University

LECTURE Series

Volume 3 4

Superdiffusions an d Positiv e Solutions o f Nonlinea r

Partial Differentia l Equation s

E. B . Dynki n

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/ulect/034

EDITORIAL COMMITTE E Jerry L . Bon a (Chair ) Eri c M . Friedlande r Nigel J . Higso n Pete r Landwebe r

Ranee Brylinsk i

2000 Mathematics Subject Classification. P r imar y 60-02 ; Secondary 31B05 , 35J60 , 60J60 .

For addi t iona l informatio n an d upda te s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / u l e c t - 3 4

Library o f Congres s Cataloging-in-Publicatio n D a t a

Dynkin, E . B . (Evgeni i Borisovich) , 1924 -Superdiffusions an d positiv e solutions of nonlinear partia l differentia l equation s / E . B . Dynkin .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 34 ) Includes bibliographica l reference s an d indexes . ISBN 0-8218-3682- X (alk . paper ) 1. Differentia l equations , Nonlinear . 2 . Differential equations , Partial . 3 . Diffusion processes .

I. Title . II . Universit y lectur e serie s (Providence , R.I. ) ; 34.

QA372.D96 200 4 515/.353—dc22 2004055059

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10 9 8 7 6 5 4 3 2 1 0 9 0 8 0 7 0 6 0 5 0 4

Contents

Preface

Chapter 1 . Introductio n 1 1. Trac e theor y 1 2. Organizin g th e boo k 3 3. Notatio n 4 4. Assumption s 4 5. Note s 6

Chapter 2 . Analyti c approac h 9 1. Operator s GD and KD 1 0 2. Operato r VD and equatio n Lu = xp(u) 1 1 3. Algebrai c approac h t o th e equatio n Lu = ip(u) 1 5 4. Choque t capacitie s 1 6 5. Note s 1 7

Chapter 3 . Probabilisti c approac h 1 9 1. Diffusio n 2 0 2. Superprocesse s 2 4 3. Superdiffusion s 2 8 4. Note s 3 3

Chapter 4 . N-measure s 3 5 1. Mai n resul t 3 6 2. Constructio n o f measures N x 3 6 3. Application s 4 0 4. Note s 4 7

Chapter 5 . Moment s an d absolut e continuit y propertie s o f superdiffusions 4 9

1. Recursiv e momen t formula e 4 9 2. Diagra m descriptio n o f moments 5 4 3. Absolut e continuit y result s 5 6 4. Note s 5 9

Chapter 6 . Poisso n capacitie s 6 1 1. Capacitie s associate d wit h a pai r (/c,ra ) 6 1

iv CONTENT S

2. Poisso n capacitie s 6 2 3. Uppe r boun d fo r Cap(r ) 6 3 4. Lowe r boun d fo r Cap x 6 7 5. Note s 6 9

Chapter 7 . Basi c inequalit y 7 1 1. Mai n resul t 7 1 2. Tw o proposition s 7 1 3. Relation s betwee n superdiffusion s an d conditiona l diffusion s i n

two ope n set s 7 2 4. Equation s connectin g P x an d N x wit h 11 ^ 7 4 5. Proo f o f Theore m 1. 1 7 6 6. Note s 7 7

Chapter 8 . Solution s wp ar e a-moderat e 7 9 1. Pla n o f the chapte r 7 9 2. Thre e lemma s o n th e conditiona l Brownia n motio n 8 0 3. Proo f o f Theore m 1. 2 8 2 4. Proo f o f Theorem 1. 3 8 3 5. Proo f o f Theore m 1. 5 8 4 6. Proo f o f Theorems 1. 6 an d 1. 7 8 5 7. Note s 8 6

Chapter 9 . Al l solutions ar e cr-moderat e 8 9 1. Pla n 8 9 2. Proo f o f Localization theore m 9 0 3. Sta r domain s 9 3 4. Note s 10 1

Appendix A . A n elementar y propert y o f the Brownia n motio n

J.-F. L e Gal l 10 3

Appendix B . Relation s betwee n Poisso n an d Besse l capacitie s

I. E . Verbitsk y 10 7

Notes 11 1

References 11 3

Subject Inde x 11 7

Notation Inde x 119

Preface

This boo k i s devote d t o th e application s o f th e probabilit y theor y t o the theor y o f nonlinea r partia l differentia l equations . Mor e precisely , w e investigate th e clas s U o f al l positiv e solution s o f th e equatio n Lu — ip(u) in E wher e L i s a n ellipti c differentia l operato r o f th e secon d order , E i s a bounded smoot h domai n i n R d an d ip is a continuously differentiat e positiv e function.

The progres s i n solvin g thi s proble m til l th e beginnin g o f 200 2 wa s de -scribed i n th e monograp h [D] . [W e us e a n abbreviatio n [D ] fo r [Dy02]. ] Under mil d condition s o n ^ , a trac e o n th e boundar y dE wa s associate d with ever y u EW . Thi s i s a pai r (T , v) wher e T i s a subse t o f dE an d v i s a a-finite measur e o n dE\T. [ A point y belong s t o T i f ip f(u) tend s sufficientl y fast t o infinit y a s x — * y.\ Al l possibl e value s o f th e trac e wer e describe d and a 1- 1 correspondenc e wa s establishe d betwee n thes e value s an d a clas s of solution s calle d cr-moderate . W e sa y tha t u i s a-moderat e i f i t i s the limi t of a n increasin g sequenc e o f moderat e solutions . [ A moderat e solutio n i s a solution u suc h tha t u < h where Lh = 0 in E.\ I n th e Epilogu e t o [D] , a cru -cial outstandin g questio n wa s formulated : Are all the solutions a-moderate? In th e cas e o f th e equatio n Au = u 2 i n a domai n o f clas s C 4 , a positiv e an -swer t o thi s questio n wa s give n i n th e thesi s o f Mselat i [Ms02a]— a studen t of J.-F . L e Gall. 1 Howeve r hi s principa l tool—th e Brownia n snake—i s no t applicable t o mor e genera l equations . I n a serie s o f publication s b y Dynki n andKuznetsov [Dy04b] , [Dy04c] , [Dy04d] , [Dy04e] , [DK03], [DK04] , [Ku04] , Mselati's resul t wa s extended , b y usin g a superdiffusio n instea d o f the snake , to th e equatio n Au — ua wit h 1 < a < 2 . Thi s require d a n enhancemen t o f the superdiffusio n theor y whic h ca n b e o f interes t fo r anybod y wh o work s on application s o f probabilisti c method s t o mathematica l analysis .

The goa l o f thi s boo k i s t o giv e a self-containe d presentatio n o f thes e new developments . Th e boo k ma y b e considere d a s a continuatio n o f th e monograph [D] . In th e firs t thre e chapter s w e giv e a n overvie w o f th e theor y presented i n [D ] withou t duplicatin g th e proof s whic h ca n b e foun d i n [D] . The boo k ca n b e rea d independentl y o f [D] . [I t migh t b e eve n usefu l t o rea d the firs t thre e chapter s befor e readin g [D]. ]

In a serie s o f paper s (includin g [MV98a] , [MV98b ] an d [MV04] ) M . Mar -cus an d L . Vero n investigate d positiv e solution s o f the equatio n Au = u a b y

The dissertatio n o f Mselat i wa s publishe d i n 200 4 (se e [Ms04]) .

VI PREFACE

purely analyti c methods . Bot h analyti c an d probabilisti c approache s hav e their advantage s an d a n interactio n betwee n analyst s an d probabilist s wa s important fo r th e progres s o f the field. I take thi s opportunit y t o than k M . Marcus an d L . Veron fo r keepin g m e informe d abou t thei r work .

The Choque t capacitie s ar e one of the principa l tools in the study o f the equation Au = u a. Thi s clas s contain s th e Poisso n capacitie s use d i n th e work o f Dynki n an d Kuznetso v an d i n thi s boo k an d th e Besse l capacitie s used b y Marcu s an d Vero n an d b y othe r analysts . I a m ver y gratefu l t o I. E. Verbitsky who agreed to write Appendix B, where the relations between the Poisso n an d Besse l capacitie s ar e established , thu s allowin g t o connec t the wor k o f both groups .

I a m indebte d t o S . E. Kuznetsov wh o provided m e with severa l prelim -inary draft s o f hi s pape r [Ku04 ] use d i n Chapter s 8 an d 9 . I a m gratefu l to hi m an d t o J.-F . L e Gal l an d B . Mselat i fo r man y helpfu l discussions . It i s m y pleasan t dut y t o than k J.-F . L e Gal l fo r permissio n t o includ e i n the boo k a s Appendi x A hi s not e whic h clarifie s a statemen t use d bu t no t proved i n Mselati' s thesi s (w e use i t i n Chapte r 8) .

I a m especiall y indebte d t o Yuan-chun g She u fo r readin g carefull y th e entire manuscrip t an d suggestin g man y correction s an d improvements .

The researc h o f the autho r reporte d i n thi s boo k wa s supported i n par t by th e Nationa l Scienc e Foundation Gran t DMS-0204237 .

References

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Subject Inde x

branching exi t Marko v [BEM ] system , 24

Branching exi t Marko v [BEM ] syste m canonical, 2 5

Choquet capacities , 1 6 comparison principle , 1 2 Conditional L-diffusion , 2 2

diffusion wit h killin g ra t £,2 1

envelope o f r.c.s , 2 6 exhausting sequence , 4 extended mea n valu e property , 1 4

Green functio n gD{x,y), 1 0 Green operato r GD, 1 0

harmonic functions , 1 h-transform, 2 2

Infinitely divisibl e rando m measures , 3 6

kernel, 4

L-diffusion, 2 0

(L,^)-superdiffusion, 2 8 linear boundar y functional , 3 0 log-potential, 2 9 Luzin spac e

measurable, 3 7 topological, 3 6

Markov property , 2 5 mean valu e property , 1 2 moderate solutions , 1 moment measures , 5 6 multiplicative system s theorem , 3 9

normalized surfac e are a j(dy), 1 0 N-measures, 3 5

operator 7r , 1 5

Poisson capacitie s Cap , Cap x , 6 2 Poisson kerne l kn(x,y), 1 0 Poisson operato r K D , 1 1 Poisson rando m measur e wit h intensit y

71, 57

random close d se t (r.c.s.) , 2 6 random measure , 2 4 range o f supe r process, 2 7

smooth domain , 5 star domain , 8 9 stochastic boundar y valu e SBV , 28 , 40 straightening o f the boundary , 5 subcritical an d supercritica l value s o f a ,

62 subsolution, 1 2 supersolution, 1 2

trace, 3 transition density , 2 0

(j-moderate solutions , 1 (£, VO-superprocess, 2 5

117

This page intentionally left blank

Notation Inde x

£ , 4

£( • ) , 4 &£, 4 £ M , 3 7

Bn(x,K), 7 9

C ( D ) , 4 C A ( D ) , 4

Cfc(£>), 4 C f c ' A (D) , 4

C+ , 1 5

2>i, 4

D*, 7 1 d i a m ( B ) , 4 d ( z , £ ) , 4

£ + , 5 £ o , 5

£ « ( # ) , 7 9 S(y\ 6 1 &M, 62 £ r ( * V ) , 8 7

E, 6 3 <£, 8 9

£ i , 8 9

</,M>,4

f C D , 2 5

J ^ D D , 2 5

• F e s - , 3 0 J ^ D S - , 3 0

.Fa, 3 0

/ii/, 1

hv, 7 2

W, 1 W(-), 1 3 Hi, 1

fti(-), 1 3

/CD, 1 0 Kv, 6 1

K D , 1 1 /C, 1 6 K, 6 1

M(-M M c ( £ ) , 2 6

•A/i,2 A ^ , 1 4

M, 1 A/f, 1 3

O, 1 6

O x , 3 6

P ( . ) , 4

7 1 B , 2 7 ft*, 3 8

S, 2 7 S o , 2 7 Sup, 2

Tr, 3

ur, 2 u„ , 13 , 1 4

U, 1

w(-), i i Wo(-), 1 4 Wi, 1 W ( - ) , 1 6

Vb, 1 1

W K , 2 wr , 2

y , 2 4 y t / , 4 2

119

120 NOTATIO N INDE X

34,36

Zv, 31 ZV,T2 Ziu, 28 Z, 24 Z,x, 36 3,30

l(dy), 10 Sv, 4 7T, 15

IE, 22 n£, 22 n£, 22 n«,22 ns, 22 ns, 72 /»(«). 4 ¥>(a;,K), 79 <p(x,r),62 $(u), 74

e, is e, 2 €.4

Titles i n Thi s Serie s

34 E . B . Dynkin , Superdiffusion s an d positiv e solution s o f nonlinea r partia l differentia l equations, 200 4

33 Kristia n Seip , Interpolatio n an d samplin g i n space s o f analyti c functions , 200 4

32 Pau l B . Larson , Th e stationar y tower : Note s o n a cours e b y W . Hug h Woodin , 200 4

31 Joh n Roe , Lecture s o n coars e geometry , 200 3

30 Anatol e Katok , Combinatoria l construction s i n ergodi c theor y an d dynamics , 200 3

29 Thoma s H . Wolf f (Izabell a Lab a an d Caro l Shubin , editors) , Lecture s o n harmoni c analysis, 200 3

28 Ski p Garibaldi , Alexande r Merkurjev , an d Jean-Pierr e Serre , Cohomologica l invariants i n Galoi s cohomology , 200 3

27 Sun-Yun g A . Chang , Pau l C . Yang , Karste n Grove , an d J o n G . Wolfson , Conformal, Riemannia n an d Lagrangia n geometry , Th e 200 0 Barret t Lectures , 200 2

26 Susum u Ariki , Representation s o f quantu m algebra s an d combinatoric s o f Youn g tableaux, 200 2

25 Wil l ia m T . Ros s an d Harol d S . Shapiro , Generalize d analyti c continuation , 200 2

24 Victo r M . Buchstabe r an d Tara s E . Panov , Toru s action s an d thei r application s i n topology an d combinatorics , 200 2

23 Lui s Barreir a an d Yako v B . Pesin , Lyapuno v exponent s an d smoot h ergodi c theory , 2002

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations ,

2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d

modular functors , 200 1

20 Aliso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f point s o n surfaces , 199 9

17 Marce l Berger , Riemannia n geometr y durin g th e secon d hal f o f th e twentiet h century , 2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h

notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f th e symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Probeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 Stephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 And y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4 6 Dus a McDuf f an d Die tma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

TITLES I N THI S SERIE S

1 Michae l H . Preedma n an d Fen g Luo , Selecte d application s o f geometr y t o low-dimensional topology , 198 9