Long range trap models on Z and quasistable processes

TL;DR

This paper studies the long-term behavior of a long-range trap model on the integer lattice, revealing a quasistable process as the scaling limit and providing aging results for the process.

Contribution

It introduces the quasistable process as a new scaling limit for long-range trap models with inhomogeneous jump rates, extending understanding of their asymptotic behavior.

Findings

Derives the scaling limit as a time-changed stable process involving local time and an independent stable subordinator.

Establishes aging properties and an integrated aging result for the process.

Provides detailed analysis of the process's long-time dynamics and limit behavior.

Abstract

Let $\mathcal X=\{\mathcal X_t:\, t\geq0,\, \mathcal X_0=0\}$ be a mean zero $\beta$-stable random walk on $\mathbb{Z}$ with inhomogeneous jump rates $\{\tau_i^{-1}: i\in\mathbb{Z}\}$, with $\beta\in(1,2]$ and $\{\tau_i: i\in\mathbb{Z}\}$ a family of independent random variables with common marginal distribution in the basin of attraction of an $\alpha$-stable law, $\alpha\in(0,1)$. In this paper we derive results about the long time behavior of this process, in particular its scaling limit, given by a $\beta$-stable process time-changed by the inverse of another process, involving the local time of the $\beta$-stable process and an independent $\alpha$-stable subordinator; we call the resulting process a quasistable process. Another such result concerns aging. We obtain an (integrated) aging result for $\mathcal X$.