Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance integral(s boolean AND t)(o) u(a)[(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) (s(h) + t(h) - 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 - (s(2) log s + t(2) log 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.