For 0 < α ≤ 2 and 0 < H < 1, an
α-time fractional Brownian motion is an iterated process
Z = {Z(t) = W(Y(t)), t ≥ 0}
obtained by taking a fractional Brownian motion
{W(t), t ∈ ℝ} with Hurst index
0 < H < 1 and replacing the time parameter with a
strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that such
processes have natural connections to partial differential equations and, when
Y is a stable subordinator, can arise as scaling limit of randomly
indexed random walks. The existence, joint continuity and sharp Hölder conditions in the
set variable of the local times of a d-dimensional
α-time fractional Brownian motion
X = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ..., Xd(t)),
where t ≥ 0 and
X1, ..., Xd
are independent copies of Z, are investigated. Our methods rely on the
strong local nondeterminism of fractional Brownian motion.