UNIFORM PACKING DIMENSION RESULTS FOR MULTIPARAMETER STABLE PROCESSES

In this article, authors discuss the problem of uniform packing dimension of the image set of multiparameter stochastic processes without random uniform Holder condition, and obtain the uniform packing dimension of multiparameter stable processes. If Z is a stable （N, d, α）-process and αN ≤ d, then the following holds with probability 1 Dim Z（E）=α Dim E for any Borel setE ∈B（R ＋^N）, where Z（E）={x：E←t∈E,Z（t）=x}, Dim （E） denotes the packing dimension of E.

A NOTE ON MULTIFRACTAL PACKING DIMENSION OF MEASURES

The relations between the multifractal packing dimension of Borel probability measures and the asymptotic behavior of the function Ф＊（x）=limsup logv（B（x,r））-qlogu（B（x,r））/logr are discussed and some applications are given.

Packing Dimension of Space-time Anisotropic Gaussian Random Fields

Let X={X(t)∈R^(d),t∈R^(N)}be a centered space-time anisotropic Gaussian random field whose components are independent and satisfy some mild conditions.We study the packing dimension of range X(E)under the anisotropic(time variable)metric space(R^(N),ρ)and(space variable)metric space(R^(d),τ),where E⊂R^(N) is a Borel set.Our results generalize the corresponding results of Estrade,Wu and Xiao(Commun.Stoch.Anal.,5,41-64(2011))for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.

ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS

In this paper,we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions.More precisely,we show that,given a real number 1≤β≤2,any real-valued continuous function in C([0,1])can be decomposed into a product of two real-valued continuous functions,each having a graph of Hausdorff dimensionβ.In addition,a product decomposition result for the packing dimension is obtained.This work answers affirmatively two questions raised by Verma and Priyadarshi[14].

DIMENSION OF POLAR SETS FOR BROWNIAN SHEET

Let W={W(t);t ∈R_+~N} be the d-dimensional N-parameter Brownian Sheet. Sufficient conditions for a compact set F C Rd \ {0} to be a polar set for W are proved. It is also proved that if 2N≤d, then for any compact set E (?)R_>~N ,inf {dimF : F ∈B(Rd), P{W(E) ∩F≠(?)}>0} = d- 2DiroE, and if 2N > d, then for any compact set F C Rd \ {0}, inf{dim E : E ∈ B(R_>~N), P{W(E)∩F≠(?)}>0}=d/2-DimF/2,where B(Rd) and B(R_>~N) denote the Borel σ-algebra in Rd and R_>~N respectively, and dim and Dim are Hausdorff dimension and Packing dimension respectively.

THE DIMENSIONS OF THE RANGE OF RANDOM WALKS IN TIME-RANDOM ENVIRONMENTS

Suppose {Xn} is a random walk in time-random environment with state space Z^d, ｜Xn｜ approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.

THE PACKING MEASURE OF A CLASS OF GENERALIZED SIERPINSKI CARPET

For 1/4< a <(?)/4, let S1(x) =ax, S2(x)=1-a+ax, x∈[0,1]. Ca is the attractor of the iteratedfunction system {S1,S2}, then the packing measure of Ca×Ca isPs(a)(Ca×Ca) = 4·2s(a)(1-a)s(a),where s(a) = -loga4.

The Multifractal Formalism for Measures, Review and Extension to Mixed Cases

The mulfifractal formalism for single measure is reviewed. Next, a mixed generalized multifractal formalism is introduced which extends the multifractal formalism of a single measure based on generalizations of the Hausdorff and packing measures to a vector of simultaneously many measures. Borel-Cantelli and Large deviations Theorems are extended to higher orders and thus applied for the validity of the new variant of the multifractal formalism for some special cases of multi-doubling type measures.

MULTIFRACTAL FORMALISMS:BOXED VERSUS CENTERED INTERVALS

There are mainly two approaches to the multifractal analysis of measures. The first one,which is used in applications and in studying problems arising from dynamical systems,uses a hierarchy of boxes. The second one,which is more satisfactory from the viewpoint of geometric measure theory,uses more intrinsic concepts. This article is an account of a work by J.Barral,F.Ben Nasr,and J.Peyriére [3] which provides a bridge between these two theories.

Hitting Probabilities and Fractal Dimensions of Multiparameter Multifractional Brownian Motion

The main goal of this paper is to study the sample path properties for the harmonisable-type N-parameter multifractional Brownian motion, whose local regularities change as time evolves. We provide the upper and lower bounds on the hitting probabilities of an （N, d）-multifractional Brownian motion. Moreover, we determine the Hausdorff dimension of its inverse images, and the Hausdorff and packing dimensions of its level sets.

Polar Sets and Relative Dimension for Generalized Brownian Sheet

Let W^～=^～{W^～（t）; t∈ R＋^N） be a d-dimensional N-parameter generalized Brownian sheet. Necessary and sufficient conditions for a compact set E × F to be a polar set for （t,W^～（t）） are proved. It is also proved that if 2N ≤αd, then for any compact set E ∩→ R〉^N,d-2/α Dim E≤inf{dim F：F∈B（R^d）,P{W^～（E）∩F≠0}〉0}≤d-2/βDimE,and if 2N〉αd, then for any compact set F∪→R^d／{0},α/2（d-DimF）≤inf{dimE：E∈B（R〉^N）,P{W^～（E）∩F≠0}〉0}≤β/2（d-DimF）,where B（R^d） and B（R〉^N） denote the Borel σ-algebra in R^d and in R〉^N respectively, dim and Dim are Hausdorff dimension and Packing dimension respectively.

The Fractal Dimensions of the Level Sets of the Generalized Iterated Brownian Motion

Let{W1（t）, t∈R＋} and {W2（t）, t∈R＋} be two independent Brownian motions with W1（0） = W2（0） = 0. {H （t） = W1（｜W2（t）｜）, t ∈R＋} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H（t） = x} are established for any 0 T ≤ 1.

On the Projections of the Mutual Multifractal Rényi Dimensions

In this paper,we compare the mutual multifractal Renyi dimensions to the´mutual multifractal Hausdorff and pre-packing dimensions.We also provide a relationship between the mutual multifractal Renyi dimensions of orthogonal projections´of a couple of measures(µ,ν)in R^(n).As an application,we study the mutual multifractal analysis of the projections of measures.

Dimensional Results for the Moran-Sierpinski Gasket

In this paper, the dimensional results of Moran-Sierpinski gasket are considered. Moran-Sierpinski gasket has the Moran structure, which is an extension of the Sierpinski gasket by the method of Moran set. By the technique of Moran set, the Hausdorff, packing, and upper box dimensions of the Moran-Sierpinski gasket are given. The dimensional results of the Sierpinski gasket can be seen as a special case of this paper.

SOME GEOMETRIC PROPERLIES OFBROWNIAN MOTION ON SIERPINSKI GASKET

Let {X(t), ≥ 0} be Brownian motion on Sierpinski gasket.The Hausdorff and packingdimensions of the image of a compact set are studied. The uniform Hausdorff and packingdimensions of the inverse image are also discussed.

Polar Functions of Multiparameter Bifractional Brownian Sheets

Let B^H,K ： （B^H,K（t）, t ∈R＋^N} be an （N,d）-bifractional Brownian sheet with Hurst indices H = （H1,..., HN） ∈ （0, 1）^N and K = （K1,..., KN）∈ （0, 1]^N. The characteristics of the polar functions for B^H,K are investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of B^H,K is presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov＇s entropy index for B^H,K are obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.

On Multifractal of Cantor Dust

We consider quasi-self-similar measures with respect to all real numbers on a Cantor dust. We define a local index function on the real numbers for each quasi-self-similar measure at each point in a Cantor dust, The value of the local index function at the real number zero for all the quasi-self-similar measures at each point is the weak local dimension of the point. We also define transformed measures of a quasi-self-similar measure which are closely related to the local index function. We compute the local dimensions of transformed measures of a quasi-self-similar measure to find the multifractal spectrum of the quasi-self-similar measure, Furthermore we give an essential example for the theorem of local dimension of transformed measure. In fact, our result is an ultimate generalization of that of a self- similar measure on a self-similar Cantor set. Furthermore the results also explain the recent results about weak local dimensions on a Cantor dust.

Polar Functions and Intersections of the Random String Processes

Let {us （x） ： s 〉 0, x ∈ JR} be a random string taking values in ]Rd. The main goal of this paper is to discuss the characteristics of the polar functions of {us （x） ： s ≥ 0, x ∈ JR}. The relationship between a class of continuous functions satisfying the HSlder condition and a class of polar-functions of {us（x） ： s 〉 0, x ∈ R} is presented. The Hausdorff and packing dimensions of the set that the string intersects a given non-polar continuous function are determined. The upper and lower bounds are obtained for the probability that the string intersects a given function in terms of respectively Hausdorff measure and capacity.

Moduli of Continuity of a Class of N-parameter Gaussian Processes and Their Fast Points

We study the moduli of continuity of a class of N-parameter Gaussian processes and get some results on＇the packing dimension of the set of their fast points.