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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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