Time-squeezing and time-expanding transformations in harmonic force fields

Abstract

A variety of real life phenomena exhibit complex time-inhomogeneous nonlinear diffusive motion in the presence of an external harmonic force. In capturing the characteristics of such dynamics, the class of Ornstein–Uhlenbeck processes, with its physical time appropriately modulated, has long been regarded as the most relevant model on the basis of its mean reversion property. In this paper, we contrast two similar, yet definitely different, time-changing mechanisms in harmonic force fields by systematically deriving and presenting a variety of key properties all at once, that is, whether or not and how those time-changing mechanisms affect the characteristics of mean-reverting diffusion through sample path properties, the marginal probability density function, the asymptotic degeneracy of increments, the stationary law, the second-order structure, and the ensemble- and time-averaged mean square displacements. Some of those properties turn out rather counter-intuitive due to, or irrespective of, possible degeneracy of time-changing mechanisms in the long run. In light of those illustrative comparisons, we examine whether such time-changing operations are worth the additional operational cost, relative to physically relevant characteristics induced, and deduce practical implications and precautions from modeling and inference perspectives, for instance, on the experimental setup involving those anomalous diffusion processes, such as the observation start time and stepsize when measuring mean square displacements, so as to preclude potentially misleading results and paradoxical interpretations.

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

“…Under the influence of an external linear damping force, distinct initial states and times may end up with distinct outcomes in the form of disagreement between the ensemble and time averaged mean square displacements, suggesting weak ergodicity breaking [21]. In a similar manner, weak ergodicity breaking has also been observed for super-, normal-and subdiffusive positions modeled by Ornstein-Uhlenbeck processes incorporating subordination and integration over time simultaneously [58], as well as Lévy flights modeled by time-squeezed Ornstein-Uhlenbeck processes [59].…”

Stochastic Processes With Applications in Physics and Insurance

“…To further explore the topic, two representative time-changing mechanisms, time changing by a subordinator and an inverse subordinator, hereafter referred to as time-squeezing and time-expanding transformations, are thoroughly examined and contrasted [1]. We systematically derive a series of properties based upon the Ornstein–Uhlenbeck process after time-squeezing and time-expanding transformations, including the sample path properties, the marginal probability density function, degeneracy of increments, the stationary law, the second-order structure and mean square displacements.…”