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

Abstract

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 $\{W(t), t\in \RR{R} \}$ with Hurst index $0<H<1$ and replacing the time parameter with a strictly $\alpha$-stable Lévy process $\{Y(t),t\ge 0\}$ in $\RR{R}$ 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 Hölder conditions in the set variable of the local times of a $d$-dimensional $\alpha$-time fractional Brownian motion $X=\{X(t),t\in {\mathbb{R}}_{+}$\} defined by $X(t)=(X_1(t),...,X_d(t)),$ where $t\ge 0$ and $X_1,...,X_d$ are independent copies of $Z$, are investigated. Our methods rely on the strong local ' nondeterminism of fractional Brownian motion.

arXiv comment29 pages, typos corrected in section 2