Large Deviations for Brownian Motion on the Sierpinski Gasket

Gerard Ben Arous^{a} , Takashi Kumagai ^{b,} ^{∗}

aDepartment of Mathematics, EPFL 1015 Ecublens, Lausanne, Switzerland bGraduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Abstract

We study large deviations for Brownian motion on the Sierpinski gasket in the short time limit. Because of the subtle oscillation of hitting times of the process, no large deviation principle can hold. In fact our result shows that there is an infinity of different large deviation principles for different subsequences, with different (good) rate functions. Thus, instead of taking the time scaling ϵ → 0, we prove that the large deviations hold for ϵ_{n}^{z} ≡ (2/5)^{n}z as n → ∞ using one parameter family of rate functions I^{z} (z ∈ [2/5, 1)). As a corollary, we obtain Strassen-type laws of the iterated logarithm. AMS classification: 60F10; 60J60; 60J80 Keywords: Large deviation; Diffusion; Sierpinski gasket; Fractal; Branching process

1 Introduction

In this paper, we obtain some large deviation results in the short time regime (Schilder-type large deviations) for Brownian motion on the Sierpinski gasket. Let E be the Sierpinski gasket on R^{2} with an intrinsic geodesic metric d, called a shortest path metric, and let X^{x}(t) be Brownian motion on E starting at x ∈ E. This process exhibits sub-diffusive behavior in the sense that E^{x}[d(X(t), x)] ≍ t^{1/d_{w}} for small t > 0 where d_{w} = log 5/ log 2 > 2 and f(t) ≍ g(t) means f(t)/g(t) is bounded from above and below by some positive constants. For fixed T > 0, let Ω_{x} ≡ C_{x}([0, T] → E) = {ϕ ∈ C([0, T] → E) : ϕ(0) = x} with uniformly continuous topology. Let P_{ϵ}^{x} be the law for X^{x}(ϵt). Then, our main theorem is the following.

∗Corresponding author. E-mail: kumagai@i.kyoto-u.ac.jp

Theorem 1.1 For each z ∈ [2/5, 1), A ⊂ Ω_{x},

− inf_{ϕ∈IntA} I_{x}^{z}(ϕ) ≤ lim inf_{n→∞}((2/5)^{n}z)^{1/(d_{w}−1)} log P_{(2/5)^{n}z}^{x}(A)
≤ lim sup_{n→∞}((2/5)^{n}z)^{1/(d_{w}−1)} log P_{(2/5)^{n}z}^{x}(A) ≤ − inf_{ϕ∈ClA} I_{x}^{z}(ϕ).

Here {I_{x}^{z}}_{z∈[2/5,1)} is a sequence of rate functions defined as follows for each ϕ ∈ Ω_{x},

I_{x}^{z}(ϕ) = \{ \begin{matrix}∫_{0}^{T} (ϕ̇(t))^{d_{w}/(d_{w}−1)}F (z/ϕ̇(t))dt, & ϕ is absolutely continuous, \\ ∞, & otherwise, \end{matrix}

(1.1) where F is a periodic non-constant positive continuous function with period 5/2 and ϕ̇(t) ≡ lim_{s→t} \frac{d(ϕ(s),ϕ(t))}{|s−t|} for t ∈ [0, T]. This result tells us that the large deviation result does not hold when one takes the time scaling ϵ to 0. Instead, for each fixed z, it holds via the sequence ϵ_{n}^{z} ≡ (2/5)^{n}z as n → ∞. This result is closely related to the following estimate for the transition probability density p_{t}(x, y) of X obtained in Kumagai (1997),

− lim_{n→∞}((2/5)^{n}z)^{\frac{1}{dw−1}} log p_{(2/5)^{n}z}(x, y) = d(x, y)^{\frac{dw}{dw−1}} F (\frac{z}{d(x, y)}),

for x, y ∈ E where F is the same periodic function as above. This F is defined as a Legendre transform of some limiting function of a Laplace transform of some hitting time of X and a ”tiny” oscillation of the hitting times makes F non-constant. When A = {f ∈ Ω_{x} : f(T) = y}, inf{I_{x}^{z}(ϕ) : ϕ ∈ A} is attained independently of z by the path(s) which moves on the geodesic(s) between x and y homogeneously. Thus ”the most probable path” should be this path, but the energy (action functional) of the path depends on time sequences determined by z. For Brownian motion on a smooth Riemannian manifold, it is well-known (see Varadhan (1967)) that Schilder-type large deviations hold with I_{x}(ϕ) = \frac{1}{2} ∫_{0}^{T} |ϕ̇(t)|^{2}dt (this rate function is recovered from (1.1) by taking d_{w} = 2, F ≡ constant). Our result shows an interesting contrast to this fact. In Section 2, we will briefly present Brownian motion on the Sierpinski gasket and the apriori estimates on its hitting times and transition probabilities.

In Section 3, we will show some properties of our rate functions and sketch the proof of our main theorem. The corresponding large deviation results for pinned Brownian motion is also introduced. In Section 4 we will obtain a Strassen-type law of the iterated logarithm as an application of our main theorem. Part of this research was done while the second author was visiting EPFL. Also, the authors thank H. Osada for fruitful comments.

2 Brownian motion on the Sierpinski gasket

2.1 The Sierpinski gasket and its Dirichlet form

Let {Ψ_{i}}_{i=1}^{3} be a family of affine maps on R^{2} with contraction rate 1/2 where the fixed point of Ψ_{i} (1 ≤ i ≤ 3) is (0, 0), (1, 0) and (1/2, \sqrt{3}/2) respectively. Then, there exists unique non-void compact set E such that E = ∪_{i=1}^{3}Ψ_{i}(E) which is called the (2-dimensional) Sierpinski gasket. Let F^{(0)} = \{(0, 0), (1, 0), (1/2, \sqrt{3}/2)\}. For A ⊂ R^{2}, define Ψ_{i_{1},...,i_{n}}(A) = Ψ_{i_{1}} ∘…∘Ψ_{i_{n}}(A). We will call the set Ψ_{i_{1},...,i_{n}}(F^{(0)}) an n-cell and Ψ_{i_{1},...,i_{n}}(E) an n-complex. Set

F^{(n)} = ⋃_{i_{1},...,i_{n}=1}^{N} Ψ_{i_{1},...,i_{n}}(F^{(0)}), F^{(∞)} = ⋃_{n=0}^{∞} F^{(n)}.

Taking closure, E can be recovered: E = Cl(F^{(∞)}). We next introduce an intrinsic metric on the gasket which we call a shortest path metric. For x, y ∈ F^{(m)}, let

π_{m}(x, y) = {π_{m} : π_{m} is an m-walk in E from x to y which

(2.1) does not contain multiple points}.

Here π_{m} = {p_{k}, p_{k+1}}_{k=1}^{l} is an m-walk if l ∈ N, p_{k} ∈ F^{(m)} for 1 ≤ k ≤ l and p_{k} and p_{k+1} are in the same m-cell for 1 ≤ k ≤ l − 1. For π_{m} = {p_{k}, p_{k+1}}_{k=1}^{l} ∈ π_{m}(x, y), we say the length of π_{m} is l and denotes it by |π_{m}| = l. Now we define the distance on F^{(m)} as follows:

d_{F^{(m)}}(x, y) = min_{π∈π_{m}(x,y),π={(p_{k},p_{k+1})}_{k=1}^{|π|}} ∑_{k=1}^{|π|} 2^{−m}.

Then, we can easily prove the following (see Fitzsimmons, Hambly and Kumagai (1994) for the proof under more general situations).

Lemma 2.1

1) d_{F^{(m)}}(x, y) = d_{F^{(m+1)}}(x, y) if x, y ∈ F^{(m)}.

2) For any choice of p, q ∈ E, define d(p, q) by d(p, q) = lim_{n→∞} d(p_{n}, q_{n}), where p_{n}, q_{n} ∈ F^{(∞)} and p_{n} → p, q_{n} → q as n → ∞. Then d is well defined and d is a metric on E.

We now define the Dirichlet form on the gasket. Let f, g ∈ l(F^{(∞)}) = {f : F^{(∞)} → R} and define

E_{n}(f, g) = \frac{1}{2} ∑_{1≤k_{1}...k_{n}≤3} ∑_{x,y∈F^{(0)}}(5/3)^{n}(f(Ψ_{k_{1},...,k_{n}}(x)) − f(Ψ_{k_{1},...,k_{n}}(y)))

×(g(Ψ_{k_{1},...,k_{n}}(x)) − g(Ψ_{k_{1},...,k_{n}}(y))). (2.2)

This is the energy of the network on F^{(n)} with conductance (5/3)^{n} on each n-complex. The sequence of quadratic forms has the following consistency:

inf{E_{n}(f, f) : f|_{F^{(n−1)}} = v} = E_{n−1}(v, v) for all v ∈ l(F^{(n−1)}).

By this,

E_{n}(f|_{F^{(n)}}, f|_{F^{(n)}}) ≤ E_{n+1}(f|_{F^{(n+1)}}, f|_{F^{(n+1)}}) for all f ∈ l(F^{(∞)}).

For f ∈ l(F^{(∞)}), define F = {f : sup_{n} E_{n}(f, f) < ∞} and E(f, f) = lim_{n→∞} E_{n}(f, f)

for f ∈ F. Then, from Fukushima (1992), Kigami (1993), Kusuoka (1993), we have the following.

Theorem 2.2

1) Any function in F can be extended uniquely to a continuous function on E (thus we can consider F ⊂ C(E) = {f : f is a continuous function on E}). Further, (E, F) is a local regular Dirichlet form on L^{2}(E, μ) (μ is a normalized Hausdorff measure) which has the following properties.

(2.3)  E(f, g) = ∑_{i=1}^{3} \frac{5}{3}E(f ∘ Ψ_{i}, g ∘ Ψ_{i}) for all f, g ∈ F
(2.4)  sup_{x∈E} |f(x)| ≤ c_{2.1} · E_{(1)}(f, f) for all f ∈ F

where E_{(β)}(·, ·) ≡ E(·, ·) + β(·, ·)_{L^{2}(E,μ)} for β > 0 and c_{2.1} > 0 is a constant.

2) E_{(β)} admits a positive symmetric continuous reproducing kernel g_{β}(·, ·) which is bounded and equi-uniform continuous w.r.t. β > 0 on E. The corresponding diffusion is point recurrent.

As we have a local regular Dirichlet form, there is a corresponding diffusion process X(t). Because of the geometric characterization of this process in Section 8 of Barlow and Perkins (1988), we will call this process Brownian motion on the gasket.

2.2 Apriori estimates

In the following, we will explain estimates for Brownian motion on E which are already obtained in Barlow and Perkins (1988), Kumagai (1997). \underline{Hitting time estimates} Let X(t) be Brownian motion on E and let

W_{n} = inf{t ≥ 0 : X(t) ∈ F^{(n)} \ {X(0)}}.

In this case, W_{0} is a limit random variable of a supercritical branching process divided by its mean. Setting ϕ(s) = E^{0}[exp(−sW_{0})], the following equation holds (see Barlow and Perkins (1988) Section 2):

ϕ(5z) = \frac{ϕ(z)^{2}}{4 − 3ϕ(z)}. (2.5)

Using this fact, we have the following.

Proposition 2.3 (Barlow and Perkins (1988) Section 3) There exist positive constants c_{2.2} ∼ c_{2.9} such that

(2.6)  c_{2.2} exp(−c_{2.3}s^{1/d_{w}}) ≤ ϕ(s) ≤ c_{2.4} exp(−c_{2.5}s^{1/d_{w}})
(2.7) c_{2.6} exp(−c_{2.7}s^{−1/(d_{w}−1)}) ≤ P^{0}(W_{0} ≤ s) ≤ c_{2.8} exp(−c_{2.9}s^{−1/(d_{w}−1)})

for all s > 0.

Let L(s) = −s^{1/d_{w}} log ϕ(s). Then, we further have the following.

Proposition 2.4 (Barlow and Perkins (1988) Section 3, Kumagai (1997) Section 4) k(s) ≡ lim_{n→∞} L(s · 5^{n}) > 0 exists and it is not a constant. Moreover, s^{1/d_{w}}k(s) is strictly concave and real analytic on R_{+}.

This proposition means that there is an oscillation of hitting times so that one cannot take c_{2.3} = c_{2.5} in (2.6).

Heat kernel estimates Let F̄(y) = sup_{s}{k(s)s^{1/d_{w}} − ys} and F (y) = y^{1/(d_{w}−1)}F̄(y). Remark that by Proposition 2.4 and by the properties of the Legendre transform, we see that F̄(y) is monotone decreasing and strictly convex. Also F (\frac{5}{2}y) = F (y) and F is not constant.

Theorem 2.5 (Barlow and Perkins (1988) Theorem 1.5, Kumagai (1997) Theorem 1.2) 1) There exists a jointly continuous transition density p_{t}(x, y) for X(t) w.r.t. the Hausdorff measure μ on E which satisfies the following.

c_{2.10}t^{− \frac{ds}{2}} exp (−c_{2.11}Ψ(d(x, y), t)) ≤ p_{t}(x, y) ≤ c_{2.12}t^{− \frac{ds}{2}} exp (−c_{2.13}Ψ(d(x, y), t))

for all 0 < t < 1, x, y ∈ E where Ψ(z, t) = (z^{d_{w}}t^{−1})^{1/(d_{w}−1)}, d_{s} = 2 log 3/ log 5 and c_{2.10} ∼ c_{2.13} are positive constants.

2) The following holds for all z > 0, x, y ∈ E. (The right hand side is 0 when d(x, y) = 0.)

− lim_{n→∞}((2/5)^{n}z)^{\frac{1}{dw−1}} log p_{(2/5)^{n}z}(x, y) = d(x, y)^{\frac{dw}{dw−1}} F (\frac{z}{d(x, y)}). (2.8)

For each fixed z, this convergence is compact uniform for x, y w.r.t. d.

As a corollary to this theorem, we see that Varadhan-type estimates does not hold for the heat kernel of Brownian motion on the gasket.

Corollary 2.6 (Kumagai (1997) Corollary 4.1) There is no function f : E × E → R_{+} which satisfies the following for some bounded function G:

− lim_{t→0} G(t)t^{\frac{1}{dw−1}} log p_{t}(x, y) = f(x, y) ∀x, y ∈ E. (2.9)

3 Large deviations for Brownian motion on the

Sierpinski gasket

For ϕ ∈ Ω_{x}, we say ϕ is absolutely continuous if for each ϵ > 0, there exists δ > 0 such that ∑_{i=1}^{n} d(ϕ(t_{i}), ϕ(t_{i−1})) < ϵ for any n and any disjoint collection of intervals {(t_{i−1}, t_{i})}_{i=1}^{n} in [0, T] whose lengths satisfy ∑_{i}(t_{i}−t_{i−1}) < δ. It can be proved by routine arguments that if ϕ is absolutely continuous, then ϕ̇(t) ≡ lim_{s→t} \frac{d(ϕ(s),ϕ(t))}{|s−t|} exists for a.e. t ∈ [0, T], ϕ̇ ∈ L^{1}([0, T], dt) and ∫_{0}^{T} ϕ̇(t)dt is a length of the path {ϕ(s) : 0 ≤ s ≤ T }. Now, for z > 0 and ϕ ∈ Ω_{x}, define I_{x}^{z}(ϕ) as (1.1) where we consider 0^{1/(d_{w}−1)}F (∞) = 0. When ϕ ∈ C([0, T] → E) (no restriction for ϕ(0)), we denote the corresponding rate function as I^{z}(ϕ).

Remark 3.1 The definition of the rate function in p 226 of Kumagai (1997) was wrong unless E is a line. But the results in the paper hold using the rate function introduced here.

We next give some notation. For Δ : 0 = t_{0} < t_{1} < t_{2} ⋯ < t_{m} = T and ϕ ∈ Ω_{x}, we set Π_{Δ}ϕ = {ϕ(t_{1}), ⋯, ϕ(t_{m})}. Also, define ϕ_{Δ} ∈ Ω_{x} by taking points {ϕ(t_{j})} and joining the successive ones by geodesics with natural parametrization. If there are more than one geodesics between two such points, it is immaterial which one is chosen. Thus, ϕ_{Δ} is piecewise geodesic and ϕ_{Δ}(t_{j}) = ϕ(t_{j}) (0 ≤ j ≤ m). We then have the following.

Lemma 3.2

a) On C([0, T] → E), it holds that

inf_{{ϕ(α)=a, \atop ϕ(β)=b}} I^{z}(ϕ) = (\frac{d(a, b)^{d_{w}}}{β − α})^{1/(d_{w}−1)}F (\frac{z(β − α)}{d(a, b)})

where the infimum is attained by the geodesic on the gasket.

b) On C_{x}([0, T] → E), it holds that

inf_{{ϕ(t_{i})=x_{i}, \atop i=1,···,m}} I_{x}^{z}(ϕ) = I_{x}^{z}(ϕ_{Δ}) = ∑_{i=1}^{m}(\frac{d(x_{i}, x_{i−1})^{d_{w}}}{t_{i} − t_{i−1}} )^{1/(d_{w}−1)}F (\frac{z(t_{i} − t_{i−1})}{d(x_{i}, x_{i−1})})

where Δ : 0 = t_{0} ≤ t_{1} ≤ ⋯ ≤ t_{m} ≤ T, x_{0} = x, x_{1}, ⋯, x_{m} ∈ E and ϕ_{Δ} is piecewise geodesic with ϕ_{Δ}(t_{j}) = x_{j} (0 ≤ j ≤ m).

Proof. Note that b) is an obvious extension of a). For a), it is enough to prove for the case α = 0, β = T as otherwise the infimum is attained by the path which does not move in the intervals [0, α] and [β, T]. Now, for each ϕ ∈ Ω_{x} which is absolutely continuous, set D(ϕ) = ∫_{0}^{T} ϕ̇(t)dt. Then,

∫_{0}^{T} (ϕ̇(t))^{d_{w}/(d_{w}−1)}F (\frac{z}{ϕ̇(t)})dt = ∫_{0}^{T} (ϕ̇(t))^{d_{w}/(d_{w}−1)}(\frac{z}{ϕ̇(t)})^{1/(d_{w}−1)}F̄(\frac{z}{ϕ̇(t)})dt

= z^{1/(d_{w}−1)}D(ϕ) ∫_{0}^{T} F̄(\frac{z}{ϕ̇(t)})ϕ̇(t)\frac{dt}{D(ϕ)}
≥ z^{1/(d_{w}−1)}D(ϕ)F̄(\frac{1}{D(ϕ)} ∫_{0}^{T} \frac{z}{ϕ̇(t)}ϕ̇(t)dt) = z^{1/(d_{w}−1)}D(ϕ)F̄(\frac{zT}{D(ϕ)})
≥ z^{1/(d_{w}−1)}d(a, b)F̄(zT/d(a, b)) = (\frac{d(a, b)^{d_{w}}}{T})^{1/(d_{w}−1)}F (zT/d(a, b)).

Here we use Jensen’s inequality in the first inequality and the second inequality is because D(ϕ) ≥ d(a, b) and F̄ is monotone decreasing. As F̄ is strictly convex, the equalities hold if and only if ϕ̇(t) = constant and D(ϕ) = d(a, b). That is the geodesic with natural parametrization. We thus obtain the result.

Using the results, we see for ϕ ∈ Ω_{x} and 0 ≤ α ≤ β ≤ T

I_{x}^{z}(ϕ) ≥ (\frac{d(ϕ(α), x)^{d_{w}}}{α})^{1/(d_{w}−1)}F (\frac{zα}{d(ϕ(α), x)})

+(\frac{d(ϕ(β), ϕ(α))^{d_{w}}}{β − α})^{1/(d_{w}−1)}F (\frac{z(β − α)}{d(ϕ(β), ϕ(α))})

≥ M(\frac{d(ϕ(β), ϕ(α))^{d_{w}}}{β − α})^{1/(d_{w}−1)}

where M ≡ min_{2/5≤x≤1} F (x) > 0. Thus,

d(ϕ(α), ϕ(β)) ≤ M′I_{x}^{z}(ϕ)^{(d_{w}−1)/d_{w}}(β − α)^{1/d_{w}} (3.1)

for some M′ > 0. For ψ, ϕ ∈ Ω_{x}, define ∥ψ − ϕ∥ = sup_{0≤t≤T} d(ψ(t), ϕ(t)).

Lemma 3.3

1) The function I_{x}^{z}(ϕ) is lower semi-continuous. Further, for every N > 0, {ϕ : I_{x}^{z}(ϕ) ≤ N} is compact.

2) If C ⊂ Ω_{x} is closed in Ω_{x}, then

lim_{δ→0} inf_{ϕ∈C_{δ}} I_{x}^{z}(ϕ) = inf_{ϕ∈C} I_{x}^{z}(ϕ)

where C_{δ} = {ϕ ∈ Ω_{x} : ∥ϕ − ψ∥ < δ for some ψ ∈ C}.

Proof. For the lower semi-continuity, it is enough to show that if I_{x}^{z}(ϕ_{n}) ≤ N and ∥ϕ_{n} − ϕ∥ → 0, then I_{x}^{z}(ϕ) ≤ N. But this can be proved similarly to Varadhan (1980), p. 157, noting that x ↦ x^{d_{w}/(d_{w}−1)}F (z/x) = z^{1/(d_{w}−1)}xF̄(z/x) is monotone increasing and strictly convex. Next, (3.1) shows that {ϕ : I_{x}^{z}(ϕ) ≤ N} is uniformly bounded and equi-continuous. As it is closed by the lower semicontinuity of I_{x}^{z}, 1) follows from Ascoli-Arzéla’s theorem. 2) can be proved in the same way as in Varadhan (1980), p. 159, using (3.1) and 1).

We are now ready to prove Theorem 1.1. In fact, it can be proved following the argument of the corresponding proof in Varadhan (1967). In the following, we state key lemmas of the proof for readers’ convenience. Set ϵ_{n}^{z} = (2/5)^{n}z.

Lemma 3.4 Let C ⊂ Ω_{x} be a closed set of the form Π_{Δ}^{−1}A, where A ⊂ E^{m} is closed. Then

lim sup_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}(C) ≤ − inf_{ϕ∈C} I_{x}^{z}(ϕ).

Proof. Using Theorem 2.5 and Lemma 3.2, this can be proved in the same way as Lemma 3.1 of Varadhan (1967).

For m ∈ N, let Δ_{m} : 0 = t_{0} < t_{1} < t_{2} ⋯ < t_{m} = T be an equally spaced partition, i.e. t_{j} = jT/m (0 ≤ j ≤ m).

Lemma 3.5 For every δ > 0,

lim sup_{m→∞} lim sup_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}(∥ϕ − ϕ_{Δ_{m}}∥ ≥ δ)) = −∞.

Proof. Remark that W_{n} = W_{0}/5^{n} where W_{n}, W_{0} are hitting times appeared in Section 2. By this, one has the corresponding estimates of (2.6), (2.7) for W_{n}. Using this, the proof is the same as Lemma 3.2 of Varadhan (1967).

By these lemmas, we can prove the third inequality of Theorem 1.1. Indeed, it is enough to prove the inequality when A is closed. Let I_{x}^{z,δ}(ω) = inf_{ω′:∥ω′−ω∥<δ} I_{x}^{z}(ω′) and define T_{δ} = inf_{ω∈C_{δ}} I_{x}^{z}(ω). If ω ∈ C then I_{x}^{z,δ}(ω) ≥ T_{δ} and therefore

P_{ϵ_{n}^{z}}^{x}[C] ≤ P_{ϵ_{n}^{z}}^{x}[I_{x}^{z,δ}(ω) ≥ T_{δ}] ≤ P_{ϵ_{n}^{z}}^{x}[∥ω − ω_{Δ_{m}}∥ ≥ δ] + P_{ϵ_{n}^{z}}^{x}[I_{x}^{z}(ω_{Δ_{m}}) ≥ T_{δ}].

From Lemma 3.5,

lim sup_{m→∞} lim sup_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}[∥ω − ω_{Δ_{m}}∥ ≥ δ] = −∞.

As the set (C ⊂){I_{x}^{z}(ω_{Δ_{m}}) ≥ T_{δ}} is equal to

\{ω : ∑_{i=1}^{m}(\frac{d(ω(t_{i}), ω(t_{i−1}))^{d_{w}}}{t_{i} − t_{i−1}} )^{1/(d_{w}−1)}F (\frac{z(t_{i} − t_{i−1})}{d(ω(t_{i}), ω(t_{i−1}))}) ≥ T_{δ}\},

we see from Lemma 3.4 that

lim sup_{m→∞} lim sup_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}[I_{x}^{z}(ω_{Δ_{m}}) ≥ T_{δ}] ≤ −T_{δ}.

Combined these facts with lim_{δ→0} T_{δ} = inf_{ϕ∈A} I_{x}^{z}(ϕ), which comes from Lemma 3.3 2), we obtain the third inequality of Theorem 1.1.

Next comes the lemma for the lower bound.

Lemma 3.6 Let f ∈ Ω_{x}, V = {ϕ ∈ Ω_{x} : ∥ϕ − f∥ < δ} where δ > 0. Then

lim inf_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}(V) ≥ −I_{x}^{z}(f).

Proof. Using Theorem 2.5, Lemma 3.2 and the third inequality of Theorem 1.1, this can be proved in the same way as Lemma 3.4 of Varadhan (1967).

By this lemma, we can prove the first inequality of Theorem 1.1. Indeed, it is enough to prove the inequality when A is open. For f ∈ A, take a sphere V around f contained in A. Then, by the above lemma,

lim inf_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}(A) ≥ lim inf_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}(V) ≥ −I_{x}^{z}(f).

As this is true for all f ∈ A, we have the result. This concludes the proof of Theorem 1.1.

Finally in this section, we mention the large deviations for pinned Brownian motion (c.f. Hsu (1990), Fujita and Watanabe (1989)). For x ≠ y ∈ E, let P_{ϵ}^{x,y} be the pinned measure P_{ϵ}^{x}[·|X^{x}(ϵT) = y]. We set ϵ_{n}^{z} = (2/5)^{n}z as before and define f_{T}^{z} (x, y) = (\frac{d(x,y)^{dw}}{T})^{1/(d_{w}−1)}F (\frac{zT}{d(x,y)}).

− inf_{{ϕ∈IntA \atop ϕ(T)=y}} \{I_{x}^{z}(ϕ) − f_{T}^{z} (x, y)\} ≤ lim inf_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x,y}(A)

≤ lim sup_{n→∞}(ϵ_{n}^{z})^{1/(d_{w}−1)} log P_{ϵ_{n}^{z}}^{x}(A) ≤ − inf_{{ϕ∈ClA \atop ϕ(T)=y}} \{I_{x}^{z}(ϕ) − f_{T}^{z} (x, y)\}.

Let p̂(u, v, z_{1}, z_{2}) be the probability density of the pinned process under P^{·}( | X(T) = y) = P^{·,y}, i.e.

p̂(u, v, z_{1}, z_{2}) = P^{·,y}(X(v) ∈ dz_{2}|X(u) = z_{1})/μ(dz_{2}),

for 0 ≤ u < v ≤ T, z_{1}, z_{2} ∈ E. Then

p̂(u, v, z_{1}, z_{2}) = p_{v−u}(z_{1}, z_{2})p_{T −v}(z_{2}, y)/p_{T −u}(z_{1}, y).

Replacing p_{t}(x, y) by p̂(u, v, z_{1}, z_{2}) in the proof of Theorem 1.1 with suitable modifications of the lemmas, the proof of Theorem 3.7 can be given in the same way. We thus omit it.

4 Law of the iterated logarithm

In this section, we will mention the Strassen-type law of the iterated logarithm. Define Ê = ∪_{n=1}^{∞}2^{n}E and call it the unbounded Sierpinski gasket. As is mentioned in Fukushima (1992), Fitzsimmons, Hambly and Kumagai (1994), we can construct Brownian motion on Ê via Dirichlet forms. By the same argument mentioned in the last section, we can show that Theorem 1.1 holds for Brownian motion on Ê. For Brownian motion starting at 0, set

ξ_{n}(t, ω) = 2^{−n}X(5^{n}(log n)^{1−d_{w}}t, ω) ∼^{L} X((log n)^{1−d_{w}}t, ω).

Then, we can prove the following theorem using Theorem 1.1 by a simple modification of the proof of Theorem 1.17 in Stroock (1984).

Theorem 4.1 For P^{0}-a.a. ω, the sequence {ξ_{n}(·, ω)}_{2}^{∞} has the following properties:

1) {ξ_{n}(·, ω)}_{2}^{∞} is precompact in C_{0}([0, T] → Ê).

2) If {ξ_{n′}(·, ω)}_{2}^{∞} is a convergent subsequence of {ξ_{n}(·, ω)}_{2}^{∞} and ψ is its limit, then min_{z} I_{0}^{z}(ψ) ≤ 1.

3) If ψ ∈ C_{0}([0, T] → Ê) with max_{z} I_{0}^{z}(ψ) ≤ 1, then there is a subsequence of {ξ_{n}(·, ω)}_{2}^{∞} which converges to ψ. In particular, if Φ : C_{0}([0, T) → Ê) → R is a continuous function, then

(4.1) P^{0}( sup_{ψ∈K_{∗}} Φ(ψ) ≤ lim sup_{n→∞} Φ(ξ_{n}(·)) ≤ sup_{ψ∈K^{∗}} Φ(ψ)) = 1,

where K_{∗} = {ϕ ∈ C_{0}([0, T] → Ê) : max_{z} I_{0}^{z}(ϕ) ≤ 1} and K^{∗} = {ϕ ∈ C_{0}([0, T] → Ê) : min_{z} I_{0}^{z}(ϕ) ≤ 1}.

Taking Φ(ξ) = sup_{0≤s≤1} d(ξ(s), 0) in (4.1), we can easily see 0 < sup_{ψ∈K_{∗}} Φ(ψ) ≤ sup_{ψ∈K^{∗}} Φ(ψ) < ∞ using (3.1). Thus, we have for some c_{4.1}, c_{4.2} > 0,

(4.2) P^{0}(c_{4.1} ≤ lim sup_{n→∞} sup_{0≤s≤1} d(ξ_{n}(s, ω), 0) ≤ c_{4.2}) = 1.

Now we quote a powerful 0 − 1 law due to Barlow and Bass (1999) Theorem 8.4.

Theorem 4.2 (Barlow and Bass (1999)) Suppose Γ is a tail event: Γ ∈ ∩_{t}σ{X_{u} : u ≥ t}. Then, either P^{x}(Γ) is 0 for all x or else it is 1 for all x.

Using this, we have the classical law of the iterated logarithm.

Theorem 4.3 There exists C > 0 such that

lim sup_{t→∞} \frac{sup_{0≤s≤t} d(X_{s}, X_{0})}{t^{1/d_{w}}(log log t)^{1−1/d_{w}}} = C P^{x}-a.s., x ∈ Ê.

Proof. Note that d(ξ_{n}(s, ω), 0) = d(X(5^{n}(log n)^{1−d_{w}}s), 0)/2^{n} and t ≍ 5^{n}(log n)^{1−d_{w}} if and only if 2^{n} ≍ t^{1/d_{w}}(log log t)^{1−1/d_{w}}. Then the result is an immediate consequence of (4.2) and Theorem 4.2.

Remark 4.4

1) Theorem 4.3 can be obtained more directly and easily using hitting time estimates in Barlow and Perkins (1988) and Theorem 4.2. In fact, the following was already mentioned by Fukushima and Shima (unpublished).

c_{4.3} ≤ lim sup_{t→∞} \frac{sup_{0≤s≤t} |X_{s} − X_{0}|}{t^{1/d_{w}}(log log t)^{1−1/d_{w}}} ≤ c_{4.4} P^{x}-a.s., x ∈ Ê

for some c_{4.3}, c_{4.4} > 0.

2) In Bass and Kumagai (1998), the result of this section is generalized to diffusion processes whose heat kernels have Aronson-type estimates. In Fukushima, Shima and Takeda (1998), Bass and Kumagai (1998), several laws of the iterated logarithm for diffusion processes on fractals are studied.

References

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