w e write L P(/J>) i f we need t o specify th e measure /j,. I f the measure is clear in the context, we sometimes write , e.g., L p([0, oo)). L p function s ar e usually rea l valued. T o specify a spac e o f values , w e write , e.g. , L P(/J;K).

• A point i n Rd i s denoted b y x = ( x 1 , . . . , xd) wit h superscript . A partia l derivativ e o f function / i s denoted b y ^ j o r simpl y by djf.

• Sij i s Kronecker' s delta . • aAb stand s fo r the minimum of a and 6 , aVb for the maximum .

Bibliography

We restric t ourselve s t o closel y relate d references . Refer e t o [16 , 2 1 , 7 ] fo r detailed references . S. Aida , Stochasti c analysi s o n loo p spaces , Sugaku Expositions 1 3 (2000) , 197-214. J. M . Bismut , Th e Atiyah-Singe r theorems , A probabilisti c approach . I . Th e index theorem , J. Fund. Anal, 5 7 (1984) , 56-99 . J. M . Bismut , Th e Atiyah-Singe r theorems , A probabilisti c approach . II . Th e Lefschetz fixed poin t formulas , J. Fund. Anal., 5 7 (1984) , 329-348 . N. Boulea u an d F . Hirsch , " Dirichlet forms and analysis on Wiener space" Walter d e Gruyter , Berlin , 1991 . L. Gross , Abstrac t Wiene r spaces , Proceedings of Fifth Berkeley Symp. Math. Statist. Prob. II, Part 1, 31-41 , Univ . Calif . Press , Berkeley , 1965 . L. Gross , Logarithmic Sobole v inequalities , Amer. J. Math., 9 7 (1975) , 1061-1083. N. Iked a an d S . Watanabe , " Stochastic differential equations and diffusion processes" Secon d edition , Kodansha/North-Holland , Tokyo/Amsterdam , 1989. I. Karatza s an d S . E . Shreve , u Brownian motion and stochastic calculus ," Second edition , Springer-Verlag , Ne w York , 1991 . S. Kusuoka , Mor e recen t theor y o f Malliavi n calculus , Sugaku Expositions 5 (1992), 155-171 . S. Kusuoka , Stochasti c analysi s a s infinite-dimensiona l analysis , Sugaku Ex-positions 1 0 (1997) , 183-194 . S. Kusuok a an d D . W . Stroock , Application s o f th e Malliavi n calculus , Par t I, Proceedings of the Taniguchi Intern. Symp. on Stochastic Analysis, Kyoto and Katata, 1982, ed . b y K . ltd , 271-306 , Kinokuniya , Tokyo , 1984 . S. Kusuok a an d D . W . Stroock , Application s o f th e Malliavi n calculus , Par t II, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 3 2 (1985) , 1-76 . S. Kusuok a an d D . W . Stroock , Application s o f th e Malliavi n calculus , Par t III, J. Fac. Set. Univ. Tokyo Sect. IA Math., 3 4 (1987) , 391-442 . P. Malliavin , Stochasti c calculu s o f variation an d hypoellipti c operators , Pro-ceedings of Intern. Symp. SDE, Kyoto, 1976, ed . by K. Ito , 195—263 , Kinoku-niya, Tokyo , an d Wiley , Ne w York , 1978 . P. Malliavin , C fc-hypoellipticity wit h degeneracy , Stochastic Analysis, ed . b y A. Friedma n an d M . Pinsky , 199-214 , 327-340 , Academi c Press , Ne w York , 1978. P. Malliavin , " Stochastic analysis," Springer-Verlag , Berlin , 1997 .

179

180 BIBLIOGRAPHY

[17] P . A . Meyer , Note s su r le s processu s d'Ornstein-Uhlenbeck , Seminaire de Prob., XVI, ed . par J. Azema et M. Yor, Lecture Note s in Math., 92 0 (1982), 95-133, Springer-Verlag , Berlin .

[18] P . A. Meyer , Quelque s resultat s analytique s su r le semi-groupe d'Ornstein -Uhlenbeck e n dimensio n infinie , Theory and application of random fields, Proceedings of IF IP-WG 7/1 Working conf. at Bangalore, ed . by G. Kallian -pur, Lectur e Note s i n Cont . an d Inform. Sci. , 49 (1983) , 201-214 , Springer -Verlag, Berlin .

[19] P . A. Meyer , Transformation s d e Ries z pou r le s lois gaussiennes , Seminaire de Prob., XVIII, ed . par J. Azem a e t M. Yor, Lecture Note s i n Math. , 1059 (1984), 179-193 .

[20] J . Norris , Simplifie d Malliavi n calculus , Seminaire de Prob. XX , Lectur e Notes i n Math. , vol . 1204, 101-130, Springer , Berlin , 1986.

[21] D . Nualart , "The Malliavin calculus and related topics,'' 1 Springer-Verlag , Berlin, 1995.

[22] D . Revu z an d M . Yor , "Continuous martingales and Brownian motion, 1'' Third edition , Berlin , Springer-Verlag , 1999.

[23] I . Shigekawa , Derivative s o f Wiene r functional s an d absolut e continuit y o f induced measures , J. Math. Kyoto Univ., 2 0 (1980) , 263-289 .

[24] D . W. Stroock , Th e Malliavin calculu s an d its application s t o secon d orde r parabolic differentia l operators , I , II, Math. System Theory, 1 4 (1981), 25-65, 141-171.

[25] D . W . Stroock , Th e Malliavi n calculus , functiona l analyti c approach , J . Fund. Anal, 4 4 (1981) , 217-257 .

[26] H . Sugita , Positiv e generalize d Wiene r function s an d potentia l theor y ove r abstract Wiene r spaces , Osaka J. Math., 2 5 (1988) , 665-696 .

[27] A . S . Ustiinel , "An introduction to analysis on Wiener space", Springer -Verlag, Berlin , 1995.

[28] S . Watanabe , Malliavin' s calculu s i n term s o f generalize d Wiene r function -als, Theory and application of random fields, Proceedings of IFIP-WG 1/1 Working conf. at Bangalore, ed . by G . Kallianpur , Lectur e Note s i n Cont . and Inform . Sci. , 49 (1983) , 284-290 , Springer-Verlag , Berlin .

[29] S . Watanabe , "Lectures on stochastic differential equations and Malliavin calculus," Tat a Institut e o f Fundamenta l Research , Springer-Verlag , Berlin , 1984.

[30] S . Watanabe , Analysi s o f Wiene r functional s (Malliavi n calculus ) an d it s applications t o hea t kernels , Annals of Prob., 1 5 (1987) , 1-39 .

[31] S . Watanabe, Stochasti c analysi s an d its applications, Sugaku Expositions, 5 (1992), 51-69 .

[32] G . N. Watson, "A Treatise on the Theory of Bessel Functions," Cambridg e University Press , Cambridge , 1944.

Index

abstract Wiene r space , 7 asymptotic expansion , 17 6

Borel a-field , 2 , 5 Brownian motion , 1 , 2 2 Burkholder-Davis-Gundy inequality ,

48

Cameron-Martin space , 6 , 1 9 Cameron-Martin theorem , 8 canonical realization , 2 Cauchy operator , 3 9 Cauchy process , 6 4 Cauchy semigroup , 3 9 Chapman-Kolmogorov equation , 2 1 characteristic function , 2 classical Wiene r space , 7 convolution operator , 2 1 convolution semigroup , 3 9 covariance, 2

eigenfunctions, 2 9 eigenvalue, 2 9 Euclidean inne r product , 3 exponential integrability , 1 1

Fernique theorem , 1 1 finite variation , 5 Frechet differentiation , 2 6 fundamental solution . 14 4

Gateaux derivative , 2 6 Gateaux differentiate , 2 6 Gaussian distribution , 2 , 5 Gaussian measure , 5 , 7 Gaussian process , 5 generalized Wiene r functional , 9 4 generator, 14 3

Green function , 3 7

//-derivative, 2 6 //-differentiable, 2 6 //-function, 7 3 heat kernel , 14 4 Hermite polynomial , 1 3 Hilbert-Schmidt class , 1 6 Hilbert-Schmidt inne r product , 1 6 Hille-Yosida theory , 2 5 hitting time , 6 2 hypercontractivity, 3 1 hypoellipticity, 15 1

independent increment , 1 index theory , 17 6 integration b y part s formula , 11 6 interpolation property , 8 0 invariant measure , 2 3 Ito's formula , 4 Ito-Wiener expansion , 1 6

kernel expression , 1 6 Kolmogorov's criterion , 2 2 Kolmogorov's extensio n theorem , 2 2

large deviatio n theory , 17 6 Levy's stochasti c area , 17 7 Littlewood-Paley theory , 4 7 Littlewood-Paley-Stein inequality , 5 8 local martingale , 3 locally squar e integrabl e continuou s

martingale, 3 locally square integrabl e martingale ,

47 logarithmic Sobole v inequality , 3 3 loop space , 17 8

181

182 INDEX

Malliavin's covarianc e matrix , 11 3 martingale, 3 maximal ergodi c inequality , 5 4 mean, 2 measurable function , 1 Meyer's equivalence , 8 2 moment inequality , 9 0 multiple Wiene r integral , 16 , 18 , 4 5 multiplier, 3 9

natural inclusion , 6 non-degenerate

in th e sens e o f Hormander , 16 2 in th e sens e o f Malliavin , 11 3

operator norm , 3 9 Ornstein-Uhlenbeck process , 2 2 Ornstein-Uhlenbeck semigroup , 24 ,

25

subordination, 37 , 3 9 supremum norm , 4 symmetric group , 1 7 symmetrization, 1 7

topological cr-field , 5 trace, 2 7 trace class , 2 6 transition probability , 2 1

uniform ellipticity , 14 6

Wick product , 1 8 Wiener integral , 5 Wiener integra l o f orde r 1 , Wiener measure , 2 , 1 1 Wiener process , 1 , 5 Wiener space , 2

path, 1 path space , 2 Poisson semigroup , 3 9 polynomial, 2 4 probability space , 1 process o f bounde d variation , 4

quadratic variation , 4

random variable , 1 reproducing Hilber t space , 7 resolvent, 4 0 Riesz representatio n theorem , 5

sample path , 1 Schrodinger operator , 17 7 semimartingale, 4 signed measure , 5 Sobolev space , 86 , 8 8 spectral decomposition , 3 9 spectrum, 3 9 stationary increment , 2 step function , 3 stochastic integral , 2 , 5 , 6 stochastic oscillator y integral , 177 stochastic process , 1 stopping time , 3 Stratonovich's symmetri c integral , 14 1 strongly continuou s contraction semi -

group, 2 5

Titles i n Thi s Serie s

224 Ichir o Shigekawa , Stochasti c analysis , 200 4

223 Masatosh i Noumi , Painlev e equation s throug h symmetry , 200 4

222 G . G . Magaril-Il'yae v an d V . M . Tikhomirov , Conve x analysis :

Theory an d applications , 200 3

221 Katsue i Kenmotsu , Surface s wit h constan t mea n curvature , 200 3

220 I . M . Gelfand , S . G . Gindikin , an d M . I . Graev , Selecte d topic s i n

integral geometry , 200 3

219 S . V . Kerov , Asymptoti c representatio n theor y o f th e symmetri c grou p

and it s application s t o analysis , 200 3

218 Kenj i Ueno , Algebrai c geometr y 3 : Furthe r stud y o f schemes , 200 3

217 Masak i Kashiwara , D-module s an d microloca l calculus , 200 3

216 G . V . Badalyan , Quasipowe r serie s an d quasianalyti c classe s o f

functions, 200 2

215 Tatsu o Kimura , Introductio n t o prehomogeneou s vecto r spaces , 200 3

214 L . S . Grinblat , Algebra s o f set s an d combinatorics , 200 2

213 V . N . Sachko v an d V . E . Tarakanov , Combinatoric s o f nonnegativ e

matrices, 200 2

212 A . V . Mel'nikov , S . N . Volkov , an d M . L . Nechaev , Mathematic s o f

financial obligations , 200 2

211 Take o Ohsawa , Analysi s o f severa l comple x variables , 200 2

210 Toshitak e Kohno , Conforma l field theor y an d topology , 200 2

209 Yasumas a Nishiura , Far-from-equilibriu m dynamics , 200 2

208 Yuki o Matsumoto , A n introductio n t o Mors e theory , 200 2

207 Ken'ich i Ohshika , Discret e groups , 200 2

206 Yuj i Shimiz u an d Kenj i Ueno , Advance s i n modul i theory , 200 2

205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 1

204 A . M . Vinogradov , Cohomologica l analysi s o f partia l differentia l equations an d Secondar y Calculus , 200 1

203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d

coding, 200 2

202 V . P . Maslo v an d G . A . Omel'yanov , Geometri c asymptotic s fo r

nonlinear PDE . I , 200 1

201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1

200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1

199 Shigeyuk i Morita , Geometr y o f characteristi c classes , 200 1

198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology ,

2001

197 Kenj i Ueno , Algebrai c geometr y 2 : Sheave s an d cohomology , 200 1

196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes , 2001 195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1

TITLES I N THI S SERIE S

194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1

193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1

192 Yu . P . Solovyo v an d E . V . Troitsky , C* -algebras an d ellipti c

operators i n differentia l topology , 200 1

191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n

geometry, 200 0

190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces ,

2000

189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e

phenomena, 200 0

188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f

random variable s an d rando m processes , 200 0

187 A . V . Fursikov , Optima l contro l o f distribute d systems . Theor y an d

applications, 200 0

186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r

theory 1 : Fermat' s dream , 200 0

185 Kenj i Ueno , Algebrai c Geometr y 1 : From algebrai c varietie s t o schemes ,

1999

184 A . V . Mel'nikov , Financia l markets , 199 9

183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9

182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d

conservation law s fo r differentia l equation s o f mathematica l physics , 199 9

181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups .

Part 2 , 199 9

180 A . A . Milyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d

optimal control , 199 8

179 V . E . Voskresenski i , Algebrai c group s an d thei r birationa l invariants ,

1998

178 Mi t su o Morimoto , Analyti c functional s o n th e sphere , 199 8

177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8

176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e

Hamiltonian system s wit h simpl e singula r point s (topologica l aspects) , 199 8

175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8

174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d

elliptic systems , 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l properties o f distribution s o f stochasti c functionals , 199 8

For a complet e lis t o f title s i n thi s series , visi t th e AMS Bookstor e a t www.ams.org/bookstore / .