α-time fractional Brownian motion: PDE connections and local times^∗

¹ Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA
www.duc.auburn.edu/˜ezn0001/. nane@stt.msu.edu
² Department of Mathematical Sciences, 201J Shelby Center, University of Alabama in Huntsville, Huntsville, AL 35899, USA
http://webpages.uah.edu/˜dw0001. dongsheng.wu@uah.edu
³ Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, MI 48824, USA
http://www.stt.msu.edu/˜xiaoyimi. xiao@stt.msu.edu

Received: 14 October 2010

Abstract

For 0 < α ≤ 2 and 0 < H < 1, an α-time fractional Brownian motion is an iterated process Z =  {Z(t) = W(Y(t)), t ≥ 0}  obtained by taking a fractional Brownian motion  {W(t), t ∈ ℝ} with Hurst index 0 < H < 1 and replacing the time parameter with a strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ 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 ∈ ℝ₊} defined by X(t) = (X₁(t), ..., X_d(t)), where t ≥ 0 and X₁, ..., X_d are independent copies of Z, are investigated. Our methods rely on the strong local nondeterminism of fractional Brownian motion.