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

Abstract

Let $τ= (τ_i : i \in {\Bbb Z})$ denote i.i.d.~positive random variables with common distribution $F$ and (conditional on $τ$) let $X = (X_t : t\geq0, X_0=0)$, be a continuous-time simple symmetric random walk on ${\Bbb Z}$ with inhomogeneous rates $(τ_i^{-1} : i \in {\Bbb Z})$. When $F$ is in the domain of attraction of a stable law of exponent $α<1$ (so that ${\Bbb E}(τ_i) = \infty$ and X is subdiffusive), we prove that $(X,τ)$, suitably rescaled (in space and time), converges to a natural (singular) diffusion $Z = (Z_t : t\geq0, Z_0=0)$ with a random (discrete) speed measure $ρ$. The convergence is such that the ``amount of localization'', $\E \sum_{i \in {\Bbb Z}} [¶(X_t = i|τ)]^2$ converges as $t \to \infty$ to $\E \sum_{z \in {\Bbb R}} [¶(Z_s = z|ρ)]^2 > 0$, 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 $μ^{(ε)} \to μ$ (in a sufficiently strong sense).