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

TL;DR

This paper explores $ ext{α}$-time fractional Brownian motion, revealing its connections to PDEs, scaling limits of random walks, and analyzing the existence and properties of its local times using strong local nondeterminism.

Contribution

It introduces and studies $ ext{α}$-time fractional Brownian motion, establishing its PDE connections, scaling limits, and detailed local time properties, which are novel contributions.

Findings

Processes have natural PDE connections.

Can arise as scaling limits of random walks.

Established existence and regularity of 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.