## 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

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*) = (*X*_{1}(*t*), *...*, *X*_{d}(*t*)),
where *t* ≥ 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.

Mathematics Subject Classification: 60G17 / 60J65 / 60K99

Key words: Fractional Brownian motion / strictly*α*-stable Lévy process / *α*-time Brownian motion / *α*-time fractional Brownian motion / partial differential equation / local time / Hölder condition.

*© EDP Sciences, SMAI, 2012*