Description: This article discusses fluctuation-dissipation process without a time scale. Abstract: We study the influence of a dissipation process on diffusion dynamics triggered by fluctuations with long-range correlations. We make the assumption that the perturbation process involved is of the same kind as those recently studied numerically and theoretically, with a good agreement between theory and numerical treatment. As a result of this assumption the equilibrium distribution departs from the ordinary canonical distribution. The distribution tails are truncated, the distribution border is signaled by sharp peaks, and, in the weak dissipation limit, the central distribution body becomes identical to a truncated Lévy distribution.