  We study the stochastic dynamics of a two-dimensional particle assuming that
the components of its position are two coupled random-acceleration processes
evolving in a confining parabolic potential and are the subjects of independent
Gaussian white noises with different amplitudes (temperatures). We determine
the standard characteristic properties, i.e., the moments of position's
components and their velocities, mixed moments and two-time correlations, as
well as the position-velocity probability density function (pdf). We show that
if the amplitudes of the noises are not equal, then the particle experiences a
non-zero (on average) torque, such that the angular momentum L and the angular
velocity W have non-zero mean values. Both are
  (irregularly) oscillating with time t, such that the characteristics of a
rotational motion are changing their signs. We also evaluate the pdf-s of L and
W and show that the former has exponential tails for any fixed t, and hence,
all moments. In addition, in the large-time limit this pdf converges to a
uniform distribution with a diverging variance. The pdf of W possesses heavy
power-law tails such that the mean W is the only existing moment. This pdf
converges to a limiting form which, surprisingly, is completely independent of
the amplitudes of noises.
