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The Slow Bond Random Walk and the Snapping Out Brownian Motion | | Dirk Erhard
; Tertuliano Franco
; Diogo S. da Silva
; | | Date: |
20 May 2019 | | 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 $alpha
n^{-eta}/2$, where $alphageq 0$ and $etain [0,infty]$ are the
parameters of the model. We prove here a functional central limit theorem for
the random walk with a slow bond: if $eta<1$, then it converges to the usual
Brownian motion. If $etain (1,infty]$, then it converges to the reflected
Brownian motion. And at the critical value $eta=1$, it converges to the
snapping out Brownian motion (SNOB) of parameter $kappa=2alpha$, 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. | | Source: | arXiv, 1905.8084 | | Services: | Forum | Review | PDF | Favorites | |
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