### Abstract

Let τ = (τ_{i} : i ∈ Z) denote i.i.d. positive random variables with common distribution F and (conditional on τ) let X = (X_{t} : t ≥ 0, X_{0} = 0), be a continuous-time simple symmetric random walk on Z with inhomogeneous rates (τ_{i}^{-1} : i ∈ Z). When F is in the domain of attraction of a stable law of exponent α < 1 [so that E(τ_{i}) = ∞ and X is subdiffusive], we prove that (X, τ), suitably rescaled (in space and time), converges to a natural (singular) diffusion Z = (Z_{t} : t ≥ 0, Z_{0} = 0) with a random (discrete) speed measure ρ. The convergence is such that the "amount of localization," E ∑_{i∈Z}[P(X_{t} = i|τ)]^{2} converges as t → ∞ to E ∑_{z∈R}[P(Z_{s} = z|ρ]^{2} > 0, which is independent of s > 0 because of scaling/self-similarity properties of (Z, ρ). The scaling properties of (Z, ρ) are also closely related to the "aging" of (X, τ). Our main technical result is a general convergence criterion for localization and aging functionals of diffusions/walks Y^{(ε)} with (nonrandom) speed measures μ^{(ε)} → μ (in a sufficiently strong sense).

Original language | English (US) |
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Pages (from-to) | 579-604 |

Number of pages | 26 |

Journal | Annals of Probability |

Volume | 30 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2002 |

### Keywords

- Aging
- Disordered systems
- Localization
- Quasidiffusions
- Random walks in random environments
- Scaling limits
- Self-similarity

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Probability*,

*30*(2), 579-604. https://doi.org/10.1214/aop/1023481003