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Hitting Probabilities and Fractal Dimensions of Multiparameter Multifractional Brownian Motion

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Acta Mathematica Sinica,English Series 2013年第9期29卷 1723-1742页

作者： zhen long chen School of Statistics and Mathematics Zhejiang Gongshang University

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.

关键词： Multifractional Brownian motion hitting probability inverse image level set Hausdorff dimension packing dimension

Polar Functions of Multiparameter Bifractional Brownian Sheets

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Acta Mathematicae Applicatae Sinica 2009年第2期25卷 255-272页

作者： zhen-long chen College of Statistics and Mathematics Zhejiang Gongshang University Hangzhou 310018 China

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.

关键词： Bifractional Brownian sheet polar function Hausdorff dimension packing dimension capacity

Hitting Probabilities and the Hausdorff Dimension of the Inverse Images of a Class of Anisotropic Random Fields

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Acta Mathematica Sinica,English Series 2015年第12期31卷 1895-1922页

作者： zhen long chen Quan ZHOU School of Statistics and Mathematics Zhejiang Gongshang University

Let X = {X（t）：t ∈ R^N} be an anisotropic random field with values in R^*** certain conditions on X,we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel-Riesz *** also obtain the Hausdorff dimension of its inverse image,and the Hausdorff and packing dimensions of its level *** results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.

关键词： Anisotropic random field non-linear stochastic heat equations spatially homogeneous Gaussian noise hitting probabilities Hausdorff dimension inverse image

Dimension Results for Space-anisotropic Gaussian Random Fields

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Acta Mathematica Sinica,English Series 2019年第3期35卷 391-406页

作者： Wen Qing NI zhen long chen Wei Gang WANG School of Statistics and Mathematics Zhejiang Gongshang University School of Science Jimei University

Let X = {X(t) ∈ R^d, t ∈ R^N} be a centered space-anisotropic Gaussian random field whose components satisfy some mild conditions. By introducing a new anisotropic metric in R^d, we obtain the Hausdorff and packing dimension in the new metric for the image of X. Moreover, the Hausdorff dimension in the new metric for the image of X has a uniform version.

关键词： Hausdorff dimension packing dimension gaussian random field uniform dimension

Hausdorff-type Measures of the Sample Path of Gaussian Random Fields

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Acta Mathematicae Applicatae Sinica 2005年第4期21卷 623-636页

作者： zhen-long chen San-yang Liu School of Statistics and Computation Science Zhejiang Gongshang University Hangzhou 310035 China Department of Applied Mathematics Xidian University Xi'an 710071 China

Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X（t）（t ∈ R^N） be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX（[0, 1]N）, and evaluate the exact φ - mA measure of the image and graph set of X（t）. A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X（[0, 1] ^N） and GrX（[0, 1]^N） are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite （ φ A）-Hausdorff measure for X（[0, 1]^N） and GrX（[0, 1]N）.

关键词： Gaussian random fields image graph Hausdorff-type measure

Packing Dimension of Space-time Anisotropic Gaussian Random Fields

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Acta Mathematica Sinica,English Series 2021年第12期37卷 1826-1840页

作者： hen long chen Jun WANG Dong Sheng WU School of Statistics and Mathematics Zhejiang Gongshang UniversityHangzhou 310018P.R.China School of Mathematics and Finance Chuzhou UniversityChuzhou 239000P.R.China Department of Mathematical Sciences University of Alabama in HuntsvilleHuntsvilleAlabama 35899USA

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 *** 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 *** results generalize the corresponding results of Estrade,Wu and Xiao(***.,5,41-64(2011))for time-anisotropic Gaussian random fields to space-time anisotropic Gaussian fields.

关键词： Gaussian random fields anisotropic packing dimension packing dimension profile range

Hausdorff Measure of Space Anisotropic Gaussian Processes with Non-stationary Increments

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Acta Mathematicae Applicatae Sinica 2024年

作者： Jun WANG zhen-long chen Wei-jie YUAN Guang-jun SHEN School of Mathematics and Physics Anhui Polytechnic University School of Statistics and Mathematics Zhejiang Gongshang University Department of Mathematics Anhui Normal University

Let X={X(t), t∈R+}be a centered space anisotropic Gaussian process values in Rdwith non-stationary increments, whose components are independent but may not be identically distributed. Under certain conditions, then almost surely c1≤?-m(X([0, 1]))≤c2, where ? denotes the exact Hausdorff measure associated with function ?(s)=s(1/αk+∑i=1k(1-αi/αk)) log log 1/s for some 1≤k≤d,(α1,···, αd)∈(0, 1]d. We also obtain the exact Hausdorff measure of the graph of X on [0, 1].

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多参数广义布朗单的象与容度

数学学报（中文版） 2009年第2期52卷 269-280页

作者：陈振龙浙江工商大学统计与数学学院杭州310018

研究了既没有平稳增量性,也没有scaling性质的N指标d维广义布朗单象的容度问题.证明了多参数广义布朗单具有扇局部不确定性,给出了一族多参数广义布朗桥的性质.应用这些结论,得到了多参数广义布朗单象集的Lebesgue测度与容度之间的关系... 详细信息

研究了既没有平稳增量性,也没有scaling性质的N指标d维广义布朗单象的容度问题.证明了多参数广义布朗单具有扇局部不确定性,给出了一族多参数广义布朗桥的性质.应用这些结论,得到了多参数广义布朗单象集的Lebesgue测度与容度之间的关系,给出了其象集的确切容度估计.所得结果包含了布朗运动和布朗单的相应结果,也解决了Kahane提出的关于可加布朗运动的Bessel-Riesz容度问题.

关键词：多参数广义布朗单象集

球面高斯随机场像集和图集的Hausdorff测度

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数学学报（中文版） 2023年第5期66卷 951-970页

作者：桑利恒陈振龙浙江工商大学统计与数学学院杭州310018 滁州学院数学与金融学院滁州239000

本文考虑了单位球面S2上取值于Rd中的球面高斯随机场T={T(t),t∈S^(2)}的像集和图集的Hausdorff测度,其中指数α满足2 4时,Φ(r)=r^(4)/α-2log lor1/r.本文将欧氏空间相关结果推广到了球面情况.

关键词：球面高斯随机场像集图集 Hausdorff测度