The Slow Bond Random Walk And The Snapping Out Brownian Motion

We consider the continuous time symmetric random walk with a slow bond on 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 > 0 and beta is an element of [0, infinity] are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if beta is an element of [0, infinity), then it converges to the usual Brownian motion. If beta is an element of [0, infinity], 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 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.