Long-range Trap Models on $\mathbb {Z}$ and Quasistable Processes

Abstract

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 $\mathcal X$.