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

Abstract

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance δs⋀t0ua[(t − u)b + (s − u)b]du, parameters a > −1, −1, < b ≤ 1, |b| ≤ 1 + a, corresponds to fractional Brownian motion for a = 0, −1 < b < 1. The second one, with covariance (2 − h) (sh + th − 1/2[s + t)h + |s − t|h]), parameter 0 < h ≤ 4, corresponds to sub-fractional Brownian motion for 0 < h < 2. The third one, with covariance − (S2 logs + t2 logt − 1/2[(s + t)2 log(s + t) + (s − t)2 log|s − t|]), is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters. © 2007 Applied Probability Trust.