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 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 αn−β/2, where α>0 and β∈[0,∞] are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if β∈[0,1), then it converges to the usual Brownian motion. If β∈(1,∞], then it converges to the reflected Brownian motion. And at the critical value β=1, it converges to the snapping out Brownian motion (SNOB) of parameter κ=2α, 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.