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 of vertices {−1,0}, associated rate is given by αn−β/2, where α>0 and β∈[0,∞] parameters model. prove here functional central limit theorem bond: if β∈[0,1), then it converges usual Brownian motion. If β∈(1,∞], reflected And at critical value β=1, snapping out motion (SNOB) parameter κ=2α, type-process recently constructed A. Lejay in Ann. Appl. Probab. 26 (2016) 1727–1742. also provide Berry–Esseen estimates dual bounded Lipschitz metric weak convergence one-dimensional distributions, we believe be sharp.