Tempered fractional Langevin–Brownian motion with inverse β-stable subordinator

Abstract

Time-changed stochastic processes have attracted much attention and wide interest due to their extensive applications, especially in financial time series, biology and physics. This paper pays attention to a special stochastic process, tempered fractional Langevin motion, which is non-Markovian and undergoes ballistic diffusion for long times. The corresponding time-changed Langevin system with an inverse β-stable subordinator is discussed in detail, including its diffusion type, moments, Klein–Kramers equation, and the correlation structure. Interestingly, this subordination could result in both subdiffusion and superdiffusion, depending on the value of β. The difference between the subordinated tempered fractional Langevin equation and the subordinated Langevin equation with external biasing force is studied for a deeper understanding of the subordinator. The time-changed tempered fractional Brownian motion by an inverse β-stable subordinator is also considered, as well as the correlation structure of its increments. Some properties of the statistical quantities of the time-changed process are discussed, displaying striking differences compared with the original process.