多维非退化扩散过程的象集与图集的一致Hausdorff维数

设X(t)=X(0)+∫^t_0α(X(s))dB(s)+∫^t_0β( 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dimX(E,ω)=dimGRX(E,ω)=2dimE,这里dimF表示F的Hausdorff维数。

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

 Let X(t)=X(0)+∫^t_0α(X(s))dB(s)+∫^t_0β( X(s))ds be a d dimensional nondegenerate diffusion process, whereB(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_0{d×d}, for all x∈R^d, then we prove that, when d≥3:P(ω: dimX(E,ω)=dimGRX(E,ω)=2dimE, for all E∈B(0,∞)))=1,where dimF 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.