Tempered fractional Langevin-Brownian motion with inverse $\beta$-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.

arXiv comment19 pages, 5 figures

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• Langevin picture of anomalous diffusion processes in expanding medium
• A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows
• Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion