Fractional Klein–Kramers dynamics for subdiffusion and Itô formula

Abstract

Subdiffusion in the presence of an external force field has been recently described in phasespace by the fractional Klein–Kramers equation. In this paper using a subordinationmethod, we identify a two-dimensional stochastic process (position, velocity) whoseprobability density function is a solution of the fractional Klein–Kramers equation. Thestructure of this process agrees with the two-stage scenario underlying the anomalousdiffusion mechanism, in which trapping events are superimposed on the Langevindynamics. Applying an extension of the celebrated Itô formula for subdiffusion we foundthat the velocity process can be represented explicitly by a corresponding fractionalOrnstein–Uhlenbeck process. A basic feature arising in the context of this stochasticrepresentation is the random change of time of the system made by subordination. For theposition and velocity processes we present a computer visualization of their sample pathsand we derive an explicit expression for the two-point correlation function of thevelocity process. The obtained stochastic representation is crucial in constructing analgorithm to simulate sample paths of the anomalous diffusion, which in turnallows us to detect and examine many relevant properties of the system underconsideration.