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 $$ \int^{s\wedge t}_0 u^a [(t-u)^b+(s-u)^b]du, $$ 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 $$ (2-h)\biggl(s^h+t^h-\frac{1}{2}[(s+t)^h +|s-t|^h]\biggr), $$ parameter $0 < h\leq 4$, corresponds to sub-fractional Brownian motion for $0 < h < 2 $. The third one, with covariance $$ -\left(s^2\log s + t^2\log t -\frac{1}{2}[(s+t)^2 \log (s+t) +(s-t)^2 \log |s-t|]\right), $$ is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.