## Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems.(English)Zbl 1128.60025

Summary: We study three self-similar, long-range dependent 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
$(2-h)(s^h+t^h- \tfrac12 [(s+t)^h+|s-t|^h]),$
parameter $$0<h\leq 4$$, corresponds to sub-fractional Brownian motion for $$0<h<2$$. The third one, with covariance
$-(s^2\log s+ t^2\log t- \tfrac12 [(s+t)^2\log (s+t) +(s-t)^2\log|s-t|]),$
is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.

### MSC:

 60G15 Gaussian processes 60G18 Self-similar stochastic processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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