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

Abstract

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

Many small changes made to take referee's comments into account; references added