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

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

2000-09-09

arxiv.org/abs/math/0009098v3

Mathematics

Probability

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

Scientific paper

Let $\tau = (\tau_i : i \in {\Bbb Z})$ 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 ${\Bbb Z}$ 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 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).

Affiliated with

Also associated with

No associations

Rating

Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-244722