Lectures in Probability Theory and Statistics 1995

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris

1690

Springer

Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo

M. T. Barlow D. Nualart

Lectures on Probability Theory and Statistics Ecole d'Et6 de Probabilit6s de Saint-Flour X X V - 1995 Editor: P. Bernard

Springer

Authors

Editor

Martin T. Barlow Department of Mathematics University of British Columbia # 121-1984 Mathematics Road Vancouver, B.C. Canada V6T 1Z2

Pierre Bernard Laboratoire de Mathdmatiques Appliqudes UMR CNRS 6620 Universitd Blaise Pascal Clermont-Ferrand F-63177 Aubibre Cedex, France

David Nualart Department d'Estadistica Universitat de Barcelona Facultat de Matem~ttiques Gran Via de les Corts Catalanes, 585 E-08007 Barcelona, Spain Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufilahme L e c t u r e s on p r o b a b i l i t y t h c o r y a n d statistics / Ecole d'Et~ de Probabilitds de Saint-Flour X X V - 1995. M. T. Barlow ; D. Nualart. Ed.: P. Bernard. - Berlin ; Heidelberg ; N e w York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ;

Singapore ; Tokyo : Springer, 1998 (Lecture notes in mathematics ; Vol. 1690) ISBN 3-540-64620-5 Mathematics Subject Classification (199 I): 60-01, 60-02, 60-06, 60D05, 60G57, 60H07, 60J15, 60J60, 60J65 ISSN 0075- 8434 ISBN 3-540-64620-5 Springer-Verlag Berlin Heidelberg New York 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions 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. 9 Springer-Verlag Berlin Heidelberg 1998 Printed 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. Typesetting: Camera-ready TEX output by the authors SPIN: 10649898 41/3143-543210 - Printed on acid-free paper

INTRODUCTION

This volume contains lectures givcn at the Saint-Flour Summer School of Probability Theory during the period 10th - 26th July, 1995. We thank the authors for all the hard work they accomplished. Their lectures are a work of refercncc in thcir domain. The school brought together i00 participants, 29 of whom gave a lecture concerning their research work. At the end of this volume you will find the list of participants and their papers. Finally, to facilitate research concerning previous schools we give here the number of the volume of "Lecture Notes" where they can be found :

L e c t u r e N o t e s in M a t h e m a t i c s 1971: n~ 1977:n~ 1982: n~ 1988: n~ 1993: n~

-

1973: n~ 1978 : n~ 1983:n~ 1989: n~ 1994: n~

L e c t u r e N o t e s in S t a t i s t i c s 1986 : n~

-

1974:n~ 1979: n~ 1984: n~ 1990: n~ 1996:n~

-

1975 : 1980: 19851991:

n~ n~ 1986 et n~

1976:n~ 1981:n~ 1987: n~ 1992:n~

-

TABLE OF CONTENTS

Martin 1

T. BARLOW

Introduction

: "DIFFUSIONS

ON FRACTALS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2

T h e Sierpinski Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3

Fractional Diffusions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

4

Dirichlet Forms, Markov P r o c e s s e s and Electrical Networks . . . . . . . . . .

46

5

G c o m e t r y of R e g u l a r Finitely Ramified Fractals

. . . . . . . . . . . . . . .

59

6

R e n o r m a l i z a t i o n on Finitely Ramified Fractals

. . . . . . . . . . . . . . . .

79

7

Diffusions on p.c.f.s.s, sets

. . . . . . . . . . . . . . . . . . . . . . . . . .

94

8

T r a n s i t i o n Density E s t i m a t e s References

David

. . . . . . . . . . . . . . . . . . . . . . . . .

106

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

NUALART

: "ANALYSIS ON WIENER SPACE AND ANTICIPATING STOCHASTIC CALCULUS"

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

125

Derivative a n d divergence o p e r a t o r s on a G a u s s i a n space . . . . . . . . . . .

126

1.1 Derivative o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

1.2 Divergence o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

1.3 Local p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

1.4 W i e n e r chaos e x p a n s i o n s

. . . . . . . . . . . . . . . . . . . . . . . . .

133

. . . . . . . . . . . . . . . . . . . . . . . . . . .

135

1.5 T h e w h i t e noise case

1.6 S t o c h a s t i c integral r e p r e s e n t a t i o n of r a n d o m variables 2

123

. . . . . . . . . .

139

O r n s t e i n - U h l e n b e c k semigroup a n d equivalence of n o r m s . . . . . . . . . . .

141

2.1 M e h l e r ' s f o r m u l a

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

2.2 H y p e r c o n t r a c t i v i t y

2.3 G e n e r a t o r of t h e O r n s t e i n - U h l e n b e c k s e m i g r o u p

. . . . . . . . . . . . .

145

2.4 M e y e r ' s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

VLLI

3

A p p l i c a t i o n of Malliavin calculus to s t u d y p r o b a b i l i t y laws . . . . . . . . . .

155

3.1 C o m p u t a t i o n of p r o b a b i l i t y densities

155

. . . . . . . . . . . . . . . . . . .

3.2 R e g u l a r i t y of densities a n d c o m p o s i t i o n of t e m p e r e d d i s t r i b u t i o n s w i t h e l e m e n t s of ~)-oo

. . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 T h e case of diffusion processes . . . . . . . . . . . . . . . . . . . . . . 3.4 Lp e s t i m a t e s of t h e density a n d a p p l i c a t i o n s . . . . . . . . . . . . . . . 4

5

160 163 164

Support theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

4.1 P r o p e r t i e s of t h e s u p p o r t

174

. . . . . . . . . . . . . . . . . . . . . . . . .

4.2 S t r i c t p o s i t i v i t y of t h e density a n d skeleton . . . . . . . . . . . . . . . .

177

4.3 Skeleton a n d s u p p o r t for diffusion processes

182

. . . . . . . . . . . . . . .

4.4 V a r a d h a n e s t i m a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

A n t i c i p a t i n g s t o c h a s t i c calculus . . . . . . . . . . . . . . . . . . . . . . . .

188

5.1 S k o r o h o d i n t e g r a l processes . . . . . . . . . . . . . . . . . . . . . . . .

188

5.2 E x t e n d e d S t r a t o n o v i c h intcgral . . . . . . . . . . . . . . . . . . . . . .

197

5.3 S u b s t i t u t i o n f o r m u l a s . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

A n t i c i p a t i n g s t o c h a s t i c differential e q u a t i o n s

210

. . . . . . . . . . . . . . . . .

6,1 S t o c h a s t i c differential e q u a t i o n s in t h e S t r a t o n o v i c h sense 6.2 S t o c h a s t i c differential e q u a t i o n s w i t h b o u n d a r y c o n d i t i o n s

........ ........

210 215

6.3 S t o c h a s t i c differential e q u a t i o n s in t h e S k o r o h o d sense . . . . . . . . . .

217

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

DIFFUSIONS

ON FRACTALS

Martin T. BARLOW

1. I n t r o d u c t i o n . T h e notes are based on lectures given in St. Flour in 1995, and cover, in greater detail, most of the course given there. T h e word "fractal" was coined by M a n d e l b r o t [Man] in the 1970s, b u t of course sets of this t y p e have been familiar for a long time - their early history being as a collection of pathological examples in analysis. There is no generally agreed exact definition of the word "fractal", and a t t e m p t s so far to give a precise definition have been unsatisfactory, leading to classes of sets which are either too large, or too small, or both. This a m b i g u i t y is not a p r o b l e m for this course: a more precise title would be "Diffusions on some classes of regular self-similar sets". Initial interest in the properties of processes on fractals came from m a t h e m a t i c a l physicists working in t h e t h e o r y of disordered media. Certain m e d i a can be modelled by p e r c o l a t i o n clusters at criticality, which are expected to exhibit fractal-like properties. Following t h e initial p a p e r s [AO], [RT], [GAM1-GAM3] a very substantial physics l i t e r a t u r e has developed - see [HBA] for a survey and bibliography. Let G be an infinite s u b g r a p h of Z a. A simple r a n d o m walk (SRW) (X,~, n > 0) on G is j u s t the Markov chain which moves from x E G with equal p r o b a b i l i t y to each of the neighbours of x. Write p n ( x , y) = IP~(X,~ = y) for the n-step t r a n s i t i o n probabilities. If G is the whole of Z d then E(X,~) 2 = n with m a n y familiar consequences - the process moves roughly a distance of order v/n in time n, and the p r o b a b i l i t y law p,~(x, .) p u t s most of its mass on a ball of radius cdn. If G is not the whole of ~d then the movement of the process is on the average r e s t r i c t e d by the removal of p a r t s of the space. Probabilistically this is not obvious b u t see [DS] for an elegant argument, using electrical resistance, t h a t the removal of p a r t of the s t a t e space can only make the process X 'more recurrent'. So it is not unreasonable to expect t h a t for certain graphs G one m a y find t h a t the process X is sufficiently r e s t r i c t e d t h a t for some fl > 2 -

(1.1)

E ~ (X~ - x) ~ • n2/~.

(Here and elsewhere I use • to m e a n ' b o u n d e d above and below by positive constants', so t h a t (1.1) means t h a t there exist constants el, c2 such t h a t c l n 2/~ < E ~ ( X , - x) ~ < c2n~/a). In [AO] and [RT] it was shown t h a t if V is the Sierpinski gasket (or more precisely an infinite graph based on the Sierpinski gasket - see Fig. 1.1) t h e n (1.1) holds w i t h / 3 = log 5 / l o g 2.

Figure 1.1: T h e graphical Sierpinski gasket.

Physicists call b e h a v i o u r of this kind by a r a n d o m walk (or a diffusion - t h e y are not very i n t e r e s t e d in the distinction) subdiffusive - the process moves on average slower t h a n a s t a n d a r d r a n d o m walk on Z d. Kesten [Ke] proved t h a t the SRW on the 'incipient infinite cluster' C (a percolation cluster at p = Pc b u t conditioned to be infinite) is subdiffusive. T h e large scale structure of C is given by taking one infinite p a t h (the ' b a c k b o n e ' ) together with a collection of 'dangling ends', some of which are very large. K e s t e n a t t r i b u t e s the subdiffusive behaviour of SRW on C to the fact t h a t the process X spends a substantial amount of time in the dangling ends. However a g r a p h such as the Sierpinski gasket (SG) has no dangling ends, and one is forced to search for a different explanation for the subdiffusivity. This can be found in t e r m s of t h e existence of 'obstacles at all length scales'. W h i l s t this holds for the graphical Sierpinski gasket, the n o t a t i o n will be slightly simpler if we consider a n o t h e r example, the graphical Sierpinski c a r p e t (GSC). (Figure 1.2).

-t : H - H - H : H - H q

i i

,;fi fi:H:H: ; f:H-t [IIIII

q

t: : H F

. . . . . . . . .

f:H-H:H:H-H:

II

;H:F :H-F " I I I I I

III

Figure 1.2: T h e graphical Sierpinski carpet. This set can be defined precisely in the following fashion. Let H0 -- Z ~. For o~ x = (n, m) e H0 write n,m in t e r n a r y - so n -- ~i=o ni3", where ni 9 {0, 1,2}, and n~ -- 0 for all b u t finitely m a n y i. Set J~ -- { ( m , n ) : nk -- 1 and mk -- 1}, so t h a t Jk consists of a union of disjoint squares of side 3k: the square in Jk closest to t h e origin is { 3 ~ , . . . , 2 . 3 ~ - 1} • { 3 k , . . . , 2 . 3 k - 1}. Now set (1.2)

H,, = Ho- O Jk, k:l

H= 5 H,~. n=O

:~_

11111

IIII

IIII]

IIlI

-H:t

lIII

ilIi

itti

IIII iIII

IIlI

~HIIII

IlII

!

IIII

i

!~H~

IlIl

!HI::

Figure 1.3: T h e set H1 9

14' ; ~-

'.

.

.

.

!

~

!

T

F!

---

i

Li

i

i

i

Fi

.__

i

!i

i

i

i

i

i

ttt

r

ii

-L

!E:_I

t

i

1"

__.

---

!ii_I

H --~F-H

4~f-H--t:t--

.

Figure 1.4: T h e set/-/2 9 Note t h a t H A [0, 3n] 2 = Hn N [0, 3n] 2, so t h a t the difference between H and Hn will only be d e t e c t e d by a SRW after it has moved a distance of 3 n from the origin. Now let X (n) be a SRW on Hn, s t a r t e d at the origin, and let X be a SRW on H. T h e process X (~ is j u s t SRW on Z~_ and so we have (1.3)

E(X(~ 2 ~" n.

T h e process XO) is a r a n d o m walk on a the intersection of a t r a n s l a t i o n invariant subset of 7..2 w i t h 7.~_. So we expect 'homogenization': the processes n - 1 / 2 X [,~], O) t _> 0 should converge weakly to a constant multiple of Brownian motion in ~ 2 . So, for large n we should have E ( X O ) ) 2 ~ a l n , and we would expect t h a t al < 1, since the obstacles will on average t e n d to i m p e d e the motion of the process. Similar considerations suggest t h a t , writing ~on(t) = 1EO(X}n)) 2, we should have ~pn(t) " a n t

as t --* oo.

However, for small t we would expect t h a t ~n and ~0,~+1 should be a p p r o x i m a t e l y equal, since the process will not have moved far enough to detect the difference between Hn and H n + l . More precisely, if tn is such t h a t ~On(tn) = (3n) 2 then ~on

and ~o=+1 should b e a p p r o x i m a t e l y equal on [O, tn-bl ]. So we m a y guess t h a t the b e h a v i o u r of the family of functions ~o,~(t) should be roughly as follows: (1.4)

~on(t) = bn + an(t - in),

v.+l(s) = v.(s),

t >_ t,~,

0 < s < t~+l.

If we a d d the guess t h a t an = 3 - ~ for some ~ > 0 then solving the equations above we deduce t h a t tn ~ 3 (2+a)'~, bn ~- 3 2n. So ff ~o(t) = E ~ 2 t h e n as 4o(t) ~- lira= ~on(t) we deduce t h a t ~v is close to a piecewise linear function, and t h a t ~(t) • t 2/z where fl = 2 + a. T h u s the r a n d o m walk X on the graph H should satisfy (1.1) for some f~ > 2. T h e a r g u m e n t given here is not of course rigorous, b u t (1.1) does a c t u a l l y hold for t h e set H - see [BB6, BBT]. (See also [Jo] for the case of the graphical Sierpinski gasket. T h e proofs however run along r a t h e r different lines t h a n the heuristic a r g u m e n t sketched above). Given b e h a v i o u r of this t y p e it is n a t u r a l to ask if the r a n d o m walk X on H has a scaling limit. More precisely, does there exist a sequence of constants Tn such t h a t t h e processes (1.5)

( 3 - ' ~ X i t / , . ] , t > 0)

converge weakly to a non-degenerate limit as n --+ co? For the graphical Sierpinski c a r p e t the convergence is not known, though there exist m= such t h a t the family (1.5) is tight. However, for the graphical Sierpinski gasket the answer is 'yes'. Thus, for certain very regular fractal sets F C I~d we are able to define a limiting diffusion process X = ( X t , t > O, P~, x C F ) w h e r e ]?~ is for each x E F a p r o b a b i l i t y measure on fl = {w Z C([0, co), F ) : w(0) = x}. Writing T t S ( x ) = E ~ f ( X t ) for the semigroup of X we can define a 'differential' o p e r a t o r s defined on a class of functions 7:)(/:F) C C ( F ) . In m a n y cases it is reasonable to call s the Laplacian on F.

F r o m t h e process X one is able to o b t a i n information a b o u t the solutions to the Laplace a n d h e a t equations associated with L:F, the heat equation for example t a k i n g the form Ou

(1.6)

0-7 = LFu, u(0, x) = u0(x),

where u = u ( t , x ) , x E F , t > O. The wave equation is r a t h e r harder, since it is not very susceptible to probabilistic analysis. See, however [KZ2] for work on the wave equation on a some manifolds with a 'large scale fractal structure'.

T h e m a t h e m a t i c a l l i t e r a t u r e on diffusions on fractals and their associated infinitesimal generators can be divided into b r o a d l y three parts: 1. Diffusions on finitely ramified fractals. 2. Diffusions on generalized Sierpinski carpets, a family of infinitely ramified fractals. 3. S p e c t r a l p r o p e r t i e s of the ' L a p l a c i a n ' s These notes only deal with the first of these topics. On the whole, infinitely ramified fractals are significantly h a r d e r t h a n finitely ramified ones, and sometimes require a very different approach. See [Bas] for a recent survey. These notes also contain very little on spectral questions. For finitely ramified fractais a direct a p p r o a c h (see for example [FS1, Shl-Sh4, KL]), is simpler, and gives more precise information t h a n the heat kernel m e t h o d based on estimating

/F

p ( t , x , x ) d x = y ~ e-~' ~. i

In this course Section 2 introduces the simplest case, the Sierpinski gasket. In Section 3 I define a class of well-behaved diffusions on metric spaces, "Fractional Diffusions", which is wide enough to include m a n y of the processes discussed in this course. It is possible to develop their properties in a fairly general fashion, without using much of the special structure of the state space. Section 4 contains a brief i n t r o d u c t i o n to t h e theory of Dirichlet forms, a n d also its connection with electrical resistances. T h e remaining chapters, 5 to 8, give the construction and some p r o p e r t i e s of diffusions on a class of finitely ramified regular fractals. In this I have largely followed the analytic ' J a p a n e s e ' approach, developed by Kusuoka, Kigami, ~'klkushima a n d others. Many things can now be done more simply t h a n in the early probabilistic work - b u t there is loss as well as gain in a d d e d generality, and it is worth pointing out t h a t the early papers on the Sierpinski gasket ([Kusl, Go, BP]) contain a wealth of interesting direct calculations, which are not r e p r o d u c e d in these notes. A n y reader who is surprised by the a b r u p t end of these notes in Section 8 should recall t h a t some, at least, of the properties of these processes have a l r e a d y been o b t a i n e d in Section 3. c~ denotes a positive real constant whose value is fixed within each L e m m a , T h e o r e m etc. Occasionally it will b e necessary to use n o t a t i o n such as c3.5.4 - this is s i m p l y the constant c4 in Definition 3.5. c, c r, c ~ denote positive real constants whose values m a y change on each appearance. B(x, r) denotes the open ball with centre x a n d radius r, and if X is a process on a metric space F then

TA = inf{t > O : Xt E A } , Ty=inf{t>0:Xt=y}, ~-(x, r) = inf{t > 0 : Xt fd B(x, r)}. I have included in the references most of the m a t h e m a t i c a l papers in this area known to me, and so t h e y contain m a n y papers not mentioned in the text. I a m grateful to G e r a r d Ben Arous for a n u m b e r of interesting conversations on the physical conditions under which subdiffusive behaviour might arise, to Ben H a m b l y

for checking the final manuscript, and to Ann Artuso and Liz ttowley for their typing.

Acknowledgements. This research is supported by a NSERC (Canada) research grant, by a grant from the Killam Foundation, and by a EPSRC (UK) Visiting Fellowship.

2. T h e Sierpinski Gasket This is the simplest non-trivial connected symmetric fractal. The set was first defined by Sierpinski [Siel], as an example of a pathological curve; the name "Sierpinski gasket" is due to Mandelbrot [Man, p.142]. Let Go = {(0, 0), (1, 0), (1/2, ~/3/2)} = {a0, al, a2} be the vertices of the unit triangle in I~2, and let Tlu(Go) = Ho be the closed convex hull of Go. The construction of the Sierpinski gasket (SG for short) G is by the following Cantor-type subtraction procedure. Let b0, bl, b2 be the midpoints of the 3 sides of Go, and let A be the interior of the triangle with vertices {bo,bl,b2}. L e t / / 1 = H0 - A, so that HI consists of 3 closed upward facing triangles, each of side 2 -1. Now repeat the operation on each of these triangles to obtain a set H2, consisting of 9 upward facing triangles, each of side 2 -2 .

Figure 2.1: The sets//1 and H2. Continuing in this fashion, we obtain a decreasing sequence of closed non-empty ~o and set sets ( H n)==0, (2.1)

G = 5 n=0

/In.

Figure 2.2: The set H4. It is easy to see t h a t G is connected: just note that OHn C H,n for all m > n, so t h a t no point on the edge of a triangle is ever removed. Since IHnl = (3/4)nlH0 I, we clearly have t h a t IGI = 0. We begin by exploring some geometrical properties of G. Call an n-triangle a set of the form G N B, where B is one of the 3 '~ triangles of side 2 -'~ which make up Hn. Let #n be Lebesgue measure restricted to H,~, and normalized so that # n ( H n ) -- 1; that is #,~(dx) = 2- (4/3)'~1/./~ (x) dz. Let /~G = wlim#,~; this is the n a t u r a l "flat" measure on G. Note that # c is the u n i q u e measure on G which assigns inass 3 -'~ to each n-triangle. Set df=log3/log2~ 1.58... L e m m a 2.1. For x E G, O < r < l

3-1r d' <_ # a ( B ( x , r ) )

(2.2)

< lSr d'.

Proof. T h e result is clear if r = 0. If r > 0, choose n so that 2 -(~+1) < r < 2 -'~ we have n > 0. Since B(x, r) can intersect at most 6 n-triangles, it follows that

(B(x, r)) < 6.3

:

= 18(2-("+U) d, < 18r d'. As each (n + 1)-triangle has diameter 2 - ( n + U , B ( x , r ) must contain at least one (n + 1)-triangle a n d therefore

#r

> 3 -('~+1) -- 3-1(2-'~) d' > 3-1r d'.

[]

Of course the constants 3 -1 , 18 in (2.2) are not important; what is significant is t h a t the #G-mass of balls in G grow as r ds. Using terminology from the geometry of manifolds, we can say that G has volume growth given by r d~ .

Detour on Dimension. Let (F, p) be a metric space. There are a number of different definitions of dimension for F and subsets of F : here I just mention a few. The simplest of these is box-counting dimension. For e > 0, A C F , let N(A,e) be the smallest number of balls B(x, e) required to cover A. Then

(2.3)

d i m B c ( A ) = lim sup ~0

log N(A, e) log e-x

To see how this behaves, consider some examples. We take (F, p) to be ~a with the Euclidean metric.

Examples.

1. Let A -- [0,1] a C ~a. Then

N(A, e)

x e -a, and it is easy to verify

that lira log N([0, 1]d, e) = d. ~0 log e - 1 2. The Sierpinski gasket G. Since G C H~, and H,, is covered by 3 ~ triangles of side 2 - n , we have, after some calculations similar to those in Lemma 2.1, that N ( G , r ) y- (1/r)l~176 So,

dimBc(G) = 3. Let A = Q N [0,1]. Then N(A,e) dimBe({p}) = 0 for any p E A.

log3 log 2"

• ~-1, so dimBc(A)

= 1. On the other hand

We see t h a t box-counting gives reasonable answers in the first two cases, but a less useful number in the third. A more delicate, but more useful, definition is obtained if we allow the sizes of the covering balls to vary. This gives us Hausdorff dimension. I will only sketch some properties of this here - for more detail see for example the books by Falconer [Fal, Fa2]. Let h : ]~+ --* R+ be continuous, increasing, with h(0) = 0. For U C F write diam(U) -- sup{p(~, y) : z, y E U} for the diameter of V. For 6 > 0 let

~(a)--inf{E h(d(U,))

: A c gu,,

i

diam(U,) < 6}.

i

Clearly 7-/h(A) is decreasing in 6. Now let (2.4) we call T/h(-)

7/h(A) = lim 7/~(A); 610

Hausdorff h-measure. Let

B ( F ) be the Borel a-field of F.

10 Lemma

2.2. 7-/h is a measure on (F, B(F)).

For a proof see [Fal, Chapter 1]. We will be concerned only with the case h(x) = x~: we then write ~ for 7~h. Note that a ~ 7-/~(A) is decreasing; in fact it is not hard to see that 7 ~ ( A ) is either +o0 or 0 for all but at most one a. Definition

2.3. The Hausdorff dimension of A is defined by dimH(A) = inf{~ : ~ ( A )

Lemma

= 0} -- s u p { a : 7-/~(A) = + ~ } .

2.4. dimH(A) < dimBc(A).

Proof. Let a > dimBc(A). Then as A can be covered by N(A, e) sets of diameter 2e, we have 7-/~(A) < N(A,E)(2e) ~ whenever 2e < 6. Choose 0 so that d i m B c ( A ) < a - 0 < a; then (2.3) implies that for all sufficiently smMl e, N ( A , e) _< e-(~-0). So ? ~ ( A ) : 0, and thus 7/~(A) : 0, which implies that dimH(A) < a. [] Consider the set A = Q N [0, 1], and let A -- {Pa,P2,..-} be an enumeration of A. Let 5 > 0, and Ui be an open internal of length 2 - i A 6 containing Pi. Then (Ui) covers A, so that 7-/~(A) < ~ i ~ 1 (6 A 2-i) ~, and thus 7 ~ ( A ) -- 0. So dimH(A) = 0. We see therefore that dimH can be strictly smaller than dimBc, and that (in this case at least) dimH gives a more satisfactory measure of the size of A. For the other two examples considered above Lemma 2.4 gives the upper bounds dimH([0, 1] d) < d, dimH(G) < log 3 / l o g 2. In both cases equality holds, but a direct proof of this (which is possible) encounters the difficulty that to obtain a lower bound on 7 ~ ( A ) we need to consider all possible covers of A by sets of diameter less than 6. It is much easier to use a kind of dual approach using measures. 2.5. Let # be a measure on A such that #( A ) > 0 and there exist Cl < c~, ro > O, such that

Theorem

(2.5)

p ( B ( x , r ) ) < e a r ~,

xeA,

r
Then ~l"(A) >_ ci-a#(A), and d i m ~ ( A ) _> a. Proof. Let Ui be a covering of A by sets of diameter less than 6, where 25 < r0. If xi E Ui, then Ui C B(xi,diam(Ui)), so that #(Ui) _< ca diam (Ui)". So

"(Ui) ~ c11/d(A)"

diam (ui)" _> ei-1 E

E

i

i

Therefore 7-[~(A) _> c~ap(A), and it follows immediately that T/a(A) > 0, and dimH(A) _> a. [] C o r o l l a r y 2.6. dimH(G) = log 3 / l o g 2.

Proof. By Lemma 2.1 #G satisfies (2.5) with a = d I. So by Theorem 2.5 dimH(G) dl; the other bound has already been proved. [] Very frequently, when we wish to compute the dimension of a set, it is fairly easy to find directly a near-optimal covering, and so obtain an upper bound on d i m g directly. We can then use Theorem 2.5 to obtain a lower bound. However, we can also use measures to derive upper bounds on dimH.

11 T h e o r e m 2.7. L e t # be a finite measure on A such that # ( B ( x , r ) ) > C2r ~ lCor aH x 9 A, r < to. T h e n 7-l~(A) < cr and dimH(A) < a.

Proof. See [Fa2, p.61].

In particular we may note: C o r o l l a r y 2.8. I f # is a measure on A with #(A) 9 (0, ~ ) and (2.6)

c~ ~ < ~(B(x,r))

< c~r ~,

x 9 A,

r < r0

then 7-l~(A) 9 ( 0 , ~ ) and dimH(A) = a.

R e m a r k s . 1. If A is a k-dimensional subspace of R d then d i m g ( A ) = d i m B c ( A ) = k. 2. Unlike dimBc dimH is stable under countable unions: thus

i=1

i

3. In [Tri] Tricot defined "packing dimension" dimp(-), which is the largest reasonable definition of "dimension" for a set. One has dimp(A) > dimH(A); strict inequality can hold. The hypotheses of Corollary 2.8 also imply that dimp(A) = a. See [Fa2, p.48]. 4. The sets we consider in these notes will be quite regular, and will very often satisfy (2.6): that is they will be "a-dimensional" in every reasonable sense. 5. Questions concerning Hausdorff measure are frequently much more delicate than those relating just to dimension. However, the fractals considered in this notes will all be sufficiently regular so that there is a direct construction of the Hausdorff measure. For example, the measure # c on the Sierpinski gasket is a constant multiple of the Hausdorff x dj-measure on G. We note here how dimH changes under a change of metric. T h e o r e m 2.9. L e t Pl, P2 be metrics on F, and write 7-(~,i, dimH,i for the Hausdorff measure and dimension with respect to p,, i = 1, 2. (a) I f p ~ ( z , y ) < p 2 ( x , y ) for all x , y C A with p 2 ( z , y ) <_ 5o, then dimH,l(A) > dimH,2(A). (b) I f 1 A p l ( x , y ) • (1 A p 2 ( x , y ) ) ~ for some 0 > 0, then dimH,2 (A) = 0 dimH,1 (A). Proof. Write de(U) for the pj-diameter of U. If (U~) is a cover of A by sets with p2(Ui) < 5 < 50, then

E

dl(Ui) a < E i

so that ?-/~'I(A) < ~ ' 2 ( A ) . proving (a).

d2(Ui) a i

Then ?-/~'I(A) < 7/~,2(A) and dimH,l(A) > dimH,2(A),

12 (b) If Ui is a n y cover of A by sets of small d i a m e t e r , we have

Z dl(u,)~ • i

Zd2(U,) " i

Hence 7-/"'1 (A) = 0 if a n d only if 7-/e•'2 (A) = 0, a n d the conclusion follows.

[]

Metrics on the Sierpinski gasket. Since we will b e s t u d y i n g c o n t i n u o u s processes on G, it is n a t u r a l to consider the m e t r i c o n G given by t h e shortest p a t h in G b e t w e e n two points. We b e g i n w i t h a general definition. Definition

2 . 1 0 . Let A C I~d. For x, y E A set

d A ( x , y ) = inf{17 ] : 7 is a p a t h b e t w e e n x a n d y a n d 7 C A}. If dA(x, y) < oc for all x, y E A we call dA the geodesic metric on A. 2 . 1 1 . Suppose A is dosed, and that d A ( x , y ) < oc for all x, y E A. Then dA is a metric on A a n d (A, dA) has the geodesic property: Lemma

For each x, y E A there exists a map ~ ( t ) : [0, 1] --+ A such that dA(x,4~(t)) : t d A ( z , y ) ,

dA(4~(t),y) = (1 -- t)dA(x,y).

Proof. It is clear t h a t dA is a m e t r i c on A. To prove t h e geodesic property, let x , y E A, a n d D : d A ( x , y ) . T h e n for each n _> 1 there exists a p a t h %~(t), 0 < t < 1 + D such t h a t 7~ C A, [d'y~(t)l = dr, 7~(0) = x a n d 7 ~ ( t n ) : y for some D <_ tn <_ D q- n -1. I f p E [0, D] fq Q t h e n since Ix - "Yn(P)] -< P t h e sequence ('/n(p)) has a convergent subsequence. By a d i a g o n a l i z a t i o n a r g u m e n t there exists a s u b s e q u e n c e n~ such t h a t "/n, (P) converges for each p E [0, D] M Q; we c a n take = lim 7,~"

Lemma

[]

2 . 1 2 . For x , y E G, Ix -

Yl -< d a ( x , y ) _< ell x - YI'

Proof. T h e left h a n d i n e q u a l i t y is evident. It is clear from t h e s t r u c t u r e of H,~ t h a t if A, B are n - t r i a n g l e s a n d A fq B = 0, then la - b I _> (v/3/2)2 - ~

for a E A, b E B.

Let x, y E G a n d choose n so t h a t

(v /2)2 -("+a) < I x - yl <

2-".

So x, y are either in t h e s a m e n - t r i a n g l e , or in a d j a c e n t n-triangles. In either case choose z E G n so it is in t h e same n - t r i a n g l e as b o t h x a n d y.

13 Let z,, = z, a n d for k > n choose zk E G~ such t h a t x, zk are in the same ktriangle. T h e n since zk and zk+l are in the same k-triangle, and b o t h are contained in H k + l , we have da(zk,z~+l) = dH~+l(Zk,Zk+l ) < 2 - k . So,

da(z,x) < ~ da(zk,z~+l) <<21-~ < 4Ix -y]. k:n

Hence dG(x, y) < dG(x, z) + dv(z, y) < 8Ix

Y["

[]

Construction of a diffusion on the Sierpinski gasket. Let G,~ be the set of vertices of n-triangles. We can make Gn into a graph in a n a t u r a l way, by t a k i n g {x, y} to be an edge in G~ if x, y belong to the same n-triangle. (See Fig. 2.3). Write E,~ for the set of edges.

Figure 2.3: The graph G3. Let y ( n ) , k = 0 , 1 , . . . be a simple r a n d o m walk on G,~. Thus from x E G,~, the process Y('~) j u m p s to each of the neighbours of x with equal probability. ( A p a r t from the 3 points in Go, all the points in G,~ have 4 neighbours). The obvious way to construct a diffusion process (Xt,t > 0) on G is to use the graphs G,~, which provide a n a t u r a l a p p r o x i m a t i o n to G, and to t r y to define X as a weak limit of the processes Y('~). More precisely, we wish to find constants (an, n > 0) such t h a t

(2.7)

(y(n) [~.~1, t

_> 0)

(xt,~ _> 0).

We have two problems: (1) How do we find the right ( a n ) ? (2) How do we prove convergence? We need some more notation. D e f i n i t i o n 2.13. Let ,,q,~ be the collection of sets of the form G N A, where A is an n-triangle. We call the elements of ,S,~ n-complexes. For x C Gn let Dn(x) = [.J{S c

S~:zcS}. T h e key p r o p e r t i e s of the SG which we use are, first t h a t it is very symmetric, a n d secondly, t h a t it is finitely ramified. (In general, a set A in a metric space F is

14

finitely ramified if there exists a finite set B such t h a t A - B is not connected). For the SG, we see t h a t each n-complex A is disconnected from the rest of the set if we remove the set of its corners, t h a t is A f3 Gn. T h e following is the key observation. Suppose Y0( ' ) = y E G,~-I (take y r Go for simplicity), a n d let T = inf{k > 0 : y(n) e Gn-1 - {y}}. Then y ( n ) can only escape from D~-I(y) at one of the 4 points, { x l , . . . , x4} say, which are neighbours of y in the graph ( G n _ l , E n - ~ ) . Therefore Y('~) E { X l , . . . , x4 }. F u r t h e r the s y m m e t r y of the set G= (3 D=(y) means t h a t each of the events {YT(n) = xi} is equally likely. X2

X3

Xl

Y

X4

Figure 2.4: y and its neighbours. Thus

(Y(T~)=x~

y ( n ) = y ) 1= ~ ,

a n d this is also equal to ~(Y1(~-1) -- xiIY(~-1) = y). (Exactly the same argument applies if y C Go, except t h a t we t h e n have only 2 neighbours instead of 4). It follows t h a t Y(~) looked at at its visits to G ~ - I behaves exactly like y ( , ~ - l ) . To state this precisely, we first make a general definition. D e f i n i t i o n 2.14. Let T = ]~+ or Z+, let (Zt,t E 7~) be a cadlag process on a metric space F , a n d let A C F be a discrete set. T h e n successive disjoint hits by Z on A are t h e s t o p p i n g times To, T 1 , . . . defined by T0=inf{t>0:Zt

(2.8)

EA},

T~+I = i n f { t > T ~ : Z t c A - { Z T ~ } } ,

n>_O.

W i t h this notation, we can summarize the observations above. Lemma

2.15. Let (Ti)i>_o be successive disjoint hits by y(,O on G~-I.

Then

(YT(?), i _> O) is a simple random wMk on G,~-I and is therefore equal in law to

(y(,-1), i _> 0). Using this, it is clear t h a t we can build a sequence of "nested" r a n d o m walks on G,~. Let N _> 0, and let y(N),~ k _> 0 be a SRW on GN with y(N) = 0. Let

[TN,m~ 0 <_ m < N - 1 and ~ i )i>_o be successive disjoint hits by y(N) on Gin, and set

y(m) = y ( m ( T N , m ) = y (TuN,) m '

i > 0.

15 It follows from L e m m a 2.15 t h a t y(m) is a SRW on Gin, and for each 0 < n < m <_ N we have t h a t y ( m ) , s a m p l e d at its successive disjoint hits on G,~, equals y ( n ) . We now wish to construct a sequence of SRWs with this p r o p e r t y holding for 0 < n <_ m < co. This can be done, either by using the Kolmogorov extension theorem, or directly, b y building y ( g + l ) from y(N) with a sequence of i n d e p e n d e n t "excursions". T h e argument in either case is not hard, and I omit it. Thus we can construct a p r o b a b i l i t y space (fl, .T, ~), carrying r a n d o m variables (v(n) "k , n _ > 0 , k > 0 ) s u c h t h a t (a) For each n, ( y ( , 0 , k > 0) is a SRW on G= starting at 0. (b) Let T~ 'm be successive disjoint hits by Y('*) on G , , . (Here m < n). T h e n (2.9)

y(,,)(T:,m) = y(m),

i > O,

m < n.

If we j u s t consider the p a t h s of the processes y ( , 0 in G, we see t h a t we are viewing successive discrete approximations to a continuous path. However, to define a limiting process we need to rescale time, as was suggested by (2.7). W r i t e 7- = T~ '~ = min{k > 0 : [y(1)] = 1}, and set f(s) = E s ~, for s 6 [0,1]. L e m r n a 2.16. f ( , ) = s2/(4 - 3,), ET ----i f ( l ) = 5, a n d E r k < co for a / / k .

Proof. This is a simple exercise in finite state Markov chains. Let a l , a2 be the two 1 non-zero elements of Go, let b = 89 + a2), and ci -- ~ai. Writing fc(s) -- E c l s ~, a n d defining .fb, fa similarly, we have fa(S) = 1, f(8) = sfc(8),

fc(a) = 88

4- fc(S) 4- fb(s) 4- f a ( s ) ) ,

h(,) = 89

+ f,,(s)),

a n d solving these equations we o b t a i n f(s). T h e remaining assertions follow easily from this.

[]

a 1

0

b 2

a2

Figure 2.5: The graph G~.

~6 Now let Zn = Tff '~ n > 0. The nesting property of the random walks Y(~) implies that Zn is a simple branching process, with offspring distribution (p,~), where (2.10)

f(s) = ~

skpk.

k=2

To see this, note that y ( n + l ) , for T~ +1'~ _< k _< T ~ +1'~ is a SRW on G,~+I N Dn(Yi('0), and that therefore T~++l'n - T ~ +1'~ (d_)z. Also, by the Markov property, the r.v. ~ = T ~ +1''~ - T~ +1''~, i _> 0, are independent. Since Z~ --1

i=O (Z~) is a branching process. As E~-2 < o% and Ez -- 5, the convergence theorem for simple branching processes implies that 5-'~Z~ ~"% W for some strictly positive r . v . W . (See [Har, p. 13]). The convergence is easy using a martingale argument: proving that W > 0 a.s. takes a little more work. (See [Har, p. 15]). In addition, if

~o(u) = Be - ' w then ~o satisfies the functional equation (2.11)

~o(5u) = f(~oCu)),

~J(O) ----- 1 .

We have a similar result in general. Proposition

2.17. Fix m > O. The processes

r~m , Z(i) = T~'m - Ti_I

n > m

are branching processes with offspring distribution T, and Z (i) are independent. Thus there exist W(i m) such that for each m (W(im),i >_ O) are independent, 5-row, and n,m ) --+ W.(trn) a.$. 5--n ( T ? ' m - Ti_l Note in particular that E(T~ '~ ) ---- 5 n, that is that the mean time taken by y ( , 0 to cross G~ is 5 ~. In terms of the graph distance on G~ we have therefore that y(n) requires roughly 5 '~ steps to move a distance 2n; this may be compared with the corresponding result for a simple random walk on Z d, which requires roughly 4 n steps to move a distance 2 n. The slower movement of Y(n) is not surprising - - to leave Gn f) B(O, 1/2), for example, it has to find one of the two 'gateways' (1/2, 0) or (1/4, vf3/4). Thus the movement of y(n) is impeded by a succession of obstacles of different sizes, which act to slow down its diffusion.

17 Given the space-time scaling of y ( , 0 it is no surprise that we should take a s = 5 ~ in (2.7). Define X~'

= v(n)

"[5-t],

t >_ O.

I n view of the fact that we have built the y(n) with the nesting property, we can replace the weak convergence of (2.7) with a.s. convergence. T h e o r e m 2.18. The processes X '~ converge a.s., and uniformly on compact interwa/s, to a process Xt, t >__O. X is continuous, a n d X t E G for all t > O.

Proof. For simplicity we will use the fact that W has a non-atomic distribution function. Fix for now rn > 0. Let t > 0. Then, a.s., there exists i = i(w) such that i

i+1

Ew?

Ew?).

j=l

j----1

a s w j~) = l i m . _ ~ 5-" (T? '~ - T?:7) it follows that for n > n0(o~), n~m z ? '~ < 5'~t < Ti+x 9

(2.12)

Now Y ( - ) ( T ? ,~) = r, ('~) by (2.9). Since r~('~) ~ D . ( r , (~)) for T? '~ < k < T3;1, we have [l~:~]-- - Y/(m) I < 2 - ~

for all n >__no.

This implies that [X~ - X~' I < 2 -'~+1 for n, n ' > no, so that X~ is Cauchy, and converges to a r.v. Xt. Since X ~ 6 Gn, we have X t 6 G. W i t h a little extra work, one can prove that the convergence is uniform in t, on compact time intervals. I give here a sketch of the argument. Let a 6 N, a n d let

~m :

rain l
W: m) TM

T h e n ~,~ > 0 a.s. Choose no such that for n > no i

5-nTt'm-

E

W (m) < l~n ,

l
TM.

j=l n~m T h e n if i = i(t, w) is such that Wim < t < W i +m l , a n d i < a 5 m w e h a v e 5 _ ~ Ti_ 1 < --n n~m t <5 T~+2 for a l l n _ > n 0 . So, I X ~ - Yim[ < 2 -m+x for all n > no. This implies

t h a t if T,~ = ~ i = as~ 1 W (m), and S < Tin, then sup I X : O n O . If S < llm infm T m

X~'l <

2- " + 5

then the uniform a.s. convergence on the

(random) interval [0, S] follows. If s, t < g,~ and It - s I < (m, then we also have [X~ -X~n[ < 2 -m+~ for n _> no. Thus X is uniformly continuous on [0, S]. Varying a we also o b t a i n uniform a.s. convergence on fixed intervals [0, to]. []

18 Although the notation is a little cumbersome, the ideas underlying the construction of X given here are quite simple. The argument above is given in [BP], but Kusuoka [Kusl], and Goldstein [Go], who were the first to construct a diffusion on G, used a similar approach. It is also worth noting that Knight [Kn] uses similar methods in his construction of 1-dimensional Brownian motion. The natural next step is to ask about properties of the process X. But unfortunately the construction given above is not quite strong enough on its own to give us much. To see this, consider the questions (1) Is W = limn-.oo 5 - n T ~ '~ = inf{t _> 0 : X~ 9 G - {0}}? (2) Is X Markov or strong Markov? For (1), we certainly have X w 9 G - { 0 } . However, consider the possibility that each of the r a n d o m walks Y. moves from 0 to a2 on a path which does not include al, but includes an approach to a distance 2 - " . In this case we have al ~ {X~, 0 < t < W}, but X T = al for some T < W. Plainly, some estimation of hitting probabilities is needed to exclude possibilities like this. (2). The construction above does give a Markov property for X at stopping times of the form V.i z..,i= 1 w(-~) ,, j . But to obtain a good Markov process X = (Xt, t _> 0, ?~, x 9 G) we need to construct X at arbitrary starting points x 9 G, and to show that (in some appropriate sense) the processes started at close together points x and y are close. This can be done using the construction given above - - see [BP, Section 2]. However, the argument, although not really hard, is also not that simple. In the remainder of this section, I will describe some basic properties of the process X , for the most part without giving detailed proofs. Most of these theorems will follow from more general results given later in these notes. Although G is highly symmetric, the group of global isometries of G is quite small. We need to consider maps restricted to subsets. D e f i n i t i o n 2.19. Let (F, p) be a metric space. A local isometry of F is a triple (A, B, ~o) where A, B are subsets of F and ~ois an isometry (i.e. bijective and distance preserving) between A and B, and between OA and OB. Let ( X t , t > O,g~,x 9 F ) b e a M a r k o v process on F. For H C F , set TH = inf{t > 0 : X t 9 H } . X is invariant with respect to a local isometry (A, B, ~o) if

P= ( ~ ( x , ^ r o ~ ) 9

t > 0) = P~(=)

(x,^ro, 9 t > o).

X is locally isotropic if X is invariant with respect to the local isometries of F. T h e o r e m 2.20. (a) There exists a continuous strong Markov process X : (Xt, t > O, ? L x 9 G) on G.

(b) The semigroup on C(G) de~ed by Pal(x) = E= y ( x , ) is Fe//er, and is pG-symmetric:

/G f(x)P~g(x)~G(d~) = /

a(~)Pd(x)~G(dx).

19

(c) X is locally isotropic on the spaces ( a , I" - " I) and ( G, da). (d) For n > 0 let Tn,i, i > 0 be successive disjoint hits by X on Gn. Then ~('*) = ~(~) XT..~, i > 0 defines a S R W on Gn, and =[5*t] -+ X t uniformly on compacts, a.s. So, in particular (Xt, t > 0, I?~ is the process constructed in Theorem 2.18. This theorem will follow from our general results in Sections 6 and 7; a direct proof may be found in [BP, Sect. 21. The main labour is in proving (a); given this (b), (c), (d) all follow in a relatively straightforward fashion from the corresponding properties of the approximating random walks ~(n). The property of local isotropy on (G, da) characterizes X : T h e o r e m 2.21. (Uniqueness). Let (Zt, t > O, Qz, x C ~) be a non-constant locally isotropic diffusion on (G, da). Then there exists a > 0 such that (F(z,

~ .,t ___0) = ? = ( x o , ~ .,t _> 0).

(So Z is equal in law to a deterministic time change of X ) . The beginning of the proof of Theorem 2.21 runs roughly along the lines one would expect: for n > 0 let (~(=), i >_ 0) be Z sampled at its successive disjoint hits on Gn. The local isotropy of ~z implies that Y(=) is a SRW on G=. However some work (see [BP, Sect. 8]) is required to prove that the process Y does not have traps, i.e. points z such that Q* (Yt = z for all t) = 1. R e m a r k 2.22. The definition of invariance with respect to local isometries needs some care. Note the following examples. 1. Let x , y E Gn be such that D , ( z ) n Go = ao, Dn(y) C3Go = 0. Then while there exists an isometry ~ from D,~(z) n G to D=(y) n G, ~v does not map OnD=(z) ;3 G to ORDn(y) n G. (OR denotes here the relative boundary in the set G). 2. Recall the definition of Hn, the n-th stage in the construction of G, and let B , = OH,~. We have G = cl(UB,). Consider the process Zt on G, whose local motion is as follows. If Zt E H,~ - H=-I, then Zt runs like a standard 1-dimensional Brownian motion on H , , until it hits Hn-1. After this it repeats the same procedure on Hn-1 (or H,~-k if it has also hit H,,-k at that time). This process is also invariant with respect to local isometries (A, B, ~o) of the metric space (G, I" - " I). See [He] for more on this and similar processes. To discuss scale invariant properties of the process X it is useful to extend G to an unbounded set G with the same structure. Set = U 2'~G, rt=0

and let G~ be the set of vertices of n-triangles in G,~, for n > 0. We have G~ = U 2kGn+k, k=0

20 and if we define Gm = {0} for m < 0, this definition also makes sense for n < 0. We can, almost exactly as above, define a limiting diffusion .~ = (_~,, t _> 0, ~x, x E G) Oil G: =

t _> o, a s

where (Y~('~),n > 0, k _> 0) are a sequence of nested simple random walks on G=, and the convergence is uniform on compact time intervals. The process X satisfies an analogous result to Theorem 2.20, and in addition satisfies the scaling relation (2.13)

F~(22~ 9 -, t > 0) : ~2~(25t 9 -, t > 0).

Note that (2.13) implies that _~ moves a distance of roughly t l~176 Set d,~ = d~(G) -- l o g 5 / l o g 2 .

in time t.

We now turn to the question: "What does this process look like?" The construction of X, and Theorem 2.20(d), tells us that the 'crossing time' of a 0-triangle is equal in law to the limiting random variable W of a branching process with offspring p.g.f, given by f ( s ) = s 2 / ( 4 - 3 s ) . From the functional equation (2.11) we can extract information about the behaviour of ~o(u) = E e x p ( - u W ) as u --* oo, and from this (by a suitable Tauberian theorem) we obtain bounds on P ( W < t) for small t. These translate into bounds on Px(1Xt - x I > A) for large A. (One uses scaling and the fact that to move a distance in G greater than 2, X has to cross at least one 0-triangle). These bounds give us many properties of X. However, rather than following the development in [BP], it seems clearer to first present the more delicate bounds on the transition densities of .Y and X obtained there, and derive all the properties of the p r o c ~ s from them. Write ~ c for the analogue o f # a for G, and Pt for the semigroup of X. Let ~. be the infinitesimal generator of Pt. 2.23. fit and Pt have densities ~(t, x, y) and p(t, x, y) respectively. ~(t, ~, y) i~ continuous on (0, ~ ) • ~ • ~. ~(t,~,y) = ~(t,y,~) for ~ t,~,y. t -~ ~ ( t , . , y) i~ C ~ on (0, ~ ) for each (~, y). For each t, y

Theorem

(a) (b) (c) (d)

I~(t,x,y) - ~(t,x',y)] <_ Clt-Xlx - x'l d~-d',

x,x' 9 G.

(e) Fort E (O, cc), x , y C (2.14)

c2t - d ~ / ~ exp

-c3

--ty-

~ c4t -d2/d~ exp

<_~ ( t , x , y ) --c5

21 (f) For each Yo 9 G, p(t,x, yo) is the fundamental solution o f the heat equation on G with pole at Yo:

,yo) = Z (t, x, yo),

y0) =

(g) p(t, x, y) satisfies (a)-(f) above (with G replaced by G and t 9 (0, c~] replaced by t 9 (0, 1]). R e m a r k s . 1. T h e p r o o f of this in [BP] is now largely obsolete - - simpler m e t h o d s are now available, t h o u g h these axe to some extent still based on the ideas in [BP]. 2. If d f -- d and d~, = 2 we have in (2.14) the form of the transition density of Brownian m o t i o n in R a. Since d~ -- log 5 / l o g 2 > 2, the tail of the d i s t r i b u t i o n of [Xt - x[ u n d e r I?~ decays more r a p i d l y t h a n an exponential, b u t more slowly t h a n a Gaussian. It is fairly straightforward to integrate the b o u n d s (2.14) to obtain information a b o u t X . At this point we just present a few simple calculations; we will give some further p r o p e r t i e s of this process in Section 3. D e f i n i t i o n 2.24. For x 9 G, n 9 Z, let x,~ be the point in Gn closest to x in E u c l i d e a n distance. (Use some p r o c e d u r e to b r e a k ties). Let D , ( x ) = D,(x,~). Note t h a t "fiG(D,(x,~)) is either 3 - " or 2.3 - " , t h a t (2.15)

Ix - Yl -< 2 ' 2 - ~

if y 9 D , ( x ) ,

and that (2.16)

Ix - Yl >- ~-~2-('~+1)

if y C G N D , ~ ( x ) c .

T h e sets D,~(x) form a convenient collection of neighbourhoods of points in G. Note t h a t U , + z D , ( x ) = G. C o r o l l a r y 2.25. F o r x E G,

clt 2/d= <_ E ~ IX~ - x} 2 <_ c~t 2/d= ,

t > O.

Proof. We have E: [Xt - xl 2 = ~ ( y - x)2~(t, x, y)~to(dy ). Set Am = Dm(x) - Dm+l(x). T h e n

(2.17)

/A (y - x)2P(t'x'Y) c(dY)

=

e:p

22 Choose n such t h a t 5 -'~ _< t < 5 -~+1, and write am(t) for the final t e r m in (2.17). Then

E~(x, - x) ~ <_ ~

am(t) +

m:--oo

am(t) m:n

For m < n, 5 - r " / t > 1 a n d the exponential t e r m in (2.17) is dominant. After a few calculations we o b t a i n n--1

a . ( t ) _< c ( 2 - " ) ' + ' , t - ' , / ~ ~rt~-- oo

at (2h-df)/g~-d2/4~ ~ Ct (2+4f)/g~-df/d~ ~

ct 2/4~,

where we used t h e fact t h a t ( 2 - n ) d~ x t. For m > n we neglect the exponential term, a n d have

am(t) < ~ t - ~ , / ~

Z

(2-")~+~,

< ct-~lld~ (2---)2+d~ < c't21 d~. Similar calculations give the lower bound. Remarks

[]

2.26. 1. Since 2 / d ~ = log 4 / l o g 5 < 1 this implies t h a t X is subdiffusive.

2. Since ~G (B(x, r)) • r d,, for x E G, it is t e m p t i n g to t r y and prove Corollary 2.25 by the following calculation:

(2.18)

E~12~ - ~1~

=

•

/{ /o

= t 2/d~

f

r2dr

~(t,x,y)~tc(dy)

JOB(x,r)

dr r 2 rds-lt-ds/d'~ exp --c(rd~/t) l[d~'-I

(

/o

s l+a~ exp - c ( s d~)l/a~-I

(

)

)

ds = ct 2/d~.

Of course this calculation, as it stands, is not valid: the e s t i m a t e

~(B(~,~

+ ~ ) - B(x,~)) • ~ ' - ' ~

is c e r t a i n l y not valid for all r. But it does hold on average over length scales of 2 '~ < r < 2 n+l, a n d so splitting G into suitable shells, a rigorous version of this calculation m a y be o b t a i n e d - a n d this is what we did in the proof of Corollary 2.25. T h e / k - p o t e n t i a l kernel density of )~ is defined by

~,(~, y) =

e-~'~(t, ~, y) dr.

From (2.14) it follows t h a t ux is continuous, t h a t u ~ ( x , z ) <_ r dl/d~-l, and t h a t ux ~ ~ as A --* 0. Thus the process X (and also X ) "hits points" - t h a t is if

23 Ty -- inf{t > 0 : )f~ = y} then

(2.19)

< oo) > 0.

It is of course clear that X must be able to hit points in Gn - otherwise it could not move, but (2.19) shows that the remaining points in G have a similar status. The continuity of u x ( x , y) in a neighbourhood of x implies that F ' ( T : = 0) = 1, that is t h a t x is regular for {x} for all x 6 G. The following estimate on the distribution of I)ft - x[ can be obtained easily from (2.14) by integration, but since this bound is actually one of the ingredients in the proof, such an argument would be circular. Proposition

2.27. For x 6 G, A > 0, t > 0,

X[ >

),)

_< c3 e x p From this, it follows that the paths of _~ are HSlder continuous of order l/d,,, for each e > 0. In fact we can (up to constants) obtain the precise modulus of continuity of X. Set h(t) = t l/d~ ( l o g t - 1 ) (d~-l)/d~ . T h e o r e m 2.28. (a) For x E G

cl _< lim

sup

8J.0 0<s
[)(, - Xt[ _< c2, h(s - t)

I?~ - a.s.

I~-~l<S (b) T h e p a t h s of .~ are of infinite quadratic variation, a.s., and so in particalar is n o t a semimartingale. The proof of (a) is very similar to that of the equivalent result for Brownian motion in I~d. For (b), Proposition 2.23 implies that [Xt+h - Xt[ is of order h 1/d~ ; as d,o > 2 this suggests that X should have infinite quadratic variation. For a proof which fills in the details, see [BP, Theorem 4.5]. [] So far in this section we have looked at the Sierpinski gasket, and the construction and properties of a symmetric diffusion X on G (or G). The following three questions, or avenues for further research, arise naturally at this point.

24 1. Are there other n a t u r a l diffusions on the SG? 2. Can we do a similar construction on other fractals? 3. W h a t finer p r o p e r t i e s does the process X on G have? (More precisely: what a b o u t p r o p e r t i e s which the b o u n d s in (2.17) are not strong enough to give information on?) T h e bulk of research effort in the years since [Kusl, Go, BP] has been devoted to (2). O n l y a few p a p e r s have looked at (1), and (apart from a number of works on s p e c t r a l properties), the same holds for (3). Before discussing (1) or (2) in greater detail, it is worth extracting one p r o p e r t y of the SRW y ( D which was used in the construction. Let V = (V,~,n > O,I?a,a E Go) be a Markov chain on Go: clearly V is specified by the t r a n s i t i o n probabilities

p(al,aj) = Fa'(V1 = hi),

0 _< i,j <_ 2.

We take p(a,a) = 0 for a E G0, so V is d e t e r m i n e d by the three probabilities p(ai,aj), where j -- i + 1 (mod 3). Given V we can define a Markov Chain V ~ on G1 by a process we call replication. Let {b01, b02, b12} be the 3 points in G1 - Go, where bij -- 89 + aj). We consider G1 to consist of three 1-cells {hi, bij,j ~ i}, 0 < i < 2, which intersect at the points {bij}. T h e law of V I m a y be described as follows: V I moves inside each 1-cell in t h e way same as V does; if V~ lies in two 1-cells then it first chooses a 1-cell to move in, and chooses each 1-cell with equal probability. More precisely, writing Y' = (V~n,n >_ o,~a,a E G1), and p'(a, b) =

= b),

we have (2.20)

p'(ai, biy) ----p(ai, a j ) , Pl(bij,bik) = lp(aj, ak),

hi). pl(bij,ai) = ~p(aj, 1

Now let T~, k > 0 be successive disjoint hits by V I on Go, and let Uk = V~-k , k > 0. T h e n U is a Markov Chain on Go; we say t h a t V is decimation invariant if U is equal in law to V. We saw above t h a t the SRW y(0) on Go was decimation invariant. A n a t u r a l question is: W h a t other d e c i m a t i o n invariant Markov chains are there on Go? Two classes have been found: 1. (See [Go]). Let p(ao,al) = p(al,ao) = 1, p(a2 ao) - 1 2. "p-stream r a n d o m walks" ([Kuml]). Let p E (0, 1) and

p(ao,al) = p ( a l , a 2 ) =p(a2,ao) =p. From each of these processes we can construct a limiting diffusion in the same way as in T h e o r e m 2.18. T h e first process is reasonably easy to understand: essentially its p a t h s consist of a downward drift (when this is possible), and a behaviour

25 like 1-dimensional Brownian motion on the portions on G which consist of line segments parallel to the x-axis. For p > 89 Kumagai's p-stream diffusions tend to rotate in an anti-clockwise direction, so are quite non-symmetric. Apart from the results in [Kuml] nothing is known about this process. Two other classes of diffusions on G, which are not decimation invariant, have also been studied. The first are the "asymptotically 1-dimensional diffusions" of [HHW4], the second the diffusions, similar to that described in Remark 2.22, which are (G, [ . - . [)-isotropic but not (G, d~)-isotropic - s e e [He]. See also [HH1, HK1, HHK] for work on the self-avoiding random walk on the SG. Diffusions on o~her /racta/ sets. Of the three questions above, the one which has of making similar constructions on other fractals. which can arise, consider the following two fractals, by a Cantor type procedure, based on squares rather the figure gives the construction after two stages.

mml

mm

=%

received most attention is that To see the kind of difficulties both of which are constructed than triangles. For each curve

mmm

9

9

=%

Figure 2.6: The Vicsek set and the Sierpinski carpet. The first of these we will call thc "Vicsek set" (VS for short). We use similar notation as for the SG, and write Go, G 1 , . . . for the succession of sets of vertices of corners of squares. We denote the limiting set by F = Fvs. One difficulty arises immediately. Let Y~ be the SRW on Go which moves from any point z E Go to each of its neighbours with equal probability. (The neighbours of x are the 2 points y in Go with Ix - y[ = 1). Then y(0) is not decimation invariant. This is easy to see: y(0) cannot move in one step from (0, 0) to (1,1), but y(1) can move from (0, 0) to (1, 1) without hitting any other point in Go. However it is not hard to find a decimation invariant random walk on Go. Let p E [O, 1], and consider the random walk (Y~,r > 0, E~,x E Go) on Go which moves diagonally with probability p, and horizontally or vertically with probability 89( l - p ) . Let (Y~',r _> 0, E~,x 9 G1) be the Markov chain on G1 obtained by replication, and let T~, k > 0 be successive disjoint hits by Y' on Go.

26 0 ! T h e n writing f ( p ) = ~p(Y~'x = (1,1)) we have (after several minutes calcula-

tion) f(P)=

1 4-3p"

T h e equation f(p) = p therefore has two solutions: p = ~ and p = 1, each of which corresponds to a d e c i m a t i o n i n w r i a n t walk on Go. (The number ~ here has no general significance: if we h a d looked at the fractal similar to the Vicsek set, b u t b a s e d on a 5 x 5 square r a t h e r t h a n a 3 x 3 square, t h e n we would have o b t a i n e d a different number). One m a y now c a r r y through, in each of these cases, the construction of a diffusion on the Vicsek set F , very much as for the Sierpinski gasket. For p = 1 one gets a r a t h e r uninteresting process, which, if s t a r t e d from (0, 0), is (up to a constant t i m e change) 1-dimensionM Brownian motion on the diagonal {(t, t), 0 < t < 1}. It is w o r t h r e m a r k i n g t h a t this process is not strong Markov: for each x E F one can take I?* to be t h e law of a Brownian motion moving on a diagonal line including x, b u t the strong Markov p r o p e r t y will fail at points where two diagonals intersect, such as the point (89 89 For p = ] one obtains a process (Xt,t > 0) with much the same behaviour as t h e Brownian m o t i o n on the SG. We have for the Vicsek set (with p = 89 df(Fvs) = log 5 / l o g 3, dw(Fvs) = log 1 5 / l o g 3. This process was studied in some d e t a i l b y K r e b s [Krl, Kr2]. T h e Vicsek set was mentioned in [Go], and is one of the "nested fractals" of L i n d s t r 0 m [L1]. This e x a m p l e shows t h a t one m a y have to work to find a decimation invariant r a n d o m walk, a n d also t h a t this m a y not be unique. For the VS, one of the decimation invariant r a n d o m walks was degenerate, in the sense t h a t P~(Y hits y) = 0 for some z, y E Go, a n d we found t h e associated diffusion to be of little interest. But it raises t h e possibility t h a t there could exist regular fractals carrying more t h a n one " n a t u r a l " diffusion. T h e second example is the Sierpinski carpet (SC). For this set a more serious difficulty arises. T h e VS was finitely ramified, so t h a t if Yt is a diffusion on Fvs, a n d (Tk, k > 0) are successive disjoint hits on G,~, for some n > 0, then (YTk, k >_O) is a Markov chain on Gn. However the SC is not finitely ramified: if (Zt, t > 0) is a diffusion on Fsc, t h e n the first exit of Z from [0, 89 could occur anywhere on t h e line segments {($,y),01 _< y _< ~},1 {(z, 5),10 < x < ~}. It is not even clear t h a t a diffusion on Fsc will hit points in Gn. Thus to construct a diffusion on Fsc one will need very different m e t h o d s from those outlined above. It is possible, a n d has been done: see [BBI-BB6], a n d [Bas] for a survey. On the t h i r d question mentioned above, disappointingly little has been done: most known results on the processes on t h e Sierpinski gasket, or o t h e r fractals, are of roughly the same d e p t h as the bounds in T h e o r e m 2.23. Note however the results on t h e s p e c t r u m of/~ in [FS1, FS2, S h l - S h 4 ] , and the large deviation results in [Kumh]. Also, K u s u o k a [Kus2] has very interesting results on the behaviour of h a r m o n i c functions, which i m p l y t h a t the measure defined formally on G by

v(dx) --- I v / I z (~)~(dz) is singular with respect to p. There are m a n y open problems here.

27 3. F r a c t i o n a l D i f f u s i o n s . In this section I will introduce a class of processes, defined on metric spaces, which will include m a n y of the processes on fractals mentioned in these lectures. I have chosen an axiomatic approach, as it seems easier, and enables us to neglect (for t h e t i m e being!) much of fine detail in the geometry of the space. A m e t r i c space (F, p) has t h e midpoint property if for each x, y E F there exists 1 X , y). Recall t h a t the geodesic metric dG in z E F such t h a t p(z, z) = p ( z , y ) = ~p( Section 2 h a d this property. The following result is a straightforward exercise: L e m m a 3.1. (See [Blu]). Let (F, p) be a complete metric space with the midpoint property. Then for each x, y E F there ex/sts a geodesic path (7(t), 0 < t < 1) such that 7(0) = z, 3'(1) = y a n d p(~/(s),'r(t)) = ]t - s[d(z,y), 0 < s < t < 1. For this reason we will frequently refer to a metric p with the m i d p o i n t p r o p e r t y as a geodesic metric. See [Stul] for a d d i t i o n a l remarks a n d references on spaces of this type. D e f i n i t i o n 3.2. Let (F, p) be a complete metric space, and # be a Borel measure on ( F , B ( F ) ) . We call ( F , p , p ) a fractional metric space (FMS for short) if (3.1a)

(F, p) has the m i d p o i n t p r o p e r t y ,

a n d there exist d f > 0, and constants Cl,C2 such t h a t if r0 = s u p { p ( z , y ) : x , y E F } E (0, oo] is t h e d i a m e t e r of F t h e n (3.1b)

clr d' < # ( B ( x , r ) ) <_ c2 rdf

for

x E F, 0 < r < r0.

Here B ( z , r ) = {y 9 F : p ( z , y ) < r}. R e m a r k s 3.3. 1. I~d, with Euclidean distance and Lebesgue measure, is a FMS, with d I = d and r0 = cr 2. If G is the Sierpinski gasket, dG is the geodesic metric on G, and # = P c is t h e measure c o n s t r u c t e d in Section 2, then L e m m a 2.1 shows t h a t (G, dG, #) is a F M S , w i t h d I = dr(G) = l o g 3 / l o g 2 and r0 = 1. Similarly ( G , d ~ , ~ ) is a F M S with r 0 ~(~2.

3. If ( F k , d k , p k ) , k = 1,2 are F M S with the same d i a m e t e r r0 and p 9 [1, oo], then setting F = F1 x F2, d((Xl, z2), (Yl, Y2)) = (dl(Xl, y l ) P + d 2 ( z 2 , yz)P) l/p, # = Pl x #~, it is easily verified t h a t ( F , d , # ) is also a F M S with d r ( f ) = dr(F1 ) + dr(F2 ). 4. For simplicity we will from now on take either r0 = ~ or r0 = 1. We will write r 9 (0, r0] to m e a n r 9 (0, r0] n (0, ~ ) , and define r~' = ~ if a > 0 and r0 = co. A n u m b e r of p r o p e r t i e s of (F, p, #) follow easily from the definition. L e r n m a 3.4. (a) d i m H ( F ) = d i m p ( F ) = dr.

(b) F is locally compact. (c) df > 1. Proof. (a) is i m m e d i a t e from Corollary 2.8. (b) Let x 9 F, A = B ( x , 1), a n d consider a m a x i m a l packing of disjoint balls B ( x i , e ) , zi 9 A, 1 < i < m. As #(A) <_ c2, and # ( B ( z i , e ) ) >_ Cle d~, we have m < c2(cled~) -1 < cr Also A = U'~=lB(Xi,2e ). Thus any b o u n d e d set in F can be

28 covered by a finite n u m b e r of balls radius s; this, with completeness, implies t h a t F is locally compact. (c) Take x, y E F with p(x, y) = D > 0. A p p l y i n g the m i d p o i n t p r o p e r t y r e p e a t e d l y we obtain, for m = 2 k, k > 1, a sequence x -- zo,zl,... ,zm -- y with p(zi,zi+l) = D i m . Set r = D/2m: the balls B(zi,r) must be disjoint, or, using the triangle inequality, we would have p(x, y) < D. But then m--1

U B(z,, r) C B(x,D), i=0

so t h a t

rn--1

c2D d' > # ( B ( x , D ) ) > E

#(B(z,,r))

i=0

> mclD d~( 2 m ) - d s = cml--d~. If d f < 1 a c o n t r a d i c t i o n arises on letting m --* ~ .

[]

3.5. Let (F,p,#) be a fractional metric space. X = (~x, x E F, Xt, t ~ 0) is a fractional diffusion on F if Definition

A Markov process

(3.2a) X is a conservative Feller diffusion with state space F . (3.2b) X is # - s y m m e t r i c . (3.2c) X has a s y m m e t r i c t r a n s i t i o n density p(t, x, y) = p(t, y, x), t > 0, x, y E F , which satisfies, the C h a p m a n - K o l m o g o r o v equations a n d is, for each t > 0, j o i n t l y continuous. (3.2d) T h e r e exist constants (~,fl, 7, cl - c4, to = v0~, such t h a t

(3.3)

Clt

exp

<

~_ c3t -~ exp (-c4p(x, y)flTt-7) , x, y E F, 0

< t ~ to.

E x a m p l e s 3.6. 1. If F is ~d, a n d a(x) = aij(x), 1 < i, j < d, x C R d is bounded, s y m m e t r i c , measurable and uniformly elliptic, let s be the divergence form o p e r a t o r

~Oa,

x

0

zj

T h e n Aronsen's b o u n d s JAr] i m p l y t h a t the diffusion with infinitesimal generator s isaFD, witha=d/2, fl=2,7=l. 2. By T h e o r e m 2.23, the Brownian motion on the Sierpinski gasket described in Section 2 is a F D , with a = dl(SV)/dw(SG), fl = dw(SG) and 7 = 1/(fl - 1). T h e hypotheses in Definition 3.5 are quite strong ones, and (as the examples suggest) t h e assertion t h a t a p a r t i c u l a r process is an F D will usually be a substantial theorem. One could of course consider more general bounds t h a n those in (3.3) (with a correspondingly larger class of processes), b u t the form (3.3) is reasonably natural, a n d a l r e a d y contains some interesting examples. In an interesting recent series of papers S t u r m [Stul-Stu4] has studied diffusions on general metric spaces. However, the processes considered there t u r n out to have an essentially Gaussian long range behaviour, and so do not include any F D s with

29 In the rest of this section we will s t u d y the general properties of FDs. In the course of our work we will find some slightly easier sufficient conditions for a process to be a F D t h a n the b o u n d s (3.3), and this will be useful in Section 8 when we prove t h a t certain diffusions on fractais are FDs. We begin by obtaining two relations between t h e indices dl, a , fl, 7, so reducing the p a r a m e t e r space of F D s to a two-dimensional one. We will say t h a t F is a F M S ( d f ) if F is a F M S and satisfies (3.1b) with p a r a m e t e r df (and constants el, c2). Similarly, we say X is a F D ' ( d f , a, f l , 7 ) if X is a F D on a F M S ( d f ) , and X satisfies (3.3) with constants a , /3, 7. (This is t e m p o r a r y n o t a t i o n - - hence the '). It w h a t follows we fix a F M S (F, p, #), with p a r a m e t e r s r0 and d I. Lemma

3.7. L e t a , 7 , x > 0 and set I ( % x) =

~1 ~176 e -xt~ dt,

S(a,7 ,x) = E

n=0

a~e-=~'"

Then (3.4)

( a - 1 ) S ( ( ~ , 7 , a T z ) < I ( % x ) <_ (a - 1 ) S ( a , 7 , z),

and (3.5)

I ( 7 , z ) • z -1/~

(3.6)

I ( 7 , z) • z - l e - x

for z <_ 1, for x > 1,

Proof. We have

,(%,): and e s t i m a t i n g each t e r m in the sum (3.4) is evident. If 0 < x < 1 t h e n since x 1 / ~ I ( 7 , x) =

1/~ e - e ' d s --* c(7) as x ~ O,

(3.5) follows. If x _> 1 t h e n (3.6) follows from the fact t h a t =

Lemma drift.

+

--> 7 - '

3.8. ("Scaling relation"). L e t X be a F D ' ( d l , a , f l , 7 )

-->

on F . T h e n a =

Proof. From (3.1) we have p ( t , x , y ) >_ e l t - ~ e -c~ = c3t - ~

[]

for p ( x , y ) < t 1/p.

30 Set to = To~. So i r A = B(x, tl/~), and t < to

1 > ~*(p(z, xt) <_t 1/~) = fAP(t,z,y)g(dy) >_c3t-~#(A) > ct -~+d'/a. If ro = cr t h e n since this holds for all t > 0 we must have a = dl/ft. If T0 = 1 t h e n we only deduce t h a t a < dr~ft. Let now T0 = 1, let A > 0, t < 1, and A = B(x, Atl/~). We have t t ( F ) < c3.1.2, a n d therefore I=~(Xt 9

~)

< g(A)supp(t,x,y) + # ( f - A) sup p(t,x,y) yEA < c4t--ct+d~/~Ad~/~ if-

yEA ~

c~t-ae-CO x~.

Let )~ = ((df/fl)c61 log(1/t))l/~'r; then we have for all t < 1 t h a t

1 < ct-"+aH#(1 + (log(lit)) 1/#'t, which gives a contradiction unless a _> drift.

[]

T h e next relation is somewhat deeper: essentially it will follow from the fact t h a t t h e l o n g - r a n g e b e h a v i o u r o f p ( t , z, y) is fixed by the exponents d i a n d fl governing its s h o r t - r a n g e behaviour. Since 7 only plays a role in (3.3) when p(z, y)# >> t, we will be able to o b t a i n 7 in terms of df and ~ (in fact, it turns out, of/3 only). We begin by deriving some consequences of the bounds (3.3). L e m m a 3.9. Let X be a FD'(df,dl/~,13,7 ). Then (a) For t E (0, to], r > 0

lP~(p(x,X,) > T) _< Cl e x p ( - c 2 r B ' r t - ' r ) . (b) There exists c3 > 0 such that c4exp (--chT#Tt -7) < ~=(p(x, Xt) > r) forr < C3T0, t < r ~.

(c) F o r x 9 F , 0 < T < car9, /YT(X,T) = inf{s > 0 : X , • B ( z , r ) } then

(3.7)

COT~~ E~(x,r) _< C~T~.

Proof. F i x z E F , a n d set D(a, b) = {y E F : a < p(x, y) <_b}. T h e n by (3.15) c3.1.zbdt > p(D(a,b)) > c3.1.1bd~ - c3.1.2adl. Choose 8 > 2 so t h a t C3.1.1od! > 2C3.1.2: t h e n we have

(3.8)

csa ~' < ~(D(a,0a))< cga~,.

Therefore, writing Dn = D(Snr, 8 n+1 r), we have # ( D n ) x 8 nd~ provided r8 n+1 _~ r0. Now

31 /.

P~(p(x,X,) > r) =

(3.9)

p(t,x,y)it(dy)

/

B(z,r) ~ = ~n~: <- ~

p(t,z,y)it(dy)

c(re')~'t-~'/€ exp(-c~ot-"(,-O")")

n=O

= c(r~/t)d'/~s(o,p'r, clo(r~/t)D. If clo r~ > t then using (3.6) we deduce that this sum is bounded by c11exp (--c12(r~/t)'r) , while ff Cxor~ <_ t then (as P ~ (p(x, Xt) > r) < 1) we obtain the same bound, on adjusting the constant c11. For the lower b o u n d (b), choose c3 > 0 so that c3O < 1. Then It(D0) _> cr ~, and taking only the first term in (3.9) we deduce that, since r ~ > t,

~(p(~, x,) > ~) > c(~lt)~,/" exp(-c~3(~/t)~) _> c e~p(-c~ (~'/t)~). (c) Note first that

(3.10)

~(~(~,~) > t) < ~ ( x ,

e B(x,~))

= _J~(~,r)p(t, y, z)it(dz) So, for a suitable c14

_~,

yEF.

Applying the Markov property of X we have for each k _> 1

P~(T(x,r) > kel4r ~) <_2 -k,

y E F,

which proves the upper bound in (3.7). For the lower bound, note first that

]

\O<s
< P~(p(x, Xt) > r) +P~(T(x, 2r) < t,p(x, Xt) < r) Writing S = ~-(x, 2r), the second term above equals

E~l(s<,)~xs(p(x,x,_s) < ~) <

sup

supP~(p(y,x,_.) > ~),

yEOB(z,2v) s<:t

32 so that, using (a), F~(~-(x,2r) < t) < 2supsupFU(p(y, Xs) > r) s<_t yEF

(3.11)

2Cl exp(-c2(r"

/t)7).

So if 4cle -c2a~ = 1 then I?~(T(x, 2r) < ar m) _< 89 which proves the left hand side of

(3.7).

[]

R e m a r k 3.10. Note that the bounds in (c) only used the upper bound on p(t, x, y). The following result gives sufficient conditions for a diffusion on F to be a fractional diffusion: these conditions are a little easier to verify than (3.3). T h e o r e m 3.11. Let (F,p,#) be a F M S ( d I ) . Let (Yt, t > O,P*,x C F) be a #symmetric diffusion on F which has a transition density q(t, x, y) with respect to # which is jointly continuous in x, y /'or each t > O. Suppose that there exists a constant/3 > O, such that

q(t, x, y) < clt -d'/~ for all x, y E F, t E (0, to], q(t,x,y) >_ c2t-a,/~ i~p(x,y) < c3t 1/~, t e (O, to], c4r/3 < E~T(x, r) < chr ~, for x E F, 0 < r < coro,

(3.12) (3.13) (3.14)

where T(x,r) = inf{t > 0 : Yt ~ B(x,r)}. parameters dl, dl//3 ,/3 and 1/(/3 1).

Then /3 > 1 a n d Y is a F D with

-

C o r o l l a r y 3.12. Let X be a FD'(dl,df//3,/3,7) on a FMS(df) F. Then fl > 1 and 7 = 1/(/3 - 1).

Proof. By L e m m a 3.8, and the bounds (3.3), the transition density p(t, x, y) of X satisfies (3.12) and (3.13). By Lemma 3.9(c) X satisfies (3.14). So, by Theorem 3.11 /3 > 1, and X is a FD'(dl,df//3,fl,(/3 - 1)-1). Since p(t,x,y) cannot satisfy (3.3) for two distinct values of 7, we must have 7 = (/3 - 1) -1. [] R e m a r k 3.13. Since two of the four parameters axe now seen to be redundant, we will shorten our notation and say that X is a FD(df,/3) i f X is a FD'(df, dr~~3,~3,7). The proof of Theorem 3.11 is based on the derivation of transition density bounds for diffusions on the Sierpinski carpet in [BB4]: most of the techniques there generalize easily to fractional metric spaces. The essential idea is "chaining": in its classical form (see e.g. [FaS]) for the lower bound, and in a slightly different more probabilistic form for the upper bound. We begin with a some lemmas. Lemma

3.14. [BB1, L e m m a 1.1] Let ~1,~2,''" ,~n, V be non-negative r.v. such ~i. Suppose that for some p E (0, 1), a > 0,

that V > ~ (3.15)

P(~{ < t[a(~l,...,~{_l)) < P + at,

t>O.

Then (3.16)

l o g P ( V < t) <_ 2 ( p t ) -

-

1/2

-

n

log -1. P

33 Proof.

If ~? is a r.v. w i t h d i s t r i b u t i o n function PO1 <- t) = (p + at) A 1, t h e n E ( e - ~ & ] a ( ~ l , . . . , ~ , - 1 ) ) < Ee -~''7

P + f ( 1 - p ) / ~ e-~tadt

_

--

JO

< p + aA -1. So

m ( v < t) = P (e -~'v > r

<_ d'~Er - ~ v

n

G e : ~ t E e x p A E ~ i < e)'t(p + h A - l ) " 1

T h e r e s u l t follows on s e t t i n g A -- (an/pt) 1/2.

[]

R e m a r k 3.15. T h e e s t i m a t e (3.16) a p p e a r s slightly odd, since it t e n d s to + c r as P I 0. However if p = 0 t h e n from the last b u t one line of the p r o o f above we o b t a i n log P ( V <_ t) ~_ At + n log ~, a n d s e t t i n g A = n / t we d e d u c e t h a t

ate

(3.17) Lemma

log P ( V <_ t) _< n l o g ( ~ - ) . 3 . 1 6 . Let (Yt,t > O) be a diffusion on a metric space (F,p) such that,/'or

x C F~ r > O, clr ~ ~_ EX T(x,r) ~ c2r f~. Then for x C F, t > 0 ,

~(~(x,r)

< t) < (1 - el/(2'c~)) + c3r-~t.

Proof. Let x e F , a n d A = B ( x , r ) , T ----T(x,r). Since ~- _< t + (~" - t)l(~>t) we have EZ'r <_ t q- E a l ( r > t ) E Y` (T -- t)

~_ t ~-Pz(T > t) supEYT. Y

As ~- ~ ~-(y, 2r) IFU-a.s. for any y C F , we d e d u c e

clr~ < E ~ < t + ~ ( ~ > t)c2(2r)~, so t h a t C22~(T

<_ t) < (2Zc2 - el) -~- t r -f~.

[]

T h e n e x t couple of results are n e e d e d to show t h a t the diffusion Y in T h e o r e m 3.11 can r e a c h d i s t a n t p a r t s of the space F in an a r b i t r a r i l y short time.

34

L e m m a 3 . 1 7 . Let Yt be a it-symmetric diffusion with semigroup Tt on a complete metric space (F, p). If f, g >_ 0 and there exist a < b such that

/ f(x)E~ g(Yt)it(dx) = 0 for t e (a,b),

(3.18)

then f f(x)E~ g(Yt)#(dx) = 0 for all t > O.

Proof. Let (Ex, s > 0) be t h e s p e c t r a l family associated with Tt. Thus (see [FOT, p. 17]) Tt = e-XtdEx, and

I~

(f, T,g) =

/o

e-:"d(f, E ,g) =

/o

where ~, is of finite variation. (3.18) and the uniqueness of the Laplace transform i m p l y t h a t t, = O, a n d so (f, Ttg) = 0 for all t. [] 3.18. Let F a n d Y satisfy the hypotheses of Theorem 3.11. If p(x,y) < c3r 0 then ~z(Yt E B(y,r)) > 0 for all r > 0 and t > O.

Lemma

R e m a r k . T h e restriction p(z,y) < c3ro is of course unnecessary, b u t it is all we need now. T h e conclusion of T h e o r e m 3.11 implies t h a t ~ ( Y t E B(y, r)) > 0 for all r > 0 a n d t > 0, for M i x , y E F .

Proof. Suppose t h e conclusion of the L e m m a fails for x, y, r, t. Choose g E C(F, N+) such t h a t fmgdit = 1 a n d g = 0 outside B(y,r). Let tl = t/2, rl = c3(tl) ~, a n d choose f 9 C ( F , I ~ + ) so t h a t f F f d # = 1, f ( x ) > 0 and f = 0 outside A = B(x, rl). If 0 < s < t t h e n t h e construction of g implies t h a t 0

E~ g(Yt) = f~F q(s, x, x' )E x' g(Yt-,)it(dx ' ).

Since b y (3.13) q(s,x,x') > 0 for t/2 < s < t, x' 9 B(x, rl), we deduce t h a t E*'g(Yu) = 0 for x' 9 S ( x , rl), u 9 (0, t/2). Thus as s u p p ( f ) C B(x, rl)

F f (X')EX'g(Yu)dit

= 0

for all u 9 ( 1 , t / 2 ) , a n d hence, by L e m m a 3.17, for all u > 0. But by (3.13) if u = (p(x,y)/c3) ~ t h e n q(u,x,y) > 0, and by the continuity of f , g and q it follows t h a t f fE*g(Y=)eit > 0, a contradiction. []

Proof of Theorem 3.11. For simplicity we give full details of the p r o o f only in the case r o = co; the a r g u m e n t in the case of b o u n d e d 17 is essentially the same. We begin by o b t a i n i n g a b o u n d on

<

t).

Let n > 1, b = r/n, a n d define stopping times Si, i > 0, by So = 0,

Si+~ = inf{t _> Si : p(Ys,,Yt) _> b}.

35 Let ~i = Si - Si-1, i > 1. Let (~'t) be the filtration of Yt, and let ~i = J=s,. We have by L e m m a 3.16

~(~i+1 <_ tlGi) = IPYs' (T(Ys,, b) < t) < p + c6b-~t, w h e r e p 6 (0,1). As Ys,,Ys,§ = b, we have p(Yo,Ys~) <_ r, so that S , = ~ 1 ~i _< 7(Y0,r). So, by L e m m a 3.14, with a = c6(r/n) -~, < t) < 2p

-

(3.19)

.log 1P

1

= cT(r-t~nl+t~t) ~ - csn. If fl < 1 then taking t small enough the right hand side of (3.17) is negative, and letting n ~ oo we deduce IP~('r(x,r) _< t) = 0, which contradicts the fact that F~(Yt 9 B ( y , r ) ) > 0 for all t. So we have fl > 1. (If r0 = 1 then we take r small enough so that r < c3). If we neglect for the moment the fact that n 9 bt, and take n = no in (3.19) so that

then

n~o-1 = (c2s/4c~)r~t -1,

(3.20) and

_< t) _< -}csn0. So if r~t -1 >_ 1, we can choose n 6 l~ so that 1 < n < no V 1, and we obtain (3.21)

I?~(T(x,r) < t) _< c9 exp

-clo

9

Adjusting the constant c9 if necessary, this bound also clearly holds if r~t -1 < 1. Now let z , y 6 F , write r = p(x,y), choose e < r/4, and set C~ = B(z,e), z = x,y. Set Ax = {z 6 F : p(z,x) < p(z,y)}, Ay = { z : p(z,x) >_ p(z,y)}. Let u~, uy be the restriction of # to C~, Cy respectively. We now derive the upper bound on q(t, x, y) by combining the bounds (3.12) and (3.21): the idea is to split the journey of Y from C~ to C~ into two pieces, and use one of the bounds on each piece. We have (3.22)

F~'(Yt 6 C,) = [ [ q(t,x',y')#(dx')#(dy') C~C.

<_ IF~" (Yt 6 Cu,Yt/2 6 A~) + F " ( Y t 9 Cu,Yt/2 9 Ay). We begin with second term in (3.22): (3.23) P~'(Yt 9 Cu,Yt/2 9 Au) = IP~" (T(ro,r/4) < t/2,Yt/2 9 Au,Yt 9 Cu) -< F~" (7(Y0,r/4) < t/2) sup F y' (Yt/2 9 y'6A.

Cy)

36

_< ,,.(c~

exp k-c,o

)

=

exp

where we used (3.21) and (3.12) in the last b u t one line. To handle the first t e r m in (3.22) we use symmetry:

~'.(~ c c y , v u 2 ~ A~) = P'~(Y~ c C~,Yu2 e A~), and this can now be b o u n d e d in exactly the same way. We therefore have

/ / q(t,x',y')#(dx')#(dy') C~ C~ < It(C, )#(Cy)2Cll t -d'/~ exp ( - c 1 2 (r ~ It) 1/(f~-l)) , so t h a t as (3.24)

q(t,., .)

is continuous

q(t,x,y) < 2c, lt-a'/f~ exp (-c,2(rf~/t)l/(~-l)) .

T h e p r o o f of the lower b o u n d on q uses the technique of "chaining" the C h a p m a n - K o l m o g o r o v equations. This is quite classical, except for the different scaling. Fix x,y,t, and write r -- p(x,y). If r <_c3tl/~ then by (3.13)

q(t, x, y) > c2t -dr~z, a n d as exp(-(rfl/t) U(~-U) >_ e x p ( - c ~ / ( ~ - l ) ) , we have a lower b o u n d of the form (3.3). So now let r > c3t x/~. Let n _> 1. By the mid-point hypothesis on the metric p, we can find a chain x = xo,xl,...,xn = y in F such t h a t p(xi-l,xi) = r/n, 1 < i < n. Let Bi = B(xi,r/2n); note t h a t i f y i E Bi then P(yi-x,Yi) <_ 2r/n. We have by the C h a p m a n - K o l m o g o r o v equation, writing Y0 = x0, y , = y,

q(t,x,y)>__/#(dyl)... #(dy,.-1)Hq(t/n,y,-1,y,). n

(3.25)

/

Ba

B~_a

i----1

We wish to choose n so t h a t we can use the b o u n d (3.13) to estimate the terms q(t/n, Yi-1, Yi) from below. We therefore need: (3.26)

n

which holds provided (3.27)

n ~-1 _> 2 Z c [ ~ .

37 As 17 > 1 it is c e r t a i n l y possible to choose n satisfying (3.27). By (3.25) we then obtain, since # ( B i ) > c(rl2n) ds,

(3.28)

q(t, x, y) > c(rl2n) d,(',-1) (c2(tln) -dsi~) --

= c(,'/2n) -d'

= c'(~,/n) -'~,

n

(~(tln) '/~(~/2~)

((tln)-liZ(,,In ))~

d

')

n

Recall t h a t n satisfies (3.27): as r > c3t l l j we can also ensure t h a t for some cl3 > 0 (3.29)

r >_ c13(tln)l//3 '

u

so t h a t u ~-1 < 2~c~3~r~lt. So, by (3.28)

q(t,~,y) > c(tln)-~'i~c74 _> cl 5t -ds IZ exp ( n log cl4 )

>_ eist -d' i' exp

lt )'ir

) .

[]

Remarks 3.19. 1. Note t h a t the only point at which we used the "midpoint" p r o p e r t y of p is in the derivation of the lower b o u n d for q. 2. T h e essential i d e a of t h e proof of T h e o r e m 3.11 is t h a t we can obtain b o u n d s on t h e long range b e h a v i o u r of Y provided we have good enough information a b o u t the b e h a v i o u r of Y over distances of order t ll2. Note t h a t in each case, if r = p(x, y), the e s t i m a t e of q ( t , x , y ) involves splitting the j o u r n e y from x to y into n steps, where n x (r2lt) 1/(~-l). 3. B o t h the arguments for the u p p e r and lower bounds a p p e a r quite crude: the fact t h a t t h e y yield the same b o u n d s (except for constants) indicates t h a t less is thrown away t h a n might a p p e a r at first sight. T h e explanation, very loosely, is given by "large deviations". T h e off-diagonal bounds are relevant only when r 13 >> t - otherwise t h e t e r m in t h e exponential is of order 1. If r E >> t then it is difficult for Y to move from x to y by time t and it is likely to do so along more or less the shortest p a t h . T h e proof of the lower b o u n d suggests t h a t the process moves in a 'sausage' of radius t i n • t i t ~1-1. T h e following two theorems give additional bounds and restrictions on the p a r a m e t e r s d f a n d / 3 . Unlike the proofs above the results use the s y m m e t r y of the process very strongly. T h e proofs should a p p e a r in a forthcoming paper. Theorem (3.30)

3.20. Let F be a F M S ( d l ) , and X be a F D ( d f , / 3 ) on F. Then 2
T h e o r e m 3.21. Let F be a F M S ( d l ) . i = 1,2. Then/31 = /32.

l + d s.

Suppose X i are FD(df,/3i ) on F, for

R e m a r k s 3.22. 1. T h e o r e m 3.21 implies t h a t the constant 13 is a p r o p e r t y of the metric space F , a n d not just of the F D X. In p a r t i c u l a r any F D on ~d, with the

38 usual metric and Lebesgue measure, will have/3 = 2. It is very unlikely that every FMS F carries a FD. 2. I expect that (3.30) is the only general relation between/3 and d I. More precisely, set A = {(dr,/3) : there exists a F D ( d l , / 3 ) } , and F = {(df,fl) : 2 < /3 < 1 + d I } . Theorem 3.20 implies that A C I', and I conjecture that int I' C A. Since B M ( ~ a) is a F D ( d , 2), the points (d, 2) 9 A for d > 1. I also suspect that { d r : (dr,2) 9 A} = N, that is that if F is an FMS of dimension dr, and df is not an integer, then any FD on F will not have Brownian scaling. Properties of Fractional Diffusions. In the remainder of this section I will give some basic analytic and probabilistic properties of FDs. I will not give detailed proofs, since for the most part these are essentially the same as for standard Brownian motion. In some cases a more detailed argument is given in [BP] for the Sierpinski gasket. Let F be a F M S ( d l ) , and X be a F D ( d l , f l ) on F. Write T, = E ~ f ( X t ) for the semigroup of X , a n d / 2 for the infinitesimal generator of T~. D e f i n i t i o n 3.23. Set d",=fl,

2dr d e = d"" .

This notation follows the physics literature where (for reasons we will see below) d", is called the "walk dimension" and d~ the "spectral dimension". Note that (3.3) implies that p ( t , x , x ) • t -d~ 0 < t < to, so that the on-diagonal bounds on p can be expressed purely in terms of d~. Since m a n y important properties of a process relate solely to the on-diagonal behaviour of its density, d, is the most significant single parameter of a F D . Integrating (3.3), as in Corollary 2.25, we obtain:

Lemma3.24.

E~p(Xt, x) p • t p/d~ , x C F, t > O, p > O.

Since by Theorem 3.20 d", > 2 this shows that FDs are diffusive or subdiffusive. L e m m a 3.25. (Modulus of continuity). Let ~(t) = t 1/d~ (log(1/t)) (d~-D/d~. Then (3.31)

Cl < lira sup p(X~,X~) <_ c2. ~10 0<~<~<1 ~ ( t - s)

So, in the metric p, the paths of X just fail to be Hhlder (1/d,,). The example of divergence form diffusions in I~d shows that one cannot hope to have ci =- c~ in general.

39 L e m m a 3.26. (Law of the iterated logarithm - see [BP, Thin. 4.7]). Let r = tl/d~(loglog(1/t))(d~--l)/d~. There exist Cl, c2 and constants c(x) E [Cl,C2] such that lim0u p p ( ~ X o ) _ c(x) ~'-a.s. Of course, the 01 law implies that the limit above is non-random. L e m m a 3.27. (Dimension of range). dimH ({X~: 0 < t < 1}) = d I A d~.

(3.32)

This result helps to explain the terminology "walk dimension" for d,~. Provided the space the diffusion X moves in is large enough, the dimension of range of the process (called the "dimension of the walk" by physicists) is d~.

Potential Theory of ~ractional Diffusions. Let A >_ 0 and set

O ~

Then ff

v~f(x) =

E~

/?

e-~V(x,)

de

is the A-resolvent of X, ux is the density of UA:

Uxf(x) = f uA(x, V)p(dy). t] F

Write u for u0. P r o p o s i t i o n 3.28. Let A0 : l/r0. (If to : oo take Ao = 0).

(a) If d~ < 2 then u~(x,y) is jointly continuous on F x F and for )~ > Ao (3.33)

c1)~d'/2-1 exp(--c2A1/d~p(x,y)) <_u~(x,y) < c3z./2-, exp

(b) If d, = 2 and A > ~o then writing R = p(x,y)A 1/d~ (3.34)

e5 (log+(1/R) + e - ~ ' ' ) < un(x,y) <_e7 (log+(1/R) + e - * " ) .

( c ) / f de > 2 then

(3.35)

~p(~,Y)~-d' < u~o(~,V) _< ~,0p(~,V) ~~

These bounds are obtained by integrating (3.3): for (a) and (b) one uses Laplace's method. (The continuity in (b) follows from the continuity of p and the uniform bounds on p in (3.3)). Note in particular that:

4O (i) i f d s < 2 t h e n u ~ ( x , x ) < +oo and lim u x ( x , y ) = +oo. A---~0

(ii) if ds > 2 t h e n u(x, x) -- +c~, while u(x, y) < c~ for x ~ y Since the p o l a r i t y or n o n - p o l a r i t y of points relates to the on-diagonal behaviour of u, we deduce from P r o p o s i t i o n 3.28 C o r o l l a r y 3.29. (a) Ifd~ < 2 then for each x , y E F

P ~ ( X hits y) = 1. (b) I f d~ > 2 then points are polar for X . (c) I f d~ < 2 then X is set-recurrent: for ~ > 0 I?y ({t : X , 9 B(y, E)} is non-empty and unbounded) = 1.

(d) I f d~ > 2 and r0 = c~ then X is transient. In short, X behaves like a Brownian motion of dimension ds; b u t in this context a continuous p a r a m e t e r range is possible. L e m m a 3.30. (Polar and non-polar sets). Let A be a Borel set in F. (a) ]~(TA < oo) > 0 if d i m H ( A ) > df - dw, (b) A is polar for X f f d i m g ( A ) < df - dw. Since X is s y m m e t r i c any semipolar set is polar. As in the Brownian case, a more precise condition in terms of capacity is true, and is needed to resolve the critical case dim/~(A) = df - d~o. If X , X ' are i n d e p e n d e n t F D ( d y , • ) on F , and Zt = (Xt, X~), then it follows easily from the definition t h a t Z is a F D on F x F , with p a r a m e t e r s 2dr and ft. If D -~ { ( x , x ) : x 9 F } C F x F is the diagonal in F • F , then d i m H ( D ) -- df, and so Z hits D (with positive p r o b a b i l i t y ) if

df > 2 d f - d w , t h a t is if ds < 2. So (3.36)

lP~(Xt = X~ for some t > 0) > 0

if d8 < 2,

~(Xt

if d~ > 2.

and (3.37)

= X~ for some t > 0) = O

No d o u b t , as in the Brownian case, X and X ' do not collide if d, = 2. Lemma

3.31. X has k-multiple points i f a n d only if d, < 2 k / ( k - 1).

Proof. By [Rog] X has k-multiple points if and only if

fB (~,1) ui(x'Y)k'(dY) < (XD;

41

the integral above converges or diverges with

o I rkd~--(k--1)d~r -1 dr, by a calculation similar to that in Corollary 2.25.

[]

T h e b o u n d s on the potential kernel density u ~ ( x , y ) lead immediately to the existence of local times for X - see [Sha, p. 325]. T h e o r e m 3.32. I f d~ < 2 then X has jointly measurable local times ( L t9 , x E F , t >_

O) which satisfy the density of occupation formula with respect to #: (3.38)

f(X~)ds =

f(a)L~#(da),

f bounded a n d measurable.

I n the low-dimensional case (that is when d~ < 2, or equivalently d I < d,~) we can o b t a i n more precise estimates on the HSlder continuity of ux(x, y), and hence on the local times L~'. The m a i n lines of the argument follow that of [BB4, Section 4], b u t on the whole the arguments here are easier, as we begin with stronger hypotheses. We work only in the case r0 = oo: the same results hold in the case r0 = 1, with essentially the same prooofs. For the next few results we fix F , a F M S ( d f )

with r0 = 0r

and X , a

F D ( d l , d~) on F . For A C F write TA = TAo = inf{t _> 0 : X~ C A~}. Let Rx be an i n d e p e n d e n t exponential time with mean A-1. Set for A > 0

u A ( x , y ) -: E ~

f0

TM

e - ~ d L y -- E ~ LrA^R~ y

u2f(~) : f ~(~, y),(dy). JF

Let

p~(x, y) = ~ ( T ~ < ~A A R~); note t h a t

(3.39) Write uA(x, y) = u A ( x , y ) , U A = UoA, and note that u ~ ( x , y ) = u F ( x , y ) , Ux = UA. As in the case of u we write pA, px for pA, pF. As (I?~, Xt) is #-symmetric we have uA(x, y) = uA(y, X) for all x, y e f . The following L e m m a enables us to pass between bounds on u~ a n d UA. L e m m a 3.33. Suppose A C F, A is bounded, For x, y E F we have

~A(x,y) = ~ f ( x , y) + ~ (I(R~<~A)uA(XR~,y)) - E ~ (I(R~>~A)~f(x~A,y)).

42 P r o o f . From the definition of u A, u A ( x , y ) = E (L~A ;Rx < ~'A)+ E~(L~A ;Rx > ~'A)

= E=(L~ ;R~ < ~A) + E=(IIR~<,A)EXR~ L~A) y y , --L~A ;R~ > rA) + E~e(LR~ ;R~ > ~-A)- Ea~(LR~^~ =ux(x,y)+E

z

(I(R~_<~A)UA ( X , x , y ) ) - E ~ ( I ( R ~ > ~ A ) U X ( X ~ A , y ) ) .

[]

C o r o l l a r y 3.34. Let x E F, and r > 0. Then

clrd~-dl <_ uB(~,~)(x,z) < e2rd~--d~. Proof. Write A = B ( x , r ) , and let A = Or -d~, where /9 is to be chosen. We have from L e m m a 3.33, writing 9 = ~ ( x , r ) ,

uA(~, y) _< ~ ( ~ , Y) + E~ I(,~<~)~A(XR~, Y). So if ~ = sup~ ~A(~, y) then using (3.33)

(3.40)

v ~_ c3/~d'/2-1 -[-]]~Z(RA < T)V.

Let to > 0. Then by (3.10)

P~(/r < ~-) = P~(R:~ < T,~- < to) + IP~(R~ < T,~" > to) _< ~ ( R ~ < to) + P~(~- > to) < (1 - e -xt~ + ctodl/d~r dl. Choose first to so that the second term is less than 88 and then A so that the first term is also less than 88 We have to ~ r d~ ~ A-1, and the upper bound now follows from (3.40). The lower bound is proved in the same way, using the bounds on the lower tail of T given in (3.11). [] L e m . m a 3.35. There exist constants cl > 1, c2 such that ff x, y E F, r = p ( x , y ) , t 0 = r d~ then ~ ( T ~ < to < ~(x,c~r))> c~.

Proof. Set ~ -- (e/r) d~ ; we have px(x, y) > c3 exp(-c4e) by (3.33). So since p x ( z , y ) = E~ e -xT, <_ P~(T~ < t) + e - ~ , we deduce that P~(Tv < t) > c3 exp(-c4O) - exp(--0d~). As dw > 1 we can choose 0 (depending only on c3, c~ and d,o) such that P~(Tv < t) >_ ~31c exp(--c4O) ----C5. By (3.11) for a > 0

P~('r(z, an) < ~'~) <_c~ exp(--c~ad~/~d'-~)), so there exists c~ > 1 such that IP):e(T(X, Cl r ) < tO ) _<( ~.C5 . 1 So 1 l~=(Tu < to < T(X, c~r))> II~=(Tu < to) - ~=(T(X, clr) < to) >_ ~c5.

[]

43 D e f i n i t i o n 3.36. We call a function h harmonic (with respect to X ) in a n open subset A C F if s = 0 on A, or equivalently, h(Xt^TAo ) is a local martingale. 3 . 3 7 . (Harnack inequality). There exist constants cl > 1, c2 > O, such that if xo E F, and h >_ 0 is harmonic in B(xo,clr), then

Proposition

h(x) > c~h(y),

x,y e B(~o,r).

Proof. Let cl = 1 + ca.35.1, so t h a t B(x, c3.ah.lr) C B(xo,clr) if p(x, xo) < r. Fix x, y, write r -----p(x, y), and set S -- Tu A~(x, c3.ah.lr). As h(X.hs) is a supermartingaie, we have by L e m m a 3.35, h(x) > E= h ( X s ) > h(y)P~(Ty < T(X,e3.35.1r)) ~_ ca.35.2h(y).

[]

C o r o l l a r y 3.38. T h e r e exists Cl > 0 such that ff xo C F, and h > 0 is harmonic

in B(xo,r), then

h(~) >_ clh(y),

~,y e B(xo,88

Proof. This follows by covering B(xo, 88 by balls of the form B(y, c2r), where c2 is small enough so t h a t P r o p o s i t i o n 3.37 can be applied in each ball. (Note we use the geodesic p r o p e r t y of the metric p here, since we need to connect each ball to a fixed reference point by a chain of overlapping balls). [] Lemma

3 . 3 9 . Let x, y e F, r : p(x, y). IT R > r and B(y, R) c A then

Proof. We have, writing T = ~-(y,r), T -- TAo,

uA(y, y) = ~ L ~ + EYEX, L~ = u'(y, y) + E~uA(X~, y), so b y Corollary 3.34

(3.41)

E~(~(y,y)-

~(x~,y))

-- ~B(y,y) < c ~ - . , .

Set ~ ( x ' ) = uA(y,y)--uA(x',y); ~ is harmonic on A - {y}. As p(x,y) = r and p has t h e geodesic p r o p e r t y there exists z with p(y, z) = 88 p(x, z) = ~r. ~ By CoroUary 3.38, since ~ is harmonic in B(x, r),

~(~) _>c3 3~ ~ ( ~ ) . Now set r = E =' ~(X~) for x' E B. T h e n r is harmonic in B and ~ < r on B. A p p l y i n g Corollary 3.38 to r in B we deduce

r

__~r

__~C3.38A~P(Z) __~(C3.38.1)2~0(X).

Since r

-- E y (uA(y, y) -- uA(Xr, y)) the conclusion follows from (3.41).

Theorem

3.40. (a) Let ~ > 0. Then t'or x, x', y E F , a n d f E L I ( F ) , g E L ~ ( F ) ,

(3.42) (3.43) (3.44)

I ~ ( ~ , y ) - uA(x',y)l ~ clp(x,x') d~-d', IV~f(~) - v~f(~')l < ~,p(z,~')~o-d'llfll,. IV~g(~) - v~g(~')l _< c~-d./~p(~, ~,)d~-d, Ilgll~.

[]

44

Proof. Let x, x' C F, write r = p(x,x') and let R > r, A = B(x,R). uA(y, x') > pA(y, x)uA(x, x'), we have using the s y m m e t r y of X that (3.45)

=2(~, y)

-

Since

u2(~', y) < ~2(y, ~) p~(y, x)~2(~, ~') = p~(y, ~)(u~(~, x) u~(~, ~')). -

-

Thus

lu~(~,y) - ~(z',y)l -< 1~2(~,~) - u2(~,~')l. Setting ~ = O and using L e m m a 3.39 we deduce (3.46)

luA(x,y) _ ~.4(x,,y)l _< ~:d~-,~,.

So f

Iu~/(~) - UAf(~')E <_ ./~ IuA(~,Y) -- uA(~', Y)I If(Y)I.(dY) <_ c: "d~-d' llflAll~To obtain estimates for A > 0 we apply the resolvent equation in the form

u~(x, y) -- uA(x, y) - ~v%(~), where

v(x) = uA(x,y). (Note that Ilvlll = ;~-1). Thus < c : d ~ - d , + ~clrd~-~'tlvll,

2c3rd~--d:. Letting /~ --* ~ we deduce (3.42), and (3.43) then follows, exactly as above, by integration. To prove (3.46) note first that p~(y,x) = ux(y,x)/ux(x,x). So by (3.33) (3.47)

f p2(y,~)lf(y)l.(dy)<_ I l f l l ~ u ~ ( z , z ) - 1 / A u x ( y , x ) t t ( d Y ) _-I If II,,,,u~,(z, z)-~.~, -~ <__c411.fll~,\

-d'/2.

From (3.45) and (3.46) we have

1~2(~,y) - u~(x',y)l < c~ (p2(y, ~) + V2(y,~'))r~-~', and (3.44) then follows by intergation, using (3.47).

[]

T h e following modulus of continuity for the local times of X then follows from the results in [MR].

45 T h e o r e m 3.41. If d, < 2 then X has jointly continuous local times (L~,x E F,t >

0). Let ~o(u) = u(d~-d')/2(log(I/u) ) I/2. The modulus of continuity in space of L" is given by: l i m sup ~Oo<s
sup 0<s<~

IL~ - Z~l < c ( s u p ~(p(z,y)) ~eF

L~)1/2.

I~-yl<6 It follows t h a t X is space-filling: for each x, y E F there exists a r.v. T such t h a t I?~(T < cr = i a n d B ( y , 1) C {Xt,O < t < T}. T h e following P r o p o s i t i o n helps to explain why in early work m a t h e m a t i c a l physicists found t h a t for simple examples of fractal sets one has d~ < 2. (See also [HHW]). 3.42. Let F be a FMS, and suppose F is finitely ram/fled. Then if X is a FD(dI, d~) on F, d~(X) < 2. Proposition

Proof. Let F1, F2 be two connected components of F , such t h a t D = F1 ~ F2 is finite. I f D = {Yl,-..,Y,~}, fix ~ > 0 and set Ms = e T M ~

u),(Xt,yi).

i=1

T h e n M is a supermartingale. Let To = inf{t > 0 : Xt E D}, and let x0 C F1 - D. Since I?~~ E F2) > 0, we have IP~~ < 1) > 0. So oo > EX~

> EZ~

a n d thus MTD < ar a.S. So u~(XTD,Yi) < Or for each Yl E D, and thus we must have u~(yi, Yi) < ~ for some y~ E D. So, by Proposition 3.25, ds < 2. [] R e m a r k 3.43. For k = 1,2 let (Fk,dk,#k) be F M S with dimension d r ( k ) , a n d c o m m o n d i a m e t e r r0. Let F = F1 • F2, let p _> 1 and set d((xl,x2),(yl,Y2)) = (dl(xl,Yl) p + d2(x2,Y2)P) Up, # = #1 • #2. T h e n (F,d,#) is a F M S with dimension d f = d f ( 1 ) + d f ( 2 ) . Suppose t h a t for k = 1,2 X k is a FD(dl(k),dw(k)) on Fk. T h e n if X = ( X ~ , X 2) it is clear from t h e definition of F D s t h a t if dw(1) = d,~(2) = / 3 t h e n X is a F D ( d f , f l ) on F . However, if dw(1) r dw(2) then X is not a F D on F . (Note from (3.3) t h a t the metric p can, up to constants, be e x t r a c t e d from the t r a n s i t i o n density p(t, x, y) by looking at limits as t ~ 0). So the class of F D s is not stable u n d e r products. This suggests t h a t it might be desirable to consider a wider class of diffusions with densities of the form: Tt

1

where Pi are a p p r o p r i a t e non-negative functions on F • F . Such processes would have different space-time scalings in the different 'directions' in the set F given by t h e functions pi. A recent p a p e r of H a m b l y and K u m a g a i [HK2] suggests t h a t

46 diffusions on p.c.f.s.s, sets (the most general type of regular fractal which has been studied in detail) have a behaviour a little like this, though it is not likely that the transition density is precisely of the form (3.48).

Spectral properties. Let X be a FD on a FMS F with diameter r0 = 1. The bounds on the density

p(t, x, y) imply that p(t,., .) has an eigenvalue expansion (see [DaSi, Lemma 2.1]). T h e o r e m 3.44. There exist continuous functions ~ol, and Ai with 0 <_ ~o <- ~1 <_ ...

such that/or each t > 0 p(t,x,y) = ~

(3.49)

e-~'t~n(x)~n(y),

n-~O

where the sum in (3.49) is ,miformly convergent on F • F. R e m a r k 3.45. The assumption that X is conservative implies that A0 -- 0, while the fact that p(t, x, y) > 0 for all t > 0 implies that X is irreducible, so that A1 > 0. A well known argument of Kac (see [Ka, Section 10], and [HS] for the necessary Tauberian theorem) can now be employed to prove that ff N(A) = #{),i : Ai < )t} then there exists ci such that (3.50)

ClAd~ < N(A) <_c2)ta~

for ~ > ca.

So the number of eigenvalues of Z: grows roughly as )~d./2. This explains the term spectral dimension for ds.

4. D i r i c h l e t F o r m s , M a r k o v P r o c e s s e s , a n d E l e c t r i c a l N e t w o r k s . In this chapter I will give an outline of those parts of the theory of Dirichlet forms, and associated concepts, which will be needed later. For a more detailed account of these, see the book [FOT]. I begin with some general introductory remarks. Let X = ( X t , t >_0,]?~,x E F ) be a Maxkov process on a metric space F. (For simplicity let us assume X is a Hunt process). Associated with X are its semigroup (Tt, t > 0) defined by

Ttf(x) = E~ f ( X t ) ,

(4.1)

and its resolvent (U~, A > 0), given by (4.2)

Uj, f ( z ) =

/o

T t f ( z ) e -~t dt -- E ~

/o

e - J " f ( X , ) ds.

While (4.1) and (4.2) make sense for all functions f on F such that the random variables f ( X t ) , or f e-X~f(Xs)ds, are integrable, to employ the semigroup or resolvent usefully we need to find a suitable Banaeh space (B, II 9 118) of functions on F such that Tt : B ~ B, or Ux : B --, B. The two examples of importance here are

47

Co(F) and L2(F,#), where # is a Borel measure on F. Suppose this holds for one of these spaces; we then have that (Tt) satisfies the semigroup property Tt+, = TtT,,

s,t > O,

and (Ux) satisfies the resolvent equation

U~-V~=(fl-~)UoU~,

a , ~ > O.

We say (Tt) is strongly continuous ff IITtf - f]]s ~ 0 as t ~ O. If Tt is strongly continuous then the infinitesimal generator (Z:, 7)(L)) of (T,) is defined by L ; / = lirnt-l(Ttf - f), tl0

(4.3)

/ e 7)(/2),

where 7)(L) is the set of f E B for which the limit in (4.3) exists (in the space B). The Hille-Yoshida theorem enables one to pass between descriptions of X through its generator s and its semigroup or resolvent. Roughly speaking, if we take the analogy between X and a classical mechanical system, s corresponds to the equation of motion, and Tt or Ux to the integrated solutions. For a mechanical system, however, there is another formulation, in terms of conservation of energy. The energy equation is often more convenient to handle than the equation of motion, since it involves one fewer differentiation. For general Markov processes, an "energy" description is not very intuitive. However, for reversible, or symmetric processes, it provides a very useful and powerful collection of techniques. Let # be a Radon measure on F : that is a Borel measure which is finite on every compact set. We will also assume # charges every open set. We say that Tt is #-symmetric if for every bounded and compactly supported f, g, (4.4)

/ T,.f(x)g(x)#(dx) = f Ttg(x)f(x)#(dx).

Suppose now (Tt) is the semigroup of a Hunt process and satisfies (4.4). Since Ttl _< 1, we have, writing (.,-) for the inner product on L2(F, #), that

ITtf(x)I <_ (Ttf2(x)) 1/2 (Ttl(x)) 1/2 <_(TtfZ(x)) 1/2 by HSlder's inequality. Therefore

IITtflll <

IlTtf2111 = (Ttf2,

1) = (f2,Ttl) < (f2, 1) = Ilfll~,

so that Tt is a contraction on L 2 (F, #). The definition of the Dirichlet (energy) form associated with (Tt) is less direct than that of the infinitesimal generator: its less intuitive description may be one reason why this approach has until recently received less attention than those based on the resolvent or infinitesimal generator. (Another reason, of course, is the more restrictive nature of the theory: many important Markov processes are not symmetric. I remark here that it is possible to define a Dirichlet form for non-symmetric Markov processes - - see [MR]. However, a weaker symmetry condition, the "sector condition", is still required before this yields very much.)

48 Let F be a metric space, with a locally compact and countable base, and let # be a R a d o n measure on F . Set H = L2(F,#). D e f i n i t i o n 4.1. Let 7) be a linear subspace of H. A symmetric form (E, 7)) is a m a p E : 7) • 7) -~ • such t h a t (1) s is bilineax (2) s f ) _> O, f C 79.

allfl[ ~, and

For a > 0 define Ea on 7) by E a ( f , f ) -~ e ( f , f ) +

llfll~ = llfll~+ c,E(f,f)

=

write

E,~(I,f).

D e f i n i t i o n 4.2. Let (s 7)) be a symmetric form. (a) E is closed if (79, II-Ile~) is complete (b) (s is Markov if for f E 7), i f g = ( 0 V f ) A 1 t h e n g E 79 and s < E(f,f). (c) (s 7)) is a Dirichlet form if 7) is dense in LZ(F, #) a n d (s 7)) is a closed, Markov s y m m e t r i c form. Some further p r o p e r t i e s of a Dirichlet form will be of importance: D e f i n i t i o n 4.3. (E, 7)) is regular if (4.5) (4.6)

79 N Co(F) is dense in 7) in II" I1~1, a n d 7) N Co(F) is dense in Co(F) in [[. I1~.

s is local if C(f, g) = 0 whenever f, g have disjoint support. E is conservative if 1 E 7) and s = 0. s is irreducible if s is conservative and g ( f , f ) = 0 implies t h a t f is constant. T h e classical example of a Dirichlet form is t h a t of Brownian motion on Ra:

s

f) - -

1 ~

f IVfl2 d~, I e H1'2(R% ~a

L a t e r in this section we will look at the Dirichlet forms associated with finite state Markov chains. J u s t as the Hille-Yoshida theorem gives a 1 - 1 correspondence between semigroups a n d their generators, so we have a 1 - 1 correspondence between Dirichlet forms a n d semigroups. Given a semigroup (Tt) the associated Dirichlet form is o b t a i n e d in a fairly straightforward fashion. D e f i n i t i o n 4.4. (a) T h e semigroup (Tt) is iarkovian if f E L2(F,#), 0 <_ f < 1 implies t h a t 0 _< Ttf _< 1 #-a.e. (b) A Markov process X on F is reducible if there exists a decomposition F = A1UA2 with Ai disjoint a n d of positive measure such t h a t P~(Xt E Ai for all t) = 1 for x E Ai. X is irreducible if X is not reducible.

49 4.5. ([FOT, p. 23]) Let (Tt, t _> 0) be a strongly continuous It-symmetric contraction semigroup on L2(F, It), which is Markovian. For f C L 2 ( F , # ) the function ~l(t) defined by Theorem

q o f ( t ) = t - x ( f --Ttf, f),

t>0

is non-negative and non-increasing. Let V = {f 9 L 2 ( F , # ) : lim~oy(t) < oo}, tlo

C(Lf) =limbos(t), tlo

f e 7).

Then (s 7)) is a Dirichlet form. If (Z:, 79(/:)) is the infinitesimal generator of (Tt), then 7)(1:) C V, V(C) is dense in L2(F, It), and s

(4.7)

= ( - E l , g),

f E 7)(s

g e 7).

As one might expect, by analogy with the infinitesimal generator, passing from a Dirichlet form (s 7)) to the associated semigroup is less straightforward. Since formally we have Us = (a - 12)-1, the relation (4.7) suggests that (4.8)

( f , g ) = ((a - E)U~f,g) -- a(U~,f,g) + E(Usf, g) = C~(UsI, g).

Using (4.8), given the Dirichlet form E, one can use the Riesz representation theorem to define U~f. One can verify that Us satisfies the resolvent equation, and is strongly continuous, and hence by the Hille-Yoshida theorem (Us) is the resolvent of a semigroup (Tt). T h e o r e m 4.6. ([FOT, p.18]) Let (e,7)) be a Oirichlet form on L2(F,#). Then there exists a strongly continuous #-symmetric Maxkovian contraction semigroup (Tt) on L2(F,#), with infinitesimal generator (L:,7)(E)) and resolvent (U~,a > O)

such that 1: and ~ satisfy (4.7) and also (4.9)

s

+ a(f,g) -- (f,g),

f e L2(F, It), g e 7).

Of course the operations in Theorem 4.5 and Theorem 4.6 are inverses of each other. Using, for a moment, the ugly but clear notation C -- T h m 4.5((Tt)) to denote the Dirichlet form given by Theorem 4.5, we have T h m 4.6(Tam 4.5((Tt))) = (Tt), and similarly T h m 4.5(Whm 4.6 (s

= E.

R e m a r k 4.7. The relation (4.7) provides a useful computational tool to identify the process corresponding to a given Dirichlet form - at least for those who find it more natural to think of generators of processes than their Dirichlet forms. For example, given the Dirichlet form s f) = f IV f[ 2, we have, by the Gauss-Green formula, for f , g e C2(~d), ( - s =s = f v f.Vg = -- f g A f , so that s = A. We see therefore that a Dirichlet form (~,7)) give us a semigroup (Tt) on L2(F, It). But does this semigroup correspond to a 'nice' Markov process? In general it need not, but if C is regular then one obtains a Hunt process. (Recall that

50 a Hunt process X = (X,, t > 0, ~ , x E F ) is a strong Markov process with cadlag sample paths, which is quasi-left-continuous.) T h e o r e m 4.8. ([FOT, Thin. 7.2.1.]) (a) Let (~, 9 ) be a regu/ar Dirichlet form on L 2 ( F , # ) . Then there exists a #-symmetric Hunt process X = ( X t , t > O , ~ , x E F ) on F with Dirichlet form C. (b) In addition, X is a diffusion if and only i r e is local. R e m a r k 4.9. Let X = (X~,t >_ 0,I?~,x E I~2) be Brownian motion on R 2. Let A C I~2 be a polar set, so that P~(TA < oo) = 0 for each x. T h e n we can obtain a new Hunt process Y -- (X~ >_ 0, Q=, x E ]~') by "freezing" XonA. SetQ= =~*,xEA c,andforzEAletQ~(xt=x, a l l t E [0, oo)) = 1 . Then the semigroups (TX), (TY), viewed as acting on L2(~2), are identical, and so X and Y have the same Dirichlet form. This example shows that the Hunt process obtained in Theorem 4.8 will not, in general, be unique, and also makes it clear that a semigroup on L 2 is a less precise object t h a n a Markov process. However, the kind of difficulty indicated by this example is the only problem - - see [FOT, Thm. 4.2.7.]. In addition, if, as will be the case for the processes considered in these notes, all points are non-polar, then the Hunt process is uniquely specified by the Dirichlet form g. We now interpret the conditions that ~ is conservative or irreducible in terms of the process X. L e m m a 4.10. I_fs is conservative then Ttl = 1 and the associated Markov process X has infinite lifetime.

Proof. If f e 9 ( s then 0 <_ e(1 + 1+ for any e R, and so e(1, f ) = 0. Thus ( - s f ) --- 0, which implies that s --- 0 a.e., and hence that Tel = 1. [] L e m m a 4.11. / f s is irreducible then X is irreducible. Proof. Suppose that X is reducible, and that F -- A1 U A2 is the associated decomposition of the state space. Then TtlA~ = 1A~, and hence f(1A~, 1A~) = 0. As 1 # 1A~ in L2(F, #) this implies that E is not irreducible. [] A remarkable property of the Dirichlet form E is that there is an equivalence between certain Sobolev type inequalities involving E, and bounds on the transition density of the associated process X. The fundamental connections of this kind were found by Varopoulos [V1]; [CKS] provides a good account of this, and there is a very substantial subsequent literature. (See for instance [Co] and the references therein). We say (C, 7)) satisfies a Nash inequality if

(4.10)

llflll~/~(~lifll~ +E(f,/)) _>cllfl[~ +~/~, f e v.

This inequality appears awkward at first sight, and also hard to verify. However, in classical situations, such as when the Dirichlet form E is the one connected with the Laplacian on ]~d or a manifold, it can often be obtained from an isoperimetric inequality.

51 In w h a t follows we fix a regular conservative Dirichlet form (E, V). Let (Tt) be the associated semigroup on L 2 (F, #), and X = (Xt, t > 0, P~) be the Hunt process associated with E. 4.12. ([CKS, Theorem 2.1]) (a) Suppose E satisfies a Nash inequality ~ t h constants c, 6, 8. Then there exists c' = c'(c, 8) such that

Theorem

(4.11)

[[Ttlh-~ < c ' d q - ~

t > O.

(b) If (Tt) satisfies (4.11) with constants c', 6, 0 then C satisfies a Nash inequality with constants c" = c"(c', 8), 6, and O. Proof. I sketch here only (a). Let f Z :D(s

T h e n writing ft = Ttf, and

gth : h-l(/t+h - ft) - T t s

we have llg~hll~ < IIg0hll2 ~ 0 as h ~ 0. It follows that (d/dt)ft exists in L 2 ( F , ~ ) and t h a t

d

-~ft = T t s

= s

Set ~(t) = (ft, ft). T h e n

h -1 (T(t + h) - T(t)) - 2(TtEf, T t f )

= (gth, ft + ]t+h) + (TtEf, ~t+h -- ft),

a n d therefore T is differentiable, a n d for t > 0 (4.12)

~o'(t) = 2(/25, ft) = - 2 E ( f t , ft).

If f C L2(F,t~), T t f E :D(s for each t > 0. So (4.12) extends from f E :P(s

to all

f e L2(F,#). Now let f > 0, a n d Hf]]l = 1: we have ]1s

(4.13)

~'(t) -- -2c(1,, ft)) < 2~11/~11~ cllI~ll~ +4/~ = 2~i~o(t)2 -

Thus ~ satisfies a differential inequality. Set r r If r

= 1. T h e n by (4.10), for t > 0,

is the solution of r

< -2cr176

-

c~~ 1+2/~

= e-2~t~o(t). T h e n

e46t/~ < - 2 c r

1+2/~

.1+~./0 = -c~v 0 then for some a E ]~ we have, for co = co(c, 0),

Co(t) =- co(t -4- a) -0/2. If r

is defined on (0, oc), t h e n a > 0, so t h a t < cot -~

r

t > o.

It is easy to verify t h a t r satisfies the same b o u n d - so we deduce t h a t

(4.14)

IlTdll~ ~-- e2~t~b(t) <_ coe26tt-~

Now let f , g E L2+(F,#) with

Ilfll~

--

Ilgll~

f e I2+,

Ilfllx = 1.

-- 1. T h e n

(T~.tf, g) = (Ttf, Ttg) < IIT~III~IIT~glI~ <_c~S2q -~

52 Taking the s u p r e m u m over g, it follows that replacing 2t by t, t h a t

IlT2tfll~

~ c~ e~2tt-~

that is,

IlTtll1-~ _~ c2 e~tt -e/z.

[]

R e m a r k 4.13. In the sequel we will be concerned with only two cases: either 5 = 0, or 5 = I and we are only interested in bounds for t E (0, 1]. In the latter case we can of course absorb the constant e 5t into the constant c. This theorem gives b o u n d s in terms of contractivity properties of the semigroup

(Tt). If Tt has a 'nice' density p ( t , x , y ) , then IlTtlll --- sup~,yp(t,x,y), so that (4.11) gives global upper b o u n d s on p(t,., .), of the kind we used in Chapter 3. To derive these, however, we need to know that the density of Tt has the necessary regularity properties. So let F, $, Tt be as above, and suppose that (Tt) satisfies (4.11). Write Pt(x, ") for the t r a n s i t i o n probabilities of the process X. By (4.11) we have, for A E B(F), a n d writing ct = eeStt-el2,

P t ( x , A ) < cry(A)

for # - a . a . x .

Since F has a countable base (An), we can employ the arguments of [FOT, p.67] to see that

Pt(x,A,~) < ct#(A,~),

(4.15)

x E F - Nt,

where the set Nt is "properly exceptional". In particular we have #(N~) = 0 and

~z~(Xs E Nt or X s - E Nt for some s > 0) = 0 for x E F - Nt. From (4.15) we deduce that Pt(x, .) << p for each x E F - Nt. If s > 0 a n d y ( B ) = 0 t h e n P , ( y , B ) = 0 for #-a.a. y, and so

Pt+~(x,B) =

f P~(z, dy)Pt(y,B) = O, z E F - N~.

So Pt+s(x, .) << # for all s > O, x E F - Nt. So taking a a single properly exceptional set N = U,~Nt. such that x E F - N. Write F ~ -- F - N: we can reduce the state T h u s we have for each t, x a density p(t, x, .) of Pt(x, can be regularised by integration.

sequence tn + 0, we obtain Pt(x, ") << tt for all t > 0, space of X to F ~. .) with respect to #. These

P r o p o s i t i o n 4.14. (See [Y, Thm. 2]) There exists a jointly measurable transition

density p ( t , x , y ) , t > O, x , y E F' • F ~, such that P t ( x , A ) = f p ( t , x , y ) y ( d y ) for x E F',

p(t,x,u)

t > O, A E B ( F ) ,

A = p ( t , u , x ) rot all x , u , t ,

p(t + s,~,z) = [ p ( s , z , y ) v ( t , y , z ) , ( d y ) J

for all

x,z,t,s.

53 C o r o l l a r y 4.15. Suppose (E,/3) satisfies a Nash inequality with constants c, 6, O. Then, for all x , y E F ~, t > O,

p(t, x , y ) <_ c'e~t -~ We also obtain some regularity properties of the transition functions p(t, x, .). Write qt,~(Y) = p(t, x, y). P r o p o s i t i o n 4.16. Suppose (g, ~)) satisfies a Nash inequality with constants c, 6, O. Then for x E F ~, t > O, qt,~ E l)([:), and [[qt,~l[~ -< Cl J ~ t t - ~

(4.16)

s

(4.17)

qt,~) <_ c~e~tt-l-~

Proof. Since qt,~ = Tt/2qt/2,x, and qt/2: E L 1, we have qt,~ E / ) ( L ) , and the bound (4.16) follows from (4.14). Fix x, write ft = q,,~, and let ~(t) = llftll]. Then

~o"(t)= ~t (2Lf~,ft) = 4(/:ft, s

>_ O.

So, W~ is increasing and hence o < ~,(t) = ~ ( t / 2 ) +

/i

~'(s) ds _ ~ ( t / 2 ) + (t/2)~'(t). 2

Therefore using (4.13), e(It, lt)=

- ~1 , (t) ~ t - l ~ ( t / 2 ) ~ ce~tt -1-~

[]

Traces of Dirichlet forms and Markov Processes. Let X be a #-symmetric Hunt process on a LCCB metric space (F,#), with semigroup (Tt) and regular Dirichlet form (E, :D). To simplify things, and because this is the only case we need, we assume (4.18)

Cap({x}) > 0 for all x e F.

It follows that x is regular for {x}, for each x C F, that is, that

I?~(T~ = O) --1,

x C F.

Hence ([GK]) X has jointly measurable local times (L~, x e F, t > 0) such that

/0

](X~)ds =

f(x)L~#(dx),

f C L2(F,#).

54 Now let v be a a-finite measure on F . (In general one has to assume v charges no set of zero capacity, b u t in view of (4.18) this condition is vacuous here). Let At be the continuous additive functional associated with v:

At = / L~u(da), a n d let Tt = inf{s : A , > t} be the inverse of A. Let G be the closed s u p p o r t of v. Let -~t = X~,: t h e n by [BG, p. 212], .~ = (Xt,I? ~, x E G) is also a Hunt process. We call ) ( the trace of X on G. Now consider the following o p e r a t i o n on the Dirichlet form s For g E L 2 (G, v) set

~(g,g) = i n f { s

(4.19)

f ) : f i g = g}-

4.17. ("Trace theorem": [FOT, Thm. 6.2.1]). (a) (s is a regular Dirichlet form on L2(G,u). (b) .~ is u-symmetric, and has Dirichlet form (g, ~ ) .

Theorem

Thus g is t h e Dirichlet form associated with X : we call g the

trace of s (on G).

R e m a r k s 4 . 1 8 . 1. T h e d o m a i n ~ on ~ is of course the set of g such t h a t the infimum in (4.19) is finite. If g E :D then, as g is closed, the infimum in (4.19) is a t t a i n e d , by f say. If h is any function which vanishes on G c, then since ( f + Ah)l v = g, we have

s which implies s

<s

f+)~h),

AEII~

h) = 9. So, if f ~ :D(L), and we choose h C Z), then (-h, s

= O,

so t h a t s = 0 a.e. on G c. This calculation suggests t h a t the minimizing function f in (4.19) should be the h a r m o n i c extension of g to F ; t h a t is, the solution to the Dirichlet p r o b l e m

.f= g s

= 0

on G on G ~.

2. We shall sometimes write

= Tr(EIG) to denote the trace of the Dirichlet form s on G. 3. Note t h a t t a k i n g traces has t h e "tower p r o p e r t y " ; if H C G C F , then

Tr(el H) = Tr(Tr(elG) [ H). We now look at continuous time Markov chains on a finite state space. Let F be a finite set.

55 Definition

4.19.

A conductance matrix on F is a m a t r i x A = (asv), x, y E F ,

which satisfies as~>_0,

x#y,

a s y ---- a y s ,

EY asy ~- O. Set as = ~

axy -- - a z s . Let EA = ({x,y} : a~y > 0}. We say t h a t A is irreducible

if the g r a p h (F, EA) is connected. We can i n t e r p r e t the pair (F, A) as an electrical network: asu is the conductance of t h e wire connecting t h e nodes x a n d y. T h e intuition from electrical circuit t h e o r y is on occasion very useful in Markov Chain t h e o r y - - f o r more on this see [DS]. Given (F, A) as above, define the Dirichlet form E -- s with domain C(F) = { f : F --, ~ } by (4.20)

E ( f , g) ---- ~1 E ~ , y asy (f(x) - f(y))(g(x) - g(y)).

Note t h a t , writing fs = f ( x ) etc.,

e ( f , g ) = 89

Z

=~

as,(f

- :,)(gs - a N )

y~s

x

~

asyfsgs- Z

y~s

s

= - Z assf go x

s

~

azyfsgy

y~

Z Z as fxg s

y~s

y

In electricai terms, (4.20) gives the energy dissipation in the circuit (F, A) if the nodes are held at potential f. (A current l~y = as~ (f(y) - f(z)) flows in the wire connecting x a n d y, which has energy dissipation Isy (f(y) - f ( z ) )

asy(f(y) - f ( x ) ) 2. T h e sum in (4.20) also use this i n t e r p r e t a t i o n of Dirichlet (4.20) gives a 1-1 correspondence vative Dirichlet forms on C(F). Let # point.

=

counts each edge twice). We can of course forms in more general contexts. between conductance matrices a n d conserbe any measure on F which charges every

4 . 2 0 . (a) I r A is a conductance matrix, then gA is a regu/ar conservative Dirichlet form. (b) IrE is a conservative Dirichlet form on L2(F, #) then s = s for a conductance

Proposition

m a t r i x A. (c) A is irreducible ff and only/rE is irreducible.

Proof. (a) It is clear from (4.20) t h a t e is a bilinear form, and t h a t s f ) > 0. If g = 0 V (1 h f ) t h e n lax - gyl -< Ifs - fyl for all x,y, so since asy _> 0 for x # y, s is Markov. Since E ( f , f ) < c(A,p)llfl[~, II.lle, is equivalent to ll.ll2, and so E is closed. It is clear from this t h a t s is regular.

56 (b) As s is a s y m m e t r i c bilinear form there exists a symmetric m a t r i x A such t h a t F_.(f,g) = --fTAg. Let f = fo,f~ ----a l x + flly; then E(f, f ) = - ~ 2 a ~ x - 2o~a~.y _/32avv. Taking ~ = 1,/3 -- 0 it follows t h a t a ~ < 0. The Markov p r o p e r t y of s implies t h a t E(f01,f01) _~ E ( f ~ l , f ~ l ) i f a < 0. So 0 ~_ --o~2a~ -- 2oLa~y,

which implies t h a t a~y >_ 0 for x # y. Since s is conservative we have 0 -- E(f, 1) = --fTA1 for all f . So A1 = 0, and therefore ~-~ya~y = 0 for all x. (c) is now evident. [] E x a m p l e 4.21. Let # be a measure on F , w i t h # ( { x } ) = # ~ > 0 for x E F . Let us find the generator L of the Markov process associated with s = ~A on L2(F, #). Let z 9 F , g -- 1~, a n d f 9 L2(F,#). Then

E(s,g) = - : A S

=

az S(y)= Z azy(S(z)- S(Y))y

y

and using (4.7) we have, writing (., . ) , for the inner p r o d u c t on L2(F, #),

~(f,g) ---(-L f,g), = -#~L f(z).

So,

LS(z) =

(4.21)

Z(a.z/.z)(S(.)

-

:(z))

Note from (4.21) that (as we would expect from the trace theorem), changing the measure # changes the jump rates of the process, but not the jump probabilities.

ElectricalEqa/va/ence. Definition 4.22. Let (F, A) be an electrical network, and G C F. conductance m a t r i x on G, and

If B is a

E . = T (EAIG) we will say t h a t the networks (F, A) and (G, B) are (electrically) equivalent on G. In intuitive terms, this means t h a t an electrician who is able only to access the nodes in G (imposing potentials, or feeding in currents etc.) would be unable to distinguish from the response of the system between the networks (F, A) and (G, B). D e f i n i t i o n 4.23. (Effective resistance). Let Go, G1 be disjoint subsets of F . The effective resistance between G0 and G1, R(Go, G1) is defined by (4.22)

R(Go,G1) -1 = i n f { E ( f , f ) : fIBo = 0, f l - 1 = 1}.

This is finite if (F, A) is irreducible.

57 If G = {x, y}, then from these definitions we see that (F, A) is equivalent to the network (G, B), where B = (b~y) is given by

b~y = by~ -- - b ~ = -byy -- R(x, y ) - l . Let (F, A) be an irreducible network; and G C F be a proper subset. Let H -- G c, and for .f E C(F) write f -- (IH, ]G) where fH, fV are the restrictions of f to H and G respectively. If g e C(G), then if g = Tr(s

~(g,g):inf {(,T,gT)A(f;),

fg e C(H)}.

We have, using obvious notation (4.23)

(fH,g )A

-= f T A H H f H + 2 f T A H G g + g T A G G g .

g

The function fH which minimizes (4.23) is given by f g ----A~IHAHVg. (Note that as A is irreducible, 0 cannot be an eigenvalue of AHH , so AH~/ exists). Hence (4.24)

~(g, g) = gT(AGG -- AGHAHI AHc)g,

so that ~ -- CB, where B is the conductivity matrix --1

B = AGG -- AGHAHHAHG.

(4.25) E x a m p l e 4.24. ( A - y matrix defined by,

transform). Let G -- {x0, Xl, x2} and B be the conductance

b~o~1

---

a2,

b~lx 2 =

ao,

b~:~ o

=

O~1.

Let F = G U {y}, and A be the conductance matrix defined by a,,,j =0, a,~y = H i ,

i#j, 0
If the ai and Hi are strictly positive, and we look just at the edges with positive conductance the network (G,B) is a triangle, while (F,A) is a Y with y at the centre. The A - y transform is that (F, A) and (G, B) are equivalent if and only if O~0

(4.26)

al

fllfl2 -

-

80 + 81 + 82'

8~Zo -

-

Ot 2 - -

130 + H1 + H~' Z0H1 H0 + 131 + H2"

Equivalently, if S =- aoal + ala2 -4- a2a0, then (4.27)

S Hi=--,

0
58 This can be proved by elementary, but slightly tedious, calculations. The A - y transform can be of great use in reducing a complicated network to a more simple one, though there are of course networks for which it is not effective.

Proposition 4.25. (See [Kib]). Let (F,A) be an irreducible electric network, and R(z, y) = R({z}, {y}) be the 2-point effective resistances. Then R is a metric on F. Proof. We define R(z, x) = 0. Replacing f by 1 - f in (4.22), it is clear that R(z, y) = R(y, z), so it just remains to verify the triangle inequality. Let z0, zl, z2 be distinct points in F , and G = {x0, zl, z2 }. Using the tower property of traces mentioned above, it is enough to consider the network ( G , B ) , where B is defined by (4.25). Let a0 = b. . . . . and define a~, a2 similarly. Let/3o, ill,/32 be given by (4.27); using the A - Y transform it is easy to see that

R(zi, zj) =/3:,-1 + /3~-1,

i # j.

The triangle inequality is now immediate.

[]

R e m a r k 4.26. There are other ways of viewing this, and numerous connections here with linear algebra, potential theory, etc. I will not go into this, except to mention that (4.25) is an example of a Schur complement (see [Car]), and that an alternative viewpoint on the resistance metric is given in [Me6]. The following result gives a connection between resistance and crossing times. 4.27. Let (F, A) be an electrical network, let /Z be a measure on F which charges every point, and let (Xt, t > O) be the continuous time Markov chain associated with s on L2(F,/Z). Write T~ = inf{t > 0 : X t = z}. Then if z # y, Theorem

(4.28)

E~T~ + EYT~ = R(x, y)/Z(F).

R e m a r k . In view of the simplicity of this result, it is rather remarkable that its first appearance (which was in a discrete time context) seems to have been in 1989, in [CRRST]. See [Tet] for a proof in a more accessible publication.

Proof. A direct proof is not hard, but here I will derive the result from the trace theorem. Fix x,y, let G = {x,y}, and let g = s = Tr(E[G). If R = R(x,y), then we have, from the definitions of trace and effective resistance, B =

R_ 1

_R_ 1

.

Let v = /zig; the process -~t associated with (E, L2(G,v)) therefore has generator given by f(z) =

- f(z)). w~z

Writing T~, Ty for the hitting times associated with X we therefore have +

= R(/Z

59 We now use the trace theorem. If f ( x ) : l~(x) then the occupation density formula implies that

#zL zt =

1,(X,)

ds =

[{s _< t: X , = z}[.

So

At =

I'

l a ( X , ) ds,

and thus if S = inf{t >__Ty : Xt = x} and S is d e l e d

~ =

f

similarly, we have

1G(xs) ds.

However by Doeblin's theorem for the stationary measure of a Markov Chain

(4.29/

.(a) = (E~S/-1E 9

f 1G(X.lds~(F).

Rearranging, we deduce that

= (#(F)/#(G))E~S

= (.(F)/~(G))(E~ + E~.) = R.(F).

[]

C o r o l l a r y 4.28. Let H C F, z ~ H. Then

E ' T H <_R(x, H)#(F). Proof. If H is a singleton, this is immediate from Theorem 4.27. Otherwise, it follows by considering the network ( F ' , H ' ) obtained by collapsing all points in H into one point, h, say. (So f ' = ( f - H) U {h}, and a'~h = ~yEH a~y). [] Remark.

This result is actually older than Theorem 4.27 - see [Tell.

5. G e o m e t r y

of Regular Finitely Ramified Fractals.

In Section 2 I introduced the Sierpinski gasket, and gave a direct "hands on" construction of a diffusion on it. Two properties of the SG played a crucial role: its s y m m e t r y and scale invariance, and the fact that it is finitely ramified. In this section we will introduce some classes of sets which preserve some of these properties, and such that a similar construction has a chance of working. (It will not always do so, as we will see). There are two approaches to the construction of a family of well behaved regular finitely ramified fractals. The first, adopted by LindstrCm [L1], and most of the mathematical physics literature, is to look at fractal subsets of R d obtained by generalizations of the construction of the Cantor set. However when we come to study processes on F the particular embedding of F in •d plays only a small role,

60 and some quite n a t u r a l sets (such as the "cut square" described below) have no simple embedding. So one m a y also choose to a d a p t an a b s t r a c t approach, defining a collection of well behaved fractal metric spaces. This is the approach of Kigami [Ki2], and is followed in much of the subsequent m a t h e m a t i c a l literature on general fractal spaces. ( " A b s t r a c t " fractals may also be defined as quotient spaces of p r o d u c t spaces - see [Kus2]). T h e question of e m b e d d i n g has lead to confusion between m a t h e m a t i c i a n s and physicists on at least one (celebrated) occasion. If G is a graph then the n a t u r a l metric on G for a m a t h e m a t i c i a n is the s t a n d a r d graph distance d(x, y), which gives the length of the shortest p a t h in G between z and y. Physicists call this the chemical distance. However, physicists, thinking in terms of the graph G being a m o d e l of a polymer, in which the individual strands are tangled up, are interested in the Euclidean distance between x and y in some embedding of G in ~ a . Since t h e y regard each p a t h in G as being a random walk p a t h in Z a, they generally use the metric d'(x, y) = d(x, y)U2. In this section, after some initial remarks on self-similar sets in R d, I will introduce the largest class of regular finitely ramified fractals which have been studied in detail. These are t h e pc.f.s.s, sets of Kigami [Ki2], and in what follows I will follow the a p p r o a c h of [Ki2] quite closely. D e f i n i t i o n 5.1. A m a p r : R d --* ~ d is a similitude if there exists a E (0,1) such t h a t [r - r = a]x - y[ for all x, y 9 I~a. We call a the contraction factor of

r Let M _> 1, a n d let r A C ~ d set

CM be similitudes with contraction factors a i . For

(5.1)

ffJ(A) = U r

M i=1

Let ~(n) denote the n-fold composition of 9 . D e f i n i t i o n 5.2. Let 35 be the set of non-empty compact subsets of R d. For A C ~ d set 6E(A) = {x : [ x - - a ] < E for some a 9 A}. The g a u s d o r f f m e t r i c d on 35 is defined by d ( A , B ) -- inf{E > O: A C 6~(B) and B C 6e(A)}. 5.3. ( S e e / F e , 2.10.21]). (a) d is a m e t r i c on 35. (b) (35, d) is complete. ( c ) / f E N = { K 9 35: g C B(0, N)} then K N is compact in g .

Lemma

CM) be as above, with ai C (0, 1) for each 1 < i < M . Then there exists a unique F 9 35 such that F = ~ ( F ) . b-hrther, if G E 35 then @'~(G) -~ F in d. l_f G 9 35 satisfies ~ ( G ) C G then F = A~=0~('~)(G). T h e o r e m 5.4. Let ( r

Proof. Note t h a t ~ : 35 --, 35. Set a = m a x i a i < 1. If Ai, Bi E 35, 1 < i < M note that

M M -- ma~xd(Ai,Bi). d(Ui=lAi,ui=lBi )<

61 (This is clear since if ~ > 0 and Bi C 5,(A~) for each i, then 0B~ C 5e(OA~)). Thus

d(q(A), q(B)) <_m~x d(~b,(A), r = max ai d(A, B) = ad(A, B). z

So 9 is a contraction on/C, and therefore has a unique fixed point. For the final assertion, note that if ff2(G) C G, then ~ ( ' 0 ( G ) is decreasing. So N,~q?(")(G) is non-empty, and must equal F. [] E x a m p l e s 5.5. The fractal sets described in Section 2 can all be defined as the fixed point of a map ~ of this kind. 1. The Sierpinski gasket. Let {al, a2, a3 } be the 3 corners of the unit triangle, and set (5.2)

r

= ~ + ~(x - ~),

x c R 2,

1 < i < 3.

2. The Vicsek Set. Let {al . . . . ,ad) be the 4 corners of the unit square, let M -- 5, let a5 = (21-,89 and let (5.3)

r

= ~, + ~1 ( ~ - ~,),

1 < i < 5.

It is possible to calculate the dimension of the limiting set F from ( r However an "non-overlap" condition is necessary.

CM).

D e f i n i t i o n 5.6. ( r CM) satisfies the open set condition if there exists an open set U such that r 1 < i < M, are disjoint, and q ( U ) C U. Note that, since q ( U ) C U, then the fixed point F of 9 satisfies F = Nq('0(U). For the Sierpinski gasket, if H is the convex hull of {al, a2, a3}, then one can take U = int(H). T h e o r e m 5.7, Let ( e l , . . . ,r satisfy the open set condition, and let F be the fixed point of 9. Let/3 be the unique real such that M

(5.4)

Eaf

= 1.

i=l

Then d i m H ( F ) --/3, and 0 < ?/f~(F) < oo. Proof. See [Fa2, p. 119]. R e m a r k . If ai -- a, 1 < i < M, then (5.4) simplifies to M a ~ = 1, so that (5.5)

~ =

log M log a -1 "

We now wish to set up an abstract version of this, so that we can treat fractals without necessarily needing to consider their embeddings in R 4. Let (F, d) be a compact metric space, let I -- IM = { 1 , . . . , M}, and let

r

I
62 be continuous injections. We wish the copies r to be strictly smaller than F, and we therefore assume that there exists 5 > 0 such that (5.6)

d(r162

~_ (1 - 5 ) d ( x , y ) ,

x , y E F,

i E IM.

D e f i n i t i o n 5.8. (F, r 1 < i < M ) is a self-similar structure if (F, d) is a compact metric space, r are continuous injections satisfying (5.6) and M

(5.7)

f = U r i=1

Let (F, r 1 < i < M ) be a self-similar structure. We can use iterations of the maps r to give the 'address' of a point in F. Introduce the word spaces W , = I s,

W =- I N.

We endow W with the usual product topology. For w E W , , v in W , or W, let w 9 v = ( w l , . . . ,w,~,vl . . . . ), and define the left shift a on W (or W , ) by

= For w -- (wl, ...,wn) E Wn define

(5.8) It is clear from (5.7) that for each n > 1,

F= U wEW~

If a = ( a l , . . . , a M ) is a vector indexed by I, we write (5.9)

a~ = l ~ a~,,,

w E Wn.

i~l

Write A~ = r m > n) write (5.10)

for w E UnW,~, A C F. If n_> 1, a n d w E W (or W,n with w i n -= ( w l , . . . ,wn) E Wn.

L e m m a 5.9. For each w E W, there ex/sts a x ~ E F such that (5.11)

fi r

= {x~}.

n----1

Since r = Cw]~(r C r the sequence of sets (5.11) is decreasing. As r are continuous, r are compact, and therefore A N~F~I n is non-empty. But as diam(F~ln) _< (1 - 5)~diam(f), we have diam(A) = so that A consists of a single point. Proof.

in = 0, []

63 Lemma

5.10. There exists a unique map Ir : W ~ F such that

(5.12)

~(i.w) : r

w ~ W,

i ~ I.

lr is continuous and surjective. Proof. Define ~r(w) : zw, where z~, is defined by (5.11). Let w E W. T h e n for any n,

r ( i - w) E F(i.w)l, = Fi.(~l,~_i ) --- r

).

So 7r(i. w) E Amr = {r proving (5.12). If r' also satisfies (5.12) t h e n ~d(v. w) = r (Tr'(w)) for v E Wn, w E W, n > 1. T h e n r ' ( w ) E F~,in for any n > 1, SO

71" ! ~

"ft.

To prove t h a t 7r is surjective, let x E F . By (5.7) there exists Wl E IM such that x E F= 1 = r M = U~o~=zF~I~ 2. So there exists w2 such t h a t x E F ~ 2, and continuing in this way we o b t a i n a sequence w -- ( w l , w 2 , . . . ) E W such t h a t z E F,~ln for each n. It follows t h a t x = ~r(w). Let U be open in F , a n d w E 7r-l(U). T h e n F=ln M U c is a decreasing sequence of c o m p a c t sets w i t h e m p t y intersection, so there exists m with F~,[m C U. Hence V -- {v E W : v[m --- w[m} C 71"-l(U), and since V is open in W, r - l ( U ) is open. Thus lr is continuous. [] Remark (5.13) Lemma

5.11. It is easy to see t h a t (5.12) implies t h a t r(v-w) = r

v E W,,

w ~ W.

5.12. For x E F, n >_ O set

N,(x) = U{F= : w E W,,z

E F~,}.

Then { N , ( x ), n > 1} form a base of neighbourhoods of x. Proof. Fix x and n. If v E W~ and x r F+ then, since F . is compact, d(x,F~) = i n f { d ( x , y ) : y E F~} > 0. So, as Wn is finite, d(x, N n ( x ) c) = min{d(x,F~) : x r F~,v E W~} > 0. So x E i n t ( N n ( x ) ) . Since d i a m F w < ( 1 - 6 ) n d i a m ( F ) f o r w E W,~ we have d i a m N , ( x ) < 2(1 - 6 ) ' ~ d i a m ( f ) . So if g ~ x is open, N~(x) C U for all sufficiently large n. [] T h e definition of a self-similar structure does not contain any condition to prevent overlaps between the sets r i E IM. (One could even have r -- r for example). For sets in I~d t h e open set condition prevents overlaps, b u t relies on t h e existence of a space in which the fractal F is embedded. A general, a b s t r a c t , non-overlap condition, in terms of dimension, is given in [KZl]. However, for finitely ramified sets the s i t u a t i o n is somewhat simpler. For a self-similar s t r u c t u r e S = (F, r i E IM) set

B = B(S) =

U i,j,i~ j

n Fj.

64 As one might expect, we will require B ( S ) to be finite. However, this on its own is not sufficient: we will require a stronger condition, in terms of the word space W. Set Y = rc-1 (B(S)), P = 0

~r~(F)"

n=l

D e f i n i t i o n 5.13. A self-similar structure (F, r is post critically finite, or p.c.f., if P is finite. A metric space ( F , d ) is a p.c.f.s.s, set if there exists a p.c.f, self-similar structure (r 1 < i < M ) on F. R e m a r k s 5.14. 1. As this definition is a little impenetrable, we will give several examples below. The definition is due to Kigami [Ki2], who called F the critical set of 8, and P the post critical set. 2. The definition of a self-similar structure given here is slightly less general than that given in [Ki2]. Kigami did not impose the constraint (5.6) on the maps r but made the existence and continuity of 7r an axiom. 3. The initial metric d on F does not play a major role. On the whole, we will work with the natural structure of neighbourhoods of points provided by the self-similar structure and the sets F,~, w E W ~ , n >_ O. E x a m p l e s 5.15. 1. T h e Sierpinski gasket. Let al, a2, a3 be the corners of the unit triangle in IRd, and let r

= ai + 89 - a,),

x e R 2,

1
Write G for the Sierpinski gasket; it is ctear that (G, r structure. Writing ~ = (s, s , . . . ) , we have r(~)=a~,

r

r

is a self-similar

l<s<3.

So B(S)

1

1

i a

= { (a3 + i(al + a2), i( 2 + a3)}, r = {(1i), (3i), (12), (2i), (2i), (32)},

and P=

o(r)={(i),(~),(i)}.

2. T h e cut square. This is an example of a p.c.f.s.s, set which has no convenient embedding in Euclidean space. (Though of course such an embedding can certainly be found). Start with the unit square Co = [0, 1]2 . Now make 'cuts' along the line L1 = 1 1 {(if,y) : 0 < y < if}, and the 3 similar lines (L2, L3, L4 say) obtained from L1 by rotation. So the set C1 consists of Co, but with the points in the line segment ( 1 , y _ ) , ( 89 viewed as distinct, for 0 < y < i" 1 (And similarly for the 3 similar sets obtained by rotation). Alternatively, C1 is the closure of A = Co - U,4=ILI in the geodesic metric dA defined in Section 2. One now repeats this construction on each of the 4 squares of side 89which make up C1 to obtain successively C2, C3,...; the cut square C is the limit.

65 This is a p.c.f.s.s, set; one has M = 4, and if a l , . . . , a 4 are the 4 corners of [0, 1]2, then the maps r agree at all points with irrational coordinates with the

maps ~,(x) = a, + 89 - a,). We have 1 ( 1 , ~), 1 (~, 1 0), (~, 1 1)} ~), (~,1 ~), r = {(12), (2i), (22), (32), (3~), (42), (4i), (14), (12), (3i), (24), (42)},

B=((0,1

so that P = {(i), (2), (3), (4)}. Note also that ~r(12) = 7r(2i), and r(13) = ~r(3i) = ~-(2~1) --- r(42) = z, the centre of the square. In b o t h the examples above we had P = {(h),s e IM}, and P = amP for all n > 1. However P can take a more complicated form if the sets e l ( F ) , C j ( F ) overlap at points which are sited at different relative positions in the two sets.

3. Sierpinski gasket with added triangle. (See [Kum2]). We describe this set as a subset of I~2. Let {ax,a2,a3} be the corners of the unit triangle in ]r and let r = 89 hi) + h i , 1 <_ i _< 3. Let a4 = 1(al + a s + a3) be the centre of the triangle, and let r = a4 + ~(x - a4). Of course ( r 1 6 2 1 6 2 gives the Sierpinski gasket, but q / = (r r r r still satisfies the open set condition, and if F -- F(iI~) is the fixed point of qt then ( F , r ,r is a self-similar structure. Writing bl, b2, b3 for the mid-points of (as, a3), (a3, hi), (al, a2) respectively, and c~ = ~(ai 1 + bi), 1 < i < 3, we have

B = {bl,b2,b3,cl,c2,c3}, ~r-l(bl) -- {(23), (32)}, while ~r-l(cl) -- {(123), (139.), (4i)}, with similar expressions for ~r-l(bj), ~r-l(ci), j -- 2,3. So # ( F ) -- 15, and

~(r) = {(i), (2), (2), (22), (32), ~(r) = {(i), (2), (2)}.

(3i), (13), (12), (2i)},

Then P = a(F) consists of 9 points in W, and #(~r(P)) = 6.

Fig. 5.1 : Sierpinski gasket with added triangle.

4. (Rotated triangle). Let hi, bi, r 1 < i < 3, be as above. Let A E (0,1), and let Pl = )~b2 + (1 - ;~)b3, with P2, P3 defined similarly. Evidently {Px,P2,P3} is an

66

equilateral triangle; let r be the similitude such that r -- Pi. Let F = F(@) be the fixed point of ~P. If H is the convex hull of {a~,a~,a3}, then ~ ( H ) C H, so clearly F is finitely ramified, and

B -= {bl,b2,b3,pl,p2,p3}.

Fig. 5.2 : Rotated triangle with A -- 2/3. As before, 7r-l(bl) -- {(23), (32)}. Let Yl -- r then Yl lies on the line segment connecting a2 and a3. If A = r -1 (Yl) then A consists of one or two points, according to whether )~ is a dyadic rational or not. Let A = ~v, w}, where v = w if ~ D. Note that for each element u E A, we have, writing u -- ( u l , u 2 , . . . ) , that uk 9 {2,3}, k > 1. Then I r - l ( p l ) = { ( 4 i ) , ( 1 . v),(1 .w)}. I f S : W --* W is defined by 8(w) -- w', where w~ -= w, + 1 (mod 3), and

A . = {(i), ~ v , o"w}, then ~ " ( i ~) = An U 8(A~) U 02(A~). (a) A -- 89gives Example 3 above. (b) If A is irrational, then P = u~>la~(r) is infinite. This example therefore shows that the "p.c.f." condition in Definition 5.13 is strictly stronger than the requirement that the set F be finitely ramified and self-similar. (c) Let )~ = ~. Then v = w = (23). Therefore B consists of Pl and bl, with their rotations, and a(L) consists of (23), (32), (4i), (12323) and their "rotations" by 8. Hence P = {(i), (2), (3), (23), (32), (3i), (i3), (i2), (2i)}. So A ---- 32-does give a p.c.f.s.s, set. (d) In general, as is clear from the examples above, while F is finitely ramified for any A 9 (0, 1), F is a p.c.f.s.s, set if and only if A 9 Q n (0,1).

67

Fig. 5.3 : R o t a t e d t r i a n g l e w i t h )~ -- 0.721. We n o w i n t r o d u c e some m o r e n o t a t i o n . Definition

5.16. Let ( F , r

,r

be a p.c.f.s.s, set. Set for n _> 0,

p ( n ) = {w E W : ~ w

E P},

v(") = ~(P(~)). A n y set of t h e form Fw, w E W=, we call a n n-complex, a n d a n y set of t h e form Cw(V(0)) _- V(~ we call a n-cell. Lemma

5.17.

(a) L e t x E V (n). T h e n x -- Cw(Y), where y E V (~ a n d w E W,~.

(b) v(n) = U ~ w o v 2 ) Proof. (a) F r o m t h e definition, x = ~r(w- v), for w E W,,, v E W. T h e n if y - ~r(v), y E V (~ a n d b y (5.13), x = r ( w . v) = r (b) Let x E V(~

Then x = r

( g ( v ) ) , where v E P . Hence x = ~ ( w . v), a n d since

w 9 v E P(~), x E V (~). T h e other i n c l u s i o n follows from (a).

[]

We t h i n k of V (~ as b e i n g t h e " b o u n d a r y " of t h e set F . T h e set F consists of the u n i o n of M '~ n - c o m p l e x e s F~o (where w E W ~ ) , which intersect o n l y at their b o u n d a r y points. Lemma

5.18.

(a) I f w, v E W,~, w • v, then F~ M F , = V (~ M V (~

(b) ~ n > 0, ~-1 (~(p(~))) = ~-I(V(~) ) : p(n). Proof. (a) Let n ~ 1, v, w E W n , a n d x E Fw M Fv. So x = ~ ( w . u) ~ ~r(v. u ' ) for u, u ' E W. Suppose first t h a t wl ~ Vl. T h e n as F ~ C Fw,, we have x E F~,, M Fv~ C B. So w . u , v . u ' E F, a n d t h u s u = a ' ~ - l a ( w . u ) E P. Therefore ~r(u) E V (~

68 and x = r E V(~ If wl -- vl t h e n let k be the largest integer such t h a t w l k = vik. A p p l y i n g r we can t h e n use the a r g u m e n t above. (b) It is e l e m e n t a r y T h e n t h e r e exists v m > 1. H e n c e t h e r e r = ~r(u. v) So

t h a t p ( , 0 C 7r-1 ( l r ( p ( ' 0 ) ) . Let n = 0 a n d w C ~r-1 (~r(P)). e P such t h a t lr(w) = r ( v ) . As v e P , v e a ' ~ ( r ) for some exists u E Wm such t h a t u . v C ~ r - l ( B ) . However 7r(u. w) -9 B , a n d t h u s u . v 9 a. Hence v 9 P .

If n _> 1, a n d 7r(w) 9 7r(P(~)) = Y (~), t h e n 7r(w) e V(~ for some v e W~. (0) a n d t h u s 9 ~ n Fwlo __ (~ n V (~ by (a). Therefore 7r(w) 9 Vwl"'

r(w) = r where v 9 P . So r ( w ) = r ( w i n , v), a n d t h u s r ( a ~ w ) = 7r(v). By t h e case n = 0 above a ~ w 9 P , a n d hence w 9 p ( , 0 . [] R e m a r k 5 . 1 9 . Note we used the fact t h a t ~r(v.w) = ~r(v.w') implies lr(w) = r ( w ' ) , which follows from t h e fact t h a t Cv is injective. Lemma n>l.

5 . 2 0 . L e t s 9 { 1 , . . . , M } . T h e n r(~) is in e x a c t l y one n-complex, for each

Proof. Let n -- 1, a n d write x, = 7r(~). P l a i n l y x, 9 F , ; suppose x, E Fi where i # s. T h e n x~ -- r for some w 9 W. S i n c e x , -- r for a n y k_> 1, x~ = r (1r(i.w)) = 7r(s k . i . w ) , where s k = (s, s . . . . . s) 9 Wk. Since x , 9 F, MF, C B , z c - l ( x , ) 9 C. B u t therefore s k 9 i . w 9 C for each k > 1, a n d since i # s, C is infinite, a c o n t r a d i c t i o n . Now let n > 2, a n d s u p p o s e x , = ~r(~) 9 F ~ , where w 9 W,~ a n d w # s n. Let 0 < k < n - 1 b e such t h a t w = s k . a k w , a n d Wk+l # s. T h e n a p p l y i n g r to F,~ we have t h a t x, 9 F~,~w M F , ~ - ~ , which c o n t r a d i c t s the case n = 1 above. [] Let ( F , r

,r

be a p.c.f.s.s, set. For x 9 F , let

ran(x) = # {w e Wn : x e Fw} be t h e n - m u l t i p l i c i t y of x, t h a t is t h e n u m b e r of d i s t i n c t n - c o m p l e x e s c o n t a i n i n g x. P l a i n l y , i f x r U,~V (n), t h e n m n ( x ) = 1 for all n. Note also t h a t m . ( x ) is increasing.

Proposition

5 . 2 1 . For a11 x E F , n >_ 1, m,(x) < M#(P).

Proof. S u p p o s e x E Fwl M . . . M Fwh, where w i, 1 < i < k are d i s t i n c t e l e m e n t s of W,). S u p p o s e first t h a t w~ # w~ for some i # j . T h e n x E B, a n d therefore there exist v l , . . . , v k C W such t h a t ~r(wl . v t) -- x, 1 < l < k. Hence wt . v I E r for each l, a n d so # ( r ) _> k. B u t # ( P ) > M - I # ( F ) , a n d thus k <_ M # ( P ) . If all t h e w z c o n t a i n a c o m m o n i n i t i a l s t r i n g v, t h e n a p p l y i n g r we can use the a r g u m e n t above. []

69

Nested Fractals and AfIine Nested fractals. Nested fraetats were introduced by Lindstrem [L1], and affine nested fractals (ANF) by [FHK]. These are of p.c.f.s.s, sets, but have two significant additional properties: (1) T h e y are embedded in Euclidean space, (2) They have a large s y m m e t r y group. I will first present the definition of an ANF, and then relate it to that for p.c.f.s.s, sets. Let r 1 6 2 be similitudes in I~d, and let F be the associated compact set. Writing r also for the restrictions of r to F, (F, r . . . . , eM) is a self similar structure. Let W, ~r, V (~ etc. be as above. For x, y E V (~ let g~u : R d --* I~d be reflection in the hyperplane which bisects the line segment connecting x and y. As each r is a contraction, it has a unique fixed point, z, say. Let V = {zl,..., ZM } be the set of fixed points. Call x E V an essential fixed point if there exists y E V, and i # j such that r = e j ( y ) . Write V (~ for the set of essential fixed points. Set also V ( " ) = [,.J V (~ wEW.

D e f i n i t i o n 5.22.

(F,r

. . . . ,r

is an affine nested fractal if r

,r

the open set condition, # ( V (~ > 2, and (A1) (Connectivity) For any i, j there exists a sequence of 1-cells V_-{~ such t h a t i o = i ,

ik=jandV

satisfy V_-(~

!~ N V I r~ Zv--1

(h2) (Symmetry) For each x, y E V (~ n _> 0, g~y maps n cells to n ceils. (Aa) (Nesting) If w, v E W,~ and w # v then F~ n F, = --(0)Vwn V~~ In addition (F, r factor. If r

eM) is a nested ffactal if tile r all have the same contraction

has contraction factor a~, then by (5.4) d i m H ( F ) =/~, where f~ solves M

(5.14)

E

a t = 1.

i=1

If ai ----a, so that F is a nested fractal, then (5.15)

log M dimH(f) = log(l/a)"

Following Lindstrem we will call M the mass scale factors and 1 / a the length scale factor, of the nested fractM F.

70 L e m m a 5.23. L e t ( F , r ,r be an arlene nested [ractM. fixed p o i n t o f r T h e n zi r F j t'or any j ~ i. Proof. S u p p o s e t h a t zx 9 F2. T h e n by

Write zi for the

Definition 5.22(A3) F1 n F~ = VI~ N ~0),

so zl 9 ~ 0 ) , a n d Zl = r for some zi C ~(0). We cannot have i = 2, as r = z2 ~ zl. Also, if i = 1 then r would fix b o t h zl and z2, so could not be a contraction. So let i = 3. Therefore for any k _> 0, i _> 0,

~1

=

r

o

r

o

ri

Z

9 FI~ 2 3,

W r i t e Dn = {w 9 Wn : zl 9 F~}: by the above # ( D n ) >_ n. Let U be the open set given by t h e open set condition. Since F C U we have zi 9 U for each i. So zl 9 U~ for each w 9 Dn, while the open set condition implies t h a t the sets {Uw, w 9 Dn} are disjoint. So zl is on the b o u n d a r y of at least n disjoint open sets. If (as is t r u e for n e s t e d fractals) all these sets are congruent then a contradiction is almost immediate. For t h e general case of affine nested fractals we need to work a little h a r d e r to o b t a i n t h e same conclusion. Let a > 0 be such t h a t

IB(z. 1) n U[ > a

for each i.

Let a i , 1 < i < M be the contraction factors of the r Recall the n o t a t i o n n a,, : IIi=la~, , w E Wn. Set 5 = minwED, a ~ , and let /3 : m i n i a i . For each w 9 Dn let w' : w 9 1...1 be chosen so t h a t / 3 6 < a~, < 6. Then za 9 Fw, C Uw,, for each w 9 On, a n d the sets { U w , , w 9 Dn} are still disjoint. (Since @(U) C V we have U~, C Uw for each w 9 Dn). Now i f w E Dn t h e n i f j is such t h a t Zl : r

IB(~,~) n V~'I = ~ , l B ( z j , ~ / ~ , ) n Vl >_ (/3~)~IB(~j, 1) n UI >

~(Z~)~.

So

Ca#= IB(zl,5)] __ ~

IB(zl,~)n Uw,I _>ha(~36)a.

wED~ Choosing n large enough this gives a contradiction. P r o p o s i t i o n 5.24. L e t ( F , r . . . . . CM) be an atIine nested [ractal. the fixed p o i n t o f r T h e n (F, C a , . . . , CM) is a p.c.f.s.s, set, a n d

(a) V ~~ = Vr176 P : -{(~): ~, ~

(b)

[] Write zi for

v~~

(c) I[ z 9 V (0) then z is in exactIy one n - c o m p l e x for each n >_ 1. (d) Each 1-complex contains at most one element o f V (~ Proof. It is clear t h a t (F, r

CM) is a self-similar structure. Relabelling the r

we can assume ~(0) = { z l , . . . ,zk} where 2 < k < M . We begin by calculating B, F a n d P . It is clear from (A3) t h a t B = lJ(v~ ~ n s~t

V~o)).

71 Let w 9 I'. r ( a w ) 9 V-/~ 5.23 we must

Then 7r(w) 9 B, so (as ~r(w) 9 F~,) 7r(w) 9 V ~ ,

and therefore

Say ~r(aw) = z~, where s 9 {1, .., k}. Then since z, 9 F,~, by L e m m a have

= 8. S o

=

=

= z,, and therefore

r ( a 2 w ) ---- z~. So w3 = 8, and repeating we deduce that aw -- (~). { a w , w 9 I'} -- {(~), 1 < s < k}. This proves (b); as P is finite ( F , r

Therefore ,r is a

p.c.f.s.s, set. (a) is immediate, since r ( P ) = Y (~ = {lr(~)} = V (~ (c) This is now immediate from (a), (b) and Lemma 5.23. (d) Suppose Fi contains z~ and zt, where s ~ t. Then one of s,t is distinct from i suppose it is s. T h e n z~ 9 F~ M Fi, which contradicts (c). [] R e m a r k s 5.25. 1. Of the examples considered above, the SG is a nested fractal and the SG with added triangle is an ANF. The cut square is not an ANF, since if it were, the maps r : lI~d --* R a would preserve the plane containing its 4 corners, and then the nesting axiom fails. The rotated triangle fails the symmetry axiom unless A -- 1/2. The Vicsek set defined in Section 2 is a nested fractal, but the Sierpinski carpet fails the nesting axiom. 2. The simplest examples of p.c.f.s.s, sets, and nested fractals can be a little misleading. Note the following points: (a) Proposition 5.24(c) fails for p.c.f.s.s, sets. See for example the SG with added triangle, where V (~ contains the points {bl, b2, b3 } as well as the corners {a~, a~, a3 }, and each of the points bl lies in 2 distinct 1-cells. (b) This example also shows that for a general p.c.f.s.s, set it is possible to have F - V(~ disconnected even if F is connected. (c) Let Vi(~ and V(~ be two distinct 1-cells in a p.c.f.s.s, set. Then one can have # ( V i (~ M V(~ > 2. (The cut square is an example of this). For nested fractals, I do not know whether it is true that (5.16)

#(Vi (~ M V(~

_< 1

if i ~ j.

In [FHK, Prop. 2.2(4)] it is asserted that (5.16) holds for affine nested fractals, quoting a result of J. Murai: however, the result of Murai was proved under stronger hypotheses. While much of the work on nested fractals has assumed that (5.16) holds, this difficulty is not a serious one, since only minor modifications to the definitions and proofs in the literature are needed to handle the general case. 3. The s y m m e t r y hypothesis (A2) is very strong. We have (5.17)

g ~ y : V (~ ~ V (~

for all

z#y,

z,y 9

(~

The question of which sets V (~ satisfy (5.17) leads one into questions concerning reflection groups in R ~. It is easy to see that V (~ satisfies (5.17) if V (~ is a regular planar polygon, a d-dimensional tetrahedron or a d-dimensional simplex. (That is, the set V (~ = { e i , - e i , 1 < i < d} C ~d where ei = (fili,-.-,6di). I have been assured by two experts in this area that these are the only possibilities, and my web page see ( h t t p : / / w w w . m a t h . u b r c a / ) contains a letter from G. Maxwell with a sketch of a proof of this fact. Note that the cube in ~3 fails to satisfy (5.17).

72 4. Note also t h a t if F is a nested fractal in ~d, and V (~ C H where H is a kdimensional subspace, one does not necessarily have F C H. This is the case of the Koch curve, for example. (See ILl, p. 39]). E x a m p l e 5.26. (LindstrCm snowflake). This nested fractal is the "classical example", used in [L1] as an illustration of the axioms. It m a y be defined briefly as follows. Let z~, 1 < i < 6 be the vertices of a regular hexagon in R 2, and let 1 z~ = ~(Zl + . . . z6) be the centre. Set r

= z, + 89 - zi),

1 < i < 7.

It is easy to verify t h a t this set satisfies the axioms ( A 1 ) - ( A 3 ) above.

Fig. 5.4. Lindstr0m snowflake.

Measures on p.c.s

sets.

T h e s t r u c t u r e of these sets makes it easy to define measures which have good p r o p e r t i e s relative to t h e maps r We begin by considering measures on W. Let 0 = ( 0 1 , . . . ,OM) satisfy M

Z

Oi-=l,

0 <0i < 1

for each

iCIM.

i=1 n

Recall the n o t a t i o n 0~ = 1-Ii=l 0~,~ for w C Wn. We define the measure/~e on W to be the n a t u r a l p r o d u c t measure associated with the vector 0. More precisely, let ~,~ : W -~ IM be defined by ~ ( w ) = w~; then/2o is the measure which makes (~,~) i.i.d, r a n d o m variables with d i s t r i b u t i o n given by ~(~= = r) = 0~. Note t h a t for a n y n _ > 1, w E Wn, n

(5.18)

({v c w :

=

= II i=l

D e f i n i t i o n 5.27. Let /~(F) be the a-field of subsets of F generated by the sets {F~, w E W,~,n _> 1}. (By L e m m a 5.12 this is the Borel e-field). For A E B ( F ) , set =

73 T h e n for w E W~ ~t

(5.19)

#o(F~)=f~o(r-~(F~))

=~o((,:~l~=~})--o~=IIo~, i----1

In contexts when 0 is fixed we will write # for #e. R e m a r k . If ( F , r 1 6 2 is a nested fractal, then the sets r 1 < i < M are congruent, a n d it is n a t u r a l to take 0, -- M -1. More generally, for an ANF, the ' n a t u r a l ' 0 is given by where/3 is defined by (5.4). The following L e m m a summarizes the self-similarity of # in terms of the space LI(F,#). L e m m a 5.28. Let f E L 1(F, #). Then t'or n > 1

wGW~

Proof. It is sufficient to prove (5.20) in the case n : 1: the general case then follows by iteration. Write G -- F - V (~ Note that G~ M G~ = ~ if v, w E Wn a n d v ~ w. As # is non-atomic we have # ( F ~ ) = # ( G ~ ) for any w E Wn. Let f = l a ~ for some w E W~. T h e n f o r = 0 if i ~ Wl, and f o r = 1 G ~ . Thus

fIfo

0=:f fd ,

proving (5.20) for this particular f . The equality then extends to L 1 by a s t a n d a r d argument. [] We will also need related measures on the sets V (n). Let No -- # V (~ and set (5.21)

#~(x) = N0-1 ~ 0~lv(o,(X), wcw~

Fix

x E Y (~).

L e m m a 5.29. /z,~ is a probability measure on V ('~) a n d w l i m , ~ o o # n -- #e.

Proof. Since # V (0) -- No we have

~(v(~)) : Z No' Z e=~,o~(~): Z e~:~, zEV(~)

wGW~

wEW~

proving the first assertion. We may regard #n as being derived from # by shifting the mass on each ncomplex F ~ to the b o u n d a r y V(~ with an equal amount of mass being moved to

74 each point. (So a point x E V(~ obtains a contribution of 0,~ from each n-complex it belongs to). So if f : F --~ R then (5.22)

f fd#JF

f fd#n JR

_< max sup I f ( x ) - f(Y)l wEW~ x,yEF~

It follows t h a t #n--*#0.~ Symmetries of p.c.s

[] sets.

D e f i n i t i o n 5.30. Let g be a group of continuous bijections from F to F. We call G a symmetry group of F if (1) g : V (~ ~ V (~ for all g E G. (2) For each i E I, g E G there exists j E I, g~ E G such that (5.23)

g

o

r

= e j o g'.

Note that if g, h satisfy (5.23) then (goh) or162162 =

(r

o

g') o h'

=

ek o g",

for some j , k E I, g',h*,g" E ~. The calculation above also shows that if G1 and G2 are s y m m e t r y groups then the group generated by G1 and G2 is also a s y m m e t r y group. Write G(F) for the largest symmetry group of F. If G is a symmetry group, and g E G write ~(i) for the unique element j E I such that (5.23) holds. L e m m a 5.31. Let g E G. Then for each n _> 0, w E W,~, there exist v E W~, g' E G such that g o ev~ = ev o g'. In particular g : V ('q --* V (n). Proof. The first assertion is just (5.23) if n = 1. If n _> 1, and the assertion holds for all v E W,~ then if w = i 9v E W,,+I then

gor for

j E I, gl,g. E ~.

= gor

or

= ~ og' o r

= ~j o r

og",

[]

Proposition

5.32. Let (F, r . . , ~)M) be an A N F . Let G1 be the set of isometries of R a generated by reflections in the hyperplanes bisecting the line segments [zi, zj ], i ~ j, z,, zj E V (~ Let Go be the group generated by G1. Then GR = {giF : g E GO} is a s y m m e t r y group of F. Proof. I f g E G1 t h e n g : V (n) --~ V (n) for each n and hence a l s o g : F - - ~ F. Let i E I: by the s y m m e t r y axiom (A2) g(Vi(~ = V(~ for some j E I. For each of the possible forms of V (~ given in Remark 5.25(3), the symmetry group of V (~ is generated by the reflections in G1. So, there exists g' E G0 such that g o r = r o g'. Thus (5.23) is verified for each g E G~, and it follows that (5.23) holds for all g E G0. [] R e m a r k 5.33. In [BK] the collection of 'p.e.f. morphisms' of a p.c.f.s.s, set was introduced. These are rather different from the symmetries defined here since the definition in [BK] involved 'analytic' as well as 'geometric' conditions.

75 Connectivity Properties. D e f i n i t i o n 5.34. Let F be a p.c.f.s.s, set. For n > 0, define a graph structure on V (n) by taking {x, y} E En if x # y, and x, y E V~(~ for some w E W=. P r o p o s i t i o n 5.35. Suppose that (V (1), E1 ) is connected. Then (V(n), En) is connected for each n > 2, and F is pathwise connected. Proof. Suppose that ( v ( n ) , E n ) is connected, where n > 1. Let x , g E V ('~+1). If x , g E V (1) for some w E Wn, then, since (V(1),E1) is connected, there exists a path r = z o , z l , . . . ,zk = r in (V(1),E1) connecting r and r We have zi-1, zi E v(~ ,wl for some wi E W1, for each 1 < i < k. Then if z[ = r ! zi_ 1, z iI E Fw~.~ and so {z~_l,zl } E En+l. Thus x , y are connected by a path in (V (=+~),E.+I). For general x, y E V (n+l), as (V('~),En) is connected there exists a path Yo,...,Y,~ in ( v ( n ) , E n ) such that { Y i - l , y i } E En and x,yo, and Y, Ym, lie in the same n + 1-cell. Then, by the above, the points x, Y0, Y l , . . . , Ym, Y can be connected by chains of edges in En+l. To show that F is path-connected we actually construct a continuous path 7 : [0, 1] --~ F such that F = {7(t),t C [0, 1]}. Let x 0 , . . . ,XN be a path in (V(I),E1) which is "space-filling", that is such that V (1) C { x 0 , . . . , XN}. Define 7 ( i / N ) = x~, A1 = { i / N , 0 <_ i < N } . Now x o , x l E V~(~ for some w E W1. Let x0 = Yo,Yl,... ,Y,~ = xl be in a space-filling path in (V~(1),E2). Define 7 ( k / N m ) = Yk, 0 _< k < m. Continuing in this way we fill each of the sets V(1), w E W1, and so can define A2 C [0,1] such that A~ C A2, and 3'(t), t E A2 is a space filling path in the graph (V (2), E2). Repeating this construction we obtain an increasing sequence (An) of finite sets such that 3'(t), t E AT, is a space filling path in (V ('~), E,~), and UnA,~ is dense in [0, 1]. I f t C An, and t' < t < t" are such that (t',t") M An = {t}, then 3'(s) is in the same n-complex as 3'(t) for s e (t', t"). So, if t e [0,1] - A, and sn,tn E An are chosen so that sn < t < tn, (Sn, tn) M An = 0, then the points 3'(u), u E A M (s, t) all lie in the same n-complex. So defining 3'(t) = limn 3'(tn), we have that the limit exists, and 3' is continuous. The construction of 3' also gives that 3' is space filling; if w E W then for any n > 1 a section of the path, 3'(s), an < s <_ bn, s E A,~, fills V(~). It follows immediately from the existence of 3' that F is pathwise connected. [] Remark. This proof returns to the roots of the subject - the original papers of Sierpinski [Siel, Sie2] regarded the Sierpinski gasket and Sierpinski carpet as "curves". C o r o l l a r y 5.36. A n y A N F is pathwise connected. R e m a r k 5.37. If F is a p.c.f.s.s, set, and the graph (V(1),E1) is not connected, then it is easy to see that F is not connected. For the case of ANFs, we wish to examine the structure of the graphs (V (=), En) a little more closely. Let ( F , r ,r be an ANF. Then let

a:min{I x-y[:

x, y 6 V (~ x # y } ,

76 and set

E6 = {{x,y} E V(~ Ix-yl = a } ,

E'~ =

I{x,y}

E E,~:x--r162

W E W~, { x , y }tE

for some E0} ' , n _> 1.

P r o p o s i t i o n 5.38. L e t F be an A N F . (a) L e t x , y , z E V (~ be distinct points. T h e n there exists a p a t h in (V(~ connecting x and y and not containing z. (b) L e t x , y E V (~ T h e r e exists a p a t h in (V(1),E~) connecting x , y which does not contain a n y point in V (~ - {x, y}. (c) L e t x, y, x', y' E V (~ with Ix - Yl -- Ix' - Y'I. T h e n there exists g E GR such that =

g(y')

= y.

Proof. If # ( Y (~ = 2 then E0 = E~, so (a) is vacuous and (b) is i m m e d i a t e from Corollary 5.36. So suppose # ( V (~ _~ 3. (a) Since (see R e m a r k 5.25(3)) V (~ is either a d-dimensional tetrahedron, or a ddimensional simplex, or a regular polygon, this is evident. (For a proof which does not use this fact, see [L1, p. 34-35]). (b) This now follows from (a) by the same kind of argument as t h a t given in Proposition 5.35. (c) Write g[x, y] for the reflection in the hyperplane bisecting the line segment Ix, y]. Let gl = g[Y,Y'], and z = g l ( x ' ) . T h e n if z = x we are done. Otherwise note t h a t I x - Yl -- !x ' - Y'[ = [ z - y[, so if g2 = g[x~z] t h e n g2(Y) = Y. Hence gl og~ works. [] Metrics on N e s t e d FractaJs. Nested fractals, a n d A N F s , are subsets of I~d, and so of course are metric spaces with respect to the Euclidean metric. Also, p.c.f.s.s, sets have been assumed to be m e t r i c spaces. However, these metrics do not necessarily have all properties we would wish for, such as the mid-point p r o p e r t y t h a t was used in Section 3. We saw in Section 2 t h a t the geodesic metric on the Sierpinski gasket was equivalent to the E u c l i d e a n metric, b u t for a general nested fractal there may be no p a t h of finite length between distinct points. (It is easy to construct examples). It is however, still possible to construct a geodesic metric on a A N F . For simplicity, we will just t r e a t the case of nested fractals. Let (F, ( r M be a n e s t e d fractal, w i t h length scale factor L. Write d,~(x,y) for the n a t u r a l graph distance in the graph (V(~),E~). F i x x0, Y0 E V (~ such t h a t {x0,Y0} E E~, and let an = dn(xo, Yo), and b0 be the m a x i m u m distance between points in (V (~ E~). Lemma

5.39. I f x, y E V (~ then an < d,~(x,y) < bonn.

Proof. Since x , y are connected by a p a t h of length at most b0 in (V(~ the u p p e r b o u n d is evident. F i x x, y, and let k = d,~(x, y). If {x, y} E E~ then d~(x, y) = d~(xo,Yo) = as, so suppose {x,y} • E~. Choose y' E Y (~ such t h a t { x , y ' } E E~o, let H be the h y p e r p l a n e bisecting [y, y~] and let g be reflection in H. Write A, A ~ for the c o m p o n e n t s of I~d - H containing y, y' respectively. As Ix - y'[ < Ix - Yl we have x E A ~. Let x = Z o , Z l , . . . , z k = y be the shortest p a t h in (V(n),E,~) connecting x a n d y. Let j = min{i : z, E A}, a n d write z[ = z, if i < j , z[ = g(zi)

77 i f i _>j. T h e n z~, 0 < i < k is a p a t h in ( V ( n ) , E n ) connecting x and y', and so [] dn(x, y) = k > d , ( x , y') = an. Lemma

5.40. Let x, y E V (n) 9 Then for m > 0

(5.24)

a m d , ( x , y) < d , + m ( x , y) <_ boamd,(x, y).

In particular (5.25)

anam <_ a,~+m <_ boanam,

n > O, m > O.

Proof. Let k = dn(x, y), a n d let x = z0, z a , . . . , zk = y be a shortest p a t h connecting x a n d y in ( V ( " ) , E n ) . T h e n since by L e m m a 5.39 d , ~ ( z i - l , z i ) < boam, the u p p e r b o u n d in (5.24) is clear. For the lower b o u n d , let r = d~+,~(x, y), and let (zi)i~0 be a shortest p a t h in (V ("+m), En+,~) connecting x, y. Let 0 = i0, i l , . . . , is = r be successive disjoint hits by this p a t h on V (n). (Recall the definition from Section 2: of course it makes sense for a d e t e r m i n i s t i c p a t h as well as a process). We have s = d~(x, y) >_ an. T h e n since zi~_l,zij lie in the same n-cell, ij - ij-1 = dm(zi~_l,zij) >_ am, by L e m m a 5.39. So r = E ; : I ( i j - i j - 1 ) -> a , am. [] C o r o l l a r y 5.41. There exists 7 E [L, boa1] such that (5.26)

bo17 n _< an _< 7".

Proof. Note t h a t log(b0an) is a subadditive sequence, and t h a t log an is superadditive. So by the general t h e o r y of these sequences there exist 00, 01 such t h a t 00 = l i m n -1 log(b0an) = inf n -1 log(b0a~), 01 = l i m n -1 l o g ( a , ) = sup n -1 l o g ( a , ) . n---* o o

n~0

So 00 = 0x, a n d setting 7 -- ee~ (5.26) follows. To o b t a i n b o u n d s on ~/ note first t h a t as a , <_ boalan-1 we have -y < boa1. Also, ]xo - Yol <- anL-'~lxo - Yo], s o ' y _> L.

[]

D e f i n i t i o n 5.42. We call d~ = l o g T / l o g L the chemical exponent of the fractal F , a n d 7 t h e shortest path scaling factor. T h e o r e m 5.43. There exists a metric d F on F with the following properties. (a) There exists Cl < oo such that for each n >_ O, w E Wn, (5.27)

dF(x, y) (_ el"/-n for x, y E Fw,

and (5.28)

dF(X, y) > c2~/-n for x E V (n), y E N n ( x ) c.

(b) d e induces the same topology on F as the Euclidean metric.

78 (c) dF has the midpoint property, (d) The Hausdorff dimension of F with respect to the metric dF is log M

d r ( F ) - - log 7

(5.29)

Proof. W r i t e V = UnV(n). By L e m m a 5.41 for x, y E V we have (5.30)

bo17md,~(x, y) < dn+m(x, y) < boTmdn(x, y).

So (7-"~dn+m(x,y),m _> 0) is b o u n d e d above and below. By a diagonalization a r g u m e n t we can therefore find a subsequence nk ~ cx) such t h a t

d f ( x , y ) = lim 7 - ' ~ d , ~ ( x , y ) exists for each x,y 9 V. k---~ o o

So, if x, y c V(~ where w 9 W~ then

(5.31)

Co17 - ~ < dF(x, y) < c07 - ~

It is clear t h a t dR is a metric on V. Let n > 0 and y 9 V ('~). For m = n - 1 , n - 2 , . . . , 0 choose inductively Ym 9 V (m) such t h a t Ym is in the same m-cell as Y,~+I,...,Y,. T h e n _ max{dl(X',y') : z', d,,~+l(ym,Ym+l) <

9 V (1) } = c < oc.

y'

So by (5.30) d,~(y~, Ym+l) < b07 n - ( " + l ) c , and therefore oo

d(y~, y) < ~ ~

7 -~-1 = ~'7 -k

i=k

So if x, y E V are in the same k-cell, choosing xk in the same way we have (5.32)

dF(x,y) < dF(x,x~) + dF(xk,yk) + dF(yk,Y) <_c17 k,

since dk(zk, Yk) < bo. Thus dE is uniformly continuous on V • V, and so extends by continuity to a m e t r i c dF on F . (a) is i m m e d i a t e from (5.31). If x,y 9 V (n) and z # y then dF(x,y) >_ bo17 -n, This, together with (5.30), implies (b). If z, y E V ('~) t h e n there exists z 9 V ('~) such t h a t

I ~ ( ~ , z ) _ 1 ~dn(x,y)l <_ 1,

u = x,y.

So the metrics d,~ have an a p p r o x i m a t e m i d p o i n t property: (c) follows by an easy limiting argument. Let /t be the measure on F associated with the vector ~ = ( M - 1 , . . . , M - X ) . Thus #(F~,) --- M -I~~ for each w 9 W,~. Since we have d i a m d F ( F ~ ) • 7 - I ' l l , it follows t h a t , writing d I -- log M/log 7,

~sr d' _< ~(Bd~(~,r)) _< c6r d', and t h e conclusion then follows from Corollary 2.8.

~ 9 F []

79 R e m a r k 5.44. T h e results here on the metric dF are not the best possible. The construction here used a subsequence, and did not give a procedure for finding the scale factor 7. See [BS], [Kum2], [FHK], [Ki6] for more precise results.

6. R e n o r m a l i z a t i o n o n F i n i t e l y R a m i f i e d Fractals. Let ( F , r ,r be a p.c.f.s.s, set. We wish to construct a sequence y ( , 0 of r a n d o m walks on the sets V ('0, nested in a similar fashion to the r a n d o m walks on the Sierpinski gasket considered in Section 2. T h e example of the Vicsek set shows t h a t , in general, some calculation is necessary to find such a sequence of walks. As the r a n d o m walks we t r e a t will be symmetric, we will find it convenient to use the t h e o r y of Dirichlet forms, and ideas from electrical networks, in our proofs. F i x a p.c.f.s.s, set (F, (r M and a Bernouilli measure # -- P0 on F , where each 01 > 0. We also choose a vector r : ( r l , . . . , rM) of positive "weights": loosely speaking ri is the size of the set r : Fi, for 1 < i < M . We call r a resistance

vector. D e f i n i t i o n 6.1. Let D be the set of Dirichlet forms C defined on C(V(~ From Section 4 we have t h a t each element C 9 D is of the form CA, where A is a conductance m a t r i x . Let also I~ 1 be the set of Dirichlet forms on C(V(1)). We consider two operations on D: (1) R e p l i c a t i o n - i.e. extension of s 9 I~ to a Dirichlet form C n 9 lI)l. (2) D e c i m a t i o n / R e s t r i c t i o n / T r a c e . Reduction of a form C 9 D1 to a form ~ 9 D. N o t e . In Section 4, we defined a Dirichlet form (C, :D) with domain 9 C L 2 (F, #). But for a finite set F , as long as p charges every point in the set it plays no role in the definition of s We therefore will find it more convenient to define E on C(F) = { f : f - ~ R}.

D e f i n i t i o n 6.2. Given s 9 D, define for f,g 9 C(V(1)),

M

CR(f,g) = ~ r~-lC(f o r

(6.2)

or

i----1 (Note t h a t as r

: V (~ --* V (1), / o r

9 C(V(~

Define R : ]D --~ D1 by

R(c) = c R

L e m m a 6.3. Let C = CA, and let M

(6.3)

R

axy : ~

--1

l(zEVi(0))l(yEVi(o,)r i ar

).

i=1

Then (6.4)

Cn(f,g) = ~12 a~,n(f(x) - f(y))(g(x) - g(y)).

80 A R = (a~y) R is a c o n d u c t a n c e m a t r i x , and s

is the associated Dirichlet form.

R _> 0 i f x ~ y , a n d a ~R < 0 . are injective, it is clear t h a t a~y R = ay~ R is i m m e d i a t e from the s y m m e t r y of A. Writing xi = r (x) we have Also azy Proof. As the m a p s r

E

ely:

~-~r/-llv(o)(X)~

yEV(1)

1v(o,(y)ar162

i

yEV(1)

i

yEv(O)

)

so A R is a conductance matrix. To verify (6.4), it is sufficient by linearity to consider the case f -- g = 6~, z 9 Y (I). Let B = {i 9 W1 : z 9 Vi(~ I f i r B, then f o r cannot equal z. If i 9 B, then f o r = 6~,(z), where zi : r s

o

r

f

o

r

sincer So,

= C(6,,, ~z,) ----- a z , z,.

Thus s

f ) = -- ~

M r ~ l a z i z , = -- ~ ri-11vi(O)(z)acrl(z),erl(z )

iEB

=

R

--azz

~

i=l

while

X:

Zc~

So (6.4) is verified.

_

S(y))

_fTAR f

R

[]

T h e most intuitive explanation of the replication operation is in terms of electrical networks. T h i n k of V (~ as an electric network. Take M copies of V (~ and rescale the i t h one by multiplying the conductance of each wire by r~-1. (This explains why we called r a resistance vector). Now assemble these to form a network with nodes V (1), using the i t h network to connect the nodes in V~(~ T h e n g R is the Dirichlet form corresponding to the network V (1). As we saw in the previous section, for x , y E VO) there m a y in general be more t h a n one 1-cell which contains b o t h x and y: this is why the sum in (6.3) is necessary. If x a n d y are connected by k wires, with conductivities c l , . 9 9 ck t h e n this is equivalent to connection by one wire of conductance c1 + .. 9 + c~. R e m a r k 6.4. T h e replication of conductivities defined here is not the same as the replication of t r a n s i t i o n probabilities discussed in Section 2. To see the difference, consider again the Sierpinski gasket. Let V (~ -- {Zl,Z2, z3}, and Y3 be the midp o i n t of IZl,Z:], a n d define y~, Y2 similarly. Let A be a conductance m a t r i x on V (~ a n d write aij = a ~ z j . Take r l : r2 = r3 = 1. While the continuous time Markov Chains X (~ X (1) associated with s and CA R will d e p e n d on the choice of a measure on V (~ and V (D, their discrete time skeletons t h a t is, the processes X (i)

81

sampled at their successive j u m p times do not - see Example 4.21. Write these processes. We have ~Ys

(]i1(

1) E {Z2, Yl

})

y(i)

for

a12 + a31 = 2a12 + a31 + a23

On the other hand, if we replicate probabilities as in Section 2,

in general these expressions are different. So, even when we confine ourselves to symmetric Markov Chains, replication of conductivities and transition probabilities give rise to different processes. Since the two replication operations are distinct, it is not surprising that the dynamical systems associated with the two operations should have different behaviours. In fact, the simple symmetric random walk on V (~ is stable fixed point if we replicate conductivities, but an unstable one if we replicate transition probabilities. The second operation on Dirichlet forms, that of restriction or trace, has already been discussed in Section 4. Definition

6.5. For E 9 D1 let

(6.5)

T(s

Define A : D --* D by A(8) = that if O > 0,

=

T(R(8)).

Tr(EIV(~ Note that A is homogeneous in the sense

A(SE) = 8A(s E x a m p l e 6.6. (The Sierpinski gasket). Let A be the conductance matrix corresponding to the simple random walk on V (~ so that a~y=l,

x~y,

a~ =-2.

Then A R is the network obtained by joining together 3 symmetric triangular networks. If A(EA) = CB, then B is the conductance matrix such that the networks (V(1),A R) and (V(~ are electrically equivalent on V (~ The simplest way to calculate B is by the A - Y transform. Replacing each of the triangles by an (upside down) Y, we see from Example 4.24 that the branches in the Y each have conductance 3. Thus (V (1), A R) is equivalent to a network consisting of a central triangle of wires of conductance 3/2, and branches of conductance 3. Applying the transform again, the central triangle is equivalent to a Y with branches of conductance 9/2. Thus the whole network is equivalent to a Y with branches of conductance 9/5, or a triangle with sides of conductance 3/5. Thus we deduce A(EA) = EB,

where

3 B = ~A.

82 T h e example above suggests t h a t to find a decimation invariant r a n d o m walk we need to find a Dirichlet form s E ID such t h a t for some A > 0 (6.6)

h(s

= As

Thus we wish to find an eigenvector for the m a p A on ]I). Since however (as we will see shortly) A is non-linear, this final formulation is not p a r t i c u l a r l y useful. Two questions i m m e d i a t e l y arise: does there always exist a non-zero (E, A) satisfying (6.6) a n d if so, is this solution (up to constant multiples) unique? We will abuse terminology slightly, a n d refer to an E E I]) such t h a t (6.6) holds as a fixed point of A. (In fact it is a fixed point of A defined on a quotient space of :D.) Example

6.7. ("abc gaskets" - see [HHW1]).

Let m l , m2, m3 be integers with mi _> 1. Let Zl, z2, za be the corners of the unit triangle in R 2, H be the closed convex hull of {zl, z2, z3 }. Let M = m l + m2 + ma, and let r 1 < i < M be similitudes such t h a t (writing for convenience eM+j = ej, 1 < j < M ) Hi = r C H , and the M triangles Hi are arranged round the edge of H , such t h a t each triangle Hi touches only H i - a a n d H i + l . (//1 touches HM and H~. only). In addition, let za E / / 1 , z2 E Hm~+l, za E H m , + m l + l . So there are ma + 1 triangles along the edge [zl,z2], and m l + 1, m2 + 1 respectively along [z~,z3], [z3,zl]. We assume t h a t r are rotation-free. Note t h a t the triangles /-/2 and HM do not touch, unless m l = m2 = m3 = 1. Let F be the fractal o b t a i n e d by Theorem 5.4 from ( r 1 6 2 To avoid unnecessarily c o m p l i c a t e d n o t a t i o n we write r for b o t h r and r

Figure 6.1: abc gasket with mx = 4, m2 = 3, m3 = 2. It is easy to check t h a t (F, r , eM) is a p.c.f.s.s, set. Write r = 1, s = m 3 + 1 , t = m3 + m , + 1. We have ~(/~) = ~ ( ( / + 1)§ for 1 < i < m3, ~(ii) = ~ ( ( i + 1)~) form3+l
(~), (i)},

v (~ = { ~ , z z , z 3 } .

and

83 W h i l e it is easier to define F in R 2, r a t h e r t h a n abstractly, doing so has the misleading consequence t h a t it forces the triangles Hi to be of different sizes. However, we will view F as an a b s t r a c t metric space in which all the triangles Hi are of equal size, a n d so we will take ri = 1 for 1 < i < M . We now s t u d y the renormalization m a p A. If C = s E D, then A is specified by the conductivities Oil = a z 2 , z 3 ,

0~2 -~ az3,z~,,

(~3 = a z l ,z2.

Let f : ~3 ~ ]~3 be t h e renormalization m a p acting on ( a l , a2, a3). (So if A = A ( a ) t h e n A(~) = CA(f(,~))). It is easier to c o m p u t e the action of the renormalization m a p on the variables fli given b y t h e / ~ - Y , transform. So let ~o : (0, ~ ) 3 __, (0, c~) 3 be the A - y m a p given in E x a m p l e 4.24. Note t h a t ~o is bijective. Let fl = ~ ( a ) be the Y - c o n d u c t i v i t i e s , a n d write ~ = (ill, f12, f13) for the renormalized Y - c o n d u c t i v i t i e s : then = ~ ( f ( a ) ) . A p p l y i n g the A - y transform on each of the small triangles, we o b t a i n a network with nodes zl, z2, z3, Yl, Y2, Y3, where {zi, yi} has conductivity 8i, a n d if i ~ j {Yi, Yj} has c o n d u c t i v i t y 8i, and i f / ~ j, {Yi,Yi} has conductivity

fi

where k = k(i,j) is such t h a t k e {1,2,3} - {i,j}. A p p l y the /k - y transform again to {Yl,Y2,Y3}, to o b t a i n a Y, with conductivities 61, 52, 53, in the branches where 6i7 i :

S ~---71 "~2 ~- "~2 73 "~- "/371,

1 < i < 3.

Then

(6.7)

~;-i = ~{-1 + 6{-1 = 8;-i + (~+~,),~i s .

Suppose t h a t a E ( 0 , ~ ) 3 is such t h a t ~ ( a ) = Aa for some ), > 0. T h e n since ~(Oa) = 0 ~o(a) for any 0 > 0, we deduce t h a t ~ = ~(f(a)) = Aft. So, from (6.7), ~2~~2 A-I~{-I = Z;-I + ( ~ + ),.,s,

which implies t h a t ) - 1 > 1. Writing T = ] ~ 1 ~ 2 f l 3 / S , and 0 = TA(1 - A ) - ' , we therefore have ml(Z2 + 8 3 ) = 0, a n d (as (6.8)

S,T

are s y m m e t r i c in the ~i) we also o b t a i n two similar equations. Hence f12+83-0/ml,

fl3+fl, =8/m2,

fl1+82 =0/m3,

which has solution (6.9)

281 =

O(m~ 1 + m31 - m~l),

etc.

84 Since, however we need the/~i > 0, we deduce that a solution to the conductivity renormalization problem exists only if m~-1 satisfy the triangle condition, that is that (6.10)

m~- l + m ~ -1 > m l 1,

m31+ml

I >m21,

m~- l + m 2 1

> m ~ -1.

If (6.10) is satisfied, then (6.9) gives fl~ such that the associated c~ = ~ - l ( f l ) does satisfy the eigenvalue problem. In the discussion above we looked for strictly positive ~ such that ~(~) -~ ha. Now suppose that just one of the a~, ~3 say, equals 0. Then while z, and z2 are only connected via z3 in the network V (~ they are connected via an additional path in the network V (1). So, ~(a)3 > 0, and ~ cannot be a fixed point. If now al > 0, and a2 = a3 = 0 then we obtain ~(a)2 -- ~(~)3 : 0. So ~ = (1, 0, 0) satisfies ~(a) = h a for some A > 0. Similarly (0, 1, 0) and (0, 0, 1) are also fixed points. Note that in these cases the network (V (~ A(a)) is not connected. The example of the abc gaskets shows that, even if fixed points exist, they may correspond to a reducible (ie non-irreducible) E E ~). The random walks (and limiting diffusion) corresponding to such a fixed point will be restricted to part of the fractal F. We therefore wish to find a non-degenerate fixed point of (6.6), that is an s E I]) such that the network (V (~ A) is connected. D e f i n i t i o n 6.8, Let ]])~ be the set of E E D0 such that s is irreducible - that is the network (V(~ is connected. Call E E]I) strongly irreducible if C -- s and a ~ > 0 for all x :~ y. Write D ~ for the set of strongly irreducible Dirichlet forms on V(~ . The existence problem therefore takes the form: P r o b l e m 6.9. (Existence). Let ( F , r 1 6 2 Does there exist E EI]) i, A > 0, such that

(6.12

A(E)

be a p.c.f.s.s, set and let ri > 0.

:

Before we pose the uniqueness question, we need to consider the role of symmetry. Let (F, (r be a p.c.f.s.s,set, and let T/be a symmetry group of F.

Definition 6.10. C E ~ is 7-t-invariant if for each h E E(foh, goh)=s

f, g E C(V (~

r is 7-t-invariant if rh(i) = ri for all h E ~ . (Here h is the bijection on I associated with h). L e m m a 6.11. (a) Let C = EA. Then E is ~-l-invariazit if and only if: (6.13)

ah(x) h(y)

:

a~y for all x,y E V (~

h E 7~.

(b) Let s and r be ~=invariant. Then AE is 7~-invariant. Proof. (a) This is evident from the equation s

ly) -- -a~y.

85 (b) Let f E C(V(D). Then if h E 7-/,

s

o h , f o h) = Z

r~l ~(f o h o r

ohor

i

I f g C C ( V (~

--~ ~

?,~-1 ~,(f 0 ~)h(i) O h, f o eh(i) o h, )

_- ~

?.;(1i)E(f o ~bh(i), f o ~b~(i)) = ER(f,f).

then writing ~ = A(s

~(goh, goh) ~_s

if fly(o) = g then as f o hlv(o ) = g o h,

oh, f oh)=ER(f,f),

and taking the infimum over :, we deduce that for any h C ~, ~'(goh,goh) _< ~'(g,g). Replacing g by g o h and h by h -1 we see that equality must hold.

[]

If the fractal F has a non-trivial symmetry group ~ ( F ) then it is natural to restrict our attention to ~(F)-symmetric diffusions. We can now pose the uniqueness problem. P r o b l e m 6.12. (Uniqueness). Let (F, (r be a p.c.f.s.s, set, let 7"/be a symmetry group of F, and let r be 7~-invariant. Is there at most one 7g-invariant C E D i such that A(s = ),s (Unless otherwise indicated, when I refer to fixed points for nested fractals, I will assume they are invariant under the symmetry group GR generated by the reflections in hyperplanes bisecting the lines Ix, y], x, y E V(~ The following example shows that uniqueness does not hold in general. E x a m p l e 6.13. (Vicsek sets - see [Me3].) Let (F, r 1 < i < 5) be the Vicsek set - see Section 2. Write {zl, z2, z3, z4,} for the 4 corners of the unit square in R 2. For a,fl,3, > 0 let A(a,f~,~) be the conductance matrix given by a12 : a23 : a34 ---- a41 = or,

ot13 = ~ ,

a24 = - y ,

where aij : az~ z~. If 7t is the group on F generated by reflections in the lines [Zl, z3] and [z2,z4] then A is clearly 7~-invariant. Define &,/~, ~ by

A(EA)= CA(W,~, 7)" Then several minutes calculation with equivalent networks shows that

(6.14)

~ =

a ( a + 13)(a + 7) 5a 2 + 3a/3 + 3a'y + / ~ 7 '

5(" + ~) -

~,

7= 89 If (1,13,3,) is a fixed point then ( ~ , ~ , ~ ) -- (0,0~,0~,) for some 0 ~ 0, so that S o ~ - I - 5 , and this implies that f17 = 1. We therefore have that (1, fl,f~-l) is a fixed point (with A -- 89 for any f~ E (0, oo) Thus for the group 7/ uniqueness does not hold.

=~,~=~7.

86 However if we replace 7-/by the group ~n = G(F), generated by all the symmetries of the square then for EA to be QR-invariant we have to have/3 = 7. So in this case we obtain (6.15)

~ ( . + f~)2

~(a, fl) -- 5a 2 + 6~fl + ~ 2 ,

~(~,~)

+~)

~.

This has fixed points (0,~),/3 > 0, and (a, a), a > 0. The first are degenerate, the second not, so in this ease, as we already saw in Section 2, uniqueness does hold for Problem 6.12. This example also shows that A is in general non-linear. As these examples suggest, the general problem of existence and uniqueness is quite hard. For all but the simplest fractals, explicit calculation of the renormalization map A is too lengthy to be possible without computer assistance - at least for 20th century mathematicians. LindstrCm ILl] proved the existence of a fixed point E E I) si for nested fractals, but did not treat the question of uniqueness. After the appearance of [L1], the uniqueness of a fixed point for LindstrCm's canonical example, the snowflake (Example 5.26) remained open for a few years, until Green [Gre] and Yokai [Yo] proved uniqueness by computer calculations. The following analytic approach to the uniqueness problem, using the theory of quadratic forms, has been developed by Metz and Sabot - see [Me2-Me5, Sabl, Sab2]. Let l~+ be set of symmetric bilinear forms Q(f, g) on C(V (~ which satisfy

Q(1,1) = 0, Q ( f , / ) _> 0 for all f 9

C(V(~

For Q1, Q2 9 l~+ we write Q1 _> Q2, if Q2 - Q1 9 • + or equivalently if Then I3 C l~+; it turns out that we need to consider the action of A on 1~+, and not just on 13. For Q 9 l ~ + , the replication operation is defined exactly as in (6.2)

Q2(f,f) >_Ql(f,f) for all f 9 C(V(~

M

(6.16)

Qn(f,g)=Er~lQ(for162

f,g 9

i=1

The decimation operation is also easy to extend to IVY+ :

T(Qn)(g,g) = inf{QR(f,f): f 9 C(V(~ we can write

= g};

T(Q R) in matrix terms as in (4.24). We set A(Q) = T(Qn).

L e m m a 6.14. The m a p A on M+ satisfies: (a) A : M + ~ ~+, and is continuous on int(M+). (b) A(Q1 + Q2) _> A(Q1) + A(Q2). (c) A(OQ) = 0A(Q)

Proof. (a) is clear from the formulation of the trace operation in matrix terms.

87

Since the replication operation is linear, we clearly have QR = (0Q)R = ~QR. (c) is therefore evident, while for (b), ifg ~ C(V(~

+ Qf,

T(QR)(g,g) = inf{Q~ ( f , f ) + Q~ ( f , f ) : ftv(o) -:- g} > i n f { Q ~ ( f , f ) : fly(o) = g} +inf{Q2R(f,f): fly(o) = g} = T(Q1R)(g,g) + T(QR)(g,g).

[]

Note that for 8 E D i, we have E(f, f ) = 0 if only if f is constant. D e f i n i t i o n 6.15. For 81,82 C ~)i set

= inf{ E1(f, f) constant}. 82(f, f ) : f non Similarly let M(81/82) = sup ~r ~$1(f, f ) : f non constant}. Note that (6.18)

M(~/82)

=

m(82/$~) -~.

L e m m a 6.16. (a) For s

E ]I)i, 0 < m(81,82) < oo . (b) l-f El, & e D*i then m(81/~2) = M(E1/&) ff and only if82 = 481 for some

A>O.

(c) If •,E2,E3

E D i then

m(E1/e ) >_re(elf82) m(82183), M(E1/E3) _< M(81/E2) M(s

Proof. (a) This follows from the fact that Ei are irreducible, and so vanish only on the subspace of constant functions. (b) is immediate from the definition of m and M. (c) We have m(,-qx/Sa) = ii~f 81 (f, f) 82(f, f) > m(81/$2)rn(82/E3); 82(f,f)

E3(f,f)

--

while the second assertion is immediate from (6.18). D e f i n i t i o n 6.17.

[]

Define M(81s dH(81,82)----log m(Eas

81,82 E

Di"

Let pD i be the projective space D i / ~-, where 81 ~ E2 if 81 = 482. dH is called Hilbert's projective metric - see [Nus], [Me4].

88 = 0 if and only if s = )~s for some )~ > O. (b) dH is a pseudo-metric on D ~, and a metric on p D ~. (c) Ifs163163 E E) ~ then for aO,al > O,

P r o p o s i t i o n 6.18. (a) d H ( ~ 1 , s

d~(s ~0 G0 + ~1s

< ma~(dz(s E0), dz(s

In particular open balls in dH are convex. (d) (pD i, dH) is complete. Proof. (a) is evident from Lemma 6.17(b). To prove (b) note that dH(s163 > O, and that dH(s s = dH(s s from (6.18). The triangle inequality is immediate from L e m m a 6.17(c). So dH is a pseudo metric on D i. To see that dH is a metric on p D i, note that m(Ael/e2) = A-~(s163

A > 0,

from which it follows that dH(As s = dH(s s and thus dH is well defined on p/l) i. The remaining properties are now immediate from those of dH on I])i. (c) Replacing s by ('~(s163163 s we can suppose that m(s163 = m(s163 Write Mi = M ( s 1 6 3

= m.

Then if ~" = so Go + a1s

M(s

= inf aos

I

f) + a l s s

f)

~ o~0m(s163 -~- o ~ 1 m ( s Similarly M ( s

= 0~0 "4- 0~1.

~ aoMo + aiM1. Therefore

exp dH(E,~) < (~o/(~o + ~l))(Mo/m) + (~1/(~o + < B(s

m~x(Mo/~,M~/~).

It is immediate that if s E B ( s then dH(s163 is convex. For (d) see [Nus, Thin. 1.2].

Theorem

~1))(M1/~)

6.19. Let s163

+ (1 -- A)s

< r, so that []

E IDi. Then

(6.19)

m(A(s163

> m(s163

(6.20)

M(A(s163

<_ M(s163

In particular A is non-expansive in dH : dH(A(s163

(6.21)

<_ dH(s163

Proof. Suppose a < m(El,s Then Q = s - as non-constant f E C(V(~ So by Lemma 6.14 A(s

= A(Q + as

E

~ + , and Q ( f , f ) > O, for all

_> A(Q) + ah(s

89 and since A(Q) _> 0, this implies that A(s - h A ( & ) _> 0. So a < m(A(E1),A(s and thus m(s s <_ m(A(s A(s proving (6.19). (6.20) and (6.21) then follow immediately from (6.19), and the definition of dg. [] A strict inequality in (6.21) would imply the uniqueness of fixed points. Thus the example of the Vicsek set above shows that strict inequality cannot hold in general. So this Theorem gives us much less than we might hope. Nevertheless, we can obtain some useful information. C o r o l l a r y 6.20. (See [HHW1, Cor. 3.7]) Suppose s A(s - Ais i = 1, 2. Then A1 = A2.

s are fixed points satisfying

Proof. From (6.19) re(e,/e=) <_ m (A(s so that A1 _> A2. Interchanging s

= (A1 / ~=)m(s163

and s

we obtain )h = A2.

[]

We can also deduce the existence of 7-/-invariant fixed points. P r o p o s i t i o n 6.21. Let ~ be a symmetry group o f F . I r A has a fixed point El in D i then A has an Tl-invariant fixed point in D i .

Proof. Let A = {s E IDi : s is 7-/-invariant.}. (It is clear from Lemma 6.11 that A is non-empty). Then by Lemma 6.11(b) A : A -+ A. Let s E A, and write r = dH(s163 B = BdH(gl,2r). By Theorem 6.20 A : B - - * B. S o A : A N B A A B. Each of A, B is convex (A is convex as the sum of two 7~-invariant forms is 7-(-invariant, B by Proposition 6.18(c)), and so A fq B is convex. Since A is a continuous function on a convex space, by the Brouwer fixed point theorem A has a fixed point s 9 A N B, and s is ~-invariant. [] We will not make use of the following result, but is useful for understanding the general situation.

(with E1 ~ )~s

C o r o l l a r y 6.22. Suppose A has two distinct fixed points s and s for any A). Then A has uncountably many fixed points.

Proof. (Note that the example of the Vicsek set shows that 89163+ s is not necessarily a fixed point). Let F C D i be the set of fixed points. Let go,s 9 F; multiplying s by a scalar we can take m(s163 --- 1. Write R = dH(s163 If s ----As + (1 -- A)g0 then as in Proposition 6.19(c) expdn(Ex,s

) <_ (1 - A) + AM(s163

and so

d7~(s163

<_ log((1 + eR)/2).

Thus there exists 6, depending only on R, such that

A = {s 9 D i : s 9 B(s

- 6)R) N B ( s

- ~)R)}

is non-empty. Since A preserves A, A has a fixed point in A. F thus has the property: if s

s

are distinct elements of ]F then there exists s

such that 0 < d~(s163

< dn(s

9 F

90 As ]F is closed (since A is continuous) we deduce that F is perfect, and therefore uncountable. []

-

This if as far as we will go in general. For nested fractals the added structure s y m m e t r y and the embedding in R d, enables us to obtain stronger results. If (F, (r is a nested fractal, or an ANF, we only consider the set D i N {s : s is GR-invariant}, so that in discussing the existence and uniqueness of fixed points we will be considering only GR-invariant ones. Let (F, (r be a nested fractal, write G = GR and let EA be a (G-invariant) Dirichlet form on C ( V (~ CA is determined by the conductances on the equivalence classes of edges in (V (~ Co) under the action of g. By Proposition 5.38(c) if Ix-y] = Ix' - y'[ then the edges { x , y } and {x',y'} are equivalent, so that A~v = A~,y,. List the equivalence classes in order of increasing Euclidean distance, and write a l , a 2 . . . , a ~ for the common conductances of the edges. Since A = A(A) is also g-invariant, A induces a m a p A' : R~_ --* R~_ such that, using obvious notation, A(A(a)) = A(A'(a)). Set D* = {a : a l > a2 >~ ... > 0% > 0}. Clearly we have I~* C D s~. We have the following existence theorem for nested fractals. T h e o r e m 6.23. (See ILl, p. 48]). Let (F, (r Then A has a tlxed point in D*.

be a nested fractal (or an ANF).

Proof. Let s E D*, and let a l , ...ak be the associated 0, Q~, z E V (~ be the continuous time Markov chain (Y,~,n > 0, Q ~ , z E V (~ be the discrete time skeleton L ~~ ~~ 0 0 ) ' " ' " ' ~~(k) be the equivalence classes of edges 0 a i if {x, y} E E~ i). Then if {x, y} E E~ j),

conductivities. Let (Yt, t > associated with CA, and let of Y. in (V (~ ) Eo), so that A~ v

ai Q'(Y1 = V) - ~ y # ~ A~v" As cl = ~ A~ v does not depend on x (by the s y m m e t r y of V (~ the transition vgz probabilities of Y are proportional to the a i. Now let R ( A ) be the conductivity matrix on V (1) attained by replication of A. Let (Xt, t >_ O, I?~, x e V (1)) and (2~,n > 0, P~,z c VO)) be the associated Markov Chains. Let To,T1, ... be successive disjoint hits (see Definition 2.14) on V (~ by -~,~. Write .4 = A(A), and ~ for the edge conductivities given by A. Using the trace theorem, IP~(2T~ -= y) = ~ j / c l if {x, y} 9 E~ i). Now let x l , y l , y 2 (6.23)

9 V (~ with Ix - Yll < ]x - y2]. We will prove that ]~z'(2T~ = Y2) < IP*'(2T~ = Yl).

Let H be the hyperplane bisecting [Yl,Y2], let g be reflection in H , and x2 = g(xl). Let T = min{n > 0: .~,~ 9 V (~ - {xl}},

91

so that T1 = T F~l-almost surely. Set

/~(z) = E ~ i(r <.)(i~ (Xr) - ly~(2r)). Let p(x,y), x,y E V (~ be the transition probabilities of ,~. Then

(6.24

/~+~(~) = IA(~):o(~) + IAo(~) ~

p(~,y)/,,(y).

Y

Let J12 = {x E V (1) : tx - Yll < tx - Y~I}, and define J21 analogously. We prove by induction that f,~ satisfies

f=(x) > O, f,,(x) + f,~(g(x)) >_ O,

(6.25a) (6.25b)

x E J12, x E J12.

Since f0 = 1~ - ly2, a n d y l E J12, f0 satisfies (6.25). Let x E A c U J 1 2 and suppose fn satisfies (6.25). If p(x,y) > 0, and y C J~2, then x , y are in the same 1-cell so if y' = g(y), y' is also in the same 1-cell as xl and Ix - y'] < Ix - y]. So (since $A 6 •*), p(x,y') > p(x,y) and using (6.25b), as f,~(y') >_ O,

p(~, y)/n(y) + p(~, y')/,~(y')>_ p(~, y)(/,,(y)+ .:,,(g(y))_> o. Then by (6.24), fn+l(x) > 0. A similar argument implies that fn+l satisfies (6.25b). So (f,~) satisfies (6.25) for all n, and hence its limit f ~ does. Thus foc(xl) = I?~(-~T = Yl) -- ~(XT = Y2) > 0, proving (6.23). From (6.23) we deduce that ~l > ~2 > ... > ~k, so that A : D* ~ D*. As A'(Oa) = 0A'(a), we can restrict the action of A' to the set {.

> ... >

>

0, y :

=

This is a closed convex set, so by the Brouwer fixed point theorem, A ~ has a fixed point in ]D*. [] R e m a r k 6.24. The proof here is essentially the same as that in LindstrOm ILl]. The essential idea is a kind of reflection argument, to show that transitions along shorter edges are more probable. This probabilistic argument yields (so far) a stronger existence theorem for nested fractals than the analytic arguments used by Sabot [Sabl] and Metz [Me7]. However, the latter methods are more widely applicable. It does not seem easy to relax any of the conditions on ANFs without losing some link in the proof of Theorem 6.23. This proof used in an essential fashion not only the fact that V (~ has a very large symmetry group, but also the Euclidean embedding of V (~ and V (1). The following uniqueness theorem for nested fractals was proved by Sabot [Sabl]. It is a corollary of a more general theorem which gives, for p.c.f.s.s, sets, sufficient conditions for existence and uniqueness of fixed points. A simpler proof of this result has also recently been obtained by Peirone [Pe].

92 be a nested fractal. Then A has a unique ~R-invariant

T h e o r e m 6.25. Let (F, (r non-degenerate fixed point.

Definition 6.26. Let s be a fixed point of A. The resistance scaling factor of C is the unique p > 0 such that

A(E) = p-1 E. Very often we will also call p the resistance scaling factor of F : in view of Corollary 6.21, p will have the same value for any two non-degenerate fixed points.

Proposition 6.27. Let (F, (r

be a p.c.s set, let (r,) be a resistance vector, and let s be a non-degenerate fixed point of A. Then for each s E {1, ...M} such that r(~) C V (~ (6.27)

r~ p-1 < 1.

Proof. Fix 1 < s < M, l e t x = 7r(~), and let f = 1~ E C(V(~ EA(f,f) =

E A~v = IA~I. yE V (~ y#x

=- p-lEA,

Let g : lx C C(V(1)). As A(s (6.28)

Then

p-l[A~z[ = A ( E A ) ( f , f ) < EAR(g,g) :

since g is not harmonic with respect to CA R, strict inequality holds in (6.28). By Proposition 5.24(c), x is in exactly one 1-complex. So

SAR(g,g) = ~

r:-lSA(g o r

or

= r:llAzzl,

i and combining this with (6.28) gives (6.27).

[]

Since r~ = 1 for nested fractals, we deduce C o r o l l a r y 6.28. Let (F, (r

be a nested fractal. Then p > 1.

For nested fractals, many properties of the process can he summarized in terms of certain scaling factors. D e f i n i t i o n 6.29. Let (F, (r be a nested fractal, and E be the (unique) nondegenerate fixed point. See Definition 5.22 for the length and mass scale factors L and M. The resistance scale factor p of F is the resistance scaling factor of E. Let (6.29

~ = Mp ;

we call T the time scaling factor. (In view of the connection between resistances and crossing times given in Theorem 4.27, it is not surprising that ~- should have a connection with the space-time scaling of processes on F.) It may be helpful at this point to draw a rough distinction between two kinds of structure associated with the nested fractal ( F , r The quantities introduced in Section 5, such as L, M, the geodesic metric dF, the chemical exponent 7 and the dimension dw(F) are all geometric- that is, they can be determined entirely by a geometric inspection of F. On the other hand, the resistance and time scaling

93 factors p and ~- are analytic or physical- they appear in some sense to lie deeper than the geometric quantities, and arise from the solution to some kind of equation on the space. On the Sierpinski gasket, for example, while one obtains L = 7 = 2, and M = 3 almost immediately, a brief calculation (Lemma 2.16) is needed to obtain p. For more complicated sets, such as some of the examples given in Section 5, the calculation of p would be very lengthy. Unfortunately, while the distinction between these two kinds of constant arises clearly in practice, it does not seem easy to make it precise. Indeed, Corollary 6.20 shows that the geometry does in fact determine p: it is not possible to have one nested fractal (a geometric object) with two distinct analytic structures which both satisfy the s y m m e t r y and scale invariance conditions. We have the following general inequalities for the scaling factors.

Proposition 6.30.

Let (F, (r

be a nested fractal with scaling factors L, M, p, T.

Then (6.30)

L>I,

M_>2,

M>L,

T=Mp>L

2.

Proof. L > 1, M > 2 follow from the definition of nested fractals. If0 = diam(V(~ then, as VO) consists of M copies of V (~ each of diameter L -10, by the connectivity axiom we deduce M L - I O > O. Thus M > L. To prove the final inequality in (6.30) we use the same strategy as in Proposition 6.27, but with a better choice of minimizing function. Let 7-/be the set of functions f of the form f ( x ) = O x + a , where x C L~d and O is an orthogonal matrix. Set 7-/,~ = {fly(~), f E 7-/}. Let 0 = sup{C(f, f ) : f E 7-/0}: clearly 8 < ~ . Choose f to attain the supremum, and let g E 7-/ be such that f = gig(o). Then if fx = gig(l) M

p-10 = p-lE(f, f) = h(E)(f, f) < ER(g~,gl) = ~

E(gl o r

or

i=1

However, gl o r is the restriction to V (~ of a function of the form L - l O x + ai, and so E(g o r o r <_ L -20. Hence p-10 < M L - 2 8 , proving (6.30). [] The following comparison theorem provides a technique for bounding p in certain situations.

Proposition

6.31. Let ( F I , { r 1 < i < M1}) be a p.c.s set. Let Fo C 1;1, Mo ~_ M1, and suppose that ( F o , { r ~_ i _~ Mo}) is also a p.c.s set, and that V FI (~ = V Fo (~ " Let (r~k),l < i < Mk) be resistance vectors for k -- 0,1, and suppose that r~~ >_ r~ 1) for 1 < i < Mo. Let A~ be the renormalization map for (Fk, (r Mk /( r i(k)xMk h=l J" I f Ck are non-degenerate Dirichlet forms satisfying Ak(Ck) = p~-lEk, k = 0, 1, then Pl _< P0.

Proof. Since V(~ ) C V(1 ), we have, writing Ri for the replication maps associated with Fi,

Rle(;, f) > Roe(f, f),

f e

c(v(~)).

94

So AI(E) >_ A0(E) for any s 9 D. I f m =

p11s = AI(s

_> A l ( m s

m(s163

_> Ao(ms

then =

mPols

>_poIs , []

which implies that Po _> Pl.

7. D i f f u s i o n s

on p.c.f.s.s, sets.

Let (F, (r be a p.c.f.s.s, set, a n d r, be a resistance vector. We assume that the graph (V (1), E l ) is connected. Suppose that the renormalization map A has a non-degenerate fixed point s (~ = CA, so that A(s (~ : p-1s176 Fixing F, r, and s in this section we will construct a diffusion X on F, as a limit of processes on the graphical approximations V ('0. In Section 2 this was done probabilistically for the Sierpinski gasket, but here we will use Dirichlet form methods, following [Kus2, Ful, Ki2]. D e f i n i t i o n 7.1. For f 9 C(V(n)), set (7.1)

E(n)(f,f):p

n ~ r=lC(~ wEw~

This is the Dirichlet form on V (n) obtained by replication of scaled copies of s (~ where the scaling associated with the map Cw is p'~r~1. These Dirichlet forms have the following nesting property. 7.2. (a) For n > 1, Tr(s ('~-D : s (b) Is 9 C(V(")), and g = flv(--~) then $(")(f, f) _> g("-l)(g,g). (c) s is non-degenerate.

Proposition

Proof. (a) Let f 9 C(V(~)). Then decomposing w 9 W~ into v. i, v 9 W ~ - l , (T.2)

E(n)(f,f)=P n ~

r:l~'~r/1E(~176162176162176162176

vEW~_~ = pn--I

i

r:lE(1)(f~'f=)'

E yEW,,_ ~

where f~ = f o Cv E C(VO)). Now let g E C(V('~-I)). fv[v(o) = g o r = g,. As s (~ is a fixed point of A, (7.3)

inf{s

=g~}

=pinf{Rs176 =

=

Summing over v E W,~-I we deduce therefore

v

If

flv(.-l) = g

then

95 For each v C W n - 1 , let h~ E C(V (U) be chosen to a t t a i n the infimum in (7.3). We wish to define f E C(V('O) such t h a t (7.4)

f or

-- hv,

v C W~-I.

Let v E W n - 1 . We define f(r

= h~(y),

Y E V(1).

We need to check f is well-defined; b u t if v, u are distinct elements of W,~-I and x = r = r t h e n x C V (n-D by L e m m a 5.18, and so y, z E V (~ Therefore f(r

= h~(v) = g~(y) = g ( x ) = f ( r

so t h e definitions of f at x agree. (This is where we use the fact t h a t F is finitely ramified: it allows us to minimize separately over each set of the form V(1)). So E(n)(f, f ) = ~ ( n - 1 ) ( g , g ) , a n d therefore Tr (s = g'('~-U. (b) is evident from (a). (c) We prove this by induction. E (~ is non-degenerate by hypothesis. s is non-degenerate, a n d t h a t E ( " ) ( I , f ) = 0. From (7.2) we have

E(")(f,f) = p ~

r:~E("-~)(f o r

Suppose

f or

vEWx

a n d so f o r is constant for each v E W1. Thus f is constant on each I-complex, a n d as (V (1), E1 ) is connected this implies t h a t f is constant. [] To avoid clumsy n o t a t i o n we will identify functions with their restrictions, so, for example, if f E C(V(~')), a n d m < n, we will write s i n s t e a d of

E(m) (flv(~), flv(~)). D e f i n i t i o n 7.3. Set V (~ = U~=0 V(~). Let U -- { f : V (~) --* R}. Note t h a t the sequence (C(n)(f, f))~=1 is non-decreasing. Define 7)' ----{f e U : supC(n)(f, f ) < (x)}, n

s

= sups

f,g C 7)'.

E ~ is the initial version of the Dirichlet form we are constructing. Lemma

7.4. s is a s y m m e t r i c

Markov

form on 7)~.

Proof. s clearly inherits the properties of symmetry, bilinearity, and positivity from the s If f E 7)', a n d g = (0 V f ) A 1 then s <_s as the E ( ' ) are Markov. So s <s [] W h a t we have done here seems very easy. However, more work is needed to o b t a i n a ' g o o d ' Dirichlet form s which can be associated with a diffusion on F . Note the following scaling result for s

96

Lemma

7.5. For n > 1, f E 7:)',

E'(f,f)= E

(7.5)

flnr~iE'(fOCw,f O•w).

wEW~

Proof. W e

have, for m > n, f E If,

E(m)(f'f)= E

P'~r~'IE(m-n)(f~162176162

wEW~

Letting m --* oo it follows, first that f o r

E Z)~, and then that (7.5) holds.

[]

If H is a set, and f : H ---, ]~, we write (7.6)

O s c ( f , B ) = sup I f ( x ) - f(Y)l,

B C H.

~,y6B

Lemma

7.6. There exists a constant co, depending only on $, such that

f E C(V(~

_< coE(~

O s c ( f , V (~

Proof. Let /~0 = {{x,y} : A,y > 0}. As s is non-degenerate, (V(~ is connected; let N be the maximum distance between points in this graph. Set a = m i n { A , y , { x , y } e E0}. If z, y e Y (~ there exists a chain x = x o , x l , . . . , x n = y connecting x, y with n < N, and therefore,

If(z) - f(y)l 2 _<

[ f ( ~ ) - f(z~_~)l

< n~

lf(xl) - f(x,_1)l2

i=l

< n a - ' ~-~A~,_x,.,[f(zi ) - f ( x i _ l ) l 2 i=1

<_ N a - l ~(~

, f).

[]

Since V 0) consists of M copies of V (~ we deduce a similar result for V (1). C o r o l l a r y 7.7. There exists a constant cl = cl (F, r, A) such that

(7.7)

Osc(f, v (~))<_ cle(1)(f,f),

f 9 ~)'.

Proof. For i 9 W1, f 9 C(V(1)),

Osc(f, v~(~ = Osc(f o r v (~ < c0G(~

r

f or

So, as V (1) is connected,

Os~(f,y(')

< ~

Osc(f, v, (~ i

E c0r i

~ 1 6 2 o~bi) ~ ClC,(1)(f,f),

97

where cl is chosen so t h a t co <_ clpr~ 1 for each i C W1.

[]

C o r o l l a r y 7.8. Let w E W,', and x, y E V(1). Then

Osc(f,v(1)) < el ~,p

E (f,f),

Proof. We have O s c ( f , V O ) ) = O s c ( / o r g (1) < 8 ' , a n d by (7.5)

E'(f o r

f o

r

f 9

_< c1s

o C w , f o Cw). Since

< r~p-,'E'(f, f),

the result is i m m e d i a t e .

[]

D e f i n i t i o n 7.9. We will call the fixed point E (~ a regular fixed point if

ri < p

(7.8)

for

1
M.

P r o p o s i t i o n 6.27 implies t h a t (7.8) holds for any s 9 { 1 , . . . , M } such t h a t lr(k) 9 V (~ In p a r t i c u l a r therefore, for nested fractals, where every point in V (~ is of this form and r is constant, any fixed point is regular. It is not h a r d to p r o d u c e examples of non-regular fixed points. Consider the LindstrOm snowflake, b u t with ri = 1, 1 < i < 6, r7 = r > 1. Writing p(r) for the resistance scale factor, we have (by Proposition 6.31) t h a t p(r) is increasing in r. However, also by P r o p o s i t i o n 6.31, p(r) < Po, where P0 is the resistance scale factor of t h e nested fractal o b t a i n e d just from r 1 < i < 6. So if we choose r7 > P0, t h e n as r7 > P0 ~ p(rT), we have an example of an affine nested fractal with a non-regular fixed point. From now on we take s to be a regular fixed point. (See [Kum3] for the general situation). W r i t e 7 = m a x / r i / p < 1. For x, y 9 F , set w(x,y) to be the longest word w such t h a t x, y 9 F ~ . Proposition

7.10.

(Sobolev inequality). Let f 9 T~'. Then if s

is a regu/ar

fixed point (7.8)

i f ( x ) _ f(y)]2 ~ c2r~(,,y)p-l~(z,y)ls

x,y 9 V (~176

Proof. Let x, y 9 Y(,'), let w = w(z,y) and let Iwl = m. We prove (7.8) by a s t a n d a r d kind of chaining argument, similar to those used in continuity results such as Kolmogorov's lemma. (But this argument is deterministic and easier). We m a y assume n > m. Let u 9 W," be an extension of w, such t h a t x 9 V(~ such a u certainly exists, as x 9 V~( ~ Write uk = ulk for m < k < n. Now choose a sequence zk, m < k < n such t h a t z," = x, and zk 9 V~(~ for k < m < n k 9 { m , . . . , n - 1} we have zk, zk+l 9 V(~ ). So

(7.9)

If(z,,)- f(z.~)l < ~

lf(zk+1) - f(zk)l

n--1

< Z le,=m

(e:~

I/"

1. For each

98

n--1 Tuk ^_k+m~l/2 k=m As s is a regular fixed point, 7 = m a x i ri/p < 1, so the final sum in (7.9) is b o u n d e d CO by ( ~ k = m 7 k - m ) U2 = c3 < co. Thus we have

If(x) - f(zm)l ~ < c l c : ~ p - " E ' ( f , f), a n d as a similar b o u n d holds for If(y) - f(zm)[ 2, this proves (7.8).

[]

We have not so far needed a measure on F . However, to define a Dirichlet form we need some L 2 space in which the domain of ~ is closed. Let # be a p r o b a b i l i t y measure on (F, B(F)) which charges every set of the form F ~ , w E Wn. L a t e r we will take # to be the Bernouilli measure P0 associated with a vector of weights 8 E (0, ~ ) M , b u t for now any measure satisfying the condition above will suffice. As # ( F ) = 1, C(F) C L 2 ( F , # ) . Set

= {f e C ( F ) :

e(f,f) Proposition

=

flv(~) 9 :D'}

e'(flv(~),flv(~)),

f 9 :V.

7.11. (~, :D) is a dosed symmetric form on LZ(F, p).

Proof. Note first t h a t the condition on p implies t h a t if f , g 9 l) then [[f - g [ [ 2 = 0 implies t h a t f = g. We need to prove t h a t V is complete in the norm ][f[[~, -e ( f , f ) + Ilfll]. So suppose (fn) is Cauchy in [[. lie,- Since ( f , ) is Cauchy in H" [Is, passing to a subsequence there exists f 9 L 2 ( F , # ) such t h a t f,~ ~ f #-a.e. F i x x0 9 F such t h a t fn(xo) --~ f(x). T h e n since fn - f m is continuous, (7.8) extends to an e s t i m a t e on the whole of F and so If,,(x) - f,~(x)[ _< l(f,, - f,,,)(x) - (f,, - f,,,)(x0)l + I(f,, - fm)(x0)l I12 ,-,:. <( C 2

C(.]'n -- fro,

fn

-- fro) I/2

"~ If.(x0)

-

fm(x0)[.

So (f~) is Cauchy in the uniform norm, and thus there exists f 9 C(F) such t h a t f~ --* f tmiformly. Let n _> 1. T h e n as s is a finite sum, s

=

lim s m--coo

_< lim s u p e ( f m , f m ) m---*oo

_< sup Ill-lie, < ~ . m

Hence E('~)(f, f ) is bounded, so f 9 l). Finally, by a similar calculation, for any N>I, E(N)(f~ -- f, fn -- f) (_ limoo E(f,~ -- fm, f,~ -- f.',). So E(f,~ - f, fn - f) --~ 0 as n ---* oo, and thus Ill - f.II~,

--+ o.

[]

To show t h a t (C, Z~) is a Dirichlet form, it remains to show t h a t 2) is dense in

L2(F, p). We do this by s t u d y i n g the harmonic extension of a function.

99 D e f i n i t i o n 7.12. Let f e C(V(')). Recall that C('~)(f,f) -~ inf{E("+l)(g,9) : giv(-) -- f}- Let H n + i f E C(V (~+D) be the (unique, as E (n+l)) is non-degenerate) function which attains the infimum. For x G V (~) set

H~f(x) = lim H , ~ H , ~ - I . . . H , ~ + l f ( X ) ; note that (as H ~ + l f -- f on V (~)) this limit is ultimately constant. 7.13. Let g be a regular fixed point. (a) H,~f has a continuous extension to a function H~f C 7) M C(F), which satisfies

Proposition

~(H,J, Hnf) = E(n)(f, f).

(b)

f, a c C(F) E(H~f,g) -- g('~)(f,g).

(7.10)

Proof. From the definition of H,~+I, ~(n+i)(Hn+lf, H ~ + l f ) -- s

Thus

E(m)(H,~f,H~f) = C(~)(f, f ) for any m, so that H ~ f E 7)~ and

E(Hr, f, Hnf) = C(n)(f,f),

f E C(V(")).

If w E Win, and x, y E V (~) M F,o then by Proposition 7.10 (7.11)

] H u f ( x ) - H , I ( v ) ] 2 <_ c2rwp-mE(n)(f, f).

Since r~p -'~ < Tin, (7.11) implies that O s c ( H ~ f , U (~) f) Fw) converges to 0 as Iwl = m -~ co. Thus H,~f has a continuous extension H~f, and H~f E 23 since (b) Note that, by polarization, we have C (n+l) (Hn+a f, Hn+l 9) --- E (n) (f, g). Since g ( n + l ) ( H n + l f , h) = 0 for any h such that hiv(. ) = 0, it follows that E(n+l)(H,~+lf, g ) = E(n)(f,9). Iterating, we obtain (7.10). Theorem

[]

7.14. (C, 7)) is an irreducible, regular, 1ocM Dirichlet form on L 2 (F, #).

Proof. Let f E C(F). Since for any n > 1, w C Wn we have i n f f _< Hnf(x) < sup f, F~ F~

x C F~,

it follows that H~f --* f uniformly. As Hnf E 73, we deduce that 7) is dense in C(F) in the uniform norm. Hence also 7) is dense in L2(F,#). As (4.5) is immediate, we deduce that 7) is a regular Dirichlet form. If s = 0 then E(~)(f,f) = 0 for each n. Since s is irreducible, fly(,) is constant for each n. As f is continuous, f is therefore constant. Thus E is irreducible.

100

To prove that E is local, let f, g be functions in 79 with disjoint closed supports,

S/, Sg say. If $(=)(f,g) # 0 then one of the terms in the sum (7.1) must be non-zero, so there exists wn 9 W,~, and points xn 9 S I N V(~ yn 9 Sg A V(w~ Passing to a subsequence, there exists z such that xn ---+z, Yn ---* z, and as therefore z 9 S I A Sg, this is a contradiction.

[]

By Theorem 4.8 there exists a continuous #-symmetric Hunt process (Xt, t > 0, F =, x 9 F ) associated with (E, 79) and L 2 (F, #). R e m a r k 7.15. Note that we have constructed a process X = X (t') for each Radon measure # on F. So, at first sight, the construction given here has built much more than the probabilistic construction outlined in Section 2. But this added generality is to a large extent an illusion: Theorem 4.17 implies that these processes can all be obtained from each other by time-change. On the other hand the regularity of ($, 79) was established without much pain, and here the advantage of the Dirichlet form approach can be seen: all the probabilistic approaches to the Markov property are quite cumbersome. The general probabilistic construction, such as given in ILl] for example, encounters another obstacle which the Dirichlet form construction avoids. As well as finding a decimation invariant set of transition probabilities, it also appears necessary (see e.g. [L1, Chapter VI ]) to find associated transition times. It is not clear to me why these estimates appear essential in probabilistic approaches, while they do not seem to be needed at all in the construction above. We collect together a number of properties of (E, ~). Proposition

7.16. (a) For each n > 0

(7.12)

E(/'g)=

E P%'~lE(f~162176162 wGW~

(b) Fo~, / 9 79, (7.13) (7.14)

(7.15)

If(x) -/(y)l 2

<

Clr~p-nE(f,f)

f f2d# <

c2$(f,f)+

is x,y 9 Fw, w 9

Wn

(f fd,)

I ( z ) 2 < 2 fj f2d# + 2ClE(f,f),

x C F.

Proof. (a) is immediate from Lemma 7.5, while (b) follows from Proposition 7.10 and the continuity of f. Taking n = 0 in (7.13) we deduce that

(f(x) -- f(y))2 < ev~(f,f),

f 9 79.

101

So as # ( F ) = 1,

/ / ~,E(s,S).(dx).(d~,)=e,~(S,S)

=2//2rig-2 (/)' f d#

,

proving (7.14). Since f ( x ) 2 <_ 2 f ( y ) 2 + 2[f(x) - f(y)]~ we have from (7.13) that

f(x) 2 =

/

< 2f

f(x)U#(dY)

f(y)'~(dy)+ 2~ f ~(f, 1)~(dy), []

which proves (7.15). We need to examine further the resistance metric introduced in Section 4. D e f i n i t i o n 7.17. Let R(x, x) = 0, and for x ~ y set

R(x, y ) - I : inf {g(f, f ) : f(x) = O, f(y) = 1, f 6 7)}. Note t h a t

f(y)l 2

R(x,y) = sup( If(x)yff,?~ (

(7.16)

: fCV,

f non constantS.

P r o p o s i t i o n 7.18. (a) I~ = # y then 0 < R(=, y) <_ cl < o~. (b) I f w e W,~ then

(7.17)

R(x,y) < Clrwp-",

x,y e Fw.

(c) For f ~ 79 (7.18)

If(=) - f(y)l 2 -< R(=, y)E(I, I).

((t) R is a metric on F, and the topology induced by R is equa/ to the originM topology on F. Proof. Let x, y be distinct points in P. As 73 is dense in C ( F ) , there exists f E 7) with f ( x ) >_ 1, f(y) <_ O. Since g is irreducible, s > 0, and so by (7.16) R(x, y) > 0. (7.17) is immediate from Proposition 7.16, proving (b). Taking n = 0, and w to he the e m p t y word in (7.17) we deduce R(x,y) _< cl for any x, y E F, completing the proof of (a). (c) is immediate from (7.16). (d) R is clearly symmetric. The triangle inequality for R is proved exactly as in Proposition 4.25, by considering the trace of s on the set {x, y, z}.

102

It remains to show that the topologies induced by R and d (the original metric on F ) are the same. Let R(xn, x) ---. O. If e > 0, there exists f E 7:) with f ( x ) = 1 and s u p p ( f ) C Bd(X,e). By (7.16) R(x,y) >_ s > 0 for any y e Bd(X,e) c. So xn E Bd(X, ~) for all sufficiently large n, and hence d(x,~, x) --* O. If d(x,~, x) ~ 0 then writing

Nm(x)=U{F~o:wEW,,,

,

xEF~,}

we have by L e m m a 5.12 that xn E N,,(x) for all sufficiently large n. However if 7 = maxir~/p < 1 we have by, (7.17), R(x,y) < c1~/m for y E Win(x). Thus

R ( x , , z) ~ O.

[]

R e m a r k 7.19. The resistance metric R on F is quite well adapted to the study of the diffusion X on F. Note however that R(x, y) is obtained by summing (in a certain sense) the resistance of all paths from z to y. So it is not surprising that R is not a geodesic metric. (Unless F is a tree). Also, R is not a geometrically natural metric on F. For example, on the Sierpinski gasket, since r~ = 1, and p = 5/3, we have that if x, y are neighbours in (V (~), En) then

n(x, y) • (3/5?. However, for general p.c.f.s.s, sets it is not easy to define a metric which is well-adapted to the self-similax structure. (And, if one imposes strict conditions of exact self-similarity, it is not possible in general - see the examples in [Ki6]). So, for these general sets the resistance metric plays an extremely useful role. The next section contains some additional results on R. It is also worth remarking that the balls BR(x, r) = {y : R(x, y) < r} need not in general be connected. For example, consider the wire network corresponding to the graph consisting of two points x, y, connected by n wires each of conductivity 1. Let z be the midpoint of one of the wires. Then R(x, y) = I/n, while the eonductivities in the network { x , y , z } axe given by C(x,z) = C(z,y) = 2, C(x,y) = n - 1. So, after some easy calculations,

R(x,z)-

n+l 4n-1

1 >~"

Soifn=4, R ( x , y ) = x while R(x, z) = ~. Hence i f ~1 < r < l t h e b a l l B R ( x , r ) is not connected. (In fact, y is an isolated point of BR(x, 88 = { x ' : d(x,x') < 88 (Are the balls BR(x, r) in the Sierpinski gasket connected? I do not know). Recall the notation $~(f, g) = s of X. Since by (4.8) we have

g ) + a ( ] , g). Let (U~, a > 0) be the resolvent

E~(vof, g) = (f,g), if U~ has a density u~(x,y) with respect to #, then a formal calculation suggests that E~(u~(~,.),9)

= E~(u~=,9)

= (~=,g) = g(~).

We can use this to obtain the existence and continuity of the resolvent density u~. (See [FOT, p. 73]).

103

T h e o r e m 7.20.

(a) For each x 9 F there exists u s 9

(7.19)

for all

Ea(u~, f ) = f ( x )

suchthat

f 9

(b) Writing u~,(x, y) = u~(y), we have

u~(=,y) = ,~.(y,=)

:or a11 =,y e F.

(c) us(.,-) is continuous on F • F and in particular lu~(=,y) - ~ ( = , y ' ) l 2 < R(y,y')~o(=,=).

(7.20)

(d) ~.(=,y) is the resolvent density ~or X: ~or f e C ( f ) , E~

:

e-~tf(Xt)dt = U~f(z) =

/

u~(z,y)f(y)~(dy).

(e) There exists c2(a) such that (7.21)

~ ( ~ , y ) <_ c2(~),

=,y 9 F.

Proof. (a) The existence of u~ is given by a standard argument with reproducing kernel Hilbert spaces. Let x 9 F , and for f 9 7? let r = f ( x ) . Then by (7.15)

lr

= ]/(x)l2 < 211fll2 +

2ciE(f,f) < c~s

f),

where co = 2 m a x ( c l , a - 1 ) . Thus r is a bounded linear functional on the Hilbert space (7), II lifo), and so there exists a u~ 9 7) such that r

----,~',~(u~,f ) -= f(x),

f 9 7).

(b) This is immediate from (a) and the symmetry of E:

~(=)

E . ( u .Z, ~ . )y

= E.(~,~X)

= ~(y).

(c) As u~ 9 7), u~,(x,x) < oo. Since E(u~,u~) = u~,(x,x) < c~, the estimate (7.20) follows from (7.18). It follows immediately that u is jointly continuous on F • F. (d) This follows from (7.19) and linearity. For a measure u on F set

: 9 C(F) As us is uniformly continuous on F • F, we can choose u~---~# so that V ~ . f --~ V f uniformly, and vn are atomic with a finite number of atoms. Write V,~ = Vv., V = Vv. Since by (7.19) E , ( V J , g) = ~ - ~ ( { z } ) / ( z ) E o ( u : , g ) x

104

we have

Ea(Vnf - Vmf , Vnf - Vmf) =

Thus g~(V,J

- V~LV.I

- Vmf)

--, 0 as m, n ~

~,

and so, as g is closed, we .,J o

So

deduce that

$ a ( Y f, g) : Ea(U~f ,g) for all g, and hence Y f = V ~ f . (e) As R(y, y') < Cl for y, y' C F, we have from (7.20) that

(7.22)

u~(x,y) > ~ ( x , x ) - (c~,~o(x, ~)) 1/~.

Since I u,(:e, y)#(dy) = o~-1, integrating (7.22) we obtain

~o(x,x) < (cl~o(x,~)) 's~ + ~-~, and this implies that u,~(x, x) < c2(a), where c(a) depends only on a and el. Using (7.20) again we obtain (7.21). [] T h e o r e m 7.21. (a) For each x C F, x is regular for {x}. (b ) X has a jointly continuous local time ( L ~ , x e F, t > O) such that for all bounded

measurable f l(x.)es

=

~)Lt.(d~),

a.s.

Pro@ These follow from the estimates on the resolvent density u~. As u~ is bounded and continuous, we have that x is regular for {x}. Thus X has jointly measurable local times (L~,x E F , t > 0). Since X is a symmetric Markov process, by Theorem 8.6 of [MR], L~ is jointly continuous in (x, t) if and only if the Gaussian process Yx, x E F with covariance function given by EYaYb = ul(a,b)~

a,b E F

is continuous. Necessary and sufficient conditions for continuity of Gaussian processes are known (see [Tall), but here a simple sufficient condition in terms of metric entropy is enough. We have

E(Ya - yb)2 = ul(a,a) - 2ul(a,b) + ul(b,b) < clR(a,b) U2. Set r(a, b) = R(a, b) 1/2 : r is a metric on F. Write N~(e) for the smallest number of sets of r-diameter e needed to cover F. By (7.17) we have R(a, b) < c~/'~ if a, b e Fw and w E W~. So N~(ct~/n/2) < #W,~ = M ~, and it follows that N~(c) < e2c - ~ ,

where fl = 2 log M~ log ~-1. So

fo + (lOg~V~(~))~/~a~ < ~,

105

and thus by [Du, Thm. 2.1] Y is continuous.

[]

We can use the continuity of the local time of X to give a simple proof that X is the limit of a natural sequence of approximating continuous time Markov chains. For simplicity we take # to be a Bernouilli measure of the form It -- It0, where 0i > 0. Let It,, be the measure on V ('0 given in (5.21). Set

A'~ = ./~ L~It,~(dx), "r: = inf{s : A~ > t}, x;' = x v.

(a) (X?,t >_ 0 , I ? ' , x 6 V (~)) is the symmetric Markov process associated with $ ('~) and L2(V ("), It,). Theorem

7.22.

(b) X~ --+ Xt a.s. and uniformly on compacts. Proof. (a) By Theorem 7.21(a) points are non-polar for X. So by the trace theorem (Theorem 4.17) X n is the Markov process associated with the trace of $ on L 2 ( V ( ' 0 , I t , ) . But for f 6 :D, by the definition of s Tr

(elv ( - ) ) ( f , :) =

e(n)

(flv(-),fl,:~-)).

(b) As F is compact, for each T > 0, (LT, 0 < t < T, x 6 F ) is uniformly continuous. So, using (5.22), if T2 < T1 < T then A t ~ t uniformly in [0,T1], and so T~ --+ t uniformly on [0, T2]. As X is continuous, X~ ~ X uniformly in [0,T2]. [] R e m a r k 7.23. As in Example 4.21, it is easy to describe the generator Ln of X ~. Let a('O(x, y), x, y G V('*) be the conductivity matrix such that 1

E(")(f,:) = ~ E a(n)(~,V)(f(~)- f(V)) ~ z,y

Then by (7.1) we have (7.23)

a('q(x,y) ---

E

l(,,yev!0))p n r~--1A ( r

--1 ( x ) , r w --1 (y)) ,

w6W~

where A is such that E (~ -- CA, and A(x,y) = A=y. Then for f E L2(V('q,#,~), (7.24)

Lnf(x) -- # n ( { x } ) - '

E

a(n)(x'Y)( f ( y ) - f(x)).

yEV(~)

Of course T h e o r e m 7.22 implies that if (Y'~) is a sequence of continuous time Markov chains, with generators given by (7.24), then Y'~ TM ~X in D([0, oo), F).

106 8. Transition D e n s i t y Estimates. In this section we fix a connected p.c.f.s.s, set (F, ( r a resistance vector ri, and a non-degenerate regular fixed point ~A of the renormalization m a p A. Let P = P0 be a measure on F , and let X -- ( X t , t > O, IP~,z C F ) be the diffusion process c o n s t r u c t e d in Section 7. We investigate the transition densities of the process X : initially in fairly great generality, but as the section proceeds, I will restrict t h e class of fractals. We begin by fixing the vector 0 which assigns mass to the 1-complexes r in a fashion which relates # e ( r with ri. Let/3i = r l p - l : by (7.8) we have

fli < l,

(8.1)

l < i < M.

Let a > 0 be t h e unique positive real such t h a t M

(8.2)

ZZ?

-- 1.

i=I

Set

(8.3)

Oi =

fiT,

1<

i < M,

and let # = P0 be the associated Bernouilli t y p e measure on F . Write fl+ = maxi/3i, /3_ = m i n i f l i : we have 0 < fl_ < fl~
w.W=(w.v,v 9149 Lemma

1
8.1. (a) F o r A > 0 let

Wx = {w 9 W~ :/3~ _< ~,/~,~ > ~}. T h e n the sets {w 9 W , w 9 Wx} are disjoint, a n d

U wEWx

w.W=W.

Iwl}.

107

(5) Fo~ S 9 L ~(F, ~),

iSd#= ~ 8~Sf~d# wEWx

e(s, s) -- Z / 3 2 E ( f w , s~) wEWx

Proof. (a) Suppose w, w ~ E W~ and v E ( w . W) N (w I. W). T h e n there exist u, u t E W such t h a t v = w - u = w ~. u ~. So one of w, w ~ (say w) is an ancestor of the other. But if/~w _< A, fl~w > A t h e n as/3i < 1 we can only h a v e / 3 ~ , > A if w ~ = w. So if w # w ~, w 9W a n d w I 9W are disjoint. Let v E W. Then/3,1~ ~ 0 as n --~ cx~. So there exists m such that vtm e W~, a n d t h e n v e (vtm) 9W, completing the proof of (a). (b) This follows in a straightforward fashion from the decompositions given in (7.12) a n d L e m m a 5.28. [] Note t h a t / 3 _ > 0 and that (8.4)

f l A < / 3 ~ < A,

(/3_)~A ~ _ ~

< As,

w9

D e f i n i t i o n 8.2. The spectral dimension of F is defined by

ds = d~(F, EA) = 2(~/(1 + or). Theorem

(8Z)

8.3. For f 9 79,

llSll~?-~/~.<_c, (E(s,s)+ IlStl~)ItSll~/~:.

Proof. It is sufficient to consider the case f non-negative, so let f E 79 with jr > 0. Let 0 < A < 1: by L e m m a 8.1, (7.14) and (8.4) we have (8.6)

IlSll~= ~ 8,~S Ssdp wEW,~

< ~ 8,o elg(f~,, f~,) +

f~ d#

(i

w

_< eaA~+I E / 3 ~

(fw,f~,)+c2a

< c3a'~+~g($, f ) + c4,~-~

o~, J f,~ d~

fwd#

108 The final line of (8.6) is minimized if we take :~'+~ ~llfll~ then ~ < 1 and so we obtain from (8.6) that

(8.7)

llfl12~. _ cs

--

c~ IIflI~/C(f, f). If E(.f, .f)

>_

f)<~l(2,~+,)(llfll~)(<,+:)I(~<'+~),

which implies t h a t t h a t

(8.8)

11.:112~ +4/'~. < cE(/,f)ll:ll~ ia.

if E(:,:)

if

E(y,f) _> r

_< c511:112~then by (7.14)

Ilfll~ -< .: (c(f, f)

+

ilfll:,) <_ cll:ll~,

and so

2

41d.

IIf1122+*/<'. -< cll:ll211flll

(8.9)

if e(:,:)

_< c511:112~. []

Combining (8.8) and (8.9) we obtain (8.5). F r o m the results in Section 4 we then deduce Theorem

8.4. X has a transition density p(t, x, y) which satisfies

p(t, x, y) < Cl t-d~

(8.10)

(8.11)

0 < t < 1,

I p ( t , x , y ) - - p ( t , x , y / ) l 2 ~ c2t-l-d'/2.R(y,y/),

x, y E F, 0 ~ t ~ 1~

:c,y,y I E -P.

Proof. By P r o p o s i t i o n 4.14 X has a jointly measurable transition density, a n d by Corollary 4.15 we have for x, y E F , 0 < t < 1,

p ( t , x , y ) < ct-d'/2ea < clt -d~ By (4.17) the function qt,= = p(t,x,-) satisfies E(qt,=,qt,=) _< ct - 1 - a ~ qt,= E T~ a n d is continuous. Further, by Proposition 7.18

Ip(t,x,y) - p ( t , x , y ' ) l ~ <_ cR(y,y')t -d'/~-~,

and so

x , y , y ' 9 F.

Thus p(t,-, .) is j o i n t l y H61der continuous in the metric R on F .

[]

R e m a r k s 8.5. 1. As a > 0, we have 0 < d~ = 2a(1 + a) -1 < 2. 2. T h e e s t i m a t e (8.10) is good if t E (0, 1] and x close to y. It is poor if t is small c o m p a r e d with R(x, y), a n d in this case we can o b t a i n a b e t t e r estimate by chaining, as was done for fractional diffusions in Section 3. For this we need some additional p r o p e r t i e s of the resistance metric. Lemma

8.6. If v, w E Wx and v ~ w then F~ M F~ = V~(~ M V (~

Proof. This follows easily from the corresponding p r o p e r t y for V4n. Let v, w E W~, w i t h I v ] = m < _ ]w I = n , v ~ w . LetxEF.MF~. Setw'=wim;thenasFw CFw,, x E F v f q F , , , and so by L e m m a 5.17(a) x E V( ~ 1 7 6 Further, as x E F , there exists v t E W,, such t h a t v'lm = v, and x C F , , . Then x E F . , M F ~ = l/-(~ . , , :q V~(~ .

So ~ c v~ ~ n v~ ~

[]

109

D e f i n i t i o n 8.7. Set

Let G~ = (V~(~

LI

~

v o)

be the graph with vertex set V(0), and edge set EA such that

{x,y} is an edge if and only if x, y 9 V(~ for some w 9 W~. For A C F set N A ( A ) -- U { F ~ : w 9 W~, F~ M A ~ O}, /V~(x) = Y~ (Y~({x})). As we will see, N~(x) is a neighbourhood of x with a structure which is well adapted to the geometry of F in the metric R. We write N~(y) = N~ ({y}). Lemma

8.8. (a) I f x, y 9 V (~ and x ~ y then R ( x , y ) >_ c1)~.

(b) g { = , v } 9 E~ then n(=,V) < c2~. Proof. (b) is immediate from the definition of WA and Proposition 7.18(b). For (a), note first that if x 9 F then by Proposition 5.21 x can belong to at most M1 = M # ( P ) n-complexes, for any n. So there are at most M1 distinct elements w 9 WA such that x 9 F~. As V (~ is a finite set, and E(A~ is non-degenerate, there exists c3, c4 > 0 such that, (8.12)

c4 >_ R ( x , V (~ - {x}) > e3,

x 9 V (~

(Recall that this resistance is, by the construction of s the same in (F, E) as in (V(~176 Now fix x E V(~ I f w 9 W~, and x 9 V(~ let x' = r and g~ be the function on F such that g~(x') = 1, g~(y) = O, g 9 Y (~ - {x'}, and :

R

v(~

-

_>c ,

Define g " on F~, by g " -- g,, o r 1, and extend g" to F by setting g " = 0 on F - F ~ , . Now l e t g ~ = 0 i f x ~ V (~149 g =

E 'gv.

vEWx

Then g(x) = 1, g(y) = 0 if y 9 V(~

y ~ x, and

wEW~

--- E / S w

--1

l(~cF~)E(gw,gw) <_ C5A-1/1 .

110

Hence if y # x, y E V(~

we have

n(x,y) -1 <_ E(g,a) <_ A-Imlc[ 1, so that R(x,y) > c6A.

[]

R e m a r k . For x E V(~ the function g constructed above is zero outside Nx({x}). So we also have (8.13)

R(x,y) >_c6A,

x E V(~

y E Nn({x}) ~.

P r o p o s i t i o n 8.9. There exist constants c~ such that/'or x E F, A > 0, (8.14)

(8.15)

(8.16) (8.17)

BR(X,clA) C Ar~(x) C BR(X, c2A), c3~ ~ < ~ ( B R ( x , A ) ) < c,A ~

chA < n(~, ~(x)~) < ~ , c~A < n(~,B.(~,A) ~) < csA.

Proof. Let x C F. I f y C Nn({x}) then by (7.17), R(x,y) <_ cA. So i f z C Nn(x), since there exists y E Nn({x}) with z E Nn({y}), R(x,z) < c'A, proving the right hand inclusion in (8.14). If x E V~(~ then by (8.13), if c9 = c8.7.6, BR(x, egA) C NA(x). Now let x ~ V~(~ so that there exists a unique w C Wa with x E F,o. For each y E V(~ let fv(') be the function constructed in Lemma 8.8, which satisfies fy(y) = 1, fy = 0 outside N),(y), fy(z) -- 0 for each z e V(~ - {y}, and E(]y, fy) < cl0 A-1. Let f = ~ y fy: then ] ( y ) = 1 for each y e V(~ g=lF~

So if

+lEaf,

E(g,g) < ~ ( L f ) < #(Y~(~ -1 _< ~ 1 ~ - 1 . As g(=) = 1, and g(~) = 0 for ~ ~t N~(x), we have for z r N~(x) that R(x,z) -1 <_$(g,g) <_cnA -1. So BR(x, cllA) C N~,(x). This proves (8.14), and also that R(x,~rx(x) ~) > c51A. The remaining assertions now follow fairly easily. For w E W~ we have cl2A ~ < #(F~,) < c13A~. As N~(x) contains at least one A-complex, and at most M 2 # ( P ) 2 A-complexes, we have ,(N~(~)) • ~, and using (8.14) this implies (8.15). If A C B then it is clear that R(x,A) > R(x,B). So (provided)~ is small enough) if x ~ F we can find a chain x, yl, y2, y3 where y~ ~ V(~ {y~,y~+l} is an edge in E~, Y3 ~ N~(x), and x and y are in the same A-complex. Then R(x, y3) <_cA by (7.17), and so, using Lemma 8.8(b) we have R(x,y3) <_c'A. Thus R(x, Ar~,(x)~) < R(x, y3) <_c'A proving the right hand side of (8.16): the left hand side was proved above. (8.17) follows easily from (8.14) and (8.16). []

111 C o r o l l a r y 8.10. In the metric R, the Hausdorff dimension o f F is a, and further

0 < n~(F) <

o0.

Proof. This is immediate from Corollary 2.8 and (8.15).

[]

P r o p o s i t i o n 8.11. For x E F , r > 0 set ~-(x, r) = TBR(x:)o. Then

(8.18)

e : ~+x < Ex~(z,r) _< c : ~+1,

z 9 F,

r > 0.

Proof. Let B = Ba(x, r). T h e n by Theorem 4.25 and the estimates (8.15) and (8.17) Z~T(x,r) <_ # ( B ) R ( x , B c) <_ car ~+1, which proves the upper bound in (8.18). Let ( X f f , t _> 0) be the process X killed at T = TBo, and let g(x,y) be the Greens' function for X B. In view of Theorem 7.19, we can write

g(x,y) = E~L~,

x,y 9 F.

Then if f(y) = g(x, y)/g(x, x), f 9 7) and by the reproducing kernel property of g we have E(f, f) = g(x, x)-2E (g(x, .), g(x, .)) = g(x, x) -1 , and as in Theorem 4.25 g(x, x) = R(x, B ~) >_ car. By (7.18) If(x) - f(y)l 2 < R(z,y)C(f,f) <_ R(z,y)(c,r)-~ < if R(x, y) <_ 88

1

Thus f(y) >_ 89on BR(x, 88 r), and hence E ~ = f o g(x, y),(dv)

>

(B.

88

>

proving (8.18).

[]

We have a spectral decomposition of p(t, x, y). Write (f, g) = fF fgd#. T h e o r e m 8.12. There exist [unctions qo~ 9 7), Ai >_ O, i >_ O, such that (qoi, ~i) = 1,

O-- )~o < A1 <_...,and $(Wi, f) = Ai(~oi, f),

f 9 7).

The transition density p(t, x, y) of X satisfies (8.19)

p(t, x, y) = ~

e-X't~,(x)~,(y),

i=0

where the sum in (8.19) converges uniformly and absolutely. So p is jointly continuous in (t, x, y).

Proof. This follows from Mercer's Theorem, as in [DaS]. Note that ~o0 = 1 as s is irreducible and # ( F ) = 1. [] The following is an immediate consequence of (8.19)

112

C o r o l l a r y 8.13.

(a) F o r x , y E F ,

t >0,

p(t, ~, y)2 < p(t, x, ~)p(t, y, y). (b) For each x, y E F lim p(t, x, y) -- 1. t -"'* O 0

Lemma (8.20)

8.14.

p ( t , x , y ) > co t-d~

0 < t < 1,

R ( x , y ) < clt UO+~).

Proof. We begin with the case x --- y. From Proposition 8.11 and L e m m a 3.16 we deduce that there exists c2 > 0 such that ~ = ( T ( x , r ) _< t) < (1 -- 2C2) + c3tr - ~ Choose c4 > 0 such that c3tro a-1 -- c 2 if ro = c4tU(l+~). Then ~=(Xt e B n ( x , ro)) _> ~ ( T ( x , r0) _< t) _> C~. SO using Cauchy-Schwarz and the symmetry of p, and writing B = BR(X, ro), 0 < c~<_ ( / B P ( t ' x ' Y ) # ( d Y ) 2 )

<<

f.( .... ).(dy) .(B)) p(2t, ,x)

< cst~/(l+~)p(2t, x, x). Rep