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

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