Long range trap models on Z and quasistable processes

Abstract

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