Hausdorff-type measures of the sample paths of fractional Brownian motion

Abstract

Let φ be a Hausdorff measure function and let Λ be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ- m Λ associated to φ and Λ is studied. Let X(t) (t∈ R N) be fractional Brownian motion of index α in R d. We evaluate the exact φ- m Λ measure of the image and graph set of X( t). A necessary and sufficient condition on the sequence Λ is given so that the usual Hausdorff measure functions for X([0,1] N ) and Gr X([0,1] N) are still the correct measure functions. If the sequence Λ increases faster, then some smaller measure functions will give positive and finite ( φ, Λ)-Hausdorff measure for X([0,1] N ) and Gr X([0,1] N)