Hausdorff measure of the graph of fractional Brownian motion

Let X(t) (t is an element of R-N) be a fractional Brownian motion in R-d of index alpha. Let GrX([0, 1](N)) be the graph of X and let phi(1)(s) = s(N/alpha) log log1/s and phi(2)(s) = s(N+(1-alpha)s) (log log 1/s)(ad/N). It is proved that if N < alpha d, then almost surely K-1 less than or equal to phi(1)-m(Gr X([0, 1](N))) less than or equal to K-2 and if N > alpha d, then almost surely K-1 less than or equal to phi(2)-m(Gr X([0, 1](N))) less than or equal to K-2, where phi-m is the phi-Hausdorff measure and K-1, K-2 are positive finite constants. The exact Hausdorff measure of the image and graph of certain Gaussian random fields with independent fractional Brownian motion components are also obtained.