Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems

Tomasz Bojdecki, Luis G. Gorostiza, Anna Talarczyk

Published 2007-02-23, updated 2007-06-01Version 2

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance \int_0^{s\wedge t} u^a [(t-u)^b+(s-u)^b]du, parameters a>-1, -1<b\leq 1, |b|\leq 1+a, corresponds to fractional Brownian motion for a=0, -1<b<1. The second one, with covariance (2-h)(s^h+t^h-[(s+t)^h +|s-t|^h]/2), parameter 0<h\leq 4, corresponds to sub-fractional Brownian motion for 0<h<2. The third one, with covariance -(s^2\log s + t^2\log t -[(s+t)^2 \log (s+t) +(s-t)^2 \log |s-t|]/2), is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.