Diffusion Processes and their Sample Paths

CLASSICS IN MATHEMATICS Kiyosi Iti) Henry P. McKean, Jr. Diffusion Processes and their Sample Paths Classics in Ma...

Author: Kiyosi Itô | Henry P. Jr. McKean

118 downloads 1168 Views 7MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

CLASSICS IN MATHEMATICS

Kiyosi Iti)

Henry P. McKean, Jr.

Diffusion Processes

and their Sample Paths

Classics in Mathematics Kiyosi Ito Henry P. McKean, Jr.

Diffusion Processes and their Sample Paths

Springer Berlin Heidelberg New York

Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Kiyosi Ito Henry P. McKean, Jr.

Diffusion Processes and their Sample Paths Reprint of the 1974 Edition

Springer

Kiyosi It6

Henry P. McKean, Jr.

Kyoto University, RIMS Sakyo-Ku 606 Kyoto

New York University Courant Institute of Mathematical Sciences New York, NY 10012 USA

GAO

Japan

Originally published as Vol. 125 of the Grundlehren der mathematischen Wissenschaften

Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Ito, Kiosi:

Diffusion processes and their sample paths / Kiosi ItO ; Henry P. McKean, Jr. - Reprint of the 1974 ed. - Berlin ; Heidelberg New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo Springer, 1996 (Grundlehren der mathematischen Wissenschaften ; Vol. 125) (Classics in mathematics) ISBN 3-540-60629-7

NE: MacKean, Henry P.:; 1. GT

Mathematics Subject Classification (1991): 28A65, 60-02, 60G17, 60J60, 60J65, 60J70

ISBN 3-540-60629-7 Springer-Verlag Berlin Heidelberg New York

Eat

"'C

0.O

.0$

,00

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provision of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. m Springer-Verlag Berlin Heidelberg 1996

Punted in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN 10518453

41/3144- 5 4 3 2 10 - Printed on acid-free paper

K. Ito^

H. P. McKean, Jr.

Diffusion Processes and their Sample Paths

Second Printing, Corrected

Springer-Verlag

Berlin Heidelberg New York 1974

Kiyosi Ito Cornell University, Ithaca, N. Y. 14850, USA Henry P. McKean, Jr. Courant Institute of Mathematical Sciences, New York University, New York, USA

Geschaftsfuhrende Herausgeber B. Eckmann Eidgenossische Technische Hochschule Zurich

J. K. Moser Courant Institute of Mathematical Sciences, New York B. L. van der Waerden

Mathematisches Institut der Universtt t Zurich

AMS Subject Classifications (1970): 28 A 65, 60-02, 60 G 17, 60 J 60, 60 J 65, 60 J 70

ISBN 3-540-03302-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-03302-5 Springer-Verlag New York Heidelberg Berlin

'^'

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. Q by Springer-Verlag Berlin Heidelberg 1%5, 1974. Library of Congress Catalog Card Number 74-231. Printed in Germany. Offset printing: Julius Beltz, Hemsbach/Bergsu. Bookbinding Konrad Triltseh, Wurzburg. 2141/3140.54321

DEDICATED TO

P. LEVY WHOSE WORK HAS BEEN

OUR SPUR AND ADMIRATION

Computer-simulated molecular motions, reminiscent of 2-dimensional BROwNian motion. [From ALDER, B. J., and T. E. WAINWRIGHT: Molecular motions. Scientific

American 201, no. 4, 113-126 (1959)1.

Preface

.rte

ROBERT BROWN, an English botanist, observed (1828) that pollen grains suspended in water perform a continual swarming motion (see, 'L7

for example, D'ARCY THOMPSON [1: 73-771). .'.

0.°°

L. BACHELIER (1900) derived the law govering the position of a single grain performing a 1-dimensional BROwxian motion starting at

aER1 at time t=o: 1)

Pa[x(t) E db] = g(t, a, b) db

(t, a, b) E (0, + oo) x R',

C+1

where g is the source (GREEN) function

g(t, a, b) _

2)

of the problem of heat flow:

- 2 aa' au

(t>0).

I

A

au at

3)

yin t

BACHELIER also pointed out the MARKOVian nature of the BROWNian

path expressed in

P.[a]L Sx(t,) 0)

H1 = e1G

13)

with a suitable interpretation of the exponential, where 0 is the socalled generator.

III

D. RAY [2] proved that if the motion is strict Markov (i.e., if it starts afresh at certain stochastic (MARKOV) times including the passage

times ma = min (t : x (t) = a), etc.), then (SS is local if and only if the motion has continuous sample paths, substantiating a conjecture of W. FELLER; this combined with FELLER'S papers [4, 5, 7, 9] implies that the generator of a strict MARKOvian motion with continuous paths (diffusion) can be expressed as a differential operator 14)

(

05 u

lim )(a)-bl,a

u+ (b) - u' (a)

m(a.b] p,.

where m is a non-negative BOREL measure positive on open intervals and, with a change of scale, u+ (a) = bilm (b - a)-1 [u (b) - u (a)], except at certain singular points where f 3 degenerates to a differential operator of degree :5.'1. E. B. DYNKIN [1] also arrived at the idea of a strict MARKOV process, 4'W

derived an elegant formula for CS, and used it to make a simple (probabilistic) proof of FELLER's expression for (SS; the names of R. BLUMENTHAL [1] and G. HUNT [1] and the monographs of E. B. DYNKIN [6, 8]

should also be mentioned in this connection. Our plan is the following. BROwxian motion is discussed in chapters 1 and 2 and then, in chapter 3, the general linear diffusion is introduced as a strict MARKOVian

motion with continuous paths on a linear interval subject to possible

X

Preface

annihilation of mass. (3S is computed in great detail in chapter 4 using probabilistic methods similar to those of E. B. DYNKIN [5]; in the socalled non-singular case it is a differential operator

u+(b) - u+(a) - f udk ((53 u) (a) = li

15)

m

m(a,

(.. b)

b]

with u+ and m as in 14), where now k is the (non-negative) BOREL measure that governs the annihilation of mass (such generators occur somewhat disguised in W. FELLER [9]).

Given (s3 as in 14) and a standard BROwNian motion with sample paths w : t --+ x (t), if t is P. LEvY's Brownian local time and if f -1 is the

inverse function of the local time integral f (t) = f t(t, e) m(de),

16)

then the motion x (f -1) is identical in law to the diffusion attached to (0S, as will be proved in chapter 5, substantiating a 'suggestion of H. TROTTER; B. VOLKONSKII [I] has obtained the same time substitution in a less explicit form.

Given 0 as in 15), the associated motion can be obtained by killing the paths x (f -1) described above at a (stochastic) time m. with conditional law

P.[m, > t x(f-1)] = e I

17)

as is also proved in chapter 5; in the special case of the elastic Brownian motion on [0, + oo) with generator (5S = D2/2 subject to the condition

yu(0) = (1 -y)u+(0) (0 0-

E, applied to such non-negative functions, satisfies E(f) 0 1) 2) E(1) = 1

E(l1 + l,) = E(/1) + E(/,) E(/,) t E(/) /" t /

3)

4a)

E(/1)0,a,bER1

awl

g(t,a,b)=

6)

f

B

g(ti,0,b1)db1g(t,-t1,bl,b,)db,...

g(t" - t"-1, bn-1, b") db" BEB(R*),

n?!.

C=x-1(B),

«'b

...,

P, is a probability measure on xt' B (R") ; indeed, since xt' (B1) = x, 1(B,) implies B1= B,, Pt is well-defined; since xt' (B1) (1 x, 1(B,) = 0 implies

B1 fl B, _ 0 , it is clear from 5) and 1) that Pt is additive; and, since +00

r 'db 2n

..:

000

+00

8)

f g(t,a,b)db=2 .1 0

- 00

f

+ 00

+ 00

C41

d a f d b e-6''2 e-b'IQ

0

nl2

0 +OD

i

f d9 f e-''12rdr 0

0

I, which completes the proof. Pt coincides with P, on x;' B (Rr)* if 9 C t; indeed,

P, (W)

+W 9)

f g(t-s,a,c)g(s,c,b)dc -00 +00

/'

J

-00

y2n (t - s)

V2a t

j/2n s

=g(t,a,b)

* m is the number of points in 8.

dc

t>s>0,a,bER1,

1. The standard BRowNian motion

14

and taking, for example, B, E B(R"), B2E B(R"-1),

C = xl,[,l,t,...tw(B1) `

one finds 10)

g(t1, 0, b1) g(tz - t1, b1, b2) X

J

x

ba)EB, xg(13-12,b2,b3)g(tt-13,b3b4)...g(1"-1"-1,b"-1,b") X

x db, dbz db3 db4 ... db"

f

boo

g(t1, 0, b1) db1g(tz- t,, b1, bz) dbz x tom

(b1. b b,.... b,) E B,

b3, b4) d b4 ...

X f g (t3 - tz, bz, b3) d b3 g (14

g (t" -

bA_,

,

b") d b.

b, E R'

= f g(t1, 0, bl)

dbl g(19

- t1, bl, b2) db2g(t4 - tzi bz, b4) dbz .. .

B,

... g (t. - t,-1,

b"-1, b") d b.

Define P (C) = Pt (C) for C E xi 1 B (R") . P is a probability measure on C, and, as we will now prove, it is extensible to a Borel probability measure on the Borel extension B of C. Consider, for the proof, C E C such that C, ) Cz) ... and

P (Q

c1 > 0*; it is to be shown that 11 C is non-void. "Z 1 Suppose, as we can, that

11)

C" = xtoi B"),

where

t" = B" E B (R"'), m = the number of points of t", n a 1, and, noting the estimate 12) P ( <max . 1 x,, - x,l > C2 2-"/3] o

.-'12sl.%Etw

1»W

M11o 0 so great that

2' 3n 25n/6 e-ell"/a-1 nZ1 C2

13)

G1

3

Choose, also, compact B',' C Bn C B,, such that a-.

P(xt' (Bn -

14)

c13-"

cad

and 15)

max

x t ' (Bn) fl (w : 0

«.-+,52-

I x,, - x.,

ca 2-"l$) = xto (Bn).

81.f2 E to

and note that

P(ms xto (Bm))

16)

Z P (Cn) -"3 s "P (xto (Bm - Bm))

c12-3=6>0.

s

nP

(x7+ (BM - Bm))

fl xt"; (B',) is therefore never void, and, since B; was compact, we

m$n

can select a point (x (s) : s E t) E R°°, t = U t, such that

n z 1.

(x (s) : s E tn) E B;1

17)

But then 0 < S2 - S1 S 2'n, Si. s$ E t, , n Z 1 , and it follows that x (s) is uniformly continuous on each bounded sub18)

I x (s2) - x (s1) I S c2 2-n/3

set of t; in fact, if s1,s2Et,

19)

0<sa-s1 0) is the (bounded) solution

0

+00 e+1

u = u(a a) = E.

31)

J

r

e-°tl (xi) dt = RI

0

fdb**

Y2

of

(a - () u=/:

32)

in operator language, if et 0 / = is (1, ) (t > 0) and G. / = u (a > 0) , +00 :Z7

then et0 maps C(RI) into C°° (RI), (a/ot - 0) etc = 0, Ga= f e- I et*dt 0 maps C (RI) 1 : 1 onto C2 (RI), and Ga'= a - ('i. ( is said to generate the Brownian motion. G. is the so-called GREEN operator of the BRowxian motion ; see 2.4 for applications. Problem 1 after Y. KOMATSU [1]. Check that

as + 4 + a]-I

VII

2[

ea'l2 f a-b'12 db S 2[ a2 + 2 + a] -I

for

a

o.

a

[Writing g_ for the under-estimate, g+ for the over-estimate, and g

-

Cow

.M.

for the modified error integral in the middle, a s'mple computation gives g'_ _> a g - 1, g' = a g - 1, g+ S a g+ -1; therefore (g - g _ )' S S a (g g-), (g+ - g)' S a (g+ - g), and using g-, g, g+ S a-', it follows that g - g_, g+ - g cannot become negative.) See I. PETROVSKY [2: 241-244). + co

I

r2a

e-=e

y2nt

dt; see A. EitasLYI [1 (1) : 146 (27)).

0

I0IMc1ean, Diffusion processes

2

1. The standard Baownian motion

18

"5-

Problem 2. Polliimtx(i/t) = 01 = 1. 1+0 Problem 3 after P. LEvY [3: 246]. Consider a standard BROWNian motion starting at 0; the problem is to show that

t=0 t>0

a(t)=0

A

tx(1/t) and

t? 0

b (t) = c x (t/c1)

(c > 0)

ion

are likewise BROwxian motions starting at 0. [Both these motions are GAUSSian with mean 0, and E0[x (s) x (t)] = E0[s x (1 /s) t x (1 /t)]

= E. [c x (s/c') c x(t/c')) = s A t .] t4'

Problem 4. The BROWNian motion is differential in the sense that

its increments x[ai, bi) = x(bi) - x(ai) (i S n) over disjoint intervals [ai, bi) (i 5 n) of (0, + oo) are independent. + c x", check that Problem 6. Given / = co + cl x + c= x2 +

u(t, a) = G/ = V n

t"(6"l)(a) x!

O

is the solution of au

at = c u

u(0+. ) = 1.

and deduce, for the standard BROwNian motion, i.'

n 1. Eo(xt") =1.3 ... (2n-1) t' Problem 6. Give a direct proof that 31) is the bounded solution

of 32) and use this to deduce the formula +00

e-0-u+/srdt

e- j2,x

y2nt

2a 0

of the footnote to 31). '1c

Problem 7. Give a proof that almost all BROWNian paths are nowhere differentiable. R. PALEY, N. WIENER and A. ZYGMUND (1) discovered !ti

...

this; the proof suggested below is due to A. DVORETSKI, P. ERDds, and S. KAKUTANI [3].

[If the BROWNian path is differentiable at some point 0 S s S 1, then

lx(!)-x(s)l I

gk 2-" = f f k 2-w x* d t 0

= 2(n-I)12[2x(k2-") - x((k - 1) 2-") - x((k + 1) 2-")] should be Gaussian: a

f

P(g < a) =

_ db,

e-4'19

y2n

-00 rI

E (gk 2-" gJ 2 ..) - .l fk 2

/52d( = 0

(k 2-" + j 2-'")

0 44))

and hence independent (see prerequisites), and the integrated expansion i

c

S)

go f /ods

I

ALd

koddG2"42-442- ds

2

20

1. The standard BxowNian motion

should converge to x (t) (t S 1) ; the idea is to use 5) to define the BRowNian motion.

Consider for this purpose, the algebra B[0, i] of BOREL subsets of [0, 1] with the classical BOREL measure P attached, let 0 5 u S 9 be expanded as u=2-1u1+2-2us+2-3U3+...=.UIUSUs...,

a

let h be the function inverse to f e-b'12 d b f 2n, and defining - CO

V1 = u1 US u4 n[7 ...

Vp = 0 U3 ub US ...

V2 = 0 0 ua u9 ...

V4=000.Ujo... eic., let

go = h (v1)

911, = h(v9) g1/s = h (v5)

9114 = h(v3)

g3/4 = h (v4)

gs/s = It (v3)

gs18

= h (v7)

g718 = h (v8)

etc.,

and note that the g's are independent with common distribution 3). Consider now the sum t

t

em = go f fods + T X gk2-of n$m oddkofl B[x(tA(m+e)):tz0],

4)

as will be seen below. Bm+ is to be thought of as measuring the Brownian

path up to time t = m+. HUNT'S statement is that, conditional on the present position b = x (m), the future path x(t + m) (I ? 0) is a standard Brownian motion starting at b, and this Brownian motion is independent o l B.+; in the language of conditional probabilities,

m < +oo, b = x(m), B E B,

P.[wm E RI Bm+] = Pb(B)

5 a)

i. e.,

5b)

P.[A,w+mEB,mt

n (nl < s) E B,, while, if B E fl B,, then 1>t

s>t, sSt,

Bn(m<s)=B

=0

and B E Bm+. VII

Given MARKOV times ml S m2, if B E Bm,+, then B n (ms < t) = B fl (m, < t) n (m2 < t) E Bt for each t ? 0, i.e.,

COa

Bml+ C B....

6)

CMS

Given MARKOV times in and m, ? m2 ? etc. I in, if B E n Bm"' n'>l

_/M

then B n (m < t) = U B n (m, < t) E Bt for each t z 0, i. e., using 6), nZ1

n Bm"+ = Bm+.

7)

nal

Coming to the proof of 4), if m is MARKOV, then m A t is measurable Bt

for each t ? 0 as is clear from 2), and, using the fact that x (t As, w) is a B [0, + oo) x Bt measurable function of the pair (s, w)', it follows that x (M / i A s) is measurable Bt for each s ? 0. But then .

(w:x(mns) 0.

1.7. Passage times for the standard Bxowxian motion

25

Problem 2. BLUMENTHAL's 01 law (see R. BLUMENTHAL [I]). B0 is

trivial in the sense that BE B0 .

P. (B) = 0 or 1 [P. (B) = E. (B. P. (B I B0+)) = E. (B. P. (o) (B)) = P.

1.7. Passage times for the standard Brownian motion The most important MARKOV times are the passage times:

aER'.

ma=min(t:x,=a)

1)

P. LEVY [3: 221-223] has shown that [ma : a ? 0, PO] is the onesided stable process with exponent 4 and rate V2, i.e., it is differential and homogeneous, with law 2)

f b - a e-cb-a)'12 ds

PO[Mb - M. S t] = Po[mb-a S t] = 0

2n

b?a,tZ0

(see note 1).

Beginning with the proof of 2), if a z 0 and if f is the indicator function of [a, + oo), then, using 1.6.5 b) and the fact that e-a m = 0 if m = + 00, +00

3)

f

ewe P.(xi

z a) dt

0

W

fe

Eo

(x1) dt

0

+m

=Eo

f e_aef(xi) dt)

(m= ma)

M

=EO[e-'- EO( ('tee-atf[x(t,wm)]dtI Bm+)] 0 _ell

EO(e am.) E. f e

/(x1) dt

0

+00

Eo(e-am.) f e-"1Pa(xg L> a) dt

=

0

= (2a)-' EO(e-"m-) + 00

=

(2a)-1

f

e-°ìPO

(mnEdt)

0 +00

e--' PO (ma S t) dt, 0

0 m is measurable Bm+

1. The standard BROwNian motion

26

proving the celebrated refkction principle of D. AxnitE: 4)

PO (ma

t) = 2 PO (xi ? a) +cc

=2

6f=jdb f yen:

1,aZ0.

a

,M,o

2) is now immediate from the formula: +*0

t

SIN

Ya

2 f y2E!L db= af

a/

1, a 2 0,

e-a1128ds

0

and, at no extra cost, it is found that P. (mb < + oo) m 1. Because of 4),

+00 r E0(e-am-) = 2a f e-atdt r +00

5)

e-a+/ss

db

y2nt

J

a

0

f

+00

=2a

y2a

db

a

=

e-yQaa

using the footnote to 1.4-31); more directly, it is clear from 2) and the first integral in 5) that g E0 (e-am-) is the solution of 6)

9" = tag,

g (O) = 1,

g(+ 00) = 0,

and solving this gives 5). To complete the proof that ma (a L> 0) is a one-sided stable process, it is now sufficient to check its differential character. Given 0 < a < b, 7)

mb = min (t : x (l) = b)

=m+min(t:x(t+m)=b)

m=ma0)

suffers for b E da

is the POISSON measure with mean d a x v2n

P

(see note 1) ; in parti-

...

cular, mb is a sum of positive jumps. Note that (0, a) is used in 111 instead of [0, a] as in note 1; this is because ma is left continuous. The maximum function t- (t) = max x (s) (t z 0) is the inverse ,fit function of the passage time process m; it is continuous, and, as such, it looks something like the standard CANTOR function (see 2.2 and problem 5 for additional information on this point). Problem 1 after P. LEvY [3: 211]. Give a proof of the joint law PO[xtEda,maxx,EdbJ

- (._jr)"2(2b - a) e2b)'12tda db 2

[Given b-'> a, +co

VII

f e-tdtPo[xt Sa,mnxx,ZbJ 0 oo

= EOlm*b f e a!f(xt)dt1 +00 W-0

=

E0(e-.2m*)

Eb a

= e-Viab

f e-"t f d) _

1

y2a - 00

1

1 0

dc,

t > 0, 0

b, b

a.

1. The standard BRowNian motion

28 c>1

where / is the indicator of (- oo, 4]; thus, +00

f e-"idtPolxiEda, ax xsEdb, =

2e-y,2"(2b-a) dadb;

0

now use 2) and 6).]

1'+

Problem 2. Use the result of problem 1.4.3 to give a new derivation of the passage time law 5). [Given c > 0, min (t : c x (1/c') = b) = CS mb/C and mb are identical in law, and, using the homogeneous differential character of mb (b ? 0), it follows that Eo (e-O'm6) is an exponential a-bo(o), whereg (cza)/c = g (a)

(c > 0), i. e., g (a) = g (1) ya (a > 0) ; now compute the constant g (1).] ...

Problem 3. Check the arc sine law of P. LEvy [3: 216] : that, for the

greatest root b- < t of x(s) = 0,

't

('

1

P. (b_ S s) =

=? arc sin j s!

dI

J

t Z s Z 0;

0

compute also the distribution of the smallest root b+ > t of x (s) = 0 conditional on 6+00

[Po(b- Ss) =Po[mo(wf) > i - s] = f Po[x(s) E dal Pa[mo > t - s] -00 +00

=2f

a

da

y2ns

0

-

+ CO

a

t-:

y2:2is

e-.0121 d l

=a

('

t

dI

f

0

By a similar computation, Po[b+ E d b 16- = a]

(t-a)j(b-a)Sdb,2(b>t).] Problem 4. Use the formula

(27t i)-1

I+i CO

CT7

i

f

e" ei' da for the inverse LAPLACE

1 -1 00

transform to check ei'

+00

= JI e'"' y22t dt. o

(11-n)

Diagram I

_

[Working on the branch j =1'N etm/2 (a = , a I e1 ') , if I' is the closed curve pictured

1.7. Passage times for the standard BROwnian motion

29

.ti

..r

in the diagram, then (2n i) -1 f eta a-Y2n da = 0, and, letting n t + oo, it results that r 1+i0 _ (2n i) -1 f eta a-r2a da 1-ioo

-00

o

f eta+02H da + (2n i)-1 f eta-iV2jal da

_ (2n i)

o

CO

+00

+00

_

r J

e-ta sin

da = y

j

r

f

e-ta' cos

V a da =

'

e-1/2t

yens

1

J

0

0

«»,

a^_

Problem 6. The set 8 of roots of x (t) = 0 is a topological CANTOR set (closed, uncountable, of topological dimension 0, having no isolated point) 1; its LEBESGUE measure is 0. [8 is closed because the BROwNian sample path is continuous, of LEBESGUE measure 0 because +00

E0 [measure (8)] = f dt P0[x(t) = 0] = 0, 0

and of topological dimension 0 for the same reason. Because

t + mo (wi) E t as a function of t, lijmE0f a a(t+n,o(w+))]

10

= i oEo[e-k2al=(t)I] = I

implies that Po I l m (t + mo (wf)) = 0] = i , and, seeing as t + m0 (wi) E 3 t+O

Given t! > tl,

if it is < + ao, 8 is infinite. Define m = t1 + m0 P0 [x (s) has just one root between t1 and t2]

S PO (m < t2, l j ( + m0(w;+m)) > 0] S Po[l;o m0(w4) > 0] = 0, 80

i.e., 8 has no isolated point ; it follows from this that 8 is uncountable, and the proof is complete.]

Problem 6. Given a < < b, E.[eame, Mb

< ma] _

a)

sinhy2a(b - a) [e_a In , ,, ma < Mb] _ sinhy2a(b Et sinhy2a(b - a)

PE[mb <ma]=b ba -a EE

[ea m. A 1116]

PE [ma < mb] = b ` a

_

1 See ALEKSANDROV-HOPF [1: 45].

cosh V2& d coshy T a (b - a)/2

,

1. The standard BROWN Ian motion

30

and El[eam,nme]

=

cosy2ad

a < n=/2(b - a)2,

Cosy20, (b - a)/2 44A

where d is the distance between

and the midpoint of [a, b].

(Given a < < b, if m = man mb, then e-V aU - a)

= EE Le

a ma] =

Ee [e-a m, ma < mb] + E! [e-a m, 111b < ma] e- Y2 a (b - a)

and

= Et [e

a m+] = EE [e-a m, ma < mb] e- Tr i (b - a) + E f [e-a m, Mb < ma] ;

now solve for EE [e-°` m, ma < mb] and EE [e-a m, ma < ma] .]

Problem 7. Prove (b - a)'

Pa(xt E db, mo > t)

=

[e

se

(b

-e

a)' sr

V

Idb a b > 0.

[Use problem 1.]

Problem 8. Show that, for the standard BROWNian motion,

PP[x(t) E db, t < mo A Mil = X (-)"g(t, a, b")db InI?o

t>0, O 0 or e S 0; show also that t 3 t 1 ls* h (t) + 2 Ig. 1 -}-% 2 29 Il Ig. lg+

-I-

-r

I

belongs to the lower class. Problem 3. Use KOLMOGOROV's test to prove that if h E C (1, + oo) (11

is positive and if t-' h E 4. and t-'t' h E t for large t, then + 00

P° [x (t) < h (t) ,

t t + oo] = 0 or I according as f t-312 h e-A'12 t d t

+ oo.

Hinern's global law of the iterated logarithm:

P.

=1=1

x (t)

lim

t+m )121 Iga t

is a special case of problem 3; see A. HINCIN [1]. [Use the fact that [t x (111) : t > 0, P°] is a standard BROWNian motion

starting at 0; see problem 1.4.3.] 1.9. P. Levy's Hiflder condition K. L. CHUNG, P. ERDOs, and T. SIRAO [1] showed that if h E C(0, 1] ,N,

'""

is positive and if h E t and t-'t' h E 1. for small t, then P°

-1)

max

IO:S4g'

V 12 11

fo

'--

ek- 1 >

p1=

C

3 2

1-1/51

S 1415 ED (I ei -1 14) < 1415 x constant 1-2

= constant x 1 615.

I

-3-2-1 0 1 2 3 4 £71t - 1 1-+ Diagram I

2. BROWNian local times

40

KNIGHT has used his method to prove the existence (but not the smoothness) of the standard BROWNian local times of 2.8) and to discuss

the time substitution of 5.2). DONSKER's extension [1] of the DE MOIVRE-LAPLACE limit theorem

follows at no extra cost: if / is a Borel /unction on the space o/ continuous

paths x(t) : t 5 I and if it is continuous in the topology of uniform convergence at all but a negligible class of Brownian paths, then 7)

lm Pofa_/(S(

» . ao

ll

\`

=P0[a where [n t] is the greatest integer

])t51) 0. k S n) < 01

=

sin-:L

ye

(see ERDds and KAC [1], P. LEVY [2], and 2.6.17)).

2. Brownian local times 2.1. The reflecting Brownian motion

Consider the standard BROWNian motion D and recall the associated differential operator 0 = D2/2 acting on D(6) = Cs (R1) (see 1.4). Consider also the reflected BROWNian path x+(t) = Ix(t)1

1)

t Z 0,

let Bt = B [x+ (s) : s -< t] be the smallest BOREL subalgebra of B including all the events (a 19 x+ (s) < b) (s t), let D+ be the stochastic motion

[x+ (t) : t Z 0, B, P. : a ? 0], let (33+ be the differential operator 03 acting on the domain 2)

D(0') = C=[O, +oo) n (u: u+(O) = 0)1** * # (Q) means the number of objects in the class Q. u+(0) = lira e-1 [u (e) - u (0)]. 4+0

2.1 The reflecting BROwNian motion

41

and introduce the fundamental solution

g*(t,a,b)=g(t,-a,b)+g(t,+a,b)

3)

t>0,a,bz0

of au/at = 03+ V. A..11

Given t > s z 0, 4)

P. [x * (1) E d b I B,]

=P.[x+(t-s,w;)EdbI B,] =Pz(,) [x (t - s) E db]

a,b?0,

=g+(t-s,x'(s),b)db

and, using the fact that B; C B,, it is found that D+ is a MARKOV process.

D` is the reflecting Brownian motion; it stands in the same relation to (53' as the standard Brownian motion does to (53, i. e., the fundamental solution g+ of a u/d t = (53' u is the transition density of D+, just as the

fundamental solution g (= Gauss kernel) of au/at = (33 u was the transition density of D. Given a standard BROwNian

t

path starting at x (0) = a Z 0, intro-

duce, next, the new sample paths x- (t) = x (t)

5)

I

XS

zat=

tSm

= t- (t) - x (t) t > m, where m = mo min (s : x (s) = 0) and t- (t) = max (x (s) : in S s S t),

let B- = B [x- (s) : s S fl, and let D- be the motion [x- (t) : t >- 0, B, 0].

PQ:

D-

is identical in law to the

-X/4,f z

reflecting Brownian motion as will Diagram t now be proved'. Because x- (t) = x+ (t) up to the MARKOV time t = m, it is enough to check that [x- (t) : t Z 0, B, PO] and [x+ (t) : t --> 0, B, Po] are identical in law.

But, using 6)

P0(x(t) E da, t- (t) E db) = (2/n 13)1/2(2b - a) e-(26-a)'/21dadb I P. Liivv [3: 2341.

t > 0, a S b, b ? 0,*

* P. Livv [3: 211 ]: see problem 1.7.1.


42

it is clear that VII

7)

Po [x - (t) S c f B,] .:..'

= Po[t-(s) - x(t) 5 c,max o$t-,[x(6, w;)] - x(t)

c I B,I inn

r-'

= Po [x- (s) - [x (t) - x (s)] S c emSaxl [x (e, w;) - x (s)] -

- [x (t) - x (s)] s c I BI

f

d =t-s

P0[x(A)Eda,t-(A)Edb]

x-(8)-a$e

b-age

-oo S.c? 0

S c]

and since B; C B,, the result follows. 2.2. P. Levy's local time

Consider the visiting set 8+ - (1: z'(1) = 0)

(1: x (t) = 0).

$+ is a topological CANTOR set of LEBESGUE measure 0 (see problem 1.7.5).

P. LEvy [3: 239-241] discovered that it is possible to define a new time scale t+ = t+ (t) on ,3+, affording a non-trivial measure of the time the BROWNian traveler spends at 0; this t+ is local time. P. LEvY gave several

different definitions of t+ (see 4, 5, 6, 7)]; of these, the last is the most direct, but also the most difficult to establish. Consider the sample path x- (t) : t Z 0 and recall the decomposition of the passage time ma = mint : x (f) = a) (a ? 0) : +00

1)

ma = f

([0, a) X dl),

o+

where , (da x dl) is the POISSON measure with mean da x (2n 13)-11s dl (see 1.7). Because p ([0, a) X [1, + oo)) is differential in 1, it follows from 2)

Eo[p([O, a) X [e, +oo))] = a X 21ne

2.2. P. LEvy's local time

43

and the strong law of large numbers that Y(Lo.a)x[e,+oo))_a,a_> 0 =1 p 3)

[

2fne

and, substituting t-(t) = maxx(s) for a, since the difference of lSt P ([0, t- (t)) x [e, + oo)) and the number of flat stretches of the graph of t- (s) : s S t of length ze is not greater than 1, it is seen that 4)

2x the number of flat stretches of

Po [lipo

t-(s):sSt

oflengthze

=t-(t),tz01=1.*

4) shows that t- = t- (t) is a function of 8- = (t : x- (1) = 0); in addition, it is clear that t- grows on ,3- and is flat outside. Because D+ is identical in law to D-, there is a corresponding function t+ = t+ (t) of 8+ = (1: x+ (t) = 0), growing on 8+ and flat outside. V is local time; V 2-1a t+ is the mesure du voisinage of P. LEvv [3: 228].

Because t+ and t- are identical in law and the flat stretches of the graph of t+ are just the (open) intervals .,,(n 1) of the complement of g+, it is clear from 4) that 5)

Po [!im V"e x the number of intervals 8. C [0, t]

0

2

t+(t),t011,**

of length z=

and a little algebra converts 5) into 6)

PO [urn 1/iLs x the total length of the intervals

& C [0, t]

o t length < e

= t' (t), t

01

-= 1;***

thus t+ is a function of the fine structure of $+. 7)

P. LEvy [3: 239-241] also proved (2e) -1 measure (s: x; S e, s S t) = t+ (t), t ? 0] = 1. PO l

lo

c71

Here is his proof. Given 8+, the excursions e w [x (t) : t E 8.] are independent, and,

since 9 enters into the distribution of e via its length 1.3. 1

alone, 1

* P. LfivY (3: 224 - 225)

' P. I.frvv [3 : 224 - 225]. *** P. Lfivv [3: 224 - 225] .

2sJn = E. [f l p (CO, 1) x d l)] = the expected

sum of jumps of magnitude a figuring in tn,. 1 P. Livv [3: 233 - 236]; see also 2.9.

2. BRoWNian local times

44

the strong law of large numbers indicates that, for a 0. measure (s : x, S e, s s' t) 8a) measure (s: x, S e, s E& n [0, t)) can be replaced by Y E0 [measure (s: x; S e, s E 8b)

I,g+]

3 C(0,1) sac [0, t) +00

= f l (1 - e-'''It) p([0, t+(t)) x dl), 0

where p (di x dl) is the POISSON measure for the (one-sided stable) inverse function of t+, p ([0, t+ (t)) x [l, + oo)) being within I of the number of flat stretches of t+(s) : s S t of length Now use 4) to conclude that +00

f 1(1 - e-2 0/1)

9)

p([0,

t+(t)) x dl)

0

+00

= f p((0, t+(t)) x (1, -boo)) d[I(1 0

+00

f

t+ (t) d[I (1 - e - 21 0 0 ) ] = 2e t+(t)

i 0.

0

The difficult point of this proof is the jump between 8a) and 8b); although the meaning is clear, the complete justification escapes us (see, however, 2.8 for results of G. BOYLAN, F. KNIGHT, D. B. RAY and H. TROTTER that include 7) as a special case). Problem 1.

(ti. to fl $+ + 0,

o s ti < t2

bid

t+ (tz) > t+ (to

III

for almost all BROwNian paths. [Because t+ is identical in law to t-, PO [t+ (t) > 0, 1 > 0] = 1, and, if the statement were false, then e - PO [t+ (t=) = t+ (ti), (tl , to n 8+ + 0] would be >0 for some tt < 12. But e = P0[t+(12 - m, wm) = 0, ..+

m=tl+ MO (wi,) 0. [x (s) = t- (t) for 3 different times s S t implies that t is flat on an open interval that meets 8-; now use the result of problem 1, and the fact that t - is local time on 9-.] * P. Lgvv (-3: 238).

2.3. Elastic BROWNIan motion

45

Problem 3. Use the result of problem 1.7.1 to deduce the joint law P0[xI(t)Eda,t*(t)Edb]=2 a + b e-tb+6>'1tdadb a,b?0. 2nts Problem 4. Use the fact that t+ and t- are identical in law to prove ...

that the HAUSDORFF-BESICOVITCH dimension of $+ is a1/2 for almost all sample paths (see note 2.5.2 for the definition of HAUSDORFF-BESICOVITCH dimension).

[Because t- satisfies the HOLDER condition

0 St1 Sis 51,

t-(12) -t-(t1) 0,

a, b

0

is the fundamental solution of the heat flow problem

au=! asu

4a)

2 ab= y U(" 0) = u+ (t, 0) 8t

4 b)

t>0,

b>0, 1 > 0,

i. e., D' stands in the same relation to @* as the reflecting Brownian motion D+ does to W. D' is the so-called elastic Brownian motion. Because t+ is flat off 8+ and P. [maa E d t I B, M. > t] = y t+ (d t) , D' can be described as the reflecting BROwrian motion killed at b = 0

at the rate y t+(dt) : dl. Beginning with the simple MARxovian character of D', if e (0;5 e:< 1) (;'

is a BOREL function on the space of paths: I -+x(t) E Q (t S s, s, and if e' = e evaluated at the path x' (t) : t S s, then for a, b 0 and VII

Q = [0, + oo) U oo), if e+ = e evaluated at the path x+ (t) : t

s S 1, 5)

Ea [e*, x' (t) E d b]

= Ea[e+, x+(t) E db, t < ma,] = E. [e+, x+ (t) E d b, e-Yt+(t)]

= Ea[e+, x+ (t - s, w;) E db, e-Yt+(8) e-rt+(t

= Ea[e+ e-Yt+(,) Ez+(,)[x+(t - s) E db, a Yt+(t-')]] = E. [e+ a-Y t+ (+) Pz+ (,) [x' (t - s) E d b]]

= Ea[e+, s < m., Pz+(,) [x' (t - s) E d b]] = Ea[e' P.-(,)[x' (t - s) E db]],*

and the proof is complete. As to the identification of D' as the elastic BRowNian motion, it is enough to show that UI Q0

6)

+ OD

u = E. [ f e t l (xt) d t= E. f e-ate-Y t+ l (xe) d t * P00 [x (t - s) E d b] = 0.

c> 0, l E C (Rl)

2.3. Elastic BRowNian motion

47

is a solution of 7a)

u E D (W) = C2 [0, + oo) n (u: y u (0) = u+ (0))

(a-(')u=t

7b) and to invert the LAPLACE transform.

But me

8)

u (a) = E. f e ° t / (xt) d t + 0

+ao

+ Ea f eanl. f e ae e-rt+('

l

)

(m = m0)

/[x+(t, wm)] dt]

o

(

1

- Ea I f e-at / (xi) d t] h'1

Ea f e-at t (xe) d t]

l0

+

me

+ E. [e-11 m"] u (0) I.

r +00 e-a t

Ea I f

+ao

l (xi) d t + E. [ea m,] (u (0) - Eo I f eat / (x,+) dt )

0

-+°° a r2alb-al+e-Y2-. lb+al

V-

0J .+.

f

goo

+ e-fta a U (O)

/(b) db +

-2

e tab /(b) db

0

from which it is immediate that

au- 2 u"

9)

and, using the fact that [x- (t) = t- (1) - x(t) : t Z 0, PO] is a reflecting BROwNian motion with local time t- to compute u+ (0) from 8), one finds + CO

10) -u+(0) + 2f OF-b/(b) db 0

= y2au(0) =y2a Eo[ f

+b

e-rt-(t)/[t-(t) -x(t)] dt]

e-c't

0

+00

CO

e.-atdt f e ybdb faa2. 'CI

= 2oc

,J

0

-co

0

+oo

2n Is

b

= V2-,X f e-Yb db f da

2e-y2a(2b-a)/(b

-00

o

t00

v2ar

2

2Jr e - 26 f 0

L.

db,

- a)

e

ab-/tt(b - a)

2. BRowNian local times

48

u+(0) _ +2Vr2

11)

+oo

/' e_1sab/(b) db

. 0

=yu(0) as desired. 2.4. t' and down-crossings

Consider the number of times d, (1) that the reflecting BROwxian path x+ crosses down from e > 0 to 0 before time t and let us check that

Po[hme d,(t) = t+ (t), t -> 01 = 1

1)

as P. LEvY conjectured [6: 171], supposing for the proof that

(2e)'1 measure (s: x+ (s) < e, s S t), i o as was mentioned in 2.2 and will be proved in 2.8. t+ (t) =

2)

Consider for the proof, the successive returns r, < r2 < etc. of x+

to the origin via e > 0, i.e., let to = 0, tl = m, + nto (w+ ) , and 3)

+ tl (wiw_1) (n > 1) , and note that for 0 < 6 < da (mi) +I < de(ti (wre-,), w k-j ASde(mJ+l

the summands being independent, identical in law, and independent of d, (ml). Because of Po [da(rt) = n] =

4)

(1

- e) e

(n

-

0)

and the above independence, 5)

Eo [a(da (mi) + 1) (d: (m,J]

= 6(d1(mi) +1) Eo[do(ti) +1] = a(de(m,)

+1) [ 8 (1 - a)+1J

= e(da(ml) +1)

(a< 8),

i.e., [e (d, (mt) + 1) : e < 1, Po] is a positive backwards martingale, an since 6)

E0[e(d,(ml)+1)]=e I'9X (I -e)*C+l =i,

DOOB's martingale convergence theorem (see note 2.5.1) tells us that lime d, (m,) exists. $40

2.4. V and down-crossings

49

++"

Next, this limit is identified as t+ (m1) with the help of 2).

Using d in place of d, for e = 2-", it is desired to estimate d" = 2"-1 measure (s : x+ (s) < 2", s < MI) - 2-"d"(m1) = 2"m2+* if d"(ml) = 0

7)

= 2"-1 I [measure(s:x+ < 2-", rk-I S s < tk) kgr"

2-9n+1)

i/ dn(ml)=mz1.

Here the summands are independent, identical in law, and independent of d" (m,) = m, with common mean CO

f dsP0[x+(s) 0, n , = nun (n : eA = 0) , ns = min(n : is > n, , e 1), ns = nun (n : is > ns, eA = 0). etc., let # denote the number of times em : m < n crosses up from 0 to I (i.e., let # be the

number of integers ns, n4, ns, etc. < n), and let e = e, + Z ti (ei - et_,) , i-8 where ti = 1 if i lies in [0, n,] U (ns, ns] U (ns, na] etc. and ti

otherwise. (ti = 1) is the event that eA : it < i has not completed its current down-crossing; this is measurable Bi-1 so that 1)

i-Q

E(ei-ai-1l Bi-,))a0,

and, using

n 0, (SS' g = a g has two independent solutions 0 < g1 E t and 0 < gs E I, their Wronskian B = g'1 gs - g1 g''2 is constant (>0), the Green /unction

(a S b) VII

G (a, b) = G (b, a) = B-1 g, (a) gs (b)

11)

VII

satisfies 2a f G d b S 1, and is = 2 f G/ d b is the bounded solution of 4). VII

Here is a rapid proof. Choose / (a) = 0 (a S b) and / (a) > 0 (a > b) in 3) and let gab = Ea [e-am1 a-t (m')] (a < b). is satisfies

u(a) = E. °"'

12)

'~g

+00

e-'/(x)dt)

=gabu(b)

(a < b),

1

r'7

and since u (b) >0 and (a- (SS') u = / = 0 (a < b), it follows from the multiplication rule 13)

(a 0) cosh 2" [Use the additive functionals e0 and el based on the indicator functions e0 and e1 of [0, + oo) and [1, + co], let +00

u = E. l 0 (' and note that

eo(x(t)) dtl (a, P, y > 0),

If

( m,

lim

lim u(0) = E0I f e -P to eo(x (t)) dt

E0(e-Pss(ml))].

0 r^,

KAC tells us that u (0) can be expressed as

oaf

+00

2g1(0) B-1 f gs d b 0

(09 g = a g, (? = 0 - eO - y e1) ;

2.7. BESSEL processes

59

the formulas are VI!

(b S 0),

& (b) = ey2ab

(bz1,a-a+A)

gs(b) = e' Zt°+YIb

=cosh

(1-b)+V

(OSbs,

P-10) < b I BO = Pa [r (t - s) < b]

2)

r

e-I b- ass/acs-"

(2x (t - s))'1'

J

IbI 0

=0

(see problem 1) ; in brief, the Bessel motion is a simple Markov process, standing in the same relation to the Bessel operator 6a)

(SS+ = 2 (d b= +

b

1

db)

he

radial dart of ( _ (1 /j) d

acting on the domain

6b) D((3S*)=C[0,+oo)fl(u:C *uEC(0,+oo),hobau'(b)=0)t as in the X-dimensional case (see 2.1). o E Sd -1, the unit sphere. d o is the element of surface area on Sd -1.

* 0 is the angle between o and (1, o. .... 0).

*a I(&n,-) is the usual modified BESSEL function.

t limb'' u'(b) = 0 is automatic if d a 2; see 4.6. 6+o

2.7. BESSEL processes

61

6a) suggests that the d-dimensional BESSEL process r should be a standard BROWNian motion b subject to a drift at rate (d - 1)/2r,

i.e., that it should be a solution of r (t) = b (t) +

d

t

2

1 f d s/r (s) ; this 0

U)^

NIA

conjecture is correct for d z 2 (see H. P. MCKEAN. TR. [3] for the proof and additional information), and it is also true for d = 1 if t VII

lim (2E) -1 measure (s : r (s) < e, s

1) is used in place of d 2

(see H. P. MCKEAN, JR. [6]).

I

fasir(s) °

...

A. YA. HINCIN'S local law of the iterated logarithm PO[e1m

7a)

I(I =1=1 1

0 y2

>)g=

(see 1.8) and P. LEvy's HOLDER condition 7b)

P

I(s)l lim ° 15-11840 y2dlgI/6

1=t ]

(see 1.9) also hold for the d-dimensional BROWNian motion as one can

easily prove, using the fact that the projection of the d-dimensional

Flu°.w

motion on a line in Rd is a standard t-dimensional BROWNian motion and noting the obvious estimate III = supI (gr, e.) I for I E Rd and a

8a)

8b)

P ° [el+o tTIM-

P°

[

I

'V r('

81g' 1/a

D

't7

ea1 countable dense set of points e (n 1) on the surface of the unit ball. Because the angular distribution of the large changes in I(t) (t S 1) is isotropic, the same laws hold for the BESSEL process as well:

)i - 1 l = 1

(t ? 0)

Ir) - r(S)1- = 1l = 1.

-81-640 y2 d 1g 1 /d

J

...

A final point needed below is that the d 2 2 dimensional BESSEL path does not touch r = 0 at a positive time : 9)

P° [r (t) > 0, t > 0] = i ;

it is enough to discuss the 2-dimensional case. Consider for the proof two circles I a I = a and l b = b > a centered at 0, take a point E in the shell between them (a < I < b), draw about

it a small circle I g - 11 = c contained in the shell, and let e be the exit time min (t : I I (t) - I I = c) for BROWNian paths starting at


62

is invariant under rotations about E, it must be the uniform angular distribution do on Ig - 11 = c, and since the BRoWNian traveller starts afresh at time e, the 1 (0) =I I. Because Pj [I (e) E" d>)]

radial function 10)

p(III)

= P:(ma< mb) mr = min(t: I1(t)I = r) = min(t: r(t) = r)

satisfies

P1tma(wt) < mb(We )] = f P(ItI) do,

11)

i.e., it is harmonic (dp = 0) in the shell. But the projection argument used above shows p (a+) = i and P (b-) = 0, so

a t]

=

2elb-aI G(a,

E)G(E, b)

G(J, E)

e-4100- 1);

in the special case / = 0, 19) becomes 20a)

Pab(t(e, ) = 0) = 1 - elb-al-Ib-fl-If-al Pab[t(e, $) > t] = elb-al-Ib-11-11-al e-28,

and 14) follows. Because f (e) = 2 f t (e, ) / d $, it is enough /or the proof that the conditional local times are Markovian, to check that the additive /unctionals f : based on

/_=/ (rJ $) =0 (r/>J)

and /+ = 0

=l

(, 0, t > 0] = 1 (see 2.7) that the composite motion r4 [(eU - en V O] is MARKovian, and now a comparison of 24a) and 24b) for J < rj 0) maps t (t, a) into c t (tics, a/c) and changes the statements of 10)

into the corresponding statements at time i/cs, so 10) holds (with = in

place of $) at all positive times, and an extra result obtained at no extra cost is that Po[t(t, $) > 0, minx(s)

28)

#;58

t > 01 = 1 < f <maxx(s), SS8

as is immediate from Pa [rd > 0, t > 0) = 1

(d = 2,4).

Because of lim 2"-' measure (s: a < x < b, s s t), b-a+2-" I!\

t(t, a) =

29)

b.a

t(s, ) S t(t,

30a)

s S t, E R1;

)

also, in view of 10) and FuBINI's theorem,

0, and since t (s, ) as a function of s is constant near s = t for t * x (1),

b aOIt(tl,b)-t(t,a)J -0.0

30c)

t(t, ) is a continous function o/

r.+

3ob)

for each t = k 2-"

St(t-f-,e)-t(ttZ0,.ER'.

a<E - j132'" is tg 2 for some k < 2")

is the general term of a convergent sum, and this can be done using the standard BRowNian scaling x (t) x (tf cs) and RAY'S description

2. BAOWNian local times

72

of the conditional local times as follows: 34)

Q"

S 2" Po m ax Igo

l

) > 0 2-4/2 n, x (2-n) > -2.2-"'2

(2-n,

=2"Po[$goa t(1,>0n,x(1)>-2yn 2" E. [

5 t(1,

) > 0 n, ez(1)no) e2V* I'+

+00 00 (21)-1

2"+a e2y" Eo Lmax t(1, Sr) > On,

[ g°

r e-Ib-z(1)I dbl o

J

= 2"+2e2l+npo I m t(1, t) > 0 n, e > 1. x(e) > 0, 2'

e2yn po [ms t(e, ) > 0 n, x(e) > 0,

X10

S 2"+se21+"Po[ 5 ra(t)2> 40 n] S 2"+4e2)APo [msai x(t)2 > 0 nJ 2n+be2Vapo

[ m m a x x(t) > V

< constant x 2" e2 y"

J

e-eon

yen < constant x e-e" (o < a < (0 - lg4)/2). Problem 1. Check that f d t/x (t)

li

d t/x (t) exists. ssf

Ix(p >.

+00

[

f d t/x (t) = 2 f [t (1, b) - t (1, - b)] AL and this converges as b.

1

e M 0, thanks to lt(1, b) - t (I, - b) 0.

Because the scaling x (t) -> f -1 x (fl$t) maps mt into f-s mp t and t(1,1)

into fl-1t(f2t,Pl), it is clear that c(a) = flc(a/fl) = flc(1) (fi =a), and, to complete the proof, it is enough to evaluate c = c (1). Differentiating e = [acl + 1]-1 and letting a 1'0, E0[t(mt, 0)] = c

and, using E0[t(m1,1)] = E0[t(m1(wm),1, wm)] = Et[t(m1,1)] = Eo[t(m1-t, 0)]

(m = mt),

one finds 1

1

1

'!t

c=2 cldl=2 f Eo[t(mt,0)]dl=2 f Eo[t(m1,1)]dl 0

0

0 +00

EO 12

f t (m1,1) d it = Eo [measure (t : x (t) z 0, t S m1)] 0

+00

A

= f Po[m1>t,x(t)a0]dt 0 +00

+00

b

rP0[t-(t) < 1, x(t) z 0]dt = f dt f db f da 2 2-

f =1,

2n t'

e- (2b-a)=j2t

n"^

where problem 1.7.1 is used in the last step.] Problem 4. Given b > 0, '[t (t-1(t) , b) : t > 0, PO] is a homogeneous differential process with LEvy measure b-2e-111dl (b > 0) ; here t-1(t) is the inverse function of t (t, 0), i. e., t-1(t) = min (s : t (s, 0) = t).

2. BRowNian local times

74

[Use the fact that t-1 (t) is a MARKOV time and the addition rule t-1 (t) = m + t-1 (t - s, w ;) (t > s, m = t-1 (s)) to prove that t (t-1, b) is differential and homogeneous. As to the LEvy measure, y

Eo f

+00

o

J

III

= Eo [Toe-=(.0)e-4t(.b) t(di, 0)Jl

= Eo f e-.t(t.°)t(di, 0) +00

+ Eo[e-at(mb.°)] Eb .V.

f e-=tu.o)e-P1(t.b)t(dt, 0)J L0

=a-1Eo[1 -e

+Eo[e-=t(me.o)]Eb[e-Pt(m..b)]Y OLD.

and, using the result of problem 3, y is found to be [a + Now invert the LAPLACE transform, obtaining

1)

ire

f

+00

Eo[e-Bt(t-I(t),b)] = e4p(pb+1)-,

= e-4

+V+

as stated. Because f b-2e-t1bdl < + oo, this process has a finite number 0

of jumps up to time t = 1 as is also clear from the fact that the stopped BROwxian path x (t) : t;5 1 cannot come from 0 to 1 an infinite number of times.] Problems 5 and 6 below contain the proof of RAY'S description 11 d) of

Problem 5. Check that E0[a-Yit(m1,E1)-Y,t(m,.E4).. -y,1i

l,Ea]

is the reciprocal of

I + isn Ir Vi 0 - Si) +i b] = n COs-1 b

(b < t)

and

(a<s b - a or not; observe that b is the same as t - t-1(t+) (t-1 (t) = min (s : t+ (s) = t)) and hence identical in law

to t - m4 (a = t-(t) = 5 x(s)). [Using b- = the biggest root of x (t) = 0 (t < s) and b+ = the smallest root of x (t) = 0 (t > s), it is clear that Po (b+ < t 18 n [o, s]) = Po (b+ < t I b-) = Po (b+ < t i i(s)),

2.10. Application of the BESSEL process to BROwxian excursions

79

and so the MARKOVian nature of a follows from

>t

8(t)t-s+6(S)

0 B (t) x B (en : n 1), up to sets of WIENER measure 0. [Because Go+ is independent of B (en : n ? 1) and the universal

algebra B is identical to B (8) x B (en : n z 1) x B (en : n ? 1), the identification of Go+ with B(8) x B (es : is 1) follows from the obvious inclusion Go+ D B (8) x B (en : is ? 1).]

VII

2.10. Application of the Bessel process to Brownian excursions We saw in 2.9 that the scaled BROwxian excursion el (t) : 0 S t S 1 is MARKOV with

1a) P0[el(t) E db] = h(0, 0, t, b) db = 2

Vn t=(1

b2db

0

.+I

1 b)

s

db #

0<s0,a,b>0.

1 > I > s > J and 0 < a, b and using the MARKOVian

character of r+(1 - t)

J), one finds

(t

P. `lr+(1-t)EdbIr+(i-0):2 SOSs,

5)

= g+(1-t, 0, b)

g+(1 -s, o, a)

(t-s))lab I

1Iz

g+(t - s, b, a) db (

a=r+(i-s)

ab \ ( t -s\312

t-

t -t

s

b' d b

= h(s, a, t, b) db. Because e (j) and el (j) are identical in law [see 1a) and 2b)], r.+

5) implies that e (t) (1 Z J) is identical in law to e1 (t) (t ? J), and to complete the proof, it suffices to note that e and el are symme-

trical in law about t = J. Here is an application to the fine structure of the standard 1-dimensional BROwxian path.

A. DVORETSKY and P. ERDOs [1] proved that if 0 S / and if t- /(t) E t for small t, then Po [r (1) > / (t), t 4 0] = 0 or 1 according as f t- EJz / (t) d t Z + 00

6)

o+ +'3

where PO is the WIENER measure for the 3-dimensional BROwNian motion starting at 0*. Now using the definition 2.9.1) of the scaled BRowNian excursions coupled with the fact that

I

I8 I

f t-s/2/(I8sI t) dt = f t-312/dt, 0

0

one finds 7)

4 0, n ? 1] = 0 or 1 according as

Po[x(t, 1

f t-s/z /(t) dt Z + oo, 0+

where P. is now the WIENER measure for the 1-dimensional BRowNian

motion starting at 0 and the 3, (n z 1) are the open intervals of the * See 4.12 for a proof of 6).

2.11. A time substitution

81

complement of .3 = (t : x (t) = o); for example 8)

PO 1Ix(t. wi.fgjI>

1

Ig

t

192 ! ... (ig7) 1

+. , 1. 0 , n Z 1 = 0 or 1

according as e S 0 or e > 0. According to 2.7.8a), PO lira

9)

7

1+o

V

`-

r(t)

=1=

2119s

from which one deduces 10) PO

ol$

i

I, r j

J l=

Ix (t ! $.I ; V211g=

1

lo

Jj/2t1gs__ x(t , w,N) I= 9, n Z 1 = 1,

2.11. A time substitution

Given a standard 1-dimensional Bxowxian motion starting at 0, the decomposition of 2.9 can be used to prove that if f -1 is the inverse junction of f (t) = measure (s : x (s) z 0, s S t), then [x (f -1). Po] is identical in law to the reflecting Bxowxian motion [x+ xI, Po] of 2.1.

Given a Bxowxian path starting at 0, throw out the negative excursions and shift the positive excursions down on the time scale,

-1 +1

f(t) Diagram I

closing up the gaps as indicated in the diagram; the path so defined is x (f -1) , and it is clear from the picture that, for its identification ItA/11cKean, Diffusion processes

6

2. BRowwian local times

82

as a reflecting BROWNian motion, it suffices to show that W = (t x (f -1(t)) = 0) = f (8+) is identical in law to 8+ = (t : x+ (t) = 0). Because f (8+) is the closure of f t-1[0, + oo), where t-1 is the leftcontinuous inverse function min(s : t + (s) Z>= t) of the reflecting BROwxian

local time t+, it suffices to show that f (t-1) is a one-sided stable process

with exponent 1/2. But f (t-1) is t-1 minus the jumps arising from the negative excursions : 1)

1

A

f (t -1(t)) =

an CIO, t-' (I))

') (1 + en) 13 I .

+00

and, using the POISSON integral f l p ([0, t) x d l) for t-1(t) (see 1.7) 0

coupled with the fact that the coin-tossing game en (n ? 1) is independen tof ,$+, it is clear that the effect on the PoIssoN processp ([0, t) x dl) .,y

of ignoring the negative excursions is to cut its rate in half (see +00

problem 1). f (t-') is then identical in law to t-1(t/2) = f lp([0,t/2) X dl)

(t z 0), and the proof is complete.

0

An alternative proof is to note that f (t-1) is differential and homogeneous (see problem 2) and to use 1) to compute Eo [-a f (t-' (t))]

E0

lT

e-a(1 +on)18.1/2

old

2)

l8.CLO, t-'(t))

+0

f )g(

1+28

=E0eO

I

)I p((O.8)xd1)

+00

( 1 + e a1-1)

t

J t

e0 e

2

2

di 2-.P

2a

Time substitutions are treated in 5.1, 5.2, and 5.3; it will be seen that the present substitution is not accidental. Problem 1. Given a standard coin-tossing game en (n 2 1) (P [e1 = -1 ]

= P[e1 = +1] = 1/2) and an independent PoissoN process with jump size 1, rate x > 0, and jump times t1 < t2 < etc. with distribution P [t,, - 1, 1 > t] = e -"t (n a I, t0 = 0) , deduce that e (t) _ is POISSON with rate x/2.

k (1 + en)

3.1. Definition

83

[Given disjoint intervals d,, A., etc. of length d,, ds, etc.,

P [toI (1 + en) = j1, 1(1 + eR) =1s etc.l

I P( 1=n1, I J(1 +en)=11 Zt4Ed,1 =ns, non,

tha:h

%ah

n, s, .`Y

CDs,

m=t-1(s)).onehasf(t-1(t))-f(t-'(s))=f(t-1(t-s,w ),w,)(t>s); III

in addition, in is a MARKOV time because in < t means that t+ Z s

at some instant < t. Given t S s, (f

(t-'

(t)) < b) fl (m < a) E B4,

i. e., (f (t-1(1)) < b) E Bm+ (t S s) ,

and the homogeneous differential

character of f (t-1) follows on noting that the Bxowxian traveller starts

afresh at 0 at time t = in.]

3. The general 1-dimensional diffusion 3.1. Definition

"''

Roughly speaking, a 1-dimensional diffusion is a model of the (stochastic) motion of a particle with life time m. S + oo (also called killing time), continuous path x (t) : t < in.., and no memory, travelling in a linear interval Q; the phrase no memory is meant to suggest that the motion starts afresh at certain (MARKOV) times including all the constant times in = s > 0, i.e., if in is a MARKOV time, then, conditional on the present position x (in), the statistical properties of the future motion x (t + in) (t > 0) do not depend upon the past x (t) (t S in). Before stating the precise definition, introduce the following objects: Q a subinterval of the closed line. 0o

an isolated point adjointed to Q. 6*

84

3. The general 1-dimensional diffusion

a sample path w : Co. + oo) -> Q U oo with coordinates x (t) =

Cep'

w

811

x (l, w) = xj = xt (w) E Q U oo and life time mm = in,,,, (w) < + o0

such that

t<m

!-F

x(t)=x(t±)EQ

tZm,,,

= 00

esp., x (+ oo) F_ 00.

W the space of all such sample paths. the smallest BOREL algebra of subsets of W including

Be

(w: a S x(1) < b) for each choice of t s' s and a S b.

BBB+,,.

b=a.

.-.

mod

'.7

0

P. a function from Q to non-negative BoREL measures on B of total mass + 1 such that, for each BE B, Pa(B) is a BOREL function of a E Q, and, for each a E Q, P. [x (0) E d b] is the unit mass at

P the unit mass at the single path x (t) = oo (t Z 0). m a MARKOV time, i.e., a non-negative BOREL function m = m (w) S

+ 00 of the sample path such that (z v: m (w) < t) E Bt (t ? 0). x (I + m, w) (I ? 0). B,+ the BoREL algebra of events B E B such that B n (m< t) E Bt (I Z 0).

-0-

w n the shifted path x (1, wm)

ode

Q, B, P. as above are said to define a motion D with state interval Q, path space W, and universal Borel algebra B. Pa (B) is read : the proba-

...

bility of the event B for paths starting at a; note that P. [lim x (t) = x (0) = a] = P. [m > 0] 1 (a E Q). 8 10 D is said to be conservative, if P. (m < + oo) = 0 (a E Q) . D is said to be simple Markov if it starts afresh at each constant time

t Z 0, i.e., if I a)

P. (wi E B I

Pa (B)

a = x (t) ,

BE B,

or, what is the same, if 1b)

P.(x(t)EdbIB,)=P4(x(t-s)Edb) a=x(s),

D is said to be strict Markov or a diffusion if it starts afresh at each Markov time m, i.e., if 2a) E BI Bm+) = Pa(B) a = x(m), BE B,

or, what is the same, if

2b) P.(x(t+m)EdbI B,n+)=Pa(x(t)Edb) a=x(m), (note that x (m)

tZ0

oo if m = + oo).

Given an exponential holding time e with law P(e > t) = e', if w' is the sample path 3)

x' (t) = 0 (t < e)

= t - e (t ? e)

3.1. Definition

85

and if P. is chosen so as to have P0(B) = P(w' E B)

4a) 4b)

Pa[x(t)so a+t,t?0]M1

a>0,

'ay

then the motion so defined on Q = [0, + oo) is simple Markov but not strict Markov; indeed, m = inf (t: x (t) > 0) is a MARKOV time, Po[m > 0, m(wm) = 0, x(m) = 0] = 1, and, should the motion start afresh at time m, PO [m > 0, m (wm) = 0] would be PO (m > 0) PO (m = 0) = 1, which is absurd. Given a pair of motions D and D', if / is a BOREL function mapping part of Q onto Q' and if f is a BOREL function mapping (w: x (0, w) E into the dot path space such that

/[x(0, w)] = x (o, f w) x (0, w) E f-1(Qo) a E f-1(b) , P. (f-1 B') = Pp (B')

5 a)

B' E B', then If w, Pr, (b): b E Q'] is said to be a non-standard description of the dot motion as opposed to the standard description used before ; as an example, if D is the standard 1-dimensional BROwxian motion, then x+(t) = 1 x(t)I tZ0 6a) 5 b)

x - (t) = x(t)

6b)

t < MO

=t-(t)-x(t) t- (t)

x' (t) = x U-1)

6c)

tz mo

Ssl X (S)

tZ0

f (t) = measure (s : x(s) z 0, s S t) are three separate non-standard descriptions of the reflecting BROWNian motion (see 2.1 and 2.12) ; additional examples of non-standard description will be found in 2.5 (elastic BROwxian motion), 2.10 (BESSEL Goo

motion), and problem 2 below. Given a motion, the choice of description depends upon the problem in hand; for example, the non-standard descriptions 6a) and 6b) were useful for the discussion of the BROwNian local time t (1, 0) (see 2.2) cad

but the proof that a motion starts afresh at a MARKOV time is best done in standard description (see 3.9, 5.2, and the second part of 't7

problem 3 below). Problem I. Prove that 2a) includes BLUMENTHAL'S 01 law :

P.(B)=0 or 1 BEBO., (=flB.). .>o

[m = 0 is a MARKOV time, and so, for BE P.(B) = P.(B, wo E B) = E.(B, P.(wo E B I BO J) = P.(B)2.] Bo+.


86

Problem 2. Explain the correspondence between the MARKOV time m and BOREL algebra Bm+ of a) the standard 1-dimensional BROWNian motion

b) its non-standard description as a scaled BROwxian motion [x' (t) = c x(t/c') :1

0, P11 : E E R'] .

[Because B; = B [x' (s) : s S t] = Btt., if m' is MARKOV for the

scaled motion, i.e., if (m' < t) E B;

(t

0), then (c-' m' < t) =

(m' < c't) E Bt (t --> 0), i.e. m = c-' m' is MARKOV for the unscaled motion, and All

._Y

B'm.+= (B:Bn w< t) E Bt,t;-> 0) `°.

es Bm+=(B:Bn(m m (u), then t > nil (u), proving m1 (u) = nil (p); in addition, X. (ue) = x, NO

7a)

s < t - 0,

0 = m, (u) = m1(v)

m2(u+) < t - 0,

7b)

cad

proving m2(ue) = m2(ve), and adding, in (u) = m, (u) + M2 (ue) = m, (v) + m2 (ve) = in M, completing the proof. As in 1.7, the BOREL algebra 8)

Bm+=Bfl(B:Bf(m0

B,n+ = fl B[x(t A (m + 6)):t? 0] 411

in fact lob)

M, S mg,

e>0

(see problem 1 below) ; in addition, 10c)

Bm+=Bfl(B:wEB iff w;EB,t>m(w))

as will now be proved.

Given BE B, if wEB iff wiEB for 1> m(w),thenBn(m s, m (w;) > t] = Pa [m > s] Po [m > fl, proving .-.

PO [m > t] = e "l. Given x < + oo, PO [m > 0] = 1, and since m;5 m., E Bm+ (see problem 5) to show that it is enough to use (m
a) or min (t: lien x (s) =a in which ma+ #ft according as x (0) S a or x (0) > a.

We are going to prove that 3a) td-

e, =0 or 1

3 b)

e, = 0 or 1

3c)

e4=0or1

4a) 4b)

e,=ele,Se,

5)

e3 = e, e4

ei

e,es=e,e4.

Given a sample path starting at a, if m = ma+= inf (I: x (t) > a) then, m (wm) = 0 and x (m) = a in case m < + oo, so 6) e, = E. (e_m'+) = Ea [e-m, in < + oo, x (m) = a, Ea (e m

jj

(/QM) I Bm+)J

= ei ,

90


proving 3 a) ; 3 b) is clear from 7)

e2 = lim b4a Ea+ (e ma) = b m l m E.,

(em') Ee (em`)

= e' lim Ea+ (e-m2) = e2 b+a

3 c) from e4 = bjn Eb

8)

=1s{n lim Eb (e-"4) EQ (e-m.+)

= lim Eb (e_ m +) e4 = e42, bfa

4a) and 4b) are trivial; and, for the proof of 5), it is enough to cite the fact that, in case x (0) > a, ma > ma+ implies + oo = ma > ma+ = m

and to infer that el ea = bJim a i m E. l6 ma

9)

/ mI \mm.) m

hi

t

lim Ec [e-'-+ a

lim lira 1' E [G m4 ame (soma)] bia cia d+& = bli{m ova Ji m EC (e-m4) Ed (e-ma) = es e4. a

We can now prove 10a)

ei = P. (ma+ < + oo) = Pa (ma+ = 0)

lob)

e2=birPa+(me<e)

10c)

e3 = bm Pb (ma < + oo) = 8lin Pb (ma < e)

10d)

e4 =b m Pb(ma+ < + oo) =bm Pb(ma+ <e),

in which e > 0 is chosen at pleasure; indeed 10a), lob), and 10d) are clear from the 01 laws 3 a), 3 b), and 3 c) and the fact that, for a:-!9 b, P. (mb < + oo) and P. (mb < e) are monotone in a and b. 10 c) follows from 11)

0 S Pb (ma < + oo) - Eb (e-m') = Pb (ma < + oo) - Eb (e me+, ma < + oo)

= Eb (1 -

ma < + co) S Eb (1 -

Pb(maa,e4=0

and

12)

Pb (e S ma < + oo) 5 Pb (e S ma+ < + co) S Pb (e S ma+) j 0

bia, e4=1 if b>a, e4=0. if

SPb(ma+ ms) =min (t : x (t) -b, t > m,. ) dja d4a = m.+ + ma (wa.+)

3.4. Singular points

91

I".

23 = 8 combinations of the values of el, es, e4 are possible. 3 of these violate 4a), 4b), or 5), leaving 5 possible cases: ei

e.

e,

e4

1

1

1

1

1

1

0

0

0

1

0

0

0

0

0

0

1

0

0

F.,

10+

(the values of e3 follow from 4b) and 5); the blank indicates that all values 0 S e, S 4 are possible). Consider, now, the corresponding left limits el, e2, etc. and let el = ei, e2 = eQ, etc. 5$ = 25 combinations of ei, e2, e4 and ei, a", e* are possible. 3 combinations violate the fact that eiei = 4 implies e- e3 = 1 (see problem 1), leaving 22 possible cases. Problem 1. esea = I if eiei = 1. l4 L

= el = E. (e-mom) = lim E. (e-m.) = lim E. [e-"+ E m°(Wm.+)] cta eta

= Jim b jm E. [e m' -m. (44)]:5i bun Eb (a eta

ea as in 9), etc. ]

Problem 2. Give concrete examples of the 5 possible combinations of el, e2, e,, e4.

3.4. Singular points

Keeping the 01 law 1)

el* =PE(mjf =0) =0 or 1

in mind, the point e E Q is defined to be regular if ei = ei = 1, singular (e E K_ U K+) if eiei = 0, left singular ( E K_) if ei = 0, right singular ( E K+) if ei = 0, a left shunt if ei = 1 and ei = 0, a right

shunt if e-=0 and e+=I, a trap ( E K_ fl K+) if e i = e+ = 0. K_ (= the class of left singular points) is closed from the right in Q, meaning that if the left singular points bl > b2 > b3 etc. 1 a E Q, then a is

left singular also; for, in that case, Pa(mb < +oo) = 0 (a < b) because (a, b) contains points that cannot be crossed from the left. K+ (= the class of right singular points) is closed from the left in Q,

Nib

meaning that if the right singular points bl < b2 < b, etc. t a E Q, then a is right singular also; the proof is the same. K_ and K+ are Borel sets; for example, if the points, al, a2, etc. are dense in K+ and include all its left isolated points, then the fact


92

that K. is left closed in Q implies

K+=Q(1 f n1

1)J

,r.

Q - K_ U K+ (= the class of regular points) is open in the line (i.e., not just relative to Q) ; indeed, if is regular, then e i = e* = 1

which implies e$ = e8 = 1, or, what is the same, P. (mb < + oo) Pb(ma < -}- oo) > 0 for some a < f < b, and the whole neighborhood (a, b) must then be free of singular points. K_ U K+ (= the class of singular points) is then closed in Q. 3.6. Decomposing the general diffusion into simple pieces

Given a diffusion D, if either P. (Mb < -}- oo) or Pb (ma < + oo) is positive, then a and b are said to be in direct communication; if there d'°

is a finite chain of points c1, c2, etc., leading from a to b, such that c1 is in direct communication with c2, c2 with ca, etc., then a and b are said to be in indirect communication. On Q, (indirect) communication satisfies the following rules: each point communicates with itself ; if a and b communicate, then so do b and a; if a and b communicate and also b and c, then so do a and c. Q therefore splits into non-communicating classes Q* with perfect (indirect) communication inside each class and this splitting induces a splitting of D into autonomous diffusions D* = [W* = W(Q*), B* = B(W*), Ps = Pa:aEQ*].

D* is trivial if Q* is a single point 0, for then

1(t)=P0(m,>t)=P0[x(s)=0,sst] satisfies 1)

f(t + s) = Eo[ma, > s, x(s) = 0, Po(ma, (w,) > t I B,)] = f(t) f(s),

and therefore 2)

P0(m,o>t)=ea1

tZ0,05x a) are in (indirect) communication, and, making a similar argument for special singular points cl < C2 < c3 etc. f a E int Q*, it develops that the special singular

7C'

.+.

points are finite in number on each closed subinterval of Q*. Thus, the series of special singular points cuts Q* into non-overlapping intervals Q' (closed at the cuts), each with perfect (indirect) communication and no internal special singular points. Q' contains either no internal singular points or just one kind of internal shunt : for if Q' contains two opposing internal shunts cr < c2, then there is a maximal shunt ct S c3 < c2 of the same kind as cl and a minimal shunt c3 < c4 S cs of the same kind as c2, and either (c3, c4) is a special interval, violating the choice of Q' or (c3, c4) contains singular points, violating the choice of c3 or c4. Consider, for the next step, the exit time e = inf (t: x (t) D' = [x (t, we) : t z 0, Pa = P. : a E Q'] is, itself, a diffusion (see 3.9) ; moreover, a particle starting in an interval Q' = Qi performs the correvii

sponding diffusion D' = D1, up to the exit time e = el (S + oo); if e, < + oo, it enters into a new interval Q' = Q2 (never coming back .r.

°'4C

to Qj - Q2 at all), performing the corresponding diffusion DZ up to the exit time e2 (S + oo), etc. D* is thus decomposed into the simpler diffusions

D. Up to a change of the sign of the scale on the intervals Q' without internal left singular points, 9 separate cases arise:

3d) . 3g)

0

---i

3 b) ---

3 c) 0-a

3e)

3f) ---0

rat

3 a) 0-0

3h)

i

3i)

ii

and -o mark the end points not belonging to Q', .- and the end points belonging to Q* - (K- n K+) , and j the end points belonging to Q' n (K- n K+), and, in each case, Q' has perfect (indirect) communication and no internal left singular points, so that if inf Q' < a 0, i.e., direct comin which

munication prevails on int Q*; indeed, if a E intQ', then 0 < Pb (ma < + co)

for some b < a < c, because, if not, then a is a right shunt, i.e., Pa (ma+= 0) = Pa (ina- = + oo) =1, and Pa_ (ma < -I- oo) = 0 would prohibit

communication between Q' n (b < a) and Q' n (b > a) ; now a simple HEINE-BOREL covering completes the proof.

In the complicated bits below, case 3 g) will often be singled out, the other cases being left to the reader.

94

3. The general f-dimensional diffusion

3.6. Green operators and the space D

Given a diffusion D on an interval Q with endpoints 0 and 1, let B (Q) be the space of (real) BOREL functions / defined on Q U oo such

that /(oo) =0 I1/I1 < +00,

1a)

1b)

and, introducing the Green operators +00

a > 0,

Ga : / E B(Q) -> E. If c °t /(xt) dt

2)

0

let us show that u = G.1 satisfies 3)

u E B (Q)

4a) 4b)

u(a-) = u(a)

PQ[ma- = 0] = I

u (a+) = u (a)

Pa [nta+ = 0] = 1

5 a)

u(a-) exists

bim Ea_ (e-m') = I

5 b)

u(a+) exists

lbi m

6 a)

u (a-) = [1 - k, (a)] u (a)

urn Pb [ma_ < +001

u(a+) = [1 - k- (a)] u(a)

k+(a) = Pa- [ma = + 00] lim Pb[n ,, < +oo] = 1,

6b)

i

= 1,

.

a

k- (a) = P 1. [ ma = + 00] 7a)

u(1-)=0

1*Q,

bmPb[ml_0

and define 2)

CAS' = limo 45.,

III

fin'

where & is to be applied to the class D (W) of functions u E D such u that (W u = Jim ($. u exists pointwise and belongs to D and sup 40 &>O < + oo; it is to be proved that 3)

i.e., 4a) 4b)

'_

D(W) = D((s)) E [u (x.)] - u W u =1im .;0

4b) is to be compared to 3.7.11); the use of ( W. FELLER [5].

u E D(O).

was suggested by

3.8. Generators continued

101

I-.

DYNKIN's formula 3.7.7) applied to u E D((1) and m = e > 0 implies that (SS' = CSS on D (0) C D ((SS') because

1 ±m 03, u = im e r E. I f (0 u) (xr) dt] = O u

5 a)

8

8

0

and 5 b)

110. U11 s 110 U11,

and, to complete the proof, it is enough to show that D (t?) C D (0). But, if u E D (O') , then 6) Ga(a - Oi)u

=aGau - limGa (0,u a G. u - lim e 'E. I f Cal [u (x:+.) - u (xe)) dt) 0+00

f

`40

= u,

41

.

u (xr) dt - f er u(x) dt

'o$

=aGau -lime' E. {(e1 - 1) +-

]

.,C

and, since (a - (33') u E D, u = G. (a - ($Y) u E D (0) as desired. (SS' = CS can be used to give a new proof of the fact, noted at the end of 3.7, that 03 is a local operator. Because the path cannot cross up over a left singular point or down over a right singular point, it suffices to show that

i o t-I P. (mb < t) = 0

7)

a < b,

as will now be done using the method of D. B. RAY [2].

Given a 0, if / E D is 0 to the left of b and positive elsewhere and if a > 0, then y = (G`f) (b) > 0,

0 < u - y-' G,/ = E.(e-"m') E t and solves (Ju = au on (a, b), and using DYNKIN'S formula with in = n A M. A M. A Mb, it is found

that, if a < c < b, then 1 z EQ[u (xm)] - u (c) = E, f ((t3u) (xe) dt)

au(a) E, (m) tau(a)

?

as n t +oo.] 3.9. Stopped diffusion

We will prove that i/ Q' is a subinterval of Q. closed in Q. and if e is the exit time inf (t : x (t) ( Q' ), then the stopped motion D' = [x (t /\ e) :

t Z 0, Pa : a E Q'] is itself a diffusion; this fact was used in 3.5. Consider the standard description of the stopped motion with sample paths w' : t - Q', universal BOREL algebra B', MARKOV times m' = m' (w*), subalgebras Bm. + , and probabilities 1) Pa(B')=PP(B) aEQ', Bi-- (w: w*, EB'), B'EB';

the problem is to show that, if B' E Bm.+, then 2)

P: [B', x (1 + m') E d b] = E: [B', P. (m.) (x (t) E d b)] .

But, if m' = m' (w') is a MARKOV time for the stopped motion (in standard description), then ignorning paths not starting from Q', m (w) - m' (w;) is a MARKOV time for the original motion because 3) (W: in < t) = (W: m' (w,*) < t) B'=(w':m' 0]=1, as will now be proved. Given u E D (@), if f = (1 - G3) u, then

fED

9a) r

u = G, l = E.

9b)

fe-II(x,)dt + E. [e-e u (xe)]

,

and, if 10)

f' (a) = l (a) = (GI /) (a)

=0

a E Q' ,

Pa [e > 0) = I

a E Q%

Pa[e = 0] = I

a= oo,

* Use the fact that w E B' if and only if (w'); E B' for each I > W(e); see 3.2.10c).


104 then

t'ED',

11 a)

D' being the space D attached to D', and on Q', G' to = E: I T e-4 to (z) dtj

11 b)

= E. [Tie'/* [x (t /\ e)] d t l o

J

+m

=E. f e4/'(z1)di +E.I f eldt/'(xe)J l e

= E. f e-t t (zt) dt + E. [e-e (G1 /) (xe)] 0

=G11 = u.

8a) is clear from 11) ; as to 8b), 11 b) implies that on Q', 12)

CS'u'=(Gi-1)/'=G1/-/'=(b3u+/-/'=c5u P.[e>0]=1,

and 8b) follows. G is now seen to be local in the sense that, if A is the closure of a neighborhood B of a E Q:

it a is a left singular point ba c < a < b it a is non-singular, then 05 agrees on B with the generator (l' of the stopped motion PE: EA] e=inf(t:x(t)(A); this usage of local is to be compared with the usage in 3.7 and 3.8. Problem 1. What step of the proof that D' is a diffusion fails if e .ti

,,d

B = (b, a) = [a, b) = (c, b)

is a general MARKOV time? Give a concrete example.

[5) is false; indeed, if D is the standard Bxowlaian motion and t

..'

if f E C (R') is positive, then e = min (9: f / (x,) d s = 1) is a MARKOV 0

time, and if 5) held and m = I were < e, we should have e

t

e(wt)

t

1 = f t(x.) ds = f t(x.) ds + f ;(x (s. wi))ds = f /(x.)ds + 1, 0

which is absurd.]

0

0

0

105

4.1. A general view +.+

Problem 2. Given e = inf (1: x (t) ( Q'), show that, if P,(e < + oo) > 0 on Q', then the stopped conditional motion D' = (x (t A e) :I L 0, Pa(B I e
f u' ()m(d)=

31 b)

32)

J

[a. b)

[a, b)

u' (ln) m (1) = u+ (ln) - u (1) k (1) ,

u' () s+(d) =

J

J

[u(d) - u() k+ (d

(a, b) n H+

(a, b) n K+

)]

05 a u'. 0' is unambiguous because u* E D, 0 < m [a, b) (l S a < b < rr,) , and s+ (a) < s+ (b) (a < b) . A similar description of (55 holds in the general case; the expressions

r"'

become a bit complicated but nothing new happens and the proofs can be carried through using the special case explained above as a model and appealing to the splitting of 3.5 and the stopped motions of 3.9.

The reader may wish to skip the singular case at a first reading ; this means that he should read 4.2-4.7 and 5.1-5.9 and then come back and read 4.8-4.10, 5.10, and 5.11. 4.2. 0 as local differential operator: conservative non-singular case Consider the non-singular conservative case:

OSa51,

Pa(m,o=-{-00)=1 Pa(ma+=0)=1 Pa(ma_=0)=1

1)

2a) 2b)

0 0) ,

e- E j

j e - ( a) ( b t a) ,

e- (a) > - oo ( a < 1) ,

e- E J ,

It6JMcKean, Diffusion processes

s

4. Generators

114 and

a

e` (a-) = e+(a-) = e- (a) e-(a+) = e+(a+) = e+ (a)

25 a) 'N+

25b)

a > 0,

and it follows that the interval functions 23) induce the same nonpositive BOREL measure e - (d b) = e+ (d b) . Because of e3 E 1 and

e(0) = e(0+) = e(1) = e(1-) = 0, 26 a)

a (b) - e (0) ? et (b) [s (b) - s (0)] ? et () [s (b) - s (0)]

26b)

e(1) - e(b) S ei (b) [s(1) - s(b)] 5 of (}) [s(1) - s(b)]

bS b?

and using this, it develops that 1

27)

0= f Gdm.

34b)

[b, 11

But now 34c)

u[0,b) Z

fGdm CO. b)

for the same reasons, and adding 34b) and 34c), it appears that

e=u[o.b)+u[b,1)? f Gdm+ f Gdm= fGdm=e, [b,1]

[0, b)

0

i. e.,

u(b,1]= f Gdm

35)

0SbS1,

(b, 1]

leading at once to 30). Problem 1. Consider the splitting of the standard BROwxian motion

II'-

into independent . = (t: x (t) = 0), excursions en : n Z 1, and signs en : n z 1 (see 2-9); the problem is to compute the scale s (di) and the speed measure m (d i) for the skew BRowNian motion made up from

standard BRowxian 8 and en : n ? 4 and skew signs e, = ± 1, g«

4. Generators

116

P.[e,= + 1] = a, n z 1, 0 < a < 1. (a = 4 for the standard BRowxian motion .)

-a)-1 =a-1

-a) de

m(de) = 2(1

m(de) = 2a de

for for

S 0, z 0;

in fact, if 8.: n z I is a (BOREL) numbering of the open intervals of the complement of 8 and if m*1, m+1 are the skew BROwNian pas-

sage times to ± 1, then s(+1) -s(O) = Po[m*1 < m;r] =

s(+1)-s(-1) -

na1

Po[e = -1, m-1Am+1E 8n]

_ (1 -a) I Po[m-1Am+1E8.] = 1 -a, Sat

s ) - $(s (o) = 1

a and for the rest, it is sufficient to point out that the skew and standard BROwNian motions

or, what is the same, tun

agree up to the passage time to 0.] Problem 2. Consider a particle moving among the integers Z1 accordi.+

ing to the rule : wait at x (0) =I E Z1 for an exponential holding time e1 > 0, at time e1- make a unit step (± 1) with probabilities depending upon 1:

0 0, etc. W. FELLER [12) pointed out that the generator (J of such a birth and death process can be expressed in terms of a distance (scale) and positive weights (speed measure) :

°.'

(Su=

u+ - u-

m

where u - (1) =

u (1) - u (1- 1) s (1) - s (1- 1)

aye

-f-

1+1)-u(1) u'(1) _ Us(1+1)-s(l)

a > b) is a solution of

p+(d) = 0 p(,1-)=1

i.e.,

1

p+()=0 (0 < < ,) for each j< 1.]

0< 0 and mo+ < + oo, and the desired modification is [x' (1); t 0, P.], where x' (t) = x (t)

=0

t < mo A mo+ t?moAmo,

4. Generators

126

D is now modified at I also, so as to have I E Q and E,(e-m,-) = limE,-(e

4a)

btl

4b)

E,-(e

m,) =

omEb(e-m,-);

this modification will be understood below.

Given u E D (0). if PO (mo. < + oo) = 0, then (63 u) (0) = - x (0) u (0)

5 a)

x (O) = Eo (m.)-1, *

while, if PO (mo. = 0) = i and if a dependence c, u (0) + c$ (Ou) (0) = 0 prevails for each choice of u E D (0), then [c1 + c2 a) (Gs f) (0) = C2/(O) for each /ED and a > 0, and choosing f (0) = 0 < f (b) (b> o), (G` f) (0) > 0 implies cl = cs = 0; in brief, if PO (mo, = 0) = 1, then u (0) and (()u) (0) are independent for u E D ((s).

But if Po [mo+ = 0] = 1, then, as will be proved below, u+ (0) = am u(b)-u(0) s (b) - s (o)

6)

=u =

(0

+) **

exists for each u E D ((si) , and a dependence holds between u (0), u+ (0), and ((33u)(0):

(0 u) (0) m (0) = u+ (0) - u (0) k (0)

5 b)

u E D (0),

where 7a)

k (O) -

of

= + oo) po (m s (b) - s (o)

b

M (O) = boo E.(m A 7b) s(b) - s (o) complementing 4.3.1) :

***

(($ u) (e) m (d ) = u+ (d i) - u (E) k (d i)

0 < < 1.

Beginning with the proof of 6), if f E D is such that /(a) = 0 (a > j) then u = G, / E D (0) satisfies u (a) = E. (e-m}) u (j) (a S 1). Because u (J) > 0, 0 S u (a) E t (0 < a S J) ; because P0(mo+ = 0) = 1, u (0 +) = u (0) > 0; and, using 0 u (a) = u (a) (a < 1-) to deduce from 0 S u E T that O S U+ E T to the left of 1, it follows that }

k(0,.*]+m(0.f)Su(0)-1fuk(dE)+

8)

10+

e

EO(mo,) > o;

o+

S u(0)-1[u+(1) - u+(0+)] a-l - o if e - +00.

u+(0) - 0 if s(0) = -00. k(o) = m(0) = 0 if s(0) = -00.

1

1

< +oo.

4.5. 0 at an isolated singular point

127

8) implies the convergence of the integral

f

u+(j)

>j;

f 0+

o+

- u+(0+)

for each u E D (Gi) , proving the existence of u+ (0 +) ; as to the rest of 6), either s (0) > - oo, in which case u+

b

u

(0) = io s (b) - s (o) =

lizn blo [s (b) - s (o) ]-10f u+ s (d

)

= u` (0+),# or s (0)

oo,

in which case u+ (0) = 0, u + (0 +) =0 owing to

u () - u(0) = f u+ d s , and k (O) = m (0) = 0 in agreement with S b). 0+

Given s (O) > - oo, DYNKIN'S formula,

10)

P. (m,. .--

(0 u) (0) = lim b4o

9)

.

Fo ( A moo))- u (0)

(Gi u) (0) Eo (m, A moo) P9 (m& < + s (b) - s (0)

- u(b)-u(0) - PO (m1 0, g.() = I 12b) g' (0) > 0, 90+(0) = 0, go (}) = I and prove that g.(g') is the smallest (greatest) positive increasing solution of 11) subject to g (J) = 1. 12a)

[g. = B.IB. (J) and

g' =

B. 1B.

(1)

B. = g1 gs (0) - g1 (0) ga

B' = g, g' (0) - gi (0) g2

4.6. Solving 09 u = a u

133

,..

solve 11) and 12), 0 S g, S g*, and if 0 S g E f is another solution of 11) with g(]) = 1, then the obvious independence of go and g' per°.R

mits the choice of c. and c' such that g = cog. + c'g'. go S g S g' is now clear from g, S g' and g(0)Ig (0) = c' ! 0, g+(0)lg. (0) = c. ? 0, g(]) = 1 = c. + Co.] Problem 2. Compute the increasing and decreasing solutions g, and

2IbI-''D' (-1 0, t > 0) = 1 for d z 2 (see 2.7). [s(dr) = r'-d dr,

m(dr) = 2rd-' dr,

s(+oo) Z + oo (d $ 2), +00

+00

f sdm= f mds=+oo.] I

1

Problem 4. Compute the increasing and decreasing solutions gl and g. and the WRONSKIan for the BESSEL operator 6' = 2 (--am + y a )

(r > 0), and use gt to prove that lim

E,

(e-..r m.)

=

Y+ 2

Y> O,

What is the meaning of the disappearance of the & ?I

gs(a, )=K0(f i), gigs - gigs =1, EI (e

m.)

= Ko(i)/Ko(E 2a

,

K0()

* F. B. HILDEBRAND [1; 161-167) is helpful. 1 See problem 6.8.4 for a related result.

-lg $ (" f O).)

4. Generators

134

Problem 5. Give a new proof of D. RAY'S estimate glo

t-"Pa(mb St)=0

nZ1

a 0.

= s (db)lim atb s(b) - s(a)

.=+

'ti

< + oo) ] is differential, PO (l m M. = Inb,

Pojalumma=mbj=1 (OSbS1).

so ms, can be expressed as

i

05 b 5 1

f l p ([0, b) x dl)

VII

1(b) -}-

4)

+00

(0'.00) .ti

relative to the conditional law PO (B I m, < + co), where t E C [0, 1] is independent of the sample path and p (d b X dl) is a POISSON mea-

sure.' Taking p ((a, b] x + oo) to be independent of 4), differential, e-"

Iii

and PoissoN distributed with mean n ((a, b) x + oo) - -1g Pa [mb < + oo] VII

(0 S a < b S 1) as in 3 a), mb can now be expressed as 5) t (b) + f 1 p ([0, b) x dl) + oo x p ((O, b] x+ co) 0;5 b S 1 (e. +oo)

.ova...

relative to the unconditional law P0. t and n have now to be identified with the expressions in 2) and 3). Given a S b inside some interval of Q', Pa(mb z 1) is a concave function of s (a) (see problem 4.4.2), permitting the definition of

n' (b x dl) as in

a t b, 1 > 0 0 S [s (b) - s (a)]-' Pa (mb ? 1) t n' (b x [l, + co]) if l > 0 is not a jump of n' (b x dl). Using the solution g, = E.(e-° m') of 0 S b < 1, 7) (0* g,) (b) = a g, (b) 6)

VII

See the explanation below 4.8.2).

' See the note placed at the end of this section.

4.10. Passage times

145

it is immediate from +Co

a f a-al n' (b x [1, + oo]) dl

8)

0

+00

=a1 is f e°I[s(b) - s(a)]"1 P0(mb 0

f [1

= lim [s (b) - s (a)]-' atb

=

(0.+ol gi (b) -1

gs

1) dl

- e-°I] Pa(mb E dl)

(b - s (a )

g (b)

that

gi(b)

- e.--I] n' (b x dl) _ g, (b)

f [1

9)

ga(b)

(0. +ao1

and

f s (d ,) n' (f x + oo) = l +m lg gi (a) = -1gPa [mb < -f - 00] .

ti.

10)

[a. b)

v^,

But now, using 5) to check G_a[I(b)-l(a)]fo f '[1-e °']n([a,b)xdl)-n((a. blx+co)

11

05a0

((0, t] x dl) < + oo

o+

I~>

(see the note placed at the end of 1.8 for the definition of PoissoN measure).

P. LEvy's formula [1: 173 -1801: 4) E[e-a(P(':)-P((i))] +00

= exp[-a[c(ts) - c (t,)) - f (1 o+

- e-°t) n((t,, t=] x dl)] a>0, tl St=

is immediate from 1), and 3) is immediate from the fact that the left

side of 4) is > 0. The following proof of 1) is adapted from K. Iro [1]. Given p as in 2), 5) c (1) - p (1) - f l p ([0, t] x dl) t? 0 o+

is continuous, non-negative, increasing, and differential, and, as such,

it has to be independent of the sample path as will now be proved in the case I = 1. Because

rma E(lRt)

I.L. = [c(.)_c('')] Al, k S n, n? 1

S man E[1 - el-+]

S akZnx E [1

-

P(k p(!n1 )1] n)

e-L

f_P(!)

-e

D(ri), E

ale

=ma E x°)

S E[e-P(1)]-l max [E(e'()) - E(e , ale

6)

_

'(k"1)1-1

(7k

))1

4.10. Passage times

147

tends to 0 as n t + oo, it follows that if P < a < y, then 7)

- y Lm Inf

e

Z E(fk)

a4 400,5

= Jim sup H e-YE(I.k)

n1+oo ksn lim sup II [1 - a E (lnk)] nt+oo ksn lim sup 17 E (e-afwk)

nt+oo ksn

a £1Kt

=1im supE(e ":So +

nt+00

Lit,

-a £

lim infE(e k, nt+oo = lim inf jj E (e-ah*k) ..a

nt+oo ksn S lim inf II [1 - P E (lnk)] nt+oo ksn

: =e wt+ooksw -plimeup

and, letting

t a and y f a, one finds E[e ac(i)] = e

8)

-a llm

£ E(ink)

a+ +00 ksw

0,

III

which is impossible unless c(1) is independent of the sample path. Coming to the proof that P ([t1, t2) X [1 ,1s)) (t1 < ts, 0 < 11 < 12) is PolssoN distributed, if k 9 lnk = 0 or I according as (f n , n) x [ll, 14)) = 0 or z 1, 10)

'Z1

then, much as in 6), [1 - e-a] msax P (lnk = 1)

SmaxE[1 - e-aInk] ksn

k

S E [e-ap(')]-' max (E le-2

,- E t n) k

'.S

ksn tends to 0 as n t + oo, and as in 7), one finds -a Sl f 11) E le ks.

[

nt+oo

J

- urn H 5 n

= lim 17 [1 - (1 - e-a) P(lnk = 1)] nt+oo ksn

..r

=expt-(1-e-a`)n'imakYP(lnk=1)] a>0, 0

0

completing the proof in case tl = 0 and tq = 1. 1o*

4. Generators

148

+-'

Because of the meaning of ,p, it is clear that it is differential in t, and, to complete the proof that , is POISSON measure, it suffices to check the independence of m+ = ,p((O, 1) x (11,1=]) and I 1 p ([0,1) x di)

(0 < li < 10-

(0.111

Consider for this purpose,

l_k= jl,p([kn 1

12a)

,

kk) xdl)

(0.11]

12b)

1

,) x(11,10,

let f denote a (non-stochastic) subset of 1, 2, ... , n, and let f denote the (stochastic) set of integers k S n such that lnk > 0.

Because the event t = m can be expressed as the sum

UU (lnk=tk, kin), tEt I

where T_ is the class of all sets t = (t1, ..., tk,

of non-negative inte-

gers such that tk > 0 or = 0 according as k E f or not 13 a)

tt + is + ... + to = m,

13b)

it follows that

E[1+=m,e

14)

kEi

tk,kSn,e

E1

tEZ

kEf

I7P(l1'9*k=1k)17E[lnk=0,e kEi

tESC &Ef

_II1P i

(114k

tESC &Ef

I7 E

W

x

= tk) H P (ink = 0)

0. e

lift

aik]

r

17 E [i

kEf

=o

ear"k

II P (twk = 0) kEi

II P(t.+t= 0)HP(lwk=0) JI E[1..

I

P (1+,L. = I't' k S n) X

kEi

17r.0-° >a-1 S'i .

tESC

0.e = O] ,re

=r

kV

s`,

kEf

kEf

But the sets f and f figuring in 14) contain at most m members; thus, 15)

lim HE [a-a1k I Ink = 01 = 1,

nt40D kEf

4.11. Eigen-differential expansions

149

and as n t + oo, 14) goes over into E[D+ = m, e-aQ-] t X P (ln. = tx, k S n) E [e-` 0- I m« = 0]

16)

=n

= P[Z1+ = m] E(e-0- I n., = 0],

completing the proof. 4.11. Eigen-differential expansions for Green functions and transition densities

Given a non-singular diffusion modified as in 4.5. let Q' be the 1, closed at I if E0+ unit interval closed at 0 if E0 E, (e-'"'-) = El_ (e-m') = 1, and open otherwise, introduce the stopping time m = min (t : x (t) ( Q'), and let us use the idea of eigendifferential expansion to construct transition densities p(t, a, b) such that P6[x(t) E db, t < m] = p(t, a, b) m(db) i)

(t, a,b)E(0,+oo)xQ'xQ', where 2)

0 5 p (t, a, b) = p (t, b, a) is continuous on (0, + oo) X Q' X Q'

3)

p(t,a,b)=fp(s,a,c)p(t-s,c,b)m(dc) t>s>0, Q.

4a)

4b)

ap(t,a,b) _ e

a,bEQ'

p(i,a,b)*

if E0(e-mo+) = 0, b E Q p(t,1-,b)=0 if EI(e-m'-)=0, p+s (t, 0+, b) = 0 it E0 (e-mo+) = 1, E0+ (e-m,) = 0, b E Q *s ps (t, 1-, b) = 0 if El (e-m'-) = 1, Ei_ (e-ml) = 0, bEQ', p (t, 0+, b) = 0

bEQ.

4c)

W being the local differential operator described at the end of 4.7. The method is outlined only; see H. P. McKEAN, JR. [2] for the proof in a special case, S. KARLIN and J. McGREGOR [1] for a beautiful

method which can be adapted to the present needs, and H. WEYL [1]

for the classical eigen-differential expansion for STURM-LIOUVILLE operators. As a general source of information about HILBERT space, B. Sz-NAGY [1] is suggested.

* 09 is applied to a or to b. ps(t, a, b)

1im 1+4

p(t. , b) - p(t, a, b) s (s) - s (a)

;_,

p (t, a. b) - p(t, a, Pi (t, a, b) = 1im s(b) - s(e) ?+&

etc.

4. Generators

150

Given a > 0, let g, and g2 be the solutions of 5 a)

(0Y g,) (b) = a g, (b)

0Sb0,

(a,b)EQ'XQ',

and inverting 29 a) as a LAPLACE transform, 1), 2), 3), and 4) are found

to hold for the kernel o+

29b)

p(t, a, b) = f ev' e(y, a) f(dy) e(y, b) -00

(t, a, b) E (0, + oo) x Q' x Q' ;

4. Generators

154 °».

the proof of 3) is based upon the estimates 0+ _w.

f (a-y)-'f11(dy)=B-1g1(j)g2(j)0.

33 b)

34a)

u (t, a)

.r^.

at u (t, a) =

.Vi

t > 0,

33 a)

1,

Eo+ (e-m-) = 0

If 0 and I are exit or entrance, then (see table 4.6.1), the trace ...

i r (GQ) - f G (, ) m (d ) is < + oo, with the result that G. is come. pact and f (dy) is concentrated on a series of simple eigenvalues 35)

0 ? Y1 Z Ys >

etc.

1 > - oo. j - oo,nagI yw

If s (1) - s (0) and m (0, 1) are both < + oo, then -2 36)

lim -Yw nt+ao X2112

= [!}1s(db) m(db) 0

,


155

I

being the HELLINGER integral inf E V s (Qn) m (Q,J

J V s (d b) m (d b)

Q, )Q) (see H. P. McKEAN, JR. and D. B. RAY (1], and, for the L

nZI (U

classical case, COURANT and HILBERT [1]). S. KARLIN and J. McGREGOR [1, 2, 4]

have developed similar

eigendifferential expansions in connection with the birth and death processes described in problem 4.2.2.

Problem 1. Compute the FOURIER transform for the standard BROwxian motion. [e, (y, ) = cos }I21 y I

sin 21y1

e2 (y,

,

21yI

l I I (d Y) =

dy 27r y2 I -Y1

/12 = /21 - - 0,

,

is : (d Y) =

21y1dy .I 2n

Problem 2. Express the FOURIER transform for the BESSEL motion

attached to (3' = 2 ( b + d b 1 b) (d Z 2) in the special form +00 o+ h= r ei(Y,')/II(dy)f .1

-00

o

-°=

choosing for e, the solution of (CS' el) (b) = y e, (b) (0 < b < + oo) ,

eI (0) = 1, e1 (0) = 0. Evaluate p (t, a, b) using 29b) and A. ERDELYI [1: 186 (39)], and check it against 2.7.4). 2-

b) = 2

I r (Z) (V2 y I b) 12 J d

z

J

111(dy)

= 2 2 P(±)-2 l 1-2lYl2 -I dy,

2

01+61

p(t, a, b) =

s, e

2t(a

b)

'

Id

'

...

.

I (f 5 J

b),

and

ja b )

Problem 3. Given a conservative non-singular diffusion, define b.

pI = s, III

po

rf

P. (b) = f s(di)1.1 p"-s (n) m(d71)

n z 2;

the problem is to show that p (1, t, ,q) is an analytic function of 71 stn

(0 < 7) < 1) in the sense that for each t > 0 and each 0 < < 1 , with coeffinZo cients

p (1, t, 77) can be expanded as a power series Z c

/

t

C2n = @on ph(t, S , +00

c2n+1 ='i '-'

tt4)

P)3 (t, s, 113)

4. Generators

156

[Because el

= naoy"pzn, e2 = nZ0Ynpzn-i (p-i = 0), and ,

0

n

j ereIYI' Ie(y, E) f(dy) (pzn(rl),psn-i(rl)) -c G+

J eveI yIIn e°Ylrl 2[lii(dy)

na0 -,0

+ lzz(dy))

-2

0; the proof of s (1) _ +oo is the same, and

now a change of scale brings 0' into the form (' = 2 D2 - ca (Q = R1, ce = c4/c5

0) ]

Problem 5. Give a proof that p (t, a, b) > 0

(t, a, b) E (0, +oo) xQ' xQ

using the eigen-differential expansion 29b) and the CHAPMAN-KOLMOGOROV identity 3). o+

[p (t, a, a)

f eVte(y,a)f(dy)e(y,a)>0

(1>0,0 0 and a, b E Q' such that p (s, a, b) = 0, the CHAPMANKOLMOGOROV identity

(t<s)

p(s,a,b)= f p(s - t, a,c)p(t,c,b)m(dc) Q. III

coupled with p (s - t, a, a) > 0 implies p (t, a, b) = 0 (t < s). But

moo'

p (t) = p (t, a, b) is a LAPLACE transform, and, as such, has to be = 0 +00

(t > 0), contradicting f

p (t) dt = G(a, b) > 0.]

0

Problem 6 after KARLIN and McGREGOR [5]. Consider the space

of points c' E (0, 1)" (n ? 2) such that c1 < c, < etc. and, for t > 0, define e(t, a', b') to be the determinant p(t,a,,b") p(t,a,,b1) p(t,a1,b,) p (t, aa, b1)

p (t, a,, b,)

. .

.

P (t, a,, b0)

I p (t, a., b1)

p (t, a,., b,)

. .

.

p (t, a,.,

I

Consider also the joint sample path x (t) = (x1 (t), x, (t), etc.) E [0, 110 of n independent particles starting at a', let Pa. be the corresponding law, and let m be the crossing time sup (t : x1 (s) < x, (s) < etc., s < t). .41

Given Be = B1 x B. x etc. C (0, 1)" such that B1 lies to the left of B,, B. to the left of B,, etc.,

P0.[m>t,x(t)EBe] = f e(t,a',b')m(db'), Be

where m (d b') = m (d b1) m (d b,) etc. Give a direct probabilistic proof of

this using

Pa. [m>t,x(t)EB'] = Z (o) P.. [In > 1, x(t) E o Be] 0

= S' (o) (Pa. [x (t) E o Be] - Ps. [m < t, x(t) E o Be]) 0

=

f

0

Be

where o is a permutation of 1, 2, ..., n, (o) = f 1 according as o is even

or odd, and oB' = Bo1 x B., x etc., and deduce that e(t, a', b') > 0 for each t > 0 and each choice of a' and be (see KARLIN and McGREGOR [5]

for additional information and G. POLYA [1] for the original result in this direction). [Because Pa. [m > t, x (t) E oB'] = 0 unless o = 1, Pa. [m > t,

x(t) E Be] = f ti m(db') - f' (o) PQ[m < t, x(t) E oB'] is clear, and it Be

0

0

is also clear that the joint motion starts afresh at time in. But a per-


159

r-,

mutation that interchanges two of the coinciding particles is odd and has no effect on the future motion, so (o) Pa. [m < t, x (t) E o B'] = 0, 0

Pa.[m > t, x(t) E B'] = f em(db')

leaving

Bte)

't7

as desired. e (t, a', b') z 0 is now clear, and an application of the CHAPMAN-KOLMOGOROV identity e (t, a', b') = f e (t - s, a', c') e (s, c',b') x

m(dc') to the case a' = b' plus the fact that e(t, a', b') = e(t, b', a') implies e(t, a', a') > 0 for each t > 0 (if not, then e(1/2, a', ...) - 0 and Pa. [m > t12] = 0 which is not possible). Given (1, a', b') such t=1

now

that e(t, a', b') = 0, a second application of the CHAPMAN-KOLMOGOROV identity implies e(t - s, a', a') e(s, a', b') = 0 (s < t), i.e., e (s, a*, b') = 0 (s < t), and so a (s, a', b') = 0 (s > 0) as the LAPLACE transform of a (signed) measure. But for t > 0, 0 < e (t , . , .) near the diagonal a' = b' , and using the obvious HEINE-BOREL covering to select a chain ci, c2, etc. of l < + oo points leading from a' to b' such that 0 < e (t, ci , c2) , e (t , c2, c;), etc., a final application of the C,,

CHAPMAN-KOLMOGOROV identity implies e(lt, a', b') > 0, contradictIOU

ing e(., a', b') = 0 and completing the proof.] Problem 7. Check that

t-' G (1-1,) [p (t, E,

)

1>0.

p (t.

is convex in t, and so

+Cc

f. OD

af a-I" p(t,

dt>p(a f e'°rtdt, ,e)=p(a-'. it.

0

)]

0 C.'

Problem 8. D is transient if and only if +00

.--.

limG(a,b)= f p(t,a,b)dt 0] = I for t > 0, it appears that 05' is the differential operator based on the scale b

a h (et)2

1. o.,

i. 0.,

t t + oo]

t t + oo] = 0 or I

+00 OHO

according as f h (t)d a-h (9)'J2 d t converges or diverges,

or, what interests us, that 14)

Po[r(t) > JFI h(t)

i. o.,

t t +oo]

+m

_

>Z.

= r P. (r (1) E d ) PE [r (t) >'t + 1 h (t + 1)

i. o.

t t + oo]

0 00

= 0 or I according as f hd e-0/2-L' converges or diverges. NIA

When d = 1, this is KOLMOGOROV's test, and when d Z 2, it is the DVORETSKY-ERDOS test (see A. DVORETSKY and P. ERDOS [1]). 11*

5. Time changes and killing

164

The same method shows that if 0 < h(t)1.0 as t t + oo, then 15) P0[r(t) < l/i h (t) i. o., t t + oo] = 0 or I according as +00

r

ilg hl-, d:

(d

_ 2) converges or diverges.

+00

f

kd-sdi

(d >_

Because PO is unchanged by the transformation x (t) --1. t x (1 /1), it .Z'

follo.,., from 15) that if 0 < h (t) + 0 as t 4 0, then 16) PO [r (t) < ft h (t) i. o. I f 01 = 0 or I according as

f Jlghj-l a

(d = 2) converges or diverges

.Z"

0+

f hd-a

dt (d

3)

acv

(see A. DVORETSKY and P. ERDOS [1] for the case d Z 3 and F. SPIT-

ZER [1] for the case d = 2; another method leading to a cruder result will be found in problem 7.11.2). Problem 1. Give a complete proof of 15).

5. Time changes and killing 6.1 Construction of sample paths: a general view

Given a 1-dimensional diffusion, singular or not, a complete description of its generator 05 is now before us. 0 coincides on its domain

i..

with a differential operator ' of degree S 2 expressed in terms of invariants (scale, speed measure, etc.) via the formulas 4.1.8) [or 4.1.31,32,33)] and each invariant has a simple probabilistic meaning embodied in the formulas 4.1.7) [or 4.1.22, 23b, 23c, and 26)]. But the actual situation is even better, to wit, each choice of invariants specifies a differential operator CS' via the formulas 4.1.8) (or 4.1.31, 32, and 33)], and (aS' generates a non-singular (or singular) diffusion, the original invariants being expressible in terms of this

hop

end

,,!

diffusion via the formulas 4.1.7) [or 4.1.22, 23 b, 23 c, and 26)); in brief, a per/ect correspondence is obtained between differential operators (53' on the one hand and non-singular or singular diffusions on the other. The proof occupies the rest of the present chapter, but first a rough explanation of the leading ideas (time substitution, annihilation, shunt.CS

ing) in series of special but typical examples. Beginning with the simplest example, if D is a standard BROwxian motion with sample paths t - > x (t) and if a > 0 is constant, then a

$.1. Construction of sample paths: a general view

165

non-standard description of the diffusion associated with W IN = J a2 u" "''

is the scaled standard BRowNian motion x' = a x, or, what is the same in distribution, x' (t) = x (a' t) : t a 0. .fl

Now suppose 0 < a is continuous but not constant. Using the first prescription, the suggestion is that x' will be like x while it is near x' e., x' (t) = a + f a [x' (s)] d x

1)

a = x' (0) ;

0

this is the idea of K. ITO {2J. But to deal with the general non-singular U)'

diffusion, the second prescription is better; this suggests that x' is a standard BROwNian motion run with a new clock t' = t' (t) that grows like a' (6) t while x' is near 6, i. e., dt' = a2 (z') dt, 2a) or, what is the same, 2b) dt = o-'-(z') dt = a-2[x(t')] dt'. 2b) states that to is the inverse function f-' of the additive functional t

r

f (t) = f o-2 [x (s)] d s, and noting that f 9-2 (x) d s = f t (t, 6) in (d

),

0

0

°.a

where t is the standard BxowNian local time and m (d ) = 2a'2 (E) d is the speed measure of 03', it is now just a small jump to conjecture that a (non-standard) description of the diffusion on Q = R' associated with 3)

m(d)

is X* (t) = x (f-1)

4 a)

with

f = f t dm;

4b) see 5.2 for the proof.

Because the time substitution t - > f -1 depends upon the BROWNian

era.

path, it is often called a stochastic clock; the idea occurs in G. HUNT [2(2)] and a hint of it is contained in P. LEVY [3: 276]. H. TROTTER suggested the possibility to us. VOLKONSKII [1] obtained such time substitutions but did not find the local time integral for f ; see also 8.3. A second instructive example is that of the non-singular diffusion on (0, + oo) with generator 5a) 5 b)

u=

m(dl;)

m (0) (CJ' u) (0) = u* (0) ;


166

the method of time substitution still applies with

f = f t (t, ) m (d ) ,

6)

the time substitution of 2.11 for the reflecting Bxowxian motion being a special case.

.

0

Now let us explain how to construct the non-singular diffusion with killing associated with 7)

= u`(dE) -

k(dE)

III

on Q = R'. Suppose to begin with that k has a constant density x g 0 relative to m so that 4 = 0 - x with Q$ as in 3) ; then x* is just the motion x associated with 0, killed (i. e., sent off to the extra state co) after an exponential holding time tno, with distribution 8) P. [mao > t jx] = e-"e; .L'

this suggests that if x is continuous but not constant, then the chance that the dot particle, moving along the undotted path should not be annihilated before time t ought to be .N.

(l [1 - x[x(s)] ds] = e .Sr The time substitution 4) permits us to conclude that the local

9)

times

t (t, a) = lim

10)

boa

measure (s: a x(s) < b, s m(a, b)

s)

exist (5.4), so 9) can be expressed by means of a local time integral: i

f x[x(s)] ds

11)

= f t(t, )

0

and using this formula, the suggestion is that for a general killing measure k(db), the dot motion is identical in law to the undotted motion, annihilated (sent off to oo) at time m with conditional law

x]=e ftdm;

12)

the proof occupies 5.6. As the reader will easily believe, the relation between the motion on [0, -}- oo) with generator 5) and the motion associated with 13 a)

13 b)

". u =

u' (d ) - u (F) k (d F) m (d t)

m (0) (00 u) (0) = u* (0) - u (0) k (0)

* n is meant to suggest a continuous product.

5.2 Time changes : Q = R1

167

is just the same, except that the local time integral f t dk is used in 0-

(see 5.6; the elastic BROWNian motion of 2.3 is the simplest example of this). As an example exhibiting all the essential features of the conservative case, let (Y be the differential operator 14)

(®* u) (a) = J

't7

v,N

12)

u (a)

u(b)-u(a) =lim bja s(b) - s(a)

=0

"al

being the intervals of the complement of the standard '41

Q = U (ln)

CANTOR set K and s+ the shunt scale s+ (b) =

15)

(r,, A b - ln) + s+ ([0, b) fl K),

50 with the understanding that

(nZ1)

m^.

16a)

( 'uEC[0,11

16b)

uED(C?).

x' can be built up out of BROWNian motions on [ln, rn) with reflection at ln, hitched end to end, with a lag during the passage of the CANTOR

set so that for paths starting at 0 S a S 1, 17)

s+ ([a , b) f l K) = measure (s : x' (s) E K, s

t)

b = x' (I), t

M* .

The simplest method of doing this is to begin with a reflecting BROWNian

motion x on (0, + oo) stopped at I and to perform the time substitution t --> f -1 with

measure (s: l < x (s) < r'!

a<sRsb

s S m,,,) + s+([a, b) n K) VII

18) f (t) _

a.)

a = x(0), b = a V x(t), m,. = min (t: x = rn); the proof will be found in 5.10 and problems 5.10.1 and 2; see also 5.11 for a discussion of annihilation on the shunts. Creation instead of annihilation is introduced to obtain a sample path model of (3S' _ (1/2) D2 + x (x > 0) ; this model occupies 5.12 his 15.

I+.`

Â,

6.2. Time changes: Q = R1 Given a standard BROWNian motion on Q = R1 with scale s (db) = db and local times t = t (1, b) , let m (d b) be a speed measure on R1 and

let us prove that if f -1 is the inverse function of

i)

f(t) =f(t,w) = ft(t,b)m(db) Rl

t?0,

168


then [x (f -'), P.] is a (non-standard) description of the conservative diffusion with the BROwxian scale and speed measure m, i.e., with generator C`1

2)

'u = u+(db) m (d b)

Because t (t, b) E C ([0, + oo) x R') and vanishes outside minx (s) t (t1, b) (t= > t1) on a set of positive speed measure

and t (+ oo, .) = + oo (see problem 2.8.7), 4 a)

f (t:) > f (t1)

and

4b)

f (+ 00) an +oo,

and it follows that f -I has the same properties, esp., x (f -') is continuous.

Now let w' : t -- x' (t), Bi , B', and Pa (B') = P. [x (f -1) E B'] be the sample paths, BOREL algebras, and probabilities for the standard description of the motion D' = [x (f -1), P.], let w-1 : t -. x-' (1) denote the sample path x (f -1) , and let us prove that D' is simple MARKOV. Given B' E B, with indicator function e (w), e (w) = e (w-') is measurable

Because of GALMARINO'S lemma (see 3.2), it is enough to show

that if

f-'(t,u) <s

5b)

x(0, u) = x(0, v)

0

VII

5a)

S.

then 1^D

6)

a (u) = e (v).

But, if 0 S t, then 5 a) implies f -' (8, u) S s, 5 b) then implies f -1 (8, u) = f -' (6, v) because f -' (0) is a MARKOV time, and the resulting VII

x(6, u-1) = x(f-'(8, u), u) = x(f-1(8, v), s) = x(9, v-') 0 S t, 7) coupled with the fact that e' (w') is the indicator of a member of Bj implies 8)

as desired.

a (u) = e' (u-') = e' (v-1) = e (v)

5.2. Time changes: Q = R1

169

But now it follows that, with B', e', and e as above,

P:[B',x'(t+s)Edb]

9)

= E. [e* (w-1), x-1 (t + s) E db]

= E.[e(w), x(f-1(t + s)) E db] = E.[e(w), x(m + f-1 (s, wm)) E db] = E.[e (w), P' (x [f -1(s , wm), wm] E d b I B..)]

m = f-10)

'-'

= E.[e(w), Px(m)(x[f-1(s)] E db)] = E.[e' (w-1), PZ (t) (x' (s) E d b)] = E: [B*, PZ.(j)(x' (s) E db)],

1'1

and that finishes the proof of the simple MARKOV character. As to the strict MARxovian character, it is now enough (see 3.6) to show that the GREEN operators

Gal = E;

10)

e-ag /(x') dt]

= E. [°'e_h/(x_1) dt] 0

10+

map C (RI) into itself (see problems 2 and 3 below for an alternative proof).

Given a < b, if u = G;/ for / E C (R1) and if in = ms, then f-'(t + f(m)) = in + f-'(1, wm),

11)

l

r1 (m)

u(a) = EaI f e al /(x-I)dtl

12)

0

+ Ea [C-m)

f e at l[x(f-1 (tw), w)] dt

((m)

ii.

= Ea [ f e-a1/(x-1) dt + Ea[e-af(m)] u(b), 0

and letting b f a in 12), P.1 iln f (m) = f (0 +) = 0J = 1 implies u (a +) = u (a) ; the proof of u (a -) = u (a) is similar.

D' _ [x(1-'), P.] is now seen to be a conservative di//usion, and using 13 a)

ma (w-1) = min (t: x (f-1) = b) = f (nib)

together with the evaluation 13b)

EE[t(manmb,rl)]=G(E,r]) VI!


170

(see problem 1) to compute its hitting probabilities and mean exit times 14a)

pba() = P; (M; < ma) = Pe[f(mb) < f(ma)] a < < b

= Pe (Mb < ma) = b - a 14b)

eabR) = EE(maA mb) = Ee[f(ma) Af(mb)] b

Ee [f (ma A mb)] = Ee [

t (ma A mb,

(d rl)a

b

=fG

I

t,) in (d 21)

a

it appears that its scale is a positive multiple of the Brownian scale and its speed measure the reciprocal multiple of m, as desired. Problem 1. Use the formula for the standard BROwxian local time t tao

Ee f e

t (d t, rl) =

0

a>0

2

to give a direct proof of 13 b). (Et It (ma A mb, fl)] Ill

lim mEfl f e a't(dt,17)]

m=maAmb

0

+ao

e-a't(dt,71) - e-* In f e-a't(dt, j, w;,)J

= 1imEe

e-y2ale-of-Cat e- aalz(m)-.7I 1im EE ado

J

2 2a

IL

= a Ee[jx(m) -'7i = 2 [la - qI G

- 1 -riI]

b-a +I b - nj b-a

n).]

Problem 2. Prove that if m' (w') is a MARKOV time for the standard description of x (J-1), then m (w) = f -' (m' (w-1), w) is a MARKOV time '^"

...

for the standard BrowNian motion, and that if B' E Bm.4 , i.e., if B' n (m' < t) E B' (t > 0), then B =- (w: w-1 E B') E B, i.e., B n (m< t) E B1 (t Z 0) (use GALMARINO'S theorem; see 3.2). C)'

Problem 3. Give a direct proof of the strict MARxovian character of x (f -1) using the standard description, the result of problem 2, and the method used in 9).

S.3. Time changes: Q = (0, + °°)

171

5.3. Time changes: Q = [0, +oo) Given a speed measure m (d b) on Q C RI and standard BxowNian local times t, the statement of 5.1 was that if f -1 is the inverse function

f = f t dm, then the time change t

of

J-1 sends the standard

BROWNian motion into a (non-standard) description of the diffusion D'

with the BROWNian scale and speed measure in, i.e., with generator

_ u+(db)

1)

y-0

m(db)

The case Q = RI was justified; the cases 1

f e dm = -}-oo

.+.

2a)

Q= :" 0 '+0 0 )

0+ 1

2b)

Q=[0,+oo),

m(0,1) =+oo,

fEdm 01 is a (non-standard) description of

the diffusion D' with generator W. Given a BROWNian path starting in [0, -1-00), let us have a good +CO

look at f= f t dm. 0


172

Granting 7a)

P, [f (m0) < + 00] = 0 or 1 according as 1

Y> 0)

J' d m diverges or converges 0

'ti

and 7b)

Pr [f (m1) < + oo] = 0 or I according as

m[0,1)=+o0 or m[0,1)1 0),

the principal features off and f -1 (t) are as follows:

in case 2 a),

is continuous (1 < m0) , f (l1) < f (4) (t1 < is < m0) , f -' is continuous (t L> 0), f (f -1) = t (t ? 0), f -' < m0 f (m0) = + oo, (t z 0), and f (+ oo) = m0 (see diagram 1) ; f

in case 2 b) and 3 a) the same, except that f (m0) < + oo, f (m0+) _ + oo,

f(f-') = tnf(me), f -I(t) < mo (t < f(mo)), and f-'(t) = mo (t>f(mo)) (see diagram 2) ; coq

in case 3 b), f is continuous (1 ? 0), f (tl) < f (1$) (11 < t=) on each Bxowxian excursion x (1) : t E 8n (U ,3n = (t: xt > 0)) leading into

naI

r.'

(0, + oo), f is flat on each excursion x (t) : t E n (lJ 3n - (t : xi < 0)) 0Zl

leading into (-oo, 0), f-'(t) = f-'(t+) < +c (t (i Z 0), and x (f -1) is continuous (see diagram 3).

;? 0), f(f-') = t

7a) is proved as follows.

Given a Bxowxian path starting at l > > 0, m' = f (m,) is identical in law to the passage time to e of a diffusion on R' with the f-Yt

yf Diagram I

Diagram 2

Diagram 3

Bxowxian scale and speed measure m' = m on [e, + oo) (see 5.2), so E, [e-°`m; ] = g2 (e) where gq is the decreasing solution of m (d ) = ag (t; )

5.3. Time changes: Q = [0, + oo)

173

(e > 0), and since 1

g2(0+)

8)

according as

+ oo

f d m z +oo, 0

1

Pill Ono) =+00]=1 if f edm=+oo,

while P, [f (ino) < + oo] = I

0

l

1

if El[ f t(m0, )dm]= f Edm 0 in the neighborhood of 0 for l > m0. Given t Z 0, s-'> 0 All

(w:f-1(t)<s)=(w:t f (m0) (< + oo) implies

f-1(t+s)=MO =f-1(t)=in

11 a)

(see diagram 2) and 11 b)

f -1(s, w 1) = f -1 (S, w 0) = 0,

while, in the other cases, W -1(t)) -1(t)) = t

12a)

12b)

f (s + m) = f (m) + f (S, w;0

=t+f(s,w+11) (see diagrams I and 3), and therefore 13)

f-1(t + s)

=max(0:f(0) St+s)

=m+max(0:f(0+in) 0 = O (E)

in case m (0) = 0

implies the expected 16a)

m (0) > 0

u) (0) _ . o E0 (m') -1 [u (e) - u (0)] = m

16b)

(o)

u * (0) = 0

M (O) = 0.

6.4. Local times

Consider the non-standard description

D' = [x' =xP.]

f = f tdm Q

the general conservative non-singular diffusion on Q = R1 or [0, + oo) or [0, 1 ] with scale s (b) - s (a) = b - a, based on the standard Bxowhian motion with sample paths w : t -- x (t), local times t, and probabilities PQ(B). D' has a local time of

measure (s: z* (s) E db, s

1 a)

V (t, b) =

1 b)

t' (t, b) = t(f -1(t), b)

t)

m (d b)

at each of the non-trap points of Q; indeed, if / E C (Q) and if t is smaller than the dot passage time to the traps, i.e., if f-1(1) is smaller

5.4. Local times

195

than the standard BROWNian passage time to the traps f -' (+ oo), then 2)

JI(x.(s))ds 0

I

= f /(x (f -')) ds 0

1-1 M

=

f 1((x (s)) f (d s) 0

= lim nt+oo

/(x(k2-")) Q (k - 1) 2-"-, k2-")

Y k2-" < j-'(f)

= lim f nt+ooQ

I

/ (x (k 2-")) t ([(k - 1) 2-", k 2-'), b) m (d b)

k2 ' -oo l' z - 00

+ oo)

s[a,b)=s'[a,b)=b-a

b - oo and a sample path

Diagram 2 a

'L'

,..

w : t -+ x (t) starting at a point x (0) ? 1, t i f 1' is a trap of Q', then e < + oo and f +o. has the aspect of diagram I a or of diagram 1 b according as l' is exit or not, while, it l' is a shunt of Q' , then either f

p".

l= - ooorl> --- ooisashuntofQand has the aspect of diagram 2 a or of

diagram 2b according as l = l' or 1 > P; in both cases the identification of x (f -1) proceeds as in 5.3.

+ Diagram 2b .C3

As to the case C = - oo , if x (o) > - oo, then f has the aspect of diagram 2 a, and the same is true if x (0) =1' itA/McKean, Diffusion processes

oo is a shunt of Q' 12


178

0-

because then P

[mo < + oo] =1 and E

[f (mo)] = J. I j dm < + oo;

oo is a trap of Q' then f (0 +) _ + oo,

on the other hand, if x (0) = l'

x (f -1) _ - oo, and the proof is complete. Given conservative non-singular Df subordinate to D, if

tSet

f}(t)= ftdm}

6a)

at = +00

t > et "C3

as in 4), then x (f *') is identical in law to D, and using its local times it appears that if t+ =

r.+

t:e

't!

f(tj= ft+dm_

6b)

t > e,

where e is the obvious passage time of x(f ,1) , then x (f =') and x [f

.,5

f

= +00

r-+

I-..

Q_

are both identical in law to D_; this suggests the chain rule:

3

f;' (f-') aft

7)

x(0) E Q-

established below.

°a'

t

Because 0 < f-1(t) is the same as f (0) S t, 8)

f;' [f-' (t)] = max(s : f+ (s) S f _1(t))

= max (S: M. (s)] S 0, gyp y--

T Diagram 3

and so it is enough to prove that f (f +) = f _ . Given t z 0, f +1 [ f + (t)] = t unless f + is constant on a half-closed interval (tl S t < tE)

as in diagram 3; such an interval 8 corresponds to an excursion of the sample path into R' - Q+ or to its arrival at time t = tl at a trap of Q+, and subordination implies that f- is likewise constant on 8 with the result that f- [f+1 (f+ (t))] = f- (t)

9)

Because of 9), 6b) implies 10)

f [f+ (t)]

= f- [f;' (f+ (t))] = f- (t) +OU

f+ (t) s e

f+(t)>e,

and it remains to prove f- (t) = + 00

.,±

11)

in case f+ (t) > e

5.6. Killing times

179

CD-

.ti

But, since J-1 (e) z e-, it appears that e J. (c-), and hence f+(t) > e implies t > e-, which, in turn, implies f_(t) = + oo as desired.

5.6. Killing times

Consider a non-singular differential operator

u+ (a)

u+(db) - u(b)k(db) m (d b)

Jim

u (bb

0 0 and (a, b) E Q' x Q', Ea[Te_tuit t o(dl,b)] 0 /^w

f(t) -a f tdm+k

Ea[Te_'(O t(dt, b)j 0

r+ Ea !

fe-4t(f-(dt).b) '

,

-.2

and using problem 5.3.1 and the chain rule for time substitutions of 5.5, this can be identified as the GREEN function for the dot motion: VII

(a S b)

G (a, b) = B-' g, (a) g. (b)

g =gi,ga gi (0) = gl (0) [a m (0) + k (0)]

in case 2 a)

in case 2b);

g, (0) = 0

now use +00

Go =jeatp'dt.] Problem 2. Evaluate P; [m:, E di, x' E db] on (0, + co) x Q' x x Q' as i' (t, a, b) d t k (d b) using the formula of problem 1. [Given a > 0, a E Q' , and / E C [0, + oo) vanishing at 0, +00

8+.C

400

e-°'dt f p'(t,a,b)/dk 0

+00

-.l

= f Ea f e -as t(dt, b) f dk 0

0

+00

= Ea I r e-"t e-t f (d i) l (xt), 0

= E. e-ameof(Z. (mao ))

now invert the LAPLACE transform.)

Problem 3. Give a precise meaning to the statement:

jo t-1m(db) P,(m:, S t) = k(db)

b> 0.

5.6. Killing times

185

[Given 0< a < b< c, if I = (a, b] , then lim W t-1I f m (d ) PE (m S t, x (m; -) = 0) S n t-1 Pa(mo S t) m(I) = 0

in case 2b),

(to

r-.

lim t-1 fm(d) PE (m, S t, x' (m,, -) z c) µo I

S Jim t-1 Pb (mc S t) m (I) = 0, Ito 'AI

and so t-1 f m (d ) PF (m,o S t) Jim 410

I

=hint-1 Ito C

_`

a

f-1 f m (d E) f ds f p' (s. E.') k (drl) I

r

0

o+

I

=limi-1 f dsk(drt)P; [x'(s)EI] Ito 0

= k (I),

if neither a nor b is a jump of k.] Problem 4. Consider a non-singular diffusion on RI with generator u+ (d b) - u (b) k (db) CS

m(db)

(a) = u +(a)

u (b) - u (a) bya

b-a

.ti

sample paths w', killing time m, and probabilities Pa (B') , and introduce the composite motion s(D) r

where m1 +

4(t- MA_,--- MOD + m;,-1 = O if n = O and, condi-

v+0

tional on x (t) : t < mi +

Aoo

x(1)

+ m;,, the motion

xi+1(t) : t<m;,+l is identical in law to x'(t) : t<m

starting at the place l = x;, (m,, -) (see the diagram).

The problem is to show that

and to identify the composite motion as the conservative diffusion with generator (53u = t u + (d b)/m (d b) .

Diagram ,


186

[Consider the conservative motion with generator 6, sample paths local times t, and probabilities Ps (B) , and kill it at

m : t --a. x (t),

time mm with conditional law P.(m..> t B) = e-i(0 (f = f tdk) to obtain a (non-standard) description of the dot motion. With this model,

it is clear that

E.

n z 1;

f(dt), x(t) E dl]]

[e- 0, then a possible (non-standard) description is x' (t) =

3)

1 < M.

x(f-'(t))

I ? M""

= 00

f=1+ ,P3t1) P.(m.>tI B)=e P. If p3 = 0 < p2, then x' is the elastic BROwxian motion as described in 2.3.

Consider the case p, = 0 < p3, and let $ _ (1: x(t) = 0). Because f (to - f (tl) > t, - ti if (ti , to fl 8 + o, the clock f -' counts actual time oll g' = f ($) but runs too slow on 3', i. e., x' = x (f -') looks like a reflecting BROwNian motion oll g' but lingers too long at the barrier as it the going were a little sticky at that point. W. FELLER [4, 9] discovered that 1) is the most general (local) boundary condition that can be imposed upon the restriction of D'/2 to [0, + oo) ; it had not been noticed before him that conditions involving ((SS' u) (0) were possible.

FELLER studied the case in which the sample path is permitted to jump from the barrier back into (0, + oo). (SS' is then D2/2 applied to u E C'[0, +oo) subject to the conditions: 4)

8Ò

+00

pi UM - p2 u+ (0) + p3 (05' u) (0) = . f [u (1) o+

0

- u(0)) P(dl)

5P1 ,p:,p3,p(dl)

PI +p2+p3+ f IA1p(dl)=1, o+


188

and the entering sample path jumps out from 0 like the germ of the onesided differential process with generator +CO

5)

p2U+(0) + f [u(l) - u(0)]p(dl); 0+

see 7.20 for an explanation of 5), and K. ITO and H. P. McKEAN, JR. [2]

for a complete picture of the sample paths associated with W'.

S. WATANABE [1] obtained similar results for stable processes. 5.8. Ikeda's example

N. IKEDA pointed out to us an interesting example of a simple MARKovian motion with continuous sample paths which is not a diffusion (see 4.1 for the simplest such example). Consider a reflecting BROwxian motion with sample paths w : t

iW -x(t-t, - c, -tl-e . iu,

t3-t"'(c

)

nib

z(t) -x(t)

i(0) - x(0)-0 Diagram 1 a

4.1

x (t) , local time t = lim (2e)-1 measure (s: x (s) S e, s S f) and probabilities P. (B), let e (n Z 0) be independent exponential holding times with conditional law P.(e > t I B) = e-t 1) n Z 1, let t-' (t) be the inverse function t-1(t) = min (s : t (s) = t), and let k be a positive number. Define new sample paths x' as in diagram 1, and note that since 2)

P. (ti > t I B) = P. (e, > k t (t) I B) = e-k t (t).

5.8. IXEDA's example

189

the excursions alternating with the black intervals on which x' = 0 are independent copies of the elastic Brownian motion corresponding to u+(0) = ku(0). D' = [x*, P.] is simple MAxxov because P.[x (t) = 0] = 0 (t > 0) ; it cannot be strict MAxxov because Eo (e-'",+) is neither 0 nor 1 [see 3.3.3 a)].

F

I I

0)-x(0)-0 Diagram t b

+w

Given

/ E B[0, +oo),

G;/ = E.[ f e-`t /(xi) dtl o

belongs

to

J

III

D = B [0 , + oo) fl C (0, + oo) f1 (u: u (0 +) exists) *, and using the

obvious Ga - G; + (a - fl) G. G; = 0 (a, P > 0), it is clear that G; applied to D is 1 : 1 and that a generator (W : D ((Y) = G1D -+ D can be introduced as in 3.7. (i' is similar to the elastic BRowxian generator; in fact, D (05') is the class of functions u E D f l C' (0, + oo) such that 3 a)

3 b)

u+ (0+) = Cu (0+) - u (0)] k

u" (0+) exists,

and 4 a)

4b)

OAS' = 2 D2

on

(0, + oo)

(OP u) (0) = u (0+) - u (o)

Diagram 2

B(o, +oo) is the class of all bounded BoREL measurable functions defined on (o,+00).

5. Time chapges and killing

190

D. B. RAY [3] has found that the state space of a simple Markovian

motion can be ramified so as to make the sample path start afresh ,,r

at each MARKOV time; the ramification for the present case is shown in diagram 2.

5'0

'"'

Problem 1. Give the detailed evaluation of W. Problem 2. Check that the motion of diagram I is not even simple MARKOV if the initial black interval in diagram I b is suppressed. 5.9. Time substitutions must come from local time integrals

III

Given a standard BROWNian motion with local times t, it is to be proved that the time substitutions t --> f -1, f = f t d m of 5.2 could not have been otherwise; to be exact, if t - r-1(t) maps the standard BROWNian motion into the conservative diffusion with the BROWNian

scale and speed measure m on R1 and if, as in the case r = f tdm, 1)

r(t)=r(s)+r(t-s,w.)

2)

(a 5r (t) < b) E Bi

3a)

O S r(t±) = r(t) < +oo

3 b)

r (O+) = 0,

tZs t?0

then

4)

r(t)=f(t)= ftdm

t?0

up to a negligible set of BROWNian paths; a similar result holds for the time substitutions of 5.3. Consider, for the proof, an additive functional r of the standard BROWNian path as in 1), 2), 3) and introduce the sample paths 5)

x° = x(t)

t < m'O

t z m". ,

= 00

with m. distributed according to the conditional law 6a)

P. (M.0 > I B)=ir(1)

tz0.

[x°, P.] is a diffusion as the reader will prove using method of 5.6, and, as such, can be expressed as a conservative diffusion x' with local times V, killed at time m:. with conditional law. 6b)

P:(mom>tI B')=e-1'(r)

r'= ftdk, dkz0.

5.10. Shunts

991

is identical in law to [x' (t) : t < m;, m;,] , Because [x(t) : t < M., x and x' are identical in law, i.e., x' is a standard BaowNian motion

(see problem 5.6.4).

0, A E Bj,

But now x and x' can be identified, and choosing t and using 2), it follows that 10a)

E. [A , e-i fit)] = P. (A ,

E. [A ,

which implies ...

10b)

m > 1) = P. (A , m;, > t)

P.[r(t)=t'(t)= ftdk,Qt 011,

thanks to 3 a).

Now suppose that the time substitution t -+ r-' maps the standard BxowNian motion into the conservative diffusion on R' with speed measure m. Because x (r-1) has to be continuous, O J < r (t=)

(tl < 4), and

computing the mean exit time 11)

eab (E)

= Ee[min(t : x(r-1) ( (a, b))]

=

Ed [r (mm A mb)] b

= f Ee [t (ma A mb, ,I)J k (dry) a

b

= fG

a < E < b,

rj) k (drt)

a

Srj

(b

G(.r1)

as in 5.2, it follows that k = m; in brief, r = f tdm as stated. 5.10. Shunts

Consider non-overlapping subintervals Q. = [1,,, rn) of Q = [0, 1]

with scales s = s and speed measures m = m attached, let K, Q - U (t,,, r,J, let s+ be a (shunt) scale with

_

nal

I a)

0 < s+ [a, b) = s+(b) - s+ (a)

I b)

s+ U., b) = f m [ln, t) s (d)

atI B)1(x)di] = E. [Te--s e(t) f(x) dtl

p,'

Vv-,

map B(Q) into D, showing that D' is a di//usion and for the rest, it suffices to compute its invariants s', k', m', s+, k+. But, on Q,,, s' = s, k' = k and m' = m as is clear from 5.6 and 12) ; thus, s* = s+ and k' = k+ inside Q,,, and since mi = min (t : z' (t) = i pi (a) = Pa (m; < + oo) 16) o`1

= Pa(m1 < mm) = E, (e(m1))

r

-Z

al

e

t(ms.1)k(dE)1

n [1 - k+(db)] (a.ilnx+

S)

(10

f

t(m...!) k(df)Ea

fl [1k+(db)]

l)nB+

7P 11

a m1 I B))

=EaI- f tde+m1e(m1)J 0

(L

1

= Ea l f ledt1 r

Otr.

1

r

f - u 1.)

e (ma) d me

K+

m,.

a 0). Given (1, a, b) E (0, +oo ) x R2, if # (t, db) is the number of particles E db at the instant t, then 1) Ea[#(t,db)]

=

SF

n z 0 el,e,....,ew

Pa2-.< i
constant x e-14n'It, z (4, n)

.

art

Returning to 18) with this bound in hand, it develops that for I a I < n and t < t1/3 , w (1, a) < constant x ei"Y e-'13' e121 % ,

22)

'ti

111

which tends to 0 as n t + oo provided t + 12/t1 < 1 /3 t; in brief, w - 0 for small t, and this can be propagated, proving that w 0 (t < t1) . yam.,

Problem 1 after H. TROTTER (see also R. C. CAMERON and W. T. MAR-

TIN [1]). Prove that, if y = 2 and 0 =)I t, then om"

e(t,a,b)=

In j sin 0

1t + for each (e1, es, . . ., ex_ J, then # (t, R'):5. 2)

P. [# (t, RI) > 2x"1]

2x-1 so that

P. [tielex...@*-I+$-" S t]

2x-1 P. [t2-w S t] = 2x-1 E. f e-1 l

M;-."

1

= 2x-1 E. (n

I )!

fsn_1e_srds 0

I

= 2 E [e!

(n(2V)

f sx-1 a-(& +1)Ids !

0

2

(2nfe) E E. en) ( (n-1)!

Sw-1

f (s+ I). 0

Use

Yè-a+.)I

(n/e(1 + s))".

ds

m !!

,

f = 1(1+)

5.14. Explosions

-1E. 2

207

1/2

(ej

(2n/e)" fdt (n-1)! 1-t

5 E. (et) (!/e)" n!

-

0

E.(et)

V2nn '-.

and now the Maxxovian nature of the shower shows that if E. [et(t+)] < + oo for some t, then

1 = P. [# (t, R1) < + oo] = P. [# (21, R1) < + 00]

=P.[#(3t,R1) 2.

If y S 2, then 4)

E.[et(t)] < E.le

t (1 + msx z (s))s

.S8

J

+w

etll+(1al"M

S 2 V-2

nt db

- b-121

0

2fl > I and then take I < a < 2#, define

l*=0 (n=0)

_l$m$nm-e

(nZ1),

and note that the event

B+ :

e+ (2 X m-4) ? l.

(n Z 1)

2$m 2-" x (1 - El, [P0 (m (2-P + 3-0) > 31t2])

5)

x etc.

is positive if 6)

E,,[P0(m(1-fi+ 2-Q) > 2-°`)]

+ E4 [P0(m (2-"+ 3-a) > 3-")''] + etc.

< + co. * in (1) = nt (1, w) is the passage time min (1: x(1) = 1) .

209

5.15. A non-linear parabolic equation

But since a < 2/3, 7) PO (m ((n - 1)-P + n1 > n a) +00

1'

[(n-1)

n

a

e-,rsr Y2nrsdt

+x-]-: a

 2/3 > a,

1 E1

8)

_,

[2-is]

X2

,II ,II

nZ3

nys Erg

na

mZo

[t(n-°Am(l,-s))]"El.,

e

ml

nZ3

0 and completes the proof.

5.15. A non-linear parabolic equation

Given a non-negative number y, let u = u(t,1) be the solution of 1a)

OU

ar

_

1b)

t>o,IERI

+-u(1 -u)Illy

1

2

(9t=

0 0, so

(E>1) r¢¢11+

contradicting u < 1; a similar contradiction is obtained if u' (1) S 0. is < 0 is likewise impossible.] 14'

6. Local and inverse local times

212

6. Local and inverse local times 6.1. Local and inverse local times

We now take up the fine structure of the local time t(O = t (t, 0) and its inverse function t-1 (t) for a persistent non-singular diffusion D* on an interval Q containing 0 as an inside point or as a left end point, with -u+ (0) + m (0) ((li u) (0) = 0 in the second case. A number

of the statements made below hold for transient diffusions also (see esp. 6.3, 6.5, 6.6); the necessary modifications of t-yt the proofs are left to the reader. t-' turns out to be differential and homogeneous in time just as in the standard BROwNian case (see 6.2), and the connection between it and the intervals of the complement of .3 = (t: x (t) = 0)

comes through in part (see 6.3). t can also be interpreted as a HAUSDORFF measure (see 6.5b), and the down-crossing estimate of 2.4 is unaltered t/t (see 6.5 a). t-1 is useful for computing the HAUS-

t=ay

'C7

'"'

DORFF-BES,covrTC} dimension of 3 (see 6.6 and 6.7).

Remove from the path space a negligible set of sample paths so that the following hold with no exceptions: 1)

t E C [0 , + oo)

2)

mo = min (t : x (t) = 0) < + 00

3)

t(+oo) = iimot(t) = +00

4)

t (ti) > t (t1) t (t$) = t (t,) + t (1s - t, , wig)

5)

(ti , t£) ^ S + 0

t2 > t, ? 0,

define the inverse local time 6) t-' (t) = max (s : t (s) = t) and note for future use the following simple facts: 7a)

t-1 E t,

7b)

t-1(t +) = t-' (t),

7c)

t-1(0) = MO

7d)

+1+00 t-' (t) = + 00 t-1(+ oo) = lim 13

7e)

the range of t-1 is the set of points t E 8 not isolated above.

t-' is to be split into 4 parts : 8)

t-1(t) = mo + I m (0) + t_1 (t) + t+' (t) , * See problem 4.6.6 for a discussion of persistent diffusion.

IZ0,

6.1. Local and inverse local times

213

where

t=1(t) = measure (s x (s) < 0, mo S s S t-' (t))

9a) and

t hl (t) = measure (s x (s) > 0, ma S s S t-11 (1)) 9b) are differential, independent, and independent of mo.

Given t > 0, it is clear from 5) that t-1(t) = max (s, : t (S.) = t)

10)

I = max (s, s, = MO + ss, t (mo) + t (Ss' wnl,) = t) = mo + max (ss : t (ss, wme) = t) = mp + t-1(t , win)mo

is therefore independent of t41 (t) = measure (s : x(s, ww,,)

11)

0, s S t-1 (t, wm ))

and t-1 (t) - MO = measure (s : x (s,

12)

t+1(t)

0, s S t-1 (t, w*ie))

= t (t-1 (t, wme)) m (0)

= t m (0), All

'w^

establishing the decomposition 8). Consider for the proof that t- is differential, two times is ? 11 > 0, let m = t-' (tl) , and use 5) and 13) t (M) = tl to check a'+

t-1(ts) = max (s, : t (sl) = ts)

14)

I'1

i"+

= max (s1 : Sl = m + ss, t (M) + t (S2, wm) = ts) = m + max (s9 : t (SS, w+) = t2 - t,)

=m-}-t-1 ts - tl.wns Using 14) and wo(w;,) = 0, it results that 15)

t*1

(t8)

= measure [s : x (s, w+) < 0, mo S s S m]

+measure [s:x(s,w+) 0, s < t-1 (is - tl , wm)] = t*1(t,) + t-1102 - tl, wm)

But 16)

(w:m<s)=(w:1, - oo. 0


216

Coming to t+1, the time change t -> J-1(f) based on the local time integral f (t) = f t (t, e) m (d ) takes D into the diffusion D+ on [0, + oo) o+

with invariants

m+ =M

s+ = s,

8)

on

(0, +oo),

m+ (0) = 0

and since, for t> 0, f grows just to the left of 0 < s = t-1(t), one finds

i.e., f (0) < j (s) for

III

t+1(t) = W-110))

9)

= min (s : s

(t-1y))) A11

= min (s : t-1 (S)

t-11 W)

= min (s t (f -1(s)) ? t), .+a ova

proving that t+' (t) is identical (in law) to the inverse of the local time t+ (t) = t+ (l, 0) for D+ .

But now, taking y > 0 and using the GREEN function expansion of problem 5.6.1, one finds 10)

f

r +cc e-'3'8 di

r e-Y, ea'+ () dt

Eo(e--a t+'(t)) = Eo

[

+00

= Eo If

e-a' e-yt+(t) t+ (dl)]

0

= G_(0, 0),

where G_ is the GREEN function for the diffusion D_ on (0, +oo) with invariants i1)

s_ = s+,

k_ (0) = y, and since G_ = gs ga with 12)

m_ = m+, k_ (0, + oo) = 0,

g(d) = m(dj)

( > 0)

a ga W

gi (0) = y ga (0) g3+1 (0) ga (o) - ga (0) gs+ (0) = 1

I-'

it follows that 13)

gt(o)

G- (0, 0) = ga (0) ga (0) =

A(o) go(o)

yg:( )

g# (o)

1

Y

completing the proof of I b) apart from undoing a LAPLACE transform.

6.2. Lavv measures

217

Not all non-negative mass distributions n+ (dl) with f l A 1 d n+ < + 00

can arise as LEVY measures of inverse local times; in fact, it is neces-

sary that

0+ n+ddi)

14)

= fe'/z2(d,)

0 s fYY (dy) ,

-00 as will now be proved.

Consider the generator @' for the original diffusion starting from [0, +oo) and stopped at 0, write G = gl g! for its GREEN function, and 15)

y50, a,bz0,

e (y, a) f (dy) e (y, b)

for the eigen-differentials of (' where e = (e1, e2) is the solution of 16)

(;e=ye e(0)=(1,0),

a+(0)=(0,1)

and f is a BOREL measure on (- oo, 0) to 2 x 2 matrices with entries f11, /i t, /21, f!, such that 17)

/12 =

0 S /111 /!Y,

/2 1 .

fl l

/1Y = 1221

f!!.

as in 4.11. Owing to g, (0) = 0,

18)

gl (0) g! (0) = gi (0) g! (0) - gl (0) gi (0) = 1

cad

and so 19)

G. (o t, 0, e) = gi (0) g! (e) = g! (e) /g1(0) +00

= f e-"IP,(m0E dl)

> 0.

0

Inverting the transform, 19) goes over into 20)

P,(m0 E dl) = J e1Y a+(y, 0) f (dy) e(y, e) dl

e> 0,

-00

.+.

and, computing the gradient ate = 0, it follows from the definition of n+ [see I b)] that 0+

0+

21)

f e'Y a (y, 0) f(dy) e+(y. 0) = felV/11(dv)

1> 0.

-00

-co

It is an open problem to describe the class of non-negative mass distributions / (dy) on (- oo, 01 such that n(dl) = dl x f e"Y f (dy) -00

is the LEvy measure of the inverse local time of a diffusion. Problem 1. Check that

n+[l,+-0)
0), and n-, n+ , n the LEvy measures for t_', t', and t-', #(8-: 8- C [0, t), I8-1 La E)

2 a)

P. [lim

= t(t), t z

0J

=1

2 b)

P. [lim # (8+: 8'n+C [o, +0, 18+1 L> e) = t (t) , t L>-

OJ

=1

8j0

n_[E,+00)

!o

and

po[lio #(8* :8*C[o,t) 18*1?O =t(t),tz 01 = 1. ...

3)

Consider, for the proof of 2b), the counting function 4) # (t, e) = the number of jumps o/ magnitude ? e of tom' up to time t,

6.3. t and the intervals of [0, + oo) - 8

219

and keeping in mind that n+ [e, + oo) is continuous (e > 0) and that n+ (0, + oo) = + oo, let n+1 (s) = min (t : n+ [t, + oo) S s).

5)

[# (t, n+1 (s)) : s Z 0, P0] is differential and POISSON distributed. Because

E0[# (t, n+1(s))] = t n+ [n+1 (s), + oo) = t s, it is homogeneous in its temporal parameters, and thanks to the strong law of large numbers, 6)

P.(li1840 m

7)

# (t , E)

n+[E, + oo)

-

= Jim # (t, SKIM) - t, t:>- 0 = ! . J

000

But # (t (t), c) = #(8+: 8+C [0, t], 18+1 ? C) if t-1 (t (t), 0) = t, i. e., if t E is not isolated from above (in 8), and since # ($+ : $+ C [0, t), 18+ 1 z e) and t(t) are increasing in t, since .8 ,,a

;:n

8)

is closed and perfect, and since t is continuous and flat outside 8, the strong law PO [Jim # (8+ : 8 C [o, t) I 8+I ? E) .-.

9)

= t (t), t-1(t) =I, t Z 0J = 1,

acs

which is clear from 7), implies the desired 2b). The proofs of 2a) and 3) are similar. We also saw in 2.2.6) that for the standard BROwxian motion, 10)

1`m

iE x the total length of the intervals of [0, t] - 8 t> 0, o t length < e= t (t)

in which V2e/n is the expected sum of the jumps of t-1 of magnitude <e per unit time. A natural conjecture is 1i)

Ii m

(f l n+ (dl)1 -1 x the total length of the inkrvals .+ C [0, t] ti,

0

o l length < e= t (t) i.e., substituting t-1 (1) in place of t and n+1 (s) in place of e, '1 -1 (t) ri+l (1) 12)

Jim

+t+=

f l n+ (dl)

t Z 0,

f 1# (t, dl)

E Jim

ftiao +OD f n-+1(O)dO

=t

t?0,


220

in which the i) are the jump times of the (homogeneous) Poisson process # (1, n-.1 (s)) : s ? 0.

Because of strong law of large numbers, ....

lim n-1 S. = t-1,

13)

at OD

and it is a simple matter to check 12) from 13) if n+1 (s) is a power of s

times a function f (s) such that f (t s) - f (s) as s f + oo for each t > 0 (see 6.7.4) for the special case: ds = d E, dm = 2 JE 10 d t (fi > -1), do = constant x l-0+°> dl, a = (fi + 2)-'). But 11) is false in general as the counter example of the next section proves. 6.4. A counter example: t and the intervals of [0, +oo) - 8. 6.3.11) is false in general: for example, it is possible to have 0 < cl < e n+' (s) < c= < -I- oo as s t + oo and then 6.3.11) would force lim sup e'* X e-" S c,/c, < + oo, violating the fact that the law of at+oo kZn e'" X e-'+ is independent of n and has as its support the whole of [1, + oo) .

kan Consider for the construction of such n+ , the function y, (t) _

(1/2) f OF (s + 1)-1 ds and its inverse function 0 and let CS be the 0

generator with the natural scale and the speed measure dm = (1/4) (01'dE on (0, + oo).

Consider the increasing and decreasing solutions g, and g= of (si g = a g with g , g, - g1 g+, = 1, gl (0) = 1, and gi (0) = 0. +m

gs = g, () f g1E drl satisfies g$ < 0, 05 g3 = a g,, and g8 (0) =

= gE (0) _ -1, so that g= = g=, or what interests us, +00

..r

i)

g:(0) = f gi-2dE. 0

Given t > 0, 0 < ip'(t) E l so that 0 < 0'(1) E t and 0"(t) > 0. Now is convex so that Ig 0 (t) is concave and 0" (0') -2 < 0-1. Besides y,' (0+) = + oo so that 0'(0+) = 0, and putting g, (0) = g, (E), it follows from 0 g, = a g, and the facts just cited that y, (et)

+.+

2)

g4" (0) + 0" (0') -' g4, (0) = 4 g. (0)

g, (0+) = I g (E) i g(0+) =1im 40 W(e)

lim4a g, (e) O' (r) = 0 .{o

6.4. A counter example: t and the intervals of (0, +0o) - 8

221

and

0> o.

g4< 91 0, measure [8 fl [0, t-I ft))] = measure (s: x (s) = 0, s S t-I (t)) ,c°,

2)

= t (t-I (t)) M (O) =I M (O) > 0

dip

for each 1 > 0, so that y = 1. Consider now the case m (O) = 0 and let 8 stand for the range (t-I (t) : t > 0). 3) %9 and 8 - 9B is countable, so that dim (8) = y I See note 2.5.2 for the definition of numbers.

dimension

6.7. Comparison tests

225

is BoREL, and it follows from the properties of the POISSON integral +Cc

f l p ([0, t] x dl) for t-1 that y = dim (9) is measurable over 0

fl B[p (dt x dl) : t Z 0, l S n-lj and so is constant, thanks to the

n21 differential character of p and the KOLMOGOROV 01 law.

Problem 1. y = dim 8 = dim 8 n [0, e) for each e > 0 for paths starting at 0. 6.7. Comparison tests C7'

Consider y = dim8 as a function of the speed measure m. Because the time substitution t -. e I maps t-1 (t) -. E-1 t-1 (e t) and 3 -+ e-1 $,

it is clear that dim ,3 = dim (t-1(1) : i S 1) = dime-' = dim (t-1(e i) :1 VII

1)

1)

= li m dim 3 fl [0, t-1e)] ,

and since t-1(e)

'ti

mo as e 4 0, y is a local /unction of m, i.e., it depends only on the behavior of m in the immediate neighborhood of 0. Given a pair of diffusions D and D' with speed measures m and

m' g, m, it is to be proved that y'

The proof is not hard: in

y.

fact, using the time substitution

t - f -1(t) (f = f t d m') mapping x into x' = x (J-'), it is clear that J-1 maps 8' = (t: x' = 0) into $ _ (1: x = 0), and since f (f-1) = t, it follows that 8' C f (,3). Using 2)

S f t(dt, )m(d ) =dt

f (dl) = f t(dt, )m'

and the result of 0. FROSTMAN cited in the note placed at the end of this section, it follows that 3)

dim,3' S dim f (,3) d

sup (a

elfin o

,l e(f(8))-1 1(8)xf(8)

S sup ra : inf e

O

r

J

c(8)-1 8x8

S sup (a: eino

a (l aa, eb b) < + oo)

e(da)e(db) 11(a) - f (b) la

1

< +00 l

f e'daa)e(db) < +oo)

e(8)-1 8x8 = dim (8). completing the proof. 1tA/McKean, Diffusion processes

15


226

CA9

VII

At no extra cost, it is found that y' S y it m' - m near 0. V1 and y will be computed for the special case:

P> -i:

Q=R1,

.;.

4)

''^

the result is that t-1 is stable with characteristic exponent y = dim 8 _ = (f + 2)-1, and this stock of dimension numbers will lead to a comparison test, permitting us to read off the dimension number of the zeros of most diffusions encountered in practice. 'C"`

Given 9 > -1 and a > 0, the solution ga = ga(a, ) of 5)

a g1 (() = 4 1 t I

-1

ga (t)

ga E 1, ga (+ oo) = O, gs (0) = 1,

is invariant under the substitution

- /cy

a - C X,

6)

Y = ( + 2)-1, c > 0;

i. e., 7)

ga (a,

) = g2(1, a''

),

and therefore

(dl) = 2 f (1 - e'°`) n+ (dl) è7

.'!

f (1

8)

0

0

-2 g2+ (a,0)

2a yga(1,0)

g1(a, 0)

a

ga(1, 0)

- 0,

àrt

all

ago

E0(

is - ti-°°t(ds) t(di)

!

/1

= Eai f

t-1(1)) 1

f

11

.°i

8n10,

+3g

(Y'

VII

9)

op,

proving that t-1 is stable with characteristic exponent y = ( + 2)-1. Computing p (t) = p (t, 0, 0) = constant x ty-1 from 8), we find that the HAUSDORFF function h = p-1 p ®p of 6.5 b is a constant multiple of P. dim 8 S y is now clear from 6.5 b.2), and since

It-'(s) -t-1(1)1-°dsdtI 1

+00

1

= constant f ds f di f o

Eo(e-0t-1(It-aI))

0

o 1

0-1 d9

1

+oo

= constant f d s f d t f O-1 d9 e-CO-t--t 1 t -, I ey o

0 1

0

1

= constant f f Is - t1a"y ds dt

= (;5) constant x 1 I" d E near = 0, then y = dim 8 z (S) (fi + 2)-1; the proof is clear. Additional information on the computation of dimension numbers can be found in BLUMENTHAL and GETOOR [1].

Problem 1. Can y = 0 serve as the dimension number of the visiting set 3 of a diffusion? [Given s(d) = d. and m (d E) = e-IEI-' d , m (d E) S i d near = 0 for each fi > -1, proving y inf (fl + 2) -1 = 0.] v>-1

Note 1. Dimension numbers and fractional dimensional capacities. We need the result of O. FROSTMAN [1], that for compact BC [0, + oo),

l-

dim (B) = sup (fi: inf f e aa) eb db) < + co) , 0

1)

where e runs over the class of non-negative Borel measures of total mass

+1 loaded up on B. Here is the proof.

Given ft > 0 such that + oo > f la - b I -a compact Q C B such that e (Q) > 0 and

e (d a) a (d b) , select

l l a- b -O e (d b) 5 xinf f Ia-bl-Pe(da)e(db)

for

fl 3, the path cannot meet A for large times and + ool = I as is clear on letting A f Rd, while if d = 2, then mn < + oo for each n 1, lim M. = + oo, and selecting Pa [gf m I x (t)

"f+00

-I'

?

7.4. GREHNian domains and GRaaN functions

237

a disc D C R2, the obvious estimate sup P. (mD = + Co) = y < 1 coupled with 4)

OA

y = sup lim E. [mD >

Pz(m.)(mD = + oo)]

OA 0+0o

Y sup"t"o lim P. (MD > m*) 04

= Y2

justifies

Po(mD = +oo) = Eo[Pzlmedl (mD = +0 0 )0,

5)

proving that the path meets each dksc an infinite number of times as desired.

7.4. Greenian domains and Green functions

D C R4 (d Z 2) is said to be a domain if it is open and connected. D is said to be a GREExian domain if, in addition, there is a function G = G (a, b) = K + H defined on D x D and < + oo off the diagonal such that 1) G ? 0, G(a, b) = G(b, a),

2)

3)

K=- (23r) -l lg I b- al

- r(2 -1) lb - al=-d

d= 2

d

4n4M

d> 2 ,

d6H =dbH = 0.

4)

Given a GREENian domain D, there is a smallest such function G = G'. G' is the so-called GREEN function of D. G' (a, b) should be considered as the potential at the point a due to a unit positive electrical charge placed at b inside the grounded shell 8D. G' has a probabilistic as well as an electrical significance : 2G' (a, b) d b +CO

is

the expected time f d t Pa [maD > t, x (t) E d b] that the standard 0

BROwNian path starting at a spends in db before hitting 8D, and this aspect of G' is used below for the proof of the basic formula +Cc

5)

f g'(t,a,b)dt=G'(a,b)

if D is Greenian

0

M +oo if D is not Greenian where (a, b) E D x D, and g' is the absorbing BROwNian kernel: 6)

2g' (t, a, b)db=P.[moD>t,x(t)Edb]

t > 0, (a, b) E D x D

7. BxowNian motion in several dimensions

238

(that g' exists as a BOREL function

is

clear from the estimate

P.[MOD >t, x (1) E db] S 2g d b*; it will be found below that g' can be modified so as to be smooth and unambiguous). Because D = Rd (d > 2) is GREENian (its GREEN function is 4 X-dJ2 r (Z t) b - a 12-d) , each d > 2 dimensional domain is

-

GREENian.

D = R2 is not GREENian; in fact, it will appear that the 2-dimensional

GREENian domains are those for which P. [mOD< + oo] = 1 (see problem 7.8.3 where this is seen to be the same as R2 - D having positive logarithmic capacity). Here is the proof for d = 2 (see G. HUNT for a similar proof); the d > 2 dimensional case is much easier and can be left to the reader. _ g' is the first item of business. Given a subdomain B of D such that B C D, if (t, a, b) E (0, + oo) X B x B, then

Pa[x(i2-11 t)EB,i t, x (t) E d b] . and using the fact that the GAUSS kernel g is continuous on (0, + eo) x R2 x R2 and . 0 as ,.-,

(unambiguous) densities

8)

f g(2-" t, a, b,) 2db1 g(2-" t, bl, b2) 2db2 ... g(2-t,

b)

he" -,

=g(t,a,b)-

r--,

i0,

as it should be.) v.%

Problem 2. Compute the GREEN function for the domain D of problem I (see COURANT and HILBERT [1: 328-333). 7.7. Potentials and hitting probabilities

Given a GREENian domain D with GREEN function G, if B is a compact subset of D or an open subset with compact closure B C D, then the hitting probability pB = P. [MB < mOD] is the potential f G d eB .,r

of a Borel measure eB ? 0 concentrated on 8B; indeed, pB can be identified as the Newtonian electrostatic potential of B, meaning for compact B, that it is the greatest of the potentials f G do < 1 (e 0) , 'VA

B

1 G. DoETSCR [I: 203-216).

7. Bxowxian motion in several dimensions

248

new

and for open B, that it is the smallest majorant of the potentials

fGde S1 (ez0,ACB) A

We give the proof in the case d = 3, D = R3, G = (4x) -' Ib - a I-1. Given bounded B C R°, P. [mB (wf) < + oo] 4 0 as t t + oo because 111im Ix(t)I

= +00 =1

(see 7.3), and introducing the measures 1)

a>0,

e,(db)=e-'2dbxPb[ntB 0, where C (B) is the NEwTONian capacity of B defined in the next section. 7.8. Newtonian capacities

Given a GREENian domain D and B C D as in 7.7, the NEwTONian

electrostatic capacity of B (relative to D) is defined to be the total mass C(B) of the NEwToNian electrostatic distribution eB; if B is compact and if 8D is considered to be grounded, then C (B) is the greatest

positive electrical charge that can be placed on B and still have the potential of the corresponding electrostatic field S 1. C (B) has the following properties: 1)

2) 3)

4)

C(A) S C(B) C (A U B) + C (A fl B) S C (A) + C (B),

A C B,

C(B) S C(OB) = C(B), B compact,

C (B) = dinnf C (A) OVOR

d)B 5)

C(B) = sup C (A)

B open,

A aompad

ACB

6)

C(mB) = Imld-'C(B)

if D = Rd (d > 2) and m is a rigid motion of Rd (I m 1) or a magnification by the factor Im I > 0. GAUSS [1] observed that if the potential f G do of a signed meaB

sure e is non-negative at each point of D, then the total (algebraic)

7.8. N$wToxian capacities

251

charge e (B) is likewise non-negative, as is clear on selecting some open

A B and noting that e (B) = f PA de = f p d ed ? 0 (use 7.4.2)]. B

8A

GAUSS's observation, combined with PA S PB (A C B), proves 1). 2) is likewise clear from

tea;

7) PAUB - PB = P. [mA U B < m8 D, mB Z m] S P. [mA < ma D, 11A n B ? MOD] gyp.

=PA-PAnB, ii.

'L7

and using PB S POB = pa, another application of GAUSS's trick proves 3). Given compact B and open At B such .that limed = e exists, AtB

p= f G d e S 1

, e (D

-0B) = 0, and so C(A) te(8B) S C (B) S m C (A),

_

GAS

proving 4).

Given open B, compact A t B, and open D ) B, PA t pB, and so

C (A) = f PD deA = f PA deD t f PB deD = f PDdeB = C (B), proving 5). 6) is immediate from the fact that G is a constant multiple of b -a 1II_d

if D=Rd (d> 2). ...

G. CHOQUET [1: 147-1531 proved an infinite series of inequalities similar to 2). Here is G. HUNT'S neat derivation [2 (1): 53]. Given A, B1, B2, . . ., B. C D and using I for subsets of 1 , 2 , ... , n,

I I for the number of integers in I, and Bl for U B1, it is found that lei

OSP.[mAUB.<MOD, lSn,MA ZnI D]

8)

= P. [MA U B. < MOD, 1 S n] - P. [mA < MOD]

_ -msn (-)n` 11XP .[mAUB1< MOD]* - P.[nA< MOD] mX (-) Ì PA U Bl - PA,

I

and an application of GAUSS' trick establishes CHOQUET's result:

0 S - S' (-)m I C(A U BI) - C(A);

9)

mgn

X11-m

this reduces to 2) for n = 2 on replacing A, B1, B2 by A fl B, A - B,

B-A.

Problem 1. Let G = GD (a, b) denote the GREEN function of a bounded domain D C Rd (G m 0 outside D x D). S. BERGMAN and M. SCHIFFER [1: 368] showed that GD,UD1 + GD1nD, a GD1 + GD,.

woo

P.(f E,,,) = - E (-)" E P.(E1) is used at this point; this is the dual

"$w "S, Ill-" of the classical inclusion-exclusion formula of Po1ncARB (see W. Fnu R[3:61-62]).

7. BROWNian motion in several dimensions

252

Use the method of the proof of 2) to prove ...nDt, ? 0. GDiuD,...uDR + X (-)k E GDSt fl D, a 1Sk$n 11 0, then

e0,e(B)=1

f Gdede>C(B)-I BxB

unless e = C (B) -1 X eB .

[Given e !!0 with e (B) =1 and t> 0,

2

f G de de - t e (B) BxB

- C(B) with the > sign unless to = eB; putting .7m

f G deB deB BxB

t = C(B) completes the proof if C(B) > 0, and if C(B) = 0, f G de de BxB + oo is immediate from 12

2BxRfGdedeie(B)=t.] 7.10. Wiener's test Given compact B C R4 (d 2), the event mB = 0 is measurable Ba, , and so (use BLUMENTHAL'S 01 law) P. [mB = 0] = 0 or 1. WIENER'S test states that 1)

PQ[mB=O]=0or1 * C(B)-1 = +00 if C(B) - 0.


256 according as

In C (Bn)

(d = 2)

n,> 1

converges or diverges

n21

2" J-2, C (B.)

(d

3)

where C is NEwroxian capacity relative to a spherical surface I b - a I = y Z 1 (y < + co in case d = 2) and B. is the intersection of B with the spherical shell 2-A-1 < I b - a I < 2-n (n Z 1) .*

Consider for the proof the case d = 3, a = 0, y = + oo. 2_,,n Given n ' 2"C(B") 0

3)

f Gdede*

e(B)-1 BXB

.y1

(see problem 7.9.1), coupled with the fact that G for small b - a 1, now justifies the estimates C (A1)

4)

-(2x)'' lgI b - a

C (A=)

and

C(A)-1< f -IgIb-aldede

5)

AxA

< -2lg[length of A] < 2lg(2"+') for nt + oo, where d e is the element of arc length divided by the total length of A, and using diagram 1, it is found that 1/2n

6)

< C(A) S C(A11) = C(8A1q) S C(A1 U A9) S C(A1) + C(A,)

< 3C(A,) 5 3C(Bn) n t +00. causing WIENER'S sum Z n C (Be) to diverge. 1) is immediate from this.

na1 VII

mar.

Coming to the probabilistic proof, let Z. be the event that, for the BROwNian motion starting at 0, x(t) : 0 < t S n-' winds about 0 and cuts itself in the sense that the (closed) complement of the unbounded part of R2 - (x(t) : 0 S t < n-1) contains a disc IS I < e. ins

* G is the GREEN function of 151 < 1.

7.11. Applications of WIENER'S test

259

PO (ZI) > 0 (see diagram 2), and using the measure-preserving scaling x (t) -- Yn x (1/n), it is found that 7)

0 < Po (ZI)

=

P0[ yn x (t/n) : 0 < t < 1 = Po [x (t) : 0 < t < n-'

winds about 0 and cuts itsel/] winds about 0 and cuts itself] i.'

nZI n t +00.

= PO (Za) I PO (+aZ fl

z'a)

But fl Z,a E Bo., and using BLUMENTHAL'S 01 law, 7) implies M ,Z 1

P0(aZ1Za) = 1 = P0[mB = 0],

8)

thanks to the topological fact that the BROwNian path cannot wind about 0 and cut itself for small times without meeting B.

Coming to d ? 3 dimensions, take a positive function h E C [0, + oo) such that b-' h (b) 10 as b 4 0, let B be the thorn

Diagram 2

B:(b2,bt,...,bd)ERd, b;+..-+ba_1Sh(bd),bd?0, o0,

and let us use WIENER'S test to establish the alternative P0[mB = 0] = 0 or I according as 9)

Jig (h (b)Ib) I-1 db

(d=3)

0 ago

converges or diverges;* I

I (h(b)/b)4_3

db

(d> 3)

0

for example, in the 3-dimensional case, if h(b) = e_1Igbl', then

1 ore < 1, III

10) f I lg (h (b)/b) I -1 db converges or diverges according as e 0

""'

and, in the d > 3 dimensional case, if h = b I lg b I', then 1

11) f(h(b)/b)h1_3_. bconverges ordiverges according as e < 0 or e z 0.** 0

pox

* See K. ITO and H. P. McKEAN [1] for a similar alternative for the d-dimensional random walk. ** 11) is to be compared to POINCARIi'S test (see problem 7.10.1). 17*

7. Bitowrtian motion in several dimensions

260

Because b-1 h (b) 10 as b 10, B. = B fl (2-'" -1 S Jb I S 2-"') is long and thin for large m, and this makes it possible to obtain favorable

estimates of the capacity of B. in terms of the capacities of the ellipsoids: bi h(2-`-1)i

+

+

bid-,

h(2-0-1)'

- 3.2---')' S 1 + (ba 2-20-4

and

+... +

E+' 2nh(2 .)s

+ (ba - 3.2---2)' 2nh(21-)'

2- 2W-2

S

1

Iv'

E_:

E+ ) B,,, ) E_ . G. CHRYSTAL [1: 30] presents a neat proof that the NEWTONian capacity (relative to Rd) of the ellipsoid

W

e?

is a constant multiple of the reciprocal of the elliptic integral dt

e= 0

'

y(el + 1) (ell + t) ... (ea + t)

and using .v..

e ,.+ 2 (ell)s-d

12)

(d = 3)

2

.-.

N17

d-3

(d> 3) e'/e..j0

el=es= Vat

tad

to estimate the bounds C (E_) S C (B,,,) S C (E+) as m t + ao, it appears that WIENER'S sum ' 2-'"(d-2) C(B.) diverges or converges according as '"a1 0.0

`-'

can

13)

Ig [2m h (2-`)] -1

(d = 3) diverges or converges.

Z [2`

MZ1

h (2-+*)]d-3

(d > 3)

9) is immediate from this. Incidentally, the proof beginning with 7) above shows that the 2dimensional BROwxian path has an infinite number of double points; the same holds in 3 dimensions but not in 4 or more, as A. DVORETSKY,

a'0

P. Eiu) s, and S. KAKUTANI [1] proved, using the fact that a small segment [x(s) : f :!g s < t + h] of the BROwNian path has a positive capacity in 2 and 3 dimensions but not in 4. DVORETSKY, ERDOs, and KAKUTANI [2] also proved that the 2-dimensional BRoWNian path has

an infinite number of n-fold multiple points for each n = 2, 3, 4, etc.

7.12. DIRICHLET problem

261

and, in collaboration with S. J. TAYLOR [1], that the 3-dimensional BROwxian motion has no triple point; some open problems are to prove a) that the n-fold multiple points of the 2-dimensional BROwxian path have dimension 2 for each n z 2, and b) that the double points of the 3-dimensional BROwxian path have dimension 1. Problem 1. Check that for the 3-dimensional thorn llg.cls

VII

0 S C< eni's Po [m.B = 0] = 0 or i according as e according as e > 1 or not. This answers a question of A. DVORETSKY and P. ERDOS [1: 367]; see also 4.12 for the more precise alternative of F. SPITZER [1: 188]. t

(f (ig.-... 1

-'do

+00 f c1

converges or diverges according as e > 1 or not,

proving the first statement. Consider, for the second part, the projec-

0.I

0..N

tions x$ (x') of the 3-dimensional BROwxian motion onto c = 0 (a = b = 0)

and let &(t) = e- j

19"

(t < e;'). x2(xl) is a 2 (1) dimensional BROwxian motion, and using the law of the iterated logarithm for x', I

o-1

I (t) it is clear that, for e = 1, I xs (1) S f [lx' (1) F S / [1/3 t lgs 11 ] for an infinite number of times t l 0. On the other hand, e > I implies Po [+ x2 (t) I > / [ x' (t) 1), 1 .0] = 1, and using the independence of x' and xs, it follows that, if 0 < Po [ x2 (t) < &(t) an infinite number o l times t t 0], then it will be possible to select tl > 1= > etc. 4 0 such that P. [I x1 (Q I < t., i.o.] = 1. But limP0[1x'(t) 1 < 1] = 0, so this is absurd, and the proof is complete.] t1o

7.12. Dlrichlet problem Consider the Classical DIRICHLET problem for a bounded domain D C RI (d ? 2) : given f E C (8 D), to find a function u E C (D) such that

d u = 0 inside D and lim u(a) = f (b) at each Point b E 8D. a-+b aED

H. LEBESGUE [2: 350-352] discovered that the DIRICHLET problem

is not well-posed in d > 2 dimensions; his example, called Lebesgue's thorn, follows. * Ig, - Ig, Igs = lg(Ig). etc. ; el = e. es = e, etc.

7. BRowNian motion in several dimensions

262

Consider a d L> 3 dimensional spherical surface, push a sharp thorn into its side, deforming it into the surface OD of diagram 1, and think of the domain D enclosed as a little chamber and the tip of the thorn

as a heater, the walls (8D) being held at temperature / E C (8D) (0!5.- tS 1). Consider an extreme case: / m 0 save near the tip of the thorn, at which point f = 1. As t t +oo, the temperature u inside D should converge to the solution of the DIRICRLET problem

with u =./ on D. But the heat radiated from the thorn is proportional to its surface area, and if its tip is sharp enough, a person sitting in the chamber will be cold no Diagram t

matter how close he huddles to the heater; in other words, lim u (a) < 1 = /(0) 1) a-+O

aED

(see COURANT and HILBERT [2: 272-274] for a simple proof of this). Define b E 8 D to be singular if Pb [mRd _ D > 0] = 1, and imitating

J. DooB [2], let us prove that u (a) = Ea (/[x (moD)])

2)

aED

is the solution of the modified Diriehlet problem : to find a bounded /unction u such that d u = 0 inside D and lim u (a) = f (b) at each non-singular a--+b

aED

point b E BD.0

Given a BROWNian path starting at

n

a E D, if m is the passage time to the surface of a little closed sphere B C D centered at a then m8D= in .-f- m8D(m+) m 0

and u (a) = Es(u [x (m)]) is the arithmetic average of u over 8B, i. e., u is harmonic Diagram 2 inside D. Consider next a non-singular point b E 8D(Pb[mRd _D = 0] = 1), let BI be the sphere I b - a I < el, let G be the GREEN function of B,, and select B' : b - a I S es (es < eI) as in diagram 2. Because P. [mB,-D < MOB] is the potential f G de of a non-negative BOREL measure on 0 (B, --- D), it satisfies r < MOB,] = 1 . lim PS[mB,-D < m,3 ,) Z Pb[mB,-D 3) aED

and making es and eI j 0 in that order, it is seen that 4)

s m u(a)

(b)

aED N. WIENER [2] discovered that the modified DIRICHLET problem (with singular point defined in terms Of WIENER'S test) is well-posed. WIENER'S Solution

is the same as the older solution of 0. PERRON [1].

263

7.12. DIRICHLET problem

completing the proof that u is a solution of the modified DIRICHLET problem.

(if possible) a second solution v, select domains D. t D with -D. CGiven D and ODn of class C= and use POINCARE'S test (see 7.10) to check

that a ED,

un(a) = Ea(v[x(maD,)])

5)

converges to v (b) as a E D. approaches b E BD,,. Because the DIRICHLET problem in its classical form is well-posed for such smooth domains,

un = v inside Dn, and granting 6)

P. [x (MOD) singular] = 0,

the fact that v is bounded and approaches i at the non-singular points of OD implies 7)

v = nt+oo lim u = E. ( lim v[x(maDJ]) et+ao

r0.

= E.(/[x(maD)]) = u. Coming to the proof of 6), select open spheres D1 C D C D2 C D3 as in diagram 3 (DI C D C D C D2 C Ds C D3) and, for e> 0,

Diagram 3

VII

let B, be the points of OD at which the electrostatic potential p = P, [mD,-D < MOD, A maD.] is S I - e. B. is compact, U B, is a>o

the set of singular points of aD, p, = P. [MB. < maD, A maD,] S P S I - E on B,, and using GAUSS' principle (see 7.9), it is found that 8)

C(B,) = f Gde,de,= f p, de,< (1 -e)C(B,), B. x B,

B.

where G is the GREEN function of D. - Dl and e, is the electrostatic distribution for B,. But then C (B,) = 0 (e > 0), .K.

9)

Pa[x(maD) singular, MOD < MOD,]

=limP4[x(man)E B., maD= mB.< maD,] .4o S lim .to P4[mB < maD, A maD,]

=0

aED - DI,

and 6) follows on letting D1 shrink to a point. Coming back to LEBESGUE'S thorn, its complete explanation is now before us; it is enough to prove that if b E 8D is singular and if

7. BROwxian motion in several dimensions

264 ..n

f E C (8D) is 1 at b and 0, it is not difficult to see that Ga f =

E' I f e-°t f(x') dt] is continuous on the closed disc; in addition, u

;-'

up,

G; -- G; + (a - fl) GO, G; = 0 (a, fl> 0) as usual, so that G; C' (D) D((a3')

is independent of a, and if Ga f = 0 for a single a > 0,

then 0 = lim a Ga f = f on the open disc, with the result that at+OO

0=a(G; f)(1,0)

1)

f

= E(1. e) [a

e'1 f[1. 0 (1)] f (dt) 1

(1:r(t)-1)

= E(1. e)

Te-- f f [1, 9 (1)] t (dt)

la 0

+00

Nf(1,0)E(1e)fare-°(t(dt)

at +00

u

-f(1,9), proving that Ga : C' (D) -. D ((1) is 1 : I. A generator 05 can now be defined as usual. D(05*) turns out to be the class of functions u E C (D) such that

= -8ul8n

r t,, aEQ, BE B. b E x(ts); the problem is to prove that the space-time motion with state space Q' = [0, + oo) x Q and probabilities Pa. (B') = PQ(w;+ E B') W: t -. (t, x(t)), a' = (s, a) E Q' is simple MARKOV with probabilities no longer depending upon time, i.e.,

that at the expense of adding time as a new coordinate, the temporal dependence of the probabilities can be ignored.

7.15. Spherical Bitowxian motion and skew products

269

Problem 2. Given a smooth positive function u' = v' (t, a) of the pair (t, a) E [0, + oo) x R', if t - x(t) is the sample path of a standard 1-dimensional BROWN.ian motion with probabilities Pa(B) and if, for t a 0, f -1 is the inverse function of the unique solution f = f (s) of

s Z 0,

f(s) = f v-2 [t + f (r), x(r)] dr 0

then the time-dependent motion with probabilities Pa (w+ E B) = P. (x (f -1) E B)

(t. a) E [0, + oo) x R1

au

a

is simple MARKOV and u (t, a) a Pta(x (t1) < b) solves at = I2 us as (t < t1) , while the associated space-time motion is a diffusion with generator

u)(a')=at + 2°= "u

a' =(t,a)E[0,+oo)xR1

(see VOLKONSKII [2] for an ingeneous alternative approach to the above time change).

7.15. Spherical Brownian motion and skew products Define the spherical Brownian motion BM (Sd) to be the diffusion

on the spherical surface S':lxI = I C Rd+1 with generator 1/2 the spherical LAPLACE operator d = dd, closed up as in problem 7.2.1: Ad = (sinp)1-d

1)

a aOF

(sin9)d-1

97 = colatitude,

a a4p

+ (sing')-2dd-1

d1 = 8'/08'.

z.?

PM(

BM (S1) is the projection modulo 2n of the standard 1-dimensional BROWNIan motion BM (R1) onto the unit circle S1. BM (S') is constructed as follows. Given a circular BROWNian motion BM (S1) with sample paths t ---- 0(t) and an independent LEGENDRE process LEG (2) on [0, n] with generator (sin

2)

sin q' a (0 4p (t), the additive functional (clock) t 3)

T W = f [singq(s)]-'ds 0

converges (t Z 0) if 0 < 97 (0) < rc, and the skew product 4)

x = [q', 0 (1)]

is a diffusion since 0 begins afresh at time 1.

7. Biowxian motion in several dimensions

270

00v

Computing its generator 0, if / is the product of a smooth function of colatitude e- = e_ (qq) vanishing at 0 and n and a smooth function of longitude e+ = e+ (0), then as t + 0, 5)

E. X Ee[/(x)] = E,[e- (9' (t)) Es [e+ (e (I))]]

= E,

[e- (9' (t)) [e+ (6) + 2 e+ (e) I] ] + o (t) (sin 97) -1

= e- (97) e+ (8) +

a

sin p

- e- (4) e+ (9) t

z

+ e- (9') 4e+ (0) (sin 97)-'t + o (1). tea

2 functions, and the desired identification of =A //2 for such

a^v

the skew product and BM (S') follows at once. BM (S4) can be factored in a similar manner as the skew product of the LEGENDRE process LEG (d) with generator 6)

(0 < 9' < n)

(sing7)1-d as (sing7)d-1 a

a and an independent spherical BROWNian motion BM (Sd-1) run with t

the clock I = f (sin q )-2 ds. 0

BM (Rd) can also be factored as the skew product of its radial (BESSEL) part BES(d) with generator (a=

d-i

2ar1 + r

7)

a

(r > o)

ar)

and an independent spherical BROWNian motion BM (Sd -1) run with e

the clock I = f r (s)-2 d s ; the proof is the same. 0

The skew product formula BES (2) x BM (S') = BM (R') can be used to check the result of F. SPITZER [1: 194] that for 2-dimensional BROwNian paths not starting at 0, the total algebraic angle q7 (t) swept out up to time t satisfies a

8)

ufm P. (29p(t)/lgt S a) = (()-1

f

-00

I

+

b

Because BM (S') = BM (R') modulo 2n, q7 is identical in law to a 1-dimensional BROwNian motion BM (R') run with an independent BESSEL clock f r,2 ds; this justifies [_.j_Jr;2ds 9)

E.[eiaP(t)] = E.

[e.--

0

7.1 S. Spherical BROWNian motion and skew products

271

and taking a LAPLACE transform, it is seen that 10)

- a$2 f r(e)-1d

+00

+00

/ = f e-Bt E. [ei0 0)] di = E. [ J

e-Ot e

o

d t]

0

0

P>0

is a radial function and satisfies

[ - 2 ars8 + r a - rs 8

1

1

11)

as

r> 0.

l=1

Computing the GREEN function for 11) and solving, it develops that 12)

/(a) =Klal(t!IPa) f IIaI( 2fl b)2bdb + 0

+oo

f

+ I,

and consulting A. ERDELYI [1 (1): 284 (56)] for the inverse LAPLACE transform, +00

13)

E.[e"c(l)] _ (2t)-1 fe

all + b, Yt

Ilal(ab/t) 2bdb

a = r(0)> 0

0

into which we substitute 2a/lg t in place of a, make t t oo, and conclude as follows :

+00 (2t)-Ir

14)

E.[eix2q(I)I 1 =

e

ai+b' 21

I2Iail1at(a b/t) 2bdb

0

f

+00 e-b"12

121.1/191 -j ) b d b

0 e- IaI

_

f etb(1 + b2)-1 db.

S. BOCHNER [2] gives an instructive discussion of spherical differential processes; the spherical BROWNian motion is a special case. A. R. GALMARINO [1] has proved that the most general d ? 2 dimensional diffusion which is isotropic in the sense that its law is unchanged

by the action of the (full) rotation group 0(d) can be expressed as a skew product of its radial motion and the spherical BROwxian motion BM (Sd-1) run with a stochastic clock f which is an additive functional of the radial motion. Problem 1. Given a 2-dimensional BROWNian motion BM (R2), use time changes and skew products to check that if we take the excursions

7. BROWNIan motion in several dimensions

272

leading into the disc Es: r < 1 and rotate them clockwise as in the picture, closing up the gaps, then the resulting motion is identical in law to the reflecting BROwNian motion BM + (E2) with generator (53+ = J A,

u+(1,0) =0, 0S 0 0] must be the uniform spherical distribution. 18*

276

7. BROwNian motion in several dimensions

Given 1>11=1 -s1>ts= '- s,> ... >t,=l -s,>O and letting 1-(1) = 1(t, w'-g) (0;5 t S 1), it is not difficult to see that

Polo+(1 -sl)Ed01,0+(1 -s,)EdO2,..., 0+(1 -s,JEdO,Jr(1):OS1S 1,0+(1)]

9)

tn), wij, 0n, 0n-1) do.-,

= don

102 -t3),wy),03,02d0, P(1(tl - t,), wj), 0,, 01) d01 P(1(1 - t1), wt,) 01, 0+(1)) = P(1-(SJ. 0+(1), 01) dO1 p (I- (S2) - I- (s1) ,

01 ,

.. P(I-(se) - I-(Sn-1),

00 d8,

0n-1, on) don.

But this is just the statement that, conditional on r(t) : 0 S t' 1 A

and 0+(i) , 0+ (1 - s) : 0 S s S 1 is identical in law to 0(1-(s)):0!5'-s 0 is a version of x (t) with

nal

respect to P°, y is constant by BLUMENTHAL'S 01 law, and BIRKHOFF'S

ergodic theorem implies that y = E° (e,) i.e., 5)

P°

Ie(n")

..r

,

Eo(ei)]

= 1.

7. BRowNian motion in several dimensions

278

try

Using the additive property of e and the isotropic property of x, one can easily see that 2n

6) E0(el) =E0[e(ml) - e(m°)] = 2n

0

2n

nE(2,e)(e(m, A

zn f d6 EE(i. o)(e(m$)) 0

2n

f d6 [fR G.,((i, 0), b)e(db)+lim fG1((2. 0), b) e(db) a

Ò.

2n

fe(db) 2n f d@ [Go1((1, 0), b) +, to G,n((2, 0), b)] de,

R

...

where G0, is the GREEN function of the domain Ia I < 2 and G1s that

of I < Iaio s Z fl. Given A E A, p = P. (A) satisfies A p = 0 so that p is constant, and we infer that P. (A) - 0 or I. e (/) is measurable A and therefore constant as regards w, and its invariance under rotations forces e w = f f do, as claimed; see 8.7 for an amplification of this method.]

s'

A

Problem 4. Consider the group 1' of non-euclidean (PoINCAxi?) motions of the open unit disc E2. Give a proof that l leaves unchanged the law of the geometrical curve x (0, ma$.) that the standard BROwNian traveller sweeps out for t < ma8e and find the most general topological map of the open disc onto itself that does this. [Given a non-euclidean motion g and a standard BRowNian path, g OA A map) is a standard BROwxian motion stopped on 8E2, up to a change of time scale, and so the law of x [0, max,) is invariant under g. Given a topological mapping g1: E2 - E2 leaving the law of x[0, mar) unchanged, select a non-euclidean motion g2 : g1 (0) -- 0 and consider 9:,

gs = gs g1Ib

E B] = 2n f IeIP b12 dfl

Pa[gs x(maE) E B) =

(b =gsa)

B .U.

,L"

is harmonic for each arc B CI 8E2. i6 -l b1, is therefore harmonic in a VII

for each 0S # < 2x. Ie a - all is likewise harmonic in b for each 0 S a< 2x (use gal in place of g3), and it follows that to each s

a E [0, 2x) corresponds a fi E [0, 2x) such that Ie'= a constant multiple of Because gs (0) = 0, the constant is 1, and I

b12 .

I

le'0

I-1als

min

Osa f haD, (a, d b) eD (b), 0

and using the RIEsz decomposition [see 7.5], eD (a) is found to be the potential f GD (a, b) m(d b) * of a non-negative charge distribution m, positive on opens. Using the composition rule GD(a, b) = GD,(a, b) + f haD,(a, d6) and the resulting formula for eD, : 5)

6)

b)

a, b E D1 C D

eD, (a) = eD (a) - f haD, (a, d b) eD (b)

- f h0D,(a,

= f GD(a, b) m(db) = f GD,(a, b) m (d b),

one finds that m does not depend upon D. 0 GD is the GREEN function of D.

f GD(E, b) m(db)

7. Beovmian motion in several dimensions

284

As to 2), if (31 applied to u, E D ((b) is S - 1 on D, if u E D (@),

and if U2=U+ n u, , then O$ u2 5 0 on D for t Qt 1 10 u llo and for D, C D, DYNKIN'S formula implies -OD, I00

-A_

v (a) - f hoD1 (a, db) v (b) = -Ea f CS v dt

7)

0

0

aED, v=u, or u=. A second application of the RIEsz decomposition now permits us to express u (a) - f haD (a, d b) u(b) as the potential of a signed charge distribution e", and since non-negative potentials come from nonnegative total charges, the bounds 8a)

a f GD (a , b) m (d b) = a eD (a)

x am Df 45 u

mOD VII

S Ea f @ u dt = f haD(a, db) u(b) j u(a)

f GD(a, b) a"(db)

0

S P eD (a) = P f GD (a, b) m (db)

sup GS U D

show that a m (D) S -e" (D) S P m (D), and 2) follows at once because ® u is continuous. CAD

8b)

b.>

H. P. McKEAN, JR. and H. TANAKA [1] described the speed measures

of a wider class of motions with BROWNian hitting probabilities; in the .,,

...

present case, it can be proved that m is a speed measure if and only if each of its potentials eD = f GD dm is continuous on D and tends to 0 at the non-singular points of 8D, and each of its null sets Z is thin in the sense that, for the standard BROWNian motion,

P.[Zfl(x(t):t>o)=f]=1. .ti

Given such a measure, it is natural to hope that the associated diffusion will be the composition of the standard BROWNian motion with a stochastic clock f -1, e. g., the inverse function of the HELLINGER integral 9)

measure (s : x(s) E db, s

f

W

J

t) m(db)S

2db

Re VI'

and something close to this is true (see H. P. McKEAN, JR and H. TANAKA [1], R. BLUMENTHAL, R. GETOOR, and H. P. MCKEAN, JR [1],

1"1

and also 8.3). Because the 2-dimensional BROWNian motion is persistent, so is the motion attached to (SS in case d = 2. In d:->-' 3 dimensions it drifts * measure (s: x (s) E db, s 5 t) is not a multiple of LEBESGUE measure in dimensions d 2, i. e., local times cannot be defined.

7.19. Diffusion with BROwNian hitting probabilities

285

out to oo. P. (m00 < + oo)* = 1 if Rd can be split into A U B such that

f IbI2-dm(db) < +oo

10)

d

.ti

and B is thin at oo in the sense that, for the BRowNian motion, P. [x(t)EB i. o., It +oo] =0. 11) P. (mw < + oo) - 0 otherwise. WIENER'S test can be used to decide if 11) holds or not. Problem 1. Given a standard 2-dimensional BRowxian motion with components a and b, let t be the local time l m (2e)'1 measure (s : 0 < b (s) < e , s S t) ; the problem is to compute the speed measure e'

and the generator

C33'

of the motion D' : x' = x (f -1) (f = t + 2t)

(see T. UENO [1] for a different description of D'). [e' (d a x d b) = 2dadb on I b l> 0; also, e' (d a x0) is translation invariant and, as such, a constant multiple of the 1-dimensional LEBESGUE measi

f [i + E-' x the indicator of 0 S b< E](x,) ds

ure da. Because f = lim

e40 0

and

f /2dadb+ f /2da

R'

R' x 0

= lim f [1 + E-' x the indicator of 0 S b < e] / 2dadb, 40 R'

e' (da X 0) = 2da, and it follows that if u E D (W) and if A x B is an open rectangle meeting R' x 0, then, up to functions of class C' (A x B), - u = f GCS' 1! 2da on A x B, where G is the GREEN function of X

AxB

B. Given a E A, it is immediate that

A

l` o (2,-)-l [u (a, e) - 2 u (a, 0) + u (a, - E) ] = -1 Zm (2

f

E)-r

J 2n

AxO

= 1` 0

J (a-a')=+e' R'x0

l

= (CAS"

)'

lg (a (a a }, + -c2 t ' u (a', 0) 2da'

(5S' u(a', 0) 2da'

u) (a, 0)-

D ((J) can now be described as the class of functions u E C (R2) such

that

u' =du/2 =1 0 6-1 [u (a, e) - 2u (a, 0) + u (a, -E)]

0//R1x0 on R' X O

is of class C (R2), A being the local BROWNian generator of 7.2.3). 0' u = u' for u E D (05*).] * m0

(:: x (t -) = oo)


286

7.20. Bight-continuous paths

Consider the general motion of 7.1, but now let the sample path be merely right-continuous. (3 can be introduced as before and DYNKIN'S formula applies:

u) (a) = li mEa(m)-1[Ea[u(xm)] - u(a)]

1)

where m is the exit time inf (t: x (t) & B) from a small closed neighborhood B of a. P6(xm E db) need not be concentrated on 8B because the

path is permitted to jump out, and this is reflected in the fact that (K can be a global operator. Consider, as the simplest case, the 1-sided differential process on [0, + oo) with LEvY formula +00

2)

Eo[e-"s(t)]

= exp (_ t [ma + f (1 - e-°1) n(dl)]) `` 0+ +00

mz0,n(dl)z0' f (1 -e')n(dl) e) and

4b)

((5i u) (0) = li+m 1-1

Eo[e-Y'

) - 1]

+00

_-my + f 0+

(e-y1-1)n(dl)

implies that (1 - e-1) n.(dl) converges as e l 0 to the measure n (dl) _ (1 - e-1) s (dl) 5) n' (0) =' M.

1>0

287

7.20. Right-continuous paths r..

But if u E CI (®), then u(1) - u (0)

1>0 A

u' (l) -

6)

1=0

=u+(0) belongs to C [0, + oo) so that +00

(CSS u) (0)

7)

ii o f [u (l) - u (0)] n.(dl) 0

- e-1) ne(dl) = f u' n' (dl) +00

+00 .`t

_ li m f u' (l) (1 +00

= m "+(0) + f [u (1) - u (0)] n (d l) , 0

and using (CSi u) (a) = ((5S ua) (0), it is seen that

(0 u) (a) = m u+(a) + f (u (b + a) - u(a)] n(db)

8)

a ? 0,

0

completing the identification of ( 3 restricted to C' (Ci) . Problem I after E. B. DYNKIN [4: 58]. Check that for the i-sided

stable process with exponent 0 < a e

8!1

2 ea

Eo(m.)=r(a+1) +00

[Given y > 0. f Eo[eyZ(')] di 0

+00

+00

= f e-y1 f dt PO (x(t) E dl) = 2y-' 0

0

+00

= 2r(a)-I f e-y11'- I dl, 0 +00

f dt Po(x(t) E dl) = 2r(a)-I P` dl, 0

and it follows that if f E C [0, + oo) vanishes near + oo and to the left * DYNRIN'S evaluation of E,(m.) contains an incorrect constant.

7. BROwrrian motion in several dimensions

288

of e, then (Go. /) (0) = Eo [11(x) dtJ

= 2r(a)-' f as-' J(a) da a..

= Eo [T/(x(i -m.)) dt +00

= f PO [x (m.) E d a] (Go + I) (a) 0

+00

+00

=f Po[x(m.)Eda]2r(x)-' f

(b-a)a-'/(b)db

a

0

+00

b

=2r(a)-1 f Idb f (b-a)°-'PO [x(m.)Eda], 0

0

oho

or, what interests us, b

f (b - a)°-' P. (x (m.) E d a) = b" -' or 0 according as b ? e or b < e;

now take LAPLACE transforms on both sides and solve. Coming to E. (in.), ``)

it is clear that the scaling x(t) -- e x (t/e°) preserves the measure PO (B). in, is therefore identical in law to a nil, E0 (m.) = £ ' E0(m1) , and the W.?

evaluation E0(ml) = 2/r(a + 1) follows on putting u = e-i (l ? 0) and comparing

=im

t-1 Eo[e-'()

-

"a'

((5 u) (0) and

(G. u) (0) _ -lim E.(m.)-1 Eo[e z(m`) .j0

- i]

f e-'- 1 81v

+CIO

=limE0 (m')-1 sinks a

dl

(1 - e)°

440

1

s

=f el' - l dl/Eo(ml)1'(a) I'(i - a) +00

0

_ -i/Eo(m1) ar(a).] 7.21. Riesz potentials

We know that the NEWroNian potentials

f lb-al2-de(db)

1)

e?O,dz3

are linked to the d-dimensional BxowNian motion via the formula 00

f (22r t)-d/2 g-16 -111'/29

2) 0

dt = constant x Ib - a12-d.

7.21. Ri esz potentials

289

What about the fractional-dimensional (RiEsz) potentials

fib - aIy-de(db)

3)

e

0

0 0, e(R' - B) = 0, T(a)

1

8.1. Similar diffusions

291

of an interval B = (a, b].

[e(db)

= dbf(1 - b)a 2 r(1 - a) is the solution of 1

e(db)

= 1 (0 < a < 1), III

2

1'(a) J (b - a)' a

C, -,[0, 1] is the total charge e[0, 1] = 1/2r(2 - a), and

Cl-a[O,1)= 21'(2` a)

C1[a,b]=C1-a[0,l]=11

t

1=b-a.l

'V'

Problem 3. Given an isotropic d (z 2)-dimensional stable process with exponent 0 < a < 2 and E0 [e` 711(0) = e-1 jrja/2 (y E Rd) , prove

that for compact B C Rd, P. (mB < + oo) is the (d - a)-dimensional electrostatic potential, i.e., the greatest potential r (d-a p(a)

2

di22a_1r

(2)

fIb_ad_ti10160

Problem 4 after PoLYA and SZEGO [1: 39]. Prove that 1 n 2n

a-8

a

J I (1 - r2)2(2 - 2r 000

costo -}- r2)

2

r2 sinty dr dtp d8

2rr(2)r(2 2 a)

05 ES 1

and conclude that (3 - a)-dimensional capacity C3_a (B) of the 3-dimensional ball B:! b I S 1 is

=21-s,r(3

2

a)r(1 2 ")

Draw a picture of the electrostatic distribution e3-a(db) - V

2a-s(1 -

IbJ2)--12db

fbI s I

r(2 Za)r(3 Za)

for a near 0, a = 1, and a near 2 and give a probabilistic interpretation of its changing shape.

8. A general view of diffusion in several dimensions 8.1. Similar diffusions Given a (conservative) diffusion D on a space Q as described in 7.1,

its generator Ci can be expressed in terms of the hitting probabilities 19*

8. A general view of diffusion in several dimensions

292

and mean exit times 1a)

h(a,db) =hoD(a,db) =P.[x(maD)Edb, m,D 1) with standard BRowxian 'L7

hitting probabilities, the mean exit times eD = E. (maD) are the potentials f G dm of a (positive) speed measure m, independent of D (G is n ,-.

cud

the classical GREEN function of D). Each uED(0) can be expressed in the small as the potential f G (a, b) e° (d b) of a signed measure e" (d b) plus a harmonic function, and (SS u = - e"(db) 1) m(db)

C1.

(see 7.19).

T. UENO [1] discovered another case in which (3i can be expressed as a differential operator -e"(db)/m(db). Consider a persistent diffusion, i.e., suppose that 2a)

D + 0 open,

P, (MD < -}- 00)

and add the technical conditions f h(., db) /(b) E C(Q

2b)

- 8D)

/E B(8D)

OD

P.(maD,<MOD) >0

2c)

a(D,UD2, D,f1D2=fa, Q - D, U D2 connected.

Given D, and D. as in 2c) with compact boundaries 8D, and BD2, if 3)

m,=MOD,

m2=min t:x(t)EODs, t>m,)

ms = min (1: x(t) E 49D,, t > ms)

m4=min(t:x(t)E8D2,t> ma) etc., then

P. [MI < m=< m3< etc. t +oo] - 1,

and the hitting places x(ms._,) (n Z 1) and x(m,,,) (n Z 1)

294


on 8D1 and 8D2 constitute ergodic MARKOV chains with one-step transition probabilities 4a) 4b)

db) - p, (a, db)

df)

a,rhaD,(a, d$)hoD,($, db) = p2(a, db).

A condition of K. YOSIDA is satisfied, permitting the introduction of stable distributions e1 and e2: f el (d a) p, (a, d b) = e1(d b)

5 a)

el (8D1) = 1

aD,

e. (d a) P. (a, d b) = e, (d b)

5 b)

es (8D=)

ab

and it turns out that 6)

m(db) = f e1(da) Ea[measure(t: x(t) E db, t < MOD,)] OD,

+ f e,(da) E0[measure(5: x(t) E db, t < MOD,)] OD,

is positive on opens, finite on compacts, and stable:

t > 0; A

f m(da) Pa[x(t) E db] = m(db)

...

7)

v

moreover, a second such stable measure is a constant multiple of m (see also G. MARUYAMA and H. TANAKA [2] and HAS'MINSKIi [1] for 6) and 7), and E. NELSON [1) for a different approach to stable measures). ._.

Given D C Q with mean exit time E. (MOD) < + oo, UENO proves that the GREEN measure 8)

G (a, d b) = Ea[measure (t : x (t) E d b, t < MOD)] cod

factors on D x D into the product of a Green /unction G (a, b) and the stable measure m (d b), and expresses O u as -e"(d b)/m (d b) , using 9)

f G (a, b) e" (d b)

_ - f G (a

,

b) (SS u m (d b)

mOD

_ -Ea f (0 u) (x,) dt u

= u (a) -

f h (a. d b) u (b). OD

G (a, b) should depend on the road map alone, but this is not proved

in general; it need not be symmetric (see 8.4 for an account of the potentials of such unsymmetrical GREEN functions).

8.3. Time substitutions

295

Bearing UENO's result in mind, it is plausible that each diffusion win

should have a positive (speed) measure m (d b) and GREEN functions G (a, b) !v.

...

C1.

depending upon its road map alone such that 0 u = -e" (d b)/m(d b) . e" (db) will depend upon u E D ((b) and the road map alone; see 8.5 for a probabilistic formula for it. 8.3. Time substitutions

Given two diffusions with the same road map, common GREEN functions, sample paths x and x', and generators (b3 U=- m (db)

1 a)

u=-

I b)

u E D(@)

m dab)

u E D (Csl ) ,

it is evident that there should be a time substitution i -> f -' mapping

x -> x', as conjectured in 8.1; in fact, if m' (db) = 1(b) m(db) with 0 < / E C (Q) and if t --> f -' is the classical time change with f (t) t

= f l (x1) d s, then it is a simple matter to show that x (f -') is identical 0

in law to x*. But if m' is singular to m, the above prescription loses its sense, and the HELLINGER integral 2)

f (t) _ ( measure (s : x(s) E db, s S t) m(db) Q

which appears hopeful at first glance, turns out to be intractable, though it seems that it should converge. B. VOLKONSKII [1] in the 1-dimensional case and H. P. McKEAN, JR.

and H. TANAKA [1] in the case of BROwNian hits in d Z 1 dimensions

vii

got over the obstacle by approximating i -> f -1 by suitable classical time substitutions, and R. BLUMENTHAL, R. GETOOR, and H. P. McKEAN, JR. [1] used the same method for a wide class of motions including the general diffusions.. McKEAN and TANAKA noted that, in the case of BROwxian hits, the mean hitting times e' are excessive in the sense of G. HUNT: 140, a E D 3) E.[eD(xt), t< maD] t eD(a) and defined f for t < maD as the limit of 4)

f. (t) = f c-' [eD (x (s)) -

(en (x.) , e < moD)] d s

0

for suitable e 10, the idea being that since 5)

0, u = --'[E. (u (x,), e < map) - u]

u E D (O)


296

converges on D as e 10 to 0 it and since 0' eD = - f , it should be that 6)

o05eDdm=limo0 eDdm'=-dm'

1

and [` {t)

7)

-

J Q

measure (s : x(s) E db, s S I) 0, eo m(db) m(db)

should converge to the HELLINGER integral 2). M. G. SuR [1] has put

this idea into an elegant and useful form. Because the 1-dimensional BROwNian motion has local times, all positive measures are speed measures, but in d ? 2 dimensions the

orb

situation is not so simple; for example, in the case of standard BROWNian

hits, a speed measure has to have continuous potentials and its null sets have to be thin for the BROWNian motion in the sense that if Z2 are open and if m( ,n Zn) = 0, then n21

aEnZn

Pa[mos.j0,nt +oo] 31

nZ1

(see H. P. McKEAN, JR. and H. TANAKA [1]).

8.4. Potentials A few of the deeper properties of potentials due to G. HUNT [2 (3)] end

will now be explained; it will be seen that the standard BROWNian motion was too simple an example because the symmetry of its transition

densities obscured the significance of the backward motion described below.

Given a transient diffusion on a non-compact space Q with a positive stable measure e and continuous transition densities p(t, a, b) such that 1a)

.fe(da)p(t,a,b)

1,*

Q

Ib)

fp(t-s,a,E)p(s,E,b)e(de) Q

G (a, b)

goo

I e)

f+00p(t, a, b) dl < + o0 off the diagonal of Q X Q

0

as in the 3-dimensional Baowxian case, the backward kernel p*(t, a, b) p(t, b, a) satisfies f p*(t, a, b) e(db)

2a)

Q

f p*(t - s, a, e;) p*(s,, b) e(d) = p*(t, a, b),

2b) Q

* 1a) is almost automatic because e is stable.

8.4. Potentials

297

i.e., the p*(t, a, b) can be considered as the transition densities of a conservative backward motion starting afresh at each instant t z 0; backward because if (positive) infinite probabilities are permitted, then the stationary backward motion with probabilities 3 a)

P*[x(tl) E dbl, x(t,) E db2,

.

. ., x(t,,) E

e(db,) p*(t8 - tl, bl, by) e(dbs) ...

p*(tn - to-,, bn-1, b,,) e(dbn)

oo

15 b)

x e$ t

...

15a)

[see problem 7.15.4]. 15 b) should be considered as a reflecting BROwNian

au motion because 14b) can be expressed as = 0, where ao means differentiation along the oblique line pointing at -45° to the outwardpointing normal.

As to the original problem, similar boundary conditions au (1 - p) m u = £1 u' should occur, and the corresponding p an + motions should be expressible in terms of a reflecting barrier motion au (an = 0), a distribution t (t, d B) of (reflecting barrier) local time on Q%

and a boundary motion associated with Z*; see A. VENTCEL [1], T. UENO [2], N. IKEDA [1], K. SATO [1], and K. SATO and H. TANAKA [1]

for some information about this. A nice problem is to explain the meaning of reflecting BROwNian motion on the (MARTIN) closure of a GREENian domain D C R3 (see J. DOOB [81 for information on this point). 8.6. Elliptic operators

Consider a nice differentiable manifold Q and, on it, an elliptic 0

differential operator £1 expressed in local coordinates as 1

2

ideij

ax£aX3

aXI

C generates a diffusion if it has HOLDER continuous coefficients and if

e = [eii] is symmetric and positive definite at each point of Q; to be

8.7. FELLER'S little boundary and tail algebras

303

0

precise, Li agrees on C2(Q) with the generator of a diffusion which is permitted to run out to the point oo before time + oo if Q is non-compact (see PocoRZELSxI [1]). K. ITO [3] proved that the desired motion satisfies the stochastic differential equation d x = ye (x) d b + /(z) dt II

2)

o

up to the time it leaves the neighborhood on which 1) holds. V e is the w

o

w

O

II

A

symmetric positive definite root of e, / = (/1, /2...., /d), b(t) is a standard d-dimensional lROwxian motion, and it is supposed that the e-1 changes according

o

w

o

II

HOLDER exponent of the coefficients = 1. Under a change of local coordinates x -+ to the rule : ..'

3)

, axe axi k,1 dekl -ax, ax;

1

.

eif

so

e,t d xx d x3

4)

induces a RIEMANNian distance on Q, and La can be expressed as the o

LAPLACE-BELTRAMI operator

d=

1

N

II

t

S)

2

2

.

T,

e!

a

ax, I

e,j

a

Ve

8x1

o

m

II

N

z

II

°

plus a first order differential operator (vector field) / corresponding to a drift (see, for example, E. NELSON [1]). 2) now simplifies to dx = d b + /(x) dl, b being the (BROWN ian) motion associated with (1/2) A. Because the differential operator (n u = -eu(db)lm(db) of 8.2 shares with elliptic operators of degree S 2 the property that 0 u S 0 at a local minimum of u E D(05), it is to be hoped that some feature of the above geometrical picture can be salvaged in a general setting. o

o

w

Besides the papers cited above, the reader can consult HAs'MINSKII [1], NELSON [2], YOSIDA [3], and 8.7 for additional information about ellipo

o

O

tic operators. Problem 1. Prove that if a differential operator Li with continuous A

.

coefficients has the property that Li is z 0 at a local minimum of m

u E C°° (Rd), then it is elliptic of degree

2 with no part of degree 0.

w

rhaD(a, db)1(b) E C(D)

m

1)

b

G

8.7. Feller's little boundary and tail algebras Given a diffusion on a space Q such that

/E B(8D),

let B be the tail algebra fl B [x (l) t Z n] modulo the ideal of negligible n 1 events. :


304

Given B E B', P = P.(B) satisfies (a) = P. (u'm E B) = f han (a, d b) p (b)

2)

a E D,

i.e., p E C (Q), and because of

p = f e-t P. (u E B) dt = Ge p,

3)

p is a solution of

OSp51

4b)

p E D((tS)

4c)

c3p=0.

VII

4a)

On the other hand, if p is a solution of 4), then E.[p(x1) I B.] = E.(.)[p(xe-.)]

5)

p (x.) + E.(.) [ T(O p) (xe) do]

t Z s, °IN

= p(x,)

i. e., p (xe) (t ? 0) is a bounded martingale; as such, it converges to a B'

measurable limit p as t t + oo, and one gets 6a)

p(xe) = E.(P I B,) = E=ca(P)

t? 0;

in particular, if p = P. (B) (B E B'), then 6b)

:lim p (xe) =iturn P. (B I Be) = P. (B I B) +00 4 00

= the indicator of B. 'S'

G. HUNT pointed out to us that 6) expresses a 1 : 1 correspondence

between the events BE B' and the extreme points of the (compact

='°

Bpi

Cs.

convex) class of solutions of 4). W. FELLER [8] had used this correspondence to express B' as an algebra of subsets of a topological space Q' attached to Q as its boundary. Q' is what we are calling FELLER's little boundary (see J. FELDMAN [11 for its relation to the MARTIN boundary). Given a non-constant solution p of 4), it is possible to choose a b on B, and if the motion is persistent (i. e., if P. [mD < + 00] M 1 for each open D C Q),

then the sample path enters both A and B i. o. as t t + oo, and the bounded martingale p (xi) (I Z 0) cannot converge because p (xt) < a and p (x1) > b i. o.; in brief, p must be constant and the 01 law 7)

must hold.

P. (B) = 0 or l

BE B'

8.7. FELLER's little boundary and tail algebras

305

Now let us suppose that Q = Rd and that our generator 03 is an elliptic operator: 8)

i S 8x, axi J de{i

with continuous coefficients and let us pose the problem: to find conditions on the quadratic form (e , ) = e11 e + etc. such that the 01 law 7) holds, i. e., such that all the solutions of 4) are constant; in more detail, let V. and y_ be the greatest and smallest eigen-values of e at a point and let us ask what is the maximal rate of growth of y, /y- such that 9a) the motion is persistent, 9 b) all the solutions of 4) are constant, 9c) all the non-negative solutions of Z p = 0 are constant?

It is known that 9c) holds if y+/y_ is bounded. There is also the extraordinary result of S. BERNSTE[N [1] that 9b) holds if d = 2 and if y_ > 0 at each point. BERNSTEIN also gave the simple example

El=(2+4bl)a=+4baa b+aW in

which 9c) fails (p = en-b') and y. /y- - 4b4 as bt +oo; see

E. HOPF [1] for other examples connected with BERNSTEIN's result, and for more up to date information, J. MOSER [1].

ltb/McKean, Diffusion processes

20

C!'

Bibliography Z,.

AM - Ann. of Math.; HAMS- Boll. Amer. Math. Scc. ; 1 JM - Illinois J. Math. ; PNAS - Proc. Nat. Acad. Sc!. U.S.A.; TAMS - Trans. Amer. Math. Soc. ; TV - Year. Veroyatnost. i ee Primenen.

ALEKSANDROV, P., and H. Horn: [I] Topologie, 1. Berlin 1935. BACHELIER, L.: [1] Theorie de la speculation. Ann. Sci.1 sole Norm. Sup. 17, 2t-86 (1900).

goo

BERGMAN, S., and M. SCEIPI*ER: [1] Kernel functions and elliptic differential equations in mathematical physics. New York 1953. BERNSTEIN, S.: (1] Ober ein geometrisches Theorem and seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus. Math. Z. 26, 551-558 (1927). .°w

Basicovrrcn, A. S.: [1] On linear sets of points of fractional dimension. Math. Ann. 1II4 161-198 (1929). - [2] On existence of subsets of finite measure of sets of infinite measure. Indagationes Math. 14, 339-344 (1952). °"'

..,

BESicoviTCH, A. S., and S. J. TAYLOR: [1] On the complementary intervals of a linear closed set of zero Lebesgue measure. J. London Math. Soc. 29, 449-459 Ian

vii

(1954). ova

~7o cue

.N.

BEURLING, A., and J. DENY: (1) Dirichlet spaces. PNAS 45, 208-215 (1959). BISHOP, E., and K. DE LEEUw: [1] The representations of linear functionals by

i°.

measures on sets of extreme points. Ann. Inst. Fourier 9, 305-331 (1959). BLUMENTHAL, R.: (1] An extended Markov property. TAMS 85, 52-72 (1957). BLUMENTHAL, R., and R. GETOOR: [1] Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10, 493-516 (1961). BLUMENTHAL, R., R. GErooR, and H. P. McKEAN, JR.: [1] Markov processes with Goo

tea

identical hitting distributions. IJM 6, 402-420 (1962). .!^

v.i

BOCHNER, S.: [I] Diffusion equation and stochastic processes. PNAS 35, 368-370 (1949).

°7,

l21

Inn

- [2] Positive zonal functions on spheres. PNAS 40, 1141-1147 (1954). 0

BOREL, E.: [I] Les probabilites denombrables et leurs applications arithmetiques. Rend. Circ. Mat. Palermo 27, 247-271 (1909). '°+

BOYLAN, E.: [1] Local times for a class of Markov processes. IJM 8, 19-39 (1964). 0%0

CAMERON, R. H.: [1] The generalized heat flow equation and a corresponding Poisson formula. AM 59, 434-462 (1954). °a°

CAMERON, R. H., and W. T. MARTIN: [1] Evaluations of various Wiener integrals ..,

.+D

by use of certain Sturm-Liouville differential equations. BAMS 51, 73-90 (1945).

CARATHEODORY, C.: (I] Funktionentheorie. Basel 1950.

HIV

,x'

.::

ian motions. J. Math. Soc. Japan It, 263-274 (1959).

eat

C.7

goo

via

CHOQUET, G.: [1] Theory of capacities. Ann. Inst. Fourier, Grenoble 5, 131-295 (1953-54). CHRYSTAL, G.: [1] Electricity. Encyclopaedia Britannica, 9th ed. 8, 3-104 (1879). CHUNG, K. L., P. ERDdS. and T. SIRAo: [1] On the Lipschitz's condition for Brown-

Bibliography

307

,m,

c)'

OZ'

CHUNG, K. L., and W. FELLER: [1] On fluctuation in coin-tossing. PNAS 35, 605-608 (1949). CIESIELSKI, Z.: (1) Holder condition for realizations of Gaussian processes. TAMS 99, 403-413 (1961). COURANT, R., and D. HILBERT: [1] Methoden der mathematischen Physik 1. Berlin 1931.

- [2) Methoden der mathematischen Physik 2. Berlin 1937. 1'M

COURANT, R., and A. HURWITZ: [1] Funktionentheorie. Berlin 1929.

`"Y

-ton

t!1

DANIELL, P. J.: (1) A general form of integral. AM 19, 279-294 (1917-18). DERMAN, C.: [1] Ergodic property of the Brownian motion process. PNAS 40, 1155-1158 (1954). DoBLIN, W.: [1] Sur deux problLmes de M. Kolmogoroff concernant les chalnes d6nombrables. Bull. Soc. Math. France $6, 210-220 (1938). DoE-rscH, G.: [1] Theorie and Anwendung der Laplace-Transformation. Berlin 1937. :°,

°u'

+'"

DONSKER, M. D.: [1] An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc. No. 6 (1951).

;^,

.`S

O',

000

':f

^V'

°n4

0.0

'.9

Sao

r.°

".woo

ado

DooB, J. L.: [1) Stochastic processes. New York 1953. - [2] Semi-martingales and subharmonic functions. TAMS 77, 86-121 (1954). - [3] Martingales and one-dimensional diffusion. TAMS 78, 168-208 (ms)- (4) A probabilistic approach to the heat equation. TAMS 80, 216--280 (1955). - (5] Brownian motion on a Green space. TV 2, 3-33 (1957). - [6] Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431-458 (1957). - (7) Discrete potential theory and boundaries. J. Math. Mech. 8,433-458 (1959). - [8] Boundary properties of functions with finite Dirichlet integrals. Ann. Inst. Fourier Grenoble 12, 573-622 (1962).

o'-;

9.2

Doos, J. L., J. SNELL, and It. WILLIAMSON: [1] Application of boundary theory to 'C.

.0.'47

0.2

...

sums of independent random variables. Contributions to Probability and Statistics, 182-197. Stanford University Press 1960.

O°>

P+1

!!1

...

ray

DVORETSKY, A., and P. ERD6s: [1] Some problems on random walk in space. Proc. Second Berkeley Symposium on Math. Stat. and Probability, 353-367. University of California Press 1951. DvoRETSKY, A.. P. ERDSs, and S. KAKUTANI: [1] Double points of Brownian

Hxycne0wo

.^.

motion in n-space. Acta Scientarum Matematicarum (Szeged) 12, 75-81 (1950).

.-.

"-'

...

- (2) Multiple points of paths of Brownian motion in the plane. Bull. Res. Council Israel 3, 364-371 (1954). - [3] Nonincreasing everywhere of the Brownian motion process. Proc. Fourth Berkeley Symp. on Math. Stat. and Probability II, 103-116. Univ. of Calif. Press. 1961. G''

DVORETSKY, A., P. ERD6S, S. KAKUTANI, and S. J. TAYLOR: [1] Triple points of

--o

'L!

aO`^

Brownian paths in 3-space. Proc. Camb. Phil. Soc. 53, 856-862 (1957). DYNKIN, E. B.: [1] Continuous one-dimensional Markov processes. Dokl. Akad. Nank SSSR 105, 405-408 (1955).

'°w

i.,

.5C

- (2) Infinitesimal operators of Markov random processes. Dokl. Akad. Nauk SSSR 105, 206-209 (1955). - (3) Markov processes and semi-groups of operators. TV 1, 25-37 (1956). - [4] Infinitesimal operators of Markov processes. TV 1, 38-60 (1956). - [5] One-dimensional continuous strong Markov processes. TV 4, 3-54 (1959). - (6) Principles of the theory of Markov random processes. Moscow-Leningrad 1959. 20*

Bibliography

308

DYNKIN, E. B.: (7) Natural topology and excessive functions connected with a Markov process. Dokl. Akad. Nauk SSSR 127, 17-19 (1959). - [8] Markov processes. Moscow 1963. DYNKIN, E. B., and A. YusKEVIC: (1) Strong Markov processes. TV 1, 149-155 Òn

(1956).

à+

...

EINSTEIN, A., (1) Investigations on the theory of the Brownian movement. New York 1956. ERDHLYI, A. (Bateman manuscript project): (1) Tables of integral transforms. 1, 2. New York 1954. - [2] Higher transcendental functions. 1.2; 3. New York 1953; 1955. ERDBS, P.: [1] On the law of the iterated logarithm. AM 43, 419-436 (1942). ERD6S, P., and M. KAC: [1] On the number of positive sums of independent random variables. BAMS 53, 1011-1021 (1941). ewe

W!1

a-.

FELDMAN, J.:

[1] Feller and Martin boundaries for countable sets. IJM 6,

e+1

C";

7;'E

.°w

'^+

356-366 (1962). FFLLER, W.: (1) Zur Theorie der stochastischen Prozesse (Existent- and Eindeutigkeitssatze). Math. Ann. 133, 133-160 (1936). - [2] On the general form of the so-called law of the iterated logarithm. TAMS 54, 373-402 (1943)[3) An introduction to probability theory and its applications. New York 1957 (2nd ed.). IQ.

- [4] The parabolic differential equations and the associated semi-groups of ...

transformations. AM 55, 468-519 (1952). - [5] The general diffusion operator and positivity preserving semi-groups in

r..

'ti

q.°

one dimension. AM 60, 417-436 (1954). [6] Diffusion processes in one dimension. TAMS 77, 1-31 (1954). - [7] On second order differential operators. AM 61, 90-105 (1955). - (8) Boundaries induced by non-negative matrices. TAMS 83, 19-54 (1956). - [9] Generalized second order differential operators and their lateral conditions. IJM 1, 459-504 (1957)[10] On the intrinsic form for second order differential operators. IJM 2, 1-18 (1958).

C1.

we?

,-,

[11] Differential operators with the positive maximum property. IJM 3, 182 - 186 (1959). [12] The birth and death processes as diffusion processes. J. Math. Pures Appl. 38, 301-345 (1959).

C''

- (13] On a generalization of Marcel Riesz' potentials and the semi-groups .^.

!r+

generated by them. Comm. Semi. Math. Univ. Lund Suppl. Vol. 72-81 (1952). FROSTMAN, 0.: [1] Potentiel d'dquilibre et capacitd des ensembles avec quelques applications a la th6orie des fonctions. Medd. Lunds Univ. Mat. Sem. 3, 1-11S (1935). ,,.

GALMARINO, A. R.: [1] Representation of an isotropic diffusion as a skew product.

'L!

.°w

...

.v.

a00

J'0

.a1

Z. Wahrscheinlichkeitstheorie 1, 359-378 (1963). GAUSS, C. F.: (1) Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhaltnis e des Quadrats der Entfernung wirkenden Anziehungs- and AbstoDungskrafte. Ostwalds Klassiker der exakten Wissenschaften, No. 2. Leipzig 1889. HALMOS, P. R.: [1] Measure theory. New York 1950. HARRIS, T. E. and H. Rossixs: [1] Ergodic theory of Markov chains admitting an infinite invariant measure. PNAS 39, 860-864 (1953). HAS'MINSKII, R. Z.: (1) Ergodic properties of recurrent diffusion processes and stabilization of the problem to the Cauchy problem for parabolic equations. TV 5, 196-214 (1960).

Bibliography

309

HAUSDORFF, F.: [1] Dimension and AuBeres Mats. Math. Ann. 7$, 157-179 (1918).

urn

HEINS, M.: [1] On a notion of convexity connected with a method of Carleman.

J. d'Anal. Math. 7, 53-77 (1959). tai

fix

HILDEBRAND, F. B.: [1] Advanced calculus for engineers. New York 1949.

!ti

.pr

^.a

a"1

'--'

HILLE, E.: [1] Representation of one-parameter semi-groups of linear transformations. PNAS 28, 175-178 (1942). HINCIN, A. YA.: [1) Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Ergebn. Math. 2, No. 4, Berlin 1933. HoPr, E.: [1] Bemerkungen zu einem Satze von S. Bernstein aus der Theorie der elliptischen Differentialgleichungen. Math. Z. 23, 744-745 (1929). - (2) Ergodentheorie. Ergebn. Math. 5, No. 2, Berlin 1937. HUNT, G.: [1] Some theorems concerning Brownian motion. TAMS 81, 294-319 (1956).

n0°

yip

off.,

I;;

'n7

-:z

- (2] Markoff processes and potentials. t, 2, 3. IJM 1, 44-93 (1957); 1, 316-369 (1957); 2., 151-213 (1958). - [3) Markoff chains and Martin boundaries. IJM 4, 313-340 (1960). IKEDA, N.: [1] On the construction of two-dimensional diffusion processes satisfying Wentzell's boundary conditions and its application to boundary value problem. Mem. Coll. Sci. Univ. Kyoto, A. Math. 33, 368-427 (1961). ITS, K.: (1) On stochastic processes. 1. Japanese J. Math. 18, 261-301 (1942). - (2] On a stochastic integral equation. Proc. Japan Acad. 22, 32-35 (1946). - [3] Stochastic differential equations on a differentiable manifold. Nagoya Math. J. It 35-47 (1950). - [4) Multiple Wiener integral. J. Math. Soc. Japan 3, 157-169 (1951). - [5] On stochastic differential equations. Mem. Amer. Math. Soc. No. 4 (1931). ITS, K., and H. P. McKEAN, )R.: (1) Potentials and the random walk. IJM 4, 119-132 (1960). - [2) Brownian motion on a half line. IJM 1, 181-231 (1963). KAc, M.: [1] On some connections between probability theory and differential and integral equations. Proc. Second Berkeley Symposium on Math. Stat. and Probability, 189-215. University of California Press 1951. ego

0;,,

'C1

Nam

+'1

q,;

!5q.

Hvz

.-.

KAKUTANI, S.:[I] On Brownian motion in n-space Proc. Acad. Japan 26, 648-652 (1944).

h'7

- [2] Two-dimensional Brownian motion and harmonic functions. Proc. Acad. Japan, 29, 706-714 (1944). - [3] Random walk and the type problem of Riemann surfaces. Contribution to the theory of Riemann surfaces, 95-101. Princeton University Press 1961.

o°w-

KALLIANPUR, G., and H. RoBBINS: [1) Ergodic property of the Brownian motion

process. PNAS 3*, 525-533 (1953). KAMETANI, S.: [1] On Hausdorff's measures and generalised capacities with some ."'

.°0

of their applications to the theory of functions. Jap. J. Math. 19, 217-257 °.'

(1945).

I91

'CS

KARLIN, S., and J. McGREGOR: (1] The differential equations of birth-and- death-

process" and the Stieltjes moment problem. TAMS 85, 489-546 (1957). r-,

- [2] Random walks. IJM 3, 66-81 (1959).

- [3] Coincidence probabilities. Pacific J. Math. 9, 1141-1164 (1959). - [4] A characterization of the birth and death processes. PNAS 45, 375-379 (1959). °,,.

eat

v

- [5] Classical diffusion processes and total positivity. Technical Report No. 1, Appl. Math. and Stat. Lab., Stanford University, California 1960. KEL ocG, O. D.: [l] Foundations of potential theory. Grundlehren der Math. Wfssenschaften 31, Berlin 1929.

310 !:J

Bibliography

KNIGHT, F.: (1] On the random walk and Brownian motion. TAMS 103, 218-228 (1962).

- [2] Random walks and a sojourn density process of Brownian motion. TAMS 191, 56-86 (1963). KOLMOGORov, A. N.: [1] Uber die analytischen Methoden in der Wahrscheinlicho'1

keitsrechnung. Math. Ann. 104, 415-458 (1931). - [2] Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergebn. Math. 2, No. 3. Berlin 1933.

are

>°O

KOLMOGORov, A. N., I. PsTRovsK1, and N. PISCOUNOV : [1] Etude de 1'6quation ;V.

.,

app

WATANABE, S.: [1] On stable processes with boundary conditions. J. Math. Soc. Japan 14, 170-198 (1962). WATANABE, T.: [1] On the theory of Martin boundaries induced by countable 01,,

Markov processes. Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 33, 39-108 (1960).

WsYL, H.: [1] 10ber gew8hnliche Differentialgleichungen mit Singularitaten and

die zugeh8rigen Entwicklungen willkiirlicher Funktionen. Math. Ann. $8, 220-269 (1910). ?+A

- [2] Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig vii

gestalteten elastischen KOrpers. Rend. Cir. Mat. Palermo 39, f-50 (1915). WHITMAN, W.: Some strong laws for random walks and Brownian motion. TAMS,

to appear. WIENER, N.: [1] Differential space. J. Math. Phys. 2, 131-174 (1923). ,a°

._.

- (2] The Dirichlet problem. J. Math. Phys. 3, 127-146 (1924). - (3] Generalized harmonic analysis. Acta Math. 33, 117-258 (1930).

'CS

end

App

C°7

YOSIDA, K.: [1] On the differentiability and the representation of one-parameter semi-group of linear operators. J. Math. Soc. Japan 1, 15-21 (1948). - [2) Integration of Fokker-Planck's equation in a compact Riemannian space. Ark. Mat. 1, 71-75 (1949). - [3] A characterization of second order differential operators. Proc. Japan Acad. 31, 406-409 (1955). w...

- [4] Fractional powers of infinitesimal generators and the analyticity of the semi-groups generated by them. Proc. Japan Acad. 36, 86-89 (1960).

r,,

List of Notations a

a LAPLACE transform variable. a BOREL algebra. B [x (t, w) : a t < b] the smallest BOREL algebra measuring x (1): a the algebra B [x, : s 1]. B, B

C (Q) C" (Q)

w00

.-.

the algebra (B : B n (m < t) E B4, t ? 0). the smallest BOREL algebra measuring all open subsets of Q. the space of bounded BOREL measurable functions t: Q -i R1. the space of bounded continuous functions / : Q - R1. !gyp

B(Q) B (Q)

fib

BM+

t < b.

the space bounded, n-times continuously differentiable functions

/:Q-.R1. C. (Q)

the space of continuous functions vanishing at oo.

C (B) d

dimension.

D

dD D D (CS) A

the NEwroxian capacity of B (see 7.8).

the space of functions / E B (Q) such that / (a f) _ /(a) if P.[m,= = 0] = I in chapters 3-6 (see 3.6), and, in chapters 7-8, a domain (= connected open set). the boundary of the domain D. a diffusion. the domain of W. the LAPLACE operator.

the uniform distribution on the surface of a ball. the unit disk. the expectation of / based on P.. E. (/) the expectation of / times the indicator function of A. E. (/, A) e(dy, a, b) = e f e an eigendifferential (see 4.11). a spectral measure (see 4.11). f(dy) the left derivative. /- (a) the right derivative. /+ _ /* (a) the BOREL measure based on the interval function /(da) do

.,a

E=

-7-

/(a,b] =/(b+)-/(a+) coo

..7

a local time integral (see 5.2). f = f (t) g = g (t, a, b) the GAUSS kernel (2rr g)_h12 exp [- (b - a)2/21]. the increasing and decreasing solutions of (5y g = a g (see 4.6). 91-92 G = G(a, b) a GREEN function (see 2.6, 4.11, and 7.4). a GREEN operator (see 3.6). G. a generator (see 3.7). 0 the classical harmonic measure of B C d D as seen from a E D (see 7.12). h (a, B) 1"'

Ah

the HAUSDORFF measure based on h (see note 2.5.2).

K. X A

kt

.C:

I = I(t)

the left shunts (see 3.4). the right shunts (see 3.4). the killing rate for K_ fl K, (see 4.8). a killing measure (see 4.3). the killing measures for shunts (see 4.8). the killing functional (see 5.6).

K_

314

List of Notations the HAUSDORFF-BESICOVITCH a-dimensional measure (see note 2.5.2).

Ls (Q, m)

the space of m-measurable functions/ with 11f IN = V f f dm < + oo. modulo null functions.

nay

Aa

:iii-1

m

a speed measure (see 4.2).

a MARKOV time (see 1.6 and 3.2).

in

the passage time to a. the entrance time into B (see problem 7.1.1). the killing time (see 2.3).

Me

ma mCO

a LEVY measure (see note 1.7.1).

P. (B)

the probabilities for paths starting at a. P.(B) as a function of a.

r-,

n

'Z1

P. (B) P = p (I. a, b) a transition probability kernel (see 4.11).

a PoissoN measure figuring in the Livy decomposition of an increasing differential process (see note 1.7.1) or a projection (see 4.11).

the state space of a diffusion. the d-dimensional EvcLtnean space. s = s (a) a scale (see 4.2). st the shunt scales (see 4.8). S' the d-dimensional unit sphere C R'+1 I time. t = t (t, a) = t (1, a, w) a local time (see 2.2 and 5.4). W a space of sample paths. w : I -. z(1) a sample path. the shifted path : x, (w;) = x,., (w) . W.* the stopped path x, (w;) = xn, (w) . W., x(1, w) = x(t) the value of the sample path w at S. ZI the set of all integers. 8 _ (I: x (t) = 0) a visiting set. (n i) the (open) intervals of CO, oo) - 8.

#

'ti

.Ow

.Ow

bpi

CA'

Q Rd

the smaller of a and b. the greater of a and b. the number of objects in the class C.

aAb

aVb # (C) n

for m=0, 1....,n.

m

1gsl = lg(lgl),

Ig(19.-19) (n a 3).

sup Jf Q

f fsdm. a t b

a increases to b. infinitely often.

fEt fE.

an increasing (= non-decreasing) function. a decreasing (= non-increasing) function. III

a decreases to b.

i. o.

III

ajb

(x(t) < h(t), t j 0) means that there exists l1 > 0 such that. x(t) < h(t) for .,.

0 12 . the empty set. an extra state (see 2.3).

f CO

the ends of the line.

..N

00

Index

additive functionals of Brownian mo-

t = local time, a = (9: x - 0) K70

d = dimension, 0 - generator,

Brownian motion, reflecting on CO, +oo) ,

tion as local time integrals

as I xI 4f

as t- - x 41 'a'

individual ergodic theorem (d = 228. (d 277 KAC's formula 54 applications of KAC'S formula 5Z 58

as x(f"I) 81 (see also local time, zeros) reflecting in a disc, application to NEUMANN problem 264 17o

190

application

(see also time substitutions)

to boundary condi-

.,.

tions for A12 273. 301

additive functional used as clock of skew product 269

skew 115

space-time 266

arc sine law (see standard 1-dimensional BROWNian motion, random

application to DIRICHLET problem for 8/at _d/2 267 KOLMOGOROV'S test 268

walk)

8pp

Are'

spherical 202 connection with projection 281

application to BRowNian local ex-

f (2r)-1 61

G3' as radial part of 4 60 passage times 62, f

,,.

,..

proof that path has no roots 6L 62, 1U Blumenthal's 01 law (d a > 25, 85, (d > 1 233 Borel-Cantelli lemmas 3 boundaries FELLER'S little 303 CPC

MARTIN 300

posing of boundary conditions on 301

Brownian motion, conditional (- tied) -, covering 219

-, elastic 4. -, Filler's 186

282

standard 1-dimensional 12

as solution of r = b + (d - 1) X

Si

individual ergodic theorem 28L arc sine laws 28, 5L 58 approximation by coin-tossing 38 CHUNG-ERDOS-SIRAO test 36

construction of P. LivY 12 construction of N. WIENER 21 excursions 7-5

excursions described in terms of tied BEssaL processes 79 0 and GREEN operators 11

M°pod5.

times 63 application to BROWNIan cursions 79

stereographic F..

backward motion 291 Bessel process 52

HOLDER condition 36

individual ergodic theorem 228 KAc's formula for additive functionals of 54 KALLIANPUR-ROBBINS law 229 KOLMOGOROV's test 31 161

LAMPERTI's renewal process based

on 8 78 law of iterated logarithm 3.3, 36 local times 63 local times, conditional, as diffusion 63

Index

316

Brownian motion, standard 1-dimensional, MARKOV times 22

MARKOVIan property of 6 22 maximum function 27, 4J moments IS non-differentiability iS passage times 25

Brownian motion, standard severaldimensional, WIENER'S test, 255 WIENER's test applications of 259

(see also boundaries, capacities, diffusion, DIRICHLET problem,

excessive functions, harmonic functions, potentials)

reflection principle of D. ANDRE 26 pap

scaling i8 time reversal 1.8

capacities, fractional-dimensional, connection with HAUSDORFF-

winding of sample path 229

e50

KALLIANPUR-ROBBINS law 277

winding, SPITZER'S law of 270

(see also standard several-dimen-

distribution of eigenvalues of 4 224 DVORETSKY-ERDdS-SPITZER

test

1-64

excessive (= superharmonic) functions 2.3 0 = J/2 234

expressed in terms of RIESz measure 24S GREEN function of (a - 4L2) u 233

GREEN function of J L / and GREENian domains 231 law of iterated logarithm 61

multiple points 261 NEwTONian electrostatic poten-

tials expressed as BROwxian hitting probabilities 241 POINCARi'S test 257

wandering out to 00 236

LAPLACE limit theorem)

Choquet's inequalities for NEWTONian capacities 251 Chung-Erdos-Sirao test for BROWNian motion 36 coincidence probabilities 158 coin-tossing (see random walk) communication 92

eon

sional BROWNIan motion)

-, standard several-dimensional 23 BEssEL process as radial part of 60 DIRICHLET problem for d 262

central limit theorem (see de MOIVRE,/1

a0.

multiple points 260 visiting each disc i. o. D6 WIENER'S test applied to tip of simple curve 257

-,

CHOguET's inequalities 251 SPITZER'S interpretation of (d > 252 (see also potentials, WIENER'S test) Riesz (fractional-diensional), connection with stable processes 2900 .:1

GREEN functions and GREENian domains 231 individual ergodic theorem 277 KAKUTANI'S test and logarithmic capacities 252

252

-, natural of G. HUNT 298 -, Newtonian 250

°b-

tions to conformal mapping and

BESICOVITCH dimension 227

-, logarithmic, and KAKUTANI's test

t,.

zeros 29

-, standard 2-dimensional, area of sample path = 0 233 covering motions with applica-

concave functions 113

relative to a linear form 122 creation and annihilation of mass 200 construction of explosions 206

explosion and extinction probabilities as solutions of nonlinear parabolic equations 209 crocodile, Littkwood's 268

de Moivre-Laplace

limit theorem, DONSKER's extension of 412 HINCIN's proof 10

proof using STIRLING's approximation 12 descriptions 85 differential processes, isotropic 288 connection with RIEsz potentials 290

0 as fractional power of A12 289 -, with increasing paths, associated

Livv formula and POISSON measure 3t 146

Index

di/fusion,

several-dimensional,

stable

mass distribution 294 time substitutions 29S

ado

differential processes, with increasing paths, BROwNian passage times

317

oo]]

as 25 (Si 286

(see also BROwNian motion, elliptic

loo

L'.

0'a

zoo

tai

'"`

33

diffusion, general 1-dimensional 83 coincidence probabilities 158 communication 22 decomposition into simple pieces

differential operators, excessive functions, skew products, spacetime motions) Dirichlet problem, for A 261 application to BESSEL hitting probabilities 62

4)0

inverse local time as 212 one-sided stable process as

harmonic measure expressed as

92

BROwxian hitting probabilities

DYNKIN's formula for Qs 99

264 LEBESGUE'S thorn 261 LITTLEWOOD'S crocodile 268

0 98

PoissoN formula 249. 300 ""'

woo

Coo

POINCARH'S test 257

'6J

general view of ($ as differential operator 105

solution in terms of BRowxian motion 262

stn

GREEN functions 154 GREEN operators 94

e'3

etc. 1g9 excursions 223

eigendifferential expansions for (bS,

WIENER'S test 262

-, for 8/a t - d/2 266

local times 174

connection with KOLMOGOROV'S

sex

MARKOV times 86 MARKOVian property of 84

test 268 solution in terms of space-time BRowNian motion 11

matching numbers 889

Dynhin's formula for 0 (d = 1) 99. I^'

S. BERNSTEIN'S

theorem 35

exit and/or entrance 19

=per

endpoints, FELLER's classification of, as 1311

entrance 19$ ergodic theorem, individual, for BROWN-

a.

Diffusion Processes and their Sample Paths

Diffusion Processes and Fertility Transition

Controlled Diffusion Processes

Multidimensional Diffusion Processes

Schroedinger diffusion processes

Multidimensional diffusion processes

Multidimensional Diffusion Processes

Diffusion, Markov processes and martingales. Ito calculus

Schrödinger diffusion processes (Probability and its Applications)

Schrödinger Diffusion Processes (Probability and its Applications)

Stochastic differential equations and diffusion processes

Stochastic Processes and Their Applications

Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes

Multidimensional stochastic processes as rough paths

High Dimensional Nonlinear Diffusion Stochastic Processes

High Dimensional Nonlinear Diffusion Stochastic Processes

Diffusion Processes During Drying of Solids

Multidimensional Diffusion Processes (Classics in Mathematics)

Diffusion processes in advanced technological materials

Sample

Markov Point Processes and Their Applications

Stochastic Differential Equations and Diffusion Processes, Second Edition

Stochastic Differential Equations and Diffusion Processes, second edition

Kinetic Processes: Crystal Growth, Diffusion, and Phase Transitions in Materials

Stochastic Differential Equations and Diffusion Processes (North-Holland mathematical library)

Stochastic Differential Equations and Diffusion Processes, Second Edition

Stochastic Differential Equations and Diffusion Processes, Second Edition

Stochastic Differential Equations and Diffusion Processes, (Second Edition)

Kinetic Processes: Crystal Growth, Diffusion, and Phase Transitions in Materials

Growth and diffusion phenomena

Diffusion Processes and their Sample Paths

Diffusion Processes and Fertility Transition

Controlled Diffusion Processes

Multidimensional Diffusion Processes

Schroedinger diffusion processes

Multidimensional diffusion processes

Multidimensional Diffusion Processes

Diffusion, Markov processes and martingales. Ito calculus

Schrödinger diffusion processes (Probability and its Applications)

Schrödinger Diffusion Processes (Probability and its Applications)

Stochastic differential equations and diffusion processes

Stochastic Processes and Their Applications

Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes

Multidimensional stochastic processes as rough paths

High Dimensional Nonlinear Diffusion Stochastic Processes

High Dimensional Nonlinear Diffusion Stochastic Processes

Diffusion Processes During Drying of Solids

Multidimensional Diffusion Processes (Classics in Mathematics)

Diffusion processes in advanced technological materials

Sample

Markov Point Processes and Their Applications

Stochastic Differential Equations and Diffusion Processes, Second Edition

Stochastic Differential Equations and Diffusion Processes, second edition

Kinetic Processes: Crystal Growth, Diffusion, and Phase Transitions in Materials

Stochastic Differential Equations and Diffusion Processes (North-Holland mathematical library)

Stochastic Differential Equations and Diffusion Processes, Second Edition

Stochastic Differential Equations and Diffusion Processes, Second Edition

Stochastic Differential Equations and Diffusion Processes, (Second Edition)

Kinetic Processes: Crystal Growth, Diffusion, and Phase Transitions in Materials

Growth and diffusion phenomena

Recommend Documents