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

Let $\tau = (\tau_i : i \in {\Bbb Z})$ denote i.i.d.~positive random variables with common distribution $F$ and (conditional on $\tau$) let $X = (X_t : t\geq0, X_0=0)$, be a continuous-time simple symmetric random walk on ${\Bbb Z}$ with inhomogeneous rates $(\tau_i^{-1} : i \in {\Bbb Z})$. When $F$ is in the domain of attraction of a stable law of exponent $\alpha 0$, which is independent of $s>0$ because of scaling/self-similarity properties of $(Z,\rho)$. The scaling properties of $(Z,\rho)$ are also closely related to the ``aging'' of $(X,\tau)$. Our main technical result is a general convergence criterion for localization and aging functionals of diffusions/walks $Y^{(\epsilon)}$ with (nonrandom) speed measures $\mu^{(\epsilon)} \to \mu$ (in a sufficiently strong sense).