Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization and aging in one dimension

Abstract

Let τ = (τi : i ∈ Z) denote i.i.d. positive random variables with common distribution F and (conditional on τ) let X = (Xt : t ≥ 0, X0 = 0), be a continuous-time simple symmetric random walk on Z with inhomogeneous rates (τi-1 : i ∈ Z). When F is in the domain of attraction of a stable law of exponent α < 1 [so that E(τi) = ∞ and X is subdiffusive], we prove that (X, τ), suitably rescaled (in space and time), converges to a natural (singular) diffusion Z = (Zt : t ≥ 0, Z0 = 0) with a random (discrete) speed measure ρ. The convergence is such that the "amount of localization," E ∑i∈Z[P(Xt = i|τ)]2 converges as t → ∞ to E ∑z∈R[P(Zs = 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).