Dimension dependent properties of subdiffusions in damping force fields from an inference perspective

We investigate the fractional Fokker-Planck equation subject to a damping force with an emphasis on its dimension dependent properties. We reveal a variety of surprising properties of its solution through the lens of the probability density function of the corresponding stochastic process with nonlinear mean square displacements, such as existence, singularity, regularity, modality, stationarity and second-order structure, which are largely dependent on the dimension and the random clock. Taking into account that the trajectory information is most often collected from multidimensional systems, the discovered facts have the potential to play important roles as key foundations and alerts for inference, model identification and prediction, when departing from the well-understood univariate framework.