Long range trap models on Z and quasistable processes

Let $\mathcal{X}=\{{\mathcal{X}}_t:t\ge 0,{\mathcal{X}}_0=0\}$ 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 $\mathcal{X}$.