Aging uncoupled continuous time random walk limits

Ofer Busani

doi:10.1214/16-ejp3802

Abstract

Aging is a prevalent phenomenon in physics, chemistry and many other fields. In this paper we consider the aging process of uncoupled Continuous Time Random Walk Limits (CTRWLs) which are Levy processes time changed by the inverse stable subordinator of index $0 < \alpha < 1$. We apply a recent method developed by Meerscheart and Straka of finding the finite dimensional distributions of CTRWL, to obtaining the aging process’s finite dimensional distributions, self-similarity-like property, asymptotic behavior and its Fractional Fokker-Planck equation(FFPE).

Highlights

Continuous time random walks (CTRW) are widely used in physics and mathematical finance to model a random walk for which the waiting times between jumps are random which in many cases better describes phenomena in these fields
[6] Barkai and Cheng considered the Aging Continuous Time Random Walk (ACTRW) which is an uncoupled CTRW with iid power law waiting times, that started at t = 0 and is observed at t = t0
Since the distribution of the increments of the Continuous Time Random Walk Limits (CTRWLs) is closely related to the two dimensional distributions, their study is quite cumbersome

Summary

Introduction

Continuous time random walks (CTRW) are widely used in physics and mathematical finance to model a random walk for which the waiting times between jumps are random which in many cases better describes phenomena in these fields. In [6] Barkai and Cheng considered the Aging Continuous Time Random Walk (ACTRW) which is an uncoupled CTRW with iid power law waiting times, that started at t = 0 and is observed at t = t0 They found the one dimensional distribution of the process Xtt0 which they referred to as the ACTRW, for t0 and t large. In [19], it was shown that Ntα = NEt where Nt is a Poisson process and Et is the inverse of a standard stable subordinator of index 0 < α < 1 independent of Nt. Since the distribution of the increments (and the aging process) of the CTRWL is closely related to the two dimensional distributions, their study is quite cumbersome. CTRWLs in a larger state space that renders these processes Markovian We use this method to find the finite dimensional distributions of the process Ytt0 , its asymptotic behavior, self-similarity-like property and its FFPE

Finite dimensional distribution of CTRWL

Aging self similarity

Asymptotic behavior and the Fractional Fokker-Planck equation