多维非退化扩散过程的象集与图集的一致Packing维数

设B(t)=(B(t))=(B1(t),B2(t),…,BN(t))为N维Brown运动,设α(x)=(αij(x),1(≤)I(≤)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(ωDimX(E,ω)=DimGRX(E,ω)=2DimE,(A)E∈B0,∞))=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.

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

Let $X(t)=X(0)+\int_0^t\alpha(X(s)){\rm d}B(s)+\int_0^t \beta(X(s)){\rm d}s$ be a{\it{d}}-dimensional nondegenerate diffusion processes, where$B(t)$ is a Brownian Motion. If $\alpha(x)$ and $ \beta(x) $ are bounded continuous and satisfy Lipschitz conditions on $ R^d$, and $a(x)=\alpha(x)\alpha(x)^*$ is uniformly positive definite, that is, for some positive constant $c_0$ such that\ $ a(x)\geq c_0I_{d\times d}$, for all $x\in R^d$, then we prove that when $ d\geq 3$, one have $$P(\omega: {\rm Dim} X(E,\omega)= {\rm Dim} GRX(E,\omega)=2{\rm Dim} E,\forall E\in {\mathcal{B}}0,\infty))=1, $$where {\rm Dim}$F$ denote the Packing dimension of $F$ for $F \subset R^l(l\geq 1)$, and $X(E,\omega)=\{X(t,\omega): t\in E\},GRX(E,\omega)=\{(t,X(t,\omega)): t\in E\}, \ \omega \in {\it \Omega}$.