Packing-dimension Profiles and Fractional Brownian Motion

Fractal Measures of the Sets Associated to Gaussian Random Fields

This paper summarizes recent results about the Hausdorff measure of the image, graph and level sets of Gaussian random fields, the packing dimension and packing measure of the image of fractional Brownian motion, the local times and multiple points of Gaussian random fields. Some open problems are also pointed out.

The Packing Measure of the Trajectories of Multiparameter Fractional Brownian Motion

Let X = {X(t), t ∈ RN} be a multiparameter fractional Brownian motion of index α (0 < α < 1) in R. We prove that if N < αd , then there exist positive finite constants K1 and K2 such that with probability 1, K1 ≤ φ-p(X([0, 1] )) ≤ φ-p(GrX([0, 1] )) ≤ K2 where φ(s) = s/(log log 1/s), φ-p(E) is the φ-packing measure of E, X([0, 1] ) is the image and GrX([0, 1] ) = {(t,X(t)); t ∈ [0, 1]N} is the g...

Packing Measure of the Sample Paths of Fractional Brownian Motion

Let X(t) (t ∈ R) be a fractional Brownian motion of index α in Rd. If 1 < αd , then there exists a positive finite constant K such that with probability 1, φ-p(X([0, t])) = Kt for any t > 0 , where φ(s) = s 1 α /(log log 1 s ) 1 2α and φ-p(X([0, t])) is the φ-packing measure of X([0, t]).

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

Let BH = {BH (t), t ∈ RN } be an (N, d)-fractional Brownian sheet with index H = (H1, . . . , HN ) ∈ (0, 1)N . The uniform and local asymptotic properties of BH are proved by using wavelet methods. The Hausdorff and packing dimensions of the range BH ([0, 1]N), the graph GrBH ([0, 1]N) and the level set are determined.