  The dynamics of a linear system embedded in a heat bath environment and
subject to non- Gaussian noise is studied. Higher order cumulants in coordinate
space are derived and their impact on the dynamics and on asymptotic steady
state distributions is analyzed. In the quantum regime non-Gaussian properties
are present in the reduced density in coordinate representation which in energy
representation exist on a transient time scale only due to symmetry. Within an
exactly solvable model our results provide insight into mechanisms of linear
detectors as sensors for non- Gaussian noise at high and low temperatures.
