Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the class of polar functions are studied. Our theorem 1 improves the previous results of Graversen and Legall. Theorem2 solves a problem of Legall (1987) on Brownian motion.
设X(t)(t∈R^N)是d维分式Browa运动,本文研究X(t)的k重点集的Hausdorff维数。证明了:若P_1,…,P_k是R^N中内部不空的紧集,P=multiply from i=1 to k P_i, L_k(P)={x∈R^d|存在(t_1,…,t_k)∈P,使X(t_1)=…=X(t_k)=x},则当N≤ad,Nk>(k-1)ad时,P{dim L_k(P)=Nk/a-(k-1)d}>0,当N>ad时,P{dim L_k(P)=d}>0。当N≤ad时,对R^N\{0}中互不相交的紧集E_1,…,E_k得到了dim(X(E_1)∩…∩X(E_k))的一个上界和dim(X(E_1)∩X(E_2))的下界,从而当k=2时,证明了Testard猜想。
