For 0 < α ≤ 2 and 0 < H < 1, anα-time fractional Brownian motion is an iterated processZ = {Z(t) = W(Y(t)), t ≥ 0} obtained by taking a fractional Brownian motion {W(t), t ∈ ℝ} with Hurst index0 < H < 1 and replacing the time parameter with astrictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that suchprocesses have natural connections to partial differential equations and, whenY is a stable subordinator, can arise as scaling limit of randomlyindexed random walks. The existence, joint continuity and sharp Hölder conditions in theset variable of the local times of a d-dimensionalα-time fractional Brownian motionX = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ...,Xd(t)), where t ≥ 0 andX1, ..., Xdare independent copies of Z, are investigated. Our methods rely on thestrong local nondeterminism of fractional Brownian motion.
