Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brown...Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.展开更多
Let {W(s), s∈R_+~N} be the N-parameter Wiener process. Define the N-parameter Ornstein-Uhlenbeck process (OUP) as follows. X^(N,1)(t)=e^(-a.t)[ x_0+σ∫_[0,t]e^(a.d)dW(a)]. Make X^(N,d)={(X^1(t),…,X^d(t)), t∈R_+~N}...Let {W(s), s∈R_+~N} be the N-parameter Wiener process. Define the N-parameter Ornstein-Uhlenbeck process (OUP) as follows. X^(N,1)(t)=e^(-a.t)[ x_0+σ∫_[0,t]e^(a.d)dW(a)]. Make X^(N,d)={(X^1(t),…,X^d(t)), t∈R_+~N}, which is called (N, d) OUP, where X^i(1≤i≤d) are copies of the N-parameter OUP and are independent. As a result, for d<2N,|ImX|_d>0, a.s; d≥2N, there is |ImX|_d=0 a.s., where |·|_d denotes the d-dimensionalLebesgue measure.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10721091)
文摘Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.
基金Project supported by the National Natural Science Foundation of China.
文摘Let {W(s), s∈R_+~N} be the N-parameter Wiener process. Define the N-parameter Ornstein-Uhlenbeck process (OUP) as follows. X^(N,1)(t)=e^(-a.t)[ x_0+σ∫_[0,t]e^(a.d)dW(a)]. Make X^(N,d)={(X^1(t),…,X^d(t)), t∈R_+~N}, which is called (N, d) OUP, where X^i(1≤i≤d) are copies of the N-parameter OUP and are independent. As a result, for d<2N,|ImX|_d>0, a.s; d≥2N, there is |ImX|_d=0 a.s., where |·|_d denotes the d-dimensionalLebesgue measure.
