Universality in the coarse-grained fluctuations for a class of linear dynamical systems

The overdamped dynamics of linearly coupled elements are studied in the stochastic force-field of a reservoir. The stable equilibrium state is fixed by the reservoir, and the stochastic force field induces fluctuations around the stable state. The dynamics of the system is investigated in the case when the stochastic force-field is a multivariate Gaussian noise. For a class of practically relevant problems we prove that the coarse-grained properties of the fluctuations are not influenced by the interactions among the elements. Compact results are given for such coarse-grained quantities and an application for the price fluctuation of an economic product is discussed.