Let Λ = {λ_k} be an infinite increasing sequence of positive integers withλ_k → ∞. Let X = {X(t), t ∈ R^N} be a multi-parameter fractional Brownian motion of index (0 【α 【 1) in R^d . Subject to certain hypot...Let Λ = {λ_k} be an infinite increasing sequence of positive integers withλ_k → ∞. Let X = {X(t), t ∈ R^N} be a multi-parameter fractional Brownian motion of index (0 【α 【 1) in R^d . Subject to certain hypotheses, we prove that if N 【 αd, then there exist positivefinite constants K_1 and K_2 such that, with unit probability, K_1 ≤ φ - p_Λ(X([0,1])~N) ≤ φ -p_Λ(G_rX([0,1])~N)) ≤ K_2 if and only if there exists γ 】 0 such that ∑ from k=1 to ∞ of1/λ_k~γ = ∞, where φ(s) = s^(N/α)(loglog 1/s)^(N/2(α)), φ - p_Λ(E) is the Packing-typemeasure of E,X([0, 1]) N is the image and G_rX([0, 1]~N ) = {(t,X(t)); t ∈ [0,1]~N} is the graph ofX, respectively. We also establish liminf type laws of the iterated logarithm for the sojournmeasure of X.展开更多
基金Supported by the National Natural Science Foundation of China (No.10471148)Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei(No.2003A005)
文摘Let Λ = {λ_k} be an infinite increasing sequence of positive integers withλ_k → ∞. Let X = {X(t), t ∈ R^N} be a multi-parameter fractional Brownian motion of index (0 【α 【 1) in R^d . Subject to certain hypotheses, we prove that if N 【 αd, then there exist positivefinite constants K_1 and K_2 such that, with unit probability, K_1 ≤ φ - p_Λ(X([0,1])~N) ≤ φ -p_Λ(G_rX([0,1])~N)) ≤ K_2 if and only if there exists γ 】 0 such that ∑ from k=1 to ∞ of1/λ_k~γ = ∞, where φ(s) = s^(N/α)(loglog 1/s)^(N/2(α)), φ - p_Λ(E) is the Packing-typemeasure of E,X([0, 1]) N is the image and G_rX([0, 1]~N ) = {(t,X(t)); t ∈ [0,1]~N} is the graph ofX, respectively. We also establish liminf type laws of the iterated logarithm for the sojournmeasure of X.
