Titles in This Series
Volume 6 Eugen e B. Dynkin An introduction to branching measure-valued processe s 1994 5 Andre w Bruckner Differentiation o f real function s 1994 4 Davi d Ruelle Dynamical zeta functions fo r piecewise monotone maps of the interval 1994 3 V . Kumar Murty Introduction to Abelian varieties 1993 2 M . Ya. Antimirov , A. A. Kolyshkin, and Remi Vaillancourt Applied integral transform s 1993 1 D . V. Voiculescu, K. J. Dykema, and A. Nica Free random variables 1992 This page intentionally left blank An Introductio n t o Branchin g Measure-Valued Processe s This page intentionally left blank https://doi.org/10.1090/crmm/006 Volume 6 C CR M g MONOGRAP H SERIES Centre de Recherches Mathematique s Universite de Montrea l
An Introductio n t o Branchin g Measure-Valued Processe s
Eugene B . Dynki n
The Centr e d e Recherche s Mathematique s (CRM ) o f th e Universite d e Montrea l was create d i n 196 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M i s supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , and th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) o f Montreal , whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique .
American Mathematical Society Providence, Rhode Island US A The productio n o f this volum e wa s supporte d i n par t b y th e Fond s pou r l a Formatio n de Chercheur s e t l'Aid e a l a Recherch e (Fond s FCAR ) an d th e Natura l Science s an d Engineering Researc h Counci l o f Canada (NSERC) .
1991 Mathematics Subject Classification. Primar y 60J80 ; Secondar y 60J25 .
Library o f Congres s Cataloging-in-Publicatio n Dat a Dynkin, E . B . (Evgeni T Borisovich), 1924 - An introductio n t o branchin g measure-value d processes/Eugen e B . Dynki n p. cm . — (CR M monograp h series ; v . 6 ) Includes bibliographica l reference s an d index . ISBN 0-8218-0269- 0 (acid-free ) 1. Branchin g processes . 2 . Markov processes . I . Series . QA274.76.D96 199 4 519.2'34—dc20 94-1625 7 CIP
Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provided th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addressed t o the Manage r o f Editorial Services , America n Mathematical Society , P.O . Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o [email protected] . The owne r consent s t o copyin g beyon d tha t permitte d b y Section s 10 7 o r 10 8 o f th e U.S . Copyright Law , provide d tha t a fe e o f $1.0 0 plu s $.2 5 pe r pag e fo r eac h cop y b e pai d directl y t o the Copyrigh t Clearanc e Center, Inc. , 222 Rosewood Drive , Danvers, Massachusetts 01923 . Whe n paying thi s fe e pleas e us e the cod e 1065-8599/9 4 to refe r t o thi s publication . Thi s consen t doe s not exten d t o othe r kind s o f copying , suc h a s copyin g fo r genera l distribution , fo r advertisin g o r promotional purposes , fo r creatin g ne w collectiv e works , o r fo r resale .
© Copyrigh t 199 4 by the America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensure permanenc e an d durability . t«$ Printed o n recycle d paper . This volum e wa s typese t usin g A^fS-T^K., the America n Mathematica l Society' s T^i macr o system , and submitte d t o th e America n Mathematica l Societ y i n camera-read y form b y the Centr e d e Recherche s Mathematiques . 10 9 8 7 6 5 4 3 2 1 9 9 98 9 7 96 9 5 9 4 Contents
Preface i x Chapter 0 . Super-Brownia n Motio n an d Partia l Differentia l Equation s 1 1. Brownia n motio n an d th e Laplac e equation 1 2. A nonlinear equatio n 1 3. Super-Brownia n approac h to (2.1 ) 2 4. Super-Brownia n motio n 2 5. Rang e o f X 3 6. Pola r set s fo r the Brownia n an d super-Brownia n motion s 3 7. Pola r set s on the boundary 4 8. Marti n boundar y theor y fo r equatio n Lv = v a 5 9. Solution s i n R d \ {0 } 6 10. Positiv e solution s o f Av = v 2 i n the unit dis k 7
Chapter 1 . Introductio n 9 1. Branchin g measure-value d processe s 9 2. Superprocesse s 1 3 3. Critica l BM V processe s an d superprocesse s 1 7 4. Superprocesse s wit h a finit e bas e space 1 9 5. Superdiffusion s an d partia l differentia l equation s 2 1
Chapter 2 . Marko v Processe s 2 5 1. Marko v transition function s an d Marko v semigroup s 2 5 2. Marko v processe s 2 6 3. Remarkabl e classe s o f functions associate d wit h a right proces s 2 8 4. Additiv e functional s o f a Hunt proces s 3 1 5. Additiv e martingales . Lev y measure 3 8 6. Stoppin g distribution s an d randomize d stoppin g times 4 3
Chapter 3 . Constructio n o f Superprocesses 4 5 1. Branchin g 4 5 2. Branchin g particl e system s 4 6 3. Superprocesse s 4 8 4. Th e existenc e theore m 5 2
Chapter 4 . Genera l Feynman-Ka c formul a 6 3 1. Centra l resul t 6 3 2. Derivatio n o f the classica l Feynman-Ka c formul a 6 7 3. Applicatio n o f the Lev y measure 6 8 viii CONTENT S
Chapter 5 . Chang e o f Parameters i n Superprocesse s 7 1 1. Chang e o f parameters 7 1 2. Genera l existenc e theore m 7 3 3. Superprocesse s wit h finite first moment s 7 4 Chapter 6 . Structur e o f Branching Measure-Value d Processes 7 9 1. Introductio n 7 9 2. Relation s betwee n processe s £ and X 8 0 3. Preliminar y for m o f log-Laplace equatio n 8 7 4. Linearit y o f (af) an d L 8 9 5. Approximatio n o f (af) 9 3 6. Localizatio n o f (af) an d L 9 6 7. Characteristi c o f continuous branchin g 10 1 8. Characteristi c o f discontinuous branchin g 10 5 Chapter 7 . Historica l Note s an d Comment s 10 9 1. Definitio n o f BMV processe s 10 9 2. Constructio n o f BMV processe s 11 0 3. Structur e o f BMV processe s 11 2 4. Branchin g measure-value d processe s a s solutions o f martingal e problem 11 3 Appendix. Element s o f Stochastic Calculu s 11 5 1. Martingale s an d relate d topic s 11 5 2. Stochasti c integratio n wit h respec t t o semimartingale s 12 0
References 12 3 Index 12 7
Index o f Notatio n 133 Preface
For abou t a hal f o f century , tw o classe s o f stochasti c processe s - Gaussia n processes an d processe s wit h independen t increment s - hav e playe d a n importan t role i n th e developmen t o f stochasti c analysi s an d it s applications . Durin g th e last decade , th e thir d clas s - branchin g measure-value d (BMV ) processe s - ha s also attracted a great dea l o f efforts o f many investigators. A common feature o f all three classes is that thei r finite-dimensional distribution s ar e infinitely divisibl e an d therefore a powerful analyti c tool o f the Laplac e (o r Fourier) transfor m i s available for thei r investigation . Al l three classes , i n an infinite-dimensiona l setting , provid e means fo r stud y o f physical system s wit h infinit e man y degree s o f freedom . Finite-dimensional distributions o f a Gaussian process can be expressed throug h its first and secon d moments. I n the case of a process with independent increments , the celebrated Levy-Khinchi n formul a allow s to write the characteristic functio n o f Xtl,... ,Xt n. Fo r BM V processes , a n analogou s rol e i s played b y th e log-Laplac e equation - a n equatio n involvin g th e Laplac e transfor m o f X t. W e canno t writ e such an equation fo r the mos t genera l BM V process but a substantial progres s ha s been made recently i n this direction. Ou r main objective i s to present th e results of this work. Firs t w e construct a large class o f BMV processes, called superprocesses , by passing to the limit fro m branchin g particle systems. The n w e prove that, unde r certain restrictions , a general BM V i s a superprocess. W e restrict ourselve s to th e so-called critica l cas e whe n th e mathematica l expectatio n o f the tota l mas s o f X t does not change in time [th e subcritical case when this expectation can decrease but not increase can also be treated i n a similar way]. W e impose regularity assumption s which ar e commonl y use d i n theory o f Markov processe s o n X an d o n the proces s £ describin g th e spatia l motio n o f infinitesima l part s o f th e clou d (w e cal l i t th e projection o f X). Finally , w e introduce a condition whic h i s slightly stronge r tha n the finiteness o f the secon d moment s o f the total mass . If the projectio n £ o f X i s a diffusio n wit h generato r L , the n th e log-Laplac e equation fo r X involve s th e operato r Lu — ip(u) wher e a nonlinea r functio n i\) de - scribes the branching mechanism . Th e clas s o f admissible function s i/; contain s th e family tp(u) — ua, 1 < a < 2 . W e cal l X a superdiffusion . I f £ i s th e Brownia n motion, then X i s called the super-Brownian motion . Pat h propertie s o f the super - Brownian motio n ar e well-known du e to the wor k o f Dawson, Perkins , L e Gall a.o . Partial differential equation s involving the operator AU — I/J(U) have been studied in - dependently b y analysts: Loewne r an d Nirenberg , Friedman , Brezis , Veron, Bara s and Pierr e a.o . Th e recentl y discovere d connection s betwee n th e probabilisti c an d analytic theorie s provid e powerfu l tool s fo r investigatin g hittin g probabilitie s an d other feature s o f superdiffusions . O n th e othe r hand , thes e connection s lea d t o a better understandin g o f such phenomena a s blowing up o f solutions o f PDE a t th e
ix x PREFAC E boundary o f a domain, characte r o f isolated singularitie s etc . Thi s directio n i s the subject o f a recent expository paper [27 ] and w e touch it only slightly. However , w e open the boo k b y a brie f nontechnica l introductio n t o the super-Brownia n motio n and t o it s connections with nonlinea r PDE . ACKNOWLEDGEMENTS. Thi s boo k i s based o n the Andr e Aisenstad t lecture s given at th e Centre d e recherches mathematiques, Universit e d e Montreal i n 1992 - 93. It s content s als o reflec t m y lecture s a t th e Universit y o f Tennessee , Knoxvill e (the Barrett Memoria l Lectures, March 1993 ) and at the Euler International Math - ematical Institute, St. Petersburg (th e Kolmogorov Semester, April and June 1993) . I a m greatl y indebte d t o S . E . Kuznetso v wh o ha s rea d an d carefull y edite d the entir e manuscript . Ou r numerou s discussion s lea d to significan t improvement s of th e content s an d th e for m o f presentation . Hi s contribution s wer e crucia l i n Chapter 6 (base d o n th e joint pape r [28]) . H e mus t als o b e credite d fo r th e fina l versions o f Theorems 4.1. 1 and 4.1.2 . My thank s ar e du e als o t o She u Yuan-Chun g wh o suggeste d man y valuabl e comments an d corrections . Appendix
Elements o f Stochasti c Calculu s
We present her e elements o f the theory o f martingales, loca l martingales, semi - martingales an d stochasti c integratio n wit h respec t t o thes e processes . W e als o introduce a concept o f a measure-valued martingale .
1. Martingale s an d relate d topic s 1.1. Basi c structure s i n a filtere d probabilit y space . Le t J b e a subse t of the extended rea l lin e [—00 , +00]. A family o f a-algebras Tt,t G J i n ft i s called a filtratio n o f a probabilit y spac e (ft , T, P) i f Tt C T an d T s C T t fo r s < t. The collectio n (Q,Tt,T, P) i s called a filtered probabilit y space . A stochasti c process i s a functio n X : J x ft— > R suc h that, fo r ever y t, X t i s .F-measurable. If , for ever y £ , Xt i s ^-measurable, the n w e say that X i s adapted (t o the filtratio n Tt)- X i s progressivel y measurabl e if , fo r ever y t, it s restrictio n t o J * x ft i s Bt x ^-measurable. Her e Jt = J C\ [— 00, t] and Tt i s the Bore l cr-algebr a i n J t. Al l adapted righ t continuou s processe s ar e progressivel y measurable . Two processe s X an d Y ar e indistinguishable i f there i s a se t ft 0 suc h tha t P(fto) = 0 an d X t(u) = Y t(u) fo r al l t e J,00 £ ft 0. I n th e genera l theor y o f stochastic processes , indistinguishabl e processe s are , usually , identified . A function r:ft—>JUooisa stoppin g tim e i f {r < t} G Tt fo r ever y t G J. The pre- r a-algebr a T T consist s o f al l set s A € T suc h tha t A P i {r < t} G Tt for 1 all t G J. I f X i s progressively measurable , then X T i s .?>-measurable. A proces s X i s calle d quasi-lef t continuou s i f X Tn—- > X T P-a.s . fo r ever y sequence o f stopping times r n] r. A stochastic process X t i s a martingale i f it i s adapted, i f all Xt ar e integrabl e and i f
(1.1) E{X t I T 8} = X s a.s . fo r al l s,t G J. X i s a submartingale (supermartingale ) i f (1.1 ) hold s wit h = replace d b y > (<)• X belong s to class (D) i f the famil y X T wit h r runnin g ove r all stopping time s is uniformly integrable . Denote b y V th e ^-algebr a i n J x ft generate d b y al l adapted lef t continuou s processes. A proces s X i s calle d predictabl e i f i t i s indistinguishabl e fro m a V- measurable process. B y replacing here left continuou s functions b y right continuou s ones, w e define th e cr-algebr a O an d th e clas s o f optional processes . 1 We se t X T = 0 i f r = 0 0 and 0 0 ^ J .
115 116 APPENDIX. ELEMENT S O F STOCHASTI C CALCULU S
A substantia l par t o f the results i n the general theor y o f stochastic processe s depends o n the following condition s whic h ar e called the usual hypotheses : 1. P i s complete; 2. Tt contai n al l sets o f P-measure 0 ; 3. Tt i s right continuous , i.e. , Tt coincide s wit h the intersection o f Tu ove r all u > t). Under these hypotheses, every martingale i s indistinguishable fro m a right con - tinuous martingal e wit h lef t limit s an d therefor e i t i s possible t o assum e tha t al l martingales unde r consideratio n hav e these properties. Also : 1.1.A. A predictable proces s X t o f clas s (D) has , a.s., lef t limit s X t_ fo r all t i f and only i f the limit o f EX Tn exist s fo r ever y increasin g sequenc e o f stoppin g times r n. I t i s lef t continuou s i f and onl y if , i n addition , \\m.EX Tn = ^[limX Tn] [10, VI.4 9 and 50(f)] . l.l.B. Tw o predictable processe s X an d Y o f class (D) ar e indistinguishabl e 2 if and onl y i f EX T = EY T fo r al l predictable stoppin g time s r. [10 , IV.8 6 and 87(b)]. l.l.C. Al l predictable processes are optional and all optional processes are pro- gressively measurabl e [10 , IV.67 and 64].
l.l.D. A n optional proces s X t o f clas s (D) has , a.s., righ t limit s X t+ fo r all t i f and only i f the limit o f EX Tn exist s fo r every decreasin g sequenc e o f stoppin g times r n. I t i s right continuou s i f and only if , i n addition, \m\EX Tri = .E[limX rn] [10, VI.48] . l.l.E. Tw o optional processe s X an d Y o f clas s (D) ar e indistinguishable i f and onl y i f EXT = EYT fo r all stopping time s r [10 , IV.86 and 87(b)]. There is a standard constructio n calle d the augmentation whic h allows to get a filtere d probabilit y spac e (Sl,Tt,T,P) subjec t t o the usual hypothese s startin g from an y right continuou s filtratio n Tt o f a probability spac e (Q,T, P): th e space (Q,T,P) i s the completio n o f (Q,T,P) an d Tt i s the minima l cr-algebr a whic h contains Tt an d all sets A £ T suc h that P(A) = 0. In Section 1 and 2 we assume that the usual hypotheses are fulfilled. I n Section 1 we suppose that J = [a , b] i s a closed interval. I t is really important tha t J contain s its righ t end . Th e assumption tha t i t i s closed fro m th e lef t i s not substantial. I n Section 2 J i s an open interval . 1.2. Doob-Meye r decomposition . LEMMA 1.1 . Every positive right continuous submartingale X belongs to class (D). PROOF. Fo r every stoppin g time r ,
P{Xb | TT} > XT a.s .
The famil y i n th e lef t sid e i s know n t o b e uniforml y integrabl e [10 , V.14 ] and therefore th e family {X T} i s also uniforml y integrable . •
THEOREM 1.1 . Let X be a right continuous supermartingale of class (D). Then there exists a unique predictable right continuous increasing process At with A a = 0 2 A stoppin g tim e i s predictable i f Yt = lt> T i s a predictabl e process . Unde r th e usua l hypothesis thi s i s equivalent t o the existence o f a sequence o f stopping time s n < T 2 < • • • < r n < • n-l r (1.2) A t=At-Ar = \im^E{(Xu - X u+1) \ T U-} 2 = 0 weakly in L l(P) as A runs over a standard sequence of partitions A — {r = to < 3 t\ < • - - < t n = t] of the interval [r , t}. The convergence in (1.2 ) is strong if A is continuous. The following condition is necessary and sufficient for continuity of A: EX Tn— » EXT for every sequence of stopping times r n\ r. PROOF. Withou t an y loss of generality we can assume that X i s positive (other - r wise w e replace i t b y X t — N t wher e N t i s a right continuou s versio n o f E{Xb\J t}- The firs t par t o f th e theore m i s prove d i n [56 ] (cf . [10 , VII. 8 an d 21 ] an d [39 , Theorem 1.6.12]) . Th e secon d part follow s fro m 1.1.A . REMARK. Th e decompositio n X = M -A established i n Theorem 1. 1 i s called the Doob-Meyer decompositio n o f a super- martingale X. I t ca n no t hol d i f X doe s no t belon g to clas s (D). Indeed , i f M i s a martingale, the n \M\ i s a positive submartingale. S o is A. B y Lemm a 1.1 , M, A and M — A ar e o f class (D). A predictable right continuou s process A o f bounded variatio n i s called a com- pensator o f X i f X — A i s a martingale. Th e existenc e o f compensators fo r righ t continuous supermartingale s o f clas s (D) follow s fro m Theore m 1.1 . Clearly , com - pensators exist als o for difference s o f two such supermartingales. [Th e compensator A o f X i s determined uniquel y b y a n additional assumptio n A a = 0. ] 1.3. Martingal e measures . Le t (Z,Z) b e a measurabl e Luzi n space . A family o f random variable s Xt(B),t G J, B G Z i s calle d a martingale measur e if, fo r ever y t an d a; , i t i s a measur e i n B, and , fo r ever y B, i t i s a martingal e i n t, THEOREM 1.2 . If X t(B) is a supermartingale measure on (Z,Z) and if X t(Z) belongs to class (D), then there exists a measure T on J x Z such that M — X — T is a martingale measure and, for every B, the process T((a,£] x B) is predictable. PROOF. Ou r statemen t follow s triviall y fro m Theore m 1. 1 i f Z i s finit e o r countable. Ever y uncountabl e Luzi n measurabl e spac e i s isomorphic t o M++ wit h the Bore l cr-algebra . Let A t(u, v] be the compensator o f the supermartingale X t{u, v] and let A t(v) = A(a,v]. I f u, v, the n X t(v) = X t(u) + X t(u,v] an d therefor e A t(v) i s indistingui - shable fro m At(u) + A t(u, v]. Sinc e the compensator i s defined onl y up to indistin - guishability, w e ca n assum e tha t A t(ri) < A t(r2) fo r al l rationa l r\ < r 2 an d al l t. Denot e b y A t(s+) th e limi t o f A t(r) a s r | s alon g the se t o f the rationals . Le t rn [ s. Fo r ever y stopping tim e r , EAT(s+) = lim EAT(rn) = li m EXT(a,rn] = EX T(a,s] = EA T(s). 3 We sa y tha t a sequenc e A n o f partition s i s standard i f Ai C A 2 C • • • C A n C • • • an d th e union o f An i s everywhere dens e i n [r,t]. I n future , w e us e short writin g Y = lini A Y\ t o indicat e that Y n = li m Y\ fo r ever y standar d sequenc e o f partitions o f a give n interval . 118 APPENDIX. ELEMENT S O F STOCHASTI C CALCULU S Since A t(s) an d A t(s+) ar e predictable i n t, the y coincid e ou t o f a se t Q s o f mea - sure 0 . Pu t g{t,s) = A t(s+) fo r UJ £ Q si g(t,s) — 0 fo r UJ G Q s. Clearly , g i s monotone increasin g an d righ t continuou s i n both argument s an d X*(a , s] — g(t, s) is a martingale . Ther e exist s a measur e T suc h tha t T([a , £ ] x [0 , s]) = g{t,s). I t satisfies Theore m 1.2 . D 1.4. Quadrati c variatio n process . Square-integrabl e martingale s M for m a Hilber t spac e dJl 2 with the nor m 2 \\M\\m^[E{M )}^ <^. As usual, M an d M' ar e identified i f HM-M'^ = 0 which implies that M an d M' are indistinguishable [10 , V.22] . Therefor e ever y element o f Wl can be represente d by a righ t continuou s martingal e wit h lef t limit s an d w e wil l conside r onl y suc h martingales. I f M e 9JT 2, then M 2 i s a positive submartingale. W e apply Theore m 1.1 t o th e supermartingal e X = — M2 whic h belong s t o clas s (D) b y Lemm a 1.1 . There exist s a uniqu e integrabl e predictabl e righ t continuou s increasin g proces s A (with AQ = 0 ) suc h that —M 2 + A i s a martingale . Th e proces s A i s denoted b y (M) an d i t i s called the quadrati c variatio n proces s fo r M. Not e tha t n-l 2 (1-3) (Mr t=]im^E{(Mu-Mu+1) \^tt.} i=0 weakly i n I/ 1(P) a s A runs ove r a standar d sequenc e o f partitions A = { r = to < t\ < • • • < t n = t} o f the interva l [r , t]. Th e convergenc e i n (1.3 ) i s strong i f (M) i s continuous. Indeed , th e conditiona l expectatio n i n (1.3 ) i s equal t o E{{Ml+1-Ml)\?ti-} 2 and w e get (1.3 ) b y applying (1.2 ) to X t = -M t . The nex t resul t follow s easil y fro m th e secon d part o f Theorem 1.1 . THEOREM 1.3 . The process (M) is continuous if at least one of two conditions is satisfied: 1.4.A. M is continuous. I.4.B. The filtration Tt has no time discontinuity, i.e., T T ~ \j' J~Tn for every increasing sequence of stopping times r n | r. The join t quadrati c variatio n fo r tw o martingale s M , M' G 9Jt2 i s define d b y the formul a (M, M') = /i(M + M')\ ~(\(M- M')\ . 1.5. Continuou s an d purel y discontinuou s martingales . Al l discontinu- ities o f M e DJl 2 ar e jumps. Moreover , 2 (1.4) E^(AM sf<\\M\\ w seJ where AM S = M s - M s_ an d M a_ = 0 (se e [10 , VII.42bis]) . We denot e b y 9Jt 2 th e se t o f al l continuou s M € 9J1 2 and w e put M e Wl^ i f 2 2 2 M G M an d (M,N) = 0 fo r al l N e M C. Element s o f m d ar e calle d purel y discontinuous (square-integrable ) martingales . 1. MARTINGALE S AND RELATE D TOPIC S 119 2 THEOREM 1.4 . Every M e DJl has the unique representation C d c 2 d 2 (1.5) M = M + M where M eM c, M G m d. We have (M) = (M c) + (M d). The martingale M d is equal to the sum of its compensated jumps in the following sense. Put (1.6) A £,t = {s:s Zt(e) = Y,AMs. Ae.t Then, for every t, d (1.7) M = li m \Zt(e) - Z t(e)] in P-probability. where Z(e) means the compensator of Z(e). 4 (See [10 , VII.4 3 and 44]) . 2 REMARK. I f M e 9Jt i s purely discontinuous , the n M2-]T(AM,)2 S I 2 2 \H\ C(M) = E fJ H sd{M)& < oc. 'J 2 THEOREM 1.5 . There exists a mapping H — > H-M of'C%(M) to 9JX such that: 1.6.A. (H-M) t = H aMa for all t if H t = 0 on (a , b]. If H s = l s>uF where F is a bounded Tu-measurable function, then (H-M)t = F(M t-MtAu). 2 2 I.6.B. // H, H' e C P(M), M e m and if Y = H • M, Y' = E' • M, then EYbY{ = EJ H 8H'8d(M)8. Conditions 1.6.A ,B determine H • M uniquely. 4To prove the existence o f a compensator, w e represent Z(e) a s the difference o f two increasin g processes Z+(e) an d Z~(e) b y collectin g i n sum (1.6 ) separatel y positiv e an d negativ e terms. B y (1-4), 2 E[Z+(e) + Z-(e)} < -E ^(M s - M s_) 6 A e,t is finite an d therefor e Z^ (e) an d Z~ (e) ar e submartingales o f class (D). 120 APPENDIX. ELEMENT S O F STOCHASTIC CALCULU S The martingal e H • M i s called the stochastic integra l o f H with respec t to M an d it is denoted by t IHsdMs. a In the next sectio n we extend the stochastic integration to a larger class of processes M. 2. Stochasti c integratio n wit h respec t t o semimartingale s 2.1. Loca l martingales . I t i s more convenien t t o choos e fo r the domai n of integratio n a tim e interva l [a , b) ope n fro m th e right. Al l such interval s can be transforme d t o each othe r b y continuous monoton e mappings . Therefor e we can, withou t an y loss o f generality, follo w traditio n an d take a s J th e half-lin e R+ = [0 , ex)). Fo r all processes X o n R+ we set X_ = 0. A right continuou s adapte d proces s i s a local martingal e i f there exis t mar - tingales M n ove r (0 , oo] and stopping time s r n— > o o a.s. suc h tha t (2.1) X t = M? fo r all *e[0,r n]. Condition (2.1 ) can be replaced by the following one: X TriAt ar e uniformly integrabl e martingales (cf . [10 , VI.27]) . Indeed , i f M i s a martingale o n [0, oo], then M rAt i s a uniforml y integrabl e martingal e o n [0 , oc). O n the other hand , ever y uniforml y integrable martingal e o n [0, oo) can be continued to a martingale o n [0, oo].5 Every martingale M i s a local martingale becaus e the processes M tAn ar e uni- formly integrabl e martingales . A loca l martingal e i s locally square-integrabl e i f martingales M n i n (2.1 ) can be chosen square-integrable . LEMMA 2.1 . If all jumps of a local martingale M are bounded by a constant c, then M is locally square-integrable. Let e > 0. Every local martingale M can be represented as the sum of local martingale U with jumps bounded by e and a local martingale V of finite variation.^ The firs t statemen t i s true becaus e M tAT i s bounded i f all jumps o f M are bounded an d if T = inf{ £ : \M t\ > c}. The second statemen t follow s fro m [10, VI.85]. The concept o f the quadratic variation process (M) and the decomposition o f a square-integrable martingale s into a continuous and purely discontinuous parts ca n be extended , i n an obvious way , to all local martingale s whic h ar e locally square - integrable. B y using the decomposition i n Lemma 2.1 , we define th e continuous part M c fo r every loca l martingale M. 2.2. Semimartingales . A process X i s a semimartingale i f it admit s a decomposition (2.2) X = M + A 5The definitio n o f a loca l martingal e i n [10, VI.27] i s slightly wide r tha n our s but both are equivalent i f Xo i s integrable. 6Following [10 , VI.51], w e say that a functio n / : R+— + R has finite variatio n i f it i s a function o f bounded variatio n o n every finite interva l (0,t\. Thi s i s equivalent t o the condition that / ca n be represented a s the difference o f two monoton e increasin g functions . 2. STOCHASTI C INTEGRATIO N WIT H RESPEC T T O SEMIMARTINGALE S 12 1 where M i s a loca l martingal e an d A i s an adapte d righ t continuou s proces s o f finite variation . Th e decomposition (2.2 ) is not unique bu t the continuous par t o f M i s determined uniquel y b y X. I t i s called the continuous martingal e par t o f X an d it is denoted b y X c. We list som e properties o f semimartingales: 2.2.A. Al l right continuou s submartingales and supermartingales are semimar- tingales [10 , VII. 12]. 2.2.B. Suppos e that stoppin g times r n— > o o a.s. I f XTriAt ar e semimartingales, then s o is X [10 , VII.26]. 2.3. Locall y bounde d predictabl e processes . W e say that a process H is bounded i f there i s a constant c such that \H t(uj)\ < c for all t and UJ. A process H i s locally bounde d i f there exis t a sequenc e o f stopping time s r n | o o and constants c n suc h tha t \H t(uj)\ < c n fo r al l UJ an d all t < r n(uj). Al l predictable right continuou s processe s with lef t limit s are locally bounded [10 , VII.32]. 2.4. Th e stochastic integral . H>X= I [ HidXH s H Jo of a locally bounde d predictabl e proces s H wit h respec t t o a semimartingale X i s defined (u p to indistinguishability) i n the followin g THEOREM 2.1 . For an arbitrary semimartingale X, there exists a unique linear mapping H — > H • X from the space of locally bounded predictable processes to the space of semimartingales with the properties: 2.4.A. (H • X)t = H0X0 for all t if H = 0 on R++. If H s = \ S>UF with a bounded Tu-measurable function F, then (H-X)t = F(X t-XtAu). 2.4.B. The pointwise convergence of Hn to H implies the locally uniform con- vergence in probability7 of Hn • X to H • X if there exist stopping times r ^ | o o and constants Ck such that (2.3) \H?(uj)\ 2.4.F. If Y — H • X, then processes Y t — Yt- and Ht(Xt — X t~) are indistin- guishable (recall that Xo _ = lo- = 0). 7 Processes Y n converg e to Y locall y uniforml y i n probability if , n lim P suP|ya -ya|>e = 0 .s We only sketc h th e constructio n o f the stochasti c integra l an d refe r fo r proof s to [10 , VIII.3.9 an d 14] . The integra l H • X i s constructed, first , fo r bounde d predictabl e H an d the n it i s define d fo r locall y bounde d H b y a natural passag e t o th e limit . B y Lemm a 2.1, a n arbitrar y semimartingal e i s the su m o f a proces s A o f finite variatio n an d a locall y square-integrabl e loca l martingal e M. W e se t H • X = H • A - f H • M where H • A i s the ordinar y Lebesgue-Stieltje s integra l an d onl y H • M remain s t o be defined . I f M i s a square-integrable martingale , the n w e apply Theore m 1.5 . I f M i s a locally square-integrable loca l martingale, then w e get H • M a s the limi t o f n H • M wher e M ™ = M TnAt ar e squar e integrabl e martingale s an d r n— > oo . It follow s fro m 2.4. B that, fo r a wide class o f integrands, the stochastic integra l can b e obtained a s the limi t o f Riemannian sums . THEOREM 2.2 . If H is an adapted right continuous process with left limits, then rt n-l (2.4) / H a-dX3=]imy2Hti(Xti+l-Xti) in probability. (Cf. [10 , VII . 15]. Fo r the meanin g o f lini A se e footnote 3. ) 2.5. Ito' s formula . 1 n THEOREM 2.3 . Suppose that X ,... ,X are semimartingales in the same fil- tered probability space. Let X = (X 1,..., X n) and let f be a function of class C 2 in R n. Then (2.5) f(X t) = f(X 0) + [ [VA/(X 8_)dXJ + \ V DiDjfiXs-WX^X*) Jo L i l a x - Y, \f( s) - f(Xs-) - J2 Dif(X s-)AX, S 1. D . R . Adam s an d N . G . Meyers , Bessel potentials. Inclusion relations among classes of ex- ceptional sets, Indian a Universit y Math . J . 2 2 (1973) , 873-905. 2. P . Baras and M . Pierre, Singularities eliminables pour des equations semi-lineaires, Ann . Inst . Fourier, Grenobl e 3 4 (1984) , 185-206 . 3. , Problemes paraboliques semi-lineaires avec donnees mesurees, Appl . Anal . 1 8 (1984) , 111-149. 4. H . Brezis and L . Veron, Removable singularities of some nonlinear equations, Arch . Rational . Mech. Anal . 7 5 (1980) , 1-6 . 5. R . Courant , K . Priedrich s an d H . Levy, Uber die partiellen Differenzengleichungen der math- ematischen Physik, Math . Ann . 10 0 (1928) , 32-72 . 6. D . A . Dawson , Stochastic evolution equations and related measure processes, J . Multivar . Anal. 3 (1975) , 1-52 . 7. , The critical measure diffusion, Z . Wahr. verw . Geb . 40 (1977) , 125-145 . 8. , Measure-valued Markov processes, Ecol e d'Et e d e Probabilite s d e Sain t Flour , 1991 , Lecture Note s i n Math. , vo l 154 1 (1993) , Springer . 9. D . A. Dawson and E . A. Perkins, Historical processes, Mem . Amer. Math. Soc, vol . 454, 1991. 10. C . Dellacheri e an d P.-A.Meyer , Probabilites et potentiel, Hermann , Paris , 1975 , 1980 , 1983 , 1987. 11. J . L . Doob , Stochastic processes, Wiley , Ne w York , 1953 . 12. , Classical potential theory and its probabilistic counterpart, Springer , Ne w York, Berlin , Heidelberg, Tokyo , 1984 . 13. E . B . Dynkin , Theory of Markov processes, Pergamo n Press , Oxford , London , Ne w York , Paris, 1960 . 14. , Markov processes, Springer , Berlin , Gottingen , Heidelberg , 1965 . 15. , Regular Markov processes, Uspekh i Mat . Nau k 2 8 (1973) , 35-64 ; Marko v Processe s and Relate d Problem s o f Analysis , Londo n Math . Soc . Lectur e Not e Ser. , vo l 54 , Cambridg e Univ. Press , London , Ne w York , pp . 187-218 . 16. , Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times, Asterisqu e 157—15 8 (1988) , 147-171 . 17. , Superprocesses and their linear additive functionals, Trans . Amer . Math . Soc . 31 4 (1989), 255-282 . 18. , Regular transition functions and regular superprocesses, Trans . Amer . Math . Soc . 31 6 (1989), 623-634 . 19. , Three classes of infinite dimensional diffusions, J . Funct . Anal . 8 6 (1989) , 75-110 . 20. , A probabilistic approach to one class of nonlinear differential equations, Probab . Theor . Relat. Field s 89 (1991) , 89-115 . 21. , Branching particle systems and superprocesses, Ann . Probab . 1 9 (1991) , 1157-1194 . 22. , Path processes and historical processes, Probab . Theor. Relat. Fields 90 (1991) , 89-115. 23. , Additive functionals of superdiffusion processes, Rando m Walks , Brownia n Motio n and Interactin g Particl e Systems , Progres s i n Probabilit y (Ric k Durrett , Harr y Kesten , eds.) , vol. 28 , Birkhauser, Boston , Basel , Berlin , 1991 . 24. , Superdiffusions and parabolic nonlinear differential equations, Ann . Probab. 20 (1992) , 942-962. 25. , Birth delay of a Markov process and the stopping distributions for regular processes, Probab. Theor . Relat . Field s 9 4 (1993) , 399-411 . 123 124 REFERENCES 26. , On regularity of superprocesses, Probab . Theor . Relat . Field s 95 (1993) , 263-281 . 27. , Superprocesses and partial differential equations, Ann . Probab . 21 (1993) , 1185-1262 . 28. E . B . Dynkin , S . E . Kuznetso v an d A . V . Skorokhod , Branching measure-valued processes, Probab. Theor . Relat . Field s (t o appear) . 29. N . El Karoui and S . Roelly-Coppoletta, Proprietes de martingales, explosion et representation de Levy-Khintchine d'une classe de processus de branchement a valeurs mesures, Stochasti c Process. Appl . 3 8 (1991) , 239-266 . 30. S . N . Ethie r an d T . G . Kurtz , Markov processes. Characterization and convergence, Wiley , New York , 1986 . 31. W . Feller , An introduction to probability theory and its applications, II , Wiley , Ne w York , London, Sidney , 1966 . 32. , Diffusion processes in genetics, Proc . o f th e Secon d Berkele y Symposiu m o n Mathe - matical Statistic s an d Probability , Universit y o f Californi a Press , Berkele y an d Lo s Angeles , 1951, pp. 227-246 . 33. P . J . Fitzsimmons , Construction and regularity of measure-valued Markov branching pro- cesses, Israe l J . Math . 6 4 (1988) , 337-361 . 34. , On the martingale problem for measure-valued branching processes, Semina r o n Sto - chastic Processes , 199 1 (E . Cinlar , K . L . Chung , M . J . Sharpe , eds.) , Birkhauser , Boston , 1992. 35. A . Friedman, Partial differential equations of parabolic type, Prentice-Hall , Englewoo d Cliffs , N.J., 1964 . 36. A . Gmir a an d L . Veron, Boundary singularities of solutions of some nonlinear elliptic equa- tions, Duk e Math . J . 6 4 (1991) , 271-324 . 37. T . E. Harris, The theory of branching processes, Springer , Berlin , Gottingen, Heidelberg , 1963 . 38. N . Ikeda , M . Nagasaw a an d S . Watanabe , Branching Markov processes, J . Math . Kyot o 8 (1968), 233-278 ; II, J . Math . Kyot o 8 (1968) , 365-410; III, J . Math . Kyot o 9 (1969) , 95-160 . 39. N . Iked a an d S . Watanabe , Stochastic differential equations and diffusion processes, Nort h Holland, Amsterdam , Oxford , Ne w York , Kodansha , Tokyo , 1981. 40. M . Jirina , Continuous branching processes with continuous state space, Czesh . J . Math . 8 (1958), 292-313. 41. , Branching processes with measure-valued states, 3r d Prague Conference , 1964 , pp. 333 — 357. 42. S . Kakutani, On Brownian motions in n-space, Proc . Imp . Acad . Toky o 20 (1944) , 648—652 . 43. , Two-dimensional Brownian motion and harmonic functions, Proc . Imp . Acad . Toky o 20 (1944) , 706-714 . 44. O . Kallenberg , Random measures, Academi c Press , London , Ne w York , Paris , Sa n Diego , 1983. 45. A . N . Kolmogoro v an d N . A . Dmitriev , Branching stochastic processes, Dokl . Akad . Nau k SSSR 56 , 1 (1947) , 7-10 . 46. S . E. Kuznetsov , Nonhomogeneous Markov processes, J . Sovie t Math . 2 5 (1984) , 1380-1498 . 47. , Regularity properties of a supercritical superprocess (1993 ) (preprint) . 48. J . Lamperti , Continuous state branching processes, Bull . Amer . Math . Soc . 7 3 (1967) , 382 - 386. 49. J . F . L e Gall , Brownian excursions, trees and measure-valued branching processes, Ann . Probab. 1 9 (1991) , 1399-1439 . 50. , A class of path-valued Markov processes and its applications to superprocesses (1991 ) (preprint). 51. , Les solutions positives de Au = u 2 dans le disque unite (1993 ) (preprint) . 52. , Hitting probabilities and potential theory for the Brownian path-valued process, Ann . Inst. Fourie r (Grenoble ) 4 4 (1994 ) (t o appear) . 53. R . Sh. Liptzer and A. N. Shiryaev, Martingale theory, Nauka , Moscow, 198 6 (Russian); Englis h transl., Kluver , 1989 . 54. C . Loewne r an d L . Nirenberg , Partial differential equations invariant under conformal or projective transformations, Contribution s t o Analysi s (L . Ahlfor s e t al , eds.) , 1974 , pp. 255 - 272. 55. N . G . Meyers , A theory of capacities for potentials of functions in Lebesgue classes, Math . Scand. 2 6 (1970) , 255-292 . 56. K . M . Rao , On decomposition theorems of Meyer, Math . Scand . 2 4 (1969) , 66-78 . REFERENCES 125 57. D . Revu z an d M . Yor , Continuous martingales and Brownian motion, Springer , Berlin , Ne w York, 1991 . 58. S . Roelly-Coppoletta , A criterion of convergence of measure-valued processes: application to measure branching processes, Stochastic s 1 7 (1986) , 43-65. 59. H . Rost , The stopping distributions of a Markov process, Invent . Math . 1 4 (1971) , 1-16 . 60. W . Rudin , Real and complex analysis, McGraw-Hill , Ne w York , 1987 . 61. Yu . M . Ryzho v an d A . V . Skorokhod , Homogeneous branching processes with finite number of types and continuously varying mass, Teor . Veroyatnost . i Primenen. 1 5 (1970) , 722-726 . 62. M . Sharpe , General theory of Markov processes, Academi c Press , Sa n Diego , 1988 . 63. Y . C. Sheu, A Hausdorff measure classification of G-polar sets for the superdiffusion, Probab . Theor. Relat . Field s 9 5 (1993) , 521-533. 64. , On path properties of superdiffusions, Dissertation , Cornel l University , 1993 . 65. , Removable boundary singularities for solutions of some nonlinear differential equations, Duke Mat h J . (submitted) . 66. M . L . Silverstein , A new approach to local times, J . Math . Mechan . 1 7 (1968) , 1023-1054 . 67. , Continuous state branching semigroups, Z . Wahrsch. Verw . Gebiete 1 4 (1969) , 96-112. 68. D . W. Strooc k and S . R . S . Varadhan, Multidimensional diffusion processes, Springer , Berlin , Heidelberg, Ne w York , 1979 . 69. L . Veron , Singular solutions of some nonlinear elliptic equations, Nonlinea r Anal . 5 (1981) , 225-242. 70. S . Watanabe , A limit theorem of branching processes and continuous state branching pro- cesses, J . Math . Kyot o Univ . 8 (1968) , 141-167 . 71. , On two dimensional Markov processes with branching property, Trans . Amer . Math . Soc. 13 6 (1969) , 447-466 . 72. H . W. Watso n an d F . Galton , On the probability of the extinction of families, J . Anthropol . Inst. Grea t Britai n an d Irelan d 4 (1874) , 138-144 . This page intentionally left blank Index A additive functio n 3 8 adapted 3 8 right continuou s (RCA ) 3 8 additive functiona l 3 1 admissible 4 9 characteristic o f 3 1 continuous 3 1 corresponding t o a n exi t rul e 8 5 equivalent 3 1 linear (LAF ) 18 , 82 measure-valued (MVAF ) 3 5 natural 3 1 signed 3 1 additive martingal e 3 8 continuous 4 0 measure-valued 4 0 purely discontinuou s 4 0 square integrabl e 3 9 admissible metri c 9 8 special 10 1 associated linea r semigrou p 1 1 augmentation 11 6 B Baras, P . i base spac e 9 finite 19 , 10 9 one point 21 , 109 Bessel capacity 4 Brezis, H. i , 2 branching a-brancing 1 6 binary 1 6 characteristic o f 1 7 continuous 1 0 discontinuous 1 0 local 1 6 quadratic 1 6 127 INDEX branching particl e syste m 4 6 branching measure-value d (BMV ) proces s 9 critical 1 7 deterministic 1 0 subcritical 1 7 supercritical 1 7 branching propert y 9 week 109 , 11 0 strong 10 9 Brownian motio n 2 8 C capacity 3 characteristic o f continuous branchin g 1 7 characteristic o f discontinuous branchin g 1 7 class (D) 11 5 compensator 39 , 40, 85, 11 7 continuous martingal e part o f (/*,Xt) 8 6 of X 12 1 Courant, R . 1 critical cas e i critical Hausdorf f dimensio n 4 D Dawson, D.A . i , 10 , 11 2 Dawson-Watanabe superproces s 16 , 11 2 diffusion 2 5 Feller's 1 0 generator o f 2 8 Dirichlet proble m 1 Dmitriev, N.A . 10 9 Doob, J.L. 5 Doob-Meyer decompositio n 39 , 11 6 E El Karoui, N . 11 4 exit rul e 29 , 80 signed (SER ) 2 9 extinct clou d 2 F Feller, W. 10 9 Feller's diffusio n 1 0 Feynman-Kac formul a 14 , 63 classical 6 7 filtered spac e 27 , 11 5 filtration 11 5 fine topolog y 2 8 finely open se t 2 2 Fitzsimmons, RJ. 77 , 11 2 Friedman, A . i Friedrichs, R . 1 function, adapte d t o £ 38 INDEX 129 finely continuou s 2 8 invariant 8 0 L-continuous 2 8 nearly Bore l 27 , 37 of finite variatio n 12 0 R-continuous 2 8 RL-continuous 2 8 special (SP ) 2 9 functional, admissibl e 4 9 generating 4 5 Laplace 5 0 G Galton-Watson proces s 10 9 Gmira, A . 5 H Hausdorff dimensio n 4 Hunt, G.A . 5 Hunt proces s 2 8 I infinitely-divisible distributio n 1 2 Ito's formul a 12 2 J Jifina, M . 10 9 K Kakutani, S . 1 kernel 14 , 52 stochastic 4 5 substochastic 4 6 killing 1 4 Kolmogorov, A.N . 10 9 Kuznetsov, S.E . 77 , 11 2 L Lamperti, J . 109 , 11 0 Laplace equation 1 Laplace functiona l 5 0 Laplace transform 1 2 L-continuity 8 1 Le Gall, J.F. i , 5 , 7 , 11 2 Levy, P. 1 Levy-Khinchin formul a 1 2 Levy measure 41, 71 modified 4 1 lifting 8 2 linear additiv e functiona l (LAF ) 8 2 localization 49 , 84 Loewner, C . i , 1 log-Laplace equatio n 1 0 differential equatio n 2 0 log-Laplace semigrou p 1 0 INDEX (^-localization 8 4 Luzin spac e 2 7 M Markov proces s 2 6 in a finite stat e spac e 1 9 with mas s creation 1 6 Markov propert y 2 7 Markov semigrou p 2 6 Martin boundar y 5 martingale 11 5 over f 3 8 local 12 0 locally squar e integrabl e 12 0 purely discontinuou s 11 8 ^-adapted 3 8 X-martingale 8 0 ^-martingale 27 , 38 martingale measur e 11 7 measure-valued additiv e functiona l (MVAF ) 3 5 Mokobodzki, G . 3 4 N Nirenberg, L . i , 1 P partition o f unity 10 2 Perkins, E.A . i , 11 2 Pierre, M . i Poisson kerne l 4 Poisson measur e 4 5 polar se t 3 process, adapted 11 5 bounded 12 1 branching 9 conservative 1 0 indistinguishible 11 5 locally bounded 12 1 of quadratic variatio n 11 8 optional 11 5 predict ible 11 5 progressively measurabl e 11 5 quasi-left continuou s 28 , 11 5 right 2 7 split-time 2 9 with mas s creation 1 6 ^-predictable 2 7 projection 17 , 18 , 82 purely discontinuou s martingal e part o f (/*,Xj ) 8 6 Q quadratic variatio n proces s 11 8 INDEX 131 R Radon-Nikodym theore m 3 6 random field 2 2 range o f X 3 range o f £ 3 R-continuity 8 1 reconstructable cr-algebr a 2 7 right continuou s additiv e functio n (RCA ) 3 8 Roelly, S . 11 4 Rost theore m 4 4 it!-polar se t 3 ifo-polar se t 5 r-stopping tim e 2 7 S semigroup, linea r associate d wit h X 1 1 log-Laplace 1 0 Markov 2 6 semimartingale 12 0 Sheu, Y.C. ii , 5 signed exi t rul e 2 9 Silverstein, M.L . 109 , 110 , 11 1 special functio n 2 9 standard sequenc e o f partitions 11 7 stochastic integra l 120 , 12 1 stopping distributio n 4 3 stopping time 81 , 115 predict ible 11 6 randomized 4 3 strong Marko v propert y 2 7 Stroock, D.W . 11 3 subcritical cas e i submartingale 11 5 submartingale measur e 11 7 subprobability 2 5 super-Brownian motio n 1 6 superdiffusion 2 1 time-homogeneous 2 4 supermartingale 11 5 supermartingale measur e 11 7 superprocess 14 , 49 as a random field 2 2 enhanced 2 3 time-homogeneous 1 6 with parameters (£ , K, if;) 49 -0-superprocess 2 0 T total famil y 9 3 total mas s process 80 , 82 INDEX transition functio n 2 5 U universal completio n 2 9 usual hypothese s 1 6 V Varadhan, S.R.S . 11 3 Veron, L . i , 2 , 5 , 7 W Watanabe, S . 110 , 11 2 Watson, H.W . 10 9 Index o f Notatio n A? 8 4 a* 8 4 B9 bB9 bPB9 pB9 BMV9 C1'1 3 0 Ct 10 , 1 3 (D) 11 5 Dt 10 , 1 3 (E,B) 9,2 6 €/ 9 8 Ft 10 , 1 3 JTT 11 5 F 133 INDEX O F NOTATIO N 9tt2 11 8 2 m c us 2 m d us MVAF 35 O 11 5 -* r, x ^ P Q Px 2 , 22 P„ 2 , 22 V 11 5 P(£) 9 8 Vr 4 0 ft 2 7 R+ 1 0 R++ 1 0 RCA 3 8 5 1 0 5d98 SER29 SP29 SP+ 2 9 SQ86 SQ+ 8 6 T22 Tt 10 , 1 3 T[ 18 , 80 vt 1 0 r Vt 9 r wt n XT 2 , 22 Z+ 4 5 7(r, •;*,- ) 1 2 75(z,cft/)52 7+73 7_ 7 3 |7|73 r( i i r*(a;,dy) 7 5 <$*(£) 9 x* 80 Ur (fi,^,^[r1t],IIr,M)27 (Q,JT,jrt,p)n6