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

Let tau = (tau(i) : i is an element of Z) denote i.i.d. positive random variables with common distribution F and (conditional on tau) let X = (X-t : t greater than or equal to 0, X-0 = 0), be a continuous-time simple symmetric random walk on Z with inhomogeneous rates (tau(i)(-1) : i is an element of Z). When F is in the domain of attraction of a stable law of exponent alpha < 1 [so that E(tau(i)) = infinity 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 greater than or equal to 0, Z(0) = 0) with a random (discrete) speed measure p. The convergence is such that the "amount of localization," ESigma(iis an element ofZ)[P(X-t = i\tau)](2) converges as t --> infinity to ESigma(zis an element ofR) [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)) --> mu (in a sufficiently strong sense).