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 Rd. 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]N)) ≤ ϕ-p(GrX([0, 1]N)) ≤ K2 where ϕ(s) = sN/α/(log log 1/s)N/(2α), ϕ-p(E) is the ϕ-packing measure of E, X([0, 1]N) is the image and GrX([0, 1]N) = {(t,X(t)); t ∈ [0, 1]N} is the graph of X, respectively. We also establish liminf and limsup type laws of the iterated logarithm for the sojourn measure of X