Lecture 2: Hausdorff dimension results and hitting ...auffing/XiaoL2.pdfity is characterized by...

Lecture 2: Hausdorff dimensionresults and hitting probabilities

Yimin Xiao

Michigan State University

Northwestern University, July 11–15, 2016

Yimin Xiao (Michigan State University) Lecture 2: Hausdorff dimension results and hitting probabilitiesNorthwestern University, July 11–15, 2016 1

Outline

1 Fractal properties of Gaussian random fields

2 Hitting probabilities and Riesz-type capacity

3 Brownian motion and thermal capacity

Geometric properties of Gaussian fields

Given an (N, d)-random field X = X(t), t ∈ RN, it is ofinterest to study the geometric properties of the followingrandom sets:

Range X([0, 1]N

X(t) : t ∈ [0, 1]N

Graph GrX([0, 1]N

(t,X(t)) : t ∈ [0, 1]N.

Level set X−1(x) =

t ∈ RN : X(t) = x

Excursion set X−1(F) =

t ∈ RN : X(t) ∈ F, ∀F ⊂

Multiple points, etc.

Excursion sets

Let X be real-valued. For any u ∈ R,

Au(X,T) = t ∈ T : X(t) > u

is called an excursion set of X above the level u.This is closely related to the exceedance probability

supt∈T

X(t) > u,

which is very important in many applications.

If X(t) is smooth, we use integral geometry to charac-terize the topological structures of sets generated byX. See the books by Adler and Taylor (2007), Azaısand Wschebor (2009).

If X(t) is not smooth, one uses fractal geometry tostudy the random sets generated by X.

2.1 Hausdorff dimension and Capacity

For any metric ρ on RN , any β > 0 and E ⊆ RN , theβ-dimensional Hausdorff measure in the metric ρ of E isdefined by

Hβρ (E) = lim

δ→0inf ∞∑

(2rn)β : E ⊆

∞⋃n=1

Bρ(rn), rn ≤ δ

where Bρ(r) denotes an open ball of radius r in the metricspace (RN, ρ).The corresponding Hausdorff dimension of E is defined by

dimρHE = inf

β > 0 : Hβ

ρ (E) = 0.

The Bessel-Riesz type capacity of order α on the metricspace (RN, ρ) is defined by

Cαρ (E) =

µ∈P(E)

∫ ∫fα(ρ(u, v)

)µ(du)µ(dv)

where P(E) is the family of probability measures carriedby E and the function fα : (0,∞)→ (0,∞) is defined by

fα(r) =

r−α if α > 0,log(

er∧1

)if α = 0,

1 if α < 0.(1)

ρ will be omitted if it is the Euclidean metric.

2.1 Hausdorff dimensions of the range andgraph

Let X = X(t), t ∈ RN be a Gaussian field in Rd definedby

X(t) =(X1(t), . . . ,Xd(t)

), t ∈ RN, (2)

where X1, . . . ,Xd are independent copies of a centered GFX0.Recall that the canonical metric of X0 is

dX0(s, t) =

√E(X0(s)− X0(t)

Many fractal properties of X are determined by the behav-ior of dX0(s, t) as s→ t.

We will make use of the following condition:(C1). ∃ positive constants c1 and c2 such that for all s, t ∈

I(= [ε, 1]N),

N∑j=1

|sj − tj|2Hj ≤ dX0(s, t)2 ≤ c2

N∑j=1

|sj − tj|2Hj,

where 0 < H1 ≤ . . . ≤ HN ≤ 1 are constants.Hence the canonical metric dX0(s, t)

∑Nj=1 |sj − tj|Hj,

which will be denoted by ρ(s, t) below.

2.1 Hausdorff dimensions the range and graph

Theorem 2.1 (Ayache and X. 2005; X. 2009)Assume Condition (C1) holds. Then almost surely

dimH X([0, 1]N

)= min

N∑j=1

dimHGrX([0, 1]N

)= min

1≤k≤N

Hj+ N − k + (1− Hk)d;

N∑j=1

The proof of these results rely on regularity properties of Xand the connection of Hausdorff dimension and capacity.

The above results are significantly different from theisotropic case, as well as the space-anisotropic case.If d = 1, then

dimHGrX([0, 1]N

)= N + 1− H1 a.s.

Hence one needs d > 1 to recover all the parametersH1, . . . ,HN .To determine dimHX(E) for an arbitrary Borel set E ⊂RN , one needs to use a different Hausdorff-type di-mension; see Wu and X. (2007), X. (2009).

2.2 Hitting probabilities

Let X = X(t), t ∈ RN be a random field with values inRd. We consider the following intersection problems:

(1) For Borel sets E ⊆ RN and F ⊆ Rd, when can onehas

P(X(E) ∩ F 6= ∅) > 0? (3)

(2) [k-multiple points] Given disjoint sets E1, . . . ,Ek ⊆RN , when does

P(X(E1) ∩ · · · ∩ X(Ek) ∩ F 6= ∅) > 0? (4)

Some history about (1) and (2)In the case when E = [a, b], (a, b ∈ RN), a necessary andsufficient condition for (1) in terms of certain kind of ca-pacity of F has been established for X being

Brownian motion (Kakutani, 1944)Levy processes (Port and Stone, 1971)Some multiparameter Markov processes (Fitzsimmonsand Salisbury, 1989)The Brownian sheet (Khoshnevisan and Shi, 1999)Additive Levy processes (Khoshnevisan and Xiao, 2002,2003, 2009)SPDEs (Dalang and Nualart, 2004, Dalang, Khosh-nevisan and Nualart, 2007, ... )

Question (1) include intersections of the graph set and levelsets:

Let GrX(E) = (t,X(t)) : t ∈ E be the graph of Xon E. Then (1) is equivalent to

P(GrX(E) ∩ (E × F) 6= ∅) > 0.

Take F = 0, then (1) is equivalent to

P(X−1(0) ∩ E 6= ∅) > 0.

For general E ⊆ R and F ⊆ Rd, a necessary and suf-ficient condition for (1) in terms of “thermal capac-ity” of E × F was established for Brownian motionW = W(t), t ≥ 0 by Watson (1978). This is theonly known complete characterization in this general-ity.

The Hausdorff dimension W(E)∩ F is recently deter-mined by Khoshnevisan and Xiao (2015).

In the special case of F = 0, the hitting probabil-ity is characterized by Khoshnevisan and Xiao (2002)for a large class of additive Levy processes, and byKhoshnevisan and Xiao (2007) for the Brownian sheet.

Question (2) is related to existence of self-intersections.

When F = Rd, then (2) gives existence of k-multiplepoints.

Levy processes (Khoshnevisan and Xiao, 2005): F = Rd,general E1, . . . ,Ek

The Brownian sheet: Dalang et al (2012), Dalang and Mueller(2014): F = Rd, E1, . . . ,Ek are intervals.

No results for general F, E1, . . . ,Ek.

In the rest of this talk, we present some results from thefollowing three papers:

Bierme, H., Lacaux, C. and Xiao, Y. (2009). Hittingprobabilities and the Hausdorff dimension of the in-verse images of anisotropic Gaussian random fields.Bull. London Math. Soc. 41, 253–273.

Khoshnevisan, D. and Xiao Y. (2015). Brownian mo-tion and thermal capacity. Ann. Probab. 43, 405–434.

Dalang, R., Mueller, C. and Xiao Y. (2015). Polarityof points for Gaussian random fields.http://arxiv.org/pdf/1505.05417v1.pdf.

Let X = X(t), t ∈ RN be a Gaussian field in Rd definedby (2) that satisfies the following conditions:

(C1). ∃ positive constants c1 and c2 such that for all s, t ∈I(= [ε, 1]N),

c1 ρ(s, t) ≤ dX0(s, t) ≤ c2 ρ(s, t),

where 0 < H1 ≤ . . . ≤ HN ≤ 1 are constants,(C2). ∃ c3 > 0 such that for all s, t ∈ I,

Var(X0(t)

∣∣X0(s))≥ c3

N∑j=1

|sj − tj|2Hj.

Hitting probabilities and Riesz capacity

The following result was motivated by Dalang, Khosh-nevisan and Nualart (2007) and X. (1999) for fractionalBrownian motion.Theorem 2.2 [Bierme, Lacaux and X. (2009)]If X is defined by (2) such that X0 satisfies Conditions (C1)and (C2) on I. Then ∀ Borel set F ⊂ Rd,

c4 Cd−Q(F) ≤ P

X(I) ∩ F 6= ∅≤ c5Hd−Q(F), (5)

where Q =∑N

j=11Hj

, Cd−Q is (d − Q)-dimensional Rieszcapacity andHd−Q is (d−Q)-dimensional Hausdorff mea-sure.

Proof of Theorem 2.2: the upper bound

For proving the upper bound in (5), we make use of a cov-ering argument and the following lemma.

Lemma 2.1 [Bierme, Lacaux and X. (2009)]Assume the conditions of Theorem 2.2 hold. For any con-stant M > 0, there exist positive constants c and δ0 suchthat for all r ∈ (0, δ0), t ∈ I and all x ∈ [−M,M]d,

infs∈Bρ(t,r)∩I

∥∥X(s)− x∥∥ ≤ r

≤ c rd. (6)

In the above Bρ(t, r) = s ∈ RN : ρ(s, t) ≤ r denotes theclosed ball of radius r in the metric ρ in RN .

Proof of the upper bound

Only the case of d > Q needs a proof. Choose and fixan arbitrary constant γ > Hd−Q(F). By the definition ofHd−Q(F), there is a sequence of balls B(yj, rj), j ≥ 1 inRd such that

F ⊆∞⋃

B(yj, rj) and∞∑

(2rj)d−Q ≤ γ. (7)

Notice that

F ∩ X(I) 6= ∅

⊆∞⋃

B(yj, rj) ∩ X(I) 6= ∅

Only the case of d > Q needs a proof. Choose and fixan arbitrary constant γ > Hd−Q(F). By the definition ofHd−Q(F), there is a sequence of balls B(yj, rj), j ≥ 1 inRd such that

F ⊆∞⋃

B(yj, rj) and∞∑

(2rj)d−Q ≤ γ. (7)

Notice that

F ∩ X(I) 6= ∅

⊆∞⋃

B(yj, rj) ∩ X(I) 6= ∅

For every j ≥ 1, we divide the interval I into c r−Qj intervals

of side-lengths r−1/H`j (` = 1, . . . ,N). Hence I can be cov-

ered by at most c r−Qj many balls of radius rj in the metric

ρ.It follows from Lemma 2.1 that

B(yj, rj) ∩ X(I) 6= ∅≤ c rd−Q

j . (9)

Combining (8) and (9) we derive that P

F∩X(I) 6= ∅≤

cγ. Since γ > Hd−Q(F) is arbitrary, the upper bound in (5)follows.

B(yj, rj) ∩ X(I) 6= ∅≤ c rd−Q

j . (9)

F∩X(I) 6= ∅≤

B(yj, rj) ∩ X(I) 6= ∅≤ c rd−Q

j . (9)

F∩X(I) 6= ∅≤

Proof of Theorem 2.2: the lower boundWe make use of the following lemma.

Lemma 2.2 [Bierme, Lacaux and X. (2009)]There exists a positive and finite constant c such that for allε ∈ (0, 1), s, t ∈ I and x, y ∈ Rd, we have∫

R2dexp

(− 1

2(ξ, η)

(εI2d + Cov

(X(s),X(t)

))(ξ, η)T

)e−i(〈ξ, x〉+〈η, y〉)dξdη ≤ c(

maxρ(s, t), ‖x− y‖)d .

Here I2d denotes the identity matrix of order 2d,Cov

(X(s),X(t)

)denotes the covariance matrix of the ran-

dom vector (X(s),X(t)).Yimin Xiao (Michigan State University) Lecture 2: Hausdorff dimension results and hitting probabilities

Northwestern University, July 11–15, 2016 23/ 51

Proof of the lower bound

Without loss of generality, we assume C0(F) > 0 and Fis compact. Let M > 0 be a constant such that F ⊆[−M,M]d.We only consider the critical case of d = Q. By definitionof capacity, there is a Borel probability measure ν0 on Fsuch that

E0(ν0) :=

e‖x− y‖ ∧ 1

)ν0(dx)ν0(dy)

≤ 2C0(F)

For all integers n ≥ 1, we consider a family of randommeasures νn on I defined by

∫Ig(t) νn(dt)

(2πn)d/2 exp(−

n∥∥X(t)− x

∥∥2

)g(t) ν0(dx) dt

exp(− ‖ξ‖

2n+ i〈ξ,X(t)− x〉

)g(t) dξν0(dx)dt,

where g is an arbitrary measurable, nonnegative functionon I.

Denote the total mass of νn by ‖νn‖ := νn(I). We verifythe following two inequalities hold:

E(‖νn‖

)≥ c4 and E

(‖νn‖2) ≤ c5E0(ν0), (11)

where the constants c4 and c5 are independent of ν0 and n.

By (11) and the Paley-Zygmund inequality, one can showthat there is an event Ω0 with probability at least

c24/(2c5 E0(ν0))

such that for every ω ∈ Ω0, νn(ω), n ≥ 1 has a subse-quence that converges weakly to a finite positive measureν which is supported on X−1(F) ∩ I.Then, we have

X(I) ∩ F 6= ∅≥ P

‖ν‖ > 0

2c5 E0(ν0).

This proves the lower bound.

Conjecture: Hd−Q(F) in (5) can be replaced by Cd−Q(F).

Recently, Dalang , Mueller and Xiao (2015) verified thisfor the case of F = x. They proved that, if d = Q, thenfor every x ∈ Rd,

X(I) ∩ x 6= ∅

= P∃t ∈ I : X(t) = x

Their method is based on an argument of Talagrand (1998).

For any Borel set F ⊆ Rd, consider the inverse image

X−1(F) =

t ∈ RN : X(t) ∈ F.

Theorem 2.3 [Bierme, Lacaux and X. (2009)]Let X be as in Theorem 2.2 and let F ⊆ Rd be a Borelset such that

∑Nj=1

1Hj> d − dimHF. Then with positive

probability,

(X−1(F) ∩ I

)= min

1≤k≤N

k∑j=1

Hj+ N − k − Hk(d − dimHF)

Theorem 2.3 [Bierme, Lacaux and X. (2009)]In particular, if t

∑Nj=1

1Hj> d. Then for every x ∈ Rd,

(X−1(x) ∩ I

)= min

1≤k≤N

k∑j=1

Hj+ N − k − Hkd

with positive probability.

2.3 Brownian images and thermal capacity

Let W := W(t)t≥0 denote standard d-dimensional Brow-nian motion where d ≥ 1, and let E and F be compactsubsets of (0 ,∞) and Rd, respectively.

We consider the following questions:1 When is P(W(E) ∩ F 6= Ø) > 0?2 What is dimH(W(E) ∩ F)?

Note that

W(E) ∩ F 6= Ø = (t,W(t)) ∈ E× F for some t > 0.

Problem 1 is an interesting problem in probabilistic poten-tial theory.

Conditions for P(W(E) ∩ F 6= Ø) > 0

Necessary and sufficient condition in terms of “thermal ca-pacity” for P(W(E)∩F 6= Ø) > 0 were proved by Waston(1978) and Doob (1984).Waston and Taylor (1985) provided a simple-to-use condi-tion:

P(W(E) ∩ F 6= Ø)

> 0, if dimH(E × F ; %) > d,= 0, if dimH(E × F ; %) < d.

In the above, dimH(E × F ; %) is the Hausdorff dimensionof E × F using the metric

% ((s , x) ; (t , y)) := max(|t − s|1/2, ‖x− y‖

As a by-product of our main result, we obtain an improvedversion of the result of Waston (1978) and Doob (1984).Theorem 2.4Suppose F ⊂ Rd is compact and has Lebesgue measure 0.Then

PW(E) ∩ F 6= ∅ > 0 ⇐⇒∃ µ ∈ Pd(E × F) such that E0(µ) <∞,

where Pd(E × F) is the collection of all probability mea-sures µ on E × F such that µ(t × F) = 0 for all t > 0and the energy E0(µ) is defined by

E0(µ) :=

∫∫e−‖x−y‖2/(2|t−s|)

|t − s|d/2 µ(ds dx)µ(dt dy).

Hausdorff dimension of dimH(W(E) ∩ F)

If F = Rd, then dimHW(E) = mind, 2dimHE a.s.If E = R+, then

dimH(W(R+) ∩ F) =

dimHF if d = 1;2 + dimHF − d if d ≥ 2.

For compact sets E ⊂ (0,∞) and F ⊂ R (d = 1),Kaufman (1972) obtained ‖dimH(W−1(F)∩E)‖L∞(P),where ‖ · ‖L∞(P) denotes the L∞(P)-norm. However,this does not provide information on dimH(W(E)∩F).Hawkes (1978) considered the problem for an α-stableLevy process in R with 0 < α < 1.We solve this problem completely for Brownian mo-tion (and Levy stable processes).

Hausdorff dimension of dimH(W(E) ∩ F)

To compute ‖dimH (W(E) ∩ F)‖L∞(P), we distinguish twocases: |F| > 0 and |F| = 0, where | · | denotes theLebesgue measure.

Theorem 2.5 [Khoshnevisan and X. (2015)]If F ⊂ Rd (d ≥ 1) is compact and |F| > 0, then

‖dimH (W(E) ∩ F)‖L∞(P) = mind , 2dimHE. (12)

If dimHE > 12 and d = 1, then P|W(E) ∩ F| > 0 > 0.

Proof of Theorem 2.5

1 Thanks to the uniform Holder continuity of W(t) onbounded sets, we have

dimH (W(E) ∩ F) ≤ mind , 2dimHE, a.s.

This implies the upper bound in (12).2 For proving the lower bound in (12), we construct a

random measure on W(E) ∩ F and use the capacityargument.

3 The last part is proved by showing that the constructedrandom measure on W(E) ∩ F has a density functionalmost surely.

Theorem 2.6 [Khoshnevisan and X. (2015)]If F ⊂ Rd (d ≥ 1) is compact and |F| = 0, then∥∥dimH (W(E) ∩ F)

∥∥L∞(P)

= supγ ≥ 0 : inf

µ∈Pd(E×F)Eγ(µ) <∞

where Pd(E × F) is the collection of all probability mea-sures µ on E × F such that µ(t × F) = 0 for all t > 0,and

Eγ(µ) :=

∫∫e−‖x−y‖2/(2|t−s|)

|t − s|d/2 · ‖y− x‖γµ(ds dx)µ(dt dy).

Theorem 2.6 [Khoshnevisan and X. (2015)]If F ⊂ Rd (d ≥ 1) is compact and |F| = 0, then∥∥dimH (W(E) ∩ F)

∥∥L∞(P)

= supγ ≥ 0 : inf

where Pd(E × F) is the collection of all probability mea-sures µ on E × F such that µ(t × F) = 0 for all t > 0,and

Eγ(µ) :=

∫∫e−‖x−y‖2/(2|t−s|)

|t − s|d/2 · ‖y− x‖γµ(ds dx)µ(dt dy).

The co-dimension argument

Recall that the two common ways to compute the Haus-dorff dimension of a set are

Use a covering argument for obtaining an upper boundand a capacity argument for lower bound;

The co-dimension argument.

The “co-dimension argument” was initiated by S.J. Taylor(1966) for computing the Hausdorff dimension of the mul-tiple points of a stable Levy process in Rd. His method wasbased on potential theory of Levy processes.

Let Zα = Zα(t), t ∈ R+ be a (symmetric) stable Levyprocess in Rd of index α ∈ (0, 2] and let F ⊂ Rd be aBorel set. Then

P(Zα((0,∞)) ∩ F 6= ∅) > 0⇐⇒ Capd−α(F) > 0,

where Capd−α is the Riesz-Bessel capacity of order d− α.

The above result and Frostman’s theorem lead to the stochas-tic co-dimension argument: If dimHF ≥ d − 2, then

dimHF = supd − α : Zα((0,∞)) ∩ F 6= ∅= d − inf

α > 0 : F is not polar for Zα

[The restriction dimHF ≥ d − 2 is caused by the fact thatZα((0,∞)) ∩ F = Ø if dimHF < d − 2.]

This method determines dimHF by intersecting F using afamily of testing random sets.

Hawkes (1971) applied the co-dimension method for com-puting dimHX−1(F) of a stable Levy process.

Families of testing random sets:ranges of symmetric stable Levy processes;

fractal percolation sets [Peres (1996, 1999)];

ranges of additive Levy processes [Khoshnevisan andX. (2003, 2005, 2009), Khoshnevisan, Shieh and X.(2008)].

Hitting probability of random fields

We prove Theorem 2.6 by checking whether or not W(E)∩F intersects the range of an additive Levy stable process.

Let X(1), . . . ,X(N) be N isotropic stable processes with com-mon stability index α ∈ (0 , 2]. We assume that the X(j)’sare independent from one another, as well as from the pro-cess W, and all take their values in Rd.

We assume also that X(1), . . . ,X(N) have right-continuoussample paths with left-limits and

E[ei〈ξ,X(k)(1)〉

]= e−‖ξ‖

α/2, ∀ ξ ∈ Rd.

]= e−‖ξ‖

α/2, ∀ ξ ∈ Rd.

]= e−‖ξ‖

α/2, ∀ ξ ∈ Rd.

Define the corresponding additive stable process Xα :=Xα(t × t), t × t ∈ RN

Xα(t × t) :=N∑

X(k)(tk), ∀ t × t = (t1, . . . , tN) ∈ RN+.

(15)Khoshnevisan, X. and Zhong (2003) showed that for anyBorel set G ⊂ Rd,

P(Xα(RN

+) ∩ G 6= Ø)

= 0 if dimHG < d − αN,> 0 if dimHG > d − αN.

The key ingredient for proving Theorem 2.6

Theorem 2.7If d > αN and F ⊂ Rd has Lebesgue measure 0, then

W(E) ∩ Xα(RN+) ∩ F 6= Ø

⇐⇒ Cd−αN(E × F) > 0.

Here Cγ is the capacity corresponding to the energy form(14): for all compact sets U ⊂ R+ × Rd and γ ≥ 0,

Cγ(U) :=

µ∈Pd(U)Eγ(µ)

. (17)

Proof of Theorem 2.6Lower bound: Denote

∆ := supγ ≥ 0 : inf

. (18)

If ∆ > 0 and we choose α ∈ (0 , 2] and N ∈ Z+ 0 <d − αN < ∆. Then Cd−αN(E × F) > 0. It follows fromTheorem 2.3 and (16) that

P dimH (W(E) ∩ F) ≥ d − αN > 0. (19)

Because d − αN ∈ (0 ,∆) is arbitrary, we have

‖dimH(W(E) ∩ F)‖L∞(P) ≥ ∆.

Upper bound: Similarly, Theorem 2.7 and (16) imply that

d−αN > ∆ ⇒ dimH (W(E) ∩ F) ≤ d−αN a. s. (20)

Hence ‖dimH(W(E) ∩ F)‖L∞(P) ≤ ∆ whenever ∆ ≥ 0.This proves Theorem 2.6.

Proof of Theorem 2.7To prove the sufficiency

Cd−αN(E × F) > 0 =⇒P

W(E) ∩ Xα(RN+) ∩ F 6= Ø

we define, for every µ ∈ Pd(E × F) and ε > 0, the occu-pation measure Zε(µ) by

Zε(µ) =

∫[1,2]N

E×Fµ(ds dx) φε(W(s)−x)φε(Xα(u)−x),

whereφε(y) =

1εdIB(0,ε)(y).

The proof is based on computing E[Zε(µ)] and E[Zε(µ)2].Yimin Xiao (Michigan State University) Lecture 2: Hausdorff dimension results and hitting probabilities

For proving the necessity, we assume

W(E) ∩ Xα(RN+) ∩ F 6= Ø

and construct a probability measure µ ∈ Pd(E × F) suchthat Ed−αN(µ) <∞.

If W(E) ∩ Xα(RN+) is replaced by the range of a Levy pro-

cess, then we can use a stoping time argument and thestrong Markov property.

The current random field case is much harder. We omit thedetails.

An explicit formula

Theorem 2.8 [Khoshnevisan and X. (2016)]If d ≥ 2 and dimH (E × F ; %) ≥ d, then

‖dimH (W(E) ∩ F)‖L∞(P) = dimH (E × F ; %)− d. (21)

RemarksEq (21) does not always hold for d = 1: For E :=[0 , 1] and F = 0, we have dimH(W(E) ∩ F) = 0a.s., whereas dimH(E × F ; %)− d = 1.When F ⊂ Rd satisfies |F| > 0, it can be shown that

dimH (E × F ; %) = 2dimHE + d.

Hence (12) coincides with (21) when d ≥ 2.Yimin Xiao (Michigan State University) Lecture 2: Hausdorff dimension results and hitting probabilities

The proof replies on the following “uniform dimension re-sult” of Kaufman (1968): If W(t), t ∈ R+ is a Brownianmotion in Rd with d ≥ 2, then

dimHW(G) = 2dimHG, ∀ Borel sets G ⊂ R+ = 1.

It is sufficient to show that for all compact sets E ⊂ (0,∞)and F ⊂ Rd,∥∥dimH

(E ∩W−1(F)

)∥∥L∞(P) =

dimH (E × F ; %)− d2

(22)When d = 1, the lower bound of (22) was found first byKaufman (1972).

Thank you

Lecture 2: Hausdorff dimension results and hitting ...auffing/XiaoL2.pdfity is characterized by...

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