行政院國家科學委員會專題研究計畫 成果報告
碎形與自相似性的隨機理論(V)
研究成果報告(精簡版)
計 畫 類 別 ： 個別型
計 畫 編 號 ： NSC 95-2115-M-002-012-
執 行 期 間 ： 95 年 08 月 01 日至 96 年 10 月 31 日
執 行 單 位 ： 國立臺灣大學數學系暨研究所
計 畫 主 持 人 ： 謝南瑞
計畫參與人員： 博士後研究：西鄉達彥
報 告 附 件 ： 出席國際會議研究心得報告及發表論文
處 理 方 式 ： 本計畫可公開查詢
中 華 民 國 96 年 12 月 13 日
行政院國家科學委員會補助專題研究計畫
成 果 報 告 □期中進度報告碎形與自相似性的隨機理論
(V)
計畫類別：
個別型計畫 □ 整合型計畫
計畫編號：
NSC 95-2115-M-002-012
執行期間： 2006 年 08 月 01 日至 2007 年 10 月 31 日
計畫主持人：
謝南瑞
共同主持人：
計畫參與人員： 西鄉達彥
成果報告類型(依經費核定清單規定繳交)：□精簡報告 □完整
報告
本成果報告包括以下應繳交之附件：
□赴國外出差或研習心得報告一份
□赴大陸地區出差或研習心得報告一份
出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
執行單位：
台大數學系
中 華 民 國 96 年 12 月 12 日
1行政院國家科學委員會專題研究計畫成果報告
碎形與自相似的隨機理論（V）
計畫編號：
NSC 95-2115-M-002-012
執行期限：95 年 08 月 01 日至 96 年 10 月 31 日
主持人：謝南瑞 執行機構及單位名稱:台大數學系
一、中英文摘要
這是同一研究主題第 5 年的精簡報告。
關鍵詞：分生過程、布朗運動，Levy 過程，豪氏維度。
This is a brief report of the fifth year of a same research subject.
Keywords: branching processes,Brownian motions, Levy processes, Hausdorff
dimensions.
二、本年進度
In this year, I mainly investigate the sample paths of Levy processes and Levy-driven
Ornstein--Uhlenbeck processes, I study the fractal property of the processes.
三、成果自評
We obtain the level set dimension of some multiparameter Levy processes, and the
result is appeared in Probability Theory and Related Fields.
出席國際會議心得報告
（附發表之論文）
我於 2007 年 8 月 12 日至 19 日赴 Denmark Copenhagen University 數學
研究所，出席由該所主辦之第 5 屆李維過程國際研討會。此會議是該研究所之
歐盟共同研究計劃下的研討會，除英國學者外，尚有美國、法國、日本、大陸
學者與會。
我在會議中報告了有關 Multiparameter Levy Processes Level Set
Hausdorff Dimension 的研究成果。會中會後並
與相關學者就隨機碎形多所討論。
DOI 10.1007/s00440-007-0060-7
Hausdorff dimension of the contours of symmetric
additive Lévy processes
Davar Khoshnevisan · Narn-Rueih Shieh · Yimin Xiao
Received: 9 March 2006 / Revised: 30 January 2007 © Springer-Verlag 2007
Abstract Let X1,. . . , XN denote N independent, symmetric Lévy processes on Rd. The corresponding additive Lévy process is defined as the following N-parameter random field on Rd:
X(t) := X1(t1) + · · · + XN(tN) (t ∈ RN+). (0.1) Khoshnevisan and Xiao (Ann Probab 30(1):62–100, 2002) have found a neces-sary and sufficient condition for the zero-setX−1({0}) of X to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of X−1({0}) which hold with positive probability in the case that X−1({0}) can be
non-void.
The research of D. Kh. and Y. X. was supported by the United States NSF grant DMS-0404729. The research of N.-R. S. was supported by a grant from the Taiwan NSC.
D. Khoshnevisan (
B
)Department of Mathematics, The University of Utah, 155 S. 1400 E. Salt Lake City, UT 84112–0090, USA e-mail: davar@math.utah.edu
URL: http://www.math.utah.edu/∼davar N.-R. Shieh
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan e-mail: shiehnr@math.ntu.edu.tw
Y. Xiao
Department of Statistics and Probability, Michigan State University, A-413 Wells Hall, East Lansing, MI 48824, USA
e-mail: xiao@stt.msu.edu
Here we prove that the Hausdorff dimension ofX−1({0}) is a constant almost surely on the event{X−1({0}) = ∅}. Moreover, we derive a formula for the said constant. This portion of our work extends the well known formulas of Horo-witz (Israel J Math 6:176–182, 1968) and Hawkes (J Lond Math Soc 8:517–525, 1974) both of which hold for one-parameter Lévy processes.
More generally, we prove that for every nonrandom Borel set F in(0, ∞)N, the Hausdorff dimension ofX−1({0}) ∩ F is a constant almost surely on the event{X−1({0}) ∩ F = ∅}. This constant is computed explicitly in many cases. Keywords Additive Lévy processes· Level sets · Hausdorff dimension Mathematics Subject Classification (2000) 60G70· 60F15
1 Introduction
Let X1,. . . , XN denote N independent symmetric Lévy processes on Rd, all starting from 0. We construct the N-parameter random fieldX := {X(t)}t∈RN
+
on Rdas follows:
X(t) := X1(t1) + · · · + XN(tN), (1.1) where t := (t1,. . . , tN) ranges over RN+. Thus,X is called a “symmetric additive Lévy process,” and has found a number of applications in the study of classical Lévy processes [20–23]. Occasionally we follow the notation of these references and denote the random fieldX also by X1⊕ · · · ⊕ XN.
Consider the level set at x,
X−1({x}) :=t∈ (0, ∞)N: X(t) = x
for x∈ Rd. (1.2)
By defining X−1({x}) in this way we have deliberately ruled out the points
t∈ ∂RN+withX(t) = x, where ∂RN+:= {t ∈ [0, ∞)N: t
i= 0 for some 1 ≤ j ≤ N} denotes the boundary of RN+, since the problems for the latter can be reduced to the level sets of additive Lévy processes with fewer parameters.
Khoshnevisan and Xiao [22] assert that, under a mild technical condition, X−1({0}) = ∅ with positive probability if and only if a certain function is
locally integrable. Moreover, the function is easy to describe: It is the density function ofX(|t1|, . . . , |tN|) at x = 0.
As a by-product of their arguments, Khoshnevisan and Xiao [22] produce bounds on the Hausdorff dimension ofX−1({0}) as well. In fact, they exhibit two numbersγ ≤ γ , both computable in terms of the Lévy exponents of X1,. . . , XN, such that
Originally, the present paper was motivated by our desire to have better information on the Hausdorff dimension ofX−1({0}) in the truly multiparame-ter setting N ≥ 2. Recall that when N = 1, X is a Lévy process in the classical sense, and in addition,
either P
X−1({0}) = ∅= 1 or PX−1({0}) is uncountable= 1. (1.4) This is a consequence of the general theory of Markov processes; see Prop-osition 3.5 and Theorem 3.8 of Blumenthal and Getoor [6, pp. 213 and 214]. Moreover, it is known exactly whenX−1({0}) is uncountable and, in general, X−1({0}) differs from the range of a subordinator in at most countably-many
places. Consequently, a nice formula for dimHX−1({0}) can be derived from the result of Horowitz [12] on the Hausdorff dimension of the range of a subordi-nator. For a modern elegant treatment see Theorem 15 of Bertoin [4, p. 94]. See also Hawkes [11] where dimHX−1({0}) is described solely in terms of the Lévy exponent ofX.
We were puzzled by why the extension of the said refinements to N≥ 2 are so much more difficult to obtain. For example, the issue of when{0} is regular for itself—i.e., (1.4)—becomes much more delicate once N ≥ 2. (We hope to deal with this matter elsewhere.) Thus, it is not obvious—nor does it appear to be true—that dimHX−1({0}) is a.s. a constant.
In the present paper we prove that, under a mild technical condition, the Hausdorff dimension ofX−1({0}) is a simple function of ω. In fact, it is a con-stant a.s. on the set whereX−1({0}) is non-trivial.
We are even able to find a nice formula for the Hausdorff dimension of the zero setX−1({0}), a.s. on the event that it is nonempty. See Theorem1.1below. It can be shown that when N = 1 our formula agrees with the one-parame-ter findings of Horowitz [12] and Hawkes [11]. Moreover, suppose X−1({0}) were replaced by the closure ofX−1({0}) in (0, ∞)N, then our derivations show that the same formula holds almost surely on the event that the said closure is nonempty.
The remainder of the Introduction is dedicated to developing the requisite background needed to describe our dimension formula precisely.
Let1,. . . , Ndenote the respective Lévy exponents of X1,. . . , XN. That is, for all 1≤ j ≤ N, ξ ∈ Rd, and u≥ 0,
E eiξ·Xj(u) = exp−uj(ξ) . (1.5)
We recall that the functions1,. . . , Nare real, non-negative, and symmetric. We say thatX is absolutely continuous if
Rd exp ⎛ ⎝−u 1≤j≤N j(ξ) ⎞ ⎠ dξ < ∞ for all u > 0. (1.6)
Define for all t∈ RN, (t) := 1 (2π)d Rd exp ⎛ ⎝− 1≤j≤N |tj|j(ξ) ⎞ ⎠ dξ. (1.7) This defines on RN\{0}; is uniformly continuous and bounded away from {0}, and (0) = ∞. As a consequence of the results of Khoshnevisan and Xiao [22] we have P X−1({0}) = ∅> 0 ⇐⇒ PX−1({0}) ∩ (0, ∞)N= ∅> 0 ⇐⇒ ∈ L1 loc(RN), (1.8)
where ¯A denotes the closure of A and ∈ L1loc(RN) means ∈ L1([−T, T]N) for every T> 0. Define x to be the Euclidean 2norm of x. Then the following is our main result:
Theorem 1.1 If X1,. . . , XNare symmetric absolutely continuous Lévy processes in Rd, then almost surely on{X−1({0}) = ∅},
dimHX−1({0}) = sup ⎧ ⎪ ⎨ ⎪ ⎩q> 0 : [0,1]N (t) t q dt< ∞ ⎫ ⎪ ⎬ ⎪ ⎭. (1.9) Suppose, in addition, that there is a constant K> 0 such that
(t) ≤ (K t , . . . , K t ) for all t ∈ (0, 1]N. (1.10) Then,
dimHX−1({0}) = N − lim sup t→0
log(t)
log(1/ t ). (1.11)
When N = 1, (1.10) holds automatically, and so (1.9) and (1.11) coincide [11,12]. We will show in Example3.6that when N> 1, formula (1.11) does not hold in general; an extra condition such as (1.10) is necessary.
Compared to the one-parameter case, the proof of Theorem1.1is consider-ably more complicated when N> 1. This is mainly due to the fact that classical covering arguments produce only (1.3) in general. Thus, we are led to a different route: We introduce a rich family of random sets with nice intersection proper-ties, and strive to find exactly which of these random sets can intersectX−1({0}). There is a sense of symmetry about our arguments, since everything is described in terms of additive Lévy processes; the said random sets are constructed by means of introducing auxiliary additive Lévy processes. This argument allows
us to establish a formula for the Hausdorff dimension ofX−1({0}) ∩ F for every nonrandom Borel set F⊂ (0, ∞)N. See Theorem3.2and the examples in Sect.3. The idea of introducing random sets to help compute dimension seems to be due to Taylor [32, Theorem 4]. Since its original discovery, this method has been used by many others; in diverse ways, and to good effect [2,3,5,8–
10,14,16–20,24–30].
We conclude Sect.1 by introducing some notation that is used throughout and consistently.
• For every integer m ≥ 1, and for all x ∈ Rm, x :=x21+ · · · + x2m
1/2
, |x| := max
1≤j≤m|xj|, and
[x] := |x1| + · · · + |xm|. (1.12) They respectively denote the2,∞, and1norms of x.
• Multidimensional “time” variables are typeset in bold letters in order to help the reader in his/her perusal.
• For all integers m ≥ 1 and s, t ∈ Rm
+, we write
s≺ t iff t s iff si≤ ti for all 1≤ i ≤ m. (1.13) • Let m ≥ 1 be a fixed integer and q ≥ 0 a fixed real number. Suppose f : Rm → R+is Borel measurable, andµ is a Borel probability measure on Rm. Then,
I(q)f (µ) :=
f(x − y)
x − y qµ(dx) µ(dy). (1.14) When f ≡ 1 and q > 0, this is the q-dimensional Bessel–Riesz energy of µ, which will be denoted by I(q)(µ).
• For any Borel set G ⊂ Rm, let P(G) denote the collection of all Borel probability measures on G. The q-dimensional Bessel–Riesz capacity of G is defined by Cq(G) := inf µ∈P(G)I (q)(µ)−1. (1.15) • If f : RN\{0} → R
+, then we define the upper index and lower index of f (at 0∈ RN) respectively as
ind(f ) := lim sup
x →0
logf(x)
log(1/ x ), ind(f ) := lim inf x →0
logf(x)
Consequently, Theorem1.1asserts that if (1.10) holds then a.s. on the event thatX−1({0}) = ∅,
dimHX−1({0}) = N − ind(). (1.17)
2 Background on additive Lévy processes 2.1 Absolute continuity
We follow Khoshnevisan and Xiao [22] and call the following function the Lévy exponent ofX. It is defined as follows. For ξ ∈ Rd,
(ξ) := (1(ξ), . . . , N(ξ)) . (2.1) In this way, we can write
E
eiξ·X(t)
= e−t·(ξ) forξ ∈ Rdand t∈ RN+. (2.2) Also, we declareX to be absolutely continuous if the function ξ → exp{−t·(ξ)} is in L1(Rd) for all t ∈ (0, ∞)N.
If any one of the Xj’s is absolutely continuous, then so isX. A similar remark continues to apply if Xj is replaced by an additive process based on a proper, nonempty subset of{X1,. . . , XN}. However, it is possible to construct counter-examples and deduce that the converse to these assertions are in general false.
Here and throughout, we assume, without fail, that
X is absolutely continuous. (2.3) It is possible to check that this is equivalent to the absolute-continuity condition (1.6) mentioned in Sect. 1.
We may apply the inversion theorem and deduce that X(t) has a density function pt(•) for all t ∈ (0, ∞)N. Moreover, for all x∈ Rdand t∈ (0, ∞)N,
pt(x) = 1 (2π)d Rd cos(ξ · x) e−t·(ξ)dξ. (2.4) Let RN= be the set of all t∈ RN such that(|t1|, . . . , |tN|) ∈ (0, ∞)N. We abuse the notation slightly and also use pt(x) to denote the density function of
X(|t1|, . . . , |tN|) for all t ∈ RN=. Evidently, p is continuous on RN= × Rd and for all t∈ RN=,
sup
x∈Rd
pt(x) = pt(0) = (t). (2.5)
Throughout, we consider the probabilities: r(x ; t) := 1 (2r)dPX(|t1|, . . . , |tN|) − x ≤r , r(t) := r(0 ; t), (2.6)
valid for all r> 0, x ∈ Rd, and t= (t1,. . . , tN) ∈ RN. Evidently, for all t∈ RN=,
lim r→0+r(t) = (t), sup x∈Rd r(x ; t) ≤ (t). (2.7)
The first statement follows from the continuity of x → pt(x), and the second
from (2.5). Similarly, we have
lim
r→0+r(x ; t) = pt(x), (2.8)
valid for all t∈ RN=. 2.2 Weak unimodality
We follow Khoshnevisan and Xiao [22] and say that a Borel probability measure µ on Rkis weakly unimodal (with constantκ) if for all r > 0,
sup
x∈Rk
µB(x ; r)≤ κ µB(0 ; r), (2.9) where B(x ; r) :=y ∈ Rk : |x − y| ≤ r. Evidently, we can chooseκ to be its optimal value, κ := sup r>0xsup∈Rk µB(x ; r) µB(0 ; r) < ∞, (2.10) where 0/0 := 1.
SinceX is a symmetric additive Lévy process, Corollary 3.1 of Khoshnevisan and Xiao [21] implies that the distribution of X(t) is weakly unimodal with constant 16dfor all t∈ (0, ∞)N. Equivalently, the growth of the functionrof (2.6) is controlled as follows:
sup
x∈Rd
This and Lemma 2.8(i) of Khoshnevisan and Xiao [22] together imply the following “doubling property”:
2r(t) ≤ 32dr(t) for all t ∈ RN. (2.12) Another important consequence of weak unimodality is that t → r(t) is “quasi-monotone.” This means that if s≺ t and both are in (0, ∞)N, then
r(t) ≤ 16dr(s) for all r> 0. (2.13) See Lemma 2.8(ii) of Khoshnevisan and Xiao [22].
3 Some key results and examples
Khoshnevisan and Xiao [22, Theorem 2.9], have proven that ∈ L1
loc(RN) iff P
X−1({0}) = ∅> 0. (3.1)
They proved also that the same is true forX−1({0}) ∩ (0, ∞)N. This was men-tioned earlier in Sect.1of the present paper; see (1.8). In addition, Khoshnevisan and Xiao [22] have computed bounds for the Hausdorff dimension ofX−1({0}) in the case that is locally integrable. The said bounds are in terms of γ and ¯γ, where γ := sup ⎧ ⎪ ⎨ ⎪ ⎩q> 0 : [0,1]N (t) t q dt< ∞ ⎫ ⎪ ⎬ ⎪ ⎭, ¯γ := inf q> 0 : lim inf t →0 (t) t q−N > 0 . (3.2)
First, we offer the following. Lemma 3.1 It is always the case that
0≤ γ ≤ ¯γ ≤ N −d
2. (3.3)
If, in addition, (1.10) holds, then also, γ = inf q> 0 : lim sup t →0 (t) t q−N > 0 . (3.4)
Thus, in light of (1.16), we arrive at the following consequence:
Proof of Lemma3.1 By definition, 0 ≤ γ . Also, if q > ¯γ then there exists a positive and finite A such that(t) ≥ A t q−Nfor all t∈ [0, 1]N. Consequently,
[0,1]N(t) t −qdt ≥ A
[0,1]N t −Ndt = ∞. It follows that q ≥ γ . Let q ↓ ¯γ
to deduce that ¯γ ≥ γ .
In order to prove that ¯γ ≤ N − (d/2), we first recall that j(ξ) = O( ξ 2) as ξ → ∞ [7, eq. (3.4.14), p. 67]. Therefore, there exists a positive and finite constant A such that|s · (ξ)| ≤ A s (1 + ξ 2) for all ξ ∈ Rd and s∈ RN. Consequently, for all s∈ RN,
(s) ≥ Rd e−A s (1+ ξ 2)dξ = A e−A s s d/2 , (3.6) where Adepends only on d and A. This yields ¯γ ≤ N − (d/2) readily.
It remains to verify (3.4) under condition (1.10). From now on, it is convenient to define temporarily, θ := inf q> 0 : lim sup t →0 (t) t q−N > 0 . (3.7)
If 0< q < θ, then (t) = o( t q−N), and for all > 0 and for all sufficiently large n,
{2−n−1< t ≤2−n} (t)
t q− dt= O(2−n ) as n → ∞. (3.8) Consequently, the left-most terms form a summable sequence indexed by n. In other words, for all > 0, t → t −q(t) is integrable on neighborhoods of the origin in RN. We have proved that q ≤ γ + . Let ↓ 0 and q ↑ θ to find thatθ ≤ γ . [This does not require (1.10).]
If 0< q < γ and (1.10) holds, then ∞ > { t ≤1} (t) t q dt= 1≤n<∞ {2−n−1< t ≤2−n} (t) t q dt. (3.9) Thus, lim n→∞ {2−n−1< t ≤2−n} (t) t q dt= 0. (3.10) But the preceding integral is at least 2nq(2−n,. . . , 2−n) times the volume of {t ∈ RN
+ : 2−n−1< t ≤ 2−n}. This follows from the coordinate-wise
(2−n,. . . , 2−n) = o
2−n(q−N)
as n→ ∞. (3.11) From this we conclude also that for the constant K> 0 in (1.10),
(K2−n,. . . , K2−n) = o
2−n(q−N)
as n→ ∞. (3.12) We appeal to (1.10) to deduce that
(K2−n,. . . , K2−n)
2−(n−1)(q−N) ≥ A2−n−1sup< t ≤2−n (t)
t q−N, (3.13) where A is positive and finite, and depends only on N. This and (3.12) prove that q< θ, whence it follows that γ ≤ θ. The converse bounds has already been
proved.
We are ready to present the main theorem of this section. This theorem is new even whenX is an ordinary Lévy process [i.e., X := X and N = 1]. Theorem 3.2 Let X denote an N-parameter symmetric, absolutely continuous additive Lévy process on Rd. Choose and fix a compact set F ⊂ (0, ∞)N. Then, almost surely on{X−1({0}) ∩ F = ∅},
dimH
X−1({0}) ∩ F
= sup0< q < N : I(q) (µ) < ∞ for some µ ∈ P(F)
. (3.14) Remark 3.3 The proof of Theorem3.2implies that the Hausdorff dimension of X−1({0}) ∩ F has the same formula, almost surely on {X−1({0}) ∩ F = ∅}.
In order to have a complete picture it remains to know whenX−1({0}) ∩ F is nonempty with positive probability. This issue is addressed by Corollary 2.13 of Khoshnevisan and Xiao [22] as follows:
P
X−1({0}) ∩ F = ∅> 0 ⇐⇒ PX−1({0}) ∩ F = ∅> 0
⇐⇒ there exists µ ∈ P(F) such that I(0)(µ) < ∞. (3.15) (The weak unimodality assumption of Khoshnevisan and Xiao [22], Corollary 2.13 is redundant in the present setting; see Corollary 3.1 of Khoshnevisan and Xiao [21].) It follows from (3.15) that X−1({0}) ∩ F = ∅ a.s. whenever
dimHF < ind().
The following is an immediate consequence of Theorem3.2, used in conjunc-tion with Frostman’s theorem [15], Theorem 2.2.1, p. 521. Note that, here and in the sequel, dimHE< 0 means E = ∅.
Corollary 3.4 If the conditions of Theorem3.2are met, then for all nonrandom compact sets F⊂ (0, ∞)N,
dimHF− ind() ≤ dimH
X−1({0}) ∩ F≤ dimHF− ind(), (3.16) almost surely on{X−1({0}) ∩ F = ∅}.
Khoshnevisan and Xiao [22, Theorem 2.10], have proved the following under the assumption thatX is absolutely continuous and symmetric:
(1) For all C> c > 0, P γ ≤ dimH X−1({0}) ∩ [c, C]N≤ ¯γ> 0. (3.17) (2) If there is a K> 0 such that (t) ≤ (K t , . . . , K t ), then
P
dimH
X−1({0}) ∩ [c, C]N= γ> 0. (3.18) Thus, Corollary3.4improves (3.17) and (3.18) in several ways.
We end this section with some examples showing applications of Theorems
1.1and3.2
Example 3.5 Let X1,. . . , XN be N independent, identically distributed sym-metric Lévy processes with stable components [31]. More precisely, let X1(t) =
(X1,1(t), . . . , X1,d(t)) for all t ≥ 0, where the processes X1,1,. . . , X1,dare assumed
to be independent, symmetric stable processes in R with respective indices α1,. . . , αd∈ (0, 2]. Let X be the associated additive Lévy process in Rd. ThenX is anisotropic in the space-variable unlessα1= · · · = αd.
It can be verified thatX satisfies the conditions of Theorem3.2and for all
t∈ (0, 1]N, (t) = Rd exp ⎛ ⎝− 1≤j≤N tj 1≤k≤d |ξk|αk ⎞ ⎠ dξ t − 1≤k≤d(1/αk). (3.19)
In the above and sequel, “f(t) g(t) for all t ∈ T” means that f (t)/g(t) is bounded from below and above by constants that do not depend on t∈ T. It follows from Corollary3.4that for every compact set F ⊂ (0, ∞)N,
dimH X−1({0}) ∩ F= dimHF− d k=1 1 αk , (3.20) almost surely on{X−1({0}) ∩ F = ∅}.
The same reasoning implies that ifX is an additive α-stable process in Rd [i.e., if X1,. . . , XNare symmetricα-stable Lévy processes in Rd], then for every compact set F⊂ (0, ∞)N, dimH X−1({0}) ∩ F= dimHF− d α, (3.21) almost surely on{X−1({0}) ∩ F = ∅}.
Next we consider additive Lévy processes which are anisotropic in the time-variable.
Example 3.6 Suppose X1,. . . , XN are N independent symmetric stable Lévy processes in Rdwith indicesα1,. . . , αN∈ (0, 2], respectively. Let X be the addi-tive Lévy process in Rddefined byX(t) = X1(t1) + · · · + XN(tN). Because, for every 1≤ j ≤ N and fixed ti (i= j), the process R+ tj → X(t) is [up to an independent random variable] anαj-stable Lévy process in Rd,X = {X(t)}t∈RN
+
is anisotropic in the time-variable.
The following result is concerned with the Hausdorff dimension of the zero setX−1({0}). For convenience, we assume
2≥ α1≥ · · · ≥ αN> 0. (3.22) Define k(α) := min ⎧ ⎨ ⎩ =1,. . . , N : 1≤j≤ αj> d ⎫ ⎬ ⎭, (3.23) where min∅ := ∞. In particular, k(α) = ∞ if and only if 1≤j≤Nαj≤ d. Theorem 3.7 LetX = {X(t)}t∈RN
+ be the additive Lévy process defined above. Then, P{X−1({0}) = ∅} > 0 if and only if k(α) is finite. Moreover, if k(α) < ∞, then almost surely on{X−1({0}) = ∅},
dimHX−1({0}) = N − k(α) + 1≤j≤k(α)αj− d
αk(α) . (3.24) First, we derive a few technical lemmas. The first is a pointwise estimate for. Lemma 3.8 Under the preceding conditions, for all t∈ (0, 1]N,
(t) 1
1≤j≤N|tj|d/αj
. (3.25)
Proof For any fixed t∈ (0, 1]N we let i∈ {1, . . . , N} satisfy |ti|1/αi = max1≤j≤N
|tj|1/αj. Because(t) ≤
(t) ≤ A |ti|d/αi ≤
A
1≤j≤N|tj|d/αj
, (3.26)
where A and A< ∞ do not depend on t ∈ (0, 1]N.
For the other bound we use (3.22) to deduce the following:
(t) = 1 (2π)d Rd exp ⎛ ⎝− 1≤j≤N |tj|1/αj ξ αj⎞ ⎠ dξ ≥ 1 (2π)d Rd exp ⎛ ⎝− 1≤j≤N |ti|1/αi ξ αj⎞ ⎠ dξ ≥ 1 (2π)d { ξ ≥|ti|−1/αi} exp−N |ti|α1/αi ξ α1 dξ. (3.27)
A change of variables then shows that
(t) ≥ A
1≤j≤N|tj|d/αj
, (3.28)
where A > 0 does not depend on t ∈ (0, 1]N. The lemma follows from (3.26)
and (3.28).
Our second technical lemma follows directly from Lemma 10 of Ayache and Xiao [1] and its proof.
Lemma 3.9 Let a, b, c≥ 0 be fixed. Define for all u, v > 0,
Ja,b,c(u, v) := 1 0 dt (u + ta)b(v + t)c. (3.29) Define for all u, v> 0,
¯Ja,b,c(u, v) := ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u−b+(1/a)v−c, if ab> 1, v−clog 1+ vu−1/a , if ab= 1, 1+ v−ab−c+1, if ab< 1 and ab + c = 1. (3.30)
Then, as long as u≤ va, we have Ja,b,c(u, v) ¯Ja,b,c(u, v).
Proof of Theorem3.7 It can be verified that the additive processX satisfies the symmetry and absolute continuity conditions of Theorem1.1
According to Lemma3.8, we have that for all q≥ 0, [0,1]N (t) t q dt [0,1]N 1 1≤j≤Ntd/αj j |t|q dt = [0,1]N−1 J(d/α1),1,q ⎛ ⎝ 2≤j≤N tjd/αj, 2≤j≤N tj ⎞ ⎠ dt. (3.31) This means that the left-most term converges if and only if the right-most one does. We apply induction on N, several times in conjunction with Lemma3.9, to find that [0,1]N(t) dt = ∞ if and only if k(α) = ∞. Therefore, in accord with
Khoshnevisan and Xiao [22], k(α) < ∞ if and only if P{X−1({0}) = ∅} > 0. This proves the first part of Theorem3.7.
It remains to prove that γ equals to the right-hand side of (3.24). This is proved by appealing, once again, to (3.31), Lemma3.9, and induction [on N]. The details are tedious but otherwise elementary. So we omit them. 4 Proof of Theorem3.2
Our proof of Theorem3.2is technical and long. We will carry it out in several parts.
Throughout the remainder of this section we enlarge the probability space enough that we can introduce symmetric,α-stable Lévy processes {Sj}∞j=1—all taking values in RN—such that S1, S2,. . . are i.i.d., and totally independent of
X1,. . . , XN. We choose and fix an integer M≥ 1, and define S to be the addi-tive stable process S1⊕ · · · ⊕ SM. That is,S(t) = S1(t1) + · · · + SM(tM) for all
t = (t1,. . . , tM) ∈ RM+. The parameters 0 < α < 2 and M ≥ 1 will be deter-mined at the end of the proof of Theorem3.2. For the sake of concreteness we normalize each Sjas follows:
E
eiξ·Sj(u)
= exp−u ξ α forξ ∈ RNand u≥ 0. (4.1) Let G be a nonrandom measurable subset of RN. According to Theorem 4.1.1 of Khoshnevisan [15, p. 423], P{G ∩ S(RM+) = ∅} > 0 if and only if CN−αM(G) > 0. Owing to the independence of S and X we can apply the pre-ceding with G := X−1({0}) ∩ F to find that P{X−1({0}) ∩ F ∩ S(RM
+) = ∅} > 0
iff CN−αM(X−1({0}) ∩ F) > 0 with positive probability. The Frostman theo-rem of potential theory Khoshnevisan [15, Theorem 2.2.1, p. 521], asserts that the Hausdorff dimension of X−1({0}) ∩ F is the critical β ∈ (0, N) such that Cβ(X−1({0}) ∩ F) > 0. Because α and M can be chosen as we like, the computa-tion of dimH(X−1({0}) ∩ F) is thus reduced to deciding when, and exactly when, X−1({0}) ∩ S(RM
+) ∩ F is nonempty with positive probability. The main
positivity of P{X−1({0}) ∩ F ∩ S(RM
+) = ∅}. See Propositions4.7and4.8below.
Once we have this, the formula for the Hausdorff dimension ofX−1({0}) ∩ F follows from the preceding arguments that involve the Frostman theorem.
For all Borel probability measuresµ on RN+, and for every > 0, define J (µ) := 1 (2 )d+N RM + ⎛ ⎜ ⎜ ⎝ RN + 1{|X(s)|≤ , |S(t)−s|≤ }µ(ds) ⎞ ⎟ ⎟ ⎠ e−[t]dt. (4.2) It might help to recall that[t] denotes the 1-norm of t.
4.1 Some moment estimates
For all x∈ Rd, we let Pxdenote the law of x+ X. Similarly, for all y ∈ RN, we define Qyto be the law of y+S. These are actually measures on canonical “path spaces” defined in the usual way; see Khoshnevisan and Xiao [22, Sect. 5.2], for details. Without loss of much generality, we can think of the underlying probability measure P as P0× Q0.
On our enlarged probability space, we view Px × Qy as the joint law of (x + X, y + S). Define Lk to be the Lebesgue measure on Rk for all integers k≥ 1. Then we can construct σ-finite measures,
PLd(•) := Rd Px(•) dx and QLN(•) := RN Qy(•) dy, (4.3)
together with corresponding expectation operators, EP[f ] := Rd f dPLd and EQ[f ] := RN f dQLN. (4.4)
We are particularly interested in theσ -finite measure PLd× QLNand its
corre-sponding expectation operator EP×Q.
It is an elementary computation that for all s∈ RN+and t∈ RM+, the distribu-tion of(X(s), S(t)) under PLd× QLN isLd× LN. In particular,
PLd× QLN
{|X(s)| ≤ , |S(t) − s| ≤ } = (2 )d+N. (4.5) Thus, we are led to the following formula: For all Borel probability measuresµ on RN+and every > 0,
EP×Q[J (µ)] = 1. (4.6)
Proposition 4.1 If N > αM then there exists a finite and positive constant A— depending only on(α, d, N, M)—such that for all Borel probability measures µ on RN+and all > 0, EP×Q (J (µ))2 ≤ A (s− s) µ(ds) µ(ds) max|s− s|N−αM, N−αM. (4.7) Proof Combine Lemma 5.6 of Khoshnevisan and Xiao [22] with (2.11) of the present paper to find that for all s, s∈ RN+and > 0,
PLd
|X(s)| ≤ , |X(s)| ≤ ≤ (64 )dP|X(s) − X(s)| ≤
= 128d 2d
(s− s). (4.8) The last line follows from symmetry; i.e., from the fact thatX(s) − X(s) has the same distribution asX(r), where the jth coordinate of r is |sj− sj|. Thanks to (2.7) we obtain the following:
PLd
|X(s)| ≤ , |X(s)| ≤ ≤ 128d 2d(s − s). (4.9)
We follow the implicit portion of the proof of the preceding to find that for all x, y∈ RN, t, t∈ RM+ and > 0, QLN |S(t) − x| ≤ , |S(t) − y| ≤ = E ⎡ ⎢ ⎣ RN 1{|z+S(t)−x|≤ , |z+S(t)−y|≤ }dz ⎤ ⎥ ⎦ . (4.10)
We change the variables to find that QLN |S(t) − x| ≤ , |S(t) − y| ≤ = |z|≤ P|z + S(t) − S(t) − (y − x)| ≤ dz ≤ (2 )NP|S(t) − S(t) − (y − x)| ≤ 2 . (4.11) Thus, EP×Q (J (µ))2 ≤ 32d (2 )N RN+ RN+ (s − s)F (s − s) µ(ds) µ(ds), (4.12)
where F (x) := RM + RM + P|S(t) − S(t) − x| ≤ 2 e−[t]−[t]dtdt, (4.13)
for all x∈ RNand > 0. Because [t] + [t] = [t − t] + 2[t ∧ t] for all t, t∈ RM+, F (x) = |z−x|≤2 RM + RM + ft−t(z)e−[t−t ]−2[t∧t] dt dtdz, (4.14)
where f is the generalized “transition function,” fu(z) :=
P{S(|u1|, . . . , |uN|) ∈ dz}
dz for u∈ R
Mand z∈ RN. (4.15)
A computation based on symmetry yields RM+ RM+ ft−t(z)e−[t−t]−2[t∧t]dt dt= RM+ fu(z)e−[u]du := υ(z). (4.16)
In order to see the first equality, we write the double integral as a sum of integrals over the 2Mregions:
Dπ = (t, t) ∈ RM + × RM+ : ti≤ tiif i∈ π and ti> tiif i /∈ π , (4.17) whereπ ranges over all subset sets of {1, 2, . . . , M} including the empty set. It can been verified that the integral over Dπequals 2−M RM
+fu(z)e
−[u]du. Hence
(4.16) follows.
The functionυ(z) in (4.16) is the one-potential density ofS [15, pp. 397, 406]. We cite two facts aboutυ:
1. υ(z) > 0 for all z ∈ RN, and is continuous away from 0 ∈ RN. This is a consequence of eq. (3) of (loc. cit., p. 406) and Bochner’s subordination (loc. cit., p. 378).
2. If N > αM, then for all R > 0 there exists a finite constants A > A > 0 such that
A
|z|N−αM ≤ υ(z) ≤ A
|z|N−αM whenever|z| ≤ R. (4.18) Moreover, Acan be chosen to be independent of R> 0. This follows from (1), together used with Proposition 4.1.1 of (loc. cit., p. 420).
It follows from (4.14), (4.16) and (4.18) that for all x∈ RNand > 0, F (x) ≤ A z∈RN: |z−x|≤2 dz |z|N−αM ≤ A(2 )Nmin ) 1 |x|N−αM, 1 N−αM * . (4.19)
Here, Ais positive and finite, and depends only on(N, M, α). The proposition is a ready consequence of this and symmetry; see (4.12). We mention the following variant of Proposition4.1. It is proved by the same argument, without using (4.9).
Proposition 4.2 If N > αM then there exists a finite and positive constant A— depending only on(α, d, N, M)—such that for all Borel probability measures µ on RN+and all > 0, EP×Q (J (µ))2 ≤ A (s− s) max|s− s|N−αM, N−αM µ(ds ) µ(ds). (4.20)
Next we define two multi-parameter filtrations [15, p. 233]. First, defineXj to be the filtration of the Lévy process Xj, augmented in the usual way. Also, defineSkto be the corresponding filtration for Sk. Then, we consider
X (s) := + 1≤j≤N Xj(sj) and S (t) := + 1≤k≤M Sk(tk), (4.21) as s and t range, respectively, over RN+and RM+. It follows from Theorem 2.1.1 of Khoshnevisan [15, p. 233], thatX is an N-parameter commuting filtration. Similarly, S is an M-parameter commuting filtration. Theorem 2.1.1 of the same reference (p. 233) can be invoked, yet again, to help deduce thatF is an (N + M)-parameter commuting filtration, where
F (s ⊗ t) := X (s) ∨ S (t) for s ∈ RN
+and t∈ RM+. (4.22)
We need only the following consequence of commutation; it is known as Cairoli’s strong(2, 2)-inequality [15, Theorem 2.3.2, p. 235]: For all f ∈ L2(P),
E ⎡ ⎣ sup s∈QN+, t∈QM + E,f | F (s ⊗ t)-2 ⎤ ⎦ ≤ 4N+MEf2. (4.23)
(Qk+denotes the collection of all x∈ Rk+such that xjis rational for all 1≤ j ≤ k.) Moreover, and this is significant, the same is true if we replace E by EP×Q; i.e., for all f ∈ L2(P
EP×Q ⎡ ⎣ sup s∈QN+, t∈QM + E,f | F (s ⊗ t)- 2 ⎤ ⎦ ≤ 4N+ME P×Q f2 . (4.24)
A proof is hashed out very briefly in Khoshnevisan and Xiao [22, p. 90]. Proposition 4.3 Suppose R> 0 is fixed. Choose and fix s ∈ [0, R]Nand t∈ RM+. Then, there exists a positive finite constant A= A(α, d, N, M, R) such that for all Borel probability measuresµ that are supported on [0, R]N,
EP×Q[J (µ) | F (s ⊗ t)] ≥ A e−[t] ss (s− s) max|s− s|N−αM, N−αM µ(ds ), (4.25) (PLd× QLN)-almost everywhere on {|X(s)| ≤ /2, |S(t) − s| ≤ /2}. Proof Define χ :=PLd× QLN |X(s)| ≤ , |S(t) − s| ≤ F (s ⊗ t). (4.26) Owing to the Markov random-field property of Khoshnevisan and Xiao [22, Proposition 5.8], whenever s s and t t, we have
χ =PLd× QLN |X(s)| ≤ , |S(t) − s| ≤ X(s), S(t) = PLd |X(s)| ≤ X(s)· QLN |S(t) − s| ≤ S(t). (4.27) We apply Lemma 5.5 of Xiao [22] to each term above to find that(PLd× QLN
)-almost everywhere,
χ = P|X(s) − X(s) + z| ≤ . z=X(s) ×P|S(t) − S(t) − s+ w| ≤ .
w=S(t). (4.28) Because s s and t t, the distributions of X(s)−X(s) and S(t)−S(t) are the same as those ofX(s−s) and S(t−t), respectively. Therefore, (PLd×QLN)-a.e.
on{|X(s)| ≤ /2, |S(t) − s| ≤ /2}, χ ≥ P|X(s) − X(s)| ≤ /2· P|S(t− t) − (s− s)| ≤ /2 ≥ 1 32d+N P (s − s ; t− t), (4.29) where P (s−s; t−t):=P|X(s)−X(s)|≤ · P|S(t−t)−(s−s)|≤ . (4.30)
For the last inequality in (4.29), we have applied (2.12) to both processesX and S. This implies that
EP×Q[J (µ) | F (s ⊗ t)] ≥ 1 32d+N(2 )d+N t∈RM +: tt ⎛ ⎝ ss P (s− s ; t− t) µ(ds) ⎞ ⎠ e−[t] dt, (4.31) (PLd× QLN)-almost everywhere on {|X(s)| ≤ /2, |S(t) − s| ≤ /2}.
Recall from (4.16) the one-potential density υ of S. According to the Fubini–Tonelli theorem, for all x∈ RN,
t∈RM+: tt P|S(t− t) − x| ≤ e−[t]dt ≥ e−[t] RM + P{|S(u) − x| ≤ } e−[u]du= e−[t] z∈RN: |z−x|≤ υ(z) dx. (4.32)
Thanks to (1) and (2) [confer with the paragraph following (4.16)], we can find a finite constant a> 0—not depending on ( , t)—such that as long as |x| ≤ R,
t∈RM +: tt P|S(t− t) − x| ≤ e−[t]dt ≥ ae−[t](2 )Nmin ) 1 |x|N−αM, 1 N−αM * . (4.33) [Compare with (4.19).] The proposition follows from (4.31) and (4.33) after a few lines of direct computation. We can use the earlier results of Khoshnevisan and Xiao [22] to extend Prop-osition4.3further, which will be useful for proving Proposition4.8. In light of the existing proof of Proposition4.3, the said extension does not require any new ideas. Therefore, we will not offer a proof. However, we need to introduce a fair amount of notation in order to state the extension in its proper form.
Any subsetπ of {1, . . . , N} induces a partial order on RN+as follows: For all
s, t∈ RN+,
s≺πt means that
si≤ ti for all i∈ π, and si> ti for all i∈ π.
(4.34)
For everyπ ⊆ {1, . . . , N}, 1 ≤ j ≤ N, and u ≥ 0, define Xπ j (u) := ⎧ ⎨ ⎩ σXj(v) 0≤v≤u if j∈ π, σXj(v) v≥u if j∈ π. (4.35) As is customary,σ (· · · ) denotes the σ-algebra generated by the parenthesized quantities. For allπ ⊆ {1, . . . , N} and t ∈ RN+define
Xπ(t) := +
1≤j≤N
Xjπ(tj). (4.36) It is not hard to check thatXπis an N-parameter filtration in the partial order ≺π. That is,Xπ(s) ⊆ Xπ(t) whenever s ≺π t.
For allπ ⊆ {1, . . . , N}, s ∈ RN+, and t∈ RM+, define
Fπ(s ⊗ t) := Xπ(s) ∨ S (t). (4.37) By Lemma 5.7 in Khoshnevisan and Xiao [22],Fπ is an (N + M)-parameter commuting filtration. It follows that, for all f ∈ L2(P) and π ⊆ {1, . . . , N},
E ⎡ ⎣ sup s∈QN +, t∈QM+ E,f Fπ(s ⊗ t)-2 ⎤ ⎦ ≤ 4N+MEf2. (4.38)
Also, for all f ∈ L2(PLd× QLN) and π ⊆ {1, . . . , N},
EP×Q ⎡ ⎣ sup s∈QN+, t∈QM + E,f Fπ(s ⊗ t)- 2 ⎤ ⎦ ≤ 4N+ME P×Q f2 . (4.39)
Note that when π = {1, . . . , N}, (4.38) and (4.39) are the same as (4.23) and (4.24), respectively. However, the more general forms above have more con-tent, as can be seen by considering other partial ordersπ than {1, . . . , N} [or ∅].
We are ready to present the asserted refinement of Proposition4.3.
Proposition 4.4 Suppose R> 0 is fixed. Choose and fix s ∈ [0, R]Nand t∈ RM+. Then, there exists a positive finite constant A = A(α, d, N, M, R) such that for all Borel probability measuresµ that are supported on [0, R]N, and for allπ ⊆ {1, . . . , N}, EP×Q,J (µ) | Fπ(s ⊗ t) -≥ Ae−[t] sπs (s− s) max|s− s|N−αM, { N−αM µ(ds ), (4.40) (PLd× QLN)-almost everywhere on {|X(s)| ≤ /2, |S(t) − s| ≤ /2}.
4.2 More moment estimates
Consider a compact set B ⊂ (0, ∞)M with nonempty interior. For any Borel probability measureµ on RN+ and a real number > 0, we define a random measure on RN+by J B,µ(C) := 1 (2 )d+N B ⎛ ⎝ C 1{|X(s)|≤ , |S(t)−s|≤ }µ(ds) ⎞ ⎠ dt, (4.41) where C⊆ RN+denotes an arbitrary Borel set.
The following is the analogue of (4.6) under the probability measure P. Lemma 4.5 Choose and fix a compact set B⊂ (0, ∞)Mwith nonempty interior and a real number R> 1. Then, there exists a positive and finite number A such that for all Borel probability measuresµ on T := [R−1, R]N,
lim inf →0+ E JB, µ(T) > A. (4.42)
Proof Thanks to the inversion formula, the density function ofX(s) is continu-ous for every s∈ (0, ∞)N. Also, the density ofS(t) is uniformly continuous for each t∈ (0, ∞)M. By Fatou’s lemma,
lim inf →0+ E JB, µ(T) ≥ B ⎛ ⎝ T (s)ft(s) µ(ds) ⎞ ⎠ dt ≥ LN(B) inf s∈T(s) · infs∈Tinft∈Bft(s). (4.43)
Recall that ft(s) is the density function of S(t). It remains to prove that the two
infima are strictly positive. The first fact follows from the monotonicity bound,
inf s∈T(s) = ) 1 R,. . . , 1 R * = Rd exp ⎛ ⎝−1 R 1≤j≤N j(ξ) ⎞ ⎠ dξ, (4.44) and this is positive. The second fact follows from Bochner’s subordination [15, p. 378], and the fact that the cube T is a positive distance away from the axes
of RN+.
The analogue of Proposition4.1follows next.
Proposition 4.6 Choose and fix R > 1 and a compact set B ⊂ (0, ∞)M with nonempty interior. Let K : RN+ × RN+ → R+ be a measurable function. If N > αM, then there exists a finite and positive constant A—depending only
on(α, d, N, M, B, R)—such that for all Borel probability measures µ on T = [R−1, R]Nand all > 0, E ⎡ ⎣ T T K(s, s) JB,µ (ds)J B,µ(ds) ⎤ ⎦ ≤ A T T (s − s)K(s, s) |s − s|N−αM µ(ds) µ(ds ). (4.45) In particular, we have sup >0E JB, µ(T) 2 ≤ A I(N−αM)(µ). (4.46) Proof We use an argument that is similar to that of Khoshnevisan and Xiao [22, Lemma 3.4]. For all s, s ∈ RN+ define s∧ s to be the N-vector whose jth coordinate is min(sj, sj). We write
Z1:= X(s ∧ s), Z2:= X(s) − X(s ∧ s), (4.47)
and
Z3:= X(s) − X(s ∧ s). (4.48)
Then, it is easy to check that(Z1, Z2, Z3) are independent. Therefrom we find
that P{|X(s)| ≤ , |X(s)| ≤ } is equal to P{|Z1+ Z2| ≤ , |Z1+ Z3| ≤ } = Rd P{|z + Z2| ≤ , |z + Z3| ≤ } ps∧s(z) dz ≤ (s∧ s) Rd P{|z + Z2| ≤ , |z + Z3| ≤ } dz. (4.49)
See also (2.7). After we apply the Fubini–Tonelli theorem and then change variables [w := z + Z2], we find that P{|X(s)| ≤ , |X(s)| ≤ } is at most
(s∧ s) {|w|≤ } P{|w + Z3− Z2| ≤ } dw ≤ (2 )d(s∧ s)P {|Z 3− Z2| ≤ 2 } ≤ 32d(2 )2d(s∧ s)(s− s). (4.50)
The last inequality is a consequence of (2.11), because Z3− Z2= X(s) − X(s)
has the same distribution asX(r), where the jth coordinate of r is rj := |sj− sj|. In other words, for all > 0 and s, s∈ [1/R, R]N,
P|X(s)| ≤ , |X(s)| ≤
(2 )2d ≤ A1(s− s), (4.51)
where A1:= 32d(1/R, . . . , 1/R).
Now consider t, t∈ B and s, s∈ [1/R, R]N. For all > 0, P|S(t) − s| ≤ , |S(t) − s| ≤
= P|W1+ W2− s| ≤ , |W1+ W3− s| ≤
, (4.52) where W1:= S(t∧t), W2:= S(t)−W1, and W3:= S(t)−W1. A little thought
shows that(W1, W2, W3) are independent. Moreover, the density function of W1
is ft∧t. Therefore, P|S(t) − s| ≤ , |S(t) − s| ≤ = RN P|z + W2− s| ≤ , |z + W3− s| ≤ ft∧t(z) dz. (4.53)
Because the density function ft∧tis maximized at the origin,
P|S(t) − s| ≤ , |S(t) − s| ≤ ≤ ft∧t(0) RN P|z + W2− s| ≤ , |z + W3− s| ≤ dz. (4.54)
Next we argue—as we did earlier in order to derive (4.49) and (4.50)—to deduce that P|S(t) − s| ≤ , |S(t) − s| ≤ ≤ ft∧t(0) {|x|≤ } P|x + W2− W3− (s− s)| ≤ dx ≤ (2 )Nf t∧t(0) P |W2− W3− (s− s)| ≤ 2 = (2 )Nf t∧t(0) P |S(t) − S(t) − (s− s)| ≤ 2 = (2 )Nf t∧t(0) {|z−(s−s)|≤2 } ft−t(z) dz. (4.55)
Now, ft∧t(0) = 1 (2π)N RN e−[t∧t]· ξ αdξ = A [t∧ t]M/α, (4.56) where A := (2π)−N
RNexp(− x α) dx is positive and finite. Since t, t∈ B and
B is strictly away from the axes of RM+. Therefore, there exists a finite constant A1—depending only on the distance between B and the axes of RM+—such that
B B P|S(t) − s| ≤ , |S(t) − s| ≤ dtdt ≤ A1(2 )N {|z−(s−s)|≤2 } B B ft−t(z) dtdt dz ≤ A2(2 )N {|z−(s−s)|≤2 } ⎛ ⎝ B ft(z) dt ⎞ ⎠ dz, (4.57) where A2 is another finite constant that depends only on: (a) the distance
between B and the axes of RM+; and (b) the distance between B and infin-ity; i.e., sup{|x| : x ∈ B}. We can find a constant A3—with the same
depen-dencies as A2—such that exp(−[t]) ≥ A−13 for all t ∈ B. This proves that
Bft(z) dt ≤ A3υ(z) for all z ∈ RN. It follows that
B B P|S(t) − s| ≤ , |S(t) − s| ≤ dtdt ≤ A2A3(2 )N {|z−(s−s)|≤2 } dz |z|N−αM. (4.58)
See (4.18). From this and (4.19) we deduce that B B P|S(t) − s| ≤ , |S(t) − s| ≤ dtdt ≤ AA2A3(2 )2Nmin ) 1 |s− s|N−αM, 1 N−αM * . (4.59) This and (4.51) together imply that
E ⎡ ⎣ T T K(s, s) JB,µ (ds)J B,µ(ds) ⎤ ⎦
≤ A T T (s − s)K(s, s) max|s− s|N−αM, N−αM µ(ds) µ(ds ), (4.60)
where Adepends only on(α, d, N, M, R, B). This proposition follows. 4.3 Proof of Theorem3.2
Our proof of Theorem3.2rests on two further results. Both are contributions to the potential theory of random fields, and determine when a given time set F ⊂ RN+is “polar” simultaneously for the range ofS and for the level-sets of X.
Proposition 4.7 Choose and fix a compact set F ⊂ (0, ∞)N. If N > αM and I(N−αM)(µ) is finite for some µ ∈ P(F), then X−1({0}) ∩ F ∩ S(RM+) = ∅ with positive probability.
Proof Since F ⊂ (0, ∞)N is compact, there exists R > 1 such that F ⊆ T = [R−1, R]N. Suppose I(N−αM)
(µ) < ∞ for some Borel probability measure µ on F. Then there exists a continuous function ρ : RN → [1, ∞) such that
lims→s0ρ(s) = ∞ for every s0∈ R
Nwith at least one coordinate equals 0 and (s − s)ρ(s − s)
|s − s|N−αM µ(ds) µ(ds) < ∞. (4.61) See Khoshnevisan and Xiao [22, p. 73], for a construction ofρ.
For a fixed compact set B ⊂ (0, ∞)Mwith nonempty interior, consider the random measures{J B,µ} >0 defined by (4.41). If J B,µ(T) > 0 then certainly X−1(U ) ∩ F ∩ S(B) = ∅, where U := {x ∈ Rd: |x| ≤ }.
It follows from Lemma4.5, Proposition4.6and a second moment argument [13, pp. 204–206], that there exists a subsequence{J B,nµ} which converges weakly to a random measureν such that
P{ν(T) > 0} ≥ a 2 1 a2 > 0, (4.62) where a1:= inf 0< <1E J B,µ(T) > 0 and a2:= sup >0E JB, µ(T) 2 < ∞. (4.63) Moreover, E ρ(s − s) ν(ds) ν(ds) ≤ A (s − s)ρ(s − s) |s − s|N−αM µ(ds) µ(ds). (4.64)
This and (4.61) together imply that almost surely
ν{s ∈ T : sj= a for some j } = 0 for all a ∈ R+. (4.65)
Therefore, we have shown that
inf µ∈P(F)I (N−αM) (µ) < ∞ ⇒ P X−1({0}) ∩ F ∩ S(B) = ∅> 0 ⇒ PX−1({0}) ∩ F ∩ S(RM +) = ∅ > 0. (4.66) Now we need to make use of some earlier results of Khoshnevisan and Xiao [22,20] and Khoshnevisan et al. [23] to remove the closure signs in (4.66). First, since the density function ofS(t) (t ∈ (0, ∞)M) is strictly positive everywhere, a slight modification of the proof of Lemma 4.1 in Khoshnevisan and Xiao [20], Eqs. 4.9–4.11 implies that for every Borel set /F⊆ RN,
/
F∩ S(RM+) = ∅ a.s. ⇐⇒ LN/F S(RM+)= 0 a.s. (4.67) On the other hand, Proposition 5.7 and the proof of Lemma 5.5 in Khoshnevisan et al. [23] show thatLN/F S(R+M)= 0 a.s. is equivalent to CN−αM(/F) = 0, whereCβdenotes theβ-dimensional Bessel–Riesz capacity.
By applying the preceding facts to /F = X−1({0}) ∩ F, we conclude that (4.66) implies thatCN−αM(X−1({0}) ∩ F) > 0 with positive probability. This and Theorem 4.4 of Khoshnevisan and Xiao [20] together yield,
P X−1({0}) ∩ F ∩ S(RM +) = ∅ > 0. (4.68)
We have proved the following:
inf µ∈P(F)I (N−αM) (µ) < ∞ ⇒ P X−1({0}) ∩ F ∩ S(RM +) = ∅ > 0. (4.69) It remains to prove that (4.69) still holds when X−1({0}) is replaced by X−1({0}). This can be done by proving that the random measure ν is supported
onX−1({0})∩F ∩S(RM+). For this purpose, it is sufficient to prove that for every δ > 0, ν(D(δ)) = 0 a.s., where D(δ) := {s ∈ T : |X(s)| > δ}. However, because of (4.65), the proof of the last statement is the same as that in Khoshnevisan and Xiao [22, p. 76]. The proof of Proposition4.7is finished. Proposition 4.8 Choose and fix a compact set F ⊂ (0, ∞)N. If N > αM and I(N−αM)(µ) is infinite for all µ ∈ P(F), then X−1({x}) ∩ F ∩ S(RM+) = ∅ almost surely, for all x∈ Rd.
Remark 4.9 It follows from this proposition and Theorem 4.4 of Khoshnevi-san and Xiao [20] (or Theorem 4.1.1 of Khoshnevisan [15, p. 423]) that, under the above conditions,CN−αMX−1({x}) ∩ F= 0 a.s., for every x ∈ Rd. Hence
dimHX−1({x}) ∩ F≤ N − αM a.s. This is the argument for proving the upper
bound in Theorem3.2.
Proof By compactness, F ⊆ [1/R, R]N for some R> 1 large enough. We fix this R throughout the proof. Also throughout, we assume that for allµ ∈ P(F),
I(N−αM)(µ) = ∞. (4.70)
Let us assume that the collection of all (x, y) ∈ Rd × RN for which the following holds has positive(Ld× LN)-measure:
P
X−1({x}) ∩ F ∩y⊕ S[0, R]M= ∅> 0, (4.71) where y⊕ E := {y + z : z ∈ E} for all singletons y and all sets E. The major portion of this proof is concerned with proving that (4.71) contradicts the earlier assumption (4.70).
Note that (4.71) is equivalent to the statement that for all(x, y) in a set of positive(Ld× LN) measure,
P−x× Qy
X−1({0}) ∩ F ∩ S[0, R]M= ∅> 0. (4.72) For all s∈ [0, R]N, t∈ RM+, and > 0 consider the event,
G( ; s, t) := |X(s)| ≤ 2, |S(t) − s| ≤ 2 . (4.73) According to Proposition4.4, for all s∈ [0, R]N, t∈ RM+, > 0, and µ ∈ P(F),
π⊆{1,...,N} EP×Q,J (µ) | Fπ(s ⊗ t) -≥ Ae−[t] (2 )d P|X(s) − X(s)| ≤ max|s− s|N−αM, N−αM µ(ds ) · 1 G( ;s,t) = Ae−[t] (s− s) max|s− s|N−αM, N−αM µ(ds) · 1G( ;s,t), (4.74)
(PLd× QLN)-almost everywhere. (This uses only the fact that given s, s∈ RN+
we can findπ ⊆ {1, . . . , N} such that sπs.)
Fix > 0. It is possible to see that on the same underlying probability space we can find extended random variables σ( ) ∈ (QN+ ∩ F) ∪ {∞} and
τ( ) ∈ (QM
+ ∩ [0, R]M) ∪ {∞}, where QN+∩ F and QM+ ∩ [0, R]Mdenote
