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

Abstract

For $0<\alpha \leq 2$ 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 $\{W(t), t\in \RR{R} \}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $\alpha$-stable L\'evy 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 H\"older conditions in the set variable of the local times of a $d$-dimensional $\alpha$-time fractional Brownian motion $X = \{X(t), t \in \R_+$\} 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.