多维非退化扩散过程的象集与图集的一致Packing维数

杨新建

系统科学与数学 ›› 2007, Vol. 27 ›› Issue (5) : 669-675.

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系统科学与数学 ›› 2007, Vol. 27 ›› Issue (5) : 669-675. DOI: 10.12341/jssms10239
论文

多维非退化扩散过程的象集与图集的一致Packing维数

    杨新建
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The Uniform Packing Dimensions for the Image Sets and Graph Setsof the Nondegenerate Multidimensional Diffusion Processes

    Yang Xinjian
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摘要

设B(t)=(B(t))=(B1(t),B2(t),⋯,BN(t))为N维Brown运动,设α(x)=(αij(x),1\leqi≤d,1≤j≤N), β(x)=(βi(x),1≤i≤d),x∈Rd,1≤d≤N, α(x)和β(x)有界连续和满足Lipchitz条件,且存在常数c0>0,使得对每个x∈Rd, a(x)=α(x)α(x)∗的每个特征根都不小于c0. 设dX(t)=α(X(t))dB(t)+β(X(t))dt, 设d≥3.
可以证明
P(ω:DimX(E,ω)=DimGRX(E,ω)=2DimE,\q∀E∈B[0,∞))=1.这里 X(E,ω)={X(t,ω):t∈E},GRX(E,ω)={(t,X(t,ω)):t∈E},DimF表示F的Packing维数.

Abstract

Let X(t)=X(0)+∫0tα(X(s))dB(s)+∫0tβ(X(s))ds be a{\it{d}}-dimensional nondegenerate diffusion processes, where
B(t) is a Brownian Motion. If α(x) and β(x) are bounded continuous and satisfy Lipschitz conditions on Rd, and a(x)=α(x)α(x)∗ is uniformly positive definite, that is, for some positive constant c0 such that\ a(x)≥c0Id×d, for all x∈Rd, then we prove that when d≥3, one have P(ω:DimX(E,ω)=DimGRX(E,ω)=2DimE,∀E∈B[0,∞))=1,where {\rm Dim}F denote the Packing dimension of F for F⊂Rl(l≥1), and X(E,ω)={X(t,ω):t∈E},GRX(E,ω)={(t,X(t,ω)):t∈E}, ω∈Ω.

关键词

扩散过程 / Brown运动 / 象集 / 图集 / Packing维数.

Key words

Diffusion process / Brownian motion / image set / graph set / packing dimension.

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杨新建. 多维非退化扩散过程的象集与图集的一致Packing维数. 系统科学与数学, 2007, 27(5): 669-675. https://doi.org/10.12341/jssms10239
Yang Xinjian. The Uniform Packing Dimensions for the Image Sets and Graph Setsof the Nondegenerate Multidimensional Diffusion Processes. Journal of Systems Science and Mathematical Sciences, 2007, 27(5): 669-675 https://doi.org/10.12341/jssms10239
中图分类号： 60J60    60J65   
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