Symmetric Markov Processes (Lecture Notes in Mathematics, 426) 3540070125, 9783540070122

Citation preview

Lecture Notes in Mathematics Edited by A. Dold and 8. Eckmann

426 Martin L. Silverstein

Symmetrie Markov Processes

Springer-Verlag Berlin · Heidelberg · NewYork 1974

Prof. Martin L. Silverstein University of Southern California Dept. of Mathematics University Park Los Angeles, CA 90007/USA

Library of Congress Cataloging in Publication Data

1939Silverstein, Martin L Symmetric Markov processes. (Lecture notes in mathematics ; 426)

Bibliography: p.

1. Markov processes. 2. Potential, Theory of. I. Title. II. Series. Lecture notes in mathematics (Berlin) ; 426. 510'.8s [519.2'33]74-22376 QA3.L28 no. 426 [QA274.7]

AMS Subject Classifications (1970): 60J25, 60J45, 60J50 ISBN 3-540-07012-5 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07012-5 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.

© by Springer-Verlag Berlin · Heidelberg 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

DEDICA TED

TO

W. FELLER

Introduction This monograph is concerned with symmetric Markov processes and especially with Dirichlet spaces as a tool for analyzing them. The volume as a whole focuses on the problern of classifying the symmetric submarkovian semigroups which dominate a given one. The main results are contained in Chapter III and especially in Section 20. A modified reflected space is determined by a boundary together with an intensity for jumping to

t:,

t:,

rather than to the dead point.

Every dominating semigroup which is actually an extension is subordinate to at least one modified reflected space.

The extensions subordinate to a

given modified reflected space are classified by certain Dirichlet spaces which live on the appropriate

t:,.

When the intensity for jumping to

t:,

vanishes identically, the subordinate extensions all have the same local generator as the given one.

The most general dominating semigroup

which is not an extension is obtained by first suppressing jumps to the dead point and/or replacing them by jumps within the state space and then taking an extension. Some general theory is developed in Chapter I. A decomposition of the Dirichlet form into killing 11 , 11

11

jumping

11

and "diffusion" is

accomplished in Chapter II. Examples are discussed in Chapter N. Each chapter is prefaced by a short summary. The main prerequisite is familiarity with the theory of martingales as developed by P. A. Meyer and his school. Little is needed from the theory of Markov processes as such, except from the point of view of motivation.

VI

For a treatment of classification theory in the context of diffusions we refer to [20] and [30]. In fact it is M. Fukushima 's paper [20] that inspired our own research in this area and his influence is apparent throughout the volurne. For more inforrnation on "sample space constructions" for extensions of a given process we refer to Freedman's book [52] where current references to the literature can be found. The expert typing was dorre by Elsie E. Walker at the University of Southern California.

Notations

g:

Throughout the volume

is a separable locally compact

Hausdorff space and dx is a Radon measure on Q which charges

quasi

every nonempty open set. The indicator of a setwill be denoted both by lA and I(A). a

condi.tion

and

$ I(X t

E

The integral of a function

over the set determined by

such as "X E ru will be denoted both by $[X Er: s] t t r)

s.

The measure which is absolutely continuous with

respect to a given measure cp • f..! •

s

f..! and has density cp will often be represented

The subcollection of bounded functions in ,!; will be denoted by Jb.

All functions arereal valued.

2 2 In particular L (dx) or L (X,dx) is the

real Hilbert space of square integrable functions on the measure space

(g:, dx)

and

C com (Q),

C (:lQ 0

are the collections of real valued

respectively with compact support and

continuous functions on

~

"vanishing at infinity."

Questions of measurability are generally taken

for granted-thus functions are usually understood to be measurable with respect to the obvious sigma algebra.

Table of Contents I.

General Theory _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

3

1. Transience and Recurrence_ _ _ _

_

3

_ _ _ _ _ _ _ _ _ _ _

20

2. Regular Dirichlet Spaces _ _

_ _ _ _

_ _ _

3. Same Potential Theory _ _ _ _ _ _ _

_ _ _ _ _

_ _

_ _

5. An Approximate Markov Process __ _ _

_ _ _ _ _

7. Balayage _ _ _ _ _ _

_ _ _ _

_

_

_

_ _ _ _

_ _ _ _ _ _ _

8. Random Time Change _ _

_

_

_ _ __

_

_ _ _ _ _

_ _ __ _ _ _ _ _ _ _

Decomposition of the Dirichlet Form _ _ _ - - - 9. Potentials in the Wide Sense_ _ _ _ _ _ _

-

_

24

39

_

61

_ _

69

_ _

_

78

_

_ _ _ _

84

-

- - -

-

97

_ _ _

98

_ _ _ _ _

10.

TheLevyKernel _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

102

11.

The Diffusion F o r m _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

112

J _ _ _ _ _ _ _ _ _ _ _ _ __

126

12. Characterization of III.

_ _

4. Construction of Processes_ _ _ _ _ _ _ _ _

6. Additive Functi0nals _ _ _

II.

_

_ _ _ _ _ _

x and

Structure Theory_ _ _ _ _ _ 13.

Preliminary Formula _ _ _

_ _ _ _ _

_ _

_ __ _ _ _ _ _

_ _ _ _ _ _ __ _ _ _ _ _ ___

14. The Reflected Dirichlet Space _ _ _ _

_ _ _

_ _ _ __ _ _ _ _

130 133 143

15. First Structure Theorem _ _ _ _ _ _ _ _ _ _ _ _ _ _ ·- _ _

152

16. The Recurrent Case _ _ _ _ _ _ _

_ _ _ __

158

17. Scope of First Structure Theorem- __ - - - - - - - - - - -

165

18. The Enveloping Dirichlet Space _ _ _ _ _ _ _ _ _ _

173

_ _

_ _ _ _

_ _ _

_

X 19. Equivalent Regular Representations __ _ _ _ _ _ _ _ _ _ _ _

178

20. Second Structure T h e o r e m - - - - - - - - - - - - - - - - -

183

Third Structure Theorem_---- _ _ - - - - - - - - -

216

Examples _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

220

_ _

222

21.

N.

22.

Diffusions with Bounded Scale; No Killing _ _ _ _ _ _ _

23.

Diffusions with Bounded Scale; Nontrivial Killing __ - _ - -

225

24. Unbounded Scale _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

237

Infinitely Divisible Processes _ _ _ _ _ _ _ _ _ - - - - - -

248

25.

_ _ _

254

27. General Markov Chains _ _ _ _ _ _ _ _ _ _ _ - - - - _

258

26. Stable Markov Chains _ _ _ _ _ _ _ _ _ _ _

_ _ _

Chapter I. General Theory This chapter unifies and extends some of the results in [ 44] and [46]. In Section l we establish the connection between the submarkov property for symmetric semigroups for Dirichlet spaces

(~, E).

P

t

and the contractivity property

This was first discovered by A. Beurling

and J. Deny [l] but apparently it was M. Fukushima who first appreciated its significance for Markov processes. Also

in Section l we introduce

what seems to be the appropriate notion of "irreducibility" and we distinguish the transient and recurrent cases. We define the extended Dirichlet space

F

= (e)

by completing

__F relative to the

E form alone

(that is, without adding a piece of the standard inner product) and we show that

(~(e)

,E) is an honest Hilbert space when

(~,E)

is transient.

In Section 2 we show how a given Dirichlet space can be transformed into a regular one by introducing an appropriate modification of the state space.

Our construction differs only slightly from Fukushima's in [21].

Some potential theory for regular Dirichlet spaces is developed in Section 3 and used in Section 4 to construct a "decent" Markov process. The main result was first established by Fukushima [22]. Our approach differs from his in that we avoid Ray resolvents and quasi-homeomorphisms. In Section 5 we adapt G. A. Hunt' s construction of "approximate Markov chains" to our situation. In Section 6 we introduce various additive functionals, some of which are used to develop a theory of balayage in Section 7. In Section 8 we study random time change. We show that the time

2

I.2 changed process is symmetric relative to the "time changing measure" and we identify the time changed Dirichlet space. One immediate application is that if

(~, E)

is in the extended space

is recurrent then the constant function l

J (e)

and the norm E(l,l) = 0.

In particular

&(e) is not a Hilbert space which complements the result in Section l for the transient case.

3

J.

Transience and Recurrence

LJ, Definition. A symmetric (submarkovian) reso1vent on L 2 (dx) is a family of bounded symmetric linear operators

symmetric

G

u

f

> 0 whenever

f > 0 and

(v-u) G G

symmetric

U V

2 G , u > 0 on L (dx) satisfying. u

uG f < 1 whenever f < l.

u -

.///

Of course the inequalities in 1.1.1 are in the almost everywhere sense. It follows from 1. 1.1 tagether with the Cauchy-Schwarz inequality that each uGu

2 is a contraction on L (dx).

(Irrdeed this is the case on Lp(dx), 1

Consider now a given symmetric resolvent thecommonrangeofthe G

u

2 isdensein L (dx).

! Gu , u

~ p~ + ... )

> 0 l and as sume that

Itfollowsfrom 1.1.2 that

there exists a unique non-positive definite self adjoint operator A

2 on L (dx)

suchthat (1.1)

G

u

(u- A)

-1

u

> 0.

For t > 0 let (1. 2)

with the right side defined by the usual operator calculus. that the Pt

It is easy to check

satisfy the following conditions. 2 is a symmetric contraction on L (dx).

1. 2.1,

Each

1.2.2.

PtPs

1.2.3.

Limt~

1.2.4.

Gu =

Pt+s 0

for

s, t > 0.

2 Pt= 1 in the strong operator topology on L (dx).

J0"" dt e -ut Pt

for u

> 0.

4

there

whenever f > 0

P tf _:: 0

f < 1 whenever

and

1.2

f < 1. I I I

there is no difficulty in interpreting the integral in 1. 2. 4

Because of 1. 2. 3

Ta establish l. 2. 5 it suffices to apply Laplace inversion

as a Riemann integral.

as in [16, XIII. 4] to 1. 1. 1. Let

/_A be the unique nonnegative definite self adjoint square root of -A

and put (1. 3)

=

f

Jdx /-A

E(f,g)

Also for u

>0

domain / -A f(x)

/-A g(x)

f, g in

f.

put

{1.4)

Eu(f,g)

We will refer to the pair

Jdx

E(f, f)

+ u

(f,E)

as the associated Dirichlet space.

hx).

The following

is quite useful for analyzing (f, E). 2 For general f in L (dx)

Lemma l.l.

J dx {f(x) -

(1/t) increases as

t

~

0 and u

increases as f

is

in F

u t

Ptf(x) } f(x)

oo •

Jdx

{f(x) - u GJ(x)} f(x)

Each of these expressions remains bounded if and only if

and in this case

(1. 5)

E(f, f)

= Lim

(1. 6)

E(f, f)

=

tJ

0

Lim t u

(1/t)

00

u

Jdx {f(x) -

Ptf(x)} f(x)

Jdx {f(x) - u G u f(x)}

f(x).

III

5

Proof.

cg:"':dx),

1.3

By the spectra1 theorem there exists a measure space 'U.

an isometry

2

""'L 2 (~'"':dx),

L (:g:,dx)

2 suchthat f in L 0.

V(b)

I

t2

and (1):)(1-e-ta)

I

(u)

{ a(u+a) - ua}

I

(u+a)

2

2 2 = a / (u+a) •

Then

and it suffices to show that

C1earl y

v(+ 0

is negative.

sufficentl y small

Finally

e-b-be-b-e-b=- be-b is negative and we are done. / / / It is elementary that

products

Eu.

We say that

g

f

is a Hilbert s pace relative to any of the inner

A deeper result can be established with the help of Lemma 1.1. f if there exist ~verywhere

is a normalized contraction of

defined versions of (l. 9)

cp

then

l 2 - b(1-b) + (1-b + zb ) - 1 = -

~~ (b)

ua/(u+a)

For the fir st expres sion it suffices to

cp (t) = (1ltl (1-e-ta).

For the second expression 1et

cp 1(t) = { tae -ta - 1 + e-ta}

the expressions

f

!g(x) I ~

With our hypotheses on

and

g such that

lf(x) I

lg(x) - g(y) I ~

~ there exists for u

>0

lf(x) - f(y) I • a symmetric measure

6

g: x g:

Gu (dx, dy) on

1.4

suchthat

Jdx f(x) Gug(x) for of

JJ Gu (dx, dy) f(x)g(y) f

f in

Then for

for

g a normalized contraction

and for

f

(1. ~0)

Jdx{ g(x) =

- uGug(x)} g(x)

Jdx l

(x) {1 - uGu l(x)}

+ u

Jdx g 2 (x) { 1-uGu 1(x)}

JJ Gu (dx, dy) { g 2 (x)

+~ u

- g(x)g(y)}

JJ Gu(dx, dy) { g(x)

- g(y)}

2

and it follows that

Jdx {g -

Gug(x)} g(x) ~

Thus by Lemma 1.1 g be1ongs to f

l. 3 .1.

f

1. 3. 2.

If f is in

f- Guf(x)} f(x).

and E(g, g) < E(f, f).

We summarize in

is a Hilbert space relative to any of the inner products

also g belongs to f

f

(1. 4).

and if g is a normalized contraction of f then

and E(g, g) ~ E (f, f).

In general any pair with E

Jdx{

(f, E)

a bilinear form on F

if it satisfies l. 3.1 and l. 3. 2.

with

f

III

2 a dense linear subset of L (dx) and

will be called a Dirichlet space

~

2 L (dx)

We have shown above that the Dirichlet space

2 associated with a symmetric submarkovian resolvent on L (dx) is a Dirichlet 2 space on L (dx).

We now prove conversely that every Dirichlet space (f,E)

2 on L (dx) is associated with a unique submarkovian resolvent. 2 Let(f,E) beaDirichletspaceon L (dx). unique bounded operator from

2 L (dx)

into

f

For u>O let Gu bethe

determined by

7

Jdx cp (x) g(x)

(l.ll)

for

({J

L (dx). Let

2 in L (dx)

.f.

g in

and

It is easy to verify the resolvent identity

and that for u > 0 the operator uGu is a symmetric contraction on

l.l. 2

2

l.S

In place of 1.1.1 we prove the following more genera1

result.

T be any mapping from the rea1s to the reals satisfying

(1.12)

I TcY -

=0

TO

~

I


0 Eu (f - u Gu cp , f -u Gucp)

Jdx f l(x) + u cp (x) Gucp (x) -2f(x) (x) 2 = E(f, f) + u Jdx I f(x) - cp (x) r + u Jdx ! u cp (x) Gucp (x) - cp (x) l.

= E(f, f) + u

Cf!

and therefore the functional

P (f)

= E(f, f) + u

:fJx

-cp (x)

u Gucp.

has the unique minimum f T uGucp = u Gucp.

! f(x'

l2

But in general

iP (Tf)

~

p (f)

and so

W e summarize these results in

Let

Theorem 1.2.

I Gu,

u >0

2 with common rangedensein L (dx). 2

Dirichlet space on L (dx).

l

2 be a submarkovian resolvent on L (dx)

Then the pair Cf,El defined by (1.3) is a

2 Conversely every Dirichlet space on L (dx)

i~

2 associated in this way with a unique submarkovian reso1vent on L (dx) with dense common range.

Remark l.

III

Our restriction to resolvents having dense range and

therefore to Dirichlet spaces for Theorem 1.2.

2 F which aredensein L (dx) is not necessary

2 In the general case we need only replace L (dx) by the

8

1.6 closure of the range of the resolvent operators. can be weakened considerably. Borel set in the sense of (39].

Remark 2.

Also our assumptions about

It suffices for example that

~

X

be an absolute

III

The connection between a submarkovian resolvent

Gu' u > 0

and a submarkovian semigroup Pt' t > 0 is a familiar tool in the theory of Markov pmces ses. To our knowledge it was first used systematically by W. Fell er in his fundamental papers on diffusions and solutions of the Kolmogorov equations. The connection between symmetric resolvents and Dirichlet spaces is less well The basic idea goes back to Beurling and Deny [1].

known.

However it seems

that M. Fukushima [20] first appreciated its significance in the context of Markov processes.

III

We continue to work now with a Dirichlet space Cf,E), as in Theorem 1.2 and its associated resolvent general f > 0

with

f

n

! Gu,u

>0

l

and semigroup

! Pt,t

>0

l·

For

define

any sequence of square integrable functions which increase to f

almost

everywhere. Also define

when the right side converges.

It then follows from symmetry and from l. 2. 5

that the extended operators are contractions on Lp(dx) ,

l ~ p ~

+ "'.

the operators uGu in a similar way and note that by l.l.l the uGu are contractions on Lp(dx) ,

< p < + "'.

Extend

9 1 in L (dx)

Lemma 1. 3, For f

Limt t 0

Jdx I f(x)- Ptf(x)!

0. / / /

We use Rota's well known device [41].

Proof.

w

!.7

Irom the half line [o, ", > into the augmented space

be the usual trajectory variables and let

0 be the set of maps

Let

:g: u 1a l ,

let xt, t > o

t? be the unique measure on

0 such

that jt?(dw) f (x (w)) ... fn(Xt (w)) 0 0 n

for 0 < t 1 < • · • < tn on

0

by Xs'

generated by s

> t.

and for x

0

f , .•. , fn _::: 0

0

and for t > 0 let

are uniformly integrable on

:g:.

f in

1.2.4 to Lp(dx},

1_::: p _:::

A of

~

Ptf(x), t

>0

2 1 L (dx}nL (dx).///

The same argument gives an Lp result for 1 _::: p < +

the weak* topology on L"' (dx).

=~ - A

and therefore the functions

But this proves the lemma because it follows

Also it follows from symmetry that

c

er- algebra

are uniformly integrable on

inmeasureas tso for

froml.2.3 that Ptf,..f

A

.70 be the

.7 t be the er -algebra generated

P tf(X ), t > 0 2 0

0 in the sense of Hunt [27, Section I. 6]

A subset

Let

For f in L\X)

and it follows that the functions

Remark.

:g:.

on

co.

Pt is continuous in t with respect to

Once this is established it is routine to extend

+ "' • ///

is proper invariant if neither A nor its complement

is dx null and if PtlA < lA for all t > 0.

It follows from

10

symmetry that

A is proper invariant if and only if its complement

A

c

1.8 is.

2 f belongs to L (dx) then

Moreover if

J

+ dx l

Thus if f belongs to

F

Ac

then so does

space by restricting everything to

f(x) {1 f(x)- Ptl c f(x)} Ac A

lA f

A.

and we can obtain a new Dirichlet

This suggests that an appropriate

generalization of irreducibility for Markov chains is

1.4.

Condition

ill'

Irreducibility.

There exist

no proper

From now on we assume that this condition is satisfied.

invariant sets./// Our feeling is that

this restriction will be harmles s in practice. Next we apply the techniques associated with the Hopf decomposition to distinguish the transient and recurrent cases. is the book of Foguel (17]. in the discrete time case.

Our source for these techniques

Webegin by adapting Garsia' s well known argument Let

f be in L 1(dx)

and for n > 0 define

z-n

(1.13)

S f(x) n

J0

dt Ptf(x) kZ-n

S f(x) n,m

max1 < k < m

J0

The integrals are well defined in the L

dt P l(x)

1

sense because of Lemma l. 3.

Clearly

11

1.9 k2 -n

I0

dt Ptf

= sn f + p 2 -n

(k-1)2 -n

J0

dt Ptf

and therefore

s n,m f 0]

J dx (Si.'n,m f) + -

>

and after passage to the limit

J ,,

(1.15)

[s···

f >

n

Since

in L

1

2n Snf

where

"* f

E

=

o]

U:=O

1

n (S "'

n, m

f) +

> 0

> 0.

as n t "'

[s:

2-

m t "'

dx Snf(x)

in L

jdx P

also

(Again we are using the concept of

f > 0 ].

uniform integrability as formu1ated in [2 7, I. 6 ]. ) Thus after multiplying (1.15) by 2n and passing to the limit n t "'

Lemma 1. 4.

we obtain

1 Let f be in L (dx)

and let k2-n

E =

fx

:- supn

2: O,

k >1

~

dt Ptf(x) > 0

J.

12

IE dx f(x)

Then

>

-

l.lO

o.lll

We deiine the Green•s operator G by

{1.16)

Gf

when it makes sense. Corollary 1. 5.

Let f, g > 0 almest everywhere be in L 1 {dx).

[Gf < + oo] = [Gg < + ce] Proof.

Let A

I

B

almost everywhere.

(Gf= "', Gg 0 almost

But if f :::_ 0 is nontrivial then by irreducibility G f > 0 almost 1

13

l.ll Gf 2:_ G G f 1

everywhere and since

1

and nontrivial in L (dx).

f > 0

of the choice of

it follows that the outcome is independent Thus the following makes

sense.

l. 5.

The Dirichlet space

Definition.

0~::,

E) is transient

if

Gf finite almost everywhere for all f in

1 L (dx) and recurrent if almost everywhere

Gf = + "' forall f>O

1 andnontrivialin L (dx).///

For h > 0

r

Jo stays bounded as t t Theorem 1 6

t+h

h ds p sl -

t

ds p

s

l

There follows

co •

If

J

Cf, E)

is recurrent, then Ptl

1 almost everywhere for

every t > 0. / / / Of course the converse to Theorem 1. 6 is false.

(Consider for example

standard Brownian motion in Rd for d > 3.) f belongs to the extended Dirichlet space

1. 6 Definiti.Qn.

exists a sequence fn is

almost

l. 6. 2.

rf

n

such that

is Cauchy relative to E.

fn ,..f

Condition 1. 6.1

1. 6 1'

f

E(fn' fn)

almost everywhere on

-z. I/I

can be replaced by the apparently weaker is bounded independent of

n. / / /

f (e)

if there

14

!.12 To prove this we adapt the proof of a well known theorem of Banach and Saks [40, p. 80].

fi n l

We assume that

satisfies

the Ces'aro means of a subsequence satisfy 1. 6,1. computation

we temporarilly introduce

by first identifying functions in completing.

cp weak1y in

:r

:r.

and we show that

To avoid excessive

the Hilbert space formed from

whose difference has E

We use the same symbol for functions in

equivalence classes in f n-+

f

:r

1.6.1'

J'

norm zero and then and their corresponding

After se1ecting a subsequence we can assume that

and after again se1ecting a subsequence we can assume

that

E(cp - fm, cp - fn) < (1/n) C1early l. 6.1

for

n > m.

will follow if we show that

(1.17) But the left side of (l. 17)

2 n (1 n ) L:k=l E(cp -fk, cp -fk)

I

and we are clone.

Remark,

There is an alternative noncomputational argument which uses

the fact that the weak closure of a convex subset of a Hilbert space is also the strong closure,

III

If

15

Lemma 1. 7. (i)

Let f be in ,I(e)

The limit

Limn t

approximating sequence

E(fn, fn)

co

! f n !.

I fn l

and let

1.13 be as in Definition 1. 6.

is independent of the choice of the

Therefore E

extends uniquely to

by continuity and

(1.18)

E(f, f)

(ii)

(1.19)

=

Lim t n

c:o

E(f

f ). n, n

The expressions

(1/t}

Jdx

{1-Ptl(x)} hx)

+ ~(1/tJJfPt(dx, dy){f(y)-

f(x)}

2

(1.19 1 }

arefinite for u, t

Proof.

> 0 and increase to E(f, f) as t

By the triangle inequality

will follow from (ii) increase to

t .

t

= Limn

J0

t

and u t"'

co E(fn'

if we show that the expressions

fn)

respectively./I I

exists.

Thus

(1.19) and (1.19')

We give the argument only for (1.19) and temporarilly

introduce the specia1 notation Et(f, f} for (1.19). Et(f

The estimate

-f , f -f ) < E(f -f , f -f ) mnmnmnmn

is valid by Lemma 1.1 and it follows with the he1p of Fatou' s 1emma that

By the triangle inequality Et(f, f) 1 jEI (f, f) -

1

E~

(fn' fn)

I~

is finite and 1

Lim supm t

"'E~fm -fn' fm -fn)

(i)

16

1.14 In particular Et(fn' fn)

Et(f, f) increase as

-1>

Et(f, f)

t 10

and so again by Lemma l.l the expressions

and are dominated by .e.

Finally convergence

to .e follows from the estimate

l/ +./ Et(fn ,fn)- Et(f,fl/ ,f n n )- Et(fn n l/ +I E(fn,f -< /.e-E(fn,f E(f -f,f -f)+/E(f,f)-Et(f,f)/.111 0 then actually Gcp _::: M

on the set where

Remark 2.

and E(l, 1)

Hilbert space relative to E.

almost everywhere./11

0. Thus

f(e)

cannot be made into a

This is also true for dx unbounded, but the

proof must wait until Section 8. Remark__l.

almost everywhere

E) is recurrent and if dx is bounded then by Theorem l. 6

If

the function 1 is in f

III

f relative to the E norm.

Therefore in the

transient case these operators extend by continuity to contractions on From now on we take these extensions for granted. strong Operator topology on f(e)

u

=

J0 dt e -ut p

Pt

It follows from the spectral theorem that the operators

and u Gu are contractive on

G

f

We summarize in

cp > 0

Moreover if

bounded on the set where rp > 0 then Gp f

Also it follows

Of course the limit function

Theorem1.9 . .!f (f,E) is transient then as inner product.

and to cp _:. 0 in

immediately

subsequence converges almost everywhere.

E

in f (e)

This extends immediately to f

as

t ~0

and

f

(e)'

Also Pt' u Gu-+ 1 in the u t "'•

the identity

00

topology on f(e)

t

is valid for the extension, and Pt -+ 0 in the strong operator

as t

tce.lll

19

1.17

Remark 4.

We have recently received a preprint from M. Fukushima

entitled "Almost polar sets and an ergodie theorem. " Among other things this paper deals with transience and recurrence in a nonsymmetric setting.

III

20 Dirichlet Dirichlet Dirichlet

.f.J.

Definition.

f n

2.1.1. 2 .l. 2. open s et

The Dirichlet space

ccom(~)

is uniformly densein ccomqp and That is

dx is everywhere dense. G.

2 (f, E) on L (dx) is regular if

III

Condition 2.1 .2

dense in .f.

1

for any nonempty

is harmless since we can always replace

by the

~

(.f, E) is regular and transient then also

Notice that if

support of dx.

JG dx > 0

E

.F n Ccomqp is Edensein (f, E)

We consider now a Dirichlet space is dense in

2

L (dx)

on

2 L (dx)

suchthat

and we construct a regular Dirichlet space

modifying the state space

f

by

Our construction differs only slightly from one

~-

first given by Fukushima [21]. This s ection is es sentially a r epetition of the appendix in [ 44]. Note first that if f in to

Jh

f

is bounded almost everywhere, then

and so the subcollection of f in

form an algebra.

f

f

2

belongs

which are integrable and bounded

Obviously this algebra is dense in

f

and since

f

itself

is separable (since it is the domain of a self adjoint operator on a separable Hilbert space) there exists a subset

2.2 .1.

~O

2.2.2.

~O

of J"

satisfying

is countaLle is an algebra over the rationals

2.2.3.

Every f in

2.2.4.

~O

is densein

The uniform closure

is integrable and bounded,

~O

~

f

(and therefore in

2 L (dx)

)./II

is a commutative Banach algebra and so the

well known techniques associated with the Gelfand transform of ~ are available.

21

2.2

Y be the collection of real valued functions

Let

2. 3.1.

y (fl
$: with full measure.

a unique mapping

f in

flo·

for f in

2.4.2.

J?.

satisfying

~

2.4.1.

for some f in

is then a separable locally

compact Hausdorff space which is compact if and only if l

let

not identically

on

00

y(af+bg)

2. 3. 3.

J?

f, g in

zero, which satisfy for

y

There exists

such that

{i x) (f) = f(x) for f in

J?o

and for x

in

:lfio.

Clearly

there exists a unique Borel measure dy

ly

dy cp(y)

=

on Y

is Borel measurable and so such that

22

for nonnegative cp that

dy

is

mapping of

Radon.

It follows from the integrability of cp

We use the same symbol

;!? onto c

in

B

0

i to denote the natural

Thus

(Y).

0 -

jf (y)

Clearly for

2.3

'f.

on

y (f)

f

E

:!?•

y

E

~·

f in ;!? and for any polynomial P iP(f)

and since any T

=

P(if)

satisfying (L 12)

is continuous and therefore can be

approximated uniformly by polynomials on compact sets, also iT(f) For

f, g _:_ 0

I dy

in

j E

~

= I dx

j f(ix) j f(ix)

= I dx

jx(f) j x(g)

= I dx

jx(fg)

= I dx

f(x)g(x)

j

be defined on

The desired

onto

is an isometry from

i :I?o by

jE(j f, i g)

product

T(if).

jf(y) jg(y >

and it follows that Let

=

= E(f, g).

Dirichlet space is the closure of i B

0

relative to the inner

23

2.4 tagether with the continuous extension of

To establish

to this closure.

E

regularity for this Dirichlet space it on1y remains to checkthat

dy

is dense.

For this it suffices to show that

n~=1 ! X

meas.

(2 .1)

:

e > 0 of

for any choice of

I fi

in

y

y

and of

is fa1se then there exist po1ynomia1s Pm Pm (fl' •.. , fn)

> 0

f , ••. , fn in 1

~.

If (2.1)

in n indeterminates such that

converges uniform1 y to

and it follows that gh be1ongs to in

l

(x) - '{ (fi) ) < e

~

whenever

h

does.

h

Since every

:§1 can be represented

and since

max ~=

by polynomia1s in

follows

that y (h)

1 I fi

- y (fi)

I

I f.1

- '{ (f.)

l

1

can be uniform1y approximated

i=l, ... , n

0 for all hin

J:?,

not containing the constant term, it

which possibility has been ruled out

by hypothesis.

Remark l. Remark 2.

The proof of (2.1) given on page 71 of[44] is incorrect.lll Clear1y the final state space

However by Theorem 2.1 in [22] for

is related to

y

be a

any

1'

J

depends on the choice of

:§1 0 .

resulting from a different choice

"capacity preserving quasi-homeomorph ism"

(see [22] for the precise definition) which is enough to guarantee that

1

are identical from the point of view of the processes constructed below.

return to this subject in Section 19.

III

and We

1

I

24 3. Theory Potential Theory Theory

Throughout this section 2 on L (dx). E 1. )

(.f, E) is a transient regular Dirichlet space

(The recurrent case can always be handled by replacing E

with

The point of view taken here goes back at least to H. Cartan [3,4] for the

classical Dirichlet spaces associated with the Laplacian and Brownian motion.

The general formulation in terms of regular Dirichlet spaces is

J. Deny.

due to A. Beurling and

(See [1].)

The results themselves were

first established by Fukushima [22] using an indirect a pproach. This section differs only slightly from sectionsection is in f(e)

and

f in f(e) isapotential if E(f, g)

g ~ 0

(i)

f is a potential.

(ii)

There exists a Radon measure

E(f, g) for

lf (e) n

g in

E(f+g, f+g)

(iv)

f,?:. u Guf f > Ptf

Proof.

g

=

f ( e )'

1-1 suchthat

J 1-1 (dx) g(x)

ccom (~).

(iii)

(v)

> 0 whenever

almost everywhere. / / /

Th .::..::;.:e:_.:cfo:::l::.:lc:::oc.:w:..:.i:.:cn:.....::a:::r...:e:....:::e.:.q.::u;:.iv.:..a:.:.l::ce::.:n:.:.t:.....:.fo::::..::..r__:__:i:::n

section

section

Section l in [ 44].

> E(f, f) whenever for all u

for all t

That (ii) implies

g

is in f ( e)

and

g > 0.

> 0.

> 0. / / / (i) is trivial.

To prove that (i)

implies (ii)

let f be a potential and consider the nonnegative linear functional I defined Oll

g in

~

n ccom

(~)

by I(g) = E(f, g).

If g n

decrease to 0

pointwise

then by Dini' s theorem they do so uniformly and after comparing to a fixed

25

3.2

n C com ()~)

11;

nonnegative g in

I(gn) ! 0.

we see that

Thus

which is _:: 1 on the support of

g1

follows by the Daniell approach to

(ii)

integration (as presented for examp1e in (34] ).

That

(i)

implies (iii)

follows from (3 .1)

E(f+ tg, f

for

t

=1

small.

+ tg)

= E(f, f)

+ 2t E(f, g) + t 2 E(g, g)

implies (i) follows from (3 .1) for t > 0

and that (iii)

sufficiently

Equivalence of (iv) and (v) is easily established with the help of

Laplace inversion.

Jdx

(3. 2)

That (i) impli es

j f(x) - u Gu f(x) ] cp (x)

cp > 0

which is valid for

(iv) follows from

as

in Theorem 1. 9.

That (v)

implies (i)

follows from (3. 3)

E(J

s

0

upon dividing by

Jds

du P f, g) u

s

j f(x)

P f(x) J g(x) 8

and passing to the limit

s l 0.

(Thc identity (3. 3)

follows in the same way as (1. 20).) / / /

Corollary 3, Z. If

(iii)

If f

E(/f/ ,

lrl

Every potential is nonnegative.

f, g are potentials, then so is min(f, g).

(ii)

Proof.

(i)

isapotential then so is min(f,c) for

(i)

follows since

l

E(f, f)

+ Ed

rl - f, Ir/

+

I

-fl

+ 2E(f,

E( f/ - f,

I f/

-f)

> E(/f/, if/ + E(/f/ -f,

/f/

-f)

.:: E(f, f)

c > 0. / / /

Ir! - f)

26

and therefore

E(

lfl - f,

= 0.

lfl - f)

Conclusions

directly from Lemma 3.1- (iv) or (v).

Lemma 3. 3.

(ii)

and

3.3

(iii) follow

III

Let g be a potential in F

and let f

satisfy 0 < f _:: g

and in addition

all t > 0

from or

u G

u

Then f is a potential in F

f

O.

and E(f, f) _:: ~·

I II

Again we consider only the first alternative.

Proof. follows from

Lemma 1.1 and the estimate

Jdx {f(x)

2

- Ptf(x)} f(x)

Jdx f(x)
0 suchthat

J~ (dx)f(x) for f in

has finite energy if there exists

~

l

2

c {E(f, f)}

2

The collection of all such measures is denoted by 'fl1.

27

3.4 cp 2:_ 0 such that cp • dx belongs to

The collection of Borel by '!!! o

'lll is denoted

II/ then there exists a unique potential, written

belongs to 'lJ?

Clearly if i.J. N i.J., suchthat

(3.4)

for

E(Ni.J., g)

g in

f

(e)

n

C com

qp.

= j 1-L (dx)g(x) We introduce the special notation

(3. 5)

Important

and call E(i.J.) the energy of i.J..

'lJ?

compactnes s properties of

are summarized in

Lemma 3.4. (i) indeed f (ii)

be a sequence in 'l/1.

_!! Ni.J.n converges weakly to f in f (e) then f is a potential and Ni.J. where

.!f

is the vague limit of the

i.J.

is bounded and if

E(i.J.n)

and Ni.J.n -+ Ni.J. (iii)

I I-ln I

Let

weakly in

J' (e)

1-l n • i.J.

I-ln -+ 1-l vaguely then

is in

'!!!

•

'lll is complete relative to the energy metric E.

/II

This lemma is an immediate consequence of regularity.

We omit the proof.

To make further progress we must validate (3.4) for general g in I'(e) which means in particular that we must represent g by a refinement which is specified up to

j.L

equivalence for every

i.J. in 'l/1

The main tool for this

is a capacity as sociated with E. J..,2..

Definition.

For G an open subset of Cap(G)

= inf

E(f, f)

~

let

28

as

runs over the functions in

f

g:

subset of

G

A

For

a general Borel

let inf Cap(G)

Cap{A) as

= + "' •

If no such f exist let Cap(G)

on G.

3. 5

suchthat f > 1 almost everywhere

E(e)

runs over the open supersets of

supersets

A

Borel set A

A. / / / is

if Cap(A)

= 0.

A

general

set is polar if it is a subset of a Borel set which is polar.///

subset

g:

Let G be an open subset of G

(i)

f

Th_......_e_r_e_e_x.;;;ic:..s.:..ts:;_;a.;_:u::::;ru.;::·.:;oq~u"'e-f:..:u:::n:.:cc:..:t==i.;:o.::n:.__,P_ __::i:::n

f (e)

is minimal a.mong f in (ii) (iii)

0 < p

G

~

l

>

satisfying f

G

p

+

co •

G G (e) .::.s.::.u:..:c;:;h:....::::th:.:ca::.t:__..;E:::..>.;(p"-~'-"p--')

almost everywhere on G. _G=l a 1mos t everywh ere on G • P

almost everywhere and

pG is a potential and indeed

Cap(G)
1 almost

f in f(e)

W is convex and closed and

Clearly

(i)

follows directly.

Concclusion (ii) follows upon noting that if f belongs to W then so does min (f, 1) and max(f, 0). and if g E(f,g)

~

~

prove

To

on G then E(f+tg, f+tg)

0

note first that if g is in

(iii) ~

E(f, f) for all t > 0

(e)

and therefore

It only remains to adapt the proof of Lemma 3.1 considering

0.

. /II

restrictions to

Proposition 3. 6.

For

v in 'lr 1

{3. 6)

f

v(G)
J

and for

G

an open subset of

~

1

2 ! Cap(G) l 2 . ///

If PG were in C com qp this wou1d follow from (3. 4).

In general this is

29

3.6

false and instead we must approximate with the help of ~emma

Let

3, 7,

v be in

'11/. 0

belongs to

(ii)

'11/

for t > 0,

Lim t J 0 (1/t) (1-Pt) Nv (x) dx"' v both vague1y and relative to the

energy metric E,

I II

For T > 0

Proof. T

Therefore

T+t

t

10

ds P

s

(1-Pt)Nv

(1-Pt) Nv is in

10 ds '11/

0

P Nvs

1T

ds P Nv s

and

t

J0

ds P Nv. s

and the lemma follows with the help of Lemma 3.4.

III

Now Proposition 3. 6 follows from v (G)

J

dx(l-Pt) Nv (x) < Lim inftl 0 (1/t) G

Lim t!O (1/t) E(pG, G(l-Pt)Nv) E(pG, Nv)

and the Cauchy-Schwa rz inequality. Corollary 3. 8.

III

If v belongs to

'11/ then v charges no polar set.

III

Next we establish some properties of Gap which permit the application of Choquet' s general theory.

30

3.7 Proposition 3 ._2_,

Cap(G ) 1

(i)

G

and p n t p (iii)

G

(i) is clear. g

For

open

G , G2 1

To prove(ii) observe first that if m

G

(3. 7)

If Cap(G) is finitethenalso

G

G

> E(g, g) + E(p m-g, p m_ g)

and we conclude that g

=p

G

m

G

p

and so the

If supn Cap(Gn)

everywhere sense.

But also

).

> E(p m.p

E(g, g)

G G E(p m, p m)

Cap(Gn) 2: Cap(A)

the sequence

and it suffices to take

Let Gn

vn be as in

For each n clearly

G. fl

any vague limit point of

(Actually the proof of Proposition 3. 9-(ii) shows that the entire vn converges vaguely.

)//I

32

3.9 Now we are ready to introduce refined Versions.

Now

A property is valid q_uasi-everywhere (abbreviated q. e.)

Definition.

Two functions are quasi-equivalent if they differ

if the exceptional set is polar.

only on apolar set, that is, if they are equal quasi-everywhere.lll Now Definition.

fn-+ f quasi-uniformly if there exists a decreasing

sequence of open sets

Um with Cap(Um) I 0

on ~- Um for each

m.lll

Now

A function f

on

~

suchthat fn -+f

is quasi-continuous on an open set G if there

exists a decreasing sequence of open sets

Um with Cap(Um) l 0

f is defined and continuous on G - U ll1 for each m.

Theore1Jh3.1J.

uniformly

(i) Each f in

f(e) has

suchthat

III

a representative uniquely

specified up to to quasi-equivalence such that

(3.9)

f is quasi-continuous on f(e)

(3.9)

Cap

(3.10) for

fx

(f(x))

> e

Proof.

and for

iJ

l