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LI 2.0.5.1.. iwLnéZlqofl .1 11.6 5 \I} {’53: a. .7 r . . ..r\......t. . v. ,H , . £3... um... .wfi. . , ., . v; 1.5.: a. .. 5...“. .3: This is to certify that the dissertation entitled Geometric Properties of Anisotropic Gaussian Random Fields presented by Dongsheng Wu has been accepted towards fulfillment of the requirements for the PhD. degree in Mathematics \ ages/Kt) M'ajor Professor’s Signature 4/26j2006 Date MSU is an Affirmative Action/Equal Opportunity Institution LIBRARY Michigan State University u...—.-...--. ..d. ---v-.--- PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE JUN 2 0 2007 .0 B {3’ 0 7 M 2] 01 M11 2/05 p:/ClRC/DateDue.indd-p.1 Geometric Properties of Anisotropic Gaussian Random Fields By Dongsheng Wu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2006 ABSTRACT Geometric Properties of Anisotropic Gaussian Random Fields By Dongsheng Wu This dissertation is mainly focused on the geometric properties of two kinds of anisotropic Gaussian random fields: fractional Brownian sheets and the random string processes. Fractional Brownian sheets arise naturally in many areas, including in stochas- tic partial differential equations and in studies of the most visited sites of symmetric Markov processes. We prove that fractional Brownian sheets have the property of sec- torial local non-determinism. By introducing a notion of dimension, called Hausdor'jjr dimension contour, we determine the Hausdorff dimensions of the images of fractional Brownian sheets for arbitrary Bore] sets. Then we provide sufficient conditions for the images to be Salem sets or to have interior points. The random string processes are specified by a stochastic partial differential equa- tion with different initial conditions [Funaki (1983), Mueller and Tribe (2002)]. “’0. determine the Hausdorff and packing dimensions of the level sets and the sets of dou- ble times of the random string processes. We also consider the Hausdorff and packing dimensions of the ranges and graphs of the strings. We conclude this dissertation by describing some of our ongoing projects. To my family. iii ACKNOWLEDGMENTS I wish to express my sincere gratitude to my advisor, Professor Yimin Xiao, for his invaluable guidance. It would have been impossible for me to finish this work without the uncountable number of hours he spent sharing his knowledge and discussing various ideas throughout the study. The best way to express my thanks is, in turn, to treat my students with the same kindness and respect in the future. I am obliged to Professors Chichia. Chin, Huyi Hu, Shlomo Levental and Zhengfang Zhou, in my thesis committee. Thank you for your detailed comments and time. I would also like to thank the following people for all their help and valuable corn- munications: Professor Davar Khoshnevisan of University of Utah, Professor Wenbo Li of University of Delaware, and Professor Renming Song of University of Illinois, among many others. My special gratitude goes to Professor Mark Meerschaert of University of Otago, New Zealand, who kindly provides financial support from National Science Founda- tion grant (DMS-0417869) for my study in the year 2005-2006. I thank all the people who helped me. Last but not least, I would like to give my thanks to my wife Yuzhi and our daughter Huijia (Emily), whose patient love enabled me to complete this work. iv TABLE OF CONTENTS Introduction 1 Definitions and Preliminaries 1.1 Hausdorff measure and Hausdorff dimension .............. 1.2 Packing measure and packing dimension ................ 1.3 Fourier dimension and Salem set ..................... 1.4 Local times ................................ 1.5 The Brownian sheet ............................ 2 Geometric Properties of Fractional Brownian Sheets 2.1 Introduction ................................ 2.2 Sectorial local nondeterminisrn ...................... 2.3 Dimension results for the images ..................... 2.4 Uniform dimension results for the images ................ 2.5 Salem set ................................. 2.6 Interior points ............................... 3 Fractal Properties of the Random String Processes 3.1 Introduction ................................ 3.2 Dimension results of the range and graph ................ 3.3 Existence of the local times and dimension results for level sets . . . . 3.4 Hausdorff and packing dimensions of the sets of double times ..... 4 Concluding Remarks BIBLIOGRAPHY ©OOG3AIA 11 13 13 18 23 38 61 70 91 91 102 114 123 133 136 Introduction The study of the geometric properties of random sets, such as the image, graph, level set and so on, associated with stochastic processes and random fields is an important subject in probability as well as in other mathematical fields. On one hand, the random sets contain very rich information about the fine structure of the sample paths. On the other hand, they provide examples of sets with special properties very much needed in other branches of mathematics such as harmonic analysis, which are very hard to construct otherwise. The random sets are usually ”thin” in a sense that their Lebesgue measure is 0 and they are not smooth or surface-like. The commonly used tools in studying the geometric properties of these sets are Hausdorff dimension and Hausdorff measure. The first result of this kind was due to S. J. Taylor (1953), where he studied the Hausdorff dimension of the image set B ( [0, 1]) of a Brownian motion B (t) (t 6 R” in Rd (d 2 2) and proved that dimH B ([0, 1]) = 2 as. by using the potential theory developed by O. Frostman (1935) and M. Riesz (1938). Since then, many results have been obtained for Lévy processes, that is, processes with independent and stationary increments. The main ingredient is the strong Markov property, see Xiao (2004) for a survey. For non—Markovian processes, there was even difficulty in finding the Hausdorff dimension of the random sets associated to these processes since many classical techniques fail to apply (cf. Adler (1981) and Kahane (1985a)]. To overcome these difficulties in studying of the sample path properties of Gaussian processes, Berman (1973) introduced a concept, called local nondeterminism, and he proved that if a Gaussian process has the property of local nondeterminism, then many deep results for Brownian motion can be extended to the Gaussian process. People have developed different types of local nondeterminism for various Gaussian processes and random fields for different purposes, we refer to Xiao (20051)) for a survey. We mention here that most of the Gaussian random fields studied in the past few decades were isotropic. In general, an anisotropic Gaussian random field, that is a Gaussian random field which has different probabilistic and analytic behaviors along different directions, is not locally nondeterministic. Therefore, people had a need to develop new techniques in studying the geometric properties of the anisotropic random fields. This dissertation focuses on the geometric properties of two kinds of anisotropic Gaussian random fields: fractional Brownian sheets and the random string processes. It is organized as following. In Chapter 1, we collect definitions and properties of fractal measures and fractal dimensions, local times, and the Brownian sheet, which will be used in the subsequent chapters. Chapter 2 studies the geometric properties of fractional Brownian sheets. The objective of this chapter is to characterize the anisotropic nature of an (N, (1)-fractional Brownian sheet B H in terms of its Hurst index H. We prove the following results: (1) B H is sectorially locally nondeterministic. (2) By introducing a notion of “dimension” for Borel measures and sets, which is suitable to describe the anisotropic nature of B H , we determine dimHBH (E) for an arbitrary Borel set E C (0, OO)N. (3) When 3(a) is an (N, (1)-fractional Brownian sheet with index (CY) = 2 (Oz, . . . , Oz) (0 < Oz < 1), we prove the following uniform Hausdorff dimension result for its image sets: If IV S ad, then with probability one, 1 r diInHB<“>(E) = — dimHE for all Borel sets E C (0, 00)“. a If [V > ad then the uniform dimension result fails to hold. In this case we estab- lish uniform dimensional properties for the (N, d) fractional Brownian sheet B (a) with ad < 1, which extends the results of Kaufman (1989) and Khoshncvisan, W711 and Xiao (2005) for the 1—dimensi0nal Brownian motion and the Brownian sheet, respectively. (4) We provide sufficient conditions for the image BH (E) to be 3 Salem set or to have interior points. The results in (2) to (4) describe the geometric and Fourier analytic properties of BH . They extend and improve the previous theorems of Mountford (1989), Khosh- nevisan and Xiao (2004) and Khoshncvisan, Wu and Xiao (2005) for the Brownian sheet. In Chapter 3, we investigate the fractal properties of the random string processes. We continue the research of Mueller and Tribe (2002) by determining the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string processes. we also consider the Hausdorff and packing dimensions of the range and graph of the strings. Finally, we conclude this dissertation by describing some of our ongoing projects in Chapter 4. CHAPTER 1 Definitions and Preliminaries This chapter contains basic definitions and facts of Hausdorff dimension, packing dimension, Fourier dimension, local times and the Brownian sheet, which will be used in subsequent chapters. Throughout this dissertation. the underlying parameter space 18 R or IR+ — [0, 00)? A typical parameter, t 6 1R5 is written as t = (t1, . . . ,tf), or as (C), if t1: ---=t{ = C. For any S,t E RI such that S]- < tj (j = 1, . . . ,6), we define the cl “ d int rv l ( r rectan 1e) '1: [S t] - [It) [8' t) We alwa " write m. f ' , 058 e a O ,.. g, (S , —— j=1 )3 J . 1 (3'5 ., f 01 Lebesgue’s measure on Re, no matter the value of the integer 5. We use (o, ) and I - I to denote the ordinary scalar product and the Euclidean norm in Re respectively, no matter the value of the integer 2, We will use K to denote an unspecified positive and finite constant which may not be the same in each occurrence. More specific constants in Chapter i Section j are numbered as Ki,j,1= K2333, . . . , and so on. 1.1 Hausdorff measure and Hausdorff dimension Let cI) be the class of functions (15 : (0, d) —> (O, 1), which are right continuous, monotone increasing with ¢(0+) '2 0 and satisfy the following ”doubling” property: 4 There exists a finite constant [(1,171 > 0 for which d)(2.s) 05(8) (5 E;I(LL1, IDF()<:S‘< 5. (1.1) For d) E (I), the (fi-Hausdorff measure of E C_: RN is defined by (X; gb — m(E) = ~1€i_1+16inf { 295(25): E _C_ LJIO(.rJ-,7'J-), 7‘j < 5}, (1.2) J 1: where 0(1‘, 7‘) denotes the open ball of radius 7" centered at :12. Q5 — m is a metric . v . outer measure and every Borel set In IR] is d) — Tn measurable (cf. Rogers (1970)]. The Hausdorff dimension of E is defined by dimHE = inf {a > 0: s0 — m(E) = 0} = sup {(1 > 0: s“ — m(E) : 00}. If 0 < S” — m(E) < 00, then E is called an a-set. If there exists (£5 E (I) with 0 < 45 — m(E) < 00, then g5 is called an exact Hausdorff measure function for E. Here are some basic properties of Hausdorff dimension: (1). Monotonicity: if E g F, then diInHE S dimHF. (2). Hausdorff dimension is 0-stable dimH U E, = sup dimH En. ":1 n2] we refer to Falconer (1990) or Mattila (1995) for more details on Hausdorff mea- sure and Hausdorff dimension. CJ'I The Hausdorff dimension of a Borel measure it on IRA (or lower Hausdorff dimen- sion as it is sometimes called) is defined by dimHu = inf {dimHF2 p(F) > 0 and F g RN is a Borel set}. (1.3) Hu and Taylor (1994) proved the following characterization of dilnnyz if ,LL is a . N fimte Borel measure on IR then dime = sup {7 2 0 : lim sup r'7p(O—(t, r)) = 0 r——+0t (1.4) for u-ae. t 6 RN} I / where 5(t, 7') = {S E RN 1 I8 - i| S 7'.) It is easy to see that. for every Borel , I set E C IR”, we have dimHE = sup {dimH/rz ,u E M:(E)}, (1.5) where MEIE) denotes the family of finite Borel measures on E with compact. support in E. 1.2 Packing measure and packing dimension Packing measure was introduced by Taylor and Tricot (1985) as a dual concept to Hausdorff measure. Like Hausdorff measure and Hausdorff dimension, it is a very useful tool in analyzing fractal sets and in studying the sample path properties of stochastic processes and random fields. For (I) E (I), define the set function (D -- P(E) on RN by (,7) — P(E) = ling sup { Z q‘)(2rj) 25(rzfj, 7‘,) are disjoint, 3' (1.6) 1'} E E, ’I'J' < 5}. (,5 — P is not an outer measure because it fails to be countably subadditive. However, . . . , N it IS a. premeasure, and therefore one can get. a metric outer measure (,7) — p on R by defining o—p(E) = inf Zo—P(En): E g Us), . (1.7) (f) — p(E) is called the (1)-packing measure of E. Every Borel set. in IR“ is <9 - p measurable. The packing dimension of E is defined by dimPE = inf {a > 0: s" — p(E) : 0} =sup{a>0: s“—p(E)=oo}. The packing dimension of [1, denoted by (limp [1, can be defined by replacing dimH F in (1.3) by dimP F. There is a similar identity for dimP/L as (1.4) for dimHu; see Hu and Taylor (1994, Corollary 4.2) and Falconer and Howroyd (1997) for further information. For any 8 > 0 and any bounded set F g Rd, let N (F, e) = smallest number of balls of radius 5 needed to cover F. Then the upper box-counting dimension of F is defined as —.—— . IO N F, 5 dunBF = hm sup g ( ' ). (1.8) 5.4) — log 5 The packing dimension of F can also be defined by CC dimPF = inf { sup dimBFn : F Q U Fn}. (1.9) n. 1221 Further information on packing dimension can be found in Tricot (1982), Falconer (1990) and Mattila (199.5). 1.3 Fourier dimension and Salem set Let us recall from Kahane (1985a, b) the definitions of Fourier dimension and Salem set. Given a. constant ,6 _>_ O, a Borel set F C Rd is said to be an rMfl-set if there exists a probability measure 1/ on F such that. momma—3) as t—wo. (1.10) where I) is the Fourier transform of V. Note that if [3 > 61/2, then (1.10) implies that f; E L2(Rd) and, consequently, F has positive d-dimensional Lebesgue measure. For any Borel set F C Rd, its Fourier dimension dimFF is defined by dimFF = sup {7 Z 0: F is an Mpg-set }. (1.11) It follows from Frostman’s theorem [cf Kahane (1985a, Chapter 10)] and the fol- lowing formula for the ’7-energy of V: Liv):(,|v(£>|21t|‘“’d§ (1.12) \3 [see Kahane (1985a, Ch. 10)] that dimFF S dimHF for all Borel sets F C Rd. we say that a Borel set F C Rd is a Salem set if dimFF = dimH F. Such sets are of importance in studying the problem of uniqueness and multiplicity for trigonometric series: see Zygmund (1959. Chapter 9) and Kahane and Salem (1994) for further information. 1 .4 Local times Let X (t) be a Borel vector field on RN taking values in Rd. For any Borel probability measure )1 on E, the image measure of [,L X under X is defined as the following pX(o):/r{t€E: X(t)€o}. If if X is absolutely continuous with respect to the Lebsgue measure md in Rd, then X is said to have a local time on E. The local time I ”((13) is defined to be the Radon— Nikodym derivative d/Lx/dm,d(1‘) and it satisfies the following occupation density formula: for all Borel measurable functions f : Rd -+ IR+, / f(X(8)) Mds) = “some. (1.13) E Rd If we choose )1 as the Lebesgue. measure 777W in RN, then the corresponding image measure is called the occupation measure of X on E, denoted by #3 Similarly, if [15 << 771,1. then X is said to have local times on E, defined by dug/d'rl'l(1(;l') and denoted by l (I, E). We call :1: the space writable. and E the time variable. Sometimes, we write l(.’1?, t) in place of l(.’E, [0, t]). It is clear that if X has local times on E, then for every Borel set I g E, l (:13, I) also exists. S . . fi . . 1) E _. N . . h. T1 .1 . . ., uppose we x a rectang c — Hj=1[a], a, + J]. ien, Vt renexer we can . . . N choose a versron of the local tnne, still denoted by l (.13, H Ia], a,- + t 1]), such i=1 . . . , . d N .‘ . that it IS a continuous functlon of (:13, t1, . . . , tN) E R X sz-llov hj]. X is said to have jointly continuous local times on E. When a. local time is jointly continuous, l (.13, O) can be extended to a finite Borel measure supported on the level set Xg‘oc) = {t e E: X(t) 21:}, see Adler (1981) for details. In other words, local times often act as a natural measure on the level set of X. Therefore, they are useful in studying the fractal properties of level sets and inverse images of the vector field X. We refer to Berman (1972), Adler (1977), Ehm (1981), Rosen (1984), Monrad and Pitt (1987) and Xiao (1997) in this regard. For more information on local times of random, as well as non—random functions, we refer to Geman and Horowitz (1980), Geman et al (1984), Xiao (1997) and Dozzi (2003). 10 1.5 The Brownian sheet Let (S, S, It) be a O—finite measure space and let. C={AES:,u(A) 1R is called a Gaussian noise with control measure ,u if (i). VA 6 C, W(A) ~ N (0,,u(A)). (ii). VA, B E C, A m B = 0, we have I'V(A U B) = I'V(A) + IV(B). (iii). VA, B E C, A f) B =2 (0, I/I/(A), I'V(B) are independent. For a given U-finite measure space (S, 5, ,LL), there exists a Gaussian noise on C with ,u as its control measure. Let L‘Zm) ——— (f: S —+ R [S |f(t)|2u(dt) < oo] We can define the stochastic integral for all f E L201) with respect to IV by 1(f) = / f(t)W(dt)- Furthermore, it’s easy to prove the following isotropic property of the stochastic integral: 1E[I(f)1(g)] = / f(t)g(t)u(dt), fag 6 Ln). (1.14) 11 When (87 S? I”) 2 (RN: BURN): mA’): W0 call W a Gaussian. white noise. Vile define a real valued Gaussian random field B0 = {B()(l), l E Rf} by 30(1) = W ([0, t]), t 6 eff. Then Bo is called the (N , 1)-Brownian sheet. It can be derived that N IE {30(1)BO(5)] = Hmin(tj, 5,), t, s e aid i=1 Let B1, . . . , Bd be (l independent copies of B0. Then the (N, (l) Brownian sheet B = {B(t),t E Rf}, defined by B”(t) -_— (Bf’(t),...,Bf(t)), te RN, (1.15) is a n'iultiparameter extension of Brownian motion and is one of the most important. Gaussian random fields. We refer to Khoshnevisan (2002) for details 011 the Brownian sheet and other multiparameter processes. 12 CHAPTER 2 Geometric Properties of Fractional Brownian Sheets 2.1 Introduction For a given vector H = (H1, . . . , HN) (0 < Hj < 1 forj = 1, . . . , 1V), an (N, 1)-fra.ctional Brownian sheet B51 = {B61(t), t E RN} with Hurst index H is a real-valued, centered, Gaussian random field with covariance function given by- IE[Bl'(S)Bth(t)] N 1 (2.1) = H§(15112Hj + ltjlmj — 153' "M2110, but 6 RN. j=1 It follows from (2.1) that Ba, (t) = 0 as. for every t E RN with at least. one zero coordinate. When H1 = - - - = HN = 01 E (0,1), we will write H = (a). It is known that B51 has the following two important stochastic integral repre- sentations: 13 0 Moving average representation B”(t) :~;1/:/:g(.ts)W(ds), (2.2) where IV = {W(S), 8 E RA} is a standard meal—valued Brmvnian sheet and = fi [((t, — .sJ-)+)Hj—1/2 — ((‘Sjl-tlHj—l/Ql i=1 with 8+ 2: max{s, 0}, and where K H is the normalizing constant. given by //[Hi so that lE[(B(§1(t))2] 2‘ “17:1 ltjlgHj for allt E RN. Here (1) = (1,1,...,1)eaN. o Harmonizable representation 351:;K /N an )wun) (2.3) A where W is the Fourier transform of the white noise in RN and N . W(A) : H exp(ztj)\j) -- 1, - .1. j=1 1A.? 1H3 +2 and where KH is the normalizing constant so that IE [(B6{(t))2] = 119;, It, |2Hj for all t E RN. Let B {1 , . . . , B (51 be d independent copies of B61. Then the (N, d)-fractional 14 Brownian sheet with Hurst index H 2 (H1, . . . , Hat) is the Gaussian random field B” '2 {BH(t) : t E RN} with values in Rd defined by 8%) = (Bi’tt). . . . . 3.7m). t 6 RV. (2.4) Note that if lv 2 1, then B H is a fractional Brownian motion in Rd with Hurst index H1 E (0, 1); if N > 1 and H = (1/2), then B” is the (N, (1)-Brownian sheet, denoted by B. Hence B H can be regarded as a natural generalization of one parameter fractional Brownian motion in Rd to (N, (1) Gaussian random fields. as well as a generalizatitm of the Brownian sheet. Another well known generalization is the multiparameter fractional Brownian motion X = {X (l), t E RN}, which is a. centered (N, (1)-Gaussian random field with covariance function 1 _ lE[Xi(s)XJ-(t)] = 56).,(|s|2111+|t|2111—|s —t|2H1), Vs,t E RN, (2.5) where 0 < H1 < 1 is a constant and 675 = 1 ifi = j and 0 ifi 75 j. The main difference between B H and the multiparameter fractional Brownian motion is that the latter is isotropic and self-similar with stationary increments while B H is anisotropic and does not have stationary increments. It follows from (2.1) that B H is opcrator—sclf—similar in the sense that for all constants K > 0, {BH(KAt), t e RN} .53: {KN BH(t), te RN}, (2.6) where A = (aij) is the N X N diagonal matrix with (1,-2- = l/H,‘ for all 1 S i S N 15 and 0.)]- : 0 ifi 74 j and Y g Z means that the two processes have the same finite dimensional distrilmtions. The anisotropic property and the operator—sclf-similarity (2.6) of BH make it a possible model for bone structure [Bonami and Estrade (2003)] and aquifer structure in hydrology [Benson et al. (2004)]. Many authors have studied various properties of fractional Brownian sheets. For example, Dunker (2000), Mason and Shi (2001), Belinski and Linde (2002), Kiihn and Linde (2002) studied the small ball probabilities of an (N, 1)-fractional Brownian sheet B61. Mason and Shi (2001) also computed the Hausdorff dimension of some exceptional sets related to the oscillation of the sample paths of B61. Ayache and Taqqu (2003) derived an optimal wavelet series expansion for the fractional Brownian sheet B61; see also Kiihn and Linde (2002), Dzhaparidze and van Zanten (2005) for other Optimal series expansions for B6! . Xiao and Zhang (2002) studied the existence of local times of an (N, (1)-fractional Brownian sheet B H and proved a sufficient condition for the joint continuity of the local times. Kamont. (1996) and Ayache ( 2002) studied the box dimension and the Hausdorff dimension of the graph set of an (N, 1)-fractional Brownian sheet B51. Recently, Ayache and Xiao (2005) have investigated the uniform and local as- ymptotic properties of B H by using wavelet methods, and determined the Hausdorff and packing dimensions of the range BH ([0, 1]N), the graph GI‘BH ([0, 1]N) and the level set L1, = {t E (0,OO)N I BH(t) = 2?}. Their results suggest that, due to the anisotropy of BH in t, the sample paths of BH are more irregular than those of the Brownian sheet or (N, (1)-fractional Brownian motion. Hence it is of interest to further describe the anisotropic properties of B H in terms of the Hurst index H 2: (H1, . . . , HN). The main objective of this chapter is to investigate the geometric and Fourier analytic properties of the image B H (E) of a Borel set E C (0, OO)N. In Section 2.2, we prove that fractional Brownian sheets satisfy a type of “sectorial local nonde- 16 terminism” [see Theorem 2.1], extending a result of Khoshnevisan and Xiao (2004) for the Brownian sheet. This property plays an important role in this chapter and it will be useful in studying other problems such as local times of fractional Brownian sheets. In Section 2.3, we determine the Hausdorff dimension of the image BH (E) Unlike the well-known cases of fractional Brownian motion or the Brownian sheet, dimH BH (E) can not be determined by dimHE alone due to the anisotropy of B H [see Example 2.1]. To solve the problem, we introduce a new concept of “dimen- sion” [we call it Hausdorfl dimension contour] for finite Borel measures and Borel sets; and we prove that dimH B H (E) can be represented in terms of the Hausdorff dimensional contour of E and the Hurst index H. We believe that the concept of Hausdorff dimension contour is of independent interest because it carries more information about the geometric properties of Borel measures and sets than Hausdorff dimension does; and it is the appropriate notion for studying the image BH (E) and the local times of the fractional Brownian sheet BH on E, as shown by the results in Sections 2.3, 2.5 and 2.6. Therefore it would be interesting to further study Hausdorff dimension contours and to develop ways to determine them for large classes of fractal sets. In Section 2.4, we study the uniform dimension results for the images of the (N, (1)-fractional Brownian sheet B (a) with index H 2 (Oz), we prove the following uniform Hausdorff dimension result for its images: if N S ad, then with probability one, 1 dimHBi”>(E) = —dimHE for all Borel sets E C (0, oo)N. (2.7) a This extends the results of lVIountford (1989) and Khoshnevisan, Wu and Xiao (2005) 17 for the Brownian sheet. Our proof is based on the sectorial local nondeterminism of B (a) and is similar to that of Khoshnevisan, Wu and Xiao (2005). When N > ad, the uniform dimensional results do not hold anymore. We derive weaker uniform dimensional properties for the (N, d)-fractional Brownian sheet B (a) With ad < 1. Our results are extensions of the results of Kaufman (1989) and Khoshnevisan, Wu and Xiao (2005) for the 1-dimensional Brownian motion and Brownian sheet, respectively. Let u be a probability measure carried by E and let 1/ = #311 be the image measure of p under the mapping t H B H (t). In Section 2.5, we study the asymp- totic properties of the Fourier transform 1?“) of V as E —> 00. In particular, we show that the image BH (E) is a Salem set whenever 8(H, E) S d, see Section 2.3 for the definition of 8(H, E). These results extend those of Kahane (1985a, b) and Khoshnevisan, Wu and Xiao (2005) for fractional Brownian motion and the Brownian sheet, respectively. In Section 2.6, we prove a sufficient condition for BH (E) to have interior points. This problem is closely related to the existence of a continuous local time of B H on E [of Pitt (1978), Geman and Horowitz (1980), Kahane (1985a, b)[. Our Theorem 2.8 extends and improves the previous result of Khoshnevisan and Xiao (2004) for the Brownian sheet. 2.2 Sectorial local nondeterminism Recently Khoshnevisan and Xiao (2004) have proved that the Brownian sheet sat- isfies a type of “sectorial” local nondeterminism and applied this property to study geometric properties of the Brownian sheet; see also Khoshnevisan, Wu and Xiao (2005). In the following we prove that the (N, 1)-fracti0nal Brownian sheet 361 = {851 (t),t 6 RN } satisfies a similar type of sectorial local nondeterminism. This 18 property will play an important role in this chapter, as well as in studying the 10— cal times, self-intersections and other sample path properties of fractional Brownian sheets. Theorem 2.1 (Sectorial LND) For any fixed positive number 8 6 (0,1), there exists a positive constant [(221, depending on. E, H and 1V only, such that for all positive integers n 2 I, and all u, t1, . . . , t" E [5, OC)N, we have N' Var (351w) I 851(3),” .,B(§1(t")) 2 K224 2 min [uj — tflQHj, OSkSn 9:1 (2.8) where t0 = 0. Remark 2.1 The method we use for proving Theorem 2.1 is different from that in Khoshnevisan and Xiao (2004) because fractional Brownian sheets have no in- dependent increments but the Brownian sheet does. Our proof is mainly based on the harmonizable representation (2.3) of 3:? and is reminiscent to Kahane (1985a, Chapter 18) or Xiao (2005a). Proof: Let g E {1, . . . , N} be fixed and denote 7‘5 E minogkg-u [W — tfl. Firstly, we prove that there exists a positive constant IQ such that the following inequality holds: Var (135;1 (u) | 35%), . . . , 351m) 2 KNEW. (2.9) Summing over 6 from 1 to N in (2.9), we get (2.8). In order to prove (2.9), we work in the Hilbert space setting and write the con- ditional variance in (2.9) as the square of the L2(lfD)-distance of 361(u) from the 19 subspace generated by {35(0), . . . , 3610"». Hence it suffices to show that for all a), E R, H, 2 113(5),? (11) — 2,1,3]! ((1‘)) 2 Kflf’”. (2.10) 1:21 It follows from the harmonizable representation (2.3) of 361 that Tl IE(BJ’ (u) —- 2; Mates) 2 = K‘2/ H RN 2 K’2 H RN 2 k-1 (to) — Z airao) an k—l N " (2.11) (exp(iuj)\j) — 1) j=1 n N —— a), (exp(it§/\j) — 1) k=1 2 fohldA, i=1 where N fHO‘) = HIM—23H- j=1 Now for every j = 1, . . . , N, we choose a. bump function dj() 6 COO(R) with values in [0,1] such that (SJ-(0) = 1 and strictly decreasing in [ - [ near 0 [e.g., on (—E, 8)] and vanishes outside the open interval (—1, 1). Let 0;- be the Fourier transform of (53'. It is known that @(Aj) is also in C00 (R) and decays rapidly as )13- —> 00. Also, the Fourier inversion formula gives (SJ-(s1) =<2vrr1 / exp(_.s,,\,)zs;(1,)d1,. (2.12) 20 Let 5;,(85 71) = 7‘E16g'(S[/1'[), then by (2.12) and a change of variables, we have. (l{(8[;7‘() = (27T)_1/OXp(-—iS(/\()0\((7‘[)\1)(lAp (2.13) IR By the definition of 77;, we have 5p(Up; Ty) = 0 and (Mug — if; 7‘15) 2 0 for all k = 1, . . . ,n. Hence it follows from (2.12) and (2.13) that E/fiéN (fi(exp(iuj)\jl _ 1) _ ’7' a), fi(exp(it§)\j) — 1)) j=l k=1 j=1 N N A A X 1—1exp<—iu.j)\j)(l—Iéijjl)(5”(7‘{.’)‘l’)dA = (27TlN[ 17105110) — 5j(uj))) (5((0; rt) - (Sift-Ir; 70)] (214) —' (2703] [1:1 (W(EII (510% — if) "' 5100») X (61>(Uf — tf; Tr) — 6/?(Up; 7‘p)):[ 21 On the other hand, by the Cauchy-Schwarz inequality, (2.11) and (2.14), we have Is/,,N n N (exp(luJ-Aj)—1)akH(exp(itj-/\—1) kzl j=1 2 fohld)‘ 2.512 1:1 1 2 x 6 M) r /\ dA (RA/fie) [(fi ( My ‘ “l n 2 : It}, 1E(B,§’(u) — ZakBH(tk)) N 2 —2Hg— —2 xr A: dA t N]()\fH )\).7].;[1[A 6]( J)[ 2 :: Kragg 7’;2H€_21E(Bgl(tl) — nakB(]{(tk)> . (2.15) Combining (2.14) and (2.15) yields (2.9). This finishes the proof of the theorem. . . . r r Given Gausslan random variables 21, . . . , Z", we denote by COV(51, . . . , J“) their covariance matrix. The following formula relates the determinant of COV(Zl, . . . , Zn) with conditional variances: n det Cov(Z1, . . . , Z") = Var(Z1) HVar(Zk[Z1, . . . , Zk_1). (2.16) k=2 Hence the inequality (2.8) can be used to estimate the joint distribution of the Gaussian random variables 361(t1), . . . , 36.1“”) It is for this reason why the sectorial local nondeterminism is essential in this chapter and in studying local times of fractional Brownian sheets. The following simple result will be needed in Section 2.6. 22 Lemma 2.1 Let n 2 1 be a fired integer. Then for all t1, . . . , l” E [5, OO)N such that t], . . . , t?” are all distinct for some j E {1, . . . , N }, the Gaussian random variables Baal), . . . , 3610f") are linearly independent. Proof: Let t1, . . . , t" E [5, OO)N be given as above. Then it follows from Theorem 2.1 and (2.16) that detCov(B§I(t1), . . . , 8510”» > 0. This proves the lemma. 2.3 Dimension results for the images In this section. we study the Hausdorff dimension of the image set. BH (E) of an arbitrary Borel set E C (0, OO)N. When E = [0,1]N or any Borel set with positive Lebesgue measure, this problem has been solved by Ayache and Xiao (2005). However, when E C (0, OO)N is a fractal set, the Hausdorff dimension of B H (E) can not be determined by dimHE and H alone, as shown by Example 2.1 below. This is in contrast with the cases of fractional Brownian motion or the Brownian sheet. To solve this problem, we will introduce a new notion of dimension, namely, H ausdorfir dimension contour, for finite. Borel measures and Borel sets. It turns out the Hausdorff dimension contour of E is the natural object in determining the Hausdorff dimension and other geometric properties of BH (E) for all Borel sets E. We start with the following Proposition 2.2 which determines dimH B H (E) when E belongs to a special class of Borel sets in Rf. Proposition 2.2 Let BH = {BH(t),t 6 RN} be an. (N, (1)-fractional Brownian sheet with index H 2 (H1, . . . , HN). Assume that Ej (j = 1,... ,N) are Borel sets in (0,00) satisfying the following property: 3] {j1, . . . ,jN_1} C {1, . . . , N} 23 such that diInHEj]c = dimPEjk fork = 1, . . . ,lV — 1. Let E = E1 X - - - X EN C (0, OO)N, then we have 7V ‘ d' E,- dimHBH(E):min{d; 21—214}, as. (2.17) i=1 j For proving Proposition 2.2, we need the next two lemmas that are due to Ayache and Xiao (2005). Lemma 2.2 Let B” = {BH(l),t E RN} be an (N, (1)-fractional Brownian sheet with index H 1: (H1, . . . , HN). For all T > 0, there exist a random variable A1 = A1(w) > 0 of finite moments of any order and an event (if of probability 1 such that for every w E Q}. sup [BH(S’W) — BHinH < A1(w). (2.18) N ‘ - — _ s,t€[0,T]N 23:1 [8)" — tleJ\/10§ (3 + [52' — t_,-| 1) Lemma 2.3 Let Béq 2 {350), t E RN} be an (N, 1)-fractional Brownian sheet with index H 2 (H1, . . . , HN), then for any 0 < 5 < T, there exist positive and finite constants K2331 and K232 such that for all S,t E [5, T]N, N N K2,“ 2 lsj — 11,1250 3 1E[(Bg’(s) — B§(t))2] _<_ K233 2 Is,- — tjl2Hj- j=1 j=1 (2.19) Proof of Proposition 2.2: As usual, the proof of (2.17) is divided into proving the upper and lower bounds separately. We will show that the upper bound in (2.17) follows from the modulus of continuity of the fractional Brownian sheet and a covering 24 argument, and the lower bound follows from Forstman’s Theorem [see e.g., Kahane (1985a, Chapter 10)] and Lemma 2.3. For simplicity of notation, we will only consider the case N = 2 and dim“ E1 = dimP E1. The proof for the general case is similar. Upper bound: By the U-stability of (11an and (1.9). it is sufficient to prove that for every Borel set E 2 E1 X E2, (2.20) (T— E d“ E. dimHBH(E) _<_ min {d; —l—I—n—B——l + Eli—i}, as. H1 H2 For any V71 > CliIIlBEl, ",2 > diIIlHEg, we choose and fix ”)5 E (dimH E2, 72). Then there exists an 7‘0 > 0 such that, for all 7‘ S 7‘0, E1 can be covered by 1V(E1,7‘) S 7‘—71 many small intervals of length 7'; and there exists a covering [Um n _>_ 1} of Egsuch that 7‘", I: [U,,[ S 7'0 and 90 I Z 79,2 g 1. (2.21) n=1 For every n 2 1 and any constant (5 E (0,1) small enough, let {Vnm I 1 S m S N") be .Nn :2 N(El, 7‘£H2_0)/(H1—b)) intervals of length T£H2—6)/(H1_6) which cover E1. Then the rectangles {me x U" I n 2 1,1 S m S N,,} form a covering of E1 X E2, that is, f 00 An E c U U V... x U... n=1 m=1 and thus B”(E) c U B”(V,.,,, x U,,). (2.22) It follows from Lemma 2.2 that, almost surely, B” (an >< U”) can be cov- ered by a ball of radius K 73132—6. By this and (2.22), we have covered B H (E) as. by balls of radius K7‘£[2_6 (n = 1,2, . . .). Moreover, recalling that _. H.—6 H—(S NnSTn’H 2 )/( 1 ),we have 00 Nn - A n=1m=1 00 . - - (HQ-5l(‘vl/Hl+’)2/H9) 3 Z r,;11<”2-"">/<”1—°) . r. ‘ ~ (2.23) .3 [- H H .n . M8 ”-3 8 ll H Now we choose (5 > 0 small enough so that H1—6 H1 H2 2' 72 — 71(H2 — (Slf Then (2.21) and (2.23) imply that 00 N "‘ ‘71/H1+’72/H2 Z Z (r5276) 3 1. (2.24) n=l m=1 It should be clear that (2.20) follows from (2.24). Lower bound: Choose 71,72 such that 0 < ’71 < dimHEl, 0 < 72 < 26 dimH E2 and— 11%7-+ < d, then there exist probability measures 01 on E1 and 02 on E2 such that (d (dt) (d (dt / / 01( 81) 01(11)<1 00/ / U2( 82) 02(722) < 00. (225) E1 E1 [51 -t1| E2 E2 [32"t2l Let 0 = 01 X 02. Then 0 is a probalnlity measure on E. By Lemma 2.3 and the fact that “/1/H1 + ’72 / H 2 < d, we have E/13./13|BWU:::(:[)(I:lilHi+v2/112 gK/f 801(dsllal(dt1)02(d32)02(dt2) [51 H «H ' —t1|2H1 +|82_ t2|2H2) (71/ 1+72/ 2)/2 (2.26) S K/1531/E101(d81)01(dt1) / / 02(d82)02(dt2) E2 52 (I81 —1.1”1 +13. —1.|H2)“'1/”1+12/H2 By an inequality for the weighted arithmetic mean and geometric mean with 131 = H271/(H271+ H172) and 52 = 1 — .31 = H1e/2/(H2'n + f11’72):we have [81-t1[H1+[82 — tng2 Z 181]51_t1[H1+ 32 I82 —152|H2 (2-27) 2'81_t1|H1131[82 _ t2|H2132 . Therefore, the last denominator in (2.26) can be bounded from below by 27 [81 — “[71 [82 — tgllf‘)‘. It follows from this and (2.25), (2.26) that IE o(ds)o(dt) < 00 E E [811(5) _ BH(t)[71/H1+72/H2 This yields the lower bound in (2.17), and the proof of Proposition 2.2 is completed. The following simple example illustrates that, in general, dimHE alone is not enough to determine the Hausdorff dimension of BH (E). Example 2.1 Let BH 2 {BH(t),t E R2} be a (2, (1)-fractional Brownian. sheet ”with index H = (H1,H2) and H1 < H2. Let E 2 E1 X E2, F = E2 X E1, where E1 C (0,00) satisfies dilnH E1 = dimPEl and E2 C (0, 00) is arbitrary. It is well known that under the condition ClimH E1 = dimP E1, dimHE = dimHE1 + CllIIlHE-z = diInHF, [cf Falconer (1990, [2.94)]. However, by Proposition 2.2 we have d' E d' E dimHBH(E) = min {(1, Eff—i + Jim—Hi}. H1 H2 H1 H2 We see that dimHBH(E) 75 dimH BH(F) unless dimHE1 = dlmHE2. d' E d' E dimHBH(F)=min{d; —l—IE-H——2-+—un—H-—l}. Example 2.1 shows that in order to determine dimHBH (E), we need to have more information about the geometry of E than its Hausdorff dimension. We have found it is more convenient to work with Borel measures carried by E. 28 Recall that dlmHfl only describes the local behavior of ,u. in an isotropic way [(‘f. (1.4)] and is not quite informative if p is highly anisotropic as what we are dealing with in this chapter. To overcome this difficulty, we introduce the following new notion of “dimension" for E C (0, OO)V that is natural for studying BH(E). Definition 2.1 Given a Borel probability measure a on RN, we define the set A“ g Rf by “here R(t 7‘) Z H3321ltj— 7‘1/flj, tj + Tl/Hj] (l'll(iH—1:(Hil',. . . , 3%). Some basic properties of A“ are summarized in the following lemma. For simplic- ity of notation, we assume H] = min{Hj I l S j S N}. Lemma 2.4 A” has the following properties: (i) The set A“ is bounded: N —l N A,, g (A: (A1,...,)\N) 6 1a. : (A,H )5 Fl (2.29) 1 (ii) V5 E (0,1JN, A E A”. B O /\ E A!“ where ,6 O A = ()31A1,. . ”yap/AN) is the Hadamard product offl and A. (iii) Auis conver, i.e. VA, 77 E A” and 0 < b < 1, b)‘ + (l - b)77 E A”. (iv) For every a E (O, OO)N, sup/WA” (a, A) is achieved on the boundary of A”. 29 Because of (iv) and its importance in this chapter, we call the boundary of A“, denoted by 8A”, the Hausdorfir dimension contour of p. Proof: Suppose A = (A1, . . . , AN) 6 Al). Then it (RU, 7)) lim sup _1 MAJ! ) r—+0+ = 0 for ,a-ae. t 6 RN. (2.30) Fix a t 6 RN such that (2.30) holds. Since for any (1 > 0, the ball U(t, (l) centered at t with radius a can be covered by RU, aHl). It follows from (2.30) that U t. limsupu< (:12) + a,H1 = 0 for u-ae. t 6 RN. (231) r—+() It follows from (1.4) and (2.31) that dining. 2 H 1(A, H _1). Hence we have (A, H_]) _<_ [VT/H1. This proves (i). Statements (ii) and (iii) follow directly from the definition of A“. To prove (iv), we note that for every a E (0, OO)N, Property (i) implies sup A6 A); (a, A) < 00. On the other hand, the function A I-—> (a, /\) is increasing in each Aj. Hence sup AeAl, (a, A) must be achieved on the boundary of A“. o o o i r As examples, we mention that if TH 18 the Lebesgue measure on R1, then N A. N 1 A,,,={()\1,...,)\N)esz ZEL B H(t) is defined by War) = a{t E R1” : B”(t) e F} for all Borel sets F c IR". We will make use of the following result. Proposition 2.3 Let BH 2 {BH(t),t E RN} be an (N, (1)-fractional Brownian sheet with index H E (0, 1)N. Then for every Borel probability measure [1 on R1 dimHuBn S 3,,(H) /\ d as. (2.34) where 8,,(H) = supAEA“ (ff—1, /\). Remark 2.4 Applying a moment argument [see, e.g., Xiao (1996)] and the sectorial local nondeterminism of B H , we can actually prove that the equality in (2.34) holds. 31 Since this result is not needed in this chapter and its proof is rather long, we omit. it. Proof: To prove the upper bound in (2.34), note that, withmit loss of generality, we may and will assume the support of ,u is contained in [5, TJN for some 0 < E < T < 00. Furthermore, if 5;1.(H) Z (1, then dlInH/LBH S 8/,(H) /\ (1 holds trivially. Therefore, we will also assume that 8;,(H) < (I. Now, for any 0 < ’7 < diIIlHiigu, by (1.4) we have . n U , lim sup #8 ( (u p)) = 0 for /1.Bu-a.e. u 6 Rd. (2.35) p—+0 P7 This is equivalent to . 1 .v hm sup —+ ]N H{|BH(.-)—B"(t)|sp} ,u(d.s) = O for ,u-ae. t E R1. /)->0 p7 [e.T (2.36) Note that Vdj E (0, Hj) (j = 1, . . . , N), Lemma 2.2 implies that V8, t E [5, Tl” N |B”(s) — BH(t)| s K 2 ls,- —— tj|Hr61 as. (2.37) i=1 It. follows from (2.36) and (2.37) that almost surely . 1 . . 11m sup ——7- y(R(t,p1/(HJ'6J))) = 0 for ,u-ae. t E R1. (2.38) p—+0 P Now we choose (51, . . . ,6N in the following way: V6 E (0, H1), H- 6126 and (SJ-z—I-{lcl forj=2,...,N. (2.39) 1 32 Then, (2.38) implies 1 ,. lim sup :p(R(t,p”1/("1’6))) = 0 for n-ae. t E R1. (2.40) IHO p' we claim that "y g 8,,(H). In fact, if ”y > 8/,(H), then there exists ,3 E A), . -1 . . _ . ‘ A N . ‘. . such that ()3, H ) < 7. Since. ,3 E A”, there 1s a set A C [c, T] With posltive y-measure such that t r)) lim sup :1) > 0 for every t E A. (2.41) r—20 72b3,}! Now we choose (5 > 0 small enough such that H1 — H1 _ 6(8, H”) > 0. (2.42) ,7 Then (2.41) and (2.42) imply that for every t E A. 1 lim sup 7,1020, le/(H1—6))) p—+O p 1 u(R(t,pH1/(H1—6))) (2.43) :limsup _1 x -1 :00. p—»0 p7~H1/(H1—6) This contradicts (2.40). Therefore, we have proved that ’y S 3,,(H). Since 7 < diInH/ign is arbitrary, we have dimH/iBH g 3,,(H) /\ d, as. (2.44) This finishes the proof of Proposition 2.3. 33 The following corollary follows directly from Proposition 2.3 and (2.32). It is related to Theorem 3.1 in Ayache and Xiao (2005). Corollary 2.1 Let m be the Lebesgue measure an R1. then N 1 dimHmBu g 2 ‘1;— /\ (1,0..8. (2.45) j=1 J For any Borel set E C (0, OO)N, we define ME) = U A“. (2.46) ,ieM‘ctus) Recall that M;(E) is the family of finite Borel measures with compact support in E. Similar to the case of Borel measures, we call the set UNEMHEfiA“ the Hausdorff . . r d1mensron contour of E. It follows from Lemma 2.4 that. for every a E (0, OO)A. the supremum sup AEM E) (A, a) is determined by the Hausdorff dimension contour of E. The following is the main result of this section. Theorem 2.2 Let BH be an (N, (1)-fractional Brownian sheet with indea: H E (0, 1)N, Then, for any Borel set E C (07 OO)N, dimHBH (E) = s(H, E) A d as, (2.47) where 5(H, E) = supyeME) (A,H”1). We need the following lemmas to prove Theorem 2.2. 34 Lemma 2.5 Let E C RN be an. analytic set and let f 2 RN —-> Rd be a continuous nnction. ['0 S 7' < (1111] E , then there wrists a com Jact set E (_2 E such H I 0 that T < dime(E0). Proof: The proof is the. same as that of Lemma 4.1 in Xiao (1997), with packing dimension replaced by Hausdorff dimension. Lemma 2.6 Let E C (0, OO)N be an analytic set. Then for any continuous func- tion f : RN —> R", dime(E) = sup {dimH/Jf : a E MEL(E)}. (2.48) Proof: For any a e M5(E), we have a, e M3.(f(E)). By (1.5), we have dime(E) = sup {dimHz/z 1/ E Mg(f(E))}, (2.49) which implies that dime(E) Z sup {dimanz a E M$(E)}. (2.50) To prove the reverse inequality, let. T < dimH f (E) By Lemma 2.5, there exists a compact set E0 C E such that dimH f (E) > 7'. Hence, by (2.49), there exists a finite Borel measure V E M24 f (E0)) such that dimHI/ > 7'. It follows from Theorem 1.20 in Mattila (1995) that there exists a E MEKEO) such that 1/ 2 it), which implies sup{dimHaf : a E M;(E)} > 7. Since 7' < dime(E) is 35 arbitrary, we have dime(E) S sup{diman: a E M}(E)}. (2.51) Equation (2.48) now follows from (2.50) and (2.51). Proof of Theorem 2.2: First we prove the lower bound: dimHBH(E) Z 5(H, E) /\ d as. (2.52) For any 0 < ’7 < 8(H, E) /\d, there exists a Borel measure ,u with compact support in E such that ’7 < 8,,(H) /\ (1. Hence we can find X = ( ’1, . . . , )va) E A“ such I A . that y < 21:1 l—IJJ— /\ d and R t. ,. lim sup M = 0 for h-ae. t E R1. (2.53) I _. For E > 0 we define -1 E5 = {t e E: y(R(t,r)) g TN” > for all 0 < r g a}. (254) Then (2.53) implies that ,u(E5) > 0 if E is small enough. In order to prove (2.52), it suffices to show dimH BH(E5) Z ’7 as. 36 The proof of the latter is standard: we only need to show Eh/ (Bile “(18)/1(th wm' (2‘55) By Lemma 2.3 and the fact. that y < d, we have R] / HM(E) = — dimHE for all Borel sets E c (0, oo)N. (2.61) a 38 Our proof of Theorem 2.3 is reminiscent to that. of Khoslmevisan, \th and Xiao (2005) for the Brownian sheet. The key step is provided by the following lemma, which will be proven by using the sectorial local nondeterminisrn of B <“l. Lemma 2.7 Assume N S ad and let 6 > 0 and 0 < 20 — (5 < L3 < 20' be given constants. Then with probability 1. for all integers 71. large enough, there do not ear‘ist more than 212M distinct points of the form. tj == 11—" kl , where h‘j E {1. 2, . . . , 4n}N, such that B(ti) — B<“>(ti) < 32‘” fori 7A j. (2.62) Proof: Let A” be the event that there do exist more than 27m}! distinct points of the form 4‘"th such that (2.62) holds. Let 1V" be the number of n-tuples of distinct t1, . . . ,t" such that (2.62) holds. Then [271011 + 1] An 9 Nn 2 Tl So, IE Nn PfAn) _ ( ) (2.63) [2"‘5" + 1] Ti 39 In order to estimate lE(l\,,), we write it. as 13(an = El 2 Z ' ‘ ' Z 1{(2.62) holds}] t1 t2 tn distinct = Z :2 IP{ |B<“>(t") — B<0>(tf) < 3 - 2*“, vii 7i j g n} t1 t2 t" distinct (2.64) Now we fix n — 1 distinct points t1, . . . , t”‘1 and estimate the following sum first: Zr{ (Ewe) — B<“’>(tj) < 3 - 2—"1‘3, Vi 7A j g n}. (2.65) t" Note that for fixed t1 = k14_", . . . , ill—l = ”‘44—", there are at most (n — l)N points 7'" = (Till, . . . , fair) defined by u_j '_ Tg—t) forsomej—1,...,n—1. We denote the collection of Tu’s by F" 2 {Tu}. Clearly, t1, . . . , tn—l are all included in F". It follows from Theorem 2.1 that, for every t" E F", there exists 7"" E P" such that Var (Bt“'>(t">IBt“>(t1), . . . , 3326124)) 2 K2.4,1 It" — WP. (266) 40 < > . . o )0 . . 0' In this case, smce Bi l, . . . , 3:! > are the independent copies of BO , a standard conditioning argument and (2.66) yield IP’{ (BMW) ._ B(tj) < 3 - 2‘”, Vi #j g n} 3 IP’{ |B<“>(i") — B<0>(ii) < 3 - 2"“, W #J' S n - 1) (2.67) 3 . 2—7243 d X . (Kg/3, tn _ 7w.) If t" E P”. then we use the trivial bound IP{ |B<">(i') — B<">(ti) < 3 . 2””, Vi 7e) 3 n} (2.68) 3 IP{ |B<“>(t") — B<”>(tf)l < 3 - 2"”, Vi 7s 3' g n — 1}. Hence, by combining (2.67) and (2.68), we obtain 2 r{ )B<">(ti) — B<“>(tj) < 3 - 2‘”, Vi #j g n} tn g r{ [B<">(ti) — B<"‘>(tj)l < 3 - 2"”, Vt 741' s n — 1} (2.69) 3.2-".6 d X[ Z K2‘4'2(|t"—T“n|a) +(n—1)N]. 41 Note that 3 2—7m’ d 2_ nt (1 Z (Itn, Tunl‘ 0:) Z :7- “(ltn_ Tula) tn. ¢ 1"" TH E P" t”# _ ,. 1 < 3d _ 2—71Jd (2.70) - _ Z E It”. _ Tuitid 7.11 E 1"" tn 7g 7.1!. 3 Am (n — 1)-V+12~<2a—x3>d, where, in deriving the last inequality, we have used the fact that if [V _<_ (rd then for all fixed 7'", 1 Z til U ad S K . 220nd in" t1). 7é 7.“ I w T I Plug (2.70) into (2.69), we get ZIP{ |B<”>(t") — B<“>(t3') t' < 3 - 2"“, Vi 9e) 3 n) 3 IP{ (BMW) — BMW) < 3 - 2"“, v2: ,2 j g n -1} (2-71) X [(2744 (n _ 1)l\r+l 212(2()'—,3)d' Therefore, by iteration, we obtain 2 Z'”Z IP{ |B<“>(t") — BMW) < 3 - 2””, Vi #2 s n} distinct 2 s Kits [(-n — 1)!]N+12'" (20‘5”, (2.72) 42 which implies 2 lE(]Vn) S [(3435 (TL _ 1)n(N+l) 2n (2(1—d)d. (273) By (2.63), (2.72) and the elementary inequality 7 2726(1 +1 n5d n n.2t5d [ l 2 (2 +1) 2 2 TL we obtain 7 a 2 c / ~ lP’(A.,,) < 34.19601. — 1)"(A +2) 2" (hp/"Md. (2.74) Since 0 < 2C1 — ,5 < 6, by (2.74), we get anlD(/-ln) < 00. Hence the Borel-Cantelli Lemma implies that ll} (limn/ln) = 0. This completes the proof of our lemma. For n = 1, 2, . .. and k = (k), . . . ,kN), where each ki E {1, 2, . . . ,47‘}, define 1,: = {t e [0,11N : (k,- - 1)4-" g i.- 3 ear" for all i = 1,...,N}. (2.75) The following lemma is a consequence of Lemma. 2.7 and the modulus of continuity of the fractional Brownian sheet [cf. Lemma 2.2]. Lemma 2.8 Let 6 > 0 and 0 < 20 — (5 < ,8 < 2a. Then with probability 1, for all large enough n, there exists no ball 0 C Rd of radius 2”"5 for which B—1(O) intersects more than 211M cubes 1:. 43 N ow we are ready to prove Theorem 2.3. Proof: For simplicity, we shall only prove (2.61) for all Borel sets E g [0, ”N. By Lemma 2.2, we know that almost surely B (”)(t) satisfies a uniform Holder condition on [0,1]N of any order smaller than a. This and Theorem 6 in Ka- hane (1985a) [or Proposition 2.3 in Falconer (1990)] implies that lP’{dimHB(E) S idimHE for every Borel set E C [0,1]N} = 1. To prove the lower bound we need only to show that almost surely for every compact set F Q Rd, dimH{t e [0,1]N : B<“')(t) e F} g adimHF. (2.76) This follows from Lemma 2.8 and a simple covering argument as in Khoshnevisan, Wu and Xiao (2005). Therefore, the proof of Theorem 2.3 is finished. When .N > ad, (2.61) no longer holds. In fact, when lV > ad, the level sets of B (a) have dimension N -— ad > 0 [Ayache and Xiao (2005), Theorem 5, p. 434]. ‘1 (0). In the following, we prove two weaker forms of uniform result for the images of Therefore, (2.61) is obviously false for F 1: (3(a)) the (N, d)-Brownian sheet B(”) with ad < 1. (Of course, ad < N in this case.) They extend the results of Kaufman (1989) and Khoshnevisan, Wu and Xiao (2005). Theorem 2.4 Let ad < 1, then with probability 1 for every Borel set F _C_ (0, 1]N, 1 . dimHB<“)(F + t) = min {d, —dimHF} for almost all t E [0, 1]”. a (2.77) 44 Define HR(IL‘) ;= Rd1[_1 11.10%) V1.- 6 a", R > 0. (2.78) Also define 12(1331) 1=/ N HR. (BW-r + t) - B<">(y + 0) dt {OJ} (2.79) V1“2>0,ir:,yE[e,1]N. The following is the key to our proof of Theorem 2.4. Sectorial LND plays an important role in the proof of the next lemma. Lemma 2.9 For all :1), y E [S,1]N, R > 1 and integers p = 1, 2, . . ., IE Klaus. W] : Wigwam — xl‘ad". (2.80) Proof: The pth moment of 13(33, y) is equal to RPd/---/|:IP{ max 1937) [0,1]“ d (2.81) < 112-1)] dt1--- dtp. Béa>(:r + ti) - Bi“) (y + ti) We will estimate the above integral by integrating in the order dip , dip—1, . . . , dtl. First let t1, . . . , tp _1 E [0, I]N be fixed and assume, without loss of generality, that —1 all coordinates of t1, . . . ,tp are distinct. Define o,- := 33") (:1: + t’) — 85,")(y + t") Vi = 1, . . . ,p. (2.82) 45 we begin by estimating the comlitional probabilities ’P(t”) :2 1P{ |Q,,| < R‘1 max |f2,:| < R—1 }. (2.83) lSiSp—l Because BS”) is sectorially LND, we have Var(f2p |Q,~, lgigp— 1) (a) i. (a) i - __ ZVar 9,, B0 (:1:+t),B0 (y+t),1£2£p 1 (2.84) N Z K248 Z min {W + 17k 7 [ka — yt-IQO}, k=1 where [(2.48 > 0 is a constant which depends on 8 [we have used the fact that LE;- + t1}: 2 E for every 1 S k S Al] and 1”“ :2 1313151406: —ttl2”, live + ti: — yk — t1.|'~’“), _ _ (2.85) " == 13315:, (It? - til”). ly;r + t’; — an. — ttl‘z“). Observe that for every 1 S k S N, we have . .- . 2a ’Uk + 6k 2 1(:1:+t’) -Béa>(y+tJ)| < 2'( ’5") 1‘” dt l n0(w), 124179) S K2.4,10any - aIl'“ V132 y E [5, llN- (2-100) 50 Let 0 be an integer such that 6 > 21/0 and consider the set. Q,” g [02 ”N defined by (2,, :2 {9""k: 10,: 0, 1, . . . , 9I, I) = 1, . . . , N}. (2.101) The number of pairs it, y E Q" is at most KQZN". Hence for u > 1, Lemma 2.9 implies that ll” {1274(23, y) > unN|y - :rl‘a'd for some 3:, y 6 Q2, (1 [8,1]N} (2.102) s eINIIK-é’,400(p!)N(we)‘I. By choosing p I: n, u :2 K143“) 6N, and owing to Stirling’s formula, we know that the probabilities in (2.102) are summable. Therefore, by the Borel-Cantelli lemma, as. for all n large enough, 12n(332 y) S K2,4,16any — SCI—"d V332 y E Qn 0 [50 llN- (2-103) Now we are ready to prove (2.100). This is a trivial task unless nN 2—"d < Iy — :Clad, which we assume is the case. For LE, y E [E , llN, we can find LT: and 3] E Qn_1fl[€, 1]N so that la? - 53' S Wgfln‘ and ly — g| S WG—7", respectively. ) By the modulus of continuity of BS“ , we see that. [211.(113, y) S 12.,._1(i,g) for all n large enough. On the other hand, by (2.103) and the assumption ”N 2_nd < 51 ly __ mlod , we have 1202—1(i.9)s(n—1)le— 9| “"(.E + t) (.13, t E (O, llN). By Fr(_)stman’s Theorem, in order to prove dimH BéO>(F + t) Z 7], it. // V’lfldu )V’ (dv) < 00. (2106) 2d lu — up Now we follow Kaufman (1989), and note that the left-hand side is equal to // 1321+ 751”“22’30 0|" =n/O / HR(BO(2+0)_BO(0+2))Rn—d—12(22)0(00)2R 3 1+ / / HR (33“)(2 + t) — Bf,“>(y + a) R’HH ”(41.2) 11(dy) dB. 1 (2.107) suffices to prove that 52 To prove that the last integral is finite for almost all t E [0. llN, we integrate it over [0, 1]N and prove that ///:O 11293:?!) H"”"‘1dR/1(d~r)u(dy) < 00. (2.108) 1 We split the above integral over D = {(I, y) 2 LC — yl S R—5} and its complement. and denote them by J1 and J2, respectively. Since ()1 X ,u)(D) S [(234,17 RTE and 7] E (0, 2%). we have J1< K2 4 47 / R—%+"-1dR < 00. (2.100) 1 On the other hand, I17 — yl“" < R for all (:17, y) E DP. l\loreover, by (2.100). 144(1, y) < c(w)(log R)"'|;z: — yl‘m’. It follows that J2_ < K241 18(“1 M/ d y)/ Ell—d—l (10g R)N (IR '13:?“ IJT—y|_a , 10 N 1 —:r K2,4,19(w) // g ( ”y . ” Mammy) < oo Ix - yl‘”7 (2.110) where the last inequality follows from (2.105). Combining (2.109) and (2.110) gives (2.108). This completes the proof of Theorem 2.4. Theorem 2.5 Let ad < 1, then "with probability 1, md(B<”>(F + 15)) > 0 for almost allt E [0, ”N, for every Borel set F C (0, ”N with dimHF > ad. Proof: Since dimHF > ad, there exists a Borel probability measure [1 on F f/MM IS—tlE‘d )l III/2(u)l2 du is bounded above by 2 (4.4 [we-"10 1120012 du z Z 211 f” 4Z(2IIB(1: + t) — 2"B<“>(y + t)) de) My)- ,R . 17.20 + (2.114) 54 Consequently, it suffices to show that :2" / N /2N0(2IIB(2:+1) ——2”B<“>(y+t)) mammal: ":0 [on Rt (2.115) is finite. To this end, we define J(:1:,y,n) ;=/ V 717 (2"B(:17 + t) — 2”B<”>(y + 1)) dt. (2.116) 10.11" Lemma 2.10 There erlst positive and finite constant K1430 and ,d such that, with probability 1, for all n 2 11(0)) and W 2 II — yl Z [(243202—71/an1/a, 110:, y .4), s <2 + (2)-"1a: — yr“. (2117) Proof: It suffices to prove that there are positive constants K2421, K2422 and ,3 such that for all integer n 2 1 and VlV 2 I2: — yl Z [(294)202—71/0711/0, E W, we] : K§,'1,24n"'2~II2‘-"I(2 + (0% — 3112"“. (2.118) Then (2.117) will follow from a Borel-Cantelli argument as in the proof of Theorem 2.4. 55 Note that. the moment in (2.118) can be written as 2n ,1 2n _ _IE lElI:— H1( 1;, ((2718 a) 413+tj) _ 271.B(y+tj)) dt , 01127)." ”\S‘n j: 1 (2.119) where t I: (ll, . . . ,t2") and 271 N . k=1 (321 (2.120) '14 +1? — t? - y[' > 7‘" Vj 75 k}, and 7“,, I: K 14302—7” “(n + 1)1/0‘, where [(24.20 > 0 is a constant whose value will be determined later. We consider the integral over 8,, first; it can be rewritten as Ell / Hem: 0 + tI)>>2.o(e') dam] Sn 2nd 2n = l, [Mew[—-ZVII(202"B‘“< +tI)— Bé“’(y +tI)1)] 2n >< 1111(8) dealt. i=1 (2.121) 56 where {E := (fl, . . . ,{2"). Note that for every t E S", there is a k E {1, . . . , 271} such that ltk -- til > ’I',, 1., MIL“! for allj 74 k and II + tk — tj — yl > '1‘" for all 0 S j S 272. Since ngI E B, there exists 60 E {1, . . . , d} such that Ifékol Z (2fll)_1. we derive that 211 Var(:r + tj) — 2"B((,”>(y + tj)]) i=1 leg-m- +1I’),B§,“>(y + H), 1' # k) 1 (I ' (1 ' - (l I - _>_ 212222" var(B,<, )(a: + t*)|B,<, >(:: + tI), v) ,2 k; Bf, >(y + H), v)) N Z K1423 22" 2 111111 {'12r — #12” , 11111—11; — yf —' tilQH} (3:1 1Sj¢k§2n _ , I) Z K242322n7‘20 Z 1934231133,,(71 + 1)". 71, (2.122) In the above K1423 > 0 is a constant depending on E and again we have used the fact that Jig + t? Z 5 for every 1 S é S N. By combining (2.121) and (2.122), we obtain 2n IE / / Hexpeei2"IB<“>(x+tI)—2IB>¢<50d£dt Sn R2ndj=l S exp(—195124722). (2.123) Now, we consider the integral in (2.119) over T" I: [0, 112‘V7'\S,,, which can be written as Tu : {t 6 [0,1]?“ : We 6 {1,...,2n}, W E {1,...,N}, 3344 7e k s.t. [if - tf"1| g r" . , 16.2 or 3352 75 k s.t. I.’L‘[ + t]; — yr] — t) I S M} 2n N . : t E [0, 1]2N” : min tk —- ti.“ S 7'” (11.1%1: [ I k=1€=1 k 16.2 17p+tg —yg—t€ ISrn}). (2.124) U {t E [0,1]2N" : min 1L2¢k W ' ' QAN . horn (2.124), we can see that T" is a umon of at most (4n) " sets of the form: Aj : t E [0,1]2N" : max Z[ + t? — t?“ S r" , (2.125) ISL'S271 ‘ leSN' wherej :2 (jug I 1 S k S 2n, 1 S f S N) has the property that j“. 75 k and where Z5 2 0 01‘ mg — y. The following lemma from Khoshnevisan, \qu and Xiao (2005) estimates the Lebesgue measure of T". Lemma 2.11 For any positive even number 777., all 2:], . . . , ZL E 1R. every sequence {[1, . . . ,KL} g {1, . . . , L} satisfying [j # j, and for each 7‘ E (0,1), we have mL {3 E [0,1]L : kEI{111aXL} |z;4 + 5k — 8ka S r} g (2r)L/2. (2.126) 9..., Now, it follows from (2.124), (2.125) and Lemma 2.11 that. . , _, T < 4 2Nn 2 4 Nn 2 127 771215; n( In.) _ ( Tl) ( 7n.) - ( . ) We proceed to estimate the integral in (2.119) over Tn. It is bounded above by 2n /.,I=[H i=1 123(271B((1)(I + tj) _ 211.B((1)(y + tj))l] dt (2.128) =/ 1mm; 1),] dt+/ 1mm; D;,]dt=:11+12, Tn Tn where igjgzn D" := { max |B(.2: + v) — B<“>(y + tI‘)| > 20—5)"). (2.129) A Since ’(l’is a rapidly decreasing function, we derive from (2.127) that 11 _<. m2Nn(Tn) P(D71)8XP{_K2.4.2571} _<_ (4702“ (2r.)I"" exp{—K2,4,25n} (2.130) _ N712 0' exp( —K2,4,25n} . 1 n. N 'i 2 '— ==1<2426(”) ’(_+0)2 59 Note that. K2435 > 0 can be chosen arbitrarily large. 11 is very small. On the other hand, by the Cauchy-Schwarz inequality, [2 is at most / 1P{|B<”>(a: + tI’) — BWy + tI')| g 2“(1‘5)'",Vj=1,...,2n}dt T" S ‘/[(‘) 112Nn 1T" (t) d x (1»{|B§,“>(x +11) — 3M4 + tI)l s 2-<1—IIIII.VJ° = 1,. . . ,2n}) dt (T ) (N—ml)/N S (m n n ) { / . 2N [011215111 ' . N/a (id/N (IP{|B(§“>(1: + t-I) — Bé“>(y + MI 3 241—5)", Vj}) dt} 2 N n K. I 1: —'Il, _+(1—2£)d —2and ,. (2.131) where the last inequality follows from (2.127) and (2.98) in Remark 2.5 with p = n and where K1423 > 0 is a. constant depending on a, d and N only. Combining (2.119), (2.123) with K2435 large, (2.128), (2.130) and (2.131). we obtain 1 , 2 A _ : lE[J(;c,y,n)2"] _<- K2l,4,21n}\2’4’22n —n (0+(1 2-)d) II _ yl—2cmd. (2132) N+ad—2a 2( — fl+(1—2=)d) We. choose and fix 0 < E < ——2dd__° This guarantees that 2 2 ” for (1 some constant [3 > 0. By using (2.132), the Borel-Cantelli lemma and the modulus of continuity of B, we can derive (2.117) in the same way as in the proof of Theorem 2.4. 60 Now we conclude the proof of Theorem 2.5. Thanks to Lemma 2.10, we have _ p.)d (d ZilfllJffnyIllfldfl((111)32’12+I3)_nUfllyr)lll 1U) 10‘ ad 2 1, our proof of Lemma 2.9 breaks down. See (2.96), where the integral on I j3 can not be neglected anymore. In this general case, we do not know whether Theorems 2.4 and 2.5 remain valid. The following question was raised by Kaufman (1989) for Brownian motion in R. It is still open, and we reformulate it for the fractional Brownian sheet with Hurst index (a). Question 2.7 Suppose N > Did. Is it true that, with probability 1, B<”>(F + t) has interior points for some t E [0,1]N for every Borel set F C (0, OO)N with dimHF > ad"? 2.5 Salem set Let X = {X (t), t E RN} be a centered Gaussian random field with values in Rd. W'hen X is an (N, d)-fractional Brownian motion of index 7 E (0, 1), Kahane (1985a, b) studied the asymptotic properties of the Fourier transforms of the image measures of X and proved that, for every Borel set E C RN with dimHE S 761, X (E) is a Salem set almost surely. Kahane (1993) further raised the question of studying the Fourier dimensions of other random sets. Recently, Shieh and Xiao 61 (2005) extended Kahane’s results to a large class of Gaussian random fields with stationary increments and Khoshnevisan, Wu and Xiao (2005) proved similar results for the Brownian sheet. In this section, we study the asymptotic properties of the Fourier transforms of the image measures of the (N, (1)—fractional Brownian sheet BH. The main result of this section is Theorem 2.6 below, whose proof depends crucially on the ideas of sectorial local nondeterminism and Hausdorff dimension contour. Moreover, by combining Theorems 2.2 and 2.6 we show that, for every Borel set E C (0, OO)N, BH (E) is almost surely a Salem set whenever 3(H, E) S (1. Recall that 8(H, E) is defined in Theorem 2.2. Let 351 be an (N, 1)-fractional Brownian sheet with index H = (H4,...,HN). Let 0 < e < The fixed. For alln 2 2. t1, . . .,t",sl, . . .,s“ e E C [8, TlN, denote S = (81, . . . , s”), t = (t1, . . . , t") and \r(s, t) = 1E[ (B§(t’I) —- 35930)] 2. (2.134) 1621 For S E E" and T‘ > 0, we define n ‘n N _ — . . . . ,_ "j .l/H' F(s,r)—U Ufl{uEE.|uJ 33-131 J}. 11:1 iN=1j=1 N This is a union of at most n rectangles of side-lengths 27‘1/H1,. . . ,27‘1/HN, centered at (8111, . . . , 82,3). Let G(s,r) = {t = (t1, . . .,t"’): tk E F(s,r) for 1 S k S n}. (2.135) 62 The following lemma is essential for the proof of Theorem 2.6. Lemma 2.12 There earists a positive constant K 2,5, 1. depending on E, T, H, N only, such that for allT E (0, E] and allS, t E E" with t E C(S, 7‘). we have \II(S, t) 2 Proof: Since 12 E C(S, 7‘), there exist k0 E {1, . . . , n} and jg E {1, . . . , N} such that tfg- ' > TV H19 for all k— —' 1,. .,n. It follows from (2.3) that 8.10 \Il(s,t) = Inf/N where n N N 2 2(1—I(exp(itj)\.-)1)—(H exp( is W )—1)) 3'2] j=1 (2.136) fH()\ :HIAJ' l—2HJ' —1 Let 6j(-) E COCOR) (1 S j S 1V) be the bump functions in the proof of Theorem 2.1. We. define r -1 H —l H‘ 530 (7‘10): 7 / 106]- 0(7 / 307710) and 5§(U.7') = 5-151034%“), ifj # 30- 63 fHO‘) db- Then bv using the Fourier inversion formula ag 1111 we have ‘T , . ___ _.1 __ - _ . ’?_ 1/ [I _ 0 (“10) (271) [Rexm zu,,,A,,,)o,,,(7~/101,0)d1m and similar identities holds for (if-(11d) with ] ¢ ]0 (‘0) = 0 and 65(tj0) = 0 for allj 31é jg. Since 7‘ E (0,8) we have (530/.(1-0 k __ , __ Similarly, ($300,170 ,0 — j ) — 0 for all k — 1, . . ., 71. Hence, . 1. (i ( 3.1.1,... . .4fi4. .. A 4.4)] k=1 j=1 T1 H 1.16%) .. I 1,404.. m N x Hexp(—it§”)\,) (116% 3'21 #10 n N' _ N A k (tj))) 67‘ It I t'b — (Hen. III—I.) ‘ 44()., (rs—ta— .,,,( 3)) k=1 #10 n N' _(gflyV (H(6j(tj0_sl§)_(tk0)))(610() . (,10_ 8k0_) 610W») k=1 #10 2(27r)INr€ (—N— 1),], -1/Hj0. (2.137) In the above all the terms in the first sum are non-negative and the second sum equals 0. 64 On the other hand, by the Calichy-Schwarz inequality, (2.136) and (2.137), we get. 2 A A 2 6] (5A1) (5.1'11(""1/”j0 A10) l dA 1 N J2 K211: . S H (t’S)/111NfH(/\) 1;!) J K2111“, S) g-2(N-1>—?Zj¢joH.1 7~-‘2-‘2/”111 N XII/(AleZHJ-H i=1 R = K2,.5.27'—2_2/Hj0‘11(t1S)- (2.138) A 2 511*.) (1A1 Square the both sides of (2.137) and combine it with (2.138), the lemma follows. . . 1V ' For any Borel probability measure 11 on R+ . let I/ = 11.311 be the image measure of 11 under B H . The Fourier transform of U can be written as 9111 = (exp1z1eBH1t111u1dt1 12.1391 4 The following theorem describes the asymptotic behavior of 3(6) as g —> 00. Contrast to the results for the fractional Brownian motion and the Brownian sheet mentioned above, the behavior of l/(§) is anisotropic. Theorem 2.6 Let BH 2 {BH(t),t E RN} be an (N, d)-fractional Brownian sh eet. Assume that, for every j = 1, . . . , N, the function Tj I R+ --> R+ is non- dedreasing such that 73(0) 2 0 and Tj(27‘) S K253 Tj (7) for all T Z 0 [i.e.. Tj sagisfies the doubling property]. [flu is a Borel probability measure on [5, T]N such that N 1109111)) 3 K2“ HTML/”"1 v t e Rf, (2.140) 1:1 where [{(t, T) 2 UN 1:1ltj — 7‘1”,ij + Tl/Hll. Then there exists a positive and finite constant Q such that almost surely, lim sup lmal < 00. (2.141) lél—mc \/(l—Ijv=1,rj(|€l—1/Hj)) loge m Proof: The argument is similar to that of Kahane (1985a). First note that by considering the restriction of p on subsets of its support and the linearity of the Fourier transform, we see that, without loss of generality, we may and will assume (1 is supported on a Borel set E C [5, TlN with diamE < E 1/ H 1 [we have assumed that H1 = min{Hj I l S j S .N}]. The reason for this reduction will become, clear below. For any positive integer n 2 1, (2.139) yields 11211915112") ——— IE (11” j expue, 213W) — BH1sk11>1m1ds1w1dt1 _ 1 2 n n — [111N413] exp ( — élél \Il(s, t)) a (ds)a (dt), (2.142) where p"((lS) ‘2 [1((181) ' ' ' #(dsnl‘ 66 Let S E [5, Tln‘lV be fixed and we write j , exp ( — —11121I/1s 11).. 11111 ch 1:Xp( (-§|€|2\P(st))p"(dt) (2.143) 1 . Since )a is supported on E with diamE < E 1/ H 1, the above summation is taken over the integers m such that 7‘2"" S 8. Hence we can apply Lemma 2.12 to estimate the integrands. By (2.140), we always have 1 I ‘ TI eXp —-I€|2‘1’(S.t) W(dt) 3 (K2... 11” rid/H”) ' fa“, ( 2 ) 4 H 1 (2.144) Given 6 E Rd\{0}, we take 7“ :- Ifrl. It follows from Lemma 2.12, the doubling property of functions Tj, and (2.140) that 1 n j _ exp ( — —1112\Il1s,t1)p 1dt1 G(s,r2m)\C(s.r2m 1) 2 N n 1 ’ "l- ’ Tn ‘ 5 exp ( — 5 12.411112112 112) - (1...,411” fine 11/”11) i=1 N n N 1 H r 2m ern S (19.54” HTJ( / 3)) exp(— A2552 ) 1‘2 53- i=1 (2.145) 67 Note that 1+Zexp(— 11,2552”) I'é"."1"oo N _ - E H l/V(Hi=1Tj(|Zl l/H’) log9|z| Therefore (2.141) follows from (2.149) and Lemma 1 of Kahane ( 1985a, p.252). This finishes the proof of Theorem 2.6. Theorem 2.7 Let BH 2 {BH(t),t E RN} be an (N, (1)-fractional Brownian 68 sheet with Hurst index H E (0, 1)N. Then for every Borel set E C (O, oo)N with 8(H, E) S Cl, BH(E) is almost surely a Salem. set with Fourier dimension s(H, E). Proof: It follows from (2.47) and the fact that dimFF S dimHF for all Borel sets F C Rd that for every Borel set E C (0, 00)N satisfying 8(H, E) S d, we have dimFBH(E) S dimHBH(E) = 8(H, E) as. To prove the reverse inequality, it suffices to show that if 8(H, E) S d then for all 'y E (0, 3(H, E)) we have dimFBH 2 ’y as. Note that for any 0 < ’y < 8(H, E), there exists a Borel probability measure a with compact support in E such that ”y = {B(t),t E RN} be an (N, d) -fractional Brownian sheet, and let T I R+ —) R+ be a non-decreasing function satisfying T(0) = 0 and 69 the doubling property. If [1 is a probability measure on [8, TJN such that 11(B(a:, r)) g K’215,8T(2T), Va: 6 Riv, r 2 0. (2.151) Then there e:1:ists a positive and finite constant 9 such that lim sup |V(€)| < 00. (2.152) 1...... 711111-110110119111 Moreover, for every Borel set E C (0, 0C1)N with dimHE S ad, B<“>(E) is almost surely a Salem set with Fourier dimension dimH E/a. 2.6 Interior points By using the Fourier analytic argument. of Kahane (1985a). it is easy to show the following: If a Borel set E C (0, 0(3)N carries a probability measure a such that 1 /E/E (211:1 lSj - tj|2H1)d/2 ”(dig/1(a) < 00’ (2103) then almost surely, B H (E) has positive d—dimensional Lebesgue measure. In par- ticular, it follows from the proof of Theorem 2.2 that, if E C (0, 00)N satisfies 3(H, E) > (1, then BH (E) has positive d-dimensional Lebesgue measure. It is a natural question to further ask when BH (E) has interior points. This question for Brownian motion was first considered by Kaufman (1975), and then extended by Pitt (1978) and Kahane (1985a, b) to fractional Brownian motion and by Khoshnevisan and Xiao (2004) to the Brownian sheet. Recently, Shieh and Xiao (2005) proved similar results under more general conditions for a large class of Gaussian random 70 fields. In the following, we prove that a condition similar to that. in Shieh and Xiao (2005) is sufficient for B” (E) to have interior points almost surely. This theorem extends and improves the result of Khoshnevisan and Xiao (2004) menti111ned above. Theorem 2.8 Let B” = {BH(t),t E RN} be an (.N, Cl) ~fractional Brownian sheet with index: H E (0, 1)N. If a Borel set E C (0, 0(3)N carries a probability measure 11 such. that 1 sup / . tEIRf RN (23:1 l3j _ ,jlznj)d/2 + 1 x loggwm (Z ) 11(ds) S K2111 3:. 1e.- — 1.1”“ (2.154) for some finite constants K21“ > 0 and’)! > N, where log+ :1: = max{1, log it}, then BH (E) has interior points almost surely. From Theorem 2.8 we derive the following corollaries. Corollary 2.4 IfE C (0, OO)N is a Borel set with 8(H, E) > d, then BH(E) as. has interior points. Proof: It follows from the proof of Theorem 2.2 that, if 8(H, E) > Cl, then there is a Borel probability measure 11 on E satisfying (2.154). Hence the conclusion follows from Theorem 2.8. Corollary 2.5 Let Em) = {B(t), t E RN} be an (N, (1)-fractional Brownian sheet with Hurst index H = (a). If a Borel set E C (0, 0C1)N carries a probability 71 measure 11 such that 1 , . 1 , SUP f. “—7.1 bail/+1)” ( )11(d8) S K2112 (2155) 1611113 111;r ls— l 5‘11 or some initc constants [(2.62 > 0 and > 1V. Then BM> E has interior '7 points almost surely. The existence of interior points in BH (E) is related to the regularity of the local times of B H on E. In order to prove Theorem 2.8, we will need to use the following continuity lemma of Garsia (1972). Lemma 2.13 Assume that p(u) and \Il(u) are two positive increasing functions on [0,00), p(u) l 0 as u l 0, W(u) is convex and \I/(u) T 00 as u T 00. Let D denote an open hypercube in Rd. If the function f(.1‘) I D 1—+ R is measurable and (D f): /D /D\II (pW (IT _ Elf/C(36) )drdy < 00. (2.156) then after modifying f(1‘) on a set of Lebesgue measure 0. we have A [.2 lI-yl |f(:c) — f(y)| g 8/0 \Il_1(1d)dp(u ) for aux... e D. (2.157) we take the function p(u) in Garsia’s lemma as follows: Let ’7 be the constant 72 in (2.154) and define Q fiuzQ PW) = log"7 (e/u), if 0 < u g 17 (2.158) 'yu—y-l-l, ifu>1. Clearly, the function p(u) is strictly increasing on [0, OO) and p(u) l 0 as u l 0. Proof of Theorem 2.8: First note that, since ,u is a Borel probability measure on E, without loss of generality, we can assume that E is compact. Hence there are constants 0 < E < T < 00 such that E g [8, TlN. Since BH (E) is a compact subset of Rd, (1.13) implies that {I : l,,(.’1:) > 0} is a subset of BH(E). Hence, in order to prove our theorem, it is sufficient to prove that the local time l “(23) has a version which is continuous in :13; see Pitt (1978, p.324) or Geman and Horowitz (1980, p.12). This will be proved by deriving moment estimates for the local time l II and by applying Garsia’s continuity lemma. Secondly, as in Khoshnevisan and Xiao (2004), we may and will assume that the Borel probability measure p in (2.154) has the following property: For any constant C>0and€=1,...,N, ,Lt{t=(t1,...,tN)€E2 thC}=0. (2.159) Otherwise, we can replace BH by an (N — 1, (1)-fractional Brownian sheet EH azld prove the desired conclusion for BH (E5), where E5 is the set in (2.159) with positive u-measure. 73 Consider the set En defined by {t=(t1,...,t")€E": tiztiforsomeiaéj and 1_<_£'§N}. (2.160) It follows from (2.159) and the Fubini—Tonelli theorem that [1"(En) = 0. The following lemma provides estimates on high moments of the local time, which is the key for finishing the proof of Theorem 2.8. Lemma 2.14 Let ft be a Borel probability measure on E C [5,TJN satisfying (2.154) and (2.159) and let p(u) be defined by (2.158). Then for every hypercube D C Rd there artists a finite constant [(253 > 0. depending on N, Cl, ’7, ,u and D only, such that for all even integers “It 2 2, E] [D (p Ii—yllfb) )lndfdyq‘mww 10% WW (2161) We now continue with the proof of Theorem 2.8 and defer the proof of Lemma 2.14 to the end of this section. Let \Il(u) = uexp(u9 ), where 6 E (57 i) is a constant. Then ‘11 is increasing and convex on (0, 00). It follows from Jensen’s inequality, the Fubini—Tonelli theorem and Lemma 2.14 that for all closed hypercubes D C Rd and all integers n with 74 6+1/n<1, 113/0 /D (bill: )3}|l/%)I)ina+ldmdy ,, 0+l/n SW” LOWE/i / l0 (11,.|I_yl/,%|) drdy} < Kn(n')(0+1/n)(10gn)u(1\+1) (0+1/n) < K26M( )M 108“ NH) 671: (2.162) where [(26.4 is a finite constant. depending on N, d, 6, D and 1(2303 only. Expanding \I’ (u) into a power series and applying the inequality (2.162), we derive IE11.3//D\11(P(l35—yl;\(/yC—3)|)(Lazy :gn! IE/D/D (bl(|:—yl/\/3))|)W+ldxdy (2.163) S K165 < 00, the last inequality follows from the fact that N 6 < 1. Hence Garsia’s lemma implies that there are positive and finite random variables A] and A2 such that for almost. all :13, y E D with Ia: — yl S e’1 7 llu(x)‘_ln(3/)l S folI—yl‘l’ (:22) debt) 3 A2[log(1/lrc — yl)] "Tl/9’. Note that, by our choice of 9, we have 9 > 1/7 and hence B H has almost surely 75 a local time l#(.1:) on E that is continuous for all :1; E D. Finally, by taking a. sequence of closed hypercubes {Dm n _>_ 1} such that Rd 2 LJ0C ":1D", we. have proved that almost surely l #(17) is continuous for all (I? E Rd. This completes the. proof of Theorem 2.8. It remains to prove Lemma 2.14. Our proof relies on the sectorial local nondeter- minism of B H and on an argument which improves those in Kl‘ioshncvisan and Xiao (2004) and Shieh and Xiao (2005). We will need several lemmas. Lemma 2.15 is essentially due to Cuzick and DuPreez (1982), where the extra condition on g is dropped in Khoshnevisan and Xiao (2004). Lenuna 2.16 is a slight i’nodifieation of Lemma 4 in Cuzick and DuPreez (1982). Lemma 2.15 Let Z}, (k = 1, . . . ,7'L) be linearly independent, centered Gaussian variables. Ifg I R —+ R+ is a Borel measurable f'amrt-ion. then [Rnyh'il exp (—%Var((v7 Z») do 27,. (n—l)/2 -’>C 1 = ( gl/‘z / 9(Z/01) exp (722) dz, (2.164) where 02 = Var 21 22, . . . , Zn and Q = det COV Z1,.. . ,2” is the deter- 1 minant of the covariance matrix. 0f 21, . . . , Zn. Lemma 2.16 [fa 2 (22/2, then 00 2 / log” 3: exp ( — 1%) dz: 3 J7? log” a. (2.165) 1 Consider the non-decreasing function A(u) = 2 min{1, u} on [0, 00). Later 76 we will make use of the elementary inequality lei“ — 1| 5 A(|u|), Va 6 IR. (2.166) Lemma 2.17 Assume h(y) is any positive and non-decreasing function on [0, 00) such that h(0) = 0, y"/h"(y) is non-decreasing on [0, 1], and fix h‘2(y)dy < 00. Then there exists a constant K21625- such that for all integers n 2 1 and U E (0, oo), 00mm» . __ 1 * dy 3 K” ,h ",(—) 2.167 /0 h” (y) 2,61‘ + 'U ( ) where h+(y) = min{1, h(y)} so that h;"(y) = max{1, hi" (31)}. Proof: The proof is the same as that of Lemma 3 in Cuziek and DuPreez (1982). The following result is about the function p(u) defined by (2.158). Lemma 2.18 Let p(u) be defined as in (2.158). Then for all 0 > 0 and integers n 2 1, 2 / pl” (2) exp(—v—ytv g Kits: [logmn + log? (3)]. (2.168) 0 ”U 2 a Proof: Since _n 10g“ (5). if 0 < :1: < 1, p+ (33) = 1 if 3: 2 1 77 and logi(:ry) S 20 (log: I + log: y) for all Oz 2 0, we deduce that the integral in (2.168) is at most ,2 2 / exp ( — L) dv + 2’”/ log:7(v) exp(—1L) do 0/121 2 a/v) — exp("i)] du “n(dt)da:dy (D): [D )1)/E" Andy" 1 2963739)) x exp [—-2—Var(:(2t’", BH(tL)))]du u"(dt) dy. k=l (2.173) In the above, we have made a change of variables and D9 D = {.r — y : :13, y E D}. By our assumptions on ,u, we see that the integral in (2.173) with respect to a" can be taken over the set E"\E,,, where E” is defined by (2.160). Now we fix t E E"\E,,, a sequencej = (j1,.. . ,j,,) E {1, . . . ,d}" and define Mn(t) E Mn K221: minl lt’" — 25.2””. (2.179) 82 In order to estimate the sum in (2.179) as a function of t”, we intro- duce N permutations F1, . . . ,I‘N of {1, . . . ,n — 1} such that for every 7:: 1,...,PJ, 2f‘(1)”‘2)< < t,’ (2.180) This is possible since t? (1 g k _<_ n — 1, 1 g t’ g N) are distinct. For W0 ) convenience, we denote t, — —5 and tr [(7') 2T for all 1 S t _<_ N. For every sequence (21,...,i,\r) E {1,...,n — 1}N, let 72-1,...2“. = (#1161), . . . , thVUNU be the “center” of the rectangle N' 1 1,I‘7(2’7)_ (17D (77) r{(27-1) r7(i7) r((77+1) 17(77) Hl—Il — (t7 —t7 )1” +507 _tlr )) 7:1 (2.181) with the convention that the left-end point of the interval is 5 whenever 27 = 1; and the interval is closed and its right-end is T whenever i7- = n — 1. Thus the rectangles {Ii,,...,,:,(,} form a partition of [5, T]N. For every t” E E, let I 21,...12- N be the unique rectangle containing t”. Then (2.179) yields the following lower bound for of, (t): Hf ‘ltll . (2.182) For every I: = 1, . . . ,n — 1, we say that 1212...,2N cannot see tk from 83 direction 8 if . ,,-. 1 ,2 72h- 1 1 72' .2 . t? 62 [t§7(7)__2_(tf7(7)_t§7(7 1))a t?“‘)+§(t{[({+1)—t:((’))]. (2.183) we emphasize that if 12,2... 2,. cannot see tk from all N directions, then . 1 [22 — 22| _>_ 5121,51 |7f — 2;," for 3.11 1 g 7 g N. (2.184) Thus t" does not contribute to the sum in (2.179). More precisely, the latter means that A 2 - k m QHI.’ 01: (t) 2 K1619 2 mm It); — t2» . 7:1 maékm (2.180) The right hand side of (2.185) only depends on t1, . . . , t"’1, which will be denoted by Eat). Because of this, t" is called a “good” point for 1217...“, when (2.183) holds for every t = 1, . . . , N. Let 1 S k S n - 1. If 12,222,, sees the point tk from a direction and tk 75 r,,,...,2-N, then it is impossible to control 0,? (t) from below as in (2.182) or (2.185). We say that t" is a “bad” point for 12,222,, [Note that by definition n fié 6"213...,2,] It is important to note that, because of (2.180), the rectangle 121,...” can only have at most N bad points tk (1 S k S n — 1), i.e., at most one in each direction. We denote the set 84 of bad points for 121,...” by 8" . = {1 S k S n — 1 : tk is a bad point for I,,3...,2:,,} 11"... IN and denote its cardinality by #(G",-,,.. ). Then #(9"2,... N) S N. Now we choose the constants B], . . . , 6,, [they depend on the sequence (21, . . . ,iN)] as follows: 7’37, 2 0 if tk is a bad point for [21,...22-N; [37, = 1 if tk is a good point for 1,7,... and ° 71A? fin : 1 + #(enil-m-LV) Clearly, 73,, S N + 1. By Lemma 2.17 and Lemma 2.18, we have gfi/DGD/(A71(;l1::::2(?)7|:||//53(t)) exp ( - (—-'—u;' )2) do" dy ex — —— 27 OOAn(UE/7'n/0"(: ))y do _, 2) < Kn 11.13" (111( ) p(_ g) — 2,6,10 [11 17+ *2) 8XP 2 'n 7 , N 1 e S 2‘671‘l10g M 77+102< H (0(2))l' In the above, we have also used the fact that p(|yl/\/d) 2 p(|yjI/\/d) for allj= 1,...,d. If t" is a good point for 12;, then by the monotonicity of the ,iNi 85 function A we have 7 7-7"" 7:17.777” 7(- W) 7 /.../ A" 7 777(— W) 7 When tk is a bad points for Ii“. ...-N, we use the inequality A(u) S 2 to obtain 3k < 2"/ jexp(—(u “$352) fl)du3 dy S ’26’12. (2.188) DQD Since there are at most N bad points for 1,; their total contribu- l-H‘ 927V, tion to Mn(t) is bounded by a constant K2713, which depends on D, d and N only. 86 Combining (2.177), (2.186), (2.187) and (2.188), we derive. that Mn(t) ’7) 2611 1 (A+111\+1) e '<‘([detCov (BH( t1), BH( ))]d/2 [ 0g ” + 0g cum 0 An k t A 2 1/12. H2N{/ / (u LJAyJL/OA( )) exp ( _ (111k) )dtljk {lg} D D 19"(lyl/f) 2 26.15 :[detCov(B”(t1),. 351m 1))14/2 u a UK? ‘2 l/n sti/n D/ QiZIX/c‘ii ))exp(_( 2) )d” dyi 1(‘\'+1)7 1\+1)‘7 6 . omit) [0g "+ 094+ Mt) E321 ..... 611 ..... X (2.189) Note that Condition (2.154) implies 10g( (\+1) '14_(n+10g \‘+ ‘7( ):|#dtn) [i1 1N 011(t)d d[ + 0,1(t) ( ,u..f 1 S K2616 / ' ‘ . d/‘Z Ii1,...,1'N (29:1 it? _ tiuzUIME) x[log( (NW1 Ln+log(‘\+l) < 1 _ ‘) l)]p(dt") 2:1 it? — t§‘(’[)l~Hz S K2617 IOEUHIM n. (2.190) Integrating M,,(t) as a function of t,, with respect to p on I - and 21 ..... NV 87 using (2.189) and (2.190), we obtain Tl .(1\'+1)A1, K2618 108 n idetCOV(BH(tl)n Bé’(t"-l))]d/2, U 50/ A" 1- ” ..._( (”2) 1.1/'2} (2.191) /, Mlxtmwt >< 11-- It is important that the right hand side of (2.191) depends on t1, . . . , tl"1 only and is similar to (2.177). Summing (2.191) over all the sequences (11,. . .,z’N) E {1, . . .,n — 1}N, we derive that the integral AG- in (2.175) is bounded by dt” 1) Malt") 1(11. gUV-l—lh N/E #( 2.6181 7711‘; 72-1 [(letCOV( (BH (t1).. ..,B01{(tn—l))]d/2 a... {/wD/An(u:LJ(k1:Jl;5§)(t))€Xp(_ (9121.)2) dafkdy}l/n, O71 ..... Note that, for different sequences (11,. .. 1,N,) the index sets 9;: ..... V. may be the same. We say that a set 9" Q {1, . . . ,n — 1} is admissible if it is the set of bad points for some 1,;1, .It can be seen that every admissible set 9" has the following properties: (i) #(9") S N [recall that there are at most N bad points for each Iilsu-aix'V]; 88 (ii) Denote by x(9”) the number of sequences (2'1, . . . ,z'x) such that 9?] ..... ,N — —(-3". If #(9") = p, then x(@") g an’I’. It follows from (i), (ii) and an elementary combinatorics argument that ZX(GH) 2: Z pXHen ] = A... (R fg<6x>dxdt Hence IV j( f ) can be represented as Wis) = f0 A mam-(Mt). Note that l’V( f) is ft-measurable whenever f is supported on [0, t] x R. Recall from l\x’Iueller and Tribe (2002) that a solution of (3.1) is defined as an ft-adapted, continuous random field {ut(2:) : t Z 0, :1: 6 IR} with values in Rd satisfying the following properties: (i) u()(') 6 Sex}, almost surely and is adapted to .70, where Eexp : UA>0£A and a: {f€C(R,R")= |f($)le”"”' -—>0 as W ~00}; (ii) For every t > 0, there exists A > 0 such that u5() 6 5A for all s _<_ t, almost surely; (iii) For every t > 0 and :1: E IR, the following Green’s function repre- 92 sentation holds t W(JI) = (@0451? - y)u6(y)dy + (CH-(:13 - y) W(dy dr): (3-2) where G,(.27) = 71—; exp(—%) is the fundamental solution of the heat equation. we call each solution {ut(:r) : t Z 0, :c 6 R} of (3.1) a random string process. with values in Rd, or simply a. random string as in Mueller and Tribe (2002). Note that, whenever the initial conditions uo are deter- ministic, or are Gaussian fields independent of fo, the random string processes are Gaussian. We refer to Mueller and Tribe (2002) and Fu- naki (1983) for information on stochastic partial differential equations (SPDEs) related to the random string processes. Funaki (1983) investigated various properties of the solutions of semi-linear type SPDEs which are more general than (3.1). In par- ticular, his results [cf Lemma 3.3 in Funaki (1983)] imply that every solution {ut(:r) : t Z 0, a: E R} of (3.1) is Holder continuous of any order less than % in space and i in time. This anisotropic property of the process {ut(:1:) : t Z 0, :1: E R} makes it a very interesting ob- ject to study. Recently Mueller and Tribe (2002) have found necessary and sufficient conditions [in terms of the dimension d] for a random string in Rd to hit points or to have double points of various types. 93 They have also studied the question of recurrence and transience for {ut(:1:) : t Z 0, .2: E R}. Note that, in general, a random string may not be Gaussian, a powerful step in the proofs of Mueller and Tribe (2002) is to reduce the problems about a general random string process to those of the stationary pinned string U— — {Ut(1),t > 0, x 6 IR}, obtained by taking the initial functions U0(') in (3.2) to be defined by U0(:1:)-—-/ /(G., ()23—2 —,.zG( ))’w7(dzdr), (3.3) where W is a space-time white noise independent of the white noise W. One can verify that U0 2 {U0(.r) : [E E R} is atwo-sided Rd valued Brownian motion satisfying (10(0) 2 O and lE[(U0(:r) — U0(y))2] = II — y]. We assume, by extending the probability space if needed, that U0 is fO-measurable. As pointed out by Mueller and Tribe (2002), the solution to (3.1) driven by the noise W(az, s) is then given by U(2: )= /Gt(2:— z)U0(z) dz+/0 /G ,.(:1:—z)W(dzdr) 2/000 (Gt+,.( (.2: — 2) 62(2)) W(dzdr) +/0 [GALE — z)W(dzdr). (3.4) A continuous version of the above solution is called a stationary pinned string. The components {UtJ(:1:) : t Z O, a: 6 IR} forj : 1, . . . ,d are indepen- 94 dent and identically distributed Gaussian processes. In the following we list some basic properties of the processes {U} (.L) : t Z 0, :1: E IR}, which will be needed for proving the results in this chapter. Lemma 3.1 below is Proposition 1 of Mueller and Tribe (2002). Lemma 3.1 The components {UtJ(;z:) : t Z 0, :1: E IR} {j = 1, . . .,d) of the stationary pinned string are mean-zero Gaussian random fields with stationary increments. They have the following covariance structure: forx,yEIR, t 20, 2 was» — (13(2)) I =11: — gt (3.5) andfor allzr,yElR and0£s — my] = (t — s)1/2F(la: — yl(t — 5W), (3.6) where 1 F(a) = (277)—1/2+-2-AAG1(a—Z)Gl(a—z')(|z|+lz'l— z—z'l) dzdz'. F(:c) is a smooth function, bounded below by (27r)"1/2, and F(:r)/|2:| —-> 1 as |x| ——> 00. Furthermore there exists a positive constant KgJJ such 95 that for all s,t E [0, 00) and all 2:, y E IR, K3,1,1(|x—y|+|t—sll/2) g 112[(U{(x)_og(y))2] g 2(|2:—y|+|t—s|‘/2). (3.7) It follows from (3.6) that the stationary pinned string has the fol- lowing scaling property [or operator-self-similarity]: For any constant e > 0, {c’lUC4t(c2;r) : t 2 0,2: E IR} g {Ut(:r) : t 2 0, :1; E IR}, (3.8) where 1—1— means equality in finite dimensional distributions; sec Corol- lary 1 in Mueller and Tribe (2002). We will also need more precise information about the asymptotic property of the function F (21:) By a change of variables we can write it as F(:r) = —(27r)—1/2+%A/RG1(z)G1(z')(|z—:c|+|z’—:r.|) dzdz’. (3.9) Denote the above double integral by H (:13) Then it can be written as H(a:) 2 401(2) :5 —— :z:|dz. (3.10) The following lemma shows that the behavior of H (T) is similar to that of F(:r.), and the second part describes how fast H(.r)/|:r| ——> 1 as 96 Lemma 3.2 There exist positive constants K312 and K3113 such that A’3?1:2(IIL‘ — yl + It _ 8'1/2) S It _ Sll/2H(|I _ yllt — 3|_1/2) (311) S K311410513 — yl ‘1' ll — 8'1/2). Moreover, we have the limit: lim |H(;1:) — ml 2 0. (3.12) I—>OC Proof: The inequality (3.11) follows from the proof of (3.7) in Mueller and Tribe (2002, p.9). Hence we only need to prove (3.12). By (3.10), we see that for :1: > 0, H(2:) ——:1c 2 /RIG1(::)(|2 —:1:|—2:)dz =/:O(z—2:1:)G1(z)dz—/x zGl(z)dz. —00 (3.13) Since the last two terms tend to 0 as :1: —> oo, (3.12) follows. The following lemmas indicate that, for every 3' E {1,2, . . . ,d}, the Gaussian process {Utj (:12), t 2 0,1: E IR} satisfies some preliminary forms of sectorial local nondeterminism; see Section 2.2 for more information on the latter. Lemma 3.3 is implied by the proof of Lemma 3 in Mueller and Tribe (2002, p.15), and Lemma 3.4 follows from the proof of Lemma 97 4 in Mueller and Tribe (2002, p.21). Lemma 3.3 For any given 5 E (0,1), there exists a positive constant K3,”, which depend on 8 only, such that Var (03(1) U30») 2 13,1402: — yl + It — 511/2) (3.14) for all (t,x), (s,y) E [555”] x [—s"1,e‘1]. Lemma 3.4 For any given constants e E (0,1) and L > 0, there exist a constant K315 > 0 such that Var (UTAH) — Utj1(x1)lUg2(y2) — (131%)) Z K3.1.5(l131— yil + l132 — y2|+|t1— sill/2 + ”2 — 32'1/2) (3.15) for all (tk,xk), (sk,y;,.) E [5,5‘1] x [—E—1,€_1}, where k E {1,2}, such that |t2 — t1| 2 L and [32 — 51' Z L. Note that in Lemma 3.4, the pairs t1 and t2, 51 and 32, are well separated. The following lemma is concerned with the case when t1 = t2 and 81: 82. Lemma 3.5 Let e E (0,1) and L > 0 be given constants. Then there 98 exist positive constants ho E (0, g) and K3,”, such that Var (U,j(.r2) — UlfIll U592) — U300) (316) Z K33136- (I8 —t|1/2 +|1131 — ml ‘1’ liEQ — ygl) for all s,t E [5,5—1] with Is—tl _<_ ho and all xk, y;c E [—5—1,5—1}, where k E {1,2}, such that lxg — x1| Z L, lyg —— ml 2 L and Ix;c — ykl g g— for k=1,2. Remark 3.1 Note that, in the above, it is essential to only consider those s,t E [5, 5‘1] such that Is — t| is small. Otherwise (3.16) does not hold as indicated by (3.5). In this sense, Lemma 3.5 is more restrictive than Lemma 3.4. But it is sufficient for the proof of Theorem 3.7. Proof: Using the notation similar to that in Mueller and Tribe (2002), we let (X, Y) 2 (113a,) — oz (.11), Ug(y2) — Ug(y1)) and write 0% = IE(X2), 0% = IE(Y2) and p3”, = IE[(X — Y)2]. Recall that, for the Gaussian vector (X, Y), we have (0X — 0y)2) ((UX + 0Y)2 — ngX) 40,2, Var(X|Y) = (pfiw _ . (3.17) Lemma 3.1 and the separation condition on 23;, and yk imply that both 0% and 0?, are bounded from above and below by positive constants. Similar to the proofs of Lemmas 3 and 4 in Mueller and Tfibe (2002), we only need to derive a suitable lower bound for Pity- By using the 99 identity (a—bi—c—d)2 2 (a—b)2+(c—d)2+(a—d)2+(b—c)2—(a—e)2——(b—d)2 and (3.5) we have Pier = lt — 5|1/2F(IT2 — 112W — 81—1/2) + It —‘ «SP/2190311“ 171W — 31—1/2) (3.18) + m — 1:1! — It — 8|1/2F(|:r2 — glut — st”) +1311 — y2l — It — 5|1/2F0131— 92”t "' Sl—l/Ql' By (3.9), we can rewrite the above equation as Pity = It — 511/2H(|~T2 — y2||t — 31—1/2) + It — sII/‘ZHoyl — aunt — sI‘I/Q) (3.19) + I232 — x1| — |t — sll/2H(|x2 — ylllt — s|”1/2) +|y1— y2| — It “ 3l1/2H0331 — 312W " 31—1/2). Denote the algebraic sum of the last four terms in (3.18) by S and we need to derive a lower bound for it. Note that, under the conditions of our lemma, |x2 — yll 2 -§- and I131 — ygl 2 %. Hence Lemma 3.2 implies that, for any 0 < 6 < K312 / 2, there exists a constant ho E (0, %) such 100 that _. 5 / It ‘- 5|1/2H(|$2 — yillt — 5| 1’0) 3 I332 — w! + g It — 811’? (320) whenever It — s _<_ ho; and the same inequality holds when |:r2 — y1| is replaced by lxl — ygl. It follows that 1/2 3 2 (11172 ‘171l — |$2 —y1|+|y1—y2I—IIE1-yzl)~5lt—8l (3.21) z —6 |t — sll/2, because the sum of the four terms in the parentheses equals 0 under the separation condition. Combining (3.18), (3.19) and (3.11) yields K312 2 Pier. 2 (It — 5|1/2 + I231 — y1|+|$2 "— 3120 (3??) whenever xk, y;C (k = 1, 2) satisfy the above conditions. By (3.5), we have (0X — 0y)2 3 c(|y1 -— x1] + I232 — y2|)2. It follows from (3.17) and (3.22) that (3.16) holds whenever |y1 — x1| + |x2 — y2| is sufficiently small. Finally, a continuity argument as in Mueller and Tribe (2002, p.15) removes this last restriction. This finishes the proof of Lemma 3.5. The present chapter is a continuation of the paper of Mueller and Tribe (2002). Our objective is to study the fractal properties of various 101 random sets generated by the random string processes. In Section 3.2, we determine the Hausdorff and packing dimensions of the range u ([0, 1]?) and the graph Gru ([0,1]?) We also consider the Hausdorff dimension of the range u(E), where E g [0, 00) x IR is an arbitrary Borel set. In Section 3.3, we consider the existence of the local times of the random string process and determine the Hausdorff and packing dimensions of the level set Lu 2 {(t,x) E (0,00) x IR : ut(x) = u}, where u E IR". Finally, we conclude this chapter by determining the Hausdorff and packing dimensions of the sets of two kinds of double times of the random string in Section 3.4. 3.2 Dimension results of the range and graph In this section, we study the Hausdoff and packing dimensions of the range u([0,1]2) = {ut(x) : (t,x) E [0,1]2} C IR" and the graph Gru([0,1]2) = { ((t,x),ut(x)) : (t,x) E [0,1]2} C 1R2”. Theorem 3.1 Let {ut(:r) : t 2 0, x E IR} be a random string process taking values in IRd. Then with probability 1, dimHu([0,1]2) = min {d; 6} (3.23) 102 and 2+§d1f1§d<4, dimHGru([0,1]2) = 3 +511 ,f 4 g d < 6, (3.24) 6 if 6 S d. Proof: Corollary 2 of Mueller and Tribe (2002) states that the distributions of {111(1) : t Z 0, x E IR} and the stationary pinned string U = {U,(x) : t _>_ 0, x E IR} are mutually absolutely continuous. Hence it is enough for us to prove (3.23) and (3.24) for the stationary pinned string U = {U,(:r) : t Z 0, x E IR}. This is similar to the proof of Theorem 4 of Ayache and Xiao (2005). We include a self-contained proof for reader’s convenience. As usual, the proof is divided into proving the upper and lower bounds separately. For the upper bound in (3.23), we note that clearly dimHU ([0, 1]2) g d as, so we only need to prove the following inequal- ity: dimHU([O,1]2) g 6 as. (3.25) Because of Lemma 3.1, one can use the standard entropy method for estimating the tail probabilities of the supremum of a Gaussian process to establish the modulus of continuity of U = {U,(x) : t Z 0, x E IR}. See, for example, Kono (1975). It follows that, for any constants 0 < 71 < '71 < 1/4 and 0 < 72 < ”y; < 1 / 2, there exist a random variable 103 A > 0 of finite moments of all orders and an event 91 of probability 1 such that for all 62 E (21, US ' . —— U '. sup l ,(yw) {(I’wII S A(6o). (3.26) I . (any).(t..1-)e[o.1]2 I8 — tI "1 + Ix — y|12 Let w E Q] be fixed and then suppressed. For any integer n 2 2, we divide [0, 1]2 into n6 sub-rectangles {Rm} With sides parallel to the axes and side-lengths n“1 and n‘2, respectively. Then U ([0, 1]?) can be covered by the sets U(R,,,,-) (1 S i 3 716). By (3.26), we see that the diameter of the image U (Rm) satisfies diamU(R,,,,:) S K311 n71”, (3.27) where 6 = max{1 — 471,1 — 27.3}. We choose 7’} E (71, 1/4) and 73 E (7.2, 1 / 2) such that 1 1 (1—6) (—+—) >6. ’71 ’72 Hence, for 7 = 711- + $2, it follows from (3.27) that 6 n 2: IdiamU(R ,,,- )I g K323 n6 n’(1_6)1 —+ 0 (3.28) 2'21 104 as n —> 00. This implies that dimHU(I0,1I2) g ’y as. By letting 71 T 1/4 and 72 T 1 / 2 along rational numbers, respectively, we derive (3.25). Now we turn to the proof of the upper bound in (3.24) for the sta- tionary pinned string U. We will show that there are three different ways to cover GrU (I0, 1I2), each of which leads to an upper bound for dimHGrU(IO, 1]?) O For each fixed integer n 2 2, we have n6 GrU( ([0,) 1]2 _UR. (3.29) It follows from (3.27) and (3.29) that GrU([0, 1]?) can be covered —1+6 by n6 cubes in IR?” with side-lengths K3,2,3n and the same argument as the above yields dimHGrU([O,1I2) g 6 as. (3.30) 0 Observe that each Rm,- x U (Rm) can be covered by (",1 cubes in IR2+d of sides n'4, where by (3.26) —1+6 ‘1 , 11 5,1,1 S A3.2.4712 X ( > - 105 Hence GrU ([0, 1I2) can be covered by n6 x 6“ cubes in IR2+d with sides n74. Denote U1: 2 +(1— “/1)Cl. Recall from the above that we can choose the constants “/1, 7'1 and 7’2 such that 1 — 6 > 471. Therefore n6 X 6,“ X (Tl-4),,” S K3‘215n—(1—6—471M —> 0 as n —> 00. This implies that dimHGrU(I0,1I2) g 771 almost surely. Hence, dimHGrU([O,1I2) g 2 + 3d, 5.3. (3.31) 0 We can also cover each Rm x U (RM) by tn; cubes in IR?” of sides n‘z, where by (3.26) n—1+6 d €n,2 S K326 < _2 > - n Hence GrU ([0, 1]?) can be covered by n6 X 67,,2 cubes in lR2+d with sides n‘2. Denote 772 = 3 + (1 — 72)d. Recall from the above that we can choose the constants 72, 7'1 and 7’2 such that 1 — (5 > 272. 106 Therefore n6 x 6.2 x Wt)”2 s 1(3,2,7n—“—“‘2"2)d -—> 0 as n —+ 00. This implies that diInHGrU(I0,1I2) S 172 almost surely. Hence, 1 dimHGrU([0,1I2) g 3 + 5d, 5.3. (3.32) Combining (3.30), (3.31) and (3.32) yields 3 1 dimHGrU([0,1I2) 3 min {6, 2 + 1d, 3 + 2d}’ as. (3.33) and the upper bounds in (3.24) follow from (3.33). To prove the lower bound in (3.23), by Frostman’s theorem it is sufficient to show that for any 0 < ’y < min{d, 6}, 1 8., : / / E( ) dsdydtdx < 00. 3,34 [0.112 16.112 lUs(y) — Utaw ( ) See, e.g., Kahane (1985a, Chapter 10). Since 0 < 'y < d, we have 0 < IE(IEI"I) < 00, where E is a standard d—dimensional normal vector. Combining this fact with Lemma 3.1, we have 1 1 1 1 1 8,. g K323] ds/ dt/ dy/ dx. (3.35) 107 Recall the weighted arithmetic-mean and geometric-mean inequality: for all integer n 2 2 and x,- Z 0, {3, > 0 (i = 1,...,n) such that I'_ .3,- = 1 we have 21—1 . ’ Tl n )3 r b [Iatg§:aa. 93m i=1 i=1 Applying (3.36) with n = 2, )31 = 2/3 and )32 = 1/3, we obtain 2 , 1 Is —t|1/2+ Ia: —y| 2 gls —— ill/2+ glic— y) 2 Is -— tl‘/3I:c —— 1111/3. (337) Therefore, the denominator in (3.35) can be bounded from below by Is — t|7'/G|x -— y|“’/6. Since 7 < 6, by (3.35), we have 8., < 00, which proves (3.34). For proving the lower bound in (3.24), we need the following lemma from Ayache and Xiao (2005). Lemma 3.6 Let a, 6 and 77 be positive constants. For a > 0 and b > 0, let 1 dt J:= J(a,b)=/0 (a+t")l’(b+t)"° (3.38) Then there exist finite constants K329 and K33“), depending on 01, 6, 77 only, such that the following hold for all reals a, b > 0 sat- isfying (ll/a S K3239 b. (i) ifafl >1, then 1 J S K3210 (7:07— b"; (3.39) 108 (ii) if 01,8 2 1, then 1 < I. ' J _ A3,2,10 b’l log (1 + ba’l/a); (3.40) (iii) if0 < (113 <1 and 01/3 + 77 #1, then , 1 J S A3.2.10 (W +1) (341) IV Now we prove the lower bound in (3.24). Since dirnHGrU ([0, IV) dimHU([0,1I2) always holds, we only need to consider the cases 1 g d < 4 and 4 S d < 6, respectively. Since the proof of the two cases are almost identical, we only prove the case when 1 S d < 4 here. Let 0 < 7 < 2 + %d be a fixed, but arbitrary, constant. Since 1 S d < 4, we may and will assume 7 > 1 + d. In order to prove dimHGrU(I0,1]2) 2 7 as, again by Frostman’s theorem, it is sufficient to show dsdydtdx 97 = 2 2 IE 9 7/2 < oo. [0.11 [0.1] (Is —1t|2 + la: - yl~ + lUs(y) - Ut(1r)|2) (3.42) Since 7 > d, we note that for a standard normal vector E in IR‘ll and any number a E IR, eI 1 (a2 + IEI2) —~—d 7/2I S K3,2,11a (’ )1 109 see e.g. Kahane (1985a, p.279). Consequently, by Lemma 3.1, we derive that ' dsd' dtdx 94, S A3.2.12 / 2/ 2 (52 ~~d ' [0.1] [0.1] (Is—tl1/2+ Ix—yl) (IS—ti + I'l3—3/I)7 (3.43) By Lemma 3.6 and a change of variable and noting that d < 4, we can apply (3.41) to derive -1 1 1 . < K . d , dt 9” “ ”It/6 I/o (ti/2 +x)‘1/2(t +x)rd l 1 S K312114/0 ($(l/-1+7-dl +1) (1T < 00’ where the last inequality follows from 7 — %d - 1 < 1. This completes (3.44) the proof of Theorem 3.1. By using the relationships among the Hausdorff dimension, packing dimension and the box dimension [see Falconer (1990)], Theorem 3.1 and the proof of the upper bounds, we derive the following analogous result 011 the packing dimensions of u([0, 1]?) and Gru([0, 1I2). Theorem 3.2 Let {u,(x) : t Z 0, x E IR} be a random string process taking values in IR‘I. Then with probability 1, dimpu([0,1I2) = min (d; 6} (3.45) 110 T‘TQ‘m-s" and 2+§d if1§d<4, dimpGruflO, 112) = 3 +511 if 4 g d < 6, (3-46) 6 zf6gd. Theorems 3.1 and 3.2 show that the random fractals u(IO, II?) and Gru([0,1I2) are rather regular because they have the same Hausdorff and packing dimensions. Now we will turn our attention to find the Hausdorff dimension of the range u(E) for an arbitrary Borel set E Q [0, 00) x IR. For this purpose, we mention the related results for an (N, d)- fractional Brownian sheet B" 2 {BH (t) : t E IRQ'} with Hurst index H 2 (H1, . . . , HN) E (0, 1)N [cf Section 2.3]. What the random string process {u,(x) : t 2 0, x E IR} and a (2,d)-fractional Broanian sheet BH with H = (i, %) have in common is that they are both anisotropic. As Section 2.3 pointed out, the Hausdorff dimension of the image BH (F) cannot be determined by dimHF and H alone for an arbitrary fractal set F, and more information about the geometry of F is needed. To capture the anisotropic nature of B”, we have introduced a new no- tion of dimension, namely, the Hausdorfir dimension contour, for finite Borel measures and Borel sets and showed that dimHBH (F) is deter- mined by the Hausdorff dimension contour of F. It turns out that we can use the same technique to study the images of the random string. 111 We start with the following Proposition 3.2 which determines dimHu(E) when E belongs to a special class of Borel sets in [0, 00) X IR. Its proof is the same as that of Proposition 2.2. Proposition 3.2 Let {u,(x) : t 2 0, x E IR} be a random string in RI. Assume that E1 and E2 are Borel sets in [0,00) and IR, respectively, which satisfy dimHEl 2 dimPEl or dimHE2 2 dimPEg. Let E 2 E1 X E2 C [0, 00) X IR, then we have dimHu(E) 2 min {d; 4dimHE1 + 2dimHE2} , as. (3.47) In order to determine dimHu(E) for an arbitrary Borel set E C [0, 00) X IR, we recall from Section 2.3 the following definition. Denote by MERE) the family of finite Borel measures with compact support in E. Definition 3.1 Given [1 E ME(E), we define the set A,, C_: IR2+ by 7"—'*0+ 7~4/\l+2)\2 A, = {A = (1., A2) 6 113,; in?“ (”I”): 2) = 0, (3.48) for u-ae (t,x) E [0, 00) X IR}, where R((t, x). r) 2 It — r4, t + r4] X Ix — r2, :5 + r2]. The properties of set A“ can be found in Lemma 2.4. The boundary of A,“ denoted by BA,“ is called the Hausdorff dimension contour of u. 112 Define A(E) = U A,,. #EME-(E) and define the Hausdorff dnnenslon contour of E by U116 MU E) (9A,, It can be verified that, for every ,8 E (0,0c)2, the supremum sup,€A(E) (A,13) is achieved on the Hausdorff dimension contour of E (Lemma 2.4). Theorem 3.3 Let u 2 {11,(x) : t Z 0, x E IR} be a random string process with values in IRd. Then, for any Borel set E C [0, 00) X IR, dimHu(E) 2 min Id; s(E)}, as. (3.49) where s(E) 2 sup,€A(E)(4)\1 + 2A2). Proof: By Corollary 2 of Mueller and Tribe (2002), one only needs to prove (3.49) for the stationary pinned string U 2 {Ut(x) : t 2 0, x E IR}. The latter follows from the proof of Theorem 2.2. 113 3.3 Existence of the local times and dimension re- sults for level sets In this section, we will first give a sufficient condition for the existence of the local times of a random string process on any rectangle I E A, where A is the collection of all the rectangles in [0, 00) X IR with sides parallel to the axes. Then, we will determine the Hausdorff and packing dimensions for the level set Lu 2 {(t,x) E [000) X IR : u,(x) 2 u}, where u E IR! is fixed. The following theorem is concerned with the existence of local times of the random string. Theorem 3.4 Let {ut(x) : t Z 0, x E IR} be a random string process in W. If d < 6, then for every I E A, the string has local times {l(u, I), u E IRd} on I, and l(u, I) admits the following L2 representa- tion: l(u, I) 2 (27r)_d/dexp(—i(v, u)) [exp(i(v, ut(x)))dtdxdv,‘v’ u E IRd. IR I (3.50) Proof: Because of Corollary 2 of Mueller and Tribe (2002), we only need to prove the existence for the stationary pinned string U 2 {Ut(x): t 2 0, x E IR}. Let I E A be fixed. Without loss of generality, we may assume 114 I 2 Is, 1I2. By (21.3) in Geman and Horowitz (1980) and using the char- acteristic functions of Gaussian random variables, it suffices to prove the integral ,7 (I) defined by ‘/Idtdx/Idsdy [11“ du /111d IIEexp(i(u, Ut(x)) + i(v, U,(y))) Idv (3.51) is finite. Since the components of U are i.i.d., it is easy to see that —d/2 j(I) 2 (27r)d/Idtdx/I IdetCov(U,1(x), U91(y))I dsdy. (3.52) By Lemma 3.3 and noting that I 2 Is, 1I2, we can see that detCov(U,j(x), Uj(y)) 2 Var(U8j(y))Var(U,j(x)IUg(y)) S (3.53) 2 K3,3,1(I.I‘ — yI ‘1‘ It — 8I1/2). The above inequality, (3.37) and the fact that d < 6 lead to 1 1 1 1 J(I) S K333] / Is — tI‘d/fidtds/ / Ix — yI‘d/6dxdy < oo, (3.54) which proves (3.51), and therefore Theorem 3.4. Remark 3.3 It would be interesting to study the regularity proper- ties of the local times l(u,t), (u E IRd,t E [0,00) X IR) such as joint 115 in” . .' "ta-5*. _rau.“ — continuity and moduli of continuity. One way to tackle these problems is to establish sectorial local nondeterminism [cf Section 2.2] for the stationary pinned string U 2 {Ut(x) : t _>_ 0, x E IR}. This will have to be pursued elsewhere. Some results of this nature for certain isotropic Gaussian random fields can be found in Xiao (1997). Mueller and Tribe (2002) proved that for every u E IRd, IP’{ut(x) 2 u for some (t,x) E [000) X IR} > 0 if and only if d < 6. Now we study the Hausdorff and packing dimen— sions of the level set Lu 2 {(t,x) E [0, 00) X IR : ut(x) 2 u}. Theorem 3.5 Let {ut(x) : t Z 0, x E IR} be a random string process in IRd with d < 6. Then for every 11 E IRd, with positive probability, 2—ldi 1 n_(1“6l} (3.57) (~9:yla(t:-T)€Rn.t’ S IID{IU(T,,,{) — uI S n‘ll76l} + exp(—cums) S K333 n—(l—M- In the above we have applied Lemma 3.1 and the Gaussian isoperimet- ric inequality [cf. Lemma 2.1 in Talagrand (1995)] to derive the second inequality. Since we can deal with the cases 1 S d < 4 and 4 S d < 6 almost identically, we will only consider the case 1 S d < 4 here and leave the case 4 S d < 6 to the interested readers. Define a covering {RIM} of Lu O [5, 1]2 by HI“ = Rnx if 11 E U(Rn.[) and RIM 2 0 otherwise. Note that each RIM can be covered by n2 4 squares of side length n‘ . Thus, for every n 2 2, we have obtained 117 a covering of the level set Lu (‘1 [5,1]2 by squares of side length n‘4. Consider the sequence of integers n 2 2" (k _>_ 1), and let N;c denote the minimum number of squares of side-length 2"“ that are needed to cover Lu 0 [5, 1I2. It follows from (3.57) that IE(N),.) _<_ K3332“-22k-2‘k(1‘5)d = 113,3,3 2k(8-<1-5>“>. (3.58) By (3.58), Markov’s inequality and the Bore-Cantelli lemma we derive that for any 6’ E (0, 6), almost surely for all 16 large enough, N, g K3333 268—022"). ‘ (3.59) By the definition of box dimension and its relation to dimP Icf. Falconer (1990)], (3.59) implies that dimP (LuflIe, 1I2) S 2—(1—6’)d/4 as. Since 5 > 0 is arbitrary, we obtain the desired upper bound for dimP (Lu 0 [5,1I2) in the case 1 S d < 4. Since dimHE S dimPE for all Borel sets E C IR2, it remains to prove the following lower bound: for any 5 E (0, 1), with positive probability 2 2—idif1Sd<4 61111,.(Lu m [5,1] ) 2 (3.60) 3—§d if 4Sd<6. We only prove (3.60) for 1 S d < 4. The other case is similar and is 118 omitted. Let 6 > 0 such that 7:22- i(1+6)d> 1. (3.61) Note that if we can prove that there is a constant K 334 > 0 such that IP’{dimH(Lu r) [5,]1] ) > 7} > K334. (3.62) then the lower bound in (3.60) will follow by letting 6 I 0. Our proof of (3.62) is based on the capacity argument due to Kahane Isee, e.g., Kahane (1985a)I. Similar methods have been used by Adler ( 1981), Testard (1986), Xiao (1995), Ayache and Xiao (2005) to various types of stochastic processes. Let MI be the space of all non-negative measures on [0,1]? with finite 7-energy. It is known [cf. Adler (1981)] that MI is a complete metric space under the metric Hull =/, /,(’“lt 4144444 d” . (3.63) 2 It- 3|? + lx- ylz)” We define a sequence of random positive measures .1174 on the Borel sets 119 of [5,1]2 by , 1.. — 2 [134(0)=/(27r11)d/2exp(—nl(t(T2) u| )dtdx C 2 2 C A .1 exp ( — Lg;— +1(§,U3(.1:) — u))dg dtdx, (3-64) V C E B([5, 1]2). It follows from Kahane (1985a) or Testard (1986) that if there are positive constants K335 and K335, which may depend on 11, such that BMW“) 2 K335, IE(||/-"n”2) S K3313, (3-65) Rial/171”?) < +00: (3'66) where ||,u.,3|| = ,un([s,1]2), then there is a subsequence of {11,3}, say {#744}, such that 11,”: —+ p in MI and p is strictly positive with proba- bility 2 K 32,335 / (2K 3,3,6). It follows from (3.64) and the continuity of U that )1 has its support in Lu F] [8,1]2 almost surely. Hence Frostman’s theorem yields (3.62). 120 It remains to verify (3.65) and (3.66). By Fubini’s theorem we have lE-Hllflnll — '5 5|? - ,. . -/[€1]2/Rdexp(:: u))CXP( 27 )EGXP(Z<€,U1(J)))d£dtdIL = exp(— u))exp(—l(n_1+0‘ (t, :c))|€|2) dédtdr [51]2 Rd 2 32 IuI'Z Z - did. L312 (n- 1+20% 15, 3:) <1) eXp( 2(n-1+a2(t,3:))) T .142 lulz > — dt 2: K — [[5,1]2 (12+027r(t, 73)) ex xp( 202(t,7:)) 335’ (3.67) where 020,1") 2 1E [(Ut1 (I))2] Denote by [M the identity matrix of order 2d and by Cov(U3(y),U,(:I_:)) the covariance matrix of the Gaussian vector (U3(y),U3(7:)). Let F = 7741241 + Cov(U3.(y),U3(7:)) and let (§,n)’ be the transpose of the row vector (6,77). As in the proof of (3.51), we 121 apply (3.14) in Lemma 3.3 and the inequality (3.36) to derive EUIIUHII?) 2/15112/1531} [Id/dexp(—t<€+n,u)) x exp ( — —(5 n) 1‘(€,77)') dédndsdydtdw 1 ——u.u F-l u,u' dsd dtdrr /11/[=11\fli?t_ (2<,> (1)11 d 0 small enough, say, L. h L for all t2 6 [C12, a2 + h] and t1 6 [(11, (11+ h]. We will use this assumption together with Lemma 3.4 to prove Theorem 3.6 below. We denote the collection of the hypercubes having the above properties by J. The following theorem gives the Hausdorff and packing dimensions of the Type I double times of a random string. Theorem 3.6 Let u = {u3(.7:) : t 2 0, a: E IR} be a random string process in Rd. Ifd 2 12, then L132 2 (b (1.5. Ifd < 12, then, for every J E J, with positive probability, » 4—id iflgd<8, dimH(L33nJ) = dimP(LL3nJ) = (3.73) 6—%d if 8_<_d<12. Proof: The first statement is due to Mueller and Tribe (2002). Hence, we only need to prove the dimension result (3.73). Thanks to Corollary 2 of Mueller and Tribe (2002), it is suffi- cient to prove (3.73) for the stationary pinned string U. This will be done by working with the zero set of the (4, (1)-Gaussian field 125 AU 2 {AU(t1,a:1;t2,1:2)} define by (3.71). That is, we will prove (3.73) with L12 replaced by the zero set (AU)_1(0). The proof is a modification of that of Theorem 3.5. Hence, we only give a sketch of it. For an integer n 2 2, we divide the hypercube J into n12 sub-domains Twp = R3”, x R1213» where RIM”, R2,], C (0, 00) X IR are rectangles of side lengths n‘4h and 71’2h, respectively. Let 0 < 6 < 1 be fixed and let Tip be the lower-left vertex of Hip (k = 1, 2). Then the probability IP{0 E AU(T,,_I,)} is at most IP{ mgx IAU(t1,.7:1; t2, 182) — ill/(814 .711; 82, y2II 3.71—(1—6b0 E AU(TTMP)} + IP{ m’ax IAU(t11,a:1;t2,7: — AU(31,y1; 52,312” > 73—0-07} 3 1141411165,, 43,,” g 44-0-6) + IP{mgxIAU(t1,:1:1;t2,:1:2) — AU(31,y1; 52,y2)I > 71-0—07}: (3.74) where D = {(t1,:r1;t2,a:2), (51,311; 82, 312) E T344}. By the definition of J, we see that AU (7312,12: 73,12) is a Gaussian ran- dom variable with mean 0 and variance at least K Ll/z. Hence the first term in (3.74) is at most K3,4,1n‘(1‘5)d. 126 On the other hand, note that to IAU(81,y1;82,y2) — AU(t1,I1;t2,I2)I S X Z k=1 U8k(yk) - Unix/(H: we have IP{1113XIAU(t11$Iit27$2) — AU(517y1;827y2)I > ”_(1—6)} 71"(1‘6) C 2 S ZP{ max k IU-sk(yk) - Utk($k)l > (SksykllfksfmeRhJ, } (3.75) S exp(—K342 71%) a I where the last inequality follows from Lemma 3.1 and the Gaussian isoperimetric inequality [cf Lemma 2.1 in Talagrand ( 1995)]. Combine (3.74) and (3.75), we have 117(0 6 AU(T,,,,)} g K3,4,1n’(1’5)‘1+ exp(—[(3437426) (3.76) Hence the same covering argument as in the proof of Theorem 3.5 yields the desired upper bound for dimP ((AU)‘1(0) F) J). This proves the upper bounds in (3.73). Now we prove the lower bound for the Hausdorff dimension of (AU)‘1(0) I) J. We will only consider the case 1 S d < 8 here and leave the case 8 _<_ d < 12 to the interested readers. 127 Let 6 > 0 such that 1 - 7:24—Z(1+b)d> 2. (3.77) As in the proof of Theorem 3.5, it is sufficient to prove that there is a constant K343 > 0 such that lP’{dimH (L192 0 J) 2 ’7} Z K3143. (3.78) Let NI be the space of all non-negative measures on [0,1]4 with finite y-energy. Then NI is a complete metric space under the metric H II =/ / jdtldcrldtzdxau(dsldyldsdm (3.79) see Adler (1981). We define a sequence of random positive measures u" on the Borel set J by A t , 't. 2 143(0): /C(27rn)d/2exp(_ ”I U( 17:13 233:2)I )dtldxldtgdxg =//dexp(—2n-l——€l2+i,(§AU(t1,:1:1;tg,:1:2)))dédtldxldtgdxg. (3.30) It follows from Kahane (1985a) or Testard (1986) that (3.78) will follow 128 if there are positive constants K344 and K343 > 0 such that IE(IIV-nII) 2 [(3.143 IE(||1/,,|I2) S K3153 (381) Ewwm)<+m, as” where III/"II = V,,,(J). The verifications of (3.81) and (3.82) are similar to those in the proof of Theorem 3.5. By Fubini’s theorem we have IE(III/)nll 2 —-—//d €Xp ()ISI IEGXp (Mg A(/(t1,£€1;t2, .172») dgdtldl‘ldtgdl‘g = [I/Rd exp ( — §£(n—lld + Cov(AU(t1, x1; t2, x2)))€’) d6 dtldxldbdwz _ / (277)“2 — J \/det( n"lld+ Cov(AU(t1,:r1;t23$2))) >/J (27r)d/2 J\/det( 14 + Cov( (AU(t1,$1; @1332») dt1d131dt2d1‘2 dtldflildtzdftg I: K33434' wsm Denote by Cov(AU(sl,y1; 32,y2),AU(t1,:c1; t2,7:2)) the covariance ma- trix of the Gaussian vector (AU(31,y1; 52, 3,12), AU(t1, 7:1; t2,:c2)) and let P = 71—11271 + COV(AU(51: yl; 82, .92), A(1(151, 1131; t2: 172))- 129 Then by the definition of J and (3.15) in Lemma 3.4, we have IE(III/nil?) = 1 I ///d /(1 exp ( — 5(5a77)F(€,U))(15177d51dy1d82dy2dt1d$1dt2dgy2 J J IR IR 4112:: 0and b1 < R2. We choose h > 0 small enough, say, b—b h< 2312K. Then |;1:2 — x1| > n for all 3:2 6 [b2, ()2 + h] and £1 E [(21, b1 + h]. We denote the collection of all the cubes K having the above properties by K. 131 By using Lemma 3.5 and a similar argument as in the proof of Theo- rem 3.6, we can prove the following dimension result on L112. We leave the proof to the interested readers. Theorem 3.7 Let u = {ut(I) : t Z 0, :1: 6 IR} be a random. string process in Rd. Ifd Z 8, then L113 = (b as Ifd < 8, then for every K 6 IC, with positive probability, 3—i—d if1§d<4, dimH (L113 0 K) = dimp (L112 0 K) = 4—gd if 4§d<8. (3.86) Remark 3.4 Rosen (1984) studied k-multiple points of the Brownian sheet and multiparameter fractional Brownian motion by using their self-intersection local times. It would be interesting to establish similar results for the random string processes. 132 CHAPTER 4 Concluding Remarks In this dissertation, we have studied the geometric properties of two kinds of anisotropic Gaussian random fields. 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