Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

The power spectral density (PSD) of any time-dependent stochastic processXt is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT  ¥.Alegitimate question iswhat information on the PSDcan be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes thePSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensionalBrownian motion (BM) we calculate the probability density function of a single-trajectory PSDfor arbitrary frequency f, finite observation timeTand arbitrary number k of projections of the trajectory on different axes.We show analytically that the scaling exponent for the frequency-dependence of the PSDspecific to an ensemble ofBMtrajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail.Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncatedWiener representation ofBM, and the case of a discrete-time lattice randomwalk.Wehighlight some differences in the behavior of a single-trajectory PSDforBMand for the two latter situations.The framework developed herein will allow formeaningful physical analysis of experimental stochastic trajectories.