$α$-Time Fractional Brownian Motion: PDE Connections and Local Times

For $0<α\leq 2$ and $0<H<1$, an $α$-time fractional Brownian motion is an iterated process $Z = \{Z(t)=W(Y(t)), t \ge 0\}$ obtained by taking a fractional Brownian motion $\{W(t), t\in \RR{R} \}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $α$-stable Lévy process $\{Y(t), t\geq 0 \}$ in $\RR{R}$ independent of $\{W(t), t \in \R\}$. 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 $X = \{X(t), t \in \R_+$\} defined by $X(t)=\big(X_{1}(t),..., X_{d}(t) \big),$ where $t\geq 0$ and $X_{1},..., X_{d}$ are independent copies of $Z$, are investigated. Our methods rely on the strong local ' nondeterminism of fractional Brownian motion.

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