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Let $\tau = (\tau_i : i \in {\Bbb Z})$ denote i.i.d.~positive randomvariables 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 $\alpha<1$ (so that${\Bbb E}(\tau_i) = \infty$ and X is subdiffusive), we prove that $(X,\tau)$,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$\rho$. The convergence is such that the ``amount of localization'', $\E\sum_{i \in {\Bbb Z}} [\P(X_t = i|\tau)]^2$ converges as $t \to \infty$ to $\E\sum_{z \in {\Bbb R}} [\P(Z_s = z|\rho)]^2 > 0$, which is independent of $s>0$because of scaling/self-similarity properties of $(Z,\rho)$. The scalingproperties of $(Z,\rho)$ are also closely related to the ``aging'' of$(X,\tau)$. Our main technical result is a general convergence criterion forlocalization and aging functionals of diffusions/walks $Y^{(\epsilon)}$ with(nonrandom) speed measures $\mu^{(\epsilon)} \to \mu$ (in a sufficiently strongsense).

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