Finite-dimensional approximation for the diffusion coefficient in the simple exclusion process

We show that for the mean zero simple exclusion process in $\mathbb {Z}^d$ and for the asymmetric simple exclusion process in $\mathbb{Z}^d$ for $d\geq3$, the self-diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices. The proof depends on a similar stability property of the Sobolev inner product associated with the operator.Comment: Published at http://dx.doi.org/10.1214/009117906000000449 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems

We study the cut-off phenomenon for a family of stochastic small perturbations of a one dimensional dynamical system. We will focus in a semi-flow of a deterministic differential equation which is perturbed by adding to the dynamics a white noise of small variance. Under suitable hypothesis on the potential we will prove that the family of perturbed stochastic differential equations present a profile cut-off phenomenon with respect to the total variation distance. We also prove a local cut-off phenomenon in a neighborhood of the local minima (metastable states) of multi-well potential.Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1510.09207; text overlap with arXiv:math/0601771 by other author