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 $\mathit{\alpha }{\mathit{n}}^{-\mathit{\beta }}/2$, where $\mathit{\alpha }>0$ and $\mathit{\beta }\in [0,\infty ]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $\mathit{\beta }\in [0,1)$, then it converges to the usual Brownian motion. If $\mathit{\beta }\in (1,\infty ]$, then it converges to the reflected Brownian motion. And at the critical value $\mathit{\beta }=1$, it converges to the snapping out Brownian motion (SNOB) of parameter $\mathit{\kappa }=2\mathit{\alpha }$, which is a Brownian type-process recently constructed by A. Lejay in Ann. Appl. Probab. 26 (2016) 1727–1742. 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.