The Slow Bond Random Walk and the Snapping Out Brownian Motion.

We consider a continuous time symmetric random walk on the integers, whose 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 and \beta are 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>1, 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 k = 2 \alpha, which is a Brownian type-process recently constructed by Lejay (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.
Talk based on a joint work with D. Erhard and D. Silva.