Some Extensions of Fractional Brownian Motion and Sub-Fractional Brownian Motion Related to Particle Systems
Tomasz Bojdecki, Institute of Mathematics, University of Warsaw
Luis G Gorostiza, Centro de Investigacion y de Estudios Avanzados, Mexico
Anna Talarczyk, Institute of Mathematics, University of Warsaw
Abstract
In this paper we study three self-similar, long-range dependence, Gaussian
processes. The first one, with covariance
∫0min(s,t) ua [(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
-(s2log s + t2log t -(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.
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