Additive L´ evy Processes

Assume Ft and Fs are conditionally independent of F_tfs for all s, t ∈ R^N+, where t f s = (min(ti, si))_1≤i≤N. Then

E

 sup

04s4t

X(t)



≤

 p p − 1

N p

E[X(t)^p] ∀ t ∈ R^N+ and p > 1. (3.29) Proof. We refer to Chapter 7 of [15], Theorem 2.3.2 for a proof.

3.3 Additive L´ evy Processes

An N -parameter random field X := {X(t) : t ∈ R^N+} on R^d is an additive L´evy process if each sample path of X satisfies:

X(t) = X1(t1) + · · · + X_N(t_N) ∀ t = (t₁, . . . , t_N) ∈ R^N+, (3.30) where X1, . . . , XN are independent one-parameter L´evy processes on R^d. Let Ψ1, . . . , ΨN

denote the characteristic exponent of X₁, . . . , X_N, respectively. Then the characteristic function of X is E[e^iξ·X(t)] = exp(−PN

j=1tjΨj(ξ)) for all ξ ∈ R^d. The associated process X = { ee X(t) : t ∈ R^N} of X is defined by

X(t) :=e

N

X

j=1

sgn(t_j)X_j(|t_j|) ∀ t ∈ R^N. (3.31)

Note that eX is indexed by all of R^N. Moreover,

∀ s, t ∈ R^N+ : eX(t − s) has the same distribution as X(t) − X(s), (3.32)

33 and the Fourier transform of eX(t) is given by

Ee^{iξ· eX(t)} = exp



−

N

X

j=1

|t_j|Ψ_j(sgn(t_j)ξ)



 ∀ξ ∈ R^d. (3.33)

We will investigate the sample paths properties of additive L´evy processes. The potential theory developed in Khoshnevisan, Xiao, and, Zhong [23] is crucial in our investigation. For the sake of completeness, we reproduce them here. In the remaining part of this section, we will fix an additive L´evy process X.

Let (Ω, F , P) be the underlying probability space and let λd be the Lebesgue measure on R^d. We will first extend the probability measure P to a σ-finite measure Pλd such that the additive L´evy process X is “uniformly” distributed under Pλd. For each x ∈ R^d, define Px to be the law of x + X, in particular,

Px{X(t) ∈ A} = P{x + X(t) ∈ A} ∀ t ∈ R^N+, A ∈ B(R^d). (3.34) Furthermore let Ex be the corresponding expectation operator. Finally, we define the sigma-finite measure Pλd, and a corresponding expectation operator, Eλd, via

Pλd{Ω₀} = Z

R^d

Px{Ω₀} dx ∀ Ω₀∈ F (3.35)

Eλd[Z] = Z

R^d

Ex[Z] dx ∀ Z : Ω → R+ bounded measurable. (3.36) Remark 3.24. For each A ∈ B(R^d), by applying Fubini’s theorem and a change of variable under the Lebesgue measure λd, we obtain

Pλd{X(t) ∈ A} = E

Z

R^d

1{x+X(t)∈A}(x) dx



= λ_d(A), (3.37) for all t ∈ R^N+. This shows that the distribution of X under Pλd is λd. Furthermore, since λ_dis σ-finite, Pλd is also σ-finite.

Remark 3.25. Let G be a sub-σ-algebra of F . For each random variable Y with Eλd[|Y |] <

∞, let Eλd[Y |G] denote the conditional expectation of Y given G under the measure Pλd, that is, Eλ_d[Y |G] is G-measurable, Eλ_d[Y |G] ∈ L¹(Pλ_d), and Eλ_d[Y 1Ω0] = Eλ_d[Eλ_d[Y |G]1Ω0] for all Ω₀ ∈ G. By standard integration theory (the Radon-Nikod´ym theorem), Eλd[Y |G]

exists and is unique up to a Pλd-zero event.

Remark 3.26. The conditional expectations under Pλd have all the nice properties of con-ditional expectations under probability measures. For example, we show here a concon-ditional

Fubini’s theorem. Let G be a sub-σ-algebra and Y (t) be a random function from R^N to R^d. Then for a fixed nonrandom Borel probability measure µ on R^N and all Ω₀ ∈ G,

Eλd filtration is the inclusion property with respect to the ordering in the index set. Thus F^A := {F^A(t)}_04(A)t4(A)∞ is a valid filtration for each fixed A. In fact, this generalize the multiparameter filtration defined in the previous section as the multiparameter filtration defined there is simply F^Π. Moreover the filtration F^A has the conditional independence property.

Lemma 3.27. Let F^A be the filtration with fixed A ⊂ Π. Then for all s, t ∈ R^N+, F^A(s) and F^A(t) are conditionally independent given F^A(sf(A)t), where

(sf(A)t)i =

(min(si, ti), if i ∈ A,

max(si, ti), if i ∈ A^C. (3.41) Proof. We refer to Chapter 7, Theorem 2.4.1 of [15] for a proof. Although Theorem 2.4.1 of [15] is a result for a Brownian sheet, the same method works for an additive L´evy process.

In fact, the structure of an additive L´evy process is simpler than a Brownian sheet and so is the proof.

35 Now we can combine Pλ_d and F^A to generalize the multiparameter martingales defined in the previous section. Since the filtration F^A and the conditional expectation under Eλd are well-defined, the generalization is straightforward. In particular, a multiparameter process M := {M (t)}_04(A)t4(A)∞is a submartingale with respect to the filtration F^Aunder the σ-finite measure Pλd if

1. M is adapted to F^A;

2. M (t) ∈ L¹(Pλd), that is, Eλd[|M (t)|] < ∞, for all 0 4(A)t 4(A)∞; and 3. Eλd[M (t)|F^A(s)] ≥ M (s), Pλd-almost surely for all 0 4(A) s 4(A) t.

In the classical potential theory for one-parameter L´evy processes, the semigroup {P_t}_t≥0 plays an important role ([14], Lecture 9). In the case of additive L´evy processes, we define a similar family of operators P = {P_t}_t∈RN on L^∞(R^d) by

(P_tf )(x) := E[f ( eX(t) + x)] ∀ f ∈ L^∞(R^d) or f : R^d→ R+, for all x ∈ R^d. (3.42) Our first observation is the Markov property of X in terms of P_t.

Lemma 3.28 ([23], Proposition 3.2). Suppose that A ⊆ Π and that s 4(A) t, where s and t are both in R^N+. Then, for every measurable function f : R^d→ R+,

Eλd[f (X(t))|F^A(s)] = (Pt−sf )(X(s)), (3.43)

Pλd-a.s., where Ptis defined in (3.42).

Proof. For an arbitrary positive integer m, consider measurable functions f, g, h1, . . . , hm: R^d→ R+ and t, s, s¹, . . . , s^m ∈ R^N+ such that t <(A) s <(A)s^k for all 1 ≤ k ≤ m. Then,

The first equality follows from Fubini’s theorem; we apply a change of variable y = x + X(s) to obtain the second one; and the third equality follows from Fubini’s theorem and independent increments of X. Furthermore, according to (3.42), we have

Eλd

where we apply Fubini’s theorem to obtain the second equality, change of variable x = y − X(s) to obtain the third one; and (3.36) for the last one.

We are interested in the “local time” that an additive L´evy process spent in a given set.

In order to quantify these “local times,” we first define the occupation measure Oµ for any given Borel probability measure µ on R^N+:

Oµ(A) = Z

R^N+

1A(X(s)) µ(ds) ∀ A ∈ B(R^d), (3.46)

where 1A(·) is the indicator function of A and B(R^d) denotes the Borel σ-algebra of R^d. By standard integration theory, we can extend the occupation measure to a random linear functional on L¹(µ):

Oµ(f ) = Z

R^N+

f (X(s)) µ(ds) ∀ f ∈ L¹(µ). (3.47)

In particular, the Fourier transform of the occupation measure cOµexists, and it is given by

Ocµ(ξ) = Z

R^N+

e^iξ·X(s)µ(ds) ∀ ξ ∈ R^d. (3.48)

The following lemma computes the second moment of the Fourier transform of cOµ.

37 Lemma 3.29 ([23], Lemma 2.4). For all Borel probability measure µ on R^N+,

E and sgn(·) is the usual sign function.

Proof. Since

and |eiξ·(X(s)−X(t))| = 1, we can apply Fubini’s theorem twice to obtain

E[k cOµk²_L2(R^d)] =

Finally (3.49) follows from this and (3.33).

The following two technical lemmas are useful in estimating moments of random vari-ables under Pλ_d.

where we used Fubini’s theorem to derive the third equality and a change of variable y = X(s) + x for the last one. We can apply Fubini’s theorem again to compute the Fourier transform of Pt−sg:

P\t−sg(ξ) = E desired result by applying Plancherel’s theorem to the last term of (3.54) and combining it with (3.55).

Proof. First assume f ≥ 0. Then Fubini’s theorem shows that Eλ_d[|Oµ(f )|²]

where we used Lemma 3.30 to derive the second equality. Finally for a general f ∈ L¹(R^d) ∩ L²(R^d), we note that Eλd[|Oµ(f )|²] < ∞. Therefore we can use Fubini’s theorem to derive the same result.

39 is separable, thanks to Theorem 3.17. The following lemma gives an upper bound for the second moment of the martingale M^A_µf under the measure Pλd. and Corollary 3.23 show that,

Eλ_d

Since the above inequality is true for all n ≥ 1, we have

Eλd

Finally, (3.60) follows from Lemma 3.31.

PACKING DIMENSION AND ADDITIVE L´ EVY PROCESSES

For a nonrandom Borel set F ⊆ R^N+, the random image set X(F ) usually exhibits fractal structure, and it is natural to calculate the Hausdorff dimension and packing dimension of X(F ). When N = 1, there is a rich literature regarding the Hausdorff dimension dim_H(X(F )) of X(F ), see [20] and its extensive reference. However, there have been few results regarding the packing dimension dim_P(X(F )) of X(F ). In a recent paper, Khoshnevisan, Schilling, and Xiao [19] defined a new family of packing dimension profile Dim_κ and found that

dim_P(X(F )) = Dim_κ(F ) a.s. (4.1)

When N > 1, a formula regarding the Hausdorff dimension dim_H(X(F )) is given by Yang [40]. However, there is little known about the packing dimension dim_P(X(F )).

The packing dimension profile defined in [19] is probabilistic but has analytic significance as well. In fact it can be regarded as an extension of the packing dimension profile defined by Falconer and Howroyd [6] and Howroyd [10]. However this generalization only works for subsets of R, since the packing dimension profile in [19] is defined on the “time domain,”

that is, the half real line R+ = [0, ∞). It is natural to ask whether we can generalize it to R^N for all N ≥ 1 and still keep the connection to the packing dimension profile of Falconer and Howroyd [6] and Howroyd [10].

In this chapter, we extend the packing dimension profile defined in [19] to higher dimensions and derive a multiparameter version of (4.1). We also show that our definition gives probabilistic interpretations of the packing dimension profile of Falconer and Howroyd [6] and Howroyd [10]. The extension from N = 1 to N ≥ 1 is not straightforward since the structure of R^N is richer than R if N > 1. In particular, the result of [19] relies on a stopping time argument while we can not define stopping times for multiparameter processes in general. In order to overcome this difficulty, we employ the potential theory developed in Chapter 3 for additive L´evy processes.

41

4.1 Box-dimension Profiles

Let X = {X(t)}_t∈RN

+ be an additive L´evy process on R^d. Its associated process eX = { eX(t) : t ∈ R^N} is defined by

X(t) :=e

N

X

j=1

sgn(t_j)X_j(|t_j|) ∀ t ∈ R^N. (4.2)

For every x ∈ R^d, let |x| := max_1≤k≤d|x_k| denote the l^∞ norm of x. Then B(x, ε) := {y ∈ R^d: |x − y| < ε} is the l^∞open ball centered at x with radius ε > 0. For every ε > 0, define κ_ε(t) := P{| eX(t)| ∈ B(0, ε)}. (4.3) It follows that κε(t) is nondecreasing in ε for every fixed t. Moreover, Lemma 3.12 ensures that {X_j(t_j)}_tj≥0 is continuous in probability for all 1 ≤ j ≤ N . Thus, κ_ε(t) is continuous in t for every fixed ε. We consider the family of functions κ := {κ_ε}_ε>0.

Definition 4.1. For every bounded Borel set F ⊆ R^N, the box-dimension profile dim_κ(F ) of F with respect to the family κ is defined as

dim_κ(F ) := sup (

η > 0 : lim

ε↓0

ε^−η inf

µ∈P(F )

Z Z

κ_ε(t − s)µ(ds)µ(dt) = 0 )

, (4.4)

whereP(F ) denotes the collection of all Borel probability measures supported on F . Remark 4.2. When N = 1, we have

κε(t) = P{|X(|t|)| ∈ B(0, ε)} = P{|X(t)| ∈ B(0, ε)}, (4.5) for all ε > 0 and t ≥ 0. This shows that our definition of box dimension profile generalizes that of Khoshnevisan, Schilling, and Xiao [19]. For this reason we will use the same notations for dimension profiles as those in [19].

A routine check shows that dim_κ has the following properties:

(i) dim_κ is monotone; i.e., dim_κ(F ) ≤ dim_κ(G) if F ⊆ G;

(ii) dim_κ is finitely stable; i.e., dim_κ(∪ⁿ_l=1G_l) = max_1≤l≤ndim_κ(G_l);

(iii) dim_κ is translation invariant; i.e., dim_κ(F ) = dim_κ(t + F ) for all t ∈ R^N, where t + F := {t + s : s ∈ F }.

We may also express dim_κ(F ) in potential-theoretic terms: The following technical lemma simplifies the calculation of Z_κ(ε; F ).

Lemma 4.3. For every bounded Borel set F ⊆ R^N, let F denote the closure of F . Then wherePf(F ) denotes the collection of all finitely-supported [discrete] probability measures on F .

Proof. It suffices to prove

ν∈infPf(F ) as the converse inequality is trivial. Choose and fix any µ ∈P(F ). For every n ≥ 1, define

In(¯k) := k¹

43 It follows immediately that νn ∈Pf(F ). By the uniform continuity of κε(t) and (4.13), we have

n→∞lim    

Z Z

κε(t − s)νn(ds)νn(dt) − Z Z

κε(t − s)µ(ds)µ(dt)    

= 0. (4.15)

This further implies that inf

ν∈Pf(F )

Z Z

κ_ε(t − s)ν(ds)ν(dt) ≤ Z Z

κ_ε(t − s)µ(ds)µ(dt). (4.16)

Since µ ∈P(F ) is arbitrary, we can take inf over P(F ) on the right hand side of the above inequality to complete the proof.

Lemma 4.4. For every bounded Borel set F ⊆ R^N, let F denote the closure of F . Then

dim_κ(F ) = dim_κ(F ). (4.17)

Proof. SincePf(F ) ⊆P(F ) ⊆ P(F ), Lemma 4.3 implies that inf

µ∈P(F )

Z Z

κ_ε(t − s)µ(ds)µ(dt) = inf

µ∈P(F )

Z Z

κ_ε(t − s)µ(ds)µ(dt). (4.18)

Then (4.17) follows from Definition 4.1.

Remark 4.5. The above lemma implies that dim_κ is not σ-stable in general. For example, consider F = Q^N+∩[0, 1]^N. On one hand, for every t ∈ R^N the only Borel probability measure supported on {t} is the Dirac measure δ_t. Then we can compute dim_κ({t}) by definition and obtain dim_κ({t}) = 0. On the other hand dim_κ(F ) = dim_κ(F ) = dim_κ([0, 1]^N) according to Lemma 4.4. We will show in Corollary 4.17 that if κ is determined by a d-dimensional isotropic α-stable process X with 0 < α ≤ 2, then dim_κ([0, 1]^N) = Dim^FH_d/α([0, 1]^N), where Dim^FH_s denotes the s-dimensional packing dimension profile of Falconer and Howroyd [6]. Since Dim^FH_d/α([0, 1]^N) = min(d/α, N ) (Falconer and Howroyd [6], P286), we see that dim_κ(F ) > sup_t∈F dim_κ({t}).

4.1.1 An Equivalent Definition of Box-dimension Profiles

In general it is hard to use Definition 4.1 to compute the box dimension profile of a given set since the hitting probability P{| eX(t)| ∈ B(0, ε)} is only explicitly computable when X has nice properties. However, we may express the box-dimension profile in terms of the characteristic exponent Ψ of the underlying additive L´evy process X.

For every Borel probability measure µ on R^N, and for all ξ ∈ R^d, define the “energy An application of Fubini’s theorem shows

E_µ(ξ) =

The following proposition uses the characteristic exponent of X to compute box-dimension profiles. It is a generalization of Theorem 2.6 of Khoshnevisan, Schilling, and Xiao [19] to the multiparameter setting. The proof of Theorem 2.6 in [19] works here and needs only minor changes. See also Khoshnevisan and Xiao [21], proof of Theorem 1.1.

Proposition 4.6. For every compact set F ⊆ R^N, dim_κ(F ) = sup Proof. For all ε > 0, let fε denote the [scaled] P´olya distribution:

fε(ξ) :=

Then its Fourier transform is given by fˆε(x) := an application of Plancherel’s theorem to the right-hand side of (4.24) shows that the following holds for all ε > 0 and t ∈ R^N:

45 Then the following elementary inequality,

1 − cos(2u)

This shows that the right hand side of (4.21) is less than or equal to dim_κ(F ).

In order to establish the converse estimate we introduce a Cauchy process {C(λ)}_λ≥0, independent of X, whose coordinate processes C₁, . . . , C_d are i.i.d. standard symmetric Cauchy processes in R. Thus the characteristic function of C(λ) is E[exp{ix · C(λ)}] = e^{−λPdl=1|xl|} for all x ∈ R^d. Since

e^{−(k/ε)Pdl=1|xl|}≤ 1B(0,ε)(x) + e^−k1B(0,ε)^c(x) ∀ ε, k > 0 and x ∈ R^d, by replacing x with eX(t) and taking expectation on both sides, we get

κ_ε(t) ≥ Eexp iX(t) · C(k/ε)
 − ee ^−k

|t − s|, and integrate with respect to µ(ds)µ(dt) to find that Z Z be arbitrary small, we get the desired estimate.

4.2 Packing Dimension Profiles

We can regularize the box-dimension profile dim_κ defined in (4.4) to produce a family of packing-type dimension profiles.

Definition 4.7. For every Borel set F ⊆ R^N, the packing dimension profile Dim_κ(F ) of F with respect to the family κ is defined as

Dim_κ(F ) := inf

One can verify that Dim_κ has the following properties:

(i) Dim_κ is monotone; i.e., Dim_κ(F ) ≤ Dim_κ(G) if F ⊆ G;

(ii) Dim_κ is σ-stable; i.e., Dim_κ(∪^∞_n=1Gn) = sup_n≥1Dim_κ(Gn).

We can also associate Dim_κ to a “packing measure with respect to the family κ.” In order to do that, we borrow some ideas from Howroyd [10].

Definition 4.8. For every Borel set F ⊆ R^N and real number δ > 0, we say that a sequence of triples (ω_i, t_i, ε_i)_i≥1 is a (κ, δ)-packing of F if for all i ≥ 1: (i) ω_i ≥ 0; (ii) t_i ∈ F ; (iii) 0 < εi≤ δ; and (iv)P

j≥1ωjκεj(ti− t_j) ≤ 1.

Definition 4.9. For every constant s > 0, the s-dimensional packing measure P^s,κ(F ) of F ⊆ R^N with respect to the family κ is defined as

where P₀^s,κ is a premeasure defined by

P₀^s,κ(F ) := lim

Definition 4.10. For every Borel set F ⊆ R^N, the packing dimension profile P-dim_κ(F ) of F with respect to the family κ is defined as

P-dim_κ(F ) := inf{s > 0 : P^s,κ(F ) = 0}. (4.35)

47 The two packing dimension profiles defined in (4.31) and (4.35) coincide. This can be shown by adapting the proof of Theorem 26 of Howroyd [10]. For completeness we record it here.

First we introduce an equivalent formulation of Zκ(ε; F ) (see (4.7) for definition) for fixed ε > 0 and bounded Borel set F ⊆ R^N. A sequence of pairs (ω_i, t_i)^k_i=1is a size ε weighted

with the convention that sup ∅ := 0. Then the following technical lemma holds.

Lemma 4.11. For every bounded Borel set F ⊆ R^N and real number ε > 0,

N_κ(ε; F ) = 1/Z_κ(ε; F ). (4.37)

Proof. On one hand, for all η < N_κ(ε; F ), (4.36) implies the existence of a size ε weighted κ-packing (ω_i, t_i)^k_i=1 of F such that Pk

i=1ω_i > η. Let W = Pk

i=1ω_i and define a discrete probability measure µ on {ti}^k_i=1 by µ(ti) = ωi/W . It follows immediately that µ ∈P(F ).

Then elementary calculation shows that

Z_κ(ε; F ) ≤

(vj, sj)^l_j=1 by removing from the sequence (u^∗_i, ti)^k_i=1 all those pairs (u^∗_i, ti) with u^∗_i = 0, then we see that (v_j/J, s_j)^l_j=1 is a size ε weighted κ-packing of F . It follows

N_κ(ε; F ) ≥

l

X

j=1

v_j J = 1

J. (4.40)

Since J ≤Pk

i,j=1κε(ti− t_j)ωiωj < η, we have Nκ(ε; F ) ≥ 1/Zκ(ε; F ). This completes the proof.

Proposition 4.12. For every Borel set F ⊆ R^N, we have

Dim_κ(F ) = P-dim_κ(F ). (4.41)

Proof. Consider an arbitrary bounded Borel set E ⊆ R^N. For each ε > 0, let (ω_i, t_i)^k_i=1 be a size ε weighted κ-packing of E. Then (ωi, ti, ε)^k_i=1 is trivially a (κ, ε)-packing of E. It follows from this fact, (4.34), and (4.36) that for all s > 0,

P_ε^s,κ(E) ≥ ε^sNκ(ε; E) = ε^s Zκ(ε; E)
−1

, (4.42)

where we used Lemma 4.11 to derive the last equality. Now if 0 < η < Dim_κ(F ), then for every sequence of bounded Borel sets {F_k}_k≥1 with F ⊆ ∪_k≥1F_k, we can find some F_n such that dim_κ(Fn) > η. Combining with (4.42) and (4.6), we get P₀^η,κ(Fn) > 0. Therefore P-dim_κ(F ) ≥ η. Let η ↑ Dim_κ(F ) to obtain Dim_κ(F ) ≤ P-dim_κ(F ).

On the other hand, if 0 < η < P-dim_κ(F ), then for every sequence of bounded Borel sets {F_k}_k≥1 with F ⊆ ∪k≥1Fk, we can find some Fn such that P₀^η,κ(Fn) > 0. For every fixed θ satisfying 0 < θ < η, we can find some a such that 0 < a < P₀^η,κ(Fn). Since Pε^η,κ(Fn) decreases to P₀^η,κ(F_n) as ε ↓ 0, we have P_ε^η,κ(F_n) > a for each ε > 0. Then (4.34) implies the existence of a (κ, ε)-packing (ωi, ti, ri)^∞_i=1 of Fn such that P∞

i=1ωir_i^η > a. For each m ≥ 1, define

Km = X

{i:2^−m<ri≤2^1−m}

ωi. (4.43)

Then there must be a positive integer M (ε) such that

K_{M (ε)}> 2^{(M (ε)−1)θ}(1 − 2^θ−η)a, (4.44) since otherwise P∞

m=1Km2^(1−m)η ≤ P∞

m=12^(m−1)θ(1 − 2^θ−η)a2^(1−m)η = a. Let {i(j)}^J_j=1 be a subsequence such that 2^{−M (ε)} < r_i(j) ≤ 2^{1−M (ε)} for all 1 ≤ j ≤ J and PJ

j=1ω_i(j) ≥ 2^{(M (ε)−1)θ}(1 − 2^θ−η)a. The fact that (ωi, ti, ri)^∞_i=1 is a (κ, ε)-packing of Fn implies that

49 P

j≥1ωjκrj(ti− t_j) ≤ 1 for all i ≥ 1. Since κε(t) is nondecreasing in ε for fixed t and r_i(j) ≤ 2^{1−M (ε)} for all 1 ≤ j ≤ J , we see that (ω_i(j), t_i(j))^J_j=1 is a size 2^{−M (ε)} weighted κ-packing of Fn. Therefore,

Nκ(2^{−M (ε)}; Fn) ≥ 2^{(M (ε)−1)θ}(1 − 2^θ−η)a. (4.45) Since M (ε) ↑ ∞ as ε ↓ 0, by taking logarithms and then limsup as ε ↓ 0, we obtain

dim_κ(Fn) = lim

ε↓0

log Z_κ(2^{−M (ε)}; F_n)

log 2^{−M (ε)} = lim

M →∞

log N_κ(2^−M; F_n)

− log 2^−M ≥ θ. (4.46) Let θ ↑ η to find Dim_κ(F ) ≥ P-dim_κ(F ). This completes the proof.

The connection between packing dimension profile Dim_κ and the family of measures {P^s,κ}_s≥0 can be used to prove the following technical lemma, which will be used later.

Lemma 4.13. If a Borel set F ⊆ R^N satisfies Dim_κ(F ) > s for some s ≥ 0, then there exists a compact set K ⊆ F such that Dim_κ(K ∩ G) ≥ s for all open sets G ⊆ R^N with K ∩ G 6= ∅.

Proof. Since Dim_κ(F ) > s, for every collection of bounded Borel sets {Fn}_n≥1 with F ⊆

∪_n≥1Fn, we can find some Fn such that Dim_κ(Fn) > s. This implies that P^s,κ(Fn) = ∞.

Then we can use the proof of Theorem 22 of Howroyd [10] to complete the argument.

4.2.1 Main Result

With the box-dimension profile and packing dimension profile defined so far, we are able to present our main result regarding the packing dimension of images of additive L´evy processes.

Theorem 4.14. Let X := {X(t) : t ∈ R^N₊} be an additive L´evy process in R^d and let dim_M denote the upper box-counting dimension. Then for all nonrandom bounded Borel sets F ⊆ R^N+:

dim_M(X(F )) = dim_κ(F ) a.s.; and (4.47)

dim_P(X(F )) = Dim_κ(F ) a.s. (4.48)

We will prove this theorem in the next section.

4.3 Proof of the Main Theorem

In this section we strive to prove Theorem 4.14. The proof relies on finding upper bounds for the probabilities of the events that the underlying additive L´evy process hits small balls.

In order to estimate such probabilities, we adapt the methods of section 4 of Khoshnevisan, Xiao, and Zhong [23].

4.3.1 Hitting Probabilities

For every δ > 0 and Borel set F ⊆ R^N+, let F^δ denote the closed δ-enlargement of F , that is, F^δ := {y ∈ R^N+ : dist(y, F ) ≤ δ}, where dist(y, F ) := inf_x∈F|x − y|. Let λ_d denote the Lebesgue measure on R^d. For fixed ε > 0 and Borel set F ⊆ R^N+, we want to estimate Eλ_d (X(F ))^ε
. It turns out that this quantity is equal to the probability of the event that the underlying additive L´evy process hits small balls.

Proposition 4.15. For every bounded Borel set F ⊆ R^N+ and small number ε > 0, we have

Eλ_d (X(F ))^ε
 ≤ Cε^d

Z_κ(ε; F ), (4.49)

where Zκ(ε; F ) is defined in (4.7) and the constant C depends only on N and d.

Proof. We use the idea of the proof for Theorem 2.1 of [23]. Choose and fix some ∆ /∈ R^N+. For each δ > 0, let T^δdenote some measurable, (Q^N+∩ F^δ) ∪ {∆}-valued function on Ω such that T^δ 6= ∆ if and only if |X(T^δ)| < ε/2. Then for each k > 0 define a set function

µ^δ,k(·) := Pλd{T^δ ∈ ·, T^δ6= ∆, |X(0)| ≤ k}

Pλ_d{T^δ 6= ∆, |X(0)| ≤ k} . (4.50) Since B(0, ε) is open and X_j is right continuous for all 1 ≤ j ≤ N , an application of Fubini’s theorem implies that

Pλ_d{T^δ 6= ∆, |X(0)| ≤ k} = Pλ_d{X(F^δ) ∩ B(0, ε/2) 6= ∅, |X(0)| ≤ k}

= Z

[−k,k]^d

P{X(F^δ) ∩ B(−x, ε/2) 6= ∅} dx

= Eλ_d (X(F^δ) ⊕ B(0, ε/2)) ∩ [−k, k]^d
.

(4.51)

Since the last term is positive and finite, we see that µ^δ,k is a probability measure on F^δ. Now for every A ⊆ Π and nonnegative measurable or bounded measurable function f : R^d → R, consider the multiparameter martingale {M^A_µδ,kf (t)}_t∈RN

+ defined in (3.58).

Then

51

Pλ_d-almost surely, where the last equality follows from Lemma 3.28. We obtain the following by taking a supremum over all positive rational numbers and noticing that T^δ∈ Q^N+∪ {∆}

and |X(T^δ)| < ε/2 on {T^δ 6= ∆, |X(0)| ≤ k}:

Pλd-almost surely. Summation over all A ⊆ Π gives X

Pλd-almost surely. Note that the null set is independent of the choice of ε > 0. By squaring and taking Pλ_d-expectations on both sides of the above inequality, and noticing the special choice of µ^δ,k, we obtain the following:

Eλd

where the last inequality follows from Jensen’s inequality. On one hand, the elementary inequality (Pn

k=1ak)² ≤ nPn

k=1a²_k shows that

Eλd

where we used Lemma 3.32 to derive the last inequality. On the other hand, we choose our f to be the Fourier transform of the ε-scaled P´olya distribution, that is,

fε(x) =

where a⁺:= max(a, 0) for all real numbers a. Then the Fourier transform of fε is given by fˆε(ξ) = particular, combining with (4.19), we get

Z application of Prohorov’s theorem ([15], Chapter 6, Theorem 2.5.1), together with the continuity of infxPt−sf_ε/2(x), imply that there exists some µ ∈ P( ¯F ), such that, along

53 Here ¯F denotes the closure of F . Now we combine (4.55), (4.56), (4.60), (4.61) and (4.62) to get

Eλ_d (X(F ))^ε/2
 · Z Z

|x|≤ε/2inf P_t−sf_ε(x)µ(dt)µ(ds) ≤ 16^N(2π)^−d(8C₀)^dε^d, (4.63) where C0 =R

Ry⁻⁴(1 − cos y)²dy < ∞.

Finally, we notice that 1 − (2ε)⁻¹|z| ≥ ¹₂ whenever z ∈ B(0, ε). Consequently,

2^−d1B(0,ε)(z) ≤ f_ε(z) ∀ z ∈ R^d. (4.64) Let z := eX(t − s) + x, using (3.42) and taking expectation to find

P_t−sf_ε(x) = E[fε( eX(t − s) + x)]

≥ 2^−dE[1B(0,ε)( eX(t − s) + x)] = 2^−dP{| eX(t − s) + x| ≤ ε}

≥ 2^−dP{| eX(t − s)| < ε/2} · 1{|x| ≤ ε/2},

(4.65)

where the last line follows from the triangle inequality. Take an infimum over |x| ≤ ε/2 to get

inf

x∈R^d,|x|≤ε/2Pt−sfε(x) ≥ 2^−dP{| eX(t − s)| < ε/2} = 2^−dκ_ε/2(t − s). (4.66) We combine the above inequality with (4.63) and change ε/2 to ε to obtain

Eλ_d (X( ¯F ))^ε
 ≤ 16^N(2π)^−d(8C0)^d2^d(2ε)^d

inf_{ν∈P( ¯F )}RR κ_ε(t − s)ν(dt)ν(ds). (4.67) The proposition is proved, thanks to Lemma 4.3.

4.3.2 Proof of Theorem 4.14

With the help of Proposition 4.15 we can adapt the proof of Theorem 2.7 of Khosh-nevisan, Schilling, and Xiao [19] for one-parameter L´evy processes to the current multipa-rameter setting.

Proof of Theorem 4.14: (4.47). Recall that the upper box dimension of a bounded Borel set E ⊂ R^d is defined by

dim_M(E) := lim sup

r↓0

log K_E(δ)

− log δ , (4.68)

where K_E(δ) is the Kolmogorov capacity of E, that is, K_E(δ) is the maximal number m of points x₁, . . . , x_m in E such that min_i6=j|x_i − x_j| ≥ δ. It follows immediately that δ^dKE(δ) ≤ λd(E^δ) for all δ > 0, where E^δ is the closed δ-enlargement of E. In fact, we can

find k := KE(δ) points x1, . . . , x_k in E such that B(x1, δ/2), . . . , B(x_k, δ/2) are disjoint and sufficiently small ε > 0. We combine this with (4.69) and use the Markov inequality to obtain

P{KX(F )(ε) ≥ ε^−(η+θ)} = O(ε^θ) (ε ↓ 0), (4.70) for all θ ∈ (0, 1). We apply (4.70) with ε = 2⁻ⁿ and use Borel-Cantelli lemma in order to get

K_{X(F )}(2⁻ⁿ) = O(2^n(η+θ)) (n → ∞) a.s. (4.71) A standard monotonicity argument (see for example (3.14) of [5]) shows that with probabil-ity one dim_M(X(F )) ≤ η + θ. Now we first let θ ↓ 0 and then η ↓ dim_κ(F ) (along countable sequences) to deduce that

dim_M(X(F )) ≤ dim_κ(F ) a.s. (4.72)

In order to prove the converse inequality, we use a result from Khoshnevisan and Xiao [22]. According to Theorem 4.1 of [22], the upper box dimension of a bounded Borel set E can also be computed by

dim_M(E) = sup

Then for any η < dim_κ(F ), (4.4) shows that there exist a sequence of positive numbers {ε_n}_n≥1 and a sequence of probability measures {µ_n}_n≥1 ⊂ P(F ) such that εn ↓ 0 as This, Fatou’s lemma, and (4.74) together shows that

lim

55 Therefore, (4.73) implies that dim_M(X(F )) ≥ η almost surely. Let η ↑ dim_κ(F ) to see that dim_M(X(F )) ≥ dim_κ(F ) almost surely.

In order to prove equation (4.48) we need the following technical lemma.

Lemma 4.16. For all compact sets F ⊆ R^N+, dim_P(X(F )) = dim_P(X(F )) almost surely, where ¯E denotes the closure of a set E.

Proof. It is straightforward to see that dim_P(X(F )) ≥ dim_P(X(F )) almost surely. Then we recall that X(t) =PN

j=1X_j(t_j) for t ∈ R^N+ and each X_j is a one-parameter L´evy process.

Notice that with probability one

{t ∈ F : X is discontinuous at t} ⊆

N

[

j=1

t ∈ F : X_j(t_j) 6= X_j(t_j−) . (4.77)

Furthermore, for each 1 ≤ j ≤ N , let Y^j(t) := P

1≤l≤N,l6=jX_l(t_l) + X_j(t_j−). Since there are at most countably many jumps for each Xj, and dim_P is σ-stable, it follows that for all 1 ≤ j ≤ N ,

dim_P {Y^j(t) : t ∈ F, Xj(tj) 6= Xj(tj−)}
=

dim_P {X(t) : t ∈ F, X_j(tj) 6= Xj(tj−)}
≤ dim_P(X(F )) a.s.

(4.78)

Since X(F ) ⊆ X(F ) ∪ (∪^N_j=1{Y^j(t) : t ∈ F, X_j(t_j) 6= X_j(t_j−)} almost surely, the σ-stability of dim_P implies that dim_P(X(F )) ≤ dim_P(X(F )) almost surely. This completes the proof.

Proof of Theorem 4.14: (4.48). Definition 4.7 implies that for each η > Dim_κ(F ) there exists a sequence of bounded Borel sets {F_n}_n≥1 such that

F ⊆

∞

[

n=1

F_n and sup

n≥1

dim_κ(F_n) < η. (4.79)

Since X(F ) ⊆ ∪_n≥1X(F_n), (4.47) implies that dim_P(X(F )) ≤ sup

n≥1

dim_M(X(F_n)) = sup

n≥1

dim_κ(F_n) < η a.s. (4.80) Let η ↓ Dim_κ(F ) to obtain dim_P(X(F )) ≤ Dim_κ(F ) almost surely.

Next we prove the converse inequality, that is, dim_P(X(F )) ≥ Dim_κ(F ) almost surely.

Next we prove the converse inequality, that is, dim_P(X(F )) ≥ Dim_κ(F ) almost surely.

In document Doctor of Philosophy (Page 41-105)