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
Let {W (t), t ∈ R}, {B(t), t ∈ R +} be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iter...Let {W (t), t ∈ R}, {B(t), t ∈ R +} be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (X 1(t),…, X d (t)) and X 1(t),…, X d (t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q ? (0, ∞), the exact Hausdorff measures of the image X(Q) = {X(t): t ∈ Q} and the graph GrX(Q) = {(t, X(t)): t ∈ Q} are established.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.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No. 10131040)China Postdoctoral Science Foundation.
文摘Let {W (t), t ∈ R}, {B(t), t ∈ R +} be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (X 1(t),…, X d (t)) and X 1(t),…, X d (t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q ? (0, ∞), the exact Hausdorff measures of the image X(Q) = {X(t): t ∈ Q} and the graph GrX(Q) = {(t, X(t)): t ∈ Q} are established.
