Research Article

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

Erkan Nane^a1, Dongsheng Wu^a2 and Yimin Xiao^a3

^a1 Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA. www.duc.auburn.edu/˜ezn0001/. nane@stt.msu.edu

^a2 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

^a3 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

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.

(Received October 14 2010)

(Online publication March 09 2012)

Key Words:

• Fractional Brownian motion;
• strictlyα-stable Lévy process;
• α-time Brownian motion;
• α-time fractional Brownian motion;
• partial differential equation;
• local time;
• Hölder condition.

Mathematics Subject Classification:

• 60G17;
• 60J65;
• 60K99

Footnotes

^∗  Research partially supported by NSF grant DMS-1006903.