Equivalence of the Fractional Fokker-Planck and Subordinated Langevin Equations: The Case of a Time-Dependent Force

Abstract

A century after the celebrated Langevin paper [C.R. Seances Acad. Sci. 146, 530 (1908)] we study a Langevin-type approach to subdiffusion in the presence of time-dependent force fields. Using a subordination technique, we construct rigorously a stochastic Langevin process, whose probability density function is equal to the solution of the fractional Fokker-Planck equation with a time-dependent force. Our model provides physical insight into the nature of the corresponding process through the simulated trajectories. Moreover, the subordinated Langevin equation allows us to study subdiffusive dynamics both analytically and numerically via Monte Carlo methods.