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

For $0<\alpha \le 2$ and $0<H<1$, an $\alpha$-time fractional Brownian motion is an iterated process $Z=\{Z(t)=W(Y(t)),t\ge 0\}$ obtained by taking a fractional Brownian motion $\{W(t),t\in \text{RR}R\}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $\alpha$-stable L'evy process $\{Y(t),t\ge 0\}$ in $\text{RR}R$ independent of $\{W(t),t\in \mathbb{R}\}$. 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 H"older conditions in the set variable of the local times of a $d$-dimensional $\alpha$-time fractional Brownian motion $X = {X(t), t \in \R_+$\}definedby$ X(t)=\big(X_{1}(t),…, X_{d}(t) \big), $where$t\geq 0$and$X_{1},…, X_{d}$areindependentcopiesof$Z$, are investigated. Our methods rely on the strong local ’ nondeterminism of fractional Brownian motion.