Asymptotic properties of Brownian motion delayed by inverse subordinators

TL;DR

This paper analyzes the long-term behavior of a Brownian motion time-changed by the inverse of a subordinator, revealing mixing properties, martingale characteristics, and classical limit laws relevant to anomalous diffusion in physics.

Contribution

It provides the first rigorous proof of mixing for the stationary increments of such subdiffusive processes and derives key probabilistic properties and formulas.

Findings

Proves mixing property for stationary increments of the process

Establishes martingale properties and generalized Feynman-Kac formula

Derives laws of large numbers and iterated logarithm for the process

Abstract

We study the asymptotic behaviour of the time-changed stochastic process $\vphantom{X}^f\!X(t)=B(\vphantom{S}^f\!S (t))$, where $B$ is a standard one-dimensional Brownian motion and $\vphantom{S}^f\!S$ is the (generalized) inverse of a subordinator, i.e. the first-passage time process corresponding to an increasing L\'evy process with Laplace exponent $f$. This type of processes plays an important role in statistical physics in the modeling of anomalous subdiffusive dynamics. The main result of the paper is the proof of the mixing property for the sequence of stationary increments of a subdiffusion process. We also investigate various martingale properties, derive a generalized Feynman-Kac formula, the laws of large numbers and of the iterated logarithm for $\vphantom{X}^f\!X$.