Tempered fractional Langevin-Brownian motion with inverse $\beta$-stable subordinator

TL;DR

This paper investigates the properties of tempered fractional Langevin motion and its time-changed version via inverse -stable subordinator, revealing diverse diffusion behaviors and differences from the original process.

Contribution

It provides a detailed analysis of the tempered fractional Langevin system under inverse -stable subordination, including diffusion types, moments, and correlation structures, highlighting novel behaviors.

Findings

Subordination can induce subdiffusion or superdiffusion depending on .

Time-changed processes show significant differences in statistical properties from original processes.

Correlation structures of increments are notably altered by the inverse -stable subordinator.

Abstract

Time-changed stochastic processes have attracted great attention and wide interests 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 inverse $\beta$-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 $\beta$. 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 subordinator. The time-changed tempered fractional Brownian motion by inverse $\beta$-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.