Scaling limit of boundary random walks: A martingale problem approach

TL;DR

This paper proves that certain boundary random walks, when scaled appropriately, converge to a broad class of Brownian-type processes on the half-line, using martingale problem techniques and analyzing local time behavior.

Contribution

It introduces a novel approach to establish the scaling limit of boundary random walks to Brownian-type processes via martingale problems and local time asymptotics.

Findings

Convergence of boundary random walks to Brownian motion in the Skorokhod topology.

Asymptotic behavior of local time for boundary random walks.

Central limit theorem for Brownian-type limit processes.

Abstract

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under appropriate scaling, the process converges to the general Brownian motion in the $J_1$-Skorokhod topology. The main novelty of our approach lies in a result on the asymptotic behavior of the local time of the boundary random walks, allowing us to derive a CLT result for several Brownian-type limit processes on the half-line.