Dimension results of multifractional Brownian sheets

Abstract

Let {B̂H(t)(t)}={B̂H(t)(t),t∊R+N} be an (N,d)-multifractional Brownian sheet with N Hurst functionals Hℓ(tℓ)∊CKℓ(R+,(0,1)) for some Kℓ∊(0,1)(ℓ=1,…,N) defined by a harmonizable representation. Under some regularity conditions on Hℓ(tℓ)’s, we establish the result on the modulus of continuity of the sample paths and determine the Hausdorff and packing dimensions of the range B̂H(t)([a,b]) and the graph GrB̂H(t)([a,b]) for [a,b]⊂[ε,T]N, where 0<ε<T<∞, for the multifractional Brownian sheet.