L.R.G. Fontes, M. Isopi, C.M. Newman
Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension
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ABSTRACT. Let $\tau = (\tau_i : i \in \z)$ 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 $\z$ with inhomogeneous rates
$(\tau_i^{-1} : i \in \z)$. When $F$ is in the domain of attraction
of a stable law of exponent $\a<1$ (so that $\E(\tau_i) = \infty$
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'',
$\E \sum_{i \in \z} [\P(X_t = i|\tau)]^2$ converges as $t \to \infty$ to
$\E \sum_{z \in \r} [\P(Z_s = z|\rho)]^2 \,>\,0$, 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
$Y^{(\e)}$ with (nonrandom) speed measures $\me \to \mu$
(in a sufficiently strong sense).