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

## TL;DR

This paper studies a random walk with a slow bond on the integer lattice, showing different limiting behaviors depending on the bond's rate parameter, including convergence to Brownian motion, reflected Brownian motion, or the snapping out Brownian motion.

## Contribution

It establishes a functional central limit theorem for the slow bond random walk, identifying the limiting process as Brownian motion, reflected Brownian motion, or snapping out Brownian motion at critical parameters.

## Key findings

- Converges to Brownian motion for eta<1.
- Converges to reflected Brownian motion for eta>1.
- At eta=1, converges to 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 $\alpha n^{-\beta}/2$, where $\alpha\geq 0$ and $\beta\in [0,\infty]$ are the parameters of the model. We prove here a functional central limit theorem for the random walk with a slow bond: if $\beta<1$, then it converges to the usual Brownian motion. If $\beta\in (1,\infty]$, 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 in Lejay, A., The snapping out Brownian motion. Ann. Appl. Probab., 26(3):1727--1742, 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.