Fast Sets and Points for Fractional Brownian Motion

Transcription

1 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Fast Sets and Points for Fractional Brownian Motion By Davar Khoshnevisan* & Zhan Shi University of Utah & Université ParisVI Summary. In their classic paper, S. Orey and S.J. Taylor compute the Hausdorff dimension of the set of points at which the law of the iterated logarithm fails for Brownian motion. By introducing fast sets, we describe a converse to this problem for fractional Brownian motion. Our result is in the form of a limit theorem. From this, we can deduce refinements to the aforementioned dimension result of Orey and Taylor as well as the work of R. Kaufman. This is achieved via establishing relations between stochastic co-dimension of a set and its Hausdorff dimension along the lines suggested by a theorem of S.J. Taylor. Keywords and Phrases. Fast point, fast set, fractional Brownian motion, Hausdorff dimension. A.M.S Subject Classification. 6G15, 6G17, 6J Introduction Suppose W, W t; t é is standard one dimensional Brownian motion starting at. Continuity properties of the process W form a large part of classical probability theory. In particular, we mention A. Khintchine s law of the iterated logarithm see, for example, [21, Theorem II.1.9]: for each t é, there exists a null set N 1 t such that for all ω N 1 t, lim sup h + W t + h W t h ln ln1/h = Later on, P. Lévy showed that t é N 1 t isnotanullset.indeed,heshowed the existence of a null set N 2 outside which lim sup h + sup 6 r 6 1 W r + h W r h ln1/h = * Research supported by a grant from the National Security Agency

2 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math See [13, p. 168] or [21, Theorem I.2.7], for example. It was observed in [18] that the limsup is actually a limit. Further results in this direction can be found in [2, p. 18]. The apparent discrepancy between 1.1 and 1.2 led S. Orey and S.J. Taylor to further study the so called fast or rapid points of W. To describe this work, for all λ>, define F 1 λ to be the collection of all times t é at which lim sup h + W t + h W t h ln1/h é λ 2. The main result of [18] is that with probability one, dim F 1 λ =1 λ One can think of this as the multi-fractal analysis of white noise. Above and throughout, dima refers to the Hausdorff dimension of A. Furthermore, whenever dima is strictly negative, we really mean A =?. OreyandTaylor s discovery of Eq. 1.3 relied on special properties of Brownian motion. In particular, they used the strong Markov property in an essential way. This approach has been refined in [3, 4, 11], in order to extend 1.3 in several different directions. Our goal is to provide an alternative proof of Eq. 1.3 which is robust enough to apply to non-markovian situations. We will do so by i viewing F 1 λ as a random set and considering its hitting probabilities; and ii establishing within these proofs links between Eqs. 1.2 and 1.3. To keep from generalities, we restrict our attention to fractional Brownian motion. With this in mind, let us fix some α ], 2[ and define X, Xt; t é to be a one dimensional Gaussian process with stationary increments, mean zero and incremental standard deviation given by, Xt Xs 2 = t s α/2. See 1.8 for our notation on L p P norms. The process X is called fractional Brownian motion with index α hereforth written as f BMα. We point out that when α =1,X is Brownian motion. Let dim M E denote the upper Minkowski dimension of a Borel set E R 1 ; see references [17, 24]. Our first result, which is a fractal analogue of Eq. 1.2, is the following limit theorem: Theorem 1.1. Suppose X is fbmα and E [, 1] is closed. With probability one, Xt + h Xt lim sup sup h + t E h α/2 6 2 dim M E. 1.4 ln1/h On the other hand, with probability one, Xt + h Xt sup lim sup t E h + h α/2 é 2dimE. 1.5 ln1/h

3 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Loosely speaking, when α = 1, Theorem 1.1 is a converse to 1.3. For all λ é, define F α λ to be the collection of all closed sets E [, 1] such that Xt + h Xt lim sup sup h + t E h α/2 é λ 2. ln1/h One can think of the elements of F α λ asλ fast sets. Theorem 1.1 can be recast in the following way. Corollary 1.2. Suppose X is fbmα and E [, 1] is closed. If dim M E < λ 2,thenE F α λ almost surely. On the other hand, if dime é λ 2,then E F α λ. Remark An immediate consequence of Theorem 1.1 is the following extension of 1.2: lim sup h + Xt + h Xt sup 6 t 6 1 h α/2 = 2, a.s. ln1/h When α ], 1], this is a consequence of [15, Theorem 7]. When α ]1, 2[, the existence of such a modulus of continuity is mentioned in [16, Section 5]. A natural question is: can one replace E by a random set? The first random set that comes to our mind is the zero set. When α = 1, the process is Brownian motion. Its Markovian structure will be used to demonstrate the following. Theorem 1.3. Suppose W is Brownian motion. Let Z, { s [, 1] : W s = }. Then, with probability one, lim sup sup h + t Z W t + h =sup h ln1/h t Z lim sup h + W t + h h ln1/h =1. Thus, the escape of Brownian motion from zero is slower than Lévy s modulus 1.2. Next, we come to dimension theorems; see 1.3 for an example. Define the λ fast points for f BMα as follows: F α λ, { Xt + h Xt t [, 1] : lim sup h + h α/2 é λ } ln1/h In particular, F 1 λ denotes the λ fast points of Brownian motion. In [9], R. Kaufman has shown that for any closed E [, 1] and every λ>, with probability one, dime λ 2 6 dim E F 1 λ. 1.7

4 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Moreover, there exists a certain fixed closed set E [, 1] for which the above is an equality. Our next aim is to show that the inequality in 1.7 is an equality for many sets E [, 1], and that this holds for all f BMα s. More precisely, we offer the following: Theorem 1.4. Suppose X is fbmα, E [, 1] is closed and λ>. Then, with probability one, dime λ 2 6 dim E F α λ 6 dim P E λ 2, where dim P denotes packing dimension. See [17] and [24] for definitions and properties of dim P. In particular, 1.3 holds for the fast points of any fractional Brownian motion. Moreover, when E = Z and α = 1, we have the following dimension analogue of Theorem 1.3. Note that F 1 λ isthesetoffastpointsofw as defined earlier. In other words, it is defined by 1.6 with X replaced by W. Theorem 1.5. Suppose W is Brownian motion and Z, { s [, 1] : W s = }. Then, for all λ>, with probability one, dim Z F 1 λ = 1 2 λ2. A natural question which we have not been able to answer is the following: Problem Does Theorem 1.5 have a general analogue for all fbmα s? The proofs of Theorems 1.1, 1.3, 1.4 and 1.5 rely on parabolic capacity techniques and entropy arguments. The entropy methods follow the arguments of [18] closely. On the other hand, the parabolic capacity arguments rely on relationships between the Hausdorff dimension of random sets and stochastic co dimension see 2. The latter is a formalization of a particular application of [23, Theorem 4], which can be found in various forms within the proofs of [1, 7, 14, 19]. We suspect our formulation has other applications. In 3, we demonstrate 1.4 while 1.5 and the first inequality i.e., the lower bound of Theorem 1.4 are derived in 4. The proof of the upper bound of Theorem 1.3 appears in 5; the upper bounds of Theorems 1.4 and 1.5 can be found in 6 and 7, respectively; and the lower bounds for Theorems 1.3 and 1.5 are proved simultaneously in 8. We conclude the Introduction by mentioning some notation which will be utilized throughout this article. Define the function ψ as ψh, 2ln1/h, h ], 1[.

5 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math By Φ we mean the tail of a standard normal distribution, i.e. Φx 1, e u2 /2 du, x ], [. 2π x Furthermore, Ω, F, P denotes our underlying probability space. For any real variable Z on Ω, F, P andforeveryp> we write the L p P norm of Z by, Z p, Ω Zω p Pdω 1/p. 1.8 Throughout, X denotes f BMα for any α ], 2[. However, when we wish to discuss Brownian motion specifically i.e., when α = 1, we write W instead. In accordance with the notation of Theorem 1.3, Z will always denote the zero set of W restricted to [, 1]. Finally, the collection of all atomless probability measures on a set E is denoted by P + E. Remark. Since the first circulation of this paper, many of the gaps in the inequalities of this paper have been bridged. For instance, in Theorem 1.1, both constants of 1.4 and 1.5 can be computed. This and related material can be found in [1]. Acknowledgements. Much of this work was done while the first author was visiting Université Paris VI. We thank L.S.T.A. and Laboratoire de Probabilités for their generous hospitality. Our warmest thanks are extended to Steve Evans, Mike Marcus, Yuval Peres and Yimin Xiao. They have provided us with countless suggestions, references, corrections and their invaluable counsel. In particular, it was Yuval Peres who showed us the rôle played by packing dimensions as well as allowing us to use his argument cf. [1] in the proof of Theorem Preliminaries on Dimension In this section, we discuss a useful approach to estimating Hausdorff dimensions of random sets via intersection probabilities. Let S 1 denote the collection of all Borel measurable subsets of [, 1]. We say that E :Ω S 1 is a random set, if1l E ω :Ω S 1 ω, E {, 1} is a random variable in the product measure space. An important class of random { } sets are the closed stochastic images E, S[, 1], St; t [, 1],whereS is a stochastic process with càdlàg sample paths. Let us begin with some preliminaries on Hausdorff dimension; see [8, 16, 24] for definitions and further details. Given s é andaborelsete [, 1], let Λ s E denotethes dimensional Hausdorff measure of E. Recall that the Hausdorff dimension dime ofe is defined by: dime, inf { s>: Λ s E < }. When it is finite, Λ s E extends nicely to a Carathéodory outer

6 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math measure on analytic subsets of [, 1]. By a slight abuse of notation, we continue to denote this outer measure by Λ s. Suppose E is a random set. Since we can economically cover E with intervals with rational endpoints, dime is a random variable. We will use this without further mention. Typically, computing upper bounds for dime is not very difficult: find an economical cover I j ofe, whose diameter is h or less and compute j I j s. Obtaining good lower bounds for dime is the harder of the two bounds. The standard method for doing this is to utilize the connections between Hausdorff dimension and potential theory. For any µ P + E andallβ>, define, A β µ, µ[t h, t + h] sup sup <h 6 1/2 t [h,1 h] h β. 2.1 It is possible that A β µ =. We need the following connection between Hausdorff dimension and potential theory; while it is only half of Frostman s lemma of classical potential theory, we refer to it as Frostman s lemma for brevity. Lemma 2.1. Frostman s Lemma; [8, p. 13] Suppose E S 1 β<dime. Then there exists µ P + E for which A β µ <. satisfies Thus, a method for obtaining lower bounds on dime is this: find a probability measure µ which lives on E and show that A β µ <. Ifthiscanbe done for some β>, then dime é β. In general, this is all which can be said. However, if the set E in question is a random set in the sense of the first paragraph of this section, there is an abstract version of [23, Theorem 4] which can be used to bound dime from below; see also [1]. We shall develop this next. Define the upper stochastic co-dimension co-dim of a random set E by co-dime, inf { β [, 1] : G S 1 with dimg >β, PE G? =1 }. 2.2 In order to make our definition sensible and complete, we need to define inf?, 1. Remark In applications, we often need the following fact: if G S 1 satisfies dimg > co-dime, then PE G? =1. In this section we present two results about stochastic co-dimension, the first of which is the following. Theorem 2.2. Suppose E is a random set. Then, for all G S 1, dime G é dimg co-dime, a.s. As mentioned earlier, Theorem 2.2 is an abstract form of a part of [23, Theorem 4]. This kind of result has been implicitly used in several works. For

7 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math example, see [7, 14, 19, 2]. To prove it, let us introduce an independent symmetric stable Lévy process S γ, Sγ t; t é of index γ ], 1[. The following two facts are due to J. Hawkes; cf. [6]. Lemma 2.3. Suppose G S 1 satisfies dimg < 1 γ. Then, with probability one, S γ [, 1] G =?. Lemma 2.4. Suppose G S 1 satisfies dimg > 1 γ. Then, P S γ[, 1] G? >. Furthermore,on S γ [, 1] G?, dim S γ [, 1] G =dimg+γ 1, a.s. 2.3 Historically, the above results are stated with S γ [, 1] replaced by S γ [, 1]. By symmetry, semi-polar sets are polar for S γ. Therefore, the same facts hold for S γ [, 1]. We can now proceed with Theorem 2.2. Proof of Theorem 2.2. Without loss of generality, we can assume that the compact set G satisfies dimg > co-dime. With this reduction in mind, let us choose a number γ ], 1[ satisfying, γ>1 dimg+co-dime. 2.4 Choose the process S γ as in the earlier part of this section. Since γ>1 dimg, it follows from Lemma 2.4 that κ, P Sγ [, 1] G? >. By 2.3, κ = P Sγ [, 1] G?, dim S γ [, 1] G =dimg+γ 1 6 P Sγ [, 1] G?, dim S γ [, 1] G > co-dime, where we have used 2.4 in the last inequality. In view of Remark 2.1.1, κ is bounded above by P Sγ [, 1] G E?. Applying Lemma 2.3 gives κ 6 P Sγ [, 1] G E?, dimg E é 1 γ Sγ [, 1] G?, dimg E é 1 γ 6 P = κ P dimg E é 1 γ. The last line utilizes the independence of S γ and E. Sinceκ>, it follows that for all γ satisfying 2.4, dimg E é 1 γ, almost surely. Let γ 1 dimg+ co-dime along a rational sequence to obtain the result. Next, we present the second result of this Section. It is an immediate consequence of the estimates of [1, Section 3] and Theorem 2.2 above.

8 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Theorem 2.5. Suppose E n ; n é 1 is a countable collection of open random sets. If sup n é 1 co-dime n < 1, then co-dim n=1 In particular, P n é 1 E n? =1. E n =supco-dime n. n é 1 Informally speaking, this is a dual to the fact that for all F n S 1 n = 1, 2,..., dim n=1 F n =supn é dimf n. 3. Theorem 1.1: Upper Bound Define the set of near fast points as follows: for all λ, h >, { F α λ, h, t [, 1] : sup Xs Xt é λh α/2 ψh }. 3.1 t 6 s 6 t+h Next, for any R, > 1, all integers j é 1 and every integer 6 m<r j +1, define I m,j, [ mr j, m +1R j]. 3.2 Finally, define for the above parameters, P λ,α m,j, P I m,j F αλ, R j?. 3.3 The main technical estimate which we shall need in this section is the following: Lemma 3.1. Let X be fbmα, whereα ], 2[. For all λ>, ε ], 1[, >1 and all R>1, thereexistsj 1 = J 1 ε, α,, λ, R [2, [ such that for all j é J 1 and all m é, P λ,α m,j 6 R λ2 1 εj. Remark Part of the assertion is that J 1 does not depend on the choice of m. Proof. By stationarity and scaling, P λ,α m,j = P sup sup Xs + t Xs é λψr j. 6 t 6 R 1 j 6 s 6 1 We obtain the lemma by applying standard estimates and [12, Lemma 3.1] to the Gaussian process Xs + t Xt; s, t é. The proof of the upper bound is close to its counterpart [18]; cf. the first part of Theorem 2 therein.

9 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Proof of Theorem 1.1: upper bound. Recall 3.1 and 3.2. Consider a fixed closed set E [, 1]. Fix R, > 1andλ>. Define for all integers m é, J k, 1l {I?}1l m,j Fαλ,R j {I m,j E?}. 3.4 j é k m é By 3.3 and Lemma 3.1, for all ε ], 1[, there exists J 1 = J 1 ε, α,, λ, R [2, [ such that for all k é J 1, J k 1 = P λ,α m,j 1l {I m,j j é k m é E?} 6 R λ2 1 εj 1l {I m,j j é k m é E?} 6 R λ2 1 εj M R j ; E, j é k where Mε; E denotes the ε capacity of E. That is, it is the maximal number of points in E which are at least ε apart; see [5]. On the other hand, by definition, for all δ ], 1[, there exists J 2 = J 2 δ, R, [2, [, such that for all j é J 2, M R j ; E 6 R j1+δ dimm E. Hence, for all k é J 1 J 2, J k 1 6 R j λ 2 1 ε 1+δ dim M E. j é k It may help to recall that J 1 J 2 depends only on the parameters λ, α, R,, δ, ε. Let us pick these parameters so that λ 2 1 ε >1 + δ dim M E. It is easy to see that for this choice of parameters, k J k 1 <. By the Borel Cantelli lemma, with probability one, there exists a finite random variable k such that for all k é k, J k =. In other words, with probability one, for all j é k, F α λ, R j E =?. Rewriting the above, we see that with probability one, for all j é k,sup t E sup t 6 s 6 t+r Xs Xt j 6 λr jα/2 ψr j. Take any h 6 R k.thereexistsajék,sothatr j 1 6 h 6 R j. It follows that for all h 6 R k, sup Xt + h Xt 6 sup sup Xs Xt t E t E t 6 s6 t+r j 6 λr jα/2 ψr j 6 λr α/2 h α/2 ψh, since ψ is monotone decreasing. This implies that whenever λ 2 1 ε >1 + δ dim M E, Xt + h Xt lim sup sup h + t E h α/2 6 λr α/2, a.s. ψh Along rational sequences, let ε, δ +,, R 1 + and λ 2 this order to see that with probability one, Xt + h Xt lim sup sup h + t E h α/2 6 dim M E. ψh dim M E in

10 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math This proves the desired upper bound. 4. Theorems 1.1 and 1.4: Lower Bounds For each closed set E [, 1] and for every µ P + E andh, λ >, define, I µ h; λ, µdt1l {Xt+h Xt>λh α/2 ψh}. 4.1 The key technical estimate of this section is the following: Lemma 4.1. Suppose E [, 1] is compact, µ P + E. For any ε ], 1[ and λ>, there exists a small h ], 1[ such that for all h ],h [, I µ h; λ 2 2 µ [t h 1 ε,t+ h 1+ε ] I µ h; λ ε +4 sup 1 h 1 ε 6 t 6 1 h 1 ε Φ λψh. Proof. Since X has stationary increments, I µ h; λ 1 = Φ λψh. 4.2 We proceed with the estimate for the second moment. Define, a, λψh, Xs + h Xs U,, h α/2 Xt + h Xt V,, h α/2 ρ, UV 1. Then, ignoring the dependence on h, s, t, where, Q 1, Q 2, 2 Q 3, 2 µdt µdt µdt t t I µ h; λ 2 2 = Q 1 + Q 2 + Q 3, 4.3 µds1l {ρ 6ln1/h 2 }PU é a, V é a, µds1l {ρ>ln1/h 2 }1l {t s>2h} PU é a, V é a, µds1l {ρ>ln1/h 2 }1l {t s 6 2h}PU é a, V é a.

11 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math We estimate each term separately. The critical term is Q 1.Writex + =maxx, for any x. Ifρ<1/4, 1 PU é a, V é a = 2π 1 ρ 2 6 = a a 1 2π 1 ρ 2 a 1 ρ 2 1 4ρ + Φ a a exp exp 1 4ρ + 1 ρ 2 2. x2 + y 2 2ρxy 21 ρ 2 dx dy 1 4ρ+ x 2 + y 2 21 ρ 2 According to Mill s ratio for Gaussian tails [22, p. 85], for any x>1, 1 x 2 2πx exp x2 2 6 Φx 6 1 2πx exp x2 2 dx dy. 4.4 Therefore, using the fact that µ is an atomless probability measure, we have, 1 r 2 Q 1 6 sup 1 4r + Φ 1 4r + 2 λψh 1 r 2 r: 1<r 6ln1/h 2 [ ] r 2 3/2 6 Φλψh sup 1 r: 1<r 6ln1/h λψh r + 2 λψh 2 4r + r 2 exp 1 r 2. Since ψh 2 ln1/h 2 = o1 as h goes to, this leads to: where Bh; λ is such that, for any λ>, Q 1 6 Bh; λ [ Φλψh ] 2, 4.5 lim Bh; λ = h + To estimate Q 3, use the trivial inequality P U é a, V é a 6 P U é a, to see that Q sup µ [t 2h, t] Φλψh 2h 6 t sup µ [t 2h, t +2h] Φλψh h 6 t 6 1 2h Finally, we need to approximate Q 2. Directly computing, note that when s< t 2h, ρ = t s h α + t s + h α 2 t s α 2h α = 1 2 t s α [ 1 h h t s α + 1+ h α ] 2. t s

12 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math By Taylor expansion of 1 ± x α,weseethatforall x 6 1 2, 1 x α +1+x α 2= αα 1 [ 1 ξ 1 α 2 +1 ξ 2 α 2] x 2, 2 where ξ i for i =1, 2. In particular, for all x 6 1 2, 1 x α +1+x α α x 2. { } 2 α. In other words, when s < t 2h, ρ 6 2h/t s On the other hand, if we also know that ρ > ln1/h 2, it follows that for all h small, t s 6 2h ln1/h 2/2 α 6 h 1 ε. Since PU é a, V é a 6 Φλψh, we obtain the following: for all ε>, there exists h 1 ], 1[ such that for all h ],h 1 [, Q sup µ [t h 1 ε,t+ h 1 ε ] Φλψh. h 1 ε 6 t 6 1 h 1 ε Together with 4.7, we obtain: for all ε>, there exists h 2 ], 1[ such that for all h ],h 2 [, Q 2 + Q sup µ [t h 1 ε,t+ h 1 ε ] Φλψh. h 1 ε 6 t 6 1 h 1 ε Combining this with and , we obtain the result. Now we can prove the lower bounds in Theorems 1.1 and 1.4. Proof of Theorems 1.1 and 1.4: lower bounds. By Frostman s lemma Lemma 2.1, for any β<dime, there exists a µ P + E, such that for all h ], 1[ small, sup µ [t h, t + h] 6 h β. h 6 t 6 1 h For such a µ, uselemma4.1toseethatforallε ], 1[ and λ>, there exists h 3 ], 1[ such that whenever h ],h 3 [, I µ h; λ 2 2 I µ h; λ ε + 4h1 εβ 1 Φλψh. According to Mill s ratio for Gaussian tails see 4.4, for any λ>, there exists a small h 4 >, such that for all h ],h 4 [, Φλψh é h λ2 /4λ ln1/h. Hence, for all h ],h 3 h 4 [, I µ h; λ 2 2 I µ h; λ ε +16λh 1 εβ λ2 ln1/h. 1

13 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math By choosing λ such that 1 εβ >λ 2 we can deduce that for all h ],h 3 h 4 [, I µ h; λ 2 2 I µ h; λ ε. Applying the Paley Zygmund inequality [8, p. 8], we see that P Iµ h; λ > 1 é 1+2ε. 4.8 We are ready to complete the proof. Define, A α λ, h, { t [, 1] : Xt + r Xt } sup >λ r 6 h r α/2 ψr This is the collection of all h approximate λ fast points. Observethatfor each h>, A α λ, h isanopensubsetof[, 1]. If I µ h; λ > µ P + E, then E A α λ, h?. By 4.8, we see that as long as 1 εβ >λ 2,thenfor all h ],h 3 h 4 [, P E Aα λ, h? é1 + 2ε 1.Notethatifh6 h,then A α λ, h A α λ, h. Hence, for all λ 2 [, dime[, P A α λ, h E?, h > = 1. By Theorem 2.5, for all λ 2 [, dime[, P h> A α λ, h E? =1. Since h> A αλ, h =F α λ, we have shown that co-dim F α λ 6 λ 2. Unravelling the notation, this implies Eq. 1.5 i.e., the lower bound in Theorem 1.1. It also implies the lower bound in Theorem Theorem 1.3: Upper Bound For, R > 1, and j é 1, define, Gj, { 6 m 6 R j : W mr j 6 2 jr j ln R }. 5.1 The notation is motivated by the following description: we think of I m,j see 3.2 as good if m Gj. Otherwise, I m,j is deemed bad. We also recall Eq. 3.1 with α = 1 thus replacing X by W in 3.1. In analogy with the definition of J k see 3.4, we define, J k, j é k m Gj 1l {I m,j F1λ,R j?}. 5.2 By the independence of the increments of W, {I m,j F 1λ, R j?} is independent of {m Gj}. Recalling 3.3, we see that J k 1 = j é k m é P m Gj P λ,1 m,j.

14 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Since W 1 has a probability density which is uniformly bounded above by 1, for all 6 m 6 R j, j ln R P m Gj 6 4 m Next, fix ε ], 1[. By Lemma 3.1, there exists J 3, J 1 ε, 1,,λ,R [2, [, such that for all j é J 3 and all m é, P λ,1 m,j 6 1 εj. Using 5.3, we see that R λ2 for all k é J 3, J k j é k [R j ] m= j ln R m 1 R λ2 1 εj. If we choose λ 2 1 ε > /2, then a few lines of calculations reveal that k J k 1 <. By the Borel Cantelli lemma, there exists a finite random variable k 1, such that with probability one, for all k é k 1, J k =. In particular, with probability one, for all j é k 1, 1l {m Gj}1l {I m,j F1λ,R j?} =. 5.4 By Lévy s modulus of continuity for W cf. 1.2, there exists a finite random variable k 2, R, such that with probability one, for all j é k 2, R, 1l {m Gj} é 1l {I m,j Z?}. 5.5 Eq. 5.4 shows that with probability one, for all j é k 3, k 1 k 2, R, F 1 λ, R j Z =?. That is, almost surely, for all j é k 3, sup t Z sup W t + s 6 λr j/2 ψr j. t 6 s6 t+r j If h 6 R k3,thereexistsj é k 3, such that R j 1 6 h 6 R j. By monotonicity, sup t Z sup W t + s 6 λr j/2 ψr j 6 λr 1/2 h 1/2 ψh. t 6 s 6 t+h In particular, we have shown that as long as λ 2 1 ε >/2, then almost surely, lim sup h + W t + h sup t Z h 1/2 ψh 6 λr1/2. Along rational sequences and in this order, let, R 1 +, ε + and λ to see that with probability one, lim sup h + W t + h sup 1 t Z h 1/2 ψh 6. 2

15 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math This proves the upper bound in Theorem Theorem 1.4: Upper Bound Recall Eqs. 1.6 and 3.1. We begin with a regularization scheme which is used later in our good covering for dimension calculations. Lemma 6.1. For all integers k é 1 and all reals θ ], 1[ and R>1, F α λ F α θλr α/2,r j+1. j é k Proof. From first principles, it follows that for all β < λ, F α λ h> <δ<h F αβ,δ. Let δ<h<1, say R j 6 δ 6 R j+1. For all t F α β,δ, sup Xs Xt é βr jα/2 ψr j+1 =βr α/2 R j 1α/2 ψr j+1. t 6 s6 t+r j+1 We have used the monotonicity properties of ψ. This implies that t F α βr α/2,r j 1. The result follows. We are prepared to demonstrate the upper bound in Theorem 1.4. Proof of Theorem 1.4: upper bound. Fix R, > 1, θ ], 1[ and recall I m,j from 3.2. From Lemma 6.1, it is apparent that for any integer k é 1, we have the following covering of F α λ: F α λ j=k m=1 I m,j F αθλr α/2,r j By 3.3 and Lemma 3.1, for all ε>, there exists J 4 = J 4 ε, α,, λ, R, θ [2, [, such that for all j é J 4, P I m,j F αθλr α/2,r j+1? 6 R 1 εθ2 λ 2 R 2αj. 6.2 For any s é and every integer k é 1, define, J k s, I s m,j 1l {I m,j E?} 1l {I j é k m 6 R j +1 m,j FαθλR α/2,r j+1?}. By 6.2, for all k é J 4, Jk s 6 1 M 1 + R j 1 ; E R sj R 1 εθ2 λ 2 R 2αj. j é k

16 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Recall from 3 thatmε; E istheε capacity of E. Note that if we enlarge J 4 further, then for all β> dim M E andallj é J 4, we can also ensure that M 1 + R j 1 ; E 6 R βj. Suppose β> dim M E and s > β 1 εθ 2 λ 2 R 2α. 6.3 It follows that k Jk s 1 <. By the Borel Cantelli lemma, lim k J k s =, a.s. From 6.1 and the definition of Hausdorff dimension, it follows that for any s satisfying 6.3, Λ s E F α λ 6 lim J ks =, k Therefore, almost surely, a.s. dim E F α λ 6 s. Letting β dim M E, ε,, R 1, θ 1 in this order and along rational sequences, we obtain dim E F α λ 6 dim M E λ 2, a.s. By [17, pp. 57 and 81], for every G S 1, dimg = dim P G = inf sup dimg i, G= i=1 Gi i inf sup dim M G i, G= i=1 Gi where the G i s are assumed to be bounded. Thus, i dim E F α λ 6 dim P E λ 2, a.s., which is the desired upper bound. 7. Proof of Theorem 1.5: Upper Bound Recall the notation of 3 and 6, and Gj from 5.1. By Lemma 6.1 and 5.5, for any, R > 1andθ ], 1[, there exists a finite random variable K such that almost surely for all k é K, Z F 1 λ I m,j F 1θλR 1/2,R j j é k m Gj Next, we show that the above is a fairly economical covering. Since W has independent increments, for any s>, I s m,j P I m,j F 1θλR 1/2,R j+1?, m Gj j é k 6 m<r j +1 = j é k 6 m<r j +1 I m,j s P I m,j F 1θλR 1/2,R j+1? P m Gj.

17 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math By 5.3, j é k 6 m<r j j é k [R j ]+1 m= I I m,j m,j s P I m,j F 1θλR 1/2,R j+1?,m Gj s P I m,j F 1θλR 1/2,R j+1 j ln R? m 1. Using Lemma 3.1, we see that for all ε ], 1[, there exists J 5 = J 5 ε, θ,, λ, R [2, [, such that for all k é J 5, I s m,j P I m,j F 1θλR 1/2,R j+1?, m Gj j é k 6 m<r j j é k R sj R 1 εθ2 λ 2 R 1 j [R j ]+1 m= j ln R m 1. In particular, if s > 2 1 εθ2 λ 2 R 1, 7.2 then, lim k j é k m Gj I m,j s 1l {I m,j F1θλR 1/2,R j+1?} =, a.s. Thanks to 7.1, we can deduce that for any s satisfying 7.2, Λ s Z F 1 λ =, almost surely. In particular, almost surely, dim Z F 1 λ 6 s. Letε, θ 1 and R, 1 in this order to see that with probability one, dim Z F 1 λ λ 2. This is the desired upper bound. 8. Proofs of Theorems 1.3 and 1.5: Lower Bounds The main result of this section is the following which may be of independent interest. Theorem 8.1. Fix a compact set E [, 1]. Then, dime >λ = P Z F 1 λ E? =1. As the lower bounds in Theorems 1.3 and 1.5 follow immediately from the above, the rest of this section is devoted to proving Theorem 8.1. Suppose E [, 1] is compact, µ P + E andh, λ >. Define,

18 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math J µ h; λ, µds1l { W s <h} 1l {W s+h W s>λh 1/2 ψh}. For all µ P + E andanyh, β >, define the following: S h µ, S h µ, h µdt sup, 6 s 6 h t s sup 6 s 6 1 s s+h 1 s µdt t s. The proof of Theorem 8.1 is divided into several steps. In analytical terms, our first result is an estimate, uniform in s [,h], for the 1/2 potential of a measure µ restricted to an interval [s, h]. Indeed, recalling A β µ from 2.1, we have the following: Lemma 8.2. Suppose µ P + E. Then for all β>1/2 and all h>, S h µ 6 S h µ 6 2eβ 2β 1 A βµ h β 1 2, 8.1 2eβ 2β 1 A βµ h β Proof. Without loss of generality, we can assume that A β µ < for some β>1/2 otherwise, there is nothing to prove. We proceed with an approximate integration by parts: for all 6 s 6 h, h s µdtt s 1/2 = j= s+h se j s+h se j 1 µdtt s 1/2 6h s 1/2 e j+1/2 µ[s +h se j 1,s+h se j ] j= 6 e 1/2 h s β 1/2 A β µ exp jβ 1 2, j= which yields 8.1. The estimate 8.2 can be checked in the same way. Our next two lemmas are moment estimates for J µ. Lemma 8.3. Suppose µ P + E is fixed. For every ε>, thereexistsan h ε >, such that for all h ],h ε [,andallλ>, Jµ h; λ 1 é 2 πe 1 εh Φ λψh.

19 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Proof. By the independence of the increments of W, J µ h; λ 1 = Φ λψh µds P W s 6 h. A direct calculation reveals that if r ], 1[, P W 1 6 r é 2/πe r. Therefore, by Brownian scaling, for every h ], 1[, Jµ h; λ 2 é 1 é µds s πe h Φ λψh h 2 2 πe µ[h2, 1] h Φ λψh. The lemma follows upon taking h> small enough. Lemma 8.4. Fix µ P + E. Suppose A β µ < for some β>1/2. Then, for all h, λ >, Jµ h; λ { e2β 2β 1 2 A2 βµ 2 h β+3/2 Φ λψh + h 2 Φ 2 } λψh. Proof. To save space, for all h, t é, define, h W t, W t + h W t. By the independence of the increments of W, Jµ h; λ 2 2 =2 µdt1l { W t 6 h}1l { h W t é λh ψh} 1/2 t µds1l { W s 6 h}1l { h W s é λh 1/2 ψh} 1 =2Φ λψh t µdt1l { W t 6 h} µds1l { W s 6 h}1l { h W s é λh 1/2 ψh} 6 2Φ λψh [ T 1 + T 2 ], 8.3 where, T 1, T 2, h µdt1l { W t 6 h} µdt1l { W t 6 h} t h + t We will estimate T 1 and T 2 in turn. µds1l { W s 6 h}1l { h W s é λh 1/2 ψh}, 1 µds1l { W s 6 h}. t h + 1 1

20 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math We will need the following consequence of Gaussian laws: for all s, h >, P W s 6 h 6 s 1/2 h. 8.4 First, we estimate T 1.Notethat, t h T 1 = µdt µdsp W t 6 h, W s 6 h, h W s é λh 1/2 ψh. h 8.5 Suppose t [h, 1] and s [,t h]. Then s 6 s + h 6 t, and we have, P W t 6 h W r; r 6 s + h = P W t W s + h+w s + h 6 h W r; r 6 s + h 6 sup P W t s h+ζ 6 h. ζ R On the other hand, W t s h has a Gaussian distribution. By unimodality of the latter, P W t 6 h W r; r 6 s + h 6 P W t s h 6 h. This actually is a particular case of T.W. Anderson s inequality for general Gaussian shifted balls. Using 8.5 and the principle of conditioning, T 1 6 h By 8.4, µdt t h P 6 Φ λψh µdt µds P W t s h 6 h W s 6 h, h W s é λh 1/2 ψh h t h T 1 6 h 2 Φ λψh µdt h µdsp W t s h 6 h P W s 6 h. t h 1 µds. st s h Changing the order of integration, we arrive at the following estimate: T 1 6 h 2 Φ λψh h 6 h 2 Φ λψh S 2 1µ µds 1 s s+h µdt t s h 6 4e2β 2β 1 2 A2 βµh 2 Φ λψh, 8.6 by Lemma 8.2. Next, we estimate T 2. By another unimodality argument, for all t é s and all h>, P W s 6 h, W t 6 h 6 P W s 6 h P W t s 6 h.

21 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Applying 8.4, P W s 6 h, W t 6 h 6 h 2 / st s. Therefore, t T 2 6 h 2 1 µdt µds t h + st s = h 2 T 2,1 + T 2,2, 8.7 where, h T 2,1, T 2,2, h µdt µdt t t t h 1 µds st s To estimate T 2,1, reverse the order of integration: T 2,1 = by Lemma 8.2. Similarly, h µds s h s 1 µds. st s µdt t s 6 S 2 h µ 6 4e2β 2β 1 2 A2 βµh 2β 1, 8.8 T 2,2 = µds s 6 S 1 µ S h µ s+h 1 6 4e2β 2β 1 2 A2 βµh β 1/2. s µdt t s Since h<1andβ > 1/2, using the above, 8.8 and 8.7, we arrive at the following: T 2 6 8e2β 2β 1 2 A2 βµh β+3/2. Use this, together with 8.6 and 8.3 in this order to get the result. We are ready for the main result of this section: Proof of Theorem 8.1. For λ, h >, recall 4.9 and define, Zh, { s [, 1] : W s <h }. Path continuity of W alone implies that Zh A 1 λ, h isanopenrandomset. We estimate the probability that it intersects E. By Frostman s lemma, for all β < dime, there exists µ P + E such that A β µ <. Let us fix a µ corresponding to an arbitrary but fixed choice of β satisfying: λ <β<dime

22 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math Applying Lemmas 8.3 and 8.4 to this choice of µ, weseethatforallε>, there exists h ε >, such that for all h ],h ε [, Jµ h; λ 2 [ 1 2 h β 1/2 ] 1 Jµ h; λ 2 é γ ε,β Φ λψh +1, 2 where, γ 2β ε,β, 12 1 ε 2 4πe 1+2β A 2 β µ. According to Mill s ratio for Gaussian tails see 4.4, we can pick h λ so small that for each and every h ],h λ [, Φ λψh é h λ2 /4λ ln1/h. Therefore, for all h ],h ε h λ [, Jµ h; λ 2 1 Jµ h; λ 2 é γ ε,β [8λh β λ2 1/2 1. ln1/h +1] 2 By 8.9, lim inf h + J µ h; λ 2 J 1 µ h; λ 2 é γ 2 ε,β >. By the Paley Zygmund inequality [8, p. 8], lim inf P Jµ h; λ > é γ ε,β >. h + Note that the event J µ h; λ > implies that Zh A 1 λ, h intersects E. Hence, lim inf P Zh A 1 λ, h E? é γ ε,β >. h + However, as h, the random open set Zh A 1 λ, h decreases to Z F 1 λ. Adapting Theorem 2.5 to the positive probability case, we can conclude that P Z F 1 λ E? é γ ε,β >. Note that the only requirement on E was that dime >λ 2 + 1/2. Since Hausdorff dimension is scale invariant, we see that for any real number s ], 1[, P Z F 1 λ s 1 E? é γ ε,β, { } where s 1 E, r/s : r E. We finish the proof by showing that this probability is actually 1. Fix s ], 1[ and observe from Brownian scaling that Z F 1 λ has the same distribution as Z F 1 λ [,s] in the sense that for all G S 1, P Z F 1 λ G? = P Z F 1 λ [,s] sg?. In particular, for all s ], 1[, P Z F 1 λ [,s] E? é γ ε,β >, Note that Z F 1 λ E [,s]isincreasingins. Thus, P s ],1[ { Z F 1 λ E [,s]? } é γ ε,β. Let C, s ],1[ Z F 1 λ E [,s]?.observethatc is measurable with respect to the germ field of W at and we have just argued that PC é γ ε,β >. By Blumenthal s 1 law, PC = 1. Since Z F 1 λ E? C, theresult follows for F 1.

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24 SÉMINAIRE DE PROBABILITÉS XXXIV, Lec. Notes in Math [17] P. Matilla Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability, Cambridge University Press, Cambridge. [18] S. Orey and S.J. Taylor How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc., 28, [19] Y. Peres Remarks on intersection equivalence and capacity equivalence. Ann. Inst. Henri Poincaré: Physique Théorique, 64, [2] E. Perkins and S.J. Taylor 1988, Measuring close approaches on a Brownian path, Ann. Prob., 16, [21] D. Revuz and M. Yor Continuous Martingales and Brownian Motion, second edition. Springer, Berlin. [22] G.R. Shorack and J.A. Wellner Empirical Processes with Applications to Statistics. Wiley, New York. [23] S.J. Taylor Multiple points for the sample paths of the symmetric stable process, Z. Wahr. ver. Geb., 5, [24] S.J. Taylor The measure theory of random fractals. Math. Proc. Camb. Phil. Soc., 1, Davar Khoshnevisan Department of Mathematics University of Utah Salt Lake City, UT U.S.A. davar@math.utah.edu Zhan Shi Laboratoire de Probabilités UniversitéParisVI 4, Place Jussieu Paris Cedex 5, France shi@ccr.jussieu.fr