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

W. Barreto-Souza () and L. R. G. Fontes ()
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W. Barreto-Souza: Universidade de São Paulo
L. R. G. Fontes: Universidade de São Paulo

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1500-1519

Abstract: Abstract 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 .

Keywords: Trap model; Stable random walks; Scaling limit; Stable process; Stable subordinator; Aging; 60K35; 60K37 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-014-0548-x

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