非退化扩散过程的水平集和极集

设X(t)(t∈R )是一个d维非退化扩散过程．本文得到了比原有结果更一般的非退化扩散过程极性的充分条件,证明了对任意u∈Rd,紧集E(0, ∞),有若d=1,则对任意紧集F(?)R, 若d≥2,则对任意紧集E ∈(0, ∞), 其中B(Rd)为Rd上的Borel σ-代数,dim和Dim分别表示Hausdorff维数和Packing 维数．

The Level Sets And The Polar Sets For Nondegenerate Diffusion Processes

Let $X(t) (tin R_+)$ be the $d$-dimensional nondegeneratediffusion processe. More generalized sufficient conditions than the former onesfor a compact set $Fsubset R^d$ to be a Polar set are proved.It is also proved that for any $uin R^d$, any compact $Ein R_+$,$$P{{rm dim}(X^{-1}({u})cap E)=maxBig{0,{rm dim}E-frac{d}{2}Big}>0,$$and if $dgeq 2$, then for any compact set $Esubset R_+,$$$inf{{rm dim}F:Fin{cal B}(R^d),P(X(E)cap Fnot= phi )>0}=d-2{rm Dim}E,$$and if $d=1$, then for any compact set $Fsubset R,$$$inf{{rm dim}E:Ein {cal B}(R),P(X(E)cap Fnot= phi )>0}=frac{1}{2}-frac{{rm Dim}F}{2},$$where ${cal B}(R^d)$ denotes the Borel $sigma $-algebra in$R^d $, and dim and Dim are Hausdorff dimension andPacking dimension respectively.