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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

0 st u a [(t-u) b +(s-u) b ]du,

parameters a>-1, -1<b1, |b|1+a, corresponds to fractional Brownian motion for a=0, -1<b<1. The second one, with covariance

(2-h)(s h +t h -1 2[(s+t) h +|s-t| h ]),

parameter 0<h4, corresponds to sub-fractional Brownian motion for 0<h<2. The third one, with covariance

-(s 2 logs+t 2 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.

60G15Gaussian processes
60G18Self-similar processes
60J80Branching processes
60K35Interacting random processes; statistical mechanics type models; percolation theory