The Slow Bond Random Walk and the Snapping Out Brownian Motion

Abstract

We consider the continuous time symmetric random walk with a slow bond on $\mathbb{Z}$, which rates are equal to $1/2$ for all bonds, except for the bond of vertices $\{-1,0\}$, which associated rate is given by $\alpha n^{-\beta }/2$, where $\alpha \ge 0$ and $\beta \in [0,\infty ]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $\beta <1$, then it converges to the usual Brownian motion. If $\beta \in (1,\infty ]$, then it converges to the reflected Brownian motion. And at the critical value $\beta =1$, it converges to the snapping out Brownian motion (SNOB) of parameter $\kappa =2\alpha$, which is a Brownian type-process recently constructed in Lejay, A., The snapping out Brownian motion. Ann. Appl. Probab., 26(3):1727--1742, 2016. We also provide Berry-Esseen estimates in the dual bounded Lipschitz metric for the weak convergence of one-dimensional distributions, which we believe to be sharp.