Multifractal formalisms for

the local spectral and walk dimensions

By BEN M. HAMBLY^{1}

Department of Mathematics, University of Bristol, University Walk, Bristol,

BS8 1TW, UK.

                    e-mail: b.hambly@bris.ac.uk

JUN KIGAMI

Graduate school of Informatics, Kyoto University, Kyoto 606-8501, Japan.

                   e-mail: kigami@i.kyoto-u.ac.jp

and TAKASHI KUMAGAI^{2}

Graduate school of Informatics, Kyoto University, Kyoto 606-8501, Japan.

                  e-mail: kumagai@i.kyoto-u.ac.jp

Abstract We introduce the concepts of local spectral and walk dimension for fractals. For a class of finitely ramified fractals we show that, if the Laplace operator on the fractal is defined with respect to a multifractal measure, then both the local spectral and walk dimensions will have associated nontrivial multifractal spectra. The multifractal spectra for both dimensions can be calculated and are shown to be transformations of the original underlying multifractal spectrum for the measure, but with respect to the effective resistance metric.

1 Introduction

Multifractal analysis was introduced in the physics literature (see for example [10]) to provide a finer description of fractal phenomena which displayed a range of power law scalings. A mathematically rigorous version has been developed in a number of papers, see for example [5], [6], [1], [17], [18]. We give a brief discussion. The local dimension of a measure μ at a point x in some fractal set K is defined to be, if the limit exists,

1Research partially supported by a Royal Society Industry Fellowship.

2Research partially supported by Grant-in-Aid for Scientific Research (B)(2) 10440029.

dim_{loc}(x) = lim_{r→0} \frac{log μ(B_{r}(x))}{log r},

where B_{r}(x) is a ball of radius r about x. The sets of interest in multifractal analysis are E_{α} = {x ∈ K : dim_{loc}(x) exists and equals α}. The multifractal spectrum is then defined to be the function f(α) = dim(E_{α}), for some notion of dimension. A multifractal formalism has been developed which relates the function f(α) to the Legendre transform of the moment sums of the measure.

Analysis on fractals has shown that there are other ‘dimensions’ which have an essential role to play in the study of the Laplace operator on a fractal. We will consider firstly the spectral dimension d_{s} (or fracton dimension), which characterises the scaling in the eigenvalue counting function, N(x), for the Laplace operator on the fractal, in that

d_{s} = lim_{x→∞} \frac{2 log N(x)}{log x}.

This definition shows that the spectral dimension is a global property of the fractal set (for a discussion of the behaviour of this counting function see [16]). Secondly we tackle the walk dimension d_{w}, the scaling exponent for the mean square displacement of a diffusing particle (see [13] for a physical discussion).

We will define local versions of these dimensions for a Laplace operator with respect to a self-similar measure and develop a corresponding multifractal analysis for a class of finitely ramified fractals. In order to give the spectral dimension a local interpretation we use the short time asymptotics of the heat kernel at a particular point in the fractal. Let P_{t} be the transition semigroup generated by the Laplace operator on the fractal. This has a density (the heat kernel) with respect to the self-similar measure, denoted by p_{t}(x, y). If we consider the diagonal of this density, then, in the case of fractals such as the Sierpinski gasket, when the self-similar measure is the Hausdorff measure, it is known that there exist constants 0 < c_{1}, c_{2} such that,

c_{1}t^{−d_{s}/2} ≤ p_{t}(x, x) ≤ c_{2}t^{−d_{s}/2} for all 0 < t < 1, x ∈ K.

If the set does not have as many symmetries as the Sierpinski gasket, or the Laplace operator is defined with respect to a measure which has a non-trivial multifractal spectrum, the exponent in these upper and lower bounds may depend on the particular point x. This leads to the idea of the local spectral dimension, defined (when it exists) as

d_{s}(x) = lim_{t↓0} \frac{2 log p_{t}(x, x)}{− log t}.  (1.1)

Note that as the Laplace-Stieltjes transform of the eigenvalue counting measure is the trace of the heat semigroup there is a relationship between the global spectral dimension and the local spectral dimension.

To define the local walk dimension, we will consider T_{B_{r}(x)}, the time taken for the diffusion, whose infinitesimal generator is the Laplace operator (with respect to the self-similar measure) on the fractal, to exit B_{r}(x). Then the local walk dimension is defined to be (when it exists),

d_{w}(x) = lim_{r→0} \frac{log E^{x}(T_{B_{r}(x)})}{log r}.  (1.2)

As in the case of the usual multifractal analysis we can ask about the dimension of the subsets of the fractal for which there is a particular local spectral or walk dimension. Let

J_{γ} = {x ∈ K : d_{s}(x) exists and equals γ},  (1.3)

and define the multifractal spectrum for the local spectral dimension to be g(γ) = dim_{H}(J_{γ}), where we denote the Hausdorff dimension by dim_{H}. For the local walk dimension we can consider sets

W_{γ} = {x ∈ K : d_{w}(x) exists and equals γ},  (1.4)

and define the corresponding multifractal spectrum to be h(γ) = dim_{H}(W_{γ}).

In this paper we will consider a class of finitely ramified fractals, called p.c.f. self-similar sets and show that both of these multifractal spectra are well defined. A p.c.f. self-similar set K consists of a triple (K, S, {F_{s}}_{s∈S}) where each F_{s} is a contraction map on K such that K = ∪_{i∈S}F_{i}(K) and S = {1, 2, ⋯, N} is a set of indices for the contraction maps. Typical examples are as follows.

Example 1.1 (1) Line segment:

K = [0, 1]; S = {1, 2}; F_{1}(x) = x/2, F_{2}(x) = x/2 + 1/2 (x ∈ R).

(2) Sierpinski gasket:

K = 2-dimensional Sierpinski gasket (Figure 1); S = {1, 2, 3}; F_{1}(x) = x/2, F_{2}(x) = (x − a)/2 + a, F_{3}(x) = (x − b)/2 + b (x ∈ R^{2}),

where we set a = (1, 0), b = (1/2, \sqrt{3}/2). Similarly, the n-dimensional Sierpinski gasket is a p.c.f. self-similar set.

(3) Tree-like set:

K = Figure 2; S = {1, 2}; F_{1}(z) = βz̄, F_{2}(z) = (1 − |β|^{2})z̄ + |β|^{2} (z ∈ C),

where we set β such that |β| < 1, |1 − β| < 1 and Im β ≠ 0.

Figure 1 here  Figure 2 here

Figure 1: Sierpinski gasket Figure 2: Tree-like set

We will show that the above multifractal spectra can be obtained as simple transformations of the multifractal spectrum (in the resistance metric) of the underlying measure.

Let μ be a Bernoulli measure on the fractal with scale factors {μ_{i} : ∑_{i} μ_{i} = 1} which has a multifractal spectrum f(α). Our main result, Theorem 3.4, shows that if the Laplace operator with respect to μ is constructed from a regular harmonic structure, then the multifractal spectrum of the local spectral dimension is given by

g(γ) = f(\frac{γ}{2 − γ}) for \frac{γ}{2 − γ} ∈ Supp(f).  (1.5)

We note that for p.c.f. fractals with regular harmonic structure the local spectral dimension is always less than 2. We also show in Theorem 4.1 that under the same assumptions the multifractal spectrum for the local walk dimension is given by

h(γ) = f(γ − 1) for γ − 1 ∈ Supp(f).  (1.6)

The multifractal analysis of the exponents for more general fractals and nonregular p.c.f. fractals will be treated in [11]. The techniques that we use here are a simple extension of those used in the usual multifractal analysis of measures coupled with precise local estimates on the heat kernel.

We can relate the multifractal spectra g, h to the scaling in the moment measures. If the harmonic structure is regular with resistance scale factors ρ_{i} > 1 (for all i ∈ S), we can define β(q) as the unique real number such that

∑_{i} μ_{i}^{q}ρ_{i}^{−β(q)} = 1.  (1.7)

The multifractal spectrum of the measure (in the resistance metric) is given by a Legendre transform f(α) = inf_{q}{β(q) + αq}. The multifractal spectra for the spectral and walk dimensions can also be obtained by taking a suitable Legendre transform of β and these results are given in Theorems 3.4 and 4.1.

Note that by the Einstein relation (see [13]) d_{w} = d_{f} + d_{r}, where d_{r} is a resistance exponent, hence in the resistance metric we anticipate that d_{w} = d_{f} + 1. However, at the local level there exist points x such that d_{w}(x) ≠ d_{f}(x) + 1. The relationship between the spectral, walk and fractal dimensions is that d_{s}/2 = d_{f}/d_{w} and this shows heuristically why the transformations are of the above form.

For the case of the line segment (Example 1.1 1)), our main result can be explained as follows. For 0 < p < 1, let μ be a Bernoulli measure on [0, 1] such that μ(F_{i_{1}} ∘ ⋯ F_{i_{n}}([0, 1])) = μ_{i_{1}} ⋯ μ_{i_{n}} for all i_{1}, ⋯ i_{n} ∈ {1, 2} where μ_{1} = p, μ_{2} = 1 − p (this measure is called the de Rham p-measure). When p = 1/2, it is the Lebesgue measure on [0, 1]. Let X_{t} be a diffusion process which corresponds to the self-adjoint operator \frac{d}{dx} \frac{d}{dμ} on L^{2}([0, 1], μ). This process can be considered as a (singular) time changed Brownian motion on [0, 1]. Define the local spectral dimension and the local walk dimension for this process as in (1.1), (1.2), and set J_{γ}, W_{γ} as in (1.3), (1.4). Using the moment scalings we can set

β(q) = \frac{log (p^{q} + (1 − p)^{q})}{log 2},

(from (1.7) with ρ_{1} = ρ_{2} = 2) and find an explicit formula for the multifractal spectrum of the de Rahm p-measure,

f(α) = \frac{log((\frac{− log((1−p)2^{α})}{log(p2^{α})})^{η} + (\frac{− log((1−p)2^{α})}{log(p2^{α})})^{η−1})}{log 2} + α\frac{log(\frac{− log((1−p)2^{α})}{log(p2^{α})})}{log p/(1 − p)},

where η = log(p)/ log(p/(1 − p)). Using the transformation of f in (1.5), (1.6) we have explicit formulas for g, h and

dim_{H}J_{γ} = g(γ) ∨ 0, dim_{H}W_{γ} = h(γ) ∨ 0.

Here dim_{H} is the Hausdorff dimension with respect to the resistance metric (which is equivalent to the Euclidean metric in this case). When p = 1/2, g(γ) ∨ 0 = δ_{1}(γ) and h(γ) ∨ 0 = δ_{2}(γ), which agrees with the fact that d_{s} ≡ 1, d_{w} ≡ 2 for Brownian motion. The main point of our result is that when p ≠ 1/2, the local behavior of d_{s}(x) and d_{w}(x) are highly dependent on x and we can describe some of this dependence through the multifractal analysis presented here.

The paper is organized as follows. In Section 2 we introduce the class of fractals that we will consider and define their Laplace operators and diffusion processes via Dirichlet forms. We also state some transition density estimates which we discuss in more detail in the appendices. The result for the local spectral dimension is given in Section 3 and for the local walk dimension in Section 4.

2 Dirichlet forms on p.c.f. self-similar sets and

their heat kernel estimates

In this section, we briefly introduce p.c.f. self-similar sets, an abstract formulation of finitely ramified fractals. We also introduce their Dirichlet forms, as there is a one to one correspondence (under some conditions) between the Dirichlet forms and the diffusion processes and Laplace operators on these sets. Finally we will state on-diagonal transition density estimates for the associated diffusion process. For more details and discussions of these sets we refer the reader to [14], [2]. Readers who are not familiar with this kind of abstract setting may just consider the concrete examples indicated in Example 1.1.

We first introduce the notion of a self-similar structure.

Definition 2.1 Let K be a compact metrizable space and for each s ∈ S ≡ {1, 2, ⋯, N}, let F_{s} : K → K be a continuous injection. The triple L = (K, S, {F_{s}}_{s∈S}) is said to be a self-similar structure on K if there exists a continuous surjection π : S^{N} → K such that π ∘ σ̃_{s} = F_{s} ∘ π for every s ∈ S. Here σ̃_{s} : S^{N} → S^{N} is defined as σ̃_{s}w = sw.

In this paper, for each w ∈ S^{N} we denote the i-th element in the sequence by w_{i} and write w = w_{1}w_{2}w_{3} · · ·. For w ∈ S^{n}, we write F_{w} = F_{w_{1}} ∘F_{w_{2}} ∘⋯∘F_{w_{n}} and K_{w} = F_{w}(K). In particular, K_{s} = F_{s}(K) for s ∈ S. Note that an iterated function system and its invariant set (see [6], p 29) is a self-similar structure in the sense of Definition 2.1 by taking π(w) = ∩_{n≥1}F_{w_{1}} ∘ F_{w_{2}} ∘ ⋯ ∘ F_{w_{n}}(K) .

Definition 2.2 Let L = (K, S, {F_{s}}_{s∈S}) be a self-similar structure on K. The critical set of L is defined by C(L) = π^{−1}(∪_{s,t∈S,s≠t}(K_{s} ∩ K_{t})) and the post critical set of L is defined by P (L) = ∪_{n≥1}σ^{n}(C(L)). L is called post critically finite (p.c.f. for short) if P = P (L) is a finite set.

Thus P is the set of addresses of all pre-images of intersection points and, as this is finite, we see that p.c.f. self-similar sets correspond roughly to the class of finitely ramified fractals as used in the physics literature. Throughout the paper we will only consider the p.c.f. self-similar sets which are connected. For m ≥ 0, let

P^{(m)} = ∪_{w∈S^{m}}wP, V_{m} = π(P^{(m)}) and V_{∗} = ∪_{m≥0}V_{m}.

Thus V_{m} is a set of points approximating the fractal and V_{0} is the boundary of the fractal. Let μ be a self-similar measure on K with μ(F_{i}(K)) = μ_{i} > 0 for i ∈ S where μ_{i} satisfies ∑_{i} μ_{i} = 1.

In [14], one of the authors introduced the notion of p.c.f. self-similar sets and constructed local regular Dirichlet forms (E, F) on them. In order to construct a Dirichlet form, the existence of a regular harmonic structure was assumed. That is the existence of a non-degenerate fixed point for a non-linear renormalization map. There are many fractals which satisfy this assumption (typical examples are those in Example 1.1). We will make this assumption for p.c.f. self-similar sets and introduce the results we need concerning the basic properties of their Dirichlet forms.

The resistance R(p, q) between points p, q ∈ K can be defined using the Dirichlet form (E, F), by

R(p, q) = (inf{E(f, f) : f(p) = 0, f(q) = 1})^{−1} ,

where we set inf ∅ = ∞. The function R(·, ·) determines a metric, which we call the (effective) resistance metric, on K.

Theorem 2.3 ([14],[15]) There exists a local regular Dirichlet form (E, F) on L^{2}(K, μ) which has the following property:

|f(p) − f(q)|^{2} ≤ R(p, q)E(f, f)  for f ∈ F, p, q ∈ K,  (2.1)
E(f, g) = ∑_{i=1}^{N} ρ_{i}E(f ∘ F_{i}, g ∘ F_{i})  for f, g ∈ F,  (2.2)

where ρ_{i} > 1 (i ∈ S). Further, if we set E_{(β)}(·, ·) = E(·, ·) + β(·, ·)_{L^{2}(K,μ)} for β > 0, then, E_{(β)} admits a positive symmetric continuous reproducing kernel g_{β}^{K}(·, ·).

It is not difficult to show the following approximate scaling property

c_{2.1}ρ_{w}^{−1}R(p, q) ≤ R(F_{w}(p), F_{w}(q)) ≤ c_{2.2}ρ_{w}^{−1}R(p, q),  (2.3)

for all p, q ∈ K, w ∈ S^{m}, m ≥ 0 for some constants c_{2.1}, c_{2.2} > 0. Here and in the following, we will write ρ_{w} = ρ_{w_{1}}ρ_{w_{2}} ⋯ ρ_{w_{m}} for w = w_{1}w_{2} ⋯ w_{m}.

For the cases of Example 1.1, one can construct Dirichlet forms satisfying (2.1) and (2.2) with the following ρ_{i}’s. 1) line segment: ρ_{1} = ρ_{2} = 2. (The classical Dirichlet integral E(f, g) = \frac{1}{2} ∫_{0}^{1} ∇f(x)∇g(x)dx together with the classical Sobolev space with the Neumann boundary condition is the corresponding Dirichlet form.) 2) Sierpinski gasket: ρ_{i} = 5/3, i ∈ {1, 2, 3} ( see [14]). 3) Tree-like set: ρ_{1} = 1/α, ρ_{2} = 1/(1 − α^{2}) for any 0 < α < 1 ( see [14]).

We now give a rough outline of the construction of the Dirichlet form. The first step is to construct a sequence of Dirichlet forms E_{m} on l(V_{m}) ≡ {f : V_{m} → R} which has a consistency property and is monotone increasing w.r.t. m (this is a form of decimation invariance and follows from our fixed point assumption).

The limit (E, F) is then defined firstly as a form on l(V_{∗}) by

F = {f ∈ l(V_{∗}) : lim_{m→∞} E_{m}(f|_{V_{m}}, f|_{V_{m}}) < ∞},
E(f, g) = lim_{m→∞} E_{m}(f|_{V_{m}}, g|_{V_{m}}) for f, g ∈ F,

and then, using the metric R, F is embedded into the space of continuous functions on K. It can be shown that this is a subspace of L^{2}(K, μ) for any Borel measure μ. The Laplace operator can then be obtained from the Gauss-Green formula. For more details see [14] and [15]. We remark again that there is a one to one correspondence between local regular Dirichlet forms and diffusion processes (up to quasi-everywhere starting point), [9]. The corresponding diffusion process {X^{x}(t)} (x ∈ K is the starting point) has the following self-similarity property,

X^{F_{i}(x)}(t/t_{i}) =^{d} F_{i}(X^{x}(t)) for x ∈ K, i ∈ S, 0 ≤ t ≤ inf{s ≥ 0 : X^{x}(s) ∈ V_{0}},

where t_{i} = ρ_{i}/μ_{i} and X =^{d} Y means X and Y have the same law.

In [4], one of the authors obtained estimates on the short time behaviour of the transition density of the diffusion corresponding to (E, F) on L^{2}(K, μ). (Note that the transition density is the fundamental solution of the heat equation). Here we state part of this result; more details can be found in Appendix A and B. Let μ̂ be any Bernoulli measure on K so that μ̂(F_{i}(K)) = μ̂_{i} (i ∈ S) and μ̂(V_{0}) = 0 where μ̂_{i} ≥ 0 satisfies ∑_{i} μ̂_{i} = 1. Also define d_{s}(μ̂)/2 as follows,

d_{s}(μ̂)/2 = \frac{∑_{i} μ̂_{i} log(1/μ_{i})}{∑_{i} μ̂_{i} log(ρ_{i}/μ_{i})}.  (2.4)

Theorem 2.4 ([4]) There exists a jointly continuous heat kernel p_{t}(x, y) of (E, F) on L^{2}(K, μ) such that

lim_{t→0} \frac{log p_{t}(x, x)}{log t} = −d_{s}(μ̂)/2 for μ̂-a.e. x ∈ K.

3 Multifractal structure of the local spectral di-

mension

3.1 The multifractal analysis of self-similar measures

In this subsection, we recall the usual multifractal analysis of self-similar measures in the context of p.c.f. self-similar sets with regular harmonic structures. See Section 11 of [6] for a description of the idea of multifractal analysis. In [6] (also in Theorem 3.2 of [5]), a strong separation condition is assumed on K and the objects treated are Cantor like sets. However, as in [1], we can work with the case when the components just touch, by relaxing the strong separation condition to the so called open set condition, that our p.c.f. self-similar sets satisfy (the condition in [1] corresponds to the open set condition for deterministic self-similar sets). Let B(x, δ) = {p ∈ K : R(p, x) < δ} denote the ball of radius δ in the resistance metric and let diam_{R}(A) denote the diameter of the set A with respect to R. Recall that for each w ∈ S^{n} we have

c_{1}(ρ_{w})^{−1} ≤ diam_{R}(K_{w}) ≤ c_{2}(ρ_{w})^{−1} and μ(K_{w}) = μ_{w},

where we denote ρ_{w} = ρ_{w_{1}···w_{n}} and μ_{w} = μ_{w_{1}···w_{n}}. For each q ∈ R, define β = β(q) as the (unique) real number satisfying

∑_{i=1}^{N} μ_{i}^{q}ρ_{i}^{−β(q)} = 1.  (3.1)

Now, define f : [α_{min}, α_{max}] → R to be the Legendre transform of β, given by

f(α) = inf_{−∞<q<∞}{β(q) + αq},  (3.2)

where −α_{min}, −α_{max} are the slopes of the asymptotes of the convex function β. We also assume that log(1/μ_{i})/ log ρ_{i} is not the same for all i ∈ S, as otherwise the following result (Theorem 3.3) is easy to deduce. In this case β is strictly convex and the infimum in (3.2) is attained at a unique q = q(α). By differentiation, this occurs when

α = −\frac{dβ}{dq} ,  (3.3)

so that f(α(q)) = β(q) + αq. Note that if any one of the three quantities q ∈ R, β ∈ R and α ∈ (α_{min}, α_{max}) is given, the other two are determined by (3.1) and (3.3). In particular,

α = \frac{∑_{i} μ_{i}^{q}ρ_{i}^{−β(q)} log(1/μ_{i})}{∑_{i} μ_{i}^{q}ρ_{i}^{−β(q)} log ρ_{i}}.  (3.4)

We thus see that

α_{min} = min_{i} \frac{log(1/μ_{i})}{log ρ_{i}} and α_{max} = max_{i} \frac{log(1/μ_{i})}{log ρ_{i}}.  (3.5)

For given q and β = β(q), define a probability measure ν on K by

ν(K_{w}) = μ_{w}^{q}ρ_{w}^{−β},  (3.6)

for each w ∈ S^{n}, n ∈ N. For each α ≥ 0, define

E_{α} = {x ∈ K : lim_{r→0} log(μ(B(x, r))/ log r = α},
Ê_{α} = π(\{w ∈ S^{N} : lim_{n→∞} \frac{∑_{i=1}^{n} log(1/μ_{w_{i}})}{∑_{i=1}^{n} log ρ_{w_{i}}} = α\}),

where π : S^{N} → K is the surjection defined in Definition 2.1. By the strong law of large numbers and (3.4), we have ν(Ê_{α}) = 1. Also, for each x = π(w) ∈ Ê_{α},

lim_{n→∞} \frac{log(ν(K_{w_{1}···w_{n}}))^{−1}}{log ρ_{w_{1}···w_{n}}} = β(q) + αq = f(α(q)).  (3.7)

For w ∈ S^{N}, denote w|n = w_{1} ⋯ w_{n}. Let

A_{ν} = \{x = π(w) ∈ K : lim_{r→0} \frac{log(ν(B(x, r))}{log r} = lim_{n→∞} \frac{log(ν(K_{w|n}))^{−1}}{log ρ_{w|n}} \},

and define A_{μ} in the same way by changing ν to μ. We prepare two basic lemmas for later use. See [1], [5], [6], [11] for the proof.

Lemma 3.1 ν(A_{ν}) = ν(A_{μ}) = 1

Lemma 3.2 Let E ⊂ K and let μ be a finite measure. If μ(E) > 0 and there is a constant a > 0 such that

lim_{r→0} log μ(B(x, r))/ log r = a, for all x ∈ E,

then dim_{H} E = a and μ is the Hausdorff measure on E (up to a constant multiple).

We are now ready to explain a theorem on the multifractal analysis of selfsimilar measures on p.c.f self-similar sets. In the case when log(1/μ_{i})/ log ρ_{i} is the same for all i ∈ S (therefore α_{min} = α_{max} ≡ α_{0}), we will define f(α) = (dim_{H} K)δ_{α_{0}}(α). Then the following holds.

Theorem 3.3 If α ∈ [α_{min}, α_{max}], then dim_{H} E_{α} = f(α). If α ∉ [α_{min}, α_{max}], then dim_{H} E_{α} = 0.

Proof: For α ∈ (α_{min}, α_{max}), the proof of dim_{H} E_{α} ≥ dim_{H}(E_{α} ∩ Ê_{α} ∩ A_{ν}) = f(α) is straightforward using (3.7), Lemma 3.1, Lemma 3.2 and the fact ν(Ê_{α}) = 1. dim_{H} E_{α} ≤ f(α) can be proved in the same way as Lemma 4.2 of [1]. The case α ∉ (α_{min}, α_{max}) is also routine (see, for example, Lemma 4.3 of [1] for α ∉ [α_{min}, α_{max}] and Theorem 2.14 of [5] for α ∈ {α_{min}, α_{max}}).

3.2 The main theorem

In this subsection, we will prove our main theorem on the multifractal structure of the local spectral dimension for diffusion processes with respect to regular harmonic structures on p.c.f. self-similar sets. In order to do this we must consider the set of points with the same local spectral dimension, thus for each γ ≥ 0, let

J_{γ} = \{x ∈ K : − lim_{t→0} \frac{2 log p_{t}(x, x)}{log t} = γ\}.  (3.8)

Also we require a suitable spectrum and for this purpose we extend the multifractal spectrum f for the measure defined in (3.2) to a function f̄ on the real line by setting

f̄(α) = \{ \begin{matrix}f(α), & α ∈ [α_{min}, α_{max}], \\ 0, & otherwise. \end{matrix}  (3.9)

Now define our candidate for the spectrum by g(γ) = f̄(γ/(2 − γ)). We note that

Supp g = [\frac{2α_{min}}{α_{min} + 1}, \frac{2α_{max}}{α_{max} + 1}] ⊂ (0, 2),

where the last inclusion is due to the regularity of the harmonic structure we assumed in the construction of the Dirichlet form.

We are now ready to state the main result of this paper which provides a multifractal formalism for the local spectral dimension.

Theorem 3.4 1. Let K be a p.c.f. self-similar fractal with a regular harmonic structure, then

dim_{H} J_{γ} = g(γ) for γ ≥ 0.

2. Let β(q) be defined as the number such that

∑_{i} μ_{i}^{q}ρ_{i}^{−β(q)} = 1.

The multifractal spectrum is given by

g(γ) = inf\{β(q) + \frac{γ}{2 − γ} q\} for γ ∈ [\frac{2α_{min}}{α_{min} + 1}, \frac{2α_{max}}{α_{max} + 1}].

As we will see, the proof follows from the standard techniques of multifractal analysis. We first show the lower estimate. Set γ(α) = 2α/(1+α) for each α ≥ 0 so that g(γ(α)) = f̄(α). Proposition 3.5

dim_{H} J_{γ(α)} ≥ f(α) for α ∈ (α_{min}, α_{max}).

Proof. We consider the measure ν constructed from the set of weights {μ_{i}} as in (3.6). Recall that for each given q ∈ R we have (3.4). On the other hand, if we take μ̂ = ν in Theorem 2.4, then, by (2.4), we have

d_{s}(ν) = \frac{2 ∑_{i} μ_{i}^{q}ρ_{i}^{−β} log(1/μ_{i})}{∑_{i} μ_{i}^{q}ρ_{i}^{−β} log(ρ_{i}/μ_{i})} = 2α/(1 + α).

Thus, by Theorem 2.4, we have

ν(J_{γ(α)}) = 1.

As J_{γ(α)} ∩ Ê_{α} ∩ A_{ν} ⊂ J_{γ(α)}, we obtain dim_{H} J_{γ(α)} ≥ dim_{H}(J_{γ(α)} ∩ Ê_{α} ∩ A_{ν}) = f(α) using Lemma 3.1 and Lemma 3.2.

We next show the upper estimate. As mentioned in the last subsection, α ∈ (α_{min}, α_{max}) and γ are determined by q ∈ R. We thus consider α = α(q), γ = γ(q) as functions of q. For q < 0, set

L̄_{q} = \{x ∈ K \ V_{∗} : lim inf_{t→0} −\frac{2 log p_{t}(x, x)}{log t} ≥ γ(q)\},
L_{q} = π(\{w ∈ S^{N} : lim inf_{n→∞} \frac{∑_{i=1}^{n} log(1/μ_{w_{i}})}{∑_{i=1}^{n} log ρ_{w_{i}}} ≥ α(q)\}).

For q ≥ 0, set

Ū_{q} = \{x ∈ K \ V_{∗} : lim sup_{t→0} −\frac{2 log p_{t}(x, x)}{log t} ≤ γ(q)\},
U_{q} = π(\{w ∈ S^{N} : lim inf_{n→∞} \frac{∑_{i=1}^{n} log(1/μ_{w_{i}})}{∑_{i=1}^{n} log ρ_{w_{i}}} ≤ α(q)\}).

Note that the definition of Ū_{q} uses lim sup, while the other quantities are defined using the lim inf. We now state the upper estimate.

Proposition 3.6 1) For each q < 0, dim_{H} L̄_{q} ≤ dim_{H} L_{q} ≤ f̄(α(q)).

2) For each q ≥ 0, dim_{H} Ū_{q} ≤ dim_{H} U_{q} ≤ f̄(α(q)).

Proof. We first prove 1). Using Proposition B.4, we have

−\frac{log p_{t}(x, x)}{log t} ≤ \frac{∑_{i=1}^{n} log(1/μ_{w_{i}})}{∑_{i=1}^{n}(log(1/μ_{w_{i}}) + log ρ_{w_{i}})} − \frac{c_{B.1}}{log t},

for x = F_{w|n}(x̄_{n}) ∈ K \ V_{∗} and t_{w|n}^{−1} < t < t_{w|(n−1)}^{−1}, where we use t_{i} = ρ_{i}/μ_{i} (i ∈ S). From this estimate, and the fact that γ(α) is monotone increasing on R_{+}, we have L̄_{q} ⊂ L_{q} and the first inequality holds. The second inequality can be proved in the same way as Theorem 2.5 in [5].

For 2), using Theorem A.1 1), we have

lim sup_{t→0} −\frac{log p_{t}(x, x)}{log t} ≥ lim sup_{n→∞} \frac{1}{−\frac{log c_{A.1}}{n} + 1 + \frac{c_{A.2}p_{n}(x)}{n}} \frac{1}{n} ∑_{i=1}^{Λ_{n}(x)} log(1/μ_{w_{i}})
≥ (lim sup_{n→∞} \frac{1}{1 + \frac{c_{A.2}p_{n}(x)}{n}} ) (lim inf_{k→∞} \frac{∑_{i=1}^{k} log(1/μ_{w_{i}})}{∑_{i=1}^{k} log t_{w_{i}}}) ,

where x = F_{w|n}(x̄_{n}) ∈ K \ V_{∗}, w|n ∈ Λ_{n} for each n ∈ N. See Appendix A for the definition of Λ_{n}, Λ_{n}(x) and p_{n}(x). The second inequality holds because, if Λ_{n}(x) = k, then ∑_{i=1}^{k−1} log t_{w_{i}} ≤ n < ∑_{i=1}^{k} log t_{w_{i}}. Using (A.1), one can take a subsequence {n_{i}}_{i} such that p_{n_{i}}(x) ≤ L. Thus lim sup_{n→∞} 1/(1 + c_{A.2}p_{n}(x)/n) = 1. From this, and the fact that γ(α) is monotone increasing on R_{+}, we have Ū_{q} ⊂ U_{q} and the first inequality holds. The second inequality can be proved in the same way as Theorem 2.3 in [5] (Theorem 2.3 in [5] is stated for the lim sup, but with a simple modification, the same conclusion holds for the lim inf).

Proof of Theorem 3.4. For the first part, we obtain the result for α ∈ (α_{min}, α_{max}) by combining Proposition 3.5, Proposition 3.6 and the fact that V_{∗} is a countable set. The case α ∉ (α_{min}, α_{max}) can be done similarly. The second part is a simple consequence of the fact that g is a transform of f.

Remark 3.7 When K is [0, 1]^{d} (d > 1) or the Sierpinski carpet, (which are not finitely ramified, thus not p.c.f. self-similar sets), we have the following partial result. For simplicity, we take K = [0, 1]^{d} (the carpet case is similar, using the results in [4], [3]). K is a self-similar set in the sense of Definition 2.1 with {F_{i}}_{i∈S}, S = {1, 2, ⋯, 2^{d}}, V_{0} = ∂[0, 1]^{d} where the F_{i}’s are affine contractions on R^{d} with the contraction rate 1/2. Let E be the Dirichlet integral and F the usual Sobolev space on L^{2}(K, m) (m is the Lebesgue measure). Then (E, F) is the Dirichlet form on L^{2}(K, m) which corresponds to Brownian motion. This Dirichlet form has the self-similarity property (2.2) with ρ_{i} ≡ ρ_{K} = 2^{2−d} for all i ∈ S. Let μ be a self-similar measure on K with μ(F_{i}(K)) = μ_{i} > 0 for i ∈ S where μ_{i} satisfies ∑_{i} μ_{i} = 1. In [4] it is proved that, when μ_{i} < ρ_{K} for all i ∈ S, (E, F̃) can be a local regular Dirichlet form on L^{2}(K, μ) where F̃ is naturally defined from F. Note that changing the measure for the Dirichlet form (to a mutually singular one) corresponds to a (singular) time change for the associated diffusion process via some positive continuous additive functional (see Section 5, 6 of [9] for details). It is also proved in [4] that Theorem A.1, Theorem A.2 and Theorem 2.4 hold in this setting.

Instead of the resistance metric (which we do not have in the current setting), we use the Euclidean metric. Accordingly, we must modify the definitions slightly. Define β = β(q) as the (unique) real number such that ∑_{i=1}^{m} μ_{i}^{q}2^{−β(q)} = 1, and let α_{min} = min_{i} log(1/μ_{i}), α_{max} = max_{i} log(1/μ_{i}). Set f, f̄ and J_{γ} as in (3.2), (3.9) and (3.8). Let a_{K} = log ρ_{K}/ log 2 unless ρ_{K} = 1 (when we work with the resistance metric, a_{K} = 1) and define g̃(γ) = f̄(a_{K}γ/(2 − γ)). When ρ_{K} = 1, we define g̃(γ) = (dim_{H} K)δ_{2}(γ). We then have the following result by the same proof as that of Proposition 3.5.

dim_{H} J_{γ} ≥ g̃(γ) for γ ∈ R_{+} \ \{\frac{2α_{min}}{α_{min} + a_{K}} , \frac{2α_{max}}{α_{max} + a_{K}} \}.

We cannot obtain the upper estimate for dim_{H} J_{γ} because we do not have precise enough heat kernel estimates on V_{∗}, which is not a countable set. (Note that V_{∗} is a set of points where at least one of the components is a binary rational.) For the same reason, we do not have an estimate for γ ∈ \{\frac{2α_{min}}{α_{min}+a_{K}} , \frac{2α_{max}}{α_{max}+a_{K}} \}.

4 Multifractal structure of the local walk di-

mension

In this section, we will provide a multifractal formalism for the local walk dimension. Let {X_{t}^{x}}_{t≥0,x∈K} be the diffusion process such that X_{0}^{x} = x, which corresponds to a Dirichlet form on the p.c.f. self-similar set with a regular harmonic structure. Let

W_{γ} = {x ∈ K : lim_{ϵ→0} log E^{x}T_{B(x,ϵ)}/ log ϵ = γ},

where T_{A} = T_{A}(X_{t}^{x}) = inf{t ≥ 0 : X_{t}^{x} ∈ A^{c}} for each Borel set A and the metric is the resistance metric. Our main result in this section is the following.

Theorem 4.1 1. Let K be a p.c.f. self-similar fractal with a regular harmonic structure, then

dim_{H} W_{γ} = f̄(γ − 1) for γ ≥ 1.

2. Let β(q) be defined as the number such that

∑_{i} μ_{i}^{q}ρ_{i}^{−β(q)} = 1.

The multifractal spectrum is given by

h(γ) = inf{β(q) + (γ − 1)q} for γ ∈ [α_{min} + 1, α_{max} + 1].

The proof is basically similar to those in Section 3 and we just sketch it. In the following, we use the notation in Appendix A. Firstly, we will give estimates for the exit time from the neighbourhood of a point. They can be proved in the same way as Lemma 4.1 in [4] and Lemma 3.5 in [12].

Lemma 4.2 There exists c_{4.1}, c_{4.2} > 0 such that

E^{x}[T_{∂D_{Λr} (x)}] ≤ c_{4.1}e^{−r} for x ∈ K, r > 0,  (4.1)
E^{x}[T_{∂D_{Λr} (x)}] ≥ c_{4.2}e^{−(r+l)} for x ∈ K \ ( ⋃_{x_{i}∈∂D_{Λr} (x)} D_{Λ_{r+l}}(x_{i})), r, l > 0. (4.2)

Lemma 4.3 ν(\{x = π(w) ∈ K : lim_{ϵ→0} \frac{log E^{x}T_{B(x,ϵ)}}{log ϵ} = lim_{n→∞} \frac{log(μ(K_{w|n}))^{−1}}{log ρ_{w|n}} + 1\}) = 1.

Proof. First, note that by a Borel-Cantelli argument, we have lim sup_{r→∞} \frac{p_{r}(x)}{log r} ≤ c_{1} for ν − a.e. x, by choosing c_{1} > 0 large enough (this can be proved in a similar way to Proposition 3.3 of [4]). Thus,

lim_{r→∞} \frac{p_{r}(x)}{r} = 0  ν − a.e. x.  (4.3)

Now, using (4.2) and (4.3), the following can be easily deduced (the proof is similar to that of Theorem 3.1 of [5]),

lim sup_{ϵ→0} \frac{log E^{x}T_{B(x,ϵ)}}{log ϵ} ≤ lim sup_{n→∞} \frac{log(μ(K_{w|n}))^{−1}}{log ρ_{w|n}} + 1,  (4.4)

for all x = π(w) ∈ K. On the other hand, we have by the definition of p_{r}(x),

B(x, ρ_{σ}^{−1}ρ_{∗}^{p_{r}(x)}) ⊂ D_{Λ_{r}}(x) for x ∈ K \ V_{∗},  (4.5)

where σ ∈ Λ_{r} is chosen so that K_{σ} = D_{Λ_{r}}(x) and ρ_{∗} ≡ min_{i} ρ_{i}^{−1} < 1. Thus, we have c_{4.1}e^{−r} ≥ E^{x}T_{B(x,ϵ′)} using (4.1) where ϵ′ ≡ ρ_{σ}^{−1}ρ_{∗}^{p_{r}(x)}. As c_{2}t_{σ}^{−1} ≥ e^{−r} > t_{σ}^{−1}, we obtain

\frac{log E^{x}T_{B(x,ϵ′)}}{log ϵ′} ≥ \frac{−r + log c_{4.1}}{log ρ_{σ}^{−1} + p_{r}(x) log ρ_{∗}} ≥ \frac{1 + \frac{log μ_{σ}+log c_{2}}{log ρ_{σ}^{−1}} + \frac{log c_{4.1}}{log ρ_{σ}^{−1}}}{1 + \frac{p_{r}(x)}{log ρ_{σ}^{−1}} log ρ_{∗}} . (4.6)

For each ϵ small enough, take r ∈ N so that c_{3}ρ_{σ}^{−1}ρ_{∗}^{c_{2} log r} < ϵ ≤ ρ_{σ}^{−1}ρ_{∗}^{c_{2} log r} with some small c_{3} > 0 where σ ∈ Λ_{r}(x). By the definition of ϵ′ and (4.3), we have B(x, ϵ) ⊂ B(x, ϵ′) and log ϵ′/ log ϵ → 1 as ϵ → 0 for ν-a.e. x. Thus, using (4.6), (4.3) and the above facts, we have

lim inf_{ϵ→0} \frac{log E^{x}T_{B(x,ϵ)}}{log ϵ} ≥ lim inf_{|σ|→∞}(\frac{log μ_{σ}}{log ρ_{σ}^{−1}} + 1) ν − a.e. x = π(σ) ∈ K. (4.7)

Combining (4.4), (4.7) and the fact ν(Ê_{α}) = 1, we obtain the result.

Sketch of the proof of Theorem 4.1. Using Lemma 4.3, the proof of dim_{H} W_{γ} ≥ f̄(γ − 1) is similar to that of Proposition 3.5. Using (4.4) and (4.6), the proof of dim_{H} W_{γ} ≤ f̄(γ − 1) is similar to that of Proposition 3.6.

A Appendix A

In this appendix, we will state the estimates on p_{t}(x, x) obtained in [4]. Let Λ_{n} be defined by

Λ_{n} = {w = w_{1} ⋯ w_{k} ∈ ∪_{i≥0}S^{i} : t_{w_{1}} ⋯ t_{w_{k−1}} ≤ e^{n} < t_{w_{1}} ⋯ t_{w_{k}}}.

As before, for w = w_{1} ⋯ w_{k}, we abbreviate t_{w} = Π_{i=1}^{k}t_{w_{i}}, ρ_{w} = Π_{i=1}^{k}ρ_{w_{i}} etc. Set V_{Λ_{n}} = π(∪_{w∈Λ_{n}}wP). For x ∈ K and l ≥ 0, let

D_{Λ_{l}}(x) = {C : C is a Λ_{l}-complex which contains x},

where a Λ_{l}-complex is defined to be a set of the form F_{w}(K), w ∈ Λ_{l}, and ∂D_{Λ_{l}}(x) = cl(K \ D_{Λ_{l}}(x)) ∩ D_{Λ_{l}}(x). Now, for x ∈ K \ V_{∗}, define

p_{r}(x) = inf{k : x ∉ D_{Λ_{r}(x)+k}(∂D_{Λ_{r}}(x))},

where Λ_{r}(x) is a length of the word of the Λ_{r}-complex to which x belongs. Since ∩_{k}D_{k}(A) = A if A ⊂ K is closed, we have p_{r}(x) < ∞. For the Sierpinski gasket, p_{r}(x) = inf{k : x_{Λ_{r}(x)+1} ≠ x_{Λ_{r}(x)+k}}. Let L = max{l : e > (min_{k∈S} t_{k})^{l}}. Then p_{r}(x) has the following property.

If p_{r}(x) > L + 1, then p_{r+1}(x) < p_{r}(x).  (A.1)

Theorem A.1 ([4]) There exists a jointly continuous heat kernel p_{t}(x, y) w.r.t. (E, F) and also constants c_{A.1}, …, c_{A.7} > 0 such that the following holds.

1) (Lower estimate) For each x ∈ K \ V_{∗} and t ≤ c_{A.1}e^{−n′} (n′ = n + c_{A.2}p_{n}(x)),

c_{A.3}{μ(D_{Λ_{n}}(x))}^{−1} ≤ p_{t}(x, x).

2) (Upper estimate) For each x ∈ K\V_{∗} and for each t which satisfies c_{A.4}e^{−n′′} ≤ t ≤ c_{A.5}e^{−m′−c_{A.6} log n′} (n′ = n + c_{A.2}p_{n}(x), m′ = m + c_{A.2}p_{m}(x), n′′ = n′ + c_{A.6} log n′) for some m ≤ n,

p_{t}(x, x) ≤ c_{A.7}\{ min_{{w∈Λ_{n′′} \atop Fw(K)⊂D_{Λm} (x)}} μ_{w}\}^{−1}.

Now, as in Section 2, let μ̂ be any Bernoulli measure on K so that μ̂(F_{i}(K)) = μ̂_{i} (i ∈ S) and μ̂(V_{0}) = 0 where μ̂_{i} ≥ 0 satisfies ∑_{i} μ̂_{i} = 1.

Theorem A.2 ([4]) For any μ̂, there exist constants c_{A.8}, c_{A.9}, c_{A.10} > 0 and h : K → [0, 1] such that

1) h(x) > 0 μ̂ − a.e.,

2) For t < h(x), if e^{−(n+1)} ≤ t ≤ e^{−n}, then

c_{A.9}(log \frac{1}{t})^{−c_{A.8}}μ(D_{Λ_{n}}(x))^{−1} ≤ p_{t}(x, x) ≤ c_{A.10}(log \frac{1}{t})^{c_{A.8}}μ(D_{Λ_{n}}(x))^{−1}.

(A.2)

Theorem 2.4 is now a corollary of this theorem.

B Appendix B

In this appendix, we will introduce a simpler proof of the on-diagonal upper estimate for the heat kernel of the Dirichlet form on the p.c.f. self-similar set with regular harmonic structure. The method is to use the self-similarity of the reproducing kernel. Moreover, this estimate is sharper than that obtained in Theorem A.1 2). Such argument is based on the Dirichlet-Neumann bracketing which was used in [8] to obtain the asymptotic behaviour of the eigenvalue counting function.

For f ∈ l(K), define σ_{i} : l(K) → l(K) (1 ≤ i ≤ N) by

σ_{i}f(ω) = f ∘ F_{i}(ω) for ω ∈ K.

Definition B.1 For 1 ≤ i ≤ N, we define the following.

1) E_{m}^{i}(f, g) is a symmetric form on l(F_{i}(V_{m})) defined by

E_{m}^{i}(f, g) = ρ_{i}E_{m}(σ_{i}f, σ_{i}g).

2) We define a subspace F_{i} ⊂ l(F_{i}(V_{∗})) by

F_{i} = {f|f ∈ l(F_{i}(V_{∗})), lim_{m→∞} E_{m}^{i}(f, f) < ∞},

and a symmetric form E^{i} on F_{i} by

E^{i}(f, g) = lim_{m→∞} E_{m}^{i}(f, g) for f, g ∈ F_{i}.

By definition, σ_{i}F_{i} = F. Now, let

F_{0} = {f ∈ F : f|_{V_{0}} = 0},
F̂ = {f ∈ L^{2}(K, μ) : ∃f_{i} ∈ F_{i}, f|_{K_{i}\V_{1}} = f_{i} ( for all i ∈ S)},
F̂_{0} = {f ∈ F : f|_{V_{1}} = 0}.

Set Ê(f, g) = ∑_{i} E^{i}(f_{i}, g_{i}) for f, g ∈ F̂. Then, (Ê, F̂) is a regular local Dirichlet form on L^{2}(K, μ). Note that

F̂_{0} ⊂ F_{0} ⊂ F ⊂ F̂.  (B.1)

Let g_{λ}^{0}, g_{λ} be the λ-order reproducing kernels for (E, F_{0}), (E, F) respectively. Also, let ĝ_{λ}^{0}, ĝ_{λ} be the λ-order reproducing kernels for (E, F̂_{0}), (Ê, F̂) respectively. Note that as the harmonic structure is regular, the former kernels are continuous on K × K and the latter are continuous on ∪_{i∈S}K_{i} × K_{i}.

Lemma B.2 For each x ∈ K \ V_{1},

ĝ_{λ}^{0}(x, x) ≤ g_{λ}^{0}(x, x) ≤ g_{λ}(x, x) ≤ ĝ_{λ}(x, x).

Proof. Noting that

\frac{1}{g_{λ}(x, x)} = inf_{u∈L_{x}} E_{λ}(u, u),

where L_{x} = {u ∈ F : u(x) ≥ 1} (similar formulae also hold for ĝ_{λ}^{0}, g_{λ}^{0}, ĝ_{λ}), we obtain the result using (B.1).

Lemma B.3 For x ∈ K \ V_{1}, we have

g_{λ}^{0}(x, x) = ρ_{i}ĝ_{t_{i}λ}^{0}(F_{i}(x), F_{i}(x)),
g_{λ}(x, x) = ρ_{i}ĝ_{t_{i}λ}(F_{i}(x), F_{i}(x)).

Proof. This can be proved in the same way as [7] Proposition 2.8.

Iterating Lemma B.2 using Lemma B.3, we have for w|n ∈ S^{n} and x̄_{n} ∈ K \ V_{0},

g_{λ}^{0}(x̄_{n}, x̄_{n}) ≤ ρ_{w|n}g_{t_{w|n}λ}^{0}(F_{w|n}(x̄_{n}), F_{w|n}(x̄_{n}))
≤ ρ_{w|n}g_{t_{w|n}λ}(F_{w|n}(x̄_{n}), F_{w|n}(x̄_{n})) ≤ g_{λ}(x̄_{n}, x̄_{n}). (B.2)

On the other hand, for x ∈ K \ V_{∗} and λ > 1, take n so that t_{w|(n−1)} ≤ λ < t_{w|n} where w|n ∈ S^{n} is the first n letters of the word labeling the complex which x belongs to. Taking

d_{s}^{n}(x) = 2 log(1/μ_{w|n})/ log t_{w|n},

(thus 1 − d_{s}^{n}(x)/2 = log ρ_{w|n}/ log t_{w|n},) we have, by our choice of λ, that

c_{1}ρ_{w|(n−1)}g_{t_{w|n}}(x, x) ≤ λ^{1−d_{s}^{n}(x)/2}g_{λ}(x, x) ≤ ρ_{w|n}g_{t_{w|(n−1)}}(x, x),

for some c_{1} > 0 independent of x, n. If we write x = F_{w|n}(x̄_{n}) and apply (B.2) with λ = 1 to both sides of the above inequality, we have

c_{2}g_{1}^{0}(x̄_{n}, x̄_{n}) ≤ λ^{1−d_{s}^{n}(x)/2}g_{λ}(x, x) ≤ c_{3}g_{1}(x̄_{n−1}, x̄_{n−1}) ≤ c_{4},  (B.3)

where c_{4} = c_{3} max_{x∈K} g_{1}(x, x).

Proposition B.4 There exists c_{B.1} > 0 such that if x = F_{w|n}(x̄_{n}) ∈ K \ V_{∗} and t_{w|n}^{−1} ≤ t < t_{w|(n−1)}^{−1}, then

p_{t}(x, x) ≤ c_{B.1}t^{−d_{s}^{n}(x)/2}.

Proof. As g_{λ}(x, x) = ∫_{0}^{∞} e^{−λt}p_{t}(x, x)dt, we have

c_{4}λ^{d_{s}^{n}(x)/2−1} ≥ g_{λ}(x, x) ≥ p_{a}(x, x) ∫_{0}^{a} e^{−λt}dt = p_{a}(x, x)\frac{1 − e^{−λa}}{λ} ,

where we apply (B.3) for the first inequality and the monotone decreasing property of p_{t} in the second inequality. Taking a = λ^{−1} = t, we obtain the result.

Note that the argument in this section works only for the processes that have bounded reproducing kernels. Unfortunately, the proof of the lower estimate cannot be simplified by this method (see [4], [15] for the methods of the lower estimate).

Acknowledgment. Part of this research was done while the authors were visiting the Isaac Newton Institute, Cambridge, during the program on Mathematics and Applications of Fractals. We thank the organizers of the program, in particular Professor M.T. Barlow and Professor K. Falconer, for fruitful discussions and for making our collaboration possible.

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