Thick Points for Spatial Brownian Motion: Multifractal Analysis of Occupation Measure

Amir Dembo^{∗} Yuval Peres^{†} Jay Rosen^{‡} Ofer Zeitouni^{§}

Abstract

Let T (x, r) denote the total occupation measure of the ball of radius r centered at x for Brownian motion in IR^{3}. We prove that sup_{|x|≤1} T (x, r)/(r^{2}| log r|) → 16/π^{2} a.s. as r → 0, thus solving a problem posed by Taylor in 1974. Furthermore, for any a ∈ (0, 16/π^{2}), the Hausdorff dimension of the set of “thick points” x for which lim sup_{r→0} T (x, r)/(r^{2}| log r|) = a, is almost surely 2 − aπ^{2}/8; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for Brownian occupation measure. Analogous results hold for Brownian motion in any dimension d > 3. These results are related to the LIL of Ciesielski and Taylor (1962) for the Brownian occupation measure of small balls, in the same way that Lévy’s uniform modulus of continuity, and the formula of Orey and Taylor (1974) for the dimension of “fast points”, are related to the usual LIL. We also show that the lim inf scaling of T (x, r) is quite different: we exhibit non-random c_{1}, c_{2} > 0, such that c_{1} < sup_{x} lim inf_{r→0} T (x, r)/r^{2} < c_{2} a.s. In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of ‘limsup type’.

1 Introduction

For any Borel measurable function f from 0 ≤ t ≤ T to IR^{d} we denote by μ_{T}^{f} its occupation measure:

μ_{T}^{f} (A) = ∫_{0}^{T} 1_{A}(f_{t}) dt

for all Borel sets A ⊆ IR^{d}. Throughout, B(x, r) denotes the ball in IR^{d} of radius r centered at x, and {W_{t}}_{t≥0} denotes Brownian motion in IR^{d}, d ≥ 3.

In the last decade, much insight into the structure of various measures has been gained from their multifractal analysis. A general introduction to this analysis can be found in Olsen [11], Reidi [20] and Falconer [6]; certain important random measures were analyzed by Hu-Taylor [7], Taylor [27], Perkins-Taylor [19], Lawler [10], Shieh-Taylor [23] and Jaffard [8].

∗Research partially supported by NSF grant #DMS-9403553.

†Research partially supported by NSF grants #DMS-9404391 and #DMS-9803597 ‡Research supported, in part, by grants from the NSF and from PSC-CUNY.

§This research was supported, in part, by a US-Israel BSF grant. Research at MSRI is supported in part by NSF grant DMS-9701755.

Consider Brownian occupation measure μ_{T}^{W} in IR^{d}, d ≥ 3. It is well known that for almost all Brownian paths W, the pointwise Hölder exponent

Hölder(μ_{T}^{W} , x) := lim_{ϵ→0} \frac{log μ_{T}^{W} (B(x, ϵ))}{log ϵ}  (1.1)

takes the value 2 for all points x in the range {W_{t} | 0 ≤ t ≤ T }. In particular, the usual multifractal spectrum a ↦ dim{x ∈ IR^{d} : Hölder(μ_{T}^{W} , x) = a} vanishes for all a ≠ 2, a > 0. Indeed, this fact was crucial in Kaufman’s work [9], written long before the term “multifractal” was invented.

Rather than being the end of the story, this means that standard multifractal analysis must be refined to capture the delicate fluctuations of occupation measure under scaling; the problem of obtaining such a refined analysis was posed by Hu and Taylor [7, Pg. 287] in 1997, but it is closely linked to problems posed by Taylor [25] in 1974. Our main results, Theorems 1.1 and 1.3 below, resolve these problems.

The correct scaling for studying the fluctuations of occupation measure was already indicated by Taylor [25]; more details were given by Perkins-Taylor [17, Lemmas 2.3 and 2.5], who showed that there exist absolute constants 0 < c_{1} < c_{2} < ∞, such that almost surely for all points x ∈ {W_{t}| 0 ≤ t ≤ T } and all positive ϵ ≤ ϵ_{0}(ω),

c_{1}ϵ^{2}/| log ϵ| ≤ μ_{T}^{W} (B(x, ϵ)) ≤ c_{2}ϵ^{2}| log ϵ| .  (1.2)

(As they point out, the lower bound is immediate from Lévy’s uniform modulus of continuity.)

Our main result describes the multifractal nature, in a fine scale, of “thick points” for the occupation measure of Brownian motion in IR^{d}, d ≥ 3. (We call a point x ∈ IR^{d} on the Brownian path a thick point if x is in the set considered in (1.3) for some a > 0; similiarly, t > 0 is called a thick time if it is in the set Thick_{a} considered in (1.4) for some a > 0 and T > 0.)

Theorem 1.1 With d ≥ 3, let q_{d} denote the first positive zero of the Bessel function J_{d/2−2}(x). (See [30] for information on q_{d}; in particular, q_{3} = π/2.) Then, for any T ∈ (0, ∞] and all 0 < a ≤ 4/q_{d}^{2},

dim\{x ∈ IR^{d} | lim sup_{ϵ→0} \frac{μ_{T}^{W} (B(x, ϵ))}{ϵ^{2}| log ϵ|} = a\} = 2 − aq_{d}^{2}/2 a.s.  (1.3)

Equivalently, for any T ∈ (0, ∞] and all 0 < a ≤ 4/q_{d}^{2} ,

dim\{0 ≤ t < T | lim sup_{ϵ→0} \frac{μ_{T}^{W} (B(W_{t}, ϵ))}{ϵ^{2}| log ϵ|} = a\} = 1 − aq_{d}^{2}/4 a.s.  (1.4)

Denote the set in (1.4) by Thick_{a}. Then Thick_{a} ≠ ∅ at the critical value a = 4/q_{d}^{2}.

For comparison purposes, recall three fundamental results on Brownian increments:

(i) The large increments at a fixed time t, are governed by Khinchin’s classical LIL:

lim sup_{ϵ→0} \frac{W_{t+ϵ} − W_{t}}{(2ϵ log | log ϵ|)^{1/2}} = 1 a.s.

(ii) The dimension of certain exceptional fast points was determined by Orey-Taylor [12]:

∀a ∈ [0, 1], dim \{0 ≤ t < T | lim sup_{ϵ→0} \frac{W_{t+ϵ} − W_{t}}{(2ϵ| log ϵ|)^{1/2}} = a\} = 1 − a^{2} a.s.

(This can be viewed as a multifractal decomposition of white noise.)

(iii) Lévy’s uniform modulus of continuity governs the largest increments overall:

lim_{ϵ→0} sup_{0≤t≤T} \frac{W_{t+ϵ} − W_{t}}{(2ϵ| log ϵ|)^{1/2}} = 1 a.s.

The three statements above hold in any dimension d ≥ 1. Next, we indicate their analogues for Brownian occupation measure in dimension d ≥ 3; only the first of these was previously known.

(i’) The limsup asymptotic behavior of Brownian occupation measure around a fixed time t, is governed by the LIL of Ciesielski-Taylor [3, Theorem 3]: for any T ∈ (0, ∞] and t ≤ T,

lim sup_{ϵ→0} \frac{μ_{T}^{W} (B(W_{t}, ϵ))}{ϵ^{2} log | log ϵ|} = \frac{2}{q_{d}^{2}} a.s.  (1.5)

(ii’) The dimension of exceptional thick times is given by (1.4) above.

(iii’) Our results (1.7) and (1.9) give the largest occupation measure possible for a small ball.

Further remarks on Theorem 1.1.

• Perhaps more significant than the numerical values obtained in (1.3) and (1.4) is the insight gained, while proving these results, about the manner by which the “thick points” on the Brownian path arise. The key to our proof of Theorem 1.1 is a localization phenomenon for transient Brownian motion: the balls of radius ϵ that have the largest occupation measure (of order ϵ^{2}| log ϵ|), accumulate most of this measure in a surprisingly short time interval (of length at most ϵ^{2}| log ϵ|^{b} for some b, e.g. b = 6 works); see Section 3 where this localization is established. The localization phenomenon breaks down in dimension d = 2, where the correct scaling of occupation measure, and the techniques needed to establish it, are quite different. We have obtained the corresponding results for the planar case, and will present them separately; we emphasize that the current paper concerns only d ≥ 3.

• Given the localization phenomenon, there are several possible approaches to the proof of the lower bound in (1.4). Our proof relies on a general lower bound on Hausdorff measure of random fractals “of limsup type”, Theorem 2.1. This general bound sharpens similar estimates obtained by Orey-Taylor [12], Hu-Taylor [7], Deheuvels-Mason [4] and Shieh-Taylor [23]; of course, our work owes a substantial debt to these earlier papers.

• For any x ∉ {W_{t} | 0 ≤ t ≤ T } and ϵ small enough, μ_{T}^{W} (B(x, ϵ)) = 0. Hence, the equivalence of (1.3) and (1.4) is a direct consequence of the uniform dimension doubling property of Brownian motion, due to Kaufman [9] (see also, [17, Eqn. (0.1)]).

• Let Λ_{d}^{−1} denote the first eigenvalue of the (Dirichlet) half-Laplacian in the unit ball of IR^{d−2}. As the spherically symmetric fundamental solution for the Laplacian eigenvalue problem in B(0, 1) is J_{d/2−2}(\sqrt{λ}|x|), the required Dirichlet boundary conditions imply that Λ_{d}^{−1} = q_{d}^{2}/2 (see for example [3, (2.15)]). The appearance of (d − 2) here is due to the celebrated identity of Ciesielski-Taylor [3, Theorem 2].

To indicate the qualitative difference between the sets of thick points and the most familiar random fractals associated with Brownian motion (the range, the graph, and the level sets) we present the following proposition; for the definition and properties of packing dimension dim_{P}, see [28] or [6].

Proposition 1.2 Let the notation of Theorem 1.1 be in force. For all 0 < a < 4/q_{d}^{2}, the union Thick_{≥a} := ∪_{b≥a}Thick_{b} has the same Hausdorff dimension as Thick_{a} a.s., but its packing dimension a.s. satisfies dim_{P}(Thick_{≥a}) = 1. Equivalently,

dim_{P}\{x ∈ IR^{d} | lim sup_{ϵ→0} \frac{μ_{T}^{W} (B(x, ϵ))}{ϵ^{2}| log ϵ|} ≥ a\} = 2 a.s.  (1.6)

Remark. The importance of comparing the Hausdorff and packing dimensions of a set was stressed in the survey Taylor [26]. By a more involved argument, it can be shown that Thick_{a} itself also has packing dimension 1 for 0 < a ≤ 4/q_{d}^{2}.

The next theorem solves two problems posed by Taylor in 1974 (see [25, Pg. 201]).

Theorem 1.3 Let {W_{t}} be a Brownian motion in IR^{d}, d ≥ 3. Then, for any R ∈ (0, ∞) and any T ∈ (0, ∞],

lim_{ϵ→0} sup_{|x|≤R} \frac{μ_{T}^{W} (B(x, ϵ))}{ϵ^{2}| log ϵ|} = 4q_{d}^{−2} a.s.  (1.7)

Furthermore, for any k ∈ (0, ∞) and any T ∈ [k, ∞],

lim_{ϵ→0} inf_{t∈[0,k]} \frac{μ_{T}^{W} (B(W_{t}, ϵ))}{ϵ^{2}/| log ϵ|} = 1 a.s.  (1.8)

Remarks:

• Our proof shows that for any T ∈ (0, ∞],

lim_{ϵ→0} sup_{0≤t≤T} \frac{μ_{T}^{W} (B(W_{t}, ϵ))}{ϵ^{2}| log ϵ|} = 4q_{d}^{−2} a.s.  (1.9)

• Combining (1.3) and (1.7) we see that

sup_{x∈IR^{d}} lim sup_{ϵ→0} \frac{μ_{∞}^{W}(B(x, ϵ))}{ϵ^{2}| log ϵ|} = 4q_{d}^{−2} a.s.

In particular, the sets in (1.3) and (1.4) are a.s. empty for any a > 4q_{d}^{−2}, T ∈ (0, ∞].

Computation of Laplace transforms is an important component of a complete multifractal analysis, and it was also the starting point of our investigation. Pemantle, Peres and Shapiro [14] showed that ∫_{0}^{1} μ_{1}^{W} (B(W_{t}, ϵ)) /ϵ^{2} dt, the pathwise first moment of the ratio μ_{1}^{W} (B(W_{t}, ϵ)) /ϵ^{2}, remains bounded almost surely as ϵ → 0. The following theorem provides a pathwise asymptotic formula for the moment generating function of that ratio. In one sense, it is finer than Theorem 1.1, since it yields a precise estimate of the total duration in [0, 1] that the Brownian particle spends in balls of radius ϵ that have unusually high occupation measure (see Corollary 1.5 below). Such an estimate (which is an analogue in our setting of the “coarse multifractal spectrum”, cf. Reidi [20]), cannot be inferred from Theorem 1.1.

Theorem 1.4 Denote by μ̄W̄ ; ∞ the total occupation measure for a two-sided Brownian motion {W̄_{t}}_{−∞}^{∞} in IR^{d}, d ≥ 3. Then for each θ < q_{d}^{2}/2,

lim_{ϵ→0} ∫_{0}^{1} e^{θμ_{1}^{W} (B(W_{t},ϵ))/ϵ^{2}} dt = IE (e^{θμ̄W̄ ; ∞ (B(0,1))}) a.s.  (1.10)

Remarks:

• We note by [3] that

IE (e^{θμ̄W̄ ; ∞ (B(0,1))}) = (IE (e^{θμ_{∞}^{W} (B(0,1))}))^{2} = \frac{1}{∏_{j=1}^{∞} (1 − \frac{2θ}{q_{d,j}^{2}} )^{2}}  (1.11)

for each θ < q_{d}^{2}/2, where {q_{d,j}}_{j≥1} are the positive zeros of the Bessel function J_{d/2−2}(x), enumerated in increasing order. It is clear that the right hand side diverges as θ ↑ q_{d}^{2}/2 = q_{d,1}^{2}/2. The case d = 3 is particularly explicit because then q_{3} = π/2 and the right hand side of (1.11) simplifies to cos^{−2}(\sqrt{2θ}) (c.f. [3]).

• Let τ denote a random variable uniform on [0, 1], which is independent of the Brownian path W. Then, (1.10) implies in particular that for almost every Brownian path W, the ratio μ_{1}^{W} (B(W_{τ}, ϵ)) /ϵ^{2}, a random variable in τ, converges in law as ϵ → 0 to the total occupation time μ̄W̄ ; ∞ (B(0, 1)) of the unit ball by a two-sided Brownian motion W̄.

Next, we state the promised corollary of Theorem 1.4, which is analogous to the coarse multifractal spectrum.

Corollary 1.5 Let {W_{t}} be a Brownian motion in IR^{d}, d ≥ 3, and denote Lebesgue measure on IR^{1} by Leb. Then, for any a ∈ (0, 4/q_{d}^{2}),

lim_{ϵ→0} \frac{log Leb\{0 ≤ t ≤ 1 | μ_{1}^{W} (B(W_{t}, ϵ)) ≥ aϵ^{2}| log ϵ|\}}{log ϵ} = aq_{d}^{2}/2 a.s.

The thick points considered in Theorem 1.1 are centers of balls B(x, ϵ) with unusually large occupation measure for infinitely many radii, but these radii might be quite rare. The next theorem shows that for the balls B(x, ϵ) to have unusually large occupation measure for all small radii ϵ and the same center x, what constitutes “unusually large” must be interpreted more modestly. Define

I_{d}(a) := \frac{a}{4}(max\{0, d − 2 − \frac{2}{a}\})^{2} ,  (1.12)

and let

C_{d} := inf\{a : I_{d}(a) = 2\} = \frac{2}{d − 2\sqrt{d − 1}} .  (1.13)

(The equality on the right is easily verified.)

Then

Theorem 1.6 For {W_{t}} a Brownian motion in IR^{d}, d ≥ 3, and a ∈ (0, C_{d}],

dim\{x ∈ IR^{d} | lim inf_{ϵ→0} \frac{μ_{∞}^{W}(B(x, ϵ))}{ϵ^{2}} ≥ a\} ≤ 2 − I_{d}(a) a.s.  (1.14)

and this can be strengthened to

dim_{P}\{x ∈ IR^{d} | lim inf_{ϵ→0} \frac{μ_{∞}^{W}(B(x, ϵ))}{ϵ^{2}} ≥ a\} ≤ 2 − I_{d}(a) a.s.  (1.15)

where dim_{P} denotes packing dimension. Moreover,

\frac{1}{d} ≤ sup_{x∈IR^{d}} lim inf_{ϵ→0} \frac{μ_{∞}^{W}(B(x, ϵ))}{ϵ^{2}} ≤ C_{d} a.s.  (1.16)

Remarks:

• In particular, replacing the lim sup by lim inf in (1.3) and (1.4) yields an a.s. empty set for any a > 0.

• The new assertion in (1.16) is the right hand inequality; the inequality on the left is an immediate consequence of Theorem 9 of Perkins [16] concerning “Brownian slow points”.

• It is an open problem to determine exactly the dimension appearing in (1.14) and the precise asymptotics in (1.16).

• That the upper bound (1.14) on Hausdorff dimension applies to packing dimension as well is in sharp contrast with Theorem 1.1 and Proposition 1.2. Intuitively, the reason for this contrast is that for a point to be in the set considered in (1.3), it only needs to satisfy a certain condition at infinitely many scales, so that set can appear large at other scales; these scales can be used to pack many disjoint balls with centers in the set. Points considered in (1.14), however, must satisfy a (less stringent) condition at all scales.

The next section contains a discussion of fractals “of limsup type” and a general lower bound (Theorem 2.1) for their Hausdorff measure. In Section 3 we prove the crucial Localization Lemma 3.1. The results of those two sections are applied in Section 4 to establish the lower bounds on Hausdorff dimension in Theorem 1.1 and Proposition 1.2. The complementary upper bounds in Theorem 1.1 are proved in Section 5. Combining these bounds with the Localization Lemma 3.1, we prove Theorem 1.3 in Section 6. Section 7 is devoted to the proof of Theorem 1.4, with Corollary 1.5 proved in Section 8. Theorem 1.6 is proved in Section 9. At the end of the paper we present some open problems. Analogous results for transient symmetric stable processes will appear in [5].

2 Random fractals of limsup type

Suppose that for each n ≥ 1, a finite union A(n) of intervals of length λ_{n} is given. Assume that λ_{n} → 0 as n → ∞, and that the number of intervals comprising A(n) grows like a negative power of λ_{n}. We call A := lim sup A(n) = ⋂_{n=1}^{∞} ⋃_{k=n}^{∞} A(k) a fractal of limsup type. We will be interested in situations in which the A(n) are random, and in hypotheses on their distribution which will allow us to obtain dimension bounds on A. The main result of this section, Theorem 2.1, provides a general framework for obtaining lower bounds on the Hausdorff measure of random fractals of limsup type.

Random sets that are (well approximated by) random fractals of limsup type include:

• The fast points of Orey-Taylor [12];

• The initial points of exceptional Brownian excursions considered by Barlow-Perkins [1];

• The close approaches on the Brownian path measured by Perkins-Taylor [18];

• The paths in a family tree where a tree-indexed random walk has positive burst speed, see Benjamini-Peres [2];

• Times where the Strassen functional LIL fails, see Deheuvels-Mason [4];

• Sets arising in multifractal analysis of stable subordinators (studied by Hu-Taylor [7] and by Shieh-Taylor [23]).

• The sets Thick_{a} in Theorem 1.1 .

Such random sets differ qualitatively from the random fractals most frequently encountered (e.g. ranges, graphs, levels sets and slow points of Brownian motion). For instance, the packing dimension of sets of limsup type is typically full, hence larger than their Hausdorff dimension; see Corollary 2.4. In particular, that corollary implies that the sets of fast points of [12] have packing dimension 1 (The assertion to the contrary in [26, Pg. 401] is wrong).

Three general methods have been employed to establish lower bounds for Hausdorff dimension of random fractals of limsup type. (These methods were used earlier for other sets).

• Orey-Taylor [12] constructed a Frostman measure directly, using estimates on binomial probabilities. Their method is expounded by Deheuvels-Mason [4]. This elegant method requires strong independence assumptions “within levels”, and it is difficult to refine it to handle sets defined by an equality, like Thick_{a}, rather than an inequality. Orey-Taylor [12, Pg. 185] state that this can be done for the random fractals of limsup type which they consider, the Brownian fast points, by “tightening their argument”, but extending this to more general situations seems quite hard.

• Intersection properties with an independent random set (the range of a stable subordinator) were used by Barlow-Perkins [1] and Perkins-Taylor [18]; random Cantor sets arising from fractal percolation as in [15] could also be used. Here independence assumptions can be replaced by correlation bounds, but, as above, handling sets like Thick_{a} is unwieldy.

• A powerful method based on estimation of energy integrals was used by Hu-Taylor [7] and Shieh-Taylor [23]. Below we sharpen and extend this method, and show that it yields good estimates of Hausdorff measure, while requiring only mild correlation hypothesis.

Let D_{n} denote the collection of dyadic intervals {[(i − 1)2^{−n}, i2^{−n}]}_{i=1}^{2^{n}}. For any increasing function φ : [0, 1] → [0, ∞) with φ(0) = 0, let H^{φ}(A) denote the Hausdorff measure of a set A in the gauge φ (see, e.g., [26] for the definition).

Theorem 2.1 Suppose that for every n ≥ 1, a collection of {0, 1} valued random variables {Z_{I}}_{I∈D_{n}} is given, so that p_{n} := P(Z_{I} = 1) is the same for all I ∈ D_{n}. Let

A(n) = ∪{I ∈ D_{n} | Z_{I} = 1} and A := lim sup A(n) = ⋂_{n=1}^{∞} ⋃_{k=n}^{∞} A(k) .

For I ∈ D_{m}, with m < n, define

M_{n}(I) = ∑_{J∈D_{n},J⊂I} Z_{J} .

Choose ζ(n) ≥ 1 such that

Var(M_{n}(I)) ≤ ζ(n)IE(M_{n}(I)) = ζ(n)p_{n}2^{n−m} .  (2.1)

Let φ(r) be a gauge function which is regularly varying of index α ∈ (0, 1) as r ↓ 0. (I.e., φ(r) = r^{α}L(r) where L(cr)/L(r) → 1 as r ↓ 0 for any c > 0). If

\frac{2^{−n}ζ(n)}{φ(2^{−n})p_{n}} → 0  (2.2)

then H^{φ}(A) > 0 a.s.

Remarks:

• We emphasize that no independence or correlation assumptions are made relating Z_{I} and Z_{J} for I and J of different lengths.

• H^{φ}(A) > 0 immediately implies that dim(A) ≥ α.

• Theorem 2.1 can be applied to the “fast points” and “thick points” of a variety of processes; the only essential requirements are stationarity of increments, suitable decay of correlations and (for discontinuous processes) bounds on the jump probabilities. The non-vanishing of Hausdorff measure is proved in Theorem 2.1, rather than merely a bound on dimension, in order to handle the sets Thick_{a}, rather than just their unions Thick_{≥a} = ∪_{b≥a}Thick_{b}, in (1.4). (See the remark following the statement of Corollary 4.1).

• Let φ_{n} := 1/φ(2^{−n}). Beyond the obvious fact of some exponential growth of φ_{n}, our proof uses only the following simple consequence of the assumption that φ(r) is regularly varying of index α ∈ (0, 1) as r ↓ 0: for some C < ∞ that does not depend on n,

∑_{m=1}^{n} φ_{m} ≤ Cφ_{n}  and  ∑_{m=n}^{∞} 2^{−m}φ_{m} ≤ C2^{−n}φ_{n}.  (2.3)

• We will apply Theorem 2.1 below to prove Theorem 1.1. In that application, we will take φ(r) = r^{1−γ}| log_{2}(r)|^{13} with p_{n} ≥ 2^{−nγ} for some 0 < γ < 1 and ζ(n) = n^{12}, where throughout this paper, log_{2} stands for the logarithm to the base 2.

• Theorem 2.1, which is formulated for random fractals of limsup type in [0, 1], has an obvious generalization to random ‘fractals of limsup type’ in [0, 1]^{d}. In this setup we can take φ(r) to be any gauge function which is regularly varying of index α ∈ (0, d) as r ↓ 0, and replace (2.1) and (2.2) by Var(M_{n}(I)) ≤ ζ(n)IE(M_{n}(I)) = ζ(n)p_{n}2^{d(n−m)} and 2^{−dn}ζ(n)/(φ(2^{−n})p_{n}) → 0 respectively. The proof of such a generalization is basically identical to the proof of Theorem 2.1.

To establish Theorem 2.1 we need two lemmas. The first one is a version of the well-known connection between energy and Hausdorff measure. For the reader’s convenience, we include the brief proof.

Lemma 2.2 Fix an increasing gauge function φ such that φ(0) = 0. Suppose that B is a Borel set in [0, 1], and ν is a probability measure on B. If the dyadic energy

E_{φ}(ν) := ∑_{m=1}^{∞} ∑_{J∈D_{m}} \frac{ν(J)^{2}}{φ(2^{−m})}

of ν is finite, then H^{φ}(B) > 0.

(In fact H^{φ}(B) = ∞, but that is unimportant for our purpose). See [13] for the connection of E_{φ}(ν) to more traditional expressions for energy. Proof: Let

Ψ(x) := ∑_{m=1}^{∞} ∑_{J∈D_{m}} \frac{ν(J)}{φ(2^{−m})}1_{J}(x) .

Since ∫_{B} Ψ(x) dν = E_{φ}(ν), taking C = 2E_{φ}(ν), the set B_{C} := {x ∈ B | Ψ(x) ≤ C} has ν(B_{C}) ≥ 1/2. The restriction ν_{C} of ν to B_{C} satisfies ν_{C}(J) ≤ Cφ(2^{−m}) for every J ∈ D_{m} for all m. Since any interval I ⊂ [0, 1] can be covered by three shorter dyadic intervals, it follows that ν_{C}(I) ≤ 3Cφ(|I|) for any interval I. Hence, if A is any countable collection of intervals with B_{C} ⊆ ∪_{A}I, then

\frac{1}{2} ≤ ν(B_{C}) ≤ ∑_{A} ν_{C}(I) ≤ 3C ∑_{A} φ(|I|)

which implies that 1/(6C) ≤ H^{φ}(B_{C}).

Alternatively, the a.s. finiteness of Ψ, in conjunction with [22], imply that H^{φ}(B) = ∞. □

The following lemma, which, roughly speaking, controls the “quadratic variation” of the random sets A(n), is the key to the proof of Theorem 2.1. Recall that φ_{n} = 1/φ(2^{−n}), and note that by (2.2), for any ℓ we can choose an integer n(ℓ) > ℓ such that

\frac{φ_{n(ℓ)}ζ(n(ℓ))}{2^{n(ℓ)}p_{n(ℓ)}} ≤ 2^{−3ℓ}.  (2.4)

Lemma 2.3 Let the assumptions of Theorem 2.1 be in force. There exist an a.s. finite random variable ℓ_{0}(ω) and a constant C_{3}, such that if ℓ ≥ ℓ_{0}(ω) and n = n(ℓ), then for all D ∈ D_{ℓ}, we have

|M_{n}(D) − IEM_{n}(D)| < \frac{1}{2}IEM_{n}(D) ,  (2.5)

and

∑_{m=ℓ}^{n(ℓ)} φ_{m} ∑_{J∈D_{m}, J⊂D} \frac{M_{n}(J)^{2}}{(2^{n−ℓ}p_{n})^{2}} ≤ C_{3}φ_{ℓ} .  (2.6)

Proof: For m ≤ n and J ∈ D_{m}, denote

Δ_{n}(J) := M_{n}(J) − IEM_{n}(J) .

Also, for ℓ ≤ n and D ∈ D_{ℓ}, set

Υ_{n}(D) := ∑_{m=ℓ}^{n} φ_{m} ∑_{J∈D_{m}, J⊂D} Δ_{n}(J)^{2} .

For J ∈ D_{m}, the assumption (2.1) gives IE[Δ_{n}(J)^{2}] ≤ ζ(n)p_{n}2^{n−m}. Therefore,

∀D ∈ D_{ℓ}, IEΥ_{n}(D) = ∑_{m=ℓ}^{n} φ_{m}2^{m−ℓ}ζ(n)p_{n}2^{n−m} = 2^{n−ℓ}ζ(n)p_{n} ∑_{m=ℓ}^{n} φ_{m} .

By (2.3), we then have

IEΥ_{n}(D) ≤ C2^{n−ℓ}p_{n}^{2}φ_{n}ζ(n)/p_{n} .

Thus, by (2.4), since n = n(ℓ),

IE ∑_{D∈D_{ℓ}} \frac{Υ_{n}(D)}{(2^{n−ℓ}p_{n})^{2}} ≤ C2^{−ℓ} .

Since the right-hand side is summable in ℓ, we conclude that the summands inside the last expectation tend to 0 a.s. as ℓ → ∞. In particular, there exists ℓ_{0}(ω) < ∞ such that for all ℓ ≥ ℓ_{0}(ω) and D ∈ D_{ℓ}, we have

Υ_{n}(D) ≤ (2^{n−ℓ}p_{n})^{2} = [IEM_{n}(D)]^{2} .  (2.7)

To deduce (2.5), observe that

Δ_{n}(D)^{2} ≤ φ_{ℓ}^{−1}Υ_{n}(D) ≤ φ_{ℓ}^{−1}[IEM_{n}(D)]^{2} < \frac{1}{4}[IEM_{n}(D)]^{2} .

Next, we calculate

∑_{J∈D_{m}, J⊂D} \frac{[IEM_{n}(J)]^{2}}{(2^{n−ℓ}p_{n})^{2}} = ∑_{J∈D_{m}, J⊂D} 2^{2(ℓ−m)} = 2^{ℓ−m} .

Therefore, by (2.3),

∑_{m=ℓ}^{n} φ_{m} ∑_{J∈D_{m}, J⊂D} \frac{[IEM_{n}(J)]^{2}}{(2^{n−ℓ}p_{n})^{2}} = 2^{ℓ} ∑_{m=ℓ}^{n} 2^{−m}φ_{m} ≤ Cφ_{ℓ} .  (2.8)

Rewrite (2.7) in the form

∑_{m=ℓ}^{n} φ_{m} ∑_{J∈D_{m}, J⊂D} \frac{Δ_{n}(J)^{2}}{(2^{n−ℓ}p_{n})^{2}} = \frac{1}{(2^{n−ℓ}p_{n})^{2}} ∑_{m=ℓ}^{n} φ_{m} ∑_{J∈D_{m}, J⊂D} Δ_{n}(J)^{2} ≤ 1 .  (2.9)

Since

M_{n}(J)^{2} = [IEM_{n}(J) + Δ_{n}(J)]^{2} ≤ 2[IEM_{n}(J)]^{2} + 2Δ_{n}(J)^{2} ,

adding the inequalities (2.8) and (2.9) yields (2.6), for a suitable constant C_{3}. □ Proof of Theorem 2.1. We use freely the terminology introduced in the statement of Lemma 2.3. With ℓ_{0} = ℓ_{0}(ω) as in the lemma, define inductively ℓ_{k+1} := n(ℓ_{k}) for k ≥ 0. For D ∈ D_{ℓ_{k−1}} with k ≥ 1, write

Q_{k} := IEM_{ℓ_{k}}(D) = 2^{ℓ_{k}−ℓ_{k−1}}p_{ℓ_{k}} ,

and note that by (2.5),

∀k ≥ 1 ∀D ∈ D_{ℓ_{k−1}} , \frac{1}{2}Q_{k} ≤ M_{ℓ_{k}}(D) ≤ 2Q_{k} .  (2.10)

Summing this over D ∈ D_{ℓ_{k−1}} gives

∀k ≥ 1 M_{ℓ_{k}}([0, 1]) ≤ 2^{ℓ_{k−1}+1}Q_{k} .  (2.11)

To establish the theorem, we will construct a (random) probability measure ν, supported on ∩_{k≥1}A(ℓ_{k}) ⊂ A, such that E_{φ}(ν) < ∞ a.s. To specify ν, it suffices to define ν(J) consistently for all binary intervals J. Start by assigning the leftmost interval in D_{ℓ_{0}} full measure, i.e., set ν[0, 2^{−ℓ_{0}}] := 1. Continue inductively: If J ∈ D_{m} with ℓ_{k−1} < m ≤ ℓ_{k}, and J ⊂ D with D ∈ D_{ℓ_{k−1}}, define

ν(J) := \frac{M_{ℓ_{k}}(J)ν(D)}{M_{ℓ_{k}}(D)} .  (2.12)

It is straightforward to verify that this assignment is consistent and that ν is supported on ∩_{k≥1}A(ℓ_{k}). For k ≥ 2 and J as in (2.12), two applications of (2.10) and the bound

ν(D) ≤ \frac{Z_{D}}{min_{D̃∈D_{ℓ_{k−2}}} M_{ℓ_{k−1}}(D̃)} ,

give

ν(J) ≤ \frac{2M_{ℓ_{k}}(J)ν(D)}{Q_{k}} ≤ \frac{4M_{ℓ_{k}}(J)Z_{D}}{Q_{k}Q_{k−1}} .  (2.13)

Now we apply Lemma 2.3. For k ≥ 2 and D ∈ D_{ℓ_{k−1}},

∑_{m=ℓ_{k−1}}^{ℓ_{k}}φ_{m} ∑_{J∈D_{m}, J⊂D}ν(J)^{2} ≤ \frac{16Z_{D}}{Q_{k−1}^{2}} ∑_{m=ℓ_{k−1}}^{ℓ_{k}}φ_{m} ∑_{J∈D_{m}, J⊂D} \frac{M_{ℓ_{k}}(J)^{2}}{Q_{k}^{2}} ≤ \frac{16C_{3}Z_{D}}{Q_{k−1}^{2}} φ_{ℓ_{k−1}} ,

(2.14)

by the definition of Q_{k} and (2.6). Summing this over all D ∈ D_{ℓ_{k−1}}, and then using (2.11) with k − 1 in place of k, we obtain

∑_{m=ℓ_{k−1}}^{ℓ_{k}} φ_{m} ∑_{J∈D_{m}} ν(J)^{2} ≤ \frac{16C_{3}M_{ℓ_{k−1}}([0, 1])}{Q_{k−1}^{2}} φ_{ℓ_{k−1}} ≤ \frac{C_{4}2^{ℓ_{k−2}}}{Q_{k−1}} φ_{ℓ_{k−1}}

≤ \frac{C_{4}2^{2ℓ_{k−2}}}{2^{ℓ_{k−1}}p_{ℓ_{k−1}}} φ_{ℓ_{k−1}} ≤ C_{4}2^{−ℓ_{k−2}} ,  (2.15)

where the last step used (2.4) and the fact that ζ ≥ 1. As the right-hand side of (2.15) is summable in k, we conclude that

E_{φ}(ν) = ∑_{m=0}^{∞} φ_{m} ∑_{J∈D_{m}} ν(J)^{2} < ∞ a.s.

By Lemma 2.2 , this completes the proof. □

The next corollary will be used to prove Proposition 1.2 in Section 4. If K ⊂ [0, 1], then we write N_{m}(K) for the number of intervals in D_{m} that intersect K.

The only property of the packing dimension dim_{P} we need, is that if a closed set K ⊂ [0, 1] satisfies

lim sup_{m→∞} \frac{log_{2} N_{m}(K ∩ V)}{m} ≥ η  (2.16)

for any open set V that intersects K, then dim_{P}(K) ≥ η. See Tricot [29] or Falconer [6, Prop. 3.6].

Corollary 2.4 In the setting of Theorem 2.1, the random set A = lim sup A(n) satisfies dim_{P}(A) = 1 a.s.

Proof: Using the notation of Theorem 2.1 and Chebycheff’s inequality, we have for m < n and J ∈ D_{m} that

P[M_{n}(J) = 0] ≤ \frac{VarM_{n}(J)}{[IEM_{n}(J)]^{2}} ≤ \frac{ζ(n)}{2^{n−m}p_{n}} = \frac{ζ(n)φ_{n}}{2^{n}p_{n}} · \frac{2^{m}}{φ_{n}} ≤ \frac{2^{m}}{φ_{n}}

where the last step used (2.2) with n sufficiently large. The assumption that φ is regularly varying of index α > 0 certainly implies that φ_{n} ≥ 2^{αn/2} for large n. Therefore, if m is large enough and n > 8m/α, then

∑_{J∈D_{m}} P[M_{n}(J) = 0] ≤ 2^{−αn/4} ,

and consequently

∑_{m} P[ ⋃_{n>8m/α} ⋃_{J∈D_{m}}{M_{n}(J) = 0}] < ∞.

The Borel-Cantelli Lemma then implies that almost surely, for all sufficiently large m,

∀n > 8m/α, ∀J ∈ D_{m}, M_{n}(J) > 0 .  (2.17)

To complete the proof, we will use the notation introduced in the proof of Theorem 2.1, and show that the compact set A^{∗} := ∩_{k≥1}A(ℓ_{k}) satisfies dim_{P}(A^{∗}) = 1 a.s. (note that A^{∗} ≠ ∅ by our proof of Theorem 2.1). Without loss of generality we can assume that ℓ_{k+1} > 8ℓ_{k}^{2}/α for all k. Fix an open set V that intersects A^{∗}. For all sufficiently large k, there is an interval I(k) ∈ D_{ℓ_{k}} actually contained in the intersection of V with ∩_{j=1}^{k}A(ℓ_{j}). Hence, if I := {J ∈ D_{ℓ_{k}^{2}} | J ⊆ I(k)}, each interval J ∈ I is also contained in the intersection of V with ∩_{j=1}^{k}A(ℓ_{j}). We now show that in fact each J ∈ I has non-empty intersection with A^{∗} := ∩_{k≥1}A(ℓ_{k}). By applying (2.17) with m = ℓ_{k}^{2} and n = ℓ_{k+1}, we deduce that every interval J ∈ I contains an interval J_{k+1} ∈ A(ℓ_{k+1}). Then, from (2.17) it follows that every interval J ∈ I contains an infinite nested sequence of intervals J_{r} ∈ A(ℓ_{r}), r ≥ k + 1, hence has non-empty intersection with A^{∗} := ∩_{k≥1}A(ℓ_{k}) as claimed. Since |I| = 2^{ℓ_{k}^{2}−ℓ_{k}} we have that N_{ℓ_{k}^{2}}(A^{∗} ∩ V) ≥ 2^{ℓ_{k}^{2}−ℓ_{k}} for all large k, and this implies that dim_{P}(A^{∗}) = 1 a.s. by the criterion in (2.16). □

3 Localization

Throughout this section, c, c′ denote positive, finite constants, independent of ϵ, the values of which may change from line to line, using the notation a ∼ b if lim_{ϵ→0} a/b = 1.

To derive lower bounds on the Hausdorff dimension of the sets appearing in Theorem 1.1, as well as for proving (1.7), it is crucial to be able to consider the occupation measure of a ball of radius ϵ over a small time interval (of length δ_{ϵ} which tends to zero with ϵ), rather than over an interval of constant length.

Surprisingly, it turns out that with only a small loss in probability, we can work with rather short time intervals; the following lemma makes this precise.

Lemma 3.1 (The Localization Lemma) Let {W_{t}} be a Brownian motion in IR^{d}, d ≥ 3. Write h(r) := r^{2}| log r|, and θ^{∗} := Λ_{d}^{−1} = q_{d}^{2}/2. Finally, denote δ_{ϵ} := ϵ^{2}| log ϵ|^{6} and β_{ϵ} := 1 − 2| log ϵ|^{−2}. Then for some 0 < c < ∞, we have

p_{ϵ} := P(μ_{δ_{ϵ}}^{W} (B(0, ϵβ_{ϵ})) ≥ ah(ϵ)) ≥ cϵ^{aθ^{∗}} .

We did not attempt to optimize the powers of | log ϵ| appearing in the definitions of δ_{ϵ} and β_{ϵ}. Nevertheless, to appreciate the sharpness of this lemma, recall that by [3], c.f. (3.4) below,

P(μ_{∞}^{W}(B(0, ϵ)) ≥ ah(ϵ)) ∼ c′ϵ^{aθ^{∗}} .

Proof: Define

T = T (ϵ) := inf{s ≥ 0 : |W_{s}| = ϵ| log ϵ|^{2}} .

By Brownian scaling, we deduce the existence of positive constants c_{1}, c_{2} such that

P(T > δ_{ϵ}) = P( sup_{t∈[0,| log ϵ|^{2}]} |W_{t}| ≤ 1) ∼ c_{1} exp(−c_{2}| log ϵ|^{2}) .  (3.1)

Therefore,

p_{ϵ} ≥ P(ϵ^{−2} ∫_{0}^{T} 1_{{|W_{s}|<ϵβ_{ϵ}}}ds ≥ a| log ϵ|; T ≤ δ_{ϵ})
≥ P(ϵ^{−2} ∫_{0}^{T} 1_{{|W_{s}|<ϵβ_{ϵ}}}ds ≥ a| log ϵ|) − P(T > δ_{ϵ})  (3.2)

By (3.1) and (3.2), the lemma will be proved once we establish that

P(ϵ^{−2} ∫_{0}^{T} 1_{{|W_{s}|<ϵβ_{ϵ}}}ds ≥ a| log ϵ|) ≥ cϵ^{aθ^{∗}} .  (3.3)

To see (3.3), denote by τ_{d−2} the hitting time of the unit sphere in IR^{d−2} by Brownian motion, and define

I = ϵ^{−2} ∫_{0}^{∞} 1_{{|W_{s}|<ϵβ_{ϵ}}}ds ,
I^{T} = ϵ^{−2} ∫_{0}^{T} 1_{{|W_{s}|<ϵβ_{ϵ}}}ds .

Recall that, using [3] for the first equality,

\frac{P(∫_{0}^{∞} 1_{\{|W_{s}|<1\}}ds ≥ x)}{e^{−xθ^{∗}}} = \frac{P(τ_{d−2} ≥ x)}{e^{−xθ^{∗}}} →_{x→∞} c .  (3.4)

Therefore, using Brownian scaling and (3.4),

P(I ≥ a| log ϵ|) = P((β_{ϵ}ϵ)^{−2} ∫_{0}^{∞} 1_{{|W_{s}|<ϵβ_{ϵ}}}ds ≥ β_{ϵ}^{−2}a| log ϵ|)
= P(τ_{d−2} ≥ β_{ϵ}^{−2}a| log ϵ|)
∼ c exp ( − θ^{∗}a| log ϵ|/(1 − 2| log ϵ|^{−2})^{2}) ∼ cϵ^{aθ^{∗}} . (3.5)

Let now T′ := inf{t > T : |W_{t}| < ϵ}, and define

I_{T′} = ϵ^{−2} ∫_{T′}^{∞} 1_{{|W_{s}|<ϵβ_{ϵ}}}ds .

Then, I = I^{T} 1_{{T′=∞}} + (I^{T} + I_{T′})1_{{T′<∞}} so that

P(I^{T} ≥ z; T′ = ∞) = P(I ≥ z) − P(I^{T} + I_{T′} ≥ z; T′ < ∞) .  (3.6)

Let Ĩ be an independent copy of I for a Brownian motion whose expectation when starting at v we denote by ĨE^{v}. Using symmetry and the strong Markov property we have

P(I^{T} + I_{T′} ≥ z; T′ < ∞) = IE (ĨE^{W_{T ′}} {Ĩ ≥ z − I^{T} }; T′ < ∞)
≤ IE (ĨE {Ĩ ≥ z − I^{T} }; T′ < ∞)
= ĨE {IE (I^{T} ≥ z − Ĩ; T′ < ∞)}
= ĨE {IE (IE^{W_{T}} (T_{B(0,ϵ)} < ∞); I^{T} ≥ z − Ĩ)}
= | log ϵ|^{−2(d−2)}P(I^{T} + Ĩ ≥ z)
≤ | log ϵ|^{−2(d−2)}P(I + Ĩ ≥ z)  (3.7)

where T_{B(0,ϵ)} = inf{t ≥ 0 : W_{t} ∈ B(0, ϵ)} denotes the first hitting time of B(0, ϵ).

Let τ̃_{d−2} denote an independent copy of τ_{d−2}, and let q_{τ} denote their common law. Then, for some constant C independent of z, which may change from line to line,

P(τ_{d−2} + τ̃_{d−2} > z) = P(τ_{d−2} > z) + ∫_{0}^{z} P(τ̃_{d−2} > z − y)q_{τ}(dy)
≤ C[ exp(−zθ^{∗}) + ∫_{0}^{z} exp(−(z − y)θ^{∗})q_{τ}(dy)]
≤ C exp(−zθ^{∗}) + C ∫_{0}^{z} exp(−zθ^{∗})dy
= C(1 + z) exp(−zθ^{∗}) ,  (3.8)

where the third line came from integration by parts. Hence, by the same argument as in (3.5), for some c > 0 and any ϵ > 0 small enough,

P(I + Ĩ ≥ a| log ϵ|) ≤ ca| log ϵ|ϵ^{aθ^{∗}} .

Since 2(d − 2) > 1, the inequality (3.3) follows from (3.5), (3.6), (3.7) and the above. □

4 Proof of the lower bound and critical case in Theorem 1.1

The following corollary of Theorem 2.1 and the Localization Lemma will yield the desired lower bound. From its proof we will also obtain Proposition 1.2 concerning packing dimension.

Recall that θ^{∗} = Λ_{d}^{−1} = q_{d}^{2}/2 denotes the first eigenvalue of the Dirichlet half-Laplacian in the unit ball of IR^{d−2}.

Corollary 4.1 Let T ∈ (0, ∞] and a ∈ (0, 2Λ_{d}). Denote h(ϵ) = ϵ^{2}| log ϵ|, and consider the set of “thick times”

Thick_{≥a} = \{0 ≤ t < T | lim sup_{ϵ→0} \frac{μ_{T}^{W} (B(W_{t}, ϵ))}{h(ϵ)} ≥ a\} .

Let γ = aθ^{∗}/2 ∈ (0, 1) and φ(r) = r^{1−γ}| log_{2} r|^{13}. Then H^{φ}(Thick_{≥a}) > 0 a.s.

Derivation of the lower bound in Theorem 1.1: Assuming for the moment the upper bounds on dimension obtained in Section 5, we may infer that dim(Thick_{a}) = 1−γ as follows (cf. the argument in [12, Pg. 185]). The inequality dim(Thick_{≥(a+1/n)}) ≤ 1−(a+1/n)θ^{∗}/2 of Section 5 implies that H^{φ}(Thick_{≥(a+1/n)}) = 0, and since Thick_{a} = Thick_{≥a} − ∪_{n=1}^{∞}Thick_{≥(a+1/n)}, Corollary 4.1 shows that H^{φ}(Thick_{a}) > 0 which in turn implies that dim(Thick_{a}) ≥ 1 − γ . Using once again the upper bound from Section 5 then completes the proof that dim(Thick_{a}) = 1 − γ . □ Derivation of the critical case in Theorem 1.1: We now show that Thick_{4/q_{d}^{2}} ≠ ∅; perhaps surprisingly, this can be done by a “soft” argument. For h > 0 and a < 4/q_{d}^{2}, consider the set of approximate thick times

Thick(a, h) := ⋃_{ϵ∈(0,h)} \{0 < t < T | \frac{μ_{T}^{W} (B(W_{t}, ϵ))}{ϵ^{2}| log ϵ|} > a\} .

For any a < 4/q_{d}^{2} and h > 0, it follows from (1.4) and the Markov property of Brownian motion, that Thick(a, h) is a.s. dense in [0, T], and it is easy to check that Thick(a, h) is an open set. Thus fixing sequences a_{n} ↑ 4/q_{d}^{2} and h_{n} ↓ 0, Baire’s category theorem implies that

∩_{n}Thick(a_{n}, h_{n}) ≠ ∅ .

Finally, inspection shows that this intersection coincides with Thick_{≥4/q_{d}^{2}}, which in turn coincides with Thick_{4/q_{d}^{2}} by the remark following Theorem 1.3. □ Proof of Corollary 4.1: Since we are proving a lower bound, we may assume that T is finite; by Brownian scaling, it is enough to consider T = 2. Take ϵ_{n} = n^{3}2^{−n/2}, n = 1, 2, … and β_{ϵ_{n}} = 1 − 2| log ϵ_{n}|^{−2} as in the Localization Lemma. With I = [t, t + 2^{−n}] ∈ D_{n}, define Ĩ = [t, t + n^{12}2^{−n}], and let

Z_{I} = 1 iff ∫Ĩ 1_{{|W_{s}−W_{t}|<ϵ_{n}β_{ϵn}}}ds ≥ ah(ϵ_{n}) .

By Lévy’s uniform modulus of continuity, there exists an a.s. finite random variable n_{0}(ω), such that for all n ≥ n_{0}(ω),

sup\{|W_{t} − W_{t′}| : t, t′ ∈ [0, 1], |t − t′| ≤ 2^{−n}\} ≤ 2\sqrt{2^{−n} log(2^{n})} .

Therefore, for all n > n_{0}(ω), if I ∈ D_{n} and Z_{I} = 1, then ∫Ĩ 1_{{|W_{s}−W_{t′}|<ϵ_{n}}}ds ≥ ah(ϵ_{n}) for every t′ ∈ I. The set A defined in Theorem 2.1 satisfies A ⊂ Thick_{≥a} a.s. (we have taken T = 2 rather than T = 1 to avoid boundary effects here). The Localization Lemma, Lemma 3.1, shows that for I ∈ D_{n}, and all n large enough, p_{n} = P(Z_{I} = 1) ≥ 2^{−aθ^{∗}n/2}. Thus, Corollary 4.1 will be established once we verify the variance condition (2.1). For intervals I, J ∈ D_{n} the variables Z_{I} and Z_{J} always satisfy Cov(Z_{I}, Z_{J}) ≤ IE(Z_{I}) = p_{n}, and if dist(I, J) > n^{12}2^{−n}, then Z_{I} and Z_{J} are independent. Therefore, fixing m < n and D ∈ D_{m}, each I ∈ D_{n} satisfies Cov(Z_{I}, M_{n}(D)) ≤ n^{12}p_{n}. Consequently

Var(M_{n}(D)) = ∑_{I∈D_{n}, I⊂D} Cov(Z_{I}, M_{n}(D)) ≤ 2^{n−m}n^{12}p_{n} .

Hence, Theorem 2.1 may be applied (with p_{n} ≥ 2^{−γn} and ζ(n) = n^{12}) to yield the conclusion. □ Proof of Proposition 1.2: In the course of the proof of Corollary 4.1, we showed that for a ∈ (0, 2Λ_{d}), the set Thick_{≥a} contains a set of the form A = lim sup A(n) that satisfies the hypothesis of Theorem 2.1. Thus, the assertion dim_{P}(Thick_{≥a}) = 1 follows immediately from Corollary 2.4. Finally, we may deduce (1.6) from the uniform doubling of packing dimension by spatial Brownian motion, established by Perkins-Taylor [17, Cor. 5.8]. □

5 The upper bound in Theorem 1.1

In this section we establish the upper bound for (1.3), thus completing the proof of Theorem 1.1.

Let {W_{t}}_{t≥0} be a Brownian motion in IR^{d}, d ≥ 3, and h(ϵ) = ϵ^{2}| log ϵ|. Set

z_{T} (x, ϵ) := μ_{T}^{W} (B(x, ϵ))/h(ϵ),

with z(x, ϵ) = z_{∞}(x, ϵ). In this section we show that

dim{x ∈ B(0, k) | lim sup_{ϵ→0} z(x, ϵ) ≥ a} ≤ 2 − aΛ_{d}^{−1}  (5.1)

a.s. for all a ≤ 2Λ_{d}, k ∈ [1, ∞). Using z(x, ϵ) ≥ z_{T} (x, ϵ), and considering the countable union over k = 1, 2, …, will then complete the proof of the upper bound on the dimension of sets in (1.3).

Fix k ∈ [1, ∞) and δ ∈ (0, 1/5). Choose a sequence ϵ_{n} ↓ 0 as n → ∞ in such a way that ϵ_{1} < e^{−2} and

h(ϵ_{n+1}) = (1 − δ)h(ϵ_{n}).  (5.2)

Since, for ϵ_{n+1} ≤ ϵ ≤ ϵ_{n} we have

z(x, ϵ_{n}) = \frac{h(ϵ_{n+1})}{h(ϵ_{n})} \frac{μ_{∞}^{W}(B(x, ϵ_{n}))}{h(ϵ_{n+1})} ≥ (1 − δ)z(x, ϵ)  (5.3)

it is easy to see that for any a > 0,

{x ∈ B(0, k) | lim sup_{ϵ→0} z(x, ϵ) ≥ a} ⊆ D_{a} := {x ∈ B(0, k) | lim sup_{n→∞} z(x, ϵ_{n}) ≥ (1−δ)a}.

Let {x_{j} : j = 0, 1, …, K_{n}}, with x_{0} = 0, denote a maximal collection of points in B(0, k) such that inf_{ℓ≠j} |x_{ℓ} − x_{j}| ≥ δϵ_{n}. Let A_{n} be the set of j , 0 ≤ j ≤ K_{n}, such that

μ_{∞}^{W}(B(x_{j}, (1 + δ)ϵ_{n})) ≥ (1 − 2δ)ah(ϵ_{n}).

We will shortly prove that for any a > 0,

IE|A_{n}| ≤ c′ϵ_{n}^{(1−4δ)aθ^{∗}−2}.  (5.4)

Assuming this for the moment, fix a ≤ 2/θ^{∗} and let V_{n,j} = B(x_{j}, δϵ_{n}). For any x ∈ B(0, k) there exists j ∈ {0, …, K_{n}} such that x ∈ V_{n,j} and B(x, ϵ_{n}) ⊆ B(x_{j}, (1+δ)ϵ_{n}). Consequently, ∪_{n≥m} ∪_{j∈A_{n}} V_{n,j} forms a cover of D_{a} by sets of maximal diameter 2δϵ_{m}. Since V_{n,j} have diameter 2δϵ_{n}, it follows from (5.4) that for γ = 2 − (1 − 5δ)aθ^{∗} > 0,

IE ∑_{n=m}^{∞} ∑_{j∈A_{n}} |V_{n,j}|^{γ} ≤ c′(2δ)^{γ} ∑_{n=m}^{∞} ϵ_{n}^{δaθ^{∗}} < ∞ .

Thus, ∑_{n=m}^{∞} ∑_{j∈A_{n}} |V_{n,j}|^{γ} is finite a.s. implying that dim(D_{a}) ≤ γ a.s. Taking δ ↓ 0 completes the proof of the upper bound (5.1), subject only to (5.4) which we now prove.

Let σ_{j} = inf{t ≥ 0 : W_{t} ∈ B(x_{j}, (1 + δ)ϵ_{n})}. By the strong Markov property, and [3] (c.f. (3.4)), for some c = c(δ, a, d) < ∞ and all n

P(μ_{∞}^{W}(B(x_{j}, (1 + δ)ϵ_{n})) ≥ (1 − 2δ)ah(ϵ_{n}))
= P(IE^{W_{σ_{j}} −x_{j}} (μ_{∞}^{W}(B(0, (1 + δ)ϵ_{n})) ≥ (1 − 2δ)ah(ϵ_{n})) ; σ_{j} < ∞)
≤ P(IE (μ_{∞}^{W}(B(0, (1 + δ)ϵ_{n})) ≥ (1 − 2δ)ah(ϵ_{n})) ; σ_{j} < ∞)
≤ cϵ_{n}^{(1−4δ)aθ^{∗}}P(σ_{j} < ∞)

where the first inequality is due to symmetry. Recall that

P(σ_{j} < ∞) = (\frac{(1 + δ)ϵ_{n}}{|x_{j}|})^{d−2} ∧ 1.

Hence, for some c_{1} = c_{1}(δ, a, d), c′ = c′(δ, a, d, k) < ∞ and every n,

IE|A_{n}| = ∑_{j=0}^{K_{n}} P(μ_{∞}^{W}(B(x_{j}, (1 + δ)ϵ_{n})) ≥ (1 − 2δ)ah(ϵ_{n}))  (5.5)

≤ c_{1}ϵ_{n}^{(1−4δ)aθ^{∗}−2}(1 + ∫_{|x|≤k} \frac{1}{|x|^{d−2}} dx) ≤ c′ϵ_{n}^{(1−4δ)aθ^{∗}−2}

which completes the proof of (5.4) and consequently the proof of Theorem 1.1. □

6 Solution of Taylor’s 1974 problems

Proof of Theorem 1.3: We begin by proving (1.7). To this end, fix T ∈ (0, ∞), δ ∈ (0, 1/4) and a < 2Λ_{d} = 2/θ^{∗} such that η = 2 − (1 + δ)aθ^{∗} > 0. Choose a sequence ϵ_{n} ↓ 0 as in (5.2), noting that for ϵ_{n} ≤ ϵ ≤ ϵ_{n−1} and any x ∈ IR^{d},

(1 − δ)z_{T} (x, ϵ_{n}) ≤ z_{T} (x, ϵ) ≤ (1 − δ)^{−1}z_{T} (x, ϵ_{n−1}) .  (6.1)

Let δ_{ϵ} = ϵ^{2}| log ϵ|^{6}, N_{n} = [T/δ_{ϵ_{n}}], and t_{i,n} = iδ_{ϵ_{n}} for i = 0, …, N_{n} − 1. Writing W_{s}^{t} = W_{s+t} − W_{t} it follows that

inf_{ϵ∈[ϵ_{n},ϵ_{n−1}]} sup_{t∈[0,T]} z_{T} (W_{t}, ϵ) ≥ (1 − δ) max_{i=0}^{N_{n}−1} Z_{i}^{(n)} ,

where Z_{i}^{(n)} = μ_{δ_{ϵn}}^{W^{t_{i,n}}}(B(0, ϵ_{n}))/h(ϵ_{n}) are i.i.d. and by Lemma 3.1, for some c = c(T) > 0 and all n large enough,

P(max_{i=0}^{N_{n}−1} Z_{i}^{(n)} ≤ a) ≤ (1 − p_{ϵ_{n}})^{N_{n}} ≤ e^{−cϵ_{n}^{−η}} .

Since ϵ_{n}^{η} is summable, applying Borel-Cantelli, then taking δ ↓ 0 and a ↑ 2Λ_{d}, we see that a.s.

lim inf_{ϵ→0} sup_{t∈[0,T]} z_{T} (W_{t}, ϵ) ≥ 2Λ_{d} = 4q_{d}^{−2}

With S_{k}(ω) = inf{t : |W_{t}| ≥ k} ∧ T ∈ (0, ∞) a.s. and T ↦ z_{T} (x, r) monotone non-decreasing, it follows that a.s.

lim inf_{ϵ→0} sup_{|x|≤k} z_{T} (x, ϵ) ≥ lim inf_{ϵ→0} sup_{t∈[0,S_{k}(ω)]} z_{S_{k}(ω)}(W_{t}, ϵ) ≥ 4q_{d}^{−2} .

Turning to the proof of the corresponding upper bound, fix k ∈ (0, ∞), δ ∈ (0, 1/5) and let a = (2 + δ)/((1 − 4δ)θ^{∗}) > 2/θ^{∗}. Considering the sequence ϵ_{n} of (5.2) and the sets A_{n} as in Section 5, it follows from (5.4) that

∑_{n=1}^{∞} P(|A_{n}| ≥ 1) ≤ ∑_{n=1}^{∞} IE|A_{n}| ≤ c′ ∑_{n=1}^{∞} ϵ_{n}^{δ} < ∞

By Borel-Cantelli, it thus follows that a.s. A_{n} is empty for all n ≥ n_{0}(ω). By the construction of Section 5 the latter event implies that

lim sup_{ϵ→0} sup_{|x|≤k} z_{∞}(x, ϵ) ≤ a .

Taking δ ↓ 0 for which a ↓ 2/θ^{∗} = 4q_{d}^{−2}, we conclude that a.s.

lim sup_{ϵ→0} sup_{|x|≤k} z_{∞}(x, ϵ) ≤ 4q_{d}^{−2} ,

as needed to complete the proof of (1.7).

The left side of (1.8) is monotone in T and by Brownian scaling its law depends only on T/k. Therefore, it suffices to consider k = 1 and the extreme values T = 1 and T = ∞. Fix δ > 0 and ϵ_{n} = (1 − δ)^{n}. (Note, this is different from the ϵ_{n} used above!). Using the notation

ẑ_{T} (x, ϵ) := \frac{μ_{T}^{W} (B(x, ϵ))}{(ϵ^{2}/| log ϵ|)} ,

it follows that for any ϵ ∈ [ϵ_{n}, ϵ_{n−1}] and x ∈ IR^{d}

\frac{n − 1}{n}(1 − δ)^{2}ẑ_{T} (x, ϵ_{n}) ≤ ẑ_{T} (x, ϵ) ≤ \frac{n}{n − 1}(1 − δ)^{−2}ẑ_{T} (x, ϵ_{n−1}) .

Thus, it suffices for (1.8) to show that for any fixed δ ∈ (0, 1/5) both the lower bound

lim inf_{n→∞} inf_{t∈[0,1]} ẑ_{1}(W_{t}, ϵ_{n}) ≥ (1 − δ)^{5}  (6.2)

and the upper bound

lim sup_{n→∞} inf_{t∈[0,1]} ẑ_{∞}(W_{t}, ϵ_{n}) ≤ (1 + δ)^{5} ,  (6.3)

hold a.s.

Our first task in proving (6.2) is to get a good upper bound on the probability of small occupation measure. If μW̄ ; [−a,b](B(0, ϵ_{n})) denotes the occupation measure of a two-sided IR^{d}-valued Brownian motion W̄ in B(0, ϵ_{n}) during the time interval [−a, b] with a, b ≥ 0, then μW̄ ; [−a,b](B(0, ϵ_{n})) ≤ γ implies that τ̄_{d}(ϵ_{n}) ∧ a + τ_{d}(ϵ_{n}) ∧ b ≤ γ, where

τ̄_{d}(ϵ) = inf{t ≥ 0 : |W̄_{−t}| ≥ ϵ} , τ_{d}(ϵ) = inf{t ≥ 0 : |W̄_{t}| ≥ ϵ}.

Taking γ = (1−δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}|, then a∧b ≥ (1−δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}| together with Brownian scaling shows that

P (μW̄ ; [−a,b](B(0, ϵ_{n})) ≤ (1 − δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}|) ≤ P (τ̄_{d}(1) + τ_{d}(1) ≤ (1 − δ)^{2}/| log ϵ_{n}|) .

(6.4)

Since P(τ_{d}(1) ≤ x) = P(sup_{0≤t≤x} |W_{t}| ≥ 1), it is well known, see [28, Lemma 6.4], that for 0 < x < 1

c_{1}x^{1−d/2}e^{−.5/x} ≤ P(τ_{d}(1) ≤ x) ≤ c_{2}x^{1−d/2}e^{−.5/x}.  (6.5)

This estimate leads, as in the proof of [28, Lemma 6.5], to

P (τ̄_{d}(1) + τ_{d}(1) ≤ x) ≤ e^{−2(1−δ)/x}  (6.6)

for any δ > 0 and x ≤ x(δ). Hence, whenever a ∧ b ≥ (1 − δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}|,

P (μW̄ ; [−a,b](B(0, ϵ_{n})) ≤ (1 − δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}|) ≤ ϵ_{n}^{2/(1−δ)}  (6.7)

for all n ≥ n_{0}(δ), which is the good upper bound we need. In particular, using W̄_{s}^{t} = W̄_{t+s} − W̄_{t} for the time-shifted path, this shows that for all n ≥ n_{0}(δ),

P(ẑ_{1}(W_{t}, ϵ_{n}) ≤ (1 − δ)^{2}) = P (μW̄^{t} ; [−t,1−t](B(0, ϵ_{n})) ≤ (1 − δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}|) ≤ ϵ_{n}^{2/(1−δ)}

(6.8)

provided that

(1 − δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}| ≤ t ≤ 1 − (1 − δ)^{2}ϵ_{n}^{2}/| log ϵ_{n}|  (6.9)

On the other hand, if 0 ≤ t ≤ 1 but condition (6.9) does not hold, (i.e. for t close to 0 or 1), we can no longer use the good upper bound (6.8), but must work with the following bound which comes from (6.5):

P(ẑ_{1}(W_{t}, ϵ_{n}) ≤ (1 − δ)^{2}) ≤ P (τ_{d}(1) ≤ (1 − δ)^{2}/| log ϵ_{n}|) ≤ ϵ_{n}^{.5/(1−δ)^{2}}  (6.10)

for all n ≥ n_{1}(δ), some n_{1}(δ) < ∞.

To apply these estimates for proving (6.2) take k = k(δ) = 20(1 − δ)^{2}/δ^{2} to be an integer, ρ_{n} = (1 − δ)^{2}ϵ_{n}^{2}/(| log ϵ_{n}|k(δ)) = δ^{2}ϵ_{n−1}^{2}/(20| log ϵ_{n}|), N_{n} = ⌊ρ_{n}^{−1}⌋ and t_{i,n} = iρ_{n}, i = 1, …, N_{n}. On the one hand, by Lévy’s uniform modulus of continuity, we have that a.s. for some finite n_{0} = n_{0}(ω) ≥ δ^{−1} and all n ≥ n_{0},

max_{i=1}^{N_{n}} sup_{|s|<ρ_{n}} |W_{t_{i,n}+s} − W_{t_{i,n}}| < δϵ_{n−1} ,

which implies that

inf_{t∈[0,1]} ẑ_{1}(W_{t}, ϵ_{n−1}) ≥ (1 − δ)^{3} min_{i=1}^{N_{n}} ẑ_{1}(W_{t_{i,n}}, ϵ_{n}) .  (6.11)

On the other hand, we see that condition (6.9) is satisfied by all but the first and last k points of the form t = t_{i,n}, i = 1, …, N_{n}. Hence, using the good upper bound (6.8) for those t_{i,n}, and the bound (6.10) for the remaining 2k t_{i,n}’s we have

P(min_{i=1}^{N_{n}} ẑ_{1}(W_{t_{i,n}}, ϵ_{n}) ≤ (1 − δ)^{2}) ≤ ∑_{i=1}^{N_{n}} P(ẑ_{1}(W_{t_{i,n}}, ϵ_{n}) ≤ (1 − δ)^{2})

≤ 2kϵ_{n}^{.5/(1−δ)^{2}} + N_{n}ϵ_{n}^{2/(1−δ)} ≤ ϵ_{n}^{2δ}. (6.12)

Since ϵ_{n}^{2δ} is summable, combining (6.11) and (6.12) yields (6.2) by an application of the Borel-Cantelli Lemma.

Turning to prove (6.3), let now γ_{n} = (1 + δ)^{5}ϵ_{n}^{2}/(2| log ϵ_{n}|), ρ_{n} = ϵ_{n}^{2−5.6δ} and n large enough for ρ_{n} ≥ γ_{n}. (Our choice of the constant 5.6 will become clear at the end of the proof). Consider the event A = A^{+} ∩ A^{−}, where

A^{+} = {τ_{d}((1 + δ)ϵ_{n}) ≤ γ_{n}, inf_{s∈[0,ρ_{n}]} |W_{τ_{d}((1+δ)ϵ_{n})+s}| ≥ ϵ_{n}, |W_{τ_{d}((1+δ)ϵ_{n})+ρ_{n}}| ≥ ϵ_{n}^{1−δ}}

and

A^{−} = {τ̄_{d}((1 + δ)ϵ_{n}) ≤ γ_{n}, inf_{s∈[0,ρ_{n}]} |W̄_{−τ̄_{d}((1+δ)ϵ_{n})−s}| ≥ ϵ_{n}, |W̄_{−τ̄_{d}((1+δ)ϵ_{n})−ρ_{n}}| ≥ ϵ_{n}^{1−δ}} .

By the strong Markov property and symmetry,

P(A^{+}) = P (P^{W_{τ_{d}((1+δ)ϵn)}} (|W_{ρ_{n}}| ≥ ϵ_{n}^{1−δ}, inf_{s∈[0,ρ_{n}]} |W_{s}| ≥ ϵ_{n}) ; τ_{d}((1 + δ)ϵ_{n}) ≤ γ_{n})
= P(τ_{d}((1 + δ)ϵ_{n}) ≤ γ_{n})P^{x_{0}} (|W_{ρ_{n}}| ≥ ϵ_{n}^{1−δ}, inf_{s∈[0,ρ_{n}]} |W_{s}| ≥ ϵ_{n}) ,  (6.13)

for any x_{0} with |x_{0}| = (1 + δ)ϵ_{n}.

By Brownian scaling, P(τ_{d}((1 + δ)ϵ_{n}) ≤ γ_{n}) = P(τ_{d}(1) ≤ (1 + δ)^{3}/(2| log ϵ_{n}|), so that using (6.5) and (1 + δ)^{−3} = 1 − 3δ + O(δ^{2}) we get

c_{3}ϵ_{n}^{1−2.9δ} ≤ P(τ_{d}((1 + δ)ϵ_{n}) ≤ γ_{n}) ≤ c_{4}ϵ_{n}^{1−3.1δ}

for some c_{3}, c_{4} > 0, δ small and all n large enough. Since

P^{x} (inf_{s≤0} |W_{s}| < ϵ) = (\frac{ϵ}{|x|})^{d−2} ,  (6.14)

whenever |x| > ϵ, we have, with |x_{0}| = (1 + δ)ϵ_{n},

P^{x_{0}} (inf_{s≤0} |W_{s}| ≥ ϵ_{n}) = 1 − (1 + δ)^{−(d−2)}  (6.15)

hence

1 − (1 + δ)^{−(d−2)} ≤ P^{x_{0}} ( inf_{s∈[0,ρ_{n}]} |W_{s}| ≥ ϵ_{n}) ≤ 1,  (6.16)

while

P^{x_{0}} (|W_{ρ_{n}}| ≤ ϵ_{n}^{1−δ}) = P (|x_{0} + ϵ_{n}^{1−2.8δ}W_{1}| ≤ ϵ_{n}^{1−δ})
= P (|ϵ_{n}^{2.8δ}(x_{0}/ϵ_{n}) + W_{1}| ≤ ϵ_{n}^{1.8δ}) → 0

since |x_{0}/ϵ_{n}| = 1 + δ, independent of n. Putting this all together and noting that P(A) = P(A^{+})P(A^{−}) = P(A^{+})^{2} shows that

cϵ_{n}^{2−5.8δ} ≤ P(A) ≤ c′ϵ_{n}^{2−6.2δ}  (6.17)

for c, c′ > 0 independent of n.

With t_{i,n} = 4iρ_{n} and N_{n} = [(4ρ_{n})^{−1}] = [0.25ϵ_{n}^{−2+5.6δ}], set A_{i} = A ∘ θ_{t_{i,n}}, that is, the event A for the shifted path W^{t_{i,n}} (W̄^{t_{i,n}}). By the strong Markov property, for any i = 1, …, N_{n},

P(ẑ_{∞}(W_{t_{i,n}}, ϵ_{n}) ≥ (1 + δ)^{5}|A_{i}) ≤ 2 max_{|x_{0}|≥ϵ_{n}^{1−δ}} P^{x_{0}}(inf_{t≥0} |W_{t}| < ϵ_{n}) ≤ 2ϵ_{n}^{(d−2)δ} ,

where (6.14) was used in the second inequality. Hence, by the independence of the events {A_{i}}_{i=1}^{N_{n}},

P(min_{i=1}^{N_{n}} ẑ_{∞}(W_{t_{i,n}}, ϵ_{n}) ≥ (1 + δ)^{5})

≤ (1 − P(A))^{N_{n}} + ∑_{i=1}^{N_{n}} P(ẑ_{∞}(W_{t_{i,n}}, ϵ_{n}) ≥ (1 + δ)^{5}, A_{i})

≤ e^{−P(A)N_{n}} + ∑_{i=1}^{N_{n}} P(ẑ_{∞}(W_{t_{i,n}}, ϵ_{n}) ≥ (1 + δ)^{5} | A_{i})P(A_{i})

≤ e^{−cϵ_{n}^{−.2δ}} + c′ϵ_{n}^{−.6δ}ϵ_{n}^{(d−2)δ}

≤ e^{−cϵ_{n}^{−.2δ}} + c′ϵ_{n}^{.4δ}

and (6.3) follows by an application of the Borel-Cantelli Lemma. (One can see now the reason for choosing 5.6 above. With more care, we could have chosen any 5 ≤ q < 6). This completes the proof of Theorem 1.3. □

7 Almost sure convergence of exponential moments

Proof of Theorem 1.4: For any Borel function f : [a, b] → IR^{d}, we use μ_{a,b}^{f} to denote its occupation measure:

μ_{a,b}^{f}(A) = ∫_{a}^{b} 1_{A}(f_{t}) dt

for all Borel set A ⊆ IR^{d}. We use the abbreviations μ_{T}^{f} = μ_{0,T}^{f} and μ̄_{T}^{f} = μ_{−T,T}^{f}.

As a first step in proving Theorem 1.4, we rewrite things so that we deal only with occupation measures of B(0, 1). Writing W_{t}^{ϵ} = ϵ^{−1}W_{ϵ^{2}t} and W_{s}^{ϵ,t} = W_{t+s}^{ϵ} − W_{t}^{ϵ} with similar notation for W̄ we have

μ_{1}^{W} (B(W_{ϵ^{2}t}, ϵ)) = ∫_{0}^{1} 1_{{|W_{s}−W_{ϵ^{2}t}|≤ϵ}} ds = ϵ^{2} ∫_{0}^{1/ϵ^{2}} 1_{{|W_{ϵ^{2}s}−W_{ϵ^{2}t}|≤ϵ}} ds
= ϵ^{2} ∫_{0}^{1/ϵ^{2}} 1_{{|W_{s}^{ϵ}−W_{t}^{ϵ}|≤1}} ds = ϵ^{2} ∫_{0}^{1/ϵ^{2}} 1_{B(0,1)}(W_{s}^{ϵ} − W_{t}^{ϵ}) ds

and consequently

∫_{0}^{1} e^{θμ_{1}^{W} (B(W_{t},ϵ))/ϵ^{2}} dt = ϵ^{2} ∫_{0}^{1/ϵ^{2}} e^{θμ_{1}^{W}(B(W_{ϵ^{2}t},ϵ))/ϵ^{2}} dt
= ϵ^{2} ∫_{0}^{1/ϵ^{2}} exp (θ ∫_{0}^{1/ϵ^{2}} 1_{B(0,1)}(W_{s}^{ϵ} − W_{t}^{ϵ}) ds) dt (7.1)
≤ ϵ^{2} ∫_{0}^{1/ϵ^{2}} exp (θ ∫_{−∞}^{∞} 1_{B(0,1)}(W̄_{s}^{ϵ} − W̄_{t}^{ϵ}) ds) dt

= ϵ^{2} ∫_{0}^{1/ϵ^{2}} exp (θ ∫_{−∞}^{∞} 1_{B(0,1)}(W̄_{s}^{ϵ,t}) ds) dt
= ϵ^{2} ∫_{0}^{1/ϵ^{2}} e^{θμ̄W̄^{ϵ,t} ; ∞(B(0,1))} dt.

Hence for each θ < q_{d}^{2}/2 and any subsequence ϵ_{m} → 0, in order to show that

lim sup_{m→∞} ∫_{0}^{1} e^{θμ_{1}^{W} (B(W_{t},ϵ_{m}))/ϵ_{m}^{2}} dt ≤ IE (e^{θμ̄W̄ ; ∞ (B(0,1))})  a.s.  (7.2)

it suffices to show that

lim_{m→∞} ϵ_{m}^{2} ∫_{0}^{1/ϵ_{m}^{2}} e^{θμ̄W̄^{ϵm,t} ; ∞(B(0,1))} dt = IE (e^{θμ̄W̄ ; ∞ (B(0,1))})  a.s.  (7.3)

For any 1 < p < 2 such that pθ < q_{d}^{2}/2, (7.3) will follow with ϵ_{m} = m^{−2/(p−1)} from the Borel-Cantelli lemma, Chebycheff’s inequality and the following lemma.

Lemma 7.1 For any θ < q_{d}^{2}/2, there exists c = c_{d,θ} finite, such that for all n,

∥\frac{1}{n^{2}} ∫_{0}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; ∞(B(0,1))} dt − \frac{1}{n^{2}} ∫_{0}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt∥_{1} ≤ cn^{−(d/2−1)},  (7.4)

and for any 1 < p < 2 such that pθ < q_{d}^{2}/2, there exists c = c_{p,d,θ} finite, such that for all n,

∥\frac{1}{n^{2}} ∫_{0}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt − IE (e^{θμ̄W̄ ; ∞ (B(0,1))}) ∥_{p} ≤ cn^{−(1−1/p)}.  (7.5)

We next show that for any 1 < p < 2 such that pθ < q_{d}^{2}/2, and with ϵ_{m} = m^{−2/(p−1)}

lim inf_{m→∞} ∫_{0}^{1} e^{θμ_{1}^{W} (B(W_{t},ϵ_{m}))/ϵ_{m}^{2}} dt ≥ IE (e^{θμ̄W̄ ; ∞ (B(0,1))})  a.s.  (7.6)

Note that for any n ≤ t ≤ n^{2} − n

∫_{0}^{n^{2}} 1_{B(0,1)}(W_{s}^{n^{−1}} − W_{t}^{n^{−1}}) ds = ∫_{−t}^{n^{2}−t} 1_{B(0,1)}(W̄_{s}^{n^{−1},t}) ds
≥ ∫_{−n}^{n} 1_{B(0,1)}(W̄_{s}^{n^{−1},t}) ds = μ̄W̄^{n^{−1},t} ; n (B(0, 1)) .

Hence from (7.1),

∫_{0}^{1} e^{θμ_{1}^{W}(B(W_{t},n^{−1}))/n^{−2}} dt

= \frac{1}{n^{2}} ∫_{0}^{n^{2}} exp (θ ∫_{0}^{n^{2}} 1_{B(0,1)}(W_{s}^{n^{−1}} − W_{t}^{n^{−1}}) ds) dt

≥ \frac{1}{n^{2}} ∫_{n}^{n^{2}−n} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt .

(7.6) then follows by using Lemma 7.1 as before and noting that

∥\frac{1}{n^{2}} ∫_{0}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt − \frac{1}{n^{2}} ∫_{n}^{n^{2}−n} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt∥_{1}

≤ ∥\frac{1}{n^{2}} ∫_{0}^{n} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt∥_{1} + ∥\frac{1}{n^{2}} ∫_{n^{2}−n}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt∥_{1}

≤ \frac{1}{n^{2}} ∫_{0}^{n} ∥e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))}∥_{1} dt + \frac{1}{n^{2}} ∫_{n^{2}−n}^{n^{2}} ∥e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))}∥_{1} dt

≤ 2n^{−1}∥e^{θμ̄W̄ ; ∞ (B(0,1))}∥_{1} .

Since μ_{1}^{W} (B(W_{t}, ϵ)) is monotone in ϵ and lim_{m→∞} ϵ_{m+1}/ϵ_{m} = 1, the proof of Theorem 1.4 now follows from (7.2), (7.6) and a simple interpolation argument. Proof of Lemma 7.1: (7.4) will follow from

∥\frac{1}{n^{2}} ∫_{0}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; ∞(B(0,1))} dt − \frac{1}{n^{2}} ∫_{0}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt∥_{1}

≤ \frac{1}{n^{2}} ∫_{0}^{n^{2}} ∥e^{θμ̄W̄^{n^{−1},t} ; ∞(B(0,1))} − e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))}∥_{1} dt

and the following lemma.

Lemma 7.2 For any θ < q_{d}^{2}/2, there exists c = c_{d,θ} finite such that for any ϵ > 0,

∥e^{θμ̄W̄ ; ∞ (B(0,1))} − e^{θμ̄W̄ ; 1/ϵ(B(0,1))}∥_{1} ≤ cϵ^{d/2−1}.

As for (7.5), we first rewrite

\frac{1}{n^{2}} ∫_{0}^{n^{2}} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt = \frac{1}{n} ∑_{k=0}^{n−1} \frac{1}{n} ∫_{kn}^{(k+1)n} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt

= \frac{1}{n} ∑_{k=0}^{n−1} I_{n,k}

where

I_{n,k} = \frac{1}{n} ∫_{kn}^{(k+1)n} e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))} dt.

Unraveling the definitions we see that for each fixed n, the I_{n,k}; 0 ≤ k ≤ n are identically distributed, and I_{n,k} is measurable with respect to the σ-algebra generated by {W̄_{t+s}^{n^{−1}}−W̄_{t}^{n^{−1}} ; kn ≤ t ≤ (k+1)n; −n ≤ s ≤ n}. Hence I_{n,k}, I_{n,k′} are independent as soon as |k − k′| ≥ 3. Thus we can write

\frac{1}{n} ∑_{k=0}^{n} I_{n,k} = \frac{1}{n} ∑_{k=0}^{n/3} I_{n,3k} + \frac{1}{n} ∑_{k=0}^{n/3} I_{n,1+3k} + \frac{1}{n} ∑_{k=0}^{n/3} I_{n,2+3k}

where each of the three sums on the right hand side is now a sum of i.i.d. random variables. Furthermore

IE (I_{n,k}) = \frac{1}{n} ∫_{kn}^{(k+1)n} IE (e^{θμ̄W̄^{n^{−1},t} ; n(B(0,1))}) dt = IE (e^{θμ̄W̄ ; n (B(0,1))}) .

Using Lemma 7.2, to complete the proof of Lemma 7.1 it now suffices to note that for any 1 < p < 2 such that pθ < q_{d}^{2}/2 we have the following bounds, where the first inequality comes from the Marcinkiewicz-Zygmund inequality (see for example [24, Pg. 341], where our condition pθ < q_{d}^{2}/2 guarantees that I_{n,k} ∈ L^{p}), and the second inequality comes from the fact that |a + b|^{p/2} ≤ |a|^{p/2} + |b|^{p/2} (since p < 2):

IE (|\frac{1}{n/3} ∑_{k=0}^{n/3} (I_{n,i+3k} − IE (I_{n,i+3k})) |^{p}) ≤ \frac{c}{n^{p}} IE (| ∑_{k=0}^{n/3} (I_{n,i+3k} − IE (I_{n,i+3k}))^{2} |^{p/2})
≤ cn^{−(p−1)}

for i = 0, 1, 2. Proof of Lemma 7.2: Let p_{r}(x) = (2πr)^{−d/2} exp(−|x|^{2}/2r) and u^{0}(x) = ∫_{0}^{∞} p_{r}(x)dr = \frac{c_{d}}{|x|^{d−2}} denote the zero-potential of W_{t}. Let Λ_{d} denote the norm of

Kf(x) = ∫_{B(0,1)} u^{0}(x − y)f(y) dy

considered as an operator from L^{2} (B(0, 1), dx) to itself, observing that Λ_{d}^{−1} = q_{d}^{2}/2 is the first eigenvalue of the half-Laplacian in the unit ball of IR^{d−2} with Dirichlet boundary conditions. While u^{0} ∉ L^{2}(B(0, 1), dx) for d > 3, we always have u^{0} ∈ L^{1}(B(0, 1), dx), hence K^{i}u^{0} ∈ L^{1}(B(0, 1), dx) for all i. Further, each application of K lowers the degree of divergence by 2, so that K^{i}u^{0} ∈ L^{2}(B(0, 1), dx) for sufficiently large i, (i = [d/2] will do). In particular, for some κ_{d} < ∞ and all i

∫_{B(0,1)} K^{i}u^{0}(x)dx ≤ κ_{d}(Λ_{d})^{i} .  (7.7)

Setting t_{0} = 0, x_{0} = 0, we first bound the following moments for m = 1, 2, …

\frac{1}{m!}IE (∫_{1/ϵ}^{∞} 1_{B(0,1)}(W_{r}) dr\{∫_{0}^{∞} 1_{B(0,1)}(W_{s}) ds\}^{m−1})

= \frac{1}{m} ∑_{i=1}^{m} ∫_{B(0,1)^{m}} ∫_{ϵ^{−1}≤t_{i}^{0≤t_{1}≤...≤tm<∞}} ∏_{j=1}^{m} p_{t_{j}−t_{j−1}}(x_{j} − x_{j−1}) dt_{1} ⋯ dt_{m} dx_{1} ⋯ dx_{m}

= \frac{1}{m} ∑_{i=1}^{m} ∫_{B(0,1)^{m}} ∫_{ϵ^{−1}≤∑_{j=1}^{i} r_{j}} ∏_{j=1}^{m} p_{r_{j}}(x_{j} − x_{j−1}) dr_{1} ⋯ dr_{m} dx_{1} ⋯ dx_{m}

≤ ∑_{i=1}^{m} ∫_{B(0,1)^{m}} ∫_{(mϵ)^{−1}≤r_{i}} ∏_{j=1}^{m} p_{r_{j}}(x_{j} − x_{j−1}) dr_{1} ⋯ dr_{m} dx_{1} ⋯ dx_{m}

= ∑_{i=1}^{m} ∫_{B(0,1)^{m}} ∏_{j≠i^{j=1}}^{m} u^{0}(x_{j} − x_{j−1}) ∫_{(mϵ)^{−1}}^{∞} p_{r_{i}}(x_{i} − x_{i−1}) dr_{i} dx_{1} ⋯ dx_{m}

≤ (∫_{(mϵ)^{−1}}^{∞} p_{r}(0) dr) ∑_{i=1}^{m} ∫_{B(0,1)^{m}} ∏_{j≠i^{j=1}}^{m} u^{0}(x_{j} − x_{j−1}) dx_{1} ⋯ dx_{m}

= k_{d}(mϵ)^{d/2−1}[(1, K^{m−1}1)_{B(0,1)} + ∑_{i=2}^{m} ( ∫_{B(0,1)} K^{i−2}u^{0}(x_{i−1})dx_{i−1})(1, K^{m−i}1)_{B(0,1)}]

≤ c_{d}ϵ^{d/2−1}m^{d/2}Λ_{d}^{m−1} ,

where (·, ·)_{B(0,1)} denotes the inner product in L^{2}(B(0, 1), dx) and (7.7) was used in the last inequality. With c_{d} independent of m and g_{d,θ} = IE(e^{θμ_{∞}^{W} (B(0,1))}) finite, it follows that

∥e^{θμ̄W̄ ; ∞ (B(0,1))} − e^{θμ̄W̄ ; 1/ϵ(B(0,1))}∥_{1}

= IE (e^{θμ̄W̄ ; ∞ (B(0,1))}) − IE (e^{θμ̄W̄ ; 1/ϵ(B(0,1))})

= {IE (e^{θμ_{∞}^{W} (B(0,1))})}^{2} − {IE (e^{θμ_{1/ϵ}^{W}(B(0,1))})}^{2}

≤ 2g_{d,θ} |IE (e^{θμ_{∞}^{W} (B(0,1))}) − IE (e^{θμ_{1/ϵ}^{W}(B(0,1))}) |

≤ 2g_{d,θ}θ IE (|μ_{∞}^{W} (B(0, 1)) − μ_{1/ϵ}^{W} (B(0, 1)) |e^{θμ_{∞}^{W} (B(0,1))})

= 2g_{d,θ} ∑_{m=0}^{∞} \frac{θ^{m+1}}{m!} IE (∫_{1/ϵ}^{∞} 1_{B(0,1)}(W_{r}) dr\{∫_{0}^{∞} 1_{B(0,1)}(W_{s}) ds\}^{m})

≤ 2θg_{d,θ}c_{d}ϵ^{d/2−1} ∑_{m=0}^{∞}(m + 1)^{d/2+1}(θΛ_{d})^{m} ≤ c_{d,θ}ϵ^{d/2−1}

for any θ < Λ_{d}^{−1}, as needed to complete the proof of Lemma 7.2 and hence of Theorem 1.4. □

8 The coarse multifractal spectrum

Proof of Corollary 1.5: The upper bound in Corollary 1.5 is an immediate consequence of (1.10) and Chebycheff’s inequality. Turning to the corresponding lower bound, fix a ∈ (0, 2/θ^{∗}). Choosing δ ∈ (0, 1/4) such that η = 2 − aθ^{∗}(1 + 3δ) > 0 and ϵ_{n} as in (5.2), leads (see (6.1)) to

lim inf_{ϵ→0} \frac{log Leb\{0 ≤ t ≤ 1 | z_{1}(W_{t}, ϵ) ≥ a\}}{log ϵ} ≥ lim inf_{n→∞} \frac{log Leb\{0 ≤ t ≤ 1 | z_{1}(W_{t}, ϵ_{n}) ≥ \frac{a}{1−δ} \}}{log ϵ_{n−1}} .

(8.1)

Let W_{s}^{t} = W_{t+s} −W_{t}, δ_{n} = ϵ_{n}^{2}| log ϵ_{n}|^{6} and β_{n} = 1−2| log ϵ_{n}|^{−2}. The random variables Y_{i}^{(n)} = μ_{δ_{n}}^{W^{iδn}}(B(0, β_{n}ϵ_{n}))/h(ϵ_{n}), i = 1, …, δ_{n}^{−1} − 1 are i.i.d. The Localization Lemma implies that for some c > 0 and all n large enough,

p_{n}^{∗} := P(Y^{(n)} ≥ a/(1 − δ)) ≥ cϵ_{n}^{aθ^{∗}(1+2δ)} .

Thus, by standard tail estimates for the Binomial(δ_{n}^{−1} − 1, p_{n}^{∗}), for all n large enough,

P(|{i : Y_{i}^{(n)} ≥ a/(1 − δ)}| ≤ ϵ_{n}^{−η}) ≤ exp(−ϵ_{n}^{−η}) ,

since (1 − δ)^{−1} ≤ 1 + 2δ. It follows that a.s., for all n ≥ n_{0}(ω, δ, a),

|{i : Y_{i}^{(n)} ≥ a/(1 − δ)}| ≥ ϵ_{n}^{−η} .  (8.2)

Taking ρ_{n} = ϵ_{n}^{2}/| log ϵ_{n}|^{6}, by Lévy’s uniform modulus of continuity, we have that a.s. for some n_{1} = n_{1}(ω) < ∞ and all n ≥ n_{1},

max_{i=1}^{δ_{n}^{−1}−1} sup_{|s|<ρ_{n}} |W_{iδ_{n}+s} − W_{iδ_{n}}| < (1 − β_{n})ϵ_{n} ,

which together with (8.2) implies that a.s. for any n ≥ n_{2}(ω, δ, a),

Leb{0 ≤ t ≤ 1 | z_{1}(W_{t}, ϵ_{n}) ≥ a/(1 − δ)} ≥ ρ_{n}|{i : Y_{i}^{(n)} ≥ a/(1 − δ)}| ≥ ϵ_{n}^{2−η+δ} .

In view of (8.1), we have a.s.

lim inf_{ϵ→0} \frac{log Leb\{0 ≤ t ≤ 1 | z_{1}(W_{t}, ϵ) ≥ a\}}{log ϵ} ≥ (1 − 2δ)^{−1/2}(2 − η − δ) .

To complete the proof consider δ ↓ 0, for which 2 − η − δ → aθ^{∗}. □

9 Large occupation measure at all scales

Proof of Theorem 1.6: For k ∈ (1, ∞), T < ∞, let Γ_{k} = {x : |x| ∈ [1/k, k]} and

D_{a} := \{x ∈ Γ_{k} | lim inf_{ϵ→0} \frac{μ_{T}^{W} (B(x, ϵ))}{ϵ^{2}} ≥ a\} .

(We work with the annulus Γ_{k} rather than the ball B(0, k) because the basic bound we will use, Lemma 9.2, blows up at the origin).

Fix δ > 0 and let b = 1 + δ > 1. Set η_{n} = 2^{−n} and δ_{n} = η_{n}^{1−b^{−1}} for n = 1, 2, . . .. Let {x_{j} : j = 1, …, K_{n}}, K_{n} ≤ c(δ, k, d)η_{n}^{−d}, denote a maximal collection of points in Γ_{k} such that inf_{ℓ≠j} |x_{ℓ} −x_{j}| ≥ δη_{n}. Let H_{n} = H_{n}(a, δ, T) be the set of j , 1 ≤ j ≤ K_{n}, such that

inf_{ϵ∈[η_{n},δ_{n}]} \frac{μ_{T}^{W} (B(x_{j}, bϵ))}{ϵ^{2}} ≥ \frac{a}{b}.  (9.1)

We will shortly prove that for any γ > 0 we can find δ > 0 such that for some c = c(a, δ, T) < ∞ and all n,

IE|H_{n}| ≤ cη_{n}^{I_{d}(a)−2−γ}  (9.2)

where I_{d}(v) is defined in (1.12). Assuming this for the moment, let U_{n,j} = B(x_{j}, δη_{n}). Then, for any x ∈ Γ_{k} there exists j ∈ {1, …, K_{n}} such that x ∈ U_{n,j} and B(x, ϵ) ⊆ B(x_{j}, ϵ+δη_{n}) ⊆ B(x_{j}, bϵ) for all ϵ ≥ η_{n}. If x ∈ D_{a} then a.s. for some m_{1}(ω, x, b) < ∞ and all n ≥ m_{1},

inf_{ϵ∈[η_{n},δ_{n}]} \frac{μ_{T}^{W} (B(x, ϵ))}{ϵ^{2}} ≥ \frac{a}{b} .

Therefore, ∪_{n≥m} ∪_{j∈H_{n}} U_{n,j} forms a 2δη_{m}-cover of D_{a} for any m ≥ 1. Since U_{n,j} has diameter 2δη_{n}, it follows from (9.2) that

IE ∑_{n=m}^{∞} ∑_{j∈H_{n}} |U_{n,j}|^{2−I_{d}(a)+2γ} = ∑_{n=m}^{∞} IE|H_{n}|(2δη_{n})^{2−I_{d}(a)+2γ} ≤ c_{2} ∑_{n=m}^{∞} η_{n}^{γ} < ∞. (9.3)

Thus, ∑_{n=m}^{∞} ∑_{j∈H_{n}} |U_{n,j}|^{2−I_{d}(a)+2γ} is finite a.s. implying that dim(D_{a}) ≤ 2−I_{d}(a)+2γ a.s. for any T < ∞, γ > 0. Since a.s. there exists T_{k} = T_{k}(ω) finite, such that |W_{t}| ≥ (k + 1) for any t ≥ T_{k}, obviously a.s. also

dim(\{x ∈ Γ_{k} | lim inf_{ϵ→0} \frac{μ_{∞}^{W}(B(x, ϵ))}{ϵ^{2}} ≥ a\}) ≤ 2 − I_{d}(a) + 2γ .

Taking γ ↓ 0 and considering the countable union over k = 1, 2 … completes the proof of (1.14).

To get our upper bound on packing dimension, denote by D^{m}(a/b) the set of points x ∈ IR^{d} such that for all n ≥ m, we have

inf_{ϵ∈[η_{n},δ_{n}]} \frac{μ_{T}^{W} (B(x, ϵ))}{ϵ^{2}} ≥ \frac{a}{b} .

Clearly, ∪_{j∈H_{n}}U_{n,j} forms a 2δη_{n}-cover of D^{m}(a/b) for any n ≥ m. Thus, from (9.3)

lim_{n→∞} ∑_{j∈H_{n}} |U_{n,j}|^{2−I_{d}(a)+2γ} = 0 a.s.  (9.4)

Denote by N (A, ϵ) the minimal cardinality of a collection of balls of radius ϵ that covers A. Recall that \overline{dim}_{M}(A), the upper Minkowski dimension of a set A (also known as the upper box-counting dimension), may be defined by

\overline{dim}_{M}(A) = lim sup_{ϵ→0} \frac{log N (A, ϵ)}{| log ϵ|} ;  (9.5)

see [6, (3.5)]. From (9.4) we may deduce that \overline{dim}_{M}(D^{m}(a/b)) ≤ 2−I_{d}(a)+2γ. Since b > 1, necessarily

D_{a} ⊂ ∪_{m≥1}D^{m}(a/b) ,  (9.6)

and the upper bound dim_{P}(D_{a}) ≤ 2 − I_{d}(a) + 2γ a.s. follows by [6, Prop. 3.8]. This completes the proof of (1.15).

We next recall that I_{d}(v) of (1.12) is strictly increasing in v ≥ 2/(d − 2), whereas I_{d}(C_{d}) = 2. Hence, fixing a > C_{d}, we may and shall fix γ > 0 such that I_{d}(a)−2−γ > 0. Then, by (9.2), for any δ > 0 sufficiently small

∑_{n=1}^{∞} P(|H_{n}| ≥ 1) ≤ ∑_{n=1}^{∞} IE|H_{n}| ≤ c_{1} ∑_{n=1}^{∞} η_{n}^{I_{d}(a)−2−γ} < ∞.

Thus, by Borel-Cantelli, it follows that a.s. H_{n} is empty for all n ≥ m_{2}(ω), implying that the sets D_{a} are a.s. empty for all T < ∞. Since a.s.

lim inf_{ϵ→0} \frac{μ_{∞}^{W}(B(0, ϵ))}{ϵ^{2}} = 0

(see [28, Theorem 6.8]), taking k ↑ ∞ and a ↓ C_{d} completes the proof of (1.16) and hence of Theorem 1.6, subject only to (9.2).

The first step in the proof of (9.2) is the following simple lemma (see [21] for the definition and properties of Bessel processes).

Lemma 9.1 Let Z = ∫_{0}^{T} U_{s}^{−2}ds with {U_{s} : s ∈ [0, T]} the Bessel process of index d′ = d/2 − 1 > 0, starting at U_{0} = u ∈ (0, k]. Then, for any α ∈ (0, d′], b > 1, there exist c = c(b, T, d′, k) < ∞ such that

IE_{(d′)}^{u}(e^{(d^{′2}/2−α^{2}/2)Z}1_{inf_{s∈[0,T]} U_{s}≤v}) ≤ cv^{2α/b}u^{−(d′−α)−2α/b} .  (9.7)

Proof: Let P_{(ν)}^{u}(·) denote the law of the Bessel process {U_{s} : s ∈ [0, T]} of index ν > 0 starting at U_{0} = u. Recall that for any index ν > 0,

dU_{s} = (ν + 1/2)\frac{ds}{U_{s}} + dB_{s} , U_{0} = u > 0 ,

where B_{s} is a one dimensional Brownian motion. In particular, dP_{(d′)}^{u}/dP_{(α)}^{u} exists for any d′ ≥ α > 0 and is given by the Girsanov transformation as (see [21, Pg. 419]),

\frac{dP_{(d′)}^{u}}{dP_{(α)}^{u}} = (\frac{U_{T}}{u})^{d′−α}e^{−(d^{′2}/2−α^{2}/2) ∫_{0}^{T} U_{s}^{−2}ds} .

In particular, by Hölder’s inequality, for q = b/(b − 1),

IE_{(d′)}^{u}(e^{(d^{′2}/2−α^{2}/2)Z}1_{inf_{s∈[0,T]} U_{s}≤v}) = u^{α−d′}IE_{(α)}^{u}(U_{T}^{d′−α}1_{inf_{s∈[0,T]} U_{s}≤v})
≤ u^{α−d′}P_{(α)}^{u}(inf_{s≥0} U_{s} ≤ v)^{1/b}IE_{(α)}^{u}(U_{T}^{q(d′−α)})^{1/q}
≤ u^{α−d′}(\frac{v}{u})^{2α/b}IE_{(α)}^{k}(U_{T}^{q(d′−α)})^{1/q}

where the last inequality follows using the fact that the Bessel process of index α has scale function −x^{−2α} [21, Pg. 415], and for the right-most expectation we used a simple comparison argument. A further comparison argument shows that we can take c = IE_{(d′)}^{k}((1 ∨ U_{T})^{qd′})^{1/q} < ∞. □

The next step in proving (9.2) is to establish the following consequence of Lemma 9.1.

Lemma 9.2 For any T < ∞, b > 1, k > 1, there exists c < ∞ such that for any a > 0, α ∈ (0, d′], η > 0, δ = η^{1−b^{−1}}, |x| ∈ (0, k],

P( inf_{ϵ∈[η,δ]} \frac{μ_{T}^{W} (B(x, bϵ))}{ϵ^{2}} ≥ \frac{a}{b}) ≤ cη^{ab^{−4}(d^{′2}−α^{2})+2α/b}|x|^{−(d′−α)−2α/b} .  (9.8)

Proof: Fix T, a, b, k, α, η, δ and x as in the statement of the lemma. Observe that U_{s} = |W_{s} − x| is a Bessel process of index d′, starting at U_{0} = |x| ∈ (0, k]. Clearly

{μ_{T}^{W} (B(x, v)) > 0} = { inf_{s∈[0,T]} U_{s} < v}  (9.9)

Setting Z = ∫_{0}^{T} U_{s}^{−2}ds, also

b^{2}Z = ∫_{0}^{T} ∫_{b^{−1}U_{s}}^{∞} \frac{2dϵ}{ϵ^{3}} ds = ∫_{0}^{T} ∫_{0}^{∞} 1_{\{|W_{s}−x|≤bϵ\}} \frac{2dϵ}{ϵ^{3}} ds
= ∫_{0}^{∞} \frac{2dϵ}{ϵ^{3}}μ_{T}^{W} (B(x, bϵ)) ≥ ∫_{η}^{δ} \frac{2dϵ}{ϵ^{3}}μ_{T}^{W} (B(x, bϵ)) .  (9.10)

If

inf_{ϵ∈[η,δ]} ϵ^{−2}μ_{T}^{W} (B(x, bϵ)) ≥ \frac{a}{b}

then μ_{T}^{W} (B(x, bη)) > 0 and

∫_{η}^{δ} \frac{2dϵ}{ϵ^{3}}μ_{T}^{W} (B(x, bϵ)) ≥ \frac{a}{b} ∫_{η}^{δ} \frac{2dϵ}{ϵ} = −2ab^{−2} log η .

Thus, for v = bη and λ = (d^{′2} − α^{2})/2 ≥ 0, by (9.9), (9.10) and Chebycheff’s inequality,

P( inf_{ϵ∈[η,δ]} \frac{μ_{T}^{W} (B(x, bϵ))}{ϵ^{2}} ≥ \frac{a}{b}) ≤ P_{(d′)}^{|x|}(Z ≥ −2ab^{−4} log η, inf_{s∈[0,T]} U_{s} ≤ v)
≤ η^{2λab^{−4}}IE_{(d′)}^{|x|}[e^{λZ}1_{inf_{s∈[0,T]} U_{s}≤v}]

We thus obtain (9.8) by applying Lemma 9.1. □

We now return to complete the proof of (9.2). For b > 1 and α ∈ (0, d′] let

f_{a}(b, α) = ab^{−4}(d^{′2} − α^{2}) − d + 2α/b .

By Lemma 9.2, for some c, c_{1}, c_{2} < ∞ independent of n,

IE|H_{n}| = ∑_{j=1}^{K_{n}} P( inf_{ϵ∈[η_{n},δ_{n}]} \frac{μ_{T}^{W} (B(x_{j}, bϵ))}{ϵ^{2}} ≥ \frac{a}{b})

≤ cη_{n}^{ab^{−4}(d^{′2}−α^{2})+2α/b} ∑_{j=1}^{K_{n}} |x_{j}|^{−(d′−α)−2α/b}
≤ c_{1}η_{n}^{ab^{−4}(d^{′2}−α^{2})+2α/b−d}(1 + ∫_{{|x|≤k}} |x|^{−(d′−α)−2α/b} dx)
≤ c_{2}η_{n}^{f_{a}(b,α)},  (9.11)

using (d′ − α) + 2α/b < d′ + α ≤ d − 1.

Setting α = d′ − θ for θ ∈ [0, d′), in which case d^{′2} − α^{2} = θ(d − 2 − θ), we see that

f_{a}(b, α) = ab^{−4}θ(d − 2 − θ) − d(1 − b^{−1}) − (2θ + 2)/b.  (9.12)

Observe that I_{d}(a), defined in (1.12), can also be written as

(max{0, a(d − 2) − 2})^{2}/4a ,

whence

I_{d}(a) = sup_{0≤θ<(d−2)/2}{aθ(d − 2 − θ) − 2θ} ,  (9.13)

and the supremum in (9.13) is attained at θ = max{0, (d − 2)/2 − a^{−1}}. Comparing this with (9.12) we see that

lim_{b↓1} sup_{α∈(0,d′]} f_{a}(b, α) = I_{d}(a) − 2  (9.14)

which completes the proof of (9.2) and hence of Theorem 1.6. □ Some unsolved problems:

1. Determine exactly the dimension appearing in (1.14) and the precise asymptotics in (1.16).

2. Does the set considered in (1.14) have equal Hausdorff and packing dimensions?

3. By arguments similar to those in the proof of (1.16), we can show that there exist non-random constants c̃_{d} > 0, C̃_{d} < ∞ such that

c̃_{d} ≤ inf_{t∈[0,1]} lim sup_{ϵ→0} \frac{μ_{∞}^{W}(B(W_{t}, ϵ))}{ϵ^{2}} ≤ C̃_{d} a.s.  (9.15)

More precisely, the upper bound here is proved just like the lower bound in (1.16), while the lower bound can be inferred from Perkins [16] or from a branching process argument. As in (1.16), it is an open problem to determine the optimal constants in (9.15).

Acknowledgements We thank N.-R. Shieh and S. J. Taylor for useful discussions and for providing us with a preprint of [23], and J. Pitman for a useful discussion of the Ciesielski-Taylor identity. We are indebted to MSRI, to the organizers of the program on stochastic analysis there, and in particular to S. Evans and R. Williams, for making our collaboration possible.

References

1. M. T. Barlow and E. Perkins, Levels at which every Brownian excursion is exceptional, Seminar on probability XVIII, 1–28, Lecture Notes in Math. 1059, Springer, Berlin-New York, 1984.

2. I. Benjamini and Y. Peres, Tree-indexed random walks on groups and first passage percolation, Probab. Theory Related Fields 98 (1994), 91–112.

3. Z. Ciesielski and S. J. Taylor, First passage and sojourn times and the exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103 (1962), 434–452.

4. P. Deheuvels and D. M. Mason, Random fractal functional laws of the iterated logarithm, preprint.

5. A. Dembo, Y. Peres, J. Rosen and O. Zeitouni, Thick points for transient symmetric stable processes, in preparation.

6. K. J. Falconer, Fractal geometry: mathematical foundations and applications. John Wiley & Sons, 1990.

7. X. Hu and S. J. Taylor, The multifractal structure of stable occupation measure, Stoch. Proc. Appl. 66 (1997), 283–299.

8. S. Jaffard, The multifractal nature of Lévy processes, preprint.

9. R. Kaufman, Une propriété metriqué du mouvement brownien, C. R. Acad. Sci. Paris, 268 (1969), 727–728.

10. G. Lawler, The frontier of a Brownian path is multifractal, preprint.

11. L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), 82–196.

12. S. Orey and S. J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail?, Proceed. Lond. Math. Soc. 28 (1974), 174–192.

13. R. Pemantle and Y. Peres, Galton-Watson trees with the same mean have the same polar sets. Ann. Probab. 23 (1995), 1102–1124.

14. R. Pemantle, Y. Peres and J. W. Shapiro, The trace of spatial Brownian motion is capacity-equivalent to the unit square, Probab. Theory and Related Fields 106 (1996), 379–399.

15. Y. Peres, Intersection-equivalence of Brownian paths and certain branching processes, Comm. Math. Phys. 177 (1996), 417–443.

16. E. A. Perkins, On the Hausdorff Dimension of Brownian Slow points, Zeits. Wahrschein. verw. Gebeite 64 (1983), 369–399.

17. E. A. Perkins and S. J. Taylor, Uniform measure results for the image of subsets under Brownian motion, Probab. Theory Related Fields 76 (1987), 257–289.

18. E. A. Perkins and S. J. Taylor, Measuring close approaches on a Brownian path, Ann. Probab. 16 (1988), 1458–1480.

19. , The multifractal structure of super-Brownian motion, Ann. Inst. Henri Poincaré 34 (1998), 97–138.

20. R. Reidi, An improved multifractal formalism and self-similar measures, J. Math. Anal. Applic. 189 (1995), 462–490.

21. D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, New York, 1991.

22. C. A. Rogers and S. J. Taylor, Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961), 1–31.

23. N.-R. Shieh and S. J. Taylor, Logarithmic multifractal spectrum of stable occupation measure, preprint.

24. D. W. Stroock, Probability theory, an analytic view. Cambridge University Press, 1993.

25. S. J. Taylor, Regularity of irregularities on a Brownian path, Ann. Inst. Fourier (Grenoble) 39 (1974), 195–203.

26. , The measure theory of random fractals, Math. Proceed. Cambr. Phil. Soc. 100 (1986), 383–406.

27. , Super Brownian motion is a fractal measure for which the multifractal formalism is invalid, Symposium in Honor of Benoit Mandelbrot (Curagao, 1995). Fractals 3 (1995), 737–746.

28. S. J. Taylor and C. Tricot, Packing measure, and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), 679–699.

29. C. Tricot, Two definitions of fractal dimension, Math. Proc. Camb. Phil. Soc. 91 (1982), 57–74.

30. G. N. Watson, A treatise on the theory of Bessel functions. Cambridge University Press, 1922.

Amir Dembo  Yuval Peres
Departments of Mathematics and of Statistics Dept. of Statistics
Stanford University  UC Berkeley
Stanford, CA 94305  Berkeley, CA 94720
amir@stat.stanford.edu  peres@stat.berkeley.edu

Jay Rosen  Ofer Zeitouni
Department of Mathematics  Dept. of Electrical Eng.
College of Staten Island, CUNY  Technion
Staten Island, NY 10314  Haifa 32000, Israel
jrosen3@idt.net  zeitouni@ee.technion.ac.il