Super- and subdiffusive positions in fractional Klein–Kramers equations

Abstract

The Ornstein–Uhlenbeck process is a Gaussian process with applications in various fields, originally as a model for the velocity of a Brownian particle subject to friction force. Its cumulative integral over time thus describes the position of the particle, which however remains overly regular, for instance, concerning Gaussianity and linear mean square displacement. Doubts cast on such over-regularity has led to two generalizations through time-expanding transformation: one with the velocity occasionally paused resulting in superdiffusive positions, the other one with the position trapped once in a while causing subdiffusive positions. The aim of the present work is to systematically derive and present various results to contrast those two similar yet different generalizations. In the framework of the Klein–Kramers equation, those two time-changing mechanisms make only a slight difference in what part of the operator the fractional derivative acts on, whereas by taking the probabilistic approach, we reveal that they are definitively distinct models through a variety of new findings in the theory of the super and subdiffusive positions. We expect our findings to provide an effective means for illustrative comparisons on the relevance of those modeling frameworks.