Irregular Gyration of a Two-Dimensional Random-Acceleration Process in a Confining Potential

We study the stochastic dynamics of a two-dimensional particle whose coordinates are described by two coupled one-dimensional random-acceleration processes, that evolve in a confining parabolic potential and are subject to independent Gaussian white noises with different amplitudes (temperatures). We first determine standard characteristics: the mixed moments of positions and velocities, as well as the position-velocity probability density function (PDF) and those of its kinetic and potential energies. Going then beyond these standard characteristics, we consider the emerging rotational motion of the particle around the origin: We show that if the amplitudes of the noises are not equal, 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 which both are (irregularly) oscillating with time t. We 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 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. However, this PDF converges to a well-defined large-time limit manifesting the possibility of stabilizing phenomenon even in frictionless driven systems. Surprisingly, the limit is independent of the amplitudes of noises.