BROWNIAN MOTION ON A RANDOM RECURSIVE ... - Project Euclid

The Annals of Probability 1997, Vol. 25, No. 3, 1059 ᎐ 1102

BROWNIAN MOTION ON A RANDOM RECURSIVE SIERPINSKI GASKET 1 BY B. M. HAMBLY University of Edinburgh We introduce a random recursive fractal based on the Sierpinski gasket and construct a diffusion upon the fractal via a Dirichlet form. This form and its symmetrizing measure are determined by the electrical resistance of the fractal. The effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion. The main result is to obtain uniform upper and lower bounds for the transition density of the Brownian motion on the fractal in terms of this metric. The bounds are not tight as there are logarithmic corrections due to the randomness in the environment, and the behavior of the shortest paths in the effective resistance metric is not well understood. The results are deduced from the study of a suitable general branching process.

1. Introduction. The study of diffusion on fractals has concentrated on fractals with spatial symmetry. The randomization used in w 14x preserved this property, which was crucial to the analysis. In this work we introduce a random recursive fractal with statistical self-similarity and extend the theory of diffusion on fractals to a simple family of Sierpinski gaskets from this broader class of fractals. The fractal is based on the Sierpinski gasket as this is a simple multiply connected fractal and provides an instructive example where the behavior of random recursive fractals can be seen. A more general class of random recursive fractals based on nested fractals, w 23x , or affine nested fractals w 8x , could be constructed but provides more work in the construction of a diffusion. Recently T. Hattori w 15x has considered processes on scale irregular gaskets, another class of sets which are not exactly self-similar and include the homogeneous random fractals considered in w 14x . The random recursive Sierpinski gasket is a random recursive construction in the sense of w 25x , w 12x or net fractal in the terminology of w 6x . A family of sets of contraction maps is used recursively to generate the fractal. We will consider sets from the family of Sierpinski gaskets defined in w 14x , examples of which are shown in Figure 1. The fractal SGŽ ␯ . from the family has dimension d f s logŽ ␯ Ž ␯ q 1.r2.rlog ␯ and can be described by a ␯ Ž ␯ q 1.r2ary tree in which the nth generation branches of the tree correspond to the sets in the fractal at the nth level of construction.

Received September 1994; revised November 1996. 1 Research partially supported by UK SERC and Raybould Fellowship, University of Queensland. AMS 1991 subject classifications. Primary 60J60; secondary 60J25, 60J65. Key words and phrases. Brownian motion, random fractal, Dirichlet form, spectral dimension, transition density estimate, general branching process.

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B. M. HAMBLY

FIG. 1. The first two stages in the construction of three fractals from the family of Sierpinski gaskets.

The random recursive gasket can be described using a random tree. A branch is associated with each subset of the fractal of a given size and each splits into more branches, according to a simple Galton᎐Watson process, which describe the evolution of that component of the fractal. This tree will provide a means of discussing the geometry of the fractal. A version of the random recursive gasket formed from SGŽ2. and SGŽ3. is shown in Figure 2. In this fractal the graph formed from the vertices and edges of the triangles after n iterations of the contractions has no well defined length scale. However, such a scale will be imposed using the electrical resistance of the fractal. The resistance of each edge in the graph approximation to the fractal is determined by the need to maintain a unit resistance across the graph of unit side length. The contraction maps are then iterated until the resistance of each edge is approximately eyn . This provides a sequence of graphs Gn , convergent to the fractal, with the property that each edge has roughly the same resistance, and hence a random walker will move along any edge with roughly equal probability. The construction of a diffusion process on the random recursive gasket will be accomplished using Dirichlet forms. The method uses the connection between reversible Markov chains and electrical networks. By our choice of resistance the graphs Gn form a compatible sequence of networks in the sense of w 20x . The resistance also gives rise to a natural measure ␮ on the fractal which is equivalent to the Hausdorff measure in the effective resistance metric. From the associated sequence of Dirichlet forms and the corresponding Markov chains we can construct a natural diffusion process, X t associated with the limiting Dirichlet form Ž E , D Ž E .. on L2 Ž ␮ . which will be called ‘‘Brownian motion’’ on the fractal G. Let Ž ⍀, F , ⺠. be a probability space of infinite trees Žspecified in Section 2.. For each tree there is a corresponding fractal, and a diffusion process can be constructed via its Dirichlet form. THEOREM 1.1. For each ␻ g ⍀ there exists a continuous symmetric strong Markov process  X t : t G 04 on the space GŽ ␻ .. The properties of this process can be deduced with a further assumption, that there are only a finite number of possible gasket types. Then the results

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BROWNIAN MOTION ON A RANDOM FRACTAL

FIG. 2. The graph formed from the first stages in the construction of a random recursive Sierpinski gasket.

that we will state, concerning the properties of processes on random recursive gaskets, will be ⺠-almost sure results on the space of fractals ⍀. In order to determine the properties of the fractal we introduce a general branching process which describes the behavior of the sequence of approximating graphs. This will enable us to show that the box counting dimension is equal to the Hausdorff dimension. A feature of the geometry of the fractal is the existence of points in the fractal with widely differing ancestry, as the proportions of triangle types in each branch do not converge uniformly over the fractal. We also obtain uniform estimates on the measures of triangles in Gn . The effective resistance between two points x, y g G can be defined in terms of the Dirichlet form as d r Ž x , y . s Ž inf  E Ž f , f . : f g D Ž E . , f Ž x . s 0, f Ž y . s 1 4 .

y1

.

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In w 20x this was shown to be a metric, and it will be the appropriate metric with which to discuss the analytic properties of the fractal. Recent work of w 18x shows that the similarity dimension of the fractal provides a useful intrinsic notion of dimension. This is the Hausdorff dimension of the set in the effective resistance metric. The similarity dimension ␣ is defined as the unique solution to the following equation: N

½

␣ s s: ⺕

Ý ris s 1 is1

5

,

where ri is the resistance of the ith component, of which there are N. For the case of the Sierpinski gasket based upon the two generators SGŽ2. and SGŽ3., chosen with probability p and q, respectively, the similarity dimension is the number s which satisfies s

s

7 3 p Ž 35 . q 6 q Ž 15 . s 1.

From work in w 19x , where it is shown for P.C.F. self-similar sets, the similarity dimension can be written in terms of the spectral dimension as

␣s

ds 2 y ds

.

In w 20x this relationship is conjectured to hold in much greater generality. We will denote by ␣␯ the corresponding parameter associated with SGŽ ␯ .. The ␣␯ are seen to be increasing in ␯ , as for this family of gaskets the spectral dimension increases to 2 as ␯ ª ⬁ w 27x . We write K for the maximum value of ␯ . In the case where ␯ s 2, the similarity dimension of the Sierpinski gasket SGŽ2. is log 3rlogŽ5r3.. The effective resistance metric does not reflect the geometry of the fractal very well. In order to establish heat kernel estimates, we need to know how the shortest paths scale with effective resistance. The results in Section 7 show that there is a constant ␬ such that, if a n denotes the number of steps in the shortest path on Gn required to cross the fractal G, then

Ž 1.1.

n n ␬ 2 s exp Ž log 2rlog Ž 5r3 . . F lim inf a1r F lim sup a1r F ␬. n n

nª⬁

nª⬁

Transition density estimates for Brownian motion on fractals were first obtained in w 4x . The techniques were refined in w 3x , and applied to nested fractals in w 21x and affine nested fractals in w 8x . We will show how these techniques can be extended to incorporate the random recursive gaskets which are not spatially homogeneous. The spatial inhomogeneity in the structure leads to logarithmic corrections for the heat kernel. To express the results, we define the function g aŽ z . s Žlog z k 1. a, z ) 0, and the positive

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BROWNIAN MOTION ON A RANDOM FRACTAL

exponents b2 s

␣ ␣2

y 1,

␩ s b2 q 2 ␰ , ␤2 s

␣

,

␰ s b 2 q bK ,

␩1 s ␩␤ 2 q ␰ ,

␩ ˆ s ␣␰ q bK ,

bK s 1 y

log ␬ 2

␣ q 1 y log ␬ 2

,

␤␧ s

␣K

log Ž ␬ q ␧ .

␣ q 1 y log Ž ␬ q ␧ .

for ␧ ) 0.

Combining the effective resistance metric and the similarity dimension, we will obtain the following transition density estimates. THEOREM 1.2. Ži. For each ␻ g ⍀, there exists a jointly continuous real-valued function pt Ž x, y . on Ž0, 1x = GŽ ␻ . = GŽ ␻ .. Žii. There exist constants c1.1Ž ␻ ., c1.2 Ž ␻ . such that pt Ž x , y . F c1 .1 Ž ␻ . ty ␣ rŽ ␣q1. g b 2Ž 1rt .

1r Ž ␣ q1 .

= exp Ž yc1 .2 Ž ␻ . DU Ž x , y, t .

␤2

Ž logq DU Ž x , y, t . .

y␩ 1

.,

0 - t F t 0 , ᭙ x , y g GR , ⺠-a.s., where D Ž x, y, t . s d r Ž x, y rtg␩ Ž1rd r Ž x, y ... For ␧ ) 0 there exist constants c1.3 Ž ␻ ., c1.4 Ž ␻ . such that U

. ␣q1

pt Ž x , y . G c1 .3 Ž ␻ . ty ␣ rŽ ␣q1. g␩ˆ Ž 1rt .

ž

y1r Ž ␣ q1 .

= exp yc1 .4 Ž ␻ . D# Ž x , y, t .

␤␧

= Ž logq Ž D# Ž x , y, t . . .

␤␧ ␩ ˆ

ž ž logq

dr Ž x , y . t

log ␬ ␧y1

2

// /

,

0 - t - t 0 , x , y g GR , ⺠-a.s., . ␣q1

where D#Ž x, y, t . s d r Ž x, y borhood in GŽ ␻ . and t 0 - 1.

rtgy ␩ˆ Ž1rt . and GR is a suitably small neigh-

REMARKS. Ži. The bounds are not tight. The fact that even the on-diagonal bounds are not tight is due to the local inhomogeneities in the structure. There will almost always exist neighborhoods in the fractal which have atypical behavior. Žii. The exponents for the leading term in the exponential, d r Ž x, y . ␣q1 ty1 , differ in the bounds as it has not been possible to establish the existence of a shortest path exponent in the effective resistance metric for these fractals. In order to do this, we would need the existence of the limit in Ž1.1.. Žiii. The estimates for the transition density on the homogeneous random Sierpinski gasket w 14x differ as the nth term in the constructing sequence is

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B. M. HAMBLY

equivalent to choosing about e ␣ n random variables in the random recursive model. The set of trees which are used to construct the homogeneous random gaskets form a null set in the space of all random recursive gaskets. Živ. The transition density estimates for the Brownian motion on the Sierpinski gasket in the effective resistance metric can be recovered from the above. We set ␣ s ␣ 2 s ␣ K s log 3rlogŽ5r3., so that all the correction exponents are 0. Also the shortest path exponent log ␬ s d c s log 2rlogŽ5r3. will control the path uniformly at each level, allowing us to remove the ␧ and the final term in the lower bound Žfrom the proof. to obtain pt Ž x , y . ; c1 ty ␣ rŽ ␣q1. exp yc2 Ž d r Ž x , y .

ž

␣q1 y1 d c r Ž ␣ q1yd c .

t

.

/,

᭙ x , y g G, 0 - t - 1, where the constants c1 , c 2 differ for the upper and lower bounds. 2. A random recursive Sierpinski gasket. We define a class of random sets that have statistical self-similarity, following w 11x , w 12x , w 25x or alternatively w 6x . The fractal is formed recursively from the family of contractions described in w 14x . At each stage a member of the set of possible contraction maps is chosen according to a probability distribution, for each copy of the original set. We begin by constructing a space of trees, which controls the number of components of the fractal and their position. We then form the random recursive gasket by considering the limit set obtained by associating triangles with the branches of the tree. Let In s D nks0 ⺞ k and I s D k Ik be the space of arbitrary length sequences. We write i, j for concatenation of sequences. For a point i g I _ In denote by w ix n the sequence of length n such that i s w ix n , k for a sequence k. We write j F i, if i s j, k for some k, which provides a natural ordering on branches. Also denote by < i < the length of the sequence i. The infinite random tree, T, is a subset of the space I, defined as the sample path of a Galton᎐Watson process. Let the root be T0 s I0 , the empty sequence. Let Ui , i g T be positive i.d. random variables, indicating the number of offspring of an individual, with probability distribution P Ž Ui s ␯ Ž ␯ q 1 . r2 . s p␯ ,

␯ s 2, 3, . . . .

Then i g T if w ix n g Tn ; In for each n F < i < , where w ix n g Tn if we have the following: 1. w ix ny 1 g Tny1. 2. There is a k: 1 F k F Uwi x ny 1 such that w ix ny1 , k s w ix n . There is also a labeling of the tree given by the associated random variables Vwi x n s ␯ if Uwix n s ␯ Ž ␯ q 1 . r2. Then the pair ŽUi , Vi .; i g T 4 denotes a labeled tree. There is a natural probability space associated with these trees given by Ž ⍀, F , ⺠., where the

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BROWNIAN MOTION ON A RANDOM FRACTAL

␴-algebras are Fn s ␴ Ž Ui ; i g Tny1 Ž ␻ . . ,

⬁

Fs

D Fn ,

ns1

and the probability measure, ⺠, is determined by the Galton᎐Watson process with offspring distribution  p␯ : ␯ s 2, . . . 4 . Taking expectations with respect to this probability space of all constructing trees will be denoted by ⺕. For these random recursive fractals, the branching process will be supercritical with no possibility of extinction. For example in the case of the Ž2, 3.-gasket GŽ ␻ ., shown in Figure 2, we have generating function for the offspring distribution f Ž u. s pu 3 q qu 6 and Vi s 2 if Ui s 3, Vi s 3 if Ui s 6. We now define a sequence of sets to be attached to the branches and we drop the reference to the underlying probability space. Let E s E0 be the unit equilateral triangle, and let G 0 denote the complete graph on the vertices of E0 . Let ⌿ ␯ s ␺ i␯ , i s 1, 2, . . . ,

½

␯ Ž ␯ q 1.

5

2

,

denote the family of contraction maps of type ␯ , which involve dividing the side of the equilateral triangle by ␯ . Then set Ei , i g Tn , geometrically similar to E, to be V

V

Ei s ␺ i Ž E . s ␺ wixwi1x 1 ⭈⭈⭈ ␺ wixwinx n Ž E .

ž ž

//.

A random gasket can then be defined by GŽ ␻ . s

⬁

F D

Ei .

ns1 igTnŽ ␻ .

The Hausdorff dimension of the set GŽ ␻ . can be found by applying the results of w 6x , w 25x and is given by,

Ž 2.1.

½

d f Ž G Ž ␻ . . s inf ␣ :

⬁

Ý

␯s2

␯ Ž ␯ q 1. 2

p␯

1

␣

ž / 5 s1

␯

a.s.

REMARK. The homogeneous random fractal of w 14x , generated by a set of trees which is a null set in ⍀, has Hausdorff dimension df s

Ý⬁␯s2 p␯ log Ž ␯ Ž ␯ q 1 . r2 . Ý⬁␯s2 p␯ log ␯

.

We can also use w 12x to find the exact Hausdorff measure function for this random recursive construction. LEMMA 2.1. The exact Hausdorff measure function for GŽ ␻ . is given by h Ž t . s t d f Ž log < log t < . where d f s d f Ž GŽ ␻ ...

1y d fr2

, ⺠-a.s.

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B. M. HAMBLY

PROOF. We apply w 12x , Theorem 5.2, which, by w 12x , Theorem 5.4 will follow if n

E

Hi0 ) 1,

žÝ / is1

where n is the maximum number of new components and the Hi , the ratio of the component sizes, take only finitely many values. Also a condition on nondegeneracy is needed: n

P

žÝ

Hid f / 1 ) 0.

/

is1

For the case of a finitely ramified fractal, the first condition is trivial and the second follows provided there is a p␯ with 0 - p␯ - 1. Thus the Hausdorff measure function is given by ␪

h Ž t . s t d f Ž log < log t < . , where y1

␪

½

s sup a: a ) 1,

␯ Ž ␯ q 1. 2

1

ž / ␯

d fr Ž1y1ra .

5

F 1, ␯ s 2, 3, . . . .

Evaluating this gives the result. I The random recursive Sierpinski gasket G will be the set considered from now on. For a given realization of the constructing tree, let  Gn 4n G 0 be a sequence of graphs formed from the vertices of the triangles such that Gn ; Gnq1. For each n, Gn ; G⬁ , and the random fractal G can be recovered as G s clŽ G⬁ .. The corresponding set formed from the union of the triangles in Gn will be denoted by En . We will not explicitly refer to the underlying probability space when referring to the fractal or associated graph approximations. The results obtained will depend on this realization of the fractal and hence constants followed by a Ž ␻ . will denote a constant dependent on the environment. For x g Gn , we will denote by NnŽ x . the points in the graph Gn directly connected to x by an edge. For x g G _ G⬁ , let ^ nŽ x . denote the n-level triangle containing x, and let NnŽ^ nŽ x .. denote the n-level triangles having a common vertex with ^ nŽ x .. Then let DnŽ x . s ^ nŽ x . j NnŽ^ nŽ x .., the n-neighborhood of the n-triangle containing the point x. For a point x g Gn we define DnŽ x . to be the union of the n-triangles containing x. We may also write ^ nŽi. for i g T _ T⬁ , the triangle associated with the branch w ix n and then En s D i g T ^ nŽi.. n 3. General branching processes. In order to discuss the random gasket, we introduce a description of the set via a general branching process. The random tree used to construct the fractal is just the realization of a Galton᎐Watson process. It contains information about the number of sets at each level but not about the size of the sets at a given level. This can be

BROWNIAN MOTION ON A RANDOM FRACTAL

1067

incorporated by using a general branching process. We begin by discussing a strictly supercritical general branching process without referring to the specific process needed in the discussion of fractals. The general branching process is described in w 16x and w 1x . The set-up is as follows. We have a reproduction point process ␰ Ž t ., which describes the birth events as well as a life-length ␶ and a function ␾ on w 0, ⬁. called a random characteristic of the process. We make no assumptions about the joint distributions of these quantities. The individuals in the population are ordered according to their birth times ␴n . As we can have multiple births, this will not be a strictly increasing sequence. We denote the attributes of the nth individual by Ž ␰ n , ␶n , ␾n , . . . .. At time 0 we have an initial ancestor so that ␴ 1 s 0. The process that we wish to consider is written as Z␾ Ž t . s

Ý

n, ␴nFt

␾n Ž t y ␴n . .

An example of a random characteristic is

␾ Ž t . s I␶ ) t4 , so that Z ␾ Ž t . is the total number of individuals alive at time t. This process, for the current population size, will be denoted by z t . We also define the mean reproduction measure mŽ t . s E␰ Ž t .. We will assume that mŽ0. s 0 and in the supercritical case that it is Malthusian, so there exists an ␣ ) 0 such that ⬁

H0

ey␣ t m Ž dt . s 1.

We also assume that each individual has at least two offspring so there is no possibility of extinction and the process will grow rapidly. We will write m␣␾ Ž t . s E Ž ey␣ t Z ␾ Ž t . . , for the discounted mean of the process with random characteristic ␾ . We now introduce a martingale, analogous to the standard branching process martingale, which will enable us to discuss the asymptotic growth of this process. This plays an important role in the discussion of random recursive fractals. Let An s ␴ Ž Ž ␰ k , ␶ k , ␾ kŽ1. , . . . . : 1 F k F n . . Observe that the birth time of individuals is determined by their parents’ reproduction process, so that the birth times ␴ k are Aky1 measurable. Now define Rn s

⬁

Ý

ey␣ ␴ l Il is a child of 1 . . . n4 .

lsnq1

Then we have the following theorem.

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B. M. HAMBLY

THEOREM 3.1 Žw 1x , Chapter VI, Theorem 4.1.. The quantity  R n 4⬁ns1 is a nonnegative martingale with respect to An and W s lim R n exists. nª⬁

Also W ) 0 if and only if E

ž

⬁

H0

⬁

ey␣ t ␰ Ž dt . log

/ ž

H0

ey ␣ t ␰ Ž dt . - ⬁.

/

Otherwise W s 0, a.s. We will also need to look at various characteristics of the process and so we will state here w 1x , Chapter VI, Theorem 5.1 and its Corollary 5.3. THEOREM 3.2. Let ␾ i , i s 1, 2 be characteristics with sample functions that are right continuous and satisfying the following: Ži. EŽsup t G 0 eya t ␾ i Ž t .. - ⬁ for some 0 - a - ␣ , i s 1, 2; Žii. ␣ ) 0, lim t ª⬁ eyb t mŽ t . - ⬁ for some 0 F b - ␣ . Then lim tª⬁

Z ␾1 Ž t . Z␾2 Ž t .

s

m␣␾ 1 Ž ⬁ .

a.s.,

m␣␾ 2 Ž ⬁ .

and also lim ey␣ t Z ␾ i Ž t . s Wm␣␾ i Ž ⬁ .

Ž 3.1.

tª⬁

a.s. i s 1, 2.

It will be useful for estimating properties of the process to have estimates on the tails of the random variable W. For the left tail upper bound the approach will be similar to that used in w 2x and w 13x , where a loose estimate on the distribution function can be used with the branching structure to get an improved estimate. LEMMA 3.3. For a uniformly integrable family of nonnegative random variables  X a : a g C 4 there exist constants c 3.1 - 1 and c 3.2 such that P Ž X a - sE Ž X a . . F c 3 .1 q c 3 .2 s

᭙ s ) 0, ᭙ a g C .

PROOF. Let Ya s X arEŽ X a . and observe that, by uniform integrability, for ␧ ) 0 there exists a K such that EŽ Ya ; Ya ) K . - ␧ . Then 1F

K

H0

Ž 1 y P Ž Ya F s . . ds q ␧ ,

1 y ␧ F K Ž 1 y P Ž Ya F K . . and thus P Ž Ya F s . F 1 y as desired. I

1y␧ K

q

1y␧ K2

s,

s ) 0,

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BROWNIAN MOTION ON A RANDOM FRACTAL

We now use this estimate with a decomposition of the branching process martingale. Let Sn s inf m: ␴m G n4 , the number of births when the first birth occurs after time n. Then there will be more than b n offspring alive at this time, where b is the minimum family size born in a time unit Žassumed strictly bigger than one.. Thus the weighted sum of the offspring up to the stopping time Sn will be bigger than the sum of the weighted sums of the offspring of the first b m individuals run for time n y m, giving bm

Ž 3.2.

R Sn G

Ý exp Ž y␣␴S . R S m

ny m ,

i.

is1

As R n is a convergent martingale and Sn ª ⬁ as n ª ⬁, we can let n ª ⬁ in Ž3.2. to get bm

Ž 3.3.

WG

Ý Wi ey␣ m . is1

Using this we have the following result on the tail of the random variable W. LEMMA 3.4. There exist constants c 3.3 , c 3.4 such that if ␥ s log b, P Ž W - ␦ . F c 3 .3 exp Ž yc3 .4 ␦y␥ rŽ ␣y ␥ . . ,

␦ ) 0.

PROOF. By Lemma 3.3 we have a loose estimate for W of the form P Ž W - ␦ E Ž W . . F c 3 .1 q c 3 .2 ␦ . We can write P Ž Wey␣ m - ␦ . F c 3 .1 q ce ␣ m ␦ . Now apply Ž3.3. and w 2x , Lemma 1.1, to obtain log P Ž W - ␦ . F ycb m q C Ž b m e ␣ m␦ .

Ž 3.4.

s ycb m 1 y

ž

C c

1r2

Ž e Ž ␣y ␥ . m␦ .

1r2

/

.

We can now choose an appropriate value of m to obtain the bound. That is, let m 0 s logŽ C 2r4c 2␦ .rŽ ␣ y ␥ .. Then, for m s w m 0 x , e Ž ␣y ␥ . m␦ F

1 C 4

ž / c

2

.

Hence 1y

C c

Ž e Ž ␣y ␥ . m␦ .

1r2

Gˆ c

and ycb m F ycU␦y␥ rŽ ␣y ␥ . . We can now substitute this in Ž3.4. to obtain the result for small values of ␦ . By choice of c 3.3 we can extend the bound to all ␦ ) 0. I

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B. M. HAMBLY

In the case of the random recursive Sierpinski gasket generated from two types, the general branching process can be described explicitly. This is a Bellman᎐Harris or age dependent branching process as the offspring are born at the moment of death. In the case of this fractal, the life length is either log 3, when six offspring are produced, or log 2, when three appear. Thus from a single individual at time 0, there evolves a branching process, z t , where individual x has life and offspring according to

Ž ␭x , ␰x . s

½

Ž log 2, 3 Itslog 24 . Ž log 3, 6 Itslog 34 .

with probability p, with probability q.

Thus the number of individuals in the branching process z n , at time n, is the number of sets of diameter at least eyn in the fractal. Using the general branching process, results about the Hausdorff dimension and the upper bound for the Hausdorff measure function can be recovered. However, we will approximate the fractal via graphs with the conductance determining the construction of the graph, as this determines the appropriate metric to use on the fractal. Let ␥␯ denote the conductance of SGŽ ␯ . Žit is easy to calculate that ␥ 2 s 5r3, ␥ 3 s 15r7; see w 14x. . In this case the appropriate life length and reproduction point process are

Ž 3.5.

Ž ␭ x , ␰ x . s log ␥␯ ,

ž

␯ Ž ␯ q 1. 2

Itslog ␥␯ 4

/

with probability p␯ .

Thus the fractal G is approximated by the graph Gn given by Gn s

D ␺ i Ž G0 . ,

igT˜n

where T˜ is the family tree of the general branching process. This is a sequence of cut sets of the tree T. It is this approximate graph that will be considered from now on. We will now assume that the number of components that can be chosen is finite. Thus the random variable Vi g w 2, K x where we let K denote the maximal family of maps. As a corollary to Lemma 3.4, we state the general branching process result for the process used here. Let ␣␯ denote the Malthusian parameter corresponding to the general branching process of only type ␯ , that is, the similarity dimension of the fractal SGŽ ␯ ., which is ␣␯ s logŽ ␯ Ž ␯ q 1.r2.rlog ␥␯ . As noted before, ␣␯ s d s Ž ␯ .rŽ2 y d s Ž ␯ .. and as d s Ž ␯ .­2 as ␯ ª ⬁, we see that ␣␯ is increasing in ␯ . COROLLARY 3.5.

For the general branching process determined by Ž3.5.,

P Ž W - ␦ . F c 3 .3 exp Ž yc3 .4 ␦ ␣ 2 rŽ ␣ 2y␣ . .

᭙ ␦ ) 0.

PROOF. This follows by calculating the minimum family size possible in a time unit. I

BROWNIAN MOTION ON A RANDOM FRACTAL

1071

We can now provide an estimate on the right tail of W for the general branching process which is used here. First, observe that we can decompose W at the time of the first birth, U1

Ý ␥yV ␣ Wi .

Ws

Ž 3.6.

1

is1

In order to use large deviations, we must estimate the moment generating function, M Ž ␪ . s EŽexpŽ ␪ W ... Observe that the number of offspring is bounded over a unit time interval and hence the moment generating function will exist. Using Ž3.6., this satisfies the following inequality: M Ž ␪ . s E Ž exp Ž ␪ W . . , K

s

Ý

␯s2

p␯ M Ž ␪␥␯y ␣ .

␯ Ž ␯ q1 .r2

F max M Ž ␪␥␯y ␣ .

␯ Ž ␯ q1 .r2

2F ␯ FK

.

By iterating this inequality, we have

Ž 3.7.

M Ž ␪ . F max M Ž ␪␥␯y ␣ n .

Ž ␯ Ž ␯ q1 .r2 . n

2F ␯ FK

.

LEMMA 3.6. There exist constants c 3.5 , c 3.6 such that M Ž ␪ . F c 3 .5 exp Ž c 3 .6 ␪ ␣ K r ␣ .

᭙ ␪ G 0.

PROOF. Let c 3.6 be defined by c 3 .6 s

sup ␪ g w 1, ␥ K x

␪y␣ K r ␣ log M Ž ␪ . .

Assume that the result holds for ␪ g w 1, c x for some c ) ␥ K . Now consider the interval w c, ␥ 2␣ c x . Then, for all ␯ g w 2, K x , ␥␯y␣␪ g w 1, c x . By Ž3.7. and the fact that the ␣␯ are increasing in ␯ , we have M Ž ␪ . F max M Ž ␪␥␯y ␣ .

␯ Ž ␯ q1 .r2

2F ␯ FK

F max exp Ž c 3 .6 Ž ␪␥␯y␣ . 2F ␯ FK

␣ Kr␣

Ž ␯ Ž ␯ q 1. r2. .

F max exp Ž c 3 .6 ␪ ␣ K r ␣␥␯␣␯y␣ K . 2F ␯ FK

F exp Ž c 3 .6 ␪ ␣ K r ␣ . . By repeatedly applying this procedure, we can extend the interval from ␪ g w 1, ␥ K x to ␪ G 1. For ␪ - 1, we use the fact that M Ž ␪ . is increasing in ␪ and set c 3.5 s e c 3.6 to give the result. I THEOREM 3.7. There exist constants c3.7 , c 3.8 such that P Ž W ) ␦ . F c 3 .7 exp Ž yc3 .8 ␦ ␣ K rŽ ␣ Ky␣ . .

᭙ ␦ ) 0.

1072

B. M. HAMBLY

PROOF. Using Markov’s inequality, P Ž W ) ␦ . s P Ž exp Ž ␪ W . ) exp Ž ␪␦ . . F exp Ž y␪␦ . M Ž ␪ . F c 3 .5 exp Ž y␪␦ q c 3 .6 ␪ ␣ K r ␣ . , and now optimize over ␪ . I 4. The Dirichlet form and the diffusion. A natural method for constructing processes on finitely ramified fractals is via their Dirichlet forms Žsee w 10x , w 17x , w 20x. . We assign electrical conductances to each edge in the graph Gn and use this to determine the approximating Markov chains and invariant measure on the set. We consider the sequence Gn of approximating graphs constructed from the general branching process. Let the corresponding family tree be denoted by T˜n . The conductance assigned to the edges of the triangle ^ nŽi., i g T _ T˜ny 1 is the birth time of the individual corresponding to the triangle, that is,

␳n Ž i . s

⬁

Ł Ž ␥␯ . k Ž ␯ . , i n

x , y g Gn l ^ n Ž i . ,

␯s2

where k ni Ž ␯ . s

n Ži.y1

Ý js1

IVwi x s ␯ 4 , j

m

½

n Ž i . s inf m:

Ł ␥V

js1

wi x j

5

G en .

We will then write the conductance of edge Ž x, y . in the graph Gn as ␳nŽ x, y . s ␳nŽi., if x, y g Gn l ^ nŽi.. We can also define the weight on each node x g Gn as ␳nŽ x . s Ý y g N n Ž x . ␳nŽ x, y . and the associated measure ␮ n on the graph by

␮n Ž x . s

␳n Ž x . Ý x g G n ␳n Ž x .

,

x g Gn .

By our construction of the graph Gn , the conductance ␳nŽ x, y . is bounded above by e n and below by cU e n. The random walk moving according to the conductance of each edge in the graph is the natural random walk on a graph. For a continuous time random walk, the time for it to take each step is also determined by the conductance, being exponential with rate ␳nŽ x .. The natural Laplacian associated with the graph is determined by this conductance, and we write it as ⌬n f Ž x. s

Ý Ž f Ž y . y f Ž x . . ␳n Ž x , y . . ygNnŽ x .

The Laplacian on the fractal is obtained by taking the limit of this sequence as it is suitably normalized by our choice of conductance.

1073

BROWNIAN MOTION ON A RANDOM FRACTAL

Now let ␶n s Ý x g G n ␳nŽ x . and rewrite ⌬ n f Ž x . s ␶n

Ý Ž f Ž y. y f Ž x. . ygNnŽ x .

␳n Ž x , y . ␳n Ž x .

␮n Ž x . ,

and then the Dirichlet form for the sequence of chains is defined as En Ž f , g . s y Ž f , ⌬ n g . s

1 2

␶n Ý Ž f Ž x . y f Ž y . . Ž g Ž x . y g Ž y . .

␳n Ž x , y . ␳n Ž x .

␮n Ž x . .

Note that the Laplacian is determined when we fix the measure ␮ , and any finite measure with full support could be used. We choose this measure as it will turn out to be equivalent to the ␣-dimensional Hausdorff measure in the effective resistance metric. The crucial property of this sequence of quadratic forms is that it is nested. This allows the construction of the process to be completed. LEMMA 4.1. Let the graph G 0 have edge conductivities c x y s c for all edges xy. If the graph G1 is of type ␯ then, if the conductivity placed on each new edge is of the form ␥␯ c, the network conductivity is unchanged. PROOF. That there is a resistance which leaves the network conductivity unchanged follows as these are nested fractals w 23, 26x . As they have three essential fixed points, there is a unique fixed point for the conductivity map; see w 14x for the explicit case of SGŽ ␯ .. I A consequence of this result is that the sequence of Dirichlet forms EnŽ f, f . is monotone increasing. To see this write Q nq 1 for the q-matrix, the generator of the Markov chain corresponding to the random walk on the graph Gnq 1. Then the Ž n q 1.-level Dirichlet form can be written as Enq 1Ž f, g . s yf t Q nq 1 g. If we write Q nq 1 s

A nq 1 Cnq 1

Bnq1 , Dnq1

where A nq 1 determines transitions on Gn to itself Žit is diagonal for nested T fractals.; Bnq 1 and Cnq1 s Bnq1 give transitions from Gn to Gnq1 and vice versa, and Dnq 1 denotes transitions from Gnq1 _ Gn to itself. Then, by the choice of ␥␯ and hence ␳nŽ x ., the generator on Gn satisfies

Ž 4.1.

T y1 Dnq1 Bnq1 . Q n s A nq1 y Bnq1

This is the compatibility condition of w 20x and ensures that the forms have the property that

Ž 4.2.

Eny1 Ž g , g . s inf  En Ž f , f . : f < F ny 1 s g 4

for g g C Ž Fny1 . .

1074

B. M. HAMBLY

Thus the sequence of forms is monotone increasing and there exists a limiting bilinear form Ž E , D Ž E .. given by D Ž E . s f : sup En Ž f , f . - ⬁ ,

½

5

n

E Ž f , f . s lim En Ž f , f . , nª⬁

f g DŽ E . ,

In order to prove we have a Dirichlet form, we need some further results. We extend the definition of ␳nŽ x . to all x g G by setting ␳nŽ x . s ␳nŽi., if x g intŽ^ nŽi... Define an n-harmonic function h to be the function which satisfies ^ n h s 0. Following w 17x and w 22x it can be shown that this function is unique and can be extended uniquely to a function which is harmonic on the fractal G. For f g C Ž G . we write Hn f for the function which takes the values of f on Gn and is harmonic everywhere else. From w 20x there is an effective resistance defined by the limiting Dirichlet form d r Ž x , y . s Ž inf  E Ž f , f . : f Ž x . s 0, f Ž y . s 1 4 .

y1

,

which is a metric on G. LEMMA 4.2.

For x, y g ⭸ ^ n then there exist constants c 4.3 , c 4.4 such that c 4 .3 eyn F d r Ž x , y . F c 4 .4 eyn .

PROOF. This follows from the construction of the Dirichlet form. By definition we have for all f with f Ž x . s 0, f Ž y . s 1 with y g NnŽ x ., ce n F ␳n Ž x , y . F En Ž f , f . F E Ž f , f . . Thus there is a constant such that y1 Ž inf  E Ž f , f . : f Ž x . s 0, f Ž y . s 14 . F c4 .4 eyn .

For the lower bound, we take the n-harmonic function f 1 which is 1 at y and 0 at all other points of Gn . Then inf  E Ž f , f . : f Ž x . s 0, f Ž y . s 1 4 F E Ž f 1 , f 1 . s En Ž f 1 , f 1 . F ce n , and we have the lower bound. I Using the limiting form we have the following result from w 20x :

Ž 4.3.

f Ž x. y f Ž y.

2

F dr Ž x , y . E Ž f , f .

᭙ f g D Ž E . , ᭙ x , y g G,

which allows us to control the continuity of functions in the domain. LEMMA 4.3.

For any ^ nŽ z . we have that for all f g D Ž E ., sup

f Ž x . y f Ž y . F c 4 .2 ␳n Ž z .

y1 r2

EŽ f, f .

x , yg^ nŽ z .

The proof follows by combining Lemma 4.2 and Ž4.3..

1r2

.

BROWNIAN MOTION ON A RANDOM FRACTAL

1075

THEOREM 4.4. The bilinear form Ž E , D Ž E .. is a local regular Dirichlet form on L2 Ž GŽ ␻ ., ␮ .. An immediate corollary of this result is the existence of a diffusion process upon the fractal. COROLLARY 4.5. For each ␻ g ⍀, there exists a continuous, strong Markov diffusion process X t␻ on the set GŽ ␻ ., which is symmetric with respect to the measure ␮ ␻. The law of the process X t␻ on GŽ ␻ . started from the point x g GŽ ␻ . will be denoted by P x, ␻ , though we will usually suppress the ␻ . REMARK. By Lemma 4.1, the addition of new nodes and edges with the given conductivities within a given triangle only affect the fractal through the vertices of that triangle. This decomposition into separate pieces allows this technique to be used for any sequence of cutsets of the constructing tree. PROOF OF THEOREM 4.4. Fix a fractal GŽ ␻ .. To prove that the limiting form is a Dirichlet form, observe that the Markov property is inherited from the Markov property for the Dirichlet forms in the approximating sequence and the closability can be proved as in w 17x , Theorem 7.2, or w 22x , Theorem 4.14. Let f g C Ž G . and take f n s Hn f g F l C Ž G .. Then from Lemma 4.3 we see that F l C Ž G . is dense in C Ž G . in the sup norm and as EnŽ f, f . s E Ž Hn f, Hn f . we can prove the density in F. Thus the form is regular. The local property can be proved in the same manner as w 17x . I 5. The geometry of the fractal. We begin by computing the other dimensional indices of the fractal with respect to the Euclidean metric. The Hausdorff dimension is given in Ž2.1.. The box counting dimension of a set A is formed by covering the set with balls of fixed radius, ␧ , finding the minimum number N␧ Ž A., and setting d u b s lim sup ␧ x0

log N␧ Ž A . ylog ␧

.

LEMMA 5.1. The box counting dimension of G is given by du b s d f , ⺠-a.s. PROOF. Let ␧ s eyn . We regard the sequence of approximating graphs as those formed by constructing each set until its length is less than eyn . Then each set in En is of side length at least eyn and at most Keyn . By using the population size of the general branching process z n , corresponding to the fractal in the Euclidean metric, the set G can certainly be covered by

Ž 5.1.

Neyn Ž G . F K 2 z n

1076

B. M. HAMBLY

balls of size eyn . ŽThat this is in fact close enough to the minimum number of balls required will follow from the fact that value of the box counting dimension obtained is equal to the Hausdorff dimension.. Now d u b s lim sup log Neyn Ž G .

1rn

,

nª⬁

and by a simple application of Ž3.1. with characteristic given by ␾ Ž t . s IŽ␶ ) t . we see that c1 Ž ␻ . e sn - z n - c 2 Ž w . e sn ,

⺠-a.s.

with s in the required form. Thus, letting n ª ⬁, and using Ž5.1., gives d u b s lim sup nª⬁

F lim sup nª⬁

½

1 n

log Neyn Ž G .

1 n

s s s inf u:

Ž log c2 Ž ␻ . K 2 q sn . , K

Ý

␯ Ž ␯ q 1 . p␯ 2

␯s2

⺠-a.s.

5

eyu log ␯ s 1 s d f ,

⺠-a.s.

as desired. I Thus the geometric dimensions coincide and the set is fractal in the sense of Taylor w 28x . The natural measure for the diffusion on this set is determined not by the Hausdorff measure in the Euclidean metric but by the Hausdorff measure in the effective resistance metric. The dimensions of the set can now be stated in terms of the effective resistance metric. THEOREM 5.2. The Hausdorff dimension of the set GŽ w . in the effective resistance metric is given by

½

␣ s d rf s d ur e s s:

K

Ý

␯s2

␯ Ž ␯ q 2. 2

5

p␯ ␥␯ys s 1 ,

⺠-a.s.

PROOF. Following Lemma 5.1, using Lemma 4.2 and the general branching process z n in which the lifetimes are determined by the conductance, we have that d u b F ␣ . For the lower bound we use the mass distribution given by the measure ␮ ˜ obtained by putting weight 1rz n on each set in En . From the growth of the general branching process, z n F c1We ␣ n for large n. Thus for ␧ ) 0 there exists a constant c 2 such that ␮ ˜ ŽU . F c2 < U < r␣y ␧, for all sets U with diameter in the effective resistance metric < U < r - eyn . By w 6x , Theorem 4.2, the lower bound on the Hausdorff dimension in the effective resistance metric will be ␣ as desired. I The behavior of the measure ␮ introduced in Section 4 can be described.

BROWNIAN MOTION ON A RANDOM FRACTAL

1077

LEMMA 5.3. There exist constants c5.1Ž ␻ ., c5.2 Ž ␻ . such that c5 .1 Ž ␻ . e Ž ␣q1. n F

␳n Ž x . F c5 .2 Ž ␻ . e Ž ␣q1. n

Ý

᭙ n G 0.

xgGn

PROOF. To see this we use the general branching process with random characteristic given by

␾ Ž t . s e ␴ I␶ - t4 . As the births in the general branching process are determined by the conductivity, each individual alive at time t corresponds to a subset of the fractal with conductance given by its birth time e ␴. Thus Zn␾ s

Ý

␳n Ž x . .

xgGn

By the assumption of boundedness of the components, there exists a constant cU such that for all individuals alive at time t, their conductivity e ␴ i satisfies cU e t F e ␴ F e t. Now we can estimate the growth of the characteristic. For the upper bound we write z n for the population size of the general branching process at time n; then there is a constant c such that Zn␾ s

Ý e ␴ I␶ ) ny ␴ 4 i

i

i

i

F e n zn F cWe Ž ␣q1. n ,

⺠-a.s.

for large enough n. The lower bound is treated similarly. I This allows us to prove the following result for the measure ␮. Let SnŽ y . ␻ s ␻ nŽ y .q1 ␻ nŽ y .q2 . . . be the shift on sequences y in the tree ␻ . THEOREM 5.4. The measures ␮ n converge weakly to a measure ␮ equivalent to the ␣-dimensional Hausdorff measure in the effective resistance metric on the fractal G. PROOF. The weak convergence follows from Lemma 5.3 and the definition of ␮ n . For any ^ nŽ x . ; G we have

␮ Ž ^ n Ž x . . s lim

Ý y g ^ n Ž x .l G m ␳m Ž y .

mª⬁

Ý y g G m ␳m Ž y .

s lim ␳n Ž x . mª⬁

F ce n lim

mª⬁

Ý y g G my n Ž S nŽ x .␻ . ␳myn Ž y . Ý y g G m ␳m Ž y .

W Ž SnŽ x . ␻ . e Ž ␣q1.Ž myn. W Ž ␻ . e Ž ␣q1. m

1078

B. M. HAMBLY

for some constant c. Thus

␮ Ž ^ n Ž x . . F c5 .3 Ž ␻ . ey␣ n ,

Ž 5.2.

where the constant is given by c 3.5 Ž ␻ . s c sup ^ n W Ž SnŽ x . ␻ .Wy1 Ž ␻ .. We denote the corresponding lower bound constant by c5.4 Ž ␻ .. Then for any ␧ ) 0 there exist c1 , c 2 such that c1 F ␮ Ž ^ n Ž x . . e Ž ␣ q ␧ . n F ␮ Ž ^ n Ž x . . e Ž ␣ y ␧ . n F c 2

᭙ x g G, ᭙ n G 0.

By using a density theorem, w 7x , Proposition 4.9, we see that the measure ␮ must be equivalent to the ␣-dimensional Hausdorff measure in the effective resistance metric on the fractal. I We now obtain finer estimates on the measure of triangles ^ n . These will be crucial to the later estimates on the transition density for the Brownian motion. THEOREM 5.5. There exist constants c5.5 Ž ␻ ., c5.6 Ž ␻ . such that for all n ) 0 and x g GŽ ␻ .,

Ž 5.3.

c5 .5 Ž ␻ . Ž log n .

yb 2 y ␣ n

e

F ␮ Ž^ n Ž x . . F c5 .6 Ž ␻ . Ž log n .

bK

ey ␣ n ,

⺠-a.s.

There exist constants c5.7 Ž ␻ ., c5.8 Ž ␻ . such that for all n ) 0, c5 .7 Ž ␻ . nyb 2 ey␣ n F min ␮ Ž ^ n . F max ␮ Ž ^ n .

Ž 5.4.

^n

^n

F c5 .8 Ž ␻ . n

b K y␣ n

e

,

⺠-a.s.

PROOF. For Ž5.3. consider Ž5.2. in which the constants c5.3 Ž ␻ . and c5.4 Ž ␻ . can be written in terms of W. For the lower bound, consider a given x g G, then for all n,

Ž 5.5.

␮ Ž^ n Ž x . . G c

W Ž SnŽ x . ␻ . WŽ ␻.

ey ␣ n .

As a function of n the random variable W is a constant. Thus all we need is an estimate on the size of the fluctuations in W Ž SnŽ x . ␻ ., each of which has the same distribution as W. For the lower bound, we use the estimates on the tail of W obtained in Corollary 3.5 and the first Borel᎐Cantelli lemma. As ⺠ Ž W Ž Sn ␻ . F ␦ . F c 3 .3 exp Ž yc3 .4 ␦ ␣ 2 rŽ ␣ 2y␣ . .

᭙ n,

if we set ␦ s cŽlog n.Ž ␣ 2y␣ .r ␣ 2 , then ⺠ Ž W Ž Sn ␻ . F c Ž log n .

yb 2

. F c3 .3 nyc

3 .4 c

.

1079

BROWNIAN MOTION ON A RANDOM FRACTAL

Hence, as ⺠

ž

W Ž Sn ␻ . W

F

c W

Ž log n .

yb 2

/

s ⺠ Ž W Ž Sn ␻ . F c Ž log n .

yb 2

.,

we see that

Ý⺠ n

ž

W Ž Sn ␻ . W

F

c W

Ž log n .

yb 2

/

F c 3 .3

Ý nyc

3 .4 c

- ⬁,

n

by the choice of c. Thus W Ž Sn ␻ . W

G c Ž ␻ . Ž log n .

yb 2

⺠-a.s.

,

For the upper bound we follow the same approach and use Theorem 3.7 to obtain the upper bound on the fluctuations in W Ž Sn ␻ .. For Ž5.4. we need to estimate the maximum and minimum of W Ž SnŽ x . ␻ . over level n. As the number of possible triangles on level n is bounded above by e ␣ K n and each W Ž SnŽ x . ␻ . is independent and identically distributed we can use the following: P max W Ž Sn ␻ . ) ␦ F 1 y Ž 1 y P Ž W Ž Sn ␻ . ) ␦ . .

ž

/

^n

e ␣K n

F e ␣ K n P Ž W Ž Sn ␻ . ) ␦ . F c n with c - 1, if ␦ F c1 n b K , by choice of c1 in Theorem 3.7. In which case max W Ž Sn ␻ . F c5 .8 n b K , ^n

⺠-a.s.

as desired. As above we use the first Borel᎐Cantelli lemma to obtain the result. Similar methods yield for the lower bound. I We conclude with another indication of the type of bad behavior that will occur within the fractal. Define the ratio of types of map at level n for a branch by 1 n ␩n Ž i . s , i g T _ T⬁ . ÝI n js0 Vwi x js24 For each i g T _ T⬁ , ␩nŽi. ª p 2 , pointwise; however, the convergence is not uniform. LEMMA 5.6.

If p␯ ) 2r␯ Ž ␯ q 1. for ␯ G 3, then lim

inf ␩n Ž i . s 0

a.s.

sup ␩n Ž i . s 1

a.s.

nª⬁ igT_T⬁

If p 2 ) 1r3, then lim

nª⬁ igT_T ⬁

1080

B. M. HAMBLY

PROOF. We consider the case p 3 ) 1r6 as the other case follows in the same way. The process which generates type 3 triangles, Zˆn , has generating function f Ž u. s 1 y p 3 q p 3 u 6 and is supercritical for p 3 ) 1r6. Hence it has an extinction probability pe s f Ž pe . - 1. Let ␧ ) 0 and consider n

P

ž

inf ␩n Ž i . ) ␧ s P

/

igT_T⬁

ž

Ý IV js1

) n ␧ , ᭙ i g T _ T⬁ .

/

wi x j s24

For this event to occur, we require that, for each of the individuals alive at generation w n ␧ x , all their Zˆn offspring must become extinct. As the minimum number of individuals at this generation is 3w n ␧ x , P

ž

inf ␩n Ž i . ) ␧ F pe3

/

igT_T⬁

w n␧ x

n

F pec ,

for c ) 1. By the Borel᎐Cantelli lemma, as inf Ý P ž igT_T

␩n Ž i . ) ␧ - ⬁,

/

⬁

n

then P

ž

inf ␩n Ž i . ) ␧ i.o. s 0.

/

igT_T⬁

This holds for each ␧ , which gives the result. I 6. Resolvent and local time for the process. The Dirichlet form has an associated resolvent which is approximated by the sequence of resolvents associated with the sequence of forms. In this section we will approximate the process by the sequence of Markov chains associated with the resolvents. Let X tn denote the Markov chain on Gn moving according to the conductivities assigned to the edges of Gn . By construction this is the process on G watched only when it is in Gn Žsee w 14x. . Let the local time at x for the chain X n be denoted by Lxt Ž X n . s

t

H0 I

 X sn sx4

ds.

We normalize the local time by writing ⌳xt Ž X n . s

Lxt Ž X n .

␮n Ž x .

᭙ x g Gn .

We will write TAn s inf t: X tn g Ac 4 for the exit time from a set A. The normalized resolvent densities for the Markov chain killed upon leaving an open connected set A are defined as rAn Ž x , y . s E x

TAn

H0

I X snsy4

␮n Ž y .

ds s E x⌳ Ty An Ž X n .

᭙ x , y g Gn .

We begin by proving some elementary results for the sequence of resolvents.

1081

BROWNIAN MOTION ON A RANDOM FRACTAL

LEMMA 6.1.

Ž 6.1. Ž i . Ž 6.2. Ž ii . Ž 6.3. Ž iii.

rAn Ž x , y . F

␦ Ž x, y.

q

␳m Ž x .

zgNmŽ x .

rAn Ž x , y . s rAk Ž x , y . rAn

Ž x, y. s

rAn

␳m Ž x , z .

Ý

Ž y, x .

␳m Ž x .

x , y g Gm ;

rAn Ž z , y . ,

᭙ k G n, x , y g Gn ; ᭙ x , y g Gn .

The proof is essentially the same as w 14x , Lemma 7.1. From Lemma 6.1Žii. we can work with the limiting resolvent. Let rA Ž x , y . s lim rAn Ž x , y . nª⬁

᭙ x , y g Gm , ᭙ m.

For each n, by the strong Markov property of the process, we can write rAn Ž x , y . s ␺An Ž x , y . rAn Ž y, y . , where ␺An Ž x, y . s P x ŽTyn - TAn . with Tyn s inf t: X tn s y4 . Thus continuity properties of the resolvent can be recovered from these first passage time probabilities. The existence of the limit of the sequence of first passage time probabilities follows from the convergence of the resolvents, which is a consequence of the convergence of the Dirichlet forms; hence

␺A Ž x , y . s lim ␺An Ž x , y . s

Ž 6.4.

nª⬁

rA Ž x , y . rA Ž y, y .

.

In order to get bounds on the passage time probabilities, we need bounds on ratios of resolvents. The methods to be used will be the same as those in w 4x . First, we bound the resolvent on the diagonal. LEMMA 6.2. There are constants c6.1 , c6.2 such that c6 .1 rA Ž x , x . G , x g G⬁ l A, ^ m Ž x . ; A, Ž 6.5. ␳m Ž x . c6 .2 rA Ž x , x . F q max rA Ž z , z . ᭙ x g G⬁ . Ž 6.6. ␳m Ž x . zg ⭸ ^ mŽ x . PROOF. For Ž6.5., let A s Dm Ž x .; then, doing some simple calculations on the interior of one possible neighborhood, shown in Figure 3, rA Ž a, a . s Ž ␳m Ž a . . rA Ž b, a . s

1 4

y1

q

2 ␳m Ž a, b .

␳m Ž a .

rA Ž b, a . ,

Ž rA Ž a, a . q rA Ž b, a . . ,

rA Ž c, a . s rA Ž d, a . s rA Ž e, a . s 0. Solving this gives rA Ž a, a . s

3 3 ␳m Ž a . y 2 ␳m Ž a, b .

s c Ž ␳m Ž a . .

y1

,

1082

B. M. HAMBLY

FIG. 3.

A neighborhood of x for the random recursive gasket.

and over all the interior points, there are constants C, Cˆ such that C Ž ␳m Ž a . .

y1

F rA Ž ⭈, ⭈ . F CˆŽ ␳m Ž a . .

y1

.

A similar calculation can be done for each possible neighborhood. For points in Gm _ Gmy1 the equations will be the same as for a triangle of that type, the only complications occur at the higher level vertices, such as a in Figure 2, where we introduce the terms dependent upon the environment. As there are only a finite number of neighborhoods, though there may be a large number, we can always find a constant c6.1 s min c: rAŽ⭈, ⭈ . G cŽ ␳m Ž a..y1 4 , which gives the bound for a g Gn . Then rAŽ x, x . G rAŽ z, x . s rAŽ x, z ., by Ž6.4. and symmetry, for z g ⭸ ^ m Ž x .. So rA Ž x , z . s

Ý

ag ⭸ ^ mŽ x .

Ž 6.7.

G

min

ag ⭸ ^ mŽ x .

P x Ž X T ^ s a . rA Ž a, z . m

rA Ž a, z .

G c6 .1 Ž ␳m Ž a . .

y1

a g Dm Ž x . l Gm _ Gmy1 .

,

For Ž6.6., we follow w 4x , Lemma 5.4, in a similar fashion to the lower bound. I Let

Ž 6.8.

␾ A Ž x , y . s 1 y ␺A Ž x , y . s

rA Ž y, y . y rA Ž x , y . rA Ž y, y .

.

LEMMA 6.3. There exist constants c6.3 , c6.4 such that if ^ m ; A, then y1

Ž 6.9.

␾A Ž x , z . rA Ž z , z . F c6 .3 ␳m Ž z .

Ž 6.10.

rA Ž y, y . y rA Ž z , z . F c6 .4 ␳m Ž z .

x g ^ m , z g ⭸ ^ mŽ x . ,

, y1

,

y g ^ m , z g ⭸ ^ mŽ y. .

1083

BROWNIAN MOTION ON A RANDOM FRACTAL

PROOF. For Ž6.9., we recall Ž6.1. and rearrange to get

␳m Ž z , b .

Ý bgNmŽ z .

␳m Ž z .

Ž rA Ž z , z . y rA Ž b, z . . F ␳m Ž z . y1 ,

z g Gm .

Thus for a g Nm Ž z ., we have

␳m Ž z , a . ␳m Ž z .

Ž rA Ž z , z . y rA Ž a, z . . F

␳m Ž z , b .

Ý

F ␳m Ž z .

Ž rA Ž z , z . y rA Ž b, z . .

␳m Ž z .

bgNmŽ z . y1

a, z g Gm l ^ m Ž x . .

,

Rearranging gives, for a, z g Gm l ^ m Ž x .,

Ž 6.11.

␾A Ž a, z . rA Ž z , z . s rA Ž z , z . y rA Ž a, z . F ␳m Ž z , a .

y1

F c1 ␳ m Ž z .

y1

.

To extend this to the interior of the triangle we use the fact that rAŽ⭈, z . is harmonic within ^ m and thus attains its extreme values on the boundary. So for x g ^ m and z g ⭸ ^ m Ž x . the result holds. For Ž6.10., use Ž6.6., rA Ž y, y . F c6 .2 ␳m Ž z .

y1

q

max

zg ⭸ ^ mŽ y .

᭙ y g G⬁ .

rA Ž z , z .

Then using Ž6.11. and rAŽ a, a. G rAŽ z, a. s rAŽ a, z ., we get rA Ž y, y . y rA Ž z , z . F c 2 ␳m Ž z .

y1

,

and we have the result. I We now have a fixed upper bound on the resolvent diagonal, when the set A s Dm Ž x .. COROLLARY 6.4. There exist constants c6.5 , c6.6 such that for x g G⬁ ,

Ž 6.12.

c6 .5 ␳m Ž x .

y1

F rD m Ž x . Ž x , x . F c6 .6 ␳m Ž x .

y1

.

For the proof, the lower bound is given in Lemma 6.2. The upper is similar to w 4x , Lemma 5.8. Now we obtain a loose bound on the denominator in Ž6.8.. LEMMA 6.5. There exists a constant, c6.7 , such that, rA Ž y, y . y rA Ž x , y . F c6 .7 d r Ž x , y .

᭙ x , y g G⬁ .

PROOF. Let m be such that eym y1 F d r Ž x, y . F eym . By Lemma 4.2 and our choice of graph, ^ m Ž x . l ^ m Ž y . / ⭋, and we can choose z g ⭸ ^ m Ž x . l

1084

B. M. HAMBLY

⭸ ^ m Ž y .. Then rA Ž y, y . y rA Ž x , y . s ␾A Ž x , y . rA Ž y, y . F Ž ␾A Ž x , z . q ␾A Ž z , y . . rA Ž y, y . s ␾A Ž x , z . rA Ž z , z . q Ž 1 q ␾A Ž x , z . .Ž rA Ž y, y . y rA Ž z , z . . q ␾A Ž z , y . rA Ž y, y . y rA Ž y, y . q rA Ž z , z . . Using ␾AŽ z, y . rAŽ y, y . y rAŽ y, y . q rAŽ z, z . s ␾AŽ y, z . rAŽ z, z . we get that

Ž 6.13.

rA Ž y, y . y rA Ž x , y . F 2 ␾A Ž x , z . rA Ž z , z . q 2 Ž rA Ž y, y . y rA Ž z , z . . .

By Lemma 6.3 we have bounds on both terms in this expression, rA Ž y, y . y rA Ž x , y . F 2 c6 .3 Ž ␳m Ž z . . s C ␳m Ž z .

y1

y1

q 2 c6 .4 ␳m Ž z .

y1

.

Now by the choice of m, we get ␳m Ž z .y1 F ceym F c6.7 d r Ž x, y ., the required upper bound. I Thus the resolvent is uniformly continuous in both variables and can be extended uniquely to a continuous function on G = G. The estimates obtained above will hold for all x, y g G and we then have the following corollary. COROLLARY 6.6. Ži. The process X t hits points, P x ŽTx s 0. s 1. Žii. The fine topology on G is the ordinary topology. For the proof, these results can be obtained from the above estimates in the same way as w 4x , Corollary 5.14. We end this section with some estimates on exit times from neighborhoods. LEMMA 6.7. There exist constants c6.8 Ž ␻ ., c6.9 Ž ␻ ., such that ⺠-a.s. for each x g G and all m ) 0,

Ž 6.14.

c6 .8 Ž ␻ . Ž log m .

yb 2 yŽ ␣ q1. m

e

F E x TD m Ž x . F c6 .9 Ž ␻ . Ž log m .

bK

eyŽ ␣q1. m ,

and constants c6.10 Ž ␻ ., c6.11Ž ␻ . such that ⺠-a.s. for all m ) 0, c6 .10 Ž ␻ . myb 2 eyŽ ␣q1. m F inf E x T ^ m Ž x .

Ž 6.15.

xgG

F sup E x T ^ m Ž x . F c6 .11 Ž ␻ . m b K eyŽ ␣q1. m . xgG

1085

BROWNIAN MOTION ON A RANDOM FRACTAL

PROOF. The exit time from an m-level triangle can be written E x TD m Ž x . s E x

HD Ž x . L m

y TD mŽ x .

␮ Ž dy . s

HD

rD m Ž x . Ž x , y . ␮ Ž dy . . mŽ x .

By Corollary 6.4, rD m Ž x .Ž x, y . s rD m Ž x .Ž y, x . F rD m Ž x .Ž x, x . F c6.7 ␳m Ž x .y1 . Then E x TD m Ž x . F c6 .6 ␳m Ž x .

y1

␮ Ž Dm Ž x . . F c6 .9 Ž ␻ . Ž log m .

bK

eyŽ ␣q1. m ,

by Theorem 5.5 Ž5.3.. For the lower bound we use the Holder continuity of the resolvent. If we ¨ write rD m Ž x . Ž x , y . s rD m Ž x . Ž y, y . y Ž rD m Ž x . Ž y, y . y rD m Ž x . Ž x , y . . , then, applying Corollary 6.4 and Lemma 6.5, we see that there are c1 , k such that for y g ^ mq k Ž x ., r D m Ž x . Ž x , y . G c1 ␳ m Ž x .

y1

.

Then E x T ^ m Ž x . G c1 ␳ m Ž x .

y1

␮ Ž ^ mqk Ž x . . ,

and applying Theorem 5.5 gives the desired result. For the second result we apply the same approach but use Ž5.4. from Theorem 5.5. I From this result we can obtain the following results on the oscillations in the mean exit times. COROLLARY 6.8. There exist constants c6.12 , c6.13 , c6.14 , c6.15 , dependent on ␻ such that E 0 T ^ m Ž0.

c6 .12 Ž ␻ . F lim inf

eyŽ ␣q1. m Ž log m .

mª⬁

F lim sup mª⬁

yb 2

E 0 T ^ m Ž0. eyŽ ␣q1. m Ž log m .

bK

F c6 .13 Ž ␻ . ,

⺠-a.s.

and c6 .14 Ž ␻ . F lim inf r x0

F lim sup r x0

E 0 TB r Ž0. r ␣q1 Ž log log Ž 1rr . .

yb 2

E 0 TB r Ž0. r ␣q1 Ž log log Ž 1rr . .

bK

F c6 .15 Ž ␻ . ,

⺠-a.s.

PROOF. The first result is a simple consequence of Ž6.14.. For the second result observe that Bc 4.3 r Ž0. ; ^ m Ž0. ; Bc 4.4 r Ž0. if r s eym . I

1086

B. M. HAMBLY

7. Shortest paths and crossing times. Now that we have some preliminary resolvent estimates, we can obtain an estimate on the crossing time of a triangle. This will then be used to obtain some more resolvent estimates and then estimates on the transition density using the techniques developed in w 3x and w 8x . We use w 2x , Lemma 1.1 and Lemma 4.3, to show how the mean crossing times determine the tail behavior of the actual crossing time. Let T0n Ž ␻ . s inf  t G 0: X t g Gn 4 , n Tiq1 Ž ␻ . s inf  t ) Tin Ž ␻ . : X t g Gn Ž ␻ . _  XT in Ž ␻ . 4 4 , n T˜in Ž ␻ . s Tin Ž ␻ . y Tiy1 Ž ␻.,

denote the sequence of crossing times for the process on Gn . A simple extension of Lemma 6.7 leads to the following lemma. LEMMA 7.1. There exist constants c 7.1Ž ␻ ., c 7.2 Ž ␻ ., such that ⺠-a.s. for all m ) 0, c 7 .1 Ž ␻ . myb 2 eyŽ ␣q1. m F inf E x T1m F sup E x T1m xgG

Ž 7.1.

xgG

F c 7 .2 Ž ␻ . m b K eyŽ ␣q1. m .

We now give an elementary technical lemma which will be useful. LEMMA 7.2. for some a, b,

If, for x, y ) e, there exist positive constants c1 , c 2 such that b

c1 y F x a Ž log x . F c 2 y, then there exist positive constants c 3 , c 4 such that c 3 y 1r a Ž log y .

yb ra

F x F c 4 y 1r a Ž log y .

ybra

᭙ x , y ) e.

The effective resistance metric does not take into account the geometry of the shortest paths on the fractal. We determine bounds on the growth rate in n of the number of steps in the shortest path on Gn across the unit fractal. For x, y g Gn , m ) n, let ⌸ n, m Ž x, y . denote the set of all paths from x to y on Gm . Let < ␲ < denote the number of steps in the path ␲ and let a n , m Ž x , y . s inf  < ␲ < : ␲ g ⌸ n , m Ž x , y . 4 be the number of steps in the shortest path on Gm between x, y g Gn . LEMMA 7.3. There exists a constant ␬ such that, for each pair x, y g G 0 ,

␬ 2 s exp Ž log 2rlog Ž 5r3 . . F lim inf a0 , m Ž x , y .

1rm

mª⬁

F lim sup a0 , m Ž x , y .

1rm

F ␬,

⺠-a.s.

mª⬁

PROOF. The lower bound is just the trivial worst case bound. The upper bound uses a branching argument. The following subbranching inequality

BROWNIAN MOTION ON A RANDOM FRACTAL

1087

holds: a 0, m Ž x , y .

Ž 7.2.

a0 , mqn Ž x , y . F

᭙ m, n ) 0,

a m , mqn Ž x iy1 , x i .

Ý is1

where a m, mqnŽ x iy1 , x i . s a m, mqn s a0, n in distribution Žand we suppress the reference to the points x, y .. We then follow the proof of a similar result in w 5x . First, taking expectations and using the independence, we have a submultiplicative sequence and hence there exists ␬ such that lim ⺕ Ž a0 , n .

1rn

nª⬁

s inf ⺕ Ž a 0 , n .

1rn

n)0

s ␬.

By inequality Ž7.2. we have a0, n k

a 0 , Ž nq1. k F

Ý

a0 , k Ž i . .

is1

Then consider a branching process Zn with offspring distribution given by a0, k , so that a0, Ž nq1. k F Znq1. Now let u ) ␬ and consider ⺠ Ž a 0 , n k G u n k . F ⺠ Ž Zn G u n k . F ⺠ Ž Zn G c n . , where c s u k G m s ⺕Ž a0, k . G ␬ k . From the boundedness of a0, k , using the ideas of Lemma 3.6 and Theorem 3.7, we have an exponential tail for the limiting random variable of the standard branching process martingale, W s lim n ª⬁ Znrm n. Thus there are constants c1 , c 2 such that

Ž 7.3.

⺠ Ž Zn ) ␦ m n . F ˆ c⺠ Ž W ) ␦ . F c1 exp Ž yc2 ␦ .

᭙ ␦ ) 0.

Hence ⺠ Ž Zn ) c n . F c1 exp Ž yc2 Ž crm .

n

.,

and thus

Ý ⺠ Ž a0 , n k G u n k . - ⬁

᭙ u ) ␬.

n

We can approximate the general case using, for all v ) 0, ⺠ Ž a 0 , n kqj G v n kqj . F ⺠ Ž a0 , n kqj G v n k . ,

j s 1, . . . , k

and, for all c ) 0, ⺠ Ž a 0 , n kqj G c . F ⺠ Ž a0 , Ž nq1. k G c . ,

j s 1, . . . , k.

Thus we have k

Ý ⺠ Ž a0 , n G v n . s Ý Ý ⺠ Ž a0 , n kqj G v n kqj . n

n js1

F

Ý Ý ⺠ Ž a0 , n kqj G v n k . n

j

F k Ý ⺠ Ž a0 , Ž nq1. k G v n k . - ⬁, n

for all v ) ␬ . Hence

n lim sup n ª⬁ a1r 0, n

F ␬ , ⺠-a.s. I

1088

B. M. HAMBLY

The shortest paths on Gm between points on G 0 will not necessarily be straight. Thus we can find an upper bound on ␬ using a general branching process which describes the growth of the number of steps in the straight path. In this case the appropriate life length and reproduction point process are

Ž ␭ x , ␰ x . s Ž log ␥␯ , ␯ Itslog ␥␯ 4 . with probability p␯ . The parameter ␬ F ␬ ˆ, the Malthusian parameter for the branching process, is given by K

½

␬ˆ s s:

Ý ␯␥␯ys p␯ s 1

␯s2

5

.

We need a uniform bound on the length of the shortest path between two points in the fractal, where the step size is determined by the resistance metric. LEMMA 7.4. For ␧ ) 0 there exists a constant c 7.3 such that for all i, n G 0, if x, y g Gi with y g Ni Ž x ., then n

a i , nqi Ž x , y . F c 7 .3 Ž i q 1 . Ž ␬ q ␧ . ,

⺠-a.s.

nŽ . PROOF. By the previous lemma we know that a1r 0, n x, y converges and we wish to determine how the convergence depends on x, y. We begin by considering the behavior in n:

Ž 7.4. ⺠ sup

ž

n

a i , nqi Ž x , y .

Ž␬q␧.

n

)␦ F

/

⬁

n

Ý ⺠ Ž ai , nqi Ž x , y . ) ␦ Ž ␬ q ␧ . . . ns0

We know that a i, nqi Ž x, y ., for x, y g Gi with x g Ni Ž y ., is a bounded discrete random variable equal in distribution to a 0, nŽ x 0 , y 0 ., where x 0 , y 0 g G 0 , and that

␬ s lim ⺕ Ž a0 , k .

1rk

.

kª⬁

Hence for ␧ ) 0 there exists a k 0 such that ⺕Ž a0, k . - Ž ␬ q ␧ . k for all k ) k 0 . Fix such a k and let Zn be the branching process with offspring distribution a0, k . Then a 0, Ž nq1. k F Znq1 and, as in Ž7.3., we have ⺠ Ž a0 , n k ) ␦ Ž ⺕ a0 , k .

n

. F ⺠ Ž Zn ) ␦ m n . F c1 exp Ž yc2 ␦ . .

1089

BROWNIAN MOTION ON A RANDOM FRACTAL

From this we can deduce that for any l, which we can write l s nk q j, we have ⺠ Ž a0 , l ) ␦ ⺕ Ž a0 , k .

lrk

. s ⺠ ž a0 , n kqj ) ␦ ⺕ Ž a0 , k . nqjrk / F ⺠ a 0 , Ž nq1. k ) ␦ Ž ⺕ Ž a0 , k . .

ž

F c1 exp yc2 Ž ⺕ Ž a0 , k . .

ž

nq 1

jrky1

␦

Ž ⺕ Ž a0 , k . .

jrky1

/

/

F c1 exp Ž yc3 ␦ . . Using this in Ž7.4. we have ⺠ sup

ž

n

a0 , n

Ž␬q␧.

n

)␦ F

/

⬁

Ý

⺠

ns0

F

⬁

Ý

žŽ

ž

⺠

ns0

F

⬁

Ý

a0 , n

␬q␧.

n

)␦

a0 , n

Ž ⺕ Ž a0 , k . .

nrk

ž ž

c1 exp yc2 ␦

ns0

/

)␦

Ž␬q␧.

Ž ⺕ Ž a0 , k . .

nrk

/

n

␬q␧

Ž ⺕ Ž a0 , k . .

n

1rk

//

F c 3 exp Ž yc4 ␦ . , by choice of k ) k 0 . Finally, ⺠

ž

max sup x , ygG i

n

a i , nqi

Ž␬q␧.

n

)␦ F

/

⺠ sup

Ý x , ygG i

ž

n

a i , nqi

Ž␬q␧.

n

)␦

/

F c5 exp Ž 2 ␣ i . exp Ž yc4 ␦ . . Hence there exists a constant c6 such that ⺠

ž

max sup x , ygG i

n

a i , nqi

Ž␬q␧.

n

) c6 i F exp Ž yc 7 i . ,

/

and Borel᎐Cantelli gives the result. I From Lemma 7.4 we can control the number of steps in the shortest path. First, we extend the definition of ⌸ n, m Ž x, y . to all x, y g G by setting ⌸ m Ž x, y . s ⌸ m, m Ž x m , ym ., where x m , ym g Gm are given by the lower left corners of ^ m Ž x . and ^ m Ž y ., respectively. Hence we can define a m Ž x , y . s inf  < ␲ < : ␲ g ⌸ m Ž x , y . 4 . THEOREM 7.5. For ␧ ) 0, there exists a constant c 7.4 such that for all x, y g G with d r Ž x, y . F eyk and for each m there exists a path ␲ s  x 0 , . . . , x