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 $X = )X_t : t \geq 0, X_0 = 0)$, be a continuous-time simple symmetric random walk on $\mathbf{Z}$ 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 $Z = (Z_t : t \geq 0, Z_0 = 0)$ with a random (discrete) speed measure $\rho$. The convergence is such that the “amount of localization,” $\mathbb{E} \sum_{i \in \mathbb{Z}}[\mathbb{P}(X_t = i|\tau)]^2$ converges as $t \to \infty$ to $\mathbb{E} \sum_{z \in \mathbb{R}}[\mathbb{P}(Z_s = z|\rho)]^2 > 0$, 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).