多维非退化扩散过程的象集与图集的一致Hausdorff维数

The Uniform Hausdorff Dimensions for the Image Sets and Graph Sets of the Nondegenerate Multidimensional Diffusion Processes

设 X(t) =X(0 ) +∫t0 α(X(s) ) d B(s) +∫t0 β(X(s) ) ds为一 d(d≥ 3)维非退化扩散过程 .令X(E) ={ X(t) :t∈E} ,GRX(E) ={ (t,X(t) ) :t∈ E} ,该文证明了 :对几乎所有 ω: E B([0 ,∞ ) ) ,有dim X(E,ω) =dim GRX(E,ω) =2 dim E,这里 dim F表示 F的

Let X(t)=X(0)+∫\+t\-0α(X(s)) d B(s)+∫\+t\-0β(X(s)) d s be a d dimensional nondegenerate diffusion process, where B(t) is a Brownian motion. If α(x) and β(x) are bounded continuous on R\+d and satisfying Lipschitz condition, and a(x)=α(x)α(x)\+* is uniformly positive definite, that is for some positive constant C\-0, a(x)≥C\-0I\-\{d×d\} , for all x∈R\+d , then we prove that, when d≥3:P(ω: dim X(E,ω)= dim GRX(E,ω)=2 dim E, for all E∈B(\[0,∞)))=1, where dim F denotes the Hausdorff dimension of F for FR\+l(l≥1) , and X(E,ω)={X(t,ω): t∈E},GRX(E,ω)={(t, X(t,ω)): t∈E}, ω∈Ω.