Diffusions on Fractals

DIFFUSIONS ON FRACTALS

Martin T Barlow

University of British Columbia

Vancouver BC

Canada VT Z

barlowmathub cca

CONTENTS

Intro duction

The Sierpinski Gasket

Fractional Diusions

Dirichlet Forms Markov Pro cesses and Electrical Networks

Geometry of Regular Finitely Ramied Fractals

Renormalization on Finitely Ramied Fractals

Diusions on pcfss Sets

Transition Density Estimates

References

Intro duction

The notes are based on lectures given in St Flour in and cover in greater

detail most of the course given there

The word fractal was coined by Mandelbrot Man in the s but of course

sets of this typ e have b een familiar for a long time their early history b eing as a

collection of pathological examples in analysis There is no generally agreed exact

denition of the word fractal and attempts so far to give a precise denition have

b een unsatisfactory leading to classes of sets which are either to o large or to o small

or b oth This ambiguity is not a problem for this course a more precise title would

b e Diusions on some classes of regular selfsimilar sets

Initial interest in the prop erties of pro cesses on fractals came from mathematical

physicists working in the theory of disordered media Certain media can b e mo delled

by p ercolation clusters at criticality which are exp ected to exhibit fractallikeprop

erties Following the initial pap ers AO RT GAMGAM a very substantial

physics literature has develop ed see HBA for a survey and bibliography

Let G b e an innite subgraph of Z A simple random walk SRW X n

on G is just the Markovchain whichmoves from x G with equal probabilityto

each of the neighb ours of xWrite p x y P X y for the nstep transition

n n

probabilities If G is the whole of Z then E X n with many familiar con

sequences the pro cess moves roughly a distance of order n in time n and the

probabilitylaw p x puts most of its mass on a ball of radius c n

n d

If G is not the whole of Z then the movement of the pro cess is on the average

restricted by the removal of parts of the space Probabilistically this is not obvious

but see DS for an elegant argument using electrical resistance that the removal

of part of the state space can only make the pro cess X more recurrent So it is

not unreasonable to exp ect that for certain graphs G one may nd that the pro cess

X is suciently restricted that for some

E X x n

Here and elsewhere I use to mean b ounded ab ove and b elowby p ositive con

stants so that means that there exist constants c c suchthatc n

E X x c n In AO and RT it was shown that if G is the Sierpinski

gasket or more precisely an innite graph based on the Sierpinski gasket see Fig

then holds with log log

Figure The graphical Sierpinski gasket

Physicists call b ehaviour of this kind by a random walk or a diusion they are

not very interested in the distinction subdiusive the pro cess moves on average

slower than a standard random walk on Z Kesten Ke proved that the SRWon

the incipient innite cluster C a p ercolation cluster at p p but conditioned to

b e innite is sub diusive The large scale structure of C is given by taking one

innite path the backb one together with a collection of dangling ends some of

whicharevery large Kesten attributes the sub diusive b ehaviour of SRWonC

to the fact that the pro cess X sp ends a substantial amount of time in the dangling

ends

However a graph such as the Sierpinski gasket SG has no dangling ends and

one is forced to search for a dierent explanation for the sub diusivity This can

b e found in terms of the existence of obstacles at all length scales Whilst this

holds for the graphical Sierpinski gasket the notation will b e slightly simpler if we

consider another example the graphical Sierpinski carp et GSC Figure

Figure The graphical Sierpinski carp et

For This set can b e dened precisely in the following fashion Let H Z

x n m H write nm in ternary so n n where n f gand

i i

n for all but nitely many iSet

J fm n n and m g

k k k

so that J consists of a union of disjoint squares of side the square in J closest

k k

k k k k

to the origin is f gf gNowset

H H J H H

n k n

Figure The set H

n n

Note that H H so that the dierence b etween H and H

n n

will only b e detected byaSRW after it has moved a distance of from the origin

NowletX be a SRWonH started at the origin and let X be a SRWonH

The pro cess X is just SRWonZ and so wehave

E X n

The pro cess X is a random walk on a the intersection of a translation invariant

subset of Z with Z So we exp ect homogenization the pro cesses n X

t should converge weakly to a constantmultiple of Brownian motion in R So

a nandwewould exp ect that a since for large n we should have E X

the obstacles will on average tend to imp ede the motion of the pro cess

weshouldhave Similar considerations suggest that writing tE X

t a t as t

n n

However for small t wewould exp ect that and should b e approximately

n n

equal since the pro cess will not havemoved far enough to detect the dierence

between H and H More preciselyift is such that t then

n n n n n n

and should b e approximately equal on t So wemay guess that the

n n

behaviour of the family of functions t should b e roughly as follows

tb a t t t t

n n n n n

s s s t

n n n

If we add the guess that a for some then solving the equations ab ove

we deduce that

n n

t b

n n

So if t E X then as t lim twe deduce that is close to a

t n n

piecewise linear function and that

t t

where Thus the random walk X on the graph H should satisfy for

some

The argumentgiven here is not of course rigorous but do es actually

hold for the set H see BB BB See also Jo for the case of the graphical

Sierpinski gasket The pro ofs however run along rather dierent lines than the

heuristic argumentsketched ab ove

Given b ehaviour of this typ e it is natural to ask if the random walk X on H

has a scaling limit More precisely do es there exist a sequence of constants such

that the pro cesses

X t

converge weakly to a nondegenerate limit as n For the graphical Sierpinski

carp et the convergence is not known though there exist such that the family

is tight However for the graphical Sierpinski gasket the answer is yes

Thus for certain very regular fractal sets F R we are able to dene a limiting

x x

diusion pro cess X X t P x F whereP is for each x F a probability

measure on f C F xg Writing T f xE f X for

t t

the semigroup of X we can dene a dierential op erator L dened on a class of

functions D L C F In many cases it is reasonable to call L the Laplacian

F F

on F

From the pro cess X one is able to obtain information ab out the solutions to

the Laplace and heat equations asso ciated with L the heat equation for example

taking the form

L u

uxu x

where u ut x x F t The wave equation is rather harder since it is not

very susceptible to probabilistic analysis See however KZ for work on the wave equation on a some manifolds with a large scale fractal structure

The mathematical literature on diusions on fractals and their asso ciated in

nitesimal generators can b e divided into broadly three parts

Diusions on nitely ramied fractals

Diusions on generalized Sierpinski carp ets a family of innitely ramied frac

tals

Sp ectral prop erties of the Laplacian L

These notes only deal with the rst of these topics On the whole innitely

ramied fractals are signicantly harder than nitely ramied ones and sometimes

require a very dierentapproach See Bas for a recent survey

These notes also contain very little on sp ectral questions For nitely ramied

fractals a direct approach see for example FS ShSh KL is simpler and

gives more precise information than the heat kernel metho d based on estimating

pt x xdx e

In this course Section intro duces the simplest case the Sierpinski gasket In

Section I dene a class of wellb ehaved diusions on metric spaces Fractional

Diusions which is wide enough to include many of the pro cesses discussed in

this course It is p ossible to develop their prop erties in a fairly general fashion

without using much of the sp ecial structure of the state space Section contains

a brief intro duction to the theory of Dirichlet forms and also its connection with

electrical resistances The remaining chapters to give the construction and

some prop erties of diusions on a class of nitely ramied regular fractals In this

Ihave largely followed the analytic Japanese approach develop ed by Kusuoka

Kigami Fukushima and others Many things can now b e done more simply than in

the early probabilistic work but there is loss as well as gain in added generalityand

it is worth p ointing out that the early pap ers on the Sierpinski gasket Kus Go

BP contain a wealth of interesting direct calculations which are not repro duced

in these notes Any reader who is surprised by the abrupt end of these notes in

Section should recall that some at least of the prop erties of these pro cesses have

already b een obtained in Section

c denotes a p ositive real constant whose value is xed within each Lemma

Theorem etc Occasionally it will b e necessary to use notation suchasc this

is simply the constant c in Denition c c c denote p ositive real constants

whose values maychange on each app earance B x r denotes the op en ball with

centre x and radius r and if X is a pro cess on a metric space F then

T infftX Ag

A t

T infftX y g

y t

x r infft X B x r g

Ihave included in the references most of the mathematical pap ers in this area

known to me and so they contain many pap ers not mentioned in the text Iam

grateful to Gerard Ben Arous for a number of interesting conversations on the

physical conditions under which sub diusivebehaviour might arise to Ben Hambly

for checking the nal manuscript and to Ann Artuso and Liz Rowley for their

typing

Acknow ledgements This researchis supp orted by a NSERC Canada research

grant bya grant from the Killam Foundation and by a EPSRC UK Visiting

Fellowship

The Sierpinski Gasket

This is the simplest nontrivial connected symmetric fractal The set was rst

dened by Sierpinski Sie as an example of a pathological curvethe name Sier

pinski gasket is due to Mandelbrot Man p

Let G f g fa a a g b e the vertices of the unit

triangle in R andletH uG H b e the closed convex hull of G The con

struction of the Sierpinski gasket SG for short G is by the following Cantortyp e

subtraction pro cedure Let b b b b e the midp oints of the sides of G and

let A be the interior of the triangle with vertices fb b b g Let H H Aso

that H consists of closed upward facing triangles eachofside Now rep eat

the op eration on each of these triangles to obtain a set H consisting of upward

facing triangles eachofside

Figure The sets H and H

Continuing in this fashion we obtain a decreasing sequence of closed nonempty

sets H and set

G H

Figure The set H

It is easy to see that G is connected just note that H H for all m n

n m

so that no p oint on the edge of a triangle is ever removed Since jH j jH j

we clearly have that jGj

We b egin by exploring some geometrical prop erties of G Call an ntriangle a

n n

set of the form G B where B is one of the triangles of side which make

up H Let b e Leb esgue measure restricted to H and normalized so that

n n n

H that is

n n

dx x dx

n H

Let wlim this is the natural at measure on G Note that

G n G

is the unique measure on G which assigns mass to each ntriangle Set

d log log

Lemma For x G r

d d

f f

B x r r r

n n

Proof The result is clear if r If r cho ose n so that r

wehave n Since B x r can intersect at most ntriangles it follows that

n n

B x r

n d d

f f

As eachn triangle has diameter B x r must contain at least one

n triangle and therefore

n n d d

f f

B x r r

Of course the constants in are not imp ortantwhat is signican t

is that the mass of balls in G growasr Using terminology from the geometry

of manifolds wecansay that G has volume growth given by r

Detour on Dimension

Let F b e a metric space There are a numb er of dierent denitions of

dimension for F and subsets of F here I just mention a few The simplest of these

is boxcounting dimension For A F let N A bethesmallestnumber

of balls B x required to cover A Then

log N A

dim A lim sup

log

Toseehow this b ehaves consider some examples WetakeF tobeR with the

Euclidean metric

d d d

Examples Let A R Then N A and it is easy to verify

that

log N

lim d

log

The Sierpinski gasket G Since G H andH is covered by triangles

n n

of side wehave after some calculations similar to those in Lemma that

log log

N G r r So

log

dim G

log

Let A Q Then N A sodim A On the other hand

dim fpg forany p A

We see that b oxcounting gives reasonable answers in the rst two cases but

a less useful numb er in the third A more delicate but more useful denition is

obtained if we allow the sizes of the covering balls to varyThisgives us Hausdor

dimension I will only sketch some prop erties of this here for more detail see for

example the b o oks byFalconer Fa Fa

Let h R R b e continuous increasing with h For U F write

diamU supfx y x y U g for the diameter of U For let

n o

H A inf h dU A U diamU

i i i

i i

Clearly H A is decreasing in Nowlet

h h

A H AlimH

we call H Hausdor hmeasure LetB F b e the Borel eld of F

Lemma H is a measure on F B F

For a pro of see Fa Chapter

We will b e concerned only with the case hxx we then write H for H

Note that H A is decreasingin fact it is not hard to see that H Aiseither

or for all but at most one

Denition The Hausdor dimension of A is dened by

dim Ainff H Ag supf H Ag

Lemma dim A dim A

H BC

Proof Let dim A Then as A can b e covered by N A sets of diameter

wehave H A N A whenever Cho ose so that dim A

then implies that for all sucien tly small N A So

H A and thus H A which implies that dim A

Consider the set A Q and let A fp p g b e an enumeration of

ALet and U b e an op en internal of length containing p ThenU

i i i

and thus H A So dim A covers A so that H A

We see therefore that dim can b e strictly smaller than dim and that in this

H BC

case at least dim gives a more satisfactory measure of the size of A

For the other two examples considered ab ove Lemma gives the upp er b ounds

dim d dim G log log In b oth cases equality holds but a direct

H H

pro of of this which is p ossible encounters the diculty that to obtain a lower b ound

on H Awe need to consider all p ossible covers of A by sets of diameter less than

Itismuch easier to use a kind of dual approach using measures

Theorem Let b e a measure on A such that A and there exist c

r suchthat

B x r c r x A r r

A and dim A Then H A c

ProofLetU be a covering of A by sets of diameter less than where r If

x U thenU B x diam U so that U c diam U So

i i i i i i i

X X

diam U c U c A

i i

Therefore H A c A and it follows immediately that H A and

dim A

Corollary dim Glog log

Proof By Lemma satises with d Soby Theorem dim G

G f H

d the other b ound has already b een pro ved

Very frequently when we wish to compute the dimension of a set it is fairly

easy to nd directly a nearoptimal covering and so obtain an upp er b ound on

dim directly We can then use Theorem to obtain a lower b ound However

we can also use measures to derive upp er b ounds on dim H

Theorem Let b e a nite measure on A suchthat B x r c r for all

x A r r ThenH A anddim A

ProofSeeFa p

In particular wemay note

Corollary If is a measure on A with A and

c r B x r c r x A r r

then H A and dim A

RemarksIfA is a k dimensional subspace of R then dim Adim A

H BC

Unlikedim dim is stable under countable unions thus

BC H

dim A supdim A

H i H i

In Tri Tricot dened packing dimension dim which is the largest rea

sonable denition of dimension for a set One has dim A dim Astrict

P H

inequality can hold The hyp otheses of Corollary also imply that dim A

See Fa p

The sets we consider in these notes will b e quite regular and will very often

satisfy that is they will b e dimensional in every reasonable sense

Questions concerning Hausdor measure are frequently much more delicate than

those relating just to dimension However the fractals considered in this notes will

all b e suciently regular so that there is a direct construction of the Hausdor mea

sure For example the measure on the Sierpinski gasket is a constantmultiple

of the Hausdor x measure on G

We note here howdim changes under a change of metric

Theorem Let b e metrics on F and write H dim for the Hausdor

measure and dimension with resp ect to i

a If x y x y for all x y A with x y then dim A

dim A

b If x y x y for some then

dim A dim A

H H

Proof Write d U for the diameter of U If U isa cover of A by sets with

j j i

U then

X X

d U d U

i i

A H A and dim A dim A so that H A H A Then H

H H

proving a

b If U is anycover of A by sets of small diameter wehave

X X

d U d U

i i

Hence H A if and only if H A and the conclusion follows

Metrics on the Sierpinski gasket

Since we will b e studying continuous pro cesses on G it is natural to consider

the metric on G given by the shortest path in G between twopoints We b egin with

a general denition

Denition Let A R For x y A set

d x y inf fj j is a path b etween x and y and Ag

If d x y for all x y A we call d the geodesic metric on A

A A

Lemma Supp ose A is closed and that d x y for all x y AThen

d is a metric on A and A d has the geo desic prop erty

A A

For each x y A there exists a map t A such that

d x t td x y d ty td x y

A A A A

Proof It is clear that d is a metric on A Toprove the geo desic prop ertylet

x y A and D d x y Then for each n there exists a path t

A n

t D such that A jd tj dt x and t y for

n n n n n

some D t D n If p D Q then since jx pjp the sequence

n n

p has a convergent subsequence By a diagonalization argument there exists

a subsequence n such that pconverges for each p D Q we can take

k n

lim

Lemma For x y G

jx y j d x y c jx y j

Proof The left hand inequality is evident

It is clear from the structure of H that if A B are ntriangles and A B

then

ja bj for a A b B

Let x y G and cho ose n so that

p p

n n

jx y j

So x y are either in the same ntriangle or in adjacent ntriangles In either case

cho ose z G so it is in the same ntriangle as b oth x and y n

Let z z and for kncho ose z G suchthatx z are in the same k

n k k k

triangle Then since z and z are in the same k triangle and b oth are contained

k k

in H wehave d z z d z z So

k G k k H k k

d z x d z z jx y j

G G k k

k n

Hence d x y d x z d z y jx y j

G G G

Construction of a diusion on the Sierpinski gasket

Let G b e the set of vertices of ntriangles We can make G into a graph

n n

in a natural wayby taking fx y g to b e an edge in G if x y b elong to the same

ntriangle See Fig Write E for the set of edges

Figure The graph G

k b e a simple random walk on G Thus from x G the Let Y

n n

pro cess Y jumps to each of the neighb ours of x with equal probability Apart

from the p oints in G all the p oints in G have neighb ours The obvious way

to construct a diusion pro cess X t on G is to use the graphs G which

t n

provide a natural approximation to G and to try to dene X asaweak limit of the

pro cesses Y More preciselywe wish to nd constants n such that

Y t X t

Wehavetwo problems

Howdowe nd the right

Howdowe proveconvergence

We need some more notation

Denition Let S b e the collection of sets of the form G A where A is an

ntriangle We call the elements of S ncomplexesFor x G let D x fS

n n n

S x S g

The key prop erties of the SG whichwe use are rst that it is very symmetric

and secondly that it is nitely ramied In general a set A in a metric space F is

nitely ramied if there exists a nite set B such that A B is not connected For

the SG we see that each ncomplex A is disconnected from the rest of the set if we

remove the set of its corners that is A G

The following is the key observation Supp ose Y y G take y G

for simplicity and let T inffk Y G fy gg Then Y can only

escap e from D y at one of the p oints fx x g saywhich are neighbours of

y in the graph G E Therefore Y fx x gFurther the symmetry

n n

of the set G D y means that each of the events fY x g is equally likely

n n i T

xx23

x 1 yx4

Figure y and its neighb ours

Thus

n n

P Y x Y y

n n

and this is also equal to PY x jY y Exactly the same argument

applies if y G except that wethenhave only neighb ours instead of It

n n

follows that Y lo oked at at its visits to G b ehaves exactly like Y To

state this preciselywerstmake a general denition

Denition LetT R or Z letZ t T b e a cadlag pro cess on a metric

space F and let A F b e a discrete set Then successive disjoint hits by Z on A

are the stopping times T T dened by

T inf ft Z Ag

T inf tT Z A fZ g n

n n t T

With this notation we can summarize the observations ab ove

Lemma Let T b e successive disjoint hits by Y on G Then

i i n

Y i is a simple random walk on G and is therefore equal in lawto

Y i

Using this it is clear that we can build a sequence of nested random walks

N N

on G Let N and let Y k beaSRWonG with Y Let

n N

m N and T b e successive disjointhitsby Y on G and set

i m

N m

Y T Y i Y

i i

T i

It follows from Lemma that Y is a SRWon G and for each n m N

m n

wehavethatY sampled at its successive disjoint hits on G equals Y

Wenow wish to construct a sequence of SRWs with this prop erty holding for

n m This can b e done either by using the Kolmogorov extension

N N

theorem or directlyby building Y from Y with a sequence of indep endent

excursions The argument in either case is not hard and I omit it

Thus we can construct a probability space F P carrying random variables

Y n k such that

a For each nY k is a SRWonG starting at

b Let T b e successive disjointhitsby Y on G Here m n Then

i m n Y Y T

i i

If we just consider the paths of the pro cesses Y in Gwe see that we are

viewing successive discrete approximations to a continuous path However to dene

a limiting pro cess we need to rescale time as was suggested by

Write T minfk jY j g and set f sE s for s

Lemma f ss s E f and E for all k

Proof This is a simple exercise in nite state Markovchains Let a a be the two

nonzero elements of G let b a a and c a Writing f sE s

i i c

and dening f f similarlywehave f s

b a a

f ssf s

s f sf sf sf s f s

c b a c

s f sf s f s

c a b

and solving these equations we obtain f s

The remaining assertions follow easily from this

a 1

b 1 c

0 ba2 2

Figure The graph G

NowletZ T n The nesting prop erty of the random walks Y

implies that Z is a simple branching pro cess with ospring distribution p where

n n

s p f s

nn nn

To see this note that Y for T k T is a SRWonG

i i

nn nn

D Y and that therefore T T Alsoby the Markov prop erty

i i i

nn nn

the rv T T i are indep endent Since

i i

n i

Z isabranching pro cess

As E and E the convergence theorem for simple branching

pro cesses implies that

Z W

for some strictly p ositiverv W See Har p The convergence is easy using

a martingale argument proving that W as takes a little more work See

Har p In addition if

uEe

then satises the functional equation

uf u

Wehave a similar result in general

Prop osition Fix m The pro cesses

nm nm

Z T T n m

i i

are branching pro cesses with ospring distribution and Z are indep endent

m m

i are indep endent such that for each m W Thus there exist W

i i

W W and

nm nm

W as T T

i i i

n n

Note in particular that E T that is that the mean time taken by Y

to cross G is In terms of the graph distance on G wehave therefore that

n n

n n n

Y requires roughly steps to move a distance this ma y b e compared with

the corresp onding result for a simple random walk on Z which requires roughly

n n

steps to move a distance

The slower movementof Y is not surprising to leave G B for

example it has to nd one of the two gateways or Thus the

movementofY is imp eded by a succession of obstacles of dierent sizes which

act to slowdown its diusion

Given the spacetime scaling of Y it is no surprise that we should take

in Dene

Y X t

In view of the fact that wehave built the Y with the nesting prop ertywe

can replace the weak convergence of with as convergence

Theorem The pro cesses X converge as and uniformly on compact inter

vals to a pro cess X t X is continuous and X G for all t

t t

Proof For simplicitywe will use the fact that W has a nonatomic distribution

function Fix for now m Let t Then as there exists i i such that

i i

X X

m m

W t W

j j

nm nm

As W lim T T it follows that for n n

j j j

nm nm

T tT

i i

m m n

nm nm nm

D Y for T k T Now Y T Y by Since Y

i i i i i

wehave

n m

jY Y j for all n n

n n m n

This implies that jX X j for n n n so that X is Cauchyand

t t t

converges to a rv X Since X G wehave X G

t n t

With a little extra work one can prove that the convergence is uniform in ton

compact time intervals I give here a sketch of the argument Let a N and let

W min

Then as Cho ose n such that for n n

T W i a

i j

m m m n

Then if i it is such that W t W and i a wehave T

i i

n n m m

t T for all n n So jX Y j for all n n This implies

that if T W andST then

m m

n n m

sup jX X j

t t

for all n n n If Slim inf T then the uniform as convergence on the

m m

random interval S follows If s tT and jt sj then we also have

m m

n n m

jX X j for n n Thus X is uniformly continuous on S Varying

t s

a we also obtain uniform as convergence on xed intervals t

Although the notation is a little cumb ersome the ideas underlying the con

struction of X given here are quite simple The argumentaboveisgiven in BP

but Kusuoka Kus and Goldstein Go who were the rst to construct a diusion

on G used a similar approach It is also worth noting that Knight Kn uses similar

metho ds in his construction of dimensional Brownian motion

The natural next step is to ask ab out prop erties of the pro cess X But unfor

tunately the construction given ab ove is not quite strong enough on its own to give

us much To see this consider the questions

Is W lim T infft X G fgg

n t

Is X Markov or strong Markov

For we certainly have X G fgHowever consider the p ossibility that each

of the random walks Y moves from to a on a path which do es not include a but

n n

includes an approach to a distance In this case wehave a f X t W g

but X a for some TW Plainly some estimation of hitting probabilities is

needed to exclude p ossibilities like this

The construction ab ovedoesgiveaMarkov prop ertyfor X at stopping times of

the form W But to obtain a go o d Markov pro cess X X t P x

Gwe need to construct X at arbitrary starting p oints x G and to show that in

some appropriate sense the pro cesses started at close together p oints x and y are

This can b e done using the construction given ab ove see BP Section

However the argument although not really hard is also not that simple

In the remainder of this section I will describ e some basic prop erties of the

pro cess X for the most part without giving detailed pro ofs 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

Denition Let F b e a metric space A local isometry of F is a triple

A B whereA B are subsets of F and is an isometry ie bijective and distance

preserving b etween A and B and b etween A and B

Let X t P x F beaMarkov pro cess on F For H F setT

t H

inf ft X H g X is invariant with respect to a local isometry A B if

x x

t t P X P X

tT tT

B A

X is local ly isotropic if X is invariant with resp ect to the lo cal isometries of F

Theorem a There exists a continuous strong Markov pro cess X X t

P x G on G

b The semigroup on C G dened by

P f xE f X

t t

is Feller and is symmetric

Z Z

f xP g x dx g xP f x dx

t G t G G

c X is lo cally isotropic on the spaces G jj and G d

d For n let T i b e successive disjointhitsby X on G Then Y

ni n

X i denesaSRWonG andY X uniformly on compacts as So

T n t

in particular X t P is the pro cess constructed in Theorem

This theorem will follow from our general results in Sections and a direct

pro of may b e found in BP Sect The main lab our is in proving agiv en this

b c d all followinarelatively straightforward fashion from the corresp onding

prop erties of the approximating random walks Y

The prop erty of lo cal isotropyonG d characterizes X

Theorem Uniqueness Let Z t Q x G b e a nonconstant lo cally

isotropic diusion on G d Then there exists a suchthat

x x

Q Z t P X t

t at

So Z is equal in law to a deterministic time change of X

The b eginning of the pro of of Theorem runs roughly along the lines one

e e

would exp ect for n let Y i b e Z sampled at its successive disjoint

e e

hits on G The lo cal isotropyofZ implies that Y is a SRWonG However

n n

some work see BP Sect is required to prove that the pro cess Y do es not have

traps ie p oints x such that Q Y x for all t

Remark The denition of invariance with resp ect to lo cal isometries needs

some care Note the following examples

Let x y G be such that D x G a D y G Then while there

n n n

exists an isometry from D x G to D y G do es not map D x G

n n R n

to D y G denotes here the relative b oundary in the set G

R n R

Recall the denition of H thenth stage in the construction of G and let

B H Wehave G clB Consider the pro cess Z on G whose lo cal

n n n t

motion is as follows If Z H H then Z runs like a standard dimensional

t n n t

Brownian motion on H until it hits H After this it rep eats the same pro cedure

n n

on H or H if it has also hit H at that time This pro cess is also invariant

n nk nk

with resp ect to lo cal isometries A B of the metric space G jj See He

for more on this and similar pro cesses

To discuss scale invariant prop erties of the pro cess X it is useful to extend G

to an unb ounded set G with the same structure Set

G G

e e

and let G b e the set of vertices of ntriangles in G for n Wehave

n n

G G

n nk

and if wedeneG fg for m this denition also makes sense for n We

e e e e

can almost exactly as ab ove dene a limiting diusion X X t P x G

on G

e e

X lim Y t as

t n

e e

where Y n k are a sequence of nested simple random walks on G

and the convergence is uniform on compact time intervals

The pro cess X satises an analogous result to Theorem and in addition

satises the scaling relation

x x

e e

P X t P X t

t t

log log

Note that implies that X moves a distance of roughly t in time t

Set

d d Glog log

w w

Wenow turn to the question What do es this pro cess lo ok like

The construction of X and Theorem d tells us that the crossing time of

a triangle is equal in law to the limiting random variable W of a branching pro cess

with ospring pgf given by f ss s From the functional equation

we can extract information ab out the b ehaviour of uE expuW as u

and from this by a suitable Taub erian theorem we obtain b ounds on PW t

for small t These translate into b ounds on P jX xj for large One uses

scaling and the fact that to move a distance in G greater than X has to cross at

least one triangle These b ounds give us many prop erties of X However rather

than following the development in BP it seems clearer to rst present the more

delicate b ounds on the transition densities of X and X obtained there and derive

all the prop erties of the pro cess from them Write e for the analogue of for G

G G

e e e e

and P for the semigroup of X LetL b e the innitesimal generator of P

t t

Theorem P and P have densities pet x y and pt x y resp ectively

t t

e e

a pet x y is continuous on G G

b pet x y pet y x for all t x y

c t pet x y is C on for each x y

d For each t y

d d

w f

jpet x y pet x yjc t jx x j x x G

e For t x y G

jx y j

d d

f w

c t exp c pet x y

jx y j

d d

f w

c t exp c

f For each y G pet x y is the fundamental solution of the heat equation on

G with p ole at y

pet x y Lpet x y pe y

g pt x y satises af ab ovewithG replaced by G and t replaced

by t

Remarks The pro of of this in BP is now largely obsolete simpler metho ds

are nowavailable though these are to some extent still based on the ideas in BP

If d d and d wehave in the form of the transition densityof

f w

Brownian motion in R Since d log log the tail of the distribution of

jX xj under P decays more rapidly than an exp onential but more slowly than

a Gaussian

It is fairly straightforward to integrate the b ounds to obtain information

ab out X At this p ointwe just present a few simple calculationsw e will give some

further prop erties of this pro cess in Section

e e

Denition For x G n Zlet x be the pointinG closest to x in

n n

Euclidean distance Use some pro cedure to break ties Let D xD x

n n n

n n

Note that e D x is either or that

G n n

jx y j if y D x

and that

n c

if y G D x jx y j

The sets D x form a convenient collection of neighb ourho o ds of p oints in G Note

that D xG

nZ n

Corollary For x G

x d d

w w

t E jX xj c t c t

ProofWehave

E jX xj y x pet x y e dy

t G

Set A D x D x Then

m m m

y x pet x y e dy

m d d m m

f w

c t exp c t

m d d d m d

f f w w

t exp c t c

n n

Cho ose n suchthat t and write a t for the nal term in

Then

X X

E X x a t a t

t m m

m mn

For mn t and the exp onential term in is dominant After a few

calculations we obtain

n d d d

f f w

a t c t

d d d d d d d d d

f w f w f w f w w

ct ct ct

n d

t For m n we neglect the exp onential where we used the fact that

term and have

X X

d d m d

f w f

a t ct

mn mn

d d n d d

f w f w

ct c t

Similar calculations give the lower b ound

Remarks Since d log log this implies that X is sub diusive

Since e B x r r for x G it is tempting to try and prove Corollary

by the following calculation

Z Z

pet x y e dy r dr E jX xj

G t

Bxr

d d d d t

f f w w

dr r r t exp cr

d d d d

w w w f

t ds ct exp cs s

Of course this calculation as it stands is not valid the estimate

e B x r dr B x r r dr

is certainly not valid for all r But it do es hold on average over length scales of

n n

r and so splitting G into suitable shells a rigorous version of this

calculation may b e obtained and this is what we did in the pro of of Corollary

The p otential kernel densityofX is dened by

e pet x y dt u x y

d d

f w

From it follows that u is continuous that u x x c and that

u as Thus the pro cess X and also X hits p oints that is if

T inf ftX y g then

y t

P T

It is of course clear that X must b e able to hit p oints in G otherwise it could not

move but shows that the remaining p oints in G have a similar status The

continuityof u x y in a neighb ourho o d of x implies that

P T

that is that x is regular for fxg for all x G

The following estimate on the distribution of jX xj can b e obtained easily

from byintegration but since this b ound is actually one of the ingredients in

the pro of such an argumentwould b e circular

Prop osition For x G t

x d

t P jX xj c exp c

t c exp c

From this it follows that the paths of X are Holder continuous of order d

for each In fact we can up to constants obtain the precise mo dulus of

continuityof X Set

d d d

w w w

log t htt

Theorem a For x G

e e

jX X j

s t

c lim sup c P as

hs t

jtsj

e e

b The paths of X are of innite quadratic variation as and so in particular X

is not a semimartingale

The pro of of a is very similar to that of the equivalent result for Brownian motion

in R

For b Prop osition implies that jX X j is of order h as d this

th t w

suggests that X should have innite quadratic variation For a pro of whichllsin

the details see BP Theorem

So far in this section wehavelooked at the Sierpinski gasket and the construc

tion and prop erties of a symmetric diusion X on G or G The following three

questions or avenues for further research arise naturally at this p oint

Are there other natural diusions on the SG

Can we do a similar construction on other fractals

What ner prop erties do es the pro cess X on G have More precisely what

ab out prop erties which the b ounds in are not strong enough to give

information on

The bulk of research eort in the years since Kus Go BP has b een devoted

to Only a few pap ers havelooked at and apart from a number of works

on sp ectral prop erties the same holds for

Before discussing or in greater detail it is worth extracting one prop erty

of the SRW Y whichwas used in the construction

Let V V n P a G beaMarkovchain on G clearly V is sp ecied

by the transition probabilities

pa a P V a i j

i j j

Wetake pa a fora G so V is determined by the three probabilities

pa a where j i mo d

i j

Given V we can dene a Markov Chain V on G by a pro cess wecallreplication

Let fb b b g b e the p oints in G G where b a a We consider

ij i j

G to consist of three cells fa b j ig i whichintersect at the p oints

i ij

fb g The lawofV may b e describ ed as follows V moves inside each cell in

the way same as V do esif V lies in two cells then it rst cho oses a cell to

moveinandcho oses each cell with equal probability More precisely writing

P a G and n V V

p a b P V b

wehave

p a b pa a

i ij i j

pa a p b a pa a p b b

j k ij i j i ij ik

k NowletT k b e successive disjointhitsby V on G and let U V

k k

Then U is a Markov Chain on G wesay that V is decimation invariant if U is

equal in lawtoV

Wesawabove that the SRW Y on G was decimation invariant A natural

question is

What other decimation invariant Markovchains are there on G

Two classes have b een found

See Go Let pa a pa a pa a

pstream random walks Kum Let p and

pa a pa a pa a p

From each of these pro cesses we can construct a limiting diusion in the same

way as in Theorem The rst pro cess is reasonably easy to understand essen

tially its paths consist of a downward drift when this is p ossible and a b ehaviour

like dimensional Brownian motion on the p ortions on G which consist of line

segments parallel to the xaxis

For p Kumagais pstream diusions tend to rotate in an anticlo ckwise

direction so are quite nonsymmetric Apart from the results in Kum nothing is

known ab out this pro cess

Two other classes of diusions on G which are not decimation invariant have

also b een studied The rst are the asymptotically dimensional diusions of

HHW the second the diusions similar to that describ ed in Remark which

are G j jisotropic but not G d isotropic see He See also HH HK

HHK for work on the selfavoiding random walk on the SG

Diusions on other fractal sets

Of the three questions ab ove the one which has received most attention is that

of making similar constructions on other fractals To see the kind of diculties

which can arise consider the following two fractals b oth of which are constructed

by a Cantor typ e pro cedure based on squares rather than triangles For eachcurve

the gure gives the construction after two stages

Figure The Vicsek set and the Sierpinski carp et

The rst of these we will call the Vicsek set VS for short We use similar

notation as for the SG and write G G for the succession of sets of vertices

of corners of squares We denote the limiting set by F F One diculty arises

immediatelyLetY b e the SRWonG whichmoves from any p oint x G to each

of its neighb ours with equal probability The neighb ours of x are the p oints y in

G with jx y j Then Y is not decimation invariant This is easy to see

Y cannot move in one step from to but Y can movefrom to

without hitting any other p ointin G

However it is not hard to nd a decimation invariant random walk on G Let

p and consider the random walk Y r E x G onG whichmoves

p diagonally with probability p and horizontally or vertically with probability

Let Y r E x G b e the Markovchain on G obtained by replication and

r p

let T k b e successive disjointhitsby Y on G

Then writing f pP Y wehave after several minutes calcula

tion

f p

and p eachofwhich The equation f p p therefore has two solutions p

corresp onds to a decimation invariantwalk on G The number here has no

general signicance if we had lo oked at the fractal similar to the Vicsek set but

based on a square rather than a square then wewould have obtained a

dierentnumber

One maynow carry through in each of these cases the construction of a dif

fusion on the Vicsek set F very much as for the Sierpinski gasket For p one

gets a rather uninteresting pro cess which if started from is up to a constant

time change dimensional Brownian motion on the diagonal ft t t gIt

is worth remarking that this pro cess is not strong Markov for each x F one can

take P to b e the lawofa Brownian motion moving on a diagonal line including

x but the strong Markov prop ertywillfailatpoints where two diagonals intersect

such as the p oint

one obtains a pro cess X t with much the same b ehaviour For p

as the Brownian motion on the SG We have for the Vicsek set with p

d F log log d F log log This pro cess was studied in some

f VS w VS

detail by Krebs Kr Kr The Vicsek set was mentioned in Go and is one of the

nested fractals of Lindstrm L

This example shows that one mayhavetowork to nd a decimation invariant

random walk and also that this may not b e unique For the VS one of the decima

tion invariant random walks was degenerate in the sense that P Y hits y for

some x y G and we found the asso ciated diusion to b e of little interest But it

raises the p ossibility that there could exist regular fractals carrying more than one

natural diusion

The second example is the Sierpinski carp et SC For this set a more serious

diculty arises The VS was nitely ramied so that if Y is a diusion on F

t VS

and T k are successive disjointhitson G for some n then Y k

k n T

isaMarkovchain on G However the SC is not nitely ramied if Z t is a

n t

could o ccur anywhere on the diusion on F then the rst exit of Z from

line segments f y y g fx x g It is not even clear that a

diusion on F will hit p oints in G Thus to construct a diusion on F one

SC n SC

will need very dierent metho ds from those outlined ab ove It is p ossible and has

b een done see BBBB and Bas for a survey

On the third question mentioned ab ove disapp ointingly little has b een done

most known results on the pro cesses on the Sierpinski gasket or other fractals

are of roughly the same depth as the b ounds in Theorem Note however the

results on the sp ectrum of L in FS FS ShSh and the large deviation results

in Kum Also Kusuoka Kus has very interesting results on the b ehaviour of

harmonic functions which imply that the measure dened formally on G by

dxjrf j xdx

is singular with resp ect to There are many op en problems here

Fractional Diusions

In this section I will intro duce a class of pro cesses dened on metric spaces

which will include many of the pro cesses on fractals mentioned in these lectures I

havechosen an axiomatic approach as it seems easier and enables us to neglect

for the time b eing much of ne detail in the geometry of the space

A metric space F hasthemidpoint property if for each x y F there exists

z F suchthatx z z y x y Recall that the geo desic metric d in

Section had this prop erty The following result is a straightforward exercise

Lemma See Blu Let F b e a complete metric space with the midp oint

prop erty Then for each x y F there exists a geo desic path t t such

that x y and st jt sjdx y s t

For this reason we will frequently refer to a metric with the midp oint prop erty

as a geodesic metric See Stu for additional remarks and references on spaces of

this typ e

Denition Let F b e a complete metric space and b e a Borel measure

on F B F WecallF a fractional metric space FMS for short if

a F has the midp oint prop erty

and there exist d and constants c c such that if r supfx y x y

F g is the diameter of F then

d d

f f

b c r B x r c r for x F r r

Here B x r fy F x y rg

Remarks R with Euclidean distance and Leb esgue measure is a FMS

with d d and r

If G is the Sierpinski gasket d is the geo desic metric on Gand is

G G

the measure constructed in Section then Lemma shows that G d isa

e is a FMS with FMS with d d G log log and r Similarly G d

f f

If F d k are FMS with the same diameter r and p then

k k k

p p p

setting F F F dx x y y d x y d x y

it is easily veried that F d isalsoaFMSwithd F d F d F

f f f

For simplicitywe will from nowontake either r or r We will write

r r to mean r r and dene r if and r

Anumb er of prop erties of F follow easily from the denition

Lemma a dim F dim F d

H P f

b F is lo cally compact

c d

Proof a is immediate from Corollary

B x and consider a maximal packing of disjoint balls b Let x F A

B x x A i m As A c and B x c wehave

i i i

m d

Also A B x Thus any b ounded set in F can b e m c c

covered by a nite numb er of balls radius this with completeness implies that F

is lo cally compact

c Take x y F with x y D Applying the midp oint prop erty rep eatedly

we obtain for m k a sequence x z z z y with z z

m i i

Dm Set r Dm the balls B z rmust b e disjoint or using the triangle

inequalitywewould have x y D But then

B z r B x D

so that

c D B z r B x D

d d d

f f f

m cm mc D

If d a contradiction arises on letting m

Denition Let F be a fractional metric space A Markov pro cess

X P x F X t is a fractional diusion on F if

a X is a conservativeFeller diusion with state space F

b X is symmetric

c X has a symmetric transition density pt x y pt y x t x y F

which satises the ChapmanKolmogorov equations and is for each t jointly

continuous

d There exist constants c c t r suchthat

c t exp c x y t pt x y

c t exp c x y t x y F t t

d d

Examples If F is R andaxa x i j d x R is b ounded

symmetric measurable and uniformly elliptic let L be the divergence form op erator

L a x

x x

i j

Then Aronsens b ounds Ar imply that the diusion with innitesimal generator L

is a FD with d

By Theorem the Brownian motion on the Sierpinski gasket describ ed in

Section is a FD with d SGd SG d SGand

f w w

The hyp otheses in Denition are quite strong ones and as the examples

suggest the assertion that a particular pro cess is an FD will usually b e a substantial

theorem One could of course consider more general b ounds than those in with

a corresp ondingly larger class of pro cesses but the form is reasonably natural

and already contains some interesting examples

In an interesting recent series of pap ers Sturm StuStu has studied diusions

on general metric spaces However the pro cesses considered there turn out to have

an essentially Gaussian long range b ehaviour and so do not include any FDs with

In the rest of this section we will study the general prop erties of FDs In

the course of our work we will nd some slightly easier sucient conditions for a

pro cess to b e a FD than the b ounds and this will b e useful in Section when

weprove that certain diusions on fractals are FDs Webeginby obtaining two

relations b etween the indices d so reducing the parameter space of FDs

to a twodimensional one

We will say that F is a FMSd ifF is a FMS and satises b with

parameter d and constants c c Similarlywesay X is a FD d if

f f

X isaFDonaFMSd and X satises with constants This is

temp orary notation hence the

It what follows wexaFMSF with parameters r and d

Lemma Let x and set

I x e dt

n x

S x e

Then

S x I x S x

and

I x x for x

I x x e for x

Proof Wehave

I x e dt

and estimating each term in the sum is evident

If x then since

x I x e ds c as x

follows

If x then follows from the fact that

u x

e x ux du as x xe I x

Lemma Scaling relation Let X be a FD d on F Then

ProofFrom wehave

c t for x y t pt x y c t e

Set t r SoifA B x t and t t

x d

P x X t pt x y dy c t A ct

If r then since this holds for all twemust have d Ifr then

we only deduce that d

Let now r let t and A B x t Wehave F c

and therefore

x x c

P X AP X A

t t

pt x y A sup pt x y F Asup

y A y A

d d c

f f

c t c t e

log t then w ehave for all t that Let d c

ct logt

which gives a contradiction unless d

The next relation is somewhat deep er essentially it will follow from the fact

that the longrange b ehaviour of pt x y isxedby the exp onents d and govern

ing its shortrange b ehaviour Since only plays a role in when x y t

we will b e able to obtain in terms of d and in fact it turns out of only

We b egin by deriving some consequences of the b ounds

Lemma Let X be a FD d d Then

f f

a For t t r

P x X r c exp c r t

b There exists c such that

c r t P x X r for rc r tr c exp

c For x F r c r if x r inffs X B x r g then

c r E x r c r

ProofFixx F and set D a bfy F a x y bgThenby b

d d d

f f f

c b D a b c b c a

Cho ose sothatc c thenwehave

d d

f f

c a D a a c a

n n nd n

Therefore writing D D r r wehave D provided r r

n n

Now

pt x y dy x X r P

B xr

pt x y dy

i d d n

f f

cr t exp c t r

cr t S c r t

If c r t then using we deduce that this sum is b ounded by

c r c exp t

while if c r t then as P x X r we obtain the same b ound on

adjusting the constant c

For the lower b ound b cho ose c sothatc Then D cr

and taking only the rst term in we deduce that since r t

d x

x X r cr t expc r t P

c expc r t

c Note rst that

y y

P x r t P X B x r

pt y z dz

B xr

d d

f f

ct r

So for a suitable c

P x r c r y F

Applying the Markov prop ertyofX wehave for each k

y k

P x r kc r y F

whichproves the upp er b ound in

For the lower b ound note rst that

x x

P x r tP sup x X r

x x

x r tx X r x X r P P

t t

Writing S x r the second term ab ove equals

y x X

x X r sup sup P y X r E P

tS ts

y Bxr

so that using a

x y

P x r t sup sup P y X r

st y F

c exp c r t

c a x

which proves the left hand side of So if c e then P x r ar

Remark Note that the b ounds in c only used the upp er b ound on pt x y

The following result gives sucient conditions for a diusion on F to b e a

fractional diusion these conditions are a little easier to verify than

Theorem Let F be a FMSd Let Y t P x F be a

f t

symmetric diusion on F which has a transition density q t x y with resp ect to

whichisjointly continuous in x y for each t Supp ose that there exists a

constant such that

q t x y c t for all x y F t t

q t x y c t if x y c t t t

c r E x r c r for x F r c r

where x r inf ft Y B x r g Then and Y is a FD with

parameters d d and

f f

Corollary Let X be a FD d d on a FMSd F Then

f f f

and

Proof By Lemma and the b ounds the transition density pt x y ofX

satises and By Lemma c X satises So by Theorem

and X is a FD d d Since pt x y cannot satisfy

f f

for two distinct values of wemust have

Remark Since two of the four parameters are now seen to b e redundant we

will shorten our notation and say that X is a FDd ifX is a FD d d

f f f

The pro of of Theorem is based on the derivation of transition density

b ounds for diusions on the Sierpinski carp et in BB most of the techniques

there generalize easily to fractional metric spaces The essential idea is chaining

in its classical form see eg FaS for the lower b ound and in a slightly dierent

more probabilistic form for the upp er b ound We b egin with a some lemmas

Lemma BB Lemma Let V b e nonnegative rv such

that V Supp ose that for some p a

P tj p at t

i i

Then

ant

log P V t n log

p p

Pro ofIf is a rv with distribution function P t p at then

E e j Ee

p e adt

p a

t V V t

e Ee P V tP e e

t n t

e p a e E exp

p exp t

The result follows on setting anpt

Remark The estimate app ears slightly o dd since it tends to as

p However if p then from the last but one line of the pro of ab ovewe obtain

log P V t t n log and setting nt we deduce that

ate

log P V t n log

Lemma Let Y t b e a diusion on a metric space F such that for

x F r

c r E x r c r

Then for x F t

x r t c c c r t P

ProofLetx F andA B x r x r Since t t wehave

x x Y

E t E E t

x y

t P t sup E

As y r P as for any y F we deduce

x x

c r E t P tc r

so that

c P t c c tr

The next couple of results are needed to show that the diusion Y in Theorem

can reach distant parts of the space F in an arbitrarily short time

Lemma Let Y be a symmetric diusion with semigroup T on a complete

t t

metric space F Iff g and there exist ab such that

f xE g Y dx for t a b

then f xE g Y dx for all t

Proof Let E b e the sp ectral family asso ciated with T Thus see FOT

e dE and p T

Z Z

t t

f T g e df E g e d

where is of nite variation and the uniqueness of the Laplace transform

imply that and so f T g for all t

Lemma Let F and Y satisfy the hyp otheses of Theorem If x y

c r then P Y B y r for all r and t

Remark The restriction x y c r is of course unnecessary but it is all we

need now The conclusion of Theorem implies that P Y B y r for all

r and t for all x y F

Proof Supp ose the conclusion of the Lemma fails for x y rt Cho ose g C F R

such that gd and g outside B y r Let t t r c t and

cho ose f C F R sothat fd f x andf outside A B x r

If s t then the construction of g implies that

x x

q s x x E g Y dx E g Y

ts t

Since by q s x x for t s t x B x r we deduce that

E g Y for x B x r u t Thus as suppf B x r

f x E g Y d

for all u t and hence by Lemma for all u But by if

u x y c then q u x y and by the continuityof f g and q it follows

that f E g Y d a contradiction

Proof of Theorem For simplicitywe give full details of the pro of only in the

case r the argumen t in the case of b ounded F is essentially the same We

begin by obtaining a b ound on

P x r t

Let n b rn and dene stopping times S i by

S S infft S Y Y bg

i i S t i

We Let S S i Let F b e the ltration of Y and let G F

i i i t t i S

haveby Lemma

x Y

Y b t p c b t P tjG P

S i i

where p As Y Y bwehave Y Y r so that S

i S S S n

i i n

Y r So by Lemma with a c rn

log P x r t p n log c r n t

c n c r n t

If then taking t small enough the right hand side of is negative and

letting n we deduce P x r t whichcontradicts the fact that

P Y B y r for all t So wehave If r then we take r small

enough so that rc

If we neglect for the moment the fact that n N and take n n in so

that

c n c n tr

then

n c c r t

and

c n x r t log P

So if r t we can cho ose n N so that n n and we obtain

P x r t c exp c

Adjusting the constant c if necessary this b ound also clearly holds if r t

Nowletx y F write r x y cho ose r and set C B z

z x y SetA fz F z x z yg A fz z x z ygLet

x y x

b e the restriction of to C C resp ectively

y x y

Wenowderive the upp er b ound on q t x y by combining the b ounds

and the idea is to split the journey of Y from C to C into two pieces and

x y

use one of the b ounds on eachpieceWehave

Z Z

q t x y dx dy P Y C

t y

C C

x y

x x

Y C Y A P Y C Y A P

t y y t y x

t t

We b egin with second term in

x x

P Y C Y A P Y r tY A Y C

t y y y t y

t t

Y C P Y r t sup P

y A y

c C t C c exp c

y y x x

C C c t exp c r t

x y

where we used and in the last but one line

To handle the rst term in we use symmetry

y x

Y C Y A Y C Y A P P

t x x t y x

t t

and this can now b e b ounded in exactly the same wayWe therefore have

Z Z

q t x y dx dy

C C

y x

c r t C C c t exp

x y

so that as q t is continuous

q t x y c t exp c r t

The pro of of the lower b ound on q uses the technique of chaining the

ChapmanKolmogorov equations This is quite classical except for the dierent

scaling

Fix x y t and write r x y If r c t then by

q t x y c t

and as expr t expc wehavea lower b ound of the form

So nowletrc t Let n By the midp ointhyp othesis on the metric

we can nd a chain x x x x y in F such that x x rn

n i i

i n Let B B x rnnote that if y B then y y rn We

i i i i i i

haveby the ChapmanKolmogorov equation writing y x y y

Z Z

q t x y dy dy q tn y y

n i i

B B

Wewishtocho ose n so that we can use the b ound to estimate the terms

q tn y y from b elow We therefore need

i i

r t

n n

which holds provided

n c

As it is certainly p ossible to cho ose n satisfying By wethen

obtain since B crn

d n d

f f

q t x y crn c tn

d d

f f

crn c tn rn

c rn tn rn

Recall that n satises as rc t we can also ensure that for some c

c tn

so that n c r t So by

d n

q t x y ctn c

c t exp n log c

c t exp c r t

Remarks

Note that the only p ointatwhichwe used the midp oint prop ertyof is in the

derivation of the lower b ound for q

The essential idea of the pro of of Theorem is that we can obtain b ounds on

the long range b ehaviour of Y provided wehave go o d enough information ab out the

behaviour of Y over distances of order t Note that in each case if r x y

the estimate of q t x y involves splitting the journey from x to y into n steps

where n r t

Both the arguments for the upp er and lower b ounds app ear quite crude the

fact that they yield the same b ounds except for constants indicates that less is

thrown away than might app ear at rst sight The explanation very lo oselyis

given by large deviations The odiagonal b ounds are relevant only when r t

t then it is dicult otherwise the term in the exp onential is of order If r

for Y to movefrom x to y bytimet anditislikely to do so along more or less the

shortest path The pro of of the lower b ound suggests that the pro cess moves in a

sausage of radius rn tr

The following two theorems give additional b ounds and restrictions on the

parameters d and Unlike the pro ofs ab ove the results use the symmetry of the

pro cess very strongly The pro ofs should app ear in a forthcoming pap er

Theorem Let F be a FMSd andX be a FDd on F Then

f f

are FDd on F for Theorem Let F be a FMSd Supp ose X

f i f

i Then

Remarks Theorem implies that the constant is a prop ertyofthe

metric space F and not just of the FD X In particular anyFDonR with the

usual metric and Leb esgue measure will have It is very unlikely that every

FMS F carries a FD

I exp ect that is the only general relation b etween and d More precisely

set

A fd there exists a FDd g

f f

andfd d g Theorem implies that A and I

f f

conjecture that int A Since BM R isa FDd the p oints d A for

d I also susp ect that

fd d Ag N

f f

that is that if F is an FMS of dimension d and d is not an integer then anyFD

f f

on F will not have Brownian scaling

Prop erties of Fractional Diusions

In the remainder of this section I will give some basic analytic and probabilistic

prop erties of FDs I will not give detailed pro ofs since for the most part these

are essentially the same as for standard Brownian motion In some cases a more

detailed argumentisgiven in BP for the Sierpinski gasket

Let F be a FMSd and X be a FDd on F Write T E f X for

f f t t

the semigroup of X and L for the innitesimal generator of T

Denition Set

d d

w s

This notation follows the physics literature where for reasons we will see b elow

d is called the walk dimension and d the sp ectral dimension Note that

w s

implies that

pt x x t t t

so that the ondiagonal b ounds on p can b e expressed purely in terms of d Since

many imp ortant prop erties of a pro cess relate solely to the ondiagonal b ehaviour

of its density d is the most signicant single parameter of a FD

Integrating as in Corollary we obtain

x p pd

Lemma E X x t x F t p

Since by Theorem d this shows that FDs are diusive or sub diusive

d d d

w w w

Then log t Lemma Mo dulus of continuity Let tt

X X

s t

c c lim sup

t s

jtsj

So in the metric the paths of X just fail to b e Holder d The example

of divergence form diusions in R shows that one cannot hop e to have c c in

general

Lemma Law of the iterated logarithm see BP Thm Let t

d d d

w w w

There exist c c and constants cx c c such log log t t

that

X X

lim sup cx P as

Of course the law implies that the limit ab ove is nonrandom

Lemma Dimension of range

dim fX t gd d

H t f w

This result helps to explain the terminology walk dimension for d Provided

the space the diusion X moves in is large enough the dimension of range of the

pro cess called the dimension of the walk byphysicists is d

Potential Theory of Fractional Diusions

Let and set

u x y e ps x y ds

Then if

x s

U f xE e f X d

s s

is the resolventof X u is the densityof U

u x y dy U f x

Write u for u

Prop osition Let r Ifr take

a If d then u x y is jointly continuous on F F and for

d d

s w

c exp c x y u x y

d d

s w

x y c exp c

b If d and then writing R x y

c R c R

log R e u x y c log R e c

c If d then

d d d d

w f w f

u x y c x y c x y

These b ounds are obtained by integrating for a and b one uses

Laplaces metho d The continuity in b follows from the continuityof p and

the uniform b ounds on p in Note in particular that

i if d thenu x x and lim u x y

ii if d thenux x whileux y for x y

Since the p olarity or nonp olarityofpoints relates to the ondiagonal b ehaviour

of uwe deduce from Prop osition

Corollary a If d then for each x y F

P X hits y

b If d then p oints are p olar for X

c If d then X is setrecurrent for

P ft X B y g is nonemptyandunb ounded

d If d and r then X is transient

In short X behaves like a Brownian motion of dimension d but in this con text a

continuous parameter range is p ossible

Lemma Polar and nonp olar sets Let A b e a Borel set in F

a P T if dim A d d

A H f w

b A is p olar for X if dim A d d

H f w

Since X is symmetric any semip olar set is p olar As in the Brownian case a

more precise condition in terms of capacity is true and is needed to resolvethe

critical case dim Ad d

H f w

If X X are indep endent FDd on F andZ X X then it follows

f t t

easily from the denition that Z is a FD on F F with parameters d and If

D fx xx F gF F is the diagonal in F F then dim D d and

H f

so Z hits D with p ositive probability if

d d d

f f w

that is if d So

P X X for some t if d

t s

and

P X X for some t if d

t s

No doubt as in the Brownian case X and X do not collide if d

Lemma X has k multiple p oints if and only if d kk

Proof By Rog X has k multiple p oints if and only if

u x y dy

B x

the integral ab oveconverges or diverges with

kd k d

w f

r r dr

by a calculation similar to that in Corollary

The b ounds on the p otential kernel density u x y lead immediately to the

existence of lo cal times for X see Sha p

Theorem If d then X has jointly measurable lo cal times L x F t

which satisfy the density of o ccupation formula with resp ect to

Z Z

f X ds f aL da f b ounded and measurable

In the lowdimensional case that is when d or equivalently d d

s f w

we can obtain more precise estimates on the Holder continuityof u x y and

hence on the lo cal times L The main lines of the argument follow that of BB

Section but on the whole the arguments here are easier as we b egin with stronger

hyp otheses Wework only in the case r the same results hold in the case

r with essentially the same pro o ofs

For the next few results we x F a FMSd with r and X a

FDd d onF For A F write

f w

T infft X A g

A A t

Let R b e an indep endent exp onential time with mean Set for

A x s y x

u x y E e dL E L

A A

U f x u x y dy

Let

A x

x y P p T R

y A

note that

A A A A

u x y p x y u y y u y y

F A A A A

and note that u x y u x y x y U U Write u x y u

A A A F x

U U As in the case of u we write p p for p p AsP X issymmetric

A A

wehave u x y u y x for all x y F

The following Lemma enables us to pass b etween b ounds on u and u

Lemma Supp ose A F A is b ounded For x y F wehave

x B A B x A

y y E u X u x y u x y E u X

R R

A A

Pro of From the denition of u

x y A x y

R R E L u x y E L

A A

x x X y

E L R E E L

R A

y y

x x y

R E L R E L L

A A

R R

x A x

u x y E u X y E u X y

R R

A A

Corollary Let x F andrThen

d d B xr d d

w f w f

c r u x x c r

Proof Write A B x r and let r where is to b e chosen Wehave

from Lemma writing x r

A x A

u x y u x y E u X y

So if v sup u x y then using

d x

v c P R v

Let t Then by

x x x

P R P R t P R t

x x

P R t P t

d d

f w

d t

r e ct

Cho ose rst t so that the second term is less than and then so that the rst

term is also less than Wehave t r and the upp er b ound now follows

from

The lower b ound is proved in the same way using the b ounds on the lower tail

of given in

Lemma There exist constants c c such that if x y F r x y

t r then

P T t x c r c

Proof Set r wehave p x y c expc by So since

x T x t

p x y E e P T te

we deduce that

x d

P T t c expc exp

As d we can cho ose dep ending only on c c and d suchthatP T

w w y

c expc c By for a t

d d x d

w w w

c expc a P x aR R

c So x c r t so there exists c such that P

x x x

P T t x c r P T t P x c r t c

y y

Denition We call a function h harmonic with resp ect to X inanopen

subset A F if Lh on Aorequivalently hX is a lo cal martingale

Prop osition Harnack inequality There exist constants c c

such that if x F andh is harmonic in B x c r then

hx c hy x y B x r

Proof Let c c so that B x c r B x c r ifx x r Fixx

y write r x y and set S T x c r As hX is a sup ermartingale

y S

wehaveby Lemma

x x

hx E hX hy P T x c r c hy

S y

Corollary There exists c such that if x F andh is harmonic

in B x r then

hx c hy x y B x r

Proof This follows bycovering B x r by balls of the form B y c r where c

is small enough so that Prop osition can b e applied in each ball Note weuse

the geo desic prop erty of the metric here since we need to connect eachballtoa

xed reference p ointbyachain of overlapping balls

Lemma Let x y F r x y IfRr and B y R A then

A A d d

w f

u y y u x y c r

Proof Wehave writing y r T T

A y y B y A y X

u y yE L u y yE u X y E E L

so by Corollary

y A A B d d

w f

E u y y u X y u y y c r

A A

Set x u y y u x y is harmonic on A fy gAsx y r and has

r x z r By Corollary the geo desic prop erty there exists z with y z

since is harmonic in B x r

z c x

Nowset x E X for x B Then is harmonic in B and on B

Applying Corollary to in B we deduce

y c z c z c x

y A A

Since y E u y y u X y the conclusion follows from

Theorem a Let Thenforx x y F andf L F g L F

d d

w f

ju x y u x yjc x x

d d

w f

j c x x jU f x U f x jjf jj

d d d

s w f

jjg jj jU g x U g x j c x x

Proof Let x x F write r x x and let R r A B x R Since

A A A

u y x p y xu x x wehave using the symmetry of X that

A A A A A

x x y xu y x p x y u x y u u

A A A

p y x u x x u x x

Thus

A A A A

ju x y u x yjju x x u x x j

Setting and using Lemma we deduce

A A d d

w f

ju x y u x yjc r

A A A A

jU f x U f x j ju x y u x yjjf y jdy

d d

w f

c r jjf jj

To obtain estimates for we apply the resolvent equation in the form

A A A

u x y u x y U v x

where v xu x y Note that jjv jj Thus

A A A A A A

x yjju x y u x yj jU v x U v x j x y u ju

d d d d

w f w f

c r c r jjv jj

d d

w f

c r

Letting R we deduce and then follows exactly as ab ove by

integration

Toprove note rst that p y xu y xu x x So by

Z Z

u y xdy y xjf y jdy jjf jj u x x p

A A

jjf jj u x x

c jjf jj

From and wehave

A A A A d d

w f

ju x y u x yjc p y xp y x r

and then follows byintergation using

The following mo dulus of continuity for the lo cal times of X then follows from the results in MR

x F t Theorem If d then X has jointly continuous lo cal times L

d d

w f

Let uu logu The mo dulus of continuityinspaceof L

is given by

x y

jL L j

s s

lim sup sup c sup L

x y

st st xF

jxy j

It follows that X is spacelling for each x y F there exists a rv T such

that P T and

B y fX t T g

The following Prop osition helps to explain why in early work mathematical

physicists found that for simple examples of fractal sets one has d See also

HHW

Prop osition Let F b e a FMS and supp ose F is nitely ramied Then if

X is a FDd d on F d X

f w s

Proof Let F F be two connected comp onents of F suchthatD F F is

nite If D fy y gx and set

u X y M e

t i t

Then M is a sup ermartingale Let T infft X D g and let x F D

D t

x x

Since P X F wehave P T So

x x

E M E M

and thus M as So u X y for each y D and thus wemust

T T i i

D D

have u y y for some y D So by Prop osition d

i i i s

Remark For k letF d b e FMS with dimension d k and

k k k f

common diameter r Let F F F letp andsetdx x y y

p p p

d x y d x y Then F d is a FMS with dimension

d d d Supp ose that for k X is a FDd k d k on F Then

f f f f w k

if X X X it is clear from the denition of FDs that if d d

w w

then X is a FDd on F However if d d then X is not a FD on

f w w

F Note from that the metric can up to constants b e extracted from the

transition density pt x y by lo oking at limits as t So the class of FDs is not

stable under pro ducts

This suggests that it might b e desirable to consider a wider class of diusions

with densities of the form

i i i

x y pt x y t exp t

where are appropriate nonnegative functions on F F Such pro cesses would

have dierent spacetime scalings in the dierent directions in the set F given

by the functions Arecent pap er of Hambly and Kumagai HK suggests that i

diusions on pcfss sets the most general typ e of regular fractal which has b een

studied in detail haveabehaviour a little like this though it is not likely that the

transition density is precisely of the form

Sp ectral prop erties

Let X beaFDonaFMSF with diameter r The b ounds on the density

pt x y imply that pt has an eigenvalue expansion see DaSi Lemma

Theorem There exist continuous functions and with

i i

such that for each t

pt x y e x y

n n

where the sum in is uniformly convergenton F F

Remark The assumption that X is conservative implies that while

the fact that pt x y for all t implies that X is irreducible so that

Awell known argument of Kac see Ka Section and HS for the necessary

Taub erian theorem can now b e employed to prove that if N f g

i i

then there exists c such that

d d

s s

c N c for c

So the numberofeigenvalues of L grows roughly as This explains the

term spectral dimension for d

Dirichlet Forms Markov Pro cesses and Electrical Networks

In this chapter I will give an outline of those parts of the theory of Dirichlet

forms and asso ciated concepts which will b e needed later For a more detailed ac

count of these see the b o ok FOT I b egin with some general intro ductory remarks

Let X X t P x F beaMarkov pro cess on a metric space F For

simplicity let us assume X is a Hunt pro cess Asso ciated with X are its semigroup

T t dened by

T f xE f X

t t

and its resolventU given by

Z Z

s t x

e f X ds T f xe dt E U f x

s t

While and make sense for all functions f on F such that the random

variables f X or e f X dsareintegrable to employ the semigroup or re

t s

solvent usefully we need to nd a suitable BanachspaceB kk of functions on

F suchthat T B B orU B B The two examples of imp ortance here are

C F andL F where is a Borel measure on F Supp ose this holds for one

of these spacesw e then havethatT satises the semigroup prop erty

T T T s t

ts t s

and U satises the resolvent equation

U U U U

WesayT is strongly continuous if kT f f k as t If T is strongly

t t B t

continuous then the innitesimal generator L D L of T is dened by

Lf limt T f f f DL

where D L is the set of f B for which the limit in exists in the space B

The HilleYoshida theorem enables one to pass b etween descriptions of X through

its generator L and its semigroup or resolvent

Roughly sp eaking if we take the analogy b etween X and a classical mechanical

system L corresp onds to the equation of motion and T or U 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 dierentiation

For general Markov pro cesses an energy description is not very intuitive

However for reversible or symmetric pro cesses it provides a very useful and p ow

erful collection of techniques Let b e a Radon measure on F that is a Borel

measure which is nite on every compact set We will also assume charges ev

ery op en set Wesay that T is symmetric if for every b ounded and compactly

supp orted f g

Z Z

T f xg xdx T g xf xdx

t t

Supp ose nowT is the semigroup of a Hunt pro cess and satises Since

T wehave writing for the inner pro duct on L F that

jT f xj T f x T x T f x

t t t t

byHolders inequality Therefore

kT f k T f f T f kf k kT f k

t t t t

so that T is a contraction on L F

The denition of the Dirichlet energy form asso ciated with T is less direct

than that of the innitesimal generator its less intuitive description maybeone

reason why this approach has until recently received less attention than those based

on the resolvent or innitesimal generator Another reason of course is the more

restrictive nature of the theory many imp ortant Markov pro cesses are not symmet

ric I remark here that it is p ossible to dene a Dirichlet form for nonsymmetric

Markov pro cesses see MR However a weaker symmetry condition the sector

condition is still required b efore this yields very much

Let F b e a metric space with a lo cally compact and countable base and let

b e a Radon measure on F SetH L F

Denition Let D b e a linear subspace of H A symmetric form E D isa

map E DD R such that

E is bilinear

E f f f D

For dene E on D by E f f E f f kf k and write

kf k kf k E f f E f f

Denition LetE D b e a symmetric form

a E is closed if D kk iscomplete

b E D isMarkov if for f Difg f then g Dand E g g Ef f

c E D isaDirichlet form if D is dense in L F and E D is a closed Markov

symmetric form

Some further prop erties of a Dirichlet form will b e of imp ortance

Denition E D isregular if

DC F is dense in D in kk and

DC F is dense in C F in kk

E is local if E f g whenever f g have disjoint supp ort

E is conservative if D and E

E is irreducible if E is conservativeand E f f implies that f is constant

The classical example of a Dirichlet form is that of Brownian motion on R

E f f jrf j dx f H R

Later in this section wewilllookattheDirichlet forms asso ciated with nite state

Markovchains

Just as the HilleYoshida theorem gives a corresp ondence b etween semi

groups and their generators so wehavea corresp ondence b etween Dirichlet

forms and semigroups Given a semigroup T the asso ciated Dirichlet form is

obtained in a fairly straightforward fashion

Denition a The semigroup T isMarkovian if f L F f

implies that T f ae

b A Markov pro cess X on F is reducible if there exists a decomp osition F A A

with A disjoint and of p ositive measure such that P X A for all t for

i t i

x A X is irreducible if X is not reducible i

Theorem FOT p Let T t b e a strongly continuous symmetric

contraction semigroup on L F whichis Markovian For f L F the

function t dened by

tt f T f f t

f t

is nonnegative and nonincreasing Let

D ff L F lim t g

E f f lim t f D

Then E D is a Dirichlet form If L D L is the innitesimal generator of T

then D L D D L is dense in L F and

E f gLf g f DLgD

As one might exp ect by analogy with the innitesimal generator passing from

a Dirichlet form E D to the asso ciated semigroup is less straightforward Since

formally wehave U L the relation suggests that

f g LU f g U f gE U f gE U f g

Using given the Dirichlet form E one can use the Riesz representation theorem

to dene U f One can verify that U satises the resolvent equation and is

strongly continuous and hence by the HilleYoshida theorem U is the resolvent

of a semigroup T

Theorem FOT p Let E D be a Dirichlet form on L F Then

there exists a strongly continuous symmetric Markovian contraction semigroup

T on L F with innitesimal generator L D L and resolvent U

such that L and E satisfy and also

E U f gf gf g f L F gD

Of course the op erations in Theorem and Theorem are inverses of each

other Using for a moment the ugly but clear notation E Thm T to

denote the Dirichlet form given by Theorem wehave

Thm Thm T T

t t

and similarly Thm Thm E E

Remark The relation provides a useful computational to ol to identify the

pro cess corresp onding to a given Dirichlet form at least for those who nd it more

natural to think of generators of pro cesses than their Dirichlet forms For example

given the Dirichlet form E f f jrf j wehave by the GaussGreen formula

R R

for f g C R Lf gE f g rfrg g f so that L

We see therefore that a Dirichlet form E D give us a semigroup T on

L F But do es this semigroup corresp ond to a nice Markov pro cess In gen

eral it need not but if E is regular then one obtains a Hunt pro cess Recall that

a Hunt pro cess X X t P x F is a strong Markov pro cess with cadlag

sample paths which is quasileftcontinuous

Theorem FOT Thm a Let E D b e a regular Dirichlet form on

L F Then there exists a symmetric Hunt pro cess X X t P x F

on F with Dirichlet form E

b In addition X is a diusion if and only if E is lo cal

Remark Let X X t P x R beBrownian motion on R Let

A R b e a p olar set so that

P T for each x

Then we can obtain a new Hunt pro cess Y X Q x R by freezing

x x c x

X on A Set Q P x A and for x A let Q X x all t

X Y

Then the semigroups T T viewed as acting on L R are identical and so

t t

X and Y have the same Dirichlet form

This example shows that the Hunt pro cess obtained in Theorem will not in

general b e unique and also makes it clear that a semigroup on L is a less precise

ob ject than a Markov pro cess However the kind of diculty indicated by this

example is the only problem see FOT Thm In addition if as will b e

the case for the pro cesses considered in these notes all p oints are nonp olar then

the Hunt pro cess is uniquely sp ecied by the Dirichlet form E

Wenowinterpret the conditions that E is conservative or irreducible in terms

of the pro cess X

Lemma If E is conservative then T and the asso ciated Markov pro cess

X has innite lifetime

Proof If f DL then E f f forany R and so E f

Thus Lf which implies that L ae and hence that T

Lemma If E is irreducible then X is irreducible

Proof Supp ose that X is reducible and that F A A is the asso ciated de

comp osition of the state space Then T and hence E As

t A A A A

in L F this implies that E is not irreducible

A remarkable prop erty of the Dirichlet form E is that there is an equivalence

between certain Sob olev typ e inequalities involving E and b ounds on the transition

density of the asso ciated pro cess X The fundamental connections of this kind were

found byVarop oulos VCKS pro vides a go o d account of this and there is a very

substantial subsequent literature See for instance Co and the references therein

WesayE D satises a Nash inequality if

E f f ckf k f D kf k jjf jj

This inequality app ears awkward at rst 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 R or a manifold it can often b e obtained from an isop erimetric

inequality

In what follows we x a regular conservative Dirichlet form E D Let T be

the asso ciated semigroup on L F and X X t P b e the Hunt pro cess

asso ciated with E

Theorem CKS Theorem a Supp ose E satises a Nash inequality

with constants c Then there exists c c c such that

kT k c e t t

b If T satises with constants c then E satises a Nash inequality

with constants c c c and

ProofIsketch here only a Let f DL Then writing f T f and

t t

g h f f T Lf

th th t t

wehave kg k kg k ash It follows that ddtf exists in L F

th h t

and that

f T Lf LT f

t t t

Set tf f Then

t t

h t h t T Lf T f g f f T Lf f f

t t th t th t th t

and therefore is dierentiable and for t

tLf f E f f

t t t t

If f L F T f DL for each t So extends from f DLtoall

f L F

Nowletf and kf k wehave jjf jj Thenby for t

t E f f jjf jj ckf k t ct

t t t t

Thus satises a dierential inequality Set te t Then

t c t e c t

If is the solution of c then for some a R wehave for c c c

tc t a

If is dened on then a so that

t c t t

It is easy to verify that satises the same b ound so we deduce that

t t

kT f k e t c e t f L kf k

Nowletf g L F withkf k kg k Then

T f gT f T g kT f k kT g k c e t

t t t t t

Taking the supremum over g it follows that kT f k c e t that is

replacing t by t that

e t kT k c

Remark In the sequel we will b e concerned with only two cases either

or and we are only interested in b ounds for t In the latter case we

can of course absorb the constant e into the constant c

This theorem gives b ounds in terms of contractivity prop erties of the semigroup

T If T has a nice density pt x y then kT k sup pt x y so that

t t t

gives global upp er b ounds on pt of the kind we used in Chapter To

derive these however we need to know that the densityof T has the necessary

regularity prop erties

So let F E T b e as ab ove and supp ose that T satises Write P x

t t t

for the transition probabilities of the pro cess X By wehave for A BF

and writing c ce t

P x A c A for aa x

t t

Since F has a countable base A we can employ the arguments of FOT p to

see that

P x A c A x F N

t n t n t

where the set N is prop erly exceptional In particular wehave N and

t t

P X N or X N for some s

s t s t

for x F N From we deduce that P x for each x F N If

t t t

sandB then P y B for aa y and so

P x B P x dy P y B x F N

ts s t t

So P x for all s x F N So taking a sequence t we obtain

ts t n

a single prop erly exceptional set N N such that P x for all t

n t t

x F N Write F F N we can reduce the state space of X to F

Thus wehave for each t x adensityp t x ofP x with resp ect to These

can b e regularised byintegration

Prop osition See Y Thm There exists a jointly measurable transition

density pt x y t x y F F suchthat

pt x y dy for x F tABF P x A

pt x y pt y x for all x y t

pt s x z ps x y pt y z dy for all x z t s

Corollary Supp ose E D satises a Nash inequality with constants c

Then for all x y F t

pt x y c e t

We also obtain some regularity prop erties of the transition functions pt x

Write q y pt x y

Prop osition Supp ose E D satises a Nash inequality with constants c

Then for x F t q DL and

kq k c e t

E q q c e t

tx tx

Proof Since q T q and q L wehave q DL and the b ound

tx tx

t tx tx

follows from

Fix xwritef q and let tkf k Then

t tx t

t Lf f Lf Lf

t t t t

So is increasing and hence

tt s ds t t t

Therefore using

t t t ce t E f f

t t

Traces of Dirichlet forms and Markov Pro cesses

Let X be a symmetric Hunt pro cess on a LCCB metric space F with

semigroup T and regular Dirichlet form E D To simplify things and b ecause

this is the only case we need we assume

Capfxg for all x F

It follows that x is regular for fxg for each x F that is that

P T x F

Hence GK X has jointly measurable lo cal times L x F t such that

Z Z

dx f L F f xL f X ds

Nowlet be a nite measure on F In general one has to assume charges

no set of zero capacity but in view of this condition is vacuous here Let A

be the continuous additive functional asso ciated with

da A L

and let inffs A tg b e the inverse of A Let G b e the closed supp ort of

t s

e e e

Let X X then by BG p X X P x G is also a Hunt pro cess

t t

WecallX the trace of X on G

Now consider the following op eration on the Dirichlet form E For g L G

set

E g ginffE f f f j g g

Theorem Trace theorem FOT Thm

e e

a E D is a regular Dirichlet form on L G

e e e

b X is symmetric and has Dirichlet form E D

e e e

Thus E is the Dirichlet form asso ciated with X we call E the trace of E on G

e e

Remarks The domain D on E is of course the set of g such that the inmum

in is nite If g D then as E is closed the inmum in is attained

by f say If h is any function whichvanishes on G then since f hj g we

have

E f f Ef h f h R

whichimpliesE f h So if f DL and wecho ose h D then h Lf

so that Lf aeon G

This calculation suggests that the minimizing function f in should b e

the harmonic extension of g to F that is the solution to the Diric hlet problem

f g on G

Lf on G

We shall sometimes write

E TrEjG

to denote the trace of the Dirichlet form E on G

Note that taking traces has the tower prop ertyif H G F then

TrEjH Tr TrEjG H

Wenowlookatcontinuous time Markovchains on a nite state space Let F b e a nite set

Denition A conductance matrix on F is a matrix A a x y F

which satises

a x y

a a

xy yx

a a LetE ffx y g a gWesaythatA is irreducible Set a

xy xx A xy x

y x

if the graph F E is connected

Wecaninterpret the pair F A as an electrical network a is the conductance

of the wire connecting the no des x and y Theintuition from electrical circuit theory

is on o ccasion very useful in Markov Chain theory for more on this see DS

Given F Aasabove dene the Dirichlet form E E with domain C F

ff F Rg by

E f g a f x f y g x g y

Note that writing f f x etc

X X

a f f g g E f g

xy x y x y

y x

X X X X

a f g a f g

xy x x xy x y

x x

y x y x

X X X

a f g a f g

xx x x xy x y

x x

y x

X X

a f g f Ag

xy x y

x y

In electrical terms gives the energy dissipation in the circuit F Aif

f y f x ows in the no des are held at p otential f A current I a

xy xy

the wire connecting x and y whichhas energy dissipation I f y f x

a f y f x The sum in counts each edge twice We can of course

also use this interpretation of Dirichlet forms in more general contexts

gives a corresp ondence b etween conductance matrices and conser

vative Dirichlet forms on C F Let be anymeasureonF whichcharges every

p oint

Prop osition a If A is a conductance matrix then E is a regular conser

vative Dirichlet form

b If E is a conservativeDirichlet form on L F then E E for a conductance

matrix A

c A is irreducible if and only if E is irreducible

Proof a It is clear from that E is a bilinear form and that E f f If

g f thenjg g jjf f j for all x y so since a for x y

x y x y xy

E is Markov Since E f f cA jjf jj jjjj is equivalentto jjjj and so E is

closed It is clear from this that E is regular

b As E is a symmetric bilinear form there exists a symmetric matrix A such

that E f gf Ag Letf f then

x y

E f f a a a

xx xy yy

Taking it follows that a The Markov prop ertyof E implies that

E f f Ef f if So

a a

xx xy

which implies that a for x y Since E is conservativewehaveE f

f A for all f SoA and therefore a forall x

c is now evident

Example Let b e a measure on F with fxg for x F Let

us nd the generator L of the Markov pro cess asso ciated with E E on L F

Let z F g andf L F Then

X X

E f gg Af a f y a f z f y

zy zy

y y

and using wehave writing for the inner pro duct on L F

E f gLf g Lf z

Lf z a f x f z

xz z

x z

Note from that as wewould exp ect from the trace theorem changing

the measure changes the jump rates of the pro cess but not the jump probabilities

Electrical Equivalence

Denition Let F Abean electrical network and G F If B is a

conductance matrix on G and

E TrE jG

B A

we will say that the networks F A and G B are electrical ly equivalent on G

In intuitive terms this means that an electrician who is able only to access the

no des in G imp osing p otentials or feeding in currents etc would b e unable to

distinguish from the resp onse of the system b etween the networks F AandG B

Denition Eective resistance Let G G b e disjoint subsets of F The

eective resistance b etween G and G R G G is dened by

R G G inf fE f f f j fj g

B B

This is nite if F A is irreducible

If G fx y g then from these denitions we see that F Aisequivalentto

the network G B where B b isgiven by

b b b b R x y

xy yx xx yy

Let F A b e an irreducible network and G F b e a prop er subset Let

H G and for f C F write f f f where f f are the restrictions of

H G H G

f to H and G resp ectivelyIfg C G then if E TrE jG

T T

E g g inf f g A f C H

Wehave using obvious notation

T T T T T

f g A f A f f A g g A g

HH H HG GG

H H H

A g Note that The function f which minimizes is given by f A

HG H H

exists Hence as A is irreducible cannot b e an eigenvalue of A soA

A g E g gg A A A

HG GG GH

so that E E where B is the conductivity matrix

A B A A A

HG GG GH

Example Y transform Let G fx x x g and B b e the conductance

matrix dened by

b b b

x x x x x x

Let F G fy gand A b e the conductance matrix dened by

a i j

x x

i j

a i

x y i

If the and are strictly p ositive and we lo ok just at the edges with p ositive

i i

conductance the network G B is a triangle while F AisaYwithy at the

centre The Y transform is that F AandG B are equivalent if and only if

EquivalentlyifS then

This can b e proved by elementary but slightly tedious calculations The Y

transform can b e of great use in reducing a complicated network to a more simple

one though there are of course networks for which it is not eective

Prop osition See Ki Let F A b e an irreducible electric network and

R x y R fxg fy g b e the p oint eective resistances Then R is a metric on F

Proof WedeneR x x Replacing f by f in it is clear that

R x y R y x so it just remains to verify the triangle inequalityLetx x x

b e distinct p oints in F andG fx x x g

Using the tower prop erty of traces mentioned ab ove it is enough to consider

the network G B where B is dened by Let b and dene

x x

similarly Let b e given by using the Y transform it is easy to

see that

R x x i j

i j

The triangle inequalityisnow immediate

Remark There are other ways of viewing this and numerous connections

here with linear algebra p otential theoryetc I will not go into this except to

mention that is an example of a Schur complement see Car and that an

alternative viewp oint on the resistance metric is given in Me

The following result gives a connection b etween resistance and crossing times

Theorem Let F A b e an electrical network let be a measure on F

whichcharges every p oint and let X t b e the continuous time Markovchain

asso ciated with E on L F Write T inf ftX xg Then if x y

A x t

x y

E T E T R x y F

y x

Remark In view of the simplicity of this result it is rather remarkable that its

rst app earance whichwas in a discrete time context seems to have b een in

in CRRST See Tet for a pro of in a more accessible publication

Proof A direct pro of is not hard but here I will derive the result from the trace

theorem Fix x y letG fx y g and let E E TrEjG If R R x y then

wehave from the denitions of trace and eective resistance

R R

e e

Let j the pro cess X asso ciated with E L G v therefore has generator

G t

given by

Lf z R f w f z

w z

e e e

Writing T T for the hitting times asso ciated with X we therefore have

x y

y x

e e

T R T E E

x x y y

Wenow use the trace theorem If f x x then the o ccupation density

formula implies that

L X ds jfs t X z gj

z z s s

X ds A

G s t

and thus if S infft T X xg and S is dened similarlywehave

y t

X ds S

G s

However by Do eblins theorem for the stationary measure of a MarkovChain

x x

GE S E X dsF

G s

Rearranging we deduce that

x x y

E S E T E T

y x

F G E S

x y

e e

RF E T E T F G

y x

Corollary Let H F x H Then

E T R x H F

Proof If H is a singleton this is immediate from Theorem Otherwise it

follows by considering the network F H obtained by collapsing all p oints in H

into one p oint hsaySoF F H fhg and a a

y H

Remark This result is actually older than Theorem see Tel